Laplace Thermistor Models Describing Electrothermal Feedback in Organic Semiconductor Devices Matthias Liero

Weierstrass Institute for Applied Analysis and Stochastics

On p(x)-Laplace thermistor models describing electrothermal feedback in organic semiconductor devices Matthias Liero

joint work with Annegret Glitzky, Thomas Koprucki, Jürgen Fuhrmann (WIAS Berlin) Axel Fischer, Reinhard Scholz (IAPP, TU Dresden) Miroslav Bulìček (Charles University, Prague) Introduction

Organic semiconductors

⌅ Carbon-based materials conducting electrical current C60 pentacene ⌅ Used in smartphone and TV displays, photovoltaics, and lighting applications

⌅ OLED: whole area emits light, ﬂexible

Problem Inhomogeneities in large-area OLEDs Position [cm] 10000 2 2000 – 6000 cd/m Position [cm] 10000 Current [mA]: Luminance inhomogeneities emerge 1000 ] 900 ⌅ 2 1000 m Curre n8t0 [0mA]: /

d 17000 c

[ 1A ] 9600 2 e 1000 500 m c 800 / n

when operating at high currents d 7400 a c [ n

i 100 6300 e

m 5200 c u Current n 100 L 400 a

n 10 i 0.01A 300 100 15 cm m 200 u 100 ⌅ Strong self-heating effects Luminance @ 1A L 10 10 15 cm

7100 Current [mA]: 1000

] 900

Counter-intuitive nonlinear phenomena C 60 ⌅ 60 — 75 °C ° 70 Curre n8t0 [0mA]: [

e 17000 r ]

u 9600

t 50

C 60 a ° 8500 r [

e 400

e 700 p r 40 1A u 6300 m

t 50 e a 5200 r T

e 4100

p 30 40 3100 m

e 200 T

Current 100 320 PDE model needed! 0 2 4 0.01A6 8 10 12 14 16 18 10 Position [cm] ) Temperature @ 1A 20 0 2 4 6 8 10 12 14 16 18 Figure 3. ProfilesMeasurements: of the a) A. luminance Fischer (IAPP), and b) Adv. the temperatureFunc.Po sMat.ition [24c alongm ](2014) the 3367–3374 larger symmetry axis of the Tabola lighting panel 6 Electrothermal modeling of large-areaas indicated OLEDs by the dashed line in Fig. 2. Reproduced with permission. Copyright (c) 2014 Wiley-VCH. MeasurementsFigure 3. Profiles of the a) luminance by and A. b) theFischer temperature along the larger symmetry axis of the Tabola lighting panel as indicated by the dashed line in Fig. 2. Reproduced with permission.6 Copyright (c) 2014 Wiley-VCH. either the cathode or anode with values corresponding to the sheet resistance of the electrodes. More details on the simulation are described in Ref.6 either the cathode or anode with values corresponding to the sheet resistance of the electrodes. More details on The result for a large area lighting panel is given in Fig. 4. Up to 500 mA, the homogeneity of the lighting the simulation are described in Ref.6 p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14,panel 2016 is preserved,Page but for larger1 (15) currents, the light emission concentrates at the corners of the active area. By · · · comparingThe result the·for central a large positions area lighting of eachpanel picture, is givenone can in clearly Fig. 4. see Up that to 500 the mA, luminance the homogeneity does not further of the rise lighting with panelthe applied is preserved, current, but as also for larger observed currents, in experiment. the light Theemission eect concentrates can be investigated at the corners in more of detail the active when area. the local By comparingdierential the resistance central positions of each picture, one can clearly see that the luminance does not further rise with the applied current, as also observed in experiment. Thed eVectij can be investigated in more detail when the local Rij = (1) dierential resistance dIij dVij is evaluated at each point (i,j) of the thermistorR network.ij = Figure 5 b) presents a table of relevant scenarios.(1) dI Please note that the total IV curve of the OLED is assumedij to be monotonically increasing (dV/dI > 0), e.g. isdue evaluated to an external at each series point resistance. (i,j) of the The thermistor normal case network. is described Figure by 5 an b) increase presents of a local table voltage of relevantVij and scenarios. currents PleaseIij with note their that external the total counterparts IV curveV ofand the OLEDI. At low is assumed current densities,to be monotonically substantial increasing self-heating (d doesV/dI not> 0), occur, e.g. dueand toall an curves external for di serieserent resistance. positions onThe the normal substrate case coincide is described as indicated by an increase in Fig. of 5 a).local This voltage alsoV impliesij and currentsthat the Isheetij with resistance their external of the transparentcounterparts electrodeV and I does. At not low noticeably current densities, aect the substantial current flow. self-heating However, does upon not elevated occur, andself-heating, all curves the for situation dierent changes.positions onOne the can substrate show that coincide the current-voltage as indicated in characteristic Fig. 5 a). This of also an impliesOLED tendsthat the to sheetshow S-shapedresistance negative of the transparent dierential electrode resistance does (S-NDR) not noticeably which strongly aect the depends current on flow. the However,position on upon the elevated lighting self-heating,panel.6 Regions the situation most far changes.away from One the can edges show first that enter the an current-voltage operation mode characteristic at which the of an local OLED voltage tends drop to showdecreases S-shaped although negative the current dierential density resistance is still increasing, (S-NDR) whichas shown strongly in Fig. depends 4 at a current on the of positionI =0.85 on A. the Two lighting areas panel.operating6 Regions in regions most II arisefar away at inner from positions the edges of first the lighting enter an panel, operation and they mode expand at which until the they local merge voltage at higher drop decreasescurrents. although the current density is still increasing, as shown in Fig. 4 at a current of I =0.85 A. Two areas operating in regions II arise at inner positions of the lighting panel, and they expand until they merge at higher This raises the question why these local NDR regions first occur at central positions but not at the edges currents. where the highest power dissipation and temperatures are observed. One has to consider the interaction between all ofThis the raises thermistor the question devices why in the these array. local They NDR are regions not only first coupled occur atelectrically central positions by the sheet but notresistance at the of edges the whereelectrodes the highest but also power thermally dissipation by the and glass temperatures substrate. Thus,are observed. neighboring One hasthermistors to consider are the able interaction to exchange between heat, all of the thermistor devices in the array. They are not only coupled electrically by the sheet resistance of the electrodes but also thermally by the glass substrate. Thus, neighboring thermistors are able to exchange heat, Zero-dimensional model with Arrhenius law, non-Ohmic current-voltage relation, and global heat balance

