Quantum oscillations of resistivity in bismuth nanowires
We studied the influence of uniaxial deformation on the transport properties of bismuth wires in the wide range of temperatures. Measurements of the resistance of bismuth nanowires with several diameters and different quality reveal oscillations on the dependence of resistance under uniaxial strain...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1169692017-05-19T03:03:28Z Quantum oscillations of resistivity in bismuth nanowires Condrea, E. Gilewski, A. Наноструктуры при низких температурах We studied the influence of uniaxial deformation on the transport properties of bismuth wires in the wide range of temperatures. Measurements of the resistance of bismuth nanowires with several diameters and different quality reveal oscillations on the dependence of resistance under uniaxial strain at T = 4.2 K. Amplitude of oscillations is significant (38%) at helium temperature and becomes smearing at T = 77 K. Observed oscillations originate from quantum size effect. Evaluation of period of oscillations allows us to identify the groups of carriers involved in transport. Calculated periods of 42.2 and 25.9 nm satisfy approximatively the ratio 2:1 for two experimentally observed sets of oscillations from light and heavy electrons. 2010 Article Quantum oscillations of resistivity in bismuth nanowires / E. Condrea, A. Gilewski // Физика низких температур. — 2010. — Т. 36, № 3. — С. 316-320. — Бібліогр.: 17 назв. — англ. 0132-6414 PACS: 73.63.Nm, 73.90.+f http://dspace.nbuv.gov.ua/handle/123456789/116969 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Наноструктуры при низких температурах Наноструктуры при низких температурах |
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Наноструктуры при низких температурах Наноструктуры при низких температурах Condrea, E. Gilewski, A. Quantum oscillations of resistivity in bismuth nanowires Физика низких температур |
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We studied the influence of uniaxial deformation on the transport properties of bismuth wires in the wide range of temperatures. Measurements of the resistance of bismuth nanowires with several diameters and different quality reveal oscillations on the dependence of resistance under uniaxial strain at T = 4.2 K. Amplitude of oscillations is significant (38%) at helium temperature and becomes smearing at T = 77 K. Observed oscillations originate from quantum size effect. Evaluation of period of oscillations allows us to identify the groups of carriers involved in transport. Calculated periods of 42.2 and 25.9 nm satisfy approximatively the ratio 2:1 for two experimentally observed sets of oscillations from light and heavy electrons. |
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Condrea, E. Gilewski, A. |
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Condrea, E. Gilewski, A. |
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Condrea, E. |
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Quantum oscillations of resistivity in bismuth nanowires |
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Quantum oscillations of resistivity in bismuth nanowires |
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Quantum oscillations of resistivity in bismuth nanowires |
| title_fullStr |
Quantum oscillations of resistivity in bismuth nanowires |
| title_full_unstemmed |
Quantum oscillations of resistivity in bismuth nanowires |
| title_sort |
quantum oscillations of resistivity in bismuth nanowires |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2010 |
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Наноструктуры при низких температурах |
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http://dspace.nbuv.gov.ua/handle/123456789/116969 |
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Quantum oscillations of resistivity in bismuth nanowires / E. Condrea, A. Gilewski // Физика низких температур. — 2010. — Т. 36, № 3. — С. 316-320. — Бібліогр.: 17 назв. — англ. |
| series |
Физика низких температур |
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AT condreae quantumoscillationsofresistivityinbismuthnanowires AT gilewskia quantumoscillationsofresistivityinbismuthnanowires |
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2025-07-08T11:23:52Z |
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2025-07-08T11:23:52Z |
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1837077714882789376 |
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© E. Condrea and A. Gilewski, 2010
Fizika Nizkikh Temperatur, 2010, v. 36, No. 3, p. 316–320
Quantum oscillations of resistivity in bismuth nanowires
E. Condrea1,2 and A. Gilewski2
1Institute of Electronic Engineering and Industrial Technologies, Academy of Sciences of Moldova
3/3 Academiei Str. MD 2028, Kishinev, Republic of Moldova
2International Laboratory of High Magnetic Fields and Low Temperatures, 53-421 Wroclaw, Poland
E-mail: condrea@lises.asm.md
Received October 12, 2009
We studied the influence of uniaxial deformation on the transport properties of bismuth wires in the wide
range of temperatures. Measurements of the resistance of bismuth nanowires with several diameters and different
quality reveal oscillations on the dependence of resistance under uniaxial strain at T = 4.2 K. Amplitude of oscil-
lations is significant (38%) at helium temperature and becomes smearing at T = 77 K. Observed oscillations ori-
ginate from quantum size effect. Evaluation of period of oscillations allows us to identify the groups of carriers
involved in transport. Calculated periods of 42.2 and 25.9 nm satisfy approximatively the ratio 2:1 for two expe-
rimentally observed sets of oscillations from light and heavy electrons.
