Modeling of organic light emitting diodes: from molecular to device properties

Pascal Kordt,1, ∗ Jeroen J. M. van der Holst,1 Mustapha Al Helwi,2, 3 WolfgangKowalsky,3 Falk May,2 Alexander Badinski,4 Christian Lennartz,2 and Denis Andrienko1, †

1Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany2BASF SE, GVE/M-B009, 67056 Ludwigshafen, Germany

3IHF Institut, Technische Universitat Braunschweig, Brunswick, Germany4BASF SE, GVM/S-B009, 67056 Ludwigshafen, Germany

(Dated: December 1, 2014)

We review the progress in modeling of charge transport in disordered organic semiconductors onvarious length-scales, from atomistic to macroscopic. This includes evaluation of charge transferrates from first principles, parametrization of coarse-grained lattice and off-lattice models, andsolving the master and drift-diffusion equations. Special attention is paid to linking the length-scalesand improving the efficiency of the methods. All techniques are illustrated on an amorphous organicsemiconductor, DPBIC, a hole conductor and electron blocker used in state of the art organic lightemitting diodes (OLEDs). The outlined multiscale scheme can be used to predict OLED propertieswithout fitting parameters, starting from chemical structures of compounds.

CONTENTS

I. Introduction 1

II. Microscopic modeling 3A. Morphology 4B. Charge transfer 4C. Site energies 5D. Reorganization energy 6E. Electronic couplings 6

III. Coarse-grained models 6A. Morphology 7B. Site energies 7C. Neighbor list 8D. Electronic couplings 9E. Charge mobility and model validation 9

IV. Lattice models 9

V. Charge dynamics 10A. Master equation 10B. Kinetic Monte Carlo 11

VI. Device simulations 11A. Drift-diffusion modeling 11B. Lattice Monte Carlo 13C. Off-lattice Monte Carlo 14

VII. Experiment 14

VIII. Materials 14

IX. Conclusions and outlook 16

Acknowledgments 16

∗ kordt@mpip-mainz.mpg.de† denis.andrienko@mpip-mainz.mpg.de

References 16

I. INTRODUCTION

The discovery of electrical conductivity in conjugatedpolymers [1] has marked the birth of the field of or-ganic semiconductors, providing a new class of materialssuitable for manufacturing of field effect transistors [2],light emitting diodes [3], and photovoltaic cells [4]. Cost-efficiency, enhanced processability, and chemical tunabil-ity have immediately been recognized as distinct proper-ties of organic semiconductors [5–8]. Since then, organiclight emitting diodes (OLEDs) have been incorporatedinto active-matrix displays and nowadays are producedon an industrial scale [3, 9]. At the same time, whiteOLEDs are considered as promising candidates for light-ing applications [9, 10].In spite of this progress, there is still a need to im-

prove lifetimes and efficiencies, especially of blue phos-phorescent OLEDs, a task which requires designing newmaterials and optimizing device architectures. This is,of course, impossible without understanding elementaryprocesses occurring in a device. In a phosphorescentOLED, schematically shown in fig. 1, electrons and holesare injected into the transport layers, which ensures theirbalanced delivery to the emission layer (EML). To allowfor triplet-harvesting mediated by spin-orbit coupling,the EML consists of an organic semiconductor (host)doped by a transition-metal coordinated compound, theemitter (guest). The excitation of the emitter can beachieved either by an energy transfer process, i.e., by theformation of an exciton on a host molecule with sub-sequent energy transfer to the dopant, or by a directcharge transfer process. In the latter case, one of thecharge carriers is trapped on the emitter and attracts acharge of opposite sign, forming a neutral on-site exciton.Hence, an interplay between the hole, electron, and exci-ton mobilities, relative energy offsets, energetic disorder,and recombination rates determines the charge/exciton

2

N N

BCP

Si Si

O

BTDF

3

N

N

Ir

3

O

Ir

N

N

1.98

DPBIC

TBFMI

FIG. 1. Multilayered structure of an OLED with corresponding energy levels and chemical structures. Electrons/holes moveupwards/downwards in energy due to an applied voltage. Higher electron/hole energy levels correspond to smaller electron affini-ties/ionization potentials. In this OLED, DPBIC (Tris[(3-phenyl-1H-benzimidazol-1-yl-2(3H)-ylidene)-1,2-phenylene]Ir) is usedin the hole-conducting layer; TBFMI (Tris[(1,2-dibenzofurane-4-ylene)(3-methyl-1/1-imidazole-1-yl-2(3/1)-ylidene)]Ir(III)) isthe emitter (guest); BTDF (2,8-bis(triphenylsilyl)dibenzofurane) is the host material and 2,9-dimethyl-4,7-diphenyl-1,10-phenanthroline (bathocuproine, BCP) is the electron conductor.

profiles in a device and therefore its current–voltage–luminescence characteristics. In principle, it is possibleto reconstruct these profiles by using charge-transportand exciton-transport models and fitting the experimen-tal current–voltage curves [11–13]. This approach, how-ever, does not provide a link to the chemical compositionor morphological order and hence does not allow to pre-screen organic compounds prior to their synthesis.

In this situation, computer simulations can help bylinking device efficiency to the morphological order andmolecular structures. A frequently used ansatz incorpo-rates simulations of atomistic morphologies using clas-sical force-fields, evaluation of charge/exciton transferrates with first principles methods, and a subsequentanalysis of the solution of the master equation [14–23].This microscopic model is, however, computationally de-manding: a layer of a few hundrer nanometers thicknessrequires simulation boxes of ∼ 107 atoms, even if periodicboundary conditions are used in the directions perpen-dicular to the applied field.

To remedy the situation, extensions to this schemehave been proposed, as depicted in fig. 2. The moststraightforward solution is to parametrize a lattice modelby matching certain macroscopic properties of the atom-istic and lattice models. For example, if charge carriermobility is the property of interest, one can use the para-metric dependencies provided by the family of Gaussiandisorder models (GDM) [24–28]. Note that a similar ap-proach is often employed to interpret experimental data,i.e., to fit the measured mobility as a function of ap-plied field, charge carrier density, and temperature. Inour case simulated instead of experimentally measuredmobility values are used. This strategy, however, has

two significant drawbacks. First, it relies on approxima-tions made to construct the underlying models, whichmay not hold for the system of interest. Second, themodel parametrization is usually performed in stationaryconditions. It is therefore not immediately obvious thatthe same model can be used to describe non-equilibrium(transient) properties of the system.

An alternative approach is to construct an off-latticemodel by matching mesoscopic system properties, suchas distributions of molecular positions and orientations,electronic couplings, and site energies [29, 30]. Onceparametrized, this model will not require explicit simu-lations of atomistic morphologies and quantum-chemicalevaluations of rates and can therefore be used to simulatelarge systems.

In this review we illustrate both approaches on anamorphous phase of DPBIC, the chemical structure ofwhich is shown in fig. 1, a compound used in hole-conducting and electron-blocking layers in blue phospho-rescent OLEDs [21, 31]. In particular, we show how ma-terial properties such as density, radial distribution func-tions, ionization potentials and electron affinities, ener-getic disorder, charge mobility and eventually current–voltage characteristics can be extracted from simulations.

