Magnetism Induced by Periodically Driven Non-Magnetic Impurities on Surfaces with Spin-Orbit Coupling

Abstract

We investigate the response of the Rashba spin-orbit system to a time-periodic scalar potential, in order to determine whether an induced magnetization exists. We approach this by employing the Floquet-Green function method within the Keldysh formalism, computing the non-equilibrium steady state of the system. We find that, even in the absence of an external magnetic field, the system evolves into a state with an oscillating magnetization density that is remarkably rich in structure. We provide a detailed physical interpretation of the results by performing a Fourier decomposition in non-local momentum-space, which helps to uncover the physical origin of the induced magnetic field in terms of Fermi surface spin polarization and the system's dynamical character.

Figures (8)

• Figure 1: Induced charge density $\delta \tilde{n} \equiv (n-n_0)/n_0$ (a), in-plane magnetization components $(\tilde{m}_x, \tilde{m}_y) \equiv (m_x /n_0, m_y/ n_0)$ (b), and the standard deviation over one period $\langle\!\langle \tilde{m}_y \rangle \!\rangle_T(\bm{r})$ (c). All densities are normalized by the unperturbed value $n_0$ and evaluated at time $-T/2$. Spatial coordinates are expressed in units of the Fermi wavelength $\lambda_{\mathrm{F}}/2 = 2\pi/(k_{\mathrm{F}+} + k_{\mathrm{F}-})$. Panels (a1) and (a2) show, respectively, the two-dimensional distribution of the induced charge and its cross section along the positive $x$-axis (dashed line in the former). Panels (b1) and (b2) show the in-plane magnetization: (b1) displays its modulus and direction as arrows representing the normalized vector $\tilde{m}_x \bm{\hat{\imath}} + \tilde{m}_y \bm{\hat{\jmath}}$, and (b2) shows its cross section along the positive $x$-axis. Panels (c1) and (c2) depict the time-averaged standard deviation $\langle\!\langle \tilde{m}_y \rangle \!\rangle_T(\bm{r})$ in the two-dimensional plane and its cross section, respectively. A small blue circle at the origin marks the location of the impurity. Panels (a1), (b1) and (c1) share a radial extension of $2.5 \lambda_\rm{F}$ in order to facilitate visualization. Likewise, the vertical scale in panel (a2) is reduced so that induced charge far from the impurity is noticeable.
• Figure 2: Absolute value of momentum resolved lesser Green's functions $G^{<}_0$ (a), $G^{<}_x$ (b) and $G^{<}_y$ (c) for initial momentum $\bm{k_0}$ close to the inner Fermi contour (left-hand side panels (a1), (b1) and (c1)) and the outer Fermi contour (right-hand side panels (a2), (b2) and (c2)), respectively. All the functions are evaluated at time $-T/2$, but the results at other snapshots are very similar (see Sec. VI of Supplemental Material suppmat).
• Figure 3: Figure 3. Fourier transform of the cross section of the induced charge (a) and the $y$ component of the induced magnetization (b) evaluated at time $-T/2$ (arbitrary units). Panel (a) shows the results for two representative values of the Fermi energy: $\varepsilon^{(1)}_\rm{F} = 0.03~\mathrm{a.u.}$ (magenta) and $\varepsilon^{(2)}_\rm{F} = 0.04~\mathrm{a.u.}$ (blue). Panel (b) presents data for two representative values of the Rashba parameter: $\alpha^{(1)}_\rm{R} = 0.02~\mathrm{a.u.}$ (blue) and $\alpha^{(2)}_\rm{R} = 0.04~\mathrm{a.u.}$ (magenta), used to analyze the peak position dependence on the spin-orbit coupling strength. Vertical dashed lines in panel (a) indicate the value of $k_{\mathrm{F}+} + k_{\mathrm{F}-}$ for each $\varepsilon_\rm{F}$, while in panel (b), they mark the location of the peaks sensitive to $\alpha_\rm{R}$.
• Figure S1: Representation of Rashba Hamiltonian and system parameters: $m^*=1 \; \rm{a.u.}$, $\alpha_\rm{R} = 0.04 \; \rm{a.u.}$, $\varepsilon_\rm{F} = 0.03\;\rm{a.u.}$ and $\omega_0 = 0.004 \;\rm{a.u.}$. a) Dispersion relation of spin up (red) and down (blue) bands. $\varepsilon_\rm{F}$ is the Fermi energy and the orange shaded area illustrates an arbitrary frequency Brillouin zone, whose height is given by the fundamental frequency $\omega_0$. b) Rashba bands at the Fermi level, given by two concentric circles of radii $k_{\rm{F}+}$ and $k_{\rm{F}-}$. The arrows depict the spin orientation, showcasing the spin-momentum locking of the Hamiltonian.
• Figure S2: Maximum value of $\langle\!\langle \tilde{m}_{y} \rangle \!\rangle_T \,(\bm{r})$ as a function of the fundamental frequency $\omega_0$. We use $\omega_0 = 0.004$ a.u. in the calculations of the main text.