Wide Bandgap Semiconductor Spintronics by Vladimir Litvinov

8158aab1dfcd6f1-261x361.jpg Author Vladimir Litvinov
Isbn 9789814669702
File size 9MB
Year 2016
Pages 196
Language English
File format PDF
Category physics


Wide Bandgap Semiconductor Spintronics This page intentionally left blank 1BO4UBOGPSE4FSJFTPO3FOFXBCMF&OFSHZ‰7PMVNF Wide Bandgap Semiconductor Spintronics editors Preben Maegaard Anna Krenz Wolfgang Palz Vladimir Litvinov The Rise of Modern Wind Energy Wind Power for the World CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2016 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20160308 International Standard Book Number-13: 978-981-4669-71-9 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To my wife, Valeria, and my children, Natasha and Vlady This page intentionally left blank Contents Preface 1. GaN Band Structure 1.1 1.2 1.3 1.4 Symmetry Hamiltonian Valence Band Structure Linear k-Terms in Wurtzite Nitrides 2. Rashba Hamiltonian xi 1 1 4 13 16 21 2.1 Bulk Inversion Asymmetry 2.2 Structure Inversion Asymmetry 2.3 Microscopic Theory of Rashba Spin Splitting in GaN 22 24 29 3.1 Spontaneous and Piezoelectric Polarization 3.2 Remote and Polarization Doping 3.3 Rashba Interaction in Polarization-Doped Heterostructure 3.4 Structurally Symmetric InxGa1–xN Quantum Well 3.4.1 Rashba Coefficient in Ga-Face QW 3.4.2 Rashba Coefficient in N-Face QW 3.4.3 Inverted Bands in InGaN/GaN Quantum Well 3.5 Experimental Rashba Spin Splitting 38 41 44 50 53 54 57 58 4.1 Double-Barrier Resonant Tunneling Diode 4.1.1 Current–Voltage Characteristics 4.1.2 Spin Current 64 64 66 3. Rashba Spin Splitting in III-Nitride Heterostructures and Quantum Wells 4. Tunnel Spin Filter in Rashba Quantum Structure 37 63 viii Contents 4.1.3 Tunnel Transparency 4.1.4 Polarization Fields 4.1.5 Spin Polarization 4.2 Spin Filtering in a Single-Barrier Tunnel Contact 4.2.1 Hamiltonian 4.2.2 Boundary Conditions and Spin-Selective Tunnel Transmission 5. Exchange Interaction in Semiconductors and Metals 5.1 5.2 5.3 5.4 Direct Exchange Interaction Indirect Exchange Interaction Three-Dimensional Metal: RKKY Model RKKY Interaction in One and Two Dimensions 5.4.1 1D-Metal 5.4.2 2D Metal 5.5 Exchange Interaction in Semiconductors 5.6 Indirect Magnetic Exchange through the Impurity Band 5.7 Conclusions 6. Ferromagnetism in III-V Semiconductors 6.1 Mean-Field Approximation 6.2 Percolation Mechanism of the Ferromagnetic Phase Transition 6.3 Mixed Valence and Ferromagnetic Phase Transition 6.3.1 Magnetic Moment 6.4 Ferromagnetic Transition in a Mixed Valence Magnetic Semiconductor 6.4.1 Hamiltonian and Mean-Field Approximation 6.4.2 Percolation 6.5 Conclusions 68 72 74 76 76 78 85 86 88 92 95 96 98 100 103 105 109 110 115 118 118 125 126 130 132 Contents 7. Topological Insulators 7.1 7.2 7.3 7.4 7.5 135 Bulk Electrons in Bi2Te3 Surface Dirac Electrons Effective Surface Hamiltonian Spatial Distribution of Surface Electrons Topological Invariant 137 140 145 150 154 8.1 Spin-Electron Interaction 8.2 Indirect Exchange Interaction Mediated by Surface Electrons 8.3 Range Function in Topological Insulator 8.4 Conclusions 160 8. Magnetic Exchange Interaction in Topological Insulator Index 159 165 172 176 179 ix This page intentionally left blank Preface The field of spintronics is currently being explored in various directions. One of them, semiconductor spintronics, is of particular recent interest since materials developed for electronics and optoelectronics are gradually becoming available for spinmanipulation-related applications, e.g., spin-transistors and quantum logic devices allowing the integration of electronic and magnetic functionalities on a common semiconductor template. The scope of this book is largely concerned with the spintronic properties of III-V Nitride semiconductors. As wide bandgap III-Nitride nanostructures are relatively new materials, particular attention is paid to the comparison between zinc-blende GaAsand wurtzite GaN-based structures where the Rashba spinorbit interaction plays a crucial role in voltage-controlled spin engineering. The book also deals with topological insulators, a new class of materials that could deliver sizeable Rashba spin splitting in the surface electron spectrum when implemented into a gated device structure. Electrically driven zero-magnetic-field spin-splitting of surface electrons is discussed with respect to the specifics of electron-localized spin interaction and voltage-controlled ferromagnetism. Semiconductor spintronics has been explored and actively discussed and various device implementations have been proposed along