Low-energy quantum scattering induced by graphene ripples

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by  Daqing Liu

Low-energy quantum scattering induced by graphene ripples
Daqing Liu1,3a,)Xin Ye1, Shuyue Chen1, Shengli Zhang3, Ning Ma2,3b)
6
1 School of Mathematics and Physics, Changzhou University, Changzhou, 213164, China
1
0 2 Department of Physics, Taiyuan University of Technology, Taiyuan, 030024, China
2
3 School of Science, Xi’an Jiaotong University, Xi’an, 710049, China

n
a
J

6

] Abstract
l
l Wereportaquantumstudyofthecarrierscatteringinducedbygrapheneripples. Crucialdifferencesbetween
a
h thescattering inducedbytherippleandordinary scattering were found. Incontrast tothelatter, in which the
- Bornapproximation isvalid forhigh-energyprocess, theformer isvalidfor thelow-energy processwith aquite
s
broad energy range. Furthermore, in polar symmetry ripples, the scattering amplitude exhibits a pseudo-spin
e
m structure,anadditionalfactorcosθ/2,whichleadstoanabsenceofbackwardscattering. Wealsoelucidatethat
thescattering cross sections are proportional to theenergy cubed of the incident carrier.
.
t
a keywords: Graphene ripple, Quantum scattering, Born approximation
m
1 Introduction
- straints E 3eV around the Dirac points.
d | |≪
It is well known that both experimental and the-
n
o Graphene[1,2],whichis a singleatomiclayerofcarbon, oretical studies have revealed that graphene is always
c is a two-dimensional crystal that can be embedded into corrugated and covered by ripples, which can be ei-
[ a three-dimensional space. In the past decade, its iso- ther intrinsic [11,12] or induced by roughness of sub-

1 lation has attracted a substantial amount of theoretical strate [13,14]. Although graphene ripples complicate
v andexperimentalresearch,mostofwhichstemmedfrom the system, they also extend their application because
7 the peculiar behavior of graphene carriers. Many appli- additionaladjustableparametersareassociatedwiththe
3
cations focus on the flat graphene, in which the move- ripples [2,15].
0
1 ment of carriers was represented by the Dirac Hamilto- For instance, incident carriers are scattered or de-
0 nian with massless fermion in 2+1 dimensional space- flected by the nonzero curvature induced by graphene
. time [3–5]. ripples. However, in such process the quantum behav-
1
0 Iorio et al. [6–10] studied the Weyl-gauge symme- iors of the carriers should be considered. In fact, many
6 tryofgrapheneHamiltoniananditsapplicationtograv- questions remain, such as the role of the pseudospin in
1 ity research. Their results revealed that if one accepts the scattering process.
:
v the flat space-time description of conduct electron, one Katsnelsonand Geim [16] have studied the influence
i mustalsoacceptacurvedspace-timedescriptionbecause on the electronic quality by the corrugation scattering
X
througha Weyl redefinition ofthe fields, actions are the effect using a fractal dimension theory. They showed
r
a same. that when the corrugationgraphene can be depicted by
However, the Dirac description of graphene is only a fractal dimensional surface, i.e. the height-correlation
valid for low energies and small momentum around the function behaviors as < [(h[r] h(0))2 > r2H, the ef-
− ∝
Dirac points ka << 1, where a 10 10m is the near- fective potential is a long-range one, such as Eq. (15)
l l −
∼
estcarbon-to-carbondistance. Inotherwords,theDirac in Ref. [16]. However,when grapheneripples cannot be
description of graphene is valid for carrier energies sat- depictedbyfractaldimensiontheory,thedetailstudyon
isfying E = ~v k E = ~v /a 3eV. Then, in thequantumscatteringinducedby,forinstance,isolated
F l F l
≪ ≃
the following, we restrict all the discussions in the con- ripple, is needed.
a)Correspondingauthor: [email protected]
b)Correspondingauthor: [email protected]
1
In this manuscript we report a quantum study on The fielbein fields ea (x,y), which satisfy eaebη =
µ µ ν ab
the carrier scattering induced by graphene ripples. Sur- g withtheconstantm·etricη =diag 1, 1 ischo-
µν a,b
{− − }
prisingly, our study showed a crucial difference between sen as e1 =e2,
2 1
the scattering induced by the graphene ripple and or- · ·
dinary scattering. In the latter, the Born approxima- zx2√zx2+zy2+1+zy2 zxzy(√zx2+zy2+1−1)
tionwasvalidinthehigh-energyprocess,whereasinthe ea = zx2+zy2 zx2+zy2 .
