Current-Induced Torque Driven
Ferromagnetic Resonance in
Magnetic Microstructures
Dong Fang
Department of Physics
University of Cambridge
A dissertation submitted for the degree of
Doctor of Philosophy
Downing College, December 2010
Abstract
Current-Induced Torque Driven Ferromagnetic Resonance in
Magnetic Microstructures
Dong Fang
This Dissertation explores the interaction between the magnetisation and
an alternating current in a uniform ferromagnetic system. Diluted magnetic
semiconductors (Ga,Mn)As and (Ga,Mn)(As,P) have been studied. Due to
their strong spin-orbit coupling and well-understood band-structure, these
materials are well-suited to this investigation. The combined effect of spin-
orbit coupling and exchange interaction permits the alternating current to
induce an oscillating current-induced torque (CIT) on the magnetisation. In
the frequency range close to the natural resonance frequency of the magnetic
moments (gigahertz), CIT can excite precessional motion of the magnetisa-
tion, a process known as ferromagnetic resonance (FMR).
CIT can be parameterised by an effective magnetic field. By analysing
the lineshape of the measured FMR signals, the magnitude and orienta-
tion of this effective field have been accurately determined. Moreover, the
current-induced fields in these ferromagnetic materials have been observed
with symmetries of the Dresselhaus, and for the first time, Rashba spin-orbit
coupling. A new class of device-scale FMR technique, named as CIT-FMR,
has been established in this Dissertation, with the advantage of simple device
structure (only a resistor is required) and scalability (measurements has been
performed on devices sized from 4 µm down to 80 nm). This technique is
not only limited to magnetic semiconductors, but can also be transferred to
study other ferromagnetic systems such as ultrathin metal films.
Finally, the CIT-FMR technique is employed to study the magnetic anisotropy
in individual (Ga,Mn)As and (Ga,Mn)(As,P) micro-devices. Devices down
to 80 nm in width have been measured in (Ga,Mn)(As,P), which show strong
strain-relaxation-induced anisotropy, larger than any previously reported
cases on (Ga,Mn)As. Furthermore, due to the tensile-strain on the (Ga,Mn)(As,P)
epilayers, the anisotropy field due to patterning-induced strain-relaxation in
these devices is observed to take the opposite direction compared to that in
the compressively-strained (Ga,Mn)As samples.
Preface
The experimental work presented in this Dissertation was performed under
the supervision of Dr. Andrew Ferguson over the period between October
2007 and October 2010 at the Cavendish Laboratory, University of Cam-
bridge, and the Hitachi Cambridge Laboratory, for the awarding of the de-
gree of Doctor of Philosophy.
The research has been funded by the following organisations: Cambridge
Overseas Trusts, Hitachi Cambridge Laboratory, Downing College, EU Grant
FP7-214499 NAMASTE.
Declaration
This Dissertation is the result of my own work and includes nothing which is
the outcome of work done in collaboration except where specifically indicated
in the text. It has not been submitted in whole or in part for the award of a
degree at this or any other University, and does not exceed 60,000 words in
length.
Dong Fang
Downing College, December 2010
To my family
Acknowledgements
The work described in this Dissertation would not have been made pos-
sible without the help of many people, directly and indirectly, whether they
know it or not. I would like to start by thanking all the people in my life,
who have supported me, and have helped me to become the person that I
am.
In particular, I first want to thank my supervisor, Andrew Ferguson, for
giving me the opportunity to be his student and for mentoring me for the
past 3 years. Andrew carries a wealth of knowledge which he enthusiastically
shares with his students; I have always enjoyed discussions with him, and
the special sensation which I feel after these discussions, knowing that I have
learned something new again — I have learned an enormous amount from
him in the past few years. On top of this, I also thank Andrew for finding
the funds to put so many cool toys in the lab, and putting me in charge
of them, and giving me the freedom to explore, learn, and...make mistakes
(sorry again about the low-temperature amplifier that I blew up, Andrew!).
In all sincerity, I always feel extremely fortunate to spend my graduate career
under such a great teacher — I could not have asked for more.
I would also like to thank my workmate Hidekazu Kurebayashi. Over the
past one year and a half or so, Hide has always been a source of knowledge
and support, and I have always enjoyed the countless conversations that we
had on things not just physics-related. On the other hand, I feel bad that
Hide has to listen to all my moans (sorry Hide!). Outside the lab, Hide is
a good personal friend; and I would like to extend my gratitude to Hide’s
family — his wife Shoko and their baby girl Mei. It has been a great pleasure
watching Mei growing up.
I would also like to extend my appreciation to my colleagues from the
Hitachi Cambridge Laboratory. In particular, I want to thank Jo¨rg Wunder-
lich, who has an enormous amount of knowledge in the field of magnetism
and spintronics, and has given me many insightful suggestions over the years.
Besides, his passion for science is inspiring to everybody around him. An-
other special thanks goes to Kai-you Wang, who has now left Cambridge and
back in China running his own group. When I first started my PhD, Kai-you
helped me solve many practical problems with my experiments, and taught
me how to do 4-point measurements (I really didn’t know anything when I
started!), how to use the semiconductor parameter analyser, and how to de-
sign my devices — I wouldn’t have gone anywhere without his help. I would
also like to thank the lab manager of HCL, David Williams, for running the
lab so that we can freely pursue our experiments, and also for his hospitality
for inviting me to his annual house BBQs, which have always been one of
the highlights of the year. I also want to thank Elisa De Ranieri, who is two
years senior than me, and has always been helpful along the way.
MRC-wise, I would like to thank Henning Sirringhaus for running the
group and providing us with such state-of-the-art facilities. Thanks to Ra-
doslav Chakalov for his irreplaceable role in running the clean room, so that
we can carry out our device fabrications without worry; well, we still worry
about equipments breaking down, but thanks to Radoslav, we just don’t have
to worry about fixing them. I also want to thank Andrew Irvine, for teaching
me the art of lithography, and for always being a source of expertise in nano-
fabrication. I would also like to extend my appreciation to the supporting
staff in the group: Ron Hodierne, who does almost everything from sorting
out the mails to fixing the network servers, so that our building doesn’t fall
apart; the (former) group secretaries Christina Simmons and Melanie Hills,
whose workloads may have easily been doubled because of me, when I went
on a spending spree with Andrew’s funds and made countless purchase or-
ders; and our cleaner Xiu Ping Zhang, who has been a good friend, and has
made me delicious lunches.
I also want to thank the supporting staff from around the department. I’d
like to specially thank Dan Cross, who has been providing me with two dewars
of liquid helium a week for the past two years (it looks as if I drink them!). I
also want to thank the guys in the Cavendish Mechanical Workshop: Peter
Norman, who always helps me with my designs; Nigel Palfrey, who heads the
student workshop and teaches me to use all the big machines in the workshop
(it’s a shame that I never become good at them); Brian Camps, the welding
expert and always nice to chat with; and Adam Brown, who masterly crafted
the coldfinger on the dipstick, one of many things he made for me.
Outside Cambridge, I would like to thank Tomas Jungwirth, Karel Vy´borny´,
and Liviu Zaˆrbo from the Institute of Physics in Prague, for their contribu-
tions to the theoretical side of my work; Richard Campion, Arianna Casir-
aghi and Bryan Gallagher from the University of Nottingham, for providing
us with top-quality wafers.
I would also like to thank Cambridge Overseas Trusts, Hitachi Cambridge
Laboratory, and Downing College for financially supporting me during my
PhD. We also acknowledge support from EU Grant FP7-214499 NAMASTE.
I could not have survived my days as a graduate student without the sup-
port of many good friends. I would like to thank Mijung Lee, for organising
many (mostly drinking-related) social events in the office, and for being a
wonderful hostess and organising the many more parties outside the office,
and more importantly for being a great friend. I’d also like to thank Mijung’s
daughter Misol, who is at the extremely active age of 4, and has always been
a pleasant company in and outside the lab (well of course, she comes into the
lab with her mom, not just by herself). Special thanks to my good friends
Gueorgui Nikiforov and Sebastian Schoefer, who have been my dinner bud-
dies, movie buddies and travel buddies for the past few years. I also want to
thank Shrivalli Bhat, for her wonderful personality and positive attitude in
life, which is something I should learn. College-wise, I want to thank Helen
Lightfoot and her husband Jo¨rg Sedelmeier, for being my good friends over
the past 3 years, and for letting me be the usher in their beautiful wedding
(and I really loved the penguin suit!).
Special thanks to my many Chinese friends in and outside the lab (well,
some of them are from Taiwan, but we are not too political here, are we?).
My special thanks goes to Jui-fen Chang, who is the most hardworking person
that I have ever met, and her working attitude is almost legendary. She has
been a role model to me, as well as a wonderful friend. I would also like to
thank Sam Owen, who started one year earlier than me and introduced me
to the life at Cambridge, and has always offered me guidance and help for my
work along the way. A big thank you to Jianpu Wang for his hospitality and
for welcoming me to his house parties to celebrate every Chinese New Year,
which makes life away from home a lot easier to cope with. Along this line I
would like to extend my thanks to Xiulai Xu, Jinjin Wang, Xiaoyang Cheng,
Ni Zhao and Chuan Liu, who have made my time at Cambridge simply more
enjoyable. I also want to thank my housemates Liang Zhang, Ming Han and
Zhi Wang, who have made our house feel like “home”, which is the first time
I have felt this way since I came to study in England nearly 8 years ago. And
they help push me into finishing my PhD, by annoyingly asking me about
my progress on writing-up every single day.
My deepest gratitude, however, goes to my parents, who raised and nur-
tured me, and then (rather unwillingly) let me go to the other side of the
world, so that I can have my own space and can freely explore and make
my own decisions in life. And they offer guidance instead of constraints, and
have supported me and loved me unconditionally along my way. No matter
where I am in the world, I would never feel alone; and I love them very much.
Respect.
Contents
1 Introduction 1
1.1 Diluted magnetic semiconductors . . . . . . . . . . . . . . . . 2
1.2 Context of this Thesis . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Electrical manipulation of the magnetisation . . . . . . 4
1.2.2 Mapping the magnetic anisotropy in individual nano-
magnets . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Background 12
2.1 III–V diluted magnetic semiconductors . . . . . . . . . . . . . 12
2.1.1 (Ga,Mn)As and its basic properties . . . . . . . . . . . 12
2.1.2 Defects in (Ga,Mn)As and their treatment . . . . . . . 14
2.2 Magnetic anisotropy . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Shape anisotropy . . . . . . . . . . . . . . . . . . . . . 17
2.2.2 Magnetocrystalline anisotropy . . . . . . . . . . . . . . 17
2.2.3 Magnetic anisotropy in (Ga,Mn)As systems . . . . . . 19
2.2.4 Control of magnetic anisotropy . . . . . . . . . . . . . 20
2.3 Magneto-transport . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.1 Anisotropic magnetoresistance . . . . . . . . . . . . . . 24
2.3.2 Anomalous Hall effect . . . . . . . . . . . . . . . . . . 29
2.4 Magnetisation dynamics . . . . . . . . . . . . . . . . . . . . . 29
2.4.1 Motion of magnetic moments . . . . . . . . . . . . . . 29
2.4.2 Ferromagnetic resonance . . . . . . . . . . . . . . . . . 32
2.4.3 Detection of magnetic motion . . . . . . . . . . . . . . 35
2.5 Spin-orbit interaction in semiconductors . . . . . . . . . . . . 38
ix
CONTENTS
3 On-chip driven ferromagnetic resonance on individual (Ga,Mn)As
microdevices 42
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Devices and experimental methods . . . . . . . . . . . . . . . 43
3.2.1 Device preparation . . . . . . . . . . . . . . . . . . . . 43
3.2.2 Measurement apparatus . . . . . . . . . . . . . . . . . 44
3.2.3 Measurement technique . . . . . . . . . . . . . . . . . . 47
3.3 Data and analysis . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.1 On-chip FMR using a waveguide . . . . . . . . . . . . 50
3.3.2 Mapping the anisotropy in the nanodevices . . . . . . . 55
3.3.3 Vector magnetometry using FMR . . . . . . . . . . . . 58
3.3.4 Effects of annealing . . . . . . . . . . . . . . . . . . . . 58
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5 Appendix – Bond pad design . . . . . . . . . . . . . . . . . . . 62
4 Ferromagnetic resonance driven by current-induced torque
in magnetic microstructures 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Devices and experimental methods . . . . . . . . . . . . . . . 66
4.3 Data and analysis . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.1 Current-induced torque driven FMR . . . . . . . . . . 69
4.3.2 Quantifying the current-induced torque . . . . . . . . . 71
4.3.3 Theoretical understanding on the current-induced torque 74
4.3.4 Effect of strain on the current-induced torque . . . . . 77
4.3.5 Effect of an electrical bias on the FMR signals . . . . . 80
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5.1 Symmetric Lorentzian component Vsym . . . . . . . . . 84
4.5.2 Calibrating the microwave current using Joule heating 87
4.5.3 Vector magnetometry on the (Ga,Mn)(As,P) devices . 87
4.5.4 Magnetic properties of the (Ga,Mn)As microdevices . . 92
4.5.5 Effects of annealing . . . . . . . . . . . . . . . . . . . . 98
4.5.6 Gating experiments on ultrathin (Ga,Mn)As films . . . 103
x
CONTENTS
5 Mapping the magnetic anisotropy in (Ga,Mn)(As,P) nanos-
tructures 106
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2 Devices and experimental methods . . . . . . . . . . . . . . . 108
5.3 Data and analysis . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3.1 Realisation of FMR in 80 nm-wide devices . . . . . . . 108
5.3.2 Mapping the magnetic anisotropy . . . . . . . . . . . . 111
5.3.3 Strain-relaxation in devices of different sizes . . . . . . 112
5.3.4 FMR signatures in devices of different sizes . . . . . . . 114
5.3.5 Effects of growth-strain on HU . . . . . . . . . . . . . . 116
5.3.6 Lithography-control over the magnetisation . . . . . . 116
5.4 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . 119
5.5 Appendix – Magnetic anisotropy control with PMMA (pre-
liminary results) . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6 Closing remarks 128
A Derivation of the FMR lineshape 131
B Measurement apparatus 136
C Lithography recipes for the fabrication of microdevices 140
References 143
xi
Chapter 1
Introduction
The charge and spin of electrons in solid state lay the foundation of today’s
information technology [1]. Integrated circuits and high-frequency devices,
using the charge of electrons in semiconductors, have enjoyed great success
in the field of data processing and communications, and have led to products
such as computers and mobile phones which are fundamental in today’s soci-
ety. On the other hand, hard drives and magneto-optical disks, utilising the
spin of electrons in ferromagnetic materials, have revolutionised the informa-
tion storage industry, and have allowed knowledge to be spread and passed
on not only on papers, but in digital formats with multimedia contents.
But what would happen if both the charge and spin of electrons can be
manipulated simultaneously in one device? Spin-based electronics, or “spin-
tronics”, is such an emerging technology which exploits the spin states of elec-
trons as well as making use of their charge states. Through the incorporation
of spin degree of freedom into conventional charge-based electronics, devices
which perform both real-time information processing (logic) and data stor-
age (memory) can be realised, with the advantage of non-volatility, increased
processing speed, reduced power consumption, and increased integration den-
sities compared with conventional semiconductor/magnetic material-based
electronics [2].
1
1.1 Diluted magnetic semiconductors
a cb
Figure 1.1: Three types of semiconductors, adapted from [1]. a, A magnetic
semiconductor, in which a periodic array of magnetic elements is present.
b, A diluted magnetic semiconductor (DMS), an alloy between nonmagnetic
semiconductors and magnetic elements (such as manganese). c, A nonmag-
netic semiconductor, which contains no magnetic moments.
1.1 Diluted magnetic semiconductors
Diluted magnetic semiconductors (DMSs) are created by doping a nonmag-
netic semiconductor host with magnetic impurities such as manganese (Fig-
ure 1.1). These materials retain their semiconductor properties, such as
sensitivity to changes in carrier concentration, and thus can be controlled by
electrical gating; in addition, they exhibit ferromagnetic behaviours due to
the ordering among the diluted magnetic moments mediated by the charge
carriers. Because of this unity of semiconducting and ferromagnetic proper-
ties in a single material, DMSs are promising candidates for integrating the
functionalities of logic and memory in one device.
Historically, magnetic semiconductors such as europium chalcogenides
and semiconducting spinels have been extensively studied in the late 1960s
and early 1970s [1]. However, the crystal structures of these materials are dif-
ferent from that of Si or GaAs; furthermore, the growth of these compounds
is notoriously difficult [1].
DMSs are first realised in the late 1970s on II–VI semiconductors such as
CdTe and ZnTe by doping with Mn. These compounds are relatively easy to
2
1.2 Context of this Thesis
prepare, and the doping can reach very high level. However, because the in-
teraction among the doped magnetic ions is predominantly antiferromagnetic
exchange, these materials are usually antiferromagnetic or paramagnetic [1].
Only in the late 1990s carrier-induced ferromagnetic transition in (Cd,Mn)Te
has been shown, but only at very low temperatures (T = 1.8 K) [3]. This
renders these materials less attractive for applications.
The most widely studied DMS to date is the III–V group magnetic semi-
conductor (Ga,Mn)As, which is first successfully synthesised in 1996 [4].
Since the host semiconductor GaAs is already being used in a wide range of
applications, including mobile phones (microwave transistors) and compact-
disks (semiconductor lasers), there is little doubt that adding ferromagnetism
would open up a wide range of new applications. In particular, the magneti-
sation in (Ga,Mn)As has been observed to respond to strain [5–8], electrical
gating [9–12] and light [13–16], making it possible for nonmagnetic manipu-
lation of (Ga,Mn)As nanomagnets.
1.2 Context of this Thesis
In modern computers, logic operations are performed by manipulating the
charges in semiconductor devices, hence they require continuous supply of
electric currents. Whereas memory operations are achieved by controlling the
magnetisation orientation in nanoscale ferromagnets using a magnetic field,
which is generated by a current-carrying coil (Ampe`re’s circuital law). The
magnetic memory media have the advantage of being non-volatile; however,
generating a magnetic field using an electric current is a slow and energy
consuming process, and this has prevented real-time information processing
from being carried out in ferromagnetic devices. Moreover, the magnetic field
is difficult to localise, incompatible with the downscale of magnetic elements
required for achieving higher storage density.
One of the major quests in spintronics research is to find ways to ma-
nipulate the magnetisation direction in a ferromagnet without the use of an
external magnetic field. Such technology would integrate the best qualities
of two disparate worlds: the impressive computing power of semiconductor
3
1.2 Context of this Thesis
logic, and the unparallel storage capacity of magnetic memory. In this The-
sis, two of the fundamental issues concerning future spintronic devices are
addressed, which are related to the manipulation and the stability of the
magnetisation in a nanoscale ferromagnet. The experiments in this Thesis
are based on III–V diluted magnetic semiconductors.
1.2.1 Electrical manipulation of the magnetisation
A big achievement towards this goal has been the discovery of spin-transfer
torque (STT) [17–19], which occurs when spin-polarised currents traverse
in non-collinear magnetic systems1. STT has been intensively studied for
the past 10 years, and has been exploited to achieve current-induced mag-
netisation reversal [20–23] and current-induced domain-wall motion [24–27]
in submicron ferromagnets. These discoveries could lead to important data
storage devices, such as magnetic random access memory (MRAM) [28, 29]
and racetrack memory [30, 31].
In addition, STT has been employed to drive the magnetisation into pre-
cessional motion [23, 32, 33], which could lead to novel microwave sources
and resonators. Moreover, a ferromagnetic resonance (FMR) technique based
on STT has also been demonstrated [34, 35], which allows characterisation
on the magnetic properties of the nanomagnets. Figure 1.2 illustrates the
concept of STT in a spin-valve structure.
However, what happens when a current travels through a uniform ferro-
magnetic system? Although non-collinear systems could give rise to STT,
uniform ferromagnets are more favourable for applications because of their
simple device structures. Recent theoretical work [37–40] suggests that spin-
torque effects could also be intrinsic to ferromagnetic materials. This leads
to the case of “current-induced torque” (CIT), which appears in uniform fer-
romagnets and provides a radically new mechanism for the manipulation of
the magnetisation in these devices.
1A non-collinear magnetic system contains structures such as spin-valves, magnetic
tunnel junctions (MTJs), or domain walls.
4
1.2 Context of this Thesis
a b
Figure 1.2: An illustration of the spin-transfer torque effect in spin-valve
device structures, adapted from [36]. a, A single magnetic layer with a spin-
polarised electron passing through it. The magnet transmits and scatters
the collinear component of the spin (s‖) and absorbs the transverse compo-
nent (s⊥). The angular momentum of the transverse spin s⊥ is therefore
transferred to the magnetic moments m in the layer and causes them to pre-
cess. b, The spin-polarised electrons are created when an electric current
passes through the pinned layer of a spin-valve, which can then act on the
magnetisation in the free layer.
5
1.2 Context of this Thesis
The current-induced torque in uniform ferromagnets has its origin in two
quantum mechanical effects: the spin-orbit interaction resulting from special
relativity, and the exchange interaction between carrier spins and magnetic
moments.
When a current passes through materials lacking inversion symmetry, a
non-equilibrium accumulation of spins occurs due to the relativistic spin-
orbit coupling. This phenomenon is called “current-induced spin polarisa-
tion” and has been observed in several nonmagnetic semiconductor devices
[41–43]. Furthermore, theoretical work has pointed out that band-structure
effects in semiconductors cause spin-orbit coupling to act like an effective
magnetic field [44, 45], which in turn can affect the electron spins [46–49].
For example, it has been experimentally demonstrated that the oscillating
effective magnetic field resulting from an oscillating current can resonantly
drive the precessional motion of spins, and create electron spin resonance
(ESR) in 2-dimensional electron gases (2DEG)1 or semiconductor quantum
wells [50–54]. In addition, static current-induced effective magnetic fields
have also been experimentally confirmed in semiconductor heterostructures
[55, 56], quantum wells [57], ultrathin metal layers [58, 59] and the interface
between insulating oxides [60].
The above theoretical and experimental studies mostly focus on the current-
induced spin polarisation in nonmagnetic materials. When this effect occurs
in a ferromagnet, the effective magnetic field then combines with the s–d
exchange interaction, which couples the conduction electron spins to the
magnetisation. As a result, a current passing through a uniformly magne-
tised ferromagnetic layer would exert an exchange-mediated effective field on
the magnetisation, thereby producing a torque, as illustrated in Figure 1.3.
This current-induced torque therefore provides direct electrical means for
manipulating the magnetisation in nanomagnets.
1To be more precise, this type of resonance experiment is referred to as “electrical
dipole spin resonance” (EDSR). Since the conventional spin resonance transitions (such
as in ESR) are usually from magnetic dipole transitions, whereas under the influence of
spin-orbit coupling, electrical dipole transitions are allowed [50].
6
1.2 Context of this Thesis
heff
m
hcit
Figure 1.3: Microscopic origin of the s–d-mediated current-induced torque,
adapted from [58]. Conduction electrons moving perpendicular to an elec-
trical field E have their spin (s) tilted by the effective magnetic field (heff),
exerting a torque on the localised moments (m) through the exchange cou-
pling (Jsd). This current-induced torque can be parameterised by a magnetic
field hcit.
The first experimental demonstration of CIT-induced magnetisation re-
versal is reported in 2009 by Chernyshov et al. [39]. The model ferromagnetic
semiconductor (Ga,Mn)As has been chosen for their experiment, because of
its strongly spin-orbit-coupled band-structure [61]. In their experiment, the
magnetisation switching behaviour in a disk-shaped device is altered by the
application of a large direct current (±0.7 mA); moreover, this alteration is
asymmetric and depends on the direction of the current with respect to the
crystal axes, both pointing to the existence of the current-induced torque.
Similar magnetisation reversal experiments have since been reported in 2010
on ultrathin ferromagnetic metals Pt/Co/AlOx [58, 59], in which the current-
induced field is observed to be ∼ 20 times larger than that in (Ga,Mn)As
[58].
Although CIT in ferromagnetic materials has been demonstrated in mag-
netisation reversal experiments, there has been no study on its effect on
the dynamics of the magnetic motion, in contrast to the large number of
ESR experiments in nonmagnetic semiconductors using an oscillating effec-
7
1.2 Context of this Thesis
tive magnetic field [50–54], and the systematic studies of the magnetisation
dynamics under the influence of STT [23, 32–35, 62, 63].
The main focus of this Thesis is the CIT-FMR experiment described in
Chapter 4. It is the first experimental demonstration of an oscillating current-
induced torque in ferromagnetic systems. FMR in microstructures patterned
on magnetic semiconductors (Ga,Mn)As and (Ga,Mn)(As,P) are driven by
a microwave frequency CIT, and coherent magnetic motion is detected elec-
trically using a frequency mixing effect. By analysing the FMR lineshape,
the magnitude and orientation of the current-induced effective field hcit have
been determined with unprecedented accuracy (within 5 µT). Moreover, the
effective field has been observed with signatures from different types of spin-
orbit interactions (Dresselhaus and Rashba), which cannot be distinguished
in the magnetisation switching experiments [39, 64].
Further measurements also reveal that depending on whether the epilayer
is compressively or tensile-strained, the directions of the CIT fields change
sign. And the strength of the CIT fields can also be influenced by changes in
the carrier concentration (by varying Mn doping levels). In addition, as the
structures are made even smaller, significant strain-relaxation occurs which
reduces the magnitude of hcit. These observations suggest possible methods
for customising the spin-orbit coupling in future spintronic devices.
1.2.2 Mapping the magnetic anisotropy in individual
nanomagnets
In modern data storage media, the digital information, encoded as 0s and
1s in binary format, is represented by the orientations of the magnetisation
in nanomagnets. The magnetic anisotropy determines the stable directions
of the magnetisation as well as the energy barrier that confines it. There-
fore it is vital for magnetic memory units to have well-defined anisotropy, so
that the binary information can be unambiguously defined, and is able to
withstand the thermal fluctuations in the memory units’ operational envi-
ronment. Furthermore, it would be highly beneficial to be able to engineer
the magnetic anisotropy to serve the application needs.
8
1.2 Context of this Thesis
Patterning magnetic materials into nanoscale devices provides a control-
lable way to modify the magnetic anisotropy. For example, in magnetic
metals, device patterning produces the shape anisotropy, which becomes im-
portant for devices with size less than a few microns. In DMSs, the mag-
netocrystalline anisotropy is strain-related, and this is greatly modified in
nanoscale devices due to the strain-relaxation effect [26, 65–67].
Ferromagnetic resonance (FMR) spectroscopy is a long-standing tech-
nique for characterising magnetic systems. It has been applied since 1950s to
many metallic materials (for reviews, see Ref. [68–70]) and has now been ex-
tended to the study of diluted magnetic semiconductors such as (Ga,Mn)As
and (Ga,Mn)(As,P) [71–77]. Among the parameters which can be learned
with this technique are the magnetocrystalline anisotropy constants, the g-
factor, the Gilbert damping factor, the spin stiffness (via spin wave excitation
detection), and even the homogeneity of the magnetic films [78].
So far FMR-based studies on the magnetic anisotropy in (Ga,Mn)As have
focused on whole wafers, and only one publication has been on the properties
of individual submicron devices [67]. This is because the conventional FMR
spectroscopy lacks the sensitivity to detect the magnetic motion in small de-
vices1. However being able to employ the power of FMR spectroscopy in the
study of individual magnetic micro- and nanostructures is highly desirable.