V ↵ I(V , T )=IrefF (T ) (↵ 1) Vref h i Eact 1 1 F (T )=exp Voltage [V] k T T B a Thermistor behavior with regions 1 h ⇣ ⌘i (T Ta)=I(V , T ) V of negative differential resistance ⇥th · Fischer et al. PRL, 2013

Positive feedback loop in organic semiconductors Organic semiconductors: Temperature-activated hopping transport of charge carriers

energy

energy levels

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 2 (15) · · · · Zero-dimensional model with Arrhenius law, non-Ohmic current-voltage relation, and global heat balance

Positive feedback loop in organic semiconductors Organic semiconductors: Temperature-activated hopping transport of charge carriers

Fischer et al., Org. Elec., 2012

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 2 (15) · · · · Positive feedback loop in organic semiconductors Organic semiconductors: Temperature-activated hopping transport of charge carriers

Fischer et al., Org. Elec., 2012

Zero-dimensional model with Arrhenius law, non-Ohmic current-voltage relation, and global heat balance

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 2 (15) · · · · Inhomogeneities in large-area OLEDs

⌅ OLEDs consist of various layers and have huge aspect ratios

⌅ Optical transparent top electrode has large sheet resistance

⌅ Sheet resistance leads to signiﬁcant lateral potential drop

Spatially resolved models needed. p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 3 (15) · · · · with electrical conductivity :⌦ R Rd [0, ) given by ⇥ ⇥ ! 1 p(x) 2 ' (x, T , ')=0(x)F (x, T ) |r | r Vref/`ref h i E (x) 1 1 F (x, T )=exp act kB T Ta PDE model can be motivated from equivalenth circuit⇣ model ⌘i s.t. ﬁnite-volume discretization of PDE system correspond to Kirchhoff’s circuit rules (see Fischer et al. 2014, talk by A. Fischer, Wed)

PDE thermistor model [L.–Koprucki–Fischer–Scholz–Glitzky, 2015]

We consider elliptic system consisting of current-ﬂow equation for electrostatic potential ' and heat equation for temperature T

(x, T , ') ' =0 r · r r (x) T = ⌘(x)(x, T , ') ' 2 r · r r |r |

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 4 (15) · · · · PDE model can be motivated from equivalent circuit model s.t. ﬁnite-volume discretization of PDE system correspond to Kirchhoff’s circuit rules (see Fischer et al. 2014, talk by A. Fischer, Wed)

