Anomalous spatial shifts in interface electronic scattering
AAnomalous spatial shifts in interface electronic scattering
Zhi-Ming Yu, Ying Liu, and Shengyuan A. Yang Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore
The anomalous spatial shifts at interface scattering, first studied in geometric optics, recentlyfound their counterparts in the electronic context. It was shown that both longitudinal and trans-verse shifts, analogous to the Goos-H¨anchen and Imbert-Fedorov effects in optics, can exist whenelectrons are scattered at a junction interface. More interestingly, the shifts are also discovered inthe process of Andreev reflection at a normal/superconductor interface. Particularly, for the casewith unconventional superconductors, it was discovered that the transverse shift can arise solelyfrom the superconducting pair potential and exhibit characteristic features depending on the pair-ing. Here, we briefly review the recent works in this field, with an emphasis on the physical pictureand theoretical understanding.
I. INTRODUCTION
The analogy between electronics and optics has in-spired many breakthroughs in both fields. One commonphenomenon for both is the presence of scattering whenelectrons/photons hit an interface. As a fundamentalphysical process, such interface scattering provides anessential mechanism for modulating the propagation ofelectrons/photons, which in turn constitutes the founda-tion for the electronic/optical device design.Probably the first knowledge we learn about interfacescattering is on the reflection of a light beam at a flat optical interface (i.e., an interface between different op-tical media). The effect is summarized by the laws ofreflection and mathematically described by the Fresnelequations. Specifically, the laws state that: (i) The inci-dent beam, the interface normal, and the reflected beamlie in the same plane; (ii) the incident beam and the re-flected beam are on opposite sides of the normal (i.e.,the reflection is specular), and they make the same an-gle with the normal. In addition, for a sharp interface,it is tacitly understood that the incident beam and thereflected beam meet at the same point on the interface.These are illustrated in Fig. 1(a).These laws have been known since ancient Greek times.However, later studies showed that the they need revisionin certain cases. In a work published in 1947 [1], Goosand H¨anchen pointed out that when the light beam un-dergoes a total reflection, the incident and the reflectedbeams may not meet at the same point on the inter-face, rather, there generally exists a longitudinal spa-tial shift between them within the plane of incidence[see Fig. 1(b)]. This shift, known as the Goos-H¨anchenshift, has been studied and verified in many differentcontexts, and is established as a powerful technique toprobe interface properties in optics, acoustics, and atomicphysics [2].More interestingly, the works by Fedorov (1955) andImbert (1972) challenged the statement (i) in the laws ofreflection [3, 4]. They found that although the law mayhold well for a non-polarized light beam, for a circularlypolarized light, however, the plane of reflection (definedby the reflected beam and the normal) may be different optical interfaceGH shift IF shift incident light beamoptical interface (a) (b) reflected light beam
FIG. 1. Illustrations showing (a) the usual picture of reflec-tion for a light beam at an optical interface, obeying the lawsof reflection; and (b) in certain cases, the reflected light beamacquires an anomalous spatial shift, including the longitudinal(GH effect) and the transverse (IF effect) components. from the plane of incidence, i.e., there exists a transverse spatial shift for the reflection [see Fig. 1(b)]. This shiftis known as the Imbert-Fedorov shift.The Goos-H¨anchen shift and the Imbert-Fedorov shiftwere initially derived based on the Maxwell equations,which are the classical description of electromagneticwaves. Hence, the two effects can be regarded as gen-eral wave phenomena. In 2004, Onoda et al. offered newinsight into these effects [5], by developing semiclassicalequations of motion for the light wave packet analogousto the Chang-Sundaram-Niu equations for electrons [6–8]. The Imbert-Fedorov shift was reproduced from theequations of motion for an interface scattering process.At that time, the electron spin Hall effect was a hottopic [9, 10], hence the opposite Imbert-Fedorov shiftsfor opposite circular polarizations was interpreted as anoptical spin Hall effect [5]. This rekindled interest inthese optical effects in recent years [11], and the advancein optical measurement technique enabled quantitativecomparisons between theory and experiment [12, 13].Since electrons also exhibit wave behavior in propaga-tion, as described by the Schr¨odinger equation, one nat-urally wonders whether analogies of the optical anoma-lous shifts exist for electronic interface scattering. Infact, proposals of the longitudinal (Goos-H¨anchen like)shift for electrons appeared quite early, at least since the1970s [14–16]. Since the early 2000s, the longitudinal a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n shift was actively explored in the two-dimensional (2D)electron gas with spin-orbit coupling (SOC) due to thesurge of research interest in spintronics [17–19], and thenin graphene and 2D material heterostructures [20–25].The research is strongly motivated by the rapid progressof the experimental techniques in fabricating high-qualityjunctions and in manipulating electrons in these micro-structures [26]. There emerged a field of electron optics,which targets at the accurate control of electron propa-gation as in optics [27–29].On the other hand, the electronic analog of the Imbert-Fedorov shift was revealed only recently. In 2015, Jiang et al. [30] and Yang et al. [31], via different approaches,predicted the presence of the transverse shift for electronsin a special type of 3D materials—the Weyl semimetals.They showed that the Berry curvatures play a key role inthe effect, and the shift leads to a chirality Hall effect in aWeyl semimetal junction [31]. Later, the effect was usedto explain the high mobility observed for Weyl semimet-als, and was extended to the closely related multi-Weylsemimetals [32]. The possible effect of topological Fermiarcs of a Weyl semimetal on the shift was investigatedrecently [33].In all the above-mentioned effects, the incident andthe scattered beams are of the same kind of particles,i.e., an electron is scattered as an electron, and a pho-ton is scattered as a photon. Yet there exists a spe-cial kind of scattering process occurring at a normal-metal/superconductor (NS) interface—the Andreev re-flection [34, 35], in which an incident electron is reflectedback as a hole. Here, the incident and the scattered par-ticles are of different identity, and even the electric chargeis changed in scattering. One also notes that the Andreevreflection may also violate the statement (ii) in the lawsof reflection: the reflected beam and the incident beamare typically on the same side of the normal, i.e., it is aretroreflection. With these nontrivial features, it is thusintriguing to ask whether the anomalous shifts also hap-pen in Andreev reflections. This question is answered inthe affirmative by Liu, Yu, and Yang in 2017 [36]. Theyshowed that the effect generally exists for a heterojunc-tion consisting of 3D metal with SOC and a conventional s -wave superconductor. In subsequent works, Liu et al. [37] analyzed in detail the longitudinal shift for such asystem. Remarkably, Yu et al. [38] showed that a siz-able transverse shift could be induced solely by uncon-ventional pairings on the superconductor side, and ex-hibits characteristic behaviors corresponding to the sym-metry of the pair potential. Most recently, the transverseshift was also proposed for the crossed Andreev reflectionprocess [39], a nonlocal scattering process for which theincident electron and the scattered hole are at differentnormal metal terminals connected to a single piece ofsuperconductor [40, 41].In this paper, we provide a brief review on these re-cent advancements in the study of the anomalous shiftsin electronic systems. We shall be focusing on the trans-verse shift (although the longitudinal shift in Andreev re- zyx θ R θ I θ T Ψ T Ψ I Ψ R H R H L FIG. 2. Schematic figure showing the basic setup for investi-gating the anomalous spatial shifts in interface scattering. flection will be included). As for the studies on the longi-tudinal shift in electronic systems, we refer the readers tothe previous review paper Ref. [42]. Excellent reviews onthe shifts in the optical context can be found in Ref. [43].
II. BASIC SETUP
We first discuss the basic setup for investigating theanomalous spatial shifts in interface scattering. Certainlywe need an interface between two different media, wherethe electronic scattering occurs. The interface is assumedto be flat and clean, and without loss of generality, weassume it is located at z = 0, as illustrated in Fig. 2.The whole system is assumed to be extended along x and y directions, which amounts to saying that the systemdimension in these two directions is much larger thanthe particle wavelength (and also the anomalous shift).The two media on the two sides of the interface are de-scribed by two model Hamiltonians H L and H R , respec-tively. An interfacial barrier may be modeled by addingterms such as hδ ( z ) to the model. The junction is as-sumed to be clean (in other words, the system dimensionis assumed to be within a mean free path), such thatan incident particle does not experience other (disorder)scattering except the scattering at the interface. Due tothe translational symmetry in x and y , the transversewave vector k (cid:107) = ( k x , k y ) for the incident particle will bea conserved quantity during scattering.For systems with a rotational symmetry along z , with-out loss of generality, one can assume the plane of inci-dence is the x - z plane (see Fig. 2). Then the longitudinalshift is in the x direction, and the transverse shift is inthe y direction. However, if the system does not possessthe rotational symmetry, the result will depend on theorientation of the plane of incidence. This is the case forthe systems with anisotropic pair potentials, where oneneeds to specify a rotation angle α for the incident planewith respect to the crystal axis (see Fig. 8).The anomalous positional shifts are defined for later-ally confined particle beams. They cannot be defined zx Graded Interface Sharp Interface (a) (b) zx FIG. 3. Schematic figure showing the scattering at (a) agraded interface and (b) a sharp interface. for the unconfined plane wave states. Hence, one typi-cally assumes an incident particle beam Ψ I coming fromthe left media, computes the reflected beam Ψ R and thetransmitted beam Ψ T , and compares their center posi-tions at the interface to obtain the shifts in reflection andin transmission.The setup described here can be easily extended toinclude more than one interfaces. For example, in thestudy of crossed Andreev reflection, one considers a nor-mal/superconductor/normal (NSN) sandwich structurewith two interfaces (see Fig. 11).Before proceeding, we have a few remarks regardingthe comparison between the longitudinal shift and thetransverse shift. First, the longitudinal shift allows aquite intuitive understanding. Consider a graded inter-face, where the left medium H L is smoothly interpolatedto the right medium H R . Assuming the right mediumdoes not support a propagating mode for the particle(e.g., it may have a spectral gap at the given particleenergy), then the beam will be adiabatically reflectedback, and its trajectory is schematically illustrated inFig. 3(a). The longitudinal shift can be regarded asdue to the bending of the trajectory in the graded in-terface region. When the width of the graded interfaceregion approaches zero, the interface becomes a sharpone. Then the lateral propagation along the interfaceis enabled through the evanescent modes at the inter-face which decays into the right medium, as illustratedin Fig. 3(b). In comparison, the transverse shift does nothave such a simple classical picture, hence appears to bemore nontrivial.Second, due to the above discussion, the longitudinalshift is not well defined for a graded interface, because itsvalue depends on the location taken in the graded region[see Fig. 3(a)]. The longitudinal shift is only well definedfor a sharp interface. In salient contrast, the transverseshift is well defined for both cases, since it is the differencebetween the incident plane and the scattered plane.Third, since the longitudinal shift is within the planeof incidence, it can be studied by simply taking a 2D sub-system (without considering the y dimension in Fig. 2).In contrast, the study of the transverse shift must requirea 3D system, i.e., it is a genuine 3D phenomenon.The first and the third points above might offer a possi-ble explanation for why the transverse shift was revealedmuch later than the longitudinal shift in electronic sys- tems. III. APPROACHES
In previous works on the anomalous shifts, three differ-ent approaches have been adopted. Each approach hasits own applicability, advantage, and limitation. In thefollowing, we review these three approaches.
A. Quantum scattering approach
The standard and the most general approach is thescattering approach. This is also the traditional approachadopted for studying the anomalous shifts in geometricoptics [43].In this approach, one directly solves the scattered par-ticle beam from the incident beam Ψ I . The calculationis facilitated by expanding the beam wave function usingthe eigenmodes of the system, i.e., the scattering basisstates ψ . Explicitly, one can write Ψ I = (cid:90) d k (cid:48) w ( k (cid:48) − k ) ψ I k (cid:48) ( r ) , (1)where the incident basis state ψ I (the partial wave) is la-beled by its wave vector k (cid:48) . Since we require the beam tobe laterally confined, due to the uncertainty principle, itmust consist of a spread of basis states, which is describedby the beam profile w . To have a well defined trajectory,one also needs the profile w to be peaked around an aver-age wave vector k . The specific form of w does not affectthe final result of the shifts. In calculations, one usuallychooses w to have a Gaussian form: w ( q ) = (cid:89) i w i ( q i ) , (2)where w i ( q i ) = ( √ πW i ) − e − q i / (2 W i ) , (3)and W i is the width for the i -th component.When the incident beam hits the interface, it will bescattered. For a concrete discussion, let’s assume thatthere are a single channel for reflection and a single chan-nel for transmission. (The generalization to cases withmultiple channels is straightforward.) Then there will bea reflected beam Ψ R and a transmitted beam Ψ T .How do we find the reflected beam Ψ R ? By using theexpansion in Eq. (1), we only need to know how eachpartial wave ψ I k (cid:48) is reflected at the interface, which pre-sumably is already known when we obtain the scatteringbasis states at the first place. Recall that a scatteringbasis state for the system takes the form of ψ k = (cid:40) ψ I k + r ( k ) ψ R k , z < ,t ( k ) ψ T k , z > . (4)This means that the incident partial wave ψ I is reflected(transmitted) as ψ R ( ψ T ) with an amplitude r ( t ). Con-sequently, the reflected beam can be obtained as Ψ R = (cid:90) d k (cid:48) w ( k (cid:48) − k ) r ( k (cid:48) ) ψ R k (cid:48) ( r ) . (5)The anomalous spatial shift (including both longitu-dinal and transverse components) can then be obtainedby comparing the center positions of Ψ R and Ψ I at theinterface. For example, if taking H L = (cid:126) k / (2 m ) tobe the simple 3D isotropic electron gas model, for theconfiguration in Fig. 2, by expanding the phase of theamplitude r to the first order around k y , one can findthat Ψ R ∝ e − W y (cid:2) y + ∂∂k (cid:48) y arg( r ) (cid:12)(cid:12) k (cid:107) (cid:3) / . (6)Compared with the incident beam Ψ I ∝ e − W y y / , onefinds that the reflected beam acquires a transverse shift δy R = − ∂∂k (cid:48) y arg( r ) (cid:12)(cid:12)(cid:12) k (cid:107) , (7)in the y direction. Similarly, the longitudinal shift can beobtained as δx R = − ∂∂k (cid:48) x arg( r ) (cid:12)(cid:12) k (cid:107) for this simple model.The expressions for the shifts [like Eq. (7)] depend onthe model, and can become more complicated when thequantum state has some internal spin/pseudospin degreeof freedom [20]. Following similar analysis, one can alsofind the shifts for the transmitted beam Ψ T .This approach is based on the analysis of the very fun-damental quantum scattering problem. It is quite gen-eral. Unlike the semiclassical approach to be reviewedin Sec. III B which requires the scattering potential tobe slowly varying over the particle wavelength [31], thequantum scattering approach here does not suffer fromthis constraint. There is no semiclassical approxima-tion involved. Particularly, it applies for sharp inter-faces and for cases when the particle wavelength is rela-tively large (like for doped semiconductors or semimetals[20, 30, 31]).In practice, it is easier to deal with sharp interfaces,when using the scattering approach, because the modematching at the interface needed to obtain the scatteringamplitudes can be done straightforwardly. For gradedinterfaces, one has to resort to techniques such as thetransfer matrix method to carry out the calculation.Finally, we mention that the expression in Eq. (7) isquite suggestive. It shows that the shift is connected tothe variation of the phase angle of the scattering ampli-tude versus the wave vector. In other words, a finite spa-tial shift can result from the different phase shifts for thedifferent partial waves in scattering. In addition, whenthe waves have spin/pseudospin degree of freedom, therecould be additional contributions from such internal de-gree of freedom. B. Semiclassical approach
