Mechanisms of electron scattering in uniaxially deformed n-Ge‹Sb, Au› single crystals
Temperature dependencies for concentration and the Hall mobility of electrons for the n-GehSbi and n-GehSb, Aui single crystals uniaxially deformed along the crystallographic directions [100] and [111] are obtained on the basis of piezo-Hall effect measurements. A deformation-induced increase of th...
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irk-123456789-1574772019-06-21T01:28:45Z Mechanisms of electron scattering in uniaxially deformed n-Ge‹Sb, Au› single crystals Luniov, S.V. Nazarchuk, P.F. Zimych, A.I. Udovytska, Yu.A. Burban, O.V. Temperature dependencies for concentration and the Hall mobility of electrons for the n-GehSbi and n-GehSb, Aui single crystals uniaxially deformed along the crystallographic directions [100] and [111] are obtained on the basis of piezo-Hall effect measurements. A deformation-induced increase of the Hall mobility of electrons for n-GehSb, Aui single crystals at the uniaxial pressure along the crystallographic direction [100] has been revealed. A comparison of the obtained experimental results with the corresponding theoretical calculations of temperature dependencies of the Hall mobility showed that the obtained effect occurs at the expense of the reduction probability of electron scattering on the fluctuational potential. Its amplitude depends on the temperature and on the value of the uniaxial pressure. It has also been shown that an increase of the Hall mobility for the n-GehSb, Aui single crystals uniaxially deformed along the crystallographic direction [111] with an increasing temperature turns out to be insignificant and is observed only for the uniaxial pressures P < 0.28 GPa. A decrease of the Hall mobility of electrons at the expense of the deformational redistribution of electrons among the valleys of the germanium conduction band with different mobility should be taken into account in the present case. The Hall mobility magnitude for the uniaxially deformed n-GehSbi single crystals is determined only by the mechanisms of phonon scattering and we have not observed the effect of the growth of the Hall mobility with an increase of temperature or the magnitude of uniaxial pressure. This demonstrates a secondary role of the fluctuational potential in the present case. На основi вимiрювань п’єзо-холл-ефекту одержано температурнi залежностi концентрацiї та холiвської рухливостi електронiв для одновiсно деформованих вздовж кристалографiчних напрямкiв [100] та [111] монокристалiв n-GehSbi та n-GehSb, Aui. Виявлено деформацiйно-iндуковане зростання холiвської рухливостi електронiв для монокристалiв n-GehSb, Aui при одновiсному тисковi вздовж кристалографiчного напрямку [100]. Порiвняння одержаних експериментальних результатiв з вiдповiдними теоретичними розрахунками температурних залежностей холiвської рухливостi показали, що даний ефект виникає за рахунок зменшення ймовiрностi розсiяння електронiв на флуктуацiйному потенцiалi, амплiтуда якого залежить вiд температури та величини одновiсного тиску. Показано, що зростання холiвської рухливостi для одновiсно деформованих монокристалiв n-GehSb, Aui вздовж кристалографiчного напрямку [111] є незначним i спостерiгається лише для одновiсних тискiв P < 0.28 ГПа. В даному випадку необхiдно також враховувати зменшення холiвської рухливостi електронiв за рахунок деформацiйного перерозподiлу електронiв мiж долинами зони провiдностi германiю з рiзною рухливiстю. Для одновiсно деформованих монокристалiв n-GehSbi величина холiвської рухливостi при таких самих умовах визначається лише механiзмами фононного розсiяння i ефект зростання холiвської рухливостi при збiльшеннi температури або величини одновiсного тиску не спостерiгався. Це свiдчить про другорядну роль флуктуацiйного потенцiалу в даному випадку. 2019 Article Mechanisms of electron scattering in uniaxially deformed n-Ge‹Sb, Au› single crystals / S.V. Luniov, P.F. Nazarchuk, A.I. Zimych, Yu.A. Udovytska, O.V. Burban // Condensed Matter Physics. — 2019. — Т. 22, № 1. — С. 13702: 1–10. — Бібліогр.: 22 назв. — англ. 1607-324X PACS: 72.20.Fr, 72.10.-d DOI:10.5488/CMP.22.13702 arXiv:1903.11496 http://dspace.nbuv.gov.ua/handle/123456789/157477 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Temperature dependencies for concentration and the Hall mobility of electrons for the n-GehSbi and
n-GehSb, Aui single crystals uniaxially deformed along the crystallographic directions [100] and [111] are obtained on the basis of piezo-Hall effect measurements. A deformation-induced increase of the Hall mobility
of electrons for n-GehSb, Aui single crystals at the uniaxial pressure along the crystallographic direction [100]
has been revealed. A comparison of the obtained experimental results with the corresponding theoretical calculations of temperature dependencies of the Hall mobility showed that the obtained effect occurs at the expense of the reduction probability of electron scattering on the fluctuational potential. Its amplitude depends
on the temperature and on the value of the uniaxial pressure. It has also been shown that an increase of the
