Asymmetric spin wave dispersion due to a saturation magnetization gradient
Pablo Borys, Oleg Kolokoltsev, Naser Qureshi, Martin L. Plumer, Theodore L. Monchesky
AAsymmetric spin wave dispersion due to a saturationmagnetization gradient
P. Borys,
1, 2, ∗ O. Kolokoltsev, N. Qureshi, M. L. Plumer,
2, 3 and T. L. Monchesky Instituto de Ciencias Aplicadas y Tecnolog´ıa,Universidad Nacional Aut´onoma de M´exico,Ciudad Universitaria 04510, Mexico Department of Physics and Atmospheric Science,Dalhousie University, Halifax, Nova Scotia, Canada B3H 4R2 Department of Physics and Physical Oceanography,Memorial University of Newfoundland, St. John’s,Newfoundland and Labrador, Canada A1B 3X7 (Dated: December 1, 2020)
Abstract
We demonstrate using micromagnetic simulations and a theoretical model that a gradient in thesaturation magnetization ( M s ) of a perpendicularly magnetized ferromagnetic film induces a non-reciprocal spin wave propagation and, consequently an asymmetric dispersion relation. The M s gradient adds a linear potential to the spin wave equation of motion consistent with the presence ofa force. We consider a transformation from an inertial reference frame in which the M s is constantto an accelerated reference frame where the resulting inertial force corresponds to the force fromthe M s gradient. As in the Doppler effect, the frequency shift leads to an asymmetric dispersionrelation. Additionally, we show that under certain circumstances, unidirectional propagation ofspin waves can be achieved which is essential for the design of magnonic circuits. Our resultsbecome more relevant in light of recent experimental works in which a suitable thermal landscapeis used to dynamically modulate the saturation magnetization. PACS numbers: 75.30.Ds, 75.78.-n, 75.70.Ak, 75.76.+j a r X i v : . [ c ond - m a t . o t h e r] N ov pin waves, collective excitations in magnetic media, transport information without anyparticle motion, e.g., electric charge, and are hence free of the undesired Joule heating.Magnonics -the field that study the behavior of spin waves and their quanta magnons, hasreceived much attention as a plausible complement to conventional semiconductor electronicsfor data transport and processing . For magnonic devices to be relevant, they need to beminiaturized to the nanoscale which in turn needs spin wave wavelengths on the nanometerscale where the exchange interaction dominates over dipole/magnetostatic energies. It wasonly recently that excitation and measurement of exchange spin waves in films was finallypossible opening a wide range of technological paths . Contrary to long wavelength,magnetostatic dominated spin waves that can exhibit non-reciprocal propagation, exchangespin waves are isotropic in their propagation due to the quadratic form of its dispersionrelation. Several mechanisms have been proposed to make the dispersion asymmetric sinceanisotropic propagation is key to the design of magnonic circuitry. Examples of such mech-anisms include, induced Dzyaloshinskii-Moriya Interaction , dipolar coupling , and anexternal magnetic field . Following the ideas behind graded-index optics, a continuousmodulation of the magnetic parameters has been recently proposed to control spin wavepropagation . For example, it has been shown that a gradual modulation of the satura-tion magnetization ( M s ) created with thermal landscapes can steer spin waves and changetheir dispersion relation as they propagate .In this work, we use micromagnetic simulations to demonstrate that exchange spin wavesdo not propagate reciprocally in a perpendicularly magnetized ferromagnetic thin film inwhich Ms varies linearly along the length of the film. To understand the origin of thephenomenon, we solve the linearized Landau-Lifshitz (LL) equation motion analytically.The linear variation in M s along the x -direction creates an effective linear potential V ( x ) inthe spin wave equation of motion. We transform to a non-inertial frame of reference wherethe inertial force cancels the force associated with the linear spin wave potential and allowsthe LL equation to be solved in the familiar constant M s condition. However, when wetransform back to the inertial frame, there is a frequency shift due to the acceleration of theexcitation source similar to what happens in the Doppler effect that broadens the spin wavedispersion. It is this Doppler shift of the spin waves that is the origin of the non-reciprocalbehavior. 