Kinetic Monte Carlo Study of the Sensitivity of OLED Efficiency and Lifetime to Materials Parameters

Kinetic Monte Carlo study of sensitiviy of OLED efficiency and

lifetime to materials parameters

Citation for published version (APA):

Coehoorn, R., Eersel, van, H., Bobbert, P. A., & Janssen, R. A. J. (2015). Kinetic Monte Carlo study of sensitiviy

of OLED efficiency and lifetime to materials parameters.

Advanced Functional Materials

(13), 2024-2037.

https://doi.org/10.1002/adfm.201402532

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DOI:

10.1002/adfm.201402532

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and excitonic processes and determining

the values of the relevant material-speciﬁ c

parameters, ii) supporting the layer stack

design process by providing an efﬁ cient

route for developing understanding of the

beneﬁ ts and disadvantages of the use of

novel materials and layer stack concepts.

The third (vertical) axis in the ﬁ gure indi-

cates examples of the performance charac-

teristics which may be studied using the

simulations. We envisage that simulations

can be used to clarify the sensitivity of the

OLED device performance to materials

and layer stack parameters, and can pro-

vide insights in the functioning of OLEDs

beyond the capability of experiments. For

example by a much more precise con-

trol of the device structure, by providing

insight in all processes at an ultimate

(molecular-scale) spatial resolution and at

the shortest relevant time scales, and by

providing the opportunity to explore the

performance under conditions which are

experimentally not realizable. A long-term goal of our research

is therefore to develop and apply a simulation method which

can address any combination of the three application types,

and which provides a platform within which novel insights

as obtained in each of the ﬁ elds indicated in the ﬁ gure can be

readily adopted.

So far, commonly used OLED simulations methods are based

on one-dimensional drift-diffusion approaches (see Coehoorn

and Bobbert and references therein)

[ 2 ]

in which the charge

transport and excitonic processes are included in a semi-empir-

ical manner. However, it is well-known that as a result of the

energetic disorder, resulting from the structural disorder, the

current density in OLEDs is ﬁ lamentary, rather than uniform.

[ 3 ]

We have shown from molecular-scale three-dimensional kinetic

Monte Carlo (3D-KMC) modelling that, as a consequence, the

emission takes place on certain preferred molecular sites, rather

than being laterally uniform.

[ 4 ]

It is thus necessary to develop a

full 3D-KMC model for accurately simulating the effect of the

loss processes, such as triplet-polaron quenching and triplet-tri-

plet annihilation in phosphorescent OLEDs. These losses lead

to an efﬁ ciency decrease with increasing current density and

luminance (roll-off of the internal quantum efﬁ ciency,

IQE

We have recently extended the 3D-KMC method to include

the charge transport and all excitonic processes in an inte-

gral manner.

[ 5 ]

The simulations were found to provide a good

description of the roll-off of archetypical green and red emitting

OLEDs studied intensively in the literature,

[ 6 ]

and were used to

Kinetic Monte Carlo Study of the Sensitivity of OLED

Efﬁ ciency and Lifetime to Materials Parameters

Reinder Coehoorn , * Harm van Eersel , Peter A. Bobbert , and René A. J. Janssen

The performance of organic light-emitting diodes (OLEDs) is determined by

a complex interplay of the optoelectronic processes in the active layer stack.

In order to enable simulation-assisted layer stack development, a three-

dimensional kinetic Monte Carlo OLED simulation method which includes

the charge transport and all excitonic processes is developed. In this paper,

the results are presented of simulations including degradation processes in

idealized but realizable phosphorescent OLEDs. Degradation is treated as a

result of the conversion of emitter molecules to non-emissive sites upon a

triplet-polaron quenching (TPQ) process. Under the assumptions made, TPQ

provides the dominant contribution to the roll-off. There is therefore a strong

relationship between the roll-off and the lifetime. This is quantiﬁ ed using a

“uniform density model”, within which the charge carrier and exciton densi-

ties are assumed to be uniform across the emissive layer. The simulations

give rise to design rules regarding the energy levels, and are used to study the

sensitivity of the roll-off and lifetime to various other materials parameters,

including the mobility, the phosphorescent dye concentration, the triplet

exciton emissive lifetime and binding energy, and the type of TPQ process.

DOI: 10.1002/adfm.201402532

Prof. R. Coehoorn, H. van Eersel

Philips Research Laboratories

High Tech Campus 4, 5656AE , Eindhoven

The Netherlands

E-mail: reinder[email protected]

Prof. R. Coehoorn, H. van Eersel,

Dr. P. A. Bobbert, Prof. R. A. J. Janssen

Department of Applied Physics

Eindhoven University of Technology, P.O. Box 513

5600MB , Eindhoven , The Netherlands

1. Introduction

Organic light-emitting diode (OLED) technology for large area

lighting is presently developing from design-focused special-

lighting applications towards general-lighting applications for

homes and ofﬁ ces. Future products are expected to show added

functionality such as ﬂ exibility, transparency in the off-state,

and color-tunability. Continuous improvements of the system

power efﬁ ciency and long-term stability arise from the use of

novel organic semiconductor materials, novel stack architectures

and methods for improved light outcoupling.

[ 1 ]

The development of OLED active layer stacks is supported

by the use of simulation methods. As indicated schematically

in Figure 1 , two types of applications may be distinguished:

i) revealing the mechanism of the relevant charge transport

Adv. Funct. Mater. 2015, 25, 2024–2037

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develop a design rule for an optimal power efﬁ ciency, resulting

from a small

IQE

-roll-off and a low voltage.

In this paper, we demonstrate how 3D-KMC simulations

can also be used to predict the dependence of the lifetime

of phosphorescent OLEDs on the materials and layer stack

parameters. For that purpose, we have included in the simula-

tions irreversible changes of the molecular sites, according to a

chosen scenario, and simulate the resulting time-dependence

of the emission. Whereas a wide variety of degradation mecha-

nisms can be operative,

[ 7,8 ]

recent work has indicated that in

phosphorescent OLEDs triplet-polaron quenching followed by

defect formation can play a major role.

[ 9,10 ]

We investigate two

degradation scenarios in which upon triplet-polaron quenching

the dye molecule which is involved is with a certain probability

converted in a non-emissive molecule. These explicit 3D-KMC

lifetime simulations can be computationally intensive. We

have therefore also studied to what extent it is already possible

to predict lifetime trends using the loss due to exciton-polaron

quenching as obtained from results of the steady-state simula-

tions. As a ﬁ rst step towards that goal, we focus on the case of

monochrome OLEDs with a single emissive layer (EML), and

show how under the simplifying assumption that the charge

carrier density and exciton generation are uniform across the

thickness of the EML the roll-off can be expressed in terms of

the materials and layer stack parameters. The approach is a

reﬁ nement of earlier work,

[ 11,12 ]

by including the charge car-

rier density dependence of the mobility and by including a

mechanistic expression for the triplet-polaron quenching rate.

As a second step, it is shown how using this uniform den-

sity model (UDM) variations of the lifetime with layer stack

parameters, such as the EML thickness or dye concentration,

can then be predicted from the roll-off. Within the degrada-

tion scenarios considered, the absolute lifetime is proportional

to the inverse of the probability that upon a quenching event

degradation takes place. That is the only additional parameter

which is required. Within the framework of our approach, it

can be determined by one or a few “calibration” experiments.

We stress that the UDM only provides a starting point for

analyses of roll-off and lifetime. The simulations show that

changes of the shape of the charge carrier density and emis-

sion proﬁ les with the voltage and current density also play an

important role.

The paper is organized as follows. Section 2 contains an

overview of the simulation method and a simulation example.

