The Device Physics of Organic Solar Cells a Dissertation

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

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/mq955kd8880

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Michael McGehee, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Aaron Lindenberg

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Alberto Salleo

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University 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 semiconductors. 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 mobility 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 recombination rate to show that recombination proceeds from an equilibrated population 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 studies 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 resolved terahertz spectroscopy) and the geminate pair lifetime (measured using transient absorption or transient photoluminescence). . . 23 2.3 Required local mobilities for 90% field-independent IQE for the specified 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 electrons 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 −4 2 2 hole mobility varied from 4x10 cm /Vs up to 4 cm /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 2 simulation with τct =10 ns and µe = µh = 1 cm /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 geminate separation using KMC and extracting the geminate pair lifetime

as a function of the value of τct input into the simulation for electron 2 and hole mobilities of 0.01, 0.1, 1 and 10 cm /Vs (µe = µh). The lines are a guide to the eye. The horizontal line represents a typical measured 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 reduced 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 disorder 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 parameters 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 indicate 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- 2 lar cell at Jsc producing a current of 10 mA/cm equally distributed between an electron and hole current...... 100 4.5 The extracted series resistance of each P3HT annealing condition as a function of device thickness, showing an approximately linear trend vs. thickness with a annealing temperature dependent slope...... 106 4.6 The slope of the series resistance vs. thickness curves plotted against the P3HT hole mobility showing how the series resistance in these devices appears to be due to transport in pure P3HT regions . . . . . 107 4.7 Experimental data (points) and fit to our expression for P3HT:PCBM solar cells annealed at 0C...... 108 4.8 Experimental data (points) and fit to our expression for P3HT:PCBM solar cells annealed at 48C...... 109 4.9 Experimental data (points) and fit to our expression for P3HT:PCBM solar cells annealed at 71C...... 110 4.10 Experimental data (points) and fit to our expression for P3HT:PCBM solar cells annealed at 88C...... 111 4.11 Experimental data (points) and fit to our expression for P3HT:PCBM solar cells annealed at 111C...... 112 4.12 Experimental data (points) and fit to our expression for P3HT:PCBM solar cells annealed at 148C...... 113

xv 4.13 The total amount of photocurrent produced in each device in the P3HT:PCBM annealing series...... 114

4.14 The extracted Vbi parameter for the P3HT:PCBM series. The solid lines are the actual built-in potential estimated from the crossing point between light and dark IV curves. The dashed lines are the fit parameters.116 4.15 The photocarrier dark carrier recombination coefficient for our P3HT:PCBM device series, expressed as the fraction of recombination that proceeds via this mechanism at the maximum power point...... 117 4.16 Photocarrier - Photocarrier Recombination coefficient for our P3HT:PCBM device series, expressed as the fraction of recombination that proceeds via this mechanism at the maximum power point...... 118 4.17 The inverse proportionality of the photocarrier-photocarrier recombination 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 21 given a constant N0 = 1x10 . 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 limits of organic photovoltaics (OPV), which are an emerging technology for harnessing solar energy using semiconducting plastics rather than the traditional inorganic materials 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 polarizable material, like inside a semiconductor, their attraction is screened by the polarization of neutral atoms around each charge carrier. The degree to which this screening reduces their attraction is quantiﬁed by the relative dielectric constant of the material. Silicon, for example, has a dielectric constant near CHAPTER 1. INTRODUCTION 3

12, whereas organic semiconductors typically have dielectric constants between 3 and 4. This means that electrons and holes in organic semiconductors feel attracted to each other 3-4 times stronger than in Silicon, where they are considered to be basically free. This strong attraction makes the electrons and holes tend to pair up into an overall charge neutral species called an exciton and semiconductors where excitons play a large role in the dynamics are called excitonic semiconductors.

Since organic semiconductors are excitonic, the initial electron/hole pair created by absorbing a photon is not free but instead a tightly bound singlet exciton. This exciton is overall charge neutral so it can only move slowly via energy transfer and diffusion. A large part of the design of organic solar cells is driven by how to split this exciton into a free electron and hole during its short lifetime. The canonical way to achieve this in organic solar cells is by using a heterojunction between two different organic semiconductors, a donor and an acceptor. The materials are chosen to have different electron affinities, providing an energetic driving force for an exciton that reaches the donor/acceptor interface to dissociate into a Charge Transfer (CT) state, which is an electron/hole pair that resides on nearby molecules. However, this state is still not free since the electron and hole in a CT state can still have a significant attraction. The rest of the process by which CT states split into fully free charges is discussed in Chapter 2. For reasons that are not yet fully understood, the diffusion length of an exciton in an organic semiconductor is limited to several tens of nanometers at most. This means that there must be a donor/acceptor interface within 10 nm of each location where a photon could be absorbed so that the exciton generated from that absorption can be split before it recombines. In order to absorb the majority of incident light however, the total device must be several hundreds of nanometers thick, making a simple bilayer (donor on top of acceptor) device architecture inefficient since only a small slice of the device near the interface can contribute to photocurrent. These conflicting constraints led to the introduction of the bulk heterojunction architecture, where partially immiscible donor and acceptor materials are mixed in a solvent, cast and allowed to dry. The materials undergo a partial phase separation during the CHAPTER 1. INTRODUCTION 4

Mixed Donor Acceptor Region

EVac EA IP - Energy +

Polymer Fullerene

Figure 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. drying process leading to small domains of pure donor, pure acceptor and molecularly mixed regions containing both donor and acceptor materials as shown schematically in Figure 1.1. The current state of the art in organic solar cells is to use a thin (typically 100-200 nm) bulk heterojunction absorber layer placed between two electrodes with different work functions to create an electric field across the device. The electric field helps move electrons and holes toward separate contacts more quickly than diffusion alone would be able to accomplish.

1.2 Basic Solar Cell Device Physics

The simplest way to understand a solar cell is as a light-powered electron pump (shown schematically in Figure 1.2). Solar cells have a reservior of electrons that are largely immobile but can be excited into a mobile state by absorbing a photon. Once excited these electrons are free to move either by diﬀusion or by drifting in an electric CHAPTER 1. INTRODUCTION 5

Reservoir of Electrons Conduction Band Recombination Light

External Circuit Photogeneration

ReservoirReservoir of of Holes Holes Valence Band

Electron Hole

Figure 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.

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 concentrations leads to the species having different electrochemical potentials, which can be exploited to perform work in an external circuit. Sometimes this process is simplified 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 description 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 semiconductor and the conduction band provides mobile electronic states that these valence band electrons can be excited into. This is shown in the right of Figure 1.2. CHAPTER 1. INTRODUCTION 6

ﬃnity

Electron A

Active Layer Hole Contact Electron Contact

Ionization Potential Voltage Electric Field

Position Position

Figure 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.

The solar cells that we will discuss in this thesis are all fabricated using a p- i-n architecture, where an undoped semiconductor is placed between two contact materials that have different work functions, leading to the creation of an electric field across the intrinsic active layer as electric charges move from the low work-function contact to the high work function contact during device fabrication. This electric field is critical for pin device functioning and must be included in any discussion of their device physics. A schematic of a PIN device is shown in Figure 1.3. The fundamental relation that describes solar cell behavior is that the current that can be extracted from a solar cell is equal to the photogenerated current produced by CHAPTER 1. INTRODUCTION 7

absorbing sunlight minus any recombination losses that occur when an electron and hole meet and annihilate each other inside the device. So,

J(V ) = q [G(V ) − R(V )] (1.1)

where J(V) is the current measured leaving the solar cell, G(V) the rate at which electrons are being excited in the solar cell as a function of the operating voltage, R(V) is the rate at which electrons and holes are recombining and q is the charge of an electron. In most solar cells, including organic solar cells, G actually has no voltage dependence, so the above equation simpliﬁes to:

J(V ) = q [G − R(V )] (1.2)

The goal of solar cell device physics is to understand R(V) using physical models that allow us to relate it to material and architectural properties. It is impossible to completely eliminate recombination since the condition of detailed balance requires that any device that absorbs light also emits light, so there is a lower bound on R(V) set by an unavoidable amount of radiative recombination that is always present in all solar cells. Nonetheless, most solar cell materials do not operate near this radiative limit and there is substantial work to be done to minimize R(V). Succinctly put, the goal of this thesis is to explain why G is voltage independent in organic solar cells and to derive an expression for R(V).

1.2.1 Electrons, Holes and Quasi-Fermi Levels

There are two diﬀerent ways to quantify the density of excited electrons and holes in a solar cell. One can just directly measure the density of electrons at a point in the device and report that number in units like electrons/cm−3. One could also equivalently report the chemical potential of the electrons at that same position in electron-volts. There is a one-to-one relationship between chemical potential and CHAPTER 1. INTRODUCTION 8

carrier density, so the two methods are equally appropriate for specifying how many electrons are present at a point in the device. Depending on the context, one or the other representation might be more useful. For example, we will see below that the operating voltage of a solar cell is speciﬁed in terms of the chemical potentials of electrons and holes. Recombination, however is typically expressed more easily in terms of the densities of electrons and holes. The link between chemical potential and carrier density comes from realizing that electrons in the conduction band, for example, relax into equilibrium very quickly among the conduction band states, so the excited electrons are always distributed among the conduction band states in a Fermi-Dirac distribution. Similarly the holes are always distributed among the accessible valence band states in a Fermi-Dirac distribution. So, if we know the density of electronic states as a function of energy, g(E), we can relate the chemical potential of electrons to the density of electrons using:

Z ∞ 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:

Z ∞ 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 reasons and some mathematical simplicity, device physicists do not talk about chemical potentials but instead quasi-Fermi levels. The quasi-Fermi level for electrons (Efn) is just another name for the chemical potential of electrons, however the quasi-Fermi level for holes (Efp) is deﬁned to have the opposite sign as its chemical potential. So,

Efn = µe (1.5)

Efp = −µp (1.6) CHAPTER 1. INTRODUCTION 9

ﬃnity ﬃnity Acceptor Electron A Acceptor Electron A

Donor Ionization Potential Donor Ionization Potential

Figure 1.4: Two example band diagrams showing the quasi-Fermi level for electrons as 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 electrons 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 ) Z L R(V ) = kn(x, V )p(x, V ) dx (1.7) 0 CHAPTER 1. INTRODUCTION 10

Equation 1.7 says that locally the rate of recombination is just proportional to the local density of electrons and holes. We want to know the total rate of recombination, so we need to integrate this recombination density over the thickness L of the solar cell. All of the solar cells we deal with in this thesis will be symmetrical in two dimensions so we only have to integrate over one dimension.

1.2.3 Quasi-Fermi Levels and Operating Voltage

In order to understand organic solar cells, it is important to make the connection between external parameters that you control and the microscopic internal parameters that you do not directly observe but drive the behavior of the device, i.e. the quasi- Fermi levels. The connection comes from realizing that an electron very near the electron extracting contact is locally in equilibrium with the reservoir of electrons in the contact since it is easy for electrons to be exchanged between the contact and the active layer. Similarly a hole very close to the hole extracting contact is locally in equilibrium with the electrons in that contact. This is important because the difference in electrochemical potential between the hole and electron extracting contacts is what we measure when we connect a volt meter to our solar cell and it is what we control when we force the voltage across the solar cell to be a specific value, by connecting the cell to a battery for example. So, the operating voltage that we measure on the solar cell is equal to the difference in the electron and hole quasi-Fermi levels at the two contacts:

qV = Efn(0) − Efp(L) (1.8)

In general, measuring the operating voltage does not tell us the quasi-Fermi level splitting throughout the entire device, it merely tells us the splitting measured at two separate points as shown in Figure 1.4. In order to determine the quasi-Fermi level splitting throughout the device, we need to use drift-diﬀusion modeling in order to relate the shape of the quasi-Fermi levels to currents in the device. Knowledge of the current being drawn, the illumination level and the voltage applied are enough to calculate the electron and hole quasi-Fermi levels as we do in Chapter 4. CHAPTER 1. INTRODUCTION 11

1.2.4 Maximum Power Point

Equation 1.8 states that as we increase the voltage on our solar cell, we are also increasing the splitting between the electron and hole quasi-Fermi levels. Previously we saw that there is a monotonic relationship between carrier density and quasi-Fermi level, so this means that increasing the voltage on the device results in more charge carriers being present in the active layer. This in turn leads to the n*p product increasing, which means that recombination must necessarily increase with voltage. So, every solar cell faces a tradeoff. In order to increase the power output, you would like to operate the solar cell at a higher voltage. However, as you increase the operating voltage you begin to lose current according to equation 1.7. The power output is proportional to the product of J and V, which are changing in opposite ways so there will be an optimal voltage Vmpp that maximizes the power output. This is called the maximum power point voltage. In order to describe the power output of a solar cell, researchers use a combination of the maximum current (Jsc) the device can produce when V=0, the maximum voltage (Voc) the device can produce when J=0 and a reduction factor (FF ) that is determined by the ratio of JmppVmpp and JscVoc. FF is called the fill factor and is a number between 0 and 1. The power output of any solar cell can be specified by the product of these three quantities:

P = JscVocFF (1.9)

The distinction between Jsc, Voc and FF is useful because there are typically different materials parameters and architectural tradeoffs that determine each one, so they can be thought of as quantifying three different aspects of a given solar cell’s operation. CHAPTER 1. INTRODUCTION 12

1.3 Organic Solar Cell Device Physics

1.3.1 Charge Transfer States

In most solar cells, the only electronic species of interest are electrons and holes, which are assumed to move basically independently of each other except for occasion- ally recombining when they are close by. This is because inorganic semiconductors have high dielectric constants, which means that they eﬀectively screen the Coulomb attraction between electrons and holes so that they are barely attracted to each other at all. In contrast, organic solar cells have low dielectric constants, usually between 3-5 so they do not screen Coulombic attractions well. This means that the energy of an electron-hole pair that is, say one nanometer apart could be hundreds of meV lower than that same pair 20 nm apart because you need to account for their attrac- tive interaction energy. When and electron and hole are next to each other in an organic solar cell, with the electron typically on an acceptor molecule and the hole on a nearby donor molecule, the pair is said to be in a Charge Transfer state since if they were to recombine it would be by transferring charge from one molecule to another. We will see in Chapters 2 and 3 that the energetics of Charge Transfer states plays a key role in determining how organic solar cells function.

1.3.2 Polarons

In inorganic semiconductors, electrons and holes are pictured as moving basically freely among the atoms that compose the crystalline semiconductor. This is because there is little interaction between the electronic excitations and the vibrational modes of the crystals so they can be treated independently. However, in organic semiconductors, there is a strong interaction between nuclear coordinates and electronic ones. This results in molecules reorganizing themselves into diﬀerent physical conformations when an electron or hole resides on them. As charge carriers move then, we need to picture them dragging around a local polarization and reorganization of the nearby molecules. This combined vibrational and electronic excitation is called a polaron. In this thesis we will not discuss polaronic eﬀects in any great detail but just mention CHAPTER 1. INTRODUCTION 13

them here. We will use the terms electron or hole and negative or positive polaron interchangeably in this work.

1.3.3 Energetic Disorder

In many solar cells, the valence and conduction bands are treated as delta functions that have many electronic states at essentially the same energy. However, in organic solar cells the electron aﬃnity and ionization potentials are much more diﬀuse. There are three essential causes for this:

1. Dipolar disorder - Organic semiconductors are composed of polarizable molecules with static and induced dipole moments. Since the local environment of each molecule varies slightly due to random ﬂuctuations in molecular orientation and density, the dipoles also ﬂuctuate in strength and orientation. This leads to large scale inhomogeneities in the electrostatic potential of the solar cell, giving a Gaussian shape to the energy levels.

2. Conformational disorder - A hole on an extended donor molecule can lower its energy by delocalizing along the length of the molecule. If there is a break in the conjugation of the molecule, however, the delocalization process is arrested at that break. So, the local conjugation length of each molecule sets the local energy of an electron (negative polaron). This conjugation length varies from place to place in the solar cell active layer since molecules pack in slightly diﬀerent ways, leading to twists and turns in the molecules.

3. Traps - Reactions between organic molecules and other impurities in the organic solar cell active layer can create defect states that have diﬀerent energies from the original molecules. If these energies are lower than the unreacted molecules, then electrons or holes will preferentially reside in a trap state and these states also serve to spread out the distribution of available electronic states for electrons and holes.

