Topological insulators based on HgTe

First, we describe the energy spectrum of a QW based on mercury telluride in more detail. Figure 1 shows a qualitative view of the bottom energies of the main dimensional-quantization subbands in such a well as a function of its thickness. As can be seen, the behavior of the spectrum fundamentally depends on the well thickness, and it can be conventionally divided into three regions. The first region is $d_{w}>d_{c}$, where a direct-band-gap 2D insulator is realized. Its band gap decreases as the thickness increases, to collapse at a critical well thickness dc, which is equal to 6.3–6.5 nm, depending on the QW orientation and deformation. As $d_{w}$ increases further, the second region appears, which contains a 2D insulator but with inverse bands. Finally, if $d_{w}>15–16nm$, a semimetal state [40, 41] emerges due to the overlap of hole-like bands H1 (conduction band) and H2 (valence band). Because the discussion in what follows is focused on the properties of 2D TIs, of interest for us is only the second region, in which a 2D TI is realized. The energy spectrum of this TI calculated in [42] for (100) and (013) surfaces is shown in Fig. 2a. As can be seen, the basic characteristics of this spectrum only weakly depend on the orientation of the surface. The critical thickness is in both cases $d_{c}=6.2–6.3nm$, and the 2D TI state with the largest band gap, which is characterized by the simplest s–p inversion, is realized at a QW thickness of 8.2–8.5 nm. The band gap width is in this case approximately 30 meV.

The dispersion law for edge and bulk states for a QW with the thickness 8.5 nm and orientation (013) is shown in Fig. 2b, which well illustrates all the features of the spectrum of a 2D TI based on the HgTe QW: the linear Dirac spectrum of edge current states and a parabolic band-gap spectrum of bulk states. We note that the edge states exist not only in the band gap but also at energies that correspond to the allowed bulk bands. Figure 2b also clearly shows the anticrossing of the edge states with the bulk states in the lower part of the band gap due to lower surface symmetry (013).

The vast majority of experimental samples used in the studies reviewed here are made on the basis of QWs with a given thickness of 8 or 8.3 nm and orientation (013). This orientation is chosen because, on the one hand, the presence of steps on suchlike surfaces ensures a more equilibrium growth of the HgTe and HgCdTe layers, which reduces the concentration of various point and dislocation defects, and, on the other hand, as shown in Fig. 2, the energy spectrum of a 2D TI does not substantially depend on orientation.

It is also of importance that in specifying the QW thickness, its exact value for a given sample may not correspond to the specified growth thickness, and deviations from it by several tenths of a nanometer are possible due to the heterogeneity of the atomic beam density in the process of molecular beam epitaxy.

The QW itself does not enable a full-fledged study of the state of a specifically 2D TI, because two conditions must be satisfied for its transport response to be observable: it is necessary to ensure that the Fermi level is located in the bulk band gap, and the clearest, most reliable, and simplest way to detect edge states is needed. The first condition may be satisfied using the field-effect transistor structure schematically shown in Fig. 3a. The second condition is specifically discussed in detail in Section 3.3.

Two more operations are required to produce a field-effect transistor based on the HgTe QW: low-temperature growth of the dielectric layer and the subsequent deposition of a metal gate on it. Either a pyrolytic $SiO_{2}$ layer or a double $SiO_{2}+Si_{3}N_{4}$ layer grown at temperatures of $80–100^{\circ}C$ was used as a dielectric, and the Ti/Au layer served as a gate. We note that there are other methods to grow dielectric layers, but we do not discuss them here.

We now briefly describe the conditions under which the transport response of the investigated 2D TI was measured. The measurements were carried out in the temperature range 0.2–10 K in magnetic fields up to 15 T using the standard phase-sensitive detection scheme at frequencies of 6–12 Hz and currents 0.01–10 nA, depending on the type of experiment, to exclude the effects of heating of the electronic subsystem.

Figure 3 shows the dependences of the dissipative and Hall components of the resistance tensor on the gate voltage, typical of a macroscopic sample made on the basis of an 8 nm HgTe QW. It is clearly seen that the resistance is small (of the order of $100\Omega/\Box$) at displacements that correspond to the location of the Fermi level (EF) in the conduction band, passes through a maximum (equal to about $300k\Omega$ in this case) that corresponds to $E_{F}$ occurring in the middle of the bulk gap, and then begins to decrease, reaching values of several $k\Omega/\Box$, when the Fermi level enters the valence band. The point of the maximum of $\rho_{xx}$ is commonly referred to as the charge neutrality point (CNP). The dependence $\rho_{xy}(V_{g})$ in turn exhibits a well-pronounced plateau at the Landau level filling factors i = 1 and i = 2 on the electron side, passes through zero at the CNP, and has the opposite sign in the valence band, but the plateaus are no longer observed due to significantly lower (by an order of magnitude) hole mobility. The absence of the Hall signal at the CNP indicates that there are no mobile charges in the QW. Thus, this point fully justifies its name.

We note that strictly speaking, a zero of the Hall signal and especially the maximum of resistance are not direct evidence of the absence of charge in the well. Therefore, caution is needed in every determination and analysis of the CNP. Another feature of $\rho_{xy}(V_{g})$ curves is of importance: their half-width is significant (about 1 V). This feature indicates that the density of states inside the bulk band gap is quite high, a feature that has been overlooked in the vast majority of studies on 2D TIs. This is discussed in more detail in Section 3.4.

The dependence shown in Fig. 3b essentially says nothing about edge transport, because the measurement does not allow eliminating the influence of the bulk. Of crucial importance for determining edge transport is measuring it in a nonlocal geometry.

We now make some preliminary remarks regarding the essence of the resistance of the 2D TI edge channel. We compare this resistance with that measured in the QHE regime, which is also a kind of 2D TI.