In Section 3, the uniform density model is developed, and

analytical expressions are given for the dependence of the

roll-off and lifetime on the material parameters, neglecting

exciton dissociation and exciton diffusion. In Sections 4 and

5, the sensitivity of the roll-off and lifetime, respectively, to

various materials parameters is studied from 3D-KMC simu-

lations. Two design rules for the most optimal energy level

structures are established, and the sensitivity of the roll-off to

the emissive lifetime of the dye molecules and to the triplet

binding energy is examined. The UDM is shown to provide a

fair prediction of the OLED lifetime for small dye concentra-

tions, for which exciton diffusion to already degraded sites

may be neglected. Section 6 contains a summary, conclusions

and outlook.

2. Simulation Method

In this section, ﬁ rstly an overview is given of the 3D-KMC sim-

ulation method and of the simulation parameters used in this

paper (subsection 2.1). Subsequently, more detailed descrip-

tions are given on the treatment of charge transport, exci-

tonic processes and degradation (subsections 2.2, 2.3 and 2.4,

respectively). Finally, as an example the simulations are applied

in subsection 2.5 to a symmetric three-layer OLED, for which

the roll-off curve, emission proﬁ le, carrier density proﬁ le and

time-dependent decrease of the emission due to degradation

is shown. In Section 4, the energy level structure assumed for

this example will be shown to quite optimally fulﬁ ll two design

rules leading to a small roll-off, and in Section 5, the sensitivity

Adv. Funct. Mater. 2015, 25, 2024–2037

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Figure 1. Application areas of OLED device simulations (horizontal and diagonal axes) and examples of the performance characteristics which may be

studied (vertical axis). TADF denotes Thermally Assisted Delayed Fluorescence.

[ 7,8 ]

Simulation studies for a variety of performance characteristics, layer

stack designs and materials can help to develop an improved material-speciﬁ c understanding of the relevant physical processes. Realizing progress in

that understanding is needed to make simulation-assisted OLED stack development more powerful.

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of the OLED lifetime when varying the Ir-dye concentration in

this OLED will be discussed.

2.1. Simulation Parameters—Overview

Within the 3D-KMC simulation method developed, the

molecules are treated as point sites, without any internal struc-

ture. The sites are placed on a simple cubic grid with a lattice

parameter a = N

−1/3

, with N

the average molecular site density.

Any layer sequence and mixing of different types of molecules

in each layer can be deﬁ ned, such as random mixing, aggre-

gation of similar molecules,

[ 13 ]

mixed interface layers, or, for

example, a concentration gradient in the emissive layer.

[ 14,15 ]

any moment in time, a site in the box is either empty, or occu-

pied by one electron, one hole, or by one electron and one hole.

When at a site a compound species (exciton, one electron and

one hole) is present, the exciton is further speciﬁ ed as either

a singlet or a triplet exciton. Typically, for each set of external

conditions (voltage, temperature) ﬁ ve parallel simulations are

carried out for nominally equal boxes, but with different dis-

order realizations, with a lateral area of 50 × 50 sites. From the

time-resolved and time-averaged results as obtained for these

simulation boxes, the statistical uncertainty can be judged.

The simulations are speciﬁ ed by four sets of parameters:

i) parameters deﬁ ning the system structure (thicknesses

and composition of all sub-layers), ii) electrical parameters,

iii) excitonic parameters, iv) the parameters specifying the

external conditions (voltage, V , and temperature, T ). The

charge transport and excitonic processes are included in a

manner as described by Mesta et al.

[ 4 ]

and by van Eersel et al.

[ 5 ]

A brief description is given in the next subsections. Table 1

gives an overview of the basic electrical and excitonic param-

eters, and their values used as a default in the simulations dis-

cussed in this paper. We note that the code allows for various

reﬁ nements, not mentioned further here, such as non-isotropic

mobilities, non-Gaussian shapes of the density of states, and

long-distance (e.g., Förster-type) exciton-polaron quenching

and exciton-exciton annihilation. Furthermore, it allows for

carrying out simulations under special conditions, such as

steady-state or time-resolved photoluminescence

[ 11,16,17 ]

electroluminescence conditions.

[ 18 ]

All simulations were done

for T = 298 K.

The accessible material and device parameter range is

determined by practical (real time, CPU time, memory) limi-

tations. The total required real simulation time depends on

the level of time-averaged, spatially resolved or time-resolved

detail needed and the number of nominally equal OLEDs with

different disorder realizations run in parallel. When consid-

ering ﬁ ve 50 nm × 50 nm of such OLEDs in parallel, the IQE

roll-off curve simulations discussed in this paper are obtained

typically in 3−10 days on a 3.0 GHz core [Intel Xeon X5675],

depending on the current density. Simulations in which the

fraction of quenched excitons or the fraction of radiatively

decaying excitons is small (at a very small current or very large

current density, respectively) are most demanding, since a

large number of events needs to be recorded to gather enough

statistics on the most rare event. The lifetime results discussed

in this paper are based on simulations over periods ranging

from a few days up to two months. Achieving sufﬁ cient sta-

tistics is slowed down when considering larger device dimen-

sions, larger energy barriers at interfaces, enlarged charge

carrier or exciton hop distances (due to enlarged wavefunction

decay lengths or Förster radii, at large ﬁ elds, or in very dilute

systems), enlarged density of states widths or trap depths, and

enlarged exciton lifetimes (so that at a given time more exci-

tons are present in the device).

2.2. Charge Transport

The hop rate

of electrons and holes over a distance R and

from a molecule with energy E

(initial state energy) to a mole-

cule with E

= E

+ Δ E (ﬁ nal state energy) is assumed to be given

by the Miller-Abrahams formalism. When the hopping occurs

between two molecules of types A and B the rate is assumed to

be given by:

exp 2 exp

1,A 1,B

ννν α

[]

()

=−−−

Δ−Δ

⎛

⎝

⎜

⎞

⎠

⎟

(1)

where

1, X

is the hopping attempt rate between two molecules

of type X at a distance a (ﬁ rst neighbor), for the case Δ E = 0,

and

≡ 1/

is the inverse of the wavefunction decay length (

The square-root form of the prefactor used arises when consid-

ering the transfer integral due to the overlap of the two expo-

nentially decaying wavefunctions. The expression applies to

any pair of molecules, also to molecules in different layers. So

no special parameters are introduced to describe the hop rates

across internal interfaces. In this paper, materials with spatially

uncorrelated Gaussian disorder are considered. The hole and

electron site energies are taken from a random Gaussian

distribution with a material-speciﬁ c mean value and with a

material-speciﬁ c standard deviation

, respectively,

modiﬁ ed by i) the Coulomb interaction energy with the other

Adv. Funct. Mater. 2015, 25, 2024–2037

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Table 1. Overview of the electrical and excitonic parameters used in the

3D-KMC simulations, and the default values used in this paper.

Parameter Description Value

Hopping attempt rate to the ﬁ rst neighbor

3.3 × 10

−1

Width (standard deviation) of the Gaussian

density of states

0.10 eV

Site density

−3

Wavefunction decay length 0.3 nm

Relative dielectric constant 3.0

T, b

Triplet exciton binding energy 1.0 eV

D,0

Dexter transfer prefactor

1.6 × 10

−10

−1

F,diff

Förster radius for exciton diffusion between

dye molecules

1.5 nm

rad

Radiative decay rate dye molecules

0.544 µs

−1

Non-radiative decay rate dye molecules

0.181 µs

−1

Values as obtained experimentally for the orange-red emitter Ir(MDQ)

(acac) in

-NPD matrix, assuming a radiative decay efﬁ ciency

rad

= 0.75, intermediate

between the values obtained from analyses assuming randomly oriented or mainly

parallel emission dipole moments.

[ 19 ]

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charge-carriers and with its image charges in the electrodes

and by ii) the electrostatic potential difference due to the ﬁ eld

arising from the applied voltage. As in the transport processes

studied holes or electrons are removed or added, the material-

speciﬁ c mean energy level values for holes and electrons

should be termed properly as a mean ionization potential or

a mean electron afﬁ nity, respectively. However, we indicate

these energies instead as a highest occupied molecular orbital

(HOMO) energy, E

HOMO

, or a lowest unoccupied molecular

orbital (LUMO) energy, E

LUMO

, respectively, as is common prac-

tice in the literature.

