Berry phase in strained InSb whiskers
Strain influence on the longitudinal magnetoresistance for the n-type conductivity InSb whiskers doped by Sn to concentrations 6·10¹⁶–6·10¹⁷ сm⁻³ were studied at temperatures from 4.2 to 50 K and magnetic field up to 10 T. The Shubnikov–de Haas oscillations at low temperatures were revealed in the...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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nasplib_isofts_kiev_ua-123456789-1764922025-02-10T01:12:37Z Berry phase in strained InSb whiskers Фаза Беррі в деформованих ниткоподібних кристалах InSb Фаза Берри в деформированных нитевидных кристаллах InSb Druzhinin, A. Ostrovskii, I. Khoverko, Yu. Liakh-Kaguy, N. Rogacki, K. Низькотемпературний магнетизм Strain influence on the longitudinal magnetoresistance for the n-type conductivity InSb whiskers doped by Sn to concentrations 6·10¹⁶–6·10¹⁷ сm⁻³ were studied at temperatures from 4.2 to 50 K and magnetic field up to 10 T. The Shubnikov–de Haas oscillations at low temperatures were revealed in the strained and unstrained samples with all range doping concentration. Some peaks of the longitudinal magnetoresistance split as a doublet in the InSb whiskers with doping concentration in the vicinity to metal-insulator transition. Taking into account peak splitting giant g-factor from 30 to 60 was defined for strained and unstrained samples. The magnetoresistance oscillation period of the InSb whiskers doesn’t differ under strain for all doping concentration, but Fermi energy increases and electron effective mass mс decreases and consists 0.02 m₀. Berry phase presence was also revealed in strained n-InSb whiskers that shows their transition under a strain to topological insulator phase. Досліджено вплив деформації на поздовжній магнітоопір ниткоподібних кристалів InSb з провідністю n-типу, легованих оловом в концентраціях 6·10¹⁶–6·10¹⁷ сm⁻³ , при температурах від 4,2 до 50 К та магнітних полях до 10 Тл. При низьких температурах осциляції Шубнікова–де Гааза виявлено в деформованих й недеформованих зразках у всьому діапазоні концентрацій допування. Деякі піки поздовжнього магнітоопору розщеплюються в дублети в ниткоподібних кристалах InSb з концентрацією допанта, близькою до переходу метал–ізолятор. Беручи до уваги розщеплення піків, для деформованих та недеформованих зразків визначено гігантський g-фактор від 30 до 60. Період осциляцій магнітоопору ниткоподібних кристалів InSb не змінюється в деформованому стані для всіх концентрацій допанта, але енергія Фермі зростає, а ефективна маса електрона mс зменшується і становить 0,02 m₀. Присутність фази Беррі було також виявлено в деформованих ниткоподібних кристалах n-InSb, які демонстрували перехід в фазу топологічного ізолятора під дією деформації. Исследовано влияние деформации на продольное магнитосопротивление нитевидных кристаллов InSb с проводимостью n-типа, легированных оловом в концентрациях 6·10¹⁶–6·10¹⁷ сm⁻³ , при температурах от 4,2 до 50 К и магнитных полях до 10 Тл. При низких температурах осцилляции Шубникова–де Гааза обнаружены в деформированных и недеформированных образцах во всем диапазоне концентраций допирования. Некоторые пики продольного магнитосопротивления расщепляются в дублеты в нитевидных кристаллах InSb с концентрацией допанта, близкой к переходу металл– изолятор. Принимая во внимание расщепление пиков, для деформированных и недеформированных образцов определен гигантский g-фактор от 30 до 60. Период осцилляций магнитосопротивления нитевидных кристаллов InSb не изменяется в деформированном состоянии для всех концентраций допанта, но энергия Ферми возрастает, а эффективная масса электрона mс уменьшается и составляет 0,02 m₀. Присутствие фазы Берри было также обнаружено в деформированных нитевидных кристаллах n-InSb, которые демонстрировали переход в фазу топологического изолятора под действием деформации. 2018 Article Berry phase in strained InSb whiskers/ A. Druzhinin, I. Ostrovskii, Yu. Khoverko, N. Liakh-Kaguy, K. Rogacki // Физика низких температур. — 2018. — Т. 44, № 11. — С. 1521-1527. — Бібліогр.: 27 назв. — англ. 0132-6414 https://nasplib.isofts.kiev.ua/handle/123456789/176492 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Низькотемпературний магнетизм Низькотемпературний магнетизм |
| spellingShingle |
Низькотемпературний магнетизм Низькотемпературний магнетизм Druzhinin, A. Ostrovskii, I. Khoverko, Yu. Liakh-Kaguy, N. Rogacki, K. Berry phase in strained InSb whiskers Физика низких температур |
