The Berry phase in graphene and graphite multilayers
We discuss the electron energy spectra and the Berry phases for graphene, a graphite bilayer, and bulk graphite allowing for a small spin-orbit interaction. If an electron orbit in the Brillouin zone surrounds several Dirac points (band-contact lines in graphite), one can find relative signs of th...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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nasplib_isofts_kiev_ua-123456789-1175572025-02-09T10:01:31Z The Berry phase in graphene and graphite multilayers Mikitik, G.P. Sharlai, Yu.V. Graphene and graphite multilayers We discuss the electron energy spectra and the Berry phases for graphene, a graphite bilayer, and bulk graphite allowing for a small spin-orbit interaction. If an electron orbit in the Brillouin zone surrounds several Dirac points (band-contact lines in graphite), one can find relative signs of the Berry phases generated by these points (lines) by taking into account this interaction. 2008 Article The Berry phase in graphene and graphite multilayers / G.P. Mikitik, Yu.V. Sharlai // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1012-1019. — Бібліогр.: 37 назв. — англ. 0132-6414 PACS: 73.22.Dj;03.65.Sq;03.65.Vf https://nasplib.isofts.kiev.ua/handle/123456789/117557 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Graphene and graphite multilayers Graphene and graphite multilayers |
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Graphene and graphite multilayers Graphene and graphite multilayers Mikitik, G.P. Sharlai, Yu.V. The Berry phase in graphene and graphite multilayers Физика низких температур |
| description |
We discuss the electron energy spectra and the Berry phases for graphene, a graphite bilayer, and bulk
graphite allowing for a small spin-orbit interaction. If an electron orbit in the Brillouin zone surrounds several
Dirac points (band-contact lines in graphite), one can find relative signs of the Berry phases generated
by these points (lines) by taking into account this interaction. |
| format |
Article |
| author |
Mikitik, G.P. Sharlai, Yu.V. |
| author_facet |
Mikitik, G.P. Sharlai, Yu.V. |
| author_sort |
Mikitik, G.P. |
| title |
The Berry phase in graphene and graphite multilayers |
| title_short |
The Berry phase in graphene and graphite multilayers |
| title_full |
The Berry phase in graphene and graphite multilayers |
| title_fullStr |
The Berry phase in graphene and graphite multilayers |
| title_full_unstemmed |
The Berry phase in graphene and graphite multilayers |
| title_sort |
berry phase in graphene and graphite multilayers |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2008 |
| topic_facet |
Graphene and graphite multilayers |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/117557 |
| citation_txt |
The Berry phase in graphene and graphite multilayers
/ G.P. Mikitik, Yu.V. Sharlai
// Физика низких температур. — 2008. — Т. 34, № 10. — С. 1012-1019. — Бібліогр.: 37 назв. — англ. |
| series |
Физика низких температур |
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| first_indexed |
2025-11-25T15:43:42Z |
| last_indexed |
2025-11-25T15:43:42Z |
| _version_ |
1849777639413252096 |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10, p. 1012–1019
The Berry phase in graphene and graphite multilayers
G.P. Mikitik and Yu.V. Sharlai
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: mikitik@ilt.kharkov.ua
E-mail: sharlai@ilt.kharkov.ua
Received March 28, 2008
We discuss the electron energy spectra and the Berry phases for graphene, a graphite bilayer, and bulk
graphite allowing for a small spin-orbit interaction. If an electron orbit in the Brillouin zone surrounds sev-
eral Dirac points (band-contact lines in graphite), one can find relative signs of the Berry phases generated
by these points (lines) by taking into account this interaction.
PACS: 73.22.Dj Single particle states;
03.65.Sq Semiclassical theories and applications ;
03.65.Vf Phases: geometric; dynamic or topological.
Keywords: Berry phase, graphite bilayer.
Introduction
In the last few years graphene, i.e. a graphite monolayer,
has attracted considerable attention thanks to its unusual
physical properties, see, e.g., the recent review [1] and re-
ferences therein. In particular, a phase shift of the Shub-
nikov–de Haas oscillations and a half-integer quantum Hall
effect were observed in this two-dimensional material [2,3].
