Modeling and Numerics for Two Partial Differential
Equation Systems Arising From Nanoscale Physics
Daniel Brinkman
Darwin College
DAMTP, Center for Mathematical Sciences
A thesis submitted for the degree of
Doctor of Philosophy
December 2012
I dedicate this thesis to my Fiance´e, whose love and support were always appreciated,
even from 4000 miles away.
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.
Daniel Brinkman
ii
Acknowledgements
This work would not have been possible without the help and support of my colleagues,
friends, and family.
First and foremost I would like to thank my supervisors Professor Peter Markowich
and Professor Klemens Fellner. Professor Markowich introduced me to the topics con-
tained within my thesis and helped me to find both coauthors and funding through his
KAUST grant. Professor Klemens Fellner was instrumental in directing my day-to-day
research beginning from my first week in Cambridge. Even after he accepted a position
at the University of Graz, he continued to help me find direction both in my overall
career development and the research which makes up Chapter 2 of this thesis.
I am very grateful to the post-doctoral researchers in the Applied Partial Differential
Equations group at the University of Cambridge for many helpful discussions, especially
Marie-Therese Wolfram and Clemens Heitzinger. Marie’s assistance was vital for my
understanding of the finite-element method used in Chapter 2 of this thesis (especially
with regards to the use of NetGen and NGSolve). Clemens provided excellent and
immediate hands-on direction in the project associated with Chapter 3 of the thesis.
I would also like to thank the other PhD students of Peter Markowich at the Uni-
versity of Cambridge: Daniel Marahrens, Jan Pietschmann, and Alex Lorz. I would
especially like to thank my officemate Daniel for many interesting discussions, mathe-
matical and otherwise. I would also like to thank Professor Peter Olver for kindling my
love of PDEs as an instructor and research advisor during my undergraduate years.
Outside of the world of mathematics, I would like to thank my flatmate Matt for
helping me maintain perspective (or at least extending my awareness to PhDs in sciences
other than math and physics) and for many evenings of videogames and deep conversa-
tions. I would also like to thank my friends from the USA who took the time to cross
the Atlantic and visit me: Nic, Tyler, Ben, Amanda, Katrina, Jon, and Kory.
None of this would have been possible without the support of my family. I would
like to thank my parents, Tom and Diane, for their support and encouragement to leave
my home country and my comfort-zone. I would also like to thank my sister, Beth, for
helping me to feel connected to home through various online conversations.
Last but certainly not least I would like to thank my fiance´e Denise for all of the
love and support she has given me over the last three and a half years. Thank you for
encouraging me to follow my dreams and pursue my studies far far away from home.
iii
Abstract
The mathematical field of analysis of partial differential equations has undergone
rapid changes in recent years. Consistent improvement of mathematical computation
allows more and more questions to be addressed in the form of numerical simulations.
At the same time, novel materials arising from advances in physics and material sciences
are creating new problems which must be addressed. This thesis focuses on two such
materials, organic semiconductors and graphene.
In Chapter 2 we derive a generalized model for organic photovoltaic devices – so-
lar cells based on organic semiconductors. After selecting an appropriate self-consistent
model, we derive asymptotic estimates for the operation of the device in several regimes.
We use these estimates to partially justify several assumptions used in the formulation
of simplified models which have already been discussed in the literature. Furthermore,
we simulate such devices using a 2D hybrid discontinuous Galerkin finite element scheme
and compare the numerical results to our asymptotics. We then discuss the potential
applicability of the simulations to real-devices by identifying which regimes correctly
model physical device behavior and discussing the limitations of the model choice. Fi-
nally, several perspectives are given on proving existence and uniqueness of the model.
In Chapter 3 we derive a convergent second-order finite difference numerical scheme
for simulation of the 2D Dirac equation. We demonstrate the convergence of the numer-
ical scheme with several examples for which explicit solutions are known and consider
how errors appear in the simulations. We furthermore extend the Dirac system with
Poisson’s equation (in 3D) for the electrical potential and investigate the application to
electrons in graphene. In particular, we show that the numerical scheme captures several
interesting physical effects which have been predicted to appear in graphene.
Contents
1 Introduction 9
1.1 Organic Photovoltaic Devices . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.1 Classical Semiconductor Theory . . . . . . . . . . . . . . . . . . . 11
1.1.2 Physics of Organic Devices . . . . . . . . . . . . . . . . . . . . . . 16
1.1.3 Mathematical Modeling of the Organic System . . . . . . . . . . . 20
1.2 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.2.1 Physics of Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.2.2 The Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Photovoltaics 31
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.1 OPV Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 Scaling and Coefficient Modeling . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.1 Dissociation Term . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.2 Device Parameter Values . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.3 Steady State Equations . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Sample Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4.1 Internal Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4.2 IV Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5 1D Stationary State Asymptotics . . . . . . . . . . . . . . . . . . . . . . . 53
2.5.1 Large Applied Field (Constant µ) Approximation . . . . . . . . . . 54
2.5.2 Unipolar Approximation . . . . . . . . . . . . . . . . . . . . . . . . 61
2.6 Discussion of Numerical Examples . . . . . . . . . . . . . . . . . . . . . . 63
2.6.1 A Note On Exciton Diffusion . . . . . . . . . . . . . . . . . . . . . 64
1
2.6.2 Electron/Hole and Exciton Densities at SC, OPP and OV . . . . . 66
2.6.3 Shunt Resistance and Asymptotics of the IV Curves . . . . . . . . 74
2.6.4 Damping and Iteration Convergence . . . . . . . . . . . . . . . . . 77
2.7 Improving the Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.7.1 Implicit Definition of J . . . . . . . . . . . . . . . . . . . . . . . . 83
2.7.2 Approximating the M i . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.7.3 Improved Asymptotic IV . . . . . . . . . . . . . . . . . . . . . . . 86
2.8 Obtaining a Unipolar Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.8.1 The Small η Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.8.2 Zero-Interface Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.8.3 Possible Routes for Improvement . . . . . . . . . . . . . . . . . . . 97
2.9 Extended 2D Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
2.10 Perspective: Proving Existence and Uniqueness . . . . . . . . . . . . . . . 105
2.10.1 Fixed Point Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 108
2.10.2 Entropy Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.10.3 Duality Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.A Explicit Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
2.A.1 Asymptotic W 1,∞ Bounds for X . . . . . . . . . . . . . . . . . . . 119
2.A.2 Calculating JOC for Nonsymmetric Devices . . . . . . . . . . . . . 124
2.B Coding Implementation Details . . . . . . . . . . . . . . . . . . . . . . . . 125
2.B.1 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
2.B.2 Solving the PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3 Graphene 129
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.2 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.2.1 The Scaled Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.2.2 Dirac Equation in Two Spatial Dimensions . . . . . . . . . . . . . 132
3.2.3 Derivation of Dirac Model for Graphene . . . . . . . . . . . . . . . 134
3.2.4 Adding Poisson’s Equation . . . . . . . . . . . . . . . . . . . . . . 134
3.2.5 The Main Model Equations . . . . . . . . . . . . . . . . . . . . . . 135
3.3 The Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.3.1 The Finite-Difference Scheme . . . . . . . . . . . . . . . . . . . . . 136
3.3.2 Adding a Consistent Potential . . . . . . . . . . . . . . . . . . . . 144
2
3.3.3 Scaling for Graphene Devices . . . . . . . . . . . . . . . . . . . . . 145
3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
3.4.1 Wave Equation Results . . . . . . . . . . . . . . . . . . . . . . . . 146
3.4.2 Results for Particles with Mass . . . . . . . . . . . . . . . . . . . . 148
3.4.3 Adding the Electric Potential . . . . . . . . . . . . . . . . . . . . . 155
3.4.4 Adding the Magnetic Potential . . . . . . . . . . . . . . . . . . . . 157
3.4.5 Self-Consistent Fields . . . . . . . . . . . . . . . . . . . . . . . . . 158
4 Conclusions 162
4.1 Photovoltaic Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.2 Graphene and the Dirac Equation . . . . . . . . . . . . . . . . . . . . . . 164
3
List of Figures
1.1 Electron and Hole Illustration . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Exciton and Polaron Pair Illustration . . . . . . . . . . . . . . . . . . . . . 17
1.3 Bulk Heterojunction Schematic . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 OPV Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Sample Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3 Current-Voltage Curves with Power . . . . . . . . . . . . . . . . . . . . . . 51
2.4 Current-Voltage Comparison for Constant kd . . . . . . . . . . . . . . . . 52
2.5 OPV Sample Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.6 Changing µX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.7 Short Circuit Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.8 Short Circuit Boundary Layer Solution . . . . . . . . . . . . . . . . . . . . 70
2.9 Optimal Power Point Solution . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.10 Open Circuit Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.11 Current-Voltage Curve for Shunt Current . . . . . . . . . . . . . . . . . . 75
2.12 Asymptotic Current-Voltage Curve . . . . . . . . . . . . . . . . . . . . . . 76
2.13 Current-Voltage Comparison for Constant kd . . . . . . . . . . . . . . . . 77
2.14 Short Circuit Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.15 Optimal Power Point Convergence . . . . . . . . . . . . . . . . . . . . . . 79
2.16 Open Circuit Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.17 Convergence Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.18 Required Iterations Depending on α . . . . . . . . . . . . . . . . . . . . . 82
2.19 Required Iterations for kd Constant . . . . . . . . . . . . . . . . . . . . . . 82
2.20 Asymptotic IV Curve from the M i . . . . . . . . . . . . . . . . . . . . . . 87
2.21 Second Order Asymptotic IV Curve . . . . . . . . . . . . . . . . . . . . . 88
4
2.22 Very Accurate Asymptotic IV . . . . . . . . . . . . . . . . . . . . . . . . 90
2.23 Carrier Concentrations on the Interface . . . . . . . . . . . . . . . . . . . 92
2.24 Minority Electron Concentration at Open Circuit . . . . . . . . . . . . . . 93
2.25 Asymptotic IV with Jn and Jp . . . . . . . . . . . . . . . . . . . . . . . . 99
2.26 Piecewise Linear Interface Plot . . . . . . . . . . . . . . . . . . . . . . . . 101
2.27 2D Sample Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
2.28 Dynamic Work Function Plots . . . . . . . . . . . . . . . . . . . . . . . . 103
2.29 Exciton Trapping Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.30 Sinusoidal Device Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.31 Meshing Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.1 1D Traveling Wave Example . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.2 1D Symmetric Wave Example . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.3 2D Wave Equation Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.4 2D Simulation with Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3.5 Traveling Wave with Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.6 Symmetric Wave with Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.7 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3.8 Convergence Order Evidence . . . . . . . . . . . . . . . . . . . . . . . . . 152
3.9 Discretization Error Demonstration . . . . . . . . . . . . . . . . . . . . . . 153
3.10 Traveling Wave Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
3.11 Conservation of Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
3.12 Veselago Lensing over Time . . . . . . . . . . . . . . . . . . . . . . . . . . 156
3.13 Veselago Lensing for Various V . . . . . . . . . . . . . . . . . . . . . . . . 156
3.14 Beam Splitting over Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
3.15 Beam Splitting for Various V . . . . . . . . . . . . . . . . . . . . . . . . . 158
3.16 Traveling Wave with Magnetic Field . . . . . . . . . . . . . . . . . . . . . 159
3.17 Effect of Changing λ2 with Self-Consistent Potential . . . . . . . . . . . . 159
3.18 Beam Splitting with Self-Consistent Potential . . . . . . . . . . . . . . . . 160
3.19 Beam Splitting Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 161
3.20 Effect of Domain-Size on Self-Consistent Potential . . . . . . . . . . . . . 161
5
Nomenclature
General
χI The characteristic function of I
∂
∂t Partial derivative with respect to time
∇· Divergence operator
∇ Gradient operator
ν Unit normal vector
Ω Domain of the problem
Γ The boundary (∂Ω)
C1 The space of continuously differentiable functions
i
√
−1
L2 The Lebesgue space of order 2
Chapter 2 (Photovoltaics)
n Electron concentration
Jn Electron current
p Hole concentration
Jp Hole current
J Total electric current
6
U Work function potential
V The electric potential
E The electric field, −∇V
X Exciton (or polaron pair) concentration
α Iterative damping parameter
r Relative electric permittivity
ΓD The part of Γ with mostly Dirichlet conditions
ΓN The part of Γ with mostly Neumann conditions
λ Debye length
µn Electron mobility
µp Hole mobility
µX Exciton mobility
∇ν Normal derivative (with respect to I)
cr Bimolecular recombination rate
c′r Rate of excition formation due to bimolecular recombination
d Interface width
ESC The typical electric field at short circuit
G Exciton generation rate
h Dipole charge separation length
I Interface region of the device (in the domain sense)
IC The complement of I with respect to Ω; the bulk of the device
kd Exciton dissociation rate
kr Exciton recombination rate
7
Vdiff Potential difference
Chapter 3 (Dirac)
ψ Wave vector
∆x The step size in the x-direction
 Electric permittivity
Φ Electric potential
A Magnetic vector potential
c Wave speed (not equal to speed of light for graphene)
m Electron mass
8
Chapter 1
Introduction
The role of mathematics in modern science cannot be overstated. Every discipline takes
advantage of mathematical models and statistical tests, with numerical simulations be-
coming ever more common. As the science begins to demand more and more mathe-
matical sophistication, classical mathematical techniques are applied to more and more
interesting problems. New mathematical tools can be applied to garner new insights to
old problems, and to answer new questions which require a high level of sophistication.
With this backdrop, partial differential equations have become increasingly valu-
able. The broad array of problems which can be modeled using variations of classical
PDEs is staggering. Simple variants of the prototypical elliptic, parabolic, and hyper-
bolic PDE systems can be used to model phenomena ranging from electronic devices
to crowd dynamics to the structure of our universe. Adding to the complexity are the
rich applications which result from coupling multiple PDEs into a system. Such sys-
tems can consist of several similar equations with complicated coupling mechanics, or of
completely different types of equations.
This thesis focuses on two systems of the second type, which couples the elliptic
Poisson’s equation for the electrical potential to evolution equations governing the mo-
tion of the particles, which then in turn govern the potential. The lack of a specific
time-scale in the elliptic equation requires special consideration in order to apply appro-
priate numerical schemes. The most prevalent of such techniques are iterative. Solving
Poisson’s equation can be done using well-understood elliptic techniques at each time
step, allowing us to insert a “known” potential into the evolution equations in order to
solve systems which can be essentially arbitrarily non-linear in the potential.
9
Despite this underlying structural similarity and the common application to novel
nanoscale electronic devices, the problems addressed in this thesis have very little in
common. In Chapter 2 we deal with a coupled system of parabolic second-order drift-
diffusion-convection equations modeling the behavior of organic photovoltaic devices
using finite element methods for numerical simulation. In Chapter 3 we deal with a
coupled system of hyperbolic first-order advection equations modeling the behavior of
a sheet of graphene using finite difference methods. In this introduction, each problem
will be introduced from both a physical and mathematical perspective, including an
overview of the current literature and a discussion of the role of this work within the
larger scheme of the problems.
1.1 Organic Photovoltaic Devices
Solar energy has been touted as an energy generating alternative to fossil fuels for ages.
First patented for electricity generation near the end of the 19th century [94], solar
power had already been used (for instance as a method for heating water) for many
years. Modern commercial solar cells are all largely based on semiconductor technologies,
the first of which was patented in 1946 [66]. The patent was submitted in 1941, thus
predating all forms of nuclear technology. However, difficulties in nanoscale fabrication
lead to solar energy lagging behind nuclear power for many years. Recent concerns about
the safety and environmental impact of nuclear waste have lead to exceptional renewed
interest in solar power.
Other forms of renewable energy sources also exist. Hydroelectric power and wind
power, in particular, have received considerable interest. However, the environmen-
tal impact of such technologies are also questioned. Whether disrupting the mating
and migration of fish species or killing endangered birds, the impact of harnessing the
transportation media of animals is fairly controversial. In contrast, solar cells can be
installed on artificial structures which are already in place, with significantly reduced
environmental impact.
The ubiquitous semiconductor solar cell does have several disadvantages, however.
High efficiency cells often rely on rare elements which only result from environmentally
damaging mining processes, and some reports indicate that the total carbon cost of
producing, transporting, and installing the devices outweighs the carbon savings from
the clean energy. Furthermore, since classical semiconductor devices tend to be etched
10
onto a rigid substrate, they are by nature inflexible and somewhat unwieldy.
New research into organic photovoltaic (OPV) devices is intended to rectify both of
these difficulties. Consisting of polymers which can be readily synthesized in the bulk
from common materials, these devices can be loosely called plastics. Often flexible and
significantly cheaper than traditional devices, OPVs have the potential to revolutionize
the solar industry. Unfortunately, these devices currently have solar efficiencies which
are significantly below those of modern inorganic semiconductor devices (i.e. at present
8.3% compared to 27.6% for “one-sun” Gallium Arsenide and 42.3% for concentrator
cells [37]). However, significant research from a variety of disciplines has focused on
methods for improving these efficiencies.
Many mathematical models have been proposed to study the behavior of bilayer
OPV cells as simple implementable organic devices (for instance [5, 13, 15, 21, 50]). A
shared mathematical basis for these models comes from extensive literature on inorganic
semiconductor models. We begin by presenting the classical mathematical system for
inorganic semiconductor devices, then present the necessary extensions to apply the
model to OPVs.
1.1.1 Classical Semiconductor Theory
The PDE Model System
In classical semiconductor theory there are two charge carriers. Free electrons (n) are
the negative charge carriers, but are only considered at sufficiently high energies (the
conduction band) enabling them to move throughout the device. Lower energy levels are
nearly completely filled and thus usually display no dynamics. However, if an electron
is missing from one of these lower levels (the valence band), it creates the dynamics of a
positively charged particle called a hole (p) – see Figure 1.1. In a real device, the equilib-
rium state can be modified chemically. A region with extra electrons is called n-type and
a region with extra holes is called p-type. Because the carriers are electrically charged,
they move in accordance to the local electric potential (V ). The electric potential exerts
a force on charged particles proportional to its negative gradient (E = −∇V ) which is
called the electric field.
For inorganic devices, the standard macroscopic models couple two drift-diffusion-
reaction equations for the behavior of the free charge carrier densities (i.e. electrons
and holes) in a system with Poisson’s equation governing the self-consistent electrical
11
n-TypeNormal
Electric Field Electron Drift
Excited State
p-TypeHole Drift
En
er
gy
Position
Figure 1.1: Schematic illustration of electron and hole transport for the conduction band
(top line) and valence band (lower line) of a classical semiconductor with an applied field
tilting the bands. Electrons are red circles, holes are black outlines. The blue area is a
forbidden energy region called the band gap. Note that this simplified schematic ignores
the effects of the self-consistent potential, which would cause the bands to “bend“ in the
n-type and p-type materials.
potential [74, 81]. The usual drift-diffusion system for classical semiconductors (also
called van-Roosbroeck’s system [89]) is as follows:
∇ · (∇V ) = q(n− p− C), (1.1a)
∂n
∂t
= ∇ · (Dn∇n− µnn∇V )−R, (1.1b)
∂p
∂t
= ∇ · (Dp∇p+ µpp∇V )−R; (1.1c)
where the variables and parameters will be explained in the following paragraphs.
It is possible for a free electron to fill a hole, resulting in the “loss” of both an
electron and a hole. This process, called recombination, is taken to occur at rate R,
which depends on the concentrations n and p. The reverse process is also possible, but
requires an input of energy equal to the band gap (the difference between the valence
band and the conduction band – the blue region in Figure 1.1). When this energy is
provided by light, we call the process photogeneration, which gives an additional negative
contribution to R.
Within the semiconductor, the influence of the electric potential on the charge carriers
is modeled by a drift coefficient proportional to µ∇V . The electrical current in the device
12
is given by the sum of the electron current, Jn, and the hole current, Jp, given by:
Jn = (Dn∇n− µnn∇V ) Jp = −(Dp∇p+ µpp∇V ).
The mobility µ and the diffusion constant D account for how the material properties
govern the motion of the charges. Even for general semiconductors several models exist
for the dependence of µ on other variables in the system, especially the electric field [31].
For most applications, D is taken to be proportional to µ (the Einstein relation) [74].
The interesting behavior of semiconductors arises because the electric potential is
not specified, but is strongly influenced by the charged carriers via Poisson’s equation.
Each of the charge carriers has the same fundamental charge q which contributes to the
electric potential modified by the electrical permittivity . The electrical permittivity
is a material property, so in a classical semiconductor with only one material (often
silicon), it is usually a constant.
Semiconductor devices (such as transistors and diodes) form the basic components
of most modern technology and are usually created by creating spatial inhomogeneity
via doping – adding stationary charges (ions) to the system. This enters the system as
a doping concentration C which is spatially dependent and can be positive or negative
(depending on the nature of the doping). When C is very negative, it tends to repel
the negative free electrons and creates a p-type region which consists predominantly of
free holes; the corresponding electron-rich region is called an n-type region (as shown in
Figure 1.1). Modifying how boundary conditions are applied and the geometry of the
n- and p-regions allows for incredibly varied systems. For further information on the
mathematical system, please refer to [55].
P-N Diodes and Solar Cells
The device which most resembles the OPV case is the P-N diode, which consists of
exactly one p-region and one n-region, with a single (not necessarily linear or planar)
interface between them (see, for instance Figures 1.2 and 2.1). An extensive mathemat-
ical treatment of such devices is given in [74, Chapter 4]. Such a device is often used to
limit current to a single specific direction. An applied potential in the forward direction
(corresponding to Vapp > 0 in Chapter 2), results in a current roughly following Ohm’s
law. However, an applied potential in the reverse direction (corresponding to Vapp < 0)
gives only a small leakage current. When such a device has charges generated within the
13
device (say, from a negative contribution −G to R resulting from particles generated by
light), this small leakage current is supplemented by the photogenerated current.
Because the device is designed with two intrinsically different energy levels (usually
as a result of the choice of contact metals), it is possible for this current to provide power
to the system. The power of a circuit is given by the product of the current and the
applied potential. For a device with an internal (or built-in) potential Vint, the power
generated by the device is given by the formula
P = −JVapp = J(Vint + Vdiff).
Here Vdiff is the potential difference in the device, which corresponds to the total potential
difference between the two metallic contacts. Since the direction of the photogenerated
current is usually determined by the potential difference Vdiff and NOT the applied
potential Vapp, there exists a regime of positive applied potentials where the current is
negative and the generated power is positive. As explained in the previous paragraph,
for a typical P-N diode J has the same sign as Vapp and thus power is only consumed
by the device, not produced. Note that the power consumption regime can correspond
to using the device as a light-emitting diode (LED). In the case where R corresponds
(at least partially) to light emission, the diode will produce light when a positive field is
applied.
Unfortunately, the notation for these quantities is not uniform throughout the liter-
ature. In particular, for the rest of this thesis, we generally treat Vdiff as the dependent
variable, as it represents the overall boundary data for the system. However, in many
other references (including [74]) Vapp is treated as the dependent variable. We choose
to look at the potential difference mainly because it simplifies the notation for the 1D
asymptotics discussed in Section 2.5.1. The other possible source of confusion is the sign
of the potential quantities. Because the convention for a 1D P-N diode model is for the
P material to be on the left and for the potential difference to be the difference of the P
material and the N material, the potential difference is negative when the potential on
the right is higher than the potential on the left. To reduce the possible confusion, for
this work we will take zero boundary conditions on the right boundary and take the left
hand boundary condition to be Vdiff . For more details, see Figure 2.1 and the discussion
at the beginning of Section 2.5.1.
14
Mathematical Analysis
Much is known about classical semiconductor systems, including existence and unique-
ness results for the steady state. The steady state is of particular interest for DC devices
which generally operate on time scales much longer than the duration of the transient
(i.e. the switching-on) dynamics. In particular, [55] uses an iterative fixed point method
to prove the existence of solutions on a domain Ω ⊂ R2 of the steady state system for
a general class of devices with moderate assumptions on the form of C(x), µ(x,∇V ),
and R(n, p). The boundary of the device Γ = ∂Ω is split into two disjoint parts, with
homogeneous Neumann boundary conditions on one part (ΓN ) and Dirichlet boundary
conditions on the remainder (ΓD). The same reference also demonstrate uniqueness for
small applied voltages. For general doping parameters, it is possible to create a device
with non-unique solutions, but this is not expected to occur for bilayer devices (such as
the P-N diodes of the previous section or the organic bilayer devices considered below).
Existence and uniqueness results are also known for the parabolic time-dependent
system 1.1. One of the most powerful tools for such calculations is the entropy of the
system. Under similar assumptions to those above, one can show that the entropy
functional:
∫
Ω
n log
n
n∞
− (n− n∞) + p log
p
p∞
− (p− p∞) +

