A quantitative numerical model of multilayer vapor deposited organic light emitting

A quantitative numerical model of multilayer vapor-deposited organic light emitting
diodes
J. Staudigel, M. Stößel, F. Steuber, and J. Simmerer
Citation: Journal of Applied Physics 86, 3895 (1999); doi: 10.1063/1.371306
View online: http://dx.doi.org/10.1063/1.371306
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/86/7?ver=pdfcov
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A quantitative numerical model of multilayer vapor-deposited organic light
emitting diodes
J. Staudigela)
Siemens AG, Corporate Technology, ZT MF 1, Paul-Gossen-Strasse 100, D-91052 Erlangen
and Universita¨t Erlangen-Nu¨rnberg, Institut fu¨r Angewandte Physik, D-91058 Erlangen, Germany
M. Sto¨ßel and F. Steuber
Siemens AG, Corporate Technology, ZT MF 1, Paul-Gossen-Strasse 100, D-91052 Erlangen
and Universita¨t Erlangen-Nu¨rnberg, Institut fu¨r Werkstoffwissenschaften VI, D-91058 Erlangen, Germany
J. Simmerer
Siemens AG, Corporate Technology, ZT MF 1, Paul-Gossen-Strasse 100, D-91052 Erlangen, Germany
͑Received 22 February 1999; accepted for publication 8 June 1999͒
A one-dimensional numerical model for the quantitative simulation of multilayer organic light
emitting diodes ͑OLEDs͒ is presented. It encompasses bipolar charge carrier drift with
field-dependent mobilities and space charge effects, charge carrier diffusion, trapping, bulk and
interface recombination, singlet exciton diffusion and quenching effects. Using field-dependent
mobility data measured on unipolar single layer devices, reported energetic levels of highest
occupied and lowest unoccupied molecular orbitals, and realistic assumptions for experimentally not
direct accessible parameters, current density and luminance of state-of-the-art undoped
vapor-deposited two- and three-layer OLEDs with maximum luminance exceeding 10000 cd/m2
were successfully simulated over 4 orders of magnitude. For an adequate description of these
multilayer OLEDs with energetic barriers at interfaces between two adjacent organic layers, the
model also includes a simple theory of charge carrier barrier crossing and recombination at organic–
organic interfaces. The discrete nature of amorphous molecular organic solids is reflected in the
model by a spatial discretization according to actual molecule monolayers, with hopping processes
for charge carrier and energy transport between neighboring monolayers. © 1999 American
Institute of Physics. ͓S0021-8979͑99͒02218-5͔
I. INTRODUCTION
Ever since the first appearances of efficient organic light
emitting diodes ͑OLEDs͒,1–4
their properties and the mecha-
nisms of the physical processes involved were subject to ana-
lytical and numerical investigation. Basic understanding of
OLED properties still remains a crucial point for the devel-
opment of highly efficient devices, which meet the require-
ment specifications of typical target applications.
Several theoretical models for OLEDs have been pro-
posed in the past, each starting from different assumptions to
simplify the complex interplay of mechanisms involved. Due
to the large variety of known OLED setups, suitable materi-
als, and mechanisms governing their behavior, distinct mod-
els were developed to explain the experimental findings.5–17
The proposed models comprise injection limited current
flow,5,6,9
the transition from injection to bulk limitation,7,10,13
and purely space charge limited current ͑SCLC͒
flow,11,12,15,16
either on monolayer5,9,12–17
or multilayer6–8,11
devices.
The present article comprises the numerical modeling of
undoped multilayer vapor-deposited OLEDs based on small
functional molecules. For this purpose, a numerical rather
than an analytical model was chosen for an adequate imple-
mentation of the discrete nature of the amorphous molecular
organic solids. The simulation results are compared with ex-
perimental data in order to gain further insight into the
mechanisms governing the behavior of our processed
OLEDs, with special respect to the implications for device
optimization.
In Sec. III, we will present dark current transients of
hole- and electron-only single layer samples which, in addi-
tion to a quantitative agreement between measured steady
state current densities and predictions from space charge lim-
ited current theory, prove space charge limitation of the cur-
rent flow.18
Thus, the assumption of ohmic contacts both for
hole and electron injection in our OLED model, which will
be presented in detail in Sec. II, is experimentally verified.
The electro-optical characteristics of our undoped vapor-
deposited two- and three-layer OLEDs showing a maximum
luminance of more than 10 000 cd/m2
, were successfully
simulated over 4 orders of magnitude in current density and
luminance. Starting from measured field-dependent electron
and hole mobilities ͑which were derived from our own ex-
perimental data based on the evaluation of transient SCLC
peaks18
͒, reported data on energetic levels of lowest unoccu-
pied and highest occuppied molecular orbitals ͑LUMOs,
HOMOs͒,19
and realistic assumptions for experimentally not
direct accessible parameters, we will deduce the built-in po-
tential, average trap depths and trap concentrations of our
processed two-layer devices by a comparison of simulateda͒
Electronic mail: joerg.staudigel@erls.siemens.de
JOURNAL OF APPLIED PHYSICS VOLUME 86, NUMBER 7 1 OCTOBER 1999
38950021-8979/99/86(7)/3895/16/$15.00 © 1999 American Institute of Physics
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and measured current densities in Sec. III. The energetic dis-
order parameter at the interface between hole transport layer
͑HTL͒ and electron transport layer ͑ETL͒, which governs the
probability of bulk exciton formation from interface recom-
bination, is then derived from the comparison of simulated
and measured luminance data.
The experimentally observed improvements in the
electro-optical characteristics from two- to three-layer de-
vices ͑the latter comprising an additional HTL͒, such as re-
duced current density and enhanced luminance, can then be
understood in terms of different charge carrier accumulation
zones arising from different energetic barriers for charge car-
riers at the organic–organic interfaces. Detailed spatial dis-
tributions of electric field, charge carrier densities and singlet
exciton density inside the organic layers, which will be dis-
cussed in Sec. IV, yield new physical insight into the differ-
ent mechanisms governing the properties of state-of-the-art
two- and three-layer devices.
II. THE MODEL
Most models for the simulation of OLEDs start with the
implicit or explicit assumption of a spatially continuous dis-
tribution of the variables of state.6–17
In those cases where no
analytic solution for the steady state can be derived from the
differential equation system used for describing the physical
processes, variables of state and equation system are only
discretized to gain numeric solutions for the steady state un-
der given boundary conditions, with no direct size and posi-
tion relation between supporting points and molecules.13–16
In this paper, we will choose a different approach which
partly reflects the discrete nature of amorphous molecular
organic thin films, since the diameters of commonly used
organic molecules are only between one and two orders of
magnitude smaller than actual layer thicknesses.
In our model we describe the organic layers as a stack of
discrete molecule monolayers sandwiched between an anode
and a cathode. The distance between the centers of two
neighboring monolayers is expected to be equivalent to the
average diameter dM of the molecules. The use of an aver-
aged molecule diameter in spite of different molecule species
present in the organic layers can be justified by the fact that
all molecules are of comparable size. Each molecule is as-
sumed to have six next neighbors: four in the same mono-
layer and one in each of the two neighboring monolayers,
respectively. These more or less arbitrary assumptions of
molecule arrangement are helpful in order to simplify nu-
meric algorithms, but have definitely no impact on the physi-
cal relevance of the simulation results, since effects arising
from the spatial disorder of the molecules are implemented
using field- and temperature-dependent charge carrier mo-
bilities.
The z axis is defined to be oriented normal to the elec-
trode surfaces, starting from the anode ͑see Fig. 1͒. Lateral
invariance is assumed, which means that the variables of
state vary only along the z axis, whereas their values in a
certain monolayer parallel to the electrodes remain spatially
constant. This reduction to a one-dimensional description
hinders, e.g., the simulation of effects due to laterally inho-
mogeneous charge carrier injection or localized high current
flow due to point defects, but simplifies actual calculations to
a great extent.
The variables of state are given by the vector x
ϭ(x1
,x2
, . . . ,xN
), with xi
denoting all variables of state of
concern in the ith monolayer above the anode ͑see Fig. 1͒.
Extrinsic parameters, which have an influence on the system
behavior, are given in y. The system time evolution is then
given as a function of x and y
‫ץ‬x
‫ץ‬t
ϭf͑x,y͒. ͑1͒
There are two obvious ways to calculate the steady state
solution xss of this equation. The first way is the direct cal-
culation of xss by solving the equation f(xss ,y)ϭ0 under
given boundary conditions.15,16
This is straightforward for
linear equation systems, whereas for OLEDs this direct so-
lution can only be approximated, since Eq. ͑1͒ represents in
our case a system of nonlinear equations.
Our approach will follow the second way, i.e., the direct
calculation of the time evolution until the steady state xss is
achieved.13,14
Basically, the numeric calculation of a continu-
ous time evolution is not accessible; consequently, discrete
time steps have to be performed. A simple discretization in
time, given by xЈϭxϩf(x,y)•⌬t, with x and xЈ denoting the
variables of state before and after the time step ⌬t, forces
unacceptable small time steps because of an under- or over-
estimation of the actual change during the time step. This is
due to the fact that the transformation of f(x,y) in f(xЈ,y)
during the time step is neglected. This effect can be compen-
sated in first order approximation by an iterative calculation
of the time steps, with an iteration given by20
xnϩ1Ј ϭxϩ 1
2 •͓f͑x,y͒ϩf͑xnЈ ,y͔͒•⌬t. ͑2͒
The iteration starts at x0Јϭx, and the variables of state after
the time step are given by xЈϭlimn→ϱxnЈ . For practical cal-
culations the iteration can be stopped at nϭ3. The value of
⌬t is adjusted after each time step in a way that the expected
maximum relative change during the next time step in any of
the variables of state remains small. If this next time step
yields a higher than expected change in any of the variables
of state, the time step will be repeated with a reduced ⌬t.
