Local sublattice symmetry breaking for graphene with a centro-symmetric deformation

3.4 MB·English

by  Martin Schneider

Local sublattice symmetry breaking for graphene with a centro-symmetric
deformation
M. Schneider,1 D. Faria,2 S. Viola Kusminskiy,1 and N. Sandler3,∗
1Dahlem Center for Complex Quantum Systems and Institut fu¨r Theoretische Physik,
Freie Universita¨t Berlin, Arnimallee 14, 14195 Berlin, Germany
2Instituto de F´ısica, Universidade Federal Fluminense, Nitero´i, Av. Litoraˆnea sn 24210-340, RJ-Brazil
3Department of Physics and Astronomy, Nanoscale and Quantum Phenomena Institute,
Ohio University, Athens, Ohio 45701-2979, USA
(Dated: April 20, 2015)
5
We calculate the local density of states (LDOS) for an infinite graphene sheet with a single
1
centro-symmetric out-of-plane deformation, in order to investigate measurable strain signatures on
0
graphene. We focus on the regime of small deformations and show that the strain-induced pseudo-
2
magnetic field induces an imbalance of the LDOS between the two triangular graphene sublattices
r in the region of the deformation. Real space imaging reveals a characteristic six-fold symmetry
p
patternwherethesublatticesymmetryisbrokenwithineachfold,consistentwithexperimentaland
A
tight-bindingobservations. Theopengeometrywestudyallowsustomakeuseoftheusualcontin-
7 uum model of graphene and to obtain results independent of boundary conditions. We provide an
1 analytic perturbative expression for the contrast between the LDOS of each sub-lattice, showing a
scalinglawasafunctionoftheamplitudeandwidthofthedeformation. Weconfirmourresultsby

] a numerically exact iterative scattering matrix method.
l
l
a PACSnumbers: 72.80.Vp,73.23.-b,72.10.Fk,77.80.bn
h
-
s
e Introduction.— Graphene under strain has been largely mental test-bed to explore the interplay between highly
m discussed in the literature and explored for different ge- tunable magnetic fields and Dirac fermions. For exam-
. ometries, with particular features providing alternative ple,ascanningtunnelingmicroscope(STM)tiphasbeen
at routes to confine and control its charge carriers1–3. Sig- used not only to probe samples, but also to continuously
m nificant development in the theoretical description of deform graphene nanomembranes, demonstrating elec-
strained graphene elucidated how its electronic proper- tronic confinement due to non-uniform pseudomagnetic
-
d tiesaremodifiedonstrainedsurfaces. Atthemicroscopic fields10. For a similar experimental setup, Mashoff et al.
n level,ageneraldeformationisdescribedbymodifications obtained atomically resolved STM images of stable and
o
in the atomic positions which reflects in the Hamilto- lifted regions of graphene11. Whereas a hexagonal ar-
c
nian as local changes in the hopping parameter4,5. In rangement of the carbon atoms was found at unstrained
[
the continuum model these changes appear as an effec- regions,asexpectedformonolayergraphene11,withinthe

2
tive gauge field, and electrons with momentum around strainedareaatriangularpatternofbrightspotswasob-
v
the Dirac valleys move in the deformed region as in the served, signaling a symmetry breaking between A and B
1
presence of a pseudomagnetic field6. Strain also pro- sublatticesinsomeregions. Atthetimetheauthorsspec-
8
9 duces a deformation potential, i.e., a scalar field simi- ulated the effect to be caused by an instability in which
2 lar to a local chemical potential that can affect electron the different sublattices acquire a zigzag configuration
0 dynamics6. Very recently, measurements in high-quality with respect to the substrate. However, a local sublat-
. graphene samples on particular substrates suggested a tice rearrangement requires energies that are prohibitive
1
0 strong connection between random fluctuations in strain within the regime of STM imaging making such scenario
5 and transport properties7. ratherunlikely. Atomistictight-bindingmodels16–18have
1 predictedthedevelopmentofsuchasymmetrybutacon-
v: The use of strain effects to engineer graphene elec- tinuum, symmetry-based description remains missing.
