Quantum oscillations as the tool for study of new functional materials (Review Article)
We present an overview of our recent results on quantum magnetic oscillations in new functional materials. We begin with the Lifshitz and Kosevich approach for quasi-2D layered materials and obtain general formulas for the oscillatory parts of the grand thermodynamic potential and magnetization. The...
Saved in:
| Published in: | Физика низких температур |
|---|---|
| Date: | 2014 |
| Main Authors: | , , , , |
| Format: | Article |
| Language: | English |
| Published: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2014
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/119456 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Quantum oscillations as the tool for study of new functional materials (Review Article) / V.P. Gusynin, V.M. Loktev, I.A. Luk’yanchuk, S.G. Sharapov, A.A. Varlamov // Физика низких температур. — 2014. — Т. 40, № 4. — С. 355-366. — Бібліогр.: 43 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-119456 |
|---|---|
| record_format |
dspace |
| spelling |
Gusynin, V.P. Loktev, V.M. Luk’yanchuk, I.А. Sharapov, S.G. Varlamov, A.A. 2017-06-07T03:53:42Z 2017-06-07T03:53:42Z 2014 Quantum oscillations as the tool for study of new functional materials (Review Article) / V.P. Gusynin, V.M. Loktev, I.A. Luk’yanchuk, S.G. Sharapov, A.A. Varlamov // Физика низких температур. — 2014. — Т. 40, № 4. — С. 355-366. — Бібліогр.: 43 назв. — англ. 0132-6414 PACS 72.15.Jf, 72.20.Pa, 71.70.Di https://nasplib.isofts.kiev.ua/handle/123456789/119456 We present an overview of our recent results on quantum magnetic oscillations in new functional materials. We begin with the Lifshitz and Kosevich approach for quasi-2D layered materials and obtain general formulas for the oscillatory parts of the grand thermodynamic potential and magnetization. Then we consider the oscillations of the Nernst–Ettingshausen coefficient which consists of thermal and magnetization parts. The difference between normal and Dirac carriers is also discussed. To conclude we consider a model for multilayer graphene which allows to calculate exactly the Berry phase which remains undetermined in the Lifshitz–Kosevich approach. The magnetic oscillations of the density of states and capacitance for different number of the carbon layers are described. This work is supported partially by the European FP7 program, Grant No. SIMTECH 246937. The work of V.P.G., V.M.L. and S.G.Sh. is supported also by the grant STCU #5716-2 “Development of Graphene Technologies and Investigation of Graphene-based Nanostructures for Nanoelectronics and Optoelectronics”. V.P.G. and S.G.Sh. acknowledge the support of the joint Ukrainian-Russian SFFR-RFBR grant No. F53.2/028. The work of IL was done in frame of International Training Network FP7-ITNNOTEDEV. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Новые достижения фермиологии Quantum oscillations as the tool for study of new functional materials (Review Article) Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Quantum oscillations as the tool for study of new functional materials (Review Article) |
| spellingShingle |
Quantum oscillations as the tool for study of new functional materials (Review Article) Gusynin, V.P. Loktev, V.M. Luk’yanchuk, I.А. Sharapov, S.G. Varlamov, A.A. Новые достижения фермиологии |
| title_short |
Quantum oscillations as the tool for study of new functional materials (Review Article) |
| title_full |
Quantum oscillations as the tool for study of new functional materials (Review Article) |
| title_fullStr |
Quantum oscillations as the tool for study of new functional materials (Review Article) |
| title_full_unstemmed |
Quantum oscillations as the tool for study of new functional materials (Review Article) |
| title_sort |
quantum oscillations as the tool for study of new functional materials (review article) |
| author |
Gusynin, V.P. Loktev, V.M. Luk’yanchuk, I.А. Sharapov, S.G. Varlamov, A.A. |
| author_facet |
Gusynin, V.P. Loktev, V.M. Luk’yanchuk, I.А. Sharapov, S.G. Varlamov, A.A. |
| topic |
Новые достижения фермиологии |
| topic_facet |
Новые достижения фермиологии |
| publishDate |
2014 |
| language |
English |
| container_title |
Физика низких температур |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
We present an overview of our recent results on quantum magnetic oscillations in new functional materials. We begin with the Lifshitz and Kosevich approach for quasi-2D layered materials and obtain general formulas for the oscillatory parts of the grand thermodynamic potential and magnetization. Then we consider the oscillations
of the Nernst–Ettingshausen coefficient which consists of thermal and magnetization parts. The difference between normal and Dirac carriers is also discussed. To conclude we consider a model for multilayer graphene which allows to calculate exactly the Berry phase which remains undetermined in the Lifshitz–Kosevich approach. The magnetic oscillations of the density of states and capacitance for different number of the carbon layers are described.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/119456 |
| citation_txt |
Quantum oscillations as the tool for study of new functional materials (Review Article) / V.P. Gusynin, V.M. Loktev, I.A. Luk’yanchuk, S.G. Sharapov, A.A. Varlamov // Физика низких температур. — 2014. — Т. 40, № 4. — С. 355-366. — Бібліогр.: 43 назв. — англ. |
| work_keys_str_mv |
AT gusyninvp quantumoscillationsasthetoolforstudyofnewfunctionalmaterialsreviewarticle AT loktevvm quantumoscillationsasthetoolforstudyofnewfunctionalmaterialsreviewarticle AT lukyanchukia quantumoscillationsasthetoolforstudyofnewfunctionalmaterialsreviewarticle AT sharapovsg quantumoscillationsasthetoolforstudyofnewfunctionalmaterialsreviewarticle AT varlamovaa quantumoscillationsasthetoolforstudyofnewfunctionalmaterialsreviewarticle |
| first_indexed |
2025-11-25T10:04:45Z |
| last_indexed |
2025-11-25T10:04:45Z |
| _version_ |
1850509647856795648 |
| fulltext |
© V.P. Gusynin, V.M. Loktev, I.A. Luk’yanchuk, S.G. Sharapov, and A.A. Varlamov, 2014
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 4, pp. 355–366
Quantum oscillations as the tool for study
of new functional materials
(Review Article)
V.P. Gusynin
1
, V.M. Loktev
1,2
, I.A. Luk’yanchuk
3
, S.G. Sharapov
1
, and A.A. Varlamov
4
1
Bogolyubov Institute for Theoretical Physics, 14-b Metrologicheskaya Str., Kiev 03680, Ukraine
E-mail: vloktev@bitp.kiev.ua
2
National Technical University of Ukraine “KPI”, 37 Peremogy Ave., Kiev 03056, Ukraine
3
Laboratory of Condensed Matter Physics, University of Picardie Jules Verne, Amiens 80039, France
4
CNR-SPIN, Viale del Politecnico 1, University “Tor Vergata”, I-00133 Rome, Italy
Received December 2, 2013
We present an overview of our recent results on quantum magnetic oscillations in new functional materials.