FIG. 1. Discretization used for the organic layers. Beginning at the top, the
direction of the electric field F for a positively biased ITO anode can be
seen. Further, starting from the ITO anode, the molecule monolayers, shown
as dots, are numbered from 1 to N. The numbering of the interfaces between
two neighboring monolayers is also shown, whereby 0 denotes the anode–
organic and N the organic–cathode interface. At the bottom, the z axis,
starting at the anode, the thickness dM of a molecule monolayer and the total
thickness L of the organic layers are shown.
3896 J. Appl. Phys., Vol. 86, No. 7, 1 October 1999 Staudigel et al.
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In the following the variables of state x and their pro-
posed change during a time step ⌬xϭf(x,y)•⌬t will be
identified with variables of state and corresponding processes
in OLEDs. The result will be a nonlinear equation system,
which is of the same type as an equation system obtained
from a spatial and temporal discretization of a second order
partial differential equation system. In normal cases such
equation systems, including drift and diffusion terms, need
boundary conditions to ensure that the obtained solutions are
independent of the chosen discretization. In our special case,
however, there exists a self-regulation mechanism, namely
the decrease of the electric field at the electrodes ͑causing a
decrease of the charge carrier injection͒ due to space charges
in the bulk material. This self-regulation allows, in combina-
tion with the fixed spatial discretization in molecule mono-
layers, the reduction of the boundary condition problem to an
initial value problem, namely the state of the system at the
beginning, making the calculation of the time evolution a
whole lot easier.
In Secs. II A–II L the model will be presented in detail.
Sections II B–II I refer to the mechanisms governing the
electronic characteristics of OLEDs, whereas the implemen-
tation of the optical characteristics will be outlined in Secs.
II J and II K.
A. Variables of state
The total amount of charge carriers in a real device is
divided into free and trapped charge carriers in our OLED
model.11,13,15
Free charge carriers are expected to have an
average mobility equivalent to the mobilities obtained from
dark current transients.18
Trapping of charge carriers is as-
sumed to occur either due to structural disorder or
impurities.12
For an adequate description of OLED behavior consid-
ering these assumptions, the variables of state x must at least
comprise the free and trapped hole densities pf and pt , the
free and trapped electron densities nf and nt , and the singlet
exciton concentration s. pf
i
, pt
i
, nf
i
, nt
i
, and si
denote the
according densities in the ith monolayer above the anode.
Triplet excitons are not expected to be of any major influ-
ence on device properties, which includes the neglect of
triplet–triplet annihilation processes.21
The extrinsic parameters y are given by the applied volt-
age V and the temperature T.
B. The electric field
The applied voltage V causes the existence of an electric
field in the organic layers. The average electric field F¯ for a
total layer thickness L is given by
F¯ ϭ
VϪVbi
L
, ͑3͒
where Vbi denotes the built-in potential due to different work
functions of the electrode materials and possible dipole lay-
ers at the electrode–organic interfaces. The voltage V is de-
fined to be positive for a positively biased anode.
Free and trapped charge carriers in OLEDs consisting of
the later on mentioned organic materials ͑see Fig. 4͒ are to a
major extent excess charge carriers, so that their influence on
the local electric field Fi
in the ith monolayer has to be
considered. The electric field Fif
i
at the interface between two
adjacent monolayers ͑see Fig. 1͒ must consider all excess
charges, given by the excess charge density
i
ϭe•͓͑pf
i
Ϫpf,0
i
͒ϩ͑pt
i
Ϫpt,0
i
͒Ϫ͑nf
i
Ϫnf,0
i
͒Ϫ͑nt
i
Ϫnt,0
i
͔͒, ͑4͒
with the elementary charge e and the intrinsic free and
trapped charge carrier densities pf,0
i
, pt,0
i
, nf,0
i
, and nt,0
i
, in
the organic monolayers below and above:
Fif
i
ϭ
dM
2␧0
͚ͫjϭ1
i j
␧j Ϫ ͚jϭiϩ1
N j
␧j ͬϩ
␳
␧¯␧0
. ͑5͒
␧0 denotes the dielectric constant, while ␧i
and ␧¯ denote the
local and the averaged relative dielectric constant. The elec-
tric field in the ith organic layer is then given as the average
between the electric fields at the enclosing interfaces:
Fi
ϭ
Fif
iϪ1
ϩFif
i
2
. ͑6͒
The anode surface charge density ␳ in Eq. ͑5͒ is adjusted
for every time step to maintain the condition
͚iϭ1
N
Fi
•dMϭVϪVbi , ͑7͒
which in other words means that space charges will be com-
pensated by additional electrode surface charges to maintain
the total potential drop in the organic layers.
C. Charge carrier drift
The local electric field in the organic layers is respon-
sible for the drift of the free charge carriers. In amorphous
molecular organic solids such transport phenomena are com-
monly described as hopping processes. Holes are transported
in the HOMOs, whereas electrons are transported in the LU-
MOs.
Such hopping processes occur randomly and must there-
fore be described on a microscopic level using Monte Carlo
methods.22
Nevertheless, for a huge amount of hopping pro-
cesses, in our case from one monolayer to the neighboring
monolayers, statistics allow the use of average parameters
describing the hopping processes.22
This averaging of the
microscopic charge carrier mobility distribution leads to a
nondispersive description of the charge carrier transport.
Nondispersive charge carrier transport has been observed for
typical electric fields during OLED operation in the hole and
electron transporting materials which are investigated in our
simulation and experimental work, thus justifying this
simplification.23,24
All the following formulas for the trans-
port mechanisms will therefore use average parameters. The
relations will be given for holes only, since the formulas for
electrons have equivalent forms.
The average waiting time ␶¯p
i
between two consecutive
hopping processes ͑for a molecule arrangement as described
above͒ can be expressed as the quotient between the jump
3897J. Appl. Phys., Vol. 86, No. 7, 1 October 1999 Staudigel et al.
14.139.194.12 On: Sun, 09 Nov 2014 17:37:16

distance, given as the molecule diameter, and the average
local charge carrier velocity of the molecule at the initial
position i
␶¯p
i
ϭ
dM
␮p
i
Fi . ͑8͒
The average local hole mobility ␮p
i
is a function of the local
electric field and of the temperature T, as well as it is a direct
function of the position i due to the stacking of different
organic materials in multilayer OLEDs. As a simplification,
the mobility corresponding to the molecule at the final posi-
tion after the hopping process ͑either at position iϪ1 or i
ϩ1͒ is assumed not to have any influence on the average
waiting time ␶¯p
i
. All effects arising from the energetic and
spatial disorder in organic solids are packed into the field and
temperature dependence of the charge carrier mobilities. A
suitable previously reported relation for this dependence will
be presented later on.22
The change of the free hole density in the ith monolayer
due to hopping processes from the ith into the (iϩ1)th
monolayer during the time step ⌬t is given by ⌬pdrift
i→iϩ1
ϭ⌬t/␶¯p
i
•pf
i
, where a substition of ␶¯p
i
according to Eq. ͑8͒
yields the final form
⌬pdrift
i→iϩ1
ϭͭ␮p
i
Fi
dM
pf
i
•⌬t, Fi
Ͼ0
0, Fi
р0. ͑9͒
⌬pdrift
i→iϪ1
is represented by an equivalent form. The total
change of the free hole density in the ith monolayer due to
hopping processes results from the sum of all hopping pro-
cesses involving this monolayer
⌬pdrift
i
ϭ⌬pdrift
iϪ1→i
Ϫ⌬pdrift
i→iϪ1
ϩ⌬pdrift
iϩ1→i
Ϫ⌬pdrift
i→iϩ1
. ͑10͒
The total change ⌬ndrift
i
of the free electron density can be
derived to an equivalent form to Eq. ͑10͒.
D. Charge carrier diffusion
In addition to the charge carrier drift parallel to the di-
rection of the electric field, a random charge carrier move-
ment due to thermally stimulated hopping processes with no
preferential direction occurs. This free charge carrier diffu-
sion can be described equivalently to Eq. ͑9͒, however, by
excluding the direction limitation due to the electric field
⌬pdiff
i→iϩ1
ϭ
v¯diff,p
i
dM
pf
i
•⌬t ͑11͒
Again, only the free hole density will be derived, since the
free electron diffusion is equivalent. The average local dif-
fusion velocity v¯diff,p
i
can be obtained from the diffusion co-
efficient Dp
i
and the molecule diameter
v¯diff,p
i
ϭ
Dp
i
dM
. ͑12͒
The Einstein relation between diffusion coefficient and field
independent charge carrier mobility18
Dpϭ
␮p•kBT
e
, ͑13͒
with kB denoting the Boltzmann constant, is in this general
form not applicable for OLEDs. The observed increase in the
charge carrier mobility due to the electric field, arising from
the lowering of the potential barrier for a hopping process,22
exists only in the drift direction, whereas in the reverse di-
rection a decrease must be expected. Additionally, the elec-
tric field itself decreases the probability of hopping processes
against the drift direction, since the charge carriers gain po-
tential energy according to the total potential difference
along the jump path, which is for typical electric fields and
molecule diameters in the order of magnitude of kBT.
The above mentioned effects, which favor a diffusion in
the drift direction, will be partly considered in the model by
a proposed diffusion mobility, defined as
␮diff,p
i→iϩ1
͑Fi
͒ϭͭ␮p
i
͑Fi
͒, Fi
Ͼ0
␮p
i
͑0͒, Fi
р0, ͑14͒
with ␮diff,p
i→iϪ1
(Fi
) analogous to this definition. Using Eqs.
͑11͒, ͑12͒, and ͑13͒, the change in the free hole density due
to thermally assisted jumps from the ith into the (iϩ1)th
monolayer can be expressed as
⌬pdiff
i→iϩ1
ϭ
␮diff,p
i→iϩ1
͑Fi
͒•kBT
e•dM
2 pf
i
•⌬t, ͑15͒
with ⌬pdiff
i→iϪ1
having an analogous form. The total change of
the free hole density in the ith monolayer due to diffusion
can again be gained from the sum over all changes involving
this monolayer
⌬pdiff
i
ϭ⌬pdiff
iϪ1→i
Ϫ⌬pdiff
i→iϪ1
ϩ⌬pdiff
iϩ1→i
Ϫ⌬pdiff
i→iϩ1
. ͑16͒
The total change of the free electron density ⌬ndiff
i
is equiva-
lent to Eq. ͑16͒.
