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Bipolar charge and current distributions in organic light-emitting diodes

J. C. Scott, S. Karg, and S. A. Carter

Citation: Journal of Applied Physics 82, 1454 (1997); doi: 10.1063/1.365923

View online: http://dx.doi.org/10.1063/1.365923

View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/82/3?ver=pdfcov

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Bipolar charge and current distributions in organic light-emitting diodes

J. C. Scott,a)

IBM Research Division, Almaden Research Center, 650 Harry Road, San Jose, California 95120

S. KargPhysikalisches Inst., Universitat Bayreuth, 95440 Bayreuth, Germany

S. A. CarterDepartment of Physics, University of California, Santa Cruz, California 95064

Received 17 January 1997; accepted for publication 22 April 1997

The electron and hole charge distributions and the luminance profile in organic light-emitting diodes

OLEDs depend upon the bulk properties of the emissive layer, as well as on the injection

characteristics at the anode and cathode interfaces. We address the problem of separating the

relative contributions of hole injection, electron injection, and recombination to the overall

performance of single layer OLED devices. Using the approach of Parmenter and Ruppel J. Appl.

Phys. 30, 1548 1959, and including Langevin recombination, expressions are derived for the

currentvoltage and radiancecurrent dependencies in terms of electron and hole mobility,

luminescence yield, and a current balance factor. When one carrier dominates the current flow,

as in many practical cases, it is possible to obtain a simple asymptotic relationship which permits a

test of the assumptions required to obtain the analytic solution. Experimental data from

poly2-methoxy-52-ethylhexoxy-phenylenevinylene diodes fabricated with various anode and

cathode materials are evaluated in the context of this analytical approach. 1997 American

Institute of Physics.S0021-89799703315-X

I. INTRODUCTION

Organic and polymeric light-emitting diodes OLEDs

have become the focus of considerable attention since the

development of efficient multilayer organic devices1 and the

subsequent discovery of electroluminescence in several

classes of conjugated polymer.2 4 In spite of significant ad-

vances in the understanding of the qualitative behavior of

OLEDs, a fully quantitative description of charge injection at

the electrodes, charge transport in the bulk, and electron

hole recombination has yet to emerge. The analysis of bipo-lar, space-charge-limited current flow dates back at least to

the 1950s, notably to work at the RCA labs.5 7 The difficulty

of the problem arises from the necessity of determining self-

consistently the electric fields at the cathode and anode in-

terfaces, which simultaneously dictate the injection currents

and satisfy the electrostatics of the hole and electron densi-

ties. Since neither the interfacial injection characteristics for

example, Ohmic, Schottky8 or FowlerNordheim9 are

known a priori, and since the mobility of both signs of

charge is known for very few materials of interest, it is by no

means straightforward to design a series of experiments

which will permit such a complete and self-consistent de-

scription. Indeed, the experimentalist is frequently faced withconsiderable uncertainty regarding the chemical and mor-

phological structure of the interface which may depend not

only on preparation conditions, but also on the electrical his-

tory of the device.

Several recent papers have addressed the transport prob-

lem for OLEDs, taking diametrically opposite approaches.

For example, Parker10 used the electric field dependence

to deduce that in poly2-methoxy-52-ethylhexoxy-

phenylenevinylene MEH-PPV devices, especially

those fabricated to show monopolar transport, the current

voltage (IV) characteristics are controlled by Fowler

Nordheim tunneling at the interfaces. Davids et al.11 modi-

fied the FowlerNordheim equations to account for the low

mobility of carriers in amorphous organic materials. In con-

trast, Blom and coworkers12 have analyzed IVdata in mo-

nopolar devices and conclude that the IVbehavior is bulk-

controlled, with space-charge-limited hole current and trap-

limited electron current. Both approaches appear to describethe relevant data equally well, which may reflect the fact that

both injection and bulk limitation are active in the current

regimes of interest and that the power law behavior of space-

charge currents passes smoothly into the exponential law for

tunneling. There may also be inherent difficulties in extrapo-

lating data from hole-only and electron-only devices, which

may be particularly sensitive to electrode preparation, to

truly bipolar OLEDs. For example, a bottom calcium cath-

ode is almost certainly oxidized in even the best glove box

and depositing a gold top anode may lead to the diffusion of

Au atoms into the polymer film. The approach taken in the

current article permits, in principle, the derivation of the re-

lation between interface field and injection current. Unfortu-nately in practice, the absence of a set of well-characterized

material parameters prohibits such a completely quantitative

evaluation.