By using this mechanistic approach, we circumvent the

need to describe the charge transport by means of complicated

expressions for the mobility and diffusion coefﬁ cient, which

are used in one-dimensional drift-diffusion simulations. The

mobility, e.g., depends on the charge carrier density, electric

ﬁ eld and temperature, and is affected by the shape of the den-

sity of states, the type of disorder, Coulomb interaction effects

in accumulation regions near interfaces, and the presence of

guest or trap molecules.

[ 2 ]

Furthermore, the mobility may effec-

tively be time (or frequency) dependent due to charge carrier

relaxation

[ 20,21 ]

and it is effectively layer thickness dependent

below a certain critical length scale.

[ 22 ]

In a uniform layer, the mobility at any ﬁ eld, temperature and

carrier density is determined by four material parameters: a ,

and

. These parameters can be determined from dedicated

transport studies of single-carrier single-layer devices, as shown

for several polymeric

[ 23–25 ]

and small-molecule materials.

[ 26,27 ]

The lattice parameter a which is obtained from such studies, is

quite similar to the value expected from the experimental molec-

ular site density. In this paper, we take a = 1 nm,

= 0.1 eV,

and

= 3.3 × 10

−1

, which are typical values for small-mol-

ecule OLED materials (see van Eersel et al.

[ 5 ]

for a motivation

of the choice for

). The wavefunction decay length

is only

relevant in dilute host-guest systems in which guest-guest

hopping over distances larger than a is signiﬁ cant. From such

studies, we deduce that

is larger than the value of 0.1 × a (i.e.,

0.1 nm for a = 1 nm) which has been used in earlier modeling

studies of transport in one-component materials.

[ 4,28,29 ]

In this

paper, we take

= 0.3 nm, the value used by van Eersel et al.

to describe the roll-off in Ir(ppy)

and PtOEP based OLEDs.

[ 5 ]

Within a more reﬁ ned approach, the hop rates should be

described using Marcus theory.

[ 30 ]

However, as an additional

parameter would be involved (the reorganization energy), to

which the shape of the current voltage-curves will be quite

insensitive,

[ 31 ]

we have not adopted that approach. Based on

similar arguments (see also Cottaar et al.),

[ 31 ]

a simple cubic

grid is used in all cases instead of treating the grid structure as

a degree of freedom.

2.3. Excitonic Processes

Exciton generation (dissociation) is treated as a result of a hop of

a charge carrier to (away from) a site at which already a carrier of

the opposite polarity resides, using Equation (1) , albeit that Δ E

includes now a gain (loss) equal to the on-site singlet or triplet

exciton binding energy, E

S(T),b

. This takes into account that the

actual on-site exciton binding energy is ﬁ nite, whereas within

the point-site model used the on-site electron-hole Coulomb

interaction would be inﬁ nite. In this paper, we focus on phos-

phorescent OLEDs and take the triplet binding energy equal for

all sites. Exciton generation and dissociation are thus a natural

consequence of the electron-hole attraction; no additional rate

coefﬁ cients are introduced. The Coulomb interaction energy

between an electron and hole on a nearest neighbor site

(a charge-transfer exciton) is within our formalism equal to

CT,b

= e /(4

a ), with e the electron charge,

the vacuum per-

mittivity and

the relative dielectric constant (taken equal to 3),

that is, 0.48 eV. In a homogeneous material, a signiﬁ cant dis-

sociation rate will thus be obtained unless the exciton binding

energy is well above E

CT,b

. In Section 4, we study the sensitivity

of the roll-off to E

T, b

. In this paper, we assume that all excitons

which are generated are triplets. As we consider only devices in

which almost all the excitons are formed on the phosphorescent

dye molecules, so that singlet-to-triplet conversion due to inter-

system crossing is fast, that is an excellent approximation.

Radiative and non-radiative decay are assumed to be mono-

exponential processes with rates Γ

rad

and Γ

, respectively,

which may be deduced from the measured total decay rate

Γ = Γ

rad

+ Γ

≡

−1

(with

the effective decay time) and the

radiative decay efﬁ ciency

rad

(which may be taken equal

to the photoluminescence efﬁ ciency) using the expressions

rad

Γ and Γ

= (1 –

rad

) Γ.

The exciton transfer rate from a site of type A to a site of type

B, at a distance R , is described as a sum of contributions due to

Förster transfer, with a rate

F,AB

F,diff ,AB

=Γ

⎛

⎝

⎜

⎞

⎠

⎟

(2)

with

F,diff,AB

the Förster radius for exciton diffusion between

the two types (A, B) of molecules involved, and a Dexter contri-

bution given by

exp 2 exp

D D,0A D,0B

kkk R

[]

=−−

Δ+Δ

⎛

⎝

⎜

⎞

⎠

⎟

(3)

where

D,0,A(B)

are material-type speciﬁ c prefactors,

is the

inverse of the wavefunction decay length, and Δ E is the dif-

ference between the ﬁ nal and initial state exciton energies.

Such an energy difference can occur when the two mol-

ecule types are different, or as a result of excitonic disorder.

In this paper, excitonic disorder is neglected. Only few

studies on triplet exciton diffusion in phosphorescent host-

guest systems have been carried out. Based on the work of

Kawamura et al.,

[ 32 ]

we take R

F,diff

= 1.5 nm. It has also been

argued that exciton diffusion between Ir-dye molecules is

predominantly due to Dexter transfer.

[ 33,34 ]

In view of this

uncertainty, we take the Dexter prefactors such that at a dis-

tance equal to the Förster radius both rates are approxi-

mately equal. At much larger distances the diffusion is

then predominantly due to the Förster process, whereas at

distances of 1–2 nm also Dexter transfer contributes.

The detailed mechanisms of triplet-polaron quenching

(TPQ) and triplet-triplet annihilation (TTA) are not well known.

In this paper, we describe both processes as immediate when

the two interacting species are at a distance a . A distinction

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is made between TPQ processes in which the triplet is trans-

ferred to the site at which the polaron resides (“TPQ-p pro-

cess”) or one in which the polaron hops to the site at which

the triplet resides (“TPQ-t process”) (see Figure 2 ). Making this

distinction does not affect the roll-off. However, it can affect

the degradation rate (see Section 5). After a TTA process one

triplet exciton is assumed to be left, randomly chosen to be on

one of the dye sites. Under these assumptions, no free para-

meters describing TPQ and TTA are introduced. The TPQ rate

is then determined predominantly by the polaron density and

the polaron diffusion coefﬁ cient (see the next section), as for

the dilute systems considered the triplet diffusion coefﬁ cient

is relatively small. The TTA rate is at the low voltages consid-

ered for lighting applications very small, in view of the small

triplet volume density obtained for the commonly used short

radiative lifetime phosphorescent Ir-based emitters, and the

small triplet diffusion coefﬁ cient.

[ 5 ]

In a forthcoming paper, we

investigate to what extent TPQ still provides the dominant con-

tribution to the roll-off when describing TPQ and TTA as long-

range Förster-type processes with Förster radii up to 5 nm,

[ 35 ]

suggested, for instance, for the case of TTA from the work of

Reineke et al.

[ 16 ]

2.4. Degradation

In this paper, we describe OLED degradation as a result of the

conversion of emitter sites to non-emissive sites with an inﬁ -

nite value of Γ

when on that site a TPQ process occurs, with

a probability p

degr

. All other parameters are kept constant. The

effect of degradation is only included when mentioned explic-

itly. In such a case, we ﬁ rst carry out a steady-state 3D-KMC

simulation without degradation ( p

degr

= 0), in order to obtain

equilibrated charge and exciton distributions. Subsequently,

we continue the simulation under the condition p

degr

= 1. Our

approach thus employs the largest possible acceleration, in

order to make the simulations feasible. We ﬁ nd that such an

approach is permitted, as simulations using a smaller value of

degr

lead to a decrease of the decay rate which is proportional

to p

degr

. We remark that p

degr

is in reality expected to be a very

small number. The decay of the emission intensity is found to

show in general a stretched-exponential-like shape, as is also

observed experimentally. The lifetime is shorter for the case of

TPQ at the site where the triplet resides (TPQ-t) than for the

case of TPQ at the site where the polaron resides (TPQ-p), as

only in the former case all TPQ processes occur on an emitter

molecule (see Section 5).