A comparison of drift and diffusion velocities for typical
electric fields in OLEDs makes clear that the major part of
the total charge carrier transport is governed by the charge
carrier drift.
E. Charge carrier injection
In our proposed OLED model the injecting contacts for
holes and electrons are assumed to be ohmic, since current
limitation due to barriers at the electrode–organic interfaces
is not acceptable for high-performance OLEDs. Naturally,
this condition has to be verified experimentally both for an-
ode and cathode before starting the simulation ͑see Sec. III͒.
Generally, an ideally injecting ohmic contact can be
implemented by a huge amount of charge carriers waiting at
the electrode–organic interfaces ͑position 0 for holes and N
ϩ1 for electrons, see Fig. 1͒. One possible description for
the change of the free hole density in the first monolayer is
⌬pdrift
0→1
ϭͭ␮p
1
F1
dM
panode•⌬t, F1
Ͼ0
0, F1
р0, ͑17͒
where panode should be at least one order of magnitude larger
than the maximum of the hole density in the organic layers.
The actual amount of injected holes is governed by the elec-
tric field F1
in the first monolayer, which will be decreased
14.139.194.12 On: Sun, 09 Nov 2014 17:37:16

due to space charges ͑but will remain positive͒ and limit
further charge injection. The electron injection ⌬ndrift
Nϩ1→N
at
the cathode is equivalent to Eq. ͑17͒.
The diffusion of charge carriers from the electrodes into
the organic bulk material is not implemented. Arguments
showing that this diffusion is not relevant can be derived
from the charge injection model proposed by Arkhipov and
co-workers.10
In their model for charge injection in OLEDs,
the number of charge carriers effectively entering the organic
bulk is significantly reduced by the image potential caused
by the charge carriers themselves, which forces the majority
of the carriers to jump back into the electrode after entering
the first organic monolayer. The charge carriers are only able
to overcome the image potential in the presence of a positive
electric field. The diffusion length of charge carriers from the
electrodes is therefore significantly reduced in comparison to
diffusion lengths neglecting the image potential. The as-
sumed absence of charge carrier diffusion from the elec-
trodes into the adjacent organic layers is forced in the model
by ⌬pdiff
0→1
ϭ0 and ⌬ndiff
Nϩ1→N
ϭ0.
The control of the amount of injected charge carriers
through the charge carriers present in the organic layers pro-
vides an efficient controlling mechanism in the simulation
for a proper time evolution, thus making forced boundary
conditions not neccessary. One initial condition, preferably
an ‘‘empty’’ device without any excess charge carriers, is
sufficient to obtain physical relevant results as outlined
above.
F. Bulk recombination
The recombination of holes and electrons to excitons
occurs due to attractive Coulombic interaction between the
two different charge carrier species. The change of the free
hole and electron density during the time step ⌬t due to the
recombination of the free charge carriers in the ith mono-
layer can be described as
⌬Rpfnf
i
ϭ␥•pf
i
nf
i
•⌬t, ͑18͒
where ␥ is the recombination coefficient in accordance with
Langevin’s theory, which is known to be appropriate for dis-
ordered organic systems25
␥ϭ
e•͑␮p
i
ϩ␮n
i
͒
␧i
␧0
. ͑19͒
G. Charge carrier trapping
Charge carrier traps are states with energetic positions in
between the energetic levels of the HOMO and the LUMO.
Trap states may have various origins, e.g., material impuri-
ties or structural disorder.12
The occupation of trap states,
which are uncharged when empty, depends on the free
charge carrier densities and the trap depth, given as the en-
ergetic distance of the trap state to the next transport site.
For shallow trap states arising from structural disorder in
the organic layers, recombination of free charge carriers with
trapped ones is assumed to be radiative with the same effi-
ciency as for recombination in between free charge carriers.
The decrease of the free hole density, respectively the
increase of the trapped hole density, in the ith monolayer
during the time step ⌬t is given by
⌬pt
i
ϭ
ͫ␮p
i
•͉ͩFi
͉ϩ
kBT
e•dM
ͪ
vp
i
ͬ•␴t•ͫpf
i
•͑Nt,p
i
Ϫpt
i
͒Ϫpt
i
•͑NHOMOϪpf
i
ӍNHOMO
͒•expͩϪ
Et,p
i
kBT
ͪͬ•⌬t. ͑20͒
The trap states are supposed to be not degenerated. The ve-
locity vp
i
of the holes in the ith monolayer is approximated
as the sum of the drift and diffusion velocity. The cross
section for charge carrier trapping ␴t is estimated from the
geometric area occupied by a single molecule. Trapping is
supposed to possibly occur on the regarded molecule itself
plus on its four next neighbors in the same monolayer, thus
yielding the estimation ␴tϭ5•dM
2
.
Equation ͑20͒ reflects that the trapping of holes is pro-
portional to the free hole density and to the concentration of
empty traps Nt,p
i
Ϫpt
i
. The detrapping, which is proportional
to the density of trapped holes and ͑to a good approximation͒
to the density NHOMO of free states in the HOMO, occurs as
a function of the trap depth Et,p
i
only thermally activated.
Considering every single molecule as a state, the density
of states in the HOMO can be approximated to
NHOMOϭdM
Ϫ3
, ͑21͒
which is essentially the density of molecules in organic sol-
ids.
The presence of free electrons can significantly decrease
the number of trapped holes due to recombination processes.
The decrease of the free electron density and of the trapped
hole density due to recombination processes according to
Langevin’s theory is given by
⌬Rptnf
i
ϭ
e•␮n
i
␧i
␧0
•pt
i
nf
i
•⌬t. ͑22͒
The density of trapped holes pt,ss
i
in the steady state can
be derived from the condition ⌬pt
i
Ϫ⌬Rptnf
i
ϭ0 to
pt,ss
i
ϭNt,p
i
•
ͫ1ϩ
NHOMO
pf
i •expͩϪ
Et,p
i
kBT
ͪ
ϩ
e•␮n
i
␧i
␧0
•nf
i
␴t•␮p
i
•͉ͩFi
͉ϩ
kBT
e•dM
ͪ•pf
i
ͬ
Ϫ1
. ͑23͒
The calculation of the time evolution can be stopped when
the relative deviation between the actual and the steady state
trapped charge carrier densities becomes insignificant
͑р0.1% in our simulation͒.
The formula given by Lampert and Mark18
for the uni-
polar ͑hole-only͒ case can be derived from Eq. ͑23͒ for nf
i
14.139.194.12 On: Sun, 09 Nov 2014 17:37:16

ϭ0, that means in the absence of free electrons. The defini-
tion of the commonly used quasi-Fermi level, given here for
holes by18
EF,p
i
ϭEHOMO
i
ϪkBT•ln͑pf
i
/NHOMO͒, ͑24͒
normally allows the description of the hole trap states occu-
pation using a Fermi distribution.18
Due to the additional
recombination term in Eq. ͑23͒, however, this definition
makes no sense in the bipolar case. EF,p
i
as defined by Eq.
͑24͒ can then no longer be interpreted as the local Fermi
level which governs the occupation of the states.
For the trapping of electrons equivalent relations to the
above mentioned can be derived. Again, the definition of a
quasi-Fermi level EF,n for electrons makes no sense in the
presence of free holes. It is also possible to implement con-
tinuous trap distributions by a discretization in single trap
levels, where each trap level has to be treated separately.
H. Internal interfaces
Interfaces between two adjacent organic layers have a
major influence on current flow and device efficiency. The
actual processes at internal interfaces are not fully under-
stood yet, so we will propose a simple model of the interface
processes in the first step. In the following, the organic ma-
terials are assumed not to interact directly. This means that
no particular surface states or charge-transfer complex for-
mation are considered, since there is no experimental evi-
dence for the existence of such states in electroluminescence
spectra for the material combinations used.
In our vapor-deposited multilayer OLEDs the energetic
levels of the HOMO and of the LUMO, starting from the
anode, decrease from organic layer to organic layer.19
There-
fore, charge carriers have to ‘‘climb’’ a staircase on an en-
ergetic scale before they reach the recombination zone. Each
interface represents an energetic barrier, which has to be
crossed by the free charge carriers. The height of such a
barrier ͑⌬Ep for holes͒ is then given as the difference be-
tween the energetic levels of the HOMOs of the two organic
layers ͑see Fig. 2͒.
A hole arriving at the interface is assumed to have two
ways to cross the interface. On one hand, it can perform a
thermally assisted jump over a ͑due to the potential drop in
the electric field͒ decreased energetic barrier ⌬EpЈ , which is
proposed to be of the form
⌬EpЈϭ⌬EpϪe•
Fj
ϩFjϩ1
2
•dM ͑25͒
͑see dashed arrows in Fig. 2͒ or, on the other hand, recom-
bine directly with an electron from the other side of the in-
terface. The interface recombination, that means the decrease
of the hole density in the jth monolayer, which is equivalent
to the decrease of the electron density in the (jϩ1)th mono-
layer, is assumed to be similar to the bulk recombination,
with the difference that holes and electrons are spatially
separated by the interface
⌬Rpfnf
if
ϭ
e•͑␮p
j
ϩ␮n
jϩ1
͒
␧¯␧0
•pf
j
nf
jϩ1
•⌬t, ͑26͒
where ␧¯ denotes the averaged relative dielectric constant in
these two monolayers.
For thermally assisted barrier crossing, the energetic dis-
order of the density of states ͑DOS͒ has to be considered.
The energetic disorder at the interface arises from a similar
effect as the energetic disorder in the bulk, namely an inho-
mogeneous broadening of the DOS due to interactions be-
tween neighboring molecules,22
which may be superposed by
an additional disorder due to the interdiffusion of the differ-
ent molecule species.
