Search Results
Multiband k $cdot$ p model and fitting scheme for ab initio-based electronic structure parameters for wurtzite GaAs
We develop a 16-band k · p model for the description of wurtzite GaAs, together with a novel scheme to determine electronic structure parameters for multiband k · p models. Our approach uses low-discrepancy sequences to fit k · p band structures beyond the eight-band scheme to most recent ab initio data, obtained within the framework for hybrid-functional density functional theory with a screened-exchange hybrid functional. We report structural parameters, elastic constants, band structures along high-symmetry lines, and deformation potentials at the Γ point. Based on this, we compute the bulk electronic properties (Γ point energies, effective masses, Luttinger-like parameters, and optical matrix parameters) for a ten-band and a sixteen-band k · p model for wurtzite GaAs. Our fitting scheme can assign priorities to both selected bands and k points that are of particular interest for specific applications. Finally, ellipticity conditions can be taken into account within our fitting scheme in order to make the resulting parameter sets robust against spurious solutions.
Influence of the carrier reservoir dimensionality on electron-electron scattering in quantum dot materials
We calculated Coulomb scattering rates from quantum dots (QDs) coupled to a 2D carrier reservoir and QDs coupled to a 3D reservoir. For this purpose, we used a microscopic theory in the limit of Born-Markov approximation, in which the numerical evaluation of high dimensional integrals is done via a quasi-Monte Carlo method. Via a comparison of the so determined scattering rates, we investigated the question whether scattering from 2D is generally more efficient than scattering from 3D. In agreement with experimental findings, we did not observe a significant reduction of the scattering efficiency of a QD directly coupled to a 3D reservoir. In turn, we found that 3D scattering benefits from it’s additional degree of freedom in the momentum space
On the Darwin--Howie--Whelan equations for the scattering of fast electrons described by the Schrödinger equation
The Darwin-Howie-Whelan equations are commonly used to describe and simulate the scattering of fast electrons in transmission electron microscopy. They are a system of infinitely many envelope functions, derived from the Schrödinger equation. However, for the simulation of images only a finite set of envelope functions is used, leading to a system of ordinary differential equations in thickness direction of the specimen. We study the mathematical structure of this system and provide error estimates to evaluate the accuracy of special approximations, like the two-beam and the systematic-row approximation.
Beyond just ``flattening the curve'': Optimal control of epidemics with purely non-pharmaceutical interventions
When effective medical treatment and vaccination are not available, non-pharmaceutical interventions such as social distancing, home quarantine and far-reaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptible-exposed-infectious-recovered) model and continuous-time optimal control theory, we compute the optimal non-pharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet competing requirements: First, the minimization of disease-related deaths, and, second, the establishment of a sufficient degree of natural immunity at the end of the measures, in order to exclude a second wave. Moreover, the socio-economic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a single-intervention scenario that goes beyond heuristically motivated interventions and simple "flattening of the curve". Careful analysis of the computed control strategy reveals, however, that the obtained solution is in fact a tightrope walk close to the stability boundary of the system, where socio-economic costs and the risk of a new outbreak must be constantly balanced against one another. The model system is calibrated to reproduce the initial exponential growth phase of the COVID-19 pandemic in Germany.
On thermodynamic consistency of a Scharfetter-Gummel scheme based on a modified thermal voltage for drift-diffusion equations with diffusion enhancement
Driven by applications like organic semiconductors there is an increased interest in numerical simulations based on drift-diffusion models with arbitrary statistical distribution functions. This requires numerical schemes that preserve qualitative properties of the solutions, such as positivity of densities, dissipativity and consistency with thermodynamic equilibrium. An extension of the Scharfetter-Gummel scheme guaranteeing consistency with thermodynamic equilibrium is studied. It is derived by replacing the thermal voltage with an averaged diffusion enhancement for which we provide a new explicit formula. This approach avoids solving the costly local nonlinear equations defining the current for generalized Scharfetter-Gummel schemes.
