Dipole-exchange spin waves in unsaturated ferromagnetic nanorings with interfacial Dzyaloshinski–Moriya interactions

Abstract

A theoretical analysis is made for the quantized spin waves in single-layered ferromagnetic nanorings with the added effect of interfacial Dzyaloshinski–Moriya interactions (DMI). A microscopic, or Hamiltonian-based, formalism is employed that includes competing terms for the symmetric (bilinear) exchange interactions, the antisymmetric DMI, the magnetic dipole–dipole interactions, and applied magnetic field. It is found that, in our model, the transition field value between vortex and onion states is shifted by the inclusion of DMI effects. Significantly, the spin-wave frequencies are also modified with the effects being largest in the onion state close to the transition field. We present combined analytical and numerical results obtained for the static and dynamical magnetization, including the frequencies and amplitudes (with relative phase) of the spin waves when interfacial DMI is present.

Spin Waves in Ferromagnetic Nanorings with Interfacial Dzyaloshinskii–Moriya Interactions: II. Directional Effects

“…where d is the axial unit vector of the DMIs and J DM n,m is the interaction strength (which satisfies the antisymmetry property J DM n,m = −J DM m,n on interchange of the site labels). We assume nearest-neighbour exchange only, and we adopted the same sign convention as in [32] by labelling the value as J DM when the in-plane vector separation has a positive x or z component, and −J DM otherwise. An interior spin site in the nanoring will have four neighbours (with two neighbours of J DM and two of −J DM ), but spin sites near the lateral edges will have fewer neighbours, leaving the possibility of unmatched + and/or − pairs.…”

“…An interior spin site in the nanoring will have four neighbours (with two neighbours of J DM and two of −J DM ), but spin sites near the lateral edges will have fewer neighbours, leaving the possibility of unmatched + and/or − pairs. There are consequences for both the static and dynamic properties of the nanorings, particularly at the edges, and these were investigated in [32] when the axial vector d is perpendicular to the plane of the nanoring. Here, we focus on the novel effects arising when d is in the plane, either along x (perpendicular to B 0 ) or along z (parallel to B 0 ).…”

“…These coupled finite-difference equations can be solved numerically using iterative techniques, as described in earlier similar works (see, e.g., [31,32]). Briefly, this involves choosing a trial initial configuration for the set of angles.…”

“…Having discussed the static magnetic properties, we now examine the magnetization dynamics in terms of the spin waves (SWs) in the nanorings. Following the approach in [31,32], for each spin at site n, we carry out a matrix transformation from the global (x, y, z) axes to a new set of local axes (X n , Y n , Z n ) aligned with that spin such that the new Z n axis is along its equilibrium direction. Following [31,32], we rewrite the previous spin Hamiltonian in terms of boson operators using the Holstein-Primakoff transformation [34] applied relative to the local axes.…”

“…Initially, the SW modes were investigated without DMIs [31]. Subsequently, a separate calculation was reported for nanorings with interfacial DMIs (or i-DMIs) included [32]. The latter paper (henceforth referred to as I), which is the forerunner to the present work, studied the case in which the directional properties of the interfacial DMIs were varied through making different choices for the axial vector.…”