Spin-orbit coupling induced ultrahigh-harmonic generation from magnetic dynamics

The generation of a nonlinear high-frequency response in solids from powerful optical pumps has gained momentum over the past decade. High-harmonic generation (HHG) in solids can be obtained from strong-field laser excitation, usually restricted to optical frequencies and limited both in amplitude and in harmonic order. Here, we demonstrate that high-harmonic emission can be achieved by exploiting conventional spin pumping, without the need for optical excitation. Considering a noncentrosymmetric (ferro- or antiferro-)magnet excited at a frequency $\omega$, we demonstrate the emergence of HHG in two main regimes: (i) In the perturbative regime, where a weak spin-orbit interaction is considered, the carrier pumping features a number of harmonics with a cutoff order $n_{\max }<10$. (ii) When the spin-orbit coupling strength is close to, or higher than, the $s\text{−}d$ exchange energy, a strongly nonlinear regime resulting from resonantlike spin-flip scattering occurs leading to the emission of a large number of harmonics. This is in sharp contrast to conventional pumping, where the corresponding time-dependent currents simply oscillate with the frequency of the magnetic drive $\omega$. Our proposal enables the enhancement of both spin and charge dynamics by orders of magnitude. This effect could be used to trigger high-frequency emission deep in the terahertz regime.

Figure 1

Illustration of the spin pumping setup. (a) Schematics of the dc and ac spin pumping: A magnetic order precessing around its rest direction $\mathbf{z}$ induces a spin current composed of a dc component, ${\mathcal{J}}_{s,\mathrm{dc}}^{\left|\right|}$ (black), and an ac (green) component, ${\mathcal{J}}_{s,\mathrm{ac}}^{\perp }$, whose spin polarization is aligned on $\mathbf{m}\times {\partial }_t\mathbf{m}$. We neglect the contribution from the imaginary part of the spin mixing conductance ($\sim {\partial }_t\mathbf{m}$). Via spin-orbit coupling, these spin currents are converted into a dc and an ac charge current flowing along $\mathbf{y}$ and $\mathbf{z}$, respectively. (b) Corresponding simulation setup: The pumped charge current is collected along the direction about which the magnetic order precesses.

Figure 2

Spin-flip-scattering-driven high-harmonic generation: In the presence of a precessing magnetic order (black curve) the spin-orbit interaction leads to spin-flip scattering. Subsequently, a phase corresponding to the fundamental frequency is accumulated in the electronic wave function. Under adiabatic magnetization dynamics, the spin-flip time ${\tau }_{sf}$ becomes very small compared with the period of the precessing magnetic order, and higher harmonics appear as a consequence of subsequent spin-flip scattering events during one fundamental cycle.

Figure 3

(a)–(d) Time domain current signal as a function of Rashba strength: Time-dependent currents for different spin-orbit coupling strengths are displayed. Here, an exchange coupling of $J=500\mathrm{meV}$ is taken, and the precession angle is set at $\theta ={10}^{\circ }$. Considering a lattice spacing of 5 Å, the value of ${\alpha }_{\mathrm{R}}$ at ${\alpha }_{\mathrm{R}}{k}_{\mathrm{F}}/J=1$ corresponds to 3.57 eV Å, which is very close to its value in Bi/Ag alloys [26] or in other bismuth-based topological insulators [27].

Figure 4

(a)–(d) Frequency domain charge current as a function of Rashba strength: Fourier amplitudes of the charge current are shown at different spin-orbit coupling strengths in both perturbative and nonperturbative regimes. The antiferromagnetic dynamics parameters are the same as in Fig. 3.

Figure 5

Charge current amplitude of the harmonic spectrum as a function of spin-orbit coupling strength: The current amplitude (in logarithmic scale) is shown as a function of the harmonic order as well as the Rashba strength. The frequency of the dynamics as well as the precession angle are the same as in Fig. 3.

Figure 6

Emission bandwidth and charge current amplitude as a function of the driving frequency: (a) Maximum number of harmonics as a function of the inverse of the fundamental frequency. (b) Amplitude of selected harmonics as a function of the driving frequency. The Rashba strength is tuned to the maximally excited regime.