Antiferromagnetic Domain Wall as Spin Wave Polarizer and Retarder
AAntiferromagnetic Domain Wall as Spin Wave Polarizer and Retarder
Jin Lan, ∗ Weichao Yu, ∗ and Jiang Xiao
1, 2, 3, † Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China Institute for Nanoelectronics Devices and Quantum Computing, Fudan University, Shanghai 200433, China (Dated: June 7, 2017)
As a collective quasiparticle excitation of the magnetic order in magnetic materials, spin wave,or magnon when quantized, can propagate in both conducting and insulating materials. Like themanipulation of its optical counterpart, the ability to manipulate spin wave polarization is not onlyimportant but also fundamental for magnonics. With only one type of magnetic lattice, ferromag-nets can only accommodate the right-handed circularly polarized spin wave modes, which leaves nofreedom for polarization manipulation. In contrast, antiferromagnets, with two opposite magneticsublattices, have both left and right circular polarizations, and all linear and elliptical polarizations.Here we demonstrate theoretically and confirm by micromagnetic simulations that, in the presence ofDzyaloshinskii-Moriya interaction, an antiferromagnetic domain wall acts naturally as a spin wavepolarizer or a spin wave retarder (waveplate). Our findings provide extremely simple yet flexibleroutes toward magnonic information processing by harnessing the polarization degree of freedom ofspin wave.Introduction.
Spintronics extends the physical limit of conventional electronics by harnessing the electronic spin, an-other intrinsic degree of freedom of an electron besides its charge. [1, 2] As a collective excitation of orderedmagnetization in magnetic systems, spin wave (or magnon when quantized) carries spin angular momentumlike the spin-polarized conduction electron. Different from the spin transport by conduction electrons, thepropagation of spin wave does not involve physical motion of electrons. Therefore, spin wave can propagatein conducting, semiconducting, and even insulating magnetic materials without Joule heating, a troublingissue faced by the present day silicon-based information technologies. [3] The dissipation in the spin wavesystem is mainly due to the magnetic damping, which is much weaker than the electronic Joule heating,especially in magnetic insulators. [4] Magnonics is a discipline of realizing energy-efficient informationprocessing that uses spin wave as its information carrier. [5–7] Besides its low-dissipation feature, thewave nature and the wide working frequency range of spin wave make magnonics a promising candidate ofupcoming beyond-CMOS (Complementary Metal Oxide Semiconductor) computing technology. a r X i v : . [ c ond - m a t . o t h e r] J un Similar to the intrinsic spin property of elementary particles, polarization is an intrinsic property of wave-like (quasi-)particles such as photons, phonons, and magnons. It is more natural to encode informationin the polarization degree of freedom than other degrees of freedom such as amplitude or phase. Forinstance, photon polarization has been widely used in encoding both classical and quantum information,[8] and manipulation of photon polarization is essential for applications in photonics. [9] The phononpolarization has also been used in encoding acoustic information for realizing phononic logic gates, [10, 11]and for controlling magnetic domain walls. [12] In contrast, most magnonic devices proposed or realizedso far mainly use the spin wave amplitude [13–18] or phase [19–23] to encode information, and spin wavepolarization is rarely used except in very few cases. [24] The lack of usage of polarization in magnonicshas reasons. With only one type of magnetic lattice, ferromagnets can only accommodate the right-circularpolarization, hence there is simply no freedom in polarization manipulation. This situation is similar tothe case of half-metal, which has only one spin spieces. [25] In other words, ferromagnet is a half-metalfor spin wave. To overcome this disadvantage, a straightforward approach is to use antiferromagnets inmanipulating spin wave polarization. In antiferromagnets, due to the two opposite magnetic sublattices, spinwave polarization has complete freedom as