tronic properties has also been explored in several ex-
i Similar patterns were observed in earlier works but re-
X periments in the last years8–13. As one of the most
mained unexplained also19. These results indicate an
relevant findings, Levy et al. were able to show the
r incomplete understanding of the fundamental electronic
a presence of pseudo Landau levels generated by giant
propertiesofgraphenesampleswherelocalmanipulation
pseudomagnetic fields induced by homogeneous strain in
produce effective inhomogeneous gauge and scalar fields.
graphene nanobubbles8. This experimental confirmation
that strain can have striking effects on the electronic In this work, we approach this problem by investigat-
properties of graphene has been followed by other ex- ing the electronic properties of a graphene sheet in the
periments that explore the effect of different geometries presence of an axially symmetric out-of-plane deforma-
as a path to control graphene electromechanically10–13. tion. The strain produced by such distortion is repre-
A generic deformation of a graphene sheet can cause in- sentedbyapseudomagneticfieldwithtrigonalsymmetry
homogeneous strain, which results in an effective non- and embodies a good approximation to standard exper-
uniform pseudomagnetic field and provides an experi- imental configurations, while still allowing for analytical
0,8
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tion of confinement of electrons due to this deformation. 1   Resources & Contro1l1.l0e.r0s  
  Shared by You 00.5.5  
InparticularwecalculatetheLDOSandshowthatano- Out[11]= 0   Shared with You 00  
ticeable imbalance in the distribution of charge density -1   Trash -0.05.5  
between the two graphene nonequivalent sublattices ap- -1.10.0
pears even for small deformations, providing a possible (c) x -2-2 -1 0 1 2r---6/4026244022646b
explanation for the experimentally observed sublattice
asymmetry. We perform exact numerical calculations
and show that these results are well described within an
analytical perturbative approach for small deformations.
We analyze the dependence of the maximum LDOS con-
trast between sublattices on energy and strain strength,
providing a scaling dependence with the parameters of
the deformation. Finally, we also show that the effective
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scalar field introduced by strain minimally modifies the FIG.1: (Coloronline)Schematicviewofthegraphenelattice
predicted sublattice asymmetry. with a magnified out-of-plane deformation. (b) Pseudomag-
Model.— The electronic properties of undeformed netic field created by a deformation with a Gaussian height
profile as in (a). (c) Spatial profile of the LDOS for sublat-
graphene are, for low energies and large system sizes,
ticeAinthepresenceofabump,seeEq.(13). Bright(dark)
governed by two copies of a two-dimensional (2D) Dirac
spots indicate an increase (decrease) of LDOS compared to
Hamiltonian H = v p·σ where v ≈ 106m/s is the
0 F F the undeformed graphene sheet. For sublattice B, the effect
velocity of graphene electrons, p the electronic momen-
is exactly opposite.
tumaroundtheK(K’)point,andσ =(σ ,σ )arePauli
x y
matrices reflecting the pseudospin degree of freedom as-
sociated with the sublattice structure of the honeycomb
For the radial symmetric deformation, the associated
lattice14. The strain is produced by a mechanical defor-
pseudomagnetic field B = ∇ × A shares the trigonal
mationmodeledwithaheight-profileh(r)thatiscentro-
symmetry of the graphene lattice,
symmetricandiswrittengenericallyash(r)=Ah (r/b),
0
where h0 contains the radial profile, and the parameters (cid:126) −β
Aandbdescribeamplitudeandeffectiveradiusofthede- Bz(r)= αb0(r/b)sin3θ (3)
e 2ab
formation. In the following, to illustrate our results, we (cid:18) (cid:19)
consider the case of a Gaussian bump with height pro- where the spatial profile is given by the function b (x)=
0
file h0(x) = e−x2. Note however that our results below 2f(x)−f(cid:48)(x). FortheGaussian-shapeddeformation, one
are qualitatively valid for a generic profile h with axial x
0 has b (x)=8x3e−2x2.
