Strain and electric field modulation of the electronic structure of bilayer graphene
B. R. K. Nanda and S. Satpathy
Department of Physics & Astronomy, University of Missouri, Columbia, Missouri 65211, USA
共Received 31 May 2009; revised manuscript received 1 October 2009; published 29 October 2009
兲
We study how the electronic structure of the bilayer graphene 共BLG兲 is changed by electric field and strain
from ab initio density-functional calculations using the linear muffin-tin orbital and the linear augmented plane
wave methods. Both hexagonal and Bernal stacked structures are considered. We only consider interplanar
strain where only the interlayer spacing is changed. The BLG is a zero-gap semiconductor like the isolated
layer of graphene. We find that while strain alone does not produce a gap in the BLG, an electric field does so
in the Bernal structure but not in the hexagonal structure. The topology of the bands leads to Dirac circles with
linear dispersion in the case of the hexagonally stacked BLG due to the interpenetration of the Dirac cones,
while for the Bernal stacking, the dispersion is quadratic. The size of the Dirac circle increases with the applied
electric field, leading to an interesting way of controlling the Fermi surface. The external electric field is
screened due to polarization charges between the layers, leading to a reduced size of the band gap and the
Dirac circle. The screening is substantial in both cases and diverges for the Bernal structure for small fields as
has been noted by earlier authors. As a biproduct of this work, we present the tight-binding parameters for the
free-standing single layer graphene as obtained by fitting to the density-functional bands, both with and without
the slope constraint for the Dirac cone and keeping the hopping integral up to four near neighbors.
DOI: 10.1103/PhysRevB.80.165430 PACS number共s兲: 81.05.Uw, 73.22.⫺f
I. INTRODUCTION
Recently, bilayer graphene 共BLG兲 has been shown to pos-
sess a band gap in the presence of an external electric field,
1,2
an effect that has important implications for transport across
the graphene layers and for possible device applications.
While the freestanding BLG is a zero band-gap
semiconductor
3,4
like the single layer graphene 共SLG兲,an
applied electric field through an external gate induces an
asymmetric potential between the graphene layers, which in
turn opens a gap between the valence and the conduction
bands in the Bernal structure.
4–8
It has been recently noted
10
that graphene exhibits the hexagonal stacking surprisingly
often, surprising because of the higher energy of the hexago-
nal structure. In the hexagonal structure, the two graphene
layers are stacked vertically on top of one another, while in
the Bernal structure, one layer is rotated with respect to the
other as indicated in Fig. 1, so that half of the atoms are
directly above the carbon atoms in the first layer, while the
remaining half lie on top of the hexagon centers. The differ-
ence in symmetry between the Bernal and the hexagonal
structures leads to substantial differences in the band struc-
ture, especially, in the formation of a band gap, an effect we
study in this paper.
To tune the field-induced band gap of the BLGs for prac-
tical applications in nanoelectronic devices, it is important to
examine the factors having strong influence on its electronic
structure. Electronic structure of BLG was shown to be sen-
sitive to the interlayer spacing due to coupling between the
graphene layers.
11,12
Hence, a uniaxial strain, which changes
the interlayer spacing, could modulate the field-induced band
gap.
13,14
The other factor that controls the gap is the screen-
ing of the electric field. An applied electric field causes un-
equal charge distribution among the graphene layers which
in turn creates a polarized electric field in the opposite
direction,
7
thereby reducing the effective electric field. A de-
tail investigation through density-functional calculations is
required to understand the effect of the electric field and
strain on the electronic structure.
While the electronic structure of the Bernal BLG has been
theoretically examined, such studies have not been per-
formed for the hexagonal structure to our knowledge. In this
paper, we report results from density-functional calculations
and tight-binding models to examine the modification of
electronic structure of both Bernal and hexagonal stacked
BLG by electric field and strain. In the Bernal structure, the
screening of the external electric field substantially reduces
the band gap as shown earlier, while in the hexagonal struc-
ture the screening is not as strong, as explained from a
simple tight-binding model. In the Bernal structure, where a
gap can be opened up by the electric field, strain can be used
in addition to modify its magnitude. We find that the gap can
be increased considerably by reducing the interlayer spacing
~
B
~
A
Hexagonal
3.8
Bernal
Interla
y
er S
p
acin
g(
Å
)
3.0 3.4
15
25
35
−5
5
(c)
Energy (meV/atom)
~
A
~
B
(a) (b)
Γ KM
−5
0
5
10
−10
Γ KM
Energy
(
eV
)
A
B
Bernal Stackin
g
Hexa
g
onal Stackin
g
P
A
B
LAPW
++
E
−−
FIG. 1. Band structure of Bernal stacked 共a兲 and hexagonal
stacked 共b兲 BLG without any strain or electric field as obtained
from the LMTO calculations. Fig. 1共c兲 shows the computed total
energy per carbon atom as a function of the interlayer spacing.
