Density of states of Dirac–Landau levels in a gapped graphene monolayer under strain gradient
We study a gapped graphene monolayer in a combination of uniform magnetic field and strain-induced uniform pseudomagnetic field. The presence of two fields completely removes the valley degeneracy. The resulting density of states shows a complicated behavior that can be tuned by adjusting the streng...
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irk-123456789-1752222021-02-01T01:26:24Z Density of states of Dirac–Landau levels in a gapped graphene monolayer under strain gradient Shubnyi, V.O. Sharapov, S.G. Наноструктуры при низких температурах We study a gapped graphene monolayer in a combination of uniform magnetic field and strain-induced uniform pseudomagnetic field. The presence of two fields completely removes the valley degeneracy. The resulting density of states shows a complicated behavior that can be tuned by adjusting the strength of the fields. We analyze how these features can be observed in the sublattice, valley and full density of states. The analytical expression for the valley DOS is derived. 2017 Article Density of states of Dirac–Landau levels in a gapped graphene monolayer under strain gradient / V.O. Shubnyi, S.G. Sharapov // Физика низких температур. — 2017. — Т. 43, № 10. — С. 1508-1514. — Бібліогр.: 30 назв. — англ. 0132-6414 PACS: 73.22.Pr, 71.70.Di http://dspace.nbuv.gov.ua/handle/123456789/175222 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Наноструктуры при низких температурах Наноструктуры при низких температурах |
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Наноструктуры при низких температурах Наноструктуры при низких температурах Shubnyi, V.O. Sharapov, S.G. Density of states of Dirac–Landau levels in a gapped graphene monolayer under strain gradient Физика низких температур |
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We study a gapped graphene monolayer in a combination of uniform magnetic field and strain-induced uniform pseudomagnetic field. The presence of two fields completely removes the valley degeneracy. The resulting density of states shows a complicated behavior that can be tuned by adjusting the strength of the fields. We analyze how these features can be observed in the sublattice, valley and full density of states. The analytical expression for the valley DOS is derived. |
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Shubnyi, V.O. Sharapov, S.G. |
| author_facet |
Shubnyi, V.O. Sharapov, S.G. |
| author_sort |
Shubnyi, V.O. |
| title |
Density of states of Dirac–Landau levels in a gapped graphene monolayer under strain gradient |
| title_short |
Density of states of Dirac–Landau levels in a gapped graphene monolayer under strain gradient |
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Density of states of Dirac–Landau levels in a gapped graphene monolayer under strain gradient |
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Density of states of Dirac–Landau levels in a gapped graphene monolayer under strain gradient |
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Density of states of Dirac–Landau levels in a gapped graphene monolayer under strain gradient |
| title_sort |
density of states of dirac–landau levels in a gapped graphene monolayer under strain gradient |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2017 |
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Наноструктуры при низких температурах |
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http://dspace.nbuv.gov.ua/handle/123456789/175222 |
| citation_txt |
Density of states of Dirac–Landau levels in a gapped graphene monolayer under strain gradient / V.O. Shubnyi, S.G. Sharapov // Физика низких температур. — 2017. — Т. 43, № 10. — С. 1508-1514. — Бібліогр.: 30 назв. — англ. |
| series |
Физика низких температур |
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AT shubnyivo densityofstatesofdiraclandaulevelsinagappedgraphenemonolayerunderstraingradient AT sharapovsg densityofstatesofdiraclandaulevelsinagappedgraphenemonolayerunderstraingradient |
| first_indexed |
2025-07-15T12:28:52Z |
| last_indexed |
2025-07-15T12:28:52Z |
| _version_ |
1837715982370471936 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 10, pp. 1508–1514
Density of states of Dirac–Landau levels in a gapped
graphene monolayer under strain gradient
V.O. Shubnyi
Department of Physics, Taras Shevchenko National University of Kiev
6 Academician Glushkov Ave., Kiev 03680, Ukraine
S.G. Sharapov
Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine
14-b Metrolohichna Str., Kiev 03680, Ukraine
E-mail: sharapov@bitp.kiev.ua
Received February 20, 2017, published online August 27, 2017
We study a gapped graphene monolayer in a combination of uniform magnetic field and strain-induced uni-
form pseudomagnetic field. The presence of two fields completely removes the valley degeneracy. The resulting
density of states shows a complicated behavior that can be tuned by adjusting the strength of the fields. We ana-
lyze how these features can be observed in the sublattice, valley and full density of states. The analytical expres-
sion for the valley DOS is derived.
