Nonlinear spin-motive force driven by mixed-space quantum geometry

Abstract

Spin-motive force, i.e., the electric current induced by magnetization dynamics, is theoretically studied beyond the Thouless-pump paradigm. In contrast to the linear-response regime, where the induced current is purely AC, we show that spin-motive force acquires both a DC component and a second-harmonic component at nonlinear order in magnetization dynamics. We further clarify that both contributions originate from the geometric properties of electronic bands -- quantum geometry defined in the mixed parameter space $({\boldsymbol k}, {\boldsymbol m})$ spanned by electron's momentum ${\boldsymbol k}$ and magnetization ${\boldsymbol m}$. By applying the theory to a Luttinger model, we demonstrate that our mechanism yields a finite nonlinear current even in the insulating regime, and the resulting electrical signal is measurable in a conventional current-measurement setup. Our findings offer a new operating principle of AC-to-DC conversion with magnetic materials, highlighting the pivotal role of the $({\boldsymbol k}, {\boldsymbol m})$-mixed space quantum geometry in magnetization-dynamics-induced electric currents.

Figures (2)

• Figure 1: (a) The band structure of $\hat{\cal H}_{0,\bm k}$ on the $k_{x} = k_{y}$ line. (b) Density of states. The gray shade represents energy region where band is gapped. The model parameters are set as $\gamma_1 = 3.0t_{0}a^2$, $\gamma_2 = 2.0t_{0}a^2$, $\gamma_3 = 1.5t_{0}a^2$, $\Delta = 2.0t_{0}$, $E_{\rm R} = 0.08t_{0}a$, $E_{\rm D} = 0.3t_{0}a$, and $J_{\rm c} = 0.5t_{0}$, where $t_{0}$ is the energy unit and $a$ is the lattice constant. The direction of the equilibrium magnetization is set along the $x$-direction ; $\bm m_0 = \bm e_x$.
• Figure 2: Calculation results of the electric current along the $x$-axis under the magnetization precession around the axis $\bm m_0 = \bm e_x$. Upper (lower) panels show the response coefficients arising from the Berry curvature (Berry connection polarizability). Panels (a,d) show the direct-current component, while (b,e) and (c,f) show the $\sin 2\omega t$ and $\cos 2\omega t$ components of second-harmonic generation, respectively. The gray shade represents the energy region where the band is gapped. The colored lines in each panel represent the contributions of the geometric quantities that arise when one expands the summation for the magnetization direction, $\alpha$ and $\beta$, in the expression of the electric current, Eqs. (\ref{['eq:jDC_general']}) and (\ref{['eq:jSHG_general']}), explicitly.