Magnetospectroscopy of two-dimensional HgTe-based topological insulators around the critical thickness

Recently, a new class of materials, so-called topological insulators, has emerged. These are systems characterized by the inversion of the electronic band structure and also by a certain strength of the spin-orbit interaction. HgTe/Cd${}_x$Hg${}_{1-x}$Te quantum wells represent a prominent example. They can change to the topological insulator phase from the conventional insulator phase when the thickness of the quantum well is increased over the critical thickness $d_c$ = 6.3 nm. Here, we report on a far-infrared magnetospectroscopy study of a set of HgTe/Cd${}_x$Hg${}_{1-x}$Te quantum wells with different thicknesses from below to above the critical value $d_c$. In quantizing magnetic fields up to 16 T, both intraband and interband transitions have been clearly observed. In the widest quantum well with inverted band structure, we confirm the avoided crossing of the zero-mode Landau levels observed earlier in similar structures. In both noninverted quantum wells close to the critical thickness, we report unambiguously on the square root dependence of the transition energy on the magnetic field, as expected in the single-particle model of massless Dirac fermions. The obtained results are compared with the allowed transition energies between Landau levels in the valence and conduction bands calculated using the 8 × 8 Kane model.

Figure 1

Schematic drawing of the quantum wells showing the cadmium rate ($x$) in the samples from the buffer to the surface. $d$ is the QW thickness.

Figure 2

(a) Hall resistance measured for three different pairs of Hall contacts. Consecutive numbers denote plateau indices. Insert shows the geometry of the microstructure patterned on A for transport measurements. The wire ends are connected to large contact pads (not shown here). Indium contacts are soldered to the pads. (b) Longitudinal resistance $R_{xx}\equiv R_{82,63}$, as compared with the nonlocal resistance $R_{\mathrm{nloc}}\equiv R_{12,610}$ ($R_{12,610}$ denotes that the current is passed between probes 1 and 2, and the voltage is measured between 6 and 10).

Figure 3

Calculated dispersion of Landau levels for sample A. The expected dominant absorption transitions are denoted by arrows and greek letters. Depending on the LL filling factors, the α, α′, β, and γ lines should be observable in all samples with band inversion.

Figure 4

Measured transmission spectra of sample A in a relevant range of magnetic fields. The α transition shows a double peak behavior close to a critical magnetic field $B_c$ = 6 T. The spectra are vertically shifted for the sake of clarity.

Figure 5

Peak positions of the observed transitions in sample A, plotted as a function of the applied magnetic field. The shaded areas are the reststrahlen bands. Solid lines show the expected excitation energies following the calculation presented above with adjusted parameters. Dashed lines show the same energies for parameters from Ref. 10. The α transition (squares and circles) shows a double component character in the vicinity of the crossing fields $B_c$. The intraband β transition (stars) is also seen above 8 T. The cyclotron resonance (CR) absorption is linear with $B$. The last transition visible from 4 up to 11 T is usually denoted as $\gamma$ transition (triangles). The full symbols correspond to experimental points obtained at higher magnetic fields in Montpellier Teralab.

Figure 6

Peak positions of the observed transitions in sample B, as a function of the applied magnetic field. The double component of the α transition is not seen close to the expected crossing field $B_c$. The intraband β transition is seen above 4.5 T and the cyclotron resonance (CR) absorption is linear with $B$. The $\gamma$ transition is also visible from 4.5 to 11 T.

Figure 7

Calculated dispersion of Landau levels for sample C. The expected dominant absorption transitions are denoted by arrows and greek letters. The α, $\alpha -$, β, β−, and γ− lines have energies that are in the spectral range of our measurements.

Figure 8

Measured transmission spectra of sample D. The notation of the observed lines is in agreement with the transitions assignment used in Fig. 7. U${}_1$ corresponds to an unknown transition line. The shaded areas are the reststrahlen bands. The spectra are vertically shifted for the sake of clarity.

Figure 9

(a) Peak positions of the observed transitions in sample D. The β (empty stars)$,$ and $\alpha -$ (empty pentagons) transitions are observed in good agreement with theory represented by the solid lines. Two other transitions called U${}_1$ and U${}_2$ are observed (represented, respectively, by empty triangles and squares). (b) Peak positions of the transitions as a function of the square root of the magnetic field. The U${}_1$ transition exhibits a square root dependence on B up to 4 T.

Figure 10

(a) Peak positions of the observed transitions in the noninverted sample C. The $\alpha -$ (pentagons) and β (stars) transitions are observed in good agreement with theory represented by the straight lines. Another transition called U${}_1$ represented with triangles is observed at low energies. (b) Peak positions of the transitions as a function of the square root of the magnetic field. The U${}_1$ transition exhibits a square root dependence on B up to 5 T.