Supporting Information for Suppressing Klein tunneling in graphene using a one-dimensional array of localized scatterers

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1 Supporting Information for Suppressing Kein tunneing in graphene using a one-dimensiona array of ocaized scatterers Jamie D Was, and Danie Hadad Department of Chemistry, University of Miami, Cora Gabes, Forida 334,USA Corresponding author: jwas@miamiedu

2 d y^ FIG Scattering of an incident Dirac x^ pane wave spinor of energy E = hν F k 0, φ ± K inc r) = k, from a ν F k X e i k r ±e iθ k one-dimensiona array of ocaized cyindricay symmetric scatterers in graphene The +K φ inc r) unit ce for the scattering array consists of a singe scatterer with the position of the n th scatterer given by r n = ndŷ In the foowing, the basic theory for intravaey mutipe scattering in graphene is presented and appied to the probem of a pane wave scattering from an infinite, one-dimensiona array of ocaized scatterers First, the basic formaism for cacuating the scattering soutions for a pane wave scattered from a finite number of ocaized scatterers in graphene is presented Next, the scattering wave functions, ψ ± K r), for the singe and two scatterer cases are expicity presented The theory is then extended to the case of scattering from an infinite, one-dimensiona array of ocaized scatterers in graphene where expicit expressions for the refection and transmission coefficients are provided Finay, the anaogous theory for scattering from an infinite, one-dimensiona array of ocaized scatterers in a two-dimensiona eectron gas DEG) is derived for comparison to the graphene resuts I INTRAVALLEY MULTIPLE SCATTERING FORMALISM In the foowing, we briefy review the basic formaism for intravaey mutipe scattering in graphene[, where the scattering soutions are expanded about either the + K or K Dirac points, where K = 4π 3 9b x and b = 4Å is the carbon-carbon bond ength in graphene Let φ ± K inc r) be an

3 incident Dirac pane wave spinor of energy E = hν F k = hν F λ normaized to unit fux aong the xdirection, φ K inc r) = k ν F k X e k r e iθ or φ K inc r) = k ν k F k X e k r e iθ with a wave k K vector given by k = k cosθ k ) x+k sinθ k )ŷ = k X x+k Y ŷ for θ k π, π ), and λ = π k K is the waveength of the incident wave In a cacuations, hν F = J-m In the foowing, a scattering potentia consisting of a inear arrangement of N + identica scatterers aong the ŷ- direction is considered with the scatterers positions indexed by the integer n [ N, N, r n = ndŷ where d is the spacing between adjacent scatterers [Fig The tota wave function for a Dirac pane wave spinor incident to the inear array of N + scatterers, ψ ± K r), can be written as: ψ ± K r) = φ ± K inc r) + N n= N 4i hν F k s Ĝ,± K r, r n,e) T,± K ψ r n) ) where s is the scattering ampitude of the th partia wave, + are the number of partia waves that are incuded in cacuation of ψ ± K r) in Eq ) T,± K ψ± K r)) r= r n, and where H ) with 0, T,± K ψ r n) Ĝ,± K r, r n,e) = i+ k H) k ρ n )e iθ n ±ih ) + k ρ n )e i+)θ n 4 hν F ±ih ) + k ρ n )e i+)θ n H ) k ρ n )e iθ n = ik ˆL +[H ) 0 k ρ n ) ±ˆL + [H) 0 k ρ n ) 4 hν F ±ˆL + + [H) 0 k ρ n ) ˆL [H ) 0 k ρ n ) z) is a hanke function of order, ρ n = r r n, e ±iθ n = r r n) x±iŷ), and T,± K is the -partia wave t matrix operator given by: T,± K = ˆL 0 0 ˆL + where ˆL ± = ik e ±iθ r ± i r θ ) = ik x ± i y ), and T 0,± K = is the identity matrix In writing Eq ), the scattering ampitudes were taken to satisfy the reationship s = s +), which is a consequence of the scattering potentia being identica over both inequivaent attice sites in graphene For the numerica cacuations performed in this work, each scatterer was modeed as a cyindricay symmetric step potentia with an effective radius of r s, ie, the potentia for the n th scatterer was given by V 0 Θ r r n ) where Θ r r n ) = if r r n r s and Θ r r n ) = 0 3 ρ n ) 3)

