Numerical Methods for Innovative Semiconductor Devices

Publications

Articles in Refereed Journals

• M. O'Donovan, P. Farrell, T. Streckenbach, Th. Koprucki, S. Schulz, Multiscale simulations of uni-polar hole transport in (In,Ga)N quantum well systems, Optical and Quantum Electronics, 54 (2022), pp. 405/1--405/23, DOI 10.1007/s11082-022-03752-2 .
  Abstract
  Understanding the impact of the alloy micro-structure on carrier transport becomes important when designing III-nitride-based LED structures. In this work, we study the impact of alloy fluctuations on the hole carrier transport in (In,Ga)N single and multi-quantum well systems. To disentangle hole transport from electron transport and carrier recombination processes, we focus our attention on uni-polar (p-i-p) systems. The calculations employ our recently established multi-scale simulation framework that connects atomistic tight-binding theory with a macroscale drift-diffusion model. In addition to alloy fluctuations, we pay special attention to the impact of quantum corrections on hole transport. Our calculations indicate that results from a virtual crystal approximation present an upper limit for the hole transport in a p-i-p structure in terms of the current-voltage characteristics. Thus we find that alloy fluctuations can have a detrimental effect on hole transport in (In,Ga)N quantum well systems, in contrast to uni-polar electron transport. However, our studies also reveal that the magnitude by which the random alloy results deviate from virtual crystal approximation data depends on several factors, e.g. how quantum corrections are treated in the transport calculations.

• B. Takács, Y. Hadjimichael, High order discretization methods for spatial-dependent epidemic models, Mathematics and Computers in Simulation, 198 (2022), pp. 211--236, DOI 10.1016/j.matcom.2022.02.021 .
  Abstract
  In this paper, an SIR model with spatial dependence is studied and results regarding its stability and numerical approximation are presented. We consider a generalization of the original Kermack and McKendrick model in which the size of the populations differs in space. The use of local spatial dependence yields a system of integro-differential equations. The uniqueness and qualitative properties of the continuous model are analyzed. Furthermore, different choices of spatial and temporal discretizations are employed, and step-size restrictions for population conservation, positivity, and monotonicity preservation of the discrete model are investigated. We provide sufficient conditions under which high order numerical schemes preserve the discrete properties of the model. Computational experiments verify the convergence and accuracy of the numerical methods.

Contributions to Collected Editions

• D. Abdel, N. Courtier, P. Farrell, Volume exclusion effects in perovskite charge transport modeling, in: 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2022), Turin, Italy, 2022, J. Piprek, P. Bardella, eds., IEEE, 2022, pp. 107--108, DOI 10.1109/NUSOD54938.2022.9894826 .

• Y. Hadjimichael, O. Marquardt, Ch. Merdon, P. Farrell, Band structures in highly strained 3D nanowires, in: 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2022), Turin, Italy, 2022, J. Piprek, Bardella Paolo, eds., IEEE, 2022, pp. 119--120, DOI 10.1109/NUSOD54938.2022.9894837 .

• S. Piani, W. Lei, L. Heltai, N. Rotundo, P. Farrell, Data-driven doping reconstruction, in: 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2022), Turin, Italy, 2022, J. Piprek, P. Bardella, eds., IEEE, 2022, pp. 109--110, DOI 10.1109/NUSOD54938.2022.9894774 .

• J. Moatti, P. Farrell, Comparison of flux discretizations for varying band-edge energies, in: 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2022), Turin, Italy, 2022, J. Piprek, P. Bardella, eds., IEEE, 2022, pp. 103--104, DOI 10.1109/NUSOD54938.2022.9894742 .

• M. O'Donovan, P. Farrell, T. Streckenbach, Th. Koprucki, S. Schulz, Carrier transport in (In,Ga)N quantum well systems: Connecting atomistic tight-binding electronic structure theory to drift-diffusion simulations, in: 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), Turin, Italy, 2022, J. Piprek, P. Bardella, eds., IEEE, 2022, pp. 97--98, DOI 10.1109/NUSOD54938.2022.9894745 .

Preprints, Reports, Technical Reports

• D. Abdel, N.E. Courtier, P. Farrell, Volume exclusion effects in perovskite charge transport modeling, Preprint no. 2965, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2965 .
  Abstract, PDF (863 kByte)
  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, Preprint no. 2958, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2958 .
  Abstract, PDF (855 kByte)
  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.

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.

• D. Abdel, 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.

• D. Abdel, 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.

• CH. Merdon, Raviart--Thomas enriched Scott--Vogelius FEM for the Navier--Stokes equations, European Finite Element Fair 2023 (EFEF2023), Department of Mathematics University of Twente (DAMUT), Enschede, Netherlands, May 10, 2023.

• D. Abdel, Volume exclusion effects in perovskite charge transport modeling (online talk), 22th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD) (Online Event, September 12 - 16, 2022, Politecnico di Torino, Italy.

• Y. Hadjimichael, O. Marquardt, Ch. Merdon, P. Farrell, Band structures in highly strained 3D nanowires, 22th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD) (Online Event), Italy, September 12 - 16, 2022.

• P. Farrell, Challenges for non-Boltzmann drift-diffusion charge transport simulations in semiconductors, International Conference on Boundary and Interior Layers, November 28 - December 2, 2022, Universidad de Buenos Aires, Argentina, December 1, 2022.

• P. Farrell, Data-driven solutions of ill-posed inverse problems arising from doping reconstruction in semiconductors, AI4Science Konferenz, Rabat, Morocco, December 12 - 16, 2022.

• P. Farrell, Numerical methods for innovative semiconductors devices, Leibniz-Institut für Innovative Mikroelektronik (IHP), July 26, 2022.

• D. Fritsch, Identifying and efficiently computing band-edge energies for charge transport simulations in strained materials, MATH+ Day 2022, Berlin, November 18, 2022.

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.

• 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, Preprint no. arXiv.2209.11657, Cornell University, 2022, DOI 10.48550/arXiv.2209.11657 .
  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.

• S. Piani, P. Farrell, W. Lei, N. Rotundo, L. Heltai, A weighted hybridizable discontinuous Galerkin method for drift-diffusion problems, Preprint no. arXiv:2211.02508, Cornell University, 2022, DOI 10.48550/arXiv.2211.02508 .

• S. Piani, P. Farrell, W. Lei, N. Rotundo, L. Heltai, Data-driven solutions of ill-posed inverse problems arising from doping reconstruction in semiconductors, Preprint no. arXiv:2208.00742, Cornell University, 2022, DOI 10.48550/arXiv.2208.00742 .