the photon polarization. Thus, antiferromagnet is a normal metalfor spin wave with all possible polarizations. In comparison with the ferromagnets, besides gaining the fullfreedom in polarization, antiferromagnets also have numerous other advantages, such as the much higheroperating frequency (up to THz) and having no stray field, etc . [26, 27] Due to these merits, antiferromagnetis regarded as a much better platform for magnonics than ferromagnet [24, 28–30]. In order to unleash thefull power of antiferromagnets in magnonics, however, it is highly desirable to have a simple yet efficientway in manipulating spin wave polarization, ideally in a similar fashion as in its optical counterpart.In general, to be able to manipulate polarization with full flexibility, two basic devices are indispensable, i.e. the polarizer and retarder (waveplate). [9, 31] The former is to create a particular linear polarization,and the latter is to realize the conversion between circular and linear polarizations. The actual realizationof these two building blocks relies on the types of the (quasi-)particle in question. For instance, for photon,an array of parallel metallic wires functions as an optical polarizer, [9] and a block of birefringent materialwith polarization-dependent refraction indices acts as an optical waveplate. [31]Here, we show theoretically and confirm by micromagnetic simulations that, utilizing the Dzyaloshinskii-Moriya interaction (DMI) existing in the symmetry-broken systems, [32, 33] an antiferromagnetic domainwall serves as a spin wave polarizer at low frequencies and a retarder at high frequencies. Due to the extremesimplicity and tunability of a magnetic domain wall structure, the manipulation of spin wave polarizationin magnonics becomes not only possible but also simpler and more flexible than its optical counterpart.
Figure 1.
Schematics of an antiferromagnetic domain wall with its in-plane and out-of-plane modes.
Thedomain wall profile is indicated by the thick red/blue arrows for the two sublattices. The Bloch sphere on the leftshows that the magnetization m (red arrow) in a domain wall traces a longitude on a Bloch sphere in the y - z planefrom north pole to south pole. The green (at a ) and orange (at b ) arrows indicate the the in-plane (at a ) and out-of-plane(at b ) spin wave excitations upon a Bloch type antiferromagnetic domain wall. The in-plane and out-of-plane modesoscillate along longitude and latitude on the Bloch sphere, and they are connected to linear y - and x -polarizationinside domains (where m i (cid:107) ˆ z ), respectively. Results.
Model.
We consider a domain wall structure in an one-dimensional antiferromagnetic wire along ˆ x direction as shown in Fig. 1, where the red/blue arrows denote the magnetization direction m , for the twomagnetic sublattices of an antiferromagnetic domain wall. The domain wall profile is taken as Bloch type,where the magnetization rotation plane ( y - z plane) is perpendicular to the domain wall direction ( ˆ x ). Themagnetization dynamics in the antiferromagnetic wire can be described by two coupled Landau-Lifshitz-Gilbert (LLG) equations for each sublattice [34, 35], ˙ m i ( r , t ) = − γ m i ( r , t ) × H eff i + α m i ( r , t ) × ˙ m i ( r , t ) , (1)where i = 1 , denote the two sublattices, γ is the gyromagnetic ratio, α is the Gilbert damping constant.Here γ H eff i ( r , t ) = Km zi ˆ z + A ∇ m i + D ∇× m i − J m ¯ i (with ¯1 = 2 and ¯2 = 1 ) is the effective magnetic fieldacting locally on sublattice m i , where K is the easy-axis anisotropy along ˆ z , A and D are the Heisenbergand Dzyaloshinskii-Moriya (DM) exchange coupling constant within each sublattice, and J is the exchangecoupling constant between two sublattices. The Heisenberg exchange coupling tends to align neighboringmagnetization in parallel. While the DM exchange coupling, existing in magnetic materials or structureslack of spatial/inversion symmetry, [32, 33] prefers magnetization to rotate counter-clockwise about an axis( x -axis in our case) determined by the symmetry broken direction.We denote the static profile of the antiferromagnetic domain wall along x -axis as m ( x ) = − m ( x ) ,and the stagger order n ( x ) ≡ ( m − m ) / θ cos φ , sin θ sin φ , cos θ ) , where θ ( x ) and φ ( x ) are the polar and azimuthal angle of n with respect to ˆ z (see Fig. 1). No matter DMI is present ornot, an antiferromagnetic domain wall in Fig. 1 always takes the Walker type profile like its ferromagneticcounterpart with θ ( x ) = − x/ ∆ )] and φ = const . , [16, 30, 36] where ∆ = (cid:112) A/K is thecharacteristic domain wall width. (See Supplementary Figure 1, Supplementary Note 1 and SupplementaryMovie 1) The effect of DMI is to determine the domain wall type and chirality: when DMI is absent, thedomain wall is free to take either the Bloch type ( φ = π/ ) or Néel type ( φ = 0 ) configuration, or anymixture of the two with < φ < π/ ; in the presence of DMI, the domain wall becomes chiral [37, 38]and is pinned to the Bloch type as shown in Fig. 1.With the static profile m i ( x ) , let m i ( x, t ) = m i ( x ) + δ m i ( x, t ) and δ m i ( x, t ) = m θi ( x, t )ˆ e θ + m φi ( x, t )ˆ e φ be the dynamical spin wave excitation upon the static m i ( x ) , where ˆ e θ and ˆ e φ are the localtransverse (polar and azimuthal) directions with respect to n ( x ) . Since ˆ e θ lies in the magnetization rota-tion plane (the y - z plane), we call the excitation in ˆ e θ the in-plane mode. Similarly, the excitation in ˆ e φ is perpendicular to the rotation plane and is called the out-of-plane mode (see Fig. 1). By eliminating thestatic profile from the LLG equation (1), the spin wave dynamics is governed by the following linearizedLLG equations: ˙ m φ ∓ = − (cid:20) − A ∂ ∂x + V K ( x ) + J ∓ J (cid:21) m θ ± , (2) ˙ m θ ± = + (cid:20) − A ∂ ∂x + V K ( x ) + J ± J + V D ( x ) (cid:21) m φ ∓ , (3)where m φ,θ ± = m φ,θ ± m φ,θ . The effects of the inhomogeneous domain wall texture are transformedinto two effective potentials V K ( x ) and V D ( x ) : [16, 36] V K ( x ) = K [1 − ( x/ ∆ )] arises due to theeasy-axis anisotropy along ˆ z , and V D ( x ) = ( D/ ∆ )sech( x/ ∆ ) is due to the combined action of DMI andthe inhomogeneous magnetic texture. Equations (2)(3) can be regrouped into two independent sets for ( m φ + , m θ − ) and ( m φ − , m θ + ) , whose solutions are two linearly polarized spin wave modes oscillating in the(in-plane) ˆ e θ -direction and (out-of-plane) ˆ e φ -direction, respectively, as depicted in Fig. 1. Deep inside eachdomain (where n (cid:107) ± ˆ z ), the two transverse directions ˆ e θ and ˆ e φ coincide with ˆ y and ˆ x , therefore we alsocall the in-plane (out-of-plane) modes y -polarized ( x -polarized) modes.The influences of a domain wall on spin wave propagation are all captured by the effective potentials V K and V D . In the short wavelength limit (WKB approximation), using equations (2)(3), we may define thelocal dispersion at position x for chosen polarization and wavevector k :in-plane: ω IP ( k, x ) = ω θ ( k, x ) = (cid:112) [2 J + Ak + V K ( x ) + V D ( x )] [ Ak + V K ( x )] , (4)out-of-plane: ω OP ( k, x ) = ω φ ( k, x ) = (cid:112) [2 J + Ak + V K ( x )] [ Ak + V K ( x ) + V D ( x )] , (5)which gives the local spin wave gap as ω IP/OPG ( x ) ≡ ω IP/OP (0 , x ) , below which no in-plane/out-of-plane modeis allowed to propagate. In the absence of DMI ( V D = 0 ), the in-plane and out-of-plane modes are degenerate( ω IP = ω OP ). This degeneracy is lifted by the introduction of DMI (the V D term in equations (4)(5)), withwhich the local spin wave gap for the out-of-plane modes is higher than that for the in-plane modes inside thedomain wall ( ω OPG ( x ) ≥ ω IPG ( x ) ) (see the plots of the local spin wave gap in Fig. 2a). Deep inside each domain( | x | (cid:29) ∆ , V K → K, V D → ), no matter whether DMI is present or not, both dispersions in equations (4)(5)reduce to the standard antiferromagnetic dispersion ω ( k ) = (cid:112) (2 J + K + Ak )( K + Ak ) with a spinwave gap ω G = (cid:112) (2 J + K ) K . [34, 39] It is seen that the dispersions inside domains is not modified byDMI, this is because that DMI modifies the spin wave dispersion with a linear term in wavevector onlywhen the wavevector has component (anti-)parallel to the magnetization direction, [40] and in our case thespin wave wavevector ( ˆ x ) is perpendicular to the magnetization direction ( ˆ z ). Polarization-dependent scattering.