| description |
Strain influence on the longitudinal magnetoresistance for the n-type conductivity InSb whiskers doped by
Sn to concentrations 6·10¹⁶–6·10¹⁷ сm⁻³ were studied at temperatures from 4.2 to 50 K and magnetic field up
to 10 T. The Shubnikov–de Haas oscillations at low temperatures were revealed in the strained and unstrained
samples with all range doping concentration. Some peaks of the longitudinal magnetoresistance split as a
doublet in the InSb whiskers with doping concentration in the vicinity to metal-insulator transition. Taking into account peak splitting giant g-factor from 30 to 60 was defined for strained and unstrained samples. The
magnetoresistance oscillation period of the InSb whiskers doesn’t differ under strain for all doping concentration, but Fermi energy increases and electron effective mass mс decreases and consists 0.02 m₀. Berry phase
presence was also revealed in strained n-InSb whiskers that shows their transition under a strain to topological
insulator phase. |
| format |
Article |
| author |
Druzhinin, A. Ostrovskii, I. Khoverko, Yu. Liakh-Kaguy, N. Rogacki, K. |
| author_facet |
Druzhinin, A. Ostrovskii, I. Khoverko, Yu. Liakh-Kaguy, N. Rogacki, K. |
| author_sort |
Druzhinin, A. |
| title |
Berry phase in strained InSb whiskers |
| title_short |
Berry phase in strained InSb whiskers |
| title_full |
Berry phase in strained InSb whiskers |
| title_fullStr |
Berry phase in strained InSb whiskers |
| title_full_unstemmed |
Berry phase in strained InSb whiskers |
| title_sort |
berry phase in strained insb whiskers |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2018 |
| topic_facet |
Низькотемпературний магнетизм |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/176492 |
| citation_txt |
Berry phase in strained InSb whiskers/ A. Druzhinin, I. Ostrovskii, Yu. Khoverko, N. Liakh-Kaguy, K. Rogacki // Физика низких температур. — 2018. — Т. 44, № 11. — С. 1521-1527. — Бібліогр.: 27 назв. — англ. |
| series |
Физика низких температур |
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 11, pp. 1521–1527
Berry phase in strained InSb whiskers
A. Druzhinin1,2, I. Ostrovskii1,2, Yu. Khoverko1,2, N. Liakh-Kaguy1, and K. Rogacki2
1Lviv Polytechnic National University, 12 S. Bandera Str., Lviv 79013, Ukraine
E-mail: druzh@polynet.lviv.ua
2Institute of Low Temperature and Structure Research PAS, 95 Gajowicka, Wroclaw, Poland
Received May 23, 2018, published online September 26, 2018
Strain influence on the longitudinal magnetoresistance for the n-type conductivity InSb whiskers doped by
Sn to concentrations 6·1016–6·1017 сm–3 were studied at temperatures from 4.2 to 50 K and magnetic field up
to 10 T. The Shubnikov–de Haas oscillations at low temperatures were revealed in the strained and unstrained
samples with all range doping concentration. Some peaks of the longitudinal magnetoresistance split as a
doublet in the InSb whiskers with doping concentration in the vicinity to metal-insulator transition. Taking in-
to account peak splitting giant g-factor from 30 to 60 was defined for strained and unstrained samples. The
magnetoresistance oscillation period of the InSb whiskers doesn’t differ under strain for all doping concentra-
tion, but Fermi energy increases and electron effective mass mс decreases and consists 0.02 m0. Berry phase
presence was also revealed in strained n-InSb whiskers that shows their transition under a strain to topological
insulator phase.
Keywords: InSb whiskers, longitudinal magnetoresistance oscillations, doping concentration, g-factor, Berry phase.
1. Introduction
The features of strain-induced effects were studied in the
classical semiconductors (silicon, germanium) with doping
concentration in the vicinity to the critical concentration of a
metal-insulator transition (MIT) at low temperatures [1–4].
At the same time, there are data on the strain-induced MIT,
which is associated with radical rearrangement of the con-
duction band as a result of strain due to the reducing of elec-
tron effective mass [5]. The studying of strain-stimulated
effects in semiconductors of the A3B5 group is equally im-
portant, including further using of such effects in microelec-
tronic devices. Some semiconductor compounds are also
very strain-sensitive, for example, n-GaSb (G [111] = – 226),
p-InSb (G [111] = 212) [6].