We begin with the Lifshitz and Kosevich approach for quasi-2D layered materials and obtain general formulas
for the oscillatory parts of the grand thermodynamic potential and magnetization. Then we consider the oscilla-
tions of the Nernst–Ettingshausen coefficient which consists of thermal and magnetization parts. The difference
between normal and Dirac carriers is also discussed. To conclude we consider a model for multilayer graphene
which allows to calculate exactly the Berry phase which remains undetermined in the Lifshitz–Kosevich ap-
proach. The magnetic oscillations of the density of states and capacitance for different number of the carbon lay-
ers are described.
PACS: 72.15.Jf Thermoelectric and thermomagnetic effects;
72.20.Pa Thermoelectric and thermomagnetic effects;
71.70.Di Landau levels.
Keywords: quantum oscillations, magnetization, Nernst coefficient, grapheme, graphite, quasi-two-dimensional
metal.
Content
1. Introduction .......................................................................................................................................... 355
2. Lifshitz–Kosevich approach for 2D–3D dimensional crossover .......................................................... 356
2.1. Magnetic oscillations.................................................................................................................... 358
3. Giant Nernst–Ettingshausen coefficient oscillations in quasi-2D metals ............................................ 358
4. Exact calculation of DOS and capacitance in chiral multilayer graphene ........................................... 362
5. Conclusion ........................................................................................................................................... 365
References ................................................................................................................................................ 365
1. Introduction
Quantum oscillations of physical parameters of metallic
and semimetallic systems in high magnetic fields is the
powerful tool to study the electronic properties of new
functional materials, which often are low-dimensional and
with nontrivial spectra. Wealth of parameters such as elec-
tronic spectrum, scattering mechanism, geometry of Fermi
surface, etc. can be extracted from the shape, period, and
phase of quantum oscillations. The first experimental evi-
dence of magnetic moment oscillations as function of ap-
plied field was presented by de Haas and van Alphen in
1930 [1] and independently theoretically predicted by Lan-
dau [2] as a consequence of the electronic level quantiza-
tion in magnetic field. Further theoretical description of
quantum oscillations was developed in 30’s in papers of
Peierls [3] for 2D systems and Shoenberg [4] and Landau
[5] for 3D systems.
However these studies were based on the simple metal
model with quasifree electrons, described by the effective
electron mass. The breakthrough in exploration of complex
electron spectra occur in 1954 after the seminal work of
V.P. Gusynin, V.M. Loktev, I.A. Luk’yanchuk, S.G. Sharapov, and A.A. Varlamov
356 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 4
Lifshitz and Kosevich (LK) [6] who generalized the Landau
and Shoenberg approach for the case of arbitrary spectrum
by suggestion to use the dependence of the Fermi-surface
cross-section S on the chemical potential : = ( )S S as
the relevant parameter of the theory. In this case the cyclo-
tron mass of the electron = (1/2 ) /m dS d naturally gene-
ralizes the effective electron mass for the case of non-
parabolic spectra. Basing on LK theory, measurements of S
at different field orientations permitted to reconstruct the
geometry of Fermi surface and corresponding electronic
spectra for many metallic and semimetallic materials. In
addition, it was recognized that complimentary information
can be extracted from the study of quantum oscillations of
another quantities such as longitudinal and Hall resistivity,
thermopower, Nernst coefficient, etc. [7,8].
Intensive study of highly-anisotropic and low-dimen-
sional materials, films, multilayers and superlattices posed
the challenging question, how to adopt the Landau–Pierls
theory for the dimensional crossover in such systems with
arbitrary spectrum. In this article we present our recent re-
sults where we generalize the LK approach for systems with
2D–3D crossover in application to such exciting subject as
detection and study of Dirac fermions in graphite and
graphene. After introduction of the generalized LK formal-
ism we obtain the general crossover formula for quantum
oscillation of magnetization and Nernst coefficient.
Finally we consider the model of multilayer graphene
which allows to calculate exactly the Berry phase which
remains undetermined in the Lifshitz–Kosevich approach.
The magnetic oscillations of the density of states (DOS)
and capacitance for different number of the carbon layers
are also studied in this model.
This work is devoted to 60th anniversary of a remarkable
work of Kharkov physicists Ilya Lifshitz and Arnold Kose-
vich [6] which resulted in the creation of the fermiology.
2. Lifshitz–Kosevich approach for 2D–3D dimensional
crossover
We consider the quasi-2D layered system with corru-
gated cylinder Fermi surface and dispersion
( ) = ( ) 2 sin ,zp
p t dp (2.1)
where = ( , )zpp p is the momentum ( = | |)p p and
d is the distance between layers. Such a model is general
enough since it accounts for the arbitrary type of dispersion
( )p in plane and allows to describe the 2D–3D di-
mensional crossover by variation of the interplane hopping
parameter t from 2 = 0Dt to 3 .D Ft
When the magnetic field || zH is applied, the Landau
quantization in perpendicular direction takes place and the
quantized electronic spectrum is written as
( ) = ( ) 2 sin , = 0, 1, ,z
n z B
p
k n t d H n (2.2)
where the last term presents the Zeeman splitting. To find
( )n we can use the LK approach and present the prob-
lem in terms of Fermi-surface cross-section area ( )S of
the orbital electron motion that is quantized in magnetic
field in accordance to Lifshitz–Onsager condition [6,9]:
( ) = ( ) 2 .
eH
S n
c
(2.3)
The correction factor to quasiclassical approximation,
= 1/2( / ) ,m m contains the orbital and spin parts
[10]. The orbital part was assumed in the original LK
publication [6] to be in between 0 and 1 with specification
that =1/2 for parabolic quasi-free electronic spectrum
2( ) = /2 .p mp More detailed calculations of Falkovsky
[10] revealed that for large class of semimetallic spectra,
parameter can take the definite value of either 0 or 1/2.
