Publications

Articles in Refereed Journals

  • 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.

  • 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.

  • D. Abdel, N.E. Courtier, P. Farrell, Volume exclusion effects in perovskite charge transport modeling, Optical and Quantum Electronics, 55 (2023), pp. 884/1--884/14, DOI 10.1007/s11082-023-05125-9 .
    Abstract
    Due to their flexible material properties, perovskite materials are a promising candidate for many semiconductor devices such as lasers, memristors, LEDs and solar cells. For example, perovskite-based solar cells have recently become one of the fastest growing photovoltaic technologies. Unfortunately, perovskite devices are far from commercialization due to challenges such as fast degradation. Mathematical models can be used as tools to explain the behavior of such devices, for example drift-diffusion equations portray the ionic and electric motion in perovskites. In this work, we take volume exclusion effects on ion migration within a perovskite crystal lattice into account. This results in the formulation of two different ionic current densities for such a drift-diffusion model -- treating either the mobility or the diffusivity as density-dependent while the other quantity remains constant. The influence of incorporating each current density description into a model for a typical perovskite solar cell configuration is investigated numerically, through simulations performed using two different open source tools.

  • D. Abdel, C. Chainais-Hillairet, P. Farrell, M. Herda, Numerical analysis of a finite volume scheme for charge transport in perovskite solar cells, IMA Journal of Numerical Analysis, published online on 10.6.2023, DOI 10.1093/imanum/drad034 .
    Abstract
    In this paper, we consider a drift-diffusion charge transport model for perovskite solar cells, where electrons and holes may diffuse linearly (Boltzmann approximation) or nonlinearly (e.g. due to Fermi-Dirac statistics). To incorporate volume exclusion effects, we rely on the Fermi-Dirac integral of order −1 when modeling moving anionic vacancies within the perovskite layer which is sandwiched between electron and hole transport layers. After non-dimensionalization, we first prove a continuous entropy-dissipation inequality for the model. Then, we formulate a corresponding two-point flux finite volume scheme on Voronoi meshes and show an analogous discrete entropy-dissipation inequality. This inequality helps us to show the existence of a discrete solution of the nonlinear discrete system with the help of a corollary of Brouwer's fixed point theorem and the minimization of a convex functional. Finally, we verify our theoretically proven properties numerically, simulate a realistic device setup and show exponential decay in time with respect to the L2 error as well as a physically and analytically meaningful relative entropy.

  • R. Finn, M. O'Donovan, P. Farrell, J. Moatti, T. Streckenbach, Th. Koprucki, S. Schulz, Theoretical study of the impact of alloy disorder on carrier transport and recombination processes in deep UV (Al,Ga)N light emitters, Applied Physics Letters, 122 (2023), pp. 241104/1--241104/7, DOI 10.1063/5.0148168 .
    Abstract
    Aluminum gallium nitride [(Al,Ga)N] has gained significant attention in recent years due to its potential for highly efficient light emitters operating in the deep ultra-violet (UV) range (<280 nm). However, given that current devices exhibit extremely low efficiencies, understand- ing the fundamental properties of (Al,Ga)N-based systems is of key importance. Here, using a multi-scale simulation framework, we study the impact of alloy disorder on carrier transport, radiative and non-radiative recombination processes in a c-plane Al 0.7 Ga0.3 N/Al0.8 Ga0.2 N quantum well embedded in a p-n junction. Our calculations reveal that alloy fluctuations can open "percolative" pathways that promote transport for the electrons and holes into the quantum well region. Such an effect is neglected in conventional and widely used transport sim- ulations. Moreover, we find that the resulting increased carrier density and alloy induced carrier localization effects significantly increase non-radiative Auger--Meitner recombination in comparison to the radiative process. Thus, to suppress such non-radiative process and poten- tially related material degradation, a careful design (wider well, multi-quantum wells) of the active region is required to improve the effi- ciency of deep UV light emitters.