Several different approaches to device-scale FMR have recently been in-
troduced. A combination of microwave excitation and local detection is re-
quired. The excitation by a microwave magnetic field can be performed in a
standard cavity [79] or a waveguide [80–82]. Alternatively a local excitation
technique can be employed, for example using a current-carrying waveguide
closely aligned to the magnetic nanostructure [83–87]. In addition, for sam-
ples in a spin-valve geometry, the spin-transfer torque can be used to drive the
magnetisation [34, 35, 63]. Device-level detection of FMR can be achieved
1The magnetic anisotropy can also be determined by measuring the saturation mag-
netisation along different crystalline axes using a superconducting quantum interference
device (SQUID). However, this technique also suffers from sensitivity issues when measur-
ing small structures.
9
1.2 Context of this Thesis
electrically by a frequency mixing effect based on the anisotropic magne-
toresistance (AMR) [79–81, 83–87] or tunnelling magnetoresistance (TMR)
[34, 63], which allows the sample itself to be the detector (see Section 2.4.3
for details). Furthermore, magnetisation precession can be detected using fo-
cused magneto-optical effects or x-rays [67, 88]. As well as these device-level
measurements of FMR, scanning probe FMR techniques based on magnetic
force microscopy are available [89], allowing spatial variations of magnetic
anisotropy to be mapped within materials.
In Chapter 3, an all-electrical, on-chip driven FMR experiment on sub-
micron (Ga,Mn)As stripes is described, using a method similar to that used
by Costache et al. [83, 84]. With this technique, the magnetisation of the
ferromagnet is driven by a local microwave magnetic field, which is gener-
ated electrically by a waveguide placed next to the device. The magnetic
motion is also detected electrically by utilising the frequency mixing effect.
The magnetic anisotropy data measured on these nano-bars corroborate the
published results by Hoffmann et al. [67], who have employed an electrically-
driven, optically-detected FMR setup. In addition, analysis on the FMR
lineshape allows one to determine the orientation of the microwave excita-
tion field, and this proves to be crucial to the understanding of the CIT-FMR
experiments.
The CIT-FMR technique, established in Chapter 4, is a highly scalable
method for the characterisation of individual ferromagnetic microstructures.
Moreover, since the magnetisation is driven by the microwave frequency
CIT which appears inside the device, no additional waveguide structure
is required, greatly simplifying the fabrication process (only a rectangular
mesa structure with bond pads is needed). The magnetic anisotropy in the
(Ga,Mn)As micro- and nano-devices are characterised using CIT-FMR (given
in Chapter 4.5.4), and the observations are consistent with earlier studies us-
ing SQUID and magneto-transport measurements [26, 65, 66].
Besides (Ga,Mn)As, a newly synthesised III–V group ferromagnetic semi-
conductor (Ga,Mn)(As,P) has drawn research interests in the field of semi-
conductor spintronics [77, 90–94]. Because of the crystal structure, the
(Ga,Mn)(As,P) epilayers can be grown under tensile-strain [77], different
10
1.2 Context of this Thesis
from the majority of the (Ga,Mn)As wafers being studied which are under
compressive-strain [1, 95]. However, so far there has been no study on the
magnetic properties of nano-devices patterned on (Ga,Mn)(As,P).
In Chapter 5, CIT-FMR is employed to investigate the magnetic anisotropy
in individual (Ga,Mn)(As,P) micro- and nanostructures for the first time. It
has been observed that the patterning-induced strain-relaxation has the re-
verse effect on the magnetic anisotropy in these samples, compared with
(Ga,Mn)As devices under compressive-strain [26, 65–67].
11
Chapter 2
Background
This Thesis explores the dynamic behaviour of the magnetisation under the
influence of a driving torque oscillating at frequencies close to the magneti-
sation’s natural resonance frequency. The experiments have been performed
on the diluted magnetic semiconductors (Ga,Mn)As and (Ga,Mn)(As,P) and
their characteristics are introduced in Section 2.1. A property related to all
ferromagnetic materials, the magnetic anisotropy, is then discussed in Sec-
tion 2.2.
Methods for characterising ferromagnetic systems include dc transport
measurements and magnetisation dynamics analysis, which are reviewed in
Sections 2.3 and 2.4, respectively. Finally, the effect of spin-orbit coupling in
semiconductors is introduced in Section 2.5.
2.1 III–V diluted magnetic semiconductors
2.1.1 (Ga,Mn)As and its basic properties
(Ga,Mn)As is created by doping the host GaAs with Mn during the crystal
growth. In order to reach the high concentrations of Mn ions required for
creating ferromagnetism in these materials (∼ 1%), the crystal growth condi-
tion has to be driven away from its equilibrium temperature, using a so-called
low-temperature molecular-beam-epitaxy technique (LT-MBE) [95].
12
2.1 III–V diluted magnetic semiconductors
Figure 2.1: (Ga,Mn)As lattice with Mn atoms in substitutional (MnGa) and
interstitial (MnI) positions, adapted from [61]. The latter is a common type
of defects in (Ga,Mn)As epilayers.
X-ray diffraction experiments reveal that the (Ga,Mn)As crystal has a
zinc-blende structure, much like its GaAs host (Figure 2.1). This is because
when Mn dopants are introduced in the crystal growth process, they tend
to occupy substitutional positions at cation sites and participate in crystal
bonding with As similar to the substituted Ga atoms. Substitutional MnGa
atoms provide local magnetic moments with zero angular momentum and
spin s = 5/2. Because of the missing valence 4p electron, the MnGa atoms
act as acceptors providing one hole per ion [61].
In the absence of holes, the magnetic interaction among Mn has been
shown to be antiferromagnetic in fully carrier compensated (Ga,Mn)As [1],
which indicates that the ferromagnetic order is hole-induced. The doping
level of Mn ions is extremely important, and it has been observed experi-
mentally that ferromagnetism arises when the concentration of Mn reaches
∼ 1% [96], and the system is near the Mott insulator-to-metal transition1 [61].
At these large Mn concentrations, the localisation length of impurity-band
1The Mott insulator-to-metal transition happens when rc, the average distance be-
tween Mn impurities (or between holes bound to the Mn ions), is comparable in size to
the impurity effective Bohr radius a∗ = E~/(m∗e2), where m∗ is the effective mass near
the top of the valence band [61].
13
2.1 III–V diluted magnetic semiconductors
states is extended to a degree that allows them to mediate ferromagnetic
exchange interaction between Mn moments, even though the moments are
dilute.
2.1.2 Defects in (Ga,Mn)As and their treatment
The non-equilibrium growth of (Ga,Mn)As epilayers by LT-MBE causes a
substantial number of meta-stable impurity states to form. Two common
types of defects are Mn ions occupying the tetrahedral interstitial sites (MnI)
rather than the substitutional Ga sites (see Figure 2.1), and As atoms on
cation sites (AsGa, antisite defects). The number of such unintended defects
increases with higher Mn doping because of the tendency of the material to-
wards self-compensation, even under non-equilibrium growth conditions [61].
Both impurities act as donors and can have severe impact on the electrical
and magnetic properties of the ferromagnetic films. For example, the intersti-
tial MnI ions contribute to local moment compensation, and cause structural
changes in the crystal; whereas the antisite AsGa atoms can result in hole
compensation [61].
These defects, especially the interstitial MnI atoms, can be removed by
post-growth annealing. Early studies indicate that annealing at tempera-
tures ∼ 400oC reduces the number of electrically and magnetically active
ions in the (Ga,Mn)As layer [97]. However, it is later found that annealing
at temperatures close to the growth temperature (∼ 200oC) results in the
removal of MnI ions by out-diffusion and passivation at the sample surface,
and leads to an increase in the conductivity and Curie temperature TC of the
ferromagnet film [11, 98–101]. The effect of annealing is more pronounced
in (Ga,Mn)As wafers with more interstitial MnI ions, i.e. with higher Mn
doping.
Figure 2.2a shows the effect of annealing on the magnetisation1 M and
resistivity ρ of a (Ga,Mn)As sample [100]. Temperature-dependent measure-
ment on the spontaneous magnetisation M using superconducting quantum
interference device (SQUID) demonstrates that for the as-grown sample, M
1M represents the magnitude of the magnetisation vectorM in a system, i.e.M = |M|.
14
2.1 III–V diluted magnetic semiconductors
T (K)
ρ
Ω
(m
c
m
)
a b
Figure 2.2: The effect of annealing on the Curie temperature of (Ga,Mn)As
and (Ga,Mn)(As,P) films. a, The magnetisation and resistivity as a func-
tion of temperature for a (Ga1−x,Mnx)As sample with x = 0.0597 (circles,
as-grown; triangles, annealed), adapted from [100]. b, The temperature-
dependent resistivity measured on a (Ga0.94,Mn0.06)(As0.9,P0.1) sample.
vanishes for temperatures above ∼ 75 K; while after being annealed at 250oC
for 2 h, this temperature rises to 110 K [99]. The resistivity of the sample ρ
is also found to be lowered in the annealed sample.
Figure 2.2a also demonstrates that there exists a correlation between the
resistivity ρ of the (Ga,Mn)As sample and its temperature T . This can be
understood as a result of the competition between the increase in phonon
energy and decrease in ferromagnetic order as the sample temperature is
raised. It has been confirmed that the origin of the broad peak seen in
the ρ(T ) curve is due to a singular behaviour as T → T+C [102], and this
singularity can be found by differentiating ρ with respect to T . It is observed
in SQUID measurements that the singularity in ∂ρ/∂T coincides with TC ,
especially in annealed, defect-free samples [102]. This is particularly useful
for determining TC in patterned samples, whose small dimensions usually
prevent their magnetisation from being directly measured by SQUID.
In another magnetic semiconductor (Ga,Mn)(As,P), post-growth anneal-
ing treatment becomes more crucial. It has been found that the as-grown
(Ga,Mn)(As,P) epilayers exhibit low TC (below 30 K) and are electrically in-
15
2.2 Magnetic anisotropy
θH
θ
φH
φ
y // [010]
x // [100]
z // [001]
H0
M
Film plane
Figure 2.3: The coordinate system used for defining the magnetic anisotropy
energies. Here H0 is the external magnetic field, andM is the magnetisation
vector in the system.
sulating at low temperatures, both pointing towards high degrees of carrier
compensation in these wafers [92]. After annealing at 180oC for 48 h, the
conductivity at low temperatures dramatically increases and TC rises above
100 K, as illustrated in Figure 2.2b: The resistivity ρ at 6 K drops by 8 times
in the annealed sample, accompanied by the recovery of TC .
2.2 Magnetic anisotropy
Magnetic anisotropy describes the preference for the magnetisation to lie in
a particular direction with respect to crystallographic axes in a sample. An
energy term of this kind is referred to as the magnetic anisotropy energy. In
ferromagnetic semiconductors, the anisotropy can be caused by sample shape,
crystalline symmetry and stress. Figure 2.3 defines a coordinate system used
for describing the various anisotropy energies.
A quantitative measure of the strength of the magnetic anisotropy is
the field Hi needed to saturate the magnetisation in a hard direction, and
Hi is called the “anisotropy field”. The anisotropy energy per unit volume
(i.e. energy density) needed to saturate a magnetic material in a particular
16
2.2 Magnetic anisotropy
direction is given by the following generalised equation [103]:
Ki = µ0
∫ Ms
0
H(M)dM
1st order−→ µ0MsHi
2
(2.1)
Here Ms is the saturation magnetisation, and Ki has the unit of J/m
3. The
concept of the anisotropy field Hi and its linear relation to the anisotropy
energy density Ki are used throughout this Thesis.
2.2.1 Shape anisotropy
The shape anisotropy, also called magnetic dipolar anisotropy, describes the
relation between the demagnetisation energy and the shape of a sample. It is
caused by the long-range magnetic dipolar interactions. In a thin magnetic
film, the magnetic energy density due to shape anisotropy can be written as
[104]:
ES =
1
2
µ0M
2
s cos
2 θ (2.2)
Equation 2.2 suggests that in-plane alignment of magnetic moments usu-
ally prevails in ferromagnetic thin films (i.e. θ = 90◦) as it leads to a lower
energy state. However, for DMSs such as (Ga,Mn)As and (Ga,Mn)(As,P),
perpendicular-to-plane magnetic orders have also been reported [1, 92], due
to the much stronger strain-dependent magnetocrystalline anisotropy present
in these materials.
2.2.2 Magnetocrystalline anisotropy
Besides the shape anisotropy, the direction of the spontaneous magnetisation
in ferromagnetic systems also has preference over particular crystalline axes.
This is described as the magnetocrystalline anisotropy. Two common types
are the uniaxial and cubic anisotropy, with the latter sometimes referred to
as biaxial anisotropy in epitaxially-grown (Ga,Mn)As and (Ga,Mn)(As,P)
films.
17
2.2 Magnetic anisotropy
Uniaxial anisotropy The preference of the magnetisation in a ferromag-
net over a single axis is described by the uniaxial anisotropy. The energy
in a system with uniaxial anisotropy is minimised when the magnetisation
lies along the easy axis. For example, cobalt is a hexagonal crystal and the
hexagonal axis (z-direction in Figure 2.3) is the direction of easy magnetisa-
tion. The uniaxial anisotropy energy density in cobalt (EU) can be described
by a power series [105]:
EU = KU0 +KU1 sin
2 θ +KU2 sin
4 θ +KU3 sin
6 θ + . . . (2.3)
where KUi are uniaxial anisotropy energy constants. Note that KU0 does not
carry any anisotropy properties as it is independent of the orientation of the
magnetisation. KU1 > 0 indicates that the easy axis is along the hexagonal
axis. Careful analysis of the magnetisation orientation curves indicates that
for most purposes, it is sufficient to keep only terms up to KU2 [103].
Cubic anisotropy In a cubic system, such as iron, the cubic edges define
the directions of easy magnetisation. There is an additional energy cost if the
sample is magnetised in an arbitrary direction. The energy density associated
with this, referred to as the cubic anisotropy (EC), is described by powers of
the the direction cosines of M along the three cubic edges (α1, α2, α3). Due
the the high symmetry of cubic crystals, a low-order approximation of EC is
[103]:
EC = KC0 +KC1
(
α21α
2
2 + α
2
2α
2
3 + α
2
3α
2
1
)
+KC2
(
α21α
2
2α
2
3
)
+ · · · (2.4)
Here αi = Mi/Ms. It is known from experiments that higher order terms in
Equation 2.4, in most cases even the KC2 term, are negligible [106].
When KC1 > 0, such as in Fe and Co, the first term in Equation 2.4
becomes minimum in the [100] direction, indicating that this is the magnetic
easy axis; whereas for KC1 < 0, such as in Ni, the easy axis is along the [111]
direction. Using the trigonometric functions of spherical coordinates (θ, ϕ),
the coefficient of KC1 in Equation 2.4 reduces to [103]:
sin4 θ cos2 ϕ sin2 ϕ+ cos2 θ sin2 θ =
1
4
(sin4 θ sin2 2ϕ+ sin2 2θ) (2.5)
18
2.2 Magnetic anisotropy
2.2.3 Magnetic anisotropy in (Ga,Mn)As systems
The magnetocrystalline anisotropy of bulk (Ga,Mn)As is cubic due to its
zinc-blende structure [107]. The magnetic easy axes are along the three
symmetrically equivalent <100> or <111> crystal directions. However, the
pure bulk properties do not apply to the epitaxially-grown wafers, in which
the substrate plays an important role in defining the magnetic anisotropy.
In particular, the vast majority of work has been done on (Ga,Mn)As films
grown on [001]-oriented substrates, and their properties are the focus of this
Section.
The (Ga,Mn)As epilayers are strained in the growth direction as a re-
sult of lattice mismatch and exhibit an additional perpendicular-to-plane
anisotropy. For example, for a (Ga,Mn)As film grown on a substrate with a
smaller lattice constant (such as GaAs), lattice mismatch causes the ferro-
magnetic layer to be compressively-strained, leading to uniaxial anisotropy
with an in-plane easy axis [95]. Whereas when a InGaAs buffer layer is
used which has a larger lattice constant than (Ga,Mn)As, the magnetic film
grown on it is tensile-strained and has a perpendicular-to-plane easy axis
[1]. More recently, it has been demonstrated that by the incorporation of
phosphorus atoms at the As sites, the resulting (Ga,Mn)(As,P) epilayers can
become tensile-strained for P concentrations higher than 6%, leading to a
perpendicular-to-plane easy axis [77, 92–94].
The in-plane anisotropy of compressively-strained (Ga,Mn)As layers is
dominated by the competition between the biaxial anisotropy along the cubic
edges1 and the uniaxial anisotropy with easy axis along either the [11¯0] or
[110] directions [107]. In some materials an additional uniaxial anisotropy
along [010] is also observed [108]. The anisotropy energies are temperature-
dependent: The biaxial anisotropy scales with the magnetisation as M4,
while the uniaxial anisotropy scales as M2. As the temperature approaches
TC and M decreases, the dominant anisotropy term changes from biaxial to
uniaxial [109]. It has also been observed that low temperature annealing
1For (Ga,Mn)As epilayers grown on [001]-oriented GaAs substrates, the biaxial
anisotropy leads to magnetic easy axes along the [100] and [010] directions.
19
2.2 Magnetic anisotropy
(190oC) causes the easy axis of the uniaxial anisotropy to rotate from [11¯0]
to [110] direction [110].
The magnetic anisotropy can be characterised by measuring the rema-
nent magnetisation along various crystalline orientations using SQUID. Al-
ternatively, it can be inferred from magneto-transport measurements (see
Section 2.3), from magneto-optical experiments [111], or from FMR spec-
troscopy (see Section 2.4).
2.2.4 Control of magnetic anisotropy
It is important from an application perspective to be able to manipulate the
magnetic anisotropy, and therefore the direction of the magnetisation, of a
ferromagnetic device via a nonmagnetic parameter. The change in magnetic
anisotropy caused by lattice mismatch during the crystal growth phase, de-
scribed in the previous Section, cannot satisfy the need of a spintronic device
in an operational environment. Several different approaches towards device-
scale control of magnetic anisotropy is reviewed in this Section.
From the large impact of growth-induced strain on the magnetocrystalline
anisotropy, it is conceivable that adding mechanical strain via a piezoelec-
tric transducer can achieve electrical control of the magnetic anisotropy in
(Ga,Mn)As devices. This has been experimentally demonstrated: Upon the
application of a high voltage (∼ 150 V), the mechanical strain generated by
a piezoelectric transducer is enough to cause rotation of the magnetisation in
a (Ga,Mn)As sample attached to it [5–8]. Figure 2.4 demonstrates the effect
of mechanical stress on the magnetisation switching curves measured on a
(Ga,Mn)As device [7].
Since the ferromagnetism in (Ga,Mn)As is carrier-mediated, it is there-
fore expected that by changing the carrier (hole) concentration, the magnetic
anisotropy can also be affected. Although (Ga,Mn)As behaves like a dirty
metal at Mn concentrations of a few percent, the carrier density in these
20
2.2 Magnetic anisotropy
a b
μ0 0H (mT)
Figure 2.4: Magnetisation control using a piezoelectric transducer, adapted
from [7]. a, Schematic of the experimental setup. The (Ga,Mn)As device is
patterned into a Hall bar for transport measurements (see Section 2.3). b,
Transverse AMR measured across the device while the external field H0 is
swept. The curves are shifted by charging the transducer.
materials is usually 2 – 3 orders of magnitude lower than in metals1, making
them suitable for electrical gating. These experiments are first performed on
(In,Mn)As and (Ga,Mn)As samples using a metal-insulator-semiconductor
device structure (i.e. a field-effect transistor, or FET) [9, 10, 112, 114]. The
application of an electric field causes hole depletion/accumulation in the mag-
netic semiconductor, changing its anisotropy and rotating the direction of
the magnetisation. Furthermore, sizeable carrier change in (Ga,Mn)As with
a moderate electric field has also been demonstrated in an all-semiconductor
FET device, with a n-type, highly Si-doped GaAs layer acting as the gate
electrode [11, 12]. Figure 2.5 demonstrates the effect of gating on a (In,Mn)As
FET device [112].
The above two methods are in principle compatible with both whole
wafers and individual devices. In addition, it is also noticed that in patterned
(Ga,Mn)As devices, the lithographically-defined edges allow the lattice to
1In (Ga,Mn)As with Mn doping of a few percent, the atomic concentration of Mn is in
the order of nMn ∼ 1019 − 1021 cm−3 (1% of Mn corresponds to nMn of 2.2× 1020 cm−3),
and the hole density is p ∼ 1018 − 1020 cm−3. Note that nMn and p are not the same
because of carrier compensation. In metals the charge density is usually 1022 cm−3 [113].
21
2.2 Magnetic anisotropy
a
b
Figure 2.5: Field-effect control of the hole-induced ferromagnetism in a
(In,Mn)As FET, adapted from [112]. a, Schematic of the role of electri-
cal gating on the change in carrier concentration in (In,Mn)As. The arrows
schematically show the magnitude of the magnetisation. b, The Hall resis-
tance RHall (see Section 2.3.2) across the device under different gate biases.
Application of VG = 0, +125 and −125 V results in qualitatively different
field dependence of RHall. Inset: The same curves shown at higher magnetic
fields.
22
2.2 Magnetic anisotropy
ξ
Figure 2.6: Simulation results of the spatial dependence of the strain coef-
ficient ξ due to lattice-relaxation in a narrow bar with thickness ÷ width =
0.4 and compressive growth-strain, adapted from [115].
partially relax and thus reduce the total strain on the sample. Numerical
simulations have shown that this effect occurs over a length scale of a few
hundred nm [26, 66, 115], and magneto-optical Kerr microscopy has provided
direct observations [67]. In bar-shaped (Ga,Mn)As samples, patterning-
induced strain-relaxation introduces an additional uniaxial anisotropy, with
easy axis perpendicular to the direction of strain-relaxation (i.e. along the
long axis of the bar). In micrometre-wide devices, this additional anisotropy
can act to rotate the magnetic easy axes from their as-grown positions [26];
in very narrow bars (submicron), this anisotropy becomes more pronounced
and has been demonstrated to dominate over the intrinsic anisotropy ener-
gies, over all temperatures up to TC [65, 67]. Figure 2.6 shows the simula-
tion results of strain-relaxation in a compressively-strained (Ga,Mn)As stripe
[115].
So far the influence of strain-relaxation on the magnetic anisotropy has
only been investigated on compressively-strained (Ga,Mn)As devices. In
Chapter 5, FMR experiments on individual tensile-strained (Ga,Mn)(As,P)
nano-bars confirm that the growth-strain has a major impact on the strain-
relaxation-induced anisotropy, leading to reversal of its easy axis.
23
2.3 Magneto-transport
2.3 Magneto-transport
Magneto-transport in ferromagnetic systems can be broken down into ordi-
nary effects depending on the applied magnetic field alone, which arise from
the Lorentz force acting on the charge carriers; and extraordinary effects
which depends on the relative orientations between the magnetisation and
the current, and originate from the spin-orbit and exchange interactions.
Two important transport effects are introduced in this Section, namely
the anisotropic magnetoresistance (AMR) and the anomalous Hall effect
(AHE). Their measurements provide means to detect the direction of the
magnetisation vector M, to measure the magnitude of the magnetic field re-
quired for switching M from one easy axis to another (switching field), and
to quantify the magnetic anisotropy constants.
Figure 2.7a sketches a Hall bar structure which is commonly used for
four-probe transport measurement of AMR and AHE: While the current
flows from 8 to 4, the longitudinal and transverse voltages Vxx and Vxy are
measured. This allows the respective resistivity ρxx and ρxy to be calculated.
Alternatively, two-probe measurement (using only contacts 8 and 4) is also
commonly performed for determining the longitudinal resistivity ρxx, pro-
vided that resistance from the contacts is not important. Figure 2.7b shows
a variation of the two-probe measurement, in which the current flows radially
from the central contact to the edge in all directions (a Corbino disk struc-
ture). This removes the crystalline effect from the transport measurement
results.
2.3.1 Anisotropic magnetoresistance
The phenomenon of magnetoresistance (MR) was discovered more than 150
years ago by Lord Kelvin, when he found experimentally that iron and nickel
samples exhibit higher resistivity ρ‖ when the magnetisation M is saturated
in the direction of a current flow (M ‖ I), and lower resistivity ρ⊥ when
M is saturated perpendicular to the current (M ⊥ I). AMR is defined as
the relative change in resistivity between the two configurations: AMR =
24
2.3 Magneto-transport
Vxx Vxy
1
7 6 5
4
32
8 j
a b
Figure 2.7: Sketches of standard device structures for transport measure-
ments: a, Hall bar and b, Corbino disk (The yellow parts represent metal
contacts).
(ρ‖ − ρ⊥)/ρ‖. In (Ga,Mn)As it is found that ρ‖ < ρ⊥ [116, 117], opposite to
that observed in metal systems.
For an isotropic sample in a single domain state, the longitudinal and
transverse resistivity ρxx and ρxy can be expressed as [116, 118]:
ρxx = ρ⊥ − (ρ⊥ − ρ‖) cos2 ϑ (2.6)
ρxy = −(ρ⊥ − ρ‖) cosϑ sin ϑ (2.7)
where ϑ is the angle between M and I. Note that the transverse AMR is
also referred to as planar Hall effect [118].
Figure 2.8 shows the typical resistance measured on a (Ga,Mn)As Hall
bar when a constant external magnetic field H0 is rotated in-plane with
the sample. In this case the applied field H0 is much stronger than the
saturation magnetisation Ms of the material so that M ‖ H0 for all ϑ. The
angle dependence described by Equations 2.6 and 2.7 is clearly observed.
When the single domain assumption is not applicable, the measured AMR
has two contributions: spin-orbit interaction and non-cubic symmetry in-
duced by epitaxial strain [119]. This leads to observations that the ρxx(ϑ)
and ρxy(ϑ) curves not only depend on the angle between M and I (non-
crystalline AMR), but also on the directions of I and M with respect to
the crystal axes (crystalline AMR) [117, 120]. For systems exhibiting cubic
25
2.3 Magneto-transport
ϑ (deg.)
ρ
ρ
μ
Ω
c
m
x
x
0
-
(
)
ρ
μ
Ω
c
m
x
y
(
)
a
b
Figure 2.8: Measurements on the longitudinal and transverse resistivity on
a Hall bar (3.4% Mn) in a rotating magnetic field of 0.6 T at 4.2 K, adapted
from [116]. a, The longitudinal resistivity ρxx−ρ0, where ρ0 = (ρ‖+ρ⊥). The
solid line is the fitted results according to Equation 2.6. b, The transverse
resistance ρxy and fitted results according to Equation 2.7 (solid line).
26
2.3 Magneto-transport
[100] and uniaxial [110] anisotropy, the AMR components can be described
as [120, 121]:
ρxx − ρave
ρave
= CI cos 2ϑ+ CU cos 2φ (2.8)
+CC cos 4φ+ CI,C cos(4φ− 2ϑ)
ρxy − ρave
ρave
= CI sin 2ϑ− CI,C cos(4φ− 2ϑ) (2.9)
where φ is the angle between M and the [110] crystal direction, and ρave
is the average value of the longitudinal resistivity as the magnetic field is
rotated through 360o. CI represents the non-crystalline AMR coefficient, CU
and CC are the uniaxial and cubic crystalline AMR coefficients respectively,
and CI,C is a crossed non-crystalline/crystalline term. The crystalline AMR
components in (Ga,Mn)As devices have been experimentally identified and
extensively studied in Ref. [120–122].