scatteringbytypicalgrapheneripples,the Bornapprox- ·µ  zxzy(√zx2+zy2+1−1) zy2√zx2+zy2+1+zx2 
z2+z2 z2+z2
x y x y
imation showed an opposite validity but with a quite  (3)
broad energy range provided the Dirac description of This choice guarantees that when graphene is flat, fiel-
grapheneisvalid. We furthermoreshowedthat, ina po- bein matrices are consistently unitary transformations.
lar symmetry ripple, the scattering process emerged an Because the affine connection are defined by Γµ =
νλ
explicitpseudospinstructure,inwhichanadditionalfac- 1gρµ(g +g g ),theaffineconnectionsareread
2 ρν,λ ρλ,ν− νλ,ρ
tor cosθ/2 was exhibited in the expression of scattering as
amplitude, whichleadsto the absenceofbackwardscat-
tering. Compared to ordinary quantum scattering, the Γ = Γµ = 1+zxzx2zx+xzy2 1+zxzx2zx+yzy2 ,
scatteringcrosssectionswereproportionaltothe energy 1 1ν zyzxx zyzxy
1+z2+z2 1+z2+z2 !
cubed of the incident carrier. Furthermore, the Hamil- x y x y
zxzxy zxzyy
tonianalsoshowedthatifthegraphenewasbentinonly Γ = Γµ = 1+zx2+zy2 1+zx2+zy2 . (4)
onedirection,its propertiesdid notchangeifthere is no 2 2ν 1+zyz2zx+yz2 1+zyz2zy+yz2 !
x y x y
phase transition because suitable parameters could be
rearranged. In flat space, the gamma matrices are
2 Hamiltonian in the ripple γ1˙ = 0 1 , γ2˙ = 0 −i ,
1 0 i 0
(cid:18) − (cid:19) (cid:18) − (cid:19)
graphene 1 0
γ0 = β = . (5)
0 1
(cid:18) − (cid:19)
Inthisstudy,weassumethatthegrapheneisnotconsis-
tentlyflat,thatis,thecurvatureofgrapheneisnotzero. Whereas in ripple graphene they are defined as γµ =
eµγa,
To depict the curvature, we first formulate graphene a
metric.
A curve graphene can be considered a two- 0 √1+zzxx2+zy2+izy
dimensional surface embedded in a three-dimensional γ1 = zx+izy ,
space. The surface is defined by the function z(r),  izy−√1+zzxx2+zy2 0 
rwh=ic(hxi,syt)hieshtheieghcotowrditihnarteisopnecint tzo=th0e zpl=an0e,palsansehoawnnd  zx−0izy √1+zzyx2+zy2−izx
in Eq. (1):dz2 =z2dx2+z2dy2+2z z dxdy, (1) γ2 =  −izx−√1+zzyx2+zy2 zx+0izy . (6)
x y x y  zx−izy 
 
where z = ∂z, z = ∂z, z = ∂2z etc. When the spin connection coefficient is defined as
x ∂x y ∂y xy ∂x∂y ωab = eaσ(eb Γλ eb ), and the spin connection is
The space-time metric is (in this study we set ~ = deλfined as Ω·σ=,µ −1ωabµ[σγ ·,λγ ], we find
vF 1) µ 8 µ a b
≡
1 0 0 zxzxzy2−+zzy2zxx 0
Ω = Ω x y ,
gµν = 00 −zz(xx(,xy,)yz)2(+x,1y) −zxz(x(,xy,)yz)y2(+x,1y) , 1 0 zyzxzxx2−+zzyx2zxy !
x y y
 −(cid:0) (cid:1) − (2) zxzzyy2−+zzy2zxy 0
(cid:0) (cid:1) Ω = Ω x y , (7)
where the zeroth component on 2+1 dimensional space- 2 0 zyzxzy2−+zzx2zyy !
time, t, does not mix with space components. Thus, in x y
the following, the zeroth component is ignored if there i(√1+z2+z2 1)
is no confusion. with Ω= 2√1+xz2+yz−2 .
