An Experimental and Theoretical
Investigation into Mg-ion Battery
Electrodes using Nuclear Magnetic
Resonance Spectroscopy
Jeongjae Lee
Supervisor: Professor Clare Grey
Advisor: Dr Siân Dutton
Department of Chemistry
University of Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
Trinity College May 2019

To my loving parents and my grandparents, who always supported me in doing science

Declaration
This dissertation is the result of my own work and includes nothing which is the outcome of
work done in collaboration except as declared in the Acknowledgement and specified in the
text.
It is not substantially the same as any that I have submitted, or, is being concurrently
submitted for a degree or diploma or other qualification at the University of Cambridge or
any other University or similar institution except as declared in the Acknowledgement and
specified in the text. I further state that no substantial part of my dissertation has already
been submitted, or, is being concurrently submitted for any such degree, diploma or other
qualification at the University of Cambridge or any other University or similar institution
except as declared in the Acknowledgement and specified in the text.
This dissertation contains fewer than 65,000 words including appendices, bibliography,
footnotes, tables and equations and has fewer than 150 figures.
Jeongjae Lee
May 2019

Acknowledgements
No thesis (good or bad) is made alone, and luckily I have had the chance to work with one of
the most brilliant minds in this field:
Firstly, I would like to thank my supervisor, Professor Clare Grey, for all her help in
guiding me through the work presented in this thesis. This thesis would not have existed
without her insights and support, both scientific and non-scientific. In the darkest hours of
the night where things seemed hopeless, the help from Clare led me out into the bright sun
and kept me running; and I cannot thank her enough for being my supervisor despite the
frequent deviation from my path.
A significant portion of this thesis is on the synthesis and magnetism of inorganic
compounds. On this ground, the help of my advisor, Dr Siân Dutton, was indispensable.
From advices in synthesis techniques to discussion on exotic magnetism, her comments and
insights made this thesis sitting at the crossroads between physics and chemistry.
Different people guided me with different parts of this thesis. I would like to thank Dr
Ieuan Seymour for his guidance in introducing me to many fields of computational chemistry,
including (but not limited to!) ab initio calculations and transition state searching. Dr
Bartomeu Monserrat performed and analysed some of the electronic structures in Chapter 5
and I am grateful to him for all his help and advice. I would also like to thank Dr Zigeng Liu,
who has guided me in my attempts in performing electrochemistry and also did most of the
NMR work which formed the basis of Chapter 5. Professor Andrew Pell, who is one of the
most brilliant NMR spectroscopists I have met (with a sense of British humour!), gave the
insight and helped me setting up many advanced NMR experiments performed in this study.
I would like to thank all four for their kindness in helping me learn and perform.
The members of various Grey Group teams also need to be properly acknowledged:
Evan Keyzer, Hugh Glass, Josh Lewis, Sunita Dey (Team Mg)–despite me working on a
completely different project from them using NMR, their comments and ideas have much
helped in shaping of this thesis.
Insights (and the caffeination) from weekly coffee meetings in Team Theory was helpful
in many computational work performed: I would like to thank Derek Middlemiss, David
viii
Halat, Roberta Pigliapochi, Erlendur Jónsson, Céline Merlet, John Kattirtzi, and Matthew
Cliffe for all their discussions over a cup of coffee every Friday afternoon.
Many of my Korean friends in and out of the Grey lab have been a big emotional aid to
my 9-year life in Cambridge: Gunwoo Kim, Yumi Kim, Yuji Kim, Donghwi Ko, Mia Kim,
Seung-woo Kim, Sanggil Han, Minkoo Ahn. Some as co-workers and others as good friends,
I have had the honour to receive their contributions to both scientific and non-scientific parts
of my life.
Special thanks must be given to Paromita Mukherjee and Michael Hope for proofreading
and providing helpful comments on the chapters, as well as training me on the SQUID
(Paromita) and providing a sample of VO2 (Michael).
Finally, I would like to thank all Grey group members. I really enjoyed working in the
Grey Group for the past five years with all the amazing science and occasional cheese.
The computational work carried out in this thesis was performed on three different
clusters: the group cluster Odyssey, Scientific Data & Computing Center at Brookhaven
National Laboratory, and ARCHER, the UK national supercomputing cluster. I am grateful
to all the admins in their efforts in managing the cluster.
Abstract
This thesis presents a combined experimental and theoretical approach on studying Mg-ion
battery electrode materials, where Nuclear Magnetic Resonance (NMR) spectroscopy plays
a central role in identifying the local structure and dynamics of the magnesium ions. Density
Functional Theory (DFT) techniques are used extensively to (i) calculate and rationalise
the observed NMR shifts, (ii) provide insights into the dynamics involved in such electrode
materials, and (iii) guide the synthesis of candidate electrode materials.
This work begins by a systematic study of 25Mg solid-state NMR in paramagnetic oxides,
where the presence of transition metals makes them suitable for applications in high-voltage
cathode materials. DFT methods for predicting and rationalising the paramagnetic NMR
shifts are developed, with experimental verifications on synthesised samples. Feasibility
of using advanced NMR pulse sequences such as Rotor-Assisted Population Transfer and
Magic Angle Turning is demonstrated on such systems to afford enhanced resolution and
sensitivity.
This approach of combined NMR and DFT techniques is then applied to two of magne-
sium vanadates for high-voltage cathode applications. In particular, DFT-based thermody-
namic energies are used to rationally design the synthetic steps leading to the said vanadate
materials, followed by DFT prediction of the migration barriers. The prepared material
was subject to experimental characterisation using NMR and diffraction techniques, with an
initial cycling data in an electrochemical cell.
In the final part, a combined experimental and ab initio investigation on Mg3Bi2, a promis-
ing Mg-ion battery anode material, is presented. Previous reports on variable-temperature
25Mg NMR spectroscopy is validated by DFT calculations on the migration barrier and
defect energetics. Mechanistic insights on the migration mechanism are presented using
the hybrid eigenvector-following transition state searching method, where the relativistic
effects of heavy bismuth is shown to influence the migration barrier. We show that the defect
formation energy of a Mg vacancy is critical in the apparent Mg diffusion barrier, which is
heavily influenced by sample preparation conditions.

List of Publications
Chapter 3 contains materials from the publication:
Lee, J.; Seymour, I. D.; Pell, A. J.; Dutton, S. E. and Grey, C. P. A systematic study of
25Mg NMR in paramagnetic transition metal oxides: applications to Mg-ion battery materials.
Phys. Chem. Chem. Phys. 2017, 19, 613–625.
Chapter 5 contains materials from the publication:
Lee, J.; Monserrat, B.; Seymour, I. D.; Liu, Z.; Dutton, S. E. and Grey, C. P. An ab initio
investigation on investigation on the electronic structure, defect energetics, and magnesium
kinetics in Mg3Bi2. J. Mater. Chem. A 2018, 6, 16983–16991.
Liu, Z.; Lee, J.; Xiang, G. Glass, H. F. J.; Keyzer, E. N.; Dutton, S. E. and Grey, C. P.
Insights into the electrochemical performances of Bi anodes for Mg ion batteries using 25Mg
NMR spectroscopy. Chem. Commun. 2017, 4, 743–746.
Other publications not included in this thesis:
Keyzer, E. N.; Lee, J.; Liu, Z.; Bond, A. D.; Wright, D. S.; and Grey, C. P. A general
synthetic methodology to access magnesium aluminate electrolyte systems for Mg batteries.
J. Mater. Chem. A 2019, 7, 2677-2685.
Li, Q.; Liu, Z.; Zheng, F.; Liu, R.; Lee, J.; Xu, G.-L.; Zhong, G.; Hou, X.; Fu, R.;
Chen, Z.; Amine, K.; Mi, J.; Wu, S.; Grey, C. P. and Yang, Y. Identifying the Structural
Evolution of the Sodium Ion Battery Na2FePO4F Cathode. Angew. Chem. Int. Ed. 2018, 57,
11918–11923.
Cliffe, M. J.; Lee, J.; Paddison, J. A. M.; Schott, S.; Mukherjee, P.; Gaultois, M.
W.; Manuel, P. Sirringhaus, H.; Dutton, S. E. and Grey, C. P. Low-dimensional quantum
magnetism in Cu(NCS)2: A molecular framework material. Phys. Rev. B 2018, 97, 144421.
Pecher, O.; Halat, D. M.; Lee, J.; Liu, Z.; Griffith, K. J.; Braun, M. and Grey, C. P.
Enhanced efficiency of solid-state NMR investigations of energy materials using an external
automatic tuning/matching (eATM) robot. J. Magn. Reson. 2017, 275, 127–136.

Table of contents
List of figures xvii
List of tables xxv
1 Introduction 1
1.1 Introduction to Mg-ion Batteries . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 The Working Principles of Rechargeable Batteries . . . . . . . . . 1
1.1.2 Motivation for Secondary Mg-ion Batteries . . . . . . . . . . . . . 3
1.1.3 Challenges of Developing MIB Systems . . . . . . . . . . . . . . . 4
1.2 25Mg NMR Spectroscopy Applied to MIBs . . . . . . . . . . . . . . . . . 5
1.2.1 Motivation and Challenges for Using 25Mg NMR for Studying MIBs 5
1.2.2 Previous Reports on 25Mg NMR Spectroscopy for MIB Materials . 6
1.3 Investigated Oxide Structures . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Mg6MnO8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 Spinel Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Outlook and Structure of This Thesis . . . . . . . . . . . . . . . . . . . . . 17
2 Background Theory 19
2.1 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 The Hartree-Fock Approximation . . . . . . . . . . . . . . . . . . 19
2.1.2 The Hohenberg-Kohn-Sham Approach . . . . . . . . . . . . . . . 21
2.1.3 The Hubbard-U Method . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.4 Hybrid Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.5 Basis Sets in DFT Calculations . . . . . . . . . . . . . . . . . . . 25
2.1.6 Solid-state DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Relativistic Quantum Chemistry: a Brief Introduction . . . . . . . . . . . . 26
2.2.1 Relativity and the Dirac Equation . . . . . . . . . . . . . . . . . . 26
xiv Table of contents
2.2.2 Relativistic Corrections: the Mass-velocity and Darwin Terms . . . 27
2.2.3 Relativistic Corrections at the Core and Valence: Spin-orbit Coupling 30
2.2.4 The Zeroth-Order Regular Approximation . . . . . . . . . . . . . . 30
2.2.5 Handling of Relativistic Corrections: VASP versus CASTEP . . . . 31
2.3 Transition State Searching: The HEF Method . . . . . . . . . . . . . . . . 32
2.3.1 Transition States: Definition and the Motivation for Searching Them 32
2.3.2 The Hybrid Eigenvector-following Method . . . . . . . . . . . . . 33
2.4 Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.1 Physical Basis of NMR Interactions . . . . . . . . . . . . . . . . . 35
2.4.2 Solid-state NMR and Magic Angle Spinning . . . . . . . . . . . . 42
2.4.3 Pulse Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5 Magnetism and the Paramagnetic NMR Shifts . . . . . . . . . . . . . . . . 50
2.5.1 The Basics of Electron Magnetism . . . . . . . . . . . . . . . . . . 50
2.5.2 Magnetic Measurements with a SQUID Magnetometer . . . . . . . 51
2.5.3 Electron-Nucleus Interactions in Paramagnetic Systems . . . . . . 52
2.5.4 Ab initio Calculation of the Magnetic Coupling Parameter J . . . . 55
2.5.5 Ab initio Calculation of Paramagnetic NMR Parameters from the
Curie-Weiss Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.6 Carbothermal Reaction: Thermodynamics . . . . . . . . . . . . . . . . . . 57
2.6.1 Using Carbon as a Reducing Agent . . . . . . . . . . . . . . . . . 57
2.6.2 Ab initio Computation of G for Solids . . . . . . . . . . . . . . . . 59
2.7 X-ray Diffraction Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.7.1 Powder Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.7.2 Rietveld Refinement . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides 63
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.1 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.2 Magnetic Measurements . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.3 25Mg NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.4 Ab Initio Calculations . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3.1 Diffraction and Magnetic characterisation . . . . . . . . . . . . . . 69
3.3.2 25Mg NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Table of contents xv
3.3.3 DFT Calculation of NMR and Magnetic Parameters . . . . . . . . 90
3.3.4 Shift Mechanism and the Fermi Contact Pathways . . . . . . . . . 92
3.4 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4 Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion
Battery Cathode Material 97
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2.1 Materials Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2.2 Magnetic Measurements . . . . . . . . . . . . . . . . . . . . . . . 99
4.2.3 25Mg NMR Spectroscopy . . . . . . . . . . . . . . . . . . . . . . 99
4.2.4 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . 100
4.2.5 Electrochemical Testing . . . . . . . . . . . . . . . . . . . . . . . 102
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.3.1 Calculation of Mg-ion Migration Barrier . . . . . . . . . . . . . . 103
4.3.2 Rational Design of Synthetic Steps by DFT . . . . . . . . . . . . . 109
4.3.3 Characterisation of MgV2O5 . . . . . . . . . . . . . . . . . . . . . 115
4.3.4 Electrochemical Cycling of MgV2O5 . . . . . . . . . . . . . . . . 122
4.3.5 Computation of Magnetic and NMR Parameters . . . . . . . . . . . 126
4.3.6 Characterisation of MgV2O4 Prepared Through the CTR method . . 130
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.4.1 Implications of the CTR Method for Other Oxides . . . . . . . . . 137
4.4.2 Discussion on the Initial MgV2O5 Electrochemistry . . . . . . . . . 137
4.4.3 MgV2O4: Sample Dependence on the Preparation Method . . . . . 139
4.5 Conclusion and Further Work . . . . . . . . . . . . . . . . . . . . . . . . . 140
5 An Investigation on the Electronic Structure, Defect Energetics, and Magne-
sium Kinetics in Mg3Bi2 143
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.2.1 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . 144
5.2.2 Calculation of NMR Parameters . . . . . . . . . . . . . . . . . . . 145
5.2.3 Transition State Searching . . . . . . . . . . . . . . . . . . . . . . 146
5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.3.1 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.3.2 Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . 148
xvi Table of contents
5.3.3 Calculation of the 25Mg NMR Shifts . . . . . . . . . . . . . . . . . 151
5.3.4 Defect Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.3.5 Mg Migration Kinetics . . . . . . . . . . . . . . . . . . . . . . . . 156
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6 Conclusion and Further Work 163
References 167
Appendix A Post ball-mill analysis of MgV2O5 179
A.1 SEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
A.2 XRD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Appendix B EDX data of lithium counterelectrode from charged VO2 cell 183
List of figures
1.1 A schematic illustration of a MIB system under work. . . . . . . . . . . . . 2
1.2 Structure of Mg6MnO8, shown with oxygen polyhedra around the metal ions. 10
1.3 Compositional phase diagram (at 0 K) of the Mg–V–O system, calculated
using the Materials Project.[65, 66]. PBE+U functional was used with
U = 3.25 eV for V. Stable phases at 0 K are also labelled on the diagram.
MgV2O5 exists at only 0.026 eV above the hull. . . . . . . . . . . . . . . . 14
1.4 Structures of the α-, δ -, and metastable ε-polymorphs of MgV2O5. Repro-
duced from Sai Gautam et al.[67] . . . . . . . . . . . . . . . . . . . . . . . 16
2.1 Total energy as a function of electron count in the system. Black line gives
the analytical DFT energy, red line gives the exact energy, and the blue
line gives the parabolic energy difference given by the U-correction. Figure
adopted from [81]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Schematic illustration of the HEF method for TS searching. Detailed expla-
nations are presented in the main text. Saddle point figure was adopted from
Wikipedia under a CC-BY 3.0 license. . . . . . . . . . . . . . . . . . . . . 34
2.3 Schematic illustration of the pulse-acquire experiment on a 2-level system.
Both spin state and vector formalisms are shown. . . . . . . . . . . . . . . 36
2.4 Schematic energy levels (not drawn to scale) in the presence of quadrupolar
coupling, illustrated for γ < 0 and I = 5/2 (the case for 25Mg). . . . . . . . 41
2.5 Schematic illustration of MAS experiments. . . . . . . . . . . . . . . . . . 43
2.6 (a) Hahn-echo pulse sequence. (b) RAPT-echo pulse sequence. (c) QMAT
pulse sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.7 Illustration of different spin density transfer mechanisms in TM−O−Li
bonds, with 90◦ and 180◦ bond angles. Figure reproduced from Carlier et al.
[144] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
xviii List of figures
2.8 Illustration of calculating J by a broken symmetry supercell approach. In (a),
the total energy equals Ea = E0+8J1. In (b), the total energy equals Eb = E0.
Subtracting the energies yields J1 = (Ea−Eb)/8. . . . . . . . . . . . . . . 55
2.9 Ellingham diagram, reproduced from the original construction by Elling-
ham.[145] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.1 25Mg bond pathway (P) contributions and TM–TM J couplings. . . . . . . 68
3.2 X-ray powder diffraction pattern and Rietveld refinement data for Mg6MnO8.
The positions of allowed reflections are indicated by the tick marks. . . . . 70
3.3 Inverse magnetic susceptibility per mol Mn, 1/χ , as a function of temperature
for Mg6MnO8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4 X-ray powder diffraction pattern and Rietveld refiment data for MgCr2O4.
The positions of allowed reflections are indicated by the tick marks. . . . . 74
3.5 Inverse magnetic susceptibility per mol Cr, 1/χ , as a function of temperature
for MgCr2O4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.6 X-ray powder diffraction pattern for the SS-MgV2O4. The positions of
allowed reflections are indicated by the tick marks. . . . . . . . . . . . . . 76
3.7 X-ray powder diffraction pattern for MgMn2O4. Collected data (red), refined
data (black), and diffrences (lower panel) are shown. The positions of
allowed reflections are indicated by the tick marks. 7.9 wt % of Mg6MnO8
phase was detected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.8 Inverse magnetic susceptibility per mol Mn, 1/χ , as a function of temperature
for MgMn2O4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.9 (a) 25Mg spin echo spectrum of Mg6MnO8 (14 kHz MAS, 81712 transients),
with fitted central transition lineshape including contributions from both the
paramagnetic shift anisotropy and the quadrupolar interaction (parameters are
listed in Table 3.7). (b) Magic angle turning (MAT) spectrum of Mg6MnO8
(20 kHz MAS, 128 slices in the F1 dimension with 2.23 µs delay increment,
3072 transients acquired in each slice). RAPT pulses were applied before
the MAT pulses to enhance the signal-to-noise ratio. . . . . . . . . . . . . . 81
List of figures xix
3.10 (a) 25Mg spin echo signal intensities of Mg6MnO8 with increasing offset
frequency of the saturating pulse trains in the RAPT experiment. (b) En-
hancement of spin echo signal intensity using the RAPT pulse. Saturating
pulses were applied at a modulation frequency of 270 kHz. Both experiments
were performed at MAS spin rate of 14 kHz. 1024 transients were acquired
in each spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.11 25Mg NMR spectra of Mg6MnO8, recorded with the rotor-synchronised
Hahn echo and Double Frequency Sweep (DFS) pulse sequences. Signal-
to-noise enhancement of around 1.5 is observed for the isotropic shift. A
36 kHz-strength DFS sweep pulse starting from an offset of 1000 kHz and
ending at 100 kHz was applied for 2040 µs. 51200 transients were acquired
in each case with recycle delays of 0.01 s. . . . . . . . . . . . . . . . . . . 83
3.12 Simulated 25Mg NMR spectra of Mg6MnO8, with different Euler angles β
between the anisotropic hyperfine tensor and the EFG tensor in the simulation. 84
3.13 25Mg spin echo spectrum of MgCr2O4 at a MAS spin rate of 10 kHz. 20480
transients were acquired. The broad feature seen around 2700 ppm is due to
probe background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.15 25Mg spin echo spectrum of SS-MgV2O4 (solid-state route) at MAS spin
rate of 10 kHz. 505600 transients were acquired with recycle delays of 0.1 s. 86
3.16 25Mg spin echo spectrum of MgMn2O4 at MAS spin rate of 14 kHz. Mg6MnO8
secondary phase signals are shown with asterisks (*). 3550928 transients
were acquired. (a) Comparison between experimental and fitted spectrum.
Fitted parameters are listed in Table 3.7. (b) Enhancement of spin echo signal
intensity with RAPT pulse sequence. RAPT modulation pulses were applied
at 240 kHz. 1024000 transients were acquired in each spectrum. . . . . . . 88
3.17 3-dimensional spin density maps. Yellow denotes positive spin; blue denotes
negative spin. Bounding box refers to the unit cell and solid lines connecting
atoms refer to the metal-oxygen bonds. Bond pathway contributions P to the
spin density are shown in red. . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1 Determination of rotationally invariant Hubbard U of V4+ in MgV2O5 by a
linear response method by varying the site potential α . SC and non-SC refer
to self-consistent and non-self-consistent calculations, respectively. . . . . . 103
xx List of figures
4.2 (a) Energy profiles of a single Mg vacancy hop between 6-fold coordinated
(stable) and 5-fold coordinated (metastable) sites. For the HSE06 calcula-
tions, only a handful of points along the PBE+U diffusion path was selected
to calculate the single point energies. All energies are referenced to their
lowest respective state. (b) Metastable 5-coordinated, (c) transition state, and
(d) stable 6-coordinated Mg site along the diffusion pathway. Colour scheme:
orange(Mg), red(V). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3 (a) Local geometry around the Mg in its transition state, fully magnesiated
MgV2O5. (b) Local geometry around the Mg in its transition state, fully
demagnesiated V2O5. All units are in Å. Mg–O bond distances closer than
3.2 Å are represented by a bond. . . . . . . . . . . . . . . . . . . . . . . . 107
4.4 (a) Energy profiles of a single Mg hop between 6-fold coordinated (stable)
and 5-fold coordinated (metastable) sites. For HSE06 calculations, only a
handful of points along the PBE+U diffusion path was selected to calcu-
late the single point energies. All energies are referenced to their lowest
respective state. (b) Metastable 5-coordinated, (c) transition state, and (d)
stable 6-coordinated Mg site along the diffusion pathway. Colour scheme:
orange(Mg), red(V). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.5 Reaction scheme to synthesise MgV2O5 and MgV2O4 via the CTR approach. 110
4.6 Free energy ∆G vs. temperature T plot, or Ellingham diagram, of the var-
ious oxidation reactions involved in syntheses of MgV2O5 and MgV2O4
phases, calculated from DFT. All values are under standard 1 atm partial
pressures. The two carbon lines cross at T = 973 K (700 ◦C). To allow a
direct calculation of ∆G, the values are normalised with respect to 1 mol of O2.111
4.7 DFT-based free energy ∆G of the CTR reactions to produce MgV2O5 and
MgV2O4 phases at 900 K as a function of CO2 partial pressure ρCO2 . The
two lines cross at ρCO2 = 10
−2 atm. The values are normalised with respect
to 1 mol of MgV2O6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.8 DFT-based free energy ∆G of the oxidation reactions involving MgV2O5
and MgV2O4 phases at 1200 K as a function of O2 partial pressure ρO2 .
Equilibrium ρO2 of Cu oxidation is also shown as a dashed line at 10
−8 atm.
The values are normalised with respect to 1 mol of MgV2O6. . . . . . . . . 115
4.9 Refined powder X-ray diffraction data of MgV2O5 as prepared by the CTR
method. Refined parameters are shown in Table 4.1. Reflections for MgV2O5
(50979) and VO2 (34033) are taken from the ICSD. . . . . . . . . . . . . . 117
List of figures xxi
4.10 Thermogravimetric data of the magnesium acetate (nominally tetrahydrate)
precursor used in the synthesis of MgV2O6. . . . . . . . . . . . . . . . . . 118
4.11 (a) 25Mg spin echo spectrum of the as-synthesised MgV2O5, measured
under 20 kHz MAS and a 0.1 s recycle delay. Fitted parameters are shown
in Table 4.3. (b) Structure of MgV2O5, showing the 6-fold Mg coordination
environment as polyhedra. . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.12 Enhancement profile of the integrated signal intensity in MgV2O5 using the
RAPT pulse sequence, as a function of the Gaussian offset νoff. All samples
were measured under 20 kHz MAS and a 0.1 s recycle delay. . . . . . . . . 120
4.13 25Mg spectra of as-synthesised MgV2O5, measured with a normal spin echo
and a RAPT-spin echo pulse sequences. Both were measured under 20 kHz
MAS and 0.1 s recycle delay. . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.14 25Mg spectra of ball-milled MgV2O5, measured under normal spin echo
and RAPT-spin echo. Both were measured under 20 kHz MAS and a 0.1 s
recycle delay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.15 (a) Electrochemical cycling data of ball-milled MgV2O5, charged vs Li. (b)
Glass fibre separator and lithium disc after disassembly of the coin cell
shown in (a). (c) Powder X-ray diffraction data of charged cathode. Broad
background at 20◦ is due to Kapton sample holder. . . . . . . . . . . . . . 124
4.16 (a) Electrochemical cycling data of VO2, charged vs Li. (b) Lithium disc
after disassembly of the coin cell. (c) Powder X-ray diffraction data and Le
Bail fit of charged VO2. Broad background at 20◦ is due to Kapton sample
holder. Self-standing films were provided by Michael Hope. . . . . . . . . 125
4.17 (a) Zero field cooled molar susceptibility χ and (b) Zero field cooled inverse
molar susceptibility 1/χ of MgV2O5 as measured by SQUID magnetometry.
Linear fitting was performed from 200 K to 300 K. . . . . . . . . . . . . . 127
4.18 (a) Illustration of the V–V exchange interactions Jn up to the fourth nearest
neighbour in MgV2O5. Dark blue colour refers to vanadium. (b) Dominant
exchange mechanisms for the J1 (left) and J2,3 (right). . . . . . . . . . . . . 129
4.19 X-ray powder diffraction pattern for CTR-MgV2O4 (carbothermal route).
The positions of allowed reflections are indicated by the tick marks. . . . . 132
4.20 Inverse magnetic susceptibility per mol V, 1/χ , as a function of temperature
for the CTR-MgV2O4 sample prepared via a carbothermal route. . . . . . . 133
xxii List of figures
4.21 25Mg spin echo spectrum of CTR-MgV2O4 (carbothermal route) at MAS
spin rate of 10 kHz. Positions of the isotropic resonances are indicated.
628400 transients were acquired with recycle delays of 0.1 s. . . . . . . . . 135
5.1 (a) Crystal structure of Mg3Bi2. Positions of tetrahedrally coordinated Mg
(orange sphere, Tet) and octahedrally coordinated Mg (green sphere, Oct)
are shown. Bismuth atoms sit on the line vertices (not shown for clarity).
(b) Schematic illustration of the Mg3Bi2 structure (on the left) showing the
possible diffusion pathways involving the tetrahedral and the octahedral
interstitial sites (‘Tet Int’ and ‘Oct Int’). . . . . . . . . . . . . . . . . . . . 147
5.2 Density of states plot of Mg3Bi2 using the PBE and HSE06 exchange-
correlation functionals. Local atomic DOS projections inside the sphere
defined by the Wigner-Seitz radii (1.63 and 1.52 Å for Bi and Mg, respec-
tively) are also shown. All energies were referenced to the highest occupied
state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.3 Band structure plots of Mg3Bi2 using the PBE (left) and HSE06 (right)
exchange-correlation functionals without (grey) and with (red) spin-orbit
coupling (SOC). The Fermi level is located at the zero of energy. . . . . . . 150
5.4 Density-of-states (DOS) plot for Mg3Bi2, generated using CASTEP. Three
cases are considered: nonrelativistic (NR), scalar relativistic (SR) using the
ZORA method, and full relativistic (FR) including the SOC. All energies
were referenced to the highest occupied state. Site- and orbital-decomposed
DOS under SOC is not supported in the version of CASTEP used. . . . . . 152
5.5 Relaxed cell structures using the (a) scalar relativistic ZORA approximation,
and (b) nonrelativistic Schrödinger equation, projected along the a-direction.
Here, Mg(oct) sits on layer A and Mg(tet) sits on layer B. . . . . . . . . . . 153
5.6 Diffusion profile of Mg ion for selected pathways as illustrated in Figure
5.1b. (a) Scalar relativistic, (b) full relativistic, with SOC included. Energies
are referenced to the bulk energy of each 120-atom supercell with one Voct
defect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
A.1 (a) SEM image of pristine MgV2O5 before ball-milling. (b) SEM image of
MgV2O5 after ball-milling. . . . . . . . . . . . . . . . . . . . . . . . . . . 180
A.2 (a) XRD pattern of pristine MgV2O5 before ball-milling. (b) XRD pattern
of MgV2O5 after ball-milling. . . . . . . . . . . . . . . . . . . . . . . . . 181
List of figures xxiii
B.1 EDX data of the lithium disc from the coin cell after charging of VO2. . . . 183

List of tables
3.1 Synthesis conditions of samples. . . . . . . . . . . . . . . . . . . . . . . . 64
3.2 Rietveld refined parameters from the PXRD data of Mg6MnO8. . . . . . . 70
3.3 Magnetic characterisation data of compounds studied. µeff refers to the
effective magnetic moment in Bohr magneton (µB) per TM ion, Θ refers to
the Weiss temperature, J1 refers to the nearest neighbour exchange coupling
constant extracted with Equation 2.73, and TN refers to the Néel temperature.
Theoretical spin-only values for the MgMn2O4 compositions determined by
phase fractions and occupancy refinements are also shown. . . . . . . . . . 72
3.4 Rietveld refined parameters from the PXRD data of MgCr2O4. . . . . . . . 74
3.5 Rietveld refined parameters from the PXRD data of MgV2O4 prepared via a
solid-state method (SS-MgV2O4). . . . . . . . . . . . . . . . . . . . . . . 76
3.6 Rietveld refined parameters from the PXRD data of MgMn2O4. . . . . . . 78
3.7 Ab-initio calculated and experimentally fitted paramagnetic NMR parameters of
Mg6MnO8, MgCr2O4, MgV2O4, and MgMn2O4. Hyb20 and Hyb35 refer to the de-
gree of Hartree-Fock exchange energy (see methods). Values marked with asterisks
(*) were fixed in the fitting. For MgV2O4, the experimental Weiss constant was
obtained from literature.[164] EFG tensor eigenvalues and anisotropic hyperfine
tensor components are near zero for MgV2O4 and MgCr2O4 and Euler angles are
not reported for these systems. Detailed magnetic characterisation data are reported
in Table 3.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.8 Mulliken spin population analysis of transition metal d orbitals in selected
spinel compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.1 Rietveld refined parameters from the PXRD data of MgV2O5 as prepared by
the CTR method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
xxvi List of tables
4.2 Magnetic parameters of MgV2O5 determined by experiment and DFT calcu-
lations. Experimental parameters were determined by SQUID magnetometer
(Figure 4.17); J-couplings are as denoted on Figure 4.18a. . . . . . . . . . . 126
4.3 NMR parameters of MgV2O5 determined by experiment and DFT calcula-
tions. Experimental parameters are fitted from the spectrum in Figure 4.11a
(anisotropies could not be determined from the spectrum). Herzfeld-Berger
convention is used for the shift anisotropy. . . . . . . . . . . . . . . . . . . 130
4.4 Rietveld refined parameters from the PXRD data of MgV2O4 prepared via a
carbothermal method (CTR-MgV2O4). . . . . . . . . . . . . . . . . . . . . 132
4.5 Estimated hyperfine shifts and the intensity ratio for the Mg ions in the
MgV2O4 spinel structure neighbouring V3+ and V4+ ions. The shifts ‘ref-
erenced’ from the 1845 ppm (SS-MgV2O4) or 1861 ppm (CTR-MgV2O4)
are shown. Integrated relative intensities are shown for the experimental
spectrum and also calculated assuming a random distribution of the V4+ ions. 135
5.1 DFT-predicted cell parameters of Mg3Bi2 using the PBE and HSE06 exchange-
correlation functionals using CASTEP and VASP. NR, SR and FR refer to
nonrelativistic, scalar relativistic, and full relativistic (i.e. explicit spin-orbit
coupling) calculations, respectively. ZORA method was used for SR cal-
culation on CASTEP. Experimental lattice constants are from Lazarev et
al.[224] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.2 Scalar relativistic (SR) and full relativistic (FR) Bader charge analysis
of Mg3Bi2 using the Perdew-Burke-Ernzerhof (PBE) and Heyd-Scuseria-
Ernzerhof (HSE06) exchange-correlation functionals. Only the valence
charge is calculated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.3 Calculated NMR parameters of the two Mg sites (octahedral and tetrahedral)
in Mg3Bi2 using the GIPAW method. Core electrons are treated with two
levels of theory: scalar relativistic (SR) ZORA and nonrelativistic (NR)
Schrödinger. δiso, Ω, and η refer to the chemical shift tensor expressed in
the Herzfeld-Berger convention. CQ and η refer to the quadrupolar coupling
parameter and asymmetry, respectively. . . . . . . . . . . . . . . . . . . . 153
List of tables xxvii
5.4 Scalar relativistic (SR) and full relativistic (FR) ab initio formation energies
∆E f of various stoichiometric and non-stoichiometric defects in Mg3Bi2
using the PBE functional. Defect notations follow the convention of Kroger
and Vink with neutral sign omitted for clarity.[218] All calculations assumed
non-charged defects (see text) with cell dimensions fixed to simulate a dilute
limit. Vacancy defect energies are referenced to the respective metals. For
Frenkel defects, two scenarios where the Mg sits on a nearby (nn) or far
interstitial sites were considered. Asterisks(*) indicate that the resulting
structure was unstable and reverted back to the starting structure. All values
are quoted in electron-volts. Only some of the calculations were performed
under the 120-atom supercell condition after an initial screening with the
40-atom supercell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.5 Effective migration barriers Eeffa estimated through DFT and VT NMR tech-
niques. VT NMR data are taken from Liu et al.[40] For the NMR measure-
ments, Mg3Bi2 prepared through electrochemical insertion and mechanical
milling were considered. SR and FR refer to scalar relativistic and full
relativistic calculations, as described in the text. . . . . . . . . . . . . . . . 161

