Chaos in Dirac electron optics: Emergence of a relativistic quantum chimera

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1 Suppeentary Inforation for Chaos in Dirac eectron optics: Eergence of a reativistic quantu chiera Hong-Ya Xu, Guang-Lei Wang, Liang Huang, and Ying-Cheng Lai Corresponding author: Y.-C. Lai Ying-Cheng.Lai@asu.edu CONTENTS I. Basics 1 II. Mutichanne eastic scattering theory for two-diensiona assess Dirac ferions - S-atrix approach III. S-atrix for eccentric annuar shaped ring scatterer 6 IV. Cacuation of wavefunctions 7 V. Idea centric case: anaytic resuts 8 VI. Vaidation of the S-atrix approach 10 A. Syetry constraints 10 B. The case of ξ 0 1 VII. Fu data set for the pot Fig. c in the ain text 1 VIII. Feasibiity of experienta ipeentation 13 References 14 I. BASICS The starting point of our anaysis is the effective ow-energy Haitonian of graphene or graphene-ike systes with Dirac cones: H = v F s 0 σ p + s 0 σ 0 V gate r s z σ 0 M r, S1.1 where the identity atrix s 0 and the Paui atrix s z act on the rea eectron spin space whie the Paui atrices σ = σ x,σ y and the identity atrix σ 0 define the subattice pseudospin. The first ter in Eq. S1.1 characterizes the pristine Dirac cone band dispersion with a four-fod degeneracy at a Dirac point: two for the subattice pseudospin and two for the rea eectron spin. Since s z σ 0,H = 0, it is equivaent to two copies of Dirac-ike Haitonian indexed by the spin quantu nuber s = ±: H s = H 0 +V gate r sm r, S1. 1

2 where H 0 = v F σ p is effectivey the fundaenta Dirac-Wey Haitonian describing the twodiensiona free-space assess Dirac ferions. The Haitonian H s acts on two-coponent pseudospinor waves for the assess Dirac quasipartices beonging to the rea spin state s in graphene or siiar aterias. The ast two ters in Eq. S1.1 represent the appied gate and exchange potentia, respectivey. In the ain text, the cacuations are for the scattering of such quasipartices fro the step potentia that can ead to spin-resoved ray-path defined cassica dynaics in the short waveength iit. The scattering process is of the reativistic type for assess Dirac ferions. In the foowing Secs. II-V, we deveop an S-atrix based schee to sove the reativistic quantu scattering probe, which is vaidated coputationay in Sec. VI. In Sec. VII, we provide a detaied deonstration of the phenoenon of enhanced spin poarization as shown in Fig. c in the ain text. II. MULTICHANNEL ELASTIC SCATTERING THEORY FOR TWO-DIMENSIONAL MASS- LESS DIRAC FERMIONS - S-MATRIX APPROACH The ain theoretica too that we epoy to investigate the roe of chaos in Dirac eectron optics is the forais of stationary quantu scattering for two-diensiona assess Dirac ferions, where the scatterer has an irreguar shape and a finite range. The scattering process is assued to be eastic. The fundaenta quantity of interest is the scattering S- atrix, fro which a physicay reevant quantities characterizing the scattering process can be deduced. In the free space, the syste is governed by the stationary Dirac-Wey equation H 0 χ = hv F σ kχ = Eχ, for which the pane-wave soutions for energy E = α hv F k is given by e ik r, where k = χ k r = 1 1 αe iθ k S.3 S.4 k x + k y, α = sgne and θ k = arctank y /k x characterize the propagating direction parae to the wavevector k for E > 0. For E < 0, the two directions are anti-parae with each other. In the poar coordinates r = rcosθ,sinθ, the corresponding spinor cyindrica waves with given anguar oentu and energy are k h r,θ = Z kr iαz +1 kre iθ e iθ, S.5 where Z is the -th order Besse or Hanke function of the physicay reevant kind. In particuar, under the tie convention e ie/ ht and for positive energy E > 0, we have k h H kr = ih e iθ, S.6a +1 kreiθ as the cyindrica wave basis of the spinor waves of the incoing type, and k h + H 1 kr = ih 1 e iθ, +1 kreiθ S.6b