2
(∇V −∇V∞)
2 (1.2)
is positive and monotonously decreasing in t. The steady-state variables n∞, p∞, and
V∞ are the solution to the time-independent steady-state problem. Using standard
regularity results for elliptic and parabolic PDEs, it is thus possible to prove that a
unique solution exists for domains Ω ⊂ R2,3 [32]. With a few additional assumptions on
the structure of the coefficients and artificial trapping potentials (instead of more general
boundary conditions), it is also possible to prove exponential decay of the solution to
the steady state solution [23].
Existence and uniqueness of the system with more general coefficients (particularly
µ and R) is also possible, but the proof becomes much more technical because a simple
gradient-flow structure is not evident. However, it is still possible to prove global exis-
tence and uniqueness as well as convergence to a unique equilibrium through fixed point
methods [97].
Numerical simulation of semiconductor devices has been an extremely active field
for decades, with extensive work done at Bell Labs as transistor devices became vitally
15
important for modern technology. A 1964 article by Gummel [39] described the most
ubiquitous method for resolving the difficulty with simultaneously simulating the elliptic
Poisson’s equation with the parabolic reaction-diffusion systems for n and p by iterating
their solutions (now called Gummel iteration or the Gummel map). For an extensive
consideration of the numerical simulation of such systems see [8].
1.1.2 Physics of Organic Devices
The basic mathematical structures of organic photovoltaic devices are nearly identical to
the system for silicon semiconductors. However, the physical processes are significantly
more complicated, in large part because organic materials are not generally good semi-
conductors. More specifically, the band gap in organic polymers is significantly larger
than that of typical inorganic semiconductors. Because most light cannot excite an elec-
tron from the valence band to the conduction band, it is incapable of directly creating
electrons and holes [41] (as adding the term −G to the recombination rate R in System
1.1 would imply). All is not lost, however, because the light is still capable of creating
an excited electron energy state which remains bound to the hole it left behind. This
excited state acts like a neutral particle and is called an exciton [44]. The first organic
bilayer solar cell to take advantage of this alternate light generation mechanism was
produced in 1985 by Tang [86].
Excitons and Polaron Pairs
The dynamics of these excitons have been studied in various materials for many years.
Perhaps the earliest comprehensive analysis of their behavior is given in a 1934 paper
by Lars Onsager [67] (unrelated to the work for which he was awarded the 1968 Nobel
prize in chemistry). In particular, he posited the existence of charge-transfer-states
(excitons) – bound pairs of oppositely charged particles which could either recombine
or dissociate with specific rates. His work focused on explaining why the resistance
of electrolytic fluids did not follow the usual Ohm’s law. However, his method was
extended by paired 1983 papers [7, 36] for the specific case of organic materials. In
this context, the recombination refers to the the electron falling back into the valence
band, emitting energy in the form or light or heat, whereas dissociation refers to the
(productive) creation of free charge carriers. For a detailed examination of dissociation
rate modeling, see Section 2.2.1.
16
Because of the large energy gap in silicon, an exciton will almost always recombine
unless an external force acts to separate the pair. Thus in a region with homogeneous
properties, excitons will usually not create free charges. Luckily for the field of organic
photovoltaics, photogenerated excitons are not stationary, but move within the device.
In a typical device they might diffuse 10 − 50nm before recombining [58]. This allows
us to design a device which contains regions with strong external forces, for instance
the interface between two polymers with properties analogous to the p-type and n-type
silicon discussed in Section 1.1.1. At the intersection of two such materials, the exciton
settles in a configuration for which the bound electron and hole each reside in a material
which is energetically favorable (see Figure 1.2). In such a state, the particles become
much more stable, and eventually dissociate (as opposed to recombining) with very
high probability [3]. Such a configuration is often called a polaron pair. In solid state
physics, a polaron consists of a quasi-particle coupled to a localized electromagnetic
wave. Because our polaron pair has a preferred direction, it creates a dipole moment
near the interface. These dipoles can change the intrinsic energy levels of the device and
lead to complicated electronic structure [90].
n-TypeNorTyp
m-aTlE-ctirToT- 
FdDfFp -dDfFp
xSaEoT- iTHNgT-
tiNEg 
PTHp
xHpaogT-
Figure 1.2: Schematic illustration of an organic photovoltaic bilayer, showing excitons
and polaron pairs. Electrons are shown in red, holes in black. The drawing is not
to scale, but the charge separation distance truly is longer for polaron pairs than for
excitons. Compare to Figure 1.1 where electrons and holes exist only in n and p-type
materials respectively.
17
Device Geometry
Given that the majority of exciton dissociation occurs at the interface between the n-
type and p-type regions of the device, it is natural to consider ways in which to increase
the length of the interface. In particular, if excitons generally diffuse at least 10nm [58],
the optimal device would have an interface within 10nm of any given point. Because it
is difficult to create coherent polymer layers thinner than 50nm, an alternative to the
traditional flat bilayer device is necessary.
One possibility is to use two polymers which tend to phase-separate as a common
solvent evaporates. Two groups demonstrated such bulk heterojunction (BHJ) devices
independently in 1995 [40, 99]. These devices have the benefit of increasing the number
of excitons which reach an interface and thus dissociate. This number is often measured
by the internal quantum efficiency, the percentage of absorbed photons which yield
escaping charges. The problem facing BHJs is the possible creation of “islands” of one
type of polymer (see Figure 1.3). A charge created in an energetically favorable such
region can become trapped and will not productively reach the contacts of the device.
Carefully optimizing the feature size of the BHJ can mitigate this result, with recent
devices exhibiting an internal quantum efficiency of nearly 100% [69].
Another strategy is to construct interdigitated polymers layer-by-layer so that “fin-
gers” of each polymer protrude into the bulk of the opposite material [57]. This main-
tains the charge-transport advantages of a bilayer, since each generated charge still has
a clear path to the contacts of the device – see Figure 1.3. More recent advantages
in nano-fabrication have allowed for bulk processes which create small features such as
nano-columns in a regular pattern across the device [42]. However, it is still extremely
difficult to design such interfaces in practice, especially in the bulk. Nevertheless, such
interfaces are extremely common in modeling because they represent a periodic struc-
ture governed by simple parameters (i.e. the frequency and amplitude of the interface
oscillations). See, for instance [13, 95].
Coefficient Modifications
Although the addition of excitons is the most obvious modification to the classical semi-
conductor system, it is by far from the only such change. The electric-field dependent
mobilities included in classical semiconductor systems become even more complicated
for organic semiconductors. The Poole-Frenkel mobility discussed above [31] can still
18
nInterdigitated Bilayer
Bulk Heterojunction 
Figure 1.3: Schematic illustration of an interdigitated organic photovoltaic bilayer and
a bulk heterojunction device (with medium-sized features). The red material is a n-
type material and the black material is a p-type material. Note that for the BHJ some
material might be isolated from the corresponding electrode. Compare to the structure
Figure 1.2.
be used, but the justification is less clear because in general organic systems are not
crystalline. A similar e
√
|E| form can still be derived [49], but experimental effects in-
dicate that the regime for which these forms are applicable is not particularly broad.
For example, temperature and charge concentration effects can become quite important
for small electric fields [70]. Such effects can be even more important if self-heating is
considered, adding another Poisson’s equation for the temperature in the device [28].
We do not consider these effects in our model, especially since the exact forms for such
modifications are not well-developed. See Section 2.2 for comprehensive details on the
coefficient choices we make in our model.
Another potential issue with the modeling is the behavior of the boundary conditions
of the device. In the first organic mono-layer devices, the charge injection from the metal
contacts into the polymer device played a very important role [86]. However, bilayer
devices show less impact from the metal-organic interfaces and instead depend heavily
on the organic-organic interface [41]. For the majority of our simulations, we use the
nonlinear organic-metallic contact boundary conditions given by [80]. These have shown
great use in a variety of other papers on OPV modeling, including the majority of the
19
drift-diffusion modeling papers cited in this thesis (including [5, 13, 21, 47, 91]), but we
show in Section 2.4 that they have little impact on the simulated behavior of the device.
1.1.3 Mathematical Modeling of the Organic System
The long diffusion ranges of the excitons above suggest supplementing System (1.1) with
a reaction-diffusion equation for the excitons:
∂X
∂t
= ∇ · (Dp∇X) + cR− (kd + kr)X, (1.3)
where X is the concentration of excitons in the device, kd is the rate at which excitons
dissociate and kr is the rate at which excitons recombine. The recombination rate R for
free electrons and holes (which usually depends on n and p) is the same R which appears
in the continuity equations for n and p. The coefficient c represents the portion of re-
combining free electrons and holes that create excitons; the remaining portion represents
radiative losses, generating either heat or re-emitted light. The term kdX must be added
to the right-hand side of each of the continuity equations to represent the generation of
free charges via exciton dissociation. Polaron pairs can be treated with similar dynamics
(up to the meaning of the diffusion term if they are trapped on an interface).
Drift-Diffusion Models
Drift-diffusion equations have been used with different degrees of generality in a variety
of articles on mathematical modeling of OPV devices. In fact, every drift-diffusion model
for organic photovoltaic devices in the literature can be classified based on its treatment
of the excitons and polarons in the device. Because the main contribution of the excitons
to the dynamics of the device is only through the generation of electrons n and holes p,
the primary importance of the model is the behavior of the term kdX throughout the
device. There are several different alternatives treated in the literature. It is particularly
important to distinguish between whether the model considers excitons, polaron pairs,
both individually, or both combined. In particular, if the model focuses on a BHJ, then
the interface regions are indistinguishable from the non-interface regions and the two
types of charge pairs are indistinguishable.
Because the dissociation of polaron pairs is expected to be the main contribution to
free charge carriers in an organic device, it is sensible to consider a model where the
only charge generation comes from polaron pairs. The most influential article with this
20
type of modeling is the 2002 paper by Barker, Ramsdale, and Greenham [5]. The main
downside of their analysis is that it applies only to a homogeneous bilayer device in one
spatial dimension. Although this does allow for very general forms for the impact of
the polaron pairs on the system (including what seems to be the only consistent dipole
treatment given in the literature), it also ignores several interesting details. In particular,
the formula for the polaron pair concentration becomes an ODE, and in the steady state
the concentration can be explicitly written as a function of the other variables. They do
extend the model for transient photocurrents in [46], but a 2D extension is still required
in order to consider phenomena related to polaron pair orientation.
The other possibility for a bilayer device is to simulate only the excitons in the
device. In this case, the dissociation rate can be made dependent on the background
properties of the material being investigated. The most influential work of this type is
described in two 2006 papers by Buxton and Clark [12, 13]. In [13], they describe their
2D finite difference numerical scheme and apply it to sinusoidal interfaces with different
characteristic amplitudes and frequencies. In [12], they first simulate a realistic 2D
polymer morphology and then use their model to simulate the characteristics of a typical
disordered polymer system (the only such drift-diffusion modeling for such complicated
interfaces). They then compare the resulting current-voltage characteristics arising from
a realistic system and the ideal interdigitated system above. The key to the modeling is
that they include a contribution from energy differences in the polymers as part of the
electric field in the device (what we call U in Chapter 2). This allows for the use of an
electric-field dependent dissociation rate which is small for the average field expected in
the bulk of the device. For more details about this modeling, see Section 2.2.1.
Modeling only one type of bound electron-hole pairs is especially reasonable when
simulating BHJ devices. Because it can be more or less impossible to distinguish whether
an individual point consists of the n-type polymer or the p-type polymer, it becomes
reasonable to simply consider averaged characteristics of the system. In particular,
physical observations on sample polymer blends can be used to determine the mobilities
and dissociation rates throughout the device. In this regime, excitons and polaron pairs
are indistinguishable and only one type of particle needs to be simulated. For instance,
the polaron pair modeling above can be used with a bulk concentration of polaron pairs
which are assumed to be stationary (since they will always rest on some type of local
interface), both in the steady state [21, 50] or considering transient phenomenon [47].
21
Alternatively, once can calculate the average creation and dissociation of polaron pairs
as a function of the optical spectrum of the incoming light and use this simulation to
calculate a spatially dependent generation term G for free carriers [51].
The most general perspective is to consider separate dynamics for both excitons and
polaron pairs. To this end, Section 2.1.1 below justifies the creation of a system for
which Equation 1.3 simulates the behavior of both types of bound particles depending
on the location within the device. Taking 1nm as a typical charge separation distance
for an exciton, we assume that bound states within 1nm of the interface are polaron
pairs and that bound states in the bulk are excitons. By smoothing the interface over a
thin region, we allow polaron pairs to diffuse along the interface (as opposed to the work
of [95], which again treats the polaron pair concentrations as an ODE). We consider the
model in 2D, generalizing the effects of [5], including a dipole effect generated by dipoles
on the interface. In particular, we demonstrate that our dipole model exactly replicates
[5] in 1D.
Asymptotic Results
Another common tactic in the literature is to begin with the full drift-diffusion models
presented above and then study simplified asymptotic systems. Of particular note are
early works focused on organic light emitting diodes [19]. These devices are essentially
equivalent to the organic bilayers presented above (see Figure 1.2), except that current is
forced into the device (as discussed for P-N diodes above). Because there is no photogen-
eration term, the model becomes much simpler. In particular, since electrons and holes
are only injected at the appropriate contact, the minority carriers (i.e. electrons in the
hole-transporting material and holes in the electron-transporting layer) are negligible. If
we further assume that the applied field dominates the device so strongly that diffusion
is negligible, then the system (1.1) is explicitly solvable [60].
The assumption that the minority carrier concentrations are zero is extremely preva-
lent in the organic photovoltaic literature. It simplifies many of the calculations, in
particular since the right-hand-side of Poisson’s equation (1.1a) becomes the density of
the majority carrier. As we show in Section 2.5.2, we can thus write the whole system
(1.1) as decoupled PDEs which only interact on the interface of the device. The zero-
current (open circuit) case was given in the literature by [15], while the explicit results in
Section 2.5.2 for nonzero currents were developed independently in [73]. They further-
22
more extend the modeling to include boundary conditions at a thin interface between
the n-type and p-type regions and then solve the resulting equations in several different
current regimes.
A major goal of our asymptotic calculations in Section 2.5 is to justify the zero mi-
nority carrier limit for general organic photovoltaic devices. We give numerical evidence
supporting this qualitative conclusion in Section 2.6. However, we also show in Section
2.8 that a physically relevant strict limit of this kind cannot be obtained from a bulk
drift-diffusion model of the type we have chosen. Motivated by the large-field approxi-
mations in [60], we develop an asymptotic scheme (including minority carriers) based on
a strong electric field in the device (see Section 2.5.1). This gives reasonable results for
large potential differences, but begins to fail as the potential difference approaches zero.
Of particular interest is the response of the current in the device, J , to the potential
difference Vdiff (see Figure 2.3). Of specific importance is the short circuit current JSS,
the open circuit voltage VOC (the point where the current is zero), and the fill factor.
The fill factor represents the ratio of the optimal power output to the ideal power output
of the device [22]. The ideal power output is JSS ∗ (Vint + VOC), these being the largest
current and the largest applied potential in the energy-generating regime. We see in
Section 2.6.3 that our asymptotic result matches the numerical results (discussed below)
well for negative fields and approaching VOC. To better simulate the intermediate region,
we show in Section 2.7 that exactly integrating the PDE system yields simple expressions
which can determine important quantities for device operation. In particular, we can
then use Gummel iteration to approximate the electric potential and charge carriers
in the device and we obtain excellent current-voltage characteristic results in Section
2.7.3. However, we also demonstrate in Section 2.8 that our model gives quantitatively
unrealistic results at VOC, even though it demonstrates the correct qualitative behavior.
Finite Element Numerical Simulation
Following the example of the majority of the literature, we choose to simulate the device
in the steady state regime. In Section 2.3 we implement the model equations using
a hybrid discontinuous Galerkin [16, 17] finite element discretization. These elements
allow for discontinuities or near-discontinuities along edge boundaries, providing an ideal
situation for the interface between the n-type and p-type regions within the device. We
solve the system by iterating solution of Poisson’s equation with solution of the continuity
23
equations (Gummel iteration [39]). Although potentially slower than Newton-methods
used elsewhere [20], Gummel iteration is known to be extremely robust. We create
the mesh using NetGen [78] and the related NGSolve wrapper. The matrix equations
are solved using the PARDISO solver [76, 77]. The same combination has been used
extensively for other reaction-diffusion problems [52, 96]. For details on the numerical
implementation, see Appendix 2.B.
In Section 2.9 we discuss the use of our system for complicated interfaces such as those
discussed in [12] and [95]. The use of finite elements allows for complicated interfaces
to be implemented directly. However, choosing to have a thin interface region around
the interface introduces numerous difficulties with implementation in the current form,
especially with regards to the dipole term and interfaces which are not continuously
differentiable (such as the interdigitated bilayer in Figure 1.3).
Existence and Uniqueness Results
Based on the extensive literature on classical semiconductor devices, there have been
several results on the existence and uniqueness of drift-diffusion models for organic semi-
conductor devices. However, because of the variety of models presented in the literature,
few general results are given. Several particular difficulties exist in the organic modeling
case, most of which are mathematically intractable. In particular, the convection and
diffusion coefficients become quite complicated. The mobilities µ and the permittivity 
are usually bounded away from zero for physical models, so uniform ellipticity is usually
not problematic. However, the two materials in the system usually have different values
for the coefficients, so for a bilayer with a sharp interface, discontinuous mobilities must
be considered. Although difficult, this can be dealt with by using approximating systems
and taking an appropriate weak limit [30]. Likewise, the mobilities given in physics lit-
erature are not a-priori bounded from above. Because this is generally unphysical (since
very strong electric fields will cause a breakdown of the material), it is usually artificially
bounded by a very large saturating mobility (which is never reached in simulations).
Existence is proven for the steady state and transient regimes in [21]. However, they
have µX = 0 and require that the other coefficients in the exciton equation (1.3) are
constant. In particular, this avoids potential feedback difficulties of kd(E) and allows
the exciton equation to be solved explicitly in the steady state. With an explicit formula
for the excitons, the system fits within the framework of [74] and the proof of existence
24
is straightforward. A more general work given in [35] allows for a countable number
of defect states with a similar structure to the exciton equation. They again use an
approximation method to handle the worst nonlinearities, and also use an entropy similar
to Equation (1.2). In many respects, their model is close to the one we consider, but
once again they do not include field dependent rates or exciton dissociation.
In Section 2.10, we give several perspectives on proving the existence and uniqueness
of our system. In particular, we consider how the fixed point methods might be extended
to cover our case, examine modifications to the entropy in order to control excitons which
can diffuse, and consider how duality methods could be applied to the problem.
Modeling Extensions
There are several possibilities which we do not consider here but which could provide
interesting extensions to the current modeling. Some coefficient generalizations have
already been discussed above and fit perfectly into the scheme we present below. One
such topic is optical modeling. For most devices light is only incident from one direction
and is absorbed at different rates in different materials [71]. Since this type of absorption
profile is static, it would only require a preprocessing step to calculate the (now spatially
dependent) value of G (as in [51, 95]). Likewise, including mobilities which depend on
the charge concentrations would require only minor changes to our numerical implemen-
tation. Adding consistent temperature effects (as in [28]) would be less straight-forward,
requiring another Poisson’s equation in the system. However, this too could fit into our
framework, likely being implemented with an additional iteration step using the previous
values of n, p, and V .
In addition, there are also extensions which would require an entirely different imple-
mentation system. For the simple organic bilayer considered in Section 2.6, the interface
region is isolated from the rest of the device and simulation in the bulk is relatively well
understood. However, the behavior of excitons on the interface is less clear, and could
benefit from extended small-scale modeling techniques. One such possibility would be
the use of Monte Carlo simulations.
Because the drift-diffusion systems discussed above all rely upon local concentrations
being well-defined, they are not necessarily applicable to devices which have features on
a very small scale. In particular, BHJ devices can have features on the same scale over
which quantum mechanical dissociation of excitons can occur near the interface. For
25
such simulations, Monte Carlo statistical modeling can be used to understand exciton
dissociation [65]. Such simulations can be extended and used for an entire OPV de-
vice [56], showing the importance of both the local interface orientation and the size
of the features. Further work by the same authors [38] indicates that larger features
result in better efficiencies (at least in the applicable range of 4nm-16nm), mirroring the
experimental results discussed above for BHJs.
1.2 Graphene
Graphene is an allotrope of carbon, consisting of a single 2D hexagonal lattice. The more
common graphite (common in pencils around the world) consists of stacks of millions
of layers of graphene. Writing with a graphite pencil leaves behind tens or hundreds of
graphene sheets along the paper. Although studied mathematically as an auxiliary to the
study of graphite rods over sixty years ago [92], graphene was ignored as a 2D material
until quite recently. Long considered thermodynamically unstable, recent advances in
fabrication techniques have resulted in a relatively simple recent method for isolating
graphene [64], creating significant interest in the material.
1.2.1 Physics of Graphene
In recent years graphene has received attention due to its interesting mechanical, optical,
and electronic properties. Although the bulk mechanical properties are difficult to test
using the sample sizes available, there is no doubt that graphene exhibits impressive
electronic qualities (for a review, see [59]). The electronic properties of graphene are
interesting due to both useful applications and because they provide a test-bed for the
Dirac equation [82].
The extensive list of interesting properties is one of the main reasons that Andre
Geim and Konstantin Novoselov were awarded the 2010 Nobel Prize in physics [1]. Their
seminal 2004 article, [61], demonstrated the experimental properties of graphene for the
first time. The next year, they published their technique for isolating single sheets of
graphene from bulk graphite, [64]. In that same year, they also published an article, [63],
discussing the possibility of using graphene to experimentally test phenomena predicted
by the Dirac equation (albeit at longer time-scales). This allows for exciting tests of
quantum mechanical properties which are not possible (given current technology) for
26
fundamental particles. From these three articles, research in the area has exploded,
with study proceeding in an astounding number of directions.
The electronic properties of graphene have been studied extensively from a variety
of vantage points. Largely due to the low resistance of graphene, precise measurements
can be made for previously infeasible applications or for challenging environments. In a
pair of 2007 papers, two such directions are considered. In [62], the quantum hall effect
is demonstrated in graphene at room temperature, a substantial improvement over the
liquid-helium temperatures previously required. In [75], detection of single gas molecules
by graphene sheets is demonstrated. Practical applications have also abounded, with
particular emphasis on high-frequency transistors [54, 98]. Such devices could prove
key to future improvement in computing speeds, with the potential to replace the aging
complementary metal oxide semiconductor architecture. Other research has indicated
that because electron transport is nearly ballistic, it should be possible to observe optical
effects in graphene [14]. In particular, we investigate possible applications to beam
splitters and Veselago lenses in Section 3.4.3.
The geometry of graphene is also nontrivial. In fact, some effects (especially the
Hall effect) in graphene ribbons depend non-trivially on the relationship between the
hexagonal structure of graphene and the geometry of the device [59]. Furthermore,
graphene is not necessarily limited to flat geometries [68]. Carbon nanotubes can be
considered as graphene sheets rolled into a tube (with different properties depending
on the structure relationship discussed above). Such results emphasize the need for
modeling to determine the stable structures which will be formed in different regimes [6]
as well as to predict their electronic and optical properties.
1.2.2 The Dirac Equation
Many interesting studies have focused on the behavior of low-energy electrons in neutral
graphene [63, 82]. In this regime, one can show that the tight-binding model for the
graphene lattice yields a Hamiltonian identical to that of the 2D Dirac equation (albeit
with a velocity vf ≈ .01c).
Using the trigonal symmetry of the graphene lattice, it is well known [92] that the
tight-binding model yields the Hamiltonian
E(k) = ±γ
√
3 + 2 cos (
√
3kya) + 4 cos (
√
3kya/2) cos (3kxa/2),
27
where k = (kx, ky) is the wave vector, γ is a hopping frequency and a is the lattice
spacing (assuming symmetric energy levels for convenience) [59]. Because each atom
has one free electron, the negative branch of the square root is usually completely filled,
and the minimum energy states will be the minimums of the positive branch. It is easy to
verify that six such minimum momenta K exist, oriented at the corners of the hexagonal
Brillouin zone of graphene. For wave vectors k = K + q with |q| << 1, we find
E(k) =
3at
2
|q|+O(|q|2)
where E(K) = 0 due to the symmetry assumptions above.
This dispersion relation is identical to the dispersion relation of the 2D Dirac equation
with the speed of light, c replaced by the velocity vf = 3aγ/2. One can therefore derive
the Hamiltonian for the system and verify that indeed we obtain the Dirac Hamiltonian
[82]. Thus for the low-energy wave vectors sufficiently close to the Dirac points K, we
can use the 2D Dirac equation to simulate the behavior of electrons in graphene.
With this motivation, we turn to the mathematical modeling of the Dirac equation:
i~
∂ψ
∂t
= (cα¯ · P¯ + βmc2)ψ
for the wave vector ψ, where t is time, c is the characteristic speed (usually the speed of
light), P¯ is the spatial momentum operator (with three components), and m is the mass.
The coefficients αi and β must obey structural assumptions in order for the equation to
correctly model physical behavior. The momentum operator P¯ is primarily determined
by choosing position-space as a basis. Furthermore, coupling with an electromagnetic
field alters both the time derivative and the momentum operator. Choosing the usual
minimal coupling [79] we obtain:
i~
∂ψ
∂t
=
(
cα¯ · (−i~∇+
e
c
A¯) + βmc2 + eΦ
)
ψ,
where e is the fundamental charge, Φ is the electric potential, and A is the magnetic
vector potential.
First introduced by Paul Dirac in 1928 [24], the eponymous Dirac equation models
the relativistic motion of electrons and other spin-1/2 fermions. It now forms the basis
of huge sections of theoretical physics and is a cornerstone of modern study of quantum
mechanics. The most novel aspect of the equation is that in 3D the coefficients (αi and
β) are 4 × 4 matrices, so the equation is actually a system of four partial differential
28
equations. This was particularly unexpected since only two “types” of electrons (spin up
and spin down) are physically observed and the other two equations seem to indicate the
existence of electrons with negative energy. Dirac himself proposed one solution to this
apparent paradox by speculating that the unobserved states were completely filled with
electrons and that we only observe “holes” where such electron states are empty [25].
This provides the nomenclature of electrons and holes used extensively in semiconductor
theory (as in Chapter 2 of this thesis). Global existence and uniqueness of results for the
electromagnetic system (for instance coupled with Maxwell’s equations) is well known
[29].
Although some work has been published addressing the Dirac equation with electro-
magnetic fields from a numerical perspective, it has largely focused on the 3D regime,
often using a fluid-dynamics framework to speed computation [83, 84] or specifically in
the semi-classical and non-relativistic regimes [45]. Further work has been done specif-
ically in 2D, but it has largely focused on illustrating the phenomena associated with
quantum mechanics [87, 88] rather than the simulation of graphene as a 2D material.
For a 2D material, we only have to deal with two systems, because the orthogonality
conditions on α and β can be satisfied by 2x2 matrices (for instance by the 2× 2 Pauli
matrices). After scaling (see Section 3.2), we obtain the following system:
∂tu
1 = −
(
∂xu
2 + iAxu
2 − i∂yu
2 +Ayu
2)− i(1 + V )u1,
∂tu
2 = −
(
∂xu
1 + iAxu
1 + i∂yu
1 −Ayu
1)+ i(1− V )u2.
We show in Section 3.2.2 that a na¨ıve homogenization of the 3D Dirac equation with
one isotropic direction results in two coupled copies of the 2D Dirac equation (provided
that the electric potential and magnetic fields also obey consistency requirements).
By choosing the Coulomb gauge [79] and further assuming that the magnetic field is
transverse and dominated by the applied magnetic field, we show in Section 3.2.4 that we
can couple the system to a self-consistent electric potential by introducing the (scaled)
Poisson’s equation:
∆V = −λ˜2
(
|u˜1|2 + |u˜2|2
)
,
where |ui|2 is the concentration of particles described by the ith wave component of the
wave vector.
We create a numerical scheme for solving the equation in several parts in Section
3.3. First, we choose to use the method of Gummel [39] to split the solution of Poisson’s
29
equation and the continuity equations. This has the added advantage that the scheme
works identically for the constant-potential case (without Poisson’s equation). For the
continuity equations, we note that for a given potential, the equations are linear and can
be split into two individually solvable parts (using the method of Strang [85] to obtain
the combined solution). The term proportional to ∓i(1 ± V ) can be solved exactly
as an exponential. For V (x, t), the integral in the exponent is effectively calculated by
Riemann summation on the time steps. Because the exponent is complex, both solutions
correspond to complex rotation (and therefore conserve mass). We solve the remaining
portion using a Crank-Nicolson type implicit scheme [18]. By choosing center-difference
spatial derivatives, we can ensure that the scheme conserves the necessary quantities
explicitly and prevent artificial dissipation. Similar results for the Schro¨dinger equation
are well-known in the literature [43].
For Poisson’s equation, we solve using standard fourth-order compact finite-difference
methods [33, 93]. It is important to note that Poisson’s equation must still be solved
in three dimensions. Even though the charges are constrained to a plane, the electro-
magnetic energy is not confined. In particular, taking a cubic domain, the boundary
conditions on the planes parallel to the graphene sheet can greatly impact the operation
of the device. We demonstrate this effect in Section 3.4.5 and provide a connection to
the role of gates and back-gates in the design of physical devices.
In Section 3.3.1, we prove that our scheme is consistent and of second order. We
first show that the Crank-Nicolson scheme itself (corresponding to the zero mass and
zero potential case) is von Neumann stable and thus convergent [53]. We then extend
the results to the case of a constant potential and demonstrate that the scheme is still
convergent and second order.
In Section 3.4 we show the results of implementing our scheme with a Matlab script.
We show that our scheme accurately approximates simplified tests cases where the ex-
plicit solution is known and give numerical evidence of the second order convergence.
We also demonstrate results in cases where explicit solutions are not known and demon-
strate that our scheme can correctly capture interesting effects in graphene, specifically
beam splitting and Veselago lensing [14].
30
Chapter 2
Photovoltaics
This section is the result of collaborative work with Klemens Fellner, Marie-Therese
Wolfram, and Peter A. Markowich. Some of the material appeared in our accepted pub-
lication [10] and more should appear in a second publication [9] (in prep). Professor
Markowich was responsible for the initial direction of the project, and Professor Fellner
was instrumental in deciding the final form and motivating the second article. Professor
Fellner also provided significant motivation for some of the calculations, although all of
the material below was obtained by myself. Dr. Wolfram helped with the implementa-
tion of the computer code, accounting for approximately 30% of the C++ code. The
simulations were done by myself alone.
2.1 Introduction
The search for cheap, environmentally sustainable energy solutions has led to intense
interest and research in the area of organic photovoltaic (OPV) devices. Currently,
these devices have solar efficiencies which are significantly below those of modern in-
organic semiconductor devices (i.e. at present 8.3% compared to 27.6% for “one-sun”
Gallium Arsenide and 42.3% for concentrator cells [37]), currently limiting commercial
implementation.
A main difference of OPV devices (in contrast to inorganic PV) is the generation
mechanism of free charge carriers, which typically occurs through the generation and
dissociation of so called excitons, excited energy states created by incoming light. (We
shall discuss excitons and the generation of free charge carriers in more detail below.)
31
Many mathematical models have been proposed to study the behavior of bilayer OPV
cells as simple implementable organic devices: Refs. [5, 13, 15, 21, 50]. These models
involve various approximations of the generation of free charges. A mathematical basis
for the these models comes from extensive literature on inorganic semiconductor models.
For these devices, the standard macroscopic models couple two drift-diffusion-reaction
equations for the behavior of the free charge carrier densities (i.e. electrons and holes)
in a system with Poisson’s equation governing the self-consistent electrical potential
[74]. Much is known about such systems, including existence and uniqueness results
for the steady state (see e.g. [74, 55] and the references therein). The steady state
is of particular interest for photovoltaic devices which are expected to generate power
on time scales much longer than the duration of the transient (i.e. the switching-on)
dynamics, observed to occur on the microsecond scale [47]. For a brief overview of
classical semiconductor theory and standard notation, see Section 1.1.
2.1.1 OPV Model
Drift-diffusion-reaction type models for OPV devices must crucially take into account
the specific material properties of inorganic and organic semiconductor materials, which
involves in particular electric field dependent mobilities and specific recombination and
dissociation rates (see Section 2.2 below). A second crucial amendment has to model
the exciton dynamics and couple it appropriately to the original system.
We shall thus consider equations for the four main components, i.e. the concentra-
tions of electrons (n), holes (p), and excitons (X) and the electric potential in the device
(V ), which must be calculated self-consistently.
Charge Carrier Densities
The evolution of the free charge carrier densities n and p shall be described by typ-
ical drift-diffusion-recombination equations and an additional source term due to the
excitons:
q
∂n
∂t
= ∇ · (qDn∇n− qµnn∇(U + V ))− qRnp + qkdX, (2.1)
q
∂p
∂t
= ∇ · (qDp∇p+ qµpp∇(U + V ))− qRnp + qkdX. (2.2)
Here q denotes the positive fundamental charge, Dn, Dp and µn, µp are the diffusion
coefficients and mobilities of n and p, respectively, Rnp is the recombination rate of
32
free electrons and free holes, and kd is the rate of dissociation of excitons into a pair
consisting of a free electron and hole. Functional expressions for diffusion coefficients,
mobilities and recombination/dissociation rates in organic semiconductor materials will
be specified in Section 2.2.
For a bilayer OPV device formed of two different organic semiconductor materials (see
Figure 2.1), the potential U (which is not an electric potential and doesn’t contribute to
the electric field) models an additional convection term that comes from the differences
in the energy levels of the two materials. In an organic device, the energy levels involved
in charge transfer are the Highest Occupied Molecular Orbital (HOMO) and the Lowest
Unoccupied Molecular Orbital (LUMO). The HOMO roughly corresponds to the valence
band for classical semiconductors, and the LUMO resembles the conduction band (as in
Figure 1.1). In general, the HOMO-LUMO gap corresponds to the band-gap.
Although this modeling is based extremely closely on various models in the literature
(especially [5] and [13]), several deep problems exist with regards to the behavior of the
minority carriers. We discuss these limitations in Section 2.8. In particular, minority
carriers in the bulk can carry a nontrivial, nonphysical current - violating the principle
of detailed balance in forward bias. This difficulty extends deep into the formulation
of the model and since no simple fix is apparent, we carefully investigate the model as
given before focusing specifically on its deficiencies.
Excitons
For most organic materials, the HOMO-LUMO gap is too large for a photon to create
free electrons and holes. Instead, it creates a bound electron/hole pair, a so-called
exciton. In the bulk material, these excitons usually recombine without producing free
carriers (with rate kr, see below). However, at an interface between two materials, a free
electron can be generated in the material with a lower energy LUMO while a free hole is
generated in the material with a higher energy HOMO. This reduced energy gap allows
excitons to dissociate into free holes and electrons more readily at an interface between
two materials. This effect is included in the model equations (2.1) and (2.2) by the term
U , which represents the energy shift between the two materials. (Note that using the
same U in both equations implies that each material has the same bulk HOMO-LUMO
gap. This assumption simplifies the asymptotics in Section 2.5, but could probably be
eliminated to account for more general materials.)
33
Excitons are typically called polaron pairs (also referred to as charge-transfer states
or coulomb bound pairs) when aligned across the interface and are much more stable
with respect to recombination. On the other hand, the aligned polaron pairs will either
be pushed together or pulled apart by an electric field approximately parallel to the
charge separation vector. Accordingly the dissociation rate (kd) will be heavily field
dependent. The combination of these two effects (reduced polaron pair recombination
and field-driven dissociation) can lead to very high quantum efficiency (proportion of
light creating free charges) for polaron pairs at an interface under an appropriately
applied external potential [3]. Note that this is an internal quantum efficiency for polaron
pair dissociation, not an external quantum efficiency, which further includes how many
of the incoming photons actually create polaron pairs.
Following the above discussion we shall assume that every exciton becomes imme-
diately a polaron pair at the interface (see further explanation below). We therefore
consider the following diffusion equation for the evolution of a combined density of ex-
citons and polaron pairs (to be identified by their position), which lacks a convection
term since the excitons are electrically neutral:
∂X
∂t
= ∇ · (DX∇X) + cRnp +G− kdX − krX. (2.3)
The coefficient G is the photo-generation rate for excitons, kr is the rate of geminate re-
combination of the excitons, and c represents the proportion of recombining free charges
that form excitonic states (as opposed to recombination leading to emission of light, etc).
Note that although a dipole will not drift, an applied electric field will generally induce
a torque on the dipole. Because the difference in material properties will generally force
the dipoles to align to the interface, we assume that the torque generated by the electric
field is negligible. In particular, this is justified for a device in which the interface is lin-
ear (or planar) and the dominant electric field term is perpendicular to the interface (as
in Figure 2.1). See Section 2.9 for some discussion of the difficulties when the interface
is no longer linear.
This equation, although apparently quite straightforward, can be quite complicated
depending on the expected behavior of the coefficients. Because the device has two
materials, the interface between the two materials is expected to be quite thin. It is
originally tempting to immediately take the width of this interface to zero, but several
issues arise with this approximation. See Section 2.8 for more details. We furthermore
discuss in Section 2.6.1 that our chosen model for the polaron-pair diffusion results in
34
x_0 x_mx_l x_r x_L
d
p n
ΓN
ΓN
ΓD ΓD
0
U
Figure 2.1: Schematic of the 2D device giving labels to the quantities used in the asymp-
totics and a graphical representation of the function U . For the numerics we take U to
be piecewise quadratic connecting the values at either side so that U ∈ C1. Note that
for the asymptotics in Section 2.5 the vertical direction of the first diagram is suppressed
and the problem becomes 1D.
35
non-physical behavior with regards to polaron-pairs diffusing away from the interface (a
process generally not physically observed. We discuss one possible model extension for
dealing with this difficulty in Section 2.9.
Poisson’s Equation
Finally, we have Poisson’s equation for the self-consistent electric potential:
0∇ · (r∇V ) = q(n− p− hχI∇νX), (2.4)
where 0 denotes the vacuum permittivity and r is the relative permittivity of the
material. Furthermore, ∇ν represents the derivative normal to the direction of the
material interface I and χI denotes the indicator function of the interface region while
h is the separation length of the charge pairs of polaron pairs in the interface region. In
fact, the extra term proportional to ∇νX is the field contribution due to the alignment
of the polaron pairs at the material interface. The gradient results from taking the
continuum limit of a sum of Dirac delta primes assumed to be aligned normal to the
interface (corresponding to point dipoles [48]). The exciton distribution X includes both
polaron pairs and excitons, with their identity based on their location in the system.
Polaron pairs are split over the interface such that the electron and hole (although
still bound) each lie in an energetically favorable material. This has the dual effect of
aligning a sheet of dipoles (resulting in the aforementioned term in Equation (2.4)) and
of making polaron pair states extremely energetically favorable to excitons [3]. We thus
assume that any X in an interface region I are polaron pairs and that any X ∈ IC
(the complement of I) are excitons. In actual devices, organic interfaces are not sharp,
and can be blended over a variety of length scales. For our model, we take I to be
the region within d = 1nm of the theoretical sharp interface. Note that if we take I
to be a straight line parallel to the contacts of the device and assume homogeneity in
the parallel direction(s), we exactly recover the model in [5]. No boundary conditions
govern the edges of the interface region, but the sharp change in material parameters
(as given in Table 2.1) do, however, generate fairly major changes in the charge carrier
concentrations.
36
Domain and Boundary Values
For the remainder of this thesis, we will consider the domain Ω ∈ R2. We take Ω to be
rectangular, with Neumann boundary segments ΓN on the top and bottom edges of the
domain and Dirichlet boundary segments on the right and left edges (see Figure 2.1).
Unless otherwise noted (as in Sections 2.9 and 2.10) we take the interface I to be linear
and parallel to the Dirichlet contacts.
We supplement the system (2.1) to (2.4) with the following boundary conditions:
{
V = VD, n = nD, p = pD on ΓD,
∇νV = 0, ∇νn = 0, ∇νp = 0 on ΓN ,
and ∇νX = 0 on Γ (2.5)
where the Dirichlet data (VD, nD, pD) are constant on each Dirichlet boundary segment.
The Dirichlet boundary conditions for ΓD in Equation (2.5) correspond to the conducting
ends of the device. The difference in the values for the potential V at the two Dirichlet
contacts correspond to the potential difference in the device (in Volts) whose offset does
not affect the dynamics of the device because only the electric field E = −∇V enters
the other equations. The usual notation for a photovoltaic device follows the convention
scheme of a diode, but in the working case current flows in the direction opposite to
that of a usual diode. For this reason, a consistent applied potential is given at the left
boundary of the device. We thus take the potential at the right side of the device to
be zero, and take the potential at the left side to be the potential difference. Note that
this notation has the convenient consequence that the average field in the device has the
same sign as the potential difference.
For the boundary conditions of electrons and holes one has to take into account that
the HOMO/LUMO levels of the device determine different energies for the electrons and
holes implying that the concentrations of n and p at the metallic contacts can be very
different. Thus, for majority carriers in their energetically favorable material (p to the
left of the interface and n to the right as indicated in Figure 2.1), we take the boundary
conditions given by Scott and Malliaras in [80]. For the minority carriers (n to the left
and p to the right), homogeneous Dirichlet boundary conditions constitute a very good
approximation as the Scott-Malliaras boundary conditions give values approximately
seven orders of magnitude smaller than typical concentrations (see Table 2.1 in Section
2.2.2). Note that although the minority carrier boundary conditions are taken to be
homogeneous Dirichlet, this does not imply that the minority concentrations are zero
(the case considered, for instance by [20, 73]). However, they are usually several orders
37
of magnitude smaller than the majority carrier – see the discussion at the beginning of
Section 2.5.2. Further complicating the situation is that even though they are several
orders of magnitude smaller, the minority carriers can still play a significant role in the
device – see Section 2.8.
On the insulated boundary ΓN we take homogeneous Neumann conditions for all of
the variables because no particles should be able to pass into the insulator and it should
be electrically neutral. For the excitons, we take homogeneous Neumann conditions
on the whole boundary Γ because the excitons are not be able to transmit from the
organic material into the inorganic contacts. Because the exciton equation contains no
drift term, the exciton flux is exactly DX∇X and homogeneous Neumann conditions
correspond to no-flux boundary conditions.
The first half of this chapter presents the results previously published in [10]. In
Section 2.2 we will scale the physical parameters of the equations and detail the specific
expressions for the mobilities, dissociation rates and recombination rates in an organic
semiconductor material. Next, in Section 2.3 we present the numerical system used
for the 2D simulations. In Section 2.5 we analyze the 1D equations in two physically
relevant asymptotic regimes – large potential difference and small minority carriers.
Moreover, in 2.6 we plot the calculated and asymptotic concentrations in the device
at three important points in the device operation – short circuit, optimal power point,
and open circuit. We interpret these results in terms of their relation to the asymptotic
results and furthermore show the current-voltage characteristic curves generated by both
the numerical and asymptotic methods.
We proceed to consider an alternative asymptotic scheme which focuses on implicitly
defining the carriers in the device via integral equations. In Section 2.7 we demonstrate
that this allows us to generate asymptotic curves which closely match the numerical
results presented earlier. We furthermore use this system to demonstrate the physical
ramifications of the minority carriers in Section 2.8. We intend to extend these results
for a future publication [9]. Finally, we discuss the possibility of extending our numerical
simulations to more complicated geometries (Section 2.9) and give several perspectives
on proving the existence and uniqueness of the system (Section 2.10).
38
2.2 Scaling and Coefficient Modeling
In this section, we scale the system presented in Section 2.1.1 and discuss appropriate
forms of the various coefficients defined therein. We begin by non-dimensionalizing the
system, then proceed by defining the functional forms of the mobility and the dissociation
terms, before finally giving physical values for all of our parameters.
We will scale (2.1) - (2.3) and (2.4). The basic scaling introduces the following
dimensionless quantities:
V =
kbT
q
V˜ = UTV˜ , x = Lx˜, (n, p,X) = Nr(n˜, p˜, X˜), µ = µ0µ˜, (2.6)
where UT is the thermal voltage, L is a characteristic length (usually on the order of
the device length), Nr is a reference concentration, and µ0 is a reference mobility. We
assume the Einstein’s relation D = UTµ. Although this might not be justified for some
organic materials, it greatly simplifies the equations and as of yet a generally accepted
alternative model has not been found. We further introduce the dimensionless quantities:
λ2 =
0UT
qL2Nr
, T =
L2
µ0UT
, (2.7)
where T denotes a reference time scale corresponding to the diffusion time of a free
charge carrier in the bulk of the device. Note that the Debye length λ is typically not a
small parameter – see Table 2.1.
For an OPV device under light we follow [5] and choose the Langevin recombination
rate Rnp =
q(µn+µp)np
0r
, which rescales with the reference time T as follows:
L2
Nrµ0UT
Rnp =
1
λ2
(µ˜n + µ˜p)n˜p˜
r
= crn˜p˜.
Setting c′r = ccr this gives the following system:
λ2∇ · (r∇V˜ ) = n˜− p˜−
hχI
L
∇νX˜ (2.8a)
∂n˜
∂t˜
= ∇ · (µn∇n˜− µnn˜∇(U + V˜ ))− crn˜p˜+ kdTX˜ (2.8b)
∂p˜
∂t˜
= ∇ · (µp∇p˜+ µpp˜∇(U + V˜ ))− crn˜p˜+ kdTX˜ (2.8c)
∂X˜
∂t˜
= ∇ · (µX∇X˜) + c
′
rn˜p˜+GT − kdTX˜ − krTX˜. (2.8d)
39
For the charge carrier mobilities µn and µp we shall apply the Poole-Frenkel model
[31] postulating an exponential dependence on the square-root of the electric field E˜ =
−∇V˜ :
µn = µn(0)e
γn
√
|E˜|, µp = µp(0)e
γp
√
|E˜|. (2.9)
2.2.1 Dissociation Term
It is relatively common in the literature [13, 50] to derive exciton and polaron pair dis-
sociation terms based upon the theory of electrolytic fluids outlined in [7, 36]. Several
inconsistencies arise in the treatment which become especially important when we at-
tempt to consider the zero-width interface limit. In this case, the polaron pairs will
exist only on the interface, and thus they only enter the continuity equations as a dis-
tribution. However, since they will change the field locally around the interface, most
forms of kd(E) will be ill-posed functions of distributions (often two delta distributions).
Special care needs to be taken for the dissociation rate if an eventual zero-width model
is to be successful.
In [13], we have the following integral for the average polaron dissociation rate:
kd(E) =
∫ ∞
0
kˆd(E, a)f(a)da = C
∫ ∞
0
1
a
e−a
2/a20da
where kˆd(E, a) a specific dissociation rate depending on the electric field and the charge
separation distance a. Here C depends only on E and temperature (and other physical
constants, not a). The immediate problem is that this integral diverges for the a → 0
limit. The proposed solution is to define a “physical lower bound” (usually a0/2 = .5
Angstroms) corresponding to the minimal possible exciton separation width. Although
this solves the divergence issue, the result is highly dependent on the value of this lower
bound (which is not physically well understood).
This integral is cited from [36] for the distribution of what they call the charge-
transfer radius, a. This represents the distance between the electron and the hole in
the exciton state which lies on an interface between two different polymers. Aside from
notational confusion between a, a0 and r, the more interesting issue is that the integral
in this paper is not calculated directly from kd, but rather from a derived quantity
Pˆ (E, a) =
kd(E)
kd(E) + kr
(2.10)
40
for the “the probability for a hole escape”. This is the probability that the decaying par-
ticles will dissociate productively (with rate kd) into a free electron and hole rather than
recombining (with rate kr). Using this notation, the integral over separation distances
is given by:
P (E) =
∫ ∞
0
Pˆ (E, a)f(a)da = C1
∫ ∞
0
a2
C2a3 + 1
e−a
2/a20da
for some constants C1 and C2. This integral converges.
In [50] they rearrange equation (2.10) and then insert this relationship into the steady
state equation for X˜:
0 = ∇ · (µX∇X˜) + c
′
rn˜p˜+GT − (kd + kr)TX˜
= ∇ · (µX∇X˜) + c
′
rn˜p˜+GT − (1 +
1− P
P
)kdTX˜
Since they only treat polaron pairs stuck to the interface (and not bulk excitons), the
diffusion term also disappears (µX = 0) and they can solve the equation algebraically
for kdTX˜ to obtain
kdTX˜ = P (GT + c
′
rn˜p˜).
One can then insert this relationship into the continuity equations for n and p. This
decouples X from the equations for n, and p. Although this simplifies the problem, it
is not a feasible direction for a model in higher dimensions in which the polaron pairs
themselves are free to move along the interface.
Given the difficulties with this type of integral kd kernel, we take the polaron pair
dissociation rate kd given by [5] as a function of M , which is a scaled square-root of
electric field E:



k−d (M) =
2kd(0)
M
(
eM
(
1− 1M
)
+ 1M
)
,
k+d (M) =
4kd(0)
M2
(
1− e−M
2/4
)
,
M =
1
λ
√
|E|
L2UTNrpir
,
where the +,− superscripts represent positive and negative fields (with respect to the
interface normal). Note that this formula is continuous with respect to E, but is not
continuously differentiable at |E| = 0. As discussed before, we consider all bound pairs
within 1nm of the interface to be polaron pairs, and thus this formula only applies in
the interface region. For the excitons in the bulk we take a constant dissociation rate
since excitons are not generally aligned to the electric field. Note that this means that
41
although kd is continuous with respect to E, it has spatial discontinuities at the interface
edges corresponding to the increased stability of polaron pairs compared to excitons -
see Table 2.1. In Section 2.6 we also consider a field-independent dissociation rate and
discuss the impact of such a change on the device characteristics.
2.2.2 Device Parameter Values
The physical scaling parameters we take are:
V = UTV˜ = .0258V V˜
E = UTE˜/L = 2.58× 10
5V/m E˜
x = Lx˜ = 100nm x˜
n = Nrn˜ = 10
20m−3 n˜.
The values for the various constants are taken from [5] (exceptG which is converted to
a spatial density instead of an area density). Table 2.1 collects all the used dimensionless
parameters. For the field dependent parameters we also give the values at the short
circuit electric-field ESC = 3.5 × 106 (with both signs where applicable), corresponding
to the short circuit potential difference of ∓.5V . This gives a qualitative understanding
of the range of the The expressions kout and kin refer to the rate-values in- and outside
of the interface region.
Table 2.1: Values for physical parameters.
(h/L) = .01 (d/L) = .01 r = 4
λ2 ≈ 1.43 T = .00386 E˜SC = 13
GT ≈ 16990 cr = c′r ≈ .6987 U(xr)− U(xl) ≈ 6
kd,outT ≈ 1 kr,inT ≈ 3.86 kr,outT ≈ 3864
kd,in(0)T ≈ 386 kd,in(+ESC)T ≈ 178 kd,in(−ESC)T ≈ 2763
µ0 = 10−10 (µ1/µ0) ≈ .01
µn(0) = 3 γn = .788 µn(ESC) ≈ 53.3
µp(0) = 1 γp = .153 µp(ESC) ≈ 1.75
n(xL), p(x0) ≈ .04 n(x0), p(xL) ≈ 4× 10−7
The values of µX and kd,out are not easy to calculate physically and no consensus
seems to exist in the literature for their values, but we take the given values based on
reasonable estimates given in the sources listed above.
42
2.2.3 Steady State Equations
For usual device operation, a solar cell will be producing current steadily for time-
scales on the order of hours. Any transient behavior occurs over such short time-scales
(i.e. microseconds [47]) that we neglect them for modeling the long-term behavior and
efficiency of the device.
Therefore, for the remainder of the chapter (except Section 2.10 which contains some
parabolic regularity results), we will consider only the steady state equations where we
drop all the tildes and absorb the time-scaling T into the rates:
λ2∇ · (r∇V ) = n− p−
hχI
L
∇νX (2.11a)
−∇ · (µn∇n− µnn∇(U + V )) = −crnp+ kdX (2.11b)
−∇ · (µp∇p+ µpp∇(U + V )) = −crnp+ kdX (2.11c)
−∇ · (µX∇X) = c
′
rnp+G− kdX − krX. (2.11d)
We expect that this system of equations will have a unique steady state solution.
If we insist on the physically reasonable requirement that kd and µ are smooth and
bounded from above (corresponding to physical device saturation), then the first three
equations fit very nearly into the standard semiconductor framework. See Section 2.10.1
for details.
Concerning the steady state solutions, we emphasize that the light generation term
G > 0 generally implies a non-zero right-hand-side in Equations (2.11b) and (2.11c)
and thus non-constant drift-diffusion fluxes of n and p on the left-hand-side. Thus, we
expect a steady flux of electrons and holes away from the interface with nearly negligible
bimolecular recombination (due to the work function considerations discussed in the
introduction). Even for G = 0, we will only recover constant stationary fluxes when the
equation
c′r
cr
kdX − kdX − krX = 0
is satisfied. This can hold in particular cases such as X = 0 and cr =
c′rkd
kd+kr
, but these
are not physically realistic given our device parameters. As a consequence, the model
system (2.11) is not designed to accurately predict the behavior of the device in the dark
G = 0.
43
2.3 Numerical Scheme
We present a numerical scheme based on a Gummel-type iteration [39], which is well-
suited to deal with the nonlinear dependence of the equations for n, p, and X on the
electric field (E = −∇V ). At each Gummel-iteration step we calculate first the electric
potential V from the current state of n, p,X and then use this new electric potential to
update n, p,X. We can write this as follows after the kth step:
− λ2∇ · (r∇Vk+1) = pk − nk +
h
L
∇Xk
−∇ · (µn(Ek+1)∇nk+1 − µn(Ek+1)nk+1∇(Vk+1 + U)) + crnk+1 pk = kd(Ek+1)Xk
−∇ · (µp(Ek+1)∇pk+1 + µp(Ek+1)pk+1∇(Vk+1 + U)) + crnk pk+1 = kd(Ek+1)Xk
−
µ1
µ0
∇ · (µX∇Xk+1) + (kd(Ek+1) + kr)Xk+1 = c
′
rnkpk +G.
(2.12)
Note that we use a semi-implicit discretization for each equation. Specifically, we
implicitly discretize the flux and mass terms in (2.12), but use semi-implicit discretization
for the recombination term crnp and an explicit form for the dissociation term kdX. This
has the benefit of decoupling the system so that n, p, and X can be effectively updated
in parallel since they only require the previously updated values for V . Furthermore,
each variable n, p, X is treated completely implicitly within its equation, which greatly
reduces numerical instabilities arising from the errors in the previous step.
The iteration continues until a preset L2 error between the two latest solutions is
achieved. In practical numerical simulation, we have found this Gummel iteration to
converge well except for high positive values for the applied potential difference (which
constitutes the non-working regime for a photovoltaic device). This case, which is char-
acterized by opposite convection terms coming from the work function and the electric
potential, leads to large values for n and p near the interface which prevent the Gum-
mel iteration from converging – see Section 2.6.4. To alleviate this difficulty, we add a
damping parameter α which interpolates between old and new solutions. More precisely,
denoting with V ′k, n
′
k, p
′
k, X
′
k the solutions of the above system (2.12), the damped scheme
is:
Vk+1 = αV
′
k+1 + (1− α)Vk, nk+1 = αn
′
k+1 + (1− α)nk,
pk+1 = αp
′
k+1 + (1− α)pk, Xk+1 = αX
′
k+1 + (1− α)Xk.
(2.13)
44
In particular, note that since V ′k+1 and Vk both satisfy linear boundary conditions, so
does any convex combination of V ′k+1 and Vk. This is equally true for the boundary
conditions for n,p, and X, which are fixed for a given potential Vk+1.
Our numerical experiments in the case of positive applied potentials show that it is
possible to optimize the convergence by tuning the damping parameter. In fact it turns
out that using a small damping α ≈ .01 for the first few (e.g. three) iteration steps serves
to correct the initial guess and prevents the concentrations from going negative due to
badly chosen initial data. Further optimization on the damping, such as a potentially
dynamic choice of α, is considered in Section 2.6.4.
Note also that the damped scheme could be modified to represent the parabolic
system with a modified Euler method. If the current values (Vk, nk, pk, Xk are taken
with coefficient 1, then α corresponds to the time step. As discussed above, however, the
right-hand side is evaluated semi-implicitly and the equivalence of the iteration scheme
to a specific time-dependent model is not immediate.
Space-Discretization
Our choice of a space-discretization has to take into account the two major difficulties
inherent in our system. First of all, the material interfaces will cause very strong changes
in the densities n, p, and X in the interface region. Secondly, depending on the electric
field, the equations can be either convection or diffusion dominated. Furthermore, we
want our scheme to easily generalize to complicated 2D interfaces.
The hybrid discontinuous Galerkin (HDG) finite element method is well-suited to
handle these challenges. The novel idea for the HDG method is to have separate degrees
of freedom on the elements u (an element in the usual sense) and on the facets (bound-
aries) of the elements uF . We assume that our domain is decomposed into a mesh T con-
sisting of triangles with a set of facets F consisting of the facets (edges) of these triangles.
The space we use for each of the equations is V ′ = {(u, uF ) : u ∈ H1(Ω) ∩H2(T ), uF ∈
L2(F)}. Our solution space is (V, Vf , n, nf , p, pf , X,Xf ) ∈ V = V ′ × V ′ × V ′ × V ′ and
we denote the approximate solutions to the four equations as uV , un, up, uX with corre-
sponding facet functions uV f , unf , upf , uXf . The facet elements allow for upwind-type
calculations for convection-dominated behavior, as well as allowing for large changes of
behavior at material boundaries. The functions within the elements are approximated
by polynomials of a given order (seven for the results given in Section 2.6). Both in-
45
creasing the polynomial order and refining the mesh improve the properties of a solution
after a given number of iterations at the price of larger computational costs.
A solid introduction to HDG methods is [52], specifically the first section on the
general convection-diffusion case. The HDG method can more precisely be called a
hybridized symmetric interior penalty discontinuous Galerkin method. The relationship
to standard discontinuous Galerkin methods runs quite deeply. The stabilization and
symmetrization terms have close analogs in traditional DG methods, see for instance
[4, 16, 17].
We rewrite the system (2.12) in the suitable HDG weak form below. For notational
convenience, we write the variable without the subscript indicating the equation step,
and instead put a bar above the old value of the variable, ie Vk+1 → V and Vk → V̂ ).
Furthermore, we define the variable
(n/p)up =



u ∇νV > 0
uF ∇νV < 0
so that for an outflow boundary we choose the value given by the bulk variable, but
for an inflow boundary we choose the facet value. The additional term of µ(E) in the
convection term has been removed without changing (n/p)up since it is a strictly positive
scalar. Note that as before a subscript of ν represents a normal derivative.
We seek (V, Vf , n, nf , p, pf , X,Xf ) ∈ V such that:
∑
T∈T
(∫
T
λ2r∇V · ∇uV dx−
∫
∂T
λ2r
∂uV
∂n
(V − Vf )ds−
∫
∂T
λ2r
∂V
∂n
(uV − uV f )ds
+
∫
∂T
A
h
λ2r(V − Vf )(uV − uV f )ds
)
=
∑
T∈T
∫
T
(p̂− n̂+ h/L∇xX̂)uV dx+
∫
ΓD
τ(VD − V )ds
∑
T∈T
(∫
T
µn(E)∇n · ∇undx−
∫
∂T
µn(E)
∂un
∂n
(n− nf )ds−
∫
∂T
µn(E)
∂n
∂n
(un − unf )ds
+
∫
∂T
A
h
µn(E)(n− nf )(un − unf )ds−
∫
T
µn(E)∇V n∇undx+
∫
∂T
(µn(E)∇V )nn
upunds
+
∫
∂Tout
(µn∇V )n(nf − n)unfds+
∫
T
crnp̂undx
)
=
∑
T∈T
∫
T
kd(E)X̂undx+
∫
ΓD
τ(nD − n)ds
46
∑T∈T
(∫
T
µp(E)∇p · ∇updx−
∫
∂T
µp(E)
∂up
∂n
(p− pf )ds−
∫
∂T
µn(E)
∂p
∂n
(up − upf )ds
+
∫
∂T
A
h
µp(E)(p− pf )(up − upf )ds+
∫
T
µp(E)∇V p∇updx−
∫
∂T
(µp(E)∇V )νn
upupds
−
∫
∂Tout
(µp∇V )ν(pf − p)upfds+
∫
T
crn̂pupdx
)
=
∑
T∈T
∫
T
kd(E)X̂updx+
∫
ΓD
τ(pD − p)ds
∑
T∈T
µ1
µ0
(∫
T
µX∇V · ∇uXdx−
∫
∂T
µX
∂uX
∂n
(X −Xf )ds−
∫
∂T
µX
∂X
∂n
(uX − uXf )ds
+
∫
∂T
A
h
µX(X −Xf )(uX − uXf )ds+
∫
T
(kd(E) + kr)XuXdx
)
=
∑
T∈T
∫
T
(c′rn̂p̂+Gx)uXdx
for all test functions (uV , uV f , un, unf , up, upf , uX , uXf ) ∈ V .
It is important to note that quantities of the form (Vf − V ) are equal to zero for
the true continuous solution, so they all drop out in the analysis (this is the “interior
penalty” discussed earlier). These terms are added to the numerics to symmetrize the
Laplacian. The integrals multiplied by terms of the form (Vf−V )(uV −uV f ) are also zero
and are included with a factor of A = 10 to stabilize the system. This term is usually
called α in the literature, but we use A here since α refers to our damping parameter.
We take the boundary conditions given in equation (2.5). However, implementing
Dirichlet conditions directly will cause issues for the stability of the system. Instead, we
define a parameter τ = 1010 >> 1. and regularize the boundary condition u = ud giving
(u + 1/τ∇νu = ud) (on ΓD). In the weak formulation this gives a penalty term of the
form
∫
ΓD
τ(u− ud)ds. The Neumann conditions are all homogeneous, and thus add no
terms to the weak formulation.
All of the numerical examples in Section 2.6 consider the two-dimensional computa-
tional domain Ω = [0, 1.5] × [0, 0.2] (in the scaled variables from Section 2.2). In order
to compare with the one-dimensional asymptotic analysis given in Section 2.5, we take
the domain interfaces (see Figure 2.1) to be straight vertical lines located at x0 = 0,
xl = 0.49, xm = 0.5, xr = 0.51, and xL = 1.5 respectively. In Sections 2.4 and 2.9, we
show example plots for other interfaces which oscillate around these linear boundaries.
The mesh generation was accomplished using the Netgen program created by Scho¨berl
[78]. In particular, the mesh generation automatically preserves the internal interfaces
allowing all coefficients to be defined piecewise depending on the properties of the local
material. The numerics were done using NGSolve, a solver wrapper for the Netgen pro-
47
gram. We also included the inverse solver PARDISO to assist with the non-symmetric
matrix calculations [76, 77]. See [96] for more details about using the NGSolve program
on convection-diffusion equations and Appendix 2.B for details on our implementation.
2.4 Sample Results
Here we present sample results from our numerical scheme. Of primary interest is
understanding the qualitative behavior of the device. In the next section we will identify
asymptotic regimes with which to understand the device analytically. Because we seek
to model a physical device, we seek asymptotics which approximate the results in both
a mathematical sense and a physical sense. From a mathematical perspective, the goal
is to completely understand the behavior of the model outlined in Section 2.1.1. From
an engineering perspective, we would like to understand the operational characteristics
of the device. The most important of these factors can be understood by plotting the
current though the device versus the potential difference – an IV Curve. (Note that the
term “IV curves” uses I instead of J for the current. In general, J is used to prevent
confusion with i =
√
−1. For our purposes it also frees I to be used for the interface
region.)
2.4.1 Internal Variables
First we consider the results for the device at short circuit. As we will discuss in the
following section, this operating voltage is specifically important because it indicates the
maximal amount of current that a device can produce. Plots of the concentrations n and
p and the two components of the electric field Ex and Ey are given in Figure 2.2. Here
the interface is sinusoidal, with the unit normal defined analytically. Details on creating
a nonlinear interface with constant width are given in Section 2.9 and Appendix 2.B.
Several important aspects are immediately obvious from the plots. First, all of
the significant effects seem to happen orthogonal to the interfaces. Near the metallic
contacts we see boundary layers forming, but no effects in the direction parallel to
the boundary. Near the organic-organic interface, I, we again see that the primary
effects occur perpendicular to the interface. This suggests that it might be possible to
understand the dominant behavior of a 2D device with 1D asymptotics. In Section 2.5
we proceed to analyze the 1D system under this assumption.
48
Figure 2.2: Sample simulation results for a 2D device at short circuit with a sinusoidal
interface. The first row shows the electron and hole concentrations and the second row
shows the electric field in the x and y directions (to the left and right respectively for
both rows). Note that the electric field in the x direction dominates (the field in the y
direction is two orders of magnitude smaller). In addition, the field stays within 10% of
the estimated value ESC = 13 given in Table 2.1.
49
Further evidence for the 1D model is suggested by the electric field plots in the second
row of Figure 2.2. The electric field in the y direction, Ey is approximately two orders of
magnitude smaller than Ex. In addition, it is nearly zero away from the interface. Since
the convection effects near the interface will be dominated by U , Ey will not be critical
to the device function. In addition, we see that Ex ≈ ESC (as defined in Table 2.1). The
field varies in the p-region of the device, but remains within 10% of the average field
value ESC. In Section 2.5.1 we use this observation to expand the results in terms of
deviations from the average field.
Additionally, as expected based on the physics of the device, n is very small in the
p-type region to the left of the device and p is very small in the n-type region to the right
of the device (refer back to Figure 1.2). This suggests taking the values of the minority
carriers to be zero (as considered in Section 2.5.2). We consider this assumption (which
is very common in the literature, especially for asymptotic results [5, 15, 73]) more
carefully in Section 2.8, showing that it cannot be derived from our chosen model in a
physically consistent way.
2.4.2 IV Curves
One of the most important characteristics of an OPV device is the effect of changing
the applied voltage on the current in the device. We show this relationship by plotting
the current as a function of potential difference using multiple simulations with different
boundary values. We pick values for Vdiff and then use these values to calculate the
boundary values for n and p according to [80]. Due to the built-in potential of the
metallic contacts at the OPV device, we have that zero applied voltage leads to an
approximate potential difference over the device of Vdiff = −19.3 (which we take to be
the left edge of the IV curves here and below).
Figure 2.3 plots the current for our typical OPV device green) with respect to the
potential difference (as well as the power (blue), which is discussed below). This ref-
erential IV curve compares very closely with a second IV curve (red) computed after
replacing the stated boundary conditions (from [80]) by homogeneous Dirichlet boundary
conditions for each carrier at both Dirichlet boundaries. These conditions (taken for sim-
plicity only) demonstrate that the observable device behavior is relatively independent
of boundary conditions.
Both IV curves show how the produced current decreases (in magnitude) as the
50
-350
-300
-250
-200
-150
-100
-50
 0
-20 -15 -10 -5  0  5  10  15
 0
 500
 1000
 1500
 2000
 2500
 3000
 3500
 4000
 4500
C
ur
re
nt
Po
w
er
 G
en
er
at
ed
Potential Difference
Normal Boundary Conditions
Power Generated
Zero Boundary Conditions
Figure 2.3: A comparison of the IV curves obtained using the boundary conditions
outlined above in Table 2.1 above and homogeneous Dirichlet conditions for the charge
carriers and a plot of the generated power density for the usual boundary conditions.
Note that the power corresponds to the areas of the rectangles with opposite corners at
the lower-left and the plotted current at each potential difference. We recall from Table
2.1 that the the scaled internal voltage is taken to be Vint = 19.3. A different internal
voltage would significantly alter the generated power curve.
51
potential difference increases. In particular we observe a distinctive kink. This is an effect
of the dependence of kd,in(E) on the self-consistent electric field E. The kink arises at the
potential V ≈ −3 for which the E-field touches zero from below at the interface. Note
that the E-field touching zero happens for a potential V smaller than zero because the
holes on the left side of the device (i.e. the majority charges) increase self-consistently
the negative applied E-field towards the interface. This argument can be verified by
plotting the IV curve obtained from a constant dissociation rate kd,in = kd,in(ESC). A
plot comparing this result to the asymptotic results given in Section 2.5.1 is shown in
Figure 2.4. Additionally, the effect is significantly reduced by decreasing the exciton
mobility µX (see Section 2.6.1).
-350
-300
-250
-200
-150
-100
-50
 0
-20 -15 -10 -5  0  5  10
Cu
rre
nt
Potential Difference
Constant k_d Numerics
Constant k_d Asymptotics
Figure 2.4: A comparison of the IV curves obtained using the numerical and asymptotic
schemes with kd,in = kd,in(ESC) = 2763 to be constant. Reprinted below as Figure 2.13.
The power density generated by a device is given by P = J(Vint +Vdiff). As shown in
Figure 2.3, the power (in blue) is zero for two specific characteristic points in the device:
Vdiff = −Vint (short circuit) and J = 0 (open circuit). A third characteristic point, the
optimal power point, is defined by the voltage point with the maximal value of P . For
potential differences outside of the regime shown in Figure 2.3 the power generated is
negative, indicating that the device requires power input to operate and is no longer in
the working regime for a photovoltaic device. An upper bound for the maximum power
is given by the product of the short circuit current and the open circuit voltage (these
52
being the maximum current and maximum voltage for the device in the working regime).
The fill factor denotes the ratio of the maximum obtainable power and this upper bound
[22]. Geometrically this is represented by the largest possible rectangle contained under
the IV curve and the smallest rectangle containing the IV curve. For the plots given
in Figure 2.3 the maximal power point occurs at a potential difference of V ≈ −3 with
a fill factor of 39.5%. This fill factor is in good agreement with other simplified organic
devices [37]. In general, fill factors vary heavily depending on the device, and are notably
improved for multijunction devices.
2.5 1D Stationary State Asymptotics
In this section we discuss two asymptotic regimes for analyzing System (2.11) in 1D.
First, we discuss the large-field approximation. Unlike previous results [60], we do not
eliminate the diffusion fluxes, but rather switch to the Slotboom variables (quasi-Fermi-
levels) and solve the differential equation system asymptotically in the high-field limit.
This gives reasonable results for large potential differences, but begins to fail as the
potential difference approaches zero. A more intricate approximation scheme with the
same general structure is presented in Section 2.7. Although the latter scheme gives
results which more accurately represent the numerical simulations, the large-potential
asymptotics still give interesting results (especially for the short-circuit case) and warrant
individual consideration.
In Section 2.5.2 we discuss the zero-minority carrier limit which is partially justified
in the large-field limit discussed in Section 2.5.1. The zero-current (open circuit) case
was given in the literature by [15], while the explicit results here were developed inde-
pendently in [73]. See Section 2.8 for a discussion of the limitations of the zero-minority
carrier limit.
We plot these results along with numerical results in Section 2.6. Several issues
become apparent as we carefully analyze the numerical and asymptotic results side-
by-side. We discuss possible issues with exciton and polaron pair diffusion in Section
2.6.1. Possible resolution of this issue is further discussed in Section 2.9 with regards to
extending the numerical simulations to full 2D problems.
53
2.5.1 Large Applied Field (Constant µ) Approximation
We begin by considering the steady state equations in 1D:
λ2rVxx = n− p−
hχI
L
Xx (2.14a)
−(µnnx − µnn(U + V )x)x = −crnp+ kdX (2.14b)
−(µppx + µpp(U + V )x)x = −crnp+ kdX (2.14c)
−µXXxx = c
′
rnp+G− kdX − krX, (2.14d)
together with the boundary conditions
n(x0) = n0 p(x0) = p0 V (x0) = Vdiff Xx(x0) = 0
n(xL) = nL p(xL) = pL V (xL) = 0 Xx(xL) = 0
where we have assumed that the r and µX are spatially homogeneous.
In reverse bias, which constitutes most of the operating state of an OPV bilayer
device, a negative voltage Vdiff < 0 is given at the left boundary of the device. Note that
with this notation, Vdiff is the potential difference in the device. For working photovoltaic
cells, this potential difference is the combination of the potential applied to the device
and a built-in potential from the metallic contacts.
With a constant relative permittivity r we introduce the Debye length λ2D := λ
2εr ≈
5, and rescale the potential according to this potential difference, i.e. V → |Vdiff |V which
rewrites Equation (2.14a) as
Vxx = ε
(
n− p−
hχI
L
Xx
)
, ε :=
xL − x0
λ2D|Vdiff |
, V (x0) = −1, V (xL) = 0. (2.15)
Since in a working device a typical short circuit voltage is about −0.5 Volts [5], which
is |Vdiff | ≈ 19.3 in our units, we find that physically ε ≈ 0.01.
Hence (assuming for the moment that ε hLXx  1, where
h
L is the fraction of interface
width to device width) we expect that the electric potential V is in zeroth ε-order
dominated by the potential difference and thus that the electric field is constant in the
zeroth order of ε, i.e.
V 0(x) = Vdiff − E
0(x− x0), E
0 :=
Vdiff
xL − x0
= ±
1
λ2D
. (2.16)
We shall see in the following that the approximation (2.16) remains consistent with
the assumption ε hLXx  1 after being carried over to the equations for the charge
carriers and the excitons .
54
Next, we remark that with the zeroth order approximation E0, the mobilities µn and
µp are also constant in the zeroth order. For  < .01 (the physical short-circuit value),
we calculate:
µn = µn(0)e
γn
√
|E0| = µn(0)e
γn
√∣
∣
∣
∣
1
λ2Dε
∣
∣
∣
∣

