Multifractal Conductance Fluctuations of Helical Edge States.

Two-dimensional topological insulators are characterized by the bulk gap and one-dimensional helical states running along the edges. The theory predicts the topological protection of the helical transport from coherent backscattering. However, the unexpected deviations of the conductance from the quantized value and localization of the helical modes are generally observed in long samples. Moreover, at millikelvin temperatures significant mesoscopic fluctuations are developed as a function of the electron energy. Here we report the results of an experimental study of the transport in a HgTe quantum well with an inverted energy spectrum that reveal a multifractality of the conductance fluctuations in the helical edge state dominated transport regime. We attribute observed multifractality to mesoscopic fluctuations of the electron wave function or local density of states at the spin quantum Hall transition. We have shown that the mesoscopic two-dimensional topological insulator provides a highly tunable experimental system in which to explore the physics of the Anderson transition between topological states.

Transport properties of a 1000 nm HgTe film: the interplay of surface and bulk carriers.

We report on systematic study of transport properties of a 1000 nm HgTe film. Unlike thinner and strained HgTe films, which are known as high-quality three-dimensional topological insulators, the film under study is much thicker than the limit of pseudomorphic growth of HgTe on a CdTe substrate. Therefore, the 1000 nm HgTe film is expected to be fully relaxed and has the band structure of bulk HgTe, i.e. a zero gap semiconductor. Additionally, the system is characterized by the bands inversion, so that the two-dimensional topological surface states (TSSs) are expected to exist. To check this claim we studied classical and quantum transport response of the system. We demonstrate that by tuning the top-gate voltage one can change the electron-dominating transport to the hole one. The highest electron mobility is found to be more than300×103 cm2 Vs-1. The system exhibits Shubnikov-de Haas (SdH) oscillations with a complicated pattern and shows up to five independent frequencies in corresponding Fourier spectra. These Fourier peaks are attributed to the TSSs, Volkov-Pankratov states and spin-degenerate bulk states in the accumulation layer near the gate. The observed peculiarities of the quantum transport are the strong SdH oscillations of the Hall resistance, and the suppressed oscillatory response of the TSSs.

Engineering topological phases in triple HgTe/CdTe quantum wells.

Quantum wells formed by layers of HgTe between Hg[Formula: see text]Cd[Formula: see text]Te barriers lead to two-dimensional (2D) topological insulators, as predicted by the BHZ model. Here, we theoretically and experimentally investigate the characteristics of triple HgTe quantum wells. We describe such heterostructure with a three dimensional [Formula: see text] Kane model, and use its eigenstates to derive an effective 2D Hamiltonian for the system. From these we obtain a phase diagram as a function of the well and barrier widths and we identify the different topological phases composed by zero, one, two, and three sets of edge states hybridized along the quantum wells. The phase transitions are characterized by a change of the spin Chern numbers and their corresponding band inversions. Complementary, transport measurements are experimentally investigated on a sample close to the transition line between the phases with one and two sets of edges states. Accordingly, for this sample we predict a gapless spectrum with low energy bulk conduction subbands given by one parabolic and one Dirac subband, and with edge states immersed in the bulk valence subbands. Consequently, we show that under these conditions, local and non-local transport measurements are inconclusive to characterize a sole edge state conductivity due to bulk conductivity. On the other hand, Shubnikov-de Haas (SdH) oscillations show an excellent agreement with our theory. Particularly, we show that the measured SdH oscillation frequencies agrees with our model and show clear signatures of the coexistence of a parabolic and Dirac subbands.

Distinction between electron states formed at topological insulator interfaces with the trivial phase and vacuum.

In this paper, we show that electron states formed in topological insulators at the interfaces topological phase-trivial phase and topological phase-vacuum may possess different properties. This is demonstrated on an example of heterostructures based on thick topological Hg1-xCdxTe films, in which the PT-symmetric terahertz photoconductivity is observed. It is shown that the effect originates from features of the interface topological film-trivial buffer/cap layer. The PT-symmetric terahertz photoconductivity is not provided by electron states formed at the interface topological film-vacuum.

Non-local terahertz photoconductivity in the topological phase of Hg1-xCdxTe.

We report on observation of strong non-local photoconducitivity induced by terahertz laser pulses in non-zero magnetic field in heterostructures based on Hg1-xCdxTe films being in the topological phase. While the zero-field non-local photoconductivity is negligible, it is strongly enhanced in magnetic fields ~ 0.05 T resulting in appearance of an edge photocurrent that exceeds the respective dark signal by orders of magnitude. This photocurrent is chiral, and the chirality changes every time the magnetic field or the electric bias is reversed. Appearance of the non-local terahertz photoconductivity is attributed to features of the interface between the topological film and the trivial buffer.

Anomalous Decay of Quantum Resistance Oscillations of 2D Helical Electrons in Magnetic Field.

