Topological Phase Transitions in the Photonic Spin Hall Effect

The recent synthesis of two-dimensional staggered materials opens up burgeoning opportunities to study optical spin-orbit interactions in semiconducting Dirac-like systems. We unveil topological phase transitions in the photonic spin Hall effect in the graphene family materials. It is shown that an external static electric field and a high frequency circularly polarized laser allow for active on-demand manipulation of electromagnetic beam shifts. The spin Hall effect of light presents a rich dependence with radiation degrees of freedom, and material properties, and features nontrivial topological properties. We discover that photonic Hall shifts are sensitive to spin and valley properties of the charge carriers, providing an unprecedented pathway to investigate spintronics and valleytronics in staggered 2D semiconductors.

Figure 1

Schematic representation of the system under study. The inset exhibits the top and side views of staggered graphene family materials. The lattice constant and staggering length values are $a=3.86$, 4.02, and 4.7 Å, and $\ell =0.23$, 0.33, and 0.4 Å for silicene, germanene, and stanene, respectively [24].

Figure 2

(a) Electronic band structure of graphene family semiconductors for spin down ($s=-1$) and up ($s=+1$). $n$ and $\mathcal{C}$ are the number of closed gaps and Chern number, respectively. Parameters are $\{e\ell E_z/{\lambda }_{\mathrm{SO}},\mathrm{\Lambda }/{\lambda }_{\mathrm{SO}}\}=\{0,0\}$ (left), $\{0,1\}$ (middle), and $\{0,2\}$ (right). Real (solid) and imaginary (dashed) components of (b) ${\sigma }_{xx}$ and (c) ${\sigma }_{xy}$, for the phases described in (a). The layer is neutral, $k_{B}T={10}^{-2}{\lambda }_{\mathrm{SO}}$, and $\hslash \mathrm{\Gamma }=0.002{\lambda }_{\mathrm{SO}}$.

Figure 3

Phase diagram of (a) ${\mathrm{\Delta }}_{\mathrm{IF}}$ and (b) ${\mathrm{\Delta }}_r$ for suspended neutral staggered monolayers. The electronic phases are the quantum spin Hall insulator (QSHI), the spin-valley polarized metal (SVPM), the band insulator (BI), the single Dirac cone (SDC), the spin-polarized metal (SPM), the anomalous quantum Hall insulator (AQHI), and the polarized-spin quantum Hall insulator (PS-QHI). (c) ${\mathrm{\Delta }}_{\mathrm{IF}}$ (blue) and ${\mathrm{\Delta }}_r$ (red) as a function of $\mathrm{\Lambda }$ for $\mu =0$. (d) ${\mathrm{\Delta }}_r$ vs $\mathrm{\Lambda }$ for $\mu /{\lambda }_{\mathrm{SO}}=0.05$ (blue), 0.1 (red), and 0.2 (black). The beam is $s$ polarized; $\hslash \omega =0.1{\lambda }_{\mathrm{SO}}$, $\theta =\pi /4$, $d_0=\hslash c/{\lambda }_{\mathrm{SO}}$, and $e\ell E_z/{\lambda }_{\mathrm{SO}}=0.5$ in (c) and (d) [36].

Figure 4

(a)–(d) Photonic spin Hall shifts ${\mathrm{\Delta }}_{\mathrm{SHEL}}^{-}$ (solid) and ${\mathrm{\Delta }}_{\mathrm{SHEL}}^{+}$ (dashed) as a function of different parameters for a suspended neutral monolayer and $s$ polarization. The dotted curves correspond to the results in Eq. (4). (e) ${\mathrm{\Delta }}_{\mathrm{IF}}$ and (f) ${\mathrm{\Delta }}_r$ vs wavelength for a $p$-polarized beam impinging on a stanene coated SiC substrate. In all panels $\{e\ell E_z/{\lambda }_{\mathrm{SO}},\mathrm{\Lambda }/{\lambda }_{\mathrm{SO}}\}=\{0,0\}$ (QSHI, blue), $\{0,1\}$ (SPM, black), $\{0,2\}$ (AQHI, red), and $\{0.5,1.5\}$ (SDC, orange).