Hall mobility for the n-GehSb, Aui single crystals uniaxially deformed along the crystallographic direction [111]
with an increasing temperature turns out to be insignificant and is observed only for the uniaxial pressures
P < 0.28 GPa. A decrease of the Hall mobility of electrons at the expense of the deformational redistribution
of electrons among the valleys of the germanium conduction band with different mobility should be taken into
account in the present case. The Hall mobility magnitude for the uniaxially deformed n-GehSbi single crystals is
determined only by the mechanisms of phonon scattering and we have not observed the effect of the growth
of the Hall mobility with an increase of temperature or the magnitude of uniaxial pressure. This demonstrates
a secondary role of the fluctuational potential in the present case. |
| format |
Article |
| author |
Luniov, S.V. Nazarchuk, P.F. Zimych, A.I. Udovytska, Yu.A. Burban, O.V. |
| spellingShingle |
Luniov, S.V. Nazarchuk, P.F. Zimych, A.I. Udovytska, Yu.A. Burban, O.V. Mechanisms of electron scattering in uniaxially deformed n-Ge‹Sb, Au› single crystals Condensed Matter Physics |
| author_facet |
Luniov, S.V. Nazarchuk, P.F. Zimych, A.I. Udovytska, Yu.A. Burban, O.V. |
| author_sort |
Luniov, S.V. |
| title |
Mechanisms of electron scattering in uniaxially deformed n-Ge‹Sb, Au› single crystals |
| title_short |
Mechanisms of electron scattering in uniaxially deformed n-Ge‹Sb, Au› single crystals |
| title_full |
Mechanisms of electron scattering in uniaxially deformed n-Ge‹Sb, Au› single crystals |
| title_fullStr |
Mechanisms of electron scattering in uniaxially deformed n-Ge‹Sb, Au› single crystals |
| title_full_unstemmed |
Mechanisms of electron scattering in uniaxially deformed n-Ge‹Sb, Au› single crystals |
| title_sort |
mechanisms of electron scattering in uniaxially deformed n-ge‹sb, au› single crystals |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2019 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157477 |
| citation_txt |
Mechanisms of electron scattering in uniaxially deformed n-Ge‹Sb, Au› single crystals / S.V. Luniov, P.F. Nazarchuk, A.I. Zimych, Yu.A. Udovytska, O.V. Burban // Condensed Matter Physics. — 2019. — Т. 22, № 1. — С. 13702: 1–10. — Бібліогр.: 22 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT luniovsv mechanismsofelectronscatteringinuniaxiallydeformedngesbausinglecrystals AT nazarchukpf mechanismsofelectronscatteringinuniaxiallydeformedngesbausinglecrystals AT zimychai mechanismsofelectronscatteringinuniaxiallydeformedngesbausinglecrystals AT udovytskayua mechanismsofelectronscatteringinuniaxiallydeformedngesbausinglecrystals AT burbanov mechanismsofelectronscatteringinuniaxiallydeformedngesbausinglecrystals |
| first_indexed |
2025-07-14T09:54:11Z |
| last_indexed |
2025-07-14T09:54:11Z |
| _version_ |
1837615653417123840 |
| fulltext |
Condensed Matter Physics, 2019, Vol. 22, No 1, 13702: 1–10
DOI: 10.5488/CMP.22.13702
http://www.icmp.lviv.ua/journal
Mechanisms of electron scattering in uniaxially
deformed n-Ge〈Sb, Au〉 single crystals
S.V. Luniov1, P.F. Nazarchuk1, A.I. Zimych1, Yu.A. Udovytska1, O.V. Burban2
1 Lutsk National Technical University, 75 Lvivska St., 43018 Lutsk, Ukraine
2 Volyn College of the National University of Food Technologies, 6 Cathedral St., 43016 Lutsk, Ukraine
Received January 5, 2019, in final form February 4, 2019
Temperature dependencies for concentration and the Hall mobility of electrons for the n-Ge〈Sb〉 and
n-Ge〈Sb, Au〉 single crystals uniaxially deformed along the crystallographic directions [100] and [111] are ob-
tained on the basis of piezo-Hall effect measurements. A deformation-induced increase of the Hall mobility
of electrons for n-Ge〈Sb, Au〉 single crystals at the uniaxial pressure along the crystallographic direction [100]
has been revealed. A comparison of the obtained experimental results with the corresponding theoretical cal-
culations of temperature dependencies of the Hall mobility showed that the obtained effect occurs at the ex-
pense of the reduction probability of electron scattering on the fluctuational potential. Its amplitude depends
on the temperature and on the value of the uniaxial pressure. It has also been shown that an increase of the
Hall mobility for the n-Ge〈Sb, Au〉 single crystals uniaxially deformed along the crystallographic direction [111]
with an increasing temperature turns out to be insignificant and is observed only for the uniaxial pressures
P < 0.28 GPa. A decrease of the Hall mobility of electrons at the expense of the deformational redistribution
of electrons among the valleys of the germanium conduction band with different mobility should be taken into
account in the present case. The Hall mobility magnitude for the uniaxially deformed n-Ge〈Sb〉 single crystals is
determined only by the mechanisms of phonon scattering and we have not observed the effect of the growth
of the Hall mobility with an increase of temperature or the magnitude of uniaxial pressure. This demonstrates
a secondary role of the fluctuational potential in the present case.
Key words: Hall effect, transport phenomena, electrical properties, impurities
PACS: 72.20.Fr, 72.10.-d
1. Introduction
The development of modern electronics is associated with the search and development of new mate-
rials or the improvement of the properties of the existing ones. At present, a semiconductor material such
as germanium is a promising material for creating various electronic devices and sensors such as diodes,
triodes, dosimeter devices, transducers, Hall sensors, detectors of infrared radiation [1, 2]. Technologies
of strained germanium find their practical use in nanosized transistor structures and nanophotonics [3–7].
The use of n-Ge single crystals as a material for channels NMOSFET transistors allows one to increase
their gain factor and the tunnel current in relation to those transistors whose channel is made of n-Si
[3–5]. The purposeful impact of Germanium lattice deformation opens prospects for the creation of
fundamentally new elements of nanophotonics on its basis [6, 7]. However, almost all scientific publica-
tions on nanoelectronics, in particular, works [3–7] which are devoted to the investigation and modelling
of physical properties of strained germanium and nanostructures on its basis are restricted only to the
examination of the deformation impact and the related effects on the atoms of the germanium lattice.
However, the impact of the defective structure of germanium single crystals, created by controllable and
uncontrollable impurities, was not taken into account. These impurities are introduced at the synthesis
of such nanostructures and determine the degree of compensation for the present single crystals. Such
impurities can create both shallow and deep energy levels in the band gap of germanium and significantly
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
13702-1
https://doi.org/10.5488/CMP.22.13702
http://www.icmp.lviv.ua/journal
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S.V. Luniov et al.
affect the functional characteristics of semiconductor devices on its basis and also largely determine
the percentage of the output of suitable products. Theoretical models of the deformation influence on
different kinetic and optical effects are very scarce both for germanium and in other semiconductors in the
presence of deep energy levels of impurities. Today, the theory of deep levels in semiconductors, unlike
the theory of shallow levels is based only on their semi-empirical models use [8]. The practical value of
such investigations is connected with the fact that the impurity centers with deep energy levels determine
the light-emitting diodes spectra. They are centers of rapid recombination, create additional regions of
photosensitivity, strongly affect the tensosensitivity of semiconductors. Therefore, investigations of the
impact of the alloyed impurities with deep levels on electrical and optical properties of the deformed
germanium single crystals are urgent both from fundamental and applied points of view. Surely, these
investigations, in their turn, will allow scientists and engineers to provide certain scientific and metho-
dical recommendations concerning technological conditions of synthesis and modelling of germanium
properties and strained nanostructures on its basis.
2. Results and discussion
So, we were measuring the temperature dependences of the Hall constant and the electrical conducti-
vity for uniaxially deformed n-Ge single crystals, alloyed by Sb and Au impurities. Uniaxial deformation
was attached along the crystallographic directions [100] and [111]. The crystallographic orientation
of the investigated single crystals was determined using an X-ray machine, with an accuracy of 10′.
Germanium samples for research were cut in the form of a rectangular parallelepipeds, 0.8×0.8×10 mm
in size. Measurements were conducted for n-Ge samples of two groups. n-Ge〈Sb〉 samples of the first
group were alloyed only by Sb impurity, NSb = 5 · 1014 cm−3 concentration. n-Ge〈Sb, Au〉 samples of
the second group were alloyed by Sb impurity, NSb = 9.8 · 1014 cm−3 concentration and Au impurity,
NAu = 5.05 · 1014 cm−3 concentration. The temperature control was carried out with the help of the
copper-constantan differential thermocouple. During the experiment, the accuracy of the measurement
of temperature was ±1 K. As it is known [9], impurity of Sb is a shallow donor. It forms in the band gap
of germanium shallow energy level EC − 0.0096 eV. Au impurity forms a deep donor level EV + 0.04 eV
and three acceptor levels EV + 0.15 eV, EC − 0.2 eV and EC − 0.04 eV in Ge. The piezo-Hall effect
study for the n-Ge〈Sb〉 single crystals uniaxially deformed along the crystallographic directions [100]
and [111] shows that concentration of electrons at temperatures T > 77 K is equal to the concentration
of alloying impurity Sb and does not depend on the temperature. This is explained by the fact that
Figure 1. Temperature dependencies of the Hall mobility for the n-Ge〈Sb〉 single crystals at different
values of the uniaxial pressure along the crystallographic directions [100] and [111]: 1 – 0 GPa; 2 –
0.34 GPa (for the crystallographic direction [100]); 3 – 0.85 GPa (for the crystallographic direction
[100]); 4 – 0.47 GPa (for the crystallographic direction [111]); 5 – 0.96 GPa (for the crystallographic
direction [111]).
13702-2
Mechanisms of electron scattering
Figure 2. Temperature dependencies of the concen-
tration of electrons for the n-Ge〈Sb, Au〉 at different
values of the uniaxial pressure along the crystallo-
graphic direction [100]: 1 – 0 GPa; 2 – 0.29 GPa;
3 – 0.59 GPa; 4 – 0.88 GPa.
Figure 3. Temperature dependencies of the concen-
tration of electrons for the n-Ge〈Sb, Au〉 at different
values of the uniaxial pressure along the crystallo-
graphic direction [111]: 1 – 0 GPa; 2 – 0.28 GPa;
3 – 0.69 GPa; 4 – 0.97 GPa.
the shallow donors of Sb will be completely ionized for such temperatures. Experimental results of
temperature dependencies of the Hall mobility for n-Ge〈Sb〉 single crystals uniaxially deformed along
the crystallographic directions [100] and [111] are presented in figure 1. As can be seen in figure 1,
the Hall mobility at the deformation along the crystallographic direction [100] does not depend on the
magnitude of the uniaxial pressure. There is no deforming redistribution of electrons between the minima
of germanium conduction band which will be shifting upward (according to the scale of energies) at the
deformation with the same speed for the present case [10]. The deforming redistribution of electrons
between three minima of germanium conduction band with higher mobility (which ascend upwards) and
one minimum with smaller mobility (which descends down according to the scale of energies at the
deformation) will take place at the uniaxial deformation n-Ge〈Sb〉 along the crystallographic directions
[111]. In this case, the concentration of electrons in three minima with higher mobility will decrease.
However, the concentration of electrons in the minima with smaller mobility will increase [10]. This
will lead to a decrease of the average Hall mobility of electrons with an increase of the uniaxial pressure
magnitude. This fact explains the obtained experimental results (figure 1, curves 4 and 5). The growth
of the concentration of electrons in the germanium conduction band with an increasing temperature
for n-Ge〈Sb, Au〉 single crystals (figure 2 and figure 3) is explained by thermal ionization of the deep
acceptor level of gold EC − 0.2 eV.
Moreover, for the given single crystals, as can be seen in figure 2 and figure 3 (curves 2−4), the
concentration of electrons increases when the magnitude of the uniaxial pressure increases. This is
associated with a decrease of ionization energy of the level EC − 0.2 eV at the deformation [11].
A research of the hydrostatic pressure influence on the position for the deep level of gold EC − 0.2 eV
in germanium single crystals has been conducted by the authors of works [12, 13]. The baric coefficient
magnitude for ionization energy of the given level which had been calculated in work [12], turned out to
be understated, taking into account the experimental results of work [13]. This is explained by the fact that
the authors of work [12] did not take into account the impact of fluctuation potential in their calculations
[13, 14]. As it is known [15–18], such a potential arises in the alloyed compensated semiconductors
or in semiconductors with radiation-induced defects when the concentration of free charge carriers is
small in comparison with the concentration of ionized impurity centres or charged radiation defects. The
amplitude of this potential can be quite significant which will lead to modulation of density of states
for charge carriers and, as a result, to a decrease of Hall mobility. For the given case, Hall mobility of
electrons can be written as follows [17, 18]:
µH = µH0 exp
(
−
∆
kT
)
, (1)
13702-3
S.V. Luniov et al.
Figure 4. Temperature dependencies of the ampli-
tude of fluctuation potential for the uniaxially de-
formed n-Ge〈Sb, Au〉 single crystals at different
values of the uniaxial pressure along the crystal-
lographic direction [100]: 1 – 0 GPa; 2 – 0.29 GPa;
3 – 0.59 GPa; 4 – 0.88 GPa.
Figure 5. Temperature dependencies of the ampli-
tude of fluctuation potential for the uniaxially de-
formed n-Ge〈Sb, Au〉 single crystals at different
values of the uniaxial pressure along the crystal-
lographic direction [111]: 1 – 0 GPa; 2 – 0.28 GPa;
3 – 0.69 GPa; 4 – 0.97 GPa.
where µH0 is Hall mobility for uncompensated semiconductor, ∆ is amplitude of fluctuation potential. In
accordance with [16],
∆ =
q2N
2
3
εn
1
3
, (2)
where N is the total concentration of charged impurities or defects, ε is dielectric penetration, n is the
concentration of electrons in the conduction band, q is electron charge.
