On the theory of the Schrödinger equation with the full set of relativistic corrections
All relativistic corrections to the Scrödinger equation which determine the interlink between spin and orbit of moving particles, are directly calculated from the Dirac equation using the spin invariant operators. It is shown that among the second order corrections there are not only the well-known...
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irk-123456789-1761692021-02-04T01:30:35Z On the theory of the Schrödinger equation with the full set of relativistic corrections Eremko, A.A. Brizhik, L.S. Loktev, V.M. Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова All relativistic corrections to the Scrödinger equation which determine the interlink between spin and orbit of moving particles, are directly calculated from the Dirac equation using the spin invariant operators. It is shown that among the second order corrections there are not only the well-known Darwin and Thomas terms, but also the new ones. Only with the account of the latter corrections the energies found with the obtained spin-orbit interaction operator, coincide with the energies of the Dirac equation exact solution. The problem of electron spectrum in the quantum well type structures is studied in details and the physical reasons for the appearance of spin-orbit interaction operators in the Dresselhaus or Rashba form, are analyzed. 2018 Article On the theory of the Schrödinger equation with the full set of relativistic corrections / A.A. Eremko, L.S. Brizhik, V.M. Loktev // Физика низких температур. — 2018. — Т. 44, № 6. — С. 734-746. — Бібліогр.: 19 назв. — англ. 0132-6414 PACS: 03.65.Pm, 03.65.Ta, 73.20.At http://dspace.nbuv.gov.ua/handle/123456789/176169 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
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Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова Eremko, A.A. Brizhik, L.S. Loktev, V.M. On the theory of the Schrödinger equation with the full set of relativistic corrections Физика низких температур |
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All relativistic corrections to the Scrödinger equation which determine the interlink between spin and orbit of moving particles, are directly calculated from the Dirac equation using the spin invariant operators. It is shown that among the second order corrections there are not only the well-known Darwin and Thomas terms, but also the new ones. Only with the account of the latter corrections the energies found with the obtained spin-orbit interaction operator, coincide with the energies of the Dirac equation exact solution. The problem of electron spectrum in the quantum well type structures is studied in details and the physical reasons for the appearance of spin-orbit interaction operators in the Dresselhaus or Rashba form, are analyzed. |
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Eremko, A.A. Brizhik, L.S. Loktev, V.M. |
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Eremko, A.A. Brizhik, L.S. Loktev, V.M. |
| author_sort |
Eremko, A.A. |
| title |
On the theory of the Schrödinger equation with the full set of relativistic corrections |
| title_short |
On the theory of the Schrödinger equation with the full set of relativistic corrections |
| title_full |
On the theory of the Schrödinger equation with the full set of relativistic corrections |
| title_fullStr |
On the theory of the Schrödinger equation with the full set of relativistic corrections |
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On the theory of the Schrödinger equation with the full set of relativistic corrections |
| title_sort |
on the theory of the schrödinger equation with the full set of relativistic corrections |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2018 |
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Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
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| citation_txt |
On the theory of the Schrödinger equation with the full set of relativistic corrections / A.A. Eremko, L.S. Brizhik, V.M. Loktev // Физика низких температур. — 2018. — Т. 44, № 6. — С. 734-746. — Бібліогр.: 19 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT eremkoaa onthetheoryoftheschrodingerequationwiththefullsetofrelativisticcorrections AT brizhikls onthetheoryoftheschrodingerequationwiththefullsetofrelativisticcorrections AT loktevvm onthetheoryoftheschrodingerequationwiththefullsetofrelativisticcorrections |
| first_indexed |
2025-07-15T13:50:56Z |
| last_indexed |
2025-07-15T13:50:56Z |
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1837721148012363776 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6, pp. 734–746
On the theory of the Schrödinger equation with the full
set of relativistic corrections
A.A. Eremko1, L.S. Brizhik1, and V.M. Loktev1,2
1Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine
14-b Metrolohichna Str., Kyiv 03143, Ukraine
2National Technical University of Ukraine “KPI”, 37 Peremohy Ave., Kyiv 03056, Ukraine
E-mail: eremko@bitp.kiev.ua; brizhik@bitp.kiev.ua; vloktev@bitp.kiev.ua
Received December 12, 2017, published online April 25, 2018
All relativistic corrections to the Scrödinger equation which determine the interlink between spin and orbit of
moving particles, are directly calculated from the Dirac equation using the spin invariant operators. It is shown
that among the second order corrections there are not only the well-known Darwin and Thomas terms, but also
the new ones. Only with the account of the latter corrections the energies found with the obtained spin-orbit in-
teraction operator, coincide with the energies of the Dirac equation exact solution. The problem of electron spec-
trum in the quantum well type structures is studied in details and the physical reasons for the appearance of spin-
orbit interaction operators in the Dresselhaus or Rashba form, are analyzed.
PACS: 03.65.Pm Relativistic wave equations;
03.65.Ta Foundations of quantum mechanics; measurement theory;
73.20.At Surface states, band structure, electron density of states.
Keywords: Dirac equation, Schrödinger equation, spin-orbit interaction, spin splitting, two-dimensional electrons.
1. Introduction
Study of the spin-orbit interaction (SOI) is one of the
main streams of modern solid state physics [1] which leads
to important practical applications, such as an effective
control tool of spin-polarized carrier states in spintronics
devices. In particular, spin spliting arising from Rashba
SOI [2] allows manipulating spin in semiconducting
heterostructures by electric field [1,3]. On the other hand, SOI
is also the source for some interesting physical phenomena
such as spin current and Hall effects. At last, SOI determines
the peculiarities of a new class of condensed systems, so
called topological dielectrics. According to Rashba’s recent
remark [4], SOI as the notion and physical reality, “goes glob-
al”, deeply penetrating into many areas of the fundamental
science or technical applications, providing new phenomena,
on which future technologies will be based.
Functioning of spintronics devices at ambient condi-
tions (atmospheric pressure and room temperature) require
strong enough spin splitting, and, hence, large SOI [3–7].
Among such materials there are two-dimensional (2D) or
quasi-2D systems, such as layered structures and hetero-
structures, crystal surfaces, interfaces, thin films (up to
monomolecular or monoatomic width). The interest to
low-dimensional electron phenomena as a whole, has start-
ed growing from 1970-s, but during last 5–7 years it has
increased dramatically [8] because of their potential per-
spectives in using in the devices of the future generation.
In systems with broken inversion symmetry, Rashba
splitting of 2D electron and hole bands takes place in the
result of the SOI constant finite value. With decreasing
system dimensionality number of carrier spatial degrees of
freedom also reduces. Charge particle motion in one or two
directions becomes finite and SOI proves to be one of the
factors which determines the mobile carriers states.
It is generally adopted that the problem of the relation
of particle propagation and its spin state, described by SOI,
has been solved. This includes formal derivation of SOI
and understanding of physical reasons determining its ex-
istence. Starting with the pioneer papers of Dresselhaus [9]
and Rashba [10], account of SOI in electron structure of
crystals is based on the solutions of the Schrödinger–Pauli
equation H Eψ = ψ for two-component electron wave
function (spinor) ψ with the Hamiltonian
[ ]( )
2
(0)1 ˆ= ( ) ,
2 SOH V V
m
−+ + λ ×
p r p ∇ σ (1)
2
(0)
2 2= ,
4SO m c
λ
© A.A. Eremko, L.S. Brizhik, and V.M. Loktev, 2018
On the theory of the Schrödinger equation with the full set of relativistic corrections
which defines their eigen energies E . Here ( )V r is the po-
tential, in which an electron moves, ˆ = i−p ∇ is its mo-
mentum operator, and σ̂ is the spin operator (operator of
the intrinsic momentum ˆ( /2) σ), whose components are
represented by Pauli matrices ˆ jσ ( = , ,j x y z ). The last
term in the Hamiltonian (1) is known as Thomas correction
and is usually called SOI operator (Darwin term is omitted
in (1) as comparatively small).
As it is generally known, the fundamental basis for
studying electron states is Dirac theory from which in a
natural way the existence of an electron spin /2 and Fer-
mi-statistics for electrons follow. Electron spectra, calcu-
lated within this theory, practically coincide with their ob-
servable values. Expanding Dirac equation (DE) with
respect to the degrees of the ratio /p mc, where p is char-
acteristic momentum, m is mass and c is the light speed,
one can calculate relativistic corrections to the non-
relativistic Schrödinger equation (SchE). In other words,
SchE with the Hamiltonian (1) is the limit of DE when
particle's rest energy, 2mc , significantly exceeds all other
energy scales. In this sense, SOI operator can be consid-
ered as one of the relativistic corrections of the order 21/c
to non-relativistic Hamiltonian [11–14].
