Upload
others
View
0
Download
0
Embed Size (px)
Citation preview
THE DEVICE PHYSICS OF ORGANIC SOLAR CELLS
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF MATERIALS
SCIENCE AND ENGINEERING
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Timothy M. Burke
May 2015
http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/mq955kd8880
© 2015 by Timothy Matthew Burke. All Rights Reserved.
Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.
ii
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Michael McGehee, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Aaron Lindenberg
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Alberto Salleo
Approved for the Stanford University Committee on Graduate Studies.
Patricia J. Gumport, Vice Provost for Graduate Education
This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.
iii
Abstract
Organic solar cells are photovoltaic devices that use semiconducting plastics as the
active layer rather than traditional inorganic materials such as Silicon. Like any
solar cell, their efficiency at producing electricity from sunlight is characterized by
three parameters: their short-circuit current (Jsc), open-circuit voltage (Voc) and fill
factor (FF ). While the factors that determine each of these parameters are well-
understood for established solar technologies, this is not the case for organic solar
cells. The short-circuit current is much higher than we would expect given the strong
attraction between electrons and holes in organic semiconductors that should lead
to fast recombination, preventing the carriers from being collected as current. In
contrast, the open-circuit voltage is much lower than we would expect based on the
traditional relationship between optical absorption and voltage in inorganic semicon-
ductors. Finally, the fill factor is highly variable from device to device and typically
gets much worse as the cells are made thicker.
In this work we develop a novel and general framework for understanding the
short-circuit current, open-circuit voltage and fill factor of organic solar cells. The
concept that turns out to unify all three aspects of device operation is the idea that
electrons and holes move rapidly enough relative to their lifetimes to equilibrate with
each other in the statistical mechanics sense before recombining. Previously, it had
been thought that such equilibration was impossible because of the low macroscopic
mobilities of charge carriers in organic solar cells.
We first show using Kinetic Monte Carlo simulations that the charge carrier mobil-
ity is 3-5 orders of magnitude higher on short length scales and immediately after light
absorption by comparing simulated results to experimental terahertz spectroscopy
iv
data. Combining this high mobility with experimental lifetime data fully rationalizes
high charge carrier generation efficiency and also explains how carriers can live long
enough to be affected by strong inhomogeneities in the energetic landscape of the
solar cell, which also improves charge generation.
Turning to Voc, we use the same concept of fast carrier motion relative to the re-
combination rate to show that recombination proceeds from an equilibrated popula-
tion of Charge Transfer states. This simplification permits us to develop an analytical
understanding of the open-circuit voltage and explain numerous puzzling Voc trends
that have been observed over the years.
Finally, we generalize our equilibrium result from open-circuit to explain the entire
IV curve and use it to show how the low fill-factor of organic solar cells is not caused,
as is often thought, by a voltage dependent carrier generation process but instead
by low macroscopic charge carrier mobilities and the presence of dark charge carriers
injected during device fabrication.
Taken together, these results represent the first complete theory of organic solar
cell operation.
v
Acknowledgments
No work is done in isolation. I would like to gratefully acknowledge the collabora-
tion and input from both the McGehee and Salleo group memebers, especially Jon
Bartelt, Sean Sweetnam and Eric Hoke. I would further like to thank my advisor
Mike McGehee for his advice and support during my PhD as well as the love and
support of my fiancee, Saumya Sankaran.
vi
Contents
Abstract iv
Acknowledgments vi
1 Introduction 1
1.1 What is an Organic Solar Cell . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Basic Solar Cell Device Physics . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Electrons, Holes and Quasi-Fermi Levels . . . . . . . . . . . . 7
1.2.2 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Quasi-Fermi Levels and Operating Voltage . . . . . . . . . . . 10
1.2.4 Maximum Power Point . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Organic Solar Cell Device Physics . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Charge Transfer States . . . . . . . . . . . . . . . . . . . . . . 12
1.3.2 Polarons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.3 Energetic Disorder . . . . . . . . . . . . . . . . . . . . . . . . 13
2 The Short-Circuit Current 15
2.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Current Understanding and Background . . . . . . . . . . . . . . . . 15
2.3 Core Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 KMC Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 PL Decay Simulation Details . . . . . . . . . . . . . . . . . . . . . . . 28
2.7 Converting Hopping Rates to Mobility Values . . . . . . . . . . . . . 29
vii
2.8 Dependence on Mobility, Lifetime and Morphology . . . . . . . . . . 30
2.9 Dependence of Pesc on Local Mobility and Lifetime . . . . . . . . . . 31
2.10 Dependence of Pesc on Morphology . . . . . . . . . . . . . . . . . . . 33
2.11 The Impact of Energetic Disorder . . . . . . . . . . . . . . . . . . . . 35
2.11.1 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . 35
2.12 Independence from Bulk Mobility . . . . . . . . . . . . . . . . . . . . 38
2.13 Exponential Decay of Photoluminescence . . . . . . . . . . . . . . . . 38
3 The Open-Circuit Voltage 41
3.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Background Information . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 The Temperature Dependence of Voc Leads Us Beyond Langevin Theory 47
3.5 Reduced Langevin Recombination Implies Equilibrium . . . . . . . . 48
3.6 Equilibrium Simplifies the Understanding of Voc . . . . . . . . . . . . 52
3.7 Effects of an Energy Cascade in 3-Phase Bulk Heterojunctions . . . . 56
3.8 The Role of Energetic Disorder . . . . . . . . . . . . . . . . . . . . . 59
3.9 Experimental Observations Explained by the Model . . . . . . . . . . 62
3.10 Explaining the Magnitude of the Voltage Loss . . . . . . . . . . . . . 63
3.11 Opportunities for Improving Voc . . . . . . . . . . . . . . . . . . . . . 66
3.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.13 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.13.1 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . 68
3.13.2 FTPS measurements . . . . . . . . . . . . . . . . . . . . . . . 69
3.14 Why We Expect the CT State Distribution to be Gaussian . . . . . . 70
3.15 Inhomogeneously Broadened Marcus Theory Absorption . . . . . . . 71
3.16 Relating CT State Density and Chemical Potential . . . . . . . . . . 72
3.17 Defining an Effective Density of CT States . . . . . . . . . . . . . . . 75
3.18 The Voltage Dependence of τct . . . . . . . . . . . . . . . . . . . . . . 77
3.19 The Low Temperature Limit of Voc . . . . . . . . . . . . . . . . . . . 78
3.20 The Light Ideality Factor . . . . . . . . . . . . . . . . . . . . . . . . . 79
viii
3.21 The Langevin Reduction Factor . . . . . . . . . . . . . . . . . . . . . 80
3.22 CT State Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.23 The Applicability of Chemical Equilibrium to Electrons and Holes . . 81
3.24 Deriving our Result Directly From the Canonical Ensemble . . . . . . 84
4 The Fill Factor 92
4.1 The Myth of the Intrinsic Organic Solar Cell . . . . . . . . . . . . . . 93
4.2 Why Dark Carriers Matter . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4 The Carrier Distribution in an OPV Device . . . . . . . . . . . . . . 96
4.5 Recombination Away from Open-Circuit . . . . . . . . . . . . . . . . 99
4.5.1 Classifying Recombination Types . . . . . . . . . . . . . . . . 101
4.6 Using These Results to Understand Organic Solar Cells . . . . . . . . 104
4.7 Validating Our Expression Using P3HT:PCBM . . . . . . . . . . . . 104
4.7.1 Correcting for Series Resistance . . . . . . . . . . . . . . . . . 105
4.7.2 Correcting for Shunt Resistance . . . . . . . . . . . . . . . . . 108
4.7.3 P3HT:PCBM Data Fits Our Expression . . . . . . . . . . . . 108
4.7.4 The Photocurrent Term . . . . . . . . . . . . . . . . . . . . . 114
4.7.5 The Built-in Potential . . . . . . . . . . . . . . . . . . . . . . 115
4.7.6 Photocarrier - Dark Carrier Recombination . . . . . . . . . . . 117
4.7.7 Photocarrier - Photocarrier Recombination . . . . . . . . . . . 118
4.7.8 Dark - Dark Recombination . . . . . . . . . . . . . . . . . . . 120
4.7.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.8 Molecular Weight Variations in PCDTBT . . . . . . . . . . . . . . . 121
4.9 Apparent Field Dependent Geminate Splitting . . . . . . . . . . . . . 123
4.9.1 Time Delayed Collection Field Measurements . . . . . . . . . 125
4.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.11 Additional Theoretical Background . . . . . . . . . . . . . . . . . . . 130
4.11.1 Properly Counting States in the Presence of Disorder . . . . . 130
4.11.2 The Link Between Voltage and Carrier Density . . . . . . . . 131
Bibliography 138
ix
List of Tables
2.1 Lifetime and mobility values that were required in previous KMC stud-
ies to predict 90% geminate splitting at short circuit conditions (field
of 105 V/cm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Literature measurements for local mobility (measured using time re-
solved terahertz spectroscopy) and the geminate pair lifetime (mea-
sured using transient absorption or transient photoluminescence). . . 23
2.3 Required local mobilities for 90% field-independent IQE for the speci-
fied device morphologies. . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Conversion of reported hopping rates into local mobility values. . . . 30
2.5 Extracted escape probabilities for mixed regions between 3.2 and 9.6
nm wide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Extracted CT state distribution centers and standard deviations with
experimental Voc measurements for comparison. All raw data except
for RRa P3HT is from literature.[98] . . . . . . . . . . . . . . . . . . 62
3.2 The potential increases that could be obtained from improvements to
each of the material parameters that affects Voc. . . . . . . . . . . . . 66
3.3 Tabulated Langevin Reduction Factors from Literature . . . . . . . . 80
3.4 Reported measurements related to the CT state lifetime in literature 81
4.1 Extracted Photocurrent and Short-circuit Currents for PCDTBT:PCBM
devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2 Extracted Photocurrent and Short-circuit Currents for p−DTS(FBTTh2)2-
PC71BM devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
x
List of Figures
1.1 A schematic view of the molecular and energy landscape of a three
phase organic solar cell showing the pure and mixed regions as well as
the variation in local energy levels among the various phases. . . . . . 4
1.2 (left) A blackbox view of a solar cell, showing reservoirs of electrons
and holes with photoexcitation and recombination pathways (right) A
typical solar cell built using a semiconducting material. . . . . . . . . 5
1.3 (top left) A schematic of a pin device stack showing the electron and
hole contacts and the intrinsic active layer. (top right) An electronic
band structure showing the slope in the electron affinity and ionization
potentials of the active layer caused by the electric field. (bottom) The
electric field and correspond electric potential as a function of position
across the active layer. . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Two example band diagrams showing the quasi-Fermi level for elec-
trons as a blue dashed line and the quasi-Fermi level for holes as a red
dashed line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 (left) Schematic of a BHJ solar cell including the mixed region. Po-
tential shifts in the local energetic landscape at the border between
the donor, mixed and acceptor phases are shown in detail. EA is the
electron affinity, IP is the ionization potential. (right) A 2D schematic
of the Kinetic Monte Carlo simulation method showing the rates for
hopping and recombination. . . . . . . . . . . . . . . . . . . . . . . . 16
xi
2.2 a) The field dependent dissociation of geminate pairs in a mixed region
3.2nm wide with the electron mobility fixed at 4x10−5 cm2/Vs and the
hole mobility varied from 4x10−4 cm2/Vs up to 4 cm2/Vs, τct is fixed at
5 ns. The dashed lines are without an energetic offset, the solid lines
with a 200 meV energetic offset. b and c) The separation distance
evolution between the electron and hole in a typical geminate splitting
simulation with τct =10 ns and µe = µh = 1 cm2/Vs. b ended in
recombination, c in splitting. . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Calibration curve mapping measured geminate pair decay lifetimes to
nearest-neighbor recombination lifetimes produced by simulating gem-
inate separation using KMC and extracting the geminate pair lifetime
as a function of the value of τct input into the simulation for electron
and hole mobilities of 0.01, 0.1, 1 and 10 cm2/Vs (µe = µh). The lines
are a guide to the eye. The horizontal line represents a typical mea-
sured bulk heterojunction CT photoluminescence lifetime of 4 ns.[101] 26
2.4 Variation in geminate splitting is accounted for by variation only in
the product of the carrier mobility and lifetime, not their individual
values. The same data is plotted on semilog and log-log axes to aid
examination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Simulation of geminate splitting for different mixed regions, showing
how each one is fit with a single value for Pesc for all different mobility
and lifetime combinations. The green/red divide shows an upper bound
on splitting efficiency with Pesc = 1. The inset shows the same data
on a linear y-axis when splitting is likely. . . . . . . . . . . . . . . . . 34
2.6 Difference in splitting behavior for a trilayer with a 4.8 nm mixed region
when the mixed region is modeled as a homogenous region and a 50:50
blend of donor and acceptor molecules without disorder. . . . . . . . 36
xii
2.7 A simulation of a single region with 80 meV (FWHM) of Gaussian
disorder in each energy level and the electron held fixed at the origin.
The symbols are the simulated data and the lines are the fit to the data
with our model using a single value of Pesc to explain each morphology,
independent of the mobility and lifetime. . . . . . . . . . . . . . . . . 37
2.8 Simulation of geminate splitting with the bulk mobility artificially re-
duced by a factor of 10,000 (dashed lines with circles) and not reduced
(solid lines with squares), with 80 meV of energetic disorder showing
that bulk mobility does not affect the geminate splitting probability. . 39
2.9 Simulated PL decay curves for a fixed lifetime of 10 ns and various
electron and hole mobilities showing that the decays remain exponential. 40
3.1 (left)The sources of open-circuit voltage losses from the optical gap in
an organic solar cell and various energy levels in the device to which
they correspond. The specific losses for exciton splitting (electron
transfer), the CT state binding energy and free carrier recombination
are based on previous literature reports. The loss due to interfacial dis-
order is presented in this work and the magnitude of the recombination
loss is explained. (right)Schematic band diagram of an organic solar
cell at open-circuit showing the relationship between the quasi-Fermi
levels for electrons (Efn) and holes (Efp), E0 and the open-circuit volt-
age. (Voc). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 (left) Kinetic scheme describing the recombination process in organic
solar cells. (right) The difference in recombination rate and predicted
Voc between the reduced Langevin recombination expression and the
equilibrium approximation as a function of the Langevin Reduction
Factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 Schematic showing how the density of available CT states, gct(E), com-
bined with knowledge of the CT state chemical potential, µct, permits
the calculation of the number of filled CT states, Nct. . . . . . . . . . 54
xiii
3.4 Two example energy diagrams showing a solar cell with and without
an energy cascade between mixed and aggregated phases. . . . . . . . 57
3.5 The carrier density in each phase assuming a IP-IP and EA-EA offset
between the donor and acceptor materials of 150 meV each. . . . . . 58
3.6 Fits to the temperature dependence of Ectexp for MDMO-PPV:PCBM,
P3HT:PCBM and AFPO3:PCBM (1:1 and 1:4 blend ratios). (left) The
extracted Ect and reorganization energies for a blend of regiorandom
P3HT:PCBM showing that they are both linear in 1/T and have very
similar slopes (104.3 meV disorder is extracted from the slope of the
CT State Energy and 104.1 meV for the reorganization energy, fit in-
dependently). (right) The temperature dependent Ect measurements
taken from literature.[98] The data points are the experimental fit pa-
rameters at each temperature and the lines are 1/T fits to the data. . 61
3.7 (left) A 2D schematic showing the effect of CT state delocalization on
the number of CT states in an organic solar cell. Grey circles indi-
cate molecules and dashed lines show different delocalization lengths.
(right) The expected voltage difference (V) between Ect,exp/q and Voc
for a 100 nm thick active layer with a Jsc of 10 mA/cm2. A constant
molecular density of 1021 cm−3 [1 nm−3] is used with 32 CT states per
molecule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.8 Simplified OPV device schematic. . . . . . . . . . . . . . . . . . . . . 84
4.1 (left)An IV curve where recombination is purely described by a single
exponential function, resulting in a device with a high Fill Factor.
(right)A typical IV curve for an organic solar cell, where recombination
is not a simple exponential function of voltage, resulting in a device
with a low Fill Factor and reduced efficiency. . . . . . . . . . . . . . . 93
xiv
4.2 (left) The band diagram of an organic solar cell at equilibrium in the
dark showing how the built-in potential causes a tilt to the energy levels
which leads to carrier accumulation near the contacts of the solar cell.
(right) Schematic dark electron and hole density in an organic solar
cell as a function of position with approximately correct magnitudes
showing how there is a very large carrier density near the two solar cell
contacts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3 The required fermi level and charge carrier density profiles in order to
have a constant current in an intrinsic semiconductor device. . . . . . 98
4.4 The energy bands and quasi-fermi level positions for an organic so-
lar cell at Jsc producing a current of 10 mA/cm2 equally distributed
between an electron and hole current. . . . . . . . . . . . . . . . . . . 100
4.5 The extracted series resistance of each P3HT annealing condition as
a function of device thickness, showing an approximately linear trend
vs. thickness with a annealing temperature dependent slope. . . . . . 106
4.6 The slope of the series resistance vs. thickness curves plotted against
the P3HT hole mobility showing how the series resistance in these
devices appears to be due to transport in pure P3HT regions . . . . . 107
4.7 Experimental data (points) and fit to our expression for P3HT:PCBM
solar cells annealed at 0C. . . . . . . . . . . . . . . . . . . . . . . . . 108
4.8 Experimental data (points) and fit to our expression for P3HT:PCBM
solar cells annealed at 48C. . . . . . . . . . . . . . . . . . . . . . . . . 109
4.9 Experimental data (points) and fit to our expression for P3HT:PCBM
solar cells annealed at 71C. . . . . . . . . . . . . . . . . . . . . . . . . 110
4.10 Experimental data (points) and fit to our expression for P3HT:PCBM
solar cells annealed at 88C. . . . . . . . . . . . . . . . . . . . . . . . . 111
4.11 Experimental data (points) and fit to our expression for P3HT:PCBM
solar cells annealed at 111C. . . . . . . . . . . . . . . . . . . . . . . . 112
4.12 Experimental data (points) and fit to our expression for P3HT:PCBM
solar cells annealed at 148C. . . . . . . . . . . . . . . . . . . . . . . . 113
xv
4.13 The total amount of photocurrent produced in each device in the
P3HT:PCBM annealing series. . . . . . . . . . . . . . . . . . . . . . . 114
4.14 The extracted Vbi parameter for the P3HT:PCBM series. The solid
lines are the actual built-in potential estimated from the crossing point
between light and dark IV curves. The dashed lines are the fit parameters.116
4.15 The photocarrier dark carrier recombination coefficient for our P3HT:PCBM
device series, expressed as the fraction of recombination that proceeds
via this mechanism at the maximum power point. . . . . . . . . . . . 117
4.16 Photocarrier - Photocarrier Recombination coefficient for our P3HT:PCBM
device series, expressed as the fraction of recombination that proceeds
via this mechanism at the maximum power point. . . . . . . . . . . . 118
4.17 The inverse proportionality of the photocarrier-photocarrier recombi-
nation coefficient to the P3HT hole mobility after correcting for the
variation in electron mobility . . . . . . . . . . . . . . . . . . . . . . . 119
4.18 The reverse saturation current density extracted from our fits. . . . . 120
4.19 The raw IV curve data and fits for PCDTBT:PCBM solar cells reported
in literature[59]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.20 The inverse photocarrier-photocarrier recombination coefficient plotted
against the measured PCDTBT:PCBM hole mobility. . . . . . . . . . 124
4.21 Experimental IV curve data (points) and fits (lines) for a small molecule
solar cell blended with PC71BM. The raw data is from Proctor et al
[75]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.22 The density of charge carriers as a function of the quasi-fermi level
given a constant N0 = 1x1021. The dashed lines show the analytic
approximation given in Equation 4.29. . . . . . . . . . . . . . . . . . 132
4.23 The ratio of charge carriers in a disordered device compared to a non-
disordered device as a function of the quasi-fermi level location. . . . 133
4.24 The ratio of charge carriers in a disordered device to a fully ordered
device calculated using Equation 4.29. . . . . . . . . . . . . . . . . . 135
xvi
4.25 The average charge carrier density (of one type) in the device as a
function of applied voltage for three different levels of disorder. The
device’s bandgap is 1.7eV. Solid lines correspond to a built-in voltage
at short circuit of 1.2V, dashed lines correspond to a built-in voltage
of 1V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
xvii
Chapter 1
Introduction
This thesis describes a framework for understanding the operating principles and lim-
its of organic photovoltaics (OPV), which are an emerging technology for harnessing
solar energy using semiconducting plastics rather than the traditional inorganic ma-
terials like Silicon or Galium Arsenide. Like any solar cell, the efficiency of an organic
solar cell can be characterized by its short-circuit current, open-circuit voltage and fill
factor. The main portion of the thesis is broken down into 3 chapters around each of
these topics describing the materials properties that determine each parameter. Fi-
nally, a concluding chapter brings all of the concepts together and talks about future
efforts to improve the performance of organic solar cells.
This introductory section covers background details about how organic solar cells
are made and function at a high level for readers that are not familiar with them. It
also builds the necessary semiconductor physics needed to understand the specialized
equations developed later specifically for organic solar cells. For readers that already
have some familiarity with organic solar cells, we cover some common misconceptions
that are prevalent in the OPV community so that the reader is not later surprised by
our results.
1
CHAPTER 1. INTRODUCTION 2
1.1 What is an Organic Solar Cell
An organic solar cell is, first of all, a solar cell, which is a device that produces current
from sunlight by exciting electrons in a semiconductor from an almost filled set of
energy levels to a basically empty set of energy levels. These excited electrons and
the holes they leave behind are both charged mobile species that are free to move
around the solar cell. Electrons and holes, however, have a finite lifetime, since when
any electron and hole meet, the pair can recombine, which results in the loss of the
energy associated with that excitation. The goal of solar cell design is to find some
way of coercing electrons to travel preferentially in one direction while the holes move
in the opposite direction. This leads to a build-up of electrons and holes on opposite
sides of the device, creating a voltage potential that can be used to perform work in
an external circuit.
The organic part of an organic solar cell refers to the type of semiconductor
used. Rather than employing an inorganic semiconductor like Silicon or Gallium
Arsenide, organic solar cells use molecular semiconductors like conjugated polymers
or small molecules. The delocalized π and π∗ molecular orbitals inherent to conjugated
molecules provide the necessary filled and empty bands of states required to make the
material semiconducting. Compared with inorganic materials, there are two major
differences:
1. Organic semiconductors can be designed using synthetic chemistry. Rather than
being stuck with the elements in the periodic table, organic chemists can create
novel semiconducting materials with tailored properties.
2. Organic semiconductors are excitonic. An excitonic semiconductor is one that
does not effectively screen the interaction between nearby electrons and holes.
Since electrons and holes are oppositely charged, they should be attracted to
each other and indeed they are. However, when they are placed in a polariz-
able material, like inside a semiconductor, their attraction is screened by the
polarization of neutral atoms around each charge carrier. The degree to which
this screening reduces their attraction is quantified by the relative dielectric
constant of the material. Silicon, for example, has a dielectric constant near
CHAPTER 1. INTRODUCTION 3
12, whereas organic semiconductors typically have dielectric constants between
3 and 4. This means that electrons and holes in organic semiconductors feel
attracted to each other 3-4 times stronger than in Silicon, where they are con-
sidered to be basically free. This strong attraction makes the electrons and
holes tend to pair up into an overall charge neutral species called an exciton
and semiconductors where excitons play a large role in the dynamics are called
excitonic semiconductors.
Since organic semiconductors are excitonic, the initial electron/hole pair created
by absorbing a photon is not free but instead a tightly bound singlet exciton. This
exciton is overall charge neutral so it can only move slowly via energy transfer and
diffusion. A large part of the design of organic solar cells is driven by how to split this
exciton into a free electron and hole during its short lifetime. The canonical way to
achieve this in organic solar cells is by using a heterojunction between two different
organic semiconductors, a donor and an acceptor. The materials are chosen to have
different electron affinities, providing an energetic driving force for an exciton that
reaches the donor/acceptor interface to dissociate into a Charge Transfer (CT) state,
which is an electron/hole pair that resides on nearby molecules. However, this state
is still not free since the electron and hole in a CT state can still have a significant
attraction. The rest of the process by which CT states split into fully free charges is
discussed in Chapter 2.
For reasons that are not yet fully understood, the diffusion length of an exciton
in an organic semiconductor is limited to several tens of nanometers at most. This
means that there must be a donor/acceptor interface within 10 nm of each location
where a photon could be absorbed so that the exciton generated from that absorption
can be split before it recombines. In order to absorb the majority of incident light
however, the total device must be several hundreds of nanometers thick, making a
simple bilayer (donor on top of acceptor) device architecture inefficient since only
a small slice of the device near the interface can contribute to photocurrent. These
conflicting constraints led to the introduction of the bulk heterojunction architecture,
where partially immiscible donor and acceptor materials are mixed in a solvent, cast
and allowed to dry. The materials undergo a partial phase separation during the
CHAPTER 1. INTRODUCTION 4
Donor AcceptorMixed Region
+
-
Energy
EAIP
EVac
Polymer Fullerene
Figure 1.1: A schematic view of the molecular and energy landscape of a three phaseorganic solar cell showing the pure and mixed regions as well as the variation in localenergy levels among the various phases.
drying process leading to small domains of pure donor, pure acceptor and molecularly
mixed regions containing both donor and acceptor materials as shown schematically
in Figure 1.1.
The current state of the art in organic solar cells is to use a thin (typically 100-200
nm) bulk heterojunction absorber layer placed between two electrodes with different
work functions to create an electric field across the device. The electric field helps
move electrons and holes toward separate contacts more quickly than diffusion alone
would be able to accomplish.
1.2 Basic Solar Cell Device Physics
The simplest way to understand a solar cell is as a light-powered electron pump
(shown schematically in Figure 1.2). Solar cells have a reservior of electrons that are
largely immobile but can be excited into a mobile state by absorbing a photon. Once
excited these electrons are free to move either by diffusion or by drifting in an electric
CHAPTER 1. INTRODUCTION 5
Reservoir of Electrons
Reservoir of Holes
Light
Reservoir of Holes
Recom
bination
Phot
ogen
erat
ion
ExternalCircuit
Electron Hole
Conduction Band
Valence Band
Figure 1.2: (left) A blackbox view of a solar cell, showing reservoirs of electrons andholes with photoexcitation and recombination pathways (right) A typical solar cellbuilt using a semiconducting material.
field preferentially in one direction. Similarly, the hole left behind by the excited
electron is also a mobile charged species that can move in the opposite direction.
The application of light then results in a build-up of electrons on one side of the
solar cell and holes on the other side. This difference in excited electron and hole con-
centrations leads to the species having different electrochemical potentials, which can
be exploited to perform work in an external circuit. Sometimes this process is sim-
plified down to the statement that light excites electrons, which are then collected at
a specific contact and channeled through an external circuit. This high-level descrip-
tion works to explain how solar cells are able to produce current, but by neglecting
the natural build-up of electrons and holes in the devices, it cannot provide insight
into what sets the operating voltage of the cell and hence cannot say how efficient
the device will be since the power output of a solar cell is the product of its current
and voltage. Indeed, it is typically much more complicated to understand the voltage
output of a solar cell than it is to understand its current.
Typically, solar cells are built from solid-state semiconducting materials where the
reservoir of electrons is provided by the basically filled valance band of the semicon-
ductor and the conduction band provides mobile electronic states that these valence
band electrons can be excited into. This is shown in the right of Figure 1.2.
CHAPTER 1. INTRODUCTION 6
Elec
tron
Con
tact
Active Layer
Hol
e C
onta
ct
Elec
tric
Fie
ld
Position
Volta
ge
Position
Electron Affinity
Ionization Potential
Figure 1.3: (top left) A schematic of a pin device stack showing the electron andhole contacts and the intrinsic active layer. (top right) An electronic band struc-ture showing the slope in the electron affinity and ionization potentials of the activelayer caused by the electric field. (bottom) The electric field and correspond electricpotential as a function of position across the active layer.
The solar cells that we will discuss in this thesis are all fabricated using a p-
i-n architecture, where an undoped semiconductor is placed between two contact
materials that have different work functions, leading to the creation of an electric field
across the intrinsic active layer as electric charges move from the low work-function
contact to the high work function contact during device fabrication. This electric
field is critical for pin device functioning and must be included in any discussion of
their device physics. A schematic of a PIN device is shown in Figure 1.3.
The fundamental relation that describes solar cell behavior is that the current that
can be extracted from a solar cell is equal to the photogenerated current produced by
CHAPTER 1. INTRODUCTION 7
absorbing sunlight minus any recombination losses that occur when an electron and
hole meet and annihilate each other inside the device. So,
J(V ) = q [G(V )−R(V )] (1.1)
where J(V) is the current measured leaving the solar cell, G(V) the rate at which
electrons are being excited in the solar cell as a function of the operating voltage,
R(V) is the rate at which electrons and holes are recombining and q is the charge
of an electron. In most solar cells, including organic solar cells, G actually has no
voltage dependence, so the above equation simplifies to:
J(V ) = q [G−R(V )] (1.2)
The goal of solar cell device physics is to understand R(V) using physical models
that allow us to relate it to material and architectural properties. It is impossible to
completely eliminate recombination since the condition of detailed balance requires
that any device that absorbs light also emits light, so there is a lower bound on R(V)
set by an unavoidable amount of radiative recombination that is always present in all
solar cells. Nonetheless, most solar cell materials do not operate near this radiative
limit and there is substantial work to be done to minimize R(V).
