New optimal control problems in density functional theory motivated by photovoltaics

We present and study novel optimal control problems motivated by the search for photovoltaic materials with high power-conversion efficiency. The material must perform the first step: convert light (photons) into electronic excitations. We formulate various desirable properties of the excitations as mathematical control goals at the Kohn-Sham-DFT level of theory, with the control being given by the nuclear charge distribution. We prove that nuclear distributions exist which give rise to optimal HOMO-LUMO excitations, and present illustrative numerical simulations for 1D finite nanocrystals. We observe pronounced goal-dependent features such as large electron-hole separation, and a hierarchy of length scales: internal HOMO and LUMO wavelengths $<$ atomic spacings $<$ (irregular) fluctuations of the doping profiles $<$ system size

Uniform convergence to equilibrium for a family of drift–diffusion models with trap-assisted recombination and the limiting Shockley–Read–Hall model

In this paper, we establish convergence to equilibrium for a drift–diffusion–recombination system modelling the charge transport within certain semiconductor devices. More precisely, we consider a two-level system for electrons and holes which is augmented by an intermediate energy level for electrons in so-called trapped states. The recombination dynamics use the mass action principle by taking into account this additional trap level. The main part of the paper is concerned with the derivation of an entropy–entropy production inequality, which entails exponential convergence to the equilibrium via the so-called entropy method. The novelty of our approach lies in the fact that the entropy method is applied uniformly in a fast-reaction parameter which governs the lifetime of electrons on the trap level. Thus, the resulting decay estimate for the densities of electrons and holes extends to the corresponding quasi-steady-state approximation

Regularity of random elliptic operators with degenerate coefficients and applications to stochastic homogenization

We consider degenerate elliptic equations of second order in divergence form with a symmetric random coefficient field $a$. Extending the work of the first author, Fehrman, and Otto [Ann. Appl. Probab. 28 (2018), no. 3, 1379-1422], who established the large-scale $C^{1,\alpha}$ regularity of $a$-harmonic functions in a degenerate situation, we provide stretched exponential moments for the minimal radius $r_*$ describing the minimal scale for this $C^{1,\alpha}$ regularity. As an application to stochastic homogenization, we partially generalize results by Gloria, Neukamm, and Otto [Anal. PDE 14 (2021), no. 8, 2497-2537] on the growth of the corrector, the decay of its gradient, and a quantitative two-scale expansion to the degenerate setting. On a technical level, we demand the ensemble of coefficient fields to be stationary and subject to a spectral gap inequality, and we impose moment bounds on $a$ and $a^{-1}$. We also introduce the ellipticity radius $r_e$ which encodes the minimal scale where these moments are close to their positive expectation value

Variance reduction for effective energies of random lattices in the Thomas-Fermi-von Weizsäcker model

In the computation of the material properties of random alloys, the method of 'special quasirandom structures' attempts to approximate the properties of the alloy on a finite volume with higher accuracy by replicating certain statistics of the random atomic lattice in the finite volume as accurately as possible. In the present work, we provide a rigorous justification for a variant of this method in the framework of the Thomas–Fermi–von Weizsäcker (TFW) model. Our approach is based on a recent analysis of a related variance reduction method in stochastic homogenization of linear elliptic PDEs and the locality properties of the TFW model. Concerning the latter, we extend an exponential locality result by Nazar and Ortner to include point charges, a result that may be of independent interest