x y
The Hamiltonian is a very complex 2 2 matrix, as shown in Eq. (8),
×
2
H = H =0,
11 22
i( 1+z2+z2 1)(z z z z )
i zy x y − x yy− xy y
H = ( iz + )(∂ )+
12 −zx+izy{ − x 1+z2+z2 y − q 2(z2+z2) 1+z2+z2
x y x y x y
q q
i(z z z z )( 1+z2+z2 1)
zx x xy− xx y x y −
(iz + )(∂ ) ,
y 1+z2+z2 x− 2(z2+z2) q1+z2+z2 }
x y x y x y
q q
i( 1+z2+z2 1)(z z z z )
i zy x y − x yy− xy y
H = ((iz + )(∂ + )
21 −zx izy{ x 1+z2+z2 y q 2(z2+z2) 1+z2+z2 −
− x y x y x y
q q
i(z z z z )( 1+z2+z2 1)
zx x xy− xx y x y −
(iz )(∂ + )). (8)
y− 1+z2+z2 x 2(z2+z2) q1+z2+z2
x y x y x y
q q
However, the complexity of the Hamiltonian should not be intimidating. If graphene is only bent in x-direction,
that is, z =z =z =0, the Hamiltonian becomes
y xy yy
0 ∂ i∂x
− y− √1+z2
H = x .
′  ∂ i∂x 0 
y − √1+z2
x
 
In the above expression, if we make a substitution x x and y y , where y = y and x = x 1+z2dx,
→ ′ → ′ ′ ′ 0 x
the Hamiltonian is the same as the one in flat graphene. The physics effect is a Fermi velocity renormalization in
R p
x-direction, v vF .
F → √1+z2
x
Such bend leads to an effective potential. To illustrate this point we consider a bent graphene depicted by
0, I:x 0
≤
z(x) = cx, II:0<x<d and normal incident conduction electrons (with energy k > 0) in the left side. In

cd, III:x d

≥
this case z =c at region II and z =0 at regions I and III. Wave functions in three regions are
x x

1 1 1 1
ψ = eikx+r e ikx,
I √2 1 √2 1 −
(cid:18) (cid:19) (cid:18) − (cid:19)
1 1 1 1
ψ =a ei√1+c2kx+r e i√1+c2kx
I √2 1 √2 1 −
(cid:18) (cid:19) (cid:18) − (cid:19)
and
1 1
ψ =t eikx,
III √2 1
(cid:18) (cid:19)
respectively. Comparing above results with Eqs. (25)-(27) in Ref. [3], we find that the bend leads to an effective
square well potential,
(√1+c2 1)k, 0<x<d,
V(x)= − −
0, otherwise.
(cid:26)
One has a similar result obviously at the case of oblique incidence.
For simplification, we assume in the following that the ripple is polar symmetrical, that is, in the polar coordi-
nates, z =0, where θ is the polar angle with respect to the x-axis.
θ
In this situation Hamiltonian turns into
0 e iθ[ i ∂ ∂θ + i (1 1 )]
− −√1+z2 r− r 2r − √1+z2
H = r r (9)
 eiθ[ i ∂ + ∂θ + i (1 1 )] 0 
−√1+z2 r r 2r − √1+z2
r r
 
3
3 General consideration
In the following, we assume that the ripple is not
only small but also local, that is, z 1 at r < a and
r
z dropsoffrapidlyatr a. The Ham≪iltoniancanthen To study the scattering induced by graphene curvature,
r
be decomposed as, ≥ the Green function was used, which satisfies,
k i∂ +∂ δ(r) 0
H =H +V, (10) x y G(r)= ,
0 i∂ ∂ k 0 δ(r)
x y
(cid:18) − (cid:19) (cid:18) (cid:19)
(14)
where where k is the energy of the carrierswith momentum k.