The reduction of the amplitude of fluctuation potential for the uniaxially deformed n-Ge〈Sb, Au〉
single crystals (figure 4 and figure 5), taking into account the expression (2) and experimental results
(figure 2 and figure 3), is caused by an increase of the concentration of electrons in the germanium
conduction band at the expense of magnification of the temperature or magnitude of the uniaxial pressure.
This, in its turn, explains the anomalous temperature dependencies of Hall mobility for the
n-Ge〈Sb, Au〉 single crystals uniaxially deformed along the crystallographic direction [100] (figure 6).
As it follows from figure 6, Hall mobility does not depend on uniaxial pressure at high temperatures
because the amplitude of fluctuation potential is small and weakly depends on deformation (figure 4).
For the case of uniaxial pressure along the crystallographic direction [111] (figure 7), the deformation-
induced increase of the Hall mobility at low temperatures was not observed. Only at T < 140 K and
P > 0.69 GPa (figure 7, curves 3 and 4) Hall mobility slightly increased when uniaxial pressure grew.
In this case, as mentioned above, it is also necessary to take into account a decrease of the Hall mobility
of electrons at the expense of the deformation redistribution of electrons between the valleys of the
germanium conduction band with different mobility. Corresponding theoretical calculations which are
presented in figure 6 and figure 7 (solid and dashed curves) were carried out by us for the quantitative
estimation of the Hall mobility magnitude for the uniaxially deformed n-Ge〈Sb, Au〉 single crystals.
According to [19], the isoenergetic surfaces of the germanium conduction band are ellipsoids of
rotation, with an axis of symmetry which coincides with the crystallographic direction [111]. Then, the
mobility of charge carriers for an arbitrary direction can be determined from the ratio:
µ = µ⊥ sin2 θ + µ‖ cos2 θ , (3)
where θ is the angle between the examined direction and the main axis of the ellipsoid; µ⊥ and µ‖ is the
mobility of charge carriers across and along the axis of the ellipsoid.
In accordance with (1), for n-Ge single crystals undeformed and those uniaxially deformed along the
13702-4
Mechanisms of electron scattering
Figure 6.Temperature dependencies of theHall mo-
bility for the n-Ge〈Sb, Au〉 single crystals at differ-
ent values of the uniaxial pressure along the crystal-
lographic directions [100]: 1 – 0GPa, 2 – 0.29GPa,
3 – 0.59 GPa, 4 – 0.88 GPa (experimental results);
5 – 0GPa, 6 – 0.29GPa, 7 – 0.59GPa, 8 – 0.88GPa
(solid curves are theoretical calculations taking into
account the fluctuation potential); 9 – 0 GPa, 10 –
0.29 GPa, 11 – 0.59 GPa, 12 – 0.88 GPa (dashed
curves are theoretical calculations without taking
into account the fluctuation potential).
Figure 7.Temperature dependencies of theHall mo-
bility for the n-Ge〈Sb, Au〉 single crystals at differ-
ent values of uniaxial pressure along the crystal-
lographic directions [111]: 1 – 0GPa, 2 – 0.28GPa,
3 – 0.69 GPa, 4 – 0.97 GPa (experimental results);
5 – 0.97 GPa, 6 – 0.69 GPa, 7 – 0.28 GPa, 8 –
0 GPa (solid curves are theoretical calculations tak-
ing into account fluctuation potential); 9 – 0 GPa,
10 – 0.69 GPa, 11 – 0.28 GPa, 12 – 0.97 GPa
(dashed curves are theoretical calculations without
taking into account the fluctuation potential).
crystallographic direction [100]
µ0 =
1
3
µ‖ +
2
3
µ⊥ . (4)
Under uniaxial pressure n-Ge along the crystallographic direction [111], one minimum i.e., the main
axis of the isoenergetic ellipsoid which is oriented along the axis of deformation, will descend according
to the scale of energies by a value [19]
∆E1 = −
(
Ξd +
1
3
Ξu
)
(S11 + 2S12) P −
1
3
ΞuS44P, (5)
and the other three minima will ascend by a value
∆E2 = −
(
Ξd +
1
3
Ξu
)
(S11 + 2S12) P +
1
9
ΞuS44P, (6)
which will lead to the emergence of the energy gap between them ∆E1,2 =
4
9ΞuS44P.