On the other hand, oscillation phenomena of magneto-
resistance as Shubnikov–de Haas (SdH) effect [7,8] were
earlier revealed in the InSb compounds at low tempera-
tures down to liquid helium that were induced by the uni-
axial strain. These strain-induced effects are a complicated
and complex problem. In this case, the InSb whiskers with
doping concentration corresponding to the MIT are a good
model for studying the influence of strain, due to their
morphology, structural perfection and high mechanical
strength.
The strain influence on the hole effective mass mс at
temperature 4.2 K was also shown in the doped InSb sam-
ples by authors [9]. The effective mass decreases down to
0.017 me under the largest biaxial compressive strain 1.05 %
due to the energy separation between two bands [10].
Moreover, according to recent reports in bismuth-based
materials, the emergence of the Berry phase (Aaronov–
Bohm oscillations) was observed as a manifestation of
topological isolators [11,12]. The Berry phase arises also
in strained InSb materials as a result of strong spin–orbit
coupling [13]. It can be expected that the use of the strain-
induced MIT in the InSb whiskers, whose conductivity is
due to surface states, will enable the control of the material
properties.
The aim of present paper is study magnetoresistance
oscillations in n-type InSb whiskers with tin doping con-
centration in the vicinity to metal-insulator transition under
influence of compressive strain at low temperatures. The
strain was shown to introduce the InSb whisker in topo-
logical insulator state, which accompanied with Berry
phase appearance.
2. Experimental procedure
The n-type InSb whiskers with tin doping concentration
grown by chemical transport reaction method were the
object for the longitudinal magnetoresistance studies. The
InSb whiskers with a length of 2–3 mm and a diameter of
30–40 µm were selected for the investigations. Au contacts
to n-type conductivity InSb whiskers with the diameter
10 µm form an eutectic with whiskers under the pulsed
© A. Druzhinin, I. Ostrovskii, Yu. Khoverko, N. Liakh-Kaguy, and K. Rogacki, 2018
A. Druzhinin, I. Ostrovskii, Yu. Khoverko, N. Liakh-Kaguy, and K. Rogacki
welding. The contact technique allows to measure whisker
longitudinal magnetoresistance using four contacts to the
sample.
The InSb whiskers were strained by the mounting them
on substrates with a thermal expansion coefficient different
from that of InSb material. The similar experimental meth-
od with using the thermal strain estimation of the p-type
conductivity silicon whiskers on different substrates, were
shown in work [1]. Copper substrates were used in order to
achieve the uniaxial compression strain (ε = –3.8·10–3 rel.
units) at temperature 4.2 K. The thermal strain of the InSb
whiskers in the <111> direction was calculated at tempera-
tures 4.2–50 K.
The helium cryostat was used to study the low-tem-
perature magnetoresistance for the n-type conductivity
InSb whiskers in the range 4.2–50 K. The whisker magne-
tic properties were studied in magnetic fields 0–10 T crea-
ted by Bitter magnet with time scanning of 1.75 T/min.
The stabilized electric current from 1 to 10 mA depending
on InSb whisker resistance was created using the Keithley
224 source. Temperature measuring were carried out with
use a Cu–CuFe thermocouple.
3. Experimental results and their discussions
Three groups of the n-type conductivity InSb whiskers
with doping concentration (Sn) that correspond to different
approximation to critical concentration of the metal-
insulator transition were selected for studies of the
magnetoresistance:
i) with doping concentration that corresponds to the
MIT (2·1017 сm–3);
ii) with high doping concentration that corresponds to
the metal side of the MIT (6·1017 сm–3);
iii) with doping concentration shifted into the insulator
side of the MIT (6·1016 сm–3).
The strain influence on the longitudinal magneto-
resistance in the n-type conductivity InSb whiskers at low
temperatures down to 4.2 K were studied in the range of
magnetic field 0–10 T. The results of the studies for strained
and unstrained samples doped by tin to different concentra-
tion in the vicinity to the MIT are shown in Figs. 1–3.
The longitudinal magnetoresistance peaks of the n-
type InSb whiskers with various doping concentrations
6·1016–6·1017 сm–3 were revealed in the temperature
range 4.2–40 K as in unstrained and also in strained sam-
ples (Figs. 1–3). The decreasing of maximum peak ampli-
tudes with temperature increasing was shown in magnetic
field range 0–10 T. The values of magnetic field induc-
tion at temperature 4.2 K that corresponds to the longitu-
dinal magnetoresistance peaks for the unstrained and
strained InSb whiskers were presented in the Tables 1
and 2, respectively.
