Dependence of the anisotropy parameter of drag thermo-emf on the impurity concentration in the n-type germanium and silicon crystals
Features of the concentration dependences of the anisotropy parameter of thermo-emf of electron-phonon drag M in germanium and silicon crystals of n-type conductivity were found in a wide range of charge carrier concentrations. Insensitivity of the anisotropy parameter M to the presence of impuritie...
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| Опубліковано в: : | Semiconductor Physics Quantum Electronics & Optoelectronics |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2017
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| Цитувати: | Dependence of the anisotropy parameter of drag thermo-emf on the impurity concentration in the n-type germanium and silicon crystals / G.P. Gaidar, P.I. Baranskii // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2017. — Т. 20, № 1. — С. 123-128. — Бібліогр.: 24 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860279365762285568 |
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| author | Gaidar, G.P. Baranskii, P.I. |
| author_facet | Gaidar, G.P. Baranskii, P.I. |
| citation_txt | Dependence of the anisotropy parameter of drag thermo-emf on the impurity concentration in the n-type germanium and silicon crystals / G.P. Gaidar, P.I. Baranskii // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2017. — Т. 20, № 1. — С. 123-128. — Бібліогр.: 24 назв. — англ. |
| collection | DSpace DC |
| container_title | Semiconductor Physics Quantum Electronics & Optoelectronics |
| description | Features of the concentration dependences of the anisotropy parameter of thermo-emf of electron-phonon drag M in germanium and silicon crystals of n-type conductivity were found in a wide range of charge carrier concentrations. Insensitivity of the anisotropy parameter M to the presence of impurities in the germanium crystals up to the concentrations of ∼ 10¹⁵ cm⁻³ was found, whereas in silicon, with increasing the doping level, the monotonic decrease in this parameter was observed. The significantly lower absolute values of the parameter M were obtained for the silicon crystals as compared with the corresponding values of this parameter for the germanium ones. The physical nature of the identified effects was explained.
|
| first_indexed | 2026-03-21T13:44:12Z |
| format | Article |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 1. P.123-128.
doi: https://doi.org/10.15407/spqeo20.01.123
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
123
PACS 61.82.Fk
Dependence of the anisotropy parameter of drag thermo-emf on the
impurity concentration in the n-type germanium and silicon crystals
G.P. Gaidar1, P.I. Baranskii2
1Institute for Nuclear Research, National Academy of Sciences of Ukraine,
47, prospect Nauky, 03680 Kyiv, Ukraine; e-mail: gaydar@kinr.kiev.ua
2V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine,
45, prospect Nauky, 03028 Kyiv, Ukraine
Abstract. Features of the concentration dependences of the anisotropy parameter of
thermo-emf of electron-phonon drag M in germanium and silicon crystals of n-type
conductivity were found in a wide range of charge carrier concentrations. Insensitivity of
the anisotropy parameter M to the presence of impurities in the germanium crystals up to
the concentrations of ∼ 1015 cm-3 was found, whereas in silicon with increasing the
doping level the monotonic decrease in this parameter was observed. The significantly
lower absolute values of the parameter M were obtained for the silicon crystals as
compared with the corresponding values of this parameter for the germanium ones. The
physical nature of the identified effects was explained.
Keywords: germanium, silicon, thermoelectromotive force (thermo-emf), tenso-thermo-
emf, anisotropy parameter of thermo-emf, anisotropy parameter of mobility, charge
carrier concentration.
Manuscript received 12.10.16; revised version received 08.02.17; accepted for
publication 01.03.17; published online 05.04.17.
1. Introduction
Investigations of semiconductors under extreme
conditions (in the strong quantizing magnetic and
electric fields, at very low temperatures, at high
pressures) make it possible to obtain the important
information concerning their properties and charac-
teristics [1]. The studies under conditions of the strong
uniaxial deformation are of particular interest, since
these experiments allow changing (relatively easy and in
a wide range) the essential characteristics of semi-
conductor crystals [2-5]. For example, the many-valley
n-Ge and n-Si crystals can be converted into one- and
two-valley states by application of uniaxial compression,
which can provide a real opportunity to measure such
characteristics as the deformation potentials, the
anisotropy parameters, etc [6-10].
