publications

Preprints

2022

1. Fine properties of geodesics and geodesic lambda-convexity for the Hellinger–Kantorovich distance

   Matthias Liero, Alexander Mielke, and Giuseppe Savaré

   Weierstrass Institute Berlin Preprint, No. 2956 2022

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   We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger–Kantorovich problem (HK), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the Hamilton–Jacobi equation arising in the dual dynamic formulation of HK, which are sufficiently strong to construct a characteristic transport-dilation flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of HK geodesics and to derive the convex behaviour of their Lebesgue density along the transport flow. Finally, exact conditions for functionals defined on the space of measures are derived that guarantee the geodesic lambda-convexity with respect to the Hellinger–Kantorovich distance.

2021

1. Homogenization of a porous intercalation electrode with phase separation

   Martin Heida, Manuel Landstorfer, and Matthias Liero

   Weierstrass Institute Berlin Preprint, No. 2905 2021

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   In 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.

Journal Articles

2022

1. A coarse-grained electrothermal model for organic semiconductor devices

   Annegret Glitzky, Matthias Liero, and Grigor Nika

   Math. Methods Appl. Sci., 45, pp. 4809–4833, 2022

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   We 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.

2. Analysis of a hybrid model for the electro-thermal behavior of semiconductor heterostructures

   Annegret Glitzky, Matthias Liero, and Grigor Nika

   J. Math. Anal. Appl., 507(2), pp. 125815, 2022

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   We 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.

2021

1. Dimension reduction of thermistor models for large-area organic light-emitting diodes

   Annegret Glitzky, Matthias Liero, and Grigor Nika

   Discr. Cont. Dynam. Systems Ser. S, 14(11), pp. 3953–3971, 2021

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   An 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 p(x)-Laplace thermistor model. The OLED consists of several thin layers that scale differently with respect to the multiscale parameter epsilon>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.

2. An existence result for a class of electrothermal drift-diffusion models with Gauss–Fermi statistics for organic semiconductors

   Annegret Glitzky, Matthias Liero, and Grigor Nika

   Analysis and Applications, 19, pp. 275–304, 2021

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   This 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 mobility 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.

3. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence

   Thomas Frenzel, and Matthias Liero

   Discr. Cont. Dynam. Systems Ser. S, 14(1), pp. 395–425, 2021

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   The 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.

4. Electrothermal Tristability Causes Sudden Burn-In Phenomena in Organic LEDs

   Anton Kirch, Axel Fischer, Matthias Liero, Jürgen Fuhrmann, Annegret Glitzky, and Sebastian Reineke

   Advanced Functional Materials, 31, pp. 2106716, 2021

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   Abstract Organic 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.

5. Analysis of a bulk-surface thermistor model for large-area organic LEDs

   Annegret Glitzky, Matthias Liero, and Grigor Nika

   Port. Math., 78(2), pp. 187–210, 2021

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   The 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 threedimensional bulk glass substrate and two semilinear 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.

2020

1. Drift–diffusion simulation of S-shaped current–voltage relations for organic semiconductor devices

   Duy Hai Doan, Axel Fischer, Jürgen Fuhrmann, Annegret Glitzky, and Matthias Liero

   Journal of Computational Electronics, 19, pp. 1164–1174, Sep 2020

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   We 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.

2. Introducing pinMOS Memory: A Novel, Nonvolatile Organic Memory Device

   Yichu Zheng, Axel Fischer, Michael Sawatzki, Duy Hai Doan, Matthias Liero, Annegret Glitzky, Sebastian Reineke, and Stefan C. B. Mannsfeld

   Advanced Functional Materials, 30, pp. 1907119, Sep 2020

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   Abstract In recent decades, organic memory devices have been researched intensely and they can, among other application scenarios, play an important role in the vision of an internet of things. Most studies concentrate on storing charges in electronic traps or nanoparticles while memory types where the information is stored in the local charge up of an integrated capacitance and presented by capacitance received far less attention. Here, a new type of programmable organic capacitive memory called p-i-n-metal-oxide-semiconductor (pinMOS) memory is demonstrated with the possibility to store multiple states. Another attractive property is that this simple, diode-based pinMOS memory can be written as well as read electrically and optically. The pinMOS memory device shows excellent repeatability, an endurance of more than 104 write-read-erase-read cycles, and currently already over 24 h retention time. The working mechanism of the pinMOS memory under dynamic and steady-state operations is investigated to identify further optimization steps. The results reveal that the pinMOS memory principle is promising as a reliable capacitive memory device for future applications in electronic and photonic circuits like in neuromorphic computing or visual memory systems.