2.5. Example: Mixed Matrix Symmetric OLED

As a ﬁ rst example, we discuss the results of 3D-KMC simu-

lations for a symmetric OLED with a layer structure (anode |

HTL (40 nm) | EML (20 nm) | ETL (40 nm) | cathode), with

a mixed-matrix emissive layer containing equal concentra-

tions of the hole transport layer (HTL) and electron transport

layer (ETL) materials (host molecules) and 4 mol% of emitter

molecules (guest). The energy level structure is shown in

Figure 3 a, and the simulation parameters as given in Table 1

are used. The shallow LUMO and deep HOMO energies of the

HTL and ETL, respectively, ensure perfect carrier blocking, so

the recombination efﬁ ciency is 100%. We already showed for

similar single-matrix OLEDs that the carefully chosen guest

trap depth (0.2 eV for electrons and holes) leads to a quite

optimal balance between a small roll-off and a low operating

voltage.

[ 5 ]

First, the OLED performance under steady-state

conditions (no degradation) is discussed (Figures 3 b–e). Sub-

sequently, the decrease of the emission in the presence of deg-

radation is shown (Figure 3 f).

Figure 3 b shows the roll-off of the IQE as a function of the

current density and (inset) the current–voltage ( J ( V )) curve in

the range 3–6 V. The full curve is a ﬁ t to the roll-off using the

often-used empirical expression

IQE

rad

⎛

⎝

⎜

⎞

⎠

⎟

(4)

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Figure 2. The two types of triplet-polaron quenching processes considered: a) TPQ-p process, and b) TPQ-t process. Empty circles: neutral molecules

in the ground states. Filled circles: molecules on which a polaron resides, in the polaronic ground state. Stars (initial conﬁ guration): neutral molecules

in the lowest triplet exciton state. Stars (intermediate conﬁ guration): polaron in an excited state.

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with J

the current density at which the IQE has been

reduced by a factor 2, and m an exponent which increases with

increasing steepness of the roll-off around J

. It may be seen

that this expression provides an excellent ﬁ t. Alternatively, the

IQE may be written as

IQE

= [Γ/(Γ + Γ

)]

rad

, with Γ

the effective

rate of all loss processes due to quenching (and annihilation).

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Figure 3. 3D-KMC simulation example, as discussed in Section 2.5. a) Energy level structure of the OLED considered. b) Current density dependence

of internal quantum efﬁ ciency. The full curve is a ﬁ t using Equation ( 4) . c) Current density dependence of the relative quench rate. d) Hole and electron

concentration proﬁ les at 3 V. The full curves are guides-to-the-eye. The molecular layers in the EML are at a distance of 41 to 60 nm from the anode,

and the cathode is at a distance of 101 nm from the anode. e) Emission proﬁ le at 3 V. The full curve is a guide-to-the-eye. f) Time-dependence of the

normalized emission at 6 V (degradation study), assuming only TPQ-t processes and assuming p

degr

= 1, with and without exciton diffusion (closed

and open symbols, respectively). The dashed curve is a stretched-exponential ﬁ t to the simulation results without diffusion, using the parameter values

discussed in the text. In (b,c) the statistical uncertainty of the data points is smaller than the size of the data points. In (d–f) the statistical uncertainty

may be judged from the variability with the position and time, respectively.

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It follows from Equation ( 4) that the relative quench rate, Γ

/ Γ,

may be expressed as

rad

IQE 50

=−=

⎛

⎝

⎜

⎞

⎠

⎟

(5)

A double logarithmic plot of the quantity [(

rad

IQE

) − 1]

should then yield a straight line with a slope m and an x -axis

zero-crossing at J = J

. We call this a “quench rate plot”. Pos-

sible deviations from Equation ( 4) for small current densities

can be seen more clearly from such a plot than from the roll-off

curve. Figure 3 c shows that Equation ( 4) is indeed well obeyed

within the current density range studied. The ﬁ t parameters

are J

= 10.0 kA/m

and m = 0.80. Such a ﬁ t makes it possible

to quite accurately derive the roll-off at small current densities,

where the simulation results are somewhat less accurate due

to relatively larger statistical ﬂ uctuations. The current density

= (1/9)

1/ m

, which is a more commonly used measure for

the roll-off,

[ 12 ]

is equal to 642 A/m

. This value is higher than

the highest values of J

for phosphorescent OLEDs reported to

date, which range up to 300 A/m

[ 12 ]

We attribute the improved

roll-off in the simulated device to the effect of carrier trapping

on the mobility in the EML. Figure 3 d,e shows the charge car-

rier concentration and emission proﬁ les, respectively, at 3 V.

As may be seen from the ﬁ gures, strong accumulation of elec-

trons at the anode-side of the EML and of holes at the cathode-

side of the EML is prevented, and the emission proﬁ le is quite

uniform across the thickness of the EML. A study in Section

4 of the effect of the guest trap depth supports this picture.

We ﬁ nd that in the current density range studied the roll-off is

almost completely due to TPQ. We found a similar conclusion

for Ir(ppy)

and PtOEP based devices, where the small contri-

bution of TTA was attributed to the relatively small triplet con-

centration in the devices at the current densities studied.

[ 5 ]

Figure 3 f shows for the same device the decrease of the

emission at 6 V as a function of the time during a degrada-

tion simulation, assuming only TPQ-t processes and for p

degr

1. The excited polaron is then in all cases located on the dye

molecule, so that each TPQ process leads to degradation of the

dye site involved. The simulations were carried out with and

without triplet exciton diffusion. Switching off diffusion makes

the simulations computationally less intensive, so that for the

same total simulation (CPU) time the simulated time is much

larger. Under these strongly accelerated conditions, the time at

which the emission is reduced to 90% of the t = 0 value (LT90)

is (5.5 ± 1.0) and (7 ± 2) µs, respectively. For the small emitter

concentration used, the effect of diffusion of excitons to already

degraded sites is thus quite small. The ﬁ gure shows that a

stretched exponential function (dashed curve, I ∝ exp[−( t /

life

)

with a (1/e) lifetime

life

≈ 80 µs and

≈ 0.91) provides a rea-

sonable, although not perfect, ﬁ t. In Section 4 we show that dif-

fusion becomes more important at larger dye concentrations.

Simulations of this type are expected to provide a prediction of

the real lifetime for any other dye concentration and EML thick-

ness after extrapolation of the simulation results to smaller cur-

rent densities, and after a “calibration” experiment to deduce

the actual value of p

degr

3. The Uniform Density Model—The Relationship

Between Roll-Off and Lifetime

In this section, it is ﬁ rst shown how within the framework of

the uniform density model (UDM) simpliﬁ ed expressions may

be obtained for the dependence of the IQE roll-off on the mate-

rial and device parameters (subsection 3.1). The analysis leads

to a shape of the roll-off curves which is consistent with the

empirical roll-off given in Equation ( 4) , and obtained from the

3D-KMC simulations shown in Figure 3 . In subsection 3.2, it is

shown how, under the assumption that upon a triplet-polaron

quenching process subsequently (with a small probability) a

degradation process takes place, the lifetime is related to mate-

rial and device parameters. The analysis suggests that the expo-

nent which gives the superlinear decrease of the lifetime with

increasing current density in an accelerated lifetime experi-

ment may be obtained from the shape of the roll-off curve. In

Sections 4 and 5, the predictions from the model will be com-

pared to simulation results for various OLED layer structures.