Succinctly put, the goal of this thesis is to explain why G is voltage independent
in organic solar cells and to derive an expression for R(V).
1.2.1 Electrons, Holes and Quasi-Fermi Levels
There are two different ways to quantify the density of excited electrons and holes
in a solar cell. One can just directly measure the density of electrons at a point
in the device and report that number in units like electrons/cm−3. One could also
equivalently report the chemical potential of the electrons at that same position in
electron-volts. There is a one-to-one relationship between chemical potential and
CHAPTER 1. INTRODUCTION 8
carrier density, so the two methods are equally appropriate for specifying how many
electrons are present at a point in the device. Depending on the context, one or
the other representation might be more useful. For example, we will see below that
the operating voltage of a solar cell is specified in terms of the chemical potentials
of electrons and holes. Recombination, however is typically expressed more easily in
terms of the densities of electrons and holes.
The link between chemical potential and carrier density comes from realizing that
electrons in the conduction band, for example, relax into equilibrium very quickly
among the conduction band states, so the excited electrons are always distributed
among the conduction band states in a Fermi-Dirac distribution. Similarly the holes
are always distributed among the accessible valence band states in a Fermi-Dirac
distribution. So, if we know the density of electronic states as a function of energy,
g(E), we can relate the chemical potential of electrons to the density of electrons
using:
n =
∫ ∞−∞
g(E)f(E, µe, T ) dE (1.3)
where n is the density of electrons, µe is the chemical potential of electrons, f is
the Fermi-Dirac distribution and T is the temperature. Similarly, we can define a
relation for the holes:
p =
∫ ∞−∞
g(E)f(E, µh, T ) dE (1.4)
where p is the density of holes and µp is their chemical potential. For historical rea-
sons and some mathematical simplicity, device physicists do not talk about chemical
potentials but instead quasi-Fermi levels. The quasi-Fermi level for electrons (Efn)
is just another name for the chemical potential of electrons, however the quasi-Fermi
level for holes (Efp) is defined to have the opposite sign as its chemical potential. So,
Efn = µe (1.5)
Efp = −µp (1.6)
CHAPTER 1. INTRODUCTION 9
Acceptor Electron Affinity
Donor Ionization Potential
Acceptor Electron Affinity
Donor Ionization Potential
Figure 1.4: Two example band diagrams showing the quasi-Fermi level for electronsas a blue dashed line and the quasi-Fermi level for holes as a red dashed line.
This convention is used because one can represent the quasi-Fermi levels for elec-
trons and holes on the same diagram whereas it is more difficult to visualize their
chemical potentials. It is common to represent the operating condition of a solar cell
by specifying the valence and conduction band energies as well as the quasi-Fermi
levels on what is known as a band diagram. The closer the quasi-Fermi level for
electrons is to the conduction band or electron affinity of the material, more electrons
are present at that point in the device. Two examples are shown in Figure 1.4.
1.2.2 Recombination
Recombination between electrons and holes is typically pictured as an irreversible
chemical reaction between electrons and holes where they annihilate each other and
return to the ground state. According to the Law of Mass Action, then its rate should
be proportional to the product of the electron and hole densities at the same location
in the solar cell. So,
R(x, V ) = kn(x, V )p(x, V )
R(V ) =
∫ L
0
kn(x, V )p(x, V ) dx (1.7)
CHAPTER 1. INTRODUCTION 10
Equation 1.7 says that locally the rate of recombination is just proportional to the
local density of electrons and holes. We want to know the total rate of recombination,
so we need to integrate this recombination density over the thickness L of the solar
cell. All of the solar cells we deal with in this thesis will be symmetrical in two
dimensions so we only have to integrate over one dimension.
1.2.3 Quasi-Fermi Levels and Operating Voltage
In order to understand organic solar cells, it is important to make the connection
between external parameters that you control and the microscopic internal parameters
that you do not directly observe but drive the behavior of the device, i.e. the quasi-
Fermi levels. The connection comes from realizing that an electron very near the
electron extracting contact is locally in equilibrium with the reservoir of electrons in
the contact since it is easy for electrons to be exchanged between the contact and the
active layer. Similarly a hole very close to the hole extracting contact is locally in
equilibrium with the electrons in that contact.
This is important because the difference in electrochemical potential between the
hole and electron extracting contacts is what we measure when we connect a volt
meter to our solar cell and it is what we control when we force the voltage across the
solar cell to be a specific value, by connecting the cell to a battery for example. So,
the operating voltage that we measure on the solar cell is equal to the difference in
the electron and hole quasi-Fermi levels at the two contacts:
qV = Efn(0)− Efp(L) (1.8)
In general, measuring the operating voltage does not tell us the quasi-Fermi level
splitting throughout the entire device, it merely tells us the splitting measured at two
separate points as shown in Figure 1.4. In order to determine the quasi-Fermi level
splitting throughout the device, we need to use drift-diffusion modeling in order to
relate the shape of the quasi-Fermi levels to currents in the device. Knowledge of
the current being drawn, the illumination level and the voltage applied are enough
to calculate the electron and hole quasi-Fermi levels as we do in Chapter 4.
CHAPTER 1. INTRODUCTION 11
1.2.4 Maximum Power Point
Equation 1.8 states that as we increase the voltage on our solar cell, we are also
increasing the splitting between the electron and hole quasi-Fermi levels. Previously
we saw that there is a monotonic relationship between carrier density and quasi-Fermi
level, so this means that increasing the voltage on the device results in more charge
carriers being present in the active layer. This in turn leads to the n*p product
increasing, which means that recombination must necessarily increase with voltage.
So, every solar cell faces a tradeoff. In order to increase the power output, you
would like to operate the solar cell at a higher voltage. However, as you increase the
operating voltage you begin to lose current according to equation 1.7. The power
output is proportional to the product of J and V, which are changing in opposite
ways so there will be an optimal voltage Vmpp that maximizes the power output. This
is called the maximum power point voltage.
In order to describe the power output of a solar cell, researchers use a combination
of the maximum current (Jsc) the device can produce when V=0, the maximum
voltage (Voc) the device can produce when J=0 and a reduction factor (FF ) that is
determined by the ratio of JmppVmpp and JscVoc. FF is called the fill factor and is a
number between 0 and 1. The power output of any solar cell can be specified by the
product of these three quantities:
P = JscVocFF (1.9)
The distinction between Jsc, Voc and FF is useful because there are typically
different materials parameters and architectural tradeoffs that determine each one,
so they can be thought of as quantifying three different aspects of a given solar cell’s
operation.
CHAPTER 1. INTRODUCTION 12
1.3 Organic Solar Cell Device Physics
1.3.1 Charge Transfer States
In most solar cells, the only electronic species of interest are electrons and holes,
which are assumed to move basically independently of each other except for occasion-
ally recombining when they are close by. This is because inorganic semiconductors
have high dielectric constants, which means that they effectively screen the Coulomb
attraction between electrons and holes so that they are barely attracted to each other
at all. In contrast, organic solar cells have low dielectric constants, usually between
3-5 so they do not screen Coulombic attractions well. This means that the energy
of an electron-hole pair that is, say one nanometer apart could be hundreds of meV
lower than that same pair 20 nm apart because you need to account for their attrac-
tive interaction energy. When and electron and hole are next to each other in an
organic solar cell, with the electron typically on an acceptor molecule and the hole on
a nearby donor molecule, the pair is said to be in a Charge Transfer state since if they
were to recombine it would be by transferring charge from one molecule to another.
We will see in Chapters 2 and 3 that the energetics of Charge Transfer states plays a
key role in determining how organic solar cells function.
1.3.2 Polarons
In inorganic semiconductors, electrons and holes are pictured as moving basically
freely among the atoms that compose the crystalline semiconductor. This is because
there is little interaction between the electronic excitations and the vibrational modes
of the crystals so they can be treated independently. However, in organic semicon-
ductors, there is a strong interaction between nuclear coordinates and electronic ones.
This results in molecules reorganizing themselves into different physical conformations
when an electron or hole resides on them. As charge carriers move then, we need to
picture them dragging around a local polarization and reorganization of the nearby
molecules. This combined vibrational and electronic excitation is called a polaron.
In this thesis we will not discuss polaronic effects in any great detail but just mention
CHAPTER 1. INTRODUCTION 13
them here. We will use the terms electron or hole and negative or positive polaron
interchangeably in this work.
1.3.3 Energetic Disorder
In many solar cells, the valence and conduction bands are treated as delta functions
that have many electronic states at essentially the same energy. However, in organic
solar cells the electron affinity and ionization potentials are much more diffuse. There
are three essential causes for this:
1. Dipolar disorder - Organic semiconductors are composed of polarizable molecules
with static and induced dipole moments. Since the local environment of each
molecule varies slightly due to random fluctuations in molecular orientation
and density, the dipoles also fluctuate in strength and orientation. This leads
to large scale inhomogeneities in the electrostatic potential of the solar cell,
giving a Gaussian shape to the energy levels.
2. Conformational disorder - A hole on an extended donor molecule can lower its
energy by delocalizing along the length of the molecule. If there is a break in
the conjugation of the molecule, however, the delocalization process is arrested
at that break. So, the local conjugation length of each molecule sets the local
energy of an electron (negative polaron). This conjugation length varies from
place to place in the solar cell active layer since molecules pack in slightly
different ways, leading to twists and turns in the molecules.
3. Traps - Reactions between organic molecules and other impurities in the organic
solar cell active layer can create defect states that have different energies from
the original molecules. If these energies are lower than the unreacted molecules,
then electrons or holes will preferentially reside in a trap state and these states
also serve to spread out the distribution of available electronic states for elec-
trons and holes.
The presence of energetic disorder does not qualitatively change the relationship
between quasi-Fermi level and carrier density, but it does need to be taken into
CHAPTER 1. INTRODUCTION 14
account and we will do so in Chapter 3 when we consider the open-circuit voltage of
an organic solar cell.
Chapter 2
The Short-Circuit Current
2.1 Preface
This chapter is adapted with permission from published work by the author in Ad-
vanced Materials[12].
2.2 Current Understanding and Background
The best Organic Photovoltaics (OPV) with Bulk-Heterojunction (BHJ) morpholo-
gies based on partially phase separated donor:acceptor blends now have over 9% power
conversion efficiency and field-independent internal quantum efficiencies over 90%.[40]
However, an incomplete understanding of how free charges are photogenerated in
BHJ devices hinders the rational design of better materials still needed for OPV to
reach commercial viability. Recent attention has turned to the ubiquitous molecular
mixing between fullerenes and polymers, which results in a molecularly-mixed re-
gion in BHJ systems along with the typically pictured aggregated donor and acceptor
phases.[67, 20, 103] A schematic of this three-phase morphology is shown in Figure 2.1.
Understanding the role of the amorphous mixed region in charge generation is impor-
tant since it makes up a large fraction of the film volume in many polymer-fullerene
BHJ systems. In P3HT:PCBM solar cells, for example, a study found that only about
50% of the P3HT was aggregated while the high performing system PTB7:PC71BM
15
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 16
Donor AcceptorMixed
Region
+
-
Energy
EAIP
EVac
Polymer Fullerene
- + +
+
τct
-
τhop
Figure 2.1: (left) Schematic of a BHJ solar cell including the mixed region. Potentialshifts in the local energetic landscape at the border between the donor, mixed andacceptor phases are shown in detail. EA is the electron affinity, IP is the ionizationpotential. (right) A 2D schematic of the Kinetic Monte Carlo simulation methodshowing the rates for hopping and recombination.
was found to consist entirely of an amorphous mixed region with embedded PC71BM
clusters.[20, 91] Work by many groups has shown that the presence and composition
of the mixed region can have a dramatic impact on device performance.[8, 37, 89]
This impact could be due to the fact that the mixed region has been reported to
have energy levels that are shifted with respect to the aggregated phases, producing
an energy cascade that assists in free charge generation.[37, 48, 52, 86] For example,
PCBM has been shown to have a 100-200 meV shift in electron affinity upon aggre-
gation and P3HT, the prototypical OPV donor, displays a 300 meV change in optical
bandgap between amorphous and crystalline regions.[91, 48, 82]
2.3 Core Simulation Results
In this chapter, we study the role of the mixed region in assisting geminate splitting
using Kinetic Monte Carlo (KMC) simulations of idealized trilayer (pure donor/mixed
region/pure acceptor) morphologies. We find that efficient geminate separation effi-
ciency is predicted by KMC when fast, local (monomer-scale) charge carrier mobilities
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 17
are taken into account. Additionally, we demonstrate that a 200 meV energetic offset
between the mixed and pure regions in the simulated trilayer devices greatly decreases
the local mobilities and Charge Transfer state lifetimes required for efficient change
generation.
Excitons in BHJ systems are known to dissociate at the heterojunction between
the donor and acceptor materials into a hole (positive polaron) residing on the donor
and an electron (negative polaron) residing on the acceptor.[19, 53] However, due
to the low dielectric constant of organic semiconductors, these charges are not free
and instead form a coulombically bound radical pair with a binding energy that is
calculated to be around 350 meV (assuming r = 4 and a typical intermolecular spac-
ing of 1 nm).[19] This geminate pair needs to become separated by‘ approximately
12 nm before its binding energy is equal to the thermal energy at room tempera-
ture, the point at which the charges are typically considered to be free, although
entropic considerations as well as the presence of disorder could reduce this distance
to about 5 nm.[19] In either case, the formation of free charges is a kinetic com-
petition between the rate at which geminate pairs split via a combination of drift
and diffusion, which is determined by the electron (µe) and hole (µh) mobilities, and
the rate at which they recombine when they are on neighboring molecules, which
has a characteristic lifetime τct. When the electron and hole are adjacent to each
other and could immediately recombine, the pair is said to form a Charge Transfer
(CT) state.[97] Given values for µ and τct, one can predict the fraction of photons
that result in free charges either by using the analytical Onsager-Braun theory or
by simulating and averaging many individual electron and hole trajectories using the
Kinetic Monte Carlo technique.[78, 84, 73] A troubling issue is that when one uses
experimental values for the bulk mobility in BHJs on the order of 10-4 cm2/Vs or
lower and an estimate for τct obtained from photoluminescence decay or transient
absorption measurements (1-10 ns), the predicted device quantum efficiency is typ-
ically less than 10% at short circuit conditions and increases significantly when one
simulates a device under reverse bias by adding a strong electric field.[19, 71] This
inefficient, field-dependent splitting is characteristic of a process where the mobility
and lifetime of the geminate pairs are not large enough for the charges to separate
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 18
on their own. When splitting does occur, it is primarily due to the built-in field
of the BHJ overcoming the pairs binding energy and pulling the electron and hole
apart. The strength of this field decreases at forward bias, making charge generation
less efficient as the cell approaches open circuit and reducing the fill factor. Thus,
while this model can explain the poor performance of low-efficiency OPV material
systems like MDMO-PPV:PCBM, which do show field-dependent geminate splitting,
it stands in sharp contrast to the field-independent internal quantum efficiencies near
or above 90% observed experimentally in champion polymer systems like PCDTBT,
PTB7 and PBDTTPD.[8, 19, 47, 65, 72, 60]
The inability to reconcile experimental quantum efficiency measurements of high-
performing systems with Monte Carlo simulations has led many groups to propose
additional theories about what factors the simulations are lacking that could explain
the discrepancy. One current theory is that efficient geminate splitting requires the
presence of excess thermal energy, although this is under debate and sub-bandgap
quantum efficiency measurements suggest that excess energy is not necessary in some
systems.[5, 46, 35, 94] Previous Kinetic Monte Carlo studies have also investigated
the potential effects of charge carrier delocalization, energetic disorder, molecular
dipoles and dielectric reorganization and found that, while each can improve gem-
inate splitting, none were able to account for a 90%, field-independent IQE with-
out assuming a value for τct that is orders of magnitude longer than experimentally
reported.[37, 19, 71, 26, 81]
Our simulation environment is similar to that previously reported and is described
in Section 2.5, but a brief summary is useful to aid in interpreting the results.[73] The
KMC algorithm simulates geminate splitting by iteratively tracking the progress of
many individual electron and hole trajectories as the carriers hop along a three di-
mensional lattice of sites that represent donor or acceptor molecules (see Figure 2.1).
When multiple kinetic processes could occur in competition, KMC chooses one at
random in such a way that faster processes occur correspondingly more often. The
rate of hopping events was determined by the Miller Abrahams (M-A) hopping ex-
pression, since it is computationally simple and has been used extensively to model
geminate splitting.[73, 71, 26, 106, 14, 66] The M-A model assumes that energetically
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 19
downhill hops proceed at a constant rate while uphill hops are thermally activated:
k =
k0 exp(−∆E
kT
)if ∆E > 0
k0 if ∆E ≤ 0(2.1)
The energy term includes contributions both from the electric field and the Coulomb
potential and can be written, following Peumans as:
E =−q2
4πε0εrreh− qF · reh + ULUMO(re)− UHOMO(rh) (2.2)
where q is the elementary charge, ε the dielectric constant, F the electric field
and reh the geminate pair separation vector.[73] U specifies the energy levels of the
electron and hole lattice sites. It is important to note that, at this nanometer length
scale, hole transport from the mixed to aggregated regions could occur along a single
polymer chain, potentially resulting in extremely high local hole mobilities.[70] To
study this, we fixed the electron mobility at 4x10−5 cm2/Vs in our simulations and
varied the hole mobility to investigate the combination of a 3 phase morphology and
fast local hole motion. To make the simulation amenable to analytical analysis, we
modeled each region as a homogenous average material without energetic disorder.
In the Supplemental Information we show simulations that include energetic disorder
and a mixed region composed of a blend of donor and acceptor molecules that only
transport one type of charge carrier (Figures 2.4, 2.6, 2.7 and 2.8). These additions to
the model affect the results in a much smaller manner than the effects we emphasize
in the main text. We did find, though, that all the simulations depended sensitively
on the choice of average carrier mobility and recombination lifetime, as illustrated
in Figure 2.2. For a fixed mixed region width of 3.2 nm and CT state lifetime of 5
ns, the apparent effect of the energy cascade, measured as the difference in splitting
efficiency between its presence and absence, varies from imperceptible when µh =
4x10−4 cm2/Vs to fully accounting for ¿90% field-independent geminate splitting
when µh = 4 cm2/Vs. Thus, before presenting the results, a discussion is in order of
what is known experimentally about µ and τct.
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 20
103 104 105 106 107
Applied Field / V cm−1
0.0
0.2
0.4
0.6
0.8
1.0
Disso
ciation P
robabili
ty
a)Mobility
0.0004
0.04
4 cm2/Vs
0.0 0.2 0.4 0.6 0.8 1.0Simulation Time / ns
0.0
0.2
0.4
0.6
0.8
1.0
Gem
inate
Pair S
epara
tion / n
m0
4
8
12b)
0 1 2 3 4 5 6 7 80
4
8
12c)
Figure 2.2: a) The field dependent dissociation of geminate pairs in a mixed region3.2nm wide with the electron mobility fixed at 4x10−5 cm2/Vs and the hole mobilityvaried from 4x10−4 cm2/Vs up to 4 cm2/Vs, τct is fixed at 5 ns. The dashed linesare without an energetic offset, the solid lines with a 200 meV energetic offset. band c) The separation distance evolution between the electron and hole in a typicalgeminate splitting simulation with τct =10 ns and µe = µh = 1 cm2/Vs. b ended inrecombination, c in splitting.
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 21
Group Mobility[cm2/V s]
Lifetime [ns] 90% IQE Pre-dicted
Field Indepen-dent
Janssen 2005[71] 3x10−5 1000 Yes NoGroves 2008[36] 2x10−3 2000 Yes NoDeibel 2009[26] 3x10−5 10000 Yes YesWojcik2010[106]
5x10−3 100 Yes No
Groves 2013[37] 7x10−4 100 Yes Yes
Table 2.1: Lifetime and mobility values that were required in previous KMC studiesto predict 90% geminate splitting at short circuit conditions (field of 105 V/cm).
Previous KMC studies have tended to use mobility values designed to reproduce
bulk diode mobilities measured in BHJ devices, with values on the order of 10−3-10−4
cm2/Vs. Table 2.1 reports the mobilities and lifetimes required in those studies to
predict 90% IQE at short circuit conditions. Carrier mobility in KMC simulations
is specified by giving an absolute rate for hops between lattice sites (units of hops
per second). A standard result for three-dimensional random walk simulations relates
this rate to the diffusion coefficient, which is linked to the experimentally measurable
mobility using the Einstein relation (see Section 2.7 for complete details). It is impor-
tant to note, however, that charge transport in disordered organic semiconductors is
not characterized by a single mobility across all length scales.[70, 57] Long-range, bulk
mobility is limited by sparse, deep traps whereas short-range mobility is determined
by the charges intrinsic hopping rates.[70, 57] Consequently, the mobility value mea-
sured in a space-charge-limited current measurement or time of flight measurement
is lower than that measured by time resolved microwave conductivity, which is lower
still than that measured by time resolved terahertz conductivity (TRTC).[8, 102, 28]
As one probes shorter length scales, the carrier mobility increases since the probabil-
ity of it encountering a trap during the measurement is lower. Studies have shown
that only the high frequency terahertz conductivity gives information directly on the
intrinsic hopping rate, while the other techniques report values limited by slow but
infrequent processes (compared to the hopping rate).[102]
A complete device simulation that includes all of the mechanisms by which high
local mobilities naturally decay into low bulk mobilities over longer length scales
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 22
should fully reproduce this hierarchical behavior, but for focused simulations solely of
geminate splitting the question arises as to whether bulk mobility values or single-hop
terahertz mobility values are more appropriate. The answer depends on what role the
mobility parameter is playing in the simulation. To elucidate this role, the separation
as a function of time between two typical geminate pairs (τct = 10 ns, µe = µh = 1
cm2/Vs) is plotted in Figure 2.2. As can be seen, the charges, due to their strong
binding energy, spend the majority of their time right next to each other, with brief,
relatively infrequent separations. Each of these separations, which we call splitting
attempts, can end either with the charges becoming free or with them again becoming
nearest neighbors, reforming the CT state. Recombination is assumed to be a nearest-
neighbor process, so once the charges take one hop apart they cannot recombine until
they first meet each other again. The probability that the charge carriers, once they
are no longer nearest neighbors, separate completely without meeting again turns out
to be largely independent of both the carrier mobility and the recombination lifetime
(see Figure 2.4, 2.7 and 2.8). It is independent of τct since recombination only happens
between nearest neighbors. It is independent of µ since the mobility is modeled as
being isotropic so the carrier mobility just sets the timescale for each hop, it does not
make the carriers more likely to hop in one direction (toward each other, reducing
their separation) than in another direction (away from each other, increasing their
separation).
We call the probability that an electron and hole successfully escape from their
mutual attraction in a single splitting attempt Pesc. Since Pesc does not depend
on either the carrier mobility or lifetime, it must be a constant determined by the
device’s energetic landscape (see Figure 2.5 and 2.7). The geminate splitting efficiency
is determined by the number of splitting attempts each geminate pair makes, on
average, before recombining and the probability that any single attempt is successful.
The number of attempts is set by the product of µ and τct since when the carriers
are nearest neighbors, they can either recombine or attempt to split again. The
probability of them recombining is set by the kinetic competition between the rate of
a single hop apart, set by µ, and the rate of recombination, set by 1/τct.
We conclude that the carrier mobility in a KMC simulation of geminate splitting
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 23
Morphology Local Mobility[cm2/V s]
Geminate Pair Lifetime [ns]
P3HT/PCBM[17, 2, 24, 69] 0.1 - 30 3AFPO-3/PCBM[69] 0.73 - 1 n/aZnPc/C60[7] 0.4 n/aTQ1/PCBM[74] 0.1 n/aPF10TBT/PCBM[101] n/a 4
Table 2.2: Literature measurements for local mobility (measured using time resolvedterahertz spectroscopy) and the geminate pair lifetime (measured using transientabsorption or transient photoluminescence).
primarily sets the branching ratio between recombination and another splitting at-
tempt when the electron and hole are nearest neighbors. Once the carriers are no
longer next to each other, whether they continue to split until they are free depends
mainly on the energetic landscape. The fact that geminate splitting does not de-
pend on the average value of the mobility for more than a single hop means that
the appropriate mobility value is not the bulk mobility but the value for a single
carrier hop, which is given by TRTC measurements. Put another way, using the bulk
mobility will dramatically underestimate the number of splitting attempts each gem-
inate pair makes but will reproduce the bulk mobility over long length scales. Using
the terahertz mobility will correctly predict the geminate splitting efficiency but will
overestimate the bulk mobility if combined with a simplified morphology.
Choosing the correct mobility value is critically important because the TRTC mo-
bilities of BHJ material systems (shown in Table 2.2) are between 0.1 and 30 cm2/Vs,
which is 2-5 orders of magnitude larger than the bulk mobility values. This explains
why previous authors were forced to assume long, physically unlikely, recombination
lifetimes to reproduce experimental geminate splitting efficiencies (see Table 2.1). Be-
cause the geminate splitting efficiency depends on the product µτct, an underestimate
of µ results in an overestimate of τct by the same amount in order that the product
of the two be sufficiently large to ensure many splitting attempts per geminate pair.
Having established the appropriate range of values for µ from experiments re-
ported in literature, we now do the same for τct, which specifies the rate of recom-
bination for electrons and holes that reside on neighboring molecules. This rate is
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 24
not the same as the free carrier lifetime measured with a technique like transient
photovoltage (TPV). TPV lifetimes are dominated by the rate at which already-free
carriers encounter each other rather than the rate at which they recombine once they
have become nearest neighbors. It is this latter rate that is needed for KMC simula-
tions. There are far fewer reports of CT state (nearest neighbor) lifetimes, which are
primarily measured using time-resolved photoluminescence (PL) decay or transient
absorption.[17, 101] Transient absorption measurements for P3HT and a variety of
fullerenes yield lifetimes between 3 and 6 ns.[17] PL decay measurements of the CT
state in PF10TBT:PCBM blends give a lifetime of 4 ns.[101] PL decay measurements
are particularly interesting since the technique is directly sensitive to the population
of geminate pairs and the decay constant gives the rate at which that population
is depleted. However, in high performing BHJ systems, geminate pairs are almost
always depopulated by splitting into free charges rather than by recombination. So,
the measured polaron pair lifetime is determined by the timescale for recombination
and the timescale for dissociation into free carriers, with the timescale for dissoci-
ation dominating the measured response. To extract τct from these measurements
we simulated PL decay curves using KMC for a range of mobilities and recombina-
tion lifetimes and calculated from each combination a prediction for the measured
lifetime. Our simulations show that the decay remains exponential, as observed ex-
perimentally (Figure 2.9), but with a modified decay constant. Figure 2.3 shows a
calibration curve that maps PL lifetimes to CT state recombination lifetimes that can
be input into a KMC simulation. For low mobilities, when geminate recombination
is likely, the lifetime obtained by a PL experiment and τct are similar. However, for
high mobilities, such as the local mobilities present in BHJ solar cells, the measured
lifetime approaches a limiting value set by the mobility. It is interesting to note that
the two measured lifetimes reported in Table 2.2 (3 and 4 ns) are in good agreement
with the limiting values we predict for mobilities between 1 and 10 cm2/Vs, again
reinforcing that local mobilities in BHJ solar cells are on this order and that these
are the appropriate values to use when simulating geminate separation. For mobili-
ties between 0.1 and 1 cm2/Vs, the reported PL decay lifetime of 4 ns would imply
an intrinsic CT state lifetime on the order of 1-10 ns. If the carrier mobility were
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 25
higher, τct could also be longer, which would serve to increase the geminate splitting
efficiency, so this is a conservative underestimate.
So far, we have established that geminate splitting in BHJ solar cells, as simulated
using KMC is determined by the number of splitting attempts per geminate pair (set
by the product µτct) and the probability that any given attempt is successful (a
constant of the energetic landscape we denoted Pesc). We can now examine the effect
of the mixed region, which alters the energetic landscape, on geminate splitting. The
presence of an energy cascade between mixed and aggregated regions means that
once a carrier crosses from a mixed to an aggregated region, it is energetically very
unlikely to cross back, making the carriers effectively become free after crossing the
width of the mixed region, not after traveling 12 nm, as would be predicted with no
energy cascade. Reducing the width of the mixed region allows one to systematically
increase Pesc, thereby greatly improving geminate splitting. Using an estimate for
τct of 10 ns, and values for Pesc obtained from KMC simulations of mixed region
widths between 3.2 and 9.6 nm, we can calculate what terahertz mobility would be
required for a 90% field-independent IQE in each situation. The results are shown in
Table 2.3. For devices with terahertz mobilities above 11 cm2/Vs we would expect a
greater than 90% field independent IQE even without an energy cascade. For lower
mobilities down to 0.2 cm2/Vs (the low end of the range reported in literature for
OPV materials), we would still predict greater than 90%, field-independent IQE, but
this high IQE requires a sufficiently thin mixed region with an energy cascade to
reduce the distance geminate pairs have to travel before splitting. The results are
reported for τct = 10 ns, however since we have shown that the splitting efficiency
depends on the product µτct only, if τct were 10 times shorter, the required mobility
would simply be 10 times higher.
