The electronic properties of bilayer graphene
Edward McCann
Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
Mikito Koshino
Department of Physics, Tohoku Universi ty, Sendai, 980-8578, Japan
We review the electronic properties of bilayer graphene, beginning with a description of the tight-
binding model of bilayer graphene and the derivation of the effective Hamiltonian describing m assive
chiral quasiparticles in two parabolic bands at low energy. We take into account five tight-binding
parameters of the Slonczewski-Weiss-McClure model of bulk graphite plus intra- and interlayer
asymmetry between atomic sites which induce band gaps in the low-energy spectrum. The Hartree
mod el of screening and band-gap opening due to interlayer asymmetry in the presence of ex ternal
gates is presented. The tight-binding model is used to describe optical and transport properties
including the integer quantum Hall effect, and we also discuss orbital magnetism, phonons and the
influence of strain on electronic properties. We conclude with an overview of electronic interaction
effects.
CONTENTS
I. Introduction 1
II. Electronic band structure 2
A. The crystal structure and the Brillouin
zone 2
B. The tight-binding model 3
1. An arbitrary crystal structure 3
2. Monolayer g raphene 4
3. Bilayer graphene 5
4. Effective four-band Hamiltonian near the
K points 7
C. Effective two-band Hamiltonian at low
energy 7
1. General procedure 7
2. Bilayer graphene 8
D. Interlayer coupling γ
1
: massive chiral
electrons 8
E. Interlayer coupling γ
3
: trigonal warping and
the Lifshitz transition 9
F. Interlayer coupling γ
4
and on-site parameter
∆
′
: electron-hole asymmetry 9
G. Asymmetry between on-site energie s: band
gaps 9
1. Interlayer asymmetry 9
2. Intralayer asymmetry between A and B
sites 10
H. Next-nearest neighbour hopping 10
I. Spin-orbit coupling 10
J. The integer quantum Hall effect 11
1. The Landau level spectrum of bilayer
graphene 11
2. Three types o f integer qua ntum Hall
effect 12
3. The role of interlayer asymmetry 13
III. Tuneable band gap 13
A. Experiments 13
B. Hartree model of screening 13
1. Electrostatics: asymmetr y parameter,
layer densities and external gates 13
2. Calculatio n of individual layer densities 14
3. Self-consistent screening 15
IV. Transpor t properties 15
A. Introduction 15
B. Ballistic transport in a finite system 16
C. Transport in disordered bilayer graphene 18
1. Conductivity 18
2. Localisation effects 20
V. Optical properties 20
VI. Orbital magnetism 21
VII. Phonons and strain 22
A. The influence of strain on elec trons in
bilayer gr aphene 22
B. Phonons in bilayer graphene 22
C. Optical phonon anomaly 23
VIII. E lectronic interactions 24
IX. Summary 25
References 26
I. INTRODUCTION
The production by mechanical exfoliation of isolated
flakes of graphene with excellent conducting proper ties
[1] was soon followed by the observation of an unusual
sequence of plateaus in the integer quantum Hall effect in
monolayer graphene [2, 3]. This confirmed the fact that
charge car riers in monolayer graphene are ma ssless chiral
quasiparticles with a linear dispersion, as described by a
Dirac-like effective Hamiltonian [4–6], and it prompted
an explo sion of interest in the field [7].
2
The integer quantum Hall effect in bilayer graphene [8]
is arguably even more unusual than in monolayer because
it indicates the presence of massive chiral quasiparticles
[9] with a para bolic dispersion at low energy. The effec-
tive Hamiltonian of bilayer graphene may be viewed as
a generalisation of the Dirac-like Hamiltonia n of mono-
layer graphene and the seco nd (after the monolayer) in a
family of chiral Hamiltonians that appear at low energy
in ABC-stacked (rhombohedral) multilayer graphene [9–
15]. In addition to interesting underlying physics, bilayer
graphene holds potential for elec tronics applications, not
least because of the possibility to control both carrier
density and ene rgy band gap through doping or gating
[9, 10, 16–2 0].