It is worth to recall that 2D electrons can be modeled as
the states determined by the quantum well (QW) formed
by a layer of the heterostructure or by a surface of the in-
terface, with the potential changing in one direction only,
namely in the perpendicular to the plane, z -direction. Free
motion of carriers in xy -plane is characterized by the 2D
wave vector = x x y yk k⊥ +k e e ( je are corresponding unit
vectors), and 1D SchE determines discrete eigen states in
the QW in which each energy level creates 2D electron
band ( )nE ⊥k . Taking into account that the equality
= zV e∇ is valid for asymmetric QW with the asymmetry
arising from the electric field that is perpendicular to the
QW plane, one can derive a model with the operator
ˆ= [ ]R BR zH ⊥λ ×e k σ , which is called Rashba SOI where
the parameter BRλ ∝ is Bychkov–Rashba constant. For a
long time this SOI (see Eq. (1)) a priori was considered as
the only one and as the one appropriate for all physical
situations.
Nevertheless, as it has been shown in Refs. 15, 16, there
can exist several possible solutions of the DE which are
different from each other and correspond to different spin
states of a particle. More precisely, they correspond to dif-
ferent mutual directions of the spin quantization axis and
the direction of the momentum. It has turned out that there
is a finite number of various situations, among them also
the situation which is different from the Rashba case, the
difference between which is controlled by the so called
spin invariants. These invariants commute with the Dirac
Hamiltonian but do not commute between themselves [17].
This fact has allowed to obtain the general solution of the
DE and to calculate all relativistic corrections to it in a full
agreement with the analytical general solution. In these
papers, however, the SchE which includes such corrections
has not been derived. In the present paper we find the ex-
plicit form for SOI terms in the SchE which lead to the
same energy corrections that follow from the exact solu-
tion found in [18].
The paper is organized as follows. In Sec. 2 the essen-
tial information of the quantum field theory which allows
to write down the Dirac Hamiltonian in the second quanti-
zation representation is given. In Section 3 it is shown that
with the accuracy of the second order of the ratio
2| | / 1V mc the Hamiltonian of particles and antiparti-
cles, linked via external potential field ( )V r , can be trans-
formed to the Hamiltonian of non-interacting electrons and
positrons as it takes place in the case of free particles. In
Sec. 4 we describe the operator invariants controlling spin
states of relativistic particles. Section 5 deals with the tran-
sition in the electron Hamiltonian to the non-relativistic
limit with account of all relativistic corrections among
which there are the new ones. The case of the potential in
the form of the QW is considered in Sec. 6 using the re-
sults of the previous section. In particular, the Hamiltonian
is derived which describes the states of 2D electrons. Their
states corresponding to the basic spin invariants and to the
generalized invariant are studied. It is also analyzed when
and how in the general approach the spin-orbit band split-
ting arises in the Rashba or Dresselhaus form.
The paper is dedicated to the outstanding theoretical
physicist Alex Abrikosov whose contribution to physics
in general, and low temperature physics and physics of
low-dimensional systems, in particular, is impossible to
overestimate. The paper was prepared to Alex Abrikosov’s
90-years jubileum, but in view of his recent unexpected
demise, turned out to be a tribute to the memory of the
great researcher and extraordinary personality.
2. Relativistic Hamiltonian
Let us start with the DE for a particle in the external
field:
2 ˆˆˆ ˆ= ( )ei c V I mc
t c
∂Ψ − + + β Ψ ∂
p A r α
where e is elementary charge, A is vector-potential of the
external electromagnetic field, ( )V r , similar to (1), is the
potential, ˆ ˆ= j j
j
α∑ eα is vector matrix whose components
ˆ jα ( = , ,j x y z ) together with the unit matrix Î and matrix
β̂ are hermitian Dirac matrices (DM), and, finally,
( )1 2 3 4( ; ) = TtΨ ψ ψ ψ ψr is a 4-component function,
known also as a bispinor, or 4-spinor (here and below a
symbol ∧ (‘hat’) is used over matrices and matrix opera-
tors, only).
According to the quantum field theory, the DE is the
Euler–Lagrange equation which follows from the variation
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 735
A.A. Eremko, L.S. Brizhik, and V.M. Loktev
of the Lagrange functional density . It depends on the
two 4-component variables, bispinors Ψ, namely, on its
components µψ ( = 1, 2, 3, 4µ ), and † ˆ=Ψ Ψ β is the Dirac
conjugated bispinor (see, e.g., Refs. 11, 12). The transition
to the Hamilton form is provided by introducing generali-
zed momenta,
†ˆ= = = , = ,i i
t
∂ ∂Ψ
Ψβ Ψ Ψ
∂∂Ψ
which are canonically conjugate to the components of the
bispinor Ψ, and Hamilton functional density,
† † ˆ= = ( , ) ( , ),Di t H t
t
∂Ψ
Ψ − Ψ Ψ
∂
r r
in which the operator
2 ˆˆ ˆˆ ˆ= ( )DH c V I mc
c
− + + β
p A rα
(2)
is the Dirac Hamiltonian. Subsequently, the spinor field
operator is reduced to the integral
† ˆ= = ( ) ( )DH d H dΨ Ψ∫ ∫r r r r
where the spatial integration is carried out over the whole
volume. Here the bispinor Ψ is considered not as the
Schrödinger wave function, but as an amplitude of some
physical field which is called ‘a spinor field’ whose com-
ponents are q-numbers in the meaning that the inequality
∗ ∗
ν µ µ νψ ψ ≠ ψ ψ takes place.
Consider, first, particle dynamics in the absence of the
magnetic field, = 0A . The Dirac Hamiltonian, (2), can be
represented in the form of the sum of two terms
(0)ˆ ˆ ˆ=D DDH H V+ , one of which is the Hamiltonian of a free
particle,
(0) 2 ˆˆ ˆ ˆ= ,DH c mc+ βpα (3)
and the second one accounts for particle interaction with
the external field, ˆ ˆ= ( )DV V Ir . This transforms the spinor
field Hamiltonian to the form:
( )(0)† †ˆ= ( ) ( ) ( ) ( ) ( ) .DH H V dΨ Ψ +Ψ Ψ∫ r r r r r r (4)
Any bispinor, ( )Ψ r , can be expanded over the complete
ortho-normalized bispinor system, in particular, the one for
free particles, i.e., bispinors which satisfy the equation
(0) (0) (0)ˆ ( ) = ( )DH EΨ Ψr r .
Since a free particle momentum is conserved, it is con-
venient to undertake the transition to the momentum repre-
sentation and to use Fourier components of the eigen
bispinor (0)Ψ , i.e., to expand bispinor ( )Ψ r over plane
waves in the cube with the side L (L →∞ ):
( )(0)
3/2
1( ) = e .i
L
Ψ Ψ∑ kr
k
r k (5)
Here = j j
j
k∑k e with = (2 / )j jk L nπ ( = , ,j x y z ) where
the integer numbers jn take values from −∞ to ∞. In such
presentation momentum operator is a c-number (wave vec-
tor), ˆ ⇒p k , and the bispinor components ( )(0)Ψ k are
determined from the equation
( ) ( )(0) (0) (0)ˆ ( ) = ,DH EΨ Ψk k k
(0) 2 ˆˆ ˆ( ) = .DH c mc+ βk k α (6)
In a block form the 4-line matrices are expressed via the
2-line ones [11,12]
2
2
ˆˆ 00 ˆˆ = , = ,
ˆˆ 0 0
I
I
β −
σ
α
σ
(7)
where σ̂ is the vector operator, whose components are giv-
en by the Pauli matrices, and 2̂I is a unit matrix of the se-
cond order. It is convenient also to write the bispinor in the
block-form, too: (0) ( ) = ( ( ) ( ))T
u dΨ ψ ψk k k , where
1 2= ( )Tuψ ψ ψ and 3 4= ( )Tdψ ψ ψ are the upper and low-
er spinors of the bispinor, respectively. Within this scheme
Eq. (6) takes a simple form,
2
2
2
2
ˆ ˆ
= .
ˆˆ
u u
d d
mc I c
E
c mc I
ψ ψ ψ ψ −
k
k
σ
σ
(8)
Its solution can be found, in particular, using the well-
known Foldy–Wouthuysen (FW) unitary transform (see,
e.g., Refs. 13, 14). The following four ortho-normalized
eigen bispinors are the solutions of Eq. (8) (or Eq. (6)):
( )
( )
( ) ( )
( ) ( ) ( ) ( )
,
(0)
,
,2
,2(0)
,
,
ˆ= , = ,
ˆ
= , = ,
p
p p
p
a
a a
a
A E Ec
mc
c
mcA E E
σ
σ
σ
σ
σ
σ
χ
Ψ ≡ ε⋅ χ ε +
⋅ − χ
ε +Ψ ≡ −ε
χ
k
k
k k kk
k
k
kk k k
σ
σ
( )
( )
2
= .