PACS: 73.22.Pr Electronic structure of graphene;
71.70.Di Landau levels.
Keywords: Landau levels, graphene, strain.
1. Introduction
The carbon atoms in monolayer graphene form a honey-
comb lattice due to sp2 hybridization of their orbitals. Since
the honeycomb lattice is not a Bravais lattice, one has to
consider the honeycomb lattice as a triangular Bravais lat-
tice with two atoms per until cell. Thus one naturally ar-
rives at a two-component spinor wave function of the
quasiparticle excitations in graphene (see Ref. 1 for some
analogies with a real spin). These components reflect the
amplitude of the electron wave function on the A and B
sublattices. The two-component form of the wave function
along with the band structure results in the Dirac form of
the effective theory for graphene.
The Dirac fermions had shown up the celebrated magne-
to-transport and STS properties of graphene (see Refs. 2–4
for the reviews). Recently STM/STS measurements al-
lowed not only to observe relativistic Landau levels, but
also to resolve directly their sublattice specific features. By
resolving the density of states (DOS) on A and B
sublattices of a gapped graphene, it was experimentally
confirmed [5] that the amplitude of the wave function of
the lowest Landau level (LLL) is unequally distributed
between the sublattices depending on its energy sign.
In the presence of a gap ∆ driven by inversion symmetry
breaking, the LLL splits into two levels with the energy
0 = sgn ( )E eBη∆ , where =η ± distinguishes inequivalent
K and 'K points of the Brillouin zone and an external mag-
netic field = = (0,0, )B∇×B A is applied perpendicularly
to the plane of graphene along the positive z axis [6]. Here
= | |e e− is the electron charge and A is the vector electro-
magnetic potential. The corresponding amplitudes of the wave
function of the positive energy electron-like, 0+ , and nega-
tive energy hole-like, 0−, levels are on A and B sublattices.
In other words, the individual sublattices are valley polar-
ized for the LLL [7].
An exciting opportunity for manipulating the amplitudes
of the wave function on the sublattices opens due to the
close connection between the impact of deformation and
external electromagnetic field on the electronic structure
of graphene. Change in hopping energy between A and B
atoms induced by strain can be described by a vector po-
tential pmA analogous to the vector potential A of the ex-
ternal magnetic field (see Refs. 8, 9 for a review).
The corresponding field, =pm pm∇×B A , is called pseu-
domagnetic field (PMF), as it formally resembles the real
magnetic field, with one crucial distinction that it is directed
oppositely in K and 'K valleys. This implies that the LLL
© V.O. Shubnyi and S.G. Sharapov, 2017
Density of states of Dirac–Landau levels in a gapped graphene monolayer under strain gradient
breaks the electron-hole symmetry, with the LLL energy,
0 = sgn ( )pmE B∆ for both K and 'K points. Furthermore,
the states corresponding to the LLL are sublattice polarized,
as they reside exclusively on either A and B sublattice [7].
While the formation of the LLL is associated with zero
modes and does not require a homogeneous PMF, to form
higher Landau levels a uniform PMF is needed [10]. This
is in fact the main challenge [11] for the implementation of
strained graphene, although recently there has been some
progress both in experiment [12–14] and in theory [15].
In the presence of either external magnetic field or de-
formation, the higher energy levels from K and 'K points
remain degenerate. This degeneracy is lifted when both
strain and magnetic field are present. One of the interesting
consequences of the lifting is that for | | | |pmB eB> , the Hall
conductivity is oscillating between 0 and 22 /e h [16].
The latest experiments [14] show that it is possible to
create a homogeneous PMF of order of a few Tesla. There-
fore, there is a good chance that the STS/STM measure-
ments of the Dirac–Landau levels similar to that done in
Ref. 5 are now possible on strained graphene. Thus the
purpose of the present work is to study the DOS (including
the sublattice resolved) in a combination of a constant PMF
pmB created by non-uniform strain and magnetic field B .
In particular, we will look for the specific effects related to
the presence of nonzero gap ∆ and lifting of the degenera-
cy between K and 'K that can be observed in STS meas-
urements.
The paper is organized as follows. We begin by pre-
senting in Sec. 2 the model describing gapped monolayer
graphene in the combination of PMF and magnetic field. In
Sec. 3 we provide the definitions of the valley, sublattice
and full DOS in terms of the Green’s function decomposed
over Landau levels. The corresponding DOS are written in
Sec. 4 as the sums that in the case of the valley DOS can
be calculated analytically. The results for the DOS in the
various regimes are discussed in Sec. 5 and conclusions are
given in Sec. 6.
2. Model
We consider gapped monolayer graphene in the contin-
uum approximation described by the effective Hamiltonian
0
=
0 '
H
H
H
K
K
. (1)
The full Hamiltonian (1) acts on wave function with
four components
= ,
'
'
•
•
ψ
ψ
Ψ ψ
ψ
K
K
K
K
(2)
where • and denote, respectively, A and B sublattices
and we followed the notations of Refs. 3, 6 with exchanging
the sublattices in the 'K valley. Thus the Hamiltonian (1)
includes two blocks corresponding to K and 'K valleys
3= ,F pm
eH i
c
− ∇ − − + τ ∆
K τ A Av (3)
3= .' F pm
eH i
c
− − ∇ − + − τ ∆
K τ A Av (4)
Here 1 2= ( , )τ ττ and 3τ are Pauli matrices acting in the
sublattice space, Fv is the Fermi velocity, the gap ∆ corre-
sponds to the energy difference 2∆ between the A and B
sublattices, A and pmA are the the electromagnetic and
strain induced vector potentials, respectively. We neglect
the spin splitting, because for commonly used strengths of
magnetic field the Zeeman splitting is small compared to
the distance between the Landau levels. For a fixed direc-
tion of external magnetic field, the corresponding to A
term in the Hamiltonian breaks time-reversal symmetry,
while the pmA term breaks the inversion symmetry and
leaves time-reversal symmetry unbroken.
With the x-axis aligned in the zigzag direction, the
strain-induced vector potential reads [17,18] (see also the
reviews [8,9])
0
= ,
22
xx yy
pm
xy
u u
ua
− βκ
−
A (5)
where = 0
= ln / ln | 3a at aβ −∂ ∂ ≈ is the dimensionless elec-
tron Grüneisen parameter for the lattice deformation, t the
nearest-neighbour hopping parameter, 1/ 3κ ≈ is a parame-
ter related to graphene’s elastic property [17], a is the
length of the carbon-carbon bond, ( 0a is the length of the
unstrained bond), and iju with , = ,i j x y is the strain tensor
as defined in classical continuum mechanics [8,9]. We also
assume that the deformation is a pure shear, so that
= 0xx yyu u+ , and there is no scalar potential term in the
Hamiltonian.
The sign of the PMF depends on the valley, and, for ex-
ample, in K valley,
0
1= ( ) ,
2pm x xy y xx yyB u u u
a
βκ − ∂ + ∂ −
(6)
whereas it has the opposite sign in 'K valley, because pmA
enters Eqs. (3) and (4) with the opposite signs. Equation (6)
illustrates the main problem in this field of research, viz. a
uniform PMF can only be created by a non-uniform strain
[11]. As was already stated in the Introduction, considering
the experimental progress achieved in the field [14], we
restrict ourselves to a constant PMF. Thus we arrive at the
model with two independent K points characterized by the
following combinations of the fields, = / pmB eB c B± ± . A
more complicated, but analytically intractable case with a
combination of a constant magnetic and inhomogeneous
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 10 1509
V.O. Shubnyi and S.G. Sharapov
pseudomagnetic fields was considered in Ref. 19, where
a circularly symmetric strain is induced by a homogeneous
load.
