Manifestation of deformation effect in band spectra in crystals with inhomogeneous bonding
The effect of the shear strains on the energy spectrum of the strongly anisotropic SbSI crystal has been investigated by group-theoretical method in combination with the Pikus method of invariants. The first-principles local density approximation has been implemented to determine the band structu...
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| Zitieren: | Manifestation of deformation effect in band spectra in crystals with inhomogeneous bonding / D.M. Bercha, K.Z. Rushchanskii, I.V. Slipukhina, I.V. Bercha // Condensed Matter Physics. — 2003. — Т. 6, № 2(34). — С. 229-236. — Бібліогр.: 19 назв. — англ. |
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irk-123456789-1207042017-06-13T03:06:06Z Manifestation of deformation effect in band spectra in crystals with inhomogeneous bonding Bercha, D.M. Rushchanskii, K.Z. Slipukhina, I.V. Bercha, I.V. The effect of the shear strains on the energy spectrum of the strongly anisotropic SbSI crystal has been investigated by group-theoretical method in combination with the Pikus method of invariants. The first-principles local density approximation has been implemented to determine the band structure of the crystal. Ab initio calculations of the band structure have shown an exact localization of twofold degenerate maximum of the valence band in the T point. It turned out that the shear strains result in the band topology changes in the vicinity of the T point and the k -linear term appears in the corresponding dispersion law. Теоретико-груповим методом в поєднанні з методом інваріантів Пікуса досліджено вплив зсувових деформацій на енергетичний спектр сильно анізотропного кристала SbSI. Для визначення зонної структури кристала було застосовано розрахунки з перших принципів з використанням локального наближення. Дослідження зон- ної структури вказали на точну локалізацію двократно виродженого максимума валентної зони в точці Т. Виявилося, що зсувові деформації приводять до зміни топології зон в околі точки Т, а у відповідному законі дисперсії з’являється k -лінійний член. 2003 Article Manifestation of deformation effect in band spectra in crystals with inhomogeneous bonding / D.M. Bercha, K.Z. Rushchanskii, I.V. Slipukhina, I.V. Bercha // Condensed Matter Physics. — 2003. — Т. 6, № 2(34). — С. 229-236. — Бібліогр.: 19 назв. — англ. 1607-324X DOI:10.5488/CMP.6.2.229 PACS: 71.15.Ap, 71.15.Hx, 71.15.Mb, 61.50.Ah http://dspace.nbuv.gov.ua/handle/123456789/120704 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The effect of the shear strains on the energy spectrum of the strongly
anisotropic SbSI crystal has been investigated by group-theoretical method
in combination with the Pikus method of invariants. The first-principles local
density approximation has been implemented to determine the band structure
of the crystal. Ab initio calculations of the band structure have shown
an exact localization of twofold degenerate maximum of the valence band
in the T point. It turned out that the shear strains result in the band topology
changes in the vicinity of the T point and the k -linear term appears in the
corresponding dispersion law. |
| format |
Article |
| author |
Bercha, D.M. Rushchanskii, K.Z. Slipukhina, I.V. Bercha, I.V. |
| spellingShingle |
Bercha, D.M. Rushchanskii, K.Z. Slipukhina, I.V. Bercha, I.V. Manifestation of deformation effect in band spectra in crystals with inhomogeneous bonding Condensed Matter Physics |
| author_facet |
Bercha, D.M. Rushchanskii, K.Z. Slipukhina, I.V. Bercha, I.V. |
| author_sort |
Bercha, D.M. |
| title |
Manifestation of deformation effect in band spectra in crystals with inhomogeneous bonding |
| title_short |
Manifestation of deformation effect in band spectra in crystals with inhomogeneous bonding |
| title_full |
Manifestation of deformation effect in band spectra in crystals with inhomogeneous bonding |
| title_fullStr |
Manifestation of deformation effect in band spectra in crystals with inhomogeneous bonding |
| title_full_unstemmed |
Manifestation of deformation effect in band spectra in crystals with inhomogeneous bonding |
| title_sort |
manifestation of deformation effect in band spectra in crystals with inhomogeneous bonding |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120704 |