Note that when r , each component of the
H0 = eiθ( i∂0 + ∂θ) e−iθ(−i0∂r− ∂rθ) (11) Green function should b→eha∞ve as e√ikrr. We finally read
(cid:18) − r r (cid:19) the Green function as
is the Hamiltonian in flat graphene and an effective po- G(r,θ)= k −iH01(kr) e−iθH11(kr) , (15)
tential induced by graphene ripples 4 eiθH1(kr) iH1(kr)
(cid:18) 1 − 0 (cid:19)
whereH1(kr)isthefirstkindi’thorderHankelfunction.
iz2 0 e iθ( 1 +∂ ) i
V = r − 2r r . (12) Using the asymptotic behaviorof the firstkind Han-
2 (cid:18) eiθ(21r +∂r) 0 (cid:19) kel functions, we know that at kr ,
→∞
willTchaeurseesacraett∂e-rtienrgmesffeincttphreopeoffretcitoinvaelptootkean,tiwalh,icwhhhicahs G(r,θ)→ 8kπe−34πie√ikrr e1iθ e−1iθ . (16)
r (cid:18) (cid:19)
two results: First, in a large momentum the perturba-
Assuming that the incident carriers move along
tional treatment is invalid; second, the unitarity of the
the x-axis, the incident wave function is ψ (r ,θ ) =
result should be broken, that is, there will be no the i ′ ′
optical theorem in the scattering process. eikr′cosθ′ 1 in position space or ψ (p) =
√2 1 i
Notice that if we make a unitary transformation (cid:18) (cid:19)
1
(inFtWo HT0′tr=anUsfHor0mUa†.tioPna)rtUic,utlhaerlyH,abmyillteottniinagnUH0=wei2illθσtu3r=n iδs(pi√n−2cki0d)e(cid:18)nt1wa(cid:19)veinvemctoomr.entum space, where k0 = (k,0)
ei2θ 0 , where σ is the generator of the rota- AccordingtotheBornapproximation,thescattering
0 e−i2θ ! 3 wave function is
tion with respect to z-axis, we have
ψs(r,θ)= r′dr′dθ′G(r,θ;r′,θ′)V(r′)ψi(r′,θ′). (17)
0 ∂ +i∂θ + 1 Z
H = i r r 2r ,
0′ − (cid:18) ∂r−i∂rθ + 21r 0 (cid:19) eWˆ ke.fuBrethcaeurseintwreodfuocceusaonsclyatotenritnhge wlimavietatvieocntorr kf =,
r
which is the same as that in Refs. [17,18] (for instance, k r r′ r→∞kr keˆr r′ =kr kf r′. → ∞
| − | → − · − ·
Regarding the Green function, at the leading order,
Eq. (9) in Ref. [17]) up to a factor i. Therefore, al-
− we have
thoughourHamiltonianisnotthesameastheonegiven
itnioRn.efIsn.[t1h7e,1p8r],estehnetysatruedoy,nltyheupbatsoeaoFfWpsTeudtroa-nspsfionromrais- G(r,θ;r′,θ′)→ 8kπe−34πieikr√−irkf·r′ e1iθ e−1iθ .
ψ r (cid:18) (cid:19)
a constantvector A , whereas in Refs. [17,18] the (18)
ψ
(cid:18) B (cid:19) By letting q=kf k0, we obtain
base of pseudo-spinor is a position-dependent vector. −
reprFersoemntsthaepexaprtriecslseiownitohfHen0e,rtghyekwa>ve0fumncotvioinng, walhoinchg ψs(r,θ) = e√ikrr 8√√kπe−π4i e1iθ e−1iθ
kˆ =(cosθ0,sinθ0) is e(cid:18) iθ′ iqr′(1 +(cid:19)ikr cosθ )
eikrcos(θ−θ0) 1 Z zr2′dr′dθ′(cid:18) e−iθ′−−iq·r·′(122+ikr′′cosθ′)′ (cid:19).
ψ(r,θ)= √2 eiθ0 . (13) (19)
(cid:18) (cid:19)
The calculations result in ψ :
s
If θ =0, that is, if the particle moves along the x-axis,
0
the wave function will be reduced to eikrcosθ 1 . ψ (r,θ)= eikr 1 e−iθ/2 F(θ)cosθ, (20)
√2 1 s √r √2 eiθ/2 2
(cid:18) (cid:19) (cid:18) (cid:19)
4
where F(θ) = k2πkqeπ4i dr′zr2′J1(qr′), J1(qr) is the distance between carbon-carbonatom. For a smoothing
ripple, z = 0 at point r = 0 and z drops off rapidly
first-order Besseql function and q =2ksinθ. r r
R 2 when r a, the inequalityis alwayssatisfied. However,
Here we emphasize physics interpretations of Eq. ≫
for a ripple induced by defect, z = 0 at point r = 0
(20). The factor eikr stands for circular wave spreading r 6
√r and the left hand side of the inequality becomes into
douectwreaarsdin,gthbeeahmavpiolirtuadserof1w/2hbicuhtdneoctrreas1esisaastrtr−i1b/u2t.eTdhtoe 4ba22 lnaa0, which is about ab22 if we choose a/a0 ∼102.