Finally Mikitik and Sharlai [11] demonstrated that this
factor is the topologically protected quantum number that
is intimately related to so-called Berry phase and permits
separate the electronic states on two classes: so-called Di-
rac fermions (DF) with = 0 and normal carriers (NC)
with =1/2. Proposed in [12] method of measurement of
via the phase of quantum oscillation permitted to detect
the DF in graphite [12,13] and in graphene [14,15]. The
spin contribution 1/ 2( / )m m with = 1 leads to the
Zeeman spin-splitting of quantum oscillation and permits
to measure the electron effective mass .m Below we omit
this factor for brevity since it can be easily reconstructed at
the final stage of calculation.
As was already mentioned, the advantage of LK approach
is that the quantum oscillation can be calculated in the very
general form, without the detailed specification of the spec-
trum ( )p since the corresponding spectrum quantization
can be expressed via orbital cross section (2.3) as
1
( ) = 2 ( )
e
n n H
S c
.
In what follows we shall use the LK formalism to cal-
culate the quantum oscillations for spectrum (2.1) with
arbitrary dependence ( ),p generalizing the method
that was proposed in [12] for 2( ) = /2 .p p m
To calculate the quantum oscillations of thermodynam-
ical parameters we seek for the field and temperature de-
pendences of the thermodynamic potential of the system
(we assume the Boltzman constant kB = 1)
= ( ) ln 1 e ,TT g d (2.4)
where ( )g is the density of states. Following [16] we
present it via the imaginary part of the space-averaged
electron Green function
,
1 1
( ) = sgn ( ) Im ,
( ) sgn ( )n zn kz
g
k i
(2.5)
Quantum oscillations as the tool for study of new functional materials
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 4 357
where is the impurity-provided level broadening and
( )n zk is given by (2.4). Summation is performed over
the set of one-electron quantum states that is parameterized
by the Landau level quantum number n and wave vector kz.
Presenting the summation over , zn k as
/
=0/
1
2 2
d
z
nd
dkeH
c
and using the Poisson summation over n
=0 =
( ) = ( )exp(2 ) , 1 < < 0,
n l a
f n f il d a
one arrives to the expression
/
/
1
( ) =
2 2
d
z
d
dke
g H
c
2
= 1 2
exp(2 )
.
2 ( ) 2 cos
l
z
il
d
e
S H t k d
c
(2.6)
Here the lower limit of the last integral was extended to
. Integration over the variable can be substituted by
the sum of the simple pole residues, located at the points
k that are provided by condition
2 ( ) = ( 2 cos )k z
e
H S i t k d
c
( ) 2 cos .z
S
S i t k d (2.7)
Finally one can obtain
/
2
=/
1 1 1 1
( ) = exp 2 ( )
2 2 2
d
z
ld
dkS c
g il S
d H e
| | 1
exp exp 2 2 cos .
2
z
l c S c S
il t k d
H e H e
(2.8)
Now, using the integral representation of Bessel function
0
1
( ) = exp(2 cos )
2
J x ix d ,
integrating over ,zk and re-arranging the terms one can
present ( )g in the final form
2 2
1 2 1
( ) =
m m
g
d d
0
=1
( )
exp cos 2 ,
2B Bl
t c S
l J l l
H H e H
(2.9)
where Bohr magneton = /2 ,B e mc and is defined in
Table 1.
The thermodynamical potential contains the non-
oscillating, ,c and oscillating, , parts. The first con-
tribution is easily calculated from (2.4) and (2.9) by inte-
gration by parts
2 2
0
1 1 1
= ( ) ( )
2
c S d O T
d
. (2.10)
For the oscillating part we expand ( )S in vicinity of
as ( ) ( ) 2 ( )S S m and, after integration by
parts, one obtains
Table 1. Parameters of general LK formalism for the particu-
lar cases of normal carriers (NC) and Dirac fermions (DF) with
parabolic and linear energy dispersions
2 /2p m and v|p |
correspondingly
Parameter NC DF
(p ) 2 /2p m v|p |
(n)
1
2
c n (2 cn)
1/2
S( ) 2 m
2
/v2
* *
1
, | | /
2
dS
m m m
d
m /v2
1 1
2 2
m
m
2
1
2 mv
1
*
, 2c c B
eH
H
m c
eH/cm ev2H/c
( )
2
c
c S
H
e
1
2 B
2
1
4 Bmv
2
B
T
H
– –
2
2
2
2m
z
S dS
td
dp
4 m td
2
4 td
2
/v2
V.P. Gusynin, V.M. Loktev, I.A. Luk’yanchuk, S.G. Sharapov, and A.A. Varlamov
358 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 4
0
=1
1 1 1
= exp
B Bl
e t
HT l J l
d c l H H
0
exp2 ( )
( ) 1
Im exp 2 .
2
1 exp
c m
il
c S e Hi l d
e H
T
(2.11)
The last integral can be calculated by expansion of the
lower limit to (valid for )T ) and by using the
relation
exp
=
1 exp sinh
i y i
dy
y
.
Finally one finds
2 2
2 2 2
=1, = 1
1 ( )
= , ,
22
c
l
l
m l
H
l
(2.12)
with ( ) = / sinhl l l , and
, = =l l lH i
0
2 ( )
2 exp 2 .
2c c
t c S
J l i l
e H
(2.13)
Here we reconstructed the spin variable = 1. The di-
mensional crossover is driven by parameter
= 2 /( ): 1cx t x in quasi-2D case and 1x in 3D
case.
2.1. Magnetic oscillations
Knowledge of the field and temperature dependences of
oscillating part of thermodynamical potential ( , )H T
permits to calculate the quantum oscillations of various
thermodynamical parameters of the system, and, in particu-
larly, oscillating magnetization
,
= =
T
M
H
2 2 2
=1, = 1
1 ( ) 1 ( )
( , ).
2(2 )
l
l
e S l
H
c m l
(2.14)
Using the limit expansions 1
0( ) 1,xJ x
1
0( )xJ x
1/2(2/ ) cos( /4)x x we, following [18], present the
quantum oscillation of magnetization in 2D ( 1)x and
3D ( 1)x cases as
2 3 2
1 /
=
Ry4
B
D
B
S m
M
a
=1, = 1
( ) /
e sin 2
2
l
HB
Bl
l S m
l
l H
(2.15)
and
1/21/2 2
3 3 3 2
2 /
=
Ry Ry8
B B
D
B z
HS m S
M
a p
3/2
=1, = 1
( ) /
e sin 2 ,
2 4
l
HB
Bl
l S m
l l
Hl
(2.16)
where 2D magnetization is normalized per unit of the layer
area and 3D magnetization per unit of volume. The atomic
units: Bohr radius 2 2= /( )Ba me and Rydberg energy
4 2Ry = /2me were used to simplify the dimensional
analysis of the resulting expressions.
It is interesting to note, that Eq. (2.15) can be written in
a simple form at low temperatures 2 < BT H when
( ) 1l :
2 3 2
1 /
=
Ry2
B
D
B
S m
M
a
= 1
/
sin 2
21
arctan .
2 /
exp cos 2
2
B
B B
S m
H
S m
H H
(2.17)
In case of NC this expression was obtained in [16] and in
case of DF in [17]. For more detailed analysis of dimen-
sion- and temperature-induced crossover of magnetic oscil-
lations we refer to publication [18].
3. Giant Nernst–Ettingshausen coefficient oscillations
in quasi-2D metals
In this section written basing mainly on the results of
Ref. 19, we will apply the general relations obtained above
in order to explain the giant Nernst–Ettingshausen (NE)
coefficient oscillations observed recently on the graphene
samples [20,21]. Let us start from recalling of the basic
notions.
The Nernst–Ettingshausen effect in metals [22] can be
considered as a thermoelectric counterpart of the Hall ef-
fect. It consists of the induction of an electric field yE
normal to the mutually perpendicular magnetic field zH
and temperature gradient .xT All electric circuits are
supposed to be broken: = = 0x yJ J and heat flow along
Quantum oscillations as the tool for study of new functional materials
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 4 359
y axis to be absent (adiabatic conditions). Quantitatively
the NE effect is characterized by the NE coefficient
= ,
( )
y
x z
E
T H
(3.1)
which varies by several orders of magnitude in different
materials ranging from about 7mV/(K T) in bismuth
(where the effect has been observed by NE for the first
time in 1886) up to 510 mV/(K T) in some metals [23].
NE effect remained poorly understood until 1948 when
Sondheimer [24], basing on the classical Mott formula for
the thermoconductivity tensor, calculated for a degener-
ated electron system, linking it to the energy derivative of
the Hall angle = /xy xx which allowed to reveal a cor-
relation between NE and Hall effects. Within this model
was found to be independent on the magnetic field in weak
fields and to decrease as 2H in the region of classically
strong fields, when the cyclotron frequency c is larger
then the inverse electron mean free time 1.
In 1959 Bergeron and co-authors experimentally ob-
served the giant oscillations of the NE coefficient in
strong magnetic fields in zinc [25]: the phenomenon ap-
parently related to crossing of the electronic Fermi energy
by Landau levels (LL). Simular to de Haas–van Alphen
(dHvA) oscillations of magnetization and Shubnikov–de
Haas (SdH) oscillations of conductivity, in the NE oscilla-
tions the corresponding quantizing fields = nH H are giv-
en by Lifshitz–Onsager condition (2.3). Later, in 1964,
Obraztsov [26] suggested that so-called magnetization cur-
rents (i.e., electric currents induced due to inhomogeneous
distribution of magnetization in the sample) can contribute
supplementary to the NE effect.
Very recently, the NE effect has been measured [20,21]
and theoretically analyzed [27,28] in graphene. Surprising-
ly, it has been found that changes its sign at = nH H in
graphene while it has maxima at these intersections in zinc
[25] and bismuth [29]. Zhu et al. [30] demonstrated that
such untypical behavior of ( )H observed in graphene is
not reproduced in graphite. They concluded that piling of
multiple graphene layers leads to a topological phase tran-
sition in the spectrum of charge carriers, so that graphite
behaves as a 3D crystal despite of its apparent structural
anisotropy and of similarity of its electronic properties to
those of graphene.
Another challenging property of quantum oscillations is
the possibility to distinguish between two types of charge
carriers, having already mentioned topologically different
parameter [11]: =1/2 for the NC with the parabolic
2D dispersion and linear LL quantization and = 0 for the
DF having the linear two-branch spectrum and 1/2n LL
quantization.
Below we use a simple thermodynamic approach to the
description of the NE effect which allows linking the oscil-
lations of the NE coefficient to the presented above oscilla-
tions of the chemical potential. Both thermal (Sondheimer)
and magnetization contributions to the Nernst coefficient
are evaluated analytically for a quasi-2D electronic system
with either parabolic or Dirac spectrum. In the 2D limit for
the Dirac spectrum we recover the behavior of the NE co-
efficient observed in graphene [20,21] while the recent
data of Zhu et al. [30] on graphite are fitted better assum-
ing a quasi-2D character of the system. Very interestingly,
the amplitude of the NE oscillation is found to decrease as
a function of the Fermi energy in system of Dirac fermi-
ons, while it increases with Fermi energy for carriers hav-
ing a parabolic 2D dispersion.
As it was mentioned above the NE coefficient is meas-
ured in the absence of the electric current flowing through
the system along the temperature gradient. The system is at
thermal equilibrium and can be characterized by a constant
electrochemical potential ,e where is supposed to
be the temperature dependent chemical potential and is
the electrostatic potential. In the thermodynamic approxi-
mation [31,32], the effect of the temperature gradient is
reduced to appearance of an effective electrostatic potential
and, henceforth, an electric field in x direction = / .xE e
In this way, the problem is reduced to the classical Hall
problem, which allows to obtain the thermal contribution
to the NE coefficient:
therm
2
= ,xx d
dTe nc
(3.2)
where xx is the diagonal component of the conductivity
tensor, n is the concentration of carriers. This simple for-
mula reproduces the result of Sondheimer for a normal
metal [32].
An additional contribution to NE coefficient appearing
due to the spatial dependence of magnetization in the sam-
ple [26] can be found from the Ampere law:
mag = ( /4 ) ,cj B where
mag
j is the magnetization cur-
rent density, the magnetic induction = 4 ,B H M H is
the spatially homogeneous external magnetic field, M is
the magnetization, which can be temperature and, hence-
forth, coordinate dependent. One can readily express the
magnetization current as
mag = ( / )yj c dM dT T and the
corresponding contribution to the electric field in y direc-
tion (Nernst field) as
mag mag= ,y yy yE j where yy is the
diagonal component of the resistivity tensor. The “magnet-
ization” contribution to the NE coefficient reads
mag = .