2
mc
A
ε +
εk
k
k
(9)
where Ak is the normalization coefficient, and
( ) 2 4 2 2 2= .m c cε +k k (10)
The spinors ,ν σχ ( = ,p aν ) are not fully determined be-
cause the bispinors (9) satisfy Eq. (8) at arbitrary spinors.
The bispinor (0)
,p σΨ in Eq. (9) corresponds to the posi-
tive eigen value pE , and bispinor (0)
,a σΨ — to the negative
one < 0aE , which are degenerate. The number σ in
736 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
On the theory of the Schrödinger equation with the full set of relativistic corrections
Eq. (9) takes two values, which are assigned to the two
arbitrary chosen spinors of the degenerate state. Therefore,
the four eigen bispinors (9) (0)
,ν σΨ , where each index, ν and
σ , takes two values, form a complete ortho-normalized
system. The condition of their ortho-normalization
(0)(0) †
, , ,,( ) = ′ ′ν σ ν ν σ σ′ ′ν σΨ Ψ δ δ directly leads to the ortho-
normalization of the corresponding pair of the spinors
†
, , ,=′ ′ν σ ν σ σ σχ χ δ . Index σ has the meaning of the spin
number and can be assigned the values = 1σ ± or = ,σ ↑ ↓.
The relation between the two ortho-normalized spinors
with different values of σ is given by the Kramers rela-
tion: , ,
ˆ= KKν −σ ν σχ χ , where ˆ ˆ=K yK i K− σ is the Kramers
operator, which includes K -operation of the complex con-
jugation.
This consideration shows the principal possibility of
expanding any bispinor ( )Ψ r over the bispinors (9):
( ) ( ) ( )( )†(0) (0)
, , ,,3/2
,
1= e .i
p aa b
L
σ σ σ− σ
σ
Ψ Ψ + Ψ∑ kr
k k
k
r k k (11)
Here we have used the notations accepted in quantum field
theory for the creation and annihilation operators,
† †
, ,/a bσ σk k , , ,/a bσ σk k of a particle/antiparticle with the
wave vector k and spin number σ , respectively. The phys-
ical requirement of positive eigen energy values of the
Hamiltonian (4) determines Fermi commutation rules (see,
e.g., Ref. 12):
† † † †
, , , ,, , , ,= 1, = 1,a a a a b b b bσ σ σ σσ σ σ σ+ +k k k kk k k k
with all other pairs of these operators mutually anti-
commuting.
Substituting expression (11) in Eq. (4) and taking into
account that
[ ]
/2
1 2 ,3 1 2
/2
1 exp ( ) =
L
L
i d
L −
− δ ≡∫ k kk k r r
, , ,1 2 1 2 1 2
,k k k k k kx x y y z z
≡ δ δ δ
where ,1 2
δk k is Kronecker symbol, one derives the Hamil-
tonian in the occupation number representation,
( )
( )
( )
( )
( )† †
, ,, ,,3
, ,
( )† †
, ,, , ,3
,
( ) † †
,, ,3
,
( )
, ,,3
,
1= ( ) ,
1( ) ,
1 ,
1 , ,
p
a
p a
a p
H a a a a
L
b b b b
L
a b
L
b a
L
′ ′σ σσ ′ σσ σ
′ ′σ σ
− σ − σ− σ ′ ′ ′σ σ − σ
′ ′σ
−
′ σ ′ ′σ σ − σ
′ ′σ
−
′ ′− σ σ′σ σ
′ ′σ
′ε + −
′−ε + +
′+ +
′ +
∑ ∑
∑
∑
∑
k kk k
k k
k kk k
k
k k
k
k k
k
k k k
k k k
k k
k k
(12)
where
( ) ( ) ( )( ) ( )
†( ) (0)(0)
,, ,, = , = , ;V p aν
ν σ′ ′σ σ ν σ′ ′ ′− Ψ Ψ νk k k k k k
( ) ( ) ( )( ) ( )
†( ) (0)(0)
,, ,, = ,p a
p aV−
σ′ ′σ σ σ′ ′ ′− Ψ Ψk k k k k k (13)
( ) ( )( ) ( )( ) ( )
, ,, = , , = e ( ) .a p p a iV V d
∗− − −
′ ′σ σ σ σ′ ′ ∫ krk k k k k r r
From the above it follows that the operator (12) is the sum
of the three terms, p a p aH H H V −= + + , which are the
Hamiltonians of particles, pH , antiparticles, aH , and oper-
ator p aV − which describes their mutual transformation un-
der the scattering in the external potential. After transform-
ing the product of creation and annihilation operators to the
standard form, the Hamiltonian (12) becomes positively
determined, except, according to quantum field theory pos-
tulates [12], the infinite additive constant, i.e., energy of
the state in the absence of any particles, vacuum state,
from which energies of all elementary excitations of the
spinor field, particles and antiparticles, are calculated.
The operator p aV − describes mixing of particle and anti-
particle states in the external field, and necessarily has to be
taken into account. The transformation which separates parti-
cle and antiparticle states in the Hamiltonian (12) exactly, is
not known. Nevertheless, in the case of interaction (13) the
perturbation theory can be used with any required accuracy.
3. Approximate renormalization
An important problem of non-relativistic physics is ap-
proximate separation of particle and antiparticle states.
This is possible for potentials that satisfy the inequality
2| ( ) |V mcr . In such a case the operator p aV − can be con-
sidered as a perturbation and one can use the canonical
transformation of the Hamiltonian 0H H V= + λ with pa-
rameter λ characterizing the smallness of the perturbation,
to the new representation exp ( ) exp ( )H S H S= −λ λ in
which the value of the non-diagonal part of the Hamiltoni-
an exceeds the given accuracy. For small perturbations,
1λ , the exponent can be expanded into the series with
respect to the degrees of λ, and the Hamiltonian after the
Schrieffer–Wolff transformation [19] takes the form
[ ] [ ]
2
e e , ,
2
S SH H H H S H S S−λ λ λ
= = + λ + +… .
To diagonalize this Hamiltonian up to the given accura-
cy, e.g., up to nλ , it is convenient to search the operator S
( †= S− ) in the form =1= n j
jjS Sλ λ∑ . In particular, for
diagonalization up to the second order 2λ , it is enough to
preserve only the first term in the operator Sλ and to
choose it from the condition
[ ]0 1, 0.V H S+ =
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 737
A.A. Eremko, L.S. Brizhik, and V.M. Loktev
Taking into account the explicit form of the operator
p aV V −= in the Hamiltonian (12), we get
( )
( )( ) ( )
( )( ) ( )
3
1
, , ,
† (0)(0)
, , ,,
† (0) † †(0)
, ,, ,
=
( ) ( )
.
a p
p a
V
S L
b a
a b
−
′ ′σ σ
′ ′σ − σ σ′σ
σ ′ σ ′ ′σ − σ
′−
×
′ε + ε
′× Ψ Ψ −
′− Ψ Ψ
∑
k k
k k
k k
k k
k k
k k
k k
(14)
From the last expression it follows that the renormalization
with the second order accuracy is correct if the inequality
max min2
| | 1, | |= .
2
V V V V
mc
− (15)
is fulfilled. It, as a rule, is valid for any non-relativistic
potential.
Therefore, up to terms 2λ the operator (12) reduces to
the form
[ ]
2
4
0 1 1, ( ),
2 p aH H H S H H Oλ
= + +…= + + λ
in which the states of particles and antiparticles turn out to be
independent and can be considered using the Hamiltonians
( ) ( )( )
†
,,
,
( ) ( ) †
,,, ,3
,
= ( )
1 , , ,
p
p p
H a a
W a a
L
σσ
σ
′ ′σ′ ′ σσ σ σ σ
′ ′σ
ε +
′ ′ + +
∑
∑
kk
k
kk
k
k
k k k k
(16)
( ) ( )( )
†
,,
,
( ) ( ) †
,,, ,3
,
= ( )
1 , ,
a
a a
H b b
W b b
L
σσ
σ
′ ′σ′ ′ σσ σ σ σ
′ ′σ
ε −
′ ′ − − − −
∑
∑
kk
k
kk
k
k
k k k k
(17)
for particles and antiparticles, respectively. In the above
equations the amplitudes ( )
, ( , )ν
′σ σ ′k k are given by the ex-
pressions (13) and notations
( ) ( ) ( )
[ ][ ]
( )( ) ( ) ( )( ) ( )
( ) ( )
( )
1 1, 3
,1 1
1
1 1
†† (0) (0) (0)(0)
, 1 1 ,, ,1 1
( ) ( )
, ,
1, =
2
( ) 2 ( ) ( )
( ) ( ) ( ) ( )
,
, = ,
p
p pa a
a p a
W V V
L
W W
′σ σ
σ
σ ′σσ σ
→
′ ′σ σ σ σ
′ ′− − ×
′ε + ε + ε
× ×
′ε + ε ε + ε
′× Ψ Ψ Ψ Ψ
′ ′
∑
k
k k k k k k
k k k
k k k k
k k k k
k k k k
(18)
are used. In the Hamiltonian aH , the product of creation
and annihilation operators is written in the normal form
with the change → −k k in the sum, and the energy of the
vacuum is deduced.
