Articles in Refereed Journals

  • 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. Marquart, 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), 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.

  • 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 Probability, 130 (2021), pp. 065702/1--065702/13, DOI 10.1063/5.0059014 .

  • I. Selmer, P. Farrell, I. Smirnova, P. Gurikov, Comparison of finite difference method and finite volume method simulations for a mass transport model describing the supercritical drying kinetic of gel particles in a packed bed, Gels, 6 (2020), pp. 45/1--45/26, DOI 10.3390/gels6040045 .

  • D. Uebel, S. Kayser, T. Markurt, O.C. Ernst, Th. Teubner, T. Boeck, Fast Raman mapping and in situ TEM observation of metal induced crystallization of amorphous silicon, CrystEngComm, 22 (2020), pp. 7983--7991, DOI 10.1039/D0CE00960A .
    Crystalline silicon is grown onto an amorphous silicon (a-Si) seed layer from liquid tin solution (steady state liquid phase epitaxy, SSLPE). To investigate the crystallization of embedded a-Si during our process, we adapted Raman measurements for fast mapping, with dwell times of just one second per single measurement. A purposely developed imaging algorithm which performs point-by-point gauss fitting provides adequate visualization of the data. We produced scans of a-Si layers showing crystalline structures formed in the a-Si matrix during processing. Compared to scanning electron microscopy images which reveal merely the topography of the grown layer, new insights are gained into the role of the seed layer by Raman mapping. As part of a series of SSLPE experiments, which were interrupted at various stages of growth, we show that plate-like crystallites grow laterally over the a-Si layer while smaller, randomly orientated crystals arise from the a-Si layer. Results are confirmed by an in situ TEM experiment of the metal-induced crystallization. Contrary to presumptions, initially formed surface crystallites do not originate from the seed layer and are irrelevant to the final growth morphology, since they dissolve within minutes due to Ostwald ripening. The a-Si layer crystallizes within minutes as well, and crystallites of the final morphology originate from seeds of this layer.

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.

  • D. Abdel, P. Farrell, J. Fuhrmann, Comparison of Scharfetter--Gummel schemes for (non-)degenerate semiconductor device simulation, in: Proceedings of the 20th International Conference on Numerical Simulation of Optoelectronic Devices -- NUSOD 2020, J. Piprek, K. Hinzer, eds., IEEE Conference Publications Management Group, Piscataway, 2020, pp. 107--108.

  • S. Schulz, D. Chaudhuri, M. O'Donovan, S. Patra, T. Streckenbach, P. Farrell, O. Marquardt, Th. Koprucki, Multi-scale modeling of electronic, optical, and transport properties of III-N alloys and heterostructures, in: Physics and Simulation of Optoelectronic Devices XXVIII, B. Witzigmann, M. Osiński, Y. Arakawa, eds., 11274 of Proceedings of SPIE, San Francisco, 2020, pp. 1127416/1--1127416/10, DOI 10.1117/12.2551055 .

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.

  • B. Takács, Y. Hadjimichael, High order discretization methods for spatial-dependent SIR models, Preprint no. 2805, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2805 .
    Abstract, PDF (4433 kByte)
    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.

  • P. Farrell, S. Kayser, N. Rotundo, Modeling and simulation of the lateral photovoltage scanning method, Preprint no. 2784, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2784 .
    Abstract, PDF (3689 kByte)
    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.

  • D. Abdel, P. Vágner, J. Fuhrmann, P. Farrell, Modelling charge transport in perovskite solar cells: Potential-based and limiting ion depletion, Preprint no. 2780, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2780 .
    Abstract, PDF (1051 kByte)
    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.

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 vacancy depletion (online presentation), 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 (online presentation), 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.

  • D. Abdel, Comparison of Scharfetter--Gummel schemes for (non-)degenerate semiconductor device simulation (online talk), 20th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD) (Online Event), September 14 - 18, 2020, Turin, Italy, September 16, 2020.

  • P. Farrell, Numerical methods for innovative semiconductor devices (online talk), Technische Universität Dresden, Institut für Wissenschaftliches Rechnen, May 27, 2020.

  • S. Kayser, The lateral photovoltage scanning method (LPS): Understanding doping variations in silicon crystals (online talk), 20th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD) (Online Event), September 14 - 18, 2020, Turin, Italy, September 16, 2020.

  • N. Dropka , P. Farrell, S. Kayser, N. Rotundo, Numerics for innovative semiconductor devices -- An outlook, German Conference on Crystal Growth, Munich, March 11 - 13, 2020.

  • P. Farrell, D. Peschka, Challenges in drift-diffusion semiconductor simulations, Finite Volumes for Complex Applications IX (Online Event), Bergen, Norway, June 15 - 19, 2020.