We consider the simplest example: a two-terminal conductor of length L and width W with ohmic contacts L (Left) and R (Right) in the cases where transport through it is maintained by the 2D TI edge states in the ballistic mode (Fig. 4a) or in the QHE mode in filling the degenerate Landau ground level (Fig. 4b). We start with the first option. The current transport is maintained in this case by two single-mode quantum wires with removed spin degeneracy, located on the lower and upper edges of the conductor. The state that bears the electrochemical potential of the left contact $\mu_{L}$ is located on the same edge of the sample where the oppositely directed state having the potential $\mu_{R}$ is localized.

Exactly the same reasoning applies to the opposite edge of the sample, and we therefore actually have two single-mode ballistic wires connected in parallel, each of which has a conductance equal to $e^{2}/h$. Then the measured conductance $G_{12,12}=eI_{12}/(\mu_{L}-\mu_{L})$ is equal to $2e^{2}/h$. The situation in the QHE regime is completely different: the current is transported by spatially separated states, i.e., those states localized at opposite edges of the conductor. One of them bears the electrochemical potential of the left contact $\mu_{L}$, and the other, of the right contact, $\mu_{R}$, and the measured conductance has the same magnitude $2e^{2}/h$.

We now consider a 2D conductor with a resistivity $\rho_{xx}$ of length L and width W with contacts 1–4, as shown in Fig. 5. If the current passes through contacts 1 and 2, and the voltage is measured at contacts 3 and 4, the resistance $R_{1234}=V_{34}/I_{12}$ is by the order of magnitude equal to [43].

$R_{1234}\approx\rho_{xx}exp\left(-\frac{\pi}{W}\right)$

As the ratio of the conductor length to its width increases, the resistance decreases exponentially for a trivial reason: only an exponentially small part of the total current reaches contacts 3 and 4. This is the configuration that corresponds to the measurement of nonlocal resistance.

We now consider the situation where a band gap emerges in the bulk of the conductor. No current flows through it in the case of a conventional insulator. However, if it is a TI, the entire current flows through the edge states, because they are delocalized. In the case of ballistic transport, we then obtain $R_{1234}=\frac{h}{4e^{2}}$

and, in the case of diffusive transport, a weak linear decrease of the resistance, $R_{1234}=R_{L}\frac{W^{2}}{L+W}$

where $R_{L}$ is the edge-wire resistivity. It is hence apparent that a comparative analysis of the local and nonlocal responses enables unambiguous determination of the presence of edge transport and hence the edge states that make its occurrence possible. An example displayed in Figure 6 shows a typical measurement of local ($R_{loc}$) and nonlocal ($R_{nonloc}$) resistances for a sample (whose topology is shown in the inset in Fig. 6) made on the basis of an HgTe QW 8 nm in thickness. At first glance, the behavior of these resistances is qualitatively similar and coincides with that of the $\rho_{xx}(V_{g})$ dependence shown in Fig. 3. But a more careful comparative analysis of $R_{loc}$ and $R_{nonloc}$ reveals a significant difference between their behaviors: while the $R_{loc}$ values are sufficiently large at all gate voltages, including those that correspond to the location of the Fermi level in the allowed band, $R_{nonloc}$, as it should be, is close to zero at the specified voltages. However, $R_{nonloc}$ becomes comparable to Rloc in the vicinity of the CNP, i.e., when the Fermi level is located in the center of the bulk gap. Just this property of the 2D TI transport is a direct indication of the existence of charge transport along the edge.

The first experiments on edge transport were carried out in [8–10], where ballistic edge transport in an HgTe QW with a thickness of 7–8 nm was demonstrated using submicron-size samples. It was next shown in [11] that transport along edge states in these QWs also exists on macroscopic scales, of the order of 1 mm, but this time in the diffusive regime. The experiments in [44, 45] should also be noted, in which edge states were visualized, thus confirming their presence.

The easiest way to analyze edge transport is to use the Kirchhoff relations under the assumption that there is no bulk conductivity. An equivalent circuit in the case of a standard Hall bridge with two current and four potential contacts is shown in Fig. 7. It is easy to see that the resistance between contacts i and j can be expressed in terms of the resistance between all other contacts or in terms of the distance between them (the latter reflects the proportionality of the resistance of the edge wire to its length): $R_{m,n}^{i,j}=\frac{h}{e^{2}}\frac{L_{n,m}L_{i,j}}{lL}$

where the current flows through the contacts i and j, and $L_{i,j}(L_{m,n}$) denote the length of the edge states located under the metal gate that do not include these contacts, and L and l are the edge state circumference and the mean free path, respectively. This simple formula enables predicting the relation between local and nonlocal resistances in any configuration, which no longer depends on the mean free path and therefore on the presence or absence of backscattering. Formula (1) in the ballistic transport limit is the Landauer–Büttiker relation. The resistances of ballistic 6- and 4-pin bridges calculated in this way are in good agreement with experimental data. A comparison of the results calculated using Eqn (1) with measurements of samples exhibiting diffusion transport revealed a significant disagreement, which increases as the edge state length increases [13]. It is natural to assume that if the regime deviates from the ballistic one, it is necessary to take not only backscattering between the edge states but also their scattering into the bulk into account. The Kirchhoff rules are not suitable for a quantitative analysis of such a deviation. The problem can be resolved by introducing two phenomenological parameters $\gamma$ and $g$ for the rate of scattering between the edge states and the edge state and the bulk [46].

We recall that different edge states running towards each other are associated with different spins. The distribution of the bulk potential is determined in this case using the Laplace equation and the corresponding boundary conditions. The distribution of the edge state potential is found using the balance equation [47]. Both the edge and bulk potentials that belong to different spin states are mixed in the contact region, which in our case is a 2D electron gas outside the gate area (which covers only the central part of the Hall bridge).

It is of importance that the results calculated in this model are independent of the mechanisms of microscopic scattering between the edge states or the edge channels and the bulk. Possible scattering mechanisms are described below.