In the absence of DMI ( D = V D = 0 ), the dispersions and the localspin wave gaps for in-plane and out-of-plane polarization in equations (4)(5) are identical, thus the scatteringbehavior of spin waves by the domain wall is independent of polarization. Furthermore, the potential V K ( x ) is the well known reflectionless Pöschl-Teller type potential well. [16, 18, 36, 41] Consequently, regardlessof its polarization, the incident spin wave experiences no reflection by the domain wall, and only accumulatea common phase delay. However, when DMI is present ( D, V D (cid:54) = 0 ), the dispersions and local spin wavegaps for the in-plane and out-of-plane modes are different. Considering that the inter-sublattice exchangecoupling J is the dominating energy scale in antiferromagnet, the in-plane gap ω IPG is barely affected by V D ( (cid:28) J ). On the contrary, the out-of-plane gap ω OPG is elevated by V D > and reaches its maximum value ω D inside the domain wall. This effect of DMI in equations (4)(5) is equivalent to an effective hard-axisanisotropy along the out-of-plane ( ˆ x ) direction inside the domain wall [41, 42], which suppresses the out-of-plane excitation. Consequently, the domain wall can still be regarded as reflectionless for the in-plane y -polarization, but becomes a potential barrier of height ω D for the out-of-plane x -polarization (see Fig. 2(a)).Therefore, the x - and y -polarized modes are scattered differently: At low frequencies ( ω G < ω < ω D ),the y -polarization experiences no (or little) reflection but the x -polarization hits a potential barrier and isstrongly reflected; At higher frequencies ( ω > ω D ), both polarization transmit almost perfectly, but the x -and y -polarization no longer share the same phase delay.The polarization-dependence of the spin wave scattering behaviors by the domain wall are confirmedby micromagnetic simulations based on the LLG equations (1) and Green’s function calculations based onlinearized LLG equations (4)(5). (See Supplementary Note 3 for details) Using these methods, we calculatethe spin wave transmission probabilities T x,y through the domain wall for the linear x - and y -polarization, T x T y ω G ω D ω ω T r an s m ss i on Gap Polarizer Retarder δϕ xy ω / π ( GHz ) P ha s e D i ff e r en c e ( π ) a bc Figure 2.
Polarization-dependent spin wave scattering by an antiferromagnetic domain wall. a.