The DOS in the interfacial monolayers j and jϩ1 are
assumed to have a Gaussian shape distribution equivalent to
the energetic bulk disorder proposed by Ba¨ssler,22
given for
holes by
⌫͑E͒ϭ
NHOMO
␴ͱ2␲
•expͩϪ
E2
2␴2ͪ, ͑27͒
with E denoting the energetic distance to the energetic posi-
tion of the maximum of the DOS, and ␴ denoting the ener-
getic DOS width, which is assumed to be equal at both sides
of the interface.
The zero-field steady state value for the energetic dis-
tance of energetically relaxed charge carriers to the maxi-
mum in the DOS ͑given as EHOMO
i
, respectively, ELUMO
i
)
after a large number of hopping processes can be calculated
to ␴2
/kBT.22
Since the charge carriers gain an energy portion
eFi
dM ͑typically on the order of kBT) during every hopping
process, that means they can enter the DOS in the next
monolayer at a higher energy level, and since the number of
hopping processes needed to cross such thin layers is rather
limited, the assumption of a totally relaxed zero-field occu-
pation of the DOS cannot be justified in OLEDs. Instead, we
FIG. 2. Proposed barrier crossing and recombination processes at the inter-
face between two different organic materials. Under the condition of ener-
getic barriers ⌬Ep and ⌬En due to steps in the energetic positions of the
HOMO and the LUMO at the interface, holes and electrons are, on one
hand, able to cross the reduced energetic barriers ⌬EpЈ and ⌬EnЈ by a single
thermally stimulated hopping process ͑dashed arrows͒, or, on the other hand,
they recombine directly at the interface ͑dotted arrows͒.
14.139.194.12 On: Sun, 09 Nov 2014 17:37:16

assume a homogeneous occupation of the DOS at the inter-
face, reflecting an insufficient energetic relaxation of the
charge carriers within the DOS in the bulk.
The probability for a charge carrier to enter the neigh-
boring DOS at a higher energetic level is assumed to be
thermally activated, whereas a downward jump is assumed to
occur instantaneously.22
The total probability for a charge
carrier to cross an interface with a reduced energetic barrier
⌬EЈ is then given as
G͑⌬EЈ͒ϭc• ͵Ϫϱ
ϱ
dE ⌫͑E͒ ͵Ϫϱ
ϱ
dEЈ ⌫͑EЈ͒
•ͭexpͩϪ
EЈϩ⌬EЈϪE
kBT ͪ, EЈϩ⌬EЈϾE
1, EЈϩ⌬EЈрE
ͮ,
͑28͒
with E and EЈ denoting the energetic distance to the maxi-
mum of the DOS before and after the interface ͑defined by
the charge carrier jump direction͒. The constant c, which
depends only on ␴, is given by the condition G(0)ϭ1. In the
case of a negative barrier in jump direction, G(⌬EЈ) is set to
one. In Fig. 3, the dependence of the probability G on the
energetic barrier height is plotted for different values of the
energetic disorder ␴. Equation ͑28͒ is only applicable as long
as the occupation of the states in the monolayer before the
interface is not altered, that means only for ⌬EЈӷ␴. Never-
theless, this description for the interface crossing probability
yields, on a phenomenological basis, reasonably good results
even for ⌬EЈϷ␴, as will be shown in Sec. III.
The expected changes of the free hole density in front of
an interface at position j due to transport in the bulk,
⌬pdrift
j→jϩ1
and ⌬pdiff
j→jϩ1
, must both be multiplied with the
probability G(⌬EpЈ) for interface barrier crossing. This may
lead to an accumulation of holes before the interface with
drastic effects on the electric field distribution in the organic
layers in the case of a high interface barrier.
Naturally, all the above mentioned relations for holes are
also applicable in an equivalent form for electrons, for which
we assume the same energetic disorder ␴ as for holes.
I. Current density
The current density, one of the simulation results di-
rectly comparable with experimental data, can be derived for
unipolar single layer samples from the changes of the charge
carrier densities and of the electric field. For the calculation
of the current density due to hole transport in the bulk, the
expression
jtransport,p
i
ϭe•ͫ⌬pdrift
i→iϩ1
Ϫ⌬pdrift
i→iϪ1
⌬t
ϩ
⌬pdiff
i→iϩ1
Ϫ⌬pdiff
i→iϪ1
⌬t
ͬ•dM ͑29͒
can be used, representing a formula given by Lambert et al.
which was modified to be suitable for the proposed model.18
Additionally, the displacement current density has to be con-
sidered as long as the steady state is not achieved:
jdisplacement
i
ϭ␧i
␧0•
‫ץ‬Fi
‫ץ‬t
. ͑30͒
In multilayer OLEDs the contributions of both hole and
electron transport, the time evolution of the electric field, as
well as the bulk and probably interface recombination, have
to be added to yield the total current density.
J. Singlet excitons
The recombination of holes and electrons in the organic
layers results in the formation of singlet and triplet excitons
in a ratio of 1:3 due to spin statistics.26
In the proposed
model, excitons are supposed to be Frenkel excitons rather
than charge-transfer excitons or Mott–Wannier excitons,
which means that they are spatially limited to one ͑excited͒
molecule.
Efficient light emission due to radiative decay of excited
molecules can only be expected from singlet excitons in non-
doped OLEDs, since the diffusion length of triplet excitons
is due to their long lifetime large enough to favor nonradia-
tive recombination at the electrodes. The implementation of
exciton diffusion and decay will therefore be limited to sin-
glet excitons.
The change of the singlet exciton density si
in the ith
monolayer during the time step ⌬t due to singlet exciton
formation and normal decay ͑partly radiative͒ can be ex-
pressed as
⌬srad
i
ϭ
1
4
•⌬Rpfnf
i
Ϫ
si
␶s
•⌬t, ͑31͒
where ␶s denotes the singlet exciton lifetime. Exciton forma-
tion at shallow traps arising from structural disorder have to
be considered additionally ͑⌬Rpfnt
i
and ⌬Rptnf
i
͒.
Exciton and therefore excitation energy transport is sup-
posed to occur only due to exciton hopping processes to next
neighbor molecules.21
In our model, radiative energy transfer
is not implemented. The one-dimensional diffusion velocity
vdiff of singlet excitons can be derived from the average
number N¯ hop of hopping processes during the lifetime ␶s to
FIG. 3. Probability G(⌬E) of a thermally assisted hopping processes over
an energetic barrier ⌬E for different values of the energetic width ␴ of a
proposed Gaussian density of states ͑DOS͒ distribution at both sides of the
interface.
14.139.194.12 On: Sun, 09 Nov 2014 17:37:16

vdiffϭ
N¯ hop•dM
␶s
. ͑32͒
The change of the singlet exciton density due to hopping
processes from the ith into the (iϩ1)th monolayer is then
given by
⌬sdiff
i→iϩ1
ϭ
vdiff
dM
•si
•⌬t. ͑33͒
The total change ⌬sdiff
i
including all changes involving the
ith monolayer is equivalent to Eq. ͑16͒.
For diffusion across an interface in a direction with de-
creasing energetic distance between HOMO and LUMO no
limitation is expected, whereas diffusion in reverse direction
is supposed to occur similar to the charge carrier interface
barrier crossing, i.e., the probability is given according to Eq.
͑28͒. The energetic barrier for singlet exciton interface cross-
ing is assumed to be the difference between the energetic
distances between the HOMO and the LUMO on both sides
of the interface. The energetic disorder is supposed to be the
same as for charge carriers.
There are several possible reasons for nonradiadive de-
cay of singlet excitons. Only a small fraction of all possible
quenching mechanisms will be implemented, since most of
them are not expected to be of major importance.
First, the singlet exciton quenching at the electrodes is
already partly implemented by the diffusion terms ⌬sdiff
1→0
and ⌬sdiff
N→Nϩ1
. An additional quenching mechanism at the
electrodes exists due to interdiffusion of electrode material
into the organic layers. This effect is estimated by
⌬sel
i
ϭͫexpͩϪ
zi
Lq
ͪϩexpͩϪ
LϪzi
Lq
ͪͬ•
si
␶q
•⌬t, ͑34͒
with Lq denoting the length of the interdiffusion zone of the
electrode materials, ␶q representing a reduced lifetime ͑pro-
posed to be about one order of magnitude shorter than the
actual life time ␶s͒, and zi
denoting the position of the ith
monolayer ͑see Fig. 1͒.
Second, a quenching process supposed to be of major
importance is the exciton quenching due to the presence of
charge carriers. For the change of the singlet exciton density
due to reactions with free holes the following expression is
proposed:
⌬Rspf
i
ϭ͑vdiffϩvp
i
͒•␴q•si
pf
i
•⌬t, ͑35͒
with vp
i
defined in Eq. ͑20͒, and ␴q denoting the reaction
cross section. For exciton quenching at trapped holes and at
free and trapped electrons similar equations are used.
The dissociation of excitons in the presence of high elec-
tric fields as reported in the literature27
is not considered,
since the total effects are expected to be similar to those
arising from the quenching at charge carriers, which also
increase with increasing electric field. Therefore, the absence
of this dissociation mechanism will be partly compensated
by using an increased quenching cross section ␴q .
The third quenching mechanism implemented arises
from the fact that only part of the charge carriers, which
recombine at the internal interfaces ͑see Fig. 2͒, form exci-
tons in the bulk material. Excitons formed at an interface
with energetic barriers for holes and electrons have an en-
ergy corresponding to the distance between the energetic
level of the HOMO before (EHOMO
j
) and the level of LUMO
after (ELUMO
jϩ1
) the interface, which is less than the differ-
ences between the HOMO and the LUMO in the correspond-
ing bulk materials ͑see Fig. 2͒. Such an interface exciton can
either recombine nonradiatively at the interface ͑see dotted
arrows in Fig. 2͒ or form a ͑potentially radiative͒ bulk exci-
ton. For this second possibility, either the hole has to enter
the bulk material after or the electron has to enter the bulk
before the interface ͑see dashed arrows in Fig. 2͒. The prob-
ability for this charge carrier interface crossing and, there-
fore, for bulk exciton generation is given ͑with field-reduced
barrier heights͒ by G(⌬EnЈ) for the bulk before and by
G(⌬EpЈ) for the bulk after the interface. To ensure that the
correct number of bulk excitons is produced, the generation
probability has to be multiplied by the factor 1ϪG(⌬EpЈ)
•G(⌬EnЈ)/2. Otherwise, e.g., in the case of small barriers
with G(⌬EpЈ)Ӎ1 and G(⌬EnЈ)Ӎ1, the number of generated
bulk excitons would become twice the number of formed
interface excitons. Naturally, the ratio of singlet to triplet
excitons has to also be considered.