Experimentally the transverse resistivity ρxy has been used to determine
the magnetic anisotropy constants: In the case that the applied field is
smaller than Ms, the magnetisation vector M does not generally follow H0,
but instead stays in a direction which minimises the total free energy of the
system. Figure 2.9 illustrates the typical results from such measurements
[118].
In a (Ga,Mn)As device, the primary contributions to the total energy
are the magnetocrystalline anisotropy energies. Combining Equations 2.3,
2.4 and 2.5, and considering only the in-plane anisotropy (i.e. θ = 90o in
Figure 2.3), the free energy density E is given as [10, 116, 118]:
E = µ0
MH4‖
8
sin2 2ϕ+µ0
MH2‖
2
sin2(ϕ−45o)−µ0MH0 cos(ϕH−ϕ) (2.10)
Here M = |M|; H4‖ and H2‖ are the first order in-plane biaxial and uniaxial
anisotropy fields respectively, they are defined in terms of the anisotropy
energies Ki according to Hi = 2Ki/µ0Ms (cf. Equation 2.1). [110] is the
easy axis when H2‖ > 0. The third term in the expression corresponds to
the Zeeman energy. The small shape anisotropy associated with the device
is neglected.
27
2.3 Magneto-transport
Field angle (deg.)
ρ
x
y
Figure 2.9: The transverse resistivity ρxy on a (Ga,Mn)As Hall bar in a
group of fixed-magnitude field rotation measurements, adapted from [118].
At large H0, the magnetisation is saturated in the direction of the field, i.e.
M ‖ H0, and the measured ρxy(ϑ) agrees with Equation 2.7. At small H0
(< 1000 Oe), M no longer follows H0, instead only stays along the easy axes
where the total energy is minimum, and the rotation of H0 causes M to
switch from one easy axis to the next. The hysteresis loops of magnetisation
switching are clearly observed (The red and blue lines are from clockwise and
counterclockwise rotations respectively).
28
2.4 Magnetisation dynamics
By determining the angle ϕ between M and the crystalline axis using
Equation 2.7, and then imposing the conditions ∂E/∂ϕ = 0 and ∂2E/∂2ϕ >
0 on Equation 2.10 (i.e. M always lies in the direction which minimises the
total energy in the system), the in-plane anisotropy constants H4‖ and H2‖
can be determined.
2.3.2 Anomalous Hall effect
The Hall resistivity measured in a magnetic semiconductor can be described
as [107]:
ρHall = RoB +RsM (2.11)
where the first term is from the ordinary Hall effect with the ordinary Hall
coefficient Ro, caused by the deflection of charge carriers by a magnetic field;
the second term is from the anomalous Hall effect (AHE), which is described
by the product of the magnetisation M and the spontaneous Hall coefficient
Rs arising from the spin-orbit interaction.
Careful AHE measurements can be used to determine the saturation mag-
netisation Ms in (Ga,Mn)As samples, and this allows determination of TC by
examining the temperature-dependence of Ms [117, 123], as demonstrated in
Figure 2.10.
2.4 Magnetisation dynamics
So far only the response of the magnetisation to a dc magnetic field has been
discussed. It is therefore natural to ask: How does M respond when it is un-
der the influence of an oscillation magnetic field, with frequency approaching
the natural resonance frequency of M?
2.4.1 Motion of magnetic moments
A classical approach to study the precession of magnetic moments in an
external field is to consider a macroscopic magnetisation vector M which
includes all individual magnetic moments (macrospin model). This is valid
29
2.4 Magnetisation dynamics
Figure 2.10: The anomalous Hall effect measurements, adapted from [107].
Magnetic field dependence of the Hall resistivity (ρHall) in a (Ga,Mn)As de-
vice (5.3% Mn) at different temperatures. The saturation magnetisation Ms
is determined for each curve. Inset: Plotting the temperature-dependence of
Ms reveals the Curie temperature TC .
30
2.4 Magnetisation dynamics
Heff
M
-M x Heff
Heff
M
-M x Heff
M x ( M/ t)∂ ∂
a b
Figure 2.11: Schematic of magnetisation dynamics. a, The magnetisation
vector M (brown) precesses counterclockwise around the effective magnetic
field Heff (green) following the LL equation (grey trace, described by Equa-
tion 2.12). b, Adding the phenomenological damping term (α/Ms)M ×
(∂M/∂t) (orange), described in the LLG equation (Equation 2.14), causes
the magnetic motion to relax towards the direction of Heff.
in ferromagnetic materials since all magnetic moments are correlated. A
dynamic model for the motion of M was proposed by Landau and Lifshitz
(LL) in 1935:
∂M
∂t
= −γµ0M×Heff (2.12)
Here Heff is the effective magnetic field. It consists of the applied magnetic
field H0 and the internal fields (such as anisotropy, demagnetisation, etc.),
and determines the equilibrium orientation of M.
According to the LL model,M is connected to a total angular momentum
which is subject to a torque, resulting in its precessional motion. However,
the behaviour predicted by Equation 2.12 pictures an undamped precession
with a constant cone angle, and is not consistent with experimental observa-
tions: The magnetisation always aligns with the direction of minimal energy
after a finite time. Landau and Lifshitz amended this inconsistency by adding
a phenomenological dissipation term:
∂M
∂t
= −γµ0M×Heff − λ
γM2s
M× (M×Heff) (2.13)
31
2.4 Magnetisation dynamics
where λ > 0 is a phenomenological damping constant and has the dimen-
sion of frequency. In Equation 2.13 the magnitude of the magnetisation is
conserved according to |M| =Ms.
While Equation 2.13 can account for experimental observations, the most
commonly used equation of motion is the Landau-Lifshitz-Gilbert (LLG)
equation. It consists of the same precessional term as in the LL equation
(Equation 2.12) but with a different damping term. This phenomenologi-
cal dissipation term after Gilbert (1955) models a “viscous” damping and
depends on the time derivative of the magnetisation. The LLG equation is
written as [124]:
∂M
∂t
= −γµ0M×Heff + α
Ms
M× ∂M
∂t
(2.14)
where α is the phenomenological dimensionless Gilbert damping parameter.
2.4.2 Ferromagnetic resonance
Ferromagnetic resonance (FMR) is a condition in which steady-state preces-
sion ofM occurs in a ferromagnetic system. To counter the effect of damping,
the magnetisation is subject to a driving torque usually from an ac magnetic
field.
FMR is closely related to two other phenomena, namely the electron
spin resonance (ESR) and nuclear magnetic resonance (NMR). In these two
situations, magnetic resonance occurs when the angular frequency (ω) of
the ac magnetic field and the static magnetic field H0 satisfy the following
condition:
ω = −γµ0H0 (2.15)
where γ is the gyromagnetic ratio of the electrons (ESR) or the nuclei (NMR).
The steady-state precession of magnetic moments in an external magnetic
field is known as Larmor precession.
In a ferromagnetic system, the presence of the internal magnetic field
makes the condition for Larmor precession more complex. In particular, the
rotational axis of M now lies along the direction the effective magnetic field
32
2.4 Magnetisation dynamics
Heff (Figure 2.11b), which is determined by the functional derivative of the
total free energy density E with respect to M:
Heff = − 1
µ0
∇ME (2.16)
For (Ga,Mn)As epilayers under tetragonal distortion, the total free energy
density1 E is expressed as (see Figure 2.3 for the coordinate system) [74]:
E =
1
2
µ0M
{
−2H0[cos θ cos θH + sin θ sin θH cos(ϕ− ϕH)] (2.17)
+M cos2 θ −H2⊥ cos2 θ − 1
2
H4⊥ cos
4 θ
−1
2
H4‖
1
4
(3 + cos 4ϕ) sin4 θ −H2‖ sin2 θ sin2
(
ϕ− π
4
)}
Here the first term describes the Zeeman energy; the second term is the de-
magnetisation energy (shape anisotropy, as in Equation 2.2); and the remain-
ing are magnetocrystalline anisotropy energy terms, in which H2⊥ and H4⊥
represent the perpendicular-to-plane uniaxial and cubic anisotropy fields, re-
spectively (cf. Equations 2.3, 2.4 and 2.5); H2‖ and H4‖ are the in-plane
uniaxial and cubic anisotropy fields, respectively. As usual, the anisotropy
fields Hi are defined in terms of the energies Ki according to Hi = 2Ki/µ0Ms.
In the case of submicron devices, the energy from the additional uniaxial
anisotropy caused by strain-relaxation (see Section 2.2.4) can be described
as [67]:
EU = −µ0MsHU
2
sin2 θ sin2(ϕ− ϕU) (2.18)
where ϕU denotes the angle of the easy axis of HU with respect to the [100]
direction.
Solving the LLG equation (Equation 2.14) using the effective magnetic
field Heff allows all aspects of the magnetisation dynamics to be investigated.
Ferromagnetic resonance spectroscopy therefore is an extremely powerful tool
for studying the magnetic properties of ferromagnetic thin layers.
1Equation 2.17 is the most general expression of the total free energy density in a
(Ga,Mn)As system. The expression given by Equation 2.10 is a simplified version.
33
2.4 Magnetisation dynamics
The condition for Larmor precession in ferromagnets can be directly de-
rived from the free energy density E following an approach developed by Smit
and Beljers [125], also independently by Suhl [126]. For a specific orientation
of the applied field (θH , ϕH), the resonance condition is given as the partial
derivatives of E in spherical polar coordinates [69, 126]:(
ω
γ
)2
=
1
M2s sin
2 θ
[
∂2E
∂θ2
∂2E
∂ϕ2
−
(
∂2E
∂θ∂ϕ
)2]
(2.19)
Combining Equations 2.17 and 2.19, the ferromagnetic resonance condi-
tion is given as [73, 74]:(
ω
γ
)2
= µ20[(Hres × a1 + b1)(Hres × a2 + b2)− b23] (2.20)
Here
a1 = cos θ cos θH + sin θ sin θH cos(ϕ− ϕH) (2.21)
b1 = −
[
M −H2⊥ +H2‖ cos2
(
ϕ+
π
4
)]
cos 2θ
+H4⊥
cos 2θ + cos 4θ
2
+H4‖
cos 4θ − cos 2θ
2
3 + cos 4ϕ
4
(2.22)
b2 = −(M −H2⊥) cos2 θ +H4‖ sin2 θ
(
cos 4ϕ− cos2 θ3 + cos 4ϕ
4
)
+H4⊥ cos
4 θ −H2‖
{
sin 2ϕ+
[
cos θ cos
(
ϕ+
π
4
)]2}
(2.23)
b3 =
1
2
cos θ
(
3
2
H4‖ sin 4ϕ sin
2 θ +H2‖ cos 2ϕ
)
(2.24)
It is customary to define Meff = M −H2⊥ since the two terms always appear
together. Note that Equation 2.20 can be greatly simplified if the mag-
netisation is saturated along H0, i.e. θ = θH and ϕ = ϕH . This can be
easily achieved in (Ga,Mn)As as Ms is generally low (a few tens of mT)
due to the diluted magnetic moments in the material. By measuring the
angle/frequency dependence of the resonance field Hres, the anisotropy con-
stants can be determined.
The strain-relaxation-induced anisotropy HU is not included in the above
equations, since its general form is rather complex and is rarely used. In
Section 3.3, an expression for HU specific to the experiments described in
this Thesis is provided.
34
2.4 Magnetisation dynamics
H0
Figure 2.12: Conventional cavity-based FMR spectroscopy, adapted from
[78]. The microwave radiation is transferred from the source to the sample,
and the reflected wave is sent to a diode detector.
2.4.3 Detection of magnetic motion
In a conventional FMR experiment, a magnetic material is exposed to a
microwave electromagnetic radiation at a fixed frequency ω typically in the
gigahertz (GHz) range. By sweeping the external field H0, the magnetic
material is driven through the resonance condition (Equation 2.20). On res-
onance, strong absorption of the incident radiation occurs and a peak appears
in the energy absorption spectrum. Figure 2.12 shows a schematic setup of
conventional FMR spectroscopy, which utilises a combined microwave waveg-
uide/cavity to transfer the radiation from the microwave source to the sam-
ple, and subsequently transfer the reflected wave to a diode detector.
However, this energy-absorption-based FMR is not sensitive enough to
be applied on micrometre-scale sample. To perform device-scale FMR, other
approaches for signal detection have recently been introduced, which include
magneto-optical Kerr effect [67], x-rays [88], and an electrical method based
on a frequency mixing effect [34, 35, 79, 81, 83–87].
The frequency mixing effect has been employed in the experiments de-
scribed in this Thesis. This technique relies on the fact that the precessing
35
2.4 Magnetisation dynamics
Vdc
Time
R(t)
I(t)
V(t)
Figure 2.13: Principle of the frequency mixing effect. The precessing mag-
netisationM causes a time-dependent change R(t) in the device’s resistance,
and the ac current I(t) inside the sample also oscillates at the same frequency.
Product of the two gives rise to an ac voltage component varying at double
frequency and a dc voltage component (photovoltage).
magnetisation causes a time-dependent change in the sample’s resistance1
∆R(t) from the equilibrium value R0. The resistance oscillates at the same
frequency as the driving torque, and when combined with the time-dependent
current inside the device I cos(ωt), gives rise to a dc voltage across the sam-
ple. This dc voltage Vdc, referred to as “photovoltage” in this Thesis, allows
the magnetic motion in the micro-devices to be detected. Figure 2.13 illus-
trates schematically the frequency mixing effect.
The electrical detection method has a few advantages: Firstly, it only
requires a voltmeter which can be integrated seamlessly into the rest of the
FMR setup (as shown in the experiment setup in this Thesis). There is no
requirement for additional optical or x-ray detection suite. Secondly, the
1The resistance change can be due to the tunnelling/giant magnetoresistance
(TMR/GMR) in a MTJ/spin-valve device [34, 35], or domain wall resistance in a nanowire
[127], or the AMR in a uniform magnetic structure [79, 81, 83–87].
36
2.4 Magnetisation dynamics
technique is compatible with individual nanoscale devices (measurements on
sub-100 nm devices have been demonstrated in Ref. [35] and also in Chap-
ter 5), in which the energy absorption signature is too small for conventional
FMR spectroscopy to detect.
If the magnetisationM precesses under the driving torque of a microwave
frequency excitation field hmw = (hx, hy, hz)e
iωt, and the current inside the
sample1 is I = (Ix, 0, 0)e
iωt, the ferromagnetic resonance curve detected in
the photovoltage spectrum Vdc can be described by solving the LLG equation
(Equation 2.14). Its lineshape is found to be a combination of symmetric and
anti-symmetric Lorentzian functions:
Vdc = Vsym
∆H2
(H0 −Hres)2 +∆H2 + Vasy
∆H(H0 −∆H)
(H0 −Hres)2 +∆H2 (2.25)
with the amplitudes of the two components given as:
Vsym(ϑ) =
I∆R
2
Asym sin(2ϑ)hz (2.26)
Vasy(ϑ) =
I∆R
2
Aasy sin(2ϑ)(hx sinϑ+ hy cosϑ) (2.27)
Here ϑ is the angle between the magnetisation vector M and the current I.
The terms Asym and Aasy are the scalar amplitudes of the magnetic suscep-
tibility (Ai = χi/Ms). In the case that the magnetisation is saturated, i.e.
M ‖ H0, they can be expressed as:
Asym =
γ(Hres + b1)(Hres + b2)
ω∆H(2Hres + b1 + b2)
(2.28)
Aasy =
(Hres + b1)
∆H(2Hres + b1 + b2)
(2.29)
where b1 and b2, defined in Equations 2.22 and 2.23, are terms containing
the anisotropy energies of the system. The derivations for Equations 2.25 –
2.29 are given in Appendix A.
1It is assumed here that the current I(t) and the driving field hmw(t) are in-phase. A
phase difference (ψ) between the two terms results in an imaginary part in the expression
of Vdc, see Appendix A.
37
2.5 Spin-orbit interaction in semiconductors
E
H
E’
a b
Figure 2.14: The effect of an electric field to a moving electron in: a, the
rest frame of the electric field; and b, the rest frame of the electron. In the
electron’s own reference frame, the electric field is moving, and generates an
effective magnetic field according to Lorentz transformation. This effective
magnetic field then couples to the electron spins.
2.5 Spin-orbit interaction in semiconductors
As introduced in Section 1.2.1, spin-orbit interaction offers a pathway to spin
manipulation using electric fields alone. For a semiconductor in the presence
of spin-orbit coupling, the one-particle Hamiltonian can be written as [38, 50]:
H = HF +HSO (2.30)
HF =
p2
2m
+ V (2.31)
HSO =
~
4m2c2
[∇V × ~p ] · ~σ (2.32)
where HF represents the free-electron Hamiltonian and HSO described the
interaction between the carrier spin and the carrier momentum. Here V is the
electric potential and∇V is the effective local electric field seen by the charge
with momentum operator ~p. ~σ is the electron spin operator. In materials
with holes as charge carriers, such as (Ga,Mn)As and (Ga,Mn)(As,P), ~σ
should be replaced by the total angular momentum ~J [39].
One can think of HSO as a consequence of special relativity, where an
electric field transforms into a magnetic field in the reference frame of a
moving electron and influences its spin, as illustrated in Figure 2.14.
38
2.5 Spin-orbit interaction in semiconductors
The Hamiltonian described by Equation 2.30 is invariant under time-
reversal operations. This implies that if both the electron wavevector ~k and
its spin ~s = ~σ/2 reverse signs, the energy of the conduction band must stay
the same, i.e. E
(
~k, ↑
)
= E
(
−~k, ↓
)
. This degeneracy is called Kramers
degeneracy1 [50]. Consider the following two cases:
• If the Hamiltonian possesses inversion symmetry, i.e. E
(
~k, ↑
)
= E
(
−~k, ↑
)
,
since the inversion operation reverses ~k but not ~s, it is found that
E
(
~k, ↑
)
= E
(
~k, ↓
)
for all ~k, i.e. the bands are always spin degenerate
in a system with inversion symmetry.
• If the Hamiltonian lacks inversion symmetry, the relation E
(
~k, ↑
)
=
E
(
−~k, ↓
)
still holds, but for ~k 6= ~0 the bands are no longer necessarily
degenerate and spin-splitting occurs.
Dresselhaus spin-orbit interaction If a crystal has bulk inversion asym-
metry (BIA), for example a zinc-blende structure such as in GaAs, Dressel-
haus [44] has shown that the spin-orbit coupling is proportional to k3 and
has the following form:
HD = γ0
[
σxkx
(
k2y − k2z
)
+ σyky
(
k2z − k2x
)
+ σzkz
(
k2x − k2y
)]
(2.33)
where γ0 is a parameter measuring the strength of coupling.
When such as crystal is subject to strain at the interface, this introduces
structure inversion asymmetry (SIA) which yields a spin-orbit coupling term
linear in k [48]:
H4 = C4 [σxkx(εyy − εzz) + σyky(εzz − εxx) + σzkz(εxx − εyy)] (2.34)
Here εij are components of the Cauchy strain tensor which are defined
as:
εij =
1
2
(
∂ui
∂rj
+
∂uj
∂ri
)
(2.35)
1The Kramers degeneracy can be broken by the application of a magnetic field, in
which case the time-reversal symmetry is destroyed.
39
2.5 Spin-orbit interaction in semiconductors
where ~r is the equilibrium position of an atom in the crystal and ~u (~r ) is its
displacement from the equilibrium position. Note that the strain tensor is
symmetric, since an anti-symmetric tensor would correspond to the rotation
of the crystal.
Rashba spin-orbit interaction In addition to the Dresselhaus spin-orbit
coupling (Equations 2.33 and 2.34), in asymmetric quantum wells, or in the
presence of deformation, Bychkov and Rashba have derived a different form
of SIA spin-splitting [45]:
HR = α
(
~k × zˆ
)
· ~σ (2.36)
where α is the Rashba coefficient and the growth direction zˆ is the symmetry-
breaking axis.
In deformed crystals, the Rashba coefficient α contains the off-diagonal
strain elements1 {εxy, εyz, εzx} (sometimes also referred to as “shear” strain
components), and the Hamiltonian can be written as [48, 128, 129]:
HR = C3 [σx(εzxkz − εxyky) + σy(εxykx − εyzkz) + σz(εyzky − εzxkx)] (2.37)
The Rashba spin-orbit coupling therefore introduces an effective magnetic
field perpendicular to ~k, and its relative orientation to ~k remains constant in
the plane normal to zˆ in contrast to the Dresselhaus term [130].
Figure 2.15 illustrates the orientations of the effective magnetic fields
resulting from the linear Dresselhaus and Rashba spin-orbit interactions in
a zinc-blende crystal with respect to ~k.
1The off-diagonal strain components {εxy, εyz, εzx} can be induced by an uniaxial
strain along the [110] crystal axis, for example [50].
40
2.5 Spin-orbit interaction in semiconductors
a bk // [110]y
k // [110]x
k // [110]y
k // [110]x
Figure 2.15: Orientations of the effective magnetic fields due to spin-orbit
coupling, adapted from [54]. a, The directions of the effective field resulting
from the linear Dresselhaus term, with respect to ~k on a unit circle. b, The
orientations of the field due to the Rashba term.
41
Chapter 3
On-chip driven ferromagnetic
resonance on individual
(Ga,Mn)As microdevices
3.1 Introduction
Conventional ferromagnetic resonance (FMR) spectroscopy has been exten-
sively utilised in the study of magnetic anisotropy in bulk (Ga,Mn)As wafers
[71–76]. However, cavity-based FMR is not sensitive enough for detecting
the magnetic motion in micron-scale ferromagnets.
In this Chapter, FMR experiments on individual submicron (Ga,Mn)As
devices are reported. An all-electrical, device-scale FMR technique, originally
developed on permalloy systems, is employed [83, 84], which uses a current-
carrying waveguide to generate a local excitation field to drive the magnetic
precession, and employs the frequency mixing effect (Section 2.4.3) to detect
the magnetic motion.
Similar locally driven FMR experiments on individual (Ga,Mn)As nanos-
tructures have also been reported [67], in which the magnetic motion is de-
tected using time-resolved scanning Kerr microscopy with a spatial resolution
of 500 nm. The all-electrical FMR technique presented in this Chapter, on
the other hand, allows measurements on devices down to sub-100 nm in size.
42
3.2 Devices and experimental methods
3.2 Devices and experimental methods
3.2.1 Device preparation
The (Ga1−x,Mnx)As wafers are grown on GaAs(001) substrates with LT-
MBE by Richard Campion and coworkers at the University of Nottingham.
Films with Mn concentrations of 3% and 6% have been used for the experi-
ments.
In the first step of device fabrication, e-beam lithography1 and reactive
ion etching have been used to create isolation trenches on the magnetic films,
which define the microstructures. The trenches are 200 nm-wide and are
etched into the GaAs substrates for approximately 150 nm. The micro-bars
are patterned along either the [100] or [010] crystalline directions. They
also contain two large bond pads on each end, which are covered with Cr/Au
ohmic contacts (20/200 nm-thick) made from a second step of e-beam lithog-
raphy and evaporation. The full recipe for the device fabrication is given in
Appendix C.
A Cr/Au coplanar stripline waveguide (CPS) is lithographically placed
next to the device. It is patterned on an area where the (Ga,Mn)As epilayer
has been removed with reactive ion etching, leaving only the semi-insulating
GaAs substrate. The CPS is designed with 50 Ω characteristic impedance2 to
match that of the microwave transmission line. Close to the sample (∼ 2 µm),
the ground and central lines of the CPS are shorted together. This way when
a microwave current flows through the waveguide, a microwave magnetic field
hmw is generated (Ampe`re’s circuital law) in the vicinity of the (Ga,Mn)As
microstructure, which in turn induces an oscillating Eddy current I cosωt
inside the sample. From the position of the waveguide, it is clear that the
microwave excitation is mainly out-of-plane, i.e. hmw = (0, 0, hz)e
iωt, where
zˆ ‖ [001] (Figure 3.1c). Since the device and the CPS are defined at different
stages of the fabrication process, careful alignment is required to ensure that
the shorted-end is as close to the micro-bar as possible, in order to maximise
1The e-beam lithography of the devices is performed by Andrew Ferguson.
2The equations for calculating the characteristic impedance in CPS and CPW can be
found in Ref. [131].
43
3.2 Devices and experimental methods
the driving field. Figure 3.1a shows the scanning electron microscopy (SEM)
images of a device.
3.2.2 Measurement apparatus
The sample chips are glued1 onto carriers which are specially designed for
the FMR experiments (Figure 3.2a). The sample carrier is a printed-circuit
board (PCB), which is designed to contain 10 dc lines for measuring the
photovoltage, and a coplanar waveguide (CPW) for coupling with a SMA
connector for microwave current input. The CPW also has an open end
for coupling with the coplanar stripline waveguide on the chip. The reason
for the three-step transfer of microwave current (SMA → CPW → CPS)
is because that the precision required for making a CPS waveguide with
50 Ω impedance is beyond the capability of PCB-tooling machines (but well
within the reach of lithographical methods). The CPS waveguide on the
sample chip is wire-bonded to the CPW on the carrier. Multiple bonds are
made between the two waveguides in order to reduce the inductance of the
bond-wires (∼ 1 nH/mm), as shown in Figure 3.2b.
The FMR experiments are performed in a varying-temperature cryostat
(manufactured by Cryogenic Limited). The samples are cooled to 6 K us-
ing He vapour with temperature variation less than 0.05 K. The cryostat is
equipped with a 3-coil superconducting magnet, with each magnet able to
generate a magnetic field up to 1 T. This allows 3D rotation of the applied
magnetic field H0 during the experiment without physically disturbing the
sample.
Measurements are performed using a dipstick custom made specially for
this cryostat. Figure B.1 shows a photo of the dipstick. In order to minimise
thermal conduction, the body of the dipstick is made of stainless steel. Inside
1GE Varnish (also known as IMI-7031) is used as the adhesive for attaching sam-
ples onto the carriers, due to its good thermal conductivity, compatibility with cryogenic
temperatures, and ease of removal (with methanol/ethanol).
44
3.2 Devices and experimental methods
a b
100 mμ
2 mμ
[100]
[010]
[110][110]
CPS
micro-bar
hmw
current
substrate
c
M
H0
φ
Figure 3.1: a, SEM images of the sample chip and an enlarged photo on
the micro-bar (cyan false colour), which is 500 nm-wide and 10 µm-long. A
Cr/Au coplanar stripline waveguide with 50 Ω impedance is lithographically
placed next to the device (yellow false colour). It is placed directly on the
semi-insulating GaAs surface. b, Vector diagram of M and the external
magnetic field H0. ϕ is the angle between M and the [100] direction. c,
Cross-section of a sample. A current inside the shorting wire generates a
magnetic field perpendicular to the (Ga,Mn)As device, according to Ampe`re’s
law.
45
3.2 Devices and experimental methods
26mm
39mm
SMA
CPW
Sample
DC
connector
a b
Figure 3.2: a, The sample carrier used in the FMR experiments. b, Micro-
graph of a prepared sample.
there are 10 constantan1 wires for carrying dc signals, and 24 copper wires
for powering the electronic components mounted on the coldfinger which is
attached to the end of the dipstick. The coldfinger is made of oxygen-free
copper. This material is chosen because of its softness for tooling and good
thermal conductivity. The top part of the coldfinger is designed to be large
enough to host a wide range of high frequency components, such as amplifiers,
couplers and bias-tees. The sample carrier is mounted at the bottom of the
coldfinger, together with a Hall sensor and a temperature sensor2.