Since the goal of this manuscript is to explain how geminate pairs split, not to get
exact results for a particular material system, we have not taken into account that
the splitting probability would depend on where in the mixed region the geminate
pair formed. We find that Pesc depends primarily on the distance the fastest carrier
needs to travel to reach an energy cascade, so if the carriers were formed near a pure
fullerene domain, rather than in the center of the mixed region, the hole would have
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 26
10-1 100 101 102 103
CT State Recombination Lifetime / ns
10-1
100
101
102
103
Measured PL Decay Lifetime / ns
4 ns
0.01 cm2/Vs0.1 cm2/Vs1 cm2/Vs10 cm2/Vs
Figure 2.3: Calibration curve mapping measured geminate pair decay lifetimes tonearest-neighbor recombination lifetimes produced by simulating geminate separationusing KMC and extracting the geminate pair lifetime as a function of the value ofτct input into the simulation for electron and hole mobilities of 0.01, 0.1, 1 and 10cm2/Vs (µe = µh). The lines are a guide to the eye. The horizontal line represents atypical measured bulk heterojunction CT photoluminescence lifetime of 4 ns.[101]
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 27
Morphology Pesc Required Mobility τct = 10 nsNo Mixed Region 1.4x10−4 11 cm2/Vs9.6 nm Mixed Region 2.9x10−4 5.1 cm2/Vs8 nm Mixed Region 3.5x10−4 4.2 cm2/Vs6.4 nm Mixed Region 6x10−4 2.5 cm2/Vs4.8 nm Mixed Region 1.3x10−3 1.2 cm2/Vs3.2 nm Mixed Region 6.5x10−3 0.23 cm2/Vs
Table 2.3: Required local mobilities for 90% field-independent IQE for the specifieddevice morphologies.
to travel twice as far to reach a pure polymer domain, and Pesc could be found by
considering a mixed region twice as wide but with the geminate pair formed in the
center. On the other hand, if the pair formed near a pure polymer domain, the hole
would cross the energy cascade almost immediately. Choosing to have the geminate
pair form in the center of the mixed region provides a consistent way to evaluate the
impact of mixed region width on geminate splitting.
2.4 Conclusion
The question of how BHJ solar cells are able to efficiently generate free charges has
persisted for over a decade and many groups have discovered important parts of the
explanation like the beneficial role of disorder and the impact of local polarizability
and charge carrier delocalization on reducing the geminate pair binding energy. In
this communication we build on their work by showing that experimentally measured
local charge carrier mobilities and lifetimes in BHJ systems are in the range required
for efficient geminate splitting. The picture that emerges of what makes a good BHJ
solar cell is a high local charge carrier mobility, long CT state decay lifetime and,
when µτct is not high enough on its own, a three-phase structure with an energy
cascade for either the electron or the hole that increases the probability that a single
geminate pair splitting attempt is successful. The combination of these three classes
of effects explains how some bulk heterojunctions are able to generate free charges so
efficiently. Looking back at Figure 2.2, it also explains the wide variability in device
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 28
performance from system to system since missing any one of these characteristics can
be the difference between high, field-independent and low, field-dependent geminate
splitting. Importantly, the commonly measured device parameters of bulk mobil-
ity and transient absorption lifetimes are shown not to be directly linked to charge
generation. Instead we have shown how terahertz mobilities and corrected CT state
photoluminescence lifetimes can be used to provide more accurate measurements of
the parameters that do determine the efficiency of free charge generation in BHJ solar
cells.
2.5 KMC Simulation Details
All KMC simulations were performed using custom KMC code written by the authors.
The First Reaction Approximation was not used. Only single geminate pairs were
simulated at a time with open boundary conditions. The world was generated on
demand so there was no limit on the size of the simulated lattice. A lattice constant
of 8 angstroms was used. The dielectric constant was set at 4 and the temperature
was set at 300K. Each combination of morphology, lifetime, mobility and field was
averaged for at least 10,000 trials and up to 200,000 trials when necessary to capture
rare events. For trilayer simulations, the geminate pair was assumed to be formed
in the center of the mixed region. The Miller-Abrahams mobility model was used to
calculate carrier hopping rates. Each material region was assumed to be homogenous
and disorder was not simulated in order to make the simulation amenable to analytical
analysis.
2.6 PL Decay Simulation Details
100,000 individual geminate pairs were simulated for each combination of lifetime and
mobility and only those that ended in geminate recombination were selected. The time
for each recombination event was calculated, binned and histogrammed to produce a
simulated PL decay curve. This was fit with a single exponential function to extract
the measured lifetime, which was plotted as a function of the actual lifetime input into
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 29
the simulation. The simulated decays were well fit by single exponential functions
(Figure 2.9). A homogenous morphology was used. The same trilayer morphology
described above was tried as well and the results were not greatly sensitive to the
change (not shown).
2.7 Converting Hopping Rates to Mobility Values
To calculate the carrier mobility, the absolute hopping rate between lattice sites is
needed. In the Miller-Abrahams (MA) model, this is the hopping rate prefactor. In
Marcus theory, the hopping rate can be calculated as:
ν = ν0 exp
(−λ4kT
)(2.3)
where λ is the molecular reorganization energy and ν0 is the hopping prefactor. The
mobility is calculated in the low field regime where the landscape is assumed to be
isoenergetic. Once the hopping rate is known, the mobility is related to it by:
µ =νa2
0
6kT(2.4)
where a0 is the lattice constant, which was 1 nm in the previous studies reported
below. In this work we followed Peumans[73] and used 8 angstroms. Two groups
specified the hopping rate using an exponential term:
ν = nu0 exp (−2γa0) (2.5)
where γ is a localization radius and a0 is the lattice constant. The conversions are
summarized in Table 2.4. Wojcik et al. specified the mobility directly in their work.
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 30
Group Model ν0 [s−1] γ [A−1] λ [meV] ν [s−1] Mobility [cm2/Vs]Janssen MA 1e13 0.5 n/a 4.5e8 2.9e−5
Groves Marcus 3.4e13 n/a 750 2.5e10 1.6e−3
Deibel MA 1e13 0.5 n/a 4.5e8 2.9e−5
Wojcik MA n/a n/a n/a n/a 5e−3
Groves Marcus 1e11 n/a 250 9e9 5.8e−4
Table 2.4: Conversion of reported hopping rates into local mobility values.
2.8 Dependence on Mobility, Lifetime and Mor-
phology
KMC simulations are stochastic but their average behavior is deterministic. At each
step in the simulation, the next event is chosen from a list of possibilities at random
according to their rate constants such that each follows a Poisson distribution. When
a geminate pair is formed as nearest neighbors, there are 11 possible first steps. Either
the electron or the hole can jump to one of its 5 nearest neighbor lattice sites that
are not occupied by the other carrier or the pair could recombine. The probability
that a hop is made, rather than recombination is:
p =g(ν0,e + ν0,h)τct
g(ν0,e + ν0,h)τct + 1(2.6)
where p is the probability that the geminate pair takes at least 1 hop apart.
It depends on the product of the mobility prefactors for the electron and hole, the
nearest neighbor recombination lifetime and a numerical constant g, which contains
information about the functional form of the mobility model and the local energetic
landscape. This result holds for any mobility model with a prefactor including both
Marcus Theory and the Miller-Abrahams model (the numerical value of the g factor
would change between the two models but the rest of the expression is the same, so
the dependence on the mobility and recombination lifetime is the same).
The expression represents the competition between two rates: the rate of a single
hop apart and the rate of recombination. If geminate pairs that make a single hop
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 31
apart have a fixed probability of splitting completely, independent of their mobility
and lifetime, then the rate for geminate splitting is just the rate for a single hop apart
multiplied by the fixed probability that the pair continues on to split completely (Pesc),
i.e. if the particles take a single hop apart once per second and 1 in 10 times that
results in splitting, the particles split, on average, once in 10 seconds.
p =g(ν0,e + ν0,h)τctPesc
g(ν0,e + ν0,h)τctPesc + 1(2.7)
This expression, then, predicts the geminate splitting efficiency once the numerical
values for g and Pesc are known. We argued that Pesc is independent of lifetime because
recombination is a nearest neighbor process and Pesc describes behavior when the
charges are not nearest neighbors. We also argued that it is independent of mobility
since the isotropic mobility does not bias the charges to hop toward each other versus
away from each other; that bias is provided by the energetic landscape and the value
of the mobility just sets the timescale for how fast all of the hops are. We will
now verify that Pesc is basically independent of the value of mobility and lifetime by
performing simulations for many values of mobility and lifetime and fitting them to
the above expression with the parameter g*Pesc as the fitting parameter.
2.9 Dependence of Pesc on Local Mobility and Life-
time
A trilayer simulation with a mixed region width of 3.2 nm was performed. The
electron and hole mobilities and lifetime were independently varied (in normalized
units) from 1 to 1x104 in 5 steps. The splitting efficiency was plotted as a function of
(µe+ µh)* for each of the 125 different combinations on the same plot (Figure 2.4) and
fit to the above equation with a single value of Pesc across 6 orders of magnitude in the
mobility-lifetime product (the dashed line). All splitting efficiencies are reported for
a field of 103 V/cm, well below the range where the field plays a role in the splitting
process.
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 32
0.0 0.2 0.4 0.6 0.8 1.0
Mobility-Lifetime Product
0.0
0.2
0.4
0.6
0.8
1.0G
em
inate
Split
ting E
ffic
iency
101 102 103 104 105 106 1070.0
0.2
0.4
0.6
0.8
1.0
101 102 103 104 105 106 10710-4
10-3
10-2
10-1
100
Figure 2.4: Variation in geminate splitting is accounted for by variation only in theproduct of the carrier mobility and lifetime, not their individual values. The samedata is plotted on semilog and log-log axes to aid examination
This means that for a fixed morphology, the same value of Pesc describes the
probability that geminate pairs, once separated by at least one hop, eventually become
completely separated, implying that Pesc is independent of both the mobility and the
lifetime. The geminate efficiency does still depend on the value of the mobility and
lifetime but only their product because that sets the number of splitting attempts
per geminate pair as described in the manuscript and derived at the end of this
document. Note the sensitivity of the results around the mobility (hopping rate)
lifetime product of 4x104 [unitless]. This inflection point (on a log-log axis) indicates
the point at which the average geminate pair begins to live long enough to split
and the wider spread in the data at that point could be indicative that, in this
sensitive regime, there is a dependence of Pesc on the ratio of the electron and hole
mobility, though not its absolute magnitude. This dependence has been seen by
previous authors in other KMC studies and is particularly important in the presence
of energetic disorder.[38, 106]
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 33
Morphology Fit Parameter (A) Extracted Pesc3.2 nm Mixed Region 1.5e−4 6.5e−3
4.8 nm Mixed Region 3.2e−5 1.3e−3
6.4 nm Mixed Region 1.5e−5 6.0e−4
8 nm Mixed Region 8.7e−6 3.5e−4
9.6 nm Mixed Region 7.2e−6 2.9e−4
No Energetic Offset 3.5e−6 1.4e−4
Table 2.5: Extracted escape probabilities for mixed regions between 3.2 and 9.6 nmwide.
2.10 Dependence of Pesc on Morphology
We have shown that Pesc is independent of the mobility and lifetime. It remains to be
shown that it is determined by the morphology. To do this, trilayer simulations were
performed with mixed regions 4-12 layers wide (3.2-9.6 nm). The electron mobility
was fixed at 1 (normalized units) and the hole mobility varied from 1 to 1e5 in 6
steps. The lifetime was varied from 0.01 to 100 in 5 steps. Mobility and lifetime
were independently varied for each thickness. The result is shown in Figure 2.5
below. Each thickness was fit with a single value for g*Pesc for all mobility and
lifetime combinations, shown in the dashed lines. The fit is excellent across 8 orders
of magnitude in the mobility lifetime product. The inset shows the region where
splitting is likely on a linear y-axis. To be clear, the data was fit to the expression:
ηgem =A(νe + νh)τct
A(νe + νh)τct + 1(2.8)
where A is the only fitting parameter. Data for each mixed width was fit separately
and fitting was done using a nonlinear curve fitting routine on a log scale due to the
fact that the data span many orders of magnitude. The extracted fit parameters are
given in Table 2.5.
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 34
10-1 100 101 102 103 104 105 106 107
Hopping Rate-Lifetime Product [unitless]
10-6
10-5
10-4
10-3
10-2
10-1
100
Gem
inate
Split
ting E
ffic
iency 4 Layers
6 Layers
8 Layers
10 Layers
12 Layers
103 104 105 106 1070
1High Efficiency Region
Figure 2.5: Simulation of geminate splitting for different mixed regions, showinghow each one is fit with a single value for Pesc for all different mobility and lifetimecombinations. The green/red divide shows an upper bound on splitting efficiencywith Pesc = 1. The inset shows the same data on a linear y-axis when splitting islikely.
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 35
2.11 The Impact of Energetic Disorder
The results shown in this manuscript are based on simulations without energetic
disorder in order to facilitate analysis. In this section we verify that these same
results hold in the presence of energetic disorder as well as in blend materials. Our
goal is to show that the geminate splitting probability depends only on the product of
the hopping rate, lifetime and a constant that is a function of the energetic landscape
even in blended and disordered materials. To show that our results hold in these
three cases, we performed simulations of a homogenous trilayer like those reported in
the main text, the same trilayer with a 50:50 blend of donor and acceptor molecules
in the mixed region and a single layer with the energy levels chosen from a Gaussian
distribution, as is typically done to model a disordered density of states. The results
are shown in Figure 2.6 and 2.7. We fit the data as described previously with a
single value of Pesc for each morphology. The figures show that while changing the
morphology does change the value of Pesc, as expected since the energetic landscape is
changing, the dependence on the local mobility and lifetime remains the same. This
shows that our conclusion that TRTC mobilities should be used in KMC simulations
is applicable to blended and disordered materials as well as homogenous ones.
2.11.1 Simulation Details
Simulations for Figure 2.6 were performed for a mixed region 6 layers wide (4.8 nm)
and the results compared for a homogenous mixed region and a random 50:50 blend
of donor and acceptor with no disorder. For the blend, 10 different environments
were each averaged over 10,000 trials. For the homogenous region 100,000 trials were
averaged.
In order to investigate the impact of disorder, a single region was modeled with the
HOMO and LUMO energy levels chosen from a Gaussian distribution with FWHM
of 80 meV. A single morphology was generated and 10,000 trials were performed and
averaged. The electron was modeled as fixed at the origin and the geminate pair was
injected with an equilibrated energy σ2
kTbelow the center of the distribution, where σ
is the standard deviation of the Gaussian distribution.
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 36
103 104 105 106 107 108
Hopping Rate-Lifetime Product [unitless]
0.0
0.2
0.4
0.6
0.8
1.0
Dis
soci
ati
on P
robabili
lity
50:50 Blend
Homogenous
Figure 2.6: Difference in splitting behavior for a trilayer with a 4.8 nm mixed regionwhen the mixed region is modeled as a homogenous region and a 50:50 blend of donorand acceptor molecules without disorder.
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 37
103 104 105 106 107 108 109
Hopping Rate-Lifetime Product [unitless]
0.0
0.2
0.4
0.6
0.8
1.0
Dis
soci
ati
on P
robabili
lity
Simulation
Model Fit
Figure 2.7: A simulation of a single region with 80 meV (FWHM) of Gaussian disorderin each energy level and the electron held fixed at the origin. The symbols are thesimulated data and the lines are the fit to the data with our model using a singlevalue of Pesc to explain each morphology, independent of the mobility and lifetime.
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 38
2.12 Independence from Bulk Mobility
It could be argued that since the bulk mobility is a function of the energetic landscape
and the hopping rate, there should be a way to invert this relation and write the
geminate splitting rate as a function of the bulk mobility. We agree that this could,
in principle, be possible, but it becomes extremely difficult when the bulk mobility
is limited by very slow, infrequent processes since then it provides little information
on the hopping rate, which is not rate limiting. To show this, we performed the
same simulations on a disordered region detailed in the previous section but when
the electron and hole were not nearest neighbors, we artificially reduced the hole
mobility by an arbitrary factor of 10,000. This means that when the electron and
hole were nearest neighbors, the mobility was high, but when they were not, the
mobility was 4 orders of magnitude lower. This would have a dramatic impact on the
bulk mobility since almost every single hop is 104 times slower. As expected from
our model, however, this had no effect on the geminate splitting efficiency as shown
in Figure 2.8. The solid lines are the simulation without artificially reduced bulk
mobilities and the circles are with reduced bulk mobilities.
This directly shows that the bulk mobility does not matter in geminate splitting
and is important only insofar as it provides insight into the local mobility. However
it is very difficult to extract the local mobility from the bulk mobility as we detail
throughout this manuscript, which is why we recommend that the directly measured
local mobility values obtained from time-resolved terahertz conductivity be used in-
stead. This also directly shows that overestimating the bulk mobility does not affect
the result since it can vary by 4 orders of magnitude without affecting the simulation.
2.13 Exponential Decay of Photoluminescence
Simulated photoluminescence from a KMC simulation in a single homogenous region
with the electron and hole mobilities both set to the values in the legend and the
nearest neighbor recombination lifetime set at 10 ns. As can be seen in Figure 2.9,
the decays remain exponential even though the decay constant changes.
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 39
10-1 100 101 102
Mobility [cm2 /Vs]
0.0
0.2
0.4
0.6
0.8
1.0
Dis
soci
ati
on P
robabili
ty
Normal Bulk Mobility
Reduced Bulk Mobility
Figure 2.8: Simulation of geminate splitting with the bulk mobility artificially reducedby a factor of 10,000 (dashed lines with circles) and not reduced (solid lines withsquares), with 80 meV of energetic disorder showing that bulk mobility does notaffect the geminate splitting probability.
CHAPTER 2. THE SHORT-CIRCUIT CURRENT 40
0 10 20 30 40 50 60 70 80Simulation Time / ns
10-3
10-2
10-1
100
Norm
aliz
ed P
L D
eca
ys
/ a.u
.
µ=0.01 cm2/Vs
µ=0.1 cm2/Vs
µ=1 cm2/Vs
µ=10 cm2/Vs
Figure 2.9: Simulated PL decay curves for a fixed lifetime of 10 ns and various electronand hole mobilities showing that the decays remain exponential.
Chapter 3
The Open-Circuit Voltage
3.1 Preface
This chapter is adapted with permission from published work by the author in Ad-
vanced Energy Materials[13].
3.2 Introduction
Organic solar cells (OPV) have the potential to become a low-cost technology for
producing large-area, flexible solar modules that are ideal for tandem, portable and
building-integrated applications. However, they are not yet commercially competi-
tive due to their low power conversion efficiencies (10%) relative to those of silicon
(25%).[6] Thus, a key challenge confronting the field of OPV is raising the power
conversion efficiency (PCE). Since the PCE of a solar cell is the product of its short-
circuit current (Jsc), open-circuit voltage (Voc) and fill factor (FF ), we can divide
this task into three separate components.
High performance organic solar cells have internal quantum efficiencies (IQE) near
100% indicating that the devices are able to efficiently photogenerate charges.[8, 40]
However, they have low open-circuit voltages and typically cannot be made optically
thick while maintaining high fill factors.[34, 77] For comparison, the best silicon solar
cell has a bandgap of 1.1 eV and an open-circuit voltage of 0.71 V, corresponding to
41
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 42
a difference between the bandgap and qVoc of only 0.40 eV.[6] In contrast, one of the
best performing organic solar cells, PTB7:PC71BM, has an optical gap of 1.65 eV
and an open-circuit voltage of 0.76 V, a difference of 0.89 eV.[39] The lower qVoc of
organic solar cells relative to their optical gaps directly translates into lower power
conversion efficiencies.[25]
Some of this voltage loss is known to occur during the charge generation process
when the initial photoexcitation produced by absorbing light is split at the heteroin-
terface between donor and acceptor materials to form a Charge Transfer state, which
is an interfacial electronic state composed of an electron in the acceptor material and
a nearby hole in the donor material that can directly recombine back to the Ground
State.[54] In order to provide a driving force for this exciton splitting process to oc-
cur, donor and acceptor materials are typically chosen to have electron affinities that
differ by 0.1 to 0.3 eV, which also reduces qVoc by the same amount.[19, 43] Since
the voltage loss between optical absorption and CT state formation is thought to be
a necessary tradeoff in order to efficiently split excitons, Voc is often referenced to
the CT state energy rather than the optical gap.[34, 19, 43, 98] Even by this met-
ric, however, the voltage is still quite low, with almost all organic solar cells having
qVoc between 0.5 and 0.7 eV below the CT state energy.[34, 97] In this work we ex-
plain why the open-circuit voltage of organic solar cells has remained persistently low
and develop a theory that provides guidance on how to improve it. Our key results
and the relevant energy levels for understanding Voc are summarized schematically in
Figure 3.1.
3.3 Background Information
In order to understand Voc we will need to build a model that describes how electrons
and holes recombine in organic solar cells and how this process depends on voltage.
Since our goal is to develop an understanding of Voc that will allow for the rational
design of organic solar cells with improved voltages, the theory must not only explain
the available experimental data, but also provide useful insights that can guide the
future design of materials. For example, would slightly raising the dielectric constant
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 43
Singlet States
CT States
Free ChargesExciton Splitting100 - 300 meV
Interfacial Disorder75 - 225 meV
Recombination500 - 700 meV
CT State Binding Energy0 - 350 meV
Voc Energy
Eopt
E0
Ect
Ect,exp
Equilibrium
Acceptor Electron Affinity
Donor Ionization Potential
E0
Efn
Efp
qVoc
Ener
gy
Position
Figure 3.1: (left)The sources of open-circuit voltage losses from the optical gap in anorganic solar cell and various energy levels in the device to which they correspond. Thespecific losses for exciton splitting (electron transfer), the CT state binding energy andfree carrier recombination are based on previous literature reports. The loss due tointerfacial disorder is presented in this work and the magnitude of the recombinationloss is explained. (right)Schematic band diagram of an organic solar cell at open-circuit showing the relationship between the quasi-Fermi levels for electrons (Efn)and holes (Efp), E0 and the open-circuit voltage. (Voc).
of organic semiconductors have a significant or marginal impact on Voc?[16, 15] Is
there an open-circuit voltage tradeoff in using energy cascades to improve charge
separation?[88, 12] Will raising the mobility of charge carriers in order to improve the
fill-factor also cause a decrease in open-circuit voltage by making carriers encounter
each other more frequently?[64] Finally, is Voc low simply because of the large amounts
of energetic disorder present in OPV materials?[9] The theory we develop in this work
will allow us to answer all of these questions. It will be useful in our discussion to
refer to two distinct but related quantities: Ect and E0. Ect is the average energy of
all of the CT states in an organic solar cell and E0 is the average difference between
the Electron Affinity (EA) of the acceptor material and the Ionization Potential (IP)
of the donor at the interface between the two. Since organic solar cells are disordered,
there is not one single value for either the CT state energy or the EA-IP difference;
instead we have to work with average quantities. We specify that E0 should be
averaged only over the interfacial/mixed portions of the device since both the EA
and IP are known to be different in aggregated versus mixed regions of many organic
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 44
solar cells and we wish to compare E0 with the energy of a CT state that only forms
at an interface.[88, 48] If there were no interaction between the electron and hole in
the CT state, Ect would equal E0. In general they are related by:
Ect = E0 − EB (3.1)
where EB is the average CT state binding energy.[54, 15] We can estimate EB based on
the dielectric constant of organic semiconductors and the average separation between
the electron and hole in the CT state (rct) using Coulombs Law:
EB =q2
4πεrct(3.2)
where q is the charge of an electron and is the dielectric constant of the material.
Experiments have estimated average CT state separations between 1 and 4 nm and
organic semiconductors typically have relative dielectric constants between 3 and 5
so we would expect values for EB between 70 and 480 meV.[15, 4, 30] Recently, Chen
et al developed a technique to measure EB and reported values between 0 and 350
meV for seven different polymer-fullerene systems, which compares well with our
simple calculation.[15] The reason we emphasize the distinction between Ect and E0
is because a large body of work has established that in optimized organic solar cells
recombination is a two-step process. The electron and hole first meet at the interface
between the donor and acceptor materials and form a CT state, which then either
recombines or dissociates back into free carriers.[56, 49, 104, 21] We will find that
we can determine whether recombination is limited by the rate at which free carriers
form CT states or the rate at which those CT states recombine by analyzing if Voc
correlates more strongly with E0 or Ect. So it is important to establish that the two
numbers are distinct and that the difference can be measured experimentally.[98, 15]
In either case, since recombination involves one electron and one hole, the Law of
Mass Action states that its rate is proportional to the product of the electron and
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 45
hole concentrations (n and p respectively):
R = knp (3.3)
where R is the rate of recombination per unit volume and k is a proportionality
constant. Under open circuit conditions, where the quasi-Fermi levels are flat, we can
directly relate the product np (though not the individual concentrations n or p) to
the voltage of the solar cell (see Figure 3.1 for variable definitions):
np = N0 exp
(Efn − Ec
kT
)∗N0 exp
(Ev − Efp
kT
)(3.4)
np = N20 exp
(qVoc − E0
kT
)(3.5)
where N0 is the density of electronic states in the device, typically taken to be around
1021 cm−3 (1 nm-3) for organic semiconductors, EC is the acceptor Electron Affinity
and EV is the donor Ionization Potential.[38, 36] The built-in potential of the solar cell
and the possible presence of band-bending do not affect this result since they change
EC and EV in the same manner, canceling out in the expression for np. In general,
k must be measured experimentally, however in certain limiting cases an analytical
expression can be found. One such case is Langevin recombination, where every time
an electron and hole meet, they recombine.[56, 18] In this limit the recombination
rate constant has been shown to be:
klan =q(µe + µh)
ε(3.6)
where µe is the electron mobility and µh is the hole mobility. Langevin recombina-
tion has been experimentally validated for organic Light Emitting Diodes (OLEDs)
and is often also applied to OPV.[56, 11, 105, 93, 36, 76] However, for organic
solar cells it overpredicts the measured recombination rates by a material system
and temperature dependent factor as high as 104 for P3HT:PCBM though typ-
ically between 10 and 100.[56, 76, 55] Device modelers account for this discrep-
ancy by introducing a Langevin reduction factor that artificially lowers klan until
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 46
it agrees with experiment.[56] In the absence of a better alternative, most researchers
have described recombination in organic solar cells in terms of reduced Langevin
recombination.[56, 76, 55] However, the theory has not been able to provide useful
guidance on how to improve Voc. For example, based on Equation 3.6 we would ex-
pect that raising the charge carrier mobilities would reduce Voc by making free carriers
recombine quicker. It is difficult to test this prediction experimentally since we do
not have precise control over the charge carrier mobilities but it is typically observed
that organic solar cell efficiencies actually improve with higher mobilities because
the Fill Factor increases without a corresponding loss in open-circuit voltage.[77, 80]
Langevin theory would also imply that slight changes in dielectric constant should
have a negligible effect on Voc. Recalling that the open-circuit voltage of any solar
cell depends logarithmically on the recombination rate, Equation 3.6 says that chang-
ing the dielectric constant from 3 to 5 should only improve the open-circuit voltage
by:[56]
∆Voc =kT
qln
(5
3
)(3.7)
This would mean that the OPV community should not look to slight dielectric
constant increases as a meaningful way to improve Voc. In contrast, Chen et al recently
showed that changing r from 3 to 5 modified the measured open-circuit voltage by
hundreds of mV and that the dependence of Voc on r was approximately linear.[15]
A linear dependence of Voc on r means that recombination must actually depend
exponentially on the dielectric constant. Several other authors have also altered the
dielectric constant of an organic solar cell by methods such as modifying the polymer
sidechains or adding a high-dielectric-constant additive. All of these studies found
large (greater than 100 mV) open-circuit voltage gains for slight dielectric constant
improvements, which is inconsistent with a logarithmic dependence of Voc on εr.[58, 16]
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 47
3.4 The Temperature Dependence of Voc Leads Us
Beyond Langevin Theory
Before presenting our model, we would like to review what is known about the tem-
perature dependence of Voc because it strongly hints at what needs to be added to
complete the theory. Looking at Equation 3.5, we can see that Langevin recombi-
nation predicts that Voc should depend on E0. Equating the recombination current
with the short-circuit current to solve for Voc gives:
qVoc = E0 − kT log
(qN2
0LklanJsc
)(3.8)
where L is the thickness of the device and Jsc is its short-circuit current. Equa-
tion 3.8 implies that looking across material systems we should see strong correlations
between Voc and E0 in each system. In fact, while Voc does tend to increase with E0,
the trends in open-circuit voltage across a large number of material systems are best
described by changes in CT state energy, not by changes in E0.[34, 97, 15, 95] Given
that Voc has been shown to be linearly related to Ect across many systems with de-
viations less than 200 meV and that the difference between Ect and E0 varies by
more than 300 meV, it would be very difficult to explain the observed dependence of
Voc on Ect if it actually depended on E0 instead.[34, 15] Another consequence of this
dependence is that if we cool an organic solar cell down to cryogenic temperatures,
Langevin theory predicts that Voc will approach E0 (details in the SI). In fact, when
extrapolated to 0K, Voc does not approach E0 but instead converges to the CT state
energy.[98, 15, 44] Since the temperature dependent experiments are performed on a
single solar cell and not by comparing different material systems, there is no scatter
in the data and the discrepancy is very clear.
Intuitively, if every time free carriers meet they recombine, there is no way for the
value of Ect to affect their behavior since by the time the carriers are close enough
to experience Ect their fate is already determined. On the other hand, if the carriers
were able to form CT states several times and split before finally recombining then an
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 48
equilibrium could exist between CT states and free carriers, in which case Ect would
be critically important because the density of CT states would be proportional to a
Boltzmann factor involving Ect. This raises the question of whether Langevin theory
mispredicts the recombination rate in organic solar cells because it overestimates the
frequency with which free carriers meet each other or because only a small fraction of
those encounters lead to recombination. Several authors have explored this issue and
shown with Kinetic Monte Carlo simulations that klan actually does a surprisingly
good job of predicting how often carriers encounter each other, even in disordered
material systems, which agrees with the fact that the expression works reasonably well
for OLEDs.[93, 36] This implies that the Langevin reduction factor must be necessary
because not every encounter between free carriers results in recombination. Recent
experimental work has confirmed this hypothesis by showing that the low energy CT
states that would be formed by free carriers encountering each other have the same
high splitting efficiency as higher energy CT states formed during the photogeneration
process.[94] The CT state splitting process has also been investigated using detailed
Kinetic Monte Carlo simulations, which show that carriers actually have a very low
chance of recombining during any given encounter.[12, 50, 37, 45, 1] The likely reason
that Langevin recombination works for OLEDs is because those systems have been
specifically designed for free carriers to efficiently find each other and recombine.