3 as the outgoing type, where H 1 and H denote the Hanke functions of the first and second kind, respectivey. For the scattering probe iustrated in Fig. 1 in the ain text, the stationary wavefunction outside the scatterer generay can be decoposed into two parts - incoing and outgoing waves: Ψ = Ψ in + Ψ out. S.7 In the spinor cyindrica wave basis for assess Dirac ferions with positive energy, the incoing wave can be written as Ψ in = a k h, S.8 and the outgoing wave can be expressed as Ψ out = a S k h +, S.9 where the coefficients a are deterined to yied a desired kind of incoing test wave, S denotes the transition apitude for an incoing cyindrica wave k h scattered into an outgoing one k h +. This defines the S-atrix with and covering a possibe anguar oentu channes. We thus have H Ψr,θ = kr a ih e iθ H + 1 S kr +1 kreiθ ih 1 e i θ, +1 kreiθ J kr = a ij +1 kre iθ e iθ H + a 1 S.10 S δ kr ih 1 e i θ. +1 kreiθ To be concrete, we assue the incident wave to be a pane wave given by χ kin r,θ = 1 1 e iθ e ikin r = 1 1 e ikr cosθ θ, k in e iθ for assess Dirac ferions with positive energy E = hv F k and incident wavevector k in = kcosθ,sinθ that akes an ange θ with the x axis. This defines the incident propagating direction as shown in Fig. 1 in the ain text. We have χ kin = i e iθ J kr ij +1 kre iθ e iθ, S.11 where the Jacobi-Anger expansion e izcosθ = i J ze iθ has been used. Given the coefficients a = a θ = i e iθ and with the definition T S δ, we get Ψr,θ = χ kin + a T 3, S.1 H 1 kr ih 1 e i θ. +1 kreiθ S.13

4 Far away fro the scatterer center, i.e. kr 1, the asyptotic wavefunction can be written as i Ψ = χ k in + f θ,θ 1 kr 1 ir e iθ e ikr, S.14 where f is the scattering apitude for two-diensiona assess Dirac ferions, which is reated to the differentia cross section through the tota cross section through dσ dθ σθ,θ = f θ,θ, σ t θ = dθ f θ,θ, S.15a S.15b the transport cross section through σ tr θ = dθ1 cosθ f θ,θ, S.15c and the skew cross section through σ sk θ = dθsinθ f θ,θ. S.15d It foows fro Eqs. S.13 and S.14 that Finay, we obtain Defining f θ,θ 1 ir e iθ e ikr = i a kr 1 T f θ,θ = i f θ = we rewrite the scattering apitude as H 1 kr ih 1 e i θ. +1 kreiθ πk a θ T i e i θ. a θ T i = a θ S δ i, S.16 S.17 f θ,θ = i πk f θ e iθ, which, when substituted into Eqs. S.15a-S.15d, eads to convenient suation fors of the various cross sections in ters of f θ and eventuay the scattering atrix eeents S as σθ,θ = πk f θ e iθ = πk,, a a S δ S δ i e i θ, σ t θ = 4 k f θ = 4 k, a T T a, S.18a S.18b 4

5 σ tr θ = σ t θ 4 k R f f+1 = σ tθ 4 k R ia T T a, S.18c, and σ sk θ = 4 k I f f+1 = 4 k I ia T T a, S.18d, where T T +1, = T,+1. A the scattering cross sections are functions of θ that defines the direction of the incident wave with respect to the x axis. Averaging over a the incident directions θ, we obtain the cross sections that are independent of the ange θ as and σ t = 1 π dθ σ t θ = 4 k 1 π, σ tr = σ t 4 k i R π, Fro the definition σ sk = 1 k dθ a θ T T a θ = 1 k, T, S.19a dθ a θ T T a θ = 1 k { T R it T,+1},, S.19b I it T δ = 1 k I it T,+1. S.19c,, T = S δ, i.e. T = S I, we can cacuate the characteristic cross sections once the scattering S-atrix is obtained. In addition to the cross sections, associated with the S-atrix, another quantity of interest is the Wigner-Sith deay tie 1, defined as τe = i htr S S, S.0 E which characterizes the tepora aspects of the scattering process. The deay tie is reated to the density of states 3 through ρe = τe/π h. By definition, the transport cross section ost appropriatey characterizes the transport property, which deterines the transport reaxation tie τ tr according to the Feri s goden rue with its reciproca given by 1 = n c v F σ tr, S.1 τ tr where n c is the concentration of identica scatterers that are assued to be sufficienty diute so that utipe scattering effects can be negected. If the syste diension is arger than the ean-free path L = v F τ tr, fro the seicassica Botzann transport theory, we obtain the conductivity of the syste as G G 0 = k F v F τ tr = where G 0 = e /h is the conductance quantu. 5 k n c σ tr, S.

6 III. S-MATRIX FOR ECCENTRIC ANNULAR SHAPED RING SCATTERER We perfor an expicit cacuation of the S-atrix for the scatterer of annuar shape defined by two disks of different radii R 1,R < R 1 with a finite reative dispaceent ξ of the disk centers, as shown in Fig. 1a in the ain text. For convenience, we adopt the convention that the unpried coordinates are defined by choosing the origin as the center of the arger disk O whie the pried ones have their origin at the sa disk center O. Appying the standard S-atrix forais, we obtain the wavefunction outside the eccentric annuar scatterer, i.e., r > R 1, in the unpried poar coordinates r = r,θ as Ψ I r = a 0 k 0 h = + S = k 0 h 1, S3.3 where S denotes the S-atrix eeents in ters of the two given channes indexed by and, respectivey, and the coefficients a 0 are chosen to yied a desired kind of incident test wave. Let k 0h a 0 k 0h and S a 0 S, and so Ψ I r = k 0 h = + S = k 0 h 1. S3.4 The wavefunction in the annuar region r > R and r < R 1 can be expressed in the unpried coordinates as Ψ II r = = = a 1 k 1 h + = S od k 1 h 1, S3.5 where the resuting atrix S od S od characterizes the scattering fro the off-centered sa inner disk and is non-diagona. Making use of the addition property of the Besse functions, we obtain the foowing reation S od = U 1 S cd U, S3.6 where the transforation atrices U = U µ = J µ k 1 ξ and U 1 = U 1 = J k 1 ξ are responsibe for the eccentric dispaceent/deforation, and S cd = S cd δ is the diagona scattering atrix for the centered inner disk scatterer in the pried coordinates with its eeents S cd given by S cd = α 1H +1 k 1R J k R α H k 1 R J +1 k R α 1 H 1 +1 k 1R J k R α H 1 k 1 R J +1 k R. S3.7 The S-atrix of the whoe scatterer can thus be deterined by the atching conditions at the outer boundary r = R 1. In Eqs. S3.3 and S3.5, k 0,1 h 1, denote the basic spinor waves consisting of the expanding basis indexed by the anguar oentu in the poar coordinates and are expicity given in Eq. S5.44a. In particuar, for a given incident spinor wave with anguar oentu, wavefunction atching for each oentu vaue yieds a 0 H k 0 R 1 δ + a 0 S H 1 k 0 R 1 = a 1 H k 1 R 1 + a 1 Sod H1 k 1 R 1, S3.8a 6

7 iα 0 a 0 H +1 k 0R 1 δ + a 0 S H 1 +1 k 0R 1 = iα 1 a 1 H +1 k 1R 1 + a 1 Sod H1 +1 k 1R 1. S3.8b Defining atrices and X 1, = H 1, k 0 R 1 δ, Y 1, = H 1, +1 k 0R 1 δ, S3.9a x 1, = H 1, k 1 R 1 δ, y 1, = H 1, +1 k 1R 1 δ, S3.9b we can rewrite the above equations in the foowing copact for A 0 X + A 0 SX 1 = Ax + AS od x 1, S3.30a α 0 A 0 Y + A 0 SY 1 = α 1 Ay + AS od y 1, S3.30b with the coefficient atrices A 0 = a 0 δ and A = a 1. Soving the above equations, we arrive at S = Y α 0 α 1 X T Y 1 α 0 α 1 X 1 T, S3.31 where T = F 1 G with the conventions F = x + S od x 1,G = y + S od y 1 and band indices α 0,1 = ±1. Substituting the S-atrix given in Eq. S3.31 into Eq. S3.30a, we obtain atrix A consisting of the expansion coefficients a 1 in the annuar region as A = A0 X + A 0 SX 1 x + S od x 1. S3.3 IV. CALCULATION OF WAVEFUNCTIONS Inside the inner disk region, i.e., r < R, the wavefunction in the pried poar coordinates r = r,θ with origin at the sa disk center O is Ψ III r,θ = J k r b is J +1 k r e iθ e iθ. S4.33 The expansion coefficients b can be deterined by the atching condition at the inner boundary r = r ξ = R between Ψ r,θ and Ψ 1 r,θ. To do so, it is convenient to reforuate the wavefunction inside the annuar region in the pried coordinates. Using the reations S od = J S cd J, δ = J J, S4.34a S4.34b 7