The presence of energetic disorder does not qualitatively change the relationship between quasi-Fermi level and carrier density, but it does need to be taken into CHAPTER 1. INTRODUCTION 14

account and we will do so in Chapter 3 when we consider the open-circuit voltage of an organic solar cell. Chapter 2

The Short-Circuit Current

2.1 Preface

This chapter is adapted with permission from published work by the author in Ad- vanced Materials[12].

2.2 Current Understanding and Background

The best Organic Photovoltaics (OPV) with Bulk-Heterojunction (BHJ) morphologies 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 region 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 important 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

Mixed Donor Acceptor τhop Region τct EVac EA IP - - - + +

Energy + +

Polymer Fullerene

Figure 2.1: (left) Schematic of a BHJ solar cell including the mixed region. Potential 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. 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 aggregation 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 efficiency is predicted by KMC when fast, local (monomer-scale) charge carrier mobilities CHAPTER 2. THE SHORT-CIRCUIT CURRENT 17

are taken into account. Additionally, we demonstrate that a 200 meV energetic offset between the mixed and pure regions in the simulated trilayer devices greatly decreases the local mobilities and Charge Transfer state lifetimes required for efficient change generation. Excitons in BHJ systems are known to dissociate at the heterojunction between the donor and acceptor materials into a hole (positive polaron) residing on the donor and an electron (negative polaron) residing on the acceptor.[19, 53] However, due to the low dielectric constant of organic semiconductors, these charges are not free and instead form a coulombically bound radical pair with a binding energy that is calculated to be around 350 meV (assuming r = 4 and a typical intermolecular spac- ing of 1 nm).[19] This geminate pair needs to become separated by‘ approximately 12 nm before its binding energy is equal to the thermal energy at room temperature, 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 competition 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 typically less than 10% at short circuit conditions and increases significantly when one simulates a device under reverse bias by adding a strong electric field.[19, 71] This inefficient, field-dependent splitting is characteristic of a process where the mobility and lifetime of the geminate pairs are not large enough for the charges to separate CHAPTER 2. THE SHORT-CIRCUIT CURRENT 18

on their own. When splitting does occur, it is primarily due to the built-in field of the BHJ overcoming the pairs binding energy and pulling the electron and hole apart. The strength of this field decreases at forward bias, making charge generation less efficient as the cell approaches open circuit and reducing the fill factor. Thus, while this model can explain the poor performance of low-efficiency OPV material systems like MDMO-PPV:PCBM, which do show field-dependent geminate splitting, it stands in sharp contrast to the field-independent internal quantum efficiencies near or above 90% observed experimentally in champion polymer systems like PCDTBT, PTB7 and PBDTTPD.[8, 19, 47, 65, 72, 60] The inability to reconcile experimental quantum efficiency measurements of high- performing systems with Monte Carlo simulations has led many groups to propose additional theories about what factors the simulations are lacking that could explain the discrepancy. One current theory is that efficient geminate splitting requires the presence of excess thermal energy, although this is under debate and sub-bandgap quantum efficiency measurements suggest that excess energy is not necessary in some systems.[5, 46, 35, 94] Previous Kinetic Monte Carlo studies have also investigated the potential effects of charge carrier delocalization, energetic disorder, molecular dipoles and dielectric reorganization and found that, while each can improve geminate splitting, none were able to account for a 90%, field-independent IQE without 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 dimensional 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 expression, since it is computationally simple and has been used extensively to model geminate splitting.[73, 71, 26, 106, 14, 66] The M-A model assumes that energetically CHAPTER 2. THE SHORT-CIRCUIT CURRENT 19

downhill hops proceed at a constant rate while uphill hops are thermally activated:

 −∆E k0 exp if ∆E > 0 k = kT (2.1) k0 if ∆E ≤ 0

The energy term includes contributions both from the electric ﬁeld and the Coulomb potential and can be written, following Peumans as:

−q2 E = − qF · reh + ULUMO(re) − UHOMO(rh) (2.2) 4π0rreh

where q is the elementary charge, 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 2 when µh = 4 cm /Vs. Thus, before presenting the results, a discussion is in order of what is known experimentally about µ and τct. CHAPTER 2. THE SHORT-CIRCUIT CURRENT 20

1.0 1.012 Mobility a) b) 8 0.8 0.0004 0.8 0.04 4 0.6 4 cm2/Vs 0.6 0 12 0.4 0.4 c) 8 0.2 0.2 4 Dissociation Probability Dissociation

0.0 nm / Separation Pair Geminate 0.00 103 104 105 106 107 0.00 1 0.22 0.43 4 0.65 6 0.87 1.08 Simulation Time / ns Applied Field / V cm−1

Figure 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 −4 2 2 varied from 4x10 cm /Vs up to 4 cm /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 2 geminate splitting simulation with τct =10 ns and µe = µh = 1 cm /Vs. b ended in recombination, c in splitting. CHAPTER 2. THE SHORT-CIRCUIT CURRENT 21

Group Mobility Lifetime [ns] 90% IQE Pre- Field Indepen- [cm2/V s] dicted dent Janssen 2005[71] 3x10−5 1000 Yes No Groves 2008[36] 2x10−3 2000 Yes No Deibel 2009[26] 3x10−5 10000 Yes Yes Wojcik 5x10−3 100 Yes No 2010[106] Groves 2013[37] 7x10−4 100 Yes Yes

Table 2.1: Lifetime and mobility values that were required in previous KMC studies to 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 important 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 measured 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 probability of it encountering a trap during the measurement is lower. Studies have shown that only the high frequency terahertz conductivity gives information directly on the intrinsic hopping rate, while the other techniques report values limited by slow but infrequent processes (compared to the hopping rate).[102] A complete device simulation that includes all of the mechanisms by which high local mobilities naturally decay into low bulk mobilities over longer length scales CHAPTER 2. THE SHORT-CIRCUIT CURRENT 22

should fully reproduce this hierarchical behavior, but for focused simulations solely of geminate splitting the question arises as to whether bulk mobility values or single-hop terahertz mobility values are more appropriate. The answer depends on what role the mobility parameter is playing in the simulation. To elucidate this role, the separation as a function of time between two typical geminate pairs (τct = 10 ns, µe = µh = 1 cm2/Vs) is plotted in Figure 2.2. As can be seen, the charges, due to their strong binding energy, spend the majority of their time right next to each other, with brief, relatively infrequent separations. Each of these separations, which we call splitting attempts, can end either with the charges becoming free or with them again becoming nearest neighbors, reforming the CT state. Recombination is assumed to be a nearest- neighbor process, so once the charges take one hop apart they cannot recombine until they ﬁrst meet each other again. The probability that the charge carriers, once they are no longer nearest neighbors, separate completely without meeting again turns out to be largely independent of both the carrier mobility and the recombination lifetime

(see Figure 2.4, 2.7 and 2.8). It is independent of τct since recombination only happens between nearest neighbors. It is independent of µ since the mobility is modeled as being isotropic so the carrier mobility just sets the timescale for each hop, it does not make the carriers more likely to hop in one direction (toward each other, reducing their separation) than in another direction (away from each other, increasing their separation). We call the probability that an electron and hole successfully escape from their mutual attraction in a single splitting attempt Pesc. Since Pesc does not depend on either the carrier mobility or lifetime, it must be a constant determined by the device’s energetic landscape (see Figure 2.5 and 2.7). The geminate splitting eﬃciency is determined by the number of splitting attempts each geminate pair makes, on average, before recombining and the probability that any single attempt is successful.

The number of attempts is set by the product of µ and τct since when the carriers are nearest neighbors, they can either recombine or attempt to split again. The probability of them recombining is set by the kinetic competition between the rate of a single hop apart, set by µ, and the rate of recombination, set by 1/τct. We conclude that the carrier mobility in a KMC simulation of geminate splitting CHAPTER 2. THE SHORT-CIRCUIT CURRENT 23

Morphology Local Mobility Geminate Pair Lifetime [ns] [cm2/V s] P3HT/PCBM[17, 2, 24, 69] 0.1 - 30 3 AFPO-3/PCBM[69] 0.73 - 1 n/a ZnPc/C60[7] 0.4 n/a TQ1/PCBM[74] 0.1 n/a PF10TBT/PCBM[101] n/a 4

Table 2.2: Literature measurements for local mobility (measured using time resolved terahertz spectroscopy) and the geminate pair lifetime (measured using transient absorption or transient photoluminescence). primarily sets the branching ratio between recombination and another splitting attempt 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 depend 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 geminate 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 mobilities 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 reported in literature, we now do the same for τct, which specifies the rate of recombination for electrons and holes that reside on neighboring molecules. This rate is CHAPTER 2. THE SHORT-CIRCUIT CURRENT 24

not the same as the free carrier lifetime measured with a technique like transient photovoltage (TPV). TPV lifetimes are dominated by the rate at which already-free carriers encounter each other rather than the rate at which they recombine once they have become nearest neighbors. It is this latter rate that is needed for KMC simulations. 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 dissociation dominating the measured response. To extract τct from these measurements we simulated PL decay curves using KMC for a range of mobilities and recombination lifetimes and calculated from each combination a prediction for the measured lifetime. Our simulations show that the decay remains exponential, as observed experimentally (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 mobilities between 0.1 and 1 cm2/Vs, the reported PL decay lifetime of 4 ns would imply an intrinsic CT state lifetime on the order of 1-10 ns. If the carrier mobility were CHAPTER 2. THE SHORT-CIRCUIT CURRENT 25

higher, τct could also be longer, which would serve to increase the geminate splitting efficiency, so this is a conservative underestimate. So far, we have established that geminate splitting in BHJ solar cells, as simulated using KMC is determined by the number of splitting attempts per geminate pair (set by the product µτct) and the probability that any given attempt is successful (a constant of the energetic landscape we denoted Pesc). We can now examine the effect of the mixed region, which alters the energetic landscape, on geminate splitting. The presence of an energy cascade between mixed and aggregated regions means that once a carrier crosses from a mixed to an aggregated region, it is energetically very unlikely to cross back, making the carriers effectively become free after crossing the width of the mixed region, not after traveling 12 nm, as would be predicted with no energy cascade. Reducing the width of the mixed region allows one to systematically increase Pesc, thereby greatly improving geminate splitting. Using an estimate for

τct of 10 ns, and values for Pesc obtained from KMC simulations of mixed region widths between 3.2 and 9.6 nm, we can calculate what terahertz mobility would be required for a 90% field-independent IQE in each situation. The results are shown in Table 2.3. For devices with terahertz mobilities above 11 cm2/Vs we would expect a greater than 90% field independent IQE even without an energy cascade. For lower mobilities down to 0.2 cm2/Vs (the low end of the range reported in literature for OPV materials), we would still predict greater than 90%, field-independent IQE, but this high IQE requires a sufficiently thin mixed region with an energy cascade to reduce the distance geminate pairs have to travel before splitting. The results are reported for τct = 10 ns, however since we have shown that the splitting efficiency depends on the product µτct only, if τct were 10 times shorter, the required mobility would simply be 10 times higher. Since the goal of this manuscript is to explain how geminate pairs split, not to get exact results for a particular material system, we have not taken into account that the splitting probability would depend on where in the mixed region the geminate pair formed. We find that Pesc depends primarily on the distance the fastest carrier needs to travel to reach an energy cascade, so if the carriers were formed near a pure fullerene domain, rather than in the center of the mixed region, the hole would have CHAPTER 2. THE SHORT-CIRCUIT CURRENT 26

103 0.01 cm2/Vs 0.1 cm2/Vs 2 10 1 cm2/Vs 10 cm2/Vs

101 4 ns 100

Measured PL Decay Lifetime ns / PLDecay Measured 10-1 10-1 100 101 102 103 CT State Recombination Lifetime / ns

Figure 2.3: Calibration curve mapping measured geminate pair decay lifetimes to nearest-neighbor recombination lifetimes produced by simulating geminate 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 2 cm /Vs (µe = µh). The lines are a guide to the eye. The horizontal line represents a typical measured bulk heterojunction CT photoluminescence lifetime of 4 ns.[101] CHAPTER 2. THE SHORT-CIRCUIT CURRENT 27

Morphology Pesc Required Mobility τct = 10 ns No Mixed Region 1.4x10−4 11 cm2/Vs 9.6 nm Mixed Region 2.9x10−4 5.1 cm2/Vs 8 nm Mixed Region 3.5x10−4 4.2 cm2/Vs 6.4 nm Mixed Region 6x10−4 2.5 cm2/Vs 4.8 nm Mixed Region 1.3x10−3 1.2 cm2/Vs 3.2 nm Mixed Region 6.5x10−3 0.23 cm2/Vs

Table 2.3: Required local mobilities for 90% ﬁeld-independent IQE for the speciﬁed device morphologies.

to travel twice as far to reach a pure polymer domain, and Pesc could be found by considering a mixed region twice as wide but with the geminate pair formed in the center. On the other hand, if the pair formed near a pure polymer domain, the hole would cross the energy cascade almost immediately. Choosing to have the geminate pair form in the center of the mixed region provides a consistent way to evaluate the impact of mixed region width on geminate splitting.

2.4 Conclusion

The question of how BHJ solar cells are able to efficiently generate free charges has persisted for over a decade and many groups have discovered important parts of the explanation like the beneficial role of disorder and the impact of local polarizability and charge carrier delocalization on reducing the geminate pair binding energy. In this communication we build on their work by showing that experimentally measured local charge carrier mobilities and lifetimes in BHJ systems are in the range required for efficient geminate splitting. The picture that emerges of what makes a good BHJ solar cell is a high local charge carrier mobility, long CT state decay lifetime and, when µτct is not high enough on its own, a three-phase structure with an energy cascade for either the electron or the hole that increases the probability that a single geminate pair splitting attempt is successful. The combination of these three classes of effects explains how some bulk heterojunctions are able to generate free charges so efficiently. Looking back at Figure 2.2, it also explains the wide variability in device CHAPTER 2. THE SHORT-CIRCUIT CURRENT 28

performance from system to system since missing any one of these characteristics can be the difference between high, field-independent and low, field-dependent geminate splitting. Importantly, the commonly measured device parameters of bulk mobility and transient absorption lifetimes are shown not to be directly linked to charge generation. Instead we have shown how terahertz mobilities and corrected CT state photoluminescence lifetimes can be used to provide more accurate measurements of the parameters that do determine the efficiency of free charge generation in BHJ solar cells.

2.5 KMC Simulation Details

All KMC simulations were performed using custom KMC code written by the authors. The First Reaction Approximation was not used. Only single geminate pairs were simulated at a time with open boundary conditions. The world was generated on demand so there was no limit on the size of the simulated lattice. A lattice constant of 8 angstroms was used. The dielectric constant was set at 4 and the temperature was set at 300K. Each combination of morphology, lifetime, mobility and ﬁeld was averaged for at least 10,000 trials and up to 200,000 trials when necessary to capture rare events. For trilayer simulations, the geminate pair was assumed to be formed in the center of the mixed region. The Miller-Abrahams mobility model was used to calculate carrier hopping rates. Each material region was assumed to be homogenous and disorder was not simulated in order to make the simulation amenable to analytical analysis.

2.6 PL Decay Simulation Details

100,000 individual geminate pairs were simulated for each combination of lifetime and mobility and only those that ended in geminate recombination were selected. The time for each recombination event was calculated, binned and histogrammed to produce a simulated PL decay curve. This was ﬁt with a single exponential function to extract the measured lifetime, which was plotted as a function of the actual lifetime input into CHAPTER 2. THE SHORT-CIRCUIT CURRENT 29

the simulation. The simulated decays were well ﬁt by single exponential functions (Figure 2.9). A homogenous morphology was used. The same trilayer morphology described above was tried as well and the results were not greatly sensitive to the change (not shown).

