Giant Magneto-Optical Kerr Effect and Universal Faraday Effect in Thin-Film Topological Insulators

Topological insulators can exhibit strong magneto-electric effects when their time-reversal symmetry is broken. In this Letter we consider the magneto-optical Kerr and Faraday effects of a topological insulator thin film weakly exchange coupled to a ferromagnet. We find that its Faraday rotation has a universal value at low frequencies ${\theta }_{\mathrm{F}}={tan}^{-1}\alpha$, where $\alpha$ is the vacuum fine structure constant, and that it has a giant Kerr rotation ${\theta }_{\mathrm{K}}=\pi /2$. These properties follow from a delicate interplay between thin-film cavity confinement and the surface Hall conductivity of a topological insulator’s helical quasiparticles.

Figure 1

Faraday and Kerr spectra for a single vacuum-TI interface. ${\theta }_{\mathrm{F}}$ (red/grey) and ${\theta }_{\mathrm{K}}$ (black) at three different carrier densities for $\Delta =20\mathrm{meV}$ and the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ parameters $ϵ=29$, $\mu =1$, $v=5\times {10}^5\mathrm{m}{\mathrm{s}}^{-1}$. We take ${\epsilon }_c=0.175\mathrm{eV}$, half of the bulk band gap.

Figure 2

Faraday and Kerr spectra for a TI thin film. ${\theta }_{\mathrm{F}}$ and ${\theta }_{\mathrm{K}}$ as a function of frequency at a series of $\Delta$ values for a 30 nm-thick ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ thin film with the Fermi level inside the Dirac gap on both top and bottom surfaces. Notice that ${\theta }_{\mathrm{K}}$ values for $\omega \to 0$ lie well beyond the range of this figure.

Figure 3

Low-frequency Kerr rotation. Kerr rotation as a function of frequency $\omega /{\epsilon }_c$ and gap $\Delta /{\epsilon }_c$, showing a “waterfall” feature with ${\theta }_{\mathrm{K}}$ exhibiting a constant value $\simeq -1.56=-\pi /2$ at low frequencies $\omega \lesssim 0.02{\epsilon }_c$. Dashed line shows ${\omega }_{\mathrm{K}}$ from Eq. (10). The approximate formula is in good agreement with numerical results.

Figure 4

Proposed experimental setup for observing the topological Kerr and Faraday effects. A topological insulator thin film is sandwiched between two sets of ferromagnets (FM) with opposite magnetizations (indicated by straight arrows) stacked together, which induce an exchange field at the topological insulator surface.