However, it should be noted that under study of
semiconductors in conditions of strong deformation the
galvanomagnetic measurements became widespread [11-
13], while the experimental works devoted to the study
of thermoelectric and thermomagnetic phenomena under
these conditions appear quite rare [14, 15]. The studies,
especially at low temperatures in the region of drag
effect of electrons by phonons, can provide information
not only concerning the structure of the energy spectrum
of carriers and the character of their scattering but also
about the phonon-phonon interaction mechanism. In
addition, these studies will make it possible to determine
a number of important characteristics and parameters of
crystals, namely: the anisotropy parameter of thermo-
emf, the effective masses of charge carriers, the impurity
concentration, the relaxation time of the phonon-phonon
interaction and others.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 1. P.123-128.
doi: https://doi.org/10.15407/spqeo20.01.123
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
124
The anisotropy parameter of drag thermo-emf of
electrons by phonons
phphM ⊥αα= || (1)
is one of the most important parameters of the kinetics
theory for electronic processes in the many-valley
semiconductors, along with the anisotropy parameter of
mobility ||μμ= ⊥K [14, 16, 17]. Here, ph
||α , ph
⊥α are
the phonon components of drag thermo-emf along and
across the long axis of isoenergetic ellipsoid,
respectively; ||μ , ⊥μ are the mobilities of charge
carriers along and across the long axis of this ellipsoid,
respectively.
In contrast to the values of the anisotropy
parameter of mobility K that describes the electronic
subsystem, the anisotropy parameter of thermo-emf M
characterizes the phonon subsystem in a certain way.
This parameter is found from the thermoelectric
measurements, whereas the parameter K is found from
the experimental data on tensoresistance.
The value of parameter M for n-Ge was found by
various authors and even with using of the different
techniques, but mainly for the conditions of phonon
scattering (see, e.g., [18, 19]). In consideration of
differences in the values of M, which are available in the
individual papers, we can assume that the most probable
value of this parameter is equal to 9 ± 1 for the
sufficiently pure n-Ge crystals. With account that
germanium and silicon doped with impurities over a
wide range of concentrations are used in the electronic
engineering, it is also necessary to know the value of
parameter M in the mixed scattering region, when we
calculate the thermoelectric and thermomagnetic effects
in these crystals by using the anisotropic scattering
theory.
The aim of this study was to investigate the
anisotropy parameter of drag thermo-emf M of electrons
by phonons in germanium and silicon of n-type
conductivity over a wide range of concentrations, which
includes both mixed and predominantly phonon
scattering.
2. Theoretical information
Germanium and silicon are characterized by isotropism
of the kinetic coefficients in the natural (i.e.,
mechanically relaxed) state due to cubic symmetry of
these crystals. Consequently, the kinetic phenomena in
these crystals (including thermo-emf) are described by
the scalar quantities under the named conditions. In
particular, thermo-emf coefficient (Seebeck coefficient)
also is a scalar on the macrolevel (i.e., on the level of
crystal). The situation changes in the uniaxially
elastically deformed Ge and Si crystals, and the Seebeck
coefficient becomes a tensor quantity [20].
On the example of n-Ge many-valley
semiconductor, let us consider the method for
determination of the anisotropy parameter for thermo-
emf in the area of electron-phonon drag by using the
results of measuring the diagonal components of thermo-
emf tensor.
