Monte-Carlo Simulation of ChargeTransport in Disordered Organic

Photoconductors

Heiko Schmitt

June 2011

supervisors Dr. Nils Reinke, Martin Neukom

Abstract

After the Nuclear disaster has ouccured in Fukushima on March 2011, the demand for renewable energyis increasing (cf. [Tag11b]). As a consequence of this event, Basel-City has decided to force ownersof a house to install a self-financed solar collector if the house has ideal conditions for installing such aplant (cf. [Tag11a]). But can Solar power meet the expectations? There are organic and inorganic pho-toconductors on the market. Experts expect much more efficiencies of thin-film organic devices whichare cheaper in production than their counterpart. "The chemical flexibility for modifications on organicsemiconductors as well as the perspective of low cost, large-scale production drives the research in thisfield in academia and industry" [HS04]. And Monte Carlo (MC) simulation is a reasonable approachto achieve these performance improvements of disordered organic photoconductors. An advantage ofthis technique is that it can be considered as an idealized experiment carried out on samples of arbitrar-ily adjustable computational parameters. Thus it allows to determine which parameters are required toreproduce the properties of a real world sample and to check the validity of approximations (cf. [Bae93]).

The goal of this Bachelor Thesis is to simulate charge transport in disordered organic photoconductors bydesigning and implementing a Monte-Carlo simulator according to Bässler’s model published in 1993(cf. [Bae93]). The difficulty in building the presented Monte-Carlo simulator was first to design aprogram that runs in a reasonable computational time and second to find out parameter configurationsthat were not described by Bässler (cf. sec. 5.1).

It is shown that the higher the applied voltage the faster the electron-hopping to the cathode. This findingcan also be seen in the visit probabilitiy distribution of an electron (cf. fig. 5.12 and fig. 5.13). We couldalso emulate the drift of carriers to energetic levels over time without any applied voltage (cf. fig. 5.8).In addition, it was simulated that there is an correlation between voltage and current (cf. sec. 5.3).

iii

Abstract

Zürcher Fachhochschule

Erklärung betreffend das selbständige Verfassen einer Bachelorarbeit an der School of Engineering

Mit der Abgabe dieser Bachelorarbeit versichert der/die Studierende, dass er/sie die Arbeit selbständig und ohne fremde Hilfe verfasst hat. (Bei Gruppenarbeiten gelten die Leistungen der übrigen Gruppenmitglieder nicht als fremde Hilfe.)

Der/die unterzeichnende Studierende erklärt, dass alle zitierten Quellen (auch Internetseiten) im Text oder Anhang korrekt nachgewiesen sind, d.h. dass die Bachelorarbeit keine Plagiate enthält, also keine Teile, die teilweise oder vollständig aus einem fremden Text oder einer fremden Arbeit unter Vorgabe der eigenen Urheberschaft bzw. ohne Quellenangabe übernommen worden sind. Bei Verfehlungen aller Art treten die Paragraphen 46 (Unredlichkeit und Verfahren bei Unredlichkeit) der ZHW Prüfungsordnung sowie die Bestimmungen der Disziplinarmassnahmen der Hochschulordnung in Kraft.

Ort, Datum: Unterschriften:

………………………………… ………………………………………………………...

…………………………………………………………

…………………………………………………………

Das Original dieses Formulars ist bei der ZHAW-Version aller abgegebenen Bachelorarbeiten zu Beginn der Dokumentation nach dem Abstract bzw. dem Management Summary mit Original-Unterschriften und -Datum (keine Kopie) einzufügen.

iv

Contents

Abstract iii

1. Introduction 1

2. Fundamentals 32.1. Organic Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2. Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3. Hopping Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4. Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3. Numerical Model 73.1. Hopping Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1.1. getPossibleDestCells() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.2. Jumprate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.3. CumHist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.4. Visit Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2. Influence of Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3. Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3.1. Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3.2. Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4. Monte Carlo Simulator 134.1. Imitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2. Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.1. Influence of Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2.2. Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5. Results 175.1. Validation of MC Simulator with Bässler . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.1.1. Validation of the Java method nextGaussian() . . . . . . . . . . . . . . . . . . . 185.1.2. Validation of Equation for Jump Probability . . . . . . . . . . . . . . . . . . . . 195.1.3. Validation of the MC Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.2. Filaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3. Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.4. Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.5. Tips for Building a Monte Carlo Simulator . . . . . . . . . . . . . . . . . . . . . . . . . 32

6. Conclusion and Future Work 33

A. Appendix 35A.1. Matlab Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35A.2. Java Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

v

Contents

Bibliography 37

vi

1. Introduction

The basic understanding of the details of the photogeneration, photophysics, electron transfer and chargerecombination of thin-film organic photoconductors have to be developed to meet the stability and relia-bility goals of such devices (cf. [Law93]). Performing MC simulations is an effective approach to do so(cf. [Bae93]).

"Today the electron and hole transport in polymer-fullerene based solar cells could be im-proved by allowing a relatively short route for holes to reach the anode and hence increasingthe effective hole mobility." [HYH+11]

"On the basic research side, a rich variety of new concepts have been uncovered due theinterplay between electronic and geometric structure. On the applied research side, whilenot destined to replace silicon-based technologies, organic semiconductors promise the ad-vent of fully flexible devices for large-area displays, solid-state lighting, radio frequencyidentification tags or solar cells. Major chemical companies worldwide, such as Sumitomo,DuPont, Solvay, BASF, Ciba, and Merck to name but a few, are now involved in developingefficient sources of chemicals for organic electronics applications." [CCdSF+07]

The value of this work is that it gives a framework to realize a MC simulator in an object-orientedcomputer language. It is shown that MC simulations can be performed on just one machine with today’scomputer power. Tips were elaborated which programming snippet should be avoided and used (cf. sec.5.5). The simulated filaments, current densities and conductivities shall show the technical feasibility ofMC simulation (cf. sec. 5.2 and 5.3).