These properties of graphene are due to the specific feature
of its electron energy spectrum: At the point K of its Bril-
louin zone the electron and hole bands � e h, ( )k touch one an-
other, while in the vicinity of this point they split linearly in
the wave vector k, see Fig. 1. In other words, this spectrum
looks like the spectrum of relativistic Dirac particles, and so
the band-contact point in graphene received the name Dirac
point. This type of the spectrum generates a nonzero value
of the so-called Berry phase [4], the Berry phase modifies
the semiclassical quantization condition for the electron en-
ergy in a magnetic field [5], and this modification leads to a
shift of the Landau levels in graphene as compared to the
levels for the usual parabolic spectrum. It is this shift of the
Landau levels that explains the above-mentioned properties
of graphene. One may expect that these considerations ba-
sed on the semiclassical approach fail for the lowest Landau
levels, i.e., when the quantum number n ~ 1. But in the case
of linear dependence �( )k the electron spectrum in a mag-
netic field can be found exactly, see, e.g., Refs. 6,7, and it is
turned out that the exact and semiclassical spectra coincide
for all n, including n � 0 [6]. Thus, in the case of graphene
one can apply considerations based on the Berry phase with-
out any restrictions.
In this paper we first review results concerning the
Berry phase in metals, and then discuss the Berry phase in
graphene, in a graphite bilayer, and in a bulk graphite that
can be considered as a sample with a sufficiently large
number of the layers. Moreover, in this paper we shall an-
alyze the Berry phase taking into account the spin-orbit
interaction since this interaction is important for under-
standing of the Berry phase in graphite multilayers.
The Berry phase in metals
According to Berry [4], if a Hamiltonian of a quantum
system depends on parameters, and if the parameters un-
dergo adiabatic changes so that they eventually return to
© G.P. Mikitik and Yu.V. Sharlai, 2008
K
e
h
ky
kx
K
Fig. 1. The dispersion law �( , )k kx y of the electrons (e) and holes
(h) in graphene near the point K of its hexagonal Brillouin zone.
their original values, the wave function of the system can
acquire the so-called geometrical phase in addition to the
familiar dynamical one. This additional phase (the Berry
phase) differs from zero when the trajectory � of the sys-
tem in the parameter space is located near a point at which
the states of the system are degenerate. In analyzing this
situation, Berry assumed that the Hamiltonian of the sys-
tem is a Hermitian matrix which is linear in deviations of
the parameters from the point, and he presented his final
result in the pictorial form. He found that such the point
can be considered as a «monopole» in the parameter space
when the geometrical phase is calculated. In other words,
the point «generates» a field which coincides in the form
with that of the monopole, and the flux of this Berry field
through the contour � gives the geometrical phase of the
system.
Berry’s result is applicable to the semiclassical elec-
tron trajectories in crystals, with the Brillouin zone
playing the role of the parameter space [8]. However, in
three-dimensional crystals with the inversion symmetry
and a weak spin-orbit interaction, the Berry phase of the
electrons has the specific features [5] which are due to the
fact that the electron states are invariant under the simul-
taneous inversion of time and spatial coordinates. This
invariance permits one to transform the Hermitian Ha-
miltonian of the electron into the real form for any point
of the Brillouin zone. As a consequence, the character of
the energy-band degeneracy in the three-dimensional
crystals differs from that considered by Berry. Now the
electron energy bands � l ( )k contact along lines in the
Brillouin zone rather than at points, and the lines need not
be symmetry axes [9]. In other words, Berry’s monopoles
in the k space disappear. However, as it was shown in our
paper [5], the above-mentioned band-contact lines play
the role of infinitely thin «solenoids» which generate the
Berry field with the flux ��, and the situation similar to
the Aharonov–Bohm effect [10] occurs: Although this
field is zero outside the solenoids, but if the electron orbit
surrounds a contact line of its band with some other band,
the flux threads the orbit, and the electron acquires the
Berry phase�B � ��when it moves around this line. It is
clear that in this case the Berry phase does not depend on
the shape and the size of the electron orbit but is specified
only by its topological characteristics (there is a linking
of the orbit with the band-contact line or not). In the case
of a two-dimensional metal the band-contact lines trans-
form into the Dirac points, and if the electron orbit � sur-
rounds Dirac points, each of these points contributes ��
to�B . When there are no Dirac points inside the orbit, the
Berry phase is equal to zero.