1
100ε
µp = µn(0)e
γp
√
|E0| = µp(0)e
γp
√∣
∣
∣
∣
1
λ2Dε
∣
∣
∣
∣

1
ε
and thus 1/(µnµp) O(ε2).
Thus, by rescaling
µnn→ n and µpp→ p, (2.17)
we obtain with cr = O(1) the following rescaled charge carrier equations
−(nx − n(V + U)x)x = −O(ε
2)np+ kdX, n0 := n(x0) 1, nL := n(xL) = O(1),
(2.18)
−(px + p(V + U)x)x = −O(ε
2)np+ kdX, p0 =: p(x0) = O(1), pL := p(xL) 1.
(2.19)
(Note that with this scaling, the scales for n and p in Figure 2.2 become identical, with
shared maximum values µnn ≈ µpp ≈ 60.)
Moreover, the rescaled equation for the excitons is
−µXXxx = O(ε
2)np+G− (kd + kr)X, Xx(0) = Xx(xL) = 0, (2.20)
where µX  1 is small.
Letting ε go to zero, we can solve this equation explicitly in terms of hyperbolic sines
and cosines. This allows us to check the consistency the approximation (2.16) and the
rescaling (2.17) with the underlying assumption ε hLXx  1. (See Appendix 2.A.1 for
detailed calculations.) With kr,out ≈ 4 · 103 and kd,out ≈ 1 outside the interface and with
kr,in ≈ 1 and kd,in ≈ 3 · 103 (for typical E0 < 0) inside the interface we have:
‖X0x‖L∞([x0,xL]) ≈
G
√
µX
(
1
√
kd,in + kr,in
−
1
√
kd,out + kr,out
)
(2.21)
and thus ε hL‖X
0
x‖L∞([x0,xL]) ≈ O(ε).
55
Zeroth Order Charge Carriers
Now, we proceed to study the zeroth order solutions n0 and p0 of (2.18) and (2.19).
Introducing the zeroth order fluxes
J0n := n
0
x − n
0ϕx, J
0
p := −(p
0
x + p
0ϕx), with ϕ = V
0 + U,
and neglecting the O(ε2) recombination term, the equations (2.18) and (2.19) integrate
immediately as
J0n(x) = J
0
n0−F (x), J
0
p (x) = J
0
p0 + F (x), (2.22)
F (x) :=
∫ x
x0
kd(y)X
0(y)dy.
Utilizing the Slotboom variables J0n = e
ϕ(n0e−ϕ)x and J0p = −e
−ϕ(p0eϕ)x, also called
quasi-Fermi levels, we can explicitly solve for n0 and p0:
n0(x) = n0e
ϕ(x)−ϕ0 + J0n0
∫ x
x0
eϕ(x)−ϕ(y)dy −
∫ x
x0
F (y)eϕ(x)−ϕ(y)dy (2.23)
= n0e
ϕ(x)−ϕ0 + J0n0Φn(x)−Fn(x),
p0(x) = p0e
ϕ0−ϕ(x) − J0p0
∫ x
x0
eϕ(y)−ϕ(x)dy −
∫ x
x0
F (y)eϕ(y)−ϕ(x)dy, (2.24)
= p0e
ϕ0−ϕ(x) − J0p0Φp(x)−Fp(x),
where ϕ0 := ϕ(x0) and we define
Φn(x) :=
∫ x
x0
eϕ(x)−ϕ(y)dy, Φp(x) :=
∫ x
x0
eϕ(y)−ϕ(x)dy,
Fn(x) :=
∫ x
x0
F (y)eϕ(x)−ϕ(y)dy, Fp(x) :=
∫ x
x0
F (y)eϕ(y)−ϕ(x)dy.
Next, we can determine the parameters J0n0, J
0
p0 by evaluating the boundary condi-
tions n(xL) = nL and p(xL) = pL. Upon rearrangement, we have with ϕL := ϕ(xL):
J0n0 =
nL − n0eϕL−ϕ0 + Fn(xL)
Φn(xL)
, J0p0 =
−pL + p0eϕ0−ϕL −Fp(xL)
Φp(xL)
, (2.25)
and obtain explicit formulas for n0 and p0:
n0(x) = n0
(
eϕ(x)−ϕ0 − eϕL−ϕ0
Φn(x)
Φn(xL)
)
+ nL
Φn(x)
Φn(xL)
+ Fn(xL)
Φn(x)
Φn(xL)
−Fn(x)
(2.26)
p0(x) = p0
(
eϕ0−ϕ(x) − eϕ0−ϕL
Φp(x)
Φp(xL)
)
+ pL
Φp(x)
Φp(xL)
+ Fp(xL)
Φp(x)
Φp(xL)
−Fp(x) (2.27)
56
which satisfy both boundary conditions since Φn/p(x0) = Fn/p(x0) = 0.
These equations, although explicit, remain difficult to understand in terms of leading
order behavior due to the the competing exponential behavior of their terms. However,
using the fact that U is constant outside of the interface, one can explicitly calculate the
integrals Φn and Φp to the zeroth order (see Section 2.7.2 for explicit results).
In the following, we use these formulas to identify the leading contributions to J0n0
and J0p0 and thus n
0 and p0. More precisely, we introduce the parameters
1
δ1
:= eϕ(xl)−ϕ0 ,
1
η
:= eϕ(xr)−ϕ(xl),
1
δ2
:= eϕL−ϕ(xr) (2.28)
and observe that δ2 = δ
−2
1 for the considered device geometry since we have 2(xl−x0) =
xL−xr (see Fig 2.1). For the rest of this section we identify δ1 = δ. With ϕx = −E0+Ux
and E0 < 0 as in the working device, these parameters are small: δ, η ≈ 10−3 for
E0 = ESC with the numerical values given in Table 2.1.
However, using these parameters directly in (2.26) and (2.27) does not directly yield
insights into the behavior of n0 and p0. Because Φn/p(x) can change over many orders of
magnitude, the behavior of n0 and p0 depends highly on competing exponential terms,
none of which can be easily eliminated. One possible way to proceed splits the domain
[x0, xL] = [x0, xl] ∪ (xl, xr) ∪ [xr, xL] into three areas: left of the interface, the interface,
and right of the interface. For each of these areas, we would obtain formulas for n0 and
p0 of the form of (2.23) and (2.24) in terms of the values n0(xl), n0(xr) and p0(xl), p0(xr)
and the boundary terms n0, nL and p0, pL. Then, imposing continuity of n0 and p0 and
continuity of the fluxes J0n0 and J
0
p0 at x = xl and x = xr would allow us to determine
the values n0(xl), n0(xr) and p0(xl), p0(xr). This is quite similar to the method which
we will discuss below in Section 2.7.
However, a simpler way to proceed considers the zeroth order currents, which are
sufficient to understand the short circuit electric current. Using the explicit formulas for
(the zero order) Φn and Φp in terms of δ and η we have from (2.25):
J0n0 =
nL −
n0
ηδ3 + Fn(xL)
1
E0
1
ηδ2
(
1− 1δ
)
+ 1E0−Ux(θ)
1
δ2
(
1− 1η
)
+ 1E0
(
1− 1δ2
) , (2.29)
J0p0 =
−pL + p0ηδ3 −Fp(xL)
− 1E0 ηδ
2 (1− δ)− 1E0−Ux(θ)δ
2 (1− η)− 1E0 (1− δ
2)
, (2.30)
where Ux(θ) denotes a mean value of Ux within the interface.
57
Because Jn+Jp = const, the sum J0n0 +J
0
p0 can be used to calculate the total current
output of the device (as predicted by the asymptotics). We shall plot and discuss the
asymptotic IV curve in Section 2.6. Note that for the given plots therein we take U to
be piecewise linear so that Ux(θ) is explicitly defined.
Short Circuit Current
As mentioned earlier, the sign of E0 determines if the parameters η, δ are large or small.
For the short circuit current, we have E0 ≈ −13 < 0 and η, δ  1. In order to determine
the lowest order terms of these equations, we must also calculate a more precise form
of Fn/p(xL) and thus its order. First we remark that F (x) =
∫ x
x0
kd(y)X0(y)dy ≈ 102
(since kd,inX0 ≈≈ 104 on the interface with width 10−2 and kd,outX0 ≈ 10 outside the
interface with device length xL − x0 ≈ 1). Thus, one can verify that the exponentials of
ϕ(x) will determine the leading contributions to the integrals Fn/p(xL). More precisely,
since ϕ(x) is strictly increasing in the E0 < 0 regime (recall that ϕx = −E0 + Ux) the
leading order contributions derive from
Fn(xL) ≈
∫ xl
x0
F (y)eϕL−ϕ(y)dy, Fp(xL) ≈
∫ xL
xr
F (y)eϕ(y)−ϕLdy.
Applying the definitions of Equation (2.28), integration by parts and the mean value
theorem yields
∫ xl
x0
F (y)eϕL−ϕ(y)dy =
1
ηδ2
∫ xl
x0
F (y)e−E
0(xl−y)dy
=
1
ηδ2
[
F (y)
e−E
0(xl−y)
E0
]xl
x0
−
1
ηδ2
∫ xl
x0
F ′(y)
E0
e−E
0(xl−y)dy
=
1
ηδ2
F (xl)
E0
−
1
ηδ2
F ′(θn)
(E0)2
∫ xl
x0
e−E
0(xl−y)dy,
for a mean value θn ∈ (0, xl). Together with similar calculations for Fp(xL) and θp ∈
(xr, xL), we obtain
Fn(xL) ≈
1
ηδ2
F (xl)
E0
−
kd,outX0(θn)
(E0)2
1
ηδ2
(
1−
1
δ
)
+O(η−1δ−2)
Fp(xL) ≈ −
kd,outX0(θp)
(E0)2
(
1− δ2
)
−
F (xL)
E0
+
F (xr)
E0
δ2 +O(δ2)
58
Inserting only the dominating terms O(η−1δ−3) into (2.29) and O(1) into (2.30)
yields:
J0n0 ≈ E
0n0 −
kd,outX0(θn)
E0
J0p0 ≈ E
0pL −
kd,outX0(θp)
E0
− F (xL)
J0tot ≈ E
0(n0 + pL)−
kd,out
E0
(
X0(θn) +X
0(θp)
)
−
∫ xL
x0
kdX
0(y)dy. (2.31)
The third term is the easiest to understand, because it is simply the total number
of charges generated by the dissociation of excitons in the device. In the working case
the majority of the excitons are expected to be produced by the light and thus the final
term represents the photogenerated current in the device.
The first term, E0(n0 + pL), gives a current which is proportional to the field, indi-
cating that it represents a resistor. For a usual working device this term will be small
(n0 + pL ≈ 10−6 in Section 2.2) since it refers to energetically unfavorable injection of
minority carriers into the device. Because the term does not depend on the length of
the device, we interpret it as a shunt resistance (parallel to the device). We observe
from Equations (2.29) and (2.30) that we have another current term with the form of a
shunt resistance: ηδ3E0(nL + p0). This factor is negligible in the short circuit case, but
becomes large as we move to positive fields, where δ3 > 1/η. We investigate the total
shunt resistance further in Section 2.6.
Note that in this form Ohm’s law becomes J = Eσ where σ, the conductivity, is
the reciprocal of the resistance. Thus a small value of (n0 + pL) would correspond to
high shunt resistance, which is characteristic of a good device. In particular, a lower
shunt resistance will reduce the open circuit voltage as charges can move across the
device freely without piling up at the interface. This effect is (barely) visible in Figure
2.3, where the zero boundary condition curve reaches J = 0 slightly later than the curve
with non-zero boundary conditions. This corresponds to a slightly increased open-circuit
voltage – one consequence of increasing the shunt resistance.
The second term of Equation 2.31 represents the interaction of the bulk excitons
on the system. For usual device parameters, kd,out  kd,in (for the parameters given
in Section 2.2, we have kd,out = 10−3kd,in). Since X0 ≈ Gkd,in on the interface and
X0 ≈ Gkr,out outside the interface, we can neglect the contributions from outside the
interface (recall that θn ∈ (0, xl) and θp ∈ (xr, xL)) compared to the contribution from
within the interface (as represented by the integral over the interface region).
59
Thus the most important contribution to the current for negative fields is:
J0approx = −
∫ xr
xl
kd,inX
0(y)dy. (2.32)
This approximation works very well in the short circuit case (as examined in Section
2.6, in the discussion preceding Figure 2.7), where it replicates J0tot within 2% and is
actually closer to the numerically calculated J .
Note that J0approx comes entirely from the J
0
p0 term. Because we have evaluated J
0
n
and J0p at the point x0, we expect that the hole current will be dominant. Using Equation
(2.22) we can calculate J0nL and J
0
pL, and note that the only difference from the currents
at x0 is that the −F (xL) term appears in the electron current. Thus, as expected, in the
region where the electrons are favored, the primary contribution to the current comes
from the electrons and the total current is conserved. This dominance of the current of
the majority carriers (which is expected from the physics) may cease to hold outside of
the reverse-bias regime – see Section 2.8.
First Order Terms
Since we have explicit forms for n0, p0, and X0, we can integrate twice to calculate V 1
(with an additional linear term to account for the two boundary values of V ). Normally
we don’t plot V since it is dominated by its boundary terms near short circuit. However,
E1 = −V 1x can also be calculated explicitly (by a single integration) and we include the
first-order field in the plots in Section 2.6. The explicit form is not very illuminating and
thus not written here. Note that we need to add a constant value to the field to ensure
that its integral gives the correct potential difference in the device (corresponding to the
slope of the aforementioned linear term).
In theory we could use the second order form for the potential to calculate the next
order n1 and p1 solutions. In fact, this is more or less the essence of the iteration scheme
outlined in Section 2.3. However, without a constant field, it is no longer possible to
explicitly integrate the continuity equations (especially given the sub-exponential forms
for the electron and hole mobilities, which become constant in the zeroth order) and
we cannot guarantee that the crnp term remains small. Instead, we will next present
another method for asymptotic calculations for the carrier densities. Later, in Section
2.7, we show that integrating the n and p equations implicitly can yield reasonable second
order approximations for the current J0 without requiring explicit calculation of n and
60
p.
2.5.2 Unipolar Approximation
The discussion of the previous section, in particular Equation (2.32), can be summarized
by the statement that the electron and hole fluxes J0n and J
0
p are approximately constant
outside the interface but feature a strong variation over the interface with a magnitude
of F (xr)− F (xl) =
∫ xr
xl
kd,inX0(y)dy.
As a consequence, using the Equations (2.26) and (2.27), it follows that the zero-
order electron and hole densities n0 and p0 vary also strongly over the interface. In fact,
we obtain for the hole density p0:
p0(xl) =
F (xL)− F (xl)
−E0
+O(δ) +O(pL)
p0(xr) =
F (xL)− F (xr)
−E0 + Ux(θin)
+
kd,inX0(θin)
(E0 − Ux(θin))2
+
kd,outX0(θout)
E0(−E0 + Ux(θin))
+O(η, δ2) +O(pL),
where θin ∈ (xl, xr) is a position within the interface used by the mean value theorem
and θout ∈ (xr, xL) is a corresponding position outside of the interface. The values of η,
δ and pL are (as mentioned above) small in the working regime. Then, since Ux(θ) 1
for θ ∈ (xl, xr) and because the dominant part of the integral F (x) comes from the
interface region, we see that
p0(xl)
p0(xr)
≈ 103, in the working case E0 < 0,
as a consequence of the work function gap over the interface. A similar result can be
obtained for n0 giving n0(xr) n0(xl).
Note that in some devices one might expect this ratio to be even larger. However,
from Boltzmann’s statistics we would expect a ratio of e∆U = e6, in reasonable agreement
to the results above. In addition, if we narrow the interface while keeping the work
function difference ∆U and the generation term given in (2.32) constant, we tend to
increase the value of Ux(θin). Thus in some sense the smallness of p0(xr) increases
naturally by reducing the size of the interface. See Section 2.8.2 for more details regarding
the thin-interface limit.
The previous estimates quantify the observation that a bilayer device is operated such
that in one material, n 1 (here to the right of the interface) and in the other material
61
p  1 (here to the right of the interface) because of the nature of the HOMO/LUMO
levels of the different polymers. We can thus ask for an asymptotic simplification of
System (2.14) in the bulk-material away from the interface.
In order to proceed with such an approximation, we observe furthermore that the
ratio p
0(xl)
kd,outX0
≈ 10 − 20, i.e. that the boundary value of the majority charger carrier
p0(xl) given at the left of the interface dominates over the generation of holes from
excitons in the bulk material left of the interface. In fact the above factor would even be
larger in more realistic exciton models compared to (2.14), which misses a term modeling
a electrochemical trapping of excitons (due to stability of polaron pairs on the interface
[3]) yielding further decreased the exciton concentrations (see Section 2.9 for a discussion
of including this potential). We shall thus entirely neglect here the photogeneration term
of charge carriers outside the interface (kd,outX0).
Neglecting the minority charge carriers and excitons outside the interface, we consider
the following simplified models for the majority charge carriers:
−λ2DVxx = λ
2
DEx = p, px − pE =
−Jp0
µp
, (2.33)
in the p-material at the left of the interface (with the Debye length λ2D = λ
2r) and
−λ2DVxx = λ
2
DEx = −n, nx + nE =
Jn0
µn
,
for the n-material at the right of the interface. Here Jp0 and Jn0 are the constants arising
from integration.
For the rest of this section we shall focus only on the hole density p. The analysis
of the electron equation is analogous. We remark that the Equations (2.33) constitute
a generalization of the system considered in Appendix A of [15], where the zero-current
case Jp0 = 0 was analyzed. For the present case with non-zero current we assume first
that µp is constant and exploit then a first-integral of the equation to obtain
Ex −
1
2
E2 =
−Jp0
λ2Dµp
x+ Cp
where Cp is another integration constant. Using the substitution x = −2u and E = yu/y
(see e.g. [15]), we obtain
yuu = (κu− 2Cp)y
where κ = −4Jp0
λ2Dµp
. With the further substitution z = 3
√
κ
(
u− 2Cpκ
)
, this becomes the
familiar Airy Equation, yzz = yz and we can write down the formula for y explicitly in
62
terms of Airy Functions. Since E = yu/y, it is more convenient to write the form of
V = − log (y) (derived independently in [73]):
V = − log
[
αAi
[
3
√
κ
(
−
x
2
−
2Cp
κ
)]
+ βBi
[
3
√
κ
(
−
x
2
−
2Cp
κ
)]]
(2.34)
where we can express α and β in terms of the integration constants and the boundary
values. Note that for the n material we take the substitution x = 2u and −Jp → Jn
positive and everything else proceeds identically.
This asymptotic result gives an explicit form for the charge carriers in the bulk in 1D.
However, for bilayer systems away from the interface, the system generally approximates
a 1D system since the bulk material acts primarily as a charge-transport layer (see the
discussion surrounding 2.2). Since we explicitly account for the self-consistent potential,
this formulation works much better than our previous asymptotics when we have large
carrier concentrations (especially for less negative potential differences). In theory we
could combine these equations in the bulk with a numerical calculation of the interface,
simplifying the computations. However, in practice, we can change the mesh (or grid)
to account for the reduced complexity of the problem in the bulk (see Figure 2.5) and
the gain would not necessarily be significant.
The main disadvantage of this formulation is that it relies upon the boundary data
of the system. Since it doesn’t model the interface, it is thus incapable of predicting the
current in the device based on the device parameters, and requires J as a parameter.
In fact, we need all of the boundary values p0, E0, Jp0 in order to calculate the field and
the hole concentration in the p-region of the device, but generally only p0 is given. The
other values, E0 and Jp0, both require calculating the solution of the whole problem or
at least calculating a model of the interface and using asymptotic approximations of the
boundary values E0 and Jp0. Alternatively, we can take the values p(xl), E(xl), Jpl and
again calculate the values in the bulk. Either method gives extremely accurate results
for the cases in which J  0, which we show in Figures 2.7 - 2.9. A treatment of the
results coupled with interface conditions on the boundary is given in the recent work
[73].
2.6 Discussion of Numerical Examples
In the following we present selected numerical examples of the steady state device be-
havior of the organic photovoltaic bilayer device plotted in Figure 2.1. In particular, we
63
shall investigate the steady states calculated from different applied potentials.
Using the scheme outlined in Section 2.3 we use the damped-Gummel iteration given
in (2.12) and (2.13) with α = .01 for the first three steps (as a sort of preconditioning)
and α = .6 until convergence (see Section 2.6.4). All of the plots show a mesh consisting
of 720 elements, greatly refined near the interface, and use polynomial interpolants of
order 7 for the bulk elements – see Figure 2.5. Refining the mesh and increasing the
interpolation order both give better accuracy, in particular for the steep concentration
changes over the interface (see, i.e. Figures 2.7, 2.9 and 2.10). Note that the geometry
of the test cases is taken to be homogeneous in the y-direction and we plot only a sample
x-cross section which can be compared with the one-dimensional asymptotics.
Figure 2.5: Mesh used for all of the plots in Section 2.6
2.6.1 A Note On Exciton Diffusion
It is clear from the previous sections that the behavior of the polaron pairs on the
interface is key to the overall current generated in the device. In fact, the approximate
current given by (2.32) is simply a weighted integral of X over the interface. Many
models for the operation of photovoltaic devices disregard the field dependence of kd
entirely, and doing so with our model still yields qualitatively realistic results (see e.g.
Figure 2.4). Since many of the coefficients of the exciton equation are not physically
well-understood, this begs the question of whether or not they significantly impact the
characteristics of the model. We show in Figure 2.6 that changing the exciton and
polaron pair mobility µX does in fact have an impact on the qualitative nature of the
current-voltage characteristics. This effect is not trivial – we show in Appendix 2.A.1
that L∞ bounds on X are independent of µX . However, given that the equilibrium
values for polaron pairs (X inside the interface) are higher than the equilibrium values
64
for excitons, reducing µX supplies significantly more polaron pairs near the edges of the
interface.
-350
-300
-250
-200
-150
-100
-50
 0
-20 -15 -10 -5  0  5  10  15  20
C
ur
re
nt
Potential Difference
mu=.01
mu=.001
mu=.0001
mu=.01 outside, .0001 inside
Figure 2.6: Numerical current-voltage characteristics for our given µX = .01, .001, and
.0001. Note that the Short Circuit current and open circuit voltage are similar, but that
the fill factor is significantly better for the smaller mobility. Also plotted is an IV curve
for µX = .0001 inside the interface and µX = .01 outside the interface. In a physical
device, increasing the mobility of the excitons in the bulk should result in many more
polaron pairs being produced and thus much more current.
This result, while not necessarily surprising mathematically, does indicate non-physical
behavior. Excitons near the interface should be trapped on the interface as polaron pairs.
This was modeled by choosing a small µX , but clearly the value given in Table 2.1 was
not small enough, as decreasing it further results in the changes shown in Figure 2.6.
Furthermore, there is also experimental evidence that increasing the bulk exciton mo-
bility in a bilayer should increase the current [58]. However, our numerical results show
that reducing the exciton mobility in the bulk by two orders of magnitude does not
significantly change the current-voltage characteristic. This suggests that increasing the
65
diffusion in the bulk is not increasing the number of excitons or polaron pairs dissociat-
ing. In particular, we note in the plots of the exciton densities in the next section that
we do not observe a reduction in the exciton concentration near the interface.
If the excitons were truly moving preferentially towards the interface, we would ex-
pect a non-zero flux ∇X towards the interface, probably over a length-scale comparable
to the exciton diffusion-length (≈ .1 in scaled units). In hindsight, this is not necessarily
surprising. For µX very small, the concentration of polaron pairs will be very close to
the equilibrium value Gkd+kr and the behavior of the excitons will have little or no effect
on the polaron pair concentration. We discuss a possible solution to this issue in Section
2.9.
Thus the results below thus focus on the effect the excitons and polaron pairs have
on the dynamics of the system (especially the current) as opposed to the particulars of
the exciton modeling.
2.6.2 Electron/Hole and Exciton Densities at SC, OPP and OV
In this section, we study the details of the charge density and exciton distributions at
particular characteristic points of an IV curve: i) Short Circuit (SC), ii) Optimal Power
Point (OPP) and iii) Open Circuit Voltage (OV). The short circuit current is the current
at zero applied voltage, or a potential difference of Vdiff = −19.3. The optimal power
point is the potential difference for which the power is maximized, which is Vdiff = −3
in Figure 2.3. The open circuit voltage is the point where J = 0, which occurs for
Vdiff = 12.
For each of these points, we will plot the numerical solutions for n (green ×, right
y-axis) and p (red +, left y-axis) in the same chart. Note that we have rescaled n and
p by µn and µp in the asymptotics section, and must appropriately scale them back for
the plots. The main effect of this rescaling is that n and p once again have different
scales (as in Figure 2.2 in Section 2.4). On the same figure, we also plot the zero-order
asymptotics derived in Section 2.5.1 (in blue) and the unipolar asymptotics derived in
Section 2.5.2 (in pink). The unipolar results reproduce the numerical solution in the
bulk very well when complemented with the numerical data from the simulation of the
interface (which is not part of the unipolar approximation). Specifically, we use the
boundary data values p(x0), E(x0), and J for the holes and n(xL), E(xL) and J for the
electrons. We observe that in all three cases – SC, OPP and OC – the minority carriers
66
(n to the left of the interface and p to the right of the interface) are strongly dominated
by the majority carriers (as discussed above in Section 2.5.2). We also note, however,
that small concentrations do not necessarily imply small currents – see Section 2.8.
Next, we plot the numerical solutions of the exciton concentration (in the lower
left plot) and the electric field (in the lower right plot). The exciton concentration is
compared with the explicit results in Section 2.5.1 and the electric field is compared with
the first-order asymptotic approximation (near the end of Section 2.5.1. The unipolar
asymptotics do not include exciton behavior to compare with the numerical results.
Furthermore, we refrain from plotting the unipolar approximation to the electric field
outside the interface (which is discontinuous over the interface since the field is calculated
independently on the two sides of the device). However, good agreement similar to the
charge carrier densities can be achieved.
We also list the numerically calculated current given at each voltage. The current
predicted by the zero-order asymptotics is the sum of J0nl and J
0
pl given in Equation (2.29)
and (2.30). However, because this predicted current assumes that E < 0, it cannot give
accurate results in the open circuit case.
Short Circuit
The case with the largest (negative) potential difference Vdiff is the short circuit case
(which has zero applied voltage). This is due to the built-in potential arising from the
work functions of the anode and cathode metals. For the device parameters laid out in
Section 2.2, this occurs for Vdiff ≈ −19.3.
Since short circuit considers the largest negative potential difference it is thus the
case for which the zero-order asymptotics derived in Section 2.5.1 are best suited. This
is confirmed by Figure 2.7 which compares the numerical solutions of n, p, X and E
with the corresponding asymptotics. Note that we see a dramatic change in n, p, X at
the interface, and that either electrons or holes are largely dominant on one side of the
device. Moreover, note that the slope of the electric field is significantly larger on the
left-side of the device due to the larger concentration of holes on the left-side compared
to the smaller concentration of electrons on the right-side (Figure 2.7 plots n and p on
different scales).
The numerically calculated current (as listed in Figure 2.7) is J = −334.2. The
current predicted by Equations (2.29) and (2.30) is J0 = −345.15. Although not in
67
 0
 2
 4
 6
 8
 10
 12
 14
 16
 0  0.2  0.4  0.6  0.8  1  1.2  1.4
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
H
ol
e 
C
on
ce
nt
ra
tio
n
E
le
ct
ro
n 
C
on
ce
nt
ra
tio
n
Position
Holes (Numerics)
Electrons (Numerics)
Zeroth Order Asymptotics
Unipolar Asymptotic Fit
 4.2
 4.4
 4.6
 4.8
 5
 5.2
 5.4
 5.6
 5.8
 6
 6.2
 6.4
 0  0.2  0.4  0.6  0.8  1  1.2  1.4
Exc
iton
 Co
nce
ntra
tion
Position
ExcitonsZeroth Order Asymptotics
-13.8
-13.6
-13.4
-13.2
-13
-12.8
-12.6
 0  0.2  0.4  0.6  0.8  1  1.2  1.4
Ele
ctri
c F
ield
Position
Electric Field (x-direction)First Order Asymptotics
Figure 2.7: Plots of n, p, X and E for the short circuit case Vdiff = −19.3. Note the
different scales for n and p, and that the total variation of E is less than 10% of its
average value. J = −334.2. The electrons and holes are each dominant in one half of
the device, and the asymptotic results provide good agreement to the numerical results
in all cases.
68
exact agreement with the numerically calculated current for the short circuit case, this
constitutes a reasonable match for the zeroth order term. Furthermore, the lowest order
current (in δ, η) predicted by (2.32) is Japprox = −339.32, also in excellent agreement for
the short circuit case. However, the current given by the asymptotics does not depend
on the potential as dramatically as the simulations suggest, most likely indicating that
the δ  1 assumption begins to fail.
Another observed feature of Figure 2.7 are the boundary layers for n to match the
boundary data at xL and for p to match the boundary data at x0. The values of n and
p at their majority side of the interface are predicted by the asymptotics to be n(xr) =
0.526 and p(xl) = 15.6, in good agreement with the numerically calculated values. We
emphasize that Equation (2.32) in Section 2.5.1 predicts that the concentrations of n
and p at the interface are predominantly a consequence of the leading order flux-changes
over the interface J0approx = −
∫ xr
xl
kd,inX0(y)dy and are essentially independent of the
boundary conditions p0 and nL. We can confirm this statement by using the interface
values as the boundary conditions (i.e. n(xL) = 0.526 and p(x0) = 15.6). We plot
the results in Figure 2.8 without visible difference in the concentrations of the charge
carriers at the interface compared to Figure 2.7. In particular note that the current and
the maximal values of n and p do not change significantly.
Optimal Power Point
A second key characteristic point for device operation is the optimal power point. This
is the point where the power from the device is maximized. As shown in Figure 2.3, this
occurs for a voltage difference of Vdiff ≈ −3. This is just below the voltage for which the
resulting self-consistent field touches zero from below and the field-dependence of kd(E)
causes a kink in the IV curve. The concentrations n, p and X are plotted along with
the E-field in Figure 2.9. In the plot of the E-field we see a slightly more pronounced
effect (compared to SC) of the excitons on the electric field (the small bump located at
the interface), but again this effect is small compared with the overall changes in the
field.
Comparing Figure 2.9 to Figure 2.7 we see that n and p again are primarily located on
their specific sides of the device (p to the left, n to the right) with an even larger change
in concentrations across the interface. In contrast to Figure 2.7 (which features boundary
layers for n and p), the concentrations of n and p in Figure 2.9 appear approximately
69
 0
 2
 4
 6
 8
 10
 12
 14
 16
 0  0.2  0.4  0.6  0.8  1  1.2  1.4
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
Ho
le 
Co
nc
en
tra
tio
n
El
ec
tro
n C
on
ce
ntr
ati
on
Position
Holes (Numerics)
Electrons (Numerics)
Zeroth Order Asymptotics
Unipolar Asymptotic Fit
Figure 2.8: Plot of n and p for the short circuit case with artificial boundary conditions,
nL = 0.526 and p0 = 15.6. J = −332.528, compare these plots to those for the correct
boundary conditions in Figure 2.7. The deviation of p from the constant solution results
from the self-consistent potential. The effect is less pronounced for the electrons because
the concentration is much lower.
70
 0
 10
 20
 30
 40
 50
 60
 70
 80
 90
 0  0.2  0.4  0.6  0.8  1  1.2  1.4
 0
 2
 4
 6
 8
 10
 12
 14
 16
 18
H
ol
e 
C
on
ce
nt
ra
tio
n
E
le
ct
ro
n 
C
on
ce
nt
ra
tio
n
Position
Holes (Numerics)
Electrons (Numerics)
Zeroth Order Asymptotics
Unipolar Asymptotic Fit
 0
 5
 10
 15
 20
 25
 30
 0  0.2  0.4  0.6  0.8  1  1.2  1.4
Exc
iton
 Co
nce
ntra
tion
Position
ExcitonsZeroth Order Asymptotics
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
 0
 0  0.2  0.4  0.6  0.8  1  1.2  1.4
Ele
ctri
c F
ield
Position
Electric Field (x-direction)First Order Asymptotics
Figure 2.9: Plots of n, p, and E respectively for the optimal power point OPP Vdiff = −3.
The resulting current is J = −241.585. The exciton contribution from Poisson’s equation
is visible as a bump in the field plot. We see that the zeroth order asymptotics begin to
diverge from the numerical and unipolar solutions as the potential difference increases.
71
linear in their majority sides. It is not clear whether this approximate linearity is a
characteristic of the optimal power point or coincidental. Our numerical simulations,
however, seem to suggest that this is indeed the case and we may conjecture that it is
caused by a certain balance between drift and diffusion terms. Moreover, note that the
scales for n and p remain different, but are considerably more balanced than in Figure
2.7.
Figure 2.9 also plots the zeroth order asymptotics, which are still reasonable. More-
over, we see excellent fits from the unipolar asymptotic behavior of the quasi-linear
behavior of the n and p. The increasing discrepancy of the zeroth order asymptotics
suggests the increasing importance of the nonlinearity of the self-consistent E-field. This
can be confirmed by using E0 instead of the numerical values E(x0) and E(xL) in the
unipolar approximation which is then no longer nearly as accurate.
Open Circuit
The third point of importance for device operation is the open circuit voltage. This is
the point at which the current through the device is zero (analogous to an open circuit
in electronics). For our simulations this occurs at approximately Vdiff ≈ 12. In physical
units this corresponds to a potential difference of 0.31 Volts. Considering the internal
voltage of our typical device, this corresponds to an open circuit voltage of 0.81 Volts,
larger than expected for a typical OPV (possibly due to the effects outlined in Section
2.8 below).
The plots in Figure 2.10 show a strong alteration of the concentrations n and p
compared to the previous Figure 2.7 and 2.9. The hole concentration, p, changes from
p(xl) ≈ 370 to p(xr) ≈ 1.4 over the interface, confirming that our numerical scheme is
capable of capturing large changes across the interface (predicted to be ≈ 103 by the
zeroth order asymptotics). Note that for this case the maximum values of n and p are
of about the same order albeit plotted on different scales (by a factor four).
The behavior of the E-field throughout the device is dominated by the large values of
n and p with very little effect of the excitons at the interface. Note that we do not see a
very good match of the asymptotic electric field, which is to be expected. In particular,
since the zeroth asymptotics ignore the quadratic recombination term (crnp), we are
neglecting this loss term for n and p. This is particularly important within the interface,
where the concentrations intersect at n ≈ p ≈ 18 (recall that p is strongly decreasing
72
 0
 200
 400
 600
 800
 1000
 1200
 1400
 0  0.2  0.4  0.6  0.8  1  1.2  1.4
 0
 50
 100
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 250
 300
 350
H
ol
e 
C
on
ce
nt
ra
tio
n
E
le
ct
ro
n 
C
on
ce
nt
ra
tio
n
Position
Holes (Numerics)
Electrons (Numerics)
Zeroth Order Asymptotics
 0
 10
 20
 30
 40
 50
 60
 0  0.2  0.4  0.6  0.8  1  1.2  1.4
Exc
iton
 Co
nce
ntra
tion
Position
ExcitonsZeroth Order Asymptotics
-10
-5
 0
 5
 10
 15
 20
 0  0.2  0.4  0.6  0.8  1  1.2  1.4
Ele
ctri
c F
ield
Position
Electric Field (x-direction)First Order Asymptotics
Figure 2.10: Plots of n, p, and E respectively for the (nearly) Open Circuit Vdiff = 12
case. J = 0.23. (The true Open Circuit Voltage is within 0.05 of 12, but further precision
is not numerically relevant). Note the different scales for n and p. The qualitative
behavior of the zeroth order asymptotics is nearly correct, but they no longer closely
match the numerical results. In particular, the estimated value for p is off by at least a
factor of 2 nearly everywhere as the majority carrier.
73
from the left of the interface to the right while n is strongly increasing). This leads
n and p to be overestimated. Note, however, that the asymptotics correctly reproduce
the qualitative behavior of the carrier concentrations with exponential growth from the
boundaries of the device to the interface.
Finally we remark that the presented numerical scheme is capable of continuing
further into forward-bias. However, the above model and in particular the parameters are
primarily focused on the working-case, and we do not necessarily expect realistic results
for the strong forward-bias regime. As the number of polaron pairs on the interface
increases (due both to a reduction in kd and an increase in c′rnp), we observe that
many of them diffuse away from the interface. This corresponds physically to polaron
pairs becoming excitons, a transformation which is not generally physically observed
(see Section 2.6.1). We discuss attempts to rectify this problem in Section 2.9.
2.6.3 Shunt Resistance and Asymptotics of the IV Curves
A comparison of the IV curves in Figure 2.3 indicates clearly that the nonlinear boundary
conditions proposed in [80] appear to have very little effect on the IV curve. The
resulting difference is barely noticeable until we reach open circuit (although it does
grow as we pass into the forward bias regime). We can study this discrepancy more
clearly by simulating the OPV device in the dark with G = 0. The resulting current
is plotted in Figure 2.11 and represents the shunt current in the device. Recalling the
asymptotic currents Equation (2.29) and (2.30), we can see the increase in the shunt
current for E  0 (and thus δ  1) proportional to the terms containing nL and p0. In
the large δ limit we recover a term of the form: (nL + p0)E. This has the usual form of
a resistor following Ohm’s law with (scaled) resistivity 1/(nL + p0).
In Figure 2.11 we show heuristically the equivalence of the shunt current (green ×)
to the difference between the nonlinear Dirichlet and homogeneous Dirichlet boundary
conditions (red +). The difference shows exactly the effect of the boundary conditions
given by [80] compared to zero boundary conditions. The blue line represents the zeroth
order asymptotic fit for the current with G = 0. The apparent noise for negative fields
is a result of rounding errors in taking the difference of the currents from two different
simulations. Choosing a smaller tolerance parameter for the convergence and calculating
the current to higher precision (to eliminate errors of magnitude .1 for J ≈ −350) will
reduce the apparent noise. Averaging over slightly different potential differences (i.e.
74
-2
 0
 2
 4
 6
 8
 10
 12
-20 -15 -10 -5  0  5  10  15
Cu
rre
nt
Potential Difference
BC Difference
Shunt Current
Asymptotic Fit
Figure 2.11: An IV curve showing the similarity of the shunt current produced by
injected carriers with no light generation and the result of subtracting the IV curve for
zero boundary conditions from the usual IV curve. The zeroth-order asymptotic fit is
also included. See Figure 2.3 for individual plots of these curves.
-19.99, -20, and -20.01) also reduces the apparent noise.
For a device in the working case the boundary conditions are relatively unimportant,
but as we move into the forward bias regime, these boundary conditions become more
important. The given shunt current is reminiscent of the exponential growth of the
dark-current for a photovoltaic device in forward bias. Although both currents result
from taking G = 0, our model is not designed to reproduce this dark current, which is
normally generated by an equilibrium concentration term in the recombination rate – for
instance cr(np− n∞p∞). With light generation, this term is negligible compared to the
exciton contribution and thus is not included in our modeling assumptions. However,
carefully choosing boundary conditions would allow our model to replicate the physically
observed diode behavior in the forward bias regime.
Next, Figure 2.12 plots the asymptotic IV curve predicted by the sum Jn0 + Jp0 of
Equations (2.29) and (2.30). Although the asymptotic approximation is questionable
for small potential differences (as stated previously) we do observe a qualitatively real-
istic IV curve in Figure 2.12. In particular we observe a similar open circuit voltage
and short circuit current as well as a significantly better fill-factor of 87%. The high
75
fill factor for the asymptotic curves is expected, since the asymptotics neglect the re-
combination current, which should be the main cause of power loss for a bilayer device.
Indeed, by ignoring the recombination term, the asymptotics describe what is called the
photogenerated current, the idealized current generated by the device. We show in Sec-
tion 2.8 that the common VOC for the numerical and asymptotic results is symptomatic
of unphysical behavior near open circuit.
-350
-300
-250
-200
-150
-100
-50
 0
 50
-20 -15 -10 -5  0  5  10
Cu
rre
nt
Potential Difference
Asymptotic IV Curve
Constant k_d Asymptotics
Figure 2.12: The IV Curve predicted by the asymptotics in Section 2.5.1. Note that it
has a similar open circuit voltage and short circuit current, but has a better fill factor
than the numerical simulations.
A comparable increase in fill-factor is observed in Figure 2.4 (reprinted below as
Figure 2.13), where we plot a numerical IV curve assuming a constant dissociation rate
kd,in  kr,in. Moreover, Figure 2.13 plots the corresponding asymptotic IV curve with
good agreement. This indicates that a primary loss of efficiency is due to the exciton
recombination term, which becomes non-negligible near the interface when approaching
the open circuit voltage. This quantifies the physical intuition that device efficiency is
increased by forcing the polaron pairs to dissociate more efficiently into free carriers.
For the numerical simulation the new fill factor is a much improved 60%, whereas the
asymptotics are virtually unchanged from the standard model shown in Figure 2.12.
76
-350
-300
-250
-200
-150
-100
-50
 0
-20 -15 -10 -5  0  5  10
Cu
rre
nt
Potential Difference
Constant k_d Numerics
Constant k_d Asymptotics
Figure 2.13: A comparison of the IV curves obtained using the numerical and asymptotic
schemes with kd,in = kd,in(ESC) = 2763 to be constant.
2.6.4 Damping and Iteration Convergence
Here we present the evidence that our damped iteration scheme indeed converges. First,
we show log-plots of the L2-norm difference between the solutions at consecutive iteration
steps. After showing the convergence for the simulations given in Section 2.6, we proceed
to discuss the role of the damping parameter in the convergence.
L2 Errors for the Iteration Scheme
Here we give plots of the L2 norm of the difference between the subsequent solutions
at each step of the numerical iteration scheme. In all of the plots the first three steps
have nearly constant error because of the small value of the damping parameter α = .01.
After the initial stage, we expect exponential convergence.
If our undamped solution u′ is always the true solution u∗, then from Equation (2.13),
we can calculate the error E by
E1 = u1 − u0 = α (u
∗ − u0) Ek+1 = uk+1 − uk = α (u
∗ − uk) .
77
We then note that
Ek+1 = uk+1 − uk
= α
(
u∗ −
(
uk−1 + α
(
u∗ − uk−1
)))
= (1− α)α (u∗ − uk−1)
and by induction we have
Ek+1 = (1− α)
kE1. (2.35)
For constant α, we would thus expect the error for a “perfect” damped scheme to decrease
exponentially with base (1 − α). Since our u′ is only the current best-estimate for the
equilibrium solution, we do not expect this relationship to hold. In particular, for the
undamped case α = 1, this result would be completely meaningless.
However, this equation gives a good approximate decay rate for comparison and
we proceed to show the result for α = .6 as a cyan line in Figures 2.14 - 2.16. For
comparison, we also plot the L2-norm difference between consecutive steps for V (red),
n (green), p (blue), and X (pink).
Short Circuit
We show the error plots for short circuit in Figure 2.14. After approximately three steps,
we see exponential decay of the L2 errors, indicating good convergence to a steady state
solution. For short circuit, the errors for p are the limiting factor of the scheme. Note
that the concentrations of p are roughly thirty times larger than the concentrations of
n (see Figure 2.7) and that this difference probably leads to the absolute L2 error being
roughly 30 times larger for p.
Optimal Power Point
We show the error plots for the optimal power point in Figure 2.15. The early stages are
similar to those in Figure 2.14, but the intermediate stage lasts for six points instead of
three. In addition, the convergence is no longer better than the damping estimate.
Note that the ”noise” in the six intermediate points looks quite similar to the con-
vergence shown for open circuit in Figure 2.16.
78
 1e-06
 1e-05
 0.0001
 0.001
 0.01
 0.1
 1
 10
 100
 1000
 10000
 0  5  10  15  20  25  30
Lo
g(L
^2 
Err
or)
Step
V
n
p
X
Alpha-Limited
Figure 2.14: Log-plot of the L2 norm of the difference between subsequent steps for the
short circuit simulation shown in Figure 2.7.
 1e-08
 1e-06
 0.0001
 0.01
 1
 100
 10000
 0  5  10  15  20  25  30  35  40  45
Lo
g(L
^2 
Err
or)
Step
V
n
p
X
Alpha-Limited
Figure 2.15: Log-plot of the L2 norm of the difference between subsequent steps for the
optimal power point simulation shown in Figure 2.9.
79
Open Circuit
We show the error plots for open circuit in Figure 2.16. The convergence is not as uniform
as the short circuit and optimal power point cases (Figures 2.14 and 2.15). However, we
still see roughly exponential decay of the L2 errors, indicating good convergence.
 1e-08
 1e-06
 0.0001
 0.01
 1
 100
 10000
 1e+06
 0  5  10  15  20  25  30
Lo
g(L
^2 
Err
or)
Step
V
n
p
X
Alpha-Limited
Figure 2.16: Log-plot of the L2 norm of the difference between subsequent steps for the
open circuit simulation shown in Figure 2.10.
Choice of α
The essential reason for the damping is that the iteration scheme will overcompensate
for the current error. For example, if n and p are too large, then the −crnp term will be
too large and (all else being equal), n and p will drop below their equilibrium values on
the next step. If this effect is too strong, then the simulation can alternate between two
states or even diverge. In Figure 2.17 we show two possible scenarios where the iteration
scheme does not converge. The first plot shows the iteration switching between two
points in solution space (i.e. the L2 norm of the difference between the steps converges
to a constant). The second plot shows the iteration diverging. The errors jump to 1080
on the next step before failing completely on the following step.
The damping parameter prevents this overshoot and instead allows the system to
converge faster than it might otherwise. However, if the nonlinearities are small (for
80
 0.1
 1
 10
 100
 1000
 10000
 100000
 0  10  20  30  40  50  60  70  80  90  100
Lo
g(L
^2 E
rror
)
Step
V
npX
 1
 10
 100
 1000
 10000
 100000
 1e+06
 1e+07
 1e+08
 0  5  10  15  20  25
Lo
g(L
^2 E
rror
)
Step
V
npX
Figure 2.17: Log-plots of the L2 norm of the difference between subsequent steps for
two simulations which fail with no damping (α = 1). The first is for the optimal power
point (Vdiff = −3), the second is for Vdiff = 6.
instance because crnp 0), then the damping will simply propagate according to Equa-
tion (2.35). In particular, as the asymptotics from Section 2.5.1 show, we might expect
that damping to be unnecessary at short circuit, but vital as we approach forward bias.
Assuming we have chosen an appropriate mesh and element orders, the computa-
tional time for a single iteration is relatively constant. Thus we can evaluate the opti-
mality of α by comparing the number of iterations required for the scheme to reach a
given error tolerance. In Figure 2.18 we show the number of iterations required for the
simulation to reach the desired error tolerance. Note that for α = 1 (cyan), there are
missing points. The maximum number of allowed iterations is 100, and reaching this
point indicates that the simulation did not converge. For one of these points (Vdiff = 3),
the simulation was still converging (albeit very very slowly).
It is interesting to note that the simulations begin to fail at the optimal power point
Vdiff = −3. We can test the role that the field dependence of kd has on this behavior by
comparing the iteration count for simulations with the usual kd and with kd = kd(ESC).
The results of this are shown in Figure 2.19. Note that we see no equivalent bump in
the iteration count near the open circuit-voltage but have the same jump for potential
differences just larger than zero. However, past the zero field point the constant kd
approximation begins to take significantly longer to converge. This is likely because for
E > 0, kd(E) < kd(0). We test this claim by running the simulation again with an
increased value of kd(0) and note that it always takes more iterations to converge.
81
 0
 10
 20
 30
 40
 50
 60
 70
 80
-20 -15 -10 -5  0  5  10  15
Ite
rat
ion
 st
ep
s t
o C
on
ve
rge
nc
e
Potential Difference
alpha=.6
alpha=.7
alpha=.8
alpha=.9
alpha=1
Figure 2.18: Plot of the number of iterations required for convergence for various α.
 0
 10
 20
 30
 40
 50
 60
 70
 80
-20 -15 -10 -5  0  5  10  15
Ite
rat
ion
 st
ep
s t
o C
on
ve
rge
nc
e
Potential Difference
k_d normal
k_d constant
k_d higher
Figure 2.19: Plot of the number of iterations required for convergence for the usual value
of kd, kd = kd(ESC) constant, and with kd(0) =
kd(ESC)
kd(0)
kd(E) with α = .8.
82
2.7 Improving the Asymptotics
The main objective of this section is to understand the physical characteristics of a
photovoltaic device. Section 2.6 demonstrates that our large-field-approximation works
well for the reverse-bias case, encompassing the short circuit current point. However, to
fully understand the operating characteristics, we also need to understand the optimal
power point and the open circuit case. At open circuit, the current is by definition zero,
so we only really need to estimate the applied potential for which it occurs. For the
optimal power point, we seek to estimate the power output of the device, which requires
estimating both the field and the current.
By slightly rewriting the integrals Φ and F without assuming that the field is con-
stant, we can derive an integral form for the current and the values of n and p. While
not explicit, we can use the integrals to extend the asymptotics by manipulation of the
exact current before inserting the asymptotic approximations. Furthermore, by consid-
ering the order of the integrals, we can also derive a simple approximation for the open
circuit voltage which matches our numerical simulations.
This result suggests deep problems with the physical applicability of our model near
open circuit, however, especially with regards to the minority carriers. In Section 2.8 we
quantify these difficulties and discuss why a bulk drift-diffusion model cannot reasonably
approximate the unipolar models discussed in the literature near open circuit.
2.7.1 Implicit Definition of J
We recall the definitions of Φ and F (in Section 2.5.1), but without rescaling n and p
by µn and µp respectively:
Φn(x) :=
∫ x
x0
eϕ(x)−ϕ(y)
µn
dy, Φp(x) :=
∫ x
x0
eϕ(y)−ϕ(x)
µp
dy,
Fn(x) :=
∫ x
x0
F (y)eϕ(x)−ϕ(y)
µn
dy, Fp(x) :=
∫ x
x0
F (y)eϕ(y)−ϕ(x)
µp
dy,
where F (y) also still contains the crnp term:
F (x) :=
∫ x
x0
(
kd(y)X
0(y)− crn(y)p(y)
)
dy.
Exploiting the symmetry of n and p, we can define integral quantities M i and M iF
and manipulate the expressions exactly before examining lowest order asymptotic terms.
83
The M will depend implicitly on the true values of V , n, p, and X, but the calculations
we derive here are exact. We split Φ and F across the regions Ω1 = [x0, xl], Ω2 = [xl, xr]
and Ω3 = [xr, xL] corresponding to the region to the left of the interface, the interface
region, and the region to the right of the interface. We have
Φ(xL) = e
±ϕ(xL)(M1 +M2 +M3)
F(xL) = e
±ϕ(xL)(M1F +M
2
F +M
3
F )
where
M1 =
∫ xl
x0
e∓ϕ(y)
µ(y)
, M2 =
∫ xr
xl
e∓ϕ(y)
µ(y)
, M3 =
∫ xL
xr
e∓ϕ(y)
µ(y)
,
M1F =
∫ xl
x0
F (y)e∓ϕ(y)
µ(y)
, M2F =
∫ xr
xl
F (y)e∓ϕ(y)
µ(y)
, M3F =
∫ xL
xr
F (y)e∓ϕ(y)
µ(y)
.
This notation has two important qualities. First, it allows us to proceed identically for
n and p (changing only the sign in the exponential). Second, we have simple expressions
for the quantities at xl and xr since they correspond with the integral endpoints. We
have:
Φn/p(xl) = e
±ϕ(xl)M1 Φ(xr), = e
±ϕ(xr)(M1 +M2),
Fn/p(xl) = e
±ϕ(xl)M1F , F(xr) = e
±ϕ(xr)(M1F +M
2
F ).
Inserting these expressions into Equations (2.29) and (2.30) (assuming zero boundary
conditions) yields:
Jn0 =
M1F +M
2
F +M
3
F
M1 +M2 +M3
, Jp0 = −
M1F +M
2
F +M
3
F
M1 +M2 +M3
. (2.36)
For justification of the zero-boundary-value assumption, see Figure 2.3 and the following
discussion.
Since the sign of Jp0 occurs twice – once in (2.14c) and once in (2.30), we have
n, p(xl) = e
±ϕ(xl)
(
M1F +M
2
F +M
3
F
M1 +M2 +M3
M1 −M1F
)
= e±ϕ(xl)
M1(M2F +M
3
F )− (M
2 +M3)M1F
M1 +M2 +M3
, (2.37)
and
n, p(xr) = e
±ϕ(xr)
(
M1F +M
2
F +M
3
F
M1 +M2 +M3
(M1 +M2)− (M1F +M
2
F )
)
= e±ϕ(xr)
(M1 +M2)M3F −M
3(M1F +M
2
F )
M1 +M2 +M3
. (2.38)
84
2.7.2 Approximating the M i
We begin by approximating the order of magnitude of the M i using the mean value
theorem in order to best approximate the value in the case of a large applied field. If
the applied field and work function difference are the dominating terms, then the value
of ϕ′(x) will take characteristic values in each of the three regions of the device, Ω1, Ω2,
and Ω3. Defining El, Em, and Er as mean values for −V ′ in these domains and Wm as
a mean value for −U ′ in the interface region, we have:
M1n =
e−ϕ(xl) − e−ϕ(x0)
µn(El)El
, M1p =
eϕ(x0) − eϕ(xl)
µp(El)El
,
M2n =
e−ϕ(xr) − e−ϕ(xl)
µn(Em)(Em +Wm)
, M2p =
eϕ(xl) − eϕ(xr)
µp(Em)(Em +Wm)
,
M3n =
e−ϕ(xL) − e−ϕ(xr)
µn(Er)Er
, M3p =
eϕ(xr) − eϕ(xL)
µp(Er)Er
,
and
M1Fn =
F (xl)e−ϕ(xl)
µn(El)El
−
kd,lXl − crnlpl
µn(El)E2l
(
e−ϕ(xl) − e−ϕ(x0)
)
,
M2Fn =
F (xr)e−ϕ(xr)
µn(Em)(Em +Wm)
−
F (xl)e−ϕ(xl)
µn(Em)(Em +Wm)
−
kd,mXm − crnmpm
µn(Em)(Em +Wm)2
(
e−ϕ(xr) − e−ϕ(xl)
)
,
M3Fn =
F (xL)e−ϕ(xL)
µn(Er)Er
−
F (xr)e−ϕ(xr)
µn(Er)Er
−
kd,rXr − crnrpr
µn(Er)E2r
(
e−ϕ(xL) − e−ϕ(xr)
)
,
and
M1Fp =−
F (xl)eϕ(xl)
µp(El)El
−
kd,lXl − crnlpl
µp(El)E2l
(
eϕ(xl) − eϕ(x0)
)
M2Fp =−
F (xr)eϕ(xr)
µp(Em)(Em +Wm)
+
F (xl)eϕ(xl)
µp(Em)(Em +Wm)
−
kd,mXm − crnmpm
µp(Em)(Em +Wm)2
(
eϕ(xr) − eϕ(xl)
)
M3Fp =−
F (xL)eϕ(xL)
µp(Er)Er
+
F (xr)eϕ(xr)
µp(Er)Er
−
kd,rXr − crnrpr
µp(Er)E2r
(
eϕ(xL) − eϕ(xr)
)
.
Note that if V and U are piecewise-linear, then the equations above are exact, and
in particular we get exact expressions for the Φ and F defined in Section 2.5.1. There
is an apparent discontinuity near E = 0, but if ϕ is constant, then the exponentials are
85
constant and the limit of the expression approaches the correct value. A sign change
of E in Ω will result in the failure of the mean value theorem, but we note that this is
not the case for the three important cases (short circuit, optimal power point, and open
circuit) presented in Section 2.6.
Using the δ, η defined in Equation (2.28), we can also estimate the order of the M
integrals:
M1n,M
1
Fn = O(δ
−1
2 η
−1δ−11 ) +O(δ
−1
2 η
−1), M1p ,M
1
Fp = O(δ2η) +O(δ2ηδ1),
M2n,M
2
Fn = O(δ
−1
2 η
−1) +O(δ−12 ), M
2
p ,M
2
Fp = O(δ2) +O(δ2η),
M3n,M
3
Fn = O(δ
−1
2 ) +O(1), M
3
p ,M
3
Fp = O(1) +O(δ2).
2.7.3 Improved Asymptotic IV
The key limitation is that the approximation of the previous section ceases to be useful
when E ≈ 0 anywhere in the device, because the mean value expressions depend strongly
on the evaluation point. Since the electric field appears in the denominator, small
changes around zero will have a large impact. Since these were the explicit forms used
in Section 2.6, we conjecture that we should be able to improve our asymptotic IV
curves simply by numerically evaluating the M integrals. Although we still ignore the
recombination (since we have no estimate on n and p) we can obtain further insights
into the current-voltage characteristics of the device. The results are shown in Figure
2.20.
Since the numerically integrated (green) line in Figure 2.20 still uses the linear field
approximation, the placement of the kink at Vdiff is unsurprising, because this is the
point where E = 0 and thus kd changes form. In order to calculate the kink at the
correct position, we need to better approximate the electric field. However, judging
by the difference between the two asymptotic IV curves, we have little reason to trust
our previous calculations of n and p as we move towards Vdiff = 0. It would thus be
preferable to calculate values of n and p based on the M i integrals, but these only give
us the values of n and p at the points xl and xr. Calculating the right-hand-side of
Poisson’s equation at any other point will require the evaluation of at least one double
integral, and therefore integrating Poisson’s equation twice to calculate the second-order
potential would result in computationally inefficient quadruple integrals (not to mention
then inserting these fields back into the double integrals for the MF !).
86
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Cu
rre
nt
Potential Difference
Numerical Results
Numerically Integrated M
Linear Asymptotics
Figure 2.20: Numerical current-voltage characteristics plotted with the linear asymp-
totics from Section 2.5.1 and numerical integration of the M i integrals (with no recom-
bination). Note that the numerical integration predicts the kink in the IV curve, but at
the wrong position.
Instead, since we are particularly interested in the optimal power point, we observe
that n and p are approximately piecewise linear (consider Figure 2.9). Since we can
calculate n and p at both xl and xr using the M integrals we’ve already evaluated (care
of Equations (2.37) and (2.38)), calculating the linear approximation of n and p and
integrating twice can be done analytically. Using the resulting approximation for E to
again numerically integrate the M i results in the asymptotic curve shown in Figure 2.21.
We refer to these calculations as “second order” because they use the approximations
derived from the original M i to calculate a new value for the electric potential ϕ and
then use this value to calculate the current.
We see that the linear-approximation for n and p allows us to obtain a much more
accurate IV curve in the optimal power point regime, but that we lose accuracy spec-
tacularly as we approach open circuit. In fact, we significantly underestimate the open
circuit voltage, even though it was correctly approximated by both of the asymptotic
curves presented in Figure 2.20. Heuristically, this seems to be expected since n and p
are not linear near VOC. By assuming that n and p were linear, we have thus made sig-
nificant errors in the short circuit and open circuit regimes. At short circuit, the values
87
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nt
Potential Difference
Numerical Results
First Order M
Second Order M
Figure 2.21: Numerical current-voltage characteristics plotted with the first and second-
order numerical integration of the M i integrals. Note that the second-order curve is a
better approximation until Vdiff = 0, but then fails dramatically.
of n and p are nearly constant on each side of the interface (modulo a boundary layer
near the contacts). Thus we at most underestimate n and p by a factor of two. However,
as we move towards open circuit, n and p decay away from the interface exponentially,
and we significantly overestimate them. Thus the field changes more dramatically and
we underestimate the open circuit voltage.
We note also that the necessary approximations for these second order M i integrals
are not trivial. If the field is not constant, then neither is kd,in, and we can no longer
explicitly calculate the equilibrium value of the excitons. Instead of trying to numerically
evaluate X using the new field (resulting in undesirable triple integrals), we use the
explicit version of X (as given in Appendix 2.A.1) using the local value of kd,in(E) at
each point. This undoubtedly creates additional errors, but because the fit in Figure
2.21 is quite good, the error was deemed acceptable. Another option would be to use
the equilibrium values of Xin and Xout (assuming µX → 0), but this leads to a worse
fit. This is most likely related to the exciton mobility discussion in Section 2.6.1.
88
2.8 Obtaining a Unipolar Limit
One of the main goals of considering the full system is to understand the role of the
minority carriers. Ideally we would like to find a limit in which our bulk model asymp-
totically approaches a unipolar model in some limit. The most sensible limits are the
large work function limit and the thin-interface limit. We will see that although the
minority carrier concentrations become quite small in these limits, the minority carrier
current does not. From the drift-diffusion perspective we can take either the concen-
tration or the flux to be zero, but if we take both conditions then the system becomes
overdetermined.
Let us consider what happens if we take the original model equation with no minority
carriers and no bulk exciton dissociation. Since kdX = crnp = 0 in the bulk, we have
that
F (x) =



