Dirac-screening stabilized surface-state transport in a topological insulator

We report magnetotransport studies on a gated strained HgTe device. This material is a threedimensional topological insulator and exclusively shows surface state transport. Remarkably, the Landau level dispersion and the accuracy of the Hall quantization remain unchanged over a wide density range ($3 \times 10^{11} cm^{-2}<n<1 \times 10^{12} cm^{-2}$). This implies that even at large carrier densities the transport is surface state dominated, where bulk transport would have been expected to coexist already. Moreover, the density dependence of the Dirac-type quantum Hall effect allows to identify the contributions from the individual surfaces. A $k \cdot p$ model can describe the experiments, but only when assuming a steep band bending across the regions where the topological surface states are contained. This steep potential originates from the specific screening properties of Dirac systems and causes the gate voltage to influence the position of the Dirac points rather than that of the Fermi level.


Introduction
The discovery of two-(2D) [1][2][3][4][5] and three-dimensional (3D) topological insulators (TIs) [6,7] has generated strong activity in the condensed matter community. A main difficulty concerning 3D topological insulators is the intrinsic doping, which does not allow to selectively access the surface states in a transport experiment. Although Bi 2 Se 3 and Bi 2 Te 3 are at the focus of current research, their transport properties are dominated by parallel bulk conductance due to intrinsic bulk doping in these systems. Alternatively, thick ( 40 nm) layers of HgTe, which have extremely low background doping, also become topological insulators when epitaxially grown under coherent strain [6,8]. While unstrained HgTe is a semimetal, which is charge-neutral when the Fermi energy is at the touching point between the light-and heavy-hole Γ 8 bands, a band gap of around 20 meV opens at the Γ-point when the material is grown epitaxially on a CdTe substrate, which has a lattice constant that is 0.3 % larger than that of HgTe.
We have recently shown [8] that such a strained HgTe layer exhibits a quantized Hall conductance, providing evidence that the topological surface state dominates the transport in such structures. Here, we report on experiments on new devices where the density is controlled by an external gate voltage. Since top and bottom surfaces are differently affected by the top gate, gating allows us to obtain the same densities on both surfaces where the only-odd quantum Hall plateaus expected from a Dirac band structure are clearly resolved.
Much more strikingly, we find that the quantum Hall response is totally dominated by the Dirac-like surface states for a very wide range of gate voltages (and thus carrier densities), much larger than expected for a small band gap material like strained HgTe. The absence of any notable bulk conductance has obvious and very positive implications for studying superconducting and magnetic proximity effects in strained HgTe. We tentatively explain our observations as resulting from the unusual screening characteristics of a Dirac band structure [9,11], which strongly modifies the band bending across the heterostructure.
Using a 6-band k·p description with an appropriately adapted potential, we find that the band bending allows keeping the chemical potential in the band gap for all experimentally accessible gate voltages, mainly by shifting the Dirac points of the surface states.

Experiment
The experimental results are obtained on a 70 nm thick HgTe layer which is grown fully strained on a CdTe [001] substrate. The sample is structured into a standard Hall bar geometry of 200 µm width and a distance of 600 µm between neighboring Hall contacts.
The entire sample is covered by a multilayer insulator consisting of 11 alternating, 10 nm thick SiO 2 and Si 3 N 4 layers. A 10/100 nm Ti/Au electrode stack is deposited on top of the insulator. A gate voltage V g between typically +5 V and -5 V can be applied to control the carrier density in the HgTe layer. Fig. 3 a) shows a micrograph of the sample. Magnetotransport measurements in fields up to 16 T have been done in a dilution refrigerator system at base temperature of ≈ 20 mK. For gate voltages between -1 V and +5 V the sample exhibits n-type conductance. In this regime we observe clear quantum Hall plateaus and corresponding Shubnikov-de-Haas oscillations. For gate voltages below -1 V the Hall slope changes sign, indicating the onset of p-type conductance. In this gate voltage region, we can no longer observe clear quantum Hall plateaus for this sample, presumably due to a reduced mobility of the charge carriers. The longitudinal magneto-resistance traces of Fig. 1 b) -some of which are plotted in more detail in Figs. 2 a) -c) -provide additional information on the properties of the surface states. One notes, especially for V g = -0.5 V and 0 V, that the high field Shubnikov-de Haas oscillations display an alternating width. The straightforward interpretation is that they result from two different 2DEGs with a different mobility. One expects a lower mobility for the carriers near the gate oxide, where roughness and ionized dopants will enhance the carrier scattering rate, than for carriers that are further removed from the surface. The observation thus implies that the top and bottom surface states behave as more or less independent carrier systems; if they were strongly coupled the mobilities should be equal. observations, we conclude that the transport is completely dominated by the surface states even at carrier densities above 10 12 cm −2 . Additionally, we find that the accuracy of the quantum Hall quantization, while not perfect, is independent of gate voltage over the whole range studied. Fig. 3 c) shows this behavior in more detail. This again implies that the surface transport is not perturbed and bulk transport can be neglected over the complete gate voltage range.
Further means to substantiate that we almost exclusively observe transport through both top and bottom surface states comes from Fig. 3  in general different densities. The obtained fan charts contain more information on the character of these 2DESs. For example, the two fans exhibit nearly equidistant spacing at fixed magnetic field. This is true for all magnetic fields, also for very low ones where one might expect deviations due to an unresolved Zeeman splitting. We conclude that the data show no indication of an additional spin splitting. Both fans represent a single spin species, as expected for the Dirac surface states of a topological insulator. Furthermore, by labeling the plots at high magnetic field with the observed quantum Hall index number, we can reconstruct the Landau level crossings and filling factors at lower magnetic fields and for all gate voltages. For V g = −1 V we recover the sequence of odd filling factors discussed above, which is evidence of the Dirac character of the 2DESs. Moreover, we find that the slopes of the various Landau level dispersions scale with filling factor, meaning that our identification of the fan chart is reasonable. Finally, for the same Landau index, the two fans exhibit a different slope. This is readily interpreted in terms of the gating efficiency. While for the red fan a small change in V g leads to a strong change of the related carrier density, the blue fan is much less affected. All of these observations indicate that we indeed observe the quantum Hall effect of two separate Dirac surfaces, one close to the control gate (top surface) and one at a larger distance (bottom surface). Further, from the slopes of the two fans in Fig. 3  in studies on double 2DESs by Eisenstein and co-workers [13]. The model of Ref. [12] actually allows for similar compressibilities for both layers (as has been pointed out in Ref. [14]), but for our device (where the density change of top and bottom surface differs by a factor around 0.5) this would imply a lower limit for the HgTe dielectric constant of = 400, very far removed from the literature value = 21 [10]. We will show below that the observed behaviour can be explained by the presence of steep potentials across the structure, resulting from the unusual screening by a Dirac metal.

