Cyclotron-resonance-assisted photocurrents in surface states of a three-dimensional topological insulator based on a strained high-mobility HgTe film

We report on the observation of cyclotron-resonance-induced photocurrents, excited by continuous wave terahertz radiation, in a three-dimensional topological insulator (TI) based on an 80-nm strained HgTe film. The analysis of the photocurrent formation is supported by complementary measurements of magnetotransport and radiation transmission. We demonstrate that the photocurrent is generated in the topologically protected surface states. Studying the resonance response in a gated sample, we examined the behavior of the photocurrent, which enables us to extract the mobility and the cyclotron mass as a function of the Fermi energy. For high gate voltages, we also detected cyclotron resonance (CR) of bulk carriers, with a mass about two times larger than that obtained for the surface states. The origin of the CR-assisted photocurrent is discussed in terms of asymmetric scattering of TI surface carriers in the momentum space. Furthermore, we show that studying the photocurrent in gated samples provides a sensitive method to probe the cyclotron masses and the mobility of two-dimensional Dirac surface states, when the Fermi level lies in the bulk energy gap or even in the conduction band.

Figure 1

(a) Cross section of the investigated structures. (b) An x-ray reciprocal space map near the asymmetric (113) reflection. The diffracted intensity is plotted on a color-coded logarithmic scale. The CdTe film is relaxed with respect to the GaAs substrate because the line connecting their maxima points to the origin of the reciprocal space. The HgTe and CdHgTe films are fully strained with respect to the CdTe, as their peaks lie on a straight line along the $z$ axis perpendicular to the sample surface (r. l. u.:- reciprocal lattice unit).

Figure 2

Experimental setups of (a) the photocurrent and radiation transmission measurements in squared-shaped samples and (b) photocurrent measurements in cross-shaped samples.

Figure 3

Magnetotransport data obtained in ungated cross-shaped [(a)–(d)], gated cross-shaped [(b) and (e)], and gated Hall-bar (f) samples. The panel (a) shows the ${\rho }_{xx}$ and (b) the ${\rho }_{xy}B$ dependence for $T=4.2$ and 40 K. (c) Electron $N_s$ and hole $P_s$ densities and (d) mobilities (${\mu }_e$ and ${\mu }_p$) at different temperatures, extracted from the Drude model. (e) $U_g$ dependence of ${\rho }_{xx}$ at $T=4.2$ and 40 K. The valence and conduction band edges are marked by arrows “$E_v$” and “$E_c$,” respectively. (f) Electron and hole densities of bulk ($N_s^{\mathrm{total}}$ and $P_s^{\mathrm{bulk}})$ and surface states ($N^{\mathrm{top}}$ and $N^{\mathrm{bottom}}$). The data are obtained at $T=1.9$ K on the same sample as that used in Ref. [24] but at a different cooldown. While the overall characteristic of the data is the same, we note that the position of the gap is shifted on the gate voltage scale. This is ascribed to cooldown-dependent charge trapping in the insulator. For the photocurrent experiments described below we made sure that the transport experiments were done under the same conditions as the cyclotron resonance experiments, i.e., had the charge neutrality point fixed and has the same value for all investigated samples.

Figure 4

(a) Normalized transmissivity of the square-shaped HgTe sample at 40 K as a function of magnetic field. The data are given for right (${\sigma }^{+}$) and left (${\sigma }^{-}$) circularly polarized light. The inset shows the averaged transmissivity traces obtained for different temperatures. (b), (c) Show the photosignal normalized by the radiation power ($U_x/P$) for ${\sigma }^{+}$ and ${\sigma }^{-}$ and for linearly polarized radiation, respectively. Full lines in all panels show fits by the Lorentzian function.

Figure 5

(a) The band structure of (013)-grown 80-nm strained HgTe film calculated by $\mathit{k}·\mathit{p}$ model. Solid and dashed lines show the band dispersion for two perpendicular in-plane crystallographic directions: $\mathit{k}\parallel \left[100\right]$ (solid curves) and $\mathit{k}\parallel \left[03\overline{1}\right]$ (dashed curves). Calculations are carried out for built-in electric field $E_z=0.5$ kV/cm which results in the energy splitting of the top and bottom surface states, shown by thick red solid and dashed curves. Inset shows calculated cyclotron masses for surface states as a function of energy. (b) Cyclotron masses obtained from the transmission measurements for (${\sigma }^{+}$) light and $f=2.54$ THz. Dots and triangles show the results for two CR lines.

Figure 6

Normalized CR radiation-induced photocurrent measured for different radiation frequencies in ungated cross-shaped samples. Note that the data are $y$ shifted by 0.2 for and 0.4 for $f=1.62$ and 0.69 THz, respectively. Full lines show fits by the Lorentzian function. Inset shows the first $B_{\text{CR}1}$ (squares) and second $B_{\text{CR}2}$ (circles) resonance positions as a function of frequency. Full and open triangles show the data of Ref. [39] for top and bottom surface states measured in similar strained HgTe film of 70 nm width.

Figure 7

(a) Magnetic field dependencies of the photocurrent measured for magnetic field tilted by an angle $\mathrm{\Theta }$ and linearly polarized light. The inset shows the resonance field $B_z^{\text{CR}}=B_{\text{CR}}*\text{cos}\left(\mathrm{\Theta }\right)$ as a function of angle $\mathrm{\Theta }$.

Figure 8

(a), (b) Magnetic field dependence of the photocurrent measured for the gated cross sample $\#1$ at different gate voltages $U_g$ at $T=40$ and 20 K, respectively. Full lines show fits using a Lorentzian. Note that the data for various $U_g$ are $y$ shifted for clarity. In panel (b) the data for $U_g=2$ and 3 V are multiplied by a factor 4. The inset in panel (b) shows $B_{\text{CR}1,2}$ determined from $J_x\left(B\right)$ for both temperatures. (c) Gate voltage dependencies of cyclotron masses obtained from the position of CR resonances $B_{\text{CR}1,2}$ determined from $J_x\left(B\right)$ and $J_y\left(B\right)$. (d) Magnetic field dependence of the photocurrent measured at low temperature and high magnetic fields.

Figure 9

Magnetic field dependence of the photosignal measured for two in-plane directions ($x$ and $y$) obtained for gated cross sample $\#2$ ($U_g=0$). The inset presents the normalized signal $U_x^{\prime }$ calculated as one-half of the sum (dashed line) and the difference (solid line) for signals measured at negative and positive magnetic fields $U_x^{\prime }=[U_x(B>0)\pm U_x(B<0)]/2$.

Figure 10

The vectors of photocurrent generation rate $\mathit{G}$ and the steady-state photocurrents $\mathit{j}$ for two opposite magnetic field directions [(a) and (b)]. The direction of photocurrent $\mathit{j}$ in the magnetic field is declined from the direction of $\mathit{G}$ by the Hall angle $arctan\left({\omega }_c\tau \right)$. Figure illustrates that, for the given parameters, $j_x$ changes its sign while $j_y$ converses its sign when the magnetic field direction is reversed.

Figure 11

Wave functions of edge states in (013)-grown 80-nm strained HgTe film. Calculations are carried out for the electron energy $E=11$ meV, the wave vector components $k_x>0$ and $k_y=0$, and the built-in electric field $E_z=2$ kV/cm.