On the possible reason for superconductivity strengthening in multiwall carbon nanotubes
Basing on the structural peculiarity of two-layer graphene which consists of translational
 and energetical nonequivalency of carbon atoms from different sublattices, it is shown that the
 density of long-wave electronic states at the Fermi level is finite (in contrast to the monolay...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
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| Cite this: | On the possible reason for superconductivity
 strengthening in multiwall carbon nanotubes / Yu.B. Gaididei, V.M. Loktev // Физика низких температур. — 2006. — Т. 32, № 11. — С. 1458–1462. — Бібліогр.: 22 назв. — англ. |
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| citation_txt | On the possible reason for superconductivity
 strengthening in multiwall carbon nanotubes / Yu.B. Gaididei, V.M. Loktev // Физика низких температур. — 2006. — Т. 32, № 11. — С. 1458–1462. — Бібліогр.: 22 назв. — англ. |
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| description | Basing on the structural peculiarity of two-layer graphene which consists of translational
and energetical nonequivalency of carbon atoms from different sublattices, it is shown that the
density of long-wave electronic states at the Fermi level is finite (in contrast to the monolayer
graphene). It is suggested that the same may be the reason why the critical temperature of
superconducting transition in multiwall nanotubes more than ten times higher than in single-wall
nanotubes.
|
| first_indexed | 2025-12-07T16:58:05Z |
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| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, No. 11, p. 1458–1462
On the possible reason for superconductivity
strengthening in multiwall carbon nanotubes
Yu.B. Gaididei and V.M. Loktev
Bogolyubov Institute for Theoretical Physics, 14 B Metrologichna Str., Kiev 01413, Ukraine
E-mail: vloktev@bitp.kiev.ua
Received April 17, 2006
Basing on the structural peculiarity of two-layer graphene which consists of translational
and energetical nonequivalency of carbon atoms from different sublattices, it is shown that the
density of long-wave electronic states at the Fermi level is finite (in contrast to the monolayer
graphene). It is suggested that the same may be the reason why the critical temperature of
superconducting transition in multiwall nanotubes more than ten times higher than in single-wall
nanotubes.
PACS: 71.23.–k, 71.21.Ac, 81.05.Uw
Keywords: graphene, quantum Hall effect, unconventional Hall effect.
1. Introduction
Since the discovery of fullerene [1] (see also review
papers [2–4]) there is an increasing interest in the
investigation of fullerene-like structures which be-
sides carbon nanotubes (CNT) [5] include also single-
and multilayer graphene [6]. Carbon nanotubes are
very perspective both from fundamental and applied
view points. CNT may be used as the elements of
different nanoelectronic devices, constant-force nano-
springs [7], containers to hold in storage of light gases
(in particular hydrogen), pressure transducers and so
on. On the other hand, CNT are known as model
systems for the study of one-dimensional electronic
transport. Depending on their diameter and helicity,
carbon nanotubes are either semiconducting or me-
tallic [4]. It is well known that the usual Fermi liquid
which is described by the Landau–Silin theory is
expected to be unstable in one-dimensional (1D)
metals with the formation of a correlated Tomona-
ga–Luttinger liquid state where the electron—elect-
ron repulsive interaction prevents the appearance of
superconducting state to a greater extent than in con-
ventional 3D metals.
Investigations of the magnetic and transport pro-
perties of ropes of single-walled small-diameter carbon
nanotubes (SWNT) [8] and SWNT embedded in a
zeolite matrix [9] revealed that a few angstrom tubes
exhibit superconducting behavior manifest as an an-
isotropic Meissner effect, with a superconducting gap
and fluctuation supercurrent. It turned out that the
superconducting transition in SWNT occurs at very
low temperature, less than 0.55 K [8]. It is quite
natural to connect such a low critical temperature
with the low-dimensional character of CNT*, with
strong quantum and thermal fluctuations and with
proximity to Peierls instabilities which drive the
system to an insulating state. Therefore the report
[10] that multiwalled carbon nanotubes (with up to 9
shells) exhibit superconductivity with an unexpect-
ably high transition temperature Tc � 12 K, which is
approximately 30 times greater than Tc reported for
SWNT [8] has attracted considerable attention. The
value of the superconducting gap � � 115. meV which
© Yu.B. Gaididei and V.M. Loktev, 2006
* It is worth to emphasize that carbon nanotubes can not be considered as genuine one-dimensional objects. In 1D systems
the processes of electron—electron, electron—phonon and electron—impurity scattering involve only one, «longi-
tudinal» component of the electron wave vector while in nanotubes these processes include also the «transversal» or
orbital component of the quasimomentum.
is in excellent agreement with the BCS canonical ratio
2 3 5�/k TB c � . , the observed behavior of the critical
current as a function of the temperature and external
magnetic field is also in good qualitative agreement
with the Ginzburg–Landau critical current behavior.
It can be therefore said with confidence that the
observed superconductivity in multiwalled carbon na-
notubes is of the BCS type and that the multishell
structure of the nanotubes plays a crucial role.
The aim of this work is to show that the big
difference of the critical temperature of single- and
multiwalled nanotubes may be due to a qualitatively
different behavior of the density of states of low-
energy charge carriers. As a model we consider mono-
and bi-layer graphenes whose structure is similar to
the structure of graphit sheets rolled-up in nanotubes.
It is a great honour and pleasure for us to devote
this work to the 100 year jubilee of outstanding
physicist-experimentalist and marvellous person, An-
tonina Fedorovna Prikhot’ko, who did so much for
our formation as scientists. It is worth to note also
that being a bright and leading figure in the low tem-
perature physics, she always supported new lines of
inquiry in this area and was one of devotees of the
quest of new superconductors with unusual and un-
expectable features.
2. Hamiltonian and electron spectrum
We employ the tight-binding model to describe the
band structure of single- and bi-layer graphenes.
Basing on the well developed theory of molecular
crystals with several molecules per unit cell (see, e.g.,
[11]). the electronic Hamiltonian of bi-layer graphene
can be repsented in the form
H a a b b� � �� ����
� � � �
�
( )n n n n
n
� � �� ��1
2
t a a b bn m n m n m
n m
� � � � � �
� �
,
,
( )
� �� �
�
�1
2
t a b b a| |
( )
( )n n n n
n
� � � �
� � �
, (1)
where �� is the on-site energy of �-electron at the site
n� (the vector n gives the position of the elementary
cell, the index �( , )� 1 2 characterizes the sublattices);
the matrix element tn m� �
describes an intra-layer
electron hopping (tunneling) between the sites n�
and m�, and t| | characterizes an inter-layer electron
transfer; the Fermi-operators an�
� and bn�
� describe a
creation of an electron at the site n� of the first and
second layer, respectively (the index of the electron
spin is omitted for brevity). It is worth noting that
the last term of the Hamiltonian (1) is chosen in the
form which takes into account that the sheets in the
bi-layer graphene are shifted with respect to each
other by the vector ( )a a1 2 2� / , where a j are the
primitive vectors of graphene sheet [12]. In con-
sequence of such an unusual geometry the atoms in
each sheet become energetically (as well as trans-
lationally) nonequivalent*.
By applying the Fourier transform
a
N
a i
� �
( )k
n
n
kn� �1
e ,
b
N
b i
� �
( )k
n
n
kn� �1
e , (2)
where k is the wave vector and N is the number of
cells in the sheet, the Hamiltonian (1) can be pre-
sented in the form
H t a a b b� � �� �� { ( )[ ( ) ( ) ( ) ( )]1 1 2 1 2k k k k k
k
h.c.} +
� � �� �
�
�t a a b b2
12
( )[ ( ) ( ) ( ) ( )]
, ,
k k k k k
k
� � � �
�
� �� ��t a b b a| | [ ( ) ( ) ( ) ( )]1 2 2 1k k k k
k
, (3)
where
t t i i
1 1 1 1 1 1 2( ) ( ); ( )k k k ka ka� � � �� � e e ; (4)
t t2 2 2( ) ( )k k� � ;
�2 1 2 1 2( ) cos cos cos ( )k ka ka k a a� � �
(5)
are the Fourier-components of so-called resonance
interactions which form the dispersion of electro-
nic bands. The functions t1( )k and t2( )k given by
Eqs. (4) and (5) represent the intra- and inter-
sublattice electron dispersion, respectively. The latter
leads to a splitting which in the theory of molecular
excitons is called the Davydov splitting (see, e.g.,
[11]). The parameters t1 and t2 in Eqs. (4) and (5)
are the hopping integrals between nearest and next
nearest neighbors, respectively; a1 3 0� ( , ), a2 �
� ( , )3 2 3 2/ / are the primitive vectors of the rhombic
lattice with the distance between the nearest neigh-
On the possible reason for superconductivity strengthening in multiwall carbon nanotubes
Fizika Nizkikh Temperatur, 2006, v. 32, No. 11 1459
* Note that the same phenomenon takes place in multi-wall carbon nanotubes consisting of several rolled-up graphit sheets
in the cylindrical form. Here the atoms which belong to different cylindrical surfaces are energetically nonequivalent.
bors equal to unity. By introducing the «even» and
«odd» linear combinations of electron operators
1 1 2
1
2
( ) [ ( ) ( )]k k k� �a b ;
�1 1 2
1
2
( ) [ ( ) ( )]k k k� a b ;
2 2 1
1
2
( ) [ ( ) ( )]k k k� �a b ;
�2 2 1
1
2
( ) [ ( ) ( )]k k k� a b
(6)
and applying the unitary transformations
�� �� �
�
( ) ( ) ( )
,
k k k�
�
�u
12
;
�
� �� �
�
( ) ( ) ( )
,
k k k�
�
�v
12
;
(7)
with
| ( )|
( )
| ( )| | ( )|
( )
( )
v
t
1
2
2
1
22
�
�
�
�
�
�
�
k
k
k k
�
�
;
| ( )|
| ( )|
( ) | ( )| | ( )|( ) ( )
v
t
t
2
2 1
2
2
1
22
�
�
�
�
�
�
�
k
k
k k k� �
,
(8)
one can diagonalize the Hamiltonian (3) and obtain
the electron eigenenergies in the form
� ��
� � � �
�
�
�
�
�
�
�
� �( ) | | | |
( ) ( ) | ( )|k k kt
t t
t2
2
1
2
2 2
, (9)
� ��
� � � � �
�
�
�
�
�
�
�
� �( ) | | | |
( ) ( ) | ( )|k k kt
t t
t2
2
1
2
2 2
. (10)
In the mono-layer case which can be obtained from
Eqs. (9) and (10) by neglecting the interlayer hopp-
ing integral: t| | � 0, the electron spectrum consists of
two symmetric branches which in fact, are the Da-
vydov bands. They intersect and have the same energy
� ��� 3 22t / at six points of the Brillouin zone
which can be obtained by rotations of the vectors
K12 2 9 3 3, ( )( , )� ��/ by � �/3. By virtue of the fact
that the number of states is the same below and above
this energy, in graphene there exists a rather rare for
solid state physics situation (first apparently noticed
in Ref. 13) when the two-dimensional metal has not a
Fermi line (as usually in two-dimensional metals) but
a set of Fermi points. As a result of such a pecularity
of the spectrum, in the vicinity of Fermi points charge
carriers are characterized by a massless Dirac-like
dispersion law which in particularly manifests itself in
an unconventional Hall effect [14,15].
In the bi-layer graphene structure the electron
spectrum consists of four bands, two of them touch at
the points K j ( ... )j � 1 6 (they do not intersect as in
the mono-layer case) and two others are nondege-
nerate for all wave vectors k. Assuming for example
that t2 0� , and t| | � 0, it is easy to see that the band
with the energy ��
�( )( )k is the conduction band of
bi-layer graphene.
Note also that the expressions similar to given by
Eqs. (9) and (10) (but without intra-sublattice hopp-
ing) was obtained in Ref. 16 (see also [17]). It follows
from them, in particularly that the Hall effect ac-
quires some new features. However their discussion is
beyond the scope of the paper.
3. Superconducting gap
Let us now consider the superconducting phase
transition in mono- and bi-layer graphene. To this end
we augment the Hamiltonian (3) with the operator of
attractive electron-electron interaction. There is a
strong belief (see, e.g., [18]) that phonon exchange
holds responsible for attractive electron-electron in-
teraction in metallic carbon nanotubes. The derivation
of the operator of the phonon mediatated electron-
electron interaction is well known (see, e.g., [19,20]).
We take this operator in the form
H
N
Vint
,
( , ) ( ) ( ) ( ) ( )�
�
�
�
�
�1
k q k k q q
k q
, (11)
where the parameter V( , )k q characterizes the strength
of the electron—electron interaction;
�
� ( )k (
�( )k )
is the creation (annihilation) operator of a carrier
with the wave vector k and the spin � � � �, . By using
the standart procedure [19,20] it is easy to obtain
that the order parameter
�( ) ( , ) ( ) ( )k k q q q
q
� � ��
�
�N V1
(the gap in the electron spectrum) is determined by
the equation
�
�
�
( ) ( , )
( )
[ ( ) ] | ( )|( )
k k q
q
q qq
�
�
�
�
�
1
2 2 2N
V
F� �
,
(12)
where the symbol �� � �� denotes a thermodynamic
mean value and �F is the Fermi energy which in our
case is doping controlled.
In the simplest approximation when V V( , )k q �
the gap is k independent � �( )k � , and Eq. (12) takes
the form
1460 Fizika Nizkikh Temperatur, 2006, v. 32, No. 11
Yu.B. Gaididei and V.M. Loktev
1
8
1 1
2 2 2� � �
d
V
F
k
k
�
�
�
[ ( ) ] | |cond �
, (13)
where
�cond( ) | | | |k k k� � �
�
�
�
�
� � �
�
�
�
�
�
9
8
1
2
1
2
3
22
2
2
1
2
t t t t (14)
is the energy of charge carriers in the vicinity of the
node points K j [12,17]. By introducing the density of
quasiparticle states � �( ) (see Appendix for details)
Eq. (14) can be transformed to the form
d
V
FF
�
� �
� �
( )
( ) | | �
�
�
� 2 2
2
�
,
which for a given � �( ) (see Eq. (A.2)) gives the
expression
� ��
�
�
!
"!
#
$
!
%!
2
9 1
2
D
F
t
V t
exp
( )| |
�
�
, (15)
for the superconducting gap in the one-electron spect-
rum. Here �D is the Debye frequency.
Comparing the superconducting gap � �& bi in the
bi-layer graphene which is given by Eq. (15) with the
superconducting gap � �& mono in the mono-layer
graphene which can be also obtained from Eq. (15) by
putting t � 0 one can see that the ratio
�
�
bi
mono
�
�
�
!
"!
#
$
!
%!
exp
9
1
1
2�
�
�
�
t
V F
, (16)
where � �& t/ in the low-doping regime may be ex-
ponentially big. This result is a direct consequence of
the qualitative difference between the spectra of
charge carriers in single- and bi-layer graphene.
While in the mono-layer graphene the spectrum is
massless and the density of of states vanishes when
the concentration of carriers decreases (� ' 0) in the
bilayer graphene the carriers have a finite effective
mass and the density of states in the limit (� ' 0) is
finite.
4. Summary
We have shown that the interlayer carrier transfer
in graphene may be of crucial importance in creation
of superconducting condensate. It is interesting to
notice that in the same way the low-energy beha-
vior of the density of states plays a pivotal role
in electron-hole pairing and forming the exciton di-
electric phase graphene. In the mono-layer graphene
such a phase can form only in the presence of strong
magnetic field which effectively reduces to 1D the
charge carrier motion and in this way enhances Cou-
lomb attraction (magnetic catalysis [21]). In the
bi-layer graphene the presence of interlayer hopping
makes inessential applying the magnetic field.
In our opinion, the strengthening of supercon-
ductivity in multiwalled nanotubes may be also due to
the interlayer electron hopping. The charge carriers in
multiwalled nanotubes have a finite effective mass
and as a result the density of states is finite as � �' F .
It follows also from our model that the critical
temperature in a mono-layer graphene (singlewalled
nanotube) increases when the doping increases. More-
over the critical temperatures of mono- and bi-layer
graphenes (as well as the critical temperatures of
single- and multiwalled CNT) equalize when �F t�� | | .
The above developed theory does not take into
account the Coulomb interaction between electrons. It
is well known, however, that this interaction it de-
creases the critical temperature of conventional 3D
superconductors. The decisive factor here is a screen-
ing of Coulomb interaction in metals. In the case
under the consideration we meet with a qualitatively
new situation when not only the low-dimensional
character of charge transport (quasi 1D in CNT and
2D in graphene) is important but the size and the
shape of the objects (radius of CNT, relative shift of
sheets in graphene and so on) as well. Quite recent
studies of the screening of Coulomb interaction in
non-polar systems [22] showed that in low dimensions
and small finite-size systems this screening deviates
strongly from conventionally assumed. For example,
in one dimension at small and intermediate distances
the Coulomb interaction is anti-screened thereby
strongly reducing the gradient of the Coulomb in-
teraction. It is also shown in [22] that in finite-size
systems due to weak dependence of the effective
Coulomb interaction on the distance between the
electrons the correlation effects are reduced and the
mean-field approach works well because the corre-
lation effects are reduced and the mean-field approach
works well. Note also that increasing the number of
walls in CNT (number of layers in graphene) is to
some extent equivalent to increasing of their di-
mensionality. So the superconducting effects are en-
hanced in these systems.
Appendix A
The aim of this Appendix is to calculate the density
of quasiparticle states
� �
�
( � �( ) [ ( )]� �
1
4 2
dk kcond (A.1)
On the possible reason for superconductivity strengthening in multiwall carbon nanotubes
Fizika Nizkikh Temperatur, 2006, v. 32, No. 11 1461
for the case of bi-layer graphene. Here �cond( )k is
given by Eq. (14) and the two-dimensional character
of the system is taken into account. From Eq. (A.1)
we get
� �
�
� � �( )
max
� � � �
�
�
�
�
�
�
�
1
4 2
9
8 2
3
2
0
2 2
2
1
2
dk
t t
k
t t
k 2
2k
�
�
�
�
�
�
�
�
,
where the maximal wave vector kmax is determin-
ed from the condition that the total number of states
in the conduction band is equal to the number
of elementary cells N. By using the new variable
k x t / / t /� [ ( )] ( )| |
2
1
22 3 2 , we obtain
� �
�
(( ) [( )( )]
||
max
� � ��
4
9 2
2
t
dxx x x x x
t /
x
,
where
x / t /t � �( )[ ( )1 2 2 1
2
2
� � � �( ) ( )( ) ]| | | |
4 8 21
4
2
2
1
2
2
2t /t t /t t / t�
are the roots of the equation �cond( )k � 0 and the
notation x t / tmax | | max( )� �2 2
1
2 2k is introduced. As
a result we get
� �
� �
( )
( / ) ( / )| |
�
� �
)
*
+
+
+
,
-
.
.
.
2
9
1
1
1 2 42 2 1
2
2 1
2t t t t t t
.
(A.2)
From Eq. (A.2) we obtain that for the mono-layer
graphene (when the interlayer hopping is absent,
t| | )� 0 the density of states takes the form
� �
� �
�
1
2 2 1
2
0 1
2
2
9
1
1
1 2
L
t t /t t
( )
( )
�
�
)
*
+
+
,
-
.
.
�
� .(A.3)
In the two-layer case (when one can neglect the
intra-sublattice hopping t2) we have the following
expression for the density of states
� �
�
�
2
1
2 0
1
2
4
9
2L
t
t
t
t
( ) ( )|| |
| |
� � /� . (A.4)
Comparing Eqs. (A.3) and (A.4), we see that the
ratio
� � � � �
1 2
0
L L/ t( ) ( )| / | |� /
is small as long as � �0 ��F t| | .
Similar enhancement of density of states one can
expect, as it was mentioned above, under the transi-
tion from singlewalled to multiwalled nanotubes.
Indeed, it is easy to see that due to a massless
character of the carrier dispersion in the singlewalled
nanotube the density of states � �NT
SW /t( ) / 1 1 while
in the multiwalled nanotube the carriers have a finite
effective mass and the density of states
� � �NT
MW /( ) / 1 , fastly growing for small �.
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1462 Fizika Nizkikh Temperatur, 2006, v. 32, No. 11
Yu.B. Gaididei and V.M. Loktev
|
| id | nasplib_isofts_kiev_ua-123456789-120896 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T16:58:05Z |
| publishDate | 2006 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Gaididei, Yu.B. Loktev, V.M. 2017-06-13T09:24:08Z 2017-06-13T09:24:08Z 2006 On the possible reason for superconductivity
 strengthening in multiwall carbon nanotubes / Yu.B. Gaididei, V.M. Loktev // Физика низких температур. — 2006. — Т. 32, № 11. — С. 1458–1462. — Бібліогр.: 22 назв. — англ. 0132-6414 PACS: 71.23.–k, 71.21.Ac, 81.05.Uw https://nasplib.isofts.kiev.ua/handle/123456789/120896 Basing on the structural peculiarity of two-layer graphene which consists of translational
 and energetical nonequivalency of carbon atoms from different sublattices, it is shown that the
 density of long-wave electronic states at the Fermi level is finite (in contrast to the monolayer
 graphene). It is suggested that the same may be the reason why the critical temperature of
 superconducting transition in multiwall nanotubes more than ten times higher than in single-wall
 nanotubes. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур High-Temperature Superconductivity On the possible reason for superconductivity strengthening in multiwall carbon nanotubes Article published earlier |
| spellingShingle | On the possible reason for superconductivity strengthening in multiwall carbon nanotubes Gaididei, Yu.B. Loktev, V.M. High-Temperature Superconductivity |
| title | On the possible reason for superconductivity strengthening in multiwall carbon nanotubes |
| title_full | On the possible reason for superconductivity strengthening in multiwall carbon nanotubes |
| title_fullStr | On the possible reason for superconductivity strengthening in multiwall carbon nanotubes |
| title_full_unstemmed | On the possible reason for superconductivity strengthening in multiwall carbon nanotubes |
| title_short | On the possible reason for superconductivity strengthening in multiwall carbon nanotubes |
| title_sort | on the possible reason for superconductivity strengthening in multiwall carbon nanotubes |
| topic | High-Temperature Superconductivity |
| topic_facet | High-Temperature Superconductivity |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120896 |
| work_keys_str_mv | AT gaidideiyub onthepossiblereasonforsuperconductivitystrengtheninginmultiwallcarbonnanotubes AT loktevvm onthepossiblereasonforsuperconductivitystrengtheninginmultiwallcarbonnanotubes |