| citation_txt |
Manifestation of deformation effect in
band spectra in crystals with
inhomogeneous bonding / D.M. Bercha, K.Z. Rushchanskii, I.V. Slipukhina, I.V. Bercha // Condensed Matter Physics. — 2003. — Т. 6, № 2(34). — С. 229-236. — Бібліогр.: 19 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT berchadm manifestationofdeformationeffectinbandspectraincrystalswithinhomogeneousbonding AT rushchanskiikz manifestationofdeformationeffectinbandspectraincrystalswithinhomogeneousbonding AT slipukhinaiv manifestationofdeformationeffectinbandspectraincrystalswithinhomogeneousbonding AT berchaiv manifestationofdeformationeffectinbandspectraincrystalswithinhomogeneousbonding |
| first_indexed |
2025-07-08T18:26:14Z |
| last_indexed |
2025-07-08T18:26:14Z |
| _version_ |
1837104289358544896 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 2(34), pp. 229–236
Manifestation of deformation effect in
band spectra in crystals with
inhomogeneous bonding
D.M.Bercha 1,2 , K.Z.Rushchanskii 1 , I.V.Slipukhina 1 ∗,
I.V.Bercha 1
1 Uzhgorod National University,
54 Voloshyn Str., 88000 Uzhgorod, Ukraine,
2 Institute of Physics, Pedagogical University,
16a Rejtana Str., 35–310 Rzeszów, Poland
Received October 25, 2002, in final form April 3, 2003
The effect of the shear strains on the energy spectrum of the strongly
anisotropic SbSI crystal has been investigated by group-theoretical method
in combination with the Pikus method of invariants. The first-principles local
density approximation has been implemented to determine the band struc-
ture of the crystal. Ab initio calculations of the band structure have shown
an exact localization of twofold degenerate maximum of the valence band
in the T point. It turned out that the shear strains result in the band topology
changes in the vicinity of the T point and the k -linear term appears in the
corresponding dispersion law.
Key words: ab initio, deformations, SbSI
PACS: 71.15.Ap, 71.15.Hx, 71.15.Mb, 61.50.Ah
The inhomogeneous bonding in the strongly anisotropic crystals with a compli-
cated unit cell results in some peculiarities of their energy band structure caused
by external effect. Such crystals are the crystals with distinguished structural units
in the form of layers or chains. Some of them belong to the group of ferroelectrics-
semiconductors. The SbSI crystal is one of the representatives of the mentioned
group.
As it is well known, the main characteristic of semiconductors is their energy
spectrum. The changes in the energy spectrum under external effects are crucial in
the manifestation of such effects on the electronic physical properties. One of the
above mentioned factors is the effect of external strains.
The kp-theory and its modern modifications make it possible to investigate dis-
persion laws of electrons and holes for strained and unstrained crystals. Nevertheless,
∗E-mail: iv@iss.univ.uzhgorod.ua
c© D.M.Bercha, K.Z.Rushchanskii, I.V.Slipukhina, I.V.Bercha 229
D.M.Bercha et al.
taking into account the symmetry only, we cannot define the points of absolute band
extrema. For the clarification of the situation it is necessary to carry out the band
structure calculations focusing a special attention on the vicinity of the points, in
which the group-theoretical analysis permits the existence of the extrema in E(k)
dependence.
There exist several numerical calculations of the energy band structure of the
SbSI crystal with the symmetry of D16
2h (Pnam) space group [1–3]. Because of the
ambiguity of these results we carried out our own calculations of the band structure
of the SbSI crystal in the paraelectric phase. It should be noted that a more conve-
nient coordinate system to describe the SbSI crystal is that of the Pnam group, since
this crystal exhibits a phase transition to the ferroelectric phase describable in terms
of the Pna2 (C9
2v) space group [4], which is a subgroup of the Pnam group. There
are twelve atoms in the unit cell of the SbSI crystal which create two translationally
nonequivalent chains. Lattice constants of this crystal in the coordinate system of
the Pnam group are as follows: a1 = 8.52 Å, a2 = 10.13 Å, a3 = 4.10 Å. The atomic
coordinates are taken from [5].
