Designing Rashba-Dresselhaus effect in magnetic insulators

One of the major strategies to control magnetism in spintronics is to utilize the coupling between electron spin and its orbital motion. The Rashba and Dresselhaus spin-orbit couplings induce magnetic textures of band electrons called spin momentum locking, which produces a spin torque by the injection of electric current. However, joule heating had been a bottleneck for device applications. Here, we propose a theory to generate further rich spin textures in insulating antiferromagnets with broken spatial inversion symmetry (SIS), which is easily controlled by a small magnetic field. In antiferromagnets, the ordered moments host two species of magnons that serve as internal degrees of freedom in analogy with electron spins. The Dzyaloshinskii-Moriya interaction introduced by the SIS breaking couples the two-magnon-degrees of freedom with the magnon momentum. We present a systematic way to design such texture and to detect it via magnonic spin current for the realization of antiferromagnetic memory.


I. INTRODUCTION
Development of tunable magnetic structure has long been a key issue to efficiently control magnetic moments via electric excitations using electric fields or current injections 1 . Besides the conventional domain walls that appear in real space, particular focus is given on emergent spin textures in reciprocal space, called "spin momentum locking". Two well-known classes are Rashba-2-4 and Dresselhaus-types 5 that take the vortex-and anti-vortex profile of spin structures, respectively, along the closed Fermi surfaces. There, the spin-orbit (SO) interaction induced by the spatial inversion symmetry (SIS) breaking shifts the energy bands odd-in-momentum k, and locks the spin moments they host. Since k distinguishes the electronic state of matter, such spin texture allows the selection of magnetic moment the state/current carries. This has brought about fundamentally important and technologically promising phenomena including spin Hall effect [6][7][8][9] , spin-orbit torque 10,11 , and Rashba-Edelstein effect [12][13][14] .
In insulating magnets, the excitation is carried by the quasiparticle called magnon, which represents a quantum mechanical spin precession propagating in space. Such propagation is predominantly mediated by the standard magnetic exchange interaction JS i · S j , between spins, S i and S j . In a uniform ferromagnet, a simple exchange interaction, J(< 0), generates non-degenerate quadratic magnon bands. When the SIS is broken, an antisymmetric spin exchange called Dzyaloshinskii-Moriya (DM) interaction 15,16 , D · (S i × S j ), appears, depending on the crystal symmetry. This term bends the propagation of magnons in space, in a similar manner to the cyclotron motion of electrons in the presence of magnetic flux. Thus, when D is parallel to the magnetization, magnon bands in a ferromagnet become asymmetric, reflecting the "nonreciprocal" propagation [17][18][19][20][21][22][23] . Nevertheless, the phenomena related to magnons in non-centrosymmetric ferro or ferrimagnets lacks abundance, its investigation being still far behind the electronic Rashba system. This is apparently because the magnon has no internal degrees of freedom like the spins of electrons.
In antiferromagnets (J > 0) with a doubled magnetic unit cell, the magnon bands are folded and a doubly-degenerate linear dispersion appears at Γ-point. What if we regard the two different species of magnons each belonging to the degenerate band as an analogue of the electronic spin degrees of freedom? If this degrees of freedom couples to the magnon momentum via DM interaction, such that the SO does in electron systems, one may expect as rich phenomena as those of the Rashba-electronic systems in insulators as well. So far, however, non-descript bipartite antiferromagnets did not show any magnon-based phenomena such as the a thermal Hall effect already found in the kagome and pyloclore magnets [24][25][26][27][28][29][30][31] , indicating that the situation is not as simple. In this article, we demonstrate that a typical two-dimensional (2D) antiferromagnet can afford spinmomentum locking with a variety of textures not even found in the electronic systems. We classify its condition by constructing a series of 2D antiferromagnets based on the building blocks of one-dimensional(1D) antiferromagnetic chain. The relative directions of the D-vector, spin anisotropy, and magnetic field turn out to be the factors that enable the fine control of magnetic textures.