yyc dM
H dT
(3.3)
The Eqs. (3.2) and (3.3) reveal the essential physics of
Nernst oscillations in the quantizing magnetic fields. In
particular, one can see that the NE coefficient is dependent
on the diagonal components of conductivity and resistivity
tensors. Their oscillations as a function of magnetic field
constitute the SdH effect. In graphene and graphite the
V.P. Gusynin, V.M. Loktev, I.A. Luk’yanchuk, S.G. Sharapov, and A.A. Varlamov
360 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 4
giant Nernst oscillations have been observed in the regime
where the SdH effect is negligibly weak [20,21,30]. This is
why one should attribute the giant NE coefficient oscilla-
tions to the remaining factors in the Eqs. (3.2) and (3.3),
namely to the temperature derivatives of the chemical po-
tential and magnetization, /d dT and / ,dM dT respec-
tively. Remarkably, to evaluate these quantities no sup-
plementary knowledge of the transport properties of the
system is needed. These derivatives can be expressed in
terms of the thermodynamic potential of the system
( , , )H T as
1
2 2
2
=
T
d
dT T
and
2
= .
dM
dT T H
(3.4)
The corresponding expression for the oscillating (de-
noted by tilde) part of is given by Eq. (2.12). One can
present it in the LK form [6], using the Fermi-surface cross
section ( )S at = 0,zp the electron cyclotron mass ,m
the cyclotron frequency ,c and the phase as the pa-
rameters that do not depend on the particular form of
( ).p Their specific values for the parabolic and Dirac
spectra are given in the Table 1 above. Below we omit the
index implying the spin averaging.
In the experimental configuration of the NE effect, the
number of particles n is fixed, so that [16]
2
, ,
( ) ( ) ( )
= = 2 = const.
2H T H T
S
n
(3.5)
This relation implicitly determines the dependence of on
H and T for the given n. In the present derivation we
assume a Lorentzian broadening of LL with a constant .
Such approximation can be justified for c F in the
case of 3D system. In 2D systems it is expected to be valid
only in the low-field regime c
–1
. The oscillating parts
of the chemical potential and magnetization can be ex-
pressed using Eq. (3.4) as
{1}
{0}
Im
= , = ,
1 2Re
d dM n d
dT dT H dT
(3.6)
where
{ } ( )
=1, = 1
1
= ( ) ( , )
2
l F
l
l H (3.7)
and
( ) ( )x is the derivative of the order of = 0, 1 of
the function . One can see from Eqs. (3.3) and (3.6) that
the NE coefficient oscillates proportionally to the deriva-
tive of magnetization over temperature. This shows an im-
portant link between NE and dHvA oscillations, which is
universal and independent on the dimensionality of the
system and of the type of carriers.
It is convenient to express the NE coefficient as
therm mag
0= = ( ) ( )H H (3.8)
with 0 ( )H and ( )H being the background and the os-
cillating parts. The background part can be evaluated in the
Drude approximation as [32]
2
0 2
1
( ) = .
6 1 ( )F c
T
H
m c
(3.9)
Taking into account the magnetization currents leads to the
correction of the order of 2( )F with respect to
Sondheimer result described by Eq. (3.9).
The oscillating part of the Nernst coefficient can be
written using Eqs. (3.2), (3.3) and (3.6) as
{1}
{0}
Im
( ) = 2 ( ) ,
1 2Re
H H (3.10)
with
2 2
( ) ( )
( ) = .xx xxH cn H
H
e nc H
(3.11)
In the Drude approximation for NC
2
Drude 2 2
1 2( )1
( ) = .
( ) 1 ( )
c
c c
H
m c
(3.12)
Equation (3.10) describes oscillations of the NE effect in
the most general form. It is valid for any type of the disper-
sion ( )p if , .T t
The 2D case: graphene. We start analysis of the
Eq. (3.10) from the pure 2D case when = 0.t In the low-
temperature limit
22 < cT in Eq. (2.12) 1, hence
2 2( ) 1 ( )/6.l l For < 0.02m m and H = 10 T (typi-
cal in graphene experiments) this yields T < 10 K. Since
m m we neglect also the Zeeman splitting, assuming
that = = 0 for NC and = =1/2 for DF. The se-
ries {0} and {1} in Eq. (3.10) in this case can be
summed exactly which gives
3
(2 )
( )
( ) sin 2
2 2
( , ) = .
2 ( )3
cosh cos2
2
D
c
c
c S
H
T e H
H
c S
e H
(3.13)
In the experimental configuration corresponding to the
measurement of the NE effect in graphene, the number of
particles n is fixed.
Quantum oscillations as the tool for study of new functional materials
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 4 361
Hense the chemical potential in the 2D case can be deter-
mined from the equation
2 2 2 2
( )
sin 2
( ) 2
= 2 arctan .
(2 )
( )
e cos 2
2
c
c
c S
S m e H
n
c S
e H
(3.14)
This equation can be inverted for ( ):S
2
2
sin 2
( )
= arctan .
2
e cos 2c
c n
c S c n e H
e H e H
c n
e H
(3.15)
Equation (3.15) yields the dependence ( , ).n H Substitut-
ing it to Eq. (3.13) after some cumbersome algebra one can
find the oscillating part of the Nernst coefficient explicitly:
3
(2 ) 2 ( )
( , ) = sin 2 ,
23
sinh
D
c
c
T H c n
n H
e H
(3.16)
that is a strongly oscillating function. It crosses zero at the
intersections of LL and chemical potential, given by the
condition = kH H defined by (2.3). The field depended
factor ( )H is governed by magnetoresistance and is given
by Eq. (3.11). At 1c where SdH oscillations are small,
( )H can be roughly estimated using the Drude approxi-
mation (3.12). In particular, approaching the limit 1c
and assuming /2 we obtain that ( ) /( )H m c
and the amplitude of NE oscillations is giant in comparison
with the background:
(2 )
0( / ) .D
F c At higher fields
>1,c in the quantum Hall regime, the shape of oscilla-
tions of the NE coefficient is affected by strong variation of
the magnetoresistance and Dingle temperature. This can be
taken into account by substitution of the corresponding
magnetoresistance and Dingle temperature field dependence
into Eqs. (3.10), (3.11).
The profiles of 2D NE oscillation as function of H
and n for DF and NC given by Eq. (3.16) are presented
in Fig. 1. Both our theory for DF and experiment in
graphene [20,21] show a sine-like profile of the signal
whose amplitude slightly decreases with increasing n.
This tendency contradicts to the earlier theoretical predic-
tions of the classical Mott formula [20] that has been de-
rived for a Boltzmann gas of electrons. In contrast, the am-
plitude of NE oscillations increases with increasing n for
the NC in a qualitative agreement with the Mott formula.
Quasi-2D and 3D cases. In order to describe the NE ef-
fect in the general quasi-2D case when 0t the Bessel
function in the Eq. (2.13) should be taken into account.