The first step was to perform a target/actual comparison between an already existing applet, that simulatesgraphically the Miller-Abrahams hopping model of electrons in disordered organic photoconductors, andthe Monte-Carlo model of Bässler (cf. [Sch10] and [Bae93]). The next step, which represents the mainemphasis of this Bachelor Thesis, included designing the Monte-Carlo model and its implementationand validation (cf. ch.4 and 5). Finally, it was to determine the probability that 100x100x100 lattice-cells were visited by an electron and to illustrate the path that an electron walks in a specific number oftime-steps (cf. sec. 5.2).

In chapter 2, a general overview of organic semiconductors and important derivations of parameters forthe MC simulator are presented. The next chapter 3 explains the used numerical model with its hoppingprobability-, voltage- and current-equations. Important algorithms are also documented in chapter 3.Section 4.1 documents the design of the imitated MC simulator. Finally, the last chapter 5 presents theobtained results. A general conclusion of this work can be found in chapter 6.

"In the future, organic thin-films with structural order as high as possible will be true candidates fortechnical electronic applications in contrast to organic single crystals" [N.04]. In 2011, Lunt and Bulovicfabricated organic photovoltaics at the Massachusetts Institute of Technology that are highly transparentto visible light. "A series-integrated array of these transparent cells are capable of powering electronicdevices and are therefore an appropriate architecture for power-generating windows" [LB11]. As cathodematerial in future production of organic photovoltaics, Marshall et al. suggest single-layer graphenewhose main advantage is its transparency in comparison to conventional photovoltaic electrode materials

1

1. Introduction

(cf. [CGK+11]).

The author thanks Mr. Prof. Dr. Thaler for studying, revising and suggesting code for the presentedMC simulator (cf. sec. 5.5 and alg. 3), Mr. Prof. Dr. Burkhard Stiller of the Communication SystemsGroup (CSG) at University of Zurich for hosting a powerful number cruncher with 28 cores, Mr. Gohlfor supporting statistical issues and the supervisors Mr. Dr. Reinke, Neukom and Züfle of the Instituteof Computational Physics at Zurich University for Applied Sciences for answering questions of physicalorigin.

2

2. Fundamentals

In this chapter the fundamentals for the numerical model in chapter 3 are elaborated.

2.1. Organic Semiconductors

"Our understanding of devices based on disordered organic semiconductors, within whichhopping conduction takes place, is far less developed than our understanding of transportin devices based on crystalline inorganic semiconductors. [...] Organic semiconductors arepresently used in a wide variety of devices, such as organic light-emitting diodes, OLEDs,organic field-effect transistors and organic photovoltaic cells." [vdHUR+09]

"An organic semiconductor is an organic material with semiconductor properties. Thereare two major overlapping classes of organic semiconductors: organic charge-transfer com-plexes and various linear-backbone conductive polymers derived from polyacetylene. Tun-nelling, localized states, mobility gaps and phonon-assisted hopping contribute to conduc-tion. Like inorganic semiconductors, organic semiconductors can be doped what meansa treatment of the material with halogen. Doping of a disordered organic semiconductor,on the one hand, increases the concentration of charge carriers and lifts up the Fermi levelbut, on the other hand, creates additional deep Coulombic traps of the opposite polarity."[Wik11b]

By Arkhipov et al., it was shown that the doping efficiency strongly depends upon the energy disorderand external electric field (cf. [APVB05]). Typical current carriers in organic semiconductors are holesand electrons. "The charge carriers are either injected into the organic semiconductors from metal orconducting oxide electrodes in the case of light-emitting diodes or field-effect transistors or generatedwithin the materials in the case of solar cells via photon-induced charge separation at the interface be-tween electron-donor and electron-acceptor components". [CCdSF+07]. Charge transport depends onthe ability of the charge carriers to pass from one molecule to another. When the molecules of the organicsemiconductor have π-conjugate systems, electrons can move by hopping via electron cloud overlaps.To be precise, the charge transport in organic semiconductors is dependent on π-bonding orbitals andquantum mechanical wave-function overlap. In disordered organic semiconductors, there is limited π-bonding overlapping between molecules and conduction of charge carriers what is described by quantummechanical tunnelling. The transport process is commonly referred to as hopping transport. Furthermore,hopping of charge carriers from molecule to molecule depends upon the energy gap between HOMO andLUMO levels. Carrier mobility is reliant upon the abundance of similar energy levels for the electrons orholes to move to and hence will experience regions of faster and slower hopping. This can be affected bythe temperature and also electric field across the system (cf. [Wik11b]). There are two competitive pro-duction techniques used today: either wet solution processing or dry thermal evaporation of the organicconstituents (cf. [HS04]).

"An important difference to inorganic solid-state semiconductors lies in the generally poorcharge-carrier mobility [...], which has a large effect on the design and efficiency of organic

3

2. Fundamentals

semiconductor devices. However, organic semiconductors have relatively strong absorptioncoefficients, which partly balances low mobilities, giving high absorption in even < 100nmthin devices. Another important difference to crystalline, inorganic semiconductors is therelatively small diffusion length of primary photoexcitations (called excitons) in these ratheramorphous and disordered organic materials. These excitons are an important intermediatein the solar energy conversion process, and usually strong electric fields are required to dis-sociate them into free charge carriers, which are the desired final products for photovoltaicconversion. This is a consequence of exciton binding energies usually exceeding those of in-organic semiconductors. These features of organic semiconducting materials lead generallyto devices with very small layer thicknesses of the order ≤ 100nm". [HS04]

To sum up, transport effects in organic semiconductors occur as a result of charge transport betweenimpurity states (cf. [ME60]). That is the reason why Bässler has based his calculations for the jumprateVij on the energetic levels of source-cell ci and destination-cell cj in equation (3.2).

The most popular charge-transport models are Polaron Models and Disorder Models (cf. [CCdSF+07]).Polaron Models consider the disorder energy to be unimportant relative to the intramolecular deformationenergy. Disorder Models assume that the coupling of the charge to intra- (or inter- ) molecular modes isweak (cf. [Bae93]).