Organic solar cells, on the other hand, have been specifically designed to prevent this
process.
3.5 Reduced Langevin Recombination Implies Equi-
librium
The suggestion that most Charge Transfer states reseparate has been made before as
an explanation for the Langevin Reduction factor and detailed numerical models have
been constructed to explore its impact.[36, 41] For example, Hilczer and Tachiya were
able to accurately reproduce the temperature dependence of the Langevin reduction
factor with a model that allowed CT states to split back into free carriers.[41] In this
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 49
work we would like to take the idea one step further. If free carriers form CT states
and split much faster than they recombine, there should be time for equilibrium to
be reached between the population of free carriers and the population of interfacial
CT states. In this limit it does not matter how quickly carriers move, a certain
fraction of them will always be in CT states and that fraction can be calculated
using Boltzmann statistics and a knowledge of the free energy difference between
free carrier states and Charge Transfer states. In order to see if such a description
is appropriate, however, we must first investigate how close to equilibrium the free
carrier and CT state populations are in an organic solar cell. When carriers meet
and split 10,000 times before recombining, there is clearly time for equilibrium to be
established between the two populations; however, it is not obvious that the same
is true when they only meet and split 10 times. We can answer this question using
a kinetic model. Figure 3.2 shows the recombination process schematically with all
of the relevant rates labeled. Without making any assumptions about whether free
carriers and CT states are in equilibrium with each other, we can write down rate
equations describing the interactions between the two populations:
dnctdt
= kmnp− (kr + ks)nct (3.9)
dn
dt= −kmnp+ ksnct +G (3.10)
dp
dt= −kmnp+ ksnct +G (3.11)
where nct is the density of CT states, kr is the (average) rate constant at which
CT states recombine, ks is the rate constant at which CT states split back into free
carriers, km is the rate constant at which free carriers meet and G is the rate at which
free carriers are being generated. Since the solar cell is in steady state, we know
that n and p are being replenished, either by injected carriers from the contacts or by
photogenerated carriers, at precisely the same rate that the CT states are recombining
so G = krnct. Solving for steady state leads to:
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 50
nctnp
=km
kr + ks(3.12)
We can also define the equilibrium density of CT states (neqct ) we would expect if
kr were much slower than ks as:
neqctnp
=km
kr + ks(3.13)
Since we argued before that the Langevin reduction factor (γ) primarily measures
the fraction of free carrier encounters that lead to recombination, we can use it to
relate kr and ks:
γ =kr
kr + ks(3.14)
kr =γ
1− γks (3.15)
If the rate of CT state recombination is much faster than CT states splitting back
into free carriers then γ approaches 1 and Langevin theory applies. In the other
limit, γ approaches 0 and equilibrium holds between free carriers and CT states. To
quantify how close to equilibrium free carriers and CT states are we can compare
the recombination rate that we would expect at equilibrium (Req) with the reduced
Langevin recombination expression (Rlan):
Rlan = γkmnp (3.16)
Req = krneqct =
γ
1− γkmnp (3.17)
Figure 3.2 plots both recombination expressions as a function of γ. The goal
is to determine how small γ needs to be before an equilibrium description becomes
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 51
CT States
Free Charges
km
ks
kr
Ground State0.0 0.2 0.4 0.6 0.8 1.0
Langevin Reduction Factor
0.0
0.5
1.0
1.5
2.0
Reco
mbin
ati
on R
ate
[a.u
.]
Reduced Rate
Equilibrium Rate
0.0 0.2 0.4 0.6 0.8 1.0
Langevin Reduction Factor
0
20
40
60
80
100
Equili
bri
um
Volt
age E
rror
[mV
]
Figure 3.2: (left) Kinetic scheme describing the recombination process in organicsolar cells. (right) The difference in recombination rate and predicted Voc betweenthe reduced Langevin recombination expression and the equilibrium approximationas a function of the Langevin Reduction Factor.
appropriate. We find that once the Langevin reduction factor is smaller than about
0.1, the reduced Langevin recombination rate is extremely close to the rate expected
if the free carriers were fully in equilibrium with Charge Transfer states. Furthermore,
since Voc depends on recombination in a logarithmic fashion, even a solar cell with
a Langevin reduction factor of 0.5 would have an open circuit voltage that deviates
from the equilibrium prediction by less than 20 mV and in fact γ must be very close to
1 before the equilibrium picture breaks down. Almost all organic solar cell materials
have γ 0.2, so we can treat bimolecular recombination as occurring from a population
of free carriers in equilibrium with CT states (tabulated reduction factors and more
discussion of this point is presented in the SI).[76] This means that we do not need a
complicated numerical model to estimate ks and calculate γ in order to understand
the open-circuit voltage of organic solar cells; we can instead just write down the
density of CT states based on the requirement that they be in equilibrium with free
carriers.
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 52
3.6 Equilibrium Simplifies the Understanding of
Voc
When chemical or electronic species are in equilibrium with each other the fundamen-
tal requirement is that there must not be a thermodynamic driving force to convert
one species into another. In the case of CT states, it means that the free energy
gained by creating one additional CT state must be exactly equal to the free en-
ergy lost by destroying a free electron and free hole, otherwise nature could lower its
free energy by simply converting one more electron/hole pair into a CT state or vice
versa and this reaction would spontaneously happen. This is a very general condi-
tion for equilibrium that holds both for electrons and holes as well as for atoms and
molecules. It underlies the Law of Mass Action and the calculation of equilibrium
constants for chemical reactions. In chemistry, the free energy of a species is often
called its chemical potential. In solid-state physics, the free energy of an electron is
called its quasi-Fermi level. By convention, however, the quasi-Fermi level of holes is
defined to have the opposite sign as its free energy, which is why holes float in semi-
conductor band diagrams. In short, equilibrium between electronic species allows us
to relate their quasi-Fermi levels since this is the quantity that measures their molar
free energies and at equilibrium it is their free energy that must be equal, not, for
example, their concentrations.
So, equilibrium between CT states and free carriers requires that the chemical
potential of the CT states (µct) be equal to the difference of the electron and hole
quasi-Fermi levels for their molar free energies to be equal:
µct = Efn − Efp (3.18)
For further discussion of the relationship between quasi-Fermi levels and chemical
potentials, readers are directed to a lengthy treatment by Wurfel, who validated and
used the same approach to relate the quasi-Fermi levels of electrons and holes with
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 53
the chemical potential of photons in order to derive the semiconductor electrolumi-
nescence spectrum.[107] For readers who prefer an alternative derivation that does
not require introducing chemical potentials, we arrive at the same result in the final
section of this chapter directly from the Canonical Ensemble in statistical mechanics
by considering the many-particle partition function of electron-hole pairs in an or-
ganic solar cell. We specified the the chemical potential of the CT state population
using Equation 3.18 because we know that at open-circuit the difference between the
electron and hole quasi-Fermi levels is constant across the device and given by qVoc:
Efn − Efp = qVoc = µct (3.19)
Equation 3.19 means that equilibrium between free carriers and CT states gives
us a way to directly relate the open-circuit voltage to the chemical potential of the
CT states, letting us calculate the number of CT states without needing to know how
many free carriers there are in the device, how quickly they are moving or what the
energetic landscape for those free carriers looks like.
Now that we know the chemical potential of the CT states, we can determine
how many are occupied (Nct) by integrating over the density of possible CT states,
gct(E) (see Figure 3.3). Intuitively, one can think of µct as measuring the amount of
free energy the system can use to populate CT states. It makes sense then that by
combining this information with knowledge of how much energy it takes to occupy
each CT state and how many possible CT states there are, i.e. the density of states,
you can calculate the total number of populated states. For readers familiar with
the standard expressions relating electron and hole quasi-Fermi levels to electron and
hole densities, the result for CT states is exactly analogous. The precise functional
form for all three expressions is typically derived from the grand canonical ensemble
in statistical mechanics and worked out step by step for the case of CT states in a
later section. Here we quote the result:
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 54
Filled StatesOccupation Function
µct
CT State DOS0.8
0.6
0.4
0.2
0.0
0.2
0.4
Energ
y [
eV
]
Figure 3.3: Schematic showing how the density of available CT states, gct(E), com-bined with knowledge of the CT state chemical potential, µct, permits the calculationof the number of filled CT states, Nct.
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 55
Nct =
∫ ∞−∞
gct(E) exp
(µct − EkT
)dE (3.20)
As a first approximation, we show in a subsequent section that the CT state
distribution should have a Gaussian shape, as is typical for inhomogenously broadened
energy levels, which means this integral can be computed analytically (see references
and calculation in the SI).[9, 42, 51, 10] If the standard deviation of the CT state
distribution is σct and its center is Ect then:
Nct = fN0 exp
(σ2ct
2(kT )2
)exp
(qVoc − Ect
kT
)(3.21)
where f is the volume fraction of the solar cell that is mixed or interfacial. Each of
these CT states recombines with an average lifetime τct = 1/kr, so the recombination
current in the solar cell can be written as:
Jrec =qNctL
τct=qfN0L
τctexp
(σ2ct
2(kT )2
)exp
(qVoc − Ect
kT
)(3.22)
where L is the thickness of the solar cell. Now that we have an expression for
recombination as a function of Voc, we can invert it and solve for Voc since at open-
circuit Jrec = Jsc:
qVoc = Ect −σ2ct
2kT− kT log
(qfN0L
τctJsc
)(3.23)
Similar expressions relating Voc and Ect but excluding the effects of disorder have
been derived previously by various methods including detailed balance relationships
and solar cell equilibrium with a black body.[44, 99, 79, 31] The benefit of our approach
is that by explicitly considering an illuminated organic solar cell with interfacial dis-
order and an arbitrary energetic landscape for free carriers we remove any ambiguity
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 56
about when the result is applicable, show how it is equivalent to reduced Langevin
recombination and connect all of the input parameters directly with concrete material
properties that can either be measured or calculated. This last point is critical as it
will allow us to explain why qVoc is so consistently 0.5 to 0.7 eV below the measured
CT state energy in almost all organic solar cells, despite the widely varying electronic
properties among those different systems. Our result shows that, in the absence of
device imperfections like contact pinning or shunts, Voc is determined solely by the
degree of mixing in the device, the energy of the center of the CT state distribution,
the degree of energetic disorder in the mixed region and the CT state lifetime. The
CT state lifetime describes the rate at which CT states directly recombine either
radiatively or nonradiatively. It is distinct from the free carrier lifetime that could be
measured in a transient photovoltage experiment as we discuss below.
3.7 Effects of an Energy Cascade in 3-Phase Bulk
Heterojunctions
One of the reasons we derived our expression for Voc in terms of quasi-Fermi levels
instead of free carrier densities is because it makes it clear that there is no dependence
of Voc on the energy levels of free carriers, i.e. E0 appears nowhere in our expression
for Voc and we did not need to make any assumptions about the energetic landscape
for free carriers in order to derive it. This is not to say that the energetic landscape is
unimportant for solar cell operation, just that our theory shows it does not affect the
numerical value of the open-circuit voltage. When calculating the potential efficiency
of a solar cell material, one is typically not interested in Voc in isolation but in the
difference between the optical gap and qVoc since a device with a smaller optical
gap absorbs more light and can compensate for its lower voltage with additional
photocurrent, increasing the overall efficiency. To use an extreme example, silicon
solar cells have lower open-circuit voltages than many OPV devices, but this does
not mean that organic solar cells are more efficient. So, if one is able to decrease
the optical gap of an organic solar cell without affecting the CT state energy, then,
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 57
AcceptorDonor Mixed
Efn
No Energy Cascade Energy Cascade
AcceptorDonor Mixed
LUMOs
HOMOs
Efp
Figure 3.4: Two example energy diagrams showing a solar cell with and without anenergy cascade between mixed and aggregated phases.
our theory says that within certain limits discussed below, the photocurrent should
increase without a corresponding decrease in the open-circuit voltage. A potential
way to achieve this would be by introducing controlled energy cascades. To explore
what happens at open-circuit in a three-phase bulk-heterojunction with an energy
cascade, let us consider two example situations as shown in Figure 3.4. In one case
we have an organic solar cell that is one third mixed, one third aggregated acceptor
and one third aggregated donor but has uniform energy levels for free carriers in all
of the phases. In the other case, we have an energy cascade where the mixed region
is identical to the first case but the aggregated regions have energy levels that are
shifted by 100 meV each. In both cases we will consider E0 = 1.7 eV, Ect = 1.5 eV,
N0 = 1021 cm−3 and 80 meV of Gaussian disorder in each of the energy levels. For
clarity we will ignore the built-in potential so that the carrier densities are constant
in each phase and calculated using Equation 3.5. The presence of a built-in potential
does not change our conclusion it just makes the calculation less intuitive. We want
to determine the density of free carriers and CT states as well as the recombination
rate and free carrier lifetime at an open-circuit voltage of 0.9 V.
Without the energy cascade, we calculate the average free electron and hole den-
sities to be 1.6x1016 cm−3 and the density of CT states to be 3.6x1012 cm−3. In a
100-nm-thick device with a CT state lifetime of 500 ps, this would correspond to a
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 58
1014
1015
1016
1017
1018C
arr
ier
Densi
ty [
cm−
3]
Electrons
Holes
Figure 3.5: The carrier density in each phase assuming a IP-IP and EA-EA offsetbetween the donor and acceptor materials of 150 meV each.
recombination current of 12 mA/cm2. Even though the CT state lifetime is only 500
ps, there are 4,390 times more free carriers than CT states, so each carrier, on average,
has only a 1 in 4,390 chance of occupying a CT state. Since transient photovoltage
measures the lifetime of the average carrier, one would measure a free carrier lifetime
of:
500 ps ∗ 4390 = 2.2 us (3.24)
With the energy cascade, the density of CT states and free carriers in the mixed
region is unchanged since Ect and E0 are unchanged but there are now many more free
carriers in the aggregated regions so that the average density of carriers has increased
to 4x1017 cm−3. The recombination current is the same since both the number of CT
states and their lifetimes are the same, which means the free carrier lifetime must
have increased substantially to 53 us since the odds of each free carrier occupying a
CT state has decreased to 1 in 105,000. If we only had access to information on
the free carrier densities and lifetimes, for example through charge extraction and
transient photovoltage measurements, we would conclude that the solar cell with the
energy cascade had substantially reduced recombination since both the free carrier
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 59
lifetime and the density of free carriers at open-circuit increased significantly.[85, 22]
However, the actual amount of recombination is the same in the two solar cells and
the presence of the energy cascade neither increased nor decreased Voc. This is one
of the consequences of equilibrium between free carriers and CT states and it also
implies that traps and energetic disorder outside of the mixed region, which would
have a similar effect to an energy cascade, do not impact the open-circuit voltage. Put
another way, we are saying that for a given solar cell Ect and E0 will be related to each
other because both involve the EA - IP difference. However, if one keeps Ect constant
but varies E0 (using an energy cascade, for example), Voc will not change. On the
other hand if one keeps E0 constant but varies Ect (by modifying the CT state binding
energy, for example), Voc will change to track the variation in Ect. So, the important
variable that determines Voc is Ect, not E0. If one changes E0 and in-so-doing also
changes Ect (by changing the donors IP, for example), then Voc will, of course, also
change. However, it changes because of the change in Ect, not the change in E0. We
can use this effect to our advantage by introducing energy cascades that broaden the
optical absorption without affecting the CT state energy to increase the photocurrent
without sacrificing voltage.[88] For example, both of the solar cells that we discussed
above have the same open-circuit voltage but the one with the energy cascade could
achieve this voltage with a 200 meV smaller optical gap, increasing the short-circuit
current. This extra current comes at the expense of a reduced EA-EA offset between
aggregated donor/acceptor phases, but provided the offset remains large enough to
drive exciton splitting, there should be no impact on charge generation and energy
cascades could be used as a way to recover some of the voltage lost due to overly
large EA-EA offsets.
3.8 The Role of Energetic Disorder
Looking at equation 1.19 and noting that the energetic disorder could easily be 100
meV, our model implies that we should expect significant variations in the difference
between Voc and Ect based on differences in the amount of interfacial energetic disor-
der, which, in contrast to free carrier disorder, is predicted to affect the open-circuit
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 60
voltage by setting the width of the CT state distribution. This would seem to be
in contradiction to the experimental finding that qVoc is almost always 0.5 to 0.7 eV
below the CT state energy so we need to briefly discuss the relation between what we
call Ect and what is measured experimentally. Experimental values for Ect are typ-
ically extracted by sensitively measuring the optical absorption of an organic solar
cell below its optical gap.[98, 96, 32] CT states weakly absorb light so they appear as
a low-energy shoulder in the absorption spectrum of organic solar cell blends. Since
the absorption of the CT states is vibrationally broadened, one cannot directly infer
the energy of a CT state from the energy of the light that it absorbs. Instead, Marcus
Theory is used to calculate the energy of the state based on its absorption spectrum.
Marcus Theory describes the vibrational broadening of a single absorber in terms
of its reorganization energy, λ, and has been very successful in fitting the CT state
absorption spectrum in many OPV material systems.[98, 94, 92] This is somewhat
surprising since we do not expect to have a single CT state in organic solar cells but
rather an inhomogenously broadened distribution of CT states as described earlier.
Thus, the absorption of the CT states is better described by the Marcus Theory ab-
sorption expression for a single CT state integrated over the distribution of states.
When the distribution is Gaussian in shape, the resulting inhomogenously broadened
absorption turns out to be identical to that of a single Marcus Theory absorber with
an effective energy Ect,exp and reorganization energy λexp given by (derivation in SI):
Eexpct = Ect −
σ2ct
2kT(3.25)
λexp = λ+σ2ct
2kT(3.26)
This result explains why it is possible to successfully fit the CT state absorption
as if it were a single state, but it also means that the experimentally measured CT
state energy already incorporates the presence of energetic disorder. In Figure 3.6
we verify this prediction by measuring the CT state absorption of a 1:4 Regiorandom
P3HT:PCBM blend as a function of temperature using Fourier Transform Photocur-
rent Spectroscopy (FTPS).[96] We find that both Ect,exp and λexp are linear in 1/T
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 61
4 6 8 10 12
1000/Temperature [K−1 ]
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
CT S
tate
and R
eorg
aniz
ati
on E
nerg
ies
[eV
]
Apparent CT Energy
Apparent Reorganization Energy
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
1000/Temperature [K−1 ]
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
P3HT:PCBM 1:1
MDMO-PPV:PCBM 1:4
APFO3:PCBM 1:4
APFO3:PCBM 1:1
Figure 3.6: Fits to the temperature dependence of Ectexp for MDMO-PPV:PCBM,P3HT:PCBM and AFPO3:PCBM (1:1 and 1:4 blend ratios). (left) The extractedEct and reorganization energies for a blend of regiorandom P3HT:PCBM showingthat they are both linear in 1/T and have very similar slopes (104.3 meV disorder isextracted from the slope of the CT State Energy and 104.1 meV for the reorganizationenergy, fit independently). (right) The temperature dependent Ect measurementstaken from literature.[98] The data points are the experimental fit parameters ateach temperature and the lines are 1/T fits to the data.
with opposite slopes that are very similar in magnitude, consistent with our theoret-
ical prediction. Fits to the data yield values for σct of 104.3 and 104.1 meV from the
CT State and Reorganization Energies, respectively.
In Figure 3.6 we use this new tool to extract the interfacial energetic disorder from
previously published temperature dependent measurements of Ect,exp.[98] We find σct
for MDMO-PPV, P3HT and AFPO3 blended with PCBM to be between 60 and 75
meV. The results are summarized in Table 3.1. Using Equation 3.26 we can now
simplify our expression for Voc to:
qVoc = Eexpct − kT log
(qfN0L
τctJsc
)(3.27)
and see that the dependence of Voc on interfacial disorder is exactly masked by
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 62
Material System Ect σct Eexpct Voc Ect − qVoc Eexp
ct − qVoc[eV] [meV] [eV] [V] [eV] [eV]
P3HT:PCBM 1:1 1.24 75 1.14 0.61 0.61 0.53RRa P3HT:PCBM 1:4 1.66 104 1.44 0.83 0.83 0.61MDMO-PPV:PCBM 1:4 1.52 75 1.42 0.84 0.68 0.58APFO3:PCBM 1:4 1.73 71 1.64 1.05 0.68 0.59APFO3:PCBM 1:1 1.74 64 1.68 1.09 0.65 0.59
Table 3.1: Extracted CT state distribution centers and standard deviations withexperimental Voc measurements for comparison. All raw data except for RRa P3HTis from literature.[98]
the experimental techniques used to measure Ect.
3.9 Experimental Observations Explained by the
Model
Our model predicts that Voc should increase linearly as we lower the temperature
of the solar cell and appear to converge to Ect,exp when extrapolated to 0K as seen
experimentally and in contrast to the predictions of Langevin recombination. It
also explains why Ect,exp - qVoc 0.6 eV for many systems that have been studied
even though they had different amounts of energetic disorder since only interfacial
energetic disorder matters and the available techniques to measure Ect happen to be
affected by interfacial disorder in precisely the same way as Voc. We see why Voc
is exponentially dependent on the dielectric constant, since that sets the CT state
binding energy, which determines, at equilibrium, what fraction of free carriers will be
in a CT state via a Boltzmann factor. We also see why the highly variable energetic
landscape for free carriers, including ubiquitous energy cascades between aggregated
and mixed regions, does not impact the difference between Ect,exp and Voc since the
number of populated CT states at equilibrium depends only on the CT state energy
and the open-circuit voltage.[88] Finally, we see that the carrier mobility does not
affect Voc because the recombination process is not limited by the rate at which free
carriers find each other.
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 63
3.10 Explaining the Magnitude of the Voltage Loss
We now turn to the empirical result that qVoc is almost always 0.5 to 0.7 eV below
Ect,exp.[34] To compare our model with experiment we need to estimate the expected
ranges of all of the necessary input parameters. We start with the volume fraction of
interfaces and mixed regions in organic solar cells. On the high side, we have solar
cells like regiorandom P3HT that are completely amorphous, so the solar cell could
be 100% mixed. On the low side we have low-donor-content cells (1-10% mixed) and
bilayers. Even a perfect 100 nm bilayer would still be approximately 1% interface
(1 nm of donor/acceptor molecules involved in an interface in a 100 nm thick active
layer), so we conclude that organic solar cells are between 1% and 100% mixed. We
also need to know the CT state recombination lifetime. This quantity is difficult to
measure experimentally since the distribution of CT states excited in the transient
experiments used to measure CT state recombination rates is far from equilibrium,
meaning that the average lifetime of those CT states may differ from that of the
equilibrium distribution that exists at steady state. We discuss this point at more
length in the SI and present tabulated lifetimes from literature. In this section we
summarize the available estimates of τct from a variety of experimental and theoretical
methods. Ultrafast pump-push measurements and photoluminescence studies tend
to report lifetimes between 100 ps and several ns.[17, 101, 5] Quantum chemical
calculations on a P3HT:PCBM analog predict 500 ps for one model and as fast as
90 ps for a different interface conformation.[62, 61] Further calculations have shown
that the donor/acceptor interface is actually dynamic on the timescale of 10 ns so
even if a particular interfacial conformation would lead to very slow recombination,
the interface will explore enough conformations within 10 ns to find one that allows
for fast recombination.[61] Given the experimental and computational variability, we
consider a range of lifetimes between 10 ps and 10 ns, keeping in mind that at the low
end of the lifetime range we do not necessarily expect there to be time for complete
equilibrium to develop between free carriers and CT states. However, as we showed in
Figure 3.2, we do not actually need full equilibrium for the predictions of our theory
to be accurate; we simply need τct to be nonnegligible such that CT states dissociate
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 64
several times before recombining, which we generally know to be the case since the
solar cells were able to photogenerate free carriers in the first place.[94] In principle
we also need to know the degeneracy of the Charge Transfer states. It is tempting to
assume that there is one CT state for each pair of nearest neighbor donor/acceptor
molecules, however a range of experimental and theoretical work has shown that
CT states form between non-nearest neighbor molecules as well due to long-range
couplings between non-adjacent molecules.[30, 63, 83, 87] This effect is shown in
Figure 3.7 and is very important because if you consider only CT states forming
between molecules 1 nm apart, you might expect 3 CT states per acceptor molecule
since 3 of its 6 nearest neighbors in a simple cubic, 50:50 blend of donor or acceptor
molecules would be donors. On the other hand if you increase the interaction distance
to 2 nm, you would have 33 molecules with which each acceptor can interact and 16
CT states. At 3 nm it would be 113 molecules and 56 CT states per acceptor. In
general the density of CT states increases like the cube of the CT state delocalization
length. Previous authors have discussed the importance of CT state delocalization for
improving charge generation. [30] Here we add that increased delocalization is also
likely to limit the open-circuit voltage by providing more pathways through which
recombination can occur, implying a design tradeoff that will need to be optimized.
We find good agreement with experimental Voc measurements at 32 CT states per
acceptor molecule (an approximate delocalization length of 2.5 nm). Answering the
question of precisely how many CT states are formed at each interface would be an
important candidate for future quantum chemical calculations.
Figure 3.7 explores the expected difference between Voc and Ect,exp for a 100-
nm-thick active layer with a short-circuit current of 10 mA/cm2 across the range
of plausible material parameters that we found in the preceding paragraphs. The
key point to take away from Figure 3.7 is that almost all combinations of material
parameters will result in an open-circuit voltage between 0.5 and 0.7 V below Ect,exp,
explaining why this empirical rule has worked so well. This is a consequence, however,
of the range of CT state lifetimes and degrees of mixing observed in organic solar
cells. More precisely, we could say that the reason why qVoc is almost always 0.5 to
0.7 eV below Ect,exp is because the CT state recombination lifetime is rarely higher
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 65
1 nm
2 nm
20 40 60 80 100
Degree of Mixing [%]
10-11
10-10
10-9
10-8
CT S
tate
Reco
mbin
ati
on L
ifeti
me [
s]
0.500
0.550
0.600
0.650
Figure 3.7: (left) A 2D schematic showing the effect of CT state delocalization onthe number of CT states in an organic solar cell. Grey circles indicate moleculesand dashed lines show different delocalization lengths. (right) The expected voltagedifference (V) between Ect,exp/q and Voc for a 100 nm thick active layer with a Jsc of10 mA/cm2. A constant molecular density of 1021 cm−3 [1 nm−3] is used with 32 CTstates per molecule.
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 66
Parameter Improvement Strategy Voc IncreaseReduce volume fraction of mixed phase from 50% to 1% 100 mVIncrease CT state lifetime from 100 ps to 10 ns 120 mVDecrease interfacial disorder from 100 to 50 meV 150 mVDecrease CT state binding energy from 200 to 50 meV 150 mVDecrease number of CT states per interface from 30 to 3 60 mV
Table 3.2: The potential increases that could be obtained from improvements to eachof the material parameters that affects Voc.
than 10 ns since this is the timescale for dynamic interfacial reconfiguration and it is
never lower than 10 ps since this would prevent the photogeneration of free carriers.
Three orders of magnitude of change in CT state lifetime corresponds to a 180 mV
difference in Voc at 300K since the CT state lifetime affects the voltage logarithmically.
Because the exciton diffusion length in organic photovoltaic materials is typically
less than 30 nm, the community has not been able to explore orders of magnitude
differences in donor/acceptor mixing ratios. It has been observed, however, that in
dilute blends, where you can measure the interfacial area by the strength of the CT
state absorption, Voc does depend logarithmically on interfacial area in agreement
with our expression.[100]
3.11 Opportunities for Improving Voc
In addition to explaining why the open-circuit voltage of organic solar cells is low even
though their internal quantum efficiencies can be quite high, this study also provides
a framework in which to identify and rank opportunities to raise Voc. Table 3.2
summarizes the potential gains in open-circuit voltage that could be achieved by
improving each of the terms that appears in our expression for Voc. Since many of
the parameters appear in a logarithm, they would need to be changed by orders of
magnitude to significantly enhance the open-circuit voltage. However, both the degree
of interfacial disorder and the CT state binding energy are outside of the logarithm,
implying that the largest voltage gains are likely to come from reductions in those
two parameters.
