Investigation of the physical properties of multi-component solid solutions Hg₁₋x₋y₋zAxByCzTe
This paper presents theoretical research on the basic band parameters and galvanomagnetic phenomena in multicomponent solid solutions Hg₁₋x₋yAxByCzTe, Hg₁₋x₋zCdxZnzTe and Hg₁₋x₋y₋zAxByCzTe, resulting in the empirical formulae for the energy gap and the intrinsic carrier concentration of these materi...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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Ostapov, S.E. Gorbatyuk, I.N. Zhikharevich, V.V. 2017-06-12T15:15:03Z 2017-06-12T15:15:03Z 2005 Investigation of the physical properties of multi-component solid solutions Hg₁₋x₋y₋zAxByCzTe / S.E. Ostapov, I.N. Gorbatyuk, V.V. Zhikharevich // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 1. — С. 30-35. — Бібліогр.: 12 назв. — англ. 1560-8034 PACS: 72.20.Dp; 72.20.Fr; 72.20.My https://nasplib.isofts.kiev.ua/handle/123456789/120649 This paper presents theoretical research on the basic band parameters and galvanomagnetic phenomena in multicomponent solid solutions Hg₁₋x₋yAxByCzTe, Hg₁₋x₋zCdxZnzTe and Hg₁₋x₋y₋zAxByCzTe, resulting in the empirical formulae for the energy gap and the intrinsic carrier concentration of these materials in a wide range of temperatures and compositions. The effective mechanisms of charge carrier scattering, concentration and activation energy of acceptor impurities have been determined. The results of theoretical research are in a good agreement with the experimental and literature data. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Investigation of the physical properties of multi-component solid solutions Hg₁₋x₋y₋zAxByCzTe Article published earlier |
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Investigation of the physical properties of multi-component solid solutions Hg₁₋x₋y₋zAxByCzTe |
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Investigation of the physical properties of multi-component solid solutions Hg₁₋x₋y₋zAxByCzTe Ostapov, S.E. Gorbatyuk, I.N. Zhikharevich, V.V. |
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Investigation of the physical properties of multi-component solid solutions Hg₁₋x₋y₋zAxByCzTe |
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Investigation of the physical properties of multi-component solid solutions Hg₁₋x₋y₋zAxByCzTe |
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Investigation of the physical properties of multi-component solid solutions Hg₁₋x₋y₋zAxByCzTe |
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Investigation of the physical properties of multi-component solid solutions Hg₁₋x₋y₋zAxByCzTe |
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investigation of the physical properties of multi-component solid solutions hg₁₋x₋y₋zaxbyczte |
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Ostapov, S.E. Gorbatyuk, I.N. Zhikharevich, V.V. |
| author_facet |
Ostapov, S.E. Gorbatyuk, I.N. Zhikharevich, V.V. |
| publishDate |
2005 |
| language |
English |
| container_title |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| format |
Article |
| description |
This paper presents theoretical research on the basic band parameters and galvanomagnetic phenomena in multicomponent solid solutions Hg₁₋x₋yAxByCzTe, Hg₁₋x₋zCdxZnzTe and Hg₁₋x₋y₋zAxByCzTe, resulting in the empirical formulae for the energy gap and the intrinsic carrier concentration of these materials in a wide range of temperatures and compositions. The effective mechanisms of charge carrier scattering, concentration and activation energy of acceptor impurities have been determined. The results of theoretical research are in a good agreement with the experimental and literature data.
|
| issn |
1560-8034 |
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https://nasplib.isofts.kiev.ua/handle/123456789/120649 |
| citation_txt |
Investigation of the physical properties of multi-component solid solutions Hg₁₋x₋y₋zAxByCzTe / S.E. Ostapov, I.N. Gorbatyuk, V.V. Zhikharevich // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 1. — С. 30-35. — Бібліогр.: 12 назв. — англ. |
| work_keys_str_mv |
AT ostapovse investigationofthephysicalpropertiesofmulticomponentsolidsolutionshg1xyzaxbyczte AT gorbatyukin investigationofthephysicalpropertiesofmulticomponentsolidsolutionshg1xyzaxbyczte AT zhikharevichvv investigationofthephysicalpropertiesofmulticomponentsolidsolutionshg1xyzaxbyczte |
| first_indexed |
2025-11-26T09:53:41Z |
| last_indexed |
2025-11-26T09:53:41Z |
| _version_ |
1850618243984654336 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 1. P. 30-35.
