Probing topological phase transitions via quantum reflection in the graphene family materials

We theoretically investigate the quantum reflection of different atoms by two-dimensional (2D) materials of the graphene family (silicene, germanene, and stanene), subjected to an external electric field and circularly polarized light. By using Lifshitz theory to compute the Casimir-Polder potential, which ensures that our predictions apply to all regimes of atom-2D surface distances, we demonstrate that the quantum reflection probability exhibits distinctive, unambiguous signatures of topological phase transitions that occur in 2D materials. We also show that the quantum reflection probability can be highly tunable by these external agents, depending on the atom-surface combination, reaching a variation of 40% for rubidium in the presence of a stanene sheet. Our findings attest not only that dispersive forces play a crucial role in quantum reflection, but also that the topological phase transitions of the graphene family materials can be comprehensively and efficiently probed via atom-surface interactions at the nanoscale.

Figure 1

(a) Atomic specimen being reflected by a suspended 2D material of the graphene family. The system is under the influence of a static electric field $E_z$ and it is shined on by a circularly polarized light, characterized by the parameter $\mathrm{\Lambda }$. (b) Topological phase diagram of the 2D graphene family materials. Horizontal dashed gray lines show the paths used in this work to explore this diagram. The acronyms of each topological phase are as follows: Band insulator (BI; $C=0$), quantum spin Hall insulator (QSHI; $C=0$, indexed by the nontrivial ${\mathbb{Z}}_2$ index [29]), polarized-spin quantum Hall insulator (PS-QHI; $C=-1$), and anomalous quantum Hall insulator (AQHI; $C=-2$), with $C$ standing for the corresponding Chern numbers.

Figure 2

Results for QR of a Rb atom with energy $E={10}^{-4}$ neV by a suspended sheet of pristine stanene (${\lambda }_{\text{SO}}=50$ meV, chemical potential $\mu =0$, and inverse of the scattering time $\mathrm{\Gamma }={10}^{-4}{\lambda }_{\text{SO}}/\hslash$, as described in Appendix pp2). (a) QR probability as a function of the electric field for different values of the laser parameter. (b) Percentage modification in the QR probability caused by the electric field.

Figure 3

(a) The probability $R$ of a Rb atom suffering QR from a stanene sheet as a function of its incident energy $E$. (b) Relative modification in the CP energy between the Rb atom and the 2D surface caused by the applied electric field as a function of the distance $z$. (c) The plot of $Q\left(z\right)$ as a function of the distance $z$ for the incident energy $E={10}^{-4}$ neV chosen in the intermediate energy regime between the classical and the quantum limit of QR [see panel (a)]. In all plots, $\mathrm{\Gamma }={10}^{-4}{\lambda }_{\text{SO}}/\hslash$, $\mu =0$, and $\mathrm{\Lambda }=0$.

Figure 4

(a) The probability $R$ of a Rb atom suffering QR from a stanene sheet as a function of its incident energy $E$. (b) Relative modification in the CP energy between the Rb atom and the 2D surface caused by the applied electric field as a function of the distance $z$. (c) The plot of $Q\left(z\right)$ as a function of the distance $z$ for the incident energy $E={10}^{-4}$ neV chosen in the intermediate energy regime between the classical and the quantum limit of QR [see panel (a)]. In all plots, $\mathrm{\Gamma }={10}^{-4}{\lambda }_{\text{SO}}/\hslash$, $\mu =0$, and $\mathrm{\Lambda }=0$.

Figure 5

Results for QR of a Rb atom with energy $E={10}^{-4}$ neV by a suspended sheet of pristine germanene (${\lambda }_{\text{SO}}=20$ meV, $\mu =0$, and $\mathrm{\Gamma }={10}^{-4}{\lambda }_{\text{SO}}/\hslash$). (a) QR probability as a function of the electric field for different values of the laser parameter. (b) Percentage modification in the QR probability caused by the electric field.

Figure 6

Results for QR of a Rb atom with energy $E={10}^{-4}$ neV by a suspended sheet of pristine silicene (${\lambda }_{\text{SO}}=2$ meV, $\mu =0$, and $\mathrm{\Gamma }={10}^{-4}{\lambda }_{\text{SO}}/\hslash$). (a) QR probability as a function of the electric field for different values of the laser parameter. (b) Percentage modification in the QR probability caused by the electric field.

Figure 7

Evolution of the electronic spectrum of the Hamiltonian of Eqs. (B1) and (B2), along path I of Fig. 1, i.e., $\mathrm{\Lambda }/{\lambda }_{\mathrm{SO}}=0$, and (a) $e\ell E_z/{\lambda }_{\mathrm{SO}}=0$, (b) $e\ell E_z/{\lambda }_{\mathrm{SO}}=0.5$, (c) $e\ell E_z/{\lambda }_{\mathrm{SO}}=1$, (d) $e\ell E_z/{\lambda }_{\mathrm{SO}}=1.5$, and (e) $e\ell E_z/{\lambda }_{\mathrm{SO}}=2$. The left panels show the spectrum for the valley $K$ ($\eta =+1$) and the right panels show the spectrum for the valley $K^{\prime }$ ($\eta =-1$). The black dashed line depicts the spectrum for the spin-$\uparrow$ sector ($s=+1$), and the red dotted line depicts the spectrum for the spin-$\downarrow$ sector ($s=-1$). The topological critical point is represented in panel (c), where the electronic spectrum becomes gapless, and it separates the QSHI phase [panels (a) and (b)] and the BI phase [panels (d) and (e)].

Figure 8

Panels (a) and (b): Real (solid) and imaginary (dashed) parts of ${\sigma }_{xx}\left(\omega \right)$ and ${\sigma }_{xy}\left(\omega \right)$ for some combinations of parameters $\{e\ell E_z/{\lambda }_{\text{SO}},\mathrm{\Lambda }/{\lambda }_{\text{SO}}\}$. A: $\{0,0\}$, B: $\{0,1\}$, and C: $\{0,2\}$, as indicated in the topological phase diagram of panel (f). Panels (c) and (d): The same as before but for A: $\{0,0\}$, D: $\{1,0.5\}$, and E: $\{2,0\}$, as in panel (f). We set $\mathrm{\Gamma }={10}^{-4}{\lambda }_{\text{SO}}/\hslash$ and $\mu =0$ in all cases. Panels (e) and (f): Topological phase diagram of the graphene family materials described by the Hamiltonian (B1). Panel (e) shows the phase diagram with the acronyms and their respective Chern numbers, while panel (f) sketches the set of parameters used in panels (a)–(d). A detailed discussion of this phase diagram and the optical conductivities can be found in Refs. [21, 30, 79].

Figure 9

Results for a Na atom with incident energy $E={10}^{-3}$ neV. QR probability as a function of the applied electric field by a suspended sheet of pristine (a) stanene, (c) germanene, and (e) silicene. Percentage modification in the QR probability due to the electric field by a suspended sheet of pristine (b) stanene, (d) germanene, and (f) silicene. In all cases different curves refer to different values of the laser parameter.