Effect of symmetry center losses on energy bands and carrier kinematics in Zn₃As₂ and Cd₃As₂
The comparative analysis of the band structure and carrier kinematics for Zn₃As₂ and Cd₃As₂ has been executed. The influence of presence and absence of symmetry center in different crystalline phases of the above materials is explored. The direct and indirect solutions of dispersion equations were u...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2006
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| Цитувати: | Effect of symmetry center losses on energy bands and carrier kinematics in Zn₃As₂ and Cd₃As₂ / G. Chuiko, N. Don, V. Martyniuk, D. Stepanchikov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 2. — С. 17-22. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-1214262017-06-15T03:05:29Z Effect of symmetry center losses on energy bands and carrier kinematics in Zn₃As₂ and Cd₃As₂ Chuiko, G. Don, N. Martyniuk, V. Stepanchikov, D. The comparative analysis of the band structure and carrier kinematics for Zn₃As₂ and Cd₃As₂ has been executed. The influence of presence and absence of symmetry center in different crystalline phases of the above materials is explored. The direct and indirect solutions of dispersion equations were used for the analysis. The results of researches are presented in the form of graphic dependences. The splitting of energy bands caused by the symmetry center loss is estimated. Such splitting is maximal along any directions normal to the main crystalline axis. The principle possibility of the carrier separation by spin with the use of found distinction for the modules of their velocities is shown. 2006 Article Effect of symmetry center losses on energy bands and carrier kinematics in Zn₃As₂ and Cd₃As₂ / G. Chuiko, N. Don, V. Martyniuk, D. Stepanchikov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 2. — С. 17-22. — Бібліогр.: 5 назв. — англ. 1560-8034 PACS 71.20.-b, 71.18.+y http://dspace.nbuv.gov.ua/handle/123456789/121426 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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The comparative analysis of the band structure and carrier kinematics for Zn₃As₂ and Cd₃As₂ has been executed. The influence of presence and absence of symmetry center in different crystalline phases of the above materials is explored. The direct and indirect solutions of dispersion equations were used for the analysis. The results of researches are presented in the form of graphic dependences. The splitting of energy bands caused by the symmetry center loss is estimated. Such splitting is maximal along any directions normal to the main crystalline axis. The principle possibility of the carrier separation by spin with the use of found distinction for the modules of their velocities is shown. |
| format |
Article |
| author |
Chuiko, G. Don, N. Martyniuk, V. Stepanchikov, D. |
| spellingShingle |
Chuiko, G. Don, N. Martyniuk, V. Stepanchikov, D. Effect of symmetry center losses on energy bands and carrier kinematics in Zn₃As₂ and Cd₃As₂ Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Chuiko, G. Don, N. Martyniuk, V. Stepanchikov, D. |
| author_sort |
Chuiko, G. |
| title |
Effect of symmetry center losses on energy bands and carrier kinematics in Zn₃As₂ and Cd₃As₂ |
| title_short |
Effect of symmetry center losses on energy bands and carrier kinematics in Zn₃As₂ and Cd₃As₂ |
| title_full |
Effect of symmetry center losses on energy bands and carrier kinematics in Zn₃As₂ and Cd₃As₂ |
| title_fullStr |
Effect of symmetry center losses on energy bands and carrier kinematics in Zn₃As₂ and Cd₃As₂ |
| title_full_unstemmed |
Effect of symmetry center losses on energy bands and carrier kinematics in Zn₃As₂ and Cd₃As₂ |
| title_sort |
effect of symmetry center losses on energy bands and carrier kinematics in zn₃as₂ and cd₃as₂ |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/121426 |
| citation_txt |
Effect of symmetry center losses on energy bands and carrier kinematics in Zn₃As₂ and Cd₃As₂ / G. Chuiko, N. Don, V. Martyniuk, D. Stepanchikov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 2. — С. 17-22. — Бібліогр.: 5 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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2025-07-08T19:52:49Z |
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| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 2. P. 17-22.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
17
PACS 71.20.-b, 71.18.+y
Effect of symmetry center losses on energy bands
and carrier kinematics in Zn3As2 and Cd3As2
G. Chuiko, N. Don, V. Martyniuk, and D. Stepanchikov
Kherson National Technical University, Department of General and Applied Physics
24, Berislavskoye shosse, 73008 Kherson, Ukraine, e-mail: gp47@mail.ru
Abstract. The comparative analysis of the band structure and carrier kinematics for n-
Cd3As2 and p-Zn3As2 has been executed. The influence of presence and absence of
symmetry center in different crystalline phases of the above materials is explored. The
direct and indirect solutions of dispersion equations were used for the analysis. The
results of researches are presented in the form of graphic dependences. The splitting of
energy bands caused by the symmetry center loss is estimated. Such splitting is maximal
along any directions normal to the main crystalline axis. The principle possibility of the
carrier separation by spin with the use of found distinction for the modules of their
velocities is shown.
