Articles in Refereed Journals

  • 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 .
    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.

  • D. Abdel, P. Vágner, J. Fuhrmann, P. Farrell, Modelling charge transport in perovskite solar cells: Potential-based and limiting ion depletion, Electrochimica Acta, 390 (2021), pp. 138696/1--138696/12, DOI 10.1016/j.electacta.2021.138696 .
    From Maxwell--Stefan diffusion and general electrostatics, we derive a drift-diffusion model for charge transport in perovskite solar cells (PSCs) where any ion in the perovskite layer may flexibly be chosen to be mobile or immobile. Unlike other models in the literature, our model is based on quasi Fermi potentials instead of densities. This allows to easily include nonlinear diffusion (based on Fermi--Dirac, Gauss--Fermi or Blakemore statistics for example) as well as limit the ion depletion (via the Fermi--Dirac integral of order-1). The latter will be motivated by a grand-canonical formalism of ideal lattice gas. Furthermore, our model allows to use different statistics for different species. We discuss the thermodynamic equilibrium, electroneutrality as well as generation/recombination. Finally, we present numerical finite volume simulations to underline the importance of limiting ion depletion.

  • D. Abdel, P. Farrell, J. Fuhrmann, Assessing the quality of the excess chemical potential flux scheme for degenerate semiconductor device simulation, Optical and Quantum Electronics, 53 (2021), pp. 163/1--163/10.
    The van Roosbroeck system models current flows in (non-)degenerate semiconductor devices. Focusing on the stationary model, we compare the excess chemical potential discretization scheme, a flux approximation which is based on a modification of the drift term in the current densities, with another state-of-the-art Scharfetter-Gummel scheme, namely the diffusion-enhanced scheme. Physically, the diffusion-enhanced scheme can be interpreted as a flux approximation which modifies the thermal voltage. As a reference solution we consider an implicitly defined integral flux, using Blakemore statistics. The integral flux refers to the exact solution of a local two point boundary value problem for the continuous current density and can be interpreted as a generalized Scharfetter-Gummel scheme. All numerical discretization schemes can be used within a Voronoi finite volume method to simulate charge transport in (non-)degenerate semiconductor devices. The investigation includes the analysis of Taylor expansions, a derivation of error estimates and a visualization of errors in local flux approximations to extend previous discussions. Additionally, drift-diffusion simulations of a p-i-n device are performed.

  • S. Kayser, P. Farrell, N. Rotundo, Detecting striations via the lateral photovoltage scanning method without screening effect, Optical and Quantum Electronics, 53 (2021), pp. 288/1--288/10, DOI 10.1007/s11082-021-02911-1 .
    The lateral photovoltage scanning method (LPS) detects doping inhomogeneities in semiconductors such as Si, Ge and Si(x)Ge(1-x) in a cheap, fast and nondestructive manner. LPS relies on the bulk photovoltaic effect and thus can detect any physical quantity affecting the band profiles of the sample. LPS finite volume simulation using commercial software suffer from long simulation times and convergence instabilities. We present here an open-source finite volume simulation for a 2D Si sample using the ddfermi simulator. For low injection conditions we show that the LPS voltage is proportional to the doping gradient as previous theory suggested under certain conditions. For higher injection conditions we directly show how the LPS voltage and the doping gradient differ and link the physical effect of lower local resolution to the screening effect. Previously, the loss of local resolution was assumed to be only connected to the enlargement of the excess charge carrier distribution.

  • D. Chaudhuri, M. O'Donovan, T. Streckenbach, O. Marquardt, P. Farrell, S.K. Patra, Th. Koprucki, S. Schulz, Multiscale simulations of the electronic structure of III-nitride quantum wells with varied Indium content: Connecting atomistic and continuum-based models, Journal of Applied Physics, 129 (2021), pp. 073104/1 -- 073104/16 (published online on 18.02.2021), DOI 10.1063/5.0031514 .