2.7 Converting Hopping Rates to Mobility Values

To calculate the carrier mobility, the absolute hopping rate between lattice sites is needed. In the Miller-Abrahams (MA) model, this is the hopping rate prefactor. In Marcus theory, the hopping rate can be calculated as:

−λ ν = ν exp (2.3) 0 4kT 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:

νa2 µ = 0 (2.4) 6kT where a0 is the lattice constant, which was 1 nm in the previous studies reported below. In this work we followed Peumans[73] and used 8 angstroms. Two groups speciﬁed the hopping rate using an exponential term:

ν = nu0 exp (−2γa0) (2.5)

where γ is a localization radius and a0 is the lattice constant. The conversions are summarized in Table 2.4. Wojcik et al. speciﬁed the mobility directly in their work. CHAPTER 2. THE SHORT-CIRCUIT CURRENT 30

−1 −1 −1 2 Group Model ν0 [s ] γ [A ] λ [meV] ν [s ] Mobility [cm /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:

g(ν + ν )τ p = 0,e 0,h ct (2.6) g(ν0,e + ν0,h)τct + 1

where p is the probability that the geminate pair takes at least 1 hop apart. It depends on the product of the mobility prefactors for the electron and hole, the nearest neighbor recombination lifetime and a numerical constant g, which contains information about the functional form of the mobility model and the local energetic landscape. This result holds for any mobility model with a prefactor including both Marcus Theory and the Miller-Abrahams model (the numerical value of the g factor would change between the two models but the rest of the expression is the same, so the dependence on the mobility and recombination lifetime is the same). The expression represents the competition between two rates: the rate of a single hop apart and the rate of recombination. If geminate pairs that make a single hop CHAPTER 2. THE SHORT-CIRCUIT CURRENT 31

apart have a ﬁxed probability of splitting completely, independent of their mobility and lifetime, then the rate for geminate splitting is just the rate for a single hop apart multiplied by the ﬁxed probability that the pair continues on to split completely (Pesc), i.e. if the particles take a single hop apart once per second and 1 in 10 times that results in splitting, the particles split, on average, once in 10 seconds.

g(ν + ν )τ P p = 0,e 0,h ct esc (2.7) g(ν0,e + ν0,h)τctPesc + 1

This expression, then, predicts the geminate splitting efficiency once the numerical values for g and Pesc are known. We argued that Pesc is independent of lifetime because recombination is a nearest neighbor process and Pesc describes behavior when the charges are not nearest neighbors. We also argued that it is independent of mobility since the isotropic mobility does not bias the charges to hop toward each other versus away from each other; that bias is provided by the energetic landscape and the value of the mobility just sets the timescale for how fast all of the hops are. We will now verify that Pesc is basically independent of the value of mobility and lifetime by performing simulations for many values of mobility and lifetime and fitting them to the above expression with the parameter g*Pesc as the fitting parameter.

2.9 Dependence of Pesc on Local Mobility and Life- time

A trilayer simulation with a mixed region width of 3.2 nm was performed. The electron and hole mobilities and lifetime were independently varied (in normalized units) from 1 to 1x104 in 5 steps. The splitting eﬃciency was plotted as a function of

(µe+ µh)* for each of the 125 diﬀerent combinations on the same plot (Figure 2.4) and

fit to the above equation with a single value of Pesc across 6 orders of magnitude in the mobility-lifetime product (the dashed line). All splitting efficiencies are reported for a field of 103 V/cm, well below the range where the field plays a role in the splitting process. CHAPTER 2. THE SHORT-CIRCUIT CURRENT 32

1.0 100

0.8 10-1

0.6

10-2

0.4

10-3 0.2 Geminate Splitting Efficiency 0.0 10-4 10.001 102 103 0.2104 105 100.46 107 101 0.6102 103 1040.8 105 106 11.007 Mobility-Lifetime Product

Figure 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

This means that for a fixed morphology, the same value of Pesc describes the probability that geminate pairs, once separated by at least one hop, eventually become completely separated, implying that Pesc is independent of both the mobility and the lifetime. The geminate efficiency does still depend on the value of the mobility and lifetime but only their product because that sets the number of splitting attempts per geminate pair as described in the manuscript and derived at the end of this document. Note the sensitivity of the results around the mobility (hopping rate) lifetime product of 4x104 [unitless]. This inflection point (on a log-log axis) indicates the point at which the average geminate pair begins to live long enough to split and the wider spread in the data at that point could be indicative that, in this sensitive regime, there is a dependence of Pesc on the ratio of the electron and hole mobility, though not its absolute magnitude. This dependence has been seen by previous authors in other KMC studies and is particularly important in the presence of energetic disorder.[38, 106] CHAPTER 2. THE SHORT-CIRCUIT CURRENT 33

Morphology Fit Parameter (A) Extracted Pesc 3.2 nm Mixed Region 1.5e−4 6.5e−3 4.8 nm Mixed Region 3.2e−5 1.3e−3 6.4 nm Mixed Region 1.5e−5 6.0e−4 8 nm Mixed Region 8.7e−6 3.5e−4 9.6 nm Mixed Region 7.2e−6 2.9e−4 No Energetic Oﬀset 3.5e−6 1.4e−4

Table 2.5: Extracted escape probabilities for mixed regions between 3.2 and 9.6 nm wide.

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:

A(νe + νh)τct ηgem = (2.8) A(νe + νh)τct + 1 where A is the only fitting parameter. Data for each mixed width was fit separately and fitting was done using a nonlinear curve fitting routine on a log scale due to the fact that the data span many orders of magnitude. The extracted fit parameters are given in Table 2.5. CHAPTER 2. THE SHORT-CIRCUIT CURRENT 34

100 4 Layers

10-1 6 Layers 8 Layers

10-2 10 Layers 12 Layers High Efficiency Region 10-3 1

10-4

10-5 0 103 104 105 106 107 Geminate Splitting Efficiency 10-6 10-1 100 101 102 103 104 105 106 107 Hopping Rate-Lifetime Product [unitless]

Figure 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. CHAPTER 2. THE SHORT-CIRCUIT CURRENT 35

2.11 The Impact of Energetic Disorder

The results shown in this manuscript are based on simulations without energetic disorder in order to facilitate analysis. In this section we verify that these same results hold in the presence of energetic disorder as well as in blend materials. Our goal is to show that the geminate splitting probability depends only on the product of the hopping rate, lifetime and a constant that is a function of the energetic landscape even in blended and disordered materials. To show that our results hold in these three cases, we performed simulations of a homogenous trilayer like those reported in the main text, the same trilayer with a 50:50 blend of donor and acceptor molecules in the mixed region and a single layer with the energy levels chosen from a Gaussian distribution, as is typically done to model a disordered density of states. The results are shown in Figure 2.6 and 2.7. We ﬁt the data as described previously with a single value of Pesc for each morphology. The ﬁgures show that while changing the morphology does change the value of Pesc, as expected since the energetic landscape is changing, the dependence on the local mobility and lifetime remains the same. This shows that our conclusion that TRTC mobilities should be used in KMC simulations is applicable to blended and disordered materials as well as homogenous ones.

2.11.1 Simulation Details

Simulations for Figure 2.6 were performed for a mixed region 6 layers wide (4.8 nm) and the results compared for a homogenous mixed region and a random 50:50 blend of donor and acceptor with no disorder. For the blend, 10 diﬀerent environments were each averaged over 10,000 trials. For the homogenous region 100,000 trials were averaged. In order to investigate the impact of disorder, a single region was modeled with the HOMO and LUMO energy levels chosen from a Gaussian distribution with FWHM of 80 meV. A single morphology was generated and 10,000 trials were performed and averaged. The electron was modeled as ﬁxed at the origin and the geminate pair was σ2 injected with an equilibrated energy kT below the center of the distribution, where σ is the standard deviation of the Gaussian distribution. CHAPTER 2. THE SHORT-CIRCUIT CURRENT 36

1.0

0.8

0.6

0.4

0.2 50:50 Blend Dissociation Probabilility Homogenous 0.0 103 104 105 106 107 108 Hopping Rate-Lifetime Product [unitless]

Figure 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. CHAPTER 2. THE SHORT-CIRCUIT CURRENT 37

1.0

0.8

0.6

0.4

0.2 Simulation Dissociation Probabilility Model Fit 0.0 103 104 105 106 107 108 109 Hopping Rate-Lifetime Product [unitless]

Figure 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. CHAPTER 2. THE SHORT-CIRCUIT CURRENT 38

2.12 Independence from Bulk Mobility

It could be argued that since the bulk mobility is a function of the energetic landscape and the hopping rate, there should be a way to invert this relation and write the geminate splitting rate as a function of the bulk mobility. We agree that this could, in principle, be possible, but it becomes extremely difficult when the bulk mobility is limited by very slow, infrequent processes since then it provides little information on the hopping rate, which is not rate limiting. To show this, we performed the same simulations on a disordered region detailed in the previous section but when the electron and hole were not nearest neighbors, we artificially reduced the hole mobility by an arbitrary factor of 10,000. This means that when the electron and hole were nearest neighbors, the mobility was high, but when they were not, the mobility was 4 orders of magnitude lower. This would have a dramatic impact on the bulk mobility since almost every single hop is 104 times slower. As expected from our model, however, this had no effect on the geminate splitting efficiency as shown in Figure 2.8. The solid lines are the simulation without artificially reduced bulk mobilities and the circles are with reduced bulk mobilities. This directly shows that the bulk mobility does not matter in geminate splitting and is important only insofar as it provides insight into the local mobility. However it is very difficult to extract the local mobility from the bulk mobility as we detail throughout this manuscript, which is why we recommend that the directly measured local mobility values obtained from time-resolved terahertz conductivity be used instead. This also directly shows that overestimating the bulk mobility does not affect the result since it can vary by 4 orders of magnitude without affecting the simulation.

2.13 Exponential Decay of Photoluminescence

Simulated photoluminescence from a KMC simulation in a single homogenous region with the electron and hole mobilities both set to the values in the legend and the nearest neighbor recombination lifetime set at 10 ns. As can be seen in Figure 2.9, the decays remain exponential even though the decay constant changes. CHAPTER 2. THE SHORT-CIRCUIT CURRENT 39

1.0 Normal Bulk Mobility Reduced Bulk Mobility 0.8

0.6

0.4

0.2 Dissociation Probability

0.0 10-1 100 101 102 Mobility [cm2 /Vs]

Figure 2.8: Simulation of geminate splitting with the bulk mobility artiﬁcially reduced 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. CHAPTER 2. THE SHORT-CIRCUIT CURRENT 40

100 µ =0.01 cm2/Vs µ =0.1 cm2/Vs µ =1 cm2/Vs -1 10 µ =10 cm2/Vs

10-2 Normalized PL Decays / a.u.

10-3 0 10 20 30 40 50 60 70 80 Simulation Time / ns

Figure 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. 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 heterointerface 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 occur, 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 explain why the open-circuit voltage of organic solar cells has remained persistently low and develop a theory that provides guidance on how to improve it. Our key results and the relevant energy levels for understanding Voc are summarized schematically in Figure 3.1.

3.3 Background Information

In order to understand Voc we will need to build a model that describes how electrons and holes recombine in organic solar cells and how this process depends on voltage.

Since our goal is to develop an understanding of Voc that will allow for the rational design of organic solar cells with improved voltages, the theory must not only explain the available experimental data, but also provide useful insights that can guide the future design of materials. For example, would slightly raising the dielectric constant CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 43

ﬃnity Acceptor Electron A

E Singlet States fn qVoc Eopt E Exciton Splitting Free Charges 0 100 - 300 meV E E0 fp CT State Binding Energy 0 - 350 meV Energy E Interfacial Disorder Equilibrium ct 75 - 225 meV Ect,exp Recombination CT States Donor Ionization Potential 500 - 700 meV

Voc Energy Position

Figure 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 disorder 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 voltage. (Voc).

of organic semiconductors have a significant or marginal impact on Voc?[16, 15] Is there an open-circuit voltage tradeoff in using energy cascades to improve charge separation?[88, 12] Will raising the mobility of charge carriers in order to improve the fill-factor also cause a decrease in open-circuit voltage by making carriers encounter each other more frequently?[64] Finally, is Voc low simply because of the large amounts of energetic disorder present in OPV materials?[9] The theory we develop in this work will allow us to answer all of these questions. It will be useful in our discussion to refer to two distinct but related quantities: Ect and E0. Ect is the average energy of all of the CT states in an organic solar cell and E0 is the average difference between the Electron Affinity (EA) of the acceptor material and the Ionization Potential (IP) of the donor at the interface between the two. Since organic solar cells are disordered, there is not one single value for either the CT state energy or the EA-IP difference; instead we have to work with average quantities. We specify that E0 should be averaged only over the interfacial/mixed portions of the device since both the EA and IP are known to be different in aggregated versus mixed regions of many organic CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 44

solar cells and we wish to compare E0 with the energy of a CT state that only forms at an interface.[88, 48] If there were no interaction between the electron and hole in the CT state, Ect would equal E0. In general they are related by:

Ect = E0 − EB (3.1)

where EB is the average CT state binding energy.[54, 15] We can estimate EB based on the dielectric constant of organic semiconductors and the average separation between the electron and hole in the CT state (rct) using Coulombs Law:

q2 EB = (3.2) 4πrct where q is the charge of an electron and is the dielectric constant of the material. Experiments have estimated average CT state separations between 1 and 4 nm and organic semiconductors typically have relative dielectric constants between 3 and 5 so we would expect values for EB between 70 and 480 meV.[15, 4, 30] Recently, Chen et al developed a technique to measure EB and reported values between 0 and 350 meV for seven different polymer-fullerene systems, which compares well with our simple calculation.[15] The reason we emphasize the distinction between Ect and E0 is because a large body of work has established that in optimized organic solar cells recombination is a two-step process. The electron and hole first meet at the interface between the donor and acceptor materials and form a CT state, which then either recombines or dissociates back into free carriers.[56, 49, 104, 21] We will find that we can determine whether recombination is limited by the rate at which free carriers form CT states or the rate at which those CT states recombine by analyzing if Voc correlates more strongly with E0 or Ect. So it is important to establish that the two numbers are distinct and that the difference can be measured experimentally.[98, 15] In either case, since recombination involves one electron and one hole, the Law of Mass Action states that its rate is proportional to the product of the electron and CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 45

hole concentrations (n and p respectively):

R = knp (3.3) where R is the rate of recombination per unit volume and k is a proportionality constant. Under open circuit conditions, where the quasi-Fermi levels are ﬂat, we can directly relate the product np (though not the individual concentrations n or p) to the voltage of the solar cell (see Figure 3.1 for variable deﬁnitions):

E − E E − E np = N exp fn c ∗ N exp v fp (3.4) 0 kT 0 kT qV − E np = N 2 exp oc 0 (3.5) 0 kT where N0 is the density of electronic states in the device, typically taken to be around 21 −3 10 cm (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:

q(µ + µ ) k = e h (3.6) lan where µe is the electron mobility and µh is the hole mobility. Langevin recombination 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 typically between 10 and 100.[56, 76, 55] Device modelers account for this discrepancy by introducing a Langevin reduction factor that artiﬁcially lowers klan until CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 46

it agrees with experiment.[56] In the absence of a better alternative, most researchers have described recombination in organic solar cells in terms of reduced Langevin recombination.[56, 76, 55] However, the theory has not been able to provide useful guidance on how to improve Voc. For example, based on Equation 3.6 we would expect 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 changing the dielectric constant from 3 to 5 should only improve the open-circuit voltage by:[56]

kT 5 ∆V = ln (3.7) oc q 3

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 r.[58, 16] CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 47

3.4 The Temperature Dependence of Voc Leads Us Beyond Langevin Theory

Before presenting our model, we would like to review what is known about the temperature 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 recombination predicts that Voc should depend on E0. Equating the recombination current with the short-circuit current to solve for Voc gives:

2 qN0 Lklan qVoc = E0 − kT log (3.8) Jsc

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 deviations less than 200 meV and that the diﬀerence between Ect and E0 varies by more than 300 meV, it would be very diﬃcult to explain the observed dependence of

Voc on Ect if it actually depended on E0 instead.[34, 15] Another consequence of this dependence is that if we cool an organic solar cell down to cryogenic temperatures,

Langevin theory predicts that Voc will approach E0 (details in the SI). In fact, when extrapolated to 0K, Voc does not approach E0 but instead converges to the CT state energy.[98, 15, 44] Since the temperature dependent experiments are performed on a single solar cell and not by comparing different material systems, there is no scatter in the data and the discrepancy is very clear. Intuitively, if every time free carriers meet they recombine, there is no way for the value of Ect to affect their behavior since by the time the carriers are close enough to experience Ect their fate is already determined. On the other hand, if the carriers were able to form CT states several times and split before finally recombining then an CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 48

equilibrium could exist between CT states and free carriers, in which case Ect would be critically important because the density of CT states would be proportional to a

Boltzmann factor involving Ect. This raises the question of whether Langevin theory mispredicts the recombination rate in organic solar cells because it overestimates the frequency with which free carriers meet each other or because only a small fraction of those encounters lead to recombination. Several authors have explored this issue and shown with Kinetic Monte Carlo simulations that klan actually does a surprisingly good job of predicting how often carriers encounter each other, even in disordered material systems, which agrees with the fact that the expression works reasonably well for OLEDs.[93, 36] This implies that the Langevin reduction factor must be necessary because not every encounter between free carriers results in recombination. Recent experimental work has confirmed this hypothesis by showing that the low energy CT states that would be formed by free carriers encountering each other have the same high splitting efficiency as higher energy CT states formed during the photogeneration process.[94] The CT state splitting process has also been investigated using detailed Kinetic Monte Carlo simulations, which show that carriers actually have a very low chance of recombining during any given encounter.[12, 50, 37, 45, 1] The likely reason that Langevin recombination works for OLEDs is because those systems have been specifically designed for free carriers to efficiently find each other and recombine. Organic solar cells, on the other hand, have been specifically designed to prevent this process.

