Kramers Weyl Semimetals as Quantum Solenoids and Their Applications in Spin-Orbit Torque Devices
KKramers Weyl Semimetals as Quantum Solenoids and Their Applications inSpin-Orbit Torque Devices
Wen-Yu He, ∗ Xiao Yan Xu, and K. T. Law † Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China (Dated: October 21, 2019)Recently, Kramers Weyl semimetals (KWS) with chiral lattice structures and Weyl points pinnedat time-reversal invariant momenta have been discovered. In this work, we show that both the topo-logical and symmetry properties of KWS are important for their applications in spintronic devices.More specifically, due to the chiral lattice symmetry, KWS exhibit a longitudinal magnetoelectric re-sponse in which the charge current induced spin and orbital magnetization is parallel to the directionof the current. This feature allows KWS to act as nanoscale quantum solenoids with both orbitaland spin magnetization. Importantly, near the Kramers Weyl points, the orbital contribution to themagetization can be large due to the large Berry curvatures near the Weyl points. As a result, whenelectrons of KWS are injected into a ferromagnet, both the orbital and spin angular momentumcarried by the electrons can induce torques for magnetization switching in the ferromagnet. Wefurther show that KWS can be used for new designs of spin-orbit torque devices.
Introduction
Weyl semimetals are nodal topological materials char-acterised by isolated band touching points, called Weylpoints, in 3D momentum space [1–4]. Due to the Weylpoints, which act as the monopoles of the Berry cur-vature in the momentum space [1–4], Weyl semimetalsexhibit many exotic properties such as the chiral mag-netic effect [5–8], the presence of topologically protectedFermi arcs states [1–4, 9, 10], unconventional quantumoscillations [11], and novel optical phenomena [12–14].Recently, a new type of Weyl semimetals called KramersWeyl semimetals (KWS) in chiral crystals have been dis-covered [15–22]. Chiral crystals are crystals which lackinversion, mirror and improper rotation symmetries. Asa result, a chiral crystal has a definite handedness andcan be described by the 11 chiral point groups. It wasshown that generally, due to the low lattice symmetry,band splittings appear away from time-reversal invariantpoints in momentum space and result in Kramers Weylpoints pinned at time-reversal invariant momenta. How-ever, it is not clear how the properties of KWS are dis-tinct from that of other Weyl semimetals with non-chiralpoint group symmetry.In this work, we point out a unique property of KWS,that is an electric field applied along the principal sym-metry axis of the crystal would induce spin and orbitalmagnetization which is parallel to the applied electricfield. This is in sharp contrast to the case of all other non-centrosymmetric Weyl semimetals which give zero mag-netoelectric response if the electric field is applied alongthe principal symmetry axis. Interestingly, the orbitalmagnetization is significantly enhanced near the Weylpoints due to the large Berry curvatures. Therefore, ifthe electrons which carry both orbital and spin angularmomentum are injected into a ferromagnetic layer, thetorque induced by the electrons can cause magnetizationswitching in the ferromagnetic layer as shown in Fig.1(a). The KWS based device is different from the magnetictunnelling devices based on spin transfer torque [23, 24]shown in Fig.1(b) and the spin-orbit torque devices [23–26] shown in Fig.1(c). The detail comparison of the threedesigns is given in the Discussion Section.In the following sections, we demonstrate that the spe-cial form of the spin-orbit coupling (SOC) of KWS isresponsible for generating the spin and orbital magne-tization for the Bloch electrons. In the presence of anelectric field, the non-equilibrium distribution of the elec-trons would result in a net spin [27] and orbital magneti-zation [28, 29]. The general form of the magnetoelectricresponse is determined by the symmetry of the chiralcrystal and KWS can be classified into three subclasses.In the cases of KWS such as β − RhSi, CoSi and AlPtwhich had been studied experimentally with cubic pointgroup symmetries T and O [18–22], the induced magne-tization is parallel to the applied electric field. Unlikea classical solenoid in which the magnetic field directionis determined by the structure of the solenoid, KWS ex-hibit the longitudinal magnetoelectric