Topological insulators are new states of quantum matter which cannot be adiabatically connected to conventional insulators and semiconductors. They are characterized by a full insulating gap in the bulk and gapless edge or surface states which are protected by time-reversal symmetry. These topological materials have been theoretically predicted and experimentally observed in a variety of systems, including HgTe quantum wells, BiSb alloys, and Bi2Te3 and Bi2Se3 crystals. Theoretical models, materials properties, and experimental results on two-dimensional and three-dimensional topological insulators are reviewed, and both the topological band theory and the topological field theory are discussed. Topological superconductors have a full pairing gap in the bulk and gapless surface states consisting of Majorana fermions. The theory of topological superconductors is reviewed, in close analogy to the theory of topological insulators.

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Topological insulators and superconductors

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.

Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells

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 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.

Quantum Spin Hall Insulator State in HgTe Quantum Wells

The quantum Hall effect (QHE), one example of a quantum phenomenon that occurs on a truly macroscopic scale, has attracted intense interest since its discovery in 1980 and has helped elucidate many important aspects of quantum physics. It has also led to the establishment of a new metrological standard, the resistance quantum. Disappointingly, however, the QHE has been observed only at liquid-helium temperatures. We show that in graphene, in a single atomic layer of carbon, the QHE can be measured reliably even at room temperature, which makes possible QHE resistance standards becoming available to a broader community, outside a few national institutions.

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Room-Temperature Quantum Hall Effect in Graphene

http://absimage.aps.org/image/MAR07/MWS_MAR07-2006-005868.pdf

Room Temperature Quantum Hall Effect in Graphene

The band structure of semimagnetic ${\mathrm{Hg}}_{1\ensuremath{-}y}{\mathrm{Mn}}_{y}\mathrm{Te}∕{\mathrm{Hg}}_{1\ensuremath{-}x}{\mathrm{Cd}}_{x}\mathrm{Te}$ type-III quantum wells (QW's) has been calculated using an eight-band $\mathbit{k}∙\mathbit{p}$ model in an envelope function approach. Details of the band structure calculations are given for the Mn-free case $(y=0)$. A mean-field approach is used to take the influence of the $sp\text{\ensuremath{-}}d$ exchange interaction on the band structure of QW's with low Mn concentrations into account. The calculated Landau level fan diagram and the density of states of a ${\mathrm{Hg}}_{0.98}{\mathrm{Mn}}_{0.02}\mathrm{Te}∕{\mathrm{Hg}}_{0.3}{\mathrm{Cd}}_{0.7}\mathrm{Te}$ QW are in good agreement with recent experimental transport observations. The model can be used to interpret the mutual influence of the two-dimensional confinement and the $sp\text{\ensuremath{-}}d$ exchange interaction on the transport properties of ${\mathrm{Hg}}_{1\ensuremath{-}y}{\mathrm{Mn}}_{y}\mathrm{Te}∕{\mathrm{Hg}}_{1\ensuremath{-}x}{\mathrm{Cd}}_{x}\mathrm{Te}$ QW's.

https://arxiv.org/pdf/cond-mat/0409392.pdf

Band structure of semimagnetic Hg 1 − y Mn y Te quantum wells

A variety of BeMgZnSe–ZnSe‐ as well as BeTe‐based quantum‐well structures has been fabri‐ cated and investigated BeTe buffer layers improve the growth start on GaAs substrates drasti‐ cally compared to ZnSe/GaAs The valence‐band offset between BeTe and ZnSe has been determined to be 09 eV (type II) Due to the high‐lying valence band of BeTe, a BeTe–ZnSe pseudograding can be used for an efficient electrical contact between p‐ZnSe and p‐GaAs BeMgZnSe quaternary thin‐film structures have reproducibly been grown with high struc‐ tural quality, and rocking curve widths below 20 arcsec could be reached Quantum‐well structures show a high photoluminescence intensity even at room temperature

Molecular‐beam epitaxy of beryllium‐chalcogenide‐based thin films and quantum‐well structures

The E0 band gap energies and the lattice constants of zinc‐blende Zn1−xMgxSe alloys grown by molecular beam epitaxy in the composition range of 0≤x≤0.95 are determined. A nonlinear dependence on the composition is observed for both the band‐gap energies and the lattice con‐ stants of the ternary alloys. To our knowledge this is an initial report of a bowing in the lattice constant of a ternary II–VI alloy. Considering the bowings, the band‐gap energy and the lattice constant of zinc‐blende MgSe are extrapolated to be about 4.0 eV and 5.91 A, respectively.

E0 band‐gap energy and lattice constant of ternary Zn1−xMgxSe as functions of composition

Rashba spin splitting has been observed in the first conduction subband of n-type modulation-doped HgTe quantum wells (QW's) with an inverted band structure via an investigation of Shubnikov--de Haas oscillations in gated Hall bars. In accordance with calculations, no spin splitting was observed in the second conduction subband $(E2),$ but an obvious Rashba splitting is present in the first heavy-hole-like conduction subband $(H1)$ that displays a large dependence on gate voltage. Self-consistent Hartree calculations of the band structure based on an $8\ifmmode\times\else\texttimes\fi{}8\mathbf{k}\ensuremath{\cdot}\mathbf{p}$ model are compared with experiment, which enables us to understand and quantitatively describe the experimental results. It has been shown that the heavy-hole nature of the $H1$ conduction subband greatly influences the spatial distribution of electrons in the QW and also enhances the Rashba spin splitting at large electron densities. These are unique features of type III heterostructures in the inverted band regime. The $\ensuremath{\beta}{k}_{\ensuremath{\Vert}}^{3}$ dispersion predicted by an analytical model is a good approximation of the self-consistent Hartree calculations for small values of the in-plane wave-vector ${k}_{\ensuremath{\Vert}}$ and has consequently been employed to describe the spin splitting of the $H1$ conduction subband rather than the commonly used $\ensuremath{\alpha}{k}_{\ensuremath{\Vert}}$ dispersion for the conduction subband in type I heterojunctions. The relative magnitude of Rashba splitting in the $H1$ and $E2$ subbands as well as the splitting of the $H1$ subband for different well widths are also presented.

Rashba splitting in n-type modulation-doped HgTe quantum wells with an inverted band structure

Beryllium chalcogenides have a much higher degree of covalency than other II–VI compounds. Be containing ZnSe based mixed crystals show a significant lattice hardening effect. In addition, they introduce substantial additional degrees of freedom for the design of wide gap II–VI heterostructures due to their band gaps, lattice constants, and doping behavior. Therefore, these compounds seem to be very interesting materials for short wavelength laser diodes. Here, we report on the first fabrication of laser diodes based on the wide band gap II–VI semiconductor compound BeMgZnSe. The laser diodes emit at a wavelength of 507 nm under pulsed current injection at 77 K, with a threshold current of 80 mA, corresponding to 240 A/cm2.

G. Landwehr

Papers

Band structure of semimagnetic Hg 1 − y Mn y Te quantum wells

Molecular‐beam epitaxy of beryllium‐chalcogenide‐based thin films and quantum‐well structures

E0 band‐gap energy and lattice constant of ternary Zn1−xMgxSe as functions of composition

Rashba splitting in n-type modulation-doped HgTe quantum wells with an inverted band structure

Laser diodes based on beryllium-chalcogenides