The Berry phase is defined up to 2�, and in this context
the phases�B � � and�B � �� are equivalent. But when
the electron orbit surrounds several of the band-contact
lines (the Dirac points), it may be useful to know relative
signs of the Berry phases generated by each of the line
(the Dirac point). This can be done using the approach of
Ref. 11 in which the effect of a small spin-orbit interac-
tion on the Berry phase was investigated. It is important
that if this interaction is weak, it does not change the
Berry phase, but it fixes the sign of the phase for a given
direction of the electron spin [11,12]. Here we present the
formulas that enable one to calculate the Berry phase�B
in this case.
Let an electron be in the band � 0( )k . Consider an orbit
� of this electron in the Brillouin zone. The Berry phase
for the orbit � and at a fixed direction of the electron spin
is given by [5,8]
� �
�
B d� � k , (1)
where � is the intraband matrix element of the periodic
(in k) part of the coordinate operator in the crystal mo-
mentum representation [13],
� ( ) ( ) ( )*k r r rk k�
i d u uk0 0 , (2)
and u k r0( ) is the periodic factor in the electron Bloch
wave function of the band 0, � k kr k r0 0( ) exp( ) ( )� i r u .
Note that the Berry phase changes its sign for the opposite
direction of the electron spin since it follows from sym-
metry considerations [11] that � �up down� � for the spin
directed «up» and «down».
The Berry phase manifests itself in the Onza-
ger–Lifshitz–Kosevich quantization condition [14,15] for
the Landau levels � n of the electron in the magnetic field H,
S k
eH
c
nn H( , ) ( )�
�
��
2
�
, (3)
where S is the cross-section area of the closed orbit in the
k space; k H is the component of k along the magnetic
field H; n is a large integer (n � 0); e is the absolute value
of the electron charge, and the constant � is given by the
formula [5]:
�
�
� �
1
2 2
�B . (4)
The direction of the integration in Eq. (1) is now deter-
mined by the vector sign( )[ ]
*
m /H k�� �� 0 where
m
S k H
*
( , )
�
�
�
�
2
2�
�
�
is the cyclotron mass which is positive for electrons and
negative for holes, and hence sign( )
*
m � �1. The direc-
tion of the integration becomes essential when one wants
to determine not only the value of the Berry phase but also
its sign.
The meaning of formula (4) is simple: When the elec-
tron makes a complete circuit in its orbit, the change of
The Berry phase in graphene and graphite multilayers
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1013
the phase of its wave function consists of the usual semi-
classical part �cS/eH, the shift �� associated with the
so-called turning points of the orbit where the semiclas-
sical approximation fails, and the Berry phase. Equating
this change to 2�n, one arrives at Eqs. (3), (4). Thus, when
the electron orbit links to the band-contact line, one ob-
tains � � 0 or � �1 instead of the usual value � �1 2/ . Note
that the quantity ��B /2� in Eq. (4), in fact, describes the
electron g factor that differs from zero even at infinitesi-
mal spin-orbit interaction if the orbit surrounds the band-
contact line (the Dirac point) [16].
The quantity � can be calculated for any electron-band
structure if an appropriate k p� Hamiltonian is known [11]:
� �
�
�
�
�
�
�
�
�
i S
S
k 00
, (5)
where S( )k is the matrix reducing the k p� Hamiltonian to
the diagonal form, S
is the Hermitian conjugate matrix,
and the subscript means that one has to consider the diag-
onal matrix element corresponding to the band 0 dis-
cussed here. This formula follows from Eq. (2) if one
takes into account the definition of the matrix S,
S u u dij i j( ) ( ) ( )*k r r rk� 0 .
