Inhomogeneous luminance in organic light emitting ...

JOURNAL OF APPLIED PHYSICS 100, 114513 共2006兲

Inhomogeneous luminance in organic light emitting diodes related to electrode resistivity Kristiaan Neyts,a兲 Matthias Marescaux, and Angel Ullan Nieto ELIS, Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium

Andreas Elschner and Wilfried Lövenich H.C. Starck GmbH, c/o Bayer AG, Geb. B202, D-51368 Leverkusen, Germany

Karsten Fehse, Qiang Huang, Karsten Walzer, and Karl Leo Institut für Angewandte Photophysik, TU Dresden, D-01062 Dresden, Germany

共Received 2 August 2006; accepted 19 September 2006; published online 12 December 2006兲 In organic light emitting diodes 共OLEDs兲 with transparent electrodes, the luminance usually becomes inhomogeneous if the size of the pixel increases above 10 mm. A theoretical model for inhomogeneous voltage and luminance in OLEDs is provided together with an approximate analytical solution for the problem in case of cylindrical symmetry. Experimental observations of inhomogeneous luminance are compared with numerical simulations based on the theoretical model, proving the applicability of the approximations made in the theoretical model. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2390552兴 I. INTRODUCTION

Organic light emitting diodes 共OLEDs兲 are serious candidates for bright, efficient, flexible, and cheap displays or lighting units.1 For 共back兲lighting applications, OLEDs are aiming at large areas with homogeneous brightness. The resistivity of the transparent electrode will lead to an appreciable lateral voltage drop if the lateral current flow reaches a certain value. The luminance of the OLED depends on the voltage over the organic layers and may also become inhomogeneous. The conductivity of the electrode material and the size of the pixel play an important role in this inhomogeneity. The issue of voltage drop over resistive electrodes is important for many kinds of devices with transparent electrodes and dc current operation: OLEDs,1–3 solar cells,4 and electrochromic devices.5 In general, lateral inhomogeneities can be reduced by reinforcing the transparent electrode material with a metallic wire grid. In this paper, we present a theoretical description for voltage and brightness inhomogeneity due to a lateral voltage drop. The differential equations can be solved analytically for a simplified cylindrical geometry. The inhomogeneity of the luminance is observed experimentally as a function of the applied voltage. Finally, a comparison is made between the observed inhomogeneity and a numerical simulation based on the theoretical model.

100 nm兲 are much smaller than the lateral dimensions of the OLED device 共⬎100 ␮m兲. In the model we now make the following approximations. •

The potential in the electrode materials is practically independent of the z coordinate, thus Vb共x , y兲 and Vt共x , y兲. • The current density 共A / m2兲 in the emitting layer is approximately perpendicular to the substrate and independent of the z coordinate: jz共x , y兲. • In OLEDs, the current density is a function of the voltage difference over the layer, jz = jOLED共Vb − Vt兲. •

共1兲

The luminance L is proportional with the current, which means that the efficiency ␩ 共cd/A兲 is constant, L共x,y兲 = ␩ jz共x,y兲.

共2兲

The current density in the electrode layers is proportional to the electric field, jt,b = − ␴t,b grad Vt,b .

共3兲

The conservation of the current flux in a stationary situation requires

II. THEORETICAL MODEL

For the electrical modeling of the OLED we consider three layers 共see Fig. 1兲: a bottom electrode 共usually transparent兲, an emitting layer, and a top electrode 共usually reflective兲, with respective thicknesses db, de, and dt. It is important to note that the thicknesses of these layers 共order of a兲

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FIG. 1. OLED device with conductive top and bottom electrodes and indication of thicknesses, voltages, and current densities 共dimensions not to scale兲.

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© 2006 American Institute of Physics

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div共dtjt兲 = jz , 共4兲

div共dbjb兲 = − jz .

Eliminating the current density components jtx, jty, jbx, jby, and jz from these equations leads to the following set of differential equations for the electrode voltages: ⵜ2Vt = − Rt jOLED共Vb − Vt兲, ⵜ2Vb = Rb jOLED共Vb − Vt兲,

共5兲

with R关⍀兴 the sheet resistance of the electrodes, Rt,b =

1 . ␴t,bdt,b

共6兲

These differential equations have to be completed by boundary conditions of the following form: 共a兲 boundaries of electrodes connected with a voltage source, Vt,b共x , y兲 = Vt0,b0 and 共b兲 boundaries of electrodes which are not contacted, grad共Vt,b兲 · nt,b = 0, with nt,b normal to the electrode boundary. In the regions 共x , y兲 where only one electrode is present, or an additional insulator covers the OLED, the differential equations for the electrode voltages 共5兲 become ⵜ2Vt = 0, 共7兲

ⵜ2Vb = 0.