A specific feature of the model under consideration is that transport in various systems is described in the model in a universal way. This model was applied first to a 2D system demonstrating the QHE, when the chiral edge state that belongs to the last Landau level is mixed with the bulk level, resulting in a significant nonlocal response [47]. The model also successfully described the quantum transport of the zeroth Landau level of DFs in graphene, which form edge modes running towards each other [48]. The bulk transport can be described in both of these cases as the transport of a 2D electron in a quantizing magnetic field. Edge transport occurs in a TI when the bulk is an insulator in the absence of a magnetic field, and describing bulk transport requires a different approach. It has been suggested that bulk transport is determined by Gaussian tails of the density of states due to the presence of a fluctuation potential arising from fluctuations in the QW thickness and an impurity potential. Based on this assumption, a description was provided for both local and nonlocal transport in the presence of scattering both between the edge states running towards each other and between the edge states and the bulk using $\gamma$ and $g$ as fitting parameters [46].

The Gaussian broadening of the density of states was found in this case using the mobility of electrons and holes at the bottom of the corresponding bands. The proposed model, which takes the leakage of current into the bulk into account, describes the dependence of the resistance on the density of charge carriers fairly well (Fig. 8). Indeed, if the Fermi level is located in the center of the forbidden band, current leakages through the bulk are minimal, and the resistance is determined by the edge transport, i.e., scattering between edge states. As the Fermi level approaches the valence or conduction band, the contribution of the bulk due to the scattering of edge states into the bulk, as well as the contribution due to an increase in the bulk conductivity itself, increases, and the total resistance decreases. The width of the resistance peak is then determined by the velocity of the Fermi level motion through the tails of the density of states in the topological insulator band gap, and, to describe the peak width observed in the experiment, it is necessary to assume a high density of states inside the band gap, only several times lower than the bulk one.

In this section, we analyze measurements of the temperature dependence of the resistance of a 2D TI in the diffusive transport mode. Such measurements are of importance for two reasons: first, it is necessary to determine the activation band gap and compare its value with the calculated one, while the second, more fundamental, reason is due to the edge channel of a 2D TI being an almost perfect 1D conductor, a fact that at first glance allows testing all predictions of numerous theories of 1D conductivity.

Typical measured temperature dependences at high (T ¿ 4 K) temperatures for the samples described in Sections 3.1–3.4 are displayed in Fig. 9. The inset in Fig. 9a shows the topology of the samples. It is clearly seen that at temperatures higher than T = 30 K, an activation increase in resistance is observed, which is followed by complete saturation of the R(T) dependence, and, if the temperature decreases to 4 K, no significant temperature dependence is observed.

The activation increase in resistance is associated with the freezing of bulk conductivity, the activation energy for the dependence shown in Fig. 9 being approximately 200 K. We note that the activation energy can vary significantly, depending on the sample, in the range 200–400 K. Calculations of the energy spectrum for the wells under study shown in Fig. 2 yield a gap of about 30 meV, which fits into the specified activation energy range. We note, however, that suchlike measurements actually determine the mobility band gap, which heavily depends on disorder, whose significant role is suggested by the large spread of the activation energy.

At temperatures above 20–30 K, the temperature dependence of the resistance of the 2D TI based on HgTe QWs with a thickness of 8–9 nm in the regime of diffusive transport reflects the bulk properties, more precisely, the size of the mobility band gap in the bulk. As was noted above, as the temperature decreases further to 4–5 K, the resistance ceases to change and, when transport is actually maintained by edge states, a metallic behavior of the resistance is observed. To check whether this (metallic) behavior persists at lower temperatures, measurements were carried out using a dilution refrigerator at temperatures down to 40 mK [49]. Figure 9b, where the results of these measurements are displayed, shows that the character of the temperature dependence barely changes: in the temperature range 4–1 K, only a very weak ($10\%$) increase in resistance is observed, and further, at temperatures down to 40 mK, there is no temperature dependence whatsoever.

To date, various approaches have been proposed to explain this behavior of resistance, but so far all of them have failed to provide a comprehensive explanation. The model of metal droplets proposed in [50, 51] seems to be preferable. According to this model, an electron moving along an edge state enters such a droplet, and backscattering can occur inside it as a result of inelastic scattering, which leads to suppression of ballistic transport and to the conductance values lower than the conductivity quantum. However, the temperature dependence predicted by this model disagrees with the experimental data: as the temperature decreases, the edge conductance should increase due to the suppression of inelastic processes, and at temperatures tending to zero, a conductance close to $e^{2}/h$ should be observed, while in the experiment it is practically independent of temperature in the range from 20 to 0.2 K. Perhaps this shows that along with impurity disorder, the structural disorder caused by fluctuations in the QW thickness are to be taken into account.

We first describe the response of edge transport in a 2D TI to a normal magnetic field. Figure 10a shows a typical dependence of local and nonlocal resistances on the magnetic field at the CNP. It can be seen that the qualitative behavior of the two dependences is the same, i.e., in both local and nonlocal geometry, positive magnetoresistance (MR) is observed in magnetic fields up to 2 T, which is followed by its rapid decrease and subsequent equally rapid growth. The displayed results thus clearly show the edge nature of the MR. Figure 10b shows the behavior of linear magnetoconductivity (MC) in more detail at temperatures of 4.2 and 1.6 K (data of local measurements are quoted in this case). It can be seen that MC weakly depends on temperature and is about $10–15\%$ in magnetic fields of about 1 T. This linear MC behavior was predicted in [52], where it was explained by the suppression of topological protection from back scattering by the magnetic field.