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 67
A promising route to improving the open-circuit voltage of organic solar cells could
be engineering the donor and acceptor molecules to dock in preferred orientations in
order to reduce conformational disorder at the interface.[33] As an example of the
effect of conformational disorder on Voc we compared the interfacial disorder in re-
giorandom and regioregular P3HT blended with PCBM. The regiorandom blend was
found to have 104 meV of interfacial disorder compared with 75 meV in the regioregu-
lar blend (see Figure 3.6). According to our model, this slight reduction in interfacial
disorder contributes approximately 100 mV to the open-circuit voltage of the re-
gioregular blend. In this case however, the increase in Voc due to reduced disorder is
overshadowed by the fact that the center of the CT state distribution for the regioreg-
ular blend is 0.4 eV lower in energy than the regiorandom blend due to the well-known
differences in polymer Ionization Potential in the two systems, so the overall open-
circuit voltage is lower for regioregular P3HT than for regiorandom P3HT.[90] The
measured values of 63-104 meV of interfacial energetic disorder imply that 77-210
mV of open-circuit voltage are lost to this effect in the five systems studied. Further
increases in Voc could come from reductions in the CT state binding energy either by
designing molecules with increased amounts of wavefunction delocalization or from
raising the bulk dielectric constant of the active layer. While it may seem that large
increases in dielectric constant would be needed for significant improvements in Voc,
Chen et al have suggested that even a dielectric constant near 5 could be enough to
largely eliminate the CT state binding energy, presumably because as the dielectric
constant increases the CT states also become more delocalized, which further reduces
their binding energy.[15] Another way to improve the open-circuit voltage would be
to increase the CT state lifetime. The lifetime is known to be dominated by non-
radiative transitions with an electroluminescence quantum efficiency typically worse
than 10-6.[98] This means that 6 orders of magnitude of improvements in CT state
lifetime are possible but there is currently not a clear understanding of precisely what
mechanism is leading to such fast nonradiative recombination. Future studies focused
on this point could help recover some of the more than 360 mV of Voc currently lost
to this effect. We speculate that perhaps the dynamic nature of the donor/acceptor
interface plays a large role in allowing CT states to find configurations that lead
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 68
to fast nonradiative recombination. In that case, rigidly locking the donor/acceptor
conformation could be key to increasing the radiative quantum efficiency and hence
Voc.
3.12 Conclusions
We have shown that the available experimental evidence strongly points toward a
model of recombination in organic solar cells where free carriers are in equilibrium
with CT states. This description simplifies understanding the recombination process
and enabled us to directly link the low open-circuit voltage of organic solar cells to a
combination of their high degree of mixing, short CT state lifetimes, large amounts
of interfacial energetic disorder and low dielectric constants leading to high CT state
binding energies. We quantify the impact of each of these parameters and physically
explain both the dependence of qVoc on Ect,exp and the generally observed 0.5 to
0.7 eV difference between them. Our work shows that there is significant practical
potential for improving Voc, provided we target the right parameters. For example,
reducing interfacial energetic disorder and the CT state binding energy could raise Voc
by hundreds of mV without requiring any change to the CT state lifetime or degree
of mixing. The picture of Voc that emerges is one of a quantity that is limited mainly
by the microscopic details of the interface between donor and acceptor molecules. By
optimizing this interface, the OPV community has the opportunity to significantly
enhance the efficiency of organic solar cells through increases in open-circuit voltage.
3.13 Experimental Details
3.13.1 Sample Preparation
Substrates used for FTPS samples were ITO-coated glass (Xinyan Technologies,
LTD.). Substrates were immersed in a detergent solution of 1:9 extran:deionized
water solution then scrubbed with a brush. Samples were then sonicated in the
detergent solution, rinsed with deionized water, sonicated in acetone, sonicated in
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 69
isopropanol, and blown dry with nitrogen. Substrates were stored in an oven held at
115 C. Immediately before depositing films onto substrates, substrates were exposed
to a UV-ozone plasma for 15 minutes. PC60BM was purchased from Solenne BV.
RRa-P3HT was obtained from Reike. A solution of 1:4 wt:wt RRa-P3HT:PC60BM
was prepared in chloroform at a polymer concentration of 4 mg/ml, and was heated
and stirred at 70 C overnight. The RRa-P3HT:PC60BM film was deposited in a nitro-
gen filled glovebox (H2O and O2 levels typically ¡ 10 ppm) onto prepared substrates
via spin-coating at 1000 RPM for 45 seconds with a ramp speed of 500 RPM/sec. Top
electrodes consisting of 7nm of calcium and then 250nm of aluminum were deposited
via thermal evaporation (approximately 1x10−7 torr).
3.13.2 FTPS measurements
Temperature dependent FTPS measurements of the 1:4 RRaP3HT:PCBM sample
were performed using a Nicolet iS50R FT-IR spectrometer, with signal amplified
using a Stanford Research Systems Model SR570 Low-Noise Current Pre-Amplifier.
Samples were mounted on the cold finger of a Janis Research Company ST-100H
cryostat. Thermal paste was used to maintain good thermal contact between the
cold finger and the sample. Sample temperature was controlled using a LakeShore
331 Temperature Controller. The sample was measured at several temperatures from
82K to 300K. Before each measurement, the sample temperature was set to the de-
sired value with the temperature controller and then allowed to stabilize until less
than 0.05K variation in temperature was observed. The photocurrent spectrum was
then recorded with no band pass filter, and with two bandpass filters which blocked
all transmission of light with wavenumber larger than approximately 13800 cm-1 and
12088 cm-1, respectively. The three resulting spectra were stitched together, prior-
itizing the spectra generated with the lowest wavenumber bandpass filter, to create
a photocurrent spectrum for the sample. Charge Transfer Parameter Determination
Values of Ect,exp and λexp were determined for each temperature independently. To
determine Ect,exp and λexp, the sub-bandgap absorption was fit to Marcus Theory
absorption expression shown in the SI using a linear least squares fitting procedure.
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 70
Under the assumption that Marcus Theory is a good description of the CT absorption,
the fit was restricted to the portion of the sub-bandgap absorption whose natural log
had a linear first derivative (i.e. (d(ln(E*(E))/dE is linear).
3.14 Why We Expect the CT State Distribution
to be Gaussian
Charge Transfer states are composed of an electron and a hole in separate materials;
therefore we should be able to relate the distribution of CT states to the Electron
Affinity and Ionization Potential of the two materials. Since these are energetically
disordered materials, the EA and IP take a range of values at different positions in
the film.
CT(E) = Acceptor EA(E)−Donor IP(E)− EB (3.28)
The distribution of CT states then can be described in terms of the distribution
of free carrier energy levels of the acceptor and donor materials, modified by an
interaction energy EB. Since the low energy portions of the energy levels of organic
semiconductors are usually described as having a Gaussian shape, and the difference of
any two Gaussians is always a Gaussian distribution even in the presence of arbitrary
correlations between the two distributions, we can say that the relevant low energy
portion of Acceptor EA(E) - Donor IP(E) should be Gaussian in shape.[6, 8] We would
expect that EB will also be a distribution of values since there will be conformational
and dipolar disorder at the interface between the donor and acceptor materials. Since
there are a large number of interactions that set EB, we can invoke the Central Limit
Theorem to argue that EB should be normally distributed as well. Thus to first
approximation, CT(E) should have a normal distribution.
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 71
3.15 Inhomogeneously Broadened Marcus Theory
Absorption
From Marcus Theory, one can calculate the absorption spectrum of a single molecular
excitation using the expression:[40]
α(E,E0) =f
E√
4πλkTexp
(−(E0 + λ− E)2
4λkT
)(3.29)
where λ is the reorganization energy of the molecule, E0 is the energy of the relaxed
excited state and f is the electronic coupling. When there are N identical molecules
that all have the same energy levels, the combined absorption expression is simply
N ∗ α(E,E0). However, since we have an inhomogeneously broadened distribution
of CT states absorbing light, we should integrate this expression over the density of
states to get the actual absorption:
αct(E) =
∫ ∞−∞
α(E,E ′)g(E ′) dE′ (3.30)
where g(E) is the distribution of CT states, which we will assume is Gaussian:
g(E;Ect) =Nct
σct√
2πexp
(−(E − Ect)2
2σ2ct
)(3.31)
Performing the above integration yields:
αct(E) =fNct
E√
2π√σ2ct + 2λkT
exp
(−(Ect + λ− E)2
2(σ2ct + 2kTλ)
)(3.32)
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 72
While this appears distinct from the original expression for Marcus Theory Ab-
sorption, we can put it in the same form by making the following identifications:
λ′ = λ+σ2ct
2kT(3.33)
Eexpct = Ect −
σ2ct
2kT(3.34)
The simultaneous modification of the reorganization energy and the CT state
energy cancel in the numerator of the exponential function while putting the denom-
inator into the correct form for Marcus Theory. Thus, any Gaussian distribution
of Marcus Theory absorbers will be indistinguishable from a single Marcus Theory
absorber since the functional form of the absorption expression is identical. However,
both the reorganization energy and CT state energy will be temperature dependent
in the case of an inhomogeneously broadened distribution, allowing us to distinguish
between homogeneous and inhomogeneous broadening using temperature dependent
measurements.
3.16 Relating CT State Density and Chemical Po-
tential
Previously in this chapter, we calculate the number of occupied CT states based on
knowledge of the chemical potential for CT states, µct, and the density of states.
First, we assume that the CT states do not interact with each other so that each
CT state can be treated independently. We make this assumption since, as we will
show in the next section, the density of CT states is much lower than the density of
interfaces, so each CT state should be formed far from any other. This independence
assumption means that we can consider each interfacial site in isolation. Formally, it
means that we can decompose the grand canonical partition function into a product
of partition functions for isolated interfaces. Consider then, an interfacial site where
a CT state could form. If there is no CT state occupying the interface, the energy
associated with the interface is 0. If there is a CT state, the energy of the interface is
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 73
ε, the CT state energy associated with that interface. To make the derivation more
analogous to the corresponding result for electrons and holes we assume that it is
energetically very unfavorable for two CT states to form at the same interface since
the electron and hole portions of the CT states would repel each other. We will show
later that our result does not actually depend on this assumption since during solar
cell operation there are far fewer occupied CT states than interfaces so the issue of
double occupancy does not play a role.
We want to determine the odds of that interface being occupied given that the
chemical potential of the interface is µct. There are two necessary results from sta-
tistical mechanics that we will need. First, we need the idea of the grand canonical
ensemble, which is the mathematical entity that determines the behavior of a system
(our single interface) at equilibrium when it is allowed to exchange energy and parti-
cles with a reservoir (the reservoir of free electrons and holes). The grand canonical
partition function is defined as:
ξ =∑
s∈states
exp
(−E(s) + µN(s)
kT
)(3.35)
where E(s) is the energy of a state, s, of the system, µ is the chemical potential and
N(s) is the number of particles in the system, which unlike in the canonical ensemble
is not fixed.
Given an expression for ξ, we can calculate the expected value of the number of
particles in the system as:
〈N〉 = kT∂ ln ξ
∂µ(3.36)
The proof of these results is in any standard statistical mechanics text.
Since there are only two possible states of our interface, occupied or free, the
grand partition function is particularly simple:
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 74
ξct = 1 + exp
(−ε+ µctkT
)(3.37)
The odds of this interface being occupied then, is given by:
〈N〉 = kT∂
∂µctln
[1 + exp
(µct − εkT
)](3.38)
Performing the differentiation leads to the standard Fermi-Dirac distribution, just
as it does for electrons and holes:
〈N〉 =1
exp(e−µctkT
)+ 1
(3.39)
Note that in this case, we arrive at a Fermi-Dirac distribution not because we
assumed that the CT states were fermions but simply because we assumed it was
energetically unfavorable for multiple CT states to form at the same interface making
such configurations effectively inaccessible. Now, since we know the odds of any given
interface being occupied is very small since we calculated the maximum number of
CT states during solar cell operation in Section 6 and found it to be orders of mag-
nitude smaller than the number of interfaces, we can simplify ¡N¿ into a Boltzmann
distribution since the exponential term in the denominator must be much larger than
1 for 〈N〉 to be much smaller than 1. Had we made a different assumption about
whether multiple CT states could form at the same interface it would have resulted
in the same simplified expression in the low CT state concentration limit relevant for
organic solar cell operation so that assumption turned out to be unimportant. The
odds of any interface being occupied then (Pct), given its energy E and the chemical
potential of CT states µct is:
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 75
Pct = exp
(µct − EkT
)(3.40)
Since we have a distribution of interfaces with different energies given by gct(E)
as discussed in a previous section, the total number of occupied interfaces, i.e. the
total number of CT states, is found by adding up the probability that each individual
interface is occupied, which we express as an integral:
nct = ∫∞−∞ gct(E) exp
(µct − EkT
)dE (3.41)
This is the result that we quoted previously. We evaluate the integral in the next
section.
3.17 Defining an Effective Density of CT States
In the previous section we show that you can write the density of CT states in terms of
the energetic distribution of interfacial states and a Boltzmann-like factor containing
the chemical potential. In this section we show how to calculate the resulting integral.
Our goal is to find a way to determine the total number of occupied CT states given
their chemical potential, µct, which is given by:
nct(µct) =
∫ ∞−∞
gct(E)f(E, µct) dE (3.42)
where gct(E) is the arbitrary distribution of CT states and f(E, µct) is the Fermi-
Dirac distribution function. When µct is far from the energy of the majority of the
CT states, we can replace the Fermi-Dirac distribution with a Boltzmann distribution
with very little error as we indicated in the previous section. The condition on µct
for this approximation to be valid is given by Neher for a Gaussian density of states
as:[8]
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 76
µct < Ect −σ2ct
kT(3.43)
Since we have shown σct is less than 110 meV for organic solar cell interfaces, this
means that µct must be more than 470 meV away from the center of the CT state
distribution for the Boltzmann approximation to hold. In operation up to one sun,
organic solar cells have µct greater than 0.5 eV away from Ect so the Boltzmann ap-
proximation clearly holds near room temperature. We can rewrite the above integral
then as:
nct(µct) =
∫ ∞−∞
gct(E) exp
(µct − EkT
)dE (3.44)
Now one can break up the exponential function into two parts, one of which has
no E dependence so it comes out of the integrand:
nct(µct) =
[∫ ∞−∞
gct(E) exp
(Ect − EkT
)dE
]exp
(µct − EctkT
)(3.45)
Here we chose to define an arbitrary reference energy to describe the CT state
distribution using its average Ect. We could have picked any other point. The integral
in brackets has no dependence on the Fermi level position and so it is simply a
constant, which is what we define as the effective density of states Nct. While this
can be done for any density of states, only for some special cases is there an analytical
expression for the result. For a Gaussian distribution, Nct is given by:
Nct = fN0 exp
(σ2ct
2(kT )2
)(3.46)
where fN0 is the total number of CT states and the exponential factor captures
the fact that the lower energy portion of the distribution is far more likely to be
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 77
populated than Nct states all located exactly at Ect. Another way to present this
result is that a Gaussian density of states (DOS) is equivalent to a delta function
DOS located at:
Eexpct = Ect −
σ2ct
2kT(3.47)
This is the same derivation that is used to define the effective density of valence
and conduction band states for inorganic semiconductors.
3.18 The Voltage Dependence of τct
Since in quasi-equilibrium low energy CT states are much more likely to be populated
than higher energy CT states and these states are likely to have a different natural
lifetime, we need to ask if we would expect the average CT state lifetime to vary as
a function of voltage since different voltages could potentially result in different pop-
ulations of the CT state distribution. To answer this question we need to rigorously
define τct. τct is the average recombination lifetime of all populated CT states at a
given voltage, so:
1/〈τct〉 =1
Z
∫ ∞−∞
gct(E)
τct(E)exp
(µct − EkT
)dE (3.48)
where Z is a normalization factor defined as:
Z =
∫ ∞−∞
gct(E) exp
(µct − EkT
)dE (3.49)
For the same reason that we can define an effective CT state density, Nct, in the
previous section, we can define an average CT state lifetime 〈τct〉 in a voltage inde-
pendent manner since we can pull the CT state quasi-Fermi level out of the above
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 78
integrals and see that it cancels out. Another way to state this result is that in Boltz-
mann statistics, the energetic distribution of populated CT states does not change
with voltage, rather the entire distribution is simply scaled by a voltage dependent
constant. Since the ratio of populated CT states at different energies doesn’t change,
their potentially different lifetimes do not contribute in different manners at different
voltages and so we can express the average lifetime as a voltage-independent constant
regardless of how complicated τct(E) might be. This has been confirmed experimen-
tally by CT state electroluminescence measurements that show the normalized EL
distribution from the CT states in 10 different OPV systems is voltage independent
until you go into forward bias far enough that the Boltzmann approximation breaks
down.[34]
3.19 The Low Temperature Limit of Voc
The open-circuit voltage of all solar cells is temperature dependent and typically
increases as the temperature of the cell is decreased. For Langevin recombination, we
would expect the solar cell open-circuit voltage to obey the following relation:
Voc = E0 − kT log
(qN2
0LklanJsc
)(3.50)
Thus as the solar cell is cooled down, the open-circuit voltage should approach
E0 linearly. It is tempting to think that Voc will equal E0 at 0K, but in reality,
nonidealities not captured in the above equation prevent this from actually occurring,
so the verification of the relation is done by linearly extrapolating back to 0K from
the high-temperature regime in which the equation holds. The principal assumptions
that break down at low temperature and lead to deviations from the above equation
are:
1. Temperature dependent current production. Even if the initial photogeneration
step in organic solar cells is temperature independent, the subsequent transport
is not and so at low temperature the internal quantum efficiency of the solar cell
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 79
could decrease, meaning that Jsc in the above equation would be temperature
dependent.[77]
2. Non-selective contacts. As Voc increases, it may be the case that the voltage
becomes pinned to the built-in potential of solar cell. This can happen if Vbi
is less than Ectexp (or E0 in the Langevin limit). This will manifest itself as a
roll-off in Voc at low temperature where the predicted Voc would be higher than
Vbi so Voc instead approaches Vbi at low temperatures.[39]
3. Breakdown of the Boltzmann Approximation. The above equation is only valid
for temperatures and voltages for which we can use the Boltzmann approxi-
mation instead of the Fermi-Dirac distribution. While this approximation is
very good in inorganic solar cells even at cold temperatures, the high degree
of energetic disorder in organic solar cells makes the assumption break down
well above 0K. We provide a specific criterion in the above section on Defining
an Effective Density of CT states but as a rule of thumb, we find in numerical
simulations that it typically begins to break down around 150-200K.
3.20 The Light Ideality Factor
Like Langevin Recombination, our model predicts that a plot of Voc vs kT*log(Jsc)
should have a slope of 1. This value is called the light ideality factor. Many OPV
materials systems have been observed to have light ideality factors very near 1.[7-13]
A few material systems, though, have been observed to have light ideality factors
other than 1, indicating that in those systems there are loss mechanisms that are
either not proportional to np or that the quasi-Fermi levels are not flat at open-
circuit.[97, 15] In some cases the additional mechanism has been identified to be non-
selective contacts.[39] In other cases it is unclear what the origin is, however light
ideality factors higher than 1 may be caused by many factors including trap-assisted
recombination or simply the presence of shunts in the solar cell.[98, 88]
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 80
System Reduction Factor NotesPCPDTBT:PC70BM[12] 0.77, 0.83 Measured with two techniquesPCPDTBT:PC70BM[64] 0.2 No DIOPCPDTBT:PC70BM[64] 0.07 3% DIOF-PCPDTBT:PC70BM[64] 0.14 No DIOF-PCPDTBT:PC70BM[64] 0.04 1% DIOF-PCPDTBT:PC70BM[64] 0.03 3% DIOmono-DPP:PCBM[43] 0.11 Solution processed small moleculebis-DPP:PCBM[43] 0.03 Solution processed small moleculeP3HT:PCBM[9] 0.1 As-castRRa-P3HT:PCBM[48] 4x10−4 Regiorandom P3HTP3HT:PCBM[12] 0.06 Annealed at 170C for 2 minutes
Table 3.3: Tabulated Langevin Reduction Factors from Literature
3.21 The Langevin Reduction Factor
Table 3.3 summarizes all of the measurements of the Langevin Reduction factor at
room temperature that we were able to find in the literature.
The report comparing PCPDTBT:PC70BM with the fluorinated version of the
same polymer is particularly interesting since they report both a decrease in the
Langevin Reduction factor upon fluorination as well a corresponding increase in gem-
inate separation and a reduced field-dependence for the geminate separation process.
This is consistent with the Langevin reduction factor measuring the likelihood of a
CT state splitting into free carriers, since lower values should imply lower geminate
as well as nongeminate recombination.
3.22 CT State Lifetimes
Measuring the back electron transfer rate at a heterointerface in organic solar cells is
very difficult. In this section we summarize the lifetime measurements that have been
performed with different techniques. Each measurement technique has its own partic-
ular complications but taken together we believe they support the general statement
that the average CT state recombination lifetime is somewhere between 100 ps and
10 ns for most organic solar cell materials.
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 81
System Lifetime MethodP3HT:F8BT[4] 1.3 ns Pump-push photocurrent decayP3HT:PCBM[4] 840 ps Pump-push photocurrent decayPCPDTBT:PC70BM[4] 400 ps Pump-push photocurrent decayMDMO-PPV:PC70BM[4] 580 ps Pump-push photocurrent decayPFB:F8BT[4] 6.8 ns Pump-push photocurrent decayP3HT:PCBM (nonannealed)[30] 780 ps Pump-push photocurrent decayP3HT:PCBM (annealed)[30] 660 ps Pump-push photocurrent decayMDMO-PPV:PC70BM (1:1)[30] 600 ps Pump-push photocurrent decayMDMO-PPV:PC70BM (1:2)[30] 460 ps Pump-push photocurrent decayMDMO-PPV:PC70BM (1:4)[30] 480 ps Pump-push photocurrent decayP3HT with 5 fullerene types[56] 3-6 ns Polaron transient absorption decayP3HT:PCBM (4:1)[49] 500 ps Terahertz SpectroscopyP3HT:PCBM (1:1)[49] 450 ps Terahertz Spectroscopy
Table 3.4: Reported measurements related to the CT state lifetime in literature
3.23 The Applicability of Chemical Equilibrium to
Electrons and Holes
The method we used to derive our main result in this work is to apply concepts from
chemical equilibrium, i.e. the notion of equating chemical potentials of reactants and
products to find out what concentrations of each you will have when the reaction is
allowed to equilibrate. These are very general concepts that are applied frequently to
electrons and holes in solar cells. However, it is typically not explicitly stated that this
is what is being done. In this section we briefly review some of the standard results
that rely on the same method we use in this paper to provide additional support for
our use of it.
The most commonly used result is often simply called the Law of Mass Action,
which states that at equilibrium the product of the electron and hole concentrations
in a (non-degenerately doped) semiconductor is equal to a constant:
np = n2i (3.51)
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 82
This result comes from the equilibrium condition of the following chemical reac-
tion:
n + p←−→ nothing
The forward direction of the reaction is recombination and the reverse direction is
thermal generation. The reaction simply says that electron-hole pairs can annihilate
each other or be formed from nothing. Since the chemical potential of nothing is
0 (assuming there to always be an excess of valence band electrons that could be
promoted to the conduction band), the equilibrium condition of this reaction is:
µe + µh = 0 (3.52)
where µe is the chemical potential of the electrons and µh is the chemical potential
of the holes. As discussed in the main text, the chemical potential of electrons is
equal to its quasi-Fermi level (Efn) and the chemical potential of holes is equal to the
opposite of its quasi-Fermi level (Efp). Substituting these relations into the above
equation yields:
Efn − Efp = 0 (3.53)
This implies that that Efn = Efp, i.e. that the two quasi-Fermi levels are equal
to each other, which in this case we simply call the Fermi level, Ef . Using equation
1.4 in the main text we can relate Efn and Efp to electron and hole densities so that:
np = N20 exp
(Efn − Ec
kT
)exp
(Ev − Efp
kT
)(3.54)
np = N20 exp
(−(Ec − Ev)
kT
)(3.55)
np = N20 exp
(−E0
kT
)(3.56)
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 83
where we have used the result that Efn = Efp when electrons and holes are in
equilibrium with each other. The point of rederiving this basic result is to show how,
as the name suggests, it is actually a product of the same type of analysis that we use
in this paper, demonstrating that the analysis is applicable to electronic excitations
in solar cells.
As we discussed in the main text, the logic of setting up a chemical reaction
between electronic excitations and looking for the equilibrium condition was used by
Wurfel to relate electron and hole densities to photon densities by considering the
radiative recombination (or generation) reaction:
n + p←−→ photon
Similarly, Wurfel considered nonradiative recombination by allowing electrons and
holes to react with phonons:
n + p←−→ N · (phonons)
The reverse direction of this reaction is simply thermal generation of electron/hole
pairs.
As another example, materials scientists commonly consider defect reactions with
associated chemical potentials and equilibrium concentrations for crystalline defects
like interstitials or vacancies.[104] All of these techniques are built on the same foun-
dation of calculating the distribution of states that minimizes a systems free energy,
but using the concept of chemical potentials, or molar free energies or quasi-Fermi
levels to simplify the associated mathematics.
Seen in this context, our discussion of the consequences of CT state equilibrium
with free electrons and holes does not require novel methods of analysis, its simply
that we had not before had reason to think that CT states were close to equilibrium
with free carriers. Our work just analyzes the equilibrium condition of the following
reaction and its implications:
n + p←−→ CT
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 84
Acceptor EA
Donor IP
E0qVoc
Ef
q(Vbi - Voc)
Figure 3.8: Simplified OPV device schematic.
3.24 Deriving our Result Directly From the Canon-
ical Ensemble
In the preceding sections, we presented our expression for recombination and hence
Voc as a consequence of aligning chemical potentials across a chemical reaction in-
volving an electron/hole pair forming a CT state and resplitting. This presentation is
mathematically simple and generally applicable, which is why we chose it, but some
readers may prefer a more physical approach. In this section we derive the same
result directly from the many-particle partition function for a simplified organic solar
cell model that turns out to be analytically solvable.
A schematic for the simplified device model that we will use in this derivation is
shown in Figure 3.8. The key simplifications that we make are:
1. We do not include the effects of energetic disorder in either the donor Ionization
Potential or the acceptor Electron Affinity, modeling both energy levels as delta
functions with N0 states per unit volume. We do include a Gaussian density of
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 85
CT states with center energy Ect and width σct so that the reader can see the
effect of interfacial disorder on recombination.
2. We ignore any potential band bending and assume that the electric field across
the device is uniform.
3. We assume that the device is finely intermixed so that any location in the device
could host either an electron or a hole.
4. We reduce the Coulomb interaction between electrons and holes to a simple
nearest neighbor interaction. When an electron and hole are in a CT state,
their energy is assumed to be reduced by EB, the CT state binding energy,
and when they are not in a CT state, they are assumed to not feel each others
presence at all.
5. We assume that the active layer is overall charge neutral (n = p).
6. We assume that the charge carrier distribution is precisely uniform in the plane
of the device (though not in the direction in which the electric field is applied)
so that we can use periodic boundary conditions in our derivation.
7. We assume that the carrier density is low enough that an electron only ever
interacts with one hole at a time. This assumption lets us expand the partition
function into a sum over electron-hole pairs.
Of these assumptions, 4, 5, 6 and 7 are key to the derivation. 1-3 can be relaxed at
the cost of additional mathematical complexity without affecting either our method
of solution or our ability to express the result in terms of elementary functions. We
include assumptions 1-3 since the goal of this section is to present an alternative
view on an expression that we have already derived quite generally using chemical
potentials, so we dont want to introduce additional mathematical complexities that
obscure the core idea of the exposition.
Our first task is to calculate the probability that a given electron is part of a Charge
Transfer state at any instant in time. Because the CT states are in equilibrium with
free charges, we know that we can express this probability given the many-particle
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 86
partition function of the system and knowledge of the free energy difference between
bound CT states and free carriers. The free energy difference includes an entropic
component since two carriers have many more spatial arrangements that result in
them not being nearest-neighbors than being nearest neighbors. In a box with volume
V, there are V2 two particle states but only V CT states. We will need a way to
incorporate this effect into our expression for recombination. Starting with what we
know to be true:
Z =∑
s∈states
exp
(−E(s)
kT
)(3.57)
where Z is the system’s partition function, which is just a sum over all possible
configurations of all electrons and holes in the device. To make progress, we will
assume that we only need to take into account the interactions between an electron
and its nearest hole, i.e. 3-body effects are unimportant. This means that we can
expand the above equation into a sum over pairs of electrons and holes. Further,
we will assume that the carriers are uniformly distributed in the plane of the device
so that we can introduce periodic boundary conditions and replace the sum over all
pairs of electrons and holes with a single pair of one electron and one hole over a
small portion of the device. If the electron (and hole) density is n per unit volume,
this means that the area (in the plane of the device) in which we would expect to
find a single electron and a single hole is:
A =1
Ln(3.58)
So, this is the problem that we need to solve: given one electron and one hole in
a rectangular solid with area A and height L, under a uniform voltage potential V =
Vbi - Voc applied along the L direction, what fraction of the time will the electron and
hole be located next to each other. To further simplify, we will assume that there are
only two possible states: fully bound CT state and fully free carriers. Not making
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 87
this simplification leads to the partition function containing the Exponential Integral,
which is not expressible in terms of elementary functions. With these assumptions,
the partition function reduces to:
Z =∑ct
exp
(−EctkT
)+∑s∈free
exp
(−E(s)
kT
)(3.59)
Our strategy for evaluating Z will be to convert the summations to integrals and
directly compute them. Working term-by-term we have for the first term:
∑ct
exp
(−EctkT
)≈∫ ∞−∞
gct(E) exp
(−EkT
)dE (3.60)
We have already evaluated this integral in the section on defining an effective
density of CT states, so we just quote the result here:
∑ct
exp
(−EctkT
)≈ NctAL exp
(−Ect +
σ2ct
2kT
kT
)= NctAL exp
(−Ectexp
kT
)(3.61)
where Nct is the density of CT states per unit volume and AL is the volume of the
periodic cell of the solar cell that we are considering, which is a function of carrier
density.
Turning to the free carrier term in the partition function, we have that the energy
of an electron-hole pair due to their positions in the applied voltage potential is given
by:
E(ze, zh) = V (zh − ze) + E0 (3.62)
where ze and zh are the electron and hole positions along the electric field axis
and E0 is the energy required to create an electron-hole pair from the Ground State.