symmetry. 0
In addition to the gauge field, the electrons are also
The effect of such deformation on the electronic prop-
exposed to a scalar potential proportional to the trace
erties in the continuum limit are described within the
of the strain tensor. In our model it is represented by
theory of elasticity. For an out-of-plane deformation the
V(r) = −g αf(r/b) with g = 3eV as a typical value16.
strain tensor of elasticity15 is derived from the height s s
The low-energy electronic properties in the presence of
profile h according to (cid:15) = 1∂ h∂ h, which in polar co-
ij 2 i j the deformation are hence described by
ordinates (r,θ) reads
cos2θ sinθcosθ H =vF[p+eA(r)]·σ+V(r) (4)
(cid:15)=αf(r/b) , (1)
sinθcosθ sin2θ
(cid:18) (cid:19) In this article we consider a bump that is smooth on
whereα=A2/b2 setsthestrengthofthestrain,whileits the scale of the lattice constant, such that a coupling
spatial distribution is contained in the function f(x) = betweenthevalleyscanbeneglected16,18,20,21. Moreover,
1[h(cid:48)(x)]2. For the Gaussian profile, one has f(x) = weconsideraninfinitegraphenesheetcontainingasingle
2 0 deformation, hence our results are independent of finite
2x2e−2x2.
size effects and boundary conditions16–18,20,21.
In the presence of the deformation, electrons experi-
Perturbation theory.— In this section we present ana-
ence the strain as a gauge field
lytic results for the change in the LDOS produced by
A(r)=−egvv αf(r/b) −cosisn22θθ , (2) twhiethscaaptteerrtiunrgbaotfiveelecatprpornosacohffitnhreeadlefsopramcea.tiWoneocbotnasiindeedr
F (cid:18) (cid:19) therefore small deformations that allow for an expansion
where we chose the zigzag direction to lie along the x- in the parameter α.
axis2. The coupling constant is gv = β(cid:126)vF/2a ≈ 7eV5, From now on we set (cid:126) = vF = 1, and work around
being β = |∂logt/∂loga| ≈ 3, and t and a the hopping the K valley. The effect of the K’ valley is discussed at
parameter and the lattice constant of graphene. the end of this section. We split the Hamiltonian in the
3
kinetic part H and the perturbation V, are interested in the different sublattice occupations, we
0
further introduce the sublattice-resolved LDOS
0 A (r)
V(r)=eA(r)·σ = − , (5)
A+(r) 0 νA/B(ε,r)= (cid:104)Ψm(r)|PA/B|Ψm(r)(cid:105) (11)
(cid:18) (cid:19)
m
(cid:88)
where we defined A (r) = e(A (r) ± iA (r)) =
± x y where P are projectors on the respective sublattice
A/B
−β αf(r/b)e∓2iθ. We neglect the scalar potential in A/B. Forundeformedgraphene,evaluatingEq.(11)with
2a
t(cid:16)his p(cid:17)art, we will include its effect in the next section. thefreestates|Φ(m0)(r)(cid:105)producesthewell-knownvalueof
Let us start with the states of the Dirac equation in
the absence of the deformation. Here, we take circular ν(0) (ε,r)= ε J (εr) 2 = ε , (12)
A,B 4π |m−1/2| 4π
waves as a set of basis states, m
(cid:88)(cid:2) (cid:3)
ε e−iθ/2J (εr) for the LDOS per sublattice.
|Φ(m0)(r)(cid:105)= 4πeimθ isgn(m)eiθ|/m2J−1/2| (εr) , We now want to discuss the effect of the deformation
(cid:114) (cid:18) |m+1/2| (cid:19) on the LDOS. Specifically we address the limit εb (cid:28) 1,
(6)
which is the relevant case for experiments with a radius
where ε denotes the energy of the Dirac fermions (which
of a few lattice constants. In this case, one may simplify
we assume to be positive, for simplicity), and m is a
the results by using the asymptotic expressions of the
half-integer index labeling the states according to their
BesselandHankelfunctionsforsmallargumentsr (cid:46)b22.