A. Model and Formulation
We consider a quantum spin system with nearest neighbor exchange interaction, J, spin anisotropy, Λ, the DM interaction, D ij , and a uniform external magnetic  1. (a)-(e) Schematic illustration of the 1D magnetic chain with easy-plane spin anisotropy (Λ > 0) and easy-axis anisotropy (Λ < 0); (a) Ferromagnetic chain with uniform DM interaction along three different directions. (b)-(d) Antiferromagnetic chain with uniform and (e) staggered DM interactions. Magnon bands calculated for h = 0 and D = 0 is shown, where only the ferromagnetic case (a) shows nonreciprocity. When h = 0, the antiferromagnetic moments cant and (b)-(d) falls onto either of I−IV cases (panel (f)). Circle/cross symbols denote the case where the spin momentum locking and nonreciprocity occurs/does not occur. Triangle denotes the case where the magnitude of S k± varies continuously with k but points in the same direction. (g) Antiferromagnetic honeycomb lattice, which is understood as a combination of ferroagnetic chains with uniform DM with magnetic moments belonging to different sublattices pointing in the opposite direction (parallel to D).
field, h. The general form of the Hamiltonian is given as, where S Λ i is the spin moment in the Λ-direction. For Λ > 0 the spin moments have an easy-plane anisotropy, and for and Λ < 0, an easy-axis one, and are pointing in-plane or parallel to Λ-axis, respectively. The DM interaction emerges when the midpoint of two magnetic sites lacks the inversion symmetry. There are two different ways of aligning the D-vector; Defining the spin indices that couples to D ij in an order, i → j, the D-vectors can take either a uniform or a staggered configuration along that direction (e.g. see the +δ-direction in Fig.1). The former breaks the global SIS of the crystal, whereas the latter keeps the site-centered SIS.
We perform a spin-wave analysis on Eq.(1) starting from the magnetically ordered ground state. Here, we present the overview of the analysis for a 2D easy plane antiferromagnet. The case of 1D antiferromagnets is basically the same. The starting point is the canted antiferromagnetic classical order with spins on two sublattice A and B. To include cases where the magnetic moments are aligned off the parallel/antiparallel direction, we apply a local unitary transformation 32 , U † HU , and set the z-axis of the spin operator space to the direction of the magnetic moments, M A and −M B , independently for the two magnetic sublattices A and B. in Figs.2(a) and 3. The operator for the unitary transformation is given as where, r j , is the spatial corrdinate of site-j, and the ordering wave vector Q satisfies e iQrj = ±1 for sublattices A and B, respectively. The spin operators are now set antiparallel, and the standard Holstein-Primakoff transformation is given for the two sublattices as, Then, we introduce the Fourier transformation where k = (k x , k z ) is the two dimensional vector in reciprocal space. The spin wave Hamiltonian is written finally in the quadratic form with a vectors consisting of four species The matrix representation of the Hamiltonian, H SW (k), is diagonalized analytically using the paraunitary matrix 33 . The actual diagonalization is done by solving the following eigenvalue equation, The corresponding magnon dispersion consisting of two branches is given in the analytical form as, where ξ k , P k , Q k are the function of the components of H SW (k), and satisfy ξ −k = ξ k , The local spectral weight used in Eq.(9) shortly is given as,

B. 1D systems
Let us start by reminding of ferromagnets J < 0; The uniform ferromagnet shows a gapless single cosine dispersion which is parabolic at k = 0 (see the top panel of Fig.1), and by the easy axis anisotropy, Λ < 0, the gap opens. Figure 1(a) shows three different ways of introducing uniform DM interaction, and among them, only the vector D parallel to the ordered magnetic moments couples to the orbital motion of magnons and contributes to the nonreciprocity 17,18 , i.e., the bands are shifted off the center as, ω(k) = ω(−k). There, the offdiagonal (xy) element of Eq.(1) is transformed in the language of bosons as, , namely the magnons acquire a phase 24,25 , ϕ = atan(D/J), during propagation, which is known to generate a thermal Hall effect [24][25][26][27][28] . The net phase they carry when going around the closed path is regarded as a fictitious magnetic flux that generates the cyclotron motion of magnons 34 . As a result, the total momentum of magnon bands is shifted to −k direction, which is observed in MnSi 35 .
The uniform antiferromagnet has two-fold degenerate magnon branches that cross at k = 0 (see the top panel of Fig.1), each belonging to sublattice A and B. In the case of easy-axis anisotropy, Λ < 0, these branches shift in opposite directions by introducing D parallel to the magnetic moments 36,37 , which is understood as a combination of the two ferromagnetic chains we saw in the top panel of Fig.1(a). (While here, the reciprocity recovers as the A and B bands are symmetric about k = 0.) Contrastingly in Fig.1(c), the DM interactions remain irrelevant and the dispersion does not change. Namely, the collinearly aligned spins do not respond to D perpendicular to them.