In a general case, the experimentally measured
values of the coefficient of differential thermo-emf α can
be represented as a sum of the electronic (diffusion) eα
and phonon phα components [21]:
phe α+α=α . (2)
Taking into account that, in the elastically
deformed along the crystallographic direction [111]
n-Ge, the thermo-emf can be represented as a second-
rank tensor in the laboratory system of coordinates
(associated with the axes of the isoenergetic ellipsoid,
located on the deformation axis) [17]
33
22
11
00
00
00
ˆ
α
α
α
=α , (3)
where 2211 α=α and 33α are the diagonal terms of the
thermo-emf tensor, then phe
111111 α+α=α and
phе
333333 α+α=α .
It is noted that under the uniaxial elastic
deformation of n-Ge along [111] the energy minimum,
oriented in this direction, is shifted down along the
energy scale, while the remaining three minima are
shifted upwards. Let us denote the concentration of
charge carriers in the minimum, which is lowered, by N1
and the concentration of charge carriers in any from
three minima, which is raised, by N2.
It can be shown [22] that, at the arbitrary in
magnitude mechanical load X on the n-Ge crystal, the
following expression will take place under the condition
of X
r
|| J
r
|| [111] (J – current):
3
181
3
8
3333 +
γ+
+
γ+
⋅α=α−α ⊥ K
MKM
phe , (4)
where T
XX
Tk
S
ee
N
N
u 120.09
4
1
2
44
−
Ξ
−
===γ is the ratio
of the carrier concentrations in ellipsoids for arbitrary
values of the mechanical stress X and temperature T; Ξu
is the constant of the shear deformation potential; S44 is
the compliance coefficient (for n-Ge S44 = 1.46⋅10–11 Pa–1).
The anisotropy parameter of electron mobility K within
the separate isoenergetic ellipsoid is given by the
expression
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 1. P.123-128.
doi: https://doi.org/10.15407/spqeo20.01.123
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
125
2
1
2
3
0||
−
ρ
ρ
=
μ
μ
= ∞⊥K , (5)
where ρ0 and ( )X
X
ρ=ρ
∞→
∞ lim are the resistivity of
undeformed (at Х = 0) and uniaxially elastically
deformed (at Х → ∞) crystal, respectively [ρ∞
corresponds to the saturation region of the function
ρ = ρ (Х)].
Eq. (2) shows that the phonon components of
thermo-emf without pressure (Х = 0) ph
0α and in
saturation (Х → ∞) ph
∞α are equal to the experimentally
measured values of thermo-emf (α0 and α∞) without
electronic component ( e
4α and e
1α – for the case of the
non-deformed and strongly deformed crystal,
respectively):
eph
400 α−α=α ,
pheph
||1 α≡α−α=α ∞∞ . (6)
The electronic (diffusion) component of thermo-
emf e
Nα can be determined by the Pisarenko formula
[23]:
( )
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡ π
+=α 3
0
2/3*22ln2
hn
Tkm
e
ke
N , (7)
where n0 is the charge carrier concentration; e – electron
charge; k – Boltzmann constant; T – temperature; h –
Planck constant; 3 2
||
2/3*
⊥= mmNm – effective mass
of the density of states; N – number of the isoenergetic
ellipsoids, particularly, for n-Ge
⎩
⎨
⎧
=≥
=
=
K77andGPa6.0at1
,0at4
ТХ
Х
N . Since the
electronic component of thermo-emf eα is essentially
independent of the mechanical load X, then
eee α≈α≈α 14 .
Using the directly measured values of α∞, one can
obtain, according to Eq. (6), the value of the longitudinal
phonon component eph
1|| α−α=α ∞ . And using Eq. (1),
one can find the transverse phonon component:
Mphph
||α=α⊥ . (8)
It follows from Eqs (6) and (8) that
Mphphе ⋅α=α=α−α ⊥∞ || . (9)
In the absence of uniaxial mechanical load (Х = 0)
on the investigated sample, one can obtain from Eq. (4)
(if removing the indices 33) the following expression:
12
2
00 +
+
⋅α=α−α=α ⊥ K
MKpheph . (10)
Equation (10) binds (through the anisotropy
parameters K and M) the phonon thermo-emf of the
whole crystal (at Х = 0) with one of the components of
phonon thermo-emf in a single isoenergetic ellipsoid
ph
⊥α .