The creation and annihilation operators of parti-
cles/antiparticles in the expansion (11) have the indeces k
and σ , which have the meaning of the quantum numbers
with σ corresponding to the spinor / ,p a σχ in bispinors (9),
while the spinors themselves are not determined uniquely.
Their arbitrariness in Eq. (9) indicates that in the general
case the Hamiltonian (12) is ‘invariant’ with respect to the
choice of spinors. In particular, without loss of generality,
one can choose the following pair of the orthogonal spinors
= (10)T
↑χ , = (01)T↓χ , i.e., to use the simplest spin func-
tions related to the initial coordinate system, which are
eigen functions of the operator ˆ zσ .
For free particles such choice is not important, since the
energy is independent of the spin variable, and any spinor
can be expressed via the two other spinors. Therefore, one
can use solutions (9) with spinors / ,p a σχ which can be
chosen according to convenience.
Below we shall show that spin polarization in concrete
spin states is tightly related to the wave vector, characteriz-
ing spatial motion. This relation is described by some new
effective interaction called SOI. According to expressions
(13) and (18), this interaction is directly determined not only
by the symmetry of the field in which particle propagates,
but also by its spin when the particle is scattered by this field
(see the Hamiltonian (12) and expressions (16), (17)).
4. Spin invariants
As it has been reminded above, particle spin states, i.e.,
concrete form of the spinors in Eq. (9), can be found using
spin invariants. For example, the existence of these invari-
ants has allowed to find new spin states of quasi-2D elec-
trons [15,16]. When a particle is free, its dynamics can be
characterized by several invariants [17]: vector of magnetic
spin polarization,
( )( )1 1 ˆˆ ˆ ˆ ˆˆˆ = = ,
mc mc
µ+ × − × ≡p k kµ Σ Γ Σ Γ (19)
vector of spin polarization,
( )( )
1 1
ˆˆ ˆˆ ˆˆ ˆ= = ,
mc mc
+ρ + ρ ≡
p k k
Ω Ω (20)
helicity ˆ ˆˆ=h pΣ, and vector of electric spin polarization
ˆˆˆ = − ×p Ω . It is easy to see that the latter two invariants
can be represented in the form ˆ ˆ ˆ=h pµ and ˆˆˆ = − ×p , re-
spectively. In formulas (19) and (20) we have taken into
account that ˆ =⇒p p k .
The three vector DMs, according to (7), have the block
form
2
1
2
ˆ ˆ0 0ˆ ˆ= , = ,
ˆ ˆ0 0
ˆˆ 00ˆ ˆ= , = .
ˆˆ0 0
i
i
I
I
−
ρ −
σ σ
Σ Γ
σ σ
σ
Ω
σ
(21)
738 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
On the theory of the Schrödinger equation with the full set of relativistic corrections
The matrix Σ̂ determines the spin operator ˆˆ = ( /2)σ Σ
[11,12,14], and its projections, ˆ ˆ ˆ=j k liΣ − α α , have different
spatial indeces, which form a cyclic permutation. The other
projections are given by the expressions ˆˆ ˆ=j jiΓ − βα ,
ˆˆ ˆ=j jΩ βΣ , and, finally, the DM 1ˆ ˆ ˆ ˆ= j k liρ − α α α has the
same three cyclic indeces.
Since the invariants commute with the Hamiltonian,
(0)ˆ
DH , their eigen bispinors are compatible with the eigen
bispinors of Eq. (6), or matrix equation (8). This means
that the bispinors (9) satisfy the equalities
( ) ( ) ( ) ( )( ) (0) (0)
, , , ,
ˆ = ,j
jsν σ ν σ ν σΨ Ψk k k k
where ( )ˆ ( )j k is one of the invariants (including Cartesian
projections in the case of vector matrices), and ( ), ,js ν σ k is
the corresponding eigen value. Namely this equality de-
fines the concrete form of the spinors ,ν σχ in (9). As the
operators (19), (20) do not commute, each of them corre-
sponds to its own pair of eigen spinors.
Worth mentioning, any arbitrary linear combination of
these invariants also commutes with the Hamiltonian (0)ˆ
DH
and can be considered as some generalized invariant
( ) ( )gen
ˆ ˆˆ= .µ +r k r k µ (22)
Then, choosing ( ) =µ µr k r , = 0r or ( ) = 0µr k ,
( ) =r k r , one gets both expressions (19) and (20), re-
spectively. A possible dependence of the coefficients
( )µr k and ( )r k on the momentum =p k can be used to
find the two other invariants. Indeed, if ( )µr k k , and
= 0r , one can get from (22) gen
ˆˆ ˆˆ= = h≡k k µ Σ ; if
( ) = 0µr k , and ( ) j×r k k e , gen
ˆ ˆ j , which recon-
structs j -projection of the operator ̂ . Therefore, choosing
various coefficients in the operator (22), we can represent
any invariants and their linear combinations.
Using the expansion (11) and the explicit forms of the
bispinors (9) and matrices (21), we come to the following
expression for the invariant (22):
( )
†
gen gen
† †† †
, , , , , ,, ,
, ,
ˆ= =
ˆ ˆ ,p p p a a a
d
a a b b′ ′ ′σ σ σ σ σ − σσ ′− σ
′σ σ
Ψ Ψ
= χ χ + χ χ
∫
∑ k kk k
k
r
r r
I
σ σ
(23)
where
( )
( )
2 2
/ 2 2 2
2 2
2 2
( )= ( )
( )
( ) .
( )
p a
c
mc mc mc
c
mc mc
µ
µ
ε − ± ε +
± + ε +
r kkr r k k
k
r k
r k k
k
(24)
In view of the fact that the operator (23) splits into the sum
( ) ( )
gen gen gen= p a+I I I , the invariants for particles ( )
gen
pI and
antiparticles ( )
gen
aI can be diagonalized with respect to the
spin variables independently. This can be performed using
the bispinors ,ν σχ , which are eigen bispinors of the matri-
ces ˆνr σ, and, hence, satisfy the equations
( ), , , ,ˆ = | | , = ( , ).p aν ν σ ν ν σχ σ χ νk kr r kσ (25)
The direction of the vector /p ar determines the quanti-
zation axis of the spin /ˆ p aσ of particles/antiparticles and at
p a≠r r the corresponding axes do not coincide. Moreover,
the matrices ˆνr σ turn out to be spin invariants of free mo-
tion which in the coordinate representation under the change
ˆ⇒k p are independent invariants for particles ( )
gen ˆ=p
prI σ
and antiparticles ( )
gen ˆ=a
arI σ , where ˆ( ).ν ν≡r r p
In the Cartesian coordinate system arbitrary vectors
( )µr k and ( )r k in Eq. (24) can be represented as de-
scribed ( ) = ( ) ( ) ( )x y zx y zν ν ν ν+ +r k k e k e k e ( = ,ν µ ),
where je ( = , ,j x y z ) are the basis vectors. This means that
the spin polarization of particles is determined by six inde-
pendent parameters, that are coordinates of these two vectors.
On the other hand, to describe polarization of free parti-
cles which are characterized by the wave vector k , it is
sometimes convenient to make a transition in the momen-
tum space to a local reper which is given by the three mu-
tually orthogonal unit vectors 1e , 2e and 3e that are related
to a unit vector 1 2 3= = / | |×e e e k k . It is easy to see that
the vectors 1e and 2e , remaining orthogonal are fixed with
the accuracy of rotations around the propagation axis
3e k . Then each of the vectors (24) has also another ex-
pansion, 1 2 3( ) = ( ) ( ) ( )ν ν ν νξ + η + ζr k k e k e k e . Respective-
ly, vector pr in Eq. (24) which determines electron spin
quantization axis in this local basis, takes the form
( ) ( )
( ) ( ) ( )
2
2
1 2 3
( )=
.
p
mc
mc
µ
µ µ
ε −
+ + ×
× ξ + η + ζ
kr r k r k
k e e k k e
Note that in the states corresponding to the invariant
( )
gen
pI with independent on k components ( ) =ν νξ ξk ,
( ) =ν νη ηk , ( ) =ν νζ ζk ( = ,ν µ ) in a local system,
among which there are vanishing components,
= = = 0µ µξ η ζ , spin quantization axis is determined by
the vector 1 2 3=p µξ + η + ζr e e e . This means that in
such a basis particle spin orientation is the same for all
momenta. In a particular case, when = = 0ξ η , = 1µζ ,
this corresponds to a spiral state in which particle spin and
momentum are parallel. In the case = 0µζ and = 1η a
particle’s spin and momentum are orthogonal. Such spin
state corresponds to the projection of the operator ̂ .