So far, we discussed the standard cases already studied in experiments 23,35,37 . Additionally, we consider the easy-plane antiferromagnet in Fig.1(d). When Λ > 0, one of the modes becomes gapped (see Fig.1(d)), which is responsible for the in-plane stretching mode. The gap of the remaining mode opens when the in-plane rotational symmetry is broken by h. Again, when we set D ⊥ M A , M B (upper panel), the magnon band structures remain unchanged. If we rotate M A and M B within the easy plane off the direction parallel to D, the energy band is slightly modified by D. These cases becomes important when we apply a field.
Notice that for the staggered DM interaction in Fig.1(e), the nonreciprocal shift of energy bands does not appear even though D M A , M B , indicating that the global SIS breaking is required for such phenomena to occur.
By the application of h, the antiferromagnetic moments M A and M B are canted off the collinear alignment and gain a net moment in the field direction. The excitation against this weak ferromagnetic element couples to D depending on the relative angle between h and D, which falls onto either of I−IV in Fig.1(f). The dispersion shown in case I is actually observed experimentally in noncentrosymmetric antiferromagnet, α-Cu 2 V 2 O 7 37 . At finite field, magnons start to carry spin moment that has a net value opposite to h ∝ M A + M B ; The one at each k is evaluated as 38 , using the local spectral weight at k of ω ± -band, d A/B (ω ± (k)), on sublattice A/B (see Eq. (8)). Equation (9) immediately shows that the necessary condition to vary both the direction and the amplitude of S k,± in momenum space is to have M A and M B linearly independent 38 (see Supplementary material). Although this condition is fulfilled for cases II-IV in Fig.1(f), only case II shows a distinct spin momentum locking. This is understood as follows; In case II, the stretching modes of M A and M B couple to D in an asymmetric manner, as the projected element of M A and M B onto the direction of D is not equal, i.e. there exists some k that fulfills d A (ω ± (k)) = d B (ω ± (k)). Thus, the necessary and sufficient condition for the spin momentum locking is to have linearly independent M A and M B , which also have different magnitude of component parallel to D. We also provide the symmetry operation that has one-to-one correspondence with the breaking this condition in the case of antiferromagnets with two-sublattices (see Supplementary material). The reason why the spin momentum locking was not observed in antiferromagnets was possibly because the collinear spins do not satisfy the necessary condition, and even when the spins are canted off the collinear one, it is only case II that fulfills the sufficient condition, which is apparently the situation not straightforward to realize systematically. In the previous arguments 38 , the spin momentum locking included cases where the amplitude of S k,± varies with k while pointing in the same direction, represented by Case III and IV in Fig.1(f) marked with triangle. Here, however, we explicitly denote only the spin configurations that do not simply point in a single direction as "spin momentum locking". In case I, S k,± is quantized at each band, except at the band degenerate point, which is definitely excluded from the spin momentum locking.
The 1D chains discussed above serve as building blocks to construct 2D antiferromagnets, and this scheme opens up a path to realize abundant spin momentum locking and nonreciprocity, which were difficult to realize within previous considerations on antiferromagnets. We deal with the square lattice antiferromagnet shortly as a representative example. Besides, there are few examples of antiferromagnet studied in terms of SO physics on the honeycomb lattice [39][40][41][42][43] . The crystal stucture of the correponding materials generally keeps the SIS, in which case only a DM interaction between next nearest neighboring sites belonging to the same magnetic sublattices is active, which form a staggard configuration of D-vectors in all. In our scheme, this structure is regarded as the assembly of two different species of 1D ferromagnets each with uniform DM interaction, whose D-vectors point in the opposite directions, as shown in Fig.1(g). The thermal Hall effect of magnons appears 42,43 by acquiring ϕ in respective two sublattices, in the same context to the topological Haldane model 44 . The nonreciprocity is also observed, which is simply explained within our scheme as the combination of those of the two sublattice ferromagnets in Fig.1(a), while they accompany the spontaneous toroidal ordering instead of a simple antiferromagnetic order 39 .

A. Square lattice antiferromagnet
On the basis of the classification of the role of DM in the 1D antiferromagnets shown in Fig.1, the 2D or even 3D antiferromagnets are easily designed by combining them, where one could activate or cancel the role of D on the magnon excitations by controlling the magnetic field angle. Here, we show one of the interesting cases with global SIS breaking, where the spin momentum locking together with the nonreciprocity is observed.