8 and assuing = + n, we have k 1 h + = S od k 1 h 1 k δ 1 h = = = J + J k 1 h k J J 1 n h +n + Scd n J S cd J k 1 h 1 + S cd J k 1 h 1, n J n k 1 h 1 +n { } Making use of the Graf s addition theore 4 for the Besse functions Z J,H 1, : Z kr e i θ = J n kξz +n kre i +nθ, n we can rewrite the Eq. S4.35 in the pried coordinates as where k 1 h + = S od k 1 h 1 k 1 h1, = =., S4.35 J k1 h + S cd k1 h1, S4.36 1, H k 1 r iα 1 H 1, +1 k 1r e i θ. e iθ S4.37 Substituting this expression into Eq. S3.5, we obtain the wavefunction for the annuar region in the pried coordinates as Ψ II r,θ = a 1 J k1 h + S cd k1 h1 k1 h + S cd k 1 h1, S4.38 where ã 1 = = ã 1 a 1 J k 1 ξ. Iposing the continuity of the wavefunction at r = R, we get S4.39 b = ã 1 H k 1 R + S cdh1 k 1 R. S4.40 J k R With the expansion coefficients a 1, ã 1, and b so deterined and the scattering atrices S,S od,s cd obtained in the reevant regions via Eqs. S3.3, S4.39, S4.40 and Eqs. S3.31, S3.6, S3.7, respectivey, we can cacuate the wavefunctions accordingy, which together give the fu wavefunction in the entire space. V. IDEAL CENTRIC CASE: ANALYTIC RESULTS For the centric case, i.e., ξ = 0, we can obtain the anaytic soutions of the scattering probe via the standard technique of partia wave decoposition. In particuar, due to the circuar rotationa 8

9 syetry and hence conservation of the tota anguar oentu, the partia waves outside the annuar scatterer r > R can be written as ψ I = k 0 h + S k 0 h 1. Inside the annuar region R 1 < r < R, the waves are ψ II = A k1 h + S cd k 1 h 1 S5.41 S5.4 and ψ III = B k χ, S5.43 in the inner disk region r < R, where k 0,k 1,k = E 0, E 0 V 1, E 0 V / hv, and k 0,1 h 1, = H 1, k 0,1 r iα 0,1 H 1, +1 k 0,1re iθ e iθ, J χ = k r iα J +1 k re iθ e iθ, S5.44a S5.44b with α 0,1, = ±1 being the band index defined as the signs of E 0,E 0 V 1,, respectivey, and = 0,±1,±, denote the orbita anguar oentu. The scattering apitudes S,S cd and the expansion coefficients A,B can be deterined fro the boundary conditions ψ I R 1,θ = ψ II R 1,θ;ψ II R,θ = ψ III R,θ, eading to the foowing inear atrix equation H k 1 R J k R H 1 k 1 R 0 α 1 H +1 k 1R α J +1 k R α 1 H 1 +1 k A 0 1R 0 B H k 1 R 1 0 H 1 k 1 R 1 H 1 k 0 R 1 C = 0 H k 0 R 1, α 1 H +1 k 1R 1 0 α 1 H 1 +1 k 1R 1 α 0 H 1 +1 k 0R 1 S α 0 H +1 k 0R 1 S5.45 where C A S cd. Fro the standard quantu scattering theory, we have that S is an eeent of the S-atrix for the concentric circuar scatterer, which is diagona in the basis of anguar oentu states. Soving Eq. S5.45, we obtain the coefficients as A = H k 0 R 1 + H 1 k 0 R 1 S H k 1 R 1 + H 1 k 1 R 1 S cd ; B = A H k 1 R + H 1 whie the scattering apitudes for the whoe scatterer are given by k 1 R S cd, S5.46a J k R S = α 0x H +1 k 0R 1 α 1 y H k 0 R 1 α 0 x H 1 +1 k 0R 1 α 1 y H 1 k 0 R 1, S5.46b where x = H k 1 R 1 + H 1 k 1 R 1 S cd and y = H +1 k 1R 1 + H 1 +1 k 1R 1 S cd with S cd given by Eq. S3.7. 9