  • B. Spetzler, D. Abdel, F. Schwierz, M. Ziegler, P. Farrell, The role of vacancy dynamics in two-dimensional memristive devices, Advanced Electronic Materials, published online on 08.11.2023, DOI 10.1002/aelm.202300635 .
    Abstract
    Two-dimensional (2D) layered transition metal dichalcogenides (TMDCs) are promising memristive materials for neuromorphic computing systems as they could solve the problem of the excessively high energy consumption of conventional von Neumann computer architectures. Despite extensive experimental work, the underlying switching mechanisms are still not understood, impeding progress in material and device functionality. This study reveals the dominant role of mobile defects in the switching dynamics of 2D TMDC materials. The switching process is governed by the formation and annihilation dynamics of a local vacancy depletion zone. Moreover, minor changes in the interface potential barriers cause fundamentally different device behavior previously thought to originate from multiple mechanisms. The key mechanisms are identified with a charge transport model for electrons, holes, and ionic point defects, including image-charge-induced Schottky barrier lowering (SBL). The model is validated by comparing simulations to measurements for various 2D MoS2-based devices, strongly corroborating the relevance of vacancies in TMDC devices and offering a new perspective on the switching mechanisms. The insights gained from this study can be used to extend the functional behavior of 2D TMDC memristive devices in future neuromorphic computing applications.

  • P. Farrell, J. Moatti, M. O'Donovan, S. Schulz, Th. Koprucki, Importance of satisfying thermodynamic consistency in optoelectronic device simulations for high carrier densities, Optical and Quantum Electronics, 55 (2023), pp. 978/1--978/12, DOI 10.1007/s11082-023-05234-5 .
    Abstract
    We show the importance of using a thermodynamically consistent flux discretization when describing drift-diffusion processes within light emitting diode simulations. Using the classical Scharfetter--Gummel scheme with Fermi--Dirac statistics is an example of such an inconsistent scheme. In this case, for an (In,Ga)N multi quantum well device, the Fermi levels show steep gradients on one side of the quantum wells which are not to be expected. This result originates from neglecting diffusion enhancement associated with Fermi--Dirac statistics in the numerical flux approximation. For a thermodynamically consistent scheme, such as the SEDAN scheme, the spikes in the Fermi levels disappear. We will show that thermodynamic inconsistency has far reaching implications on the current-voltage curves and recombination rates.

Contributions to Collected Editions

  • R. Finn, M. O'Donovan, P. Farrell, T. Streckenbach, J. Moatti, Th. Koprucki, S. Schulz, Theoretical investigation of carrier transport and recombination processes for deep UV (Al,Ga)N light emitters, in: 23nd International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2023), P. Bardella, A. Tibaldi, eds., IEEE, 2023, pp. 83--84, DOI 10.1109/NUSOD59562.2023.10273485 .
    Abstract
    We present a theoretical study on the impact of alloy disorder on carrier transport and recombination rates in an (Al,Ga)N single quantum well based LED operating in the deep UV spectral range. Our calculations indicate that alloy fluctuations enable percolative pathways which can result in improved carrier injection into the well, but may also increase carrier leakage from the well. Additionally, we find that alloy disorder induces carrier localization effects, a feature particularly noticeable for the holes. These localization effects can lead to locally increased carrier densities when compared to a virtual crystal approximation which neglects alloy disorder. We observe that both radiative and non-radiative recombination rates are increased. Our calculations also indicate that Auger--Meitner recombination increases faster than the radiative rate, based on a comparison with a virtual crystal approximation.

Preprints, Reports, Technical Reports

  • D. Abdel, A. Glitzky, M. Liero, Analysis of a drift-diffusion model for perovskite solar cells, Preprint no. 3073, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3073 .
    Abstract, PDF (541 kByte)
    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, Preprint no. 3064, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3064 .
    Abstract, PDF (6517 kByte)
    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.

Talks, Poster

  • D. Abdel, N.E. Courtier, P. Farrell, Modelling and simulation of charge transport in Perovskite solar cells, SIAM Conference on Computational Science and Engineering (CSE23), Amsterdam, Netherlands, February 26 - March 3, 2023.

  • Y. Hadjimichael, Efficient implementation of implicit Runge--Kutta methods with downwind-biased operators., SIAM Conference on Computational Science and Engineering (CSE23), Minisymposium MS419 ``Structure-Preserving Time-Stepping Methods for Differential Equations", February 26 - March 3, 2023, Society for Industrial and Applied Mathematic, Amsterdam, Netherlands, March 3, 2023.

  • P. Farrell, Charge transport in Perovskites devices: modeling, numerical analysis and simulations, Workshop on Applied Mathematics: Quantum and Classical Models, Università degli Studi di Firenze, Dipartimento di Matematica e Informatica 'Ulisse Dini', Italy, November 29, 2023.

  • P. Farrell, Modeling and numerical simulation of two-dimensional TMDC memristive devices, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Tokyo, Japan, August 20 - 25, 2023.