angular momentum. J (x) denotes the Bessel function
n Upon retaining only the leading contribution for small
of n-th order. We chose a normalization such that
energies,onefindsforthecorrectionsδν =ν −ν(0)
A,B A,B A,B
to leading order in α
dr(cid:104)Φ(0)(ε;r)|Φ(0)(ε(cid:48);r)(cid:105)=δ δ(ε−ε(cid:48)). (7)
m n nm
(cid:90) δν (ε,r) δν (ε,r) βA2
A =− B =− sin3θg(r/b) (13)
Our goal is to find the scattering state |Ψ (r)(cid:105), that ν (ε,r) ν (ε,r) ba
m A B
replaces |Φ(0)(r)(cid:105) when the bump is present. This is de-
m with the function
termined by the Lippmann-Schwinger equation
1 x
g(x)= dyy3f(y) (14)
|Ψm(r)(cid:105)=|Φ(m0)(r)(cid:105)+ dr(cid:48)G(r,r(cid:48))V(r(cid:48))|Ψm(r(cid:48))(cid:105) (8) x3 (cid:90)0
(cid:90)
To leading order in α, one can replace ν(0) by ν in
which contains the Green’s function of graphene, A,B A,B
the denominator of Eq. (13). Notice that the relative
|Φ(0)(r)(cid:105)(cid:104)Φ(−)(r(cid:48))|, r <r(cid:48) LDOS correction has no dependence on energy. Thus,
G(r,r(cid:48))=−iπ m m (9) the deformation changes the local occupation in the dif-
m (cid:40)|Φ(m+)(r)(cid:105)(cid:104)Φ(m0)(r(cid:48))|, r >r(cid:48), ferent sublattices in opposite directions, and their spatial
(cid:88)
distributionsharesthesymmetryoftheunderlyingpseu-
where we defined domagnetic field with a radial distribution governed by
thefunctiong(x). SpecificallyforaGaussianheightpro-
|Φ(±)(r)(cid:105)= ε eimθ e−iθ/2H|(m±−)1/2|(εr) . file, one finds
m (cid:114)4π (cid:32)isgn(m)eiθ/2H|(m±+)1/2|(εr)(cid:33) 1
(10) g(x)= 1−e−2x2(1+2x2+2x4) . (15)
4x3
Hµ(±)(x) = Jµ(x)±iYµ(x) are Hankel functions of first (cid:104) (cid:105)
and second kind. A derivation of Eq. (9) is given in the The spatial distribution of the change in LDOS for sub-
Supplementary Material22. lattice A according to Eq. (13) is shown in Fig. 1 (c) for
For small deformations, we solve the Lippmann- a Gaussian deformation of typical dimensions.
Schwinger equation in the Born approximation, and re- AquantityofexperimentalrelevanceistheLDOScon-
place the scattering state |Ψ (r)(cid:105) on the right-hand side trast between sublattices, defined as
m
of the equation by the unperturbed state |Φ(m0)(r)(cid:105). Note |ν −ν |
that our perturbative approach is valid for g α (cid:28) ε. C =2 A B (16)
s,v ν +ν
The explicit form of the resulting scattering states is A B
shown in the Supplementary Material22. The trigonal which is plotted as a function of the radial distance in
symmetry of the pseudomagnetic field underlies the cou- Fig. 2 (a). Note that, for a fixed width b of the de-
pling between angular momentum states differing by 3. formation, Eq (13) indicates that the contrast C scales
The LDOS is obtained by calculating ν(ε,r) = quadratically with the amplitude of a centro-symmetric
(cid:104)Ψ (r)|Ψ (r)(cid:105). The new states are properly nor- deformation. This scaling is shown for a Gaussian defor-
m m m
malized to leading order in α, as the linear in α correc- mation in Fig. 3 and compared with the exact numerical
(cid:80)
tion is orthogonal to the unperturbed state. Since we results presented in the next section.