If n1 is a concentration of electrons in a descending minimum, and n2 in three minima which ascend
under uniaxial pressure P ‖ J ‖ [111], then the total concentration of electrons in the conduction band
of germanium is as follows:
n = n1 + n2 . (7)
For the nondegenerate electron gas [20]
n1 = 2
(
2πm1kT
~2
) 3
2
e
EF−∆E1
kT , n2 = 2
(
2πm2kT
~2
) 3
2
e
EF−∆E2
kT . (8)
Then,
n2
n1
=
(
m2
m1
) 3
2
e−
∆E1,2
kT = A , (9)
13702-5
S.V. Luniov et al.
wherem1,m2 represent the effectivemass of the density of states for the givenminima. For the isoenergetic
surface, which is an ellipsoid of rotation, the effective mass of the density of states is as follows:
m = N
2
3
(
m‖m2
⊥
) 1
3
, (10)
where N denotes the number of equivalent ellipsoids, µ‖ and m⊥ are components for the tensor of the
effective mass of an electron along and across the axis of the ellipsoid.
In accordance with the (3), the mobility of electrons under the uniaxial pressure P ‖ J ‖ [111] in the
descending minimum is equal to
µ1 = µ‖ . (11)
Hence, for the three minima which ascend according to the scale of energies,
µ2 =
8µ⊥ + µ‖
9
. (12)
From expressions (7) and (9) we find
n1 =
n
A + 1
, n2 =
An
A + 1
. (13)
Then, for an arbitrary value of the uniaxial pressure P ‖ J ‖ [111], the conductivity n-Ge is as
follows:
σP = qnµ = q (n1µ1 + n2µ2) . (14)
Taking into account expressions (11)−(14), the mobility of electrons at the uniaxial pressure n-Ge
along the crystallographic direction [111] is equal to
µ =
µ1 + Aµ2
A + 1
. (15)
As it is known [21], electron scattering on optical phonons is also possible for germanium, in addition
to their scattering on acoustic phonons and ions of impurity. Electron scattering on optical phonons
is caused by the interaction of electrons with phonons, whose frequencies correspond to temperature
TC1 = 430 K (intravalley scattering) and intervalley scattering on acoustic phonons with characteristic
temperaturies TC2 = 320 K.
Intervalley electron scattering and electron scattering on optical phonons are described by a scalar
relaxation time τj :
1
τj
= ajϕj , (16)
where
aj =
Ξ2
j
(
m j
d
) 3
2
√
2πρ~2
(
kTCj
) 1
2
(
T
TCj
) 1
2
,
ϕj(x) =
1
e
TC j
T − 1
[(
x +
TCj
T
) 1
2
+ e
TC j
T θ
(
x,
TCj
T
) (
x −
TCj
T
) 1
2
]
,
m j
d is the effective mass of the density of states for electrons of the conduction band, Ξj is a constant of
intervalley or optical deformation potential; ρ is a density of the crystal; TCj is characteristic temperature
of j phonon; x = ε
kT is a dimensionless energy of electron; θ
(
x; TC j
T
)
is a step function. For intervalley
scattering
m j
d =
(
m‖ jm2
⊥j
) 1
3
(
Z j − 1
)
, (17)
13702-6
Mechanisms of electron scattering
where m‖ j , m⊥j are a longitudinal and transverse component of the tensor of the effective mass for
electrons which are in the ellipsoid of j type; Z j is the number of equivalent ellipsoids of the conduction
band of j type.
For intravalley electron scattering on optical phonons
m j
d = (m‖ jm
2
⊥j)
1
3 Z
2
3
j , (18)
Expressions for components of the relaxation-time tensor τa, j
‖
and τa, j⊥ under conditions of mixed
electron scattering on acoustic phonons and ions of impurity [22] can be written based on the theory of
anisotropic scattering:
τa,i
‖
=
a‖
√
kBT
3
2
·
x
3
2
x2 + b0
, τa,i⊥ =
a⊥
√
kBT
3
2
·
x
3
2
x2 + b1
. (19)
(Expressions for a‖ , a⊥ , b0 , b1 are presented in appendix A).
Then, in the most general case, the scattering of electrons on the acoustic phonons, ions of impurity,
optical phonons and acoustic phonons, which are responsible for the intervalley electron scattering,
expressions for components of the relaxation-time tensor can be presented as follows [21]:
1
τ‖
=
1
τa,i
‖
+
1
τ1
+
1
τ2
,
1
τ⊥
=
1
τa,i⊥
+
1
τ1
+
1
τ2
, (20)
where τa, j
‖
, τa,i⊥ , τ1 , τ2 are longitudinal and transverse components of the relaxation-time tensor for
scattering on acoustic phonons and ions of impurity, respectively; τ1 , τ2 are the relaxation time for
intervalley scattering and scattering on optical phonons.