3. Green’s function, sublattice and valley resolved DOS
Although it is straightforward to obtain the DOS direct-
ly from the solution of the corresponding Dirac equation,
we rely on the Green’s function (GF) machinery that au-
tomatically takes into account the degeneracy of levels and
avoids the necessity to work with different directions of
fields separately. Since the K points in the model (1) are
independent, we will use the GF’s corresponding to the
separate K points. In particular, we are interested in the
translation invariant part G of the GF that allows to derive
both the DOS and the transport coefficients. Its derivation
using the Schwinger proper-time method and decomposi-
tion over Landau–level poles has been discussed in many
papers (see, e.g., Refs. 20–23). Here we begin with the
translation invariant part for K point written in the Mat-
subara representation (we set = = = 1Bc k in what fol-
lows)
2
| |
2 2
=0
( , , )
( , ) = e ( 1) ,
( ) ( )
= (2 1) ,
p
B n n
n n
G B i
G i
i M
m T
− ∞
++
+
ω
ω −
ω −
ω π +
∑K p
p (7)
where T is the temperature,
2 2= 2 | |n FM n B±
±∆ + v (8)
are the energies of the relativistic Landau levels at K and
'K points ( =η ±), respectively, and the function
2
3 3
2( , , ) = ( ) (1 sgn ( ))
| |n n
pG B i i B L
B+ +
+
ω ∆τ + ω + τ −
p
2
3 1
2(1 sgn ( ))
| |n
pB L
B+ −
+
− − τ −
2
1
1
24 ( ) .
| |F x y y x n
pp p L
B−
+
− τ + τ
v (9)
Here ( )nL zα are the generalized Laguerre polynomials, and
0( ) ( )n nL z L z≡ ( 1
1 0L− ≡ ). When deriving GF from the known
wave-functions, the Laguerre polynomials originate from
the integration of two Hermite polynomials with proper
weights. Looking at the structure of the GF (7), one can see
that the projectors 3= (1 sgn ( )) / 2P B± +± τ take into ac-
count that, for example, for > 0B+ , the states on A and B
sublattices involve nL and 1nL − , respectively. The most
general expression of the propagator in the presence of B ,
pmB and various types of the gaps is provided in [24].
The corresponding contribution of the K point to the
DOS per spin and unit area on A and B sublattices reads
2
, 2
1( ) = [ ( 0, ) ( 0, )]
2 (2 )
ii iiA B
d pD G i G i
i
ε ε − − ε +
π π∫
K KK p p (10)
with = 1,2i for A and B sublattices, respectively. It fol-
lows from Eqs. (3) and (4) that
( , ) = ( , , )'
F FG i G B B+ −ω → − ∆ → −∆ →
K Kp v v (11)
and
2
, 2
1( ) = [ ( 0, ) ( 0, )]
2 (2 )
' ''
ii iiB A
d pD G i G i
i
ε ε − − ε +
π π∫
K KK p p (12)
with = 1,2i for B and A sublattices, i.e., exchanging the
sublattices. While the valley resolved DOS presents a theo-
retical interest and will also be considered below, the STS
measurements allow to observe the full DOS involving two
valleys on each sublattice
, , ,( ) = ( ) ( ).'
A B A B A BD D Dε ε + εK K (13)
We will also consider the valley resolved but summed
over sublattices DOS
, , ,( ) = ( ) ( )' ' '
BAD D Dε ε + εK K K K K K (14)
which presents interest for valleytronics. Finally, the full
DOS can also be found by summing the valley resolved DOS
( ) = ( ) ( ).'D D Dε ε + εK K (15)
4. Expressions for numerical and analytical calculation
of the DOS
Using the integral [25]
2 2 2
2
2 ( 1) | |exp =
| | | | 2 2(2 )
n
n
d p p p BL
B B
−
− ππ
∫ (16)
and evaluating the discontinuity of the GF we arrive at the
final result
, (0) , ( 1),
, , ,( ) = ( ) ( ).n
A B A B A BD D D′ ′′ ≥ε ε + εK K K KK K (17)
Here , (0)
, ( )A BD ′ εK K is the LLL contribution to the valley and
sublattice resolved DOS and , ( 1)
, ( )' n
A BD ≥ εK K is the corre-
sponding contribution from the Landau levels with 1n ≥ .