3.5 Reduced Langevin Recombination Implies Equi- librium

The suggestion that most Charge Transfer states reseparate has been made before as an explanation for the Langevin Reduction factor and detailed numerical models have been constructed to explore its impact.[36, 41] For example, Hilczer and Tachiya were able to accurately reproduce the temperature dependence of the Langevin reduction factor with a model that allowed CT states to split back into free carriers.[41] In this CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 49

work we would like to take the idea one step further. If free carriers form CT states and split much faster than they recombine, there should be time for equilibrium to be reached between the population of free carriers and the population of interfacial CT states. In this limit it does not matter how quickly carriers move, a certain fraction of them will always be in CT states and that fraction can be calculated using Boltzmann statistics and a knowledge of the free energy diﬀerence between free carrier states and Charge Transfer states. In order to see if such a description is appropriate, however, we must ﬁrst investigate how close to equilibrium the free carrier and CT state populations are in an organic solar cell. When carriers meet and split 10,000 times before recombining, there is clearly time for equilibrium to be established between the two populations; however, it is not obvious that the same is true when they only meet and split 10 times. We can answer this question using a kinetic model. Figure 3.2 shows the recombination process schematically with all of the relevant rates labeled. Without making any assumptions about whether free carriers and CT states are in equilibrium with each other, we can write down rate equations describing the interactions between the two populations:

dn ct = k np − (k + k )n (3.9) dt m r s ct dn = −k np + k n + G (3.10) dt m s ct dp = −k np + k n + G (3.11) dt m s ct

where nct is the density of CT states, kr is the (average) rate constant at which

CT states recombine, ks is the rate constant at which CT states split back into free carriers, km is the rate constant at which free carriers meet and G is the rate at which free carriers are being generated. Since the solar cell is in steady state, we know that n and p are being replenished, either by injected carriers from the contacts or by photogenerated carriers, at precisely the same rate that the CT states are recombining so G = krnct. Solving for steady state leads to: CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 50

n k ct = m (3.12) np kr + ks

eq We can also deﬁne the equilibrium density of CT states (nct ) we would expect if kr were much slower than ks as:

neq k ct = m (3.13) np kr + ks

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:

k γ = r (3.14) kr + ks γ k = k (3.15) r 1 − γ s

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) γ R = k neq = k np (3.17) eq r ct 1 − γ m

Figure 3.2 plots both recombination expressions as a function of γ. The goal is to determine how small γ needs to be before an equilibrium description becomes CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 51

2.0 100

80 1.5 Free Charges 60 CT States km 1.0

40 ks

0.5 20 k Reduced Rate r Recombination Rate [a.u.] Equilibrium Rate Equilibrium Voltage Error [mV] 0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Ground State Langevin Reduction Factor Langevin Reduction Factor

Figure 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. appropriate. We ﬁnd that once the Langevin reduction factor is smaller than about 0.1, the reduced Langevin recombination rate is extremely close to the rate expected if the free carriers were fully in equilibrium with Charge Transfer states. Furthermore, since Voc depends on recombination in a logarithmic fashion, even a solar cell with a Langevin reduction factor of 0.5 would have an open circuit voltage that deviates from the equilibrium prediction by less than 20 mV and in fact γ must be very close to 1 before the equilibrium picture breaks down. Almost all organic solar cell materials have γ 0.2, so we can treat bimolecular recombination as occurring from a population of free carriers in equilibrium with CT states (tabulated reduction factors and more discussion of this point is presented in the SI).[76] This means that we do not need a complicated numerical model to estimate ks and calculate γ in order to understand the open-circuit voltage of organic solar cells; we can instead just write down the density of CT states based on the requirement that they be in equilibrium with free carriers. CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 52

3.6 Equilibrium Simpliﬁes the Understanding of

Voc

When chemical or electronic species are in equilibrium with each other the fundamental 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 energy 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 condition 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 semiconductor band diagrams. In short, equilibrium between electronic species allows us to relate their quasi-Fermi levels since this is the quantity that measures their molar free energies and at equilibrium it is their free energy that must be equal, not, for example, their concentrations. So, equilibrium between CT states and free carriers requires that the chemical potential of the CT states (µct) be equal to the difference of the electron and hole quasi-Fermi levels for their molar free energies to be equal:

µct = Efn − Efp (3.18)

For further discussion of the relationship between quasi-Fermi levels and chemical potentials, readers are directed to a lengthy treatment by Wurfel, who validated and used the same approach to relate the quasi-Fermi levels of electrons and holes with CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 53

the chemical potential of photons in order to derive the semiconductor electroluminescence 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 organic solar cell. We specified the the chemical potential of the CT state population using Equation 3.18 because we know that at open-circuit the difference between the electron and hole quasi-Fermi levels is constant across the device and given by qVoc:

Efn − Efp = qVoc = µct (3.19)

Equation 3.19 means that equilibrium between free carriers and CT states gives us a way to directly relate the open-circuit voltage to the chemical potential of the CT states, letting us calculate the number of CT states without needing to know how many free carriers there are in the device, how quickly they are moving or what the energetic landscape for those free carriers looks like. Now that we know the chemical potential of the CT states, we can determine how many are occupied (Nct) by integrating over the density of possible CT states, gct(E) (see Figure 3.3). Intuitively, one can think of µct as measuring the amount of free energy the system can use to populate CT states. It makes sense then that by combining this information with knowledge of how much energy it takes to occupy each CT state and how many possible CT states there are, i.e. the density of states, you can calculate the total number of populated states. For readers familiar with the standard expressions relating electron and hole quasi-Fermi levels to electron and hole densities, the result for CT states is exactly analogous. The precise functional form for all three expressions is typically derived from the grand canonical ensemble in statistical mechanics and worked out step by step for the case of CT states in a later section. Here we quote the result: CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 54

0.4

0.2

0.0

0.2

Energy [eV] 0.4

µct 0.6

0.8 CT State DOS Occupation Function Filled States

Figure 3.3: Schematic showing how the density of available CT states, gct(E), combined with knowledge of the CT state chemical potential, µct, permits the calculation of the number of ﬁlled CT states, Nct. CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 55

Z ∞ µct − E Nct = gct(E) exp dE (3.20) −∞ kT

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:

σ2 qV − E N = fN exp ct exp oc ct (3.21) ct 0 2(kT )2 kT

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:

2 qNctL qfN0L σct qVoc − Ect Jrec = = exp 2 exp (3.22) τct τct 2(kT ) kT

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:

2 σct qfN0L qVoc = Ect − − kT log (3.23) 2kT τctJsc

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 disorder and an arbitrary energetic landscape for free carriers we remove any ambiguity CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 56

about when the result is applicable, show how it is equivalent to reduced Langevin recombination and connect all of the input parameters directly with concrete material properties that can either be measured or calculated. This last point is critical as it will allow us to explain why qVoc is so consistently 0.5 to 0.7 eV below the measured CT state energy in almost all organic solar cells, despite the widely varying electronic properties among those diﬀerent systems. Our result shows that, in the absence of device imperfections like contact pinning or shunts, Voc is determined solely by the degree of mixing in the device, the energy of the center of the CT state distribution, the degree of energetic disorder in the mixed region and the CT state lifetime. The CT state lifetime describes the rate at which CT states directly recombine either radiatively or nonradiatively. It is distinct from the free carrier lifetime that could be measured in a transient photovoltage experiment as we discuss below.

3.7 Eﬀects of an Energy Cascade in 3-Phase Bulk Heterojunctions

One of the reasons we derived our expression for Voc in terms of quasi-Fermi levels instead of free carrier densities is because it makes it clear that there is no dependence of Voc on the energy levels of free carriers, i.e. E0 appears nowhere in our expression for Voc and we did not need to make any assumptions about the energetic landscape for free carriers in order to derive it. This is not to say that the energetic landscape is unimportant for solar cell operation, just that our theory shows it does not affect the numerical value of the open-circuit voltage. When calculating the potential efficiency of a solar cell material, one is typically not interested in Voc in isolation but in the difference between the optical gap and qVoc since a device with a smaller optical gap absorbs more light and can compensate for its lower voltage with additional photocurrent, increasing the overall efficiency. To use an extreme example, silicon solar cells have lower open-circuit voltages than many OPV devices, but this does not mean that organic solar cells are more efficient. So, if one is able to decrease the optical gap of an organic solar cell without affecting the CT state energy, then, CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 57

No Energy Cascade Energy Cascade

LUMOs

Efn Efp

HOMOs

Donor Mixed Acceptor Donor Mixed Acceptor

Figure 3.4: Two example energy diagrams showing a solar cell with and without an energy 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, 21 −3 N0 = 10 cm 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 densities to be 1.6x1016 cm−3 and the density of CT states to be 3.6x1012 cm−3. In a 100-nm-thick device with a CT state lifetime of 500 ps, this would correspond to a CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 58

] 18

3 10 − m

c 17 [

10 y t i

s 16

n 10 e D

r 1015 e

i Electrons r r

a 14 Holes

C 10

Figure 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. recombination current of 12 mA/cm2. Even though the CT state lifetime is only 500 ps, there are 4,390 times more free carriers than CT states, so each carrier, on average, has only a 1 in 4,390 chance of occupying a CT state. Since transient photovoltage measures the lifetime of the average carrier, one would measure a free carrier lifetime of:

500 ps ∗ 4390 = 2.2 us (3.24)

With the energy cascade, the density of CT states and free carriers in the mixed region is unchanged since Ect and E0 are unchanged but there are now many more free carriers in the aggregated regions so that the average density of carriers has increased to 4x1017 cm−3. The recombination current is the same since both the number of CT states and their lifetimes are the same, which means the free carrier lifetime must have increased substantially to 53 us since the odds of each free carrier occupying a CT state has decreased to 1 in 105,000. If we only had access to information on the free carrier densities and lifetimes, for example through charge extraction and transient photovoltage measurements, we would conclude that the solar cell with the energy cascade had substantially reduced recombination since both the free carrier CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 59

lifetime and the density of free carriers at open-circuit increased significantly.[85, 22] However, the actual amount of recombination is the same in the two solar cells and the presence of the energy cascade neither increased nor decreased Voc. This is one of the consequences of equilibrium between free carriers and CT states and it also implies that traps and energetic disorder outside of the mixed region, which would have a similar effect to an energy cascade, do not impact the open-circuit voltage. Put another way, we are saying that for a given solar cell Ect and E0 will be related to each other because both involve the EA - IP difference. However, if one keeps Ect constant but varies E0 (using an energy cascade, for example), Voc will not change. On the other hand if one keeps E0 constant but varies Ect (by modifying the CT state binding energy, for example), Voc will change to track the variation in Ect. So, the important variable that determines Voc is Ect, not E0. If one changes E0 and in-so-doing also changes Ect (by changing the donors IP, for example), then Voc will, of course, also change. However, it changes because of the change in Ect, not the change in E0. We can use this effect to our advantage by introducing energy cascades that broaden the optical absorption without affecting the CT state energy to increase the photocurrent without sacrificing voltage.[88] For example, both of the solar cells that we discussed above have the same open-circuit voltage but the one with the energy cascade could achieve this voltage with a 200 meV smaller optical gap, increasing the short-circuit current. This extra current comes at the expense of a reduced EA-EA offset between aggregated donor/acceptor phases, but provided the offset remains large enough to drive exciton splitting, there should be no impact on charge generation and energy cascades could be used as a way to recover some of the voltage lost due to overly large EA-EA offsets.

3.8 The Role of Energetic Disorder

Looking at equation 1.19 and noting that the energetic disorder could easily be 100 meV, our model implies that we should expect significant variations in the difference between Voc and Ect based on differences in the amount of interfacial energetic disorder, which, in contrast to free carrier disorder, is predicted to affect the open-circuit CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 60

voltage by setting the width of the CT state distribution. This would seem to be in contradiction to the experimental finding that qVoc is almost always 0.5 to 0.7 eV below the CT state energy so we need to briefly discuss the relation between what we call Ect and what is measured experimentally. Experimental values for Ect are typically 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 absorption 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 exp an effective energy Ect,exp and reorganization energy λ given by (derivation in SI):

σ2 Eexp = E − ct (3.25) ct ct 2kT σ2 λexp = λ + ct (3.26) 2kT

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- exp rent Spectroscopy (FTPS).[96] We ﬁnd that both Ect,exp and λ are linear in 1/T CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 61

1.6 1.7

1.4 1.6

1.2 1.5

1.0 1.4

0.8 1.3 P3HT:PCBM 1:1 MDMO-PPV:PCBM 1:4 APFO3:PCBM 1:4 0.6 1.2 APFO3:PCBM 1:1

0.4 1.1 Apparent CT Energy

CT State and Reorganization Energies [eV] Apparent Reorganization Energy 0.2 1.0 4 6 8 10 12 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 1 1 1000/Temperature [K− ] 1000/Temperature [K− ]

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 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 independently). (right) The temperature dependent Ect measurements taken from literature.[98] The data points are the experimental fit parameters at each temperature and the lines are 1/T fits to the data. with opposite slopes that are very similar in magnitude, consistent with our theoretical 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:

exp qfN0L qVoc = Ect − kT log (3.27) τctJsc

and see that the dependence of Voc on interfacial disorder is exactly masked by CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 62

exp exp Material System Ect σct Ect Voc Ect − qVoc Ect − qVoc [eV] [meV] [eV] [V] [eV] [eV] P3HT:PCBM 1:1 1.24 75 1.14 0.61 0.61 0.53 RRa P3HT:PCBM 1:4 1.66 104 1.44 0.83 0.83 0.61 MDMO-PPV:PCBM 1:4 1.52 75 1.42 0.84 0.68 0.58 APFO3:PCBM 1:4 1.73 71 1.64 1.05 0.68 0.59 APFO3: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 with experimental Voc measurements for comparison. All raw data except for RRa P3HT is from literature.[98]

the experimental techniques used to measure Ect.