In the following sections, we present the details of the

analysis necessary to determine the profiles of electric field

and electron and hole densities across the thickness of a

singlelayer OLED. The model is based on the solution of the

bipolar injection problem due to Parmenter and Ruppel

PR.5 Their approach contains several assumptions: the dif-

fusive current due to carrier density gradients can be ne-aElectronic mail: [email protected]

1454 J. Appl. Phys. 82 (3), 1 August 1997 0021-8979/97/82(3)/1454/7/$10.00 1997 American Institute of Physics[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to

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glected relative to drift in the applied electric field; deep

traps for both electrons and holes are ignored; and the mo-

bility of the carriers is assumed to be independent of electric

field. One must question the validity of each of these as-

sumptions in the present case, and it is one of the goals of

this article to evaluate the extent to which each of them

holds. In addition, we make one more assumption: recombi-

nation kinetics can be described by the Langevin

equation.13,14 This is not absolutely essential mathematically,

but it greatly simplifies the evaluation of the integrals which

appear in the PR solution, and in addition reduces by one the

number of materials parameters which need to be deter-

mined.

We cast the theoretical predictions in a form which is

amenable to comparison with experimental data, namely

currentvoltage and radiancecurrent relations. In contrast

to the photoconductive materials which motivated the RCA

work, the connection between theory and experiment is im-

proved in the present case by the fact that there is an addi-

tional experimental observable, namely the intensity of the

emitted light. The boundary conditions of the PR solution

enter in the form of a current balance factor which is

directly related to the radiance of the device. Hence in the

present case, the emission of the diode itself helps to deter-

mine the boundary conditions and thus further reduces the

number of undetermined parameters. Further, when one sign

of carrier dominates the current, true in many cases of prac-

tical interest, then a function of current, voltage, radiance,

and sample thickness is found which depends only on con-

stant materials parameters. We use the constancy of this

function to assess the validity of the model and its solution,

independent of the boundary conditions i.e., of the charge

injection mechanisms. It is found that the relationship in-

deed holds reasonably well and therefore we conclude that

the assumptions and approximations which permit analytic

solution of the problem are not too severe.The manuscript is arranged as follows. The next section

summarizes the PR solution and describes the extensions

made on the basis of Langevin recombination and observable

radiance. In Sec. III we reanalyze data from several samples

on the basis of the predictions obtained. Section IV contains

a discussion of the relevance and limitations of the results

and the conclusions are ennumerated in Sec. V.

II. THEORY

For the convenience of the reader, and to establish nota-

tion, we reproduce here the essential points of the PR paper.5

The relevant equations give the electron and hole currents

JnXNeeF, 1a

JpXPehF 1b

as functions of the charge densities N and P , and electric

field F. Here we use capital letters for physical dimen-

sionedvariables and will use lower case for reduced, dimen-

sionless variables. In steady state, the total current, JJnJp, is independent of position, X, between the cathode

(X0) and anode (XL). The charge carrier mobilities,

e and h, are taken to be constants, independent of electric

field. Electronhole recombination is described by bimolecu-

lar kinetics:

1

e

Jn

XrN P

1

e

Jp

X . 2a,b

For Langevin recombination, which is expected to apply in

this case where charge transport is due to hopping, the re-

combination coefficient is given by14

r e0

eh 2e00 . 3

Here, is the dielectric constant of the organic material, and

this equation defines0 , the recombination mobility. Lastly,

the electric field satisfies the Poisson equation

F

x

e

0NP . 4

This form implies that all carriers are mobile and that none

are frozen in deep traps. On the other hand, shallow trapping

events may be included in the definition of a trap-controlled

mobility, where the carriers are in equilibrium with a trap

distribution which is no wider than a few times their thermalenergy.

