Scattering of Dirac electrons from a skyrmion: emergence of robust skew scattering
SScattering of Dirac electrons from a skyrmion: emergence of robust skew scattering
Cheng-Zhen Wang, Hong-Ya Xu, and Ying-Cheng Lai
1, 2, ∗ School of Electrical, Computer and Energy Engineering,Arizona State University, Tempe, Arizona 85287, USA Department of Physics, Arizona State University, Tempe, Arizona 85287, USA (Dated: February 10, 2020)We study electron scattering from a closed magnetic structure embedded in the top surface of atopological insulator (TI). Outside of the structure there is a uniform layer of ferromagnetic insulator(FMI), leading to a positive effective mass for the Dirac electrons. The mass inside the structure canbe engineered to be negative, leading to a skyrmion structure. The geometric shape of the structurecan be circular or deformed, leading to integrable or chaotic dynamics, respectively, in the classicallimit. For a circular structure, the relativistic quantum scattering characteristics can be calculatedanalytically. For a deformed structure, we develop an efficient numerical method, the multiple mul-tipole method, to solve the scattering wavefunctions. We find that, for scattering from a skyrmion,anomalous Hall effect as characterized by strong skew scattering can arise, which is robust againststructural deformation due to the emergence of resonant modes. In the short (long) wavelengthregime, the resonant modes manifest themselves as confined vortices (excited edge states). Theorigin of the resonant states is the spin phase factor of massive Dirac electrons at the skyrmionboundary. Further, in the short wavelength regime, for a circular skyrmion, a large number of an-gular momentum channels contribute to the resonant modes. In this regime, in principle, classicaldynamics are relevant, but we find that geometric deformations, even those as severe as leading tofully developed chaos, have little effect on the resonant modes. The vortex structure of the resonantstates makes it possible to electrically “charge” the skyrmion, rendering feasible to manipulate itsmotion electrically. In the long wavelength regime, only the lowest angular momentum channelscontribute to the resonant modes, making the skew scattering sharply directional. These phenom-ena can be exploited for applications in generating dynamic skyrmions for information storage andin Hall devices.
I. INTRODUCTION
This paper is devoted to studying relativistic quan-tum scattering of Dirac electrons in systems involvingmagnetism. There are two motivations. Firstly, quan-tum scattering of spin-1/2 fermions is fundamental todeveloping two-dimensional (2D) Dirac material baseddevices. Secondly, magnetic materials have been efficientcarriers of information and the physics of magnetic tex-tures has been a topic of significant interest. In gen-eral, in quantum scattering, the nature of the underlyingclassical dynamics can play a role. For example, con-sider electronic scattering from a 2D electrical potentialdomain generated by an external gate voltage. In theclassical limit of zero wavelength, the electrons are pointparticles and the domain is effectively a 2D billiard sys-tem in which electrons move along straight lines and arereflected when “hitting” the boundary. For a circulardomain, the classical dynamics are integrable. However,for a deformed domain, e.g., a stadium shaped domain,the classical dynamics can be ergodic in the phase space.In this case, there is sensitive dependence on initial con-dition because two nearby trajectories will diverge fromeach other exponentially - the hallmark of chaos. Sincegeometric deformations are inevitable in applications, itis necessary in the study of quantum scattering to take ∗ [email protected] into account the nature of classical dynamics. Especially,it is useful to consider deformed domains to uncover thepossible effects of classical chaos on quantum scattering.We employ the setting of a two-dimensional (2D),closed magnetic structure embedded in a uniform layerof ferromagnetic insulating (FMI) materials on the top ofa 3D topological insulator (TI). Outside of the structure,due to the FMI layer and the proximity effect, the elec-trons obey the Dirac equation with a positive mass. Themass of the closed structure can be engineered to be nega-tive, making it a skyrmion [1–4]. The skyrmion structurecan be deformed so that the classical particle motions in-side are chaotic. The massive Dirac electrons moving onthe surface of the TI are scattered by the structure. Thesystem thus not only provides a setting for exploring newphysics associated with scattering of Dirac electrons froma magnetic skyrmion for applications (e.g., in spintron-ics), but also represents a paradigm to study the effectsof classical chaos on relativistic quantum scattering inthe presence of magnetism.To be systematic and general, we consider the caseswhere the magnetic structure on the top of TI can be ofeither the skyrmion or the non-skyrmion type. The struc-ture can simply be a circle, in which case the classicaldynamics are integrable, or it can be deformed from thecircular shape, e.g., a stadium, where there is fully devel-oped chaos in the classical limit. For a circular structure,the various scattering cross sections can be obtained an-alytically from the standard partial wave analysis. For adeformed structure, we adopt an efficient method, the a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b multiple multipole (MMP) method in optics, to solv-ing the scattering wavefunctions of the two-componentDirac fermion in the magnetic system. We focus ontwo regimes: the short wavelength regime where the sizeof the magnetic structure is larger than the wavelengthso that the underlying classical dynamics are relevant,and the long wavelength regime where the structure sizeis comparable or smaller than the wavelength. Thereare two main results. Firstly, a skyrmion can lead tostrong skew scattering due to the emergence of resonantmodes that manifest themselves as confined vortices in-side the skyrmion in the short wavelength regime or con-fined edge states in the long wavelength regime. Theresonant modes are the result of mass sign change acrossthe skyrmion boundary. For a circular skyrmion, in theshort wavelength regime, a large number of angular mo-mentum channels contribute to the resonant modes andelectron charging arises, providing a way to electricallymanipulate the skyrmion motion. In the long wavelengthregime, only the lowest angular momentum channels con-tribute to the resonant states, leading to strongly direc-tional skew scattering with implications in developingHall devices. The second result is that classical chaosgenerated by geometrical deformations has little effecton the scattering from a skyrmion. The scattering phe-nomena uncovered for the circular case are thus robust.The immunity of the scattering dynamics to severe de-formation of the skyrmion structure is advantageous forspintronic device applications.This paper is organized as follows. In Sec. II, we pro-vide the background of our research in terms of magneticskyrmion, TIs, the “marriage” between skyrmion and TI,and relativistic quantum chaos. In Sec. III, we describethe Hamiltonian and outline the methods (analytic andnumerical). In Sec. IV, we demonstrate the emergenceof robust resonant states in scattering from skyrmion forboth integrable and chaotic classical dynamics. In Sec. V,we develop a partial-wave decomposition based analysisfor resonances associated with scattering correspondingto classical integrable dynamics. In Sec. VI, we sum-marize the main findings and discuss experimental fea-sibility and open issues. Finally, in Appendix, we detailthe developed MMP method for numerically calculatingthe scattering wave functions associated with deformeddomain hosting chaotic dynamics in the classical limit. II. BACKGROUND
Magnetic skyrmion.
Generally, a skyrmion is aparticle-like magnetic excitation with a swirling topolog-ical 2D spin texture, i.e., the spin at the core and thespin at the perimeter point are in opposite directions [1–4]. The small size of the skyrmions and the possibilityof moving them with electrical currents of small density( ∼ A/m ) make them promising candidates for spin-tronic storage or logic devices [1, 2]. Skyrmions have beenexperimentally observed in chiral magnets [5, 6] as a re- sult of the competition between the Dzyaloshiskii-Moriya(DM) interactions, Heisenberg exchange, and Zeemaninteractions. It has been demonstrated that metallicskyrmions can be driven by spin transfer torque (STT)from the electric current [7–9]. Optical skyrmion lat-tices have been achieved in an evanescent electromagneticfield [10]. In addition, the topological spin Hall effect hasbeen demonstrated in which a pure transverse spin cur-rent is generated from a skyrmion spin texture [11–15]. Topological insulators.
TIs are quantum materialswith surface states residing in the bulk insulating gap [16,17]. The edge states are topologically protected and arerobust against nonmagnetic disorders due to a strongspin-momentum locking. The electron motions on thesurface follow the 2D linear dispersion with a single band-touching Dirac point and are described by the Diracequation. In spite of the strong spin-momentum lock-ing, the surface electronic states are sensitive to magneticperturbations. That is, the electrons will be scatteredoff upon encountering a magnetic structure on the sur-face of the TI. The interaction between the topologicalsurface states and magnetic materials in a quasi-one di-mensional setting has been studied [18–20] where, due tothe spin-momentum locking, the exchange coupling be-tween the magnetization and the surface electronic statescan lead to intriguing phenomena such as anomalousmagnetoresistance and unconventional transport behav-iors [21, 22]. The interaction can also lead to nonlinear oreven chaotic dynamics in the evolution of magnetizationof the FMI [23, 24]. For example, complicated dynam-ics can emerge in the magnetization switching due to aHall-current-induced effective anisotropic field [18, 25–27] and steady self-oscillations can arise in an FMI/TIheterostructure [28–30]. A quite recent computationalstudy has revealed phase locking in the magnetizationdynamics of two FMIs on the top of a 3D TI [31].
Skyrmion and TI.
Efforts in improving thermal effi-ciency and better manipulating skyrmions have led to the“marriage” between skyrmion and TI, where skyrmionsarise on the surface of a TI. Electric charging of magneticvortices on the surface of a TI was investigated [32], andthe confinement state in the skyrmion structure on thesurface of a TI was discovered, paving the way to drivingskyrmion motion using an applied electric field [33]. Elec-tron skew scattering induced by the skyrmion structureon the TI surface was also studied [34]. Quite recently,the combination of two skyrmions with opposite windingnumbers, called skyrmionum in an FMI/TI heterostruc-ture was observed in the physical space [35–37]. Theo-retically, fluctuation-induced N´eel and Bloch skyrmionson the surface of a TI have been predicted [38].Previous studies focused on scattering of electronsfrom radially symmetric skyrmion structures. Deformedskyrmion structure has been studied in recent years. Forexample, it was found that Majorana modes are robustagainst skyrmion deformations [39]. Quantum engineer-ing of Majorana fermions in deformed skyrmion struc-ture was also studied [40, 41] and deformed (elongated)skyrmions were used for stabilization and control of Ma-jorana bound states in proximity to an s-wave super-conductor [42]. Shape dependent resonant modes havebeen discovered recently in skyrmions in magnetic nan-odisks [43].
Relativistic quantum chaos.
The study of the mani-festations of classical chaos in relativistic quantum sys-tems was pioneered by Sir Michael Berry and his collab-orator [44] and recently emerged as an interdisciplinaryfield of research [45, 46] with applications to Dirac mate-rial systems [47, 48]. In contrast to the traditional field of(nonrelativistic) quantum chaos [49, 50] where classicalchaos often bears strong signatures in the correspondingquantum systems, such “fingerprints” tend to be weak-ened in the relativistic quantum counterparts [51, 52].For example, in scattering (e.g., electronic transportthrough a quantum dot structure), chaos tends to smoothout fluctuations in scattering matrix elements, quantumtransmission, or conductance [53–58] if the quantum be-haviors are governed by the Schr¨odinger equation. How-ever, in two-dimensional (2D) Dirac materials such asgraphene, strong fluctuations of the quantum scatteringcharacteristics can persist to certain extent in spite ofclassical chaos [59, 60]. Another example is a 2D de-formed ring with a line of magnetic flux through the cen-ter, where Schr¨odinger electrons are localized but Diracelectrons can keep circulating along the edges of the ringdomain, generating a superpersistent current in spite offully developed classical chaos in the domain [61] - a phe-nomenon that can be exploited for creating a robust rel-ativistic qubit [62]. Quite recently, the weakening of themanifestations of chaos in spin-1/2 Dirac fermion sys-tems was studied [52] using the approach of out-of-time-ordered correlator [63]. It has also been revealed that, forscattering in spin-1 Dirac-Weyl fermion systems, a classof robust resonant modes can emerge that defy classicalchaos completely [51].
III. MODEL AND METHOD
We place an FMI thin film (e.g., Cu OSeO ) on the topof a TI with a single magnetic structure at the center ofthe thin film, as schematically illustrated in Fig. 1. Themotions of the surface electrons are affected by the struc-ture with the magnetization vector n ( r ). The Hamilto-nian of the system is H = v F (ˆ p × σ ) z − ∆ s n ( r ) · σ , (1)where v F is the Fermi velocity, ˆ p = − i ∇ is the momen-tum operator, σ = ( σ x , σ y , σ z ) are the Pauli matrices,and ∆ s ( >
0) is the spin-splitting energy from the ex-change interaction between the electron and the magne-tization. In the polar coordinates r = ( r, θ ), for a circularstructure, the magnetization vector can be parameterizedas n ( r ) = [ − sin θ (cid:112) − n z ( r ) , cos θ (cid:112) − n z ( r ) , n z ( r )] . (2) (a) TI TI 𝑥 𝑦 𝑧 (b) Δ 𝑠 𝑛 𝑧 = 10𝐸 = 10 𝐸 = 20
𝐸 𝑥 𝒆 − Δ s n z = 10 skyrmion Δ 𝑠 n z = −10 FMI
FIG. 1. Schematic illustration of electron scattering from askyrmion structure in a thin FMI film deposited on the top ofa TI. (a) The band structure of the FMI/TI heterostructure.Outside (inside) of the skyrmion structure, the mass corre-sponding to the band gap is positive (negative). (b) Illustra-tion of electron scattering behavior from the skyrmion struc-ture. For electronic states outside and inside of the skyrmion,the associated spin direction is different due to the oppositesigns of mass.
For a deformed magnetic structure, there is swirling spintexture with magnetic moment points up on the edge anddown in the center [64]. The out-of-plane component ofthe magnetic texture n z ( r ) acts as a Dirac mass term,which opens a gap in the electronic band structure. Thein-plane component n || can lead to an emergent magneticfield in the form B ( r ) = c ∆div n || ( r ) e (cid:126) v F . For a swirling skyrmion structure, the emergent magneticfield B is zero and the in-plane component can be gaugedaway [33, 34]. In this case, the “hard-wall” approxima-tion n z ( r ) = ± n = 1, n = − E ± = ± (cid:113) (cid:126) v F ( k x + k y ) + ∆ n z , (3)as shown in Fig. 1(a). While the energy dispersion curveinside of the skyrmion appears similar to that outside ofskyrmion, the spin direction is different for the electronicstate due to the opposite signs of mass. An electron willthen go through a scattering process in this 2D system.Because of the breaking of the time reversal symmetry,skew scattering will arise.For a circular magnetic structure, the scattering wave-function and the related behavior can be solved ana-lytically using the partial-wave decomposition method(Sec. V). For a deformed skyrmion, analytic solutionsof the scattering wavefunction are not feasible. We havedeveloped an MMP based method, which has its originin optics [65–69] and has recently been extended to scat-tering of pseudospin-1 particles [51]. The basic idea is toassume two sets of fictitious poles along and in the vicin-ity of the entire boundary of the magnetic structure: oneoutside and another inside of the boundary. Each poleemits a wave in the form of Hankel function (sphericalwave in the far field). The transmitted wavefunction ateach point inside of the scatterer can be expressed as thesuperposition of the waves emitted by the poles outsideof the scatterer. Similarly, the refracted wavefunctionat each point outside of the scatterer can be written asthe combination of the waves emitted by the poles in-side of the scatterer. The incident plane wave as wellas the reflected and transmitted waves are matched onthe boundary to enable the poles to be determined, andthe expansion coefficients can be obtained by solving thematrix eigenfunctions. (The details of the MMP methodadopted to scattering from a magnetic structure are givenin Appendix) We validate the method by comparing theMMP solutions with the analytic solution based on par-tial wave expansion for a circular skyrmion. Overall, theMMP method is effective and efficient for solving boththe near- and far-field scattering problem for a magneticscatterer of arbitrary shape.In our calculation, we use the dimensionless quantityobtained via considerations of the scales of the physicalquantities involved. In particular, the energy scale inthe FMI/TI heterostructure is on the order of meV. Infree space with zero mass, the wavevector correspond-ing to the energy of 1 meV is k ∼ meV / ( (cid:126) v F ) =3 . × − /nm . We take the dimensionless radius ofthe magnetic structure (circular shape) to be R = 1,which corresponds to a real structure of size of 100 nm.We then set the dimensionless energy corresponding to 1meV to be kR = 0 . / . ≈
33 meV.
IV. EMERGENCE OF ROBUST RESONANTSTATES IN SCATTERING FROM SKYRMIONA. Short wavelength regime - resonant vorticesand edge modes
We concentrate on regime where the wavelength of theincoming Dirac electron is smaller than the size of themagnetic structure so that the classical dynamics insidethe structure are relevant. We consider a circular struc-ture as well as a deformed structure that leads to chaosin the classical limit to identify any effect of chaos on theelectron scattering behavior.
Far-field behavior . Far away from the scattering center,for unit incident density the spinor wavefunction can bewritten asΨ I = Ψ inc + Ψ ref ≈ C (cid:18) i (cid:126) v F kE − m (cid:19) e ikr cos θ + C (cid:18) e − iθ i (cid:126) v F kE − m (cid:19) f ( θ ) √ r e ikr (4)where C is the normalization factor, k = (cid:113) k x + k y is theelectron wavevector, m = ∆ s n and m = ∆ s n are themass terms outside and inside of the magnetic structure, f ( θ ) denotes the 2D far-field scattering amplitude in thedirection defined by angle θ with the x -axis. For a cir-cular structure, f ( θ ) can be obtained analytically. Fora chaotic structure, once the reflection function is calcu-lated from the MMP method, f ( θ ) can be obtained. Thedifferential cross section is dσdθ = | f ( θ ) | . (5)The transport and skew cross sections are defined, re-spectively, as σ tr = (cid:90) π dθ | f ( θ ) | (1 − cos θ ) (6)and σ skew = (cid:90) π dθ | f ( θ ) | sin θ. (7)Figures 2(a) and 2(b) show, respectively, the skew scat-tering and transport cross sections as a function of in-cident electron energy, for a skyrmion (negative valueof m ) of circular shape (upper panel) and stadiumshape (lower panel) of the same area π in dimensionlessunits. The stadium shape is chosen because of its mir-ror symmetry for the incident plane waves so as to avoidan unnecessary complication: mixing of skew scatteringand back-scattering (or reflection). For both skyrmionshapes, there are sharp resonant peaks in the skew crosssection in the lower energy range close to the gap - anindication of the emergence of anomalous Hall effect asso-ciated with Dirac electron scattering from the skyrmion.As the incident energy is increased, the peak height isreduced but its width becomes larger, as a larger en-ergy value corresponds to less distortion in the energy-momentum dispersion with the mass gap. Note thatthere is little difference in the skew scattering cross sec-tion curves for the two skyrmion shapes, indicating thatthe nature of the classical dynamics hardly affects thescattering. For the curves of the transport cross section,as shown in Fig. 2(b), its value decreases with increasingenergy. For low energy values, the valleys in the trans-port cross section correspond exactly to the skew scatter-ing peaks. Sharp peaks also exist in the backscatteringcross section curve. Similar to the skew cross section, thenature of the classical dynamics has no appreciable effect.The results in Fig. 2 indicate that skyrmion skew scat-tering is robust against geometric deformations that areso severe as to change the classical behavior completely:from integrable dynamics to chaos. FIG. 2. Skew scattering and transport cross sections ver-sus incident electron energy in the short wavelength regime.(a) Skew scattering cross section versus the energy. The redand blue curves correspond to a circular and stadium-shapedskyrmion, respectively. The mass values are m = 10 and m = −
10. (b) Backscattering cross section as a function ofelectron energy for the two skyrmion shapes as in (a). In eachpanel, the red curve has been shifted upwards by an amountspecified by the horizontal red-dashed line for better visual-ization and comparison with the blue curve.
Near-field behavior . To understand the origin of the de-formation (chaos) independent far-field scattering (trans-port) behavior, we study the near-field scattering be-havior by examining the probability density and thecurrent density distribution associated with some spe-cific energy state. In particular, the probability den-sity is given by P = Ψ † Ψ, where Ψ = ( ψ , ψ ) T isthe wavefunction, and the probability current operatoris ˆ J = ∇ p H = v F ( σ y , − σ x ). The current density can beobtained as J = ( J x , J y ) = v F [2( iψ ψ ∗ ) , − ψ ψ ∗ )] . (8)The probability density distribution of the spin- z com-ponent is given by (cid:104) σ z (cid:105) = | ψ | − | ψ | . We choose a representative energy value correspondingto a skew scattering cross section peak: E = 11 .
225 for
FIG. 3. Probability and current density distribution for se-lected vortex states. (a) The probability distribution for scat-tering from a circular skyrmion for m = 10, m = −
10, and E = 11 . z component (color coded) density distribution in the cir-cular skyrmion region. (c,d) The corresponding probability,current and spin distribution for scattering from a stadium-shaped skyrmion for m = 10, m = −
10, and E = 11 . the circular skyrmion and E = 11 .
42 for the stadium-shaped skyrmion - marked as the red and blue stars inFig. 2(a), respectively. The probability and the cur-rent density distributions are shown in Fig. 3. Fromboth skyrmion structures, there are scattering resonantstates, as shown in Figs. 3(a) and 3(c). The resonantpatterns correspond to weak backscattering but strongerskew scattering cross sections, indicating that these areeffectively quasi-confined states. Further insights into thecontribution of the resonant states to skew scattering canbe gained by examining the current density distribution(marked as arrows) and the spin- z component densitydistribution (color coded) in the 2D skyrmion structure,as shown in Figs. 3(b) and 3(d). We see that the confinedresonant states form vortices with counter-clockwise cur-rents. There is also an out-of-plane spin component alongthe positive z direction. The vortices have an apparentdirectionality, so they can affect the skew scattering di-rection and magnitude. The vortices are formed by in-terference of waves reflected from the boundary and arerobust against boundary deformation. As a result, thenature of the classical dynamics, integrable or chaotic,has no significant effect on scattering.In addition to the confined vortex states inside of theskyrmion structure, another form of confined states arisesalong the skyrmion boundary, as shown in Figs. 4(a)and 4(c), for scattering from a circular and a stadium-shaped skyrmion, respectively. There is strong confine-ment of the scattering wavefunction near the boundarywith clockwise current and spin- z component along thenegative z axis direction, as shown in Figs. 4(b) and 4(d).The edge states correspond to sharp resonant peaks inthe backscattering cross section marked as the filled cir-cles in Fig. 2(b). For the circular skyrmion, the edgestates have no corresponding sharp peaks in skew scatter- FIG. 4. Wavefunction probability and current density dis-tribution associated with selected edge states. (a) The proba-bility distribution for scattering from a circular skyrmion for m = 10, m = −
10, and E = 11 . z component(represented by colors) density distribution. (c,d) The prob-ability and spin distributions associated with scattering froma stadium-shaped skyrmion for m = 10, m = −
10, and E = 10 . ing. For the stadium-shaped skyrmion, the edges statescorrespond to sharp valleys in the skew scattering crosssection. B. Long wavelength regime - resonant modes nearthe boundary
10 15 20 E -0.4-0.200.20.4 σ s k e w (a) 10 15 20 E σ t r (b) FIG. 5. Characteristics of Dirac electron scattering from amagnetic skyrmion in the long wavelength regime. (a,b) Skewscattering and backscattering cross sections versus energy, re-spectively. The red and blue curves correspond to a circularand stadium-shaped skyrmion, respectively. The mass valuesare m = 10 and m = −
10. In each panel, the red curve hasbeen shifted upward for a proper amount for better visualiza-tion and comparison with the blue curve.
Far-field behavior . We consider the regime where theskyrmion size is smaller than the electronic wavelength: R (cid:28) /k . This can be realized by setting the areaof the skyrmion structure to be 0 . π for both circular( R = 0 .
1) and stadium-shaped skyrmions. In this longwavelength regime, for a deformed skyrmion structure, the MMP method is still effective for calculating the far-field cross sections and the near-field state distribution.Representative results on the skew scattering and trans-port cross sections versus the incident energy are shownin Fig. 5. Different from the scattering behaviors in theshort wavelength regime, the oscillations of the skew scat-tering cross section with energy are weak. For example,in the energy range 10 < E <
20, only one smooth peakappears. There is hardly any difference in the scatter-ing characteristics between the two skyrmion structures,which is understandable as any structural differences arenot resolved in the long wavelength regime. Because oflack of appreciable oscillations, there is directional skewscattering over a large energy range - a desired feature inHall device applications.
Near-field behavior . We examine the state associatedwith the energy value that leads to the lowest skew scat-tering cross section: E = 12 .
072 for the circular and E = 11 .
46 for the stadium-shaped skyrmion, and therespective probability density distributions are shown inFigs. 6(a) and 6(c). The states are concentrated in thevicinity of the boundary, which are different from the vor-tex states observed in the short-wavelength regime. Theedge states thus represent a different type of resonantstates with directional current, as shown in Figs. 6(b)and 6(d). It can be seen that the current direction isdownward at the edge, contributing to skew scattering.The spin- z component is along the negative z direction. FIG. 6. Wavefunction probability and current density dis-tributions for selected states for scattering in the long wave-length regime. (a,b) The probability distribution and in-planecurrent together with the spin-z component density distribu-tions, respectively, for scattering from a circular skyrmion for m = 10, m = −
10, and E = 12 . m = 10, m = −
10, and E = 11 . FIG. 7. Effects of varying mass on Dirac electron scatter-ing in the short wavelength regime. The area of the mag-netic structure is π . (a) Skew scattering cross section versusthe electron energy for a circular structure for mass values m = − , − , , ,
9, represented by the red, orange, green,blue and purple solid curves, respectively. In each panel, thecurves have been shifted upward for better visualization andcomparison, where each horizontal dashed line denotes thezero reference point. The mass outside of the magnetic struc-ture is m = 10. (b) The corresponding curves for a stadium-shape structure with the same mass values as in (a). C. Further demonstration of strong skewscattering from a skyrmion structure
To further demonstrate the shape-independent skewscattering behavior of Dirac electrons from a magneticstructure, we study the effects of changing the mass ofthe skyrmion texture. To be concrete, we set m > m values. In thissetting, there is a skyrmion for m < m > m = 10 and m = − , − , , ,
9. Itcan be seen that, among the five cases, the resonant os-cillations of the cross section with energy last longer for m = −
9. On the contrary, for m = 9 (non-skyrmion),the oscillations diminish rapidly as the energy is in-creased. These behaviors hold regardless of whether theunderlying classical dynamics are integrable or chaotic.Overall, a large difference between the masses inside andoutside of the magnetic structure can lead to strongerand long-lasting resonant modes and, consequently, tomore pronounced skew scattering. Figures 8(a) and 8(b)show the probability density distribution for m = 9 and m = −
9, respectively, for the circular magnetic struc-ture. The corresponding results for the stadium-shapedstructure are shown in Figs. 8(c) and 8(d). For bothstructures, there are resonant modes for m = − FIG. 8. Probability density distribution for selected states inthe circular and stadium-shaped structure for different massesin the short wavelength regime. (a) Circular skyrmion struc-ture ( m = −
9) for E = 10 . m = 9) for E = 10 . m = −
9) for E = 10 . m = 9) for E = 10 . the magnetic structure is of the skyrmion type) but notfor the case of m = 9. FIG. 9. Skew scattering for different mass values of themagnetic structure in the long wavelength regime. The areaof the structure is π/
100 and the mass outside of the struc-ture is m = 10. (a) For a circular structure, skew scatteringcross section for m = − , − , , ,
9, represented by the red,orange, green, blue and purple solid curves, respectively. Ineach panel, the curves have been shifted upward for better vi-sualization and comparison, with the horizontal dashed linesdenoting the zero reference point. (b) The corresponding re-sults for a stadium-shaped magnetic structure.
In the long wavelength regime, regardless of the shapeof the magnetic structure (circular or stadium-shaped),the skew scattering cross section decreases as the rela-tive mass difference is reduced, as shown in Fig. 9 for m = − , − , , ,
9. Figure 10 shows representative res-onant states for the circular and stadium-shaped struc-ture for m = ±
9. Again, when the magnetic structure isof the skyrmion type, skew scattering is strong, makingthe scattering electrons directional. However, when thestructure is not of the skyrmion type, skew scattering isweak.
FIG. 10. Probability density distribution for the states corre-sponding to the minimum of the skew scattering cross sectionin circular and stadium-shaped magnetic structures in thelong wavelength regime: (a) a circular skyrmion structurefor m = − E = 12 . m = 9 and E = 12 . m = − E = 11 .
53, and (d)a stadium-shaped non-skyrmion structure for m = 9 and E = 11 . V. PARTIAL-WAVE DECOMPOSITION BASEDANALYSIS
Numerically, we have observed strong skew scatteringof Dirac electrons from a skyrmion structure, which isrobust against geometric deformation. We now providean analytic understanding of skew scattering based onthe method of partial wave decomposition. Consider acircular skyrmion. Key to pronounced skew scattering isthe resonant modes emerged from the scattering process.In the short wavelength regime, a large number of angu-lar momentum components are involved in the scattering,leading to a large number of resonant modes as the resultof various combinations of the angular momentum com-ponents, which are manifested as peaks in the curve ofthe cross section with the energy. In the long wavelengthregime, typically only a single resonant mode is domi-nant, implying the involvement of only the lowest severalangular momentum components. The asymmetric contri-bution from different angular momentum channels leads to the observed pronounced skew scattering. Because thecircular and stadium-shaped skyrmion structures gener-ate similar scattering behavior, the analytic results fromthe circular skyrmion case also provides an understand-ing of the emergence of strong skew scattering in thestadium-shaped skyrmion.For a circular skyrmion, the rotational symmetry stip-ulates conservation of the total angular momentum ˆ J z :[ ˆ J z , H ] = 0, and the partial wave component with to-tal angular momentum j (= ± / , ± / , ... ) in the polarcoordinates ( r, θ ) can be written as ψ j ( r ) = (cid:18) u j ( r ) e i ( j − / θ v j ( r ) e i ( j +1 / θ (cid:19) . (9)The Hamiltonian in the polar coordinates is H = (cid:126) v F (cid:32) − ∆ s n (cid:126) v F − e − iθ ∂∂r + e − iθ i∂r∂θ e iθ ∂∂r + e iθ i∂r∂θ ∆ s n (cid:126) v F (cid:33) . (10)Substituting the partial wave form in Eq. (9) into theHamiltonian Eq. (10) leads to an eigenvalue problem andconsequently to the explicit expression for the partialwaves.The transmitted wave inside of the skyrmion structure( r < R ) can be expanded in terms of the partial wavesas ψ T ( r, θ ) = C ∞ (cid:88) l = −∞ i l − B l (cid:32) J l − ( k (cid:48) r ) e i ( l − θ − (cid:126) v F k (cid:48) E − ∆ s n (cid:48) J l ( k (cid:48) r ) e ilθ (cid:33) , (11)and the reflected wave outside of skyrmion ( r > R ) canbe written as ψ R ( r, θ ) = C ∞ (cid:88) l = −∞ i l − A l (cid:18) H l − ( kr ) e i ( l − θ − (cid:126) v F kE − ∆ s n J l ( kr ) e ilθ (cid:19) , (12)where C is a normalization factor. We denote n ( n (cid:48) ) asthe magnetic moment and k ( k (cid:48) ) as the wavevector out-side (inside) of the skyrmion structure. For the incidentelectron in the free region outside of the skyrmion struc-ture, the wavefunction is ψ I = C (cid:18) i (cid:126) v F kE − ∆ s n (cid:19) e ikr cos θ . (13)Using the Jacobi-Anger identity: e iz cos θ ≡ ∞ (cid:88) l = −∞ i l J l ( z ) e ilθ , (14)we can expand the plane wave in the form ψ I = C (cid:88) l i l − (cid:18) J l − e i ( l − θ − (cid:126) v F kE − ∆ s n J l ( kr ) e ilθ (cid:19) . (15)Matching the waves at the skyrmion boundary (r = R): ψ I ( R ) + ψ R ( R ) = ψ T ( R ) , (16)we get, after some algebraic manipulation, A l = J l − ( kR ) J l ( k (cid:48) R ) − ττ (cid:48) J l ( kR ) J l − ( k (cid:48) R ) ττ (cid:48) H l ( kR ) J l − ( k (cid:48) R ) − H l − ( kR ) J l ( k (cid:48) R ) , (17)and B l = J l − ( kR ) H l ( kR ) − J l ( kR ) H l − ( kR ) H l ( kR ) J l − ( k (cid:48) R ) − τ (cid:48) τ H l − ( kR ) J l ( k (cid:48) R ) , (18)where τ = − (cid:126) v F kE − ∆ s n , and τ (cid:48) = − (cid:126) v F k (cid:48) E − ∆ s n (cid:48) . -5 0 5 j | A l | (a) -5 0 5 j | B l | (b) -5 0 5 j | A l | (c) -5 0 5 j | B l | (d) FIG. 11. Partial wave decomposition coefficients as a func-tion of total angular momentum for a circular magnetic struc-ture in the short wavelength regime. Among the quantitiesplotted, A l ’s are the coefficients for the reflected waves outsideof the structure and B l ’s are the transmitted wave coefficients.(a,b) For a skyrmion structure ( m = 10 and m = − | A l | and | B l | as a function of j , respectively, where thecorresponding state is shown in Fig. 8(a). (c,d) For a non-skyrmion structure ( m = 10 and m = 9), | A l | and | B l | versus j , respectively, where the corresponding state is shownin Fig. 8(b). Using the explicit formulas for A l and B l as given inEq. (17) and (18), respectively, we obtain the decom-position coefficients versus the total angular momentumfor R = 1. Figures 11(a) and 11(b) show, for the caseof scattering from a skyrmion structure ( m = 10 and m = − j . Figures 11(c) and 11(d) show thecorresponding results for a non-skyrmion case ( m = 10and m = 9). It can be seen that, several angular mo-mentum components contribute to the reflected wavecomponent A l , and the asymmetric distribution of theangular momentum components about zero leads to skewscattering. For the transmitted wave components, the distribution of the angular components is asymmetric aswell, leading to the emergence of resonant vortices. Forthe B l coefficients, their values for the non-skyrmion caseis much smaller than those for the skyrmion case, indicat-ing that the skyrmion structure can confine the electronsmuch more effectively than the non-skyrmion structure. -4 -2 0 2 4 j | A l | (a) -4 -2 0 2 4 j | B l | (b) -4 -2 0 2 4 j | A l | (c) -4 -2 0 2 4 j | B l | (d) FIG. 12. Transmitted and reflected partial wave coefficientsas a function of the total angular momentum for a circularmagnetic structure in the long wavelength regime. The radiusof the structure is R = 0 .
1. (a,b) | A l | and | B l | versus j for m = 10 and m = − | A l | and | B l | versus j for m = 10 and m = 9 (non-skyrmioncase), respectively, where the corresponding state is shown inFig. 10(b). Setting R = 0 . m = 10 and m = −
9) and non-skyrmion ( m = 10 and m = 9)cases, respectively. In both cases, only a single angu-lar momentum component contributes to the coefficient A l , i.e., j = − /
2, giving rise to the directionality in thescattering and a slow change in the resonant cross sectionwith the energy. The value of A l for the non-skyrmioncase is much smaller than that of the skyrmion case. Forthe transmitted coefficient B l , the angular momentumcomponent j = − / j = − / B l are much larger in the skyrmion than the non-skyrmioncase, again implying stronger confinement by resonanceand better directionality of scattering in the skyrmionstructure as compared with those in the non-skyrmionstructure.0 VI. DISCUSSION
We have investigated relativistic quantum scatteringof Dirac electrons from a closed magnetic structure em-bedded in the top surface of a 3D TI. Outside of thestructure, there is a uniform FMI layer, leading to a fi-nite but positive mass for the Dirac electron. The massof the structure itself can be engineered to be negativeor positive, where a skyrmion and a non-skyrmion struc-ture arises in the former and latter case, respectively. Inthe short wavelength regime, the nature of the classicaldynamics in the closed structure should be relevant tothe quantum scattering dynamics, according to conven-tional wisdom from the study of quantum chaos [49, 50].For a perfectly circular structure, the classical dynam-ics are integrable. For a deformed structure such as onewith the stadium shape, there is fully developed chaosin the classical dynamics. Our main findings are two.Firstly, in the short wavelength regime, classical chaoshardly has any effect on the scattering dynamics. Infact, similar behaviors in the scattering characteristicsat a quantitative level, such as the skew scattering andbackscattering cross sections, have arise for the circularand stadium-shaped structures. The diminishing effectsof classical chaos on relativistic quantum scattering froma magnetic structure are consistent with previous resultson weakened manifestations of chaos in relativistic quan-tum systems in general [51, 52, 59–61]. Secondly, strongskew scattering can arise when the magnetic structure isa skyrmion, regardless of the nature of the classical dy-namics. In the short wavelength regime, the pronouncedskew scattering is associated with resonant modes man-ifested as confined vortices inside of the skyrmion struc-ture, which are originated from the sign change in themass when the Dirac electrons travel from outside to in-side of the skyrmion structure. A partial wave analy-sis for scattering from a circular skyrmion has revealedthat a large number of angular momentum channels con-tribute to the resonant modes. We have also studied thelong wavelength regime, where the geometric details ofthe magnetic structure are unresolved so naturally thescattering process is expected to be independent of thenature of the classical dynamics. In this regime, resonantstates can still emerge as confined edge states inside ofthe magnetic structure, to which only a single angularmomentum channel contributes, leading to highly direc-tional skew scattering.In the short wavelength regime, the resonant statesmanifested as confined vortices inside of the skyrmionstructure can be exploited for electrically charging theskyrmion structure [32, 33], enabling the surface elec-trons on the TI to drive skyrmion motion with a lowcurrent and high thermal efficiency. In the long wave-length regime, the strong and robust directionality forskew scattering may be exploited for device applicationbased on the anomalous Hall effect.About experimental realization of a skyrmion struc-ture, we note that there is recent evidence of mag- netic skyrmion at the interface of the ferromagnet/TI(Cr Te /Bi Te ) heterostructure [70]. In addition, in-homogeneous Zeeman coupling can be tuned for aferromagnetic strip with strong out-of-plane magneticanisotropy [33]. For experimental control of electron scat-tering over a skyrmion structure, a quantum-dot type ofconfiguration with skyrmion structure in a finite scatter-ing region as well as with leads and contacts is neces-sary. The scattering configuration employed in our workis mainly for theoretical convenience with the goal to gaininsights into the physics of electron scattering over theskyrmion structure with classical integrable or chaoticdynamics. For this purpose, the geometrical structureof the skyrmion is chosen to be either circular for whichthe scattering cross sections can be calculated analyt-ically, or deformed for which the numerical method ofmultiple multipoles can be used to calculate the scatter-ing wave function and consequently the resonant states,the cross sections, the current and spin distribution. Ourresults provide useful hints about the scattering of spin-1/2 fermion over a skyrmion structure. If the device sizeis significantly larger than the electron wavelength, weexpect the main results to hold.A number of open issues are worth studying, suchas using spin transfer torque of the electrons to drivethe skyrmion motion, exploitation of skyrmion relatedswitches or oscillators, and scattering from multipleskyrmions that are themselves dynamic with possiblephase-locking or anti-phase locking behavior. ACKNOWLEDGMENTS
This work was supported by the Pentagon VannevarBush Faculty Fellowship program sponsored by the BasicResearch Office of the Assistant Secretary of Defense forResearch and Engineering and funded by the Office ofNaval Research through Grant No. N00014-16-1-2828.
APPENDIX: MULTIPLE MULTIPOLE (MMP)METHOD FOR SCATTERING OF DIRACELECTRONS ON THE TOP OF A TI FROM AMAGNETIC STRUCTURE
We denote the area outside and inside of the skyrmionstructure as regions I and II , respectively. The wave-function in region II can be written asΨ II ( r ) ≡ (cid:18) ψ I ψ II (cid:19) = (cid:88) m I (cid:88) l C m I l √ (cid:32) H (1) l − ( k II d m I ) e − iθ mI τ II H (1) l ( k II d m I ) (cid:33) e ilθ mI , (19)1 xy r j r m I r m II Ψ in Region I Ψ I Ψ II Region II
FIG. 13. A schematic illustration of the basics of the MMPmethod. Shown is placement of poles (fictitious sources) in-side and outside of a magnetic structure of arbitrary shape.The scattering spinor wavefunctions inside (outside) of thestructure are determined by the poles outside (inside) of thestructure. where k II = (cid:113) E − ∆ n II / (cid:126) v F ,τ II = − (cid:126) v F k II / ( E − ∆ n II ) ,d m I = | r − r m I | ,θ m I = Angle( r − r m I ) , and C m I l are the expansion coefficients. The scatteredwavefunction in region I isΨ I ( r ) ≡ (cid:18) ψ I ψ I (cid:19) = (cid:88) m II (cid:88) l C m II l √ (cid:32) H (1) l − ( k I d m II ) e − iθ mII τ I H (1) l ( k I d m II ) (cid:33) e ilθ mII , (20)where k I = (cid:113) ( E − ∆ n I / (cid:126) v F ,τ I = − (cid:126) v F k I / ( E − ∆ n I ) ,d m II = | r − r m II | ,θ m II = Angle( r − r m II ) , and C m II l are the expansion coefficients. The incidentplane wave propagating along the direction defined byan angle β with the x axis in region I is given byΨ in ( r ) ≡ (cid:18) ψ in ψ in (cid:19) = 1 √ (cid:18) − iτ I e iβ (cid:19) e i ( k x r cos θ + k y r sin θ ) . (21) Matching the boundary conditions( ψ I + ψ in ) | r j ∈ Γ = ψ II | r j ∈ Γ (22)( ψ I + ψ in ) | r j ∈ Γ = ψ II | r j ∈ Γ , (23)we get (cid:88) m II (cid:88) l C m II l √ τ I H (1) l ( k I | r j − r m II | ) e ilθ mII − (cid:88) m I (cid:88) l C m I l √ τ II H (1) l ( k II | r j − r m I | ) e ilθ mI = i √ τ I e iβ e i k I r (24)and (cid:88) m II (cid:88) l C m II l √ H (1) l − ( k I | r j − r m II | ) e i ( l − θ mII (25) − (cid:88) m I (cid:88) l C m I l √ H (1) l − ( k II | r j − r m I | ) e i ( l − θ mI = − √ τ I e iβ e i k I r , (26)which can be cast in a compact form as (cid:88) m II (cid:88) l j A Ilm II C m II l − (cid:88) m I (cid:88) l j A IIlm I C m I l = − j ψ inII (27) (cid:88) m II (cid:88) l j B Ilm II C m II l − (cid:88) m I (cid:88) l j B IIlm I C m I l = − j ψ inI (28)where j A Ilm II = 1 √ τ I H (1) l ( k I | r j − r m II | ) e ilθ mII , (29) j A IIlm I = 1 √ τ II H (1) l ( k II | r j − r m I | ) e ilθ mI , (30) j B Ilm II = 1 √ H (1) l − ( k I | r j − r m II | ) e i ( l − θ mII , (31) j B IIlm I = 1 √ H (1) l − ( k II | r j − r m I | ) e i ( l − θ mI , (32)and j ψ in = − i √ τ I e iβ e i k I r j , (33) j ψ in = 1 √ e i k I r j . (34)In principle, the set consists of an infinite number of equa-tions with an infinite number of undetermined expansioncoefficients C m II l and C m I l . To solve the system numer-ically, finite truncation is necessary. We set the totalnumber of boundary points to be J with M I and M II poles in regions I and II , respectively, and l → [ − L, L ]for all the multipoles. The process leads to the followingfinite-dimensional matrix equation: M J × N · C N × = − Y J × , (35)where N = (2 L + 1) × ( M I + M II ) = N I + N II ,2 C N × = C II − L ... C II l C II l ... C M II l ... C M II L C I − L ... C I l C I l ... C M I l ... C M I L N × ; Y J × = ψ in ... j ψ in ... J ψ in ψ in ... j ψ in ... J ψ in J × (36)and M J × N = (cid:18) A ( I ) − A ( II ) B ( I ) − B ( II ) (cid:19) (37)with A ( τ ) = A ( τ ) − L τ · · · A ( τ ) l τ A ( τ ) l τ · · · A ( τ ) lM τ · · · A ( τ ) LM τ A ( τ ) − L τ · · · A ( τ ) l τ A ( τ ) l τ · · · A ( τ ) lM τ · · · A ( τ ) LM τ ... · · · ... ... · · · ... · · · ... j A ( τ ) − L τ · · · j A ( τ ) l τ j A ( τ ) l τ · · · j A ( τ ) lM τ · · · j A ( τ ) LM τ ... · · · ... ... · · · ... · · · ... J A ( τ ) − L τ · · · J A ( τ ) l τ J A ( τ ) l τ · · · J A ( τ ) lM τ · · · J A ( τ ) LM τ . (38) B ( τ ) = B ( τ ) − L τ · · · B ( τ ) l τ B ( τ ) l τ · · · B ( τ ) lM τ · · · B ( τ ) LM τ B ( τ ) − L τ · · · B ( τ ) l τ B ( τ ) l τ · · · B ( τ ) lM τ · · · B ( τ ) LM τ ... · · · ... ... · · · ... · · · ... j B ( τ ) − L τ · · · j B ( τ ) l τ j B ( τ ) l τ · · · j B ( τ ) lM τ · · · j B ( τ ) LM τ ... · · · ... ... · · · ... · · · ... J B ( τ ) − L τ · · · J B ( τ ) l τ J B ( τ ) l τ · · · J B ( τ ) lM τ · · · J B ( τ ) LM τ . 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