10 S -+1,-+1 S a b ReS IS theory S, ReS IS d 3 Cross Sections 1 0 c σ sk θ'=0 σ t θ'= E FIG. S1. Vaidation of the S-atrix approach through the cosed-for anaytic constrains iposed by the syetry of the syste. a Pot of the diagona eeents S +1, +1 versus S, where the thick back ine is the theoretica prediction of Eq. S6.49, b rea and iaginary parts of S, c fase coorcoded ap of the agnitudes of the fu scattering atrix eeents with a proper cut-off at = ±10 the scae bar shows the fourth root of agnitudes S ; d the skew cross section σ sk purpe ine and the tota cross section σ t ight bue curve as a function of the energy for incident waves propagating parae to the syetry axis, where the vanishing skew asyetric scattering, i.e., σ sk 0, is consistent with the prediction of Eq. S6.51. Paraeters adopted for a-c are E = 70, R /R 1 = 0.6, ξ = 0.3, v 1 = 140, and v = 0. For d, the paraeters are R /R 1 = 0.6, ξ = 0.3, v 1 = 10 and v = 40. VI. VALIDATION OF THE S-MATRIX APPROACH A. Syetry constraints In spite of the ack of circuar rotationa syetry, the syste possesses a irror parity syetry, which iposes certain constrains on the S-atrix and eads to vanishing of skew asyetric scattering provided that the incident wave propagates aong the axis of the syetry. In particuar, for the configuration shown in Fig. 1a in the ain text, for spinor scattering we can expicity write the representation of the parity syetry operation as P x = σ x R y with R y, which is the refection operator that acts in the physica position space with respect to the x axis via the operations x xk x k x and y yk y k y,θ θ. As such, the syste is invariant under parity, stipuating the reation P x HP 1 x = H so that P x Ψ is sti a state of the syste with the sae of given orbita anguar energy. Under the operation of P x, the spinor cyindrica wave k h 1, 10

11 oentu corresponding to tota anguar oentu L = + 1/ can be transfored as P k x h 1, H 1, 1, kr H = iσ x R y iαh 1, e iθ = +1 iα +1 kr +1 kreiθ iαh 1, e i+1θ, kre iθ S6.47 = +1 iα k h 1, +1. Appying this reation to the resuting state Ψ 0 given in Eq. S3.3, we obtain P x Ψ I = P x a 0 Px 1 P x ψ = P x a 0 Px 1 k P 0 x h k + S 0 h 1, = = = a 0 +1 k iα 0 0 h +1 + P x S Px 1 k 0 h 1 +1 = c 0 nψ n = n c 0 k 0 n h n= = n + S nn n = k 0 h 1 n,, S6.48 with deduced identities n +1, n +1, c 0 n a 0 +1 iα 0 = P x a 0 P 1 x +1 iα 0. We thus have S nn S +1, +1 = P x S P 1 x = S. S6.49 In particuar, for =, i.e., the diagona eeents, we have S = S +1, +1. Under such constrains and using the definition of f θ given in Eq. S.17, we have f θ = a θ S δ i, = i e iθ S +1, +1 δ +1, +1 i +1 i +1, i = e iθ +1 e i+1θ S +1, +1 δ +1, +1 i +1, = e iθ a θ S, +1 δ, +1 i +1 e iθ f +1 θ. S6.50 For θ = 0 π, i.e., when the incident wave propagates parae anti-parae to the axis of the irror syetry, we obtain f = ± f +1, based on which we can rewrite the skew cross section in Eq. S.18d as σ sk θ =0π = 4 k I f f+1 { = 4 k I f 0 + = 4 k I f 0 + =0 =0 R f f f f +1 + f +1+1 f +1 }, S6.51 These basic syetry induced, exact constrains given by the cosed fors in Eqs. S6.49 and S6.51 can serve as bencharks for vaidating the S-atrix approach. Note that, whie theoreticay the diension of the S-atrix is infinite, in practice a finite truncation is needed for a given 11

