Resonant magneto-optic Kerr effect in the magnetic topological insulator $\mathrm{Cr}:({\mathrm{Sb}}_x,{\mathrm{Bi}}_{1-x}{)}_2{\mathrm{Te}}_3$

We report measurements of the polar Kerr effect, proportional to the out-of-plane component of the magnetization, in thin films of the magnetically doped topological insulator ${\left({\text{Cr}}_{0.12}{\text{Bi}}_{0.26}{\text{Sb}}_{0.62}\right)}_2{\text{Te}}_3$. Measurements of the complex Kerr angle ${\mathrm{\Theta }}_K$ were performed as a function of photon energy in the range $0.8\text{eV}<\hslash \omega <3.0\text{eV}$. We observed a peak in the real part of ${\mathrm{\Theta }}_K\left(\omega \right)$ and zero crossing in the imaginary part that we attribute to a resonant interaction with a spin-orbit avoided crossing located $\approx 1.6$ eV above the Fermi energy. The resonant enhancement allows measurement of the temperature and magnetic field dependence of ${\mathrm{\Theta }}_K$ in the ultrathin film limit, $d\ge 2$ quintuple layers (QL). We find a sharp transition to zero remanent magnetization at 6 K for $d<8$ QL, consistent with theories of the dependence of impurity spin interactions on film thickness and their location relative to topological insulator surfaces.

Figure 1

Isolating the magneto-optic Kerr effect at 6 K for a 20 QL sample. (a) ${\mathrm{\Theta }}_K$ at 6 K in $+100$ mT (R1), and $-100$ mT (R2), and at 50 K in field $-100$ mT (R3) and $+100$ mT (R4). (b) ${\mathrm{\Theta }}_K$ at 50 K subtracted from ${\mathrm{\Theta }}_K$ at 6 K in a field $+100$ mT (D1) and $-100$ mT (D2).

Figure 2

Kerr effect in a 20 QL thick sample of ${\left({\mathrm{Cr}}_{0.12}{\mathrm{Bi}}_{0.26}{\mathrm{Sb}}_{0.62}\right)}_2{\mathrm{Te}}_3$. (a) Real ${\theta }_K=\text{Re}\left({\mathrm{\Theta }}_K\right)$ and imaginary ${\varepsilon }_K=\text{Im}\left({\mathrm{\Theta }}_K\right)$ parts of the Kerr spectrum measured at 6 K with a magnetic field of 100 mT applied perpendicular to the film plane. (b) Off-diagonal component of the dielectric function ${\epsilon }_{xy}\left(\omega \right)$ as determined by analysis of the Kerr spectra shown in (a).

Figure 3

Schematic illustration of the band structure and optical transitions in magnetic TI ${\left({\text{Cr}}_{0.12}{\text{Bi}}_{0.26}{\text{Sb}}_{0.62}\right)}_2{\text{Te}}_3$. SS, VB, and CB denote surface state, valence band, and conduction band, respectively. The shaded region in CB and VB indicates the energy range of two-dimensional subbands. The SS gap is induced by FM order. The green line illustrates the first SS to second SS optical transition, while the red line illustrates transitions between subband states that evolve to the “bulk-to-bulk” (BB) transition with increasing film thickness.

Figure 4

Thickness dependence of the Kerr angle. (a) Amplitudes of ${\theta }_K$ and ${\varepsilon }_K$ at their respective peaks of 1.7 and 1.5 eV as a function of sample thickness at 6 K and 100 mT. Plotted for comparison (dashed-dotted line) is ${\theta }_K\left(d\right)$ calculated under the assumption of $d$-independent ${\epsilon }_{xy}\left(\omega \right)$. (b) Imaginary part of ${\mathrm{\Theta }}_K\left(\omega \right)$ for various thicknesses in the range from 2 to 10 QL.

Figure 5

(a) Real part of the Kerr rotation ${\theta }_K$ at a photon energy of 1.96 eV as a function of temperature for a 20 QL film. The larger amplitude curve was measured upon cooling in a 100 mT magnetic field and warming with the field left on, and the smaller amplitude curve was recorded on warming with the field turned off at 6 K, and is a measure of the remnant magnetization. (b) Remanent ${\mathrm{\Theta }}_K$ at 6 K as a function of film thickness shown as solid circles (left axis); triangles (right axis) show the temperature $T_r$ above which the remanent signal vanishes. The dashed line indicates our experimental upper bound on $T_r$.