The band structure of the SbSI crystal was obtained by ab initio calculations.
The non-local norm-conserving pseudopotentials [6] were used, calculated on the
basis of the Hammann scheme [7] and implemented in the package program �fhi98PP
[8]. In order to take into account the exchange-correlation interaction, the results
obtained by Caperley and Alder in the local approximation [9] were implemented in
the project. Perdew and Zunger’s [10] analytic fit to these results was used in our
calculations. Scalar relativistic effects were included in the pseudopotentials. The
selected parameters ensure the generation of the “ghost-free” pseudopotentials [11].
The band structure of the SbSI crystal was calculated by means of the �fhi98md
program [12]. The wave functions were expanded in plane waves with kinetic energy
up to 20 Ry, which corresponds to approximately 3700 plane waves. To obtain the
ground state of the electron system, the Joannopoulos algorithm [12] was used in
the calculations. The special k-points method [13] was implemented to carry out the
integration in the k space over the Brillouin zone (BZ). The 105 special k points
were selected in the irreducible part of the BZ to calculate the ground state of the
SbSI crystal. The chosen parameters enabled us to obtain a good convergence in
calculations. We did not take into account the spin-orbit interaction.
The band structure of the SbSI crystal is shown in figure 1. Since the SbSI
crystal is strongly anisotropic, in the Z–T and R–U directions a small dispersion of
the branches is noticeable, especially for the deep branches of the valence band. The
minimum of the conduction band is located in the S point. This result coincides with
the data noted in [3,14] when the change in the coordinate system is introduced. It
should be noted that the aforementioned band structure of the SbSI crystal was de-
termined by the semi-empirical pseudopotential method. In the spectrum presented
in figure 1 one can see a small splitting of branches in the T point caused by a weak
interaction between the chains of the crystal. This behavior is predictable [15]. The
width of the obtained indirect band gap is equal to 1.5 eV, while the experimental
estimation of this value yields Egmin ∼ 1.8 eV [4]. The most significant discrepancy
230
Manifestation of deformation effect in band spectra
Figure 1. The energy band structure of the SbSI crystal.
between the band structure of the SbSI crystal obtained in the semi-empirical pseu-
dopotential calculations and that obtained by ab initio calculations can be specified
as follows. The maximum of the valence band is not located in the Γ point, as it
is obtained in the calculations done by the semi-empirical pseudopotential method.
The distinct maxima of the valence band are observable along the Γ–Z, Γ–Y, Γ–X
directions as well as in the T point. It is evident that for low-energy values in the
valence band there exist nine separate complexes consisting of 4 branches. More-
over, their symmetry in the Γ point is the same as the one calculated on the basis
of the empty-lattice approximation. In the Γ–Z direction an interaction between the
branches can be found.
There are two translationally nonequivalent chains in the unit cell of the SbSI
crystal and this causes Davidov’s splitting over the Brillouin zone between some
energy bands. This splitting was simulated and investigated in [15]. As one can see
from the results obtained in this paper, the splitting should be relatively smaller in
the T point than the one in other directions of the Brillouin zone. Our numerical
calculations showed that in the minimal band complex, consisting of four subbands,
located in the depth of the valence band, such a splitting really exists in the T point
and it is a minimal one. The T point is one of the extrema of the valence band
and it also presents interest because, as it will be shown a little later in this paper,
the twofold degenerate dispersion dependence in Γ–Y and Γ–Z directions does not
231
D.M.Bercha et al.
split under the shear strains in the T point itself and turns into the dependence
containing a k-linear term in the vicinity of the T point. To show this we shall here
use the Pikus method of invariants [16] to obtain the dispersion law E(k) in the
vicinity of the T point for the strained and unstrained SbSI crystal.