We next consider the effect of the magnetic field in the QW plane. Figure 11a, b shows local and nonlocal resistance of the sample (whose topology is shown in the inset in Fig. 11a) as a function of the magnetic field. It is clearly seen that the same behavior is observed at all temperatures: if the magnetic field is less than 8 T, $R_{xx}$ and $R_{nL}$ monotonically decrease to values that are 1.5 to 2 times smaller than those in the zero magnetic field. The decrease then becomes faster and, at $B>10T$, $R_{xx}$ virtually saturates at a value that is already an order of magnitude smaller, while $R_{nL}$ is close to zero.

This result has been explained in the theory developed by Raichev [53]. According to this theory, if a magnetic field is applied in the QW plane, the bulk gap gradually decreases and, in magnetic fields stronger than 12 T, the bulk transforms into the state of a 2D Dirac semimetal with a corresponding emergence of bulk conductivity.

Figure 11c shows how the QW bulk transforms into a semimetal state at various degrees of disorder. Good agreement between theory and experiment is clearly seen both in the character of the dependence on the magnetic field and in its magnitude (10–11 T in the experiment and 12–14 T in calculations), which corresponds to the complete transition of the QW into a gapless state. A slight disagreement in the values of critical fields is not a surprise, because all parameters of the system (bulk parameters of HgTe and CdTe, impurity concentration, the QW thickness fluctuations) that are used in the calculations are only known with a certain accuracy.

Here, we present the results of an experimental study of the terahertz photoresistive response of a 2D TI [12] in which the regimes of ballistic and quasiballistic transport were successfully implemented. The experimental samples were microstructures of a special Hall geometry, equipped with a semitransparent Ti/Au gate (Fig. 12a) whose characteristic dimensions are comparable to the mean free path along the edge state. In particular, the microbridge width was 3.2 $\mu m$.

The terahertz resistive response (photoresistance) of the described structures was measured at a wavelength of 118 $\mu m$ in transverse and longitudinal magnetic fields up to 4 T at temperatures $T=2–4.2K$. A molecular submillimeter methanol-based laser with optical pumping by a $CO_{2}$ laser was used as a radiation source. The terahertz radiation power $P_{\lambda}$ was in the range 20–30 mW. The photoresistance (PR) was measured using a standard modulation technique at a modulation frequency of 600–700 Hz while passing the direct current I = 100 nA through the sample.

We begin the description of the experiment with analyzing the transport response of the samples studied. Figure 12b shows the dependences on the effective gate voltage ($V_{g}$ is the gate voltage and is the gate voltage that corresponds to the maximum local resistance) of the Hall resistance and the local resistance measured on the shortest part of the sample, where the distance between the potentiometric contacts (contacts 3 and 4 in Fig. 12a) was 2.8 $\mu m$. As can be seen, the resistance is small (of the order of $1k\Omega/\Box$) at displacements that correspond to the location of the Fermi level (EF) in the conduction band, passes through a maximum (equal to 13.4 $k\Omega$ in this case) at the CNP (EF simultaneously passes through the Dirac point), and then begins to decrease to reach several $k\Omega/\Box$ when the Fermi level enters the valence band. The dependence passes through zero and changes sign. Figure 12c shows the nonlocal resistance of the sample , when contacts 3–5 and 4–6 were used as respective current and potentiometric contacts. As expected, the signal of the nonlocal resistance is much smaller than that of the local one when the Fermi level is located in allowed bands. At the CNP, the nonlocal signal is almost three times higher than the local one.

We now analyze the quoted data. The local resistance at the maximum is close to $h/(2e^{2})$ (shown by the dashed line in Fig. 12b). This implies that virtually ballistic transport is realized in the smallest section of the studied Hall structure (about $10\mu m$ along the sample edge). We note that this was the first observation of such transport in QWs 8–9 nm thick after the publication of [8, 9]. The nonlocal resistance is determined by splitting the current passing through contacts 3–5 between the parts of the sample with ballistic transport and with diffusive transport. The value of therefore lies between $2h/e^{2}$ and $h/e^{2}$. The value of $h/e^{2}$ is shown in Fig. 1c by a straight dashed line.

Figure 13a displays typical measured local PR of the sample as a function of the gate voltage under the effect of about 20 mW terahertz radiation at a wavelength of 118 $\mu m$. For the convenience of comparative analysis, the same figure shows the dependence . The dependence of the nonlocal PR at the same power values is shown in Fig. 13b. The dependence is also displayed in the same figure.

We now discuss the quoted data. It is clearly seen that both local and nonlocal PRs are virtually zero when the Fermi level is located in allowed bands, to become nonzero only when the Fermi level enters the forbidden band, and the PR sign is negative, i.e., resistance of the sample decreases under the effect of radiation. Upon reaching the CNP, both dependences and pass through a maximum at which their value is $0.1–0.5\%$ of the total resistance. A more detailed comparative analysis of the curves presented in Fig. 13 shows that while the half-width of the local PR peak virtually coincides with that of the local resistance peak, the width of the PR dependence on in the nonlocal case is more than two times smaller than the same dependence for the resistance.

We next discuss the results obtained. The bulk gap for the 8 nm QWs considered in this section is 30 meV, i.e., several times higher than the photon energy for the employed wavelength of $118\mu m(\hbar\omega=10.8meV)$. Three types of transitions are then possible in our case: (1) between Dirac branches of 1D edge states; (2) between the electron Dirac branch and the conduction band; and (3) between the valence band and the hole Dirac branch. Transitions of the last two types would apparently lead to the emergence of PR maxima near the allowed bands, i.e., to the right (for transitions of the second type) or to the left (for transitions of the third type) of the CNP on the PR dependences on . This behavior is not observed in experiment. Thus, only transitions of the first type remain. A preliminary analysis of the absorption at these transitions carried out in Ref. [54] showed that dipole transitions between edge Dirac branches are forbidden, and only significantly weaker magnetic dipole transitions occur.