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 88
The voltage (V) is taken to be equal to the built-in potential minus the open-circuit
voltage.
∑s∈free
exp
(−E(s)
kT
)≈ N2
0A2 exp
(−E0
kT
)∫ L
0
∫ L
0
exp
(qV zekT
)exp
(−qV zhkT
)dzedzh
(3.63)
This is an integral over all possible combinations of electron and hole locations
in our periodic cell taking into account that electrons and holes are charged so the
energy of the pair depends on their locations in the field direction but not in the
other two dimensions. We will make the additional simplifying assumption that even
at open-circuit, qV ¿¿ kT so that we can express the result as:
∑s∈free
exp
(−E(s)
kT
)≈ (N0AL)2
(kT
qV
)2
exp
(qV − E0
kT
)(3.64)
Thus the final result for the partition function is:
Z = NctAL exp
(−Ectexp
kT
)+ (N0AL)2
(kT
qV
)2
exp
(qV − E0
kT
)(3.65)
We can now find the odds that an electron and hole will be in a CT state in the
standard way:
Pct =1
Z
∑ct
exp
(−EctkT
)(3.66)
The summation will turn into the same integral we just evaluated to determine
the CT state portion of the partition function and cancel in the numerator and de-
nominator, leaving:
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 89
Pct =
[1 +
N0
Nct
AL
(kT
qV
)2
exp
(qV − E0 + Eexp
ct
kt
)]−1
(3.67)
Based on the results from Section 6 of the main text, we know that the odds of
a free carrier occupying a CT state under normal operating conditions of a solar cell
are less than 1% so we know that the second term must be much greater than 1 and
we can simplify Pct by ignoring the 1.
Pct =Nct
ALN20
(qV
kT
)2
exp
(−qV + E0 − Eexp
ct
kt
)(3.68)
This expression says that the higher the applied field across the device, the lower
the odds of a CT state being occupied and the higher the energy of the CT state
distribution, the lower the odds of it being occupied. To find out the density of CT
states we have:
nct = Pctn (3.69)
nct =Nctn
2
N20
(qV
kT
)2
exp
(−qV + E0 − Eexp
ct
kt
)(3.70)
where we have substituted for the periodic volume (AL) in terms of the carrier
density n.
The final task is to solve for the carrier density as a function of the open-circuit
voltage. This is nontrivial because of the presence of the electric field across the
device. At every point in the device we know that the local density of electrons is
given by a Boltzmann distribution (since we are assuming that no portion of the
device is degenerate) that depends on the distance between the quasi-Fermi level for
electrons and the acceptor LUMO. We have defined this difference at the electron
extracting contact as Ef , however since the LUMO is tilted, the difference is a linear
function of position. The average density of charge carriers then is:
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 90
n(V ) =1
L
∫ L
0
n
(Ef +
qV z
L
)dz (3.71)
n(V ) =1
L
∫ L
0
N0 exp
(−Ef − qV z
L
kT
)dz (3.72)
where V is the voltage across the active layer, i.e. Vbi - Voc, and the quasi-Fermi
levels are assumed to be flat. Evaluating the integral leads to:
n(V ) =N0kT
qVexp
(−EfkT
)(3.73)
We can now substitute our expression for n(V) into our expression for nct(n, V)
to get an expression for nct in terms of voltage only.
nct = Nct exp
(qVoc + E0 − qVbi − 2Ef − Eexp
ct
kt
)(3.74)
where we have expanded V = Vbi - Voc. This expression does not yet look identical
to the one we derived in the main text because of the apparent dependence of nct on
Vbi, E0 and the specific details of the Fermi level alignment at the electron and hole
extracting contacts captured in Ef . However, looking at Figure 3.8, we can see that
Vbi, E0 and Ef must all be related and in fact:
E0 = qVbi + 2Ef (3.75)
So, actually Vbi, E0 and Ef identically cancel out in our expression for nct and we
are left with simply:
CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 91
nct = Nct exp
(qVoc − Eexp
ct
kt
)(3.76)
which agrees exactly with what we derived in the main text by aligning chemical
potentials across the n + p ←−→ CT reaction. The additional insight that we gain
from the canonical ensemble approach is obtained by considering why Vbi, E0 and
Ef cancel out in this derivation. Simply put, anything one does to E0 or Vbi will
simultaneously change both the odds of forming a CT state and the carrier density
in opposite ways that exactly cancel, which is why these terms do not impact the
density of CT states and hence do not affect Voc. This cancelation was implicit in
the chemical potential approach but we explicitly see how it occurs in the canonical
ensemble approach.
Chapter 4
The Fill Factor
In the preceding chapters, we have explained why the short-circuit current of an
organic solar cell is voltage independent, since geminate pairs move rapidly and ex-
perience an inhomogeneous energetic landscape that favors separation. We have also
explained that recombination should turn on exponentially as the voltage on the so-
lar cell approaches the experimentally measured CT state energy, which we showed
quantified the CT state DOS in the appropriate way to describe the number of CT
states in the solar cell that could immediately recombine. In this chapter, we focus
on the remaining parameter of a solar cell, the Fill Factor, which describes the shape
of the IV curve between short-circuit and open-circuit.
We begin with a question. If photocarrier generation is voltage independent and
recombination turns on exponentially, why does the IV curve of an organic solar
cell not look like an exponential function minus a constant? In other words, why is
the Fill Factor so low? A schematic comparison between an IV curve that is purely
exponential and a typical organic solar cell IV curve is shown in Figure 4.1. While
the two curves have the same short-circuit currents and similar open-circuit voltages,
the curve on the right has a significantly reduced fill factor due to its shape, causing
a corresponding reduction in efficiency.
Traditionally, the explanation for the low and highly variable Fill Factors of or-
ganic solar cells has been that the devices show field dependent photogeneration,
92
CHAPTER 4. THE FILL FACTOR 93
Voltage
Curr
ent
Voltage
Curr
ent
Figure 4.1: (left)An IV curve where recombination is purely described by a singleexponential function, resulting in a device with a high Fill Factor. (right)A typicalIV curve for an organic solar cell, where recombination is not a simple exponentialfunction of voltage, resulting in a device with a low Fill Factor and reduced efficiency.
where the yield of free carriers from the geminate splitting process is a sensitive func-
tion of the applied voltage of the solar cell[75, 23]. We have, however, shown in
Chapter 2 why that should not be the case, so we need to find another way to explain
the low Fill Factors.
In this chapter, we first come up with an analytical theory for the Fill Factor of
organic solar cells based on a novel perturbation approach and then compare that
theory extensively with experiment to show that it works remarkably well for de-
scribing real organic solar cells. Finally, we use the validated theory to explain and
understand the factors that affect the Fill Factor of an organic solar cell.
4.1 The Myth of the Intrinsic Organic Solar Cell
Before we can build a working theory, we first need to dispel what has become a
persistent myth in the organic solar cell community. Many researchers think of the
organic materials used to make organic solar cells as intrinsic semiconductors. In this
work we accept this statement as likely true and do not dispute it. However, it is then
often concluded that since organic solar cells are make from intrinsic semiconductors,
CHAPTER 4. THE FILL FACTOR 94
Acceptor Electron Affinity
Donor Ionization PotentialEner
gy
Position Length109
1010
1011
1012
1013
1014
1015
1016
1017
1018
Carr
ier
Densi
ty [
cm−
3]
Electron Density
Hole Density
Figure 4.2: (left) The band diagram of an organic solar cell at equilibrium in thedark showing how the built-in potential causes a tilt to the energy levels which leadsto carrier accumulation near the contacts of the solar cell. (right) Schematic darkelectron and hole density in an organic solar cell as a function of position with ap-proximately correct magnitudes showing how there is a very large carrier density nearthe two solar cell contacts.
they must have very low dark carrier densities. This statement is not true and based
on the false premise that the only source of dark charge carriers in a solar cell is due
to dopants present in the active layer. In fact, in the P-I-N architecture favored by
organic solar cell researchers, the majority of dark carriers are injected by the P and N
type contacts and will always be present regardless of whether or not the active layer
is intrinsic when considered as a slab of bulk semiconductor without metal contacts.
To see this simply consider the band diagram of an organic solar cell in the dark
as shown in Figure 4.2. The dashed line in Figure 4.2 shows the position of the
Fermi level in the device. The electron density at a given position in the device is an
exponential function of the distance between the Fermi level and acceptor’s Electron
Affinity, which is why there are orders of magnitude more electrons near the electron
extracting contact than in the rest of the solar cell and similarly orders of magnitude
more dark holes near the hole extracting contact simply due to the requirement that
the active layer be in equilibrium with the contacts on either side of it.
CHAPTER 4. THE FILL FACTOR 95
4.2 Why Dark Carriers Matter
Most treatments of recombination in organic solar cells ignore the presence of dark
charge carriers leading to qualitatively incorrect conclusions about recombination in
the devices. As we showed in Chapter 3, recombination is proportional to the product
of the electron and hole densities as each position in the device. Assume for a second
(as we will later show is the case for organic solar cells), that the excess electron
and hole densities in the solar cell are proportional to the light intensity with which
you are illuminating the device. Then, in the absence of dark carriers, recombination
could be expressed as:
Jrec = knp ≈ knlpl ∝ Φ2 (4.1)
where k is a proportionality constant, nl and pl are the light induced electron and
hole populations and Φ is the light intensity. Equation 4.1 says that nongeminate
recombination should be proportional to the square of the light intensity, which means
that we should be able to reduce its magnitude by decreasing the light intensity below
one sun to turn off this recombination mechanism and measurements done at low
enough light intensity should not be affected by bimolecular recombination.
Now lets consider the case with dark carriers.
Jrec = knp = k(nd + nl)(pd + pl) (4.2)
Jrec = k (ndpd + ndpl + nlpd + nlpl) ∝ C + Φ + Φ2 (4.3)
When we account for dark charge carriers, we see that there are three distinct kinds
of bimolecular recombination, with three different light intensity dependences: dark-
dark carrier recombination, dark-light carrier recombination and light-light carrier
recombination. Only the last form of recombination has a quadratic dependence
on light intensity, so only that form can be made negligible at low light intensity.
Properly considering the presence of dark carriers in organic solar cells is critical for
the correct interpretation of a range of experimental results and, as we will show in
the rest of this chapter, also key to understanding why the Fill Factors of organic
CHAPTER 4. THE FILL FACTOR 96
solar cells are both low and poorly controlled.
4.3 Methodology
Ultimately, coming up with a theory for the Fill Factor of an organic solar cell requires
having a function that calculates the density of electrons and holes everywhere in the
solar cell as function both of the voltage that is being applied to the cell and the
current being drawn. Using this expression, we could then write down the total rate
of recombination as a function of J, the driven current, and V, the applied voltage,
and be able to describe the operation of the solar cell at any point on its IV curve.
Unfortunately, the required drift-diffusion equations that describe how many charge
carriers are present in various places in the solar cell away from open-circuit are not
typically solvable analytically, making it difficult to get insight into what determines
the Fill Factor without resorting to opaque and complicated numerical simulations.
It is likely for this reason that such confusion persists in the OPV community about
what determines the Fill-Factor. In this chapter we take a different approach. It
turns out that, in the absence of recombination, one can analytically solve for the
carrier distribution throughout an organic solar cell. This, in turn, will let us calcu-
late the np product analytically to obtain the driving force for CT state formation
(as described in Chapter 3) and hence the rate of recombination. While our result
formally only holds in the limit of very little recombination (i.e. it is a perturbative
approach), we will find that in practice it describes working OPV devices quite well.
4.4 The Carrier Distribution in an OPV Device
To begin, we ask the question, if there were very little recombination in an organic
solar cell, what would the carrier distribution inside the active layer look like as a
function of the current, J, being extracted from the device and the operating voltage
V. The starting point for our calculation is the drift-diffusion equations which relate
CHAPTER 4. THE FILL FACTOR 97
a gradient in the electron or hole quasi-fermi levels to a current. The result is:
Ji(x) = µin(x)dEfidx
(4.4)
where Ji is the current density flowing at location x in 1D, µi is the macroscopic (DC)
charge carrier mobility, n(x) is the charge carrier density and Efi is the carrier quasi-
fermi level. This equation is always satisfied and allows one to calculate a (possibly
spatially varying) current given the quasi-fermi level as a function of position (since
n(x) can be derived from Efi). The rigorous derivation of the form of n(x) can be
done by forcing the derivative of Ji(x) to be zero and solving the associated nonlinear
differential equation, which fortunately turns out to be solvable analytically. However,
more insight is gained by first taking a heuristic perspective.
To begin, assume that we have a device in which a constant electron current
Je is flowing. Further suppose that the device has a built-in potential Vbi and we
are holding it at a voltage V . We want to guess how the charge carriers must be
distributed in order to guarantee a non-spatially varying current. Since there is no
recombination in the device, we know that the current must be the same everywhere.
Imagine that the device is very thick and we are looking at the charge carrier
density far from contacts. There is only one way to guarantee that the current is
constant (looking at Equation 4.4). The charge carrier density must be constant and
the slope of the quasi-fermi level must be constant. One could also imagine that
maybe the charge carrier density is varying and the slope of the quasi-fermi level is
varying inversely to just cancel it out such that the product of the two is constant,
but this cannot work because the slope of the quasi-fermi level is linked to the slope
of the carrier density so they cannot vary in opposing ways.
Since we need the carrier density to be constant, the distance between the quasi-
fermi level and the band edge must be constant, which means that the slope of the
quasi-fermi level must exactly equal the slope of the band, caused by the built-in
voltage. We can invert Equation 4.4 to find this carrier density in terms of the
CHAPTER 4. THE FILL FACTOR 98
Vbi
Efn
n(x)
x
n(0)
n(∞)
Fermi Level Carrier Density
Figure 4.3: The required fermi level and charge carrier density profiles in order tohave a constant current in an intrinsic semiconductor device.
current Je, built-in voltage Vbi, carrier mobility µe and device thickness L:
n(∞) =JeL
qVbiµe(4.5)
So, far from the electron injecting contact we know what n(x) must look like.
It must be constant and have the value n(∞). We also know what n(0) must be
since we are fixing a voltage on the device. This means fixing a position for the
electron quasi-fermi level at x = 0, which in turn fixes the charge carrier density at
the contact. In between these two extremes, the carrier density must smoothly join
the two limiting values. One could imagine this smooth joining process happening in
an arbitrary way, but in fact it must happen very simply. Assume (as we will see is
usually the case) that n(0) > n(∞), i.e. more carriers are needed at the contact to set
the voltage than are needed to sustain the current far from the contacts. We must still
have a constant current equal to Je near the contact, but this must mean that since
n(x) is too big,dEfn
dxmust be very small. In fact since n(x) depends exponentially
on Efn, the quasi-fermi level must be almost completely flat for even a slight excess
of carriers. This allows us to draw the entire quasi-fermi level profile for an organic
solar cell away from open-circuit and the result is shown schematically in Figure 4.3.
The numerical calculation showing the exact result is given in Figure 4.4. It
compares quite nicely with what we reasoned above that it must be. The analytical
CHAPTER 4. THE FILL FACTOR 99
expressions for the electron and hole densities are:
n(x) = n(0)exp
(−xl
)+ n(∞)
(1− exp
(−xl
))(4.6)
p(x) = p(L)exp
(x− Ll
)+ p(∞)
(1− exp
(x− Ll
))(4.7)
l =LkT
qVbi(4.8)
Rigorous Derivation
Je(x) = µeNe exp
(Efn − Ec − qVbix/L
kT
)dEfndx
(4.9)
We require that our current be spatially uniform so we can set dJedx
= 0 to solve for
how the quasi-fermi level must vary spatially in order to produce a constant current
throughout the device. The general expression for Efn(x) is given below.
Efn(x) = Efn0 + kT log (1 + Aecx sinh(cx)) (4.10)
A =2JeL
qVbiµeNe
exp
(Ec − Efn0
kT
)(4.11)
c =qVbi
2LkT(4.12)
where L is the thickness of the solar cell and Efn0 is the location of the quasi-Fermi
level at the extracting contact, which is fixed by the choice of electrode work function.
4.5 Recombination Away from Open-Circuit
The previous section showed how we can calculate the electron and hole densities
in an OPV device as a function of the voltage on the device and the current being
driven, which was considered constant. However, during solar cell operation, the
photocurrent is typically being generated throughout the active layer, so neither the
electron nor the hole currents will be constant. If instead we assume that there is
CHAPTER 4. THE FILL FACTOR 100
0 20 40 60 80 100
Distance [nm]
2.0
1.5
1.0
0.5
0.0
0.5
1.0
Energ
y [
eV
]
Efn(x)
Efh(x)
Figure 4.4: The energy bands and quasi-fermi level positions for an organic solar cellat Jsc producing a current of 10 mA/cm2 equally distributed between an electron andhole current.
a constant photocurrent generation rate at every point in the solar cell we come up
with the following expressions for the electron and hole densities everywhere:
n(x) = n0 exp
(−q(Vbi − V )x
kTL
)+
JphL
qµe(Vbi − V )
(1− x
L
)(4.13)
p(x) = p0 exp
(−q(Vbi − V )(L− x)
kTL
)+
JphL
qµh(Vbi − V )
(xL
)(4.14)
Equation 4.14 are divided to show the dark and light contributions to the carrier
densities. They are also simplified from the general result by assuming that q(Vbi−V )
is much greater than kT , which should be the case for all organic solar cells that are
not contact limited. In the equations, n0 and p0 are the charge carrier densities near
the corresponding extracting contacts and Vbi is the built-in potential due to the work
function difference between the two contacts. V is the voltage bias on the device.
The equations are presented in this form because it makes clear what is happen-
ing. The first term is the exponential decay of the charge density present at the
contacts and the second term is the steady state charge density needed to carry the
CHAPTER 4. THE FILL FACTOR 101
photocurrent given the electric field present in the device. Note that the first term
is current independent whereas the second term increases in magnitude with current.
When Vbi - V is not much greater than kT, there are corrections to this expression
that come from overlap between the two terms, but they should be negligible during
solar cell operation and in reverse bias.
There are 2 terms in the expression for n(x) and p(x), meaning that there will be 4
terms in the product n*p. Two of the terms correspond to majority carriers near the
contacts recombining with photogenerated carriers, one term corresponds to majority
carriers from each contact recombining with each other and one term corresponds to
photogenerated carriers recombining with each other.
4.5.1 Classifying Recombination Types
As we can see from the above expression for n(x) and p(x), there will be one recombi-
nation term independent of Jph, two terms linear in Jph and one term quadratic in Jph.
It should be stressed that in this model the geminate splitting efficiency is assumed
to be unity as we expect it should be based on our results in Chapter 1 and only
bimolecular recombination is considered. Non-perfect geminate splitting efficiencies
could be taken into account by reducing Jph.
Dark Recombination
Even with no photocurrent, recombination will occur between carriers injected from
the contacts at forward bias. This recombination term can be found by multiplying
the first parts of the expressions for n(x) and p(x) and integrating over the length of
the solar cell:
Rdark = qγLN2s exp
(σ2n + σ2
p
2(kT )2
)exp
(V − E0
kT
)(4.15)
In this expression Ns is the density of electronic states in the device (typically taken
to be 1 state per nm−3 or 1021 cm−3), σ is the disorder in the electron and hole
CHAPTER 4. THE FILL FACTOR 102
conducting material and Ebg is the effective bandgap between the LUMO of the
acceptor and the HOMO of the donor. γ is the bimolecular recombination coefficient.
The recombination rate is expressed as a current density.
On first inspection, it could be unclear where Ebg came from and why Vbi disap-
peared. The reason is because of the form of n0 and p0. We know that given the
quasi-fermi level position and disorder, we can calculate the carrier density. We need
to figure out where the Fermi level should be in equilibrium at zero bias. To do this
we assume that the electron and hole disorder is the same and invoke overall charge
neutrality for the device at equilibrium. This means that n0 = p0 and so the Fermi
level must be equally spaced between the acceptor LUMO on one side of the device
and the donor HOMO on the other side of the device. This means that the Fermi
level must be located (Ebg - Vbi)/2 away from the center of each energy level. Using
this result we can find an expression n0:
n0 = Ns exp
(σ2n
2(kT )2
)exp
(qVbi − E0
2kT
)(4.16)
The expression for p0 is the same. I would note that this relation is strictly true
only for equal amounts of electron and hole disorder. We could take into account
the actual work functions of the contacts and where those are located relative to the
HOMO and LUMO of the active layer by incorporating different values for n0 and p0.
From the expression for n0 we can see why the built-in voltage cancels out in
Rdark. The carrier concentration increases exponentially in qVbi - Ebg but also decays
exponentially in Vbi V. The net result is that Vbi does not matter for this recombi-
nation term and we recover a result similar to a typical pn junction with an ideality
factor of 1 corresponding to thermionic emission over a barrier.
Photocarrier - Dark Carrier Recombination
By combining the majority carrier term of one carrier type with the minority carrier
term of the second carrier type, we can calculate the rate at which photoinduced
carriers recombine with dark majority carriers. The expression given below is for the
CHAPTER 4. THE FILL FACTOR 103
case of equal electron and hole mobilities and disorder parameters for simplicity.
Rcontact =4L2(kT )2JphNs
q2(Vbi − V )3µexp
(σ2
2(kT )2
)exp
(qVbi − E0
2kT
)(4.17)
This expression accounts for both minority electrons recombining with majority
holes near the hole extracting contact and minority holes recombining with majority
electrons near the electron extracting contact. The key points to draw from the
analytical expression are:
1. This effect gets worse at forward bias and thicker devices because the built-in
field is lower so minority carriers are not as well confined away from the opposite
contacts.
2. This effect is exponential in the built-in potential. This dominates the inverse
cubic dependence on Vbi in the term prefactor and comes from the fact that
exponentially more carriers are present near the contacts as the Fermi-level
approaches the bands.
3. The recombination is linear in the photocurrent since the majority carrier con-
centration is unchanged.
Photocarrier - Photocarrier Recombination
Finally, we can calculate the effect of photocarriers recombining with other photocar-
riers by taking the second terms in the expressions for n(x) and p(x):
Rbulk =L3J2
phγ
6qµeµh(Vbi − V )2(4.18)
This effect is seen to increase like L3 and decrease like the built-in potential
squared. In this case it is easy to see where the dependence comes from. The two
copies of the voltage, mobility and length come from setting the required minority
carrier concentration of both the electrons and holes in order to sustain the given
current. The third copy of the length comes from integrating the recombination
volume density over the device.
CHAPTER 4. THE FILL FACTOR 104
4.6 Using These Results to Understand Organic
Solar Cells
The major insights gained from the previous section are that you can divide bimolec-
ular recombination in organic solar cells into 3 different classes that each have distinct
dependences on the amount of photocurrent being generated and the operating volt-
age of the solar cell. However, the precise form of the constant prefactors are derived
for a very idealized case that does not correspond with device operation, so those
constants are not particularly useful. However, as we will show in the rest of this
chapter, the functional forms remain applicable to actual OPV solar cells and can be
used to extract useful information from IV curves and explain why the fill factors are
typically low.
To begin, we ignore the prefactors that we have calculated in the previous sections
and just assume that we can fit an IV curve using a function that has the form of a
sum of the 3 effects that we found:
J(V ) = −Jph + A exp
(qV
kT
)+
B
(V − Vbi)2 +C
(V − Vbi)3 (4.19)
The remaining sections in this chapter will be devoted to validating and using this
expression to understand experimental IV curves from literature.
4.7 Validating Our Expression Using P3HT:PCBM
In order to see if Equation 4.19 is able to describe the wide variety of solar cell IV
curves, we first turn to the model system P3HT:PCBM. This system is interesting
because one can tune the electron mobility by 1.5 orders of magnitude and the hole
mobility by more than 3 orders of magnitude just by controlling the length of a thermal
annealing step during device fabrication. Using data from Bartelt et al (In Press), we
had access to a data set of 24 different P3HT:PCBM solar cell conditions spanning 4
thicknesses between 100 and 300 nm and 6 different annealing temperatures. In each
case, we corrected the data for series and shunt resistance as described below and
CHAPTER 4. THE FILL FACTOR 105
then fit it to our IV curve expression using a standard nonlinear optimization routine
implemented in a Python script.
4.7.1 Correcting for Series Resistance
Many experimental IV curves are heavily impacted by series resistance, making any
sort of analysis of the IV curve shape impossible without first removing the series
resistance. This is done by realizing that the effect of series resistance is to introduce
an error term in the measured voltage:
V ′ = V + IRs (4.20)
where V ′ is the measured voltage on the voltage cell, Rs is the series resistance and
V is the actual voltage across the active layer (note that I is negative in the power-
producing quadrant so the voltage on the solar cell is higher than is measured in that
case). The series resistance can be found by fitting the dark IV curve in far forward
bias to a linear function and then the light IV curve can be corrected by applying
Equation 4.20 to get the actual voltage on the active layer from the measured voltage.
As an example of how important this correction is, consider the material system
P3HT:PCBM. As we anneal the P3HT active layer, we find that we are systematically
reducing the series resistance on the solar cell, which is presumably caused in large
part by transport through a pure P3HT domain. Figure 4.5 shows the series resistance
extracted from the dark IV curves for all 24 devices in this study. Figure 4.6 shows
that this series resistance appears to be coming from transport in P3HT regions of
the solar cell since the series resistance is linearly proportional to both the thickness
of the solar cell and the P3HT hole mobility. We speculate that there is perhaps a
P3HT rich capping layer on these solar cells that causes this effect but further study
would be required to determine its precise cause. For now, we just note its existence
and correct for it using Equation 4.20.
CHAPTER 4. THE FILL FACTOR 106
50 100 150 200 250 300 350
Thickness [nm]
0
5
10
15
20
25
30
35
40
Seri
es
Resi
stance
[O
hm
s/cm
^2
]
As Cast
48C Anneal
71C Anneal
88C Anneal
111C Anneal
148C Anneal
Figure 4.5: The extracted series resistance of each P3HT annealing condition as afunction of device thickness, showing an approximately linear trend vs. thicknesswith a annealing temperature dependent slope.
CHAPTER 4. THE FILL FACTOR 107
10-7 10-6 10-5 10-4 10-3
P3HT Diode Hole Mobility [cm2 / Vs]
104
105
106
Seri
es
Conduct
ivit
y [
Sie
mens]
Conductivity Data
Linear Fit
Linear Scale
Figure 4.6: The slope of the series resistance vs. thickness curves plotted against theP3HT hole mobility showing how the series resistance in these devices appears to bedue to transport in pure P3HT regions
CHAPTER 4. THE FILL FACTOR 108
2.0 1.5 1.0 0.5 0.0 0.5 1.0
Voltage [V]
8
6
4
2
0
2C
urr
ent
[mA/c
m2
]
90 nm
146 nm
205 nm
304 nm
Figure 4.7: Experimental data (points) and fit to our expression for P3HT:PCBMsolar cells annealed at 0C.
4.7.2 Correcting for Shunt Resistance
Correcting for shunt resistance is done in the standard way by fitting a line to the
dark IV curve near 0 volts and subtracting that line from the light IV curve to remove
the shunt. This is only possible when information on the dark IV curve is available.
This correction is not as important as the series resistance correction described in the
previous section.
4.7.3 P3HT:PCBM Data Fits Our Expression
Figures 4.7- 4.12 show the fits between experimental data (points) and out fitting
function (lines) for P3HT:PCBM solar cells annealed at 0 - 148C for 10 minutes and
made with various thicknesses between 100 and 300 nm.
CHAPTER 4. THE FILL FACTOR 109
2.0 1.5 1.0 0.5 0.0 0.5 1.0
Voltage [V]
10
8
6
4
2
0
2
Curr
ent
[mA/cm
2]
114 nm
140 nm
202 nm
324 nm
Figure 4.8: Experimental data (points) and fit to our expression for P3HT:PCBMsolar cells annealed at 48C.
CHAPTER 4. THE FILL FACTOR 110
2.0 1.5 1.0 0.5 0.0 0.5 1.0
Voltage [V]
12
10
8
6
4
2
0
2
Curr
ent
[mA/cm
2]
117 nm
151 nm
229 nm
306 nm
Figure 4.9: Experimental data (points) and fit to our expression for P3HT:PCBMsolar cells annealed at 71C.
CHAPTER 4. THE FILL FACTOR 111
2.0 1.5 1.0 0.5 0.0 0.5 1.0
Voltage [V]
12
10
8
6
4
2
0
2
Curr
ent
[mA/cm
2]
104 nm
170 nm
227 nm
275 nm
Figure 4.10: Experimental data (points) and fit to our expression for P3HT:PCBMsolar cells annealed at 88C.
CHAPTER 4. THE FILL FACTOR 112
2.0 1.5 1.0 0.5 0.0 0.5 1.0
Voltage [V]
12
10
8
6
4
2
0
2
Curr
ent
[mA/cm
2]
112 nm
132 nm
211 nm
292 nm
Figure 4.11: Experimental data (points) and fit to our expression for P3HT:PCBMsolar cells annealed at 111C.
CHAPTER 4. THE FILL FACTOR 113
2.0 1.5 1.0 0.5 0.0 0.5 1.0
Voltage [V]
12
10
8
6
4
2
0
2
Curr
ent
[mA/cm
2]
127 nm
164 nm
197 nm
312 nm
Figure 4.12: Experimental data (points) and fit to our expression for P3HT:PCBMsolar cells annealed at 148C.
CHAPTER 4. THE FILL FACTOR 114
50 100 150 200 250 300 350
Thickness [nm]
0
2
4
6
8
10
12D
evic
e P
hoto
curr
ent
[mA/c
m2
]
As Cast
48C Anneal
71C Anneal
88C Anneal
111C Anneal
148C Anneal
Figure 4.13: The total amount of photocurrent produced in each device in theP3HT:PCBM annealing series.
