Resonant current-in-plane spin-torque diode effect in magnet$-$normal metal bilayers

Abstract

Via the spin-Hall effect and its inverse, in-plane charge currents in a normal metal$-$ferromagnet (N$|$F) bilayer can be used to excite and detect magnetization dynamics in F. Using a magneto-electric circuit approach, we here consider the current response to quadratic order in the applied electric field, which is resonantly enhanced for driving frequencies close to frequencies of coherent magnetization modes. Our theory can be applied to bilayers with a magnetic insulator or with a magnetic metal. It focuses on the contribution of coherent magnetization dynamics to spin currents collinear with the equilibrium magnetization direction, but also takes into account relaxation of spin accumulation via spin currents carried by incoherent magnons and conduction electrons in F.

Figures (4)

• Figure 1: Geometry of the N$|$F bilayer, consisting of a normal metals N and a metallic or insulating ferromagnet F. An in-plane electric field $\mathbf{E}(t) = E(e^{-i \omega t} + e^{i \omega t}) \mathbf{e}_x$ in N drives a coherent magnetization mode in F. In this article, we calculate the quadratic-in-$E$ current response $\mathbf{i}_{i}^{(2)}$ at frequencies $\Omega = 0$ and $\Omega = 2 \omega$ that arises from the modulation of the spin-Hall magnetoconductance by the precessing magnetization in F.
• Figure 2: Equivalent magneto-electronic circuit diagrams for the linear charge and spin current response of an N$|$F bilayer to an applied electric field in N. The left and right circuits show the corrections arising from spin currents and spin accumulations perpendicular to and collinear with the magnetization direction in F, respectively. Dashed and solid lines indicate transport of charge and spin, respectively. The relations between the spin and charge currents and the generalized potentials $\mu_{{\rm se}\parallel,\omega}$, $\mu_{{\rm m},\omega}$, and $-\hbar \omega m_{\perp,\omega}$ are given in Eqs. (\ref{['eq:nmspinrelation']})--(\ref{['eq:fspinperp']}). The "grounds" in the diagram represent the fact that spin is not a conserved quantity in N and in F.
• Figure 3: Nonlinear response coefficients $r_0$, $\hbox{Re}\, r_{2 \omega}$, and $\hbox{Im}\, r_{2 \omega}$ vs. driving frequency $\omega$ for an Au$|$Fe bilayer. Material and device parameters are taken from Ref. Reiss2021-em, reproduced as Tab. \ref{['tab:estimates']} in the appendix.
• Figure 4: Same as Fig. \ref{['fig:numerics1']}, but for a Pt$|$YIG bilayer. Material and device parameters are taken from Ref. Reiss2021-em, reproduced as Tab. \ref{['tab:estimates']} in the appendix.