Theorem about spin splitting of energy levels within Kildal-Bondar model
The semiconductor Cd₃As₂ is known as a zero-gap material like HgTe or α-Sn but with the tetragonal lattice and in various crystalline forms. One of the forms has no symmetry center, and just this form is stable under ordinary conditions. So, every of its energy bands is split into a pair of spin...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2008
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| Zitieren: | Theorem about spin splitting of energy levels within Kildal-Bondar model / O.V. Dvornik, G.P. Chuiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 3. — С. 311-313. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-1190532017-06-04T03:03:36Z Theorem about spin splitting of energy levels within Kildal-Bondar model Dvornik, O.V. Chuiko, G.P. The semiconductor Cd₃As₂ is known as a zero-gap material like HgTe or α-Sn but with the tetragonal lattice and in various crystalline forms. One of the forms has no symmetry center, and just this form is stable under ordinary conditions. So, every of its energy bands is split into a pair of spin subbands owing to the removal of the Kramers degeneration. The theory predicts that the total sum of all spin splittings will be equal to zero, whereas the modeling shows the peculiar dependences of spin splittings on the direction and modulus of the wave vector. 2008 Article Theorem about spin splitting of energy levels within Kildal-Bondar model / O.V. Dvornik, G.P. Chuiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 3. — С. 311-313. — Бібліогр.: 6 назв. — англ. 1560-8034 PACS 71.20.-b, 71.18.+y http://dspace.nbuv.gov.ua/handle/123456789/119053 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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The semiconductor Cd₃As₂ is known as a zero-gap material like HgTe or α-Sn
but with the tetragonal lattice and in various crystalline forms. One of the forms has no
symmetry center, and just this form is stable under ordinary conditions. So, every of its
energy bands is split into a pair of spin subbands owing to the removal of the Kramers
degeneration. The theory predicts that the total sum of all spin splittings will be equal to
zero, whereas the modeling shows the peculiar dependences of spin splittings on the
direction and modulus of the wave vector. |
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Article |
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Dvornik, O.V. Chuiko, G.P. |
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Dvornik, O.V. Chuiko, G.P. Theorem about spin splitting of energy levels within Kildal-Bondar model Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Dvornik, O.V. Chuiko, G.P. |
| author_sort |
Dvornik, O.V. |
| title |
Theorem about spin splitting of energy levels within Kildal-Bondar model |
| title_short |
Theorem about spin splitting of energy levels within Kildal-Bondar model |
| title_full |
Theorem about spin splitting of energy levels within Kildal-Bondar model |
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Theorem about spin splitting of energy levels within Kildal-Bondar model |
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Theorem about spin splitting of energy levels within Kildal-Bondar model |
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theorem about spin splitting of energy levels within kildal-bondar model |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/119053 |
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Theorem about spin splitting of energy levels within Kildal-Bondar model / O.V. Dvornik, G.P. Chuiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 3. — С. 311-313. — Бібліогр.: 6 назв. — англ. |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT dvornikov theoremaboutspinsplittingofenergylevelswithinkildalbondarmodel AT chuikogp theoremaboutspinsplittingofenergylevelswithinkildalbondarmodel |
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2025-07-08T15:09:13Z |
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2025-07-08T15:09:13Z |
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1837091892675739648 |
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 3. P. 311-313.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
311
PACS 71.20.-b, 71.18.+y
The theorem on the spin splitting of energy levels
within the Kildal-Bodnar model
O.V. Dvornik and G.P. Chuiko
Kherson National Technical University, Department of General and Applied Physics
24, Beryslavske Shosse, 73008 Kherson, Ukraine; e-mail: olga_dvornik@mail.ru
Abstract. The semiconductor Cd3As2 is known as a zero-gap material like HgTe or α-Sn
but with the tetragonal lattice and in various crystalline forms. One of the forms has no
symmetry center, and just this form is stable under ordinary conditions. So, every of its
energy bands is split into a pair of spin subbands owing to the removal of the Kramers
degeneration. The theory predicts that the total sum of all spin splittings will be equal to
zero, whereas the modeling shows the peculiar dependences of spin splittings on the
direction and modulus of the wave vector.
Keywords: center of symmetry, spin splitting, Kramers degeneration.
Manuscript received 27.05.08; accepted for publication 20.06.08; published online 30.09.08.