the way. Writings on this topic appear in the current literature. This book is focused on the materials science side of the question, providing a theoretical background for the most common concepts of spin-electron physics. The book is intended for graduate students and may serve as an introductory course in this specific field of solid state theory and applications. The book covers generic topics in spintronics without entering into device specifics since the overall goal of the enterprise is to give instructions to be used in solving problems of a general and specific nature. xii Preface Chapter 1 deals with the electron spectrum in bulk wurtzite GaN and the origin of linear terms in energy dispersion. Attention is paid to the symmetry and features of wurtzite spintronic materials which differentiate them from their cubic GaInAs-based counterparts. Rashba and Dresselhaus spin-orbit terms in heterostructures with one-dimensional confinement are considered in Chapter 2, where typical spin textures are discussed in relation to in-plane electron momentum. This chapter also presents the microscopic derivation of the Rashba interaction in wurtzite quantum wells that allows electron spin-splitting to be related to the material and geometrical parameters of the structure. In particular, we discuss Rashba spin splitting in a structurally symmetric wurtzite quantum well to focus on the polarization-field induced Rashba interaction. Vertical tunneling through a single barrier and a polarizationfield distorted Al(In)GaN/GaN quantum well, as a possible spininjection mechanism, is considered in Chapter 4. Chapters 5 and 6 are devoted to a detailed theoretical description of mechanisms of ferromagnetism in magnetically doped semiconductors, specifically in the III-V Nitrides. These chapters discuss the indirect exchange interaction in metals of any dimension and in semiconductors. Emphasis is placed on the specific feature of the indirect exchange interaction in a one-dimensional metal. Also, the standard mean-field approach to ferromagnetic phase transition is described, as is the percolation picture of phase transition in certain systems, for example, wide bandgap semiconductors, for which mean-field theory breaks down. The electronic properties of topological Bi2Te3 insulators are discussed in Chapter 7, where the semiconductor is taken as an example. Topological insulator film biased with a vertical voltage presents a system with voltage-controlled Rashba interaction and it is of interest in relation to possible spintronic applications. Surface electrons in the biased topological insulator are spinsplit and this affects the indirect exchange interaction between magnetic atoms adsorbed onto a surface. The calculation of indirect exchange in a topological insulator is given in Chapter 8. I would like to thank V. K. Dugaev, H. Morkoc, and D. Pavlidis for many useful discussions of the topics discussed in this book and Toni Quintana for carefully reading and correcting the text. Chapter 1 GaN Band Structure To deal with the spin and electronic properties of wurtzite IIInitride semiconductors and understand the specific features that differentiate them from zinc blende III-V materials, one has to know the energy spectrum. The energy spectrum gives us all necessary information about how electron spin is related to its momentum; and that is the key information we need in order to use the material in various spintronic applications. 1.1  Symmetry Ga(Al,In)N crystallizes in two modifications: zinc blende and wurtzite. The crystal structure of the wurtzite GaN belongs to the space group P63mc (International notation) or C ​ 4​6u  ​​  (Schönflies notation). The unit cell is shown in Fig. 1.1. In the periodic lattice potential, the electron Hamiltonian is invariant to lattice translations, so the wave function should be an eigenfunction y(r) of the translation operator: y(r + R ) = y(r )exp(ikR ), (1.1) where exp(ikR) is the eigenvalue of the translation operator, and R is the arbitrary lattice translation. This condition is the Wide Bandgap Semiconductor Spintronics Vladimir Litvinov Copyright © 2016 Pan Stanford Publishing Pte. Ltd. ISBN  978-981-4669-70-2 (Hardcover),  978-981-4669-71-9 (eBook) www.panstanford.com  GaN Band Structure consequence of symmetry only and it presents a definition of the wave vector. In an infinite crystal, the wave vector would be a continuous variable. Since we are dealing with a crystal of finite size, we have to impose boundary conditions on the wave function. This can be done in two ways. First, we may equate the wave function to zero