2.2. Monte Carlo Simulation

"Monte Carlo Simulation can be considered as an idealized experiment carried out on sam-ples of arbitrarily adjustable degree of disorder and devoid of any accidental complexity. Itallows to determine which level of sophistication is required to reproduce the properties of areal world sample and, by comparison with theory, to check the validity of approximationsin analytical treatments that are based on the same physical principles." [Bae93]

2.3. Hopping Rate Models

The hopping process, which is mentioned in section 2.1, was suggested by Conwell and Mott (cf.[ME60]). Coropceanu et al. cite two main models for the hopping rates: the model of Miller-Abrahamsand the model of Marcus (cf. [CCdSF+07]).

In the Miller-Abrahams formalism, the hopping-rate (jump-rate) Vij from site i to site j is expressed as

Vij = v0 × e−2γa×∆Ea ×

{eej−eiKT , ej > ei

1, ej < ei.(2.1)

v0 denotes the attempt hopping frequency, Rij is the distance between sites i and j, γ is the overlap factorand ei resp. ej are the site energies. The last term in 2.1 is a Boltzmann factor for a jump upward inenergy and is equal to 1 for a jump downward in energy (cf. [ME60]). "Notice that there is no otheractivation energy except the difference in electronic site energies a carrier has to overcome in order tohop." [Bae93]

4

2.4. Equations

The second model for hopping rate is the Marcus expression for semi-classical electron-transfer ratesand is expressed as

Vij =t2

h× [

π

kBTλreorg]

12 × e−

(λreorg+ej−ei)2

4λreorgkBT (cf.[Mar93]). (2.2)

The Miller-Abrahams equation is valid for weak electron-phonon (vibration) coupling and low tempera-tures. The Marcus expression is valid for large electron-phonon (vibration) couplings and high tempera-tures (cf. [CCdSF+07]).

Monte Carlo (MC) simulations of the hopping transport of single carriers in a Gaussian Density-of-States (DOS) were performed by Bässler and co-workers in 1993 (cf. [Bae93]). "For a long time, it hasbeen known that the mobility in disordered inorganic and organic materials is not only a function of thetemperature and electric field but also of the carrier density. [...] The independent-carrier assumption,made in the MC simulations by Bässler and co-workers, is then invalid and the mobility increases withincreasing carrier density, as the occupation of the deepest states by a certain fraction of the carriersreduces the effect of these states as trapping centers." [vdHUR+09]

2.4. Equations

The reduced energetic disorder parameter σ is

σ =σ

KT(cf.[Bae93]). (2.3)

The equilibration energy < ε∞ > of a carrier at zero electric field can be calculated analytically from

< ε∞ >= −σ × σ̂ = − σ2

KT(cf.[Bae93]). (2.4)

The average distance a in 3.2 is reducible in 3.1.

The probability P that a random event X occurs is defined as

P (X) =g

m. (2.5)

g is the number of favorable events and m the number of all possible events.

5

3. Numerical Model

The for the presented MC simulator used hopping model is of Miller-Abrahams type (cf. eq. (2.1)).The site energies and the overlap parameter are subject to a Gaussian distribution. All formulas andalgorithms that are listed in this chapter are relevant for the MC simulations presented in chapter 5.

3.1. Hopping Probability

The probability that a carrier jumps from a site i to any site j within a cube consisting of 7 x 7 x 7 sites is

Pij =Vij∑i 6=j Vij

(cf.[Bae93]). (3.1)

7

3. Numerical Model

3.1.1. getPossibleDestCells()

The following code shall explain the algorithm how the destination cube consisting out of 73 LatticeCellsis obtained:

Algorithm 1 Calculate possbile destination cell cube

LatticeCell[][][]destCube→ newLatticeCell[7][7][7]for i = −3→ destCube.length− 3 do

for j = −3→ destCube[i+ 3].length− 3 dofor k = −3→ destCube[i+ 3][j + 3].length− 3 dov1if pos[0] + i < 0 thenv1← LATTICE.length+ (pos[0] + i)

else if pos[0] + i > LATTICE.length− 1 thenv1← pos[0] + i− LATTICE.length

elsev1← pos[0] + i

end ifv2if pos[1] + j < 0 thenv2← LATTICE.length+ (pos[1] + j)

else if pos[1] + j > LATTICE.length− 1 thenv2← pos[1] + j − LATTICE.length

elsev2← pos[1] + j

end ifv3if pos[1] + k < 0 thenv3← LATTICE.length+ (pos[2] + k)

else if pos[2] + k > LATTICE.length− 1 thenv3← pos[2] + k − LATTICE.length

elsev3← pos[2] + k

end ifdestCube[i+ 3][j + 3][k + 3] = this.LATTICE[v1][v2][v3]

end forend for

end forreturn destCube

8

3.1. Hopping Probability

3.1.2. Jumprate

The jump-rate Vij is described as

Vij = v0 × e−2γa×∆Ea ×

{e−

ej−eiKT , ej > ei

1, ej < ei.(3.2)

Equation (3.2) is described of the Miller-Abrahams type (cf. sec. 2.3. Vij is the product of a prefactorv0, an electronic wave function overlap factor 2γa and a Boltzmann factor for jumps upward in energy.a is the average lattice distance. Bässler chooses a Gaussian for the probability density of the overlapfactor that is a complicated and strong function of the mutual molecular angles (cf. [Bae93]).

