High harmonic generation in magnetically-doped topological insulators

As a new class of condensed matter, topological insulators challenge the traditional wisdom of condensed matter physics. Doping topological insulators with magnetic elements can realize the quantum anomalous Hall state, a two-dimensional bulk insulator with nonzero Chern number. High harmonic generation (HHG) has emerged as a very promising tool to probe electronic properties. However, its application to magnetically doped topological insulators has not been realized. Here, we predict that high harmonics from six quintuple layers of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$(6QL-${\mathrm{Bi}}_2{\mathrm{Se}}_3$) carry crucial information about its structure. HHG sensitively depends on crystal symmetry and laser polarization. While in the parent compound 6QL-${\mathrm{Bi}}_2{\mathrm{Se}}_3$ all the harmonic orders are odd, once it is doped with a magnetic Cr atom, selective even and odd harmonics appear, depending on whether the laser field is in-plane or out-of-plane. With spin-orbit coupling, both even and odd harmonics appear. We find that these seemingly complicated harmonics have a simple origin: microscopic interplay between symmetry operations and laser polarization. Our finding demonstrates the unexplored power of HHG in topological insulators and will have a broad impact on future research.

Figure 1

(a) Structure of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The red rectangle is the repeated quintuple cells. Red and blue spheres represent Se and Bi atoms, respectively. Top right panel: bulk Brillouin zone (BZ) and (111) surface BZ of the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, with high symmetry points ($\mathrm{\Gamma }$, Z, F, L, $\overline{\mathrm{\Gamma }}$, $\overline{M}$, and $\overline{K}$) identified. Bottom right panel: the projection on the $ab$ plane of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. (b) Element-resolved band structure of bulk ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, where red and blue circles represent the Se and Bi atoms, respectively. (c) Surface electronic structure of 6QL-${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The red and blue circles represent the Se and Bi atoms in the top and bottom QLs, respectively. (d)–(f) High harmonic signals for 6QL-${\mathrm{Bi}}_2{\mathrm{Se}}_3$ with laser polarization along the (d) $x$, (e) $y$, and (f) $z$ axes. The photon energy of the laser field is $\hslash \omega =1.5$ eV, with duration $\tau =60$ fs and vector potential amplitude $A_0=0.03$ Vfs/Å. Signals at noise level are not shown.

Figure 2

Harmonic signals from the spin up channel of Cr-doped 6QL-${\mathrm{Bi}}_2{\mathrm{Se}}_3$ for the laser polarization along the (a) $x$, (b) $y$, and (c) $z$ axes. The inset in (a) is the structure of Cr-doped 6QL-${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The photon energy is taken as $\hslash \omega =1.0$ eV to excite the electronic states near the Fermi level. Other parameters of the laser are the same as those in Fig. 1. (d), (e), and (f) use the laser polarization along the $y$ axis. (d) Symmetry-resolved harmonic signal along the $x$ axis from the $\mathrm{\Gamma }$ point alone. Symmetry operations that enter the calculation are given above each of the curves. For clarity, all the spectra, except the bottom one, are shifted vertically. (e) Same as (d), but with the harmonic signal along the $y$ axis. (f) Same as (d), but with the harmonic signal along the $z$ axis.

Figure 3

Harmonic spectra from Cr-doped 6QL-${\mathrm{Bi}}_2{\mathrm{Se}}_3$ in the presence of spin-orbit coupling. (a) The laser polarization is along the $x$ axis. In the presence of spin-orbit coupling, the group symmetry is reduced, the inversion symmetry is broken, and even harmonics appear. The harmonic signals are also generated along the $z$ axis, which is very different from that in Fig. 2. The laser parameters are the same as in Fig. 2. (b) Laser polarization is along the $y$ axis. The results are similar to Fig. 2. (c) Laser polarization is along the $z$ axis.

Figure 4

Symmetry properties of HHG signals under two configurations $\parallel$ and $\perp$. From left to right are for the pristine structure, Cr-doped structure, and Cr-doped structure with spin-orbit coupling. The bottom panel shows the rules how harmonic signals are emitted. IS denotes the inversion symmetry, which is broken by magnetic doping. SOC breaks the in-plane reflection symmetry ($\sigma$), where the spin of Cr atoms (the arrows) plays a role.