K. Emission and luminance
The electrical characteristics can already be described
quantitatively with the above mentioned relations. To obtain
quantitatively the optical characteristics in commonly used
photometric units, the sensitivity of the human eye, the
losses due to waveguiding in the transparent substrate and
absorption in the layers, and also to the material-specific ra-
tio of radiative to nonradiative normal decay of singlet exci-
tons, have to be considered.
The internal emission, defined as the number of photons
emitted per surface and time period, is given as sum over all
monolayers
Sϭ͚iϭ1
N
␩bulk
i
•
si
␶s
•dM , ͑36͒
with ␩bulk denoting the efficiency of singlet exciton radiative
decay in the solid state.
To obtain the external emission several loss mechanisms
have to be considered, such as waveguiding and absorption.
Standard refraction theory is not applicable for the calcula-
tion of waveguiding losses inside the functional layers, since
the layer thicknesses are smaller than the wavelength of the
emitted light. To calculate at least the waveguiding losses in
the glass substrate, we assume the photons to be isotropically
emitted from the functional layers into the glass substrate, so
that a Lambertian emission characteristic inside the substrate
can be expected.28
The losses due to waveguiding in the glass substrate can
then be easily estimated under the assumption of a totally
reflecting cathode, the absence of absorption and microcavity
effects in the functional layers, and a diode area diameter
much larger than the substrate thickness. In this case, all
forward and backward emitted photons with angles relative
to the normal of the substrate surface smaller than the total
14.139.194.12 On: Sun, 09 Nov 2014 17:37:16

reflection angle ␣tϭarc sin(1/nsub) between substrate and air
(nairϭ1) will leave the substrate–air interface in the end due
to multiple reflections.29
The Lambertian emission characteristic inside the sub-
strate is proportional to cos ␣ ͑with ␣ as emission angle rela-
tive to the substrate normal͒,29
so that the fraction of photons
leaving the substrate can be calculated by an integration over
all possible emission angles to
␩wgϭ
2͐0
2␲
d␸͐0
␣t
d␣ sin ␣•cos ␣
͐0
2␲
d␸͐0
␲
d␣ sin ␣•cos ␣
ϭ
1
nsub
2 , ͑37͒
with nsub denoting the refractive index of the glass substrate.
Equation ͑37͒ differs by a factor of 2 from the previously
reported proportionality factor between internal and external
quantum efficiency 1/2nsub
2
.28
This deviation arises from the
fact that in our calculation part of the emission originally
directed backwards also leaves in the forward direction due
to the reflective cathode.
For a calculation of the conversion factor between the
total emission and the ͑more commonly used͒ emission per
solid angle in the forward direction, the angular emission
characteristic outside the substrate has to be derived. Previ-
ously reported calculations either expect a decrease28
or an
increase30
of the outside angular emission in comparison to a
Lambertian emitter for increasing ␤. Due to this discrepancy,
we will only use a simple calculation to derive the angular
emission characteristic outside the substrate under the as-
sumptions mentioned above.
The emission angle ␤ relative to the substrate normal
outside the substrate is related to the internal emission angle
␣ by sin ␤ϭnsub•sin ␣.29
Using this relation, the relative an-
gular emission into a certain solid angle inside the substrate
can be transformed into the according angular emission out-
side the substrate:
d␸ d␣ sin ␣•cos ␣
ϭd␸
d␣
d␤
d␤ sin ␣•ͱ1Ϫsin2
␣
ϭd␸
cos ␤
ͱnsub
2
Ϫsin2
␤
d␤
sin ␤
nsub
•ͱ1Ϫ
sin2
␤
nsub
2
ϭ
1
nsub
2 d␸ d␤ sin ␤•cos ␤. ͑38͒
From Eq. ͑38͒ it can be clearly seen that the angular depen-
dence of the emission outside the substrate remains Lamber-
tian.
Since for emission angles ␣ close to ␣t the number of
reflections inside the substrate increases before practically all
photons have left the substrate, the increased absorption due
to a longer path inside the layers is expected to cause a
decrease in the angular emission relative to a Lambertian
emission characteristic, which perfectly explains previously
reported experimental data.28
The conversion factor between emission per solid angle
in the forward direction and total emission can then be cal-
culated by an integration over a half sphere of the relative
angular emission per solid angle of an ideal lambertian emit-
ter to
͵0
2␲
d␸ ͵0
␲/2
d␤ sin ␤•cos ␤ϭ␲. ͑39͒
The optical characteristics of OLEDs are commonly pre-
sented in photometric units. Variables in energetic units can
be transformed using the formula29
Xphotoϭkm• ͵0
ϱ
d␭Xenergetic͑␭͒•V͑␭͒, ͑40͒
with the constant for a daylight-adopted human eye km
ϭ683 lm/W, and V(␭) denoting the relative spectral sensi-
tivity of the human eye, with a maximum value of 1 in the
green spectral region. For actual calculations we assume a
monochromatic green OLED with V(␭)ϭ1, so that Eq. ͑40͒
can be modified to yield the luminance29
Lϭ␩abs␩wg•km•S•h␯•
1
␲
, ͑41͒
where h␯ is the energy of the emitted photons, and ␩abs
comprises the efficiency decrease due to absorption and
waveguiding in the functional layers, which were excluded
for the estimation of ␩wg in Eq. ͑37͒. According to Eq. ͑39͒,
the factor 1/␲ in Eq. ͑41͒ results from the transition from
lumen ͑lm, total emission͒ to candela ͑cd, emission per solid
angle, here in the forward direction͒, since optical character-
istics of OLEDs are commonly presented in cd/m2
.
L. Test of simulation routines
All simulation routines were tested individually in com-
parison with solutions of analytically solvable problems. For
example, the analytic solutions for the current density in uni-
polar single layer samples, with a field-independent mobility
and discrete shallow or deep trap levels,18
were reproduced
quantitatively with relative deviations of less than 0.1%. The
algorithms are also able to reproduce analytically expected
unipolar space charge limited dark current transients with
correct peak heights and temporal positions.18
Test runs of the simulation routines using synthetic pa-
rameters have been performed in order to gain some insight
in to the influence of individual parameters. These investiga-
tions were mainly focused on those parameters, which are
experimentally not directly accessible. Parameters of major
importance on the device efficiency are the energetic barriers
and energetic disorder parameters at internal interfaces.
These parameters govern the probabilities for charge carrier
interface crossing and bulk exciton generation at internal in-
terfaces.
III. COMPARISON WITH EXPERIMENT
Before a quantitative simulation of OLEDs becomes
possible, the parameters characterizing the organic layers
have to be derived. The existence of ohmic contacts has to be
proved self-consistently both for holes and electrons. Infor-
mation on charge carrier mobilities can be gained from
14.139.194.12 On: Sun, 09 Nov 2014 17:37:16

pulsed measurements on unipolar single layer samples. En-
ergetic levels of HOMOs and LUMOs relative to the vacuum
level will be taken from the literature as well as from cyclo-
voltammetric measurements and molecular orbital calcula-
tions. Experimentally not direct accessible parameters such
as, e.g., the energetic disorder at the organic–organic inter-
faces, will be adjusted within realistic boundaries to gain a
proper fit to the measured electro-optical characteristics of
the processed OLEDs.
A. Sample preparation
The samples under investigation were prepared by con-
ventional vacuum vapor deposition from tungsten boats onto
indium–tin–oxide ͑ITO͒-coated glass substrates at room
temperature ͑RT͒. All processing steps were performed un-
der Ar atmosphere with reduced H2O and O2 residual gas
contents (Ͻ10 ppm). ITO-coated glass substrates are com-
mercially available from Merck–Balzers and were patterned
by standard photolithographic processes. Before deposition
of organic layers, the ITO contacts were pretreated
with an oxygen plasma. 4,4Ј,4Љ-tris͕N-͑3-methylphenyl͒
-N-phenylamino͖-triphenylamine ͑m-MTDATA͒, 4,4Ј,4Љ
-tris͕N-(1-naphthyl)-N-phenylamino͖-triphenylamine ͑1-
Naphdata͒, N,NЈ-di͑naphthalen-1-yl͒-N,NЈ-diphenyl-
benzidine ͑␣-NPD͒, obtained from SynTec GmbH, were pu-
rified by vacuum sublimation. 8-hydroxyquinoline aluminum
(Alq3) was synthesized by A. Kanitz ͑Siemens AG͒. The
molecular structures of all organic materials used are de-
picted in Fig. 4. Ag, Mg, LiF and Al are commercially avail-
able from Aldrich. All materials were thermally evaporated
at a base pressure below 1ϫ10Ϫ5
hPa. Film thicknesses
were measured using a Tencor alpha-step 200 profilometer
or, alternatively, using a Plasmos SD 2300 ellipsometer.
The unipolar hole-only samples under investigation con-
sisted of a single layer of m-MTDATA, respectively,
1-Naphdata sandwiched between ITO and Ag electrodes. Ag
was used as a hole extracting electrode in order to avoid
electron injection. The unipolar electron-only samples con-
sisted of a single Alq3 layer sandwiched between Mg and
LiF/Al electrodes, where Mg was used as an electron extract-
ing electrode to avoid hole injection. The thicknesses of the
organic layers ranged from 300 nm to 1 ␮m at deposition
rates of 1 nm/s. The LiF layer thickness of 0.5 nm was re-
producibly obtained due to deposition rates below 0.05 nm/s.