In the external field the spin operator ˆ( )νr k σ can com-
mute with the Hamiltonian (16) or (17) only at certain val-
ues of the coefficients ( )µr k and ( )r k which give the
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 739
A.A. Eremko, L.S. Brizhik, and V.M. Loktev
direction of the quantization axis with account of the con-
crete symmetry of the field.
5. General form of the Hamiltonian with the relativistic
corrections
In the previous Section we have used the ratio (15) as a
small parameter. In the non-relativistic case the inequality
/ 1k mc is also valid. Respectively, the Hamiltonians
(16) and (17) can be expanded using this ratio as a small
parameter as well. In this case the kinetic energy can be
written as
2 2 2 2
2
2 2( ) = 1 .
2 4
k kmc
m m c
ε + −
k
In view of Eq. (9), the convolution of the bispinors in
Eqs. (13) and (18) has to be expanded also:
( )( ) ( )
( ) [ ]
( )( ) ( ) ( )
( )( ) ( ) ( )
† (0)(0)
, ,
22 2
†
, ,, , 2 2 2 2
† (0) †(0)
, , ,, ,,
† (0) †(0)
, , ,, ,,
ˆ1 , = , ,
8 4
ˆ ,
2 (26)
ˆ .
2
p apa
a pap
i p a
m c m c
mc
mc
ν σ ′ν σ
′ ′ν σν σ
′ ′σ σ′ σσ
′ ′σ σ′ σσ
′Ψ Ψ
′− ′χ − + × χ ν
′ ′Ψ Ψ χ − χ
′ ′Ψ Ψ − χ − χ
kk
kk
kk
k k
k k
k k
k k k k
k k k k
σ
σ
σ
Supposing that both ratios 2| | /V mc and /k mc are of
the same order λ , one can neglect renormalization of
particles and antiparticles up to the second order, since, as
it was indicated above (see Eqs. (18) and (26)), the small-
ness of ( ) 3
, ( , )pW ′σ σ ′ λk k exceeds this accuracy. In the re-
sult, the Hamiltonian of particles (16) in the non-
relativistic approximation takes the form
( )
( )
2 2
(0) †2
,,
,
†
, ,,3
,
= 1
2
1 , ,
SO
kH k a a
m
a a
L
σσ
σ
′ ′ ′σ σ σσ
′ ′σ
− λ +
′+
∑
∑
kk
k
kk
k
k k
(27)
where their energy is counted from the rest energy 2mc .
The scattering amplitudes in the second sum, according to
(13) and (26), are given by expression
( ) ( )
( ) [ ]
†
, ,
(0)
2 (0)
,
, =
ˆ1 ,
2
SO
SO
V
i
′σ σ σ
′ ′σ
′ ′− χ ×
λ
′ ′× − − + λ × χ
k
k
k k k k
k k k k
σ
(28)
For the sake of simplicity here and below we consider par-
ticles only, therefore, the index “p” is omitted. The Hamil-
tonian for antiparticles can be derived in a similar way.
Operators †
,a σk and ,a σk in the Hamiltonian (27) are
creation and annihilation operators of an electron with the
momentum k in the spin state corresponding to the invar-
iant (23). This means that the spinors ( )σχ k in Eq. (28) are
the solutions of Eq. (25) (with = pν ), hence, they are eigen
spinors of the matrix ˆ( )r k σ with ( ) p≡r k r (see Eq. (24)).
Therefore, in the non-relativistic limit, vector ( )r k also has to
be expanded with respect to the small parameter. In particular,
up to the second order it is given by the expression
( ) (0) (0) (2)2 ,SO+ λk kr k r r (29)
( ) ( )(0) = ,µ +kr r k r k
( )( ) ( )(2) = ,µ − × × kr r k k k r k k k
which shows that in such a case the spinors also contain
the relativistic corrections.
As it has been pointed, the direction of the vector ( )r k
determines spin quantization axis for each k -state. This
axis can be obtained from the initial z -axis via the rotation
using some operator ω̂. Then, the spin functions σχ trans-
form into functions ˆ=σ σχ ωχ , and matrix ˆ( )r k σ , trans-
forms into matrix †ˆ ˆˆ ˆ( ) = ( ) zrω ω σr k kσ . In the latter ex-
pression the matrix ˆ zσ refers to a new (rotated or local)
coordinate system. Therefore, its spinors σχ are eigen
spinors for this Pauli matrix, and, hence, have the form ↑χ
and ↓χ , as it has been underlined above. From them the
spinors in the initial coordinate system can be calculated
by the action of the operator ˆ ( )ω k :
( ) ( )†ˆ= ,σ σχ ω χk k (30)
( ) ( ) ( ) ( )† †ˆ ˆ= , = .+ −↑ ↓χ ω χ χ ω χk k k k
It can be shown that the corresponding rotation (ne-
glecting the common phase multiplier) can be realized by
the above operator
( ) ( ) ( )
( )
( )
( ) ( )
( ) ( )
2
ˆˆˆ = ,
2
.
z
z
z
r
I i
r r
+
ω + +
≡ ×
k e r k d k
k
k k e r k
d k e r k
σ
(31)
Using vector r in the form (29) and choosing without the
loss of the generality (0)
kr in (29) as a unit vector, one can
write down the expansion for the matrix (31) with respect
to the parameter /k mc :
( ) ( ) ( )(0)(0) (2)ˆ ˆ ˆ2 .SOω ω + λ ωk k k
where
( )
( )
(0) (0)
(0)
2 (0)
†(0) (0)
2
ˆ1 ˆˆ = ,
2 1
ˆˆ ˆ = ,
z
I i
z
I
+
ω +
+
ω ω
k k
k
d
k
σ
(32)
740 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
On the theory of the Schrödinger equation with the full set of relativistic corrections
( )
(2) (0)(0)
(2)
2 (0)
ˆ1 ˆˆ = ,
2 1
Qz
Q I i
z
+
−
−+ ω + +
k kk
k
d d
k
σ
(33)
and, according to (29), the notations
(2) (2) (2)
(2) (0) (2)
(0) (0) (0)
( ) ( ) ( ) ( )
1= , = ,
2
= , = , = 1,2j j j j
z z
r z r
Q r
r z r
z j
±
+
±
+
×
k k k
k k k
k k k
k k k k
r r
d e r r e
(34)
are used. Matrix ˆ ( )ω k is unitary up to the required accura-
cy due to the relation
( )( ) ( ) ( ) ( )( ) ( )
† †(2) (0) (0) (2)ˆ ˆ ˆ ˆˆ= =
2
i
ω ω − − ω ωk k k k kΛ σ
(35)
in which
( )
(2) (2) (2) (0)
(2) (0)
(0) (0)= .
1 1
r z
z z
+ ×
− −
+ +
k k k k
k k
k k
d d
k d dΛ
Using equalities of vector algebra and definitions (34),
this expression can be reduced to the following form:
( )
( )
(0) (2)
(0) (2) (0)
(0)= .
1
z
z
× × +
+
k k
k k k
r e r
k r r r
k
Λ (36)
To calculate the relativistic corrections to the spinors,
we will use the expression (30) and matrix expansion (31)
taking into account that matrix (32) is unitary:
( )( ) ( )( )
( )( ) ( )( ) ( ) ( )( )
( )( )
† †(0)(0) (2)
,
† † †(0)(0) (2) (0) (0)
(0) (0)
2 ,
ˆ ˆ= 2 =
ˆ ˆ ˆ ˆ= 2 =
ˆ ˆ= , (37)
SO
SO
SOI i
σ σ
σ
σ
χ ω + λ ω χ
ω + λ ω ω ω χ
− λ χ
k
k
k k
k k k k
k
Λ σ
where the equality (35) has been used and
(0) (0) †
, ˆ= ( ( )) σσχ ω χk k .
After the substitution of obtained expression for ,σχk
into Eq. (28), the scattering amplitudes become
( ) ( )( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) [ ] ( ) ( ) ( )
†(0) (0)
, , ,
(0)
2 (0)
2
Th
Th
ˆ, = , ,
ˆ ˆ ˆ, = 1 , ,
2
, = , , ,
, = , , = .
