Browsing by Author "Liero, Matthias"
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- Item3D electrothermal simulations of organic LEDs showing negative differential resistance(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Liero, Matthias; Fuhrmann, Jürgen; Glitzky, Annegret; Koprucki, Thomas; Fischer, Axel; Reineke, SebastianOrganic semiconductor devices show a pronounced interplay between temperature-activated conductivity and self-heating which in particular causes inhomogeneities in the brightness of large-area OLEDs at high power. We consider a 3D thermistor model based on partial differential equations for the electrothermal behavior of organic devices and introduce an extension to multiple layers with nonlinear conductivity laws, which also take the diode-like behavior in recombination zones into account. We present a numerical simulation study for a red OLED using a finite-volume approximation of this model. The appearance of S-shaped current-voltage characteristics with regions of negative differential resistance in a measured device can be quantitatively reproduced. Furthermore, this simulation study reveals a propagation of spatial zones of negative differential resistance in the electron and hole transport layers toward the contact.
- ItemAnalysis of a hybrid model for the electrothermal behavior of semiconductor heterostructures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Glitzky, Annegret; Liero, Matthias; Nika, GrigorWe prove existence of a weak solution for a hybrid model for the electro-thermal behavior of semiconductor heterostructures. This hybrid model combines an electro-thermal model based on drift-diffusion with thermistor type models in different subregions of the semiconductor heterostructure. The proof uses a regularization method and Schauder's fixed point theorem. For boundary data compatible with thermodynamic equilibrium we verify, additionally, uniqueness. Moreover, we derive bounds and higher integrability properties for the electrostatic potential and the quasi Fermi potentials as well as the temperature.
- ItemAnalysis of p(x)-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Glitzky, Annegret; Liero, MatthiasWe study a stationary thermistor model describing the electrothermal behavior of organic semiconductor devices featuring non-Ohmic current-voltage laws and selfheating effects. The coupled system consists of the current-flow equation for the electrostatic potential and the heat equation with Joule heating term as source. The self-heating in the device is modeled by an Arrhenius-like temperature dependency of the electrical conductivity. Moreover, the non-Ohmic electrical behavior is modeled by a power law such that the electrical conductivity depends nonlinearly on the electric field. Notably, we allow for functional substructures with different power laws, which gives rise to a p(x)-Laplace-type problem with piecewise constant exponent. We prove the existence and boundedness of solutions in the two-dimensional case. The crucial point is to establish the higher integrability of the gradient of the electrostatic potential to tackle the Joule heating term. The proof of the improved regularity is based on Caccioppoli-type estimates, Poincaré inequalities, and a Gehring-type Lemma for the p(x)-Laplacian. Finally, Schauders fixed-point theorem is used to show the existence of solutions.
- ItemA coarse-grained electrothermal model for organic semiconductor devices(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Glitzky, Annegret; Liero, Matthias; Nika, GrigorWe derive a coarse-grained model for the electrothermal interaction of organic semiconductors. The model combines stationary drift-diffusion based electrothermal models with thermistor type models on subregions of the device and suitable transmission conditions. Moreover, we prove existence of a solution using a regularization argument and Schauder's fixed point theorem. In doing so, we extend recent work by taking into account the statistical relation given by the Gauss--Fermi integral and mobility functions depending on the temperature, charge-carrier density, and field strength, which is required for a proper description of organic devices.
- ItemA coarse‐grained electrothermal model for organic semiconductor devices(Chichester, West Sussex : Wiley, 2022) Glitzky, Annegret; Liero, Matthias; Nika, GrigorWe derive a coarse-grained model for the electrothermal interaction of organic semiconductors. The model combines stationary drift-diffusion- based electrothermal models with thermistor-type models on subregions of the device and suitable transmission conditions. Moreover, we prove existence of a solution using a regularization argument and Schauder's fixed point theorem. In doing so, we extend recent work by taking into account the statistical relation given by the Gauss–Fermi integral and mobility functions depending on the temperature, charge-carrier density, and field strength, which is required for a proper description of organic devices.