PDE thermistor model [L.–Koprucki–Fischer–Scholz–Glitzky, 2015]

We consider elliptic system consisting of current-ﬂow equation for electrostatic potential ' and heat equation for temperature T

(x, T , ') ' =0 r · r r (x) T = ⌘(x)(x, T , ') ' 2 r · r r |r | with electrical conductivity :⌦ R Rd [0, ) given by ⇥ ⇥ ! 1 p(x) 2 ' (x, T , ')=0(x)F (x, T ) |r | r Vref/`ref h i E (x) 1 1 F (x, T )=exp act kB T Ta h ⇣ ⌘i

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 4 (15) · · · · PDE thermistor model [L.–Koprucki–Fischer–Scholz–Glitzky, 2015]

We consider elliptic system consisting of current-ﬂow equation for electrostatic potential ' and heat equation for temperature T

(x, T , ') ' =0 r · r r (x) T = ⌘(x)(x, T , ') ' 2 r · r r |r | with electrical conductivity :⌦ R Rd [0, ) given by ⇥ ⇥ ! 1 p(x) 2 ' (x, T , ')=0(x)F (x, T ) |r | r Vref/`ref h i E (x) 1 1 F (x, T )=exp act kB T Ta PDE model can be motivated from equivalenth circuit⇣ model ⌘i s.t. ﬁnite-volume discretization of PDE system correspond to Kirchhoff’s circuit rules (see Fischer et al. 2014, talk by A. Fischer, Wed)

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 4 (15) · · · · PDE thermistor model non-dimensionalized form

p(x) 2 (x)F (x, T ) ' ' =0 r · 0 |r | r p(x) (x) T = ⌘(x)0(x)F (x, T ) ' r · r |r | Mixed boundary conditions

' = 'D on (x, T , ') ' ⌫ =0 on D r r · N (x) T ⌫ = (x)(T T ) on :=@⌦ r · a

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 5 (15) · · · · PDE thermistor model non-dimensionalized form

p(x) 2 (x)F (x, T ) ' ' =0 r · 0 |r | r p(x) (x) T = ⌘(x)0(x)F (x, T ) ' r · r |r | Mixed boundary conditions

' = 'D on (x, T , ') ' ⌫ =0 on D r r · N (x) T ⌫ = (x)(T T ) on :=@⌦ r · a Properties

⌅ Current-ﬂow equation is of p(x)-Laplacian type

⌅ Abrupt change of p(x) between materials: Exponent p(x) describes non-Ohmic behavior of materials, p(x)=2in electrodes (Ohmic) and p(x) > 2 in organic materials (e.g. p(x)=9.7) For ' Lp(x)(⌦)d, Joule heat term in general only in L1 ⌅ r 2

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 5 (15) · · · · PDE thermistor model non-dimensionalized form

p(x) 2 (x)F (x, T ) ' ' =0 r · 0 |r | r p(x) (x) T = ⌘(x)0(x)F (x, T ) ' r · r |r | Mixed boundary conditions ' = 'D on (x, T , ') ' ⌫ =0 on D r r · N (x) T ⌫ = (x)(T T ) on :=@⌦ r · a

(i) Effective elec. conductivity 0 L1(⌦) s.t. 0 <0 0 0 a.e. in ⌦ 2   (ii) Thermal conductivity L+1(⌦) s.t. 0 < < a.e. in ⌦ 2 ⇤   ⇤ 1 (iii) Activation energy Eact L1(⌦) 2 + (iv) Heat transfer coefﬁcient  L1(⌦), (x)d> 0 2 + (v) Light-outcoupling factor ⌘ L1(⌦), ⌘ [0, 1] a.e. in ⌦ 2 R 2 (vi) Ambient temperature Ta > 0 is constant 1, (vii) Dirichlet data 'D W 1(⌦) 2 (viii) Power-law exponent x p(x) is measurable and 7! 1

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 5 (15) · · · · Analytic result

Theorem (Bulícek-Glitzky-L.ˇ 2016 a)

There exists a weak solution to the coupled system

(Current ﬂow) (x, T, ' ) ' =0 r· |r | r (Heat ﬂow) (x) T = ⌘(x)(x, T, ' ) ' 2 r· r |r | |r | with

' = 'D on (x, T, ') ' ⌫ =0 on D r r · N (x) T ⌫ = (x)(T T ) on :=@⌦ r · a where ' W 1,p(x)(⌦) and T W 1,q(⌦) with 1 q

