Electron mobility in CdxHg₁₋xSe
Electron scattering on the short-range potential caused by interaction with polar and nonpolar optical phonons, piezoelectric and acoustic phonons, static strain, ionized impurities in CdxHg₁₋xSe (0 ⩽ x ⩽ 0.547) samples annealled in selenium vapour or in dynamic vacuum are considered. Within the...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2009
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irk-123456789-1188732017-06-01T03:05:37Z Electron mobility in CdxHg₁₋xSe Malyk, O.P. Electron scattering on the short-range potential caused by interaction with polar and nonpolar optical phonons, piezoelectric and acoustic phonons, static strain, ionized impurities in CdxHg₁₋xSe (0 ⩽ x ⩽ 0.547) samples annealled in selenium vapour or in dynamic vacuum are considered. Within the framework of the precise solution of the stationary Boltzmann equation on the base of short-range principle, temperature dependences of the electron mobility within the range 4.2 – 300 K are calculated. A good coordination of the theory to experiment in the investigated temperature range is established. 2009 Article Electron mobility in CdxHg₁₋xSe / O.P. Malyk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2009. — Т. 12, № 3. — С. 272-275. — Бібліогр.: 10 назв. — англ. 1560-8034 PACS 72.10.-d, 72.10.Fk, 72.15.-v http://dspace.nbuv.gov.ua/handle/123456789/118873 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Electron scattering on the short-range potential caused by interaction with
polar and nonpolar optical phonons, piezoelectric and acoustic phonons, static strain,
ionized impurities in CdxHg₁₋xSe (0 ⩽ x ⩽ 0.547) samples annealled in selenium vapour or
in dynamic vacuum are considered. Within the framework of the precise solution of the
stationary Boltzmann equation on the base of short-range principle, temperature
dependences of the electron mobility within the range 4.2 – 300 K are calculated. A good
coordination of the theory to experiment in the investigated temperature range is
established. |
| format |
Article |
| author |
Malyk, O.P. |
| spellingShingle |
Malyk, O.P. Electron mobility in CdxHg₁₋xSe Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Malyk, O.P. |
| author_sort |
Malyk, O.P. |
| title |
Electron mobility in CdxHg₁₋xSe |
| title_short |
Electron mobility in CdxHg₁₋xSe |
| title_full |
Electron mobility in CdxHg₁₋xSe |
| title_fullStr |
Electron mobility in CdxHg₁₋xSe |
| title_full_unstemmed |
Electron mobility in CdxHg₁₋xSe |
| title_sort |
electron mobility in cdxhg₁₋xse |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/118873 |
| citation_txt |
Electron mobility in CdxHg₁₋xSe / O.P. Malyk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2009. — Т. 12, № 3. — С. 272-275. — Бібліогр.: 10 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT malykop electronmobilityincdxhg1xse |
| first_indexed |
2025-07-08T14:49:05Z |
| last_indexed |
2025-07-08T14:49:05Z |
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1837090625890025472 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 3. P. 272-275.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
272
PACS 72.10.-d, 72.10.Fk, 72.15.-v
Electron mobility in CdxHg1-xSe
O.P. Malyk
Lviv Polytechnic National University, Semiconductor Electronics Department
12, Bandera str., Lviv 79013, Ukraine; e-mail: omalyk@mail.lviv.ua
Abstract. Electron scattering on the short-range potential caused by interaction with
polar and nonpolar optical phonons, piezoelectric and acoustic phonons, static strain,
ionized impurities in CdxHg1-xSe (0 x 0.547) samples annealled in selenium vapour or
in dynamic vacuum are considered. Within the framework of the precise solution of the
stationary Boltzmann equation on the base of short-range principle, temperature
dependences of the electron mobility within the range 4.2 – 300 K are calculated. A good
coordination of the theory to experiment in the investigated temperature range is
established.
Keywords: cadmium-mercury-selenium solid solution, charge carrier scattering.
Manuscript received 08.04.09; accepted for publication 14.05.09; published online 15.05.09.
1. Introduction
The electron scattering in the solid solution CdxHg1-xSe
was considered in [1-4] in relaxation time
approximation. The models of electron scattering by
lattice defects used in these works have an essential
shortcoming – they are long-range. In these models, it is
supposed that either charge carrier interacts with all the
crystal (electron-phonon interaction) or it interacts with
the defect potential of the impurity, the action radius of
which is approximately equal to 50–100а0 (а0 – lattice
parameter). However, such an assumption contradicts
the special relativity according to which the charge
carrier should interact only with the neighbouring crystal
region. Besides, for defects with the interaction energy
2)1,(
1
n
r
U
n
, at the distances 10а0 the potential
takes the magnitude of the second order, while all the
theories mentioned above are considered in the first
(Born) approximation. On the other hand, in [5] the
short-range models of electron scattering in CdxHg1-xTe
were proposed, in which the above mentioned
shortcomings were absent. There, it has been supposed
that the carrier interacts with the defect potential only
within the limits of one elementary cell. The purpose of
this work is to use this approach for description of the
electron scattering processes by various types of crystal
defects in CdHgSe solid solution.
2. Theory
The electron transition probability from a state k to a
state k caused by the interaction with polar optical
(PO), nonpolar optical (NPO), piezooptic (POP) and
piezoacoustic (PAC), acoustic (AC) phonons, ionized
impurity (II) was chosen from [5]:
,)()1()(
2
)()1(
)(
1
ε225
γπ64
),(
TOTOTOTO
TO
LOLO
LOLO
LO
Te
Te
4
0
2
0
410
PO
7
PO
NN
N
N
MM
MM
Ga
e
W
x
xkk
(1)
,)()1(
)(
2
)(
)1()(
1
288
),(
TOTO
TOTO
TO
LO
LOLOLO
LO
Te
Te
2
0
2
NPO
3
NPO
N
N
NN
MM
MM
Ga
E
W
x
xkk
(2)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 3. P. 272-275.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
273
,)()1()(
2
)()1(
)(
1
75
32
),(
TOTOTOTO
TO
LOLO
LOLO
LO
Tex
Te
2
0
10
PZ
2
14
292
POP
NN
N
N
MM
MM
G
ee
W xkk
(3)
,)(
21
)1(252
128
),(
2
TOLO
TeHgCd
2
0
B
10
PZ
2
0
2
14
27
PAC
cc
MMxMxG
Tkaee
W
kk
(4)
,)(
21
144
),(
2
TOLO
Te
2
ACB
3
AC
cc
MMG
ETk
W
x
kk
(5)
,)(
2
γ
),(
2
0
4
0
4
IIII
24
II
V
aNZe
W i
kk (6)
where ,, ,)1( SeCdHgHgCd MMMMxMxM x are
atom masses; G – number of unit cells in the crystal
bulk; NLO, NTO – number of longitudinal (LO) and
transverse (TO) phonons with frequencies LO and TO,
respectively; e14 – non-vanishing component of the
piezoelectric tensor; cLO, cTO – respective sound
velocities; V – crystal volume; NII – concentration of
ionized impurities; Zi – the impurity charge in electron-
charge units; EAC, ENPO – acoustic and optical
deformation potentials (EAC = 2.04 eV, ENPO = 29.8 eV),
respectively; γPO, γPZ, γII – adjustable parameters
determining the action radius of the short-range potential
(R = γ a0, 0 ≤ γPO, γPZ ≤ 0.86, 0 ≤ γII ≤ 1); 0 – dielectric
constant; e – elementary charge; Bk – Boltzmann
constant; ħ – Planck constant; (ε) – Dirac delta-
function; – carrier energy.
It should be noted that the strong power
dependence of parameters γPO, γPZ, γII sharply limits
opportunities to choose their numerical values.
To describe the electron-disorder (DIS) scattering,
the respective transition probability defined in [6] was
used.
Besides, the above mentioned scattering
mechanisms of the so-called static strain (SS) scattering
on the short-range potential was considered. According
to Fedders [7], the potential caused by the strain field
takes the following form:
2
0
14
3
0 19
)(
r
e eb
U
r , (7)
where b0 has the length units and is related to the size of
defect.
In (7) we neglected the angular dependence of
U(r). Following the short-range principle, we put
b0 = a0. To calculate the transition matrix element, we
shall use the electron plain wave function normalized
over the crystal volume:
)(
49
0
14
3
0 qRSi
q
π
V
e ea
U
krk , (8)
where kk q , )(xSi is the sine integral.
Our calculations show that the electron wave vector
(and q together with it) varies within the limits from 0 up
to 109 m-1 when the energy changes from 0 up to 10 kВТ
within the temperature range 4.2-300 K. For R 10-10 m,
this gives the estimation of 1.0)( CxSi . Then the
transition probability looks like:
)(
1π32
),(
22
0
SS
2
14
26
0
2345
SS
qV
N eeaC
W
kk ,(9)
where NSS is the concentration of strain centers.
Using the formalism of a precise solution for the
stationary Boltzmann equation [8], one can obtain the
logarithmic divergence of the integral over angular
variable . To eliminate this divergence, let’s specialize
the lower limit of the integral in a manner providing
coordination of the theory and experiment for using this
integral as an adjustable parameter:
d
0
cos1
sin
SS , (10)
where θ0 is the angle that corresponds to an adjustable
parameter γSS.
Let’s note that the similar way to choose the lower
limit of the integral is used in the Conwell-Weisskopf
method [9], when considering the electron-ionized
impurity scattering. However, the values received using
this method are too large (for the impurity concentration
1015 cm-3 the action radius of the potential is
approximately 160а0).
After that, the values n mK from a precise solution
of the stationary Boltzmann equation for this scattering
mechanism can be now obtained:
,)(1)(
2
γ32
2
2
00
3
2
B
2
0
SSSS
2
14
226
0
233
3
dff
m
Tk
NeeCaV
K
mn
gpp
hh
mn
(11)
where )(0 pf is the Fermi-Dirac function for electrons;
– Kronecker delta symbol and zero of the energy is
at the bottom of the conduction band.
It should be noted that in (11) the product NSS γSS is
used as an adjustable parameter.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 3. P. 272-275.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
274
Fig. 1. Temperature dependences of the electron mobility in
CdxHg1-xSe crystal for different x values. Solid line – mixed
scattering mode; 1, 2, 3, 4, 5, 6, 7, 8 – AC, II, NPO, PAC, PO,
POP, DIS, SS scattering modes, respectively. Experimental
data were taken from [2-4].
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 3. P. 272-275.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
275
3. Comparison of the theory and experiment
The theoretical temperature dependences of the electron
mobility μ(T) were compared to the experimental data
presented in [2-4] for CdxHg1-xSe crystals with
compositions x = 0; 0.05 (sample A1); 0.1 (sample B1);
0.2 (sample C1); 0.268 (sample 26BB2); 0.353 (sample
24AA1-1); 0.547 (sample 40EB2). All the samples were
obtained by the annealing in selenium vapour or in
dynamic vacuum. The Fermi level was obtained from
the electroneutrality equation:
AD NNp n , (12)
where NA, ND are the ionized acceptor and donor
concentrations taken from [2-4].
The material parameters used for calculation were
the same as in [2, 3, 10]. The theoretical μ(T) curves are
presented in Figs 1a-g. The solid lines represent the
curves calculated on the basis of the short-range models
within the framework of the precise solution of the
Boltzmann equation. The obtained scattering parameters
for different scattering modes are listed in Table. It is
seen that the theoretical curves well agree with
experimental data in all the investigated temperature
range. To estimate the role of different scattering
mechanisms in Figs. 1a-1g, the dotted lines represent the
appropriate dependences. It is seen that at low
temperatures (T < 60 K) the main scattering mechanism
is static strain scattering and disorder scattering (for
x > 0). At high temperatures, the contribution of the
polar optical phonon scattering becomes dominant.
Other scattering mechanisms, such as acoustic and
piezoacoustic scattering, piezooptic and nonpolar optical
phonon one, ionized impurity one, give negligibly small
contributions.
Table.
x γ PO γ PZ γ II NSSγSS×10-14 cm-3
0 0.61 0.32 0.26 1.9
0.05 0.64 0.32 0.26 2.8
0.10 0.7 0.32 0.26 4.5
0.2 0.70 0.32 0.26 2.2
0.268 0.68 0.32 0.26 2.9
0.353 0.64 0.32 0.26 2.3
0.547 0.59 0.32 0.26 7.5
4. Conclusion
On the base of the short-range principle, the electron
scattering processes with participation of various lattice
defects in the solid solution CdxHg1-xSe were considered.
A good agreement between the theory and experimental
data within the investigated temperature range was
established.
References
1. W. Szymanska, T. Dietl, Electron scattering and
transport phenomena in small-gap zinc-blende
semiconductors // J. Phys. Chem. Solids 39,
p. 1025-1040 (1978).
2. R.J. Iwanowski, T. Dietl, W. Szymanska, Electron
mobility and electron scattering in CdxHg1-xSe
crystals // J. Phys. Chem. Solids 39, p. 1059-1070
(1978).
3. T. Dietl, W. Szymanska, Electron scattering in
HgSe // J. Phys. Chem. Solids 39, p. 1041-1057
(1978).
4. D.A. Nelson, J.G. Broerman, C.J. Summers,
S.R. Whitsett, Electron transport in Hg1-xCdxSe
alloy system // Phys. Rev. 18, p. 1658-1672 (1978).
5. O.P. Malyk, Electron scattering on the short-range
potential in narrow gap CdxHg1-xTe // Mater. Sci. &
Engineering B 129, p. 161-171 (2006).
6. J.J. Dubowski, Disorder scattering in CdxHg1-xTe
mixed crystals // Phys. status solidi (b) 85, p. 663-
672 (1978).
7. P.A. Fedders, Strain scattering of electrons in
piezoelectric semiconductors // J. Appl. Phys. 54,
p. 1804-1807 (1983).
8. O.P. Malyk, Construction of the exact solution of
the stationary Boltzmann equation for the
semiconductor with isotropic dispersion law //
WSEAS Trans. Math. 3, p. 354-357 (2004).
9. E.M. Conwell, V.F. Weisskopf, Theory of impurity
scattering in semiconductors // Phys. Rev. 77,
p. 388-390 (1950).
10. A. Lehoczky, D.A. Nelson, C.R. Whitsett, Elastic
constants of mercury selenide // Phys. Rev. 188,
p. 1069-1073 (1969).
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