We review low and high field magnetotransport in 80 nm-thick strained HgTe, a material that belongs to the class of strong three-dimensional topological insulators. Utilizing a top gate, the Fermi level can be tuned from the valence band via the Dirac surface states into the conduction band and allows studying Landau quantization in situations where different species of charge carriers contribute to magnetotransport. Landau fan charts, mapping the conductivity $ σ_{xx}(V_g, B) $ in the whole magnetic field - gate voltage range, can be divided into six areas, depending on the state of the participating carrier species. Key findings are: (i) the interplay of bulk holes (spin-degenerate) and Dirac surface electrons (non-degenerate), coexisting for $ E_F $ in the valence band, leads to a periodic switching between odd and even filling factors and thus odd and even quantized Hall voltage values. (ii) A similar though less pronounced behavior we found for coexisting Dirac surface and conduction band electrons. (iii) In the bulk gap, quantized Dirac electrons on the top-surface coexist at lower B with non-quantized ones on the bottom side, giving rise to quantum Hall plateau values depending - for a given filling factor - on the magnetic field strength. In stronger $ B $ fields, Landau level separation increases, charge transfer between different carrier species becomes energetically favorable and leads to the formation of a global (i.e. involving top and bottom surface) quantum Hall state. Simulations using the simplest possible theoretical approach are in line with the basic experimental findings, describing correctly the central features of the transitions from classical to quantum transport in the respective areas of our multicomponent charge carrier system.
The double quantum well systems consisting of two HgTe layers separated by a tunnel-transparent barrier are expected to manifest a variety of phase states including two-dimensional gapless semimetal and two-dimensional topological insulator. The presence of several subbands in such systems leads to a rich filling factor diagram in the quantum Hall regime. We have performed magnetotransport measurements of the HgTe-based double quantum wells in both gapless and gapped state and observed numerous crossings between the Landau levels belonging to different subbands. We analyze the Landau level cro…
HgTe/HgCdTe quantum wells with the inverted band structure have been probed using far infrared magneto-spectroscopy. Realistic calculations of Landau level diagrams have been performed to identify the observed transitions. Investigations have been greatly focused on the magnetic field dependence of the peculiar pair of "zero-mode" Landau levels which characteristically split from the upper conduction and bottom valence bands, and merge under the applied magnetic field. The observed avoided crossing of these levels is tentatively attributed to the bulk inversion asymmetry of zinc blend compound…
We present both the experimental and theoretical investigation of a non-trivial electron Landau levels shift in magnetic field in wide ~20 nm HgTe quantum wells: Landau levels split under magnetic fields but become degenerate again when magnetic field increases. We reproduced this behavior qualitatively within an isotropic 6-band Kane model, then using semiclassical calculations we showed this behavior is due to the mixing of the conduction band with total spin 3/2 with the next well subband with spin 1/2 which reduces the average vertical spin from 3/2 to around 1. This change of the average …
Landau level spectroscopy has been employed to probe the electronic structure of the valence band in a series of p-type HgTe/HgCdTe quantum wells with both normal and inverted ordering of bands. We find that the standard axial-symmetric 4-band Kane model, which is nowadays widely applied in physics of HgTe-based topological materials, does not fully account for the complex magneto-optical response observed in our experiments - notably, for the unexpected avoided crossings of excitations and for the appearance of transitions that are electric-dipole forbidden within this model. Nevertheless, re…
The manuscript contains Comment to "Quantum Spin Hall Effect and Topological Phase Transitio in HGTe Quantum Wells" by B. Andrei Bernevig, Taylor L. Hughes and Shou-Cheng Zhang Science 314, 1757 (2006). The text proves that the quantum state, identified by these authors as the surface state located inside HgTe well is incorrect physically. The kinetic energy of the state is negative. The state cannot be identified as proper eigenfunction (vector) in the Hilbert space. Therefore the state could be used as a primer in the topological state theory, developed in the following publications.
The parallel field of a full spin polarization of the electron gas in a \Gamma8 conduction band of the HgTe quantum well was obtained from the magnetoresistance by three different ways in a zero and quasi-classical range of perpendicular field component Bper. In the quantum Hall range of Bper the spin polarization manifests in anticrossings of magnetic levels, which were found to strongly nonmonotonously depend on Bper.