Explicit expressions for these terms are
(0)
(0)
| |
( ) = ( ) ( ),
2
| |
( ) = ( ) ( ),
2
A
A
BD B
B
D B
+
+
′ −
−
ε θ δ ε − ∆
π
ε θ − δ ε − ∆
π
K
K
(18)
(0)
(0)
| |
( ) = ( ) ( ),
2
| |
( ) = ( ) ( ),
2
B
B
BD B
BD B
+
+
′ −
−
ε θ − δ ε + ∆
π
ε θ δ ε + ∆
π
K
K
(19)
1510 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 10
Density of states of Dirac–Landau levels in a gapped graphene monolayer under strain gradient
and for the Landau levels with 1n ≥ ,
, ( 1)
=1
| |
( ) = ( )
2 2
n n
nA
n n
MBD M
M
±∞
′ ≥ ±±
±
+ ∆
ε δ ε − +
π
∑K K
( ) ,
2
n
n
n
M
M
M
±
±
±
− ∆
+ δ ε +
(20)
, ( 1)
=1
| |
( ) = ( )
2 2
n n
nB
n n
MBD M
M
±∞
′ ≥ ±±
±
− ∆
ε δ ε − +
π
∑K K
( ) ,
2
n
n
n
M
M
M
±
±
±
+ ∆
+ δ ε +
(21)
where ± sign corresponds to K and ′K points. As ex-
pected, presence of PMF removes degeneracy of the levels
with 1n ≥ [16].
In Sec. 5 we compute the sublattice and valley resolved
DOS numerically on the base of Eqs. (18)–(21) by widen-
ing δ-fuction peaks to a Lorentzian shape, viz.
2 2
1 1( ) ,
( )
n
n n
M
M
δ ε − →
π ε − + Γ
(22)
where nΓ is the n th level width. Such broadening of Lan-
dau levels with a constant Γ was found to be rather a good
approximation valid in not very strong magnetic fields.
4.1. The DOS at the zero pseudomagnetic field
Setting = 0pmB we recover the well-known results that
were experimentally observed in [5]. Then Eqs. (18) and (19)
result in the sublattice DOS
(0) | |( ) = ( ),
2A
eBD ε δ ε − ∆
π
(0) | |( ) = ( ).
2B
eBD ε δ ε + ∆
π
(23)
This confirms that the LLL is valley polarized, because
each LLL contribution to the DOS comes from either K or
'K valley, as discussed in the Introduction. This feature has
to be contrasted with the valley resolved but summed over
the two sublattices DOS
, (0) , (0), (0) ( ) = ( ) ( )' ''
BAD D Dε ε + ε =K K K KK K
| |= ( sgn ( )).
2
eB eBδ ε −η∆
π
(24)
For 1n ≥ the levels at K and 'K points described by Eqs. (20)
and (21) are degenerate, but the DOS on A and B sub-
lattices differs and this effect is observable [5].
4.2. The DOS at the zero magnetic field
Setting = 0eB we obtain from Eqs. (18) and (19) that
the LLL contribution to the sublattice DOS is
(0)
(0)
| |
( ) = ( ) ( ),
2
| |
( ) = ( ) ( ).
2
pm
pmA
pm
pmB
B
D B
B
D B
ε θ δ ε − ∆
π
ε θ − δ ε + ∆
π
(25)
This confirms that the LLL is sublattice polarized, as dis-
cussed in the Introduction.
4.3. Analytical expression for the valley DOS
Although the expressions for the sublattice and valley
DOS presented in Sec. 3 are sufficient for the numerical
study presented in Sec. 5, it is always useful to have a sim-
ple analytical expression for the DOS. One can notice that
the valley DOS, Eq. (14) is the sum of delta-functions (or
Lorentzians when the level widening is taken into account),
because the sum of the weight factors ( ) / (2n nM M± ±± ∆ )
present in Eqs. (20) and (21) gives 1. This allows one to
use the results of Ref. 26, and calculate the sum over Lan-
dau levels analytically
,
2 2 2
1( ) = | | ( )
2 ( )
D B B′
± ±
Γε − θ −
π ε − ∆ + Γ
K K
2
2 2| | ( ) ln
2 | |( )
B B
B± ±
±
Γ Λ
− θ ± + Γ −
ε + ∆ + Γ
2 2( )Im ( ) .
2 | |
ii
B±
∆ − ε + Γ − ε + Γ ψ
(26)
Here ψ is the digamma function, ± sign corresponds to K
and 'K points, the width of all levels Γ is assumed to be
the same, and Λ is the cutoff energy that has the order of
bandwidth. Eq. (26) differs from Eq. (4.15) of Ref. 26 by
the first two terms. In the present case they take care of the
electron hole asymmetry of the LLL, while in [26] both K
and 'K points contribute to the full DOS. The advantage of
Eq. (26) is that it allows to consider the low field regime
when the direct numerical summation over many Landau
levels is consuming.
5. Results
Now we use Eqs. (18)–(21) to study the valley (14),
sublattice (13) and the full (15) DOS numerically. For
simplicity we assume that all Landau levels have the same
width Γ . In this case the valley DOS and then the full DOS
can also be calculated using Eq. (26). To fit real experimen-
tal data [27] it may be necessary to consider the width, nΓ ,
dependent on the Landau level index. This can be easily
done in the framework of numerical computation of the sum
over Landau levels. However, when all levels have the same
width and one is interested in the valley DOS, it is more
efficient to compute it from the analytical expression (26)
which is easier to use in the low field regime. In all numer-
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 10 1511
V.O. Shubnyi and S.G. Sharapov
ical work we take the value of the Fermi velocity,
6= 10 m/sFv that corresponds to the Landau energy scale,
2 1/2
0 = ( ) = 25.7 [ ] meVF B B T± ±ε v . The gap ∆ that lifts
the energy degeneracy of the A and B sublattices and
breaks the inversion symmetry was observed for a
graphene monolayer on top of SiC, graphite [4], and hex-
agonal boron nitride [28]. Its value ranges from 10 meV to
several tens of meV.
Figure 1 demonstrates how the full density of states,
Eq. (15), is formed by the contributions from the valley
resolved DOS, Eq. (14): left panel (a) is for | | < | |pmeB B
and the right panel (b) is for | | > | |pmeB B . The two curves
(thin solid red and thin dashed blue) in the bottom part
of the figure show the valley resolved DOS, , ( )'D εK K .
The curves for K and 'K points have the peaks corre-
sponding to the relativistic Landau levels with the energies
| |n B±± . The positions of the peaks corresponding to
the LLL with 0 =E ±∆ distinguish the cases (a) the PMF
dominated regime, | | < | |pmeB B , when both peaks have
the same sign of the energy, and (b) the magnetic field
dominated regime, | | > | |pmeB B , when the peaks have the
opposite energy sign. We checked that the same curves
also follow from the analytical expression (26). Those are
rather trivial consequences of having a superposition of
magnetic and PMF.
The full DOS ( )D ε shown by thick black curve in the
two panels obviously has two series of peaks. One could
see that in the special cases, the difference between two
curves is substantial, and resulting DOS curve has irregular
features and/or masked peaks. Depending on the values of
the effective fields B± , the Landau levels could be viewed
as a splitting of one level (in case | | | |B B+ −≈ ) or as the two
largely independent series, as for the case shown in Fig. 1.
Let us look closer at the pattern that overlapping Lan-
dau levels may create for certain values of B and pmB . The
energies of the Landau levels with indices n+, n− for K
and 'K points coincide, viz. =n nM M
+ −
if there exist
some values of B+ and B− satisfying the condition,
| | = | |B n B n+ + − − . This implies that the fraction
| | / | | = /B B a b+ − has to be rational. In terms of the initial
fields B and pmB this condition implies that
1 /= .
