THE DEVICE PHYSICS OF ORGANIC SOLAR CELLS A DISSERTATIONmq955kd8880/Burke_Dissertation... · the...

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



http://purl.stanford.edu/mq955kd8880

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


Aaron Lindenberg


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








solar cells annealed at 111C. . . . . . . . . . . . . . . . . . . . . . . . 112


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


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


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


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.


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


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


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)


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)


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.


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.


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


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


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


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


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


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.


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.


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


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


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


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


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


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]


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


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


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.


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


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.


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]


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


8 nm Mixed Region 8.7e−6 3.5e−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.


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.


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.


103 104 105 106 107 108


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.


103 104 105 106 107 108 109


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.


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.


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.


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


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



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


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


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


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]


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


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


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:


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


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.


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


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:


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.


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


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,


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


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


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


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


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


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.


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


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


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.


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.


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


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


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.


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.


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)


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


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:


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:


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]


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


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


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


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]


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.


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)


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)


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


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


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


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


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.


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:


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:


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:


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,



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.


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


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


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


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


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


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


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


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


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.


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


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.


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.


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


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.


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



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



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



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



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



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


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.


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.


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.


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)


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.


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.


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


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].


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


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.


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


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].


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):


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


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


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.


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


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


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.


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).


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.


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.


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.

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