3.1. Roll-Off

Within the TPQ-related degradation mechanisms considered in

this paper, the roll-off and lifetime are related. Even if for cer-

tain operational conditions the IQE loss due to TPQ is small,

it will thus be of interest to understand how it depends on the

material parameters. In general, the electron, hole and exciton

densities show a dependence on the position in the EML,

precluding the development of an analytical model for the efﬁ -

ciency roll-off. However, we ﬁ nd that it is nevertheless useful

to further develop a simpliﬁ ed model within which the carrier

density non-uniformity across the EML or across the part of the

EML within which the emission takes place (“emission zone”)

is neglected.

[ 11,12 ]

Within this uniform density model, an OLED

is regarded as a “chemical reaction vessel”, with “reactants”

(electrons and holes) and “products” (excitons). The interactions

between all these species can be complex, but the neglect of a

spatial dependence strongly simpliﬁ es the analysis. Figure 3 e

shows that it is indeed possible to obtain under some condi-

tions a quite uniform emission proﬁ le in the EML. In this

section, we extend the UDM by including the effect of a charge-

carrier density of the mobility and diffusion coefﬁ cient, and by

including a mechanistic description of the TPQ rate, leading to

a reﬁ ned expression for the roll-off.

We consider a phosphorescent OLED with in the emission

zone uniform and equal electron and hole volume densities,

= n

= n , and a uniform triplet density, n

. Under steady-state

conditions, the exciton generation rate is equal to the exciton

loss rates due to radiative and non-radiative decay, TPQ, TTA

and dissociation. Neglecting TTA (as the simulations shown in

Section 2.5 and shown by van Eersel et al.

[ 5 ]

suggest that TPQ is

more relevant) and dissociation, the triplet density balance may

under steady-state conditions be written as

TPQ T

knn

τε

()

=− − + =

(6)

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with k

TPQ

the TPQ rate coefﬁ cient, µ ( n ) the charge density

dependent mobility, which is assumed to be equal for holes and

electrons, and with the last (generation) term as expected from

the Langevin formula, neglecting the electric ﬁ eld dependence

of the exciton generation rate and neglecting a correction to

the mobility in bipolar devices.

[ 36 ]

The radiative decay rate will

be proportional to the triplet density, so that from Equation ( 6)

the IQE roll-off is given by

012

IQE

T TPQ

rad

TPQ

nk k n

ηη

()

(7)

Simple expressions for the current density dependent functions

n ( J ) and k

TPQ

( J ) may be obtained as follows. The condition of

charge conservation (the number of injected carriers is equal to

the number of excitons created, both per unit of time and area)

implies that

nnd

()

=×

(8)

with

d the thickness of the recombination zone (or, for a well-

designed OLED, the thickness of the EML). We assume that the

mobility may be expressed as

μμ

()

≅

⎛

⎝

⎜

⎞

⎠

⎟

(9)

where the prefactor µ

is the mobility as obtained by extrapola-

tion from the carrier density range of interest to the total site

density, N

(i.e., at a carrier concentration c ≡ n / N

equal to 1),

and with b > 0. The electric ﬁ eld dependence of the mobility has

been neglected. Equation ( 9) is exact for the case of an expo-

nential density of states (DOS),

[ 37 ]

and provides for a Gaussian

DOS a fair approximation in charge carrier density ranges with

a width of at least one order of magnitude.

[ 29 ]

In both cases,

b increases with increasing energetic disorder. An effective

carrier density dependence of the mobility arises also in host-

guest systems with a bimodal Gaussian density of states.

[ 38 ]

The

exponent b depends then on the energy distance between the

host and guest states. Combining Equations ( 8) and ( 9) leads to

⎛

⎝

⎜

⎞

⎠

⎟

(10)

Note that n does not depend on the triplet-polaron quench rate.

Triplet-polaron quenching is the result of encounters of dif-

fusing triplets and polarons. In phosphorescent OLEDs, with

typical dye concentrations of 5−10 mol%, the diffusion length

of triplets is very small (of the order of the Förster radius, which

we take 1.5 nm). TPQ is therefore predominantly due to the

diffusion of the charge carriers, so that k

TPQ

is proportional

to the charge carrier diffusion coefﬁ cient, D . From diffusion

theory,

[ 39 ]

it is known that for the case of a capture radius R

the rate coefﬁ cient is then given by

TPQ c

kn DnR

ππμ

() () ()

(11)

where the second step has been made using the Einstein equa-

tion which relates the diffusion coefﬁ cient to the mobility. The

TPQ rate coefﬁ cient is thus carrier density dependent. The

electric ﬁ eld dependence of the diffusion coefﬁ cient has been

neglected. For simplicity, we have used the standard form of the

Einstein equation, for non-interacting particles, instead of the

generalized Einstein equation which would lead to a correction

at high carrier densities.

[ 40 ]

From Equation ( 7–11) it follows that

IQE

rad

Ned

πτ

με

()

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

(12)

This simpliﬁ ed analysis leads thus to expression for the roll-off

which is consistent with the empirical expression (Equation 4 )

which is often found experimentally in OLEDs. Also the roll-

off as obtained from simulations for the devices studied in this

paper is quite well described using Equation 4 (see Section 2.5

and Section 4). The parameter values m and J

are given in

Table 2 for the general case and for the speciﬁ c case of a constant

mobility ( b = 0). If all other factors remain the same, the roll-off

can thus be reduced by i) decreasing the mobility in the EML (as

then the TPQ rate decreases), ii) increasing the emission zone

thickness (as then the polaron density decreases), iii) decreasing

the radiative decay time (as then radiative decay better outcom-

petes TPQ), and iv) reducing the TPQ capture radius.

The model suggests that the slope of the quench rate curve

will be in the range 0.5 to 1.0, depending on the (effective) dis-

order of the mobility in the EML. The value m = 0.80 found

for the example discussed in Section 2.5 is well in that range.

The effective mobility exponent b is then approximately equal

to 3. Interestingly, only a relatively weak sensitivity of the rel-

ative quench rate and J

to the mobility prefactor

is then

expected, viz. Γ

/ Γ ∝ µ

−0.2

and J

∝ µ

−0.25

3.2. Lifetime

Within the UDM, the degradation rate of an emitter molecule

in the EML is given by

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Table 2. Summary of roll-off parameters for the case of uniform and

equal carrier densities, equal electron and hole mobility functions, equal

exciton-polaron quench rates, and no dissociation. For the case

b = 0, we have µ

= µ .

General case

Constant mobility ( b = 0)

πτ

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎤

⎦

⎥

kTR32

εμ π τ

()

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degr

q qd degr degr

gp p p f

()

=× × × ×−

(13)

with

degr

( t ) the fraction of the emitter molecules which on time

t has already degraded, p

the probability that an exciton is lost

due to TPQ, p

the probability that the TPQ process occurs on a

dye site (which depends on the type of TPQ-process considered

and on the emitter density, see Section 5), and p

degr

the proba-

bility that an excited polaronic dye molecule degrades. The value

of p

degr

depends on the chemical stability of the molecule, and

must be determined experimentally. Within the framework of

the UDM, the exciton generation rate per dye site is equal to

dye

edn

××

(14)

with

dye

the volume density of emitter molecules in the EML.

If the light emission intensity I ( t ) is proportional to the fraction

of non-degraded sites, as expected in the absence of exciton dif-

fusion, it is (from Equation ( 13) ) given by

10exp 0

degr

life,nd

It f I

()

() () ()

=− = −

⎛

⎝

⎜

⎞

⎠

⎟

(15)

with a (1/e) lifetime when no exciton diffusion is included,

life,nd

, given by

life,nd

qqddegr

dye

pp p

edn

=×

××

(16)

Using

that p

= (1 −

IQE

rad

), when writing

IQE

rad

[1+( J / J

)

], where J

and m can be obtained from Table 2 , it

follows that p

≅ ( J / J

)

for small current densities ( J << J

so that then

life,nd

qd degr

dye 50

edn J

≅×

×× ×

(17)

The model thus establishes a relationship between the roll-

off and the lifetime in the absence of exciton diffusion. If

valid, Equation ( 17) can be used to predict

life,nd

already from

a steady-state simulation, instead of from a usually computa-

tionally more demanding explicit 3D-KMC lifetime simulation.

