Scaling Laws of Magnetically Driven High-order Harmonic Generation in Spin-Orbit Coupled Systems

Abstract

We investigate the scaling behavior of high harmonic generation (HHG) driven by magnetic dynamics in spin-orbit coupled systems. In contrast to optically driven HHG--where the harmonic cutoff scales as $ω^{-3}$ with the driving frequency $ω$--our time-dependent quantum transport simulations reveal a qualitatively distinct scaling law for magnetically driven HHG in the presence of Rashba spin-orbit interaction: the harmonic cutoff $n_{\mathrm{max}}$ scales as $ω^{-1}$. This fundamental difference arises from distinct excitation mechanisms--namely, spin-flip transitions driven by vectorial magnetic precession, as opposed to scalar electric fields. Furthermore, we demonstrate that the precession cone angle $θ$ serves as a crucial control parameter. Increasing $θ$ broadens the harmonic bandwidth, with peak emission achieved for nearly in-plane magnetic dynamics. Our findings establish magnetically driven HHG as a robust and tunable mechanism for nonlinear spin transport, governed by unique scaling laws with potential applications in ultrafast spintronic technologies.

Figures (6)

• Figure 1: A sketch of the studied one dimensional antiferromagnetic system, hosting a Rashba SOC. The cones depicts the staggered antiferromagnetic dynamics. The system is attached to two normal leads. Both magnetic moments oscillate with the same frequency but in opposite directions. This simple geometry is slightly different from the more general antiferromagnetic resonance setup, where the two moments precess with different cone angles. While, in the latter case both a staggered ($\mathbf{N}=(\mathbf{m}_a-\mathbf{m}_b)/2$) order and a ferromagnetic like ($\mathbf{M}=(\mathbf{m}_a+\mathbf{m}_b)/2$) contribute to the pumped spin current. In the former case, only the staggered contribution $\mathbf{m}$ is expected. Here, $a$ and $b$ stand for the two sub-lattices of the antiferromagnet. In general, the pumped spin current in the absence of SOC is given as Cheng2014$\mathcal{J}\propto \mathbf{N}\times\partial_t\mathbf{N}+\mathbf{M}\times\partial_t\mathbf{M}.$ In the actual case of a compensated antiferromagnet, only the first term survives. Once the SOC is turned on the dynamics becomes very non-linear leading to ultrahigh frequencies.
• Figure 2: The HHG spectra at different values of the s-d exchange parameter are shown. Here, $J$ is varied from the low coupling case $J=0.1\gamma$ to the extremely nonlinear regime $J=5\gamma$. The parameters of the drive are chosen such that $\hbar\omega=10^{-2}\gamma$ and $\theta=\pi/8$. The Fermi energy is set at $E_F=0$. Here, the current response is represented in a logarithmic scale.
• Figure 3: In the main panel, the Fourier transform of the charge current as a function of $J$ is shown. Here, the calculations are performed at $\alpha_{\rm R} a^{-1}= \gamma$. A resonance feature centered around $\alpha_{\rm R} k_{\rm F} = 2J$ is observed. In the left panel, the Fourier spectrum at resonance is displayed. The driving frequency, the cone opening, and the Fermi energy are set to their values in Fig. \ref{['fig:js']}. Here, the current response is represented in a logarithmic scale.
• Figure 4: The scaling of HHG in terms of the driving frequency $\omega$ is shown. The upper row shows the time domain signals. In the lower row, the corresponding frequency domain responses are displayed. In the insets of d) - f), a zoom over few harmonics is shown. One observes clearly the increase of the HHG cutoff as $\omega$ is reduced. In panel f), the vertical axis is cut in order to zoom over the harmonics near the cutoff, whose amplitudes can be up to two orders of magnitude smaller than the intensity of the harmonics in the low harmonics region.
• Figure 5: The scaling up of $n_{max}$ in terms of the driving frequency is shown. The sd exchange coupling and precession angle are set respectively at $J=\alpha_{\rm R}a^{-1}=5\gamma$ and $\theta=\pi/10$. To examine the fitting procedure used here, we define the root square mean error as ${\rm{RMSE}}^2=\langle(y_i-f_i)^2\rangle$, with $y$ and $f$ being respectively the original data and the fitting line. We also introduce the coefficient of determination $R^2$ given as $R^2=1-\sum_i(y_i-f_i)^2/\sum_i(y_i-\langle y\rangle)^2$. The value $R^2=0.99$ reflects an excellent linear fit, corresponding to an average deviation of a single harmonic (${\rm{RMSE}}=1.03$).