outside the boundaries of the crystal. This would correspond to taking the surface effects into account. If we are not interested in finite-size (or surface) effects, there is a second option: We assume that the crystal comprises an infinite number of the periodically repeated parts of volume V (volume of a crystal) and then impose the Born–von Karman cyclic boundary conditions: Figure 1.1 Unit cell of a GaN crystal. Large spheres represent Ga sites. (r + Ni bi ) = (r ), (1.2) where bi are basis vectors of the Bravais lattice. From Eqs. (1.1) and (1.2) exp(ikLi )= 1 ki = 2 m , m = 0, ±1, ..., Li i i (1.3) where Li = biNi is the linear size of the crystal of volume V in the direction bi. Thus, the wave vector takes discrete values, so all integrals over the wave vectors that may appear in the theory should be replaced by summation over the discrete variable k. In bulk materials (V  ), the wave vector is a quasicontinuous variable. The exact summation over k can be replaced by the integral Symmetry V  f (k )  (2p)  f ( k)d k. 3 k (1.4) The wave vector can be handled in much the same way as it was in a free space. However, the difference between k in free space and in the periodic lattice field is that the lattice periodicity introduces an ambiguity to the wave vector; that is it is defined up to the reciprocal lattice vector K. Formally, the reciprocal space is defined by expanding arbitrary lattice periodic function into the Fourier series: z(r ) =  z(K )exp(iKr ) K Let’s expand z(r) displaced on the lattice vector R: (1.5) z(r + R ) =  z(K )exp(iK(r + R )) (1.6)  z(K )exp(iKr ) = z(K ) exp(iK(r + R )). (1.7) exp(iKR ) = 1. (1.8) K As the displacement R cannot change z(r) due to the lattice periodicity, the condition z(r) = z(r + R) holds K K It follows from Eq. (1.7) Equation (1.8) defines the reciprocal lattice (or dual lattice) for vectors K. From Eqs. (1.1) and (1.8), we conclude that replacement k  k + K does not change the wave function, Eq. (1.1), so k and k + K are equivalent. This means that the electron kinematics in the lattice can be fully described by the wave vectors in the finite part of the reciprocal space, the first Brillouin zone (BZ). BZ for the crystals of the wurtzite family is illustrated in Fig. 1.2. The wave function that satisfies Eq. (1.1) is the Bloch function ynk (r ) = 1 unk (r )exp(ikr ), V (1.9)  GaN Band Structure which is the modulated plane wave normalized on the crystal volume V, unk(r) is the lattice periodic Bloch amplitude, and n is the band index. Figure 1.2 First Brillouin zone for wurtzite crystal. Capital letters indicate the high symmetry points of wave vector k in BZ. Symmetry dictates that the Hamiltonian is invariant to all transformations of the space group ​C4​6u  ​.​  When an element of the space symmetry group acts on a crystal, it transforms both the space coordinate r and the wave vector k. In each high symmetry point shown by capital letters in Fig. 1.2, there exists a point subgroup of rotations that leave the corresponding wave vector unchanged. Both this subgroup and the time reversal symmetry determine the energy spectrum near the k-point of high symmetry. In III-nitrides the energy spectrum that is responsible for the electrical, magnetic, and optical properties of the material, lies near the point  (k = 0) The relevant energy levels for spin-up and spin-down electrons include six valence and two conduction bands each corresponding to irreducible representations of the point group C6u [1, 2]. Below the spectrum at the Γ-point will be constructed using Luttinger– Kohn basis wave functions. 1.2  Hamiltonian Energy bands can be found as eigenvalues of the Schrödinger equation: Hy = Ey H= p2 ħ . ,c    ​  σ + VV(r) (ró) + ​ _____ V ( ). p, p ×× V(r)  2r 4​ 2m0 4mm0​ 02​c​  c2 2 �  (1.10) Hamiltonian where sx,y,z are the Pauli matrices; e, m0 are the free electron charge and mass, respectively; p = –i is the momentum operator; and V(r) is the electron potential energy in the periodic crystal field. The third term in Eq. (1.10) represents the spin–orbit interaction. Using ynk(r) from Eq. (1.9) we obtain the equation for Bloch amplitudes: � H k unk = E nk unk , H k = H0 +  2 p2 k. σ + V (r ), H1 = s . V (r ) × p, H2 = ó × Ñ V (r ). σ 2 2 2m0 4m0 c 4m0c 2 (1.11) � � H0 = 2k 2 kp + + H1 + H 2 , 2m0 m0 In order to find the eigenvalues Enk, one has to choose the full set of known orthogonal functions that create the initial basis on which we can expand the unknown amplitudes unk(r). As we are looking for the spectrum in the vicinity of the -point, the set of band edge Bloch amplitudes un0(r) can serve as the