3. Experimental proof of Joule heating-induced switched-back regions in OLEDs

   Anton Kirch, Axel Fischer, Matthias Liero, Jürgen Fuhrmann, Annegret Glitzky, and Sebastian Reineke

   Light: Science & Applications, 9, pp. 5, Sep 2020

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   Organic 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.

2019

1. Instationary drift-diffusion problems with Gauss–Fermi statistics and field-dependent mobility for organic semiconductor devices

   Annegret Glitzky, and Matthias Liero

   Comm. Math. Sci., 17, pp. 33–59, Sep 2019

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   This paper deals with the analysis of an instationary drift-diffusion model for organic semiconductor devices including Gauss–Fermi statistics and application-specific mobility functions. The charge transport in organic materials is realized by hopping of carriers between adjacent energetic sites and is described by complicated mobility laws with a strong nonlinear dependence on temperature, carrier densities and the electric field strength. To prove the existence of global weak solutions, we consider a problem with (for small densities) regularized state equations on any arbitrarily chosen finite time interval. We ensure its solvability by time discretization and passage to the time-continuous limit. Positive lower a priori estimates for the densities of its solutions that are independent of the regularization level ensure the existence of solutions to the original problem. Furthermore, we derive for these solutions global positive lower and upper bounds strictly below the density of transport states for the densities. The estimates rely on Moser iteration techniques.

2. The weighted energy-dissipation principle and evolutionary Γ-convergence for doubly nonlinear problems

   Matthias Liero, and Stefano Melchionna

   ESAIM Control Optim. Calc. Var., 25, pp. 36, Sep 2019

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   We consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter epsilon. For each of these problems, we introduce the so-called weighted energy-dissipation (WED) functional, whose minimizers correspond to solutions of an elliptic-in-time regularization of the target problems with regularization parameter delta. We investigate the relation between the Gamma-convergence of the WED functionals and evolutionary Gamma-convergence of the associated systems. More precisely, we deal with the limits delta to 0, epsilon to 0, as well as delta + epsilon to 0 either in the sense of Gamma-convergence of functionals or in the sense of evolutionary Gamma-convergence of functional-driven evolution problems, or both. Additionally, we provide some quantitative estimates on the rate of convergence for the limit epsilon to 0, in the case of quadratic dissipation potentials and uniformly lambda-convex energy functionals. Finally, we discuss a homogenization and a dimension reduction problem as examples of application.

3. Analysis of a drift-diffusion model for organic semiconductor devices

   Duy Hai Doan, Annegret Glitzky, and Matthias Liero

   Z. Angew. Math. Phys., 70, pp. 55, Sep 2019

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   We 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 statistical relations with Gauss–Fermi integrals, 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 Schauder’s fixed-point theorem.

2018

1. Balance of Horizontal and Vertical Charge Transport in Organic Field-Effect Transistors

   Franz Michael Sawatzki, Duy Hai Doan, Hans Kleemann, Matthias Liero, Annegret Glitzky, Thomas Koprucki, and Karl Leo