4 for r r n > r s To consider ony intravaey scattering and to negect intervaey scattering, r s b = 4Å For an individua scatterer, the th partia wave scattering ampitude is given[, by: s = J k r s )J + k r s ) J k r s )J + k r s ) J + k r s )H ) k r s ) J k r s )H ) + k r s ) where k = E V 0 hν F, and J z) is a besse function of the first kind of order, respectivey In a simuations, was chosen to take into account 999% of the tota scattering ampitude for an individua scatterer, ie, s 0999 s Knowedge of T,± K ψ r n) for a n and [0, competey determines ψ ± K r) in Eq ); these can be determined sef-consistenty as foows: T, ψ r m) = T, φ ± K inc r m) + n m 4) s T, [Ĝ,± K r m, r n,e) T,± K ψ r n) 5) where T, [Ĝ,± K r n, r j,e) = ik i + ) H ) k r n j )e i )θ n j ±ih ) + + k r n j )e iθ n j+ +) 6) 4 hν F ±ih ) + + k r n j )e iθ n j+ +) ) H ) k r n j )e iθ n j ) and r n j = r n r j and e ±iθ n j = r n r j ) x±iŷ) r n j In this case, Eq 5) formay represents a set of N + ) + ) equations that can be soved sef-consistenty as foows[: Define the foowing + ) coumn vectors reated to the tota and incident wave functions evauated at scatterer j [ N, N +,, N, N: T 0,± K ψ r j) T T ψ ± K r,± K ψ r j) j) = T,± K ψ r j), T φ inc,± K r j) = T max, ψ r j) T 0,± K φ ± K inc r j) T,± K φ ± K inc r j) T,± K φ ± K inc r j) T max, φ ± K inc r j) and the foowing N + ) + ) coumn vectors: T ψ ± K r N) T φ inc,± K r N) T ψ ± K r N ) T φ inc,± K r N ) T ψ ± K = T ψ ± K r 0), T φ inc,± K = T φ inc,± K r 0) T ψ ± K r N )) T φ inc,± K r N )) T ψ ± K r N) T φ inc,± K r N) 7) 8) 4

5 Using Eqs 7) and 8), Eq 5) for n = N to n = N and for = 0 to = can be written compacty as: T ψ ± K = TT) T φ inc,± K 9) where is the N + )max + ) N + ) + ) identity matrix, and TT is a N + ) + ) N + ) + ) matrix given by: 0 T T ± K r N, r N ) T T ± K r N, r N ) T T ± K r N, r N ) T T TT = ± K r N, r N ) 0 T T ± K r N, r N ) T T ± K r N, r N ) T T ± K r N, r N ) T T ± K r N, r N ) T T ± K r N, r N ) 0 where Ĝ 0,± K r n, r j,e) Ĝ,± K r n, r j,e) Ĝ,± K r n, r j,e) Ĝ r max, n, r j,e) T,± K [Ĝ0,± K r n, r j,e) T,± K [Ĝ,± K r n, r j,e) T,± K [Ĝ,± K r n, r j,e) T,± K [Ĝmax, r n, r j,e) T T ± K r n, r j ) = T,± K [Ĝ0,± K r n, r j,e) T,± K [Ĝ,± K r n, r j,e) T,± K [Ĝ,± K r n, r j,e) T,± K [Ĝmax, r n, r j,e) T max, [Ĝ0,± K r n, r j,e) T max, [Ĝ,± K r n, r j,e) T max, [Ĝ,± K r n, r j E) T max, [Ĝmax, r n, r j,e) 0) Ŝ max ) and Ŝ max = 4i hν F k s s s s s s s max 0 ) s max Using Eq 8) and Eq 0), ψ ± K r) in Eq ) can be written compacty as: ψ ± K r) = φ ± K inc r) + ĜG ± K r) T ψ ± K = φ ± K inc r) + ĜG ± K r) TT) T φ inc,± K 3) 5