1 / pm
a beB B
a b
−
+
(27)
The corresponding beating patterns for four values of
the fraction /a b are shown in Fig. 2. The lowest (green)
curve is for the simplest case, / = 1/ 2B B− + . Each second
level with coinciding energies is enhanced. The second
from the bottom (blue) curve is for / = 1/ 3B B− + . In this
case an enhancement occurs for each third level. The third
from the bottom (red) curve is for / = 2 / 3B B− + has even
more tricky pattern with the highest each third level and
each forth level of an intermediate height. The curve on the
top (black) is for / = 1/ 4B B− + .
It is instructive to represent the dependences of the full
DOS, ( )D ε , on the fields B and pmB employing the densi-
ty plot. Since a wide range of the fields is involved, its
consideration in the low field regime may demand summa-
tion over many Landau levels. Thus we use Eq. (26), where
the summation is done analytically. Figures 3(a) and (b) on
the top panel show the full DOS, ( , , )pmD B Bε as a func-
tion of energy ε in meV and magnetic field B in T for
= 0 TpmB and = 8 TpmB . Figs. 3(c) and (d) in the bottom
panel show the full DOS, ( , , )pmD B Bε as a function of
energy ε in meV and PMF pmB in T for = 0 TB and
= 8 TB . The density plot is partly overlaid with the solid (red)
and dashed (blue) curves that show position of the peaks in
the DOS originating from the Landau levels at K and 'K
points, respectively. Figure 3(a) (top left panel) describes
unstrained graphene. The Landau levels fan away from the
Dirac point at = 0ε . One can find a similar DOS map for
the STS measurements [4] of graphene on chlorinated SiO2.
In the real case the spectra are distorted at low fields due to
the substrate induced disorder and are strongly position
Fig. 2. (Colour online) The full DOS, ( )D ε , in arbitrary units as
the functions of energy 0/ε ε , where 2 1/2
0 = ( ) =F B+ε v
= 25.7 [ ] meVB T+ . The green curve is for / = 1 / 2B B− + , the
blue curve is for / = 1 / 3B B− + , the red curve is for
/ = 2 / 3B B− + and the black curve is for / = 1 / 4B B− + . The gap
= 10 meV∆ and the scattering rate = 2 meVΓ .
Fig. 1. (Colour online) The full DOS, ( )D ε , (thick solid) and the
valley-resolved DOS, , ( )'D εK K , (thin solid and thin dashed) in
arbitrary units as the functions of energy 0/ε ε , where
2 1/2
0 = ( ) = 25.7 [ ] meVF B B T+ +ε v is the Landau scale for K
valley. Left panel: (a): the fields = 5 TB and = 18 TpmB . Right
panel: (b): the fields = 10 TB and = 1 TpmB . The gap
= 50 meV∆ and the scattering rate = 10 meVΓ in the both cases.
1512 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 10
Density of states of Dirac–Landau levels in a gapped graphene monolayer under strain gradient
dependent. The density plot [4] allows to observe at higher
fields the sequence of broadened Landau levels with sepa-
rated peaks. Fig. 3(c) (bottom left panel) describes strained
graphene in zero magnetic field. It is almost identical to
Fig. 3(a) except to the LLL that in the case of strained
graphene breaks the electron-hole symmetry. Figures 3(b)
and (d) (right top and bottom panels) describe strained gra-
phene in the external magnetic field. This case was also
studied experimentally in [13], where SMT and STS meas-
urements were made on the deformed by gating graphene
drumhead.
Comparing all these panels we observe that in the pres-
ence of both PMF and magnetic field there exist regions of
intersecting Landau levels with the opposite slope that are
related to the opposite valleys. In Fig. 3(b) this is the re-
gion with | | < | |pmeB B , while Fig. 3(d) the corresponding
region is seen for | | < | |pmB eB . This behavior of Landau
levels is almost obvious in the presented case. However, in
the case of poorly resolved Landau levels this feature can
be rather helpful for proving the presence of both PMF and
magnetic field.
Finally, we illustrate in Fig. 4 how the full DOS is dis-
tributed between the sublattices. The DOS on A and B sub-
lattices, , ( )A BD ε , are shown by (dashed) green and solid
(orange) curves, respectively. Figure 4(a) (top left panel)
describes unstrained graphene in the external magnetic field.