PACS: 73.63.Nm Quantum wires;
73.90.+f Electronic structure and electrical properties of surfaces, interfaces, thin films, and low-
dimensional structures.
Keywords: bismuth; electrical resistance; quantum size oscillations; nanowires.
.
1. Introduction
A lot of the investigations concerned with various na-
nostructures is motivated partially by the very interesting
thermoelectric and magnetotransport properties of bismuth
nanowires (NWs) that make them potentially useful for
device applications. Theoretical calculations [1–3] pre-
dicted that Bi nanowires should have an enlarged thermo-
electric figure of merit, which results from the quantum
size effect, have induced extensive studies of Bi NWs.
Under the quantum size effect (QSE), several fundamental
macroscopic characteristic of solids exhibit an anomalous
dependence on reduced size. Therefore, for subsequent
applications, a precise determination of the size-dependent
parameters of investigated nanostructures is required. If the
decreased size of wires or films becomes comparable with
the electron wavelength ( ~d λ) , the transverse motion of
carriers is quantized. Thus, the energy spectrum splits into
subbands. When the discreteness of the energy subbands
becomes significant, an oscillatory behavior of electron
and hole density of states (DOS) as a function of thickness
is expected for metal films [3,4]. Oscillations of DOS are
due to variation in number of the subbands with diameter.
According to the theoretical considerations of the QSE [3–
5] all the transport coefficients which depend on the DOS
should oscillate as a function of sample thickness with the
period:
( )1/2
*/ 2 2 t Fd h m EΔ = , (1),
where *tm is the transverse effective masses, and FE is
Fermi level.
The main experimental results in the investigations of
the QSE have been obtained for thin semimetal films. The
first quantum size oscillations in the resistivity, Hall coef-
ficient, and magnetoresistance with a period of 400 Å were
observed in thin bismuth films [6]. Investigations [7] of
bismuth films in large range of thickness (200–3000 Å)
revealed that the period of the resistance oscillations varied
from 40 to 250 Å with sample thickness. The difference in
the period was attributed to the differences in the carrier
concentrations due to growth conditions.
Quantum oscillations in the resistance of bismuth-
antimony alloy films were registered under variation in
both thickness and Sb concentration at a fixed thickness
[8]. The concentration oscillations in a sample with con-
stant thickness are explained by the change of the trans-
verse quasi-momentum caused by composition variation in
the Bi-Sb alloy. An other manifestation of the QSE [9] was
observed while studying the thickness dependence of the
Quantum oscillations of resistivity in bismuth nanowires
Fizika Nizkikh Temperatur, 2010, v. 36, No. 3 317
ratio of the electron and hole density of states measured
under electric field effect (EFE). The method allows a ra-
ther precise determination of the film thickness period of
the oscillations, which is about 370 Å.
In most of the above-mentioned cases, QSE was shown
as an oscillatory behavior of the resistance dependence on
film thickness. As is pointed out in [5], a variation in the
value of the band overlap should also produce an oscillato-
ry behavior of the kinetic coefficients. The changes in band
overlap in a bulk Bi samples under deformation was de-
scribed by Brandt [10]. The influence of the deformation
on the band overlap changes was tested for bismuth films
condensed on mica substrates [11]. Observed non-mo-
notonous behavior of the resistance in bismuth films under
sagging deformation is in a good agreement with the con-
cept of QSE.