Not surprisingly, many of the properties of bilayer
graphene ar e similar to those in monolayer [7, 21]. These
include excellent electrica l conductivity with room tem-
perature mobility of up to 40, 000 cm
2
V
−1
s
−1
in air [22];
the poss ibility to tune electrical pr operties by changing
the carrier density thro ugh gating o r doping [1, 8, 16];
high thermal conductivity with room temperature ther-
mal conductivity of about 2, 800 W m
−1
K
−1
[23, 24]; me-
chanical stiffness, strength and flexibility (Young’s mod-
ulus is estimated to be about 0.8 TPa [25, 26]); tr ans-
parency with transmittance of white light of about 95 %
[27]; impermeability to gases [28]; and the ability to be
chemically functionalised [29]. Thus, as with monolayer
graphene, bilayer graphene has potential for future appli-
cations in many areas [21] including transparent, flexible
electrodes for touch screen displays [30]; high-frequency
transistor s [31]; thermoelectric devices [32]; photonic de-
vices including plasmonic devices [33] and photodetectors
[34]; energy applications including batteries [35, 36]; and
composite materials [37, 38].
It should be stressed, however, that bilayer graphene
has features that make it distinct from monolayer. The
low-energy band structure, described in detail in Sec-
tion II, is different. Like monolayer, intrinsic bilayer has
no band gap between its conduction and valence bands,
but the low-energy dispersion is quadratic (rather than
linear as in monolayer) with massive chiral quasiparti-
cles [8, 9] rather than massless ones. As there are two
layers, bilayer graphene represents the thinnest possible
limit of an intercalated material [35, 36]. It is possible
to address each layer separately leading to entir ely new
functionalities in bilayer graphene including the poss ibil-
ity to control an energy band gap of up to about 300 meV
through doping or gating [9, 10, 16–20]. Re c ently, this
band gap has been used to create devices - constrictions
and dots - by electrosta tic confinement with gates [39].
Bilayer or multilayer graphene devices may also be prefer-
able to monolayer ones when there is a need to use more
material for increased e lectrical or thermal conduction,
strength [37, 38], or optical signature [3 3].
In the following we review the electronic properties
of bilayer graphene. Section II is an overview of the
electronic tight-binding Hamiltonian and r esulting band
structure describing the low-energy chiral Hamiltonian
and taking into account different parameters that cou-
ple atomic orbitals as well a s external factors that may
change the electron bands by, for example, opening a
band gap. We include the Landau level spectrum in the
presence of a perpendicular magnetic field and the corre-
sponding intege r quantum Hall effect. In section III we
consider the opening of a band gap due to doping or gat-
ing and present a simple analytical model that descr ibes
the density-dependence of the band gap by taking into
account screening by electrons on the bilayer device. The
tight-binding model is use d to describe tr ansport prop-
erties, se ction IV, and optical pr operties, section V. We
also discuss orbital magnetism in section VI, phonons
and the influence of strain in section VII. Section VIII
concludes with an overview of electronic-interaction ef-
fects. Note that this review considers Bernal-stacked
(also known as AB-stacked) bilayer graphene; we do
not consider other stacking types such as AA-stacked
graphene [40], twisted graphene [41–46] or two graphene
sheets separated by a dielectric with, possibly, electronic
interactions between them [47–52].
II. ELECTRONIC BAND STRUCTURE
A. The crystal structure and the Brillouin zone
Bilayer graphene consists of two coupled monolayers
of carbon atoms, each with a honeycomb crysta l struc-
ture. Figure 1 shows the cry stal structure of monolayer
graphene, figure 2 shows bilayer graphene. In both cases,
primitive lattice vectors a
1
and a
2
may be defined as
a
1
=
a
2
,
√
3a
2
!
, a
2
=
a
2
, −
√
3a
2
!