Chapter 1
Introduction
1.1 Introduction to Mg-ion Batteries
1.1.1 The Working Principles of Rechargeable Batteries
The rapid increase in global energy demand requires more renewable energy sources and
subsequently novel energy storage solutions, both on a grid and a portable scale. Indeed, the
rapid development of current portable electronics such as smartphones is heavily indebted to
rechargeable Li-ion batteries (LIBs). The key to the commercial and technological success
of LIBs lies in the light weight and high oxidation potential of lithium, which is ideal for
portable devices where high energy density is the most important factor. However, the
perennial demand for higher capacity and faster charging/discharging capability has driven
the research in building ‘better batteries’.
To this end, understanding the working principles of a battery is essential. This is
schematically illustrated in Figure 1.1. Most simply put, any battery is formed of an anode,
a cathode, and an electrolyte. Energy stored on the electrodes in the form of free energy
G could be released upon redox processes of the electrode materials, resulting in electric
potentials E according to Equation 1.1:
∆G =−nFE (1.1)
where n and F refer to the electron count and Faraday constant, respectively.
Here we discuss rechargeable batteries relying on an intercalation mechanism. Upon
charging, the charge carrying ions move out of the cathode and migrate towards the anode,
resulting in the net energy being stored in the system. The opposite happens in a discharge:
2 Introduction
Fig. 1.1 A schematic illustration of a MIB system under work.
1.1 Introduction to Mg-ion Batteries 3
ions migrate from the anode towards the cathode, resulting in an energy release. Separation
of the redox centres on the two electrodes by means of an ion-conducting but electrically
insulating electrolyte ensures the generated electric energy could be passed through an
external circuit to do useful work. This poses several requirements for the materials involved
in the electrochemistry:[1]
• The intercalation must be reversible; i.e. the host structure must not be destroyed
during the insertion/de-insertion of guest ions.
• As both charging/discharging processes involve the guest ions moving in a host struc-
ture, the diffusion must be rapid in order to extract currents at a useful rate. This is com-
monly called the cycling rate C, where 1 C denotes a current density to charge/discharge
the cell in 1 hour.
• The difference in redox potentials between the anode and the cathode preferably should
be large to allow the flow of current at higher potentials, resulting in larger energy
densities.
Combination of a graphite anode with a LiCoO2 cathode satisfies all three requirements; this
is the concept of the famous ‘rocking-chair’ battery.
1.1.2 Motivation for Secondary Mg-ion Batteries
Despite the immediate success of LIBs in portable electronics and electric vehicles applica-
tions, currently several problems are faced in using lithium in a large-scale energy storage, or
even in extending the electric vehicles applications:
• High cost of lithium. Li only takes up 0.002 % of the Earth’s crust by mass,[2] and is
generally sourced from salt mines, or extracted from seawater.
• Geopolitical constraints. Li is mainly produced in certain parts of the world (Australia,
China, and South America); the production is vulnerable to geopolitics which is also
linked to the growing prices of lithium.[3]
• Dendrites and safety issues. Currently, Li metal (which gives the largest cell potential
possible) cannot be used in LIBs due to problems with dendrite growths and eventual
short-circuiting of the cell.[4, 5] This leads to safety concerns on thermal runaways
ultimately leading to battery explosions.
4 Introduction
• Low volumetric capacity. As lithium forms a monovalent cation, higher volumetric
capacity could be attained by using a higher valent cation instead.
Despite the continuing political and technological efforts in solving these problems, batteries
based on alternative chemistries other than Li have been investigated over the years. These
include those using different alkali/alkaline-Earth metals (Na-ion, K-ion, Mg-ion, K-ion)
or relying on a chemistry different to a simple intercalation mechanism (metal-sulphur,
metal-air).[6, 7, 8, 9] Of these, Mg-ion batteries (MIBs) are particularly promising due to the
following factors:
• Low cost due to the high natural abundance of Mg in the Earth’s crust (2.8 %).[2]
• Distribution of Mg is less uneven, with many of the minerals contain Mg as a compo-
nent. Also, Mg could be extracted from MgCl2 (obtained from brines) by electrolysis.
• Mg metals are commonly known not to suffer from dendritic growth.[10] This is due
to the self-diffusion of Mg atoms being faster than Li or Na on a surface[11], although
under certain conditions dendrite growth could be possible (see [12, 13, 14] for reports)
• Mg forms a divalent cation, thus resulting in a significantly higher volumetric capacity
compared to Li metals (3833 mAh/cm−3 for Mg versus 2046 mAh/cm−3 for Li).
• Unlike metal-air batteries, many technologies for LIBs can be extended to MIBs to aid
faster commercialisation.[15]
1.1.3 Challenges of Developing MIB Systems
Hence, since the 1990s many attempts were made to develop a working MIB system.[10]
The first system to show more than 2000 reversible charge-discharge cycles was reported
in 2000 using MgxMo6S8 Chevrel-type cathodes,[16] but their low redox potential (1.1 V
versus Mg/Mg2+) and low specific capacity (110 mAhg−1) seriously limit the commercial
applications to replace LIBs (3.6 V and 200 mAhg−1 using LiCoO2 cathodes). Further-
more, nonaqueous magnesium electrochemistry is still relatively poorly understood and the
development of reliable electrolytes with wide potential windows is still an active area of
research.[17]
The challenge could be largely divided into two sub-problems: electrode and electrolyte.
1.2 25Mg NMR Spectroscopy Applied to MIBs 5
Electrodes
Mg2+ has a similar size to Li+ (0.72 versus 0.76 Å for 6-fold coordinated environments).[18]
This means that the significant electrostatic interaction between the Mg2+ and O2 – ions in
the structure hampers the mobility of Mg ions. This is clearly different from the Na case,
where the same monovalent nature of Na+ enables many of the LIB materials to be used
without problems in diffusion. This means that the typical high-voltage transition metal (TM)
oxides performs poorly in MIBs.
To circumvent this issue, many previous reports have tried using chalcogenides (sul-
phides or selenides) to reduce the charge density on the anionic sublattice and weaken the
electrostatic interaction. However, this inevitably results in a lower redox potential of the
cathodes compared to the oxides, as the strongly polarising oxygen ligand contributes to the
redox reactions happening at higher potentials.
Electrolytes
Currently there is no reliable Mg electrolyte that can (i) electrochemically remove and deposit
(i.e. strip and plate) Mg reversibly on metallic surfaces, (ii) stable at high potentials to allow
use of high-voltage cathodes, and (iii) stable at high potentials (>4 V) against commonly used
current collectors (stainless steel, copper, aluminium). This makes testing of the potential
high-voltage cathodes problematic and often parasitic reactions are the dominant source of
observed capacity. While significant advances have been made in this area of research for
the past decades, the absence of a reliable electrolyte makes it difficult to test especially
high-voltage cathode materials. To this end, initial charging (de-magnesiation) is often tested
in Li-ion half cells (against Li metal as the counterelectrode) to assess the charging capacity.
For more accounts on electrolyte development the reader is referred to some recent reviews in
this aspect;[17, 10, 19] also, works on solid-state Mg-ion electrolytes using sulphide spinels
(thiospinels) have been recently reported.[20, 21]
1.2 25Mg NMR Spectroscopy Applied to MIBs
1.2.1 Motivation and Challenges for Using 25Mg NMR for Studying
MIBs
Nuclear Magnetic Resonance (NMR) is a valuable method to study battery materials as the
characteristic NMR shift, quadrupolar coupling, lineshape, and relaxation behaviour can
6 Introduction
give important information on the local structure and dynamics.[22, 23] This allows us to
characterise the individual local environments in disordered systems, which are difficult
to analyse with diffraction methods that probe long range order. In particular, solid-state
magic-angle spinning (MAS) NMR is particularly useful as electrode materials can be
studied.
NMR techniques have been successfully used in studying many battery systems including
Li-ion, Na-ion, Li-air, and Li-S.[22] Applying the same experimental approach to MIBs,
however, is challenging because of the low gyromagnetic ratio (approximately 1/20 of 1H)
and natural abundance (10 %) of 25Mg , the only NMR active nucleus. In addition, significant
quadrupolar interactions (25Mg I = 5/2; quadrupole moment Q = 0.2 barns) can lead to a
broadening of the signal and poor resolution.
Despite the prevalence of Mg in many important minerals and bioinorganic complexes,
the challenging nature of 25Mg has prevented serious solid-state NMR studies of such
compounds for many years.[24] The first 25Mg MAS spectrum was recorded in 1988.[25]
Since then, the development of high-field magnets made the sufficient resolution possible by
larger splitting of the Zeeman levels and by suppressing the quadrupolar effects relative to
the Zeeman effects.[26] Also, developments of novel pulse sequences such as qCPMG[27]
have allowed broad spectra to be recorded for local structure analysis of a number of different
diamagnetic compounds.[28, 29]
These difficulties are made more problematic when studying cycled battery cathodes,
since the samples are inherently diluted (mixing with conductive carbon, binders, etc.) and
less crystalline (many electrode formulations use nanocrystalline materials for better cycling
performances). Thus, only a few reports have been published which have used solid-state
25Mg NMR in a context of MIBs. In the following sections, we briefly discuss the prior
results and set where this thesis stands in terms of the literature.
1.2.2 Previous Reports on 25Mg NMR Spectroscopy for MIB Materials
To date, there only exists a scarce (<20) literature on solid-state 25Mg NMR spectropy applied
to MIB materials. These can be broadly divided according to their compositions: oxides,
other chalcogenides, and intermetallics.
Oxides
Perhaps the most obvious place to start, various Mg TM oxides have been investigated as
potential cathode materials for MIBs. The first observation of reversible Mg intercalation into
1.2 25Mg NMR Spectroscopy Applied to MIBs 7
such oxides was observed for MgMn2O4, where a combined 25Mg qCPMG NMR and X-ray
spectroscopic approach was used to confirm the reversible Mg chemistry into this spinel type
cathode.[30] However, the system was cycled in an aqueous electrolyte, which limits the
application to low-voltage cells; absence of high-voltage electrolytes have not seen reports
to cycle this material under nonaqueous conditions. This could also be related to water
co-insertion aiding the Mg diffusion, as was shown for V2O5 xerogels and birnessite-type
MnO2 structures.[31, 32, 33, 34]
V2O5-type cathode materials were also investigated using 25Mg NMR spectroscopy,
confirming the reversible insertion and elucidating the local structure of Mg ions upon
intercalation. Sa et al. have reported the presence of two local environments with 24.7 and
1840 ppm shifts, the second of which was attributed to the MgV2O5 phase.[34] Absence
of MgO in the pair distribution function analysis denoted the 24.7 ppm environment to be
present inside the V2O5 xerogel host, which was identified as hydrated Mg ions octahedrally
coordinated inside the V2O5 layers.
A study of 25Mg NMR in chemically magnesiated oxide structures was performed by
Wang et al.[35] It was shown that MgO is formed as a by-product in many chemically mag-
nesiated oxides (resonance of 26 ppm), including MgxTiO2, α-V2O5, and TiO2, questioning
the previously observed ‘reversible’ electrochemistry in these materials. As expected, mag-
nesiation of MnO2 have resulted in resonances around 1800 ppm, which is characteristic of
large hyperfine-type shifts typically observed in TM oxides. Thus, 25Mg NMR spectroscopy
could be used to observe the intercalation and identify any amorphous side products difficult
to detect with diffraction techniques.
Finally, 25Mg NMR was also used to study a molybdenum oxyfluoride cathode ma-
terial MgxMoO2.8F0.2.[36] Broad resonances around -60 ppm was observed in this case,
which was attributed to disordered Mg intercalation in these structures. No in-depth NMR
characterisation was performed apart from observing the signal in this case.
Other Chalcogenides
As discussed above, using chalcogenides (S, Se) as the anion to facilitate diffusion is a
commonly adopted strategy to devise novel electrode materials for MIBs. In the context
of 25Mg NMR, the fast dynamics afforded by such approach could be directly observed.
A recent report have demonstrated that fast Mg motion is present in thiospinels MgX2Z4
(X=(In, Y, Sc) and Z=(S,Se)) with a measured activation barrier of 320 to 360 meV, which
are comparable to fast Li-ion conductor garnets.[20] More importantly, variable temperature
spin-lattice (T1) relaxation rates were conducted on the sample MgSc2Se4 to determine the
8 Introduction
activation barrier of 370 meV, which is a first demonstration of using T1 relaxation rates to
investigate fast Mg-ion dynamics. These class of materials are potentially interesting for
solid-state electrolyte applications.
Other thiospinel cathodes were also investigated in this aspect. Wustrow et al. have
synthesised MgCr2S4 thiospinel structure and reported the 25Mg NMR resonance at 11220
ppm.[37] This is clearly a large downfield shift from 2862 ppm resonance of MgCr2O4,[38]
which clearly displays the more covalent nature of Cr–S–Mg bonding compared to Cr–O–
Mg, which results in a larger hyperfine shift. Attempted electrochemical cycling, however,
has resulted in structural degradation and no reversible capacity was observed. This may
be a common problem for structures containing sulphur in general, as the electrostatic
forces holding the structure intact is much smaller and degradation is more likely than the
corresponding oxides.
Chevrel cathodes MgxMo6X8 (X=S, Se), despite being the besting working cathode so
far, have not been investigated systematically with 25Mg NMR. One 25Mg spectrum of a
chemically magnesiated Chevrel cathode MgxMo6S8 is reported with a broad resonance
spanning 0 to -500 ppm, which may be attributed to the fast Mg dynamics and the asymmetric
nature of Mg sites present in the structure.[35] Negative shifts indicate that polarisation-
type hyperfine shift is predominantly present in this material. In contrast, 23Na spectra for
Chevrels cycled in Mg aluminate electrolytes Mg(THF)n[Al(OR)4]2 contaminated with Na
ions clearly show positive values of hyperfine shifts (344.8 and 783.1 ppm).[39] This is
attributed to a co-insertion of Na into the Chevrel structure (accounting for around 22 % of
the capacity) in this contaminated electrolyte. More work is needed to fully understand the
extent of NMR shift mechanisms in these materials.
Intermetallics
These class of materials which formally ‘alloys’ with Mg to form intermetallic phases are
normally used for anode materials due to their low redox potentials. In this regard, Liu et al.
have investigated bismuth anodes for MIBs, which accepts Mg to form a phase Mg3Bi2.[40]
Mg3Bi2 adopts a hexagonal lattice with two available Mg sites: octahedral and tetrahedral
(Chapter 5); two 25Mg resonances corresponding to each Mg was observed experimentally
at -6 (octahedral) and -306 (tetrahedral) ppm. An interesting aspect of this material is that
it shows fast Mg (de)-insertion kinetics: rates of up to 5 C were observed in real cells. By
use of a variable-temperature 25Mg NMR and lineshape fitting, Mg exchange rates and
the activation barrier were calculated, which shows a very small activation barrier (0.2 eV)
1.3 Investigated Oxide Structures 9
comparable to Li-ion battery electrodes (<0.3 eV). This observation forms the motivation of
Chapter 5, where this result is validated computationally by use of ab initio methods.
As Sb belongs to the same group as Bi, investigations have been performed on the alloy
Mg3Sb2 and the mixed-composition electrodes Mg3Sb2 – xBix.[41] Mg3Sb2 is isostructural
with Mg3Bi2 and the resulting NMR spectrum also showed two resonances at 50 and -106
ppm, where the downfield shifts could be attributed to the larger electronegativity of Sb
compared to Bi. Motional averaging by Mg dynamics was not observed in this material,
which could be explained by an increased crystallinity in the sample compared to the above
results by Liu et al. It was also found that Sb is also electrochemically active in this alloy,
thus reaching a complete magnesiation up to Mg3Sb2 – xBix regardless of the Sb content.
Finally, reports on 25Mg spectra in other intermetallics also have been published (not in
the context of MIBs). They may find importance as potential candidate anode materials, or
investigating Mg alloying into current collectors: Mg2Sn, MgZn2, Mg2Si, Cu2Mg, CuMg2,
and Al2CuMg.[42, 25, 43, 44]
1.3 Investigated Oxide Structures
Despite the apparent success, chalcogenide systems such as Chevrels have a major drawback
in terms of the energy density of a battery: the reduced electrostatic attraction backfires to
give low redox potentials. In addition, the gravimetric capacity is also reduced due to the
use of heavier sulphur and selenium compared to oxygen. As the energy density depends
on both the potential difference between the operating electrodes and their capacities, this
results in a lower energy density compared to other systems such as Na-ion batteries. Hence,
the seemingly contradictory goal of finding TM oxide cathodes which display low barrier of
Mg-ion diffusion need to be achieved to allow high energy density Mg-ion batteries.
Fortunately, the solution to this seemingly contradictory goal could be approached from
a structural point of view. Solid-state ionic diffusion should occur via a saddle point (energy
maxima along the diffusion pathway), commonly denoted as transition states. Local structural
similiarities between the local minima and saddle point can reduce the energy differences,
thus effectively reducing the activation barrier required for diffusion.[45, 46] In light of
this argument, systems with Mg-ions sitting in a high-energy coordination environment are
expected to show low diffusion barriers and good Mg-ion mobilities.
This is also interesting from a 25Mg NMR point of view, as the high degree of structural
distortion is likely to result in a strong quadrupolar coupling and hence broad spectra.
Developing experimental NMR techniques aided by computational predictions is thus likely
10 Introduction
Fig. 1.2 Structure of Mg6MnO8, shown with oxygen polyhedra around the metal ions.
to be valuable to investigating such compounds. Here, we briefly discuss the structural
aspects of the oxide materials under study in this thesis.
1.3.1 Mg6MnO8
First reported in 1954 by Kasper et al.,[47] Mg6MnO8 shows a defect rocksalt structure
with both Mg/Mn ordering and defect ordering: [Mg2+6/8Mn
4+
1/81/8]O
2− with the Fm3¯m
space group symmetry. A family of oxide and halides, such as Ni6MnO8 and Na6CdCl8,
adopts this structure, commonly called Murdochite or Suzuki-like phases; a high pressure
polymorph of Cu6PbO8 is also expected to adopt this structure.[48] This compound has an
interesting structure with ordered vacancies, where a vacancy is formed according to the
following defect reaction written in Kröger-Vink notation:
MnO2
2MgO−−−→Mn··Mg+2OxO+
′′
Mg (1.2)
In addition, the oxygen sublattice is slightly distorted (Figure 1.2) from the ideal rocksalt
structure due to the smaller ionic radius of Mn4+ (0.53 Å) compared to Mg2+ (0.72 Å).[49]
This should give a nonvanishing quadrupolar coupling constant for Mg, as the local envi-
ronment is no longer spherically symmetric. Mg site in this structure shows a m.mm site
symmetry, reflecting the distorted octahedra as opposed to the Mg in the MgO structure.
Although this material is not expected to display any electrochemical activity, these structural
properties make Mg6MnO8 a good reference material for performing 25Mg paramagnetic
solid-state NMR (Chapter 3).
1.3 Investigated Oxide Structures 11
1.3.2 Spinel Structures
Spinels are a group of metal oxide minerals with chemical formula AB2O4 where A and
B are metal ions. Various spinel-type TM oxides have also been investigated as cathode
materials for LIBs,[50] which makes it a natural extension to apply them for MIBs.
Spinels generally display a cubic Fd3¯m space group symmetry with the 32 oxygen
atoms occupying the 32e Wyckoff positions, forming an approximate cubic close-packed
(ccp) arrangement. Distortions from the ccp structure towards the [111] direction can be
described in terms of an oxygen parameter u (u = 0.25 for perfect ccp). As u increases, the
8a tetrahedral sites are enlarged at the expense of 16d octahedral sites.[51] In addition, there
exists empty octahedral 16c sites, which are important for cationic diffusion (vide infra).
A2+ and B3+ cations can occupy the 8a or 16d positions in the ccp lattice. If all 8A2+
ions occupy the 8a site, the structure is perfectly ordered and called a normal spinel. At the
other end we have 8A2+ ions occupying half of the 16d sites and 16B3+ ions distributed
equally among the 8a and 16d sites, called an inverse spinel. Most real spinels are somewhere
between the completely normal and inverse spinels, having some degree of disorder. This is
quantified in terms of the inversion parameter x: [A1−xBx]tet [AxB2−x]oct O4.[52]
Thermodynamics of cation distribution in spinels
The free energy of any spinel can be expressed as a function of x:[52, 51]
G(x) = H−T S
=U−T S+PV
=Uelec+UCFSE−T (Sint+Sconfig) (1.3)
Here we have assumed purely electrostatic interactions between the ions and ignored
any vibrational and volumetric contributions to the free energy. The remaining terms are
determined by electrostatic interactions (Uelec), crystal field stabilisation energy (UCFSE),
and internal (electronic) degree of freedom (T Sint).[53] Configurational entropy (Sconfig) is
maximised in the completely random distribution and is expressed as[52]
Sconfig =−kB
[
x ln(x)+(1− x) ln(1− x)+ x ln(x
2
)+(2− x) ln(1− x
2
)+C
]
(1.4)
12 Introduction
where C is a constant.* Electrostatic interactions are primarily determined by the cationic
radii r and charges. Here a balance between two factors exists: trivalent ions are better
stabilised in 6-coordinate octahedral sites by electrostatics,[54] but they cannot fully fill
the large octahedral site. The structure distorts along the [111] direction, changing the u
parameter:[51] small rB/rA results in u > 0.25, favouring the B3+ ions in octahedral sites
and the A2+ ions in the tetrahedral sites.
CFSE is important for TM ions with dn electronic configurations.[55] As the O2− ion
is a low-field ligand, almost all TM ions adopt the high-spin configuration;[53] the CFSE
can then be obtained from the tetrahedral and octahedral crystal field splitting ∆oct and
∆tet ∼ 49∆oct.[56] The CFSE effect is especially dominant for the Cr3+ ion, where d3 spin
configuration produces large CFSE and all known ACr2O4 spinels are normal.[57, 58]
The electronic degree of freedom, although small in magnitude, can be important in some
structures, e.g. MgV2O4. Here the V3+ (d2) ions can have orbital degrees of freedom only
in an octahedral configuration. This reinforces the small CFSE effect of V3+ ions, making
MgV2O4 a normal spinel.[53]
MgMn2O4
MgMn2O4, which shows a spinel structure with I41/amd symmetry, is different from the
above two compounds since the two Mn3+ (d4) ions in octahedral environments cause a
Jahn-Teller distortion of the original cubic spinel structure, resulting in a tetragonal spinel
(as evident from the space group symbol). This d4 electronic configuration is also important
for structural changes at high temperatures, combined with the inversion of the Mg and Mn
cations:
Mg2+(tet)+2Mn3+(oct)→Mg2+(oct)+Mn4+(oct)+Mn2+(tet) (1.5)
This charge disproportionation is favourable because of the smaller CFSE of tetrahedral
sites, resulting in a t32ge
2
g configuration of the inverted tetrahedral Mn atom, whereas a nearby
octahedral Mn atom tends to get oxidised to t32g state. This inversion behaviour is known to
occur around 800 ◦C,[59, 60] resulting in the following cation distribution:
[Mg2+1−xMn
2+
x ]tet[Mg
2+
x Mn
2+
x Mn
4+
2−2x]octO4 (1.6)
*As only the change in entropy as a function of x is relevant, the exact value of C is unimportant.
1.3 Investigated Oxide Structures 13
This disproportionation was studied with resistance measurements[59] and Mn X-ray
absorption studies.[30] As the Mn2+, Mg2+, and Mn4+ cations are all non Jahn-Teller active
in their respective coordination environments, increasing the temperature results in a removal
of the Jahn-Teller distortion. The sample undergoes a tetragonal to cubic transformation
above 1200 ◦C.[61]
Ab initio studies on the impact of this cation inversion on the Mg-ion mobility in
MgMn2O4 was also reported.[62] It was found that the tetrahedral (8a) Mg hopping via a
tet(8a)–oct(16c)–tet(8a) pathway could be considered as ‘open’ (<750 meV barrier) when
perfectly normal spinels are considered with no degree of cation inversion (x = 0). Barrier
heights of 475 meV and 717 meV were observed with dilute Mg and vacancy limits, re-
spectively. Upon increasing the degree of inversion, the barrier initially shows a decrease
then again increases to >1000 meV under a full degree of inversion (x = 1). In addition, the
octahedral (16d) Mg, if present, shows low migration barrier only if a nearby tetrahedral (8a)
site is either empty or occupied by a Mg. In this study, therefore, we restrict ourselves to
study only the perfectly normal spinel with 25Mg NMR.
MgV2O5
As discussed above, one way of achieving low migration barrier of Mg in oxides likely
requires unusual high-energy Mg coordination environments. This, in turn, calls the necessity
for structural flexibilities in designing the candidate host materials if one would like to
systematically investigate a particular TM ion.
Vanadium oxides are especially interesting in this regard due to the versatile structures
with different vanadium oxidation states, spanning from V2+ to V5+. Phase diagram of
the Mg–V–O system (Figure 1.3) shows the presence of numerous stable vanadium oxide
structures (at 0 K) with different averaged oxidation states; many other metastable states are
also reported in the literature. These include V2O5, V6O13, VO2, V5O9, V3O7, V4O7, V3O5,
V2O3, and substoichiometric VO1 –δ , where incorporation of metal atoms (Li, Na, Mg. . . )
gives rise to further structural versatility.[2, 63] Many of these vanadium oxides have been
studied as Li- and Na-ion battery cathodes, where the multiple possible oxidation states were
exploited to give good Li- and Na-ion storage capacity.[64]
Recently, two computational works were reported on a magnesium vanadate material
MgV2O5 which adopts a layered structure with potential 1-dimensional diffusion chan-
nels.[67, 68] This material is a natural extension from α-V2O5, which previously showed
reversible Mg insertion up to Mg0.5V2O5 at a potential of 2.3 V vs Mg/Mg2+. A full cycling
of Mg ions would be expected to yield 260 mAh/g capacity.
14 Introduction
V
2
O
3
V
MgV
2
O
4
MgO
Mg
149
V
V
2
O
5
Mg
2
V
2
O
7
O
2
VO
2
V
3
O
7
V
3
O
5
MgVO
3
Mg
3
V
2
O
8
Mg
Fig. 1.3 Compositional phase diagram (at 0 K) of the Mg–V–O system, calculated using
the Materials Project.[65, 66]. PBE+U functional was used with U = 3.25 eV for V. Stable
phases at 0 K are also labelled on the diagram. MgV2O5 exists at only 0.026 eV above the
hull.
1.3 Investigated Oxide Structures 15
α-V2O5 exhibits an AA-type stacking of the V2O5 layers. Introduction of a stacking
fault by translation of every other layer by a/2 results in a new polymorph, δ -V2O5, with
an AB-type stacking and additional puckering of the V2O5 layers (Figure 1.4). It was
shown that the V2O5 layers in the MgV2O5 structure adopt the δ -V2O5 motif,[69] which
has Mg activation barriers substantially lower than those predicted in α-V2O5 (0.6–0.8 eV
compared to 1–1.2 eV).[67] In the same paper, the intercalation phase diagram of δ -V2O5
was generated through a cluster expansion technique; it was also suggested that the V2O5
layer stacking is likely to remain metastable over a range of Mg stoichiometries, allowing
reversible cycling of this phase. However, direct electrochemical cycling and detailed NMR
characterisation of this compound has not been reported to date, despite several observations
on the 25Mg NMR of this material (without a detailed characterisation).[35, 34] This is likely
to be due to difficulties in preparing this material pure-phase, alongside the aforementioned
challenges in cycling this high-voltage cathode candidate.
Finally, we note that many of these compounds show interesting physical phenomena due
to the relatively diffuse nature of vanadium d-orbitals compared to the late TMs: VO2 shows
a metal-insulator transition at 340 K, MgV2O5 is a compound with competing magnetic
interactions in a spin-ladder structure, and LiV2O4 is an example of a ‘heavy fermion’
compound with the effective electron mass much greater than that of a free electron. Hence,
reliable ways for preparing these compounds and characterising the local structures would
be valuable, as it is in general difficult to control the precise oxidation states of these
compounds.[63]
16 Introduction
Fig. 1.4 Structures of the α-, δ -, and metastable ε-polymorphs of MgV2O5. Reproduced
from Sai Gautam et al.[67]
1.4 Outlook and Structure of This Thesis 17
1.4 Outlook and Structure of This Thesis
Despite several reports on observing 25Mg NMR spectra in MIB electrode materials being
presented before, no detailed systematic investigations on the NMR spectra have been
performed in many cases. For more complicated electrodes, it is likely that one will face
a challenge in obtaining good spectra and assigning the resonances. Ab initio methods,
combined with advanced NMR techniques, are thus needed to facilitate the understanding
and development of novel electrode materials.
Based on the motivations above, this thesis attempts to develop a combined computational
and experimental methodology in utilising 25Mg NMR in MIB research. In Chapter 2, we
first start by outlining the experimental and computational approaches used in this thesis.
In the following Chapter 3, various Mg TM oxides are investigated experimentally with
25Mg NMR, aided by computational predictions on the spectral parameters. As many of the
potential cathode materials contain paramagnetic TM ions, a systematic investigation on
paramagnetic 25Mg NMR parameters is shown, alongside the detailed report on the structural
and magnetic aspects on these samples. Application of advanced NMR pulse sequences to
enhance the sensitivity and resolution is also validated.
Following this, we turn our attention to MgV2O5, a promising candidate MIB cathode
material. Here the focus is on (i) developing synthetic approaches to this sample using
ab initio thermodynamic parameters, (ii) structural and magnetic characterisation of the
compound, (iii) validation of the 25Mg NMR techniques as developed in Chapter 3, and
finally (iv) attempts in cycling this compound in an electrochemical cell. A brief discussion
on the NMR and magnetic characterisation of MgV2O4 prepared through the same synthetic
approach is presented in the end.
In the final chapter, the focus is shifted towards an anode material Mg3Bi2. This com-
pound has been shown to display fast Mg kinetics via 25Mg NMR spectroscopy with elec-
trochemical cycling; by application of ab initio transition state searching and defect energy
calculations, we attempt to confirm the fast Mg dynamics. Investigations on the electronic and
structural aspects behind this fast ionic conduction are also presented. The thesis concludes
with the major achievements and implications for future researches in this direction.

Chapter 2
Background Theory
In this chapter, the physical grounds of the computational and experimental techniques used
in this thesis are reviewed, in three broad sections.
This chapter starts by introducing the ab initio computational techniques, including the
Density Functional Theory (DFT) and the relavitistic corrections that can be applied to the
standard DFT methods. Transition state searching approaches using ab initio methods are
then discussed.
This is followed by an introduction to the Nuclear Magnetic Resonance (NMR) spec-
troscopy, with particular focus on the solid-state and quadrupolar NMR techniques. A
separate section on the electron magnetism and the paramagnetic NMR theory is presented.
This chapter concludes with the thermodynamics behind the idea of carbothermal synthe-
sis central in Chapter 4, with a brief discussion on the X-ray diffraction as a characterisation
technique in crystalline materials.
2.1 Density Functional Theory
2.1.1 The Hartree-Fock Approximation
In principle, many of the physico-chemical behaviour of any quantum system can be described
exactly with the (time-independent) Schrödinger equation:
HˆΨ= EΨ (2.1)
20 Background Theory
where Hˆ is the Hamiltonian operator of the system, Ψ is the wavefunction, and E is the
energy corresponding to Ψ. By taking the modulus |Ψ|2 of this (complex) wavefunction, one
can extract the electron distribution and hence the related properties.
The Hamiltonian Hˆ describes all the possible interactions present in this system. In a
system containing N-electrons and M-atoms such as a molecule, however, the many-body
interactions in general cannot be solved exactly and approximations must be taken. For
an atom, the nucleus is much heavier than the electrons by a factor of 1820 (at least) and
we can consider them to be stationary in the frame of electrons (the Born-Oppenheimer
approximation) and subsequently the Hamiltonian can be written as (in atomic units)[70]
Hˆ =−
N
∑
i
1
2
∇2i −
N
∑
i
M
∑
A
ZA
riA
+
N
∑
i
N
∑
j
1
ri j
(2.2)
where the individual terms refer to the electron kinetic energy, electron-nuclear electrostatic
attraction for nuclei of charge ZA, and electron-electron repulsion. As the last many-body
term depends on the positions of other electrons, there are no exact solutions to this equation
and numerical methods are necessary. One such approach is to exploit the Fermionic nature
of electrons to variationally minimise the ground state wavefunction Ψ0; here the degree of
freedom lies in the choice of spin-orbitals. Such procedure yields the Hartree-Fock (HF)
equation in a form of
f (i)χ(xi) = εχ(xi) (2.3)
f (i) =−1
2
∇2i −
M
∑
A
ZA
riA
+ vHF(i) (2.4)
where V HF(i) describes the effective ‘averaged’ potential of all other electrons on the i-th
electron, similar to the idea of molecular field used in magnetism (Section 2.5.1). As this
equation is nonlinear, it should be solved self-consistently: the energy is minimised by
feeding in the resultant vHF into the new equation to solve the spin-orbitals χ until the
energy change falls below a set limit. This method of self-consistent field (SCF) is central to
quantum chemical calculations.
However, it needs to be stressed that the HF method does not take (Coulombic) electron
correlation into account by virtue of the fact that only an averaged potential vHF is used in
the calculation; it ignores the ‘hole’ around a single electron where the other electrons are
unlikely to be present due to the repulsive Coulomb interactions (but does take the Fermi
correlation into account). Thus, considering also the variational nature of minimisation
involved, HF energies are always higher than the ‘exact’ energies.
2.1 Density Functional Theory 21
2.1.2 The Hohenberg-Kohn-Sham Approach
One downside of the HF method is its associated computational cost and unfavourable
scaling. As the electron density could be directly obtained from the wavefunction and is
physically observable, methods using this 3-variable electron density n(r) instead of the full
3N-variable wavefunction have been investigated over years for a cheaper solution to the
Schrödinger equation.
n(r) already contains information about the total number of electrons and nuclear posi-
tions with charge ZA:[71] ∫
n(r)dr = N (2.5)
lim
riA→0
∂
∂ r
n(r) =−2ZAn(0) (2.6)
The relationship of energy to the electron density is formulated by the famous Hohenberg-
Kohn theorems.[72]
• In a system with the full Hamiltonian Hˆ = Tˆ + Vˆee + Vˆext where T is the electronic
kinetic energy, Vee is the Coulombic repulsion, and Vext is the potential energy due to
an external field, Vext and hence the ground-state energy E0[n] is a unique functional of
n(r).
• The energy functional E[n] is minimised if and only if n(r) is the true ground-state
electron density.
The Hohenberg-Kohn energy functional can be expressed as
E[n] =
∫
VNen(r)dr+
1
2
∫∫ n(r1)n(r2)
r12
dr1dr2+T [n]+Encl[n] (2.7)
The idea of Kohn and Sham was to do exact calculations on the kinetic energy and
nonclassical energy functionals T [n] and Encl[n] as far as possible, and then put the rest under
a collective ‘unknown’ functional called the exchange-correlation functional.[73] By use of a
similar variational approach to the HF method, the Kohn-Sham equations could be obtained
as:
fˆKSϕi = εiϕi (2.8)
fˆKS =−12∇
2+Veff(r) (2.9)
22 Background Theory
where ϕi are called the Kohn-Sham (KS) orbitals. From this we can extract the exact kinetic
energy of non-interacting electrons and re-write E[n] as
E[n] =
∫
VNen(r)dr− 12
N
∑
i
〈
ϕi | ∇2 | ϕi
〉
+
1
2
∫∫ n(r1)n(r2)
r12
dr1dr2+Exc[n] (2.10)
where the exchange-correlation functional Exc[n] accounts for all the unknowns in the Hamil-
tonian. It is independent of the system, and hence finding approximate functional forms of
Exc[n] is one of the major goals of DFT.
The most basic functional of these exchange-correlation functionals is the local density
approximation (LDA) functional, where the functional only depends on the electron density
n at any given point
ELDAxc [n] =
∫
n(r)εxc(n)dr (2.11)
where εxc(n)= εx(n)+εc(n) is the exchange-correlation energy of a single particle, calculated
for a homogenous electron gas of density n. Analytical expression for the exchange part
εx(n) is known, but further approximations must be employed for εc(n). This LDA functional
and its spin-polarised variant LSDA functional can accurately describe systems with a
homogeneous electron distribution such as metals. Extension of this method to include
higher order gradients in the density have resulted in generalised gradient approximation
(GGA) functionals which include a∇n(r) term and meta-GGA functionals which also include
a ∇2n(r) term.[74]
However, all DFT methods suffer from what is called a self-interaction error (SIE); this
arises due to the electrons ‘interacting’ with the mean field created by itself in the DFT
framework, thus resulting in an excessive delocalisation of the density compared to the true
density. For TM oxides where the d-electron correlation makes the system insulating (Mott
insulator), DFT methods often (incorrectly) predict a metallic groundstate. In the following
sections, we will look at two different ways of treating this error: Hubbard-U and hybrid
functionals.
2.1.3 The Hubbard-U Method
The LDA+U formalism was introduced to resolve this problem of strong correlation.[75]
In this model, the correlated electronic states (e.g. localised d-orbitals), are treated with a
Hubbard model where the electron hopping and on-site Coulombic interactions are taken
into account by two parameters t and U .[76] In the limit of U ≪ t, the electrons are more
favourable to ‘hop’ or delocalise and the DFT methods would correctly predict a metallic
2.1 Density Functional Theory 23
Fig. 2.1 Total energy as a function of electron count in the system. Black line gives the
analytical DFT energy, red line gives the exact energy, and the blue line gives the parabolic
energy difference given by the U-correction. Figure adopted from [81].
state. On the other hand, DFT performs poorly when t ≪U , as the strong on-site Coulombic
repulsion forces dictate that an electron ‘hop’ at the cost of U would be unfavourable; a
localised, insulating state then results.[77]
In the fully localised limit (FLL) formulation of this approach, the energy consists of
three components EDFT+U = EDFT +EHub +Edc. Here the Hubbard correction EHub and
the ‘double-counting’ term Edc (to compensate for the interactions already counted in the
DFT energy) are applied to the normal DFT energy EDFT. Different implementations of
this formulation exist;[78, 79] here we have used the rotationally invariant formulation of
Dudarev et al.[80] where only one parameter Ueff =U− J is used:
EDFT+U = EDFT+∑
l
U leff
2
[
(nl)2−∑
σ
Tr
[
(nlσ )2
]]
−∑
l
U leff
2
nl(nl−1) (2.12)
where nlσ are the occupation numbers of each local orbitals labelled by atomic site l and
spin σ . This formulation essentially ignores the dependence of U on the magnetic quantum
number (i.e. the orbital orientation).
An intuitive picture of this U correction could be envisaged by looking at the energy
change of a system as a function of electron counts (Figure 2.1). Exact wavefunction solutions
24 Background Theory
should result in kinks at integer occupations due to the changes in electron count changing
the effective band gap. DFT wavefunctions, however, are analytic: discontinuities in the
derivatives are less well represented in this case. This spurious curvature of the DFT energy
profile is the main reason for the underestimation of the band gap (or excessive electron
delocalisation). The U correction in this case attempts to restore the discontinuities by adding
a parabolic energy term to the DFT energy.[75]
Now the values of U could be computed simply as the spurious curvature of the energy
differences (blue line in Figure 2.1) as a second derivative term d2E/dn2 using a linear-
response theory.[82] Since it is not possible in general to control the site occupation in
plane-wave DFT implementations, the diagonal component of the response matrix χ can be
evaluated as a variation in nl due to α l , the perturbing potential on site l.
χll =
dnl
dα l
(2.13)
Then, it can be shown that d2E/dn2 =−dα/dn. Extracting the ‘noninteracting’ variation
in occupation χ0 arising from the rehybridisation of orbitals (obtained by fixing the charge
density), the U can now be calculated as
U = (χ−10 −χ−1) (2.14)
which is used in Chapter 4.
2.1.4 Hybrid Functionals
In contrast to the DFT approach, the SIE is explicitly cancelled out in the HF approach by the
exchange contribution to the energy. As LDA functional shows self-interaction error whereas
the HF exchange energy does not, incorporating a degree of HF exchange energy into LDA
functionals could cancel this error.[83] Several such ’hybrid’ functionals were developed,
such as the B3LYP[84], PBE0,[85] and HSE[86] functionals. The most popular of these
’hybrid’ functionals is the B3LYP functional which has the form
EB3LY Pxc = (1−a)ELSDx +aEHFx +bEBx + cELY Pc +(1− c)ELSDc (2.15)
where EBx is Becke’s exchange functional[87] and E
LY P
c is the Lee-Yang-Parr (LYP) cor-
relation functional.[88] B3LYP functional has been shown to give good results in many
systems.[89] In the ‘pure’ B3LYP functional a = 0.20, b = 0.72, and c = 0.81; however,
2.1 Density Functional Theory 25
recent results indicate that for solid-state magnetic calculations, a = 0.35 gives better re-
sults.[90, 91] Previous work on 6/7Li and 23Na hyperfine calculations indicate that the
experimental value lies between a = 0.20 and a = 0.35.[92, 93, 94] Hence in this study both
cases are considered.
2.1.5 Basis Sets in DFT Calculations
In general, the Hartree-Fock or Kohn-Sham equations cannot be solved directly to yield
a close-form analytical solution. Instead, the corresponding wavefunctions are usually
expressed as a linear combination of some basis functions {ηµ}:
ϕi =∑
µ
cµηµ (2.16)
This allows us to express the KS/HF equation in a matrix form, which is easily solvable
using numerical linear algebra algorithms. Two families of basis sets are widely used in DFT
programs: Gaussian-type orbitals (GTOs) and plane-waves.
GTOs have form[95]
ηGTO = Nxlymzne−αr
2
(2.17)
where xlymzn describes the rotational symmetry and e−αr2 describes the radial part. In this
approach, the basis functions are localised to a particular atom which makes it convenient
to think of them as ‘atomic orbitals’, and the overall wavefunction is expressed as a linear
combination of these basis functions: hence the name Linear Combination of Atomic Orbitals
(LCAO) method.
Another choice of basis sets is plane-waves ηPW = eik·r. In solid-state calculations
plane-waves are desirable as they explicitly contain the periodic boundary condition and
reciprocal space representations. The number of plane-waves used in the calculation is often
set by the cutoff energy where we set the energy of plane-wave ε = h¯
2
2me
|k+G|2 where G is
a reciprocal lattice vector .[96]
In paramagnetic NMR we are interested in the spin density at nuclear positions. This
is problematic as neither of the two bases can accurately represent the cusp condition at
the nucleus. This can be approximated in GTO bases by adding functions with large α . If
we wish to use plane-wave bases, pseudopotentials are normally used to represent the core
states and a more sophisticated scheme such as gauge-including projector augmented wave
(GIPAW) method is necessary to describe the core behaviour.[97]
26 Background Theory
In this study, GTO-type basis sets were used to accurately describe the core electronic
states using the CRYSTAL code (see below).
2.1.6 Solid-state DFT
In a periodic lattice we have an infinite array of primitive cells with a general lattice vector
given as g = n1a1 + n2a2 + n3a3. Since the system is periodic, any wavefunction of the
system with basis set {µ} should satisfy the Bloch’s theorem[98]
φµ(r+g; k) = eik·gφµ(r; k) (2.18)
Functions satisfying the Bloch’s theorem can be expressed as the Bloch functions
φµ(r; k) =
1√
N∑g
eik·gηgµ(r−Rµ) (2.19)
Such functions have translational symmetry in the k-space, hence we only need to
evaluate a number of sampled k-points in the irreducible Brillouin zone and then interpolate
between the k-points. Using the Bloch function we can represent the crystalline orbitals as a
linear combination
ψn(r; k) =∑
µ
cµn(k)φµ(r; k) (2.20)
which satisfies the Schrödinger equation labelled with k, evaluated in the real space. This
problem is analogous to the basis set expansion in the Kohn-Sham equation with an infinitely
large matrix. For details on approximate evaluation methods in this study using the CRYSTAL
code see [99].
2.2 Relativistic Quantum Chemistry: a Brief Introduction
2.2.1 Relativity and the Dirac Equation
Many important systems relevant to battery chemistries, such as Bi, Sn, and Pb for instance,
involve ‘heavy’ elements, i.e. elements with large proton number Z. For instance, Bi, which
is important as a high-rate MIB anode material (Chapter 5) has Z = 83; Pb, which finds
use in lead-acid batteries, has Z = 82; Sn, which is being widely investigated as secondary
battery anodes, has Z = 50. As Z is increased, the core electrons experience a significantly
2.2 Relativistic Quantum Chemistry: a Brief Introduction 27
enhanced electrostatic attraction to the nucleus, which is exhibited as a decrease in potential
energy and an increase in kinetic energy.
What this means in practice is that the electron kinetic energy Ek, and subsequently the
momentum p, ultimately reaches a level where the electron ‘velocity’ is a significant fraction
of the light velocity c. Under such conditions, the assumptions of classical electrodynamics
(Newtonian and Maxwellian) do not hold and the particles should be treated within the
framework of relativistic quantum theory. Subsequently, the Schrödinger equation should
now be extended to include the theory of relativity; this yields the Dirac equation[100](
βm0c2+ c
(
3
∑
n=1
αn pn
))
ψ(x, t) = ih¯
∂ψ(x, t)
∂ t
(2.21)
where the ψ is the electron wavefunction with the rest mass m0, c is the light velocity, pn are
the x,y,z components of the momentum. Solutions to this equation can correctly describe
the relativistic motion of the electrons. By introducing spin into the equation by means of
Pauli matrices αn and β , solutions to this equation could be obtained as a four-component
wavefunction
ψ =