The second approach is through the application of thesemiclassical theory. The semiclassical theory aims todescribe the dynamics of a quantum particle using a set ofequations of motion analogous to the Hamilton equationsin classical mechanics [44].One may ask: the uncertainty principle says that youcannot have both position and momentum well definedat the same time, then how can you write down equa-tions to describe their dynamics? Well, indeed, that istrue. When the momentum is precisely defined, like fora plane wave, the position is completely undetermined,and vice versa. To obtain a semiclassical description, weneed to make a compromise for both variables, i.e., welet each variable carry certain “acceptable” uncertainty,such that together they could satisfy the fundamentallimit posed by the uncertainty principle. This means thatwe are studying the dynamics of a particle wave packetΨ( r c , k c ). By “acceptable”, the wave packet spread isrequired to be sufficiently narrow in both position andmomentum spaces, such that its center ( r c , k c ) in phasespace can be defined.The validity of the semiclassical description requiresthe external perturbations to be smooth and slowly vary-ing in space, such that the wave packet can be viewed asa point particle. A guideline is that the length scale forthe perturbation must be much greater than the particlewavelength. Thus, regarding our current problem, thismeans that the semiclassical approach can only apply forgraded interfaces, but not for sharp interfaces.There exists a systematic way to derive the semiclassi-cal equations of motion for ( r c , k c ) from the Schr¨odingerequation for the system. Sundaram and Niu [8] showedthat the equations for a Bloch wave packet in a singleband take the general form of (setting (cid:126) = 1)˙ r c = ∂ E ∂ k c − (Ω kr · ˙ r c + Ω kk · ˙ k c ) − Ω k t , (8)˙ k c = − ∂ E ∂ r c + (Ω rr · ˙ r c + Ω rk · ˙ k c ) + Ω r t . (9)Here, E ( r c , k c ) is the energy of the wave packet, and theΩ’s are the various Berry curvatures defined in termsof the gauge potentials A known as Berry connections.For example, A q i = i (cid:104) u | ∂ q i u (cid:105) ( q = r , k ), where | u (cid:105) isthe periodic part of the Bloch state. Note that the r c dependence of | u (cid:105) comes from the dependence ofthe Hamiltonian on certain spatially varying parame-ters. For simple notations, here and hereafter, we dropthe subscript c from r c and k c whenever appropriate.Then the phase space Berry curvatures are defined asΩ k i r j = ∂ k i A r j − ∂ r j A k i (Ω kk , Ω rr , and Ω rk are simi-larly defined). Ω r i t = ∂ r i A t − ∂ t A r i arises due to certaintime dependent parameters in the Hamiltonian, whichmay lead to pumping effects [45].For the particular case with weak electric field E andmagnetic field B . The equations of motion reduce to theform derived by Chang and Niu [6, 7] (setting e = 1):˙ r = ∂ E ∂ k − ˙ k × Ω , (10)˙ k = − E − ˙ r × B . (11)Here Ω (cid:96) ≡ (cid:15) ij(cid:96) Ω k i k j / C. Symmetry argument
In the study of the optical Imbert-Fedorov shift, it wasargued that the shift must exist as a result of SOC andtotal angular momentum conservation [5]. Here, the spinis tied to the helicity of the light, and the SOC is in-herent in the Maxwell equations [11]. The total angularmomentum conservation is due to the rotational symme-try in the direction normal to the interface.This approach can also be applied for electronic sys-tems. In a crystalline solid, due to the presence of lat-tice, we do not have any continuous rotational symme-try. Nevertheless, the effective models which describe thelow-energy electrons may acquire an emergent rotationalsymmetry. For example, the isotropic electron gas model,which may describe the electrons at the conduction bandedge for some semiconductors, enjoys a full rotationalsymmetry along any axis.Assuming the model of our system possesses such a ro-tational symmetry along z (as for the setup in Fig. 2), thismeans that the total angular momentum, represented bythe operator ˆ J z , is a conserved quantity:[ ˆ H , ˆ J z ] = 0 , (12)where ˆ H is the Hamiltonian for the whole system. Here, we add hats for the symbols to stress that these are op-erators. Generally, ˆ J z has the form ofˆ J z = (ˆ r × ˆ k ) z + ˆ S, (13)where the first term is the orbital angular momentum,and the second term includes any additional contributionwhich may come from the internal degree of freedom.Now, considering a wave packet Ψ I reflected by theinterface into Ψ R , the conservation of ˆ J z means that (cid:104) Ψ I | ˆ J z | Ψ I (cid:105) = (cid:104) Ψ R | ˆ J z | Ψ R (cid:105) . (14)For the configuration in Fig. 2, this equation leads to atransverse shift in reflection: δy R = 1 k x ( (cid:104) Ψ R | ˆ S | Ψ R (cid:105) − (cid:104) Ψ I | ˆ S | Ψ I (cid:105) ) . (15)The result in Eq. (15) shows that the transverse shiftis nonzero, when the internal state (cid:104) ˆ S (cid:105) , correspondingto spin or some kind of pseudospin, changes in scatter-ing. This change in the internal state generally resultsfrom a coupling between spin (pseudospin) and orbitalmotion. Thus, intuitively, when the angular momentumassociated with spin (pseudospin) changes in scattering,the transverse shift must arise so that the change in theorbital angular momentum can compensate to ensure theconservation of the total angular momentum.When there are multiple scattering channels, the con-servation relation above applies for each channel sepa-rately. This is required when the particle is regarded asa quantized object, such that each particle is scatteringinto one of the channels with certain probability.Certainly, the applicability of this symmetry argumentapproach requires the presence of the rotational symme-try, which depends on the system. When the symmetrydoes exist, this approach will be very powerful. Fromthe discussion above, one can see that the result onlydepends on the asymptotic incident and outgoing wavepacket states away from the interface (which are deter-mined by the bulk properties), but not on the details ofthe interface nor on the detailed interaction between theparticle and the interface. It also makes no assumptionon the particle wavelength as well as the width of theinterface region. IV. TRANSVERSE SHIFT IN NORMALSCATTERING
Before 2015, most studies on interface scattering, es-pecially on the anomalous shifts in interface scattering,were done for the 2D electronic systems, possibly becausesuch systems can be well fabricated and well controlled,have high mobilities, and are simple enough yet allownontrivial effects to happen. Nevertheless, regarding theshifts, only the longitudinal shifts can occur for 2D sys-tems, but not the transverse shifts. For a review on theseworks, please see Ref. [42].The possibility of transverse shifts in electronic in-terface scattering was first proposed by two works in2015 [30, 31]. The discovery is a natural byproduct ofthe study on 3D topological materials. As we have men-tioned in Sec. III B, the transverse shift can be relatedto the nontrivial Berry curvatures, which are often thekey features of topological materials. Particularly, inso-called Weyl semimetals [50, 51], the conduction andvalence bands cross at twofold degenerate Weyl points,which behave as monopole charges for the Berry curva-ture fields. Hence, the transverse shift has been firstrevealed for such Weyl electrons in interface scattering.In a Weyl semimetal, an electron near a Weyl point (at K ) may be described by the effective model H = − iχ (cid:88) i = x,y,z v i σ i ∂ i , (16)where χ = ± v i ’s are the Fermi velocities, σ i ’s arethe Pauli matrices corresponding to a spin or pseudospindegree of freedom. Let us assume that the Weyl pointsare sufficiently separated in k -space, such that the (in-tervalley) scattering between the different points can beneglected. It is noted that Weyl model in Eq. (16) repre-sents a model with the strongest SOC, because the entireHamiltonian is of an SOC term [52].In Refs. [30, 31], the authors considered one simplesttype of interface—the interface caused by an electrostaticpotential step V ( z ). For a sharp interface, one may write V ( z ) = V Θ( z ) for the potential step, where Θ( z ) is theHeaviside step function. According to the setup in Fig. 2,the model of the system is given by H = H + V ( z ) . (17)To study the transverse shift, Jiang et al. [30] adoptedthe quantum scattering approach, while Yang et al. [31]used the symmetry argument. The different approachesreached the same result.For example, consider the symmetry argument, whichapplies when v x = v y in model (16). For this model, thetotal angular momentum operator is given byˆ J z = (ˆ r × ˆ k ) z + χ σ z , (18)such that its average over the wave packet state is givenby J z = ( r × k ) z + χ n ) z , (19)where n is the unit vector ( v x k x , v y k y , v z k z ) / E k , and E k = η (cid:113) v x k x + v y k y + v z k z are the energy dispersionsof the electron and hole ( η = ± ) bands.As sketched in Fig. 2 and Fig. 4(c), consider an inci-dent electron that has an energy E I > θ I = arctan( k Ix /k Iz ) in the x - z plane. All energyscales are assumed to be small compared to the band-width, such that the velocities change little across the I δ y ( μ m ) R y ( n m ) T δ y ( n m ) - 40 0 0.2-0.2 V (eV)4δ y ( n m ) T V c - V c ε I R - total reflectionKlein tunneling k z k x k R k I k T k R k I k R k I k T V
