Wide bangap semiconductor spitronics

Wide Bandgap
Semiconductor
Spintronics


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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

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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)
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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
​ 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  <un0 (r )|un0 (r )> = 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






GaN Band Structure

functions in the same way as in zinc blende materials with the axis
Z parallel to the [111] direction. The validity of the “quasi-cubic
approximation” will be discussed below.
In a first-order kp-approximation, the Hamiltonian matrix
can be written by making use of (1.17):


Hnn(k ) =  < un (r )|H k |un(r ) > d

(1.18)



Some of the matrix elements are equal to zero as a result of the
symmetry of the basis functions:




 Ec

 P2k–
–
2

 P2k+

 2

 Pk
1 z
H =

 0


 0


 0


 0

where

E c = D1 =
D2 =


D3 =

–

P2k+

P2k–

F

0

0

0

l

0

0

0

0

0
0

2

2

G

0

0

0

2D3

P1 k z

0

2D3
0

0

0

0

0

0

0

0

0

0

0

0

Ec

P2 k+
–

0

P2k–

2
P2 k–
2

P1 k z

2

F

0

–

0

0

2D3
P2k+
0

2

G

0


0 


0 


2D3 


0 
,

P1 k z 


0 



0 


l  (1.19)



p2
+ V iS >,
2m0

p2
p2
+ V X > = + V Y >,
2m0
2m0


2 



X > + Y >,
2 2 x
y
8m0 c 



2

2

Z> =
Z >,
2 2 z
z
4m0 c
4m02c 2

(1.20)


Hamiltonian

and k ± = kx ± iky  k exp (±ij), k2 = ​k2​x​ ​ + k​ 2​y​ ​, tan(j) = ky/kx, F

|  |

2
2 __ ​ < iS​
= D1 + D2, G = D1 – D2, P1 = _______
​    __ ​ < iS​ ____
​     ​ ​Z >, P2 = ​ _______
iz

m0√
​ 2 ​ 
m0​√2 ​ 
2
____
​     ​  ​ X > = _______
​    __ ​ < iS​ ____
​     ​  ​Y >, are momentum matrix elements, Ec
ix
iy
m0√
​ 2 ​ 
is the position of the conduction band edge, D1 and D2,3 are the
parameters of the crystal field and spin–orbit interactions,
respectively, l = Ev0 = 0 is the reference energy which would be
the valence band edge position if the crystal-field and spin–orbit
splitting were not taken into account.
Hexagonal symmetry makes the Hamiltonian (1.10) invariant
with respect to b = p/3 rotation in the basal xy plane:

|  |



|  |

x = x cos b + y sin b
y = y cos b– x sin b

That results in conditions

<Y |H0|Y > = <Y |H0|Y > =
<Y |H0|Y > = < X |H0| X >.

(1.21)
3
1
< X |H0 | X > + <Y |H0|Y >,
4
4



(1.22)

Eigenvalues of the Hamiltonian found with Det [H – E] = 0 present
the energy levels in -point (k = 0). Notations used below
correspond to those in Fig. 1.3:
Conduction band, 7c :

E c = E g + D1 + D 2 ,

Light holes band, 7v:

E 2 =

Heavy holes band,  9v :

Crystal-field split-off band, 7v :

E v 1 = D1 + D 2 ,

E 3 =

 D – D2 2
D1 – D 2
2
+  1
 + 2D3 ,
2
 2 

2
D1 – D 2  D1 + D 2 
2
– 
 + 2D3 ,
2
 2 



(1.23)

The reference energy E = 0 corresponds to the lowest valence
band Eu3 when spin–orbit interaction is absent. The line-up of
the valence bands in GaN, given in increasing order of their distance
to the conduction band are heavy holes, light holes, and split-off

band. This order is different in AlN, where the positive value of
crystal-field splitting D1 = 38 meV [3] in GaN changes to a large




10

GaN Band Structure

negative value of –219 meV in AlN. As a result, in AlN heavy and
light hole bands trade positions and the order becomes: light
holes, heavy holes, split-off band, and the position of the conduction
band is Ec = Eu2 + Eg.
At finite k matrix (1.19) becomes more populated, so it is
convenient to use the matrix identity that operates with blocks
of lower dimensions and helps in finding eigenvalues:


 A B
–1
Det
 = Det[AD – ACA B ]
C D


(1.24)

Using (1.24), we rewrite Det [H – e] = 0 and find energy bands E(k):
F (e, k z , k ) = 0,

F (e, k z , k ) = [(D1 – e)2 – D22 ][(E c – e)E + P12k z2 ] – e(D1 – e)P22k 2
+ D32[2(E c – e)(D1 + D2 – e) – P22k 2 ],
2 2

E(k ) = e(k )+  k /2m0.


(1.25)

Four solutions to Eq. (1.25) are a conduction and three valence
bands each double spin degenerate.
The initial k-dependence of the energy level is determined
by the heavy bare electron mass, so the levels are almost
dispersionless. Additional dispersion from e(k) renormalizes the
bare mass and imparts an effective mass which appears to be the
result of coupling between the level and all other levels under
consideration. Let us find effective masses of conduction electrons
at e  Ec = D1 + D2 + Eg. An exact solution to Eq. (1.25) is not
needed as the inverse effective mass can be found as the
coefficient in the k2 expansion of the exact energy:
–1
2  1
1   F  F 

 = – 2  
–
2  mcz m0   k z  e 
–1
2  1

1   F  F 

= – 2  
–
2  mc m0   k  e 



k 0
e  Ec

k 0
e  Ec

=

=

(2D2 + E g )

(2D2 + E g )(D1 + D2 + E g ) – 2D32
(D2 + E g )(D1 + D2 + E g ) – D32

P12 ,

E g [(2D2 + E g )(D1 + D2 + E g ) – 2D32 ]

P22

(1.26)


Hamiltonian

Effective masses in the valence band cannot be described by
Hamiltonian (1.19) as eight basis functions (1.17) do not form the
full set: All other energy levels are neglected. However, remote
bands cannot be neglected as they generate ​k​2i​ ​ corrections and
contribute to effective masses in all bands. In order to account
for remote levels the Löwdin method is normally used so that the
full set of basis functions is divided into two subsets: Subset A
includes energy levels located close to the Fermi energy, and subset
B includes more distant levels with the energy much larger than
the actual energy of carriers contributing to the electrical and
magnetic properties of the material:


uk (r ) = Cnk un0 (r ) +C jk u j 0 (r )

(1.27)

A
H = +
C

(1.28)

n A

jB

In the basis (1.27) the matrix Hamiltonian can be generally
written as



C
,
B

where square blocks A and B correspond to subsets A and B,
respectively, and the rectangular matrix C describes the coupling
between A and B, so it comprises matrix elements of perturbation
kp
_____
​  m  ​  + H1 + H2 on basis wave functions that belong to different
0
sets. The Schrödinger equation with the Hamiltonian (1.28) can be
written as


A
 +
C

C  f   f 
  = E  ,
B  R   R 
A

C  f 

f

(1.29)

 + function
  = E  ,
where the spinor wave
 C B  R   R  consists of sub-spinors f and
R each carrying a number of components equal to dimensions
of matrices A and B, respectively. Notation C+ stands for the
Hermitian conjugate (conjugate and transposed) matrix. Matrix
Eq. (1. 29) can be written as a system of two equations

11


12

GaN Band Structure



 Af + CR = Ef
 +
C f + BR = ER

(1.30)

Af + C(E – B )–1C + f = Ef .

(1.31)



H = A + C(E – B )–1C +

(1.32)

Solving the lower equation for R and substituting it into the
upper one, we find the Schrödinger equation for A-block wave
functions

So, the effective Hamiltonian for A-block takes the form

The second term in Eq. (1.32) gives a correction originated
from distant bands. This term is small if the energy distance
between the level from subset A and remote bands is large. The
Hamiltonian (1.32) is never used in its exact form. Instead, it is
taken into account approximately, with desired accuracy on wave
vector components. Energy corrections from remote bands enter
in diagonal elements of A-matrix, they are proportional to ​k2i​​ ​ and
contribute to effective masses. With ​k2i​​ ​ accuracy the Hamiltonian
matrix has the form

 Ec


 P2k–
–
2

 P2k+

 2

2k 2  P1 k z
=
H–
2m0 
 0


 0


 0


 0



–

P2k+

P2k–

F

–K *

–H *

H*

l

0

0

2

–K

–H
0

0
0
0

2

G

0
0

2D3

P1 k z

2D3
0

0

0

0

0

0

0

H

0

0

0

0

Ec

P2k+
–

2
P2k–
2

P1 k z

0

P2k–
2

F

–K *
H*

0

–

2D3

P2k+
2

–K
G

–H


0 


0 


2D3 

0 
,

P1 k z 


H 


*
–H 



l 


(1.33)

where D1,2,3,4 are the deformation potentials, eii are the strain
components, and coefficients A1–6 stem from remote bands
corrections to the four bands under consideration [1, 4]: