Chapter 5 - Quantum Spin Hall State in HgTe

Following the recent observation of the quantum spin Hall (QSH) effect in HgTe quantum wells, an important issue is to understand the effect of impurities on transport in the QSH regime. Using linear response and renormalization group methods, we calculate the edge conductance of a QSH insulator as a function of temperature in the presence of a magnetic impurity. At high temperatures, Kondo and/or two-particle scattering give rise to a logarithmic temperature dependence. At low temperatures, for weak Coulomb interactions in the edge liquid, the conductance is restored to unitarity with unusual power laws characteristic of a "local helical liquid," while for strong interactions, transport proceeds by weak tunneling through the impurity where only half an electron charge is transferred in each tunneling event.

The quantum spin Hall (QSH) state, observed in a zero magnetic field in HgTe quantum wells, respects the time-reversal symmetry and is distinct from quantum Hall (QH) states. We show that the QSH state persists in strong quantizing fields and is identified by counterpropagating (helical) edge channels with nonlinear dispersion inside the band gap. If the Fermi level is shifted into the Landau-quantized conduction or valence band, we find a transition between the QSH and QH regimes. Near the transition the longitudinal conductance of the helical channels is strongly suppressed due to the combined effect of the spectrum nonlinearity and enhanced backscattering. It shows a power-law decay B(-2N) with magnetic field B, determined by the number of backscatterers on the edge N. This suggests a rather simple and practical way to probe the quality of quasiballistic QSH devices using magnetoresistance measurements.

We show that the quantum spin Hall (QSH) effect, a state of matter with topological properties distinct from those of conventional insulators, can be realized in mercury telluride-cadmium telluride semiconductor quantum wells. When the thickness of the quantum well is varied, 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 methods for experimental detection of the QSH effect.

Recent theory predicted that the quantum spin Hall effect, a fundamentally new quantum state of matter that exists at zero external magnetic field, may be realized in HgTe/(Hg,Cd)Te quantum wells. We fabricated such sample structures with low density and high mobility in which we could tune, through an external gate voltage, the carrier conduction from n-type to p-type, passing through an insulating regime. For thin quantum wells with well width d < 6.3 nanometers, the insulating regime showed the conventional behavior of vanishingly small conductance at low temperature. However, for thicker quantum wells (d > 6.3 nanometers), the nominally insulating regime showed a plateau of residual conductance close to 2e(2)/h, where e is the electron charge and h is Planck's constant. The residual conductance was independent of the sample width, indicating that it is caused by edge states. Furthermore, the residual conductance was destroyed by a small external magnetic field. The quantum phase transition at the critical thickness, d = 6.3 nanometers, was also independently determined from the magnetic field-induced insulator-to-metal transition. These observations provide experimental evidence of the quantum spin Hall effect.

Both quantum spin Hall and quantum anomalous Hall states are novel states of quantum matter with promising applications. We propose junction quantum wells comprising II-VI, III-V or IV semiconductors as a large class of new materials realizing the quantum spin Hall state. Especially, we find that the bulk band gap for the quantum spin Hall state can be as large as 0.1 eV. Further more, magnetic doping would induce the ferromagnetism in these junction quantum wells due to band edge singularities in the band-inversion regime and to realize the quantum anomalous Hall state.

We present the theory and numerical evaluations of the backscattering rate determined by acceptor holes or Mn spins in HgTe and (Hg,Mn)Te quantum wells in the quantum spin Hall regime. The role of anisotropic s-p and sp-d exchange interactions, Kondo coupling, Luttinger liquid effects, precessional dephasing, and bound magnetic polarons is quantified. The determined magnitude and temperature dependence of conductance are in accord with experimental results for HgTe and (Hg,Mn)Te quantum wells.

