Nonlinear current response of Weyl semimetals to a strong dc-ac electric field in the ultraquantum regime

The linear charge transport properties of Weyl semimetals, such as negative magnetoresistance related to chiral anomaly, have been studied extensively. In this work, the nonlinear current response of Weyl semimetals to a strong dc-ac electric field in the ultraquantum regime with a strong magnetic field is explored by employing a nonperturbative treatment based on the stochastic Liouville equation. Our systematic studies of nonlinear charge transport for two types of ac fields (the cosinusoidal electric field and the periodic pulsed field) have revealed extraordinary modulation of nonlinear current response by the ac fields. For the case with the cosinusoidal electric field, we find the following: (i) in the high-frequency regime, dynamic localization (vanishing of current) and quasienergy band collapse occur under a suitable condition of $J_0\left(\frac{e{E}_{A}d}{\hslash {\omega }_0}\right)=0$, where $J_0$ is the Bessel function with $E_A$ and ${\omega }_0$ being the strength and frequency of the ac field, and $d$ denoting the lattice constant of Weyl semimetals; and (ii) in the intermediate- and low-frequency regimes, the multiple-photon-assisted transport leads to extremal values of current responses whose patterns can be tuned by the magnetic field. As for the pulsed electric field, our results show that (i) the dynamic localization and quasienergy band collapse appear under a different condition of $cos\left(\frac{e{E}_{A}d}{\hslash {\omega }_0}\right)=0$; and (ii) the influence of the ac field on the current response disappears when $\frac{e{E}_{A}d}{\hslash {\omega }_0}=m\pi$, with $m$ being an integer. The experimental conditions are also discussed and the predicted nonlinear transport effects could be observed in experiments.

Figure 1

The scheme of a Weyl semimetal under the external electric field and the magnetic field, which are parallel to the $z$ direction.

Figure 2

The sketch map of the function $f\left(\overline{t}\right)$ in our analytical derivation. The green solid line and the red dashed line represent the $f\left(\overline{t}\right)$ corresponding to the ac fields of the trigonometric function [Eq. (6)] and the periodic pulse [Eq. (7)], respectively.

Figure 3

(a) The current response $\langle j\rangle$ (in unit of $eg{v}_F$) as a function of ${\overline{\omega }}_A=ed{E}_A/\hslash {\omega }_0$. The parameters are ${\omega }_0=2$ THz, $\overline{\alpha }=1/\tau {\omega }_0=0.2$, and ${\overline{\omega }}_D=ed{E}_D/\hslash {\omega }_0=0.1,1.0$. (b) The quasienergy spectrum (in units of $\hslash v/d$) as a function of ${\overline{\omega }}_A$. ${\overline{\omega }}_D=0.1$. The other parameters are the same as those in panel (a).

Figure 4

The current (in units of $eg{v}_F$) as a function of ${\overline{\omega }}_D$ with $\overline{\alpha }=0.1$ in panel (a) and $\overline{\alpha }=0.67$ in panel (b). The black and red lines correspond to ${\overline{\omega }}_A=6.0$ and 9.3, respectively. The frequency of the cosinusoidal field is chosen as ${\omega }_0=0.6$ THz.

Figure 5

The current response $\langle j\rangle$ (in units of $eg{v}_F$) as a function of ${\overline{\omega }}_D$ for the periodic pulsed fields with different field strengths (${\overline{\omega }}_A$). The parameters are chosen as ${\omega }_0=0.6$ THz, $\overline{\alpha }=0.2$, and ${\overline{\omega }}_A=0$, 3.33, 9.4, 10.0, and 20.0. The curves for ${\overline{\omega }}_A=0$ (black curve) and ${\overline{\omega }}_A=9.4$ (blue curve) overlap.

Figure 6

The current response $\langle j\rangle$ (in units of $eg{v}_F$) as a function of ${\omega }_D=e{E}_{D}d$ for the periodic pulsed field with various frequencies ${\omega }_0$. The black, red, blue, and purple curves correspond to ${\omega }_0=0.2$, 0.4, 0.6, and 1.2 THz, respectively. Other parameters are chosen as ${\omega }_A=e{E}_{A}d=6$ THz and $1/\tau =0.12$ THz.