The paper is organized as follows: In section II we reca-pitulate the methods required to compute charge trans-fer rates in (microscopic) morphologies. Section III dealswith the construction and validation of coarse-grainedmodels. Section IV discusses how lattice models can beparametrized from an atomistic reference. In section Vwe review efficient kinetic Monte Carlo algorithms neededto simulate charge transport in systems with large ener-getic disorder. We then present device simulations and a

3

First principles

~10 n

m

~103 m

olec

ules

Microscopic models

Ground state geometries

Distributed multipoles and polarizabilities

Force-field parameters

Reorganization energies

Ionization potentials

Electron affinitiesMorphologies

Site energies

Electronic couplings

Rates

Energetic disorder

Lattice spacing

Off-lattice models

+

-

~ 100 nm

~105 – 10

6 molecules

Lattice models

~1 nm

Drift-diffusion

equations

Kinetic Monte Carlo

De

vic

e p

rop

ert

ies

Curr

ent-

voltage c

ha

racte

ristics

Charg

e d

ensity d

istr

ibutions

Curr

ent density d

istr

ibutions

Electrodes

Long-range

Coulomb

interactions

Injection barriers

Mirror images

Injection rates

Poisson equation

Forbidden events

Binary tree search

Aggregate Monte Carlo

Coarse-grained morphologies

Models for site energies

Models for electronic couplings

FIG. 2. Possible workflows of parameter-free OLED simulations: polarizable force-fields and electronic properties of isolatedmolecules obtained from first principles are used to generate amorphous morphologies and evaluate charge transfer rates insmall systems (microscopic models). Coarse-grained models are parametrized either by matching macroscopic observables,e.g., charge mobility, of the microscopic and coarse-grained (lattice) models. The resulting analytical expressions for mobilityare then used to solve drift-diffusion equations for the entire device, after incorporating long-range electrostatic effects andelectrodes. Alternatively, off-lattice models can be developed by matching distributions and correlations of site energies,electronic couplings, and positions of molecules. The master equations for this model can be solved using the kinetic MonteCarlo algorithm, yielding macroscopic characteristics of a device.

comparison to experiments in section VI. Finally, in sec-tion VIII we review computer simulation predicted mate-rial properties for a number of OLEDmaterials, includingBTDF, TBFMI, BCP, Alq3, and α-NPD.

II. MICROSCOPIC MODELING

We will begin with the computationally most demand-ing approach, which is later used as a reference for con-structing coarser charge transport models. For rela-tively small systems of circa 104 molecules it is possibleto parametrize the master equation for charge/excitontransport by combining first-principles and classical forcefield based methods [14–17]. First, for every molecule

type, the potential energy surface, reorganization en-ergy, ionization potential and electron affinity, as wellas distributed multipoles and polarizabilities are evalu-ated using first-principles calculations. These are usedto parametrize the classical (polarizable) force field andto simulate amorphous morphologies. The polarizableforce field is also employed to evaluate the electrostaticand induction interactions of a localized charge with theenvironment in a perturbative fashion [32]. Finally, elec-tronic coupling elements and rates are computed andused to formulate the master equation. The solution ofthe master equation yields charge carrier mobility andoccupation probabilities in the system.

This scheme has been applied to study transport ina number of organic semiconducting materials, such as

4

tris(8-hydroxyquinoline) aluminium (Alq3) [17, 33–35],fullerene (C60) and its derivatives [36, 37], dicyanovinyl-substituted oligothiophenes [18, 19, 38], poly(3-hexylthiophene) (P3HT) [22], and poly(bithiophene-alt-thienothiophene) (PBTTT) [23], to name a few.In the next sections we briefly recapitulate the methods

used to evaluate the ingredients of charge transfer rates.These will later be used to parametrize coarse-grainedmodels of an organic semiconductor.

A. Morphology

In general, predicting realistic molecular arrangementsof organic semiconductors is a challenging task: organicmaterials can be highly crystalline, form partially or-dered liquid crystalline, smectic, or lamellar mesophases,or be completely amorphous. Many are polymorphsand practically all have a fairly high density of defects,which has a large impact on their conductive proper-ties [18, 19]. X-ray scattering, tunneling electron mi-croscopy (TEM), and solid-state nuclear magnetic res-onance (NMR) spectroscopy are common experimentaltechniques which are used to probe molecular ordering oforganic materials [39–41]. Organic semiconductors em-ployed in OLEDs are usually amorphous, which simplifiesmodeling of their morphologies.In order to simulate large-scale morphologies, atom-

istic and coarse-grained models are often employed. Bothrely on force fields, i.e., parametrizations of the poten-tial energy surface of inter-molecular (non-bonded) andintra-molecular (bonded) interactions. While for bio-molecules a number of force fields exist and can be used“out of the box”, this is normally not the case for or-ganic semiconductors, which have extended π-conjugatedsystems. Such compounds require – at least partial –reparametrizations of existing force fields. For bondedinteractions this can be performed by scanning the cross-sections of the potential energy surface (e.g., dihedralpotentials) using first-principles methods [34, 42, 43], asillustrated for DPBIC in fig. 3, where two potential en-ergy surfaces are shown, one of the bending of a benzinering out of the ligand plane (γ) and the other one of itsrotation out of this plane (β).The situation with non-bonded interactions is more

complex: Coulomb and Lennard-Jones potentials canalso be parametrized by using first-principles calcula-tions [32]. This, however, requires distributed multi-poles and polarizabilities, as well as many-body vander Waals interactions, making force-evaluation compu-tationally demanding. As a compromise, most forcefields account for the induction contribution by adjust-ing partial charges and van der Waals parameters. Theseparametrizations are, however, state-point dependentand should be validated against experimental data.With the force field at hand, an amorphous mor-

phology of an OLED layer can be simulated by us-ing Monte Carlo or molecular dynamics (MD) tech-

niques [14, 17, 44, 45]. Several protocols were suggestedfor this purpose. One can, for example, anneal the systemabove the glass transition temperature, and then quenchto room temperature using the NPT ensemble [19]. Itis also possible to use Monte Carlo [33, 44] or molecu-lar dynamics simulations to gradually deposit the ma-terial [46]. Due to large deposition rates in moleculardynamics, simulated morphologies can be very far fromequilibrium, while the Monte Carlo technique does notcontrol the kinetics of the deposition process.As an illustration, an amorphous morphology of DP-

BIC, simulated using the annealing protocol, is shownin fig. 2. Here, the Berendsen barostat and thermo-stat [47] were used and the system was first equilibratedat 700K for 1 ns, then quenched to 300K and equili-brated for another 1.3 ns. The final density of the systemwas 1.29 g/cm3, which corresponds to a number densityof 8.50× 1020 cm−3.

B. Charge transfer

Charge transport in organic materials is intimatelylinked to the underlying material morphology: in crystalsand at low temperatures transport is coherent, chargesare delocalized, and consequently a band-like transportprevails [48–50]. At higher temperatures intramolecu-lar and intermolecular vibrations destroy the coherence,charges become localized by the dynamic disorder andtransport becomes diffusive [51–54]. In disordered mate-rials with weak electronic couplings between molecules,static disorder localizes charged states already at lowtemperatures and a thermally-activated hopping trans-port dominates [55]. Descriptions of transport in poly-meric systems are the most challenging: along chains itcan be diffusion-limited by dynamic disorder while be-tween the chains it is in the hopping limit [56, 57].Since we are dealing with amorphous materials, we

assume thermally-activated type of transport (this canbe verified experimentally by studying temperature- andfield-dependencies of mobility). The simplest expressionfor a charge transfer rate which takes into account ener-getic landscape, polaronic effects, and electronic couplingelements, is the semi-classical Marcus rate [58, 59]

ωij =2π

~

J2ij

√

4πλijkBTexp

[

− (∆Eij − λij)2

4λijkBT

]

. (1)

Here the reorganization energy, λij , reflects the effectof molecular rearrangement in response to the changeof the charge state, the electronic coupling element, Jij ,describes the strength of the electronic coupling betweentwo localized (diabatic) states, and Eij = Ei − Ej isthe driving force, where Ei is the site energy of moleculei [60].Note that the semiclassical Marcus rate is the high

temperature limit of a more general, quantum-classicalrate [61] and as such has a limited range of applicability.

5

0

5

10

-20 -15 -10 -5 0 5 10

ener

gy,kJ/m

ol

γ, deg

force fieldDFT

0

50

100

150

0 90 180 270 360

ener

gy,kJ/m

ol

β, deg

force fieldDFT

3

N

N

Ir

(b) (c)(a)

FIG. 3. Examples of the potential energy scans of the dihedral angles γ (b) and β (c), shown in (a). DFT calculations(B3LYP/6-311g(d,p)) and the force field cross-sections (after adjusting its parameters) of the potential energy surfaces areshown in (b) and (c).

