Quantum Hall effect and Landau levels in the three-dimensional topological insulator HgTe

We review low- and high-field magnetotransport in 80-nm-thick strained HgTe, a material that belongs to the class of strong three-dimensional topological insulators. Utilizing a top gate, the Fermi level can be tuned from the valence band via the Dirac surface states into the conduction band and allows studying Landau quantization in situations where different species of charge carriers contribute to magnetotransport. Landau fan charts, mapping the conductivity ${\sigma }_{xx}\left(V_g,B\right)$ in the whole magnetic field–gate voltage range, can be divided into six areas, depending on the state of the participating carrier species. Key findings are (i) the interplay of bulk holes (spin degenerate) and Dirac surface electrons (nondegenerate), coexisting for $E_F$ in the valence band, leads to a periodic switching between odd and even filling factors and thus odd and even quantized Hall voltage values. (ii) We found a similar though less pronounced behavior for coexisting Dirac surface and conduction band electrons. (iii) In the bulk gap, quantized Dirac electrons on the top surface coexist at lower B with nonquantized ones on the bottom side, giving rise to quantum Hall plateau values depending—for a given filling factor—on the magnetic field strength. In stronger $B$ fields, Landau level separation increases; charge transfer between different carrier species becomes energetically favorable and leads to the formation of a global (i.e., involving top and bottom surfaces) quantum Hall state. Simulations using the simplest possible theoretical approach are in line with the basic experimental findings, describing correctly the central features of the transitions from classical to quantum transport in the respective areas of our multicomponent charge carrier system.

Figure 1

(a) Scheme of the heterostructure cross section. (b) Sketch of the 200-µm-wide and 1100-µm-long Hall bar. (c) Longitudinal resistivity ${\rho }_{xx}\left(B\right)$ at different $V_g$ for $E_F$ in the conduction band (2, 1.5 V), gap (1 V), and valence band (–0.5, −1.5 V). (d) $V_g$ dependence of ${\rho }_{xx}$ (left axis, light blue) at $B=0$ and ${\rho }_{xy}$ (right axis, dark blue) at $B=0.5\mathrm{T}$. The vertical arrows show the charge neutrality point (CNP), top of the bulk valence ($E_V$), and bottom of the conductance band ($E_C$), respectively. Here, $E_V$ and $E_C$ mark the gate voltage at which bulk hole and bulk electron densities vanish, respectively. (e) Electron $n_s\left(V_g\right)$ and hole densities $p_s\left(V_g\right)$ extracted from the Hall slope, two-carrier Drude model, and SdH oscillations. The analysis follows Ref. [13] and yields positions of the valence and conduction band edges at $V_g\left(E_V\right)=0.45\mathrm{V}$ and $V_g\left(E_C\right)=1.2\mathrm{V}$, respectively, marked in (d). (f) Pictorial of the capacitively coupled layers in the 3D TI sample which form the basis of our electrostatic model. (g) Electron $n_s\left(V_g\right)$ and hole densities $p_s\left(V_g\right)$ as calculated with our electrostatic model. The densities which can be directly compared with (e) are shown as solid lines, the ones which are not directly accessible as dashed lines.

Figure 2

(a) Schematic band structure of strained HgTe. Conduction and valence band edges are marked by $E_C$ and $E_V$, the Dirac point, which is located in the valence band, by $E_{\mathrm{DF}}$. The topological Dirac surface states are shown in blue. Note that the position of $E_F$ within the band structure is usually different for top and bottom surfaces. (b) The quantum Hall effect in strained 80-nm HgTe, shown for different gate voltages $V_g$, can stem from surface electrons in the gap (with additional contributions from bulk electrons in the CB) and bulk holes in the VB. This is reflected by the changing sign of the Hall slope when tuning the gate voltage $V_g$ across the charge neutrality point.

Figure 3

(a) Color map of the normalized longitudinal conductivity ${\sigma }_{xx}\left(V_g,B\right)$ for magnetic fields up to $B=12\mathrm{T}$. Conductance maxima are yellow and green; filling factors for the bluish energy gaps are shown in white. Data are normalized to the average value for a given $B$ value. We use a logarithmic color scale to enhance the visibility of features throughout the entire investigated parameter space. (b) Excerpt of the data in (a), shown as the second derivative ${\partial }^2{\sigma }_{xx}/\partial V_g^2$. Yellow lines correspond, as above, to Landau levels, and green and blue regions to Landau gaps. White dashed lines separate the investigated parameter space $\left(V_g,B\right)$ into distinct sections discussed in the text. (c) Simulated density of states based on the simple model described in the Appendix. Maxima (minima) of the DoS are shown in yellow (blue) and correspond to maxima (minima) of ${\sigma }_{xx}\left(V_g,B\right)$, i.e., to LL positions (gaps). $E_V$, $E_C$, CNP, and $E_{\mathrm{DF},\mathrm{top}}$ label points on the gate voltage axis, which were matched to the experimentally obtained values. White lines represent the fan chart originating from the CNP.

