Effect of next-to-nearest neighbor hopping on electronic properties of graphene
In the tight-binding approximation, we take into account the next-to-nearest neighbor hopping in graphene that leads to nonrelativistic-like corrections in its low energy spectrum. The electronic density of states in a magnetic field is found and the fan diagram is plotted, which interpolates betw...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2008
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1175602017-05-25T03:03:32Z Effect of next-to-nearest neighbor hopping on electronic properties of graphene Suprunenko, Y.F. Gorbar, E.V. Sharapov, S.G. Loktev, V.M. Graphene and graphite multilayers In the tight-binding approximation, we take into account the next-to-nearest neighbor hopping in graphene that leads to nonrelativistic-like corrections in its low energy spectrum. The electronic density of states in a magnetic field is found and the fan diagram is plotted, which interpolates between those for the relativistic and nonrelativistic limiting cases. It is shown that the Berry phase for the system under consideration coincides exactly with its value for the relativistic system. 2008 Article Effect of next-to-nearest neighbor hopping on electronic properties of graphene / Y.F. Suprunenko, E.V. Gorbar, S.G. Sharapov, V.M. Loktev // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1033-1039. — Бібліогр.: 21 назв. — англ. 0132-6414 PACS: 71.70–d;81.05.Uw http://dspace.nbuv.gov.ua/handle/123456789/117560 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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Graphene and graphite multilayers Graphene and graphite multilayers |
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Graphene and graphite multilayers Graphene and graphite multilayers Suprunenko, Y.F. Gorbar, E.V. Sharapov, S.G. Loktev, V.M. Effect of next-to-nearest neighbor hopping on electronic properties of graphene Физика низких температур |
| description |
In the tight-binding approximation, we take into account the next-to-nearest neighbor hopping in
graphene that leads to nonrelativistic-like corrections in its low energy spectrum. The electronic density of
states in a magnetic field is found and the fan diagram is plotted, which interpolates between those for the
relativistic and nonrelativistic limiting cases. It is shown that the Berry phase for the system under consideration
coincides exactly with its value for the relativistic system. |
| format |
Article |
| author |
Suprunenko, Y.F. Gorbar, E.V. Sharapov, S.G. Loktev, V.M. |
| author_facet |
Suprunenko, Y.F. Gorbar, E.V. Sharapov, S.G. Loktev, V.M. |
| author_sort |
Suprunenko, Y.F. |
| title |
Effect of next-to-nearest neighbor hopping on electronic properties of graphene |
| title_short |
Effect of next-to-nearest neighbor hopping on electronic properties of graphene |
| title_full |
Effect of next-to-nearest neighbor hopping on electronic properties of graphene |
| title_fullStr |
Effect of next-to-nearest neighbor hopping on electronic properties of graphene |
| title_full_unstemmed |
Effect of next-to-nearest neighbor hopping on electronic properties of graphene |
| title_sort |
effect of next-to-nearest neighbor hopping on electronic properties of graphene |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2008 |
| topic_facet |
Graphene and graphite multilayers |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117560 |
| citation_txt |
Effect of next-to-nearest neighbor hopping on electronic properties of graphene / Y.F. Suprunenko, E.V. Gorbar, S.G. Sharapov, V.M. Loktev // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1033-1039. — Бібліогр.: 21 назв. — англ. |
| series |
Физика низких температур |
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2025-07-08T12:28:04Z |
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| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10, p. 1033–1039
Effect of next-to-nearest neighbor hopping on electronic
properties of graphene
Y.F. Suprunenko1,2, E.V. Gorbar1, S.G. Sharapov3, and V.M. Loktev1
1
Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine
14-b Metrolohichna Str., Kiev 03680, Ukraine
E-mail:vloktev@bitp.kiev.ua
2
Physics Department, Lancaster University, Lancaster, LA1 4YB, UK
3
Physics Department, Western Illinois University, Macomb, Illinois 61455, USA
E-mail:s-sharapov@wiu.edu
Received April 2, 2008
In the tight-binding approximation, we take into account the next-to-nearest neighbor hopping in
graphene that leads to nonrelativistic-like corrections in its low energy spectrum. The electronic density of
states in a magnetic field is found and the fan diagram is plotted, which interpolates between those for the
relativistic and nonrelativistic limiting cases. It is shown that the Berry phase for the system under consider-
ation coincides exactly with its value for the relativistic system.
PACS: 71.70–d Level splitting and interactions;
81.05.Uw Carbon, diamond, graphite.
Keywords: graphene, electron density of states, magnetic field, Berry phase.
Introduction
Since the discovery of graphene [1] and especially the
experimental observation [2,3] and theoretical prediction
[4–7] of an anomalous quantization in the quantum Hall
effect (QHE), studies of graphene present an extremely
active and even separate branch of research in the modern
condensed matter physics.
Graphene is a single-atom thick two-dimensional
plane of graphite. The honeycomb lattice of graphene
(see Fig. 1) can be considered as composed of two trian-
gular sublattices A and B. In 1947, Wallace [8] in the
tight-binding approximation considered graphene as a
building block of graphite taking into account only the
nearest neighbor �-electron hopping. According to Fig. 1,
the vectors �AB describe this hopping between two
sublattices of graphene. Wallace showed that graphene’s
2D nature and honeycomb atomic structure cause elec-
trons to move as if they have no mass (they are described
by the Dirac equation [9]). Indeed, at low energy the band
structure of graphene is formed from the �-electron car-
bon orbitals and consists of a valence (full) and conduc-
tion (empty) bands both conical in shape with vertex
meeting at a point called a Dirac point. There are two non-
equivalent pairs of such cones. The low energy spectrum
of the �-electrons is given by a relativistic-like relation
E F� � �v | |k for massless particles, where the Fermi ve-
locity vF is approximately 300 times smaller than the
speed of light c, and k is quasiparticle wavevector.
The experimental proof of the existence of Dirac fer-
mions in graphene came from the observation [2,3] of the
unconventional QHE. The reason for the occurrence of
odd integers is connected with a quantum-mechanical ef-
fect called the Berry phase [10], which equals � for
© Y.F. Suprunenko, E.V. Gorbar, S.G. Sharapov, and V.M. Loktev, 2008
�AA
�AB
A
B
Fig. 1. Graphene’s honeycomb lattice and vectors of the near-
est and next-to-nearest neighbor hoppings.
graphene. This value of the Berry phase was obtained in
the original paper Ref. 10 for a toy model. This model can
be derived from the tight-binding consideration of gra-
phene taking into account the nearest neighbor hopping
only. For a recent discussion of the Berry phase in gra-
phene see [11].
This hopping is the leading term contribution in gra-
phene. There is also the next-to-nearest neighbor hopping
denoted by the �AA (or �BB ) vectors in Fig. 1. Its effect on
the energy spectrum in a constant magnetic field was al-
ready considered in [7,12], where it was shown that such
a hopping breaks the electron-hole symmetry of the Dirac
spectrum and produces terms characteristic for a non-
relativistic problem.
In the present paper, we will consider in more detail
how the next-to-nearest neighbor hopping affects the
electronic properties of two-dimensional systems with
honeycomb lattice. In Sec. 2, we repeat the derivation of
the energy spectrum in a constant magnetic field. In
Sec. 3, we calculate the density of states (DOS). Fan dia-
grams are plotted in Sec. 4. Conclusions are given in
Sec. 5.
2. The energy spectrum
In the tight-binding approximation, the electron Ha-
miltonian in graphene reads
H t a a�� �
nm
n m
n m
,
, (1)
where vectors n and m denote certain positions on the
graphene lattice and Fermi-operators an
� and am create
and annihilate electrons at the corresponding places.
For the sake of simplicity, we omitted the spin variables
in (1). Hopping amplitude t nm is the matrix element of
the operator of kinetic energy. We will consider only
two nonzero matr ix elements t t
AB
� �n n � and t� �
� �� �t t
AA BBn n n n� � which describe the nearest and
next-to-nearest neighbor hoppings, respectively. Since
the graphene’s hexagonal lattice can be described in terms
of two triangular sublattices A and B, the parameter t is
evidently connected with intersublattice hopping and t�
with intrasublattice hopping (see Fig. 1).
At low energy, we consider states in the vicinity of the
Dirac K point, K �
�
�
�
2
3
2
3
0
�
aCC
, , where aCC � 1.42 � is
the distance between nearest carbon atoms of the different
sublattices. The constant � �3t is omitted, because it can
absorbed in chemical potential. Then we retain leading in
momentum terms and readily obtain the following effec-
tive low energy Hamiltonian defined on two-component
spinors whose upper and down components are connected
with states on the A and B sublattices, respectively:
� ( )
( )
( )
*
*
H m
k ik
k ik
m
F x y
F x y
K k
k
k
�
� �
� �
�
�
�
�
�
�
�
2 2
2 2
2
2
v
v��
�
�
�
�
��
, (2)
where the effective mass and Fermi velocity are
m
t aCC
* �
�
2
9
2
2
�
, vF
CCta
�
3
2�
. (3)
The effective Hamiltonian for low energy excitations in
the vicinity of the Dirac point K� � �
�
�
�
2
3
2
3
0
�
aCC
, is ob-
tained from (2) through the replacement k kx x� � . One
can easily find the spectrum of the Hamiltonian (2)
E
m
F� �
�
�
2 2
2
k
k
*
| |v . (4)
If t� � 0 (only the nearest neighbor hopping is retained),
then the Hamiltonian (2) together with H K k� ( ) is the con-
ventional Dirac Hamiltonian, which is now widely used
for the description of electronic properties of graphene.
This Hamiltonian has the electron-hole symmetry. Its
spectrum is given by the second term on the right-hand
side of Eq. (4) and presents the two Dirac cones (near
points K and K�) of positive and negative energies. If,
however, t� � 0, then the electron-hole symmetry is lost
and as it is evident from Eq. (4) the spectrum is not sym-
metric with respect to the change of sign of energy
E E� � . Note that in the other limiting case t � 0 (only the
next-to-nearest neighbor hopping is retained), the Ha-
miltonian (2) describes conventional nonrelativistic qua-
siparticles whose spectrum is given by the first term on
the right-hand side of Eq. (4). Of course, these limiting
cases are quite useful because they help to check calcula-
tions performed in the general case.
The Hamiltonian for electron states in a magnetic field
in the vicinity of the K point follows from (2) by using the
minimal coupling with electromagnetic field
�
( � � ) ( � � )
( � � ) (
*
*
H m
i
i
m
x y F x y
F x y
K �
� � �
� �
1
2
1
2
2 2� � � �
� �
v
v � � )� �x y
2 2�
�
�
�
�
�
�
�
�
�
, (5)
where � /� � � � �i ie c� A (obviously, the Hamiltonian for
the states in the vicinity of the K� point is obtained
through the replacement � �� �x x� � ).
It is straightforward to find the spectrum of graphene
with t and t�hoppings in a constant magnetic field perpen-
dicular to the graphene plane. The spectrum was first
obtained in [7,12] and consists of two non-symmetric
branches (see Fig. 2)
1034 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
Y.F. Suprunenko, E.V. Gorbar, S.G. Sharapov, and V.M. Loktev
E n n
eB
c
n B
B
F
( )� � �
�
� �
�
�
� ��
�
�
2
2
2
2
v , (6)
where �B eB m c� / * is the standard cyclotron frequency,
index � denotes the upper � � � and lower � � � branches,
n � 1 is the Landau level index, and we ignore the simple
Zeeman linear contribution to energy due to spin. For the
lowest Landau level n � 0, the energy
E B
0
2
�
��
(7)
does not depend on t and equals the conventional lowest
Landau level energy in the nonrelativistic case.
Note that the lower branch En
( )� of the spectrum (6)
unlike the case of the Dirac spectrum is not monotonic.
Let us assume for simplicity that n is continuous. Then,
as follows from Eq. (6), for t t tl aB CC� � � �max / 27
( /l c eBB � � is the magnetic length), that is true for
graphene in realistic magnetic fields, the energy En
( )� de-
creases for small values of n, attains a minimum at
E
m
m
F B
F
min
*
*
� � �
v
v
2 2 2
22 8
� �
, (8)
(since n is discrete, the true minimum of the spectrum (6),
in general, is higher than (8)), and then increases for
larger values of n (see Fig. 2). At low energies, we stay
close to the Dirac points that implies that | |k aCC �� 1.
This means that we should consider small values of n such
that only the decreasing part of the lower branch of the
spectrum contributes.
The eigenfunctions of higher Landau levels n � 1of the
spectrum (6) are given by the following equation below
(we use the Landau gauge A � ( , )0 xB for the vector poten-
tial of the external magnetic field and, as was mentioned
above, suppress the spin index):
�
�
�
�
n
y
n
n
n
x y
ik y
X
x
i x X
( )
( )
( , )
[ ( ) ]
( )
( )
�
�
�
� � �
exp ( )
2 1 2 1 n
( )�
��
�
�� , (9)
where
� � �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�n
n
B
B y
B
x
n l
x l k
l
( )
!
exp
1
2
1
2
2
2
�
�
�
�
�
�
�
�
H
x l k
l
n
B y
B
2
, (10)
H xn ( ) are Hermite polynomials, and
X
n
eB
c
n
eB
c
n
F
B B
F
( )�
�
�
�
� �
�
�
�
� �
v
v
2
2 2
2
2
2
�
� �
�
. (11)
Eigenfunctions in the vicinity of the K' point have the
same factor before the spinor, i x Xn n� �1( ) ( )� as the upper
component of the spinor, and �n x( ) as the lower one.
The eigenfunctions for the lowest Landau level (7) are
given by (we assume that sign( )eB 0 and again, like in
the expression (9), suppress the spin variables)
�
�
0
0
2 0
( , )
exp( ) ( )
x y
ik y xy
�
�
�
�
� . (12)
The spectrum (6) has an unusual dependence on the
Landau index n and magnetic field B because the energy
levels En
( )� cannot be represented as E f B nn
( ) ( ( ))�
� !� � ,
where f x� ( ) are arbitrary functions and ! is a constant. In
Sec. 3, we will see that this leads to the dependence of the
frequency of oscillations BF on the magnetic field B. As
well known, BF does not depend on B in the nonre-
lativistic and relativistic cases, where the energy spectra
have the canonical form E nn B� ��� ( / )1 2 and En
( )� �
� � 2 2n eB cF�v / with n � 0 1 2, , , .., respectively.
What is the reason for this unusual property of spec-
trum (6), (7)? In order to get an answer to this question,
let us first recall why the nonrelativistic and relativistic
spectra depend on B only through the combination
B n( )� ! . The nonrelativistic Hamiltonian contains the
operator ��
2, where � /� � � � �i ie c� A . Its eigenvalue is
equal to 2 1 2�eB n c( / ) /� . The Dirac Hamiltonian con-
tains the operator ��!, where ! are the Dirac matrices. At
the same time, one can equivalently determine its eigen-
v a l u e s f r o m � � �( � ) � /�� �
2 2 1 2i eB c� ! ! . S i n c e
( )i! !1 2 2 1� , the eigenvalues of i! !1 2 are � 1. Consequent-
Effect of next-to-nearest neighbor hopping on electronic properties of graphene
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1035
100 500 1000
n
0.5
1.0
1.5
2.0
2.5
3.0
0
E
/t
(±
)
Fig. 2. The spectrum of graphene in units of t, with t t� �/ .07, in
magnetic field such that
1
2 2
t
eB
c
Fv
�
� 0.05, and n is the Landau
level index. The dots and squares correspond to the positive E( )�
and negative E( )� branches of the spectrum (the square at n � 0
is absent according to Eq. (7)).
ly, eigenvalues of �( � )��
2 are 2 1 2� �eB n c eB c( / ) / /� � .
Therefore, the lowest Landau level has zero energy in the
relativistic case and higher Landau levels are apparently
twice degenerate in view of the twice degenerate spinor
part contribution � �eB c/ . Thus, the energy spectrum in
the relativistic case like in the nonrelativistic one depends
on magnetic field only through the combination B n( )� ! .
In our case, for electron states of the Hamiltonian (5),
the action of the operators � �� �x yi� and � �� �x y
2 2� on func-
tions�n
ik yye contains both square root and linear terms in B
( � � )
| |( )
� �x y n
ik y
n
ik y
i i
eB n
c
y y� � �
�
� �e e
2 1
1
�
,
( � � )
| |
� �x y n
ik y
n
ik y
i i
eB n
c
y y� � � � � �e e
2
1
�
,
( � � )
| |
( )� �x y n
ik y
n
ik yy yeB
c
n2 2 2 1� � � � �e e
�
and, consequently, eigenvalues En
( )� of the Hamiltonian
(5) cannot be represented as f B n( ( ))� ! .
Finally, we would like to mention that, strictly speak-
ing, the relativistic Hamiltonian for the electron in addi-
tion to the Dirac terms contains also an anomalous mag-
netic moment term e c FB
2 28" � ! !"�
" �/ ( )� (see, e.g.,
[13]), where "B is the Bohr magneton. According to [14],
in such a case, the energy spectrum
E c p
B
c
m c
eBn
c
n z
B� � � �
�
�
�
�
�
�
�
�
�
�
�
�2 2
2
2
2 2
2
8
2
#
"
�
e
( )* �
�
1 2/
,
where# � � is the electron spin, also does not have the con-
ventional form f B n( ( ))� ! like the energy spectrum (6).
Qualitatively, the most important feature of the spec-
trum (6), (7) is that it interpolates between the relativistic
and nonrelativistic spectra. Indeed, for �B � 0, the spec-
trum Eqs. (6), (7) reduces to a relativistic spectrum and,
for �B � $, it tends to a nonrelativistic one. Thus, the
next-to-nearest neighbor hopping changes the low energy
spectrum of quasiparticles in a system with honeycomb
lattice, although, quantitavely this change in real
graphene is quite small due to the inequality t t� ��/ 1. Note
that the degeneracy of the states of the spectrum (6), (7)
coincides with the degeneracy of states of the purely rela-
tivistic spectrum. Consequently, the next-to-nearest
neighbor hopping in graphene does not change the Berry
phase which is connected with the degeneracy of spec-
trum [15] and equals �. (For t � 0, the degeneracy of spec-
trum is the same as in the nonrelativistic problem and the
Berry phase in such a case equals zero.) Therefore, the
QHE quantization in odd integers still applies to the case
under consideration.
3. The density of states
In the absence of scattering from impurities the quasi-
particle DOS can be written as the sum of %-functions of
Landau level energies:
D E
eB
c
E E E E
n
n( ) ( ) ( )
,
( )� � � �
�
�
�
�
�
�
�
�� � �
$
�2
0
1
�
% %
�
�
�
, (13)
or, equivalently,
D E
eB
c
d
dE
E E E E
n
n( ) ( ) ( )
,
( )� � � �
�
�
�
�
�
�
�
�� � �
$
�2
0
1
�
& &
�
�
�
. (14)
Note that 2eB c/ �� is the density of the Landau levels
which includes their fourfold valley and spin degeneracy
(recall that we suppressed the spin variables contribution
to energy which can be easily restored).
Using the Poisson formula
f n f f x dx f x kx
n k
( ) ( ) ( ) ( ) cos ( )
�
$ $ $
�
$
� ' '�� � � �
1 0 01
1
2
0 2 2� dx ,
(15)
we find
& &
n
nE E E E
�
$
�� � � � � �
1
0
1
2
( ) ( )( )
� � �
�
�
�
�
�
�
�
��
$
�&
�
�( ) sin ( )E E x
k
kx
k
0 1 1
1
1
2 , (16)
for the � � � branch of the spectrum and
& &
n
nE E E E
�
$
�� � � � � �
1
0
1
2
( ) ( )( )
� � �
�
�
�
�
�
�
�
�
�
�
$
�&
�
�( ) sin ( )minE E x
k
kx
k
2
1
2
1
2
� � � � �
�
�
�
�
�
�
�
��
$
�[ ( ) ( )] sin ( )min& &
�
�E E E E x
k
kx
k
0 1
1
1
1
2 ,
(17)
for the � � � branch of the spectrum, where
x E
eB
c
E
j
F
B B
( )
( )
� � �
v
2
2
�
� �� �
�
�
� �
( )
( )
( )
( )1
2
42
2 2 2
4j
B
F B F
BeB
c
eB
c
E
�
� �
� �
�
�
�
v v ,
j �1 2, .
(18)
1036 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
Y.F. Suprunenko, E.V. Gorbar, S.G. Sharapov, and V.M. Loktev
It is easy to check that x Ej ( ) correspond to two inverse
functions n Ej ( ) which can be found from Eq. (6). The
presence of two functions x Ej ( ) is due to the fact that for
E E min (where Emin is defined in Eq. (8)), a given en-
ergy corresponds to two points in the spectrum (see
Fig. 2). The function x E1( ) denotes intersection of the
line of constant energy with the � � � branch of the spec-
trum or with the decreasing part of the � � � branch. The
function x E2( ) corresponds to intersection with the in-
creasing part of the � � � branch. Therefore, the sum over
the � � � branch of spectrum given by Eq. (16) contains
only x E1( ) unlike Eq. (17) for the � � � branch of the
spectrum which, for E E Emin ( ( � 0, contains terms with
both x E1( ) and x E2( ). For E E � 0, the line of constant
energy intersects only the increasing part of the � � �
branch, therefore, only the term with x E2( ) in (17) contri-
butes in this case.
Combining Eqs. (14), (16) and (17) we obtain the ex-
act DOS for the spectrum (6), (7)
D E
eB
c
d
dE
E E E E( ) ( ) ( )� � � �
)
*
+
,+
�
2 1
2
1
2
0 0�
& &
�
� � �
�
�
�
�
�
�
�
�
� � �
�
$
�&
�
� &( ) sin ( ) [ ( )minE E x
k
kx E E
k
2
1
2 0
1
2
� � � � �
�
�
�
�
�
�
�
�
-
.
�
$
�& &
�
�( ) ( )] sin ( )minE E E E x
k
kx
k
0 1
1
1
1
2
+
/+
.
(19)
According to the discussion in Sec. 2, since x is contin-
uous, Emin , in general, is not equal to the lowest energy
level of the spectrum (6), (7), where n is discrete. The ex-
pression in the curly brackets in (19) is nonzero for
E E min . Therefore, it is not immediately clear that the
DOS given by Eq. (19) is zero for energies between Emin
and the true lowest level of the spectrum (6),(7). To prove
this, it is useful to recall [19] that the first Bernoulli poly-
nomial in the interval 0 1� �x can be expressed as fol-
lows:
B x x
k
kx
k
1
1
1
2
1
2( ) sin ( )� � � �
�
$
��
� . (20)
For x beyond the interval [ , ]0 1 , in order to find the sum
over sin ( )2�kx functions, one can use the first Bernoulli
polynomial periodically continued beyond the interval
x 0 [ , ]0 1 which depends on the fractional part of its argu-
ment, i.e. B x x1( [ ])� . Here, [ ]x is the largest integer satis-
fying [ ]x x( . Thus, we obtain
1
2
1
2
1
1�
�
k
kx B x x x x
k�
$
� � � � � � �sin ( ) ( [ ]) [ ] .
As result, the function
x
k
kx x
k
� � �
�
$
� 1
2
1
2
1
�
�sin ( ) [ ]
has a staircase-like behavior (see Fig. 3).
It is not surprising at all that we obtained this result.
Our DOS (19) is given as a full derivative with respect to
E. Integrating it over E, one finds the number of states.
Consequently, the quantity in the curly brackets in
Eq. (19) up to a factor coincides with the number of states.
Since the spectrum in a magnetic field is discrete, this
quantity should have a staircase-like behavior. Therefore,
the DOS is not equal zero only when E crosses some en-
ergy level of the system. When this happens, the expres-
sion in the curly brackets in Eq. (19) experiences a jump
between different plateaus.
As it was mentioned at the end of Sec. 2, we should
consider only rather small values of n. This implies that
only terms with x1 should be retained in Eq. (19) and all
terms with x 2 can be omitted because they correspond to
states with very large n. Thus, we arrive at the following
final expression for the physically relevant DOS in gra-
phene:
D E
eB
c
d
dE
E E E E E E( ) ( ) ( ) [ ( )� � � � � � �)
*
,
2 1
2
1
2
0 0 0�
& & &
�
� � � � �
�
�
�
-
.
/
& &( ) ( )] [ ]minE E E E x0 1
1
2
. (21)
The customary form (see, e.g. , [16]) of x1 is
x cS E eB1 2� �� ( ) / ( )� !, where S E( ) is the semiclassical
electron orbit area in the space of wave vector k, and
! �� ��1 2 2/ /B with �B being the Berry phase. Accord-
ing to the results of the previous section, the Berry phase
for the system under consideration equals � and, conse-
quently, ! � 0. Therefore, using (18), one finds that the an-
alog of the semiclassical electron orbit in our case is
�
2
2 2
2�
S E m EmF( ) ( )* *� � �v
� � �v
v v
F
F F
m
E
m
e B
m c
2 2
2
2 2 2
4 2
1
2
4 4
( )
( )
*
* *
�
. (22)
Effect of next-to-nearest neighbor hopping on electronic properties of graphene
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1037
2 4 6 8
x
x
–
B
(x
–
[x
])
1
2
4
6
8
Fig. 3. The function x B x x� �1( [ ]).
The fact that Eq. (22) depends on magnetic field B means
that S E( ) cannot be interpreted as a semiclassical electron
orbit. This unusual form of S E( ) is connected with the un-
conventional dependence of the energy levels (6) on the
Landau level index and magnetic field discussed in the
previous section. Moreover, we will see in the next sec-
tion, where we plot the fan diagram, that the unusual char-
acter of the dependency of x1 on B is responsible for the
unconventional form of the fan diagram for our system
which is not a straight line unlike the well known cases of
nonrelativistic and relativistic systems.
Finally, we would like to note that S E( ), given by
Eq. (22), tends to the correct value of semiclassical elec-
tron orbit area �E F
2 2 2/ ( )� v [16] in the relativistic limit
t�� 0. In the nonrelativistic limit t � 0, the function S E( ),
given by Eq. (22), tends to 2 2� �Em eB c* / / ( )� �� , where
2 2�Em* / � is the correct nonrelativistic semiclassical
electron orbit area and, obviously, the second term
��eB c/ ( )� multiplied by �c eB/ ( )2� gives the standard
nonrelativistic value ! �1 2/ [16]. Note that, although Eq.
(22) tends smoothly to the relativistic and nonrelativistic
expressions in the corresponding limits, the Berry phase
does not have such a smooth behavior. It equals � for any
nonzero t and abruptly jumps to zero when t exactly
equals zero. This behavior of the Berry phase is con-
nected with its topological character [10].
4. Fan diagrams
Fan diagrams are a standard [2,17] and convenient
way to display graphically information about the spec-
trum of the system. They express the inverse magnetic
field B �1 as function of the Landau level index n at some
fixed chemical potential ".
For the � � � branch of the spectrum (6), we find
1
2
2B
e
m c
e
c
nF� �
�
�
�
�
�
�
� �
*" "
v
� �
�
�
�
�
�
�
v
v v
F
F F
e
c m
n
m
2
2 2
2
2
2 4
2
1
4
�
"
" "
* *( )
. (23)
For t � 0, Eq. (23) reduces to
1 1
2
4 1
20B
e
cm
n n
nr
� �
�
�
� �
�
�
�
�
�
�
*" �
, (24)
where in the last equality we used the relation �nr �
� 2 2" �m* / ( )� between carrier density �nr and chemical
potential " in the nonrelativistic case (� �0 hc e/ is the
flux quantum). For t � � 0, Eq. (23) gives
1 2 4
2
2
0B
e
c
n nF� �
�
�v
" � rel
, (25)
where in the last equality we expressed " via the relativis-
tic carrier density � " �rel �
2 2 2/ ( )� vF . Equations (24) and
(25) illustrate the well-known fact that the fundamental
frequency B S EF � �0
24( ) / ( )� of quantum magnetic os-
cillations for any 2D system is defined by its degeneracy
f and the concentration of electrons or holes �; so that
B fF � �0� / . For graphene f � 4 because of double-spin
and double-valley degeneracy.
For n 1, Eq. (23) implies that
1
1
2
1
1
2
2 2B
e
m c
e
c m
n O
n
F
F
� � � �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
� �
* *" "
"v
v
�
�
� . (26)
Since a constant term (the term which does not depend
on n) is absent in (26), the dependence of B �1 on n is the
same as in Eq. (25) for the relativistic case with t� � 0.
Matching slopes of the dependencies B n�1( ) in Eqs. (24),
(25), and (26) gives
� �
" "
"
rel nr� �
� � � �
�
�
�
�
�
�
�
�
�
4
1
2
10
2
2 2
� �e
m c
e
c m
F
F
* *
v
v
�
�
�
�
. (27)
Using Eq. (27), we plot in Fig. 4 the fan diagrams for
the system under consideration (23) and the nonrelati-
vistic and relativistic systems given by Eqs. (24) and (25),
respectively. The most important property of the fan dia-
gram for the system under consideration is that it is not a
straight line unlike the fan diagrams for the nonrelati-
1038 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
Y.F. Suprunenko, E.V. Gorbar, S.G. Sharapov, and V.M. Loktev
1 2 3 4 5
0.05
0.10
0.15
0
1
/B
,
1
0
(T
)
–
3
–
1
n+1/2
Fig. 4. Fan diagrams for the � � � branch of the spectrum (solid
line) with parameters t� �189. eV, t � 2.7 eV, " � 11.5 eV , the
relativistic system t� � 0, t � 2.7 eV, "r � 5 eV (dash-dotted line),
and the nonrelativistic system t � 0, t� � 1.89 eV, "nr � 65. eV
(dashed line). Integer values of n denoted by dots correspond
to the Landau level index in the spectrum.
vistic and relativistic systems separately. As we have al-
ready mentioned in the previous section, this fact is due to
the unconventional dependency of x1 on B which in its
turn is connected with the unusual dependency of the
spectrum (6) on B. According to Fig. 4, the fan diagram
for our system (solid line) interpolates between the fan di-
agrams for the nonrelativistic and relativistic systems.
Namely, it intersects the nonrelativistic fan diagram
(dashed line) at the origin and tends to the relativistic fan
diagram (dash-dotted line) for large n.
5. Discussion and conclusion
In the present work we have taken into account the
next-to-nearest neighbor hopping in graphene in the
tight-binding approximation. For the states in the vicinity
of the Dirac points, we have considered the off-diagonal
sublattice terms of the Hamiltonian up to the first order in
momentum and the diagonal sublattice terms up to the
second order. The advantage of the effective Hamiltonian
(2) is that it leads to the model exactly solvable in an ex-
ternal magnetic field whose Hamiltonian is given by
Eq. (5) and the energy spectrum by Eqs. (6), (7). For this
model, we have calculated the DOS (19) and ploted the
fan diagram (Fig. 4), which interpolates between the fan
diagrams for the relativistic and nonrelativistic systems.
We have shown that the Berry phase for the model under
consideration equals �, i.e. coincides exactly with the
Berry phase for the relativistic system.
In a recent paper [21] a related problem was studied.
There both the off-diagonal and diagonal terms of the
Hamiltonian were considered up to the third order in mo-
mentum. The corresponding energy dispersion includes
not only the contribution due to the next-to-nearest neigh-
bor hopping (see Eq. (4)), but also high-order band cor-
rections such as trigonal warping terms. It turns out, how-
ever, that in an external magnetic field the warping terms
do not contribute to the spectrum. Accordingly, the spec-
trum (6) agrees with the spectrum derived in Ref. 21 in the
large-n limit up to the terms 1 n 3 2/ . The only essential
difference is that the large-n limit misses the n-independ-
ent term under the square root in Eq. (6) which is respon-
sible for the recovery of the zero Berry phase in the
nonrelativistic, t � 0, limit.
The experimental results reported in Ref. 21 indicate
that at energies above 500 meV a deviation from the ideal
behavior of the Dirac quasiparticles is observed. How-
ever, the asymmetry between the valence and conduction
bands studied here seems to play a minor role as com-
pared to the trigonal warping of the Fermi surface.
Acknowledgements
The authors are grateful to V.P. Gusynin for stimulat-
ing discussions. The work of E.V.G. was supported by the
SCOPES project IB 7320-110848 of the Swiss NSF, the
grant 10/07-N «Nanostructure systems, nanomaterials,
nanotechnologies», the Program of Fundamental Re-
search of the Physics and Astronomy Division of the Na-
tional Academy of Sciences of Ukraine, and by the State
Foundation for Fundamental Research of Ukraine (grant
F16-457-2007).
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