Keywords: dispersion law, band structure, cadmium arsenide, zinc arsenide.
Manuscript received 24.01.06; accepted for publication 29.03.06.
1. Introduction
Cd3As2 and Zn3As2 have the equivalent polymorphic
transformations below 500 K with the loss of centers of
symmetry. As a result, the symmetry is lowered from
mmm/4 ( hD4 ) to mm4 ( vC4 ). Both crystals possess
very similar lattices with 160 atoms per a cell,
demonstrating the small tetragonal tensions:
0 < 1
2
1 −=−
a
c
η << 1 [1]. Besides, their cells can be
considered only as slightly divergent, if comparing the
pair of crystal modifications with mmm/4 ( hD4 ) and
mm4 ( vC4 ) symmetries, not only without but with the
center of symmetry, making it individual for each of
both materials, of course. These cells differ only by one
of the four layered atom packets, which create such a
cell [2].
However, these materials possess also a few
essential distinctions, despite the undoubted similarity of
their crystalline forms. Firstly, there is a difference
between the standard and inverted band structures:
0>gε for Zn3As2 and 0<gε for Cd3As2. Secondly,
there is a contrast of their types of the conductivity: p-
type and n-type, respectively [1, 2]. As a result, the
typical concentrations and carrier mobility differ within
a few orders: these are much higher for Cd3As2.
Therefore, Zn3As2 is a typical semiconductor, whereas
Cd3As2 looks like a semimetal. Thus, the proper Fermi
levels are located not only within visibly different
energy intervals but even within different energy bands
of both materials.
Recently the simple band model was proposed for
Cd3As2 [3]. On the one hand, this model is a
generalization of well-known Kane’s and Kildal’s
models [4, 5]. On the other hand, the model [3] enables
to take into account the symmetry center loss and the
small tetragonal tensions. Moreover, this theory can be
appropriately used both for semiconductors ( 0>gε )
and semimetals ( 0<gε ). Below it is used to describe
the crystals with and without the symmetry center.
These arguments suggested an idea to use the
mentioned model for the comparative analysis of two
materials within the framework of joint description. Let
us define the main goal of such analysis as the
investigation of the influence of symmetry center losses
on the feature of the energy band structures and
properties of carriers in the relevant energy bands.
