Probing the Dzyaloshinskii-Moriya Interaction via the Propagation of Spin Waves in Ferromagnetic Thin Films

Zhenyu Wang, Beining Zhang, Yunshan Cao, and Peng Yan

Phys. Rev. Applied 10, 054018 – Published 7 November 2018

Abstract

The Dzyaloshinskii-Moriya interaction (DMI) has attracted considerable recent attention owing to the intriguing physics behind it and the fundamental role it plays in stabilizing magnetic solitons, such as magnetic skyrmions and chiral domain walls. A number of experimental efforts have been devoted to probe the DMI, among which the most popular method is the Brillouin light scattering spectroscopy (BLS) to measure the frequency difference of spin waves with opposite wave vectors $\pm k$ perpendicular to the in-plane magnetization $m$ . Such a technique, however, is not applicable to the cases of $k ∥ m$ , since the spin-wave reciprocity is recovered then. For a narrow magnetic strip, it is also difficult to measure the DMI strength using BLS because of the spatial resolution limit of lights. To fill these gaps, we propose to probe the DMI via the propagation of spin waves in ferromagnetic films. We show that the DMI can cause the noncollinearity of the group velocities of spin waves with $\pm k ∥ m$ . In heterogeneous magnetic thin films with different DMIs, negative refractions of spin waves emerge at the interface under proper conditions. These findings enable us to quantify the DMI strength by measuring the angle between the two spin-wave beams with $\pm k ∥ m$ in homogeneous film and by measuring the incident and negative refraction angles in heterogeneous films. For a narrow magnetic strip, we propose a nonlocal scheme to determine the DMI strength via nonlinear three-magnon processes. We implement theoretical calculations and micromagnetic simulations to verify our ideas. The results presented here are helpful for future measurement of the DMI and for designing novel spin-wave spintronic devices.

Received 11 July 2018
Revised 11 September 2018

DOI:https://doi.org/10.1103/PhysRevApplied.10.054018

Physics Subject Headings (PhySH)

Dzyaloshinskii-Moriya interaction Ferromagnetism Magnons Spin waves Spintronics

Nonlinear waves Ultrathin films

Landau-Lifschitz-Gilbert equation

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Zhenyu Wang, Beining Zhang, Yunshan Cao, and Peng Yan^*

School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Film and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China

^*yan@uestc.edu.cn

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Vol. 10, Iss. 5 — November 2018

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Images

Figure 1
(a) Schematic illustration of an ultrathin film with width $L_{x}$ , length $L_{y}$ , and thickness $L_{z}$ . A magnetic field $H_{0}$ is applied along $+ \hat{y}$ , which makes the magnetization $m$ lying in the film plane ( $m ∥ + \hat{y}$ ). (b) The isofrequency curve calculated based on the dispersion relation Eq. (4). $k_{x}^{+}$ and $k_{x}^{-}$ are the wave vectors of spin waves propagating along the direction perpendicular to the magnetization. $v_{g}^{+}$ and $v_{g}^{-}$ are the group velocities of spin waves with $k_{y}^{\pm} ∥ m$ . $k^{+}$ and $k^{-}$ are the wave vectors of spin waves propagating along the magnetization ( $v_{g y}^{\pm} ∥ m$ ). (c) Noncollinear propagation of two spin-wave beams with opposite wave vectors ( $k_{1} = - k_{2}$ ). The spin-wave beams are excited by a sinusoidal monochromatic microwave source with $μ_{0} h_{0} = 0.1$ T and $ω / 2 π = 80$ GHz in a rectangular region (black bar). (d) The angle between the two spin-wave beams as a function of the excitation frequency for various DMI constants $D$ . The dots and curves correspond to the numerical simulation results and the analytical formula Eq. (6), respectively.
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Figure 2
(a) Schematic plot of the generalized Snell’s law for spin-wave scattering at the DMI interface. The magnetization of ultrathin film is saturated along the $+ \hat{y}$ direction. The green (lower) and magenta (upper) circles correspond to the isofrequency curves in momentum space for spin-wave propagation in no-DMI and DMI regions, respectively. (b)–(f) Refraction and reflection of spin waves scattering at the DMI interface with different incident angles: (b) $θ_{i} = - 60^{\circ}$ , (c) $θ_{i} = - 18^{\circ}$ , (d) $θ_{i} = 0^{\circ}$ , (e) $θ_{i} = 18^{\circ}$ , and (f) $θ_{i} = 60^{\circ}$ . Arrows label the group velocities of spin waves.
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Figure 3
(a) The heterogeneous ultrathin film with a DMI strip (gray region) in the center. The width of the DMI strip is $w = 50$ nm. The sinc-function field is applied in the regions of the black bar. (b) Three components of the equilibrium magnetization $m_{0}$ along the longitudinal direction at $x = 500$ nm. (c) FFT spectrum along the DMI strip center $y = 900$ nm in (a). In (c), the black curve corresponds to the analytical formula Eq. (3), while the yellow one represents the modified formula Eq. (13) with the fitting parameter $δ \approx {25.1}^{\circ}$ . The wave vectors of spin waves with 35 GHz are $k_{b 1} = 0.039 {nm}^{- 1}$ and $k_{b 2} = 0.165 {nm}^{- 1}$ , which correspond to the backward and forward spin waves, respectively.
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Figure 4
Schematic picture of nonlinear three-magnon processes in the DMI strip. In the dashed red square, it shows the three-magnon confluence of $k_{i}$ and $k_{b}$ into $k$ . In the dashed blue square, we plot the stimulated three-magnon splitting of $k_{i}$ into two modes $k_{1} = k_{b}$ and $k_{2}$ , assisted by a localized magnon $k_{b}$ (gray arrow).
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Figure 5
(a) Micromagnetic simulations of three-magnon processes. The incident and localized spin waves are excited at the lower part of the magnetic film (horizontal black bar) and the right side of the DMI strip (vertical black bar), respectively. (b) FFT spectrum at a single lattice [black dot in (a)]. (c)–(e) Spatial FFT spectra analyses for three peaks at (c) 80 GHz, (d) 45 GHz, and (e) 115 GHz, observed in (b) where the incident spin-wave frequency is $ω_{i} / 2 π = 80$ GHz and the localized spin-wave frequency is $ω_{b} / 2 π = 35$ GHz. The FFT analysis is implemented over the region inside the dashed black square with the side length 700 nm in (a).
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Figure 6
(a) Micromagnetic simulations of three-magnon processes with a different localized spin-wave mode. The incident and localized spin waves are excited at the lower part of the magnetic film (horizontal black bar) and the left side of the DMI strip (vertical black bar), respectively. (b) FFT spectrum at a single lattice cell [black dot in (a)]. Spatial FFT spectra analyses for the frequency $ω / 2 π = 70$ GHz (c) and $115$ GHz (d) are implemented over the region inside the dashed black square with the side length 700 nm in (a).
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