Magnonics of time-varying media: Giant amplification via phase-transition-driven temporal interfaces

TL;DR

This work addresses the fundamental limitation of Gilbert damping in magnonics by exploiting time-variation of the magnetic medium near a field-driven PMA–DMI phase transition to achieve large spin-wave amplification. The authors develop an analytical magnonic-impedance framework and validate it with micromagnetic simulations, revealing a damping-induced amplification controlled by an exceptional point and a slow-instability regime between $H_{ m EP}$ and $H_c$. They demonstrate three key capabilities: adiabatic temporal interfaces that suppress reflections, frequency-converting amplification via temporal slabs, and a giant, frequency-preserving amplification up to $175$-fold in a lithography-free device. The findings introduce temporal magnonics as a viable paradigm for reconfigurable, high-gain spin-wave control and frequency processing, with potential for compact, field-tunable magnonic circuits. The work also highlights the role of DMI-induced nonreciprocity and dissipation-mediated instabilities as distinctive magnonic phenomena absent in photonic or acoustic time-varying media.

Abstract

Gilbert damping-the primary obstacle limiting spin-wave propagation in magnonic devices-can be transformed from an adversary into an asset. Here we demonstrate 175-fold spin-wave amplitude amplification in ultrathin films with perpendicular magnetic anisotropy at temporal interfaces associated with a field-driven transition between a uniform in-plane state and a stripe-domain state, exceeding existing parametric and spin-torque schemes (10-50-fold) without a continuous power supply. When the in-plane bias field is swept through a critical value in the presence of finite Gilbert damping, the spin-wave dispersion undergoes dramatic softening, and the eigenfrequency crosses zero and acquires a positive imaginary part that drives exponential growth. We identify this as a damping-induced instability operating near an exceptional point-a non-Hermitian degeneracy where, counterintuitively, increased Gilbert damping enhances amplification. This mechanism exploits ingredients specific to these magnetic films: the interplay of Gilbert damping, Dzyaloshinskii-Moriya-interaction-induced nonreciprocity, and field-driven phase transitions-a combination that, to our knowledge, has no direct counterpart in photonic or acoustic time-varying platforms. Our analytical framework provides explicit design rules, while micromagnetic simulations capture the full nonlinear dynamics, including stripe-domain formation. This work establishes temporal magnonics as a new paradigm for reconfigurable, lithography-free spin-wave control.

Figures (6)

