Measuring the dispersion relations of spin wave bands using time-of-flight spectroscopy

T. Devolder, G. Talmelli, S. M. Ngom, F. Ciubotaru, C. Adelmann, and C. Chappert

Phys. Rev. B 103, 214431 – Published 21 June 2021

Abstract

We develop a generic all-inductive procedure to measure the band structure of spin waves in a magnetic thin stripe. In contrast to existing techniques, our method works even if several spin wave branches coexist in the investigated frequency interval, provided that the branches possess sufficiently different group velocities. We first measure the microwave scattering matrix of a network composed of distant antennas inductively coupled to the spin wave bath of the magnetic film. After a mathematical transformation to the time domain to get the transmission impulse response, the different spin wave branches are viewed as wave packets that reach successively the receiving antenna after different travel times. In analogy with time-of-flight spectroscopy, the wave packets are then separated by time gating. The time-gated responses are used to recalculate the contribution of each spin wave branch to the frequency domain scattering matrix. The dispersion relation of each branch stems from the absolute phase of the time-gated transmission parameter. The spin-wave wave vector can be determined unambiguously if the results for several propagation distances are combined, so as to get the dispersion relations.

Received 9 April 2021
Revised 1 June 2021
Accepted 7 June 2021

DOI:https://doi.org/10.1103/PhysRevB.103.214431

Physics Subject Headings (PhySH)

Magnetism Spin waves

Thin films

Magnetic techniques

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

T. Devolder ^1,*, G. Talmelli ^2,3, S. M. Ngom¹, F. Ciubotaru ², C. Adelmann ², and C. Chappert ¹

¹Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120 Palaiseau, France
²imec, Kapeldreef 75, 3001 Heverlee, Belgium
³KU Leuven, Department of Electrical Engineering, Kasteelpark Arenberg 10, 3001 Leuven, Belgium

^*thibaut.devolder@c2n.upsaclay.fr

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Vol. 103, Iss. 21 — 1 June 2021

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Images

Figure 1
(a) Sketch of the different families of spin wave branches, their spatial profiles within the conduit width, their effective wave vectors, and their group velocities for $w_{mag} = 4.7 μ m$ at 18 mT. (b) Cross section of the sample, with thicknesses in nm. (c) Far field optical view of the rf contacts routing the electrical signals toward the antenna. The metallic parts appear in golden color. (d) Sketch of the device and definitions of its dimensions. (e) Scanning electron micrographs of the central parts of devices with conduit widths $w_{mag} = 4.7 μ m$ (left) and $w_{mag} = 0.85 μ m$ (right).
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Figure 2
Backward transmission parameters ${\tilde{S}}_{12}$ for spin wave conduits of widths of $4.7 μ m$ (left panels) and $0.85 μ m$ (right panels) for an antenna-to-antenna distance of $r_{3} = 6.9 μ m$ . (a) and (b): Field dependence of the imaginary part. (c) and (d): Selected spectra where the multimode character is striking to the eye, manifest either as a ripple superimposed on the main envelope signal in (c) or as a kind of two-tone beating in (d). (e) Example of complex transmission parameter in the single-mode model [Eq. (4)], evaluated with vanishing damping for the dispersion relation displayed in panel (f). The dispersion relation starts at $f_{k = 0} = 4.75 GHz$ and grows with a constant (hence nondispersive) group velocity $v_{g} = 7 km / s$ . The envelope (green curve) in (e) is the antenna efficiency function [Eq. (5)].
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Figure 3
Backward transmission impulse response $s_{12} (t)$ for spin wave conduits of widths of $4.7 μ m$ (left panels) and $0.85 μ m$ (right panels) for a propagation distance $r_{3} = 6.9 μ m$ . Panels (a) and (b): Field dependence of the impulse response [note the change in color scale on the upper side of panel (a) that is necessary to better reveal the late-arriving wave packets]. Panels (c)–(g): Selected impulse responses for which the multimode character is visible from the successive arrival of several wave packets. The red curves are the raw data $s_{12} (t)$ and the black curves are time-amplified version thereof being $s_{12} (t) \times e^{t / τ}$ , where the amplification is meant to correct for the time decay of the spin waves at the theoretical rate $τ = 2 / Δ f \approx 1 ns$ . Panel (h): Calculated impulse responses for several propagation distances for a theoretical lossless spin wave mode with the dispersion relation of Fig. 2.
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Figure 4
Time-gating method to isolate the spectra of single modes in experimental data. (a) Exponentially amplified impulse response $s_{21} (t) e^{t / τ}$ for spin wave conduits of width of $4.7 μ m$ at 18 mT after a travel distance of $r_{3} = 6.9 μ m$ corresponding to the raw spectrum in (b). The attenuation rate is taken as that of FMR, i.e., $τ = 2 / Δ f$ . (c)–(f): Real parts of the time-gated ${\tilde{S}}_{12}$ spectra. In (c), the feedthrough and the long travel time were removed, effectively suppressing the base line and the high-frequency noise. In (d), the fastest wave packet is kept to isolate the contribution of a single mode. In (e), the medium-velocity wave packet is kept. In (d), the slowest wave packet is kept. The vertical scales in (c) and (d) were multiplied by 10.
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Figure 5
Method to determine the absolute phase of the transmitted spin wave signals and to disambiguate the wave vectors. (a), (b), (c): Imaginary parts of ${\tilde{S}}_{12}$ for propagation distances $r_{1}$ , $r_{2}$ , and $r_{3}$ at 18 mT for $w_{mag} = 4.7 μ m$ . The time gating was set to keep the feedthrough and the first-arriving wave packet. Zoom within panel (a): Complex signal related to the distant excitation of the $k_{x} = 0$ mode. Panel (d): Dispersion relations obtained from Eq. (13) for each propagation distance ( $r_{1}$ : black; $r_{2}$ : red; $r_{3}$ : blue). The bold curves that are almost superimposed correspond to the mathematically correct choices of the integration constants $n_{1}, n_{2}, n_{3}$ within Eq. (13). Only the first phase match is both physically and mathematically correct. The dotted lines are when errors of $2 π$ and $4 π$ are done within Eq. (13).
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Figure 6
(a) and (b): Experimental dispersion relations (bold lines) for spin wave conduits of widths 4.7 and $0.85 μ m$ . Narrow lines: Theoretical dispersion relations [Eq. (1)] of the ${DE}_{1}$ , ${DE}_{2}$ , and ${DE}_{3}$ modes. In the models, the internal fields $H_{0}$ are taken as the experimental applied fields (18 and 70 mT) minus the experimental shape anisotropy fields (6 and 30 mT) of the conduits. (c) Comparison between the antenna efficiency function [Eq. (5)] and the modulus of the transmission coefficients of the fastest mode for $r_{3} = 6.9 μ m$ . Green curve: $w = 4.7 μ m$ and $μ_{0} H_{y} = 18 mT$ . Violet curve: $w = 0.85 μ m$ and $μ_{0} H_{y} = 70 mT$ . The vertical dotted lines indicate the theoretical positions of the zeros and the maxima of $h_{x} {(k_{x})}^{2}$ .
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