According to well-known developed methodology [14]
we constructed Landau fan diagram. The Landau level (LL)
index number N to each resistance minimum (N + 1/2 to
each resistance maximum) according to experimental data
(Figs. 1–3). Landau level index N versus reversal the mag-
netic field induction 1/B was presented in Fig. 4 (a), (b). As
can be seen from the figures, data points fall on the straight
lines and the best linear fit was represented by the solid line.
An intercept of linear fit with N-index axis yielded zero
phase β = 0 in the unstrained InSb whiskers with doping
concentration 2·1017 сm–3 that indicates in Schrödinger
electron transport responsible for SdH oscillations (Fig.
4(a)). However, for the strained samples with doping con-
centrations 2·1017 and 6·1016 сm–3 the phase β = ½ was
obtained, that is shown in the Figs. 4(b) and 4(c).
Thus, the Berry phase was absent in unstrained InSb
whiskers in whole doping concentration range at low tem-
peratures and shown in the Landau fan diagrams in
Fig. 4(a). But their presence was revealed in the strained
samples only with doping concentration in the vicinity to
the MIT. So, strain influence on the magnetoresistance
behavior leads to Berry phase appearance in heavily doped
whiskers with strong spin-orbit interaction (Figs. 4(b), (c)).
We show strain influence on the longitudinal magneto-
resistance dependences for InSb whiskers with various do-
Fig. 1. (Color online) Longitudinal magnetoresistance in the un-
strained (a) and strained (b) InSb whiskers tin doped to a concentra-
tion 6·1016 сm–3 at temperatures: 4.2 (1), 13 (2), 29 (3), 40 (4) K.
1522 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 11
Berry phase in strained InSb whiskers
ping concentrations in all ranges of magnetic fields and
temperatures. The peaks corresponding to the transitions
between Landau levels with N = 1, 2, …. The number of the
longitudinal magnetoresistance peaks decreases from nine
peaks in the unstrained samples (Table 1) to six (Table 2) in
strained InSb whiskers with doping concentration 2·1017 сm–3
that corresponds to the MIT (Figs. 2(a),(b)). And for InSb
whiskers tin doped to concentration 6·1017 сm–3 in the vicini-
ty to the MIT from metal side of the transition (Figs. 3(a),(b))
the number of the magnetoresistance peaks doesn’t change
and consists of 5 in both unstrained and strained samples (see
Tables 1, 2). Five peaks with the maxima (Table 1) were
revealed on the magnetoresistance dependences at tem-
perature 4.2 K for the unstrained InSb whiskers with doping
concentration of 6·1016 сm–3 shifted into the insulator side of
the MIT (Fig. 1(a)). But in this case, the number of the peaks
increases up to nine (Table 2) for the strained samples
(Fig. 1(b)).
The strain influence on the longitudinal magneto-
resistance dependences of the InSb whiskers were studied at
low-temperature range. So, the influence of strain was mani-
fested due to the doping concentration shift relative to the
MIT that corresponds to the change of the magneto-
resistance peak number. InSb whiskers with tin concentra-
tion 2·1017 сm–3 was shifted from the MIT that corresponds
to the decreasing peak number under the strain (Fig. 2(b)).
Fig. 2. (Color online) Longitudinal magnetoresistance in the un-
strained (a) and strained (b) InSb whiskers tin doped to a concentra-
tion 2·1017 сm–3 at temperatures: 4.2 (1), 13 (2), 29 (3), 40 (4) K.
Fig. 3. (Color online) Ongitudinal magnetoresistance in the un-
strained (a) and strained (b) InSb whiskers tin doped to a concentra-
tion 6·1017 сm–3 at temperature: 4.2 (1), 13 (2), 29 (3), 40 (4) K.
