week ending PRL 118, 017203 (2017) PHYSICAL REVIEW LETTERS 6 JANUARY 2017
Breaking of Galilean Invariance in the Hydrodynamic Formulation of Ferromagnetic Thin Films
Ezio Iacocca,1,2,* T. J. Silva,3 and Mark A. Hoefer1 1Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309-0526, USA 2Department of Physics, Division for Theoretical Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden 3National Institute of Standards and Technology, Boulder, Colorado 80305-3328, USA (Received 30 June 2016; published 5 January 2017) Microwave magnetodynamics in ferromagnets are often studied in the small-amplitude or weakly nonlinear regime corresponding to modulations of a well-defined magnetic state. However, strongly nonlinear regimes, where the aforementioned approximations are not applicable, have become exper- imentally accessible. By reinterpreting the governing Landau-Lifshitz equation of motion, we derive an exact set of equations of dispersive hydrodynamic form that are amenable to analytical study even when full nonlinearity and exchange dispersion are included. The resulting equations are shown to, in general, break Galilean invariance. A magnetic Mach number is obtained as a function of static and moving reference frames. The simplest class of solutions are termed uniform hydrodynamic states (UHSs), which exhibit fluidlike behavior including laminar flow at subsonic speeds and the formation of a Mach cone and wave fronts at supersonic speeds. A regime of modulational instability is also possible, where the UHS is violently unstable. The hydrodynamic interpretation opens up novel possibilities in magnetic research.
DOI: 10.1103/PhysRevLett.118.017203
Magnetodynamics in thin film ferromagnets has been This Letter shows that the LL equation exactly maps into studied for many decades. Advances in nanofabrication and a DH system of equations, without long-wavelength and the advent of spin transfer [1,2] and spin-orbit torques [3] low-frequency restrictions. The conservative equations are have opened a new frontier of experimentally accessible analogous to the Euler equations of a compressible, isen- nonlinear physics [4–8]. Large-amplitude excitations neg- tropic fluid. The DH equations for a planar ferromagnet ate the use of typical linear or weakly nonlinear analyses admit spin-current-carrying, spatially periodic magnetiza- [9–11], necessitating instead either micromagnetic simu- tion textures termed “uniform hydrodynamic states” lations [12] or analytical approaches suited to strongly (UHSs), providing a continuous interpolation between large nonlinear dynamics. Therefore, an interpretation of the amplitude spin superflows [14–16] and small-amplitude Landau-Lifshitz (LL) equation that includes full nonlinear- spin waves. Within the DH formulation, we prove that ity, yet is amenable to analytical study, would be insightful. planar ferromagnets break Galilean invariance and elucidate A hydrodynamic interpretation was proposed by Halperin their reference-frame-dependent dynamics by identifying and Hohenberg [13] to describe spin waves in anisotropic the linear dispersion relation for spin waves propagating on ferro- and antiferromagnets. Recently, theoretical studies of top of a UHS background. Such symmetry breaking at the thin film ferromagnets with planar anisotropy have identified level of linear spin waves is striking and fundamentally a relationship to superfluidlike hydrodynamic equations different from the nontrivial speed-dependent dynamics of [14–19] supporting large-amplitude modes beyond weakly topological textures due to their inherent nonlinearity, e.g., nonlinear spin wave and macrospin modes [10,11]. However, Walker breakdown for domain wall propagation [35] and these studies are limited to the long-wavelength, low- core reversal in magnetic vortices [36]. In this Letter, we also frequency regime where linear and weakly nonlinear show that static textures can break Galilean invariance for