0 x < xl
∫ x
xl
kd,inX(y)− crn(y)p(y)dy xl ≤ x ≤ xr
F (xL) x > xr
. (2.39)
This allows us to set M1F = 0 and M
3
F = F (xL)M
3.
In particular, we then have the following expression for the currents at x = 0 (in the
hole-material):
Jn0 =
M2Fn + F (xL)M
3
n
M1n +M2n +M3n
Jp0 = −
M2Fp + F (xL)M
3
p
M1p +M2p +M3p
We use these formulas to plot an asymptotic IV curve and once again compare to
the numerical results - see Figure 2.22.
2.8.1 The Small η Limit
Because η must be very small, we can consider the limit η → 0, corresponding to
an infinitely strong work function difference between the two halves of the device. In
particular, this corresponds to a device in which it is physically impossible for electrons
to pass into the hole material and for holes to pass into the electron material.
89
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ur
re
nt
Potential Difference
Numerical Results
Asymptotic IV for no bulk exciton dissociation
Figure 2.22: The IV curve obtained from our original numerical simulations plotted
with the (lowest-order) M i asymptotic IV curve for kd,out = 0. Note that the fit is
very good, suggesting that dynamics in the bulk do not contribute to the IV curve. In
fact, because the asymptotic calculations do not consider recombination, this suggests
that our numerical IV curve is relatively independent of recombination – an unphysical
result for the bulk model. (Note, however, that this exactly matches some of the unipolar
models given in the literature [5].)
90
This has the following effect on the M orders from the previous section:
ηM1n = O(δ
−1
2 δ
−1
1 ) +O(δ
−1
2 ), M
1
p = 0,
ηM2n = O(δ
−1
2 ), M
2
p = O(δ2)
ηM3n = 0, M
3
p = O(1) +O(δ2).
For the MF (using Equation 2.39), we have:
ηM1Fn = 0, M
1
Fp = 0,
ηM2Fn = O(δ
−1
2 ), M
2
Fp = O(δ2),
ηM3Fn = 0, M
3
Fp = F (xL)M
3
p .
We now have:
Jn0 =
M2Fn
M1n +M2n
+O(η)
Jp0 = −
M2Fp + F (xL)M
3
p
M2p +M3p
+O(η).
At short circuit, we have that δ1, δ2  1, and thus:
JSSn = 0 +O(δ1) +O(δ2) +O(η)
JSSp = −F (xL) +O(δ2) +O(η)
JSS ≈ −F (xL) (2.40)
which exactly matches our calculation in Equation (2.32) when kd,out = 0. In addition,
to lowest order we have that Jn(x0) = 0, matching the intuition that there should be no
electrons (and thus no electron current) in the p-material.
However, for E > 0, we have δ1, δ2 > 1, and in forward bias we expect δ1, δ2  1. In
this case, we obtain
JFBn = O(1) +O(δ
−1
1 ) +O(δ
−1
2 )O(η)+
JFBp = −O(1) +O(δ2) +O(η).
Even with an infinitely strong work function difference, we have an O(1) electron current
in the hole material!
Although physically unrealistic, this effect is not necessarily surprising from a reaction-
diffusion standpoint. Because of the quasi-positivity of the right-hand side (kdX−crnp),
91
we generally expect that n and p will be positive everywhere. However, for a realistic
model of the work function, we expect that the concentrations n and p and their deriva-
tives ∇n and ∇p will be small on the minority side of the interface, even at open circuit.
In fact, we see this qualitatively from the numerics in Figure 2.23.
 0
 50
 100
 150
 200
 250
 300
 350
 400
 0.49  0.495  0.5  0.505  0.51
 0
 50
 100
 150
 200
 250
 300
Ho
le 
Co
nc
en
tra
tio
n
El
ec
tro
n C
on
ce
ntr
ati
on
Position
Figure 2.23: Plots of the numerically calculated values of n and p across the interface at
open circuit with the usual parameters given in Table 2.1.
This is not the entire story. Although the minority carrier concentrations are expo-
nentially suppressed at the edge of the interface, we know from Section 2.6 (Figures 2.7
and 2.8) that the current in the device is relatively independent of artificial constraints
on the concentration values. In Figure 2.24 we plot the numerically calculated concen-
tration for n in the p-type region at open circuit. We see that although the electrons
are exponentially suppressed, we still have an electric current of Jn ≈ 100. This electric
current counteracts the hole current of Jp ≈ −100 resulting in a thermodynamically
unrealistic open circuit voltage. We discuss this result in the thin-interface limit regime
in the next sections (specifically, see Figure 2.25 and the surrounding discussion).
Although a strict unipolar model will force n = Jn = 0 in the p-region, it will also
mathematically violate the conservation of current. This can be heuristically described
by assuming that the current at the interface entirely recombines (even if this is not
included in the mathematical model [5]). However, neither solution suggests compatible
mathematical and physical conditions for a unipolar model at open-circuit. Since the
92
 0
 0.2
 0.4
 0.6
 0.8
 1
 1.2
 1.4
 1.6
 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45
El
ec
tro
n C
on
ce
ntr
ati
on
Position
Minority Carrier Electrons
Figure 2.24: Plot of the numerically calculated value of the minority carrier n in the
p-type region at open circuit with the usual parameters given in Table 2.1. Note that
although n is several orders of magnitude smaller than p, it is not negligible and generates
an appreciable current.
large work function limit does not give us a reasonable unipolar model, we proceed to
consider the zero-interface limit.
2.8.2 Zero-Interface Limit
An interesting question is what limit our system approaches if we take the width of the
interface to zero.If we require that the behavior of the device in the bulk is independent
of size of the interface, then we find several necessary conditions.
Since the potential in the device is bounded, we immediately see that M2 → 0 in
the limit. The integrand is bounded and the interval of integration goes to zero. We
have to be more careful with the MF . The forcing integral F (x) is extremely important
for the characteristics of the device. For instance, from Equation 2.40 we calculate the
short circuit current Jtot ≈ F (xL) at strong reverse-bias when E  1. Since the short
circuit current should not depend on the choice of interface width in the model, we must
require that F (xL) remains constant as  → 0. Since F (xl) and F (xL) − F (xr) will
remain nearly constant based on our bulk independence assumption, we conclude that
F (x) will have a jump at the new infinitesimal interface.
93
Since F (x) is the integral of the right-hand-side (kdX − crnp), we conclude that
either kdX or crnp (or both) must weakly converge to a Dirac delta in the limit. This
is to be expected for kdX, since polaron pairs will only exist on the interface, and thus
if the total number of polaron pairs
∫
I X is conserved as the size of the interface goes
to zero, we expect that X ⇀ Cδ. Note that this separates the polaron pair density
from the exciton density, and if exciton dissociation in the bulk is non-negligible, we will
retain the usual kd,outX term in the bulk. For the crnp term, we expect to end up using
one-sided limits (from opposite sides) for n and p, corresponding to the dominance of
the majority carrier (which we quantify in Equation (2.41)).
Since the zero-interface limit F (x) is still bounded, we also have that M2F → 0 in
the limit. With these two approximations, many of our equations become significantly
simpler. As required in our assumptions, JSS from the previous section is completely
unchanged in the limit, since M2 and M2F are never dominant at short circuit.
The Jump in n and p
We can also calculate the ratio of n(xl) and p(xl) to n(xr) and p(xr). Starting from
Equations (2.37) and (2.38), we have:
n, p(xr)
n, p(xl)
= e±(ϕ(xr)−ϕ(xl))
(M1 +M2)M3F −M
3(M1F +M
2
F )
M1(M2F +M
3
F )− (M
2 +M3)M1F
= η∓1
M1M3F −M
3M1F
M1M3F −M
3M1F
= η±1. (2.41)
Recall that η = eϕ(xr)−ϕ(xl), and thus that in the limit, η is the jump in the potential
between the left and right side of the interface. Physically, the electric potential should
remain continuous, so the jump in ϕ will be due entirely to the work function gap:
ϕ(xr) − ϕ(xl) = ∆U . Thus η represents exactly the exponential of the difference in
energies and therefore the concentration jump predicted by Boltzmann statistics. We
therefore conclude that for the usual organic photovoltaic bilayers (where a large work
function gap exists between the materials and exciton dissociation in the bulk of the
device is negligible) the unipolar approximation discussed in Section 2.5.2 is justified.
(The connection with problems illustrated in the previous section is not immediate, but
we demonstrate below that the same issues with the minority carrier current appears.)
94
The Zero-Interface-Width Current
Continuing our approximations, we note that in the narrow-interface limit, we can also
explicitly calculate the open circuit voltage. We begin by assuming that near VOC our
mean value approximations are valid because the field is strictly positive. Note that
this seems reasonable since the asymptotic IV curves shown previously all match the
numerically calculated VOC with reasonable accuracy. In this case, using the results of
Section 2.7.2 (with M2 = M2F = 0) we have:
Φn(xL) =
1− δ−12 + δ
−1
2 η
−1 − δ−12 η
−1δ−11
µn(E)E
Φp(xL) =
δ1ηδ2 − ηδ2 + δ2 − 1
µp(E)E
Fn(xL) =
(
1− δ−12
)
F (xL)
µn(E)E
Fp(xL) = −
(1− δ2)F (xL)
µp(E)E
.
Using Equation (2.36) we have:
Jtot =
(1−δ−12 )F (xL)
µn(E)E
1−δ−12 +δ
−1
2 η
−1−δ−12 η
−1δ−11
µn(E)E
−
− (1−δ2)F (xL)µp(E)E
δ1ηδ2−ηδ2+δ2−1
µp(E)E
=
(
1− δ−12
)
F (xL)
1− δ−12 + δ
−1
2 η
−1 − δ−12 η
−1δ−11
+
(1− δ2)F (xL)
δ1ηδ2 − ηδ2 + δ2 − 1
=
(δ1ηδ2 − δ1η)F (xL)
δ1ηδ2 − δ1η + δ1 − 1
+
(1− δ2)F (xL)
δ1ηδ2 − ηδ2 + δ2 − 1
(2.42)
It is possible to proceed in generality, but the calculations are simpler and more
illuminating for a symmetric device. Thus for notational convenience we show the sym-
metric case here and relegate the non-symmetric case to Appendix 2.A.2. In a symmetric
device, we would expect δ1 = δ2 = δ. In this case, we have
Jtot =
(
δ2η − δη
)
F (xL)
δ2η − δη + δ − 1
+
(1− δ)F (xL)
δ2η − δη + δ − 1
= F (xL)
δη(δ − 1)− (δ − 1)
δη(δ − 1) + (δ − 1)
= F (xL)
δη − 1
δη + 1
This has several very interesting consequences. In reverse-bias it predicts that up to
order O(δη), the current is simply the net number of generated charges, F (xL), again
95
matching the calculations above. If dissociation outside of the interface is negligible, this
exactly matches the approximate current given in Equation (2.32). This reinforces the
intuition of a device in reverse bias pulling all of the charges out of the device. Even for
δ → 1, the device continues to extract all of the generated charges if η is small. This
case (which exists as a removable pole) corresponds to the zero internal-field case which
in Section 2.6 corresponds roughly to the Optimal Power Point.
The Open Circuit Voltage
The next question is how to calculate the open circuit voltage. There are two possibilities,
either F (xL) = 0 or δη − 1 = 0. Whichever one of these points occurs first will be the
observed open circuit voltage in the model. If F (xL) = 0, then Jn = Jp = 0 and we have
a consistent physical picture of an open circuit voltage. The other possibility, δ = η−1
yields a physically unrealistic case where we have equal and opposite electron and hole
currents with no net current.
Let us proceed to calculate the open circuit voltage which corresponds to this unre-
alistic current. Recalling the definition of δ and η, we have:
eϕ(xr) = eϕ(xr)−ϕ(xl) → ϕ(xl) = 0.
Using the definition of ϕ, we conclude that V (xl) = −U(xl). Because we chose ϕ(xL) =
0, we can rewrite the relationship as:
V (xl)− V (xL) = −U(xl) + U(xr).
Finally, if the width of the interface is small with respect to the device length, then
our symmetry assumption allows us to write:
V (x0)− V (xL) = 2 (U(xr)− U(xl))
and given the definition of Vdiff , we calculate the open circuit voltage:
VOC = 2∆U .
Thus we estimate that the open circuit voltage in the device is approximately twice the
work function gap between the materials at the interface. Note that for the value of
∆U = 6 given in Table 2.1, this predicts VOC = 12, which corresponds exactly to the
results in Section 2.6. This indicates that the open circuit voltages considered within do
96
not correspond to physically realistic device operation (as they violate the principle of
detailed balance).
In particular, it indicates that the point where F (xL) = 0 (if it exists) requires a
larger potential difference than the observed Vdiff = 12. Our propositions about the
behavior of the M integrals in the limit presupposed that the behavior of the device
was preserved when taking the thin-interface limit. In particular, we implicitly assumed
that F (xL) would remain constant as we took the interface width to zero. However,
observation of Figure 2.23 suggests that this will not be the case. Integrating F (xL)
over the interface gives significantly less recombination than using the values of the
carriers at the left and right sides of the interface. Furthermore, results from unipolar
models where recombination on the interface is ignored give good qualitative agreement
to our model [5].
We speculate that at open circuit the violation of the conservation of current in the
unipolar model gives precisely the same majority carrier results as the detailed-balance
violation in the two carrier model. Work on establishing this correspondence is ongoing.
Unless an extremely precise model of recombination on the interface is given, we must
either assume a-priori that the electron and hole fluxes at the interface completely cancel
in the form of recombination or accept (large) minority carrier currents.
2.8.3 Possible Routes for Improvement
In the previous two sections, we show that the minority carriers continue to play an
important role even if we take an extremely large work function or an exponentially
thin interface. In particular, an electron current exists in the hole-material, leading to
a physically unrealistic open circuit voltage. In addition, we have shown that increasing
the work function or decreasing the interface size will not correct the flaw.
One possible route for improvement is increasing the bimolecular recombination rate
cr inside the interface or decreasing the generation rate G. Increasing cr by a factor of
100 and decreasing G by a factor of 100 still yields the same problem as before – we
do not reach a point where F (xL) = 0, and we obtain an unphysical VOC. Carefully
choosing coefficients could potentially produce a physical open circuit voltage, but the
coefficients will no longer match physically observed quantities. Abandoning physical
parameters to simulate physical behavior is not a particularly attractive option.
We can also consider the general structure of the variables which would allow a
97
physically realistic open circuit voltage. If we expect minority carrier current to be zero
and continuous across the interface, then not only do we need F (xL) = 0, but we also
require that the right-hand-side of the charge carrier equations is zero. That is, we
require:
(∇ · Jn)(xl) = kd(xl)X(xl)− crn(xl)p(xl) = 0
(∇ · Jp)(xr) = kd(xr)X(xr)− crn(xr)p(xr) = 0
where kd(x) is shorthand for kd
(
E(x)
)
. If these conditions do not hold, then the minority
carrier current cannot be zero on both sides of the interface boundary.
Since we have n(xl), p(xr) 1, we will also require that kdX  1. Our chosen form
of kd(E) does not approach zero for reasonable fields, and thus we will need to enforce
that X(xl), X(xr)  1. However, we model both excitons and polaron pairs with X
and this requirement is not possible within the constraints of our model. One option
would be to enforce X(xl) = X(xr) = 0 as boundary conditions for the polaron pairs on
the interface. This, however, precludes us from considering excitons diffusing into the
interface, because if we specify zero Dirichlet conditions as well as an incoming flux the
problem becomes overdetermined.
From a physical perspective, we could simply ignore the minority carrier current by
assuming that the fluxes at the interface exactly cancel in the form of recombination,
but this is mathematically intractable for a reaction-diffusion interface, resulting in a
violation of the conservation of current. In particular, we conclude that considering
excitons in the bulk and polaron pairs which are free to diffuse within an interface
of non-zero-width does not allow for physically realistic simulation of the open circuit
voltage. Although it might be possible to design an alternative model with diffusion
excitons and a thin interface region where a work function separates electrons and holes,
this would require significant alterations to the model we have discussed here.
Although our model is incapable of describing the open circuit conditions accurately,
it still gives interesting insights into the behavior of the device up through the optimal
power point. In particular, Figure 2.25 shows that the electron current in the p-material
is zero between short circuit and the optimal power point. In particular, since the
asymptotic fit does not include recombination but still fits the numerical curves, this
confirms the usual belief that bimolecular recombination does not play an important
role in a photovoltaic device in the operating regime.
98
-350
-300
-250
-200
-150
-100
-50
 0
 50
 100
 150
-20 -15 -10 -5  0  5  10
C
ur
re
nt
Potential Difference
Total Current
Hole Current
Electron Current
Figure 2.25: Asymptotic IV curve for kd,out = 0 with J in blue (as in 2.22), Jn in gold,
and Jp in red. Note that the current is taken at the left boundary (in the hole material),
so that Jn 6= 0 corresponds to an unphysical current.
99
2.9 Extended 2D Simulations
Although the system presented in Section 2.1.1 is fully 2D, extending the implementation
from the 1D analog case to a full 2D case with a nonlinear interface is quite challenging,
especially if we want to enforce an interface which always has a width of 2nm. Of
additional difficulty is appropriately smoothing the coefficients across the interface in
a way which exactly aligns with the interface implemented by the mesh. We start
this section by discussing the specific difficulties associated with implementing the work
function U . We give a possible route for simplifying these difficulties and then show how
the same technique can be used
The most obvious extension is to create piecewise linear interfaces by extending the
simple smoothing used in Section 2.6. However, this results in severe problems at the
corners of the interface – see Figure 2.26. In particular, since a piecewise quadratic U
will have a discontinuous y derivative at the interface corners, numerical instabilities
occur. In addition, if the partial derivative Uy is nonzero where the interface intersects
with the Neumann boundary, we have unphysical conditions for the flux:
Jn|ΓN = µn∇yn− νnn∇y(U + V )) = µnn∇yU
where the last equality follows from the homogeneous Neumann conditions on n and
V . If U does not also satisfy Neumann boundaries, the electric current through the
Neumann boundaries is not zero. This also prevents us from considering simple bilayers
where the interface is not perpendicular to the Neumann boundaries.
Because the 2D NetGen interface is only directly capable of creating lines and B-
splines, the number of interfaces which can be easily described analytically are quite lim-
ited. However, one can create a nonlinear C1 interface consisting of quarter-circles (the
degenerate case for the B-Spline of three points which form an isosceles right-triangle) –
see Figure 2.27. However, quarter-circles do not provide a particularly satisfying example
since we are unable to change the proportion of the amplitude to the frequency. Instead,
we need to consider meshes with geometries that NetGen is not capable of generating
automatically.
Dynamically Generating U
One possibility for considering other meshes is to consider a sinusoidal mesh. In this
case, we can explicitly sample n points from along the curve and additionally generate n
100
Figure 2.26: Sample numerical results for the hole concentrations for a device with a
non-uniform interface at short circuit. Note that there are significant problems (i.e.
negative concentrations) near the left-facing corners of the interface. These effects arise
from the discontinuous nature of the piecewise quadratic U .
points along the left and right boundaries such that the interface has a constant width
at every sample point (see Appendix 2.B). We can then use these points to define the
edges of a triangulation and develop a new mesh. Because the equidistant points are
difficult to handle explicitly, we then choose to dynamically calculate the value for U .
We can use a diffusion equation
∇ · (D∇U) = 0
with boundary conditions at the edges of the interface region (I) corresponding to the
required maximum and minimum values of U . However, this cannot be done directly
because we need U to be C1 in the whole domain and requiring that U = ∇ηU = 0 on
both sides of I creates an ill-posed problem. Thus we need to solve the dynamic problem
for U on the whole domain. Since we only want a thin transition layer between the two
polymers near the interface, we need to change the diffusion coefficient D throughout the
device. Once again, changing the coefficient discontinuously will create a discontinuity
in the derivative of U , so we need a smooth approximation of a step function near the
interface. If this is chosen carefully, we can finally observe dynamically generated work
101
Figure 2.27: Sample numerical results for the electron and exciton/polaron pair con-
centrations for a device with a non-uniform interface at short circuit. The exciton
concentration demonstrates the shape of the interface. The strange peaks in n appear as
a result of the given C1 form of U and the concentrating effects of the electric field in the
vertical direction. Additional modeling work is necessary to obtain physically realistic
results. Note that away from the interface we see the same nearly-constant behavior
with a boundary layer near the metal contact.
102
functions such as the one shown in Figure 2.28, which uses a Gaussian with extremely
small characteristic length in the x-direction centered on the interface. However, if the
parameters are not carefully chosen the numerically calculated derivatives are not contin-
uous. The careful tuning necessary to produce a valid U makes large-scale investigation
of various interfaces extremely difficult.
Figure 2.28: Dynamically generated form for the work function U in a device with a
sinusoidal interface. The second row depicts the partial derivatives of U with respect to
x and y on the left and right respectively.
103
Exciton Trapping Potential
An interesting benefit is that we can use the method of the previous section to produce an
exciton trapping potential. In order to capture the appropriate behavior of the exciton to
polaron pair transition (as discussed in Sections 2.6.1 and 2.8), we can add a convection
term to our exciton equation:
∂X
∂t
= ∇ · (µX∇X − µX∇U
XX) + c′rnp+GT − kdX − krTX.
where UX is a confining potential for the polaron pairs. Thus polaron pairs tend to
remain on the interface and the physically unrealistic transition from polaron pairs to
excitons in suppressed. In practice, it is rather difficult to estimate a physically appro-
priate form of this function, leaving us free to use a method mirroring the generation of
U . In order to solve the polaron pair to exciton transition problem, we must have that
∇UX = O
(
µ−1X
)
so that the convection contribution is non-negligible.
We can use Poisson’s equation with the same diffusion coefficient used for generating
the work function and a right-hand side which is negative on the interface. Note that
the right-hand side does not need to be continuous, and we can simply take the negative
characteristic function of I – the right-hand-side is -1 on the interface region and zero
outside. This gives a C1 function which we can use to trap the excitons on the interface
– see Figure 2.29. Unfortunately, these functions do not provide particularly satisfying
trapping potentials, because the depth of the well is not equal along the interface. This
results because it is difficult to choose diffusion coefficients which match the interface
exactly and preserve all symmetries. As such, the regions where the interface is approx-
imately parallel to the contacts have slightly lower averaged diffusion coefficients and
thus a more negative trapping potential.
After carefully selecting the necessary coefficients, we can finally obtain results for
a sinusoidal interface in Figure 2.30. Note that for each figure, the 1D cross-sections
appear to be identical to those given in Figure 2.9. We speculate that averaging over the
solution to 1D problems with different interface widths of the device would provide an
excellent approximation to the results (this idea was generated with the authors of [10]
and [73] during a discussion of current projects). The main issue is that the problems
with the work function and exciton-trapping term are not easily solved. In fact, careful
selection of parameters is necessary for the numerical scheme to even converge.
It might be possible to handle these problems within the framework of Netgen and
104
Figure 2.29: Dynamically generated form for the exciton trapping potential in a device
with a sinusoidal interface. Note that the potential does not have a constant depth –
this is a result of the small discrepancies between the theoretical sinusoidal interface and
the versions which preserve the width of the interface.
NGSolve, but the extensive problems suggest that it might be more convenient to simply
choose a new method. Although the programs are open-source and theoretically could
be extended, this does not seem to be a feasible alternative, especially when easier
simulations are possible. This especially holds for the case where we take the interface
width to zero. In such a regime (as considered in the very recent work [20]), most of the
problems presented in this section disappear. In particular, no work function is needed
since there is simply a constant jump in the effective potential across the interface. In the
simplest case, polaron pairs can be taken with fixed position on the interface, but it would
also theoretically be possible to use the Laplace-Beltrami operator to model diffusion of
the polaron pairs along the interface. If the interface is C1, we can furthermore exactly
define the direction of dipole moments.
2.10 Perspective: Proving Existence and Uniqueness
The reduced system of two reaction-convection-diffusion equations for n and p with
Poisson’s equation for V (also called the van Roosbroeck system) has received substantial
105
Figure 2.30: Simulation results for a 2D device with a sinusoidal interface for Vdiff = −3
corresponding to the optimal power point for the flat device (compare to Figure 2.9).
The first row shows the electron and hole concentrations and the second row shows
the electric field in the x and y directions (to the left and right respectively for both
rows). The final image shows the exciton concentration in the device, which shows spikes
corresponding to the lowest points in the trapping potential shown in Figure 2.29.
106
attention as a model for semiconductor devices. For substantial background on such
systems, see Chapter 3 of [74]. Different methods have been used in various contexts
to prove analytic results for systems in increasing generality. The system presented in
Section 2.2 has several key extensions to the systems analyzed in the literature.
For convenience, we recall System (2.8) here (with the tilde notation suppressed):
λ2∇ · (r∇V ) = n− p−
hχI
L
∇νX (2.43a)
∂n
∂t
= ∇ · (µn∇n− µnn∇(U + V ))− crnp+ kdX (2.43b)
∂p
∂t
= ∇ · (µp∇p+ µpp∇(U + V ))− crnp+ kdX (2.43c)
∂X
∂t
= ∇ · (µX∇X) + c
′
rnp+G− kdX − krX. (2.43d)
One obvious extension is the addition of the exciton equation. Although a major
component, the analysis for three drift-diffusion systems instead of two is more or less
identical. We can also use a-priori estimates to show that X will only blow-up if n and
p blow up simultaneously (see Section 2.10.3) and use the same machinery as presented
in the literature [35]. The other structural change is the exciton dipole term in Poisson’s
equation. Assuming we use the same estimates for X as are used for n and p, we will
lose one derivative on the right-hand-side of Poisson’s equation. This can have dramatic
consequences depending on the smoothness necessary for the proofs.
Finally, our given models for R(n, p), µ, and kd do not fit with the technical as-
sumptions required by many analytical techniques. Most vitally, our mobilities do not
saturate for large fields and are therefore a-priori unbounded. Since this requirement is
generally used only technically for the analysis, cutting off µ for some value of the electric
field E never physically expected should be sufficient. The other potential issue with
the Poole-Frenkel [31] mobility is that it is not differentiable for E = 0. However, the
model is based upon experimental data which results from conductance experiments. In
particular, this means that they do not have data-points for E ≈ 0 [31], and it should be
reasonable to smooth µ in this regime. A similar problem may occur for kd, which is also
not differentiable for E = 0. In this case there are physical results for the dissociation at
zero field, but it is unlikely that smoothing slightly in this regime will have any effect.
The constraints on R(n, p) are potentially more limiting. Depending on the method
used for the proofs, an assumption of sublinearity at infinity may be required. For our
expression R(n, p) = crnp this clearly does not hold. However, since the exciton equation
107
contains the same term (up to a constant) with a different sign, it is likely that we could
loosen this requirement, if not abolish it completely – see for instance [35].
A final quality is the dimensionality of the system. Many of the analytical techniques
require a system in 2D. Although this is exactly the case considered for the numerical
simulations presented above, it would be interesting to find a mathematical technique
which was valid up to three dimensions in order to handle more complicated interfaces
(particularly large-feature bulk heterojunction devices).
We now present three possible methods which could be used to establish new analytic
results for the time-dependent organic photovoltaic system (2.8) and the corresponding
steady state system (2.11).
2.10.1 Fixed Point Methods
Since the steady state system presented in Section 2.2.3 is of particular interest, it would
be useful to establish the existence and uniqueness of this steady state. In particular, this
could justify our numerical iteration scheme and quantify the damping results of Section
2.6.4. In addition, if we know that a steady state solution of specific smoothness exists,
it might be possible to prove convergence to the steady state using other techniques. For
the classical semiconductor system, there are extensive results for the steady state using
fixed point methods. Of particular importance is the general fixed point result given in
[55] for solutions in H1(Ω) ∩ L∞(Ω). With several smoothing assumptions, our system
satisfies most of the assumptions contained therein.
The most important missing requirement relates to the doping profile C, which is
replaced by the dipole term hχIL ∇νX in Poisson’s equation. The requirement that C ∈
L∞ in [55] is not easily replicated for the exciton term. If we treat X analogously to n
and p, then we will only have X ∈ H1∩L∞, so∇νX ∈ L2 and we do not have obvious L∞
estimates. This limitation is likely technical and more general existence and uniqueness
theorems for elliptic PDEs (for instance in [34]) would likely provide a solution. The
problem, however, is exasperated by further requirements on the smoothness of V . If
µ(E) and kd(E) are not bounded from above for large E, then we require E ∈ L∞ and
therefore V ∈ W 1,∞. The lack of an H2 → W 1,∞ embedding in two (or more) spatial
dimensions means that even with full elliptic regularity results we will need ∇νX ∈ Lp
for p > n (the spatial dimension).
Work on understanding the general case is ongoing, but at least some smoothing
108
assumptions on µ and kd will likely be necessary to complete the proof. With kd constant
and µX = 0, the exciton equation can be solved explicitly in terms of n and p. In such a
case (with no dipoles and assuming µn and µp are bounded from above), the fixed point
theorem from [55] can be applied directly to the organic model [21]. For more general
results it may be easier to consider the dynamics of the time-dependent structure – we
consider these possibilities in the following.
2.10.2 Entropy Methods
Entropy methods are a powerful tool for the analysis of partial differential equations
[27], with such techniques being applied as early as 1985 for semiconductor equations
[32]. Given specific simplifications to the structure of the equation, one can even use
such entropy methods to prove exponential decay of the time-dependent solution to the
steady state [23].
The first three terms in the entropy are the standard terms used for the semiconductor
equations [23, 32], but we add the corresponding Boltzmann entropy of the exciton term:
F =
∫
Ω
n log
n
n∗
− (n− n∗) + p log
p
p∗
− (p− p∗) +
λ2D
2
(∇V −∇V ∗)2
+
∫
Ω
X log (X/X∗)− (X −X∗), (2.44)
where λ2D := λ
2r.
Theorem 1. Consider a local-in-time solution (V, n, p,X) to System (2.43) with no
dipole term (h = 0), a field-independent dissociation rate (kd = kd(x)), no radiative
losses (cr = c′r), and µ bounded from above. If the entropy functional F given by Equation
(2.44) is bounded, then it stays bounded for all time and thus V ∈ H1 and n, p,X ∈ L1.
Proof. We begin the proof by defining appropriate reference variables. We then show
that the entropy F satisfies:
∂F
∂t
≤ C(1 + F )
for some constant C independent of F and t. This will ensure via Gronwall inequality
that at most the entropy grows exponentially, and therefore remains bounded for all
time.
109
Reference Variables
Because the light generation term has the potential to create excitons (and there-
fore electrons and holes) from an otherwise stable system, we cannot expect exponen-
tial convergence to equilibrium. Instead, we choose arbitrary reference concentrations
n∗, p∗, V ∗ ∈ W 1,∞ which satisfy the same boundary conditions as n, p, and V . We
require that the derivatives be bounded so that we have J∗n ∈ L
∞.
Finally, we can choose
X∗ =
kd
crn∗p∗
,
so that kdX∗ − crn∗p∗ ≡ 0 in Ω. Note that this requires that crn∗p∗ 6= 0 everywhere,
and in particular, we cannot take zero-boundary conditions. However, the exponentially
small conditions discussed earlier [80] are still appropriate. This assumption is extremely
important because it allows us to copy the analytical techniques for classical semiconduc-
tors. For a photovoltaic device, we expect constant generation via the photogeneration
term G to drive extraction of charges from the device, and thus do not expect a steady
state where kdX∞ − crn∞p∞ = 0 (except in the J = 0 case – see Section 2.8).
Differentiating the Entropy
We differentiate F with respect to t and attempt to simplify and show that ∂F∂t is bounded
by some constant times F . Assuming that we can differentiate under the integral, we
have:
∂F
∂t
=
∫
Ω
nt log
n
n∗
+ pt log
p
p∗
+ λ2D(∇V −∇V
∗) · ∇Vt +Xt log
X
X∗
=
∫
Ω
(∇ · (Jn)− crnp+ kdX) log
n
n∗
+
∫
Ω
(∇ · (Jp)− crnp+ kdX) log
p
p∗
+
∫
Ω
λ2D(∇V −∇V
∗) · ∇Vt
+
∫
Ω
(µX∆X + c
′
rnp+G− kdX − krX) log
X
X∗
.
110
Rearranging, we split the derivative into four terms:
∂F
∂t
= F1 + F2 + F3 + F4
F1 =
∫
Ω
∇ · (Jn) log
n
n∗
+∇ · (Jp) log
p
p∗
F2 = −
∫
Ω
(crnp− kdX)
(
log
n
n∗
+ log
p
p∗
)
F3 =
∫
Ω
λ2D(∇V −∇V
∗) · ∇Vt
F4 =
∫
Ω
(µX∆X + c
′
rnp+G− kdX − krX) log
X
X∗
in order to simplify the notation for the following estimates. We proceed by simplifying
each of these terms before recombining them.
We begin by integrating the first term by parts:
F1 =
∫
∂Ω
Jn|ν log
n
n∗
+ Jp|ν log
p
p∗
−
∫
Ω
Jn · ∇ log
n
n∗
+ Jp · ∇ log
p
p∗
,
where the subscript ν represents a normal derivative. The boundary consists of two
parts, with either Dirichlet or homogeneous Neumann boundary conditions for n, p, and
V . At the Dirichlet boundaries n = n∗ and p = p∗ (by the definition of n∗ and p∗) so
the logarithms are zero. On the Neumann boundaries, Jn|ν = Jp|ν = 0, so the boundary
term is zero. Expanding the gradients, we have
F1 = −
∫
Ω
Jn · ∇ log
n
n∗
+ Jp · ∇ log
p
p∗
= −
∫
Ω
Jn ·
(
∇n
n
−
∇n∗
n∗
)
+ Jp ·
(
∇p
p
−
∇p∗
p∗
)
. (2.45)
The second term we simply rearrange:
F2 = −
∫
Ω
(crnp− kdX) log
np
n∗p∗
. (2.46)
For the third term, again we integrate by parts:
F3 =
∫
∂Ω
λ2D(V − V
∗)∇νVt −
∫
Ω
(V − V ∗)λ2D∆Vt.
The boundary consists of two parts, with either Dirichlet or homogeneous Neumann
boundary conditions for V . On the Dirichlet sections V − V ∗ = 0 (by the definition
111
of V ∗) and on the Neumann sections ∇νVt = 0. Thus the boundary term disappears.
Assuming we can interchange the time derivative with the spatial derivatives,
F3 = −
∫
Ω
(V − V ∗)(nt − pt)
= −
∫
Ω
(V − V ∗)(∇ · (Jn)−∇ · (Jp)).
We can now integrate by parts again:
F3 = −
∫
∂Ω
(V − V ∗)(Jn|ν − Jp|ν) +
∫
Ω
(∇V −∇V ∗) · (Jn − Jp).
On the Dirichlet boundary sections, V − V ∗ = 0, and on the Neumann boundaries the
normal fluxes are by definition zero. Thus the boundary term again cancels, yielding:
F3 =
∫
Ω
(∇V −∇V ∗) · (Jn − Jp). (2.47)
Combining equations (2.45) and (2.47), we have
F1 + F3 =
∫
Ω
Jn
n
· (n∇(U + V )−∇n) +
Jn
n∗
· (−n∗∇(U + V ∗)− n∗∇U +∇n∗)
+
∫
Ω
−
Jp
p
· (p∇(U + V ) +∇p)−
Jp
p∗
· (−p∗∇(U + V ∗)−∇p∗),
where we have added and subtracted both Jn∇U and Jp∇U to better illustrate the
following step. Using the definitions of the current we can simplify this sum:
F1 + F3 =
∫
Ω
−
J2n
µnn
+
JnJ∗n
µ∗nn∗
−
J2p
µpp
+
JpJ∗p
µ∗pp∗
.
We use Young’s inequality on the steady state flux terms:
JJ∗
µ∗u∗
≤
J2
2µ∗u
+
µuJ∗2
2µ∗2u∗2
.
For any chosen  < 2, the negative flux terms will dominate the positive contribution,
reducing the positive quadratic term to only the steady state currents. Choosing  = 1
for convenience, we have:
F1 + F3 ≤
∫
Ω
−
J2n
2µnn
+
µnnJ∗2n
2µ∗2n n∗2
−
J2p
2µpp
+
µppJ∗2p
2µ∗2p p∗2
.
However, by our choice of n∗, p∗, and V ∗ and our assumptions on µ, we have L∞ bounds
on all of the reference variables and µ, so we have:
F1 + F3 ≤
∫
Ω
−
J2n
2µnn
−
J2p
2µpp
+
∥
∥
∥
∥
J∗2n µn
2µ∗2n n∗2
∥
∥
∥
∥
∞
∫
Ω
n+
∥
∥
∥
∥
∥
J∗2p µp
2µ∗2p p∗2
∥
∥
∥
∥
∥
∞
∫
Ω
p.
112
For F4 we can integrate the Laplacian term by parts, yielding:
F4 =
∫
∂Ω
∇νX log
X
X∗
+
∫
Ω
−
µX |∇X|2
X
+
µX(∇X · ∇X∗)
X∗
+
∫
Ω
(crnp+G− kdX − krX) log
X
X∗
.
The boundary term is zero because of the Neumann conditions on X. We once again
use Young’s inequality for the reference flux term, and again choosing  = 1, we have:
F4 ≤
∫
Ω
(crnp+G− kdX − krX) log
X
X∗
−
µX |∇X|2
2X
+
µXX|∇X∗|2
2X∗2
≤
∫
Ω
(crnp+G− kdX − krX) log
X
X∗
+
∥
∥
∥
∥
µX |∇X∗|2
2X∗2
∥
∥
∥
∥
∞
∫
Ω
X.
Noting the similarity of the logarithm term to F4, we consider (temporarily moving
the ‖ · ‖ term for notational convenience):
F2 + F4 − ‖ · ‖ =
∫
Ω
(crnp+G− kdX − krX) log
X
X∗
− (crnp− kdX) log
np
n∗p∗
=
∫
Ω
(G− krX) log
X
X∗
− (crnp− kdX) log
X∗np
Xn∗p∗
=
∫
Ω
(G− krX) log
X
X∗
− (crnp− kdX) log
crnp
kdX
,
where the last equality follows from our definition of X∗.
Closing the Entropy Dissipation Relation
Putting the expressions together, we have:
∂F
∂t
≤
∫
Ω
−
J2n
µnn
−
J2p
µpp
− (crnp− kdX) log
crnp
kdX
+ (G− krX) log
X
X∗
+
∥
∥
∥
∥
J∗2n µn
2µ∗2n n∗2
∥
∥
∥
∥
∞
∫
Ω
n+
∥
∥
∥
∥
∥
µpJ∗2p
2µ∗2p p∗2
∥
∥
∥
∥
∥
∞
∫
Ω
p+
∥
∥
∥
∥
µX |∇X∗|2
2X∗2
∥
∥
∥
∥
∞
∫
Ω
X. (2.48)
The first two terms are clearly strictly negative. The third term is also strictly negative
because the logarithm is monotonic with a zero at the same point as the linear term.
Since all of the L∞ terms are bounded, we can use the Csizsa´r-Kullback inequality:
‖u− u∞‖
2
L1 ≤ C
∫
u log
u
u∞
− (u− u∞)
on the integrals to obtain
∥
∥
∥
∥
J∗2n µn
2µ∗2n n∗2
∥
∥
∥
∥
∞
∫
Ω
n+
∥
∥
∥
∥
∥
µpJ∗2p
2µ∗2p p∗2
∥
∥
∥
∥
∥
∞
∫
Ω
p+
∥
∥
∥
∥
µX |∇X∗|2
2X∗2
∥
∥
∥
∥
∞
∫
Ω
X ≤ CF
113
for some constant C which does not depend on t.
Finally, since 0 < G, kr <∞, we can use the following trivial bounds:
log
X
X∗
≤ log
1
X∗
+X − 1 and −X log
X
X∗
≤
X∗
e
to obtain:
∫
Ω
(G− krX) log
X
X∗
≤
∫
Ω
C + CX
≤ C (|Ω|+ ‖X‖L1)
≤ C + CF
where the equalities result from the aforementioned bounds, the boundedness of the
domain Ω, and the Csiza´r-Kullback inequality.
Using these inequalities with Equation (2.48) yields
∂F
∂t
≤ C(1 + F ). (2.49)
Note that entropy decay cannot be expected because of the energy introduced to the
system by the photogeneration G.
Using a Gronwall-type inequality, we see that:
(
∂F (t)
∂t
− CF (t)
)
≤ C
∂
∂t
(
F (t)e−Ct
)
≤ Ce−Ct
F (t)e−Ct − F (0) ≤
∫ t
0
Ce−Cτdτ =
[
−e−Cτ
∣
∣t
0 = 1− e
−Ct
F (t) ≤ F (t0)e
Ct +
(
eCt − 1
)
and thus for a solution with initial entropy F (t0) bounded, the entropy remains bounded
for all time. Therefore, again by the Csiza´r-Kullback inequality, we have that the L1
norms of n, p, and X are bounded for all time, completing the proof.
Conclusions
We have outlined using entropy methods to control any solutions with initially bounded
entropy. Furthermore, following the argument presented in [35], it is possible to utilize
114
the bounded entropy and full elliptic regularity to obtain L2+ regularity for the 2D
device (with no dipole term). Bootstrap arguments (presented for similar systems in
[23, 35]) should then allow for even better regularity with the feasible goal of solutions
in L∞∩H1 implying boundedness and finite energy. However, the argument is not valid
in 3D and including the dipole term also causes the machinery to break down in 2D (for
the reasons discussed in Section 2.10.1). For these cases existence of solutions is still an
open problem.
One possibility for directly obtaining higher order estimates independent of the spa-
tial dimension is using duality methods.
2.10.3 Duality Methods
Duality methods have shown great promise in deducing L2 estimates directly for reaction-
diffusion systems, especially positive systems with mass control [72]. Our system (mod-
ulo Poisson’s equation), generally fits into this abstract setting if c′r ≤ cr (no carrier
multiplication). In this case, the main difficulty is the mixed boundary conditions as-
sociated with a photovoltaic device. In particular, the boundary conditions of the dual
problem cannot be chosen so that the boundary flux terms in the estimates all disappear.
The basic idea is as follows: given a weak solution to the system, we can take linear
combinations of equations in order to obtain an inequality with only one variable on each
side. Then, by a specific choice of our test function φ, we can obtain estimates which
might otherwise have been quite difficult to obtain. In particular, since the dual method
does not rely on the usual Sobolev embeddings, it is independent of dimension. Here we
show estimates indicating that if blow-up in L2 does occur, it must occur simultaneously
for n, p, and X.
Bounding n and p by X
One direction can easily be calculated using the maximum principle (since n, p, X > 0).
Theorem 2. For any weak solution to System (2.43)
‖n‖L2 ≤ C‖kdX‖L2 and ‖p‖L2 ≤ C‖kdX‖L2.
115
Proof. For u := n or u := p, we can define v such that
ut − Lu = −crnp+ kdX,
vt − Lv = kdX,
(u− v)t − L(u− v) = −crnp ≤ 0.
If we choose v with the same boundary conditions and initial conditions as u, then by
the maximum principle we have u − v ≤ 0 and therefore u ≤ v. Standard parabolic
results on v yield Lp bounds in terms of kdX. Since n and p are nonnegative, we have:
‖n‖L2 ≤ C‖kdX‖L2 ‖p‖L2 ≤ C‖kdX‖L2 (2.50)
where C is a constant depending on the coefficients of the equations (which are functions
of ∇V but not n, p, or X) and the size of the domain.
Bounding X by n or p
The other direction is less straightforward, and we proceed using an argument based on
the dual variable φ.
Theorem 3. If we have L2 bounds on the outgoing current, the initial data, and G for
all time, then for any weak solution to System (2.43)
‖X‖L2 ≤ C (1 + ‖n‖L2) and ‖X‖L2 ≤ C (1 + ‖p‖L2) .
Proof. Let us consider the combination:
(n+X)t −∇ · (µn∇n− µnn∇V ) + µX∆X = −(cr − c
′
r)np+G− krX
≤ G− krX
where the inequality holds because n and p are nonnegative and cr ≥ c′r (not all recombi-
nation yields excitons). We rearrange the terms and proceed to use a duality argument
to bound X by n. Note that the same argument holds for p – only the sign of the
convection term changes.
We proceed as follows:
Xt − µX∆X + krX ≤ −nt +∇ · (µn∇n− µnn∇V ) +G.
116
We multiply the equation by a test-function φ and integrate over the whole domain in
space and up to time T :
∫
ΩT
Xtφ− µX∆Xφ+ krXφ ≤
∫
ΩT
−ntφ+∇ · (µn∇n− µnn∇V )φ+Gφ.
For notational simplicity, we rearrange the left hand side independently using inte-
gration by parts:
∫
ΩT
Xtφ− µX∆Xφ+ krXφ
=
∫
ΩT
−Xφt − µXX∆φ+ krXφ+
[∫
Ω
Xφ
∣
∣
∣
∣
T
0
−
∫
∂ΩT
µXφ∇X · νˆ +
∫
∂ΩT
µXX∇φ · νˆ
=
∫
ΩT
X (−φt − µx∆φ+ krφ) +
∫
Ω
X(T )φ(T )−
∫
Ω
X(0)φ(0) +
∫
∂ΩT
µXX∇φ · νˆ,
where the final identity occurs because X satisfies Neumann boundary conditions on the
domain Ω. Since we have no a-priori knowledge about the future behavior of X, we will
choose a specific test function φ which solves the problem:
−φt − µx∆φ+ krφ = Θ in ΩT
φ(T ) = 0 in Ω ∇φ · νˆ = 0 in ∂Ω
for some Θ ≥ 0. Moving the initial condition integral to the right hand side, our original
inequality now reads
∫
ΩT
XΘ ≤
∫
Ω
X(0)φ(0) +
∫
ΩT
−ntφ+∇ · (µn∇n− µnn∇V )φ+Gφ
Integrating the final integral by parts yields:
∫
ΩT
−ntφ+∇ · (µn∇n− µnn∇V )φ+Gφ
=
∫
ΩT
nφt − (µn∇n− µnn∇V ) · ∇φ+Gφ+
[∫
Ω
nφ
∣
∣
∣
∣
T
0
+
∫
∂ΩT
φ(µn∇n− µnn∇V ) · νˆ
=
∫
ΩT
nφt + n∇ · (µn∇φ) + µnn∇V · ∇φ+Gφ−
∫
Ω
n(0)φ(0) +
∫
∂ΩT
φJn · νˆ −
∫
∂ΩT
µnn∇φ · νˆ
=
∫
ΩT
n (φt +∇ · (µn∇φ) + µn∇V · ∇φ) +Gφ−
∫
Ω
n(0)φ(0) +
∫ T
0
∫
∂ΩD
φJn · νˆ.
The final boundary term is unusual in that it does not exactly cancel. On the Dirichlet
boundary we only expect the current to be zero for specific operating conditions (i.e.
117
specific boundary conditions for V ). Finally, using Ho¨lder’s inequality, we obtain:
∫
ΩT
XΘ ≤‖X(0)− n(0)‖Lp‖φ(0)‖Lq + ‖n‖Lp ‖φt +∇ · (µn∇φ) + µn∇V · ∇φ‖Lq
+ ‖φ‖Lq
∫ T
0
(
‖G(s)‖Lp + ‖Jn · νˆ‖Lp(∂ΩD)
)
ds.
From standard estimates for Parabolic PDEs, we have:
ess sup0≤t≤T
(
‖φ(t)‖H2(Ω)
)
+ ‖φt‖L2(0,T ;H10 (Ω)) ≤ C
(
‖f‖H1(0,T ;L2(Ω) + ‖φ(T )‖H2(Ω)
)
([26], Theorem 7.1.5). This proof assumes that the non-divergence form coefficients are
smooth. In particular, it would require µ ∈ C1 for the divergence case, but this technical
assumption is probably not necessary. In particular, for Θ constant and φ(T ) = 0, we
have:
‖φ(0)‖L2(Ω) + ‖φt‖L2(ΩT ) + ‖∇φ‖L2(ΩT ) + ‖∆φ‖L2(ΩT ) ≤ CΘ.
Inserting these estimates into our inequality we have:
∫
ΩT
XΘ ≤CΘ‖X(0)− n(0)‖L2 + CΘ‖n‖L2 (1 + ‖∇µn‖L∞ + ‖µn‖L∞)
+ CΘ
∫ T
0
(
‖G(s)‖L2 + ‖Jn · νˆ‖L2(∂ΩD)
)
ds
and by duality,
‖X‖L2 ≤ C
(
1 + ‖X(0)− n(0)‖L2 + ‖n‖L2 (1 + ‖µn‖W 1,∞)
)
+ C
∫ T
0
(
‖G(s)‖L2 + ‖Jn · νˆ‖L2(∂ΩD)
)
ds. (2.51)
Therefore, if we have L2 bounds on the outgoing current, the initial data, and G for
all time, then with µn, µp Lipshitz, we have:
‖X‖L2 ≤ C (1 + ‖n‖L2) and ‖X‖L2 ≤ C (1 + ‖p‖L2) (2.52)
where the final constant C depends on T and the outgoing current Jn · ν (as shown
in Equation 2.51).
Combined with the results from above, we see that the L2 norms of n, p, and X can
each be controlled in terms of the others. In particular, if any one of the equations in
the system is shown to be well-posed, Equations (2.50) and (2.52) give well-posedness
for all of the variables.
118
We have presented several results, all of which show some promise for producing more
general results in the future. Relaxing the assumptions for the fixed point or entropy
results could produce interesting results for the full system, especially if they can be
expanded to handle a field-dependent dissociation rate kd(E). Extensions to the duality
estimates could also provide interesting results independent of dimension (unlike the
entropy results presented in Section 2.10.2).
2.A Explicit Calculations
2.A.1 Asymptotic W 1,∞ Bounds for X
We seek to derive bounds for X and Xx which are valid in the asymptotic regime of
Section 2.5.1.
We begin with the exciton equation:
−µXXxx = O()np+G− (kd + kr)X Xx(0) = Xx(xL) = 0.
For notational convenience we define Kout = kd,out + kr,out and Kin = kdin + krin.
Furthermore, G is generally large (O(1/2)) so we assume that G + O()np ≈ G is
constant for the working regime. Splitting the problem into three domains, we have:
Xxx =