Following PR, we introduce several new variables:

aeh0

20LJ F2, 5

BJn/J, 6a

CJp/J. 6b

Note that this definition ofa differs from PRs A in that it is

rendered dimensionless by normalizing to the sample thick-

ness, L. Otherwise we retain the PR notation. B(x) is the

fraction of the current carried by electrons at normalized

position xX/L (0 x1) . The dimensionless mobilitiesare

ee/0 , 7a

hh/0 , 7b

where now, because of the Langevin recombination relation

eh2, 8

and

0e,h 2. 9

The set of Equations 1,2, and 4 then becomes

BC1, 10

B

x

C

x

BC

a , 11a,b

a

x 2e BeC. 12

The general solution is most conveniently written in terms of

B(x) which is the solution of the differential equation

B

xK1B 1e 1B e1, 13

1455J. Appl. Phys., Vol. 82, No. 3, 1 August 1997 Scott, Karg, and Carter[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to

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where Kis a constant of integration

K1/Ba

B cdB Be1 1B 1e. 14

Then a is given by

aKBe 1B 2e 15

and C follows from Eq. 10. The boundary conditions Bcand B a are the fractions of the current carried by electrons at

the cathode and anode, respectively.

The variables of physical interest are the charge densities

and the electric field

nNL 3j 1/2 2ee

1/2 B

a1/2, 16

pPL 3j 1/2 e2e

1/2 C

a 1/2, 17

fF/ 2e0L

2j 1/2 ae 2e

1/2

, 18

where the normalized current density is given by

jJ/ 2e2

00L

5 . 19Also of interest is the n p product which gives the profile of

recombination and hence, in the absence of self-absorption

and of enhanced interfacial quenching, of luminance

n pjBC

a . 20

In the typical experiment with OLEDs, one generally

measures the current, voltage, radiance behavior. Devices

use different metals for anode and cathode and there is there-

fore a built-in potential which is given to a first approxima-

tion by the work-function difference of the electrodes. Thebuilt-in potential must be added as a constant of integration

to the voltage, V. The currentvoltage characteristic is then

given by

VVbi

J1/2 20L

3

0eh

1/2

K3/2Bn

B cdBB3/2e1 1B 23/2e

21

and the radiance RpN PdX i.e., the emitted optical

power per unit area is given by

Rp

J

e

fB cB afb, 22

where is the mean photon energy and the fluorescence

yield, f, is the fraction of recombination events which re-

sults in an externally emitted photon. The form of Eq. 22

simply states that the quantum efficiency is given as one

might have deduced by a straightforward argument requiring

no differential equations by the product of the fluorescent

yield and the fraction of the electron current which, having

left the cathode, fails to reach the anode. It suggests the

usefulness of introducing the current balance factor bBcBa. Equivalently, by Eq. 10, bCaCc, in obvious

notation. It also provides a means of using experimental

data to deal with at least one of the boundary conditions

necessary for the explicit evaluation of the integrals in the

currentvoltage relation, Eq. 21.

It may be argued for many combinations of organic and

electrode materials that one or another sign of carrier domi-

nates the current. This is almost certainly true at very low

current levels, where only one electrode injects, but may be-

come less true at emitting current levels, since clearly both

signs of carrier must be present. In a previous publication15

we have described experiments which lead to the conclusion

that, for polyaniline/MEH-PPV/Ca structures, holes are the

dominant carrier. With these ideas as motivation, we now

evaluate the integrals in Eqs. 14 and 21 for small values

of the current balance factor, b1. Note, however, that the

equations are completely symmetric with respect to inter-

change of electrons and holes, and that one could perform an

identical analysis for small hole current. The asymptotic ex-

pansion yields an approximate currentvoltage relation

VVbi

J1/2 20L

3

0eh

1/2 1

b1/22 43eB c

1 1e B c3/2 . 23

First, let us examine the third factor containing B c1

and e2. Unless the denominator approaches zero, this has

a value of order unity, but if, for example, e1 and

Bc1 then the fraction would diverge. However, in this

case the numerator is negative, and therefore unphysical. In-

deed, one might make the hypothesis, based on the fact that

the numerator must remain positive, that for vanishingly

small electron mobility, the fraction of the current carried by

electrons at the cathode, B c, never exceeds 1/2, which is

physically reasonable. Hence the denominator will never be-

come vanishingly small and the factor remains of order

unity.

Combining Eqs. 21 and 23, we arrive at one of the

central results of this article

VVbiRp1/2

J 20L

3

0eh

e f

1/2

O 1 . 24

The combination of mobilities which enters this expression

is an effective, or reduced mobility

eh

20e

1h

11r, 25

the value of which is dominated by the lowerof the electron

and hole mobilities.

