Uniaxial strain on gapped graphene
ABSTRACT We study the effect of uniaxial strain on the electronic band structure of gapped graphene. We consider two types of gapped graphene, one which breaks the symmetry between the two triangular sublattices (staggered model), and another which alternates the bonds on the honeycomb lattice (Kekul\'e model). In the staggered model, the effect of strains below a critical value is only a shift of the band gap location. In the Kekul\'e model, as strain is increased, band gap location is initially pinned to a corner of the Brillouin zone while its width diminishes, and after gap closure the location of the contact point begins to shift. Analytic and numerical results are obtained for both the tight-binding and Dirac fermion descriptions of gapped graphene. Comment: 5 pages, 6 figures, published version
arXiv:0909.5052v2 [cond-mat.mes-hall] 10 May 2010
Uniaxial strain on gapped graphene
M. Farjama,∗, H. Rafii-Tabara,b
aDepartment of Nano-Science, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran
bDepartment of Medical Physics and Biomedical Engineering, and Research Centre for Medical Nanotechnology and Tissue Engineering, Shahid Beheshti
University of Medical Sciences, Evin, Tehran 19839, Iran
We study the effect of uniaxial strain on the electronic band structure of gapped graphene. We consider two types of gapped
graphene, one which breaks the symmetry between the two triangular sublattices (staggered model), and another which alternates
the bonds on the honeycomb lattice (Kekul´ e model). In the staggered model, the effect of strains below a critical value is only
a shift of the band gap location. In the Kekul´ e model, as strain is increased, band gap location is initially pinned to a corner of
the Brillouin zone while its width diminishes, and after gap closure the location of the contact point begins to shift. Analytic and
numerical results are obtained for both the tight-binding and Dirac fermion descriptions of gapped graphene.
Keywords: Graphene, Gap, Strain, Kekul´ e
PACS: 73.22.−f, 81.05.Uw, 71.20.−b
Recently, strain engineering of the electronic structure has
been explored as an alternative method in the design of
graphene-based electronic circuitry . The approach is based
on generating local strains to change the hopping amplitudes in
an anisotropic way, which in turn leads to the presence of ef-
fective gauge fields for the Dirac electrons . A key finding is
that for small and moderate uniaxial deformations the gapless
Dirac spectrum is robust, and a gap opens only for large defor-
mations above a particular threshold [3, 4, 5]. More precisely,
the presence of anisotropy in the tight-binding hoppings on a
honeycomb lattice makes the Dirac points approach each other
until they merge at a critical asymmetry, at which point a band
gap begins to open [6, 7, 8, 9].
For practical purposes, graphene can be considered as a
gapless semiconductor. Nevertheless, gaps of various origins
have been identified in graphene. Intrinsically, spin-orbit cou-
pling is responsible for a tiny gap, on the order of 10−3meV
[10, 11, 12, 13], and electron-electron interactions may ren-
der graphene an insulator in vacuum . Extrinsically, in-
teraction with substrates and adlayers can induce a band gap
in graphene. Epitaxial graphene has a band gap  which
has been explained as substrate-induced . Other substrates
studied include boron-nitride and Cu , Ni , and several
other metals . Band gaps induced by the adsorption of wa-
ter and ammonia molecules , and alkali metals  have
been studied theoretically. Generally, there are two ways of in-
ducing a gap in monolayer graphene. One way is the mixing of
electronic states with different pseudospins in the same valley,
and another way is the mixing of states that belong to different
Email address: email@example.com (M. Farjam)
valleys. The former can be achieved by sublattice symmetry
breaking which leaves the A and B carbon atoms in different
environments, while the latter is produced by certain transla-
tional symmetry breakings .
The purpose of the present work is to study the effect of uni-
axial strain on the band structure of gapped graphene . As
weshow,theeffectdependsontheoriginofthebandgap. Ifit is
inducedby ABsublattice symmetrybreaking,small andmoder-
ate strainsdonotchangethewidthofthe gap,but causeits loca-
tions in k-space to move in a similar way as those of the corre-
sponding Dirac points in the gapless case. On the other hand, if
couples the different valley states, a different behavioremerges.
With increasing uniaxial strain, the band gap first diminishes
and closes in its fixed location and, afterwards, the neutrality
point is shifted in reciprocal lattice as in the gapless case.
Our paper is organized as follows. Section 2 contains our
analysis of the relevant tight-binding models. Section 3 con-
tains analysis of the corresponding Dirac equations. Section 4
contains numerical results and their discussion, and Section 5
presents our conclusions.
2. Tight-binding model
The models we use can be described as different types of
strain distributions  shown in Fig. 1. Sublattice symme-
try breaking, shown in Fig. 1(a), can be associated with an
out-of-plane strain distribution. The so-called Kekul´ e distor-
tion, shown in Fig. 1(b), breaks the translational symmetry in
a way which can be described by a
surate lattice, and can be the result of an in-plane strain dis-
tribution. Finally, the quinoid distortion of graphene, shown
in Fig. 1(c), represents in-plane uniaxial strain parallel to the
Preprint submitted to Physica EMay 11, 2010
Figure 1: (a) Staggered, (b) Kekul´ e, and (c) quinoid models.
Figure 2: (a) Primitive unit cell of graphene. The vectors?δ defined as vectors
from atom A1to its nearest neighbors on the B sublattice are given by?δ1 =
(0,−1/√3)a,?δ2= (1/2,1/2√3)a and?δ3= (−1/2,1/2√3)a. (b) Kekul´ e-type
graphene primitive unit cell containing its six atom basis. The Kekul´ e distortion
is characterized by two kinds of hoppings, one set around the hexagonal ring
(red), and the other set crossing the unit cell boundaries (blue). The addition of
a uniaxial strain in the vertical direction results in four hoppings, t1, t2, t3and
t4. (c) The relationship between the two Brillouin zones. The path ΓKMΓ is
used in the band plots.
nearest-neighbor bond in the vertical direction.
binding models based on this description are defined by on-site
energies and the nearest-neighbor hoppings, but we neglect the
change in bond lengths and use the perfect honeycomblattice.
The primitive unit cell of grapheneand the nearest neighbors
of the A and B atoms in the unit cell are shown in Fig. 2(a).
The tight-binding models involve the vectors?δ from an A atom
to its nearest-nearest neighbors which are defined in the cap-
tion of Fig. 2. The primitive unit cell of the Kekul´ e model is
shown in Fig. 2(b). The Brillouinzoneof the Kekul´ e modeland
that of graphene, the larger hexagon, are depicted in Fig. 2(c),
with special symmetry points defined. In the Kekul´ e model
the points K, K′and Γ belong to the reciprocal lattice and are
therefore equivalent, which results in the coupling of the two
inequivalent valleys that is responsible for the band gap.
els. In addition to the hopping parameter t (≈ −2.7 eV for
graphene), there is the band gap of gapped graphene and the
asymmetry in hoppings caused by uniaxial strain. We denote
by ∆ the half-width of the band gap, and by δt the shift in one
of the hoppings which is in the direction of strain axis. The pa-
rameter t may be taken to be positive without loss of generality,
and it can serve as the energy scale, so that the models contain
two adjustable parameters ∆/t and δt/t.
The gapped graphene represented by Fig. 1(a) can be de-
scribed by the tight-binding model with staggered on-site en-
ergies ±∆. Including the effect of uniaxial strain, we can write
the Hamiltonian of this staggered model as
h(?k) = (t + δt)ei?k·?δ1+ tei?k·?δ2+ tei?k·?δ3.
Diagonalizing the Hamiltonian defined by Eq. (1), we find the
which show that the band gap is ≥ 2∆. The function h(?k) is the
same as in gapless graphene under strain, where zero modes
exist for δt/t < 1 and are shifted from K and K′toward M .
Settingh(?k) = 0,andtakingkya = 2π/√3andkxa = 2π/3+pxa,
to move along the KK′line, we find the shift from K to be
ǫ±(?k) = ±
We now turn to the gapped graphenebased on Kekul´ e distor-
tion, shownin Fig.1(b), whichhas zeroon-siteenergiesbuttwo
alternating hoppings, on one-third and two-thirds of the bonds,
respectively [21, 25],
t1= t +2
t2= t −1
where, as in the staggered model, ∆ is half of the energy gap.
When uniaxialstrain is applied to the Kekul´ e model, we assume
that a shift of δt is added to the hoppings that are parallel to the
strain axis. We define these hoppings as
t3= t1+ δt,
t4= t2+ δt.
The Hamiltonian of the Kekul´ e model is given by a 6 × 6
where the 3 × 3 block HABis
and HBA(?k) = H†
general form tijexp(i?k ·?δij) when linking Aiand Bjatoms, and
can be read off Fig. 2(b).
AB(?k). The matrix elements of Eq. (8) have the
The band structure of the Kekul´ e model can be obtained by
numerical calculation of the eigenvalues of Eq. (7). However,
the existence of zero modes and their locations are determined
more simply by
Since the zero modes are expected to occur on the horizontal
line through Γ, we calculate the determinant of HABfor ky= 0,
D(kx;∆/t,δt/t) = 2t1t2
1) − 2t4t2
Setting D = 0, we find
which has a solution for kxif the right-hand side is in the inter-
val [−1,1]. In particular, if δt = ∆ then t1 = t4and the right-
hand side of Eq. (11) becomes unity yielding kx= 0 which is
the Γ point, or its equivalent K points. Therefore, the gap in the
spectrum at K vanishes as δt is increased from 0 to ∆. Further
increase of δt then causes the contact points to shift away from
the K points, until they merge at M, when the right-hand side
of Eq. (11) becomes −1 and kxa = 2π/3.
3. Dirac equation
We can use the Dirac equation to describe the effect of strain
on low-energy electrons provided that both ∆/t and δt/t ≪ 1.
by setting?k =?K+? p near the K and K′points. For the staggered
model under strain we obtain
p + δt
−p + δt
where p = px+ ipyand we have used units such that ? = 1
and vF = ta√3/2 = 1. However, we restore vFexplicitly in
some of the derived results below. Here we have followed the
ψ = [ΦKA,ΦKB,ΦK′A,ΦK′B]T
for the four-dimensional spinor . Our results can be ex-
tended to the band gap due to spin-orbit interaction which can
be described with a similar Hamiltonianas Eq. (12), except that
the gaps have opposite signs for K and K′points . The
energy dispersions derived from Eq. (12) are given by
ǫ±(? p) = ±
(±px+ δt)2+ p2
and can be easily verified to be the limiting cases of Eq. (3).
The shifts of the Dirac points are px = ∓δt/vF, which agree
with Eq. (4) for δt/t ≪ 1.
For the Kekul´ e model under strain we have
p + δt
−p + δt
and the energy eigenvalues are given by
ǫ±(? p) = ±
p2x+ ∆2± δt
For δt < ∆ there is a gap of size ∆ − δt at p = 0. If δt ≥ ∆, zero
modes exist at
For δt = 0 the energy dispersions, (13) and (15), give identi-
cally gapped Dirac spectra, with a density of states (DOS) per
unit area given by
|ǫ| < ∆
However, for δt ? 0 the DOS of the staggered model remains
the same, while that of the Kekul´ e model changes as the gap
shrinks. For δt = ∆ and ǫ ≪ ∆, Eq. (15) becomes
which yields a DOS given by
4. Numerical results and discussion
Equation (11) can be solved numerically for kxas a function
of ∆ and δt. The results for ∆ = 0,t/2,t are shown in Fig. 3.
For ∆ = 0 the same curve can be obtained from Eq. (4). For
nonzero ∆, the gap first diminishes at K as δt is increased from
0 to ∆, and with further increase of δt the Dirac point moves
In Fig. 4 we make plots of Eqs. (13) and (15) for a few values
of δt, using∆ as the scale of energyand momentum. For δt = 0,
shown in Fig. 4(a), both the staggered and Kekul´ e models give
the same gapped spectrum. However, for δt = ∆, the valence
and conduction bands shift laterally in the pxdirection with-
out a change in the gap for the staggered model as in Fig. 4(b).
In contrast, for δt = ∆ the gap closes in the Kekul´ e model for
one pair of bands while the other pair are repelled by 2∆, as in
Fig. 4(c). We note that the dispersionis quadraticinstead of lin-
ear near the degeneracypoint as can be expected from Eq. (18).
Further increase of δt to 2∆, shown in Fig. 4(d), causes a shift
of the neutrality points in the pxdirection, and the dispersions
become linear where the bands cross.
Figure 5 shows the band structures from Eq. (3) for ∆ = 0
and t/10. For δt = 0, Figs. 5(a) and (b), the Dirac point and
∆ = 0
Figure 3: Location of the Dirac point in the Kekul´ e model along the KM line as
a function of δt for ∆ = 0,t/2,t. At M the values of δt/t are 1, 1.0763, 1.2996,
-4 -2 0 2 4
δt = 0
-4-2 0 2 4
δt = ∆
δt = ∆
-4 -2 0
δt = 2∆
Figure 4: Dirac fermion dispersions along the py = 0 line. (a) Gapless and
gapped Dirac fermions, with gap equal to 2∆. (b) Strain of δt = ∆ on gapless
and staggered models. (c) Strain of δt = ∆ on Kekul´ e model. (d) Strain of
δt = 2∆ on Kekul´ e model.
δt = ∆ = 0
δt = 0, ∆ =t/10
δt =t/2, ∆ = 0
(d) δt =t/2, ∆ =t/10
δt =t, ∆ = 0
δt =t, ∆ =t/10
Figure 5: Band structures from the tight-binding model. (a,c,e) show the band
structures of graphene for δt = 0, t/2, t, and (b,d,f) show the band structures
for the staggered model for the same set of strains.
the gap, respectively, are located at the K point. For δt = t/2,
Figs. 5(c) and (d), they are shifted by the same amountto some-
where along the KM line. For δt = t, Figs. 5(e) and (f), the shift
reaches the M point. The last cases are critical in that the Dirac
pointsmergeat M. We cansee thatthedispersionsarequadratic
near M on the KM line, but linear on the MΓ line, i.e., a flatten-
ing of the Dirac cones takes place which can be seen in contour
plotsof thebandstructure(asshowninRef. 6). Forδt > 1a gap
opens at the M point for graphene, and the gap of the staggered
model becomes wider.
Figure 6 shows the band structures for the Kekul´ e model
obtained from the numerical evaluation of the eigenvalues of
Eq. (7). The band structures are plotted in the extended scheme
so they can be compared with those of Fig. 5. The ΓPKMQΓ
path is implied in Fig. 2(c), where P and Q are the points where
the path crosses the Brillouin zone of the Kekul´ e model. It
must be remarkedthat this path does not enclose the irreducible
wedge of the Brillouin zone for strained graphene [5, 27], but
includes the KM line where the band crossing may occur. Fig-
ure 6(a) may be compared with Fig. 5(b) which shows the band
structure of the staggered model for the same ∆. The regions
of the gap at the K point are similar in both figures, consistent
with the Dirac equation description, but there are extra gaps in
the band structure of the Kekul´ e model at P and Q. As we have
δt = 0, ∆ =t/10
δt = ∆ =t/10
δt =t/2, ∆ =t/10
δt ≈t, ∆ =t/10
0, t/10, t/2, 1.00299t. The extended zone scheme is used.
Band structures of the Kekul´ e tight-binding model for δt =
seen in Fig. 4, the gap closes when δt = ∆ and this can also be
seen in Fig. 6(b). Here the dispersion at K is linear on the PK
line and quadratic on the KM line. A comparison of Figs. 6(c)
and (d) with Fig. 5(c) and (e) shows that in the Kekul´ e model,
after the gap closes, the behavior of the neutrality points are not
very different from that in the gapless case.
We considered two models of gapped graphene, denoted by
staggered and Kekul´ e, respectively, and studied the effect of
strain on their band structures. We found that in the staggered
model the widthof the bandgapdoes not changefor strains less
than a critical value, but its locations move following the mo-
tion of the Dirac cones in the gapless case. The effect of strain
on the band structure of the Kekul´ e model is less trivial. With
increasing strain, reflected in changes in the hoppings, the gap
beginstodiminishwithitslocationspinnedtothe K points. The
gap closes when δt = ∆, i.e., when the shift in hoppings due to
strain equals the half-width of the original gap and, afterwards,
increasing the strain makes the neutrality point to move in k
space as in the gapless case. The effects we have discussed may
be observed experimentallyin gappedgrapheneon Ni substrate
and epitaxial graphene on SiC, respectively.
M.F. acknowledges funding from the Iranian Nanotechnol-
ogy Initiative and H.R.-T. fromthe Iran National Science Foun-
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