ψ1
ψ2
ψ3
ψ4
 (2.22)
(i.e. superposition of four different components), which can describe the full behaviour of a
relativistic electron in terms of spin up/down electron and spin up/down positron (antiparticle
of electron).
In theory this equation could be solved for all particles, regardless of their relativistic
nature, to yield the correct wavefunctions and their energies. From the solid-state chemistry
point of view, however, this problem is unwieldy; often, simple nonrelativistic calculations
are performed with the relativistic ‘correction’ terms applied to the energy, which will be
discussed below.
2.2.2 Relativistic Corrections: the Mass-velocity and Darwin Terms
Most of the quantum chemical calculations have the aim of looking at ‘valence’ states,
i.e. states that participate the most in bonding interactions. As the elements for which
the relativistic corrections are meaningful have a lot of ‘core’ states that do not directly
participate in bonding, this calls for parametrising the core states. This is most often done
28 Background Theory
with pseudopotentials, as the core could safely be frozen. As generation of pseudopotentials
require a calculation of the atomic wavefunctions, different levels of corrections could be
applied while generating the wavefunctions (and subsequently the pseudopotential).
To see how this could work, let us consider the nonrelativistic expression for the electron
energy ENR:[101]
ENR = Ek +EV =
p2
2m
+V (2.23)
where p and m represent the electron momentum and mass, and V the potential energy. Under
relativistic conditions, the increase in electron ‘velocity’ means the kinetic energy Ek should
follow the relation E2k = m
2
0c
4+ p2c2 instead. Excluding the rest mass contribution m0c2, the
relativistic energy expression is shown as:
ER =
√
m20c
4+ p2c2−m0c2+V (2.24)
= m0c2
√√√√(1+( p
m0c
)2)
−m0c2+V (2.25)
Under the condition (p/m0c)≪ 1, by means of Binomial expansion the square root term
approximates to √√√√(1+( p
m0c
)2)
= 1+
1
2
(
p
m0c
)2
− 1
8
(
p
m0c
)4
+ ... (2.26)
which then gives the final energy expression (to a second order)
ER =
p
2m0
− 1
8
p4
m30c
2
(2.27)
From this we can see that the correction term for the Hamiltonian should be
Hˆ ′mv =
pˆ4
8m30c
2
(2.28)
which has the physical meaning of change in electron mass with the momentum, and is
accordingly called the mass-velocity term. Applying this as a first-order perturbation to a
2.2 Relativistic Quantum Chemistry: a Brief Introduction 29
hydrogen-like wavefunction ψn with an energy of E0n gives[101]
∆E ′mv = ⟨ψn
∣∣Hˆ ′1∣∣ψn⟩ (2.29)
=−E0n
Zα2
n2
(
3
4
− n
l+1/2
)
(2.30)
with α being the fine-structure constant, which now shows the dependence of energy on l.
For a given n, the effect would be the greatest for l = 0, i.e. the s-orbital, as they ‘approach’
the nucleus closer than any other orbitals.
The above mass-velocity correction arises from a purely relativistic point of view as-
suming a point mass/charge. From a microscopic point of view, however, electrons are
represented by a positional uncertainty on the order of its Compton wavelength λc = h/m0c,
which then gives a variation of potential inside the electron when placed under the nuclear
field.
The magnitude of this interaction can be evaluated by averaging the potential over a
sphere of λc, which yields:[101]
∆Hˆ ′d =
Ze2
4πε0
π h¯2
2m2c2
δ (x) (2.31)
and we can clearly see that this effect is nonzero only for the s-orbitals of l = 0 due to the
Dirac delta function. Evaluation of this as a perturbation for a hydrogen-like wavefunction
again yields an energy correction which is n-dependent in this case:
∆E ′d =−E0n
(Zα)2
n
(2.32)
This term, which can be thought of as an energy correction due to the positional uncertainty
of the electron at the nucleus, is again only significant for the s-orbital; this term is commonly
called the Darwin term. Combined together, the mass-velocity and Darwin terms represent
the relativistic interaction which happens to be strongest at the nucleus. Both effects are most
pronounced for the s-orbitals and have a quadratic dependence on the nuclear charge Z; thus,
these effects become stronger for the heavy atoms with large Z. As the correction for these
two effects are significant and also computationally inexpensive, most pseudopotentials/basis
sets take them into account when built. As these corrections do not involve explicit spin
vector components, they are termed ‘scalar relativistic’ corrections.
30 Background Theory
2.2.3 Relativistic Corrections at the Core and Valence: Spin-orbit Cou-
pling
As the Dirac equation is fundamentally about the spins and their behaviours under relativistic
conditions, this gives rise to a phenomenon called spin-orbit coupling (SOC). The classical
picture of an atom involves electrons circling around the nucleus which poses an electric
field E = −(x/r)dφ/dr. Under this picture, the electrons experience a magnetic field
B=−v×E/c generated by the nucleus: in the stationary frame of the electron, the positively
charged nucleus is seen to circle around it, and this accelerating charge creates a magnetic
field. This magnetic field is responsible for the angular momentum Lˆ coupled to the spin
momentum Sˆ, which can be expressed as a coupling of Sˆ and B:[101]
H ′so =−
e
mc
Sˆ ·B =− e
mc2
S · (v×x)1
r
dφ
dr
(2.33)
=− e
mc2
Sˆ · Lˆ1
r
dφ
dr
(2.34)
For a hydrogen-like atom, the electric potential φ varies as Ze2/(4πε0r), which then
gives the expected Z-dependence of Z4. Under this condition, the spin and angular momenta
are expected to be coupled to each other, thus giving the name spin-orbit coupling. Here the
quantum number J = L+S decides the energy, which results in an energy splitting between
states of same l. However, this term by definition vanishes with l = 0; other higher l-states
such as p, d, and f are significantly affected by this, resulting in a ‘fine structure’. For heavy
atoms, relativistic SOC often significantly influences the electronic structure and subsequently
their properties. However, treatment of this requires an explicit SOC Hamiltonian 2.34 which
is computationally demanding; this treatment gives the ‘fully relativistic’ solution.
2.2.4 The Zeroth-Order Regular Approximation
As seen from the derivation, validity of the mass-velocity term (Equation 2.28) entirely relies
on the approximation 2.26 holding true. It was, however, not explicitly justified why the
condition (p/m0c)≪ 1 should hold true, or even whether it does. Indeed, it was first pointed
out by Farazdel and Smith[102] that this condition may not hold true in presence of a strong
Coulomb potential V ∝−Ze/r in the vicinity of the nucleus, as the kinetic energy Ek and
subsequently momentum p will increase as far as the relativistic limit permits. Whereas this
normally has a small effect for valence properties such as bonding, it is largely problematic
for computing core properties such as chemical shielding and EFG tensors: the ‘perturbation’
2.2 Relativistic Quantum Chemistry: a Brief Introduction 31
applied by the mass-velocity term Hˆ ′mv may not be a small one. Subsequently, the computed
core states will be unreliable and so are the properties that rely on them.[103]
In this regard, a slightly better approximation to Equation 2.24 could be found by a
different kind of expansion:[104, 105]
ER =
√
m20c
4+ p2c2−m0c2+V (2.35)
=
p2c2
m0c2+
√
m20c
4+ p2c2
+V (2.36)
=
p2c2
(2m0c2−V )
(
1+ ER2m0c2−V
) +V (2.37)
=
p2c2
(2m0c2−V )
(
1+
ER
2m0c2−V
)−1
+V (2.38)
≈ p
2c2
2m0c2−V +V (2.39)
where the last approximation have used a zeroth-order expansion in ER/(2m0c2−V ) term.
Under most chemical conditions, E ≪ m0c2 (orbital energies are hundreds of eV compared
to rest mass energy 0.511 MeV); so this approximation is physically justified, unlike the
(p/m0c)≪ 1 used in the derivation for the mass-velocity term. This method is hence called
the Zeroth-Order Regular Approximation (ZORA) as the correction is ‘regular’ even for
the states close to the nucleus. The corresponding ZORA Hamiltonian can be thus written
as[106]
HˆZORA =
pˆ2c2
(2c2−V ) +
c2
(2c2−V )2 Sˆ ·∇V × pˆ (2.40)
where the scalar relativistic and the SOC terms are both included. This scalar relativistic
version of ZORA was first used to calculate the chemical shielding in 77Se and 125Te-
containing systems;[107] since then, it has been applied on various systems containing heavy
atoms such as 127I and 209Bi.[108, 109]
2.2.5 Handling of Relativistic Corrections: VASP versus CASTEP
At this point, a brief comparison between the two plane-wave codes used in this study, VASP
and CASTEP, is needed. While both codes utilise pseudopotentials to treat the core electrons,
VASP is provided with pre-generated pseudopotentials by the developers, which accounts for
the scalar relativistic mass-velocity and Darwin terms (Equations 2.28 and 2.31).[110, 111]
32 Background Theory
CASTEP, on the other hand, gives an option for the on-the-fly generation of pseudopotentials,
with different Hamiltonians available for the reference atomic calculations: Schrödinger,
Koelling-Harmon, or ZORA (only scalar relativistic). Such flexibility in pseudopotential
generation is not available in VASP; hence both CASTEP and VASP are tried for the electronic
structure calculations, especially to evaluate the effects of the relativistic corrections. Both
codes do not include spin-orbit coupling by default: this interaction needs to be switched on
in each code.
2.3 Transition State Searching: The HEF Method
2.3.1 Transition States: Definition and the Motivation for Searching
Them
A chemical reaction, whether it being an actual transformation or simply an ion hopping
inside a lattice, can be represented by a 3N-dimensional potential energy space (PES) where
the 3-dimensional coordinates of N atoms can change freely. Obviously, some of the atomic
arrangements are bound to be more stable than others, thus creating energy minima; multiple
local minima can be present alongside the one global minimum. For a single Mg atom
moving through an empty Mg sublattice in intercalation-type electrodes, for instance, all the
available Mg positions represent a possible local minimum, with the corresponding transition
pathways connecting these local minima.
For most of the time, the system sits in one of its minima since it is kept inside a stable
potential well. However, extra energies made available to the system (by means of thermal
energy or external potential, such as charging a cathode) can drive the system to walk an
uphill path in energy and ultimately make transformations happen from one state to another.
This pathway could be illustrated mathematically as a saddle point (Figure 2.2) connecting
the two minima, as the system must follow a path along an energy minimum perpendicular
to the reaction vector but ultimately go through an energy maximum (the energy barrier),
commonly referred to as the transition state (TS). Locating the transition state in any reaction
is important for (a) understanding the mechanism of reactions, as the local structure of TS
can give hints on why this particular path is chosen; and (b) calculation of activation barriers
in the reaction under concern, which is a key factor in determining the reaction rate.
Despite the importances, finding the TS in any reaction is known to be difficult since
there is no a priori knowledge of the 3N-dimensional energy landscape available to the
system. Since we already have a good knowledge of at least one local minimum involved in
2.3 Transition State Searching: The HEF Method 33
the reaction (again, stable Mg lattice positions for example) but not necessarily the pathway
connecting it to the another nearby local minimum, locating the TS and its energy then
reduces to a problem of finding the TS starting from a nearby energy minimum, down the
saddle point. Once the TS has been identified, both local minima could simply be identified
by walking the positive and negative downhills along the reaction path. Hence this problem
is called a ‘single-ended’ transition state searching, since both the endpoints are not fixed;
alternatively, one can look for the TS and the related pathway with fixing the both minima,
which is called a ‘double-ended’ TS searching. The idea of hybrid eigenvector-following
approach discussed here is a single-ended method; other popular techniques such as a nudged
elastic band (NEB) method are typically double-ended.[112]
2.3.2 The Hybrid Eigenvector-following Method
As discussed above, the problem of finding the TS starting from a local minimum requires
essentially finding an uphill path to the saddle point in one degree of freedom (maximising
the energy) while keeping the minimum energy in all other degrees of freedom. To achieve
this, knowledge of the full Hessian at any given point on the PES is typically required, as
it gives information on the local curvature around that point. However, the computational
cost associated with this approach when combined with ab initio methods is prohibitively
expensive; thus a different approach needs to be developed.
The method of hybrid eigenvector-following (HEF) TS search[113, 114] circumvents this
problem by exploiting the fact that one can often approximate the local PES to be quadratic
in nature, thus requiring only the first- and second-order derivatives. Then a variational
approach is taken to obtain the minimum eigenvalue of Hessian H without calculating the
full H and subsequently the reaction vector.
To see how this could work, let us assume an arbitrary reaction vector y at nuclear
coordinates x0 on the PES. We can define the Rayleigh-Ritz ratio, or the ‘expectation value’
for the eigenvalue λ corresponding to y as
λ (y) =
yTHy
yTy
(2.41)
where yT refers to the transpose of y. Minimising this ratio variationally to find a negative
value of λ equals finding the uphill path on the PES to yield the initial search direction. Since
calculation of the full Hessian is undesirable on an ab initio level, a quadratic approximation
34 Background Theory
y
i
y
min Gradient
search
Transi on
state
Ini al
guess
Fig. 2.2 Schematic illustration of the HEF method for TS searching. Detailed explanations
are presented in the main text. Saddle point figure was adopted from Wikipedia under a
CC-BY 3.0 license.
to the local PES is applied to compute this expectation value:
λ (y)≈ E(x0+ξy)+E(x0−ξy)−2E(x0)
(ξy)2
(2.42)
where E(x0) is the energy corresponding to x0 and ξ ≪ 1. Then a conjugate gradient
minimisation could be performed using a numerical derivative as defined below:
∂λ
∂y
=
∇E(x0+ξy)−∇E(x0−ξy)
ξ
(2.43)
The result of this is the smallest eigenvalue λ and the corresponding eigenvector y at x0.
This gives the uphill path on the point x0; once a step in this direction is taken, the system
is then minimised along the tangential direction using a gradient-based algorithm to find
the reaction path. This process, illustrated schematically in Figure 2.2, is repeated until the
gradient and the path both converge under a given limit, and the TS and reaction vector are
found as x0 and y.
2.4 Nuclear Magnetic Resonance 35
2.4 Nuclear Magnetic Resonance
The macroscopic properties and the physico-chemical behaviour of any material is closely
related to its microscopic structure at the atomic level. This requires understanding of the
local structures around the atom of interest, which can often be different from the long-range
ordered structure commonly probed by diffraction techniques. Local disorder, arising from
many different origins such as stacking faults, point defects, and ionic motions, to name a
few, significantly influences the battery performance; hence, experimental techniques to look
at atomic structures on a local level are valuable.
Nuclear Magnetic Resonance (NMR) is a very powerful technique to look at such local
structures. As we will see in the following sections, the physical origin of the various
NMR-related interactions make it suitable for observing the structure and dynamics of ions
at a local level, which is often difficult by conventional diffraction methods. Here we briefly
present the physical basis of NMR experiments, followed by more specific pulse sequences
utilised in this study.
2.4.1 Physical Basis of NMR Interactions
Here we briefly discuss the physical basis of relevant NMR interactions. Electron-nuclear in-
teractions are discussed in a separate section (Section 2.5) alongside the electron magnetism.
The Zeeman interaction
Nuclei possessing spin I can interact with an external static magnetic field B0 pointing along
the z-direction, resulting in the nuclear Zeeman interaction. Classically this can be thought
of as a spin magnetic moment precessing around the B0 direction with the Larmor frequency
ν0 = ω0/2π =−γB0/2π . The relevant nuclear Zeeman Hamiltonian for this interaction can
be expressed as[115]
Hˆ0 =−γ h¯B0Iˆz = hν0Iˆz (2.44)
where γ is the gyromagnetic ratio of the nucleus (in rads−1 T−1). For a system of spin-1/2
nuclei, this gives only two possible quantisation of the spins: |+1/2⟩ or |−1/2⟩ (relative to
the B0 field), with an energy separation ∆E = hν0. Under a thermal equilibrium, the spins
inside any sample on average occupy the two states according to the Boltzmann distribution,
which results in a net magnetisation along the z-direction.
Upon application of a radiofrequency (rf) pulse of amplitude ν1 = ω1/2π = γB1/2π on
the xy-plane which resonates with ∆E, the system can undergo a time-dependent deviation
36 Background Theory
|-1/2>
|+1/2>
rf FID
t
x
y
z
x
y
z
x
y
z
B
0
Fig. 2.3 Schematic illustration of the pulse-acquire experiment on a 2-level system. Both
spin state and vector formalisms are shown.
from the equilibrium, which is expressed as a superposition/linear combination of the two
allowed states | ± 1/2⟩. This results in a nonequilibrium state which evolves during the
length of rf pulse. If the pulse length tp is such that ω1tp = π/2, the two states are equally
populated after the pulse; thus this represents a 90◦ pulse from the vector model of NMR.
Analogously, a pulse with a length of 2× tp results in a population inversion between the
|±1/2⟩ levels. Thus, the spins undergo a forced oscillation from the B1 field, a phenomenon
observed in nutation experiments.
After the application of a pulse to perturb the population difference, the system relaxes
back to its equilibrium via a nutational motion along the field axis. This process involves a
dissipation of the energy difference as electromagnetic radiation (for the B0 fields pertinent
to NMR spectroscopy, in the rf-region) of the same frequency ν0, convoluted with a decay
function. This time-domain free-induction decay (FID) signal is recorded and Fourier trans-
formed to yield a frequency-domain spectrum, which shows the dispersion of signals with
respect to their frequencies. This forms the basis of a simple pulse-acquire (zg) experiment
in NMR spectroscopy, which is illustrated schematically in Figure 2.3.
Chemical shielding interaction
The Zeeman interaction above tells us that each isotope should exhibit a distinct Larmor
frequency ν0 in the presence of a field. Were this the only interaction, NMR could only be
used to differentiate between different isotopes which resonate at characteristic frequencies;
in order to gain insights on the different local arrangements of the same nucleus, a mechanism
which gives dispersion according to the local electronic arrangements is needed. The chemical
2.4 Nuclear Magnetic Resonance 37
shielding interaction, which is present in all systems (paramagnetic or diamagnetic) is the
first important interaction relevant to gaining chemical insights from NMR.
When placed under an external field B0, diamagnetic, or spin-paired, electrons by defini-
tion generates a local magnetic field which opposes the applied field.* This means that the
effective field Bloc at the nuclear position becomes marginally different from the applied field.
This in effect changes the Larmor frequency of the nucleus, where an increased electron
density should result in a enhanced ‘shielding’ of the nuclear spin from B0, and subsequently
a reduction in the effective Larmor frequency. This mechanism is commonly called the
‘chemical shielding’ interaction, and is the dominant mechanism for the NMR shifts for
diamagnetic systems.
The local field could be expressed in terms of the shielding tensor σ as:
Bloc = σ ·B0 (2.45)
which includes the effect of the local magnetic fields produced by electrons in the applied
field. Hence, the chemical shielding Hamiltonian can be written as
Hˆcs =−γhIˆ ·σ ·B0 (2.46)
The chemical shielding tensor can be separated into the symmetric and antisymmetric
components, and the symmetric part of the tensor is transformed into the principal axis frame
(PAF) where σ is diagonal to obtain the anisotropy parameters.[116] However, in NMR
experiments, the formalism of chemical shielding is inconvenient since it is referenced to
a fictitious nucleus of zero electron density (i.e. completely deshielded). Moreover, the
full shielding tensor components are difficult to obtain (unless a single crystal experiment
with goniometer is performed) and only the principal values could be measured; hence the
more convenient formalism of chemical shift is used, where the values of the shift tensor are
referenced to the isotropic shielding of a reference compound σref:[117]
δii =
σref−σii
1−σii (2.47)
*For elements heavier than 1H, the ‘paramagnetic’ contribution to the chemical shielding is also significant;
this arises from the electron excitation from the ground state into the high-lying orbitals. This contribution
often has the same sign as the diamagnetic term.
38 Background Theory
Under this formalism, we have a sequence of the principal values as δ11 ≥ δ22 ≥ δ33; the
anisotropy of chemical shielding can now be represented as[118]
δiso =
δ11+δ22+δ33
3
(2.48)
Ω= δ11−δ33 (2.49)
κ =
3(δ22−δiso)
Ω
(2.50)
in the Herzfeld-Berger convention. δiso gives the isotropic chemical shielding, Ω gives the
span of anisotropy (Ω≥ 0), and κ gives the skewness (i.e. axial symmetry; −1≤ κ ≤+1).
Axially symmetric shift tensor gives either κ =±1 depending on the value of δ22 relative to
δiso. The anisotropy can also be expressed in the more common Haeberlen convention, which
orders the principal values by their separation from δiso: |δzz−δiso| ≥ |δxx−δiso| ≥ |δyy−δiso|.
Using this convention, the anisotropies can be represented as[119]
δiso =
δxx+δyy+δzz
3
(2.51)
∆= δzz−δiso (2.52)
η =
(δyy−δxx)
∆
(2.53)
which is often more convenient for calculating the tensor components under spinning.
Quadrupolar interaction
Non spin-1/2 nuclei possess nonspherical nuclear charge distributions. Such quadrupolar
nuclei can interact with the traceless electric field gradient (EFG) tensor V at the nuclear
position according to
HˆQ =
eQ
6hI(2I−1) Iˆ ·V · Iˆ (2.54)
where Q is the nuclear quadrupole moment and e is the unit charge. In the PAF where V is
diagonal, it is conventional to define ηQ, the EFG asymmetry parameter in the PAF and CQ,
2.4 Nuclear Magnetic Resonance 39
the quadrupole coupling constant (usually defined in Hz).[117]
eq =Vzz (2.55)
ηQ =
Vxx−Vyy
Vzz
(2.56)
CQ =
e2qQ
h
(2.57)
From the structural point of view, CQ is a parameter which gives the degree of non-
spherical electron distribution; a large CQ represents a highly non-spherical distribution. η ,
on the other hand, represents the degree of axial symmetry present in the electron distribution.
For instance, a perfectly octahedral/tetrahedral environment should give a vanishing CQ and
η . Typical observed values of CQ for 25Mg lie between hundreds of kHz and several MHz.
When purely quadrupolar transitions are considered, e.g. in a nuclear quadrupole reso-
nance experiment in absence of any external field, this treatment of quadrupolar coupling is
sufficient on its own. In NMR experiments, however, usually the nuclear Zeeman interaction
is the dominant interaction and we must treat the quadrupole as a perturbation to the nuclear
Zeeman Hamiltonian (Equation 2.44). Usually a perturbation treatment up to second order is
sufficient; however, in nuclei with large quadrupole moment Q such as 185/187Re, higher order
effects are observed[120] and a solution considering the combined Zeeman and quadrupolar
Hamiltonian by means of numerical techniques is necessary.[121] Here, the experiment is
conducted in the laboratory frame and the sample is placed inside a static field B0. This
means that we need to consider the relative orientation between the B0 field direction and the
anisotropy tensor of interest. As the field is axially symmetric, only two angles are necessary
to express this orientation: the resonance frequency can be parametrised as ν(θ ,φ).
We now define the quadrupolar frequency νQ and the orientation-dependent quadrupolar
frequency ν ′Q:
νQ =
3CQ
2I(2I−1) (2.58)
ν ′Q = νQ
(
3cos2θ −1
2
+
ηQ
2
sin2θ cos2φ
)
(2.59)
Applying the quadrupolar Hamiltonian as a perturbation to the dominant Zeeman inter-
action gives the first- and second-order energy changes to the transition frequencies νm,m+1
from m+1→ m:
ν(1)m,m+1−ν0 = ν ′Q
(
m+
1
2
)
(2.60)
40 Background Theory
ν(2)m,m+1−ν0 =−
ν2Q
18ν0
{ [24m(m+1)−4I(I+1)+9]V+1V−1 (2.61)
[6m(m+1)−2I(I+1)+3]V+2V−2)}
where V are expressed in terms of spherical tensors (in the laboratory frame)[122]
V±1 =∓(Vzx± iVzy) (2.62)
V±2 =
1
2
(Vxx−Vyy± iVxy) (2.63)
The resulting first- and second-order energy shifts are complex, and are described for
the I = 5/2 systems in Figure 2.4 for an oriented single crystal (θ = 0 and ηQ = 0 for
illustration). A few insights from the first- and second-order energy corrections are discussed
below:
• The central transition (CT; m =−1/2 to +1/2) frequency for any half-integer I is not
affected by the first-order quadrupolar coupling (Figure 2.4).
• The second-order energy correction varies as 1/ν0, meaning that the second order
quadrupolar effect is reduced as the magnetic field gets stronger.
• There exists an isotropic term in the second-order energy correction, meaning that
the observed shift is the sum of chemical and quadrupolar shifts, even under MAS
(Section 2.4.2). Because of this, methods such as lineshape fitting or more advanced
pulse sequences are necessary to extract the isotropic chemical shift.
Nutation effects for the quadrupolar interaction
We have seen in Section 2.4.1 that rf pulses with amplitudes ν1 induce a time-dependent
perturbation to the spin system, resulting in a fluctuation of the spin populations which
exhibits itself as a nutation of the net magnetisation. Of course, this ‘perturbation’ needs to
be small relative to the main interaction in order for this approximation to hold true. For
the Zeeman interactions, this is always the case since nuclear Larmor frequencies are on
the order of ν0 ≈100-1000 MHz whereas the available rf field is typically ν1 ≈10-100 kHz.
Thus ν1 ≪ ν0 and the perturbative approach is justified.
For quadrupolar interactions, however, this is not always the case: for 25Mg which has
a CQ range of 1-10 MHz, the quadrupolar frequency νQ =0-1.5 MHz. Unlike the Zeeman
2.4 Nuclear Magnetic Resonance 41
B0=0 B0≠0, Q=0 B0≠0, Q≠0
(first order)
m = –5/2
m = –3/2
m = –1/2
m = 1/2
m = 3/2
m = 5/2
5ν
0
/2+5ν
Q
/3+10/27(ν
Q
2/ν
0
)
ν0
ν0
B0≠0, Q≠0
(second order)
3ν
0
/2-ν
Q
/3-2/45(ν
Q
2/ν
0
)
ν
0
/2-4ν
Q
/3-8/135(ν
Q
2/ν
0
)
-ν
0
/2-4ν
Q
/3-8/135(ν
Q
2/ν
0
)
-3ν
0
/2-ν
Q
/3+2/45(ν
Q
2/ν
0
)
-5ν
0
/2+5ν
Q
/3-10/27(ν
Q
2/ν
0
)
Energy levels for
I=5/2
θ=0 & η
Q
=0
Fig. 2.4 Schematic energy levels (not drawn to scale) in the presence of quadrupolar coupling,
illustrated for γ < 0 and I = 5/2 (the case for 25Mg).
42 Background Theory
interaction, the magnitude of this interaction could be comparable to the available ν1 field.
Two extremes could then be identified:[123]
• νQ ≪ ν1 (small CQ), where the pulse is strong enough to excite the whole spectrum
evenly. In such cases, the pulse is said to be ‘hard’, i.e. it excites the CT and STs
nonselectively. The system behaves in the limit of no quadrupolar interaction with the
same nutation behaviour.
• νQ ≫ ν1 (large CQ), where the pulse on resonance with the CT is unable to excite the
whole transition uniformly. Here, it is observed that the CT nutates faster than the
nonquadrupolar case by a factor of I+1/2. Selective excitation of the CT is possible
by means of a low-power pulse. The pulse in this case is said to be ‘soft’.
2.4.2 Solid-state NMR and Magic Angle Spinning
MAS for averaging anisotropic interactions in solids
In solution state NMR, rapid isotropic tumbling of molecules inside the liquid faster than the
NMR timescale, results an averaging of all anisotropic interactions (chemical shift, dipolar,
and quadrupolar); only scalar (J-)coupling and the isotropic shift (δiso) are observed in such
cases. In solid-state NMR, however, the system is stationary (in the absence of atomic motion)
on the NMR timescale. This means that the anisotropy averaged out in the solution-state
remains present.
Recall in Section 2.4.1 that we can represent the orientation of any anisotropic tensor
in a static field with two polar angles (θ ,φ). In well-ground powder samples, all possible
combination of the angles (θ ,φ) are randomly present; spectra in such cases exhibit a ‘static’
powder pattern which is a superposition of the individual resonances for each crystallite in
the system. Such spectra are influenced by any anisotropic interactions present in the system
(chemical shift, dipolar, and quadrupolar). Typically, this is undesirable since the resulting
broad spectra give poor resolution and sensitivity.
A solution to rectify this problem is to exploit the fact that the applied field is axially sym-
metric; under rapid averaging about an the axis at an angle β to B0, the anisotropic component
of any perturbation depends on the n-th order Legendre polynomials Pn(cosβ ):[124]
P0(cosβ ) = 1 (2.64)
P2(cosβ ) = (3cos2β −1) (2.65)
P4(cosβ ) = (35cos4β −30cos2β +3) (2.66)
2.4 Nuclear Magnetic Resonance 43
B
54.74°
Driving gas
Bearing gas
Spinning axis
Fig. 2.5 Schematic illustration of MAS experiments.
For the chemical shift and dipolar coupling Hamiltonians, a first-order perturbation suffices
as their magnitudes are small compared to the Zeeman Hamiltonian. This means that the
anisotropic interaction depends solely on the second-order Legendre polynomial P2(cosβ ).
Therefore, a choice of the angle β which removes this term should result in effective averaging
of the shift anisotropy. Such an angle β = arccos(
√
1/3) = 54.74◦ to the B0 axis is called
the ‘magic angle’;[125] rapid sample rotation about this axis removes the anisotropy for a
first order interaction. Mechanically, this is achieved by two types of gases: the ‘bearing’ gas
keeps the rotor afloat and the ‘drive’ gas rotates the turbine fins on the rotor. This technique
of Magic Angle Spinning (MAS) is essential to solid-state NMR spectroscopy to obtain
high-resolution spectra.
Typically, spinning frequencies νr on the order of 4 times the magnitude of the anisotropic
interaction is needed for complete averaging.[126] This represents different requirements
for νr depending on the sample. For instance, 13C typically has Ω ≃200 ppm (35 kHz at
16.4 T), whereas typical MAS rotors are capable of 5-60 kHz spinning depending on the
size; incomplete averaging of the anisotropic chemical shift interaction under slow MAS
results in spinning ‘sidebands’ which occur at multiples of νr from the centreband. The
envelope of these sidebands follow that of the original anisotropic interaction, from which
44 Background Theory
the anisotropic tensor components can be extracted to give extra information on the local
structure and dynamics.
The case of quadrupolar interactions: incomplete averaging by MAS
The case is more complicated for the quadrupolar coupling Hamiltonian HˆQ, as both the first-
and second-order perturbations need to be taken into account. The first-order perturbation, as
in the case of chemical shift, has a P2(cosβ )-dependence and is averaged out by sufficiently
fast MAS. Of course this is irrelevant to the CT as we have seen that the CT frequency is
not affected by quadrupolar coupling to first order; however, the STs undergo a significant
line-narrowing under MAS as the first-order broadening is now removed.
On the other hand, the second-order perturbation depends on both P2(cosβ ) and P4(cosβ ),
and the latter is incompletely averaged by MAS. This results in a second-order quadrupolar
interaction which gives rise to an isotropic shift according to the following equation:
δQ =−
ν2Q
30ν0
[
I(I+1)− 3
4
]
(1+
η2Q
3
) (2.67)
However, it needs to be stressed that MAS is still beneficial in this case as the P2 term is still
removed from the second-order perturbation; nonetheless, suppression of this second-order
effect has been an active area of research in solid-state NMR:
• As the second-order Hamiltonian depends on 1/ν0 (Equation 2.62), higher magnetic
fields can be used to suppress this second-order broadening. This is the most obvious
and straightforward method to obtain high-resolution spectra; however, design and
construction of strong superconducting magnet >20 T is very challenging. Higher
fields have been achieved by means of superconducting-resistive hybrid magnets (40
T),[127] or pulsed magnets (>100 T).[128]
• Examination of P4 reveals that two solutions are possible at β1 = 30.56◦ and β2 =
70.12◦. Thus, spinning at either of the angles can remove this second-order interaction.
Technologically this is achieved by two ways: Double-rotation (DOR) or Dynamic-
Angle Spinning (DAS).[129, 130] Both require specialised probes which can either
spin rotors at two different angles (DOR) or change the spinning axis while keeping
the sample rotating (DAS).
• As the above two methods require specialised hardware, pulse sequences that attempt to
correlate the other quadrupolar transitions to the CT under MAS have been developed.
2.4 Nuclear Magnetic Resonance 45
Two such examples are Multiple-Quantum Magic Angle Spinning (MQMAS), which
correlates the multiple-quantum transition to the CT, and Satellite-Transition Magic
Angle Spinning (STMAS), which correlates the satellite transitions to the CT.[131,
132] Despite the fact that both the CT and the ST are broadened by the second-order
effect, the STs have a lineshape that is narrowed by a constant factor compared to
the CT. Hence, the correlation between the CT and the ST could be determined by a
2-dimensional experiment, where the narrower STs can be correlated to the broad CTs.
However, MQMAS requires a rather strong rf pulse for effective excitations of the MQ
coherence; STMAS requires a very stringent setting of the magic angle.
2.4.3 Pulse Sequences
The pulse-acquire experiment introduced in Section 2.4.1 forms the basis of all NMR pulse
sequences. The idea of manipulating spin populations by means of rf pulses has led to the
application of a sequence of rf pulses to achieve a particular aim (e.g. enhanced resolution,
sensitivity...). In this regard, four types of pulse sequences were used in this work:
Hahn-echo
While the pulse-acquire experiment gives in general the best sensitivity for 1-dimensional
spectra, it suffers from a practical problem: the FID acquisition cannot start immediately
after the initial π/2-pulse due to probe ringdown and delays in transmitter gating. Typically,
delays of several µs are inserted to allow the pulse to decay leading to a loss in intensity; even
so the residual ringdown may still be present in the FID resulting in a rolling background
and difficulty in phasing the spectra. The Hahn-echo[133] experiment (Figure 2.6a) uses
a π-pulse as the refocusing pulse after the initial π/2-pulse in a π/2− τ−π− τ sequence.
FID acquitistion starts immediately after the second τ delay in the half-echo acquisition;
alternatively, the full echo can also be acquired. In solid-state MAS NMR, the delay τ is
typically rotor synchronised to an integer multiple of the rotor period tr.
The working principle of Hahn-echo experiments is best illustrated with the vector model
following the scenario in Section 2.4.1. After the initial π/2-pulse which rotates the net
magnetisation to the xy-plane, the magnetisation vectors undergo a dephasing about the
field axis with different offsets in the Larmor frequency (arising from various isotropic and
anisotropic interactions present). After a fixed delay τ , a π-pulse is then applied; this results
in magnetisation being refocused at the end of the second τ delay irrespective of the offset
frequency. The system is now in the exact same condition as following a simple π/2-pulse
46 Background Theory
minus any T1 and T2-relaxations that have occurred during 2τ . Acquisition of the FID can
then start at the echo top.
This technique of spin-echo is indispensable for low-γ 25Mg NMR, since the issues with
probe ringing and background at this low frequency (42.8 MHz at 16.4 T) could largely be
removed with spin-echo experiments.
Rotor Assisted Population Transfer
In this section and the following, CT signal enhancement techniques in quadrupolar NMR
are briefly discussed; for more detailed discussion the reader is referred to a recent review
paper.[134]
The RAPT pulse sequence (Figure 2.6b)[135] allows enhancements of the CT signal for
quadrupolar nuclei. The idea behind this pulse sequence is that under thermal equilibrium,
the spin levels arising due to quadrupolar interactions (Figure 2.4) are populated according to
the Boltzmann distribution, but the STs are not usually observed (unless we are specifically
probing them, as in STMAS experiments). Hence, irradiation of the spins with rf pulses that
can saturate, or invert, these satellite levels can result in a larger population difference for the
CT.
To understand how this is feasible, let us look at the case of a single crystal in a B0 field.
As the EFG tensor in single crystals can only have one possible orientation with respect
to this field, the transition frequencies determined by Equations 2.60 and 2.62 are fixed at
multiples of νQ (illustrated in Figure 2.4). Hence, the STs could be saturated by a train of
rf pulses at the offset frequencies equal to the STs, thus allowing an enhanced population
difference for the CTs. Maximum CT enhancements of up to I+1/2 could be achieved this
way.[135]
However, most of the solid-state samples of interest are in powder forms, thus having all
possible EFG orientations. This means that a dispersion of the resonances is present and rf
pulses at a fixed offset are not able to uniformly excite the resulting powder spectra under
static conditions. However, under MAS the individual ST isochromats are modulated in
frequencies by the rotational motion of the sample; thus the ST frequencies can effectively
‘sweep’ through a fixed offset multiple times. From Equations 2.58 and 2.59, we can see that
the outermost ‘edge’ of any given (static) ST powder pattern should occur at ±νedge away
from the centreband:
νedge = νQ =
3CQ
2I(2I−1) (2.68)
2.4 Nuclear Magnetic Resonance 47
and an excitation with a frequency of νmax, which is roughly half of νedge, is expected to give
a maximum saturation of the STs:
νmax =
νQ
2
=
3CQ
4I(2I−1) (2.69)
which was first experimentally demonstrated by Yao et al.[135] The first generation of the
RAPT pulse sequence used in their study employed a train of fast amplitude modulated
(FAM) pulses alternating in phase X-X¯ , which were applied at the same offset as the CT
frequency. Manipulation of the satellite level populations was achieved by tuning the pulse
lengths tp which results in an effective RAPT modulation frequency νm = 1/2tp. In the
Fourier transformed frequency dimension, this equals harmonics in odd multiples of νm;
these harmonics can then be used to selectively saturate the satellite populations by setting
νm = νmax. It was successfully demonstrated that under sufficiently slow MAS and with a
large number of pulses, this ST saturation could yield enhancements in the CT population
differences, yet still short of the expected maximum; for instance, an enhancement factor
of 2 was reported for the 27Al (I = 5/2) spectrum in diamagnetic polycrystalline albite
NaAlSi3O8.[135, 136]
While the RAPT pulse sequence can be used to enhance the CT sensitivity, another inter-
esting application of the RAPT pulse sequence is measurement of the quadrupolar coupling
parameters CQ and ηQ.[137] By varying νm, an enhancement profile can be generated which
exhibits a maximum when νQ matches νmax. Since the first-order quadrupolar lineshapes
of the STs have widths that only depend on the CQ and not on η , the enhancement returns
to unity at νedge. Additionally, the enhancement edges can be used to find out qualitative
information on ηQ as steep edges correspond to a large ηQ.
However, the range of νm that can be explored in this manner is principally limited by the
lower limit of pulse length the spectrometer can generate (only pulses longer than 0.5-1 µs
can reliably be generated from modern day spectrometers). Also, an important consideration
when saturating the satellite levels is that the pulses should be selective to the STs without
affecting the CT, or the generated population difference would be destroyed. In this regard,
Gaussian pulses at given offsets νoff could also be used for selective ST saturation; this
second generation of the RAPT pulse sequence is called Frequency Switched Gaussian
RAPT, or FSG-RAPT.[136, 137] As the probe bandwidth is proportional to the resonance
frequency, however, the available bandwidth at 25Mg frequencies is quite narrow and often
large values of νoff cannot be applied. This means that the enhancement maximum, rather
48 Background Theory
than the enhancement edge, is more straightforward to observe experimentally with smaller
values of CQ, which we investigate in Chapter 5 for MgV2O5.
For the paramagnetic samples under consideration, fast relaxation effects by the param-
agnetic electrons also need to be taken into account; as Gaussian pulses are typically longer
than normal square pulses (10-20 µs versus 2-5 µs), the additional relaxation that takes place
during this period is expected to make the RAPT enhancement less efficient when Gaussian
pulses are used for these samples.
Double Frequency Sweep
Another way of enhancing the CT population difference is to invert the STs instead of
saturating them.[138] This method is expected to yield even higher enhancements (2I if all
the satellite populations could be inverted), but comes at a cost of longer, more complex
shaped pulses to invert the magnetisation. Normal hard pulses will saturate the STs instead
of inverting them, as we have seen earlier for RAPT.
The solution is to apply a small rf field and then slowly ‘sweep’ it from off resonance to
the desired transition. Under sufficiently slow sweep conditions, the magnetisation vector
is in equilibrium with the applied rf field at all times (adiabaticity condition) and slowly
nutate from the initial equilibrium along the +z direction (off resonance), to the xy-plane
(on resonance), and to the final inverted state along the −z-direction (off resonance). This
condition is achieved by a linear frequency sweep from both high and low frequencies
towards the desired STs, hence the title Double Frequency Sweep (DFS). This typically
requires a very long low-power pulse (on the order of several ms) to meet the adiabaticity
condition; significant paramagnetic relaxation during the sweep period makes DFS less
useful than RAPT methods, as we will see later (Chapter 3).
Quadrupolar Magic Angle Turning
The QMAT experiment by Hung and Gan[139] is designed to identify the isotropic shift
from a series of spinning sidebands, as frequently seen in MAS experiments. The QMAT
pulse sequence, shown in Figure 2.6c, consists of nine π-pulses where the even-numbered
π-pulses remain fixed, but the odd-numbered π-pulses are delayed by incrementing t1. When
the MAT condition is satisfied, the net evolution of the sideband modulations sums to zero
and the centreband frequency (including the chemical shift and quadrupolar shift) evolves as
ν0t1, resulting in a resonance in the F2 dimension. This enables us to extract the isotropic
shift, ν0.
2.4 Nuclear Magnetic Resonance 49
pi/2 pi
τ
r
τ
r
acq.
x y
(a)
pi/2
τ
p
τ
p
τ
r
τ
r
acq.
x x
pi/2
n
pi/2 pi
yx
(b)
pi acq.
x xx
pipi/2 pi
x x
pipi
x x
pi pi
x x
pi pi
x
τr/5 2τr/5 3τr/5 4τr/5 τr0
(τr-t1)/10 (τr+t1)/10
(c)
Fig. 2.6 (a) Hahn-echo pulse sequence. (b) RAPT-echo pulse sequence. (c) QMAT pulse
sequence.
50 Background Theory
2.5 Magnetism and the Paramagnetic NMR Shifts
The above description of chemical shielding holds principally for diamagnetic samples,
where the paired electron density near the nucleus shields the external magnetic field. In
materials with localised unpaired electrons with spin S, however, the electron spin also
possesses a magnetic moment[140]
µe =−µBgeS (2.70)
where the Bohr magneton is defined as µB = eh¯/2me and ge = 2.0023 is the free electron
g-factor. Inside an applied magnetic field B0, these (2S+ 1)-fold degenerate spin states
split in energy by means of the electron Zeeman interaction, analogous to the nuclear spins
described above:
E(ms) = msµBgeB0 (2.71)
Transitions between these levels form the physical basis of Electron Paramagnetic Resonance
(EPR) spectroscopy. As this electron magnetism is essential to understanding the calculations
of the paramagnetic shifts central to this work, a brief discussion on the principles of electron
magnetism is presented below. Using these ideas, methods to calculate the paramagnetic
(Fermi-contact) NMR shifts are discussed in Section 2.5.5.
2.5.1 The Basics of Electron Magnetism
For a coupled n-electron spin system with spin quantum number S, the combined Zeeman
and magnetic Hamiltonian can be expressed as
Hˆ =−∑
J
∑
(i, j)
Jsis j +geµB∑
j
B0s j (2.72)
where si and s j denote the spin quantum numbers on each site. The sum over spin pairs
(si,s j) with magnetic interactions (J < 0 denotes antiferromagnetically coupled spins) is
included to express the effect of coupled electron spins. The idea of Weiss to solve this
equation to obtain the energy states is to approximate the effect of neighbouring spins
as an effective molecular field Bm f , which is proportional to the expectation value of the
total magnetisation M = ngeµB⟨ms⟩.[140] Solutions to this system feature several important
observations under a low-field approximation (molecular fields are typically much stronger
than the applied magnetic field under NMR conditions) and counting only the nearest-
neighbouring interactions (z is the number of nearest neighbouring spins):
2.5 Magnetism and the Paramagnetic NMR Shifts 51
• There exists a critical temperature
T ∗ =
2zS(S+1)
3kB
J (2.73)
above which the system loses spontaneous ordered magnetisation; the thermal energy
is large enough to arrange the electrons in a random orientation. The system is then
called ‘paramagnetic’. For J < 0 (antiferromagnetic), this temperature is called the
Néel temperature TN .
• Above T ∗, the magnetic susceptibility χ = M/B0 can be expressed as
χ =
C
T −Θ (2.74)
where
C =
ng2eµ2BS(S+1)
3kB
(2.75)
is called the Curie constant, which depends on the magnitude of electronic spins
present in the system (S) and Θ is the Curie-Weiss constant, which gives the strength
of interaction. For frustrated magnets, typically |TN | ≤ |Θ|, as it is more difficult for
the spins to order themselves.
• The scaled magnetisation Φ, which gives the fraction of magnetisation M to the
maximum possible saturated Msat, is given as
Φ=
M
Msat
=
B0µ2eff
3kBµBS(T −Θ) (2.76)
where we have substituted the spin-only moment µeff = ge
√
S(S+1)µB.
2.5.2 Magnetic Measurements with a SQUID Magnetometer
As paramagnetic NMR fundamentally depends on the response of paramagnetic electrons
inside the NMR magnet, its room temperature magnetism must be studied to allow the
link between calculation and experiment. Although various experimental methods exist to
measure the magnetic behaviour of materials, the instrument most widely used today is the
superconducting quantum interference device, or SQUID.
The working physical principle of SQUID is the Josephson effect, where a supercurrent
flow is observed across an insulator staged between two superconductors. Two identical
52 Background Theory
Josephson junctions are used; originally, current I flows through the two junctions equally.
Upon application of external magnetic flux, an extra current is induced through the coil,
resulting in a voltage difference across the two junctions. The induced field (and hence
induced voltage) oscillates as the external magnetic flux is increased, because the flux inside
the coil must be an integer multiple of the magnetic flux quantum Φ0. This allows a very
sensitive measurement of the magnetic flux.
2.5.3 Electron-Nucleus Interactions in Paramagnetic Systems
Intuitively speaking, electronic spin momentum could be understood as a small magnet which
generates a magnetic field of its own. When present close to a nuclear spin momentum, the
two spins can then couple to each other to cause a split in both energy levels, called the
hyperfine interaction.
It is important to stress that the individual electron spins fluctuate very rapidly in param-
agnetic systems to which most battery cathode materials belong. EPR experiments indicate
that this fluctuation is on the order of nanoseconds to femtoseconds,[141] which is much
faster than the typical correlation time of nuclear spins (100s of seconds to microseconds
for the NMR timescale); thus, the nuclear spin can be treated as fully decoupled from the
electronic spin transitions ms and only ‘sees’ a time-averaged state of ⟨ms⟩, which is of course
non-zero under an applied B0 field. This net electronic magnetic moment ⟨ms⟩ is expressed
as the population difference between the parallel and antiparallel moments to the field.
The magnitude and orientation of this interaction can be expressed in terms of a hyperfine
coupling tensor A with the relevant hyperfine coupling Hamiltonian defined as
Hˆh f = Sˆ ·A · Iˆ (2.77)
where Sˆ and Iˆ refer to the electron and nuclear spins. A can be further divided into the
isotropic and anisotropic components.[142] Under high fields, there exists four different
coupling mechanisms:
A = (ANRiso,FC +A
SO
iso,FC) ·1+(ANRdip+ASOdip) (2.78)
• ANRiso,FC: Nonrelativistic, isotropic through-bond (Fermi contact) shift, in which the
localised unpaired spin density can delocalise through a chemical bond pathway to the
observed nucleus
2.5 Magnetism and the Paramagnetic NMR Shifts 53
• ANRdip: Nonrelativistic, anisotropic through-space (electron-nuclear dipolar) shift, in
which the electronic spin moment is coupled to the nuclear spin moment through
dipolar coupling
• ASOiso,FC: Relativistic, isotropic pseudocontact shift, where the spin and orbital angular
momenta (if present) both contribute to the electron magnetic moment and ultimately
couple to the nuclear spin
• ASOdip: Relativistic, anisotropic dipolar contact shift, which happens via a through-
space dipolar coupling between the orbital angular momentum and the nuclear spin
momentum
Pseudocontact shift may have an effect in the shifts of MgV2O4 (S = 1 in an octahedral
crystal field) where an orbital degree of freedom is present; but previous reports have shown
that the magnitude of such pseudocontact interactions is an order of magnitude smaller than
the main Fermi contact interaction[143] and they are not considered further. This leaves us
with the Fermi contact and electron-nuclear dipolar interactions, which we will discuss in
detail below.
The Fermi contact interaction
In chemical bonding, all bonds are covalent to some extent. Even in metal oxides where one
naïvely expects an ionic bonding, some degree of covalency exists; hence, an electronic spin
can delocalise through chemical bonding (similar to the mechanism of spin-spin coupling
often observed in solution NMR), resulting in a finite unpaired spin density on nuclear
positions. This causes a small change in the local magnetic field at the nucleus, resulting in
coupling to the electronic spins. This interaction is isotropic and commonly called the Fermi
contact (FC) interaction.
As the FC interaction is a through-bond interaction, the net spin transfer depends on the
nature of the orbitals involved in bonding; in particular the symmetry of the orbitals is the
most important factor in determining the sign and magnitude of coupling. In TMs, unpaired
spins occupy the d orbitals; for the observed nucleus, however, only the s orbital is important
since it has a finite value of the wavefunction value at the nuclear position. Selected cases of
resulting spin density for the 90◦ and 180◦ bond angles are shown in Figure 2.7.
A further important point about the FC interaction is its additive property; i.e. the
observed Aiso can be expressed as a sum of different bond pathways connecting the observed
atom to the electronic spin. This leads to a rational way of explaining the FC shifts by looking
54 Background Theory
Fig. 2.7 Illustration of different spin density transfer mechanisms in TM−O−Li bonds,
with 90◦ and 180◦ bond angles. Figure reproduced from Carlier et al. [144]
2.5 Magnetism and the Paramagnetic NMR Shifts 55
(a) (b)
Fig. 2.8 Illustration of calculating J by a broken symmetry supercell approach. In (a), the
total energy equals Ea = E0+8J1. In (b), the total energy equals Eb = E0. Subtracting the
energies yields J1 = (Ea−Eb)/8.
at the bond geometry and the spin distribution, and can be used for estimating different bond
contributions to the observed shift, as explored extensively in Chapter 3.
The Electron-nuclear dipolar interaction
This interaction does not require a chemical bonding to be present between the two atoms;
rather, it describes the through-space dipolar coupling of spins via the magnetic field gener-
ated by electron spins.
The dipolar interaction varies as 1/r3, where r is the distance between the spins. As this
interaction depends on the relative orientation between the two spins, it is anisotropic and
also contributes to the observed anisotropy of the resulting NMR spectrum. Hence, the same
argument of chemical shielding anisotropy could be applied to this dipolar interaction with
the same convention on the tensor anisotropy (Section 2.4.1).
2.5.4 Ab initio Calculation of the Magnetic Coupling Parameter J
Often, one wishes to know the size of magnetic coupling J present in the system. For instance,
complete ab initio calculation of the Fermi contact shift (see Section 2.5.5) requires precise
knowledge of the magnetism at finite temperatures; this can be done either ab initio or by
experiment using a SQUID magnetometry (Section 2.5.2).
56 Background Theory
The Hamiltonian 2.72 can be rewritten, in an absence of external magnetic fields, as
Hˆ =−∑
J
∑
(i, j)
Jsis j (2.79)
Using this Hamiltonian, one can determine the J’s by a broken symmetry approach in
a supercell by calculating the DFT energies of various ferromagnetic, ferrimagnetic, and
antiferromagnetic spin configurations (Figure 2.8). Then a multivariate linear regression
can be performed to yield the interaction parameters Jn. In this definition of Hˆ, positive J
corresponds to a ferromagnetic interaction, and negative J corresponds to an antiferromag-
netic interaction. Using this value of J, the Brillouin function for magnetisation is solved
self-consistently at various temperatures to yield the susceptibility χ(T ) as a function of
temperature.[140] A Curie-Weiss fit is then performed to obtain the Weiss constant Θ.
2.5.5 Ab initioCalculation of Paramagnetic NMR Parameters from the
Curie-Weiss Law
Hyperfine shift parameters
Paramagnetic NMR parameters can be calculated ab initio using the scaling approach of Kim
et al.[92] assuming the FC as the dominant shift mechanism. This method takes Aiso and
Aaniso from a DFT calculation using a ferromagnetic spin configuration (performed at 0 K).
As the samples are in a paramagnetic regime in the temperature range of NMR experiments,
the electron magnetic moment needs to be ‘scaled’ from the (maximum) ferromagnetic value
to values at finite temperatures. This is performed by scaling the ferromagnetic moment by a
Curie-Weiss type factor Φ as defined in Equation 2.76. From this, the FC shift δiso could be
obtained as
δFC =
106Aiso
2hν0
Φ (2.80)
Aiso =
2
3
µ0µBµNgegI|Ψα−βN |2 (2.81)
where h is the Planck constant, ν0 is the nuclear Larmor frequency in Hz, ge is the free
electron g-factor, gI is the nuclear g-factor, µB is the Bohr magneton, and µN is the nuclear
magneton. Most importantly, |Ψα−βN |2 refers to the spin density at the nuclear position N
which, assuming a point nucleus model, can only be nonzero for orbitals with s-symmetry.
2.6 Carbothermal Reaction: Thermodynamics 57
In the above formalism, shift anisotropy can also be computed from the electron-nuclear
dipolar interaction tensor Ti j and scaled in the same way to obtain the anisotropic components
of the shift tensor δi j :
δi j =
106Adipi j Φ
2hν0
(2.82)
Adipi j =
Ti j
4π
µ0µBµNgegI (2.83)
After this δi j can be diagonalised to obtain the principal components δii which can be
used to calculate the shift anisotropy as detailed in Section 2.4.1.
Euler angles: relative orientation of the shift and EFG tensors
For systems with a large shift anisotropy tensor and EFG tensor (such as Mg6MnO8 in
Chapter 3), the relative orientation of the two tensors is also important for accurate fitting of
the spinning sideband manifolds. This can be represented by a combination of three Euler
angles α , β , and γ , where the angles represent rotations of the electric field gradient principal
axes to match the Aanisoii principal axes. Hyperfine and electric field gradient tensor principal
axes are ordered such that
∣∣Ayy∣∣≤ |Axx| ≤ |Azz| and |V22| ≤ |V11| ≤ |V33| (i.e. the Haeberlen
convention with setting Aiso = 0). The ZYZ convention and counterclockwise (positive)
rotation are assumed. The full rotation matrix is shown in Equation 2.84.
x
′
y′
z′
=
 cosα cosβ cosγ− sinα sinγ sinα cosβ cosγ+ cosα sinγ −sinβ cosγ−cosα cosβ sinγ− sinα cosγ −sinα cosβ sinγ+ cosα cosγ sinβ sinγ
cosα sinβ sinα sinβ cosβ

xy
z

(2.84)
2.6 Carbothermal Reaction: Thermodynamics
2.6.1 Using Carbon as a Reducing Agent
Setting fire to carbon results in a combustion reaction with atmospheric O2, producing a
mixure of CO and CO2. If, however, the carbon is mixed with a metal oxide and heated under
an anaerobic condition, the carbon can remove oxygen from the metal oxide to effectively
reduce the metal oxide to a metal. This principle, of course, has been exploited in smelting
furnaces for several centuries to extract metals from their ores, where a cheap source of
carbon such as coke is used.
58 Background Theory
Let us assume that V2O3 is to be reduced to V using this method. The corresponding
reduction reaction would read:
2
3
V2O3 −−→ 43V+O2 (2.85)
where the equation is normalised to 1 mol of oxygen. The spontaneity of this reaction then
could be expressed in terms of the free energy change. Under constant pressure conditions,
the change in Gibbs free energy, ∆G, is the relevant quantity where a negative ∆G would
give a spontaneous reaction.
The overall Gibbs free energy change could be separated into the enthalpic and entropic
components
∆G = ∆H−T∆S (2.86)
where ∆H is the enthalpy change, ∆S is the entropy change, and T is the temperature. As
this reaction involves only solids on the reactant side and releases O2 as a product, ∆S must
be positive, indicating the reaction becomes more spontaneous as the temperature increases.
Of course, chemical intuition tells that reduction of metals just by heating would require a
very high temperature, as the strong metal-oxygen bond dictates a very large enthalpic penalty
to this reaction. Hence, to reduce this energy penalty, carbon is added to the oxides and
heated to drive off CO and CO2; thus this method is named carbothermal reduction (CTR).
This produces nominally triple bonded CO and two double bonded CO2 upon oxidation:
C(s)+O2(g)−−→ CO2(g) (2.87)
2C(s)+O2(g)−−→ 2CO(g) (2.88)
As the first reaction involves the same number of gaseous species on both sides of the
equation, a flat line is anticipated for the ∆G−T plot since the T∆S contribution to ∆G is
expected to be small. Conversely, a downward sloping curve is expected for the second
reaction since ∆S > 0; both lines follow the expected trend.
More importantly, the two lines cross at a temperature around T = 973 K (700 ◦C): at
lower temperatures, the CO2-reduction is thermodynamically more favoured, whereas the
CO-reduction would be more favoured at elevated temperatures. This implies that the CTR
treatment temperature has to be kept below 973 K to generate stoichiometrically reduced
products, as the required ratio of carbon to oxygen increases from 1:2 to 1:1 above this
temperature.
2.6 Carbothermal Reaction: Thermodynamics 59
For the possible redox reactions, a convenient way of representing the temperature
dependence of ∆G is the Free energy-temperature diagram, or the Ellingham diagram. In this
diagram, ∆G is plotted as a function of T , where all the energies are normalised to 1 mol of
O2 to allow direct calculation of ∆G for the full reaction. A sample Ellingham diagram is
shown in Figure 2.9.
For transition metals with several possible oxidation states, thermodynamic considerations
could reveal conditions for a partial reduction, based on the ∆G of the reactions involved.
This idea lies at the heart of the CTR method used in this work.
2.6.2 Ab initio Computation of G for Solids
Determining the appropriate CTR reaction conditions require knowledge of G, or equally,
H and S. For many binary oxides, experimental values for the H and S are well known and
often available in tabulated forms. Many of the complex oxides, however, lack such data, and
it would normally require calorimetric measurements to determine the values of H and S.
Fortunately, the development of classical pair potential methods and the advent of quan-
tum chemical methodologies allows a reasonable estimate for these thermodynamic parame-
ters.
G for any compound can be further expanded into
G = H−T S =U +PV −T S (2.89)
where the internal energy U , thermal expansion (work against external pressure) PV , and
lattice vibration (entropy) T S terms are included. In theory, all three terms could be obtained
from any classical or ab initio calculations: U could be equated to the zero-temperature
cohesive energy of a system from a given reference point; PV could be estimated from the
volume-energy relationship by varying the external pressure, commonly called the equation
of state (EOS); T S could be obtained from the energies of lattice phonon modes at finite
temperatures (in absence of other compositional or defect-related disorders). However, in
many solids the latter two terms are often negligible compared to the U , since (i) solids have
small compressibility due to their close packed nature and (ii) phonons only represent a small
displacement of atoms from their equilibrium positions, limiting the level of disorder present
in solids. Thus the PV and T S terms could be safely neglected from the calculation to yield
the approximation G≈U . This approximation allows us to avoid calculating the EOS and
phonon energies, which are computationally expensive especially at an ab initio level.
60 Background Theory
Fig. 2.9 Ellingham diagram, reproduced from the original construction by Ellingham.[145]
2.6 Carbothermal Reaction: Thermodynamics 61
However, calculation of U with ab initio approaches require a further consideration
of the functional used. As the ∆G we are interested in represents a change in oxidation
states of the system (e.g. MgV2O6 + C −−→ MgV2O5 + CO), methods are needed to
accurately treat the localised electrons in these Mott insulators. Hubbard-U corrections,
although computationally inexpensive, are sensitive to the oxidation states and even the local
environment in which the metal ions reside; it have been shown to produce large errors in
energies for reactions involving a change in oxidation states.[66, 75, 146] Thus, the more
computationally expensive hybrid functionals are necessary not only to obtain the correct
insulating electronic states, but also to account for the changes in transition metal valences.
In this regard, CRYSTAL, a code which allows a more efficient use of hybrid functionals
especially with lattice symmetry, was used in combination with the B3LYP functional to
calculate U .
G for gases: dependence on partial pressure
G for gasesous species such as CO and CO2 needs further consideration as their partial
pressures also influence the values of G. The dependence of chemical potential µ on the
partial pressure can be expressed as
µ = µ◦+RT ln
(
ρ
ρ◦
)
(2.90)
where µ◦ is the chemical potential under standard conditions, R is the gas constant, and ρ◦ is
the standard pressure (1 atm). As ∆G represents a change in chemical potentials between the
reactants and products, this equation could be used to calculate the dependence of ∆G on ρ
at a given temperature. For instance, assuming that the reaction
V2O5(s)−−→ V2O3(s)+O2(g) (2.91)
has a standard free energy change ∆G◦, the change due to oxygen partial pressure ρO2 could
be expressed as
∆G = ∆G◦+RT ln
(
ρO2
ρ◦
)
(2.92)
Under ρO2 < 1 atm, the second term becomes negative, resulting in a change in ∆G towards
negative direction, i.e. making it more spontaneous (which corresponds with the Le Châtelier
principle).
62 Background Theory
2.7 X-ray Diffraction Techniques
2.7.1 Powder Diffraction
The majority of technologically important materials are in solid form, and many of them are
periodic crystals with repeating lattices. Upon irradiation of a wave (X-rays, electrons, or
neutrons), components of the repeating lattice can reflect the incident wave by scattering.
X-ray and electrons are scattered by the electron densities, whereas neutrons are scattered
by the nuclei or magnetic fields from unpaired electrons. The reflected waves can either
constructively or destructively interfere with each other, where the Bragg’s law gives the
criterion for constructive interferences
nλ = 2d sinθ (2.93)
where n is an integer, λ is the incident wavelength, d is the lattice spacing, and θ is
the scattering angle. In laboratory powder diffraction, usually the Bragg-Brentano θ −θ
configuration is used. In this setup the X-ray source and detector are rotated around a
fixed flat sample with random crystallite orientations by the same angle θ . As the angle
is changed, only the crystallites which satisfy the Bragg condition at the given angle will
show a constructive interference. The resulting diffraction intensities are usually plotted
against the angle 2θ , which is equivalent to projecting the 3-dimensional reciprocal space to
a 1-dimensional line.
2.7.2 Rietveld Refinement
Rietveld refinement [147] is a whole-profile method to extract structural information from
the powder diffraction data. This method is essentially a least-squared fitting process where
the sum of squared differential between the observed and calculated intensities y(obs) and
y(calc) is minimised over the observed data points i in the 2θ domain
M =∑
i
Wi
[
yi(obs)− 1c yi(calc)
]2
(2.94)
where Wi is the weighting factor and c is the scaling factor. Starting from an initial structural
model, a least-squares minimisation is performed to yield a refined model that best fits the
diffraction data.
Chapter 3
A Systematic Study of 25Mg NMR in
Paramagnetic Transition Metal Oxides
3.1 Introduction
Mg transition metal (TM) compounds are an important class of materials for uses as Mg-
ion battery cathode systems.[148] With the high redox potential and capacity afforded by
the often paramagnetic TM, such systems are promising candidates for advanced Mg-ion
batteries for e.g. automotive applications.[149, 15] In addition, they can, for example, be
potentially used as carbon capture materials[150] and electrochemical catalysts for water
oxidation.[151] Therefore, elucidating the Mg local environment is crucial for understanding
and evaluating the working principles of this important class of materials.
In light of this, the current chapter presents a baseline for a combined experimental and
theoretical approach for performing 25Mg solid-state NMR experiments in paramagnetic
TM oxides. Four Mg TM oxides which have different crystal structures are considered:
Mg6MnO8 (defect rocksalt), MgCr2O4, MgV2O4 (cubic normal spinel), and MgMn2O4
(tetragonal normal spinel). First, we report the results of their structural and magnetic
characterisation, which is followed by 25Mg NMR experiments. Then ab initio calculations
for magnetic data and paramagnetic 25Mg NMR shifts are presented. First principles results
are compared to experimental spectra to aid their interpretation. Finally, we discuss the
observed NMR shifts in terms of the Fermi contact interaction and decompose the value of
shifts in terms of the individual TM–O–Mg spin transfer pathways.
64 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
Sample Thermal profile Atmosphere
MgV2O4 1273 K 24h 5% H2 in Ar
MgCr2O4 1073 K 12h - 1473 K 24h Air
Mg6MnO8 1173 K 12h Air
MgMn2O4 453 K 12h - 773 K 36h Air
Table 3.1 Synthesis conditions of samples.
3.2 Experimental
3.2.1 Sample Preparation
Mg6MnO8, MgCr2O4, and MgV2O4 samples were prepared via solid state (SS) synthesis
from stoichiometric amounts of MgO (Sigma-Aldrich, 99.99 %), V2O3 (Sigma-Aldrich,
99.7 %), Cr2O3 (Sigma-Aldrich, 99.99 %), and MnO2(Sigma-Aldrich, 99.99 %).* Mixtures
were ball milled using ZrO2 jar and balls (SPEX M8000 high energy ball mill) for 30
minutes, pressed into pellets, and calcined according to Table 3.1. The MgMn2O4 sample
was synthesised from anhydrous citric acid (Breckland Scientific, 99 %), Mg(NO3)2 ·6H2O
(Sigma-Aldrich, 99 %), and Mn(NO3)2 ·4H2O (Sigma-Aldrich, 97 %) through a citrate
sol-gel method.[152] The organic matter was burned off at 453 K and final calcination was
performed at 773 K with one intermediate grinding. Synthesised samples were checked
using a PANalytical Empyrean powder X-ray diffractometer (Cu Kα=1.5406 Å) and the
structures were refined from their diffraction patterns by using the Rietveld method[147] as
implemented in the X’pert Highscore Plus software.
3.2.2 Magnetic Measurements
Magnetisation measurements were performed on a Quantum Design Magnetic Property
Measurement System (MPMS) with a SQUID magnetometer. Magnetic moments of zero
field-cooled samples were measured at temperatures from 2 K up to 301 K in a field of 1000
Oe to obtain the dependence of magnetic susceptibility χ(T ) = dM/dH ≈M/H (low-field
approximation) on temperature. The Weiss constant Θ was extracted by fitting the Curie-
Weiss law χ =C/(T −Θ), where C is the Curie constant at high temperatures. The effective
electron magnetic moment µeff was extracted from C by the relation µeff =
√
3kBC/NA
(Equation 2.75) where kB is the Boltzmann constant and NA is the Avogadro constant. The
*MgCr2O4 and SS-MgV2O4 samples were prepared during the previous Part III research project carried out
by the author.
3.2 Experimental 65
magnetic exchange coupling constant J1 between the nearest neighbouring spins (S) was
calculated from Θ by the mean-field relation (Section 2.5.1, Equation 2.73).
3.2.3 25Mg NMR
All NMR spectra were acquired on a 16.4 T Bruker Avance III spectrometer operating
at a Larmor frequency of 42.9 MHz for 25Mg with conventional Bruker 4 mm and 3.2
mm triple resonance MAS low-γ probes. π/2-pulse amplitudes were calibrated on solid
MgO, giving 90◦ pulse amplitudes of 30 kHz and 36 kHz for the 4 mm and 3.2 mm probes,
respectively. For quadrupolar samples, CT-selective π/6-pulses were used. All shifts were
referenced to MgO at 26 ppm.[153] Recycle delays of 0.1 s were used for all acquisitions
unless otherwise indicated. Spectra were acquired with rotor-synchronised spin echo,[133]
magic angle turning (MAT),[139] and rotor assisted population transfer (RAPT)[135] pulse
sequences as shown in Chapter 2. A FAM-type X¯-X pulse train was used to saturate the
satellite levels. The RAPT modulation frequency νm was calculated from DFT values of
CQ. Values of νm = 270 kHz (CQ = 3.6 MHz) and 240 kHz (CQ = 3.2 MHz) were used for
Mg6MnO8 and MgMn2O4, respectively. Experimental signal enhancement was measured for
varying saturation frequencies for Mg6MnO8. Fitting of the spectra was performed assuming
a central transition lineshape using the Bruker Topspin 3.0 software.
3.2.4 Ab Initio Calculations
DFT calculations
All calculations were performed in CRYSTAL09, a solid-state DFT code using a Gaussian-
type basis set to describe core states accurately. CRYSTAL has been used successfully
in calculation of 6/7Li-, 23Na-, and 31P-paramagnetic shifts.[92, 94, 93] Given that the
calculation of paramagnetic shifts depends substantially on the quality of Gaussian basis
sets, two types of basis sets were utilised: a smaller basis set (termed BS-I) for geometry
optimisations, and a more extended basis set (BS-II) for hyperfine and magnetic single-point
calculations. This dual basis set scheme was first reported by Kim et al.[92] for paramagnetic
shift calculations in periodic systems and is known to work well for 6/7Li-, 23Na-, and 31P-
paramagnetic shifts.[94, 93] BS-I sets were taken from solid-state studies of Catti et al.[154,
155, 156] and were used unchanged. BS-II sets for metal ions were taken from the Ahlrichs
set, with TZDP-derived basis for Mg and DZP-derived sets for Cr and V. Modified IGLO-III
66 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
sets[157] are adopted for O. The choice of BS-II is similar to the previous works on Li
transition metal oxides.[92, 94, 93]
All calculations were performed using the spin-polarised B3LYP functional[83, 158].
Recent results indicate that using 35% of HF exchange gives better results for solid-state
magnetic calculations.[90, 91] Previous ab initio studies on 7Li, 23Na, and 31P paramagnetic
shifts show that values obtained using 20% and 35% provide the upper and lower bounds
for the experimental shifts.[92, 94, 93] Hence the original B3LYP (termed ‘Hyb20’) and a
modified B3LYP with 35% of HF exchange (termed ‘Hyb35’) were both used in calculations.
Convergence of energy and spin density was checked with the number of sampled points
in the reciprocal space, and a 4×4×4 Monkhorst-Pack[159] sampling scheme was used for
all systems. Self-consistent field (SCF) cycles were converged to an energy difference of
10−7 Hartree limit.
Experimental cell geometries were expanded into supercells and were optimised in
CRYSTAL09 with all cell symmetries removed, except for the MgV2O4 case where the
cubic cell was used with the available space group symmetry due to instabilities in the
self-consistent field cycles. All geometry optimisations were performed under the CRYSTAL
default convergence criteria. Results from DFT calculations are reported in Table 3.7 and are
also discussed in detail in the Results section.
Magnetic parameters
Values of Θ were obtained through ab initio DFT methods and also from SQUID magnetome-
ter measurements. As determination of the magnetism is crucial to calculating paramagnetic
shifts, magnetic exchange coupling constants J were obtained from DFT calculations as
detailed in Section 2.5.4.
NMR parameters
Ab initio calculation of the total NMR shift tensor requires consideration of two contributions:
the diamagnetic orbital contribution and the paramagnetic (FC) contribution due to unpaired
electrons. Diamagnetic 25Mg shifts in inorganic solids are known to be around tens of ppm
(e.g. MgAl2O4 δiso = 48 ppm).[160] As the diamagnetic contribution (positive or negative
relative to the reference) is much smaller than the paramagnetic contribution, we have ignored
the former in the calculation of shifts.
The paramagnetic contribution to the shift tensor originates from the isotropic Aiso
and anisotropic Aaniso components of the electron-nuclear hyperfine coupling tensor. Here
3.2 Experimental 67
the isotropic component is ascribed solely to the Fermi contact (FC) mechanism, which
contributes to the isotropic shift. The anisotropic (dipolar) component contributes to the shift
anisotropy. Detailed discussion on the formalism behind the hyperfine shift calculation could
be found in Section 2.5.3.
Following the approach of Middlemiss et al., the contributions of each Fermi contact
pathway to the total spin density ∆|Ψα−βN |2 were also calculated by reversing the direction of
electronic spins on each TM site in turn.[94] In addition, the total spin density |Ψα−βN |2 was
calculated from experimental shift by Equations 2.81 and 2.82.
As 25Mg is a quadrupolar nucleus (I = 5/2), the overall observed isotropic shift δiso under
MAS conditions can be expressed as a sum of Fermi contact component δFC and second-
order quadrupolar component δQ (as defined in Equation 2.67). Hence, both contributions to
the observed shift are separately considered in this work.
68 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
(a) Mg6MnO8
(b) MgM2O4, M=Cr, V
(c) MgMn2O4
Fig. 3.1 25Mg bond pathway (P) contributions and TM–TM J couplings.
3.3 Results and Discussion 69
3.3 Results and Discussion
3.3.1 Diffraction and Magnetic characterisation
Defect rocksalt Mg6MnO8
As introduced in Chapter 1, Mg6MnO8 has an ordered defect rocksalt structure [Mg2+6/8Mn
4+
1/81/8]O
2−
with a Fm3¯m space group symmetry. The oxygen sublattice is also slightly distorted from
the ideal cubic rocksalt structure due to the smaller ionic radius of Mn4+ (0.53 Å) compared
to Mg2+ (0.72 Å).[49] The Mg site in this structure shows m.mm site symmetry, reflecting the
distorted octahedra as opposed to the undistorted octahedra in the parent MgO structure. This
should give a nonvanishing quadrupolar coupling constant, CQ, and quadrupolar asymmetry
parameter, η , for 25Mg.
The powder X-ray diffraction pattern of Mg6MnO8 and the Rietveld refinement result
are shown in Figure 3.2 and Table 3.2. The product is single phase with a diffraction
pattern that matches to the previously reported result of Mg6MnO8, showing a cubic Fm3¯m
symmetry.[47]
Transition metal (TM) sites in Mg6MnO8 are separated by Mn–O–Mg–O–Mn bond
pathways, resulting in an extended superexchange interaction between the unpaired electronic
spins of Mn4+ (Figure 3.1a). The nature (ferromagnetic or antiferromagnetic) and magnitude
of this interaction can be represented by an exchange coupling parameter J1, where we are
only considering the nearest neighbouring interactions (12 for each Mn4+). At temperatures
T > 20 K, the magnetic susceptibility data of Mg6MnO8 (Figure 3.3) show the weak Curie-
Weiss paramagnetism, with an antiferromagnetic ordering transition at the Néel temperature
of TN = 5 K. Fitting to the Curie-Weiss law, 35 < T < 300 K, yielded the Weiss constant
Θ=−21.9±0.4 K and J1 = −0.73± 0.01 K. Due to the long Mn−Mn distance (6 Å),
weak exchange coupling is expected; in this case, the coupling is antiferromagnetic (J < 0)
as expected for the extended superexchange interactions.[161] The obtained Weiss constant
is in excellent agreement with the published experimental result of −20± 5 K.[162] In
addition, the effective electron magnetic moment µeff = 3.99±0.01 µB obtained from the
Curie constant shows good agreement with the spin-only value of 3.87 µB.
70 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
 10  20  30  40  50  60  70  80  90  100
Mg6MnO8
Co
un
ts 
(ar
b. 
un
it)
2θ/degrees
observed
calculated
difference
Fig. 3.2 X-ray powder diffraction pattern and Rietveld refinement data for Mg6MnO8. The
positions of allowed reflections are indicated by the tick marks.
Mg6MnO8 100 wt % Fm3¯m space group
a / Å 8.38008(6) α / ◦ 90
b / Å 8.38008(6) β / ◦ 90
c / Å 8.38008(6) γ / ◦ 90
Atom x y z
Mg1 (24d) 0 0.25 0.25
Mn1 (4a) 0 0 0
O1 (8c) 0.25 0.25 0.25
O2 (24e) 0.229(3) 0 0
Rexp 2.41
Rwp 14.21
χ2 5.60
Table 3.2 Rietveld refined parameters from the PXRD data of Mg6MnO8.
3.3 Results and Discussion 71
 0
 50
 100
 150
 200
 0  50  100  150  200  250  300
y=mx+b
m=0.502 ± 0.001
b=11.0 ± 0.2
Θ=−21.90 ± 0.40 K
1/
χ 
(m
ol 
Oe
/em
u)
Temperature (K)
Fig. 3.3 Inverse magnetic susceptibility per mol Mn, 1/χ , as a function of temperature for
Mg6MnO8.
72 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
µ e
ff
(S
O
)/
µ B
µ e
ff
(e
xp
t)
/µ
B
Θ
/K
J 1
/K
T N
/K
Fi
tte
d
ra
ng
e
/K
M
g 6
M
nO
8
3.
87
3.
99
±
0.
01
−2
1.
9
±
0.
4
−0
.7
3
±
0.
01
5
35
–3
01
M
gC
r 2
O
4
3.
87
4.
25
±
0.
03
−4
57
±
3
−3
0.
4
±
0.
2
13
10
0–
30
1
M
gM
n 2
O
4
4.
90
5.
93
±
0.
06
−4
53
±
5
–
–
20
0–
30
1
(M
g2
+ 0.
81
M
n2
+ 0.
19
)M
n3
+ 2
O
4
5.
00
(M
g2
+ 0.
91
M
n2
+ 0.
09
)M
n3
+ 2
O
4
4.
95
Ta
bl
e
3.
3
M
ag
ne
tic
ch
ar
ac
te
ri
sa
tio
n
da
ta
of
co
m
po
un
ds
st
ud
ie
d.
µ e
ff
re
fe
rs
to
th
e
ef
fe
ct
iv
e
m
ag
ne
tic
m
om
en
ti
n
B
oh
r
m
ag
ne
to
n
(µ
B
)
pe
r
T
M
io
n,
Θ
re
fe
rs
to
th
e
W
ei
ss
te
m
pe
ra
tu
re
,J
1
re
fe
rs
to
th
e
ne
ar
es
tn
ei
gh
bo
ur
ex
ch
an
ge
co
up
lin
g
co
ns
ta
nt
ex
tr
ac
te
d
w
ith
E
qu
at
io
n
2.
73
,a
nd
T N
re
fe
rs
to
th
e
N
ée
lt
em
pe
ra
tu
re
.T
he
or
et
ic
al
sp
in
-o
nl
y
va
lu
es
fo
rt
he
M
gM
n 2
O
4
co
m
po
si
tio
ns
de
te
rm
in
ed
by
ph
as
e
fr
ac
tio
ns
an
d
oc
cu
pa
nc
y
re
fin
em
en
ts
ar
e
al
so
sh
ow
n.
3.3 Results and Discussion 73
Cubic spinel MgCr2O4
MgCr2O4 adopts the normal spinel structure with cubic Fd3¯m space group symmetry and
Mg2+ ions in the tetrahedral (8a) sites, the Cr3+ occupying the octahedral (16d) metal sites.
The thermodynamic factors controlling normal versus inverse spinel formation are previously
described in the Introduction (Section 1.3.2). In MgCr2O4 and MgV2O4 (described later in
the text), the crystal field stabilisation energy (CFSE) of the TM ions are responsible for their
preference for occupancy of the 16d site.
The powder X-ray diffraction pattern of prepared MgCr2O4 is shown in Figure 3.4. As
expected from the structural discussion above, Rietveld refinement of the pattern shows a
cubic Fd3¯m symmetry and is a normal spinel. No impurity phase is detected in the MgCr2O4
sample.
The TM sites in cubic spinels have six nearest neighbouring TM sites connected through
90◦ TM–O–TM bonds, resulting in magnetic exchange which can be represented with J1
(Figure 3.1b). For the MgCr2O4 sample under consideration, SQUID magnetometry mea-
surements were performed. From the Curie-Weiss fit (100 < T < 301 K) to the experimental
magnetic susceptibility data (Figure 3.5), Néel temperature of TN = 13 K and Weiss constant
of Θ=−457±3 K are obtained. By use of the relation 2.73, J1 =−30.4±0.2 K could be
calculated from Θ. The effective electron magnetic moment µeff = 4.25±0.03 µB obtained
from the Curie constant is slightly larger than the spin-only value of 3.87 µB, which originates
from the fact that there are still significant short-range fluctuations over the temperature range
of fitting (Θ> 300 K). The observed Weiss constant is in good agreement to the previously
reported value of −433 K.[163]
Cubic spinel MgV2O4
MgV2O4, similar to MgCr2O4, adopts the normal spinel structure with the Mg2+ ions in the
tetrahedral (8a) sites, the V occupying the octahedral (16d) metal sites. On top of the CFSE
effect, it is known that the orbital degree of freedom in the S = 1 V3+ ion also favours the
normal spinel structure.[53]
The powder X-ray diffraction pattern of MgV2O4 prepared via a conventional solid-state
reaction (denoted SS-MgV2O4 to differentiate from the sample prepared and analysed in
Chapter 4) is shown in Figure 3.6 and Table 3.5. Rietveld refinement indicates a normal
spinel with Fd3¯m symmetry as expected. 15.4±0.4 % by weight of unreacted V2O3 is also
seen in the diffraction pattern, also demonstrated by previous reports of SS-MgV2O4.[164]
This behaviour is known to occur due to a reduction of MgO and sublimation of Mg metal
74 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
 10  20  30  40  50  60  70  80
MgCr2O4
Co
un
ts 
(ar
b. 
un
it)
2θ/degrees
observed
calculated
difference
Fig. 3.4 X-ray powder diffraction pattern and Rietveld refiment data for MgCr2O4. The
positions of allowed reflections are indicated by the tick marks.
MgCr2O4 100 wt % Fd3¯m space group
a / Å 8.33242(9) α / ◦ 90
b / Å 8.33242(9) α / ◦ 90
c / Å 8.33242(9) α / ◦ 90
Atom x y z
Mg1 (8b) 0.375 0.375 0.375
Cr1 (16d) 0 0 0
O1 (32e) 0.2423(2) 0.2423(2) 0.2423(2)
Rexp 2.19
Rwp 9.76
χ2 4.46
Table 3.4 Rietveld refined parameters from the PXRD data of MgCr2O4.
3.3 Results and Discussion 75
 150
 200
 250
 300
 350
 400
 0  100  200  300
y=mx+b
m=0.438 ± 0.003
b=202.2 ± 0.6
Θ=−457 ± 3 K
1/
χ 
(m
ol 
Oe
/em
u)
Temperature (K)
Fig. 3.5 Inverse magnetic susceptibility per mol Cr, 1/χ , as a function of temperature for
MgCr2O4.
at high temperatures under a reducing atmosphere of H2; however, it is known that H2
gas flow is essential to keep the vanadium oxidation state of 3+.[165] Previous syntheses,
therefore, resorted to adding excess MgO in the starting material to compensate for this
evaporation.[165, 166]
However, in the case of the solid-state prepared SS-MgV2O4, an experimental mea-
surement of magnetic susceptibility was not performed due to the presence of the V2O3
secondary phase. More discussion on the MgV2O4 sample prepared through a different
synthesis method to eliminate this secondary phase is presented in Chapter 4, Section 4.3.6.
Tetragonal spinel MgMn2O4
As introduced in Chapter 1, MgMn2O4 shows a spinel structure with I41/amd symmetry,
which is different from the above two compounds. Since the octahedral Mn3+ (d4) ions cause
a Jahn-Teller distortion of the original cubic spinel structure, a tetragonal spinel is obtained.
Mn3+ ion disproportionates at high temperatures, resulting in an antisite defect between
the Mn and Mg ions. This inversion behaviour is known to occur around 1073 K,[59, 60]
resulting in the following cation distribution:
[Mg2+1−xMn
2+
x ]tet[Mg
2+
x Mn
2+
x Mn
4+
2−2x]octO4 (3.1)
76 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
 10  20  30  40  50  60  70  80
MgV2O4
V2O3
Co
un
ts 
(ar
b. 
un
it)
2θ/degrees
observed
calculated
difference
Fig. 3.6 X-ray powder diffraction pattern for the SS-MgV2O4. The positions of allowed
reflections are indicated by the tick marks.
MgV2O4 84.6(4) wt % Fd3¯m space group
a / Å 8.4131(1) α / ◦ 90
b / Å 8.4131(1) α / ◦ 90
c / Å 8.4131(1) α / ◦ 90
Atom x y z
Mg1 (8b) 0.375 0.375 0.375
V1 (16d) 0 0 0
O1 (32e) 0.2442(2) 0.2442(2) 0.2442(2)
Rexp 11.69
Rwp 16.33
χ2 1.40
Table 3.5 Rietveld refined parameters from the PXRD data of MgV2O4 prepared via a
solid-state method (SS-MgV2O4).
3.3 Results and Discussion 77
It is known experimentally that low-temperature synthesis of this material through a
coprecipitation or sol-gel method suppresses this inversion, resulting in an ordered normal
spinel.[152, 167] We also note that this material has been previously shown to be a promising
Mg-ion battery cathode material, being able to reversibly de-insert 25Mg ions in an aqueous
electrolyte system.[30]
As discussed above, we have attempted the synthesis of an ordered, normal MgMn2O4
spinel through a citrate sol-gel method at a relatively low temperature of 773 K. The X-ray
diffraction pattern of this sample is shown in Figure 3.7, alongside the results from the
Rietveld refinement. Refinement of the sample shows a tetragonal spinel, with 7.9±0.3 % of
Mg6MnO8 secondary phase by weight. This impurity phase could not be eliminated despite
attempts to make a homogeneous mixture of Mg and Mn in the gel, and repeated calcination
attempts with intermediate grinding. Previous syntheses of MgMn2O4 through sol-gel and
solid-state methods have also reported the presence of this secondary phase.[152, 168, 30]
This suggests that excess Mn is present in the tetrahedral (4a) site of the spinel structure in
a divalent form, Mn2+. From the starting Mg to Mn ratio of 1:2, the stoichiometry of this
sample is calculated as (Mg2+0.81Mn
2+
0.19)Mn
3+
2 O4. However, refinement of the site occupancies
shows a composition of (Mg2+0.91±0.01Mn
2+
0.09±0.01)Mn
3+
2 O4 (Table 3.6), which suggests an
overall Mn deficiency in the starting composition; this could arise from the excess hydration
of the starting Mn(NO3)2 precursor (nominally tetrahydrate). Due to the low calcination
temperature, the peaks are inherently broad and good fit could not be obtained.
Evidence for presence of this tetrahedral Mn2+ comes from the magnetic susceptibility
data (Figure 3.8). In an ordered normal MgMn2O4 spinel, the tetragonal Jahn-Teller distortion
of Mn3+ ions results in four nearest neighbour interactions J1 along the a and b axes and two
second nearest neighbour interactions J2 along the c axis (Figure 3.1c). Despite the fact that
these values are not straightforward to obtain from experiments without a good model for the
magnetic exchange, we should expect a maximum below the magnetic ordering transition
in the inverse susceptibility (1/χ) versus temperature plot, since the exchange coupling in
the octahedral sublattice is antiferromagnetic. Zero-field cooled susceptibility data, however,
clearly shows presence of a ferrimagnetic component with a minimum in the plot. This
behaviour is similar to Mn3O4, where an antiferromagnetic coupling between the tetrahedral
Mn2+ and octahedral Mn3+ causes ferrimagnetic ordering at low temperatures.[169] In
addition, the fitted effective electron magnetic moment µeff = 5.93±0.06 µB is clearly larger
than the spin-only value of 4.90 µB for the Mn3+ ion (d4), suggesting the presence of Mn2+
ion (d5). Calculated spin-only magnetic moment µeff of the (Mg2+0.81Mn
2+
0.19)Mn
3+
2 O4 and
(Mg2+0.91Mn
2+
0.09)Mn
3+
2 O4 (determined from composition and refinement, respectively) are 5.00
78 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
 20  40  60  80  100  120  140
MgMn2O4
Mg6MnO8
Co
un
ts 
(ar
b. 
un
it)
2θ/degrees
observed
calculated
difference
Fig. 3.7 X-ray powder diffraction pattern for MgMn2O4. Collected data (red), refined data
(black), and diffrences (lower panel) are shown. The positions of allowed reflections are
indicated by the tick marks. 7.9 wt % of Mg6MnO8 phase was detected.
MgMn2O4 92.1(3) wt % I41/amd space group
a / Å 5.7274(2) α / ◦ 90
b / Å 5.7274(2) β / ◦ 90
c / Å 9.2660(5) γ / ◦ 90
Atom x y z Occ
Mg1 (4a) 0 0.25 0.375 0.91(1)
Mn1 (4a) 0 0.25 0.375 0.09(1)
Mn2 (8d) 0 0 0 1.00(2)
O1 (16h) 0 0.51 0.2417(3) 1
Rexp 3.28
Rwp 5.43
χ2 2.74
Table 3.6 Rietveld refined parameters from the PXRD data of MgMn2O4.
3.3 Results and Discussion 79
 0
 50
 100
 150
 200
 0  50  100  150  200  250  300
y=mx+b
m=0.230 ± 0.003
b=102.3 ± 0.6
Θ=−453 ± 5 K1/
χ 
(m
ol 
Oe
/em
u)
Temperature (K)
Fig. 3.8 Inverse magnetic susceptibility per mol Mn, 1/χ , as a function of temperature for
MgMn2O4.
and 4.95 µB. As for the MgCr2O4 case, the large discrepancy is likely due to the short-range
fluctuations at T <Θ.
The magnetic properties of the (MgxMn1 – x)Mn2O4 solid solution have previously been
studied at T < 60 K.[169] The magnetic susceptibility is shown to be small (< 0.1 emu/mol)
and qualitatively similar for MgMn2O4 and Mg0.75Mn0.25Mn2O4 at 60 K. In this work we
are interested in χ at 320 K (MAS frictional heating) and so will approximate our sample
to MgMn2O4. Finally, we note that the experimental Weiss constant Θ = −452.5 K is
consistent with a previously reported value of −500 K,[170] with the deviation between the
two values potentially arising due to the excess Mn as discussed above. However, accurate
numerical estimation of the moment from C is unlikely as the short-range ordering is still
likely to exist at this temperature (T ≈ |TN |).
3.3.2 25Mg NMR
Mg6MnO8
The 25Mg NMR spectrum of Mg6MnO8 is shown in Figure 3.9a, at 14 kHz MAS. To
determine the isotropic shift, a MAT experiment was performed (Figure 3.9b) at 20 kHz
MAS. The MAT spectrum (performed with the RAPT enhancement scheme as below) clearly
80 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
shows a single peak with δiso = 2960 ppm in the isotropic dimension (ω2). This clearly shows
the applicability of RAPT-enhanced 2-dimensional experiments in paramagnetic systems
with poor signal-to-noise ratio. RAPT-enhanced MAT experiments could be especially useful
in determining isotropic resonances, since the conventional method of performing MAS
experiments at two different spin rates (hence different degree of frictional heating) is not
straightforward due to the strong temperature dependence of the shift tensor.
Determination of quadrupolar parameters from paramagnetic NMR spectra is considered
to be difficult, as the characteristic quadrupolar patterns are often smeared out due to
paramagnetic broadening of the spectra. We used the RAPT pulse sequence, which was
previously used to estimate 27Al quadrupolar parameters in diamagnetic samples,[137] to
estimate the quadrupolar coupling constant, CQ, of the Mg site in Mg6MnO8. For optimum
enhancement, the modulation frequency νm of saturating pulse trains alternating in phase
(+π/2, −π/2) should match the value of νQ/2 = 3CQ/40 (for I = 5/2), where νQ is the
quadrupolar frequency of the observed nucleus given by Equation 2.58.[135] By plotting
the enhancement in integrated signal intensity versus the offset frequency, we see that the
maximum enhancement occurs between ν = 250−300 kHz (Figure 3.10a). This is consistent
with the DFT prediction of ν = 277 kHz, or the quadrupolar coupling constant CQ = 3.69
MHz (Table 3.7).
Starting from these values of δiso and CQ, a spectral fitting was performed with quadrupo-
lar parameters fixed to the Hyb35 calculated values. The MAS lineshape could be fitted
assuming a paramagnetic shift anisotropy and a quadrupolar interaction and the fitted parame-
ters are shown in Table 3.7. Good fitting was obtained with the quadrupolar parameters from
DFT calculations, and no further attempts were made to fit the quadrupolar parameters. The
spectrum is dominated by the central transition (transition between spin levels 1/2↔−1/2),
which is only affected in second-order by the quadrupolar coupling, and hence (other than
the linewidth of the individual peaks within the spinning sideband manifold) is relatively
insensitive to the size of the quadrupolar interaction. The paramagnetic shift anisotropy,
arising from the dipolar coupling to the Mn4+ ions, which gives rise to a lineshape identical
to that observed from the chemical shift anisotropy, is clearly visible. We note that the
fitted δiso = 2994 ppm is slightly larger than the δiso = 2960 ppm obtained from the MAT
experiment above. As the slower MAS rate results in lower sample temperatures, an increase
in the paramagnetic shift is expected in the 14 kHz MAS case when compared to 20 kHz
MAS spectrum (i.e. the MAT experiment).
From the RAPT pulse sequence, a maximum signal-to-noise enhancement by a factor
of close to 2 is observed (Figure 3.10b). Under ideal conditions, the RAPT pulse sequence
3.3 Results and Discussion 81
(a)
(b)
Fig. 3.9 (a) 25Mg spin echo spectrum of Mg6MnO8 (14 kHz MAS, 81712 transients), with
fitted central transition lineshape including contributions from both the paramagnetic shift
anisotropy and the quadrupolar interaction (parameters are listed in Table 3.7). (b) Magic
angle turning (MAT) spectrum of Mg6MnO8 (20 kHz MAS, 128 slices in the F1 dimension
with 2.23 µs delay increment, 3072 transients acquired in each slice). RAPT pulses were
applied before the MAT pulses to enhance the signal-to-noise ratio.
82 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
(a)
(b)
Fig. 3.10 (a) 25Mg spin echo signal intensities of Mg6MnO8 with increasing offset frequency
of the saturating pulse trains in the RAPT experiment. (b) Enhancement of spin echo signal
intensity using the RAPT pulse. Saturating pulses were applied at a modulation frequency of
270 kHz. Both experiments were performed at MAS spin rate of 14 kHz. 1024 transients
were acquired in each spectrum.
3.3 Results and Discussion 83
Fig. 3.11 25Mg NMR spectra of Mg6MnO8, recorded with the rotor-synchronised Hahn
echo and Double Frequency Sweep (DFS) pulse sequences. Signal-to-noise enhancement of
around 1.5 is observed for the isotropic shift. A 36 kHz-strength DFS sweep pulse starting
from an offset of 1000 kHz and ending at 100 kHz was applied for 2040 µs. 51200 transients
were acquired in each case with recycle delays of 0.01 s.
should result in an enhancement factor of I + 1/2 = 3 for 25Mg. However, due to the
difficulty of saturating multiple satellite levels in a polycrystalline I = 5/2 system, it is
not uncommon to observe enhancement factors lower than the theoretical maximum (for
instance, an enhancement factor of 2 is reported for polycrystalline 27Al (I = 5/2) spectrum in
diamagnetic albite NaAlSi3O8).[135, 136] Despite the added difficulty of enhanced relaxation
of the spin polarisation in paramagnetic materials, the observed enhancement factor of 2
clearly shows that the RAPT pulse sequence can be used for significant improvement in
signal-to-noise ratio in paramagnetic materials.
In addition to the RAPT pulse sequence, we have also attempted a Double Frequency
Sweep (DFS) enhancement scheme[138] also frequently used for quadrupolar nuclei. Only
a signal-to-noise enhancement factor of around 1.5 is observed (Figure 3.11), as opposed
to 2 times enhancement in RAPT. This could be explained in terms of faster paramagnetic
relaxation effects present in the sample, as the DFS pulses are usually longer (around 2000
µs) to satisfy adiabaticity, whereas the RAPT pulses are shorter (around 110 µs optimised
for Mg6MnO8).
84 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
Fig. 3.12 Simulated 25Mg NMR spectra of Mg6MnO8, with different Euler angles β between
the anisotropic hyperfine tensor and the EFG tensor in the simulation.
Based on the DFT (see below) and 25Mg NMR results of Mg6MnO8, we propose that
Mg6MnO8 can be conveniently used as a model compound for paramagnetic 25Mg NMR
studies owing to a number of attractive properties: (1) it has six Mg atoms per formula unit and
good NMR sensitivity, allowing signals to be observed with only 1024 scans at 16.4 T field;
(2) it has nonvanishing quadrupolar coupling, meaning that we can study the quadrupolar
behaviour of Mg atoms, which is likely to be the case for technologically important materials;
(3) d3 electron configuration of Mn4+ in octahedral environment eliminates the need to
consider spin-orbit coupling effects in ab initio calculations to a good approximation; (4) the
large distance (6 Å) between Mn atoms makes it a weak, well-defined paramagnet, which
reduces the possible error in the paramagnetic scaling approach as taken in this work; (5) it
is easy to prepare as a pure phase through solid-state reaction of corresponding oxides in air,
although previous works have reported a sol-gel route [171] or calcination of carbonates under
oxygen atmosphere [47]. It is also interesting to note that this compound was theoretically
predicted to be a good carbon capture and storage material through a large-scale screening
study.[172]
Finally, we note that the large magnitude not normally observed in shift anisotropy tensor
(computational and experimental determination of chemical shift anisotropy in 25Mg typi-
cally shows <100 ppm magnitude[42]) makes the relative orientation between the principle
3.3 Results and Discussion 85
Fig. 3.13 25Mg spin echo spectrum of MgCr2O4 at a MAS spin rate of 10 kHz. 20480 tran-
sients were acquired. The broad feature seen around 2700 ppm is due to probe background.
components of the shift tensor and EFG tensor crucial. Figure 3.12 shows the effect of
changing the Euler angle β (the angle between the largest shift and EFG principal compo-
nents) on the simulated spectrum. As the paramagnetic contribution to the shift anisotropy
is typically much larger than the corresponding CSA for many nuclei, determination of the
Euler angles by DFT methods is crucial for accurate simulation/fitting of the spectra. This
is also evidenced by a recent study of paramagnetic 17O spectra of Li2MnO3, where the
observed and fitted spectra strongly depends on the two tensor orientations.[173]
MgCr2O4
MgCr2O4 adopts a cubic AB2O4 spinel-type structure with Mg atoms sitting on the tetra-
hedral (8a) site on the spinel lattice. Due to this tetrahedral coordination of Mg sites, the
quadrupolar coupling constants CQ are expected to be zero. The observed spectrum of
MgCr2O4 is shown in Figure 3.13. Experimentally, the vanishingly small CQ is confirmed
by the sharp isotropic peak that does not exhibit typical quadrupolar lineshape under MAS
conditions.
Tetrahedral coordination also dictates that the anisotropic (dipolar) electron-nuclear
spin interaction be zero. Despite this, small nonvanishing dipolar interaction was observed
(Ω= 840 ppm). This discrepancy may be attributed to (i) bulk magnetic susceptibility effect
which originates from inhomogeneous magnetic field due to random crystallite orientations,
86 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
Fig. 3.14
Fig. 3.15 25Mg spin echo spectrum of SS-MgV2O4 (solid-state route) at MAS spin rate of 10
kHz. 505600 transients were acquired with recycle delays of 0.1 s.
as noted in previous paramagnetic MAS NMR works,[174, 175] or (ii) local defects present
in the open spinel structure, resulting in breaking of the local symmetry. With the current
data, it is difficult to ascertain which of these is going to be dominant; it is likely that both
are responsible for the sideband intensities.
We note that Wustrow et al. have synthesised a MgCr2S4 thiospinel structure and reported
the 25Mg NMR resonance at 11220 ppm.[37] Variable temperature measurement indicates
this shift is highly temperature dependent, which is characteristic of the Fermi contact shifts.
This is clearly a large downfield shift from 2862 ppm resonance of MgCr2O4, which clearly
displays the more covalent nature of Cr–S–Mg bonding compared to Cr–O–Mg, which results
in a larger transferred spin density and a larger hyperfine shift.
MgV2O4
MgV2O4 is isostructural with MgCr2O4 at room temperature, showing a cubic symmetry.
Thus, the expectation for vanishing CQ and hyperfine tensor anisotropy Ω should also hold
true for MgV2O4.
3.3 Results and Discussion 87
The observed spectrum for solid-state synthesised MgV2O4 (SS-MgV2O4), shown in
Figure 3.15a, exhibits a sharp resonance at 1845 ppm, which demonstrates the small CQ in
this system. In line with the MgCr2O4, nonvanishing shift anisotropy Ω= 672 ppm is likely
to arise from local defect or inhomogeneous magnetic fields around the crystallites.
MgMn2O4
The 25Mg NMR spectrum of MgMn2O4 sample is shown in Figure 3.16a and the existence
of the Mg6MnO8 secondary phase is clearly seen in the NMR spectrum. Fitted parameters
(Table 3.7) reveal that nonzero anisotropic components in the hyperfine coupling tensor
are present. As the tetragonal distortion reduces the tetrahedral symmetry of Mg sites, we
should compare the anisotropy to the DFT value, which will be discussed in detail in the DFT
section below. In addition, the RAPT enhancement scheme (Figure 3.16b) also shows around
twicefold increase in intensity without any distortion to the sideband manifold, the offset
used in the sequence again being chosen on the basis of the DFT-calculated CQ value. This
clearly demonstrates the utility of RAPT pulses to paramagnetic 25Mg NMR experiments.
We now compare this result to the Kim et al., who have recently reported a 25Mg NMR
spectrum of this compound.[30] Their reported NMR spectrum at a MAS frequency of 24
kHz shows two peaks at 2980 and 2850 ppm, which they assigned to MgMn2O4, the two
peaks being assigned to discontinuities of a second-order quadrupolar lineshape with a CQ of
5.4 MHz. Considering the inverse temperature dependence of Fermi contact shifts (Equation
2.76), the MgMn2O4 resonance frequency should occur at higher shifts in the case of the
14 kHz MAS used here. For example, assuming representative rotor temperatures of 340
K and 320 K for 24 kHz and 14 kHz MAS, respectively, and using experimental value of
Θ, our observed shift of δiso = 3128 ppm at 14 kHz MAS translates to δiso = 3047 ppm
at 24 kHz MAS for our MAS probes. Thus, we assign their 2980 ppm peak to the Mg in
MgMn2O4 structure, the lower shift originating from a lower sample temperature and/or
slight differences in magnetic properties between samples. Considering the twicefold (i.e. the
maximum attainable enhancement from the Mg6MnO8 case) enhancement in signal intensity
using the RAPT pulse parameters calculated from CQ = 3.2 MHz, the extra peak at 2850 ppm
is not likely to arise from a MAS quadrupolar lineshape with CQ of 5.4 MHz as suggested
in their paper. Alternatively, we suggest that these peaks represent Mg sites in different
coordination environments to the Mn in different oxidation states, arising from the antisite
defect (Equation 1.5). We tentatively assign the peak at 2850 ppm to Mg neighbouring one
Mn4+ and eleven Mn3+, as opposed to the 2980 ppm peak where the Mg is neighbouring
twelve Mn3+.
88 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
(a)
(b)
Fig. 3.16 25Mg spin echo spectrum of MgMn2O4 at MAS spin rate of 14 kHz. Mg6MnO8
secondary phase signals are shown with asterisks (*). 3550928 transients were acquired.
(a) Comparison between experimental and fitted spectrum. Fitted parameters are listed in
Table 3.7. (b) Enhancement of spin echo signal intensity with RAPT pulse sequence. RAPT
modulation pulses were applied at 240 kHz. 1024000 transients were acquired in each
spectrum.
3.3 Results and Discussion 89
M
g 6
M
nO
8
M
gC
r 2
O
4
M
gV
2O
4
(s
ol
id
-s
ta
te
)
M
gM
n 2
O
4
H
yb
20
H
yb
35
E
xp
t
H
yb
20
H
yb
35
E
xp
t
H
yb
20
H
yb
35
E
xp
t
H
yb
20
H
yb
35
E
xp
t
M
ag
n.
J 1
/K
-0
.7
-0
.5
-0
.7
-1
9.
8
-1
4.
8
-3
0.
4
Se
e
te
xt
-5
0
-4
5.
0
-4
6.
2
–
Θ
/K
-2
2.
5
-1
6.
4
-2
1.
9
-2
97
.3
-2
22
.2
-4
56
.7
-6
00
-4
48
.3
-4
66
.8
-4
52
.5
N
M
R
|Ψ
α
−β
N
|2
/1
0−
3
×
B
oh
r−
3
19
.5
17
.4
17
.0
40
.4
34
.1
37
.9
41
.2
36
.1
36
.1
43
.0
34
.1
34
.9
∆|
Ψ
α
−β
N
|2
/1
0−
3
×
B
oh
r−
3
9.
5
8.
5
–
3.
3
3.
0
–
3.
4
3.
0
–
4.
0/
2.
6
3.
3/
1.
8
–
δ i
so
/p
pm
32
91
31
52
29
94
38
42
36
90
28
62
21
05
18
40
18
45
39
01
30
21
31
28
δ F
C
/p
pm
33
37
31
99
30
41
38
42
36
90
28
62
21
05
18
40
18
45
39
38
30
54
31
78
δ Q
/p
pm
-4
6
-4
7
-4
7
0
0
0
0
0
0
-3
7
-3
3
-5
0
Ω
/p
pm
21
10
22
97
19
30
0
0
84
0
0
0
67
2
69
8
63
2
13
40
κ
-0
.8
4
-0
.8
2
-0
.8
4
–
–
1.
0
–
–
-0
.7
5
-1
.0
-1
.0
1.
0
C
Q
/M
H
z
3.
64
3.
69
3.
7∗
0
0
0
0
0
0
3.
41
3.
22
3.
4
η Q
0.
40
0.
38
0.
38
∗
–
–
–
–
–
–
0.
0
0.
0
1.
0
α
27
0◦
∗
–
–
0◦
∗
β
90
◦∗
–
–
0◦
∗
γ
90
◦∗
–
–
42
◦∗
Ta
bl
e
3.
7
A
b-
in
iti
o
ca
lc
ul
at
ed
an
d
ex
pe
rim
en
ta
lly
fit
te
d
pa
ra
m
ag
ne
tic
N
M
R
pa
ra
m
et
er
s
of
M
g 6
M
nO
8,
M
gC
r 2
O
4,
M
gV
2O
4,
an
d
M
gM
n 2
O
4.
H
yb
20
an
d
H
yb
35
re
fe
rt
o
th
e
de
gr
ee
of
H
ar
tr
ee
-F
oc
k
ex
ch
an
ge
en
er
gy
(s
ee
m
et
ho
ds
).
V
al
ue
s
m
ar
ke
d
w
ith
as
te
ri
sk
s
(*
)w
er
e
fix
ed
in
th
e
fit
tin
g.
Fo
rM
gV
2O
4,
th
e
ex
pe
ri
m
en
ta
lW
ei
ss
co
ns
ta
nt
w
as
ob
ta
in
ed
fr
om
lit
er
at
ur
e.
[1
64
]E
FG
te
ns
or
ei
ge
nv
al
ue
s
an
d
an
is
ot
ro
pi
c
hy
pe
rfi
ne
te
ns
or
co
m
po
ne
nt
s
ar
e
ne
ar
ze
ro
fo
r
M
gV
2O
4
an
d
M
gC
r 2
O
4
an
d
E
ul
er
an
gl
es
ar
e
no
tr
ep
or
te
d
fo
rt
he
se
sy
st
em
s.
D
et
ai
le
d
m
ag
ne
tic
ch
ar
ac
te
ri
sa
tio
n
da
ta
ar
e
re
po
rt
ed
in
Ta
bl
e
3.
3.
90 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
3.3.3 DFT Calculation of NMR and Magnetic Parameters
We now discuss the ab initio results on the compounds studied above. DFT calculated values
of magnetic parameters and paramagnetic NMR shifts are shown in Table 3.7.
Mg6MnO8
Good agreements between the ab initio values of NMR and magnetic parameters are seen,
with the experimental shift lying between the Hyb20 and Hyb35 calculated values. In
particular, the anisotropy parameter Ω and κ shows good agreement, despite the difficulties
in obtaining good anisotropy parameters from powder MAS patterns in paramagnetic systems.
Considering the possible origins of the discrepancies as discussed above, the dilute nature
of paramagnetic spins in the Mg6MnO8 structure and close-packed rocksalt type structural
arrangement are likely to be responsible for this good agreement.
MgCr2O4 and MgV2O4
The exchange coupling constant J1 for MgCr2O4 determined from DFT (−19.8 K for Hyb20
and −14.8 K for Hyb35) and experiment (−30.4 K) shows that the magnitude of this
exchange is strong and antiferromagnetic as expected. The experimentally measured and
calculated J1 values, while of the same order of magnitude, differ by around a factor of 2. The
differences between the two values are ascribed to the difficulties in modelling the frustrated
pyrochlore lattice with a simple Ising-type spin model at 0 K with DFT. A Heisenberg-type
spin model would be more appropriate, but this is not supported by the current version
of CRYSTAL. Consistent with this proposal, a previous plane-wave DFT calculation on
this compound using the Heisenberg model has yielded a value close to the experimental
value, J1 = −26.5 K.[176] However, we note that the isotropic shift calculated using the
DFT-predicted spin density |Ψα−βN |2 and the experimental Weiss constant Θ=−456.7 K is
close to the ab initio value (3053 ppm, 2576 ppm, and 2862 ppm for Hyb20, Hyb35, and
experimental, respectively). This shows that accurate determination of magnetism is key to
ab initio calculation of paramagnetic shifts.
For MgV2O4, an ab initio prediction of magnetic parameters could not be made due
to SCF instabilities in the supercell structure. However, a reasonable estimate of the para-
magnetic shift could still be made from a previous experimental measurement of Weiss
constant Θ=−600 K.[164] Despite the fact that spin-orbit interactions at the d2 center may
influence the shift, calculations involving explicit spin-orbit coupling Hamiltonian could
not be performed in the present version of CRYSTAL. Despite this limitation, the shift
3.3 Results and Discussion 91
values obtained without any correction for spin-orbit coupling shows a close match with
the experimental result. Mulliken spin population analysis (Table 3.8) and the spin density
map (Figure 3.17d) of the converged wavefunction shows equal occupation of the three t2g
orbitals dxy, dyz, and dzx (which is expected, since we have preserved the cubic symmetry of
the experimental cell). As discussed in the preceding section. this is likely to arise from the
fact that the spin-orbit coupling is relatively weak for a V3+ ion (spin-orbit coupling constant
λ =104 cm-1 for free ion, 95 cm-1 when doped into Al2O3).[177] Hence the energy levels
split by spin-orbit coupling are equally occupied. It is thought that the ab initio result reflects
this average occupation in effect, producing results close to the experimental values.
Finally, we note that in both cases DFT predicts zero paramagnetic shift anisotropy,
which reflects the tetrahedral symmetry of Mg sites. As discussed above, this discrepancy is
attributed to both the bulk susceptibility effect and local defects in the open spinel structures.
MgMn2O4
As values of J1 and J2 cannot be obtained from experiments without a good model to
fit the susceptibility data, they were calculated from DFT. DFT Values of J1 and J2 for
MgMn2O4 differ roughly by an order of magnitude (J1 =−45.0 K, J2 =−5.5 K for Hyb20
and J1 = −46.2 K, J2 = −6.1 K for Hyb35). As magnitudes of exchange interactions are
known to be very sensitive to bond lengths, |J1| > |J2| as expected. In this case, the
tetragonal distortion effectively reduces the frustration and the main J1 exchange occurs
along the Mn chain (see Figure 3.17f), with J2 perturbations between the chains. This
explains the good agreement of DFT calculated Weiss constant Θ to the experimental value
(−448.3, −466.8, and −452.5 K for Hyb20, Hyb35, and experimental, respectively)
Paramagnetic NMR shifts calculated using these values also show good match to the
experimental value, where the experimental shift is between the Hyb20 and Hyb35 calculated
values. However, again we see a discrepancy in the anisotropic shift parameter, similar to the
result for other spinels MgCr2O4 and MgV2O4. The discrepancy is also attributed to inho-
mogeneous magnetic field created by random crystallite orientations and local defects. These
effects are more significant in the MgMn2O4 sample, as the low-temperature preparation
condition results in (i) smaller crystallites, which increases the magnetic field inhomogeneity,
and (ii) higher concentration of defects present in the sample.
92 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
3.3.4 Shift Mechanism and the Fermi Contact Pathways
We now turn our attention to the factors determining these large paramagnetic shifts and how
they might be rationalised in terms of electronic structures. Paramagnetic NMR shifts in vari-
ous Li transition metal oxides have been ascribed to the Fermi contact (FC) mechanism.[144]
In this framework, a superexchange-like mechanism operates between the TM t2g/eg and
the Li s orbital, resulting in small amounts of paramagnetic electron spins on the Li nucleus.
As a similar kind of mechanism is expected to be responsible for paramagnetic Mg shifts,
understanding the shift mechanism is important for rationalising both the sign and magnitude
of shifts.
As shown in Equation 2.80, factors contributing to the Fermi contact shift can be separated
into the spin density transfer |Ψα−βN |2 (in units of Bohr−3), and electron paramagnetism.
Considering the expression forΘ, we can observe that metal ions with larger S are expected to
give larger Fermi contact shifts, due to the µeff factor. This factor only depends on the formal
spin of metal ions involved. On the other hand, Fermi contact spin density transfer depends
heavily on the geometry of TM–O–Mg pathway. Here we can evaluate the contributions
of each FC pathway P as ∆|Ψα−βN |2, which sums up to |Ψα−βN |2. This is important since
pathways with similar bond lengths and angles should contribute similar amounts of spin
density to the observed nucleus. With sufficient amount of experimental and ab initio data, a
database can be constructed which can give approximate shifts for novel structures without
the need for further first principles calculations.
3-dimensional maps of electron spin density obtained from DFT wavefunctions are
shown in Figure 3.17. From the spin density maps, it is clear that a delocalisation mechanism
involving the t2g orbitals is in operation for all four compounds, resulting in positive spin
density |Ψα−βN |2 on the nucleus. This is particularly evident in the Mg6MnO8 case, where a
95◦ interaction along the TM–O–Mg pathway (P1 in Figure 3.1a) results in p orbitals with
positive spin density pointing directly towards the Mg site (Figure 3.17a). We also note
that p orbitals with negative spin density all point towards the vacancy site, which would
result in a negative shift mediated by polarisation mechanism, had this site been occupied.
Also large differences in spin densities are observed between the two crystallographically
distinct oxygen sites O1 and O2, where essentially no spin is observed on the O2 site. This
originates from the fact that O2 is only surrounded by Mg in its first coordination shell due
to the ordered cation arrangements in the defect rocksalt structure (Figure 3.17b).
The situation in spinels is more complex, as the TM–O–Mg pathway is no longer near
90◦. Here, both delocalisation (t2g− pπ − s) and polarisation (eg− pσ − s) mechanisms are
likely to operate along the P1 pathway (Figure 3.1b), resulting in a transfer of positive and
3.3 Results and Discussion 93
dxy dyz dzx dx2−y2 d2z
MgV2O4 0.625 0.625 0.625 0.056 0.056
MgMn2O4 0.934 0.934 0.919 0.143 0.855
Table 3.8 Mulliken spin population analysis of transition metal d orbitals in selected spinel
compounds.
negative spins on Mg, respectively. Hybridisation of the oxygen orbital is evident, as the
main lobe points towards the Mg atom in the case of cubic spinels MgCr2O4 and MgV2O4
(Figures 3.17c and 3.17d, respectively). In both cases where we have no ambiguity in the
t2g orbital occupation, it is clear that the delocalisation mechanism is dominant, resulting in
positive spin density on Mg.
A similar analysis could be done for tetragonal spinel MgMn2O4 where we have a
Jahn-Teller distorted Mn3+ ion (d4), but the d orbital occupancies need to be determined.
A close examination of the spin density map (Figures 3.17e and 3.17f) and Mulliken spin
population analysis (Table 3.8) reveals that the valence electron configuration for Mn is
d1xyd
1
yzd
1
zxd
1
z2 , showing occupation typical of a positive Jahn-Teller elongation. This has
important implications for the FC mechanism, as we have two distinct FC pathways P1 and
P2 as a result of the Jahn-Teller distortion. P1 lies on the crystallographic a,b-plane and P2
points along the c-axis in the crystal structure (Figure 3.1c). In both cases, the occupied
dz2 orbital on Mn
3+ is likely to contribute a positive spin density to the oxygen along the
c-axis (delocalisation), whereas the empty dx2−y2 orbital will contribute negative spin density
to the oxygen along the a-axis (polarisation). Again, examination of the spin density map
reveals that the dominant shift mechanism is also the delocalisation mechanism in both cases,
with the oxygen pz orbital playing the most significant role. Large contribution from the
dz2 − pσ − s pathway is evident. A noticeable negative spin density is observed on oxygen
positions (resulting from polarisation along the eg− pσ pathway), although this does not
contribute significantly to the shift.
By obtaining the spin density contributions ∆|Ψα−βN |2 from each FC pathways (P), mag-
nitudes of the observed shifts could be rationalised. Comparing the first-order contribution
P1 of two d3 ions, Cr3+ in MgCr2O4 and Mn4+ in Mg6MnO8 (Table 3.7), a significant
decrease in spin density transfer could be observed in the former case. In Mg6MnO8,
each Mg atom is bonded to two Mn4+ ions via two 95◦ TM–O–Mg bonds. In terms of
spin densities, pathway decomposition shows that each Mn4+ contributes approximately
P1 = 9.5×10−3 Bohr−3 (Hyb20) or 8.5×10−3 Bohr−3 (Hyb35) to Mg. In MgCr2O4 where
each Mg is bonded to 12 Cr3+, each Cr3+ contributes approximately P1 = 3.3×10−3 Bohr−3
94 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
(Hyb20) or 3.0× 10−3 Bohr−3 (Hyb35). This is rationalised by the differences in bond
angles: as shown by Carlier et al., 90◦ interactions between the TM t2g and observed nucleus
result in positive spin transfer via a delocalisation mechanism, whereas 180◦ interactions
result in negative spin transfer via a polarisation mechanism.[144] In Mg6MnO8, near 90◦
interactions dictate that the delocalisation mechanism should be the dominant spin transfer
mechanism, whereas in MgCr2O4 both delocalisation and polarisation mechanism should be
in operation due to the 121◦ bond angle. The polarisation mechanism partially cancels the
delocalisation mechanism, resulting in small spin density transfer in MgCr2O4.
In MgMn2O4, the Jahn-Teller elongation dictates that a > c. Due to the occupation of dz2
orbital and bond elongation, we expect P1 > P2. Values of P1 and P2 cannot be gained from
experiments and need to be determined ab initio. The DFT calculation shows that the first
contribution is roughly 1.5 times larger in terms of spin density contribution (P1 = 4.0×10−3
versus P2 = 2.6× 10−3 Bohr−3 for Hyb20 and P1 = 3.3× 10−3 versus P2 = 1.8× 10−3
Bohr−3 for Hyb35, respectively). As for the case of magnetic exchange coupling (see above),
magnitude of the FC interaction is very sensitive to bond distances, which explains the
difference.
3.3 Results and Discussion 95
(a) Projections of the Mg6MnO8
(001) plane along the c-direction
(b) Projections of the Mg6MnO8
(00 14 ) plane along the c-direction
(c) Projections of the MgCr2O4 cu-
bic cell along the c-direction.
(d) Projections of the MgV2O4 cu-
bic cell along the c-direction.
(e) Projections of the MgMn2O4
tetragonal cell along the a-
direction.
(f) Projections of the MgMn2O4
tetragonal cell along the c-direction.
Fig. 3.17 3-dimensional spin density maps. Yellow denotes positive spin; blue denotes
negative spin. Bounding box refers to the unit cell and solid lines connecting atoms refer to
the metal-oxygen bonds. Bond pathway contributions P to the spin density are shown in red.
96 A Systematic Study of 25Mg NMR in Paramagnetic Transition Metal Oxides
3.4 Conclusion and Outlook
In this chapter, a combined DFT and experimental NMR approach to studying 25Mg NMR
spectra in paramagnetic TM oxides is demonstrated. Application of advanced NMR pulse
sequences such as RAPT and MAT are also demonstrated to be viable for using in paramag-
netic systems. Experimentally, it is shown that signal-to-noise enhancement is possible in
paramagnetic Mg systems with RAPT pulses, allowing us to perform a 2-D MAT experiment
on samples with natural 25Mg abundances. While MAT is not strictly necessary to obtain the
isotropic shift in the Mg6MnO8 case, we note that MAT experiments were used in the 23Na
NMR of related Na-ion battery materials to succesfully deconvolute the complex spectra
comprising multiple sites and sidebands.[93] DFT calculation of 25Mg NMR parameters
using the paramagnetic scaling approach is demonstrated with good agreement with the
experiment. Comparison of the 25Mg shifts in spinel and Mg6MnO8 compounds shows that
TM-O-Mg 90◦ interaction results in stronger Fermi contact spin transfer, as expected from
previous studies.
For spinel compounds MgCr2O4 and MgV2O4, comparison of the experimental results
(NMR, SQUID magnetometry) and DFT calculations show that NMR anisotropy parameters
are difficult to determine from the experiments due to the bulk magnetic susceptibility effects.
In addition, DFT prediction of magnetism is difficult due to the magnetic frustration of these
structures. Despite these difficulties, we show that reasonable predictions of 25Mg shifts can
be made with the DFT values of spin density and the Weiss constant measured from SQUID
magnetometry.
With the aid of DFT calculations and RAPT pulse sequence as shown above, it is possible
to predict and acquire paramagnetic 25Mg spectra with signal-to-noise enhancement factors
of 2. Good agreements are observed between the DFT predicted values of magnetism and
NMR shifts, with the exception of NMR anisotropy parameters. This approach enables us to
record and interpret the paramagnetic 25Mg spectra of other complex Mg compounds, which
are more difficult to understand without the aid of DFT calculation.
Chapter 4
Carbothermal Synthesis and
Characterisation of MgV2O5, a Potential
Mg-ion Battery Cathode Material
4.1 Introduction
As introduced in Chapter 1, computational works have identified δ -MgV2O5 as a promising
MIB cathode material with a relatively low migration barrier (0.6-0.8 eV) and high redox
potential (>3 V). Two computational studies were reported on this compound which adopts a
layered structure with potential 1-dimensional diffusion channels for the Mg2+ ions.[67, 68]
This material is a natural extension from α-V2O5, which previously showed reversible Mg
insertion up to Mg0.5V2O5 at a potential of 2.3 V vs Mg/Mg2+.
Despite the promising computational predictions, no literature has been reported to
date on the synthesis and cycling of δ -MgV2O5 phase in Mg-ion cells. Apart from the
evident challenges in obtaining suitable high-voltage electrolytes, this may be partly due
to the difficulty in preparing a pure phase MgV2O5: previous reported (impure) phases
were prepared under a sealed quartz ampoule. This challenge, combined with the inherent
unscalability of the sealed tube method, calls for a reliable approach for preparing these type
of compounds.
In this regard, the present chapter attempts a comprehensive investigation of MgV2O5,
from synthesis to NMR, magnetic, and electrochemical characterisation. In the first part,
a Hybrid eigenvector-following (HEF) approach is used to calculate the Mg migration
barriers in MgV2O5 at various levels of theory to verify the previous results. Subsequently,
98
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
DFT-based computation of free energies are used to rationally design the synthetic steps
leading to MgV2O5, using carbon as the reducing agent. An extension of this method to
one other vanadate phase, MgV2O4, is also discussed. Characterisation of the product using
X-ray diffraction and 25Mg NMR spectroscopy follows the synthesis, combined with an
initial electrochemical cycling data. Finally, ab initio prediction on the magnetic coupling
and hyperfine NMR shift is presented. The chapter concludes with a discussion on the
electrochemistry of MgV2O5 and prospects of extending this approach to other candidate
cathodes for Mg-ion batteries.
4.2 Methods
4.2.1 Materials Synthesis
Here we outline the steps toward the syntheses; detailed reasoning behind the reaction
temperatures is presented in the Results section.
Syntheses of MgV2O5 and MgV2O4
MgV2O6 precursor. Polycrystalline MgV2O5 and MgV2O4 samples were prepared by
carbothermal reduction of microcrystalline MgV2O6 precursor (detailed rationale behind the
synthetic procedure are discussed in the results section). MgV2O6 precursor was synthesised
using a citrate sol-gel method. Mg(CH3COO)2 ·4H2O (Sigma-Aldrich, 98 %), V2O5 (Sigma-
Aldrich, 99.9 %), and anhydrous citric acid (Breckland Scientific, 99 %) were dissolved in a
molar ratio of 1:1:4 in deionised water. After heating under stirring to remove acetic acid
and water, the resulting gel was decomposed at 623 K for 6 hours. The collected ash was
finally annealed at 873 K for 24 hours to yield MgV2O6 as a yellow powder.
MgV2O5. The above MgV2O6 precursor was mixed intimately with a stoichiometric
quantity of high surface-area Super P carbon (Timcal) with the stoichiometry according to
the following equation:
MgV2O6+0.5C−−→MgV2O5+0.5CO2 (4.1)
The mixture was then pressed into a pellet and placed in an alumina boat under an atmosphere
of flowing Ar gas by means of a quartz tube. The pellet was subsequently heat-treated at 873
K for 24 hours followed by an additional annealing at 1173 K for 12 hours to yield MgV2O5.
The obtained MgV2O5 powder was green, consistent with previous reports.[69, 178]
4.2 Methods 99
MgV2O4. The above MgV2O6 precursor was mixed with Super P carbon (Timcal) with
the following reaction stoichiometry:
MgV2O6+2.5C−−→MgV2O4+2CO+0.5C (4.2)
A 25 % excess carbon was used in this case to improve the conductivity of final product. The
mixture was then pressed into a pellet and placed in an alumina boat under an atmosphere
of flowing Ar gas. The pellet was subsequently heat-treated at 1173 K for 24 hours to yield
MgV2O4. The obtained MgV2O4 powder was black, consistent with previous reports;[165,
166] however, presence of residual carbon may contribute to this colour. Synthesised samples
were checked using a PANalytical Empyrean powder X-ray diffractometer (Cu Kα=1.5406 Å)
and the structures were refined from their diffraction patterns by using the Rietveld method
as implemented in the X’pert Highscore Plus software.
TGA measurements
A thermogravimetric analyser (TGA/DSC 1, Mettler Toledo) was used with a nitrogen flow
of 80 mL/min and 5 K/min temperature ramping from the ambient temperature to 573 K.
The sample was loaded in a 70 µL alumina crucible.
4.2.2 Magnetic Measurements
Magnetisation measurements were performed on a Quantum Design Magnetic Property
Measurement System (MPMS) with a SQUID magnetometer. Magnetic moments of zero
field-cooled samples were measured at temperatures from 2 K up to 301 K in a field of 1000
Oe to obtain the dependence of magnetic susceptibility χ(T ) = dM/dH ≈M/H (low-field
approximation) on temperature.
4.2.3 25Mg NMR Spectroscopy
NMR experiments were performed on a 16.4 T Bruker Avance III spectrometer operating
at a Larmor frequency of 42.9 MHz for 25Mg with a Bruker 3.2mm triple resonance MAS
probe. The pulse amplitude of 33.3 kHz was calibrated on solid MgO, which also served
as a secondary shift reference of 26 ppm. CT-selective π/6-pulses were used for the rotor-
synchronised spin echo pulse sequence.
Rotor-assisted population transfer (RAPT) pulse sequence using a train of frequency-
switched Gaussian pulses (FSG-RAPT) was used.[179] In total, 40 Gaussian pulses with
100
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
5 kHz amplitude and 16 µs length were used to excite the satellite transitions; the number
of Gaussian pulses and their offset frequencies were optimised for maximum enhancement.
Spectrum fitting was performed with the Bruker Topspin 3.0 software.
4.2.4 Computational Methods
Free energy calculations
Cohesive energies of compounds MgV2O4, MgV2O5, and MgV2O6 were calculated with the
CRYSTAL17 code. Standard B3LYP functional was used with POB-TZVP basis sets.[180]
Cells were relaxed to the CRYSTAL default convergence criteria with a 4×4×4 Monkhorst-
Pack k-sampling,[181] 0.0003 a.u. root-mean-square gradient, and 10−6 hartree limit on
electronic convergence. Temperature-dependent free energies of C, O2, CO, and CO2 were
obtained from a published reference table.[182] We note that the entropic contribution to
the free energy is expected to be significant, so simple formation energies cannot be used in
these cases.
For carbon, free energies determined for the crystalline graphite were used; in practice,
high surface area carbons such as Super P are noncrystalline and may deviate from the values
determined for graphite. The difference in free energy, however, is expected to be minor
compared to the energy scales (hundreds of kJ/mol) considered here: the enthalpy for a
glassy (vitreous) carbon is measured to be only 0.4 kJ/mol higher than a graphite and the
entropy at 0 K is estimated to be only 1-3 J/mol K.[183, 184]
Computation of NMR and magnetic parameters
Calculations were performed in CRYSTAL17 using a dual-basis set scheme as previously
described.[185] A smaller basis set (BS-I) taken from works of Dovesi, Harrison, and
Bredow[186, 187, 188] was used to relax the cell to CRYSTAL17 convergence limit with
an electronic convergence criterion of 10−7 hartree. Spin-polarised B3LYP functional with
20 % and 35 % of Hartree-Fock exchange were used (Hyb20 and Hyb35, respectively).
Monkhorst-Pack k-sampling[181] of 6×2×2 was used. The resulting cell was expanded
3×1×1-times to generate a supercell for hyperfine and magnetic calculations.
Hyperfine and magnetic calculations were performed with more extensive BS-II sets taken
from Mg (TZDP), V (DZP), and O (IGLO-III),[189, 190] consistent with previous studies on
paramagnetic NMR of Mg TM oxides.[38] For magnetic exchange J calculations, a linear
regression method was used on a set of energies calculated on different spin configurations.
4.2 Methods 101
Eight such configurations were used for the regression. The resulting values of J were used
in a self-consistent mean-field theory code* to obtain the Curie-Weiss constant Θ.
Determination of the linear-response Hubbard-U parameter
To correct the self-interaction error in GGA-type functionals, we used the rotationally
invariant Hubbard U correction by Dudarev et al.[80] We determined the Ueff = U − J
through the linear-response method of Cococcioni and de Gironcoli.[82, 75] Unit cell as
determined by Millet et al.[69] was expanded 3×1×1-times to build a supercell onto which
the perturbative potential α was applied. Twelve values of α were used to determine the
changes in the d-orbital occupation.
Calculation of Mg-ion migration barrier
The hybrid eigenvector-following approach was used to locate the transition state as imple-
mented in the OPTIM code.[113, 191] A low-memory Broyden-Fletcher-Goldfarb-Shannon
scheme was used for finding the uphill path (root-mean-square gradient is less than 0.025
eV/Å). Up to five LBFGS minimisation steps are performed in the tangent direction until
the root-mean-square gradient < 10−3 eV/Å. The steepest descent pathway was found
by displacing the moving atom (Mg) by 0.1 Å from the transition state along the parallel
and antiparallel directions to the eigenvector. Local minima were found by converging the
energies to 10−3 eV.
VASP was used for the energy and force calculations. Spin-polarised PBE, PBE+U
(Ueff = 3.55 eV as fitted by a linear response method), vdW-DF2+U (vdW-DF2), and HSE06
functionals were used.[192, 86, 193, 194] For the hybrid HSE06 calculations, only a few
selected points along the reaction pathway were selected to sample the energies due to
computation cost. We note that self-consistent vdW corrections such as vdW-DF2 are
necessary to account for charge separation and transfer happening in these systems, as
evidenced by Li migration in graphites.[195] Parametrised corrections based on point charges
such as Grimme-D3 are significantly cheaper, but they do not account for the charge transfer.
A 3×1×1-expansion of MgV2O5 crystallographic cell was used to calculate the migra-
tion barrier. 520 eV plane-wave energy cutoff, 2×2×2 Monkhorst-Pack k-sampling, and
10−6 eV convergence criterion of the energy were used. The cell was relaxed to 0.01 eV/Å.
Structures were plotted using the VESTA software.
*Provided by Dr Derek Middlemiss
102
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
4.2.5 Electrochemical Testing
Due to the aforementioned difficulties in cycling materials in a Mg-ion cell, the synthesised
MgV2O5 compound were tested in Li-ion cells to investigate the de-magnesiation capacity.
The samples were mechanically milled to reduce the particle size in a high-energy shaker
mill (SPEX SamplePrep 8000M) using a ZrO2 jar and balls. The milling was conducted in
an Ar atmosphere to prevent the possible oxidation of vanadium compounds. 5 minutes of
milling was followed by 5 minutes of rest period to minimise heating effects. Six such cycles
were performed. The resulting material was checked with SEM and XRD to confirm the
reduction in particle sizes, and NMR was performed to check any changes in the sample
(Appendix A).
Self-supporting films of active material were created from 72 wt % active material, 19
wt % poly(vinyline fluoride-co-hexafluoropropylene (PVDF-HFP, Kynar), 9 wt % Super P
cabon (Timcal), and 6 drops of dibutylphthalate (DBP, Sigma-Aldrich) in acetone to create
a slurry. After thorough mixing, the slurry was cast onto a glass surface and left to dry.
The resulting film was subsequently punched, washed with diethyl ether (Sigma-Aldrich) to
remove DBP, dried under vacuum at 100 ◦C overnight, and transferred into an Ar-filled glove
box (MBraun) with H2O and O2 levels below 0.1 ppm.
2032-type coin cells (Cambridge Energy Solutions) were assembled inside an Ar-filled
glove box using stainless steel current collectors. Li metal (LTS research, 99.95 %) and 1
M LiPF6 in 1:1 ethylene carbonate:dimethyl carbonate electrolyte (Sigma-Aldrich, battery
grade) solution were used as the counterelectrode and electrolyte, respectively. Electrochem-
ical measurement was performed on a Lanhe battery cycler (Wuhan Land Electronics Co.
Ltd.) with a charging cutoff of 4.4 V.
4.3 Results
This section is divided into three parts. We start by discussing the computational work on
MgV2O5, initially focusing on validating the previous report on Mg-ion migration barrier in
this compound. Next, the carbothermal (CTR) synthetic approach (as introduced in Section
2.6) and rational design of the reaction conditions is presented based on the DFT-based
thermodynamic energies. Following this, the prepared product is then characterised with
X-ray diffraction and 25Mg NMR. Electrochemical characterisation of the sample is then
attempted. Finally, we also present a computational investigation on the magnetic and NMR
property of this compound, aided with experimental measurements on the low-temperature
4.3 Results 103
Fig. 4.1 Determination of rotationally invariant Hubbard U of V4+ in MgV2O5 by a linear
response method by varying the site potential α . SC and non-SC refer to self-consistent and
non-self-consistent calculations, respectively.
magnetism. We also briefly discuss the synthesis and characterisation (magnetic, NMR) of
CTR-prepared MgV2O4.
4.3.1 Calculation of Mg-ion Migration Barrier
Despite the previous work on MgV2O5 reporting a promising Mg-ion migration barrier, the
authors have only used the PBE functional[67] which is known for its deficiency in accounting
for the electron correlation. As charge hopping is also a crucial factor in determining the
size of energy barriers involved in divalent ion diffusion,[196] calculations based on more
advanced GGA+U and hybrid functionals would be expected to give a more accurate picture
of Mg-ion hopping in MgV2O5. Therefore, we start by determining the U-parameter used in
this approach.
Linear response determination of Hubbard U
Whereas a previous computation of the U-parameter in this system has been reported,[197]
it was based on the Liechtenstein formulation of the U-correction[78] whereas we use the
Dudarev formulation where only a single value Ueff.
104
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
Figure 4.1 shows the plot of changing the d-orbital occupation as a function of applied
site potential α . This linear response approach yields a good linear regression of the
self-consistent and non-self-consistent lines, resulting in a Ueff = 3.55 eV. This is in good
agreement to the previously determined values of U = 3.6 eV by Korotin et al.[197] and is
also close to the Ueff = 3.1 eV by Jain et al., where an averaged value over multiple vanadium
oxides were reported.[66] Hence, the determined Ueff = 3.55 eV was used for subsequent
PBE+U calculations.
Migration of a Mg vacancy
As shown in Figure 1.4, δ -MgV2O5 has a 1-dimensional diffusion channel of Mg ions along
the c-direction, which is in parallel with the direction of V2O5 ladder. Mg-ion diffusion in this
structure has been shown to have a relatively low barrier of 0.6-0.8 eV, which is remarkable
for a diffusion of divalent ions in an oxide structure. The following results attempts to
validate this result using a single-ended transition state searching with considerations of
coupled charge and ion hopping. We start with the fully discharged structure MgV2O5, in
which a Mg vacancy diffuses through the cell.
Figure 4.2a shows the energy profile of a vacancy diffusion in a fully magnesiated phase.
To evaluate the effect of functionals on the observed energy, a variety of functionals were
investigated: hybrid HSE06, GGA PBE, Hubbard-U corrected PBE+U , and van der Waals
(vdW) corrected vdW-DF2+U . The energy profiles of the latter GGA-based functionals
show similar energy barriers between 0.6-0.8 eV. The range is in line with the previous study
by Sai Gautam et al., where only a PBE functional was used due to force instabilities.[67]
Inclusion of Hubbard-U is shown to decrease the barrier height by approximately 0.1 eV,
where inclusion of vdW correction further lowers the activation barrier when compared to the
PBE+U case. Interestingly, the effect of this vdW correction is only significant for the barrier
height (calculated by the energy difference between the transition state and the local energy
minima) and does not affect the energies of both end minima significantly. Considering the
crucial role of vdW interactions in maintaining the structures in layered materials, this point
was further investigated as following.
A close examination of the PBE+U /vdW-DF2+U energy profiles (Figure 4.2a) indicate
that they could be divided into three steps: (i) initial steep rise (-4 to -2 Å integrated path
length), (ii) gradual plateau (-3 to 1 Å), and (iii) steep descent (1 to 3 Å). Structurally, steps
(i) and (iii) correspond to the ‘rocking’ motion of the VO5 octahedra with (partial) charge
hopping only, whereas the actual Mg jump takes place in step (ii).
4.3 Results 105
(a)
(b)
(c)
(d)
Fig. 4.2 (a) Energy profiles of a single Mg vacancy hop between 6-fold coordinated (stable)
and 5-fold coordinated (metastable) sites. For the HSE06 calculations, only a handful of
points along the PBE+U diffusion path was selected to calculate the single point energies.
All energies are referenced to their lowest respective state. (b) Metastable 5-coordinated, (c)
transition state, and (d) stable 6-coordinated Mg site along the diffusion pathway. Colour
scheme: orange(Mg), red(V).
106
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
This behaviour is suggestive of a combined charge and ion hopping, as jump of a divalent
Mg2+ ion to a nearby vacancy would result in a charge migration to satisfy the local charge
balance. Pure PBE calculation does not exhibit this mid plateau-like region, which is more
reminiscent of a simultaneous ion and electronic hop; this is expected from a delocalised
electron cloud due to the lack of proper electron correlation. The fact that inclusion of vdW
interaction only changes the energies of step (ii), the plateau, suggests that the rocking motion
(related to the phonon modes) and charge hopping are less affected by the vdW correction,
whereas the actual Mg migration is more heavily affected by so.
The same scenario with a hybrid HSE06 functional shows a much higher migration barrier
of around 1 eV. However, a recent study on layered LixCoO2 compounds has shown that
several properties including the lithiation voltages, relative phase stabilities, and structural
properties are better predicted with vdW-DF2+U approaches.[198] Hybrid methods tend
to localise the electron density and this would result in a reduction of long-range diffuse
interaction. As the vdW interactions are shown to be crucial in structures and insertion
properties of layered oxides, it is likely that the actual activation barrier is closer to the
vdW-DF2+U values of 0.6-0.8 eV.
Figures 4.2b–4.2d show the snapshots of the metastable 5-coordinated minimum (Figure
4.2b), 5-coordinated transition state (Figure 4.2c), and stable 6-coordinated minimum (Figure
4.2d). Analysis of the Mg–O bond lengths at the transition state (Figure 4.3a) shows that
three slightly shorter (2.02, 2.02, 1.96 Å) bonds form a trigonal coordination, with two
longer (2.39, 2.25 Å) bonds to ultimately form a 5-coordinated Mg transition state. Whereas
Sai Gautam et al. fixed both local minima using a NEB method, they predicted a 3-fold
coordinated transition state;[67] the NEB approach did not allow to predict the 5-coordinated
transition state as found in this study. The overall Mg migration pathway would be therefore
6→ 5′→ 5→ 5′→ 6, rather than 6→ 3→ 5→ 3→ 6. The discovery of a 5-fold coordinated
TS allows even smaller changes in coordination along the diffusion pathway and could be
the driving force behind such low diffusion barrier of Mg.
Migration of a Mg atom
As MgV2O5 is charged, demagnesiation takes place to yield V2O5. In terms of Mg diffusion,
this is the another limiting case in which a single Mg ion moves through the lattice, unlike
the vacancy diffusion above.
Figure 4.4a shows the energy profile for a single Mg diffusion. Since vanadium in a fully
charged V2O5 is a 3d0 cation, PBE+U was not used for this system. The plot shows a more
symmetrical migration profile, indicating a similarity in local minima between the metastable
4.3 Results 107
(a)
(b)
Fig. 4.3 (a) Local geometry around the Mg in its transition state, fully magnesiated MgV2O5.
(b) Local geometry around the Mg in its transition state, fully demagnesiated V2O5. All units
are in Å. Mg–O bond distances closer than 3.2 Å are represented by a bond.
108
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
(a)
(b)
(c)
(d)
Fig. 4.4 (a) Energy profiles of a single Mg hop between 6-fold coordinated (stable) and
5-fold coordinated (metastable) sites. For HSE06 calculations, only a handful of points along
the PBE+U diffusion path was selected to calculate the single point energies. All energies
are referenced to their lowest respective state. (b) Metastable 5-coordinated, (c) transition
state, and (d) stable 6-coordinated Mg site along the diffusion pathway. Colour scheme:
orange(Mg), red(V).
4.3 Results 109
5-coordinated site and the 6-coordinated site. This is also reflected in the small difference
between the end-point energies (0.05 eV), indicating a small energy penalty for such sites.
Perhaps the most surprising aspect of this plot is the agreement between the HSE06 and
PBE results, which is distinct from the vacancy diffusion as seen above. The exact origin of
this phenomenon is not yet clear; again coupled charge and ion migration may play a role.
However, energy differences between the local minima also shows small formation energy
(0.15 eV), again reinforcing the similarity argument in the PBE case.
Figures 4.4b–4.4d again show the snapshots of the metastable 5-coordinated minimum
(Figure 4.4b), 3-coordinated transition state (Figure 4.4c), and stable 6-coordinated minimum
(Figure 4.4d). In this case, a 3-fold coordination is predicted (Figure 4.3b), where the extra
bonds show much longer Mg–O bond lengths (2.01 vs 3.12/2.80 Å). This gives an overall
diffusion path of 6 → 3 → 5 → 3 → 6, in agreement with previous report using a NEB
approach.
Finally, we note that an accurate TS and its steepest descent energy profile could be
constructed by virtue of TS searching using the HEF method. We note that this is a feature
absent in a traditional NEB-type searching methodology, where the points along the band
does not necessarily include the TS. Improved NEB methods such as the climbing image
(CI)-NEB could be used to locate the accurate TS, but at the same time it is more expensive.
Also, the HEF method allowed TS searching under the PBE+U and vdW functionals, which
could show instabilties when used with the NEB method. The ability of this HEF approach
to sample the diffusion pathway allows us to capture the subtle effects of the ion/charge
migration on the calculated profile, allowing investigations of the detailed mechanism behind
such migrations.
4.3.2 Rational Design of Synthetic Steps by DFT
Carbothermal synthesis: the motivation
Despite many previous reports on preparation and characterisation of complex AVxOy (A=Li,
Na, Ca, Mg...)-type oxides, it is in general difficult to prepare stoichiometric compounds of
alkaline vanadium oxides in reduced V3+ and V4+ oxidation states. This is evidenced by the
subtle difference in stoichiometries of MgV2O4 samples synthesised under different partial
pressures of H2 in Ar.[166, 165] Clearly, a reliable way of preparing these complex oxides is
needed.
As detailed in Chapter 2 (Section 2.6), ab initio methods could be used to calculate the
free energy G of compounds under consideration. Under the approximation G ≈U , the
110
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
Fig. 4.5 Reaction scheme to synthesise MgV2O5 and MgV2O4 via the CTR approach.
cohesive, or formation, energy of any given structure could be calculated and the relevant ∆G
obtained for any reaction using elemental carbon as the reducing agent. Using this approach,
a synthesis scheme leading to MgV2O5 and a related compound MgV2O4 is presented in
Figure 4.5, which we now discuss.
Free energy relations under standard conditions
Figure 4.6 shows the ∆G–T diagram of reactions involved in syntheses of MgV2O5 and
MgV2O4 at standard pressures. Several important points could be noted from the diagram:
• The reactions involving magnesium vanadates
2MgV2O5+O2 −−→ 2MgV2O6 (4.3)
MgV2O4+O2 −−→MgV2O6 (4.4)
exhibit lines with similar positive gradients, as expected from the consumption of 1
mol gaseous O2 in both reactions.
• Reduction of MgV2O6 to generate MgV2O5 (green line on Figure 4.6) is thermo-
dynamically more favoured than forming MgV2O4 (purple line). This also means
the MgV2O5 phase would be thermodynamically more stable than MgV2O4 under
standard conditions.
• Oxidation of carbon has a sufficiently negative ∆G to allow formation of MgV2O5 and
MgV2O4. Below 973 K, the CO2 reduction mechanism would be responsible; the CO
mechanism takes place instead above this temperature.
These points clearly indicate that MgV2O5 could be prepared by a CTR reaction of the
MgV2O6 precursor from thermodynamic considerations. However, it should be stressed that
4.3 Results 111
Fig. 4.6 Free energy ∆G vs. temperature T plot, or Ellingham diagram, of the various
oxidation reactions involved in syntheses of MgV2O5 and MgV2O4 phases, calculated from
DFT. All values are under standard 1 atm partial pressures. The two carbon lines cross at
T = 973 K (700 ◦C). To allow a direct calculation of ∆G, the values are normalised with
respect to 1 mol of O2.
112
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
Fig. 4.7 DFT-based free energy ∆G of the CTR reactions to produce MgV2O5 and MgV2O4
phases at 900 K as a function of CO2 partial pressure ρCO2 . The two lines cross at ρCO2 =
10−2 atm. The values are normalised with respect to 1 mol of MgV2O6.
these predictions are solely based on thermodynamic arguments concerning equilibria under 1
atm partial pressures. In practice, the reactions are conducted under a flowing Ar atmosphere
where the oxygen partial pressure ρO2 is kept minimised to eliminate unwanted oxidation
of products. Flowing Ar also ensures that the reaction goes to completion by continuous
removal of CO2/CO. To account for these reaction conditions, the same free energies must
now be considered as a function of ρCO2 and ρO2 , which we now discuss.
Free energy relations under nonstandard conditions
Figure 4.7 shows the free energy plots of CTR reactions to produce MgV2O5 and MgV2O4,
now as a function of ρCO2 to simulate the continuous removal of CO2. The plot is created at
900 K to allow a stoichiometric CO2-reduction mechanism. As expected from the standard
free energies, both ∆G < 0, which becomes more negative as ρCO2 is reduced. However,
below a CO2 partial pressure of 10−2 atm, MgV2O4 becomes more favoured as a product,
which arises from more CO2 being produced in the reaction. In terms of the reaction, two
cases are possible:
4.3 Results 113
• Both MgV2O5 and MgV2O4 are likely to be produced around ρCO2 = 10
−2 atm since
both reactions are close in free energy changes. This indicates that a selective reduction
to MgV2O5 would be difficult under such circumstances.
• Under very low CO2 partial pressures ρCO2 ≪ 10−2 atm, MgV2O4 is likely to be
produced as the dominant reduction product. This could be a condition that is reached
near the completion of the reaction, where the available amount of carbon is small.
These results indicate that a mixture of MgV2O4 and MgV2O5 would be the final product.
It is difficult to accurately predict the final outcome based on purely thermodynamic consid-
erations, especially since the kinetic contribution to the reaction is not taken into account.
It is clear, however, that MgV2O4 is the more favoured product under strongly reducing
conditions, and this phase would be exclusively produced in excess of added carbon (which is
often done in CTR reactions to coat the particles with conductive carbon). Hence, MgV2O4
could be produced in a relatively straightforward fashion, where an excess (typically 25 %)
of carbon is added and the precursor heated under Ar. The product could in addition be
annealed at a higher temperature to improve crystallinity.
A two-step reaction leading to MgV2O5
Having established a potential reaction route to synthesise MgV2O4, we turn our attention
to the more challenging MgV2O5 synthesis. It is clear that a single-step CTR reaction
cannot selectively produce this phase; a two-step process is then devised. In the first step, a
stoichiometric CTR reduction is performed to yield an average vanadium oxidation state of
4+, and the whole mixture is annealed at a higher temperature to allow comproportionation
of vanadium to form MgV2O5.
Under standard 1 atm partial pressures, subtracting Equation 4.3 from Equation 4.4 yields
the comproportionation equation to generate MgV2O5:
0.5MgV2O4+0.5MgV2O6 −−→MgV2O5 (4.5)
which has a negative ∆G = −88 kJ/mol at all temperatures, indicating a possibility for
spontaneous comproportionation reaction of MgV2O4 and MgV2O6 to form MgV2O5. This
spontaneous reaction opens up a possibility for a new synthetic route to MgV2O5: combined
with the capacity for a stoichiometric reduction by the CTR method, the starting MgV2O6
precursor could in principle be reduced to MgV2O4/MgV2O5/MgV2O6 mixture with an
average vanadium oxidation state of 4+, which would then be allowed to comproportionate
114
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
to give MgV2O5. This two-step process could be conducted in a one-pot setting in which the
second annealing immediately follows the first reduction step in the same setup, i.e. without
any intermediate grinding or change in ρCO2 .
In terms of determining the appropriate reaction conditions for this two-step process to
prepare MgV2O5, several points need to be considered:
• As the one-pot reaction needs a stoichiometric reduction to yield the correct oxidation
state for vanadium, a CO2, rather than CO, reduction needs to be performed. This sets
an upper bound for the reduction temperature to below 973 K, but the temperature
still needs to be sufficiently high to overcome the kinetic barrier. For this reason, the
reduction step is performed at 873 K (600 ◦C).
• The second comproportionation step could be performed at a higher temperature,
as long as the MgV2O5 product is thermally stable in Ar. MgV2O5 is known to
decompose above 1223 K,[178] so the annealing is performed at 1173 K (900 ◦C).
• However, designing this reaction also requires a careful consideration of the oxygen
partial pressure ρO2 involved in the reaction, as the inevitable presence of oxygen in
the Ar flow can oxidise/reduce the mixture depending on the free energies.
Intuitively, one would expect MgV2O4 to be more stable under a strongly reducing (low
ρO2) condition due to its lower oxidation state. During the second annealing step, this could
result in MgV2O4 being formed instead if ρO2 is sufficiently low. In an attempt to control the
ρO2 , the whole crucible was wrapped with a Cu foil, with additional Cu foils placed before
the crucible to catch the O2. Since both foils are kept at the same temperature as the reaction
mixture, we can now investigate the equilibrium ρO2 of the oxidation reaction
4Cu+O2
1200K−−−−⇀↽ − 2Cu2O (4.6)
of this ‘sacrificial’ Cu close to the comproportionation temperature (1173 K). Referring to the
Ellingham diagram in Chapter 2 (Figure 2.4), this reaction has an equilibrium ρO2 ≈ 10−8
atm at 1200 K; this is assumed to be the lower limit of ρO2 for the reaction atmosphere, as
oxidation of copper would take place above this oxygen pressure.
Figure 4.8 shows the free energy ∆G of the oxidation reactions involving MgV2O5 and
MgV2O4 phases as a function of ρO2 at 1200 K. At ρO2 = 10
−8 atm, the reduction reaction
to form MgV2O5 would be thermodynamically more favourable than the corresponding
reaction to form MgV2O4, although both are energetically allowed. As this is the lower
bound for the oxygen partial pressure (based on Cu oxidation), the diagram also indicates
4.3 Results 115
Fig. 4.8 DFT-based free energy ∆G of the oxidation reactions involving MgV2O5 and
MgV2O4 phases at 1200 K as a function of O2 partial pressure ρO2 . Equilibrium ρO2 of Cu
oxidation is also shown as a dashed line at 10−8 atm. The values are normalised with respect
to 1 mol of MgV2O6.
that formation of MgV2O5 is thermodynamically allowed at least until 1 atm ρO2 at this
temperature. Hence, the annealing step could be predicted to result in MgV2O5, the product
we aimed for, rather than MgV2O4.
Based on these considerations, the reaction scheme to produce MgV2O4 and MgV2O5
could be finalised and is summarised in Figure 4.5 (vide supra). A final note on the precursor:
as CTR reactions happen at the particle surface, both the precursor and carbon need to
be small particles with high surface areas. This is most easily achieved by sol-gel or co-
precipitation methods to prepare the precursor, or alternatively a bulk sample could be
prepared initially and ball-milled to reduce the particle size. Commercial high surface area
carbon such as Super P could be used as the reducing agent.
4.3.3 Characterisation of MgV2O5
Powder X-ray diffraction
Figure 4.9 presents the powder X-ray diffraction data of MgV2O5 (5 g batch) prepared by
the CTR approach developed above. Rietveld refinement results are also shown in Table 4.1.
116
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
Apart from a minor VO2 impurity (1 wt %), XRD identifies the product as MgV2O5. This
clearly shows the success of the rational design strategy; it also highlights the scalable nature
of CTR method. Excess VO2 phase is likely to be present due to the hydrated magnesium
acetate precursor (nominally tetrahydrate).
To elucidate the origin of excess VO2 present in the final MgV2O5 sample, a thermo-
gravimetric analysis (TGA) of the starting magnesium acetate precursor was performed. The
resulting TGA data is shown in Figure 4.10. At 573 K, complete dehydration of the hydrated
magnesium acetate results in a plateau.[199] From this the stoichiometry of initial precursor
is estimated to be Mg(CH3CO2)2 ·4.22H2O, whereas the sample is nominally a tetrahydrate;
this would result in approximately 2 mol % of magnesium deficiency. Considering the
presence of around 1 wt % (equivalent to 2.5 mol %) of VO2 detected on PXRD, this excess
hydration is likely to be the reason behind the VO2 secondary phase.
It should be stressed that the CTR reaction is selective for vanadium reduction (more
stable MgO reduction requires a much higher reaction temperature), so this vanadium excess
does not have an influence on the carbon stoichiometry. This is evidenced by the formation
of VO2, which is likely to be a reduction product of excess V2O5. It also indicates the
possibility of using the CTR approach to prepare VO2, a phase which has been shown to
exhibit electrolyte gating behaviour on top of the metal-insulator transition at 340 K.[200]
This phase has wide potential applications in ‘smart’ windows, Mott transistors, and memory
devices.[201, 200]
25Mg NMR: signal enhancements using RAPT
Having obtained a sample of MgV2O5 through the CTR method, we now turn our attention
to the 25Mg NMR spectrum of this compound. Figure 4.11a shows the 25Mg spectrum of
the as-synthesised compound. Fitting of this spectrum gives δiso = 1763 ppm, together with
the CQ = 5.3 MHz and ηQ = 0. The spectrum is in good agreements to previously reported
25Mg spectrum of this compound,[34, 35] although no detailed NMR characterisation was
presented in these works. As expected from a distorted 6-fold Mg coordination environment,
the resonance shows a distinct MAS quadrupolar lineshape with a CQ = 5.3 MHz. This
is by far the largest CQ measured in paramagnetic Mg environments, despite the fact that
quadrupolar coupling in paramagnetic solids is often difficult to measure due the large line
broadening and loss of resolution. In MgV2O5, however, the small electron spin moment
of V4+ (S = 1/2) is likely to reduce the nuclear relaxation rate (1/T1 ∝ S(S+ 1) from the
Solomon-Bloembergen-Morgan Equation[142]) and thus result in an improved resolution.
4.3 Results 117
 10  20  30  40  50  60  70  80
MgV2O5
VO2
Co
un
ts 
(ar
b. 
un
it)
2θ/degrees
observed
calculated
difference
Fig. 4.9 Refined powder X-ray diffraction data of MgV2O5 as prepared by the CTR method.
Refined parameters are shown in Table 4.1. Reflections for MgV2O5 (50979) and VO2
(34033) are taken from the ICSD.
MgV2O5 99 wt % Cmcm space group
a / Å 3.69050(4) α / ◦ 90
b / Å 9.97075(1) β / ◦ 90
c / Å 11.01735(1) γ / ◦ 90
Atom x y z
Mg1 (4c) 0 0.8801(2) 0.25
V1 (8 f ) 0 0.2050(1) 0.0959(1)
O1 (8 f ) 0 0.0440(4) 0.1298(3)
O2 (8 f ) 0 0.2401(3) 0.5794(3)
O3 (4c) 0 0.2983(6) 0.25
Table 4.1 Rietveld refined parameters from the PXRD data of MgV2O5 as prepared by the
CTR method.
118
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
Fig. 4.10 Thermogravimetric data of the magnesium acetate (nominally tetrahydrate) precur-
sor used in the synthesis of MgV2O6.
Having measured the NMR parameters and especially the CQ, we can now apply the
RAPT pulse sequence to this sample, as used in the previous chapter for signal enhancements
in paramagnetic 25Mg spectra. In terms of the experiment, the large quadrupolar coupling
observed in this material poses a significant challenge due to two reasons:[179, 202]
• Large quadrupolar coupling necessitates the use of faster spinning to deconvolute the
spnning sidebands from the broad second-order CT lineshape. The individual ST
isochromat in RAPT experiments should ‘sweep’ through the alternating Gaussian
pulses at a fixed frequency, so faster spinning would result in less efficient saturation
of ST populations and thus lower enhancements.
• The STs are located at frequencies far apart from the CT resonance, such that the
saturating Gaussian pulses should also have large offsets from the CT frequency. This
is problematic in terms of the limited bandwidth of NMR probe circuitry, and one
would expect a rapidly decreasing enhancement when the offset is increased beyond the
available probe bandwidth. For low-γ nuclei such as 25Mg, the narrower bandwidth at
lower frequencies makes this extra challenging. Referring to the theoretical description
in Chapter 2, this also means that offset at the excitation edge (Equation 2.68), where the
signal enhancement is expected to return to 1, cannot be recorded accurately. Instead,
the CQ should be estimated from the offset at maximum enhancement (Equation 2.69).
4.3 Results 119
(a)
(b)
Fig. 4.11 (a) 25Mg spin echo spectrum of the as-synthesised MgV2O5, measured under 20
kHz MAS and a 0.1 s recycle delay. Fitted parameters are shown in Table 4.3. (b) Structure
of MgV2O5, showing the 6-fold Mg coordination environment as polyhedra.
120
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
Fig. 4.12 Enhancement profile of the integrated signal intensity in MgV2O5 using the RAPT
pulse sequence, as a function of the Gaussian offset νoff. All samples were measured under
20 kHz MAS and a 0.1 s recycle delay.
Fig. 4.13 25Mg spectra of as-synthesised MgV2O5, measured with a normal spin echo and a
RAPT-spin echo pulse sequences. Both were measured under 20 kHz MAS and 0.1 s recycle
delay.
4.3 Results 121
In this regard, Figure 4.12 shows a full enhancement profile over a range of offset
frequencies recorded on MgV2O5. As predicted in Equation 2.69, a maximum enhancement
of 3 should occur at νoff = 400 kHz (CQ = 5.3 MHz). The experimental enhancement
profile displays a maximum enhancement of 1.6 around νoff = 200 kHz, a clear indication of
inefficient ST saturation under this challenging condition.
Examination of the enhancement profile could also give information on the asymmetry
parameter ηQ. Fitting of the MgV2O5 spectrum (Table 4.3) shows a well-defined ηQ = 0,
which would be expected to result in a ‘tailing off’ slow decline in the enhancement profile.
Figure 4.12, however, shows a rather steep decline in intensity near νoff = 250 kHz, a feature
characteristic of systems with large ηQ close to unity. This discrepancy is likely to be a direct
consequence of the limited probe bandwidth at this low frequency (42.5 MHz at 16.4 T),
which also contributes to the lower enhancement factor compared to the expected factor of 3.
Despite these challenges, Figure 4.13 demonstrates a maximum enhancement of around
1.6 using the RAPT pulse sequence. It is also seen that the lineshape distortion is minimal
upon RAPT enhancement, which further confirms the applicability of RAPT in quadrupolar
NMR of paramagnetic species.
25Mg NMR of the ball-milled MgV2O5
Having established the RAPT methodology for the pristine, as-synthesised sample, we now
turn our attention to a mechanically milled sample with a reduced particle size. As typical
Mg-ion battery electrodes are prepared in reduced particle sizes with a ball-milling step, this
provides us with the sample in a similar condition to those in a cycled Mg-ion battery. Figure
4.14 shows the 25Mg spectrum of ball-milled MgV2O5 sample as analysed in Appendix A.
The spectrum shows a similar shift as the pristine sample at 1763 ppm, supporting the XRD
data that the sample has not degraded. The characteristic CT lineshape, however, has now
completely disappeared, possibly due to the reduced crystalline sizes and/or presence of
defects as created by the milling. As the quadrupolar coupling interaction is local to each
spin, which results from a local coordination environment and not the overall crystallinity,
the RAPT pulse sequence is still expected to enhance the 25Mg signals regardless of the
milling. Under these conditions, application of the RAPT pulse sequence to this material
still displays an overall enhancement factor of 1.5, illustrating the powerful sensitivity of the
RAPT pulse sequence (and NMR in general) to the local environment around the spin.
122
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
Fig. 4.14 25Mg spectra of ball-milled MgV2O5, measured under normal spin echo and
RAPT-spin echo. Both were measured under 20 kHz MAS and a 0.1 s recycle delay.
4.3.4 Electrochemical Cycling of MgV2O5
Based on the computational discussion above, Mg diffusion in the MgV2O5 host would have
a relatively low Mg diffusion barrier. Combined with the work on NMR detection of this
compound, electrochemically cycled sample could be measured to confirm the reversible
magnesiation behaviour of this material. In this section the initial cycling data are presented.
Charging behaviour of MgV2O5
To confirm the demagnesiation of MgV2O5, the compound was charged in a standard Li-ion
half-cell with Li metal counterelectrode and standard LP30 electrolyte. Ball-milled MgV2O5
was used to reduce the particle size to enhance diffusion, and the sample was checked for
structural integrity post ball-mill with powder XRD and NMR (Figures A.2 and 4.14). SEM
images indicate that the particle size is reduced to a sub-micron level (Figure A.1).
Electrochemical response for the galvanostatic charging step (C/50-rate) is presented in
Figure 4.15a. The curve displays an initial plateau at 3.8 V and shows a gradual increase in
potential until about 4.2 V, after which it displays a ‘wiggle’-like behaviour. Previous compu-
tation on this compound predicted a potential of 2.56 V vs Mg metal, which corresponds to
2.9 V vs Li metal (using the standard electrode potentials). This agrees with the observed
4.3 Results 123
open circuit voltage (OCV) of 3 V, but a significant overpotential of about 0.8 V is observed
on charging.
The onset of this wiggle happened at about 250 mAh/g capacity, close to the theoretical
capacity of 260 mAh/g (assuming a full demagnesiation of MgV2O5). The cell clearly was
charged beyond its theoretical capacity, which could arise from an electrolyte decomposition
or a structural failure. As electrolyte decomposition typically happens at a flat potential,
structural disintegration, possibly involving vanadium dissolution, was suspected.
Postmortem examination of charged cell (Figure 4.15b) revealed that the lithium metal
counterelectrode was covered in a black species and the glass fibre separator was in green
colour. Upon charging, vanadium (either from the VO2 or (Mg)V2O5) could dissolve into
the electrolyte giving a green hue for the separator, which could then plate on the lithium
counterelectrode. Possible origins for this behaviour are discussed in the next section.
XRD pattern of the charged cathode film was measured and is given in Figure 4.15c.
Almost all MgV2O5 reflections have disappeared with appearance of α-V2O5 reflections,
confirming the charging step. Several relatively narrow reflections arising from uncycled
MgV2O5 still survive (2θ =20, 25, 27, 30 degrees). However, the broad α-V2O5 reflection
indicates possible particle amorphisation upon cycling and irreversibility; further work needs
to be done with this compound to confirm this. A detailed study on cycling this compound
was not attempted due to an absence of high-voltage electrolytes and the problem of vanadium
dissolution (see below).
Vanadium dissolution of VO2
Previous XRD analysis revealed presence of VO2 in this batch of MgV2O5 (Figure 4.9).
As VO2 becomes metallic above 340 K (70 ◦C), the possibility of a facilitated vanadium
dissolution arising from VO2 was investigated by charging a pristine VO2 sample.
Figure 4.16a shows the charging curve of a VO2 self-standing film in a Li-ion cell. An
abrupt increase in potential is observed above 3.4 V, which plateaus out at 4 V. Postmortem
examination of the Li counterelectrode is also black, confirming the vanadium dissolution
from VO2. XRD analysis shows the presence of significant amount of V6O13, equally
written V12/13O2, which would have resulted from an oxidation reaction as described in the
Discussion section (Section 4.4.2); what is clear however is the formation of V6O13 as an
oxidation product of VO2 upon charging, as evidenced by the XRD pattern.
124
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
(a)
(b)
 10  20  30  40  50  60  70  80
α-V2O5
MgV2O5
Co
un
ts 
(ar
b. 
un
it)
2θ/degrees
observed
calculated
difference
(c)
Fig. 4.15 (a) Electrochemical cycling data of ball-milled MgV2O5, charged vs Li. (b) Glass
fibre separator and lithium disc after disassembly of the coin cell shown in (a). (c) Powder
X-ray diffraction data of charged cathode. Broad background at 20◦ is due to Kapton sample
holder.
4.3 Results 125
(a)
(b)
 10  20  30  40  50  60  70  80
V6O13
VO2
Co
un
ts 
(ar
b. 
un
it)
2θ/degrees
observed
calculated
difference
(c)
Fig. 4.16 (a) Electrochemical cycling data of VO2, charged vs Li. (b) Lithium disc after
disassembly of the coin cell. (c) Powder X-ray diffraction data and Le Bail fit of charged
VO2. Broad background at 20◦ is due to Kapton sample holder. Self-standing films were
provided by Michael Hope.
126
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
This work Reported
Expt PBE+U Hyb20 Hyb35 Korotin[197] Millet[203] Onoda[178]
J1 / K - 62 25 16 -30 -100 -
J2 / K - -55 -55 -40 -46 -282 -
J3 / K - -57 -74 -58 -72 - -
J4 / K - - - - -9 - -
Θ / K -277 -22 -77 -62 -339 -307 -174
Table 4.2 Magnetic parameters of MgV2O5 determined by experiment and DFT calcula-
tions. Experimental parameters were determined by SQUID magnetometer (Figure 4.17);
J-couplings are as denoted on Figure 4.18a.
4.3.5 Computation of Magnetic and NMR Parameters
Now we turn our attention to the ab initio computation of the magnetic and 25Mg NMR
parameters for MgV2O5. The magnetic parameters are of interest since the previous magnetic
measurements on this compound have reported a magnetic response characteristic of a system
with reduced dimensionality. Since the hyperfine NMR parameters also intimately depend
on the magnetism, we attempt to measure and predict the magnetism present in MgV2O5.
Magnetic interactions
The plot showing the experimentally measured magnetic susceptibility χ is displayed in
Figure 4.17a. The low-temperature data presented in figure inset is in good agreement to the
previous report by Millet et al. where a fall and then rapid rise in χ was attributed to the
reduced dimensionality of the system.[203]
Our value for the Curie-Weiss constant, Θ=−277 K, determined from the inverse sus-
ceptibility plot (Figure 4.17b) also shows good agreement to experimental values determined
by Korotin and Millet et al. Onoda et al. reported a slightly smaller Θ = −174 K; it is
likely their sample was highly nonstoichiometric (residual VO2 was visually observed) which
would have contributed to the error. However, further work on preparing a better quality
sample with no detectable impurity on the XRD must be carried out.
Table 4.2 shows the ab initio computed exchange coupling parameters Jn as determined
by the linear regression method. Literature values are also shown, where available. The
largest discrepancy is observed in the sign and magnitude of J1, the nearest neighbouring
interaction along the edge sharing square pyramids. PBE+U and both hybrids (20 % and 35
% Hartree-fock exchange) consistently show a positive (ferromagnetic) exchange for this
4.3 Results 127
 0.00075
 0.0008
 0.00085
 0.0009
 0.00095
 0.001
 0.00105
 0.0011
 0.00115
 0  50  100  150  200  250  300
χ 
(em
u/m
ol 
Oe
)
Temperature (K)
 0.001
 0.00105
 0.0011
 0.00115
 0  10  20  30
(a)
 600
 700
 800
 900
 1000
 1100
 1200
 1300
 0  50  100  150  200  250  300
y=mx+b
m=2.231 ± 0.006
b=618.5 ± 1.6
Θ=−277.21 ± 0.00 K
1/
χ 
(m
ol 
Oe
/em
u)
Temperature (K)
(b)
Fig. 4.17 (a) Zero field cooled molar susceptibility χ and (b) Zero field cooled inverse molar
susceptibility 1/χ of MgV2O5 as measured by SQUID magnetometry. Linear fitting was
performed from 200 K to 300 K.
128
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
interaction, whereas reported values using the Linear Muffin-Tin Orbital method with the
LDA+U functional show a negative (antiferromagnetic) exchange.[197]
To explain this discrepancy, we need to understand the underlying orbital overlap.
MgV2O5 is a S = 1/2 system with a 3d1 state, with the dxy orbital being the lowest in
energy.[197] This orbital lies in the pyramidal plane avoiding the oxygen, as illustrated in
Figure 4.18a. Two competing interactions are present in the case of J1: a direct exchange
between the 3dxy orbitals lead to an antiferromagnetic coupling, whereas a superexchange via
an O2 – leads to a ferromagnetic correlation superexchange. It is thought that the DFT-based
methods ‘over-bind’ the electrons to underestimate the direct exchange and overestimate
the superexchange, resulting in an overall ferromagnetic interaction. The relatively diffuse
nature of vanadium orbitals make the overlaps very sensitive to bond lengths and angles;
further investigation is necessary using different structures and functionals. This discrepancy
is most likely to be the reason behind the significantly smaller values of the Curie-Weiss
constant Θ (simulated values range from -22 to -77 K versus the experimental -277 K) when
compared to the experiment and other reported values.
The J2,3 couplings arise mostly from a superexchange mediated by an O2 – , and a
delocalisation superexchange mechanism successfully explains the antiferromagnetic nature
of these two interactions. These couplings also show good agreement to the Linear Muffin-tin
Orbital (LMTO) method results by Korotin et al.[197] Results by Millet et al.[203] were
estimated from the structural parameters by use of empirical relationships between the J and
bond length/angles, so they are likely to be less accurate when compared to first principles
methods. Onoda et al. did not report any values for J.[178]
NMR parameters
Hyperfine and quadrupolar parameters determined from experiment and DFT computations
are shown in Table 4.3. Both Hyb20 and Hyb35 results consistently overestimate the shifts.
DFT predictions in other Li-, Na-, and Mg-oxides have consistently shown an overestimation
of Hyb20 and an underestimation for Hyb35, thus defining a boundary of the experimental
shift;[38, 204, 205, 185] this is clearly not the case for MgV2O5.
A major source of this error comes from the ab initio determination of the magnetism.
The shift δiso depends inversely on the Θ: δiso ∝ 1T−Θ . As seen in the previous section, DFT
predicts a ferromagnetic interaction for the J1, which gives a significantly less negative value
of Θ and overestimates the shifts.
An additional error comes from the electronic structure itself; it is likely that the hyper-
fine coupling constant Aiso (or equally, spin density at nuclear positions |Ψα−βN |2) is also
4.3 Results 129
(a)
(b)
Fig. 4.18 (a) Illustration of the V–V exchange interactions Jn up to the fourth nearest
neighbour in MgV2O5. Dark blue colour refers to vanadium. (b) Dominant exchange
mechanisms for the J1 (left) and J2,3 (right).
130
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
Hyb20 Hyb35 Expt
δiso / ppm 2329 1991 1763
Ω 4329 4164 -
κ -0.23 -0.34 -
CQ / MHz 4.3 4.3 5.3
ηQ 0.0 0.0 0.0
Table 4.3 NMR parameters of MgV2O5 determined by experiment and DFT calculations.
Experimental parameters are fitted from the spectrum in Figure 4.11a (anisotropies could
not be determined from the spectrum). Herzfeld-Berger convention is used for the shift
anisotropy.
underestimated due to the over-binding of hybrid methods, in a manner similar to the J1
interaction. Using the experimentally determined Θ = −277 K, Hyb20 and Hyb35 gives
δiso =1478 and 1217 ppm, respectively. This confirms the over-binding of hybrid methods,
as the discrepancy in this case should come from the hyperfine coupling itself.
Hence, determination of accurate magnetism is the most crucial component of the
hyperfine shift calculations in vanadium compounds; this behaviour is observed in other
Mg-compounds (Chapter 3, Section 3.3.3) as well. The diffuse nature of vanadium makes
the prediction particularly challenging; more advanced functionals such as the meta-GGA
functional combined with plane-wave basis sets could be used for such purposes.
The computed quadrupolar parameters show a value of CQ = 4.3 MHz for both Hyb20
and Hyb35, which is smaller than the fitted value of 5.3 MHz. This result is also consistent
with the observation of over-binding of hybrid methods in this case.
4.3.6 Characterisation of MgV2O4 Prepared Through the CTR method
As shown previously in Figure 4.5, the CTR method could also provide a novel way to
prepare a spinel-type magnesium vanadate, MgV2O4. In Chapter 3, this compound was
prepared via a solid-state route (with the resulting sample containing V2O3 secondary phases)
and characterised with 25Mg NMR spectroscopy; hence we exploit this opportunity to briefly
demonstrate the applicability of the CTR method to synthesise MgV2O4 and present the
initial magnetic and 25Mg NMR characterisation of this compound.
Powder X-ray diffraction
Powder X-ray diffraction data and the Rietveld refinement of the resulting MgV2O4 prepared
through the CTR method (denoted CTR-MgV2O4) is shown in Figure 4.19 and Table 4.4).
4.3 Results 131
The diffraction pattern shows a predominant MgV2O4 phase without noticeable V2O3 im-
purity, clearly demonstrating the utility of CTR method in preparing various magnesium
vanadate compounds. SS-MgV2O4 sample exhibits broader reflections compared to the
CTR-MgV2O4, which may reflect the smaller particle size/increased degree of disorder for
this sample.
However, X-ray diffraction was insufficient to conclude any nonstoichiometry (arising
due to the hydrated magnesium acetate; see Section 4.3.3), where the refinement indicated no
significant partial occupancies (within the experimental limits of laboratory X-ray diffraction)
in both Mg and V sites (0.991(2) and 0.983(2), respectively).
Magnetic characterisation
As the CTR-MgV2O4 sample is phase-pure in contrast to the SS-MgV2O4 sample prepared
in Chapter 3, SQUID magnetometry was performed on the CTR-MgV2O4 to determine
the magnetism and possible site disorder present in this material. The resulting inverse
susceptibility plot is shown in Figure 4.20. Upon cooling under a zero magnetic field, the
cubic to tetragonal transition occurs at Ts = 60 K, followed by an antiferromagnetic ordering
at TN = 34 K (Néel temperature). These value are in slight disagreement to the previous
reports of Ts = 65 K and TN = 42 K;[165, 164] the discrepancy could originate from the V4+
in the system, arising from the slight Mg deficiency expected in the composition as explained
below.
The electron magnetic moment per mol V (from the ‘elevated’-temperature regime of
200–300 K) of µeff = 3.60±0.02 µB was determined for the CTR-MgV2O4, whereas the
spin-only moment is expected to be µeff = 2.83 µB (Table 3.3). This inconsistency could
again be ascribed to the short-range fluctuations as for the case of MgCr2O4; our result is
consistent with the report by Mamiya et al. where a value of µeff = 3.35 µB was measured
for a temperature range of 100–200 K.[164] At a higher temperature range of 450–600 K,
the reported value of µeff decreased to 2.97 µB (similar to the spin-only value of 2.83 µB),
which clearly indicates a near complete suppression of this short-range fluctuation.
When analysing the magnetic data, we should make note that the preparation of CTR-
MgV2O4 starts from a MgV2O6 precursor made under ambient air, and the Mg under-
stoichiometry of approximately 2 mol % (arising from the acetate precursor as described
in Section 4.3.3) should still be present in this precursor. In MgV2O6, the Mg deficiency
should be compensated by oxygen vacancies to form a Mg0.98V2O5.98 phase (no extra VxOy-
type secondary phases were observed in the precursor). This translates to a Mg-deficient
phase of Mg0.98V3+1.96V
4+
0.04O4 after the reduction step. This was not readily observed with
132
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
 10  20  30  40  50  60  70  80
MgV2O4
248565
Co
un
ts 
(ar
b. 
un
it)
2θ/degrees
observed
calculated
difference
Fig. 4.19 X-ray powder diffraction pattern for CTR-MgV2O4 (carbothermal route). The
positions of allowed reflections are indicated by the tick marks.
MgV2O4 100 wt % Fd3¯m space group
a / Å 8.41499(7) α / ◦ 90
b / Å 8.41499(7) α / ◦ 90
c / Å 8.41499(7) α / ◦ 90
Atom x y z Occ
Mg1 (8b) 0.375 0.375 0.375 0.991(2)
V1 (16d) 0 0 0 0.983(2)
O1 (32e) 0.2419(1) 0.2419(1) 0.2419(1) 1
Rexp 7.43
Rwp 12.26
χ2 1.65
Table 4.4 Rietveld refined parameters from the PXRD data of MgV2O4 prepared via a
carbothermal method (CTR-MgV2O4).
4.3 Results 133
 500
 550
 600
 650
 700
 750
 0  50  100  150  200  250  300
34 60
y=mx+b
m=0.615 ± 0.004
b=537.5 ± 0.8
Θ=−873.26 ± 5.4 K
1/
χ 
(m
ol 
Oe
/em
u)
Temperature (K)
Fig. 4.20 Inverse magnetic susceptibility per mol V, 1/χ , as a function of temperature for the
CTR-MgV2O4 sample prepared via a carbothermal route.
XRD (refined site occupancies correspond to a composition of Mg0.99V1.97O4) but could be
detected with magnetic measurements and 25Mg NMR (vide infra).
The nonstoichiometry argument is also consistent with a previous observation of variation
in Ts and TN along the growth direction of a single crystal by Islam et al.[165] In their study,
3 mol % of excess V2O3 was added to the crystal flux to allow a homogeneous composition.
The lower part of this single crystal exhibited Ts = 55 K and TN = 38 K, which are lower than
the corresponding Ts = 65 K and TN = 42 K measured from the middle part of the crystal.
Our values of Ts = 60 K and TN = 34 K are in agreement to this lowering of both transition
temperatures.
What is also clear in our measurement is the absence of clear spin-glass like transition at
low temperatures, which has been previously demonstrated for a site-disordered MgV2O4
with excess Mg (~3 %) on octahedral sites.[165] The presence of clear structural (Ts) and
magnetic (TN) transitions may indicate the absence of a site disorder in this compound.
In addition, the Weiss constant Θ=−873.3±5.4 K and the nearest-neighbour magnetic
interaction J1 = −109.6± 0.7 K (determined from Θ by use of the relation 2.73) is also
significantly larger than the reported values of Θ=−600 K and J1 =−75 K, which would
be consistent with the large observed value of µeff arising from the short-range fluctuations.
134
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
Mamiya et al. again reported values of Θ=−760 K (100–200 K) and −600 K (450–600
K);[164] our value agrees better with their 100–200 K data.
25Mg NMR
The 25Mg MAS spectrum of the carbothermally prepared CTR-MgV2O4 is shown in Figure
4.21. It is noticeably different from the spectrum of the solid-state synthesised SS-MgV2O4
sample (Figure 3.15): first of all, three peaks are observed at 1861, 1783, and 1713 ppm,
where the 1783 ppm peak is the dominant resonance; they are also significantly broadened
compared to the SS-MgV2O4 case. Whereas a full DFT calculation of the individual sites
could not be performed due to SCF instabilities, approximate insights into the nature of
these resonance could be gained by considering the bond pathway contribution (similar to
that considered in Section 3.3.4) in MgCr2O4 and SS-MgV2O4. Taking into account that
each A-site Mg is neighbouring twelve B-site TMs, the contribution of each Cr in MgCr2O4
approximates to 238.5 ppm per Cr3+ ion or 79.5 ppm per d-electron. Mg in SS-MgV2O4,
on the other hand, resonates at 1845 ppm, which gives 153.8 ppm per V3+ ion or 76.9 ppm
per d-electron. Hence, we can estimate that the contribution of a single d-electron on the
cubic spinel B sublattice is around 75-80 ppm. As the V4+ cations have only one d-electron
compared to two of V3+, we can expect a smaller shift for the Mg neighbouring a V4+.
Assuming that a single d-electron contributes 77 ppm to the total hyperfine shift of a Mg
site, the hyperfine shifts of Mg atoms neighbouring two and one V4+ ions could be estimated
as 1768 and 1691 ppm, respectively. These values are close to the observed shifts of 1783
and 1713 ppm, which suggests the presence of V4+ on the vanadium lattice. The results are
summarised in Table 4.5.
One can also predict the relative intensities of these resonances by assuming a random
distribution of the Mg vacancy-V4+ pair. As each Mg site is bonded to twelve V sites, a
composition of Mg0.98V3+1.96V
4+
0.04O4 (determined from the starting Mg/V ratio) should result
in a relative intensity ratio of 39:9.5:1 for the resonances at 1861, 1783, and 1713 ppm,
respectively. The experimental intensities, however, show a ratio of 0.9:7.7:1; this initial
investigation shows that the random model of V4+ distribution is not valid for this sample and
may point towards a higher concentration of the V4+ ions contained in the lattice, coupled
with some degree of ion ordering.
Lastly, the peak broadening observed for the CTR-MgV2O4 could come from a lack of
sintering present in this sample, as the pellet was pressed with carbon which ultimately gets
oxidised to leave an empty space in the pellet; it could also arise from a small degree of
quadrupolar coupling introduced by an uneven distribution of electrons due to the arrange-
4.3 Results 135
Fig. 4.21 25Mg spin echo spectrum of CTR-MgV2O4 (carbothermal route) at MAS spin
rate of 10 kHz. Positions of the isotropic resonances are indicated. 628400 transients were
acquired with recycle delays of 0.1 s.
Coordination 12V3+ 11V3+,1V4+ 10V3+,2V4+ Remarks
δexpt / ppm 1861 1783 1713 Experimental
δest,SS / ppm 1845 1768 1691 Referenced to 1845 ppm of SS-MgV2O4
δest,CTR / ppm 1861 1784 1707 Referenced to 1861 ppm of CTR-MgV2O4
Iexpt 0.9 7.7 1 Experimental
Irandom 39 9.5 1 Random distribution
Table 4.5 Estimated hyperfine shifts and the intensity ratio for the Mg ions in the MgV2O4
spinel structure neighbouring V3+ and V4+ ions. The shifts ‘referenced’ from the 1845 ppm
(SS-MgV2O4) or 1861 ppm (CTR-MgV2O4) are shown. Integrated relative intensities are
shown for the experimental spectrum and also calculated assuming a random distribution of
the V4+ ions.
136
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
ments of V3+/V4+ around the Mg positions. Without DFT calculations, however, quantitative
estimations on the magnitude of quadrupolar coupling parameters could not be made.
4.4 Discussion 137
4.4 Discussion
4.4.1 Implications of the CTR Method for Other Oxides
Many of the potential high-voltage cathode materials for Mg-ion batteries, including Mg3Nb6O11
and Mg2Mo3O8,[206] have unconventional TM oxidation states that are difficult to prepare.
The conventional way of preparing these compounds would be a comproportionation of pre-
cursors, or a (partial) reduction under a reducing gas such as H2/Ar. The comproportionation
route often requires a crucible inside an evacuated quartz ampoule (MgO reacts with SiO2 at
elevated temperatures), which poses a moderate experimental difficulty. The reduction route
under H2 cannot, in general, be used to easily control the final oxidation state of the product
as the reaction is not stoichiometric. This is most easily evidenced by the fact that MgV2O5,
a V4+ compound, is prepared via the first route; reduction under H2 would result in a V3+
compound, as previously mentioned.
As an alternative route, the CTR method could potentially be used to make products
with these unusual oxidation states as it provides a way for stoichiometric reduction starting
from a stable precursor such as V5+. When combined with computational predictions of the
thermodynamic parameters involved, the reaction scheme could be designed in a rational
manner. This approach is in principle completely general and it could potentially be a
powerful method to prepare many of these complex Li, Na, Mg, and Ca TM oxides which
may have technological importance. In addition, it could also be a valuable tool in exploring
the complex magnetism in TM oxides, as many of these compounds exhibit unusual magnetic
structures at low temperatures. The CTR method can be easily used to prepare large batches
of samples for neutron diffraction to probe the magnetic order. This could be an advantage
over the quartz ampoule route, where the amount of a single batch is determined by the size
of ampoules used. Further work is also in progress regarding the low-temperature magnetic
behaviour of S = 1/2 compounds MgV2O5 and MgVO3 prepared by the CTR method.
4.4.2 Discussion on the Initial MgV2O5 Electrochemistry
Vanadium dissolution
Upon charging/de-magnesiation of the MgV2O5 sample with a VO2 impurity, it is evident
that a vanadium dissolution takes place on the cathode, which is then replated on the lithium
anode. Whereas we have shown that VO2 can give vanadium to become oxidised to V6O13,
possible involvement of (Mg)V2O5 should not be ruled out.
138
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
Vanadium dissolution in metal vanadates during an electrochemical cycling has been
reported before, mostly for lithium vanadates. For instance, Jouanneau et al. reported
dissolution of a V3+ species upon lithiation of Li1.1V3O8, which takes place via a Li–V
site exchange, or an O2 – loss.[207] As our observation of the dissolution takes place upon
charging, however, the dissolution must take place via an oxidation of the sample. This
oxidation could happen alongside an electrolyte degradation (ethylene carbonate and dimethyl
carbonate both contain oxygen in the structure) caused by the high applied potential; either
or both VO2 and MgV2O5 may exhibit such dissolution.
Assuming an oxidised V5+ species is present as a VO43 – , a potential oxidation equation
could be written for VO2 as:
7VO2+3 [O]−−→ V6O13+VO43− (4.7)
where [O] denotes an oxygen generated from the electrolyte degradation. However, a similar
oxidation on the MgV2O5 again should not be ruled out, which will become evident when a
pure-phase sample is prepared and tested.
This vanadium dissolution poses problems for a proper analysis of MgV2O5 electrochem-
istry displayed on Figure 4.15a, for at least three reasons.
• Some of the observed capacity could come from an electrolyte decomposition, which
cannot be ruled out. This could be solved by using better electrolytes such as ionic
liquids to achieve a higher stability window.
• Vanadium plating on lithium will change the observed cell potential (standard potential
for V2+ −−⇀↽−V is −1.13 V vs SHE), thus masking the real potential. A three-electrode
cell, with the reference, working, and counter electrodes, could be used to deconvolute
the effect of Li counterelectrode.
• However, there is no way of distinguishing between the VO2 dissolution and MgV2O5
demagnesiation in terms of voltage, as both are likely to happen in a cell. Obviously,
the only solution to this problem is to prepare a sample without VO2.
However, we note that a full oxidation of VO2 up to 4.2 V would give 60 mAh/g (based
on the amount present in the sample). This is clearly smaller than the observed ‘capacity’ of
250 mAh/g (the tailing ‘wiggles’ excluded); the rest of the capacity is expected to arise from
MgV2O5, which is also evidenced by the V2O5 reflections on the diffraction pattern. Hence,
further work with a phase-pure MgV2O5 would clarify this electrochemical activity.
4.4 Discussion 139
The origin of the observed capacity and electrochemical activity
In our electrochemical cycling, a full ‘capacity’ of 250 mAh/g (the tailing ‘wiggles’ excluded)
was observed, whereas the material is expected to have a theoretical capacity of 260 mAh/g.
While this observed capacity could arise from a combination of the real capacity, electrolyte
degradation, and possible vanadium dissolution, we note the possibility of electrochemical
activity being present upon charging, as evidenced from the V2O5 reflections on the powder
diffraction of the charged sample.
We also note that a full demagnesiation of δ -MgV2O5 have resulted in a structural
transformation from the AB-type stacking of the V2O5 layers in δ -MgV2O5 to the AA-type
stacking in α-V2O5. This is in direct contrast to the computational result predicted by
Sai Gautam et al., where the δ -V2O5 type AB-stacking was expected to be metastable
over a wide Mg stoichiometry.[67] However, the application of a significant overpotential
upon charging (0.8 V in our case) may serve as the driving force for this δ → α phase
transformation; in addition, the ‘wiggles’ that extend beyond this theoretical capacity (Figure
4.15a) may be characteristic of this phase transition.
Based on these considerations, further work on preparing and cycling a pure-phase
MgV2O5 needs to be carried out. Fortunately, the potential source of this excess VO2 has al-
ready been identified: the MgV2O6 CTR precursor was prepared by a sol-gel chemistry using
a magnesium acetate precursor (nominally tetrahydrate). It is likely that the precursor was
further hydrated beyond tetrahydrate, which would result in a Mg deficiency and thus excess
vanadium, as evidenced by the TGA measurement. Thus, VO2 formation could be circum-
vented by using a stoichiometric amount of the acetate precursor. Alternatively, MgV2O6
could be prepared via a solid-state route by reacting MgO and V2O5, and subsequently
ball-milled to reduce the particle size for the CTR reaction.
4.4.3 MgV2O4: Sample Dependence on the Preparation Method
In this Chapter, we have also demonstrated the applicability of this rational design approach
to prepare MgV2O4, a material previously only synthesised through conventional solid-state
methods. Starting from the same MgV2O6 precursor, it is clear that the reaction conditions
(carbon stoichiometry, reaction temperature) could have a direct control over the final product
formed (MgV2O4, MgV2O5).
25Mg NMR spectrum of the CTR-MgV2O4 showed a remarkable difference from the
spectrum of SS-MgV2O4, where additional Mg sites were observed as two extra resonances.
The starting Mg/V composition and the paramagnetic shifts of these additional resonances
140
Carbothermal Synthesis and Characterisation of MgV2O5, a Potential Mg-ion Battery
Cathode Material
indicate the possible presence of V4+ on the lattice arising from the Mg deficiency. This
warrants a further work using X-ray spectroscopic techniques such as XANES experiments to
confirm the distribution of oxidation states; also, new samples using stoichiometric quantities
of precursors must be prepared to rule out the effects of magnesium nonstoichiometry.
However, we do note that this also opens up a potential opportunity to perform a systematic
study on the overall magnetism (using a SQUID magnetometry) and the local distribution of
this V4+ sites (as probed with 25Mg NMR) by varying the level of Mg deficiency present in
the precursor. This would not only clarify the nature and distribution of the V4+ sites, but
also help to explain the intensity ratios of the three 25Mg resonances which do not agree with
the scenario of random distribution.
4.5 Conclusion and Further Work
In this chapter, a comprehensive investigation was performed on the synthesis, characterisa-
tion, and electrochemical cycling of MgV2O5. It was shown that the carbothermal reduction
(CTR) method could be used to prepare complex vanadium oxides which are difficult to
prepare with conventional methods; computational prediction of the thermodynamic pa-
rameters allowed a rational design of reaction conditions leading to this compound, which
was experimentally demonstrated. This would open a potential way for large-scale battery
applications since many of Li-, Na-, Mg-, and Ca-transition metal oxides are shown to be
promising battery electrodes; in addition, large quantities of samples could be easily prepared
for neutron diffraction, a technique to probe the long- and short-range magnetic order in these
materials. Work is being carried out on extension of this methodology to other interesting
oxide materials.
The prepared compound was characterised with advanced NMR techniques. In particular,
application of the RAPT pulse sequence allowed a signal enhancement by a factor of 1.6,
which would afford a 2.5-fold reduction in acquisition time. The fitted NMR parameters
showed good agreement to the DFT-predicted values.
Transition state searching on Mg-ion migration showed that the barrier is indeed low for a
divalent ion, 0.6–0.8 eV, which is further reduced by the van der Waals interaction. Interplay
of charge and cation migration plays a crucial role in Mg-ion migration for this compound.
In particular, doping the system with other cations to investigate the changes in electronic
and ionic conductivity would be a natural extension from this work.
4.5 Conclusion and Further Work 141
Finally, the attempted cycling of MgV2O5 in a Li-ion cell showed some evidences for
demagnesiation capacity, despite the VO2 dissolution; further work will be carried out on
cycling a pure-phase sample to confirm reversible cycling.

Chapter 5
An Investigation on the Electronic
Structure, Defect Energetics, and
Magnesium Kinetics in Mg3Bi2
5.1 Introduction
Despite the extensive research conducted in the MIB field, it still remains extremely chal-
lenging to identify an electrolyte that is suitable for use with both high voltage cathodes and
Mg metal.[10] Whereas the previous chapters have primarily focused on developing 25Mg
NMR techniques for cathode materials, the ultimate aim of producing a working MIB system
cannot be achieved with the anode counterelectrode and electrolytes. This incompatibility of
electrolytes that are suitable for high voltage cathodes with Mg metal and vice versa has been
a major problem in developing a working MIB system. Hence, a number of studies have
focused on using an alternative anode material other than Mg to bypass this problem.[19] In
this respect, bismuth metal is a promising anode material with a low discharge voltage of 0.2
V versus Mg metal.[208, 209, 210] Bismuth alloys with magnesium to form an intermetallic
Mg3Bi2 phase, resulting in a theoretical capacity of 385 mAh/g. However, the most inter-
esting feature of bismuth anode is that it displays fast Mg ion insertion and de-insertion, a
feature not present in most other Mg-ion electrodes.[211, 208] 25Mg NMR, being an optimal
technique to study local ionic motion in solid systems, thus is a promising technique that can
be applied to study bismuth system.
As ion dynamics are properties intimately related to the defect chemistry (which in turn
depends heavily on the preparation conditions) and ultimately the electronic structure, an
144
An Investigation on the Electronic Structure, Defect Energetics, and Magnesium Kinetics in
Mg3Bi2
in-depth ab initio investigation on the Mg3Bi2 system, the end product of discharge process,
is presented. This chapter is divided into four sections: in the first section, the structural
characteristics of Mg3Bi2 is discussed, identifying the possible Mg diffusion pathways. Next,
electronic structures and defect energetics are described. Due to the well known problem of
band gap underestimation in semilocal DFT, we use three different methods (PBE, HSE06,
and G0W0) to calculate the electronic structure. In particular, the Greens function based
G0W0 method is used to give the most accurate value for the minimum band gap, overcoming
the band gap problem in semilocal DFT.
Bismuth is a heavy element where relativistic spin-orbit coupling is expected to have a
significant influence on the properties relevant to battery chemistry, as shown for lead in a
recent ab initio study of the lead-acid battery.[212] Thus, calculations were performed with
and without the explicit spin-orbit coupling Hamiltonian. Inclusion of spin-orbit coupling
was found to reduce the band gap, defect formation energies, and also the Mg migration
barrier which was calculated using the hybrid eigenvector-following approach. In addition,
inclusion of scalar relativistic effects was shown to significantly influence the lattice structure
and calculated NMR shifts.
Finally, previous data on variable-temperature 25Mg NMR data on discharged Mg3Bi2 is
compared to the ab initio data. Using these results one is able to reconcile and rationalise
the previous reports on the Mg mobility in Mg3Bi2, in which very different Mg mobilities
were assumed to occur due to the different concentrations of Mg vacancies in the samples.
We also demonstrate that the hybrid eigenvector-following method can be a very efficient
approach for locating transition states in systems where spin-orbit coupling is likely to play
an important role.
5.2 Methodology
5.2.1 Computational Details
All Density Functional Theory (DFT) calculations were performed with the VASP code[213,
214] employing the projector-augmented wave (PAW) method[215]. Spin-polarized Perdew-
Burke-Ernzerhof (PBE) and Heyd-Scuseria-Ernzerhof (HSE06) exchange-correlation func-
tionals were adopted.[216, 86] For the energy and force calculations, PAW pseudopotentials
treating the Mg 3s2 and Bi 5d106s26p3 as valence states were used, with a plane-wave basis
cutoff of 350 eV. All lattice relaxations were performed with 1.3 times the ENMAX value as
defined in the pseudopotential file. In addition, additional support grid was used for the
5.2 Methodology 145
evaluation of augmented charges (ADDGRID=.TRUE.). Self-consistent field (SCF) cycles
were converged with an energy tolerance of 10−4 eV. Monkhorst-Pack k-point sampling
of < 0.05 Å-1 was used in the Brillouin zone. For the density of states and band structure
calculations, Mg 2p states were also treated as valence states in the PAW potential and an
increased cutoff of 550 eV was used. Cellular relaxations SCF cycles were converged to a
10−6 eV limit. Γ-centered k-point sampling of < 0.03 Å-1 was used.
Single-shot G0W0 calculations* were performed with 250 frequency grid points and a
360 eV plane-wave cutoff for the response function calculations. HSE06 wavefunctions were
used as starting wavefunctions for the G0W0 calculations. Band gap convergence with respect
to the number of frequency grid points, number of empty bands, and plane-wave cutoffs were
checked. Quasiparticle energy iterations were converged to a 10−8 eV limit.
Relativistic corrections to the electronic structure were taken into account through two
levels of theory: a ‘scalar relativistic’ correction which only includes the mass-velocity and
Darwin terms of the relativistic Hamiltonian, as implemented in VASP; (see Section 2.2 and
[217]) and a ‘full relativistic’ correction that explicitly includes the spin-orbit coupling term
alongside the scalar relativistic correction.
The initial structure of Mg3Bi2 was fully relaxed until the energy differences between the
subsequent steps are converged to 10−5 eV per cell and the forces are < 0.05 eV/Å-1. For the
defect calculations, two supercells each containing 40 atoms (16 Mg, 24 Bi) and 120 atoms
(48 Mg, 72 Bi) were created and again relaxed to the above criteria. Defect notations follow
that of Kroger and Vink, with charge superscripts omitted for clarity (e.g. BiMg(oct) denotes
a neutral Bi sitting on an octahedral Mg site).[218] All defect energies were referenced to
the respective bulk metals (Mg, Bi) at the same level of theory as the defect calculations.
Only the neutral defects are considered since the system has a vanishing band gap under the
PBE level of theory, with the metallic behaviour enhanced under the inclusion of spin-orbit
coupling. Whereas the formation energies under the HSE06 level of theory (where a finite
band gap was observed) could give a chemical potential dependence, this was not possible
due to the computational resources available.
5.2.2 Calculation of NMR Parameters
NMR parameters of Mg3Bi2 using the Gauge Including Projector-Augmented Wave (GI-
PAW) method was calculated with the CASTEP 16.11 code.[219, 97, 220] Experimental
cell structures were fully relaxed (under symmetry constraints) to < 0.03 eV/Å-1 limit under
*Performed by Dr Bartomeu Monserrat (Cavendish Laboratory, University of Cambridge)
146
An Investigation on the Electronic Structure, Defect Energetics, and Magnesium Kinetics in
Mg3Bi2
three different conditions: (i) nonrelativistic (NR), with the core electrons treated with only
Schrödinger Equation; (ii) scalar relativistic (SR), with the Zeroth-order regular approxima-
tion (ZORA) to the core electrons; (iii) full relativistic (FR), with full inclusion of spin-orbit
coupling in the Hamiltonian. CASTEP keywords RELATIVISTIC_TREATMENT=Schroedinger/ZORA
were used to generate the on-the-fly pseudopotentials for the first two, whereas a j-dependent
pseudopotential available from the CASTEP developers were used for the FR calculations.
Monkhorst-Pack k-point sampling of < 0.03 Å-1 was used in the Brillouin zone with a
plane-wave cutoff of 700 eV and SCF convergence of 10−5 eV. NMR parameters were only
calculated for the NR and SR case, as CASTEP does not support a FR Hamiltonian in GIPAW
calculations. The calculated shielding σiso were converted to the shift δiso with reference to
solid MgO (26 ppm).
5.2.3 Transition State Searching
Transition state and steepest-descent pathway were found using the hybrid eigenvector-
following approach as implemented in the OPTIM code.[221, 113, 191] In brief, this method
finds the smallest eigenvalue of the Hessian matrix without calculating the full matrix, by
minimizing the Rayleigh-Ritz ratio. Here we use a low-memory Broyden-Fletcher-Goldfarb-
Shannon (LBFGS) scheme to minimize this ratio on a given point to find the uphill path.
The minimization was deemed to have occurred when the root-mean-square gradient is
smaller than 0.025 eV Å-1. Then up to five LBFGS minimization steps are performed in
the tangent space until the root-mean-square gradient is smaller than 10−3 eV Å-1. Initial
guesses of the transition state structure were produced by removing a nearby Mg atom and
putting the migrating Mg atom in the middle of the proposed diffusion path. The steepest
descent pathway from the transition state is found by displacing the atoms by 0.1 Å from the
transition state in the parallel and antiparallel directions to the eigenvector. Local minima are
then found by the LBFGS algorithm with energy convergence of 10−3 eV.
We note that the transition state geometry and eigenvectors obtained from the significantly
cheaper scalar relativistic calculations could be re-used as an initial guess for the fully
relativistic calculations. We have observed that only < 5 force evaluations are typically
needed for the relativistic calculations when this approach is taken, which significantly
reduces the computational requirement. In addition, unlike the classical nudged elastic band
(NEB) method where only a few points along the pathway are sampled, HEF method can
sample the path at an arbitrary step size. This allows us to capture the finer details of the
energy variance along the pathway.
5.3 Results and Discussion 147
5.3 Results and Discussion
5.3.1 Crystal Structure
Mg3Bi2 is the last in a series of magnesium pnictide compounds Mg3X2 (X=P, As, Sb, Bi)
and is a Zintl-type compound with closed shell ions with formal charges of Mg2+ and Bi3 – .
It adopts an anti-La2O3 type structure with hexagonal P3¯m1 space-group symmetry.[222]
The structure is schematically illustrated in Figure 5.1. In terms of ion arrangements, Mg3Bi2
incorporates two alternating layers: layer A consists of tetrahedrally coordinated Mg2+
cations (denoted Mg(tet)) and octahedral interstitial sites forming ‘covalent’ Mg2Bi22 –
sheets whereas layer B consists of octahedrally coordinated Mg2+ cations (denoted Mg(oct))
and tetrahedral interstitial sites. These interstitial sites are expected to play an important role
in ionic diffusion[223] as discussed further below.
b
c
a
Tet Mg Oct Int Tet Mg
(b)(a)
Layer A
Layer B
Layer A
Tet Int Oct Mg Tet IntOct Mg
2 1
3
Tet Mg Oct Int Tet MgOct Int
Tet Int
Oct Int
Tet Int
Fig. 5.1 (a) Crystal structure of Mg3Bi2. Positions of tetrahedrally coordinated Mg (orange
sphere, Tet) and octahedrally coordinated Mg (green sphere, Oct) are shown. Bismuth atoms
sit on the line vertices (not shown for clarity). (b) Schematic illustration of the Mg3Bi2
structure (on the left) showing the possible diffusion pathways involving the tetrahedral and
the octahedral interstitial sites (‘Tet Int’ and ‘Oct Int’).
From DFT-based lattice relaxation, we can see that the PBE functional (both on CASTEP
and VASP) consistently overestimates the experimentally determined a cell parameter[224]
at both the scalar relativistic and full relativistic levels of theory, whereas it marginally
underestimates the c parameter at the scalar relativistic level, and overestimates it at the full
relativistic level (Table 5.1). Fully nonrelativistic calculations, however, result in a distortion
in the a/c ratio; the effect of this is discussed in Section 5.3.3. The HSE06 functional again
148
An Investigation on the Electronic Structure, Defect Energetics, and Magnesium Kinetics in
Mg3Bi2
overestimates the a lattice parameter, although to a smaller extent than PBE, whereas it
underestimates the c lattice parameter. Finally, inclusion of SOC results in slight expansion
of the cell which may arise from reduced electrostatic interactions (see below).
PBE) PBE HSE06
(CASTEP) (VASP) (VASP)
NR SR FR SR FR SR FR Expt
a / Å 4.96 4.70 4.72 4.71 4.72 4.65 4.66 4.62
c / Å 6.50 7.46 7.48 7.40 7.44 7.35 7.36 7.41
Table 5.1 DFT-predicted cell parameters of Mg3Bi2 using the PBE and HSE06 exchange-
correlation functionals using CASTEP and VASP. NR, SR and FR refer to nonrelativistic,
scalar relativistic, and full relativistic (i.e. explicit spin-orbit coupling) calculations, re-
spectively. ZORA method was used for SR calculation on CASTEP. Experimental lattice
constants are from Lazarev et al.[224]
Finally, we note that Mg3Bi2 undergoes a phase transition above 703 ◦C to a defective
body-centered cubic structure similar to AgI, with excess Mg cations as the tetrahedral inter-
stitials, resulting in Mg1.5Bi.[225] As expected from the AgI structure, it shows superionic
conduction of Mg2+ cations as determined through neutron diffraction.[226] In this study,
however, we restrict the investigation to the room-temperature hexagonal phase which is
more relevant to Mg-ion batteries.
5.3.2 Electronic Structure
Despite being first reported in 1933 by Zintl[222], the precise electronic structure and the
band gap (Eg) of Mg3Bi2 has not been conclusively determined with either experiment or
theoretical calculations. In a series of work, Ferrier and co-workers have speculated that
Mg3Bi2 is semimetallic based on a conductivity-composition plot;[227, 228] Lazarev et al.
have assumed Mg3Bi2 is semiconducting with Eg = 0.1 eV;[224] Watson et al. obtained
Eg = 0.5 eV from Mg X-ray emission spectra.[229] Work on the amorphous Mg3Bi2 alloy
have shown from conductivity measurements that it is semiconducting with a band gap of
0.15 eV.[230] These varying results may arise from the difficulty in preparing these materials
stoichiometrically, due to the high vapour pressure of Mg. Previous literature describing
electronic structure calculations is also sparse: Sedighi et al. concluded Mg3Bi2 is a semi-
conductor with Eg = 0.25 eV based on Engel-Vosko Generalized Gradient Approximation
(EV-GGA),[231] whereas Imai et al., Xu et al., and Zhang et al. concluded it is a semimetal
based on pure GGA calculations.[232, 233, 234] However, all these works except the last
5.3 Results and Discussion 149
one did not consider explicit spin-orbit coupling (SOC), which has been shown to play
an important role in the electronic structure of compounds involving heavy atoms such as
bismuth.[235, 236]; in addition, the well known problem of band gap underestimation in
semilocal DFT necessitates the use of a higher level of theory to predict the accurate elec-
tronic structure of this material. Hence, we revisit the electronic structure of this compound
using state-of-the-art electronic structure methods such as hybrid functionals and many-body
perturbation theory in the G0W0 approximation, and also consider the effects of spin-orbit
coupling.
First we look at the Bader charges of Mg and Bi in the structure, shown in Table 5.2. The
HSE06 functional results in an increased ionicity in the system compared to the semilocal
PBE values. In all cases, inclusion of SOC resulted in decreased ionicity which is consistent
with the density of states data shown below.
Bi Mgoct Mgtet
PBE SR -2.10 +1.42 +1.39
PBE FR -2.06 +1.39 +1.37
HSE06 SR -2.21 +1.51 +1.45
HSE06 FR -2.18 +1.48 +1.44
Table 5.2 Scalar relativistic (SR) and full relativistic (FR) Bader charge analysis of Mg3Bi2
using the Perdew-Burke-Ernzerhof (PBE) and Heyd-Scuseria-Ernzerhof (HSE06) exchange-
correlation functionals. Only the valence charge is calculated.
Figure 5.2 shows the density of states (DOS) plots obtained for Mg3Bi2 using DFT within
the semilocal PBE and hybrid HSE06 approximations. Both methods show the same trend
comparing the results with and without SOC. As expected from the charge distribution of
Mg3Bi2, the top of valence band is strongly dominated by the bismuth 6p contribution down
to around −5 eV from the Fermi level and has a negligible Mg contribution. As expected
from the literature, the relativistic contraction of the bismuth 6s states around −12 eV from
the Fermi level results in a relatively large separation of around 5 eV between the top of
6s and the bottom of 6p states, which makes it chemically inactive (the so-called inert-pair
effect).[237]. The downshift in 6s energy by adding the full relativistic effect is around -0.15
eV; effects of similar magnitudes were observed in PbO2, an important active material in
lead-acid batteries.[212]
Figure 5.3 shows the band structure of Mg3Bi2 using DFT within the semilocal PBE
and hybrid HSE06 approximations, with and without the spin-orbit interaction. At the PBE
level, Mg3Bi2 exhibits semimetallic behaviour, consistent with the earlier reports of a type-II
nodal line semimetal at this level of theory.[234] The inclusion of spin-orbit coupling opens
150
An Investigation on the Electronic Structure, Defect Energetics, and Magnesium Kinetics in
Mg3Bi2
0
2
4
6
8
D
O
S  
( s t
a t e
s / e
V ) TotalBi s
Bi p
Mg
-15 -12 -9 -6 -3 0 3
Energy (eV)
0
2
4
6
8
D
O
S  
( s t
a t e
s / e
V )
-15 -12 -9 -6 -3 0 3
Energy (eV)
PBE PBE+SOC
HSE HSE+SOC
Fig. 5.2 Density of states plot of Mg3Bi2 using the PBE and HSE06 exchange-correlation
functionals. Local atomic DOS projections inside the sphere defined by the Wigner-Seitz
radii (1.63 and 1.52 Å for Bi and Mg, respectively) are also shown. All energies were
referenced to the highest occupied state.
Γ K M Γ A H L A-2.0
-1.5
-1.0
-0.5
0
0.5
1.0
1.5
2.0
B a
n d
 s
t r u
c t
u r
e  
( e V
)
PBE
PBE+SOC
Γ K M Γ A H L A-2.0
-1.5
-1.0
-0.5
0
0.5
1.0
1.5
2.0
B a
n d
 s
t r u
c t
u r
e  
( e V
)
HSE
HSE+SOC
Fig. 5.3 Band structure plots of Mg3Bi2 using the PBE (left) and HSE06 (right) exchange-
correlation functionals without (grey) and with (red) spin-orbit coupling (SOC). The Fermi
level is located at the zero of energy.
5.3 Results and Discussion 151
a small gap in the nodal line, but the system remains semimetallic. At the HSE06 level,
Mg3Bi2 is a semiconductor with the minimum band gap of 0.36 eV located at the Γ point.
The inclusion of spin-orbit coupling reduces the band gap to 0.17 eV, but the system remains
semiconducting. The G0W0 calculations show that the Γ-point band gap increases to 0.58 eV
without spin-orbit coupling. We expect that spin-orbit coupling would reduce the band gap
by an amount similar to that observed in HSE06, and thus Mg3Bi2 is also semiconducting at
the G0W0 level of theory.
We note that inclusion of SOC results in a reduced band gap and increased dispersion
of the Bi 6p states. Previous reports on binary Sb and Te compounds have attributed this
phenomenon to spin-orbit coupling in the anion p-orbitals: energy splitting of the degenerate
p-states result in the j = 3/2 state being split upward in energy, and the j = 1/2 state being
split downward in energy.[238] The net effect of this splitting is a reduction in the band gap.
5.3.3 Calculation of the 25Mg NMR Shifts
With a full understanding of the relativistic SOC on the electronic structure of Mg3Bi2,
DFT-based NMR parameter calculations were also performed on the Mg3Bi2 structure to
confirm the assignment and to evaluate the effect of SOC. Calculations using the CASTEP
code is shown in Table 5.3. The corresponding DOS plot in Figure 5.4 show a result
consistent with the VASP result in Figure 5.2. While the relativistic contribution of the heavy
bismuth atoms is significant for the electronic structure and ultimately the chemical shielding,
fully relativistic calculation of chemical shielding is not yet supported in this version of
CASTEP. Indeed, the chemical shifts δiso using the scalar relativistic Zeroth-Order Regular
Approximation (ZORA) method show a poor match to the experimentally observed shifts,
showing errors on the order of hundreds of ppm. However, it is seen that the ZORA method
gives a qualitatively correct prediction on the relative magnitudes of the chemical shielding:
Mg(oct) environment is more shielded (down-shifted) relative to the Mg(tet) environment,
which supports our NMR assignment. The error is likely to arise from the (i) relativistic
contribution to the chemical shift and (ii) semimetallic behaviour of Mg3Bi2 at the PBE level
of theory, which creates small partial occupancies of bands close to the Fermi level.
On the other hand, a fully nonrelativistic calculation using the on-the-fly pseudopotential
on CASTEP shows the partial extent of relativistic SOC effects for chemical shifts in Mg3Bi2,
where the errors are on the order of thousands of ppm. This extreme increase in chemical
shift of both Mg ions are likely to come again from the fact that the relativistic contraction of
Bi s-orbitals make the Bi ions ‘softer’ by screening the 6p-orbitals (i.e. make the bonding
152
An Investigation on the Electronic Structure, Defect Energetics, and Magnesium Kinetics in
Mg3Bi2
 0
 2
 4
 6
 8
 10
 12
 14
NR
D
O
S 
/ (s
tat
es
/eV
) totalBi s
Bi p
 0
 2
 4
 6
 8
 10
 12
 14
SR (ZORA)
D
O
S 
/ (s
tat
es
/eV
) totalBi s
Bi p
 0
 2
 4
 6
 8
 10
 12
 14
-12 -10 -8 -6 -4 -2  0  2
FR (SOC)
D
O
S 
/ (s
tat
es
/eV
)
Energy / eV
total
Fig. 5.4 Density-of-states (DOS) plot for Mg3Bi2, generated using CASTEP. Three cases are
considered: nonrelativistic (NR), scalar relativistic (SR) using the ZORA method, and full
relativistic (FR) including the SOC. All energies were referenced to the highest occupied
state. Site- and orbital-decomposed DOS under SOC is not supported in the version of
CASTEP used.
5.3 Results and Discussion 153
Site Level δiso / ppm Ω / ppm κ CQ / MHz η
Mg(oct)
Expt -306 - - - -
SR 64.9 204.2 -0.96 -1.23 0.0
NR 2145.7 3920.3 -0.49 3.44 0.0
Mg(tet)
Expt -6 - - - -
SR 169.6 166.5 -0.96 -2.47 0.0
NR 2157.8 2522.7 -0.51 -0.15 0.0
Table 5.3 Calculated NMR parameters of the two Mg sites (octahedral and tetrahedral) in
Mg3Bi2 using the GIPAW method. Core electrons are treated with two levels of theory:
scalar relativistic (SR) ZORA and nonrelativistic (NR) Schrödinger. δiso, Ω, and η refer to
the chemical shift tensor expressed in the Herzfeld-Berger convention. CQ and η refer to the
quadrupolar coupling parameter and asymmetry, respectively.
Fig. 5.5 Relaxed cell structures using the (a) scalar relativistic ZORA approximation, and (b)
nonrelativistic Schrödinger equation, projected along the a-direction. Here, Mg(oct) sits on
layer A and Mg(tet) sits on layer B.
154
An Investigation on the Electronic Structure, Defect Energetics, and Magnesium Kinetics in
Mg3Bi2
less ionic). However, it must also be stressed that the system is properly metallic under this
level of theory, with a significant density of states at the Fermi level (Figure 5.4); this would
likely render the calculated chemical shift to be unphysical.
EFG components show only a variation on the same magnitude between the ZORA and
Schrödinger methods. This is expected as EFG tensors are quantities more dependent on the
local atomic arrangements and less dependent on the core electron distribution compared
to the chemical shielding. Previous determination of 23Na EFG tensors in oxides have also
reported a qualitative agreement of the EFG values obtained by use of a classical point charge
approximation.[239]
However, a reduction of -94 % in the EFG tensor is observed for the Mg(tet), from 2.47
MHz (ZORA) to 0.15 MHz (Schrödinger). Such negative variation is not observed for the
Mg(oct), where the rate of change is larger and positive (1.23 MHz to 3.44 MHz; 227 %). To
better understand such different changes in the EFG tensor, the structural aspects need to be
taken into account. Figure 5.5 shows the dramatic difference in the local Mg coordination
between the relaxed cell structures using the ZORA and Schrödinger approximations. As
shown in Table 5.1, nonrelativistic Schrödinger calculations result in a large contraction of
the c-lattice parameter and slight expansion of the a-lattice parameter. The net effect of this
is a ‘squeezing’ of the cell, as shown in Figure 5.5. Due to this effect, the local coordination
around Mg is drastically changed, and the coordination octahedron around the Mg(oct) is
distorted significantly; this results in an increased CQ, but keeps η = 0 due to the axial
symmetry present from the lattice symmetry. The effect of relativistic SOC on Mg(tet) is less
obvious, but the contraction of c and elongation of a simultaneously result in a spherically
more symmetric environment, resulting in a significantly reduced CQ.
5.3.4 Defect Energetics
To investigate the possibility of different types of defects present in the Mg3Bi2 sample,
formation energies ∆E f of various stoichiometric and non-stoichiometric point defects were
calculated ab initio. The results outlined in Table 5.4 are separated into three categories:
antisite defects, Frenkel defects, and vacancy defects, which we discuss in sequence. Nonstoi-
chiometric antisite and vacancy defects were created by removing or inserting relevant atoms
from the stoichiometric cell while retaining the charge neutrality of the cell due to Mg3Bi2
being a semimetal at the PBE level of theory (i.e. charged defects are not possible under such
a situation). For instance, a Mg vacancy defect would involve removal of a neutral Mg atom
resulting in a composition of Mg23Bi16 (for a 40-atom supercell) and two electron holes on
5.3 Results and Discussion 155
the Fermi level; similarly, a Mg substitutional defect would involve a Bi atom replaced for a
Mg atom, resulting in Mg25Bi15 (again for a 40-atom supercell) and five excess electrons on
the Fermi level. Charge neutrality is automatically maintained in the antisite and Frenkel
defects since they are stoichiometric.
First, the formation of antisite defects are energetically unfavorable, with the Mg on a
Bi site having the highest formation energy of 2.92 eV. This is most likely due to the large
difference between their ionic radii. Despite the fact that no ionic radius for the Bi3 – ion is
reported, the trend can still be explained in terms of the corresponding atomic radii of Mg
and Bi (Mg 1.50 Å; Bi 1.60 Å), the difference being expected to get even larger as Mg is
oxidized and Bi is reduced. This is also supported from the Mg–Bi bond lengths in Mg3Bi2
crystal: Mgoct–Bi 3.21 Å versus Mgtet–Bi 2.92 Å. Using the Shannon radii of 0.72 and 0.57
Å for Mgoct and Mgtet, respectively,[18] Bi3 – radii of 2.49 and 2.35 Å are obtained in each
case. In addition, the unfavourable electrostatic interaction between the ions of same charge
(Mg occupying the Bi lattice site is coordinated by Mg ions, and vice versa) reinforces the
high energy cost in forming these types of defects.
The ∆E f of Frenkel-type defects (creation of a vacancy plus an interstitial) are also
shown to be relatively high for both nearby and separated vacancy–interstitial pairs; around
0.8-1.2 eV is required for their formation. However, performing structural relaxation on
some of the starting guesses with the defect Mg ion sitting on an interstitial site that is
adjacent (in the first coordination shell) to a Mg vacancy (specifically, the VMg(oct)+Mgi(tet)
(nn), VMg(tet)+Mgi(tet) (nn), and VMg(tet)+Mgi(tet) (nn) cases) resulted in the structure
reverting back to that of the pristine cell, with no defects. This indicates that these defects are
energetically unstable and they would revert back to the original structure, if formed at all.
These sites, as we will see later, may play an important role as energy maximum transition
state sites in Mg-ion diffusion.
Perhaps the most surprising results are the energies of the vacancy defects: while Mg
vacancy defects have around the same ∆E f as Frenkel defects without SOC, their magnitudes
are reduced significantly when SOC is included in the Hamiltonian. From the fully relativistic
calculation on the 120-atom supercell with SOC included, ∆E f of octahedral vacancies
VMg(oct) is found to be as low as 0.33 eV, with a slightly higher value of 0.42 eV for
tetrahedral vacancies VMg(tet). This effect is likely to be connected to the enhanced shielding
of Bi 6p-levels due to the SOC effect as explained above: as formation of a neutral Mg
vacancy should involve loss of electrons from the Bi, this deshielding should result in
a significant reduction of the formation energy. Hence, considering their low formation
156
An Investigation on the Electronic Structure, Defect Energetics, and Magnesium Kinetics in
Mg3Bi2
energies, we conclude that vacancy defects are the dominant type of defects present in
Mg3Bi2.
5.3.5 Mg Migration Kinetics
Having established the defect chemistry in Mg3Bi2, we now turn our attention to the Mg
diffusion in this structure and study the migration barriers.
As Mg vacancies were shown to have low formation energy, especially with SOC included,
we attempt to simulate the effect of vacancy diffusion following the creation of one octahedral
Mg vacancy VMg(oct). This has lower ∆E f than the tetrahedral vacancy VMg(tet) as illustrated
in the previous section. Then, a nearby tetrahedral Mg atom is removed from its original
position and placed on the guessed ‘transition state’. Finally, a hybrid eigenvector-following
approach is used to find the transition state , followed by the search for a steepest descent
path connecting the two corresponding minima.
The result presented in Figure 5.6 clearly shows a small diffusion barrier of hopping
between the Mgoct and Mgtet (Path 1 on Figure 5.1b), with 0.34 eV for the fully relativistic
calculation and 0.43 eV for the scalar relativistic calculation. In contrast, the Mgoct−Mgoct
diffusion barrier (Path 2 on Figure 5.1b) is around twice that of the Mgoct−Mgtet diffusion
barrier, indicating that the Mg diffusion must occur via octahedral-tetrahedral exchange.
This is in line with the conclusion from 25Mg NMR studies of Liu et al.[40] An alternative
exchange mechanism involving Path 3 on Figure 5.1b was also investigated, but the transition
structure searching resulted in the same transition state as for the Path 1, clearly indicating
the absence of a diffusion pathway along this line.
With the ab initio calculations of the defect creation and activation energies, we now
compare our results with available experimental data. As reported in the literature, bismuth
can be cycled reversibly to form Mg3Bi2 in a Mg-ion battery.[208] Previous work on magne-
sium ion conduction in bismuth anodes either used the Galvanostatic Intermittent Titration
Technique (GITT) on an electrochemical cell, or ex-situ variable-temperature (VT) 25Mg
NMR spectroscopy to probe Mg transport.[211, 40] In this work, however, we restrict the
discussion to the latter NMR result, due to the following limitation of GITT experiments: in
the GITT measurement, the voltage response (resulting from the relaxation) of the cell after
the application of a short current pulse is modelled with the diffusion equation to extract the
diffusion coefficient D under the assumption that the (de)insertion reaction occurs via a solid
solution (i.e. the relaxation of the potential after the current pulse is a measure of ionic trans-
port through a single phase). In-situ X-ray diffraction results, however, have clearly shown
5.3 Results and Discussion 157
∆E f
40-atom Supercell 120-atom Supercell
SR FR SR FR
Antisite defects
BiMg(oct) 1.84
BiMg(tet) 2.01
MgBi 2.92
BiMg(oct)+MgBi 2.51 2.27
BiMg(tet)+MgBi 1.17 1.15
Frenkel defects
VMg(oct)+Mgi(oct) (nn) 0.87 0.79 0.89
VMg(oct)+Mgi(oct) (far) 0.89 0.81 1.04
VMg(oct)+Mgi(tet) (nn) * * * *
VMg(oct)+Mgi(tet) (far) 1.49 1.41 1.55
VMg(tet)+Mgi(oct) (nn) * * * *
VMg(tet)+Mgi(oct) (far) 1.07 0.98 1.10
VMg(tet)+Mgi(tet) (nn) * * * *
VMg(tet)+Mgi(tet) (far) 1.17 1.15 1.61
Vacancy defects
VMg(oct) 1.09 0.40 1.03 0.33
VMg(tet) 1.17 0.50 1.12 0.42
VBi 1.88 2.02
Table 5.4 Scalar relativistic (SR) and full relativistic (FR) ab initio formation energies ∆E f of
various stoichiometric and non-stoichiometric defects in Mg3Bi2 using the PBE functional.
Defect notations follow the convention of Kroger and Vink with neutral sign omitted for
clarity.[218] All calculations assumed non-charged defects (see text) with cell dimensions
fixed to simulate a dilute limit. Vacancy defect energies are referenced to the respective
metals. For Frenkel defects, two scenarios where the Mg sits on a nearby (nn) or far interstitial
sites were considered. Asterisks(*) indicate that the resulting structure was unstable and
reverted back to the starting structure. All values are quoted in electron-volts. Only some
of the calculations were performed under the 120-atom supercell condition after an initial
screening with the 40-atom supercell.
158
An Investigation on the Electronic Structure, Defect Energetics, and Magnesium Kinetics in
Mg3Bi2
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
-3 -2 -1  0  1  2  3  4
Scalar relativistic
1 (Mgoct -Mgtet)
0.43 eV
E (
rel
. to
 V o
ct
) / 
eV
Integrated path length / Å
-4 -3 -2 -1  0  1  2  3  4
2 (Mgoct -Mgoct)
(a)
0.78 eV
Integrated path length / Å
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
-3 -2 -1  0  1  2  3  4
Full relativistic
1 (Mgoct -Mgtet)
0.34 eV
E (
rel
. to
 V o
ct
) / 
eV
Integrated path length / Å
-4 -3 -2 -1  0  1  2  3  4
2 (Mg oct-Mgoct)
(b)
0.62 eV
Integrated path length / Å
Fig. 5.6 Diffusion profile of Mg ion for selected pathways as illustrated in Figure 5.1b. (a)
Scalar relativistic, (b) full relativistic, with SOC included. Energies are referenced to the
bulk energy of each 120-atom supercell with one Voct defect.
5.3 Results and Discussion 159
that magnesiation occurs via a two-phase reaction between Bi and Mg3Bi2.[40] Thus the
diffusion coefficients extracted from GITT measurements must be treated with caution, since
the relaxation phenomena under these conditions may include multiple contributions such as
(i) redistribution of Mg ions due to the formation of metastable (possibly non-stoichiometric
solid solution) phases formed under operating conditions (and on application of an over-
potential), and (ii) redistribution of phase boundaries to minimise the interfacial energies,
etc.
We now consider transport in two cases, the first in a completely stoichiometric material
where we need to consider the energy associated with defect formation. In this case, substi-
tutional and vacancy defects are not relevant because they result in stoichiometry changes.
The second case explores diffusion in off-stoichiometry materials Mg3 – xBi2 and Mg3Bi2+y,
representing Mg vacancy and (excess) Bi substitutional defects, respectively. As no extrinsic
Mg vacancies are expected in the stoichiometric Mg3Bi2 compound, the diffusion in this case
must occur via a vacancy diffusion mechanism involving thermally generated Mg vacancies;
hence the observed D would be the diffusion coefficient of Mg vacancies. In the latter case,
nonstoichiometry dictates that extrinsic Mg vacancies must exist in the compound.
Table 5.5 compares the effective activation barriers Eeffa obtained experimentally versus
the Eeffa estimated from the ab initio calculations using the scalar relativistic (SR) and full
relativistic (FR) treatments. Looking at the SR case first, a large variation in Eeffa is observed
where a significantly lower barrier is predicted for the Mg jumps with an existing nearby
extrinsic vacancy defect (i.e. Case 2, Mg3 – xBi2 SR): 0.43 eV. In Case 1 (stoichiometric
Mg3Bi2), where generation of a thermal Mg vacancy is required for diffusion, the effective
diffusion barrier Eeffa should now include this formation of vacancy. Naturally, we would
expect an increase in Eeffa due to this inclusion of vacancy formation energy: depending on
the vacant site (tetrahedral or octahedral Mg vacancy), the Eeffa is estimated to be 1.46 eV
(octahedral) or 1.55 eV (tetrahedral).
We also observe that inclusion of spin-orbit coupling enhances the diffusion noticeably
in all three cases, where a reduction of effective activation barrier is seen (e.g. for Case 1,
SR with Voct 1.46 eV versus FR Voct 0.67 eV. This is mainly due to the vacancy formation
energy, where the FR case results in a dramatically lowered formation energy.
Comparing these results to the experiment, the effective activation barrier Eeffa in the
fully relativistic case ignoring the vacancy creation (Case 2) agrees with the Eeffa of the
ball-milled sample, determined through VT NMR. On the other hand, Eeffa obtained for the
electrochemically prepared samples (VT NMR Echem) agree well with the Eeffa assuming
a Mg vacancy creation plus Mg diffusion (Case 1). Hence, we conclude that the primary
160
An Investigation on the Electronic Structure, Defect Energetics, and Magnesium Kinetics in
Mg3Bi2
diffusion mechanism in electrochemically prepared samples (measured ex-situ, i.e. not during
battery operating conditions) should involve vacancy creation, whereas the mechanism
in mechanically prepared samples only involves the vacancy diffusion. This could be
explained by method of sample preparation: the electrochemical Mg insertion process in this
case produced samples closer to the thermodynamic equilibrium creating fewer vacancies,
whereas mechanical milling is largely a high energy process resulting in more vacancies (for
instance, vacancy formation in ZnO through milling was previously observed with HRTEM
studies[240]). This is also a known phenomenon in the synthesis of intermetallic phases:
mechanical milling can provide excess energy to the material, which can be stored in the
sample as atomistic disorders of which vacancies are one example.[241] Furthermore, as
discussed above, Mg deficiency is likely due to preferential Mg sticking to the ball-mill
components (jar, balls). The already present vacancies in mechanically prepared samples act
as potential diffusion sites for adjacent Mg ions, enhancing their diffusion.
Extending this result to a battery under operating conditions, it is important to stress
that the Mg3 – xBi2 phases formed in-situ may not be stoichiometric. The kinetics of Bi
(de)magnesiation will depend on a number of factors which include the interfacial energies
between the Bi and Mg3Bi2 phases, and transport of Mg in Bi, Mg3Bi2, and at the various
interfaces. In particular, the mechanism of demagnesiation will depend strongly on the
ease of vacancy formation in Mg3Bi2. By analogy with previous work on e.g. lithium
silicides[242] and CuTi2S4,[243]) this energy of vacancy formation may even be responsible
for setting (or strongly influencing) the overpotential observed on charge.
Finally, we note that Zintl-type A3B2 materials to which Mg3Bi2 belongs have been
identified as potentially promising thermoelectric materials.[244, 245] Mg mobility in these
materials could have important implications on the thermoelectric power generation, and
work is in progress on doping other atoms into this structure to enhance, or suppress, this
mobility.
5.4 Conclusion 161
Case Diffusion process
Eeffa / eV
SR FR
DFT
Case 1 formation and diffusion of Voct 1.46 0.67
Case 1 formation and diffusion of Vtet 1.55 0.76
Case 2 pre-formed vacancy diffusion 0.43 0.34
Experiment
VT NMR Echem 0.71
VT NMR Ball-mill 0.19
Table 5.5 Effective migration barriers Eeffa estimated through DFT and VT NMR techniques.
VT NMR data are taken from Liu et al.[40] For the NMR measurements, Mg3Bi2 prepared
through electrochemical insertion and mechanical milling were considered. SR and FR refer
to scalar relativistic and full relativistic calculations, as described in the text.
5.4 Conclusion
In conclusion, advanced electronic structure calculations show that spin-orbit coupling plays
an important role in structure and dynamics of Mg3Bi2, a promising Mg-ion battery anode
material. Inclusion of relativistic spin-orbit coupling also significantly changes the electronic
structure and the calculated NMR parameters of this material. It also lowers the formation
energies of Mg vacancy defects, which is crucial to the apparent low Mg-ion diffusion barrier.
Using an efficient single-ended hybrid eigenvector-following approach, we have calculated
the Mg migration barriers involving relativistic spin-orbit coupling which are as low as 0.34
eV for the octahedral to tetrahedral diffusion. The calculated activation barriers are in good
agreement with the previous experimental report using variable temperature 25Mg NMR
experiments for materials prepared electrochemically and via ball-milling. Stoichiometric
materials show higher activation energies, since the activation energy involves both the cost
of vacancy generation and transport. An understanding of Mg transport and the energetics
of vacancy formation are important in understanding the mechanisms for demagnesiation
of Mg3Bi2. Further work needs to be carried out on improving the Mg diffusion in similar
materials such as Sn, which have been shown to have good capacity but poor rate performance
in Mg-ion batteries.[246]

