Field Theory of Linear Spin-Waves in Finite Textured Ferromagnets

Abstract

In the context of an ever-expanding experimental and theoretical interest in the magnetization dynamics of mesoscopic magnetic structures, both in the classical and quantum regimes, we formulate a low energy field theory for the linear spin-waves in finite and textured ferromagnets and we perform its constrained canonical quantization. The introduction of a manifestly gauge invariant Lagrangian enables a straightforward application of the Noether's theorem. Taking advantage of this in the context of a broad class of axisymmetric ferromagnets of special conceptual and experimental relevance, a general expression of the conserved and quantized spin-wave total angular momentum is rigorously derived, while separate conservation and quantization of its orbital and spin components are established for a more restricted class of uniaxial exchange ferromagnets. Further particularizing this general framework to the case of axially saturated magnetic thin disks, we develop a semi-analytic theory of the low frequency part of the exchange-dipole azimuthal spin wave spectrum, providing a powerful theoretical platform for the analysis and interpretation of magnetic resonance experiments on magnetic microdots as further demonstrated in a joint paper [arxiv The Orbital Angular Momentum of Azimuthal Spin-Waves]

Figures (6)

• Figure 1: a) Schematic representation of the linear approximation of the Larmor precession. In the out-of-equilibrium regime, the instantaneous magnetization direction $\bm u = {\bm M}/M_{ S}$ (blue vector) rotates counterclockwise around the equilibrium direction $\bm u_0$ (red vector). We define the linear spin-wave $\bm m = \bm u - (\bm u \cdot \bm u_0 ) \bm u_0$, the dynamical deviation component perpendicular to $\bm u_0$ and neglect axial component of $\mathcal{O}(m^2)$. (a) In axisymmetric geometries, the conservation of total angular momentum follows from the invariance of the action on a global rotation $\delta r_J$ through an infinitesimal angle $\delta \theta$, about a natural $O_z$ axis (black arrow). This rotation admits a decomposition $\delta r_J = \delta r_L \circ \delta r_S$ as a combination of an extrinsic rotation about the origin $O$: the orbital component, and an intrinsic rotation in the local frame (red arrow): the spin component.
• Figure 2: The notion of conservation of angular momentum is contingent on an invariance under continuous global rotation about a given axis. A first necessary condition for the SW Lagrangian of a finite ferromagnet to satisfy this constraint is, of course, that the volume it occupies in space is axisymmetric. This is for instance the case for ferromagnets in the shape of a) a sphere, b) a disk or c) a ring. The same requirement applies also to its texture. While this is trivially satisfied by d) an axially uniform one, this condition is evidently also fulfilled by e.g., e) a vortex and f) a Bloch or g) Néel skyrmion.
• Figure 3: Snapshot of the spatio-temporal pattern, $\bm m (x,t)$, formed by azimuthal spin-waves with index $n_{ J}\in [-1,1]$ propagating in a disk axially magnetized with $H_0 < M_{ S}$ (cone state). The textured local equilibrium is $\bm u_0 (\bm x)$ (orange arrow); the instantaneous magnetization is $\bm u (x,t)$ (blue arrow); the dynamical magnetization vector is $\bm{m} (\bm{x},t)$ (cyan arrow) ; the torus indicates its time trajectory along a right-handed rotation around $\bm{u}_0$. By emphasizing the elliptical precession of the magnetization, we point out that the total angular momentum usually cannot be separated into a spin and an orbital component and the sole index $n_{ J}$ is a good quantum number.
• Figure 4: (a) Spatial snapshot patterns formed by azimuthal SW modes of index $n_{ J} \in [-4,+4]$ propagating in a magnetic disk uniformly magnetized along the normal direction with the north pole oriented towards the top. In the asymptotic case of $H_0 \gg M_{ S}$, the local precession becomes circular, allowing the decomposition of the total angular momentum into a spin and an orbital component. Amplitude wise, the index $|n_{ J}|$ counts the number of oscillations of the wavefront (visualized here by its radial component; see dotted line) along the periphery, while $|n_{ L}|$ counts the number of revolutions of $\bm{m}$ (see the repetition of the orientation at $0^\circ$, marked by the $\star$ symbol). Polarity wise, each pattern uniquely links the sense of gyration of its wavefront to the direction of the local precession. The large circular arrow indicates the direction of the phase gyration (see color wheel) and the small circular arrow indicates the local Larmor precession. By convention, a positive index indicates the Larmor direction: right-handed with respect to the magnetization so $n_{ J}=n_{ L}+n_{ S}$. Arranging patterns with the same $|n_{ J}|$ in columns reveals which patterns, having both opposite $n_{ J}$ and opposite frequency within a given column, are coupled by the dynamical DDI.
• Figure 5: Convergence in eigenfrequency (expressed in GHz) of the first ten radial harmonics for the exchange-dipole SWs belonging to subspace $n_{ J} = 0$, as a function of the number of basis eigenvectors retained in the truncated expansion.