The following code shall explain the algorithm how the calculation of the real distance ∆ E is realized:

Algorithm 2 Calculate real distance ∆ E

ArrayList < Integer > l← newArrayList()int[]spos← srcCell.POSint[]dpos← destCell.POSfor i = 0→ spos.length do

if spos[i] 6= dpos[i] thenl.add(i)

end ifend forInteger[]differingDimensions← newInteger[0]differingDimensions← l.toArray(differingDimensions)intnumberOfDifferingDimensions← differingDimensions.lengthdoublerealDistance← 0if numberOfDifferingDimensions == 3 thenrealDistance ← Math.sqrt(Math.pow((dpos[differingDimensions[0]] −spos[differingDimensions[0]]), 2) + Math.pow((dpos[differingDimensions[1]] −spos[differingDimensions[1]]), 2) + Math.pow((dpos[differingDimensions[2]] −spos[differingDimensions[2]]), 2))

else if numberOfDifferingDimensions == 2 thenrealDistance ← Math.sqrt(Math.pow((dpos[differingDimensions[0]] −spos[differingDimensions[0]]), 2) + Math.pow((dpos[differingDimensions[1]] −spos[differingDimensions[1]]), 2))

else if numberOfDifferingDimensions == 1 thenrealDistance ← Math.abs((dpos[differingDimensions[0]] −spos[differingDimensions[0]]))

elsereturn realDistance

end if

9

3. Numerical Model

3.1.3. CumHist

The first approach to determine a destination cell out of 73 possible destination cells consists out of threesteps. The first is to loop over a 3-dimensional array for summing up the total probability. In the nextstep, a new array anew of size totalprobability was defined of type LatticeCell and initialized withcorresponding lattice cells. Finally, a uni-formal distributed random variable was taken over anew thatdetermined the destination cell.

A faster construct to determine a destination cell was suggested by Mr. Prof. Dr. Thaler. The optimizedalgorithm works with one loop less in comparison to the first approach. The following code shall explainthe optimized algorithm:

Algorithm 3 Calculate destination cell

longcumHist[]→ newlong[7 ∗ 7 ∗ 7 + 1]LatticeCell[]ltc→ newLatticeCell[7 ∗ 7 ∗ 7]cumHist[0]→ 0intidx→ 1for i = 0→ jumpProbabilities.length do

for j = 0→ jumpProbabilities[i].length dofor i = 0→ jumpProbabilities[i][j].length doflg ← jumpProbabilities[i][j][k]if flg > 0 thencumHist[idx]← cumHist[idx− 1] + flgltc[idx− 1]← possibleDestCells[i][j][k]idx← idx+ 1

end ifend for

end forend forintindexrandom← (int)(Math.random() ∗ cumHist[idx− 1])for c = 1→ idx do

if indexrandom < cumHist[c] thenbreak

end ifend forchoosenLatticeCell← ltc[c− 1]

3.1.4. Visit Probability

The probability P that a lattice cell L was visited by E electrons during a simulation with N time-stepsand c visit-counts in L is defined as

P =c

E ×N. (3.3)

10

3.2. Influence of Voltage

3.2. Influence of Voltage

The voltage has influence on the jump-rate Vij with an additional term in the first case of equation (3.2)and is described as

Vij = e−2γa×∆Ea ×

e−(ej−ei)+(V

d×a×q)

KT , ej > ei1, ej < ei.

(3.4)

V is the applied voltage, d the distance between anode and cathode, a the average distance between two

sites and q the elementary charge of an electron. The termVd×a×qKT is unit-less, since [V]= Volt, [d]=

Meter, [a]= Meter, [q]= Coulomb and [KT]= Joule= eV.

Hence downward jumps are not accelerated by the electric field. They are assumed to always occur. Thisassumes that there is always a channel to accept the energy difference (cf. [CCdSF+07]).

3.3. Current

The current I with unit A in disordered organic photoconductors is described as

I =n

T × t. (3.5)

n is the number of electrons that reached the cathode among T time-steps. t stands for the time inseconds which an electron needed for calculating its new position, but is not clear defined by Bässler (cf.[Bae93]). He described t as

tij = xci × (∑l 6=i

Vil)−1, (3.6)

where xci is taken from an exponential distribution of random numbers. Which exponential distributionis meant is not named. We took for t an average time of 0.068489 seconds that was measured with EPTPEclipse Test & Performance Tools Platform for profiling (cf. sec. 5.5).

3.3.1. Current Density

The current density ~J , whose unit is A/cm2, is an indicator for current capability and is defined as

~J =I

A. (3.7)

I is the current (cf. 3.3). A is defined as cross-sectional area of the photoconductors.

11

3. Numerical Model

3.3.2. Conductivity

The conductivity σ indicates the ability of a material to conduct current and is defined as

σ =I

U× l

A. (3.8)

I is the current,U the applied voltage, l the distance between anode and cathode andA the cross-sectionalarea of the photoconductors. Therefore the unit of σ is A

V ×1m .

Semiconductors can have a conductivity between 10−6 and 102 Sm . Materials under 10−6 Sm are called

insulators and above 102 Sm metals (cf. [NK00]).

12

4. Monte Carlo Simulator

4.1. Imitation

In the following section, the Monte Carlo Simulator of Bässler is rebuilt (cf. [Bae93]). In section 4.2,the imitated and validated simulator is modified to simulate filaments, current density and conductivity.First the java method nextGaussian() of the java class Random was validated (cf. subsec. 5.1.1). In thenext step, the equation for calculating the jump probability that Bässler used in his MC simulation hadto be figured out (cf. subsec. 5.1.2). To be more precise, in subsection 5.1.2 it is tried to answer thequestion if a jump-rate Vij is calculated for the case i= j. The next problem was that there is no cleardeclaration for the overlap parameter in Bässler’s paper except the information that he used not only forthe energetic levels of the lattice-cells but also for the overlap parameter a Gaussian distribution. Hewrites the following about the overlap parameter: "The variance of Γij is Σ which is defined as 2

√σΓ

and is at constant 2γa= 10 during simulation." [Bae93]. "The product of wave-function overlap factor 2γand lattice constant a had been chosen to be 2γa 10." [BSAP82] In subsection 5.1.3, a solution for thequestion which mean µ and standard devation σ for the Gaussian distribution of the overlap parameter2γa have to be chosen is motivated with comparisons between figures of Bässler and emulated figures.