0. Meanwhile, the transverse shift takes thesame form as in Eq. (20).The transverse shift in reflection ( R ) can be obtainedin a similar way, given by δy R = χ ( n Rz − n Iz )2 k Rx = − χv z cot θ I E I , (21)where k Rx = k Ix and k Rz = − k Iz .The dependence of these shifts on the incident angleand the potential step is shown in Fig. 4(a) and 4(b).One notes the following points. First, the shifts are oddfunctions of the incident angle. Second, when the sym-metry argument holds, the transverse shifts would haveuniversal behaviors independent of the interface details.Particularly, the results above apply for both sharp andgraded interfaces. Third, the sign of the shifts depends onthe chirality χ . Hence, the Weyl electrons with differentchiralities should shift in opposite directions. This leadsto the proposition of the chirality Hall effect in Ref. [31].It is also noted that the shift in reflection diverges when θ I approaches perpendicular incidence. Physically, theshift cannot diverge. There are two factors that regulatethis diverging behavior. (i) The probability of reflectionis completely suppressed at perpendicular incidence dueto the reversed spin direction, so the seemingly divergingshift at perpendicular incidence cannot manifest in mea-surement. (ii) Due to the uncertainty principle, a con-fined beam must have a finite spread in the wave vector(and hence the incident angle) distribution for the par-tial waves. When approaching perpendicular incidence,the diverging behavior indicates that the different partialwaves would scatter in drastically different ways, suchthat the scattered beam would no longer be confinedand the shift would become ill-defined. Consequently,although the shift should get enhanced with decreasingincident angle, the diverging behavior at perpendicularincidence would not occur in reality.The results above can be exactly reproduced by thequantum scattering approach. As mentioned, the scat-tering approach is more general, and it applies also forcases without the rotational symmetry. For example, if v x (cid:54) = v y in the current model, one finds from the scatter-ing approach that [30] δy R = − χ v x v z v y · cot θ I E I (22)for reflection, which recovers Eq. (21) when v x → v y .In fact, for the simple Weyl model studied here, the re-sult in Eq. (22) can also be obtained from the symmetryargument after making a scaling transformation on thecoordinates: ( x (cid:48) , y (cid:48) ) = ( x (cid:112) v y /v x , y (cid:112) v x /v y ).Finally, for graded interfaces, the transverse shift canalso be obtained from the semiclassical approach. Asderived in Ref. [31], assuming the potential V and theFermi velocities v ’s are slowly varying spatially comparedto the Fermi wavelength, the semiclassical equations ofmotion for the Weyl wave packet center ( r , k ) take thefollowing form ˙ r = ∂ E ∂ k − Ω kr · ˙ r − ˙ k × Ω , (23)˙ k = − ∂ E ∂ r − ∂V∂ r + Ω rk · ˙ k . (24) For the Weyl model in Eq. (16), the momentum spaceBerry curvature is given by Ω = − χ v x v y v z k E , (25)for the electron state in the conduction band.The shift can be determined by integrating these twoequations of motion. For example, when the v ’s are con-stants and V depends on z only, the anomalous shift inthe x - y plane is given by δ (cid:96) α = − (cid:90) αI ˙ k × Ω dt, (26)where α = T for V < V c ≡ E I − (cid:113) v x k x + v y k y and α = R for V > V c . This equation shows that the shift is closelyconnected with the Berry curvature. In the presence ofrotational symmetry ( v x = v y ), one can check that itleads to the same results in Eqs. (20) and (21). How-ever, there are two important points to be noted. First,the semiclassical trajectory is unique, i.e., transmission if V < V c and reflection if V > V c . The transmission andreflection cannot occur simultaneously like in the quan-tum case. Second, the above equation cannot apply tothe Klein tunneling case because the point where E = 0requires a non-Abelian treatment.On the other hand, the spatial ( z -)variation of theFermi velocities alone can also lead to an anomalous shift.From the equations of motion, one finds that [31] δ (cid:96) α = − (cid:90) αI
11 + Ω k z z (cid:104) Ω k z ∂ E ∂k z + ( Ω × ˆ z ) ∂ E ∂z (cid:105) dt. (27)One notes that apart from the contribution due to themomentum space Berry curvature Ω , there is an addi-tional contribution entirely due to the phase space Berrycurvature Ω kr . The phase space Berry curvature is lesswell known. The transverse shift is probably the firstpredicted physical effect induced by Ω kr .From the semiclassical approach, one can see that thetransverse shift should generally exist for materials withnontrivial Berry curvatures. Hence it is not limited toWeyl semimetals. There are many different types of topo-logical semimetals discovered in recent years, which mayalso give rise to transverse shifts. For example, the trans-verse shifts in multi-Weyl semimetals have been studiedin Ref. [32], and the different behaviors for intravalleyand intervalley scattering processes have been addressed.It should be mentioned that a sizable longitudinal shiftalso exists for the Weyl electron scattering, which hasbeen investigated using the scattering approach for sharpinterfaces. Later, Jiang et al. [53] proposed that theseanomalous shifts (longitudinal and transverse) lead to ananomalous scattering probability for a Weyl wave packetscattered at defect potentials, which enhances the ra-tio between the transport lifetime and the quantum life-time. Intuitively, the anomalous shift helps the Weylelectron to circumvent the scatterer, effectively reducing optical interface potential step NS interface (a) (b) (c) circ. light e eh ╳ xy z ? SC FIG. 5. Three kinds of reflection processes. (a) A cir-cularly polarized light beam undergoes a transverse shift(Imbert-Fedorov effect) when reflected at an optical inter-face. (b) Electron with strong spin-orbit coupling, like in Weylsemimetals, acquires a transverse shift when reflected from apotential barrier. (c) An incident electron is reflected as a holein Andreev reflection from a normal-metal/superconductor(NS) interface. the strength of disorder scattering. This was suggestedas a possible explanation for the high mobility observedfor Weyl semimetal materials.
V. TRANSVERSE SHIFT IN ANDREEVREFLECTION
In the previous examples, the scattered particle andthe incident particle are of the same identity: a photonis scattered as another photon, and an electron is scat-tered as another electron. However, there is an intriguingscattering process happening at the interface between anormal-metal (N) and a superconductor (S), in which theparticle identity is changed [see Fig. 5(c)]. This is the fa-mous Andreev reflection [34, 35].In Andreev reflection, an incoming electron from theN side at excitation energy ε above the Fermi level E F is reflected back as a hole with energy ε below E F . Theprocess conserves energy and momentum but not charge:the missing charge of ( − e ) is absorbed as a Cooper pairinto the superconductor. For excitation energies belowthe superconducting gap, electrons cannot penetrate intothe superconductor, and Andreev reflection becomes thedominating mechanism for transport through the NS in-terface. Is there any transverse shift in Andreev reflection?
Atfirst sight, this seems unlikely, because the incident andthe scattered particles are of distinct identities, even theirelectric charges are opposite. However, it should be notedthat the two particles are not independent. They docorrelate coherently through the superconductor on theother side of the interface.
A. Junction with Conventional Superconductor
The question above was first addressed by Liu, Yu, andYang in 2017 [36], with an affirmative answer. They in-vestigated NS junctions consisting of a normal metal withstrong SOC and a conventional s -wave superconductor. The conclusion is that a finite transverse shift generallyexists in Andreev reflection, which is connected with theSOC in the normal metal.Mathematically, the model is not much different fromthat for the normal-state junctions, except that the scat-tering here is governed by the Bogoliubov-de Gennes(BdG) equation [35] instead of the Schr¨odinger equation.For the N side ( z < H L = (cid:20) H L − E F E F − T − H L T (cid:21) ; (28)whereas for the S side ( z > H R = (cid:20) H R − E F ∆∆ ∗ E F − T − H R T (cid:21) . (29)Here, H L and H R are the Hamiltonian for the normalstates of the two sides, T is the time reversal operator,and ∆ is the superconducting pair potential which cou-ples electron and hole excitations. The BdG Hamiltonianfor the whole system may be written as H BdG = H L Θ( − z ) + H R Θ( z ) , (30)where Θ is the Heaviside step function. The scatteringstates are solutions of the BdG equation H BdG ψ = εψ, (31)where the wave function ψ ≡ ( u, v ) T is a multicomponentspinor with u ( v ) standing for the electron (hole) state.The above treatment assumes a sharp interface. Thestep function model for the pair potential has been widelyused in literature [54–56], and it has been shown to bea good approximation to the full self-consistent solutionof the BdG equation for such junction structures [57–59].Particularly, it is accurate when there is large Fermi mo-mentum mismatch across the interfaces (which effectivelyreduces the coupling between the layers). On the S side,the mean-field requirement for superconductivity is thatthe Fermi wavelength in S should be much smaller thanthe coherence length. It should be noted that the Fermiwavelength in N is not constrained to be small. Particu-larly, when N is of a doped semiconductor or semimetal,one may have the Fermi energy on the N side comparableto ∆ ≡ | ∆ | . The possible existence of an interface bar-rier can also be described in the model by adding a term hδ ( z ) τ z , where h represents the barrier strength and τ ’sare the Pauli matrices for the Nambu space. This barriermainly affects the scattering probabilities [54].In Ref. [36], the authors demonstrated the transverseshifts using two concrete models. The first is based onthe Weyl semimetal model, by letting H L = H and H R = H − U , where H is the Weyl model in Eq. (16)and U is some constant potential offset. Here, U isneeded to fulfill the mean-field requirement for supercon-ductivity on the S side (such that E F + U (cid:29) ∆ ). Inthe BdG Hamiltonian, an electron state at k is relatedto a hole state at − k . If the T symmetry is assumed forthe system, then the reflected Weyl hole should have thesame chirality as the incident Weyl electron.Via the quantum scattering approach, the transverseshift in Andreev reflection was derived for such Weyl NSjunction model, given by δy A = χ v y v z v x (cid:18) cot θ h E F − ε − cot θ e E F + ε (cid:19) , (32)where θ e/h = arctan( k x /k e/hz ) is the incident/reflectionangle.The result can also be reached with the symmetry ar-gument when v x = v y . In the current case, one finds thatthe conserved quantity isˆ J z = (ˆ r × ˆ k ) z + χ τ ⊗ σ z . (33)Recall that τ and σ are for the Nambu and spin spaces,respectively. The conservation leads to the transverseshift δy A = χ k x ( n hz − n ez ) , (34)where n e/h = ( v x k x , v y k y , v z k e/hz ) / ( E F ± ε ) (35)is the spin polarization direction for the electron/hole.Typical behavior of the shift is shown in Fig. 6(d-e).One observes that δy A is an odd function of the inci-dent angle θ e ; it vanishes at normal incidence where n e and n h are parallel, and reaches maximum magnitudeat a critical angle θ ce , beyond which k hz becomes imagi-nary and electrons can no longer be Andreev reflected. δy A vanishes when ε (cid:28) E F or ε (cid:29) E F , because n e and n h become parallel in both limits; and its seem-ingly divergent behavior at ε → E F is reconciled by not-ing that in this limit the hole Fermi surface becomes apoint, so the reflection has a vanishingly small probabil-ity. In fact, ε = E F marks the transition point betweenAndreev retroflection ( θ e θ h >
0) and specular reflection( θ e θ h < δy A has the same sign in both regimes. Importantly,the shift is opposite for different chirality, which may gen-erates a chirality Hall effect also for the Andreev-reflectedholes, similar to that for the normal reflection [31].It should be emphasized that as long as the symmetryargument is valid, the transverse shift is independent ofthe details of the NS interface and of the S region for thecurrent setup. This indicates that the Weyl-like modelas well as SOC are not necessary for the S region; theeffect results from the SOC on the N side.Another important question is: Does the shift only ex-ist for junctions with Weyl or other topological semimet-als? From the symmetry argument, one can see that theanswer is negative. Liu, Yu, and Yang [36] demonstratedthis with a concrete example. They considered the fol-lowing model H L = 12 m L ( −∇ + M ) σ z − ivσ x ∂ x − ivσ y ∂ y , (36) ε (meV) y ( n m ) A δ -0.4 0.40 θ /π e y ( n m ) A δ E z k z k ε e z k h eh n e n h k z k x SN y A δ electronhole θ h θ e z zyx (a)(b) (c)(d) (e) FIG. 6. (a) Schematic figure showing the transverse shift δy A for an incident electron wave-packet in the x - z plane Andreevreflected at the NS interface. (b,c) Schematic figure showing(b) the BdG Fermi surfaces, and (c) spectrum at a finite k x [corresponding to the horizontal dashed line in (b)]. The solid(hollow) sphere denotes the incident electron (reflected hole)state, and the arrows indicate their spin polarization direc-tions. (d,e) Transverse shift versus (d) incident angle θ e and(e) excitation energy. Figure adapted with permission fromRef. [36]. and H R = (cid:16) − m R ∇ − U − E F (cid:17) σ , (37)where m L and m R are the effective masses for the twosides, M and v are model parameters. The advantageof H L is that it nicely interpolates between two dis-tinct phases determined by the sign of M : for M < k z axis at ±√− M with opposite chirality, which simulates a T -broken Weylsemimetal [see Fig. 7(a)]; for M >
0, the two bandsare fully separated with a gap [see Fig. 7(b)], and when E F > M/ m L , it becomes a doped semiconductor, with-out any band crossing. As for the S side, we take it tobe the simplest metallic superconductor. In this model,the SOC appears on the N side, but not the S side.For M <
0, the transverse shift exists and can be cal-culated similar to the Weyl model, as shown in Fig. 7(e).One interesting point here is that the shift can hap-pen for both intravalley and intervalley scattering pro-cesses, but with distinct dependence on the incident angle0 e M>0 y ( n m ) A δ θ /π e -0.4 0.40 -300 M<0 y ( n m ) A δ y ( n m ) A δ ( ) ( ) M<0 E F K + K ehh K + K M>0 E F eh k x k z (e)(a) (b)(f) k x k z (c) (d) M<0 M>0
FIG. 7. (a,b) Two phases of model (36) and their corre-sponding BdG Fermi surfaces are schematically shown in (c)and (d), respectively. (e) Shift δy (1) A ( δy (2) A ) for intravalley(intervalley) Andreev reflection versus the incident angle θ e for M <
0, with incident electron from the K + valley. (f)shows the corresponding result for M >
0. In (e,f), the datapoints are from scattering approach, while the curves are fromsymmetry argument. Figure adapted with permission fromRef. [36]. [see Figs. 7(c) and 7(e)]. More interestingly, the trans-verse shift still exists for the doped semiconductor case(
M >
0) [see Figs. 7(d) and 7(f)]. The value is given by δy A = 12 k x ( n hz − n ez ) , (38)with n e/hz = ± [( E F ± ε ) − v k (cid:107) ] / ( E F ± ε ) . (39)These results explicitly demonstrate the followingpoints. (i) The key ingredient here is the SOC on theN side, however, Weyl or other types of band crossingsare not necessary. (ii) The role of the S side is to enablethe electron-hole conversion. Any conventional supercon-ductor suffices and it does not require SOC. (iii) Factorssuch as intervalley scattering, interfacial barrier, Fermisurface mismatch, and spatial profile of the pair poten-tial are inessential for the shift. And when symmetryargument applies, they have no effect on the value of theshift, although they do affect the probability of the pro-cess. In addition, we emphasize again that the σ here canbe real spin or any pseudospin. A transverse shift shouldbe induced, as long as the spin state is coupled with theorbital motion and is changed in the scattering. Particu-larly, the results for the Weyl model should directly applyfor those spin-orbit-free Weyl semimetals [61, 62]. B. Junction with Unconventional Superconductor
For the junctions discussed in Sec. V A, the key ingre-dient is the SOC on the N side, which is similar to casesin optics and in normal electron scattering. The shiftwould vanish if the SOC is negligible. Meanwhile, thesuperconductor only plays a passive role, i.e., a channelfor electron-hole conversion.In a following work, Yu et al. [38] discovered that a fun-damentally new effect can appear for junctions with un-conventional superconductors, as illustrated in Fig. 5(c).There, by “unconventional”, the authors referred to su-perconductors with unconventional pair potentials.The key observation is that unconventional pair po-tentials necessarily have a strong wave-vector depen-dence [63]. This generates an effective coupling betweenthe orbital motion and the pseudospin of the Nambu(electron-hole) space. Thus, the transverse shift can ariseeven in the absence of SOC . Remarkably, Yu et al. [38]found that the behavior of the shift is sensitive to thestructure of the pair potential and manifests characteris-tic features for each pairing type, as summarized in Ta-ble I. Therefore, the effect may provide a powerful newtechnique capable of probing the structure of unconven-tional pairings.To demonstrate the effect, they took a simplest model,with H L = − m ∇ , (40)and H R = − m (cid:107) ( ∂ x + ∂ y ) − m z ∂ z − U . (41)On the S side, there are two effective mass parameters m (cid:107) and m z . This is for describing the possible anisotropyin the S material. For certain layered superconductors(like cuprates), the Fermi surface is highly anisotropicand may take a cylinder-like shape in the normal state.Such cases can be described by using a lattice model.For unconventional superconductor, the pair potential ∆in the BdG Hamiltonian would have characteristic wave Pairpotential Expression Periodin α vanish for ε > | ∆ ( α ) | No. ofSZ δ(cid:96) eT δ(cid:96) hT Chiral ∆ e iχφ k χk F sin γ (cid:30) No 0 p x ∆ cos φ k δ(cid:96) eT ≈ δ(cid:96) hT π Yes 2 p y ∆ sin φ k π d x − y ∆ cos 2 φ k π/ d xy ∆ sin 2 φ k π/ α varies from 0to 2 π . Reproduced from Ref. [38]. UnconventionalSC ℓ T γ α αxy z incidencereflection xyx ℓ Th UnconventionalSC ℓ T γ α αx incidencereflection xyx ℓ T (a) (b) δ δ FIG. 8. (a) Schematic of the NS junction set-up. In Andreevreflection, the reflection plane (green-colored) is shifted by dis-tance δ(cid:96) T from the incident plane (orange-colored) along itsnormal direction (ˆ n ), due to unconventional pairing in S. (b)Top view of the x - y plane in (a). For certain pairings, theremay also be a finite shift for normal-reflected electrons (notshown here). Figure adapted with permission from Ref. [38]. vector dependence. Often one considers the weak cou-pling limit, with E F + U (cid:29) | ∆ | , ε in the S region, sothat the wave vector for ∆’s k -dependence is fixed on the(normal state) Fermi surface of S, and ∆ only dependson the direction of the wave vector k [64].The symmetry argument can be applied for the chiral p -wave pairing case, with ∆ = ∆ e iχφ k . Here, χ = ± θ k , φ k ) are thespherical angles of k . The magnitude ∆ is assumed tobe independent of φ k but may still depend on θ k . Onefinds that the quantityˆ J z = (ˆ r × ˆ k ) − χτ z (42)resembles an effective angular momentum operator, andit commutes with the BdG Hamiltonian H BdG . For elec-trons and holes, the expectation values of the Nambupseudospin are opposite: (cid:104) ˆ τ z (cid:105) e/h = ±
1. Because thepseudospin flips in Andreev reflection, the conservationof J z must dictate a transverse shift δ(cid:96) T to compensatethis change. (Here, the plane of incidence is not assumedto be the x - z plane, so the symbol δ(cid:96) T instead of δy isused to denote the transverse shift. See Fig. 8.) Theresult is δ(cid:96) T = − χ k (cid:107) ( (cid:104) τ z (cid:105) h − (cid:104) τ z (cid:105) e ) = χk (cid:107) , (43)where k (cid:107) = k F sin γ , k F is the Fermi wave vector in N,and γ is the incident angle.This remarkable result demonstrates several points.First, the shift here is entirely due to the unconventionalpairing, which plays the role of an effective SOC thatcouples k and τ . However, the spin here is the Nambupseudospin, which is intrinsic and unique for supercon-ductors. Second, as a general advantage of the symme-try argument, as long as symmetry is preserved, the re-sult does not depend on the details of the interface (seeFig. 9). Third, the result in (43) also applies for chiralpairings with higher orbital moments ( | χ | > d + id or f + if pairings.Quantum scattering approach was adopted to studythe transverse shift for other types of pairing in Table I. (a) (b) χ = -1 χ = Δ (meV) ε
10 0 y x k F chiral p T () n m δ T ( ) nm δ FIG. 9. Transverse shift δ(cid:96) T for chiral p -wave pairing versus(a) rotation angle α (here χ = +1), and (b) the excitationenergy ε . δ(cid:96) T is independent of ε and ∆ , and its sign dependson χ . Figure adapted with permission from Ref. [38]. An important case is for the d x − y -wave pairing, with ∆ = ∆ cos(2 φ k ). Here, because the pair potential isanisotropy in the x - y plane, the transverse shift dependson the orientation of the plane of incidence. Hence, oneneeds to define a rotation angle α between the incidentplane and the crystal x axis. It was found through cal-culation that δ(cid:96) T ∝ sin(4 α )Θ( | ∆ cos 2 α | − ε ) . (44)The expression in Eq. (44) highlights the dependence onthe rotation angle α and the excitation energy ε . Thetypical behavior is shown in Fig. 10.One can observe the following key features for the shift.(i) The shift has a period of π/ α , and it flips signat multiples of π/ ε , there must appear multiple zonesin α where δ(cid:96) T is suppressed [see Fig. 10(b)]. The cen-ter of each suppressed zone coincides with a node. (iv)The shift is also suppressed when k (cid:107) is away from theFermi surface of the S side, as indicated in Fig. 10(c) and10(f), where we compare the results for a closed ellip-soidal Fermi surface and for an open cylinder-like Fermisurface. This can be understood by noticing that theeffect of pair potential diminishes away from the Fermisurface.These features encode rich information about the un-conventional gap structure, including the d -wave symme-try [feature (i)], the gap magnitude profile [feature (ii)],and the node position [feature (iii)]. Feature (iv) alsooffers information on the geometry of the Fermi surface.Thus, by detecting the effect, one can extract importantinformation about the unconventional superconductor.Real unconventional superconductor materials couldhave other complicated features, such as multiple Fermisurfaces, multiple bands with different pairing magni-tudes, and possible interface bound states [63, 65–67].How these features would affect the anomalous shifts areinteresting questions to explore. Nevertheless, the anal-ysis in Ref. [38] suggested that a nonzero shift is gener-ally expected, owing to the coupling between the Nambu2 (b)(c)(a) k F α k (cid:8749) K c k (cid:8749) k F ( ) (d) (meV) ε R e fl . P r o b . Normal Andreev K c Υ Δ ( α ) | | N S K c1 K c2 N S K c1 K c2 (e) (f ) k F k (cid:8749) Υ k (cid:8749) k F ( ) T () n m δ d x - y ++ -- k F T () n m δ T () n m δ FIG. 10. Results for d x − y -wave pairing. (a-d) are forthe S side with an ellipsoidal (closed) Fermi surface, and (e-f) are for S with a cylinder-like (open) Fermi surface. (a)Schematic figure showing the Fermi surfaces of N and S. K c denotes the maximum magnitude of transverse wave-vectoron the S Fermi surface. (b) δ(cid:96) T versus α . The green shadedregions indicate the suppressed zones, in which ε > | ∆ ( α ) | .(c) δ(cid:96) T versus k (cid:107) . Corresponding to (a), δ(cid:96) T is suppressedwhen k (cid:107) > K c , as denoted by the gray shaded region. (d)Reflection probabilities versus ε for normal and Andreev re-flections. (e) illustrates the case when the S Fermi surface is ofopen cylinder-like shape. K c and K c denote the lower andupper bounds for the transverse wave-vector on the S Fermisurface. For such case, the qualitative features in (b) and (d)remain the same. The main difference is that the shift is nowsuppressed in regions except for K c < k (cid:107) < K c , as shownin (f). Figure adapted with permission from Ref. [38]. pseudospin and the orbital motion as generated by theunconventional pair potential. Although its detailed pro-file requires more accurate material-specific modeling, itis likely that the key features for the shift (as those listedin Table I) are robust, since they are determined by theoverall characteristic associated with the symmetry ofunconventional pairings. This also helps to distinguishthe signal from the shift against random noises such asfrom the impurities or interface roughness. Finally, whenthe SOC effect is included, it can generate an additionalcontribution to the shift. However, its dependence onthe incident geometry and the excitation energy will bedistinct from that due to the unconventional pairings. SN1 incident electron N2 outgoing hole d δ CAR y θ θ e h z0 zyx FIG. 11. Schematic figure showing the process of CAR. Inthe hybrid NSN structure, an incident electron from terminalN1 is coherently scattered as an outgoing hole in terminalN2. There may exist a transverse shift ( δy CAR ) between thetwo scattering planes. Figure adapted with permission fromRef. [39].
VI. TRANSVERSE SHIFT IN CROSSEDANDREEV REFLECTION
In 2018, Liu et al. [39] extended the study to the pro-cess of crossed Andreev reflection (CAR). CAR is a non-local version of the conventional Andreev reflection [40,41]. It appears in hybrid normal-superconductor-normal(NSN) structures, as schematically illustrated in Fig. 11.When the thickness of the S layer is smaller than or com-parable to the superconducting coherence length, an elec-tron incident from the left N terminal can form a Cooperpair in S with another electron from the right N terminal,thereby coherently transmitting a hole into the right Nterminal. The process has been successfully detected inexperiment [68–70].Using the quantum scattering approach and the sym-metry argument, Liu et. al. [39] predicted that sizabletransverse shift δy CAR can also exist in CAR. They con-sidered systems where the N terminals have strong SOCdescribed by similar models as in Ref. [36] (includingWeyl model and spin-orbit-coupled-metal model), andthe S layer is of conventional s -wave superconductor. Forsuch setups, the transverse shift is resulted from the SOCin the N layers. Compared with the local Andreev reflec-tion studied in Ref. [36], a new ingredient here is thatthe two N terminals (hence the incident electron and thescattered hole) can be controlled independently. Par-ticularly, one can use doped semiconductors as the twoterminals, and make the left N terminal n -doped andthe right N terminal p -doped, realizing a so-called p S n junction [71]. For this kind of setup, one can minimizea competing transmission process—the elastic cotunnel-ing, during which an incident electron directly tunnelsthrough the structure. It was shown that the transverseshift still exists for the CAR holes, which may results ina measurable voltage signal, providing a new method fordetecting CAR in experiment.3 θ δ eeh N S e xzx z δ h x FIG. 12. Schematic figure showing the longitudinal shift innormal reflection ( δx e ) and in Andreev reflection ( δx h ) for anincident electron beam reflected from an NS interface. Thesolid and open circles indicate the electron and the hole, re-spectively. Figure adapted with permission from Ref. [37]. VII. LONGITUDINAL SHIFT IN ANDREEVREFLECTION
For completeness, here we also briefly discuss the re-search on the longitudinal shift in Andreev reflection. Aswe have mentioned in Sec. II, the longitudinal shift isessentially a 2D effect, so it can be studied using 2D sys-tems. It was noticed that a previous theoretical work [72]studied this effect for a model based on a 2D electron gas,however, the shift was found to be absent. In 2018, Liu et al. [37] showed that the result in Ref. [72] is actuallya special limiting case. The longitudinal shift does existfor the general case, and it and can be quite sizable.Liu et al. [37] studied two concrete examples. The firstis a junction based on the simple 2D electron gas model,with H L = 12 m k , (45)and H R = 12 m k − U . (46)The system is assumed to be in the x - z plane, as illus-trated in Fig. 12.The calculation of the longitudinal shift via the quan-tum scattering approach is straightforward. It was foundthat when the N side is heavily doped such that E F (cid:29) U , ∆ , ε , the Andreev reflection amplitude is given bya k -independent number r h = e − iβ . Hence, in thisregime, the longitudinal shift in Andreev reflection van-ishes: δx h = 0. This recovers the result obtained inRef. [72], which assumed this regime.However, outside of the above regime, when E F is notlarge, the shifts would generally be nonzero. Typical be-havior of this shift is shown in Fig. 13. In addition, theremay also be sizable shift for the normal reflection, asshown in Fig. 13(a),(c). It is interesting to note thatwhile the shift in Andreev reflection stays positive asshown in Fig. 13(b),(d), the shift in normal reflectioncan be made either positive or negative, depending onthe excitation energy. An explanation of this behavior isprovided in Ref. [37].The second example is an NS junction based ongraphene [73]. The graphene band structure has two /θ π /θ π /θ π /θ π (a) (b)(c) (d) -6061218 0612-100-500 0612 ( n m ) δ e x ( n m ) h δ x ( n m ) δ e x ( n m ) h δ x FIG. 13. Longitudinal shifts (a,c) in the normal reflection( δy e ), and (b,d) in the Andreev reflection ( δy h ) versus theincident angle θ for the 2DEG/superconductor model. (a-d)are for small ε < ∆ ; while (c,d) are for ε close to ∆ . Theshaded region in each figure denotes the range with | θ | > θ c ,where Andreev reflection is not allowed. Figure adapted withpermission from Ref. [37]. Dirac cones (valleys) located at the two corner points ± K of the hexagonal Brillouin zone, which are connected bythe time reversal symmetry [74]. In the BdG model, H L ( k , τ ) = v F ( τ k x σ x + k y σ y ) , (47)where τ = ± denotes the two valleys, v F is the Fermi ve-locity, the wave-vector is measured from the valley cen-ter, and σ ’s are the Pauli matrices acting on A/B sub-lattice degree of freedom. Note that T H L ( k , τ ) T − = H L ( − k , − τ ), indicating that an incident electron in onevalley is coupled to the hole in the other valley throughthe superconducting pair potential. The S region is as-sumed to be described by the same Hamiltonian (47)(but with a nonzero pair potential and a potential en-ergy offset). Physically, this may be realized by cover-ing the graphene in the S region with a superconduct-ing electrode, which induces a finite ∆ by proximity ef-fect. The potential energy offset U may be adjusted bygate voltage or by doping. This model has been used byBeenakker [60] in discussing the special specular Andreevreflection in graphene.The calculation result showed that the longitudinalshift is enhanced by the additional pseudospin degree offreedom for graphene (see Fig. 14). In addition, the shiftin Andreev reflection can also be made negative, and thesign is connected with whether the Andreev reflection isa retroreflection or a specular reflection, as illustrated inFig. 15.4 -100-50050100 0 0.2-100-500 ( n m ) δ x ( n m ) δ x e h ( n m ) δ x ( n m ) δ x e h (a) (b)(c) (d) /θ π /θ π /θ π /θ π FIG. 14. Longitudinal shifts (a,c) in the normal reflection( δx e ), and (b,d) in the Andreev reflection ( δx h ), as functionsof the incident angle for the graphene/superconductor model.(a,b) are for the case with ε < E F ; while (c,d) are for the casewith ε > E F . The shaded region in each figure denotes theregion of | θ | > θ c where the Andreev reflection is not allowed.Figure adapted with permission from Ref. [37]. (a) (b) zx zx Retro-Reflection Specular-Reflection h δ x h δ x N S N S eh eh z z FIG. 15. Schematic figures for the shifts in the two typesof Andreev reflection in the graphene/superconductor model.(a) is for the retroreflection and (b) is for the specular re-flection. Note that the shifts have opposite signs for the twocases. Figure adapted with permission from Ref. [37].