The dipstick also consists of three semi-rigid coaxial cables3 for carrying
high frequency currents. For two of the coaxial cables, both the outer sheath
1Constantan is a copper-nickel alloy usually consisting of 55% copper and 45% nickel.
Its main feature is that its resistivity stays constant over a wide range of temperatures.
The constantan wire used on this dipstick are 110 µm in diameter and has resistivity
∼ 66 Ω/m (Data from retailer’s website: www.cmr-direct.com).
2The Hall sensor is from Lakeshore Cryotronics, Inc. (Model number: HGCT-3020).
The magnetic field can be determined by measuring the Hall voltage with a four-probe
technique. The temperature sensor is also from Lakeshore Cryotronics, Inc. (Model num-
ber: CX-1050-SD-1.4L). A Lakeshore Temperature Controller 340 is used for monitoring
and controlling the temperature in the cryostat.
3The model number of the semi-rigid coaxial cables is UT-85, with outer diameter
∼ 2.2 mm.
46
3.2 Devices and experimental methods
and the inner lines are made of stainless steel. This helps to minimise thermal
conduction along the cables. However, due to the relatively large resistance of
stainless steel, the loss in these cables becomes significant at high frequencies.
Hence they are most suitable for carrying ac signals in the frequency range
of MHz. The third coaxial cable consists of stainless steel outer sheath and
a BeCu inner line, and is used for carrying signals in the range of GHz. Its
attenuation at 10 GHz is 5 dBm/m. All three cables are terminated with
SMA connectors (excellent electrical performance from dc to 18 GHz).
3.2.3 Measurement technique
Figure 3.3 shows schematically the experimental setup. The magnetic motion
in the sample is detected electrically by measuring the dc photovoltage across
the device with a voltmeter1. A lock-in technique is employed to improve the
signal-to-noise ratio in the measurement: The microwave current from the
source is alternated between “on” and “off” states by modulation with a
square wave. A lock-in amplifier is then referenced to the frequency of this
modulation, therefore records the difference in photovoltage between the two
states, V = V (Ion) − V (Ioff). This way only the additional voltage caused
by the FMR-enhanced AMR effect is measured. Figure 3.4 illustrates the
modulation square wave and the photovoltage captured by the voltmeter, as
seen on an oscilloscope.
A practical consideration related to the lock-in technique is the modu-
lation frequency. Since the microwave current causes Joule heating in the
device, on resonance, the strong absorption of microwave radiation by the
sample leads to increase in its temperature and hence changes in its resis-
tance. This is another contribution to the measured FMR photovoltage.
The sample cools down when the microwave current is periodically turned
off. Therefore the modulation frequency determines the rate of oscillation
in the sample’s temperature, and the effect of Joule heating on the FMR
1To be precise, the photovoltage is measured using the differential mode on a low-noise
voltage preamplifier (from Stanford Research Systems, model SR560), which is illustrated
as a voltmeter in Figure 3.3. The signal is magnified by 2000 times before it is fed to the
lock-in amplifier.
47
3.2 Devices and experimental methods
Ref. Signal
Lock-in
pulsed mw in
re
fe
re
n
c
e
V
Figure 3.3: Schematic of the experimental setup (not-to-scale). A mi-
crowave current is injected into the coplanar stripline waveguide (yellow),
which generates a microwave magnetic field to drive the magnetisation in
the (Ga,Mn)As microstructure (cyan). A lock-in measurement technique is
employed to improve the signal-to-noise ratio in the measurement.
48
3.2 Devices and experimental methods
Time
V
(V
)
m
o
d
V
(
V
)
d
c
μ
Figure 3.4: The modulation square wave (black) and the photovoltage de-
tected by the voltmeter (red). The square wave is 5 Vpk-pk and is offset by
2.5 V. This way the modulated microwave alternates between “on” and “off”
states. The detected photovoltage oscillates at the same frequency with the
modulation. Since lock-in amplifiers always detect signals as sine functions,
in the case of a square wave (red curve), only the signal level in the middle of
the square shape is measured, and the large spike noise seen at the beginning
of each transition is discarded.
49
3.3 Data and analysis
Device Name Wafer Condition TC (K)
A (‖ [010]) (Ga0.97,Mn0.03)As Annealed 57/71
B (‖ [010]) (Ga0.94,Mn0.06)As As-grown & Annealed 73/110
C (‖ [100]) (Ga0.94,Mn0.06)As As-grown 73/110
Table 3.1: The three devices discussed in this Section. All are 500 nm-wide
and 10 µm-long, and share the same bond pad design. Their orientations
are shown next to the device names. The Curie temperature (TC) at as-
grown/annealed states are shown, with the annealing condition being 160oC
in air for 10 h.
signal. This bolometric effect has been carefully studied on permalloy micro-
bars and reported in Ref. [85]. In the experiments presented in this Chapter,
a modulation frequency of 987.6 Hz has been chosen, and the level of noise
signal in the measured photovoltage is found to be ∼ 0.3 µV.
3.3 Data and analysis
A total number of 5 (Ga,Mn)As micro-bars have been measured in this ex-
periment. Consistent results among the devices have been obtained, and the
typical data from three devices are presented in this Section. Their charac-
teristics are listed in Table 3.1.
3.3.1 On-chip FMR using a waveguide
Similar to conventional FMR experiments, the microwave frequency is kept
fixed and the external field H0 is swept. Figure 3.5a shows the photovoltage
Vdc measured across Device A, when a driving field of 17 GHz is generated
andH0 is swept along the ϕ = 90
o direction (i.e. along [010], see Figure 3.1b).
The coherent precession of Mn spins manifests as a dip clearly seen in the pho-
tovoltage spectrum, and its lineshape can be well described by Equation 2.25.
The resonance field is deduced from the fitting to be µ0Hres = 350 mT.
50
3.3 Data and analysis
μ0 0H (T)
V
(
V
)
d
c
μ
a
b
V
(
V
)
d
c
μ
Figure 3.5: Ferromagnetic resonance observed in the photovoltage spectrum.
a, Vdc measured using an excitation field at 17 GHz and ϕ = 90
o (green
circles). The black solid line is the fitted results according to Equation 2.25.
b, The symmetric and anti-symmetric Lorentzian components deduced from
the fitting.
51
3.3 Data and analysis
The two components of the FMR peak (symmetric and anti-symmetric
Lorentzian) are shown in Figure 3.5b. They are calculated from the fitted re-
sults to Equation 2.25. For an out-of-plane driving field hmw = (0, 0, hz)e
iωt,
Equation 2.26 and 2.27 predicts that the FMR peak should contain a simple
symmetric Lorentzian lineshape, and Vasy → 0, and this is indeed observed.
However a small anti-symmetric Lorentzian component does exist. It is at-
tributed to the phase difference ψ between the Eddy current and the driving
field, caused by non-uniformity of the device and parasitic capacitive cou-
pling between the waveguide and the micro-bar. This effect is described in
Ref. [80, 83] using the following expression:
Vdc ∝ Vasy cosψ + Vsym sinψ (3.1)
For the FMR signal shown in Figure 3.5, the phase shift is estimated to be
ψ = −22o, i.e. the current leads the driving field by 22o.
Magnetic resonance condition Different frequencies are also used to
excite magnetic resonance in Device A. Figure 3.6a plots the magnetic field
sweep curves for hmw at 12, 17 and 19 GHz. In each case the FMR peak
is clearly visible in the photovoltage spectrum, and agrees well with Equa-
tion 2.25 (solid lines are the fitted results). The figure also demonstrates
that the resonance field Hres increases with frequency. This can be under-
stood by examining the condition for ferromagnetic resonance described in
Equation 2.20.
Note that in this experiment, the resonance field Hres (a few hundreds
of mT) is much larger than the saturation magnetisation of the material (a
few tens of mT), hence the magnetisation vector M is always collinear with
the applied field H0. Moreover, since H0 is only applied in-plane with the
sample (i.e. θ = 90o), the magnetic resonance condition can be simplified to:(
ω
γ
)2
= µ20(Hres +H1)(Hres +H2) (3.2)
with
H1 = Meff +
H4‖
4
(3 + cos 4ϕ) +H2‖ cos
2
(
ϕ+
π
4
)
+HU sin
2 ϕ (3.3)
H2 = H4‖ cos 4ϕ−H2‖ sin 2ϕ−HU cos 2ϕ (3.4)
52
3.3 Data and analysis
where HU represents the additional uniaxial anisotropy caused by patterning-
induced strain-relaxation. HU > 0 if its easy axis is along [010].
The frequency-dependence of Hres is fitted
1 to Equation 3.2 using a gyro-
magnetic constant characteristic for the Mn2+ spins γ = 176 GHz/T (g-factor
2). The fitted curve is shown as the red solid line in Figure 3.6b. The good
agreement between the measured Hres(f) data and Equation 3.2, together
with the good agreement between individual FMR peak and Equation 2.25,
prove that the peaks detected in the photovoltage spectra indeed come from
the coherent precession of Mn spins.
FMR linewidth The FMR linewidth ∆H (half width at half maximum)
is plotted in Figure 3.6c. It is generally decomposed into a frequency-
independent inhomogeneously-broadened part and an intrinsic damping-related
part [74]:
∆H = ∆Hinhomo +∆Hα = ∆Hinhomo +
αω
γ
(3.5)
with α being the dimensionless Gilbert damping constant. Fitting ∆H(f)
to a straight line yields ∆Hinhomo = 27.5 mT and α = 0.04. These values are
comparable with an early electrical FMR study on micron-scale (Ga,Mn)As
Hall bars with 2% Mn (∆Hinhomo = 20 mT, α = 0.021) [87].
Comparing with theoretical studies on the magnetisation relaxation in
(Ga,Mn)As [132, 133], the value of the α deduced from the experimental data
agrees within theoretical predications. For example, for (Ga,Mn)As wafers
with 2% Mn, hole density of 1 nm−3 and quasiparticle lifetime broadening
of 50 meV, α is calculated to be 0.04 [132]. In another study, α ≈ 0.01 is
derived for a (Ga,Mn)As film with 5% Mn doping and full hole polarisation
[133]. However, the carrier density in Device A has not been characterised,
1The fitted lines appeared in Figure 3.6b and Figure 3.7c are calculated by fitting to
all the in-plane measurement data (taken at different driving frequencies and field angles)
simultaneously using Equation 3.2. This is because the f-dependence ofHres can be used to
accurately determine γ, while the ϕ-dependence of Hres reveals the magnetic anisotropy;
and by combining both sets of data, all the micromagnetic parameters of the sample can
be deduced.
53
3.3 Data and analysis
b c
a
V
(
V
)
d
c
μ
μ0 0H (T)
f 
(G
H
z
)
μ0 resH (T)
Δ
H
 (
m
T
)
f (GHz)
Figure 3.6: Frequency-dependence of the FMR signals. a, Photovoltage Vdc
measured (circles) for hmw at 3 different frequencies (12, 17 and 19 GHz).
The solid lines are fitted results. The amplitudes of the resonance peaks vary
due to the frequency-dependent loss in the microwave circuit. b, Frequency-
dependence of the resonance field Hres (black circles). The red solid line is
the fitted results according to Equation 3.2, using a gyromagnetic constant
γ = 176 GHz/T. c, Frequency-dependence of the FMR linewidth ∆H (black
squares). The data are fitted to a straight line.
54
3.3 Data and analysis
and more work is needed for a more reliable comparison between experiment
and theory.
3.3.2 Mapping the anisotropy in the nanodevices
A main purpose of conventional FMR spectroscopy is to accurately determine
the magnetic anisotropy constants. Here it is demonstrated that the on-chip
driven FMR enables studying the local anisotropy in individual microdevices.
Figure 3.7a plots the photovoltage spectra measured on Device A by sweeping
H0 along different crystalline orientations ([110], [100] and [11¯0] directions,
respectively). The magnitude of Hres is observed to be ϕ-dependent. This
is understandable from Equation 3.2: Resonance in ferromagnetic systems
occurs due to the effective magnetic field Heff, which is a combination of
the external and internal fields. As the internal field (predominantly the
anisotropy field) varies with ϕ, so is the applied field Hres at which steady-
state motion of M occurs.
Figure 3.7b shows the full photovoltage spectra from an in-plane rota-
tional scan of H0. The FMR signal can be clearly recognised on top of a
non-resonant background. Several characteristics of the magnetic anisotropy
in this micro-device can be observed:
• The ϕ-dependence of Hres exhibits a pronounced fourfold symmetry,
with minima in the [100] and [010] crystal directions. This can be
explained by the intrinsic biaxial anisotropy term H4‖, with easy axes
along these directions.
• Hres(ϕ) also comprises a twofold symmetry, with Hres reaching maxima
along [11¯0], while being smaller along [110]. This is caused by the
intrinsic uniaxial anisotropy H2‖, showing an easy axis along [110] and
a hard axis along [11¯0]. H2‖ usually has its easy axis along [11¯0],
as reported in many experimental studies [10, 65, 74]; however, there
have been observations on the rotation of H2‖ towards [110] in annealed
samples [110].
55
3.3 Data and analysis
H0
[100]
[110]
[010]
[110]
H0[100]
[110]
[010]
[110]
H0
[100]
[110]
[010]
[110]
μ0 0H (T)
V
(
V
)
d
c
μ
a
μ
0
0
H
(T
)
V ( V)dc μ
φ (deg.)
[1
0
0
]
[1
1
0
]
[0
1
0
]
[1
1
0
]
c
b
μ
0
re
s
H
(T
)
[1
1
0
]
[1
0
0
]
[1
1
0
]
[0
1
0
]
[1
0
0
]
Figure 3.7: Probing the magnetic anisotropy in (Ga,Mn)As microstructures
using on-chip driven FMR. a, Vdc measured when H0 is swept in-plane along
[110], [100] and [11¯0], respectively. The inset pictures show the direction of
the field sweep. A 18 GHz microwave field is used. b, The full photovoltage
spectra for an in-plane rotational scan of H0. The colour scale represents the
magnitude of Vdc (in µV). c, ϕ-dependence of the resonance field Hres (black
circles). The red solid line is the fitted results according to Equation 3.2. The
in-plane magnetic anisotropy constants Hi can be deduced from the fitting.
• A closer examination reveals that Hres along [100] is smaller than that
along the [010] direction. However, according to the intrinsic biaxial
anisotropy H4‖, these axes should be equivalent. This indicates the ex-
istence of an additional uniaxial anisotropy HU not present in the bulk
(Ga,Mn)As epilayer. It is in fact due to strain-relaxation caused by the
lithographic patterning process (Section 2.2.3). HU has a twofold sym-
metry with an easy axis along the [010] direction, i.e. the direction of
the micro-bar, and a hard axis along the direction of strain-relaxation.
This is consistent with previous experiments on similar compressively-
strained (Ga,Mn)As wafers [26, 65–67].
To deduce the in-plane magnetic anisotropy constants Hi from the FMR
56
3.3 Data and analysis
results, the values of Hres are extracted by fitting to individual resonance
peaks in Figure 3.7b using Equation 2.25. The ϕ-dependence of Hres is plot-
ted in Figure 3.7c (black circles). A least square fitting algorithm is used to fit
the experimentally obtainedHres data to Equation 3.2 (the red solid line), and
the anisotropy fields are found to be: µ0H4‖ = 117 mT, µ0H2‖ = −30 mT,
and µ0HU = 9.3 mT (with easy axis along the [010] direction).
One important remark for the on-chip driven FMR experiment is that
the geometry of the bond pads has not been designed perfectly and this has
led to two problems:
• Since the strength of the magnetic field generated by the shorted waveg-
uide drops as 1/d2, where d is the distance away from the centre of the
short1 (Biot-Savart law), the large amount of magnetic moments in
the bond pad region are driven by hmw with varying strengths, and
their collective motion can no longer be considered as due to a single
macrospin M. This nonuniform motion of spins obscures the FMR
peak detected across the device, and could lead to broadening of the
linewidth ∆H .
• The magnetic anisotropy measured in the FMR experiment is an aver-
age over the entire (Ga,Mn)As device, i.e. micro-bar + bond pads. But
since the bond pads exhibit bulk-like magnetic properties due to their
large size, the change in magnetic anisotropy in the microstructures
due to strain-relaxation can be overshadowed. The imperfect match
between the experimental (black circles) and calculated (the red line)
results in Figure 3.7c may be a result of this.
Since Device A only contains 3% of Mn, the compressive-strain caused
by lattice mismatch is small, and the strain-relaxation-induced anisotropy
HU is negligible compared to the intrinsic anisotropy fields H4‖ and H2‖.
For the same reason, HU is expected to be more significant in (Ga,Mn)As
1According to the Biot-Savart law, the magnetic field h generated by the shorting-
wire (with length L) varies as µ0h = [µ0I/(2pid)](L/
√
L2 + d2). Hence it drops as 1/d for
d≪ L, and as 1/d2 for d≫ L.
57
3.3 Data and analysis
nano-devices with 6% Mn (Device B and C), as the wafer is under more
compressive-strain. The measurement data are presented in the Appendix
of this Chapter (Section 3.5). However, the influence from the bond pads
is more pronounced in these devices. A new design is used in the subse-
quent experiments described in this Thesis, which resolves this issue by us-
ing symmetric bond pads fully covered with metal (Cr/Au) contacts, hence
minimising their contribution to the measured signals.
3.3.3 Vector magnetometry using FMR
Following the discussions in Section 2.4.3, the lineshape of the photovoltage
is determined by the direction and magnitude of the driving field. For an
out-of-plane excitation field hmw = (0, 0, hz)e
iωt, the FMR peak should have
the shape of a symmetric Lorentzian function, with its amplitude changing
with ϑ according to (cf. Equation 2.26):
Vsym(ϑ) ∝ sin 2ϑ (3.6)
This has indeed been observed in Device A (Figure 3.8). In the exper-
iments described in the next Chapter, this vector magnetometry method
by considering the lineshape of the photovoltage is employed to analyse the
direction and magnitude of an unknown driving field hmw caused by the
current-induced torque.
3.3.4 Effects of annealing
As introduced in Section 2.1.2, annealing (Ga,Mn)As wafers at moderate
temperatures can remove the interstitial MnI defects and improve the ho-
mogeneity of the material. It has been observed in bulk (Ga,Mn)As films
that annealing leads to better defined FMR signals with narrower linewidths
[73, 74].
In Device B, the impacts of annealing on the inhomogeneous broadening
term ∆Hinhomo and the Gilbert damping constant α have been investigated.
Figure 3.9a compares the change in FMR lineshapes before and after the sam-
ple has been annealed, using the same excitation frequency of 17 GHz. The
58
3.3 Data and analysis
ϑ (deg.)
V
(
V
)
s
y
m
μ
Exp.
sin(2ϑ)
Figure 3.8: ϑ-dependence of the amplitude of the symmetric Lorentzian Vsym
(black circles). The red solid line is fitting to a sin(2ϑ) function. The shift
which appears in the data (i.e. Vsym(ϑ = 0
o) 6= 0) is due to sample mis-
alignment (estimated to be < 2o) and a uniaxial crystalline AMR term CU
associated to the 3% (Ga,Mn)As wafer (for more details, see Section 4.5.3).
sharper lineshape and reduced width in the FMR signal from the annealed
sample can be easily spotted. The resonance field Hres is also shifted, which
indicates changes in the magnetic anisotropy upon annealing (discussed in
details in Section 4.5.5).
Figure 3.9b quantitatively compares the frequency dependence of the
FMR linewidth in the as-grown and annealed samples. Firstly, the overall
FMR linewidth ∆H is smaller in the annealed device, than in the as-grown
device under the same excitation frequency. Fitting the data to straight lines
also reveals that the broadening due to wafer inhomogeneity ∆Hinhomo halves
upon annealing, reducing from 7.15 mT to 3.15 mT. The Gilbert damping
constant α also decreases from 0.019 to 0.0121. All observations confirm that
annealing helps to improve sample homogeneity and leads to more uniform
precession of Mn ions.
1The Gilbert damping constant α in Device B is found to be smaller than that in
Device A, even in the as-grown state. This is because the FMR linewidth scales approx-
imately inversely with the carrier (hole) concentration p of the material, i.e. ∆H ≈ p−1
[76].
59
3.3 Data and analysis
f (GHz)
Δ
H
 (
m
T
)
a
b
as-grown annealed
Δ
H
 (
m
T
)
f (GHz)
μ0 0H (T)
V
(
V
)
d
c
μ
Figure 3.9: a, The FMR peak measured on Device B before and after it has
been annealed (cyan squares and orange triangles, respectively), using the
same excitation frequency of 17 GHz. The solid lines are fitted data accord-
ing to Equation 2.25, from which the FMR linewidth can be determined.
b, Comparison of the frequency-dependence of ∆H between the as-grown
and annealed states of the device (black squares and triangles, respectively).
The data are fitted to straight lines to extract ∆Hinhomo (intercept) and α
(gradient).
60
3.4 Conclusions
3.4 Conclusions
In this experiment, an all-electrical, on-chip driven FMR technique has demon-
strated. It allows investigation on the magnetic properties of individual
(Ga,Mn)As microstructures. In particular, the additional uniaxial anisotropy
caused by patterning-induced strain-relaxation is observed, which has an easy
axis perpendicular to the direction of strain-relaxation.
Furthermore, FMR in these microdevices gives rise to a dc photovolt-
age due to an AMR frequency-mixing effect, whose amplitude reflects the
characteristics of the microwave driving field (orientation and magnitude).
It has been confirmed that the observed angle-dependence of the FMR am-
plitude agrees with the mathematical predictions for an out-of-plane driving
field. This vector magnetometry technique is used in the following Chapter
to investigate an unknown excitation field.
However, the true FMR peaks and the magnetic anisotropy of the mi-
crostructures are obscured by the magnetic elements in the bond pad region.
This issue is addressed in the experiments in the following Chapters (see also
Section 3.5).
61
3.5 Appendix – Bond pad design
3.5 Appendix – Bond pad design
Figure 3.10a shows the photovoltage spectra from an in-plane rotational scan
on Device C, together with a schematic of the device geometry. FMR mea-
surements exhibits a profound fourfold symmetry in Hres(ϕ) corresponding
to the intrinsic biaxial anisotropy H4‖ (82 mT). Fitting to the resonance
field Hres using Equation 3.2 reveals that there is also an additional uniax-
ial anisotropy due to patterning-induced strain-relaxation, µ0HU = −8 mT,
where the negative sign implies that HU has an easy axis along the [100]
direction, i.e. the long axis of the micro-bar.
However, the magnitude of HU is much smaller than those reported in
other studies on similar (Ga,Mn)As devices, due to the bulk anisotropy from
the large bond pads of Device C, as suggested in Section 3.3.2. For example,
Hoffmann et al. have measured devices on a (Ga,Mn)As wafer with 6% Mn
(50 nm-thick film) and have observed KU = 1.8×104 erg/cm3 (equivalent to
µ0HU = 47.6 mT) on a 400 nm-wide bar [67]. For narrower bars (200 nm-
wide), Hu¨mpfner et al. and Wenisch et al. show that µ0HU in (Ga,Mn)As
epilayers with lower Mn concentrations (4% and 2.5% respectively) is a few
tens of mT (25 and 45 mT respectively) [65, 66].
To illustrate the impact of the bond pads on the overall magnetic anisotropy,
measurement data from a micro-bar with a new bond pad geometry are also
presented here. The new bond pads are squared-shaped, so that their re-
sistance is exactly 1/20 the resistance of the micro-bars (which contain 20
squares). The size of the metal contacts (Cr/Au) are also enlarged to cover
almost the entire bond pads. Furthermore, the magnetisation precession
is driven by a more uniform1 hmw. Figure 3.10b shows the new geometry
together with the results from a rotational field scan on such a device.
The magnetic anisotropy in the device with new bond pad geometry
clearly exhibits a superposition of fourfold and twofold symmetry (due to
1FMR in this device is driven by the current-induced torque (CIT) instead of the
torque from a local microwave magnetic field. The CIT-FMR technique is the focus of
Chapter 4.
62
3.5 Appendix – Bond pad design
φ (deg.)
μ
0
0
H
(T
)
μ
0
re
s
H
(T
)
a b
Figure 3.10: Magnetic anisotropy measured from FMR experiments in two
micro-bars with different bond pad geometries. Both devices are patterned
on the same (Ga0.94,Mn0.06)As wafer (as-grown) and are both along the [100]
crystalline direction. The two devices have identical dimensions (500 nm-
wide, 10 µm-long), but with different bond pad designs: a, The geometry of
Device C and its magnetic anisotropy profile; b, A new device design and
the magnetic anisotropy measured on such a device. The schematics are not-
to-scale. In each case, the resonance field Hres (black circles) is fitted with
Equation 3.2 (red solid lines).
63
3.5 Appendix – Bond pad design
H4‖ and HU , respectively), with µ0H4‖ = 43 mT and µ0HU = −28 mT, con-
sistent with earlier studies [65–67]. This demonstrates that the bond pads
in the first design have obscured the true magnetic anisotropy in the micro-
bars. The second device geometry has since been adopted in the following
chapters.
64
Chapter 4
Ferromagnetic resonance driven
by current-induced torque in
magnetic microstructures
4.1 Introduction
Torques exerted on a ferromagnet by static or alternating magnetic fields
induce magnetisation switching or ferromagnetic resonance (FMR) which are
fundamental phenomena utilised in magneto-electronic applications as well
as basic studies of magnetic systems. As introduced in Chapter 1, a major
aim in spintronics research is to find replacement for the role of the magnetic
field in these processes.
The discovery of spin-transfer torque (STT) offers such an opportunity
[17–19]. STT is a collective quantum phenomenon in which a spin-polarised
electric current acts on spatially varying magnetisation via exchange inter-
action and, in a simplified picture, can be viewed as a macroscopic angular
momentum transfer effect. Similar to the phenomenon itself, the detection
of the spin-transfer torque is commonly performed by magneto-transport
effects, such as the tunnelling magnetoresistance (TMR), which rely on the
ferromagnetic exchange interaction and non-uniformity of the magnetic struc-
ture.
65
4.2 Devices and experimental methods
In a uniformly magnetised system, the current-induced torque (CIT) is
a result of the internal exchange field combined with a microscopic non-
collinearity of individual electron spins which originates from the relativistic
spin-orbit coupling. When an electric current traverses through materials
with a specific broken symmetry1 in their spin-orbit coupled band-structure,
a non-equilibrium spin polarisation of the carriers occurs [42, 49, 134]. It
produces a transverse component of the internal exchange field and a torque
is applied to the magnetisation vector [38, 40].
In this Chapter, a microwave frequency CIT is realised in magnetic mi-
crostructures and is utilised to electrically drive FMR. For detection the
anisotropic magnetoresistance (AMR) effect is employed, which, similar to
CIT, is a combined exchange and spin-orbit coupling phenomenon present in
uniform ferromagnets. The III–V diluted magnetic semiconductors (Ga,Mn)As
and (Ga,Mn)(As,P) are particularly favourable systems for observing and
exploring CIT-FMR because of their compatibility with advanced semicon-
ductor micro-fabrication techniques, because the carrier bands have strong
spin-orbit coupling and strong exchange interaction with the localised mag-
netic moments, and because the system can be described by a common,
model semiconductor Hamiltonian.