Modeling
In order to substantiate our interpretation of the data in Figs. 1 -3 we set up a calculation along the lines of the 6 band k·p theory of Ref. [15] omitting the split-off band which is energetically well separated from other bands. In our calculations we consider only the 70 nm strained HgTe layer and we assume hard wall boundary conditions in growth direction.
We verified that for our system hard wall boundary conditions do not give spurious solu-tions by comparison with infinite mass boundary conditions [16]. Because of the remarkable gating behavior observed in Fig. 3 d), we do not attempt to obtain a self-consistent Hartree potential as is usually done in modeling quantum wells. In fact, a self-consistent k·p calculation using the methods of Ref. [15] only yields a Fermi energy within the band gap for V g 0.5 V, in clear disagreement with the experiment. Instead, we have phenomenologically searched for an effective potential that keeps the electrochemical potential in the band gap for all gate voltages, gives an equal density for both 2DESs at V g = -1 V (i.e. the potential is symmetric at this gate voltage), and depletes both 2DESs in a similar manner as in Fig. 3 d). The resulting potential is shown in Fig. 4 d). The shape of this potential can tentatively be understood as arising from electrostatic decoupling of surface and bulk states resulting from different dielectric constants in these regions. For the bulk we have = 21, as appropriate for HgTe [10], and the potential in the region of the surface states would result from a much smaller effective dielectric constant around = 3. This number actually appears very reasonable for the surface state regions where the screening occurs in a Dirac-type band dispersion. A similar value was predicted [9] and observed [11] for graphene, where again the Dirac band structure causes unusual screening behavior. Evidently, due to the interplay of bulk and metallic states, topological insulators are more complicated than graphene. To confirm our suggestion that Dirac screening is responsible for the remarkable band bending in our devices, further microscopic modeling of the structure using the Random Phase Approximation outlined in Ref. [9] is needed.
We further simplify the model by neglecting charge transfer between top and bottom surface (e.g. through the side surfaces). Finally, we assume that the Dirac points are located in the valence band, at positions around 30-50 meV below the band edge (for V g = 0 V), consistent with our previous ARPES and THz spectropy data [8,17] as well as with the atomic tight binding calculations of Ref. [18]. Within this model, we find that the experimentally. For V g = 5 V, the Fermi level has hardly changed and is still in the gap, however the surface states are no longer degenerate and differ strongly in density. Note that because of hybridization with the valence band states, the position of the Dirac points is not directly obvious from the dispersion plots. The location of the charge carriers in the HgTe layer can be inferred from Fig. 4 c) where the probability densities of the two surface state wave functions (at the Fermi level) are plotted for two different electron densities, again corresponding to V g = −1 V and V g = 5V.

Conclusion
We investigated the effect of gating on a 3D TI system and demonstrated its extraordinary screening properties. We obtained detailed information about the individual surface states by the analysis of the n-type quantum Hall effect. We found that for the entire range of gate voltages only the Dirac surface states are visible in transport due to to their remarkable screening properties. This makes strained HgTe an ideal playground for investigating magnetic and superconducting [19] proximity effects on the topological surface state.