1. Introduction
A band model suitable for crystals with and without
symmetry center has been presented in [1]. This model
describes uniaxial semiconductors by taking into account
their lattice deformations, spin-orbital interaction, and
the splitting within the kP-approach. One of the features
of the Hamiltonian in [1] is the account of removing the
Kramers degeneration in a crystal modification without
symmetry center. This allows the partition of some
energy band into two subbands having opposite-spin
states. The knowledge about this splitting is essential for
the newest branches of electronics dealing with the
opposite-spin states in crystals or nanostructures. We are
going to prove here a general theorem on such a splitting
within the model developed in [1].
The predictions of the theorem will be illustrated
by the example of Cd3As2. We have two reasons for this:
1. This material has an inverted band structure like
those in α-Sn, HgTe or HgSe what is interesting by
itself [2].
2. This material can have few tetragonal structures,
and one of them has no symmetry center (I41cd-
12
4υC ) and is stable under normal conditions [3].
2. Theorem
It has been shown in [1] that the generalized kP-
Hamiltonian for a uniaxial material can be presented in
the rational canonical form (alias Frobenius’ form):
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
β
α
H
H
H
0
0
. (1)
It is worth to note that βα = HH if the symmetry center
exists. Each of the submatrices has size [4×4]:
⎟⎟
⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎜
⎝
⎛
−
−
−
−
=
⎟⎟
⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎜
⎝
⎛
−
−
−
−
= β
β
α
α
3
2
1
0
3
2
1
0
100
010
001
000
;
100
010
001
000
a
a
a
a
H
a
a
a
a
H . (2)
Here, ja are the coefficients of jε (j = 0, 1, 2, 3) in two
characteristic polynomials in the spherical system of
coordinates like that in [1]:
( )
( )
,)(sin))1(
3
(
3
)))1(
3
(
3
2
3
2)(sin
3
2
))(cos((sin
),(
222
2
21
2242
34
Pk
Pk
Pk
P
g
g
θη−
∆
+δ
∆
−ε×
×η−
∆
+δ∆ε⎟
⎠
⎞
⎜
⎝
⎛+
+ξ∆⎟
⎠
⎞
⎜
⎝
⎛+θξ∆η⎟
⎠
⎞
⎜
⎝
⎛−
−θη+θ−
−εδ−∆−ε−ε=θε
−
−
−
−
α
(3)
( )
( )
.)(sin))1(
3
(
3
)))1(
3
(
3
2
3
2)(sin
3
2
)()cos((sin
),(
222
2
21
2242
34
Pk
Pk
Pk
P
g
g
θη−
∆
+δ
∆
−
−εη−
∆
+δ∆ε⎟
⎠
⎞
⎜
⎝
⎛+
+ξ∆⎟
⎠
⎞
⎜
⎝
⎛+θξ∆η⎟
⎠
⎞
⎜
⎝
⎛+
+θη+θ−
−εδ−∆−ε−ε=θε
−
−
−
−
β
(4)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 3. P. 311-313.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
312
These polynomials are slightly different only for the
structures without symmetry center.
Additionally, we mention that εg, P, ∆ are three
known parameters of Kane's model [4]: the energy gap,
impulse matrix element, and spin-orbital interaction
parameter, respectively; δ is the crystal field parameter
introduced by Kildal [5]; ξ is the another crystal field
parameter which can be either non-zero in the absence of
a symmetry center or equal to zero otherwise [1]; and η
is the parameter of the uniaxial lattice deformation [1].
The energy ε is reckoned from the top of the highest
valence band. The subscripts (α, β) for the coefficient
a1α, β correlate with two opposite spin states.
The model [1] (ξ ≠ 0, η ≠ 0, δ ≠ 0) generalizes both
the model for cubic crystals [4] (ξ = 0, η = 0, δ = 0) and
the model for uniaxial crystals with symmetry center [5]
(ξ = 0, η = 0, δ ≠ 0) to the crystals without symmetry
center and with axial deformations of lattices.
Let us to denote the solutions of Eqs. (3) and (4) as
εn,α and εn,β, respectively (n = 1, 2, 3, 4). It is well known
that the sum of these solutions is equal to a3 (as it is the
trace of a submatrix). Thus, we have
δ−∆−ε=ε=ε ∑∑ βα g
n
n
n
n ,, (5)
simply for the reason that this coefficient is identical for
both polynomials. Moreover, it follows from relation (5)
that
( ) .0,, ∑∑ ==ε−ε βα
n
n
n
nn s (6)
Rule (6) is trivial if the center of symmetry exists,
because each element of the sum is zero (sn = 0).
