Numerical Methods for Innovative Semiconductor Devices

Publications

Articles in Refereed Journals

• C.L. Manganelli, D. Spirito, P. Farrell, J. Frigerio, A. De Lacovo, D. Marian, M. Virgilio, Strain engineering in semiconductor materials, physica status solidi (RLL) -- Rapid Research Letters (pss RRL), 19 (2025), pp. 2400383/1--2400383/3, DOI 10.1002/pssr.202400383 .
  Abstract
  Strain engineering has become an essential strategy in the advancement of semiconductor technologies, providing a power- ful mean to modulate the electronic, optical, and mechanical properties of materials. By introducing controlled deformation into crystal lattices, this approach enables enhanced carrier mobility, tailored bandgap energies, and improved device perfor- mance across applications in photonics, optoelectronics, and quantum technologies.

• D. Abdel, A. Glitzky, M. Liero, Analysis of a drift-diffusion model for perovskite solar cells, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 30 (2025), pp. 99--131 (appeared online in June 2024), DOI 10.3934/dcdsb.2024081 .
  Abstract
  This paper deals with the analysis of an instationary drift-diffusion model for perovskite solar cells including Fermi--Dirac statistics for electrons and holes and Blakemore statistics for the mobile ionic vacancies in the perovskite layer. The free energy functional is related to this choice of the statistical relations. Exemplary simulations varying the mobility of the ionic vacancy demonstrate the necessity to include the migration of ionic vacancies in the model frame. To prove the existence of weak solutions, first a problem with regularized state equations and reaction terms on any arbitrarily chosen finite time interval is considered. Its solvability follows from a time discretization argument and passage to the time-continuous limit. Applying Moser iteration techniques, a priori estimates for densities, chemical potentials and the electrostatic potential of its solutions are derived that are independent of the regularization level, which in turn ensure the existence of solutions to the original problem.

• Y. Hadjimichael, Ch. Merdon, M. Liero, P. Farrell, An energy-based finite-strain model for 3D heterostructured materials and its validation by curvature analysis, International Journal for Numerical Methods in Engineering, 125 (2024), pp. e7508/1--e7508/28, DOI 10.1002/nme.7508 .
  Abstract
  This paper presents a comprehensive study of the intrinsic strain response of 3D het- erostructures arising from lattice mismatch. Combining materials with different lattice constants induces strain, leading to the bending of these heterostructures. We propose a model for nonlinear elastic heterostructures such as bimetallic beams or nanowires that takes into account local prestrain within each distinct material region. The resulting system of partial differential equations (PDEs) in Lagrangian coordinates incorporates a nonlinear strain and a linear stress-strain relationship governed by Hooke?s law. To validate our model, we apply it to bimetallic beams and hexagonal hetero-nanowires and perform numerical simulations using finite element methods (FEM). Our simulations ex- amine how these structures undergo bending under varying material compositions and cross-sectional geometries. In order to assess the fidelity of the model and the accuracy of simulations, we compare the calculated curvature with analytically derived formula- tions. We derive these analytical expressions through an energy-based approach as well as a kinetic framework, adeptly accounting for the lattice constant mismatch present at each compound material of the heterostructures. The outcomes of our study yield valuable insights into the behavior of strained bent heterostructures. This is particularly significant as the strain has the potential to influence the electronic band structure, piezoelectricity, and the dynamics of charge carriers.

• M. O'Donovan, P. Farrell, J. Moatti, T. Streckenbach, Th. Koprucki, S. Schulz, Impact of random alloy fluctuations on the carrier distribution in multi-color (In,Ga)N/GaN quantum well systems, Physical Review Applied, 21 (2024), pp. 024052/1--024052/12, DOI 10.1103/PhysRevApplied.21.024052 .
  Abstract
  In this work, we study the impact that random alloy fluctuations have on the distribution of electrons and holes across the active region of a (In,Ga)N/GaN multi-quantum well based light emitting diode (LED). To do so, an atomistic tight-binding model is employed to account for alloy fluctuations on a microscopic level and the resulting tight-binding energy landscape forms input to a drift-diffusion model. Here, quantum corrections are introduced via localization landscape theory and we show that when neglecting alloy disorder our theoretical framework yields results similar to commercial software packages that employ a self-consistent Schroedinger-Poisson-drift-diffusion solver. Similar to experimental studies in the literature, we have focused on a multi-quantum well system where two of the three wells have the same In content while the third well differs in In content. By changing the order of wells in this multicolor quantum well structure and looking at the relative radiative recombination rates of the different emitted wavelengths, we (i) gain insight into the distribution of carriers in such a system and (ii) can compare our findings to trends observed in experiment. Our results indicate that the distribution of carriers depends significantly on the treatment of the quantum well microstructure. When including random alloy fluctuations and quantum corrections in the simulations, the calculated trends in the relative radiative recombination rates as a function of the well ordering are consistent with previous experimental studies. The results from the widely employed virtual crystal approximation contradict the experimental data. Overall, our work highlights the importance of a careful and detailed theoretical description of the carrier transport in an (In,Ga)N/GaN multi-quantum well system to ultimately guide the design of the active region of III-N-based LED structures.