We demonstrate evidences of electronic transport via topological Dirac surface states in a thin film of strained HgTe. At high perpendicular magnetic fields, we show that the electron transport reaches the quantum Hall regime with vanishing resistance. Furthermore, quantum Hall transport spectroscopy reveals energy splittings of relativistic Landau levels specific to coupled Dirac surface states. This study provides new insights in the quantum Hall effect of topological insulator (TI) slabs, in the cross-over regime between two- and three-dimensional TIs, and in the relevance of thin TI films …
We study the spin-dependent transmission through interfaces between a HgTe/CdTe quantum well (QW) and a metal - both for the normal metal and the superconducting case. Interestingly, we discover a new type of spin Hall effect at these interfaces that happens to exist even in the absence of structure and bulk inversion asymmetry within each subsystem (i.e. the QW and the metal). Thus, this is a pure boundary spin Hall effect which can be directly related to the existence of exponentially localized edge states at the interface. We demonstrate how this effect can be measured and functionalized fo…
We show that the Quantum Spin Hall Effect, a state of matter with topological properties distinct from conventional insulators, can be realized in HgTe/CdTe semiconductor quantum wells. By varying the thickness of the quantum well, the electronic state changes from a normal to an "inverted" type at a critical thickness $d_c$. We show that this transition is a topological quantum phase transition between a conventional insulating phase and a phase exhibiting the QSH effect with a single pair of helical edge states. We also discuss the methods for experimental detection of the QSH effect.
A Zener diode is a paradigmatic device in semiconductor-based electronics that consists of a pn junction where an external electric field induces a switching behavior in the current-voltage characteristics. We study Zener tunneling in HgTe quantum wells and graphene multilayers. We find that the tunneling transition probability depends asymmetrically on the parallel momentum of the carriers to the barrier. In HgTe quantum wells the asymmetry is the opposite for each spin, whereas for graphene multilayers it is the opposite for each valley degree of freedom. In both cases, a spin/valley current…
The quantum spin Hall (QSH) states discovered in an inverted band of InAs/GaSb and HgTe/CdTe quantum wells categorize them among the very superior candidates for topological insulators. In the presence of a magnetic field, these QSH states persist up to a magnetic field equal to the critical field, beyond which the edge states would consist of normal quantum Hall (QH) states. We provide the expression of this critical field which is found consistent with some previous literature. The critical field partitioned the spectrum into two types of quantum states, \textit{viz}. , the Quantum spin Hall…
We report on the longitudinal and Hall resistivities of a HgTe quantum well with inverted energy spectrum (dQW = 20.3 nm) measured in the quantum Hall (QH) regime at magnetic fields up to 9 T and temperatures 2-50 K. The temperature dependence of the QH plateau-plateau transition (PPT) widths and of variable range hopping (VRH) conduction on the Hall plateaus are analyzed. The data are presented in a genuine scale form both for PPT regions and for VRH regime. Estimations for the degree of the carrier localization length divergence reveal a decisive role of the long-range random potential (the …
We report the first electrical manipulation and detection of the mesoscopic intrinsic spin-Hall effect (ISHE) in semiconductors through non-local electrical measurement in nano-scale H-shaped structures built on high mobility HgTe/HgCdTe quantum wells. By controlling the strength of the spin-orbit splittings and the n-type to p-type transition by a top-gate, we observe a large non-local resistance signal due to the ISHE in the p-regime, of the order of kOhms, which is several orders of magnitude larger than in metals. In the n-regime, as predicted by theory, the signal is at least an order of …
Magnetotransport measurements are presented on paramagnetic (Hg,Mn)Te quantum wells (QWs) with an inverted band structure. Gate-voltage controlled density dependent measurements reveal an unusual behavior in the transition regime from n- to p-type conductance: A very small magnetic field of approximately 70 mT is sufficient to induce a transition into the nu = -1 quantum Hall state, which extends up to at least 10 Tesla. The onset field value remains constant for a unexpectedly wide gate-voltage range. Based on temperature and angle-dependent magnetic field measurements we show that the unusua…
We report on the observation of the quantum Hall effect at high temperatures in HgTe quantum wells with a finite band gap and a thickness below and above the critical thickness $d_\textnormal{c}$ that separates a conventional semiconductor from a two-dimensional topological insulator. At high carrier concentrations we observe a quantized Hall conductivity up to 60\,K with energy gaps between Landau Levels of the order of 25\,meV, in good agreement with the Landau Level spectrum obtained from $\mathbf{k\cdot p}$-calculations. Using the scaling approach for the plateau-plateau transition at $ν=2…
Quantum wells of HgTe doped with Mn display the quantum anomalous Hall effect due to the magnetic moments of the Mn ions. In the presence of a magnetic field, these magnetic moments induce an effective nonlinear Zeeman effect, causing a nonmonotonic bending of the Landau levels. As a consequence, the quantized (spin) Hall conductivity exhibits a reentrant behavior as one increases the magnetic field. Here, we will discuss the appearance of different types of reentrant behavior as a function of Mn concentration, well thickness, and temperature, based on the qualitative form of the Landau-level …
We report transport studies on a three dimensional, 70 nm thick HgTe layer, which is strained by epitaxial growth on a CdTe substrate. The strain induces a band gap in the otherwise semi-metallic HgTe, which thus becomes a three dimensional topological insulator. Contributions from residual bulk carriers to the transport properties of the gapped HgTe layer are negligible at mK temperatures. As a result, the sample exhibits a quantized Hall effect that results from the 2D single cone Dirac-like topological surface states.
