Photogalvanic effect and second-harmonic generation from radio to infrared wavelengths in the ${\mathrm{WTe}}_2$ monolayer

Second-order nonlinear optical responses, including photogalvanic effect (PGE) and second harmonic generation (SHG), are fundamental and important physical phenomena in nonlinear optics and optoelectronics. The PGE and SHG associated with linearly and circularly polarized light are called the linear photogalvanic effect (LPGE), circular photogalvanic effect (CPGE), linear second harmonic generation, and circular second harmonic generation, respectively. In this paper, we use the quantum kinetics under the relaxation time approximation to investigate the dependence of second-order nonlinear optical responses on the Fermi level and frequency under different out-of-plane electric fields in $T_d\text{−}{\mathrm{WTe}}_2$ monolayer from radio to infrared region. We find that the maximum frequency at which the Berry curvature dipole mechanism for the nonlinear Hall effect plays a major role is about 1 THz. From the aspect of the Fermi level, in the radio and microwave regions, the two large peaks of nonlinear conductivities occur when the Fermi level is equal to the energy corresponding to the vicinity of the gap-opening points in the band dispersion. From the aspect of frequency, in the radio region, the LPGE and SHG conductivities maintain a large constant while the CPGE conductivity almost disappears. In the microwave region, the LPGE and SHG conductivities start to decrease gradually with increasing frequency while the CPGE conductivity is large. In the infrared region, the frequency and Fermi-level dependence of second-order nonlinear optical responses is complicated. In the 125 THz–300 THz region and in the $y$ direction, the presence of dc current without the disturbance of second harmonic current under circularly polarized light may be useful for the fabrication of unique optoelectronic devices. Moreover, we illustrate that when calculating the nonlinear Hall effect or second-order nonlinear optical responses of practical materials, the theories in the clean limit fail and it is necessary to use a theory that takes into account scattering effects (e.g., relaxation time approximation). Our study is promising to promote the more accurate calculation of second-order nonlinear optical responses in practical materials.

Figure 1

[(a)–(c)] The structure of monolayer $T_d\text{−}{\mathrm{WTe}}_2$. (d) The first Brillouin zone of $T_d\text{−}{\mathrm{WTe}}_2$. The coordinates of $X$, $Y$, $Q$, and $Q^{\prime }$ are $(0.5,0)$, $(0,0.9)$, $(0,0.385)$ and $(0,-0.385){\text{Å}}^{-1}$, respectively. $Q$ and $Q^{\prime }$ are the gap-opening points [24].

Figure 2

[(a)–(c)] Band structures of monolayer $T_d\text{−}{\mathrm{WTe}}_2$ with (a) $E_{\perp }=0.2\mathrm{V}/\mathrm{nm}$, (b) $E_{\perp }=0.5\mathrm{V}/\mathrm{nm}$, and (c) $E_{\perp }=1\mathrm{V}/\mathrm{nm}$ obtained by using the six-band model. $Q$ and $Q^{\prime }$ are the gap-opening points.

Figure 3

[(a)–(c)] The $\vec{k}$-space distribution of $I_{\mathrm{BCD},\mathrm{LPGE}}^{xyy}$ for Fermi level (a), (b) ${\varepsilon }_F=0\mathrm{eV}$ and (c) ${\varepsilon }_F=0.095\mathrm{eV}$ when the frequency $\nu =1000\text{Hz}$ and $E_{\perp }=0.2\mathrm{V}/\mathrm{nm}$. The difference between (a) and (b) is that the large magnitude of $I_{\mathrm{BCD},\mathrm{LPGE}}^{xyy}$ in the rectangular region near $\mathrm{\Gamma }$ point is discarded in (b). The unit of $I_{\mathrm{BCD},\mathrm{LPGE}}^{xyy}$ is ${\text{Å}}^3/\text{eV}$.

Figure 4

(a), (b) The frequency $\nu$ and Fermi level ${\varepsilon }_F$ dependence of (a) LPGE conductivity $Re\left({\sigma }_{\left(2\right)\mathrm{BCD}}^{xyy}(\omega ,-\omega )\right)$ and (b) CPGE conductivity $Im\left({\sigma }_{\left(2\right)\mathrm{BCD}}^{yxy}(\omega ,-\omega )\right)$ for $E_{\perp }=0.2\mathrm{V}/\mathrm{nm}$. Here, the frequencies are taken as ${10}^3$, ${10}^4$, ${10}^5$, $...$, ${10}^{13}\mathrm{Hz}$. The Fermi level range is from $-0.25\mathrm{eV}$ to 0.15 eV with an interval of 0.005 eV. The relaxation time is taken as 5 ps.

Figure 5

[(a)–(c)] The frequency $\nu$ and Fermi level ${\varepsilon }_F$ dependence of LPGE conductivity $Re\left({\sigma }_{\left(2\right)\mathrm{BCD}}^{xyy}(\omega ,-\omega )\right)$ for (a) $E_{\perp }=0.2\mathrm{V}/\mathrm{nm}$, (b) $E_{\perp }=0.5\mathrm{V}/\mathrm{nm}$, and (c) $E_{\perp }=1\mathrm{V}/\mathrm{nm}$. [(d)–(f)] The frequency $\nu$ and Fermi level ${\varepsilon }_F$ dependence of CPGE conductivity $Im\left({\sigma }_{\left(2\right)\mathrm{BCD}}^{yxy}(\omega ,-\omega )\right)$ for (d) $E_{\perp }=0.2\mathrm{V}/\mathrm{nm}$, (e) $E_{\perp }=0.5\mathrm{V}/\mathrm{nm}$, and (f) $E_{\perp }=1\mathrm{V}/\mathrm{nm}$. Here, the frequency range for numerical calculations is from 5 GHz to 500 GHz with an interval of 5 GHz. The Fermi level range is from $-0.25\mathrm{eV}$ to 0.15 eV with an interval of 0.005 eV. The relaxation time is taken as 5 ps.