, (1)
where a = |a
1
| = |a
2
| is the lattice constant, the distance
between adjacent unit cells, a = 2.46
˚
A [53]. Note that
the lattice constant is distinct from the ca rbon-carbon
bond length a
CC
= a/
√
3 = 1.4 2
˚
A, which is the distance
between adjacent carbon atoms.
In monolayer graphene, each unit cell contains two
carbon atoms, labelled A and B, figure 1(a). The po-
sitions of A a nd B atoms are not equiva lent because it
is not possible to connect them with a lattice vector of
the form R = n
1
a
1
+ n
2
a
2
, where n
1
and n
2
are inte-
gers. Bilayer graphene consists of two coupled monolay-
ers, with fo ur atoms in the unit cell, labelled A1, B1
on the lower layer and A2, B2 on the upper layer. The
layers are arranged so that one of the atoms from the
lower layer B1 is directly below an atom, A2, from the
upper layer. We refer to these two atomic sites as ‘dimer’
sites because the electronic orbitals on them are cou-
pled together by a relatively strong interlayer coupling.
The other two ato ms, A1 and B2, don’t have a counter-
part on the other layer that is directly above or below
them, and are referred to as ‘non-dimer’ sites. Note that
some authors [10, 54–56] employ different definitions of
3
B
A
x
y
a
a
1
a
2
b
1
b
2
k
y
k
x
K
+
K
-
M
G
( a )
( b )
FIG. 1. (a) Crystal structure of m on olayer graphene with
A (B) atoms shown as white (b lack) circles. The shaded
rhombus is the conventional unit cell, a
1
and a
2
are primitive
lattice vectors. (b) Reciprocal lattice of monolayer and bilayer
graphene with lattice points indicated as crosses, b
1
and b
2
are primitive reciprocal lattice vectors. The shaded hexagon
is the first Brillouin zone with Γ indicating the centre, and
K
+
and K
−
showing two non-equivalent corners.
A and B sites as used here. The point group o f the bi-
layer crystal structure is D
3d
[12, 57, 58] consisting of
elements ({E, 2C
3
, 3C
′
2
, i, 2S
6
, 3σ
d
}), and it may be re-
garded as a direct product of group D
3
({E, 2C
3
, 3C
′
2
})
with the inversion group C
i
({E, i}). Thus, the lattice
is symmetric with respect to spatial invers ion symmetry
(x, y, z) → (−x, −y, − z).
Primitive rec iproca l lattice vectors b
1
and b
2
of mono-
layer and bilayer graphene, where a
1
· b
1
= a
2
· b
2
= 2π
and a
1
· b
2
= a
2
· b
1
= 0, are given by
b
1
=
2π
a
,
2π
√
3a
, b
2
=
2π
a
, −
2π
√
3a
. (2)
As shown in figure 1(b), the reciprocal lattice is an hexag-
onal Bravais lattice, and the first Brillouin zone is an
hexagon.
B 2
A 1
x
y
a
a
1
a
2
( a )
A 2
B 1
A 1
B 2
B 1
A 2
g
1
g
1
g
0
g
0
( b )
g
3
a
g
4
FIG. 2. (a) Plan and ( b) side view of the crystal structure of
bilayer graphene. Atoms A1 and B1 on the lower layer are
shown as white and black circles, A2, B2 on the upper layer
are black and grey, respectively. The shaded rhombus in (a)
indicates the conventional unit cell.
B. The tight-binding model
1. An arbitrary crystal structure
In the following, we will describe the tight-binding
model [53, 59, 60] and its application to bilayer graphene.
We beg in by considering an arbitrary crystal with trans-
lational invariance and M atomic orbitals φ
m
per unit
cell, labelled by index m = 1 . . . M. Bloch states
Φ
m
(k, r) for a given pos ition vector r and wave vector
k may be written as
Φ
m
(k, r) =
1
√
N
N
X
i=1
e
ik.R
m,i
φ
m
(r − R
m,i
) , (3)
where N is the number of unit cells, i = 1 . . . N labels
the unit cell, and R
m,i
is the position vector of the mth
orbital in the ith unit c ell.