3.9 Experimental Observations Explained by the Model

Our model predicts that Voc should increase linearly as we lower the temperature of the solar cell and appear to converge to Ect,exp when extrapolated to 0K as seen experimentally and in contrast to the predictions of Langevin recombination. It also explains why Ect,exp - qVoc 0.6 eV for many systems that have been studied even though they had different amounts of energetic disorder since only interfacial energetic disorder matters and the available techniques to measure Ect happen to be affected by interfacial disorder in precisely the same way as Voc. We see why Voc is exponentially dependent on the dielectric constant, since that sets the CT state binding energy, which determines, at equilibrium, what fraction of free carriers will be in a CT state via a Boltzmann factor. We also see why the highly variable energetic landscape for free carriers, including ubiquitous energy cascades between aggregated and mixed regions, does not impact the difference between Ect,exp and Voc since the number of populated CT states at equilibrium depends only on the CT state energy and the open-circuit voltage.[88] Finally, we see that the carrier mobility does not affect Voc because the recombination process is not limited by the rate at which free carriers find each other. CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 63

3.10 Explaining the Magnitude of the Voltage Loss

We now turn to the empirical result that qVoc is almost always 0.5 to 0.7 eV below

Ect,exp.[34] To compare our model with experiment we need to estimate the expected ranges of all of the necessary input parameters. We start with the volume fraction of interfaces and mixed regions in organic solar cells. On the high side, we have solar cells like regiorandom P3HT that are completely amorphous, so the solar cell could be 100% mixed. On the low side we have low-donor-content cells (1-10% mixed) and bilayers. Even a perfect 100 nm bilayer would still be approximately 1% interface (1 nm of donor/acceptor molecules involved in an interface in a 100 nm thick active layer), so we conclude that organic solar cells are between 1% and 100% mixed. We also need to know the CT state recombination lifetime. This quantity is difficult to measure experimentally since the distribution of CT states excited in the transient experiments used to measure CT state recombination rates is far from equilibrium, meaning that the average lifetime of those CT states may differ from that of the equilibrium distribution that exists at steady state. We discuss this point at more length in the SI and present tabulated lifetimes from literature. In this section we summarize the available estimates of τct from a variety of experimental and theoretical methods. Ultrafast pump-push measurements and photoluminescence studies tend to report lifetimes between 100 ps and several ns.[17, 101, 5] Quantum chemical calculations on a P3HT:PCBM analog predict 500 ps for one model and as fast as 90 ps for a different interface conformation.[62, 61] Further calculations have shown that the donor/acceptor interface is actually dynamic on the timescale of 10 ns so even if a particular interfacial conformation would lead to very slow recombination, the interface will explore enough conformations within 10 ns to find one that allows for fast recombination.[61] Given the experimental and computational variability, we consider a range of lifetimes between 10 ps and 10 ns, keeping in mind that at the low end of the lifetime range we do not necessarily expect there to be time for complete equilibrium to develop between free carriers and CT states. However, as we showed in Figure 3.2, we do not actually need full equilibrium for the predictions of our theory to be accurate; we simply need τct to be nonnegligible such that CT states dissociate CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 64

several times before recombining, which we generally know to be the case since the solar cells were able to photogenerate free carriers in the first place.[94] In principle we also need to know the degeneracy of the Charge Transfer states. It is tempting to assume that there is one CT state for each pair of nearest neighbor donor/acceptor molecules, however a range of experimental and theoretical work has shown that CT states form between non-nearest neighbor molecules as well due to long-range couplings between non-adjacent molecules.[30, 63, 83, 87] This effect is shown in Figure 3.7 and is very important because if you consider only CT states forming between molecules 1 nm apart, you might expect 3 CT states per acceptor molecule since 3 of its 6 nearest neighbors in a simple cubic, 50:50 blend of donor or acceptor molecules would be donors. On the other hand if you increase the interaction distance to 2 nm, you would have 33 molecules with which each acceptor can interact and 16 CT states. At 3 nm it would be 113 molecules and 56 CT states per acceptor. In general the density of CT states increases like the cube of the CT state delocalization length. Previous authors have discussed the importance of CT state delocalization for improving charge generation. [30] Here we add that increased delocalization is also likely to limit the open-circuit voltage by providing more pathways through which recombination can occur, implying a design tradeoff that will need to be optimized.

We ﬁnd good agreement with experimental Voc measurements at 32 CT states per acceptor molecule (an approximate delocalization length of 2.5 nm). Answering the question of precisely how many CT states are formed at each interface would be an important candidate for future quantum chemical calculations.

Figure 3.7 explores the expected diﬀerence between Voc and Ect,exp for a 100- nm-thick active layer with a short-circuit current of 10 mA/cm2 across the range of plausible material parameters that we found in the preceding paragraphs. The key point to take away from Figure 3.7 is that almost all combinations of material parameters will result in an open-circuit voltage between 0.5 and 0.7 V below Ect,exp, explaining why this empirical rule has worked so well. This is a consequence, however, of the range of CT state lifetimes and degrees of mixing observed in organic solar cells. More precisely, we could say that the reason why qVoc is almost always 0.5 to

0.7 eV below Ect,exp is because the CT state recombination lifetime is rarely higher CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 65

10-8 ] s [

e 0.500

2 nm m i t e f i -9 L 10

n 0.550 o i t a

1 nm n i b m o

c 0.600 -10

e 10 R

e t a t

S 0.650

T C

10-11 20 40 60 80 100 Degree of Mixing [%]

Figure 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 indicate 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. CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 66

Parameter Improvement Strategy Voc Increase Reduce volume fraction of mixed phase from 50% to 1% 100 mV Increase CT state lifetime from 100 ps to 10 ns 120 mV Decrease interfacial disorder from 100 to 50 meV 150 mV Decrease CT state binding energy from 200 to 50 meV 150 mV Decrease 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 each of the material parameters that affects Voc. than 10 ns since this is the timescale for dynamic interfacial reconfiguration and it is never lower than 10 ps since this would prevent the photogeneration of free carriers. Three orders of magnitude of change in CT state lifetime corresponds to a 180 mV difference in Voc at 300K since the CT state lifetime affects the voltage logarithmically. Because the exciton diffusion length in organic photovoltaic materials is typically less than 30 nm, the community has not been able to explore orders of magnitude differences in donor/acceptor mixing ratios. It has been observed, however, that in dilute blends, where you can measure the interfacial area by the strength of the CT state absorption, Voc does depend logarithmically on interfacial area in agreement with our expression.[100]

3.11 Opportunities for Improving Voc

In addition to explaining why the open-circuit voltage of organic solar cells is low even though their internal quantum eﬃciencies can be quite high, this study also provides a framework in which to identify and rank opportunities to raise Voc. Table 3.2 summarizes the potential gains in open-circuit voltage that could be achieved by improving each of the terms that appears in our expression for Voc. Since many of the parameters appear in a logarithm, they would need to be changed by orders of magnitude to signiﬁcantly enhance the open-circuit voltage. However, both the degree of interfacial disorder and the CT state binding energy are outside of the logarithm, implying that the largest voltage gains are likely to come from reductions in those two parameters. CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 67

A promising route to improving the open-circuit voltage of organic solar cells could be engineering the donor and acceptor molecules to dock in preferred orientations in order to reduce conformational disorder at the interface.[33] As an example of the effect of conformational disorder on Voc we compared the interfacial disorder in regiorandom and regioregular P3HT blended with PCBM. The regiorandom blend was found to have 104 meV of interfacial disorder compared with 75 meV in the regioregular 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 regioregular 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 regioregular 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 nonradiative transitions with an electroluminescence quantum efficiency typically worse than 10-6.[98] This means that 6 orders of magnitude of improvements in CT state lifetime are possible but there is currently not a clear understanding of precisely what mechanism is leading to such fast nonradiative recombination. Future studies focused on this point could help recover some of the more than 360 mV of Voc currently lost to this effect. We speculate that perhaps the dynamic nature of the donor/acceptor interface plays a large role in allowing CT states to find configurations that lead CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 68

to fast nonradiative recombination. In that case, rigidly locking the donor/acceptor conformation could be key to increasing the radiative quantum eﬃciency and hence

Voc.

3.12 Conclusions

We have shown that the available experimental evidence strongly points toward a model of recombination in organic solar cells where free carriers are in equilibrium with CT states. This description simplifies understanding the recombination process and enabled us to directly link the low open-circuit voltage of organic solar cells to a combination of their high degree of mixing, short CT state lifetimes, large amounts of interfacial energetic disorder and low dielectric constants leading to high CT state binding energies. We quantify the impact of each of these parameters and physically explain both the dependence of qVoc on Ect,exp and the generally observed 0.5 to 0.7 eV difference between them. Our work shows that there is significant practical potential for improving Voc, provided we target the right parameters. For example, reducing interfacial energetic disorder and the CT state binding energy could raise Voc by hundreds of mV without requiring any change to the CT state lifetime or degree of mixing. The picture of Voc that emerges is one of a quantity that is limited mainly by the microscopic details of the interface between donor and acceptor molecules. By optimizing this interface, the OPV community has the opportunity to significantly enhance the efficiency of organic solar cells through increases in open-circuit voltage.

3.13 Experimental Details

3.13.1 Sample Preparation

Substrates used for FTPS samples were ITO-coated glass (Xinyan Technologies, LTD.). Substrates were immersed in a detergent solution of 1:9 extran:deionized water solution then scrubbed with a brush. Samples were then sonicated in the detergent solution, rinsed with deionized water, sonicated in acetone, sonicated in CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 69

isopropanol, and blown dry with nitrogen. Substrates were stored in an oven held at 115 C. Immediately before depositing films onto substrates, substrates were exposed to a UV-ozone plasma for 15 minutes. PC60BM was purchased from Solenne BV. RRa-P3HT was obtained from Reike. A solution of 1:4 wt:wt RRa-P3HT:PC60BM was prepared in chloroform at a polymer concentration of 4 mg/ml, and was heated and stirred at 70 C overnight. The RRa-P3HT:PC60BM film was deposited in a nitrogen 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 exp Values of Ect,exp and λ were determined for each temperature independently. To exp determine Ect,exp and λ , the sub-bandgap absorption was fit to Marcus Theory absorption expression shown in the SI using a linear least squares fitting procedure. CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 70

Under the assumption that Marcus Theory is a good description of the CT absorption, the ﬁt was restricted to the portion of the sub-bandgap absorption whose natural log had a linear ﬁrst derivative (i.e. (d(ln(E*(E))/dE is linear).

3.14 Why We Expect the CT State Distribution to be Gaussian

Charge Transfer states are composed of an electron and a hole in separate materials; therefore we should be able to relate the distribution of CT states to the Electron Affinity and Ionization Potential of the two materials. Since these are energetically disordered materials, the EA and IP take a range of values at different positions in the film.

CT(E) = Acceptor EA(E) − Donor IP(E) − EB (3.28)

The distribution of CT states then can be described in terms of the distribution of free carrier energy levels of the acceptor and donor materials, modiﬁed by an interaction energy EB. Since the low energy portions of the energy levels of organic semiconductors are usually described as having a Gaussian shape, and the diﬀerence of any two Gaussians is always a Gaussian distribution even in the presence of arbitrary correlations between the two distributions, we can say that the relevant low energy portion of Acceptor EA(E) - Donor IP(E) should be Gaussian in shape.[6, 8] We would expect that EB will also be a distribution of values since there will be conformational and dipolar disorder at the interface between the donor and acceptor materials. Since there are a large number of interactions that set EB, we can invoke the Central Limit

Theorem to argue that EB should be normally distributed as well. Thus to ﬁrst approximation, CT(E) should have a normal distribution. CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 71

3.15 Inhomogeneously Broadened Marcus Theory Absorption

From Marcus Theory, one can calculate the absorption spectrum of a single molecular excitation using the expression:[40]

2 ! f −(E0 + λ − E) α(E,E0) = √ exp (3.29) E 4πλkT 4λkT

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:

Z ∞ 0 0 0 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:

2 ! Nct −(E − Ect) g(E; Ect) = √ exp 2 (3.31) σct 2π 2σct

Performing the above integration yields:

2 ! fNct −(Ect + λ − E) αct(E) = √ exp (3.32) p 2 2 E 2π σct + 2λkT 2(σct + 2kT λ) CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 72

While this appears distinct from the original expression for Marcus Theory Ab- sorption, we can put it in the same form by making the following identiﬁcations:

σ2 λ0 = λ + ct (3.33) 2kT σ2 Eexp = E − ct (3.34) ct ct 2kT

The simultaneous modiﬁcation of the reorganization energy and the CT state energy cancel in the numerator of the exponential function while putting the denominator into the correct form for Marcus Theory. Thus, any Gaussian distribution of Marcus Theory absorbers will be indistinguishable from a single Marcus Theory absorber since the functional form of the absorption expression is identical. However, both the reorganization energy and CT state energy will be temperature dependent in the case of an inhomogeneously broadened distribution, allowing us to distinguish between homogeneous and inhomogeneous broadening using temperature dependent measurements.

3.16 Relating CT State Density and Chemical Po- tential

Previously in this chapter, we calculate the number of occupied CT states based on knowledge of the chemical potential for CT states, µct, and the density of states. First, we assume that the CT states do not interact with each other so that each CT state can be treated independently. We make this assumption since, as we will show in the next section, the density of CT states is much lower than the density of interfaces, so each CT state should be formed far from any other. This independence assumption means that we can consider each interfacial site in isolation. Formally, it means that we can decompose the grand canonical partition function into a product of partition functions for isolated interfaces. Consider then, an interfacial site where a CT state could form. If there is no CT state occupying the interface, the energy associated with the interface is 0. If there is a CT state, the energy of the interface is CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 73

, 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 statistical 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 particles with a reservoir (the reservoir of free electrons and holes). The grand canonical partition function is deﬁned as:

X −E(s) + µN(s) ξ = exp (3.35) kT s∈states

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:

∂ ln ξ hNi = kT (3.36) ∂µ

The proof of these results is in any standard statistical mechanics text. Since there are only two possible states of our interface, occupied or free, the grand partition function is particularly simple: CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 74

− + µ ξ = 1 + exp ct (3.37) ct kT

The odds of this interface being occupied then, is given by:

∂ µ − hNi = kT ln 1 + exp ct (3.38) ∂µct kT

Performing the diﬀerentiation leads to the standard Fermi-Dirac distribution, just as it does for electrons and holes:

1 hNi = (3.39) e−µct exp kT + 1

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 magnitude 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 hNi to be much smaller than 1. Had we made a different assumption about whether multiple CT states could form at the same interface it would have resulted in the same simplified expression in the low CT state concentration limit relevant for organic solar cell operation so that assumption turned out to be unimportant. The odds of any interface being occupied then (Pct), given its energy E and the chemical potential of CT states µct is: CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 75

µ − E P = exp ct (3.40) ct kT

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:

µ − E n = ∫ ∞ g (E) exp ct dE (3.41) ct −∞ ct kT

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:

Z ∞ nct(µct) = gct(E)f(E, µct) dE (3.42) −∞

where gct(E) is the arbitrary distribution of CT states and f(E, µct) is the Fermi-

Dirac distribution function. When µct is far from the energy of the majority of the CT states, we can replace the Fermi-Dirac distribution with a Boltzmann distribution with very little error as we indicated in the previous section. The condition on µct for this approximation to be valid is given by Neher for a Gaussian density of states as:[8] CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 76

σ2 µ < E − ct (3.43) ct ct kT

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 approximation clearly holds near room temperature. We can rewrite the above integral then as:

Z ∞ µct − E nct(µct) = gct(E) exp dE (3.44) −∞ kT

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:

Z ∞ Ect − E µct − Ect nct(µct) = gct(E) exp dE exp (3.45) −∞ kT kT

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:

2 σct Nct = fN0 exp (3.46) 2(kT )2

where fN0 is the total number of CT states and the exponential factor captures the fact that the lower energy portion of the distribution is far more likely to be CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 77

populated than Nct states all located exactly at Ect. Another way to present this result is that a Gaussian density of states (DOS) is equivalent to a delta function DOS located at:

σ2 Eexp = E − ct (3.47) ct ct 2kT

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 populations 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:

Z ∞ 1 gct(E) µct − E 1/hτcti = exp dE (3.48) Z −∞ τct(E) kT

where Z is a normalization factor deﬁned as:

Z ∞ µct − E Z = gct(E) exp dE (3.49) −∞ kT

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 hτcti in a voltage independent manner since we can pull the CT state quasi-Fermi level out of the above CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 78

integrals and see that it cancels out. Another way to state this result is that in Boltz- mann statistics, the energetic distribution of populated CT states does not change with voltage, rather the entire distribution is simply scaled by a voltage dependent constant. Since the ratio of populated CT states at different energies doesn’t change, their potentially different lifetimes do not contribute in different manners at different voltages and so we can express the average lifetime as a voltage-independent constant regardless of how complicated τct(E) might be. This has been confirmed experimentally 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:

2 qN0 Lklan Voc = E0 − kT log (3.50) Jsc

Thus as the solar cell is cooled down, the open-circuit voltage should approach

E0 linearly. It is tempting to think that Voc will equal E0 at 0K, but in reality, nonidealities not captured in the above equation prevent this from actually occurring, so the veriﬁcation of the relation is done by linearly extrapolating back to 0K from the high-temperature regime in which the equation holds. The principal assumptions that break down at low temperature and lead to deviations from the above equation are:

1. Temperature dependent current production. Even if the initial photogeneration step in organic solar cells is temperature independent, the subsequent transport is not and so at low temperature the internal quantum eﬃciency of the solar cell CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 79

could decrease, meaning that Jsc in the above equation would be temperature dependent.[77]

2. Non-selective contacts. As Voc increases, it may be the case that the voltage

becomes pinned to the built-in potential of solar cell. This can happen if Vbi

is less than Ectexp (or E0 in the Langevin limit). This will manifest itself as a

roll-oﬀ in Voc at low temperature where the predicted Voc would be higher than

Vbi so Voc instead approaches Vbi at low temperatures.[39]

3. Breakdown of the Boltzmann Approximation. The above equation is only valid for temperatures and voltages for which we can use the Boltzmann approximation instead of the Fermi-Dirac distribution. While this approximation is very good in inorganic solar cells even at cold temperatures, the high degree of energetic disorder in organic solar cells makes the assumption break down well above 0K. We provide a specific criterion in the above section on Defining an Effective Density of CT states but as a rule of thumb, we find in numerical simulations that it typically begins to break down around 150-200K.