• R. Araya, A. Caiazzo, F. Chouly, Stokes problem with slip boundary conditions using stabilized finite elements combined with Nitsche, Computer Methods in Applied Mechanics and Engineering, 427 (2024), pp. 117037/1--117037/16, DOI 10.1016/j.cma.2024.117037 .
  Abstract
  We discuss how slip conditions for the Stokes equation can be handled using Nitsche method, for a stabilized finite element discretization. Emphasis is made on the interplay between stabilization and Nitsche terms. Well-posedness of the discrete problem and optimal convergence rates, in natural norm for the velocity and the pressure, are established, and illustrated with various numerical experiments. The proposed method fits naturally in the context of a finite element implementation while being accurate, and allows an increased flexibility in the choice of the finite element pairs.

• W. Lei, S. Piani, P. Farrell, N. Rotundo, L. Heltai, A weighted hybridizable discontinuous Galerkin method for drift-diffusion problems, Journal of Scientific Computing, 99 (2024), pp. 33/1--33/26, DOI 10.1007/s10915-024-02481-w .
  Abstract
  In this work we propose a weighted hybridizable discontinuous Galerkin method (W-HDG) for drift-diffusion problems. By using specific exponential weights when computing the L2 product in each cell of the discretization, we are able to mimic the behavior of the Slotboom variables, and eliminate the drift term from the local matrix contributions, while still solving the problem for the primal variables. We show that the proposed numerical scheme is well-posed, and validate numerically that it has the same properties as classical HDG methods, including optimal convergence, and superconvergence of postprocessed solutions. For polynomial degree zero, dimension one, and vanishing HDG stabilization parameter, W-HDG coincides with the Scharfetter--Gummel finite volume scheme (i.e., it produces the same system matrix). The use of local exponential weights generalizes the Scharfetter-Gummel scheme (the state-of-the-art for finite volume discretization of transport dominated problems) to arbitrary high order approximations.

• S. Piani, P. Farrell, W. Lei, N. Rotundo, L. Heltai, Data-driven solutions of ill-posed inverse problems arising from doping reconstruction in semiconductors, Applied Mathematics in Science and Engineering, 32 (2024), pp. 2323626/1--2323626/27, DOI 10.1080/27690911.2024.2323626 .
  Abstract
  The non-destructive estimation of doping concentrations in semiconductor devices is of paramount importance for many applications ranging from crystal growth, the recent redefinition of the 1kg to defect, and inhomogeneity detection. A number of technologies (such as LBIC, EBIC and LPS) have been developed which allow the detection of doping variations via photovoltaic effects. The idea is to illuminate the sample at several positions and detect the resulting voltage drop or current at the contacts. We model a general class of such photovoltaic technologies by ill-posed global and local inverse problems based on a drift-diffusion system that describes charge transport in a self-consistent electrical field. The doping profile is included as a parametric field. To numerically solve a physically relevant local inverse problem, we present three different data-driven approaches, based on least squares, multilayer perceptrons, and residual neural networks. Our data-driven methods reconstruct the doping profile for a given spatially varying voltage signal induced by a laser scan along the sample's surface. The methods are trained on synthetic data sets (pairs of discrete doping profiles and corresponding photovoltage signals at different illumination positions) which are generated by efficient physics-preserving finite volume solutions of the forward problem. While the linear least square method yields an average absolute l-infinity / displaystyle ell ^infty error around 10%, the nonlinear networks roughly halve this error to 5%, respectively. Finally, we optimize the relevant hyperparameters and test the robustness of our approach with respect to noise.