However, it was shown recently in [55] that a similar conclusion is not valid for the HgTe QW because the argument does not include spatial inversion violation at the boundaries of these QWs with barrier HgCdTe layers. It was found in [55] that due to the violation of inversion symmetry at these boundaries, direct dipole transitions between the edge branches are allowed, and formulas for the absorption coefficient have been derived. Thus, the experimental conclusion on the possibility of direct dipole transitions between edge branches has been confirmed theoretically in [55].

The existence of a 2D TI was discovered recently [11] in 14 nm QWs with (112) orientation. The TI state with ballistic transport at distances of the order of 10 $\mu m$ was obtained in this TI for the first time after [8, 9]. Therefore, we discuss the results of [11] in more detail.

A qualitative view of the spectrum for such a QW is shown in Fig. 14a. We note that the bulk spectrum is no longer as simple as for QWs with a thickness of 8–9 nm: two more branches of hole states emerge between the s and p states: hh2 and hh3. As a result, the band gap and edge states of importance for the experiment turn out to be located between two hole branches, hh1 and hh2, while the band gap is much smaller, about 3.3 meV.

Activation measurements using macroscopic samples (Fig. 15a, b) yield a noticeably smaller gap (1.2 meV), which is unsurprising because the experiment based on measuring the activation temperature dependence actually determines the mobility band gap, which is always less than the real band gap due to disorder caused by the fluctuation potential of impurities and the QW composition and thickness.

Figure 16 shows the results of experiments with micrometer-size samples. These results clearly demonstrate, on the one hand, the existence in such samples of both local and nonlocal transport close to the ballistic one, and, on the other hand, its apparent imperfection caused by mesoscopic fluctuations. The results presented are consistent in this regard with those obtained previously for QWs with a thickness of 7–8 nm [8, 9], where similar fluctuations were also observed. We note that ballistic transport exists in the described experiments in samples whose characteristic size is about $10\mu m$.

Thus, quasiballistic transport is possible in 2D TIs at distances that are significantly larger than the mean free path, which in the samples studied did not exceed $1\mu m$ at energies close to the conduction band bottom.

As noted in the Introduction, bulk mercury telluride, despite the inverse nature of its spectrum, cannot be classified as a TI because it is a gapless semiconductor.

However, if a uniaxial compression deformation is applied to HgTe, leading to the emergence of a band gap in the bulk, then a transformation into a 3D TI state can occur. A similar but not identical situation occurs, as shown in [19], for HgTe films grown on CdTe substrates due to the difference in the lattice constants of HgTe ($a_{HgTe}=0.646nm$) and CdTe ($a_{CdTe}=0.648nm$). The critical thickness of pseudomorphic growth that corresponds to this difference in the lattice constants is more than 100 nm, and thus films whose thickness is less than this critical value reproduce the lattice constant of the CdTe substrate. Tensile deformation occurs as a result in such films, leading to the emergence of a gap.

However, the Dirac point in the TI created in this way is located not inside the band gap but deep in the valence band. Due to hybridization with the valence band, the spectrum of surface states only contains the electron branch, which, when approaching the valence band bottom, deviates from the linear law to become quasiparabolic.

It can be seen in Figure 17a, which shows the spectrum of a film 80 nm in thickness, that as energy increases, the valence band is replaced by an approximately 15 meV indirect band gap in the bulk of the film, inside which there are surface bands of delocalized electron states. Because the thickness of the film is finite, its spectrum in the allowed bulk bands is a collection of dimensional quantization subbands with a small ( 1 meV) distance between them in the valence band and an order of magnitude larger distance in the conduction band.

A field-effect transistor structure has been produced based on such a film and is considered in this section. Two types of structures have been studied, whose schematic cross section is shown in Fig. 17a. The structures were grown using molecular beam epitaxy on a (013) oriented CdTe substrate. The bulk of both structures is an 80 nm HgTe layer. The structures differ in the order of the upper layers: the first structure (open) ends with an HgTe layer, while in the second structure (’closed’) the main layer is covered with a CdHgTe layer 20 nm in thickness. One of the main achievements of the developed technology, first described in our study [20], is high mobility (up to $5\times 10^{5}cm^{2}/Vs$) and low concentration of uncontrolled bulk impurities, which is reduced to $10^{16}cm^{-3}$. This result was achieved due to the use of a 20 nm buffer CdHgTe layer between the HgTe film and the CdTe substrate, which resulted in a sharp decrease in the number of dislocations and defects. This achievement made it possible to obtain not only unambiguous experimental confirmation of the existence of a 3D TI based on a strained HgTe film but also quantitative information on its energy spectrum and on the relative contribution of bulk holes and bulk and surface electrons to the transport response.

To perform magnetotransport measurements, we made Hall bridges $50\times 450\mu m$ in size with a distance of 100 and 250 $\mu m$ between contacts (see the inset in Fig. 3). The central part of the bridges was equipped with a metallic Ti/Au gate. The bridges were produced from both types of structures described in Section 4.1 using standard photolithography and chemical etching. As a gate dielectric, we used either a two-layer film consisting of a 100 nm SiO2 layer and an Si3N4 layer with a thickness of 100–200 nm grown using plasma chemical deposition technology at $T=100^{\circ}C$, or an 80 nm $Al_{2}O_{3}$ film grown using atomic-layer deposition technology at $T=80^{\circ}C$. We note that this technology is not much different from that used to produce field-effect transistors based on 2D TIs.