It should be noted that, as it was mentioned above at the very beginning of
this paper, there is an inhomogeneous bonding in the SbSI crystal (weak between
chains and strong between atoms of chains). It results in the symmetry lowering of
the simulated crystal comparatively with the symmetry of the unstrained crystal
because of the mathematical deformation which is necessary for the reconstruction
of the boundary conditions broken by applying the external strains [17]. The easiest
way to get sure that the k-linear terms appear under the shear strains is to compare
the extremeness of the corresponding dispersion curves along the main directions of
the Brillouin zone of the unstrained and simulated crystals (table 1). The latter are
described by different space groups. It turned out [17], that double degeneracy does
not disappear in the T point under the shear strain εxy (εxz) and in the corresponding
dispersion dependence a k-linear term appears in kz direction under the εxy strain
and in ky direction under the εxz strain.
Now let us show that the same result can be obtained by the Pikus method of
invariants modified by accounting for the terms of higher than k2 order of magnitude
when we neglect the layered character of the crystalline structure. So, we shall
consider the simulated crystal after the mathematical deformation which is described
by the same space group as the unstrained one.
As it is well known, a dispersion law E(k) in the Pikus method can be obtained
by solving a secular equation |Dij (k, ε̃) − E (k) δij| = 0, where
D (k, ε̃) =
∑
r,s
Cr
s
∑
i
Aisf
r
is (k, ε̃) ,
and Ais are linearly independent matrices transforming according to τ s (g) represen-
tation; f r
is (k, ε̃) are polynomials of the components of k and tensor of deformation ε̃.
These polynomials are transformed according to τ s∗ (g) representation (star means
a complex conjugation). The τ s (g) is the representation of the group of the wave
vector k = 0 [16] and is defined in different ways depending on the cases a), b), or
c) according to the Herring classification for the representations of the wave vector
group in a chosen special point in the Brillouin zone. The matrices Ais and the basis
functions f r
is (k, ε̃) for the T point in the presence of the shear strains εxy (εxz) are
presented in table 2.
In table 2, f = ±1 defines the parity of the f r
is (k, ε̃) functions. The D (k) matrix
for the unstrained crystal is defined as follows:
D (k) =
(
ak2
x + bk2
y + ck2
z dkykz
dkykz ak2
x + bk2
y + ck2
z
)
. (1)
After solving the corresponding secular equation, the dispersion dependence in
the case under consideration is found in the following way:
E (k) = ak2
x + bk2
y + ck2
z ±
√
d2k2
yk
2
z . (2)
232
Manifestation of deformation effect in band spectra
Symmetry groups
The extreme-
ness in main
directions x, y,
z
Strains leading
to the symme-
try lowering
Unstrained crystal D16
2h
{
x, y, z
= 0
}
εij = 0
Strained crystal
simulated by
mathematical
deformation
C2
2h
{
x, y = 0
z 6= 0
}
εxy
C5
2h
{
x, z = 0
y 6= 0
}
εxz
C5
2h
{
x, y, z
= 0
}
εyz
Table 1. The extremeness of E(k) in the T point in main directions of the Bril-
louin zone [17].
τ s (g) Ais f r
is (k, ε̃)
f = 1 f = −1
Γ1
(
1 0
0 1
)
k2
x, k
2
y, k
2
z –
Γ2
(
1 0
0 −1
)
– εxykz (εxzky)
Γ3
(
0 1
1 0
)
kykz –
Table 2. The Ais matrices and the f r
is (k, ε̃) basis functions for the T point in the
presence of εxy (εxz) shear strains.
233
D.M.Bercha et al.
Figure 2. The band topology changes in the vicinity of the highly symmetrical
T point caused by external shear strains [17]: left – unstrained crystal; right –
strained crystal.