For the case of the systems discussed in Section 2.5, the dye

concentration dependence of the lifetime predicted by Equation

( 17) will be investigated in Section 5.

Equation ( 17) furthermore leads to an algebraic dependence

of the lifetime on the current density,

life

∝ J

−n

, with a current

density acceleration exponent n = m + 1. As m was predicted to

fall in the range 0.5 to 1, n will be in the range 1.5 to 2. Such a

current density dependence is indeed often observed in acceler-

ated lifetime experiments.

[ 41 ]

4. Sensitivity of the Roll-Off to Material

Parameters

The beneﬁ ts of using the symmetric mixed-matrix OLEDs of

the type shown in Figure 3 a are found to be quite similar to

those of the otherwise identical single-matrix OLEDs studied by

van Eersel et al.

[ 5 ]

Due to the use of dyes which act as hole and

electron traps with a trap depth, Δ

, around 0.2 eV, the carriers

are slowed down in the EML, so that no charge accumulation

occurs in the EML near the interfaces. At voltages around 3 V

and below, the emission proﬁ les are quite uniform, leading to a

small IQE-loss. In this section we further analyze the sensitivity

of the roll-off to Δ

, to a hole (electron) injection barrier Δ

from

the HTL to the host in the EML (from the ETL to the host in

the EML), and to various other material parameters including

the triplet emissive lifetime, triplet binding energy and the

Förster radius for triplet diffusion. Figure 4 a shows the energy

level structure of the devices studied. The analysis leads to two

energy level design rules from which the roll-off can be mini-

mized without introducing a large voltage increase.

Figure 4 b shows the quench rate curves for devices of the

type shown in Figure 4 a, with variable guest trap depths Δ

in the range 0–0.5 eV and with Δ

= 0 eV. The simulations

were carried out for a series of voltages in the range 3–6 V, for

the parameter values given in Table 1 and for a typical Ir-dye

concentration of 4 mol%. Results for the case Δ

= 0.2 eV were

already given in Figure 3 . The ﬁ gure conﬁ rms in a quantita-

tive manner the more qualitative picture mentioned above. It

may be seen that for all values of Δ

, the empirical expression

(Equation ( 4) ) for the roll-off is well obeyed, with a value of J

which increases until it saturates for Δ

≅ 0.2 eV. The use of

deeper traps leads to a signiﬁ cant reduction of the current den-

sity at a given voltage, and is therefore unfavorable from a point

of view of optimizing the power efﬁ ciency. That may be seen

from the current density at a ﬁ xed voltage, that is, 3 V and 4 V

(indicated in the ﬁ gures by large and ﬁ lled symbols). We ﬁ nd

that variations of i) the disorder energy

, from 0.075 to 0.125 eV,

ii) the EML layer thickness d , from 10 to 30 nm (at a ﬁ xed 100 nm

total thickness), or iii) the wavefunction decay length

(from

0.3 to 0.1 nm) have no signiﬁ cant or only little effect on this

picture. For single-matrix OLEDs we ﬁ nd that for Δ

< 0.2 eV

the quench rate is a factor ≈1.5 larger than for the mixed-matrix

devices, and that saturation occurs at Δ

≅ 0.25–0.3 eV; the

quench rate curves coincide then to those shown in Figure 4 b.

We attribute this to the somewhat larger mobility in single-

matrix OLEDs, as all host molecules contribute to the transport,

so that deeper traps are needed to obtain a uniform emission

proﬁ le. A ﬁ rst design rule for symmetric OLEDs is thus that the

dye trap depth Δ

should ideally be ≈0.2 eV or 0.25–0.3 eV, for

mixed-matrix or single-matrix OLEDs, respectively.

As a second design rule, we ﬁ nd that the HTL-EML(host)

and ETL-EML(host) injection barriers Δ

should ideally be

(0.0 ± 0.1) eV. This may be concluded from Figure 4 c, which

shows the effect of variations of Δ

from −0.3 to +0.3 eV, for a ﬁ xed

value Δ

= 0.2 eV. For positive injection barriers, accumulation of

charges occurs in the transport layers, which enhances the TPQ

rate near the EML interfaces. When decreasing the barrier to

≅ 0, the decrease of the relative quench rates saturates.

Although a further reduction of Δ

does not further diminish the

roll-off, it gives rise to an increase of the voltage required to obtain

a certain current density, as dissipation occurs due to the energy

loss when the carriers enter the EML.

Figure 4 c shows that the quench rate at small current den-

sities and the slope parameter m characterizing the relative

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quench rate curves do not only depend on the transport and

excitonic parameters in the EML, but also on the injection

boundary conditions. We ﬁ nd that their effect on the shape of

the emission proﬁ les, and thereby on the roll-off, cannot be

neglected. Giving a detailed quantitative analysis is beyond the

scope of this paper; we limit ourselves here to a brief qualita-

tive discussion. In the absence of injection barriers (Δ

≤ 0 eV),

the emission proﬁ les for systems with Δ

≥ 0.2 eV are found

to be quite uniform at low voltages. However, above 3 V, only

part of the exciton generation occurs in the bulk of the EML. A

fraction of the charges passes the EML and is ﬁ nally blocked.

Charge accumulation zones develop near the EML interfaces,

and the emission originates to an increasing extent from these

interfacial zones. The gradual splitting of the emission proﬁ le

into two narrow peaks near the interfaces is expected to con-

tribute to the large slope of the relative quench rate curves.

Using Equation ( 10) , we deduce for such devices a value of

b ≅ 1.33 from an analysis of the carrier density dependence of

the current density in the center of the EML (where due to the

symmetry the electron and hole densities are equal). Within the

framework of the UDM, this value would lead to m ≅ 0.7, which

is indeed somewhat smaller than the value m ≅ 0.80 obtained

from the quench rate curves. In the presence of large barriers

for carrier injection into the EML (Δ

≥ 0.2 eV), charge accumu-

lation occurs already at small voltages near the EML interfaces.

The emission is then almost completely conﬁ ned to the two

thin interfacial zones. As a result of the large local carrier den-

sity, TPQ leads already at low current densities to a relatively

large quench rate. With increasing voltage (and current den-

sity), the effect of the injection barriers at the EML interfaces

is reduced. There is then less charge accumulation near the

interfaces, and a larger fraction of the exciton formation occurs

in the bulk of the EML, where the charge density is relatively

small. These two effects explain the small slope of the relative

quench rate curve for such cases, for example, m ≅ 0.20 for

devices with Δ

= 0.3 eV (as may be deduced from Figure 4 c).

As a next step, we have investigated the sensitivity of the

roll-off to a variation of the hopping attempt rate to the ﬁ rst

neighbor,

, throughout the entire device. This corresponds to

a variation of the mobility prefactor µ

introduced in Section 3.

The simulations conﬁ rm the expectation that for the symmetric

devices studied the relative quench rate curves are only weakly

sensitive to µ

. For simulations with

enhanced or decreased

by a factor of 100 as compared to the value given in Table 1 ,

the relative quench rates as extrapolated to J = 10 A/m

are

found to differ by only a factor ≈5.

[ 42 ]

From Equation ( 12) ,

a factor ≈6.3 would be expected when assuming the value

b = 3.0 which would follow from the observed slope parameter

m = 0.80. At larger current densities an even smaller effect of

changing

was found; the J

-values were only different by a

factor ≈2. We emphasize that this result does not imply that

the hopping attempt rates are not important to the roll-off of

OLEDs. Layer-speciﬁ c or charge carrier-speciﬁ c differences, for

example, may result in a non-uniform emission proﬁ le and

hence in an increased efﬁ ciency loss.