PACS: 72.20.Dp; 72.20.Fr; 72.20.My
Investigation of the physical properties
of multicomponent solid solutions Hg1–x–y–zAxByCzTe
S.E. Ostapov, I.N. Gorbatyuk, V.V. Zhikharevich
Yu. Fed’kovich Chernivtsi National University, 2, Kotsyubinsky str., 58012 Chernivtsi, Ukraine
Abstract. This paper presents theoretical research on the basic band parameters and
galvanomagnetic phenomena in multicomponent solid solutions Hg1–х–yCdxMnyTe,
Hg1-x–zCdxZnzTe and Hg1–x–y–zCdxMnyZnzTe, resulting in the empirical formulae for the
energy gap and the intrinsic carrier concentration of these materials in a wide range of
temperatures and compositions. The effective mechanisms of charge carrier scattering,
concentration and activation energy of acceptor impurities have been determined. The
results of theoretical research are in a good agreement with the experimental and
literature data.
Keywords: solid solutions, electronic transport, galvanomagnetic effects, band
structure.
Manuscript received 03.12.04; accepted for publication 18.05.05.
1. Introduction
Solid solutions Hg1–xCdxTe offer a number of unique
physical and electrooptical properties making such
materials attractive for development of IR detectors.
However, their wide use is prevented by the instability
of this material. It was theoretically shown by Sher [1]
that this instability is caused by rather large differences
between the atomic radii of Cd and Hg. Hence, material
stability should increase with introduction of Mn or Zn,
since their atomic radii are closer to that of Hg.
Despite the promising character of such materials as
Hg1–х–yCdxMnyTe and Hg1–x–zCdxZnzTe, their basic
parameters due to the difficulty of obtaining pure and
high-quality crystals are not adequately studied. First of
all, it is true for the energy gap, intrinsic carrier
concentration and effective mass of electrons. Even to a
greater extent, it refers to Hg1–x–y–zCdxMnyZnzTe, as long
as this is absolutely new material for which the above
parameters are unknown at all. All this is related in full
measure to charge carrier scattering mechanisms, as well
as to impurity states in crystals.
2. Theoretical calculations
The basic band parameters of multicomponent solid
solutions can be calculated using the method proposed
by S. Williams [2], where the initial solution is regarded
as a combination of three ternary solutions. But as long
as this procedure is rather awkward and does not yield
even empirical relationships, we have used a simpler
calculation method [3]. The essence of this method is
that multicomponent material is represented as a
combination of two simpler materials. Using this
procedure, the formula for energy band, for example, of
Hg1–х–yCdxMnyTe, will be as follows:
Eg(Hg1–x–yCdxMnyTe)=0.5Eg(Hg1–2хCd2хTe) +
+ 0.5Eg(Hg1–2yMn2yTe), (1)
and the energy gap of Hg1–x–zCdxZnzTe can be
calculated:
Eg(Hg1–x–y–zCdxMnyZnzTe) =
= 0.5Eg(Hg1–x–2yCdxMn2yTe) +
+ 0.5Eg(Hg1–x–2zCdxZn2zTe). (2)
Using the empirical formulae of the energy gap for
HgCdTe [4], HgZnTe [5], and HgMnTe [6], we get for
HgCdMnTe:
Eg(x,y,T) = –0.302 + 5.125⋅10–4T –
–(x + 2.287y)⋅10–3T + 1.93(x + 2.197y) –
–1.62(x2 + 2.728y2) + 0.272(12.235x3–y3), (3)
and for HgCdZnTe:
Eg(x,y,T) = –0.301 + 1.93x + 2.291⋅10–2y1/2 +
+ 2.731y – 1.62x2 + 5.35⋅10–4T (1 – 2x – 0.35y1/2 –
– 1.28y) + 3.328x3 – 1.248y2 + 2.132y3, (4)
and for Hg1–x–y–zCdxMnyZnzTe:
Eg(x,y,z,T) = –0.289 + 1.93(x + 1.96y + 1.415z) +
+ 4·10–4T(1 – 2.675x – 0.504z0.5 – 1.709z – 7.013y) –
–1.62(x2 + 1.553z2) + 3.328(x3 + 2.563z3). (5)
Calculated by the formulae (3 – 5), the energy gaps
of multicomponent semiconductor solid solutions
showed a good agreement with the experimental data
(see Table 1). The energy gap of Hg1–x–y–zCdxMnyZnzTe
samples was determined from the optical transmission
curves.