Note: More general constitutive equations for the electrical current can be considered. p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 6 (15) · · · · Sketch of proof

1. For ">0, introduce regularization (x, T, ') ' 2 1 f (x, T, '):=⌘(x) r |r | " r 1+"(x, T, ') ' 2  " r |r |

2. Solve regularized problem via Galerkin approximation (use strict monotonicity of current law)

3. For solutions ('",T") of regularized problem derive uniform estimates by testing weak formulation with suitable functions, e.g. T˜ = T ( (0, 1)) " 2 d '" p( ) C, T" W 1,q C(q), where q 1, kr k ·  k k  2 d 1 h ⌘

4. Pass to the limit " 0 in the weak formulation and identify limits exploiting ! monotonicity again p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 7 (15) · · · · ⌅ Heat equation

Numerics for thermistor scheme

Our numerical scheme is based on a hybrid ﬁnite element/volume approach, see also Bradji–Herbin 2008

J dx = J n da r· · KL ZK L K ZsKL X| s J ⇡ | KL| KL L K X| Finite volume scheme Construct approximation J of normal ﬂux J n KL · KL

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 8 (15) · · · · Numerics for thermistor scheme

Our numerical scheme is based on a hybrid ﬁnite element/volume approach, see also Bradji–Herbin 2008

J dx = J n da r· · KL ZK L K ZsKL X| s J ⇡ | KL| KL L K X| Finite volume scheme Construct approximation J of normal ﬂux J n KL · KL

⌅ Heat equation

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 8 (15) · · · · Current-ﬂux approximation (for same material region)

p(x) 2 (x, T, ' ) ' dx = (x)F (x, T ) ' ' n da r· |r | r 0 |r | r · ZK L K ZsKL X| ps 2 'L 'K KL sKL FsKL (TKL) ' ⇡ | | r T xK xL L K | | X| Averaging of ' is necessary for stable schemes on general grids ⌅ |r | ⌅ At material interface no averaging is performed to avoid artiﬁcial lateral diffusion ⌅ Needs boundary conforming Delaunay grids

Numerics for thermistor scheme

Problem: In current-ﬂow equation the conductivity depends not only on normal component ' n on Voronoi surface but also on tangential components. r · 'N

'L Given nodal values 'K construct { } 'K P 1 ﬁnite element interpolant ' T 'M b

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 9 (15) · · · · Averaging of ' is necessary for stable schemes on general grids ⌅ |r | ⌅ At material interface no averaging is performed to avoid artiﬁcial lateral diffusion ⌅ Needs boundary conforming Delaunay grids

Numerics for thermistor scheme

Problem: In current-ﬂow equation the conductivity depends not only on normal component ' n on Voronoi surface but also on tangential components. r · 'N

'L Given nodal values 'K construct { } 'K P 1 ﬁnite element interpolant ' T 'M Current-ﬂux approximation (for sameb material region)

p(x) 2 (x, T, ' ) ' dx = (x)F (x, T ) ' ' n da r· |r | r 0 |r | r · ZK L K ZsKL X| ps 2 'L 'K KL sKL FsKL (TKL) ' ⇡ | | r T xK xL L K | | X|

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 9 (15) · · · · Numerics for thermistor scheme

Problem: In current-ﬂow equation the conductivity depends not only on normal component ' n on Voronoi surface but also on tangential components. r · 'N

'L Given nodal values 'K construct { } 'K P 1 ﬁnite element interpolant ' T 'M Current-ﬂux approximation (for sameb material region)

p(x) 2 Simulation study for p(x)-Laplace equation ( u u)=1with r · |r | r Dirichlet boundary conditions

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 10 (15) · · · · ⌅ Simulation of IV curves via numerical path following methods

⌅ Solve extended system for (',T,V ) with V applied voltage

⌅ Predictor-corrector scheme with step-size control

Numerics for thermistor scheme

Problem: In voltage-controlled simulations, solutions have to jump into high conductance state, current-controlled simulations numerically unstable

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 11 (15) · · · · Numerics for thermistor scheme

Problem: In voltage-controlled simulations, solutions have to jump into high conductance state, current-controlled simulations numerically unstable

⌅ Simulation of IV curves via numerical path following methods

⌅ Solve extended system for (',T,V ) with V applied voltage

⌅ Predictor-corrector scheme with step-size control

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 11 (15) · · · · Numerics for thermistor scheme