   Phys. Rev. Applied, 10(3), pp. 034069, Sep 2018

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   High-performance organic field-effect transistors (OFETs) are an essential building block for future flexible electronics. Although there has been steady progress in the development of high-mobility organic semiconductors, the performance of lateral state-of-the-art OFETs still falls short, especially with regard to the transition frequency. One candidate to overcome the shortcomings of the lateral OFET is its vertical embodiment, the vertical organic field-effect transistor (VOFET). However, the detailed mechanism of VOFET operation is poorly understood and a matter of discussion. Proposed descriptions of the formation and geometry of the vertical channel vary significantly. In particular, values for lateral depth of the vertical channel reported so far show a large variation. This is an important question for the transistor integration, though, since a channel depth in the micrometer range would severely limit the possible integration density. Here, we investigate charge transport in such VOFETs via drift-diffusion simulations and experimental measurements. We use a (vertical) organic light-emitting transistor [(V)OLET] as a means to map the spatial distribution of charge transport within the vertical channel. Comparing simulation and experiment, we can conclusively describe the operation mechanism which is mainly governed by an accumulation of charges at the dielectric interface and the channel formation directly at the edge of the source electrode. In particular, we quantitatively describe how the channel depth depends on parameters such as gate-source voltage, drain-source voltage, and lateral and vertical mobility. Based on the proposed operation mechanism, we derive an analytical estimation for the lateral dimensions of the channel, helping to predict an upper limit for the integration density of VOFETs.

2. Evolutionary Γ-convergence of gradient systems modeling slow and fast chemical reactions

   Karoline Disser, Matthias Liero, and Jonathan Zinsl

   Nonlinearity, 31, pp. 3689–3706, Jul 2018

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   We investigate the limit passage for a system of slow and fast chemical reactions of mass-action type, where the rates of fast reactions tend to infinity. We give an elementary proof of convergence to a reduced dynamical system on the manifold of fast reaction equilibria. Then we study the entropic gradient structure of these systems and prove an E-convergence result via Gamma-convergence of the primary and dual dissipation potentials, which shows that this structure carries over to the fast reaction limit in a thermodynamically consistent way. We recover the limit dynamics as a gradient flow of the entropy with respect to a pseudo-metric.

3. Full Electrothermal OLED Model Including Nonlinear Self-heating Effects

   Axel Fischer, Manuel Pfalz, Koen Vandewal, Simone Lenk, Matthias Liero, Annegret Glitzky, and Sebastian Reineke

   Phys. Rev. Applied, 10(1), pp. 014023, Jul 2018

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   Organic light-emitting diodes (OLEDs) are widely studied semiconductor devices for which a simple description by a diode equation typically fails. In particular, a full description of the current-voltage relation, including temperature effects, has to take the low electrical conductivity of organic semiconductors into account. Here, we present a temperature-dependent resistive network, incorporating recombination as well as electron and hole conduction to describe the current-voltage characteristics of an OLED over the entire operation range. The approach also reproduces the measured nonlinear electrothermal feedback upon Joule self-heating in a self-consistent way. Our model further enables us to learn more about internal voltage losses caused by the charge transport from the contacts to the emission layer which is characterized by a strong temperature-activated electrical conductivity, finally determining the strength of the electrothermal feedback. In general, our results provide a comprehensive picture to understand the electrothermal operation of an OLED which will be essential to ensure and predict especially long-term stability and reliability in superbright OLED applications.

4. Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures

   Matthias Liero, Alexander Mielke, and Giuseppe Savaré

   Inventiones mathematicae, 211, pp. 969–1117, Mar 2018

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   We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. These problems arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a pair of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, which quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger–Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger–Kakutani and Kantorovich–Wasserstein distances.

5. Homogenization of Cahn–Hilliard-type equations via evolutionary Γ-convergence

   Matthias Liero, and Sina Reichelt

   Nonl. Diff. Eqns. Appl. (NoDEA), 25, pp. 6, Jan 2018

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   In these notes we discuss two approaches to evolutionary }}\backslashGamma }}Γ-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 }}\backslashGamma }}Γ-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 Savaré 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.

2017

1. On microscopic origins of generalized gradient structures

   Matthias Liero, Alexander Mielke, Mark A. Peletier, and D. R. Michiel Renger

   Discr. Cont. Dynam. Systems Ser. S, 10(1), pp. 1–25, Jan 2017

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   Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials. A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary Gamma-limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.

2. 3D electrothermal simulations of organic LEDs showing negative differential resistance

   Matthias Liero, Jürgen Fuhrmann, Annegret Glitzky, Thomas Koprucki, Axel Fischer, and Sebastian Reineke

   Opt. Quantum Electron., 49, pp. 330/1–330/8, Jan 2017

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   Organic 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.

3. Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions

   Miroslav Bulíček, Annegret Glitzky, and Matthias Liero

   Discr. Cont. Dynam. Systems Ser. S, 10, pp. 697–713, Jan 2017

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   We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (i) the elliptic term in the current-flow equation is of p(x)-Laplacian-type with discontinuous exponent p, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori L1 term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder’s fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.