SO
SO
BEL
BEL
V
I i
′σ σ σ ′ ′σ′ ′ ′− χ χ
λ
′ ′ ′− − + λ
′ ′ ′+
′ ′ ′ ′× −
k kk k k k k k
k k k k k k
k k k k k k
k k k k k k k k
Λ σ
Λ Λ Λ
Λ Λ Λ Λ
(38)
The Hamiltonian (27) with matrix elements ( ), ,′σ σ ′k k
in the form (38) contains all relativistic corrections up to
the second order both in the kinetic and potential energies.
Among them there are the well known corrections, such as
Darwin correction (the second term at the unit matrix), and
Thomas correction ( )Th , ′k kΛ [11–14]. Recall, the term
which contains the spin operator σ̂ in (38) is called SOI,
since it brings the interdependence between particle’s spin
and motion in a inhomogeneous potential. One can see,
however, this interaction includes not only Thomas correc-
tion, but also one more term ( ),BEL ′k k Λ which is fully
determined by the relativistic corrections to the spinors. So
far to our knowledge, the latter correction has not been
derived before and its role has not been investigated.
Comparing all relativistic corrections to the Schrödinger
Hamiltonian, it is clear that in the case of the non-relativistic
motion the correction to the kinetic energy is negligibly
small. Besides, the second order is, strictly speaking, satis-
factory only under the condition of small changes of the
fields on the distances of the order of the Compton wave-
length [12]. This condition, as a rule, is fulfilled for particles
in macroscopic (including crystal) fields, hence, the Darwin
correction in the potential energy is also small and usually
(as in Eq. (1)) is omitted. This explains why the Schrödinger
operator is adoptedly corrected with the one SOI term, only.
In particular, SOI provides relatively small, but experimen-
tally observable spin splitting of energy levels (bands) and,
essentially, determines spin quantization axis which is not
arbitrary, but depends on the form of the potential. Just the
spin polarization of particles in condensed systems is now
the subject of numerous studies and this is why investigation
of SOI effects is important.
To proceed, let us rewrite the Hamiltonian (27) in the
following form:
( ) ( )
2 2
†
,,
,
†
, ,,3
,
=
2
1 , ,
kH a a
m
V a a
L
σσ
σ
′ ′ ′σ σ σσ
′ ′σ
+
′ ′+ −
∑
∑
kk
k
kk
k
k k k k
(39)
where σ̂-dependent quantity
( ) ( ) ( )
( ) ( )( )
†(0) (0)
, , ,
†(0) (0) (0)
2, ,
ˆ, = ,
ˆ ˆ, ,SOI i
′σ σ σ ′ ′σ
σ ′ ′σ
′ ′χ χ ≈
′≈ χ + λ χ
k k
k k
k k k k
k k
Λ σ
(40)
includes the matrix of the spin-orbit scattering. The latter
contains both corrections, according to the definitions (38):
Th ( , )′k kΛ and ( , )BEL ′k kΛ . In (40) spinor (0)
,σχk , as in Eq.
(37), is a non-relativistic eigen spinor of the matrix (0) ˆkr σ .
Hence, the unit vector (0) = ( ) ( ) ( )x x y y z zγ + γ + γkr k e k e k e
determines the spin quantization axis with the guiding co-
sines ( )jγ k that are given by the sum of the corresponding
vector coefficients ( )µr k and ( )r k from Eq. (22). It is
easy to find that
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 741
A.A. Eremko, L.S. Brizhik, and V.M. Loktev
( )
( )
( )
/2
/2(0) (0)
, ,
/2
1
e
2, = e ,
1
e
2
= 1,
iz
i
iz
− ϕ
σϕ
σ σ
ϕ
+ σγ
σ
χ ϕ ϑ ≡ χ −σγ
σ ±
k
kk kk k
k
k
k
where tan = ( )/ ( )y xϕ γ γk k k . Using the equalities ( ) =xγ k
sin cos= ϑ ϕk k , ( ) = sin sinyγ ϑ ϕk kk , ( ) = cos ,zγ ϑkk one
can introduce spin variables (angles) ϑk and ϕk , on which
the spinors in the zero approximation depend:
(0) (0)
, ,
cos e sin
2 2= , = .
e sin cos
2 2
i
i
− ϕ
+ −
ϕ
ϑ ϑ −
χ χ
ϑ ϑ
k kk
k k
k kk
(41)
In fact, these variables can be considered and indeed are
the free parameters.
Such a picture is radically changed in the presence of the
external potential when these parameters become fixed. This
follows from the fact that the stationary electron states can be
found diagonalizing the Hamiltonian (39) to the form
†=H E a aν ν ν
ν
∑ (42)
where Eν are the energies of these states, and index
= ,nν σ is a set of the quantum numbers with the spin
number σ taking two values.
For diagonalization of the Hamiltonian (39) it is con-
venient to choose the spinors (41) for which the matrix
element (40) is proportional to , ′σ σδ . They diagonalize the
invariant (23), ( )
gen
pI , and in non-relativistic limit are deter-
mined as eigen spinors of the operator (0) ˆkr σ with (0)
kr giv-
en in Eq. (29). Therefore the operator ( )
gen
pI should com-
mute with the Hamiltonian (39). This condition implies the
restrictions on the free parameters values, or the vectors
( )µr k and ( )r k ; on the other side, such a condition results
in the equation which determines the vector (0)
kr that coin-
cides with the spin quantization axis and, hence, the spin
variable dependence appears.
The analysis of obtained equation depends on the struc-
ture of the Fourier transformation ( )V ′−k k of the poten-
tial and has to be performed in the general case in the cur-
vilinear coordinate system over the surface in which the
equipotential surfaces are formed, and perpendicular to
them directions are determined by the potential gradients
in each point. This allows to find the solutions at least in
the vicinity of singular points or lines that are characteristic
for this potential, if not in their whole definition area. The
main difficulty here is the dependence of the local reper
orientation on the coordinates of the studied spatial point.
This is the reason why it is impossible to find the general
expression for SOI in the coordinate space for the general
form potential. Below we consider one of the simplest but
nevertheless actual case of electrons in quasi-2D system.
6. Two-dimensional motion
Any 2D system, in fact is quasi-2D with a finite spatial
width, for example, in z direction. In such a case the po-
tential in the Hamiltonian (39) reflects a translational
symmetry in xy -plane: ( ) ( ) ( )= , = ,V V z V z⊥ ⊥ ⊥+r r r l
where
=1,2
= j j
j
l⊥ ∑l a is a translation vector in 2D lattice
structure with basis vectors ja ( jl are integer numbers).
Such potentials can be represented by Fourier series
( ) 0
0
, = ( )e = ( ) ( , ),
( , ) = ( )e
i
i
V z V z V z V z
V z V z
⊥ ⊥
⊥ ⊥ ⊥⊥
⊥
⊥ ⊥
⊥ ⊥ ⊥
≠⊥
+∑
∑
g r
g
g
g r
g
g
r r
r
(43)
where vectors
=1,2
= j j
j
n⊥ ∑g b are defined by basis vectors
jb of the corresponding reciprocal lattice with jn being
integer numbers. Expansion coefficients in Eq. (43), which
in a general case can depend on z , are determined by the
formula
( )
cell
1( ) = , e iV z V z dxdy
S
− ⊥ ⊥
⊥⊥ ∫ g r
g r
where integration is carried out over unit cell of an area
cellS .
According to (43), the potential can be represented as a
sum of two components, namely zero harmonic 0 ( )V z and
periodic in plane part ( , )V z⊥ ⊥r . In this case the Fourier
transformation of the potential is
( ) ( ) ( )2
, 0= , ,z z z zV L V k k V k k′ ⊥ ⊥ ⊥⊥ ⊥
′ ′ ′ ′− δ − + − −k kk k k k
which allows to present the Hamiltonian (39) in the form
(0)
per2DH H V= + where
( ) ( )
( ) ( )
2 2
(0) †
,,2
,
†
, 0 , ,,
,
†
per , ,,3
, ,
=
2
1 , ,
1= , .
D
z z
H a a
m
V k k a a
L
V V a a
L
σσ
σ
′ ′ ′ ′σ σ σσ⊥ ⊥
′ ′σ
′ ′ ′⊥ σ σ σσ
′ ′σ σ
+
′ ′+ δ −
′ ′−
∑
∑
∑ ∑
kk
k
k k kk
k
kk
k k
k
k k
k k k k
(44)
with ( ), ,′σ σ ′k k is given in Eq. (40) and, as above,
= z zk⊥ +k k e .
