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.
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.
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 …
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…
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 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…
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.
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 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 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…
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…
The 2007 discovery of quantized conductance in HgTe quantum wells delivered the field of topological insulators (TIs) its first experimental confirmation. While many three-dimensional TIs have since been identified, HgTe remains the only known two-dimensional system in this class. Difficulty fabricating HgTe quantum wells has, moreover, hampered their widespread use. With the goal of breaking this logjam we provide a blueprint for stabilizing a robust TI state in a more readily available two-dimensional material---graphene. Using symmetry arguments, density functional theory, and tight-binding…
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 …
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…
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 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…
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 …
The field of topological insulators (TIs) is rapidly growing. Concerning possible applications, the search for materials with an easily controllable TI phase is a key issue. The quantum spin Hall effect, characterized by a single pair of helical edge modes protected by time-reversal symmetry, has been demonstrated in HgTe-based quantum wells (QWs) with an inverted bandgap. We analyze the topological properties of a generically coupled HgTe-based double QW (DQW) and show how in such a system a TI phase can be driven by an inter-layer bias voltage, even when the individual layers are non-inverte…
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 quantum spin Hall insulator (QSHI) state has been demonstrated in two semiconductor systems - HgTe/CdTe quantum wells (QWs) and InAs/GaSb QW bilayers. Unlike the HgTe/CdTe QWs, the inverted band gap in InAs/GaSb QW bilayers does not open at the $Γ$ point of the Brillouin zone, preventing the realization of massless Dirac fermions. Here, we propose a new class of semiconductor systems based on InAs/Ga(In)Sb multilayers, hosting a QSHI state, a graphene-like phase and a bilayer graphene analogue, depending on their layer thicknesses and geometry. The QSHI gap in the novel structures can reac…
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…
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 parti…
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…
Using $\vec{k}$$\cdot$$\vec{p}$ theory, we derive an effective four band model describing the physics of the typical two-dimensional topological insulator (HgTe/CdTe quantum well) in the presence of out-of-plane in z-direction inversion breaking and in-plane confining potentials. We find that up to third order in perturbation theory, only the inversion breaking potential generates new elements to the four band Hamiltonian that are off-diagonal in spin space. When this new effective Hamiltonian is folded into an effective two band model for the conduction (electron) or valence (heavy hole) band…
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 results of experimental study of the magnetoresistivity, the Hall and Shubnikov-de Haas effects for the heterostructure with HgTe quantum well of 20.2 nm width are reported. The measurements were performed on the gated samples over the wide range of electron and hole densities including vicinity of a charge neutrality point. Analyzing the data we conclude that the energy spectrum is drastically different from that calculated in framework of $kP$-model. So, the hole effective mass is equal to approximately $0.2 m_0$ and practically independent of the quasimomentum ($k$) up to $k^2\gtrsim 0.…
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 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 quantum spin Hall (QSH) state is a topologically non-trivial state of quantum matter which preserves time-reversal symmetry; it has an energy gap in the bulk, but topologically robust gapless states at the edge. Recently, this novel effect has been predicted and observed in HgTe quantum wells. In this work we predict a similar effect arising in Type-II semiconductor quantum wells made from InAs/GaSb/AlSb. Because of a rare band alignment the quantum well band structure exhibits an "inverted" phase similar to CdTe/HgTe quantum wells, which is a QSH state when the Fermi level lies inside the…
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…
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. …
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 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 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…
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…
HgTe quantum wells with a thickness of ${\sim}$7 nm may have a graphene-like band structure and have been recently proposed to be potential candidates for quantum Hall effect (QHE) resistance standards under the condition of operation in the fields above certain critical field $B_c$, above which the topological phase (with parasitic edge conduction) disappears. We present experimental studies of the magnetoresistance of different of HgTe quantum wells as a function temperature and magnetic field, determining the critical magnetic field $B_c$. We demonstrate that for QWs of specific width $B_c$…
While the helical character of the edge channels responsible for charge transport in the quantum spin Hall regime of a two-dimensional topological insulator is by now well established, an experimental confirmation that the transport in the edge channels is spin-polarized is still outstanding. We report experiments on nanostructures fabricated from HgTe quantum wells with an inverted band structure, in which a split gate technique allows us to combine both quantum spin Hall and metallic spin Hall transport in a single device. In these devices, the quantum spin Hall effect can be used as a spin …
We present the first experimental study of the double-quantum-well (DQW) system made of 2D layers with inverted energy band spectrum: HgTe. The magnetotransport reveals a considerably larger overlap of the conduction and valence subbands than in known HgTe single quantum wells (QW), which may be regulated by an applied gate voltage $V_g$. This large overlap manifests itself in a much higher critical field $B_c$ separating the range above it where the quantum peculiarities shift linearly with $V_g$ and the range below with a complicated behavior. In the latter case the $N$-shaped and double-$N$…
The experiments in quantum spin Hall insulator candidate materials, such as HgTe/CdTe and InAs/GaSb heterostructures, indicate that in addition to the topologically protected helical edge modes these multilayer heterostructures may also support additional edge states, which can contribute to the scattering and the transport. We use first-principles calculations to derive an effective tight-binding model for HgTe/CdTe, HgS/CdTe and InAs/GaSb heterostructures, and we show that all these materials support additional edge states which are sensitive to the edge termination. We trace the microscopic…
We present direct experimental evidence for nonlocal transport in HgTe quantum wells in the quantum spin Hall regime, in the absence of any external magnetic field. The data conclusively show that the non-dissipative quantum transport occurs through edge channels, while the contacts lead to equilibration between the counter-propagating spin states at the edge. We show that the experimental data agree quantitatively with the theory of the quantum spin Hall effect.