6 where ĜG ± K r) is a N +) +) matrix, ĜG ± K [ĜG r, r r) = N ) ĜG r, r N ) ĜG r, r N ) where ĜG r, r j ) = 4i hν [ F s k 0 Ĝ 0,± K r, r j,e) s Ĝ,± K r, r j,e) s max Ĝ max, r, r j,e) 4) II SCATTERING FROM A SINGLE SCATTERER We wi first consider the probem of scattering from a singe scatter ocated at r 0 = 0 In this case, Eq 5) becomes T, ψ 0) = T, φ ± K inc T, ψ 0) = Inserting Eq 5) into Eq ) gives: 0), which gives: k ν F k X ψ ± K r) = φ ± K inc r) + 4i hν F s Ĝ k,± K r,0,e) T,± K ψ 0) = φ ± K i k s inc r) + ν F k X e i θ k e i +)θ k 5) H ) k r)e iθ 0 θ k ) + ih+ k r)e i+)θ 0 θ k ) ±e iθ k H ) k r)e iθ 0 θ k ) + ih ) + k r)e i+)θ 0 θ k ) ) 6) where r = r r 0 For k r, Eq 6) can be approximated by: ψ ± K r) = φ ± s e i π ) 4 cos + K inc r) + θ 0 θ k)) πνf k X r e i θ 0 θ k θ 0 θ ±e i k 7) As k r, the scattered wave function asymptoticay decays to zero as r initia pane wave, ψ ± K r) φ ± K inc r) eaving ony the III SCATTERING FROM TWO IDENTICAL SCATTERERS Consider two identica scatterers, one ocated at r 0 = 0 and the other at r = dŷ In this case, Eq ) becomes: ψ ± K r) = φ ± K inc r) + 4i hν F s [Ĝ,± k K r, r 0,E) T,± K ψ r 0) + Ĝ,± K r, r,e) T,± K ψ r ) = φ ± K inc r) + ĜG ± K r) TT) T φ inc,± K 8) 6

7 where 0 T T TT = ± K r, r 0 ) T T ± K r 0, r ) 0 T φ inc,± K = eik Y d T φ inc,± K r 0) T φ inc,± K r 0) ±e iθ k e iθ k ±e iθ k T φ inc,± K r k 0) = ν F k X e iθ k ±e 3iθ k e iθ k ±e i+)θ k ĜG ± K [ĜG r, r r) = ) ĜG r, r 0 ) 9) Formay, inverting TT requires finding the roots to a poynomia of order 4 + ), which must be numericay soved when > 0 As in the singe scattering case, the scattered wave function in Eq 8) scaes as r for k ρ k r and k ρ 0 k r, which gives ψ ± K r) φ ± K inc r) for k r IV SCATTERING FROM AN INFINITE ONE-DIMENSIONAL ARRAY OF SCATTERERS In the foowing, the theory for the scattering of pane waves from an infinite, periodic onedimensiona array of identica scatterers N ) with attice constant d [Figure is presented From transationa symmetry, the tota wave function satisfies the reation that ψ ± K r + ndŷ) = ψ ± K r)eik Y nd, which means that ony ψ ± K r) between d y d needs to be cacuated To cacuate ψ ± K r) in Eq ), one can use the fact that transationa symmetry impies that T,± K ψ r n) = e ik Y nd T,± K ψ r 0), in which case the tota wave function in Eq ) can be written as: ψ ± K r) = φ ± K inc r) + n= 4i hν F k s Ĝ,± K r,ndŷ,e)eindk Y T,± K ψ r 0) 0) 7

8 The various T,± K ψ ±K r 0 ) in Eq 0) are determined sef-consistenty by: T, ψ r 0) = T, φ ± K inc r 0) + n 0 s [ T,Ĝ, r 0, r n,e) e ink Y d T,± K ψ r 0) ) Eq ) gives a tota of + ) sets of equations, which can be written as: TG ) T ψ r 0) = T φ ± K inc r 0) ) where is a max +) +) identity matrix, T ψ ± K r 0) and T φ ± K inc r 0) are +) coumn vectors given by: ψ ± K r 0) T T ψ ± K r,± K ψ r 0) 0) = T,± K ψ r 0) T max, ψ r 0), T φ ± K inc r k 0) = ν F k X ±e iθ k e iθ k ±e iθ k e iθ k ±e 3iθ k e iθ k ±e +)iθ k 3) and TG is a max + ) + ) matrix given by: s 0 G 0,0 s G 0, s max G 0,max s TG = 0 G,0 s G, s max G,max 4) with G, = i + = e ik Y nd n 0 s 0 G max,0 s G max, s max G max, ) H ) k n d)e i )θ 0n ±ih ) + + k n d)e i+ +)θ 0n ±ih ) + + k n d)e i+ +)θ 0n ) H ) S k d,k Y d) ± ) + + S + +k d,k Y d) ±S + +k d,k Y d) ) S k d,k Y d) where e ±iθ 0n = i n n and S k d,k Y d) = n= k n d)e i )θ 0n 5) ) H ) k nd) e ik Y nd + ) e ik Y nd 6) 8

9 The attice sum, S k d,k Y d) in Eq 6), can be efficienty cacuated using a sma modification to a previousy pubished method[3 as: [ S k d,k Y d) = π e i π 4 k Y d) t i) + i ) + it a + ti ) + i ) + it e ik d +it dt + it e ik d +it +k Y d) ) + ) π e i π 4 +k Y d) a dt 0 0 [ t i) + i + it ) + ti ) + i + it ) e ik d +it + it e ik d +it k Y d) ) 7) In this work, the expression for S k d,k Y d) in Eq 7) was numericay integrated using MATLAB[4, where the upper imit of the integras in Eq 7) was chosen to be a = 4000 Additionay, the individua sums that comprise Ĝ,±K r,ndŷ,e) in Eq 0) can be cacuated[5 using the foowing pane wave expansion that is vaid for x 0: L ± [H) 0 k ρ n )e ik Y nd = n= d = d, and kn) X = k e ikn) Y y+kn) X x ) n= k n) X e ikn) Y y+kn) X x ) n= k n) X k n) X signx) ± ikn) k Y ) signx)e ±isignx)θ ) k n) 8) ) where k n) Y = k Y + πn d k n) ±iθ Y and e k n) = kn) X ±ikn) Y k ) i k n) Y k and e ±iθ k n) = i kn) X ±kn) Y k for k k n) Y Note that kn) X wi be rea for n N = { } { } [N min,n max where N min = k +k Y )d π and N max = k k Y )d + π, where {z} + corresponds to the smaest integer greater than z, and {z} corresponds to the argest integer ess than z In this case, ψ ± K r) in Eq 0) can be written as [for x 0: for k k n) Y, and kn) X = ψ ± K r) = φ ± K inc r) + n= s d e ikn) Y y+kn) X x ) k n) X signx)) e isignx)θ k n) ±signx)e isignx)+)θ k n) ±signx)e isignx)+)θ k n) e isignx)θ k n) T,± K ψ r 0) In Eq 9), ψ ± K r) consists of a series of pane waves for n N that are either transmitted [x > 0 or refected [x < 0 from the scattering array aong with evanescent waves aong the x-direction that are freey propagating aong the ŷ-direction for n N These types of evanescent waves have been predicted to exist in graphene for one-dimensiona quantum wes[6 for certain vaues of 9 9)