To the extent that the initial assumptions regarding dif-

fusion, trapping, the field-independence of the mobilities, the

absence of interfacial quenching, and the lack of self-

absorption are valid, then the right-hand side of Eq. 24 is

constant for a given sample of thickness L and therefore

serves as a good experimental test of these assumptions. Un-

fortunately, to our knowledge, both electron and hole mobili-

ties are known for none of the materials of experimental

interest, and it is therefore not yet possible to test fully the

validity of the result. In the next section, we examine some

experimental data in light of the above analysis and of the

materials parameters available to date.

1456 J. Appl. Phys., Vol. 82, No. 3, 1 August 1997 Scott, Karg, and Carter[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to

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Ideally, one wishes to use the available experimental

data to understand in detail the behavior of a given diode

structure. The complete solutionEqs.21and 22give the

three observable quantities I, V, and Rp in terms of sixparameters Vbi , e, h, f, B c , and B a. In principle,

several of these may be determined by independent experi-

ments. For example, the built-in voltage is related to the

work-function difference of the anode and cathode metals. It

may be even more accurately determined, in the same

sample as used for IVRp, by electroabsorption

measurements,16 or from the bias voltage which nulls the

photocurrent. The carrier mobilities can be determined by

time-of-flight methods,17 from field-effect transistor

measurements,18 or from the single carrier space-charge

limited current.19 The external fluorescent yield may be esti-

mated from the photoluminescence efficiency if one makes

the additional and perhaps questionable20 assumption thatthe singlet/triplet statistics introduce an additional factor of

1/4. If these four parameters are known, then both B c(J) and

B a(J) can be determined from the experimental data and the

solution is completely determined. Equation 18 gives the

electric field throughout the layer, and particularly at both

electrode interfaces. Hence the charge injection mechanism

could be unambiguously determined. Unfortunately, we

know of no diode structure for which such complete infor-

mation is available.

III. COMPARISON WITH EXPERIMENT

Figure 1 shows radiancecurrent data for a series of

MEH-PPV diodes with polyaniline anodes and several dif-

ferent cathode metals; Ca, Al, and Ag. The data are plotted

as external quantum efficiency (Rp/J)( e/) vs logJ. The

efficiency is, according to Eq. 22, proportional to the cur-

rent balance factor, b , since the fluorescence yield is as-

sumed to be constant. Both the current density and b are seen

to vary by several orders of magnitude. These same data are

replotted in Fig. 2 in the form suggested by Eq. 24. For

each sample, the function (VVbi)Rp1/2/J is seen to vary by

less than one order of magnitude. Note that the hysteresis

obtained during a voltage cycle is comparable to the varia-

tion itself. An apparent failure of the model is that, in spite ofthe common active layer, MEH-PPV in all three devices,

there is more than an order of magnitude difference between

the Ca and Ag samples. We return to these points below.

The order of magnitude of the constant (V

Vbi)Rp1/2/Jis between 0.1 and 10 VW1/2cm/A for each of

the samples shown in Fig. 2. The value of the right-hand side

of Eq. 24, using values see discussion section below for

both electron and hole mobility of 104 cm2/Vs, a thickness

of 100 nm, and a quantum yield of 1% is approximately

1 VW1/2cm/A. Thus the quantitative agreement between

theory and experiment is also encouraging.

In order to test the predicted thickness dependence we

replot in Fig. 3 the data of Ref. 15, Fig. 3 in the form ( V

Vbi)Rp1/2/J L 3/2, following Eq. 24. The data collapse to-

FIG. 1. External quantum efficiency of three MEH-PPV LEDs with polya-

niline anodes and different cathodes. For all three devices the thickness of

the MEH-PPV layer is 100 nm and the highest applied voltage is 5 V. FIG. 2. (VVbi)Rp1/2/Jplotted as a function of current density for the same

three devices as in Fig. 1. The values used for the built-in potentials are

given in the legend.

FIG. 3. (VVbi)Rp1/2/JL 3/2 for several polymer LEDs. These are the same

data as shown in Fig. 3 of Ref. 15.


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wards a common curve, varying within only a factor of 3

over nearly four decades in current.