Let us consider a 2D model shown in Fig.2(a) with easy plane anisotropy (Λ > 0) that follows the Hamiltonian (1); The DM vector has the in-plane and outof-plane components, D and D ⊥ , which are the uniform and staggered DM, respectively, and the former contribute to the global SIS breaking. We take the xand z-axes in the direction rotated by π/4 from the bond direction. To define the direction the vector D point toward, we additionally introduce two directions, +δ 1 and +δ 2 . By taking the indices i → j of D ij along the two sites forming bonds along the δ 1 -directions, both D and D ⊥ are determined for a chain running in the (e x + e z ) direction as shown in Fig.2(a). The D-vectors along the bonds in the (e z − e x )-direction are also defined using +δ 2 . We stress here for later convenience that the description of intrinsically the same D-vector can change according to the choice of δ-directions. Thus, converting δ in the opposite direction will transform the D-vectors up side down. A model material that fits to our system is the non-centrosymmetric spin-5/2 antiferromagnet Ba 2 MnGe 2 O 7 , with space group P42 1 m. It undergoes a Néel transition at T N = 4K into an easy plane type antiferromagnetic phase 45,46 , where a microwave non-reciprocity is indeed observed 47 . A spin-3/2 multiferroic Ba 2 CoGe 2 O 7 48,49 , possibly of space group P42 1 m, is considered to be in the same class regarding the antiferromagnetic ordering and its excitations.
For sufficiently small D compared to Λ, the magnetic easy axes is confined within the 2D plane. As easily understood from the context of 1D antiferromagnets, the staggered DM interaction, D ⊥ , contributes neither to nonreciprocity nor to spin momentum locking, but simply cants the spins in-plane. In the classial ground state, the  Fig.1(f). The h-direction corresponds to case II. (s) Texture of S k,± over the first Brillouin zone. Upper and lower panels are those of the upper and lower magnon bands (S k,+ and S k,− , respectively). The spin amplitude, |S k,± |, is given by the density plot, and the arrow indicates its direction.
ordered magnetic moments have a rotational symmetry about the y-axis, which is broken by the application of h. Then, the direction of the magnetic moments are fixed as shown in Fig.2(a), which is described by the magnetic field angle, φ against the x-axis, and the canting angle, θ.

B. Nonreciprocal magnon bands and spin momentum locking
We first examine the magnon band dispersion, which at D = 0, are no longer symmetric about the rotation. Figure 2(b) shows the one with h (e x + e z ) (φ = π/4). For a given set of δ 1 and δ 2 , the vectors D along the two bonds point in the directions perpendicular and parallel to h, respectively. More precisely, the relative relation of D and h follow case II and III in Fig.1(f), respectively, for the two chain directions. The cross-sections of bands are shown in Fig.2(c), which have the same structure as those found in cases II and III, demonstrating full consistency of the idea of decomposing the 2D antiferromagnet into 1D ones. The nonreciprocity indeed appears for k δ2 -direction as we saw in case III, and not along k δ1 . A simple condition to have a nonreciprocity in this system is to have θ = 0 (see Supplementary material).
The 2D texture of magnetic moments the magnons carry is also constructed by the combination of those of 1D chains. Figure 2(d) shows the direction and amplitude of S k,± ; the k δ1 -direction exhibits a locking as in case II, whereas k δ2 -direction corresponding to case III does not. One can see that the total net moment is opposite to h, while the spins remain unchanged about the mirror plane (k δ1 = 0). This is because Fig.2(a) actually has a mirror plane perpendicular to δ 1 -axis.

C. Magnetic field angle
We now show that the 2D antiferromagnetic magnons can afford a rich variety in the spin momentum textures. A series of panels in Fig. 3(a) show how the spin momentum locking evolves with magnetic field angle for the uniform DM interaction given on the lattice at the top panel (the same as Fig.2(a)). Even for such seemingly complicated case, one can elucidate the texture by decomposing the system into 1D chains; Let us configure the direction, δ II , that reproduce case II in Fig.3(c) along which the spin momentum locking occurs. This is done by taking the linear combination of δ 1 and δ 2 so as to have the summation of the two D 's attached to them become perpendicular to h (see the black arrow in Fig.3). The direction of δ II rotates anti-clockwise as h rotates clockwise. The other direction, δ III , that the locking does not occur is also defined perpendicular to δ II , in a way that the combination of their two D 's point toward h. We also put a constraint that the net moment points in the direction opposite to h. These considerations allow us to figure out the overall textures without detailed calculation.
