Magnetic scattering of Dirac fermions in topological insulators and graphene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Magnetic scattering of Dirac fermions in topological insulators and graphene
Alex Zazunov, Arijit Kundu, Artur H¨utten, and Reinhold Egger
Institut f¨ur Theoretische Physik, Heinrich-Heine-Universit¨at, D-40225 D¨usseldorf, Germany (Dated: December 6, 2018)We study quantum transport and scattering of massless Dirac fermions by spatially localized staticmagnetic fields. The employed model describes in a unified manner the effects of orbital magneticfields, Zeeman and exchange fields in topological insulators, and the pseudo-magnetic fields causedby strain or defects in monolayer graphene. The general scattering theory is formulated, and forradially symmetric fields, the scattering amplitude and the total and transport cross sections areexpressed in terms of phase shifts. As applications, we study ring-shaped magnetic fields (includingthe Aharanov-Bohm geometry) and scattering by magnetic dipoles.
PACS numbers: 73.50.-h, 72.80.Vp, 73.23.-b
I. INTRODUCTION
The recent theoretical prediction and subsequential ex-perimental verification of the conducting surface state ex-isting in a strong topological insulator (TI) has generateda burst of activity, reviewed in Refs. 1 and 2. In a TI,strong spin-orbit couplings and band inversion conspireto produce a unique time-reversal invariant topologicalstate different from a conventional band insulator. UsingBi Se as reference TI material, one finds a rather largebulk gap ≈ . The measured spin texture of thesurface is well described by two-dimensional (2D) mass-less Dirac fermions, where the spinor wavefunction hasprecisely two entries corresponding to physical spin. Un-der this “relativistic” description, spin and momentumare always perpendicular, and the surface state is sta-ble against the effects of weak disorder and weak inter-actions due to the underlying topological protection.
Useful insights can then already be obtained from a non-interacting disorder-free description. Massless 2D Diracfermions are also realized in single carbon monolayers of graphene , for reviews see Refs. 4, 5 and 6. The limit ofballistic transport in this 2D material seems experimen-tally within reach, and graphene experiments have re-ported characteristic Dirac fermion signatures, e.g., Kleintunneling.
In most experiments performed so far, cor-relation effects turned out to be weak, and it is again ofinterest to examine the single-particle theory in the ab-sence of disorder. In contrast to the TI surface case, thetwo entries of the spinor wavefunction in graphene corre-spond to the two atoms in the basis of graphene’s honey-comb lattice. Furthermore, there are four Dirac fermion“flavors” in graphene, due to the physical spin and the KK ′ (“valley”) orbital degeneracy. Another condensed-matter realization of Dirac fermions is given by the quasi-particles in d -wave superconductors, but to be specific,we here focus on the surface state in a TI and on mono-layer graphene.The fact that the electronic properties of both theseapplications correspond to massless 2D Dirac fermionscalls for a unified description of their transport proper- ties. In the graphene context, much theoretical effort hasbeen devoted to advancing the scattering theory of Diracfermions in electrostatic potentials, in particular forthe Coulomb impurity. In this paper, we instead studythe scattering of massless Dirac fermions by a local mag-netostatic perturbation . The model, see Eq. (2.3) below,describes, in a unified manner, the effects of spatially in-homogeneous orbital magnetic fields, exchange-mediatedfields due to adjacent ferromagnetic (FM) layers and Zee-man fields in topological insulators, as well as strain- ordefect-induced pseudo-magnetic fields in graphene. Forthe Schr¨odinger fermions realized in 2D semiconductorelectron gases, such perturbations, e.g., magnetically de-fined barriers, steps, and quantum wells, have been inves-tigated both theoretically and experimentally.
The desired magnetic field profiles were generated by de-position of lithographically patterned FM layers on topof the sample. For soft FM materials, one can change themagnetization orientation by weak magnetic fields. An-other possibility is to use a type-II superconductor filminstead of the FM layer.
Previous theory work on TIs in inhomogeneous mag-netic fields has addressed only a few setups. For thetransmission of an electron through a magnetic barrier(assumed homogeneous in the transverse direction), asa function of either exchange field or applied bias volt-age, Mondal et al. predict an oscillatory behavior oreven a complete suppression of the transmission proba-bility, and hence of the conductance. A spin valve geom-etry with two adjacent magnetic barriers, characterizedby non-collinear exchange fields, has also been studied. As a model for a classical magnetic impurity, the spin-resolved density of states was calculated for a disc-shapedmagnetic field profile.
For graphene, a vector potential perturbation canagain be due to external orbital fields, but may also de-scribe the effects of strain and dislocations or othertopological defects. Several theoretical works have ad-dressed aspects of the electronic structure and the trans-mission properties for Dirac fermions in graphene in thepresence of inhomogeneous magnetic fields. The sim-plest case is encountered for effectively 1D problems withtranslational invariance in the (say) y -direction, e.g., fora magnetic step or a magnetic barrier. For suit-able 1D magnetic field profiles, it is possible to havemagnetic waveguides (along the y -direction), whereelectron-electron interaction effects play an importantrole. Periodic magnetic fields, i.e., 1D magnetic su-perlattices, have also been addressed.
For radiallysymmetric fields, total angular momentum conservationagain simplifies the problem and gives an effective 1Dtheory. This has allowed for studies of quantum dotor antidot geometries, where true bound states,not affected by Klein tunneling, may exist. In quan-tum dot setups, interaction effects become important forstrong confinement. When the vector potential corre-sponds to an infinitely thin solenoid, we encounter anultra-relativistic Dirac fermion generalization of the cele-brated Aharonov-Bohm (AB) calculation.
This gen-eralization was discussed before, and exact resultsfor the transmission amplitude can be deduced. Recentstudies have also addressed the current induced by anAB flux and the behavior of the conductance whenthe chemical potential is precisely at the Dirac neutralitypoint. Such an AB conductance can be probed experi-mentally in ring-shaped graphene devices.