Problem: In voltage-controlled simulations, solutions have to jump into high conductance state, current-controlled simulations numerically unstable

⌅ Simulation of IV curves via numerical path following methods

⌅ Solve extended system for (',T,V ) with V applied voltage

⌅ Predictor-corrector scheme with step-size control

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 11 (15) · · · · Explanation of inhomogeneities

Inhomogeneities in large-area OLEDsRecall: Counter-intuitive phenomena in large-area OLEDs

Position [cm] 10000 2 2000 – 6000 cd/m Position [cm] 10000 Current [mA]: 1000 ⌅ High sheet resistance of optical ] 900 2 1000 m Curre n8t0 [0mA]: /

d 17000

c transparent electrode cannot explain

[ 1A ] 9600 2 e 1000 500 m c 800 / n

d 7400 a

c saturation of current [ n

i 100 6300 e

m 5200 c u Current n 100 L 400 a

n 10 i 0.01A 300 100 15 cm m 200 u 100 Luminance @ 1A L 10 10 15 cm ⌅ degradation of material can also be 7100 Current [mA]: 1000 excluded due to lower temperature ] 900

C 60

60 — 75 °C ° 70 Curre n8t0 [0mA]: [

e 17000 r

] center of panel u 9600

t 50

C 60 a ° 8500 r [

e 400

e 700 p r 40 1A u 6300 m

t 50 e a 5200 r T

e 4100

p 30 40 3100 m

e 200

T Effect is caused by interplay between

Current 100 ⌅ 320 0 2 4 0.01A6 8 10 12 14 16 18 10 Position [cm] Temperature @ 1A 20 S-NDR and heat conduction 0 2 4 6 8 10 12 14 16 18 Figure 3. ProfilesMeasurements: of the a) A. luminance Fischer (IAPP), and b) Adv. the temperatureFunc.Po sMat.ition [24c alongm ](2014) the 3367–3374 larger symmetry axis of the Tabola lighting panel 6 Electrothermal modeling of large-areaas indicated OLEDs by the dashed line in Fig. 2. Reproduced with permission. Copyright (c) 2014 Wiley-VCH. Figure 3. Profiles of the a) luminance and b) the temperature along the larger symmetry axis of the Tabola lighting panel as indicated by the dashed line in Fig. 2. Reproduced with permission.6 Copyright (c) 2014 Wiley-VCH. either the cathode or anode with values corresponding to the sheet resistance of the electrodes. More details on the simulation are described in Ref.6 either the cathode or anode with values corresponding to the sheet resistance of the electrodes. More details on The result for a large area lighting panel is given in Fig. 4. Up to 500 mA, the homogeneity of the lighting the simulation are describedp(x in)-Laplace Ref.6 thermistor models ECMI Santiago de Compostela June 14, 2016 Page 12 (15) panel is preserved, but for larger currents, the light emission concentrates· · at the corners of the active· area. By · comparingThe result thefor central a large positions area lighting of eachpanel picture, is givenone can in clearly Fig. 4. see Up that to 500 the mA, luminance the homogeneity does not further of the rise lighting with panelthe applied is preserved, current, but as also for larger observed currents, in experiment. the light Theemission eect concentrates can be investigated at the corners in more of detail the active when area. the local By comparingdierential the resistance central positions of each picture, one can clearly see that the luminance does not further rise with the applied current, as also observed in experiment. Thed eVectij can be investigated in more detail when the local Rij = (1) dierential resistance dIij dVij is evaluated at each point (i,j) of the thermistorR network.ij = Figure 5 b) presents a table of relevant scenarios.(1) dI Please note that the total IV curve of the OLED is assumedij to be monotonically increasing (dV/dI > 0), e.g. isdue evaluated to an external at each series point resistance. (i,j) of the The thermistor normal case network. is described Figure by 5 an b) increase presents of a local table voltage of relevantVij and scenarios. currents PleaseIij with note their that external the total counterparts IV curveV ofand the OLEDI. At low is assumed current densities,to be monotonically substantial increasing self-heating (d doesV/dI not> 0), occur, e.g. dueand toall an curves external for di serieserent resistance. positions onThe the normal substrate case coincide is described as indicated by an increase in Fig. of 5 a).local This voltage alsoV impliesij and currentsthat the Isheetij with resistance their external of the transparentcounterparts electrodeV and I does. At not low noticeably current densities, aect the substantial current flow. self-heating However, does upon not elevated occur, andself-heating, all curves the for situation dierent changes.positions onOne the can substrate show that coincide the current-voltage as indicated in characteristic Fig. 5 a). This of also an impliesOLED tendsthat the to sheetshow S-shapedresistance negative of the transparent dierential electrode resistance does (S-NDR) not noticeably which strongly aect the depends current on flow. the However,position on upon the elevated lighting self-heating,panel.6 Regions the situation most far changes.away from One the can edges show first that enter the an current-voltage operation mode characteristic at which the of an local OLED voltage tends drop to showdecreases S-shaped although negative the current dierential density resistance is still increasing, (S-NDR) whichas shown strongly in Fig. depends 4 at a current on the of positionI =0.85 on A. the Two lighting areas panel.operating6 Regions in regions most II arisefar away at inner from positions the edges of first the lighting enter an panel, operation and they mode expand at which until the they local merge voltage at higher drop decreasescurrents. although the current density is still increasing, as shown in Fig. 4 at a current of I =0.85 A. Two areas operating in regions II arise at inner positions of the lighting panel, and they expand until they merge at higher This raises the question why these local NDR regions first occur at central positions but not at the edges currents. where the highest power dissipation and temperatures are observed. One has to consider the interaction between all ofThis the raises thermistor the question devices why in the these array. local They NDR are regions not only first coupled occur atelectrically central positions by the sheet but notresistance at the of edges the whereelectrodes the highest but also power thermally dissipation by the and glass temperatures substrate. Thus,are observed. neighboring One hasthermistors to consider are the able interaction to exchange between heat, all of the thermistor devices in the array. They are not only coupled electrically by the sheet resistance of the electrodes but also thermally by the glass substrate. Thus, neighboring thermistors are able to exchange heat, For sufficiently high applied currents center of panel is switched back.