4
(a) a real magnetic field that has the same sign in both val-
leys). These two effects ensure that their contributions
to the sublattice occupancy contrast are identical.
Numerics.— In this section we discuss briefly our ex-
act numerical approach for the continuum model, which
is not restricted to the case of small amplitude de-
formations. The results obtained confirm our findings
described in the previous section in the corresponding
regimes. We use a slight modification of the method in-
(b)
troduced in Ref. 24, which allows for the calculation of
the scattering matrix S for an arbitrarily-shaped scalar
potential. The extension to include a vector potential is
straightforward. To calculate the LDOS integrated over
a certain (arbitrarily chosen) volume V per sublattice,
we include a fictitious additional scattering potential
ε 0 1, r∈V
V (r)= A × (17)
ε (cid:18) 0 εB (cid:19) (cid:40)0, else
FIG. 2: (Color online) LDOS contrast C as a function of Such potential locally changes the electron’s energy in
distance from the bump’s center for fixed angle θ = π/2
V by a magnitude ε in the different sublattices A/B.
A/B
(A = 0.1nm,b = 0.5nm). (a) Comparison of C between per-
TheLDOSpersublatticeintegratedoverV isthenfound
turbative (solid line, blue), and exact numerical approaches
from the scattering matrix S via
(red points) for ε = 0.5eV. (b) Different data sets obtained
numerically for ε=0.5eV,g =0 (blue), ε=0.5eV,g =3eV
s s 1 ∂S
(red), ε = 0.1eV,gs = 3eV (green), and ε = 0.9eV,gs = 3eV νj(V)= Tr S† (18)
(yellow). 2πi (cid:20) ∂εj(cid:21)εA/B=0
wherej =A,Bspecifiesthesublattice. Fig.2(a)showsa
comparison between analytical and numerical results for
realistic parameters. Note that in the region of small
amplitudes the contrast obtained with the expression
from perturbation theory follows closely the exact solu-
tiongivenbythenumericalapproach. Numericalcalcula-
tionsinthepresenceofthescalarpotentialV(r)induced
by the deformation were carried out for different values
of the phenomenological parameter g . Our results, as
s
shown in Fig. 2 (b), confirm that its effect on the con-
trast is negligible (note that, to leading order in α, V(r)
affects the occupation of both sublattices in the same
way, thus not affecting the contrast (16)). Furthermore,
FIG. 3: (Color online) Scaling of contrast C as a function the energy independent value for the contrast predicted
of amplitude for fixed Gaussian width b = 0.5nm. Curves withtheanalyticalapproach(aslongastherequirements
obtained with perturbative (solid line, blue), and exact nu- ε(cid:29)αg and εb(cid:28)1 are met) is also verified in the nu-
s,v
merical methods (points, red) for ε=0.5eV are compared. merical results as shown in Fig. 2 (b).
Conclusions.— We have shown that a centro-symmetric
deformation, with its local breaking of the lattice trans-
Toconcludethissectionlet’sdiscusstheroleofthetwo lational symmetry, produces a local sublattice symmetry
valleys K and K’ in these results. As mentioned above, breakingontheelectronicpropertiesofagraphenesheet,
the deformation is smooth enough that does not couple and a consequent LDOS contrast between sublattices.
the valleys and their contributions add directly. To see AnalyticexpressionswithintheBornapproximationpre-
thattheseareidentical,notethattheDiracHamiltonian dicttheintensityoftheLDOScontrasttobedetermined
takes the same form in both valleys when the spinors bytheamplitudeofthedeformationandtobeenergyin-
are written in the valley symmetric representation23: dependentfortherangeofvalidityoftheapproximation.
(ψ ,ψ ) around valley K, and (ψ ,−ψ ) around valley Exactnumericscarriedoutwithscatteringmatrixmeth-
A B B A
K’. Note that the components referring to A and B sub- ods confirm the validity of these results, for experimen-
lattices are interchanged between different valleys. On tallyrealisticparameters. Whileournumericalapproach
theotherhand,thepseudomagneticfieldenterstheDirac allows us in principle to treat any size of deformations,
equationwithoppositesignforeachvalley(incontrastto we concentrated here on the study of small deformations
5
and showed that there is a measurable LDOS contrast where g (r,r(cid:48)) is determined by
µ
between sublattices even in the absence of Landau lev-
els14. Thecrossovertothisregimewillbepublishedelse- 1 µ2 1
where. Our findings here provide an alternative inter- ε2+∂r2+ r∂r− r2 gµ(r,r(cid:48))= rδ(r−r(cid:48)). (A7)
pretation for recent experimental observations on STM (cid:18) (cid:19)
graphene nanomembranes25 and provide a quantitative
Forr (cid:54)=r(cid:48),g isthussatisfyingaBesseldifferentialequa-
way to guide the use of strain in the design of electronic µ
tion. We therefore write
properties of graphene flakes.
Acknowledgments.— We gratefully acknowledge sup-
g (r,r(cid:48))=a J (εr )H(+)(εr ) (A8)
port by CNPq, and DAAD (D.F.), DFG SPP 1459 and µ µ |µ| < |µ| >
the A. v H. Foundation (M.S., S.V.K.), and NSF No.
DMR-1108285 (D.F., N.S.) as well as discussions with with r = max(r,r(cid:48)), r = min(r,r(cid:48)). In this way, we
> <
R. Carrillo-Bastos, A. Georgi, M. Morgenstern and P. respect the conditions, that
Nemes-Incze. • g is regular at r →0,
µ
Appendix A: Greens functions • gµ is behaving as an outgoing wave (∝ e√iεrr) for
r → ∞ (this is the proper boundary condition for
the retarded Green function ε→ε+i0),
In this appendix, we show a derivation of the Green
function for graphene given in Eq. 9 in the main text.
The defining equation reads • gµ is continuous at r =r(cid:48).
(ε+iσ·∇)G(r,r(cid:48))=δ(r−r(cid:48)). (A1) The remaining coefficient a is determined by the jump
µ
condition for the derivative of g ,
µ
For the derivation, it is customary to introduce the aux-
iliary Green function Gs(r,r(cid:48)) via [∂ g (r,r(cid:48))]r=r(cid:48)+0 = 1, (A9)
r µ r=r(cid:48)−0 r(cid:48)
G(r,r(cid:48))=(ε−iσ·∇)G (r,r(cid:48)) (A2)
s
which yields a = π. Thus, the result for the scalar
TheadvantageoftheGreenfunctionG isthatitsatisfies µ 2i
s Green function reads
a scalar equation (i.e. the pseudospin degree of freedom
has been eliminated).
1
G (r,θ;r(cid:48),θ(cid:48))= eiµ(θ−θ(cid:48))J (εr )H(+)(εr ).
(ε2+∇2)Gs(r,r(cid:48))=δ(r−r(cid:48)) (A3) s 4i(cid:88)µ |µ| < |µ| >
(A10)
Moreover, Gs is the solution to the Helmholtz equation The Green function for graphene is then found in a
in two dimensions. We would like to express the Green straightforwardalbeitlengthycalculationfromEq.(A2).
functioninpolarcoordinates,thusweexpresstheLapla-
cian as
1 1
∆=∂2+ ∂ + ∂2, (A4) Appendix B: Scattering states
r r r r2 θ
and we write for the δ-function The scattering states in the Born approximation are
obtained by inserting Eq. 5, Eq. 6 and Eq. 9 into the
1 1
δ(r−r(cid:48))= δ(r−r(cid:48)) eiµ(θ−θ(cid:48)), (A5) following equation
r 2π
µ
(cid:88)
|Ψ (r)(cid:105)=|Φ(0)(r)(cid:105)+ dr(cid:48)G(r,r(cid:48))V(r(cid:48))|Φ(0)(r(cid:48))(cid:105). (B1)
wherethesummationisrunningoverintegerµ. Wethen m m m
expand G as (cid:90)
s
After insertion of the corresponding expressions, one
1
G (r,θ;r(cid:48),θ(cid:48))= eiµ(θ−θ(cid:48))g (r,r(cid:48)) (A6) finds
s µ
2π
µ
(cid:88)
6
|Ψ (r)(cid:105)= |Φ(0)(r)(cid:105)
m m
− iπ |Φ(+)(r)(cid:105) r(cid:48)dr(cid:48) dθ(cid:48)(cid:104)Φ(0)(r)|V(r(cid:48))|Φ(0)(r(cid:48))(cid:105)
n n m
n (cid:90)r(cid:48)<r (cid:90)
(cid:88)
− iπ |Φ(0)(r)(cid:105) r(cid:48)dr(cid:48) dθ(cid:48)(cid:104)Φ(−)(r)|V(r(cid:48))|Φ(0)(r(cid:48))(cid:105). (B2)
n n m
n (cid:90)r(cid:48)>r (cid:90)
(cid:88)
The angular integration is nonzero for n = m+3 and n = m−3. Calculating the matrix elements, one obtains the
scattering states given by
β
|Ψ (r)(cid:105)=|φ(0)(r)(cid:105)+π εα a (r)|φ(+) (r)(cid:105)+b (r)|φ(+) (r)(cid:105)+c (r)|φ(0) (r)(cid:105)+d (r)|φ(0) (r)(cid:105) (B3)
m m 4a m m+3 m m−3 m m+3 m m−3
(cid:16) (cid:17)
with
r
a (r)=−sgn(m) dr(cid:48)r(cid:48)f(r(cid:48)/b)J (εr(cid:48))J (εr(cid:48))
m |m+5/2| |m+1/2|
(cid:90)0
r
b (r)=sgn(m−3) dr(cid:48)r(cid:48)f(r(cid:48)/b)J (εr(cid:48))J (εr(cid:48))
m |m−5/2| |m−1/2|
(cid:90)0
∞
c (r)=−sgn(m) dr(cid:48)r(cid:48)f(r(cid:48)/b)H(+) (εr(cid:48))J (εr(cid:48))
m |m+5/2| |m+1/2|
(cid:90)r
∞
d (r)=sgn(m−3) dr(cid:48)r(cid:48)f(r(cid:48)/b)H(+) (εr(cid:48))J (εr(cid:48)).
m |m−5/2| |m−1/2|
(cid:90)r
(B4)
Theleadingordercorrectiontothelocaldensityofstatespersublatticeuptofirstorderonα,whenthedeformation
is present, is given by
δν β
A (ε,r)= − π εαe3iθ J (εr)H(+) (εr)a (r)
ν 4a |m−1/2| |m+5/2| m
A
m
(cid:88)
β
+ π εαe−3iθ J (εr)H(+) (εr)b (r)
4a |m−1/2| |m−7/2| m
m
(cid:88)
β
− π εαe3iθ J (εr)J (εr)c (r)
4a |m−1/2| |m+5/2| m
m
(cid:88)
β
+ π εαe−3iθ J (εr)J (εr)b (r)+c.c, (B5)
4a |m−1/2| |m−7/2| m
m
(cid:88)
with ν = ε for clean graphene. Using the asymptotic forms of the Bessel functions for small arguments
A 4π
1
J (x)≈ xn, (B6)
n 2nn!
2ln eγx , n=0
Y (x)≈ π 2 (B7)
n
(cid:40) −2n(cid:0)(n−1(cid:1))!, n≥1
πxn
one finds the dominant contribution to δν from the second line, for m=1/2 (+c.c.):
A
δν βb
A =− αsin3θg(r/b). (B8)
ν a
A
The LDOS correction for sublattice B is δν /ν =−δν /ν .
B B A A
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