To our knowledge, up to now, most of the studies on
quantum oscillations in transport properties of Bi nano-
structures are concerned with thin films. The conditions of
observation of the QSE on thickness dependences of the
kinetic coefficients of thin wires are complicated by diffi-
culties in the preparation of a series of samples with a
small increment in thickness and identical characteristics
of the bulk. Despite a lot of recently developed techniques
for preparation of nanowires [1,2,12] the bulk characteris-
tic data from different experiments depend not only on
sample cross-sectional dimensions and crystallographic
orientation, but also on sample quality and purity,
shell/matrix material and annealing treatment. It is possible
to observe the oscillations of kinetic parameters, due to
size-quantized energy spectrum on the individual cylin-
drical nanowire under certain external influence, for exam-
ple, by impurity doping or deformation.
The aim of this study was to study transport coefficients
of individual bismuth nanowires under condition of QSE
by applying uniaxial stress for tuning the electronic struc-
ture and inducing band overlap changes. According to the
condition of QSE realization ( ~ d λ), an oscillating be-
havior in transport coefficient of Bi samples is expected at
diameters below 100 nm. One should note that uniaxial
strain in Bi NWs promotes the increases of band overlap
between L-electron pockets and T-holes pocket at constant
band gap, in contrast to its decrease under QSE.
2. Experimental methods and samples
Long Pyrex-coated Bi wires were fabricated using the
same improved variant of the Taylor method. This method,
which is presently known as the glass-coated melt spinning
method, consists in the melting of a metal in a glass tube
by rf induction heating and drawing a glass capillary in
which the molten metal is entrapped.
The wire axis is at an angle of about 19° with the bisec-
tor axis C1 in the bisector-trigonal plane C1C3. This orien-
tation is the same as that observed in Bi nanowire arrays by
Z. Zhang et al. [1]. Due to high elasticity of Pyrex capilla-
ries, the limit of elastic stretching of the glass-coated Bi
wires attains ε = 3.5% (in comparison for Bi whiskers ε =
= 2.0 % [13] and for bulk Bi samples ε = 0.4% [14]). Due
to small dimensions of glass-coated NWs, it was not possi-
ble to apply mechanical loading directly to them. For the
measurements under uniaxial strain, the glass-coated wires
with d = 70–150 nm and the length 0L = 2.0–3.0 mm were
mounted on an elastic bronze ring in a special insert with
stretching device similar to the method, used for whiskers
[13]. The stretching was directed along the wire axis, i.e.,
close to the bisector axis C1. The measurements of resis-
tance were performed using two-probe method. Resistance
variation was noted as ( )0 0/ /R R R R RΔ = − where 0R
is the value of resistance in non-deformed state. Strain var-
iation was noted as ( ) 0 0/L L Lε = − where 0L is the
length of the wire in non-deformed state. Electrical con-
tacts to the wire ends were made by Wood’s alloy. Low dc
currents (0.1 A 1 A)Iμ μ≤ ≤ were used to make sure that
the voltage of the sample was a linear function of the ap-
plied current.
3. Results and discussion
The studies of the strain dependences of the resistance
( )R ε at 4.2 K for wires with various diameters revealed
the oscillating dependence of the resistance on applied
deformation in wires with a diameter of 90 nm subjected to
thermal annealing (Fig. 1). Since an oscillating behavior in
transport coefficient of Bi wires is expected at diameters
below 100 nm, we can suppose that observed oscillations
are due to QSE. To our surprise, non-discernable and non-
reproducible quantum oscillations on R(ε) were observed
in the thinner NWs with a diameter of 70 nm. The depen-
dences of resistance versus uniaxial strain ( )R ε for Bi
wires with the diameters d ≥ 100 nm do not exhibits any
oscillations and is similar to the one observed by us for
Fig. 1. Relative variation of the resistance as a function of applied
strain for 90 nm Bi nanowire measured at 4.2 K and 77 K.
4.2 K
77 K
�
R
/R
,
%
40
30
20
10
0
–10
–20
–30
–40
0 0.5 1.0 1.5 2.0
Strain, %
E. Condrea and A. Gilewski
318 Fizika Nizkikh Temperatur, 2010, v. 36, No. 3
thicker wires [15] and whiskers [13].
Figure 2 shows the temperature dependences of the
electrical resistance ( )R T for Bi NWs with various diame-
ters. Observed ( )R T dependences are consistent with pre-
vious results for Bi wires [12] and Bi nanowire arrays [1].