The agreement between the predictions of the model and

the experimental data, although far from perfect, is quite rea-

sonable and indicates that the approximations made are not

too severe. The variation with different electrode materials,

and even between batches of MEH-PPV is rather large and

needs to be explained. If we take the model at face value,

then according to Eq. 24, for a given device, (V

Vbi)Rp1/2/J is proportional to (f/r)

1/2. This quantity

may be quite sensitive to impurities which quench emission

and/or provide shallow traps which modify the mobility of

the charge carriers. Different electrode materials, particularly

the cathode which is deposited last, may be expected to af-

fect the polymer layer in different ways. Indeed, recent stud-

ies have shown that Ca and Al each react with PPV and its

derivatives2123 and moreover that the nature of the reaction

products differs between the two metals.24 It is then likely

that during deposition, metal atoms penetrate the polymer

layer to a different extent creating different concentrations of

trapping and quenching defects. Alternatively, the last factor

in Eq. 23 may be sufficiently different for the different

cathodes to account for the data. For example, a CaO inter-

face layer can serve to block holes from entering the cathode,

makingB c closer to unity. Additional work on other materi-

als is required to understand these observations in more de-

tail.

Another potential source of discrepancy lies in the pos-

sibility of an inhomogeneous current distribution and there-

fore inhomogeneous emission across the area of the device.

Since the radiance and current density enter Eq. 24 with

different exponents, the value of the left-hand side depends

on the active area of the device. Although our LEDs look

relatively uniform to the naked eye, we have not examined in

detail the pattern of emission of the devices reported here.

The data of Figs. 1, 2, and 3 were obtained from a volt-age cycle on nearly new diodes and reveal significant hyster-

esis. This behavior reflects physical and/or chemical changes

in the diode structure that have been previously noticed and

sometimes called juvenile aging.25 It is not yet clear

which parameter f, r, or Vbi is most affected in this

process, but we note that for Fig. 3, we were forced to use a

value of the built-in potential for the ITO/Ca couple of 1.8 V,

rather less than the accepted work-function difference,

b(ITO)b(Ca4.82.91.9 V, in order to prevent un-

physical negative values of the abscissa. Thus it is possible

that some of the aging is due to the development of interfa-

cial dipole layerswhich modify the built-in potential across

the bulk of the layer. This is the subject of ongoing investi-gation.

IV. DISCUSSION

We start this section with a more detailed examination of

the assumptions made during the analysis of Sec. II. In light

of the reasonably good agreement between theory and ex-

periment, it appears that these assumptions and approxima-

tions are quite well obeyed in the samples considered to date.

The field independence of the mobility is somewhat sur-

prising since most disordered organic charge transport mate-

rials exhibit a mobility26,27 which varies exponentially as the

square root of field (exp F). However, the strength ofthis dependence varies with both concentration of chargeable

sites and with temperature.28 becoming weaker at high con-

centration and even changing to negative slope above a cer-

tain material-dependent temperature.29,30

The experimental data seem to suggest that the reduced

bipolar mobility,r, is not much less than 104 cm2/Vs and

therefore that the electron mobility is not much less than the

hole mobility.31,32 This is in contradiction to many attempts

to measure the electron mobility by transient

electroluminescence33,34 EL or by time-of-flight methods35

which have led to the suggestion that e108 cm2/Vs.

However, time-of-flight mobility measurements are typically

made in the small-charge-density regime in order to mini-

mize space charge effects, and transient EL includes trap

filling effects before the steady state current is established.

When the currents flow in light-emitting diodes, it is very

likely that the steady state density is sufficiently high that all

available traps and sites of long charge residence time are

filled. Thus the relevant mobility is not the trap-controlledmobility but the trap-filled mobility which may be sev-

eral orders of magnitude greater, and may also have a weaker

field dependence.

This leads us next to the issue of neglected trapping,

which may seem to be at variance with the ideas presented in

the previous paragraph. However, the manner in which

trapped charge density affects the results of the analysis is to

alter the electric field distribution across the organic layer. If

this trapped-charge field is much less than the sum of the

applied field and the space-charge field due to mobile carri-

ers, then the effect will be negligible. One must also remem-

ber that both electrons and holes may be trapped, so that

some amount of neutralization takes place. From Eqs. 16and17, one may estimate the mobilecharge density at the

lowest current levels (105 A/cm2) to be of order

1015 cm3. Note that this is at the point where the emission

first becomes detectable, in our case at approximately

106 W/cm2. Thus the approximation of no trapping re-

quires that the concentration of uncompensated deep traps be

less than, say, 1014 cm3, which is low, but not unreason-

able.

In order to examine the magnitude of the error induced

by ignoring diffusion, consider the charge density gradients.

Within the scope of the present model, using Eqs. 16,17,

and 18, one can, in principle, estimate the relative magni-

tudes of the hitherto neglected diffusion current and the

drift current in terms of the poorly characterized diffusion

constant and mobilities. The result, as is well known,36 scales

according to j 1/2 and is therefore most important at low cur-

rent density. For both electrons at the cathode and holes at

the anode, the ratio of the drift currents is proportional to

b3/2 and therefore the question of divergence depends on the

relationship between b and j in the limit of low current. This

is still an open issue and the subject on ongoing work.

The last assumption to reexamine is that of Langevin

recombination, which is based on the concept that if two


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oppositely charged carriers, with sufficiently short mean free

paths, approach each other within the Coulomb radius,

rce2

40kT, 26

then they will inevitably recombine. For this to apply in the

present case, the Coulomb radius must be small compared to

the thickness of the sample, and also the electron and hole

densities must be small enough that there is a low probability

of finding multiple carriers within the volume of the Cou-lomb sphere. At room temperature and for a dielectric con-

stant of 3.5, typical for conjugated polymers, the Coulomb

radius is approximately 20 nm, sufficiently less than the

usual sample thicknesses which are about 100 nm. At the

very highest current levels 0.1 A/cm2 in the present work,

and up to 1 A/cm2 in other experiments, the charge densities

can be estimated Eqs. 16 and 17 to approach

1017 cm3 and therefore the number within the Coulomb

sphere is no more than 3. For most experimental data this

limit is rarely exceeded and hence the Langevin assumption

is generally valid.

The data plotted in Figs. 2 and 3 does seem to show a

consistent trend in its deviation from constancy: all thecurves are convex, and many show a maximum in mid- to

upper range of current. This is presumably due to break-

downs in the assumptions and approximations used in deriv-

ing the analytical solution. At this stage in our investigations,

it is not yet clear which particular assumptions might ac-

count for the observed shape. It is also worth mentioning that

care must be taken to eliminate or account forleakage cur-

rents at the level of tens of pA, as well as the series resis-

tance of the electrodes which may give an additional voltage

drop at high current. Lastly we mention the uncertainty in the

built-in potential. As noted above, this is clearly not just the

difference in workfunctions of the electrodes, but may in-

clude the effects of interfacial dipole layers. Moreover, weare currently investigating the possibility that it can change

with operation and may therefore account for the observed

hysteresis.

The solution given by PR embodies the boundary condi-

tions, i.e., the charge injection at the electrode interfaces, in

the parametersB c andB a. It will prove useful in subsequent

discussions to use a diagram for reference. In Fig. 4, we have

sketched the various points of interest in the (Bc,Ba) plane.

Both B c and Ba are fractions less than one. Since electrons

are injected at the cathode and may recombine before they

reach the anode, B aB c. Thus the region of physical inter-

est is the triangle bounded by B a0, B c1, and B aB c.

The origin represents the condition of hole-only current and

the point 1,1 corresponds to electron-only current. Along

the line B aBc there is no recombination ( b0) and lines

of equal b are parallel and to the right of this. The locus

B a1B cCc describes balanced injection where the frac-

tion of the current carried by electrons at the anode equals

that carried by holes at the cathode. Finally, the point 1,0

represents the ideal condition of balanced injection and com-

plete recombination. The behavior of the integrals in Eqs.

14 and 21, can be mapped onto this diagram, with the

reduced electron mobility, e, as an implicit parameter. The

problem to be solved is then to trace in detail the path taken

on this diagram by the cathode-organic-anode system as the

total current density is increased.