As can be seen from the table, the empirical
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
30
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 1. P. 30-35.
formulae (2)–(4) describe well the experimental data for
all investigated materials at various temperatures.
Based on the results obtained, the intrinsic carrier
concentration, the Fermi level position and the effective
mass of electrons for materials under study were
calculated. The calculations followed the procedure
described in [6].
Counting the energy from the bottom of conduction
band, for electron concentration in conduction band in
conformity with the Kane model [9] it can be written:
∫
∞
η−+
Φ+Φ+
π
=
0
2/12/1
)exp(1
)/21()/1(2
z
dzzzzNn c , (6)
where Nc is effective density of states in conduction
band, Φ = Eg / kBT is the reduced energy gap, η =F / kBT
is the reduced Fermi energy. In this case, the effective
mass of electrons near the bottom of the conduction
band is described by the expression:
.
)(
3/2
1
1
0
*
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
Δ+
Δ+
+=
gg
g
Pe
EE
E
Emm (7)
Here, EP=2m0P2/ħ2, Δ is the value of the spin-orbit
splitting of bands.
The hole concentration in the valence band was
calculated in the regular way:
),()
2
(4 2/1
2/3
2
*
B Φ+ηπ= F
h
Tmk
p hh (8)
where is the effective mass of heavy holes,
F1/2(η + Φ) is the Fermi – Dirac integral. The values of
the spin-orbit splitting Δ, as well as the effective mass of
heavy holes in the calculations were assumed to be
equal: Δ = 1 eV, .
*
hhm
0
* 55.0 mmhh =
The intrinsic carrier concentration was calculated
from the solution of the electroneutrality equation for
intrinsic semiconductor. Our comparison of the resulting
values ni with the experimental ones allows us to
propose the empirical formula for the intrinsic carrier
concentration inherent to all the materials considered:
ni(x,y,z,T) = (А + Вx + Сy + Dz + ET(1 + x + y + z)) ×
× 1014Eg
0.75·T 1.5 exp(–Eg / 2kT), (9)
with А, В, С, D, E coefficients shown in Table 2.
Table 1. Comparison of calculated and experimental Eg for materials studied.
Material x y z T,
K
Eg theor.,
eV
Eg exper.,
eV
Source
0.215 0.022 – 0.17 0.18
0.107 0.016 – –0.065 –0.07
0.069 0.012 – –0.125 –0.13 HgCdMnTe
0.012 0.007 –
7
–0.25 –0.26
[7]
0.07 – 0.17 0.301 0.328
0.07 – 0.2 0.383 0.374
0.07 – 0.16 0.276 0.277 HgCdZnTe
0.12 – 0.18
95
0.406 0.409
[8]
0.213 0.031 0.02 0.300 0.304
0.2 0.027 0.017 0.267 0.27 HgCdMnZnTe
0.135 0.02 0.01
300
0.14 0.145
Table 2. Coefficients of formula (6) for various materials.
Material А В С D E
Hg1–х–yCdxMnyTe 5.84 –4.42 2.87 0 2.53⋅10–3
Hg1–x–zCdxZnzTe 6.48 –4.42 0 –6.54 1.42⋅10–3
Hg1–x–y–zCdxMnyZnzTe 6.95 –4.42 2.87 –6.54 1.96⋅10–3
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
31
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 1. P. 30-35.
Fig. 1. Temperature dependences of the intrinsic carrier
concentration as compared to the experiment. Solid curves –
calculations by the formula (6).
Fig. 2. Effective mass of electrons in Hg1–x–y–zCdxMnyZnzTe,
calculated by the formula (4).
Fig. 3. Electron effective mass of totally degenerated
HgCdMnTe vs composition.