Thus, we have a system of two equations (9) and
(10) with two unknown quantities ( ph
⊥α and М).
Excluding the value of ph
⊥α from this system, the
expression for determining the anisotropy parameter of
thermo-emf in germanium crystals can be obtained:
( ) ( ) 112
2
112
2
00 −
α
α
+
=
−
α−α
α−α
+
=
∞∞
ph
ph
e
e
K
K
K
KM . (11)
Usage of a system of two equations (9) and (10)
when transferring to the study of n-Si (instead of n-Ge)
is related with deformation of this crystal in the [100]
direction (at conditions X
r
|| J
r
|| [100]) instead of the
experimental conditions X
r
|| J
r
|| [111] that are used in
the study of n-Ge.
3. Experimental results and discussion
Measurements of the thermo-emf and tenso-thermo-emf
were carried out at the temperature of 85 K on the single
n-Ge and n-Si crystals within the range of the
concentrations 1.9⋅1012…4.6⋅1017 cm–3, and measure-
ments of the Hall parameters and tensoresistance were
carried out at Т = 77 K. At the beginning, the resistivity
0ρ and thermo-emf 0α were measured without
pressure, then the n-Ge and n-Si samples were converted
into the one- and two-valley state, respectively, and the
values of ∞ρ and ∞α were measured. The mechanical
load of Х = 0.8 GPa was applied to the investigated
samples (in order to convert them into the one- and two-
valley state): in conditions of X
r
|| J
r
, T∇ || [111] for
n-Ge; at X
r
|| J
r
, T∇ || [100] for n-Si.
The typical view of the tensoresistance 0ρρX and
tenso-thermo-emf 0αα X dependences on the
mechanical load X is presented in Fig. 1 for one of the
studied silicon samples.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 1. P.123-128.
doi: https://doi.org/10.15407/spqeo20.01.123
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
126
1
2
3
4
5
0.0 0.2 0.4 0.6 0.8 1.0
1.0
1.5
2.0
2.5
3.0
1
α
X /
α
0
ρ X /
ρ 0
X, GPa
2
Fig. 1. Typical view of the dependences of tensoresistance
0ρρ X (1) and tenso-thermo-emf 0ααX (2) on the
mechanical load X for n-Si.
In order to not significantly affect on the average
sample temperature, which is given by the ambient
temperature, the temperature gradient on the sample
must be set, possibly, by the insignificant temperature
drop 12 TTT −=Δ . This temperature drop in the
working conditions will not exceed of 5 degree/cm. In
these experiments, the one-dimensionality of the heat
flow must be provided, reducing the lateral heat loss
from the studied samples to the possible minimum and
supporting the high-quality lateral thermal insulation.
Fig. 2 shows the experimentally determined
concentration dependences of the 0α and ∞α values for
n-Ge samples. Fig. 3 presents the values of the
anisotropy parameter of mobility K for the silicon and
germanium crystals obtained using the measurement
results of tensoresistance in the saturation region ( ∞ρ )
and the magnitude of this resistance without the
mechanical load ( 0ρ ) as well as Eq. (5). Thermo-emf in
the undeformed ( 0α ) and deformed ( ∞α ) states, the
value of the parameter K and the value of the electronic
component of thermo-emf ( eα ) calculated using Eq. (7)
made it possible to calculate for each of the samples of
silicon and germanium the value of the anisotropy
parameter of drag thermo-emf ( )e
phph nfM =αα= ⊥||
by Eq. (11) under the assumption that de Nn ≡ (Fig. 4).