Figure 5 a,b show the sensitivity of the quench rate curves to

the effective radiative decay time

, at a ﬁ xed value of the radia-

tive decay efﬁ ciency (

rad

= 0.75), for a triplet exciton binding

energy, E

T, b

, equal to 1.0 and 0.5 eV, respectively. Simulation

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Figure 4. a) Energy level structure of the symmetric OLEDs studied. In

the EML, a 50:50 mixture of two materials with a 4.0 eV single-particle

energy gap is used. b) Quench rate curves for OLEDs as show in (a),

with a varying energy distance Δ

between the host and guest levels in

the EML and with Δ

= 0 eV. The results shown in the ﬁ gure provide a

motivation for the ﬁ rst OLEDs design rule, discussed in Section 4. c)

Quench rate curves for OLEDs as shown in (a), with ﬁ xed energy levels

in the EML (Δ

= 0.2 eV), but with varying barrier heights Δ

from the

HTL and the ETL to the host levels in the EML. The Δ

= 0.0 eV case is

as shown in Figure 3 . The results shown in the ﬁ gure provide a motiva-

tion for the second OLEDs design rule, discussed in Section 4. In both

ﬁ gures, large and ﬁ lled symbols indicate the data points obtained at 3

and 4 V. The statistical uncertainty of the data points is for J > 10

A/m

of the order of the symbol sizes, as may be judged from a comparison

with the linear dashed guides-to-the-eye, but becomes somewhat larger

for smaller current densities.

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results for 0.75 eV (not shown) are essentially the same as for

1.0 eV. As mentioned in Section 2.4, a signiﬁ cant dissociation

rate is expected unless E

T, b

is well above E

CT,b

= 0.48 eV. This

is conﬁ rmed by simulation results. If E

T, b

is 0.75 eV or higher,

the quenching loss is essentially independent of E

T, b

. Further-

more, it is simply inversely proportional to

(dashed lines) as

expected from Equation ( 12) . The roll-off is sensitive to E

T, b

below a value of 0.75 eV, as may be seen from the J

( E

T, b

)

dependence shown in Figure 6 (closed symbols). For E

T, b

0.5 eV, the quench rate is signiﬁ cantly enhanced, in particular

at small current densities, so that the roll-off curves do not

obey Equation ( 4) anymore. The observed reduction of the

slope of the quench rate curves, to values which can be smaller

then 0.5, may be understood from an extension of the UDM

which includes dissociation.

[ 42 ]

The observation that at very

large current densities the effect of dissociation becomes less

may be explained by considering that then such a high volume

density of charge carriers is available for exciton formation

that very soon after dissociation already new pairs are formed.

At the 4 mol% dye concentration assumed in the pre-

ceding simulations, and for the small Förster radius for triplet

diffusion assumed ( R

F,diff

= 1.5 nm), triplet diffusion does

not contribute signiﬁ cantly to the TPQ rate. In Figure 7 , the

results of a study of the role of triplet diffusion are shown.

For clarity, Dexter-type diffusion has been switched off. Var-

ying R

F,diff

in steps of 1.5 nm reveals only a signiﬁ cant con-

tribution to the roll-off for R

F,diff

equal to 3 nm and larger,

for the default 4 mol% dye concentration as well as for

8 mol% devices. The ﬁ gure also shows that increasing the

dye concentration increases the sensitivity to triplet diffusion,

as expected. The effect of triplet diffusion is largest at small

current densities, so that the quench rate curves become non-

linear, as due to the carrier density dependence of the polaron

diffusion coefﬁ cient the relative role of polaron diffusion is

larger at large current densities. We conclude that for dye con-

centrations of 8 mol% or less, the uncertainty concerning the

most appropriate description of triplet diffusion is expected to

contribute only a small uncertainty to the simulation results

presented above. We note that triplet diffusion can also con-

tribute to less roll-off, viz. when in the case of a strongly non-

uniform emission proﬁ le the triplets migrate to regions with a

smaller polaron density.

[ 43 ]

Realizing OLEDs which satisfy the ﬁ rst design rule (equal

≈0.2 eV hole and electron trap depths due to the guest levels

in the EML) is in practice not always well possible, due to a

lack of suitable host materials with accurately tuned HOMO

and LUMO levels. It is therefore of interest to investigate the

sensitivity to deviations of the precise relative positions of the

host and guest energy levels. The open circles in Figure 5 show

the results of such a study, carried out for asymmetric mixed-

matrix OLEDs with Δ

1,h

= 0.3 eV hole trap depth and a Δ

1,e

0 eV electron trap depth (as shown in the inset). By choosing

unequal concentrations of the hole and electron transporting

Adv. Funct. Mater. 2015, 25, 2024–2037

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Figure 5. Current density dependence of the relative quench rate for

OLEDs as shown in Figure 3 a for various effective radiative decay times

relative to the default value (

= 1.38 µs, see Table 1 ), for a) E

T, b

1.0 eV, and b) E

T, b

= 0.5 eV. Thick curves indicate the results for

, and

dashed lines indicate the linear

dependence which is expected in the

absence of dissociation, and using the E

T, b

= 1.0 eV result as a reference.

The statistical uncertainty of the data points is of the order of the symbol

sizes, or smaller.

Figure 6. Closed circles: dependence of the 50% roll-off current density,

, on the triplet exciton binding energy, for symmetric mixed-matrix

OLEDs with an energy level structure as given in Figure 3 a (closed sym-

bols). Open circles: J

for asymmetric mixed OLEDs with the same HTL,

EML (host), and ETL energy level structure, but with a guest HOMO

and LUMO level structure as shown in the inset and with in the EML a

composition as indicated. The statistical uncertainty of the data points is

of the order of the symbol size.

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host materials in the EML (76 and 20 mol%, respectively), the

reduction of the effective hole mobility due to the trapping

of holes at dye sites is partially compensated. The simulation

results show a much stronger sensitivity of the roll-off to E

T, b

than for the case of a symmetric OLED with Δ

1,h

= Δ

1,e

= 0.2

eV (closed spheres). This may be attributed to the strongly

enhanced triplet exciton dissociation probability, as also CT

states consisting of a hole on a dye site and an electron on a

host site can be formed.

In view of the sensitivity of the roll-off to E

T, b

, it is impor-

tant to be able to obtain its value accurately from experiment.

By deﬁ nition, E

T, b

= E

LUMO

− E

HOMO

− E

. It can therefore be

obtained, at least in principle, from a determination of the

single particle energy gap, E

LUMO

− E

HOMO

, and a measurement

of the triplet energy. However, the accuracy of commonly used

methods for obtaining E

HOMO

and E

LUMO

from e.g., cyclic vol-

tammetry, from a combination of photoelectron spectroscopy

and inverse photoelectron spectroscopy, or from density func-

tional theory calculations, is a subject of current debate.

[ 5,44,45 ]

Partial information follows from the difference between the

singlet and triplet exciton energies, which for the commonly

used green emitter material Ir(ppy)

is approximately 0.4 eV.

[ 46 ]

Assuming a singlet exciton binding energy of a few tenths of an

eV, E

T, b

would then be around 0.6–0.9 eV.

5. Sensitivity of the OLED Lifetime to the Ir-Dye

Concentration

In this section, a comparison is given between the results of

explicit 3D-KMC simulations of the lifetime of OLEDs of

the type shown in Figure 3 a, for a range of Ir-dye concentra-

tions, and the prediction obtained from the UDM discussed

in Section 4. When exciton diffusion may be neglected, a fair

agreement is obtained for the TPQ-p and TPQ-t degradation

scenarios. From a comparison of the simulated lifetimes with

the experimental lifetime in state-of-the-art devices, an estimate

is made of the degradation probability per TPQ process, p

degr

Figure 8 shows the results of 3D-KMC simulations of the

lifetime of OLEDs of the type shown in Figure 3 a as a function

of the Ir-dye concentration in the EML, at a ﬁ xed voltage (6 V),

for the case of TPQ-p and TPQ-t processes (see Figure 2 ). The

simulations were carried out with the purpose to investigate the

usefulness of the UDM as a means to predict the OLED life-

time from the results of steady-state simulations, and to inves-

tigate to what extent exciton diffusion (which was neglected in

the UDM) affects the results. Typical examples of the raw simu-

lation data were already shown in Figure 2 f.