2. Basic equations and way of calculations
The dispersion law has the following form within the
model [3], if to apply the spherical system of coordinates
(k, θ, φ):
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 2. P. 17-22.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
18
( ) ( ) ( ) ( )( )( )
( ) ( ) .0sin
cossin
22
3
2
22
2
2
1
2
=−
−+−Γ
θε
θεθεε
fPk
ffPk (1)
The same expression has somewhat different form
if to use the cylindrical system of coordinates (kz, kxy, φ):
( ) ( ) ( )( )( )
( ) .022
3
2
22
2
2
1
2
=−
−+−Γ
xy
zxy
kfP
kfkfP
ε
εεε
(2)
Four polynomials for the energy ε are used in the
identical forms:
( ) ( )
,
3
2
9
2
33
2
2
2
2
⎥
⎦
⎤
⎟
⎠
⎞
⎜
⎝
⎛ Δ
+−
−
⎢
⎢
⎢
⎣
⎡
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛ Δ
−⎟
⎠
⎞
⎜
⎝
⎛ Δ
++⎟
⎠
⎞
⎜
⎝
⎛ Δ
+−=Γ
ε
η
δεεεεε
d
g
(3)
( )
2
2
1
933 η
δεεε
Δ
−⎟
⎠
⎞
⎜
⎝
⎛ Δ
++⎟
⎠
⎞
⎜
⎝
⎛ Δ
+=f , (4)
( ) 4
2 3
2 −⎟
⎠
⎞
⎜
⎝
⎛ Δ
+= ηεεεf , (5)
( )
η
ε
ε
3
2
3
d
f
Δ
= . (6)
There, Pg ,, Δε are three well-known Kane’s
parameters [4] (the energy gap, spin-splitting parameter
and matrix element of the pulse); δ is the known
parameter of the crystal field [5] and d is another
parameter of the crystal field, which describes the
absence of the symmetry center [3]; η is the scalar
factor taking into account the lattice deformation
(mentioned and described above). The zero of energy
( 0=ε ) is united either with the top of the band of heavy
holes (if 0>gε ) or with the bottom of the conduction
band (if 0<gε ).
It is noticeable that Eqs (1), (2) are independent
onϕ . Therefore, the surfaces of the equal energy are
surfaces of rotation around the main crystal axis that is
the polar axis, too [3].
The right parts of Eqs (1) and (2) have the trivial
forms as product of two factors (let it be
( ) ),,(,,, θεθε βα kPkP for Eq. (1), and ( ),,, xyz kkP εα
),,( xyz kkP εβ for Eq. (2)), because each of them is a
difference of two quadratic items. For instance, Eq. (1)
can be rewritten:
( ) 0),,(,, =θεθε βα kPkP . (7)
Here
( ) ( ) ( ) ( )(
( ) ) ( ) ,sincos
sin,,
3
2
2
2
1
2
θεθε
θεεθεα
Pkff
fPkkP
−+
+−Γ=
(8)
( ) ( ) ( ) ( )(
( ) ) ( ) .sincos
sin,,
3
2
2
2
1
2
θεθε
θεεθεβ
Pkff
fPkkP
++
+−Γ=
(9)
The polynomials ( ) ),,(,,, xyzxyz kkPkkP εε βα
are structurally similar without any doubts.
Let us note, that βα
PP = primarily if 0)(3 =εf .
These conditions might be fulfilled for any energy only
as for the crystal modifications with the symmetry center
(because 0=d in this case). Therefore, each energy
level should be twice degenerated. Obviously, it ought to
be the well-known Kramers’ degeneration. If 0≠d ,
then βα
PP ≠ and these are the conditions for the
phases without the symmetry center. Now the equations
like 0=αP and 0=βP have different solutions and the
mentioned above degeneration is over.
Factorization (7) allows decreasing the orders of
the dispersion equations. For instance, the equations
0=αP and 0=βP have the fourth order for functions
( )θε βα ,, k (or ( )xyz kk ,,βαε ). These direct solutions
might be obtained even in radicals, although proper
expressions are quite cumbersome and unbelievably
long. This circumstance seems to make no impression on
a computer, in a contrast with its users. Someone can get
indirect (implicit) solutions, too. This function
( )θεε βα ,, or anything like to ( ( )zxy kk ,ε , ( )xyxy kk ,ε )
within the cylindrical system of coordinates, for which
the equations (8), (9) are quadratic. These indirect
solutions can be also useful as the direct solutions, being
at the same time much shorter and simpler. What is
more, they can serve to verification of any conclusions
attained by using the direct solutions.
For instance, the velocity of carriers may be
obtained simply as the gradient of a direct solution for
their energies ( )θε βα ,, k :
)),((grad1
,, θε βαβα kkv
h
= . (10)
On the other hand, the same result may be obtained
from indirect solutions, or even from functions
( ) ),,(,,, θεθε βα kPkP , using the well-known
mathematical technique of the implicit derivatives.
Our calculations were leaning mostly on the
presented above collection of simple expressions as well
as on the following limited set of numerical parameters
built-in into these (see Table 1).