Mathematical modeling and analysis of PDE-models for semiconductor devices

Diese Arbeit beschäftigt sich mit der Untersuchung bestimmter Aspekte von Halbleitermaterialien. Dafür kommen sowohl analytische Methoden als auch Techniken der Optimierung zur Anwendung. Einerseits betrachten wir ein System partieller Differentialgleichungen (PDG), das die dynamischen Prozesse negativ geladener Elektronen und positiv geladener Löcher modelliert. Dieses PDG-System verallgemeinert das Shockley-Read-Hall-Modell, das Rekombinationen, Drift und Diffusion der Teilchen mithilfe eines zusätzlichen internen Energieniveaus beschreibt. Unser Hauptresultat besagt, dass die Ladungsdichten der Elektronen und Löcher mit exponentieller Rate zu ihren Gleichgewichtsverteilungen konvergieren. Diese Konvergenzrate ist unabhängig von der mittleren Verweildauer der Elektronen im zusätzlichen Energieniveau. Andererseits behandeln wir ein Problem der Materialgestaltung im Bereich der Photovoltaik. Zu einer gegebenen Verteilung positiver Kernladungen in einer photovoltaischen Zelle bestimmt sich die resultierende Elektronenverteilung als Lösung der Kohn-Sham-Gleichungen. Die Struktur der Elektronenladungsdichte kann sich aufgrund elektronischer Anregungen unter dem Einfluss von Licht ändern. Wir beweisen, dass es eine bestimmte Verteilung der Kernladungen gibt, die die Änderung der Elektronenverteilung unter dem Einfluss eines bestimmen Lichts maximiert. Eine 1D-Simulation einer Atomkette zeigt einen deutlichen Ladungstransfer für bestimmte Kernverteilungen. In zukünftigen Anwendungen könnte diese Ladungstrennung zur Erzeugung elektrischen Stroms genutzt werden. Am Ende studieren wir ein PDG-Modell für Elektronen und Löcher in einem Halbleiter unter dem Einfluss des selbstkonsistenten Potentials, das von diesen Ladungsträgern erzeugt wird. Das Hauptresultat ist der Beweis exponentieller Konvergenz zum Gleichgewicht für die entsprechenden Ladungsdichten.This thesis is concerned with the study of certain aspects of semiconductor materials both from an analytical point of view as well as from an optimization perspective. On the one hand, we focus on a system of partial differential equations (PDE) which models the dynamics of negatively charged electrons and positively charged holes inside a semiconductor. This PDE-system generalizes the Shockley-Read-Hall-model which accounts for recombination, drift and diffusion of the charged particles by means of an additional internal energy level. Our main result states that the charge densities of electrons and holes converge to their equilibrium distributions at an exponential rate. Moreover, this convergence rate is independent of the mean residence time of electrons in the additional energy level. On the other hand, we investigate a material design problem in the context of photovoltaics. Given a density of positive nuclear charges inside a photovoltaic cell, we determine the resulting electronic density by solving the Kohn-Sham equations. In short, the structure of the charge density of the electrons may change under the influence of incident light due to internal electronic excitations. We prove that there exists a certain nuclear density which maximizes the change of the electronic density under the influence of a specific light. A 1D simulation of an atomic chain reveals a pronounced charge transfer for certain nuclear densities. Within a future application, one could use this charge separation to obtain an electric current. At the end, we study a PDE-model for electrons and holes in a semiconductor including the influence of the selfconsistent potential generated by these charge carriers. As the main result, we prove exponential convergence to the equilibrium for the corresponding charge densities.vorgelegt von Michael KnielyZusammenfassungen in Deutsch und EnglischKarl-Franzens-Universität Graz, Dissertation, 2017OeBB(VLID)231563

New optimal control problems in density functional theory motivated by photovoltaics

We present and study novel optimal control problems motivated by the search for photovoltaic materials with high power-conversion efficiency. The material must perform the first step: convert light (photons) into electronic excitations. We formulate various desirable properties of the excitations as mathematical control goals at the Kohn-Sham-DFT level of theory, with the control being given by the nuclear charge distribution. We prove that nuclear distributions exist which give rise to optimal HOMO-LUMO excitations, and present illustrative numerical simulations for 1D finite nanocrystals. We observe pronounced goal-dependent features such as large electron-hole separation, and a hierarchy of length scales: internal HOMO and LUMO wavelengths < atomic spacings < (irregular) fluctuations of the doping profiles < system size

Global existence analysis of energy-reaction-diffusion systems

We establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear