Research Article

Electron–hole asymmetry of the topological surface states in strained HgTe

Andreas Jost, Michel Bendias, Jan Böttcher, Ewelina Hankiewicz, Christoph Brüne, Hartmut Buhmann, Laurens W. Molenkamp, Jan C. Maan, Uli Zeitler, Nigel Hussey, and Steffen Wiedmann
PNAS March 28, 2017 114 (13) 3381-3386; first published March 9, 2017; https://doi.org/10.1073/pnas.1611663114

  1. Edited by Subir Sachdev, Harvard University, Cambridge, MA, and approved February 6, 2017 (received for review July 15, 2016)

Significance

Topological insulators possess metallic surfaces states that are generally perceived as electrons and holes with a linear symmetric dispersion around the Dirac point. In this work, we show that this symmetry is significantly distorted in the 3D topological insulator-strained HgTe. In thermopower experiments, we show a distinctively different behavior of surface electrons and holes that originates from a strongly asymmetric dispersion of surface states. Nonetheless, the surface states themselves remain topologically protected as evidenced by the observation of an ambipolar surface quantum Hall effect and quantum oscillations in the thermoelectric response. This observation shows that the physics of topological surface states in 3D topological insulators is far richer than previously envisaged.

Abstract

Topological insulators are a new class of materials with an insulating bulk and topologically protected metallic surface states. Although it is widely assumed that these surface states display a Dirac-type dispersion that is symmetric above and below the Dirac point, this exact equivalence across the Fermi level has yet to be established experimentally. Here, we present a detailed transport study of the 3D topological insulator-strained HgTe that strongly challenges this prevailing viewpoint. First, we establish the existence of exclusively surface-dominated transport via the observation of an ambipolar surface quantum Hall effect and quantum oscillations in the Seebeck and Nernst effect. Second, we show that, whereas the thermopower is diffusion driven for surface electrons, both diffusion and phonon drag contributions are essential for the hole surface carriers. This distinct behavior in the thermoelectric response is explained by a strong deviation from the linear dispersion relation for the surface states, with a much flatter dispersion for holes compared with electrons. These findings show that the metallic surface states in topological insulators can exhibit both strong electron–hole asymmetry and a strong deviation from a linear dispersion but remain topologically protected.

Topological insulators (TIs) possess metallic surface states that display a Dirac-type dispersion relation, whereas they are insulating in the bulk (1⇓⇓–4). Despite enormous experimental effort in recent years, access to the topological surface states (TSSs) and the ability to distinguish them from bulk contributions in transport experiments remain a significant challenge. The investigation of their transport properties in the bismuth-chalcogenides compounds, for example, has been hindered by excessive electron doping, which shifts the Fermi energy into the bulk conduction band (5, 6). This energy shift can be circumvented by adding additional hole doping to bring the Fermi energy back into the band gap, and indeed, a surface quantum Hall effect (SQHE) has been observed in nearly stoichiometric BiSbTeSe2 (7).

The predicted linear dispersion of TSSs on 3D TIs is reminiscent of graphene, the prototype 2D Dirac material. Magnetothermoelectric transport experiments on graphene have shown that Sxx as a function of the applied gate voltage Vg is in quantitative agreement with the semiclassical Mott formula (8⇓–10) (i.e., thermopower is symmetric and diffusion-driven for electrons and holes), and no signatures of phonon drag have been observed, implying that electron–phonon coupling is weak in graphene (11). Although bismuth-chalcogenides are widely investigated thermoelectric materials for refrigeration and power dependence at room temperature (12), only a few studies to date address the thermopower of TSS, for example, on Bi2Se3 (13) and (Bi1−xSbx)Te3 (14). In Bi2Se3, the gate-dependent zero-field thermoelectric power is found to be in agreement with the Mott relation near the charge neutrality point at low temperatures (13). Quantum oscillations in Sxx have been observed in Bi2Te3, but their interpretation has been hindered by the presence of a significant bulk contribution (15).

In contrast to bismuth-chalcogenide compounds, where both bulk and surface states are present, strained HgTe is the material of choice to investigate transport properties that are unique to the TSS. Strained HgTe is another 3D TI with a relatively small band gap, which is naturally undoped, and can be grown with a very high quality, exceeding the mobility of comparable Bi-based systems by more than one order of magnitude (16⇓–18). This extreme purity coupled with the absence of any notable bulk conductance lead to a quantum Hall (QH) response that is dominated by the Dirac-like surface states (17). The presence of TSS is unambiguously shown here by the observation of ambipolar SQHE. Moreover, we explore in detail the nature of the TSSs in strained HgTe through low-temperature measurements of resistance and Seebeck and Nernst effect in a high magnetic field B on either side of the Dirac point. At B = 0, we show that thermopower is dominantly diffusion-driven for surface electrons in agreement with the results on Bi2Se3 (13) and graphene (8⇓–10). In contrast, thermopower for surface holes shows a significant phonon drag contribution. Accompanying band-structure calculations reveal the origin of this electron–hole asymmetry in the thermoelectric response.

Results and Discussion

Ambipolar SQHE.

In Fig. 1 A–C, we illustrate the observation of the ambipolar SQHE accompanied by Shubnikov–de Haas oscillations in the resistance Rxx at different Vg. We use a top gate to tune the charge-carrier concentration n from the electron region (n≃ 1.4⋅1012 cm−2) through the charge neutrality point deep into the hole regime (p≃−1.3⋅1012 cm−2). The observation of ambipolar SQHE is definitive evidence for 2D transport that originates in 3D TIs exclusively from the surface states. The total charge-carrier concentration extracted from the slope of the low-field Hall resistance, nHall, depends linearly on the gate voltage (SI Text). From the zero-field resistivity and nHall, we extract a carrier mobility at 0.35 K from 1.5⋅105 to 3.0⋅105 cm2/Vs for electrons and from 2.0⋅104 to 9.0⋅104 cm2/Vs for holes.

Fig. 1.
Fig. 1.

Electrical transport and ambipolar SQHE in strained HgTe. (A–C) Quantum oscillations in Rxx (black lines) and SQHE in Rxy (red lines) for different charge-carrier concentrations (several integer filling factors are labeled) at T = 0.35 K. In B, the hole charge-carrier concentrations of both surfaces are approximately equal. (D) Rxx as a function of T at a gate voltage Vg= 0 V. The metallic TSSs dominate transport for T< 65 K. (Inset) Extraction of the thermal activation gap lnRxx (1/T) of 16 meV (red line) and sample layout with the Hall bar in light green and gate and contacts in gold. B, magnetic field; Rxx, longitudinal resistance; Rxy, Hall resistance; T, temperature; T, tesla; Vg, gate voltage.

Because of the Dirac nature of charge carriers, TSSs exhibit the half-integer QH effect, where the Hall conductivity is quantized as σxyt(b)=νt(b)e2/h with νt(b)=(Nt(b)+ 1/2) as the filling factor for each surface and Nt(b) as the Landau-level index of the top (bottom) surface. The quantized Hall conductivity is accompanied by a zero conductivity σxx = 0. Therefore, the total Hall resistance is Rxy= 1/σxyt(b)=h/(Nt+Nb+ 1)e2 (7, 16⇓–18). In general, when the charge-carrier concentrations of the top and bottom surfaces are different, both odd and even integer QH plateaus appear with zeros in Rxx (Fig. 1 A, electrons and C, holes). When the charge-carrier concentrations of the surfaces are approximately equal, however, only odd integer QH plateaus [i.e., Rxy=h/(2N+1)e2] are observed (Fig. 1B) because of the existence of two degenerate Dirac systems. A more comprehensive transport analysis on another sample is presented in SI Text and illustrated in Fig. S3.

Fig. S3.
Fig. S3.

(A) Gate-voltage dependence of charge carrier concentrations extracted from the low-field Hall-resistance (nHall, black open squares) and from FFT analysis of the quantum oscillations due to the bottom (nb) and top (nt) surfaces (green and red triangles), respectively. The sum of two surface concentrations (nb+nt) matches the concentration from low-field Hall-resistance (blue dots). The lines serve as a guide to the eye. (B) Landau level index N as a function of 1/B extracted from SdH minima; the LL diagram intersects N at −1/2 as expected for Dirac fermions. (C) ρxx as a function of 1/B at Vg=−4 V for nb=nt. (D) FFT analysis for surface holes. B, magnetic field; meV, milli electron volt; ρxx, resistivity.

The dominance of the metallic surface states in the low-temperature resistance is further exemplified in Fig. 1D, where we show the temperature dependence of Rxx at Vg = 0. With decreasing temperature, Rxx increases strongly down to ∼65 K because of thermally activated bulk carriers (the Arrhenius plot in Fig. 1D, Inset gives an activation gap of Δ≃ 16 meV) that are still dominant. For T< 65 K, Rxx becomes dominated by the metallic surface states apparent in a decrease of the resistance that saturates at low temperatures.

Thermoelectric Coefficients.

Having established unambiguously the dominance of TSS in the magnetotransport, we now turn to investigate the thermoelectric response of our films. Thermopower, also referred to as the Seebeck effect Sxx, is the voltage that arises when a thermal gradient is applied along the sample. This voltage is needed to compensate for the thermally driven electron current, and thus, it does not depend directly on the scattering time τ but does depend on the derivative of τ with respect to the energy dτ/dε∝dσ/dε at the Fermi level, where σ is the conductivity (19). Thus, thermopower compared with other transport properties, such as conductivity, is more sensitive to details of the band structure. In metallic systems, thermopower originates predominantly from two different mechanisms: diffusion and phonon drag. The former arises from the nonequilibrium of the Fermi–Dirac distribution of the electrons caused by a thermal gradient. In the latter, phonons travel down the heat gradient, displacing charge carriers in their wake. Independent of the mechanism, Sxx is almost exclusively negative for electrons and positive for holes (19). In a 3D TI, extra care needs to be taken to disentangle contributions from both surface- and bulk-derived states.

We first focus on the low-field electrical and thermal transport as a function of gate voltage as presented in Fig. 2. The longitudinal resistance Rxx as a function of Vg (Fig. 2A) has a pronounced maximum at Vg≃−1.35 V, whereas the Hall resistance Rxy crosses 0 at Vg=−1.85 V (vertical dashed line in Fig. 2A) for B = 0.2 T. For higher (lower) Vg, Rxy is positive (negative) pointing toward dominant electron (hole) charge carriers. We, therefore, assign the 0 crossing of Rxy at Vg=−1.85 V to the charge neutrality point of the surface states. At this point, Rxx is ≃85% of its value at the maximum.

Fig. 2.
Fig. 2.

Low-field electrical and thermal transport of strained HgTe at 1.2 and 1.5 K, respectively. (A) Gate voltage dependence of the resistance Rxx and Hall resistance at B = 0.2 T. The position of the Dirac point where Rxy crosses zero is marked by a vertical dashed line. (B) Gate voltage dependence of the thermoelectric power Sxx at B = 0 T. (C) Gate voltage dependence of the thermopower Sxx and Nernst effect Sxy at 0.2 T: a small hump in Sxx is observed, where a gap opens at the Dirac point (vertical line). B, magnetic field; Rxx, longitudinal resistance; Rxy, Hall resistance; Sxx, thermopower; Sxy, Nernst coefficient; T, temperature; Vg, gate voltage.

The thermopower Sxx as a function of Vg at B= 0 is shown in Fig. 2B. A change in sign in Sxx is observed at Vg = −0.3 V, corresponding to a transition from dominant electron to dominant hole contributions. At positive gate voltages, Sxx is negative—as expected for electrons—and constant over a wide range of gate voltages. Its magnitude is surprisingly small, however, with a value of only a few microvolts per kelvin, comparable with that seen in correlated high-density metals (20, 21). The zero crossing in Sxx occurs at a more positive value of Vg than the zero crossing of Rxy. Below Vg=−1.0 V, Sxx starts to increase strongly up to a maximum of 37 μV/K at Vg=−4.4 V. This observation raises an important question: why is the thermopower for holes so much larger than that for electrons?

To quantify the diffusion thermopower Sd for electrons and holes, we use the Mott formula, which is given bySd=π23kB2Tq1σdσdε|εF=π23kB2Tq1σdσdnD(Vg)[1]with kB as the Boltzmann constant, q as the electron/hole charge, n as the charge-carrier concentration, and D(Vg) as the density of states at a certain gate voltage measured from the Dirac point (13). Although the conductivity σ and dσ/dn can be extracted directly from our measurements, to obtain D(Vg), we need the effective mass of surface carriers, which can be obtained independently (e.g., from band-structure calculations as shown below).

Band-Structure Calculations.

Band-structure calculations were performed based on the six-band 𝑘⋅𝑝 approach (22) for a heterostructure comprised of Cd0.7Hg0.3Te/HgTe/Cd0.7Hg0.3Te. We use an effective Hartree potential in the spirit of the Dirac screening model as introduced for electron surface states in ref. 16. To account for the large range of carrier concentration observed, we primarily dope the Dirac surface state from the n- to the p-type carrier concentrations for a large range of gate voltages, whereas the structure of the bulk bands is weakly affected. This phenomenological approach is motivated by the disagreement between self-consistent Hartree calculations (23) and our magnetotransport results in the regime of the SQHE accompanied by Rxx = 0 as well as the fact that we are able to tune the charge-carrier concentration on the top and bottom surfaces by the top gate. More details on transport are presented in SI Text. In contrast to the approach in ref. 16, we consider an additional interface potential arising from a reduced point symmetry (24), which introduces a coupling between the light-hole (LH) and heavy-hole (HH) components, even at k= 0. The interface potential is adjusted such that the position of the Dirac point in the 𝑘⋅𝑝 model coincides approximately with ab initio calculations (25) and angular-resolved photo-emission spectroscopy results (16, 26), where it was found that, for ungated HgTe, the Dirac point is located several tens of millielectronvolts below the HH band edge.

The dispersion relations E(k) are shown in Fig. 3 for Vg= 1 V and Vg=−3 V. Vg= 1 V corresponds to a large electron density, where the densities on the top and bottom surfaces differ. As a result, odd and even plateaus are observed in the Hall resistance as shown in Fig. 1A. Vg=−3 V corresponds to a small hole density, with the densities on the top and bottom surfaces being approximately the same in this case. These equal densities give rise to only odd plateaus in the Hall resistance Rxy, similar to those observed in Fig. 1B. In the dispersion relation in Fig. 3, we clearly identify the TSSs. Remarkably, the chemical potential μ remains in the bulk band gap throughout the whole range of carrier concentrations in agreement with the experimental observation of SQHE, even for large hole carrier concentrations (Fig. 1C). One of the most striking features of the band dispersions of the TSS, however, is the strong departure from a strictly linear (Dirac-like) behavior caused by the strong coupling of the TSS with the HH subbands. Such a deviation from strict linearity was observed first in angular-resolved photoemission experiments on Bi2Se3 (27). Using our combined techniques, we are able to address the intrinsic charge-carrier properties across the Fermi level by tuning the top gate voltage Vg. From the dispersion relations in HgTe (Fig. 3), we obtain an effective mass of me≈0.02m0 for surface electrons at Vg= 1 V and mh≈ 0.11m0 at Vg=−3 V for surface holes, where m0 is the free electron mass. The thermopower is now calculated at Vg= 1 V to be Sd≃−0.7μV/K, which is in excellent agreement with the experimental data, suggesting that thermopower for surface electrons is purely diffusion-driven. For holes, however, we estimate the diffusion contribution to be Sd≃ 2μV/K, more than one order of magnitude smaller than the experimental value at Vg=−3 V.

Fig. 3.
Fig. 3.

Band-structure calculations based on the six-band 𝑘⋅𝑝 approach (bulk bands are in black, and TSSs are in red and blue). E(k) for (A) electrons at Vg = 1 V and (B) holes at Vg = −3 V when the TSSs are degenerate. The dashed lines represent the chemical potential μ. E, energy; meV, milli electron volt; Vg, gate voltage.

Phonon Drag Thermopower.

To gain more insight into the mechanisms governing Sxx, we investigate its temperature dependence at different gate voltages as shown in Fig. 4. At lower temperatures, the thermopower for electrons is small and independent on Vg, with a weak linear temperature dependence consistent with a pure diffusion-driven mechanism (Fig. 4, Inset). In contrast, Sxx for holes is much larger in absolute magnitude over a wide range of temperatures and exhibits a peak at around T = 12 K. Phonon drag thermopower is known to give rise to large signals (28), with a maximum when the phonon heat conductivity is largest (28), which is at 7 K for CdTe (substrate) (29) and 13 K for HgTe (30). This correspondence implies that the enhanced thermopower in the hole-dominated region indeed originates from a significant phonon drag contribution that intriguingly is absent on the electron side. The origin of this dichotomy is, we believe, in the heavier mass of the hole carriers that is reflected in the flatter band dispersion (31). The sharp drop in Sxx suggests that, at elevated temperatures, electrons are also excited to the bulk conduction band and begin to dominate the zero-field thermopower.

Fig. 4.
Fig. 4.

Temperature dependence of the thermopower Sxx at different charge-carrier concentrations (gate voltage Vg). The thermopower is small and nearly flat for electrons (Vg= 0 and 1.9 V) and large for holes with a phonon drag peak visible at 12 K at Vg=−2.6 and −6.1 V. Inset highlights the linear dependence of Sxx as a function of temperature for electrons for Vg = 0. Sxx, thermopower; T, temperature; T, tesla.

In the presence of a small magnetic field of B = 0.2 T (Fig. 2C), Sxx shows the same overall behavior as at B = 0 but develops a small hump at Vg = −1.85 V directly at the Dirac point (vertical dashed line in Fig. 2) that likely originates from a gapping of surface states at the Dirac point when a magnetic field is applied. The Nernst effect Sxy (i.e., the voltage perpendicular to the heat gradient) is a very sensitive probe of charge carriers in solids. Sxy is small in systems with a single type of charge carriers but large when electrons and holes coexist or in the presence of high-mobility charge carriers (32). In strained HgTe, Sxy is found to have a maximum at Vg = −1.85 V exactly at the zero crossing of Rxy. The low-field Nernst effect for electrons is small at positive gate voltages (Fig. 2C). On the hole side, Sxy decreases with increasing Vg but remains large compared with the electron side as found in Sxx. We note that, in gated samples, not only the parts of the sample underneath the gate but also, the n-doped legs of the Hall bar contribute to Sxy (Fig. S2). This contribution of the legs gives rise to a constant offset in the measurement when sweeping the gate voltage that does not invalidate our conclusion (SI Text).

Fig. S2.
Fig. S2.

Layout of the structure used for the measurements: the Hall bar is in light green, and the gate and contacts are in gold. The charge current and the temperature gradient are applied along the x direction. The legs run perpendicular to the heat gradient and give no additional contribution to Sxx. The gray lines represent the equipotential lines. Sxx, thermopower; Sxy, Nernst coefficient.

Seebeck and Nernst Effect in a Quantizing Magnetic Field.

In Fig. 5 A and B, thermopower and resistivity are plotted as a function of the magnetic field up to B = 16 T at T = 1.5 K at fixed charge-carrier concentrations for electrons (Vg = +3.0 V) and holes (Vg = −4.5 V). Pronounced quantum oscillations originating from both surface electrons and holes are observed.

Fig. 5.
Fig. 5.

Quantum oscillations in thermopower and resistivity at T = 1.5 K as a function of the magnetic field at different carrier concentrations. Sxx and Rxx at (A) Vg = +3.0 V (electrons) and (B) Vg = −4.5 V (holes). The oscillations in thermopower are shifted by π/2 with respect to the resistance. (C) The thermopower for holes at different carrier concentrations as a function of B shows the evolution from the odd integer filling factors (Vg = −4.5 V) at equal carrier concentrations on either surface to odd and even filling factors (Vg = −6.0 V). B, magnetic field; Sxx, thermopower; T, temperature; T, tesla; Vg, gate voltage; ρxx, resistivity.

The oscillations in thermopower mimic those in electrical resistivity (Fig. 5 A and B). Because thermopower is proportional to the derivative of the conductivity with respect to the energy (33), the oscillations are shifted by π/2. On the electron side (Fig. 5A), the minima are very deep and even reach zero at higher fields. Likewise, on the hole side (Fig. 5B), clear quantum oscillations are observed. At Vg = −4.5 V, we only observe oscillations in Sxx that correspond to odd filling factors as expected for two degenerate Dirac systems with equal carrier concentrations. With decreasing gate voltage, even filling factors appear as the charge-carrier concentrations of the two surfaces become different (Fig. 5C). Therefore, the observed sequence of quantum oscillations in Sxx as a function of B independently confirms the presence of two surfaces on a 3D TI.

Next, we focus on the oscillatory behavior of Sxx as a function of the magnetic field. When the magnetic field is increased, the chemical potential sweeps through the Landau levels, and Sxx has a maximum for half-filled levels. Diffusion thermopower is a direct measure of the entropy of electrons per unit charge density, and at low temperatures, it has been shown that, in the disorder free limit, the amplitudes of the quantum oscillations have universal values that depend only on their index N(≃ 40μ/K forN= 1) (34). For our system, we observe that the amplitude of the quantum oscillations of the TSS is considerably larger than the expected universal values for diffusion thermopower for both electrons and holes. Our observation is in agreement with conventional 2D systems, such as GaAs heterojunctions, and suggests that the phonon drag contribution plays an essential role in a magnetic field (35, 36).

Finally, to highlight the coexistence of electron and hole carriers, we have measured the Nernst effect Sxy and thermopower Sxx along with electrical transport in a moderate magnetic field of 9 T as a function of Vg as illustrated in Fig. 6. Pronounced quantum oscillations are observed in Rxx, thermopower Sxx, and the Nernst effect Sxy for both electrons and holes accompanied by the SQHE in Rxy, which shows clear plateaus at several integer filling factors. We find the zero crossing of Rxy (charge neutrality) and the maximum in Sxy at Vg=−1 V. The zero crossing in Sxx corresponding to the transition from electrons to holes occurs at a slightly higher Vg, which was found for the maximum in Rxx.

Fig. 6.
Fig. 6.

Magnetotransport and thermopower at B=9 T as a function of Vg. (A) Rxx and Rxy exhibit quantum oscillations and a quantized Hall resistance for both electrons and holes. (B) Sxx and the Nernst effect Sxy also show clear quantum oscillations. The maximum in Sxy corresponds to the zero crossing of Rxy. The numbers are the integer filling factors. arb, arbitrary; B, magnetic field; Rxx, longitudinal resistance; Rxy, Hall resistance; Sxx, thermopower; Sxy, Nernst coefficient; T, temperature; Vg, gate voltage.

In conclusion, we have presented ambipolar electrical and thermal transport properties in high-quality 3D TI-strained HgTe in a wide range of carrier concentrations. We find that the thermoelectric response is purely diffusion-driven for surface electrons, whereas for surface holes, it is dominated by a large phonon drag contribution. This marked asymmetry in the thermoelectric response makes the TSSs in strained HgTe distinct from other Dirac systems, such as graphene. In high magnetic fields, we have observed pronounced quantum oscillations for both surface electrons and holes accompanied by the SQHE. In band-structure calculations, we have confirmed the absence of any bulk transport in our system and shown that the asymmetry of the thermoelectric response derives from a corresponding asymmetry in the dispersion of the surface states. Despite the significant departure from a simple linear dispersion, the surface states remain topologically protected. This observation shows that the physics of TSS in 3D TIs is far richer than previously envisaged.

Materials and Methods

Samples Σ1 and Σ2 (SI Text and Fig. S1) are two different pieces of the same HgTe wafer with a thickness of 104 nm grown on a CdTe (001) substrate by molecular beam epitaxy. Sample Σ3 is a 60-nm HgTe layer that is sandwiched between two Hg0.3Cd0.7Te buffer layers on top (5 nm) and bottom (100 nm), and it is also presented in SI Text. CdTe has a slightly larger lattice constant than HgTe, inducing strain on the sample. The strain opens an energy gap of Eg≃ 22 meV between the 𝚪8 LH and the 𝚪8 HH bands, turning strained HgTe into a 3D TI. For samples thicker than ∼150 nm (37), the strain relaxes by dislocations. Sample Σ4, shown in SI Text, has a thickness of 600 nm and is, therefore, a 3D semimetal (see data in Fig. S4). All samples are patterned in Hall bar geometry with a mesa of 1.2-mm length and 200-𝝁m width using argon etching. A schematic of the sample is shown in Fig. 1D, Inset. Care was taken to ensure that the legs running to the contact pads are perpendicular to the applied heat gradient to avoid additional contributions to longitudinal thermoelectric voltage by the legs. Samples Σ1, Σ2, and Σ3 are equipped with a 100-nm-thick Au top gate covering the middle of the sample and parts of the legs separated from the HgTe bulk material by a 110-nm-thick multilayer insulator of SiO2/Si3N4.

Fig. S1.
Fig. S1.

Thermopower and transport data on unstrained HgTe (sample Σ1) at 1.8 K. (A) Gate voltage dependence of the resistance Rxx and the low-field Hall resistance Rxy. (B) Sxx at 0 T and Sxx and Sxy at low field. (C) Quantum oscillations in Sxx and Rxx on the electron side: Vg=+3.0 V. (D) Quantum oscillations in Sxx and Rxx on the hole side: Vg=−4.5 V. B, magnetic field; Rxx, longitudinal resistance; Rxy, Hall resistance; Sxx, thermopower; Sxy, Nernst coefficient; T, tesla; Vg, gate voltage.

Fig. S4.
Fig. S4.

Unstrained HgTe. (A) Temperature dependence of the thermopower (compared with data taken from ref. 38; red dots) and resistivity of an unstrained sample. Small quantum oscillations are visible below B = 9 T. (B) Field dependence of resistivity and Hall resistivity. (C) Magnetic field dependence of thermopower and Nernst effect. B, magnetic field; Sxx, thermopower; Sxy, Nernst coefficient; T, temperature; T, Tesla; Vg, gate voltage; ρxx, resistivity; ρxy, Hall resistivity.

For the thermopower measurements, the sample was mounted free standing in a vacuum chamber and glued with one end to a silver stripe, which was connected to an He bath (cold sink). An electric heater was attached to the other end of the sample to apply a heat gradient, which was monitored by two matched 3-k𝛀 ruthenium oxide thermometers glued to the backside of the sample. To ensure that the sample is not thermally shorted by the gate, the heat gradient was cross-checked using the two-point resistance between the upper and lower legs as thermometers. The thermoelectric power has been measured by applying a very low-frequency (∼0.03 Hz) square wave-shaped current to the heater, and the thermoelectromotive force ΔV was recorded by nanovoltmeters.

The measurements of longitudinal and transverse electrical resistances were conducted in a four-point geometry with low constant voltage excitation to avoid self-heating. Throughout the paper, all values are given as resistance instead of units of resistivity in either 2D or 3D. To obtain the resistivity, all values have to be multiplied by the factor 1/6 for the 2D resistivity or 1/6⋅10−7 m for the 3D resistivity. Samples Σ1 and Σ2 were mounted in a 4He system with a base temperature of 1.2 K in a 16-T superconducting magnet. For the temperature dependence, sample Σ1 was mounted in a Variable Temperature Insert. Sample Σ3 was investigated in a 3He system at a base temperature of 0.3 K, whereas sample Σ4 was measured in a 4He flow cryostat with a base temperature of 1.2 K in a 37.5-T Bitter magnet.

SI Text

Data on Sample Σ2.

In Fig. S1, measurements on a different sample (sample Σ2) from the same wafer as the sample in the paper are presented. The sample is identical to the one used for the high-field transport measurements, and the overall behavior is comparable with measurements in the paper. In Fig. S1A, we show the gate voltage dependence of the resistance Rxx and the low-field Hall resistance Rxy at 0.2 T. The resistance in this sample does not have a clear single maximum at charge neutrality but rather, two maxima in the charge neutrality region at slightly higher Vg as the zero crossing of Rxy at the charge neutrality point.

Thermopower Sxx and low-field Nernst effect Sxy are presented in Fig. S1B. All features described in the text are reproduced by these measurements. The field dependence of thermopower and resistance is shown in Fig. S1 C and D. The quantum oscillations are even more pronounced in this sample. The amplitude of the thermopower on the electron side is much smaller than in sample Σ1. Most likely, the phonon drag contribution is smaller in sample Σ2, because the electron–phonon scattering is less efficient. In contrast, the oscillations on the hole side are comparable in absolute signal and shape with the sample presented in the paper. We also note that the sign change in Sxx for B = 0 and 0.2 T corresponds to the maximum in Sxy at Vg≃−1.4V.

Sample Layout and Nernst Effect.

In contrast to electrical transport, thermopower measurements do not depend on the current path along the sample but do depend on the temperature difference of two interconnected parts of the sample. Assuming that the heat gradient is uniform along the x direction (Fig. S2), only the parts of the sample below the gate contribute to the Seebeck effect Sxx. The legs running to the contacts pads are perpendicular to the heat gradient and therefore, do not contribute to thermopower.

When a magnetic field is applied, the voltage that arises perpendicular to the heat gradient is called the Nernst–Ettingshausen effect Sxy. In our structure, the Nernst signal that we measure consists of three contributions: the part of the sample below the gate, the legs running toward the contacts, and the gold pads. The contribution of the gold pads is very small (<0.5 nV/KT) and can, therefore, be neglected. The contribution of the thin legs of the Hall bar that are not covered by the gate is unknown and expected to be in the same order of magnitude as our signal. However, this contribution remains constant when we apply or sweep the gate voltage and thus, gives rise to a constant background in our measurement. The Nernst effect (Fig. S1B) of sample Σ2 shows qualitatively the same behavior as that for sample Σ1, but Σ1 has an offset of about ≃20μV/K.

High-Field Transport Properties of Topological Surface States.

In this section, we present the magneto-transport properties of the topological surface states for sample Σ3. The corresponding longitudinal and Hall resistances have been measured at 0.3 K and up to 30 T in a wide range of gate voltages. In Fig. S3A, we present the charge carrier concentrations as a function of the applied gate voltage Vg extracted either from the low-field Hall-resistance and the quantum oscillations that originate from the bottom (nb) and top (nt) surfaces, respectively. The corresponding fast Fourier transforms (FFT) of the SdH oscillations are plotted in Fig. S3D where we can extract two distinct frequencies that correspond to the top surface charge carrier density nt and the bottom surface charge carrier density nb. Both nt and nb are linear in Vg and nt(Vg) with the higher slope (red triangles) corresponding to the top surface (close to the gate). Adding up the carrier concentrations of the two surfaces (blue dots), we obtain approximately the carrier concentration extracted from the low-field Hall resistance.

Evidence of the topological nature of the surface states can also be obtained from the phase of the quantum oscillations. We define an integer N (N = 0, 1, 2,…) which is the number associated with the different minima of the oscillations. N is related to the quantum Hall index by i = m(N+1/2) with m = 2 being the number of Dirac cones for our system in the case of equal carrier concentration on both surfaces. From Fig. S3B, we extract the minima in ρxx (centre of quantum Hall plateaus in σxy) and plot N as a function of the inverse magnetic field in Fig. S3C (upper part) for nt=nb at Vg = −4 V. Extrapolation to 1/B→ 0 gives an intercept at ≃−1/2 which corresponds to a π Berry phase compared to a zero intercept for a conventional 2DES.

Transport and Thermopower on Unstrained HgTe.

To highlight the differences between strained HgTe (3D TI) and unstrained HgTe (semimetal), we have also performed transport and thermopower on unstrained HgTe. The sample (labeled Σ4) has been grown under exactly the same conditions but is 600-nm thick instead of 104-nm thick for the strained sample. If a thin layer of HgTe is grown on a CdTe substrate, HgTe will grow with the lattice parameters of the substrate, and we obtain a strained sample. When the thickness of the sample exceeds 150 nm, the strain starts to relax by dislocations (37), and a completely relaxed sample is obtained for much larger thicknesses.

The sample has a low concentration of n= 1.4⋅ 1016 cm−3 and a high mobility of μ= 3.0⋅ 104 cm2/Vs as obtained from zero and low-field transport experiments. In Fig. S4A, the temperature dependence of the resistivity is presented. With decreasing temperature, ρxx increases strongly but starts to saturate below 12 K. The thermopower Sxx (Fig. S4A) is negative (as expected for an electron-dominated material), is diffusion-driven (38), and increases strongly up to 70 K, where it saturates. The thermopower of unstrained HgTe is relatively independent of the carrier concentration, which can be seen compared with the data by Takita et al. in ref. 38. Thermopower experiments above 100 K on heavily doped n- and p-type bulk HgTe can be found in ref. 39. When a high magnetic field is applied (Fig. S4B), only small oscillations with a single frequency emerge in ρxx up to about B = 9 T, where the quantum limit is reached. Beyond the quantum limit, one more large oscillation appears at B≃ 20 T. The Hall effect also shows small oscillations at low fields and a broad plateau-like feature above 20 T. The thermopower, shown in Fig. S4C, first exhibits a small dip at 0.7 T but increases strongly in absolute value up to 15 T, with a second minimum at 4 T. For B>15 T, Sxx decreases and seemingly saturates when approaching 30 T. The Nernst effect first increases up to 13 T, then decreases at higher fields, and saturates above B = 20 T.

It is obvious that the overall behavior of unstrained HgTe differs substantially from that of strained HgTe. (i) Whereas the thermopower of the TI is nearly temperature independent for small electron concentrations and only increases considerably when electrons are thermally excited into the bulk conduction band, the thermopower of unstrained HgTe strongly increases with increasing temperature. (ii) Although the unstrained HgTe sample has a reasonable high mobility, only weak quantum oscillations are observed in transport and thermopower, and the SQHE is absent.

Acknowledgments

We thank Benoît Fauqué for fruitful discussions. This work was performed at the High Field Magnet Laboratory Radboud University/Fundamental Research on Matter, member of the European Magnetic Field Laboratory and supported by EuroMagNET II under European Union (EU) Contract 228043. The group from Universität Würzburg acknowledges funding from the German Research Foundation (The Leibniz Program, Sonderforschungsbereich 1170 “Tocotronics,” and Schwerpunktprogramm 1666), EU European Research Council Advanced Grant Program Project 3-TOP, and the Elitenetzwerk Bayern Internationales Dokoranden Kolleg “Topologische Isolatoren.” This work was supported by German Science Foundation Grant HA 5893/4-1 within Schwerpunkt Program 1666 (to J.B. and E.H.); J.B. and E.H. thank the Elite Bayern Network Graduate School “Topological insulators” for financial support. Financial support was provided by Nederlandse Organisatie voor Wetenschappelijk Onderzoek VENI Grant 680-47-424 (to S.W.).

Footnotes

  • ↵1To whom correspondence should be addressed. Email: s.wiedmann{at}science.ru.nl.

  • Author contributions: A.J., C.B., H.B., L.W.M., and S.W. designed research; A.J., M.B., C.B., J.C.M., U.Z., N.H., and S.W. performed research; A.J., J.B., E.H., and S.W. analyzed data; M.B. prepared the samples; J.B. and E.H. performed theoretical band-structure calculations; and A.J. and S.W. wrote the paper with input from all coauthors.

  • The authors declare no conflict of interest.

  • This article is a PNAS Direct Submission.

  • This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1611663114/-/DCSupplemental.

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Topological surface states on strained HgTe
Andreas Jost, Michel Bendias, Jan Böttcher, Ewelina Hankiewicz, Christoph Brüne, Hartmut Buhmann, Laurens W. Molenkamp, Jan C. Maan, Uli Zeitler, Nigel Hussey, Steffen Wiedmann
Proceedings of the National Academy of Sciences Mar 2017, 114 (13) 3381-3386; DOI: 10.1073/pnas.1611663114
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Proceedings of the National Academy of Sciences: 114 (13)
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