There is another way to introduce a uniform DM interaction even for this simplest square lattice antiferromagnet, as depicted in Fig. 3(b). To classify the two cases shown, let us introduce a polarization vector, P ij = e ij × D , where e ij points to either δ 1 or δ 2 along the bonds connecting the i-and j-th sites. In Fig.3(a), the DM vector keeps theC 4 symmetry (or space group P 4) where the bulk poralization is absent even though the global SIS is broken. It is realized in crystals with D 2d or T d point group symmetry, and is related to the Dresselhaus-type of SO interaction. The one in Fig.3(b) has theC 4 symmetry (or space group P4), where a bulk polarization is induced by the broken SIS. This kind of polarization is equivalent in symmetry to the ones induced by the field perpendicular to the plane whose gradient generates a Rashba SO coupling. Another way to look at is to perform a C 4 -rotation to δ 1 and δ 2 in the clockwise direction, and we find that D rotates in the anti-clockwise direction in the Dresselhaus-case, and clockwise in the Rashba-case.
To understand the spin momentum locking of the Rashba-type, the same discussion holds. In Fig.3(b) we define δ 2 in an opposite direction so as to have the same orientation of D as the Dresselhaus-one. This time, δ II and δ III rotate clockwise by varying the field angle φ.
As one can anticipate from the evolution of spin texture, the band profile rotates following the field-angle φ = 0 → 2π, clockwise and anti-clockwise for Dresselhaus and Rashba-types, respectively. Accordingly, the nonreciprocity appears in the (k x , k z ) = (cos φ, − sin φ)direction for the Dresselhaus-type, and in the direction perpendicular to the field for the Rashba-type. This is understood by changing the sign, δ 2 to −δ 2 , e.g. for φ = π/4 in Fig.2(b).
We briefly mention another way to look at these textures. Figure 3(d) shows the typical Dresselhaus-and Rashba-type spin texture observed in the electronic system. At h = 0, they are modified to have a net moment opposite to h. Still, the Dresselhaus-type keeps the πrotational symmetry about the k z = 0-axis, and if we consider the system with space group P42 1 m,4 and m are broken by h, but 2 1 along the x-axis is preserved. The Rashba-type keeps the mirror symmetry about the z-axis.
We finally confirm that the direction of S k,± has no k dependence if the uniform DM interaction, D is absent. This is shown in a more explicitly; when D = 0 a simple expression for S k,± is obtained as, where H 1 and H 2,k is included in Eq.(5), and satisfy H 1 > |H 2,k |. This means that only the amplitude of S k,± depends on k, and the direction of S k,± is always opposite to the sum, M A + M B ∝ h, so that S k,± altogether works simply to screen h.

IV. SUMMARY AND DISCUSSIONS
Based on the idea of constructing the 2D antiferromagnets using the building blocks of 1D antiferromagnetic chains with uniform and staggered DM interactions, and with spin anisotropy in a finite magnetic field, a strategy to generate spin momentum locking is thus provided. Previously, active discussions on the physics of magnons were given in a ferro or ferrimagnets with its magnetic moments parallel to the DM interactions; for the staggered DM interaction that does not break SIS, a topological magnon contributing to the thermal Hall effect is observed, and for uniform DM interaction, a nonreciprocity of ferromagnetic magnons were reported. Here, we clarified another aspect of magnons that the uniform DM interactions breaking global SIS in the antiferromagnet can generate a spin-texture far richer than the established Rashba and Dresselhaus electronic semiconductors. The two degenerate branches in the uniform antiferromagnets carry the magnetization pointing in the nearly opposite direction, which are regarded as the internal pseudo-spin degrees of freedom of the elementary excitations. The uniform DM interaction serves to mix these branches in an asymmetric manner, when and only when these magnetic moments are linearly independent and have differ-ent elements projected in the direction parallel to D. To activate such fictitious pseudo-SO coupling of magnons, the interplay magnetic field and spin anisotropy plays a crucial role. The resultant textures are easily controlled by the magnetic field angle.
The 2D antiferromagnet we made some demonstration is actually realized in noncentrosymmetric Ba 2 MnGe 2 O 7 . The exchange interactions that determines the energy scale of the system is J ∼ 27µeV 45 , which is by orders of magnitude smaller than the other materials of the same family, possibly allowing for the examination of the present phenomena by several experimental probes, and with a very small magnetic field of few tesla. Besides such noncentrosymmetric antiferromagnets, the interface of typical centrosymmetric antiferromagnets is expected to show magnonic spin-momentum locking. Thus, the scheme we proposed may allow a strong command of designing spin textures toward the application for antiferromagnetic spintronics 50 .
In electronic devises, a large amount of energy consumption due to the large current density required to control the magnetic textures, e.g. in spin-orbit torques applied to memories, had long been a bottleneck in realizing a microscopic circuit. The present setup enabling a directly control of spin textures in ubiquitous antiferromagnets, without making use of the dissipative electric current, should thus become a promising future replacement.