Curve 2 in Fig. 2 presents ( )R T dependence for a 90 nm
NW after thermal treatment. History of the thermal treat-
ment was not simple. Various variants of thermal treatment
were tested and the optimal one was chosen. The NWs
were annealed at 180° C for 10 h under vacuum with slow
cooling back to room temperature. An evident increase of
value of a residual resistance ratio (RRR) for the 90 nm
wire after thermal treatment testified to an improved quali-
ty of annealed NWs. This supports the idea that a semicon-
ducting behavior of ( )R T does not imply that a band gap
opens even in the thin wires, a negative TCR may be rather
an evidence of large defect density inside the as-prepared
wires.
If we attribute the presence of oscillations on ( )R ε in
Fig. 1 to the manifestation of QSE, in addition to the main
condition ( ~d λ) for QSE occurrence [4], we should con-
sider some important requirements such as a small washing
out of the discrete energy spectrum:
/h Eτ << Δ , (2 )
and a small temperature smearing:
kT E<<Δ , (3)
where τ is the relaxation time, taking into account scatter-
ing both in volume and on the surface; EΔ is the distance
between discrete energy subbands. Inequality (2) means a
high mobility and/or large free path length for carriers -
involved in transport. It seems that all the requirements for
QSE are satisfied for annealed 90 nm wire at T = 4.2 K.
The observed resistance oscillations were reproduced
on all the wires with a diameter of 90 nm from the series of
samples subjected to thermal annealing. As one can see
from Fig. 1, the ( )R ε dependence at 4.2 K exhibits a mod-
ulating profile consisting of two sets of oscillations with
different periods determined by different groups of carri-
ers. First, we supposed that two different periods of oscilla-
tions are due to contribution of electrons and holes to elec-
trical conductivity. It is known that difference in the pe-
riods of electron and hole quantum oscillations results
from the anisotropic properties of Bi.
A rough evaluation of possible periods of quantum oscilla-
tions in 90 nm bismuth wire was made by formula (1). The
value of effective transverse mass of heavy and light electrons
was used from the model of the subband structure calculated
in [1] for the Bi nanowire arrays with the same crystal orienta-
tion, corresponding to [1011]. This is our case where three
groups of carriers operate: heavy electrons from two equiva-
Fig. 2. Temperature dependence of resistance normalized to the
resistance at 300 K for as-prepared Bi NWs: d = 90 nm (1), d =
= 120 nm (3) and thermal annealed: d = 90 nm (2), d = 120 nm
(4), d = 150 nm (5).
R
(T
)/
R
(3
0
0
)
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
1
2
3
4
5
0 50 100 150 200 250 300
T, K
Fig. 3. Schematic diagram of Bi band structure at L-point and
T-point near Fermi energy level: for bulk Bi (a); for 90 nm Bi
NW calculated in model [1] in non-deformed state (b); a possi-
ble band structure for 90 nm NW at the strain ε ≥ 0.96% (c).
15 meV
–38 meV
BA C
EF
44 meV –22 meV
BA C
–16 meV
52 meV
L T
44 meV
BA C
52 meV
19 meV
a
b
c
L T
L T
EF
EF
Quantum oscillations of resistivity in bismuth nanowires
Fizika Nizkikh Temperatur, 2010, v. 36, No. 3 319
lent pockets A and B, light electrons from pockets C, and
heavy T-holes. According to [1], as the diameter of Bi nano-
wire arrays decreases below 100 nm, the band overlap be-
tween light and heavy electron pockets and T-hole pocket
decreases in the different ways, thus resulting in the splitting
of the L-point band edge as shown in Fig. 3,b. In this model,
the band edge energy of each subband is determined by the
average transverse effective mass, approximated by the ap-
propriate effective cyclotron mass. The values of effective
masses calculated by authors of [1] are: * 00.00212m m= and
* 00.00372m m= for light and heavy electrons, respectively.
Estimated periods for our 90 nm NW are: 1dΔ = 42.2 nm;
2dΔ = 25.9 nm and 3dΔ = 16.0 nm for light and heavy elec-
trons and holes, respectively.