- ItemDimension reduction of thermistor models for large-area organic light-emitting diodes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Glitzky, Annegret; Liero, Matthias; Nika, GrigorAn effective system of partial differential equations describing the heat and current flow through a thin organic light-emitting diode (OLED) mounted on a glass substrate is rigorously derived from a recently introduced fully three-dimensional φ(x)-Laplace thermistor model. The OLED consists of several thin layers that scale differently with respect to the multiscale parameter ε > 0 which is the ratio between the total thickness and the lateral extent of the OLED. Starting point of the derivation is a rescaled formulation of the current-flow equation in the OLED for the driving potential and the heat equation in OLED and glass substrate with Joule heat term concentrated in the OLED. Assuming physically motivated scalings in the electrical flux functions, uniform a priori bounds are derived for the solutions of the three-dimensional system which facilitates the extraction of converging subsequences with limits that are identified as solutions of a dimension reduced system. In the latter, the effective current-flow equation is given by two semilinear equations in the two-dimensional cross-sections of the electrodes and algebraic equations for the continuity of the electrical fluxes through the organic layers. The effective heat equation is formulated only in the glass substrate with Joule heat term on the part of the boundary where the OLED is mounted.
- ItemDrift-diffusion modeling, analysis and simulation of organic semiconductor devices(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Doan, Duy-Hai; Glitzky, Annegret; Liero, MatthiasWe discuss drift-diffusion models for charge-carrier transport in organic semiconductor devices. The crucial feature in organic materials is the energetic disorder due to random alignment of molecules and the hopping transport of carriers between adjacent energetic sites. The former leads to so-called Gauss-Fermi statistics, which describe the occupation of energy levels by electrons and holes. The latter gives rise to complicated mobility models with a strongly nonlinear dependence on temperature, density of carriers, and electric field strength. We present the state-of-the-art modeling of the transport processes and provide a first existence result for the stationary drift-diffusion model taking all of the peculiarities of organic materials into account. The existence proof is based on Schauders fixed-point theorem. Finally, we discuss the numerical discretization of the model using finite-volume methods and a generalized Scharfetter-Gummel scheme for the Gauss-Fermi statistics.
- ItemDrift-diffusion simulation of S-shaped current-voltage relations for organic semiconductor devices(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Doan, Duy Hai; Fischer, Axel; Fuhrmann, Jürgen; Glitzky, Annegret; Liero, MatthiasWe present an electrothermal drift-diffusion model for organic semiconductor devices with Gauss-Fermi statistics and positive temperature feedback for the charge carrier mobilities. We apply temperature dependent Ohmic contact boundary conditions for the electrostatic potential and discretize the system by a finite volume based generalized Scharfetter-Gummel scheme. Using path-following techniques we demonstrate that the model exhibits S-shaped current-voltage curves with regions of negative differential resistance, which were only recently observed experimentally.
- ItemDrift–diffusion simulation of S-shaped current–voltage relations for organic semiconductor devices(Dordrecht : Springer Science + Business Media B.V., 2020) Doan, Duy Hai; Fischer, Axel; Fuhrmann, Jürgen; Glitzky, Annegret; Liero, MatthiasWe present an electrothermal drift–diffusion model for organic semiconductor devices with Gauss–Fermi statistics and positive temperature feedback for the charge carrier mobilities. We apply temperature-dependent Ohmic contact boundary conditions for the electrostatic potential and discretize the system by a finite volume based generalized Scharfetter–Gummel scheme. Using path-following techniques, we demonstrate that the model exhibits S-shaped current–voltage curves with regions of negative differential resistance, which were only recently observed experimentally. © 2020, The Author(s).
- ItemAn effective bulk-surface thermistor model for large-area organic light-emitting diodes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Glitzky, Annegret; Liero, Matthias; Nika, GrigorThe existence of a weak solution for an effective system of partial differential equations describing the electrothermal behavior of large-area organic light-emitting diodes (OLEDs) is proved. The effective system consists of the heat equation in the three-dimensional bulk glass substrate and two semi-linear equations for the current flow through the electrodes coupled to algebraic equations for the continuity of the electrical fluxes through the organic layers. The electrical problem is formulated on the (curvilinear) surface of the glass substrate where the OLED is mounted. The source terms in the heat equation are due to Joule heating and are hence concentrated on the part of the boundary where the current-flow equation is posed. The existence of weak solutions to the effective system is proved via Schauder's fixed-point theorem. Moreover, since the heat sources are a priori only in $L^1$, the concept of entropy solutions is used.