The electronic wave function Ψ
j
(k, r) may be ex-
pressed as a linear superposition of Bloch states
Ψ
j
(k, r) =
M
X
m=1
ψ
j,m
(k) Φ
m
(k, r) , (4)
where ψ
j,m
are expansion coefficients. There are M dif-
ferent energy bands, and the energy E
j
(k) of the jth
band is given by E
j
(k) = hΨ
j
|H|Ψ
j
i/hΨ
j
|Ψ
j
i where H
is the Hamilto nian. Minimising the energy E
j
with re-
spect to the expansion coefficients ψ
j,m
[53, 60] leads to
Hψ
j
= E
j
Sψ
j
, (5)
4
where ψ
j
is a column vector, ψ
T
j
= (ψ
j1
, ψ
j2
, . . . , ψ
jM
).
The transfer integral matrix H and over lap integral ma-
trix S are M ×M matrices with matrix elements defined
by
H
mm
′
= hΦ
m
|H|Φ
m
′
i, S
mm
′
= hΦ
m
|Φ
m
′
i. (6)
The band energies E
j
may be determined from the gen-
eralised eigenvalue equation (5) by solving the sec ular
equation
det (H − E
j
S) = 0 , (7)
where ‘det’ stands for the determinant of the matrix.
In order to model a given system in terms of the gener-
alised eigenvalue problem (5), it is necessary to determine
the matrices H and S. We will proceed by considering
the relatively simple case of monolayer graphene, before
generalising the approach to bilayers. In the following
sections, we will omit the subs cript j on ψ
j
and E
j
in
equation (5), remembering that the numbe r of solutions
is M , the number of orbitals per unit cell.
2. Monolayer graphene
Here, we will outline how to apply the tight-binding
model to graphene, and refer the reader to tutorial-style
reviews [53, 60] for further details. We take into account
one 2p
z
orbital per atomic site and, as there are two
atoms in the unit cell of monolayer graphene, figure 1(a),
we include two orbita ls per unit cell labelled as m = A
and m = B (the A atoms and the B atoms a re each
arranged on an hexagonal Bravais lattice).
We begin by considering the diagonal e lement H
AA
of the transfer integral matrix H, equation (6), for the
A site orbital. It may be determined by substituting the
Bloch function (3) for m = A into the matrix element (6),
which results in a double sum over the positions o f the
unit cells in the crystal. Ass uming that the dominant
contribution arises from those terms involving a given
orbital interacting with itself (i.e., in the same unit cell),
the ma trix element may be written as
H
AA
≈
1
N
N
X
i=1
hφ
A
(r − R
A,i
) |H|φ
A
(r − R
A,i
)i. (8 )
This may be regarded as a summation over all unit cells of
a parameter ǫ
A
= hφ
A
(r − R
A,i
) |H|φ
A
(r − R
A,i
)i that
takes the same value in every unit cell. Thus, the matrix
element may b e simply expressed as H
AA
≈ ǫ
A
. Simi-
larly, the dia gonal element H
BB
for the B site orbital can
be written as H
BB
= ǫ
B
, while for intrinsic graphene ǫ
A
is equal to ǫ
B
as the two s ublattices are identical. The
calculation of the diagonal elements of the overlap inte-
gral ma trix S, equation (6), proceeds in the same way
as that of H, with the overla p of an orbital with itself
equal to unity, hφ
j
(r − R
j,i
) |φ
j
(r − R
j,i
)i = 1. Thus,
S
BB
= S
AA
= 1.