There exist also other rate expressions, which go beyondthe harmonic approximation for the thermal bath [62–64]and have been reported to describe experimental data ina wide temperature and field range [65].The distribution of rates for DPBIC, evaluated using

the semi-classical rate expression, eq. (1), is shown infig. 5(e). This distribution is very broad, with rates vary-ing by several orders of magnitude, which is typical foramorphous systems. Large variations are a result of pro-nounced energetic disorder (see sec. II C), as well as of abroad distribution of electronic couplings (see sec. II E).

C. Site energies

Site energy differences, or driving forces, which enterthe charge transfer reaction rate, eq. (1), can be eval-uated perturbatively: first the ionization potential (IP,hole transfer) or electron affinity (EA, electron transfer)of an isolated molecule is calculated and then the inter-action with the environment (electrostatic and polariza-tion) is accounted for,

Ei = Einti + Eel

i + Epoli + qF · ri. (2)

Here F is the external electric field, ri is the center ofmass of a molecule, and q is its charge.The IP and EA can be evaluated using first principles

methods as the energy difference of a charged molecule(in its charged geometry) and a neutral molecule (in itsneutral geometry), i.e., Eint = UcC−UnN, where the first(lowercase) index denotes the charge state and the second(uppercase) index denotes the geometry.The interaction with the environment is often com-

puted on a classical level, using distributed multi-poles [66] and polarizabilities [67–69]. For each moleculei the electrostatic interaction, Eel

i , is evaluated for all

molecules in a sphere of a certain radius (cutoff). Notethat, even in amorphous materials, an ample cutoff is re-quired to converge the electrostatic energy. For molecularassemblies with a long-range orientational or positionalorder it is often not feasible to achieve the convergenceby increasing the cutoff radius [70] and the long-rangeCoulomb interaction energy should be evaluated usingthe Ewald summation technique [69–72].

The induction interaction energy, Epoli , can be eval-

uated using the Thole model [67, 68]. For organicmolecules the Thole parameters must be adjusted to re-produce the molecular polarizability tensor. Normally,the induction contribution converges for much smallercutoffs than the electrostatic one, though special careshould be taken in order to not have artificially induceddipoles at the surface of the cutoff sphere (e.g., by choos-ing a larger electrostatic cutoff radius than the polariza-tion one).

In order to evaluate site energies (their differences drivecharge transfer reactions) for an amorphous morphologyof DPBIC, we have used an adapted version of the Ewaldsummation method for long-range electrostatic interac-tions [67, 69–72] (i.e., the electrostatic contribution iscalculated without a cutoff) and the Thole model witha cutoff of 3 nm for the induction interaction. Atomiccharges, fitted to reproduce the molecular electrostaticpotential (shown in fig. 2), were used for electrostaticcalculations, while Thole parameters were adjusted toreproduce the molecular polarizability tensor of DPBIC.The energetic landscape, shown in fig. 4(b), is spatiallycorrelated, which is due to a long-range dipolar electro-static contribution [26, 73]. This can be seen qualita-tively in the cross-section of the potential energy land-scape, fig. reffig:atomistic(b), and quantified by the theautocorrelation function, κ(r), shown in fig. 4(c). Thedistribution of site energies is approximately Gaussianwith a width of σ = 0.176 eV. Note that the Gaussian

6

site

ener

gy,

eV

0

4

8

12

16

x, nm0

4

8

12

16

y, nm

5.2

5.3

5.4

5.5

5.2

5.3

5.4

0

0.2

0.4

0.6

1 2 3 4 5 6

auto

corr

elation,κ(r)

r, nm

4.7 5 5.3 5.6 5.9

site energy, eV

(a) (b) (c)

FIG. 4. (a) Amorphous morphology of DPBIC with a highlighted one-molecular-layer thick cross-section for which the energeticlandscape is shown in (b). (c) The spatial autocorrelation function of site energies, κ(r). The inset shows the distribution ofsite energies, which is approximately Gaussian with a mean of 5.28 eV and energetic disorder of 0.176 eV.

shape of the distribution is only expected if the change inthe molecular polarizability upon charging is small [21],which is indeed the case for DPBIC: one third of thetrace of the polarizability tensor is 108.5 A3 for a neutralmolecule and 127.1 A3 for a cation (B3LYP/TZVP).

D. Reorganization energy

The reorganization energy, λij , quantifies the energeticresponse of a molecule due to its geometric rearrange-ment upon charging or discharging. It has two con-tributions: the internal (intramolecular) part, i.e., theenergy difference due to the internal molecular degreesof freedom, and the outersphere (intermolecular) part,which is due to the relaxation of the environment [60].The internal part is normally evaluated as the energydifference of the involved charged/neutral states in acharged/neutral geometry using density functional calcu-lations in vacuum [74]. The outersphere contribution canbe computed using the dieletric response function [17],polarizable force fields [75], or hybdrid quantum mechan-ics/molecular mechanics methods [76].For DPBIC the calculated (B3LYP, 6-311g(d,p)) in-

ternal reorganization energies for hole transport are0.068 eV (discharging) and 0.067 eV (charging). Here weneglect the outersphere contribution, which becomes im-portant in strongly polarizable environments.

E. Electronic couplings

Evaluation of electronic coupling elements, Jij , be-tween the initial and the final states of a charge transfercomplex (molecular dimer) requires the knowledge of di-abatic states and the Hamiltonian of a dimer. Diabaticstates are often approximated by the highest occupiedmolecular orbital (HOMO) of monomers in case of holetransport and the lowest unoccupied molecular orbital

(LUMO) for electron transport [77–79] (“frozen core” ap-proximation) or constructed using the constrained den-sity functional approach [80]. In some cases the explicitevaluation of the dimer Hamiltonian can be avoided byemploying semi-empirical methods [74, 81–83].Electronic couplings are intimately connected to the

overlap of electronic orbitals participating in a chargetransfer reaction and are therefore very sensitive to rela-tive molecular positions and orientations. Together withthe energetic landscape, they constitute one of the mainfactors which influence charge transport in organic semi-conductors [18, 19, 84, 85]. Depending on the first-principles method and approximations used, computedelectronic coupling elements can easily vary by a factorof ten [86, 87].For the amorphous morphology of DPBIC, the distri-

bution of electronic couplings of molecular pairs at a cer-tain distance is approximately Gaussian with a mean de-caying exponentially for intermolecular separations largerthan 1.1 nm, see fig. 5(d). Below this distance elec-tronic couplings are constant (on average) and are ofthe order of 10−3 − 10−2 eV. Such a behavior has alsobeen observed for amorphous mesophases of Alq3 andDCV4T [29, 30].

III. COARSE-GRAINED MODELS

The microscopic model outlined in section II, in combi-nation with the charge dynamics simulations (see sec. V),provides a direct link between the molecular structureand macroscopic observables, e.g., charge mobility. It is,however, limited to relatively small systems of ca. 104

molecules, which is far from sizes required for studying,e.g., rare events during material degradation. Furthe-more, one can have pronounced finite size effects, espe-cially in materials with large energetic disorder [29, 35].In view of this, various coarse-grained models have

been developed [29, 88, 89]. The idea behind these

7

models is to reproduce the key morphological and elec-tronic properties of the underlying materials without ex-plicit simulations of atomistic morphologies, evaluationsof electronic couplings and site energies. The link tothe chemical composition is retained by an appropriateparametrization of mesoscopic-scale quantities, e.g., dis-tributions and correlations of site energies, electronic cou-plings, and pair distribution functions of molecular posi-tions, evaluated as described in sec. II. Coarse-grainedmodels allow to study charge dynamics in systems ofmore than 106 molecules. In what follows we describehow such models can be parametrized.