It corresponds to the situation studied experimentally in [5].
We observe that the positive (negative) energy states reside
on A (B ) sublattice. Since these states are associated with
different valleys, the LLL is indeed valley polarized. Fur-
thermore, the sublattice asymmetry is also seen for higher
levels, because we took a large value of the gap = 50 meV∆ .
Fig. 4(b) (top right panel) describes strained graphene in
zero magnetic field. As it should be, the LLL is indeed
completely sublattice polarized, while higher levels are po-
larized in the same fashion as in Fig. 4(a). The PMF domi-
nated regime, | | < | |pmeB B , is shown in Fig. 4(c) (bottom
left panel). The LLL polarization is similar with Fig. 4(b).
The magnetic field dominated regime, | | > | |pmeB B , is
shown in Fig. 4(d) (bottom right panel) and it is similar to
Fig. 4(a). Figures 4(c) and (d) are computed for the same
values of the parameters as Fig. 1(a) and (b), respectively.
When both fields are present the asymmetry between the
sublattices can be enhanced even for higher levels.
We note that in the present work the sublattice asym-
metry is directly brought by the inversion symmetry gap ∆.
We established that the presence of PMF and magnetic
field further enhances this effect. It is shown in [29] that
the local sublattice symmetry can be broken just by the
deformation. This deformation is not a pure shear, so it
produces not only the PMF, but also a scalar potential.
Fig. 3. (Colour online) Density map of the full DOS ( , , )pmD B Bε
as a function of energy ε in meV, magnetic field B in T and
PMF in T . Top left panel (a): for a constant = 0 TpmB . Top
right panel (b): for a constant = 8 TpmB . Bottom left panel (c):
for a constant = 0 TB . Bottom right panel (d): for a constant
= 8 TB . The density-map is overlaid with red and blue curves
that show the position of the peaks originating from K and 'K
points. The gap = 50 meV∆ , the scattering rate = 5 meVΓ and
the Landau scale 2 1/2
0 = ( ) = 25.7 [ ] meVF B B T± ±ε v in all cases.
Fig. 4. (Colour online) The sublattice resolved DOS, ( )AD ε
(dashed) and ( )BD ε (solid) in arbitrary units as the functions of
energy 0/ε ε . Top left panel (a): in the absence of PMF, = 0pmB
and = 10 TB . Top right panel (b): in the absence of magnetic
= 0 TB , = 10 TpmB . Bottom left panel (c): the fields = 5 TB
and = 18 TpmB . Bottom left panel (d) the fields = 10 TB and
= 1 TpmB . The gap = 50 meV∆ and the scattering rate
= 10 meVΓ in all cases.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 10 1513
V.O. Shubnyi and S.G. Sharapov
6. Conclusion
In the present work we had in mind that the sublattice
resolved DOS can be measured by STS. However, the full
DOS can also be experimentally found by measuring the
quantum capacitance [27] which is proportional to the ther-
mally smeared DOS. The corresponding convolution with
a Fermi distribution is easily expressed in terms of the di-
gamma function [30], so that the presented here results can
be easily applied for this case.
In conclusion we note, that controlling the valley degree
of freedom is important for possible valleytronics applica-
tions of the new materials. In this respect a simultaneous
tuning of the strain (PMF) and magnetic field is rather use-
ful, because it allows to remove the valley degeneracy.
Thus the experimental testing of the features discussed in
this work would be helpful for development of valley-
tronics.
We gratefully acknowledge E.V. Gorbar, V.P. Gusynin
and V.M. Loktev for helpful discussions. S.G. Sh. acknow-
ledges the support from the Ukrainian State Grant for Fun-
damental Research No. 0117U00236 and the support of EC
for the RISE Project CoExAN GA644076.
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1514 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 10
1. Introduction
2. Model
3. Green’s function, sublattice and valley resolved DOS
4. Expressions for numerical and analytical calculation of the DOS
4.1. The DOS at the zero pseudomagnetic field
4.2. The DOS at the zero magnetic field
4.3. Analytical expression for the valley DOS
5. Results
6. Conclusion
|