The off-diagonal element H
AB
of the tra nsfer integral
matrix H describes the possibility of hopping between
orbitals on A and B sites. Substituting the Bloch func-
tion (3) into the matrix element (6) results in a s um over
all A sites and a sum over all B sites. We assume that
the dominant contribution arises from hopping between
adjacent sites. If we consider a given A site, say, then we
take into account the possibility of hopping to its three
nearest-neighbour B sites, la belled by index l = 1, 2, 3:
H
AB
≈
1
N
N
X
i=1
3
X
l=1
e
ik.δ
l
×hφ
A
(r − R
A,i
) |H|φ
B
(r − R
A,i
− δ
l
)i, (9 )
where δ
l
are the p ositions of three nearest B atoms rel-
ative to a given A atom, which may be written as δ
1
=
0, a/
√
3
, δ
2
=
a/2, −a/2
√
3
, δ
3
=
−a/2, −a/2
√
3
.
The sum with respect to the three nearest-neighbour
B sites is identical for every A site. A hopping parameter
may be defined as
γ
0
= −hφ
A
(r − R
A,i
) |H|φ
B
(r − R
A,i
− δ
l
)i, (10)
which is positive. Then, the matr ix element may be writ-
ten as
H
AB
≈ −γ
0
f (k) ; f (k) =
3
X
l=1
e
ik.δ
l
, (11)
The other off-diag onal matrix element is given by H
BA
=
H
∗
AB
≈ −γ
0
f
∗
(k). The function f (k) describing
nearest-neighbor hopping, equation (11), is given by
f (k) = e
ik
y
a/
√
3
+ 2e
−ik
y
a/2
√
3
cos (k
x
a/2) , (12)
where k = (k
x
, k
y
) is the in-plane wave vector. The cal-
culation of the o ff-diagonal elements of the overlap in-
tegral matrix S is similar to those of H. A parameter
s
0
= hφ
A
(r − R
A,i
) |φ
B
(r − R
B,l
)i is introduced to de-
scribe the possibility of non-zero overlap between orbitals
on adjacent sites, giving S
AB
= S
∗
BA
= s
0
f (k).
Gathering the matrix elements, the transfer H
m
and
overlap S
m
integ ral matrices of monolayer graphene may
be written as
H
m
=
ǫ
A
−γ
0
f (k)
−γ
0
f
∗
(k) ǫ
B
, (13)
S
m
=
1 s
0
f (k)
s
0
f
∗
(k) 1
. (14)
The corresponding energy may be determined [53] by
solving the secular eq uation (7). For intrinsic graphene,
i.e., ǫ
A
= ǫ
B
= 0, we have
E
±
=
±γ
0
|f (k) |
1 ∓ s
0
|f (k) |
. (15)
The para meter values are listed by Saito et al [53] as
γ
0
= 3.033 eV and s
0
= 0.129.
The function f(k), equation (12) is zero at the corners
of the B rillouin zone, two of which are non-equivalent
5
(i.e., they are not connected by a reciprocal lattice vec-
tor). For example, corners K
+
and K
−
with wave vectors
K
±
= ±(4π/3a, 0) are labelled in Figure 1(b). Such po-
sitions are called K points or valleys, and we will use a
valley index ξ = ±1 to distinguish points K
ξ
. At these
positions, the solutions (15) are degenera te, mar king a
crossing point and zero band gap between the conduction
and valence bands. T he transfer matrix H
m
is approxi-
mately equa l to a Dirac-like Hamilto nian in the vicinity
of the K point, describing massless chiral quasiparticles
with a line ar dispersion relation. These points ar e partic-
ularly important b e cause the Fermi level is located near
them in pristine graphene.