As the first stage of finding the energy spectrum let us
consider the Hamiltonian (0)
2DH only. The matrix element
(40) in (0)
2DH is calculated at = z zk⊥ +k k e and
742 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
On the theory of the Schrödinger equation with the full set of relativistic corrections
= z zk⊥′ ′+k k e . This leads to the following expressions
for Thomas part ( ) ( )[ ]Th , = z z zk k ⊥′ ′− ×k k e kΛ and ad-
ditional part ( ) ( ) ( ), = , ,BEL z zk k⊥ ⊥′ ′−k k k kΛ Λ Λ of SOI
(see (38)).
Spin states can be chosen arbitrary, and it is convenient
to take such spinors (0)
,σχk (41) that satisfy the condition
( ) ( ), ,, , = , ,z z z zk k k k′ ′σ σ ⊥ σ σ σ ⊥′ ′δk k . In view of the
matrix structure and ortho-normalization of spinors (0)
,σχk ,
this condition is fulfilled, provided, first, the zero order
spinors in (44) does not depend on zk , that means the de-
pendence of its spin quantization axis on ⊥k only. Second-
ly, diagonalization of (0)
2DH with respect to spin numbers is
performed if the spinors (0)
,σ⊥
χk are the eigen spinors of the
matrix ˆ( , )′k kΛ σ . Since (0)
,σ⊥
χk are defined as eigen spinors
of the matrix (0) ˆ
⊥kr σ, this condition is fulfilled provided the
commutator (0)ˆ ˆ[ ( , ) , ]
⊥
′ kk k rΛ σ σ is equal zero which leads to
the equality (0)( , ) = 0
⊥
′ × kk k rΛ .
Taking into account Eq. (36), Eq. (29) and condition
(0) (0)=
⊥k kr r the following expressions for the additional
correction can be obtained
( ) ( )
( )
(0)
(0)
(0)
(0)
, = , ,
, ,
,
1
BEL z z
z z z
k k
k k
z
⊥⊥
⊥⊥
⊥
⊥
′ ′× +
′×
+
+
k
k
k
k
k k r k
r e k
r
Λ λ
λ (45)
where
( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )( ) ( )
(2) (2)
, ,
2 2
, , =
= ,
=
,
= .
z z k kz z
z z z z
z z
z
z z z z
k k
k k k k
⊥ ′⊥ ⊥
⊥ ⊥
⊥ ⊥ µ ⊥ ⊥
⊥ ⊥ µ ⊥ ⊥
⊥ ⊥ µ ⊥
′ − =
′ ′ ′− + −
− +
+ −
′ − × ×
k kk r r
k k
k r k e r k e k
r k k r k k e
k r k e e r k e e
λ
λ λ
λ
λ
(46)
Therefore, in this 2D case the SOI vector (see (38)) in
Eq. (40) is
( )
( ) ( )
( )
( ) ( )
( )
(0)
(0) (0)
(0)
(0)
(0) (0)2 2
(0)
, =
=
1
1
z
z z z
z
z z
k k
z
k k
z
⊥⊥
⊥ ⊥⊥ ⊥
⊥
⊥⊥
⊥⊥ ⊥
⊥
′
× ′− × + × + + +
′× ′ ′+ − × + +
k
k k
k
k
k k
k
k k
r e k
e k r k r
r e k
r k r
Λ
λ
λ
λ
λ
from which the condition
( ) ( )
[ ] ( ) ( )( )
( ) ( ) ( )( )
(0)
(0) (0) (0)
(0) (0)2 2
, =
= 0
z z
z
z z
k k
k k
⊥
⊥ ⊥ ⊥⊥ ⊥ ⊥
⊥ ⊥ ⊥ ⊥
′ ′× − ×
× × × + − +
′ ′ ′+ − −
k
k k k
k k
k k r
e k r k k r r
k k r r
Λ
λ λ
λ λ
(47)
is obtained. It can be satisfied only when the equality
( ) = 0⊥′ kλ is valid. The latter, according to Eq. (46), im-
plies constrains on the vectors ( )µ ⊥r k and ( )⊥r k , which
have to satisfy the equations
( ) ( )= 0, = 0.z zµ ⊥ ⊥×r k e r k e
This immediately leads to expressions ( ) =⊥r k
= ( ) ( )x yx y⊥ ⊥+k e k e and ( ) = ( ) zzµ ⊥ µ ⊥r k k e . Hence,
the guiding cosines of a unit vector (cp. Eq. (29))
( ) ( )
( ) ( ) ( )
(0) = =
= x y zx y z
µ ⊥ ⊥⊥
⊥ ⊥ µ ⊥
+
+ +
kr r k r k
k e k e k e
(48)
are determined by the three parameters ( ) = ( )x x⊥ ⊥γ k k ,
( ) = ( )y y⊥ ⊥γ k k and ( ) = ( )z z⊥ µ ⊥γ k k . With such al-
lowed vectors values the term proportional to ( ')z zk k− in
relation (47), is identically equal zero. Therefore, there are
no additional conditions for the parameters ( )x ⊥k ,
( )y ⊥k , ( )zµ ⊥k , so that they are indeed arbitrary.
Substituting the found values of vectors ( )µ ⊥r k and
( )⊥r k into (45), one calculates the vector ( , )′k kΛ which
characterizes SOI in matrix (40) for 2D case:
( ) ( ) ( )
( )
[ ]
(0)
2
(0)
(0)
, = ,
= .
1
D z z
z
z
k k f
f
⊥ ⊥
⊥ ⊥
⊥
⊥
′ ′−
×
+
k
k
k
k k k r
e k r
k
e r
Λ
(49)
As seen, it provides the diagonal form of , ( , , )z zk k′σ σ ⊥ ′k
with respect to σ and each spin state is described by its
own Hamiltonian
( )
( ) ( ) ( )
2 2 2
†
, ,, ,
,
†
0 , ,, ,
=
2
1 1 ,
z
kk zzkz
z z z z kk zzkz
k
H a a
m
V k k i k k f a a
L
⊥
σ σσ ⊥⊥
⊥
′⊥ σσ ⊥⊥′
+ +
′ ′+ − − σ −
∑
∑
kk
k
kk
k
k
so that (0)
2DH Hσ
σ
=∑ .
The further diagonalization of the Hamiltonian with re-
spect to zk projections can be performed using the unitary
transformations
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 743
A.A. Eremko, L.S. Brizhik, and V.M. Loktev
( )
( ) ( )
, , , , , ,
, , , , ,
= ,
= .
k zz
z z
kz
a k a
k k
σ ν σ ν σ⊥ ⊥ ⊥
ν
∗
′ ′ν σ ν σ ν ν⊥ ⊥
ψ
ψ ψ δ
∑
∑
k k k
k k
(50)
Here the coefficients , , ( )zkν σ⊥
ψ k satisfy the equation
( )
( )
( ) ( ) ( )( ) ( )
( ) ( )
2 2 2
, ,
(0)
0 , ,
, , ,
2
1 1
= .
z
z
z z z z zSO
kz
z
k
k
m
V k k i k k f k
L
E k
⊥
ν σ⊥
⊥ ν σ⊥
′
ν σ ⊥ ν σ⊥
+
ψ +
′ ′ ′+ − − − σλ ψ =
ψ
∑
k
k
k
k
k
k
Its solution can be found after the transition to the coordi-
nate representation
, , , ,
1( ) = e ( )ik zz z
kz
z k
Lν σ ν σ⊥ ⊥
ψ ψ∑k k
which leads to the stationary 1D SchE for each spin state
( ) ( )
( ) ( )
2 22 2
(0) 0
02
, ,
( )
2 2
= .
SO
dV zd V z f
m m dzdz
z E z
⊥
⊥
σ σ⊥ ⊥
− + + + σλ ×
× ψ ψk k
k k
(51)
The fourth term in the l.h.s. of the latter equation is ap-
peared due to the SOI. Namely this non-relativistic equa-
tion determines the wave functions , , ( )zν σ⊥
ψ k and eigen
values ,= ( )E Eν σ ⊥k with account of SOI and coincides
with equation obtained in [16,18] where the general analyt-
ical solution of the DE was found for the given problem.
Notice, there can be situations when the electron spinor
components will be represented as linear combination of
functions , , ( )zν +⊥
ψ k and , , ( )zν −⊥
ψ k .
In the case of the QW potential the discrete eigen num-
bers = nν correspond to the bound states of electrons
which are trapped by a QW and propagate as free particles
in its plane. In fact, the ensemble of such electrons is a 2D
electron gas. Obviously, the solutions of Eq. (51) depend
on the QW form. The presence ( 0 0( ) ( )V z V z− = ) or ab-
sence ( 0 0( ) ( )V z V z− ≠ ) of the inverse symmetry of the QW
is the main factor of Rashba spin splitting of 2D electron
bands [1–4]:
( ) ( ) ( )
2 2
, = 0 ,
2n n SOE E f
m
⊥
σ ⊥ ⊥+ −σλ
kk k (52)
(0)= .SO QWSOaλ λ
Here function ( )f ⊥k (see (49)) introduces the explicit
dependence of the splitting on electron spin state = 1σ ± .