  • O.C. Ernst, D. Uebel, S. Kayser, F. Lange, Th. Teubner, T. Boeck, Revealing all states of dewetting of a thin gold layer on a silicon surface by nanosecond laser conditioning, Applied Surface Science Advances, 3 (2021), pp. 100040/1--100040/10, DOI 10.1016/j.apsadv.2020.100040 .
    Dewetting is a ubiquitous phenomenon which can be applied to the laser synthesis of nanoparticles. A classical spinodal dewetting process takes place in four successive states, which differ from each other in their morphology. In this study all states are revealed by interaction of pulsed nanosecond UV laser light with thin gold layers with thicknesses between 1 nm and 10 nm on (100) silicon wafers. The specific morphologies of the dewetting states are discussed with particular emphasis on the state boundaries. The main parameter determining which state is formed is not the duration for which the gold remains liquid, but rather the input energy provided by the laser. This shows that each state transition has a separate measurable activation energy. The temperature during the nanosecond pulses and the duration during which the gold remains liquid was determined by simulation using the COMSOL Multiphysics software package. Using these calculations, an accurate local temperature profile and its development over time was simulated. An analytical study of the morphologies and formed structures was performed using Minkowski measures. With aid of this tool, the laser induced structures were compared with thermally annealed samples, with perfectly ordered structures and with perfectly random structures. The results show that both, structures of the laser induced and the annealed samples, strongly resemble the perfectly ordered structures. This reveals a close relationship between these structures and suggests that the phenomenon under investigation is indeed a spinodal dewetting generated by an internal material wave function.
    The purposeful generation of these structures and the elucidation of the underlying mechanism of dewetting by ultrashort pulse lasers may assist the realisation of various technical elements such as nanowires in science and industry.

  • S. Kayser, N. Rotundo, N. Dropka, P. Farrell, Assessing doping inhomogeneities in GaAs crystal via simulations of lateral photovoltage scanning method, Journal of Crystal Growth, 571 (2021), pp. 126248/1--126248/5, DOI .

  • M. O'Donovan, D. Chaudhuri, T. Streckenbach, P. Farrell, S. Schulz, Th. Koprucki, From atomistic tight-binding theory to macroscale driftdiffusion: Multiscale modeling and numerical simulation of uni-polar charge transport in (In,Ga)N devices with random fluctuations, Journal of Applied Physics, 130 (2021), pp. 065702/1--065702/13, DOI 10.1063/5.0059014 .

  • P. Farrell, S. Kayser, N. Rotundo, Modeling and simulation of the lateral photovoltage scanning method, Computers & Mathematics with Applications. An International Journal, 102 (2021), pp. 248--260, DOI 10.1016/j.camwa.2021.10.017 .
    The fast, cheap and nondestructive lateral photovoltage scanning (LPS) method detects inhomogeneities in semiconductors crystals. The goal of this paper is to model and simulate this technique for a given doping profile. Our model is based on the semiconductor device equations combined with a nonlinear boundary condition, modelling a volt meter. To validate our 2D and 3D finite volume simulations, we use theory developed by Tauc [21] to derive three analytical predictions which our simulation results corroborate, even for anisotropic 2D and 3D meshes. Our code runs about two orders of magnitudes faster than earlier implementations based on commercial software [15]. It also performs well for small doping concentrations which previously could not be simulated at all due to numerical instabilities. Our simulations provide experimentalists with reference laser powers for which meaningful voltages can still be measured. For higher laser power the screening effect does not allow this anymore.

Contributions to Collected Editions

  • S. Schulz, M. O'Donovan, D. Chaudhuri, S.K. Patra, P. Farrell, O. Marquardt, T. Streckenbach, Th. Koprucki, Connecting atomistic and continuum models for (In,Ga)N quantum wells: From tight-binding energy landscapes to electronic structure and carrier transport, in: 2021 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), IEEE, 2021, pp. 135--136, DOI 10.1109/NUSOD52207.2021.9541461 .
    We present a multi-scale framework for calculating electronic and transport properties of nitride-based devices. Here, an atomistic tight-binding model is connected with continuum-based electronic structure and transport models. In a first step, the electronic structure of (In,Ga)N quantum wells is analyzed and compared between atomistic and continuum-based approaches, showing that even though the two models operate on the same energy landscape, the obtained results differ noticeably; we briefly discuss approaches to improve the agreement between the two methods. Equipped with this information, uni-polar carrier transport is investigated. Our calculations reveal that both random alloy fluctuations and quantum corrections significantly impact the transport, consistent with previous literature results.