We report on the far-infrared magnetospectroscopy of HgTe quantum wells with inverted band ordering at different electron concentrations. We particularly focus on optical transitions from zero-mode Landau levels, which split from the edges of electron-like and hole-like bands. We observe a pronounced dependence of the transition energies on the electron concentration varied by persistent photoconductivity effect. This is striking evidence that in addition to the already well-documented crystalline and interface asymmetries, electron-electron interactions also have a significant impact on the u…
We report on observation of an unconventional structure of the quantum Hall effect (QHE) in a $ p$-type HgTe/Cd$_x$Hg$_{1-x}$Te double quantum well (DQW) consisting of two HgTe layers of critical width. The observed QHE is a reentrant function of magnetic field between two $i=2$ states (plateaus at $ρ_{xy}=h/ie^2$) separated by an intermediate $i=1$ state, which looks like some anomalous peak on the extra-long $i=2$ plateau when weakly expressed. The anomalous peak apparently separates two different regimes: a traditional QHE at relatively weak fields for a small density of mobile holes $p_s$ …
Recent theory predicted that the Quantum Spin Hall Effect, a fundamentally novel quantum state of matter that exists at zero external magnetic field, may be realized in HgTe/(Hg,Cd)Te quantum wells. We have fabricated such sample structures with low density and high mobility in which we can tune, through an external gate voltage, the carrier conduction from n-type to the p-type, passing through an insulating regime. For thin quantum wells with well width d < 6.3 nm, the insulating regime shows the conventional behavior of vanishingly small conductance at low temperature. However, for thicker q…
We investigate an effective low energy theory of HgTe quantum wells near their mass inversion thickness in a perpendicular magnetic field. By comparison of the effective band structure with a more elaborated and well-established model, the parameter regime and the validity of the effective model is scrutinized. Optical transitions in HgTe quantum wells are analyzed. We find selection rules which we functionalize to optically manipulate edge state transport. Qualitatively, our findings equally apply to optical edge current manipulation in graphene.
This work presents theoretical demonstration of a carrier trap unit formed by dual topological insulator constrictions (TIC) on the HgTe/CdTe quantum well (QW) with inverted band structures. The sample of HgTe/CdTe QW is patterned into a Hall bar device and a topological quantum dot is created by adding split gate electrodes closely on the QW. In sharp contrast to conventional semiconductor quantum dots, the presence or absence of topological edge states in the proposed quantum hall bar system leads to distinct propagating/insulating state of the TICs with large on/off ratio. This topological …
The quantum spin Hall (QSH) state is a genuinely new state of matter characterized by a non-trivial topology of its band structure. Its key feature is conducting edge channels whose spin polarization has potential for spintronic and quantum information applications. The QSH state was predicted and experimentally demonstrated to exist in HgTe quantum wells. The existence of the edge channels has been inferred from the fact that local and non-local conductance values in sufficiently small devices are close to the quantized values expected for ideal edge channels and from signatures of the spin p…
We report on magnetospectroscopy of HgTe quantum wells in magnetic fields up to 45 T in temperature range from 4.2 K up to 185 K. We observe intra- and inter-band transitions from zero-mode Landau levels, which split from the bottom conduction and upper valence subbands, and merge under the applied magnetic field. To describe experimental results, realistic temperature-dependent calculations of Landau levels have been performed. We show that although our samples are topological insulators at low temperatures only, the signature of such phase persists in optical transitions at high temperatures…
We compute the phase diagram of the HgTe/CdTe quantum wells in the 3 dimensional (3D) parameter space of Dirac mass, Fermi level and disorder strength. The phase diagram reveals the Quantum spin-Hall, the metallic and the normal insulating phases. The phase boundary of the Quantum spin-Hall state is shown to be strongly deformed by the disorder. Taking specific cuts into this 3D phase diagram, we recover the so called topological Anderson insulator (TAI) phase, but now we can demonstrate explicitly that TAI is not a distinct phase and instead it is part of the Quantum spin-Hall phase. The calc…
