A bulk‐surface model for the electrothermal feedback in large‐area organic light‐emitting diodes

This work deals with an effective bulk‐surface thermistor model describing the electrothermal behavior of large‐area thin‐film organic light‐emitting diodes (OLEDs). This model was rigorously derived from a p(x)$p(x)$ ‐Laplace thermistor model by dimension reduction and consists of the heat equation in the three‐dimensional glass substrate and two semi‐linear equations describing the current flow through the electrodes coupled to algebraic equations that express the continuity of the electrical fluxes through the organic layers. The electrical problem lives on the surface of the glass substrate where the OLED is mounted. The source terms in the heat equation result from Joule heating and are concentrated on the part of the boundary where the current‐flow problem is formulated. Schauder's fixed‐point theorem is used to establish the existence of weak solutions to this effective system. Since the heat source terms at the surface are a priori only in L1, the concept of entropy solutions for the heat equation is worked with.

F I G U R E 1 (A): OLED positioned on a glass substrate Ω sub at  = ω × {0}.The bottom and top layer Ω 1 and Ω  represent the electrodes with Dirichlet boundaries Γ − and Γ + (blue) for the potential where the voltage is applied.(B): Domain for the effective model.The electrical current flow is described by equations on the grey area  with Ohmic contacts at  − and  + realizing the contacting of the OLED via the electrodes.The heat flow is formulated in Ω with a boundary source term at .
The present paper studies a simplified, effective bulk-surface model that was derived rigorously from a threedimensional ()-Laplace thermistor model by a dimension reduction carried out in [6].In Section 2, we introduce the geometric setting and motivate the effective system of model equations.Section 3 is devoted to the analysis of the effective bulk-surface thermistor model.After the formulation of the general assumptions, the electrical and thermal subproblems are discussed and finally the existence proof for the coupled problem is outlined.The paper is concluded by an outlook.
The effective bulk-surface thermistor model studied here was derived rigorously from a three-dimensional ()-Laplace thermistor model for vanishing layer thickness of the OLED in [6].The effective model idealizes the transport properties in the following way: (i) in both electrodes the current flow is assumed to be in horizontal direction only, (ii) in the organic layers only vertical current flow is supposed and (iii) the temperature in vertical direction of the OLED is assumed to be constant.
The full three-dimensional ()-Laplace thermistor model for organic LEDs (see [6]) is given by Following the procedure in [6], different ansatzes and physically motivated scalings are assumed in the different layers of the OLED for the flux function In the two electrodes Ω 1 and Ω  , no temperature dependence and a linear current law with sheet conductivities  − sh =  0 ℎ 1 and  + sh =  0 ℎ  ,  0 > 0 are supposed: For the organic layers Ω  ,  = 2, … , −1, Arrhenius like temperature laws with activation energy   a and power laws for the current of the form are taken into account.Here,  ref > 0 and  ref > 0 denote reference currents and voltages, respectively.The power law exponents   ∈ (1, ∞) and the activation energy   a are associated with the material of the organic layer Ω  .Note that the physical scaling of the flux function  OLED in (2) is done in such a way that on the one hand the sheet resistance of the electrodes is of order 1.On the other hand, in the organic layers a potential difference (⋅, ĥ ) − (⋅, ĥ−1 ) of order 1 with an electric field  = [(⋅, ĥ ) − (⋅, ĥ−1 )]∕ℎ  produces an electrical current of order 1.
The boundary conditions for the current flow and heat flow equation in the three-dimensional ()-Laplace thermistor model (1) are given by Let the parameter  ∶= ĥ ∕diam( ω) denote the ratio between thickness and diameter of the OLED.For the layer thicknesses ℎ  ∶= ĥ − ĥ−1 we assume ℎ  = ℎ  *    with   > 0, ℎ  * > 0,  = 1, … , .In the effective bulk-surface model derived in [6], only the third component of the flux function (  (, )) 3 ,  ∈ ℝ 3 , does not vanish in the description of the organic layers in the limit  → 0. In this limit of vanishing layer thickness, the potential  becomes affine in the organic layers and constant in vertical direction in the electrodes.Thus, it can be identified with a tuple ( 1 , … ,  −1 ).Using the short notion for the vertical current flow in the OLED layers, the effective bulk-surface model of [6] consists of two equations for the lateral current flow in the electrodes, algebraic equations that express the continuity of the vertical current flow through the interfaces between neighboring organic layers (on ), and a heat equation (in Ω) with boundary sources on The relevant unknowns are the temperature distribution  in the glass substrate and −1 potentials   , representing the potentials in the electrodes and the interfacial potentials between the different OLED layers.Note that the equations in the second line in (5) ensure the continuity of the electrical current through the interfaces.Here, ∇  indicates the surface gradient on ,   ∈ ℝ 2 denotes the outer unit normal vector on , and  a represents the ambient temperature.The boundary conditions for the current flow reduce to conditions for  1 and  −1 that are given by For the heat equation, Robin boundary conditions with the additional nonlinear surface Joule heating term   on  have to taken be into account,