2sing GPU-accelerated, micromagnetic code Mumax3 , we considered a 20 µ m × × × × y direction so that the effective width was 5376 nm.We used magnetic parameters of perpendicular materials such as Pt/CoFeB : exchangeconstant A = 15 pJ/m, uniaxial anisotropy K u = 1 MJ/m and applied field µ H = 1T, and recorded m y ( x, t ) in response to a field excitation of the form h sinc(2 πf c t )ˆ y with µ h = 50 mT and cutoff frequency f c = 500 GHz applied along the width over one cell inthe x direction positioned at the center of the film, x = 0. The dispersion curve is obtainedby performing a Fast Fourier Transform (2D-FFT) on m y ( x, t ) to get m y ( k, ω ) . We firstconsidered a constant saturation magnetization M = 1 MA/m throughout the sample andshow the dispersion as a surface plot of m y ( k, ω ) in Fig 1 (a). The dispersion curve exhibitsthe typical exchange-driven quadratic form, ω c ∝ k , in which spin waves propagating tothe right and to the left have the same frequency. The magnetization gradient was modeledby a linear variation of M s across 250 regions in the range x = [ − µ m, 8 µ m] (See Fig 1(e)). The maximum value, M s ( − µm ) = 1 . M s (8 µm ) = 0 . . In Fig1 (b) we show the dispersion curve obtained for spin waves propagating in a film with a M s gradient; additionally, solid lines indicate the theoretical dispersions corresponding tothe M s values at the edges and the middle of the film. There is a horizontal line below theferromagnetic resonance at 41 GHz related to a strong spin wave localization at the samplesedges due to the formation of a potential well in an inhomogeneous internal magnetic field .Two features contrast the constant M s case: First, there is an asymmetry in the dispersioncurve with respect to k = 0. Second, the dispersion curve is significantly broadened. Forpositive (negative) propagation, k > k < k increases, thebroadening extend from the 1 MA/m curve, white solid line, towards the 0 . . E = (cid:90) dV A ( ∇ m ) − K m z + µ Hm z , (1)3IG. 1: a) and b) are the dispersion curves found from micromagnetic simulations in theconstant M s and gradient M s case, respectively. In b) solid lines represent the theoreticaldispersions calculated using M s values at the edges and in the middle of the film. c)Dispersion curve obtained from the theoretical model with corresponding theoreticalcurves for comparison with b). e) M s linear gradient in the film used in the simulation.where K = K u − µ M s / δ m ( x , t ), around the static configuration, m = m z . To obtain theequations of motion in the long-wavelength limit we use m ( x , t ) = m z + δ m ( x , t ) to linearize ∂ m ∂t = − γ m × (cid:18) − M s δEδ m (cid:19) , (2)which can be cast as i ∂∂t m + = γ (cid:20) − AM s ∂ ∂ x + 2 K M s + µ H (cid:21) m + (3)for circularly polarized waves, m + = δm x + iδm y . The saturation magnetization varieslinearly as M s ( x ) = M + αx which converts (Eq 3) to i ∂∂ t m + = (cid:20) − β ∂ ∂ x + ω + q (cid:48) x − γµ M ( αx ) − i αxM ∂∂ t (cid:21) m + (4)where we have defined the effective mass: β = M / (4 γA ), the effective potential ω = γ (2 K /M + µ H − µ M ), and q (cid:48) = γµ α ( H − M ) /M related to the force that the4agnetization gradient exerts on the spin waves. Eq (4) is a Schr¨odinger-like equationwhere the term quadratic in x slightly modifies the linear potential and will be neglected(see Fig. 2 where the soft gray curve shows the effect of considering this term). The lastterms couples the space and time coordinates. To continue with an analytical description, wereplace the time derivative with the lowest possible spin wave frequency, the ferromagneticresonant frequency, ω . Then the equation to solve is i ∂∂ t m + = (cid:18) − β ∂ ∂ x + ω + qx (cid:19) m + , (5)where q = γµ α/M ( H − M − ω /γµ ) = − αγµ (1 + 2 K u /µ M ). It is worth notingthe importance of the space-time coupled term:, without it q = γµ α/M ( H − M ), whichwould allow a change of sign for H > M and a fixed α value. The coupled term preventsthe unphysical situation where the sign of q is not determined entirely by α .From the dispersion curve, Fig 1 (b), it is clear that a function of the form ω ( k ) is notachievable in the presence of a magnetization gradient. To obtain an analytical descriptionof the dispersion we perform a Fourier analysis of the solutions m + ( x, t ) to Eq. 5. Westart with the Landau-Lifshitz equation that describes the spin waves in a perpendicularlymagnetized magnetic film with a constant M s throughout the film, (cid:18) − β ∂ ∂x (cid:48) + ω (cid:19) n + = i ∂∂t (cid:48) n + (6)where n + ( x (cid:48) , t (cid:48) ) = n x + in y , n x and n y are spin wave components, and the dispersion can becalculated to be ω c ( k ) = 1 / (2 β ) k + ω . We then transform Eq 6 into an accelerated systemdescribed by x = x (cid:48) − / q/β ) t (cid:48) and t (cid:48) = t with the acceleration of the system given by − q/β . Under this transformation, the derivatives are ∂ (cid:48) x = ∂ x and ∂ (cid:48) t = ∂ t − ( qt/β ) ∂ x so thatthe equation in the accelerated reference frame is − β ∂ ∂x n + + ω n + + i qtβ ∂∂x n + = i ∂∂t n + . (7)Eq. 7 rightly describes the spin waves in the transformed system. However, to an observer atrest in the accelerated reference frame, there should be a potential of the form f x where f isthe inertial force producing the acceleration − q/β instead of the coupled term i ( qt/β ) ∂ x n + .Following refs. we perform a unitary transformation n + ( x (cid:48) , t (cid:48) ) = m + ( x, t ) e iqtx e iq t / (6 β ) (8)5IG. 2: Stationary Solutions. In a) the solution for five different frequencies are presented,the black solid line corresponds to the potential considered for the theoretical calculations,while the light gray curve is the complete potential ω + qx + γµ ( αx ) /M . In b) wepresent the k profile of the stationary solutions together with m y ( k, t ) from simulationsfor comparison.where m + ( x, t ) obeys Eq. 5 and effectively represents the physical situation with the poten-tial qx included.The stationary solution to Eq. 5, Ai [(2 βq ) / ( x − x )] e − iωt = Ai [ Bξ ] e − iωt , (9)is an Airy function with x = ( ω − ω ) /q , and is presented in Fig. 2 (a) for five differentfrequencies, f = 2 πω . In Fig. 2 (b) we present the k profile for the stationary solution with f = 70 GHz and compare with data obtained from a simulation in which the excitation fieldis of the form h sin(2 πf c t )ˆ y with µ h = 50 mT and frequency f = 70 GHz. As a result ofthe M s gradient, one frequency excites a band of wavenumbers which in turn broadens thedispersion relation.Using the stationary solution together with the integral representation of the Airy func-6ion, Ai [ Bξ ] e − iω ( k ) t = 12 πB (cid:90) ∞−∞ dk exp[ i ( k B + kξ − ω ( k ) t )] , (10)it is possible to construct an Airy wave packet. While we do not know ω ( k ) in the acceleratedsystem, we can transform back to the primed, inertial, reference frame in which ω ( k ) = ω c ( k ) = 1 / (2 β ) k + ω to calculate the integral, e − iqt ( x (cid:48) − qt (cid:48) / β ) e − iq t (cid:48) / β e − iω t (cid:48) πB (cid:90) ∞−∞ dk exp[ i ( k B + kξ (cid:48) − k t (cid:48) β )] , (11)using the useful formula (cid:82) ∞−∞ du exp[ i ( u / su + ru )] = 2 πe is (2 s / − r ) Ai ( r − s ). Aftertransforming back to the accelerated system, the Airy wave packet becomes m + ( x, t ) = e − iqtx e − iq t / β e − iω t Ai (cid:34) B ( x + qt β ) − (cid:18) B t β (cid:19) (cid:35) e − iB t/ (2 β )( B t / (6 β ) − B ( x + qt β )) . (12)Substitution of Eq 12 in Eq 5 verifies it is a solution. Fig 1 (b) shows the FFT of m + ( x, t )obtained from Eq 12 and displays a good agreement with the dispersion curve obtainedfrom the micromagnetic simulation, Fig 1 (b). In particular, the asymmetry and limits ofthe dispersion curve match. For higher frequencies our theoretical model appears narrowercompared to the simulations. This is because of the approximation made on the space-timecoupled term, Eq. 4. To visualize the accelerated reference frame and to compare thetheoretical and simulated accelerations, we change the place of excitation from the middleto the right edge of the film and record m y ( x, t ) for the gradient and constant M s situa-tions. Fig 3 (a) shows the recorded data for the M s gradient case and the solid white linecorresponds to the position of the front wave in the constant M s case. The spin wavespropagating in the M s gradient accelerate in the negative direction. The transformationsare x = x (cid:48) − / q/β ) t (cid:48) , t = t (cid:48) where ( x, t ) are the coordinates in the accelerated frame,and ( x (cid:48) , t (cid:48) ) are the coordinates in the inertial system. The point x = 0 corresponds to x (cid:48) = 1 / q/β ) t (cid:48) so that the accelerated frame is moving in the 1 / q/β ) t (cid:48) direction. Anobserver in the accelerated frame, should feel an inertial force in the − / q/β ) t (cid:48) directionproducing an acceleration ( − q/β ) = − . × m/s with the parameters used in thesimulations. In Fig 3 (b) we show the difference between the front waves of spin propagatingin the accelerated frame and in the inertial frame as a function of time. After fitting the7IG. 3: Spin wave acceleration. a) micromagnetic simulation of spin waves excited at theright edge of the M s gradient at t = 0, the white solid lines represents the trajectory thefront wave follows in the M s constant case. In b) we present the absolute difference ∆ x between the front wave position in the M s gradient case and the front wave position in the M s constant case as a function of time. The red solid line corresponds to the fitting.curve we find that ∆ x ( t ) = 1 / a ( t − t ) − ∆ x with an acceleration a = − . × m/s and a time t = 1 .