We study transport properties of HgTe quantum wells with critical well thickness, where the band gap is closed, and the low energy spectrum is described by a single Dirac cone. In this work, we examined both macroscopic and micron-sized (mesocopic) samples. In micron-sized samples, we observe a magnetic field induced, quantized resistance ($\sim h/2e^{2}$) at Landau filling factor $ν=0$, corresponding to the formation of helical edge states centered at the charge neutrality point (CNP). In macroscopic samples, the resistance near zero Landau level (LL) reveals strong oscillations, which we att…
Many-particle hybridization of optical transitions from zero-mode Landau levels in HgTe quantum wells
Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier（勒内·孔洛实验室）；Physics Department, M.V. Lomonosov Moscow State University（莫斯科国立大学物理系）；CEA-LETI, MINATEC Campus, DOPT（CEA-LETI）
The results of the longitudinal and Hall magnetoresistivity measurements in the Shubnikov - de Haas oscillation regime for the HgCdTe/HgTe/HgCdTe heterostructures with a wide (20.3 nm) HgTe quantum well are presented. An anomalous phase shift of magneto-oscillations is detected: in the region of spin-unsplit peaks the longitudinal resistivity maxima are located at even filling factor numbers in contradiction with a conventional situation in 2D systems. It is shown that the observed features are associated with the inverted nature of the spectrum in the investigated quantum well with the electr…
The realization of the quantum spin Hall effect in HgTe quantum wells has led to the development of topological materials which, in combination with magnetism and superconductivity, are predicted to host chiral Majorana fermions. However, the large magnetization ($\sim$ a few tesla) in conventional quantum anomalous Hall system, makes it challenging to induce superconductivity. Here, we report two different emergent quantum Hall effects in HgTe quantum wells dilutely alloyed with Mn. Firstly, a novel quantum Hall state emerges from the quantum spin Hall state at an exceptionally low magnetic f…
Topological insulators can be seen as band-insulators with a conducting surface. The surface carriers are Dirac particles with an energy which increases linearly with momentum. This confers extraordinary transport properties characteristic of Dirac matter, a class of materials which electronic properties are "graphene-like". We show how HgTe, a material known to exhibit 2D spin-Hall effect in thin quantum wells,\cite{Konig2007} can be turned into a textbook example of Dirac matter by opening a strain-gap by exploiting the lattice mismatch on CdTe-based substrates. The evidence for Dirac matter…
Dirac fermions have been studied intensively in condensed matter physics in recent years. Many theoretical predictions critically depend on the number of valleys where the Dirac fermions are realized. In this work, we report the discovery of a two dimensional system with a single valley Dirac cone. We study the transport properties of HgTe quantum wells grown at the critical thickness separating between the topologically trivial and the quantum spin Hall phases. At high magnetic fields, the quantized Hall plateaus demonstrate the presence of a single valley Dirac point in this system. In addit…
We study nonlocal resistance in an H-shaped two-dimensional HgTe/CdTe quantum well consist of injector and detector, both of which can be tuned in the quantum spin Hall or metallic spin Hall regime. Because of strong spin-orbit interaction, there always exist spin Hall effect and the nonlocal resistance in HgTe/CdTe quantum well. We find that when both detector and injector are in the quantum spin Hall regime, the nonlocal resistance is quantized at $0.25\frac{h}{e^2}$, which is robust against weak disorder scattering and small magnetic field. While beyond this regime, the nonlocal resistance …
The quantum spin Hall (QSH) effect is the property of a new state of matter which preserves time-reversal, has an energy gap in the bulk, but has topologically robust gapless states at the edge. Recently, it has been shown that HgTe quantum wells realize this novel effect. In this work, we start from realistic tight-binding models and demonstrate the existence of the helical edge states in HgTe quantum wells and calculate their physical properties. We also show that 3d HgTe is a topological insulator under uniaxial strain, and show that the surface states are described by single-component mass…
In recent years, Majorana physics has attracted considerable attention in both theoretical and experimental studies due to exotic new phenomena and its prospects for fault-tolerant topological quantum computation. To this end, one needs to engineer the interplay between superconductivity and electronic properties in a topological insulator, but experimental work remains scarce and ambiguous. Here we report experimental evidence for topological superconductivity induced in a HgTe quantum well, a two-dimensional topological insulator that exhibits the quantum spin Hall effect. The ac Josephson e…
