Bergische Universität Wuppertal


MultiBand Effective Mass ApproximationsAdvanced Mathematical Models and Numerical TechniquesEditorsMatthias Ehrhardt (Bergische Universität Wuppertal)Thomas Koprucki (WIAS Berlin)LNCSE, Vol. 94, Springer Verlag, Heidelberg, 2014, 310 pages 
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 semiclassical 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:
Multiband 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 kpSchrödinger Systems
We will present both mathematical models and stateoftheart 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 / optoelectronic 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 3040 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, optoelectronic industry: professional reference
by Bernd Witzigmann, Computational Electronics and Photonics, University of Kassel, Germany.
Kinetic and Hydrodynamic Models for Multiband 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 multiband quantum transport models, in both the purestate and mixedstate cases.
The first part of the chapter deals with purestates. Transport models are derived from the crystal periodic Hamiltonian by assuming that the lattice constant is small, so that an effective multiband 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 WannierSlater envelope functions and the other on the LuttingerKohn 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, minibands and semiclassical transport, are derived as a limit of multiband models.
In the second part of the chapter, the mixedstate (i.e. statistical) case is considered. In particular, the phasespace point of view, based on Wigner function, is adopted, which provides a quasiclassical description of the quantum dynamics. After a theoretical introduction to the WignerWeyl theory, a twoband phasespace 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 quantumfluid models, which are formulated in terms of a finite number of macroscopic moments of the Wigner function. For mixedstates, the maximumentropy closure of the moment equations is discussed in general terms. Then, details are given on the multiband case, where "multiband" is to be understood in the wider sense of "multicomponent wave function", including therefore the case of particles with spin or spinlike degrees of freedom. Three instances of such systems, namely the twoband k.p model, the Rashba spinorbit system and the graphene sheet, are examined.
Electronic Properties of IIIV Quantum Dots : Impact of Crystal Symmetry, Substrate Orientation and BandAlignment (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, typeI versus typeII, by comparing the commonanion system zbInAs/GaAs to the commoncation system zbGaSb/GaAs. Finally, the impact of large builtin fields resulting from piezo and pyroelectric charges will be exemplified for the wzGaN/A1N QDsystem.
Symmetries in multiband 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 multiband 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 13 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 multiband Hamiltonian whose kinetic part has the correct C2v symmetry is the 14band 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 spinorbit 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 singleparticle states, as well as the configuration interaction calculation of exciton, biexciton, and negative trion states in zingblend 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 biexcitons being bound or unbound.
Our work provides a very important conceptual message: With appropriate treatment of relevant effects, the multiband envelope function Hamiltonians is fully capable of capturing the right atomistic symmetry of QD structures.
Finite Elements for k.p Multiband 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.
PlaneWave 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 planewave based approaches to calculate the electronic structure of semiconductor nanostructures. We introduce semianalytical and numerical methods to achieve a planewave based description of such systems. This includes use of planewave methods to calculate not just the electronic structure but also the builtin 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 realspace, finite element model will be discussed and we will present representative examples of semiconductor nanostructures together with their elastic and electronic properties, computed from semianalytical and numerical approaches. We conclude that planewavebased methods provide an efficient and flexible approach when using k.p models to determine the electronic structure of semiconductor nanostructures.
The Multiband k.p Hamiltonian for Heterostructures: Parameters and Applications (50 pages)
by Stefan Birner, Walter Schottky Institute, Technische Universität München, Germany / nextnano^{3}.
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 typeII and typeIII band alignments. Finally, selfconsistent k.p calculations of a twodimensional hole gas in diamond for different substrate orientations are analyzed. Wherever possible, the k.p results are compared to tightbinding 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.
Transient Simulation of kpSchrödinger 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 timedependent k.pSchrödinger 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 Ztransformation respectively. Hence we will analyse the numerical error caused by the inverse Ztransformation. Since these DTBCs are nonlocal in time and thus very costly, we present approximate DTBCs, that allow a fast calculation of the boundary terms.
Discrete Transparent Boundary Conditions for Multiband Effective Mass Approximations (45 pages)
by Dirk Klindworth, Institute of Mathematics, Technische Universität Berlin, MA 64, 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 multiband 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 dband k.pmodel is introduced as most general MEMA. We derive DTBCs for the dband k.pmodel and test our results on a quantum well nanostructure.