A. Morphology

The first step in constructing a coarse-grained modelis the prediction of molecular positions. In the gen-eral case, the algorithm should reproduce all (includingmany-body) correlation functions of molecular positionsin the reference morphology. For amorphous molecu-lar arrangements, however, the pair distribution functionis sufficient to adequately describe the molecular order-ing. Thus, one has to develop an algorithm that gener-ates point positions of a specified density and reproducesthe radial distribution function, g(r), of the microscopicmodel.To do this, one can use a Poisson process [29, 30]. This

process first positions N uniformly distributed points ina volume w, N ∼ Poi (ρw), where ρ is the point density.Unphysically close contacts are removed by assigning arandom radius Rn to every point n and by deleting thepoints whose sphere is contained in the sphere of anotherpoint m. The point density is then adjusted by intro-ducing more points in the voids. This procedure is re-peated iteratively until the desired density, ρ, is reached.The algorithm yields reasonably good approximations ofg(r) [29, 30].An exact match of the radial distribution function can

be achieved by using techniques employed in soft con-densed matter simulations, namely the iterative Boltz-mann inversion (IBI) [90, 91] or inverse Monte Carlo [92–94] methods. Both rely on the Henderson theorem [95],which states that for every g(r) there is a unique (up toan additive constant) interaction potential, U(r), suchthat sampling the system in the canonical ensemble re-produces this g(r). Hence, it suffices to construct analgorithm capable of finding U(r) from a given g(r). InIBI, this is done by simulating a coarse-grained systemwith a potential which is iteratively refined at the i-thiteration as

U i+1(r) = U i(r)− kBT ln

[

gi(r)

gref(r)

]

. (3)

A potential that reproduces the reference radial distribu-tion function is a fixed point of this relation. In practice,the second term is rescaled by a factor between 0 and 1to make the algorithm numerically stable. Inverse Monte

Carlo follows the same idea, except that the correctionto the potential is rigorously derived (in a linear approxi-mation) and contains cross-correlations of the number ofparticles at different separations r [92–94]. Both methodsare implemented in the VOTCA software package [96].Fig. 5(a) shows the radial distribution functions of

a small atomistic (reference) systems of 4000 DPBICmolecules and of a much larger system of 255083 in-teracting points generated using the potential invertedwith the IBI procedure. Note that the generation ofthe coarse-grained morphology is very fast, since thereare significantly less degrees of freedom involved as wellas the smooth coarse-grained interaction potential leadsto much faster (molecular) dynamics and hence shorterequilibration times. In total, 250000 steps (1 ns) wereused to equilibrate the system of 255083 interactingpoints in the NPT ensemble, starting from a lattice.

B. Site energies

Early models for charge transport simulations assumeda Gaussian distribution of width σ of spatially uncor-related [25] or correlated [26] site energies, where σ isreferred to as the energetic disorder of the system. Inthe spatially correlated model, all molecules are assumedto have randomly oriented dipole moments, p, of fixedmagnitude. The site energies, Ei, are then evaluated as

Ei = −∑

j 6=i

qpj (rj − ri)

ε |rj − ri|3, (4)

where q is the charge, ri is the position of moleculei and ε is the material’s relative permittivity. Thissum, evaluated using the Ewald summation method [71],has a spatial correlation function κ(r) ≡ 〈corr[Ei, Ej ]〉which decays approximately as 1/r, where the averageis taken over all molecules with center-of-mass distance|rj − ri| = r.Since the spatial correlation function can deviate from

the 1/r asymptotics [29] (e.g., when the molecular dipolealso varies [19, 29]), an algorithm that can reproduce any(physically relevant) spatial correlation function is re-quired. For Gaussian-distributed site energies this can beachieved by mixing in the energy values of the neighbors:for each site A, N independent Gaussian distributed ran-dom variables XA

i ∼ N (0, 1), 0 ≤ i ≤ N , are drawn. Theenergy of the molecule is then calculated as

EA =√aXA

0 +N∑

i=1

√

biℓAi

∑

B∈S(ri,A)

XBi , (5)

where S(ri, A) denotes a sphere of radius ri aroundmolecule A and ℓAi is the number of molecules within thissphere. The first term in eq. (5) corresponds to an un-correlated energetic landscape. By adjusting a, one canchange the energetic disorder, σ, without affecting thespatial correlation function κ(r). The sum over molecules

8

0

0.5

1

1.1 1.2 1.3 1.4 1.5 1.6 1.7

connec

tion

pro

bability

r, nm

atomisticcoarse-grained

-10

-9

-8

-7

-6

-5

-4

-3

0.8 1 1.2 1.4

m(r),σ(r)

r, nm

atomisticcoarse-grained

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6

corr

elation,κ(r)

r, nm

4.7 5 5.3 5.6 5.9

energy, eV

atomisticcoarse-grained

0

1

2

0 1 2 3 4 5

radia

ldis

trib

ution

funct

ion,g(r)

r, nm

atomisticcoarse-grained

0

0.1

0.2

-5 0 5 10 15

dis

trib

ution

log10(ωij · s)

atomisticcoarse-grained

10−10

10−9

10−8

10−7

10−6

1024 1025

µ,m

2/V

s

ρ, m−3

atomisticcoarse-grained9× 107 V/m7× 107 V/m5× 107 V/m3× 107 V/m

(a) (b) (c)

(d) (e) (f)

FIG. 5. Comparison of the atomistic (17×17×17 nm3) and coarse-grained (50×50×120 nm3) models. (a) Radial distributionfunction, g(r). (b) Probability of two sites to be connected (added to the neighbor list) as a function of their separation. (c)Spatial site energy autocorrelation function, κ(r); Inset: Site energy distribution. (d) Mean m and width σ of a distribution ofthe logarithm of electronic couplings, log

10(J2/ eV2), for molecules at a fixed separation r. (e) Rate distributions. (f) Mobility

as a function of hole density, plotted for four different electric fields.

within a certain cutoff ri adds spatial correlations. Byadjusting the weighting coefficients, bi, one can reproducethe reference correlation function. It is also possible todevise an iterative scheme which converges to a set ofrequired weights.

For DPBIC, the autocorrelation functions of the refer-ence (evaluated in the atomistic system) and the coarse-grained (constructed by iteratively refining the weightingcoefficients) systems are shown in fig. 5(c). The autocor-relation function is perfectly reproduced and, in fact, isfar from the 1/r dependence assumed in lattice models.

C. Neighbor list

Since electronic coupling elements decrease exponen-tially with intermolecular separation, charge transfer re-actions normally happen only between nearest molecu-lar neighbors. This physical argument is frequently usedto reduce the size of the list of molecular pairs (neigh-bor list) for which charge transfer rates are evaluated.

In atomistic simulations the neighbor list is constructedbased on the cutoff for the shortest distance between π-conjugated molecular fragments [17]. In a coarse-grainedmodel, where the molecular information is not available,the neighbor list can be constructed by analyzing theprobability in the reference system of two molecules ata certain separation to be connected. This distributionis shown in fig. 5(b). One can see that all moleculesseparated by less than a certain minimal distance, rmin,are connected while there are no connected pairs if sep-aration exceeds rmax. In the coarse-grained model eachpair of sites is then connected based on this probabilitydistribution.

For DPBIC, this approach reproduces well both theconnectivity of the reference system (see fig. 5(b)) and thecoordination number (11.60 for the atomistic and 11.44for the coarse-grained model).

9

D. Electronic couplings

In case of electronic coupling elements, one can ana-lyze the distance-dependent distribution P [Jij(r)], i.e.,the distribution of electronic couplings at a given sepa-ration r. For amorphous systems it has been found thatthe logarithm of the coupling element, log(J2

ij(r)/eV2),

is Gaussian distributed with a mean m(r) and standarddeviation σ(r) which depend on the molecular separa-tion [29, 30]. The exponential decay of Jij(r) with inter-molecular separation results in a linear decrease of themean of the distribution for r ≥ 1.1 nm, as shown infig. 5(d). To reproduce this behavior in a coarse-grainedmodel, the values for the logarithm of the coupling aredrawn from the normal distribution, log(Jij(r)

2/eV2) ∼N

(

m(r), σ2(r))

, where m(r) and σ(r) are chosen accord-ing to the atomistic reference.

