Fluctuations of the inverted magnetic state and how to sense them

TL;DR

The paper analyzes fluctuations of the inverted magnetic state, where antimagnons (negative-energy excitations) arise when a ferromagnet is driven antiparallel to an applied field by spin-orbit torque. It develops both classical (stochastic LLG with Green's-function solutions) and quantum (Holstein-Primakoff, master equations) descriptions to quantify fluctuations driven by interfacial shot noise and bulk damping, revealing a driven phase-transition-like logarithmic divergence at a critical spin accumulation $\Delta\mu_c$ and an effective temperature $T_{eff}$ that captures enhanced antimagnon fluctuations. A dispersive qubit readout is proposed to spectroscopically resolve the antimagnon distribution via shifts in the qubit frequency $\omega_{q,n}=\omega_q-2\chi n$, providing a concrete sensing protocol alongside magnetoresistance measurements linked to $\langle\psi^*\psi\rangle|_{x=0}$. The work offers fundamental insights into antimagnon statistics, outlines experimental parameters (e.g., realistic $j_c$ ranges), and points to potential spintronic magnonic applications such as spin-wave amplification and entanglement, while highlighting future directions including domain-wall dynamics and spin-Seebeck effects in the inverted state.

Abstract

Magnons are the low-energy excitations of magnetically ordered materials. While the magnetic moment of a ferromagnet aligns with an applied magnetic field, it has been experimentally shown that the magnetic order can be inverted by injecting spin current into the magnet. This results in an energetically unstable but dynamically stabilized state where the magnetic moment aligns antiparallel to an applied magnetic field, called the inverted magnetic state. The excitations on top of such a state have negative energy and are called antimagnons. The inverted state is subject to fluctuations, in particular, as shot noise in the spin current, which are different from fluctuations in equilibrium, especially at low temperatures. Here, we theoretically study the fluctuations of the inverted magnetic state and their signatures in experimental setups. We find that the fluctuations from the injection of spin current play a large role. In the quantum regime, the inverted magnetic state exhibits larger fluctuations compared to the equilibrium position, which can be probed using a qubit. Our results advance the understanding of the fundamental properties of antimagnons and their experimental controllability, and they pave the way for applications in spintronics and magnonics, such as spin wave amplification and entanglement.

Figures (5)

• Figure 1: Set-up: A ferromagnet (FM) of thickness $d$ is coupled to a heavy metal (HM). An applied charge current $\boldsymbol{j}_c$ generates a spin current $\boldsymbol{j}_s$ in the HM, causing an interfacial spin accumulation $\Delta\mu$. Angular momentum is transferred into the FM with an efficiency governed by $\alpha_p$, inverting the magnetization $\boldsymbol{m}$ against its equilibrium position that is parallel to the applied magnetic field $\boldsymbol{H}$. Gilbert damping $\alpha$ works towards bringing $\boldsymbol{m}$ into its ground state. Temperatures present here are $T_p$ at the HM$|$FM interface and $T_\mathrm{th}$ in the FM bulk. A spin qubit is coupled with interaction strength $\chi$ for dispersive readout.
• Figure 2: Occupation $d\lambda_T^2\left\langle\psi^*\psi\right\rangle$ is plotted as a function of spin accumulation $\Delta\mu$ for temperatures $T/\omega_0 = 0.1$ (blue) and $T/\omega_0 = 100$ (green) with $T_p=T_{\mathrm{th}}=T$ and $\alpha=0.001$, $\alpha_p=0.01$ and $d/\xi = 0.09$ fixed. The orange dashed line marks the critical spin accumulation $\Delta\mu_c/\omega_0 = -(\alpha+\alpha_p)/\alpha_p$. In the regime $\Delta\mu<\Delta\mu_c$ ($\Delta\mu>\Delta\mu_c$), we linearize around the inverted state (ground state) such that excitations are of antimagnonic (magnonic) nature.
• Figure 3: Occupation $d\lambda_T^2\left\langle\psi^*\psi\right\rangle$ is plotted as a function of bulk temperature $T_{\mathrm{th}}$. (a) We compare the full dynamics with $h_l\neq0$ (solid) with neglecting fluctuations due to SOT $h_L=0$ (dashed) for three values of FM thickness $d/\xi =0.001$ (blue), $d/\xi=0.05$ (orange) and $d/\xi=0.09$ (green), with $\alpha = \alpha_pd/\xi =0.00001$, $\Delta\mu/\omega_0 = -1.1(\alpha+\alpha_p)/\alpha_p$ and $T_p=T_\mathrm{th}$. (b) We compare the full dynamics of 3 different ratios of $T_p/T_\mathrm{th}$: $T_p/T_\mathrm{th}=0.9$ (blue), $T_p/T=1$ (orange) and $T_p/T_\mathrm{th}=1.1$ (green) for $d/\xi=0.095$, $\alpha=0.001$, $\alpha_pd/\xi=0.01$ and $\Delta\mu/\omega_0 = -1.1(\alpha+\alpha_p)/\alpha_p$.
• Figure 4: We plot the effective temperature $T_{\mathrm{eff}} = 1/k_{\mathrm{B}}\beta_{\mathrm{eff}}$ [Eq. \ref{['eq:beta_eff']}] as a function of $T=1/k_B\beta$, with $T_p=T_{\mathrm{th}} = T$, for two values of spin accumulation $\Delta\mu = -1.1|\Delta\mu_c|$ (blue) and $\Delta\mu = -1.1|\Delta\mu_c|$ (orange), with $\Delta\mu_c/\omega_0 = (\alpha+\alpha_p)/\alpha_p$ and $\alpha=0.001$ and $\alpha_p=0.01$. We compare the full expression [Eq. \ref{['eq:beta_eff']}] with the small $T$ (dotted) [Eq. \ref{['eq:beta_eff_small_T']}]
• Figure 5: The steady state qubit excitation $\langle\hat{\sigma}_+ \hat{\sigma}_-\rangle$ is plotted as a function of the qubit drive frequency $\omega_d$. The qubit is coupled to a magnonic thermal state with temperature $k_\mathrm{B}T/\omega_0 = 0.55$ (orange) and an antimagnonic state with $k_\mathrm{B}T/\omega_0 = 0.55$, $\alpha=0.001$, $\alpha_p=0.1$,$\Delta\mu/\omega_0 = -1.5|\Delta\mu_c|$ (blue). For the spectroscopy, we employ the qubit frequency $\omega_q = 10\omega_0$, qubit dissipation $\gamma_q = 0.005\omega_q$, Rabi frequency $\Omega_d = \gamma/7$ and $\chi = 0.1\omega_q$.