  • P. Farrell, Modeling and numerical simulation of two-dimensional memristive devices, 22nd European Consortium for Mathematics in Industry (ECMI) Conference on Industrial and Applied Mathematics, June 26 - 30, 2023, Wrocław University of Science and Technology, Poland.

  • P. Farrell, Device physics characterization and interpretation in perovskite and organic materials (DEPERO), October 3 - 5, 2023, Eidgenössische Technische Hochshcule Zürich, nanoGe, Switzerland.

  • CH. Merdon, Raviart--Thomas enriched Scott--Vogelius FEM for the Navier--Stokes equations, Capita Selecta Seminar, SACS - Systems, Analysis and Computational Sciences, Department of Mathematics University of Twente (DAMUT), Enschede, Netherlands, May 10, 2023.

External Preprints

  • R. Finn, M. O'Donovan, P. Farrell, J. Moatti, T. Streckenbach, Th. Koprucki, S. Schulz, Theoretical study of the impact of alloy disorder on carrier transport and recombination processes in deep UV (Al, Ga)N light emitters, Preprint no. hal-04037215, Hyper Articles en Ligne (HAL), 2023.
    Abstract
    Aluminium gallium nitride ((Al,Ga)N) has gained significant attention in recent years due to its potential for highly efficient light emitters operating in the deep ultra-violet (UV) range (< 280 nm). However, given that current devices exhibit extremely low efficiencies, understanding the fundamental properties of (Al,Ga)N-based systems is of key importance. Here, using a multi-scale simulation framework, we study the impact of alloy disorder on carrier transport, radiative and non-radiative recombination processes in a c-plane Al0.7Ga0.3N/Al0.8Ga0.2N quantum well embedded in a p-i-n junction. Our calculations reveal that alloy fluctuations can open "percolative" pathways that promote transport for the electrons and holes into the quantum well region. Such an effect is neglected in conventional, and widely used transport simulations. Moreover, we find also that the resulting increased carrier density and alloy induced carrier localization effects significantly increase non-radiative Auger-Meitner recombination in comparison to the radiative process. Thus, to avoid such non-radiative process and potentially related material degradation, a careful design (wider well, multi quantum wells) of the active region is required to improve the efficiency of deep UV light emitters.

  • B. Spetzler, D. Abdel, F. Schwierz, M. Ziegler, P. Farrell, The role of mobile point defects in two-dimensional memristive devices, Preprint no. arXiv:2304.06527, Cornell University, 2023, DOI 10.48550/arXiv.2304.06527 .
    Abstract
    Two-dimensional (2D) layered transition metal dichalcogenides (TMDCs) are promising memristive materials for neuromorphic computing systems as they could solve the problem of the excessively high energy consumption of conventional von Neumann computer architectures. Despite extensive experimental work, the underlying switching mechanisms are still not understood, impeding progress in material and device functionality. This study reveals the dominant role of mobile defects in the switching dynamics of 2D TMDC materials. The switching process is governed by the formation and annihilation dynamics of a local vacancy depletion zone. Moreover, minor changes in the interface potential barriers cause fundamentally different device behavior previously thought to originate from multiple mechanisms. The key mechanisms are identified with a charge transport model for electrons, holes, and ionic point defects, including image-charge-induced Schottky barrier lowering (SBL). The model is validated by comparing simulations to measurements for various 2D MoS2-based devices, strongly corroborating the relevance of vacancies in TMDC devices and offering a new perspective on the switching mechanisms. The insights gained from this study can be used to extend the functional behavior of 2D TMDC memristive devices in future neuromorphic computing applications.

  • P. Farrell, J. Moatti, M. O'Donovan, S. Schulz, Th. Koprucki, Importance of satisfying thermodynamic consistency in light emitting diode simulations, Preprint no. hal-04012467, Hyper Articles en Ligne (HAL), 2023.
    Abstract
    We show the importance of using a thermodynamically consistent flux discretization when describing drift-diffusion processes within light emitting diode simulations. Using the classical Scharfetter-Gummel scheme with Fermi-Dirac statistics is an example of such an inconsistent scheme. In this case, for an (In,Ga)N multi quantum well device, the Fermi levels show steep gradients on one side of the quantum wells which are not to be expected. This result originates from neglecting diffusion enhancement associated with Fermi-Dirac statistics in the numerical flux approximation. For a thermodynamically consistent scheme, such as the SEDAN scheme, the spikes in the Fermi levels disappear. We will show that thermodynamic inconsistency has far reaching implications on the current-voltage curves and recombination rates.