We report the studies of high-quality HgTe/(Cd,Hg)Te quantum wells (QWs) with a width close to the critical one $d_c$, corresponding to the topological phase transition and graphene like band structure in view of their applications for Quantum Hall Effect (QHE) resistance standards. We show that in the case of inverted band ordering, the coexistence of conducting topological helical edge states together with QHE chiral states degrades the precision of the resistance quantization. By experimental and theoretical studies we demonstrate how one may reach very favorable conditions for the QHE resi…
HgTe quantum wells possess remarkable physical properties as for instance the quantum spin Hall state and the 'single-valley' analog of graphene, depending on their layer thicknesses and barrier composition. However, double HgTe quantum wells yet contain more fascinating and still unrevealed features. Here we report on the study of the quantum phase transitions in tunnel-coupled HgTe layers separated by CdTe barrier. We demonstrate that this system has a 3/2 pseudo spin degree of freedom, which features a number of particular properties associated with the spin-dependent coupling between HgTe …
In two-dimensional topological insulators, such as inverted HgTe/CdTe quantum wells, helical quantum spin Hall (QSH) states persist even at finite magnetic fields below a critical magnetic field $B_\mathrm{c}$, above which only quantum Hall (QH) states can be found. Using linear-response theory, we theoretically investigate the magneto-optical properties of inverted HgTe/CdTe quantum wells, both for infinite two-dimensional and finite-strip geometries, and possible signatures of the transition between the QSH and QH regimes. In the absorption spectrum, several peaks arise due to non-equidistan…
The influence of bulk inversion asymmetry in [001] and [013] grown HgTe quantum wells is investigated theoretically. The bulk inversion asymmetry leads to an anti-crossing gap between two zero-mode Landau levels in a HgTe topological insulator, i.e., the quantum well with inverted band structure. It is found that this is the main contribution to the anti-crossing splitting observed in recent experimental magneto spectroscopic measurements. The relevant optical transitions involve different subbands, but the electron-electron interaction induced depolarization shift is found to be negligibly sm…
We investigate the behavior of spin polarized currents in two-dimensional topological insulators (TI). Stationary solutions inside a HgTe/CdTe quantum well (QW) were obtained by Bernevig-Hughes-Zhang (BHZ) model modified by a electric and magnetic barrier inside a non-completely insulating bulk. An attenuated quantum spin Hall (QSH) effect occurs in the gaped region with an apparent Klein-like paradox. Even more interesting, for strong potential regime, the interaction between the quasiparticles and the barriers allows spin inversion of this electronic states in a distinct channel conduction. …
Using analytical formulas as well as a finite-difference scheme, we investigate the magnetic field dependence of the energy spectra and magnetic edge states of HgTe/CdTe-based quantum wells in the presence of perpendicular magnetic fields and hard walls, for the band-structure parameters corresponding to the normal and inverted regimes. Whereas one cannot find counterpropagating, spin-polarized states in the normal regime, below the crossover point between the uppermost (electron-like) valence and lowest (hole-like) conduction Landau levels, one can still observe such states at finite magnetic…
We report transport measurements in HgTe-based quantum wells with well width of 8 nm, corresponding to quantum spin Hall state, subject to in-plane magnetic field. In the absence of the magnetic field the local and nonlocal resistances behave very similar, which confirms the edge state transport in our system. In the magnetic field, we observe a monotonic decrease of the resistance with saturation of local resistance, while nonlocal resistance disappears completely, independent of the gate voltage. We believe that these evidences of metallic behavior indicate a transition to a gapless two-dime…
Recent experiments reveal that the strained bulk HgTe can be regard as a three-dimensional topological insulator (TI). Motivated by this, we explore the strain effects on the magnetotransport properties of the HgTe surface states at magnetic field. We analytically derive the zero frequency Hall and collisional conductivities, and find that the substrate induced strain associated with the surface index of carriers, can result in the well seperated surface quantum Hall plateaus and Shubnikov-de Haas oscillations. These effects can be used to generate and detect surface polarization.