In this article, we formulate a general scattering theoryapproach for massless 2D Dirac fermions in the presenceof such magnetic perturbations. In Sec. II, we introducethe model and outline its application to graphene andtopological insulators. In Sec. III, we formulate the gen-eral scattering theory, previously given for electrostaticpotentials, for the magnetic case. The scattering am-plitude and cross section are specified, and we discuss theBorn approximation in Sec. III A. For the radially sym-metric case, total angular momentum conservation allowsto express the scattering amplitude in terms of phaseshifts for given total angular momentum, see Sec. III B.In sections IV and V, we present applications of thisformalism. First, in Sec. IV, we discuss scattering bya magnetic dipole within the Born approximation. Sec-ond, in Sec. V, we consider ring-shaped field profiles . Thiscase also contains the AB solenoid in a certain limit, seeSec. V B, and we discuss how our phase-shift analysis re-covers known results for the AB effect. We also addressthe scattering resonances appearing due to quasi-boundstates in such a ring-shaped magnetic confinement. Fi-nally, some concluding remarks can be found in Sec. VI. II. DIRAC FERMIONS IN GRAPHENE ANDTOPOLOGICAL INSULATORSA. Model
In this section, we describe the 2D Dirac fermion modelfor electronic transport in graphene or the TI surfacestudied in this work. As perturbations, we first allow foran external static vector potential, A ( r ) = ( A x , A y , A z )with r = ( x, y ), which is included by minimal couplingand describes orbital magnetic fields and (for graphene) strain-induced pseudo-magnetic fields. In fact, for thoseapplications we have A z = 0, see below. In addition, forthe TI case, we allow for a Zeeman field or for exchangefields caused by nearby ferromagnets, whose componentsare contained in the field M ( r ) = ( M x , M y , M z ), whereprefactors such as the Bohr magneton or the Land´e factorare included. With the Pauli matrices σ = ( σ x , σ y , σ z )and the momentum operator p = − i ~ ( ∂ x , ∂ y , e > H = v F σ · (cid:16) p + ec A ( r ) (cid:17) + σ · M ( r ) − eV ( r ) . (2.1)The Fermi velocity in graphene is v F ≃ m/s, whilefor a TI surface state, a typical value for Bi Se is v F ≈ × m/s. The low-energy Hamiltonian in Eq. (2.1) isvalid on energy scales close to the neutrality level (Diracpoint), well below the bulk band gap for the TI case andwithin a window of size ≈ . A and the field M can now be combinedto a vector field Λ ( r ) ≡ A + cev F M , (2.2)which contains all considered “magnetic” perturbations.Interesting physics also follows in the presence of both Λ ( r ) and a scalar potential V ( r ), but we put V ( r ) = 0below.For the formulation of the scattering theory, it is conve-nient to employ cylindrical coordinates, x = r cos φ and y = r sin φ , with unit vectors ˆ e r = (cos φ, sin φ, , ˆ e φ =( − sin φ, cos φ, e z . With Λ = Λ r ˆ e r + Λ φ ˆ e φ + Λ z ˆ e z , Eq. (2.1) takes the compact form H = v F e − iφσ z / ˜ He iφσ z / , (2.3)˜ H = (cid:16) − i ~ ∂ r + ec Λ r (cid:17) σ x + (cid:18) r J z + ec Λ φ (cid:19) σ y , where the total angular momentum operator is J z = − i ~ ∂ φ + ~ σ z / . (2.4)For the case of azimuthal symmetry, ∂ φ Λ φ,r = 0, this isa conserved quantity, [ J z , H ] = 0, with eigenvalues ~ j forhalf-integer j . In Eq. (2.3) we have put Λ z = 0, which isthe case for all fields studied below.Let us now discuss how Eq. (2.3) relates to the sur-face states in a strong topological insulator , where thespin direction is tangential to the surface and perpendic-ular to momentum. For a given two-component spinorwavefunction ψ ( r ), the spin ( s ) and particle current ( j )density operators are s ( r ) = ~ ψ † (ˆ e z × σ ) ψ, (2.5) j ( r ) = v F ψ † ( σ x ˆ e x + σ y ˆ e y ) ψ, i.e., both spin and current are confined to the surfaceand obey s · j = 0. Under stationary conditions, thecontinuity equation, ∂ t ( ψ † ψ ) + P i = x,y ∂ i j i = 0, impliesthe relation X i = x,y ∂ i (cid:0) ψ † σ i ψ (cid:1) = 0 , (2.6)which is linked to the unitarity property of the scatteringmatrix. Equation (2.3) then allows to describe the fol-lowing setups for the TI surface state. First, an orbitalmagnetic field has only effects when it is oriented perpen-dicular to the surface, B orb = B z ( r, φ )ˆ e z . In cylindricalcoordinates, we can then choose some gauge for the vec-tor potential A such that B z ( r, φ ) = 1 r ( ∂ r ( rA φ ) − ∂ φ A r ) , (2.7)while A z drops out and is put to zero. Second, to describethe coupling of surface Dirac fermions to an in-plane ex-change field H ( r ) = ( H x , H y , Λ = ( c/ev F ) M with M = ( − H y , H x , H now denotes the Zeemanfield. A Zeeman or exchange field oriented along the ˆ e z direction can open a gap in the spectrum, and here weassume that such fields are not present. While the or-bital field breaks time reversal invariance, the Zeeman orexchange fields represent a time-reversal invariant per-turbation. Then Λ is determined by the magnetic fielditself, and hence is not a gauge field anymore.Next we turn to graphene , where the Pauli matrices σ are related to the two triangular sublattices constitutinggraphene’s honeycomb lattice. We assume that no spin-flip mechanisms are relevant, i.e., physical spin is con-served. We can then focus on one specific Dirac fermionflavor with fixed valley index and spin direction. Thisexcludes exchange or Zeeman fields, i.e., we put M = 0and hence Λ = A for graphene. Note that Zeemanfields in graphene are generally small compared to or-bital fields. Moreover, we consider only smoothly vary-ing vector potentials such that it is indeed sufficient to re-tain only one K point. Equation (2.3) can then describethe following cases. First, we may have an orbital mag-netic field, precisely as for the TI case. Second, pseudo-magnetic fields generated by strain-induced forces, or by various types of defects, e.g., dislocations, alsocorrespond to a vector potential, where time reversal in-variance implies that A has opposite sign at the two K points. A ( r ) can then be expressed explicitly in terms ofthe strain tensor, where the resulting pseudo-magneticfield is also oriented along the ˆ e z axis and A z = 0. Inaddition, strain causes a scalar potential V ( r ), which is,however, strongly reduced by screening effects. The com-bination of orbital and pseudo-magnetic fields may al-low to design a valley filter, since the total (orbital pluspseudo-magnetic) fields can differ significantly at both K points. B. Multipole expansion