Components of tensors of mobility can be expressed through components of tensors of relaxation
times and effective mass:
µ‖ =
e
m‖
〈
τ‖
〉
, µ⊥ =
e
m⊥
〈τ⊥〉 , (21)
〈τ‖〉 =
4
3
√
π
∞∫
0
dxx
3
2 e−xτ‖ , 〈τ⊥〉 =
4
3
√
π
∞∫
0
dxx
3
2 e−xτ⊥ . (22)
Numerical values of parameters for the energy-band structure of germanium single crystals such as
components of tensors of the acoustic potential of deformation and effective mass (Ξd = −6.4 eV, Ξu =
16.4 eV, m‖ = 1.58m0, m⊥ = 0.082m0) [22], constants of electron-phonon interaction for optical Ξ430 =
4 · 108 eV
cm and acoustic phonons Ξ320 = 1.4 · 108 eV
cm , which are responsible for the intervalley electron
scattering [21], should be taken into account for carrying out theoretical calculations. Temperature
dependencies of the Hall mobility for the uniaxially deformed n-Ge〈Sb, Au〉 single crystals (which were
obtained based on the given calculations) are presented in figure 6 and figure 7 (solid and dashed curves).
The results of comparison of the curves, which were obtained with and without taking into account
fluctuation potential (figure 6, curves 5−12), show that under low temperatures the Hall mobility signif-
icantly depends on the magnitude of the amplitude of the given potential. For the n-Ge〈Sb, Au〉 single
crystals uniaxially deformed along the crystallographic direction [111] the effect of the growth of the
Hall mobility with an increasing temperature is not significant and is observed only for the uniaxial
pressures P < 0.28 GPa (figure 7, curves 1 and 2). In the given case, the magnitude of the Hall mobility
is determined by the deformational reconstruction of the germanium conduction band and the effective-
ness of electron scattering on the fluctuation potential. The change of relative contribution of the given
mechanisms under an increasing uniaxial pressure explains the insignificant growth of the Hall mobility
under P > 0.69 GPa and temperatures T < 140 K (figure 7, curves 3 and 4).
13702-7
S.V. Luniov et al.
3. Conclusions
Electrons scattering on the ions of shallow Sb impurities for the germanium single crystals studied is
described using the Coulomb screening potential of the impurity. For deep impurities Au, this approach
does not apply, since today there are no adequate theoretical models of deep centres in semiconductors.
Therefore, it is difficult to make any estimates of the influence of such impurities on the mechanisms of
electron scattering in n-Ge〈Sb, Au〉 single crystals. However, for the investigated impurity concentrations
Sb and Au, the electron scattering on the ions of the gold impurity can be considered secondary, since
the experimental dependences of the Hall mobility of the electrons on the temperature for the uniaxially
deformed n-Ge〈Sb, Au〉 single crystals are well described based on the proposed theoretical model of
mobility. Additional doping of n-Ge〈Sb〉 single crystals by the impurity of gold leads to an increase in
the degree of compensation of such single crystals and consequently to the reduction of the screening
effect. The reduction of the effect of screening is caused by the occurrence of large-scale fluctuations of
the concentration of charged ions of doping impurities and, accordingly, the fluctuation potential, whose
amplitude depends on temperature and uniaxial pressure. Therefore, description of various electron
transfer phenomena in n-Ge〈Sb, Au〉 single crystals, for the investigated temperatures and concentrations
of doping impurities Sb and Au, can be restricted to the mechanisms of electron scattering on the acoustic
and optical phonons (intravalley scattering), acoustic phonons, which are responsible for intervalley
scattering, ions of shallow Sb impurities and fluctuation potential.
The obtained experimental results and theoretical calculations show that for single crystals
n-Ge〈Sb, Au〉 uniaxially deformed along the crystallographic direction [100], a decrease of the am-
plitude of fluctuation potential under an increase of temperature or the magnitude of uniaxial pressure
causes a growth of the Hall mobility. Its further decrease, passing through the maximum with an increas-
ing temperature, is explained by the increase of probability of electron scattering on optical phonons
and phonons, which are responsible for the intervalley electron scattering. Scattering of electrons on the
fluctuation potential is secondary in this case. The present mechanism of scattering does not appear for
the whole area of the investigated temperatures, and the Hall mobility magnitude is fully determined by
the mechanisms of the phonon scattering. The impact of deformational reconstruction of the germanium
conduction band (in addition to a change of the amplitude of the fluctuation potential at the uniaxial
pressure) should be additionally taken into account for the case of the single crystals n-Ge〈Sb, Au〉
uniaxially deformed along the crystallographic direction [111]. This mechanism causes a decrease of the
Hall mobility. The obtained temperature dependencies of the Hall mobility are determined by different
relative contributions of the given mechanisms depending on the magnitude of the uniaxial pressure.
It is important to consider the impact of fluctuation potential on the mechanisms of electron transport
in germanium both at the absence and in presence of deformation fields in modelling and developing
on its basis the electronic devices and sensors, nanostructures, alloyed by different impurities (alloyed
quantum dots Ge, heterostructures SiGe).
