Although the quantum Hall (QH) effect has been understood as a consequence of topological order for nearly four decades (1, 2), the interest in the field of topology has rekindled in the past decade owing to the successful realization of topological insulators (TIs) (3–5). These systems support gapless conducting states at the surface with an insulating bulk in the absence of an external magnetic field (6). Among the known topological systems, HgTe is a versatile material as it can be tuned from a trivial insulator to a two- or three-dimensional (2D or 3D) TI by changing thickness and strain (3, 5, 7). Confinement in a HgTe quantum well (QW) above a critical thickness results in an inverted band structure and forms a 2D TI with quantum spin Hall (QSH) edge states protected by time-reversal symmetry (3, 8). The band structure can be further engineered by alloying HgTe with magnetic or nonmagnetic atoms. Nonmagnetic dopants mainly affect the bandgap, while magnetic dopants can have more profound effects as they can break time-reversal symmetry. In case of HgTe QWs, a magnetically doped 2D TI can be realized by incorporating Mn atoms (9). In contrast to magnetically doped bismuth-based TIs (10), (Hg,Mn)Te is paramagnetic, and hence, in the absence of an external magnetic field, the magnetization is zero (11, 12). A QSH state is therefore expected when the band structure is in the inverted regime. The interest in (Hg,Mn)Te arises from the prediction of a quantum anomalous Hall (QAH) state when Mn is magnetized (9). It has been predicted that a chiral topological superconductor hosting chiral Majorana fermions can be realized when a chiral edge channel (either from a QH or a QAH state) is proximitized to an s-wave superconductor (13). However, the large magnetic fields (>1 T) required for QH effects and the large magnetization (approximately a few tesla) in conventional QAH systems (e.g., vanadium- or chromium-doped Bi-based topological systems) make it challenging to induce superconductivity and explore chiral Majorana fermions. It is thus important to find systems where the QH effects can be observed at very low magnetic fields and low magnetization.
Here, we show two emergent QH phenomena arising from different mechanisms at unusually low magnetic fields in (Hg,Mn)Te-based 2D TIs. First, when the chemical potential is tuned into the bulk gap (the “QSH regime”) of a direct-gap 2D TI, we observe a ν = −1 QH plateau (ν is the filling factor), which emerges from the QSH state in perpendicular magnetic fields as low as 50 mT. Band structure analysis demonstrates that this QH state forms due to splitting of the Dirac crossing and subsequent hybridization of one of the QSH edge states with the bulk states. The ν = −1 plateau is located outside the topological gap (QSH regime) and is therefore fundamentally different from the QAH state predicted in (9).
Second, when the chemical potential is tuned into the valence band (p-conducting) regime, we find that the transport is predominantly affected by the “camel back,” i.e., the maximum in the valence band at a large momentum that arises as a consequence of hybridization between the subbands in an inverted band structure. At finite magnetic fields, the magnetization of Mn leads to corrections in the dispersion relation via the paramagnetic exchange interaction (9, 12). The interplay between this exchange interaction and the van Hove singularity [very large density of states (DOS)], induced by the camel back, leads to rich Landau level (LL) structures. In direct-gap TIs, the large DOS at the camel back pins the chemical potential, which leads to formation of highly mobile carriers (mobility ~106 cm2 V−1 s−1) at the Γ point with an extremely low carrier density of ∼2 × 109 cm−2. This results in multiple resolved QH plateaus below 100 mT at low temperatures. Magneto-transport measurements in tilted magnetic fields support this pinning picture. In contrast, in indirect-gap TIs, the pinning mechanism permits QH plateaus in the bulk p-regime only at large magnetic fields.
The emergent QH phenomena rely critically on the preexistence of QSH states at zero field and the van Hove singularity in the DOS, both of which originate from the inverted band structure of HgTe. Since band inversion makes HgTe topological, our findings have a direct impact on the fundamental understanding of transport phenomena in all topological materials. In addition, the occurrence of emergent QH effects in (Hg,Mn)Te QWs at extremely low fields makes this system an ideal platform for interfacing with s-wave superconductors to realize chiral Majorana fermions (13).
RESULTS AND DISCUSSION
We explore the topological phase space of (Hg,Mn)Te QWs using k · p theory based on the 8 × 8 Kane Hamiltonian applied to bulk and strip geometries (with confinement in one and two directions, respectively), where the latter is necessary to describe both bulk and edge states. Despite this method being computationally heavy, we avoid using simplified band models such as the one proposed by Bernevig, Hughes, and Zhang (BHZ) (8), since they do not capture the essential physics in the valence band, i.e., the camel back. The existence of the camel back in the band structure of these QWs has been known for many decades (14, 15) and has also attracted attention recently (7, 16). The large DOS at the camel back has been shown to give rise to an interaction-induced 0.5 anomaly in the HgTe-based topological quantum point contacts (17).
The transport behavior of (Hg,Mn)Te QWs can be essentially understood from two properties of the zero-field dispersion: the band ordering at k = 0, as well as the presence and positions of additional extrema in the dispersion at finite momenta. The first property is well known for HgTe QWs, which exhibit a “trivial” band order below the critical thickness dc ≈ 6.3 nm and host an inverted band structure above dc, characterized by the QSH phase (8). Because of the influence of the Mn atoms on the band structure, this critical thickness varies with Mn concentration. In Fig. 1A, we map out the parameter space of band ordering in terms of QW thickness dQW and Mn concentration: The white and red areas indicate the trivial and the inverted regime, respectively. Second, band structures can be distinguished by having either a direct or an indirect bandgap depending on the position of the camel back. In Fig. 1A, the direct-indirect transition is represented by a dashed curve.
We focus on devices fabricated from two (Hg,Mn)Te QWs, which are 11 nm thick with a Mn concentration of 2.4% and 1.2% labeled as Dev 1 and Dev 2 in Fig. 1, respectively. The QWs are embedded between Hg0.32Cd0.68Te barriers on a lattice-matched Cd0.96Zn0.04Te substrate and patterned into Hall bars of length l = 600 μm and width w = 200 μm (see Materials and Methods for fabrication details). The Mn atoms substitute Hg atoms, isoelectrically ensuring that neither carrier doping nor degradation in mobility takes place. The k · p band structures show that Dev 1 has a direct gap (Fig. 1B), while Dev 2 has an indirect gap (Fig. 1C). We show results from Dev 1 here and from Dev 2 in the Supplementary Materials. All measurements are performed at temperature T = 20 mK.
Chemical potential in the bulk gap—QSH regime
The devices can be tuned from n– to p-regime by applying more negative gate voltage Vg, as shown in the inset of Fig. 2A. The bulk gap is around
is the normalized gate voltage with Vd being the gate voltage at maximal longitudinal resistance Rxx. For the QH studies here, we focus on macroscopic devices, where Rxx does not reach the value of h/2e2 characteristic of an ideal QSH edge configuration (18–20). Having the chemical potential in the bulk gap, we observe that an out-of-plane magnetic field B⊥ leads to an unexpected quantization of the transverse resistance Rxy to −h/e2 corresponding to a ν = −1 QH plateau. Simultaneously, Rxx drops to zero (Fig. 2A). The transition to the ν = −1 QH plateau occurs at an anomalously low value of B⊥ ≈ 85 mT. This behavior is unexpected from previous experimental investigations of undoped HgTe QWs in the QSH regime, where a transition to a ν = −1 QH plateau was only observed for B⊥ > 1.0 T (3). Theoretical investigations on the behavior of QSH edge states in magnetic fields have shown that they can survive up to a few tesla, even in the absence of protection by time-reversal symmetry (21–23). Chen et al. (23) have predicted “QAH-like” states in HgTe QWs with ν = ±1 in magnetic field–induced gaps. However, that analysis from the BHZ model suggests an onset field of the ν = −1 plateau at ~1 T, larger than the onset field that we observe.
From an Arrhenius plot of conductance as a function of temperature, we estimate a bulk gap of ~4.6 meV for Dev 1 (fig. S1), which agrees well with the theoretical value of 4.0 meV. If the magnetization of Mn could close the topological bulk gap for one of the edge states (9), a ν = −1 plateau would indicate a QAH phase. However, in our case, a magnetic field of 85 mT, corresponding to a spin polarization 〈m〉 ∼ 0.15 [Eq. (S6)], can only close a gap of ~1 meV (24). For this estimate, the combined effect of exchange interaction, Zeeman effect, and an additional orbital contribution has been taken into account (24). Since the topological bulk gap is a factor of ~5 larger than this theoretical estimate, it is unlikely that the low-field ν = −1 plateau in our (Hg,Mn)Te QW indicates a transition to the QAH phase.
The early onset of the ν = −1 QH plateau occurs for
ranging from 0.28 to −0.4 V with the onset field as low as 50 mT (Fig. 2B). We emphasize that the ν = −1 plateau (near the QSH phase) cannot occur from the conventional QH effect in a low density system for the following reasons. First, we observe the ν = −1 plateau even for small positive
(Fig. 2B) where we would conventionally expect a ν = +1 plateau. Second, in this picture, we would expect the p-density to increase for more negative
, which would imply higher fields to observe the QH plateaus. This is in direct contrast to Fig. 2B, where we observe lower onset fields for more negative
. The occurrence of QH plateaus at such low magnetic fields has not been reported so far.
To find out the microscopic origin of the low-field ν = −1 QH plateau, we have calculated dispersions in the presence of a magnetic field. Figure 2 (C and D) shows k · p band structures of a (Hg,Mn)Te sample in a strip geometry. The width w = 500 nm is smaller compared to that of the measured devices for practical reasons, but this choice will not affect our conclusions. Energies E are defined with respect to the conduction band (Γ8) of unstrained bulk HgTe (section S4). For zero magnetic field (Fig. 2C), we can distinguish the bulk bands (gray) and the edge states (red and blue). The (twofold degenerate) edge states cross (up to a small finite size gap) at the top of the valence band at the Γ point. Below E ≈ −52 meV, there is a high density of bulk valence band states, brought about by the camel back structure. The top of the valence band is at kx ≈ 0.4 nm−1, outside the plot range.
Upon application of a magnetic field, the different values of the magnetic gauge field A at the opposite edges (recall the Peierls substitution kx → kx + eAx/ħ) cause the degeneracy of the edge states to be lifted. The crossing at kx = 0 splits into two copies, one moving up and one moving down in energy (Fig. 2D). The lower-energy copy disappears rapidly due to hybridization of the edge states with the bulk. The higher-energy copy survives, even in the bulk conduction band. Hybridization between these edge states and the bulk states is (almost) absent, because they differ in orbital character as well as in wave function localization. Simultaneously, LLs form, starting near the original (zero-field) Dirac point near E = −49 meV. The topological gap, characterized by a pair of counterpropagating edge states, is reduced in magnetic fields (see Fig. 2D, green shading) (21, 24). Below this energy, the single edge state survives, indicating a transition to the magnetic gap (gap between LLs induced by the magnetic field; see Fig. 2D, yellow shading). In contrast to the QAH effect, where a ν = −1 plateau arises in the topological gap as a result of a bulk bandgap closing (for one “spin” channel) driven by the magnetization of Mn (9), the present ν = −1 plateau appears in a magnetic gap in the absence of a bulk bandgap closing. The preexistence of the QSH edge state leads to the emergence of a QH state at effectively lower magnetic fields.
Chemical potential in the bulk p-regime
The early onsets of the emergent QH plateaus also occur when the chemical potential is tuned into the bulk p-regime (schematic in the inset of Fig. 3A). In this regime, we see QH plateaus from ν = −1 to −5 at all values of
from −1 to −2 V. The well-resolved QH plateaus are visible in Rxy measured as a function of B⊥ (Fig. 3A) and in the LL fan (Fig. 3B). All the QH plateaus are observed at low values of B⊥ < 150 mT, while at certain gate voltages, resolved plateaus can be seen for B⊥ as low as 20 to 30 mT. In addition, the ν = −1 plateau persists up to exceptionally high B⊥ ∼ 9 T (fig. S3). The origin of these QH plateaus is different from those in the previous section, as we will explain below.
To understand these low-field QH plateaus, we show in Fig. 3C the calculated bulk LL fan for an infinite QW in the p-regime; the color code indicates the DOS. In the following, we assume that the DOS of each LL exhibits a Gaussian broadening as used in (12). In Fig. 3C, the white area in the valence band (E < − 52 meV) indicates that the DOS is ~400 times larger than in a single, conventional LL. This regime is formed by a very dense collection of LLs that arise due to the camel back. In particular, below B⊥ < 70 mT, LLs with small LL indices (j = −2, −1, …), resulting from highly mobile carriers at small k, can coexist at the same energy with higher LLs (j ∼ 500 to 1000), stemming from the camel back. For
, the chemical potential is pinned to the upper edge of the camel back since any small decrease of the chemical potential would result in a large increase of the p-density. Therefore, the large DOS originating from the camel back (van Hove singularity) is responsible for an early onset of QH plateaus with ν = −1 to −5 for a large range of p-densities, as demonstrated in Fig. 3C (see also section S6). This finding is in good qualitative agreement with the experimental results (Fig. 3, A and B). The small size of even plateaus in the theoretical result is a side effect of neglecting the effects of bulk-inversion asymmetry terms for which the exact strength is not known for (Hg,Mn)Te QWs (section S6 and fig. S10). In addition, pinning to the camel back can also explain the exceptionally long ν = −1 QH plateau, which is observed in the experiment (section S1).
Another consequence of the pinning mechanism is that all observed QH plateaus in the p-regime are very sensitive to the difference in energy of the valence band at k = 0 and the maximum of the camel back at B⊥ = 0 (ΔE in Figs. 3C and 4F). The onset fields of the QH plateaus decrease as ΔE decreases. This results from the fact that the upper edge of the camel back in magnetic fields (dashed white line in Fig. 3C) extrapolates to the camel back at B⊥ = 0 (cf. Figs. 1B and 3C). Consequently, we expect that any (Hg,Mn)Te TI with a direct gap and close to the direct gap–to–indirect gap transition (dashed line in Fig. 1A) exhibits a similar characteristic behavior in the p-regime. In contrast to the Mn-free case, we expect that for Mn-doped QWs, the onset fields are shifted to even smaller B⊥, since the exchange coupling increases the slope of the lowest LLs and the camel back height in magnetic fields.
Despite the macroscopic occupation of bulk states near the camel back, the bulk conduction remains suppressed at low temperatures, because bulk carriers are localized by disorder. The higher Landau plateaus are well resolved in experiments at 20 mK, but they are not robust against increased temperatures (section S1, B and D). The mobile p-carrier density as estimated from the classical Hall slope at low field (B⊥ < 20 mT) is ~2 × 109 to 3 × 109 cm−2 for the entire range of
shown in Fig. 3A (see also section S2), which agrees well with the theoretically calculated density of the highly mobile carriers lying above the camel back (p ≈ 2 × 109 cm−2). The density of localized carriers at the camel back as estimated from the known gate capacitance is 4 × 1011 to 5 × 1011 cm−2. These are crucial for efficiently screening the disorder and thus leading to highly mobile carriers. As extracted from the Drude model and the experimentally measured mobile carrier density and conductivity, the p-carrier mobility is ~6 × 105 to 9 × 105 cm2 V−1 s−1 (section S2). Thus, the van Hove singularity at the camel back is crucial for observing perfectly quantized Hall plateaus at such low carrier densities and anomalously low magnetic fields.