As it follows from Fig. 4, the value of the
parameter M for n-Ge (curve 1) remains constant [unlike
that in n-Si (Fig. 4, curve 2)] in the fairly wide
concentration range (1.9⋅1012 ≤ ne ≤ 1015 cm–3) and
1012 1013 1014 1015 1016 1017 1018
0
2
4
6
8
10
α
0 ,
α
∞
,
μV
/K
2
ne , cm-3
1
Fig. 2. Concentration dependence of thermo-emf
(at Т = 85 K) for n-Ge crystals: 1 – undeformed (four-valley);
2 – deformed (one-valley) in conditions of Х = 0.8 GPa,
X
r
|| J
r
, T∇ || [111].
1012 1013 1014 1015 1016 1017 1018
2
4
6
8
10
12
14
16
K
=
μ
⊥
/μ
||
2
ne , cm-3
1
Fig. 3. Concentration dependence (at Т = 77 K) of the
anisotropy parameter of mobility ||μμ= ⊥K in crystals: 1 –
n-Ge; 2 – n-Si.
equals approximately 9.8, which agrees well with the
known data for this parameter in the area of the
predominant phonon scattering. As opposed to the
anisotropy parameter of mobility K, which is formed by
combination of the electron scattering mechanisms
related with crystal lattice vibrations and impurity
centers, the phonon component of thermo-emf (more
precisely, its anisotropy, i.e., the ratio phphM ⊥αα= || ) is
almost independent from the concentration ne ≡ Nd [in
any case, in the investigated limits 1.9⋅1012…1015 cm–3]
and is completely determined by the vibrations of the
atoms in the lattice sites.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 1. P.123-128.
doi: https://doi.org/10.15407/spqeo20.01.123
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
127
1012 1013 1014 1015 1016 1017 1018
2
4
6
8
10
M
=
α
||ph
/
α
⊥
ph
ne , cm-3
2
1
Fig. 4. Concentration dependence (at Т = 85 K) of the aniso-
tropy parameter of drag thermo-emf phphM ⊥αα= || in crys-
tals: 1 – n-Ge; 2 – n-Si.
The further growth of the doping level (ne ≡
Nd ≥ 1015 cm–3) in germanium crystals leads to a sharp
decrease of the parameter M. The revealed effect is
associated with an increase in the efficiency of the
scattering of both electrons (which are captured by the
phonons) on the impurity ions, and phonons on the
conduction electrons in the field of ionized impurity.
Insensitivity of the parameter M to the presence of
impurities in n-Ge (within the concentration range
1012…1015 cm–3) is probably caused by the features of
the phonon subsystem in this material.
It is necessary to draw attention to the different
character of the effect of increasing the dopant concen-
tration on the electron and phonon subsystems in n-Ge
and n-Si. While changes in the anisotropy parameter of
mobility K in n-Ge and n-Si with increase of the con-
centration are qualitatively similar (Fig. 3), the changes
in the anisotropy parameter of drag thermo-emf M differ
significantly (Fig. 4). In n-Si crystals, both the parameter
M, and the parameter K are decreased within the range
of charge carrier concentrations 1012 ≤ ne ≤ 2⋅1015 cm–3
with an increase of the contribution of impurity
scattering, whereas in n-Ge crystals only the parameter
K is reduced, while the parameter M is almost constant.
In addition, the data comparison for the n-Ge and
n-Si samples, presented in Figs 3 and 4, indicates the
much higher (in absolute magnitude) values of the M and
K parameters, which are characteristic for the n-Ge single
crystals as compared with the corresponding values for
n-Si. In the first place, this is related with the higher
anisotropy of the effective mass of the charge carriers in
n-Ge, than in n-Si ( 3.19082.058.1|| ≅=⊥mm for
n-Ge and 75.4191.091.0|| ≅=⊥mm for n-Si), which
causes the appearance of substantially different
scattering conditions in the n-Ge and n-Si crystals. This
is also related with a significant difference in the
arrangement of the isoenergetic ellipsoids relative to the
crystal axes in germanium and silicon (Fig. 5).
Fig. 5. Reciprocal arrangement of the isoenergetic ellipsoids in
the conduction band of germanium (a), silicon (b) [24].