Assumptions
We fix the assumptions for the analysis of the effective model equations ( 5) -( 8), being slightly more general than the situation in Section 2 especially with respect to the geometry.
Moreover,   (, ⋅) are strictly monotone and vanish in zero, that is
Note that   0 can then be expressed by the remaining variables, Thus, it belongs to   − () =    0 () due to our choice of  0 .Therefore, also the expression   0 (,   0 ) is well-defined.
(1) Norm estimates: We test the equation    = 0 by  − (  − , 0, … , 0,   + ) ∈  and take into account the   -growth properties of   (, ⋅) stated in Assumption (IV).Note that according to (9) the term for  0 on the right-hand side has to be estimated by and Young's inequality.Since   − ,   + ∈  1 () we derive the bounds in the first line of estimates in Theorem 3.1.The bounds for ‖  ‖    () and the   -growth of   then ensure the estimates ‖  (,   )‖ (2) Unique solvability: The strict monotonicity of the functions   (, ⋅) leads to the strict monotonicity of the operator   defined in (10).Moreover, Assumption (IV) ensures that the operator   is demi-and hemi-continuous.Additionally, using again the   -growth properties of   (, ⋅) we find that the operator   is coercive, Therefore, the Browder-Minty theorem guarantees a solution to   () = 0.The uniqueness of the solution results from the strict monotonicity of   .A detailed existence proof for the current flow problem can be found in [7].□

Entropy solution to the heat equation
Note that Theorem 3.1 provides only an  1 () estimate for   (, ) and hence insufficient regularity properties of the surface source term in (8) for classical elliptic theory.Therefore, we use the concept of entropy solutions (see, e.g., [8]) for the heat equation.We consider the stationary heat equation with Robin boundary conditions and right-hand side  ∈  1 (Ω) as well as boundary data  ∈  1 (Γ), In our special situation in (6) with the boundary conditions ( 8), we have For  > 0, we introduce the cut-off functions   () ∶= max{−, min{, }},  ∈ ℝ, and consider the set of functions for all  > 0 and all  ∈  1 (Ω) ∩  ∞ (Ω).
Theorem 3.3 collects results related to entropy solutions for elliptic equations with Robin boundary conditions obtained in [7,9].

Existence result for the effective bulk-surface model
Here, we define our concept of weak solutions for ( 5)-( 8) and give our main result that concerns its solvability.
Remark 3.6.Since experiments as well as simulations for the full three-dimensional ()-Laplace thermistor model in, for example, [5] show S-shaped current-voltage relations for organic LEDs, we do not expect the uniqueness of the solution.
Proof.We give a short overview of the existence proof that is based on Schauder's fixed point theorem.We work with the set of  1 () functions being traces of  1,6∕5 (Ω) functions on the substrate Ω and being greater or equal to the ambient temperature  a a.e. on the boundary part  ⊂ Ω where the OLED is mounted, that is The constant   will be defined in ( 14).The corresponding fixed-point map  ∶  →  is defined as follows: Let T ∈ be arbitrarily given.Then Theorem 3.1 ensures a unique weak solution  = ( T) ∈   +  to the current flow problem  T  = 0 as well as the estimate ‖  ( T,)‖  1 () ≤   uniformly for T ∈  .Next, we define  =( T) to be the unique entropy solution to The unique solvability of problem ( 13) is justified by Theorem 3.3 (i).Moreover, for  and  as defined in (12) the Theorem 3.3 guarantees that for all T ∈  and ensures the lower estimate  ≥  a a.e. in Ω and a.e. on .Thus, we obtain  = ( T) ∈  .As demonstrated in [7, Lemma 5.1], the fixed-point map  is continuous with respect to the strong convergence in  1 ().Here, especially Theorem 3.3 (ii) and again the   -growth and monotonicity properties of the functions   from Assumption (IV) are important for the proof.Finally, Schauder's fixed point theorem ensures the existence result.□

OUTLOOK
Simulations of the full three-dimensional ()-thermistor model as performed for the paper [2], especially for large-area organic LEDs with several organic layers, are very time-consuming.Solving the effective model in ( 5)-( 8) is numerically cheaper since the thin layers do not have to be resolved and the lower-dimensional current-flow equations lead to a sparser system.
To understand the emergence of burn-ins in organic LEDs due to high temperatures [2], it is interesting to consider the instationary version of the thermistor system.The derivation of an effective bulk-surface model starting from a threedimensional, instationary ()-Laplace thermistor model as treated in [10] and the investigation of its analytical properties is still open.
Instead of the ()-Laplace thermistor model more accurate energy-drift-diffusion models can be used.In this case, the simple ()-Laplace model for the current flow is substituted by a drift-diffusion type model for electron and hole transport in organic semiconductors.First, this generalized van Roosbroeck model has to take into account that the statistical relation between charge-carrier densities and chemical potentials in organic semiconductor materials is given by temperature dependent Gauss-Fermi integrals.Second, mobility laws for organic materials with strong nonlinear dependence on temperature, carrier densities, and electric field strength have to be considered.In [11], we investigated a drift-diffusion based electrothermal model for organic thin-film devices including their electrical and thermal environment.
A challenging task would be to (rigorously) derive an effective bulk-surface model resulting from a drift-diffusion based electrothermal model.The aim is to obtain a limit system consisting of two equations in  for the (horizontal) current flow in the upper and lower electrode, similar to the first and third equation in (5).We expect that for each point in  a one-dimensional drift-diffusion problem for the vertical charge transport through the OLED layers has to be solved.Again, the heat flow equation has to be considered in the substrate Ω sub and should contain a nonlinear surface heat source term at .The latter is given by heating due to electron and hole currents and a generation-recombination heat.To the best of our knowledge, such a derivation has not been done before.For a rigorous derivation, one would have to overcome a lack of sufficiently strong convergences in the limit process since the mobility functions, the generationrecombination rate coefficients, as well as the statistical relations are nonlinear functions of the state variables.Moreover, also the expected/desired electrothermal limit model should be investigated from an analytical point of view.

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C K N O W L E D G M E N T S This work was partly supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy-The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).Open access funding enabled and organized by Projekt DEAL.O R C I D Annegret Glitzky https://orcid.org/0000-0003-1995-5491R E F E R E N C E S