• Figure 1: Temporal versus spatial interface scattering in spin-wave systems.a,b, Schematic comparison of wave scattering at spatial (a) and temporal (b) interfaces between media with different dispersion relations, implemented on this example for spin waves by different external magnetic fields. c,d, Corresponding scattering processes in reciprocal $(f, k)$-space with nonreciprocal dispersion relations for $\mu_0 H_1 = 300$ mT and $\mu_0 H_2 = 227$ mT marked in the background represented by blue and orange colors. At spatial interfaces (c), the frequency is conserved while the wavevector changes: reflection reverses the sign of $k$ (and thus the phase velocity), whereas transmission preserves it. At temporal interfaces (d), the wavevector is conserved while the frequency changes: the time-reflected wave reverses the sign of frequency (and thus the phase velocity), whereas the time-transmitted wave preserves it. e, Cross-sectional schematic of the CoFeB thin film showing the propagation geometry: thickness $d = 2$ nm (vertical, $z$-direction), length $L = 30$ µ m (horizontal, $x$-direction), and depth extending into the page ($y$-direction) with periodic boundary conditions. f, The temporal profile of the external magnetic field showing an instantaneous field change at $t = 10$ ns from $\mu_0 H_1 = 300$ mT to $\mu_0 H_2 = 227$ mT. Spin-wave packets are launched with their envelope centered at $x = 0$ and $t = 0$, and at each time instant the spatial profile is normalized so that the maximum spin-wave amplitude equals 1 to clearly visualize their propagation. g, The space-time evolution of spin-wave amplitude $m_z(t,x)$, showing the incident wavepacket ($v^i > 0$) splitting into the transmitted ($v^t > 0$, higher amplitude) and reflected ($v^r < 0$, lower amplitude) packets at the temporal interface. Wavepacket amplitudes are normalized independently at each time step for better readability of the figure. h,i, The two-dimensional FFT analysis of regions marked in (g). Before the time-interface (h, $t < 8$ ns), one spectral peak at the $H_1$ dispersion. After the time-interface (i, $t > 12$ ns), two peaks at matching wavevector but of opposite frequencies on the $H_2$ dispersion, confirming the wavevector conservation and the frequency reversal predicted by the temporal interface theory.
• Figure 2: Spin-wave refraction and reflection at sharp temporal magnetic interfaces.a, Spin-wave dispersion relations calculated in the micromagnetic simulations for four different magnetic field values: 350 mT (magenta), 300 mT (white), 228 mT (red), and 226 mT (blue). b,c, Time evolution of maximum out-of-plane magnetization component ($\max(m_z)$) for two representative wavenumbers: $k = 74$ rad/µ m (b) and $k = 49$ rad/µ m (c). Spin waves are initially excited at $\mu_0 H = 300$ mT at $t = 0$ and are incident upon a temporal interface at $t = 10$ ns, where the field abruptly changes to $\mu_0 H = 226$ mT (red), $250$ mT (black), or $350$ mT (green). The oscillations of amplitude at $t > 10$ ns arise from the interference between reflected and transmitted wave packets. d,e, The decomposition of total spin-wave amplitude into incident (dash-dotted), refracted (solid), and reflected (dotted) components, corresponding to panels (b) and (c), respectively, with color coding matching the field values. Decomposition details are in Methods section f,g, The wavenumber dependence of transmission coefficients $T_x$ and $T_z$ (f) and reflection coefficients $R_x$ and $R_z$ (g) for $\mu_0 H = 226$ mT (red), $250$ mT (black), and $350$ mT (green). Filled squares and open circles represent simulation results for $T_x$/$R_x$ and $T_z$/$R_z$, respectively; solid and dashed lines denote analytical model predictions (Eqs. \ref{['eq:T_R_x']} and \ref{['eq:T_R_z']}) for the respective components. h,i, The magnetic field dependence of transmission (h) and reflection coefficients (i) at fixed $k = 49$ rad/µ m. Symbols indicate micromagnetic simulation results; lines show analytical model predictions.
• Figure 3: Complex spin-wave dispersion and dynamical regimes near the exceptional point.a, Real part of frequency Re($\Omega$) versus wavevector $k$ for the conservative case ($\alpha = 0$) at four representative magnetic field values: $H < H_{\mathrm{EP}}$ (red), $H = H_{\mathrm{EP}}$ (black), $H_{\mathrm{EP}} < H < H_c$ (green), and $H > H_c$ (orange). Solid and dash-dotted curves represent the two dispersion branches $\Omega_+$ and $\Omega_-$, respectively. b, Imaginary part Im($\Omega$) versus $k$ for $\alpha = 0$. Note the symmetric logarithmic scale. For $H \geq H_{\mathrm{EP}}$, Im($\Omega$) = 0 identically. Instability occurs only for $H < H_{\mathrm{EP}}$ (red curves). c, Same as (a) for finite damping ($\alpha = 0.002$). The dispersion structure is qualitatively similar to the conservative case. d, Same as (b) for $\alpha = 