10 k Y, quantum we potentia and width For the scattering array, however, it is the periodicity of the one-dimensiona array of scatterers that generates the evanescent waves in ψ ± K r) Finay, it shoud be noted that cacuating ψ ± K r) using a pane wave expansion in Eq 9) is computationay efficient for x d since the number of pane and evanescent waves that significanty contribute to ψ ± K r) is on the order of O[N + N However, the number of evanescent waves that contribute significanty to ψ ± K r) increases dramaticay as x 0, which renders the pane wave expansion in Eq 9) as an inefficient method to cacuate ψ ± K r) In this case, the sum of hanke functions in Eq 8) shoud be expicity evauated Note aso that athough the expression for ψ ± K x x + yŷ) in Eq 9) is vaid ony for x 0, the wave function is continuous at x = 0 since im x 0 ψ ± K x x + yŷ) = im x 0+ ψ ± K x x + yŷ) for y nd, which can be seen using the hanke sum expressions for S k d,k Y d) in Eq 6) V TRANSMISSION AND REFLECTION FROM A ONE-DIMENSIONAL ARRAY OF SCAT- TERERS IN GRAPHENE From Eq 9), the transmitted wave function [x > 0 wi consist of a pane waves in ψ ± K r in Eq 9) that are scattered aong the Bragg directions, k n) = k n) X x + kn) Y ŷ for n N, ψ± K T r), which can be written as: ψ ± K T r) = T n e ikn) Y y+kn) n N X x) k ν F k n) X ±e iθ k n) where the sum in Eq 30) is over a open channes, n N, with the transmission coefficient for the n th open channe given by: s T n = δ n0 + νf d k k n) X s = δ n0 + νf d k k n) X ±e iθ k n) eiθ n) k e iθ k n) ±e i+)θ k n) ±e i+)θ k n) ±e i+)θ k n) e iθ k n) T 30) T,± K ψ r 0) T,± K ψ r 0) 3) where δ i j is the Kronecker deta δ i j = 0 for i j and δ i j = for i = j) The tota transmission probabiity is given by T tot = n N T n 0

11 Likewise, the refected wave function x < 0), ψ ± K R r), is given by: ψ ± K R r) = R n e ikn) Y y kn) X x) k n N ν F k n) X where R n = = ) s νf d k k n) X ) s νf d k k n) X e iθ k n) e iθ n) k e i+)θ k n) e iθ k n) e iθ k n) e i+)θ k n) e i+)θ k n) e iθ k n) T 3) T,± K ψ r 0) T,± K ψ r 0) 33) The transmission and refection coefficients satisfy the unitarity condition, n N R n + T n = VI SCATTERED WAVE FUNCTION FROM AN INFINITE ONE-DIMENSIONAL ARRAY OF SCATTERERS IN A TWO-DIMENSIONAL ELECTRON GAS DEG) In the foowing, we generaize previous work[7 on scattering of pane waves from a onedimensiona periodic grating in a DEG to incude higher partia waves [ > 0 for comparison with the resuts derived for graphene Consider an incident wave with effective mass m and energy h k m E = and normaized to unit fux aong the x direction, φ ac inc r) = m hk X e i k r, where k = k X x + k Y ŷ = k cosθ ) x + k k sinθ )ŷ is the wave vector k In this case, the tota wave function, ψ ac r), is given by [for x 0: ψ ac r) = φinc r) ac + = φinc r) ac + n= = e ik Y nd n= L = sign) where again k n) Y = k Y + πn d, and kn) X = k ) k and e ±iθ k n) and k n) X = i k n) where N min = Y { k +k Y )d π } [ H ) 0 k ρ n ) L sign) [ψac r 0 ) e ikn) Y y+kn) X x ) d k n) signx)) e isignx)θ k n) L sign) [ψac r 0 ) X = i kn) k n) Y X ±kn) Y k { k k Y )d π ) and e ±iθ k n) = kn) X ±ikn) Y k 34) for k k n) Y, for k k n) Y For n N = [N min,n max } is rea Furthermore, the various and N max =, + kn) X L ± [ψac r 0 ) are determined from soving the foowing equation: ac ) TG T ψ ac r 0 ) = T φ inc r ac 0 ) 35)