12 σ t ξ = 0 Idea centric case decreasing ξ E FIG. S. Vaidate the S atrix approach by showing the convergence to the integrabe case. For the case of cassicay integrabe dynaics, the agreeent between the theoreticay predicted cross section vaues the back curve cacuated fro Eqs. S5.46b and the nuerica resuts as ξ approaches zero. Paraeters are R /R 1 = 0.6, v 1 = 10, and v = 40. energy E since channes with higher anguar oenta ER/ hv cannot be excited effectivey and thus have negigibe contribution to the scattering process. Representative resuts are shown in Fig. S1. We obtain a good agreeent between the theoretica prediction and the siuation resuts fro propery truncated S-atrices. B. The case of ξ 0 Nuericay, it is straightforward to vaidate the S-atrix approach indirecty by evauating the convergence of the vaue of the cross section to the theoretica vaue for the iiting case of ξ 0 at which the cassica dynaics are integrabe. As shown in Fig. S, a good agreeent is achieved for ξ < VII. FULL DATA SET FOR THE PLOT FIG. C IN THE MAIN TEXT Figure S3a shows the spin poarization versus ξ and the Feri energy E, where the deep skybue regions in the energy doain indicating higher vaues of spin poarization becoe extended as ξ is increased and exceeds the vaue of 0.. Figure S3b shows the average spin conductivities versus ξ, where the conductivity for the spin up popuation is a deceasing function of ξ but that for 1

13 the spin down state is essentiay constant. Thus, on average the spin up partices undergo significanty stronger backward scattering as copared with the spin down partices, generating a severe spin ibaance e.g., for ξ = 0.3 and consequenty, significanty enhanced spin poarization. To appreciate the roe of deforation payed in generating a strong Dirac quantu chiera state, we cacuate the average differentia cross section σ di f f E E 1 1 E E 1 σ di f f σ di f f de versus the backward scattering ange θ for two cases: ξ = 0 and ξ = 0.3, as shown in Fig. S3c. A scheatic iustration of the generation of spin poarization is shown in Fig. S3d. FIG. S3. Spin poarization enhanceent as a resut of Dirac quantu chiera. a Coor-coded ap of spin poarization P z as a function of energy E and eccentricity ξ, b spin conductivities averaged over a given Feri energy range versus ξ, where the red curve is verticay shifted by an arbitrary aount for better visuaization, c iustration of a chaos rendered spin rheostat tuned by ξ, and d a scheatic iustration of the generation of spin poarization. VIII. FEASIBILITY OF EXPERIMENTAL IMPLEMENTATION In genera, the eergence of a Dirac chiera reies on the optica ike behavior of Dirac eectrons and Dirac cone spitting, which can be reaized in current experienta systes of graphene. In particuar, given the graphene attice constant and typica vaues of the Feri waveength e.g., λ F 0n, a Dirac description of the step potentia requires the ength scae characterizing the unction sharpness to be d 1n, which has been recenty achieved experientay for a circuar 13

14 unction geoetry 5. The size of the unction can be tuned to the icroeter scae λ F, vaidating the short waveength approxiation 6. Furtherore, the experientay achievabe strength of the exchange potentia for graphene is strong enough to enabe Dirac cone spitting at the roo teperature 7, providing a base for experientay observing the predicted Dirac quantu chiera. 1 E. P. Wigner, Phys. Rev. 98, F. T. Sith, Phys. Rev. 118, H. Schoerus, M. Marciani, and C. W. J. Beenakker, Phys. Rev. Lett. 114, D. Zwiinger, Tabe of Integras, Series, and Products Esevier Science, C. Gutierrez, L. Brown, C.-J. Ki, J. Park, and A. N. Pasupathy, Nat. Phys. 1, Y. Jiang, J. Mao, D. Modovan, M. R. Masir, G. Li, K. Watanabe, T. Taniguchi, F. M. Peeters, and E. Y. Andrei, Nat. Nanotech. 1, P. Wei, S.-W. Lee, F. Leaitre, L. Pine, D. Cutaia, W.-J. Cha, F. Katis, Y. Zhu, D. Heian, J. Hone, and J. S. M. C.-T. Chen, Nat. Mater. 15,