We calculate transport properties of HgTe quantum wells that exhibit the quantum spin Hall effect. We concentrate on the ballistic bulk contribution as a function of aspect ratio and Fermi energy. We show that the conductance and the shot noise are distinctively different for the so-called normal regime (the topologically trivial case) and the so-called inverted regime (the topologically non-trivial case). Thus, it is possible to verify the topological order of two-dimensional topological insulators such as HgTe quantum wells not only via observable edge properties but also via observable bulk properties. In addition, we show that the bulk contribution can even exceed the edge contribution for certain parameter regimes (and in all regimes for the case of the shot noise). We test the validity of our analytical approach against a tight-binding model that allows us to include random disorder numerically which is shown to have only a minor effect as long as its strength does not exceed the bulk gap.

The properties of HgTe/Hg1−x Cdx Te superlattices (SLs) and quantum wells, (QWs), are of fundamental interest; SLs are potentially useful for infrared opto-electronic applications and QWs have aroused much interest due to recent evidence that they are topological insulators with dissipation less transport in edge states in the absence of a magnetic field. Depending on the HgTe thickness , the band structure has either the normal sequence of quantum well states or an inverted one when nm, in which the electron E1 subband is a valence subband and lies below the heavy hole H1 subband. This has been shown to lead to a very large Rashba spin–orbit (s–o) splitting of up to meV in HgTe/Hg1−x Cdx Te QWs with an inverted band structure. This as well as other transport results will be presented. These properties of the HgTe/Hg1−x Cdx Te QW, large s–o splitting and an inverted band structure, have recently been shown to result in the quantum spin Hall effect. In the quantum spin Hall effect, carriers with opposite spin move in opposite directions on a given edge in an insulator in the absence of a magnetic field. Convincing evidence for non-local edge channel transport in HgTe based QWs has been recently published. The results demonstrate that quantum transport through helical edge channels is dissipation less at zero magnetic field and agree quantitatively with the theory of the quantum spin Hall effect. Images of current in the edge channels by means of scanning gate microscopy have revealed well localized scattering sites which perturb the quantum spin Hall edge states on a sub-micrometer scale. In the micrometer sized regions between the scattering sites, current appeared to propagate unperturbed Recently, images of edge channels in HgTe QWs in the quantum spin Hall regime have been obtained by means of their magnetic fields with a scanning superconducting quantum interference device (SQUID). The edge channels were observed to dominate transport in devices with edges much longer than the scattering length, and to persist even in the presence of bulk conduction.

We theoretically study the helical edge states in the quantum spin Hall (QSH) effect based on an effective 4-band tight-binding model for HgTe/CdTe quantum wells. We microscopically describe the spatial dependence of the helical edge modes in the QSH state. It is found that the helical edge states are gapped when the width of sample narrows. This is because the edge states on the opposite boundary of the sample overlap each other, and then the backward scattering channel opens up.

Zeeman splitting of conduction band in HgTe quantum wells near the Dirac point

Quantum spin Hall (QSH) effect, a fundamentally new quantum state of matter and topological phase transitions are characteristics of a kind of electronic material, popularly referred to as topological insulators (TIs). TIs are similar to ordinary insulator in terms of their bulk bandgap, but have gapless conducting edge-states that are topologically protected. These edge-states are facilitated by the time-reversal symmetry and they are robust against nonmagnetic impurity scattering. Recently, the quest for new materials exhibiting non-trivial topological state of matter has been of great research interest, as TIs find applications in new electronics and spintronics and quantum-computing devices. Here, we propose and demonstrate as a proof-of-concept that QSH effect and topological phase transitions can be realized in /InSb semiconductor quantum wells (QWs). The simultaneous incorporation of nitrogen and bismuth in InSb is instrumental in lowering the bandgap, while inducing opposite kinds of strain to attain a near-lattice-matching conducive for lattice growth. Phase diagram for bandgap shows that as we increase the QW thickness, at a critical thickness, the electronic bandstructure switches from a normal to an inverted type. We confirm that such transition are topological phase transitions between a traditional insulator and a TI exhibiting QSH effect—by demonstrating the topologically protected edge-states using the bandstructure, edge-localized distribution of the wavefunctions and edge-state spin-momentum locking phenomenon, presence of non-zero conductance in spite of the Fermi energy lying in the bandgap window, crossover points of Landau levels in the zero-mode indicating topological band inversion in the absence of any magnetic field and presence of large Rashba spin-splitting, which is essential for spin-manipulation in TIs.