Preprints, Reports, Technical Reports

• Y. Hadjimichael, O. Brandt, Ch. Merdon, C. Manganelli, P. Farrell, Strain distribution in zincblende and wurtzite GaAs nanowires bent by a one-sided (In, Al)As shell: Consequences for torsion, chirality, and piezoelectricity, Preprint no. 3141, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3141 .
  Abstract, PDF (31 MByte)
  We present a finite-strain model that is capable of describing the large deformations in bent nanowire heterostructures. The model incorporates a nonlinear strain formulation derived from the first Piola-Kirchhoff stress tensor, coupled with an energy functional that effectively captures the lattice-mismatch-induced strain field. We use the finite element method to solve the resulting partial differential equations and extract cross- sectional maps of the full strain tensor for both zincblende and wurtzite nanowires with lattice-mismatched core and one-sided stressor shell. In either case, we show that the bending is essentially exclusively determined by $varepsilonzz$. However, the distinct difference in shear strain has important consequences with regard to both the mechanical deformation and the existence of transverse piezoelectric fields in the nanowires.

Talks, Poster

• Z. Elsayed Amer, Numerical methods for a coupled drift-diffusion and Helmholtz model for laser applications, MESIGA25: Numerical Methods in Applied Mathematics, March 11 - 13, 2025, Institut für Mathematik der Universität Potsdam, March 12, 2025.

• Y. Hadjimichael, Strain distribution in zincblende and wurtzite nanowires bent by a one-sided stressor shell, Leibniz MMS Days 2025, March 26 - 28, 2025, The Leibniz Research Network "Mathematical Modeling and Simulation", Leibniz Institute for Baltic Sea Research Warnemünde (IOW), March 27, 2025.

• P. Farrell, Numerische Methoden für innovative Halbleiterbauteile, Transfer Workshop: ErUM-Scientists and Industry in Dialogue, February 6 - 7, 2025, ErUM Data Hub, Aachen, February 6, 2025.

• Z. Amer, Numerical methods for coupled drift-diffusion and Helmholtz Models for laser applications, International Conference on Simulation of Organic Electronics and Photovoltaics, SimOEP, September 2 - 4, 2024, ZHAW - Zurich University of Applied Sciences, Winterthur, Switzerland, September 4, 2024.

• Z. Amer, Numerical methods for coupled drift-diffusion and Helmholtz models for laser applications, Leibniz MMS Days 2024, April 10 - 12, 2024, Leibniz Network "Mathematical Modeling and Simulation", Leibniz Institut für Verbundwerkstoffe GmbH (IVW), Kaiserslautern, April 11, 2024.

• Y. Hadjimichael, An energy-based finite-strain model for 3D heterostructured materials, Leibniz MMS Days 2024, April 10 - 12, 2024, Leibniz Network "Mathematical Modeling and Simulation", Leibniz Institut für Verbundwerkstoffe GmbH (IVW), Kaiserslautern, April 11, 2024.

• D. Abdel, Modeling and simulation of vacancy-assisted charge transport in innovative semiconductor devices, Applied Mathematics and Simulation for Semiconductor Devices (AMaSiS 2024), September 10 - 13, 2024, WIAS Berlin, September 11, 2024.

• M. Demir, Time filtered second order backward Euler method for EMAC formulation of Navier--Stokes equations, 20th Annual Workshop on Numerical Methods for Problems with Layer Phenomena, May 23 - 24, 2024, University of Cyprus, Department of Mathematics and Statistics, Protaras, Cyprus, May 24, 2024.

• P. Farrell, Charge transport in perovskites solar cells: Modeling, analysis and simulations, Inria-ECDF Partnership Kick-Off Workshop, June 5 - 7, 2024, Inria and the Einstein Center for Digital Future, Berlin, June 7, 2024.

• J. Fuhrmann, Development of numerical methods and tools for drift-diffusion simulations, Applied Mathematics and Simulation for Semiconductor Devices (AMaSiS 2024), Berlin, September 10 - 13, 2024.

• Y. Hadjimichael, Strain distribution in zincblende and wurtzite GaAs nanowires bent by a one-sided (In,Al)As shell, Applied Mathematics and Simulation for Semiconductor Devices (AMaSiS 2024), Berlin, September 10 - 13, 2024.

• V. John, Finite element methods respecting the discrete maximum principle for convection-diffusion equations, Trends in Scientific Computing - 30 Jahre Wissenschaftliches Rechnen in Dortmund, May 21 - 22, 2024, TU Dortmund, Fakultät für Mathematik, LSIII, May 21, 2024.

• CH. Merdon, Pressure-robustness in Navier--Stokes finite element simulations, 10th International Conference on Computational Methods in Applied Mathematics (CMAM-10), June 10 - 14, 2024, Universität Bonn, Institut für Numerische Simulation, June 11, 2024.