Metal layer thicknesses were in excess of 100 nm. The size
of the samples was 4 mm2
.
The bipolar multilayer OLEDs under investigation were
either two-layer devices ͑HTL plus ETL͒ or three-layer de-
vices ͑two adjacent HTLs plus ETL͒. The ETL was in both
cases also used as an emissive layer ͑EML͒. The two-layer
device consisted of a stack of 1-Naphdata and Alq3 ͑thick-
nesses 50 nm each͒, whereas the three-layer device consisted
of a stack of 1-Naphdata, ␣-NPD, and Alq3 ͑thicknesses 40,
10, and 50 nm, respectively͒. The organic layers were sand-
wiched between ITO and LiF/Al electrodes. ITO was used as
a hole-injecting and LiF/Al as an electron-injecting contact.
The deposition rates of the organic compounds were reduced
due to decreased layer thicknesses from the above mentioned
1 to 0.2 nm/s. The active area of the OLEDs was 4 mm2
.
B. Sample characterization
The samples were characterized under steady state and,
to some extent, under pulsed conditions. Ar was used as inert
atmosphere for characterization to avoid sample degradation
due to oxygen and moisture.
Dark current transients on unipolar single-layer samples
at RT were recorded using a digital storage oscilloscope ͑Sie-
mens Oscillar D 1042͒ after applying a voltage step. Voltage
steps with steep onsets were realized using a charged capaci-
tor (1 ␮F), which was able to discharge across the sample
after closing a fast semiconductor switch ͑HARRIS HI-
201HS, switch on time 30 ns͒. The minimum achievable
onset time of the electric field in the organic layers is limited
by the RC time constant of the system, given as the product
of the capacity C of the sample thin film capacitor and the
resistivity R between sample and discharged capacitor. Us-
ing a relative dielectric constant ␧ϭ3, a layer thickness of
300 nm and a resistivity Rϭ200 ⍀ ͑resistor for current mea-
surement plus film resistivity of the patterned ITO contact͒,
the RC time constant can be estimated to ϳ 70 ns, which is,
in comparison with transit times greater than 1 ␮s, small
enough to ensure that no effect on the current transient inside
the organic layers arises.
Steady state current measurements on unipolar single-
layer samples and bipolar multilayer devices were performed
using a high voltage Keithley Source Measure Unit SMU
237. The luminance of the multilayer OLEDs was recorded
with silicon photodiodes, which were calibrated using a Mi-
nolta Chroma Meter CS-100 luminance camera.
C. Proof for ohmic contacts
One major condition for the applicability of the proposed
OLED model is the existence of ohmic contacts for both hole
and electron injection ͑see Sec. II͒. A self-consistent proof
for an ohmic contact can be deduced from a combination of
pulsed and steady state measurements of the current density
in unipolar single-layer samples. A maximum in the dark
current transient at tpeakϭ0.786•ttr after applying a voltage
FIG. 4. Molecular structures of the organic materials used for sample prepa-
ration. ͑a͒ m-MTDATA, ͑b͒ ␣-NPD, ͑c͒ Alq3 , and ͑d͒ 1-Naphdata.
14.139.194.12 On: Sun, 09 Nov 2014 17:37:16

step of height V, in addition to steady state current densities
which are quantitatively comparable to predictions from
SCLC theory,18
provide evidence of space charge limitation
of the current.31
The charge carrier transit time ttr is given by
L2
/␮V. Ohmic contacts for hole injection were previously
reported for m-MTDATA on ITO32
and were also verified by
our own experimental data for 1-Naphdata on ITO. Mea-
sured SCLC transients for 1-Naphdata can be seen in Fig. 5.
The high initial current represents the exponential tail of the
charge-up current of the sample thin film capacitor, the broad
maximum afterwards arises from space charge limitation of
the current flow. The inset of Fig. 5 shows a typical dark
current transient of an electron-only single layer Alq3
sample. Again, broad maxima could be observed. This veri-
fies, in addition to the observed quantitative agreement be-
tween measured current density and SCLC theory in the
high-field regime, an ohmic contact for electron injection for
LiF/Al on Alq3 .
Our key assumption of ideally injecting contacts for
holes and electrons is therefore absolutely justified for the
combinations of organic and electrode materials under inves-
tigation. The different proposed physical or chemical
processes,10,33,34
which may explain the ohmic contacts for
charge carrier injection, have no influence on the simulation
since only the existence of ohmic contacts ͑and not the actual
injection mechanism͒ is of importance.
D. Charge carrier mobilities
The field-dependent hole mobility ␮p in m-MTDATA
and 1-Naphdata as well as the field-dependent electron mo-
bility ␮n in Alq3 were obtained from the positions tpeak of the
maxima in the dark current transients ͑see Fig. 5͒ using the
relation18
␮ϭ0.786•
L2
tpeak•V
. ͑42͒
The exponential tail of the initial thin film capacitor
charge-up current superposes at high electric fields the cur-
rent maximum, thus hindering the determination of mobility
data. For small electric fields, the maximum in the current
transient was not clearly observable, which may be either
due to a transition to dispersive transport, or due to a loss of
the ohmic contact, or ͑in the case of Alq3) due to a decreased
signal-to-noise ratio. The absolute error in the measured mo-
bility data mainly arises from the error in the sample thick-
ness determination ͑5%͒, while the relative error results from
the inaccuracy in fixing tpeak to the broad current maxima
͑10%͒.
For the evaluation of the field dependence of the mobil-
ity data we assume a hopping mechanism in a Gaussian DOS
for the charge carrier transport according to the model devel-
oped by Ba¨ssler and co-workers, where the mobility is given
by22
␮͑F,␤͒ϭ␮0•expͫϪͩ2
3
␤␴ͪ2
ͬ
•ͭ exp͓C•͑͑␤␴͒2
Ϫ⌺2
͒•ͱF͔, ⌺Ͼ1.5
exp͓C•͑͑␤␴͒2
Ϫ2.25͒•ͱF͔, ⌺р1.5,
͑43͒
with ␤ϭ(kBT)Ϫ1
. In this equation ␴ characterizes disorder
on an energetic scale, and ⌺ is a spatial disorder parameter.
The constant C depends on the distance between adjacent
hopping sites.
For RT, only two parameters govern the field depen-
dence of the mobility:
␮͑F ͒RTϭ␮0,RT•exp͑CRT•ͱF͒. ͑44͒
The relation between the parameters ͑␮0,RT , CRT͒ and ͑␮0 ,
␴, ⌺, C͒ can be deduced from Eqs. ͑43͒ and ͑44͒ for T
ϭ300 K. The parameters ␮0,RT and CRT for the field-
dependent hole mobility in m-MTDATA and 1-Naphdata, as
well as for the field-dependent electron mobility in Alq3 used
FIG. 5. Dark current transients in unipolar single layer devices at room
temperature ͑RT͒. The main diagram shows the current transients for
1-Naphdata with a layer setup of ITO / 300 nm 1-Naphdata / Ag at different
applied voltage steps ͑15–7 V in 1 V steps, ITO biased positively͒. The
position of the transient space charge limited current ͑SCLC͒ peak allows
the determination of the hole mobility ͑see Fig. 6͒. The inset shows the dark
current transient for a single layer electron-only sample with a layer setup of
Mg / 300 nm Alq3 / 0.5 nm LiF / Al after applying a voltage step of 10 V
͑Mg biased positively͒. For electron mobilities in Alq3 obtained from the
transient SCLC peaks see Fig. 6.
FIG. 6. Charge carrier mobilities obtained from transient SCLC peaks at
RT. For m-MTDATA and 1-Naphdata the hole mobilities and for Alq3 the
electron mobility are plotted. The obtained mobilities for m-MTDATA and
Alq3 are in good agreement with previously reported data, ͑Refs. 32 and 34͒.
The typical experimental error for the mobility data is shown only exem-
plary.
14.139.194.12 On: Sun, 09 Nov 2014 17:37:16

for simulation, were therefore taken from straight line fit
curves to the measured data in ln ␮ vs F1/2
plots ͑see Fig. 6
and Table I͒. The measured hole mobilities in m-MTDATA
and electron mobilities in Alq3 are in good agreement with
previously reported data.32,24
The field-dependent hole mobility in ␣-NPD could
not be obtained from dark current transients, since no
maxima in the transients were observed. Expecting no sig-
nificant deviation from the hole mobility in
N,NЈ-bis(3-methylphenyl)-N,NЈ-diphenylbenzidine ͑TPD͒
͑the hole mobilities in m-MTDATA and 1-Naphdata are also
comparable, see Fig. 6͒, previously reported field-dependent
hole mobilities for TPD were used for simulation ͑see Table
I͒.23
Furthermore, the hole mobility of ␣-NPD is not ex-
pected to be a critical parameter, since the thickness of the
␣-NPD layer in the devices under investigation is substan-
tially smaller than the thicknesses of the adjacent 1-Naphdata
and Alq3 layers.
For the electron mobilities in m-MTDATA, 1-Naphdata,
and ␣-NPD ͑respectively, the hole mobility in Alq3͒ we as-
sumed 1% of the hole mobilities ͑respectively, the electron
mobility͒, which is a typical value in organic materials.35
E. Single-layer hole-only sample
Prior to the simulation of multilayer OLEDs, the rel-
evant numeric charge carrier transport and trapping routines
were tested on a single-layer hole-only sample with
m-MTDATA as hole transporting material ͑thickness 850
nm͒. The measured electrical characteristic shows ͑in a ln j
vs ln V plot͒ three regimes ͑see Fig. 7͒, each with a different
characteristic slope mϭ‫ץ‬ln j/‫ץ‬ln V.