Our scattering theory approach considers magneticfields [described by Λ ( r ) in Eq. (2.3)] that smoothlyvary on the scale of a lattice spacing and constitute a local perturbation, i.e., a well-defined cylindrical multi-pole expansion exists. Furthermore, Λ z = Λ r = 0 is as-sumed throughout. As we show below, for orbital fieldswe can choose a gauge where A r = 0. For strain-inducedfields, strictly speaking, the problem is not gauge invari-ant, and we cannot impose gauge conditions. However,in a more narrow sense, a gauge degree of freedom stillexists. For r → ∞ , with complex-valued coefficients α ( φ ) l,m = (cid:16) α ( φ ) l, − m (cid:17) ∗ , we then have the multipole expansionΛ φ ( r, φ ) = α Φ πr + ∞ X l =2 ∞ X m = −∞ e imφ r l α ( φ ) l,m , (2.8)where α denotes the total flux in units of the flux quan-tum Φ = 2 π ~ c/e .Let us now address the orbital magnetic field case, Λ = A , where we can exploit gauge invariance. We start froma more general situation with A r = 0, expressed as inEq. (2.8) with coefficients α ( r ) l,m , and also allow for nonzerocoefficients α ( φ ) l =1 ,m =0 . We now show that one can choosea gauge where A r = 0 and α ( φ )1 ,m =0 = 0. Indeed, gaugeinvariance implies that for arbitrary functions g ( x, y ), weare free to replace A i → A i + ∂ i g . Using a multipoleexpansion for rg ( r, φ ) with coefficients g l,m , an equivalentgauge choice thus follows by the replacement α ( φ ) l,m → α ( φ ) l,m + img l,m ,α ( r ) l,m → α ( r ) l,m − ( l − g l,m . We then choose the gauge function g l> ,m = α ( r ) l,m l − , g l =1 ,m =0 = iα ( φ )1 ,m m . In the new gauge, we arrive at Eq. (2.8) plus the radialcomponent A r = X m e imφ r α ( r )1 ,m . Using Eq. (2.7), the orbital field expansion (with r > B z ( r, φ ) = − ∞ X l =1 ∞ X m = −∞ e imφ r l +1 h ( l − α ( φ ) l,m + imδ l, α ( r )1 ,m i . The m = 0 term in A r neither generates flux nor mag-netic fields and can be omitted. Magnetic field profileswith α ( r )1 ,m = 0 arise only in time-dependent settings andwill not be studied here. As a consequence, the radialcomponent vanishes, A r = 0, and we arrive at Eq. (2.8). III. SCATTERING THEORY
For given energy E = ~ v F k , where k > Hψ = Eψ with Eq. (2.3), hasscattering solutions that we wish to obtain in the pres-ence of magnetic perturbations of the type in Eq. (2.8).The solution for E = − ~ v F k follows simply by reversingthe sign of the lower spinor component. We are thenlooking for a solution ψ ( r, φ ) = ψ in + ψ out consisting, inthe asymptotic regime r → ∞ , of a plane wave ( ∝ e ikx )propagating along the positive x -direction, ψ in ( r, φ ) = 1 √ e ikr cos φ (cid:18) (cid:19) , (3.1)plus the scattered outgoing spherical wave, ψ out ( r, φ ) = F ( φ ) e ikr √− ir (cid:18) e iφ (cid:19) . (3.2)We adopt the same normalization conventions asNovikov. From Eq. (2.5) we see that the incoming cur-rent density is j in = v F ˆ e x while (for the TI case) the spindensity is ( ~ / e y . Equation (3.2) defines the scatteringamplitude F ( φ ) for an outgoing wave deflected under thescattering angle φ . The resulting scattered current den-sity implies the standard definitions of the differential( dσ/dφ ), total ( σ tot ) and transport ( σ tr ) scattering crosssections, respectively: dσdφ = | F ( φ ) | , (3.3) σ tot = Z π dφ | F ( φ ) | = r πk Im F (0) ,σ tr = Z π dφ (1 − cos φ ) | F ( φ ) | . The second equality for σ tot expresses the 2D optical the-orem. When a random distribution of magnetic pertur-bations is present, the inverse mean free path determin-ing the conductivity is proportional to the transport crosssection. A. Born approximation
For small perturbation Λ φ ( r, φ ), one can evalu-ate the scattering amplitude within the first Bornapproximation. Strictly speaking, the long-ranged partΛ φ ∝ α/r in Eq. (2.8) can not be treated perturbatively,and in this section we assume α = 0.The unperturbed state is the incoming plane wave ψ in ,Eq. (3.1). Within lowest-order perturbation theory, thescattered wave obeys[ H − E ] ψ out = − ev F c Λ φ ˆ e φ · σ ψ in , (3.4)where H is the unperturbed Dirac Hamiltonian. Mul-tiplying both sides of Eq. (3.4) by H + E and noting that in real-space representation, the retarded Green’sfunction ( H − E ) − is given by the Hankel function H (1)0 , ψ out ( r ) = − iπ √ ~ Φ Z d r ′ H (1)0 ( k | r − r ′ | )( σ · p ′ + ~ k ) × Λ φ ( r ′ , φ ′ ) [ˆ e φ ′ · σ ] e ikr ′ cos φ ′ (cid:18) (cid:19) . The asymptotic large- ρ behavior of the Hankel function(where η = 1 , ± ) is H ( η ) ν ( ρ ) ≃ r πρ e ± i ( ρ − (2 ν +1) π/ , (3.5)which implies that ψ out for r → ∞ indeed has the form inEq. (3.2). After some algebra, we obtain the scatteringamplitude in Born approximation, F ( φ ) = √ πk Φ e − iφ/ Z ∞ rdr Z π dφ ′ sin φ ′ × e − ikr | sin( φ/ | sin φ ′ Λ φ ( r, φ ′ + φ/ . (3.6)For radially symmetric perturbations, ∂ φ Λ φ = 0, the φ ′ -integration can be done, and we obtain F ( φ ) = − πi √ πk Φ e − iφ/ (3.7) × Z ∞ rdr J (2 kr | sin( φ/ | ) Λ φ ( r ) , with the J Bessel function.
