Discretized dynamics of exchange spin wave bulk and edge modes in honeycomb nanoribbons with armchair edge boundaries

Abstract

We develop a field theory to study the dynamics of long wavelength exchange spin wave excitations on honeycomb nanoribbons characterized by armchair edge boundaries and the Néel antiferromagnetic ordering state. Appropriate boundary conditions are established by requiring that the bulk and edge spins precess with the same frequency for any given spin wave eigenmode in these systems. A set of characteristic boundary equations, common for bulk and edge spin wave modes, are hence derived. The equations of motion for the spin dynamics are then solved to determine the propagating and evanescent exchange spin wave modes. We prove in general that the bulk spin wave dynamics is discretized due to the finite width of the nanoribbon. For an isotropic magnetic nanoribbon, the Dirac cone is reduced to a single linear dispersion curve due to this discretization. The number and wavelengths of allowed bulk modes for isotropic and anisotropic nanoribbons are determined from the derived characteristic boundary equations. As witnessed by our numerical results for different examples it is shown that the characteristics of these modes depend on the width of the nanoribbon and its antiferromagnetic anisotropy. Further, anisotropic nanoribbons, even those with the slightest anisotropy, present evanescent modes with non-linear dispersion relations. The spatial variation of the amplitudes of the evanescent exchange spin waves across the finite widths of the nanoribbons, is found to be strongly dependent on the system magnetic anisotropy and its width. The developed theoretical approach is general and can be applied for nanoribbons with all types of boundary edges.

Asymmetric dynamics of edge exchange spin waves in honeycomb nanoribbons with zigzag and bearded edge boundaries

“…For the edge modes, the eigenvectors for both matrices are (1,0) and (0,1). This is fundamentally different from the (1, 1) and (−1, 1) eigenvectors obtained in our previous study for edge spin waves on honeycomb nanoribbons with armchair boundaries [38]. Consequently, unlike nanoribbons with armchair edges boundaries, nanoribbons with zigzag and bearded edges boundaries do not allow edge spin waves propagating with the same (or same direction) on both edges simultaneously.…”

“…The classical field derivation for the bulk equations of motion is presented briefly. Details can be found in our previous study on nanoribbons with armchair edges boundaries [38].…”

“…The classical field spin wave theory is known to be equivalent to quantum spin wave approaches in the linear spin wave approximation. Despite its success in the study of surface spin waves, the classical field theory has not been systematically developed for edge spin waves in 2D materials, until our recent study of long wavelength exchange spin wave modes on nanoribbons with armchair edges boundaries [38]. Our previous study highlighted the important consequences on bulk and edge exchange spin waves induced by the finite width of the nanoribbon and the magnetic exchange anisotropy.…”

A new class of nonreciprocal spin waves on the edges of 2D antiferromagnetic honeycomb nanoribbons

“…The solution Ω − is hence excluded for propagating modes, since the spin wave excitation energy is required to be positive [69]. Consequently, Ω + is generally considered as the unique energy dispersion relation for propagating spin waves in an antiferromagnet [11,51,52,58,59,61]. Remarkably, we will prove that this conclusion cannot be generalized to the edge spin waves in 2D antiferromagnetic honeycomb nanoribbons with nonzero DMI.…”

“…Later, Stamps and Camley [59] formulated equivalent exchange boundary conditions based on the requirement that the precession frequency of a boundary spin should match that of a bulk spin. Equivalently, boundary spins are required to satisfy the bulk equations of motion [11,51,52,[59][60][61][64][65][66][67][68] which yields an elegant boundary condition equation…”