The sums (3.7) can be reduced to the integrals by means of
the Poisson transformation. Then integration can be done
analytically resulting in
{0}
1/2
2=
2= 1
2 2
1 1 1
=
2 2
4
2 ( )
k
k
c
t
H
(3.17)
Fig. 1. (Color online) The normalized NE oscillations / 0 as
function of the inverse magnetic field H0/H and carriers concen-
tration n for NC and DF. The dependence
1( )H for DF has the
same profile as for NC but shifted by half period. The vertical
lines show the quantization condition (2.3).
V.P. Gusynin, V.M. Loktev, I.A. Luk’yanchuk, S.G. Sharapov, and A.A. Varlamov
362 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 4
and
{1}
3/2
2=
2= 1
2 2
( )1 1
= ,
6 2
4
( )
k
c k
k
c
HT
t
H
(3.18)
where
1 1( ) = ( )
2
k k
c
c S
H i H H
e
.
The NE coefficient is obtained by substitution of the
Eqs. (3.17) and (3.18) to Eq. (3.10). Resonances at
( ) = 2 /( )k ci H t in ( )H appear when the chemical
potential crosses the quantized slices of maximal (minimal)
cross sections of the corrugated cylinder Fermi surface
max(min) = 4 .S S tm
In the wide quasi-2D interval 2< ( ) /ct the behavior
of ( 2 ) ( )q D H close to = kH H can be studied selecting
in (3.17) and (3.18) only the resonant terms. With growth of
t the positions of zeros shift from Im ( ) = 0k H to
Im ( ) = 2 /( ).k cH t The superposition of two (for
maxS and min )S series of resonances leads to the beats in
( )H oscillations.
In the 3D limit 2> ( ) / ,ct {0}Re 1, so that
{0} can be neglected in the denominator of Eq. (3.10). In
the vicinity of = kH H one finds
(3 )
1/2 3/2
( ) 1
( ) = Re .
12 ( ) 2
( )
D
c
k
c
T H
H
t t
i H
(3.19)
We assumed here the constant and neglected Zeeman
splitting, taking , 1 = .k k The resonances in ( )H de-
scribed by Eq. (3.19) have the form of asymmetric spikes
with (3 ) (3 )
max min| | / | | 3.4D D as shown in Fig. 1. In
the Drude approximation the amplitude
(3 )
max 01/2
| | 0.29 ( )
( )
D cF H
t
(3.20)
is giant if
1/2( / ) /( ) >1.F c t
For 2D systems our calculations are valid for magnetic
fields c
–1
where one can neglect the quantum Hall
oscillations of conductivity. At higher fields the approach
of Girvin and Jonson [33], based on the generalized Mott
formula for the thermopower tensor for 2D systems, seems
to be more relevant. In 3D case the range of applicability
of our theory is given by .c F Recently Bergman and
Oganesyan [28] extended the approach of Ref. 33 to calcu-
late the off-diagonal thermoelectric conductivity xy for a
3D system at .c F Although xy constitute only the
part of NE coefficient = ( )/ ,xx xy xy yy H they
reproduce quite well the sawtooth dependence of ( ),H
measured in graphite [30] with the characteristic
1/2( )kH H divergences at resonances.
4. Exact calculation of DOS and capacitance
in chiral multilayer graphene
In order to treat the single-layer, bilayer graphene [34]
and chiral multilayer graphene (neglecting the trigonal
warping effects) with 2N layers [35–38] simultaneous-
ly we start from the Hamiltonian given in a unified form as
2
,
= ( )NH d x x
ˆ0 ( ) 0
( ),
0ˆ( ) 0
N
N
N N
p
a
p
x
(4.1)
where ˆ ˆ ˆ= ,x yp p ip p̂ is the canonical momentum op-
erator, 1 1= ( / ) ,N
N Fa v 6= 3/2 10 m/sF tav is the
Fermi velocity in graphene, 2.46 Åa is the lattice con-
stant, 0 3.16 eV, and 1 0.39 eV are tight-binding
parameters responsible for the in-layer nearest-neighbor
and interlayer hopping, respectively. While for =1N the
Hamiltonian is valid for momenta 1/ ,p a for 2N the
low-energy effective Hamiltonian (4.1) can be utilized for
momenta p up to 1= / .W Fp v The two-component spinor
field carries the valley ( = for the K and K
valleys, respectively) and spin ( = ) indices. For the
ABC-stacked multilayer graphene, the low-energy electron
states are located only on the outermost layers which we
will denote as layers 1 and N in what follows. Further, we
use the standard convention for wave functions:
1
= ( , ) ,T
A BN
whereas
1
= ( , )T
B AN
.
Here 1A and NB correspond to those sublattices in the
outermost layers 1 and N, respectively, which are relevant
for the low-energy dynamics. Obviously, for =1N the
sublattices belong to the same layer. In the Hamiltonian
(4.1) we also included the mass (gap) term which describes
interlayer asymmetry between on-site energy of the
atoms, 1A and 1,B on the first layer and of the at-
oms, NA and ,NB on the outermost layer.
In the presence of magnetic field H perpendicular to
the layers the energies of LL are [39] (see also Refs. 40
and 41)
є = , = 0, 1, , 1n n N (4.2)
and
2 2 2є = , = ( 1) ( 1), ,n n n cNM M n n n N n N
(4.3)
where cN is the cyclotron energy of multilayer ex-
pressed via the Landau scale
2= 2 | | /FL eH cv
36.3 [T] meVH for monolayer graphene and the hop-
Quantum oscillations as the tool for study of new functional materials
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 4 363
ping 1 as follows: 1
1= ( 2 / ) = / ,N N N
cN Na l L
where = / | |l c eH is the magnetic length.
The DOS is written as
2
1
(є) = { (є ) (є )D N N
l
=
2 [ (є ) (є )]}.n n
n N
M M (4.4)
Here we neglected the Zeeman splitting, because it is
smaller than the cyclotron energy (for example,
2 = 3.4 meV [T]).c B To calculate the DOS we follow
Ref. 17 and write
2 2 2 2
2
=
2 | є |
(є) = (є ) 2 (є ) =[ ]n
n N
D N M
l
2 2
2
1
sgn(є) (є )
є
{
d
dl
2 2 2 2
=1
2 [є ( 1) ( 1)] .}cN
n
n n n N
(4.5)
Using the Poisson summation formula
2
=1 =10 0
1
(0) ( ) = ( ) 2Re ( )e ,
2
ikx
n k
F F n dxF x dxF x
(4.6)
we find the sum over the LL
2
2
(є) = sgn (є)
є
d
D
dl
2 2 0
0
=1
sin (2 (є, ))1
(є ) (є, ) ,
2
k
kx NN
x N
k
(4.7)
where 0 (є, )x N is the largest and the only positive root of
the algebraic equation
1 2 2
2 2
=0
є
( ) = .