Magneto-transport in tilted magnetic fields
To further distinguish the magneto-transport in the QSH and in the bulk p-regime, we have performed measurements in tilted magnetic fields. The measurement schematic is shown in the inset of Fig. 4A where the sample is in the xy plane and the magnetic field B = (Bx, By, Bz) = (B∥, 0, B⊥) = BT(sin θ, 0, cos θ) is applied at an angle θ to the z direction. The measurements are performed for various values of θ. The current I flows along the length of the sample. We choose four values of
, which encompass the two regimes: two points in the vicinity of the QSH regime (chemical potential in the bulk gap) and two in the bulk p-regime. In the QSH regime, the onset field for the ν = −1 QH plateau depends only on B⊥ = Bz [shown for
and −0.4 V in Fig. 4 (A and B, respectively)] and not on Bx for all values of θ. This is markedly different from the p-regime, where the onset fields for all observed QH plateaus with ν = −1 to −5 are strongly dependent on θ. For larger θ, all QH plateaus (including ν = −1) appear at lower B⊥ as shown for
and −1.2 V in Fig. 4 (D and E), respectively.
The observed differences can be captured within our theoretical model. Figure 4C shows the dispersion for a strip geometry with the magnetic field at an angle θ = 60∘ and B⊥ = 100 mT. Comparing this band structure to Fig. 2D, we find that an in-plane magnetic field does not alter the band structure close to kx = 0 significantly. This agrees well with the experimental observation in Fig. 4 (A and B) indicating that the ν = −1 plateau is related to the QH state forming outside of the topological gap. In contrast, the height of the camel back is very sensitive to the in-plane magnetic field and increases for larger θ (implying that ΔE decreases), as shown in Fig. 4F. The decrease of ΔE indicates lower onset fields for the QH plateaus in the bulk p-regime, as observed experimentally in Fig. 4 (D and E). The difference between the k = 0 and the camel back regime is connected to the interplay between the exchange coupling and the variation in orbital character of the band structure as a function of momentum (section S5). A strong in-plane field dependence of the camel back can therefore only exist in Mn-doped samples.
We do not see any evidence of early onsets of QH plateaus for Dev 2. The QH plateaus for Dev 2 occur for B⊥ > 1 T (fig. S6B). Since Dev 2 has an indirect gap, this observation is in agreement with our model, which predicts an early onset of QH plateaus only for direct gap TIs close to the direct-indirect transition (Fig. 1A). This also highlights the crucial role of Mn in tuning the band structure and in realizing the emergent QH phenomena, since the two devices have the same thickness and differ only in Mn concentration.
Acknowledgments: We acknowledge discussions with S.-C. Zhang, B.A. Bernevig, C.-X. Liu, X.-L. Qi, C. Gould, and C. Brüne. We thank K. Bendias for fabricating some of the devices. Funding: We acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) in the Leibniz Program and in the projects SFB 1170 (Project ID 258499086) and SPP 1666 (Project ID 220179758), from the EU ERC-AdG program (Project 4-TOPS), from the Würzburg-Dresden Center of Excellence “Complexity and Topology in Quantum Matter” (CT.QMAT), and from the Free State of Bavaria (Elitenetzwerk Bayern IDK “Topologische Isolatoren” and the Institute for Topological Insulators). Author contributions: L.W.M. and H.B. planned the project. S.S. conducted the measurements and analyzed the data with help from A.B. and P.S. The band structure calculations were performed by W.B. and J.B. The material was grown by P.L. and L.L. The device was fabricated by P.S. All authors contributed to interpretation of the results. L.W.M., H.B., and E.M.H. supervised the project. All authors participated in writing the manuscript led by S.S., W.B., and J.B. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.