E. Charge mobility and model validation

The coarse-grained model can be validated by compar-ing the charge transfer rates, as well as the mobility–fieldand mobility–charge density dependencies of both mod-els. The rate distributions are shown in fig. 5(e) andare in good agreement with each other. The mobilityversus charge density is shown in fig. 5(f) for differentelectric fields and also agree with each other, validatingthe coarse-grained model. Small deviations are due tostatistical fluctuations in relatively small systems stud-ied (4000 molecules).

IV. LATTICE MODELS

In sec. III we have described how to parameterize acoarse-grained model based on a microscopic referenceand use it to simulate relatively large (mesoscopic) sys-tems. Alternatively, one can adopt an approach often em-ployed to analyze experimental data, where analytical ex-pressions for charge mobility are parametrized by fittingthe IV characteristics of, e.g., a simple diode. The ex-pressions themselves are derived with the help of lattice-based models, such as the Gaussian disorder model [25],the extended Gaussian disorder model (EGDM), whichincludes the effect of electron density [27], the correlatedGaussian disorder model (CDM), which takes into ac-count spatial correlations of site energies [26], or the ex-tended correlated Gaussian disorder model (EGDM) [28].In principle, some parameters of these models can be

obtained by analyzing the microscopic model directly:the energetic disorder, σ, for example, can be calculatedfrom the site energy distribution, as described in sec. II C.The lattice constant, a, can be estimated by matchingthe densities of the atomistic and lattice models, i.e.,

a = (V/N)1/3

, where V is the volume and N the num-ber of molecules. In practice, however, it is better toobtain the model parameters by fitting the mobility to

a [nm] σ [eV] µ0(300K) [m2/Vs]

microscopic 1.06 0.176 3.4× 10−12

EGDM, simulation 1.67 0.134 2.1× 10−11

ECDM, simulation 0.44 0.211 1.8× 10−13

ECDM, experiment 0.74 0.121 1.2× 10−10

TABLE I. Lattice spacing a, energetic disorder σ, and mobil-ity at zero field and in the limit of zero charge density for themicroscopic, EGDM, and ECDM models. The last row cor-responds to the experimentally measured IV -characteristicsfitted to the ECDM model.

results of atomistic or stochastic simulations. The rea-son for this is that the analytical expressions provided bythe lattice models rely on certain approximations, whichmight not hold for the microscopic model. For exam-ple, spatial correlations of site energies, present in themicroscopic model, can be accounted for by an smallerenergetic disorder in the GDM model. By comparing thetwo approaches one can then assess whether or not a par-ticular lattice model is capable of reproducing materialproperties [29] or can be used in drift-diffusion simula-tions of devices [97].

As an example, a mobility versus charge density fit hasbeen performed for an amorphous DPBIC morphology.To correct for finite-size effects at small charge densi-ties, an extrapolation procedure was used to obtain non-dispersive mobilities [35] via a simulation of charge trans-port at high temperatures and a subsequent extraction ofthe room-temperature mobility from the known mobility-temperature dependence, as shown in fig. 6(a). The ex-trapolated values, together with mobilities at charge den-sities typical for an operating OLED, were used to fitthe EGDM and ECDM models, as shown in fig. 6(b).Notably, the model with energetic correlations yields aworse fit than the uncorrelated one, even though spatialsite energy correlations are present in the system. Thishas been observed before in simulations of amorphousdicyanovinyl-substituted quaterthiophene [29]. The rea-son is that the spatial correlation function of site energiesdoes not decay as 1/r at large r, as correlated disordermodels predict. This is also reflected in the fitting param-eters, which are summarized in table I: for both EGDMand ECDM models the estimated energetic disorder dif-fers from the microscopic value by about 0.04 eV. TheEGDM underestimates the disorder, while the ECDMoverestimates it. Similarly, the microscopic zero-field mo-bility, µ0, is in between the two fitted values. The absenceof correlations in the EGDM is compensated by an effec-tively increased lattice constant, while in the ECDM thelattice constant is much smaller than the density-basedestimate. Note that both models were originally devel-oped for energetic disorder up to 0.15 eV, while here theatomistic and ECDM values are higher than this.

10

10−13

10−12

10−11

10−10

10−9

0 1

µ,m

2/V

s

10−9

10−8

10−7

1024 1025

ρ, m−3

EGDMECDM10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

0 0.001 0.002 0.003

50000 1200 300T3/2µ(T

),K

3/2

m2/V

s

1/T , K−1

T , K

non

-dis

per

sive

regi

me

dis

per

sive

regi

me

9× 107 V/m7× 107 V/m5× 107 V/m3× 107 V/m

(b)(a)

FIG. 6. (a) Extrapolation of high-temperature mobility simulations to room temperatures. For a system of 4000 moleculestransport simulations with one carrier are non-dispersive only above 1200K. Hence, mobilities in the temperature range between1200K and 50000K (circles) are used for extrapolations to non-dispersive mobility values at 300K (crosses). (b) A fit to EGDM(solid lines) and ECDM (dashed lines) models. All four dependencies are fitted simultaneously.

V. CHARGE DYNAMICS

The outcome of both coarse-grained and atomisticmodels are center-of-mass positions of molecules andcharge transfer rates between them, i.e., a directed graph.To model charge dynamics in this network, one needs tosolve the corresponding master equation. While the effi-ciency of the solver is not important for relatively smallsystems, it becomes essential for modeling transport inlarge coarse-grained systems of millions of sites. Here wesummarize several approaches which help to make thesesolvers computationally efficient.

A. Master equation

Our target is the solution of the master equation, whichdescribes the time evolution of the system

dPi(t)

dt=

∑

j

[TijPi(t)− TjiPj(t)] , (6)

where Pi is the (unknown) probability of state i and Tij

is the transition probability from state i to j. For onecharge carrier, the state of the system is given by theindex of the charged molecule and the rates are equalto charge transfer rates, i.e., Pi = pi, where pi is theoccupation probability of a site i, and Tij = ωij .

This equation can be solved analytically in some spe-cial cases. In general, however, direct numerical differ-ential equation solvers are used [98]. If more than onecharge carrier is present in the system, the number ofsystem states increases rapidly (it is proportional to thebinomial coefficient) and ordinary differential equation(ODE) solvers quickly become impractical unless addi-tional approximations are employed. One of them is themean-field approximation, which allows to rewrite themaster equation (6) in terms of site-occupation probabil-ities, pi,

dpidt

=∑

j

[ωijpi (1− pj)− ωjipj (1− pi)] . (7)

One can see that the differential equation becomes non-linear, with the 1− p term accounting for the prohibiteddouble-occupancy of a site in a mean-field way.An alternative approach to solve the master equation

is to use kinetic Monte Carlo (KMC) algorithms [99, 100],i.e., to construct a continuous-time Markov process,where the probability of a certain event to occur dependsonly on its current state. We will discuss particular im-plementations of KMC in the next sections.Since KMC does not rely on the mean-field approxi-

mation, it is possible to evaluate the error introduced byneglecting higher-order correlations. It turns out that foran energetic disorder of σ/kBT = 2 the correction due topair correlations is smaller than 3% [101]. In more disor-

11

dered systems this error increases, as well as it is not clearhow higher order correlations might affect this result.Once the equation for pi is solved, the mobility tensor,

µ, can be evaluated as

µαβ =1

ρF 2V

∑

ij

ωijpi (1− pj) rij,αFβ , (8)

where ρ is the charge density, F is the external electricfield, V is the box volume, and rij = ri − rj . It is alsopossible to compute the mobility tensor from the KMCtrajectory

µαβ =1

N

N∑

i=1

〈vi,αFβ〉F 2

, (9)

where 〈vi〉 is the velocity of the i-te carrier, 〈· · · 〉 denotesthe average over the simulation time, and the sum is per-formed over N carriers. The mobility at zero externalfield can also be calculated from the diffusion tensor us-ing the generalized Einstein relation [102] (generalizationis required due to the Fermi-Dirac and not Boltzmannstatistics of the site-occupation probability)

qD∂ρ

∂η= µρ. (10)

Here Dαβ = 〈∆rα∆rβ〉 /2τ , where τ is the total sim-ulated time, ρ the charge density, and η the chemicalpotential. Note that the Einstein relation holds only fora steady state that fulfills the detailed balance condi-tion [103] and for small fields [104]; it cannot be appliedto inhomogeneous temperature distributions [105, 106] orunrelaxed (“hot”) carriers.