N
k cN
x k (4.8)
Making the shift = ( 1)/2x z N this equation can be
rewritten as
( 1)/2 2 2
2 2
2 2
=1
є
( ) =
N
k cN
z z k (4.9)
for odd N, and
2/2 2 2
2
2 2
=1
2 1 є
=
2
N
k cN
k
z (4.10)
for even N. Hence for DOS we obtain
2
2
(є) = sgn(є)D
l
( 1)
2 2
0 0
=1
( 1)
(є ) (є, ) sin (2 (є, )) ,
є
k N
k
d
z N kz N
d k
(4.11)
where 0 (є, )z N is the largest root of the above equations.
Note the presence of the factor 1( 1)N in oscillations
which is a reflection of the presence of the Berry phase.
Indeed, we can write
Berry1 1
= ,
2 2 2
N
(4.12)
where the Berry phase Berry = N for graphene with N
layers. Thus our model calculation allows to find the value
of = mod[( 1)/2]N , where mod [ ]x is the shorthand
notation for the fractional part x modulo 1. In particular,
one can see that for =1N we obtain = 0 [10,11] which
allows to distinguish Dirac carriers in monolayer graphene.
Moreover, even the opening of the gap in the
quasiparticle spectrum does not change this value of .
In weak magnetic fields (quasiclassical regime)
Eqs. (4.8), (4.9) both give the largest root
1/
2 2
0 2
є
(є, ) = .
2 | |
N
N
c
z N
eH a
(4.13)
On the other hand, from the dispersion relation in the ab-
sence of a magnetic field,
2 2 2є = ( ) ,N
Na p we find
that the Fermi-surface cross section is
1/
2 2
2
2
( ) = = ,
N
N
S p
a
(4.14)
and the value 0 (є, )z N coincides with the argument
(є)/(2 | |)cS eH in oscillating functions in Eqs. (2.9) and
(2.12). For example, the DOS (4.11) for weak magnetic
fields can be written in the form
*
( 1)
2
=1
(є) (є)
(є) = 1 2 ( 1) cos 2 ,
2 | |
k N
k
m cS
D k
eH
(4.15)
where *(є) = (1/2 )[ (є)/ | є |] .m S It coincides with the
corresponding DOS (є)g given by Eq. (2.9) if we set the
coupling = 0,t = 0 and omit the factor 1/ .d
The knowledge of zero temperature DOS is completely
sufficient to write down the finite T thermodynamic poten-
tial and other thermodynamic quantities. The DOS and its
oscillations can be experimentally found by measuring the
V.P. Gusynin, V.M. Loktev, I.A. Luk’yanchuk, S.G. Sharapov, and A.A. Varlamov
364 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 4
quantum capacitance C [42,43] which is proportional to
the thermally smeared DOS and is given by
2( ) = є (є)[ (є)],FC e d D n 4.16)
where (є) =1/ [exp (є )/ 1]Fn T is the Fermi distribu-
tion. Assuming that the presence of impurities broadens the
LL to the Lorentzian form with a field-independent width
we can write
2 2 2 2 2 2
(є) =
(є ) (є )
N N
D
l
2 2 2 2
=1
1 1
2 .
(є ) (є )n n nM M
(4.17)
Using the representation
2 2
0
= e cos (є )
є
tdt t (4.18)
and the integral
є[ (є)]cos [(є ) ] =F nd n M t
cos [( ) ] ,
sinh ( )
n
tT
M t
tT
(4.19)
we present the capacitance in the form convenient for nu-
merical calculations:
2
2 2
( ) = ( ) ( ){
e
C NI NI
l
=1
2 [ ( ) ( )] ,}n n
n
I M I M
where the function
0
( ) = e cos ( ) =
sinh ( )
t Tt
I x dt xt
Tt
1 1
Re
2 2 2
ix
T T
(4.20)
can be expressed in terms of digamma function ( ).x In
Fig. 2 we show the dependence ( )C for = 1, 2, 3N and
for a particular case of zero gap = 0. Comparing the
behavior of the DOS as the number of layers N increases
Fig. 2. (Color online) Capacitance (per unit area in units of
C0 = 2.4 F/cm
2
) of multilayer graphene as a function of the
chemical potential for H = 3 T, = 1 meV, = 0, T = 5 K
(solid (blue) curve) and T = 10 K (dashed (red) curve). Top panel:
monolayer graphene, N = 1; middle panel: bilayer graphene, N = 2;
and bottom panel: trilayer graphene, N = 3.
Quantum oscillations as the tool for study of new functional materials
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 4 365
one can see that the contribution of the lower LL enhances
because the levels are getting denser near zero energy. Fur-
thermore, the DOS envelop function itself reflects the be-
havior of the zero field DOS given by
2 2 (1 )/
2 2
| є | (є )
(є) = (| є | ).
N N
N
D
N a
(4.21)
The strength of the peak corresponding to the zero en-
ergy LL also increases as the number of layers grows
which reflects the degeneracy of this level (see Eq. (4.4)).
5. Conclusion
We applied the LK formalizm to consider the dimen-
sional 3D–2D crossover of quantum oscillations in layered
systems with an arbitrary electronic spectrum and derived
the general formula for oscillating thermodynamic poten-
tial in this case. Such an approach permits to consider the
quantum oscillations of various thermodynamic parameters
in a unify way. In particularly, we presented the explicit
crossover formula for dHvA oscillation of magnetization
and specified it for the 2D and 3D cases. Then, we consid-
ered quantum oscillations of the NE coefficient and ob-
tained an analytical expression for 2D case with an arbi-
trary electron dispersion. This expression was used to
explain the recent experimental results in graphene. In ad-
dition it can be used to distinguish between NC and DF in
2D sysetms. We showed that the giant oscillations of the
NE coefficient, predicted and observed in a 2D case
(graphene), decrease significantly as the spectrum acquires
a 3D character (graphite). We described analytically also
the shape of NE oscillations and predicted that in all cases
the NE oscillations are proportional to the temperature
derivative of the dHvA oscillations.