approaches apply. The relaxation of these approximations infinitesimal spin wave excitations that ride on a textured along with the identification of a deep connection between background. To emphasize this novel result, we suggest a magnetodynamics and fluid dynamics brings new perspec- Brillouin light scattering experimental test where broken tives on magnetism and reveals novel physical regimes. Galilean invariance manifests itself as a spin-wave Indeed, nonlinear, dispersive physics are required to describe dispersion shift in the presence of a UHS. superfluids and exotic structures such as solitons, quantized We consider the nondimensionalized LL equation (see vortices, and dispersive shock waves (DSWs) [20–23], Supplemental Material [37]) as exemplified by Bose-Einstein condensates (BECs) ∂m ¼ −m × h − αm × m × h : ð1Þ [20–22,24–34]. To obtain an analytical description of ∂t eff eff large-amplitude magnetic textures, we introduce dispersive Damping is parametrized by the Gilbert constant α, hydrodynamic (DH) equations for a thin-film ferromagnet. m¼M=Ms ¼ðmx;my;mzÞ is the magnetization vector
0031-9007=17=118(1)=017203(6) 017203-1 © 2017 American Physical Society week ending PRL 118, 017203 (2017) PHYSICAL REVIEW LETTERS 6 JANUARY 2017 h ¼ Δm− 2 normalized to the saturation magnetization, and eff limit, i.e., j∇nj ≪ 1, jnj ≪ 1, and juj ≪ 1. As we show σmzzˆ þ h0zˆ is the normalized effective field including below, the full nonlinearity and exchange dispersion ferromagnetic exchange, Δm; total anisotropy determined included in Eqs. (3a) and (3b) are required to describe by σ ¼ sgnðMs − HkÞ, where Hk is the perpendicular the existence and stability regions of magnetic hydro- magnetic anisotropy field, such that σ ¼þ1 (σ ¼ −1) dynamic states and broken Galilean invariance. represents a material with easy-plane (perpendicular mag- Insight can be gained from the homogeneous field netic) anisotropy; and a perpendicular applied field, h0zˆ. ∇h0 → 0, conservative α → 0 limit, where Eqs. (3) become This nondimensionalization of a two-dimensional (2D) thin conservation laws for n and u. Notably, the non-negative film provides a parameter-free description of materials with deviation from vacuum ð1 − n2Þ, or fluid density, is not planar or uniaxial anisotropy. We consider an idealized case conserved. A conservation law for the momentum p ¼ nu where in-plane magnetic anisotropy is negligible; i.e., its can also be obtained, symmetry-breaking contribution only perturbs the leading ∂p ∇n∇nT order solution, similar to domain wall Brownian motion [40]. ¼ ∇ ½ð1 − n2ÞuuT þ∇Pðn; jujÞ þ ∇ ∂t · · 1 − n2 Fluidlike variables are introduced using the canonical Hamiltonian cylindrical transformation [41] nΔn þ 1 j∇nj2 n2j∇nj2 − ∇ 2 þ ; ð Þ 1 − n2 ð1 − n2Þ2 5 n ¼ mz; u ¼ −∇Φ ¼ −∇½arctan ðmy=mxÞ ; ð2Þ where the magnetic pressure is defined as where Φ is the azimuthal phase angle. We identify n jnj ≤ 1 u 1 2 2 ( ) as the longitudinal spin density and as the fluid Pðn; jujÞ ¼ ð1 þ n Þðσ − juj Þ − σ: ð6Þ velocity. There are two important features of Eq. (2). First, 2 the flow is irrotational because the velocity originates from Equations (3a) with α ¼ 0, and (5) are analogous to the a phase gradient, i.e., only quantized circulation, such as a time-reversed Euler equations expressing conservation of magnetic vortex [15], is possible. Second, Φ is undefined mass and momentum for a 2D, compressible, isentropic when n ¼ 1, corresponding to fluid vacuum. fluid with a density- and velocity-dependent pressure P. Utilizing the transformation (2) and standard vector Additionally, the one-dimensional conservative limit of calculus identities, the LL equation (1) can be exactly Eqs. (3a) and (3b) are exactly the equations describing recast as two DH equations [37] polarization waves in two-component spinor Bose gases jnj ∼ 1 ∂n ∂Φ [33,34] and, in the near vacuum ( ), long-wavelength, ¼ ∇ ½ð1 − n2Þu þ αð1 − n2Þ ; ð Þ and low-frequency