3.20 The Light Ideality Factor

Like Langevin Recombination, our model predicts that a plot of Voc vs kT*log(Jsc) should have a slope of 1. This value is called the light ideality factor. Many OPV materials systems have been observed to have light ideality factors very near 1.[7-13] A few material systems, though, have been observed to have light ideality factors other than 1, indicating that in those systems there are loss mechanisms that are either not proportional to np or that the quasi-Fermi levels are not ﬂat at open- circuit.[97, 15] In some cases the additional mechanism has been identiﬁed to be non- selective contacts.[39] In other cases it is unclear what the origin is, however light ideality factors higher than 1 may be caused by many factors including trap-assisted recombination or simply the presence of shunts in the solar cell.[98, 88] CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 80

System Reduction Factor Notes PCPDTBT:PC70BM[12] 0.77, 0.83 Measured with two techniques PCPDTBT:PC70BM[64] 0.2 No DIO PCPDTBT:PC70BM[64] 0.07 3% DIO F-PCPDTBT:PC70BM[64] 0.14 No DIO F-PCPDTBT:PC70BM[64] 0.04 1% DIO F-PCPDTBT:PC70BM[64] 0.03 3% DIO mono-DPP:PCBM[43] 0.11 Solution processed small molecule bis-DPP:PCBM[43] 0.03 Solution processed small molecule P3HT:PCBM[9] 0.1 As-cast RRa-P3HT:PCBM[48] 4x10−4 Regiorandom P3HT P3HT: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 geminate 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 particular complications but taken together we believe they support the general statement that the average CT state recombination lifetime is somewhere between 100 ps and 10 ns for most organic solar cell materials. CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 81

System Lifetime Method P3HT:F8BT[4] 1.3 ns Pump-push photocurrent decay P3HT:PCBM[4] 840 ps Pump-push photocurrent decay PCPDTBT:PC70BM[4] 400 ps Pump-push photocurrent decay MDMO-PPV:PC70BM[4] 580 ps Pump-push photocurrent decay PFB:F8BT[4] 6.8 ns Pump-push photocurrent decay P3HT:PCBM (nonannealed)[30] 780 ps Pump-push photocurrent decay P3HT:PCBM (annealed)[30] 660 ps Pump-push photocurrent decay MDMO-PPV:PC70BM (1:1)[30] 600 ps Pump-push photocurrent decay MDMO-PPV:PC70BM (1:2)[30] 460 ps Pump-push photocurrent decay MDMO-PPV:PC70BM (1:4)[30] 480 ps Pump-push photocurrent decay P3HT with 5 fullerene types[56] 3-6 ns Polaron transient absorption decay P3HT:PCBM (4:1)[49] 500 ps Terahertz Spectroscopy P3HT: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:

2 np = ni (3.51) CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 82

This result comes from the equilibrium condition of the following chemical reaction: 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:

E − E E − E np = N 2 exp fn c exp v fp (3.54) 0 kT kT −(E − E ) np = N 2 exp c v (3.55) 0 kT −E np = N 2 exp 0 (3.56) 0 kT CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 83

where we have used the result that Efn = Efp when electrons and holes are in equilibrium with each other. The point of rederiving this basic result is to show how, as the name suggests, it is actually a product of the same type of analysis that we use in this paper, demonstrating that the analysis is applicable to electronic excitations in solar cells. As we discussed in the main text, the logic of setting up a chemical reaction between electronic excitations and looking for the equilibrium condition was used by Wurfel to relate electron and hole densities to photon densities by considering the radiative recombination (or generation) reaction: n + p ←−→ photon Similarly, Wurfel considered nonradiative recombination by allowing electrons and holes to react with phonons: n + p ←−→ N · (phonons) The reverse direction of this reaction is simply thermal generation of electron/hole pairs. As another example, materials scientists commonly consider defect reactions with associated chemical potentials and equilibrium concentrations for crystalline defects like interstitials or vacancies.[104] All of these techniques are built on the same foun- dation of calculating the distribution of states that minimizes a systems free energy, but using the concept of chemical potentials, or molar free energies or quasi-Fermi levels to simplify the associated mathematics. Seen in this context, our discussion of the consequences of CT state equilibrium with free electrons and holes does not require novel methods of analysis, its simply that we had not before had reason to think that CT states were close to equilibrium with free carriers. Our work just analyzes the equilibrium condition of the following reaction and its implications: n + p ←−→ CT CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 84

Acceptor EA Ef

qVoc E0

q(V - V Donor IP bi oc)

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 involving an electron/hole pair forming a CT state and resplitting. This presentation is mathematically simple and generally applicable, which is why we chose it, but some readers may prefer a more physical approach. In this section we derive the same result directly from the many-particle partition function for a simplified organic solar cell model that turns out to be analytically solvable. A schematic for the simplified device model that we will use in this derivation is shown in Figure 3.8. The key simplifications that we make are:

1. We do not include the eﬀects of energetic disorder in either the donor Ionization Potential or the acceptor Electron Aﬃnity, modeling both energy levels as delta

functions with N0 states per unit volume. We do include a Gaussian density of CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 85

CT states with center energy Ect and width σct so that the reader can see the eﬀect of interfacial disorder on recombination.

2. We ignore any potential band bending and assume that the electric ﬁeld across the device is uniform.

3. We assume that the device is ﬁnely intermixed so that any location in the device could host either an electron or a hole.

4. We reduce the Coulomb interaction between electrons and holes to a simple nearest neighbor interaction. When an electron and hole are in a CT state,

their energy is assumed to be reduced by EB, the CT state binding energy, and when they are not in a CT state, they are assumed to not feel each others presence at all.

5. We assume that the active layer is overall charge neutral (n = p).

6. We assume that the charge carrier distribution is precisely uniform in the plane of the device (though not in the direction in which the electric ﬁeld is applied) so that we can use periodic boundary conditions in our derivation.

7. We assume that the carrier density is low enough that an electron only ever interacts with one hole at a time. This assumption lets us expand the partition function into a sum over electron-hole pairs.

Of these assumptions, 4, 5, 6 and 7 are key to the derivation. 1-3 can be relaxed at the cost of additional mathematical complexity without aﬀecting either our method of solution or our ability to express the result in terms of elementary functions. We include assumptions 1-3 since the goal of this section is to present an alternative view on an expression that we have already derived quite generally using chemical potentials, so we dont want to introduce additional mathematical complexities that obscure the core idea of the exposition. Our ﬁrst task is to calculate the probability that a given electron is part of a Charge Transfer state at any instant in time. Because the CT states are in equilibrium with free charges, we know that we can express this probability given the many-particle CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 86

partition function of the system and knowledge of the free energy difference between bound CT states and free carriers. The free energy difference includes an entropic component since two carriers have many more spatial arrangements that result in them not being nearest-neighbors than being nearest neighbors. In a box with volume V, there are V2 two particle states but only V CT states. We will need a way to incorporate this effect into our expression for recombination. Starting with what we know to be true:

X −E(s) Z = exp (3.57) kT s∈states

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:

1 A = (3.58) Ln

So, this is the problem that we need to solve: given one electron and one hole in a rectangular solid with area A and height L, under a uniform voltage potential V =

Vbi - Voc applied along the L direction, what fraction of the time will the electron and hole be located next to each other. To further simplify, we will assume that there are only two possible states: fully bound CT state and fully free carriers. Not making CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 87

this simpliﬁcation leads to the partition function containing the Exponential Integral, which is not expressible in terms of elementary functions. With these assumptions, the partition function reduces to:

X −Ect X −E(s) Z = exp + exp (3.59) kT kT ct s∈free

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:

Z ∞ X −Ect −E exp ≈ g (E) exp dE (3.60) kT ct kT ct −∞

We have already evaluated this integral in the section on deﬁning an eﬀective density of CT states, so we just quote the result here:

2 σct ! exp X −Ect −Ect + −Ect exp ≈ N AL exp 2kT = N AL exp (3.61) kT ct kT ct kT ct

where Nct is the density of CT states per unit volume and AL is the volume of the periodic cell of the solar cell that we are considering, which is a function of carrier density. Turning to the free carrier term in the partition function, we have that the energy of an electron-hole pair due to their positions in the applied voltage potential is given by:

E(ze, zh) = V (zh − ze) + E0 (3.62)

where ze and zh are the electron and hole positions along the electric ﬁeld axis and E0 is the energy required to create an electron-hole pair from the Ground State. CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 88

The voltage (V) is taken to be equal to the built-in potential minus the open-circuit voltage.

Z L Z L X −E(s) −E0 qV ze −qV zh exp ≈ N 2A2 exp exp exp dz dz kT 0 kT kT kT e h s∈free 0 0 (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:

2 X −E(s) kT qV − E0 exp ≈ (N AL)2 exp (3.64) kT 0 qV kT s∈free

Thus the ﬁnal result for the partition function is:

−E exp kT 2 qV − E Z = N AL exp ct + (N AL)2 exp 0 (3.65) ct kT 0 qV kT

We can now ﬁnd the odds that an electron and hole will be in a CT state in the standard way:

1 X −Ect P = exp (3.66) ct Z kT ct

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 denominator, leaving: CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 89

−1 " 2 exp # N0 kT qV − E0 + Ect Pct = 1 + AL exp (3.67) Nct qV kt

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.

2 exp Nct qV −qV + E0 − Ect Pct = 2 exp (3.68) ALN0 kT kt

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) 2 2 exp Nctn qV −qV + E0 − Ect nct = 2 exp (3.70) N0 kT kt

where we have substituted for the periodic volume (AL) in terms of the carrier density n. The final task is to solve for the carrier density as a function of the open-circuit voltage. This is nontrivial because of the presence of the electric field across the device. At every point in the device we know that the local density of electrons is given by a Boltzmann distribution (since we are assuming that no portion of the device is degenerate) that depends on the distance between the quasi-Fermi level for electrons and the acceptor LUMO. We have defined this difference at the electron extracting contact as Ef , however since the LUMO is tilted, the difference is a linear function of position. The average density of charge carriers then is: CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 90

1 Z L qV z n(V ) = n Ef + dz (3.71) L 0 L Z L qV z ! 1 −Ef − L n(V ) = N0 exp dz (3.72) L 0 kT

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 kT −E n(V ) = 0 exp f (3.73) qV kT

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.

qV + E − qV − 2E − Eexp n = N exp oc 0 bi f ct (3.74) ct ct kt

where we have expanded V = Vbi - Voc. This expression does not yet look identical to the one we derived in the main text because of the apparent dependence of nct on

Vbi, E0 and the speciﬁc details of the Fermi level alignment at the electron and hole extracting contacts captured in Ef . However, looking at Figure 3.8, we can see that

Vbi, E0 and Ef must all be related and in fact:

E0 = qVbi + 2Ef (3.75)

So, actually Vbi, E0 and Ef identically cancel out in our expression for nct and we are left with simply: CHAPTER 3. THE OPEN-CIRCUIT VOLTAGE 91

qV − Eexp n = N exp oc ct (3.76) ct ct kt

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 experience an inhomogeneous energetic landscape that favors separation. We have also explained that recombination should turn on exponentially as the voltage on the solar 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 organic solar cells has been that the devices show field dependent photogeneration,

92 CHAPTER 4. THE FILL FACTOR 93 Current Current

Voltage Voltage

Figure 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. where the yield of free carriers from the geminate splitting process is a sensitive function 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 describing real organic solar cells. Finally, we use the validated theory to explain and understand the factors that affect the Fill Factor of an organic solar cell.

4.1 The Myth of the Intrinsic Organic Solar Cell

Before we can build a working theory, we ﬁrst need to dispel what has become a persistent myth in the organic solar cell community. Many researchers think of the organic materials used to make organic solar cells as intrinsic semiconductors. In this work we accept this statement as likely true and do not dispute it. However, it is then often concluded that since organic solar cells are make from intrinsic semiconductors, CHAPTER 4. THE FILL FACTOR 94

ﬃnity

Acceptor Electron A 1018 1017 ]

3 16

− 10 m

c 15

[ 10

t 14 i 10 s n

Energy 13 e 10 D

r 1012 e i r

r 11

a 10

Donor Ionization Potential C Electron Density 1010 Hole Density 109 Position Length

Figure 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. they must have very low dark carrier densities. This statement is not true and based on the false premise that the only source of dark charge carriers in a solar cell is due to dopants present in the active layer. In fact, in the P-I-N architecture favored by organic solar cell researchers, the majority of dark carriers are injected by the P and N type contacts and will always be present regardless of whether or not the active layer is intrinsic when considered as a slab of bulk semiconductor without metal contacts. To see this simply consider the band diagram of an organic solar cell in the dark as shown in Figure 4.2. The dashed line in Figure 4.2 shows the position of the Fermi level in the device. The electron density at a given position in the device is an exponential function of the distance between the Fermi level and acceptor’s Electron Aﬃnity, which is why there are orders of magnitude more electrons near the electron extracting contact than in the rest of the solar cell and similarly orders of magnitude more dark holes near the hole extracting contact simply due to the requirement that the active layer be in equilibrium with the contacts on either side of it. CHAPTER 4. THE FILL FACTOR 95

4.2 Why Dark Carriers Matter

Most treatments of recombination in organic solar cells ignore the presence of dark charge carriers leading to qualitatively incorrect conclusions about recombination in the devices. As we showed in Chapter 3, recombination is proportional to the product of the electron and hole densities as each position in the device. Assume for a second (as we will later show is the case for organic solar cells), that the excess electron and hole densities in the solar cell are proportional to the light intensity with which you are illuminating the device. Then, in the absence of dark carriers, recombination could be expressed as:

2 Jrec = knp ≈ knlpl ∝ Φ (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) 2 Jrec = k (ndpd + ndpl + nlpd + nlpl) ∝ C + Φ + Φ (4.3)

When we account for dark charge carriers, we see that there are three distinct kinds of bimolecular recombination, with three diﬀerent light intensity dependences: dark- dark carrier recombination, dark-light carrier recombination and light-light carrier recombination. Only the last form of recombination has a quadratic dependence on light intensity, so only that form can be made negligible at low light intensity. Properly considering the presence of dark carriers in organic solar cells is critical for the correct interpretation of a range of experimental results and, as we will show in the rest of this chapter, also key to understanding why the Fill Factors of organic CHAPTER 4. THE FILL FACTOR 96

solar cells are both low and poorly controlled.

4.3 Methodology

Ultimately, coming up with a theory for the Fill Factor of an organic solar cell requires having a function that calculates the density of electrons and holes everywhere in the solar cell as function both of the voltage that is being applied to the cell and the current being drawn. Using this expression, we could then write down the total rate of recombination as a function of J, the driven current, and V, the applied voltage, and be able to describe the operation of the solar cell at any point on its IV curve. Unfortunately, the required drift-diffusion equations that describe how many charge carriers are present in various places in the solar cell away from open-circuit are not typically solvable analytically, making it difficult to get insight into what determines the Fill Factor without resorting to opaque and complicated numerical simulations. It is likely for this reason that such confusion persists in the OPV community about what determines the Fill-Factor. In this chapter we take a different approach. It turns out that, in the absence of recombination, one can analytically solve for the carrier distribution throughout an organic solar cell. This, in turn, will let us calculate the np product analytically to obtain the driving force for CT state formation (as described in Chapter 3) and hence the rate of recombination. While our result formally only holds in the limit of very little recombination (i.e. it is a perturbative approach), we will find that in practice it describes working OPV devices quite well.