Table 1. The magnetic field inductions of the magnetoresistance peaks in the unstrained InSb whiskers with different doping concentra-
tion at 4.2 K
Doping concentration, сm–3 The longitudinal magnetoresistance in unstrained samples Bmax, T
№ 1 2 3 4 5 6
6·1016 5.4 3.5 2.4 1.8 1.4
2·1017 11.4 7.2 5.1 4.0 3.4 2.85 2.5 2.2 2.0
6·1017 7.46 4.3 3.0 2.39 1.96
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 11 1523
A. Druzhinin, I. Ostrovskii, Yu. Khoverko, N. Liakh-Kaguy, and K. Rogacki
Heavily doped samples with concentration 6·1017 сm–3 was
also shifted from the MIT and magnetoresistance peak num-
ber decreased due to the strain (Fig. 3(b)). On the contrary,
the lightly doped samples have been shifted from the insula-
tor side of the MIT closer to the transition and the behavior
of the longitudinal magnetoresistance was changed by the
strain.
The SdH oscillations of the longitudinal magneto-
resistance were observed in high quality n-type conductivity
InSb whiskers with doping concentration in the range
6·1016–6·1017 сm–3 for both unstrained and strained samples
(Figs. 1–3). Every peak of longitudinal magnetoresistance
splits as doublet in magnetic field 0–10 T for the unstrained
InSb whiskers with doping concentration 2·1017 сm–3 that
corresponds to the MIT (Fig. 2(a)). Therefore, the SdH
magnetoresistance oscillation splits into two peaks that corre-
sponds to various quantum levels at the inductions of mag-
netic field: N = 1 at 7.2 and 5.1 T; N = 2 at 4 and 3.4 T; N = 3
at 2.8 and 2.5 T; N = 4 at 2.2 and 2 T (Table 1). But for the
strained samples with the same doping concentration there
had not been any observations of the splitting at all ranges of
magnetic fields and temperatures (Fig. 2(b)).
We show quite different situations for the samples that
correspond to both the insulator and metal side of the MIT.
The strain influence on the longitudinal magnetoresistance
of the InSb whiskers with high doping concentration
6·1017 сm–3 that corresponds to the metal side of the MIT
(Fig. 3(b)) leads to splitting a peak N = 1 at 8.3 and 6.5 T
(Table 2). Nevertheless, the splitting is absent for the un-
strained samples in magnetic field 0–10 T at liquid helium
temperature (Fig. 2(b)). The splitting of the magneto-
resistance peaks are absent for the lightly doped n-type
InSb whiskers in both unstrained and strained samples with
tin concentration 6·1016 сm–3 (Fig. 1(a), (b)).
The SdH oscillations are periodic on 1/H. The period of
the magnetoresistance oscillations P in the opposite magnet-
ic field corresponds to the quadratic dispersion law [15]:
( ) | |1/
F c
eP H
E m c
= ∆ =
, (1)
where e is the elementary charge, EF is the Fermi energy,
is the Planck constant; mс is the effective electron mass,
c is the light speed.
The longitudinal magnetoresistance period was found
due to the Eq. (1). The period of SdH oscillations slightly
differs for various doping concentration of the InSb whis-
Table 2. The magnetic field inductions of the magnetoresistance peaks in the strained InSb whiskers with different doping concentra-
tion at 4.2 K
Doping concentration, сm–3 The longitudinal magnetoresistance in unstrained samples Bmax, T
№ 1 2 3 4 5 6 7 8 9
6·1016 5.9 4.0 3.0 2.4 2.0 1.7 1.4 1.2
2·1017 4.1 3.2 2.35 2.0 1.5 1.0
6·1017 8.3 6.5 4.2 2.8 – 1.7
Fig. 4. The Landau level (LL) index N of SdH oscillation versus
reverse magnetic field (1/B) in unstained (a) and strained (b), (c)
InSb whiskers with doping concentration 2·1017 and 6·1016 сm–3,
respectively.
1524 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 11
Berry phase in strained InSb whiskers
kers. The period of the magnetoresistance oscillations is
about 0.1 T
–1 in the InSb whiskers with doping concentra-
tion that correspond to the vicinity to the MIT [16].
SdH oscillation for the unstrained semimetal-side sam-
ple differ from one in the semiconductor-side, that lead to
difference in intensity ratio for ∆ρ3 and ∆ρ2, where ∆ρ2 is
contribution to the resistance, which corresponds with two
terms as the hopping conductance on twice occupied by
electrons impurity 2∆ρ′ and the electron spin-orbit interac-
tion 2,∆ρ′′ ∆ρ3 is contribution to the resistance due to the
hopping conductance on once occupied by the electron
impurity. The scattering by electron spin-orbit interaction
is the main mechanism in the semimetals, so 2∆ρ′′ is the
largest and as result the reduction of ∆ρ3-term. So, the
splitting was shown in the semimetal side of MIT (Fig. 4).
