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

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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 2000 – 6000 cd/m2 Position [cm] 10000 Current [mA]: Luminance inhomogeneities emerge 1000 ] 900 ⌅ 2 1000 m Curre n8t0 [0mA]: / d 700 1000 c [ 1A ] 600 900 2 e 1000 500 m c 800 / n when operating at high currents d 400 700 a c [ n i 100 6300 e m 200 500 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 700 1000 r ] u 600 900 t 50 C 60 a ° 500 800 r [ e 400 e 700 p r 40 1A u 300 600 m t 50 e a 200 500 r T e 4100 p 430 10 300 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 effect concentrates can be investigated at the corners in more of detail the active when area. the local By comparingdifferential 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 effVectij can be investigated in more detail when the local Rij = (1) differential 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 seriesfferent 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, affect the substantial current flow. self-heating However, does upon not elevated occur, andself-heating, all curves the for situation different 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 differential electrode resistance does (S-NDR) not noticeably which strongly affect 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 differential 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 E0 σ 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 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 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 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 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 σ :⌦ R Rd [0, ) given by ⇥ ⇥ ! 1 p(x) 2 ' − σ(x, T , ')=σ0(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 σ(x, T , ') ' =0 r · r r λ(x) T = ⌘(x)σ(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.
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