A. Necessary appendices to calculate the relaxation time
a‖ =
πC11~
4
kΞ2
d
√
2m‖m2
⊥
·
1
Φ0a
, a⊥ =
πC11~
4
kΞ2
d
√
2m‖m2
⊥
·
1
Φ1a
,
b0 =
a‖ · Φ0i
√
kT
3
2 τ0i (kT)
, b1 =
a⊥ · Φ1i
√
kT
3
2 τ0i (kT)
,
τ0i(kT) =
√
2m⊥ε2 (kT)
3
2
πNde4√m‖
,
Φ1a = 1 +
1 + β2
β2
[
2 +
3
β2 −
3
(
1 + β2)
β3 α
]
Ξu
Ξd
+
(
1 + β2)
β4
Ξ2
u
Ξ2
d
(A + B) ,
13702-8
Mechanisms of electron scattering
A =
(
1 + β2) [1 + 15
4β2 −
3
4β3
(
5 + 3β2)α]
,
B =
C11
4C44
[
− 13 −
15
β2 +
3
(
1 + β2)
β3
(
5 + β2)α]
,
Φ0a = 1 +
2
(
1 + β2)
β2
(
1 −
3
β2 +
3
β3α
)
Ξu
Ξd
+
(
1 + β2)
β4
Ξ2
u
Ξ2
d
(D + K) ,
D =
(
1 + β2) [1 − 6
β2 −
3
2β2 (
1 + β2) + 15α
2β3
]
,
K =
C11
C44
[
2 +
15
2β2 −
3
2β3
(
5 + 3β2)α]
,
Φ0i =
3
2β3
[ (
β
1 + β2 − α
)
ln γ2 − α ln
(
1 + β2) + 2L (a) +
βγ2
2
M
]
,
M =
β2 − 1
β2 + 1
+
α
(
β2 + 1
)
β
,
Φ1i =
3
4β3
{[(
1 − β2)α − β] ln γ2 + 2
(
β2 − 1
)
L (a)
}
+
3
4β3 R ,
R = −2β2α −
(
β2 − 1
)
α ln
(
1 + β2) + γ2
2
[
β
(
1 + 3β2) + α (
3β4 + 2β2 − 1
) ]
,
α = arctgβ , β2 =
m‖ − m⊥
m⊥
, γ =
√
π~2e2n
2m‖εkT
.
L(a) = −
∫a
0 ln cos ϕdϕ is the Lobachevsky function, Nd is concentration of donor impurity, n is concen-
tration of electrons in the conduction band.
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Механiзми розсiяння електронiв в одновiсно деформованих
монокристалах n-Ge〈Sb, Au〉
С.В. Луньов1, П.Ф. Назарчук1, А.I. Зiмич1,Ю.А. Удовицька1, О.В. Бурбан2
1 Луцький нацiональний технiчний унiверситет, вул. Львiвська, 75, 43018 Луцьк, Україна
2 Волинський коледж Нацiонального унiверситету харчових технологiй, вул. Кафедральна, 6, 43016 Луцьк,
Україна
На основi вимiрювань п’єзо-холл-ефекту одержано температурнi залежностi концентрацiї та холiвської
рухливостi електронiв для одновiсно деформованих вздовж кристалографiчних напрямкiв [100] та [111]
монокристалiв n-Ge〈Sb〉 та n-Ge〈Sb, Au〉. Виявлено деформацiйно-iндуковане зростання холiвської рухли-
востi електронiв для монокристалiв n-Ge〈Sb, Au〉 при одновiсному тисковi вздовж кристалографiчного
напрямку [100]. Порiвняння одержаних експериментальних результатiв з вiдповiдними теоретичними
розрахунками температурних залежностей холiвської рухливостi показали, що даний ефект виникає за
рахунок зменшення ймовiрностi розсiяння електронiв на флуктуацiйному потенцiалi, амплiтуда якого за-
лежить вiд температури та величини одновiсного тиску. Показано, що зростання холiвської рухливостi
для одновiсно деформованих монокристалiв n-Ge〈Sb, Au〉 вздовж кристалографiчного напрямку [111] є
незначним i спостерiгається лише для одновiсних тискiв P < 0.28 ГПа. В даному випадку необхiдно та-
кож враховувати зменшення холiвської рухливостi електронiв за рахунок деформацiйного перерозподiлу
електронiв мiж долинами зони провiдностi германiю з рiзною рухливiстю. Для одновiсно деформованих
монокристалiв n-Ge〈Sb〉 величина холiвської рухливостi при таких самих умовах визначається лише ме-
ханiзмами фононного розсiяння i ефект зростання холiвської рухливостi при збiльшеннi температури або
величини одновiсного тиску не спостерiгався. Це свiдчить про другорядну роль флуктуацiйного потенцi-
алу в даному випадку.
Ключовi слова: ефект Холла, явища переносу, електричнi властивостi, домiшки
13702-10
https://doi.org/10.1088/1742-6596/121/2/022006
Introduction
Results and discussion
Conclusions
Necessary appendices to calculate the relaxation time
|