Author | Vladimir Litvinov | |

Isbn | 9789814669702 | |

File size | 9MB | |

Year | 2016 | |

Pages | 196 | |

Language | English | |

File format | ||

Category | physics |

Wide Bandgap
Semiconductor
Spintronics
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1BO4UBOGPSE4FSJFTPO3FOFXBCMF&OFSHZ7PMVNF
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
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To my wife, Valeria, and
my children, Natasha and Vlady
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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
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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
46u (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 C46u . 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 02c 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
n0
(r )d

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 IIIV nitride semiconductors. Particular attention is paid to the comparison between zinc blende GaAs- and 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 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