40 ns at which the maximum separation in the front waves,∆ x = 0 . µ m is reached. The theoretical acceleration, q/β , is lower than a by a factor of five which isattributable to the two approximations being made, namely, the quadratic term in x in thepotential that was neglected, and the space-time coupled term that was replaced with thelowest possible frequency ω . Still, the theoretical and simulated dispersion curves show agood agreement and the spin wave acceleration is clear.The situation changes when instead of exciting in the middle of the film, the excitationis made at the edges of the film. We recorded the m y ( x, t ) component in response to fieldexcitations of the same form as above but now placed at the edges of the film, x = − µ m,and x = 8 µ m, and the same for the remaining magnetic parameters. The dispersion curve ispresented in Fig 4 (a). As we only record the magnetic component, m y within the gradientregion, spin waves excited at the left edge, x = − µ m, only propagate to the right. The k > M s = 1 . M s values within the film. Similarly,spin wavesexcited at the right edge, x = 8 µ m, propagate to the left, with the k < M s = 0 . M s = 1 . k = 0 and creates a frequency gap between right and left propagatingstates that can be calculated as the difference between the ferromagnetic resonances of thedelimiting dispersion curves,∆ ω = µ γ ( M − s − M + s ) (cid:18) K u µ M + s M − s + 1 (cid:19) (13)where M ± s corresponds to the delimiting M s value for the positive or negative dispersionbranch. With our parameters we find ∆ f = ∆ ω/ π = 37 . x = ± µ m in Eq. 5 which modifies the Airy wavepacket by a shift in the argument of the Airy function and a modification of the phase bya factor e ± iB t/ (2 β ) Bx . Fig 4 (b) shows the dispersion curve obtained from the theoreticalmodel. The evident downward shift of the dispersion when compared to the simulation canbe explained in terms of the neglected quadratic x term in the potential that becomes largerat the edges of the gradient region. A key consequence of the discontinuity is that withinthe gap only one direction of propagation is permitted depending on the sign of the M s gradient. To verify, we again excite spin waves at the edges of the M s gradient region butchange the form of the excitation to a sinusoidal field h ( t ) = h sin(2 πf t )ˆ y with µ h = 50mT and a fixed frequency f = 50 GHz which is in the middle of the frequency gap. In Fig.4 (b) we present a snapshot taken at t = 3 ns: Propagation to the right is allowed whilepropagation to the left is forbidden. To compare, Fig. 4 (c) shows what happens in the M s homogeneous case where propagation is reciprocal.Our results demonstrate that a M s gradient induces a nonreciprocal propagation of spinwaves in a perpendicularly magnetized ferromagnetic film. The M s gradient is described byan additional linear potential as compared to the constant M s case. Mathematically, thelinear potential appears when transforming the constant M s case to an accelerated referenceframe with acceleration − q/β . The asymmetry in the dispersion is then explained as aDoppler effect. While non-reciprocity is observed in magnetostatic waves in thick films( ≈ µ m) , the non reciprocity presented in this work can be achieved in films thatare in the nano scale in thickness. Finally, we demonstrate that unidirectional spin wave9IG. 4: Unidirectional propagation of spin waves. a) Dispersion relation found frommicromagnetic simulations where spin waves are excited at the edges of the M s gradientregion. A discontinuity is found near k = 0. b) Snapshot of the m y component of themagntization taken at t = 3 ns after the start of the excitation where in the upper panel a M s gradient is considered and unidirectional propagation is achieved. For comparison thelower panel shows the constant M s case where reciprocal propagation is exhibited.propagation is achievable for a frequency band that depends on the M s gradient extremevalues. Unidirectional propagation of exchange spin waves is of the highest importance forthe design of magnonic computing devices. Our results are given in terms of a M s gradientthat can be achievable through different methods, e.g. ion implantation . However,the relevance of our study increases in light of recent studies in which modulation of the M s parameter is realized via a thermal landscape. We used parameters that correspond tothe expected variation of the saturation magnetization in a temperature range of 0-300 Kin Pt/CoFeB. While the underlying physical mechanism is different, in practice, achievingunidirectional propagation by reversing the M s gradient resembles the working principle ofa diode. Lastly, we have also verified that our results hold in the case where M s is constant10hroughout the film and the external magnetic field varies linearly. Note added . During the final preparation of this manuscript we became aware of recentlyreported work on similar effects through spatially varying exchange (R. Macedo et al., MMM2020 virtual conference, paper ER-04).
ACKNOWLEDGMENTS
This work was partially supported by fellowship Beca UNAM postdoctoral, Mitacs Glob-alink Research Award, National Council of Science and Technology of Mexico (CONACyT)under project 253754 and CB A1-S-22695, PAPIIT IG100519, Natural Sciences and Engi-neering Research Council of Canada (NSERC)
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