As an example, we calculate the potential distribution Vb共r兲 on a circular bottom electrode for a perfectly conducting top electrode 共Rt = 0, Vt = 0兲, assuming a linear voltage characteristic 共which is usually acceptable for a sufficiently small voltage interval兲, jOLED共Vb兲 = j0 + a共Vb − V0兲.

共8兲

Eliminating the voltage in Eq. 共5兲 yields for cylinder coordinates r2

djz d2 j z − aRbr2 jz = 0, 2 +r dr dr

共9兲

being the modified Bessel differential equation. With the boundary condition jz共0兲 = j0, we find solutions for the different quantities, jz共r兲 = j0I0共冑aRbr兲, Vb共r兲 = V0 +

jb共r兲 = j0␲r2

j0 关I0共冑aRbr兲 − 1兴, a 2I1共冑aRbr兲

冑aRbr

共10兲

,

with I0 and I1 the zero and first order modified Bessel function of the first kind.6 Usually the voltage 共and current density兲 are given for the radius rm, where the bottom electrode is in contact with a perfectly conducting material, leading to the boundary condition jz共rm兲 = jm. For this boundary condition the solution becomes

FIG. 2. Top view and side view of the experimental OLED structure, indicating the interconnection between the different conductive layers 共dimensions not to scale兲.

jz共r兲 = jm

I0共冑aRbr兲

I0共冑aRbrm兲

冋

= jm 1 −

册

a2R2b 4 aRb 2 2 + r 4兲 + ¯ . 共3rm − 4r2rm 共rm − r2兲 + 64 4 共11兲

For example, for a sheet resistance Rb = 200 ⍀, a proportionality factor a = 50/ ⍀ m2, and a circle with radius of 1 cm, we find that the luminance 共and current density兲 in the center is 20% lower than the luminance near the edge.

III. EXPERIMENT

A variation in the OLED luminance due to the electrode resistivity has been observed experimentally for an Ir共ppy兲3 emitting device7 using a bottom electrode based on a mixture of Baytron® PH500 共Ref. 3兲 and 5%共w/w兲 dimethylsulfoxide 共DMSO兲. The layer structure of the device is illustrated in Fig. 2. The cathodic top electrode is a 100 nm thick thermally evaporated aluminum square with a side width of 13 mm. For the preparation of the anodic bottom electrode Baytron® PH500 is mixed with 5% DMSO and the mixture is then spin coated at a spin speed of 1700 rpm for 30 s. Afterwards, the films are baked at 130 ° C on a hot plate for 15 min, yielding a 100 nm thick layer. The conductivity of the films is about 500 S / cm, corresponding to a sheet resistance of 200 ⍀ 关see Eq. 共6兲兴 The charge transport layers, blocking layers, and the emission layer consist of a well-studied stack of small molecules, deposited in a high vacuum environment by thermal sublimation. Details about the stack have been described elsewhere.7 The bottom electrode is deposited on a highly conductive silver layer with a square opening with side of 15 mm, centered around the top electrode. The luminance, averaged over an area of about 30 ⫻ 30 ␮m2, is measured with a photodiode as a function of the x coordinate 共the y coordinate is chosen in the center of the pixel兲, for different voltages applied between the two highly conducting materials. The resulting luminance scans are shown in Fig. 3.


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FIG. 3. Scan of the luminance 共cd/ m2兲 as a function of the distance for a section in the middle of the OLED device with side of 13 mm. The simple lines show experimental scans for four voltages 共3.3, 3.5, 3.7, and 3.9 V兲. The lines with symbols show simulated scans for the same set of voltages.