Coming back to measured quantum oscillations of resis-
tance (Fig. 1) we note the period ratio of 2:1 for two set of
observed oscillations. Two of calculated periods 42.2 nm
and 25.9 nm satisfy approximatively the ratio 2:1. The
calculated value of 16.6 nm for hole period is overesti-
mated as a result of using the value for hole transverse
mass * 00.21m m= available for bulk Bi. The absence of
quantum oscillations from holes can be caused by its small
period and non-detectable amplitude versus background.
Thus, we can suppose that quantum oscillations with large
amplitude and period of 42.2 nm occur from the light elec-
trons with smallest effective mass, and quantum oscilla-
tions with weak amplitude and period of 25.9 nm occur
from heavy electrons from pockets A and B. A significant
amplitude (38%) of resistance quantum oscillations with
1dΔ = 42.2 nm is due to the low quantum numbers of the
subbands located below Fermi energy level in the pocket C
with light electrons. The vanishing of these oscillations at
the deformation higher than 1% is suggestive of a change
in the Fermi surface topology, known as an electronic to-
pological transition (ETT) or Lifshitz transition. As a result
of our previous investigations [15] of the changes in the
Fermi surface topology under stress by means of Shubni-
kov-de Haas oscillation measurements, we have found the
upward shift of the light electron pocket C relative to the
other two pockets A and B up to its complete vanish at a
high strain. The shifted electron pocket is schematically
indicated by the dashed line in Fig. 3,c. Due to the continu-
ing increase in the band overlap between L-electron pock-
ets (A and B) and T-hole pocket, quantum oscillations with
a period of 25.9 nm are detected at a strain higher than 1%.
If we attribute the presence of the oscillations on
( )R ε to the QSE manifestation, it is natural to make a
brief analysis of correlation between the value of uniaxi-
al strain and the period of observed oscillations dΔ in
90 nm wires. Though the precise interrelation between
applied uniaxial strain and dΔ is not known, we try to
follow the influence of strain on the Fermi level shifting
through quantized subbands in electron pockets. In terms
of previous results on ETT with the vanishing of one
electron pocket, the work of extension of ε = 0.96%
should be on the order of the energy shift of electron
pocket C with strain. Since the measurements were made
in the elastic deformation range, we may use the value of
the elastic modulus for bulk Bi crystal along the bisector
direction; thus, it is possible to determine the tensile load
dependence P on the value of extension 0/L Lε = Δ
which is approximatively P = 0.38 GPa at ε = 0.96%.
A rough estimation of value of the energy shift was
made by using the values of shifting rate /dE dF of elec-
tron pockets under anisotropic deformation along the bi-
sector direction for bulk Bi crystals calculated by authors
in [17], which are: /dE dF = 0.5 meV/kg for pocket C
and /dE dF = –0.6 meV/ kg for pockets A and B. In our
case, the value of energy shift was determined at a strain of
0.96%, where quantum oscillations with large period dis-
appear. Calculated value is dE ≈ 19 meV. Because of
some involved calculation uncertainties, the value of dE ≈
≈ 19 meV is rather appreciative; nevertheless, it is reason-
able of the same order with value of Fermi energy level
and of the band overlap (16 meV) for light electron pocket
from the model of electronic band structure for 90 nm Bi
NWs, advanced by authors of [1].
4. Conclusions
Systematic measurements of the resistance of bismuth
nanowires with several diameters and different quality re-
veal oscillations on the dependence of resistance under
uniaxial strain at T = 4.2 K. Amplitude of oscillations is
significant (38%) at helium temperature and becomes
smearing at T = 77 K. Observed oscillations originate from
quantum size effect.
The absence of quantum size oscillations in resistance
dependence for 70 nm wires can be explained partially by
scarce number of light electrons responsible for oscilla-
tions with decreasing diameter [1] and partially by imper-
fection of the nanowires.
A simple evaluation of period of oscillations allows us
to identify the groups of carriers involved in transport.
Calculated periods of 42.2 and 25.9 nm satisfy approxima-
tively the ratio 2:1 for two experimentally observed sets of
oscillations from light and heavy electrons.
The importance of the quantum size effect manifesta-
tion in the resistance dependence on strain goes beyond
studying the structure of electron spectrum, it can also be
applied to investigate the spectrum of phonons. With a
view to elucidate some aspects of the practical use of na-
nowires, we plan the further investigations under strain of
the thermopower, which at low temperature may be due to
the diffusive or phonon drag mode of carrier interaction.
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