Figure 17c shows a typical dependence of the resistance $\rho_{xx}$ at B = 0 and the Hall resistance $\rho_{xy}$ at B = 1T as a function of the gate voltage $V_{g}$ at T = 1.9 K. Several maxima are observed on the $\rho_{xx}(V_{g})$ curve, the main one being close to $V_{g}=1V$. The curve is asymmetric with respect to the main maximum: the resistance to the left of the maximum is much larger than that to the right, while two side maxima are observed on the curve. The first maximum is located at $V_{g}=-5.5V$; its height is the same as that of the main maximum, and the second, much lower, maximum is located at $V_{g}=3.5V$. The $\rho_{xy}$ dependence displayed in the same figure is asymmetric with respect to $V_{g}=1V$, at which it crosses the abscissa axis. A change in the sign of $\rho_{xy}$ suggests that when the gate voltage changes, the Fermi level passes through both the valence and conduction bands. If $V_{g}>1V$, the Fermi level is located in the valence band, where, according to the spectrum shown in Fig. 17b, holes and Dirac surface electrons coexist. This phenomenon is confirmed by the large positive magnetoresistance and the nonlinear Hall effect typical of electron–hole systems.

The dependences of the hole density $P_{s}$ and mobility $\mu_{p}$ as well as the total electron density $N_{s}$ and the average mobility $\mu_{e}$ on the gate voltage are shown in Fig. 17d, e. These parameters were determined by fitting the calculated dependences $\rho_{xx}(B)$ and $\rho_{xy}(B)$ obtained in the model of classical Drude two-species magnetotransport to the dependences experimentally measured at fixed gate voltages.

We first analyze the behavior of the electron and hole densities. These densities undergo significant variations (by almost an order of magnitude), which indicates a small amount of residual impurities in the film. The CNP is located near the zero gate voltage. The CNP corresponds to the location of the Fermi level near the valence band ceiling, and the densities of bulk holes and surface Dirac fermions in it are the same.

We note once again that the Dirac point does not coincide with the CNP and is experimentally unattainable in our samples, because even at the maximum negative $V_{g}$ values there is a significant number of electrons in the system, i.e., Dirac electrons contribute to the conductivity at all gate voltages used in the experiment. Fitting based on the Drude model can no longer be considered reliable in the vicinity of the CNP. Therefore, the hole density in the CNP region can only be obtained by extrapolating the dependence $P_{s}(V_{g})$, which crosses the abscissa axis near $V_{g}=2V$. It can be assumed that at this voltage the Fermi level coincides with the valence band ceiling.

Thus, the semimetallic state of the system is realized at $V_{g}>2V$, which emerges as a result of the overlap of the bulk valence band and the surface electron band. For $V_{g}>2V$, there is a small—but the most interesting—voltage region in which transport is determined only by surface electrons (a 3D TI) followed by the start of filling the bulk electron band (the electronic metal state).

We now discuss the behavior of mobility. The hole mobility as a function of the gate voltage is represented by a curve that has a maximum with the mobility value $10^{5}cm^{2}/Vs$, and saturates if $P_{s}$ increases further. The $\mu_{e}(V_{g})$ dependence is more interesting: there is a wide maximum near $V_{g}=5V$, where $\mu_{e}(V_{g})$ is $4\times 10^{5}cm^{2}/Vs$, which is followed by a minimum at $V_{g}=-6V$. The described behavior of mobility can be associated with both the possible complete exhaustion of one of the surfaces with Dirac electrons (apparently located closer to the gate) and the beginning of the filling of the second hole subband. The valence band ceiling corresponding to the gate voltage $V_{g}=2V$ is confirmed by the temperature dependence$\rho_{xx}(V_{g})$ shown in Fig. 18b. It is clearly seen that the point $V_{g}=2V$ is a border-line point: temperature dependence is virtually unobservable to the right of it, while to the left of it resistance significantly increases as the temperature grows. This behavior is associated with the emergence of electron–hole scattering driven by the Landau mechanism [56, 57], similar to that observed in 2D semimetals [58].

This scattering apparently only occurs when the Fermi level crosses the valence band ceiling. Another feature in the $\rho_{xx}(V_{g})$ dependence is clearly exhibited at $V_{g}=4V$. Moreover, it blurs at temperatures that exceed 5 K. This feature can be associated with the beginning of the filling of the bulk electron band.

Thus, if the suggested identification of the band boundaries schematically depicted in Fig. 18a is correct, then transport due to surface states is only possible for $2<V_{g}<4V$. This picture is confirmed by the specific features of classical magnetotransport, more precisely, by the behavior of the dependence of the relative positive magnetoresistance (PMR) ($\rho_{xx}(B)/\rho_{xx}(B=0)$) on the gate voltage.

It should be kept in mind that according to the Drude model, the PMR value in the semimetal state is proportional to the sum of the electron and hole mobilities and, if two groups of electrons coexist, to the difference between the electron mobilities. In accordance with these arguments, the PMR dependence on the gate voltage exhibits a maximum near the CNP, i.e., when the densities of electrons and holes are approximately equal. A rapid decrease (by an order of magnitude) in $\rho_{xx}(B)/\rho_{xx}(B=0)$ is observed to the right of the maximum as the Fermi level moves to the band gap and holes disappear. If $V_{g}$ increases further, a monotonic decrease occurs that ends near $V_{g}=4V$, and a small maximum, now related to the emergence of bulk electrons, appears of the MR dependence on the gate voltage.

Thus, detailed analysis of the specific feature of classic transport enabled arriving at a self-consistent picture of energy bands. The band gap value found using the difference between electron densities at Vg = 4 V and Vg = 2 V proved to be 15 meV, a value that agrees well with calculations. To conclude, we add that the behavior of the Shubnikov–de Haas (SdH) oscillations presented in Section 4.3 also confirms the described picture.

In this section, we discuss the specific features of the behavior of SdH oscillations and the quantum Hall effect. Figure 19 displays the dependences $\rho_{xx}(V_{g})$ and $\rho_{xy}(V_{g})$ measured in magnetic fields up to 4 T. As the magnetic field increases, a sharp increase is observed in the maximum resistance located at the CNP at $V_{g}=1V$, where it reaches $10^{7}\Omega/\Box$ at B = 10 T (not shown in the figure). The sign of the dependences $\rho_{xy}(V_{g})$ changes at the same gate voltage. A monotonic dependence with minor inflection is only observed with a maximum field of 4 T to the left of the CNP, i.e., in the hole region. In contrast, to the right of the CNP, where the conductivity is determined by highly mobile electrons, well-pronounced QHE plateaus in the $\rho_{xy}(V_{g})$ dependence occur already at B = 2T. We note that the QHE also occurs in the $V_{g}$ regions in which Dirac and bulk electrons coexist. We now analyze this situation in more detail.