There are four classes we used in our MC simulator: Controller, LatticeCell, Electron and ElectronMover.The classes were designed with the principle of high cohesion and low coupling. A LatticeCell storesan electronic niveau with the unit eV, its position and an overlap parameter. Both the electronic niveauand the overlap parameter are Gaussian distributed. An Electron keeps track of its jump-history which isstored in a 1-dimensional array of type LatticeCell. The Controller stores all constants, a 3-dimensionalarray of type LatticeCell whose references are passed over to each Electron during its initialization (cf.fig. 4.1). For validation, we mostly calculated with 20 Electron-objects that were simulated in a for-loopwith 7 loops (cf. [Bae93] and fig. 4.2). As soon as the main-method is called, a Controller-object isinstantiated. In its constructor it initializes the LatticeCells, the Electrons and the ElectronMovers. Thenthe Controller-object starts each ElectronMover as Thread (cf. fig. 4.2). If there are 20 cores available,the Controller instantiates 20 ElectronMovers and allocats to each of them 1 Electron in an array. Wemade the experience, that the array has the advantage of scalability in comparison to only a reference toan Electron and the advantage of performance in comparison to a List. 703 LatticeCells are initialized (cf.[Bae93]). We decided to design the lattice consisting out of these LatticeCells border-less. That meansthat an Electron can jump from one corner to its diagonally opposite corner. In algorithm 1, the calcu-lation of the possible destination cells is written down. As soon as an ElectronMover is started, it callsTMAX − 1 times the method calcNewPosition() in an iteration-loop over the Electron-array. The Elec-tron runs step-by-step the methods getPossibleDestCells(), calcJumpRates(), calcJumpProbs(), calcDest-Cell() and saveElectronicNiveau(). The algorithm of calculating the real distance in calcJumpRates() isdocumented in algorithm 2. When the loop has reached its end, a new loop over the Electron-array isstarted where the ElectronMover calls with jumpHistory as parameter a synchronized method in Con-troller to save the results. If there is more than one simulation, new Electrons, new LatticeCells and newElectronMovers are initialized for running the next simulation-iteration (cf. fig. 4.2). After the last Elec-tron has saved its results, Controller calls a method to write out the collected results to a m-file whichis processed in Matlab for visualization of the obtained dataset later on 4.3. The Electron-objects are

13

4. Monte Carlo Simulator

passive and the ElectronMover-objects active. The reason for designing the Electrons as passive objectsis explained in section 5.4. There is only a concurrency problem in saving the results of an Electron inController. Carrier-carrier interaction is excluded.

Figure 4.1.: Domain Model of MC simulator

Figure 4.2.: Sequence Diagram that describes the interaction between a Controller-object and an ElectronMover-object

14

4.1. Imitation

Figure 4.3.: Sequence Diagram that describes how an Electron calculates its new position and when anElectronMover-Thread is terminated

15

4. Monte Carlo Simulator

4.2. Extensions

As mentioned in section 2.3, the mobility in disordered organic materials is also a function of the carrierdensity. Thus the independent-carrier assumption which is made by Bässler and used in section 4.1 is in-valid. Thus if cell cj is already possessing an electron, the jumprate Vij is 0 in section 4.2. The presenteddesign in section 4.1 must be changed to guarantee a correct concurrency of the ElectronMover-Threads.Consider the following scenario: There are two running ElectronMovers with each three Electrons han-dling in a lattice of size 100x100x100. ElectronMover1 starts to calculate a new position for Electron1by calling getPossibleDestCells() in Electron1. Afterwards calcJumprate() is called with a 7x7x7 cubeof possible destination cells. To the same time, ElectronMover2 has called calcJumprate() in Electron4with a 7x7x7 cube of possible destination cells that has an intersection with the destination cube of Elec-tron1. The consequence of this constellation is that there is the possibility that Electron1 and Electron4jump into the same destination cell. The best solution to solve this problem would be to synchronize allLatticeCell-objects that represent possible destination cells for an electron. That means there have to belocks on these LatticeCell-objects for the time calcJumpRate() starts till calcDestCell() ends. We decidedto implement an approach that is more ressource- and computing-time-friendly. The benefit out of the"clean" synchronization is unbalanced to the slower computing time. Thus we added two conditions incalcJumpRate() and calcDestCell(). Namely the first condition checks if cell cj is already possessing anelectron what concludes in Vij= 0. The second added condition checks if the chosen destination cell ispossessing an electron what concludes in a restart.

4.2.1. Influence of Voltage

In this subsection, the simulator presented in section 4.1 is extended by voltage. The modification ofjumprate Vij is described in section 3.2. The simulation results with a voltage-configuration can befound in section 5.2.

4.2.2. Current

The in section 4.1 presented MC simulator was modified in that way that current density and conductivitycould be simulated (cf. sec. 5.3).

All quantities were obtained by one simulation-session. Two lists were used two store the applied voltageand the correlating current after a simulation-iteration. Each simulation-iteration increased the voltage by0.25 V. It was only simulated the current I in Java. The current density and conductivity were calculatedin Matlab (cf. appendix A.1). The electrons are starting at P(x,y,z)= (x,y,100) where x and y are uniformalrandom generated. If an electron reached P(x,y,z)= (x,y,1), its calcNewPostion()-method is not calledanymore by its allocated ElectronMover (cf. fig. 4.3). By considering equation (3.4), Vij has a negativevoltage term if zcj > zci . If zcj < zci , the voltage term is positive - else 0 (zcj == zci).

16

5. Results

5.1. Validation of MC Simulator with Bässler

Bässler has used a model with a carrier-carrier-independence [Bae93]. Hence the case that a destinationcell can already possess an electron is not considered in this section. To be more precise, if cell cjpossesses an electron, the jump-rate Vij is calculated as described in equation (3.2).

17

5. Results

5.1.1. Validation of the Java method nextGaussian()

The method nextGaussian() of the java class Random exceeds the expectations of a standard-normal-distribued random variable (cf. [Wik11a]). For mean µ= 0 and standard deviation σ= 1, the inflectionpoints are at ±1 and the density ϕ(X) is symmetric about the y-axis (cf. fig. 5.1). If mean µ= 0 andstandard deviation σ= 2, each random variable of N(0,1) is multiplied by two what meets the expectations(cf. fig. 5.2).

