Nonlinear optical Hall effect of few-layered ${\mathrm{NbSe}}_2$

${\mathrm{NbSe}}_2$ is one of metallic two-dimensional (2D) transition-metal dichalcogenide (TMDC) materials. Because of broken crystal inversion symmetry, large spin splitting is induced by Ising-type spin-orbit coupling in odd-number-layered ${\mathrm{NbSe}}_2$, but absent for even-number-layered ${\mathrm{NbSe}}_2$ with the inversion symmetry. In this paper we numerically calculate nonlinear optical charge and spin Hall conductivities of few-layered ${\mathrm{NbSe}}_2$ based on an effective tight-binding model which includes $d_{z^2}$, $d_{x^2-y^2}$, and $d_{xy}$ orbitals of Nb atoms. We show that the nonlinear optical Hall conductivity for the second harmonic generation (SHG) process has a nonvanishing value in odd-number-layered ${\mathrm{NbSe}}_2$. Also, we provide a nonlinear optical selection rule in few-layered ${\mathrm{NbSe}}_2$ and their polarization dependencies. Furthermore, for the even-number-layered case, the nonlinear optical Hall currents can be generated by applying electric fields which break inversion symmetry. We also discuss that the nonlinear optical Hall effect is expected to occur in TMDC materials in general. Thus, our results will serve to design potential opt-spintronics devices based on 2D materials to generate the spin Hall current by SHG.

Figure 1

Crystal structures of different few-layered ${\mathrm{NbSe}}_2$ which consist of Nb (black) and Se (yellow) atoms. (a) Side view of the lattice structure of monolayer (A), bilayer (AB), and trilayer (ABA) ${\mathrm{NbSe}}_2$. Top views of the lattice structures of (b) monolayer and (c) bilayer and trilayer ${\mathrm{NbSe}}_2$. (d) First BZ of ${\mathrm{NbSe}}_2$. Energy band structures and DOS of (e) monolayer, (f) bilayer, and (g) trilayer ${\mathrm{NbSe}}_2$ with SOC parameter ${\lambda }_{\text{SOC}}=0.0784\mathrm{eV}$, respectively. Fermi level is set to zero.

Figure 2

Real parts of nonlinear optical spin Hall conductivities $\mathrm{Re}\left[{\sigma }_{xyy}^{\mathrm{spin}}(\omega ,\omega )\right]$ of (a) monolayer, (b) bilayer, and (c) trilayer ${\mathrm{NbSe}}_2$, respectively. By irradiating $y$-polarized light, nonlinear optical spin Hall current is generated in odd-number-layered ${\mathrm{NbSe}}_2$. Real parts of nonlinear optical charge Hall conductivities $\mathrm{Re}\left[{\sigma }_{yxx}^{\mathrm{charge}}(\omega ,\omega )\right]$ of (d) monolayer, (e) bilayer, and (f) trilayer ${\mathrm{NbSe}}_2$, respectively. Nonlinear optical charge Hall current is generated by $x$-polarized light irradiation. Red, blue, and black lines indicate several different SOC parameters. The units of nonlinear optical spin and charge Hall conductivities are $e^2$ and $e^3/\hslash$, respectively.

Figure 3

Schematics of nonlinear optical spin and charge Hall currents in few-layered ${\mathrm{NbSe}}_2$. (a) By $y$-polarized light irradiation, nonlinear optical spin Hall current is generated in $x$ direction, but nonlinear optical charge Hall current is absent in odd-number-layered ${\mathrm{NbSe}}_2$. (b) By $x$-polarized light irradiation, nonlinear optical spin Hall current disappears, but nonlinear optical charge Hall current is generated in $y$ direction in odd-number-layered ${\mathrm{NbSe}}_2$. Nonlinear optical (c) spin and (d) charge Hall currents are not generated in even-number-layered ${\mathrm{NbSe}}_2$.

Figure 4

Nonlinear optical (a) spin and (b) charge Hall conductivities of bilayer ${\mathrm{NbSe}}_2$ for several different applied electric fields. The electric fields are $F=0.2$ (blue), 0.4 (cyan), 0.6 (green), 0.8 (yellow), 1.0 (purple), and 2.0 eV (red), respectively. (a) By irradiating $y$-polarized light, the real part of nonlinear optical spin Hall current $\mathrm{Re}\left[{\sigma }_{xyy}^{\mathrm{spin}}(\omega ,\omega )\right]$ is generated in bilayer ${\mathrm{NbSe}}_2$ with the application of electric fields. (b) By irradiating $x$-polarized light, the real part of nonlinear optical charge Hall current $\mathrm{Re}\left[{\sigma }_{yxx}^{\mathrm{charge}}(\omega ,\omega )\right]$ is generated in bilayer ${\mathrm{NbSe}}_2$. The units of nonlinear optical spin and charge Hall conductivities are $e^2$ and $e^3/\hslash$, respectively.

