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: liudq@cczu.edu.cn b)Correspondingauthor: maning@stu.xjtu.edu.cn 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 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, 0ve0fumncotvioinng, 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 approximation (∂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 incident 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 (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. 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Appl. 109. Phys. 116 (2014) 245103. 7

Low-energy quantum scattering induced by graphene ripples

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