Effective lifting of the topological protection of quantum spin Hall edge states by edge coupling.

The scientific interest in two-dimensional topological insulators (2D TIs) is currently shifting from a more fundamental perspective to the exploration and design of novel functionalities. Key concepts for the use of 2D TIs in spintronics are based on the topological protection and spin-momentum locking of their helical edge states. In this study we present experimental evidence that topological protection can be (partially) lifted by pairwise coupling of 2D TI edges in close proximity. Using direct wave function mapping via scanning tunneling microscopy/spectroscopy (STM/STS) we compare isolated and coupled topological edges in the 2D TI bismuthene. The latter situation is realized by natural lattice line defects and reveals distinct quasi-particle interference (QPI) patterns, identified as electronic Fabry-Pérot resonator modes. In contrast, free edges show no sign of any single-particle backscattering. These results pave the way for novel device concepts based on active control of topological protection through inter-edge hybridization for, e.g., electronic Fabry-Pérot interferometry.

Survival of the Quantum Anomalous Hall Effect in Orbital Magnetic Fields as a Consequence of the Parity Anomaly.

Recent experimental progress in condensed matter physics enables the observation of signatures of the parity anomaly in two-dimensional Dirac-like materials. Using effective field theories and analyzing band structures in external out-of-plane magnetic fields (orbital fields), we show that topological properties of quantum anomalous Hall (QAH) insulators are related to the parity anomaly. We demonstrate that the QAH phase survives in orbital fields, violates the Onsager relation, and can be therefore distinguished from a quantum Hall (QH) phase. As a fingerprint of the QAH phase in increasing orbital fields, we predict a transition from a quantized Hall plateau with σ_{xy}=-e^{2}/h to a not perfectly quantized plateau, caused by scattering processes between counterpropagating QH and QAH edge states. This transition can be especially important in paramagnetic QAH insulators, such as (Hg,Mn)Te/CdTe quantum wells, in which exchange interaction and orbital fields compete.

Observation of the universal magnetoelectric effect in a 3D topological insulator.

The electrodynamics of topological insulators (TIs) is described by modified Maxwell's equations, which contain additional terms that couple an electric field to a magnetization and a magnetic field to a polarization of the medium, such that the coupling coefficient is quantized in odd multiples of α/4π per surface. Here we report on the observation of this so-called topological magnetoelectric effect. We use monochromatic terahertz (THz) spectroscopy of TI structures equipped with a semitransparent gate to selectively address surface states. In high external magnetic fields, we observe a universal Faraday rotation angle equal to the fine structure constant α=e2/2hc (in SI units) when a linearly polarized THz radiation of a certain frequency passes through the two surfaces of a strained HgTe 3D TI. These experiments give insight into axion electrodynamics of TIs and may potentially be used for a metrological definition of the three basic physical constants.

Magneto-optics of massive dirac fermions in bulk Bi2Se3.

We report on magneto-optical studies of Bi2Se3, a representative member of the 3D topological insulator family. Its electronic states in bulk are shown to be well described by a simple Dirac-type Hamiltonian for massive particles with only two parameters: the fundamental band gap and the band velocity. In a magnetic field, this model implies a unique property-spin splitting equal to twice the cyclotron energy: Es=2Ec. This explains the extensive magnetotransport studies concluding a fortuitous degeneracy of the spin and orbital split Landau levels in this material. The Es=2Ec match differentiates the massive Dirac electrons in bulk Bi2Se3 from those in quantum electrodynamics, for which Es=Ec always holds.

One-dimensional weak antilocalization due to the berry phase in HgTe wires.

We study the weak antilocalization (WAL) effect in the magnetoresistance of narrow HgTe wires fabricated in quantum wells with normal and inverted band ordering. Measurements at different gate voltages indicate that the WAL is only weakly affected by Rashba spin-orbit splitting and persists when the Rashba splitting is about zero. The WAL amplitude in wires with normal band ordering is an order of magnitude smaller than for wires with an inverted band structure. These observations are attributed to the Dirac-like dispersion of the energy bands in HgTe quantum wells. From the magnetic-field and temperature dependencies we extract the dephasing lengths and band Berry phases. The weaker WAL for samples with a normal band structure can be explained by a nonuniversal Berry phase which always exceeds π, the characteristic value for gapless Dirac fermions.