It did not matter how bulky and lengthy may be
some analytic expressions for a calculated result, on
occasion. The main focus of the attention for us was to
be in that point to represent such results in the forms of
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 2. P. 17-22.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
19
Table. Parameters of materials.
Parameter εg , eV eV,Δ meV, ⋅P eV,δ d, eV η
Material D4h C4v D4h C4v
Zn3As2 0.98 0.26 7.6·10-10 0.07 0 0.035 1.00218 1.00352
Cd3As2 -0.13 0.30 7.2·10-10 0.09 0 0.035 1.00471 1.00565
(a) (b)
Fig. 1. The dependences )(knε along the θ = 2
π directions: for Zn3As2 (a) and Cd3As2 (b).
graphic dependences, evident and relatively simple. That
is why this paper may be evaluated by someone as some
illustrative overloading, although only on the face of it,
how we are hoping.
3. The results and discussions
An old and good tradition requires the demonstration of
the dependences of the carrier energy versus the module
of the wave vector, or rarer of the pulse, under the
condition of an unchanged direction. It is shown in
Fig. 1 as the dependences ( )knε . It corresponds to the
rectilinear motion of a carrier with some tangential
acceleration, but in the absence of a normal acceleration.
Both graphics present only the substantial part of the set,
which consist of four energy bands, as a whole. So, the
Fermi level is located anywhere either within or nearby
presented pair of the energy bands. It is the pair of the
bands of heavy and light holes, of course. Although it
would be better to replace a term “heavy holes” by
“heavy carries” as for cadmium arsenide.
There is the visible splitting of two sub-bands with
the opposite spin states, which are located on both sides
from a middle line (the sets of cross-like or circle-like
points). These middle lines are the twice degenerated
energy bands for the structures with the symmetry
centers. This splitting is obviously caused by
disappearing the symmetry center after the
mmm/4 ( hD4 )→ mm4 ( vC 4 ) transition and, thus, it is a
result of the removal of Kramers’ degeneration.
The dependences of the magnitudes of this splitting
on the module of the wave vector as well as its
dependences on direction (the polar graphic) are shown
for all bands in Fig. 2. In this figure, shown are the
dependences for cadmium arsenide. However, the same
dependences for zinc arsenide are exceedingly similar to
those presented in Fig. 2, even in shallow details. The
opposite sign of such splitting literally “strikes the eyes”
as for one of these bands. Such a feature characterizes
the bands of light holes of both materials. The maxima
of the splitting magnitude correspond to θ = 2
π that
describes the directions normal to the main crystal axis,
whereas along this axis the splitting is absent. Even the
greatest magnitudes of them are rather moderate and do
not exceed a few hundred stakes of electron-volts,
because the same order has the relevant parameter d
(see Table).
Other, and in a manner reverse approach look as if
also reasonable from the physical point of view. That is
study of solely cyclic motion of a carrier with some
normal acceleration, but without any tangential
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 2. P. 17-22.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
20
(a) (b)
Fig. 2. The dependences of the spin splitting magnitude vs the module of the wave vector (a) and vs the direction (b), supposing
that the horizontal axis of the polar plot (b) coincides with that for the main crystal axis.
(a) (b)
Fig. 3. The polar plot of the dependences )(θε n for Zn3As2 (a) and Cd3As2 (b).
components. At that rate, everyone would be interested
in the dependences of the energy versus the direction of
the carrier wave vector under the condition of its
invariable module. These dependences are shown in
Fig. 3 as curves ( )θε n for the same pair of bands as in
Fig. 1. All symbols of Fig. 3 mean the same as those in
Fig. 1. It is well seen from comparison of the
dependences in Figs 1 and 3 that they seem to be very
different in Fig. 1, while in Fig. 3 they are practically
indistinguishable. Second observation is that energy
anisotropy on the spherical surface const=k is much
greater for the heavy carriers.
If examine the modules of the carrier velocities, it
ought to be keeping in mind, that the heavy and light
holes are the major carriers only in zinc arsenide.