4. Analysis of p(x)-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices

   Annegret Glitzky, and Matthias Liero

   Nonlinear Analysis: Real World Applications, 34, pp. 536–562, Jan 2017

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   We study a stationary thermistor model describing the electrothermal behavior of organic semiconductor devices featuring non-Ohmic current–voltage laws and self-heating 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 -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 -Laplacian. Finally, Schauder’s fixed-point theorem is used to show the existence of solutions.

2016

1. Optimal Transport in Competition with Reaction: The Hellinger–Kantorovich Distance and Geodesic Curves

   Matthias Liero, Alexander Mielke, and Giuseppe Savaré

   SIAM J. Math. Analysis, 48, pp. 2869–2911, Jan 2016

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   We discuss a new notion of distance on the space of finite and nonnegative measures on Ω⊂ℝ𝑑, which we call the Hellinger–Kantorovich distance. It can be seen as an inf-convolution of the well-known Kantorovich–Wasserstein distance and the Hellinger-Kakutani distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and some of its properties. In particular, the distance can be equivalently described by an optimal transport problem on the cone space over the underlying space Ω. We give a construction of geodesic curves and discuss examples and their general properties.

2. Point contacts at the copper-indium-gallium-selenide interface–A theoretical outlook

   Adrien Bercegol, Binoy Chacko, Reiner Klenk, Iver Lauermann, Martha Ch. Lux-Steiner, and Matthias Liero

   Journal of Applied Physics, 119, pp. 155304, Jan 2016

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   For a long time, it has been assumed that recombination in the space-charge region of copper-indium-gallium-selenide (CIGS) is dominant, at least in high efficiency solar cells with low band gap. The recent developments like potassium fluoride post deposition treatment and point-contact junction may call this into question. In this work, a theoretical outlook is made using three-dimensional simulations to investigate the effect of point-contact openings through a passivation layer on CIGS solar cell performance. A large set of solar cells is modeled under different scenarios for the charged defect levels and density, radius of the openings, interface quality, and conduction band offset. The positive surface charge created by the passivation layer induces band bending and this influences the contact (CdS) properties, making it beneficial for the open circuit voltage and efficiency, and the effect is even more pronounced when coverage area is more than 95%, and also makes a positive impact on the device performance, even in the presence of a spike at CIGS/CdS heterojunction.

3. Systems describing electrothermal effects with p(x)-Laplace like structure for discontinuous variable exponents

   Miroslav Bulíček, Annegret Glitzky, and Matthias Liero

   SIAM J. Math. Analysis, 48, pp. 3496–3514, Jan 2016

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   We consider a coupled system of two elliptic PDEs, where the elliptic term in the first equation shares the properties of the p(x)-Laplacian with discontinuous exponent, while in the second equation we have to deal with an a priori L^1 term on the right-hand side. Such systems are suitable for the description of various electrothermal effects, in particular, those where the non-Ohmic behavior can change dramatically with respect to the spatial variable. We prove the existence of a weak solution under very weak assumptions on the data and also under general structural assumptions on the constitutive equations of the model. The main difficulty consists in the fact that we have to simultaneously overcome two obstacles—the discontinuous variable exponent and the L^1 right-hand side of the heat equation. Our existence proof based on Galerkin approximation is highly constructive and therefore seems to be suitable also for numerical purposes.

2015

1. On gradient structures for Markov chains and the passage to Wasserstein gradient flows

   Karoline Disser, and Matthias Liero

   Networks & Heterogeneous Media, 10, pp. 233–253, Jan 2015

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   We study the approximation of Wasserstein gradient structures by their finite-dimensional analog. We show that simple finite-volume discretizations of the linear Fokker-Planck equation exhibit the recently established entropic gradient-flow structure for reversible Markov chains. Then we reprove the convergence of the discrete scheme in the limit of vanishing mesh size using only the involved gradient-flow structures. In particular, we make no use of the linearity of the equations nor of the fact that the Fokker-Planck equation is of second order.

