Fig. 2

Spin wave dispersion as a function of \(N_L\). The calculations are evaluated for \(d_L=5\) nm and a zero external field. (a) illustrates the case of \(N_L=2\), where the frequency shift is defined \(\Delta f=\vert f(-k)-f(k)\vert\). Highlighted modes (blue and green curves) show the two most non-reciprocal modes, namely the ones with a large \(\Delta f\). Bulk (BMs) and surface (SMs) modes are identified in the SW bandstructure illustrated in (e) and (f). Surface modes have chiral properties in the case of even \(N_L\) [see figures (a), (c), and (e)], while for odd \(N_L\), these modes behave reciprocally under the inversion of the wave vector [see figures (b), (d), and (f)]. In (c) and (f), calculated modes are compared with TetraX simulations depicted by open circles, obtaining a perfect agreement between both methods. The solid squares and circles represent highlighted modes evaluated at \(k=0\), which are discussed in Fig. 3.