PHYSICAL REVIEW B 80, 165430 共2009兲
1098-0121/2009/80共16兲/165430共7兲 ©2009 The American Physical Society165430-1
from its equilibrium value by a small amount. In the hexago-
nal structure, the gap cannot be opened up; however, an in-
teresting point for this structure is that the topology of the
bands leads to Dirac circles centered about the K point in the
Brillouin zone with linear dispersion due to interpenetration
of Dirac cones. Though the electric field does not produce a
gap, it increases the size of the Dirac circles, leading to an
interesting way of controlling the Fermi surface.
Note that the uniaxial strain on BLG, if strong enough,
could create deformations affecting the atomic positions
within the plane, which we have neglected in this paper.
Several authors have studied the effect of such deformations
on the electronic structure.
15,16
Density functional calculations presented in this paper
were performed using the linear muffin-tin orbital 共LMTO兲
Method
17
and linear augmented plane wave 共LAPW兲
method
18
using the local density approximation 共LDA兲.
While the LAPW method was used to obtained the total en-
ergies and the optimized structures, the electronic structure
was investigated using the LMTO method, which did not
produce any substantial difference as far as the band struc-
ture is concerned. To study the field modulated band struc-
ture of the BLG, an extra potential ⌬ was added in the
LMTO method to the on-site energies of the carbon s and p
orbitals of one individual layer. We used the periodic bound-
ary condition normal to the BLG planes, keeping about 10 Å
of vacuum between the different BLG layers and, further-
more, in the LMTO method, we included layers of empty
spheres at positions compatible with the symmetry of the
particular BLG structure. The symmetry compatibility was
especially important for studying the screening effects at low
electric fields.
II. TIGHT-BINDING PARAMETERS FOR A SINGLE
GRAPHENE MONOLAYER
Before we present the results for the BLG, in this section,
we present the tight-binding fitting parameters for the mono-
layer graphene obtained by fitting with the LAPW bands.
These parameters will be used in a tight binding description
of the BLG.
For the freestanding monolayer graphene, it is known that
the formation of the Dirac cones at the Fermi surface is the
outcome of the p
z
-p
z
hopping between the two inequiva-
lent carbon atoms A and B in the unit cell. Though a simple
tight-binding model involving the p
z
-p
z
interaction within the
nearest-neighbor 共NN兲 coordination explains the linear band
dispersion at the Dirac points K and K
⬘
, it is essential to
include further neighbor hoppings if an accurate description
of the band structure in the entire Brillouin zone is desired.
We find that for this, at least three NN hopping integrals
must be retained as indicated from the root-mean-square de-
viation given in Table I and the band structure of Fig. 2.
The tight-binding 共TB兲 Hamiltonian is a 2⫻2 matrix
H =
冉
h
AA
h
AB
h
AB
ⴱ
h
BB
冊
, 共1兲
where A and B denote the two carbon atoms in the unit cell.
Retaining only the first two NN hoppings, the form of the
matrix elements are
h
AA
= t
2
关2 cos共
冑
3k
x
a兲 + 4 cos共
冑
3k
x
a/2兲cos共3k
y
a/2兲兴,
h
AB
= t
1
关2 exp共− ik
y
a/2兲cos共
冑
3k
x
a/2兲 + exp共ik
y
a兲兴, 共2兲
and h
AA
=h
BB
. This can be easily generalized to further near
neighbors. The symbols t
1
, t
2
, t
3
, and t
4
are the NN hopping
parameters as shown in Fig. 2,“a” is the C–C bond length,
and the on-site energy of the carbon orbital is taken to be
zero. The slope of the Dirac cone centered at the K or the K
⬘
point, easily obtained by diagonalizing the Hamiltonian and
taking the limits, is given by the expression
TABLE I. Tight-binding NN hopping integrals obtained from
the least square fitting of the LAPW bands. Fitting was done both
with and without constraining the slope of the Dirac cone to its
LAPW value. Quality of the fit is indicated by the root-mean-square
deviation over the entire Brillouin zone.
Range of
hopping
t
1
共eV兲
t
2
共eV兲
t
3
共eV兲
t
4
共eV兲
RMS
deviation
Without slope constraint
1NN −2.625 0.42
2NN −2.910 0.160 0.22
3NN −2.840 0.170 −0.210 0.11
4NN −2.855 0.170 −0.210 0.105 0.10
With slope constraint
1NN −2.56 0.46
2NN −2.56 0.160 0.32
3NN −2.90 0.175 −0.155 0.12
4NN −2.91 0.170 −0.155 0.02 0.115
t
1
t
4
t
3
t
2
Γ KM
Energy (eV)
(
b
)
B
(
a
)
−10
−5
0
10
5
A
1NN TB fitting
3NN TB fitting
FIG. 2. 共Color online兲 Tight-binding fitting 共red-dashed and
blue-dotted lines兲 of the LAPW bands 共black-solid lines兲 for a
monolayer graphene and without the slope constraint at the Dirac
point 共see text兲. Tight-binding fitting was made for the p
z
bands by
retaining up to four nearest neighbors and the fitting parameters are
presented in Table I. While just the first NN hopping is enough to fit
the slope of the Dirac cone at the K point, at least three NNs must
be retained to describe the band structure well in the entire Brillouin
zone.
B. R. K. NANDA AND S. SATPATHY PHYSICAL REVIEW B 80, 165430 共2009兲
165430-2
⬅关d
⑀
k
/dk兴
K
= a共3t
1
/2−3t
3
+9t
4
/4兲. 共3兲
The slope at the Dirac point is about 5.4 eV·Å, as calcu-
lated from LAPW, which corresponds to the velocity of 8.2
⫻10
5
m/ sec. The TB bands were fitted to the LAPW bands
by keeping hopping integrals for a certain number of NNs
and neglecting the hopping for further neighbors. We ob-
tained two sets of optimized parameters. The first set of TB
parameters was obtained without constraining the slope of
the linear bands at the Dirac point, while the second set was
obtained by constraining the slope to the LAPW value. For
low energy properties, where the magnitude of the slope may
be important, the second set of the parameters should be
used. The results are shown in Fig. 2 and Table I. In our TB
work of the BLG below, we have retained only the first NN
interaction with the slope constraint, since this is sufficient
for our purpose as we are mainly interested in states close to
the band-gap region.
III. BILAYER GRAPHENE IN THE HEXAGONAL
STRUCTURE
As mentioned already, recent experiments suggest that the
hexagonal BLG is surprisingly common
9
in spite of its
higher energy as obtained from the total energy calculation
共Fig. 1 and Ref. 19兲 that indicate the Bernal BLG to be
higher than the hexagonal BLG by ⬃5 meV/ C atom. For
the equilibrium interlayer spacing 共d=3.51 Å兲, the band
structure for the hexagonal stacked BLG is shown in Fig.
3共a兲 and the topology of the bands near the Fermi surface is
shown in Fig. 6, which is discussed in Sec. V. Unlike the
case of the Bernal stacked BLG, here we see two interpen-
etrating Dirac cones, which intersect to form a circular Fermi
surface, the Dirac circle.
An electric field does not change the overall feature of the
band structure 关see Fig. 3共b兲兴 and in contrast to the Bernal
BLG, it does not produce a band gap because of the higher
symmetry situation in the hexagonal stacking. The main fea-
tures of the band structure can be understood from a tight-
binding model involving the nearest-neighbor
interaction
of the p
z
orbitals in the graphene plane and the
interaction
between the planes. With the four inequivalent carbon sites,
one can form the Hamiltonian as in the monolayer case and
in addition add the interlayer coupling term. One can then
expand the Hamiltonian matrix elements for a small momen-
tum around the Dirac point, which then becomes an excellent
approximation for the band-gap region. The result is
H
hex
=
冢
⌬
2
t
0
†
t
−
⌬
2
†
0
0
−
⌬
2
t
0
t
⌬
2
冣
, 共4兲
where
=
3
2
t
1
a is the Fermi velocity for the monolayer,
=k
x
+ik
y
is the complex momentum with respect to the Dirac
point K, ⌬ is the potential difference between the two layers,
t is the interlayer hopping integral 共⬃0.6 eV as extracted
from the LAPW results兲, and the basis set of the Hamiltonian
consists of the Bloch functions made out of the carbon orbit-
als located at A, A
˜
, B
˜
, and B, respectively, and in that order.
The atoms are indicated in Fig. 1. The Hamiltonians near
different K,K
⬘
points in the Brillouin zone differ from Eq.
共4兲 by a phase factor in the interlayer hopping terms
共
,
†
兲; however, the energy eigenvalues remain the same.
The electric field potential ⌬ appearing in the Hamil-
tonian Eq. 共4兲 corresponds to the final screened field experi-
enced by the electrons and not to the bare electric field which
is reduced by the polarization field due to screening, so that
⌬=⌬
ext
/
⑀
, where ⌬
ext
is the external electric field and
⑀
is the
static dielectric constant. The charge difference between the
two layers as a functions of the electric field is shown in Fig.
3共c兲.
The Hamiltonian is easily diagonalized to yield the eigen-
values
共k兲 = ⫾
⫾
k, 共5兲
where
=
冑
⌬
2
/4+t
2
共6兲
specifies the energy locations of the vertices of the Dirac
cones. The energy dispersion is sketched in Fig. 6, which
shows the two Dirac cones with linear dispersion with the
vertices of the cones separated by the energy 2
.
The radius of the circular Fermi surface, the “Dirac”
circle, where the cones intersect the zero of energy, is simply
0
=
/
. Since for small fields the screened potential ⌬ is
proportional to the applied field, the equation indicates that
2ρ
0
Å
−1
ρ
0
)(
Γ KMΓ KM
E = 0 V/nm
E = 2 V/nm
E = 2 V/nm
−2
−4
2
4
0
(a) (b)
Energy (eV)cmn (10
13
−
2
δ )
(c) (d)
0.5
2.0
1.6
1.2
0.8
0.4
E
(
V/nm
)
Hexagonal BL
G
3.0 3.5 4.
5
4.01.5 2.5 3.5
d
(
Å
)
0.24
0.23
0.26
0.25
FIG. 3. Band structure of the hexagonal stacked BLG without
共a兲 and with 共b兲 an electric field. The charge density difference
␦
n
between the layers induced by the electric field is shown in 共c兲. The
radius of the Dirac circle 共
0
兲 as a function of the interlayer spacing
is shown in 共d兲. All results were obtained from the density-
functional LMTO calculations.
STRAIN AND ELECTRIC FIELD MODULATION OF THE… PHYSICAL REVIEW B 80, 165430 共2009兲
165430-3
0
increases both with the applied electric field and the in-
terlayer hopping parameter 共t increases as the interlayer
separation decreases兲 as indicated from Figs. 3共d兲 and 4. This
is a notable feature of the hexagonal BLG, viz., that we have
an approximately circular Fermi surface with zero band gap
and that the radius of the circle can be modified by electric
field and strain. Any doped holes or electrons will form a
thin shell in the momentum space around the Dirac circle.
This may lead to interesting electronic and magnetic proper-
ties in the presence of impurities.
IV. BILAYER GRAPHENE IN THE BERNAL STRUCTURE
The effect of strain and electric field on the band structure
of the Bernal BLG is summarized in Fig. 5. As seen from
Fig. 1共a兲, for the Bernal BLG, there are four nearly parabolic
bands near the Fermi energy and two of these touch at the
Dirac point making it a zero band-gap semiconductor. How-
ever, in the presence of an electric field, these two bands split
opening up a small gap 关Fig. 5共b兲兴 consistent with earlier
density-functional results.
13,14
The magnitude of the band
gap increases linearly for small electric fields and saturates
for higher electric fields as seen from Fig. 5共d兲.
For large interlayer spacing, there is very little coupling
between the two individual sheets and the bands for the
single graphene sheet are reproduced 关Fig. 5共c兲兴. If the inter-
layer separation d is large enough 共⬎4.5 Å兲, the gap van-
ishes irrespective of the electric field. By decreasing the
magnitude of d the band gap increases substantially 关Fig.
5共d兲兴 due to stronger interlayer coupling. Quite interestingly,
if d is too small 共⬍2.7 Å兲, the band gap becomes indirect
关Fig. 5共a兲兴 due to interactions with other orbitals in addition
to p
z
. However, the indirect band gap may not be realized in
practice because of the large strain needed, which may also
lead to a possible deformation of the graphene sheets. For
reasonable layer separations, the electric field produces a
small direct gap at k points, the locus of which is nearly a
circle 共the Dirac circle兲 centered around the K or the K
⬘
points in the Brillouin zone as indicated from Fig. 5共f兲.
The main features of the band structure can again be ex-
plained by a nearest-neighbor TB model, analogous to the
case for the hexagonal structure. The resulting Hamiltonian
is
H
Bernal
=
冢
⌬
2
00
†
0
−
⌬
2
0
0
†
−
⌬
2
t
0
t
⌬
2
冣
, 共7兲
where the basis set of the Hamiltonian consists of the Bloch
functions made out of the carbon orbitals located at A, B
˜
, A
˜
,
and B, in that order. Diagonalization of the Hamiltonian
yields the four eigenvalues
⑀
共k兲 = ⫾
1
2
兵⌬
2
+4
2
†
+2t
2
⫾ 2关4
2
†
共⌬
2
+ t
2
兲
+ t
4
兴
1/2
其
1/2
. 共8兲
At the K or the K
⬘
points in the Brillouin zone, the energies
of the bands are
⑀
= ⫾
⌬
2
, ⫾
冑
⌬
2
4
+t
2
, and they disperse qua-
dratically as seen from Fig. 5共b兲, such that a gap of magni-
tude E
g
=⌬t / 共⌬
2
+t
2
兲
1/2
is produced at k points in the Bril-
louin zone that make the Dirac circle of radius
0
=⌬/ 2
.
The above expression clearly shows that E
g
increases lin-
early for low fields 共⌬Ⰶt兲 and saturates at high fields 共⌬
Ⰷt兲. This is consistent with the density-functional results for
the dependence of the band gap on the external field E and
bilayer spacing shown in Fig. 5共d兲.
ρ
0
radius
E = 2V/nm E = 4V/n
m
H
exagona
l
Γ
K
M
K’
Γ
K
M
K’
FIG. 4. 共Color online兲 Topology of the band structures of hex-
agonal BLG with equilibrium interlayer separation and in the pres-
ence of an electric field. The Fermi surface is formed by two inter-
penetrating cones, forming approximately a circle of radius
0
the
Dirac circle centered around the K and the K
⬘
points in the Bril-
louin zone. The size of the circle can be controlled by the electric
field as the right hand figure indicates for two different electric
fields.
E
S
P
E
E = 2 V/nm
0
0
E (V/nm)
4.5 Å
1234
E (V/nm)
01
E
Energy (eV)
Bandgap
(
eV
)
0.2
0.4
d = 2.5 Å
3.35 Å
3.75 Å
(V/nm)
0
1
2
3
243
(c)
E = 2 V/nm
(b)(a)
E = 2 V/nm
(d) (e)
(f)
K
B
erna
l
Dirac
Circl
e
Energy
ΓΓΓΜK ΜKKΜ
−4
−2
2
4
0
d = 2.5 Å
d = 3.35 Å
d = 4.5 Å
FIG. 5. Electric field modulated band structure of the Bernal
stacked BLG for different interlayer spacings 共a–c兲 and the varia-
tion in the band gap as a function of strain and electric field 共d兲. Fig.
5共e兲 shows the magnitude of the polarization field E
p
and the net
screened electric field E
s
as a function of the applied electric field
E. All results were obtained from the density-functional LMTO
calculations. Topology of the bands in the gap region is sketched in
Fig. 5共f兲, which shows the Dirac circles of radius
0
.
B. R. K. NANDA AND S. SATPATHY PHYSICAL REVIEW B 80, 165430 共2009兲
165430-4
V. SCREENING IN THE BLG
It has already been pointed out that the screening is diver-
gent in the Bernal structure for small applied fields between
the layers,
4,7
which is a consequence of the band topology at
low energies. In this section, we examine the screening in the
hexagonal structure and compare with the results for the Ber-
nal structure. An applied electric field E transfers electrons
from one graphene layer to the other, which produces a po-
larization field E
p
resulting in the net screened field E
s
that
the electrons see, so that we have E
s
=E−E
p
.
We compute the screening effects in three different ways
using 共a兲 density-functional LMTO method, 共b兲 the tight-
binding model, and finally 共c兲 from an analytic expression
obtained by using the linear band dispersion. Unlike the Ber-
nal structure, we find that the screening does not diverge in
the hexagonal structure.
In the LMTO calculations, an extra potential ⫾⌬/ 2 was
added to the on-site energies of the carbon orbitals of the two
layers of BLG and the band structure was determined self-
consistently. The screened potential ⌬
s
can then be directly
obtained from the final on-site energies of the carbon orbitals
by examining the band-center energies. For this purpose, the
C2s orbital is especially useful, since it acts like a core
orbital.
For the TB, as well as the analytic calculation, we com-
puted the polarization field E
P
by approximating the induced
charges in the graphene layer by a uniform sheet charge den-
sity
, which turns out to be an excellent approximation for
computing the screening. The polarization field is then given
from the Gauss’ Law in electrostatics to be E
p
=
/ 共2
⑀
兲,so
that
E
s
= E − E
p
= E −
␦
n
2
⑀
, 共9兲
where
␦
n is the difference between the sheet carrier density
of the two graphene layers and
⑀
is the dielectric constant. In
terms of the electric potential the above equation can be writ-
ten as
⌬
s
= ⌬ −
ed
␦
n
2
⑀
, 共10兲
where d is the interlayer separation. One can obtain the value
of
␦
n from the normalized eigenfunctions of the BLG Hamil-
tonian.
Diagonalizing the Hamiltonian for the hexagonal BLG
关Eq. 共4兲兴, the eigenfunctions corresponding to the upper and
the lower Dirac cones, centered at
and −
, respectively, are
found to be
兩
⫾
l
典 = p
−
冢
⫾e
−i
⫿q
+
e
−i
− q
+
1
冣
, 兩
⫾
u
典 = p
+
冢
⫾e
−i
⫾q
−
e
−i
q
−
1
冣
, 共11兲
where
=tan
−1
共k
y
/ k
x
兲, p
⫾
=共1⫾
⌬
2
兲
1/2
/ 2, q
⫾
=共2t兲
−1
共2
⫾⌬兲, and the ⫾ sign inside the ket indicates the
upper and the lower band and u共l兲 denotes the upper 共lower兲
Dirac cone. Denoting the four components of the wave func-
tion by
A
k
,
A
˜
k
,
B
˜
k
, and
B
k
共from top to bottom兲, correspond-
ing to the appropriate Bloch functions, the surface electron
density is given by
1共2兲
=2e
兺
k
occ
n
k
1共2兲
/A
cell
,
n
k
1共2兲
= 兩
A共A
˜
兲
k
兩
2
+ 兩
B共B
˜
兲
k
兩
2
, 共12兲
where A
cell
is the unit cell area,
is the band index, the
superscript 1共2兲 refers to the two individual layers, and the
factor of two in front of the summation takes care of the
electron spin.
The summation in Eq. 共12兲 need to be performed only
over the shaded region in Fig. 6, since we are interested only
in the difference
1
−
2
. The remaining portions of the two
occupied bands 兩
−
u
典 and 兩
−
l
典 cancel each other’s contribu-
tion at each k point to this difference as can be easily verified
from Eqs. 共11兲 and 共12兲. The net result is then
␦
n =
1
−
2
=
4e
兺
=1
2
冕
0
0
kdk共n
k
1
− n
k
2
兲, 共13兲
where the summation goes over the two bands below E
F
and
the integration goes up to the radius of the Dirac circle, so
that the contribution comes only from the shaded region of
Fig. 6. The factor of four in Eq. 共13兲 was inserted to account
for both the spin degeneracy and the two Dirac valleys in the
Brillouin zone at the K and K
⬘
points. Using the wave func-
tions Eq. 共11兲, the integration is easily performed to yield
␦
n =
e
⌬
2
2
. 共14兲
E
F
ψ
−
l
ψ
−
u
ψ
+
l
ξ
−ξ
K
H
exagona
l
Dirac Circl
e
E
nergy
ψ
+
u
ρ
0
FIG. 6. The topology of the band structure of hexagonal BLG.
The interpenetrating Dirac cones form a Dirac circle with radius
0
that makes the Fermi surface. The shaded region contributes to the
integral in Eq. 共13兲 in the screening calculation.
STRAIN AND ELECTRIC FIELD MODULATION OF THE… PHYSICAL REVIEW B 80, 165430 共2009兲
165430-5