We report on Terahertz detection by inverted band structure HgTe-based Field Effect Transistor up to room temperature. At low temperature, we show that nonlinearities of the transistor channel allows for the observation of the quantum phase transition due to the avoided crossing of zero-mode Landau levels in HgTe 2D topological insulators. These results pave the way towards Terahertz topological Field Effect Transistors.
The solutions for the helical edge states for an effective continuum model for the quantum spin Hall effect in HgTe/CdTe quantum wells are presented. For a sample of a large size, the solution gives the linear dispersion for the edge states. However, in a finite strip geometry, the edge states at two sides will couple with each other, which leads to a finite energy gap in the spectra. The gap decays in an exponential law of the width of sample. The magnetic field dependence of the edge states illustrates the difference of the edge states from those of a conventional quantum Hall strip of two-d…
We analyze thermally induced spin and charge transport in HgTe/CdTe quantum wells on the basis of the numerical non-equilibrium Green's function technique in the linear response regime. In the topologically non-trivial regime, we find a clear signature of the gap of the edge states due to their finite overlap from opposite sample boundaries -- both in the charge Seebeck and spin Nernst signal. We are able to fully understand the physical origin of the thermoelectric transport signatures of edge and bulk states based on simple analytical models. Interestingly, we derive that the spin Nernst sig…
We study the disorder effect on the transport properties in the HgTe/CdTe semiconductor quantum wells. We confirm that at a moderate disorder strength, the initially un-quantized two terminal conductance becomes quantized, and the system makes a transition to the novel topological Anderson insulator (TAI). Conductances calculated for the stripe and cylinder samples reveal the topological feature of TAI and supports the idea that the helical edge states may cause the anomalous quantized plateaus. The influence of disorder is studied by calculating the distributions of local currents. Base on th…
We have investigated beating patterns in Shubnikov-de Haas oscillations for HgTe/Hg_{0.3}Cd_{0.7}Te(001) quantum wells with electron densities of 2 to 3 X 10^{12} cm^{-2}. Up to 12 beating nodes have been observed at magnetic fields between 0.9 and 6 T. Zero magnetic field spin-orbit splitting energies up to 30 meV have been directly determined from the node positions as well as from the intersection of self-consistently calculated Landau levels. These values, which exceed the thermal broadening of Landau levels, k_B T, at room temperature, are in good agreement with Rashba spin-orbit splittin…
Magneto-transport measurements on gated high mobility heterostructures containing a 60 nm layer of tensile strained HgTe, a three-dimensional topological insulator, show well-developed Hall quantization from surface states both in the n- as well as in the p-type regime. While the n-type behavior is due to transport in the topological surface state of the material, we find from 8-orbital k.p calculations that the p-type transport results from massive Volkov-Pankratov states. Their formation prevents the Dirac point and thus the p-conducting topological surface state from being accessible in tra…
We report on the observation of magneto-oscillations of terahertz radiation induced photocurrent in HgTe/HgCdTe quantum wells (QWs) of different widths, which are characterized by a Dirac-like, inverted and normal parabolic band structure. The photocurrent data are accompanied by measurements of photoresistance (photoconductivity), radiation transmission, as well as magneto-transport. We develop a microscopic model of a cyclotron-resonance assisted photogalvanic effect, which describes main experimental findings. We demonstrate that the quantum oscillations of the photocurrent are caused by th…
Research on graphene has revealed remarkable phenomena arising in the honeycomb lattice. However, the quantum spin Hall effect predicted at the K point could not be observed in graphene and other honeycomb structures of light elements due to an insufficiently strong spin-orbit coupling. Here we show theoretically that 2D honeycomb lattices of HgTe can combine the effects of the honeycomb geometry and strong spin-orbit coupling. The conduction bands, experimentally accessible via doping, can be described by a tight-binding lattice model as in graphene, but including multi-orbital degrees of fre…
We propose a minimal effective two-dimensional Hamiltonian for HgTe/CdHgTe quantum wells (QWs) describing the side maxima of the first valence subband. By using the Hamiltonian, we explore the picture of helical edge states in tensile and compressively strained HgTe QWs. We show that both dispersion and probability density of the edge states can differ significantly from those predicted by the Bernevig-Hughes-Zhang (BHZ) model. Our results pave the way towards further theoretical investigations of HgTe-based quantum spin Hall insulators with direct and indirect band gaps beyond the BHZ model.