regime, approximate the mean field ∂t · |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} ∂t 3a |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} dynamics of a BEC [24,44]. This suggests that thin film spin current spin relaxation ferromagnets are ripe for the exploration of nonlinear ∂u Δn nj∇nj2 structures observed in BECs, e.g., “Bosenovas” [25,27] in ¼ ∇½ðσ − juj2Þn − ∇ þ ∂t |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} 1 − n2 ð1 − n2Þ2 attractive (σ ¼ −1), and dark solitons [30], vortices [22], and |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} velocity flux DSWs [20] in repulsive (σ ¼þ1) BECs. Some of these dispersion structures have been observed in uniaxial (dissipative drop- 1 – −∇h þ α∇ ∇ ½ð1 − n2Þu : ð Þ lets [5 7]) and planar (vortices [8]) thin film ferromagnets. |{z}0 1 − n2 · 3b As we demonstrate, hydrodynamic states are also supported. |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} potential force Consider an ideal planar thin film ferromagnet (σ ¼þ1) viscous loss and a homogeneous field. Equations (3a) admit a static Equation (3a) is reminiscent of spin density continuity (∂Φ=∂t ¼ 0) solution with nonzero fluid velocity, u ¼ uxˆ, [42] from which we identify the spin density flux as the juj < 1, n ¼ 0, and h0 ¼ 0. These are ground states known spin current as spin-density waves (SDWs) [45] or soliton lattices [15] that minimize both exchange and anisotropy energies; i.e., J ¼ −ð1 − n2Þu: ð Þ s 4 any configuration with juj < 1 is stable when m lies purely Vacuum carries zero spin current. However, maximal spin in plane. SDWs exhibit a periodic structure that affords current is reached when n ¼ 0, identified as the saturation them topological stability whereby the phase rotation can density. This implies that ferromagnetic textures (u ≠ 0Þ be unwound only by crossing a magnetic pole (jnj¼1) 2 are better spin current conductors than small-amplitude [15,37]. For a nonzero field, jh0j < 1 − u , SDWs are also 2 spin waves [43]. The hydrodynamic equivalents for the supported for any juj < 1 when n ¼ h0=ð1 − u Þ due to the fluid velocity Eq. (3b) are displayed. When n ¼j∇h0j¼0, longitudinal spin density relaxation effected by Eq. (3a). Eq. (3b) becomes ∂u=∂t ¼ α∇ð∇ · uÞ, a diffusion equation Such a relaxation maintains u and thus the topology of for the velocity; hence, α > 0 acts similar to a viscosity. and finite spin current carried by a SDW. This property is Previous works [13–16] have neglected exchange identical to that of equilibrium transverse spin currents dispersion and nonlinearity in Eqs. (3) by assuming the in other magnetic textures including domain walls and long-wavelength, near saturation density, low-velocity vortices [Ref. [15], Eq. (4) in Ref. [46]].
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For no damping, Eqs. (3) admit dynamic solutions para- where k isthewavevector,andthevelocityV reflects a metrized by the constants ðn;¯ u¯Þ, where jn¯j ≤ 1, u ¼ u¯ xˆ, Doppler shift, i.e., the velocity of an external observer with called the uniform hydrodynamic state (UHS). The fluid respect to the UHS. The dispersion relation shows that velocity u¯ is the wave number of the UHS whose frequency magnetic systems lack Galilean invariance. In other words, Ω ¼ dΦ=dt is an observer velocity V ∝ u does not generally result in a reference frame where the relative fluid velocity is zero. Ωðn;¯ u¯Þ¼−ð1 − u¯ 2Þn¯ þ h ; ð Þ 0 7 Galilean invariance is recovered near vacuum ðjn¯ ≈ 1Þ with ω ðk;VÞ¼ð2u−VÞ k jkj2 obtained from the magnetic equivalent of Bernoulli’s equa- dispersion · , i.e., exchange- 2 mediated spin waves and the BEC limit [24,44]; and for tion 2Pðn;¯ ju¯jÞ þ u¯ þ n¯ðΩ − h0Þ¼−σ [37]. Here, positive pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 u¯ implies clockwise spatial increase of the azimuthal spin superflow ðn¯ ≈ 0Þ, ω ðk;VÞ¼−V·k jkj 1þjkj . phase Φ, whereas negative Ω implies clockwise temporal Importantly, the fluid velocity