Explanation of inhomogeneities

Pattern formation in OLEDs induced by operation modes propagating through device Operation modes are characterized by local differential resistance dV dV dI 1 x = x x dIx dItot dItot ⇣ ⌘⇣ ⌘

Itot total current Ix local current through stack Vx local voltage drop

Modes for I tot " Normal V I x " x " S-NDR V I x # x " Switched-back V I x # x # N-NDR V I x " x #

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 13 (15) · · · · Explanation of inhomogeneities

Pattern formation in OLEDs induced by operation modes propagating through device Operation modes are characterized by local differential resistance dV dV dI 1 x = x x dIx dItot dItot ⇣ ⌘⇣ ⌘

Itot total current Ix local current through stack Vx local voltage drop

Modes for I tot " Normal V I x " x " S-NDR V I x # x " Switched-back V I x # x # N-NDR V I x " x # For sufﬁciently high applied currents center of panel is switched back. p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 13 (15) · · · · Conclusion

⌅ Introduced p(x)-Laplace thermistor model for electrothermal description of organic semiconductor devices

⌅ Existence of solutions via regularization and Galerkin approximation

⌅ Numerical approximation via hybrid ﬁnite-volume/element scheme and numerical path following techniques

⌅ Implemented in software protoype

⌅ Pattern formation in OLEDs can be explained by self-heating, Arrhenius law and sheet resistance

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 14 (15) · · · · References

Modeling Fischer, Pahner, Lüssem, Leo, Scholz, Koprucki, Gärtner, and Glitzky • Self-heating, bistability, and thermal switching in organic semiconductors, PRL 110, (2013) Fischer, Koprucki, Gärtner, Tietze, Brückner, Lüssem, Leo, Glitzky, and Scholz • Feel the heat: Nonlinear electrothermal feedback in Organic LEDs, AFM 24 (2014) Liero, Koprucki, Fischer, Scholz, and Glitzky • p-Laplace thermistor modeling of electrothermal feedback in organic semiconductor devices, ZAMP 66, (2015)

Analysis Glitzky and Liero • Analysis of p(x)-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices, WIAS-Preprint 2143, (2015) Bulícek,ˇ Glitzky, and Liero • Systems describing electrothermal effects with p(x)-Laplacian like structure for discontinuous variable exponents, WIAS preprint 2206, 2016 Bulícek,ˇ Glitzky, and Liero • Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions, WIAS preprint 2247, 2016

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 15 (15) · · · · Equivalent circuit models

p(x)-Laplace thermistor models ECMI Santiago de Compostela June 14, 2016 Page 16 (15) · · · ·