The dependences $\rho_{xx}(B)$ for fixed $V_{g}$ values are displayed in Fig. 19c, d. The observed picture corresponds as a whole to the dependences on the gate voltage: on the hole side, the SdH oscillations are rather weakly pronounced; on the electron side, on the contrary, deep minima are observed due to high mobility, which indicates that the QHE regime is in effect, and well-pronounced $\rho_{xy}$ plateaus are formed in magnetic fields as low as 2 T.

However, no exponentially small values are observed at the $\rho_{xx}$ minima even in large fields. This observation may be an indication of possible parallel conduction channels, for example, along the lateral surfaces of the film oriented along the applied field. The electron density determined using the position of the SdH oscillation minima in strong magnetic fields ($B>1-2T$) turned out to be equal to the density calculated using the Drude model. These values are compared in Fig. 20d. It follows from the coincidence of and that the filling factors $\nu$ are determined by the total density , i.e., the sum of the densities of Dirac and bulk electrons. Similarly, the hole densities obtained for large negative Vg from the analysis of SdH oscillations in strong fields and from fitting in the Drude model turn out to be quite close. However, in approaching the CNP, is systematically smaller than . It can be concluded based on these observations that the behavior of the QHE, when the Fermi level is located in the valence band, is driven by the difference between the densities of holes and electrons. The essentially important conclusion is that in strong magnetic fields, surface charge carriers also participate in the formation of unified Landau levels.

To analyze the behavior of the SdH oscillations even more deeply, we determined how the number N of the minima of these oscillations shown in Fig. 19c, d depends on their position on the axis of the inverse magnetic field 1/B. These dependences are plotted in Fig. 20a, b. The oscillations are weakly pronounced on the hole side (Fig. 19a, c). In magnetic fields less than 1–2 T, only oscillations with odd numbers remain discernible with magnetic fields up to B = 0.4 T, with corresponding filling factors exceeding 10. Each of the obtained dependences is well approximated by a straight line passing through the origin. The slope of this line corresponds to the differential hole–electron concentration mentioned above.

The SdH oscillations on the electron side are much more pronounced (Fig. 19d), regardless of whether the Fermi level is located in the band gap or in the conduction band. The oscillations remain discernible in magnetic fields up to 0.25 T with corresponding filling factors greater than 20. A more careful analysis shows that regions of weak and strong magnetic fields with a sharp transition between them can be found in any of the dependences displayed in Fig. 20b. The periodicity of oscillations as a function of the inverse magnetic field persists in each of the regions; however, the slopes of the lines that correspond to these regions of the N(1/B) dependence not the same. The difference between the densities found using the slope of these lines is $20–45\%$.

Thus, the presence of two regions with different slopes cannot be explained by any degeneracy lifted by the magnetic field.

On the other hand, the presence of two densities, determined by the periodicity of SdH oscillations in weak and strong magnetic fields, can be explained by the existence of two (or more) groups of carriers, each of which has its own set of Landau levels. Such a situation is quite possible if the effects of gate screening by the upper surface are taken into account. This would result in different densities of Dirac electrons on the upper and lower surfaces.

We now assume that flat bands in the system under study are formed near the zero gate voltage, and the concentrations and are equal. The band diagram of the structure under study at this gate voltage and at other $V_{g}$ is schematically depicted in Fig. 21. The flat-band situation is actually realized if $N_{s}^{top}=N_{s}^{bot}$. However, as the gate voltage increases, the density $N_{s}^{top}$ increases much faster than $N_{s}^{bot}$. The ratio of the filling rates $\alpha=(dN_{s}^{top}/dV_{g})/(dN_{s}^{bot}/dV_{g})$ of the surfaces can be estimated in the simplest way in the absence of bulk carriers, i.e., at $2\leq V_{g}\leq 4V$. The ratio is given under these conditions by the formula $\alpha=1+(e^{2}Dd_{HgTe}/\epsilon_{HgTe}\epsilon_{0})$, where D is the density of states of Dirac electrons on the upper surface and $d_{HgTe}$ and $\epsilon_{HgTe}$ are the thickness and dielectric constant of the mercury telluride film. Substituting typical values, we obtain $\alpha=3-5$.

Thus, the emergence of an electric field in the HgTe film should lead to electron densities that are different on the top and bottom surfaces. The main mechanism of electron scattering at T = 4.2 K is their scattering by residual impurities. It is then natural to assume that, of two identical groups of carriers, the higher-density group has greater mobility and less broadening of the Landau levels. The Landau levels on the top surface are less broadened in this case, and SdH oscillations start forming for this surface in weaker fields. As a result, the SdH oscillation period in weak fields only yields the electron density on the top surface, while both groups of carriers (to which bulk electrons are added at $V_{g}>4V$) are quantized in strong fields, and the oscillation period yields their total density.

The dependence determined in this way is shown in Fig. 20d. Because only surface electrons are present in the system in the range $2\leq V_{g}\leq 4V$, the relation is valid in this range, and can be determined. As expected, the experimentally found filling rate is three times lower than $dN_{s}^{top}/dV_{g}$. Another specific feature should be emphasized: a sharp bending of the dependence at $V_{g}=4V$. A decrease in the slope $dN_{s}^{top}/dV_{g}$ at $V_{g}>4V$ apparently can only be associated with a decrease in the contribution of the density of states of surface electrons to the total density of states. This conclusion is in line with the previously proposed picture of the energy spectrum in which the Fermi level enters the conduction band at the indicated gate voltage.