response in all fielddirections without the need to fabricate any spiral struc-ture. For KWS with dihedral (D n ) point groups, such asCsCuBr [18], and elemental Te, Se [30, 31], a purely lon-gitudinal response is also obtained when the electric fieldis applied along the direction of the symmetry axes. Onthe other hand, for KWS with cyclic (C n ) point groupssuch as Ca B Os [18], a longitudinal magnetoelectric re-sponse is generally accompanied by a transverse response.For realistic calculations of the magnetoelectric effect,we consider the representative KWS K Sn O in cu-bic point group, CsCuBr in dihedral point group, andCa B Os in cyclic point group with the realistic bandstructure and further apply the linear response theoryto calculate the magnetoelectric susceptibility pseudoten-sor. We obtain the realistic magnetoelectric susceptibil-ity for the three materials and show that the inducedmagnetization at a given electric field can be two to a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t a b c JM MMJ J M
KWS
Ferromagnet
Reference layer SpacerRecording layer Recording layerMetal with SOC electron
FIG. 1: The schematics of current induced magnetization switching in spintronic devices. (a) A KWS/ferromagnet heterostruc-ture. When a current J passes through the KWS, the effective magnetic field at the KWS and the electrons injected from KWSto the ferromagnetic layer provide a torque to switch the magnetization direction (M) of the ferromagnetic layer. (b) A magnetictunnelling junction. The junction is made of a reference ferromagnetic layer and the recording ferromagnetic layer separatedby a metallic or insulating spacer. The magnetization of the recording layer can be switched by the spin polarized electronscoming out of the reference layer. (c) A spin-orbit torque device [23–26] . The current induces magnetization at the metallayer through magnetoelectric effect (or the inverse spin galvanic effect). The effective magnetic field at the metal/ferromagnetinterface causes the magnetization switching in the ferromagnetic layer. three orders of magnitude larger than the magnetizationinduced in materials with the strong Rashba spin-orbitcouplings such as in Au (111) surfaces and Bi/Ag bilay-ers [32, 33]. These specific KWS were chosen becausethey exhibit large SOC at the Fermi level and they areexpected to exhibit large magnetoelectric response. ResultsEffective Hamiltonians for Kramers WeylSemimetals
In chiral crystals which respect time reversal symme-try, the energy bands are at least doubly degenerateat time-reversal invariant momenta due to the Kramerstheorem. In the absence of inversion, mirror and im-proper rotation symmetries in chiral crystals, and awayfrom the time-reversal invariant points, SOC would liftthe Kramers degeneracy in momentum space to cre-ate Kramers Weyl points [18]. To be more specific, inthe spin basis ψ k = [ φ k , ↑ , φ k , ↓ ] T , which satisfies therelation Θ ψ k = iσ y ψ − k under time-reversal operationΘ = iσ y K , the effective Hamiltonian H ( k ) can be ob-tained through standard k · p method [18]. Up to secondorder near a time-reversal invariant momentum k , thegeneral form of the Hamiltonian can be written as H ( k ) = (cid:88) i,j (cid:126) m ij k i k j + σ i (cid:126) v ij k j . (1)Here, k is measured from k , i, j = x, y, z , m ij is theeffective mass tensor, σ i are the Pauli matrices in spinspace, and v ij is the SOC pseudotensor. In chiral crys-tals, the little group at k is isomorphic to a chiral pointgroup which guarantees det ( v ) (cid:54) = 0 so that the KramersWeyl point emerges at k = 0. In the KWS, the specific forms of the SOC that cre-ates the Weyl point are determined by the point groupsymmetry as v = det (cid:16) ˆ R (cid:17) ˆ Rv ˆ R T , with ˆ R the symme-try transformation matrix. In materials within the cu-bic point group { T , O } such as K Sn O , β − RhSi, CoSi,and AlPt [18–22], at k (such as the Γ point) the littlegroup isomorphic to { T , O } forces v ij to be proportionalto the identity matrix and gives rise to the isotropic WeylHamiltonian H ( k ) = (cid:126) m k + (cid:126) v k · σ . (2)The isotropic SOC (cid:126) v k · σ arising from the high symmetrycubic point group { T , O } brings about the Weyl point at k . In the materials CsCuBr [18], elemental Te, Se [30,31], etc., the dihedral point group there has lower crystalsymmetry and allows the anisotropy to show up in theWeyl Hamiltonian