Bergische Universität Wuppertal
Fachbereich Mathematik und Naturwissenschaften
Angewandte Mathematik - Numerische Analysis


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Multi-Band Effective Mass Approximations

Advanced Mathematical Models and Numerical Techniques


Matthias Ehrhardt (Bergische Universität Wuppertal)

Thomas Koprucki (WIAS Berlin)

LNCSE, Vol. 94, Springer Verlag, Heidelberg, 2014, 310 pages

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Subject: The operation principle of modern semiconductor nano structures, such as quantum wells, quantum wires or quantum dots, relies on quantum mechanical effects. The goal of numerical simulations using quantum mechanical models in the development of semiconductor nano structures is threefold: First they are needed for a deeper understanding of experimental data and of the operation principle. Secondly, is to predict and optimize in advance qualitative and quantitative properties of new devices in order to minimize the number of prototypes needed. Semiconductor nano structures are embedded as an active region in semiconductor devices. Finally, the results of quantum mechanical simulations of semiconductor nano structures can be used by upscaling methods to deliver parameters needed in semi-classical models for semiconductor devices such as quantum well lasers. This book covers in detail all these three aspects using a variety of illustrating examples.

Purpose: Multi-band Effective Mass Approximations have been increasingly attracting interest over the last decades, since it is an essential tool for effective models in semiconductor materials. This book is concerned with several mathematical models from the most relevant class of kp-Schrödinger Systems We will present both mathematical models and state-of-the-art numerical methods to solve adequately the arising systems of differential equations.
The designated audience is graduate and Ph.D. students of mathematical physics, theoretical physics and people working in quantum mechanical research or semiconductor / opto-electronic industry that are interested in new mathematical aspects.

The book is designed to be consisting of a collection of contributed chapters. Outstanding experts working successfully in this challenging research area will be invited to contribute each a chapter of roughly 30-40 pages to this volume.

Academic Level: The principal audience is graduate and Ph.D. students of (mathematical) physics: research Lecturer of mathematical physics: teaching, research people working in semiconductor, opto-electronic industry: professional reference


by Bernd Witzigmann, Computational Electronics and Photonics, University of Kassel, Germany.


Part I: Physical Models

  Kinetic and Hydrodynamic Models for Multi-band and Quantum Transport in Crystals (51 pages)
     by Luigi Barletti, Dipartimento di Matematica, University of Florence, Italy
     and Giovanni Frosali, Dipartimento di Sistemi e Informatica, University of Florence, Italy
     and Omar Morandi, Institute of Theoretical and Computational Physics, Graz University of Technology, Austria

Abstract: This chapter is devoted to the derivation of k.p multi-band quantum transport models, in both the pure-state and mixed-state cases.

The first part of the chapter deals with pure-states. Transport models are derived from the crystal periodic Hamiltonian by assuming that the lattice constant is small, so that an effective multi-band Schrödinger equation can be written for the envelopes of the wave functions of the charge carriers. Two principal approaches are presented here: one is based on the Wannier-Slater envelope functions and the other on the Luttinger-Kohn envelope functions. The concept of Wannier functions is then generalized, in order to study the dynamics of carriers in crystals with varying composition (heterostructures). Some of the most common approximations, like the single band, mini-bands and semi-classical transport, are derived as a limit of multi-band models.

In the second part of the chapter, the mixed-state (i.e. statistical) case is considered. In particular, the phase-space point of view, based on Wigner function, is adopted, which provides a quasi-classical description of the quantum dynamics. After a theoretical introduction to the Wigner-Weyl theory, a two-band phase-space transport model is developed, as an example of application of the Wigner formalism to the k.p framework.

The third part of the chapter is devoted to quantum-fluid models, which are formulated in terms of a finite number of macroscopic moments of the Wigner function. For mixed-states, the maximum-entropy closure of the moment equations is discussed in general terms. Then, details are given on the multi-band case, where "multi-band" is to be understood in the wider sense of "multi-component wave function", including therefore the case of particles with spin or spin-like degrees of freedom. Three instances of such systems, namely the two-band k.p model, the Rashba spin-orbit system and the graphene sheet, are examined.