The comparison of the intrinsic carrier concentration
calculated by the formula (6) with the experimental data
(see Fig. 1) shows that in all three cases, there is a good
agreement with the experiment in the temperature range
50 to 350 K within a wide composition range.
The effective mass of electrons calculated by the
formula (4) follows a linear law in the entire temperature
range (Fig. 2).
The effective mass of electrons in the case of full
degeneracy was calculated as follows [9]:
,1027.8
105.32
1
23/230
4232
2
*
*
Pn
PE
m
m
i
g
e
e
−
−
⋅+
+⋅=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
− (10)
where Р is the matrix element of the angular momentum
operator. The value Р for Hg1–x–y–zCdxMnyZnzTe was
calculated by the same algorithm as the energy gap.
Fig. 3 shows the results of calculation of the effective
mass of electrons in the completely degenerated case for
Hg1–x–y–zCdxMnyZnzTe. It can be seen that, in the range
of compositions that are of practical interest (the right-
hand side of the plot), the effective mass is also almost
linear.
Similar results in the calculation of effective masses
are also obtained for other semiconductor solid solutions
studied.
3. Research on kinetic coefficients
As regard to the above research on the band parameters,
we have also calculated mobility, conductivity and Hall
coefficient in semiconductor solid solutions
Hg1-x-y-zAxByCzTe and made a comparison with the
experimental data.
The expressions for charge carrier relaxation times
restricted due to scattering by ionized impurities, polar
optical phonons, acoustic phonons and disorder
potential, respectively, have the form [10]:
к
к
FNе i ∂
ε∂
π
χ
=τ 2
im
4
2
im 2
h , (11)
кТFке ∂
ε∂χ
=τ
∗
op0
2op 2
h , (12)
ккТFкЕ ∂
ε∂πρυ
=τ 2
ac0
2
2
//
ac
1h , (13)
ккWF
N
∂
ε∂π
=τ 2
dis
0
dis
1h , (14)
where χ = χs + χ∞ is the sum of static and high-frequency
dielectric constants of material; χ* = χsχ∞ / (χs – χ∞); Ni is
the concentration of ionized impurities; N0 is the number
of atoms in the volume unit; ρ is the specific density of
material; υ// is the rate of propagation of longitudinal
acoustic waves; Е is the deformation potential constant;
W is a function that depends on the solid solution
composition and the alloy potential of scattering that in
the first approximation is equal to the difference in
crystal sublattice energy gaps; Fim, Fop, Fac, Fdis are the
functions that take into account the shielding of
scattering center potential by free carriers and contain
the Bloch multipliers taking into account the
nonparabolicity of the material band structure.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
32
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 1. P. 30-35.
Fig. 4. Conduction electron mobilities at various scattering
mechanisms as a function of temperature in the sample
Hg0.9Mn0.1Te with impurity concentrations: Na = 5·1015 сm–3,
Nd = 1·1015 сm–3. (1 – ionized impurities, 2 – polar optical
phonons, 3 – acoustic phonons, 4 – alloy potential, 5 – total
mobility).
Fig. 5. Temperature dependence of the mobility: 1 – Hg1–xMnxTe
(x = 0,091; Na = 2·1016 сm–3); 2 – Hg1–x–yCdxMnyTe (x = 0.1;
y = 0.027; Na = 3·1015 сm–3); 3 – Hg1–x–y–zCdxMnyZnzTe
(x = 0.14; y = 0.014; z = 0.01; Na = 6·1015 сm–3). For all
samples it is assumed that Nd = 1015 сm–3.
Fig. 4 shows the temperature dependence of the
mobility calculated in accord with (8)–(11).
As can be seen from the figure, the greatest
contribution to mobility formation in the temperature
range of 80 to 300 K is made by scattering by polar
optical phonons and ionized impurities. Scattering by the
alloy potential and acoustic phonons is of little
significance under these conditions, which is consistent
with the papers [11, 12]. Therefore, our subsequent
discussion will be restricted by two scattering
mechanisms mentioned above. The results of mobility
calculation under these conditions as compared to the
experiment ones are shown in Fig. 5.
It can be seen that, with the selected calculation
parameters, the results are in a good agreement with the
experimental data, which confirms the correctness of
assumptions made both with respect to band parameters
and the dominating scattering mechanisms.
