Band structure of a HgTe-based three-dimensional topological insulator

From the analysis of the cyclotron resonance, we experimentally obtain the band structure of the three-dimensional topological insulator based on a HgTe thin film. Top gating was used to shift the Fermi level in the film, allowing us to detect separate resonance modes corresponding to the surface states at two opposite film interfaces, the bulk conduction band, and the valence band. The experimental band structure agrees reasonably well with the predictions of the $\mathbf{k}·\mathbf{p}$ model. Due to the strong hybridization of the surface and bulk bands, the dispersion of the surface states is close to parabolic in the broad range of the electron energies.

Figure 1

Theoretical band structure of a 3D topological insulator HgTe. (a) The first valence band, the surface band, and the first conduction band of the 80-nm-thick HgTe layer at the charge neutrality point $n_{\mathrm{tot}}=0$ where the concentrations of electrons and holes equal $\pm 1.2\times {10}^{11}{\mathrm{cm}}^{-2}$. (b) Cross section of (a) at $E_f=0\mathrm{meV}$. Blue: holelike Fermi surface ($\partial A/\partial E<0$) from the islands in the $(\pm 1,\pm 1)$ directions. Orange: electronlike Fermi surface ($\partial A/\partial E>0$) from the surface states. (c) Fermi surface of the hole-doped sample where four islands are connected and lead to a different cyclotron picture: blue: holelike; violet: electronlike; orange: electronlike.

Figure 2

Cyclotron resonance with circularly polarized light. (a)–(c) The intensity of the transmitted radiation $|t_{+}{|}^2$ as a function of an external magnetic field for fixed frequencies as indicated. Resonance features for positive and negative fields correspond to electrons and holes, respectively. Points: experiment; solid lines: theoretical model based on Drude conductivity, Eq. (3). The absolute scales refer to the lowest curves, the others are shifted for clarity. The inset (d) shows the field dependence of the cyclotron resonance demonstrating linear behavior within the quasiclassical approximation according to Eqs. (1) and (2). (e) The oscillating part of the longitudinal resistivity at zero gate voltage. The experimental data (black circles) were fitted by a single-carrier Lifshits-Kosevich model (orange line).

Figure 3

Electrodynamic parameters of the cyclotron resonances in HgTe. (a)–(c) Holelike carriers, (d)–(f) electronlike carriers. Only the most relevant resonances which may be explained via a quasiclassical picture are shown. Colored symbols are experimental data from the fits of the spectra in Fig. 2. The black circle corresponds to the density $\frac{n_{\mathrm{SdH}}}{2}$ resulting from the Shubnikov–de Haas (SdH) analysis from Fig. 2. The lines are guides to the eye.

Figure 4

Comparison of the cyclotron masses in strained HgTe with $\mathbf{k}·\mathbf{p}$ model calculations. Compared to Fig. 3, the cyclotron masses are plotted as a function of density. This presentation allows the comparison with the theoretical model without integrating Eq. (2). (a) Holelike carriers. (b) Electronlike carriers. Full symbols: experimental values; empty symbols: theory; BS: bottom surface states; TS: top surface states; C1, C2: spin-polarized bulk conduction bands; CNP: charge neutrality point.

Figure 5

Band structure of the three-dimensional topological insulator based on a strained HgTe along the (1,1) direction. Symbols: experimental data obtained from cyclotron mass; solid lines are predictions of the $\mathbf{k}·\mathbf{p}$ model for three values of the total charge density: (a) hole doping: $n_{\mathrm{tot}}=4\times {10}^{11}{\mathrm{cm}}^{-2}$, (b) charge neutrality:- $n_{\mathrm{tot}}=0$ and (c) electron doping: $n_{\mathrm{tot}}=-4\times {10}^{11}{\mathrm{cm}}^{-2}$. Areas highlighted in yellow present the regions where it is valid to compare experimental results with theory.

Figure 6

Hartree potential and spatial distribution of the wave functions for the top and bottom surface states. Hartree potential (left axis) at neutral, electron, and hole dopings that is self-consistently determined as described in Sec. 3. The spatial probability distribution (right axis) of the surface states at $n_{\mathrm{tot}}=0$ is superimposed on the Hartree potential.