Figure 6

[(a)–(c)] The frequency $\nu$ and Fermi level ${\varepsilon }_F$ dependence of LPGE conductivity $Re\left({\sigma }_{\left(2\right)\mathrm{IB}3}^{xyy}(\omega ,-\omega )\right)$ for (a) $E_{\perp }=0.2\mathrm{V}/\mathrm{nm}$, (b) $E_{\perp }=0.5\mathrm{V}/\mathrm{nm}$, and (c) $E_{\perp }=1\mathrm{V}/\mathrm{nm}$. [(d)–(f)] The frequency $\nu$ and Fermi level ${\varepsilon }_F$ dependence of CPGE conductivity $Im\left({\sigma }_{\left(2\right)\mathrm{IB}2}^{yxy}(\omega ,-\omega )\right)$ for (d) $E_{\perp }=0.2\mathrm{V}/\mathrm{nm}$, (e) $E_{\perp }=0.5\mathrm{V}/\mathrm{nm}$, and (f) $E_{\perp }=1\mathrm{V}/\mathrm{nm}$. Here, the frequency range for numerical calculations is from 5 THz to 400 THz with an interval of 5 THz. The Fermi level range is from $-0.25\mathrm{eV}$ to 0.15 eV with an interval of 0.005 eV. The relaxation time is taken as 5 ps.

Figure 7

(a), (b) The frequency $\nu$ and Fermi level ${\varepsilon }_F$ dependence of (a) LSHG conductivity ${\sigma }_{\left(2\right)\mathrm{eff}}^{xyy}(\omega ,\omega )$ and (b) CSHG conductivity ${\sigma }_{\left(2\right)\mathrm{eff}}^{yxy}(\omega ,\omega )$ for $E_{\perp }=0.2\mathrm{V}/\mathrm{nm}$. Here, the frequencies are taken as ${10}^3$, ${10}^4$, ${10}^5$, $...$, ${10}^{13}\mathrm{Hz}$. The Fermi level range is from $-0.25\mathrm{eV}$ to 0.15 eV with an interval of 0.01 eV. The relaxation time is taken as 5 ps.

Figure 8

[(a)–(c)] The frequency $\nu$ and Fermi level ${\varepsilon }_F$ dependence of LSHG conductivity ${\sigma }_{\left(2\right)\mathrm{eff}}^{xyy}(\omega ,\omega )$ for (a) $E_{\perp }=0.2\mathrm{V}/\mathrm{nm}$, (b) $E_{\perp }=0.5\mathrm{V}/\mathrm{nm}$, and (c) $E_{\perp }=1\mathrm{V}/\mathrm{nm}$. [(d)–(f)] The frequency $\nu$ and Fermi level ${\varepsilon }_F$ dependence of CSHG conductivity ${\sigma }_{\left(2\right)\mathrm{eff}}^{yxy}(\omega ,\omega )$ for (d) $E_{\perp }=0.2\mathrm{V}/\mathrm{nm}$, (e) $E_{\perp }=0.5\mathrm{V}/\mathrm{nm}$, and (f) $E_{\perp }=1\mathrm{V}/\mathrm{nm}$. Here, the frequency range for numerical calculations is from 5 THz to 400 THz with an interval of 5 THz. The Fermi level range is from $-0.25\mathrm{eV}$ to 0.15 eV with an interval of 0.01 eV. The relaxation time is taken as 5 ps.

Figure 9

(a), (b) Schematic illustration of photocurrents that need to be measured to verify our theory in $T_d-{\text{WTe}}_2$ monolayer. (a) LPGE and LSHG. A monochromatic linearly polarized light with electric field vector along the $y$-axis incident along the normal to $T_d-{\text{WTe}}_2$ monolayer with $z=0$ in the $xy$ plane. Due to the mirror symmetry ${\mathcal{M}}_y$, the second-order currents in the $y$ direction disappear and only second-order dc and second harmonic currents exist in the $x$ direction. The second-order currents in the $x$ direction are induced by the nonzero $Re\left({\sigma }_{\left(2\right)}^{xyy}(\omega ,-\omega )\right)$ and ${\sigma }_{\left(2\right)\mathrm{eff}}^{xyy}(\omega ,\omega )$. (b) CPGE and CSHG. A monochromatic circularly polarized light incident along the normal to $T_d-{\text{WTe}}_2$ monolayer with $z=0$ in the $xy$ plane. From Eqs. (10) and (11), we know that dc and second-harmonic currents exist in both $x$ and $y$ directions under circularly polarized light. However, due to the mirror symmetry ${\mathcal{M}}_y$, when the circularly polarized light changes from LC to RC, the dc and second harmonic currents in $y$ direction will be reversed while the dc and second harmonic currents in $x$ direction keeps the same magnitude and direction. Thus the CPGE and CSHG responses [see Eqs. (A4) and (A7)] exist only in the $y$ direction, which are induced by the non-zero $Im\left({\sigma }_{\left(2\right)}^{yxy}(\omega ,-\omega )\right)$ and ${\sigma }_{\left(2\right)\mathrm{eff}}^{yxy}(\omega ,\omega )$.