4.7.4 The Photocurrent Term
The curve fits to our expression are typically quite good, especially given the simplicity
of the expression but the real value comes in analyzing trends in the fit parameters
extracted from the fits since variation in those parameters can tell us about changes in
the solar active layer as we anneal the devices. The first parameter is the photocurrent
produced by each device. This number should be the total number of extractable free
electrons and holes produced by the devices. Figure 4.13 shows the photocurrent
produced by each device.
What we can learn from Figure 4.13 is that the two low-temperature annealed
devices (0C and 48C) lose photocurrent when they are made thicker while the other
4 devices gain photocurrent with thickness, as would be expected since the devices
continue absorbing a larger fraction of the incident light until they are approximately
CHAPTER 4. THE FILL FACTOR 115
300 nm thick. The photocurrent loss in the two low-temperature annealed devices
has previously been shown to be the result of their extremely low hole mobilities
causing space charge to build up and create a depletion region narrower than the
device thickness, so that large fractions of the device have no electric field and do
not contribute to the photocurrent, see Bartelt et al Advanced Energy Materials (In
Press).
4.7.5 The Built-in Potential
In our fitting expression, the parameter of interest that we capture as Vbi is actually
the strength of the electric field in the device since that is what sets the drift velocity
of the charge carriers and hence how many photocarriers build up inside the device
during operation. In our derivation of the formula, we assumed that the electric field
was uniform over the device, so its magnitude would simply be Vbi − V divided by
the thickness of the solar cell. However, as we saw in the last section, there can be
significant space charge buildup in the devices, which means the field will not drop
uniformly over the entire solar cell, but will be concentrated in a small depletion
region. This should result in an apparent increase of the built-in potential since the
field over the portion of the device that produces photocurrent will be stronger by
the ratio of the depletion width to the thickness of the solar cell. This is exactly what
we find, as shown in Figure 4.14.
What you can see from Figure 4.14 is that the built-in potential for all but the
111 and 148C annealed devices is higher than the measured built-in potential, which
we ascribe to the known effect of space charge buildup in these devices. For the 111
and 148C devices, though, the extracted Vbi is in good agreement with the measured
values across the range of device thickness. The take-home message is that the Vbi
term in our fitting expression can tell you about the presence or absence of space
charge limitations in your devices by whether or not it agrees with the measured
built-in potential extracted from the crossing point of the light and dark IV curves.
CHAPTER 4. THE FILL FACTOR 116
1
10
100 150 200 250 300
Thickness [nm]
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Built
-in V
olt
age [
V]
As Cast
48C Anneal
71C Anneal
88C Anneal
111C Anneal
148C Anneal
Figure 4.14: The extracted Vbi parameter for the P3HT:PCBM series. The solid linesare the actual built-in potential estimated from the crossing point between light anddark IV curves. The dashed lines are the fit parameters.
CHAPTER 4. THE FILL FACTOR 117
50 100 150 200 250 300 350
Thickness [nm]
0
20
40
60
80
100
Fract
ion o
f Photo
-Dark
Reco
mbin
ati
on [
%]
As Cast
48C Anneal
71C Anneal
100 150 200 250 300 350
0
20
40
60
80
100 88C Anneal
111C Anneal
148C Anneal
Figure 4.15: The photocarrier dark carrier recombination coefficient for ourP3HT:PCBM device series, expressed as the fraction of recombination that proceedsvia this mechanism at the maximum power point.
4.7.6 Photocarrier - Dark Carrier Recombination
Figure 4.15 shows the photocarrier - dark carrier recombination term extracted from
our fitting procedure. We see two distinct classes of behavior. For the low-temperature
annealed devices, we see that this effect is dominant at low thicknesses but then be-
comes less important at larger thicknesses and it is never important for the as-cast de-
vices. We attribute this behavior to the small depletion widths for these devices. For
the high-temperature annealed devices, we see the opposite trend, with this recom-
bination mechanism playing an increasingly important role as the devices are made
thicker. This makes sense because from our analytical expression, we see that this
recombination mechanism should increase as L3 whereas photocarrier-photocarrier
recombination should only increase as L2, meaning that photocarrier-dark carrier
recombination should be of increasing importance as the devices are made thicker.
CHAPTER 4. THE FILL FACTOR 118
50 100 150 200 250 300 350
Thickness [nm]
0
20
40
60
80
100
Fract
ion o
f Photo
-Photo
Reco
mbin
ati
on [
%]
As Cast
48C Anneal
71C Anneal
100 150 200 250 300 350
Thickness [nm]
0
20
40
60
80
100 88C Anneal
111C Anneal
148C Anneal
Figure 4.16: Photocarrier - Photocarrier Recombination coefficient for ourP3HT:PCBM device series, expressed as the fraction of recombination that proceedsvia this mechanism at the maximum power point.
4.7.7 Photocarrier - Photocarrier Recombination
Figure 4.16 shows the photocarrier - photocarrier recombination coefficient. What we
see is the opposite trend we saw before where the low-T annealed device (below 71C)
show recombination dominated by photocarrier - photocarrier annihilation, whereas
the high temperature annealed devices show the opposite trend.
The Mobility Dependence of Photo-Photo Recombination
From our analytical expression, we expect that the photocarrier-photocarrier recom-
bination term should be inversely proportional to the product of the electron and
hole mobilities in our device. Since we have experimental data on those mobilities,
we can check if this prediction holds. We expect that the photocarrier-photocarrier
recombination parameter should be given by:
B ∝ 1
µeµh(4.21)
CHAPTER 4. THE FILL FACTOR 119
10-7 10-6 10-5 10-4 10-3
P3HT Diode Hole Mobility [cm2 / Vs]
10-8
10-7
10-6
10-5
10-4
Invers
e P
hoto
-Photo
Reco
mb. C
oeff
icie
nt
[a.u
.]
~300nm Devices
Linear Correspondence
Figure 4.17: The inverse proportionality of the photocarrier-photocarrier recombina-tion coefficient to the P3HT hole mobility after correcting for the variation in electronmobility
Rearranging Equation 4.21 shows that if we multiply the B parameter by the
electron mobility and invert it, the result should be proportional to the hole mobility.
Specifically,
B ∝ 1
µeµh(4.22)
1
B∝ µeµh (4.23)
1
µeB∝ µh (4.24)
The left-hand side of Equation 4.24 is plotted in Figure 4.17 against the hole
mobility for the 300 nm thick devices.
CHAPTER 4. THE FILL FACTOR 120
50 100 150 200 250 300 350
Thickness [nm]
10-11
10-10
10-9
J0 [
mA/c
m2
]
As Cast
48C Anneal
71C Anneal
88C Anneal
111C Anneal
148C Anneal
Figure 4.18: The reverse saturation current density extracted from our fits.
The Figure shows a decent linear proportionality over 3 orders of magnitude in
the P3HT hole mobility, indicating that the specific dependences of our analytical
expression on mobility may remain valid even for non-ideal organic solar cells.
4.7.8 Dark - Dark Recombination
For completeness, we show the dark-dark recombination term, which is typically
referred to as J0, the reverse saturation current density (Figure 4.18). There is not a
lot of information that we can extract from the values, however, since we showed in
Chapter 3 that this is mainly a measure of the degree of mixing in the solar cells, the
energy of the Charge Transfer state distribution and the CT state lifetime.
CHAPTER 4. THE FILL FACTOR 121
4.7.9 Conclusions
What we have shown in this section is that our analytical expression for the IV curve
of an organic solar cell is able to fit and explain the variation in P3HT:PCBM solar
cells across a wide range of mobilities and thicknesses showing that the expression is
useful for understanding actual OPV device performance. Further, we have shown
that the fit parameters extracted from our expression vary in understandable ways
and appear to have the meanings and dependence on materials parameters that we
expect from our analytical results. In the next section we will use this, now validated,
expression to look at other material systems from literature.
4.8 Molecular Weight Variations in PCDTBT
One of the key advantages of our analytical expression is that it only requires an IV
curve in order to extract powerful amounts of information about what is occurring
inside the solar cell active layer. To demonstrate this, we looked at literature data
showing a series of PCDTBT:PCBM solar cells with differing molecular weights[59].
The IV curves showed large FF and Jsc variations among the different molecular
weights but it was not clear why. We can now reanalyze those data to understand
why. The raw IV curves and fits are shown in Figure 4.19. Note that the data
was corrected for series resistance, which was non-negligible but not found to vary
significantly among the different molecular weight devices.
The first point to note is that the fits are superb, with almost no deviation between
the fits and the experimental data. The second point to note comes from comparing
the fit parameters obtained from fitting these IV curves. As can be seen in Figure 4.19,
there is significant variation in short-circuit current among the different molecular
weights. However, as reported in literature, there are not significant differences in
absorption among the devices[59]. So, we do not expect any variation in photocurrent
production, in contrast to the observed Jsc variation. Table 4.1 shows that, in fact,
there is little variation in photocurrent production among the devices. While the
short-circuit current varies by 3 mA per square centimeter, the actual amount of
CHAPTER 4. THE FILL FACTOR 122
0.0 0.2 0.4 0.6 0.8
Voltage [V]
10
8
6
4
2
0
Curr
ent
[mA/cm
2]
27.3 kDA
23.6 kDA
11.6 kDA
6.0 kDA
5.0 kDA
Figure 4.19: The raw IV curve data and fits for PCDTBT:PCBM solar cells reportedin literature[59].
CHAPTER 4. THE FILL FACTOR 123
Batch Jsc Jph [mA/cm2]5 kDA 7.8 11.36 kDA 9.1 10.7
11.6 kDA 9.5 10.623.6 kDA 10.3 11.327.3 kDA 10.7 11.4
Table 4.1: Extracted Photocurrent and Short-circuit Currents for PCDTBT:PCBMdevices.
photocurrent produced varies by less than 0.8 mA per square centimeter, and not in
any sort of discernible trend.
The question then is what is driving the difference in device performance if the
amount of photocurrent produced is the same? In this case we have access to the
PCDTBT hole mobilities for each molecular weight so we can perform the same
analysis of the photocarrier-photocarrier recombination coefficient that we did before
on P3HT. The results are shown in Figure 4.20.
We find that we can explain the differences in both FF and Jsc simply as hole
mobility dependent recombination losses, so in this case the primary impact of in-
creasing the molecular weight of the PCDTBT appears to be simply improving hole
transport which leads to increases in Jsc and FF. I would note that in this case the
recombination coefficient appears to be logarithmically dependent on the hole mo-
bility, in contrast to our expected and previously observed linear dependence. The
reason for this is currently unclear and would warrant further study.
4.9 Apparent Field Dependent Geminate Splitting
We are finally in a position to tackle the last remaining question of this work, which
is understanding solar cells that appear to have field-dependent geminate splitting.
As we explained in Chapters 2 and 3, we do not expect field-dependent geminate
splitting to occur in working organic solar cells since free carriers and CT states are
observed to be in equilibrium with each other, necessitating that geminate pairs, in
CHAPTER 4. THE FILL FACTOR 124
10-8 10-7 10-6 10-5
Measured Hole Mobility [cm2 /Vs]
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Invers
e P
hoto
-Photo
Reco
mb.
[a.u
.]
Figure 4.20: The inverse photocarrier-photocarrier recombination coefficient plottedagainst the measured PCDTBT:PCBM hole mobility.
CHAPTER 4. THE FILL FACTOR 125
Condition Jsc Jph [mA/cm2]As-Cast 8.3 15.1
Annealed 10.9 (+31%) 14DIO 13.0 (+57%) 14.6
Table 4.2: Extracted Photocurrent and Short-circuit Currents forp−DTS(FBTTh2)2PC71BM devices.
fact, have no trouble splitting and forming free carriers. Nevertheless, there are re-
ports in literature that previous authors have understood as implying field-dependent
geminate splitting[22, 75, 3, 27, 29, 68]. One example is Proctor et al[75], who stud-
ied small molecule solar cells with and without post-deposition processing steps and
found that the FF and short-circuit currents were significantly improved upon either
annealing or using a solvent additive DIO. Building on our explanation of the molec-
ular weight variation in PCDTBT, we expect that we can explain these variations
simply as different amount of nongeminate recombination.
The raw IV curves and our fits are shown in Figure 4.21. The very fact that we can
fit the data using a model that explicitly has no field-dependent geminate splitting is
fairly definitive proof that such a process is not occurring, but more evidence can be
found by considering the extracted photocurrent values for the 3 solar cells shown in
Table 4.2.
We find that even though the short-circuit currents vary by more than 60%, there
is less than 8% variation in the amount of produced photocurrent, indicating that
field dependent geminate splitting is not playing a role in these devices and, instead
the differences in FF and Jsc can be attributed to nongeminate mechanisms likely
caused by very poor hole transport in the devices without post-processing.
4.9.1 Time Delayed Collection Field Measurements
Previous authors have investigated the apparent field dependence of geminate split-
ting, often using the Time Delayed Collection Field Technique (TDCF)[22, 75, 3,
27, 29, 68] to distinguish between geminate and nongeminate recombination. In this
CHAPTER 4. THE FILL FACTOR 126
4 3 2 1 0 1
Voltage [V]
14
12
10
8
6
4
2
0
Curr
ent
[mA/c
m2
]
As-cast
Annealed
DIO Additive
Figure 4.21: Experimental IV curve data (points) and fits (lines) for a small moleculesolar cell blended with PC71BM. The raw data is from Proctor et al [75].
CHAPTER 4. THE FILL FACTOR 127
section we would like to explain why we believe that technique does not actually dis-
tinguish between geminate and non-geminate recombination. Briefly, TDCF works
by applying a voltage bias to a working organic solar cell and then illuminating it
with a pulse of light. After a delay of a few to a few dozen nanoseconds, the voltage
biased is switched to a strong negative bias, which is used to sweep out carriers from
the device very rapidly. The idea is that the prebias sets the field that carriers feel
during the geminate splitting process and all nongeminate recombination is removed
because of the strong collection bias. So, any difference in collected charge from dif-
ferent prebiases must come from differences in geminate splitting and since the only
variable being changed is the electric field during the splitting process, this must be
a field-dependent geminate splitting process.
We start by noting that TDCF does not in principle distinguish between geminate
and nongeminate recombination since fundamentally all recombination is just the
lost of an electron hole pair and the technique just measures how many electron-
hole pairs are lost due to different experimental conditions. There are two crucial
additional assumptions that enable the claim that the recombination probed by TDCF
is geminate. First, it is assumed that no nongeminate recombination can happen
before the collection bias is switched on. Second, it is assumed that the prebias and
the collection bias are able to uniformly penetrate through the entire device. Both of
these assumptions are problematic but the first assumption appears to be the most
problematic.
As we explained at the beginning of this chapter, there can be very high dark car-
rier populations near the contacts of an organic solar cell due to equilibration between
the large charge reservoirs in the metal contacts and the active layer. This means
that the lifetime of a photogenerated carrier that happens to be formed very near
the opposite contact will be very short since the average lifetime of a photogenerated
hole, for example is a function of the total electron density at that point including
both photoelectrons (ne,l) and dark electrons (ne,d):
CHAPTER 4. THE FILL FACTOR 128
τh =1
k(ne,l + ne,d)(4.25)
Equation 4.25 implies that near the contacts, photocarriers should have very short
lifetimes. Numerical estimates of the dark carrier density near the contacts are above
1018 cm−3, which is two orders of magnitude higher than the bulk carrier density,
implying that the photocarrier lifetime is two orders of magnitude shorter, which for
normal organic solar cells should be in the 1-10 ns range. Further, since the presence
of energetic disorder broadens the distribution of carrier lifetimes, just like it broadens
the distribution of carrier mobilities, there could be an appreciable number of non-
geminate recombination events even on times shorter than 1 ns. Since the speed at
which you can turn on the collection bias in a TDCF measurement is limited to the
nanosecond regime by the RC time constant of the solar cell, it is not possible to use
the technique to distinguish between geminate and non-geminate recombination on
the basis of timescale alone.
The other potential option is to distinguish between geminate and non-geminate
recombination on the basis of light intensity dependence but as we explained pre-
viously, recombination near the contacts involves a photocarrier and a dark carrier,
so it has the same light intensity dependence as geminate recombination (linear in
light intensity). Thus, TDCF cannot in principle tell the difference between geminate
and nongeminate recombination. It can simply report the presence of recombination.
Now, proceeding on the assumption that the recombination mechanism that TDCF
is probing is photocarrier - dark carrier recombination, we can also explain why it
would be field dependent. Our analytical expression for photocarrier - dark carrier
recombination has an inverse cubic dependence on electric field strength since that
sets the timescale for carriers to leave the high recombination contact region.
We would note that if TDCF were, in fact, probing geminate recombination, we
would expect a much stronger exponential field dependence as given in Onsager-
Braun theory. The observed field dependence of TDCF measurements is typically
fairly weak. So, we conclude that TDCF measurements are likely just quantifying
CHAPTER 4. THE FILL FACTOR 129
photocarrier - dark carrier recombination near the contacts of the solar cell since this
mechanism has very similar characteristics to geminate recombination, though we
stress that it is nongeminate.
There is another potential issue with TDCF measurements that we mention here
for completeness but we believe it to be of secondary importance in this instance.
TDCF assumes that the prebias and the collection bias are able to create electric
fields throughout the device and importantly that the fraction of the device that con-
tains a strong field during the collection bias phase does not depend on the prebias.
However, low-performing OPV devices, where TDCF sometimes sees field-dependent
recombination, often have space charge accumulation due to low carrier mobilities.
Thus, there is a depletion region in the device with a strong electric field over part
of the device and a very weak field over the rest of the device. The strength of the
applied electric field will modulate the size of the depletion region since it, combined
with the carrier mobilities, sets the density of space charge and hence the width of
the depletion region. So, it may also be that TDCF measurements showing field-
dependent recombination are just modulating the width of a depletion region inside
the device’s active layer where photocarriers formed in the depletion region are effi-
ciently collected and photocarriers formed outside the depletion region recombine. By
setting the prebias you control the density of space charge and therefore the depletion
width so the amount of collected charge becomes a function of the prebias and you
can observe an apparently field-dependent recombination mechanism that is just an
artifact of the measurement technique.
4.10 Conclusion
Our goal in this section was to show that we can understand the IV curves of arbitrary
organic solar cells in terms of purely bimolecular recombination losses without field-
dependent geminate splitting or other exotic effects. The key observation is that
since organic solar cells are made in PIN structures, they cannot be described as
intrinsic organic semiconductors without dark carriers. Once dark carriers are added
into the description, we are able to accurately describe the shape of OPV IV curves
CHAPTER 4. THE FILL FACTOR 130
for both high performance and low performance devices using one consistent theory.
Importantly, our theory for IV curve shape is completely compatible with our theory
for Voc and Jsc, namely that charge carriers are in equilibrium with CT states and so
the amount of recombination in a solar cell is just a function of how many carriers
are in the cell since that sets the driving force for CT state formation and hence
recombination.
We do not expect, nor do we observe, significant differences in photocurrent gener-
ation among devices with similar optical absorption spectra since nearly all geminate
pairs split. Rather, the differences observed in both short-circuit current and FF
were shown to be caused by non-geminate mechanisms, typically due to very low hole
mobilities. Thus, we have accomplished our goal of finding a single theory that can
explain the short-circuit current, fill factor and open-circuit voltage of organic solar
cells and our work is complete.
4.11 Additional Theoretical Background
4.11.1 Properly Counting States in the Presence of Disorder
In typical derivations relating carrier density and quasi-Fermi levels it is assumed that
the electronic states of the solar cell can be approximated by a lumped “effective”
density of states at the band edge of the conduction and valence bands. In organic
solar cells, the presence of Gaussian disorder means that this is not in general possible
since there are many states below the center of the material HOMO. In this section we
will show that we can still define an effective density of states but that the presence of
disorder makes this effective DOS approximately many (over 100) times larger than
for crystalline, non-disordered systems. This means that 100 times more carriers are
required to achieve the same quasi-fermi level splitting as in a highly crystalline solar
cell.
CHAPTER 4. THE FILL FACTOR 131
4.11.2 The Link Between Voltage and Carrier Density
We can think of a solar cell as just providing 2 reservoirs of excited charge carriers:
one of electrons and one of holes. The quasi-fermi level describing how filled each
reservoir is can be determined since the charge carriers are fermions, by simply filling
up electronic states from low to high energy until all of the carriers in the device have
been accommodated. At 0 Kelvin, the highest occupied state is the quasi-fermi level.
At finite temperature, thermal effects will excite some carriers above the quasi-fermi
level leaving some open states below the quasi-fermi level. At any temperature, we
know that the relation between the quasi-fermi level and the number of carriers in
the device is given by:
N(Ef ) =
∫ ∞−∞
1
1 + exp(x−Ef
kT
)g(x)dx (4.26)
where g(x) is the density of states, the number of electronic states with energy between
x and x + dx. Equation 4.26 always holds when the carriers in equilibrium, which
they always will be in the cases we are discussing.
Unfortunately, Equation 4.26 is not exactly solvable, but a convenient approxima-
tion can be made that when most states are far in energy (more than a few kT) from
the quasi-fermi level, the exponential term in the denominator will be much greater
than 1 and so the equation above reduces to:
N(Ef ) =
∫ ∞−∞
exp
(Ef − xkT
)g(x)dx (4.27)
Given the assumption that the energy levels in our device are properly described
by Gaussian distributions with a standard deviation σ, we have:
g(x) = N01
σ√
2πexp
(−(x− Ec)2
2σ2
)(4.28)
where Ec is the center of the Gaussian distribution and N0 is the total number of
electronic states per unit volume. Combining Equation 4.27 with Equation 4.28 and
CHAPTER 4. THE FILL FACTOR 132
0.5 0.4 0.3 0.2 0.1 0.0
Fermi Level Position [eV]
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
Carr
ier
Densi
ty [
cm−
3]
σ = 60 meV
σ = 80 meV
σ = 100 meV
σ = 120 meV
No Disorder
Figure 4.22: The density of charge carriers as a function of the quasi-fermi level givena constant N0 = 1x1021. The dashed lines show the analytic approximation given inEquation 4.29.
integrating analytically, one can show that the relation between quasi-fermi level and
charge carrier density is given by:
n(Ef ;Ec) = αN0 exp
(σ2
2k2T 2
)exp
(Ef − EckT
)(4.29)
At room temperature, with σ <= 80 meV and Ef more than about 0.3 below Ec, α
is approximately equal to 1 and only weakly depends on the fermi level.
As seen in Equation 4.29, the relation between charge carrier density and quasi-
fermi level is the same as in the inorganic, non-degenerately doped case, but the
density of electronic states is increased by an exponential factor dependent on the
level of energetic disorder.
Figure 4.22 shows the number of charge carriers in the device as a function of the
quasi-fermi level location. One thing to note is that there are orders of magnitude
CHAPTER 4. THE FILL FACTOR 133
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Fermi Level Position [eV]
100
101
102
103
104
105
Rati
o o
f D
isord
erd
to O
rdere
d C
arr
ier
Densi
ty [
unit
less
]
σ = 60 meV
σ = 80 meV
σ = 100 meV
σ = 120 meV
Figure 4.23: The ratio of charge carriers in a disordered device compared to a non-disordered device as a function of the quasi-fermi level location.
CHAPTER 4. THE FILL FACTOR 134
more charge carriers in the device at a given voltage because of the disorder. Another
was of saying this is that the presence of low-energy trap states means you have to
put more carriers into the device in order to reach a given voltage. We can quantify
this by taking the ratio between the disordered curves in Figure 4.22 and the ordered
curve given the fractional increase in carrier caused by disorder. This is shown in
Figure 4.23.
The key point to take away is that this penalty of higher carrier density for a
given voltage is most pronounced at lower voltages when the quasi-fermi level is more
than 0.3 eV away from the center of the band. As the voltage increases, the deviation
becomes less severe, as it must since the ordered and disordered devices have the
same total number of electronic states, they just have a larger energy spread in the
disordered case.
Figures 4.22 and 4.23 were calculated numerically without approximations, how-
ever in the region of Figure 4.23 that is flat, we can apply Equation 4.29 to predict
the carrier density penalty as a function of energetic disorder (Figure 4.24). The key
point to take away from Figure 4.24 is that there is a very large difference between
an energetic disorder of 60 meV and 100 meV but it does not change the ability to
express the carrier density as a simple function of the quasi-Fermi level. One hundred
times more carriers are present in the device with 100 meV of disorder than with
60 meV of disorder. Note that since bimolecular recombination is proportional to
n*p, this means that there would be 10,000 times more recombination with 100 meV
disorder than 60 meV, all other things being equal.
Calculating How Many Carriers Are in the Device
Given the expressions in Equation 4.7 and 4.8, we can calculate some basic properties
that will be useful in the subsequent sections: the total number of electrons and holes
in the device (that contribute to the current) as a function of current and voltage as
well as the shape of the recombination current, which is proportional to n(x) ∗ p(x).
CHAPTER 4. THE FILL FACTOR 135
0 20 40 60 80 100 120
Energetic Disorder [meV]
100
101
102
103
104
105
Exce
ss C
arr
ier
Rati
o [
unit
less
]
Figure 4.24: The ratio of charge carriers in a disordered device to a fully ordereddevice calculated using Equation 4.29.
CHAPTER 4. THE FILL FACTOR 136
The total number of electrons and holes is:
n =1
α
[n(0)(1− e−α) + n(∞)(α− 1 + e−α)
](4.30)
p =1
α
[p(L)(1− e−α) + p(∞)(α− 1 + e−α)
](4.31)
α =qVbikT
(4.32)
This expression holds for any built-in voltage and current combination, though
care must taken when computing the Vbi = 0 limit since you will have indeterminate
fractions that need to be evaluated with L’Hopital’s rule. One thing to note is that
when n(0) >> n(∞) and qVbi >> kT the above expression simplifies to:
n =n(0)
α(4.33)
p =p(L)
α(4.34)
This means that until the voltage on the device approaches the built-in voltage,
the total charge carrier density basically tracks the charge density at the contact but
with a linear correction factor α. This is shown for three different values of disorder
in Figure 4.25.
CHAPTER 4. THE FILL FACTOR 137
0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Open Circuit Voltage [V]
1015
1016
1017
1018
1019
1020
Charg
e C
arr
ier
Densi
ty [
cm−
3]
σ=60 meV
σ=80 meV
σ=100 meV
Figure 4.25: The average charge carrier density (of one type) in the device as afunction of applied voltage for three different levels of disorder. The device’s bandgapis 1.7eV. Solid lines correspond to a built-in voltage at short circuit of 1.2V, dashedlines correspond to a built-in voltage of 1V.
Bibliography
[1] Vytautas Abramavicius, Dimali Amarasinghe Vithanage, Andrius Devizis,
Yingyot Infahsaeng, Annalisa Bruno, Samuel Foster, Panagiotis E Keivanidis,
Darius Abramavicius, Jenny Nelson, Arkady Yartsev, Villy Sundstrom, and
Vidmantas Gulbinas. Carrier motion in as-spun and annealed P3HT:PCBM
blends revealed by ultrafast optical electric field probing and Monte Carlo sim-
ulations. Physical chemistry chemical physics : PCCP, 16(6):2686–92, January
2014.
[2] Xin Ai, Matthew C Beard, Kelly P Knutsen, Sean E Shaheen, Garry Rum-
bles, and Randy J Ellingson. Photoinduced charge carrier generation in a
poly(3-hexylthiophene) and methanofullerene bulk heterojunction investigated
by time-resolved terahertz spectroscopy. The journal of physical chemistry. B,
110(50):25462–71, December 2006.
[3] Steve Albrecht, Silvia Janietz, Wolfram Schindler, Johannes Frisch, Jona
Kurpiers, Juliane Kniepert, Sahika Inal, Patrick Pingel, Konstantinos Fos-
tiropoulos, Norbert Koch, and Dieter Neher. Fluorinated copolymer PCPDTBT
with enhanced open-circuit voltage and reduced recombination for highly
efficient polymer solar cells. Journal of the American Chemical Society,
134(36):14932–44, September 2012.
[4] D Amarasinghe Vithanage, A Devizis, V Abramavicius, Y Infahsaeng, D Abra-
mavicius, R C I MacKenzie, P E Keivanidis, A Yartsev, D Hertel, J Nelson,
V Sundstrom, and V Gulbinas. Visualizing charge separation in bulk hetero-
junction organic solar cells. Nature communications, 4:2334, January 2013.
138
BIBLIOGRAPHY 139
[5] Artem A. Bakulin, Akshay Rao, Vlad G Pavelyev, Paul H M van Loosdrecht,
Maxim S Pshenichnikov, Dorota Niedzialek, Jerome Cornil, David Beljonne,
and Richard H Friend. The role of driving energy and delocalized States for
charge separation in organic semiconductors. Science, 335(6074):1340–1344,
March 2012.
[6] D Aaron R Barkhouse, Oki Gunawan, Tayfun Gokmen, Teodor K Todorov,
and David B Mitzi. Device characteristics of a 10.1% hydrazine-processed
Cu2ZnSn(Se,S)4 solar cell. Progress in Photovoltaics: Research and Applica-
tions, 20(version 44):6–11, 2012.
[7] Andreas F. Bartelt, Christian Strothkamper, Wolfram Schindler, Konstantinos
Fostiropoulos, and Rainer Eichberger. Morphology effects on charge generation
and recombination dynamics at ZnPc:C60 bulk hetero-junctions using time-
resolved terahertz spectroscopy. Applied Physics Letters, 99(14):143304, 2011.