H ( k ) = (cid:88) i = x,y,z (cid:126) m i k i + (cid:126) v i σ i k i . (3)In the materials belonging to the cyclic point group, asthe crystal symmetry is further reduced, the constraintson the SOC is further reduced. The complete forms ofthe SOC pseudotensor v ij in the KWS Hamiltonian aresummarised in the Supplementary Materials for all thechiral point groups [34].The SOC in the KWS creates the Kramers Weyl pointsat the time reversal invariant momenta and allows thecoupling between the spin and kinetic momentum. Un-der an electric field, the SOC enables the charge carriersto have net magnetization and such magnetoelectric re-sponse respects the same crystal symmetry present inthe SOC. As shown below, for materials with cubic pointgroups, the simple form of the isotropic Weyl Hamilto-nian in Eq. 2 allows us to calculate the magnetoelectricsusceptibility analytically. For the KWS in dihedral andcyclic point group, the magnetoelectric responses are cal-culated numerically for selected materials. Magnetoelectric Pseudotensors and Their SymmetryProperties
In magnetoelectric effects, induced magnetization M and the applied electric field E are related by the mag-netoelectric pseudotensor α such that: M i = (cid:88) i,j α ij E j , (4)where i, j = x, y, z and α ij are elements of the magneto-electric pseudotensor α . For a generic Hamiltonian H = (cid:88) ν,ν (cid:48) , k c † ν, k H ,νν (cid:48) ( k ) c ν (cid:48) , k , (5)where c † ν, k ( c ν, k ) is the creation (annihilation) opera-tor, H ,νν (cid:48) ( k ) is the element of the Hamiltonian matrix H ( k ), α ij can be obtained from the linear response the-ory as [28, 29] α ij = − τ e (cid:126) π ) d (cid:90) BZ d k (cid:88) n M n k ,i v n k ,j df ( E n k ) dE n k . (6)In Eq.6, f ( E n k ) is the Fermi Dirac distribution func-tion, E n k is the energy dispersion of band n from theHamiltonian H ( k ), v n k ,j = ∂E n k ∂k j , d is the dimensionof the system, τ is the effective scattering time and i, j = x, y, z denote the spatial components. The totalmagnetic moment M n k = S n k + m n k carried by theBloch electrons consists of both the spin magnetic mo-ment S n k = (cid:104) φ k ,n | gµ b σ | φ k ,n (cid:105) and the orbital magneticmoment m n k = ie (cid:126) (cid:104) ∂ k φ k ,n | × [ H ( k ) − E n k ] | ∂ k φ k ,n (cid:105) .Here, µ b = e (cid:126) m e is the Bohr magneton, g is the Lande g factor which is set to be 2 in our calculations and | φ k ,n (cid:105) denotes a Bloch state. As we will show explic-itly below, the orbital magnetization is related to theBerry curvature of the Bloch states which has the form Ω n k = i (cid:104) ∂ k φ k ,n | × | ∂ k φ k ,n (cid:105) [35].The linear response theory applies to generic Hamilto-nians. However, to shed light on the general propertiesof KWS, we note that the form of α can be determinedby point group symmetries which is independent of thedetails of the Hamiltonian. The group theory analysisof α is elaborated in the Methods section as well as inRef. [36], and the general form for the chiral point groupsis provided in Table I. From the group theory point ofview, KWS can be classified into three sub-classes. ForKWS belonging to the cubic point groups { T , O } , α is proportional to the identity matrix as shown in Ta-ble I. This implies that the induced magnetization is al-ways parallel to the direction of the applied electric field. TABLE I: of Magnetoelectric susceptibility pseudotensor α for the chiral crystals in the 11 chiral point groups. α ij with i, j = x, y, z are in general the elements in α . In point groupwith symmetry, the 9 elements in α ij is no longer independent. α means α = α xx = α yy = α zz in T and O point group.In the { C , C , C , D , D , D } group α xx = α yy is thendenoted as α (cid:107) = α xx = α yy . α − means the antisymmetricelements as α − = − α xy = α yx in group { C , C , C } . Theprincipal axis of the crystal is set along z .Point group α Point group α O α α
00 0 α T α α
00 0 α D α xx α yy
00 0 α zz D α (cid:107) α (cid:107)
00 0 α zz D α (cid:107) α (cid:107)
00 0 α zz D α (cid:107) α (cid:107)
00 0 α zz C α xx α xy α xz α yx α yy α yz α zx α zy α zz C α xx α xy α yx α yy
00 0 α zz C α (cid:107) − α − α − α (cid:107)
00 0 α zz C α (cid:107) − α − α − α (cid:107)
00 0 α zz C α (cid:107) − α − α − α (cid:107)