B. Radially symmetric case
Next, we address the full (beyond Born approximation)scattering solution for radially symmetric perturbations, Λ = Λ φ ( r )ˆ e φ . In that case, the total angular momentumoperator J z in Eq. (2.4) is conserved and has eigenval-ues ~ j with j ≡ m + 1 / m ∈ Z ). We thus expandthe spinor wavefunction in terms of angular momentumpartial waves ψ m ( r ) ≡ ( f m , ig m ) T ,ψ ( r, φ ) = e − iφσ z / ∞ X m = −∞ e i ( m +1 / φ ψ m ( r ) , (3.8)where the Dirac equation yields (cid:20) − i (cid:18) ∂ r + 12 r (cid:19) σ x + m + 1 / ϕ ( r ) r σ y (cid:21) ψ m = kψ m . (3.9)The magnetic flux (in units of the flux quantum Φ ) en-closed by a circle of radius r around the origin is ϕ ( r ) ≡ πr Φ Λ φ ( r ) , (3.10)where α = ϕ ( ∞ ) in Eq. (2.8). The continuity relation(2.6) must hold for each partial wave ψ m separately, andimplies ∂ r (cid:0) rψ † m σ x ψ m (cid:1) = 0 . (3.11)Introducing dimensionless radial coordinates, ρ ≡ kr , aclosed equation for the upper component, f m ( ρ ), follows, (cid:20) ρ ∂ ρ ( ρ∂ ρ ) + 1 − (cid:18) ρ + W m + W ′ m (cid:19)(cid:21) f m = 0 ,W m ( ρ ) ≡ m + 1 / ϕ ( ρ/k ) ρ , (3.12)where W ′ m ≡ ∂ ρ W m . The lower component is obtainedfrom g m ( ρ ) = − (cid:18) ∂ ρ + 12 ρ − W m (cid:19) f m . (3.13)These relations imply a general expression for the scatter-ing amplitude F ( φ ) under radially symmetric magneticperturbations, and thus for the various cross sections inEq. (3.3). For ρ → ∞ , the term ∝ α/r in Eq. (2.8) dom-inates and the general solution to Eq. (3.12) is given interms of Hankel functions, f m ( ρ ) = a m H (1) m + α ( ρ ) + b m H (2) m + α ( ρ ) , (3.14)with complex coefficients a m and b m . The lower spinorcomponent then follows from Eq. (3.13), g m ( ρ ) = a m H (1) m + α +1 ( ρ ) + b m H (2) m + α +1 ( ρ ) . (3.15)The continuity relation (3.11) implies a m = b m e i ˜ δ m , i.e.,the outgoing wave can differ from a free spherical waveonly by a phase shift ˜ δ m , which depends on the magneticperturbation and is determined in Sec. V. Using theBessel function expansion formula e iρ cos φ = X m ∈ Z i m e imφ J m ( ρ )and the asymptotic behavior of H (1 , ν , see Eq. (3.5), wefind b m = i m e − iπα/ . (3.16)We then obtain the scattering amplitude in terms ofphase shifts as for the electrostatic case, F ( φ ) = − i √ πk X m ∈ Z (cid:0) e iδ m − (cid:1) e imφ , (3.17)but δ m includes the total flux α , δ m ≡ ˜ δ m − πα/ . (3.18) As a consequence, qualitatively different effects beyondthe electrostatic case arise, such as the AB effect. Thecross sections in Eq. (3.3) are then given by σ tot = 4 k X m sin ( δ m ) , (3.19) σ tr = 2 k X m sin ( δ m +1 − δ m ) . Scattering theory has thus been reduced to the determi-nation of the phase shifts δ m . In the electrostatic case, the phase shifts obey the symmetry relation δ m = δ − m − ,implying the absence of backscattering, F ( π ) = 0. In themagnetic case under consideration here, in general thissymmetry relation breaks down, and hence backscatter-ing is not suppressed anymore, F ( π ) = 0. This is closelyrelated to the fact that magnetic fields can confine mass-less Dirac particles. IV. MAGNETIC DIPOLES
As a first application, we analyze the scattering of 2Dmassless Dirac fermions by a fixed magnetic dipole mo-ment m located at position r = (0 , , h ), i.e., at a height h above the origin of the 2D plane. In that case no totalflux is generated, α = 0. The results of this section areobtained under the Born approximation, see Sec. III A. A. Perpendicular orientation
First, we consider a dipole moment oriented perpen-dicular to the layer, m ⊥ = m ⊥ ˆ e z , where we have theisotropic (vector potential) perturbationΛ φ ( r ) = m ⊥ r ( r + h ) / . (4.1)The Born approximation, see Eq. (3.7), yields the scat-tering amplitude F ⊥ ( φ ) = − ie − iφ/ p (2 π ) k ( m ⊥ / Φ ) e − kh | sin( φ/ | , (4.2)and the transport cross section is σ tr , ⊥ = (2 π ) k ( m ⊥ / Φ ) ˜ F ( kh ) . (4.3)Here we define the functions˜ F n ( x ) = 4 π Z dt t n e − tx √ − t , (4.4)which can be expressed in terms of hypergeometric func-tions. As shown in Fig. 1, after reaching a maximumaround kh ≃ .
31, the transport cross section (4.3) de-creases with increasing energy and approaches zero for E → ∞ . Figure 1 also shows a polar graph for the dif-ferential cross section, dσ/dφ = | F ( φ ) | , at kh = 0 . e z axis is almost isotropic. E σ t r perp. η=π/2η=π/3η=π/4η=0 FIG. 1: (Color online) Born approximation results for thetransport cross section σ tr for scattering of massless Diracfermions on a magnetic dipole. We show σ tr in units of(2 π ) ( | m | / Φ ) vs energy E in units of ~ v F /h , where h isthe distance of the dipole from the layer. The dot-dashedblue curve is for the perpendicular orientation, see Eq. (4.3),the other curves are for the parallel orientation and severalangles η (cf. legend), see Eq. (4.8). Upper right part: Po-lar plot of the differential cross section, | F ( φ ) | , for E = 0 . B. Parallel orientation
If the dipole instead points parallel to the layer, m = m k [ − sin( η )ˆ e x + cos( η )ˆ e y ] , (4.5)where η denotes an angle, we haveΛ φ ( r, φ ) = m k h sin( φ − η ) (cid:18) r ( r + h ) / − r (cid:19) , (4.6)i.e., no radial symmetry is present. Using Eq. (3.6), wenow find the scattering amplitude F k ( φ ) = − e − iφ/ p (2 π ) k ( m k / Φ ) (4.7) × cos( η − φ/ e − kh | sin( φ/ | , and thus the transport cross section σ tr , k = (2 π ) k ( m k / Φ ) h ˜ F + ( ˜ F − F ) cos ( η ) i , (4.8)where ˜ F n = ˜ F n ( kh ). This cross section depends onthe orientation η of the dipole even for small energies, kh ≪
1. When averaging over η (which is equivalent tosetting η = π/ σ tr , k = ( m k /m ⊥ ) σ tr , ⊥ /
2. Thetransport cross section (4.8) has a maximum for an η -dependent energy, see Fig. 1, and again approaches zerofor E → ∞ . The differential cross section shown in Fig. 1also reveals a pronounced angular dependence tied to theorientation of the dipole. C. Bilayer graphene
Let us briefly comment on the results under a quadraticdispersion relation as realized in bilayer graphene. Re-peating the Born approximation analysis, the scatteringamplitude is found to contain an additional cos( φ/
2) fac-tor modifying the above expressions. In fact, we findinstead of Eqs. (4.3) and (4.8): σ (BLG)tr , ⊥ = (2 π ) k ( m ⊥ / Φ ) ( ˜ F − ˜ F ) ,σ (BLG)tr , k = (2 π ) k ( m k / Φ ) (4.9) × h ˜ F − ˜ F + ( ˜ F − F + 2 ˜ F ) cos ( η ) i , with ˜ F n = ˜ F n ( kh ). The quoted expressions hold for thequadratic dispersion relation of bilayer graphene, and co-incide with the results for the conventional Schr¨odingercase. In contrast to the monolayer results for σ tr , k inEq. (4.8), the transport cross section (4.9) carries no η -dependence at low energies. The latter is a distinctivefeature of 2D massless Dirac fermions. Finally, we notethat the σ tr , k results for η = 0 in Fig. 1 coincide with thebilayer result σ (BLG)tr , ⊥ (when m ⊥ = m k ) for perpendicularorientation. V. RING-SHAPED MAGNETIC FIELDS
In this section we consider the scattering states fora radially symmetric ring-shaped magnetic field. Thescattering setup is schematically sketched in the inset ofFig. 2.
A. Infinitesimally thin ring
Let us first study the exactly solvable model of aninfinitesimally thin ring of radius R around the origin,where Λ φ ( r ) follows from Eq. (3.10) with ϕ ( r ) = α Θ( r − R ) , (5.1)where Θ is the Heaviside step function and, as before, α is the dimensionless total flux through the ring surfacearea. For the orbital field case, this implies B z ( r ) =( α Φ / πR ) δ ( r − R ). With ρ = kr and R ≡ kR , thesolution to Eq. (3.12) is f m ( ρ ) = ( a m J m ( ρ ) , ρ < R ,b m (cid:16) e i ˜ δ m H (1) m + α ( ρ ) + H (2) m + α ( ρ ) (cid:17) , ρ > R , (5.2)with b m in Eq. (3.16). The requirement of continuity of ψ m ( r ) at r = R , together with Eq. (3.13), leads to twoboundary conditions for f m . With R ± ≡ R ± + , theyread f m (cid:0) R + (cid:1) = f m (cid:0) R − (cid:1) , f ′ m (cid:0) R + (cid:1) − f ′ m (cid:0) R − (cid:1) = α R f m ( R ) , (5.3) FIG. 2: (Color online) Transport cross section σ tr (in unitsof 2 /k ) vs dimensionless flux α for a finite-width magneticring, see Sec. V C, with kR = 0 .
01 and R = 2 R . (Wehere also allow for α < σ tr = (2 /k ) sin ( πα ). Inset: Schematic scattering geometry.The plane wave (blue solid arrows) coming in along the ˆ e x direction is scattered by a ring-shaped magnetic field presentfor R < r < R (shaded region). The outgoing sphericalwave is indicated by red dashed arrows. where again f ′ = ∂ ρ f . The coefficient a m and the phaseshift ˜ δ m appearing in Eq. (5.2) then follow from theboundary conditions (5.3). In particular, when J m ( R ) =0, the phase shift ˜ δ m can be determined by evaluation ofthe logarithmic derivative L m ≡ d ln f m ( ρ = R + ) dρ = α R + J ′ m ( R ) J m ( R ) (5.4)= m + α R − J m +1 ( R ) J m ( R ) , where we used the second boundary condition inEq. (5.3). As a result, with the Neumann function Y ν ,we find tan ˜ δ m = J ′ m + α ( R ) − L m J m + α ( R ) Y ′ m + α ( R ) − L m Y m + α ( R ) , (5.5)while a m is given by a m = b m e i ˜ δ m H (1) m + α ( R ) + H (2) m + α ( R ) J m ( R ) . (5.6)Equation (5.5) stays valid beyond the thin-ring limitwhen a more general form for L m is used, see Sec. V C.For the special case J m ( R ) = 0, Eq. (5.2) implies e i ˜ δ m = − H (2) m + α ( R ) /H (1) m + α ( R ) and, using f ′ m ( R + ) = f ′ m ( R − ), a m = b m e i ˜ δ m ∂ R H (1) m + α ( R ) + ∂ R H (2) m + α ( R ) J ′ m ( R ) . Equations (5.5) and (5.6) include these relations whentaking the limit J m ( R ) → L m → ∞ . B. Aharonov-Bohm scattering amplitude
Let us first consider the R → f m ( r ) in Eq. (3.14) as f m ( r ) = 2 b m e i ˜ δ m sin( πα ) h sin( πα − ˜ δ m ) J m + α ( kr )+ ( − ) m sin(˜ δ m ) J − ( m + α ) ( kr ) i . (5.7)Imposing regularity for f m ( r ) as r → δ m = − ( πα/ m + α ). Correspond-ingly, for R →
0, the scattering amplitude (3.17) is givenby F ( φ ) = − i √ πk "(cid:0) e − iπα − (cid:1) ∞ X m = − [ α ] e imφ (5.8)+ (cid:0) e iπα − (cid:1) − [ α ] − X m = −∞ e imφ , where α = [ α ]+ { α } , with integer part [ α ] and non-integerpart 0 ≤ { α } <
1. Summation of the series in Eq. (5.8)yields F ( φ ) = − i √ πk (cid:16) πδ ( φ )[cos( πα ) −
1] (5.9)+ e − i ([ α ]+1 / φ sin( πα )sin( φ/ (cid:17) . Up to the forward scattering ( φ = 0) amplitude, Eq. (5.9)reproduces the AB result, here obtained in termsof scattering phase shifts. Note that the forward scat-tering δ -term, missing in the AB calculation, naturallyappears in our phase shift analysis and is essential forestablishing unitarity of the scattering matrix. In alternative approaches to obtain F ( φ ) for the idealsolenoid, following the original AB method, the asymp-totics of the exact wavefunction is computed from itsintegral representation. As a result, the incident wavecorresponding to Eq. (3.1) has an additional phase fac-tor e − iπα sgn(sin φ ) e − iαφ , i.e., one has a multi-valued in-coming plane wave. The precise relation between thesetwo approaches has been discussed in several works andis still under debate, albeit the difference is of lit-tle relevance to experimentally observable quantities. Inparticular, the transport cross section σ tr in Eq. (3.3)does not depend on the forward scattering amplitude atall. We conclude that our approach is able to reproducethe AB effect, σ tr = (2 /k ) sin ( πα ), with oscillations asfunction of the dimensionless flux parameter α . In par-ticular, σ tr = 0 for integer α . C. Magnetic ring of finite width
Before discussing concrete results for the scatteringamplitude and the transport cross section in the pres-ence of a ring-shaped magnetic field, we now generalizethe setup to a finite width, with R < R denoting theinner and outer radii of the ring, cf. the inset of Fig. 2.Again, Λ φ ( r ) in Eq. (3.10) is expressed in terms of adimensionless flux function ϕ ( r ). When Λ φ is a vectorpotential, the associated magnetic field B z ( r ) = B istaken uniform within the ring region and zero outside;for concreteness, we take B ≥
0. This profile allows foran exact solution, while more general smooth field pro-files can be treated within the Wentzel-Kramers-Brillouin(WKB) approximation, see Sec. V D.We use dimensionless coordinates ( ρ = kr and R , = kR , ) and flux parameters, ν , = πBR , Φ , ν ≡ ν R = ν R , α = ν − ν . (5.10)The function ϕ then reads with r = ρ/k : ϕ ( r ) = , ρ < R ,νρ − ν , R < ρ < R ,α, ρ > R . (5.11)For R → R , this reduces to Eq. (5.1). In particular, α in Eq. (5.11) again denotes the total dimensionless flux.For given j = m + 1 /