0.002$. Finite damping induces non-zero Im($\Omega$) across all field regimes. Crucially, the green curves ($H_{\mathrm{EP}} < H < H_c$) show Im($\Omega$) $> 0$---amplification in a regime that is stable for $\alpha = 0$. e, Zoom near the exceptional point at $H = H_{\mathrm{EP}}$, showing the characteristic $\sqrt{\alpha}$ splitting of the branches for $\alpha = 0.001$ (red) and $\alpha = 0.01$ (green), relative to the conservative case $\alpha = 0$ (black dashed). f, Frequency splitting at the EP ($k = k_{\mathrm{EP}}$, $H = H_{\mathrm{EP}}$) versus damping parameter $\alpha$. Red solid line: numerical solution; black dashed line: $\propto \sqrt{\alpha}$ scaling, confirming the square-root dependence characteristic of exceptional points. g, Minimum of Re($\Omega$) over all wavevectors as a function of magnetic field for three damping values. Background colors indicate dynamical regimes: strong instability (purple, $H < H_{\mathrm{EP}}$), slow instability (white, $H_{\mathrm{EP}} < H < H_c$), and damping (green, $H > H_c$). Vertical dashed lines mark $H_{\mathrm{EP}}$ and $H_c$. h, Maximum of Im($\Omega$) over all wavevectors versus magnetic field. Larger damping produces stronger amplification in the slow instability regime, demonstrating the counterintuitive enhancement of growth rate with increased dissipation.
• Figure 4: Temporal interfaces and spin-wave response to adiabatic field changes.a--i, Comparison of field-dependent spin-wave responses. For each set of three panels, the left column (a,d,g) shows magnetic field $\mu_0 H_y$ versus time, the middle column (b,e,h) show the space–time evolution of the spin-wave amplitude $m_z(t,x)$, normalized to unity at each time step, and the right column (c,f,i) displays $\max(|m_z|)$ (black curve) and FWHM (red curve, not shown in c) versus time. Panels (a--c): abrupt field change ($\tau = 0$) from $\mu_0 H_1 = 300$ mT to $\mu_0 H_2 = 225$ mT. Panels (d--f): adiabatic temporal interface ($\tau = 2$ ns) from $\mu_0 H_1 = 300$ mT to $\mu_0 H_2 = 225$ mT. Panels (g--i): adiabatic temporal interface ($\tau = 2$ ns) from $\mu_0 H_1 = 300$ mT to $\mu_0 H_2 = 224$ mT, showing increased FWHM broadening compared to (f). j, Space-time magnetization $m_z(t,x)$ zoom from panel (e), showing the frequency and phase velocity changes while preserving wavelength across the temporal interface. k, Frequency-domain analysis: before interface ($t < 8$ ns, black), spectrum peaks at $f_0 = 4.09$ GHz; after interface ($t > 12$ ns, red), negative frequency component appears at $f_1 = -0.42$ GHz with enhanced amplitude. l, Amplitude ratio $|R|/T$ versus transition time $\tau$ for final field values $H_2 = 225$--$260$ mT. Curves show an exponential decay with fitted parameter $C \approx 0.5$. m, Amplitude evolution $\max(|m_z|)$ versus time for different $H_2$ values. The curves show growth for lower fields and the decay for higher fields. n, FWHM evolution versus time showing wavepacket broadening for different $H_2$ values and for $\mu_0 H_2\ge 225$ is small.
• Figure 5: Temporal amplification slab: dynamics and parametric dependence.a, Temporal magnetic field profile showing downward $\mathrm{tanh}$ gradient ($\tau = 2$ ns) from $\mu_0 H_1 = 300$ mT to $\mu_0 H_2 = 225$ mT, constant low-field plateau ($t_\mathrm{low} = 10$ ns), and upward gradient ($\tau = 2$ ns) returning to $\mu_0 H_1$. b, Space-time evolution of spin-wave amplitude $m_z(t,x)$ for $\mu_0 H_2 = 225$ mT, normalized to unity at each time step. c, Amplitude envelope $\max(|m_z|)$ versus time for the same field value. Horizontal lines indicate temporal landmarks: $t_\mathrm{in} = t_i - \tau/2$ (slab entrance at 98% field descent) and $t_\mathrm{out} = t_i + t_\mathrm{low} + 1.5\tau$ (slab exit at 98% field recovery). d, Amplitude evolution in logarithmic scale for different low-field values $\mu_0 H_2 = 225$--$300$ mT ($\tau = 2$ ns, $t_\mathrm{low} = 10$ ns, $k = 49$ rad/µ m). Three temporal regions are marked: initial high-field propagation, constant low-field plateau, and final high-field recovery. Gray horizontal lines (a--c) and red vertical lines (d) indicate the positions of the temporal interfaces. e, Transmission coefficient $T$ versus $\mu_0 H_2$ for low-field plateau durations $t_\mathrm{low} = 0, 5, 10, 15, 20$ ns ($\tau = 2$ ns, $k = 49$ rad/µ m). f, Transmission coefficient $T$ versus $\mu_0 H_2$ for transition time duration $\tau = 1, 2, 3, 5$ ns ($t_\mathrm{low} = 10$ ns, $k = 49$ rad/µ m). g, Transmission coefficient $T$ versus spin-wave wavevector $k$ for field values $\mu_0 H_2 = 225$--$250$ mT ($\tau = 2$ ns, $t_\mathrm{low} = 10$ ns). h, Decomposition of transmission into components: $T_1$ (first interface), $T_2$ (low-field plateau), $T_3$ (second interface), with products $T_1 T_3$ and total transmission $T = T_1 T_2 T_3$ versus $\mu_0 H_2$ ($\tau = 2$ ns, $t_\mathrm{low} = 10$ ns, $k = 49$ rad/µ m).