Finally, the assumption that the weak-field part of the SdH oscillations emerges due to Dirac fermions is confirmed by the phase of these oscillations, the analysis of which for a system in the TI state ($V_{g}=3V$) is presented in Fig. 20. The formula $(1/B_{min})/\Delta_{1/B}=\nu_{tot}$ turns out to be correct for the strong-field part of the dependence. Here, $1/B_{min}$ is the location of the minima in the inverse magnetic field, $\Delta_{1/B}$ is the period of oscillations in the inverse field determined by the total density, and $\nu_{tot}$ is an integer that corresponds to the total filling factor for all types of electrons. However, if the weak-field part of the $N(1/B_{min})$ dependence is approximated by a linear function and continued until it crosses the vertical axis, then the intersection occurs at 1.63. The obtained linear dependence is described by the formula $(1/B_{min})/\Delta_{1/B}=\nu_{top}+0.63$, where $\nu_{top}$ is the electron filling factor on the top surface (determined up to an integer), and 0.63 is the phase oscillation shift. An approximation of the weak-field part by a linear dependence with the phase shift ignored (dashed lines in Fig. 20c) yields an inferior result. Thus, a phase shift of $0.63\pm 0.023$ is observed in weak-field oscillations, which is close to the predicted value of 0.5 for spin-polarized Dirac fermions [59].

As can be seen from Section 4.3, an analysis of SdH oscillations indicates the presence of their anomalous phase. However, because the transport response contains competing contributions from both surfaces of the TI, this does not allow drawing an unambiguous conclusion that the transport oscillations of the top surface are undisturbed by the contribution of the bottom one. Therefore, a conclusion on the anomalous phase being observed can only be made with some caution.

We show in this section that the capacitive spectroscopy of a 3D TI makes it possible to circumvent this difficulty and obtain more accurate and detailed information on the behavior of the SdH oscillations of 2D DFs on one (upper) surface and thus demonstrate the anomalous behavior of their phase due to rigid topological coupling of spin and momentum.

Figure 22 shows the structure of the sample under study and the equivalent circuit in which capacitive response is measured. The circuit diagram clearly shows that the capacitance of the upper DFs is separated from that of the lower DF by the capacitance of the film and therefore the capacitive response of the upper surface of the investigated TI should ’feel’ the effect of the lower one to a much lesser extent than when the transport response is measured. Detailed measurements of the capacitive SdH oscillations have confirmed this assumption. The measurements showed that when the Fermi level is located in the TI band gap, capacitive oscillations of only the upper DFs are observed, which exhibit an anomalous phase in a wide range of magnetic fields. Below, we discuss these observations in detail.

Figure 23b displays the capacitance $C(V_{g})$ as a function of the gate voltage in the absence of a magnetic field and at B = 2T. It can be seen that in the absence of the magnetic field, $C(V_{g})$ has a minimum, which, as a comparison with transport data shows (Fig. 23a) corresponds to the Fermi level passing through the bulk gap, while the application of a magnetic field results in the emergence of SdH oscillations. It is clearly seen that the oscillations are virtually absent when the Fermi level is located in the valence band and have a significant amplitude when it passes through the bulk gap and the conduction band. Such behavior shows that the DF mobility significantly increases when the Fermi level exits the valence band.

An analysis of the period of oscillations inside the band gap shows that they are determined for all magnetic fields by nondegenerate Landau levels of the DFs on the upper surface. This is evidence that, as one would expect, 2D DFs of the upper surface form a spin-polarized system, thus indicating its topological nature. Vivid evidence of this nature is the behavior of the SdH oscillation phase [59] described by the formula $\frac{1/B_{min,n}}{\Delta_{1/B}}=n+\delta$

where $B_{min}$, n is the location of the n-th minimum of oscillations and $\Delta 1/B$ is the period of oscillations in the inverse magnetic field. We can use this formula to determine the phase by linearly extrapolating the oscillation period to zero on the scale of the inverse magnetic field 1/B as a function of the oscillation number n. Due to the selectivity of capacitive spectroscopy noted above, which allows studying the DF oscillations of only the upper surface of the TI, the required set of oscillations can be obtained with high accuracy in a wide range of magnetic fields by measuring the dependences C(B) for given $V_{g}$.

Figure 24a shows the dependences of 1/B on n determined in this way. Shown for comparison in the same figure is a similar dependence but plotted based on the analysis of magnetotransport data at $V_{g}$ = 6 V, when the Fermi level is located deep in the conduction band. It is clearly seen that while the minima obtained from the analysis of the capacitance are well described by straight lines crossing the horizontal axis with the expected offset from zero, the corresponding dependence of the transport oscillation minima can only be described using two straight lines that separate the obtained dependence into weak-field and strong-field regions. In weak fields, where the magnetotransport oscillations only emerge due to the upper surface and thereby reproduce the behavior of the magnetic capacitance, the phase shift obtained from the transport data $\Delta_{transport}=0.72\pm 0.04$ yields the same value that was found from the capacitance data, $\Delta_{capacitance}=0.7\pm 0.04$. More than one carrier group is involved in strong fields in the formation of transport oscillations, and the shift extracted from data fitting in this region is already close to zero, as is expected for a conventional 2D electron system. Figure 24b shows the evolution of the phase $\delta$, extracted from the capacitance data, for all positions of the Fermi level. It is clearly seen that $\delta$ is close to 0.5 just when the Fermi level is located in the bulk gap, i.e., when capacitance oscillations are only formed by topologically stable 2D DFs of the upper surface. As soon as $E_{F}$ enters the valence band, the phase rapidly vanishes and, in contrast, gradually increases as $V_{g}$ increases, approaching unity at the maximum positive $V_{g}$. This observation can be explained by the hybridization of surface and bulk carriers deep in the conduction band.