4.4 The Carrier Distribution in an OPV Device

To begin, we ask the question, if there were very little recombination in an organic solar cell, what would the carrier distribution inside the active layer look like as a function of the current, J, being extracted from the device and the operating voltage V. The starting point for our calculation is the drift-diﬀusion equations which relate CHAPTER 4. THE FILL FACTOR 97

a gradient in the electron or hole quasi-fermi levels to a current. The result is:

dE J (x) = µ n(x) fi (4.4) i i dx where Ji is the current density flowing at location x in 1D, µi is the macroscopic (DC) charge carrier mobility, n(x) is the charge carrier density and Efi is the carrier quasi- fermi level. This equation is always satisfied and allows one to calculate a (possibly spatially varying) current given the quasi-fermi level as a function of position (since n(x) can be derived from Efi). The rigorous derivation of the form of n(x) can be done by forcing the derivative of Ji(x) to be zero and solving the associated nonlinear differential equation, which fortunately turns out to be solvable analytically. However, more insight is gained by first taking a heuristic perspective. To begin, assume that we have a device in which a constant electron current

Je is ﬂowing. Further suppose that the device has a built-in potential Vbi and we are holding it at a voltage V . We want to guess how the charge carriers must be distributed in order to guarantee a non-spatially varying current. Since there is no recombination in the device, we know that the current must be the same everywhere. Imagine that the device is very thick and we are looking at the charge carrier density far from contacts. There is only one way to guarantee that the current is constant (looking at Equation 4.4). The charge carrier density must be constant and the slope of the quasi-fermi level must be constant. One could also imagine that maybe the charge carrier density is varying and the slope of the quasi-fermi level is varying inversely to just cancel it out such that the product of the two is constant, but this cannot work because the slope of the quasi-fermi level is linked to the slope of the carrier density so they cannot vary in opposing ways. Since we need the carrier density to be constant, the distance between the quasi- fermi level and the band edge must be constant, which means that the slope of the quasi-fermi level must exactly equal the slope of the band, caused by the built-in voltage. We can invert Equation 4.4 to ﬁnd this carrier density in terms of the CHAPTER 4. THE FILL FACTOR 98

n(0) Efn

n(x) n(∞)

Vbi x

Fermi Level Carrier Density

Figure 4.3: The required fermi level and charge carrier density proﬁles in order to have a constant current in an intrinsic semiconductor device.

current Je, built-in voltage Vbi, carrier mobility µe and device thickness L:

J L n(∞) = e (4.5) qVbiµe

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 dEfn n(x) is too big, dx must be very small. In fact since n(x) depends exponentially on Efn, the quasi-fermi level must be almost completely flat for even a slight excess of carriers. This allows us to draw the entire quasi-fermi level profile for an organic solar cell away from open-circuit and the result is shown schematically in Figure 4.3. The numerical calculation showing the exact result is given in Figure 4.4. It compares quite nicely with what we reasoned above that it must be. The analytical CHAPTER 4. THE FILL FACTOR 99

expressions for the electron and hole densities are:

−x −x n(x) = n(0)exp + n(∞) 1 − exp (4.6) l l x − L x − L p(x) = p(L)exp + p(∞) 1 − exp (4.7) l l LkT l = (4.8) qVbi

Rigorous Derivation

E − E − qV x/L dE J (x) = µ N exp fn c bi fn (4.9) e e e kT dx

dJe We require that our current be spatially uniform so we can set dx = 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.

cx Efn(x) = Efn0 + kT log (1 + Ae sinh(cx)) (4.10) 2J L E − E A = e exp c fn0 (4.11) qVbiµeNe kT qV c = bi (4.12) 2LkT where L is the thickness of the solar cell and Efn0 is the location of the quasi-Fermi level at the extracting contact, which is ﬁxed by the choice of electrode work function.

4.5 Recombination Away from Open-Circuit

The previous section showed how we can calculate the electron and hole densities in an OPV device as a function of the voltage on the device and the current being driven, which was considered constant. However, during solar cell operation, the photocurrent is typically being generated throughout the active layer, so neither the electron nor the hole currents will be constant. If instead we assume that there is CHAPTER 4. THE FILL FACTOR 100

Efn(x) 1.0 Efh(x)

0.5

0.0

0.5 Energy [eV]

1.0

1.5

2.0 0 20 40 60 80 100 Distance [nm]

Figure 4.4: The energy bands and quasi-fermi level positions for an organic solar cell 2 at Jsc producing a current of 10 mA/cm equally distributed between an electron and hole 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:

−q(Vbi − V )x JphL x n(x) = n0 exp + 1 − (4.13) kT L qµe(Vbi − V ) L −q(Vbi − V )(L − x) JphL x p(x) = p0 exp + (4.14) kT L qµh(Vbi − V ) L

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 happening. The first term is the exponential decay of the charge density present at the contacts and the second term is the steady state charge density needed to carry the CHAPTER 4. THE FILL FACTOR 101

photocurrent given the electric ﬁeld present in the device. Note that the ﬁrst term is current independent whereas the second term increases in magnitude with current.

When Vbi - V is not much greater than kT, there are corrections to this expression that come from overlap between the two terms, but they should be negligible during solar cell operation and in reverse bias. There are 2 terms in the expression for n(x) and p(x), meaning that there will be 4 terms in the product n*p. Two of the terms correspond to majority carriers near the contacts recombining with photogenerated carriers, one term corresponds to majority carriers from each contact recombining with each other and one term corresponds to photogenerated carriers recombining with each other.

4.5.1 Classifying Recombination Types

As we can see from the above expression for n(x) and p(x), there will be one recombination 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:

σ2 + σ2 V − E R = qγLN 2 exp n p exp 0 (4.15) dark s 2(kT )2 kT

In this expression Ns is the density of electronic states in the device (typically taken to be 1 state per nm−3 or 1021 cm−3), σ is the disorder in the electron and hole CHAPTER 4. THE FILL FACTOR 102

conducting material and Ebg is the eﬀective bandgap between the LUMO of the acceptor and the HOMO of the donor. γ is the bimolecular recombination coeﬃcient. The recombination rate is expressed as a current density.

On first inspection, it could be unclear where Ebg came from and why Vbi disap- peared. The reason is because of the form of n0 and p0. We know that given the quasi-fermi level position and disorder, we can calculate the carrier density. We need to figure out where the Fermi level should be in equilibrium at zero bias. To do this we assume that the electron and hole disorder is the same and invoke overall charge neutrality for the device at equilibrium. This means that n0 = p0 and so the Fermi level must be equally spaced between the acceptor LUMO on one side of the device and the donor HOMO on the other side of the device. This means that the Fermi level must be located (Ebg - Vbi)/2 away from the center of each energy level. Using this result we can find an expression n0:

σ2 qV − E n = N exp n exp bi 0 (4.16) 0 s 2(kT )2 2kT

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 recombination term and we recover a result similar to a typical pn junction with an ideality factor of 1 corresponding to thermionic emission over a barrier.

Photocarrier - Dark Carrier Recombination

By combining the majority carrier term of one carrier type with the minority carrier term of the second carrier type, we can calculate the rate at which photoinduced carriers recombine with dark majority carriers. The expression given below is for the CHAPTER 4. THE FILL FACTOR 103

case of equal electron and hole mobilities and disorder parameters for simplicity.

2 2 2 4L (kT ) JphNs σ qVbi − E0 Rcontact = 2 3 exp 2 exp (4.17) q (Vbi − V ) µ 2(kT ) 2kT

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 concentration is unchanged.

Photocarrier - Photocarrier Recombination

Finally, we can calculate the eﬀect of photocarriers recombining with other photocarriers by taking the second terms in the expressions for n(x) and p(x):

3 2 L Jphγ Rbulk = 2 (4.18) 6qµeµh(Vbi − V )

This eﬀect is seen to increase like L3 and decrease like the built-in potential squared. In this case it is easy to see where the dependence comes from. The two copies of the voltage, mobility and length come from setting the required minority carrier concentration of both the electrons and holes in order to sustain the given current. The third copy of the length comes from integrating the recombination volume density over the device. CHAPTER 4. THE FILL FACTOR 104

4.6 Using These Results to Understand Organic Solar Cells

The major insights gained from the previous section are that you can divide bimolecular recombination in organic solar cells into 3 different classes that each have distinct dependences on the amount of photocurrent being generated and the operating voltage 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:

qV B C J(V ) = −Jph + A exp + 2 + 3 (4.19) kT (V − Vbi) (V − Vbi)

The remaining sections in this chapter will be devoted to validating and using this expression to understand experimental IV curves from literature.

4.7 Validating Our Expression Using P3HT:PCBM

In order to see if Equation 4.19 is able to describe the wide variety of solar cell IV curves, we first turn to the model system P3HT:PCBM. This system is interesting because one can tune the electron mobility by 1.5 orders of magnitude and the hole mobility by more than 3 orders of magnitude just by controlling the length of a thermal annealing step during device fabrication. Using data from Bartelt et al (In Press), we had access to a data set of 24 different P3HT:PCBM solar cell conditions spanning 4 thicknesses between 100 and 300 nm and 6 different annealing temperatures. In each case, we corrected the data for series and shunt resistance as described below and CHAPTER 4. THE FILL FACTOR 105

then ﬁt it to our IV curve expression using a standard nonlinear optimization routine implemented in a Python script.

4.7.1 Correcting for Series Resistance

Many experimental IV curves are heavily impacted by series resistance, making any sort of analysis of the IV curve shape impossible without ﬁrst removing the series resistance. This is done by realizing that the eﬀect of series resistance is to introduce an error term in the measured voltage:

0 V = V + IRs (4.20)

0 where V is the measured voltage on the voltage cell, Rs is the series resistance and V is the actual voltage across the active layer (note that I is negative in the power- producing quadrant so the voltage on the solar cell is higher than is measured in that case). The series resistance can be found by fitting the dark IV curve in far forward bias to a linear function and then the light IV curve can be corrected by applying Equation 4.20 to get the actual voltage on the active layer from the measured voltage. As an example of how important this correction is, consider the material system P3HT:PCBM. As we anneal the P3HT active layer, we find that we are systematically reducing the series resistance on the solar cell, which is presumably caused in large part by transport through a pure P3HT domain. Figure 4.5 shows the series resistance extracted from the dark IV curves for all 24 devices in this study. Figure 4.6 shows that this series resistance appears to be coming from transport in P3HT regions of the solar cell since the series resistance is linearly proportional to both the thickness of the solar cell and the P3HT hole mobility. We speculate that there is perhaps a P3HT rich capping layer on these solar cells that causes this effect but further study would be required to determine its precise cause. For now, we just note its existence and correct for it using Equation 4.20. CHAPTER 4. THE FILL FACTOR 106

40 As Cast 48C Anneal

35 71C Anneal 88C Anneal 111C Anneal 148C Anneal 30

10 Series Resistance [Ohms/cm^2]

0 50 100 150 200 250 300 350 Thickness [nm]

Figure 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. CHAPTER 4. THE FILL FACTOR 107

106

Linear Scale

105 Series Conductivity [Siemens]

104 Conductivity Data Linear Fit

10-7 10-6 10-5 10-4 10-3 P3HT Diode Hole Mobility [cm2 / Vs]

Figure 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 CHAPTER 4. THE FILL FACTOR 108

2 90 nm 146 nm 205 nm 304 nm 0 ] 2

m 2 c / A m [

t n e

r 4 r u C

8 2.0 1.5 1.0 0.5 0.0 0.5 1.0 Voltage [V]

Figure 4.7: Experimental data (points) and ﬁt to our expression for P3HT:PCBM solar cells annealed at 0C.

4.7.2 Correcting for Shunt Resistance

Correcting for shunt resistance is done in the standard way by ﬁtting a line to the dark IV curve near 0 volts and subtracting that line from the light IV curve to remove the shunt. This is only possible when information on the dark IV curve is available. This correction is not as important as the series resistance correction described in the previous section.

4.7.3 P3HT:PCBM Data Fits Our Expression

Figures 4.7- 4.12 show the ﬁts between experimental data (points) and out ﬁtting function (lines) for P3HT:PCBM solar cells annealed at 0 - 148C for 10 minutes and made with various thicknesses between 100 and 300 nm. CHAPTER 4. THE FILL FACTOR 109

2 114 nm 140 nm 202 nm

0 324 nm

2 ] 2 m c / A

m 4 [

t n e r r u

C 6

10 2.0 1.5 1.0 0.5 0.0 0.5 1.0 Voltage [V]

Figure 4.8: Experimental data (points) and ﬁt to our expression for P3HT:PCBM solar cells annealed at 48C. CHAPTER 4. THE FILL FACTOR 110

2 117 nm 151 nm 229 nm 0 306 nm

2 ] 2 m c 4 / A m [

n 6 e r r u C

12 2.0 1.5 1.0 0.5 0.0 0.5 1.0 Voltage [V]

Figure 4.9: Experimental data (points) and ﬁt to our expression for P3HT:PCBM solar cells annealed at 71C. CHAPTER 4. THE FILL FACTOR 111

2 104 nm 170 nm 227 nm 0 275 nm

2 ] 2 m c 4 / A m [

n 6 e r r u C

12 2.0 1.5 1.0 0.5 0.0 0.5 1.0 Voltage [V]

Figure 4.10: Experimental data (points) and ﬁt to our expression for P3HT:PCBM solar cells annealed at 88C. CHAPTER 4. THE FILL FACTOR 112

2 112 nm 132 nm 211 nm 0 292 nm

2 ] 2 m c 4 / A m [

n 6 e r r u C

12 2.0 1.5 1.0 0.5 0.0 0.5 1.0 Voltage [V]

Figure 4.11: Experimental data (points) and ﬁt to our expression for P3HT:PCBM solar cells annealed at 111C. CHAPTER 4. THE FILL FACTOR 113

2 127 nm 164 nm 197 nm 0 312 nm

2 ] 2 m c 4 / A m [

n 6 e r r u C

12 2.0 1.5 1.0 0.5 0.0 0.5 1.0 Voltage [V]

Figure 4.12: Experimental data (points) and ﬁt to our expression for P3HT:PCBM solar cells annealed at 148C. CHAPTER 4. THE FILL FACTOR 114

10 ] 2 m c / 8 A m [

t n e r

r 6 u c o t o h P

e 4 c i v e

D As Cast 48C Anneal 2 71C Anneal 88C Anneal 111C Anneal 148C Anneal 0 50 100 150 200 250 300 350 Thickness [nm]

Figure 4.13: The total amount of photocurrent produced in each device in the P3HT:PCBM annealing series.

4.7.4 The Photocurrent Term

The curve fits to our expression are typically quite good, especially given the simplicity of the expression but the real value comes in analyzing trends in the fit parameters extracted from the fits since variation in those parameters can tell us about changes in the solar active layer as we anneal the devices. The first parameter is the photocurrent produced by each device. This number should be the total number of extractable free electrons and holes produced by the devices. Figure 4.13 shows the photocurrent produced by each device. What we can learn from Figure 4.13 is that the two low-temperature annealed devices (0C and 48C) lose photocurrent when they are made thicker while the other 4 devices gain photocurrent with thickness, as would be expected since the devices continue absorbing a larger fraction of the incident light until they are approximately CHAPTER 4. THE FILL FACTOR 115

300 nm thick. The photocurrent loss in the two low-temperature annealed devices has previously been shown to be the result of their extremely low hole mobilities causing space charge to build up and create a depletion region narrower than the device thickness, so that large fractions of the device have no electric ﬁeld and do not contribute to the photocurrent, see Bartelt et al Advanced Energy Materials (In Press).

4.7.5 The Built-in Potential

In our fitting expression, the parameter of interest that we capture as Vbi is actually the strength of the electric field in the device since that is what sets the drift velocity of the charge carriers and hence how many photocarriers build up inside the device during operation. In our derivation of the formula, we assumed that the electric field was uniform over the device, so its magnitude would simply be Vbi − V divided by the thickness of the solar cell. However, as we saw in the last section, there can be significant space charge buildup in the devices, which means the field will not drop uniformly over the entire solar cell, but will be concentrated in a small depletion region. This should result in an apparent increase of the built-in potential since the field over the portion of the device that produces photocurrent will be stronger by the ratio of the depletion width to the thickness of the solar cell. This is exactly what we find, as shown in Figure 4.14. What you can see from Figure 4.14 is that the built-in potential for all but the 111 and 148C annealed devices is higher than the measured built-in potential, which we ascribe to the known effect of space charge buildup in these devices. For the 111 and 148C devices, though, the extracted Vbi is in good agreement with the measured values across the range of device thickness. The take-home message is that the Vbi term in our fitting expression can tell you about the presence or absence of space charge limitations in your devices by whether or not it agrees with the measured built-in potential extracted from the crossing point of the light and dark IV curves. CHAPTER 4. THE FILL FACTOR 116

1 2.0 As Cast 48C Anneal 1.8 71C Anneal 88C Anneal 111C Anneal 1.6 148C Anneal

1.4

1.2

1.0 Built-in Voltage [V]

0.8

0.6

0.4 100 150 200 250 300 Thickness [nm]

Figure 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. CHAPTER 4. THE FILL FACTOR 117

100 As Cast 100 88C Anneal 48C Anneal 111C Anneal 71C Anneal 148C Anneal

80 80

60 60

40 40

20 20 Fraction of Photo-Dark Recombination [%]

0 0

50 100 150 200 250 300 350 100 150 200 250 300 350 Thickness [nm]

Figure 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.