- ItemEffective diffusion in thin structures via generalized gradient systems and EDP-convergence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Frenzel, Thomas; Liero, MatthiasThe notion of Energy-Dissipation-Principle convergence (EDP-convergence) is used to derive effective evolution equations for gradient systems describing diffusion in a structure consisting of several thin layers in the limit of vanishing layer thickness. The thicknesses of the sublayers tend to zero with different rates and the diffusion coefficients scale suitably. The Fokker--Planck equation can be formulated as gradient-flow equation with respect to the logarithmic relative entropy of the system and a quadratic Wasserstein-type gradient structure. The EDP-convergence of the gradient system is shown by proving suitable asymptotic lower limits of the entropy and the total dissipation functional. The crucial point is that the limiting evolution is again described by a gradient system, however, now the dissipation potential is not longer quadratic but is given in terms of the hyperbolic cosine. The latter describes jump processes across the thin layers and is related to the Marcelin--de Donder kinetics.
- ItemElectrothermal Tristability Causes Sudden Burn-In Phenomena in Organic LEDs(Weinheim : Wiley-VCH, 2021) Kirch, Anton; Fischer, Axel; Liero, Matthias; Fuhrmann, Jürgen; Glitzky, Annegret; Reineke, SebastianOrganic light-emitting diodes (OLEDs) have been established as a mature display pixel technology. While introducing the same technology in a large-area form factor to general lighting and signage applications, some key questions remain unanswered. Under high-brightness conditions, OLED panels were reported to exhibit nonlinear electrothermal behavior causing lateral brightness inhomogeneities and even regions of switched-back luminance. Also, the physical understanding of sudden device failure and burn-ins is still rudimentary. A safe and stable operation of lighting tiles, therefore, requires an in-depth understanding of these physical phenomena. Here, it is shown that the electrothermal treatment of thin-film devices allows grasping the underlying physics. Configurations of OLEDs with different lateral dimensions are studied as a role model and it is reported that devices exceeding a certain panel size develop three stable, self heating-induced operating branches. Switching between them causes the sudden formation of dark spots in devices without any preexisting inhomogeneities. A current-stabilized operation mode is commonly used in the lighting industry, as it ensures degradation-induced voltage adjustments. Here, it is demonstrated that a tristable operation always leads to destructive switching, independent of applying constant currents or voltages. With this new understanding of the effects at high operation brightness, it will be possible to adjust driving schemes accordingly, design more resilient system integrations, and develop additional failure mitigation strategies. © 2021 The Authors. Advanced Functional Materials published by Wiley-VCH GmbH
- ItemThe elliptic-regularization principle in Lagrangian mechanics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Liero, Matthias; Stefanelli, UlisseWe present a novel variational approach to Lagrangian mechanics based on elliptic regularization with respect to time. A class of parameter-dependent global-in-time minimization problems is presented and the convergence of the respective minimizers to the solution of the system of Lagrange's equations is ascertained. Moreover, we extend this perspective to mixed dissipative/nondissipative situations, present a finite time-horizon version of this approach, and provide related Gamma-convergence results. Finally, some discussion on corresponding time-discrete versions of the principle is presented.
- ItemAn evolutionary elastoplastic plate model derived via Gamma-convergence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Liero, Matthias; Mielke, AlexanderThis paper is devoted to dimension reduction for linearized elastoplasticity in the rate-independent case. The reference configuration of the three-dimensional elastoplastic body has a two-dimensional middle surface and a positive but small thickness. Under suitable scalings we derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations (linear Kirchhoff--Love plate), which are coupled via plastic strains. We establish strong convergence of the solutions in the natural energy space. The analysis uses an abstract Gamma-convergence theory for rate-independent evolutionary systems that is based on the notion of energetic solutions. This concept is formulated via an energy-storage functional and a dissipation functional, such that energetic solutions are defined in terms of a stability condition and an energy balance. The Mosco convergence of the quadratic energy-storage functional follows the arguments of the elastic case. To handle the evolutionary situation the interplay with the dissipation functional is controlled by cancellation properties for Mosco-convergent quadratic energies
- ItemAn existence result for a class of electrothermal drift-diffusion models with Gauss--Fermi statistics for organic semiconductors(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Glitzky, Annegret; Liero, Matthias; Nika, GrigorThis work is concerned with the analysis of a drift-diffusion model for the electrothermal behavior of organic semiconductor devices. A "generalized Van Roosbroeck'' system coupled to the heat equation is employed, where the former consists of continuity equations for electrons and holes and a Poisson equation for the electrostatic potential, and the latter features source terms containing Joule heat contributions and recombination heat. Special features of organic semiconductors like Gauss--Fermi statistics and mobilities functions depending on the electric field strength are taken into account. We prove the existence of solutions for the stationary problem by an iteration scheme and Schauder's fixed point theorem. The underlying solution concept is related to weak solutions of the Van Roosbroeck system and entropy solutions of the heat equation. Additionally, for data compatible with thermodynamic equilibrium, the uniqueness of the solution is verified. It was recently shown that self-heating significantly influences the electronic properties of organic semiconductor devices. Therefore, modeling the coupled electric and thermal responses of organic semiconductors is essential for predicting the effects of temperature on the overall behavior of the device. This work puts the electrothermal drift-diffusion model for organic semiconductors on a sound analytical basis.