For small applied voltages (VϽ2 V), the slope m is
close to one, indicating ohmic conduction due to the pres-
ence of intrinsic charge carriers. Using an averaged constant
mobility ␮¯ p
low
derived from Eq. ͑44͒ and Table I for small
electric fields, the ͑dashed͒ straight line fit of the ohmic cur-
rent density18
jOhmicϭe␮¯ p
low
p0
V
L
͑45͒
to the measured data points ͑see Fig. 7͒ allows the estimation
of the intrinsic charge carrier concentration p0 in
m-MTDATA to ϳ1010
cmϪ3
.
Between 2 and 10 V the concentration of injected free
excess charge carriers becomes comparable to the concentra-
tion of filled and therefore positively charged traps, causing
the observed steep increase in the current density.18
When the concentration of filled traps becomes small in
comparison with the concentration of free holes (VϾ10 V),
the trap-free SCLC density18
jSCLCϭ
9
8
␧␧0␮¯ p
high
V2
L3 ͑46͒
should be observable. The deviation from the theoretically
expected slope mϭ2 ͑dotted straight line in Fig. 7͒ for a
constant high field mobility ␮¯ p
high
is caused by the experi-
mentally verified field dependence of the hole mobility.
Using the parameters dMϭ1 nm ͓thus NHOMO
ϭ1021
cmϪ3
, see Eq. ͑21͔͒, piϭ1010
cmϪ3
, Ntϭ1.9
ϫ1016
cmϪ3
, Etϭ0.7 eV, ␧ϭ3, and the measured field-
dependent hole mobility data in m-MTDATA from Table I, a
good agreement over 9 orders of magnitude on the current
axis with the experimental electrical characteristic is
achieved ͑see Fig. 7͒, thus confirming a realistic implemen-
tation of charge carrier transport and trapping. Since trap
levels in molecular organic solids are normally distributed in
energy,12
the deviation between experiment and simulation
in the high slope region ͑5–10 V͒ can be explained by the
fact that only one effective discrete deep trap level was used
for simulation. The relatively large trap depth ͑0.7 eV͒ indi-
cates trapping due to impurities,12
whereas the mentioned
deviation in the high slope region probably arises from the
shallow trap distribution in the HOMO, which was not
implemented here.
FIG. 7. Current density in a single layer hole-only device with m-MTDATA
as hole transporting material ͑thickness 850 nm͒ and ITO and Ag electrodes
͑ITO biased positively͒. The dots represent measured data, the dashed
straight line shows a fit to the ohmic current due to intrinsic charge carriers
͓see Eq. ͑45͔͒, while the dotted straight line indicates the trap-free SCLC
density for an assumed field-independent hole mobility according to an av-
erage of the measured mobilities in the high field regime ͓see Eq. ͑46͔͒. The
solid curve represents simulated current densities using measured field-
dependent mobility data, an intrinsic hole concentration of 1010
cmϪ3
and a
trap density of 1.9ϫ1016
cmϪ3
͑trap depth 0.7 eV͒. The slope in the high-
field regime (VϾ10 V) is correctly reproduced due to the implemented field
dependence of the hole mobility.
TABLE I. Parameters for field dependent hole and electron mobilities ␮p
and ␮n ͑partly estimated͒ in m-MTDATA, 1-Naphdata, ␣-NPD, and Alq3
used for simulation. The parameters ␮0 ,RT and CRT from Eq. ͑44͒ are given
in cm2
/V s and ͑cm/V͒1/2
, respectively.
m-MTDATA 1-Naphdata ␣-NPD Alq3
␮p ␮0 ,RT 4.79ϫ10Ϫ6
1.18ϫ10Ϫ5
6.1ϫ10Ϫ4
1.86ϫ10Ϫ8
CRT 3.39ϫ10Ϫ3
2.67ϫ10Ϫ3
1.5ϫ10Ϫ3
3.76ϫ10Ϫ3
␮n ␮0 ,RT 4.79ϫ10Ϫ8
1.18ϫ10Ϫ7
6.1ϫ10Ϫ6
1.86ϫ10Ϫ6
CRT 3.39ϫ10Ϫ3
2.67ϫ10Ϫ3
1.5ϫ10Ϫ3
3.76ϫ10Ϫ3
14.139.194.12 On: Sun, 09 Nov 2014 17:37:16

F. Multilayer devices
The knowledge of the HOMO and LUMO levels of or-
ganic materials used in OLEDs is of major importance for a
proper understanding of the device behavior. The heights of
the energetic barriers for charge carrier interface crossing
used in the model, which are parameters with a high impact
on simulation results, will be derived from these levels.
Cyclovoltammetric measurements for the first oxidation
potential of 1-Naphdata and ␣-NPD showed no significant
deviations to measurements performed on m-MTDATA and
TPD, so that for the energetic levels of the HOMOs of
1-Naphdata and ␣-NPD, reported data on m-MTDATA and
TPD can be used.19
The deviations of the energetic levels of
the LUMOs of 1-Naphdata and ␣-NPD relative to the
LUMOs of m-MTDATA and TPD19
were calculated using
MOPAC ͑none for ␣-NPD and approx. 0.4 eV lower for
1-Naphdata͒. The energetic positions of the HOMO and
LUMO of Alq3 were taken from literature.19
From the energetic levels of the HOMOs and LUMOs
͑shown in Fig. 8͒ relatively high energetic barriers at the
internal interface between HTL and ETL for the two-layer
device can be deduced ͑0.7 eV for holes and 0.8 eV for
electrons͒, whereas the HOMOs and the LUMOs in the
three-layer device form an ‘‘energetic staircase’’ with re-
duced energetic barriers 0.5 and 0.2 eV for holes, respec-
tively, 0.1 eV and 0.7 eV for electrons͒.
Figures 9, 10, and 11 show experimental data on two-
and three-layer devices. The maximum luminance is compa-
rable to previously reported data on similar state-of-the-art
undoped OLEDs.36
The extremely good rectification ratios
for the current flow in negatively biased diodes proove that
no significant amount of charge carriers is transported along
point defects, which means that leakage currents can be ne-
glected in the further discussion.
As indicated by experimental data ͑see Fig. 9͒, the as-
sumption of enhanced current flow in the three-layer device
due to the energetic staircases for charge carriers in compari-
son to the two-layer device with only one high energetic step
͑see Fig. 8͒ is not correct. Based on our proposed model, the
reasons for the decreased current flow in addition to the ob-
served significantly higher luminance of the three-layer de-
vice ͑see Fig. 11͒ will be outlined in the following.
For the simulation of the observed OLED charac-
terisitics, the field-dependent charge carrier mobilities for
1-Naphdata, ␣-NPD, and Alq3 were derived from Eq. ͑44͒
using the parameters from Table I. The values for the heights
of the energetic barriers at the internal organic–organic in-
terfaces have already been mentioned ͑see Fig. 8͒. The di-
ameter of the molecules and therefore the spacing between
the molecules was again approximated to 1 nm for all mate-
rials involved, thus yielding densities of state NHOMO and
NLUMO of 1021
cmϪ3
͓according to Eq. ͑21͔͒. An OLED with
a total organic layer thickness of 100 nm is therefore ex-
pected to consist of 100 molecule monolayers.
The relative dielectric constant ␧i
was estimated to an
average value of 3 in all organic materials involved. The
FIG. 8. Energetic positions of the highest occupied and the lowest unoccu-
pied molecular orbitals ͑HOMO, LUMO͒ relative to the vacuum level,
partly taken from literature. ͑see Ref. 19͒ The heights of the energetic bar-
riers at the organic–organic interfaces used in the simulation were taken
from the differences between the according energetic positions of the mo-
lecular orbitals. The layer thicknesses are given additionally, the total or-
ganic layer thickness for both two- and three-layer devices was fixed at 100
nm.
FIG. 9. Current densities of two- and three-layer OLEDs with materials and
layer thicknesses as shown in Fig. 8. Solid lines represent simulation data
including field dependent mobilities ͑see Table I͒, trapping ͑Nt
ϭ1018
cmϪ3
, Etϭ250 meV for holes in the HTLs and electrons in the ETL͒,
and a built-in potential Vbi of 2 V. The heights of the energetic barriers at
the interfaces were taken from Fig. 8, the energetic width of the DOS at the
interfaces was set to 120 meV for 1-Naphdata/Alq3 , 120 meV for ␣-NPD/
Alq3 , and 81 meV for 1-Naphdata/␣-NPD.
FIG. 10. Voltage dependence of the efficiency of two- and three-layer de-
vices. Solid lines represent simulation results including field-dependent bulk
exciton generation at the interfaces and exciton quenching at charge carriers.
The influence of the quenching reaction cross section ␴q between singlet
excitons and charge carriers on the efficiency is plotted additionally for the
two-layer device.
14.139.194.12 On: Sun, 09 Nov 2014 17:37:16

total outcoupling efficiency of the internally emitted photons,
considering waveguiding losses (␩wave) and absorption
(␩abs), was estimated to 30%. The fraction of radiative de-
cay ␩bulk of Alq3 singlet excitons, which is equivalent to the
fluorescence quantum yield, was also set to 30% in accor-
dance with previously reported data.37
The lifetime ␶s of
singlet excitons was set to a typical value for organic mate-
rials ͑10 ns͒,21
and the average number N¯ hop of hopping pro-
cesses during the lifetime was set to 10, which is compatible
to reported literature data.38
The width Lq of the quenching
zone at the electrodes ͓see Eq. ͑34͔͒, a parameter of minor
importance, was estimated to be 2 nm.
As unknown parameters remain the built-in potential
Vbi , trap densities Nt,p
i
and Nt,n
i
, trap depths Et,p
i
and Et,n
i
,
as well as the energetic disorders at the interfaces ␴ and the
reaction cross section ␴q for singlet exciton quenching at
charge carriers.
The dotted curve in Fig. 9 represents the simulation of
the trap-free current density in a two-layer device, which has
shown to be practically independent of the energetic disorder
at the internal interface between HTL and ETL. The built-in
potential, trap densities and trap depths were then adjusted to
fit the simulation results to the experimental data. The mea-
sured current density for a two-layer device can be repro-
duced over 4 orders of magnitude using a built-in potential
of 2 V, trap densities for holes in 1-Naphdata and for elec-
trons in Alq3 of 1018
cmϪ3
with trap depths of 250 meV.