2, the components of the Diracspinor ψ m obey Eqs. (3.12) and (3.13), with Eq. (5.11)now determining W m ( ρ ). The solutions for r < R and r > R are as in Eq. (5.2), f m ( ρ ) = ( a m J m ( ρ ) , ρ < R ,b m (cid:16) e i ˜ δ m H (1) m + α ( ρ ) + H (2) m + α ( ρ ) (cid:17) , ρ > R , (5.12)where a m and ˜ δ m are to be determined, and b m is givenin Eq. (3.16). For R < r < R , Eq. (3.12) can be solvedin terms of the confluent hypergeometric functions Φ andΨ, f m ( ρ ) = ρ | ˜ m | e − νρ / h c m Φ( ξ m , | ˜ m | ; νρ )+ d m Ψ( ξ m , | ˜ m | ; νρ ) i , (5.13) ξ m ≡ m Θ( ˜ m ) − / ν, ˜ m ≡ m − ν . The coefficients c m and d m , together with a m and thephase shift ˜ δ m in Eq. (5.12), follow by matching ψ m at r = R i =1 , . Taking into account that W m is a continuousfunction of ρ , we have f m (cid:0) R + i (cid:1) = f m (cid:0) R − i (cid:1) , f ′ m (cid:0) R + i (cid:1) = f ′ m (cid:0) R − i (cid:1) , (5.14)where the second condition follows by continuity of thelower spinor component g m . It is convenient to introducethe transfer matrix ˆ T m connecting the solutions at ρ = -8 -6 -4 -2 0 2 4 6 8 α σ t r ρ j φ FIG. 3: (Color online) The main panel is as in Fig. 2 but for kR = 1 .
32 and R = 5 R . Inset: Current density j φ ( ρ ) =ˆ e φ · j vs radial coordinate ρ for the quasi-bound state with j = 3 / α ≃ R +1 and R − , (cid:18) f m ( R − ) f ′ m ( R − ) (cid:19) = ˆ T m (cid:18) f m ( R +1 ) f ′ m ( R +1 ) (cid:19) = a m ˆ T m (cid:18) J m ( R ) J ′ m ( R ) (cid:19) . (5.15)Explicitly, the transfer matrix for the magnetic ring offinite width isˆ T m = (cid:18) Φ Ψ Φ ′ Ψ ′ (cid:19) (cid:18) Φ Ψ Φ ′ Ψ ′ (cid:19) − , (5.16)where we use the abbreviationΦ i =1 , ≡ R | ˜ m | i e − ν i / Φ( ξ m , | ˜ m | ; ν i ) , and similarly for Ψ i . We mention in passing that forthe infinitesimally thin magnetic ring in Sec. V A (where R = R = R ), the transfer matrix is ˆ T m = (cid:18) α/ R (cid:19) .For the finite-width ring, using Eq. (5.12) the phaseshift ˜ δ m is then again given by Eq. (5.5), with R → R and the logarithmic derivative L m replaced by L m = u m, u m, , (cid:18) u m, u m, (cid:19) = ˆ T m (cid:18) J m ( R ) J ′ m ( R ) (cid:19) . (5.17)With the above expressions, it is straightforward tocompute the scattering phases δ m = ˜ δ m − πα/ the scattering amplitude F ( φ ) from Eq. (3.17)and the transport cross section σ tr from Eq. (3.19).Numerical results obtained under this approach areshown in Figs. 2 and 3. First, in the main panel of Fig. 2,we show the transport cross section σ tr as a function ofthe total flux α . In this example, both radii R and R were chosen very small, such that scattering by thering is close to the one by an ideal AB solenoid. As aconsequence, we observe the AB oscillations with unitflux period. In contrast to the ideal AB result, a com-plete suppression of scattering for α ∈ Z is observed inthe finite-width ring only for α = 0, while the maximumvalue σ tr = 2 /k for half-integer α is still perfectly real-ized. In fact, the phase shift analysis in Sec. V B showsthat a given oscillation period is determined by one spe-cific m value in the ideal AB case. For the non-idealfinite-width ring, other total angular momenta also startto contribute, and this mixing effect destroy the perfectconstructive interference needed for σ tr = 0. On theother hand, the destructive interference responsible forthe maxima of σ tr at half-integer α is more robust sinceit is dominated by a single m value.In Fig. 3, we study scattering by a much larger ring.In this case, the AB effect is absent, which can be under-stood by noting that the Fermi wavelength (2 π/k ) of theparticle is now smaller than the outer circumference 2 πR of the ring. Quantum interference of waves surroundingthe obstacle in opposite directions is then largely aver-aged out, and, moreover, the wavefunction can partiallypenetrate into the ring area. However, a remarkable peakfeature at α ≈ j = 3 / j r = 0, we find a circularly oriented current, j φ = 0,which is mainly localized inside the ring ( r < R ) andrepresents a current-carrying bound state. We note thatmore quasi-bound states appear for larger α , causing ad-ditional peak features in σ tr ( α ) beyond those shown inFig. 3. D. Quasi-bound states and scattering resonances
The magnetic confinement built up by the ring-shapedfield can generate quasi-bound states, which for R → ∞ become true bound states. The quasi-bound statespectrum then causes resonances in the scattering am-plitude when the energy E = ~ v F k is varied. For giventotal angular momentum j = m + 1 /