Convergence of a finite volume scheme to the eigenvalues of a Schrödinger operator
We consider the approximation of a Schrödinger eigenvalue problem arising from the modeling of semiconductor nanostructures by a finite volume method in a bounded domain $OmegasubsetR^d$. In order to prove its convergence, a framework for finite dimensional approximations to inner products in the Sobolev space $H^1_0(Omega)$ is introduced which allows to apply well known results from spectral approximation theory. This approach is used to obtain convergence results for a classical finite volume scheme for isotropic problems based on two point fluxes, and for a finite volume scheme for anisotropic problems based on the consistent reconstruction of nodal fluxes. In both cases, for two- and three-dimensional domains we are able to prove first order convergence of the eigenvalues if the corresponding eigenfunctions belong to $H^2(Omega)$. The construction of admissible meshes for finite volume schemes using the Delaunay-Voronoï method is discussed. As numerical examples, a number of one-, two- and three-dimensional problems relevant to the modeling of semiconductor nanostructures is presented. In order to obtain analytical eigenvalues for these problems, a matching approach is used. To these eigenvalues, and to recently published highly accurate eigenvalues for the Laplacian in the L-shape domain, the results of the implemented numerical method are compared. In general, for piecewise $H^2$ regular eigenfunctions, second order convergence is observed experimentally.
Symmetries in TEM imaging of crystals with strain
TEM images of strained crystals often exhibit symmetries, the source of which is not always clear. To understand these symmetries we distinguish between symmetries that occur from the imaging process itself and symmetries of the inclusion that might affect the image. For the imaging process we prove mathematically that the intensities are invariant under specific transformations. A combination of these invariances with specific properties of the strain profile can then explain symmetries observed in TEM images. We demonstrate our approach to the study of symmetries in TEM images using selected examples in the field of semiconductor nanostructures such as quantum wells and quantum dots.
Feel the heat: Nonlinear electrothermal feedback in organic LEDs
For lighting applications, Organic light-emitting diodes (OLED) need much higher brightness than for displays, leading to self-heating. Due to the temperature-activated transport in organic semiconductors, this can result in brightness inhomogeneities and catastrophic failure. Here, we show that due to the strong electrothermal feedback of OLEDs, the common spatial current and voltage distribution is completely changed, requiring advanced device modeling and operation concepts. Our study clearly demonstrates the effect of negative differential resistance (NDR) in OLEDs induced by self-heating. As a consequence, for increasing voltage, regions with declining voltages are propagating through the device, and even more interestingly, a part of these regions show even decreasing currents, leading to strong local variation in luminance. The expected breakthrough of OLED lighting technology will require an improved price performance ratio, and the realization of modules with very high brightness but untainted appearance is considered to be an essential step into this direction. Thus, a deeper understanding of the control of electrothermal feedback will help to make OLEDs in lighting more competitive.
Mathematical models: A research data category?
Mathematical modeling and simulation (MMS) has now been established as an essential part of the scientific work in many disciplines and application areas. It is common to categorize the involved numerical data and to some extend the corresponding scientific software as research data. Both have their origin in mathematical models. In this contribution we propose a holistic approach to research data in MMS by including the mathematical models and discuss the initial requirements for a conceptual data model for this field.
3D electrothermal simulations of organic LEDs showing negative differential resistance
Organic semiconductor devices show a pronounced interplay between temperature-activated conductivity and self-heating which in particular causes inhomogeneities in the brightness of large-area OLEDs at high power. We consider a 3D thermistor model based on partial differential equations for the electrothermal behavior of organic devices and introduce an extension to multiple layers with nonlinear conductivity laws, which also take the diode-like behavior in recombination zones into account. We present a numerical simulation study for a red OLED using a finite-volume approximation of this model. The appearance of S-shaped current-voltage characteristics with regions of negative differential resistance in a measured device can be quantitatively reproduced. Furthermore, this simulation study reveals a propagation of spatial zones of negative differential resistance in the electron and hole transport layers toward the contact.