Shubnikov de Haas resistance oscillations of highly mobile two dimensional helical electrons propagating on a conducting surface of strained HgTe 3D topological insulator are studied in magnetic fields B tilted by angle θ from the normal to the conducting layer. Strong decrease of oscillation amplitude A is observed with the tilt: [Formula: see text], where ξ is a constant. Evolution of the oscillations with temperature T shows that the parameter [Formula: see text] contains two terms: [Formula: see text]. The temperature independent term, [Formula: see text], signals possible reduction of electron mean free path [Formula: see text] and/or enhancement of in-homogeneous broadening of the oscillations in magnetic field B. The temperature dependent term, [Formula: see text], indicates increase of the reciprocal velocity of 2D helical electrons: [Formula: see text] suggesting modification of the electron spectrum in magnetic fields. Results are found in good agreement with proposed phenomenological model.

Apparent PT-symmetric terahertz photoconductivity in the topological phase of Hg1-xCdxTe-based structures.

We show that the terahertz (THz) photoconductivity in the topological phase of Hg1-xCdxTe-based structures exhibits the apparent PT- (parity-time) symmetry whereas the P-symmetry and the T-symmetry, separately, are not conserved. Moreover, it is demonstrated that the P- and T-symmetry breaking may not be related to any type of the sample anisotropy. This result contradicts the apparent symmetry arguments and means that there exists an external factor that interacts with the sample electronic system and breaks the symmetry. We show that deviations from the ideal experimental geometry may not be such a factor.

Impact Ionization Induced by Terahertz Radiation in HgTe Quantum Wells of Critical Thickness.

We report on the observation of terahertz (THz) radiation induced band-to-band impact ionization in HgTe quantum well (QW) structures of critical thickness, which are characterized by a nearly linear energy dispersion. The THz electric field drives the carriers initializing electron-hole pair generation. The carrier multiplication is observed for photon energies less than the energy gap under the condition that the product of the radiation angular frequency ω and momentum relaxation time τ l larger than unity. In this case, the charge carriers acquire high energies solely because of collisions in the presence of a high-frequency electric field. The developed microscopic theory shows that the probability of the light-induced impact ionization is proportional to exp ( - E 0 2 / E 2 ) , with the radiation electric field amplitude E and the characteristic field parameter E 0. As observed in experiment, it exhibits a strong frequency dependence for ω τ ≫ 1 characterized by the characteristic field E 0 linearly increasing with the radiation frequency ω.

Topological Protection Brought to Light by the Time-Reversal Symmetry Breaking.

Recent topological band theory distinguishes electronic band insulators with respect to various symmetries and topological invariants, most commonly, the time reversal symmetry and the Z_{2} invariant. The interface of two topologically distinct insulators hosts a unique class of electronic states-the helical states, which shortcut the gapped bulk and exhibit spin-momentum locking. The magic and so far elusive property of the helical electrons, known as topological protection, prevents them from coherent backscattering as long as the underlying symmetry is preserved. Here we present an experiment that brings to light the strength of topological protection in one-dimensional helical edge states of a Z_{2} quantum spin-Hall insulator in HgTe. At low temperatures, we observe the dramatic impact of a tiny magnetic field, which results in an exponential increase of the resistance accompanied by giant mesoscopic fluctuations and a gap opening. This textbook Anderson localization scenario emerges only upon the time-reversal symmetry breaking, bringing the first direct evidence of the topological protection strength in helical edge states.

Electronic thermal conductivity in 2D topological insulator in a HgTe quantum well.

We have measured the differential resistance in a two-dimensional topological insulator (2DTI) in a HgTe quantum well, as a function of the applied dc current. The transport near the charge neutrality point is characterized by a pair of counter propagating gapless edge modes. In the presence of an electric field, the energy is transported by counter propagating channels in the opposite direction. We test a hot carrier effect model and demonstrate that the energy transfer complies with the Wiedemann Franz law near the charge neutrality point in the edge transport regime.

Landau level spectroscopy of valence bands in HgTe quantum wells: effects of symmetry lowering.

The Landau level spectroscopy technique has been used to explore 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, reasonable agreement with experiments is achieved when the standard model is expanded to include effects of bulk and interface inversion asymmetries. These remove the axial symmetry, and among other, profoundly modify the shape of valence bands.

Radiative recombination in narrow gap HgTe/CdHgTe quantum well heterostructures for laser applications.

Radiative recombination is studied in CdHgTe/HgTe QWs with bandgap in the 40-140 meV range using four-band Kane model. Calculated radiative lifetimes agree well with the photoconductivity kinetics measurements. We show that the side maxima in the valence band hinder the radiative recombination at high carrier concentrations and discuss how to overcome this effect for the development of long-wavelength lasers.

Stimulated emission in the 2.8-3.5 µm wavelength range from Peltier cooled HgTe/CdHgTe quantum well heterostructures.

We report stimulated emission in the 2.8-3.5 µm wavelength range from HgTe/CdHgTe quantum well (QW) heterostructures at temperatures available with thermoelectric cooling. The structures were designed to suppress the Auger recombination by implementing narrow (1.5 - 2 nm wide) QWs. We conclude that Peltier cooled operation is feasible in lasers based on such structures, making them of interest for spectroscopy applications in the atmospheric transparency window from 3 to 5 µm.