− − Insummary,whenka (a)4/3 and a 1,theBorn
the fact that we are working in two-dimensional space. ≪ b b ≫
approximation is valid. In suspending graphene, a and
e iθ/2
Furthermore, 1 − stands for only the parti- b are typically at the orders of 10nm and 1nm respec-
√2 eiθ/2
(cid:18) (cid:19) tively [19]. Then, the above condition means that the
clesspreadingalongk ,thecutanglebetweenwhichand
f perturbation theory is valid as long as energies of the
the x-axis is θ.
scattered carrier are far lower than 3eV. Noticing that
The most important factor is cosθ. Noting that
2 our discussion constraint is also E 3eV, we con-
cosπ = 0, we conclude that there is no back scattering | | ≪
2 clude that for typical ripples in graphene our perturba-
if a particle passes through a polar-symmetric ripple, at
tion treatment is valid in a quite broadrange. However,
least at the first order. The effective potential V pro-
ripples can be controlled by boundary conditions and
duces a quantum scattering which changes the pseudo-
makinguseofgraphenesnegativethermalexpansionco-
spindirectionsofincidentparticlesintootherones. How-
efficient[20]. Ifthecharactersizeofthecontrolledripple
ever, the scattering amplitude undergoes a modifying
is very large, for instance a 1µm and b 0.1µm. The
factor, cosθ. Noticing that when θ = π, cosθ = 1, ∼ ∼
2 2 2 − validity of the theory narrows down to E 10 3eV,
we think that the factor is due to the pseudospin struc- | | ≪ −
thatmeansthattheperturbationtheoryisnearlyinvalid
ture and the chiral conservation of carriers in graphene.
in all the energy scale.
According to the Born scattering theory, F(θ)cosθ is
2 In the above discussion we have replaced (∂z/∂r)2
the differential scattering amplitude along θ-direction.
by (b/a)2. From the mean value theorem of inte-
Now we consider the validity of Eq. (20). From the
grals, (∂z/∂r)2 should be replaced by the mean value,
perturbation theory, for the Born approximation suit-
(∂z/∂r)2 , where r depends on the ripple. We think
ability, we have an enough condition |rs s
that, as long as (∂z/∂r)2 does not vary sharply, the ap-
proximation (∂z/∂r)2 )2 (b/a)2 is correct.
d2rG(r0 r)V(r) 1, (21) |rs ∼
|| − ||≪ From Eq. (20), the differential cross section is
Z
ateachr ,where standsforthe matrixnorm. Since
0 ||·|| dσ(θ) πk3 θ
Hi1(kr)=Ji(kr)+iYi(kr)andtheincidentwaveisalong dθ = 2q2 cos2 2( drzr2J1(qr))2. (23)
x-axis, the above condition reduces into Z
k
d2rz2j(k r r)f(r) 1, (22)
8| r | 0− | |≪ 4 Examples on scattering
Z
wherej(kr)maybeJ (kr),J (kr),Y (kr)orY (kr)and
0 1 0 1
Next, we study some general properties of the scatter-
f(r) may be 1 or k. We assume that the charactersize
2r ing. First, we assume that the amount of energy of the
and the character height of the ripple is a and b respec-
incident particleisverysmall, kr 1. Inthis situation,
tively. Therefore, zr b/a. For the far field, that is, ≪
∼
kr 1 and r >a, the condition Eq. (22) turns into
0 ≫ 0 dσ(θ) πk3 θ
= cos2 ( drrz2)2, (24)
8 a dθ 8 2 r
ka ( )2/3( )4/3, (22′) Z
≪ π b
which means that the differential cross section is nearly
since under normal circumstances, we have a b. For
≫ isotropicandisproportionaltotheenergycubed. There-
the near field, the most possible divergent point is at
fore, in the quantum scattering process, there is a cru-
r =0 and it is enough to consider only the inequality
0 cial difference between graphene and ordinary material:
1 z2 in ordinary material, the Born approximation is valid
dr r 1, (22′′) for incident particle with large momentum, whereas in
4 r ≪
Z graphene, it is valid for incident particle with small mo-
where we choose the domain of the integration as a < mentum. Note that the above expressionis also valid in
0
r < a, with the infrared cutoff a 0.1nm, the nearest nearly forward scattering, θ 1 .
0 ∼ ≪ kr
5
andaverycomplexexpressionforthetotalcrosssection,
which is also shown in Fig.2.
Itisinterestingthatinasmallmomentum,thediffer-
entialcrosssectionisalsoisotropic,exceptapseudospin
factor, and the total cross section is σ = (bk)3π2b.
32
Figure 1: Differential cross sections of the Gaus-
Figure 2: The total cross section for Gaussian
sianbump (a) andthe defect cone (b) ingraphene
bumpandcone. Theverticalcoordinateisthe σ
sheet. Dotted, solidanddashedcurvescorrespond k3b4
and the horizontal ordinate is ka.
to ka=0.1, ka=1 and ka=2 respectively.
Althoughthesetwocasesdiffer,theyshowsomesim-
We now exemplify the quantum scattering process ilarities. First, the Born approximation is valid in the
by a Gaussian bump and a cone in graphene sheet. low-energy scattering process. on the contrary, in or-
We first assume that the ripple is a Gaussian bump, dinary materials, the Born approximation is generally
z = be−r2/a2 [17,18], where a is the bump radius and valid in the high-energy scattering process. Second, the
b is the bump height. We have the differential crosssec- cross sections are proportional to the energy cubed of
tion the incident particle. Third, if the energy of the inci-
dent particle is very small, the scattering cross section
dσ(θ) πk3b4 θ θ
= cos2 exp( a2k2sin2 ) (25) isnearlyisotropic,exceptthepseudospinfactor,cosθ/2,
dθ 32 2 − 2
which can be seen in Eq. (23).
and the total cross section, which is shown in Fig.2,
5 Summary
σ = π2k3b4e a2k2/2(I (a2k2)+I (a2k2)) (26)
− 0 1
32 2 2
In summary, this paper reports a quantum study on the
low-energy scattering process induced by graphene rip-
where I (x) is the i’th order of the first kind of modified
i
ples. We showed that if the graphene is bent only in
Bessel function.
one direction, its property does not change if a suitable
Forincidentparticleswithsmallmomentum,thedif-
parameter is provided. A general graphene ripple can
ferential cross section is isotropic except the pseudospin
factor,cos2 θ andthetotalcrosssectionisσ = (bk)3π2b. be equivalent to an effect potential. We also showed a
2 t 32 Born approximation of the quantum scattering induced
This result is interesting, because the cross section does
by the (polar symmetric) ripple. However, the result
not depend on the size of Gaussian bump, which is in
also showed that, unlike in the usual scattering process,
agreement with Eq. (24).
the Born approximation used in this study showed an
We then consider a most simple case of graphene
opposite validity but with a quite broad energy range
b(1 r/a) r <a
ripple z = − , which is a cone with provided the Dirac description of graphene is valid.
0 r >a
(cid:26) Different ripples exhibit different scattering be-
heightbandbottomradiusa. Thissituationcorresponds
haviour. However, all the low-energy scattering pro-
to a topological defect in which gauge fields associated
cesses showed some general behaviour. The first is that
with the defect should be applied. Here we focus only
becauseofthechiraloftheDiracfermion,thepseudospin
on the curvature effect.
factor, cosθ/2 was shown in scattering amplitude. This
We have the differential cross section
factorledtoaninterestingphenomenon,thesuppression
dσ(θ) πk3b4 θ ofbackscattering. Second,thecrosssectionwasroughly
= cos2 (1 J (qa))2,
dθ 2q4a4 2 − 0 proportionaltotheenergycubedoftheincidentparticle.
6
Finally, as long as the energy of the incident particle is [4] P. Wallace, Phys. Rev. 71 (1947) 622.
small enough, the scattering was nearly isotropic if we
[5] D.Liu, S.Zhang,E.Zhang,N.Ma,H.Chen.Euro-
ignore the pseudospin factor, cosθ/2.
phys. Lett. 89 (2010) 37002.