3. Bilayer graphene
In the tight-binding description of bilayer graphene, we
take into account 2p
z
orbitals on the four atomic sites in
the unit cell, labelled as j = A1, B1, A2, B2. Then, the
transfer integral matrix of bilayer gra phene [9, 10, 54, 61–
63] is a 4 × 4 matrix given by
H
b
=
ǫ
A1
−γ
0
f (k) γ
4
f (k) −γ
3
f
∗
(k)
−γ
0
f
∗
(k) ǫ
B1
γ
1
γ
4
f (k)
γ
4
f
∗
(k) γ
1
ǫ
A2
−γ
0
f (k)
−γ
3
f (k) γ
4
f
∗
(k) −γ
0
f
∗
(k) ǫ
B2
,
(16)
where the tight-binding parameters are defined as
γ
0
= −hφ
A1
|H|φ
B1
i = −hφ
A2
|H|φ
B2
i, (17)
γ
1
= hφ
A2
|H|φ
B1
i, (18)
γ
3
= −hφ
A1
|H|φ
B2
i, (19)
γ
4
= hφ
A1
|H|φ
A2
i = hφ
B1
|H|φ
B2
i. (20)
Here, we use the notation of the Slo nc zewski-Weiss-
McClure (SWM) model [64–67] that describes bulk
graphite. Note that definitions of the parameters used
by authors can differ, particularly with respect to sig ns.
The upper-left and lower-right square 2×2 blocks of H
b
describe intra-layer terms and are simple generalisations
of the monolayer, equation (13). For bilayer graphene,
however, we include parameters describing the on-site
energies ǫ
A1
, ǫ
B1
, ǫ
A2
, ǫ
B2
on the four atomic sites, that
are not equal in the most general case. As ther e ar e four
sites, differences between them are described by three
parameters [63]:
ǫ
A1
=
1
2
(−U + δ
AB
) , (21)
ǫ
B1
=
1
2
(−U + 2∆
′
− δ
AB
) , (22)
ǫ
A2
=
1
2
(U + 2∆
′
+ δ
AB
) , (23)
ǫ
B2
=
1
2
(U − δ
AB
) , (24)
where
U =
1
2
[(ǫ
A1
+ ǫ
B1
) − (ǫ
A2
+ ǫ
B2
)] , (25)
∆
′
=
1
2
[(ǫ
B1
+ ǫ
A2
) − (ǫ
A1
+ ǫ
B2
)] , (26)
δ
AB
=
1
2
[(ǫ
A1
+ ǫ
A2
) − (ǫ
B1
+ ǫ
B2
)] . (27)
The three independent parameters are U to describe in-
terlayer asymmetr y between the two layers [9, 10, 16–
20, 68–74], ∆
′
for an energy difference between dimer
and no n-dimer sites [54–56, 67], and δ
AB
for an energy
difference between A and B sites on each layer [63, 75].
These parameters are described in detail in sections II F
and II G.
The upper-right and lower-left square 2 × 2 blocks of
H
b
describe inter-layer coupling. Parameter γ
1
describes
coupling between pairs of orbitals on the dimer sites B1
and A2: sinc e this is a vertical coupling, the corresp ond-
ing terms in H
b
(i.e., H
A2,B1
= H
B1,A2
= γ
1
) do not
contain f (k) which describes in-plane hopping. Param-
eter γ
3
describes interlayer coupling between non-dimer
orbitals A1 and B2, and γ
4
describes interlayer c oupling
between dimer and non-dimer orbitals A1 and A2 or B1
and B2. Both γ
3
and γ
4
couplings are ‘skew’: they are
not strictly vertical, but involve a compo ne nt of in-plane
hopping, and each atom on one layer (e.g., A1 for γ
3
) has
three equidistant nearest-neighbours (e.g., B2 for γ
3
) on
the other layer. In fact, the in-plane component of this
skew hopping is analogous to nearest-neighbour hopping
within a single layer, as parameterised by γ
0
. Hence,
the skew interlayer hopping (e.g., H
A1,B2
= −γ
3
f
∗
(k))
contains the factor f (k) describing in-plane hopping.