Figure 5.1.: nextGaussian()*1+0

Figure 5.2.: nextGaussian()*2+0

18

5.1. Validation of MC Simulator with Bässler

5.1.2. Validation of Equation for Jump Probability

There are two approaches for calculating the jump probability of an electron. The first approach is thatVij with i= j is calculated and subtracted of

∑Vij . The second approach represents the assumption that

Vij with i= j and thus Pij with i= j is always 0. Looking at the equation (3.1) for the hopping probabilityPij, the sum of all possible jump-rates excludes the case Vij with i= j. Since equation (2.5) holds alsofor the calculation of the probability, Pij with i= j is not directly calculated what supports approach 2.This does not mean that an electron can not stay in its source cell ci. If

∑i 6=j Vij= 0, the carrier keeps

staying after calcNewPosition() (cf. fig. fig:sequence-diagram2).

The following figures 5.3 and 5.4 were simulated 7 times with 20 electrons, with an overlap parameter2γa= N(10,2

√10), 104 time steps and an energetic disorder σ= N(0,0.075). They shall illustrate the

difference between approach 1 and approach 2. While a constant equilibration energy level was reachedafter approximately 102 time steps in figure 5.4, there was a decreasing energetic noise over the holemeasured time in figure 5.3.

Figure 5.3.: Approach 1, Pij with i= j is calculated

19

5. Results

Figure 5.4.: Approach 2, Pij with i= j is set to 0

A simple example shall briefly demonstrate the importance of the difference between the two describedapproaches for calculating the jump probability Pij . A lattice is given of size 70x70x70 and a cubic area-part of size 7x7x7 wich represents possible destination cells. Only the source cell ci and a neighbour cellcn have a jump-rate V > 0, to be more concrete VCi = 5 and VCn = 2. In the case like in figure 5.3,where Pij with i= j is not set to 0, Piiis5

2 . In the other case Pii is 0. In conclusion it can be said that thereis a longer decay till an equilibration energy level is reached in the case of not setting Pij with i= j to 0in equation (3.1) what approach 1 praises. But this approach conflicts more with the results of Bässlerwhere an equilibration energy level is earlier reached than t= 104 (cf. fig. 5.6). Another argument forapproach 2 is the fact, that Bässler does not take the case ei= ej in equation (3.2) into account. Thus weuse the second approach.

20

5.1. Validation of MC Simulator with Bässler

5.1.3. Validation of the MC Simulator

To find out the overlap parameter that Bässler used in his simulations, figure 5.9 and 5.6 were used forvalidation. In the following table, an overview of Bässler’s and the simulated results for the equilibrationenergy level is provided. This comparison in combination with figure 5.5 shall support the set over-lap parameter 2γa= N(10, 2

√10, the approach 2 for calculating the jump probability Pij discussed in

subsection 5.1.2 and the design of the MC simulator presented in section 4.1.

σ σ̂ < ε∞ > Bässler < ε∞ > acc. to 4.1 < ε∞ > analytical0.025 1 -1 -0.05 -10.05 2 -2, -4 -1.2 -40.0625 2.5 -2.5 -1.6 -6.250.075 3 -3 -2.1 -90.0875 3.5 -3.5 -2.6 -12.250.1 4 N/A -3.4 -16

Table 5.1.: Overview of standard deviation σ of energetic level, computational standard deviation σ̂ of energeticlevel that is unit-less, the equilibration energy level < ε∞ > of Bässler, the equilibration energy level< ε∞ > calculated according to 4.1 and the analytical equilibration energy level < ε∞ > which iscalculated with equation (2.4)

21

5. Results

Figure 5.5.: Temporal decay of mean energy with varying σ̂, 104 time steps, 2γa= N(10, 2√

10) and Pij =0fori = j

22

5.1. Validation of MC Simulator with Bässler

Figure 5.6.: Temporal decay of mean energy with varying σ̂ (cf. fig. 3 in [Bae93])

23

5. Results

(a) Time step 100 (b) Time step 101

(c) Time step 102 (d) Time step 103

Figure 5.7.: Distribution of energetic levels that were obtained by 14’000 electrons at time step 1, 10, 100 and1000. The simulation was performed with σ= 0.1 and 2γa= N(10, 2

√10. For the dataset, a number

cruncher with 20 cores calculated 12 minutes and the m-file had a size of 108 MB

24

5.1. Validation of MC Simulator with Bässler

Figure 5.8.: Temporal evolution of the distribution of carrier energies in a Gaussian DOS of width σ̂= 4 over timesteps 1 to 97, an overlap parameter 2γa= N(10,2

√10) and 700 simulations with 20 electrons per

simulation. The data processing took in Matlab on an AMD Athlon 64 X2 Dual Core Processor 4hours. The same dataset like in figures 5.7 was used in the plot

Figure 5.9.: Temporal evolution of the distribution of carrier energies in a Gaussian DOS of width σ̂= 2 (cf. fig. 2in [Bae93])

If figure 5.5 is compared with figure 5.6, it first stands out that the emulated results have an energeticnoise between time steps 101 and 102. That this characteristic is valid is supported by the temporalevolution of the distribution of carrier energies which is demonstrated in figure 5.9. There is a slightlyasymmetric distribution of the carriers between time-steps 101 till 102 what can conduct to an energeticnoise. We are not sure that no lines were drawn through the evolution plots in figure 5.6. In all emulatedevolutions, a dynamic equilibrium is established what meets the expectations. In addition, we noticedthat there were two energetic levels in which the equilibrium established for σ= 0.0875 eV and σ= 0.1

25

5. Results

eV between two energetic levels. This explains the thicker lines for these standard deviations in figure5.5. We can only explain the differences in the equilibrium levels between figure 5.5 and figure 5.6respectively in table 5.1 by claiming a mess in the documented σ values of Bässler - f.i. < ε∞ > /KTfor σ̂= 2 is -4 in figure 5.9 and -2 in figure 5.6. By comparing figure 5.8 with figure 5.9, we noticea drifting of the mean to negative energetic levels in both figures over time. This can also be seen infigure 5.7. The next shared characteristic is a steeper rise on the site to positive energetic levels thanon the site to negative energetic levels between time steps 100 + 1 and 102 (cf. also fig. 5.7(b) andfig. 5.7(c)). In conclusion it can be said that the overlap paraemter is a very important factor for theevolution of the distribution of carrier energies. F.i. setting the mean µ of the Gaussian distribution forthe overlap parameter to 0, the energetic noise increases over time. Finally, the configuration with 2γa=N(10, 2

√10) and Pij = 0fori = j delivered the most matching results.