The applicationof graphenemay be extended to the
designdevicesbasedoncurvedgraphene. However,such
[6] A. Iorio, Ann. Phys. 326 (2011) 1334.
design needs carefully study of the quantum scattering
induced by graphene curvature. It is hoped that the re- [7] A. Iorio, Eur. Phys. J.(Plus) 127 (2012) 156.
sults of this study will be helpful in designing new type
[8] A. Iorio, G. Lambiase, Phys. Lett. B 716(2) (2012)
of devices that are based on curved graphene.
334.
A motivation to study ripple graphene and the elec-
tronscatteringbygraphenerippleisthatgrapheneripple [9] A. Iorio, J. Phys.: Conf. Ser. 442 (2013) 012056.
can mimic gravity structure. Here we only consider the
elastic properties of graphene. However, there are other [10] A. Iorio, G. Lambiase, arXiv:1308.0265.
types of disorders in graphene, for instance, torsion in-
[11] J.C. Meyer, A.K. Geim, M.I. Katnelson, K.S.
duced by dislocations. In general relativity, torsion ef-
Novoselov, T.J. Booth, S. Roth, Nat. 446 (2007)
fectsaresuppressedbythesmallnessofthegravitational
60.
coupling and they are negligible. But, the suppression
does not exist in graphene and we can study the torsion
[12] J.C. Meyer, A.K. Geim, M.I. Katsnelson, K.S.
effect in experimental and theoretical aspects. In this
Novoselov,D.Obergfell,S.Roth,C.Girit,A.Zettl,
manuscript we only considered the symmetric part of
Solid State Commun. 143 (2007) 101.
connection, affine connection. To study torsion effects,
one should introduce antisymmetric part of the connec- [13] M.Ishigami,J.H.Chen, W.G.Cullen,M.S.Fuhrer,
tion, symbolised by the torsion tensor, which is induced E.D. Williams, Nano. Lett. 7(6) (2007) 1643.
by edge dislocation and proportional to the density of
[14] E. Stolyarova, K.T. Rim, S. Ryu, J. Maultzsch, P.
Burgersvector. Forthedetailstudyofthetorsioneffects
Kim, L.E.Brus,T.F. Heinz, M.S.Hybertsen, G.W.
we refer to Refs. [21–23]. We think that our treatment
Flynn, Proc. Nat. Acad. Sci. 104 (2007) 9209.
is still valid if the antisymmetric part of the connection
is induced in Eq. (4). The electron scattering by edge
[15] R. Burgos, J. Warnes, L.R.F. Lima, C. Lewenkopf,
dislocation in 2+1 dimension is an interesting topic and
Phys. Rev. B 91 (2015) 15403.
it is worth further research.
[16] M.I. Katsnelson and A.K. Geim, Trans. R. Soc.
A366 (2008) 195.
Acknowledgments
[17] F. de Juan, A. Cortijo, M.A.H. Vozmediano, Phys.
Rev. B 76 (2007) 165409.
This work is supported by the National Nature Science
foundation of (Grants Nos.51176016and 11074196).
[18] M.A.H. Vozmediano, M.I. Katsnelson, F. Guinea,
Phys. Rep. 496 (2010) 109.
References
[19] M. Zwierzycki, Acta Phys. Pol. 121 (2012) 1246.
[20] W. Bao, F. Miao, Z. Chen, H. Zhang, W. Jang, C.
[1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D.
Dames, C.N. Lau, Nature Nano. 4 (2009) 562.
Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva,
A.A. Firsov, Science 306 (2004) 666.
[21] F. de Juan, A. Cortijo, M.A.H. Vozmediano, Nucl.
Phys. B828 (2010) 625.
[2] F. de Juan, M. Sturla, M.A.H. Vozmediano. Phys.
Rev. Lett. 108 (2012) 227205. [22] A.Mesaros,D.SadriandJ.Zaanen,Phys.Rev.B82
(2010) 073405.
[3] A.H.C. Neto, F. Guinea, N.M.R. Peres, K.S.
Novoselov, A.K. Geim, Rev. Mod. Phys. 81 (2009) [23] K. Viswanathan and S. Chandrasekar, J. Appl.
109. Phys. 116 (2014) 245103.
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