4. Conclusions
1. In the many-valley n-Ge and n-Si crystals, the
relationship of the thermo-emf in undeformed (α0)
and strongly deformed (α∞) states with the
transverse phonon component ( ph
⊥α ) and with the
anisotropy parameter of drag thermo-emf M was
ascertained. The relationship of the drag thermo-
emf in undeformed crystal (α0) with the anisotropy
parameter of mobility K within the individual
isoenergetic ellipsoid was revealed, too.
2. It was found that the anisotropy parameter of drag
thermo-emf M is changed monotonically in n-Si
with an increase of the charge carrier concentration
(nе), whereas in n-Ge this parameter is decreased
sharply for the concentrations above 1015 cm–3.
3. It is shown that the n-Si single crystals are
characterized by significantly smaller (in absolute
magnitude) values of the anisotropy parameter M
as compared with the corresponding values for
n-Ge. This fact is explained by lower anisotropy of
the electron effective mass and, consequently, the
substantially different scattering of electrons.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 1. P.123-128.
doi: https://doi.org/10.15407/spqeo20.01.123
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
128
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| id | nasplib_isofts_kiev_ua-123456789-214903 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2026-03-21T13:44:12Z |
| publishDate | 2017 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Gaidar, G.P. Baranskii, P.I. 2026-03-03T11:02:20Z 2017 Dependence of the anisotropy parameter of drag thermo-emf on the impurity concentration in the n-type germanium and silicon crystals / G.P. Gaidar, P.I. Baranskii // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2017. — Т. 20, № 1. — С. 123-128. — Бібліогр.: 24 назв. — англ. 1560-8034 PACS: 61.82.Fk https://nasplib.isofts.kiev.ua/handle/123456789/214903 https://doi.org/10.15407/spqeo20.01.123 Features of the concentration dependences of the anisotropy parameter of thermo-emf of electron-phonon drag M in germanium and silicon crystals of n-type conductivity were found in a wide range of charge carrier concentrations. Insensitivity of the anisotropy parameter M to the presence of impurities in the germanium crystals up to the concentrations of ∼ 10¹⁵ cm⁻³ was found, whereas in silicon, with increasing the doping level, the monotonic decrease in this parameter was observed. The significantly lower absolute values of the parameter M were obtained for the silicon crystals as compared with the corresponding values of this parameter for the germanium ones. The physical nature of the identified effects was explained. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Dependence of the anisotropy parameter of drag thermo-emf on the impurity concentration in the n-type germanium and silicon crystals Article published earlier |
| spellingShingle | Dependence of the anisotropy parameter of drag thermo-emf on the impurity concentration in the n-type germanium and silicon crystals Gaidar, G.P. Baranskii, P.I. |
| title | Dependence of the anisotropy parameter of drag thermo-emf on the impurity concentration in the n-type germanium and silicon crystals |
| title_full | Dependence of the anisotropy parameter of drag thermo-emf on the impurity concentration in the n-type germanium and silicon crystals |
| title_fullStr | Dependence of the anisotropy parameter of drag thermo-emf on the impurity concentration in the n-type germanium and silicon crystals |
| title_full_unstemmed | Dependence of the anisotropy parameter of drag thermo-emf on the impurity concentration in the n-type germanium and silicon crystals |
| title_short | Dependence of the anisotropy parameter of drag thermo-emf on the impurity concentration in the n-type germanium and silicon crystals |
| title_sort | dependence of the anisotropy parameter of drag thermo-emf on the impurity concentration in the n-type germanium and silicon crystals |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214903 |
| work_keys_str_mv | AT gaidargp dependenceoftheanisotropyparameterofdragthermoemfontheimpurityconcentrationinthentypegermaniumandsiliconcrystals AT baranskiipi dependenceoftheanisotropyparameterofdragthermoemfontheimpurityconcentrationinthentypegermaniumandsiliconcrystals |