• CH. Merdon, Pressure-robustness in Navier--Stokes finite elements simulations, Wissenschaftlicher Beirat, WIAS Berlin, September 27, 2024.

• O. Pártl, Optimization of geothermal energy production from fracture-controlled reservoirs via 3D numerical modeling and simulation, General Assembly 2024 of the European Geosciences Union (EGU), April 14 - 19, 2024, European Geosciences Union (EGU), Wien, Austria, April 15, 2024, DOI 10.5194/egusphere-egu24-4164 .

• F. Romor, Efficient numerical resolution of parametric partial differential equations on solution manifolds parametrized by neural networks, 9th European Congress on Computational Methods in Applied Sciences and Engineering, June 3 - 7, 2024, ECCOMAS, scientific organization, Lissabon, Portugal, June 4, 2024.

External Preprints

• D. Abdel, M. Herda, M. Ziegler, C. Chainais-Hillairet, B. Spetzler, P. Farrell, Numerical analysis and simulation of lateral memristive devices: Schottky, ohmic, and multi-dimensional electrode models, Preprint no. 15065, Cornell University, 2024, DOI 10.48550/arXiv.2412.15065 .
  Abstract
  In this paper, we present the numerical analysis and simulations of a multi-dimensional memristive device model. Memristive devices and memtransistors based on two-dimensional (2D) materials have demonstrated promising potential as components for next-generation artificial intelligence (AI) hardware and information technology. Our charge transport model describes the drift-diffusion of electrons, holes, and ionic defects self-consistently in an electric field. We incorporate two types of boundary models: ohmic and Schottky contacts. The coupled drift-diffusion partial differential equations are discretized using a physics-preserving Voronoi finite volume method. It relies on an implicit time-stepping scheme and the excess chemical potential flux approximation. We demonstrate that the fully discrete nonlinear scheme is unconditionally stable, preserving the free-energy structure of the continuous system and ensuring the non-negativity of carrier densities. Novel discrete entropy-dissipation inequalities for both boundary condition types in multiple dimensions allow us to prove the existence of discrete solutions. We perform multi-dimensional simulations to understand the impact of electrode configurations and device geometries, focusing on the hysteresis behavior in lateral 2D memristive devices. Three electrode configurations - side, top, and mixed contacts - are compared numerically for different geometries and boundary conditions. These simulations reveal the conditions under which a simplified one-dimensional electrode geometry can well represent the three electrode configurations. This work lays the foundations for developing accurate, efficient simulation tools for 2D memristive devices and memtransistors, offering tools and guidelines for their design and optimization in future applications.

• G. Alì, P. Farrell, N. Rotundo, Forward lateral photovoltage scanning problem: Perturbation approach and existence-uniqueness analysis, Preprint no. arXiv:2404.10466, Cornell University, 2024, DOI 10.48550/arXiv.2404.10466 .
  Abstract
  In this paper, we present analytical results for the so-called forward lateral photovoltage scanning (LPS) problem. The (inverse) LPS model predicts doping variations in crystal by measuring the current leaving the crystal generated by a laser at various positions. The forward model consists of a set of nonlinear elliptic equations coupled with a measuring device modeled by a resistance. Standard methods to ensure the existence and uniqueness of the forward model cannot be used in a straightforward manner due to the presence of an additional generation term modeling the effect of the laser on the crystal. Hence, we scale the original forward LPS problem and employ a perturbation approach to derive the leading order system and the correction up to the second order in an appropriate small parameter. While these simplifications pose no issues from a physical standpoint, they enable us to demonstrate the analytic existence and uniqueness of solutions for the simplified system using standard arguments from elliptic theory adapted to the coupling with the measuring device.

• R. Araya, A. Caiazzo, F. Chouly, Stokes problem with slip boundary conditions using stabilized finite elements combined with Nitsche, Preprint no. arXiv:2404.08810, Cornell University, 2024, DOI 10.48550/arXiv.2404.08810 .
  Abstract
  We discuss how slip conditions for the Stokes equation can be handled using Nitsche method, for a stabilized finite element discretization. Emphasis is made on the interplay between stabilization and Nitsche terms. Well-posedness of the discrete problem and optimal convergence rates, in natural norm for the velocity and the pressure, are established, and illustrated with various numerical experiments. The proposed method fits naturally in the context of a finite element implementation while being accurate, and allows an increased flexibility in the choice of the finite element pairs.