4.2 Devices and experimental methods
Device preparation 25 nm-thick (Ga1−x,Mnx)As and (Ga1−x,Mnx)(As1−y,Py)
wafers have been used in this experiment. They are prepared using LT-MBE
by Richard Campion, Arianna Casiraghi and coworkers at the University of
Nottingham. (Ga,Mn)As wafers with different Mn concentrations have been
used (3%, 6% and 12%), and the (Ga,Mn)(As,P) wafers contain 6% Mn and
10% P.
Devices of two different sizes are fabricated: 40 µm× 4 µm and 10 µm×
0.5 µm (Figure 4.1b and 4.1c). In the first fabrication step, the micro-bars
are defined on the ferromagnetic films with isolation trenches (200 nm-wide,
1The two types inversion asymmetry exploited in these experiments are bulk inversion
asymmetry (BIA) and structural inversion asymmetric (SIA).
66
4.2 Devices and experimental methods
200 mμ
10 mμ
2.5 mμ
a b
c
Figure 4.1: SEM images of the devices used in this experiment. a, The SEM
image of a sample chip, which contains two micro-bars patterned 90o-apart.
The Cr/Au ohmic contacts are highlighted in yellow false colour. b and c,
SEM graphs of individual 4 µm- and 500 nm-wide bars.
∼ 150 nm-deep), created with e-beam lithography and reactive ion etching.
The Cr/Au (20/200 nm-thick) ohmic contacts are made in a second step of
photolithography and evaporation. The full recipe for the device fabrication
is given in Appendix C. Each sample chip contains two micro-bars, patterned
90o-apart along different crystallographic axes and sharing a common signal
input. Figure 4.1a shows a SEM image of a typical sample.
Measurement technique A schematic of the experimental setup is shown
in Figure 4.2a. The same sample carrier shown in Figure 3.2 is used. The
device is wire-bonded between the open-circuit coplanar waveguide and a
low-frequency connection which also provides a microwave ground (with par-
asitic capacitance ∼ pF). Due to the large impedance mismatch1, most of the
incident signal is reflected, however a small microwave current I cos(ωt) still
flows through the sample (a few tens of µA). Within the ferromagnet, the
oscillating current-induced torque causes the magnetisation M to precess,
1A 4 µm-wide bar has nominal resistance∼ 10 kΩ, whereas the microwave transmission
line and coplanar waveguide have characteristic impedance of 50 Ω.
67
4.3 Data and analysis
Ref. Signal
Lock-in
V
pulsed mw in
re
fe
re
n
c
e
10 mμa b
Figure 4.2: a, Schematic of the CIT-FMR experiment setup (not-to-scale).
b, Micrograph of a prepared sample.
which leads to time-dependent change in the resistance. This, when com-
bined with the microwave current inside the device, produces a measurable
dc photovoltage from the frequency mixing effect (Section 2.4.3).
The lock-in measurement technique introduced in Section 3.2.3 is also
employed in the CIT-FMR experiments. A bias-tee is used to isolate the low-
frequency modulated photovoltage (987.6 Hz) from the microwave frequency
voltage from the signal generator (GHz). The potential difference across the
device is measured with a voltage preamplifier (differential mode, the signals
are amplified by 500 times). The measurements are carried out in the same
cryogenic system described in Chapter 3, at a constant temperature of 6 K.
4.3 Data and analysis
A total number of 61 devices have been measured, and consistent observations
have been made. Representative data are presented in this Section.
68
4.3 Data and analysis
4.3.1 Current-induced torque driven FMR
Magnetic resonance signal Figure 4.3a shows the photovoltage mea-
sured across a (Ga,Mn)As micro-bar (4 µm-wide, [11¯0] direction), as the ex-
ternal field H0 is swept while a fixed-frequency microwave current I cosωt is
applied. Vdc measured at different microwave frequencies are shown. In each
case, a well-defined peak can be recognised from the photovoltage spectrum,
which can be well-fitted by the solution of the LLG equation (Equation 2.25).
The resonance fieldHres, and amplitudes of the Lorentzian functions Vsym and
Vasy are then determined from the fittings.
Figure 4.3b plots the frequency-dependence of the resonance field Hres.
Fittings to Equation 3.2 suggest that Hres(f) can be well-described by the
ferromagnetic resonance condition. A gyromagnetic constant γ characteristic
for Mn2+ spins 176 GHz/T (g-factor 2) is used for the fitting. This, together
with the good agreement between the observed peaks and the fitted results
from Equation 2.25, confirms that the observed peaks in Figure 4.3a are due
to the coherent precession of Mn spins. Figure 4.4 illustrates the CIT-FMR
process.
Comparing the lineshapes of the FMR signals in Figure 4.3a and those
from the previous FMR experiment (Figure 3.6a), it is clearly recognised
that the FMR peaks in Figure 4.3a are dominated by an anti-symmetric
Lorentzian function, which is the signature of an in-plane driving field ac-
cording to Equation 2.27. Further analysis described later in this Chapter
confirms this observation.
Since the CIT-FMR experiment is performed over a wide range of fre-
quencies, the inhomogeneous (∆Hinhomo = 1.5 mT) and frequency-dependent
(0.28 mT/GHz) contributions to the damping are also determined (Fig-
ure 4.3c). The Gilbert damping is found to be α = 0.008 for this device
(see Equation 3.5), consistent with theoretical predications [132, 133].
Mapping the magnetic anisotropy Conventional FMR spectroscopy is
usually employed to measure the magnetic anisotropy. Here this functionality
is also demonstrated using the CIT-FMR technique. In Figure 4.3d the
69
4.3 Data and analysis
d
V
(µ
V
)
d
c
μ0H (T)0
a
e
0.1 0.50.40.30.2 0.6
40
-40
-20
20
0
μ0 resH (mT)
f 
(G
H
z
)
0
0
10
20
400200
μ
0
re
s
H
(m
T
)
0               90             180            270            360
450
350
400
φ (deg.)
μ
0
0
H
(m
T
)
V (µV)dc
500
300
400
[1
0
0
]
[0
1
0
]
[1
1
0
]
[1
1
0
]
0               90             180            270            360
φ (deg.)
[1
0
0
]
[1
1
0
]
[0
1
0
]
[1
1
0
]
[1
0
0
]
b
f (GHz)
0 10            20
Δ
H
 (
m
T
)
0
6
3
c
μ
0
re
s
H
(m
T
)
0               90             180            270            360
450
350
400
φ (deg.)
μ
0
0
H
(m
T
)
500
300
400
0               90             180            270            360
φ (deg.)
V (µV)dc
f g
Figure 4.3: Microwave current-induced torque driven FMR. a, Vdc measured
on a 4 µm-wide bar using I cos(ωt) at 11, 14 and 17 GHz (circles). The reso-
nance peaks are clearly observed and can be well-described by Equation 2.25
(the solid lines are fitted results). The difference in the signal strength at dif-
ferent ω is caused by the frequency-dependent attenuation of the microwave
circuit. b, The resonance field Hres as a function of the microwave fre-
quency (black triangles). The red line is the fitted results to Equation 3.2.
c, Frequency-dependence of the FMR linewidth ∆H (black squares). The
data are fitted to a straight line to extract information on the inhomogeneous
broadening ∆Hinhomo and Gilbert damping constant α. d, Vdc measured from
in-plane rotational scans of H0. The colour scale represents the magnitude
of the photovoltage. e, Angle-plot of Hres (black circles), which is extracted
by fitting to each FMR peak using Equation 2.25. The red line is a fitted
curve to Equation 3.2 for calculating the anisotropy fields. f, Vdc measured
for in-plane rotational scans of H0 on a 500 nm-wide bar. g, Angle-plot of
Hres for the 500 nm-wide bar with fittings.
70
4.3 Data and analysis
j (t)
Heff
M
τ
cit
τ
α
Figure 4.4: Concept of the current-induced torque driven ferromagnetic res-
onance (CIT-FMR). The current-induced torque (τ cit) counters the effect
of damping (τα), and leads to steady-state precession of the magnetisation:
∂M/∂t = −M×Heff.
data from an in-plane rotational scan of H0 are shown, which reveal both
uniaxial (H2‖), along the [11¯0] direction, and biaxial (H4‖) contributions to
the anisotropy. By analysing the peak positions (Figure 4.3e) these values
are found to be µ0H2‖ = 18.7 mT and µ0H4‖ = 17.4 mT.
The CIT-FMR technique is also used to measure a 500 nm-wide bar pat-
terned in the same direction (Figures 4.3f and 4.3g), giving µ0H2‖ = 68.2 mT
and µ0H4‖ = 5.7 mT. The enhanced uniaxial term along the bar demonstrates
the effect of patterning-induced strain-relaxation on the magnetic anisotropy
[26, 65–67]. This measurement shows the scalability of the CIT-FMR tech-
nique down towards the nanoscale.
4.3.2 Quantifying the current-induced torque
The key parameters of interest in this study are the direction and magnitude
of the field hcit producing the torque. These quantities can be deduced
from the angle dependence of the FMR amplitudes Vsym and Vasy, which is
described in Equations 2.25, 2.26 and 2.27 (Section 2.4.3). The measurement
data analysed in this Section are from the 4 µm-wide devices.
Since no component of Vsym is observed to behave as sin(2ϑ) within the
71
4.3 Data and analysis
a b
30 Tμ
I // [100]x
I // [010]yc d
j (10 A/cm )
6 2
μ
μ
0
c
it
h
(
T
)
150 Tμ
I // [100]x
I // [010]y
V
(µ
V
)
a
s
y
ϑ (deg.)
e f
V
(µ
V
)
a
s
y
ϑ (deg.)
j (10 A/cm )
6 2
μ
μ
0
c
it
h
(
T
)
(Ga,Mn)As (Ga,Mn)(As,P)
E (V/cm) E (V/cm)
Figure 4.5: Characterisation of the current-induced field in (Ga,Mn)As and
(Ga,Mn)(As,P) devices. a–b, Anti-symmetric Lorentzian amplitudes Vasy
measured on a group of 4 µm-wide (Ga,Mn)As bars (circles), patterned along
different crystalline directions. The solid lines are fitted results to Equa-
tion 2.27. c, Plot of the magnitude and direction of the current-induced
field hcit measured in the (Ga,Mn)As micro-bars, scaled for a current den-
sity j = 105 A/cm2. d, Similar plot for hcit measured in the 4 µm-wide
(Ga,Mn)(As,P) devices. e–f, Current density dependence of hR (Rashba)
and hD (Dresselhaus) in both (Ga,Mn)As and (Ga,Mn)(As,P) samples. A
second horizontal scale is included for the electric field, calculated from the
device resistance.
72
4.3 Data and analysis
error of the measurements1, it is concluded that the driving field hcit is pre-
dominantly in-plane. The discussions therefore focus on the amplitude of the
anti-symmetric Lorentzian Vasy.
Figure 4.5a shows the angle dependence of Vasy for the (Ga,Mn)As device
patterned in the [11¯0] direction. By fitting to Equation 2.27, it is recognised
that Vasy(ϑ) comprises a − sin(2ϑ) cos(ϑ) term, indicating that the driving
field is perpendicular to I, i.e. hcit = (0,−hy, 0)eiωt. In a [110] device (Fig-
ure 4.5a) the amplitude of Vasy also depends on sin(2ϑ) cos(ϑ) but has oppo-
site sign, indicating that the driving field has reversed, i.e. hcit = (0, hy, 0)e
iωt.
For the micro-bar along [100] (Figure 4.5b), the Vasy curve is a superposition
of sin(2ϑ) sin(ϑ) and sin(2ϑ) cos(ϑ) functions. This implies that the driv-
ing field consists of components both parallel and perpendicular to I, i.e.
hcit = (−hx,−hy, 0)eiωt, with hx > hy. Similar observation is also made in a
[010] sample, but with the field orientation: hcit = (hx,−hy, 0)eiωt.
Taken together these results indicate two contributions to the current-
induced driving field with different symmetry, hcit = hR + hD. The fields
hR and hD have angular dependence on I reminiscent of the angular depen-
dence of Rashba and Dresselhaus spin-orbit fields in the momentum space,
respectively [44, 45] (cf. Figure 2.15). This is most clearly seen by plot-
ting the dependence of the magnitude and direction of the resultant fields
on the electric current (micro-bar) orientation (Figures 4.5c). From samples
patterned in the [010] and [100] directions it is found that there exists a
field hR always perpendicular to I, having the symmetry of the momentum-
dependent Rashba spin-orbit interaction. It is also evident that hR is present
in the [110] and [11¯0] micro-bars where it acts to either enhance or reduce
the resultant hcit. The main term, hD with symmetry of the Dresselhaus
spin-orbit interaction, is parallel to I in the [010] and [100] directions, and
perpendicular to I in the [110] and [11¯0] directions.
The strength of hD is found to be linear with current density (Figure 4.5e)
and its magnitude, scaled to j = 105 A/cm2, is hD = 33 µT. Note that the
current density is determined by a bolometric effect, using the resistance to
1The ϑ-plots for the amplitude of the symmetric Lorentzian (Vsym) measured on these
devices are presented in Section 4.5.1.
73
4.3 Data and analysis
compare the heating effect of an unknown microwave current to a known
direct current (see Section 4.5.2). The magnitude of hR is 10 µT at j =
105 A/cm2 and this weaker component of the CIT field is also linear with
current density (Figure 4.5e). These values of hD and hR are evaluated from
a device in the [100] direction where hD and hR are orthogonal and can be
separately determined.
4.3.3 Theoretical understanding on the current-induced
torque
These observations are further analysed theorectically in collaboration with
Karel Vy´borny´, Liviu Zaˆrbo and Tomas Jungwirth at the Institute of Physics,
Czech Republic.
Microscopically, hD and hR can be linked to the term HC4 in the effective
Hamiltonian describing (Ga,Mn)As [39, 40, 61, 129, 135]:
H = HKL +Hexch +HC4 (4.1)
Here HKL is the inversion symmetric part of the host GaAs semiconductor
Hamiltonian, Hexch describes the exchange interaction between carrier (hole)
spins and local Mn moments, and the HC4 term is due to broken inversion
symmetry and the presence of strain (εij).
Using the four-band Kohn-Luttinger Hamiltonian model [136], the inver-
sion symmetric part HKL is written as:
HKL =
~
2
2m
[(
γ1 +
5
2
γ2
)
k2 − 2γ3(~k · ~J )2 + 2(γ3 − γ2)
∑
i
k2i J
2
i
]
(4.2)
where γ1 = 6.98, γ2 = 2.06 and γ3 = 2.93 are three independent Luttinger
parameters, ~J is the carrier total angular momentum, and ~k is the wavevector.
The carrier (hole) – local moment exchange term reads:
Hexch =
1
3
heM · ~J (4.3)
where h = JpdnMnsMn (Jpd is the exchange constant, nMn is the density of
Mn local moments and sMn = 5/2) and eM is the magnetisation unit vector.
74
4.3 Data and analysis
Finally, the term due to broken inversion symmetry and strain HC4 is
written as (cf. Equations 2.34 and 2.37):
HC4 = C4[εzz(Jyky − Jxkx) + c.p.] + C4[εxy(Jxky − Jykx) + c.p.] (4.4)
Here c.p. denotes cyclic permutations in {x, y, z}, which refer to components
along the cubic axes of the crystal; and C4 ≈ 0.5 eVnm for the GaAs host
[135]. The hD and hR components of hcit originate from the first and second
bracket term in the HC4 Hamiltonian (Equation 4.4), respectively.
The form of the strain tensor εij therefore determines the hD and hR fields.
For epitaxially-grown (Ga,Mn)As and (Ga,Mn)(As,P) films, the strain tensor
reads [115]:
εij =


e0 0 0
0 e0 0
0 0 −2c12
c11
e0

 (4.5)
Here the growth-strain is defined as e0 = (as−a0)/a0, where as and a0 are the
lattice constants of the substrate and the magnetic film, respectively. Typical
magnitudes are e0 ∼ 10−4 − 10−2 [115]. The parameters c11 and c12 are
the elastic moduli [137]. For positive (negative)1 e0, Equation 4.5 therefore
describes an expansion (contraction) along the [100] and [010] crystal axes,
accompanied by a contraction (expansion) along the [001] axis.
It is also clear from Equation 4.5 that the off-diagonal strain components
(such as εxy) which yield the hR field are not physically present in the crystal
structure of (Ga,Mn)As epilayers. This strain has been introduced, however,
in previous studies to model the in-plane uniaxial anisotropy H2‖ present in
(Ga,Mn)As [26, 50, 110, 115]. It can be represented by a tensor:
εij =

 0 κ 0κ 0 0
0 0 0

 (4.6)
In this “intrinsic” shear strain, positive (negative) κ corresponds to turning
a square into a diamond with the longer (shorter) diagonal along the [110]
1For epilayers under compressive-strain, such as (Ga,Mn)As grown on GaAs [95], e0 <
0. For tensile-strained epilayers, such as (Ga,Mn)As grown on a relaxed InGaAs buffer
[95] or (Ga,Mn)(As,P) [77], e0 > 0.
75
4.3 Data and analysis
axis. The fitted values of the intrinsic shear strain are typically several times
smaller than the diagonal, growth-induced strain [115]. This is consistent
with the observed smaller magnitude of hR than hD in the measurements
(Figure 4.5e).
The HC4 origin of the CIT is confirmed with measurements on 4 µm-
wide bars patterned on the (Ga0.94,Mn0.06)(As0.9,P0.1) epilayer, which has
an opposite sign of e0 compared with the (Ga0.94,Mn0.06)As film [77, 92].
Consistently, the observed field hD also changes sign (Figure 4.5d and 4.5f).
The specific form of the the current-induced field hcit has also been derived
by Karel Vy´borny´ and Liviu Zaˆrbo. In the linear response regime of the
experiment, it is written as [40]:
hcit = −eEJpd
µB
τ
∫
d3k
(2π)3
∑
n
〈~s 〉n,k〈vI〉n,kδ(εn,k − εF ) (4.7)
where τ is the transport relaxation time, 〈~s 〉n,k denotes the expectation value
of the carrier spin, 〈vI〉n,k the velocity component along the current direction,
εn,k are the eigenenergies of the Hamiltonian H, and εF is the Fermi energy.
Therefore hcit is proportional to the applied electric field E (as demon-
strated in Figure 4.5e and 4.5f), to the carrier – local moment exchange
constant Jpd (≈ 55 meVnm3 in (Ga,Mn)As [61]), to the transport relaxation
time τ (~/τ ≈ 100 meV [61]), and to the non-equilibrium spin-density due
to the displaced Fermi surface given by the integral in Equation 4.7. For
carrier density p ≈ 1 nm−3, the calculated CIT field (by Liviu Zaˆrbo) at
E = 103 V/cm is |hD| ≈ 60 µT, which is consistent with the experimental
values.
A consequence of the Eτ -dependence of CIT revealed by Equation 4.7 is
that a certain current density j would induce a smaller torque in a sample
with higher carrier density1 p. This can explain the different magnitude of
1According to the two expressions for current density: j = σE and j = pevd = peµE,
where p is the carrier density, vd is the drift velocity, and µ is the mobility, the electrical
conductivity is found to be σ = peµ ∝ peτ , where τ is the transport relaxation time and
is approximately linearly proportional to the mobility µ. As a result, the current density
j = σE ∝ peEτ . i.e. at fixed j, the quantity Eτ scales inversely with increasing carrier
density p.
76
4.3 Data and analysis
hD measured in (Ga,Mn)As and (Ga,Mn)(As,P) micro-bars (Figure 4.5e and
4.5f): the two materials are annealed at different conditions (190oC for 20 h
for the (Ga,Mn)As wafer and 180oC for 48 h for the (Ga,Mn)(As,P) wafer)
and even have different conductivity (typical resistance of 4 µm-wide bars
is Rxx = 10 kΩ and Rxx = 11.5 kΩ for (Ga,Mn)As and (Ga,Mn)(As,P),
respectively). However, the carrier density in the two wafers has not been
measured, hence it is difficult at this stage to quantitatively described the
different hD measured in the two materials.
4.3.4 Effect of strain on the current-induced torque
To further investigate the effect of strain on hcit, two more experiments are
performed on: (i) 500 nm-wide nano-bars fabricated on the (Ga,Mn)As and
(Ga,Mn)(As,P) epilayers, in which strong strain-relaxation occurs due to
lithographic patterning; (ii) 4 µm-wide micro-bars fabricated on (Ga,Mn)As
epilayers with various Mn doping (3%, 6% and 12%), and thus different
growth-strain.
Effect of strain-relaxation on CIT Figure 4.6a compares the magni-
tudes of hD and hR measured in both wide (4 µm) and narrow (500 nm)
(Ga,Mn)As structures. Both fields exhibit linear dependence on j. However,
it is observed that hD and hR are weaker in the 500 nm-wide samples. This
can be understood qualitatively by considering the role of strain-relaxation.
The first term in Equation 4.4 can be rearranged to become:
HC4 = C4[(εyy − εzz)Jxkx + c.p.] (4.8)
In 4 µm-wide bars with minimal strain-relaxation, εyy − εzz = e0. Now
consider strain-relaxation in a nano-bar patterned along [100], i.e. the strain
relaxes along [010], thus reducing εyy − εzz and the total Hamiltonian in the
system. Therefore for the same current density j, hcit becomes smaller in
more strain-relaxed samples.
The influence of strain on hcit is further illustrated by the tensile-strained
micro-bars patterned on (Ga0.94,Mn0.06)(As0.9,P0.1) epilayers (Figure 4.6b):
77
4.3 Data and analysis
j (10 A cm )
6 -2
E (V cm )
-1
μ
μ
0
c
it
h
(
T
)
a b
(Ga,Mn)As
(Ga,Mn)(As,P)
Figure 4.6: Magnitude of the current-induced field in 4 µm and 500 nm-
wide bars. a, j- and E-dependence of hD and hR fields (circles and triangles
respectively), measured on the 4 µm and 500 nm-wide (Ga0.94,Mn0.06)As
devices (blue and orange coloured respectively). The data are fitted to
straight lines. b, Similar j- and E-dependence graphs for hcit measured
in the (Ga0.94,Mn0.06)(As0.9,P0.1) devices.
78
4.3 Data and analysis
j (10 A cm )
6 -2
μ
μ
0
c
it
h
(
T
)
a b
Figure 4.7: Current-induced field in 4 µm-wide (Ga,Mn)As structures with
different Mn concentrations. a, j-dependence of hD in 4 µm-wide bars with
3%, 6% and 12% Mn. The data are fitted to straight lines. b, j-dependence
of hR measured in the same devices.
Both hD and hR change sign compared with the compressively-strained (Ga,Mn)As
devices; furthermore, hcit can still be reduced by introducing strain-relaxation
via lithographic patterning.
Effect of Mn doping levels on CIT (Ga,Mn)As devices with different
Mn levels (3%, 6% and 12%, respectively, in as-grown wafers) have also been
studied in the experiment. Figure 4.7 presents the results measured on a
group of 4 µm-wide bars fabricated from the different films.
Figure 4.7a shows that hD is linear in j. Moreover, the same current
density is observed to induce the largest hD in samples with 3% Mn, and the
induced field decreases with increasing Mn concentration. This is counter-
intuitive at first glance: Samples with 12% Mn should have the largest
growth-strain e0 due to lattice mismatch, therefore Equation 4.8 predicts
that hcit should be the strongest in these materials; whereas experimental
observations suggest the opposite.
This apparent contradiction can be understood by considering the effect
of increasing Mn concentration on the electrical properties of the magnetic
79
4.3 Data and analysis
semiconductor. Since the carrier density p is higher in (Ga,Mn)As films with
higher Mn doping, the quantity Eτ in these samples becomes lower at a fixed
current density j. As Equation 4.7 clearly suggests, the current-induced field
scales linearly with Eτ , it is therefore not surprising for the (Ga,Mn0.12)As
micro-bar to have the lowest hcit among the three devices.
Similar to hD, the field hR due to the Rashba spin-orbit coupling is also
found to be lower in samples with higher Mn levels. In all 3 devices, hR is
smaller than hD, consistent with theoretical findings that the diagonal strain
elements εxx,yy dominate over the off-diagonal strain εxy [115].
4.3.5 Effect of an electrical bias on the FMR signals
It has been demonstrated in spin-valves and tunnel junctions, that a dc spin-
polarised current can induce magnetic motion via the spin-transfer torque ef-
fect [23]. Furthermore, it has been observed that STT from the dc current can
induce different types of magnetic excitation [23, 32, 33], and, when combined
with a STT-FMR technique [34, 35], allows the spin-transfer “torkance”1 to
be accurately determined [63].
In the final experiment, both a direct and microwave frequency current
are injected into a (Ga,Mn)As bar, and the photovoltage is measured2 while
sweeping H0. Preliminary analysis has revealed that the dc current alters
the lineshape of the FMR signal, and this effect depends on the magnitude
and direction of the current.
Figure 4.8a shows the evolution of the FMR peak as the dc bias current is
varied. At Idc = 0, the FMR peaks is predominantly anti-symmetric resulting
from the in-plane driving field. Upon the application of a dc current, the
FMR lineshape is significantly altered and a large symmetric component Vsym
appears. The magnitudes of both Vsym and Vasy are calculated from fittings
according to Equation 2.25 and are plotted in Figure 4.8b. Bias currents
1The torkance is defined as dτ/dV , where V is the dc bias voltage on the spin-valve
or tunnel junction.
2Two high-pass filters with 100 Hz cut-off frequency are connected to each end of the
voltmeter, so that only the modulated photovoltage (at 987.6 Hz) is measured. Without
the filters, the large dc bias would overload the voltmeter.
80
4.3 Data and analysis
μ H (T)0 0
V
(
T
)
d
c
μ
a b
V
(
T
)
a
s
y
(s
y
m
)
μ
Idc (μA)
Figure 4.8: Effect of a direct current on the CIT-FMR signal. a, The mod-
ulated photovoltage Vdc measured under different bias currents. The FMR
peak is clearly visible in each spectrum. A large current-dependent back-
ground is subtracted from Vdc before it is plotted. b, The bias-dependence of
the symmetric and anti-symmetric Lorentzian components Vsym,asy, deduced
from fittings of each FMR peak in a according to Equation 2.25. The mea-
surements are performed on a 350 nm-wide bar patterned on an as-grown
(Ga0.94,Mn0.06)As wafer. H0 is swept along ϑ = 225
o.
81
4.4 Conclusions
Idc up to ±30 µA have been applied, which corresponds to bias voltage of
±2.55 V (the device has nominal resistance ∼ 85 kΩ).
It has been observed in previous FMR experiments on permalloy that a
dc bias current alters the amplitude of the FMR signal [80, 84, 85], due to a
bias-dependent term in the time-varying current I(t):
I(t) = Imw cosωt+ Idc (R(t)) (4.9)
Here the first term in the expression of I(t) is the microwave current in-
side the device; and the second term is caused by the oscillating resistance
in the sample, and hence also varies at frequency ω. Following Ohm’s law,
this contribution to the FMR signal should scale linearly with Idc. This is
indeed observed in Figure 4.8b, and the small nonlinearity at large biases
(±20 µA) can be attributed to heating. Furthermore, according to Ref. [80],
the DC bias current creates FMR signals with symmetric Lorentzian line-
shape (Figure 4.8b), with the amplitude Vsym varies as cos 2ϑ. It is therefore
important in the next stage of the experiment to measure the ϑ-dependence
of the observed Vsym component.