Figure 4

When a gate voltage is applied, the charge carriers of the system rearrange. While the exact solution requires a complicated self-consistent solution of Schrödinger and Poisson equations, most of the features observed in magnetotransport can be explained by a simple electrostatic model. Within this model, each group of carriers is modeled by a 2D system of electrons or holes of zero thickness. Each layer generates its own set of Landau levels emerging at different energies, but with common Fermi level position. Charged 2D surfaces cause, like a charged capacitor plate, 1D electric fields across the structure leading to electrostatic potentials, depending linearly on the charge. These potentials change with gate voltage, thus shifting the band edges, from which the Landau levels emanate. (a) Schematic band diagrams of the investigated HgTe 3D TI system at different gate voltages $V_g$ (VB: $V_g=\text{–}1.5\mathrm{V}$; CNP: $V_g=0.25\mathrm{V}$; gap: $V_g=0.9\mathrm{V}$; CB: $V_g=1.5\mathrm{V}$). Occupied hole states are shown in red, occupied states of surface electrons in dark blue, and bulk states in orange. The Dirac points of top and bottom surfaces in strained HgTe thin films are located below $E_V$, as shown in the diagrams. (b) Landau level spectra at different bias conditions used in the model. The corresponding band bending is shown above in (a). The model calculations include Zeeman splitting for spin-degenerate bulk states (orange for conduction band; red for valence band states). Landau fans that do not cross the Fermi energy $E_F$ (i.e., empty levels) are shown as dotted lines.

Figure 5

(a) ${\partial }^2{\sigma }_{xx}/\partial V_g^2$ for $B$ up to 3 T and $V_g>0$ covering the bulk gap and conduction band region. Dark blue lines retrace the Landau levels. The Landau fan in the gap between $V_g\left(E_V\right)$ and $V_g\left(E_C\right)$ has its virtual origin at $V_g=0.05\mathrm{V}$ and stems from the top surface electrons. A distinct change of slope is observed when $E_F$ enters the bulk bands at $V_g\left(E_V\right)$ and $V_g\left(E_C\right)$. This is due to the reduced filling rates of surface electrons in the bulk bands. The distortion of the Landau fan in region 4 at fields between 1 and 2 T is ascribed to the onset of quantization of the bottom surface electrons. (b) ${\partial }^2{\sigma }_{xx}/\partial V_g^2$ for $E_F$ in the valence band, i.e., for $V_g<V_g\left(E_V\right)$ and $B$ up to 2 T. Landau fans stemming from bulk holes (red) and top surface electrons (black) coexist in this regime. Whenever a bulk hole LL crosses a surface electron LL, the total filling factor parity changes, resulting in a shift of the SdH phase. (c) Hall conductivity ${\sigma }_{xy}\left(V_g\right)$ for $B=0.4...5\mathrm{T}$ and $E_F$ in the bulk gap. For small $B$ fields the plateau values of the individual traces increase with increasing field (dashed lines) due to superposition of quantized Hall conductivity from the top surface (constant ${\sigma }_{xy}$) and classical Hall conductivity of back surface electrons (${\sigma }_{xy}$ linear in $V_g$). At higher $B$ quantized steps appear. (d) The Hall conductivity ${\sigma }_{xy}\left(V_g\right)$ measured in the valence band. In the region of the checkerboard pattern alternating sequences of plateaus with only odd or even filling factors (dashed lines) show up. The changes in filling factor parity stem from coexisting spin-resolved topological surface states and spin-degenerate bulk holes. (e) Enlarged region of coexisting electron and hole Landau levels of (b) showing the alternating sequences of even (odd) total filling factors.

Figure 6

(a) Calculated partial DOS of top and bottom surface states (extracted from the full calculation described in the Appendix) near the bulk gap and for $B\le 3\mathrm{T}$. Yellow stands for high DOS values (i.e., LLs) while the blue color represents small DOS (i.e., the gaps between LLs). $E_V$, $E_C$, CNP, and $E_{DF,\mathrm{top}}$ label points on the gate voltage axis, which were matched to experimentally obtained values. The observed LLs exhibit, as in experiment, pronounced kinks at the band edges $(E_V,E_C)$ resulting from the abrupt change of the partial filling rates when bulk states get filled. (b) The calculated total DOS in the valence band, corresponding to the experiment shown in Fig. 5 shows, for low $B$, an unusual pattern stemming from the interplay of spin-degenerate hole LLs (red lines) and nondegenerate electron LLs (blue lines). Color code as in (a). (c) Sketch of the crossing electron (Dirac fermion) (blue) and hole LLs (red). When $E_F$ crosses a Dirac fermion LL, the filling factor changes by 1; when it crosses a hole LL it changes by 2. The black numbers give the resulting filling factors in between electron and hole LLs. (d) ${\sigma }_{xy}$ as a function of $V_g$ calculated for the different charge carrier fractions when degenerate and nondegenerate hole and electron LLs, as in (b), coexist. The total Hall conductivity ${\sigma }_{xy}$ (black line) shows, as in experiment, the switching between odd and even plateau values, depending on the parity of the electron filling factor. (e) The simulations shown here only consider the electrostatics of the LLs resulting in degenerate LLs at crossing points. However, quantum mechanics requires that such crossings be avoided. This situation is sketched here to show that in our model LL crossings lead (incorrectly) to maxima in the DOS while in experiment such crossings are characterized by a reduced density of states at the crossing point (black circle).