2. p-Laplace thermistor modeling of electrothermal feedback in organic semiconductor devices

   Matthias Liero, Thomas Koprucki, Axel Fischer, Reinhard Scholz, and Annegret Glitzky

   Z. Angew. Math. Phys., 66, pp. 2957–2977, Jan 2015

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   In large-area organic light-emitting diodes (OLEDs), spatially inhomogeneous luminance at high power due to inhomogeneous current flow and electrothermal feedback can be observed. To describe these self-heating effects in organic semiconductors, we present a stationary thermistor model based on the heat equation for the temperature coupled to a p-Laplace-type equation for the electrostatic potential with mixed boundary conditions. The p-Laplacian describes the non-Ohmic electrical behavior of the organic material. Moreover, an Arrhenius-like temperature dependency of the electrical conductivity is considered. We introduce a finite-volume scheme for the system and discuss its relation to recent network models for OLEDs. In two spatial dimensions, we derive a priori estimates for the temperature and the electrostatic potential and prove the existence of a weak solution by Schauder’s fixed-point theorem.

2013

1. Passing from bulk to bulk-surface evolution in the Allen–Cahn equation

   Matthias Liero

   Nonl. Diff. Eqns. Appl. (NoDEA), 20, pp. 919–942, Jun 2013

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   In this paper we formulate a boundary layer approximation for an Allen–Cahn-type equation involving a small parameter epsilon. Here, epsilon is related to the thickness of the boundary layer and we are interested in the limit epsilon to 0 in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L2-gradient flow with the corresponding Allen–Cahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Gamma- and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions.

2. A new minimum principle for Lagrangian mechanics

   Matthias Liero, and Ulisse Stefanelli

   J. Nonlinear Sci., 23, pp. 179–204, Jun 2013

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   We present a novel variational view at Lagrangian mechanics based on the minimization of weighted inertia-energy functionals on trajectories. In particular, we introduce a family of parameter-dependent global-in-time minimization problems whose respective minimizers converge to solutions of the system of Lagrange’s equations. The interest in this approach is that of reformulating Lagrangian dynamics as a (class of) minimization problem(s) plus a limiting procedure. The theory may be extended in order to include dissipative effects thus providing a unified framework for both dissipative and nondissipative situations. In particular, it allows for a rigorous connection between these two regimes by means of Gamma-convergence. Moreover, the variational principle may serve as a selection criterion in case of nonuniqueness of solutions. Finally, this variational approach can be localized on a finite time-horizon resulting in some sharper convergence statements and can be combined with time-discretization.

3. Gradient structures and geodesic convexity for reaction-diffusion systems

   Matthias Liero, and Alexander Mielke

   Proc. Royal Soc. London Ser. A, 371, pp. 20120346, Jun 2013

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   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.

4. Weighted inertia-dissipation-energy functionals for semilinear equations

   Matthias Liero, and Ulisse Stefanelli

   Boll. Unione Mat. Ital. (9), 6, pp. 1–27, Jun 2013

2012

1. Rigorous derivation of a plate theory in linear elastoplasticity via Γ-convergence

   Matthias Liero, and Thomas Roche

   Nonl. Diff. Eqns. Appl. (NoDEA), 19, pp. 437–457, Aug 2012

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   This paper deals with dimension reduction in linearized elastoplasticity in the rate-independent case. The reference configuration of the elastoplastic body is given by a two-dimensional middle surface and a small but positive thickness. 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 which are coupled via plastic strains. The convergence analysis is based on an abstract Γ-convergence theory for rate-independent evolution formulated in the framework of energetic solutions. This concept is based on an energy-storage functional and a dissipation functional, such that the notion of solution is phrased in terms of a stability condition and an energy balance.

2011

1. An evolutionary elastoplastic plate model derived via Γ-convergence

   Matthias Liero, and Alexander Mielke

   Math. Models Meth. Appl. Sci. (M^3AS), 21, pp. 1961–1986, Aug 2011

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   This 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.

