Resonant amplification of the inverse Faraday effect magnetization dynamics of time reversal symmetric insulators

All-optical helicity-dependent manipulations of magnetism have attracted broad attention in the context of ultrafast control of magnetic units. Here, we investigate the spin dynamics in time reversal symmetric insulators induced by strong circularly polarized light. We perform real-time time-dependent density functional theory calculations together with model Hamiltonian analyses for $\mathrm{Mo}{\mathrm{S}}_2$ and $\mathrm{W}{\mathrm{S}}_2$ monolayers, which are exemplary spin-orbit-coupled time reversal symmetric insulators. We trace the evolution of dynamical spin states, starting from the Kramers-paired electronic ground state, and find that the induced magnetization exhibits a sharp resonance peak when the applied light frequency is close to half the spin-flipping energy gap. The resonance condition is secondarily affected by the field strength and the pulse width. We suggest that low-energy time reversal broken excitations of insulators can be pursued with a sharp frequency selection as another class of ultrafast phenomena.

Figure 1

The magnetization in a spin-orbit-coupled insulator induced dynamically by a circularly polarized electric field. (a) The band structure of the graphene-Dirac model Hamiltonian with SOC. The out-of-plane spin values are represented by the color gradient in the band. The band energies are ${\varepsilon }_1$, ${\varepsilon }_2$, ${\varepsilon }_3$, and ${\varepsilon }_4$. The SOC spin-splitting energy, the band gap, and the spin-flipping energy gap at the valley top are denoted by $E_{\mathrm{SOC}}^{}$, $E_{\mathrm{gap}}^{}$, and $E_{\mathrm{gap}}^{\uparrow \downarrow }$, respectively. (b) The upper panel shows the real-time modulation of the electric field; the unit of the $y$ axis is ${10}^{-2}\mathrm{V}/\AA$. The pulse width, i.e., the FWHM, is depicted. The lower panel shows the magnetization induced by CPL $\left(\lambda =\pm 1\right)$ and LPL $\left(\lambda =0\right)$. $M_Z^{\mathrm{ind}}$ is the magnetization remnant after the laser pulse fades away. The dashed lines in the bottom panel are the response of the system when its SOC strength is reduced half. (c) The residual magnetization at various frequencies. The inset shows a magnified view near the peak region. The frequency (${\omega }_1$), corresponding to the maximum $M_Z^{\mathrm{ind}}$, is half the spin-flipping energy gap at the Bloch point k₁ in (a).

Figure 2

Induced remnant magnetizations arising from selected Kramers-paired states around the valley tops. (a) Five equidistant Kramers pairs are selected, including valley tops K and K′ (or −K). The black, red, green, blue, and yellow dots correspond to the pairs $\pm \mathbf{K}$, $\pm {\mathbf{K}}_{\mathbf{1}}$, $\pm {\mathbf{K}}_{\mathbf{2}}$, $\pm {\mathbf{K}}_{\mathbf{3}}$, and $\pm {\mathbf{K}}_{\mathbf{4}}$, respectively. The five contours represent the sets of points in the BZ with the same spin-flipping energy gap $\left({\varepsilon }_{42}\right)$ as the selected $k$-point pairs. (b) The induced magnetizations arising from the five selected pairs. The solid gray line [DOS(ω)] is the density of $\mathbf{k}$ points whose half the spin-flipping energy gap is ω, i.e., the number of $k$ points satisfying ${\varepsilon }_{42}\left(\mathbf{k}\right)=2\omega$. The data in (b) are multiplied with the DOS(ω) and replotted in (c). ${\omega }_1$ is the frequency of the red peak, as in Fig. 1. The inset shows the induced magnetization obtained from ±k₁ for a wider energy window. The smaller (larger) peak corresponds to ${\omega }_1$ ($2{\omega }_1$).

Figure 3

Effects of laser field strength on the induced magnetization. (a) Frequency scans of the induced magnetizations for various field strengths. The induced magnetization of a given field strength is normalized to its maximum magnitude ($max\left\{M_Z^{\mathrm{ind}}\right\}$), which is presented in (b). The data in (b) fitted to a quadratic fit $\left(a{x}^2\right)$ and a quadratic plus linear fit $\left(a{x}^2+bx\right)$ are depicted by magenta and blue dashed lines, respectively. (c) The induced magnetizations arising from the five selected Kramers pairs for $3E_0$. For comparison, Fig. 2 provides the equivalent results for $E_0$.

Figure 4

The responses of the valley top pairs to monochromatic cw and pulses with various widths. (a) The time profiles of the magnetic responses of the Kramers-paired valley top states (±K) to continuous monochromatic waves with various frequencies near the resonant frequency ${\omega }_0$. The same result is presented in linear and logarithmic scales in the upper and the lower panels, respectively. (b) The oscillation period obtained from (a) (solid boxes) and the fitting model (dashed black line) deduced from the first-order time-dependent perturbation theory. The shown fitting parameter α is 0.46. (c) An example of a Gaussian pulse with a pulse width of 0.6 ps (the unit of the $y$ axis is ${10}^{-2}\mathrm{V}/\AA$). (d) The magnetic responses of the Bloch state pair (±K) to laser pulses with various pulse widths. The center frequency of pulses is fixed near ${\omega }_0$ . (e) shows the remnant magnetization for each pulse. The unit is the same as (d). In all (a)–(d), the field strength is set at $E_0=0.03\mathrm{V}/\AA$.

Figure 5

The results of the RT-TDDFT calculations of the induction and amplification of magnetization. (a) The band structure of $\mathrm{Mo}{\mathrm{S}}_2$. The resonant frequency at half the spin flipping gap is indicated by the arrow. (b) A side view of $\mathrm{Mo}{\mathrm{S}}_2$ and the incident laser beam. The oscillating laser field is given by $\mathbf{E}\left(t\right)=E_0\left[sin\left(\omega t\right)\widehat{\mathbf{x}}-\lambda cos\left(\omega t\right)\widehat{\mathbf{y}}\right]f\left(t\right)$. The upper panel of (c) shows the applied pulses near the resonant $\left({\omega }_{\mathrm{res}}=1.01\mathrm{eV}\right)$ and far from the resonant $\left({\omega }_{\mathrm{nonres}}=0.50\mathrm{eV}\right)$ frequencies for $\mathrm{Mo}{\mathrm{S}}_2$ (the unit of the $y$ axis is ${10}^{-1}\mathrm{V}/\AA$). The lower panel shows the time profiles of the induced magnetizations. The upper panel of (d) shows a laser pulse with a longer duration with the same center frequency and the same field strength as in (c). The lower panel shows the time profiles of the magnetizations in $\mathrm{Mo}{\mathrm{S}}_2$ induced under left-handed CPL $\left(\lambda =+1\right)$, right-handed CPL $(\lambda =-1)$, and LPL $\left(\lambda =0\right)$. The magnetization in $\mathrm{W}{\mathrm{S}}_2$ under left-handed CPL with its resonant frequency $\omega =1.07\mathrm{eV}$ is shown as a green line.