26

5.2. Filaments

5.2. Filaments

In this section, the obtained results of the simulator extended by voltage are shown. In addition, thesimulator was optimized that a jump-rate is 0 if the destination cell is possessing an electron (cf. subsec.4.2.1).

"A carrier visits typically some 104 to 105 sites on its way across either a simulation or a real worldsample." [Bae93] Thus we decided to simulate filaments over 104 time steps what has shown most ex-pressive effects. "It is already well known that the percolative nature of the transport in a disordered or-ganic semiconductor leads to a strongly filamentary structure of the current along the percolation paths."[vdHUR+09] Neither figure 5.10 nor figure 5.11 show this strongly filamentary structure. The currentover voltage correlation shown in figure 5.15 may diminish the deviation between figure 5.11 and figure5.14. The simulation of one figure took only 3 seconds. The long lines indicate a jump from one site tothe opposite site in the lattice. What can be said is the fact that figure 5.10 shows an area around x=30, y=60 and z= 70 that was often visited by the simulated electron over time. In figure 5.11, an electron-pathhas emerged that starts at the top of the lattice and ends on the bottom. The top represents the anodeand the bottom the cathode. Thus we have a positive voltage term in equation (3.4). In figure 5.12 andfigure 5.13, we see two electrons concurrently hopping. The long lines indicate either a jump from onesite to the opposite site in the lattice or the change from the end of the first electron-path to the start ofthe second electron-path. In figure 5.12, Electron1 starts at P(x,y,z)= (3,66,91) and ends at P(x,y,z)=(50,75,48) while Electron2 starts at P(x,y,z)= (84,32,53) and ends at P(x,y,z)= (31,12,93). The colorbarshows the mapping from color to visit-probability (cf. equation (3.3)).In figure 5.13, Electron1 startsat P(x,y,z)= (67,82,69) and ends at P(x,y,z)= (53,84,41) while Electron2 starts at P(x,y,z)= (93,62,99)and ends at P(x,y,z)= (87,18,79). The allocation of the 2-dimensional array of type LatticeCell of size2x103 with the visited cells over time and the export of the dataset to the 1-dimensional arrays X, Y, Z,S and C took only 1.8 seconds (cf. appendix A.1). The values for S that are responsible for the plottedcircles and proportional to their representing cell-visit-probabilty were multiplied by 104 to achieve avalue-range between 0 and 200. We can figure out that the elctrons-paths have a higher z-range in figure5.13. We explain this finding with the applied voltage V= 10 V. Holst et al. have published a figure wherea strongly filamentary structure of the current along the percolation paths can be seen (cf. fig. 5.14).

27

5. Results

Figure 5.10.: Filament of 1 electron over 104 time-steps at applied voltage V= 0 V, an energetic noise=N(0, 0.05),an overlap parameter= N(10, 2

√10) and a lattice of size 100x100x100

Figure 5.11.: Filament of 1 electron over 104 time-steps with at applied voltage V= 10 V, an energetic noise=N(0, 0.05), an overlap parameter= N(10, 2

√10) and a lattice of size 100x100x100

28

5.2. Filaments

Figure 5.12.: Filaments of 2 electrons over 103 time-steps at applied voltage V= 0 V, an energetic noise=N(0, 0.025), an overlap parameter= N(10, 2

√10) and a lattice of size 100x100x100

Figure 5.13.: Filaments of 2 electrons over 103 time-steps with at applied voltage V= 10 V, an energetic noise=N(0, 0.025), an overlap parameter= N(10, 2

√10) and a lattice of size 100x100x100

29

5. Results

Figure 5.14.: Filament of Holst et al. at applied voltage V= 2 V (cf. fig. 3 in [vdHUR+09])

30

5.3. Current

5.3. Current

In figure 5.15, the current I is simulated with a varying applied voltage according to equation (3.5). Weexpect to obtain an exponential correlation between voltage and current (cf. [MC08]). Unfortunately, ourresult differs from this expectation. Thus more research would be needed to obtain harmonizing resultswith academia. In section 3.3, the formulas were listed to calculate current density and conductivity.

Figure 5.15.: Current versus different voltages over 104 time-steps with 140 electrons, an increasing applied volt-age from 0 V to 10 V with 0.25 V steps, an energetic noise= N(0, 0.025), an overlap parameter=N(10, 2

√10) and a lattice of size 100x100x100. Per voltage-steps was only one simulation per-

formed

5.4. Performance

First we implemented each Electron as concurrent running Thread. This approach turned out as a notoptimal solution because there was with a low number of cores a huge number of thread-switches thatcost computing time. Therefore we decided to go for a solution where each core runs an ElectronMover.The number of thread-switches decreased to a low number. If 140 electrons have to be simulated andtwo cores are available for calculation, hence two ElectronMovers are set-up with each 70 electrons tohandle.

It is not to neglect that there are also demon threds running beside the program threads. F.i. the Garbagecollection in Java normally consumes from 5% to 20% of total execution time of a properly functioningapplication (cf. [IBM11]). In section A.2, it is described how the memory size for the JVM can beincreased.

My first approach to visualize the simulated data in Matlab was to use the java-package Matlabcontrol (cf.[Mat11]). It is a Java API to interact with Matlab. To be concrete, you can shoot Matlab commands which

31

5. Results

are passed to a running Matlab-session within Java. There is a constraint that the maximum number ofinput-characters is 25’000. It turned out that writing out to a m-file and then continue processing theresults in Matlab is a very fast approach if the input-characters to Matlab are bigger than 25’000. Am-file does not store any ASCII-information. Thus its size is very small in comparsion to a text-file.Another advantage of a m-file is its import into Matlab: While importing a text-file with half a million64-bit floatingpoint numbers took a very long time, the import of a m-file with the same properties ofstored values was finished after some seconds. For exporting a dataset within Java to a m-file, we usedthe package Jmatio which can be found under http://sourceforge.net/projects/jmatio/files/.