In addition to altering the FMR amplitude, a direct current can also
generate a dc effective magnetic field hDCcit , and this has been observed to
shift the resonance position Hres in ESR experiments [57]. However, this
effect is not observed in the present device due to its large resistance and
the considerable amount of heating generated by the current. In order to
induce an observable shift in Hres, according to Figure 4.6, a dc bias current
of 1 mA is required to generate hDCcit on the order of 2 mT in a 500 nm-wide
(Ga,Mn)As bar1.
4.4 Conclusions
In this Chapter, the first experimental demonstration of an oscillating current-
induced torque (CIT) in uniform ferromagnetic systems is reported. Ferro-
magnetic resonance has been created in microstructures patterned on diluted
1In a 4 µm-wide device, a dc bias current of 5 mA is required to generated hDCcit ≈ 2 mT.
82
4.4 Conclusions
magnetic semiconductors (Ga,Mn)As and (Ga,Mn)(As,P), using a microwave
frequency CIT, and detected electrically via frequency mixing between the
microwave current and the anisotropic magnetoresistance (AMR).
The magnitude and direction of the magnetic field leading to CIT have
been measured. The results confirm that the current-induced field switches
sign depending on the growth-strain of the wafer (compressive or tensile);
furthermore, its magnitude can be controlled by lithography-induced strain-
relaxation, or by varying the carrier densities. These observations suggest
possible methods for customising the spin-orbit coupling in future spintronic
devices.
Finally, the experiments described in this Chapter have established a new
type of scalable, all-electrical FMR technique (CIT-FMR) which provides an
unprecedented method to study the nature of CIT and to perform magnetic
characterisation on ferromagnetic microstructures, which are not limited to
magnetic semiconductors, but can also be ferromagnetic metal thin films.
83
4.5 Appendices
μ0 0H (T)
V
(
V
)
d
c
μ
Figure 4.9: A photovoltage spectrum measured in CIT-FMR experiment
showing the FMR peak (squares), together with its fitted results according
to Equation 2.25 (green solid line). Inset: The symmetric (orange) and anti-
symmetric (blue) Lorentzian components of the FMR peak, with amplitudes
given as Vsym and Vasy respectively.
4.5 Appendices
4.5.1 Symmetric Lorentzian component Vsym
Figure 4.9 illustrates the two Lorentzian components (Vsym and Vasy) which
make up a FMR signal detected in the photovoltage spectrum. For an in-
plane driving field, it is expected that the FMR lineshape should be per-
fectly anti-symmetric and Vsym → 0, according to Equation 2.27. On the
other hand, for a perpendicular-to-plane driving field as in the experiments
described in Chapter 3, one expects the FMR peak to be perfectly symmetric
and Vasy → 0.
However, in most real FMR experiments, the lineshape of the resonance
signal is rarely perfect and usually contains a superposition of symmetric and
anti-symmetric Lorentzians. Among the published literatures, several paral-
lel explanations exist for the origin of this observed complex FMR lineshape:
• In device-scale FMR experiments on permalloy and iron samples, the
84
4.5 Appendices
complex resonance peak is explained by the phase difference1 (ψ) be-
tween the microwave current and the driving field [80, 81, 83], in which
case the symmetric Lorentzian contains a sinψ term whereas the anti-
symmetric Lorentzian contains a cosψ term.
• In conventional cavity-based FMR experiments, an anti-symmetric com-
ponent in the energy absorption spectrum (which should in theory be
perfectly symmetric) has been attributed to non-collinearity between
M and H0 [78], caused by the large magnetocrystalline anisotropy in
(Ga,Mn)As.
• In spin-transfer torque (STT) driven FMR experiments on magnetic
tunnel junctions (MTJs), the appearance of a complex FMR lineshape
is interpreted as arising from a strong perpendicular component of the
driving torque (or “field-like” term as opposed to the STT term) [34].
However, later experiments on similar device structures attribute this
complexity to a superposition of two normal modes from the different
ferromagnetic layers of the MTJ [36, 63]. Moreover, in STT-FMR
experiments on spin-valves, the asymmetric FMR lineshape caused by
large microwave driving current is attributed to nonlinear oscillation
effects [35].
• Besides the explanations mentioned above, inhomogeneity in the driv-
ing field and non-uniformity of the device would almost certainly play
a role in creating the complex FMR signals.
Examining the various possible causes for the complex FMR lineshape
shown in Figure 4.9, two explanations are most plausible: (i) The driving
field has a component in the z-direction: hcit = (hx, hy, hz)e
iωt, or (ii) There
1Some possible causes for the phase shift include: (i) The Eddy current inside the
sample originates from both inductive and capacitive coupling, with a phase difference of
pi/2 between the two terms, and the total effective current can be understood as possessing
a phase shift with respect to the varying resistance. (ii) The starting point of magnetisation
precession, which occurs as the resonance condition is satisfied, and the starting point
of the microwave current, which occurs as the microwave source is turned on, are not
necessarily synchronised, giving rise to a phase difference.
85
4.5 Appendices
[110]
[010] [100]
ϑ (deg.)
V
(
V
)
d
c
μ
a
b
c
d
[110]
Figure 4.10: Amplitudes of the symmetric and anti-symmetric Lorentzian
parts Vsym and Vasy measured on a group of 4 µm-wide (Ga,Mn)As bars
(solid and hollow circles, respectively), patterned along different crystalline
directions. The solid lines are fitted results to Equation 2.27.
exists a phase difference ψ between the microwave current I cosωt and the
driving field hcit.
Figure 4.10 compares the amplitudes of the symmetric and anti-symmetric
Lorentzians in a group of 4 µm-wide (Ga,Mn)As bars. No lineshape with
sin 2ϑ symmetry can be assigned to the Vsym(ϑ) function within experimen-
tal errors, hence an out-of-plane component of hcit is ruled out as the cause of
the symmetric Lorentzian in the measured FMR signals. Therefore it is con-
cluded that the current-induced field hcit is in-plane (hz = 0), and the com-
plex shape of the FMR signals is due to the phase difference ψ, which can be
determined from the ratio between Vsym and Vasy according to Equation 3.1.
For example, for the [100]-bar shown in Figure 4.10, it is estimated that the
driving field leads the current by 12o. Similar observations have also been
made on the 500 nm-wide (Ga,Mn)As devices and on the (Ga,Mn)(As,P)
micro-bars.
86
4.5 Appendices
4.5.2 Calibrating the microwave current using Joule
heating
The power of the microwave used in the CIT-FMR experiments is 20 dBm
from the signal generator. This is reduced by many dB when it reaches
the sample, due to frequency-dependent losses in the microwave circuit and
strong reflection at the transmission line/device interface. However, in order
to calculate the magnitude of the current-induced field hcit, the strength
of the microwave current inside the sample I cosωt needs to be accurately
determined. This is achieved using the bolometric effect.
The resistance of the magnetic semiconductors (Ga,Mn)As and (Ga,Mn)(As,P)
is strongly temperature-dependent (Figure 4.11a). Therefore by comparing
the resistance change caused by Joule heating due to both a known direct cur-
rent and the microwave, the level of the microwave current inside the sample
can be determined (Figure 4.11b). Note that the direct current corresponds
to the effective value of I cosωt, i.e. its root-mean-square (rms) value.
The calibrated microwave current Imw is plotted against the microwave
voltage Vmw in Figure 4.11c, in which Vmw is the rms voltage output into the
50 Ω transmission line. The ohmic behaviour exhibited by the Imw − Vmw
data confirms the validity of the calibration method.
4.5.3 Vector magnetometry on the (Ga,Mn)(As,P) de-
vices
Photovoltage from in-plane scans Figure 4.12 shows the ϑ-dependence
of the photovoltage amplitude Vasy, measured on the 4 µm-wide (Ga,Mn)(As,P)
bars patterned along the [11¯0], [110], [010] and [100] crystalline directions.
The Vasy(ϑ) curves are reversed compared to those measured in the (Ga,Mn)As
samples (Figure 4.5a and b), implying that the directions of hD and hR have
changed signs due to the tensile-strained nature of the (Ga,Mn)(As,P) wafer.
It is noticed that in the [100] and [010] bars, the Vasy curve is shifted
and Vasy(ϑ = 0
o) 6= 0. This can be explained by a large uniaxial crystalline
term CU along [110] in the magnetoresistance of the (Ga,Mn)(As,P) devices
87
4.5 Appendices
T (K) I ( A)dc µ
P (dBm)mw
I ( A)mw µ
R
(k
)
x
x
Ω
V
(V
)
m
w
a cb
T (K)
R
 (
k
)
Ω
Figure 4.11: Deducing the microwave current using Joule heating. a, The
temperature dependence of a device’s longitudinal resistance Rxx(T ), in the
temperature range where the measurements are carried out (6 K). Inset: The
sample resistance measured over the entire temperature range 2.5 – 270 K.
b, The change in sample resistance caused by Joule heating from both a
known direct current R(Idc) and the microwave current, which is recognised
only by the output power from the signal generator R(Pmw). Typical loss
on the microwave cabling is estimated to be ∼ 20 dB. c, The calibrated
Imw − Vmw curve (triangles), where Vmw is the rms voltage output into the
50 Ω transmission line. The calibration errors are smaller than the size of the
triangles in the graph (∼ 0.3 µA), and are caused by uncertainties in finding
the exact value of the direct current corresponding to each microwave power.
The I − V plots exhibits ohmic behaviour (The red solid line is a linear fit).
88
4.5 Appendices
[110]
[010] [100]
ϑ (deg.)
V
(
V
)
a
s
y
μ
a
b
c d
[110]
Figure 4.12: The photovoltage amplitude Vasy measured on 4 µm-wide
(40 µm-long) (Ga,Mn)(As,P) bars, patterned along the [11¯0], [110], [010] and
[100] crystalline directions, respectively (black squares). The green solid lines
are the fitted results to either a sin(2ϑ) cos(ϑ) function (a and b), or a super-
position of sin(2ϑ) cos(ϑ) and sin(2ϑ) sin(ϑ) functions (c and d). Note that
there exists a relatively large sin(2ϑ) contribution to the Vasy(ϑ) measured
on the [110] bar (b), possibly caused by an out-of-plane Oersted field. After
removing this contribution, the remaining data exhibits a clear sin(2θ) cos(θ)
curve (inset to b).
89
4.5 Appendices
R
(k
)
x
x
Ω
ϑ (deg.)
[100] [010]
(Ga,Mn)(As,P)
(Ga,Mn)As
a b c
d e f
Δ
Ω
R
 (
)
[100] [010]
[100] [010]
Figure 4.13: The longitudinal resistance measured on 4 µm-wide
(Ga,Mn)(As,P) and (Ga,Mn)As bars. a, Rxx(ϑ) measured on the [100] bar
(black circles). The red line is the fitted results according to Equation 2.8,
taking into account of both non-crystalline (CI) and uniaxial crystalline (CU)
AMR components. The measurement is performed by rotating a saturation
field of 0.8 T and measuring with a probing voltage of 10 mV. b, Rxx(ϑ)
measured on the [010] bar. c, The AMR coefficients CI and CU in the [100]
and [010] micro-bars determined from fittings. d – f, The longitudinal resis-
tance measured in similar 4 µm-wide (Ga,Mn)As devices, with the fitted CI
and CU terms.
(cf. Equation 2.8). Figure 4.13 compares the longitudinal resistance Rxx
measured on (Ga,Mn)(As,P) and (Ga,Mn)As micro-bars in rotating field
experiments. The large CU term in (Ga,Mn)(As,P) devices can be clearly
recognised, which is caused by the tensile-strain in the wafers.
Photovoltage from out-of-plane scans Besides in-plane rotational scans
of H0, out-of-plane field sweep measurements are also carried out on the
90
4.5 Appendices
H0
z
x
ϑ
hD
hR
Figure 4.14: Principle of an out-of-plane rotational scan experiment. For a
micro-bar patterned along the [100]/[010] direction, the component of the
current-induced field hcit due to the Rashba spin-orbit interaction hR (blue
arrow) now becomes perpendicular to the plane of rotation, whereas the one
due to the Dresselhaus interaction hD (orange arrow) remains in-plane.
(Ga,Mn)(As,P) devices1. Figure 4.14 illustrates the experimental concept of
an out-of-plane rotational scan. For a micro-bar patterned along the [100] or
[010] direction, the component of the current-induced field due to the Rashba
spin-orbit coupling hR becomes perpendicular to the plane of field rotation,
whereas the one due to the Dresselhaus interaction hD remains in-plane.
The photovoltage amplitudes from the out-of-plane rotational scan on
a [100]-bar are plotted in Figure 4.15a–b. The graphs clearly show that
the two components of the driving field, hR and hD, are now decomposed
and mapped separately into the symmetric and anti-symmetric Lorentzian
parts of the FMR signals, respectively. The amplitude of the symmetric
Lorentzian Vsym(ϑ) clearly shows a sin(2ϑ) lineshape, which confirms the
existence of an perpendicular-to-plane driving field (hR); whereas the anti-
symmetric Lorentzian part is caused by an in-plane driving field (hD), ex-
hibiting sin(2ϑ) sin(ϑ) dependence (in-plane and along the direction of the
1The out-of-plane rotational scans are only possible on (Ga,Mn)(As,P) devices using
the current apparatus. For compressively-strained (Ga,Mn)As devices, the resonance field
Hres can be greater than 1 T in out-of-plane directions, as shown in Ref [74], hence is too
large for the superconducting magnets on this cryostat to generate.
91
4.5 Appendices
bar).
In comparison, for an out-of-plane rotational scan on a [11¯0] bar, only a
perpendicular-to-plane driving field is present since hR and hD are collinear
along this direction. Figure 4.15c shows that hcit leads to predominantly
symmetric FMR peaks with Vsym(ϑ) ∝ sin 2ϑ. The anti-symmetric part of
the FMR peaks (Figure 4.15d) in this device is much smaller, and is due
to the phase difference ψ between the current and the driving field. These
observations are consistent with the derivations given in Section 2.4.3 and
Appendix A.
By comparing the amplitude of Vasy shown in Figure 4.12d and Fig-
ure 4.15a–b, it is clear that both in-plane and out-of-plane measurements
can be used for deducing the current-induced fields hR and hD. Note that
Vasy appears to be smaller in the out-of-plane scans (Figure 4.12d), since a
17 GHz microwave current is used in these experiments, which suffers higher
loss in the coaxial cable than the 10 GHz current used on the in-plane ro-
tational scans. Similar observations can also be made by comparing the
in-plane and out-of-plane scan data on the [11¯0] micro-bar (Figure 4.12a and
Figure 4.15c).
4.5.4 Magnetic properties of the (Ga,Mn)As microde-
vices
Magneto-transport measurements Figure 4.16 shows the results from
transport measurements on 4 µm-wide (Ga,Mn)As micro-bars with 3%, 6%
and 12% Mn. Using the temperature-derivative of the longitudinal resistance
Rxx [102], the Curie temperatures in the three materials are found to be
73, 132 and 169 K respectively. The results from the AMR measurements1
(Figure 4.16b) are used to determine ∆R required for calculating the current-
induced field hcit (as shown in Figure 4.7). The non-crystalline AMR term
is observed to be dominant in all three devices.
1Note that AMR is defined as (Rxx−Rave)/Rave, where Rave is the average longitudinal
resistance in the rotational measurement.
92
4.5 Appendices
ϑ (deg.)
V
(
V
)
s
y
m
μ
a
b
V
(
V
)
a
s
y
μ
[100] bar
[110] bar
c d
Figure 4.15: a, The photovoltage amplitude Vsym from an out-of-plane rota-
tional scan experiment on a 4 µm-wide (Ga,Mn)(As,P) bar, patterned along
the [100] direction (hollow triangles). The red solid line is the fitted result to
a sin(2ϑ) function. b, The amplitude of the anti-symmetric Lorentzian Vasy
measured on the same device (hollow squares). The red solid line is the fitted
results to a sin(2ϑ) sin(ϑ) function. c – d, Vsym(ϑ) and Vasy(ϑ) curves from
an out-of-plane scan on a [11¯0] bar (solid triangles and squares respectively),
together with fittings to a sin(2ϑ) function (red solid line).
93
4.5 Appendices
T (K)
d
R
/d
T
(k
/K
)
x
x
Ω
ϑ (deg.)
A
M
R
 (
%
)
a b
Figure 4.16: Transport measurements on (Ga,Mn)As micro-bars with differ-
ent Mn concentrations (3%, 6% and 12%). a, The temperature derivative
dRxx/dT in the three devices, giving TC = 73, 132 and 169 K respectively. b,
Longitudinal AMR measured in the three devices. All samples are 4 µm-wide
and are fabricated on annealed wafers (190oC in air for 20 h).
Effect of growth-strain on the magnetic anisotropy The magnetic
anisotropy in the (Ga,Mn)As and (Ga,Mn)(As,P) microstructures are also in-
vestigated using CIT-FMR. The anisotropy in tensile-strained (Ga,Mn)(As,P)
devices is found to be significantly different from that in the compressively-
strained (Ga,Mn)As samples, and is investigated in details in Chapter 5.
Figure 4.17 plots the photovoltage spectra from in-plane rotational scans
of H0, as well as the fitted resonance peak positions Hres, measured on a
group of 4 µm-wide (Ga,Mn)As samples with different Mn doping. The
large device size ensures that strain-relaxation plays an insignificant role in
defining the magnetic anisotropy. The data demonstrates that as the level of
Mn doping increases, the magnetic anisotropy changes from predominantly
biaxial along [100]/[010] (fourfold symmetry due to the zinc-blende crystal
structure) to uniaxial along [11¯0] (twofold symmetry due to strain).
The in-plane anisotropy constants Hi and the effective magnetisationMeff
are deduced by fitting Hres(ϕ) to Equation 3.2 and are listed in Table 4.1.
(Ga,Mn)As wafers with higher Mn dopings are under more compressive-
strain, due to the larger lattice mismatch with the GaAs substrates. This in
turn leads to larger H2⊥ field, which is reflected by the increase of Meff with
94
4.5 Appendices
µ0Meff (mT) µ0H4‖ (mT) µ0H2‖ (mT)
3% 373 150.7 6.6
6% 478 21.7 27.2
12% 517 5.2 61.2
Table 4.1: The in-plane anisotropy fields Hi and the effective magnetisation
Meff, determined using the CIT-FMR technique, in (Ga,Mn)As microdevices
with different Mn levels. All three devices are 4 µm×40 µm and are patterned
along the [010] crystalline direction.
Mn levels (Meff =Ms −H2⊥).
Observation of patterning-induced strain-relaxation in narrow struc-
tures Numerical calculations have shown that strain-relaxation is spatially
nonuniform and usually takes place on a length scale of several hundred
nanometres [26, 115]. This effect has been observed to play an important
role in defining the magnetic anisotropy in submicron (Ga,Mn)As devices
[65–67]. The magnetic anisotropy profiles on the 500 nm-wide bars, mapped
with the CIT-FMR technique, corroborate the findings from earlier studies.
Figure 4.18a–d plot the resonance fieldHres detected on a group of 500 nm-
wide (Ga,Mn)As bars (6% Mn). In the [100] and [010] bars (Figure 4.18a
and 4.18b), the existence of the strain-relaxation-induced term HU strongly
distorts the magnetic anisotropy profile of bulk wafers (as in Figure 4.17d),
and causes asymmetry in Hres between the [100] and [010] directions. In
the [110] and [11¯0] bars (Figure 4.18c and 4.18d), HU is collinear with the
intrinsic uniaxial anisotropy H2‖, and acts to either enhance or reduce the
magnetic energy in the [110]/[11¯0] directions.
From the figures, it is also concluded that HU has its easy axis along the
direction of the bar, i.e. perpendicular to the direction of lattice relaxation.
This is consistent with earlier studies using SQUID, magnetotransport and
FMR measurements [26, 65–67]. Table 4.2 lists the anisotropy constants Hi
in these devices deduced from fittings.
95
4.5 Appendices
φ (deg.)
μ
0
re
s
H
(T
)
3%
6%
12%
μ
0
0
H
(T
)
φ (deg.)
a b
c d
e f
[1
0
0
]
[1
1
0
]
[0
1
0
]
[1
1
0
]
[1
0
0
]
[1
1
0
]
[0
1
0
]
[1
1
0
]
[1
0
0
]
V
(
V
)
d
c
µ
Figure 4.17: Magnetic anisotropy profiles in (Ga,Mn)As microstructures with
different Mn concentrations. a – b, The photovoltage spectra from in-plane
rotational scans on a 4 µm-wide device with 3% Mn, together with the ϕ-
dependence of the resonance field Hres (black circles). The red solid line is
the fitted results according to Equation 3.2. c – d, Similar measurements
on a micro-bar with 6% Mn. e – f, Measurements on a micro-bar with 12%
Mn. All three devices are 4 µm × 40 µm and are patterned along the [010]
crystalline direction.
96
4.5 Appendices
[100] [010]
[110] [110]
μ
0
re
s
H
(T
)
φ (deg.)
a b
c d
[1
0
0
]
[1
1
0
]
[0
1
0
]
[1
1
0
]
[1
0
0
]
Figure 4.18: Magnetic anisotropy profiles on a group of 500 nm-wide
(Ga0.94,Mn0.06)As bars, patterned along [100], [010], [11¯0] and [110] crys-
talline directions. The black circles are experimental data, and the red solid
lines are fitted results according to Equation 3.2. The black arrows mark the
directions of the nano-bars.
97
4.5 Appendices
µ0H4‖ (mT) µ0H2‖ (mT) µ0HU (mT)
[100] 1.9 13.8 -32.6
[010] 8.0 23.3 18.9
[11¯0] 5.7 62.8 ‡
[110] 7.7 -24.8 ‡
Table 4.2: In-plane magnetic anisotropy constants in a group of 500 nm-wide
(Ga,Mn)As bars (6% Mn), patterned along different crystalline directions.
The strain-relaxation-induced anisotropy HU > 0 when its easy axis is along
the [010] direction; and it is not separately deduced in the [11¯0] and [110]
bars since it is collinear with H2‖.
4.5.5 Effects of annealing
Magnetic anisotropy Finally, the effects of annealing on the magnetic
anisotropy and hcit are investigated. Figure 4.19 compares the magnetic
anisotropy profiles of a 500 nm-wide (Ga0.94,Mn0.06)As bar before and af-
ter it has been annealed. The anisotropy changes from a superposition of
twofold and fourfold symmetry in the as-grown sample (due to HU and H4‖
respectively), to a pronounced twofold symmetry (due to HU) after anneal-
ing (160oC in air for 10 h). Similar measurements are also performed on the
other micro-bar from the same sample chip, and the effective magnetisation
Meff and anisotropy fields Hi are compared in Table 4.3. The changes in
magnetic anisotropy upon annealing are summarised as follows:
• The effective magnetisation Meff increases after annealing. This agrees
with observations from early FMR experiments on (Ga,Mn)As bulk
wafers (8% Mn) [73, 74]. It is due to an increase in the perpendic-
ular uniaxial anisotropy H2⊥, which can be attributed either to the
improvement of the sample homogeneity or (more likely) to an increase
of the hole concentration after the removal of interstitial MnI ions by
the annealing process [74].
• The in-plane biaxial anisotropy H4‖ is dramatically reduced by an-
nealing. This is due to decrease in lattice mismatch in the annealed
98
4.5 Appendices
φ (deg.)
μ
0
re
s
H
(T
)
a
b
as-grown
annealed
Figure 4.19: Magnetic anisotropy profiles of a 500 nm-wide (Ga,Mn)As bar
(6% Mn), before (a) and after (b) it has been annealed at 160oC in air for
10 h. The device is patterned along the [100] axis. In the figures, the black
circles are the experimental data, and the red solid lines are fitted results
using Equation 3.2.
(Ga,Mn)As epilayer, as revealed by XRD studies [138], and hence re-
duction in the compressive-strain.
• The weak intrinsic in-plane uniaxial anisotropy H2‖ grows after an-
nealing. This observation is related to a type of spin reorientation
transitions, in which H2‖ rotates its easy axis from [11¯0] direction to
[110] direction upon annealing. This effect is first noticed in SQUID
measurements [110] and later corroborated by FMR experiments [74].
• Annealing also influences the strain-relaxation-induced anisotropy HU .
This is understandable as both the strain and hole concentration are
changed upon annealing. However, the two nano-bars exhibit oppo-
site response to the annealing process, and further investigations are
therefore required.
Angle dependence of the FMR lineshape Even more intriguing is the
impact of annealing on the FMR lineshape. In Figure 4.20, ϑ-dependence of
99
4.5 Appendices
µ0Meff (mT) µ0H4‖ (mT) µ0H2‖ (mT) µ0HU (mT)
[100] bar as-grown 416.9 43 0.4 -28.2
annealed 534.2 1.9 13.8 -32.6
[010] bar as-grown 397.7 44.2 7.9 30
annealed 500.6 8 23.3 18.9
Table 4.3: The effective magnetisation Meff and in-plane anisotropy fields Hi
in two (Ga,Mn)As bars (500 nm-wide) before and after annealing (160oC in
air for 10 h). HU > 0 indicates that its easy axis is along the [010] direction.
the anti-symmetric component Vasy before and after annealing are compared.
The same 500 nm-wide (Ga0.94,Mn0.06)As bars are used.
In the as-grown sample, the Vasy(ϑ) function contains a complex line-
shape which cannot be simply explained using the equations derived in Sec-
tion 2.4.3. Furthermore, Vasy(ϑ) in both [010] and [100] bars (Figure 4.20a
and 4.20c) displays similar lineshape. Figure 4.21 investigates the contri-
bution from the crystalline AMR coefficients to the Vasy(ϑ) lineshape. It is
found that a large cubic crystalline term is required in order to properly fit
to the experimental data, with CC ≥ CI (see Figure 4.21b inset), which is
not present in the longitudinal resistance Rxx measured on this device (Fig-
ure 4.21a inset). This confirms that AMR is not the cause of the observed
Vasy(ϑ) curve. Consistent observations have been made among several as-
grown devices along the [100]/[010] directions, and this has become a great
mystery in the early stages of the CIT-FMR project.
Upon annealing, the lineshape of the Vasy(ϑ) function changes signifi-
cantly, and can now be well-described by an in-plane driving field hcit =
(hx, hy, 0)e
iωt, with hx switching sign between the [010] and [100] bars. The
AMR coefficients are largely reduced upon annealing, with CI : 3.2% →
0.57%, CU : 0.6% → 0.1% and CC: 0.36% → 0.023%. However, this cannot
account for the drastic change in the shape of the Vasy(ϑ) function. Instead,
it is attributed to the improvement in sample homogeneity by the removal
of MnI defects. This demonstrates the complexity of the current-induced
torque in as-grown, inhomogeneous ferromagnets.
100
4.5 Appendices
[010] bar
as-grown annealed
[100] bar
as-grown
annealed
ϑ (deg.)
V
(
V
)
a
s
y
μ
a b
c
d
Figure 4.20: The amplitudes of the anti-symmetric Lorentzian Vasy(ϑ) mea-
sured before and after annealing, on the [010] bar (a and b, blue circles) and
the [100] bar (c and d, orange circles). The red solid lines are fitted results
to a A1 sin(2ϑ) cos(ϑ) +A2 sin(2ϑ) sin(ϑ) function with variables A1 and A2.