IV. NUMERICAL SIMULATION AND DISCUSSION

For the numerical solution of the differential equations 共5兲, the square resistance of the top electrode is set to zero 共Rt = 0, Vt = 0兲 and Rb = 200 ⍀ for the Baytron® PH500 layer. The boundary condition for the voltage is Vb = Vappl at the edge of the Baytron® PH500 layer 共the edge of the square with side of 15 mm兲. Outside the top electrode area, the current density jz is set to zero. In reality, the efficiency of the OLED is slightly voltage dependent, but in this simulation we will neglect this effect. The efficiency is estimated from the experiment for 3.5 V, as the ratio of the measured average luminance 共436 cd/ m2兲 over the measured average current density 共16 A / m2兲, yielding 26 cd/ A. The OLED current density as a function of the voltage jOLED共Vb兲 is reconstructed from the luminance data near the edge of the pixel: the voltage is assumed to be equal to the applied voltage and the luminance is divided by the found efficiency 共26 cd/ A兲 to obtain the local current density. The resulting current density-voltage characteristic is given in Fig. 4. The partial differential equation for the voltage Vb共x , y兲 was solved with a simple iterative finite difference method, converging to a stable solution. The solutions for the bottom electrode voltage Vb共x , y兲 and the luminance L共x , y兲 are given in Fig. 5 as a function of the lateral coordinates. The simulated lateral scans for the luminance are given in Fig. 3

FIG. 5. Simulated distribution of the voltage Vb共x , y兲 over the bottom electrode over a square with side of 15 mm 共top兲. Simulated distribution of the luminance L共x , y兲 over the OLED area 共bottom兲.

together with the measurements. The good match between experimental results and simulations indicates that the inhomogeneous luminance is indeed induced by the resistivity of the Baytron® PH500 electrode layer, and that the conductivity data for the Baytron® PH500 layer can explain the observed voltage drop. Let us also compare the simulation results for the square with the analytical formula for the circular electrode. For an applied voltage of 3.9 V 共a = 94/ ⍀ m2兲, a radius of 6.5 mm, and Rb = 200 ⍀, we find at the edge jz / j0 = 1.21, which corresponds well with the simulation result 共jz / j0 = 1.19 near the edge兲. The integrated currents obtained from the simulations do not exactly match the observed experimental currents. This is because we neglected the variation in efficiency for increasing voltage over the OLED. A better fit of the integrated currents could indeed be obtained by including the variation in efficiency. For most lighting applications, a modulation of 20 % in luminance on a scale of 1 cm is probably acceptable. This means that in this case, the Baytron® PH500 transparent electrode with 200 ⍀ sheet resistance should be enforced by a metallic wire grid with holes of this dimension, as soon as large-area applications are tackled. V. CONCLUSIONS

FIG. 4. Estimated current density vs voltage characteristic jOLED共V兲, obtained from luminance data and assuming a voltage independent device efficiency of 26 cd/ A.

The experimental observations of inhomogeneous luminance have been explained by a theoretical model based on the voltage drop over the transparent electrodes due to lateral currents. The theoretical model has been implemented in a numerical simulation and the results compare well with the experiments. In addition, analytical formulas are provided for current density and voltage distribution in a circular pixel for the case in which the current density is linearly related with the local voltage.


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ACKNOWLEDGMENT

The authors would like to acknowledge the support of the European Commission through the sixth framework Integrated Project OLLA—Organic LED’s for Lighting Applications. K. Neyts, Appl. Surf. Sci. 244, 517 共2005兲. A. Elschner, F. Bruder, H.-W. Heuer, F. Jonas, A. Karbach, S. Kirchmeyer, S. Thurm, and R. Wehrmann, Synth. Met. 111–112, 139 共2000兲.

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Baytron® PH500 is a waterborne dispersion of poly共ethylenedioxythiophene兲共PEDOT兲 and poly共styrenesulfonicacid兲共PSS兲, see data sheets at www.baytron. com. 4 T. Aernouts, W. Geens, J. Poortmans, P. Heremans, S. Borghs, and R. Mertens, Thin Solid Films 403, 297 共2002兲. 5 H.-W. Heuer, R. Wehrmann, and S. Kirchmeyer, Adv. Funct. Mater. 12, 89 共2002兲. 6 http://mathworld.wolfram.com/modifiedbesselfunctionofthefirstkind.html. 7 K. Fehse, K. Walzer, G. He, M. Pfeiffer, K. Leo, W. Lövenich, and A. Elschner, Proc. SPIE 6192, 61921Z 共2006兲.
