PDE thermistor modeling of organic LEDs

In Liero et al. 2015 a stationary PDE thermistor model was derived, that describes the electrothermal behavior of organic semiconductor devices. It consists of the current-flow equation for the electrostatic potential \(\varphi\) coupled to the heat equation with Joule heat term for the temperature \(\theta\), namely \begin{align} -\nabla\cdot\big( \sigma(x,\theta,\nabla\varphi)\nabla\varphi\big)&=0,\\ -\nabla\cdot(\lambda(x)\nabla\theta)&=\big(1{-}\eta(x,\theta,\nabla\varphi)\big)\sigma(x,\theta,\nabla\varphi)|\nabla\varphi|^2. \end{align} Here, \(\eta(x,\theta,\nabla\varphi)\in[0,1]\) denotes the light-outcoupling factor and \(\lambda\) and \(\sigma\) are the thermal and electrical conductivity, respectively. The latter is given via \begin{align} \sigma(x,\theta,\nabla\varphi)&:=\sigma_0(x)F(x,\theta)\Big[\frac{|\nabla\varphi|}{V_\text{ref}/d_\text{ref}}\Big]^{p(x)-2},\\ \text{where}\quad F(x,\theta)&:=\exp\Big[{-}\frac{E_\text{act}}{k_\text{B}}\Big(\frac{1}{\theta}-\frac{1}{\theta_\text{a}}\Big)\Big] \end{align} describes an Arrhenius-like temperature law with \(E_\text{act}\) denoting the material dependent activation energy.

In particular, the first equation above is of \(p(x)\)-Laplace-type, where \(x\mapsto p(x)\) is a measurable function satisfying \(1<\mathrm{ess\,inf}_{x\in\Omega}p(x)\leq \mathrm{ess\,sup}_{x\in\Omega}p(x)<\infty.\)

The system is complemented by Dirichlet and homogeneous Neumann boundary conditions for \(\varphi\) and Robin boundary conditions for \(\theta\), viz. \begin{align} &\varphi=\varphi^{D}~\text{on }\Gamma_{D} \quad\text{and}\quad \sigma(x,\theta,\nabla\varphi)\nabla\varphi\cdot\nu=0~\text{on }\Gamma_{N},\\ &-\lambda(x)\nabla\theta\cdot\nu=\kappa(x)(\theta - \theta_{\text a})~\text{on }\Gamma:=\partial\Omega. \end{align}

Systems of the above form model materials conducting both heat and electrical current for which the electrical conductivity \(\sigma\) strongly depends on the temperature. Devices of this type are called thermistors. Organic Light-Emitting Diodes (OLEDs) are thin-film heterostructures based on organic molecules or polymers, where each functional layer has, in general, its own current-voltage characteristics and material parameters. In particular, the exponent \(p(x)\), which describes the non-Ohmic behavior of each layer, changes abruptly from one material to another. In electrodes, the typically used parameter is \(p(x)=2\), while organic layers feature significantly larger values, e.g. \(p(x)\approx9\) (see Fischer et al. 2014).

Since the activation energy \(E_\text{act}\) is positive in organic materials, a positive feedback is introduced as the electrical conductivity increases with rising temperature and in turn the power dissipation increases with the electrical current. This mechanism was identified in Fischer et al. 2014 as the cause of the appearance of different operation modes and accompanying unpleasant brightness inhomogeneities in large-area OLEDs.

The analytical difficulties of the above problem arise from two issues: First, the exponent function \(x\mapsto p(x)\) is discontinous and in general only measurable. Second, the source term in the right hand side of the heat equations is only in \(L^1(\Omega)\) for functions \(\varphi\) in the energy space associated with the \(p(x)\)-Laplacian.