Kout
µX
X − GµX x0 < x < xl
Kin
µX
X − GµX xl < x < xr
Kout
µX
X − GµX xr < x < xL
.
Each of these equations can be solved explicitly using hyperbolic trigonometric func-
tions, and using the Neumann boundary conditions allows us to write:
X =



Al cosh
(√
Kout
µX
(x− x0)
)
+ GKout x0 < x < xl
Am cosh
(√
Kin
µX
(x− xl)
)
+Bm sinh
(√
Kin
µX
(x− xl)
)
+ GKin xl < x < xr
Ar cosh
(√
Kout
µX
(x− xL)
)
+ GKout xr < x < xL
.
(2.53)
Using continuity at xl yields:
Al =
1
cosh
(√
Kout
µX
(xl − x0)
)
(
Am +
G
Kin
−
G
Kout
)
.
119
Continuity of the derivative (which follows from allowing only jump discontinuities in
Kout and Kin) yields:
Bm =
√
µX
Kin
Al
√
Kout
µX
sinh
(√
Kout
µX
(xl − x0)
)
=
√
Kout
Kin
tanh
(√
Kout
µX
(xl − x0)
)(
Am +
G
Kin
−
G
Kout
)
≈
√
Kout
Kin
(
Am +
G
Kin
−
G
Kout
)
,
where the last approximation is exact to better than 10−100 since
√
Kout
µX
(xl − x0) >> 0
for typical device values.
The more difficult approximations occur at xr. From continuity we have:
Ar =
Am cosh
(√
Kin
µX
(xr − xl)
)
+Bm sinh
(√
Kin
µX
(xr − xl)
)
+ GKin −
G
Kout
cosh
(√
Kout
µX
(xr − xL)
)
and from continuity of the derivative we have:
Ar =
Am
√
Kin
µX
sinh
(√
Kin
µX
(xr − xl)
)
+Bm
√
Kin
µX
cosh
(√
Kin
µX
(xr − xl)
)
√
Kout
µX
sinh
(√
Kout
µX
(xr − xL)
) .
Here we have a convenient cancellation of all dependence from the length of the right-
hand side as long as the device size is physically reasonable
√
Kout
µX
(xr−xL) << 0. Note
that this expression is negative because xL > xr. The hyperbolic tangent which results
from setting the two equations for Ar equal to each other and dividing the hyperbolic
trigonometric functions from the denominators is equal to -1 within many orders of
magnitude. This yields the following equality:
Am cosh
(√
Kin
µX
(xr − xl)
)
+Bm sinh
(√
Kin
µX
(xr − xl)
)
+
G
Kin
−
G
Kout
=
= −Am
√
Kin
Kout
sinh
(√
Kin
µX
(xr − xl)
)
−Bm
√
Kin
Kout
cosh
(√
Kin
µX
(xr − xl)
)
.
We can further simplify the expression by noting that for our device values
tanh
(√
Kin
µX
(xr − xl)
)
≈ 1
120
with errors on the order of 10−10. Thus we can replace the hyperbolic sine functions by
the equivalent cosines, yielding:
(Am +Bm)
(√
Kin
Kout
+ 1
)
cosh
(√
Kin
µX
(xr − xl)
)
=
G
Kout
−
G
Kin
.
We can now plug our value for Bm into this equation and solve for Am. Algebraic
manipulation yields
Am =
A′m
cosh
(√
Kin
µX
(xr − xl)
) +
G
Kin
(√
Kin
Kout
− 1
)
,
where we define
A′m =
G
Kout
− GKin(
1 +
√
Kin
Kout
)(
1 +
√
Kout
Kin
)
for notational convenience in the following calculations.
We can then immediately evaluate Bm and Al:
Bm =
√
Kout
Kin
A′m
cosh
(√
Kin
µX
(xr − xl)
) +
G
Kin
(
1−
√
Kin
Kout
)
,
Al =
G
Kout
(√
Kout
Kin
− 1
)
+ A
′
m
cosh
(√
Kin
µX
(xr−xl)
)
cosh
(√
Kout
µX
(xl − x0)
) .
Evaluating Ar still requires some simplification and assumptions:
Ar =
A′m +
√
Kout
Kin
A′m tanh
(√
Kin
µX
(xr − xl)
)
+ GKin −
G
Kout
cosh
(√
Kout
µX
(xr − xL)
)
+
G
Kin
(√
Kin
Kout
− 1
)
cosh
(√
Kin
µX
(xr − xl)
)
cosh
(√
Kout
µX
(xr − xL)
)
+
G
Kin
(
1−
√
Kin
Kout
)
sinh
(√
Kin
µX
(xr − xl)
)
cosh
(√
Kout
µX
(xr − xL)
) .
We can simplify the form for Ar by again using that the hyperbolic tangent is nearly
one, and noting the identity (coshx − sinhx) = e−x. Furthermore, we can simplify the
121
following expression:
A′m
(√
Kout
Kin
+ 1
)
+
G
Kin
−
G
Kout
=
G
Kout
(√
Kout
Kin
− 1
)
.
Putting all of this together yields:
Ar =
G
Kout
(√
Kout
Kin
− 1
)
+ GKin
(√
Kin
Kout
− 1
)
e
−
(√
Kin
µX
(xr−xl)
)
cosh
(√
Kout
µX
(xr − xL)
) .
Having explicitly calculated the coefficients A and B, Equation (2.53) now gives an
explicit formulation for the exciton concentration. This expression is long and not very
illuminating, but we can further eliminate small terms.
The hyperbolic secant term in the first expression of Al and the exponential decay
term in Ar are both small (by a factor of about 10−5). Eliminating these terms yields:
X =



G
Kout
(√
Kout
Kin
− 1
) cosh
(√
Kout
µX
(x−x0)
)
cosh
(√
Kout
µX
(xl−x0)
) + GKout x0 < x < xl
A′m cosh
(√
Kin
µX
(x−xl)
)
+
√
Kout
Kin
A′m sinh
(√
Kin
µX
(x−xl)
)
cosh
(√
Kin
µX
(xr−xl)
)
+ GKin
(√
Kin
Kout
− 1
)
e
−
(√
Kin
µX
(x−xl)
)
+ GKin
xl < x < xr
G
Kout
(√
Kout
Kin
− 1
) cosh
(√
Kout
µX
(x−xL)
)
cosh
(√
Kout
µX
(xr−xL)
) + GKout xr < x < xL
. (2.54)
122
We can also write Xx explicitly:
Xx =