Figure 5

(a) Top view of crystal structure of monolayer ${\mathrm{NbSe}}_2$ which consists of Nb (black) and Se (yellow) atoms. Green, red, and blue arrows indicate hopping vectors ${\mathit{R}}_{\mathit{i}}(i=1,2,\cdots ,6)$ pointing to nn sites, the vectors $\widetilde{{\mathit{R}}_{\mathit{j}}}(j=1,2,\cdots ,6)$ pointing to next nn sites and the vectors $2{\mathit{R}}_{\mathit{i}}$ pointing to third nn sites, respectively. $a$ is the lattice constant. (b) Side view of crystal structure of bilayer ${\mathrm{NbSe}}_2$. Interlayer hopping vector ${\mathit{R}}_{\text{inter}}$ points from the $d$ orbital of the upper layer to nn sites of the lower layer. Energy band structures and DOS of (c) monolayer, (d) bilayer, and (e) trilayer ${\mathrm{NbSe}}_2$ with SOC parameter ${\lambda }_{\text{SOC}}=0.0784\mathrm{eV}$, respectively. Fermi level is set to zero. The arrows in (c) include the two SHG processes: (i) interband transition by two-photon absorption (green arrows) and (ii) interband transition even by one-photon absorption (red arrows). Fermi surface of (f) monolayer, (g) bilayer, and (h) trilayer ${\mathrm{NbSe}}_2$, where red, blue, and green lines indicate for up-spin, down-spin, and spin-degenerated states, respectively.

Figure 6

Contour plots of integrands for nonlinear optical spin Hall conductivities of monolayer ${\mathrm{NbSe}}_2$ in the first BZ: (a) ${\mathrm{\Omega }}_{xyy}^{\mathrm{spin}}(\omega =1.5,\omega =1.5)$, (b) ${\mathrm{\Omega }}_{xyy}^{\mathrm{spin}}(\omega =2.3,\omega =2.3)$, (c) ${\mathrm{\Omega }}_{yxx}^{\mathrm{spin}}(\omega =1.5,\omega =1.5)$, and (d) ${\mathrm{\Omega }}_{yxx}^{\mathrm{spin}}(\omega =2.3,\omega =2.3)$. Same plots of integrands for nonlinear optical charge Hall conductivities of monolayer ${\mathrm{NbSe}}_2$: (e) ${\mathrm{\Omega }}_{xyy}^{\mathrm{charge}}(\omega =1.5,\omega =1.5)$, (f) ${\mathrm{\Omega }}_{xyy}^{\mathrm{charge}}(\omega =2.3,\omega =2.3)$, (g) ${\mathrm{\Omega }}_{yxx}^{\mathrm{charge}}(\omega =1.5,\omega =1.5)$, and (h) ${\mathrm{\Omega }}_{yxx}^{\mathrm{charge}}(\omega =2.3,\omega =2.3)$.

Figure 7

Energy band structures of bilayer ${\mathrm{NbSe}}_2$ with applied electric fields of (a) $F=0.2$, (b) 0.6, (c) 1.0, and (d) 2.0 eV, respectively. Fermi levels are set to (a) $-0.1370$, (b) $-0.5165$, (c) $-0.9113$, and (d) $-1.9074\mathrm{eV}$, respectively. (e) Bilayer ${\mathrm{NbSe}}_2$ with applied electric field includes six energy bands $\text{➀}$ to $\text{➅}$. At $\mathrm{\Gamma }$ point, the energy bands of $\text{➀}$, $\text{➂}$, and $\text{➃}$ are shown by solid lines. The energy bands of $\text{➁}$, $\text{➄}$, and $\text{➅}$ are shown by dashed lines.

Figure 8

(a) Energy band structure of bilayer ${\mathrm{NbSe}}_2$ with each layer having a different Fermi energy. (b) Real part of nonlinear optical spin Hall conductivity $\mathrm{Re}\left[{\sigma }_{xyy}^{\mathrm{spin}}(\omega ,\omega )\right]$ of decoupled bilayer ${\mathrm{NbSe}}_2$. (c) Real part of nonlinear optical charge Hall conductivity $\mathrm{Re}\left[{\sigma }_{yxx}^{\mathrm{charge}}(\omega ,\omega )\right]$ of decoupled bilayer ${\mathrm{NbSe}}_2$. The units of $\mathrm{Re}\left[{\sigma }_{xyy}^{\mathrm{spin}}(\omega ,\omega )\right]$ and $\mathrm{Re}\left[{\sigma }_{yxx}^{\mathrm{charge}}(\omega ,\omega )\right]$ are $e^2$ and $e^3/\hslash$, respectively.