u¯ confers a spectral shift in precession about the þzˆ axis defining forward and backward Eq. (8) due to the UHS’s intrinsically chiral topology, similar wave conditions, schematically shown in Fig. 1. This is in to the interfacial Dzyaloshinskii-Moriya interaction [48]. contrast to magnetostatic forward and backward volume The long wavelength limit of Eq. (8) leads to coincident waves established by the relative direction between their spin-wave phase and group velocities, i.e., magnetic sound wave vector and the external applied field. velocities, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The magnetization in a UHS can exhibit large angle ¯ 2 2 deviations from the þzˆ axis, making it a strongly nonlinear s ¼ 2n¯ u¯ þV ð1 − n¯ Þð1 − u¯ Þ: ð9Þ texture. Near saturation density, jn¯j ≪ 1,aUHSlimitstoa Here, we assume V collinear and opposite to u (V ¼ −V¯ xˆ). spin superflow [14–16] whereas near vacuum, n¯ ∼ 1,the Subsonic flow occurs when spin waves can propagate UHS frequency Eq. (7) becomes the exchange spin-wave 2 both forward and backward: s− < 0
FIG. 2. UHS phase diagram for (a) V¯ ¼ 0 and (b) u¯ ¼ 0 with subsonic (white), supersonic (gray), and modulationally unstable (yellow) regimes. Circles are micromagnetic calculations of the sonic curves Mu ¼ 1 and MV ¼ 1. The BEC-limit regime sonic FIG. 1. Schematic of the magnetization rotation of a UHS. The curve is dashed. Open squares represent the micromagnetically longitudinal spin density is the vertical axis limited by vacuum calculated sonic curve of a width w ¼ 20, thickness δ ¼ 1 nano- (jnj¼1) and saturation density (n ¼ 0). Forward and backward wire including nonlocal dipolar fields and T ¼ 300 K thermal wave conditions are determined by the sign of the frequency Ω. field. Selected simulation conditions are denoted by x1 to x4.
017203-3 week ending PRL 118, 017203 (2017) PHYSICAL REVIEW LETTERS 6 JANUARY 2017 independent of h0, implying that only the UHS longitudinal We use the same numerical method described above spin density and its nontrivial topology, u¯, set the supersonic with the addition of thermal fluctuations and the symmetry- transition, not the frequency Ω. It must be noted that broken breaking nonlocal dipolar fields to study the stability of a Galilean invariance causes the standard Landau criterion SDW in a nanowire of nondimensional thickness δ ¼ 1. concept u<¯ min½s [24] to give an incorrect sonic curve. In this case, the SDW topological structure completely A qualitatively distinct flow regime occurs when ju¯j > 1 collapses at the boundary shown in Fig. 2(a) by squares. and the sound velocities Eq. (9) are complex. This In contrast to a recent report where stable spin superflow corresponds to a change in the mathematical type of the was predicted only for nanowires shorter than the material long wavelength Eqs. (3) from hyperbolic (wavelike) to exchange length [18], we observe stable SDWs over a wide elliptic (potential-like). Consequently, the UHS is unstable range of parameters in phase space (Supplemental Material, in the sense that small fluctuations lead to drastic changes video 2 [37]). in its temporal evolution or modulational instability (MI) The supersonic transition in the moving frame is esti- ¯ [50,51]. Note that ju¯j < 1, jVj > 1 does not result in MI. mated by use of a numerical method described elsewhere The aforementioned regimes were validated by perform- [53]. A moving, perpendicular, localized, weak magnetic ing micromagnetic simulations with damping [12].We field spot with velocity V¯ is used to perturb a homogeneous simulate dynamics for an ideal Permalloy nanowire state in the bias field h0 ¼ n¯. The obtained sonic curve is in μ M ¼ 1 w ¼ 20 ( 0 s T) of nondimensional width with trans- good agreement with MV ¼ 1, red circles in Fig. 2(b). verse free spin boundary conditions and horizontal periodic We now explore the effect of finite-sized obstacles on a boundary conditions (PBCs). We initialize with a SDW, UHS. As observed in BECs, obstacles can generate vortices, include only local dipolar fields (zero thickness limit), and wave fronts, and DSWs in a fluid flow [20–22]. Note α ¼ 0 01 h set . . A homogeneous field 0 stabilizes the SDW that wave fronts, i.e., “spin-Cherenkov” radiation, were n¯ u¯ at a specific and a quantized that satisfies the PBC. This previously observed via micromagnetic simulations in enables us to numerically probe along a horizontal line in homogeneous (u¯ ¼ 0), thick ferromagnets in the moving the phase diagram of Fig. 2(a) by implementing a slowly V¯ ≠ 0 h reference frame ( ) [54]. The wave fronts studied here increasing 0. By inserting a point defect (a magnetic void), u¯ ¼ 0 V¯ ≠ 0 the SDW spontaneously generates spin waves when n¯ is are different, resulting from both moving ( , ) and u¯ ≠ 0 V¯ ¼ 0 large enough to cross the supersonic transition, leading to a static ( , ) reference frames, yet another mani- breakdown in the spatial homogeneity of the SDW [37]. festation of broken Galilean invariance. We illustrate these α ¼ 0 01 Because of the SDW’s topology and the PBC, the change in features with simulations where . and local dipolar symmetry is accommodated by annihilating a single 2π fields are included, shown in Fig. 3 as a gray scale map and phase rotation and reducing u¯ in a quantized fashion. vector field for n and u, respectively (see Supplemental Topologically, this is possible in planar ferromagnets by Material [37] for the corresponding in-plane magnetization crossing a magnetic pole, e.g., nucleating a vortex, as map). First, we consider the stable subsonic condition x1 shown in the Supplemental Material, video 1 [37]. This was of Fig. 2(a) for a SDW in the static reference frame also observed in wires of width w ¼ 40. The sonic curve (n;¯ u¯Þ¼ð0.1; 0.4) with a magnetic defect within a circular estimated this way is shown in Fig. 2(a) by black circles, area of π=u¯ in diameter. The static configuration in Fig. 3(a) demonstrating good agreement with Mu ¼ 1. We attribute is analogous to Bernoulli’s principle for laminar flow. any discrepancy to boundary and finite size effects [52],as A different situation occurs at the supersonic condition x2 further explored below. ðn;¯ u¯Þ¼ð0.7; 0.6Þ, Fig. 3(b). Here, the density develops a FIG. 3. Snapshots of a (a), (b) SDW flowing past a stationary magnetic defect (V¯ ¼ 0) and; (d), (e) a homogeneous state subject to a moving, localized magnetic field (V¯ ≠ 0) with longitudinal spin density n (gray scale map) and velocity field u (arrows). The simulation region is much larger than what is visible. The defect or localized magnetic field position is shown by a red circle. For subsonic conditions, (a) and (d), the flow is static and laminar. In supersonic flow, (b) and (e), a Mach cone (dashed) and static wave fronts are ob- served. Propagating vortex-antivortex pairs with cores (asterisks) generated in (b) are shown in (c), far from the defect as opposite circulations of the velocity field (back- ground u¯ ¼ 0.6 is subtracted).
017203-4 week ending PRL 118, 017203 (2017) PHYSICAL REVIEW LETTERS 6 JANUARY 2017 distinct Mach cone (dashed), delimiting static wave fronts fronts, and vortex nucleation. In addition, the form of the and the nucleation of propagating vortex-antivortex pairs, DH equations suggests the existence of coherent structures shown far from the defect in Fig. 3(c). In the moving reference such as oblique solitons and DSWs. frame, a homogeneous state is perturbed by a moving, The authors thank Leo Radzihovsky, Eric Edwards, weak, localized field. Utilizing the subsonic condition x3 Mark Keller, and Hans Nembach for beneficial discussions. ðn;¯ u;¯ V¯ Þ¼ð0.7;0;0.6Þ, the flow is laminar, Fig. 3(d) E. I. acknowledges support from the Swedish Research [cf. supersonic x2 in Fig. 2(a)]. Wave front radiation outside Council, Reg. No. 637-2014-6863. M. A. 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