Chapter 6
Conclusion and Further Work
The primary objective of this thesis has been to develop a combined experimental and
computational approach to studying potential Mg-ion battery (MIB) electrodes using 25Mg
NMR spectroscopy. Due to the challenging nature of solid-state 25Mg NMR, particular
focus was put on ab initio prediction of various NMR parameters to (i) aid the experimental
acquisition and interpretation and to (ii) enhance the sensitivity and resolution of resulting
spectra. Also, due to the inherent magnetism present in paramagnetic TM oxides, a secondary
objective of this thesis has been on investigating the magnetism of these compounds.
In Chapter 3, this combined approach was developed and validated on a series of param-
agnetic TM oxides (Mg6MnO8, MgV2O4, MgCr2O4, MgMn2O4). Ab initio predictions of
25Mg NMR parameters (hyperfine shifts, quadrupolar coupling) yield reliable results which
are verified with the 25Mg NMR experiments. In particular, the Rotor-Assisted Population
Transfer (RAPT) pulse sequence was successfully used, for the first time, to (i) enhance
the central transition intensity in quadrupoles under paramagnetic environments, and (ii)
estimate the quadrupolar coupling parameter in such paramagnetic systems where a direct
fitting is likely to be difficult due to broad lineshapes.
With the baseline set in Chapter 3 on paramagnetic 25Mg NMR, this method was applied
to MgV2O5 in Chapter 4 to observe and interpret the NMR spectra of this potential MIB
cathode material. In particular, use of the RAPT-echo pulse sequence was investigated to
yield an enhancement of 1.6 over a simple echo, which amounts to a 2.5-fold reduction in
acquisition time. More importantly, ball-milling of the sample was shown not to affect the
RAPT enhancement significantly, which shows the possibility of applying this methodology
to studying real cycled systems. Attempted cycling of this compound, however, exhibited
vanadium dissolution from the VO2 impurity phase. Possible source of this phase was
164 Conclusion and Further Work
identified as the hydrated Mg precursor; clearly a fresh sample in absence of this impurity
phase needs to be prepared and the electrochemistry tested.
In Chapter 5, the focus was shifted towards understanding the Mg dynamics in an MIB
anode material Mg3Bi2. By advanced electronic structure calculations, this material was
predicted to be a small band-gap semiconductor, in line with previous experimental reports.
DFT-based calculations on the defect creation energies indicate that the energy of defect
formation takes a significant portion of the observed activation energy; combined with the Mg-
ion migration barrier obtained through a hybrid eigenvector following method, the dramatic
differences between the mechanochemically and electrochemically prepared samples are
explained. In particular it is observed that the relativistic spin-orbit coupling in heavy bismuth
is important for fast Mg-ion conduction, a factor which is likely to be relevant for anodes
with heavy atoms (e.g. Sn, Pb). Investigations on similar systems (Mg2Sn, Mg3Sb2) using
this methodology is needed to assess the full extent of this effect.
This work also contained substantial investigations on the magnetism of these TM
oxide compounds, many of which are not straightforward to prepare and are expected to
show interesting low-temperature magnetism. In Chapter 3, we have validated the weakly
coupled magnetism present in Mg6MnO8 through a DFT-based calculation of the electron
J-coupling and low-temperature measurement of magnetism. Calculated magnetic coupling
in a magnetically frustrated MgCr2O4 spinel is compared to experiments to yield a good
agreement. Also, Mg–Mn inversion in another spinel MgMn2O4 was shown experimentally
through a combined refinement of the X-ray data and magnetic measurements. Finally, the
magnetism of MgV2O5 was investigated with DFT methods and SQUID magnetometry.
The system showed magnetic responses characteristic of one-dimensional magnetism; DFT
calculations indicate the subtle nature of V–V magnetic interactions which depend heavily
on the computational parameters used.
In this regard, initial further work must be focused on obtaining a pure-phase MgV2O5
starting from stoichiometric quantities of precursors. The sample would then be subjected
to neutron diffraction to explore the low-temperature magnetism of this material; neutron
diffraction on this sample has not been reported before. This work could also be extended
to preparing other TM oxides (e.g. MgTi2O4, MgVO3, Mg3Nb6O11) and systematically
investigating their magnetic and NMR properties. With the aid of ab initio predictions on
the carbothermal synthesis conditions taken in Chapter 4 for MgV2O5, many TM oxides
previously difficult to access could be prepared in large quantities in a straightforward
manner.
165
This thesis, overall, sets a baseline for preparing and investigating Mg-ion battery elec-
trodes by solid-state 25Mg NMR spectroscopy and ab initio calculations. The methodology
presented in this work is general and can be expected to be applied to real battery systems for
studying the local structure, confirming reversible insertion, elucidating failure mechanisms,
and investigating Mg-ion dynamics. This provides us with a new tool to study Mg-ion
batteries, which are promising systems for future green energies.