101
4.5 Appendices
ϑ (deg.)
R
(k
)
x
x
Ω
ϑ (deg.)
V
(
V
)
d
c
μ
a b
Figure 4.21: The AMR components and their contributions to the longitudi-
nal resistance and the FMR lineshape, measured on the as-grown [100] bar
(500 nm-wide). a, The longitudinal resistance Rxx measured in a field ro-
tation experiment with µ0H0 = 0.8 T (circles: experimental data, red line:
fittings using Equation 2.8). Inset: Values (in Ω) of the non-crystalline and
first order crystalline (uniaxial and cubic) AMR coefficients CI , CU and CC .
CI is found to be the dominant contribution to the Rxx(ϑ) function. b,
ϑ-dependence of Vasy (orange circles), and the black solid line is the fitted
results taking into account of both the non-crystalline and crystalline terms
in the Rxx expression. Inset: Contributions (in µV) from the non-crystalline,
uniaxial and cubic crystalline terms to Vasy(ϑ). Contrary to the values found
from a, a large cubic crystalline term is required to describe the lineshape of
Vasy(ϑ), which does not exist in the measured Rxx(ϑ) data.
102
4.5 Appendices
as-grown annealed
[110] bar
as-grown annealed
ϑ (deg.)
V
(
V
)
a
s
y
μ
a b
c d
[110] bar
Figure 4.22: Vasy(ϑ) measured before and after annealing, on a 500 nm-wide
[11¯0] bar (a and b, purple circles) and a [110] bar (c and d, green circles).
The red solid lines are fitted results to sin(2ϑ) cos(ϑ) functions.
The effects of annealing on [110] and [11¯0] nano-bars are also studied and
compared in Figure 4.22. It is recognised that hcit also possesses complexity
in the as-grown [11¯0] bar, and the measured Vasy(ϑ) curve cannot be described
by a simple sin(2ϑ) cos(ϑ) function (Figure 4.22a). In the [110] device, on
the other hand, the Vasy curve is already well-defined in the as-grown sample,
implying an excitation field hcit = (0, hy, 0)e
iωt (Figure 4.22c). Nevertheless,
the removal of defects with low temperature annealing is also observed to
improve the behaviour of hcit in both devices (Figure 4.22b and 4.22d).
4.5.6 Gating experiments on ultrathin (Ga,Mn)As films
Besides the main experiments described in this Chapter, attempts have
also been made on studying both hcit and magnetic anisotropy in gated
(Ga,Mn)As devices, with Au/dielectric/(Ga,Mn)As structures. These ex-
periments are still on-going, and limited results have been obtained so far.
103
4.5 Appendices
50 mμ
Figure 4.23: Micrograph of the gated (Ga,Mn)As device. The rectangular
channel in the centre of the graph is the 40 µm × 4 µm device, with metal
contacts on each end (Cr/Au, light yellow colour). The AlOx dielectric has
a distinct blue colour. The brown-coloured rectangular stripe covering the
device channel is the gate electrode (Cr/Au).
The (Ga0.94,Mn0.06)As films used in this study are either 5 nm- or 7 nm-
thick. These wafers have very low carrier concentration while still exhibit
ferromagnetism, making them ideal for electrical gating experiments [10, 12,
139].
Two types of dielectric have been used. The first is a ferroelectric mate-
rial, copolymer polyvinylidene fluoride with trifluoroethylene P(VDF-TrFE).
It is recently employed in a (Ga,Mn)As–FET structure and has been demon-
strated to change the Curie temperature and magneto-transport properties
of (Ga,Mn)As upon charging [139]. However, when applying CIT-FMR on
(Ga,Mn)As devices coated with ferroelectric P(VDF-TrFE), no resonance
signals can be detected. This is possibly due to P(VDF-TrFE) having a
small dielectric constant, hence causing the microwave current to leak into
the gate electrode instead of flowing into the sample.
The second type of dielectric is aluminium oxide AlOx, which is deposited
onto the (Ga,Mn)As wafer using atomic layer deposition (ALD) with a nom-
inal thickness of 40 nm (Figure 4.23). Measured at 46 K, charging the gate
electrode to Vg = ±20 V leads to change in the sample resistance by 10%,
104
4.5 Appendices
from 110 kΩ at +20 V to 91 kΩ at −20 V (data measured by Hidekazu
Kurebayashi and Philip Chow [140]). This small change in resistance sug-
gests that the gate voltage is unable to significantly accumulate/deplete the
charge carriers. As a result, no change in the magnetic anisotropy has been
measured between Vg = ±20 V within the experimental error. In the con-
trary, a recent study by Chiba et al. [10] has revealed resistance change of
more than one order of magnitude using a gate electric field of ±12 V at
T = 2 K, in a 4 nm-thick (Ga0.9,Mn0.1)As device with 31 nm ZrO2 dielectric.
Future CIT-FMR experiments using AlOx dielectric will therefore focus
on thinner devices, in which significant resistance change can be induced by
electrical gating.
105
Chapter 5
Mapping the magnetic
anisotropy in (Ga,Mn)(As,P)
nanostructures
5.1 Introduction
(Ga,Mn)(As,P) is a new member of the (III,Mn)V diluted magnetic semicon-
ductor family, first synthesised in 2007 [90]. Since the phosphorus atoms are
smaller than the arsenic atoms, the (Ga,Mn)(As,P) wafers become tensile-
strained for P concentrations higher than ∼ 6%, leading to out-of-plane mag-
netic anisotropy1 [77]. The corresponding perpendicular-to-plane easy axis
is beneficial for studies such as current-induced domain wall motion [24] and
gating-controlled ferromagnetism [9, 10, 12], in which case the magnetisa-
tion switching and domain wall movement can be directly detected using
Kerr microscopy [142, 143] and the anomalous Hall effect [9, 142]. However,
the more conventional (Ga,Mn)As can only become tensile-strained if it is
grown on a relaxed InGaAs buffer [95], and this could result in high density
of line defects, causing incoherent magnetisation switching due to multiple
1(Ga,Mn)As layers grown directly on GaAs substrates are compressively-strained due
to lattice mismatch, and this leads to in-plane magnetic easy axes for Mn concentrations
> 2%, unless the carrier density is very low [141].
106
5.1 Introduction
nucleation and domain wall pinning sites [144, 145]. Therefore the tensile-
strained (Ga,Mn)(As,P) provides a good test-bench to study magnetisation
reversal processes.
Properties of (Ga,Mn)(As,P) bulk wafers have so far been studied, and
change in magnetic anisotropy by varying the phosphorus doping concentra-
tion at the crystal growth stage has been especially investigated [77, 92–94].
In the case of (Ga,Mn)As, various methods for manipulating the magnetic
anisotropy have been established, such as carrier depletion/accumulation
via electrostatic gating [9–12], applying mechanical stress [5–8], or induc-
ing strain-relaxation by lithographic patterning [26, 65–67]. It is an aim to
investigate these effects also in the novel (Ga,Mn)(As,P) system.
In this Chapter, CIT-FMR has been applied to (Ga,Mn)(As,P) nanos-
tructures as small as 80 × 1000 × 25 nm3, and the first systematic study of
the effect of strain-relaxation on the magnetic anisotropy in these samples is
reported. The strong spin-orbit and exchange interactions in (Ga,Mn)(As,P)
provide the prerequisites for CIT-FMR, and the following insights have been
achieved:
• The tensile-strain-relaxation in patterned (Ga,Mn)(As,P) bars gener-
ates an additional in-plane uniaxial anisotropy perpendicular to the
bar direction, which becomes the dominant term in very narrow bars
(80 nm-wide). This effect is found to be larger than previously reported
in strain-relaxed (Ga,Mn)As patterned devices [65–67].
• The strain-relaxation-induced anisotropy in bars patterned from compressively-
strained material (in this case, (Ga,Mn)As wafers with the same nomi-
nal Mn concentration as the (Ga,Mn)(As,P) epilayers) has the opposite
sign.
• The patterning process can also affect the magnetic easy axis of the
epilayers, tilting it from the out-of-plane position to partially in-plane.
107
5.2 Devices and experimental methods
5.2 Devices and experimental methods
25 nm-thick (Ga0.94,Mn0.06)As and (Ga0.94,Mn0.06)(As0.9,P0.1) wafers have
been used in this experiment. They are prepared using LT-MBE by Richard
Campion, Arianna Casiraghi and coworkers at the University of Nottingham.
Figure 5.1a shows a SEM image of a 80 nm-wide, 1 µm-long bar device. It
is fabricated by patterning isolation trenches on the (Ga,Mn)(As,P) epilayers.
The microwave frequency current-induced torque is used to drive magneti-
sation precession in these devices, and signal detection is achieved using the
frequency mixing effect (Section 3.2.3). The lock-in technique is employed
to increase the measurement sensitivity, using a modulation frequency1 of
23.7 Hz. A schematic of the experimental setup is shown in Figure 5.1b.
5.3 Data and analysis
A total number of 26 (Ga,Mn)(As,P) nanodevices with various sizes and
along different crystalline directions have been measured using CIT-FMR.
Consistent results have been attained and are summarised in this Section.
Note that all devices have been annealed at 180oC in air for 48 h. This is be-
cause the as-grown (Ga,Mn)(As,P) wafers (10% P) are electrically insulating
at low-temperatures [92].
5.3.1 Realisation of FMR in 80 nm-wide devices
Figure 5.2a shows the photovoltage spectra measured on a 80 nm-wide device
(patterned along [11¯0]), for field sweeps along different crystalline directions
(green circles), using a microwave frequency of 10 GHz. In each case, the
FMR peak is clearly recognisable on the photovoltage spectrum, and can be
fitted using the solution of the LLG equation (Equation 2.25). The positions
of the resonance Hres vary due to the magnetic anisotropy.
1When performing lock-in measurements, only the signal in-phase with the reference
frequency, i.e. the X-channel on the lock-in amplifier, is recorded. It is therefore necessary
for the modulated photovoltage to be mostly in-phase with the reference frequency. For
the (Ga,Mn)(As,P) devices, a modulation frequency of 23.7 Hz meets this requirement.
108
5.3 Data and analysis
a
b
V
200nm
Figure 5.1: a, A SEM image of a 80 nm-wide (Ga,Mn)(As,P) bar. b,
Schematic of the CIT-FMR setup (not-to-scale).
109
5.3 Data and analysis
H0
[100]
[110]
[010]
[110]
H0[100]
[110]
[010]
[110]
H0
[100]
[110]
[010]
[110]
μ0 0H (T)
V
(
V
)
d
c
μ
a
φ (deg.)
μ
0
0
H
(T
)
V ( V)dc μ
[1
0
0
]
[1
1
0
]
[0
1
0
]
[1
1
0
]
c
b
μ
0
re
s
H
(T
)
d
[001]
[010]θ
H0
[100]
φ
Figure 5.2: Photovoltage spectra measured on a 80 nm-wide sample (pat-
terned along the [11¯0] axis). a, Vdc measured for H0 swept along different
crystalline directions ([110], [100] and [11¯0] respectively, as indicated by the
inset of each graph), using a 10 GHz current. The green circles are the mea-
surement data, and the black solid lines are fitted results to Equation 2.25.
b, Full photovoltage spectra for an in-plane rotational scan. The colour scale
represents the magnitude of Vdc (in µV). c, The ϕ-dependence of the reso-
nance field Hres (black circles). The red solid line is the fitted results to
Equation 2.20. d, The geometry used in the experiment.
110
5.3 Data and analysis
Figure 5.2b presents the full photovoltage spectra for an in-plane rota-
tional scan of H0 (θ = 90
o as in Figure 5.2d). The FMR peak is clearly
observed on top of a non-resonant background. The ϕ-dependence of Hres
is plotted in Figure 5.2c (black circles). By fitting the data using Equa-
tion 2.20 (red solid line), and combined with data from the out-of-plane
rotational scan (see Figure 5.6b), the magnetic anisotropy in this device is
found to be1: µ0H2⊥ = −134 mT, µ0H4⊥ = −80 mT, µ0H2‖ = −189 mT
and µ0H4‖ = 72 mT. Hence the nano-bar exhibits large in-plane uniaxial
anisotropy, with its easy axis along the [110] direction, i.e. perpendicular to
the long axis of the bar. Furthermore, since the saturation magnetisation
in this material is µ0Ms = 50 mT (measured by Arianna Casiraghi using
SQUID), the large resonance field (µ0Hres > 350 mT) ensures that the M is
always fully saturated along the direction of the applied field H0.
5.3.2 Mapping the magnetic anisotropy
The magnetic anisotropy in other 80 nm-wide bars patterned along different
crystallographic orientations has also been investigated using CIT-FMR. The
angular plots ofHres for [010] and [100] bars are presented in Figure 5.3a (blue
and green circles respectively, in-plane scans). The plot reveals a dominant
uniaxial anisotropyHU generated by the patterned-induced strain-relaxation,
with easy axis perpendicular to the bar orientation, on top of the intrin-
sic uniaxial (H2‖) and biaxial (H4‖) terms of the unpatterned film. The
anisotropy profiles Hres(ϕ) can be well fitted using Equation 3.2 (solid lines
in Figure 5.3a), and the deduced anisotropy constants Hi in the two devices
are compared in Figure 5.3b. It demonstrates that HU is nearly an order
1An iterative fitting method is adopted [74]: Firstly the in-plane Hres(ϕ) data are
analysed and fitted to Equation 2.20 using γ = 176 GHz/T (g = 2). This gives approx-
imate values of Meff, H4‖ and H2‖, which are then used as starting parameters to carry
out a weighted nonlinear least square fit to the Hres(ϕ) data from out-of-plane scans (as in
Figure 5.6b), allowing the two parameters γ and H4⊥ to vary. Using the γ value obtained
by this procedure as the input parameter for the next iteration, new values of Meff, H4‖
and H2‖ are obtained by fitting to the in-plane data. These two steps are repeated until
optimal fitting is achieved and all five parameters (Meff , H4⊥, H4‖, H2‖, and γ) converge
and remain unchanged in subsequent iterations.
111
5.3 Data and analysis
[100] bar [010] bar [110] bar [11¯0] bar
µ0H4‖ (mT) -68 -87 92 72
µ0H2‖ (mT) 58 30 286 -189
µ0HU (mT) 250 -270 ‡ ‡
Table 5.1: In-plane anisotropy constants Hi in the 80 nm-wide bars patterned
along different crystallographic orientations on (Ga,Mn)(As,P) epilayers. In
the [110] and [11¯0] bars, HU is collinear with H2‖.
of magnitude stronger than the intrinsic anisotropy terms H4‖ and H2‖. By
definition, HU > 0 if its easy axis is along the [010] crystalline direction.
Figure 5.3c compares the anisotropy in all four 80 nm-wide bars (along
[100], [110], [010] and [11¯0] crystalline axes), and the in-plane anisotropy
constants Hi are summarised in Table 5.1. In very narrow bars, the growth-
strain caused by lattice mismatch is fully relaxed, making HU the dominant
term [66]. This is demonstrated by the 80 nm-wide devices, all of which
exhibit strong uniaxial anisotropy with easy axes always perpendicular to
the corresponding bar orientations. In devices along the [110] and [11¯0] axes,
the direction of HU coincides with the intrinsic uniaxial anisotropy H2‖, and
it is therefore either enhanced or reduced by H2‖.
5.3.3 Strain-relaxation in devices of different sizes
The magnetic anisotropy in samples of different sizes has been systematically
investigated using the CIT-FMR technique. In Figure 5.4a–c, the anisotropy
profiles in bars of widths 80, 500 and 4000 nm are compared (all devices are
patterned along the [010] axis on the (Ga,Mn0.06)(As,P0.1) wafer). Recent nu-
merical calculations show that the strain-relaxation is spatially nonuniform
and takes place on a length scale of several hundred nanometres [26]. There-
fore it is not surprising that the effect of strain-relaxation on the magnetic
anisotropy becomes smaller with increasing bar width. Figure 5.4d compares
the values of HU among the three devices. In the 4 µm-wide bar, HU is very
112
5.3 Data and analysis
w = 80nm
w = 80nm
μ
0
re
s
H
(T
)
φ (deg.)
μ
0
i
H
(m
T
)
HuH2H4
[100] [010]
[100]
φ (deg.)
a b
c
μ
0
re
s
H
(T
)
[100] [010]
[110] [010] [110]
[010]
[100]
Figure 5.3: The magnetic anisotropy in 80 nm-wide (Ga,Mn)(As,P) bars
measured using CIT-FMR. a, Polar plots for Hres measured in two 80 nm-
wide devices patterned along the [010] and [100] axes (blue and green circles
respectively). The orientations of the devices are shown in the inset, and
are marked by black arrows in the main graphs. The solid lines are fits of
Equation 3.2. b, The in-plane anisotropy constants H4‖, H2‖ and HU for
the two devices, deduced from fittings. c, The anisotropy profiles in all four
80 nm-wide bars, patterned along [100], [110], [010] and [11¯0] axes.
113
5.3 Data and analysis
small (10 mT) and the device exhibits near bulk-like magnetic anisotropy (a
superposition of twofold and fourfold symmetry due to H2‖ and H4‖).
The findings are quantitatively consistent with other published results
performed on (Ga,Mn)As devices under compressive-strain, but also provide
some new insights. For example, Hoffmann et al. have measured devices on
a (Ga,Mn)As wafer with 6% Mn concentration (50 nm-thick film), and have
observed KU = 1.8 × 104 erg/cm3 (equivalent to µ0HU = 47.6 mT) on a
400 nm-wide bar [67]; while the 500 nm-wide (Ga,Mn)(As,P) bar measured
in this study reveals similar value with µ0HU = −50 mT1. For narrower
bars (200 nm-wide), Hu¨mpfner et al. and Wenisch et al. have shown that
µ0HU in (Ga,Mn)As devices with lower Mn concentrations (4% and 2.5%
respectively) are a few tens of mT (25 and 45 mT respectively) [65, 66];
whereas this experiment has demonstrated that HU can be further increased
by making the bars even narrower (µ0HU = −270 mT in the 80 nm-wide bar
shown in Figure 5.4d).
5.3.4 FMR signatures in devices of different sizes
CIT-FMR is also performed on each device using different excitation frequen-
cies. As shown by Equation 3.5, ∆H (half width at half maximum) can be
decomposed into a frequency-independent, inhomogeneously-broadened part
(∆Hinhomo) and a frequency-dependent, damping-related part (αω/γ) [74].
In Figure 5.4f, the FMR linewidths in (Ga,Mn)(As,P) samples of different
bar sizes are compared.
It is found that ∆Hinhomo is smallest in the 80 nm-wide bar (2.5 mT),
and largest in the 500 nm-wide bar (10 mT). Since ∆Hinhomo is caused by
strain gradients within the samples, it could indicate that the 80 nm-wide
device has the least variation in internal strain, whereas the 500 nm-wide bar
has the strongest. This is consistent with the magnetic anisotropy measured
in these devices (Figure 5.4a–c). In addition, the Gilbert damping constant
is observed to change inversely with sample width, from α = 0.023 in the
1Note that the sign of HU simply indicates the direction of its easy axis.
114
5.3 Data and analysis
w = 4000nm
w = 80nm
w = 500nm
μ
0
U
H
(m
T
)
φ (deg.)
μ
0
re
s
H
(T
)
d
c
a
b
[1
0
0
]
[1
1
0
]
[0
1
0
]
[1
1
0
]
[1
0
0
]
μ0 resH (T)
f 
(G
H
z
)
e f
f (GHz)
Δ
H
 (
m
T
)
Figure 5.4: Evolution of the magnetic properties with device size. a – c, The
resonance field Hres(ϕ) measured in three (Ga,Mn)(As,P) stripes with widths
of 80, 500 and 4000 nm (length:width ratio = 10:1) respectively (all device are
patterned along the [010] direction). The circles are measured data, and the
solid lines are fittings. The black arrows mark the orientations of the bars.
d, Comparison of the strain-relaxation-induced anisotropy HU in the three
devices. e, Frequency-dependence of the resonance field Hres measured in the
three devices (triangles). The solid lines are fitted results to Equation 2.20.
f, Frequency-dependence of the FMR linewidth ∆H (squares). The data are
fitted to straight lines to calculate ∆Hinhomo and α, see Equation 3.5.
115
5.3 Data and analysis
80 nm-wide bar, to α = 0.005 (500 nm-wide), and to α = 0.004 (4000 nm-
wide). This is possibly related to the quasiparticle lifetime becoming shorter
in narrower samples due to increased scattering.
5.3.5 Effects of growth-strain on HU
To further investigate the effect of the growth-induced strain on HU , ad-
ditional devices patterned from (Ga,Mn)(As,P) epilayers with 10% and 0%
phosphorus are compared, which are under tensile- and compressive-strain
respectively1. In particular, two 500 nm-wide bars are compared, both are
fabricated along the [010] orientation.
Figure 5.5a shows the anisotropy profile measured on the (Ga,Mn)As
micro-bar. It exhibits strong uniaxial anisotropy HU due to lithography-
induced strain-relaxation, with the easy axis coincide with the [010]-bar di-
rection. This is in agreement with earlier studies on (Ga,Mn)As devices us-
ing SQUID, magnetotransport and FMR techniques [26, 65–67]. To the con-
trary, for HU in the phosphorus-doped device patterned from tensile-strained
(Ga,Mn)(As,P), the easy axis is aligned perpendicular to the direction of the
bar (Figure 5.5b), consistent with observations on the 80 nm-wide bars re-
ported earlier (Figure 5.3c). This is further illustrated in Figure 5.5c by
comparing the in-plane anisotropy fields: The intrinsic magnetic anisotropy
H4‖ and H2‖ are similar in magnitude between the two devices, and have
the same sign; whereas the strain-relaxation-induced anisotropy HU changes
sign, indicating that the strain relaxes in the opposite sense between the
two bars. This clearly establishes the reverse effect of growth-strain (com-
pressive vs tensile) on the patterning-induced anisotropy HU , as illustrated
schematically in Figure 5.5d.
5.3.6 Lithography-control over the magnetisation
Finally, it is shown that the strong perpendicular anisotropy of the unpat-
terned (Ga,Mn)(As,P) epilayer (10% P) can be largely reduced by litho-
1The wafer with 0% P is the same 25 nm-thick (Ga0.94,Mn0.06)As film used in the
previous two Chapters.
116
5.3 Data and analysis
μ
0
i
H
(m
T
)
HU
a
d
GaAs
(Ga,Mn)(As,P)
GaAs
(Ga,Mn)As
strain relaxation
c
μ
0
re
s
H
(T
)
φ (deg.)
w = 500nm (Ga,Mn )As0.06
b
φ (deg.)
w = 500nm
(Ga,Mn )(As,P )0.06 0.1
μ
0
re
s
H
(T
)
[100] [010]
Figure 5.5: Comparison of HU in the (Ga,Mn)As and (Ga,Mn)(As,P) devices
(both are 500 nm-wide and patterned along the [010] direction). a, Hres(ϕ)
measured from an in-plane rotational scan on the (Ga,Mn)As micro-bar.
The circles are experimental data, and the solid line is the fitted results.
b, Hres(ϕ) measured on the (Ga,Mn)(As,P) nanodevice. c, The in-plane
magnetic anisotropy terms Hi in the two samples. d, Schematic of strain-
relaxation in the (Ga,Mn)As and (Ga,Mn)(As,P) nanostructures.
117
5.3 Data and analysis
μ
0
re
s
H
(T
)
θ (deg.)
w = 500nm w = 80nm
a b
θ (deg.)
Figure 5.6: Out-of-plane rotational scans on (Ga,Mn)(As,P) devices pat-
terned along the [11¯0] direction. a, Rotational scans of θ on a 500 nm-wide
bar at ϕ = 45o (green squares) and ϕ = 135o (blue squares). The solid
lines are fitted results according to Equation 3.2. b, Same scans of θ on a
80 nm-wide bar.
graphic patterning, so that a second easy in-plane axis occurs in the 80 nm-
wide bars. Figure 5.6 compares the results from out-of-plane rotational scans
of H0 (i.e. rotating θ at fixed ϕ) in a 500 nm and a 80 nm-wide [11¯0] bar
respectively. Two scans are performed on each device: One is for comparing
the magnetic anisotropy in the [110]–[001] plane (i.e. θ = 90o and ϕ = 45o,
cf. Figure 5.2d), whereas the other scan allows comparison of the magnetic
anisotropy in the [11¯0]–[001] plane (θ = 90o and ϕ = 135o). The magnetisa-
tion is driven at a microwave frequency of 17 GHz.
For the 500 nm-wide bar (Figure 5.6a), Hres(θ) implies that both in-plane
directions (in-plane easy and hard axes) are significantly harder than the
perpendicular-to-plane direction, leading to an overall out-of-plane easy axis.
This is consistent with the tensile-strained nature of the bulk (Ga0.94,Mn0.06)(As0.9,P0.1)
epilayer.
For the 80 nm-wide bar (Figure 5.6b), the global hard axis along the
[11¯0] orientation (θ = 90o, ϕ = 135o) becomes harder compared to the wider
500 nm bar. The resonance field Hres along the [110] direction (θ = 90
o, ϕ =
45o), however, is now comparable to the field along [001], indicating that the
[110] orientation becomes similarly easy as the perpendicular [001] orientation
and only a weak uniaxial anisotropy in the [110]–[001] plane remains on top
118
5.4 Conclusions and outlook
of the intrinsic cubic anisotropy. This indicates that the strain-relaxation
in the in-plane bar orientation is mostly relaxed as it is the case for the
perpendicular [001]-direction.
5.4 Conclusions and outlook
To summarise, the strong spin-orbit coupling and exchange interaction in
the diluted magnetic semiconductor (Ga,Mn)(As,P) enables the application
of current-induced torque driven ferromagnetic resonance (CIT-FMR). Using
this technique the magnetic anisotropy in individual micron- and nanoscale
(Ga,Mn)(As,P) devices has been investigated, and the effects of strain-relaxation
on the anisotropy in these samples have been examined.
It is important to point out the simplicity of the devices. The microwave
magnetic field and the detection technique originate from the material prop-
erties, so only a 2-terminal nano-bar (a resistor) is required, enabling simple
and rapid fabrication. The CIT-FMR technique is also applicable to other
ferromagnetic material systems. Candidates include other magnetic semicon-
ductors but also metal thin films in which the spin-orbit interaction has the
potential to play a technological role. CoPt thin films are an important mate-
rial for future study due to the recent observation of a strong current-induced
field with Rashba symmetry [58].
119
5.5 Appendix – Magnetic anisotropy control with PMMA
(preliminary results)
5.5 Appendix – Magnetic anisotropy control
with PMMA (preliminary results)
Introduction The magnetic anisotropy in (Ga,Mn)As and (Ga,Mn)(As,P)
epilayers is directly linked to the strain on the materials, and it can be manip-
ulated by either adding additional strain via a piezoelectric transducer [5–8],
or by releasing the strain via lithographic processes (as demonstrated by the
experiments in this Chapter). On the other hand, since the measurements
are always performed at cryogenic temperatures, it is conceivable that if a
material with a different thermal contraction coefficient is attached to the
ferromagnetic device, strain is induced during cooling and possibly alter the
magnetic anisotropy in the sample.