2, the correspond-ing phase shift δ m ( E ) goes through the value π/ E crosses a resonance level E r . The corresponding reso-nance width Γ r can be estimated from ( d/dE ) cot [ δ m ( E = E r )] = − / Γ r . To access these resonances, we first put Eq. (3.12) into acanonical form with separated kinetic and potential en-ergy terms. The substitution f m ( ρ ) = ρ − / ˜ f m ( ρ ) yields (cid:2) − ∂ ρ + V m ( ρ ) (cid:3) ˜ f m = ˜ f m , V m ≡ W m + W ′ m , (5.18)where V m ( ρ ) is an effective potential energy for the ra-dial motion and the lower spinor component is g m = ρ − / ( − ∂ ρ + W m ) ˜ f m . In this form, Eq. (5.18) can be E s i n δ m R E FIG. 4: (Color online) Partial cross section sin δ m vs energy E for a ring-shaped confinement as in Sec. V C. The numericalresults are for total angular momentum states with m = 1(solid black) and m = − R = 7 R and R = 0 . ℓ B with ℓ B = p c/eB , andenergies are in units of ~ v F /ℓ B . Inset: WKB results for quasi-bound state energies E r vs R (lengths in units of ℓ B ), for m = 1 (black circles) and m = − R = 0 . ℓ B . For comparison, the exact levels for infinite R from Ref. 33 are shown for m = 1 (dotted black) and m = − treated within the standard WKB approach, which repre-sents an attractive alternative to semiclassical approachesto the Dirac equation as it avoids the appearance of non-Abelian Berry phases. For a magnetic ring as inSec. V C, the effective potential V m has a hard repulsivecore for r → r < r < r . The “turning points” r , here dependon the energy E = ~ v F k under consideration. For fi-nite R , this barrier is of finite width and quasi-boundstates within the well region may exist. The classicallyforbidden region r < r < r (where r is another turningpoint) then corresponds to tunneling trajectories wherethe “particle” escapes from the well region. For R → ∞ ,the barrier becomes infinitely wide and this escape prob-ability vanishes, i.e., we obtain true bound states in thewell region. Using the radial variable r = ρ/k , Eq. (5.18)reads (cid:2) − ∂ r + U m ( r ) (cid:3) ˜ f m ( r ) = ǫ ˜ f m ( r ) , (5.19) U m ( r ) = w m + ∂ r w m , w m ( r ) = kW m ( kr ) , complex-valued “energy” ǫ ≡ k is Z r r dr p ǫ − U m ( r ) = π (cid:18) n + 12 − χ ( a )2 π (cid:19) , (5.20) χ ( a ) = 12 i ln (cid:18) Γ( ia + 1 / − ia + 1 /
2) [1 + e − πa ] (cid:19) + a (1 − ln a ) ,a = 1 π Z r r dr p U m ( r ) − ǫ, with n = 0 , , , . . . and the Gamma function Γ( z ). Thecomplex resonance values for ǫ solving Eq. (5.20) canbe found numerically. Equation (5.20) is formally exactfor the case of a parabolic barrier, but also applies foran arbitrary smooth potential and is expected to remainaccurate even for small n . We now write ǫ = ( k − iγ/ ≈ k − ikγ. For a quasi-bound level with energy E r = ~ v F k , the reso-nance width is then Γ r = ~ v F γ . Using Im χ ( a ) ≈ e − πa / a &
1, we obtainΓ r / ~ = T − k e − πa , (5.21)where the period of radial motion is T k = 2 kv F Z r r dr p k − U m ( r ) . Our numerical results for the partial cross section,sin δ m , as a function of energy, and the WKB resultsfor the corresponding quasi-bound state energies E r areshown in Fig. 4. Here we take the field profile as inSec. V C. With increasing R (keeping R fixed), newquasi-bound energy levels localized in the well region ap-pear, see inset of Fig. 4. Very good agreement with ex-act quantum calculations for the infinite barrier case( R → ∞ ) is observed, i.e., these energy levels remain ba-sically unchanged when increasing R . The only notice-able deviation from the exact spectrum of Ref. 32 is seenfor m = 0, where the potential U m =0 ( r ) creates an in-finitely attractive well for r →
0. In that case, the WKBapproximation becomes questionable in that “steep” re-gion. The main panel in Fig. 4 illustrates the sequenceof quasi-bound states present because of the magneticconfinement. The corresponding scattering resonancesappear as peaks in the transport cross section σ tr whenvarying energy or the effective flux parameter α . VI. CONCLUDING REMARKS
In this paper, we have studied scattering of masslesstwo-dimensional Dirac fermions by magnetic perturba-tions of various types. The model is applicable to quan-tum transport in monolayer graphene and for the sur-face state of strong topological insulators. The magneticfields can correspond to orbital or Zeeman fields, strain-induced fields in graphene, or exchange fields generatedby ferromagnets.The full scattering solution was discussed in detail forradially symmetric perturbations, where the scatteringamplitude can be expressed in terms of phase shifts in agiven total angular momentum channel, and within theBorn approximation for the general case. Our approachnow allows for a systematic study of the scattering ofDirac fermions on magnetostatic perturbations.As applications, we have studied scattering by mag-netic dipoles within the Born approximation, and fullynonperturbative scattering for the case of ring-shapedmagnetic fields. The Born approximation is only validwhen the perturbation has zero total flux ( α = 0). Forthe magnetic dipole, we have pointed out characteris-tic angular dependencies in the differential cross sectionthat may allow to unambiguously identify massless Diracfermions. For the ring-shaped field case, as one increasesthe lateral size ( R ) of the magnetic perturbation, wehave a crossover from the Aharonov-Bohm case to aregime dominated by scattering resonances. In the firstcase, R →
0, particle trajectories surround the flux re-gion but essentially do not penetrate it, leading to theoscillatory transport cross section σ tr ∝ sin ( πα ). Inthe second case, where the particle wavelength is smallagainst the size of the perturbation, kR >
1, the ABoscillations in σ tr ( α ) are absent. However, now quasi-bound states arise due to the magnetic confinement,causing scattering resonances which show up as peaksin σ tr ( α ).To conclude, we hope that these predictions motivatefurther theoretical work and that they will be tested ex-perimentally in the near future. Acknowledgments