References
(1) Huggins, R. A., Advanced Batteries: Materials Science Aspects; Springer-Verlag:
2009.
(2) Greenwood, N. N.; Earnshaw, A., Chemistry of the Elements, 2nd Ed.; Butterworth-
Heinemann: 1997.
(3) Olivetti, E. A.; Ceder, G.; Gaustad, G. G.; Fu, X. Joule 2017, 1, 229–243.
(4) Aurbach, D.; Zinigrad, E.; Cohen, Y.; Teller, H. Solid State Ionics 2002, 148, 405–
416.
(5) Larcher, D.; Beattie, S.; Morcrette, M.; Edström, K.; Jumas, J.-C.; Tarascon, J.-M. J.
Mater. Chem. 2007, 17, 3759–3772.
(6) Van Noorden, R. Nature 2014, 507, 26.
(7) Kim, S.-W.; Seo, D.-H.; Ma, X.; Ceder, G.; Kang, K. Advanced Energy Materials
2012, 2, 710–721.
(8) Dunn, B.; Kamath, H.; Tarascon, J.-M. Science 2011, 334, 928–935.
(9) Larcher, D.; Tarascon, J.-M. Nature chemistry 2015, 7, 19.
(10) Canepa, P.; Sai Gautam, G.; Hannah, D. C.; Malik, R.; Liu, M.; Gallagher, K. G.;
Persson, K. A.; Ceder, G. Chemical Reviews 2017, 117, 4287–4341.
(11) Jäckle, M.; Groß, A. The Journal of Chemical Physics 2014, 141, 174710.
(12) Ding, M. S.; Diemant, T.; Behm, R. J.; Passerini, S.; Giffin, G. A. Journal of The
Electrochemical Society 2018, 165, A1983–A1990.
(13) Hebié, S.; Ngo, H. P. K.; Leprêtre, J.-C.; Iojoiu, C.; Cointeaux, L.; Berthelot, R.;
Alloin, F. ACS Applied Materials & Interfaces 2017, 9, 28377–28385.
(14) Hebié, S.; Alloin, F.; Iojoiu, C.; Berthelot, R.; Leprêtre, J.-C. ACS Applied Materials
& Interfaces 2018, 10, 5527–5533.
(15) Aurbach, D.; Gofer, Y.; Lu, Z.; Schechter, A.; Chusid, O.; Gizbar, H.; Cohen, Y.;
Ashkenazi, V.; Moshkovich, M.; Turgeman, R. Journal of Power Sources 2001, 97-98,
28.
(16) Aurbach, D.; Lu, Z.; Schechter, A.; Gofer, Y.; Gizbar, H.; Turgeman, R.; Cohen, Y.;
Moshkovich, M.; Levi, E. Nature 2000, 407, 724–727.
(17) Muldoon, J.; Bucur, C. B.; Gregory, T. Angewandte Chemie International Edition
2017, 56, 12064–12084.
(18) Shannon, R. D. Acta Crystallographica A 1976, 32, 751–767.
(19) Mohtadi, R.; Mizuno, F. Beilstein Journal of Nanotechnology 2014, 5, 1291–1311.
168 References
(20) Canepa, P.; Bo, S.-H.; Gautam, G. S.; Key, B.; Richards, W. D.; Shi, T.; Tian, Y.;
Wang, Y.; Li, J.; Ceder, G. Nature Communications 2017, 8.
(21) Bonnick, P.; Blanc, L.; Vajargah, S. H.; Lee, C.-W.; Sun, X.; Balasubramanian, M.;
Nazar, L. F. Chemistry of Materials 2018, 30, 4683–4693.
(22) Pecher, O.; Carretero-González, J.; Griffith, K. J.; Grey, C. P. Chemistry of Materials
2017, 29, 213–242.
(23) Grey, C. P.; Dupré, N. Chemical Reviews 2004, 104, 4493.
(24) Freitas, J. C. C.; Smith, M. E. Annual Reports on NMR Spectroscopy 2012, 75, 25.
(25) Dupree, R.; Smith, M. E. Journal of the Chemical Society, Chemical Communications
1988, 1483.
(26) Moudrakovski, I. L. Annual Reports on NMR Spectroscopy 2013, 79, ed. by Webb,
G., 129.
(27) Larsen, F. H.; Jakobsen, H. J.; Ellis, P. D.; Nielsen, N. C. Journal of Magnetic
Resonance 1998, 131, 144–147.
(28) Sideris, P. J.; Nielsen, U. G.; Gan, Z.; Grey, C. P. Science 2008, 321, 113–117.
(29) Griffin, J. M.; Berry, A. J.; Ashbrook, S. E. Solid State Nuclear Magnetic Resonance
2011, 40, 91–99.
(30) Kim, C.; Phillips, P. J.; Key, B.; Yi, T.; Nordlund, D.; Yu, Y.-s.; Bayliss, R. D.; Han,
S.-d.; He, M.; Zhang, Z.; Burrell, A. K.; Klie, R. F.; Cabana, J. Advanced Materials
2015, 27, 3377.
(31) Sai Gautam, G.; Canepa, P.; Richards, W. D.; Malik, R.; Ceder, G. Nano Letters
2016, 16, 2426–2431.
(32) Sa, N.; Wang, H.; Proffit, D. L.; Lipson, A. L.; Key, B.; Liu, M.; Feng, Z.; Fister,
T. T.; Ren, Y.; Sun, C.-J.; Vaughey, J. T.; Fenter, P. A.; Persson, K. A.; Burrell, A. K.
Journal of Power Sources 2016, 323, 44–50.
(33) Nam, K. W. et al. Nano Letters 2015, 15, 4071–4079.
(34) Sa, N.; Kinnibrugh, T. L.; Wang, H.; Sai Gautam, G.; Chapman, K. W.; Vaughey, J. T.;
Key, B.; Fister, T. T.; Freeland, J. W.; Proffit, D. L.; Chupas, P. J.; Ceder, G.; Bareno,
J. G.; Bloom, I. D.; Burrell, A. K. Chemistry of Materials 2016, 28, 2962–2969.
(35) Wang, H.; Senguttuvan, P.; Proffit, D. L.; Pan, B.; Liao, C.; Burrell, A. K.; Vaughey,
J. T.; Key, B. ECS Electrochemistry Letters 2015, 4, A90–A93.
(36) Incorvati, J. T.; Wan, L. F.; Key, B.; Zhou, D.; Liao, C.; Fuoco, L.; Holland, M.; Wang,
H.; Prendergast, D.; Poeppelmeier, K. R.; Vaughey, J. T. Chemistry of Materials
2016, 28, 17–20.
(37) Wustrow, A.; Key, B.; Phillips, P. J.; Sa, N.; Lipton, A. S.; Klie, R. F.; Vaughey, J. T.;
Poeppelmeier, K. R. Inorganic Chemistry 2018, 57, 8634–8638.
(38) Lee, J.; Seymour, I. D.; Pell, A. J.; Dutton, S. E.; Grey, C. P. Physical Chemistry
Chemical Physics 2017, 19, 613–625.
(39) Keyzer, E. N.; Lee, J.; Liu, Z.; Bond, A. D.; Wright, D. S.; Grey, C. P. J. Mater. Chem.
A 2019, 7, 2677–2685.
References 169
(40) Liu, Z.; Lee, J.; Xiang, G.; Glass, H. F. J.; Keyzer, E. N.; Dutton, S. E.; Grey, C. P.
Chemical Communications 2017, 53, 743–746.
(41) Murgia, F.; Laurencin, D.; Weldekidan, E. T.; Stievano, L.; Monconduit, L.; Doublet,
M.-L.; Berthelot, R. Electrochimica Acta 2018, 259, 276–283.
(42) Pallister, P. J.; Moudrakovski, I. L.; Ripmeester, J. A. Physical Chemistry Chemical
Physics 2009, 11, 11487.
(43) Azevedo, L. J.; Schirber, J. E.; Switendick, A. C.; Baughman, R. J.; Morosin, B.
Journal of Physics F: Metal Physics 1981, 11, 1521–1530.
(44) Bastow, T.; Celotto, S. Solid State Nuclear Magnetic Resonance 2009, 35, 217–222.
(45) Rong, Z.; Malik, R.; Canepa, P.; Sai Gautam, G.; Liu, M.; Jain, A.; Persson, K.;
Ceder, G. Chemistry of Materials 2015, 27, 6016–6021.
(46) Xia, Y.; Yoshio, M. In Lithium Batteries: Science and Technology, Nazri, G.-A.,
Pistoia, G., Eds.; Springer US: Boston, MA, 2003, pp 361–380.
(47) Kasper, J. S.; Prener, J. S. Acta Crystallographica 1954, 7, 246.
(48) Winkler, B.; Chall, M.; Pickard, C. J.; Milman, V.; White, J. Acta Crystallographica
Section B 2000, 56, 22–26.
(49) Shannon, R. D. Acta Crystallographica A 1976, 32, 751.
(50) Lu, J.; Lee, K. S. Materials Technology 2016, 31, 628–641.
(51) O’Neill, H.; Navrotsky, A. American Mineralogist 1983, 68, 181–194.
(52) Navrotsky, A.; Kleppa, O. Journal of Inorgnic and Nuclear Chemistry 1967, 29,
2701.
(53) Sickafus, K. E.; Wills, J. M.; Grimes, N. W. Journal of the American Ceramic Society
1999, 82, 3279.
(54) Verwey, E. J. W.; Heilmann, E. L. Journal of Chemical Physics 1947, 15, 174.
(55) Burdett, J. K.; Price, G. D.; Price, S. L. Journal of the American Chemical Society
1982, 104, 92–95.
(56) Winter, M., D-block Chemistry; Oxford University Press: 2015.
(57) McClure, D. S. Journal of Physics and Chemistry of Solids 1957, 3, 311–317.
(58) Dunitz, J.; Orgel, L. Journal of Physics and Chemistry of Solids 1957, 3, 20–29.
(59) Manaila, R.; Pausescu, P. Physica Status Solidi B 1964, 6, 101.
(60) Manaila, R.; Pausescu, P. Physica Status Solidi B 1965, 9, 385.
(61) Irani, K. S.; Sina, A. P. B.; Biswas, A. B. Journal of Physics and Chemistry of Solids
1962, 23, 711–727.
(62) Sai Gautam, G.; Canepa, P.; Urban, A.; Bo, S.-H.; Ceder, G. Chemistry of Materials
2017, 29, 7918–7930.
(63) Ropp, R., Encyclopedia of the Alkaline Earth Compounds; Elsevier: 2012.
(64) Wang, Q.; Xu, J.; Zhang, W.; Mao, M.; Wei, Z.; Wang, L.; Cui, C.; Zhu, Y.; Ma, J.
Journal of Materials Chemistry A 2018, 6, 8815–8838.
170 References
(65) Jain, A.; Ong, S. P.; Hautier, G.; Chen, W.; Richards, W. D.; Dacek, S.; Cholia, S.;
Gunter, D.; Skinner, D.; Ceder, G.; Persson, K. A. APL Materials 2013, 1, 011002.
(66) Jain, A.; Hautier, G.; Ong, S. P.; Moore, C. J.; Fischer, C. C.; Persson, K. A.; Ceder,
G. Physical Review B 2011, 84, 045115.
(67) Sai Gautam, G.; Canepa, P.; Abdellahi, A.; Urban, A.; Malik, R.; Ceder, G. Chemistry
of Materials 2015, 27, 3733–3742.
(68) Zhou, B.; Shi, H.; Cao, R.; Zhang, X.; Jiang, Z. Physical Chemistry Chemical Physics
2014, 16, 18578–18585.
(69) Millet, P.; Satto, C.; Sciau, P.; Galy, J. Journal of Solid State Chemistry 1998, 136,
56–62.
(70) Szabo, A.; Ostlund, N., Modern Quantum Chemistry: Introduction to Advanced
Electronic Structure Theory; Dover Publications: 1996.
(71) Koch, W.; Holthausen, M., A Chemist’s Guide to Density Functional Theory; Wiley-
VCH: 2000.
(72) Hohenberg, P.; Kohn, W. Physical Review 1964, 136, B864–B871.
(73) Kohn, W.; Sham, L. J. Physical Review 1965, 140, A1133–A1138.
(74) Mardirossian, N.; Head-Gordon, M. Molecular Physics 2017, 115, 2315–2372.
(75) Himmetoglu, B.; Floris, A.; de Gironcoli, S.; Cococcioni, M. International Journal
of Quantum Chemistry 2014, 114, 14–49.
(76) Hubbard, J.; Flowers, B. H. Proceedings of the Royal Society of London. Series A.
Mathematical and Physical Sciences 1963, 276, 238–257.
(77) Khomskii, D., Transition Metal Compounds; Cambridge University Press: 2014.
(78) Liechtenstein, A. I.; Anisimov, V. I.; Zaanen, J. Physical Review B 1995, 52, R5467–
R5470.
(79) Anisimov, V.; Korotin, M.; Zölfl, M.; Pruschke, T.; Le Hur, K.; Rice, T. Physical
Review Letters 1999, 83, 364–367.
(80) Dudarev, S. L.; Botton, G. A.; Savrasov, S. Y.; Humphreys, C. J.; Sutton, A. P.
Physical Review B 1998, 57, 1505–1509.
(81) Cococcioni, M. In Correlated Electrons: From Models to Materials Modeling and
Simulation; Verlag des Forschungszentrum Jülich: Jülich, Germany, 2012.
(82) Cococcioni, M.; de Gironcoli, S. Physical Review B 2005, 71, 035105.
(83) Becke, A. D. Journal of Chemical Physics 1993, 98, 5648.
(84) Kim, K.; Jordan, K. D. The Journal of Physical Chemistry 1994, 98, 10089–10094.
(85) Adamo, C.; Barone, V. The Journal of Chemical Physics 1999, 110, 6158–6170.
(86) Heyd, J.; Scuseria, G. E.; Ernzerhof, M. The Journal of Chemical Physics 2006, 124,
219906.
(87) Becke, A. D. Physical Review A 1988, 38, 3098–3100.
(88) Lee, C.; Yang, W.; Parr, R. G. Physical Review B 1988, 37, 785–789.
References 171
(89) Corà, F.; Alfredsson, M.; Mallia, G.; Middlemiss, D. S.; Mackrodt, W. C.; Dovesi, R.;
Orlando, R. In Principles and Applications of Density Functional Theory in Inorganic
Chemistry II; Springer Berlin Heidelberg: Berlin, Heidelberg, 2004, pp 171–232.
(90) Feng, X.; Harrison, N. Physical Review B 2004, 70, 092402.
(91) de P. R. Moreira, I.; Illas, F.; Martin, R. Physical Review B 2002, 65, 155102.
(92) Kim, J.; Middlemiss, D. S.; Chernova, N. A.; Zhu, B. Y. X.; Masquelier, C.; Grey,
C. P. Journal of the American Chemical Society. 2010, 132, 16825.
(93) Clément, R. J.; Pell, A. J.; Middlemiss, D. S.; Strobridge, F. C.; Miller, J. K.; Whit-
tingham, M. S.; Emsley, L.; Grey, C. P.; Pintacuda, G. Journal of the American
Chemical Society. 2012, 134, 17178.
(94) Middlemiss, D. S.; Ilott, A. J.; Clément, R. J.; Strobridge, F. C.; Grey, C. P. Chemistry
of Materials 2013, 25, 1723.
(95) Boys, S. F. Proceedings of the Royal Society A: Mathematical, Physical and Engi-
neering Sciences 1950, 200, 542–554.
(96) Payne, C.; York, N. Reviews of Modern Physics 1992, 64, 1045–1097.
(97) Pickard, C. J.; Mauri, F. Physical Review B 2001, 63, 245101.
(98) Kaxiras, E., Atomic and Electronic Structure of Solids; Cambridge University Press:
2003.
(99) Quantum-Mechanical Ab-initio Calculation of the Properties of Crystalline Materi-
als; Berthier, G., Fischer, H., Fukui, K., Hall, G. G., Hinze, J., Jortner, J., Kutzelnigg,
W., Ruedenberg, K., Tomasi, J., Eds.; Lecture Notes in Chemistry, Vol. 67; Springer
Berlin Heidelberg: Berlin, Heidelberg, 1996.
(100) Dirac, P. A. M., The Principles of Quantum Mechanics; Clarendon Press: Oxford,
1930.
(101) Schwabl, F., Quantum mechanics, 4th ed; Springer: Berlin ; New York, 2007; 424 pp.
(102) Farazdel, A.; Smith, V. H. International Journal of Quantum Chemistry 1986, 29,
311–314.
(103) Jameson, C. J.; De Dios, A. C. In Nuclear Magnetic Resonance, Webb, G. A., Ed.;
Royal Society of Chemistry: Cambridge, 2007; Vol. 34, pp 57–86.
(104) Chang, C.; Pelissier, M.; Durand, P. Physica Scripta 1986, 34, 394–404.
(105) Heully, J. L. Journal of Physics B: Atomic and Molecular Physics 1982, 15, 4079–
4091.
(106) Van Lenthe, E.; Snijders, J. G.; Baerends, E. J. The Journal of Chemical Physics
1996, 105, 6505–6516.
(107) Yates, J. R.; Pickard, C. J.; Payne, M. C.; Mauri, F. The Journal of Chemical Physics
2003, 118, 5746–5753.
(108) Hamaed, H.; Laschuk, M. W.; Terskikh, V. V.; Schurko, R. W. Journal of the American
Chemical Society 2009, 131, 8271–8279.
(109) Widdifield, C. M.; Bryce, D. L. The Journal of Physical Chemistry A 2010, 114,
10810–10823.
(110) Grabowski, B.; Hickel, T.; Neugebauer, J. Physical Review B 2007, 76, 024309.
172 References
(111) Steiner, S.; Khmelevskyi, S.; Marsmann, M.; Kresse, G. Physical Review B 2016, 93,
224425.
(112) Wales, D. J., Energy Landscapes: Applications to Clusters, Biomolecules and Glasses;
Cambridge University Press: 2003.
(113) Munro, L. J.; Wales, D. J. Physical Review B 1999, 59, 3969–3980.
(114) Kumeda, Y.; Wales, D. J.; Munro, L. J. Chemical Physics Letters 2001, 341, 185–194.
(115) Keeler, J., Understanding NMR Spectroscopy; Wiley: 2011.
(116) Harris, R. K.; Becker, E. D.; Cabral De Menezes, S. M.; Granger, P.; Hoffman, R. E.;
Zilm, K. W. Solid State Nuclear Magnetic Resonance 2008, 33, 41–56.
(117) Duer, M., Introduction to Solid-State NMR Spectroscopy; Wiley: 2005.
(118) Herzfeld, J.; Berger, A. E. Journal of Chemical Physics 1980, 73, 6021.
(119) Mehring, M., Principles of High Resolution NMR in Solids; Springer Berlin Heidel-
berg: 2012.
(120) Widdifield, C. M.; Bain, A. D.; Bryce, D. L. Physical Chemistry Chemical Physics
2011, 13, 12413.
(121) Perras, F. A.; Widdifield, C. M.; Bryce, D. L. Solid State Nuclear Magnetic Resonance
2012, 45-46, 36–44.
(122) Freude, D.; Haase, J. In Special Applications, Pfeifer, H., Barker, P., Eds.; Springer
Berlin Heidelberg: Berlin, Heidelberg, 1993, pp 1–90.
(123) Man, P. P. In Encyclopedia of Analytical Chemistry, Meyers, R. A., Ed.; John Wiley
& Sons, Ltd: Chichester, UK, 2006.
(124) Freude, D. In Encyclopedia of Analytical Chemistry, Meyers, R. A., Ed.; John Wiley
& Sons, Ltd: Chichester, UK, 2006.
(125) Andrew, E. R. Philosophical Transactions of the Royal Society of London. Series A,
Mathematical and Physical Sciences 1981, 299, 505–520.
(126) Lowe, I. J. Phys. Rev. Lett. 1959, 2, 285–287.
(127) Gan, Z.; Gor’kov, P.; Cross, T. A.; Samoson, A.; Massiot, D. Journal of the American
Chemical Society 2002, 124, 5634–5635.
(128) Haase, J.; Eckert, D.; Siegel, H.; Eschrig, H.; Müller, K.-H.; Steglich, F. Concepts in
Magnetic Resonance Part B: Magnetic Resonance Engineering 2003, 19B, 9–13.
(129) Samoson, A.; Lippmaa, E.; Pines, A. Molecular Physics 1988, 65, 1013–1018.
(130) Mueller, K.; Sun, B.; Chingas, G.; Zwanziger, J.; Terao, T.; Pines, A. Journal of
Magnetic Resonance 1990, 86, 470–487.
(131) Medek, A.; Harwood, J. S.; Frydman, L. Journal of the American Chemical Society
1995, 117, 12779–12787.
(132) Gan, Z. Journal of the American Chemical Society 2000, 122, 3242–3243.
(133) Hahn, E. L. Physical Review 1950, 80, 580–594.
(134) Perras, F. A.; Viger-Gravel, J.; Burgess, K. M.; Bryce, D. L. Solid State Nuclear
Magnetic Resonance 2013, 51-52, 1–15.
References 173
(135) Yao, Z.; Kwak, H.-T.; Sakellariou, D.; Emsley, L.; Grandinetti, P. J. Chemical Physics
Letters 2000, 327, 85.
(136) Kwak, H. T.; Prasad, S.; Clark, T.; Grandinetti, P. J. Solid State Nuclear Magnetic
Resonance 2003, 24, 71.
(137) Prasad, S.; Kwak, H.-T.; Clark, T.; Grandinetti, P. J. Journal of the American Chemi-
cal Society. 2002, 124, 4964.
(138) Kentgens, A.; Verhagen, R. Chemical Physics Letters 1999, 300, 435.
(139) Hung, I.; Gan, Z. Chemical Physics Letters 2010, 496, 162.
(140) Blundell, S., Magnetism in Condensed Matter; Oxford Master Series in Condensed
Matter Physics 4; OUP Oxford: 2001.
(141) Köhler, F. H. In Magnetism: Molecules to Materials I; John Wiley & Sons, Ltd: 2003;
Chapter 12, pp 379–430.
(142) Pell, A. J.; Pintacuda, G.; Grey, C. P. Progress in Nuclear Magnetic Resonance
Spectroscopy 2019, 111, 1–271.
(143) Kurland, R. J.; McGarvey, B. R. Journal of Magnetic Resonance 1970, 2, 286–301.
(144) Carlier, D.; Ménétrier, M.; Grey, C.; Delmas, C.; Ceder, G. Physical Review B 2003,
67, 174103.
(145) Ellingham, H. J. T. Journal of the Society of Chemical Industry 1944, 63, 125–160.
(146) Chevrier, V. L.; Ong, S. P.; Armiento, R.; Chan, M. K. Y.; Ceder, G. Physical Review
B 2010, 82, 075122.
(147) Rietveld, H. M. Journal of Applied Crystallography 1969, 2, 65.
(148) Levi, E.; Gofer, Y.; Aurbach, D. Chemistry of Materials 2010, 22, 860.
(149) Novák, P.; Imhof, R.; Haas, O. Electrochimica Acta 1999, 45, 351.
(150) D’Alessandro, D. M.; Smit, B.; Long, J. R. Angewandte Chemie Internation Edition
2010, 49, 6058.
(151) Garg, N.; Menaka; Ramanujachary, K. V.; Lofland, S. E.; Ganguli, A. K. Journal of
Solid State Chemistry 2013, 197, 392.
(152) Maitra, U.; Naidu, B. S.; Govindaraj, A.; Rao, C. N. R. Proceedings of the National
Academy of USA 2013, 110, 11704.
(153) Mackenzie, K. J. D.; Smith, M. E., Multinuclear Solid-State NMR of Inorganic
Materials; Pergamon Materials Series, Vol. 6; Elsevier: 2002, pp 461–532.
(154) Catti, M.; Valerio, G.; Dovesi, R.; Causa, M. Physical Review B 1994, 49, 179.
(155) Catti, M.; Sandrone, G.; Valerio, G.; Dovesi, R. Journal of the Physics and Chemistry
of Solids 1996, 57, 1735.
(156) Catti, M.; Sandrone, G.; Dovesi, R. Physical Review B 1997, 55, 16122.
(157) Kutzelnigg, W.; Fleischer, U.; Schindler, M. In NMR-Basic Principles and Progress;
Springer-Verlag: 1990, p 165.
(158) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. Journal of Physical
Chemistry 1994, 98, 11623.
174 References
(159) Monkhorst, H. J.; Pack, J. D. Physical Review B 1976, 13, 5188.
(160) Pallister, P. J.; Moudrakovski, I. L.; Ripmeester, J. A. Physical Chemistry Chemical
Physics 2009, 11, 11487.
(161) Goodenough, J., Magnetism and the Chemical Bond; Wiley: 1963.
(162) Porta, P.; Valigi, M. Journal of Solid State Chemistry 1973, 6, 344.
(163) Dutton, S. E.; Huang, Q.; Tchernyshyov, O.; Broholm, C. L.; Cava, R. J. Physical
Review B 2011, 83, 064407.
(164) Mamiya, H.; Onoda, M. Solid State Communications 1995, 95, 217.
(165) Islam, A. T. M. N.; Wheeler, E. M.; Reehuis, M.; Siemensmeyer, K.; Tovar, M.;
Klemke, B.; Kiefer, K.; Hill, A. H.; Lake, B. Physical Review B 2012, 85.
(166) Wheeler, E. M.; Lake, B.; Islam, A. T. M. N.; Reehuis, M.; Steffens, P.; Guidi, T.;
Hill, A. H. Physical Review B 2010, 82.
(167) Yagi, S.; Ichikawa, Y.; Yamada, I.; Doi, T.; Ichitsubo, T.; Matsubara, E. Japanese
Journal of Applied Physics 2013, 52, 025501.
(168) Cabello, M.; Alcantara, R.; Nacimiento, F.; Ortiz, G.; Lavela, P.; Tirado, J. L. Crys-
tEngComm 2015, 17, 8728.
(169) Ghigna, P.; De Renzi, R.; Mozzati, M.; Lascialfari, A.; Allodi, G.; Bimbi, M.; Mazzoli,
C.; Malavasi, L.; Azzoni, C. Physical Review B 2006, 73, 184402.
(170) Muramori, K.; Miyahara, S. Journal of the Physical Society of Japan 1960, 15, 1906.
(171) Taguchi, H.; Nagao, M. Journal of Materials Science Letters 1991, 10, 658.
(172) Dunstan, M. T.; Jain, A.; Liu, W.; Ong, S. P.; Liu, T.; Lee, J.; Persson, K. A.; Scott,
S. A.; Dennis, J. S.; Grey, C. P. Energy & Environmental Science 2016, 9, 1346.
(173) Seymour, I. D.; Middlemiss, D. S.; Halat, D. M.; Trease, N. M.; Pell, A. J.; Grey, C. P.
Journal of the American Chemical Society 2016, 138, 9405–9408.
(174) Grey, C. P.; Dobson, C. M.; Cheetham, A. K. Journal of Magnetic Resonance 1992,
98, 414.
(175) Stoll, M.; Majors, T. Journal of Magnetic Resonance 1982, 46, 283.
(176) Xiang, H. J.; Kan, E. J.; Wei, S.-H.; Whangbo, M.-H.; Gong, X. G. Physical Review
B 2011, 84, 224429.
(177) Brumage, W. H.; Quade, C. R.; Dorman, C. F. Physical Review B 1995, 52, 4256–
4262.
(178) Onoda, M.; Ohyama, A. Journal of Physics: Condensed Matter 1998, 10, 1229–1236.
(179) Prasad, S.; Kwak, H.-T.; Clark, T.; Grandinetti, P. J. Journal of the American Chemi-
cal Society 2002, 124, 4964–4965.
(180) Peintinger, M. F.; Oliveira, D. V.; Bredow, T. Journal of Computational Chemistry
2013, 34, 451–459.
(181) Monkhorst, H. J.; Pack, J. D. Physical Review B 1976, 13, 5188–5192.
(182) Barin, I.; Platzki, G., Thermochemical data of pure substances, 3rd ed; VCH: Wein-
heim ; New York, 1995.
References 175
(183) Lewis, J.; Murdoch, R.; Moul, A. Nature 1969, 221, 1137.
(184) Das, S.; Hucke, E. Carbon 1975, 13, 33–42.
(185) Kim, J.; Middlemiss, D. S.; Chernova, N. A.; Zhu, B. Y. X.; Masquelier, C.; Grey,
C. P. Journal of the American Chemical Society 2010, 132, 16825–16840.
(186) Valenzano, L.; Noël, Y.; Orlando, R.; Zicovich-Wilson, C. M.; Ferrero, M.; Dovesi, R.
Theoretical Chemistry Accounts 2007, 117, 991–1000.
(187) Ruiz, E.; Llunell, M.; Alemany, P. Journal of Solid State Chemistry 2003, 176, 400–
411.
(188) Bredow, T.; Jug, K.; Evarestov, R. A. Physica Status Solidi B 2006, 243, R10–R12.
(189) Schäfer, A.; Horn, H.; Ahlrichs, R. The Journal of Chemical Physics 1992, 97, 2571–
2577.
(190) Kutzelnigg, W.; Fleischer, U.; Schindler, M. In NMR-Basic Principles and Progress;
Springer-Verlag: 1990, p 165.
(191) Kumeda, Y.; Wales, D. J.; Munro, L. J. Chemical Physics Letters 2001, 341, 185–194.
(192) Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Vydrov, O. A.; Scuseria, G. E.; Con-
stantin, L. A.; Zhou, X.; Burke, K. Physical Review Letters 2008, 100.
(193) Klimeš, J.; Bowler, D. R.; Michaelides, A. Physical Review B 2011, 83.
(194) Lee, K.; Murray, E. D.; Kong, L.; Lundqvist, B. I.; Langreth, D. C. Physical Review
B 2010, 82.
(195) Ganesh, P.; Kim, J.; Park, C.; Yoon, M.; Reboredo, F. A.; Kent, P. R. C. Journal of
Chemical Theory and Computation 2014, 10, 5318–5323.
(196) Levi, E.; Levi, M. D.; Chasid, O.; Aurbach, D. Journal of Electroceramics 2009, 22,
13–19.
(197) Korotin, M. A.; Anisimov, V. I.; Saha-Dasgupta, T.; Dasgupta, I. Journal of Physics:
Condensed Matter 2000, 12, 113–124.
(198) Aykol, M.; Kim, S.; Wolverton, C. The Journal of Physical Chemistry C 2015, 119,
19053–19058.
(199) Koga, N.; Suzuki, Y.; Tatsuoka, T. The Journal of Physical Chemistry B 2012, 116,
14477–14486.
(200) Jeong, J.; Aetukuri, N.; Graf, T.; Schladt, T. D.; Samant, M. G.; Parkin, S. S. P.
Science 2013, 339, 1402–1405.
(201) Granqvist, C. Thin Solid Films 1990, 193-194, 730–741.
(202) Yao, Z.; Kwak, H.-T.; Sakellariou, D.; Emsley, L.; Grandinetti, P. J. Chemical Physics
Letters 2000, 327, 85–90.
(203) Millet, P.; Satto, C.; Bonvoisin, J.; Normand, B.; Penc, K.; Albrecht, M.; Mila, F.
Physical Review B 1998, 57, 5005–5008.
(204) Middlemiss, D. S.; Ilott, A. J.; Clément, R. J.; Strobridge, F. C.; Grey, C. P. Chemistry
of Materials 2013, 25, 1723–1734.
(205) Clément, R. J.; Middlemiss, D. S.; Seymour, I. D.; Ilott, A. J.; Grey, C. P. Chemistry
of Materials 2016, 28, 8228–8239.
176 References
(206) Gautam, G. S.; Sun, X.; Duffort, V.; Nazar, L. F.; Ceder, G. Journal of Materials
Chemistry A 2016, 4, 17643–17648.
(207) Jouanneau, S.; Le Gal La Salle, A.; Verbaere, A.; Guyomard, D. Journal of The
Electrochemical Society 2005, 152, A1660–A1667.
(208) Shao, Y.; Gu, M.; Li, X.; Nie, Z.; Zuo, P.; Li, G.; Liu, T.; Xiao, J.; Cheng, Y.; Wang,
C.; Zhang, J.-G.; Liu, J. Nano Letters 2014, 14, 255–260.
(209) Arthur, T. S.; Singh, N.; Matsui, M. Electrochemistry Communications 2012, 16,
103–106.
(210) Murgia, F.; Stievano, L.; Monconduit, L.; Berthelot, R. Journal of Materials Chem-
istry A 2015, 3, 16478–16485.
(211) Ramanathan, M.; Benmayza, A.; Prakash, J.; Singh, N.; Mizuno, F. Journal of The
Electrochemical Society 2016, 163, A477–A487.
(212) Ahuja, R.; Blomqvist, A.; Larsson, P.; Pyykkö, P.; Zaleski-Ejgierd, P. Physical Review
Letters 2011, 106, 018301.
(213) Kresse, G.; Hafner, J. Physical Review B 1993, 47, 558–561.
(214) Kresse, G.; Joubert, D. Physical Review B 1999, 59, 1758–1775.
(215) Blöchl, P. E. Physical Review B 1994, 50, 17953–17979.
(216) Perdew, J. P.; Burke, K.; Ernzerhof, M. Physical Review Letters 1996, 77, 3865–3868.
(217) Hafner, J. Journal of Computational Chemistry 2008, 29, 2044–2078.
(218) Kröger, F.; Vink, H. In Solid State Physics; Elsevier: 1956; Vol. 3, pp 307–435.
(219) Clark, S. J.; Segall, M. D.; Pickard, C. J.; Hasnip, P. J.; Probert, M. J.; Refson, K.;
Payne, M. Z. Kristall. 2005, 220, 567–570.
(220) Yates, J. R.; Pickard, C. J.; Mauri, F. Phys. Rev. B 2007, 76, 024401.
(221) Seymour, I. D.; Chakraborty, S.; Middlemiss, D. S.; Wales, D. J.; Grey, C. P. Chem-
istry of Materials 2015, 27, 5550–5561.
(222) Zintl, E.; Husemann, E. Zeitschrift für Physikalische Chemie 1933, 21, 138–155.
(223) Elliott, S. R., The Physics and Chemistry of Solids; Wiley: 1998.
(224) Lazarev, V. B.; Marenkin, S. F.; Shevchenko, V. Y.; Goncharenko, G. I. Neorganich-
eskie Materialy 1974, 10, 1615–1616.
(225) Barnes, A. C.; Guo, C.; Howells, W. S. Journal of Physics: Condensed Matter 1994,
6, L467.
(226) Howells, W.; Barnes, A.; Hamilton, M. Physica B: Condensed Matter 1999, 266,
97–99.
(227) Ferrier, R. P.; Herrell, D. J. Philosophical Magazine 1969, 19, 853–868.
(228) Ferrier, R. P.; Herrell, D. J. Journal of Non-Crystalline Solids 1970, 2, 278–283.
(229) Watson, L. M.; Marshall, C. A. W.; Cardoso, C. P. Journal of Physics F: Metal
Physics 1984, 14, 113.
(230) Sutton, C. Solid State Communications 1975, 16, 327–330.
References 177
(231) Sedighi, M.; Arghavani Nia, B.; Zarringhalam, H.; Moradian, R. The European
Physical Journal Applied Physics 2013, 61, 10103.
(232) Imai, Y.; Watanabe, A. Journal of Materials Science 2006, 41, 2435–2441.
(233) Xu, R.; de Groot, R. A.; van der Lugt, W. Journal of Physics: Condensed Matter
1993, 5, 7551–7562.
(234) Zhang, X.; Jin, L.; Dai, X.; Liu, G. The Journal of Physical Chemistry Letters 2017,
8, PMID: 28927265, 4814–4819.
(235) Shao, D. F.; Luo, X.; Lu, W. J.; Hu, L.; Zhu, X. D.; Song, W. H.; Zhu, X. B.; Sun, Y. P.
Scientific Reports 2016, 6, 21484.
(236) Noh, H.-J.; Koh, H.; Oh, S.-J.; Park, J.-H.; Kim, H.-D.; Rameau, J. D.; Valla, T.;
Kidd, T. E.; Johnson, P. D.; Hu, Y.; Li, Q. Europhysics Letters 2008, 81, 57006.
(237) Yatsimirskii, K. B. Theoretical and Experimental Chemistry 1995, 31, 153–168.
(238) Crowley, J. M.; Tahir-Kheli, J.; Goddard, W. A. The Journal of Physical Chemistry
Letters 2016, 7, 1198–1203.
(239) Koller, H.; Engelhardt, G.; Kentgens, A. P.; Sauer, J. The Journal of Physical Chem-
istry 1994, 98, 1544–1551.
(240) Chen, D.; Wang, Z.; Ren, T.; Ding, H.; Yao, W.; Zong, R.; Zhu, Y. The Journal of
Physical Chemistry C 2014, 118, 15300–15307.
(241) Suryanarayana, C. In Pergamon Materials Series; Elsevier: 1999; Vol. 2, pp 49–85.
(242) Ogata, K.; Salager, E.; Kerr, C.; Fraser, A.; Ducati, C.; Morris, A.; Hofmann, S.;
Grey, C. Nature Communications 2014, 5, 3217–3228.
(243) Yu, H.-C.; Ling, C.; Bhattacharya, J.; Thomas, J. C.; Thornton, K.; Van der Ven, A.
Energy & Environmental Science 2014, 7, 1760–1768.
(244) Ponnambalam, V.; Morelli, D. T. Journal of Electronic Materials 2013, 42, 1307–
1312.
(245) Sun, J.; Singh, D. J. Journal of Materials Chemistry A 2017, 5, 8499–8509.
(246) Singh, N.; Arthur, T. S.; Ling, C.; Matsui, M.; Mizuno, F. Chemical Communications
2013, 49, 149–151.

Appendix A
Post ball-mill analysis of MgV2O5
A.1 SEM
SEM images were taken with TESCAN MIRA3 FEG-SEM with a secondary electron detector.
Accerlation voltage of 15 kV was used for both images.
180 Post ball-mill analysis of MgV2O5
(a)
(b)
Fig. A.1 (a) SEM image of pristine MgV2O5 before ball-milling. (b) SEM image of MgV2O5
after ball-milling.
A.2 XRD 181
A.2 XRD
(a)
(b)
Fig. A.2 (a) XRD pattern of pristine MgV2O5 before ball-milling. (b) XRD pattern of
MgV2O5 after ball-milling.

Appendix B
EDX data of lithium counterelectrode
from charged VO2 cell
Fig. B.1 EDX data of the lithium disc from the coin cell after charging of VO2.