In the case of a TPQ-t process, the excited-state polaron is

created on the molecule where the exciton was located. With a

very high probability, around 0.97 and 0.98 for Ir-dye concentra-

tions x

= 2 and 4 mol%, respectively, and for higher concen-

trations essentially 1, the excitons are located on the dye sites.

Each TPQ process then leads to the conversion of an emissive

dye molecule into a degraded (non-emissive) dye molecule, so

that p

≅ 1 in Equation ( 17) . It follows from Equation ( 17) that

the lifetime is then expected to be proportional to the volume

density of Ir-dye molecules, if all other factors remain constant.

However, the actual concentration dependence is affected by

the concentration dependence of the current density, which

is found to be minimal around x

≅ 9 mol%. At smaller con-

centrations, the dyes act predominantly as trap sites, so that

the mobility then decreases with increasing x

, whereas at

larger concentrations direct dye-dye transport is the predomi-

nant transport mechanism. The mobility then increases with

increasing x

. Figure 8 (closed spheres) shows that the pre-

dicted LT90 lifetime indeed increases for small x

, but that it

shows a maximum around 15 mol%. The lifetime as obtained

from explicit 3D-KMC simulations is for small dye concentra-

tions within a factor 2 of the predicted value, but is smaller by

a factor of approximately 4 for large concentrations. The differ-

ence at large dye concentrations is mainly due to the neglect of

exciton diffusion in the UDM, as may be concluded from the

observation that simulations within which diffusion is switched

off (small circles) agree then excellently with the prediction

from the UDM. The small discrepancy at small concentrations

might be due to deviations from a uniform emission proﬁ le,

either laterally or in the direction perpendicular to the EML.

Upon a TPQ-p process, the excited-state polaron is cre-

ated on the molecule at which the polaron was located. From

the steady-state simulations, we ﬁ nd that the probability that

TPQ occurs on a dye site ( p

) is to an excellent approximation

proportional to x

, and approximately 0.42 for x

= 0.1. As a

result, the lifetime as predicted from Equation ( 17) is approxi-

mately independent of x

(closed squares in Figure 8 ), and

larger than as expected for the TPQ-t scenario. The explicit

3D-KMC simulation results (open squares) conﬁ rm the

latter expectation. However, the x

dependence is larger than

expected. A preliminary analysis suggests that the statistics of

the occurrence of emitter pairs strongly affects the lifetime for

this case. As degradation only occurs when the exciton and

polaron reside both on a dye molecule, it is for very small

rarer than would follow from continuum statistics. That

explains the higher than expected lifetime for the 2 mol%

Adv. Funct. Mater. 2015, 25, 2024–2037

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Figure 7. Dependence of the quenching curves on the Förster radius for

triplet exciton diffusion, R

F,diff

, for two values of the Ir-dye concentration,

obtained from simulations for the OLED shown in Figure 3 a. The full and

dashed curves are guides-to-the-eye, consistent with the numerical accu-

racy of the simulation data. The statistical uncertainty of the data points

is equal to symbol sizes, or smaller, except for the data points close to

J = 10

A/m

, where it is slightly larger.

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case, and the observation of saturation of the degradation, viz.

when each cluster of Ir-dye molecules contains a degraded

molecule. The emission occurs then from the isolated dye

molecules. Giving more in-depth analyses for both scenarios

is beyond the scope of this paper.

The LT90 lifetime obtained from the simulations is in the

range 5–50 µs when taking p

degr

= 1, and for x

in the 5–10

mol% range. The current density at 6 V as obtained from the

simulations is approximately 4000 A/m

. In efﬁ cient stacked

state-of-the-art white OLEDs operated at a luminance equal

to 1000 cd/m

, the current density is much smaller, around

10 A/m

. Assuming a current density acceleration exponent

equal to 1.8 (as expected from the analysis given in Section 3.2),

an actual LT90 lifetime of 10,000 hours would then imply an

average value p

degr

of the order 10

−8

. A method for deducing

that probability from the observed luminance degradation

versus time and from transient photoluminescence studies

of degraded devices has been demonstrated by Giebink et

al.

[ 9 ]

Interestingly, for the systems studied, with in the EML a

typical blue-emitting Ir-dye molecule embedded in a 4,4′-bis(3-

methylcarbazole-9-yl)-2,2′-biphenyl (mCBP) host, a similar deg-

radation probability per triplet-polaron encounter was found,

viz. roughly 2 × 10

−9

We note that the validity of the assumption that the degrada-

tion probability p

degr

is independent of the material composition

and simulation parameters has yet to be proven. It is conceiv-

able that protective mechanisms leading to fast de-excitation of

the excited polarons depend on, for example, the dye concentra-

tion or the applied electric ﬁ eld. If needed, such mechanisms

can be included in the 3D-KMC approach developed.

6. Summary, Conclusions, and Outlook

Results have been presented of three-dimensional kinetic

Monte Carlo (3D-KMC) simulations of the degradation in phos-

phorescent OLEDs, based on scenarios in which the emissive

dye molecules are converted to non-emissive molecules upon

a triplet-polaron quenching (TPQ) process. For the simulation

parameters assumed, TPQ processes provide the predominant

contribution to the IQE roll-off with increasing current density.

Therefore, the roll-off and lifetime are related. From a sim-

pliﬁ ed model, assuming uniform charge carrier and exciton

densities across the emissive layer (EML), an analytical expres-

sion for the roll-off has been obtained in terms of the mate-

rials parameters. Within the same approach, an expression

has been obtained (Equation 17 ) from which the lifetime due

to TPQ-induced degradation can be obtained from steady-state

simulations. The model is found to provide a fair prediction of

the Ir-dye concentration dependence of the lifetime in a model

system, provided that exciton diffusion may be neglected.

The 3D-KMC model provides a means to investigate the

sensitivity of the roll-off and lifetime to the physical processes

assumed, to the materials-speciﬁ c parameters and to the layer

stack architecture. In this paper, we have explored some of

these relationships, including the sensitivity to energy level

values, the mobility, the triplet binding energy, the emissive

lifetime, energetic disorder, the mobility and exciton diffu-

sion. We have established how in idealized symmetric OLEDs

the roll-off can be minimized by following design rules for the

trap depths of the dyes in the EML and for the injection bar-

riers from the transport layers to the EML. Furthermore, it was

found that the roll-off is quite sensitive to the triplet exciton

binding energy, E

T, b

, if that would be smaller than 0.75 eV (i.e.,

within the framework of our formalism less than 0.25 eV above

the binding energy of CT-excitons with the electron and hole on

nearest neighbor sites).

It follows from our results that the development of more

accurate methods for determining the HOMO and LUMO

energy levels and E

T, b

, with an accuracy better than 0.1–0.2 eV,

would make the simulations more versatile. Furthermore,

simulation-assisted OLED lifetime studies should be carried

out in order to establish the degradation mechanism. The sim-

ulation results suggest that it would be of interest to vary in

such a study the emitter concentration as a means to make a

distinction between different degradation scenarios.

Acknowledgements

The authors would like to thank the Philips Research e-Science

department for technical support. This research was supported by the

Dutch nanotechnology program NanoNextNL.

Received: July 28, 2014

Revised: September 25, 2014

Published online: November 4, 2014

[1] S. Reineke , M. Thomschke , B. Lüssem , K. Leo , Rev. Mod. Phys.

2013 , 85 , 1245 .

Adv. Funct. Mater. 2015, 25, 2024–2037

www.afm-journal.de

www.MaterialsViews.com

Figure 8. Results of explicit 3D-KMC lifetime for the OLEDs shown in

Figure 3 a (open symbols) as a function of the Ir-dye concentration, at 6 V,

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