Quantum Hall effect from the topological surface states of strained bulk HgTe.

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 semimetallic 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.

Backscattering of Dirac fermions in HgTe quantum wells with a finite gap.

The density-dependent mobility of n-type HgTe quantum wells with inverted band ordering has been studied both experimentally and theoretically. While semiconductor heterostructures with a parabolic dispersion exhibit an increase in mobility with carrier density, high-quality HgTe quantum wells exhibit a distinct mobility maximum. We show that this mobility anomaly is due to backscattering of Dirac fermions from random fluctuations of the band gap (Dirac mass). Our findings open new avenues for the study of Dirac fermion transport with finite and random mass, which so far has been hard to access.

Quantum Hall superfluids in topological insulator thin films.

Three-dimensional topological insulators have protected Dirac-cone surface states. In this Letter we argue that gapped excitonic superfluids with spontaneous coherence between top and bottom surfaces can occur in the topological insulator (TI)-thin-film quantum Hall regime. We find that the large dielectric constants of TI materials increase the layer separation range over which coherence survives and decrease the superfluid sound velocity, but have little influence on the superfluid density or on the charge gap. The coherent state at total Landau-level filling factor νT=0 is predicted to be free of edge modes, qualitatively altering its transport phenomenology compared to the widely studied case of νT=1 in GaAs double-quantum wells.

Ballistic quantum spin Hall state and enhanced edge backscattering in strong magnetic fields.

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.

Spin-Hall effect and spin-Coulomb drag in doped semiconductors.

In this review, we describe in detail two important spin-transport phenomena: the extrinsic spin-Hall effect (coming from spin-orbit interactions between electrons and impurities) and the spin-Coulomb drag. The interplay of these two phenomena is analyzed. In particular, we discuss the influence of scattering between electrons with opposite spins on the spin current and the spin accumulation produced by the spin-Hall effect. Future challenges and open questions are briefly discussed.

Phase diagram of the spin Hall effect.

We obtain analytic formulas for the frequency-dependent spin-Hall conductivity of a two-dimensional electron gas (2DEG) in the presence of impurities, linear spin-orbit Rashba interaction, and external magnetic field perpendicular to the 2DEG. We show how different mechanisms (skew scattering, side jump, and spin precession) can be brought in or out of focus by changing controllable parameters such as frequency, magnetic field, and temperature. We find, in particular, that the dc spin-Hall conductivity vanishes in the absence of a magnetic field, while a magnetic field restores the skew-scattering and side jump contributions proportionally to the ratio of magnetic and Rashba fields.

Direct observation of the Aharonov-Casher phase.

Ring structures fabricated from HgTe/HgCdTe quantum wells have been used to study Aharonov-Bohm type conductance oscillations as a function of Rashba spin-orbit splitting strength. We observe nonmonotonic phase changes indicating that an additional phase factor modifies the electron wave function. We associate these observations with the Aharonov-Casher effect. This is confirmed by comparison with numerical calculations of the magnetoconductance for a multichannel ring structure within the Landauer-Büttiker formalism.

Spin-Hall effect in a [110] GaAs quantum well.

A self-consistent treatment of the spin-Hall effect requires consideration of the spin-orbit coupling and electron-impurity scattering on equal footing. This is done here for the experimentally relevant case of a [110] GaAs quantum well [Sih, Nature Phys. 1, 31 (2005)]. Working within the framework of the exact linear response formalism we calculate the spin-Hall conductivity including the Dresselhaus linear and cubic terms in the band structure, as well as the electron-impurity scattering and electron-electron interaction to all orders. We show that the spin-Hall conductivity naturally separates into two contributions, skew-scattering and side-jump, and we propose an experiment to distinguish between them.