The ∆ρ3-term can increase and may be measurable in the
semiconductors. Thus, the peak splitting was observed on
the magnetoresistance curves in the vicinity to the MIT
that correspond with competition of both ∆ρ2- and ∆ρ3-
terms (see Fig. 2(а)), whereas in addition to MIT, only the
∆ρ3-term dominates, which leads to the absence of the
splitting in the magnetoresistance oscillations (Fig. 1(a)
and Fig. 3(a)).
Intensity ratio for ∆ρ2 and ∆ρ3 in unstrained sample that
corresponds to the MIT would be compared. For the interval
of the fields 0–2 T, the intensity ratio Δρ2 > Δρ3 (Fig. 2(a)),
where the amplitude of the oscillations becomes noticeable.
Moreover, intensity ratio ∆ρ2 < ∆ρ3 was observed in high
magnetic fields 2–10 T inversion of takes place. We can
observe such changes in the intensity ratio due to the MIT
inducted by the magnetic field. Such type of the MIT was
shown firstly in the work [17] and developed in [18]. ∆ρ2-
term correspond with the strong spin-orbital interaction pre-
vails in weak magnetic fields. A gradual increasing in ∆ρ3-
term corresponds with hopping conductance and occurs in
the magnetic field with induction B > 2 T.
Revealed splitting in the magnetoresistance oscillations
allow us to define the g-factor. Spin splitting of the peaks
observed in SdH oscillations at magnetic field В2, when
* Bg Bµ exceeds the doping level splitting of the Landau
*1Г / .ceB m= So, we can write *1 2Г / c BeB m g B= = µ .
Since the Bohr magneton is 0/2 ,B e mµ = thus the ex-
pression was obtained
1 0*
*2
2
c
B m
g
B mm
= . (2)
The band conduction in the InSb is strongly non-
parabolic due to narrow band gap, as result the mass de-
pends on energy [19]:
* 2( ) 1c
g
Em E m
E
= +
, (3)
where E is an electron energy, Eg is a band gap, and *cbm is
an effective mass in the edge of conduction band. According
to conduction at Fermi energy, E = EF, and using the equa-
tion (3) with parameter mc = 0.014 m0 and EF = = 0.11 eV
were obtained from the analysis of SdH oscillations and
band gap consists 0.23 eV (in the InSb whiskers at helium
temperature [20]) the corresponding values have been calcu-
lated for *m = 0.03 m0 that was used for g-factor estima-
tion.
According to above consideration the graduate increasing
of the g-factor from 46 to 60 for different Landau levels that
correspond N = 1 and N = 4, respectively, were obtained.
Obtained data were in the good agreement with the giant
values of the g-factor that revealed in the InSb nanowires,
but according to the data in work [21] some discrepancy was
revealed. Taking into accoount results in the present work,
value of the g-factor decreases with increasing of the mag-
netic field induction, while in work [21] g-factor value de-
pends on the doping level. The giant amplification of spin
splitting near the MIT was also shown in the Si hetero-
structures [22]. InSb represents as narrow-band gap material.
It has the largest electron Lande g-factor, which consists 51
for all semiconductors of the A3B5 group [23]. The authors
[24] also studied the temperature dependence of the elect-
ron Landé g-factor in the InSb crystals and showed that
the g-factor value was 51 at temperature 4.2 K. The re-
sulting large magnitude of the g-factors in this work was
associated with the emergence of spin-orbit interaction in
the whiskers for field of the hopping conductance on
twice occupied by the electrons impurities.
The strain influence on the some parameters of the lon-
gitudinal magnetoresistance was observed for the InSb
whiskers at low temperatures. So, the calculated value of
g-factor consists of 50 for the Landau level N = 1 at mag-
netic field inductions 8.3 and 6.5 T in the strained InSb
whiskers with doping concentration 6·1017 сm–3, which
corresponds to the metal side of the MIT (Fig. 3(b)). The
strain influence on the effective electron mass *m of the
n-type conductivity InSb whiskers leads to the decreasing
of its value from 0.03 m0 to 0.02 m0.
The Berry phase was revealed in the strained InSb
whiskers with doping concentration in the vicinity to the
MIT. For both samples with doping concentration removed
into the insulator side of the MIT of 6·1016 сm–3 and that
corresponds to MIT of 2·1017 сm–3 the appearance of the
Berry phase was shown under strain influence in the
Figs. 4(b),(c). An explanation for their appearance was
suggested as the strain influence shift of the samples to the
vicinity to the MIT. But the Berry phase was absented in
the strained InSb whiskers with doping concentration of
6·1017 сm–3 that corresponds to metal side of the MIT be-
cause the strain removed their deeper from the transition.