Parameter QWa in Eq. (52) depends on the form of the QW
[15,16,18] and characterizes its asymmetry, which can be
intrinsic or caused by external electric field perpendicular
to xy -plane. The expression (52) without any assumption
about the connection with a spin state, is the basic for
study of SOI effects in a 2D electron gas using the Hamil-
tonian of free 2D particles,
( )(0) †
, , ,2 , ,
, ,
= .n nD n
n
H E a aσ ⊥ σσ ⊥⊥
σ⊥
∑ kk
k
k (53)
Here the operators †
, ,na σ⊥k and , ,na σ⊥k are creation and
annihilation operators of electrons with wave vector ⊥k in
spin states determined by the spinors (41) whose quantiza-
tion axis is parallel to vector (0)
⊥kr .
Usually the Hamiltonian of a 2D electron gas is written
using the operators ,a ↑⊥k and ,a ↓⊥k which are related to
the spinors (0)
↑
χ and (0)
↓
χ with the quantization axis in z -
direction of the initial Cartesian system. Naturally, in such
a case the expression (53) is not diagonal with respect to
spin = ,σ ↑ ↓. The transformation from the operators
†
, ,na σ⊥k and , ,na σ⊥k to the operators ,a ↑⊥k and ,a ↓⊥k is
performed by the unitary matrix
, ,
, ,
cos e sin
2 2= ,
e sin cos
2 2
i
i
aa
a a
− ϕ⊥ ⊥⊥
+ ↑⊥ ⊥
ϕ− ↓⊥ ⊥ ⊥ ⊥⊥
ϑ ϑ
ϑ ϑ −
k kk
k k
k k k kk
where the relations are used
, ,
(0)
,,
, ,
,
,
cos e sin
2 2=
e sin cos
2 2
= .
i
i
a a
a
a a
a
a
− ϕ⊥ ⊥⊥
+ −⊥ ⊥
σσ ⊥⊥ ϕσ ⊥ ⊥⊥
+ −⊥ ⊥
↑⊥
↓⊥
ϑ ϑ
−
χ =
ϑ ϑ
+
∑
k kk
k k
kk
k kk
k k
k
k
With account of the explicit form (49) of functions ( )f ⊥k
in the representation of the operators †
,a σ⊥k and ,a σ⊥k
( = , )σ ↑ ↓ ), the Hamiltonian (53) takes the form
( )
2 2 ,(0) (2 )† †
2 , ,
,
ˆ= 0
2
D
D SO
a
H a a E V
am
↑⊥⊥
↑ ↓⊥ ⊥ ↓⊥⊥
+ +
∑
k
k k
kk
k
(54)
which contains SOI determined by the confinement potential
[ ]( )(0)
(2 ) (0)
(0)
ˆ ˆ=
1
zD
SOSO
z
V
⊥ ⊥
⊥
⊥
×
−λ
+
k
k
k
e k r
r
e r
σ. (55)
744 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
On the theory of the Schrödinger equation with the full set of relativistic corrections
The latter expression evidently shows the mentioned above
dependence of SOI on the wave vector components and
angles
⊥
θk ,
⊥
ϕk , i.e., on the spatial and spin degrees of
freedom, respectively.
Expression (55) indicates directly that SOI is absent if
the quantization axis is parallel to one of the vectors ze or ⊥k .
As it follows from Eq. (48), in the first case the equality
(0) = = zµ⊥kr r e is valid, and the invariant (22) reduces to z-
component of invariant (19): gen
ˆ ˆ= zµ . In the second case
at (0) = ( ) = / | |⊥ ⊥ ⊥⊥kr r k k k , this invariant becomes
gen
ˆ ˆ= ⊥k .
Above the vector (0)
⊥kr has been expressed in the Cartesian
coordinate system in the form of the expansion with respect to
unit vectors xe , ye and ze . In a 2D system it is convenient to
use a local reper with another three unit vectors
1 2 1 3= / | |, = , = ,z z⊥ ⊥ ×e k k e e e e e (56)
where ( ) ( ) ( )(0)
1 1 2 2 3 3= ⊥ ⊥ ⊥⊥
γ + γ + γkr k e k e k e and the
coefficients jγ ( = 1,2,3j ), playing the role of the guiding
cosines. It follows from expression (55) that SOI attains its
maximum value at (0)
2=
⊥kr e , where 2e is defined by (56).
This corresponds to the invariant gen
ˆ ˆ= zε . On the contra-
ry, SOI vanishes, (2 )ˆ = 0D
SOV , if (0)
2⊥
⊥kr e . This convincing-
ly demonstrates the possibility of continuous changes of
the SOI value under the smooth changes of the spin state,
when, according to (22), the generalized invariant is repre-
sented in the form of a linear combination of operators µ̂
and ̂ . This combination has to be either predefined or
given for each concrete situation, and in the general case
this linear combination has fundamental meaning, only.
7. Conclusions
This study demonstrates that SOI conventionally used
in SchE, does not describe all possible spin electron states.
If the potential operator commutes with one of the spin
invariants, the SOI operator has to be generalized to take
into account all possible states. This is reflected in the fact
that the vector (0)
⊥kr is not fixed a priori, and, hence, spin
variables in the operator (54) can take arbitrary values.
Our results prove the spin lability of 2D electrons and
show that their spectrum depends on the direction of the
quantization axis which is determined by such factors as
carriers concentration, form of the potential, presence of
external fields, etc. In a free 2D electron gas the most ad-
vantageous energy state is the state corresponding to the
invariant ẑ , when SOI operator reduces to Rashba SOI, as
it has been indicated in [16]. This interaction arises when
only zero-harmonic of the potential (43) is taken into ac-
count, and, in this sense it can be considered as the zero
approximation of SOI in real 2D (or quasi-2D) systems.
The Hamiltonians (53), (54) describe free 2D electrons,
whose two-dimensional behavior in a homogeneous iso-
tropic plane is determined by the QW, which means their
localization in the potential 0 ( )V z . The full Hamiltonian
includes also the periodical part, ( , )V z⊥ ⊥r , and is given by
the expression (0)
2 per2D DH H H V→ = + , where the expres-
sion for perV is written in (44).
After the transition to the operators †
, ,na σ⊥k and , ,na σ⊥k
with the help of the unitary transformation (50), the Hamil-
tonian 2DH describes 2D electrons in a periodic field with
account of SOI in the form (55), and from the potential
perV of crystal lattice. In literature, operator (2 )ˆ D
SOV is usual-
ly called Rashba SOI, and operator which results from the
potential perV , is called Dresselhaus SOI.
The explicit calculation of SOI with account of periodic
potential requires special consideration and is not the subject
of the present paper. Nevertheless, it is worth mentioning that
calculation of quasi-particle bands in such potential even
without account of SOI is a rather difficult problem which is
based on a spatial symmetry group of the crystal and can be
done only approximately. For bulk crystals when 0 = constV
and SOI is not present in the zero approximation, spin-orbit
splitting of such bands has been analyzed in details by
Dresselhaus in Ref. 9 for zincblende structure and by Rashba
and Sheka in Ref. 10 for wurtzite-type crystals. These papers
were based on the group theory using the kp perturbation
approximation with account of point symmetries of the
Brillouin zone. The relativistic corrections to the spinors were
taken into account in the first order perturbation theory. Such
calculations are rather cumbersome and accurate calculation
of the generalized SOI based on the secondary quantization
representation taking into account periodic potential, will be
reported elsewhere.
Acknowledgements
We express our sincere thanks to E.I. Rashba for read-
ing the manuscript and useful comments. This work has
been done under the Fundamental Research Projects
No 0117U000236 and No 0117U000240 of the Depart-
ment of Physics and Astronomy of the National Academy
of Sciences of Ukraine.
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746 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
https://doi.org/10.1038/nmat3051
https://doi.org/10.1103/PhysRevLett.109.096803
https://doi.org/10.1103/PhysRevLett.108.246802
https://doi.org/10.1088/2053-1583/3/4/042001
https://doi.org/10.1103/PhysRev.100.580
https://doi.org/10.1016/j.aop.2015.07.007
https://doi.org/10.1016/j.aop.2016.03.008
https://doi.org/10.1063/1.4980861
https://doi.org/10.1103/PhysRev.149.491
1. Introduction
2. Relativistic Hamiltonian
3. Approximate renormalization
4. Spin invariants
5. General form of the Hamiltonian with the relativistic corrections
6. Two-dimensional motion
7. Conclusions
Acknowledgements
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