We thank A. De Martino for discussions and acknowl-edge financial support by the DFG Schwerpunktpro-gramm 1459. M.Z. Hasan and C.L. Kane, arXiv:1002.3895. X.-L. Qi and S.-C. Zhang, Physics Today , 33 (2010). Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A.Bansil, D. Grauer, Y.S. Hor, R.J. Cava, and M.Z. Hasan, Nature Physics , 398 (2009). A.K. Geim and K.S. Novoselov, Nat. Mat. , 183 (2007). A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S.Novoselov, and A. Geim, Rev. Mod. Phys. , 109 (2009). V.P. Gusynin, S.G. Shaparov, and J.P. Carbotte, Int. J.Mod. Phys. B , 4611 (2007). X. Du, I. Skachko, A. Barker, and E.Y. Andrei, NatureNanotech. , 491 (2008). K.I. Bolotin, K.J. Sikes, J. Hone, H.L. Stormer, and P.Kim, Phys. Rev. Lett. , 096802 (2008). N. Stander, B. Huard, and D. Goldhaber-Gordon, Phys.Rev. Lett. , 026807 (2009). A.F. Young and P. Kim, Nature Physics , 222 (2009). A. Altland and B.D. Simons,
Condensed Matter Field The-ory , 2nd edition (Cambridge University Press, 2010). D.S. Novikov, Phys. Rev. B , 245435 (2007). See, for instance, O.V. Gamayun, E.V. Gorbar, and V.P.Gusynin, Phys. Rev. B , 165429 (2009), and referencestherein. F.M. Peeters and A. Matulis, Phys. Rev. B , 15166(1993). A.M. Matulis, F.M. Peeters, and P. Vasilopoulos, Phys.Rev. Lett. , 1518 (1994). I.S. Ibrahim and F.M. Peeters, Phys. Rev. B , 17321(1995). S.J. Lee, S. Souma, G. Ihm, and K.J. Chang, Phys. Rep. , 1 (2004). For a review, see: A. Nogaret, J. Phys.: Cond. Matt. ,253201 (2010). A. Tarasov, S. Hugger, H. Xu, M. Cerchez, T. Heinzel, I.V.Zozoulenko, U. Gasser-Szerer, D. Reuter, and A.D. Wieck,Phys. Rev. Lett. , 186801 (2010). S.J. Bending, K. von Klitzing, and K. Ploog, Phys. Rev.Lett. , 1060 (1990). A.K. Geim, S.J. Bending, and I.V. Grigorieva, Phys. Rev.Lett. , 2252 (1992). S. Mondal, D. Sen, K. Sengupta, and R. Shankar, Phys.Rev. Lett. , 046403 (2010); Phys. Rev. B , 045120(2010). T. Yokoyama, Y. Tanaka, and N. Nagaosa, Phys. Rev. B , 121401(R) (2010). Q. Liu, C.-X. Liu, C. Xu, X.-L. Qi, and S.-C. Zhang, Phys.Rev. Lett. , 156603 (2009). R.R. Biswas and A.V. Balatsky, Phys. Rev. B , 233405(2010). A.F. Morpurgo and F. Guinea, Phys. Rev. Lett. , 196804(2006). M.M. Fogler, F. Guinea, and M.I. Katsnelson, Phys. Rev.Lett. , 226804 (2008). V.M. Pereira and A.H. Castro Neto, Phys. Rev. Lett. ,046801 (2009). W. Bao, F. Miao, Z. Chen, H. Zhang, W. Hang, C. Dames,and C.N. Lau, Nature Nanotech. , 562 (2009). F. Guinea, M.I. Katsnelson, and A.K. Geim, Nature Phys. , 30 (2010). M.A.H. Vozmediano, M.I. Katsnelson, and F. Guinea,Physics Reports, in press (2010); see arXiv:1003.5179v2. A. De Martino, L. Dell’Anna, and R. Egger, Phys. Rev.Lett. , 066802 (2007). A. De Martino and R. Egger, Semicond. Sci. Technol. ,034006 (2010). M. Ramezani Masir, P. Vasilopoulos, A. Matulis, and F.M. Peeters, Phys. Rev. B , 235443 (2008). L. Oroszl´any, P.K. Rakyta, A. Korm´anyos, C.J. Lambert,and J. Cserti, Phys. Rev. B , 081403(R) (2008). T.K. Ghosh, A. De Martino, W. H¨ausler, L. Dell’Anna,and R. Egger, Phys. Rev. B , 081404(R) (2008). W. H¨ausler, A. De Martino, T.K. Ghosh, and R. Egger,Phys. Rev. B , 165402 (2008). M. Tahir and K. Sabeeh, Phys. Rev. B , 195421 (2008). L. Dell’Anna and A. De Martino, Phys. Rev. B , 045420(2009). M. Ramezani Masir, P. Vasilopoulos, and F.M. Peeters,New. J. Phys. , 095009 (2009). L.Z. Tan, C.-H. Park, and S.G. Louie, Phys. Rev. B ,195426 (2010). A. Korm´anyos, P. Rakyta, L. Oroszl´any, and J. Cserti,Phys. Rev. B , 045430 (2008). M. Ramezani Masir, A. Matulis and F.M. Peeters, Phys.Rev. B , 155451 (2009). W. H¨ausler and R. Egger, Phys. Rev. B , 161402(R)(2009). Y. Aharonov and D. Bohm, Phys. Rev. , 485 (1959). S. Olariu and I.I. Popescu, Rev. Mod. Phys. , 339 (1985). S.N.M. Ruijsenaars, Ann. Phys. (N.Y.) , 1 (1983). C.R. Hagen, Phys. Rev. Lett. , 503 (1990). C.R. Hagen, Phys. Rev. D , 2015 (1990). C.R. Hagen, Phys. Rev. D , 2466 (1995). P. Giacconi, F. Maltoni, and R. Soldati, Phys. Rev. D ,952 (1996). S. Sakoda and M. Omote, J. Math. Phys. , 716 (1997). R. Jackiw, A.I. Milstein, S.Y. Pi, and I.S. Terekhov, Phys.Rev. B , 033413 (2009). M.I. Katsnelson, EPL , 17001 (2010). P. Recher, B. Trauzettel, A. Rycerz, Ya.M. Blanter, C.W.J.Beenakker, and A.F. Morpurgo, Phys. Rev. B , 235404(2007). S. Russo, J.B. Oostinga, D. Wehenkel, H.B. Heersche, S.S.Sobhani, L.M.K. Vandersypen, and A.F. Morpurgo, Phys.Rev B , 085413 (2008). T. Fujita, M.B.A. Jalil, and S.G. Tan, arXiv:1005.5088. R.G. Newton,
Scattering theory of waves and particles , 2ndedition (Springer, New York, 1982). A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski,
Methods of quantum field theory in statistical physics (Prentice Hall, Inc., Eaglewood Cliffs, New Jersey, 1963). I.S. Gradshteyn and I.M. Ryzhik,
Table of Integrals, Seriesand Products (Academic Press, New York, 1980). In the numerical evaluation, we sum over all angular mo-mentum states | j | <
40 (with j = m + 1 / | j | are difficult to compute reliablyyet average out in practice. J. Bolte and S. Keppeler, Ann. Phys. (N.Y.) , 125(1999). P. Carmier and D. Ullmo, Phys. Rev. B , 245413 (2008). V.D. Mur and V.S. Popov, JETP Lett. , 563 (1990).65