B. Kinetic Monte Carlo

The variable step size method (VSSM) [107, 108] is oneof the kinetic Monte Carlo solvers of the master equation.It is a hierarchical method, i.e., events can be groupedand VSSM can be used for both the groups and eventsin the group. In case of charge transport, for example,it is convenient to have two levels: first a charge is se-lected based on the escape rate, ωi =

∑

j ωij , and thenthe hopping destination is chosen. VSSM allows an ef-ficient treatment of forbidden events [108], without theneed to recalculate escape rates at every step. In chargetransport simulations, forbidden events are due to thesite single-occupancy constraint which leads to Fermi-Dirac statistics for the equilibrium distribution of siteoccupations, as shown in fig. 7(a) for a DPBIC morphol-ogy of 4000 molecules. As expected, both the Fermi leveland occupation probability increase with the increase ofcharge density. This is explained in fig. 7(b), where theGaussian density of states is filled up to the Fermi en-ergy, and the product of the Fermi-Dirac distribution andGaussian density of states, integrated up to the Fermi en-ergy, gives the charge density in the system.

Note that the standard implementation of the VSSMalgorithm would require about 108 KMC steps to reachthe steady state for charge flux in DPBIC, since car-riers can be trapped in aggregates of low-energy sites.This can be circumvented using an aggregate MonteCarlo (AMC) algorithm [88], i.e., by coarsening the statespace [109–113] into “super-states”. Efficient coarseningmethods that retain physically sensible trajectories arethe stochastic watershed algorithm [109, 114, 115], whichis an improved version of the watershed algorithm [116,117], and a graph-theoretic decomposition [118]. AMChas been tested on the amorphous phase of Alq3, whereit leads to about two orders of magnitude faster conver-gence [88].

KMC solvers with explicit long-range Coulomb inter-actions require rate updates at (practically) every KMCstep. Efficiency of such algorithms can be improved byusing a binary tree search and update algorithm [119]. Itupdates only those rates which are affected by a movingcarrier and has a logarithmic scaling with the number ofevents, whereas a linear search scales linearly. The bi-nary tree algorithm becomes more and more beneficialwith increasing number of charge carriers: while for 1%carrier density (in 4000 molecules) it is slower than aprimitive search, at 10% it is already faster by about afactor of 10.

VI. DEVICE SIMULATIONS

In order to simulate IV characteristics of an entiredevice, two additional ingredients must be incorporatedin the model: the effect of metallic electrodes, whichinject and collect charges, and the strongly inhomoge-neous distribution of charge density in the device. Thiscan be done on a macroscopic level, by complement-ing the drift-diffusion equations with boundary condi-tions and Gauss’s law for charge density, as describedin section VIA. An alternative approach is to extendthe Monte Carlo scheme by including the electrodes andthe Poisson equation solver for long-range electrostat-ics. This extension has already been implemented forlattices [97], see also sec. VIB, and is discussed for off-lattice models in section VIC.

To disentangle the contributions of different functionallayers of an OLED device here we will simulate a simplersingle-carrier , which is shown schematically in fig. 9(a).The experimental measurements (sec. VII) are also per-formed using this setup.

A. Drift-diffusion modeling

The drift-diffusion model assumes that local chargedensity, ρ(x), charge mobility, µ(x; ρ, F, T ), fieldstrength, F (x), and diffusion constant, D(x; ρ, F, T ), arechanging continuously. In one dimension, the corre-

12

0

0.2

0.4

0.6

0.8

1

4.7 4.8 4.9 5 5.1

pro

bability

energy, eV

n = 0.001n = 0.004n = 0.016

0

0.5

1

1.5

2

2.5

4.8 EF 〈E〉 5.4 5.6 5.8

pro

bab

ility

energy, eV

n = 0.1

Fermi Dirac distributionDOS (fit)

product

(a) (b)

FIG. 7. (a) Site-occupation probability as a function of site energy plotted for three charge densities. Points are Monte Carlosimulations in a DPBIC morphology of 4000 molecules. Solid lines are fits to the Fermi-Dirac distribution. The Fermi level(dashed lines) increases with increasing relative charge density n (charges per site). (b) Filling of the Gaussian density ofstates (DOS) for a number density of n = 0.1. The product of the Fermi-Dirac distribution and the Gaussian DOS yields theoccupation distribution (red curve). The area under this curve, evaluated up to the Fermi level is equal to the charge density.The width of the DOS matches the energetic disorder of DPBIC (0.176 eV) and is centered at 5.28 eV.

sponding drift-diffusion equation then reads

J = qρµ+ qD∂ρ

∂x. (11)

Here the first (drift) term describes transport ofholes/electrons in/against the direction of the electricfield. The second (diffusion) term is due to transportof charges against the gradient of charge density.The drift-diffusion equation is complemented by

Gauss’s law, ∂F (x)/∂x = ρ(x)/ε, where ε is the rela-tive permittivity. The dependence of mobility on chargedensity, electric field, and temperature can be obtainedfrom Monte Carlo simulations as described in section IV.µ and D are related via the generalized Einstein equa-tion [120].Boundary conditions fix the charge carrier density at

the electrodes, ρel, to the value given by a thermal equi-librium between the electrode and the organic layer,

ρel =

∫ ∞

−∞

g(E)

1 + exp[

E+∆kBT

]dE, (12)

where g(E) is the density of states.To correct for the presence of the image potential, an

equilibrium between the Fermi level of the injecting elec-trode and the maximum of the sum of the applied driv-ing bias, the local potential as obtained from the Poissonequation and the image potential is assumed. The low-ering of the injection barrier, ∆, between the Fermi levelof the electrode and of the organic site is then described

by [121]

∆′ = ∆− q

√

qFc

4πε. (13)

The field at the electrode, Fc, and the lowering of theinjection barrier, ∆′, are determined self-consistently viaan iterative procedure [122]. When the field at the elec-trode is directed into the electrode itself, i.e., Fc < 0, theinjection barrier is not lowered by the image potential.Before solving the drift-diffusion equation we have

parametrized the field and temperature dependenciesof the mobility, as described in section IV, i.e., fittedthe results of Monte Carlo simulations performed in anamorphous morphology of 4000 DPBIC molecules to theEGDM. Lattice spacing, energetic disorder, and zero-field mobility were used as fit parameters and are sum-marized in table I.The drift-diffusion equations were then solved for sev-

eral values of the injection barrier, ∆, and are shown infig. 8(a). Up to ∆ = 0.2 eV the IV curves are practicallyon top of each other. Only from 0.4 eV the injection bar-rier starts to affect the IV characteristics. One can alsosee that for larger values of ∆ (less effective injection) thecurrent density decreases by several orders of magnitude.Since in organic materials the transport of charge car-

riers is percolative and the current density in the devicehas a filamentary structure [123–125], it is not imme-diately clear how reliable drift-diffusion device modelsare [126]. This is addressed in the next section by com-paring the drift-diffusion solution to the results of latticesimulations.