basis wave functions (Luttinger–Kohn representation). Within kp-perturbation theory, the third, fourth, and fifth terms in the Hamiltonian (1.11) are being treated as a perturbation. The Hamiltonian H0 does not include the spin–orbit interaction, so we restrict our consideration to the three band edge energy levels that correspond to the irreducible representations of the point group C6u: the conduction band 1c , the double degenerate in orbital momentum valence band 6, and one more valence band 1. Relevant bands are shown in Fig. 1.3a. The levels are degenerate and the degeneracy is shown in parentheses. With account for the spin variable, the basis comprises eight bands: three valence bands and one conduction band. The Bloch amplitude can be represented as a linear combination 8 1 Cnkun0(r ),  n=1 uk (r ) =  u * (r )u (1.12) where the amplitudes un0(r) are orthogonal and normalized on a unit cell volume : n0 n0 (r )d  = nn (1.13)  GaN Band Structure (a) (b) Figure 1.3 Structure of the Γ-bands in wurtzite (a) and zinc blende (b) crystals. The left hand side of each panel shows the basis states with no spin–orbit interaction taken into account; D1 is the crystal-field energy splitting. Conduction and valence bands in GaN stem from s- and p-orbitals of Ga and N. The conduction band has spherical s-symmetry, so the Bloch amplitude at the band edge can be chosen to be a spherical s-orbital, Y00. The three valence bands obey the p-symmetry and can be chosen as linear combinations of spherical harmonics ​∧​ that are the eigenfunctions of L​ ​ z  , z-component of the orbital momentum with L = 1 (p-state). Once we choose the principal axis Z along the c-direction (see Fig. 1.1) the basis spherical harmonics can be expressed in terms of p-orbitals |X >, |Y >, |Z > shown in ​∧​ Fig. 1.4. The operator L​ ​ z  has three eigenvalues l = –1, 0, 1 and the corresponding spherical harmonics have the form: 1 1 |(X + iY )>, Y1–1 = |(X – iY )>, Y10 = |Z >. 2 2 Y11 = – L = [r × p] = –i [r × ]. The orbital momentum operator is defined as �  (1.14) (1.15) It is straightforward to check that the set (1.14) comprises ​∧​ 2l + 1 = 3 eigenfunctions of L​ ​  z with corresponding eigenvalues –1, 0, 1: Hamiltonian    –i x – y  |(X + iY )> = |(X + iY )>, x   y    –i x – y  |(X – iY )> = –|(X + iY )>, x   y    –i x – y  |Z > = 0. x   y Figure 1.4 (1.16) Electron density in s- and p-orbitals, |S >, |X >, |Y >, |Z >. Finally, not accounting for spin–orbit interaction, the basis set can be written as 1c (2): u1 = |iS >, u5 = |i S >, 1 1 |(X + iY )>, u6 = |(X – iY )>, 2 2 1 1 6 (2): u3 = |(X – iY )>, u7 = – |( X + iY )>, 2 2 1v (2): u4 = |Z >; u8 = |Z >, 6 (2): u2 = – 1 0 (1.17) where |S> = |Y00|  *  |>, and |> = 0, |> =1 are the spinor wave   functions that correspond to spin-up and down states, respectively. Equation (1.17) forms the “quasi-cubic approximation” that neglects the anisotropy of wurtzite structure and represents basis wave 

Author Vladimir Litvinov Isbn 9789814669702 File size 9MB Year 2016 Pages 196 Language English File format PDF Category Physics Book Description: FacebookTwitterGoogle+TumblrDiggMySpaceShare This book is focused on the spintronic properties of III–V nitride semiconductors. Particular attention is paid to the comparison between zinc blende GaAs- and wurtzite GaN-based structures, where the Rashba spin–orbit interaction plays a crucial role in voltage-controlled spin engineering. The book also deals with topological insulators, a new class of materials that could deliver sizable Rashba spin splitting in the surface electron spectrum. Electrically driven zero-magnetic-field spin splitting of surface electrons is discussed with respect to the specifics of electron-localized spin interaction and voltage-controlled ferromagnetism. The book covers generic topics in spintronics without entering into device specifics, since the overall goal of the enterprise is to provide theoretical background for most common concepts of spin-electron physics and give instructions to be used in solving problems of a general and specific nature. The book is intended for graduate students and may serve as an introductory course in this specific field of solid-state theory and applications.     Download (9MB) Spin Current (series On Semiconductor Science And Technology) Semiconductor Glossary: A Resource For Semiconductor Community, Second Edition Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics Quantum Mechanics For Electrical Engineers Electric Circuits And Signals Load more posts

Leave a Reply

Your email address will not be published. Required fields are marked *