G√
µXKout
(√
Kout
Kin
− 1
) sinh
(√
Kout
µX
(x−x0)
)
cosh
(√
Kout
µX
(xl−x0)
) x0 < x < xl
√
Kin
µX
A′m sinh
(√
Kin
µX
(x−xl)
)
+
√
Kout
µX
A′m cosh
(√
Kin
µX
(x−xl)
)
cosh
(√
Kin
µX
(xr−xl)
)
+ −G√
µXKin
(√
Kin
Kout
− 1
)
e
−
(√
Kin
µX
(x−xl)
)
xl < x < xr
G√
µXKout
(√
Kout
Kin
− 1
) sinh
(√
Kout
µX
(x−xL)
)
cosh
(√
Kout
µX
(xr−xL)
) xr < x < xL
.
(2.55)
Estimates
For our asymptotics, we have that Kout > Kin and
(√
Kout
Kin
− 1
)
> 0,
(√
Kin
Kout
− 1
)
< 0.
We conclude that X and |Xx| are strictly increasing on [x0, xl] and strictly decreasing
on [xr, xL]. Therefore we find that the L∞ estimates of X and Xx can be calculated
from the interface region.
From equation (2.53) we can calculate that that the condition Xx = 0 for a maximum
or minimum is achieved if and only if Am tanh
(√
Kin
µX
(x− xl)
)
+Bm = 0. Since hyper-
bolic tangent is a one-to-one function, this equation has at most one solution. Therefore,
since we know that Xx(xl) > 0 and Xx(xr) < 0, X obtains a maximum in the interface
region of the device. Furthermore, since a maximum is obtained, we know that Xxx < 0
and so X < GKin and we have a strict estimate
G
Kout
< X < GKin everywhere.
Using this inequality, we see that Xxx < 0 for xl < x < xr, and thus that the
maximum values for |Xx| occur at the points xl and xr. Since Xx is continuous, we can
easily calculate these values. Putting these results together, we have:
‖X‖L∞ <
G
Kin
, (2.56)
‖Xx‖L∞ =
G
√
µX
(
1
√
Kin
−
1
√
Kout
)
. (2.57)
123
2.A.2 Calculating JOC for Nonsymmetric Devices
For the general non-symmetric case we proceed by combining the two current terms from
Equation (2.42) over a common denominator:
Jtot =
δ1η (δ2 − 1)F (xL)
δ1η (δ2 − 1) + (δ1 − 1)
+
(1− δ2)F (xL)
(δ1 − 1)ηδ2 + (δ2 − 1)
= F (xL)
δ1η2δ2(δ2 − 1)(δ1 − 1) + δ1η(δ2 − 1)2 − δ1η(δ2 − 1)2 − (δ1 − 1)(δ2 − 1)
δ1η2δ2(δ2 − 1)(δ1 − 1) + (δ1 − 1)2ηδ2 + δ1η(δ2 − 1)2 + (δ1 − 1)(δ2 − 1)
= F (xL)
(δ1η2δ2 − 1)(δ2 − 1)(δ1 − 1)
(δ1η2δ2 + 1)(δ2 − 1)(δ1 − 1) + (δ1 − 1)2ηδ2 + δ1η(δ2 − 1)2
Note that the numerator has three apparent zeros. However, with a constant field
approximation, we have a relationship between δ1 and δ2. If we define the symmetry
ratio of the device, α, by xL − xr = α(xl − x0), then we have
δ2 = e
−E(xL−xr) = e−αE(xl−x0) = δα1 .
Inserting this into the expression for the current above gives:
Jtot = F (xL)
(δ1+α1 η
2 − 1)(δα1 − 1)(δ1 − 1)
(δ1+α1 η
2 + 1)(δα1 − 1)(δ1 − 1) + (δ1 − 1)
2ηδα1 + δ1η(δ
α
1 − 1)
2
(2.58)
and we see that both the numerator and denominator are zero for δ1 = 1, corresponding
to zero electric-field in the device. However, the current still exists in the limit. We find
this most easily by considering Jn and Jp separately and applying L’Hopital’s rule:
Jp
F (xL)
=
δ1η(δα1 − 1)
δ1η(δα1 − 1) + (δ1 − 1)
→
η(δα1 − 1) + αηδ
α
1
η(δα1 − 1) + αηδ
α + 1
→
αη
αη + 1
Jn
F (xL)
=
1− δα1
(δ1 − 1)ηδα1 + δ
α
1 − 1
→
−αδα−11
ηδα1 + (δ1 − 1)αηδ
α−1
1 + αδ
α−1
1
→
−α
η + α
putting these together, we have:
JE=0 → F (xL)
(
αη
αη + 1
−
α
η + α
)
= F (xL)
αη2 − α
αη2 + α2η + η + α
and in the small η limit, we have JE=0 → −F (xL) = JSS, identically to the symmetric
device.
Finally, we can calculate the open circuit voltage by setting the final remaining term
in the the numerator of the expression in Equation (2.58) to zero. We have:
δ1+α1 = η
−2
124
By the definitions of δ1, η, and α, we have
eV (xL)−V (xr)+V (xl)−V (x0) = e2(ϕ(xl)−ϕ(xr))
Thus we conclude using the definition of ϕ that
V (xL)− V (x0) = 2(U(xl)− U(xr)) + V (xl)− V (xr).
Since the field is assumed to be constant, in the small interface limit, V (xl) ≈ V (xr)
and thus using the definition of Vapp we recover
VOC = 2∆U
and the results from Section 2.8.2 all continue to hold for a nonsymmetric device.
2.B Coding Implementation Details
The numerical simulations in this chapter were done through modifications to the hybrid
discontinuous Galerkin element code supplied with NETGEN-4.9.14-dev and NGSolve-
4.9.14-dev [78]. In this section I will layout the necessary details to recreate the simula-
tions.
2.B.1 Meshing
The meshing for all of the simulations was done with Netgen (NETGEN-4.9.14-dev)
[78]. Netgen includes built-in functionality for drawing a closed object using lines or B-
splines in 2D. In addition, subdomains of the object can be specified in order to separate
the bulk from the interface. This is particularly useful since any coefficients can be
defined independently on each of the subdomains. Thus for the simulations in Section
2.6 we need only to define the points and lines necessary to create Figure 2.31. We
choose appropriate mesh tolerances to emphasize finer discretization near the interface
and then refine uniformly once obtaining, for instance, Figure 2.5 automatically using
Netgen.
Creating meshes with more complicated geometries is more difficult. As discussed
in Section 2.9, we need an explicit formula for the position of the interface in order to
appropriately calculate the necessary quantities in the modeling. For a mesh created
with B-splines, this is not a trivial task. Instead, since the mesh will be triangulated
125
Figure 2.31: Schematic of the figure used by NetGen [78] to create the meshes shown
in Section 2.6 and 2.9. The first structure yields the mesh given in Figure 2.5, and the
second yields the mesh shown in the simulations of Figure 2.27.
at a specific element-size anyway, we can simply generate the appropriate set of points
on our curve and create line-segments between the points with Netgen. By generating
the appropriate points for the interface and the boundaries of the interface region, we
can create an interface region fitting any functional form we desire. For instance, the
sinusoidal boundary in Figure 2.27.
2.B.2 Solving the PDE
The simulations were all done using the hybrid discontinuous Galerkin (HDG) package
of NGSolve (NGSolve-4.9.14-dev) [78]. We created inherited versions of the built-in
integrators to generate bilinear forms using the nonlinear coefficients of the model. We
then developed a solver which used these integrators iteratively, calculating the change
in the variables between each step and proceeding until either the desired tolerance was
reached or the maximum step count was exceeded.
Modified Integrators
One limitation of Netgen is that it is only able to pass one dependent variable into each
integrator. For the model given in in System (2.12) this is not a stringent limitation
because each term on the left-hand-side depends on at most one other variable. The
convection and diffusion coefficients in the n and p equations depend only on V and
the mass terms, crnp depend only on the other charge carrier. Likewise, the mass term
for the exciton equation again depends only on V . In order to implement these new
126
integrators, we can create a new specialized HDG class which inherits all of the methods
from the built-in HDG method. Because of the symmetry in the n and p equations,
we are only need to create one new class for the charge carriers. Poisson’s equation is
standard and we can use the usual integrators.
For the charge carrier Laplace and convection integrators, we needed to add the
nonlinear mobility depending on the electric field. The integrator calculates the mobility
for each integration point (determined by the order of the system). We can loop over
the degrees of freedom of the system and calculate the electric field at each integration
point using the element vector for V and the known formula for the gradients of the
reference element. This needs to be done twice, once for the bulk degrees of freedom
and once for the facets. Once we have the value of the field, we can calculate the new
value of µ for the bilinear form, completing the modifications.
The mass integrators proceed identically. For the exciton mass integrator we calculate
the local field and use it to calculate the correct value of kd at each integration point.
For the charge carriers integrator we instead calculate the local value of the opposite
charge carrier from the previous time step and use that (scaled by cr) for the coefficient
of the mass term.
For the reasons discussed in Section 2.9, we calculate U and VX as the solution of
individual PDEs before the first step for 2D simulations. Therefore they are treated by
Netgen as dependent variables and converting them into a more useful form is difficult.
Rather than trying to calculate derived quantities in the iteration procedure, we instead
create two global pointers to the two solutions. Although global variables tend to be
a terrible programming technique, we at least minimized the possible issues by never
modifying them once they were set.
Iteration Procedure
For the most part, since our problem has been expressed in an appropriate form, Netgen
will take care of the solution of each stage automatically. There are a few extra steps
necessary to accommodate splitting our bilinear form into two parts, but since the solver
will be linear, we can simply add the two forms together. Once again, the Netgen package
takes care of the necessary details.
127
For each iteration stage k:
ˆ Increment step number and set damping parameter
ˆ Loop through all integration points for every element
– Calculate and store the values of nk−1, pk−1, and ∇xXk−1
ˆ Calculate Vk using the above values for the right-hand-side
ˆ Loop through all integration points for every element
– Calculate and store the values of Ek, nk−1, pk−1, and Xk−1
– Calculate and store the forcing terms: kdXk−1, c′rnk−1pk−1
ˆ Calculate and store the error ‖Vk − Vk−1‖L2 (implemented by Netgen)
ˆ Assemble the left-hand side for n using Vk
ˆ Calculate and store both nk and the error ‖nk − nk−1‖L2
ˆ Repeat previous two steps for p and X
ˆ Loop through all facets for every element
– If the facet lies along the boundary x0
* Calculate and store the values of nk, pk, ∇nk, ∇pk, and Ek
* Add the flux through the facet to a running total of J
ˆ Print the step number and current L2 errors to a file
ˆ Break the loop if the maximum steps or desired tolerance are met
Print whether the iteration was successful or failed
– If it was successful, print the current
– Otherwise print the L2 error from the final step
128
Chapter 3
Graphene
This chapter is the result of collaborative work with Clemens Heitzinger and Peter A.
Markowich. Most of the material appears in our preprint [11]. Professor Markowich was
responsible for the initial direction of the project and Dr. Heitzinger was instrumental
in deciding the final form.
3.1 Introduction
In the past few years, graphene has been receiving significant attention as a 2D material
[2, 61, 62, 63, 75]. It characterizes a number of interesting electronic properties [59],
and much recent work, for example, has focused on creating high frequency electronic
components [54, 98]. In contrast to carbon nanotubes, where in-situ fabrication with
controlled electronic properties is hard to achieve, graphene sheets can be fabricated
relatively easily on substrates, chemically modified, and etched into the desired shape.
Amongst other properties, this leads to interesting topological effects [6, 68].
Many of these interesting properties can be understood by considering the electrons
as 2D Dirac fermions [2, 63, 68]. Although some work has been published addressing the
Dirac equation with electromagnetic fields from a numerical perspective, it has largely
focused on the 3D regime, often using a fluid-dynamics framework to speed computation
[83, 84] or specifically in the semi-classical and non-relativistic regimes [45]. Further work
has been done specifically in 2D, but it has largely focused on illustrating the phenomena
associated with quantum mechanics [88] rather than the simulation of graphene as a 2D
material.
129
In this chapter, we present a convergent 2D finite-difference scheme of second order
in both space and time for the Dirac equation with a static electric potential and a
magnetic field normal to the surface.
The chapter is organized as follows. After a discussion of the Dirac equation and its
scaling in Section 3.2, the numerical scheme is described in Section 3.3. This includes
a convergence proof based on its consistency and stability, and we further demonstrate
how to include a self-consistent model for the electric potential via Gummel iteration. In
Section 3.4, several numerical results are discussed. First the numerical scheme and its
convergence order is verified in specific cases for which we can derive an explicit solution.
We then consider numerical problems that can arise in the numerical approximation of
solutions of the Dirac equation. Finally we consider Veselago lenses, beam splitting, and
the influence of magnetic potentials and self-consistent fields.
3.2 Model Equations
3.2.1 The Scaled Equation
The Dirac equation is usually written
i~
∂ψ
∂t
= (cα¯ · P¯ + βmc2)ψ, (3.1)
where the bar denotes a vector quantity in the spatial dimensions [79]. Here ψ is a wave-
vector with four components. There are several possible choices for the α matrices. The
usual choice uses the Pauli matrices in the top-right and lower-left corner, but the sign
of the second Pauli matrix is not chosen consistently. We choose
αx :=







0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0







, αy :=







0 0 0 −i
0 0 i 0
0 −i 0 0
i 0 0 0







,
αz :=







0 0 1 0
0 0 0 −1
1 0 0 0
0 −1 0 0







, β :=







1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 −1







.
130
Using minimal coupling, we couple the equation to the magnetic field by considering
the kinetic momentum operator
P → −i~∇+
e
c
A,
where A is the magnetic vector potential, and
i~
∂
∂t
→ i~
∂
∂t
− eΦ,
where Φ is the electrical potential. With this substitution, we can explicitly write the
equations as
(
i~
∂
∂t
− eΦ
)







ψ1
ψ2
ψ3
ψ4







=







0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0







(
−i~c
∂
∂x
+ eAx
)







ψ1
ψ2
ψ3
ψ4







−







0 0 0 −i
0 0 i 0
0 −i 0 0
i 0 0 0







(
−i~c
∂
∂y
+ eAy
)







ψ1
ψ2
ψ3
ψ4







−







0 0 1 0
0 0 0 −1
1 0 0 0
0 −1 0 0







(
−i~c
∂
∂z
+ eAz
)