Opposite, the light and heavy electrons are the same in
cadmium arsenide. Let us begin with zinc arsenide.
The velocities of the both different kinds of the
holes are almost comparable by the magnitudes along
the directions, which is normal to main crystal axis.
Nevertheless, the directions that are close to the
direction of the main axis ( 0≈θ ) are characterized by
another and sharply different relations of these
velocities. The anisotropy ought to be very strong as for
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 2. P. 17-22.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
21
Fig. 4. Zinc arsenide: the polar plot of the velocity modules vs the direction for heavy holes (a) at the energies −25 and
−15 meV, as well as for light holes (b) at −45 and −55 meV. The insertions remind about typical geometry and topology of
surfaces of the equal energy.
Fig. 5. Cadmium arsenide: the polar plot of the velocity modules vs the direction for heavy carriers at the energies −15 (a) and
−40 meV (b). The inserts have the same meaning as above.
the heavy holes, which may be extremely slow when
moving along any direction close to the main axis.
Furthermore, their velocities decrease with the energy in
contrast to the light holes.
The spin splitting of the carrier velocities is
maximal along the directions normal to the main crystal
axis. In other words, the modules of velocities of holes
with an identical spin are quite not identical at motion
back and forth along almost any direction, in particular,
in the plane normal to the main crystalline axis. This is
the obvious and direct consequence of the symmetry
center loss. Opposite, the holes with the opposite spins
will be never assorted by their velocities when moving
along this axis.
Below we consider the following material –
cadmium arsenide. The data is presented by Figs 5 and 6
separately. In particular, Fig. 5 illustrates the absolutely
another behavior of the heavy carriers as compared to
that shown in Fig. 4a for zinc arsenide. It may be
explained taking into consideration first of them clear
difference between the shapes of the equal energy
surfaces (SEE) for heavy carriers.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 2. P. 17-22.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
22
Fig. 6. Cadmium arsenide: the polar plot of the velocity
modules vs the direction for light electrons at the energies 25,
50 and 120 meV, respectively.
Nevertheless, all these effects caused by the
symmetry center loss are also present in this case, of
course. Essential changing the topology and geometry of
SEE, which occurs between −15 and −40 meV cannot
abolish these general peculiarities, in spite of bringing
some individuality into them. Both parts of the previous
sentence are well visible in Fig. 5. Both the strong
anisotropy of velocities and the possibility of separation
of carriers by their spins are evidently also common
features for heavy carriers of both materials.
The light electrons from the conduction band of
cadmium arsenide demonstrate the dependences, which
are very similar to analogous curves for light holes of
zinc arsenide, in contrast to the heavy carriers. It
becomes obvious from the comparison of Figs 4 and 6.
Let us note that the influence of the symmetry centre
loss weakens with increasing the energy module so that
such influence becomes quite insignificant as for the
strong degenerated samples of both materials. In other
words, the effects of absence of the symmetry centre
need to be searched in the samples with the lower
concentration of carriers and with the Fermi energies
that are closer to the band extremes. The degenerated
samples are insensitive to it.
4. Conclusions
The main results of our analysis are as follows:
1. The loss of the symmetry centre results in the
splitting of the energy bands that are maximal
along some direction normal to the main crystalline
axis as well as for specific values of the wave
vectors as for the different energy bands.
2. The velocities of the carriers with the same spins
are clearly various in relation to the motion “back
and forth” almost along every direction. Such
features allow to separate carriers in accord with
their spins, in principle.
3. The samples with the low concentration of carriers
demonstrate the results of the absence of the
symmetry centre better. Moreover, the velocities of
heavy carriers are more sensitive in comparison
with those of the light carriers, as well as the main
carriers of p-Zn3As2 are more sensitive as
compared with the electrons of n-Cd3As2 in the
case of the absent symmetry centre.
4. The heavy carriers of both materials demonstrate
more pronounced anisotropy of characteristics for
light carriers.
Acknowledgements
Authors wish to thank Dr. Stanislaw Shutov for his
helpful participation in the processes of preparation of
this paper.
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2
II
3 BA crystals // Functional
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