  Electronic Properties of III-V Quantum Dots : Impact of Crystal Symmetry, Substrate Orientation and Band-Alignment (26 pages )
     by Andrei Schliwa, Gerald Hönig and Dieter Bimberg, Institute of Solid State Physics, Technical University Berlin, Germany.

Abstract: Electronic properties of quantum dots are reviewed based on eightband k.p theory. We will focus on the following interrelated subjects: First the role of crystallographic symmetry is evaluated. This includes the symmetry of the lattice of the substrate [wurtzite (wz) versus zinc blende (zb)] as well as different substrate orientations [zb-(001) versus zb-(111)]. Second, we discuss two different types of band alignment, type-I versus type-II, by comparing the common-anion system zb-InAs/GaAs to the common-cation system zb-GaSb/GaAs. Finally, the impact of large built-in fields resulting from piezo- and pyroelectric charges will be exemplified for the wz-GaN/A1N QD-system.

  Symmetries in multi-band Hamiltonians for semiconductor quantum dots ( ... pages )
     by Stanko Tomic, Joule Physics Laboratory, School of Computing, Science and Engineering, University of Salford, Salford M5 4WT, United Kingdom
     and Nenad Vukmirović, Scientific Computing Laboratory, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia.

Abstract: In contrast to a popular belief that multi-band envelope function k.p Hamiltonians cannot capture the right symmetry of QDs, we showed here the opposite. We showed that the inclusion of interface band mixing effects leads to the reduction of symmetry from an artificial C4v, to the correct C2v. The main manifestation of interface effects are the energy level splittings between (e1, e2), (h0, h1), and (h4, h5) states of the order of 1-3 meV in InAs/GaAs material system.
The inclusion of the additional bands beyond the standard 8 bands also leads to symmetry reduction to C2v. We have found that that the lowest order multi-band Hamiltonian whose kinetic part has the correct C2v symmetry is the 14-band k.p Hamiltonian.
This symmetry reduction originates from the coupling between the top of the valence (C5v) and the second conduction (C5c) band. The observed splittings are comparable to the ones that originate from spin-orbit coupling (these do not reduce the symmetry) and are much smaller than the ones from piezoelectric effect in strained systems.
Deriving appropriate formulas for Fourier transforms using properties of the Parseval's theorem, the expressions that enable the efficient evaluation of Coulomb integrals in inverse space without the introduction of artificial electrostatic interactions with surrounding dots were presented.
It was also shown how symmetry can be exploited to further reduce the computational effort using irreducible representation. Numerical results illustrating the application of the methods to the calculation of single-particle states, as well as the configuration interaction calculation of exciton, biexciton, and negative trion states in zing-blend and wurtzite quantum dots were given. Due to correct symmetries of the single electron states out model can capture correctly few electron phenomena like fine structure splitting's (FSS) or distinguish between conditions for the bi-excitons being bound or unbound.
Our work provides a very important conceptual message: With appropriate treatment of relevant effects, the multi-band envelope function Hamiltonians is fully capable of capturing the right atomistic symmetry of QD structures.

Part II: Numerical Methods

  Finite Elements for k.p Multi-band Envelope Equations (26 pages)
     by Ratko Veprek, ETH Zürich, Switzerland
     and Sebastian Steiger, Purdue University, West Lafayette, USA.

Abstract: This chapter applies the finite element method to the kp equations of semiconductor nanostructures. It highlights advantages over other discretization methods and discusses the crucial ingredients in order to obtain accurate, physically correct results. One particular issue, the appearance of unphysical or spurious solutions, is demonstrated to arise from the continuum equation system, not the discretization, and two causes are identified whose correct treatment leads to the elimination of such solutions.

  Plane-Wave Approaches to the Electronic Structure of Semiconductor Nanostructures (35 pages )
     by Eoin P. O'Reilly, Tyndall National Institute, Ireland
     and Oliver Marquardt, Tyndall National Institute, Ireland
     and Stefan Schulz, Tyndall National Institute, Ireland
     and Aleksey Andreev, Hitachi Cambridge Laboratory, United Kingdom.

