Optically Probing Tunable Band Topology in Atomic Monolayers

In many atomically thin materials, their optical absorption is dominated by excitonic transitions. It was recently found that optical selection rules in these materials are influenced by the band topology near the valleys. We propose that gate-controlled band ordering in a single atomic monolayer, through changes in the valley winding number and excitonic transitions, can be probed in helicity-resolved absorption and photoluminescence. This predicted tunable band topology is confirmed by combining an effective Hamiltonian and a Bethe-Salpeter equation for an accurate description of excitons, with first-principles calculations suggesting its realization in Sb-based monolayers.

Figure 1

(a) Schematic setup. The linearly polarized (LP) probe light yields a response of definite helicity, ${\sigma }^{\pm }$, that depends on the electric field, $\mathcal{E}$. (b) Bands at the $K/K^{\prime }$ valleys. The $K^{\prime }$-valley conduction band ordering and the winding numbers, $w_{1,2}$, change when $\mathcal{E}$ exceeds a critical value, ${\mathcal{E}}_c$.

Figure 2

Band-edge energies as a function of the staggered potential at the (a) $K$ and (b) $K^{\prime }$ valleys. Helicity-resolved absorption spectra for both valleys when (c) $\mathcal{E}<{\mathcal{E}}_c$ ($U=10\mathrm{meV}$) and (d) $\mathcal{E}>{\mathcal{E}}_c$ ($U=80\mathrm{meV}$), dashed vertical lines: respective $1s$ exciton energies at 81 and 72 meV. At $K^{\prime }$, the band crossing leads to a change of the winding number and, thus, to a change in helicity for the lowest $1s$ exciton absorption peak.

Figure 3

Winding patterns of the interband matrix elements of the velocity operators $v_{\pm }(\mathbf{k})$ at the $K^{\prime }$ valley with the $K^{\prime }$ point at the origin. The real (imaginary) part of $v_{\pm }(\mathbf{k})$ is given along the horizontal (vertical) direction. (a) $v_{+}(\mathbf{k})$, (b) $v_{-}(\mathbf{k})$ for $\mathcal{E}<{\mathcal{E}}_c$ and (c) $v_{+}(\mathbf{k})$, (d) $v_{-}(\mathbf{k})$ for $\mathcal{E}>{\mathcal{E}}_c$. (e) Resonant optical excitation of $1s$ excitons with ${\sigma }^{+}$ light when $\mathcal{E}<{\mathcal{E}}_c$ and Berry curvature $\mathrm{\Omega }\ne 0$ at the valleys. (f) Anomalous valley Hall effect with an applied in-plane electric field, ${\mathcal{E}}_{\mathrm{in}}$.

Figure 4

(a) Schematic of the ML SbH doped with one W atom and the $2\times 2$ supercell (black lines) used in our calculations. A related electronic structure and Berry curvature, $\mathrm{\Omega }$, for the electrical field, $\mathcal{E}$: (b) smaller ($0\mathrm{V}/\mathrm{nm}$), (c) equal ($2\mathrm{V}/\mathrm{nm}$), and (d) greater ($4\mathrm{V}/\mathrm{nm}$) than the critical value ${\mathcal{E}}_c$, displaying a CB inversion in the $K^{\prime }$ valley (highlighted in rectangles).