In the model of chiral multilayer graphene we studied
the magnetic oscillations of the density of states and capac-
itance for different number of the carbon layers and deter-
mined the Berry phase which remains undetermined in the
LK approach.
Acknowledgments
This work is supported partially by the European FP7
program, Grant No. SIMTECH 246937. The work of
V.P.G., V.M.L. and S.G.Sh. is supported also by the grant
STCU #5716-2 “Development of Graphene Technologies
and Investigation of Graphene-based Nanostructures for
Nanoelectronics and Optoelectronics”. V.P.G. and S.G.Sh.
acknowledge the support of the joint Ukrainian-Russian
SFFR-RFBR grant No. F53.2/028. The work of IL was
done in frame of International Training Network FP7-ITN-
NOTEDEV.
1. W.J. de Haas and P.M. van Alphen, Proc. Netherlands Roy.
Acad. Sci. 33, 680 (1930).
2. L.D. Landau, Z. Phys. 64, 629 (1930).
3. R. Peierls, Z. Phys. 80, 763 (1933); Z. Phys. 81, 186 (1933).
4. D. Shoenberg, Proc. Roy. Soc. A 170, 341 (1939).
5. L.D. Landau, Proc. Roy. Soc. A 170, 341 (1939) (Appendix
to article of D. Shoenberg [4]).
6. I.M. Lifshitz and A.M. Kosevich, Dokl. Akad. Nauk SSSR
96, 963 (1954); Zh. Eksp. Teor. Fiz. 29, 730 (1955).
7. I.M. Lifshitz, M.Ya. Azbel’, and M.I. Kaganov, Electron
Theory of Metals, Consultants Bureau, New York (1973).
8. D. Shoenberg, Magnetic Oscillations in Metals, Cambridge
University Press (1984).
9. L. Onsager, Philos. Mag. 43, 1006 (1952).
10. L.A. Falkovsky, Zh. Eksp. Teor. Fiz. 49, 609 (1965) [Sov.
Phys. JETP 22, 609 (1966)].
11. G.P. Mikitik and Yu.V. Sharlai, Phys. Rev. Lett. 82, 2147
(1999).
12. I.A. Luk’yanchuk and Y. Kopelevich, Phys. Rev. Lett. 93,
166402 (2004).
13. I.A. Luk’yanchuk and Y. Kopelevich, Phys. Rev. Lett. 97,
256801 (2006).
14. K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I.
Katsnelson, I.V. Grigorieva, S.V. Dubonos, and A.A. Firsov,
Nature 438, 197 (2005).
15. Y. Zhang, Y.-W. Tan, H.L. Stormer, and Philip Kim, Nature
438, 201 (2005).
16. T. Champel and V.P. Mineev, Philos. Mag. B 81, 55 (2001).
17. S.G. Sharapov, V.P. Gusynin, and H. Beck, Phys. Rev. B 69,
075104 (2004).
18. I.A. Luk’yanchuk, Fiz. Nizk. Temp. 37, 56 (2011) [Low
Temp. Phys. 37, 45 (2011)].
19. I.A. Luk’yanchuk, A.A. Varlamov, and A.V. Kavokin, Phys.
Rev. Lett. 107, 016601 (2011).
20. Y.M. Zuev, W. Chang, and P. Kim, Phys. Rev. Lett. 102,
096807 (2009).
21. J.G. Checkelsky and N.P. Ong, Phys. Rev. B 80, 081413(R)
(2009).
22. A. Von Ettingshausen and W. Nernst, Wied. Ann. 29, 343
(1886).
23. K. Behnia, M.A. Méasson, and Y. Kopelevich, Phys. Rev.
Lett. 98, 076603 (2007).
24. E.H. Sondheimer, Proc. R. Soc. London 193, 484 (1948).
25. C.J. Bergeron, C.G. Grenier, and J.M. Reynolds, Phys. Rev.
Lett. 2, 40 (1959).
26. Yu.N. Obraztsov, Sov. Phys. Solid State 6, 331 (1964).
27. V.P. Gusynin and S.G. Sharapov, Phys. Rev. B 73, 245411
(2006).
28. D.L. Bergman and V. Oganesyan, Phys. Rev. Lett. 104,
066601 (2010).
29. K. Behnia, M.A. Méasson, and Y. Kopelevich, Phys. Rev.
Lett. 98, 166602 (2007).
30. Z. Zhu, H. Yang, B. Fauque, Y. Kopelevich, and K. Behnia,
Nature Physics 6, 26 (2010).
31. M.N. Serbyn, M.A. Skvortsov, A.A. Varlamov, and V. Gali-
tski, Phys. Rev. Lett. 102, 067001 (2009).
V.P. Gusynin, V.M. Loktev, I.A. Luk’yanchuk, S.G. Sharapov, and A.A. Varlamov
366 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 4
32. A.A. Varlamov and A.V. Kavokin, Europhys. Lett. 86, 47007
(2009).
33. M. Jonson and S. Girvin, Phys. Rev. B 29, 1939 (1984).
34. J. Cserti, Phys. Rev. B 75, 033405 (2007).
35. H. Min and A.H. MacDonald, Phys. Rev. B 77, 155416
(2008).
36. Y. Barlas, K. Yang, and A.H. MacDonald, Nanotechnology
23, 052001 (2012).
37. R. Côté, M. Rondeanu, A.-M. Gagnon, and Yafis Barlas,
Phys. Rev. B 86, 125422 (2012).
38. M. Nakamura and L. Hirasawa, Phys. Rev. B 77, 045429
(2008).
39. M. Koshino and E. McCann, Phys. Rev. B 80, 165409
(2009).
40. S. Yuan, Rafael Roldán, and M.I. Katsnelson, Phys. Rev. B
84, 125455 (2011).
41. R. Côté and M. Barrette, Preprint arXiv:1310.7551.
42. L.A. Ponomarenko, R. Yang, R.V. Gorbachev, P. Blake, A.S.
Mayorov, K.S. Novoselov, M.I. Katsnelson, and A.K. Geim,
Phys. Rev. Lett. 105, 136801 (2010).
43. G.L. Yu, R. Jalil, B. Belle, A.S. Mayorov, P. Blake, F. Schedin,
S.V. Morozov, L.A. Ponomarenko, F. Chiappini, S. Wiedmann,
U. Zeitler, M.I. Katsnelson, A.K. Geim, K.S. Novoselov, and
D.C. Elias, Proc. Natl. Acad. Sci. USA 110, 3281 (2013).
|