Abstract: This chapter is dedicated to different plane-wave based approaches to calculate the electronic structure of semiconductor nanostructures. We introduce semi-analytical and numerical methods to achieve a plane-wave based description of such systems. This includes use of plane-wave methods to calculate not just the electronic structure but also the built-in strain and the polarisation potential, with the strain and the polarisation potential each having a significant influence on the electronic properties of a semiconductor nanostructure. The advantages and disadvantages of different plane wave based formulations in comparison to a real-space, finite element model will be discussed and we will present representative examples of semiconductor nanostructures together with their elastic and electronic properties, computed from semi-analytical and numerical approaches. We conclude that plane-wave-based methods provide an efficient and flexible approach when using k.p models to determine the electronic structure of semiconductor nanostructures.

Part III: Applications

  The Multi-band k.p Hamiltonian for Heterostructures: Parameters and Applications (50 pages)
     by Stefan Birner, Walter Schottky Institute, Technische Universität München, Germany / nextnano3.

Abstract: In this chapter all the various definitions of the k.p parameters available in the literature are summarized, and equations that relate them to each other are provided. We believe that such a summary for both zinc blende and wurtzite crystals on a few pages is very useful, not only for beginners but also for experienced researchers that quickly want to look up conversion formulas. Results of k.p calculations for bulk semiconductors are shown for diamond, and for unstrained and strained InAs. Several examples of k.p calculations for heterostructures are presented. They cover spurious solutions, a spherical quantum dot and heterostructures showing the untypical type-II and type-III band alignments. Finally, self-consistent k.p calculations of a two-dimensional hole gas in diamond for different substrate orientations are analyzed. Wherever possible, the k.p results are compared to tight-binding calculations. All these calculations have been performed using the nextnano software. Therefore, this contribution provides some specific details that are relevant for a numerical implementation of the k.p method.

Part IV: Advanced Mathematical Topics

  Transient Simulation of kp-Schrödinger Systems using Discrete Transparent Boundary Conditions (24 pages)
     by Andrea Zisowsky, Institute of Mathematics, Technische Universität Berlin, Berlin, Germany
     and Anton Arnold, Institute for Analysis und Scientific Computing, Vienna University of Technology, Austria
     and Matthias Ehrhardt, Bergische Universität Wuppertal, Wuppertal, Germany
     and Thomas Koprucki, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany.

Abstract: This chapter deals with with transparent boundary conditions (TBCs) for systems of Schrödinger type equations, namely the time-dependent k.p-Schrödinger equations. These TBCs have to be constructed for the discrete scheme, in order to maintain stability and to avoid numerical reflections. The discrete transparent boundary conditions (DTBCs) are constructed using the solution of the exterior problem with Laplace and Z-transformation respectively. Hence we will analyse the numerical error caused by the inverse Z-transformation. Since these DTBCs are non-local in time and thus very costly, we present approximate DTBCs, that allow a fast calculation of the boundary terms.

  Discrete Transparent Boundary Conditions for Multi-band Effective Mass Approximations (45 pages)
     by Dirk Klindworth, Institute of Mathematics, Technische Universität Berlin, MA 6-4, Germany
     and Matthias Ehrhardt, Bergische Universität Wuppertal, Wuppertal, Germany
     and Thomas Koprucki, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany.

Abstract: This chapter is concerned with the derivation and numerical testing of discrete transparent boundary conditions (DTBCs) for stationary multi-band effective mass approximations (MEMAs). We analyze the continuous problem and introduce transparent boundary conditions (TBCs). The discretization of the differential equations is done with the help of finite difference schemes. A fully discrete approach is used in order to develop DTBCs that are completely reflectionfree. The analytical and discrete dispersion relations are analyzed in depth and the limitations of the numerical computations are shown. We extend the results of earlier works on DTBCs for the scalar Schrödinger equation by alternative finite difference schemes. The introduced schemes and their corresponding DTBCs are tested numerically on an example with single barrier potential. The d-band k.p-model is introduced as most general MEMA. We derive DTBCs for the d-band k.p-model and test our results on a quantum well nanostructure.

University of Wuppertal
Faculty of Mathematics and Natural Sciences
Department of Mathematics
Applied Mathematics & Numerical Analysis Group

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