In Sec. II, we described the behavior of the voltagecurrent relation, Eq. 21, for values of B c and b near the

origin. There is a square-root divergence in V(b1/2) as

b tends to zero, for all nonzero Bc. At the origin hole-only

current the divergence is suppressed. In this case, and for

he1, the last two factors of Eq. 21 can be evaluated

exactly to yield the finite value, 2/3, corresponding to the

single carrierChilds lawsolution.19 We have confirmed by

numerical integration that the square-root divergence persists

for small b deviations along the entire line of no recombina-

tion except at the endpoints. At the point of balanced injec-

tion and complete recombination, 1,0, again the integrals

can be exactly evaluated for he1, yielding /8. These

analytical and numerical results, along with other numericalevaluations which we do not report here, reveal that the fac-

tor K and the other integral in Eq. 21 are slowly varying

functions both ofe and of position on the (Bc,Ba) diagram

except near the line of no recombination. Thus, for most

values of the current balance factor which are not too small

the currentvoltage relation is well approximated by the fa-

miliar so called Childs law relation, J V2, independent

ofb . Thus one expects to find a crossover from the predic-

tion of Eq. 24 to Childs law behavior, albeit with a nu-

merical constant different from the usual 9/8 and with an

effective mobility, as the current density is increased.

It is clear from the foregoing discussions that, in general,

there is an electron current at the anode as well as the cath-

ode, and similarly, a hole current at both electrodes. Consid-

erable attention10,11,37 has been given to the injection currents

electrons at the cathode and holes at the anode and their

dependence on field and/or voltage. A complete description

of the problem also necessitates consideration of the extrac-

tion currents. These will depend not only on the nature of

any interfacial barrier and the electric field across it, but also

on the density of carriers at each interface. The equations

given above permit the calculation of these densities and the

electric field, but for a complete solution the boundary con-

FIG. 4. Diagramatic representation of the boundary conditions appropriate

to the PR solution.B c and B a are the fractions of the total current carried by

electrons at the cathode and anode, respectively. Their difference, b , is the

current balance factor.


14.139.194.12 On: Fri, 24 Oct 2014 14:26:52


8/8

ditions must be determined self-consistently for any given

model of injection and extraction.

V. SUMMARY AND CONCLUSIONS

We have analyzed a model for the charge distributions in

OLEDs by adapting the work of Parmenter and Ruppel to

include Langevin recombination, and have taken advantage

of the fact that, to a good approximation, the recombination

rate is proportional to the observed luminous emission. The

results are contained in the two equations Eqs. 21 and22 which describe the currentvoltage and current

radiance behavior in terms of the fraction of the current car-

ried by electrons at the cathode and anode. These boundary

conditions remain to be more fully determined by further

examination of experimental data, but, by means of an

asymptotic expansion valid when one sign of carrier domi-

nates in the low current regime, we predict that the combi-

nation of variables (VVbi)Rp1/2/JL 3/2 should be constant.

The prediction is borne out to a reasonable degree of accu-

racy in experimental data obtained on MEH-PPV diodes

with various electrode materials and for various thicknesses.

Hence we conclude that the assumptions inherent in the PR

analysis, namely negligible diffusive currents, field-independent mobilities and few deep traps, are not too se-

vere.

The concept of a current balance factor arises naturally

from the PR solution, and we have shown that it is directly

proportional to the emission. We have also introduced a dia-

grammatic method for locating the relevant boundary condi-

tions and as a guide to describing the behavior of devices as

the current density increases.

In this article we have not addressed in any detail the

question of the charge injection processes at the anode and

cathode interfaces. Because of the universal nature of the

square-root singularity at small values ofb , and because the

carrier mobilities are not yet known with sufficient accuracy,it is not yet possible to use the available data to locate the

exact operating conditions on the (B c,B a) map. This there-

fore becomes the focus of future work. We also hope that the

present analysis can be used to help identify the nature of the

changes in device performance with aging by separating

electrode degradation i.e., the boundary conditions implicit

in the current balance factor and bulk degradation reflected

in the function (VVbi)Rp1/2/J which depends only on mo-

bility and fluorescence yield. Finally, it is clear that a fully

quantitative solution may need to include some of the effects

neglected here, and will have to be obtained numerically.

The results of the analytic solution should then be useful in

estimating the values of numerical parameters.

ACKNOWLEDGMENTS

We are grateful to Barbara Jones for insights into the

properties of the integrals of Eqs. 14 and 21, to Phil

Brock and Marie Angelopoulos for providing the materials

used in this work, and to Weidong Chen for critical reading

of the manuscript.

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