VIII. EXPERIMENT
The anomalous shifts in electronic systems have notbeen directly detected in experiment at the time ofthis review. Nevertheless, several possible experimentalschemes have been put forward.The most direct way is to produce a collimated electronbeam to be scattered at the interface, and to detect thetrajectory of the scattered beam. This can in principle beachieved by using local gates and collimators, as havingbeen developed in the field of electron optics [27–29].One possible way to enhance the overall effect is todesign a structure such that the beam can undergo mul-tiple times of scattering and the shifts can be accumu-lated. For example, the cylinder-shaped setup and thesandwich setup were proposed to enhance the transverseshift in normal reflection for Weyl electrons [30, 31] (see Fig. 16). A 2D SNS waveguide was proposed to en-hance the longitudinal shift in Andreev reflection [37] [seeFig. 17(a)]. The repeated shifts lead to an anomalous ve-locity, which modifies the group velocity of the waveguideconfined mode, as indicated in Fig. 17(b) and 17(c).A less challenging approach for detecting the trans-verse shift is to fabricating a junction with an interfacetilted with respect to the average flow direction of theparticles. Because the incident electrons hit the interfaceat a finite average incident angle, the average transverseshift for the outgoing particle will also be finite and hasa definite sign. The transverse shift then leads to a netflow of the scattering particles, causing accumulation ofthe particles on the top (bottom) surface near the in-terface (see Figs. 18 and 19). For Weyl semimetals, theshift depends on the chirality of the electron. If the elec-trons of opposite chiralities have equal population, therewould be no net charge accumulation on the surface, butthere is a surface chirality accumulation [31]. This canbe detected by the imbalanced absorbance of the left andright circularly polarized light (see Fig. 18). On the otherhand, if the two populations are not equal, or for the junc-tions with unconventional superconductors (see Fig. 19),a surface charge accumulation can be generated, whichcan be directly probed electrically as a voltage signal. InRef. [38], it was estimated that the voltage signal at anNS junction with unconventional superconductor can beup to mV magnitude, which can be readily detected with zyx τ=+τ=‒+ + + ‒ ‒ ‒ zxy θ (a) (b)(c) (d) θ FIG. 16. (a) Top view of an electron undergoing multiple (to-tal) reflections in a cylindric potential well of Weyl semimet-als. (b) Side view of the enhanced chirality-dependent Halleffect in (a). (a,b) are adapted with permission from Ref. [31].(c) Schematic of a chirality splitter for Weyl fermions. Re-gions I, II, and III are three Weyl semimetal layers with dif-ferent Fermi velocities. (d) Illustration of the wavepacket tra-jectory (the black arrow) of Weyl fermions in region II. Theelectrons are injected by a TEM tip. (c,d) are adapted withpermission from Ref. [30]. zx θ w S N S e δ x (a)(b) (c) ( m e V ) ( m / s ) v k (nm ) x with shift w/o shift -1 x -1 ε || FIG. 17. (a) Schematic figure showing the trajectory for anelectron confined in the SNS structure. Its propagation ve-locity along the x direction is affected by the presence of thelongitudinal shift. (b) Numerical results for the spectrum ofthe confined modes in the SNS junction. (c) Group velocities v (cid:107) for the confined modes at energy marked by the dashedline in (b). In (c), the data points are obtained from the nu-merical results in (b), the red solid (blue dashed) line is theestimation with (without) the δx e correction. Figure adaptedwith permission from Ref. [37]. current experimental accuracy (on the order of nV).For the longitudinal shift in 2D systems, a possibleexperimental setup was proposed in Ref. [37]. As shownin Fig. 20, a collimated electron beam is incident onto theinterface, and one tries to detect the reflected beam withthe collector on the other side (see Fig. 20). The bluedashed line indicates the trajectory if there was no shift.We can engineer a barrier region (the gay colored one),e.g., by using local gating, such that the usual (dashed) χ = + χ = −+ + + ‒ ‒ ‒ SN θ τ =+ τ =‒ + + + ‒ ‒ ‒ (a) (b) θ FIG. 18. (a) A Weyl semimetal junction structure. The trans-verse shift in transmission through the green colored interfaceinduces a chirality accumulation at the top and bottom sur-faces near the interface. (b) An NS junction with the N sidebeing a Weyl semimetal. The transverse shift in Adreev reflec-tion induces chirality accumulation for the reflected holes onthe top and bottom surfaces near the interface. The chiralityaccumulation can be detected by the imbalanced absorbancefor the circularly polarized lights. Figures adapted with per-mission from Refs. [31, 37]. + + + ‒ ‒ ‒ γ e N Uncon. SC Uncon. SCNV (a) (b) FIG. 19. Schematic (a) top view and (b) side view of apossible NS junction with an unconventional superconductor.Electrons are driven to the interface with a finite average inci-dent angle. The transverse shift induces a net surface chargeaccumulation near the junction on the N side, which can bedetected as a voltage difference between top and bottom sur-faces. The illustration is for the case when Andreev reflectiondominates the interface scattering. Figure adapted with per-mission from Ref. [38]. trajectory is blocked. However, with the anomalous shift,the beam can circumvent the barrier region and followthe red path to be detected by the collector. Thus, thedetection of the reflected beam at the collector will provethe existence of the shift.It should be mentioned that real interfaces in the hy-brid structures may have roughness and imperfections.For interfaces that are too disordered, it would be diffi-cult to observe the anomalous shifts, because the parti-cles are strongly scattered and even the beam trajectoryis not well defined. However, one can expect that theshifts should be robust against weak disorders. This isbecause: (i) as shown from the studies, the shifts are notstrongly oscillating functions with respect to the incidentangle, the energy, and etc., so they are not expected to beaveraged out under perturbations from weak roughnessor imperfections; (ii) typically, for doped semiconduc-tors or semimetals, the Fermi wavelength can be mademuch larger than the atomic scale. Then the atomic-scale roughness or imperfections would have negligibleeffect on the shifts, since the particle simply does not seethem. Moreover, the advance in experimental technol-ogy has made it possible to fabricate atomically sharpand clean interfaces [75, 76]. Thus, the observation ofthe proposed effects should be within the reach of the SN collector zx barrier FIG. 20. Schematic figure of a possible setup for detectingthe longitudinal shift at an NS interface. Figure adapted withpermission from Ref. [37].
IX. LOOKING FORWARD
The recent discovery of transverse shifts in normal elec-tronic scattering and Andreev reflections has opened anew arena of physics research. The effect is intriguingand significant owing to the following aspects. First, theeffect is intimately connected with geometric quantitiessuch as the Berry curvatures. Second, the effect showsuniversal and robust features independent of the inter-face details when an emergent rotational symmetry ex-ists, which is often the case in low-energy theories. Third,for the NS junctions, the rich behavior of the shift reflectsthe key features of the superconducting pair potential.The above points indicate that the effect in the normalscattering can be utilized to probe the Berry curvaturesof a material, an important task also connected to thestudy of topological materials. The shift in the Andreevreflection can provide a powerful new method to charac-terize the pair potentials of a superconductor. By map-ping out the shift dependence on the incident geometry,one can in principle extract information of the symme-try of pairing, the gap magnitude, as well as the nodalstructure.On the theory side, we expect that the study will beextended to more types of junctions, e.g., with differ- ent types of topological materials or superconductors andwith different junction configurations. This knowledgewould be useful when one wants to use the effect to char-acterize different materials. The other direction is to ap-ply this effect for designing functional devices. For exam-ple, the chirality dependence in the effect has led to theproposal of a chirality filter device in Refs. [30, 31]. Tothis end, more quantitative study of the resulting signaldue to the shift is needed, which may be done by devicemodeling and numerical simulations.On the experiment side, it is urgent to have the firstdemonstration of the effect. We expect that the longitu-dinal shift in Andreev reflection can be detected relativelyeasily, because the experimental techniques dealing with2D systems are more advanced and mature. The detec-tion of the voltage signals at the NS junction with un-conventional superconductors is also an important task(as in Fig. 19), for which the electrical detection of thevoltage signal should be easier to perform.
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