2009

1. Rate independent Kurzweil processes

   Pavel Krejčí, and Matthias Liero

   Applications of Mathematics, 54, pp. 117–145, Aug 2009

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   The Kurzweil integral technique is applied to a class of rate independent processes with convex energy and discontinuous inputs. We prove existence, uniqueness, and continuous data dependence of solutions in BV spaces. It is shown that in the context of elastoplasticity, the Kurzweil solutions coincide with natural limits of viscous regularizations when the viscosity coefficient tends to zero. The discontinuities produce an additional positive dissipation term, which is not homogeneous of degree one.

Conference Articles

2020

1. Unipolar Drift-Diffusion Simulation of S-Shaped Current-Voltage Relations for Organic Semiconductor Devices

   Jürgen Fuhrmann, Duy Hai Doan, Annegret Glitzky, Matthias Liero, and Grigor Nika

   In Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples, Aug 2020

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   We discretize a unipolar electrothermal drift-diffusion model for organic semiconductor devices with Gauss–Fermi statistics and charge carrier mobilities having positive temperature feedback. We apply temperature dependent Ohmic contact boundary conditions for the electrostatic potential and use a finite volume based generalized Scharfetter-Gummel scheme. Applying path-following techniques we demonstrate that the model exhibits S-shaped current-voltage curves with regions of negative differential resistance, only recently observed experimentally.

2017

1. A PDE Model for Electrothermal Feedback in Organic Semiconductor Devices

   Matthias Liero, Axel Fischer, Jürgen Fuhrmann, Thomas Koprucki, and Annegret Glitzky

   In Progress in Industrial Mathematics at ECMI 2016, Aug 2017

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   Large-area organic light-emitting diodes are thin-film multilayer devices that show pronounced self-heating and brightness inhomogeneities at high currents. As these high currents are typical for lighting applications, a deeper understanding of the mechanisms causing these inhomogeneities is necessary. We discuss the modeling of the interplay between current flow, self-heating, and heat transfer in such devices using a system of partial differential equations of thermistor type, that is capable of explaining the development of luminance inhomogeneities. The system is based on the heat equation for the temperature coupled to a p(x)-Laplace-type equation for the electrostatic potential with mixed boundary conditions. The p(x)-Laplacian allows to take into account non-Ohmic electrical behavior of the different organic layers. Moreover, we present analytical results on the existence, boundedness, and regularity of solutions to the system. A numerical scheme based on the finite-volume method allows for efficient simulations of device structures.

2. Modeling and Simulation of Electrothermal Feedback in Large-area Organic LEDs

   Matthias Liero, Jürgen Fuhrmann, Annegret Glitzky, Thomas Koprucki, Axel Fischer, and Sebastian Reineke

   In Proceedings of the 17th International Conference on Numerical Simulation of Optoelectronic Devices, NUSOD 2017, Aug 2017

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   We present an empirical PDE model for the description of the interplay between total current and heat flow in large-area organic LEDs, which can lead to unwanted spatial brightness in homogeneities. The model resolves the different layers in the OLED stack by taking their individual IV characteristics into account. In particular, the electrical conductivity features a field dependence to incorporate the non-Ohmic behavior of the organic materials and a temperature dependence via an Arrhenius law. We discuss the numerical scheme and present simulation results that confirm the experimentally observed S-shaped IV characteristics with regions of negative differential resistance, which cause the inhomogeneities.

3. The Hellinger-Kantorovich distance as a generalization of optimal-transport distances to scalar reaction-diffusion problems

   Matthias Liero

   In Oberwolfach Rep. 14, Aug 2017

Book Chapters

2017

1. Hybrid Finite-Volume/Finite-Element Schemes for p(x)-Laplace Thermistor Models

   Jürgen Fuhrmann, Annegret Glitzky, and Matthias Liero

   In Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems: FVCA 8, Lille, France, June 2017, Aug 2017

2008

1. Testing the Acceleration Function in Lifetime Models

   Hannelore Liero, and Matthias Liero

   In Statistical Models and Methods for Biomedical and Technical Systems, Aug 2008

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   The accelerated lifetime model is considered. First, test procedures for testing the parameter of a parametric acceleration function are investigated; this is done under the assumption of parametric and nonparametric baseline distribution. Furthermore, based on nonparametric estimators for regression functions, tests are proposed for checking whether a parametric acceleration function is appropriate to model the influence of the covariates. Resampling procedures are discussed for the realization of these methods. Simulations complete the considerations.