Photogalvanic effect in the HgTe/CdTe topological insulator due to edge-bulk optical transitions

We study theoretically the 2D HgTe/CdTe quantum well topological insulator illuminated by circularly polarized light with frequencies higher than the difference between the equilibrium Fermi level and the bottom of the conduction band (THz range). We show that electron-hole asymmetry results in spin-dependent electric dipole transitions between edge and bulk states, and we predict an occurrence of a circular photocurrent. If the edge state is tunnel-coupled to a conductor, then the photocurrent can be detected by measuring an electromotive force in the conductor, which is proportional to the photocurrent.

Figure 1

Schematic picture of optical transitions between two edge state branches and bulk states. Circularly polarized light of frequency ${\omega }_0$ exceeding the absorption threshold by $\mathrm{\Delta }\omega$ induces transitions between edge and bulk conduction band states. The different thickness of arrows depicts that probability of electric dipole transitions may depend on chirality/spin of the edge electron.

Figure 2

(a) Schematic picture of quasi-Fermi levels if Fermi level is above the Dirac point. All the spin-up and spin-down edge electrons with energies from ${\varepsilon }_F-\mathrm{\Delta }\omega$ to ${\varepsilon }_F$ are moved to the conduction band. States near the Dirac point are occupied, and, hence, spontaneous transitions from bulk to edge states are not allowed. The electric current of the edge electrons $j=G_0({\mu }_{\uparrow }-{\mu }_{\downarrow })=0$. (b) Schematic picture of quasi-Fermi levels if Fermi level is below the Dirac point and for low intensities $\mathcal{W}<{\mathcal{W}}_{\downarrow }-{\mathcal{W}}_{\uparrow }$ (region I in Fig. 3). All the spin-up electrons with energies from ${\varepsilon }_F-\mathrm{\Delta }\omega$ to ${\varepsilon }_F$ are excited by the light, and spin-down electrons in the conduction band appear mainly due to spin relaxation. Recombination of bulk spin-down electrons shifts the quasi-Fermi level of spin-down electrons above the initial Fermi level. (c) Schematic picture of quasi-Fermi levels for high intensities $\mathcal{W}>{\mathcal{W}}_{\downarrow }/2$ (region II in Fig. 3). Almost all spin-up and spin-down electrons with energies from $\varepsilon -\mathrm{\Delta }\omega$ to $\varepsilon$ are excited by the light $\left({\mu }_{\uparrow }\approx {\mu }_{\downarrow }\approx -\mathrm{\Delta }\omega \right)$. The spin imbalance at the edge states decays with intensity of the light as $\delta \mu \propto {\mathcal{W}}^{-2}$.

Figure 3

Dependence of photocurrent (solid line) and currents of spin-up/spin-down electrons (dashed lines) on light intensity if the initial Fermi level is below the Dirac point. The currents of each branch are proportional to the corresponding quasi-Fermi level shift. For the region I $\left(\mathcal{W}<{\mathcal{W}}_{\downarrow }/2\right)$ edge branches contribute to net current with the same sign, while in the region II $\left(\mathcal{W}>{\mathcal{W}}_{\downarrow }/2\right)$ the current of spin-down electrons changes its sign. Currents are measured in the units of $G_0\mathrm{\Delta }\omega$.

Figure 4

A proposal to detect photocurrent in the edge state: 2D conductor of length $L_y$ is coupled to the edge state via the tunnel contact of length $\mathrm{\Lambda }$. The EMF appears between the opposite ends of the 2D conductor.