The initial experimental results on the manipulation of magnetic anisotropy
in (Ga,Mn)(As,P) microstructures by thermal contraction are presented in
this Section. The polymer PMMA1 (polymethyl methacrylate) has been
shown to possess large thermal expansion coefficient2 and is therefore chosen
for this experiment. It is applied by simply dropping on top of a sample and
is then solidified in an oven at 80oC for 2 h. The device is then cooled down
to 6 K and its magnetic properties are investigated using CIT-FMR.
A total number of 6 devices have been tested. 2 micro-bars broke during
the cooling procedure3, which demonstrates the strong thermal contraction
of PMMA. Consistent observations have been made on the 4 devices that
have survived the initial cooling process, and representative results from two
samples are discussed. Their characteristics are listed in Table 5.2.
Ferromagnetic resonance curves Figure 5.7a shows a typical photovolt-
age spectrum measured across Device A by sweeping H0 at a fixed frequency
of 10 GHz. Two peaks can be clearly observed. Fittings to Equation 2.25
1The PMMA used for the experiment is diluted with anisole to 6% concentration (A6)
and has molecular weight of 950k.
2The linear thermal expansion coefficient of PMMA is measured to be 80× 10−6 K−1
at 300 K [146], whereas for GaAs it is 5.8× 10−6 K−1 at 300 K [147].
3Two of the working devices also broke when they were cooled down for a second time.
120
5.5 Appendix – Magnetic anisotropy control with PMMA
(preliminary results)
Device Name Condition Dimension
A (‖ [100]) Annealed 0.5 µm× 10 µm
B (‖ [010]) Annealed 4 µm× 40 µm
Table 5.2: The two (Ga0.94,Mn0.06)(As0.9,P0.1) micro-bars discussed in this
Section.
reveals that one peak centres at 760 mT with linewidth ∆H = 28 mT, and
in addition, there is a second, broader peak which centres at 450 mT with
linewidth ∆H = 42 mT.
Similar broad, low-field peaks have been previously reported in FMR
experiments on bulk (Ga,Mn)As films, and are attributed to microwave
magneto-conductivity changes in the sample [74]. In order to further in-
vestigate their properties, field sweeps at different microwave frequencies are
performed, as shown in Figure 5.7b. The data reveal that the resonance posi-
tions Hres for the narrow peaks increase with increasing frequency, in agree-
ment with the magnetic resonance condition (Equation 2.20). The broad
peaks, on the other hand, shift to lower fields at higher excitation frequen-
cies, which implies that their origins are not FMR-related (possibly due to
Joule heating caused by the microwave current), and are not discussed in
this Section.
Change in magnetic anisotropy The change in magnetic anisotropy due
to the thermal contraction of PMMA is the focus of this study. Figure 5.8
compares measurement data from in-plane rotational scans on Device A with
and without PMMA. With the PMMA coating, the ϕ-dependence ofHres dis-
plays a pronounced twofold symmetry, which clearly indicates strong uniaxial
anisotropy with easy axis along the [100] direction, i.e. along the direction of
the micro-bar (Figure 5.8a and 5.8b). This can be described by introducing
the uniaxial anisotropy term HU .
This is in stark contrast to the magnetic anisotropy of the device without
the PMMA coating. As displayed in Figure 5.8c and 5.8d as well as in
121
5.5 Appendix – Magnetic anisotropy control with PMMA
(preliminary results)
μ0 0H (T)
V
(
V
)
d
c
μ
a b
Figure 5.7: Ferromagnetic resonance signals on Device A with PMMA coat-
ing. a, The photovoltage measured using a 10 GHz current. The red circles
are experimental data, and the green lines are fittings to individual peaks
according to Equation 2.25. b, Vdc taken at different driving frequencies of
9, 10 and 12 GHz. The curves have been scaled for comparison.
122
5.5 Appendix – Magnetic anisotropy control with PMMA
(preliminary results)
the main Chapter, the additional uniaxial anisotropy HU caused by strain-
relaxation in this device has an easy axis perpendicular to the bar, i.e. in
the [010] direction, due to the tensile-strain present in the (Ga,Mn)(As,P)
epilayer.
With regard to the magnitude ofHU in the two conditions, Figure 5.8b im-
plies that HU becomes the dominant anisotropy term in the PMMA-coated
device, similar to that observed in the 80 nm-wide bars (see Figure 5.2).
Whereas Figure 5.8d shows a superposition of twofold and fourfold symme-
try (due to HU and H4‖ respectively), which is consistent with observations
made on other 500 nm-wide bars patterned on (Ga,Mn)(As,P) and also on
(Ga,Mn)As epilayers (Figure 5.5). This immense difference in the magni-
tude of HU is more established when the anisotropy constants are directly
compared in Figure 5.8e: Under the strain induced by the thermal contrac-
tion of PMMA, µ0HU is found to rise from 30 mT to −241 mT in Device A
(note that the sign of µ0HU represents the direction of its easy axis). The
sign change in the biaxial anisotropy H4‖ is possibly an artefact because the
fitting procedure is not precise enough to reproduce the correct values of the
biaxial anisotropy when the effective anisotropy is dominated strongly by the
strain-induced uniaxial contribution.
The change in sign of HU in nanoscale Device A suggests that the tensile-
strain-relaxation of the uncovered device is compensated and superimposed
by the strain executed by the PMMA when it is under thermal contraction,
which has an apparent effect of elongating the lattice constant perpendicular
to the bar. This process is illustrated schematically in Figure 5.8f.
Finally, the effect of PMMA contraction on the magnetic anisotropy in
wider bars (Device B) is investigated. As Figure 5.9b shows, the magnetic
anisotropy in the original device exhibits a superposition of twofold and
fourfold symmetries, caused by the intrinsic biaxial and uniaxial anisotropy
terms H4‖ and H2‖, and a small strain-relaxation-induced uniaxial term
µ0HU = −10.4 mT. When PMMA is added, CIT-FMR measurements re-
veal that while the intrinsic terms stays relatively unchanged, the additional
uniaxial anisotropy µ0HU is reversed (33.5 mT) due to the strain generated by
different thermal contractions of PMMA and (Ga,Mn)(As,P) (Figure 5.9a).
123
5.5 Appendix – Magnetic anisotropy control with PMMA
(preliminary results)
φ (deg.)
Device only
μ
0
0
H
(T
)
μ
0
i
H
(m
T
) HU
a
c d
e
With PMMA
GaAs
(Ga,Mn)(As,P)
f
b
μ
0
re
s
H
(T
)
Figure 5.8: Comparison of the change in magnetic anisotropy in Device A
before and after PMMA has been applied. a, Photovoltage spectra from
in-plane rotational scans of H0 on the PMMA-coated device. A 10 GHz
microwave current is used. Both the broad and narrow peaks are found to
be ϕ-dependent. b, The resonance field Hres for the PMMA-coated device
(blue circles), and the blue line is the fitted curve. The black arrow marks
the direction of the micro-bar. c – d, Magnetic anisotropy profile on the
same device before PMMA is applied (orange circles). e, Comparison of the
in-plane anisotropy constants Hi in the two conditions. f, Schematic of the
strain induced on the micro-bar due to the thermal contraction of PMMA.
Black arrows: direction of strain-relaxation in patterned bars; red arrows:
possible direction of strain applied by PMMA contraction.
124
5.5 Appendix – Magnetic anisotropy control with PMMA
(preliminary results)
μ
0
re
s
H
(T
)
φ (deg.)
a
b
With PMMA
Device only
μ
0
i
H
(m
T
)
HU
c
Figure 5.9: Comparison of the change in magnetic anisotropy of Device B
with and without PMMA. a, The resonance field Hres for the PMMA-coated
device (blue circles), detected using a 10 GHz current. The black arrow marks
the direction of the micro-bar. b, The magnetic anisotropy profile on the
same device before PMMA has been applied (orange circles). c, Comparison
of the in-plane anisotropy fields Hi for Device B in the two conditions.
The deduced anisotropy fields Hi for Device B in the two conditions are com-
pared in Figure 5.9c, where the sign change in HU is easily visible. However,
the size of HU is much smaller in this sample than that in Device A.
An out-of-plane rotational scan carried out on Device B reveals that the
sample still possesses an overall perpendicular-to-plane easy axis, hence the
microdevice is still under tensile-strain. However, no FMR signal was ob-
served when measuring the narrower Device A, therefore it is unclear at this
stage whether the strain generated by the contraction of PMMA could lead
to an additional in-plane easy axis in the narrow bars.
In order to understand the effect of PMMA contraction on the mag-
netic anisotropy, numerical simulations are needed to map the profile of the
strain applied on the devices. This is however not straightforward for the
current samples due to the non-uniformity of the PMMA coating. Future ex-
periments on devices covered with lithographically-patterned PMMA layers
should provide more insights into the polymer’s role in defining the magnetic
anisotropy.
125
5.5 Appendix – Magnetic anisotropy control with PMMA
(preliminary results)
Summary and outlook In this Section, preliminary results using an in-
situ control of anisotropy method on (Ga,Mn)(As,P) micro-bars are pre-
sented. The large thermal expansion coefficient of the polymer PMMA has
been employed to strain the ferromagnetic samples and hence alter their
magnetic properties; and it has been established that this strain could lead
to additional uniaxial anisotropy in the microdevices, which becomes domi-
nant in nanometre-wide devices and overcomes the strain-relaxation-induced
anisotropy in these samples.
However, in order to fully understand this effect, further experiments
should be performed in the following areas:
• PMMA in this study has been applied onto the device rather arbitrar-
ily, and the thickness of the polymer coating is unknown. Thus fu-
ture experiments should be systematically carried out on samples with
controlled parameters, such as using PMMA with different thickness,
concentration and even molecular weight.
• Devices patterned on both (Ga,Mn)As and (Ga,Mn)(As,P) layers should
be measured.
• More importantly, it has been observed both in (Ga,Mn)As and (Ga,Mn)(As,P)
micro-bars that the current-induced torque due to the linear Dressel-
haus spin-orbit coupling diminishes in smaller devices due to pattering-
induced strain-relaxation (Section 4.3.4). Therefore it is also intriguing
to investigate the impact on CIT when additional strain is introduced
to the system via PMMA contraction. This could even lead to ferro-
magnetic microdevices with tunable spin-orbit interaction.
• Finally, other materials can also be used to generate strain on the fer-
romagnetic devices. Potential candidates include many metals, which
have thermal expansion coefficients usually a few times that of GaAs.
A sample with a thinned substrate can be glued on top of a metal sheet,
and the difference in thermal contraction between GaAs and the metal
during cooling can therefore generate strain on the ferromagnet.
126
5.5 Appendix – Magnetic anisotropy control with PMMA
(preliminary results)
The CIT-FMR method, established in this Thesis, provides a reliable
platform for the magnetic anisotropy and spin-orbit coupling in individual
ferromagnetic devices to be investigated.
127
Chapter 6
Closing remarks
The theoretical formulation of spin-transfer torque (STT) [17, 18], and sub-
sequent experimental observations on its role in magnetisation reversal and
magnetic dynamics [19], have allowed the magnetisation to be directly con-
trolled by an electric current and opens a new range of applications, includ-
ing magnetic memory devices and high-frequency oscillators. Spin-transfer
torque occurs in non-collinear ferromagnetic systems with relatively complex
device structures.
On the other hand, the current-induced torque (CIT) in uniform fer-
romagnets has so far only been observed in magnetisation reversal experi-
ments [39, 58, 59]. The experiments described in this Thesis have expanded
the understanding of CIT in magnetic semiconductors, including its role in
magnetisation dynamics, its magnitude and orientation in diluted magnetic
semiconductors (Ga,Mn)As and (Ga,Mn)(As,P), and its relation to internal
and external parameters such as conductivity and strain. Furthermore, a
new class of device-scale ferromagnetic resonance (FMR) technique is estab-
lished, named CIT-FMR in this Thesis, which has the advantage of simple
device structure and scalability, and can be transferred to characterise the
magnetic properties and CIT in a wide range of ferromagnet systems.
Future experimental directions The CIT-FMR technique established
in this Thesis has provided a platform for investigating both the current-
induced torque (via vector magnetometry) and the magnetic anisotropy (via
128
ferromagnetic resonance) in individual submicron devices. The experiments
in this Thesis have focused on the magnetic semiconductors (Ga,Mn)As and
(Ga,Mn)(As,P), but this technique can be equally applied to study ferro-
magnetic metal thin films at room temperature, with candidates including
Fe/GaAs and CoPt systems.
Since the linear Dresselhaus spin-orbit interaction is strain-related, a
whole family of experimental investigations regarding the control of CIT
via strain can be initiated. In (Ga,Mn)As systems, the effect of strain on
the magnetic anisotropy has been extensively studied, using techniques such
as lattice mismatch (by varying doping levels and impurity contents) [1, 77,
92, 95], mechanical stress (piezoelectric transducer) [5–8], and patterning-
induced strain-relaxation [26, 65–67, 115]. In this Thesis, the first step to-
wards understanding the impact of strain-relaxation on CIT has been re-
ported; however, it will also be interesting to study the behaviour of CIT
under the stress from a piezoelectric transducer. Being able to engineer the
magnitude and orientation of CIT, both on magnetic semiconductors and
ferromagnetic metals, could have important technological implications.
In addition, since structure inversion asymmetry (SIA) can give rise to
Rashba spin-orbit coupling, it is interesting to examine the result of symme-
try breaking caused by an external electric field in the crystal (e.g. via electri-
cal gating). Previous experiments have focused on the changes in anisotropy
by gating the magnetic semiconductor materials [10, 12], but it is of equal
importance to investigate if CIT can be manipulated in such way.
A final class of experiments are related to magnetisation switching and
thus the memory functionality of future spintronic devices. According to
the LLG equation that governs the dynamics of magnetic motion, it is ex-
pected that a fast pulse Hpulse applied perpendicular to M could generate a
torque −γ(M×Hpulse) and induce magnetisation precession. Magnetisation
switching caused by this precessional motion therefore promises high energy
efficiency with ultrafast reversal time [148]. This “precessional switching”
experiment has been performed on thin film metal systems using a real field
pulse [148–154], but it would be equally interesting to apply this principle
on magnetic semiconductors using CIT. Ultimately, the aim is to drive fast
129
precessional switching in magnetic metal films using CIT, and hence realise
room-temperature, all-electrical magnetic memory units.
130
Appendix A
Derivation of the FMR
lineshape
In this Section, the expressions for the FMR signal (Equations 2.25 – 2.29
in Section 2.4.3) is derived by solving the LLG equation (Equation 2.14)
and considering the frequency mixing effect. The derivations are based on
(Ga,Mn)As and (Ga,Mn)(As,P) epilayers, but can be extended to other fer-
romagnetic metal systems by using a different expression for the magnetic
anisotropy1.
The total voltage across a device is given by Ohm’s law:
V (t) = I(t) · R(t) (A.1)
Here I(t) = I cos(ωt) is the ac current within the sample, and R(t) is the
time-dependent longitudinal resistance.
According to Equation 2.6, the longitudinal resistance of the sample is
given as:
Rxx = R⊥ −∆R cos2 ϑ (A.2)
where R⊥ is the resistance when M ⊥ I, and ∆R > 0 is the AMR coefficient
(∆R = R⊥ − R‖).
1The dominant magnetic anisotropy in ferromagnetic metal systems is the shape
anisotropy, whereas in magnetic semiconductors such as (Ga,Mn)As and (Ga,Mn)(As,P),
the magnetocrystalline anisotropy energies are more pronounced.
131
In FMR experiments, the magnetic moments precess under the driving
torque of an ac magnetic field, resulting in a time-varying angle:
ϑ(t) = ϑ+ ϑc cos(ωt− ψ) (A.3)
where ϑc describes the deviation of M from its axis of rotation (referred to as
the “cone angle” of the precession), and ψ is the phase difference between the
resistance and the ac current. An expression for the time-varying resistance
R(t) can thus be obtained by combing Equations A.2 and A.3. In the case of
small cone angle precession, the expression can be simplified by expanding
cos2 ϑ(t) up to first order:
R(t) ≈ R⊥ −∆R
[
cos2 ϑ− 2ϑc cosϑ sin ϑ cos(ωt− ψ)
]
(A.4)
Combining Equations A.1 and A.4, it is found that the total voltage V (t)
comprises terms at frequencies ω and 2ω, and a time-independent (dc) term
Vdc, which is the focus of this derivation.
To find out the expression for ϑc, the magnetisation components in the
plane of rotation need to be determined. The geometry used for the FMR
experiments is defined in Figure A.1. Since the applied magnetic field H0 is
much larger than the anisotropy of the ferromagnet (measured to be a few
tens of mT), the magnetisation vector M stays parallel to H0. A second
coordinate system x′ − y′ is defined with respect to M. In this new system,
H0 = (H0, 0, 0), and M = (Ms, my′e
iωt, mze
iωt) in the case of small angle
precession.
The dynamics of the damped magnetic motion is described phenomeno-
logically by the Landau-Lifshitz-Gilbert (LLG) equation:
∂M
∂t
= −γµ0M× (Heff + hmw) + α
Ms
(
M× ∂M
∂t
)
(A.5)
here the first term is the field torque and the second term is the Gilbert
damping. γ is the gyromagnetic ratio and α is the phenomenological Gilbert
damping constant. Heff is the effective magnetic field in the vicinity of the
magnetisation. It is a vector sum of the dc magnetic field H0, the demag-
netisation field Hdemag and the anisotropy field of the material Hani. hmw is
132
xy
z
x’y’
M
j
ϑ
H0
Figure A.1: The coordinate systems used in the derivation.
the microwave frequency magnetic field which drives the magnetic moments;
in the x′ − y′ system, it has the form:
hmw =

 hx cosϑ− hy sinϑhx sinϑ+ hy cosϑ
hz

 eiωt (A.6)
Solving Equation (A.5) and discarding high-order terms, the following set
of linearised equations are obtained [68]:
iω
γ
my′ +
(
iαω
γ
+H0 +H1
)
mz = Mshz (A.7)(
iαω
γ
+H0 +H2
)
my′ − iω
γ
mz = Ms(hx sinϑ+ hy cosϑ) (A.8)
where
H1 = Ms −H2⊥ +H2‖ cos2
(
ϕ+
π
4
)
+
1
4
H4‖(3 + cos 4ϕ) (A.9)
H2 = H4‖ cos 4ϕ−H2‖ sin 2ϕ (A.10)
are terms containing the demagnetisation and anisotropy fields of the (Ga,Mn)As
epilayers.
In (Ga,Mn)As films, due to the strong in-plane anisotropy, the preces-
sional motion of the magnetisation is highly elliptical, with the maximum
deviation from its equilibrium position occurring for M in-plane with the
sample. For small angle precession, the maximum cone angle is ϑc = my′/Ms.
133
Its value can be obtained by solving Equation (A.7) and (A.8):
ϑc = −ihzγ
ω
− [i(H0 +H1)−∆H ] [γhz(H0 +H2 + i∆H)− iω(hx sinϑ+ hy cosϑ)]
ω [−(H0 +H1 + i∆H)(H0 +H2 + i∆H) + ω2/γ2]
(A.11)
Here ∆H = αω/γ defines the resonance linewidth (half width at half maxi-
mum). The complex expression of Equation (A.11) arises from the complex
susceptibility χ.
Combining Equation (A.1), (A.4) and (A.11), a general expression for the
dc photovoltage is obtained:
Vdc = −1
2
I∆R sin(2ϑ)
{
iγhz
ω
+
1
ω [−(H0 +H1 + i∆H)(H0 +H2 + i∆H) + ω2/γ2][
i(H0 +H1)−∆H
][
γhz(H0 +H2 + i∆H)
−iω(hx sinϑ+ hy cosϑ)
]}
(A.12)
In conventional FMR experiments, the frequency of the driving field ω is
kept constant, and the external fieldH0 is swept. To determine the lineshape
of the photovoltage near resonance, consider the profile of Vdc at a small
deviation δH from Hres, i.e.
δH = |H0 −Hres| (A.13)
For uniform precession of M in a material with small damping (α ≪
1), ∆H ≪ Hres. Also since only small perturbations near the resonance is
considered, δH ≪ Hres. Substituting Equation (A.13) into (A.12) and only
keeping terms linear in δH and ∆H , the following simplified expression is
obtained:
Vdc = −1
2
I∆R sin(2ϑ)
{
iγhz
ω
− 1
ω(2H0 +H1 +H2)(δH + i∆H)[
ω(hx sinϑ+ hy cosϑ)(δH +H0 +H1 + i∆H) +
γhz
(
iδH(2H0 +H1 +H2) + i(H0 +H1)(H0 +H2)−
∆H(2H0 +H1 +H2)
)]}
(A.14)
134
The in-phase (real) component of Vdc has the form (keeping only terms
linear in α):
Re{Vdc} = Vsym ∆H
2
(H0 −Hres)2 +∆H2 + Vasy
∆H(H0 −∆H)
(H0 −Hres)2 +∆H2 (A.15)
with angle-dependent amplitudes
Vsym(ϑ) =
I∆R
2
Asym sin(2ϑ)hz (A.16)
Vasy(ϑ) =
I∆R
2
Aasy sin(2ϑ)(hx sinϑ+ hy cosϑ) (A.17)
The terms Asym and Aasy are the scalar amplitudes of the magnetic sus-
ceptibility (Ai = χi/Ms):
Asym =
γ(Hres +H1)(Hres +H2)
ω∆H(2Hres +H1 +H2)
(A.18)
Aasy =
(Hres +H1)
∆H(2Hres +H1 +H2)
(A.19)
It is noticed that the saturation magnetisation Ms does not enter the expres-
sion of Vdc explicitly. This is a major convenience as Ms cannot be directly
deduced from FMR experiments (sinceMs and H2⊥ always appear together),
and other measurements such as SQUID and VSM are required to determined
its value. The terms Asym and Aasy are also angle-dependent, as they depend
on the magnetic anisotropy of the device.
On the other hand, the 90o out-of-phase (imaginary) component of Vdc
consists of two Lorentzian components with the following amplitudes:
V ′sym(ϑ) = −
I∆R
2
Aasy sin(2ϑ)(hx sinϑ+ hy cosϑ) (A.20)
V ′asy(ϑ) =
I∆R
2
Asym sin(2ϑ)hz (A.21)
This implies that if there is any phase difference ψ between the current
and the oscillating resistance, the angular dependence of Vasy is partially
mapped onto Vsym and vice versa. This is demonstrated in Section 4.5.1.
135
Appendix B
Measurement apparatus
136
dipstick
head
vacuum
seal
coldfinger
chip
carrier
VTI
pump
lineliquid
nitrogen
metre superconducting
magnet
power
supply
stainless
steel
body
Figure B.1: Left: The dipstick used in the FMR experiments. The dc
wires (copper and constantan, both purchased from CMR-Direct) and coaxial
cables (purchased from Oxford Instruments) go through the hollow stainless
steel body of the dipstick, and connect to the sample carrier (custom made by
PCB Train, see also Figure 3.2a) which is loaded at the end of the coldfinger
(made with oxygen-free copper by the Cavendish Workshop). Right: The
dipstick when loaded into the variable-temperature cryostat (manufactured
by Cryogenics Limited).
137
24-way connector
for Cu wires
10-way connector
for Constantan wires
Hermetically-sealed
SMA connectors
Figure B.2: An enlarged photo of the top of the dipstick. It consists of
a NW-50 flange and various vacuum components (all purchased from Nor-
Cal Products, Inc.). The flange hosts one 24-way dc connector which links
the copper wires, and serves to power any electronic component installed
on the coldfinger, such as radio-frequency couplers, amplifier, temperature
sensors (purchased from Lakeshore Cryotronics) and Hall sensors (also from
Lakeshore Cryotronics). There is also a 10-way dc connectors linking the
constantan wires, and it allows measurement of the photovoltage (both con-
nectors are purchased from Oxford Instruments). Besides the dc connectors,
the flange also hosts 3 hermetically-sealed SMA connectors (purchased from
CMR-Direct).
138
Ref. Signal
Lock-in
V
a
e
d
c
b
Figure B.3: The apparatus used in the FMR experiments. a, Signal generator
(2 → 20 GHz) from Anritsu, model MG3692A. b, High-frequency bias-tee
(0.1→ 40 GHz) fromAnritsu, model K250. c, High-frequency flexible coaxial
cable (0 → 18 GHz) from Mini-Circuits, model CBL3SMQ-SM+. d, Low-
noise preamplifier from Stanford Research Systems, model SR560. e, Lock-in
amplifier from Stanford Research Systems, model SR830.
139
Appendix C
Lithography recipes for the
fabrication of microdevices
Design 1 – Devices with on-chip waveguide
Wafer cleansing: First with acetone for 5 min, then with IPA for 5 min
(applying ultrasound in both steps)
Mesa
• Spinning of resist: PMMA (A6, 950mw) for 1 min at 4000 rpm
• Pre-bake for 90 s at 180oC on a thermal hot plate
• E-beam exposure: 650 C/cm2
• Resist development: 50 s at 20oC in diluted MIBK solution (MIBK:IPA
= 1:3)
• Rinse in IPA and blow dry with N2 gas
• Oxygen plasma for 30 s
• Reactive ion etching: mixture of SiCl4 and Ar gas for 140 s (etch rate:
∼ 10 nm/min for (Ga0.88,Mn0.12)As)
Metal contacts and waveguide
140
• Spinning of resist: PMMA (A6, 950mw) for 1 min at 4000 rpm
• Pre-bake for 90 s at 180oC on a thermal hot plate
• E-beam exposure: 650 C/cm2
• Resist development: 50 s at 20oC in diluted MIBK solution (MIBK:IPA
= 1:3)
• Rinse in IPA and blow dry with N2 gas
• Prepare for evaporation: Oxygen plasma for 30 s, wash in diluted HCl
solution (HCl(32%):H2O = 1:10) for 30 s, rinse in DI water
• Evaporation: Cr: 20 nm at 4.2 A˚/s, Au: 200 nm at 2.2 A˚/s
• Lift-off: 2 h in acetone. Once successful, rinse in IPA and blow dry
with N2 gas
Design 2 – CIT-FMR devices
Wafer cleansing: First with acetone for 5 min, then with IPA for 5 min
(applying ultrasound in both steps)
Mesa
• Spinning of resist: PMMA (A6, 950mw) for 1 min at 4000 rpm
• Pre-bake for 90 s at 180oC on a thermal hot plate
• E-beam exposure: 650 C/cm2
• Resist development: 50 s at 20oC in diluted MIBK solution (MIBK:IPA
= 1:3)
• Rinse in IPA and blow dry with N2 gas
• Oxygen plasma for 30 s
• Reactive ion etching: mixture of SiCl4 and Ar gas for 140 s (etch rate:
∼ 10 nm/min for (Ga0.88,Mn0.12)As)
141
Metal contacts
• Spinning of resist: UV-III for 30 s at 5000 rpm
• Pre-bake for 60 s at 155oC on a thermal hot plate
• UV exposure for 120 s
• Post-bake for 60 s at 155oC on a thermal hot plate
• Resist development: 30 s at 20oC in CD-26 developer
• Rinse in DI water and blow dry with N2 gas
• Prepare for evaporation: Oxygen plasma for 30 s, wash in diluted HCl
solution (HCl(32%):H2O = 1:10) for 30 s, rinse in DI water
• Evaporation: Cr: 20 nm at 4.2 A˚/s, Au: 200 nm at 2.2 A˚/s
• Lift-off: 2 h in acetone. Once successful, rinse in IPA and blow dry
with N2 gas
142
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