The schematicpicture of the effective potentials (the square of the local spin wave gap) in an antiferromagnetic domain wall for the x -and y -polarized spin wave modes. The effective potential for y -polarization (the green curve) is nearly reflectionless,hence transmission probability T y (cid:39) for ω > ω G ; the effective potential for x -polarization (the orange curve) isa barrier at the domain wall due to V D , hence transmission probability T x (cid:39) for ω G < ω < ω D and T x (cid:39) for ω > ω D . The different effective potentials for x - and y -polarizations also give rise to polarization-dependent phasedelays. Therefore, the antiferromagnetic domain wall is a spin wave polarizer in the frequency range between ω G and ω D , and a spin wave retarder in the frequency range above ω D . b. The transmission probability T x,y for thelinear x - and y -polarized spin wave modes. c. The relative phase accumulation δφ xy = φ y − φ x between y - and x -polarization across the domain wall. In (b, c), the solid curves are calculated from the Green’s function method andthe dots are obtained from the micromagnetic simulations. Frequencies ω / π = 8 GHz and ω / π = 16 . GHzdenote the position of the working frequencies used in the simulations for spin wave polarizers in Fig. 3 and spin wavequarter-wave plates (retarders) in Fig. 4, respectively. as well as their relative phase accumulation δφ xy = φ y − φ x , as presented in Fig. 2(b,c). It is seen that the y -polarization transmit almost perfectly ( T y (cid:39) ) for all frequencies above the spin wave gap ( ω > ω G ). Incontrast, because of the effective potential barrier, the x -polarization experiences total reflection ( T x (cid:39) )at low frequencies ( ω < ω D ). Above the barrier ( ω > ω D ), however, the x -polarization experiences littlereflection ( T x (cid:39) ). Because of the different effective potentials experienced at the domain wall, the x -and y -polarizations develop a relative phase accumulation δφ xy , which is frequency-dependent as shown in ab c Figure 3.
Micromagnetic simulation of spin wave polarizers. a.
An antiferromagnetic domain wall in a straightwire works as a y -polarizer. The four rows from top to bottom are: i) spin wave excitations depicted on Bloch spheres,with the spin wave amplitude exagerated by 100 (200) times at the left (right) side. ii) the static magnetization profile m (red) and m (blue). The Bloch sphere on the left shows that the magnetization in the domain wall rotates in the y - z plane with x -polarization suppressed, iii-iv) the instantaneous wave form of the x and y component of the staggerorder n x,y at a selected time. b. An x -polarizer when the domain wall is at the center segment of a Z-turn wire, andthe magnetization in the damain wall rotates in x - z plane with y -polarization suppressed. c. A y -polarizer when thedomain wall is at the left segment of a Z-turn wire, and the magnetization rotates in y - z plane with x -polarizationsuppressed. For all figures, the spin wave injected from left has circular polarization and frequency ω/ π = 8 GHz ( ω in Fig. 2(b, c)). The damping coefficient in the simulation is α = 5 × − . The wave form within the domainwall region is omitted. Fig. 2(c). All these behaviors are in perfect agreement with the qualitative understanding shown in Fig. 2(a).Utilizing these particular spin wave scattering properties, an antiferromagnetic domain wall can be used asa spin wave polarizer and retarder (or waveplate) as discussed in the following.
Spin wave polarizer.
In the low frequency regime ( ω G < ω < ω D ), because the x -polarized spin wave istotally reflected, an antiferromagnetic domain wall is naturally a y -polarizer for spin wave. The functionalityof the spin wave polarizer is confirmed by a micromagnetic simulation on a straight antiferromagnetic wireshown in Fig. 3a: when a circularly-polarized spin wave, consisting of x - and y -polarization of equalamplitude, is injected from the left side of the domain wall, only the y -polarization transmits, indicatingthat the domain wall is a y-polarizer for spin wave.It appears that the polarizer demonstrated in Fig. 3a can only be a y -polarizer. In fact, an x -polarizercan be realized simply by using a ◦ Z-turn wire as shown in Fig. 3b, where the domain wall is locatedat the central segment. Because the domain wall is now along the y direction and the static magnetizationrotates from +ˆ z to − ˆ z in the x - z plane (rather than the y - z plane in the straight wire case in Fig. 3a), rolesof x -polarization and y -polarization interchange. Therefore, a domain wall located at the central segmentof a Z-turn wire only allows the x -polarization to pass (instead of the y -polarization in the straight wirecase), hence it is an x -polarizer as demonstrated in Fig. 3b. Actually, a Z-turn wire can work as either x -or y -polarizer depending on the location of the domain wall: an x -polarizer when the wall is at the centersegment (Fig. 3b), and a y -polarizer when the wall is at either the left (Fig. 3c) or right segment. Spin wave retarder.