[8] Jonathan A. Bartelt, Zach M. Beiley, Eric T. Hoke, William R. Mateker, Jes-
sica D. Douglas, Brian A. Collins, John R. Tumbleston, Kenneth R. Graham,
Aram Amassian, Harald Ade, Jean M. J. Frechet, Michael F. Toney, and
Michael D. McGehee. The Importance of Fullerene Percolation in the Mixed Re-
gions of Polymer-Fullerene Bulk Heterojunction Solar Cells. Advanced Energy
Materials, 3(3):364–374, March 2013.
[9] James Blakesley and Dieter Neher. Relationship between energetic disorder and
open-circuit voltage in bulk heterojunction organic solar cells. Physical Review
B, 84(7), August 2011.
[10] James C. Blakesley and Neil C. Greenham. Charge transfer at polymer-electrode
interfaces: The effect of energetic disorder and thermal injection on band bend-
ing and open-circuit voltage. Journal of Applied Physics, 106(3):034507, 2009.
[11] P. W. M. Blom, M. J. M. de Jong, and S. Breedijk. Temperature dependent
electron-hole recombination in polymer light-emitting diodes. Applied Physics
Letters, 71(7):930, 1997.
BIBLIOGRAPHY 140
[12] Timothy M Burke and Michael D McGehee. How high local charge carrier
mobility and an energy cascade in a three-phase bulk heterojunction enable
¿90% quantum efficiency. Advanced Materials, 26:1923–1928, December 2014.
[13] Timothy M. Burke, Sean Sweetnam, Koen Vandewal, and Michael D. Mcgehee.
Beyond Langevin Recombination: How Equilibrium Between Free Carriers and
Charge Transfer States Determines the Open-Circuit Voltage of Organic Solar
Cells. Advanced Energy Materials, 2015.
[14] Mose Casalegno, Guido Raos, and Riccardo Po. Methodological assessment of
kinetic Monte Carlo simulations of organic photovoltaic devices: the treatment
of electrostatic interactions. The Journal of chemical physics, 132(9):094705,
March 2010.
[15] Song Chen, Sai Wing Tsang, Tzung Han Lai, John R Reynolds, and Franky So.
Dielectric effect on the photovoltage loss in organic photovoltaic cells. Advanced
Materials, 26(35):6125–6131, July 2014.
[16] Namchul Cho, Cody W Schlenker, Kristina M Knesting, Patrick Koelsch,
Hin Lap Yip, David S Ginger, and Alex K Y Jen. High-dielectric constant
side-chain polymers show reduced non-geminate recombination in heterojunc-
tion solar cells. Advanced Energy Materials, 4, March 2014.
[17] Jung Hei Choi, Kyung-In Son, Taehee Kim, Kyungkon Kim, Kei Ohkubo, and
Shunichi Fukuzumi. Thienyl-substituted methanofullerene derivatives for or-
ganic photovoltaic cells. Journal of Materials Chemistry, 20(3):475, 2010.
[18] Philip C Y Chow, Simon Gelinas, Akshay Rao, and Richard H Friend. Quan-
titative Bimolecular Recombination in Organic Photovoltaics through Triplet
Exciton Formation. Journal of the American Chemical Society, 136:3424–3429,
February 2014.
[19] Tracey M. Clarke and James R Durrant. Charge photogeneration in organic
solar cells. Chemical reviews, 110(11):6736–67, November 2010.
BIBLIOGRAPHY 141
[20] Brian A. Collins, Zhe Li, John R. Tumbleston, Eliot Gann, Christopher R.
McNeill, and Harald Ade. Absolute Measurement of Domain Composition and
Nanoscale Size Distribution Explains Performance in PTB7:PC 71 BM Solar
Cells. Advanced Energy Materials, 3(1):65–74, January 2013.
[21] Sarah R. Cowan, Anshuman Roy, and Alan J. Heeger. Recombination
in polymer-fullerene bulk heterojunction solar cells. Physical Review B,
82(24):245207, December 2010.
[22] Dan Credgington and James R. Durrant. Insights from Transient Optoelectronic
Analyses on the Open-Circuit Voltage of Organic Solar Cells. The Journal of
Physical Chemistry Letters, 3(11):1465–1478, June 2012.
[23] Dan Credgington, Fiona C. Jamieson, Bright Walker, Thuc Quyen Nguyen, and
James R. Durrant. Quantification of geminate and non-geminate recombination
losses within a solution-processed small-molecule bulk heterojunction solar cell.
Advanced Materials, 24:2135–2141, 2012.
[24] Paul D. Cunningham and L. Michael Hayden. Carrier Dynamics Resulting from
Above and Below Gap Excitation of P3HT and P3HT/PCBM Investigated by
Optical-Pump Terahertz-Probe Spectroscopy. Journal of Physical Chemistry
C, 112(21):7928–7935, May 2008.
[25] Carsten Deibel and Vladimir Dyakonov. Polymer-Fullerene Bulk Heterojunc-
tion Solar Cells. Reports on Progress in Physics, 73(9):68, September 2010.
[26] Carsten Deibel, Thomas Strobel, and Vladimir Dyakonov. Origin of the Efficient
Polaron-Pair Dissociation in Polymer-Fullerene Blends. Physical Review Letters,
103(3):1–4, July 2009.
[27] George F. a. Dibb, Fiona C. Jamieson, Andrea Maurano, Jenny Nelson, and
James R. Durrant. Limits on the Fill Factor in Organic Photovoltaics: Distin-
guishing Nongeminate and Geminate Recombination Mechanisms. The Journal
of Physical Chemistry Letters, 4(5):803–808, March 2013.
BIBLIOGRAPHY 142
[28] Gerald Dicker, Matthijs de Haas, Laurens Siebbeles, and John Warman. Elec-
trodeless time-resolved microwave conductivity study of charge-carrier photo-
generation in regioregular poly(3-hexylthiophene) thin films. Physical Review
B, 70(4):045203, July 2004.
[29] Alexander Foertig, Juliane Kniepert, Markus Gluecker, Thomas Brenner,
Vladimir Dyakonov, Dieter Neher, and Carsten Deibel. Nongeminate and gem-
inate recombination in PTB7:PCBM Solar Cells. Advanced Functional Materi-
als, 24(9):1306–1311, 2014.
[30] Simon Gelinas, Akshay Rao, Abhishek Kumar, Samuel L Smith, Alex W. Chin,
Jenny Clark, Tom S. van der Poll, Guillermo C Bazan, and Richard H. Friend.
Ultrafast long-range charge separation in organic semiconductor photovoltaic
diodes. Science, 343(December):512–516, December 2014.
[31] N. C. Giebink, G. P. Wiederrecht, M. R. Wasielewski, and S. R. Forrest. Ideal
diode equation for organic heterojunctions. I. Derivation and application. Phys-
ical Review B, 82(15):155305, October 2010.
[32] L. Goris, A. Poruba, L. Hod’Akova, M. Vanecek, K. Haenen, M. Nesladek,
P. Wagner, D. Vanderzande, L. De Schepper, and J. V. Manca. Observation
of the subgap optical absorption in polymer-fullerene blend solar cells. Applied
Physics Letters, 88(5):1–3, 2006.
[33] Kenneth R Graham, Clement Cabanetos, Justin P Jahnke, Matthew N Idso,
Abdulrahman El Labban, Guy O Ngongang Ndjawa, Thomas Heumueller, Koen
Vandewal, Alberto Salleo, Bradley F Chmelka, Aram Amassian, Pierre M Beau-
juge, and Michael D McGehee. Importance of the donor:fullerene intermolecular
arrangement for high-efficiency organic photovoltaics. Journal of the American
Chemical Society, 136(27):9608–18, July 2014.
[34] Kenneth R Graham, Patrick Erwin, Dennis Nordlund, Koen Vandewal, Ruipeng
Li, Guy O Ngongang Ndjawa, Eric T Hoke, Alberto Salleo, Mark E Thompson,
Michael D McGehee, and Aram Amassian. Re-evaluating the role of sterics and
BIBLIOGRAPHY 143
electronic coupling in determining the open-circuit voltage of organic solar cells.
Advanced materials (Deerfield Beach, Fla.), 25(42):6076–82, November 2013.
[35] G Grancini, M Maiuri, D Fazzi, A Petrozza, H-J Egelhaaf, D Brida, G Cerullo,
and G Lanzani. Hot exciton dissociation in polymer solar cells. Nature materi-
als, 12(1):29–33, January 2013.
[36] C. Groves, R. A. Marsh, and Neil C. Greenham. Monte Carlo modeling of
geminate recombination in polymer-polymer photovoltaic devices. The Journal
of chemical physics, 129(11):114903, September 2008.
[37] Chris Groves. Suppression of geminate charge recombination in organic photo-
voltaic devices with a cascaded energy heterojunction. Energy & Environmental
Science, 6:1546–1551, 2013.
[38] Chris Groves, Robin G E Kimber, and Alison B Walker. Simulation of loss
mechanisms in organic solar cells: A description of the mesoscopic Monte Carlo
technique and an evaluation of the first reaction method. The Journal of chem-
ical physics, 133(14):144110, October 2010.
[39] Cheng Gu, Youchun Chen, Zhongbo Zhang, Shanfeng Xue, Shuheng Sun,
Chengmei Zhong, Huanhuan Zhang, Ying Lv, Fenghong Li, Fei Huang, and
Yuguang Ma. Achieving high efficiency of PTB7-based polymer solar cells via
integrated optimization of both anode and cathode interlayers. Advanced En-
ergy Materials, 4(8), June 2014.
[40] Zhicai He, Chengmei Zhong, Shijian Su, Miao Xu, Hongbin Wu, and Yong Cao.
Enhanced power-conversion efficiency in polymer solar cells using an inverted
device structure. Nature Photonics, 6(9):593–597, August 2012.
[41] Maria Hilczer and M Tachiya. Unified Theory of Geminate and Bulk Electron-
Hole Recombination in Organic Solar Cells. The Journal of Physical Chemistry
C, 114(14):6808–6813, April 2010.
BIBLIOGRAPHY 144
[42] Sebastian T Hoffmann, Heinz Bassler, and Anna Kohler. What determines
inhomogeneous broadening of electronic transitions in conjugated polymers?
The journal of physical chemistry. B, 114(51):17037–48, December 2010.
[43] Eric T. Hoke, Koen Vandewal, Jonathan A. Bartelt, William R. Mateker, Jes-
sica D. Douglas, Rodrigo Noriega, Kenneth R. Graham, Jean M. J. Frechet,
Alberto Salleo, and Michael D. McGehee. Recombination in Polymer:Fullerene
Solar Cells with Open-Circuit Voltages Approaching and Exceeding 1.0 V. Ad-
vanced Energy Materials, 3(2):220–230, February 2013.
[44] Ulrich Hormann, Julia Kraus, Mark Gruber, Christoph Schuhmair, Theresa
Linderl, Stefan Grob, Stephan Kapfinger, Konrad Klein, Martin Stutzman,
Hubert Krenner, and Wolfgang Brutting. Quantification of energy losses in
organic solar cells from temperature-dependent device characteristics. Physical
Review B, 88(23):235307, December 2013.
[45] Ian A. Howard, Fabian Etzold, Frederic Laquai, and Martijn Kemerink.
Nonequilibrium charge dynamics in organic solar cells. Advanced Energy Mate-
rials, 4, March 2014.
[46] Askat E Jailaubekov, Adam P Willard, John R Tritsch, Wai-Lun Chan, Na Sai,
Raluca Gearba, Loren G Kaake, Kenrick J Williams, Kevin Leung, Peter J
Rossky, and X-Y Zhu. Hot charge-transfer excitons set the time limit for charge
separation at donor/acceptor interfaces in organic photovoltaics. Nature mate-
rials, 12(1):66–73, January 2013.
[47] Fiona C. Jamieson, Tiziano Agostinelli, Hamed Azimi, Jenny Nelson,
and James R. Durrant. Field-Independent Charge Photogeneration in
PCPDTBT/PC 70 BM Solar Cells. The Journal of Physical Chemistry Let-
ters, 1(23):3306–3310, December 2010.
[48] Fiona C. Jamieson, Ester Buchaca Domingo, Thomas McCarthy-Ward, Martin
Heeney, Natalie Stingelin, and James R. Durrant. Fullerene crystallisation as a
BIBLIOGRAPHY 145
key driver of charge separation in polymer/fullerene bulk heterojunction solar
cells. Chemical Science, 3(2):485, 2012.
[49] Rene A. J. Janssen and Jenny Nelson. Factors limiting device efficiency in
organic photovoltaics. Advanced Materials, 25:1847–58, 2013.
[50] Matthew L Jones, Reesha Dyer, Nigel Clarke, and Chris Groves. Are hot
charge transfer states the primary cause of efficient free-charge generation in
polymer:fullerene organic photovoltaic devices? A kinetic Monte Carlo study.
Physical Chemistry Chemical Physics, June 2014.
[51] L. Kador. Stochastic theory of inhomogeneous spectroscopic line shapes rein-
vestigated. The Journal of Chemical Physics, 95(8):5574, 1991.
[52] Youngkyoo Kim, Steffan Cook, Sachetan M. Tuladhar, Stelios a. Choulis, Jenny
Nelson, James R. Durrant, Donal D. C. Bradley, Mark Giles, Iain McCul-
loch, Chang-Sik Ha, and Moonhor Ree. A strong regioregularity effect in self-
organizing conjugated polymer films and high-efficiency polythiophene:fullerene
solar cells. Nature Materials, 5(3):197–203, February 2006.
[53] Bernard Kippelen and Jean-Luc Bredas. Organic photovoltaics. Energy &
Environmental Science, 2(3):251, 2009.
[54] Bernard Kippelen and Jean-Luc Bredas. Organic photovoltaics. Energy &
Environmental Science, 2(3):251, 2009.
[55] Juliane Kniepert, Ilja Lange, Niels J. Van Der Kaap, L. Jan Anton Koster,
and Dieter Neher. A Conclusive view on charge generation, recombination,
and extraction in as-prepared and annealed P3HT:PCBM Blends: Combined
experimental and simulation work. Advanced Energy Materials, 4(7), May 2014.
[56] Girish Lakhwani, Akshay Rao, and Richard H Friend. Bimolecular recombina-
tion in organic photovoltaics. Annual review of physical chemistry, 65:557–81,
January 2014.
BIBLIOGRAPHY 146
[57] Frederic Laquai, Gerhard Wegner, and Heinz Bassler. What determines the
mobility of charge carriers in conjugated polymers? Philosophical transactions.
Series A, Mathematical, physical, and engineering sciences, 365(1855):1473–87,
June 2007.
[58] Sibel Y. Leblebici, Teresa L. Chen, Paul Olalde-Velasco, Wanli Yang, and Biwu
Ma. Reducing exciton binding energy by increasing thin film permittivity: An
effective approach to enhance exciton separation efficiency in organic solar cells.
ACS Applied Materials and Interfaces, 5:10105–10110, 2013.
[59] Harrison Ka Hin Lee, Zhao Li, Iordania Constantinou, Franky So, Sai Wing
Tsang, and Shu Kong So. Batch-to-Batch Variation of Polymeric Photovoltaic
Materials: its Origin and Impacts on Charge Carrier Transport and Device
Performances. Advanced Energy Materials, pages n/a–n/a, July 2014.
[60] Yongye Liang, Zheng Xu, Jiangbin Xia, Szu-Ting Tsai, Yue Wu, Gang Li, Claire
Ray, and Luping Yu. For the bright future-bulk heterojunction polymer solar
cells with power conversion efficiency of 7.4%. Advanced Materials, 22(20):E135–
8, May 2010.
[61] Tao Liu, David L Cheung, and Alessandro Troisi. Structural variability and dy-
namics of the P3HT/PCBM interface and its effects on the electronic structure
and the charge-transfer rates in solar cells. Physical chemistry chemical physics
: PCCP, 13(48):21461–70, December 2011.
[62] Tao Liu and Alessandro Troisi. Absolute Rate of Charge Separation and Re-
combination in a Molecular Model of the P3HT/PCBM Interface. The Journal
of Physical Chemistry C, 115(5):2406–2415, February 2011.
[63] Haibo Ma and Alessandro Troisi. Direct Optical Generation of Long-
Range Charge-Transfer States in Organic Photovoltaics. Advanced Materials,
26(35):6163–6167, 2014.
[64] M. M. Mandoc, L. J. A. Koster, and P. W. M. Blom. Optimum charge carrier
mobility in organic solar cells. Applied Physics Letters, 90(13):133504, 2007.
BIBLIOGRAPHY 147
[65] V. Mihailetchi, L. Koster, J. Hummelen, and P. Blom. Photocurrent Gen-
eration in Polymer-Fullerene Bulk Heterojunctions. Physical Review Letters,
93(21):216601, November 2004.
[66] Allen Miller and Elihu Abrahams. Impurity Conduction at Low Concentrations.
Physical Review, 120(3):745–755, November 1960.
[67] Nichole Cates Miller, Eunkyung Cho, Roman Gysel, Chad Risko, Veaceslav
Coropceanu, Chad E. Miller, Sean Sweetnam, Alan Sellinger, Martin Heeney,
Iain McCulloch, Jean-Luc Bredas, Michael F. Toney, and Michael D. McGe-
hee. Factors Governing Intercalation of Fullerenes and Other Small Molecules
Between the Side Chains of Semiconducting Polymers Used in Solar Cells. Ad-
vanced Energy Materials, 2(10):1208–1217, October 2012.
[68] M. Mingebach, S. Walter, V. Dyakonov, and C. Deibel. Direct and charge
transfer state mediated photogeneration in polymer-fullerene bulk heterojunc-
tion solar cells. Applied Physics Letters, 100(19), 2012.
[69] H. Nemec, H.-K. Nienhuys, Fengling Zhang, O. Inganas, Arkady Yartsev, and
V. Sundstrom. Charge Carrier Dynamics in Alternating Polyfluorene Copoly-
mer:Fullerene Blends Probed by Terahertz Spectroscopy. Journal of Physical
Chemistry C, 112(16):6558–6563, April 2008.
[70] Rodrigo Noriega, Jonathan Rivnay, Koen Vandewal, Felix P V Koch, Natalie
Stingelin, Paul Smith, Michael F Toney, and Alberto Salleo. A general re-
lationship between disorder, aggregation and charge transport in conjugated
polymers. Nature materials, 12(8):1–7, August 2013.
[71] Ton Offermans, Stefan C.J. Meskers, and Rene a.J. Janssen. Monte-Carlo simu-
lations of geminate electronhole pair dissociation in a molecular heterojunction:
a two-step dissociation mechanism. Chemical Physics, 308(1-2):125–133, Jan-
uary 2005.
[72] Sung Heum Park, Anshuman Roy, Serge Beaupre, Shinuk Cho, Nelson Coates,
Ji Sun Moon, Daniel Moses, Mario Leclerc, Kwanghee Lee, and Alan J. Heeger.
BIBLIOGRAPHY 148
Bulk heterojunction solar cells with internal quantum efficiency approaching
100%. Nature Photonics, 3(5):297–302, April 2009.
[73] Peter Peumans and Stephen R. Forrest. Separation of geminate charge-pairs at
donoracceptor interfaces in disordered solids. Chemical Physics Letters, 398(1-
3):27–31, November 2004.
[74] Carlito S Ponseca, Arkady Yartsev, Ergang Wang, Mats R Andersson, Dimali
Vithanage, and Villy Sundstrom. Ultrafast terahertz photoconductivity of bulk
heterojunction materials reveals high carrier mobility up to nanosecond time
scale. Journal of the American Chemical Society, 134(29):11836–9, July 2012.
[75] Christopher M. Proctor, Steve Albrecht, Martijn Kuik, Dieter Neher, and
Thuc Quyen Nguyen. Overcoming geminate recombination and enhancing ex-
traction in solution-processed small molecule solar cells. Advanced Energy Ma-
terials, 4(10), March 2014.
[76] Christopher M. Proctor, Martijn Kuik, and Thuc-Quyen Nguyen. Charge
carrier recombination in organic solar cells. Progress in Polymer Science,
38(12):1941–1960, December 2013.
[77] Christopher M Proctor, John A Love, and Thuc Quyen Nguyen. Mobility
guidelines for high fill factor solution-processed small molecule solar cells, July
2014.
[78] S Rackovsky and H Scher. Theory of Geminate Recombination as a Molecular
Process. Physical Review Letters, 52(6):453–456, February 1984.
[79] Barry Rand, Diana Burk, and Stephen Forrest. Offset energies at organic semi-
conductor heterojunctions and their influence on the open-circuit voltage of
thin-film solar cells. Physical Review B, 75(11):115327, March 2007.
[80] Biswajit Ray and Muhammad Ashraful Alam. Achieving Fill Factor Above 80%
in Organic Solar Cells by Charged Interface. IEEE Journal of Photovoltaics,
2(d):1–8, 2012.
BIBLIOGRAPHY 149
[81] Brian S Rolczynski, Jodi M Szarko, Hae Jung Son, Yongye Liang, Luping Yu,
and Lin X Chen. Ultrafast intramolecular exciton splitting dynamics in isolated
low-band-gap polymers and their implications in photovoltaic materials design.
Journal of the American Chemical Society, 134(9):4142–52, March 2012.
[82] Tom J. Savenije, Jessica E. Kroeze, Xiaoniu Yang, and Joachim Loos. The
formation of crystalline P3HT fibrils upon annealing of a PCBM:P3HT bulk
heterojunction. Thin Solid Films, 511-512:2–6, July 2006.
[83] Brett M Savoie, Akshay Rao, Artem A Bakulin, Simon Gelinas, Bijan
Movaghar, Richard H Friend, Tobin J Marks, and Mark A Ratner. Unequal
partnership: Asymmetric roles of polymeric donor and fullerene acceptor in
generating free charge. Journal of the American Chemical Society, 136(7):2876–
2884, February 2014.
[84] H. Scher and S. Rackovsky. Theory of geminate recombination on a lattice. The
Journal of Chemical Physics, 81(4):1994, 1984.
[85] C. G. Shuttle, A. Maurano, R. Hamilton, B. ORegan, John C de Mello, and
J. R. Durrant. Charge extraction analysis of charge carrier densities in a poly-
thiophene/fullerene solar cell: Analysis of the origin of the device dark current.
Applied Physics Letters, 93(18):183501, 2008.
[86] Magdalena Skompska and Artur Szkurat. The influence of the structural defects
and microscopic aggregation of poly(3-alkylthiophenes) on electrochemical and
optical properties of the polymer films: discussion of an origin of redox peaks in
the cyclic voltammograms. Electrochimica Acta, 46(26-27):4007–4015, August
2001.
[87] Robert A Street, Daniel Davies, Petr P Khlyabich, Beate Burkhart, and Barry C
Thompson. Origin of the tunable open-circuit voltage in ternary blend bulk
heterojunction organic solar cells. Journal of the American Chemical Society,
135(3):986–9, January 2013.
BIBLIOGRAPHY 150
[88] Sean Sweetnam, Kenneth R Graham, Guy O Ngongang Ndjawa, Thomas
Heumueller, Jonathan A Bartelt, Timothy M Burke, Wei You, Aram Amassian,
and Michael D McGehee. Characterization of the polymer energy landscape in
polymer:fullerene bulk heterojunctions with pure and mixed phases. Journal of
the American Chemical Society, September 2014.
[89] Zhi-Kuang Tan, Kerr Johnson, Yana Vaynzof, Artem A Bakulin, Lay-Lay Chua,
Peter K H Ho, and Richard H Friend. Suppressing Recombination in Polymer
Photovoltaic Devices via Energy-Level Cascades. Advanced materials, page
4131, June 2013.
[90] Wing C. Tsoi, Steve J. Spencer, Li Yang, Amy M. Ballantyne, Patrick G. Nichol-
son, Alan Turnbull, Alex G. Shard, Craig E. Murphy, Donal D C Bradley, Jenny
Nelson, and Ji Seon Kim. Effect of crystallization on the electronic energy levels
and thin film morphology of P3HT:PCBM blends. Macromolecules, 44:2944–
2952, 2011.
[91] Sarah T. Turner, Patrick Pingel, Robert Steyrleuthner, Edward J. W. Cross-
land, Sabine Ludwigs, and Dieter Neher. Quantitative Analysis of Bulk Het-
erojunction Films Using Linear Absorption Spectroscopy and Solar Cell Perfor-
mance. Advanced Functional Materials, 21(24):4640–4652, December 2011.
[92] Kristofer Tvingstedt, Koen Vandewal, and Abay Gadisa. Electroluminescence
from charge transfer states in polymer solar cells. Journal of the American
Chemical Society, 131(33):11819–11824, 2009.
[93] J. J. M. van der Holst, F. W. A. van Oost, R. Coehoorn, and P. A. Bobbert.
Electron-hole recombination in disordered organic semiconductors: Validity of
the Langevin formula. Physical Review B, 80(23):235202, December 2009.
[94] Koen Vandewal, Steve Albrecht, Eric T Hoke, Kenneth R Graham, Johannes
Widmer, Jessica D Douglas, Marcel Schubert, William R Mateker, Jason T
Bloking, George F Burkhard, Alan Sellinger, Jean M J Frechet, Aram Amas-
sian, Moritz K Riede, Michael D McGehee, Dieter Neher, and Alberto Salleo.
BIBLIOGRAPHY 151
Efficient charge generation by relaxed charge-transfer states at organic inter-
faces. Nature materials, 13(1):63–8, January 2014.
[95] Koen Vandewal, Abay Gadisa, Wibren D. Oosterbaan, Sabine Bertho, Fateme
Banishoeib, Ineke Van Severen, Laurence Lutsen, Thomas J. Cleij, Dirk Van-
derzande, and Jean V. Manca. The Relation Between OpenCircuit Voltage and
the Onset of Photocurrent Generation by ChargeTransfer Absorption in Poly-
mer:Fullerene Bulk Heterojunction Solar Cells. Advanced Functional Materials,
18(14):2064–2070, July 2008.
[96] Koen Vandewal, L. Goris, I. Haeldermans, M. Nesladek, K. Haenen, P. Wagner,
and J.V. Manca. Fourier-Transform Photocurrent Spectroscopy for a fast and
highly sensitive spectral characterization of organic and hybrid solar cells. Thin
Solid Films, 516(20):7135–7138, August 2008.
[97] Koen Vandewal, Kristofer Tvingstedt, Abay Gadisa, Olle Inganas, and Jean V.
Manca. On the origin of the open-circuit voltage of polymer-fullerene solar cells.
Nature materials, 8(11):904–9, November 2009.
[98] Koen Vandewal, Kristofer Tvingstedt, Abay Gadisa, Olle Inganas, and Jean V.
Manca. Relating the open-circuit voltage to interface molecular properties of
donor:acceptor bulk heterojunction solar cells. Physical Review B, 81(12):1–8,
March 2010.
[99] Koen Vandewal, Kristofer Tvingstedt, and Olle Inganas. Charge Transfer States
in Organic Donor-Acceptor Solar Cells. In Quantum Efficiency in Complex
Systems, Part II, volume 85 of Semiconductors and Semimetals, pages 261–
295. Elsevier, 2011.
[100] Koen Vandewal, Johannes Widmer, Thomas Heumueller, Christoph J. Brabec,
Michael D. McGehee, Karl Leo, Moritz Riede, and Alberto Salleo. Increased
open-circuit voltage of organic solar cells by reduced donor-acceptor interface
area. Advanced Materials, 26:3839–3843, 2014.
BIBLIOGRAPHY 152
[101] Dirk Veldman, Ozlem Ipek, Stefan C J Meskers, Jorgen Sweelssen, Marc M
Koetse, Sjoerd C Veenstra, Jan M Kroon, Svetlana S van Bavel, Joachim
Loos, and Rene a J Janssen. Compositional and electric field dependence of
the dissociation of charge transfer excitons in alternating polyfluorene copoly-
mer/fullerene blends. Journal of the American Chemical Society, 130(24):7721–
35, June 2008.
[102] Nenad Vukmirovic, Carlito S Ponseca, Hynek Nemec, Arkady Yartsev, and Villy
Sundstrom. Insights into the Charge Carrier Terahertz Mobility in Polyfluorenes
from Large-Scale Atomistic Simulations and Time-Resolved Terahertz Spec-
troscopy. The Journal of Physical Chemistry C, 116(37):19665–19672, Septem-
ber 2012.
[103] Paul Westacott, John R. Tumbleston, Safa Shoaee, Sarah Fearn, James H. Ban-
nock, James B. Gilchrist, Sandrine Heutz, John DeMello, Martin Heeney, Har-
ald Ade, James Durrant, David S. McPhail, and Natalie Stingelin. On the role
of intermixed phases in organic photovoltaic blends. Energy & Environmental
Science, pages 2756–2764, 2013.
[104] G. A H Wetzelaer, M. Kuik, M. Lenes, and P. W M Blom. Origin of the
dark-current ideality factor in polymer:fullerene bulk heterojunction solar cells.
Applied Physics Letters, 99(15):153506, 2011.
[105] G. A. H. Wetzelaer, M. Kuik, H. T. Nicolai, and P. W. M. Blom. Trap-assisted
and Langevin-type recombination in organic light-emitting diodes. Physical
Review B, 83(16):165204, April 2011.
[106] Mariusz Wojcik, Przemyslaw Michalak, and M. Tachiya. Geminate electron-hole
recombination in organic solids in the presence of a donor-acceptor heterojunc-
tion. Applied Physics Letters, 96(16):162102, 2010.
[107] P Wurfel. The chemical potential of radiation. Journal of Physics C: Solid
State Physics, 15:3967–3985, 1982.