It is possible to introduce an overlap integral matrix
for bilayer graphene [63]
S
b
=
1 s
0
f (k) 0 0
s
0
f
∗
(k) 1 s
1
0
0 s
1
1 s
0
f (k)
0 0 s
0
f
∗
(k) 1
, (28)
with a form that mirrors H
b
. Here, we only include
two parameters: s
0
= hφ
A1
|φ
B1
i = hφ
A2
|φ
B2
i describing
non-orthogonality of intra-layer nearest-neighbours and
s
1
= hφ
A2
|φ
B1
i describing non-orthogonality of orbitals
on dimer sites A1 and B2. In principle, it is possible
to introduce additional parameters analogous to γ
3
, γ
4
,
etc., but generally they will be small and irrelevant. In
fact, it is common practice to neg lect the overlap inte-
gral matrix entirely, i.e., replace S
b
with a unit matrix,
because the influence of parameters s
0
and s
1
describ-
ing non-orthogonality of adjace nt orbitals is small at low
energy |E| ≤ γ
1
. Then, the generalised eigenvalue equa-
tion (5) reduces to an eigenvalue equation H
b
ψ = Eψ
with Hamiltonian H
b
, equation (16).
The energy differences U and δ
AB
are usually at-
tributed to extrinsic factors such as gates, substrates
or doping. Thus, there are five independent parame-
ters in the Hamiltonian (16) of intrinsic bilayer graphene,
namely γ
0
, γ
1
, γ
3
, γ
4
and ∆
′
. The band structure pre-
dicted by the tight-binding model ha s been compared to
observations from photoemission [16], Raman [76] and
infrared spectroscopy [55, 56, 78–81]. Parameter values
determined by fitting to e xperiments are listed in Table I
for bulk g raphite [67], for bilayer graphene by Raman
[76, 77] and infrared [55, 56, 80] spectroscopy, and for
![FIG. 3. Low-energy bands of bilayer graphene arising from 2pz orbitals plotted along the kx axis in reciprocal space intersecting the corners K−, K+ and the centre Γ of the Brillouin zone. The inset shows the bands in the vicinity of the K+ point. Plots were made using Hamiltonian Hb, equation (16), with parameter values γ0 = 3.16 eV, γ1 = 0.381 eV, γ3 = 0.38 eV, γ4 = 0.14 eV, ǫB1 = ǫA2 = ∆ ′ = 0.022 eV, and ǫA1 = ǫB2 = U = δAB = 0 [80].](/figures/fig-3-low-energy-bands-of-bilayer-graphene-arising-from-2pz-24mkpfko.webp)
![FIG. 15. Schematic of the distribution of spin, valley and layer degrees of freedom in candidates for spontaneous gapped states in bilayer graphene [14, 87, 311]. They each contribute a term proportional to σz in the two-band Hamiltonian (38), with its sign depending on the valley ξ = ±1 and spin s = ±1 orientation. Arrows indicate spin orientation, located at valley K+ (solid) or K− (dashed) and on different layers (top or bottom) of a state at the top of the valence band.](/figures/fig-15-schematic-of-the-distribution-of-spin-valley-and-30muqiql.webp)
![FIG. 7. Density-dependence of the band gap Ug(n) in bilayer graphene as the back-gate voltage Vb is varied for a fixed top-gate voltage Vt [17]. The effective offset-density is n0 = 2[ε0εtVt/(eLt) + nt0]. Plots were made for screening parameter Λ = 1, using Ug = |U |γ1/ √ U2 + γ21 and numerical solution of equation (74).](/figures/fig-7-density-dependence-of-the-band-gap-ug-n-in-bilayer-297qwrle.webp)
![FIG. 12. Interband part of the dynamical conductivity of bilayer graphene plotted against the frequency ω, in (a) zero magnetic field with different εF ’s [229] (b) several magnetic fields with εF = 0 [230]. Dashed curves in (b) indicate the transition energies between several Landau levels.](/figures/fig-12-interband-part-of-the-dynamical-conductivity-of-a8q2u9m6.webp)
![FIG. 13. (a) Density of states, and (b) susceptibility of bilayer graphenes with the asymmetry gap U/γ1 = 0, 0.2, and 0.5 [243]. In (b), the upward direction represents negative (i.e., diamagnetic) susceptibility.](/figures/fig-13-a-density-of-states-and-b-susceptibility-of-bilayer-1z9bhc0t.webp)