Nevertheless, this rule would be operating even if the
symmetry center is absent and elements of the sum are
non-zeros (sn ≠ 0). Now we can formulate the following
proposition:
eV,sn
k
θ
Fig. 1. Dependences of si on the direction (θ) and modulus of
the wave vector.
Table.
εg, eV P,
eV⋅m
∆,
eV δ, eV ξ, eV η a, Å c, Å
–0.13 7.0×
×10–10 0.27 0.095 0.035 1.005648 12.6461 25.4908
Theorem. The total sum of the spin splittings of all
subbands must be zero with Hamiltonian (1).
This theorem is true under condition that the
absence of symmetry, which is the reason for removing
the degeneration by spin, does not change anyway the
sum of parameters presented by (5): εg – ∆ – δ = a3.
The numerical values of the parameters
characterizing Cd3As2 which were used in our
computations are presented in the Table according to [1].
3. Results of computations and their discussion
Equations (3), (4) allow the direct solutions. We
calculated the dependences of the elements of sum (6)
(i.e., βα ε−ε= ,, nnns ) on the modulus and direction of the
wave vector by using the numerical values of parameters
given in the table. The results are shown in Figs. 1 and 2.
The direct computer verification of the obtained
rule (6) showed their correctness within the limits of the
computing accuracy. It is clearly visible even from the
different signs of these splittings.
Each sheet of Fig. 1 which corresponds to one of
the energy bands demonstrates an extremum depending
on the modulus of the wave vector k, whereas the
dependences of the splittings on the directions are
monotonic: they simply increase from the main axis
direction (θ = 0) up to the directions normal to them
⎟
⎠
⎞
⎜
⎝
⎛ π
=θ
2
.
eV,sn
k
Fig. 2. Dependences of sn on the modulus of the wave vector.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 3. P. 311-313.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
313
The magnitudes sn of the spin splitting are
comparable with the magnitude of the parameter ξ ≈
0.035 eV. Our calculations convince us also that they
depend on this parameter much stronger than on the
parameter δ or η.
So, both the theoretical analysis and the
computation testify that the absence of a symmetry
center is not able to create such a splitting of energy
levels, for which the total sum could be non-zero, within
the model [1].
Let us also to remark that an analogous statement
can be proved as for the influence of a tetragonal
deformation on shifts of the energy levels. Indeed, such
a deformation (η) cannot change the coefficient a3 = εg –
–∆ – δ, generally speaking. It would be right if none of
them does not depend on such a deformation directly.
However, sometimes it may take place, for instance, for
δ = δ(η) [6].
Acknowledgements
The authors wish to thank Dr. Stanislaw Shutov for for
many fruitful discussions, as well as for his help in the
preparation of the manuscript.
References
1. G.P. Chuiko, V.V. Martyniuk, and V.K. Bazhenov,
Basic peculiarities of energy band spectra within
generalized Kildal's model for semiconductors with
one main axis // Semiconductor Physics, Quantum
Electronics & Optoelectronics 8(2), p. 28-31
(2005).
2. G.P. Chuiko, I.A. Teplinskaya, Topological
transition and related with it singularity in the
density of states of conduction band as features of
the inverse tetragonal semiconductor – cadmium
arsenide // Fizika, Tekhnika Poluprovodnikov
17(6), p. 1123-1125 (1983) (in Russian).
3. G.P. Chuiko, N.L. Don, and V.V. Ivchenko,
Ordering and polytypism in V
2
II
3 BA crystals //
Functional Materials 12(3), p. 454-460 (2005).
4. E.O. Kane, Band structure of indium antimonide //
J. Phys. Chem. Solids 1, p. 249-261 (1957).
5. H. Kildal, Band structure of CdGeAs2 near k = 0 //
Phys. Rev. B 10(12), р. 5082-5087 (1974).
6. G.P. Chuiko, O.V. Dvornik, Coupling between
crystal splitting of valence bands and tetragonal
deformation of lattice for compounds V
2
II
3 BA //
Fizika, khimiya tverdogo tela 3 (4), p. 682-686
(2002) (in Russian).
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