12 ac where is a max + ) + ) identity matrix, TG is a + ) + ) matrix ac with the m,n) eement given by TG ) = ac + n S m nk d,k Y d) In this case, TG can be expicity written as: ac TG = S 0 k d,k Y d) S k d,k Y d) m,n S k d,k Y d) 0 S k d,k Y d) S +k d,k Y d) S max k d,k Y d) S max 0k d,k Y d) 0 S +k d,k Y d) S max +k d,k Y d) S max+k d,k Y d) S max k d,k Y d) S k d,k Y d) 0 S k d,k Y d) S 0k d,k Y d) S max k d,k Y d) S k d,k Y d) 0 S k d,k Y d) S k d,k Y d) S k d,k Y d) S 0 k d,k Y d) 36) where S k d,k Y d) can be evauated using Eq 7), and: L [ψ ac r 0 ) L [ψ ac r 0 ) T ψ ac r 0 ) = ψ ac r 0 ), T φ inc r ac 0 ) = e i k r 0 m hk X L + [ψ ac r 0 ) L + [ψac r 0 ) e iθ k e i )θ k e i )θ k e iθ k 37) For a scattering potentia given by V ac r,e) = n= V ac E)Θ r r n ) where Θz) = for z r s and Θz) = 0 for z > r s, the scattering ampitudes are given by = J k r s )J + k r s ) J k r s )J + k r s ) k H ) + k r s )J k r s ) k J + k r s )H ) k r s ) 38) me Vac E)) m E Vac E) where k = me, and k h = for E V h ac E) or k = i for E < V h ac E) Note that = in Eq 38) For comparing the cacuations in a DEG to the cacuations in graphene, the magnitudes of k and k for the DEG were chosen to be the same as in the graphene [detais are given in the caption of Figure 3 in the main manuscript In this case, the transmitted and refected wave functions, ψ ac T are determined from Eq 34): r) for x > 0 and ψac r) for x < 0, R m ψt ac r) = Tn ac e ikn) Y y+kn) X x) n N hk n) X m ψr ac r) = R ac n e ikn) Y y kn) X x) n N hk n) X 39)

13 where the refection and transmission coefficients satisfy the unitarity condition, n N T ac n + R ac n =, and are given by: T ac n = δ n0 + R ac n = sac = d sac = m hk n) X ) m d e iθ k n) L sign) [ψac r 0 ) hk n) e iθ k n) L sign) [ψac r 0 ) X 40) Like in graphene, was chosen to take into account 999% of the tota scattering ampitude for an individua scatterer in the DEG, ie, max = 0999 = s ac ACKNOWLEDGMENTS This work was supported by the Nationa Science Foundation under CHE and from funds from the University of Miami [ Vaishnav, J Y, Anderson, J Q & Was, J D Intravaey mutipe scattering of quasipartices in graphene Phys Rev B 83, ) [ Katsneson, M I & Novoseov, K S Graphene: New bridge between condensed matter physics and quantum eectrodynamics So State Comm 43, ) [3 Yasumoto, K & Yoshitomi, K Efficient cacuation of attice sums for free-space periodic green s function IEEE Trans Antennas Propagat 47, ) [4 Mathworks Matab [5 Nicorovici, N A, McPhedran, R C & Petit, R Efficient cacuation of the green s function for eectromagnetic scattering by gratings Phys Rev E 49, ) [6 Pereira, J M, Minar, V, Peeters, F M & Vasiopouos, P Confined states and direction-dependent transmission in graphene quantum wes Phys Rev B 74, ) [7 Vaishnav, J Y, Was, J D, Apratim, M & Heer, E J Matter-wave scattering and guiding by atomic arrays Phys Rev A 76, ) 3