The effect of phase factor in SdH oscillation could be
described by the Lifshitz–Kosevich equation [25]:
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 11 1525
A. Druzhinin, I. Ostrovskii, Yu. Khoverko, N. Liakh-Kaguy, and K. Rogacki
( ), cos 2 FR B T
B
∆ρ = π + γ − δ ρ
, (4)
where R(B,T) contains the hyperbolic and exponential
terms describing the temperature and field damping of the
SdH oscillation amplitude, F is the frequency of the SdH
oscillation in 1/B term, 1/2 ( /2 )γ = − β π is the associated
Berry’s phase (divided by 2π). Berry’s phase β = 0 corre-
sponds to the trivial case, which describes unstrained
InSb magnetoresistance oscillations. A deviation from
this value to β = 1/2 indicates in the existence of Dirac
particles [26]. The phase shift δ is determined by the
dimensionality of Fermi surface and takes values 0 for
2D and 1/8 for 3D cases [27]. The fan diagrams for
strained InSb whiskers (Fig. 4(b),(c)) show that β = 1/2.
Thus, an additional phase shift δ is equal to zero con-
firming 2D nature of Dirac electrons in strained InSb
whisker and its transition to topological insulator.
The Dingle temperature TD of the n-type conductivity
InSb whiskers is defined as a ratio of the SdH oscillation
amplitudes for two successive magnetoresistance maximum
at known value of the effective electron mass mc [15]:
( )
( )
2
1
2 11
21
2
sinh
| |,
, 2
sinh
| |
c
nn n
nn c
n
kTm c
e HA T H H
HA T H kTm c
e H
++
+
π
= × π
2
1
2 1 1exp
| |
D c
n n
kT m c
e H H+
π
× −
. (5)
The Dingle temperatures for both unstrained and
strained samples with various doping concentration were
determined from the amplitude dependences of the SdH
oscillations (Figs. 1–3) on magnetic field due to the
Eq. (5). The highest values of the Dingle temperature
reached of up to 12 K in the unstrained InSb whiskers that
could be explained by their heavy doping concentration.
The strain influence on this parameter leads to its increas-
ing up to 15 K in high magnetic fields due to the decreas-
ing of the effective mass under the strain.
4. Conclusions
The longitudinal magnetoresistance in strained and un-
strained samples of n-type conductivity InSb whiskers with
doping concentration in the range 6·1016–6·1017 сm–3 in the
vicinity to the MIT from metal and insulator side of transition
at low temperatures 4.2–40 K and magnetic fields with the
induction up to 10 T were studied. According to our investi-
gations the longitudinal magnetoresistance peaks were identi-
fied as the SdH oscillations at temperatures 4.2–40 K. Larger
number of the magnetoresistance peaks for strained InSb
whiskers than ones for strained samples were visible at heli-
um temperature. Every peak of the longitudinal magneto-
resistance for unstrained InSb whiskers with doping concen-
tration that corresponds to the MIT of 2·1017 сm–3 is splitted
as doublet in the magnetic field range 2–10 T, but the split-
ting is absent in the strained samples. For unstrained samples
with doping concentration in the vicinity to the MIT from
insulator and metal sides of the transition the splitting is also
absent in whole magnetic field. However, the strain influ-
ence leads to splitting only first peak of the longitudinal
magnetoresistance for InSb whiskers with doping concen-
tration that corresponds to the metal side of the MIT of
6·1017 сm–3.
Main parameters in the strained and unstrained n-type
conductivity InSb whiskers were estimated for various
level of the doping concentration due to analysis of SdH
magnetoresistance oscillations. The values of the cyclotron
effective mass of electrons were changed from 0.03 m0 to
0.02 m0 under strain influence. Fermi energy and Dingle
temperature also change due to strain influence from 0.12
to 0.14 eV and from 12 to 15 K, respectively. According to
splitting the magnetoresistance peaks for InSb whiskers
with doping concentration that corresponds to the MIT,
giant g-factor of 50–60 was obtained, and its values were
dependent on the doping level and strain influence.
The Landau fan diagrams confirmed Berry phase pres-
ence in the InSb whiskers with doping concentration in
the vicinity to the MIT at low temperatures. Berry’s
phase β = 0 corresponds to the trivial case, which descry-
bes unstrained InSb magnetoresistance oscillations.