Temperature-Induced Topological Phase Transition in HgTe Quantum Wells.

We report a direct observation of temperature-induced topological phase transition between the trivial and topological insulator states in an HgTe quantum well. By using a gated Hall bar device, we measure and represent Landau levels in fan charts at different temperatures, and we follow the temperature evolution of a peculiar pair of "zero-mode" Landau levels, which split from the edge of electronlike and holelike subbands. Their crossing at a critical magnetic field B_{c} is a characteristic of inverted band structure in the quantum well. By measuring the temperature dependence of B_{c}, we directly extract the critical temperature T_{c} at which the bulk band gap vanishes and the topological phase transition occurs. Above this critical temperature, the opening of a trivial gap is clearly observed.

Universal Faraday Rotation in HgTe Wells with Critical Thickness.

The universal value of the Faraday rotation angle close to the fine structure constant (α≈1/137) is experimentally observed in thin HgTe quantum wells with a thickness on the border between trivial insulating and the topologically nontrivial Dirac phases. The quantized value of the Faraday angle remains robust in the broad range of magnetic fields and gate voltages. Dynamic Hall conductivity of the holelike carriers extracted from the analysis of the transmission data shows a theoretically predicted universal value of σ_{xy}=e^{2}/h, which is consistent with the doubly degenerate Dirac state. On shifting the Fermi level by the gate voltage, the effective sign of the charge carriers changes from positive (holes) to negative (electrons). The electronlike part of the dynamic response does not show quantum plateaus and is well described within the classical Drude model.

Temperature-driven massless Kane fermions in HgCdTe crystals.

It has recently been shown that electronic states in bulk gapless HgCdTe offer another realization of pseudo-relativistic three-dimensional particles in condensed matter systems. These single valley relativistic states, massless Kane fermions, cannot be described by any other relativistic particles. Furthermore, the HgCdTe band structure can be continuously tailored by modifying cadmium content or temperature. At critical concentration or temperature, the bandgap collapses as the system undergoes a semimetal-to-semiconductor topological phase transition between the inverted and normal alignments. Here, using far-infrared magneto-spectroscopy we explore the continuous evolution of band structure of bulk HgCdTe as temperature is tuned across the topological phase transition. We demonstrate that the rest mass of Kane fermions changes sign at critical temperature, whereas their velocity remains constant. The velocity universal value of (1.07±0.05) × 10(6) m s(-1) remains valid in a broad range of temperatures and Cd concentrations, indicating a striking universality of the pseudo-relativistic description of the Kane fermions in HgCdTe.

Low field magnetoresistance in a 2D topological insulator based on wide HgTe quantum well.

Low field magnetoresistance is experimentally studied in a two-dimensional topological insulator (TI) in both diffusive and quasiballistic samples fabricated on top of a wide (14 nm) HgTe quantum well. In all cases a pronounced quasi-linear positive magnetoresistance is observed similar to that found previously in diffusive samples based on a narrow (8 nm) HgTe well. The experimental results are compared with the main existing theoretical models based on different types of disorder: sample edge roughness, nonmagnetic disorder in an otherwise coherent TI and metallic puddles due to locally trapped charges that act like local gate on the sample. The quasiballistic samples with resistance close to the expected quantized values also show a positive low-field magnetoresistance but with a pronounced admixture of mesoscopic effects.

Probing Quantum Capacitance in a 3D Topological Insulator.

We measure the quantum capacitance and probe thus directly the electronic density of states of the high mobility, Dirac type two-dimensional electron system, which forms on the surface of strained HgTe. Here we show that observed magnetocapacitance oscillations probe-in contrast to magnetotransport-primarily the top surface. Capacitance measurements constitute thus a powerful tool to probe only one topological surface and to reconstruct its Landau level spectrum for different positions of the Fermi energy.

Persistence of a two-dimensional topological insulator state in wide HgTe quantum wells.

Our experimental studies of electron transport in wide (14 nm) HgTe quantum wells confirm the persistence of a two-dimensional topological insulator state reported previously for narrower wells, where it was justified theoretically. Comparison of local and nonlocal resistance measurements indicate edge state transport in the samples of about 1 mm size at temperatures below 1 K. Temperature dependence of the resistances suggests an insulating gap of the order of a few meV. In samples with sizes smaller than 10 µm a quasiballistic transport via the edge states is observed.

Transport properties of a 3D topological insulator based on a strained high-mobility HgTe film.

We investigate the magnetotransport properties of strained 80 nm thick HgTe layers featuring a high mobility of µ ∼ 4 × 10(5) cm(2)/V · s. By means of a top gate, the Fermi energy is tuned from the valence band through the Dirac-type surface states into the conduction band. Magnetotransport measurements allow us to disentangle the different contributions of conduction band electrons, holes, and Dirac electrons to the conductivity. The results are in line with previous claims that strained HgTe is a topological insulator with a bulk gap of ≈ 15 meV and gapless surface states.