HgTe quantum wells (QWs) are two-dimensional semiconductor systems that change their properties at the critical thickness dc, corresponding to the band inversion and topological phase transition. The motivation of this work was to study magnetotransport properties of HgTe QWs with thickness approaching dc, and examine them as potential candidates for quantum Hall effect (QHE) resistance standards. We show that in the case of d > dc (inverted QWs), the quantization is influenced by coexistence of topological helical edge states and QHE chiral states. However, at d ≈ dc, where QW states exhibit a graphene-like band structure, an accurate Hall resistance quantization in low magnetic fields (B ≤ 1.4 T) and at relatively high temperatures (T ≥ 1.3 K) may be achieved. We observe wider and more robust quantized QHE plateaus for holes, which suggests—in accordance with the “charge reservoir” model—a pinning of the Fermi level in the valence band region. Our analysis exhibits advantages and drawbacks of HgTe QWs for quantum metrology applications, as compared to graphene and GaAs counterparts.

The dynamical screening function of two-dimensional electron gas in a wide HgTe quantum well (QW) has been numerically modeled in this work.Calculations were performed in the Random Phase Approximation (RPA) framework and were based on the Lindhard equation.Our simulations directly incorporated non-parabolicity of bulk 2D carriers' spectrum, which was obtained by full 8-band k.p method.Known from literature are the data that transport properties of HgTe QWs can be explained by graphene-like screening.We carried out the comparison of the screening function for the Schrödinger fermions in the inverted bands HgTe QW with the appropriate screening function for graphene monolayer with the Dirac fermions.In addition, the dependences of HgTespecific screening function on temperature, scattering wave-vector and frequency have been studied with the purpose to ascertain the transport properties under high frequency radiation for the QW structures to be used as THz detectors.Plasmon frequencies of 2DEG in HgTe quantum well under study were calculated in the long-wave limit for T = 77 K.

A large bulk band gap is critical for the applications of quantum spin hall (QSH) insulators in spintronics at room temperature. Based on first-principles calculations, we predict that the methyl-functionalized III-Bi monolayers, namely III-Bi-(CH3)2 (III=Ga, In, Tl) thin films, own QSH states with band gap as large as 0.260, 0.304 and 0.843 eV, respectively, making them suitable for room-temperature applications. The topological characteristics are confirmed by s-px,y band inversion, topological invariant Z2, and the topologically protected edge states. Noticeably, for GaBi/InBi-(CH3)2 films, the s-px,y band inversion occurred in the progress of spin-orbital coupling (SOC), while for TlBi(CH3)2 film, the s-px,y band inversion happened in the progress of chemical bonding. Significantly, the QSH states in III-Bi-(CH3)2 films are robust against the mechanical strains and various methyl coverages, making these films particularly flexible to substrate choice for device applications. Besides, the h-BN substrate is an ideal substrate for III-Bi-(CH3)2 films to realize large gap nontrivial topological states.. These findings demonstrate that the methyl-functionalized III-Bi films may be good QSH effect platforms for topological electronic devices design and fabrication in spintronics.

We study the interplay between Dirac and Schrödinger fermions in the polarization properties of a two-dimensional electron gas (2DEG). Specifically, we analyze the low-energy sector of narrow-gap semiconductors described by a two-band Kane model. In the context of quantum spin Hall insulators, particularly, in Hg(Cd)Te quantum wells, this model is named the Bernevig-Hughes-Zhang model. Interestingly, it describes electrons with intermediate properties between Dirac and Schrödinger fermions. We calculate the dynamical dielectric function of such a model at zero temperature within random phase approximation. Surprisingly, plasmon resonances are found in the intrinsic (undoped) limit, whereas they are absent-in that limit-in graphene as well as ordinary 2DEGs. Additionally, we demonstrate that the optical conductivity offers a quantitative way to identify the topological phase of Hg(Cd)Te quantum wells from a bulk measurement.