Aplying strain (ε = –3.8·10–3 rel. units) at temperature
4.2 K leads to a deviation from this value to β = 1/2 indi-
cating in the existence of Dirac particles. The fan dia-
grams for strained InSb whiskers confirming 2D nature of
Dirac electrons in strained InSb whisker and its transition
to topological insulator.
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___________________________
Фаза Беррі в деформованих ниткоподібних
кристалах InSb
А. Дружинін, І. Островський, Ю. Ховерко,
N. Liakh-Kaguy, K. Rogacki
Досліджено вплив деформації на поздовжній магніто-
опір ниткоподібних кристалів InSb з провідністю n-типу,
легованих оловом в концентраціях 6·1016–6·1017 сm–3, при
температурах від 4,2 до 50 К та магнітних полях до 10 Тл.
При низьких температурах осциляції Шубнікова–де Гааза
виявлено в деформованих й недеформованих зразках у
всьому діапазоні концентрацій допування. Деякі піки по-
здовжнього магнітоопору розщеплюються в дублети в
ниткоподібних кристалах InSb з концентрацією допанта,
близькою до переходу метал–ізолятор. Беручи до уваги
розщеплення піків, для деформованих та недеформованих
зразків визначено гігантський g-фактор від 30 до 60. Період
осциляцій магнітоопору ниткоподібних кристалів InSb не
змінюється в деформованому стані для всіх концентрацій
допанта, але енергія Фермі зростає, а ефективна маса елек-
трона mс зменшується і становить 0,02 m0. Присутність
фази Беррі було також виявлено в деформованих нитко-
подібних кристалах n-InSb, які демонстрували перехід в
фазу топологічного ізолятора під дією деформації.
Ключові слова: ниткоподібні кристали InSb, осциляції по-
здовжнього магнітоопору, концентрація допування, g-фактор,
фаза Беррі.
Фаза Берри в деформированных нитевидных
кристаллах InSb
А. Дружинин, И. Островский, Ю. Ховерко,
N. Liakh-Kaguy, K. Rogacki
Исследовано влияние деформации на продольное магни-
тосопротивление нитевидных кристаллов InSb с проводимо-
стью n-типа, легированных оловом в концентрациях 6·1016–
6·1017 сm–3, при температурах от 4,2 до 50 К и магнитных
полях до 10 Тл. При низких температурах осцилляции Шуб-
никова–де Гааза обнаружены в деформированных и неде-
формированных образцах во всем диапазоне концентраций
допирования. Некоторые пики продольного магнитосопро-
тивления расщепляются в дублеты в нитевидных кристаллах
InSb с концентрацией допанта, близкой к переходу металл–
изолятор. Принимая во внимание расщепление пиков, для
деформированных и недеформированных образцов опреде-
лен гигантский g-фактор от 30 до 60. Период осцилляций
магнитосопротивления нитевидных кристаллов InSb не из-
меняется в деформированном состоянии для всех концентра-
ций допанта, но энергия Ферми возрастает, а эффективная
масса электрона mс уменьшается и составляет 0,02 m0. При-
сутствие фазы Берри было также обнаружено в деформиро-
ванных нитевидных кристаллах n-InSb, которые демонстри-
ровали переход в фазу топологического изолятора под
действием деформации.
Ключевые слова: нитевидные кристаллы InSb, осцилляции
продольного магнитосопротивления, концентрация допиро-
вания, g-фактор, фаза Берри.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 11 1527
https://doi.org/10.1126/science.1242247
https://doi.org/10.1016/j.commatsci.2015.06.032
https://doi.org/10.1126/science.1189792
https://doi.org/10.1016/j.mssp.2015.07.030
https://doi.org/10.1080/13642818408238842
https://doi.org/10.1002/pssb.200440008
https://doi.org/10.1088/0022-3727/16/4/025
https://doi.org/10.1021/nl901333a
https://doi.org/10.1016/S0038-1098(00)00361-6
https://doi.org/10.1007/978-3-319-02633-6_1
https://doi.org/10.1007/978-3-319-02633-6_1
https://doi.org/10.1007/978-3-319-02633-6_1
https://doi.org/10.1007/978-3-319-02633-6_1
https://doi.org/10.1103/PhysRevLett.82.2147
https://doi.org/10.1038/srep18674
1. Introduction
2. Experimental procedure
3. Experimental results and their discussions
4. Conclusions
|