13

10−4

10−3

10−2

10−1

100

101

102

103

0 2 4 6 8 10

curr

ent

den

sity

,A

/m2

voltage, V

KMCexperiment

T = 313 K273 K233 K

10−8

10−6

10−4

10−2

100

102

104

0 2 4 6 8 10

curr

ent

den

sity

,A

/m

2

voltage, V

T = 300 K

0.0 eV0.2 eV0.4 eV0.6 eV0.8 eV1.0 eV

(a) (b)

FIG. 8. (a) Simulated current–voltage characteristics of a single-carrier device with electrodes. Squares show the results ofMonte Carlo simulations in a cubic lattice, which are compared to drift-diffusion simulations (solid lines) using the EGDMmobility parametrized on a microscopic system. Different colors correspond to different values of the injection barrier ∆. (b)Current–voltage characteristics of the off-lattice (coarse-grained) model obtained by Monte Carlo simulations (squares) andmeasured experimentally (circles) for three different temperatures.

Drift-diffusion equations, combined with the EGDMand ECDM models, are often used for fitting to experi-mentally measured IV characteristics, in order to charac-terize a particular material. The results of such a fit, aresummarized in table I. Here we performed a simultane-ous fit to 9 different IV curves, measured using threelayer thicknesses (108 nm, 162 nm, 219 nm) and threetemperatures (233K, 293K, 313K). The model includesa fitted injection barrier (0.05 eV) and doped injectionlayers which provide the (almost) Ohmic injection, simi-lar to the experimental setup shown in fig. 9(a)).

B. Lattice Monte Carlo

To be able to use the Monte Carlo procedure fordevices, additional sites, which play the role of elec-trodes, are added. The injection/collection of a chargecarrier to/from the electrode is modeled as a hop-ping of this charge carrier between the electrode andthe lattice site to/from which the charge carrier is in-jected/collected [97, 126].Additional contributions to site energies (due to elec-

trodes) include a linear drop in the applied potential,eV (1 − x/L). Here the Fermi energy of the electron in-jecting electrode (x = L) is taken as a reference, while theapplied potential at the collecting electrode is set to V .An injection barrier, ∆, is introduced between the Fermilevel of the electrode and the one of the sites of organicmaterial, such that the energy difference to the electrodeis raised by ∆, Eel,i = −Eel+Ei+∆, while for the charge

transfer into the electrode Ei,el = Eel−Ei−∆.These con-tributions are depicted in fig. 9(d).Coulomb interactions are split into a short-range and

a long-range contribution. The short-range part is evalu-ated explicitly for all charges (and their periodic images)within separations smaller than rc. Between one and tenimages provide sufficient accuracy, such that a computa-tionally demanding Ewald summation can be avoided.The long-range contribution is evaluated by first cal-

culating the local charge carrier density, ρ(x), by av-eraging the number of charges in two-dimensional (yz)slabs oriented perpendicular to the electric field direc-tion (along the x axis). Then the electrostatic poten-tial, φ(x), is reconstructed using the Poisson equation,φ′′(x) = −ρ(x)/ε. Note that the layer-averaged poten-tial also takes into account the short-range interactionsand one needs to correct for this double counting.The current density throughout the device is evaluated

as

J =q∑

k ∆rk · FAτF

, (14)

where ∆rk · F /F is the change in position of a chargealong the applied field F , A is the surface area of theelectrodes, and τ is the total simulated time.Apart from device simulations, the lattice Monte Carlo

method can be used to benchmark the accuracy of thecontinuum drift-diffusion solver, discussed in sec. VIA.Fig. 8(a) shows IV curves obtained by Monte Carlosimulations in a cubic lattice with lattice spacing a =1.67 nm and Gaussian-distributed site energies of width

14

σ = 0.134 eV, i.e., the same parameters as in the EGDMmobility parametrization, see section IV. One can seethat lattice Monte Carlo results agree reasonably wellwith the EGDM model combined with the drift-diffusionsolver. Note that at low voltages (small drift currents)the KMC algorithm converges very slowly, which is whywe have included KMC results only for voltages higherthan 2V.

C. Off-lattice Monte Carlo

The Monte Carlo scheme on a lattice can be gener-alized to off-lattice models [127], thus helping to avoidparametrizations of mobility as well as assumptions in-corporated in lattice models. Again, the effects that needto be taken into account are Coulomb interactions andthe injection from the electrodes into the organic layer.To model the injection process, an electronic coupling

of the form Jinj = exp(−2αrij) is assumed between theelectrodes and the organic material, where α is the in-verse wavefunction decay length and rij is the distancebetween the electrode and the site to or from which thecharge carrier is injected or collected. This introducestwo parameters, α and d, where the latter is the dis-tance between the electrode and the organic site near-est to that electrode. Close to the electrodes the imagecharge effect leads to an effective lowering of the energeticdisorder [128]. This is accounted for by a factor γ whichrescales the site energy, Ei, of injection/collection sitesto E′

i = γ(Ei − 〈Ei〉) + 〈Ei〉, where 〈Ei〉 is the averagesite energy.Coulomb interactions are split into a short-range and

a long range contribution. The short-range part is eval-uated in a similar way as for the lattice, except that tento hundred images are included to improve accuracy. Forthe long-range part the total simulation box is split intothree-dimensional slabs of equal thickness, as shown infig. 9(b). All charges in a slab are summed up to ob-tain the slab-averaged charge carrier density, ρi. Theslab-averaged potential, φ(x), is then evaluated by solv-ing the Poisson equation, where the slab-averaged chargecarrier densities are used as a source term. Again, simi-lar to the lattice Monte Carlo, the layer-averaged poten-tial already takes into account short-range interactions.Thus double-counting should be corrected for, as shownin fig. 9(c).The computationally expensive update of the averaged

long-range potential and the re-evaluation of the hoppingrates is done every 100 – 1000 hops, which yields sufficientaccuracy of long-range effects.To perform charge transport simulations, we used the

coarse-grained off-lattice model of DPBIC, parametrizedas discussed in sec. III. An amorphous morphology of255083 molecules was generated in a box of 123 nm ×50 nm × 50 nm to match the experimentally used layerthickness (see sec. VII for details). Periodic boundaryconditions were employed in the directions perpendicular

to the electric field. The simulated IV curves, togetherwith the experimental data points are shown in fig. 8(b)for α = 13nm−1, d = 0.5 nm, ∆ = 0.5 eV, and γ = 0.55,and are in good agreement with each other.

VII. EXPERIMENT

In order to decouple different processes in OLEDs, ex-periments are often performed in single-layer hole-only orelectron-only devices [129]. Here we used a setup shownin fig. 9(a) with 5 nmmolybdenum trioxide (MoO3) layerson both sides of a 123 nm DPBIC layer. The function ofthe MoO3 layers is to allow band bending [130], resultingin an (almost) Ohmic injection to the DPBIC layer [131].The devices were fabricated by vacuum thermal evap-

oration of the organic material on a glass substrate pat-terned with a 140 nm thick indium tin oxide (ITO) layeras an anode. Prior to deposition, substrates were cleanedin an ultrasonic bath using subsequently detergents andde-ionized water. The samples were dried and treatedin an ozone oven to additionally clean them and modifythe ITO workfunction. A 100 nm film of thermally evap-orated Aluminium (Al) is used as a cathode. In order toexactly determine the thickness of the DPBIC layer, wesimultaneously deposited the same amount of DPBIC ona Si-Wafer mounted adjacent to the substrate and mea-sured its thickness using optical ellipsometry.The fabricated samples were placed into the oil reser-

voir of a cryostat, where the temperature was varied be-tween 233K and 313K. The experimentally measured IVcurves are shown in fig. 8 for three different temperatures,together with simulation results described in sec. VI.