Preprints, Reports, Technical Reports

  • Y. Hadjimichael, D.I. Ketcheson, L. Lóczi, Positivity preservation of implicit discretizations of the advection equation, Preprint no. 2846, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2846 .
    Abstract, PDF (1121 kByte)
    We analyze, from the viewpoint of positivity preservation, certain discretizations of a fundamental partial differential equation, the one-dimensional advection equation with periodic boundary condition. The full discretization is obtained by coupling a finite difference spatial semidiscretization (the second- and some higher-order centered difference schemes, or the Fourier spectral collocation method) with an arbitrary θ-method in time (including the forward and backward Euler methods, and a second-order method by choosing  θ ∈ [0, 1] suitably). The full discretization generates a two-parameter family of circulant matrices M ∈ ℝ mxm , where each matrix entry is a rational function in θ and ν . Here, ν denotes the CFL number, being proportional to the ratio between the temporal and spatial discretization step sizes. The entrywise non-negativity of the matrix M---which is equivalent to the positivity preservation of the fully discrete scheme---is investigated via discrete Fourier analysis and also by solving some low-order parametric linear recursions. We find that positivity preservation of the fully discrete system is impossible if the number of spatial grid points m is even. However, it turns out that positivity preservation of the fully discrete system is recovered for odd values of m provided that θ ≥ 1/2 and ν are chosen suitably. These results are interesting since the systems of ordinary differential equations obtained via the spatial semi-discretizations studied are not positivity preserving.

Talks, Poster

  • D. Abdel, P. Vágner, J. Fuhrmann, P. Farrell, Modeling and simulation of charge transport in perovskite solar cells, AMaSiS 2021: Applied Mathematics and Simulation for Semiconductors and Electrochemical Systems (Online Event), September 6 - 9, 2021.

  • D. Abdel, P. Vágner, J. Fuhrmann, P. Farrell, Modelling charge transport in perovskite solar cells: Potential-based and limiting ion depletion, SIAM Conference on Computational Science and Engineering -- CSE21 (Online Event), Texas, USA, March 1 - 5, 2021.

  • D. Abdel, Modelling charge transport in perovskite solar cells: Potential-based and limiting ion depletion, SIAM Conference on Mathematical Aspects of Materials Science MS21 (Online Event), May 17 - 28, 2021, Basque Center for Applied Mathematics, Bilbao, Spain, May 27, 2021.

  • P. Farrell, Modelling and simulation of the lateral photovoltage scanning method (online talk), European Conference on Mathematics for Industry (ECMI2021), MSOEE: Mathematical modeling of charge transport in graphene and low dimensional structures (Online Event), April 13 - 15, 2021, Bergische Universität Wuppertal, Hungary, April 13, 2021.

  • P. Farrell, Y. Hadjimichael, Ch. Merdon, T. Streckenbach, Toward charge transport in bent nanowires, AMaSiS 2021: Applied Mathematics and Simulation for Semiconductors and Electrochemical Systems (Online Event), September 6 - 9, 2021.

  • P. Farrell, S. Kayser, N. Rotundo, Modeling and simulation of the lateral photovoltage scanning method, SIAM Conference on Computational Science and Engineering -- CSE21 (Online Event), Texas, USA, March 1 - 5, 2021.

  • P. Farrell, Finite volume and meshfree strategies to solve PDEs numerically (online talk), Heriot-Watt University, Edinburgh, Scotland, UK, June 2, 2021.

  • P. Farrell, Numerical methods for innovative semiconductor devices, Conference ''Asymptotic behaviors of systems of PDE arising in physics and biology: theoretical and numerical points of view'', November 16 - 19, 2021, Université Lille, Laboratoire Paul Painlevé, France, November 17, 2021.

  • P. Farrell, Numerical methods for innovative semiconductor devices, SISSA - International School for Advanced Studies, Trieste, Italy, December 9, 2021.

External Preprints

  • 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, Preprint no. arXiv:2111.01644, Cornell University Library,, 2021.
    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.