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 becomes less important at larger thicknesses and it is never important for the as-cast devices. 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 recombination mechanism playing an increasingly important role as the devices are made thicker. This makes sense because from our analytical expression, we see that this recombination mechanism should increase as L3 whereas photocarrier-photocarrier recombination should only increase as L2, meaning that photocarrier-dark carrier recombination should be of increasing importance as the devices are made thicker. CHAPTER 4. THE FILL FACTOR 118

100 100 88C Anneal 111C Anneal 148C Anneal

80 80

60 60

40 40

20 20

As Cast Fraction of Photo-Photo Recombination [%] 48C Anneal 0 71C Anneal 0

50 100 150 200 250 300 350 100 150 200 250 300 350 Thickness [nm] Thickness [nm]

Figure 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.

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 recombination 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:

1 B ∝ (4.21) µeµh CHAPTER 4. THE FILL FACTOR 119

10-4 ~300nm Devices Linear Correspondence

10-5

10-6

10-7 Inverse Photo-Photo Recomb. Coefficient [a.u.]

10-8 10-7 10-6 10-5 10-4 10-3 P3HT Diode Hole Mobility [cm2 / Vs]

Figure 4.17: The inverse proportionality of the photocarrier-photocarrier recombination coeﬃcient to the P3HT hole mobility after correcting for the variation in electron mobility

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,

1 B ∝ (4.22) µeµh 1 ∝ µ µ (4.23) B e h 1 ∝ µh (4.24) µeB

The left-hand side of Equation 4.24 is plotted in Figure 4.17 against the hole mobility for the 300 nm thick devices. CHAPTER 4. THE FILL FACTOR 120

10-9 ] 2 m c / 10-10 A m [

0 J

As Cast 48C Anneal 71C Anneal 88C Anneal 111C Anneal 148C Anneal 10-11 50 100 150 200 250 300 350 Thickness [nm]

Figure 4.18: The reverse saturation current density extracted from our ﬁts.

The Figure shows a decent linear proportionality over 3 orders of magnitude in the P3HT hole mobility, indicating that the speciﬁc dependences of our analytical expression on mobility may remain valid even for non-ideal organic solar cells.

4.7.8 Dark - Dark Recombination

For completeness, we show the dark-dark recombination term, which is typically referred to as J0, the reverse saturation current density (Figure 4.18). There is not a lot of information that we can extract from the values, however, since we showed in Chapter 3 that this is mainly a measure of the degree of mixing in the solar cells, the energy of the Charge Transfer state distribution and the CT state lifetime. CHAPTER 4. THE FILL FACTOR 121

4.7.9 Conclusions

What we have shown in this section is that our analytical expression for the IV curve of an organic solar cell is able to ﬁt and explain the variation in P3HT:PCBM solar cells across a wide range of mobilities and thicknesses showing that the expression is useful for understanding actual OPV device performance. Further, we have shown that the ﬁt parameters extracted from our expression vary in understandable ways and appear to have the meanings and dependence on materials parameters that we expect from our analytical results. In the next section we will use this, now validated, expression to look at other material systems from literature.

4.8 Molecular Weight Variations in PCDTBT

One of the key advantages of our analytical expression is that it only requires an IV curve in order to extract powerful amounts of information about what is occurring inside the solar cell active layer. To demonstrate this, we looked at literature data showing a series of PCDTBT:PCBM solar cells with diﬀering molecular weights[59].

The IV curves showed large FF and Jsc variations among the different molecular weights but it was not clear why. We can now reanalyze those data to understand why. The raw IV curves and fits are shown in Figure 4.19. Note that the data was corrected for series resistance, which was non-negligible but not found to vary significantly among the different molecular weight devices. The first point to note is that the fits are superb, with almost no deviation between the fits and the experimental data. The second point to note comes from comparing the fit parameters obtained from fitting these IV curves. As can be seen in Figure 4.19, there is significant variation in short-circuit current among the different molecular weights. However, as reported in literature, there are not significant differences in absorption among the devices[59]. So, we do not expect any variation in photocurrent production, in contrast to the observed Jsc variation. Table 4.1 shows that, in fact, there is little variation in photocurrent production among the devices. While the short-circuit current varies by 3 mA per square centimeter, the actual amount of CHAPTER 4. THE FILL FACTOR 122

0 27.3 kDA 23.6 kDA 11.6 kDA 6.0 kDA 2 5.0 kDA

] 4 2 m c / A m [

t 6 n e r r u C

0.0 0.2 0.4 0.6 0.8 Voltage [V]

Figure 4.19: The raw IV curve data and ﬁts for PCDTBT:PCBM solar cells reported in literature[59]. CHAPTER 4. THE FILL FACTOR 123

2 Batch Jsc Jph [mA/cm ] 5 kDA 7.8 11.3 6 kDA 9.1 10.7 11.6 kDA 9.5 10.6 23.6 kDA 10.3 11.3 27.3 kDA 10.7 11.4

Table 4.1: Extracted Photocurrent and Short-circuit Currents for PCDTBT:PCBM devices. 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 increasing 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 mobility, in contrast to our expected and previously observed linear dependence. The reason for this is currently unclear and would warrant further study.

4.9 Apparent Field Dependent Geminate Splitting

We are finally in a position to tackle the last remaining question of this work, which is understanding solar cells that appear to have field-dependent geminate splitting. As we explained in Chapters 2 and 3, we do not expect field-dependent geminate splitting to occur in working organic solar cells since free carriers and CT states are observed to be in equilibrium with each other, necessitating that geminate pairs, in CHAPTER 4. THE FILL FACTOR 124

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

-8 -7 -6 -5 Inverse Photo-Photo Recomb. [a.u.] 10 10 10 10 Measured Hole Mobility [cm2 /Vs]

Figure 4.20: The inverse photocarrier-photocarrier recombination coeﬃcient plotted against the measured PCDTBT:PCBM hole mobility. CHAPTER 4. THE FILL FACTOR 125

2 Condition Jsc Jph [mA/cm ] As-Cast 8.3 15.1 Annealed 10.9 (+31%) 14 DIO 13.0 (+57%) 14.6

Table 4.2: Extracted Photocurrent and Short-circuit Currents for p − DTS(FBTTh2)2PC71BM devices. fact, have no trouble splitting and forming free carriers. Nevertheless, there are reports 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 studied 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 molecular 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 splitting, often using the Time Delayed Collection Field Technique (TDCF)[22, 75, 3, 27, 29, 68] to distinguish between geminate and nongeminate recombination. In this CHAPTER 4. THE FILL FACTOR 126

0 As-cast 2 Annealed

] DIO Additive

2 4 m c

/ 6 A m [

8 t n e

r 10 r u

C 12

4 3 2 1 0 1 Voltage [V]

Figure 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]. CHAPTER 4. THE FILL FACTOR 127

section we would like to explain why we believe that technique does not actually distinguish 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 different 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 carrier populations near the contacts of an organic solar cell due to equilibration between the large charge reservoirs in the metal contacts and the active layer. This means that the lifetime of a photogenerated carrier that happens to be formed very near the opposite contact will be very short since the average lifetime of a photogenerated hole, for example is a function of the total electron density at that point including both photoelectrons (ne,l) and dark electrons (ne,d): CHAPTER 4. THE FILL FACTOR 128

1 τh = (4.25) k(ne,l + ne,d)

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 nongeminate 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 previously, recombination near the contacts involves a photocarrier and a dark carrier, so it has the same light intensity dependence as geminate recombination (linear in light intensity). Thus, TDCF cannot in principle tell the difference between geminate and nongeminate recombination. It can simply report the presence of recombination. Now, proceeding on the assumption that the recombination mechanism that TDCF is probing is photocarrier - dark carrier recombination, we can also explain why it would be field dependent. Our analytical expression for photocarrier - dark carrier recombination has an inverse cubic dependence on electric field strength since that sets the timescale for carriers to leave the high recombination contact region. We would note that if TDCF were, in fact, probing geminate recombination, we would expect a much stronger exponential field dependence as given in Onsager- Braun theory. The observed field dependence of TDCF measurements is typically fairly weak. So, we conclude that TDCF measurements are likely just quantifying CHAPTER 4. THE FILL FACTOR 129

photocarrier - dark carrier recombination near the contacts of the solar cell since this mechanism has very similar characteristics to geminate recombination, though we stress that it is nongeminate. There is another potential issue with TDCF measurements that we mention here for completeness but we believe it to be of secondary importance in this instance. TDCF assumes that the prebias and the collection bias are able to create electric fields throughout the device and importantly that the fraction of the device that contains 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 efficiently collected and photocarriers formed outside the depletion region recombine. By setting the prebias you control the density of space charge and therefore the depletion width so the amount of collected charge becomes a function of the prebias and you can observe an apparently field-dependent recombination mechanism that is just an artifact of the measurement technique.

4.10 Conclusion

Our goal in this section was to show that we can understand the IV curves of arbitrary organic solar cells in terms of purely bimolecular recombination losses without ﬁeld- dependent geminate splitting or other exotic eﬀects. The key observation is that since organic solar cells are made in PIN structures, they cannot be described as intrinsic organic semiconductors without dark carriers. Once dark carriers are added into the description, we are able to accurately describe the shape of OPV IV curves CHAPTER 4. THE FILL FACTOR 130

for both high performance and low performance devices using one consistent theory. Importantly, our theory for IV curve shape is completely compatible with our theory for Voc and Jsc, namely that charge carriers are in equilibrium with CT states and so the amount of recombination in a solar cell is just a function of how many carriers are in the cell since that sets the driving force for CT state formation and hence recombination. We do not expect, nor do we observe, significant differences in photocurrent generation among devices with similar optical absorption spectra since nearly all geminate pairs split. Rather, the differences observed in both short-circuit current and FF were shown to be caused by non-geminate mechanisms, typically due to very low hole mobilities. Thus, we have accomplished our goal of finding a single theory that can explain the short-circuit current, fill factor and open-circuit voltage of organic solar cells and our work is complete.

4.11 Additional Theoretical Background

4.11.1 Properly Counting States in the Presence of Disorder

In typical derivations relating carrier density and quasi-Fermi levels it is assumed that the electronic states of the solar cell can be approximated by a lumped “effective” density of states at the band edge of the conduction and valence bands. In organic solar cells, the presence of Gaussian disorder means that this is not in general possible since there are many states below the center of the material HOMO. In this section we will show that we can still define an effective density of states but that the presence of disorder makes this effective DOS approximately many (over 100) times larger than for crystalline, non-disordered systems. This means that 100 times more carriers are required to achieve the same quasi-fermi level splitting as in a highly crystalline solar cell. CHAPTER 4. THE FILL FACTOR 131

4.11.2 The Link Between Voltage and Carrier Density

We can think of a solar cell as just providing 2 reservoirs of excited charge carriers: one of electrons and one of holes. The quasi-fermi level describing how filled each reservoir is can be determined since the charge carriers are fermions, by simply filling up electronic states from low to high energy until all of the carriers in the device have been accommodated. At 0 Kelvin, the highest occupied state is the quasi-fermi level. At finite temperature, thermal effects will excite some carriers above the quasi-fermi level leaving some open states below the quasi-fermi level. At any temperature, we know that the relation between the quasi-fermi level and the number of carriers in the device is given by:

Z ∞ 1 N(Ef ) = g(x)dx (4.26) x−E −∞ f 1 + exp kT 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 approximation 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:

Z ∞ Ef − x N(Ef ) = exp g(x)dx (4.27) −∞ kT

Given the assumption that the energy levels in our device are properly described by Gaussian distributions with a standard deviation σ, we have:

2 1 (x − Ec) g(x) = N0 √ exp − (4.28) σ 2π 2σ2 where Ec is the center of the Gaussian distribution and N0 is the total number of electronic states per unit volume. Combining Equation 4.27 with Equation 4.28 and CHAPTER 4. THE FILL FACTOR 132

1021 σ = 60 meV σ = 80 meV 1020 σ = 100 meV σ = 120 meV No Disorder 1019

] 10 3 − m c [

y 10 t i s n e

D 16

10 r e i r r

a 15 C 10

1014

1013

1012 0.5 0.4 0.3 0.2 0.1 0.0 Fermi Level Position [eV]

Figure 4.22: The density of charge carriers as a function of the quasi-fermi level given 21 a constant N0 = 1x10 . The dashed lines show the analytic approximation given in Equation 4.29. integrating analytically, one can show that the relation between quasi-fermi level and charge carrier density is given by:

σ2 E − E n(E ; E ) = αN exp exp f c (4.29) f c 0 2k2T 2 kT

At room temperature, with σ <= 80 meV and Ef more than about 0.3 below Ec, α is approximately equal to 1 and only weakly depends on the fermi level. As seen in Equation 4.29, the relation between charge carrier density and quasi- fermi level is the same as in the inorganic, non-degenerately doped case, but the density of electronic states is increased by an exponential factor dependent on the level of energetic disorder. Figure 4.22 shows the number of charge carriers in the device as a function of the quasi-fermi level location. One thing to note is that there are orders of magnitude CHAPTER 4. THE FILL FACTOR 133

105 σ = 60 meV σ = 80 meV σ = 100 meV σ = 120 meV

104

103

102

101 Ratio of Disorderd to Ordered Carrier Density [unitless]

100 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Fermi Level Position [eV]

Figure 4.23: The ratio of charge carriers in a disordered device compared to a non- disordered device as a function of the quasi-fermi level location. CHAPTER 4. THE FILL FACTOR 134

more charge carriers in the device at a given voltage because of the disorder. Another was of saying this is that the presence of low-energy trap states means you have to put more carriers into the device in order to reach a given voltage. We can quantify this by taking the ratio between the disordered curves in Figure 4.22 and the ordered curve given the fractional increase in carrier caused by disorder. This is shown in Figure 4.23. The key point to take away is that this penalty of higher carrier density for a given voltage is most pronounced at lower voltages when the quasi-fermi level is more than 0.3 eV away from the center of the band. As the voltage increases, the deviation becomes less severe, as it must since the ordered and disordered devices have the same total number of electronic states, they just have a larger energy spread in the disordered case. Figures 4.22 and 4.23 were calculated numerically without approximations, however in the region of Figure 4.23 that is ﬂat, we can apply Equation 4.29 to predict the carrier density penalty as a function of energetic disorder (Figure 4.24). The key point to take away from Figure 4.24 is that there is a very large diﬀerence between an energetic disorder of 60 meV and 100 meV but it does not change the ability to express the carrier density as a simple function of the quasi-Fermi level. One hundred times more carriers are present in the device with 100 meV of disorder than with 60 meV of disorder. Note that since bimolecular recombination is proportional to n*p, this means that there would be 10,000 times more recombination with 100 meV disorder than 60 meV, all other things being equal.

Calculating How Many Carriers Are in the Device

Given the expressions in Equation 4.7 and 4.8, we can calculate some basic properties that will be useful in the subsequent sections: the total number of electrons and holes in the device (that contribute to the current) as a function of current and voltage as well as the shape of the recombination current, which is proportional to n(x) ∗ p(x). CHAPTER 4. THE FILL FACTOR 135

105

104

103

102 Excess Carrier Ratio [unitless]

101

100 0 20 40 60 80 100 120 Energetic Disorder [meV]

Figure 4.24: The ratio of charge carriers in a disordered device to a fully ordered device calculated using Equation 4.29. CHAPTER 4. THE FILL FACTOR 136

The total number of electrons and holes is:

1 n = n(0)(1 − e−α) + n(∞)(α − 1 + e−α) (4.30) α 1 p = p(L)(1 − e−α) + p(∞)(α − 1 + e−α) (4.31) α qV α = bi (4.32) kT

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(0) n = (4.33) α p(L) p = (4.34) α

This means that until the voltage on the device approaches the built-in voltage, the total charge carrier density basically tracks the charge density at the contact but with a linear correction factor α. This is shown for three diﬀerent values of disorder in Figure 4.25. CHAPTER 4. THE FILL FACTOR 137

1020 σ =60 meV σ =80 meV σ =100 meV

1019 ] 3 − m c [

y 18 t 10 i s n e D

r e i r r

a 17

C 10

e g r a h C

1016

1015 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Open Circuit Voltage [V]

Figure 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. Bibliography

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