00 0 α zz Therefore, these KWS can behave as classical solenoids inall electric field directions without the need to fabricateany spiral structures.For KWS with point groups D n , a pure longitudinalmagnetization parallel to the electric field is also obtainedwhen the electric field is applied along the direction ofany of the symmetry axes. For KWS with cyclic pointgroups, in general, magnetization with both componentsparallel and perpendicular to the direction of the appliedelectric field are generated. Interestingly, for all otherWeyl semimetals without chiral point group symmetry,the magnetoelectric response is zero if the electric fieldis applied along the principal axis [34]. Therefore, thelongitudinal magnetoelectric response along the principalsymmetry axis is a very distinctive feature of KWS dueto the special spin texture of KWS which determines thespin and orbital magnetization.Combined with the effective Hamiltonian for isotropicKWS in Eq. 2, the magnetoelectric susceptibility in Eq.6 can be explicitly calculated and it shows how the SOCstrength and band dispersion will influence the magneto-electric response. M y M z a b E x E y E z M x c d -4 -2 0 2 4 k (nm -1 ) -0.0500.050.1 E ( e V ) E F (eV) V - m / u . c . )( μ b α -0.04 0.04 0.12 0.2012345 10 -7 × FIG. 2: The longitudinal magnetoelectric response ofisotropic KWS. (a) The Weyl spin texture of isotropic WeylSOC (cid:126) v k · σ at the Fermi surface from the band branch +. (b)The electrically induced magnetization is parallel to the ap-plied electric field. (c) The energy dispersion for the isotropicchiral Weyl semimetal. (d) The magnetoelectric susceptibil-ity α strength as a function of the Fermi energy E F . In thepresence of external electric field, α gives the number of Bohrmagneton ( µ b ) per unit cell (u.c.) to denote the magnetiza-tion in the material. Longitudinal Magnetoelectric Response in IsotropicKWS
We first consider an effective Hamiltonian which de-scribes a Kramers Weyl point near the Γ point in chi-ral crystals with point groups { T , O } where the isotropicWeyl Hamiltonian is H ( k ) = (cid:126) m k + (cid:126) v k · σ . At Fermienergy E F = (cid:126) m k ± (cid:126) v | k | , there are two spherical Fermisurfaces with corresponding wave vectors k F ± = k F ± ˆ k ,where k F ± = (cid:126) √ mE F + m v ∓ mv (cid:126) and ˆ k = k | k | . Thespin and orbital magnetic moments of the two Fermi sur-faces with Fermi momenta k F ± can be written as S k F ± = ± g e (cid:126) m e ˆ k F ± , and m k F ± = ev k F ± | k F ± | . (7)It is important to note that the orbital magnetic mo-ment m k F ± is proportional to the Berry curvature gen-erated by the Weyl point on the Fermi surfaces which is Ω k F ± = ∓ ˆ k F ± k ± . The spin texture on a Fermi surface isschematically shown in Fig.2(a). It is clear that withoutbreaking time-reversal symmetry, the total magnetic mo-ment of all the electrons is zero. By applying an electricfield, the steady state distribution of the electronic statecan generate a net magnetization as indicated in Eq. 6.With this special form of spin texture of an isotropic a b H N0.60.91.21.5 E ( e V ) ΓΓ E (eV) -1.2-0.600.6 V - m / u . c . ) -6 ( μ b × α FIG. 3: The isotropic longitudinal magnetoelectric responsein K Sn O . (a) The energy dispersion in the conductionband of K Sn O . The red circle labels the Kramers Weylpoint that respect the isotropic Weyl Haimltonian in Eq. 2.(b) The isotropic longitudinal magnetoelectric susceptibility α as a function of conduction band energy. The dashed lineis the analytical result from Eq. 8 with m = 1 . m e , (cid:126) v = − · nm, τ = 6ps. u.c. is short for unit cell. KWS, at the Fermi energy E F , we obtain the isotropiclongitudinal magnetoelectric susceptibility α as [34] α = − e vτ (cid:126) π (cid:112) mE F + m v (cid:18) g mm e − (cid:19) . (8)In α , the first and second terms are the spin and the or-bital contributions respectively. It is important to notethat, for hole bands with negative effective mass, thespin and orbital contribution will always add togetherto enhance the magnetoelectric response. From Table I,the magnetoelectric response of an isotropic KWS can bewritten as M = α E which indicates that the magneti-zation induced is parallel to the applied electric field as isschematically shown in Fig.2(b). From Eq. 8, it is clearthat strong Weyl SOC v , long scattering time τ and largeeffective mass m can give large magnetoelectric response.To seek a large magnetoelectric response from realis-tic materials, we note that, for example, the conductionbands of K Sn O show large spin splitting around theKramers Weyl points [18]. The K Sn O belongs to thecubic T point group and has the Kramers Weyl pointat H near the conduction band bottom. In the slightn-doped state, the electrons will occupy Fermi pocketsenclosing the Kramers Weyl point at H and the isotropicWeyl Hamiltonian can be effectively described by Eq. 2,with the effective mass m = 1 . m e and SOC strength (cid:126) v = − · nm. The band dispersion from Eq. 2is present in Fig. 2(c) and the parameters are extractedfrom the first-principle calculations depicted in Fig. 3(a).To acquire a strong magnetoelectric response, it is alsopreferable to have long scattering length as shown inEq.2. In KWS, since the Kramers Weyl points are pinnedat the time reversal invariant momenta, the electrons onopposite sides of the Weyl point have opposite spin. Asa result, the SOC suppresses elastic backscattering fromscalar impurities, similar to the case in the surface statesof topological insulators [37–39]. The Weyl SOC can en- Y S Z00.10.20.30.4 E ( e V ) Γ Γ a b
E (eV) -50510 10 -7 xxyyzz ααα× V - m / u . c . )( μ b α FIG. 4: The anisotropic longitudinal magnetoelectric effectin CsCuBr . (a) The band structure for CsCuBr . (b)The anisotropic longitudinal magnetoelectric susceptibility { α xx , α yy , α zz } as a function of the band energy. The ef-fective scattering time is taken to be τ = 6ps. u.c. is shortfor unit cell. hance the scattering time by a factor of 3 as shown inthe Supplementary Materials [34] using Born approxima-tion. Therefore, we choose a scattering time at 6ps whichis comparable to the inter Weyl point scattering time inTaAs [40]. With the effective mass, Weyl SOC strengthand effective scattering time, the isotropic magnetoelec-tric susceptibility α as a function of Fermi energy E F isevaluated as shown in Fig. 2(d). For the Fermi pocketsenclosing a single Kramers Weyl point, the isotropic mag-netoelectric susceptibility α increases with the squareroot of the Fermi energy E F .To validate the calculations using an effective Hamil-tonian in the form of Eq.2, we constructed a realistictight-binding model for K Sn O to calculate the mag-netoelectric susceptibility α as a function of Fermi en-ergy. The Wannier functions for constructing the tightbinding model are elaborated in the Methods Section.In Fig. 3(b), the numerically calculated α in a rangeof conduction band energy is shown. Below E = 0 . α as a function of E is morecomplicated. Assuming that the chemical potential liesat E = 0 . V/m is enough to generate a magne-tization of 0 . µ b per unit cell. It is two to three ordersof magnitude larger than the magnetoelectric responseof Au(111) surfaces and Bi/Ag bilayers which have largeRashba spin-orbit couplings [32, 33].It is important to note that, the electrons near theWeyl points carry both spin and orbital magnetizations.It is clear from Eq.7 that while the spin magnetization isconstant near the Weyl point, the orbital magnetizationis stronger near the Weyl point. Both the spin and or-bital angular magntization can work together for currentinduced magnetization switching in spin-orbit torque de-vices. L Y Z-0.500.51 E ( e V ) Γ a b -0.3 0 0.3 0.6 E (eV) -1012 10 -7 xxyyyzzyzz ααααα× V - m / u . c . )( μ b α FIG. 5: The KWS Ca B Os with both longitudinal andtransverse magnetoelectric response. (a) The band structurefor Ca B Os . (b) The nonvanishing magnetoelectric sus-ceptibility elements as a function of the band energy. Theeffective scattering time is taken to be τ = 6ps. u.c. is shortfor unit cell. Anisotropic Longitudinal Magnetoelectric Responsein KWS in Dihedral Point Groups
In the KWS with dihedral point group symmetry,the magnetoelectric susceptibility from Table I has thenonzero value only in the diagonal elements. This in-dicates that the magnetization generated is parallel tothe electric field if the electric field is applied along thesymmetry axes. For example, the KWS CsCuBr be-longs to the D point group and its realistic band struc-ture in Fig. 4(a) shows the Kramers Weyl points atthe time reversal invariant momenta { Γ, Y, S, Z } . Nearthe Kramers Weyl points, the bands splittings are highlyanisotropic as allowed by the D point group symmetry.The anisotropic longitudinal magnetoelectric susceptibil-ity { α xx , α yy , α zz } , as a function of the Fermi energy, arenumerically evaluated based on the realistic band struc-ture, are depicted in Fig. 4(b). Coexistence of Longitudinal and TransverseMagnetoelectric response in KWS in Cyclic PointGroups
The KWS in the cyclic point group possess a polaraxis that leaves the crystal invariant under rotation. Thissymmetry allows both the longitudinal and the transversemagnetoelectric response: an electric field along the polaraxis can generate pure longitudinal magnetization whilethe electric field inside the basal plane produces magne-tization with nonzero parallel and perpendicular compo-nents. In Ca B Os with a single C rotation axis, thespin split bands with Kramers Weyl points are shown inFig. 5(a). Based on the realistic band structure, the non-vanishing magnetoelectric susceptibility elements underthe C symmetry are numerically calculated at differentenergy. The results are shown in Fig. 5(b) and large mag-netoelectric response is also obtained due to the strongspin-orbit coupling of the material. Discussion
We note that the current induced magnetization inWeyl semimetals was first studied by Johansson et al.in TaAs which belong to the point group C v with mir-ror planes [33]. Therefore, the Weyl points appear atgeneral k points and the resulting spin polarization in-duced by an electric field is perpendicular to the directionof the electric field. This transverse magnetoelectric ef-fect is similar to the case with Rashba SOC which is aproperty of the polar point group as shown in Supple-mentary Materials [27, 34]. If the applied electric fieldis along the principal axis, however, the magnetoelectricresponse is zero in TaAs as determined by the C v pointgroup symmetry.On the other hand, the longitudinal magnetoelectricresponse in Weyl semimetals was first studied by Yodaet al. [41, 42]. However, in their models, helical hop-ping textures [41, 42] are needed for electrons to hop ina spiral manner, imitating the movement of electrons ina classical solenoid. As a result, an orbital magnetiza-tion parallel to the direction of an applied electric fieldwould be generated and the longitudinal magnetization ispresent even without spin-orbit coupling. Unfortunately,no realistic materials that possess such helical hoppingtextures are identified. In this work, the origin of thelongitudinal response is purely induced by SOC of KWS.Importantly, as shown in Eq.7, the orbital magnetizationvanishes if the spin-orbit coupling strength v is zero.Concerning the applications of KWS, due to itsunique longitudinal magnetoelectric response, theKWS/ferromagnet heterostructures can be used for newdesigns of spin-orbit torque devices as shown in Fig.1(a).It is interesting to note that a KWS can cause current in-duced magnetization switching in the ferromagnetic layerby two effects. First, as depicted in Fig.1(a), the KWScan inject electrons which carry both orbital and spinangular momentum into the ferromagnetic layer. Theinjection of angular momentum can cause magnetiza-tion switching similar to the spin transfer torque inducedmagnetization switching in magnetic tunnelling junctionsas shown in Fig.1(b). The important difference is thatthe electrons injected by KWS can carry both orbital andspin angular momentum. Particularly, close to the Weylpoint, the orbital magnetization carried by the electronscan be significant. Therefore, KWS can work as a sourceof spin and orbital angular momentum for magnetiza-tion switching. Second, when a current is passed alongthe principal symmetry axis of the KWS, a magnetiza-tion is induced at the bulk of the KWS as well as theinterface between the KWS and the ferromagnet. Thiscurrent induced magnetization can cause the magneti-zation switching of the ferromagnet layer through ferro-magnetic coupling between the KWS and the ferromag-net. This magnetization switching mechanism is simi- lar to the conventional spin-orbit torque devices shownin Fig.1(c). Therefore, due to the unique longitudinalresponse of KWS, they allow new designs of spintronicdevices.Concerning the candidate materials of KWS, besidesthe materials which have been studied experimentally re-cently such as β − RhSi, CoSi and AlPt, elemental Seand Te also have chiral lattice structure [30, 31]. In-terestingly, a few superconducting materials with chirallattice symmetry and strong spin-orbit coupling such asLi Pt B [43], Li Pd B [44], Mo Al C [45], have been ex-perimentally studied. In their normal state, these su-perconducting materials can also have spintronic appli-cations.
MethodsSymmetry Analysis for the MagnetoelectricResponse in Chiral Crystals
In the chiral crystals with magnetoelectric effect M i = (cid:80) i,j α ij E j with i, j = x, y, z , under the crystal symmetrythe magnetization transforms as M i → det (cid:16) ˆ R (cid:17) ˆ R ij M j while the electric field transforms as E i → ˆ R ij E j , whereˆ R represents the symmetry transformation operator andis an orthogonal matrix. As a result, the magnetoelectricsusceptibility α ij under the crystal symmetry respects α = det (cid:16) ˆ R (cid:17) ˆ Rα ˆ R T . (9)The chiral point groups, which do not allow improperrotations, can be divided into three sub-classes: thecubic point groups { T, O } , the dihedral point groups { D n } with n = 2 , , , C n with n = 1 , , , ,
6. In the cubic point groups { T, O } ,the multiple high order rotation axes along different di-rections would force the magnetoelectric susceptibility tobe proportional to the identical matrix. In the dihedralpoint groups { D n } with n=2, 3, 4, 6, the C n rotationaxis along z and the in-plane C rotation axis along the x -axis would eliminate all the off diagonal elements andleave only the diagonal elements in the magnetoelectricsusceptibility α = diag { α xx , α yy , α zz } . In { D , D , D } ,the principal axis would further make α xx = α yy . Inthe cyclic point groups { C n } with n = 1 , , , ,
6, thelower symmetry would allow the off-diagonal elements tocoexist with the diagonal elements. The explicit formsof the magnetoelectric susceptibility pseudotensor α ij from the symmetry analysis is shown in Table I for the11 chiral point groups. The general forms of α for allnoncentrosymmetric point groups with nonvanishing α are given in the Supplementary Materials [34]. First-principles Calculation and Wannier FunctionConstruction
We perform first-principles calculation within the den-sity functional theory framework as implemented in Vi-enna Abinitio Simulation Package(VASP) [46]. PAW [47,48] type of pseudopotential with PBE exchange func-tional [49] is used in the calculations and spin orbitcoupling is included in the pseudopotentials. After thefirst-principle calculation is done, tight binding Hamil-tonian which perfectly recovers bands near Fermi sur-face is built through Wannier function construction us-ing package Wannier90 [50, 51]. The lattice structuresof all the materials we considered are obtained from theICSD [52]. We select the time reversal invariant momentawith bands near the Fermi level to show the energy dis-persion in Fig. 3, 4, 5, and the full band dispersion alongthe symmetry path is present in Supplementary Materi-als [34].
Acknowledgement
W.-Y. He , X.-Y. Xu and K. T. Law are thankful forthe support of HKRGC through C6026-16W, 16324216,16307117 and 16309718. K. T. Law is further supportedby the Croucher Foundation and the Dr. Tai-chin LoFoundation. ∗ [email protected] † [email protected][1] X. Wan, A. M. Turner, A. Vishwanath, S. Y. Savrasov,Topological semimetal and Fermi-arc surface states in theelectronic structure of pyrochlore iridates. Phys, Rev. B , (2011).[2] L. Balents, Weyl electrons kiss. Physics , 36 (2011).[3] A. A. Burkov, L. Balents, Weyl semimetal in a topologicalinsulator multilayer. Phys. Rev. Lett. , 127205 (2011).[4] H. Weng, C. Fang, Z. Fang, B. A. Bernevig, Xi. Dai, Weylsemimetal phase in noncentrosymmetric transition-metalmonophosphides.
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Supplementary Material: Kramers Weyl Semimetals as Quantum Solenoids and Their Applicationsin Spin-Orbit Torque Devices
SUPPLEMENTARY NOTE 1: MAGNETOELECTRIC SUSCEPTIBILITY IN ISOTROPIC KWS
In the isotropic KWS, at the Fermi energy E F = (cid:126) k m ± (cid:126) v | k | , there are two spherical Fermi surfaces with the cor-responding wave vectors k F ± = k F ± ˆ k and k F = (cid:126) (cid:0) √ mE F + m v ∓ mv (cid:1) , ˆ k = k | k | . On the Fermi surfaces enclosingthe Kramers Weyl point, the spin magnetic dipole moment and orbital magnetic dipole moment are respectively S k F ± = ± g e (cid:126) m e ˆ k F , m k F ± = ± ev ˆ k F | k F | , (S1)with ± denoting the two band branches. The density of states can be obtained as N ± ( E F ) = m π (cid:126) (cid:112) mE F + m v + mv (cid:113) E F m + v ∓ mv , (S2)and the Fermi velocities there become v k F ± = (cid:114) E F m + v ˆ k F ± . (S3)At the Fermi energy, the magnetoelectric susceptibility α can be approximated through π ) d (cid:82) BZ d k → (cid:80) n N n ( E F ) (cid:82) d Ω k F ,n π with n = ± and then we can obtain α = − e vτ (cid:126) π (cid:112) mE F + m v (cid:18) g mm e − (cid:19) . (S4) SUPPLEMENTARY NOTE 2: ISOTROPIC WEYL SOC SUPPRESSED BACK SCATTERING
In the pure scalar potential scattering process H imp = (cid:90) d k d k (cid:48) ψ † k (cid:48) V k − k (cid:48) ψ k , (S5)the scattering potential can be projected onto the band basis asˆ V k − k (cid:48) = V k − k (cid:48) U † k (cid:48) U k , (S6)with U k = (cid:18) cos θ e − iφ sin θ e − iφ sin θ − cos θ (cid:19) , (S7)and θ , φ are the azimuth and polar angle of k . In the presence of large SOC, the scattering is dominated in eachband branch ( ± ) and the effective scattering potential becomesˆ V k − k (cid:48) , + = V k − k (cid:48) (cid:20) cos θ (cid:48) θ e − i ( φ (cid:48) − φ ) + sin θ (cid:48) θ (cid:21) (S8)ˆ V k − k (cid:48) , − = V k − k (cid:48) (cid:20) sin θ (cid:48) θ e i ( φ (cid:48) − φ ) + cos θ (cid:48) θ (cid:21) . (S9)In the Born approximation, the scattering in each band branch is obtained as1 τ ± = mk F ± n imp π (cid:126) (cid:90) π dϑ (cid:90) π dϕ ˆ V k − k (cid:48) , ± (1 − cos ϑ ) sin ϑ, (S10) X S R A Z YX A T Y/Z T-1-0.500.5 E ( e V ) H N P H/P N-1.5-1-0.500.511.5 E ( e V ) Γ Γ
Γ Γ
YF L I/I Z X/X Y/M N/Z F -0.6-0.4-0.200.20.40.60.8 E ( e V ) Γ Γ Γ a b c
FIG. S1: The band structure for K Sn O in (a), CsCuBr in (b) and Ca b Os in (c) with the corresponding Brillouin zone.The high symmetry path is illustrated in the Brillouin zone. The two Weyl bands in K Sn O around H at the conductionband bottom are fitted with the isotropic Weyl Hamiltonian with m = 1 . m e , (cid:126) v = − · nm. with ϕ = φ (cid:48) − φ , ϑ = θ (cid:48) − θ . In the spherical Fermi surfaces, the scattering potential ˆ V k − k (cid:48) , ± only depends on theangle between k and k (cid:48) , so (cid:90) π dϑ (cid:90) π dϕ ˆ V k − k (cid:48) , ± (1 − cos ϑ ) sin ϑ = 172105 k ± , (S11) (cid:90) π dϑ (cid:90) π dϕV k − k (cid:48) , ± (1 − cos ϑ ) sin ϑ = 163 k . (S12)As a result, the Weyl SOC reduces the scattering potential to about 0.3 and suppresses the back scattering. ∗ [email protected] † [email protected] TABLE S1: List of Magnetoelectric suceptibility pseudotensor α ij and the SOC pseudotensor v ij for crystals in the 18 gyrotropicpoint groups. Under the crystal symmetry transformation, the α ij and v ij have the same behavior. The elements in α ij and v ij is no longer independent. In T and O point group, α ij = α δ ij , v ij = vδ ij . In the group { C , C , C , D , D , D } , α (cid:107) and v (cid:107) is used to denote α xx = α yy = α (cid:107) , v xx = v yy = v (cid:107) respectively. The superscript d and - are used to denote the symmetric andanti-symmetric element respectively. The coordinate is set to have principal axis along z . δ ij is the Kronecker delta functionwith i, j = x, y, z . The chiral point groups are highlighted in blue color.Point group v ij α ij Point group v ij α ij C v xx v xy v xz v yx v yy v yz v zx v zy v zz α xx α xy α xz α yx α yy α yz α zx α zy α zz C v − v − v − − α − α − C v xx v xy v yx v yy
00 0 v zz α xx α xy α yx α yy
00 0 α zz D d v (cid:107) − v (cid:107)
00 0 0 α (cid:107) − α (cid:107)
00 0 0 C v (cid:107) − v − v − v (cid:107)
00 0 v zz α (cid:107) − α − α − α (cid:107)
00 0 α zz S v (cid:107) v d v d − v (cid:107)
00 0 0 α (cid:107) α d α d − α (cid:107)
00 0 0 C v (cid:107) − v − v − v (cid:107)
00 0 v zz α (cid:107) − α − α − α (cid:107)
00 0 α zz D v xx v yy
00 0 v zz α xx α yy
00 0 α zz C v (cid:107) − v − v − v (cid:107)
00 0 v zz α (cid:107) − α − α − α (cid:107)
00 0 T zz D v (cid:107) v (cid:107)
00 0 v zz α (cid:107) α (cid:107)
00 0 α zz C v v xy v yx v yz v zy α xy α yx T yz α zy D v (cid:107) v (cid:107)
00 0 v zz α (cid:107) α (cid:107)
00 0 α zz C v v xy v yx α xy α yx D v (cid:107) v (cid:107)
00 0 v zz α (cid:107) α (cid:107)
00 0 α zz C v − v − v − − α − α − T v v
00 0 v α α
00 0 α C v − v − v − − α − α − O v v
00 0 v α α
00 0 α0