Fig. 6 shows calculated temperature dependences of
conductivity and the Hall constant for several
Hg1-x-zCdxZnzTe samples as compared with the
experimental data.
One should note a rather high degree of coincidence
between the theoretical and experimental data. Table 3
lists the values of parameters that were used in the
calculation of theoretical dependences. The
concentration of donor impurities was assumed to be
equal to 1015 сm–3.
Table 3 gives shows the parameters of the samples
of new five-component semiconductor solid solution
Hg1–x–zCdxZnzTe, studied by modeling the kinetic and
band parameters. As can be seen, all the parameters of
these samples: energy gap, acceptor concentration,
impurity activation energy, are within the same range as
that of better studied 3-4 component semiconductors of
the same type: HgCdTe, HgMnTe, HgCdMnTe.
The samples that were made of the first ingot of this
material (the numbers begin with 1) in general have a
lower concentration of manganese and zinc, and, hence,
a narrower energy gap. The samples from the second
ingot have a wider energy gap. As is evident from Fig. 6,
exactly these samples (2.1.1 and 2.1.6) demonstrate a
change in the Hall coefficient sign at sufficiently high
temperatures (200…230 K), which testifies to a wider
energy gap and smaller concentration of intrinsic
carriers. 1.3.3 sample is the most narrow-gap. It was cut
from the crystal “tail” and have the highest acceptor
concentration.
Table 3. Parameters of Hg1–x–y–zCdxMnyZnzTe crystals.
Composition of sample
Hg1-x-y-zCdxMnyZnzTe Sample
N x y z
Na, сm–3 Eа, meV Energy gap Eg,
eV (Т = 300 K)
1.2.2
1.3.1
1.3.3
2.1.1
2.1.6
1.1.15
0.14
0.1
0.01
0.23
0.16
0.18
0.014
0.02
0.005
0.05
0.04
0.025
0.015
0.01
0.005
0.02
0.01
0.015
1·1016
3.7·1017
1·1018
7·1015
2.5·1017
2·1016
9
2
–
15
12
10
0.142
0.0949
–0.098
0.3846
0.238
0.232
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
33
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 1. P. 30-35.
Fig. 6. Temperature dependences of the Hall constant (a-b) and conductivity (c-d) for Hg1–x–zCdxZnzTe samples of different
composition. a and c: 1 – sample 1.3.1, 2 – 1.3.3; 3 – 1.2.2. b and d: 1 – sample 1.1.15, 2 – 2.1.1; 3 – 2.1.6.
The acceptor activation energy was obtained with
the assumption that we deal with one acceptor level
located not far from the valence band top and one donor
level that we consider to be fully ionized at any
temperature. As we see, the resulting values of the
activation energy are within 2…15 meV, which is
typical for this class of semiconductors.
4. Conclusions
Thus, on the basis of the research performed, the
following conclusions can be derived: the method for
calculation of the basic band parameters of
multicomponent solid solutions of HgCdTe type has
been proposed.
1. The empirical formulae have been proposed for
the calculation of the energy gap and intrinsic
carrier concentration that are of the similar form
for HgCdMnTe, HgCdZnTe, as well as
HgCdMnZnTe – a new five-component solid
solution and differ only in coefficients.
2. The effective mechanisms of charge carrier
scattering in multicomponent semiconductors of
HgCdTe type have been studied. It was shown
that the main contribution to formation of the
temperature dependence of the mobility is made
by the carrier scattering caused by polar optical
phonons and ionized impurities.
3. Temperature dependences of the Hall constant
and conductivity in HgCdMnZnTe crystals of
various composition have been calculated. The
results of theoretical calculations are in a good
agreement with the experimental data.
4. From the closest agreement between the
theoretical and experimental curves, the activation
energy and acceptor impurity concentration have
been determined.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
34
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 1. P. 30-35.
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© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
35
Material
А
В
С
D
E
Hg1–х–yCdxMnyTe
5.84
–4.42
2.87
Hg1–x–zCdxZnzTe
6.48
–4.42
0
Hg1–x–y–zCdxMnyZnzTe
6.95
–4.42
2.87
|