ψ1
ψ2
ψ3
ψ4







+mc2







1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 −1














ψ1
ψ2
ψ3
ψ4







.
131
Writing each component individually, we have
∂ψ1
∂t
=−
(
∂ψ4
∂x
+ i
eAx
mc2
ψ4 − i
∂ψ4
∂y
+
eAy
mc2
ψ4
)
−
(
∂ψ3
∂z
+ i
eAz
mc2
ψ3
)
− i
(
1 +
eΦ
mc2
)
ψ1, (3.2a)
∂ψ2
∂t
=−
(
∂ψ3
∂x
+ i
eAx
mc2
ψ3 + i
∂ψ3
∂y
−
eAy
mc2
ψ3
)
+
(
∂ψ4
∂z
+ i
eAz
mc2
ψ4
)
− i
(
1 +
eΦ
mc2
)
ψ2, (3.2b)
∂ψ3
∂t
=−
(
∂ψ2
∂x
+ i
eAx
mc2
ψ2 − i
∂ψ2
∂y
+
eAy
mc2
ψ2
)
−
(
∂ψ1
∂z
+ i
eAz
mc2
ψ1
)
+ i
(
1−
eΦ
mc2
)
ψ3, (3.2c)
∂ψ4
∂t
=−
(
∂ψ1
∂x
+ i
eAx
mc2
ψ1 + i
∂ψ1
∂y
−
eAy
mc2
ψ1
)
+
(
∂ψ2
∂z
+ i
eAz
mc2
ψ2
)
+ i
(
1−
eΦ
mc2
)
ψ4. (3.2d)
From a physics perspective, we could eliminate the dimensional parameters by going
to a different system of physical units, but this would leave the mass parameter as a
variable. Instead, we rescale space and time to eliminate all of the parameters by setting
t =:
~
mc2
t˜,
(x, y, z) =:
~
mc
(x˜, y˜, z˜).
3.2.2 Dirac Equation in Two Spatial Dimensions
Since Ax and Ay always occur as coefficients of the same component of the wave vector,
we can couple the equations on the xy-plane with a magnetic field in the z-direction.
Constraining the particles to move in the plane and setting Bx = By = 0 yields Az = 0
132
and we find
∂ψ1
∂t˜
= −
(
∂ψ4
∂x˜
+ i
eAx
mc2
ψ4 − i
∂ψ4
∂y˜
+
eAy
mc2
ψ4
)
− i
(
1 +
eΦ
mc2
)
ψ1,
∂ψ2
∂t˜
= −
(
∂ψ3
∂x˜
+ i
eAx
mc2
ψ3 + i
∂ψ3
∂y˜
−
eAy
mc2
ψ3
)
− i
(
1 +
eΦ
mc2
)
ψ2,
∂ψ3
∂t˜
= −
(
∂ψ2
∂x˜
+ i
eAx
mc2
ψ2 − i
∂ψ2
∂y˜
+
eAy
mc2
ψ2
)
+ i
(
1−
eΦ
mc2
)
ψ3,
∂ψ4
∂t˜
= −
(
∂ψ1
∂x˜
+ i
eAx
mc2
ψ1 + i
∂ψ1
∂y˜
−
eAy
mc2
ψ1
)
+ i
(
1−
eΦ
mc2
)
ψ4.
This couples ψ1 with ψ4 and ψ2 with ψ3.
We can also consider the standard representation of the 2D Dirac equation, in which
we use the 2 × 2 Pauli matrices σk instead of the 4 × 4 Dirac matrices. This gives the
equation
(i~
∂
∂t
− eΦ)U = σ1(−i~c
∂
∂x
+ eAx)U + σ2(−i~c
∂
∂y
+ eAy)U +mc
2σ3U
or, after expanding the matrices,
(i~
∂
∂t
− eΦ)
(
u1
u2
)
=
(
0 1
1 0
)
(−i~c
∂
∂x
+ eAx)
(
u1
u2
)
+
(
0 −i
i 0
)
(−i~c
∂
∂y
+ eAy)
(
u1
u2
)
+mc2
(
1 0
0 −1
)(
u1
u2
)
.
Evaluating the last equation by component and dividing by i gives
~
∂u1
∂t
= −(~c
∂
∂x
+ ieAx)u
2 − (−i~c
∂
∂y
+ eAy)u
2 − i(mc2 + eΦ)u1, (3.3a)
~
∂u2
∂t
= −(~c
∂
∂x
+ ieAx)u
1 − (i~c
∂
∂y
− eAy)u
1 + i(mc2 − eΦ)u2. (3.3b)
Dividing by mc2 and scaling x, y, and t as before yields exactly the equations for
ψ1 and ψ4 derived from the 3D Dirac equation constrained to 2 dimensions, i.e., System
(3.2). The two formulations, (3.2) and (3.3), are equivalent except that the 4-component
spinors include a second particle with the same dynamics up to a reflection in y. This
effect is explained in the following in the case of graphene.
133
3.2.3 Derivation of Dirac Model for Graphene
Using the trigonal symmetry of the graphene lattice, it is well known [92] that the
tight-binding model yields the Hamiltonian
E(k) = ±γ
√
3 + 2 cos (
√
3kya) + 4 cos (
√
3kya/2) cos (3kxa/2) (3.4)
assuming symmetric energy levels for convenience [59]. Because each atom has one
free electron, the negative branch of the square root is usually completely filled, and
the minimum energy states will be the minimums of the positive branch. It is easy to
verify that six such minimum momenta K exist, oriented at the corners of the hexagonal
Brillouin zone of graphene. For wave vectors k = K + q, we find
E(k) =
3aγ
2
|q|+O(|q|2).
This dispersion relation is identical to the dispersion relation of the 2D Dirac equation
with the speed of light, c replaced by the velocity vf = 3at/2. With this motivation,
one can derive the Hamiltonian for the system and verify that indeed we obtain the
Dirac Hamiltonian [82]. Thus for the low-energy wave vectors sufficiently close to the
Dirac points K, we can use the 2D Dirac equation to simulate the behavior of electrons
in graphene. Furthermore, depending on which K is chosen, we have time-reversal
symmetry which corresponds exactly to the reflection in y illustrated in Section 3.2.2
[59].
Because of the emphasis on the 2D equation for 2D materials such as graphene, we
restrict our attention to the 2D Dirac equations and call the unknown two-component
wave-vector
u =
(
u1
u2
)
from now on. Note, however, that the treatment below also holds when considering the
coupled system ψ1 and ψ4 constrained to the plane as discussed above.
3.2.4 Adding Poisson’s Equation
To further implement a consistent potential, we need to expand our system of equa-
tions to include an equation for the dependence of the electric potential on u (or ψ).
This can be done in several ways, depending on the degree of accuracy required in the
potential. The most accurate method would be to include Maxwell’s equations for the
134
four-potential, but since the constraint of the particles to 2D is not a-priori relativisti-
cally invariant, this is probably not the most sensible approach. With the constraint of
the particles, we expect that the magnetic field in the x and y direction should be zero,
and thus that the magnetic vector potential term Az will also be zero. If we assume that
this is the case, then the magnetic contribution as a whole from the carriers should be
small, since it will only provide a small counter-field according to Lenz’s law (at least in
the non-relativistic regime). This motivates considering only the electrical potential Φ,
which (either in the non-relativistic regime or by choosing the Coulomb Gauge [48]) is
modeled by Poisson’s equation given the wave-vectors u1 and u2, i.e.,
−∆Φ = −e(|u1|2 + |u2|2) = −e|u|2.
Note that for the 2D Dirac equation, only electrons appear in the system and the
sign on the right-hand side is consistent with this case. In the case of the coupled ψ1
and ψ4 terms, the use of the Poisson equation is an approximation, since the electrons
and positrons cannot be separated in the Dirac equation.
For graphene, the components are decoupled and only two components, namely u1
and u2, are involved for an electron near a single Dirac point. Due to its importance
for 2D materials, we consider this case for the remainder of the paper. However, the
numerical scheme continues to hold for simultaneously simulating the two coupled spinor
systems (ψ1, ψ4) and (ψ2, ψ3) independently. For this case, the two systems couple
implicitly through contributions to the right-hand side of the Poisson equation.
3.2.5 The Main Model Equations
To this point we have not chosen a scaling value for u. Because the continuity equations
for u1 and u2 are independent with respect to rescaling, our continuity equations are
invariant with respect to the scaling
u = λu˜.
However, the coefficient of Poisson’s equation will depend on λ2, and thus our system
can distinguish concentration-dependent effects.
135
For notational convenience, we define the rescaled quantities
A˜ =
eA
mc2
,
V =
eΦ
mc2
,
l2 =
e2
mc20r
,
λ˜2 = λ2l2,
which give the following dimensionless system:
∂u˜1
∂t˜
= −
(
∂u˜2
∂x˜
+ iA˜xu˜2 − i
∂u˜2
∂y˜
+ A˜yu˜2
)
− i(1 + V )u˜1, (3.5a)
∂u˜2
∂t˜
= −
(
∂u˜1
∂x˜
+ iA˜xu˜1 + i
∂u˜1
∂y˜
− A˜yu˜1
)
+ i(1− V )u˜2, (3.5b)
∆V = −λ˜2|u˜|2. (3.5c)
We use System (3.5) for u˜1, u˜2 and V˜ as the main model equations in the remainder of
the chapter, where we will also suppress the ∼ notation for notational convenience.
3.3 The Numerical Scheme
In Section 3.3.1, a finite-difference scheme for System (3.5) is developed for the case of
constant potentials and its convergence is shown. In Section 3.3.2, the numerical ap-
proximation of the self-consistent Dirac-Poisson system is discussed, and scaling suitable
for graphene based devices is discussed finally in Section 3.3.3.
3.3.1 The Finite-Difference Scheme
We split the Dirac equations into two parts and evaluate using a Strang-splitting method
[85]. The second term in both equations is purely rotational and explicitly solvable: the
solution to ∂f∂t = ikf is simply the exponential f(t) = f(0)e
ikt. We thus define the
operator α by
α∆t(u
1, u2)> = (u1e−i(1+V )∆t, u2ei(1−V )∆t)> (3.6)
as one term in our Strang splitting. The other term must account for the coupling of
the equations through the magnetic vector potential and the spatial derivative terms.
136
We implement this term using the Crank-Nicolson method, using centered differences
for the spatial derivatives
u1k+1(xi, yj)− u
1
k(xi, yj)
∆t
= −
1
2
(iAx +Ay)
(
u2k(xi, yj) + u
2
k+1(xi, yi)
)
−
1
4∆x
(
u2k+1(xi+1, yj)− u
2
k+1(xi−1, yj) + u
2
k(xi+1, yj)− u
2
k(xi−1, yj)
)
+
i
4∆y
(
u2k+1(xi, yj+1)− u
2
k+1(xi, yj−1) + u
2
k(xi, yj+1)− u
2
k(xi, yj−1)
)
, (3.7)
u2k+1(xi, yi)− u
2
k(xi, yi)
∆t
= −
1
2
(iAx −Ay)
(
u1k(xi, yi) + u
1
k+1(xi, yi)
)
−
1
4∆x
(
u1k+1(xi+1, yj)− u
1
k+1(xi−1, yj) + u
1
k(xi+1, yj)− u
1
k(xi−1, yj)
)
−
i
4∆y
(
u1k+1(xi, yj+1)− u
1
k+1(xi, yj−1) + u
1
k(xi, yj+1)− u
1
k(xi, yj−1)
)
. (3.8)
Note that we take ∆x = ∆y =: h and define
β∆t,h(u
1, u2) := (u1k+1, u
2
k+1). (3.9)
In addition to being second order convergent in space and time, we also prove below
that this selection is also free of artificial dissipation.
We can also write the equations in matrix form, using the matrices M14 and M41
defined by
M14u2k(xi, yj) =−
1
2
(iAx +Ay)u
2
k(xi, yj)
−
1
4∆x
(
u2k(xi+1, yj)− u
2
k(xi−1, yj)
)
+
i
4∆y
(
u2k(xi, yj+1)− u
2
k(xi, yj−1)
)
,
M41u1k(xi, yj) =−
1
2
(iAx −Ay)u
1
k(xi, yj)
−
1
4∆x
(
u1k(xi+1, yj)− u
1
k(xi−1, yj)
)
−
i
4∆y
(
u1k(xi, yj+1)− u
1
k(xi, yj−1)
)
so that Equations (3.7) and (3.8) can be written as
u1k+1(xi, yj)−∆tM
14u2k+1(xi, yj) = u
1
k(xi, yj) + ∆tM
14u2k(xi, yj),
u2k+1(xi, yj)−∆tM
41u1k+1(xi, yj) = u
2
k(xi, yj) + ∆tM
41u1k(xi, yj).
137
We thus have the matrix equation
(
1 −∆tM14
−∆tM41 1
)(
u1k+1
u2k+1
)
=
(
1 ∆tM14
∆tM41 1
)(
u1k
u2k
)
. (3.10)
for (3.7) and (3.8). Note that M14 and M41 are matrices which act on the discretized
components of u1k and u
2
k.
Our method thus consists of evaluating U = (u1, u2)> and then calculating the next
value Uk+1 by
Uk+1 = α∆t/2 ◦ β∆t,h ◦ α∆t/2Uk. (3.11)
Note that the determinant of the left matrix in (3.10) is 1 + C∆t2 for a constant C
independent of ∆t. Thus for ∆t small enough, the matrix will be invertible and the
method is given explicitly by
Uk+1 =
(
e−i(1+V )
∆t
2 0
0 ei(1−V )
∆t
2
)(
1 −∆tM14
−∆tM41 1
)−1
·
·
(
1 ∆tM14
∆tM41 1
)(
e−i(1+V )
∆t
2 0
0 ei(1−V )
∆t
2
)
Uk. (3.12)
Consistency of the Crank-Nicolson Scheme
In order to establish convergence of the scheme, we start with its consistency.
Proposition 1. The Crank-Nicolson scheme defined by (3.7) and (3.8) is second order
in space and time and thus consistent under the assumption that A is time-independent.
Proof. Beginning from Equations (3.7) and (3.8) we have
∆t∂tu1 + 12∆t
2∂ttu1
∆t
+
O(∆t3)
∆t
= −
1
2
(iAx +Ay)
(
2u2 + ∆t∂tu
2 +O(∆t2)
)
−
1
4∆x
(
4∆x∂xu
2 + 2∆x∆t∂xtu
2 +O(∆x3,∆x∆t2)
)
+
i
4∆y
(
4∆y∂yu
2 + 2∆y∆t∂ytu
2 +O(∆y3,∆y∆t2)
)
,
∆t∂tu2 + 12∆t
2∂ttu2
∆t
+
O(∆t3)
∆t
= −
1
2
(iAx −Ay)
(
2u1 + ∆t∂tu
1 +O(∆t2)
)
−
1
4∆x
(
4∆x∂xu
1 + 2∆x∆t∂xtu
1 +O(∆x3,∆x∆t2)
)
−
i
4∆y
(
4∆y∂yu
1 + 2∆y∆t∂ytu
1 +O(∆y3,∆y∆t2)
)
,
138
where the subscripts u1k(xi, yi) and u
2
k(xi, yi) are implied.
For the terms proportional to ∆t0 we can use the original system (3.5) to cancel
all of the terms. The terms proportional to ∆t satisfy the time derivative of (3.5) and
similarly cancel entirely. For the next order terms, we have
1
6
∆t2∂ttu
1+O(∆t3) = −
1
2
(iAx +Ay)
(
1
2
∆t2∂ttu
2 +O(∆t3)
)
−
1
4
(
∆t2∂xttu
2 +
2
3
∆x2∂xxxu
2 +O(∆x4,∆x2∆t2,∆t3)
)
+
i
4
(
∆t2∂yttu
2 +
2
3
∆y2∂yyyu
2 +O(∆y4,∆y2∆t2,∆t3)
)
,
1
6
∆t2∂ttu
1+O(∆t3) = −
1
2
(iAx −Ay)
(
1
2
∆t2∂ttu
2 +O(∆t3)
)
−
1
4
(
∆t2∂xttu
1 +
2
3
∆x2∂xxxu
1 +O(∆x4,∆x2∆t2,∆t3)
)
−
i
4
(
∆t2∂yttu
1 +
2
3
∆y2∂yyyu
1 +O(∆y4,∆y2∆t2,∆t3)
)
so that the local truncation errors in u1 and u2 are
τ1(x, t) = −
∆t2
12
∂ttu
1 −
∆x2
6
∂xxxu
2 +
i∆y2
6
∂yyyu
2,
τ2(x, t) = −
∆t2
12
∂ttu
2 −
∆x2
6
∂xxxu
1 −
i∆y2
6
∂yyyu
1,
respectively. Thus we see that our Crank-Nicolson scheme is second order accurate in
both space and time and is thus consistent.
Von Neumann Stability of the Crank-Nicolson Scheme
The second step is to establish stability.
Proposition 2. The Crank-Nicolson scheme defined by (3.7) and (3.8) is von Neumann
stable and non-dissipative.
Proof. We must prove that the eigenvalues λg of the matrix g(ξx, ξy) defined by the
ansatz
Uk+1(xp, yq) = g(ξ)Uk(xp, yq) ∀Uk(xp, yq) = (C1, C2)
>ei(hξxp+hξyq)
satisfy the bound |λg| ≤ 1 to show von Neumann stability and furthermore that |λg| = 1
to show that the scheme is non-dissipative.
139
We want to rewrite (3.10) as
Uk+1 =
(
1 −∆tM14
−∆tM41 1
)−1(
1 ∆tM14
∆tM41 1
)
Uk,
but we must show that the first matrix is invertible.
With our given ansatz, we can write the action of M by
M14e
i(hξxp+hξyq) =−
1
2
(iAx +Ay)e
i(hξxp+hξyq)
−
1
4∆x
(
ei(hξx(p+1)+hξyq) − ei(hξx(p−1)+hξyq)
)
+
i
4∆y
(
ei(hξxp+hξy(q+1)) − ei(hξxp+hξy(q−1))
)
=ei(hξxp+hξyq)
(
−
1
2
(iAx +Ay)−
1
4∆x
(
eihξx − e−ihξx
)
+
i
4∆y
(
eihξy − e−ihξy
))
=ei(hξxp+hξyq)
(
−
1
2
(iAx +Ay)−
i
2∆x
sin (hξx)−
1
2∆y
sin (hξy)
)
and similarly
M41e
i(hξxp+hξyq) = ei(hξxp+hξyq)
(
−
1
2
(iAx −Ay)−
i
2∆x
sin (hξx) +
1
2∆y
sin (hξy)
)
.
We can thus equate M14 and M41 with their eigenvalues M14λ and M
41
λ (depending,
of course, on hξx and hξy) and explicitly calculate
(
1 −∆tM14λ
−∆tM41λ 1
)−1
=
1
1−∆t2M14λ M
41
λ
(
1 ∆tM14λ
∆tM41λ 1
)
for all sufficiently small ∆t (such that ∆t2M14λ M
41
λ < 1).
We can thus directly calculate
g(ξx, ξy) =
1
1−∆t2M14λ M
41
λ
(
1 + ∆t2M14λ M
41
λ 2∆tM
14
λ
2∆tM41λ 1 + ∆t
2M14λ M
41
λ
)
. (3.13)
The eigenvalues of this matrix are
λg =
1 + ∆t2M14λ M
41
λ ± 2∆t
√
M14λ M
41
λ
∆t2M14λ M
41
λ − 1
=
(1±∆t
√
M14λ M
41
λ )
2
∆t2M14λ M
41
λ − 1
,
which give us the spectral radius of g(ξx, ξy) and thus |g|.
140
We therefore calculate the product
M14λ M
41
λ =
(
−
1
2
(iAx +Ay)−
i
2∆x
sinhξx −
1
2∆y
sinhξy
)
·
·
(
−
1
2
(iAx −Ay)−
i
2∆x
sinhξx +
1
2∆y
sinhξy
)
=−
1
4
(A2x +A
2
y)−
Ax
2∆x
sinhξx −
Ay
2∆y
sinhξy −
1
4∆x2
sinhξx
2 −
1
4∆y2
sinhξy
2
=−
(
Ax
2
−
sinhξx
2∆x
)2
−
(
Ay
2
−
sinhξy
2∆y
)2
≤ 0.
Since this term is real and non-positive, we have ∆t
√
M14λ M
41
λ = ib for some b ∈ R. Thus
the denominator terms are all strictly negative regardless of the size of ∆t (eliminating
the smallness assumption for the inversion of the first matrix). In addition, we have
|λg| =
∣
∣
∣
∣
(1± ib)2
−b2 − 1
∣
∣
∣
∣ =
|1± ib|2
1 + b2
=
1 + b2
1 + b2
= 1
and our Crank-Nicolson scheme is von Neumann stable for all ∆t, ∆x, and ∆y.
Proposition 3. For a constant magnetic vector potential A, the Crank-Nicolson scheme
defined by equations (3.7) and (3.8) is convergent.
Proof. This statement follows immediately from Propositions 1 and 2.
Consistency of the Splitting Scheme
Proposition 4. The splitting scheme given by (3.11) is consistent for V and A constant
in time and furthermore is second order in space and time.
Proof. To determine the consistency of the splitting scheme, we follow the same pro-
cedure as Section 3.3.1, but must rotate the state vectors both before and after the
Crank-Nicolson step. We define the factors
g1 := e−
i∆t
2 (1+V ) =
(
1−
i∆t
2
(1 + V )−
∆t2
8
(1 + V )2 +O(∆t3)
)
,
1
g1
= e
i∆t
2 (1+V ) =
(
1 +
i∆t
2
(1 + V )−
∆t2
8
(1 + V )2 +O(∆t3)
)
,
g4 := e
i∆t
2 (1−V ) =
(
1 +
i∆t
2
(1− V )−
∆t2
8
(1− V )2 +O(∆t3)
)
,
1
g4
= e−
i∆t
2 (1−V ) =
(
1−
i∆t
2
(1− V )−
∆t2
8
(1− V )2 +O(∆t3)
)
141
for simplicity. We can then write uk+1/g = β∆t,hguk, which follows since g is non-zero.
Since the Crank-Nicolson scheme solves uk+1 = β∆t,huk, we can use Equations (3.7) and
(3.8) with the replacement uk → guk and uk+1 → uk+1/g to find
u1k+1(xi, yj)/g
1 − g1u1k(xi, yj)
∆t
=
= −
1
2
(iAx +Ay)
(
g4u2k(xi, yj) + u
2
k+1(xi, yi)/g
4)
−
1
4∆x
(
u2k+1(xi+1, yj)/g
4 − u2k+1(xi−1, yj)/g
4 + g4u2k(xi+1, yj)− g
4u2k(xi−1, yj)
)
+
i
4∆y
(
u2k+1(xi, yj+1)/g
4 − u2k+1(xi, yj−1)/g
4 + g4u2k(xi, yj+1)− g
4u2k(xi, yj−1)
)
,
u2k+1(xi, yi)/g
4 − g4u2k(xi, yi)
∆t
=
= −
1
2
(iAx −Ay)
(
g1u1k(xi, yi) + u
1
k+1(xi, yi)/g
1)
−
1
4∆x
(
u1k+1(xi+1, yj)/g
1 − u1k+1(xi−1, yj)/g
1 + g1u1k(xi+1, yj)− g
1u1k(xi−1, yj)
)
−
i
4∆y
(
u1k+1(xi, yj+1)/g
1 − u1k+1(xi, yj−1)/g
1 + g1u1k(xi, yj+1)− g
1u1k(xi, yj−1)
)
.
The order O(∆t0) terms are
i(1 + V )u1 + ∂tu
1 =−
1
2
(iAx +Ay)
(
2u2
)
−
1
4
(
4∂xu
2 +O(∆x2)
)
+
i
4
(
4∂yu
2 +O(∆y2)
)
−i(1− V )u2 + ∂tu
2 =−
1
2
(iAx −Ay)
(
2u1
)
−
1
4
(
4∂xu
1 +O(∆x2)
)
−
i
4
(
4∂yu
1 +O(∆y2)
)
,
which are both satisfied to order O(∆x2) +O(∆y2) using system (3.5).
The order O(∆t1) terms are
∆t
2
∂ttu
1+
∆t
2
i∆t(1 + V )∂tu
1 = −
∆t
2
(iAx +Ay)
(
∂tu
2)
−
∆t
2
(
∂xtu
2 +O(∆x2)
)
+
∆t
2
(
i∂ytu
2 +O(∆y2)
)
∆t
2
∂ttu
2+
∆t
2
i∆t(1 + V )∂tu
2 = −
∆t
2
(iAx −Ay)
(
∂tu
1)
−
∆t
2
(
∂xtu
1 +O(∆x2)
)
−
∆t
2
(
i∂ytu
1 +O(∆y2)
)
.
Eliminating the factor of ∆t/2, this is the time derivative of system (3.5) and thus (again
assuming A and V are constant in time) satisfied to order O(∆x2) +O(∆y2). As in the
142
Crank-Nicolson case, the O(∆t2) terms do not cancel, and the method is still locally
second-order accurate in both space and time.
Von Neumann Stability of the Strang Splitting Scheme
Proposition 5. The Strang splitting scheme (3.11) is von Neumann Stable and non-
dissipative.
Proof. The proof follows the same strategy as the proof of Proposition 2. By combining
equations (3.12) and (3.13), we can directly calculate g(ξx, ξy) for the splitting scheme,
yielding
g(ξx, ξy) =
1
1 + b2
(
e−i(1+V )
∆t
2 0
0 ei(1−V )
∆t
2
)(
1− b2 2∆tM14λ
2∆tM41λ 1− b
2
)(
e−i(1+V )
∆t
2 0
0 ei(1−V )
∆t
2
)
=
1
1 + b2
(
(1− b2)e−i(1+V )
∆t
2 2∆tM14λ e
−i(1+V ) ∆t2
2∆tM41λ e
i(1−V ) ∆t2 (1− b2)ei(1−V )
∆t
2
)(
e−i(1+V )
∆t
2 0
0 ei(1−V )
∆t
2
)
=
1
1 + b2
(
(1− b2)e−i(1+V )∆t 2∆tM14λ e
−iV∆t
2∆tM41λ e
−iV∆t (1− b2)ei(1−V )∆t
)
=
e−iV∆t
1 + b2
(
(1− b2)e−i∆t 2∆tM14λ
2∆tM41λ (1− b
2)ei∆t
)
,
where we have once again used the negativity of the product M14λ M
41
λ resulting in the
constant b ∈ R. We can again calculate the eigenvalues of this matrix explicitly to find
λg =
e−iV∆t
1 + b2
(
(1− b2) cos ∆t± i
√
4b2 + (1− b2)2 sin ∆t2
)
and therefore
|λg|
2 =
1
(1 + b2)2
(
(1− b2)2 cos ∆t2 + 4b2 + (1− b2)2 sin ∆t2
)
=
1
(1 + b2)2
(
(1− b2)2 + 4b2
)
=
1
(1 + b2)2
(
1 + 2b2 + b4
)
= 1.
Hence our full splitting scheme is also von Neumann stable for all ∆x,∆y, and ∆t.
If we take the potentials A and V to be constant, then consistency and von Neumann
stability guarantee convergence, and our numerical scheme for the 2D Dirac equation
converges.
143
Theorem 4. The splitting scheme given by (3.11) as implemented in Equation (3.12)
is second order in space and time and convergent for A and V constant in time.
Proof. This statement follows immediately from Propositions 4 and 5.
3.3.2 Adding a Consistent Potential
To solve Poisson’s equation, we use the standard 19-point stencil with modified right-
hand side [33, 93] (taken directly from (14) in the latter):
−8(1 + α2 + β2)V (xi, yi, zi) + (4− α
2 − β2) (V (xi+1, yi, zi) + V (xi−1, yi, zi))
+(4α2 − 1− β2) (V (xi, yi+1, zi) + V (xi, yi−1, zi)) + (4β
2 − 1− α2) (V (xi, yi, zi+1) + V (xi, yi, zi−1))
+
1
2
(1 + α2) (V (xi+1, yi+1, zi) + V (xi+1, yi−1, zi) + V (xi−1, yi+1, zi) + V (xi−1, yi−1, zi))
+
1
2
(1 + β2) (V (xi+1, yi, zi+1) + V (xi+1, yi, zi−1) + V (xi−1, yi, zi+1) + V (xi−1, yi, zi−1))
+
1
2
(α2 + β2) (V (xi, yi+1, zi+1) + V (xi, yi+1, zi−1) + V (xi, yi−1, zi+1) + V (xi, yi−1, zi−1))
=
∆x2
2
(6ρ(xi, yi, zi) + ρ(xi+1, yi, zi) + ρ(xi−1, yi, zi) + ρ(xi, yi+1, zi))
+
∆x2
2
(ρ(xi, yi−1, zi) + ρ(xi, yi, zi+1) + ρ(xi, yi, zi−1))
where
α =
∆x
∆y
, β =
∆x
∆z
,
and the local error is O(h5). We call the solution to this equation γ, i.e., V = γ(ρ).
Note that the stencil is still compact, consisting only of grid points which are imme-
diately adjacent to the center point (horizontally, vertically, and diagonally). Although
the rest of our scheme is only second order, using the fourth order scheme for V does
not increase the computational complexity of the step and was thus deemed to be a
reasonable choice.
Since Poisson’s equation is not time-dependent, we cannot use Strang-type splitting
to evaluate it on half-time steps before and after the splitting scheme from the previous
section. The standard method from semiconductor theory is to solve Poisson’s equation
at the beginning of each time step and then to use this new potential to solve the
144
transport equation [39]. Therefore our self-consistent numerical method is
Vk+1 = γ(ρk),
(u1k+1, u
2
k+1) = α∆t/2 ◦ β∆t,h ◦ α∆t/2(u
1
k, u
2
k, Vk+1),
ρk+1 = −λ
2|uk+1|
2,
where initial wave-vectors u10 and u
2
0 must be specified. (Again, note that for the 2D
Dirac equation u2 and u3 do not appear. For the 3D analog, this would be the equivalent
of setting u20 = u
3
0 = 0.)
3.3.3 Scaling for Graphene Devices
Regarding the simulation of graphene based devices, the equations arising for scalings dif-
ferent from those in Section 3.2 are advantageous. We can scale using characteristic val-
ues from a graphene based device. In order to obtain an equivalent non-dimensionalized
system, the ratio of the distance and time scalings, i.e., L/T , must remain equal to c = vf .
For a device with a characteristic size L = 1µm, this corresponds to a time scaling of
T = 1ps. In graphene, the particles are massless and hence the mass term disappears,
so the scaling term, l2 for the potential is eT~ ≈ 1500.
3.4 Numerical Results
We present and discuss several numerical results obtained from our implementation of
the above numerical scheme in MATLAB. The first results are shown to verify the FD
scheme and its second-order convergence in simple cases with V = Ax = Ay = 0 where
1D explicit solutions are known. Then we proceed by demonstrating the capabilities of
our numerical scheme by applying it to situations with applied electrostatic and magnetic
potentials such as a Veselago lens and a beam splitter.
The first numerical results in this section are performed on a square 100× 100 grid
with the scaling parameters introduced in Section 3.2 and a simulation domain of scaled
size 1 × 1 so that h = ∆x = ∆y = 0.01; the time step is k = 0.01 as well. We
furthermore take periodic boundary conditions in the x and y directions for numerical
simplicity. After demonstrating several test cases, we begin altering the grid size to
investigate the discretization errors and to better implement several physical examples.
145
3.4.1 Wave Equation Results
We first find a suitable wave equation with explicit solutions to compare the numerical
results with. In the case m = V = Ax = Ay = 0, the Dirac equation reduces to
∂u1
∂t
= −
∂u2
∂x
+ i
∂u2
∂y
,
∂u2
∂t
= −
∂u1
∂x
− i
∂u1
∂y
and therefore
∂2u1
∂t2
=
(
−
∂
∂x
+ i
∂
∂y
)(
−
∂
∂x
− i
∂
∂y
)
u1 =
∂2u1
∂x2
+
∂2u1
∂y2
= ∆u1
and analogously ∂
2u2
∂t2 = ∆u
2 holds as well.
However, for the charge density ρ = |u1|2 + |u2|2, we have
∂2ρ
∂t2
=
∂
∂t
(
u1
∂u¯1
∂t
+ u¯1
∂u1
∂t
+ u¯2
∂u2
∂t
+ u2
∂u¯2
∂t
)
=u1
∂2u¯1
∂t2
+ 2
∂u1
∂t
∂u¯1
∂t
+ u¯1
∂2u1
∂t2
+ u¯2
∂2u2
∂t2
+ 2
∂u¯2
∂t
∂u2
∂t
+ u2
∂2u¯2
∂t2
=u1∆u¯1 + 2
(
−
∂
∂x
+ i
∂
∂y
)
u2
(
−
∂
∂x
− i
∂
∂y
)
u¯2 + u¯1∆u1+
+ u¯2∆u2 + 2
(
−
∂
∂x
+ i
∂
∂y
)
u¯1
(
−
∂
∂x
− i
∂
∂y
)
u1 + u2∆u¯2
=∆ρ+ i
∂u2
∂x
∂u¯2
∂y
− i
∂u¯2
∂x
∂u2
∂y
− i
∂u1
∂x
∂u¯1
∂y
+ i
∂u¯1
∂x
∂u1
∂y
,
meaning that the charge density ρ does not solve the 2D wave equation. On the other
hand, in a one dimensional homogeneous case the additional terms cancel and ρ satisfies
the 1D wave equation. We can thus verify our FD scheme and its implementation by
comparing with solutions of the wave equation for which the exact traveling wave solution
is known. We control the initial data and the velocity by taking u2(t = 0) = Cu1(t = 0)
for the y-uniform case, and then ∂ρ∂t (t = 0) = −C
∂ρ
∂x(t = 0) holds. The same procedure
also applies for the x-uniform case after replacing C by −iC. In this manner, we can
set the initial velocity by C and allow for left-traveling, right-traveling, and symmetric
waves for C = 1,−1, 0, respectively.
The corresponding results are shown in Figure 3.1 for the case uniform in y and
in Figure 3.2 for the case uniform in x. It is seen that the numerical solution indeed
approximates the exact solution.
146
 0
 0.05
 0.1
 0.15
 0.2
 0.25
 0.3
 0.35
 0.4
 0.45
 0.5
 0.55
 0  0.2  0.4  0.6  0.8  1
NumericalExplicit
 0
 0.2
 0.4
 0.6
 0.8
 1
 0  0.2  0.4  0.6  0.8  1
NumericalExplicit
 0
 0.2
 0.4
 0.6
 0.8
 1
 0  0.2  0.4  0.6  0.8  1
NumericalExplicit
Figure 3.1: Plots of the charge concentration ρ computed by the FD scheme and of
the exact solution of the 1D wave equation at times t = 0.25, t = 1, and t = 4 with
u2(t = 0) = 0 corresponding to a single traveling wave. Even after 4 periods, the wave
form is nearly identical to the initial concentration.
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
 0  0.2  0.4  0.6  0.8  1
NumericalExplicit
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
 0  0.2  0.4  0.6  0.8  1
NumericalExplicit
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
 0  0.2  0.4  0.6  0.8  1
NumericalExplicit
Figure 3.2: Plots of the charge concentration ρ computed by the FD scheme and of
the exact solution of the 1D wave equation at times t = 0.25, t = 1, and t = 4 with
u2(t = 0) = u1(t = 0) corresponding to a single traveling wave. Even after 4 periods,
the wave form is nearly identical to the initial concentration.
147
In the 2D case, we cannot use the exact solution of the wave equation for comparison.
The rotationally symmetric solution obtained by the FD scheme for u2(t = 0) = 0 is
shown in Figure 3.3.
Figure 3.3: Plots of the charge concentration ρ obtained as the numerical solution of the
Dirac equation at times t = 0.25, 1, and and 4.
Error estimates for the more complicated case m 6= 0 are shown in Figures 3.7 and 3.8
below.
3.4.2 Results for Particles with Mass
In the next step, we consider the Dirac equation in the m 6= 0 regime, while leaving the
other parameters unchanged. Now the complex rotation term is added and the long-
term behavior is completely changed. The initial behaviour of the system is close to
the massless case, but a dragging wave slowly destabilizes the system and the eventual
behavior is completely different.
We can again calculate explicit solutions by generating a second-order PDE and
calculating
∂2u1
∂t2
= −
∂
∂x
(
−
∂u1
∂x
+ iu2
)
− i
∂u1
∂t
=
∂2u1
∂x2
− i
(
−
∂u1
∂t
− iu1
)
− i
∂u1
∂t
=
∂2u1
∂x2
− u1.
A similar calculation gives an identical result for y in place of x. As before in the massless
case, the 2D case contains the additional terms mentioned above and there is no obvious
explicit solution. Figure 3.4 shows the numerical solution; the different behavior at t = 1
and especially at t = 4 is seen compared to Figure 3.3.
148
Figure 3.4: Plots of the charge concentration ρ for the m = 1 regime at times t = 0.25,
t = 1, and t = 4.
The equation in 1D can be solved explicitly using, for instance, separation of vari-
ables, yielding
u1 =
∞∑
n=−∞
C±n e
i(±
√
4pi2n2+1t−2pinx). (3.14)
Thus wave packets move with the dispersion relation
ω(k) =
√
k2 + 1.
In this case, using 5 (symmetric) terms of the Fourier expansion yields a good approxi-
mation of the Gaussian and the explicit solution. The initial data are
u1(t = 0) =
4∑
n=−4
(
C+n + C
−
n
)
e−2piinx,
∂u1
∂t
(t = 0) =
4∑
n=−4
i
√
4pi2n2 + 1
(
C+n − C
−
n
)
e−2piinx.
The sum C+n + C
−
n is calculated by the Fourier transform of the initial Gaussian dis-
tribution, and the equation for the initial velocity allows us to calculate the difference
C+n −C
−
n and thus to determine the coefficients. This explicit solution makes it possible
to verify the numerical scheme including the rotation step.
Figures 3.5 and 3.6 show results for waves traveling in the x- and y-directions, respec-
tively. Figure 3.5 shows an initially traveling wave, whereas Figure 3.6 shows an initially
stationary wave. Both Figure 3.5 and 3.6 show noticeable error at time t = 4. Since the
errors in Figure 3.5 are larger, this case is used to demonstrate that the method indeed
converges numerically – See Section 3.3.
Plots of the residual error for uniform refinements of space and time by a factor of
2 are given in Figure 3.7. It is seen that the error decreases by the expected factor of
149
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
 0  0.2  0.4  0.6  0.8  1
NumericalExplicit
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 0  0.2  0.4  0.6  0.8  1
NumericalExplicit
 0
 0.05
 0.1
 0.15
 0.2
 0.25
 0.3
 0.35
 0.4
 0  0.2  0.4  0.6  0.8  1
NumericalExplicit
Figure 3.5: Plots of |u1|2 for the numerical simulation for the 1D m = 1 regime at times
t = 0.25, t = 1, and t = 4 with the initial condition u2(t = 0) = u1(t = 0) compared to
the explicit solution given by Equation (3.14).
 0
 0.05
 0.1
 0.15
 0.2
 0.25
 0  0.2  0.4  0.6  0.8  1
NumericalExplicit
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 0  0.2  0.4  0.6  0.8  1
NumericalExplicit
 0
 0.05
 0.1
 0.15
 0.2
 0.25
 0.3
 0.35
 0  0.2  0.4  0.6  0.8  1
NumericalExplicit
Figure 3.6: Plots of |u1|2 for the numerical simulation for the 1D m = 1 regime at times
t = 0.25, t = 1, and t = 4 with the initial condition u2(t = 0) = 0 compared to the
explicit solution given by Equation (3.14).
150
≈ 4 for each refinement. The convergence order is quantified with more data by plotting
the maximum value of the absolute value of the error in Figure 3.8. Each dotted line
represents uniform refinement by factors of 2 (in both space and time) for different initial
discretizations. The results in Figure 3.7 correspond to the central dotted line. The bold
line indicates the expected second-order convergence with an arbitrary start point.
-0.1
-0.08
-0.06
-0.04
-0.02
 0
 0.02
 0.04
 0.06
 0.08
 0.1
 0  0.2  0.4  0.6  0.8  1
h=1/1600
h=1/800
h=1/400
h=1/200
h=1/100
Figure 3.7: The error in calculating |u1|2 at time t = 4 for the case in Figure 3.5 for
refined discretizations. Here ∆x = ∆y = ∆t = h = 100, 200, 400, 800, 1600. The error
decreases as the step size is reduced. The final two situations are indistinguishable on
the plot, and hence more comparisons of the absolute error are shown in Figure 3.8.
A Note on Discretization Errors
A common error seen in similar numerical schemes for the Dirac equation is a loss
of rotational symmetry, usually observed as a four-fold symmetric solution with errors
following the directions of the square grid. In particular, if the original Gaussian wave
packet is too tight with respect to the grid size, incorrectly broken symmetry is observed.
In Figure 3.9, numerical approximations of solutions that are radially symmetric are
shown at time t = .5 on a grid of size 100×100 with a time step of 0.01 for three different
initial conditions given by e
−2
(
α (xi−50)
2+(yi−50)
2
1002
)
with α = 5, 10, and 20 (corresponding
151
Log@hD
Log@ErrorD
Figure 3.8: The maximal error in calculating |u1|2 at time t = 4 (as in Figure 3.5) for
refined discretizations as a log-log plot. The expected second-order convergence is shown
as a bold line. The other plots consist of uniformly refining by a factor of 2 both the
grid ∆x = ∆y = h and the time step ∆t = k. The starting points (1/h, 1/k) are (from
top to bottom) the points (50, 200), (50, 100), (50, 50), (100, 50), and (200, 50).
to MATLAB’s gausswin(N,α) with N = 100). In addition, the solutions for the same
initial conditions but with a homogeneous spatial refinement by a factor of 2 and a
time-refinement by a factor of 10 are shown. The spatial refinement greatly reduces the
non-symmetric errors that are seen as four peaks in the ring, whereas reducing the time
step has little effect.
This symmetry-breaking effect can be explained by considering the truncation er-
rors calculated in Section 3.3.1, since a sharply peaked Gaussian will have much higher
second-order spatial derivatives than a broad profile. On the other hand, the change in
time is relatively independent of the profile so that the spatial variations account for the
majority of the error.
It is therefore recommended to use a suitable step size in the spatial discretization in
order to avoid these numerical artifacts. The suitable step size depends on the smooth-
ness of the initial condition.
152
Figure 3.9: Plots of the charge concentration ρ for the numerical simulation for rota-
tionally symmetric initial Gaussian wave packets at time t = .5. For narrower profiles,
we see broken symmetry resulting from discretization errors along the directions of the
discretizations. The columns correspond to Guassians of the form e
−2
(
α (xi−50)
2+(yi−50)
2
1002
)
with α = 5, 10, and 20. The second row shows the results for a 200×200 grid, whereas the
third row shows the results for a time discretization of 0.001. Spatial refinement reduces
the symmetry-breaking errors, whereas temporal refinement has little or no effect.
153
Plane Wave Simulation
We have already calculated the explicit solution for a wave in only one-direction in
Equation (3.14), and we can use this result to write the explicit solution for plane-
waves. As these plane waves are uniform in space, they result in a constant value for the
charge concentration ρ, while u1 and u2 rotate in the complex plane. In order to examine
such solutions more closely, we define the current densities J and the total currents F :
Jx := u¯1u
2 + u1u¯2, Fx =
∫
y
Jxdy
Jy := −iu¯1u
2 + iu1u¯2 Fy =
∫
x
Jydx.
We show plots of Fx and Fy for the simulations shown in Figures 3.2 and 3.5 in Figure
3.10.
Figure 3.10: Plots of the currents Fx (blue) and Fy (green) for the simulations given by
Figure 3.2 and Figure 3.5 respectively. For the massless case the current orthogonal to
the wave-packet remains zero for all time and we see decay in Fy corresponding to the
decay in ρ. For the second case, our original wave packet is not a pure-state and thus
we see mixing in the current. Note that the total current in the domain is conserved,
but that this plot gives the currents evaluated over lines at the center of the domain.
Although we see from Figure 3.7 that errors in |u| occur over time, our method is
very good at conserving currents. In Figure 3.11, plots of the currents Fx and Fy are
shown after 1000 time steps up to time t = 10 in order to demonstrate the stability of
the system regarding the long-time behavior of particles with mass for a uniform plane-
wave. Effects such as these depend greatly on the chosen numerical scheme are are well
known, for instance, in the case of the Schro¨dinger equation [43].
154
Figure 3.11: Plots of the currents Fx (blue) and Fy (green) for plane-wave solutions
to the Dirac equation versus time up to time t = 10 with timestep ∆t = .01 and a
100 × 100 grid. The first image shows a plane wave in the y-direction with associated
momenta of 2pi (to preserve periodicity). The second image shows a zoomed-in version
of Fx indicating that the deviation from zero does not increase over time, but instead
is completely periodic. Note that the scales are different so that the oscillations in the
second image are four orders of magnitude smaller than the constant current in the first
image.
3.4.3 Adding the Electric Potential
We can furthermore add an electric potential V . The most famous effect resulting
from adding a potential to the Dirac equation is the Klein paradox. For the 3D Dirac
Equation, particles of a specific energy regime are reflected from a potential barrier, but
particles of higher and lower energy are transmitted. The 2D Dirac Equation does not
exhibit the same behavior. In fact, the transmission is perfect for all energy barriers at
normal incidence in the 2D Dirac model [59].
In order to demonstrate the effect of the electric potential, we consider two other
interesting examples. First, we investigate the effect of a step potential for rotationally
symmetric initial data. This mirrors the simulation of the effect of a P-N junction on
electrons from a point source. In this case we hence expect the step potential to act as a
a Veselago lens [14] with a symmetric focusing point on the opposite side of the barrier.
Simulations showing the time evolution are given in Figure 3.12 and for potential barriers
of various heights in Figure 3.13.
In the second example, we consider a triangle shaped potential and simulate a beam
splitting potential. We show the potential used and a sequence of plots for various
155
Figure 3.12: Plots of the charge concentration ρ on a 300×300 grid with ∆x = ∆y = 0.01
beginning with a rotationally symmetric Gaussian supported only on the middle 100×100
segment of the domain and set to zero outside. The potential is zero in the left two
thirds of the domain and equal to 20 in the right segment. The solution is shown at
times t = 0.5, t = 0.75, and t = 1, respectively, with a time step of ∆t = 0.01. The
phenomenon of Veselago lensing is clearly seen, whereby the charge concentration to the
right concentrates at a point symmetrically opposite of the point source.
Figure 3.13: Plots of the charge concentration ρ on a 300×300 grid with ∆x = ∆y = 0.01
beginning with a rotationally symmetric Gaussian supported only on the middle 100×100
segment of the domain. The potential is zero for the left two thirds of the domain and
equal to 5, 10, and 20 in the right segment. The solutions are shown at time t = 1 with
a time step of ∆t = 0.01. Veselago lensing is again observed as in Figure 3.12.
156
time steps in Figure 3.14 and show the splitting capability and its dependence on the
magnitude of the potential in Figure 3.15.
Figure 3.14: Plots of the charge concentration ρ on a 200×100 grid with ∆x = ∆y = 0.01
beginning with a traveling wave on the left half of the domain. The potential is shown in
the bottom image. The solutions at times t = 0.5, t = 0.75, and t = 1, respectively, are
shown for a time step of ∆t = 0.01. The original wave packet is completely supported
in the V = 0 region, and the peak coincides with the potential jump at the first time
point, t = 0.5.
Figures 3.12 - 3.15 are shown for the massless case m = 0 to reflect the application
to graphene, but similar results are obtained for particles with mass (see Figure 3.18
below).
3.4.4 Adding the Magnetic Potential
The next component to be considered is the magnetic potential. The main difficulty in
appropriately applying a magnetic vector potential is that the potentials which corre-
spond to constant magnetic fields are not periodic, and thus cause issues for the periodic
boundary conditions of the Crank-Nicolson system. On the other hand, we can easily
simulate toy models with constant Ax or Ay in the analogous 1D case to insure that
157
Figure 3.15: Plots of the charge concentration ρ on a 200×100 grid with ∆x = ∆y = 0.01
beginning with a traveling wave on the left half of the domain (as in Figure 3.1) for
various heights of the potential barrier. The potential is shown in the bottom image from
Figure 3.14 and has magnitude 1, 2, and 5, respectively, for the plots. The solutions are
shown at time at t = 1.5 with a time step of ∆t = 0.01.
they are implemented correctly. For V = Ax = m = 0 uniform in y, we have
∂u1
∂t
= −
(
∂
∂x
+Ay
)
u4,
∂u2
∂t
= −
(
∂
∂x
−Ay
)
u1
and thus for Ay constant,
∂2u1
∂t2
=
(
∂
∂x
+Ay
)(
∂
∂x
−Ay
)
u1 =
∂2u1
∂x2
−A2yu
1. (3.16)
If we take Ay = 1, then we get exactly the same case as Ay = 0, m = 1 above and
we have already obtained the explicit solution in Equation (3.14). We obtain the same
results for V = Ay = m = 0 uniform in x, and can thus do the same simulations as
above to test the consistency of the implementation for A – see Figure 3.16.
3.4.5 Self-Consistent Fields
Note that up to this point we have not included the self-consistent field (V is always
given) and thus have not chosen a value for λ2. We show results for changing this
parameter (the coefficient of ρ in the Poisson equation as discussed in Section 3.2.4) in
Figure 3.17.
Since we consider a 2D Dirac model for confined transport embedded in 3D real
space, the 3D Poisson’s equation is solved with a thin sheet of charge representing the
only contribution to the right-hand side. With this extension, we extend our domain
158
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
 0  0.2  0.4  0.6  0.8  1
Ay=1 m=0Ay=0 m=1
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 0  0.2  0.4  0.6  0.8  1
Ay=1 m=0Ay=0 m=1
 0
 0.05
 0.1
 0.15
 0.2
 0.25
 0.3
 0.35
 0  0.2  0.4  0.6  0.8  1
Ay=1 m=0Ay=0 m=1
Figure 3.16: Plots of |u1|2 for the numerical simulation for the 1D in x massless regime
with Ay = 1 uniformly plotted with the results for the 1D m = 1 regime at times t = 0.25,
t = 1, and t = 4. Note that the two plots are nearly indistinguishable, verifying the
results of Equation 3.16. See Figure 3.5 for plots of the explicit results. The same results
are obtained in y but are not shown here.
Figure 3.17: Plots of the charge concentration ρ and the electric potential V on a
100 × 100 × 19 grid representing a 1 × 1 × 1 box with a charge sheet parallel to the
xy-plane in the center of the box and homogeneous Dirichlet boundary conditions for
the potential at z = 0 and z = 1. The figures use λ2 = 1, 10, 100, and 1000 respectively.
The magnitude of the potential changes proportionally, but there is little impact on the
charge density for the lower two values. Compare to the middle image in Figure 3.4 for
the V = 0 case.
159
from a 2D rectangle to a 3D rectangular prism. In addition, we must now define bound-
ary conditions for Poisson’s equation in the z-direction (corresponding to the gate and
backgate contacts for a graphene based device). Because Poisson’s equation is linear,
any uniform Dirichlet boundary conditions on the top and bottom of the box simply add
a linear contribution to the potential in the z-direction. Thus adding nonzero boundary
conditions at the top and bottom of the box acts in the same manner as adding a given
potential in the plane. This roughly corresponds physically to the difference between
doping a device and selectively applying an external potential. We simulate a beam
splitter with a self-consistent potential in Figure 3.18, again illustrating the effect of
changing λ2.
Figure 3.18: Plots of the charge concentration ρ on a 200×100×9 grid with ∆x = ∆y =
0.01 and Lz = .01. The setup is identical to that in Figures 3.14 and 3.15, except that
the potential is applied at the top of the domain and the potential at the bottom of the
domain is zero. The plots correspond to λ2 = 1, 10, 100, 1000 respectively. The large
discrepancy in the fourth image is a result of the net potential difference being negative
instead of positive. See Figure 3.19.
Decreasing the size of domain in the z-direction has a similar effect to increasing
λ2. Increasing the strength of the electric field in the z-direction reduces the potential
differences in the xy-plane following Gauss’ Law. This effect is illustrated in Figure 3.20.
160
Figure 3.19: A reproduction of the final image from Figure 3.18 (left) and an image
for the splitting case when the splitting potential is negative instead of positive. The
Poisson scaling parameter λ2 = 1000 for the left image and 1 for the right image. note
that the concentrations are very similar, suggesting that the change in behavior observed
in Figure 3.18 occurs because the net potential (from the external potential and the self-
consistent potential) is negative and the particles react accordingly.
Figure 3.20: Plots of the charge concentration ρ and the electric potential V on a
100 × 100 × 19 grid representing a box with a charge sheet parallel to the xy-plane in
the center of the box. The boxes have dimensions 1× 1× 1, 1× 1× 0.1, and 1× 1× 0.01,
respectively, with homogeneous Dirichlet boundary conditions for the potential at the
top and bottom of the box. Reducing the size of the domain counteracts the size of
λ2 = 1000. Compare to Figure 3.17.
161
Chapter 4
Conclusions
4.1 Photovoltaic Devices
Solar energy is an increasingly attractive alternative to fossil fuels and nuclear power.
Classical inorganic semiconductor solar cells have been the mainstay of the industry
for many years, but the more environmentally-friendly alternative of organic devices is
growing more and more popular. Because the efficiency of such devices lags significantly
behind their inorganic counterparts, understanding the simulation of the devices is in-
credibly important. The first step is to understand exactly which model characteristics
are necessary in order to understand the operating characteristics of such a device.
In Chapter 2 we present a self-consistent model which considers independent dynam-
ics for both excitons and polaron-pairs within a device. We develop the model based
on extending, combining, and generalizing models discussed in the current literature.
Of particular importance, we investigate the effect of consistently treating the minority
carriers in the device. We additionally include dipole effects near the interface and dif-
fusion of polaron pairs on the interface of the device – effects not widely considered in
the literature.
We develop a 2D hybrid discontinuous Galerkin finite element scheme to simulate
the model in the steady state and use the simulations to generate qualitatively realistic
current-voltage characteristics. We furthermore develop an asymptotic scheme designed
to accurately describe the behavior of the device in short circuit and optimal power
point conditions. We also show that when the minority carriers are negligible, the PDE
system can be solved explicitly in the bulk for nonzero currents – a result which had not
162
previously appeared in the literature.
We then discuss the physical ramifications of our model with regards to the simu-
lation of actual devices. Our numerical simulations suggest that the consistent dipole
model is not necessary for understanding the physical characteristics of the device. Our
model also gives minority carrier concentrations which are significantly smaller than the
majority carrier concentrations, but this does not guarantee that the minority current
is zero. In fact, we show that a continuous model for charge carriers and polaron pairs
in an interface region develops significant problems exist as the device nears open cir-
cuit. The presence of a minority carrier current corresponds loosely to na¨ıvely allowing
discontinuous currents in unipolar models. We give several conditions for models which
will not exhibit this effect, which seems to be symptomatic of the smoothness enforced
on the solutions by the reaction-diffusion framework.
We also show that a model with an interface region is extremely difficult to model
numerically in a robust way. In particular, the exact two effects which have been largely
ignored in the literature (interface dipoles and polaron pair diffusion) are extremely hard
to generalize for complicated device interfaces. We conclude that numerical models with
a thin interface provide the best opportunity for physically relevant numerical modeling
for organic devices. Such a model would allow for dipole direction to be exactly defined
via the interface normal, and allow for diffusion on the interface (which would be a
manifold of one order lower than the domain).
Finally, we give perspectives on proving the existence and uniqueness of our model
system. We demonstrate how our model fits into the fixed-point methods which have
been used extensively in the mathematical analysis of semiconductor modeling and ex-
plain how dipole-interactions provide an interesting analytical challenge. We then dis-
cuss extending entropy methods to include an exciton equation. We must again ignore
the dipole effects, but we can then develop a non-increasing entropy for the system
with some mild additional assumptions. However, the smoothness provided by such
entropy estimates is insufficient for the bootstrap argument necessary to prove global
existence of reasonable solutions. We suggest that duality methods might provide a
dimension-independent method for obtaining initial regularity results and show some
sample calculations in this direction.
163
4.2 Graphene and the Dirac Equation
The Dirac equation is one of the fundamental equations in physics. It has received
much less attention with regards to numerical methods compared to its closest relative,
the Schro¨dinger equation. This is possibly due to its counter-intuitive properties and
modeling complexity, providing complications for numerical methods. As a new and
interesting 2D material, graphene represents an ideal system for numerical simulations
with the 2D Dirac equation.
In Chapter 3 we develop a finite-difference scheme for the 2D electromagnetic Dirac
equation and we show its convergence by demonstrating its consistency and stability.
Electrostatic and magnetic potentials are included in the equation. The FD approx-
imations are of second order in both space and time. The FD method, including its
convergence order, is verified using several test cases with known results and further
results are shown for cases where no explicit solution is known. We also discuss common
numerical errors which result in a loss of rotational symmetry and explain how to avoid
them.
Having established the capabilities of our numerical scheme, we use our method for
several simulations of graphene. We consider the case of low energies, in which the
particles have wave vectors sufficiently close to the Dirac points and thus the Dirac
equation represents an accurate model system. We discuss the influence of the electro-
static potential, the magnetic potential, and self-consistent simulations. In particular
we demonstrate that our method can simulate potentials which act as Veselago lenses
or beam splitters.
164
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