5.5. Tips for Building a Monte Carlo Simulator

The following list shall contribute some pieces of advice that can be taken into account for building aMonte Carlo Simulator in Java or another computer language.

- put conditions that are checked more in if-elsif-statements first

- remove unnecessary attributes

- remove unnecessary mathematical operations, f.i. Math.abs(Math.pow(−2, 5)) toMath.pow(−2, 5)

- reduce mathematical expressions, f.i. Math.exp(a) ∗Math.exp(b) to Math.exp(a+ b)

- try to express functions as a constant, f.i.Math.pow(2, 31) to Integer.MAXVALUEor Math.pow(10, 5) to 1e5

- try to use ArrayList for unknown array dimension, f.i. for eruating number of differing dimensionsof three1-dimensional arrays of an Integer-type

- try to use the presented cumHist-construct for calculating a random variable out ofmore than one number-heaps (cf. alg. 3)

- optimize garbage collecting by increasing the memory-limit (cf. appendix A.2)

- use assert-construct for debugging and testing code (cf. appendix A.2)

- use JMatIO java package for writing out to a m-file (cf. http://sourceforge.net/projects/jmatio/files/)

- use EPTP Eclipse Test & Performance Tools Platform for profiling(cf. http://www.eclipse.org/tptp/index.html)

- use matlabcontrol package for passing matlab commands within Java to Matlab if number of input-charactersis not bigger than 25’000 (cf. http://code.google.com/p/matlabcontrol/)

32

6. Conclusion and Future Work

As future work it is recommendable to realise a simulator that runs on GPUs for obtaining faster com-puting time. Graphic cards provide much more processing cores to a lower price in comparison to CPUs.Thus a huge number of electrons could be realized as threads and run concurrently without an electron-mover. The disadvantage of an electron-mover is the absence of the possibility to simulate correctly thereal world.

For improvements of the simulator environment, Hyperion would be an interesting approach. It supportshigh-performance execution of Java programs by utilizing a Java-bytecode-to-C translator and by sup-porting parallel execution via the distribution of Java threads across the multiple processors of a cluster ofLinux machines. The Hyperion run-time system implements the Java memory model using an efficientcommunication substrate previously developed for Linux and FastEthernet switches (cf. [MMH98]).

Karl et al. suggest to include thermally dependency term in equation (3.2) for calculating Vij (cf. [N.04]).This could be a reasonable approach to obtain a simulated exponential correlation between voltage andcurrent (cf. sec. 5.3). Another approach to obtain this correlation - maybe in combination with athermally dependency term - would be to use equation (3.6) for t.

"The dream is to put electronic circuit properties into single molecules. Arrays of suchmolecules - possibly connected by conductive-polymer wires - on molecular scaffoldingswould form molecular wafers. One may speculate that reduced dimensions and the concomi-tant shrinkage in circuit size could increase the speed and dynamic memory of computers bya factor of 108. Such progress would correspond to forty years of computer technology de-velopment. Conductive polymers may become crucial for the building of such a molecularelectronics world." [NK00]

33

A. Appendix

A.1. Matlab Commands

All Matlab commands were executed on Matlab R2010b.

One curve in figure 5.5 was obtained by executing the following matalb commands:

% double_arr is a 2-dimensional array% columns represent time-steps and rows energetic levelx_all=1:1e4;x_all=uint32(x_all);y_all=mean(double_arr(1:140,x_all));semilogx(x_all,y_all);

% for drawing a line through the plotted pointsx1=power(10,0:0.1:4);x1=int32(x1);y1=mean(double_arr(1:140,x1));for i=2:length(x1)-1x_out(i-1)=(x1(i)+x1(i+1))/2;y_out(i-1)=mean(y_all(1,x1(i):x1(i+1)));endx_out=[1 x_out];y_out=[y_all(1) y_out];semilogx(x_out,y_out);

The figures in 5.7 were obtained by executing the following matalb commands:

% T is the time-step that shall be plotted% double_arr is a 2-dimensional array% columns represent time-steps and rows energetic levelhist(double_arr(1:14000,T),-20:0.3:15)

The figure 5.8 was obtained by executing the following matalb commands:

% double_arr is a 2-dimensional array% columns represent time-steps and rows energetic levelV=[];for i=1:97for j=1:14000V=[V; i double_arr(j,i)];end

35

A. Appendix

endhist3(V,[97 140])set(gcf,’renderer’,’opengl’);set(get(gca,’child’),’FaceColor’,’interp’,’CDataMode’,’auto’);

The figures in section 5.2 were obtained by executing the following matalb commands:

% X, Y, Z, S and C were loaded from a m-file% the MC simulator created the m-file with the JMatIO java packagecolordef black;plot3(X,Y,Z,’w’);xlabel(’x’);ylabel(’y’);zlabel(’z’);title(’V=1’); hold on;scatter3(X,Y,Z,S,C,’filled’,’o’);

The figures for current density and conductivity could have been obtained by executing the followingmatalb commands:

% V and I are loaded from a m-file% the MC simulator created the m-file with the JMatIO java package% 0.068489 is the needed time in seconds for one time-stepI2=I/0.068489;plot(V,I2) % adapted currentD=I2/(36*10^-12);plot(V,D) % currentDensityC=(I2./V)*(1/(6*10^-8));plot(V,C) % conductivity

A.2. Java Commands

// -Xms4000m allocates 4000 MB memory-space to the JVM heap// -ea enables assertionsjava -Xms4000m -ea -jar jarpackage.jar > cmdlineoutput.txtSystem.gc(); // calls the Garbage CollectorRuntime.getRuntime().maxMemory(); // returns max. memory sizeRuntime.getRuntime().availableProcessors();

36

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