When the external shear strain εxy is present, the D (k) matrix takes the form:
D (k) =
(
ak2
x + bk2
y + ck2
z + fεxykz dkykz
dkykz ak2
x + bk2
y + ck2
z − fεxykz
)
, (3)
and a k-linear term appears in the corresponding dispersion law:
E (k) = ak2
x + bk2
y + ck2
z ±
√
f 2ε2
xyk
2
z
+ d2k2
yk
2
z
. (4)
The dispersion law containing a k-linear term in the vicinity of the T point in
the presence of the shear strain εxz is obtained in the same way.
An analysis of the dispersion law (4) shows that within the framework of the
chosen approximation with the accuracy up to k3 (εxy (εxz) is of k2 order) there
is no splitting of the twofold degenerate energy state and two extrema of E(k)
symmetrically shifted in the kz(ky) direction appear in the vicinity of the T point
(figure 2).
So, we can make an important conclusion that the appearance of the k-linear
terms in the energy spectrum of the SbSI crystal caused by the shear strains εxy
(εxz) results in the shifts of the extrema from the T point in kz (ky) direction. It
should be noted that such shifts caused by a spin-orbit interaction are observable in
the crystals with a lack of inversion symmetry. The existence of the k-linear terms in
the dispersion law of the SbSI crystal under the shear strain effect can be manifested
in strong electric fields when the charge carriers heating takes place (such effects are
observable in Te with the energy spectrum containing a k-linear term [18]), as well
as in optical absorption spectra and in other physical effects when the peculiarities
of the energy band structure are very important. Moreover, a possibility of managing
such k-linearities is predictable. It is worth noting that in the SbSI crystal, as it is
indicated in [19], both in the paraelectric and in the ferroelectric phase, an elastic
modulus C66 is the smallest one and it is four times less in magnitude than the C44
one. That is why εxy will be the largest component of the tensor of deformation under
234
Manifestation of deformation effect in band spectra
the same strains. This leads to the possibility of considering εxykz to be the value
proportional to k2 in contrast to weakly anisotropic crystals. Adding a symmetrical
term proportional to εxyεxz to the off-diagonal elements in D(k) we shall obtain
the splitting within the same approximation. This causes a warped dependence of
the upper valence band (dotted line in figure 2, right) with the negative effective
mass. The same splitting can be obtained when the inhomogeneous bonding between
structural units is taken into account and is accompanied by symmetry lowering
after the mathematical deformation. But it can be observed when the components
of higher order of magnitude are taken into account.
K.Z.R. is very grateful to Professor M.Scheffler for his permission to utilize the
fhi98md and fhi98PP computer program.
References
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Відображення впливу деформацій в зонних
спектрах в кристалах з неоднорідним зв’язком
Д.М.Берча 1,2 , К.З.Рущанський 1 , І.В.Сліпухіна 1 ,
І.В.Берча 1
1 Ужгородський національний університет,
88000 Ужгород, вул. Волошина, 54
2 Інститут фізики, Педагогічний університет,
Польща, 35–310 Жешув, вул. Рейтана, 16a
Отримано 25 жовтня 2002 р., в остаточному вигляді – 3 квітня
2003 р.
Теоретико-груповим методом в поєднанні з методом інваріантів
Пікуса досліджено вплив зсувових деформацій на енергетичний
спектр сильно анізотропного кристала SbSI. Для визначення зонної
структури кристала було застосовано розрахунки з перших прин-
ципів з використанням локального наближення. Дослідження зон-
ної структури вказали на точну локалізацію двократно виродженого
максимума валентної зони в точці Т. Виявилося, що зсувові дефор-
мації приводять до зміни топології зон в околі точки Т, а у відповід-
ному законі дисперсії з’являється k -лінійний член.
Ключові слова: ab initio, деформації, SbSI
PACS: 71.15.Ap, 71.15.Hx, 71.15.Mb, 61.50.Ah
236
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