This large amount of discrete shallow traps, which decreases
the current density but does not alter the shape of the simu-
lation curve ͑as it would be the case, e.g., for an exponential
trap distribution11,18
͒, can be justified with structural defects
in the disordered amorphous films rather than with impuri-
ties, which would lead to deeper traps with smaller trap
densities.12
The value of 250 meV is comparable to reported
trap depths derived from thermally stimulated luminescence
measurements.39
For holes in the ETL and electrons in the
HTLs no trapping was considered, since the mobilities are
already assumed to be two orders of magnitude smaller.
In contrast to the current density, the device efficiency
depends strongly on the energetic disorder ␴ at the internal
interface. That is clear evidence for the fact that the majority
of charge carriers recombines at the interface between HTL
and ETL, where the probability for the formation of a bulk
exciton from an interface exciton depends strongly on the
interface barrier crossing probabilities of the charge carriers.
The two-layer device efficiency can be quantitatively repro-
duced by adjusting the energetic disorder ␴ to 120 meV and
the quenching cross section ␴q to 10Ϫ16
m2
͑see Fig. 10, the
influence of ␴q on the efficiency is shown additionally͒. The
obtained value for the quenching cross section ␴q is surpris-
ingly high ͑cross-section diameter equals several times dM),
which may be due to the fact that exciton dissociation in the
electric field is not considered, as already mentioned.
The increase of the device efficiency for operation volt-
ages V exceeding the onset voltage ͑equivalent to Vbi͒ arises
mainly from the electric field dependence of the bulk exciton
formation probability at the interfaces ͓see Eqs. ͑25͒, ͑27͒,
and ͑28͔͒. The efficiency decrease at higher voltages results
from exciton quenching at free and trapped charge carriers.
The ratio between the emission and this quenching mecha-
nism decreases with increasing electric field due to the in-
crease of the total number of charge carriers in the device
͓see Eqs. ͑35͒ and ͑36͔͒.
Naturally, with the correct simulation of device current
density and efficiency ͑which is a major result of our paper͒,
the luminance of the two-layer device is also reproduced
over 4 orders of magnitude ͑see Fig. 11͒.
The additional third organic layer ͑10 nm ␣-NPD͒ be-
tween the HTL and ETL is not expected to change any of the
above mentioned parameters. The charge carrier trapping in
the new second HTL layer is assumed to be same as in the
first HTL. Therefore the energetic disorders at the two
organic–organic interfaces remain as the only possible free
parameters to fit the simulation results to experimental data.
The energetic disorder at the ␣-NPD/Alq3 interface was set
to the same value used for the 1-Naphdata/Alq3 interface
͑120 meV͒, whereas the energetic disorder at the
1-Naphdata/␣-NPD interface was adjusted to 81 meV.
Using these energetic disorders, the decreased current
density, the higher device efficiency, and the increased lumi-
nance of the three-layer device can be reproduced, with a
very good agreement between simulation and experimental
data for applied voltages exceeding 4 V ͑see Figs. 9–11͒.
IV. DISCUSSION
The model is able to quantitatively reproduce the char-
acteristics of hole-only single-layer samples and of bipolar
two- and three-layer OLEDs, as the comparison with experi-
ment has shown ͑see Figs. 7, 9, 10, and 11͒. The experimen-
tally not direct accessible parameters, adjusted to fit the re-
sults to experimental data, remained within realistic
boundaries.
Extensive parameter studies have verified that only one
solution exists which simultaneously describes the behavior
of both two- and three-layer devices correctly. This solution
is defined ͑in dependence of the applied voltage͒ by the spa-
tial distribution of the values of the variables of state, namely
charge carrier densities and singlet exciton density, which
FIG. 11. Luminance of two- and three-layer OLEDs corresponding to Figs.
9 and 10. The model predictions are quantitatively in good agreement with
experimental data, particularly in the technologically relevant range for pas-
sive matrix displays from 1000 to 10 000 cd/m2
.
14.139.194.12 On: Sun, 09 Nov 2014 17:37:16

allow a physical interpretation of the different device behav-
iors.
Nevertheless, it should be clear that the choice of un-
known parameters for reproducing the electro-optical char-
acteristics of the processed two- and three-layer OLEDs is
not a singular one, since the actual solution is influenced
only by combinations of parameters. For example, the num-
ber of trapped charge carriers and, therefore, also the current
density depends simultaneously on the parameters trap
depth, trap concentration, and density of states in the HOMO
͑respectively, LUMO͒. Changes in one parameter ͑e.g., trap
depth͒ can often ͑to a certain extent͒ be compensated by
adjusting other parameters ͑e.g., trap concentration͒. The
same applies for the energetic disorder parameter ␴ at inter-
nal interfaces, which is strongly coupled to the energetic
height of the barriers for charge carrier interface crossing
͑which has a high experimental error of about 0.1 eV͒, since
only the interface barrier crossing probability G, a function
of both parameters, is of importance for the simulation.
The different electro-optical characteristics of the two-
and three-layer devices presented in this paper arise mainly
from the different spatial distribution of the charge carrier
accumulations. In two-layer OLEDs, the holes are stored in
front of the HTL/ETL interface in direct contact with the
electrons accumulated behind this interface ͑see Fig. 12͒.
Therefore interface recombination plays a major role in the
total recombination. The low device efficiency can be ex-
plained with the inefficient generation of bulk excitons at the
interface, since the energetic barriers for charge carrier inter-
face crossing are relatively high ͑see Fig. 8͒.
In contrast, in the three-layer device the holes are accu-
mulated before the interface between the two HTLs ͑since
there it is the highest barrier͒, whereas the electrons are
stored after the interface between the second HTL and the
ETL ͑see Fig. 13͒. These two charge carrier reservoirs are
separated by the thin ␣-NPD layer, thus reducing the direct
interface recombination and, due to the builtup of large space
charges reducing the electric field and the charge carrier ve-
locity in the first HTL and in the ETL ͑see Fig. 13͒, also the
current density ͑see Fig. 9͒.
The high energetic barrier for electrons at the HTL/ETL
interface causes an accumulation of electrons in the ETL
bulk near this interface, with only very few electrons enter-
ing the adjacent HTL. For charge carrier recombination, the
holes have to cross the 1-Naphdata/␣-NPD interface by nor-
mal hopping processes ͑drift and diffusion͒ to arrive at the
␣-NPD/Alq3 interface. Since the energetic barrier for holes
at this HTL/ETL interface is significantly reduced for the
three-layer device in comparison with the two-layer device,
interface recombination becomes a very effective source for
bulk excitons, thus yielding a higher device luminance in
spite of the decreased current density. Additionally, the holes
can easily enter the ETL bulk for further direct bulk exciton
formation.
FIG. 12. Spatial distribution of the electric field, the energetic postions of
the HOMO and the LUMO, the free and trapped charge carrier densities,
and the singlet exciton density along the z axis in a two-layer device for an
applied voltage of 8 V. The HTL/ETL interface is marked by the vertical
solid line. The dashed line represents the average electric field.
FIG. 13. Spatial distribution of the electric field, the energetic postions of
the HOMO and the LUMO, the free and trapped charge carrier densities,
and the singlet exciton density along the z axis in a three-layer device for an
applied voltage of 8 V. The organic–organic interfaces are again marked by
vertical solid lines. The dashed line again represents the average electric
field.
14.139.194.12 On: Sun, 09 Nov 2014 17:37:16

Summarizing both the experimental and simulation work
of this paper, two main aspects have to be outlined. First, we
have shown the relevance of our model for the study of state-
of-the-art vapor-deposited OLEDs by experimentally verify-
ing the key assumption of two ohmic contacts and by a quan-
titative reproduction of electro-optical device characterstics.
Second, from the insight into the spatial distribution of the
charge carrier accumulation zones, we have explained the
enhanced luminance and device efficiency of three-layer de-
vices relative to two-layer devices.
V. CONCLUDING REMARKS
The proposed model can also be applied to a large vari-
ety of other materials and different multilayer OLED archi-
tectures, even for the case of injection limitation for one
charge carrier species. Additionally, time-dependent phe-
nomena like luminance onset and offset can be simulated,
and even impedance spectra are possible in the case where
computer power is no limitation. The model can easily be
modified to include spatial variations in trap densities, dop-
ing of emissive layers, or extra charge carrier blocking lay-
ers.
We expect from the quantitative simulation of our
OLEDs a positive impact on further device optimization,
such as improved efficiency and operating voltage. For ex-
ample, instead of performing time-consuming experiments
for layer thickness optimization in multilayer OLEDS, a hint
for the ideal layer setup can be derived from calculations.
This method can also be used to align the operating voltages
of different colored pixels in multicolor flat panel displays
based on OLED technology.
Naturally, the simulation results depend strongly on the
parameter values used, so that the determination of unknown
parameters remains a crucial point to further ensure the
physical relevance. The severest limitation for large param-
eter studies, in order to gain some insight into the qualitative
behavior of OLEDs, is given by the long calculation times,
which are on the order of days on common Pentium II PCs
for a complete electro-optical characteristic.
In conclusion, we have shown in comparison with ex-
perimental data the applicability of our model for quantita-
tive simulations of multilayer OLED properties. We have
outlined the different mechanisms governing the behavior of
two- and three-layer devices and the implications on device
performance.
ACKNOWLEDGMENTS
The authors would like to thank J. Gudarslu, G. Witt-
mann, and J. Schumann for technical help, A. Kanitz for
synthesizing the Alq3 and performing molecular orbital cal-
culations, A. Richter from SynTec GmbH for providing the
cyclovoltammetric measurement data, and H. v. Seggern and
R. Helbig for helpful discussions. Support from the
Bundesministerium fu¨r Bildung und Forschung ͑Grant No.
BMBF-Project 01 BK 703/4͒ is gratefully acknowledged.
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