In the high frequency regime ( ω > ω D ), both x - and y -polarized spin wave modestransmit almost perfectly through the domain wall, but they accumulate different phases. As a result ofthis relative phase delay, an antiferromagnetic domain wall functions as a spin wave retarder (or a wave-plate). As shown in Fig. 2(c), the relative phase delay depends on the spin wave frequency. By choosingthe working frequency at ω/ π = ω / π = 16 . GHz, the relative phase delay between the x - and y -polarization δφ xy ( ω ) = π/ , hence the antiferromagnetic domain wall is specifically a quarter-wave plate.Fig. 4 shows a micromagnetic simulation of a right-circularly polarized spin wave passing through fourconsecutive domain walls, each of which is a quarter-wave plate. As shown in the top panel, the relativephase delay δφ xy ( x ) increases by π/ across each domain wall. By passing through these domain walls, theright-circular polarized spin wave is converted into a ◦ -linear mode, a left-circular mode, a − ◦ -linearmode, and back into a right-circular mode, successively. Discussion.
The antiferromagnetic domain wall based spin wave polarizer/retarder mimics its optical counterpart inmany aspects. For example, the spin wave polarizer follows the Malus’s law [9] as the optical polarizer, i.e. the spin wave intensity is reduced by cos θ with θ the angle between the injecting polarization and thepolarizer direction. Owing to the relative phase delay, an antiferromagnetic domain wall with DMI can beregarded as a birefringent material for spin wave. This birefringence brought by magnetic texture here onlyoccurs within the domain wall region, thus it is drastically different from the birefringence caused by the m x m y - -
500 0 500 1000 - - ( nm ) m abc Figure 4.
Micromagnetic simulation of a series of spin wave retarders.
At working frequency ω/ π = ω / π =16 . GHz, each domain wall is a quarter-wave plate. The right-circular spin waves are injected from the left side. a. The relative phase accumulation δφ xy between the x - and y -polarization components as function of position whenthe spin wave passes through four domain walls. The time traces of magnetization on a Bloch sphere in each domain(amplitudes exaggerated by times) are shown in the insets. b. The static domain wall profile m z ( x ) (red) and m z ( x ) (blue). c. The wave forms of m x and m y at a given time, showing that the relative phase changes from domainto domain. In this simulation, we artificially set the damping coefficient to be zero to show that the retarder (domainwall) itself is (nearly) reflectionless and does not influence the amplitude of spin waves. biaxial anisotropy with the polarization degeneracy lifted in the whole antiferromagnetic material. [43] Inaddition, both polarizing and retarding effects are independent of propagation direction, or equivalently, anup-to-down domain wall and a down-to-up domain wall are the same in polarizing or retarding. All thesefeatures enable us to construct magnonic circuits in a similar fashion as the well-established optical circuits.[44] Beyond the capability of its optical counterpart, due to the extreme tunability of magnetic textures, e.g. creating/annihilating or moving a domain wall, the magnonic circuit built upon magnetic domain walls canbe modulated with even greater flexibility. [16]As demonstrated above in Fig. 3, a simple geometric modification (from a straight to a Z-turn wire) can0change the polarizing behavior. This is because that the spin wave polarization is characterized in the spinspace, and it does not change when the propagation takes turns in real space. However, the rotation planeof the magnetization of a domain wall depends on the domain wall direction in real space. Therefore, thepolarization component allowed for transmission depends on the domain wall direction as seen in Fig. 3.Based on this principle, a linear polarizer along arbitrary direction can be realized straightforwardly usinga wire with a turn of appropriate angle.The polarizing and retarding effects sustains at smaller DMI strength D (See Supplementary Figure 2and Supplementary Note 2). For fixed material parameters ( K, A, J, D etc ), the working frequency rangefor the spin wave polarizer and retarder are different, i.e. a polarizer for ω < ω D and a retarder for ω > ω D .However, by modulating the material parameters (such as tuning the DMI strength as in [45, 46]), it ispossible to have the spin wave polarizer and retarder functioning in the same frequency range. Hence-forth, multiple polarizers (of different polarization direction) and retarders (of different phase delay) can beassembled into one magnonic circuit to realize more complex spin wave manipulations.The DMI has two different forms, the bulk and the interfacial form, [47] where the former is caused bythe bulk inversion symmetry breaking, such as in MnSi [48] and other B20 compounds [49], and the latteris due to the interfacial inversion symmetry breaking, such as in Co/Pt or Co/Ni bilayer. [50, 51] Here,we adopt the bulk form DMI to demonstrate the working principle of the spin wave polarizer and retarder.However, the materials or structures with interfacial form DMI would work in almost the same manner. Theonly difference is that interfacial DMI prefers a Néel type domain wall rather than a Bloch type for the bulkDMI, [37, 38] and the roles of in-plane and out-of-plane modes interchange. Besides the inhomogeneousDMI discussed above, there is also homogenous DMI that makes the antiferromagnet a weak ferromagnetby slightly misaligning the magnetizations in two sublattices. [32, 33] However, such a misalignment isextremely small because of the dominating antiferromagnetic coupling, thus is ignored in our discussions.The working principles for the spin wave polarizer and retarder work for both real antiferromagnet andartificial synthetic antiferromagnet, where the latter consists of two antiferromagnetically coupled ferro-magnetic wires. [52, 53] The domain wall structures have been observed in both types of antiferromagneticmaterials [53–55], and can be effective controlled by using the exchange bias effects. [30, 56–60] Exper-imentally, it should be more straightforward to realize the spin wave polarizer and retarder proposed hereusing the synthetic antiferromagnetic wires used for the racetrack memory. [53]In conclusion, we demonstrated that the antiferromagnetic domain wall with Dzyaloshinskii-Moriya in-teraction naturally functions as a spin wave polarizer at low frequency, and a spin wave retarder (waveplate)at high frequency. This extremely simple design enables all possible polarization manipulations for antifer-romagnetic spin waves. Our findings greatly enrich the possibility of magnonic information processing by1harnessing the polarization degree of freedom of spin wave. Methods.
Micromagnetic Simulations.
The micromagnetic simulations are performed in COMSOL Multiphysics,where the LLG equation is transformed into weak form by using the mathematical module and solved bythe generalized-alpha method. The synthetic antiferromagnet wire is composed of two ferromagnetic YIGwires with the following parameters:[16, 18, 36] the easy-axis anisotropy
K/γ = 3 . × A · m − , theintra-sublattice Heisenberg exchange constant A/γ = 3 . × − A · m , the saturation magnetization M s = 1 . × A · m − , the gyromagnetic ratio γ = 2 . × rad · m · A − · s − , and the DMI constant D/γ = 3 × − A . The inter-sublattice antiferromagnetic exchange coupling between two ferromagneticwires is J/γ = 5 × A · m − . The dipolar interaction is neglected for this antiferromagnetic environment.To obtain the transmission probability, we first verify that the spin wave perfectly transmits for D = 0 , thenthe transmission probability for a chosen polarization is extracted by taking the ratio between the spin waveamplitude for i) D = 0 and ii) D (cid:54) = 0 at the location nm away from the domain wall. Data availability.
The data that support the findings of this study are available from the correspondingauthors on request.
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This work was supported by the National Natural Science Foundation of Chinaunder Grant No. 11474065, National Basic Research Program of China under Grant No. 2015CB921400and No. 2014CB921600. J.L. is also supported by the China Postdoctoral Science Foundation under GrantNo. KLH1512074. J.L. is grateful to Ran Cheng at Carnegie Mellon University for help with preparingfigures.
Author Contributions.
J.L. did the analytical calculations. W.Y. did the micromagnetic simulations.J.L. and W.Y. contributed equally to the work. J.X. planned and supervised the study. All authors discussedthe results and worked on the manuscript.