VIII. MATERIALS

To this end, we have simulated current–voltage char-acteristics of a diode with a hole-conducting DPBIC usedas organic layer. While this example is so far the only ex-haustive illustration of multiscale simulations techniqueswhich can be used to simulate charge transport in disor-dered organic semiconductors, a number of similar meth-ods have already been used to evaluate material prop-erties of such materials [16, 17]. Here we will providea (far from complete) review of such case studies. Inparticular, we will focus on charge transport in amor-phous mesophases of DPBIC, BTDF, TBFMI, BCP, α-NPD, and Alq3, chemical structures of which are shownin fig. 10.DPBIC Tris[(3-phenyl-1H-benzimidazol-1-yl-2(3H)-

ylidene)-1,2-phenylene]Ir is used in OLEDs as a hole-transporting and electron-blocking material [132–135].Apart from IV dependencies discussed here, simula-tions of amorphous DPBIC predict a density value of1.29 g/cm3 and an energetic disorder of 0.176 eV, whichcompares reasonably well to the experimentally deter-mined value of 0.121 eV, that relies on the ECDM [28].

15

mirro

r im

ages

mirro

r images

ele

ctr

od

e

ele

ctr

od

e

FIG. 9. (a) Experimental setup for single-carrier-type measurements, here of the DPBIC hole-conduction layer. (b) Simulationsetup mimicking the device. (c) Evaluation of the long- and short-range electrostatic contributions in a device geometry. (d)Additional contributions to site energies due to electrodes: a linear drop in the applied potential and an injection barrier ∆.

In amorphous morphology the perturbative approachyields an ionization potential (IP) of 5.28 eV, in a perfectagreement with the Conductor-like screening model(COSMO) [136] giving the same value of 5.28 eV. Boththe Poole-Frenkel dependence and the absolute valueof mobility are in agreement with experimental mea-surements: For small fields the simulated hole mobilityvaries from 6.6 × 10−10 m2/Vs (ρ = 4.25 × 1023 m−3)to 3.6 × 10−8 m2/Vs (ρ = 1.36 × 1025 m−3), while theexperimental value is 9× 10−9 m2/Vs.

BTDF 8-bis(triphenylsilyl)-dibenzofuran (BTDF) isan electron-conducting and hole-blocking material [132]which, in combination with a deep blue emitter canlead to quantum efficiencies above 17% [20]. Its glasstransition temperature is at 107C◦ [20]. Moleculardynamics simulations predict an amorphous density of1.37 g/cm3 [20]). Calculations based on polarizable force-fields predict an energetic disorder of σ = 0.085 eV (elec-trons) and σ = 0.078 eV (holes). Simulated in thezero-charge density limit mobilities, 1.0 × 10−9 m2/Vsfor holes and 3.2 × 10−8 m2/Vs for electrons, are com-parable to the experimental values obtained using ad-mittance spectroscopy, 2.5 × 10−10 m2/Vs for holes and1.0 × 10−8 m2/Vs for electrons, measured at an electricfield of 9× 106 V/m (holes) 3× 107 V/m (electrons).

TBFMI Tris[(1,2-dibenzofurane-4-ylene)(3-methyl-1/1-imidazole-1-yl-2(3/1)-ylidene)]Ir(III)) (TBFMI) hasbeen used as an emitter molecule in various host-guestmixtures, for example, with BTDF. An important

quantity in such mixtures is the relative alignment ofdensities of states of the host and guest materials. Forexample, in TBFMI:BTDF the energy offset betweenthe host and the guest is very small, which is mostlydue to environmental effects. In particular, the changein molecular polarizability upon charging plays animportant role [21].

BCP 2,9-dimethyl-4,7-diphenyl-1,10-phenanthroline(bathocuproine, BCP) is a typical electron conductingand hole blocking material which is used in phosphores-cent OLEDs [137–141]. Simulations of charge transportin a crystalline phase of this material [141], using amodel based on quantum mechanic calculations of asample dimer, show mobilities of 1.8 × 10−6 m2/Vs ina reasonable agreement with a reported experimentalvalue of 1.1× 10−7 m2/Vs [142].

α-NPD N,N’-bis(1-naphthyl)-N,N’-diphenyl-1,1’-biphenyl-4,4’-diamine (α-NPD or NPB) can beused in OLEDs as a hole-transporting, electron-blocking [139, 143–147], or blue-emitting layer [148, 149],and as a host material [150, 151]. It has a relatively highglass transition temperature of 95C◦, favorable for thematerial’s stability. In the amorphous phase, a density of1.16 g/cm3 has been reported from X-ray reflectometrymeasurements [152]. IV curves in an α-NPD baseddiode were simulated by solving the drift-diffusionequations using both extended and correlated Gaussiandisorder models. Both models provide a consistentdescription of the IV characteristics at different temper-

16

DPBIC BTDF

BCP

TBFMI

N N

Si Si

O

3

O

Al

N

3

N

N

Ir

N N

Alq3 -NPD

3

O

Ir

N

N

FIG. 10. Chemical structures of different materials used inOLEDs.

atures. However, the lattice constant obtained from theECDM fit is closer to the experimentally known intersitedistance, i.e., spatial correlations of site energies in thismaterial are important. The amorphous morphologyof α-NPD has been studied by using the Monte Carloalgorithm that mimics the vacuum deposition pro-cess [44] as well as by MD simulations [153]. The latter,combined with electrostatic and induction calculations,gave energetic disorder values of σ = 0.09 − 0.11 eV(electrons) and σ = 0.14 − 0.15 eV (holes). Simulatedmobilities, obtained by applying an analytic solution ofthe master equation to the transport network, variedfrom 4.2× 10−10 m2/Vs to 1.3× 10−9 m2/Vs (holes) andfrom 1.1 × 10−6 m2/Vs to 4.1 × 10−6 m2/Vs (electrons),in a good agreement with the experimentally measuredvalue of 7.8× 10−10 m2/Vs for holes [154].

Alq3 Mer-tris(8-hydroxyquinolinato)aluminium(Alq3) was used in the first organic diode alreadyin 1987 [155] and ever since has been studied inten-sively. The amorphous morphology of Alq3, obtainedby annealing above the glass transition temperatureand subsequent cool-down [34] to room temperature,

resulted in density of 1.18 g/cm3 (Williams 99 forcefield) and 1.29 g/cm3 (OPLS force field), while MonteCarlo-based vacuum-deposition algorithm with the localrelaxation after deposition led to 1.22 g/cm3 (OPLSforce field) [44]. Experimentally reported values rangefrom 1.3 g/cm3 [156] to 1.5 g/cm3 [157]. Simulationspredict energetic disorder of 0.21 eV for holes [17].Charge transport simulations have shown that spatialcorrelation of site energies is responsible for the Poole-Frenkel behavior of the charge mobility [17] and thatfinite-size effects should be carefully accounted for insystems with large energetic disorder [35].

IX. CONCLUSIONS AND OUTLOOK

We have reviewed a set of simulation techniques usedto understand charge transport in amorphous organicmaterials. The distinct feature of these techniques is thatthey all start with chemical structures and predict, bybuilding a hierarchical set of models, macroscopic prop-erties of devices, in our case charge carrier mobility andcurrent-voltage characteristics of an organic light emit-ting diode. The advantages and shortcomings of all tech-niques have been illustrated by simulating (and validat-ing experimentally) charge dynamics in a unipolar devicewith a hole conducting material used in state of the artdiodes.We have also identified areas where substantial method

development is still required in order to achieve aparameter-free modeling of realistic devices. In par-ticular, still missing are (i) first-principles evaluationsof charge injection rates (from metal to organic mate-rial), (ii) explicit treatment of the induction interactionwhen solving the master equation (re-evaluation of ratesat every Monte Carlo step), (iii) quantitative treatmentof excited states embedded in a heterogeneous polariz-able molecular environment, (iv) descriptions of charge-exciton and exciton-exciton interactions. Advancementsin all these directions are absolutely vital for devising ac-curate structure-property relationships for organic semi-conductors.

ACKNOWLEDGMENTS

This work was partly supported by the DFG programIRTG 1404, DFG grant SPP 1355, and BMBF grantsMEDOS (FKZ 03EK3503B) and MESOMERIE (FKZ13N10723). We are grateful to Carl Poelking, AntonMelnyk, Jens Wehner, and Kurt Kremer for a criticalreading of this manuscript.

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