As it is well known [17], a frequency of de Haas–van
Alphen oscillations, or of other oscillation effects in the
magnetic field, gives an extremal cross-section area Sex
of the Fermi surface of a metal. On the other hand, a phase
of these oscillations is expressed via the constant � of the
semiclassical quantization rule. For example, the first
harmonic of the de Haas–van Alphen oscillations of the
magnetic susceptibility has the form [15],
� �cos
�cS
eH
ex
�
�
�
�
�
� , (6)
where Sex is some extremal (in k H ) cross section of the
Fermi surface of a metal, a positive � is the amplitude of
this first harmonic, and � is its phase which is given by
� �� �� �
2 (7)
with � �� � / 4 for a minimum and maximum cross-section
Sex, respectively, and � � 0 in the case of a two-dimen-
sional Fermi surface. The phase � (together with the fre-
quency) can be found by the Fourier analysis of the mag-
netic-susceptibility oscillations [18] or by plotting the fan
diagram [17]. Thus, a measurement of the phase of the de
Haas–van Alphen oscillations gives � and hence enables
one to detect a band-contact line if it penetrates the ext-
remal cross section of the Fermi surface even though this
line lies far away from the Fermi level.
In two-dimensional metals, when the quantum Hall
effect is observable, the Berry phase also manifests itself
in the Hall conductivity. In the conventional integer quan-
tum Hall effect, when the filling factor � � hcN/ egH( ) of
the Landau levels reaches an integer value n, the Hall con-
ductivity � �xy ( ) shows the plateau that is equal to� xy �
� ( )ge /h n2 . Here N is the two-dimensional density of
electrons in a metal, and g is the integer accounting for a
degeneracy of the electron spectrum (in graphene g � 4
due to two points K in the Brillouin zone and the two-fold
spin degeneracy). The nonzero Berry phase changes the
constant � and hence shifts the Landau levels in the
energy scale. This shift leads to a half-integer quantum
Hall effect: The plateaus of � xy n( ) now occurs at� xy �
�
( )( )ge /h n /2 1 2 .
The Berry phase in grapheme
Let us now consider the Berry phase in graphene taking
into account the spin-orbit interaction. The effect of the
spin-orbit interaction on the spectrum of graphene was
considered in Refs. 19,20, and the following k �p Ha-
miltonian for the electrons and holes with spin up was ob-
tained in the vicinity of the point K of the Brillouin zone:
� ( )H v k kx x y y so z�
� � � �� , (8)
where � i are the Pauli matrices describing the two bands, v
is a matrix element of the velocity operator, v �10 6 m/s, and
2 0 2� so ~ . meV is the gap caused by the spin-orbit interac-
tion. This Hamiltonian leads to the following dispersion
laws � e h, ( )k for the electrons (e) and holes (h) in graphene:
� e h so vk, ( ) ( )k � �
�2 2
� , (9)
where the signs plus and minus correspond to the elec-
trons and holes, respectively.
Hamiltonian (8) can be transformed to the diagonal
form (9) by the matrix
�
( ) ( )
S
vk
vk
e e so
so e
so e
�
�
�
�
��
�
�
��
�
1
2� �
�
��
�
�
�
�
(10)
where k k ikx y� � � . Then, formula (5) yields
�
�
�
�x
y
e e so
y
x
e e so
v k v k
�
� �
� �
2 2 2 2
2 2� � � �( )
,
( )
, (11)
Inserting this � into Eq. (1), we obtain the Berry phase
� �
�
�
B
so
e
d� �� �
�
�
��
�
�
�� � k �
�
1 (12)
for an electron orbit � that is defined by the condition
� e ( )k � �const 0. Note that this expression agrees with
the intraband contribution to the electron g factor cal-
culated for a more general two-band model of the spec-
trum in Ref. 11. Finally, in the limit of the infinitesimal
spin-orbit interaction we arrive at
� � �
�
B B so e
so
� � �
lim ( , )
0
� � . (13)
1014 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
G.P. Mikitik and Yu.V. Sharlai
In similar manner one obtains �B � � for the holes
with spin up, while for the quasiparticles with spin down
the Berry phase changes its sign.
Formula (4) gives � �1 for the electrons and � � 0 for
the holes with the spin up. Inserting the expression for the
cross-section area S E / vso� ��( ) ( )2 2 2� � into Eq. (3),
and using the obtained values of �, we arrive at the spec-
trum in the magnetic field H [21]:
� n so
e v Hn
c
� �
�2
22 �
, (14)
where � correspond to the electrons and holes, respectively,
and n goes through the values 1 , ... ! for the electrons and
0 , ...! for the holes. For the spin down the spectrum is still
given by Eq. (14), but the intervals of admissible n for the
electrons and holes are exchanged. Thus, the lowest electron
Landau level with spin down lies just at the edge � so of the
upper band, while the edge �� so of the hole band coincides
with the Landau level with spin up. All other Landau level
are two-fold degenerate in spin. We emphasize that the spec-
trum thus obtained is exact, i.e., it is valid at all integer n, in-
cluding n � 0 [6].
If the spin-orbit gap � so tends to zero, one finds that all
the Landau levels are two-fold degenerate, and they can be
described by Eq. (3) with � � 0. The Landau level n � 0 now
lies exactly at � � 0 and it is shifted in energy as compared
to the the lowest Landau level for a parabolic spectrum. As
was mentioned above, this shift leads to the half-integer
quantum Hall effect that was indeed observed in graphene
in Refs. 2,3. In these papers the constant � was also mea-
sured using Shubnikov–de Haas oscillations, and the value
� � 0 was obtained.
The Berry phase in a graphite bilayer
In the recent letter [22] McCann and Fal’ko considered
the electron-energy spectrum of a graphite bilayer in the
vicinity of the point K of its Brillouin zone and stated that
in this bilayer the low-energy electronic excitations cor-
respond to chiral quasiparticles with a parabolic disper-
sion exhibiting Berry phase 2�.
First of all we point out that the electron spectrum of the
bilayer [22] can be easily obtained from the well-known
Slonzewski-Weiss-McClure (SWM) spectrum [23] of bulk
graphite if one puts cos ." � 0 5 and � �2 5 0� � (the parame-
ters � 2, � 5 describe the interaction of the atoms in the
next-nearest-neighbor layers of graphite that are absent in
the bilayer; " is the dimensionless wave vector perpendicu-
lar to the graphite layers). The SWM model describes the
wave-vector dependence of four electron energy bands of
graphite �( )k in the vicinity of the vertical edge HKH of its
Brillouin zone. These bands can be found from the forth-or-
der secular equation:
# #det �H � �� 0 , (15)
where the Hamiltonian matrix �H has the form
�
*
*
* *
*
H
E H H
E H H
H H E H
H H H E
�
�
�
1 13 13
2 23 23
13 23 3 33
13 23 33 3
0
0
�
�
�
�
�
�
�
�
�
�
�
�
�
�
. (16)
Here the following notations have been used:
E
E
E
1 1 5
2
2 1 5
2
3 2
2
1
2
1
2
1
2
�
� �
�
� � �
� � �
�
� �
� �
�
,
,
,
$ %
H
H
H
i
i
i
13 0 4
23 0 4
33 3
1
2
1
2
� �
�
�
( ) ,
,
� � &
� � &
� & '
(
(
(
�
�
�
e
e
e
(17)
where ( is the angle between the direction of the vector k
and the K� direction in the Brillouin zone; � � 2 cos") "
and & are dimensionless wave vectors in the direction
of the z-ax i s ( i . e . HKH axis ) and in the basa l
plane, respectively: " �� ( ) ( | | )/ k / KHz2 , & �� �( )2 3/
� *( /| | )k K� ; k k kx y* �
2 2 ; k is measured from the point
K; | |�K and | |KH are the distances between the point K and
points � and H of the Brillouin zone. The parameter � 0
which describes the interaction between neighbor atoms
in a graphite layer is sufficiently large as compared to the
other parameters � i , � which describe relatively weak in-
teractions between atoms in different graphite layers. If
one puts � i � �� 0 (i � �1 5) and � 0 � � 3 2�v K /| |� �, we
arrive at the double degenerate spectrum of graphene (9)
with � so � 0. On the other hand, the spectrum of the
bilayer is obtained if one puts � �1 and � �2 5 0� � , see
Table 1. Note that in this way one can allow for the small
parameter � 4 0 044� . eV that was neglected in Ref. 22.
Table 1. Values of the parameters of SWM model (eV) that lead to
the spectrum of Ref. 22.
Parameter Value
�0 2.47
�1 0.39
�2 0
�3 0.315
�4 0
�5 0
� 0
The Berry phase in graphene and graphite multilayers
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1015
In the interval Ec ++ ++| |� �1 equations (15), (16),
and (17) lead to the approximate formula for the two
low-energy bands of electrons and holes in the bilayer
�
�
�
&e h x yk k, ( , ) � � 0
2
1
2 , (18)
that exhibits a quadratic dependence on k* discussed in
Ref. 22. Here E / /c � ( ) ( )� � �1 3 0
24 , and the signs plus and
minus correspond to the electrons and holes, respectively.
At energies | |� + Ec the role of the so-called trigonal
warping, which is characterized by the parameter � 3 in
Hamiltonian (16), (17), increases, and this warping breaks
the line � �constant in the k x � k y plane into one central
and three side pockets [22]. We emphasize here that each
of these pockets contains a point at which the electron and
the hole bands contact, and near all these points the spec-
trum is linear in k, Fig. 2, see also [24–26]. Thus, in con-
trast with graphene, in the graphite bilayer near the point K
there are four Dirac points. The central contact point coin-
cides with K, while the three side contact points are at a
distance of ( ) ( ) | | . | |3 2 0 0053 1 0
2/ / K K� � � � � �� from K
where | |�K is the distance between the point K and the
center � of the Brillouin zone.
The spectrum shown in Fig. 2 clarifies existence of the
resistivity maximum discovered in the bilayer [27]. In
graphene the universal resistivity maximum h/ e4 2 was
observed at zero magnetic field and low charge-carrier
concentration N , and this maximum was explained by ab-
sence of localization for electrons with the Dirac-type
spectrum [2]. A similar resistivity maximum was also ob-
served in the bilayer [27], and Novoselov et al. [27] em-
phasized that this observation is unexpected due to the
parabolic spectrum in the bilayer. In this context, exis-
tence of the Dirac points seems to shed a light on appear-
ance of the resistivity maximum in the bilayer.
McCann and Fal’ko derived the Berry phase for the
electron orbits in the bilayer from an effective Hamil-
tonian that leads to the parabolic spectrum (18). But in the
case of the bilayer the trigonal warping generates the
three additional side Dirac points and changes the type of
the central band-contact point. Thus, for the derivation of
the Berry-phase value to be justified, it is necessary to
consider the real symmetry of the spectrum in the bilayer
and to take into account all the band-contact points of this
spectrum.
Since the interaction between carbon atoms in a graph-
ite layer is larger than the interaction between the atoms
in different graphite layers (i.e. � �0 �� i ), we use Hamil-
tonian (8) of a single layer to include the spin-orbit inter-
action in Hamiltonian (16), (17) of the bilayer. Then, in
the leading order in the small parameter � so /�1 the matrix
elements E3 in the third and forth lines of formula (16)
should be replaced by E so3 �� and E so3
� , respec-
tively.
The calculation of the field � ( , )k kx y in graphite bi-
layer is similar to the calculation of this quantity in
1016 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
G.P. Mikitik and Yu.V. Sharlai
E
h
e
Ec ky
kxK
Fig. 2. Shown is the dispersion law �( , )k kx y of the electrons (e)
and holes (h) in the graphite bilayer near the point K. At the en-
ergy Ec all the electron cones merge. Note that without neglecting
the parameter �4 0044� . eV, the spectrum �( , )k kx y is slightly
asymmetric relative to the plane � � 0: The Dirac points in the
central and side cones are at energies � � 0 and � �2 4 1 3
2
0
3� � � �/ �
� 015. Ec, respectively. Thus, in contrast with graphene, the graph-
ite bilayer is a semimetal, and changing the Fermi level in it, one
cannot obtain the concentration of the charge carriers less than
~ 108 cm�2. Scales of the plot are distorted for clarity.
0.02
0.01
0
–0.01
–0.02
0 0.01 0.02–0.01
Fig. 3. The field �( , )k kx y (arrows) for the electron band e in the
bilayer. The length of the arrows is proportional to | |�. Small
circles mark the positions of the Dirac points in the k kx y� plane,
and the solid line shows an electron orbit with � � Ec. Note that
near the central and the side Dirac points the vector � «circu-
lates» in opposite directions. The components ki are given in the
dimensionless units & �i i/ k / K� ( )( | | )2 3 � .
graphene. Eventually we arrive at the result shown in
Fig. 3. Note that near the central and the side Dirac points
the vector � «circulates» in opposite directions, which
means the opposite signs of the Berry phases generated by
these points. The calculation of the integral (1) gives the
Berry phase � for electron orbits surrounding each of the
three side Dirac points and �� for an orbit around the cen-
tral point [12]. Since the Berry phase of an electron does
not depend on a size or a shape of its orbit in the Brillouin
zone but is determined only by the Dirac points enclosed
by the orbit, one finds the Berry phase 3 2� � �� � for
electrons with energies E E / /c� � �( ) ( )� � �1 3 0
24 2 meV
(i.e., at the electron concentration N N c� � �5 1010 cm–2)
when their orbits surround all these four points. Thus, at
� � Ec we arrive at the same Berry phase 2� as in the case
of the parabolic spectrum (18) in spite of the change in the
degeneracy of the energy bands. The difference in the
Berry phases for the parabolic and real spectra can mani-
fest itself only in the interval � + +E Ec c� .
The coincidence of the Berry phases for the parabolic
and the real spectra can be understood as follows:
McCann and Fal’ko [22] found Landau levels for spec-
trum (18) exactly, and they obtained � ,n c n n� �� ( )1
where , �c v eH/c� 2 2
1 is the cyclotron frequency (v is
proportional to � 0, see above). The expansion of this ex-
pression at large n �� 1 gives � ,n c n /� �� ( )1 2 . Com-
paring this result with the semiclassical formula
� , �n c n�
� ( ) and taking into account Eq. (4), we obtain
�B � 2�. It is clear that these considerations remain true
even with the trigonal warping of the spectrum if this
warping is sufficiently small, Ec c++ �, . Thus, we con-
clude that the Berry phase for the real spectrum has to be
equal to 2� at least at high energies � ,�� ��� c cE . But
this phase does not depend on � in the region � � Ec , and
hence the conclusion remains valid everywhere in this
region.
The Berry phase 2� leads to the half-integer �, Eq. (4),
as in the case of the usual parabolic spectrum. Hence, one
may expect to observe the conventional integer quantum
Hall effect in the graphite bilayer. Such the effect was in-
deed observed [27], but with a specific feature due to the
approximate degeneracy of the Landau levels with n � 0
and n �1.
The Berry phase in graphite
With increasing number of graphite layers, the elec-
tron energy structure of such a multilayer evolves to the
three-dimensional spectrum of bulk graphite, and the Di-
rac points (that occur in the bilayer and in multilayers at
certain values of k z ) merge and transform into the
band-contact lines. According to [25], the multilayers
with 10 or a larger number of layers is practically indistin-
guishable from the bulk material, and now we shall ana-
lyze the spectrum of graphite and the Berry phase in it.
Table 2. Values of the parameters of SWM model (eV) that lead to
the spectrum of Ref. 22.
Parameter Value
�0 3 16 0 05. .�
�1 0 39 0 01. .�
�2 � �0 020 0 002. .
�3 0 315 0 015. .�
�4 0044 0 024. .�
�5 0 038 0 005. .�
� � �0 008 0 002. .
�F � �0 024 0 002. .
As it was mentioned above, the electron energy spec-
trum of graphite is described by the SWM model [23], and
values of the parameters of this model were found suffi-
ciently accurately from the analysis of various experi-
mental data; see the review of Brandt et al. [28] and Table
2. In Fig. 4 the electron energy bands of graphite are
shown. In the edge HKH of the Brillouin zone of graphite
the bands � 2( )k and � 3( )k are degenerate. In other words,
this edge is the band-contact line. Apart from the
The Berry phase in graphene and graphite multilayers
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1017
kz
H
�
K
H
P0
ky
0
–1
K kz P0 H
Fig. 4. Dependences of the bands � i (i � �1 4) on k near the edge
HKH in graphite. Shown are the dependences � i zk( ) at k* � 0
and the dependences of � i on k k kx y* �
2 2 at some characteris-
tic values of kz. The dashed line marks the position of the Fermi
level �F . For clarity, in the construction of the figure the parame-
ter �1 has been used which is twenty times smaller than that of
Table 2. For realistic value of �1 the point P0 is much closer to the
point H than in the figure. The insert shows the Brillouin zone of
graphite and its characteristic points.
band-contact line coinciding with the edge HKH, three
additional band-contact lines of the same bands exist near
this edge [29], Fig. 5. The contact in these three side lines
is accidental [9]. It is due to the trigonal warping of the
spectrum. In the vicinity of the point K the three addi-
tional band-contact lines can be approximately found
from the equations [30]:
cos , cos3 1
4 3 1
0
2
2( &
� �
�
"� � . (19)
Note that at cos ." � 0 5 these equations give positions of
the Dirac points in the bilayer. Thus, one may say that the
four band-contact lines in graphite develop from the four
Dirac points of the bilayer, Fig. 2.
The bands � 2( )k and � 3( )k are degenerate along KH in
the interval from K to the point P0 defined by the condition
E E1 3( ) ( )" "� (i.e., by the equality cos" � | |/� 2 1� � 0 01. ),
while from P0 to the point H the degenerate bands are � 3( )k
and � 4 ( )k . The change of the band degeneracy at P0 can be
understood if one takes into account an infinitesimal spin-
orbit interaction in SWM model. Thus, the situation in gra-
phite can be described as follows: The four contact lines of
the bands � 2( )k and � 3( )k come to the point P0 from the one
side of the HKH axis, and the four contact lines of the bands
� 3( )k and � 4 ( )k come to this point from the opposite side,
and all these lines merge at the point P0. It is essential that in
the vicinity of all these lines the band splitting is linear in a
deviation of k from the lines.
The Fermi surface of graphite consists of elongated
pockets enclosing the edge HKH of its Brillouin zone.
These pockets are formed by electrons and holes which
are located near the points K and H of the Brillouin zone,
respectively, see Fig. 5. The electron pocket is formed by
electrons of the third band � 3( )k , while the hole group be-
longs to the second band � 2( )k . All the contact lines of
the bands � 2( )k and � 3( )k lie under the Fermi surface of
the electron and hole groups. When the lines pass from
the electron part of the surface to its hole part, the conical
features of the Fermi surface, the so-called «outrigger»
pieces [28], appear, Fig. 5. The lines are just axes of these
four pieces connecting the electron and hole parts. In the
magnetic field H parallel to a band-contact line the split-
ting of the Landau levels for the electron states near the
line is proportional to H , see Eq. (14). In the case of
graphite these levels are known as the so-called «leg lev-
els» investigated by Dresselhaus [31]. The existence of
the leg levels (and, in fact, of the band-contact lines in
graphite) was confirmed by magneto-optical experiments
[32,33]. Interestingly, the well-known large diamagne-
tism of graphite [34] is also caused by electron states near
the band-contact lines [35].
When the magnetic field H is directed along the z axis,
the maximum electron cross section in k z is located at " � 0,
while the maximum cross section of the holes is between the
points K and P0, viz., at cos ( ) ./" � �� � �F /6 0 452
1 2 where
�F is the Fermi energy in graphite, see Fig. 5. Thus, both
these cross sections are penetrated by the four band-contact
lines. The Berry phases of these lines coincide with the
Berry phases of the corresponding Dirac points in the graph-
ite bilayer. Hence, for both the extremal cross sections of
graphite the Berry phase is 2�, and one can expect to find the
usual half-integer � in the oscillation effects. Such the value
of � was indeed observed in Refs. 36,37.
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The Berry phase in graphene and graphite multilayers
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1019
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