- ItemExperimental proof of Joule heating-induced switched-back regions in OLEDs(London : Nature Publishing Group, 2020) Kirch, Anton; Fische, Axel; Liero, Matthias; Fuhrmann, Jürgen; Glitzky, Annegret; Reineke, SebastianOrganic light-emitting diodes (OLEDs) have become a major pixel technology in the display sector, with products spanning the entire range of current panel sizes. The ability to freely scale the active area to large and random surfaces paired with flexible substrates provides additional application scenarios for OLEDs in the general lighting, automotive, and signage sectors. These applications require higher brightness and, thus, current density operation compared to the specifications needed for general displays. As extended transparent electrodes pose a significant ohmic resistance, OLEDs suffering from Joule self-heating exhibit spatial inhomogeneities in electrical potential, current density, and hence luminance. In this article, we provide experimental proof of the theoretical prediction that OLEDs will display regions of decreasing luminance with increasing driving current. With a two-dimensional OLED model, we can conclude that these regions are switched back locally in voltage as well as current due to insufficient lateral thermal coupling. Experimentally, we demonstrate this effect in lab-scale devices and derive that it becomes more severe with increasing pixel size, which implies its significance for large-area, high-brightness use cases of OLEDs. Equally, these non-linear switching effects cannot be ignored with respect to the long-term operation and stability of OLEDs; in particular, they might be important for the understanding of sudden-death scenarios. © 2020, The Author(s).
- ItemGradient structures and geodesic convexity for reaction-diffusion systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Liero, Matthias; Mielke, AlexanderWe consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory. We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory.
- ItemHomogenization of a porous intercalation electrode with phase separation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Heida, Martin; Landstorfer, Manuel; Liero, MatthiasIn this work, we derive a new model framework for a porous intercalation electrode with a phase separating active material upon lithium intercalation. We start from a microscopic model consisting of transport equations for lithium ions in an electrolyte phase and intercalated lithium in a solid active phase. Both are coupled through a Neumann--boundary condition modeling the lithium intercalation reaction. The active material phase is considered to be phase separating upon lithium intercalation. We assume that the porous material is a given periodic microstructure and perform analytical homogenization. Effectively, the microscopic model consists of a diffusion and a Cahn--Hilliard equation, whereas the limit model consists of a diffusion and an Allen--Cahn equation. Thus we observe a Cahn--Hilliard to Allen--Cahn transition during the upscaling process. In the sense of gradient flows, the transition goes in hand with a change in the underlying metric structure of the PDE system.
- ItemHomogenization of Cahn-Hilliard-type equations via evolutionary Gamma-convergence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Liero, Matthias; Reichelt, SinaIn this paper we discuss two approaches to evolutionary Gamma-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Gamma-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savare 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for Cahn-Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Gamma-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.
- ItemHybrid finite-volume/finite-element schemes for p(x)-Laplace thermistor models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Fuhrmann, Jürgen; Glitzky, Annegret; Liero, MatthiasWe introduce an empirical PDE model for the electrothermal description of organic semiconductor devices by means of current and heat flow. The current flow equation is of p(x)-Laplace type, where the piecewise constant exponent p(x) takes the non-Ohmic behavior of the organic layers into account. Moreover, the electrical conductivity contains an Arrhenius-type temperature law. We present a hybrid finite-volume/finite-element discretization scheme for the coupled system, discuss a favorite discretization of the p(x)-Laplacian at hetero interfaces, and explain how path following methods are applied to simulate S-shaped current-voltage relations resulting from the interplay of self-heating and heat flow.