Theory of nonreciprocal spin-wave excitations in spin Hall oscillators with Dzyaloshinskii-Moriya interaction

Abstract

A two-dimensional analytical model for the description of the excitation of nonreciprocal spin waves by spin current in spin-Hall oscillators in the presence of the interfacial Dzyaloshinskii-Moriya interaction (i-DMI) is developed. The theory allows one to calculate the threshold current for the excitation of spin waves, as well as the frequencies and spatial profiles of the excited spin wave modes. It is found, that the frequency of the excited spin waves exhibits a quadratic red shift with the i-DMI strength. At the same time, in the range of small and moderate values of the i-DMI constant, the averaged wave number of the excited spin waves is almost independent of the i-DMI, which results in a rather weak dependence on the i-DMI of the threshold current of the spin wave excitation. The obtained analytical results are confirmed by the results of micromagnetic simulations.PACS numbers: 75.30. Ds, 76.50.+g, 75.78.Cd, 75.76.+j been shown that the i-DMI in STO and SHO results in the excitation of two-dimensional nonreciprocal spin waves, and, at a sufficient strength of the i-DMI, in the generation of spiral spin wave modes.The main purpose of this work is the development of an analytical model, which describes the excitation of two-dimensional nonreciprocal spin waves in a nanocontact SHO (the quasi-onedimensional case of a nanowire-based SHO has been already considered theoretically in Ref. [38]).Our approach is based on an approximate solution of the linearized Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation and, in fact, is a generalization of the Slonczewski's theory [14] to the case of the presence of the i-DMI. The developed theory allows one to calculate profiles of the excited spin waves, which are approximately described by a combination of Laguerre's polynomials and Tricomi's hypergeometric functions, as well as to calculate the excitation threshold and frequency of excited spin waves, which both become lower with the increased i-DMI strength.The paper is organized as follows. Sections II describes the model system used in this study. In Sec. III a step-by-step derivation of the analytical formalism is presented. Analytically calculated results are compared with micromagnetic modeling in Sec. IV. Finally, conclusions are given in Sec. V. II. Device under study and micromagnetic simulationsThe device under investigation is shown in Fig. 1. It is a typical SHO, consisting of a ferromagnetic/ heavy metal bilayer. The current is injected locally in the bilayer by using a gold concentrator of a double-triangular shape with a distance d between the tips. The system is biased by an external magnetic field B ext , applied in the y-z plane and making the angle θ B with the film normal (axis z) ( Fig. 1(b)). The bias magnetic field is required in order to tilt the film static magnetization from the in-plane direction, and, if the angle θ M between the static magnetization and film normal is sufficiently small, the SHO supports excitation of propagating spin waves.

Spin Waves in Thin Films and Magnonic Crystals with Dzyaloshinskii–Moriya Interactions

“…In particular, this numerical technique can be used to calculate the spectrum of spin waves, and when a DMI is present, it reveals different properties characteristic of these systems, such as the spin-wave asymmetry. In this context, simulations have successfully supported theoretical formulations [80,96,97,99,[132][133][134][135][136][137] and experimental measurements [72][73][74][75][76][77][78][79][80][81][82][83][84][85][86][87]138] on thin-film systems with a homogeneous DMI. A micromagnetic simulation is based on numerically discretizing the continuum description of the magnetic system into a mesh of magnetic moments whose arrangement depends on the discretization method.…”

Nonreciprocal Magnetoacoustic Waves in Dipolar-Coupled Ferromagnetic Bilayers

“…Furthermore, the SW dispersion relation can be nonreciprocal. For instance, the Dzyaloshinskii-Moriya interaction (DMI) induces SW nonreciprocity [15][16][17] and therefore nonreciprocal SAW propagation, as recently theoretically [18] and experimentally demonstrated for ultrathin ferromagnetic/heavy metal bilayers [12]. In addition, it was shown many years ago that the SW dispersion can be highly nonreciprocal in magnetic multilayers [19][20][21], which was very recently exploited to obtain nonreciprocal SAW transmission in a Fe-Ga-B/Al 2 O 3 /Fe-Ga-B magnetic bilayer system for frequencies up to 1.4 GHz [22].…”

Hamiltonian formalism for nonlinear spin wave dynamics under antisymmetric interactions: Application to Dzyaloshinskii-Moriya interaction

“…In particular, it is critical to investigate the variation of SW spectrum at high SW amplitudes, and calculate the three-wave nonlinear SW coefficients that determine the threshold of the SW parametric instability. This knowledge is especially important for the successful development of parametric 54 and spin-torque [56][57][58] IDMI-based devices, in which high amplitudes of SWs are easily realized.…”