Bulk-boundary correspondence in three dimensional topological insulators

We discuss the relation between bulk topological invariants and the spectrum of surface states in three dimensional non-interacting topological insulators. By studying particular models, and considering general boundary conditions for the electron wavefunction on the crystal surface, we demonstrate that using experimental techniques that probe surface states, only strong topological and trivial insulating phases can be distinguished; the latter state being equivalent to a weak topological insulator. In a strong topological insulator, only the {\it parity} of the number of surface states, but not the number itself, is robust against time-reversal invariant boundary perturbations. Our results suggest a \z definition of the bulk-boundary correspondence, compatible with the \z classification of topological insulators.Comment: TeXLive (Unix), revtex4-1, 7 pages, 3 figure

Reflection of light and heavy holes from a linear potential barrier

In this paper we study reflection of holes in direct-band semiconductors from the linear potential barrier. It is shown that light-heavy hole transformation matrix is universal. It depends only on a dimensionless product of the light hole longitudinal momentum and the characteristic length determined by the slope of the potential and doesn't depend on the ratio of light and heavy hole masses, provided this ratio is small. It is shown that the transformation coefficient goes to zero both in the limit of small and large longitudinal momenta, however the phase of a reflected hole is different in these limits. An approximate analytical expression for the light-heavy hole transformation coefficient is found.Comment: 6 pages, 2 figure

Tunable unconventional Kondo effect on topological insulator surfaces

We study Kondo physics of a spin-12 impurity in electronic matter with strong spin-orbit interaction, which can be realized by depositing magnetic adatoms on the surface of a three-dimensional topological insulator. We show that magnetic properties of topological surface states and the very existence of Kondo screening strongly depend on details of the bulk material, and specifics of surface preparation encoded in time-reversal preserving boundary conditions for electronic wavefunctions. When this tunable Kondo effect occurs, the impurity spin is screened by purely orbital motion of surface electrons. This mechanism gives rise to a transverse magnetic response of the surface metal, and to spin textures that can be used to experimentally probe signatures of a Kondo resonance. Our predictions are particularly relevant for STM measurements in PbTe-class crystalline topological insulators, but we also discuss implications for other classes of topological materials

First-principles envelope-function theory for lattice-matched semiconductor heterostructures

In this paper a multi-band envelope-function Hamiltonian for lattice-matched semiconductor heterostructures is derived from first-principles norm-conserving pseudopotentials. The theory is applicable to isovalent or heterovalent heterostructures with macroscopically neutral interfaces and no spontaneous bulk polarization. The key assumption -- proved in earlier numerical studies -- is that the heterostructure can be treated as a weak perturbation with respect to some periodic reference crystal, with the nonlinear response small in comparison to the linear response. Quadratic response theory is then used in conjunction with k.p perturbation theory to develop a multi-band effective-mass Hamiltonian (for slowly varying envelope functions) in which all interface band-mixing effects are determined by the linear response. To within terms of the same order as the position dependence of the effective mass, the quadratic response contributes only a bulk band offset term and an interface dipole term, both of which are diagonal in the effective-mass Hamiltonian. Long-range multipole Coulomb fields arise in quantum wires or dots, but have no qualitative effect in two-dimensional systems beyond a dipole contribution to the band offsets.Comment: 25 pages, no figures, RevTeX4; v3: final published versio

Tunnelling Studies of Two-Dimensional States in Semiconductors with Inverted Band Structure: Spin-orbit Splitting, Resonant Broadening

The results of tunnelling studies of the energy spectrum of two-dimensional (2D) states in a surface quantum well in a semiconductor with inverted band structure are presented. The energy dependence of quasimomentum of the 2D states over a wide energy range is obtained from the analysis of tunnelling conductivity oscillations in a quantizing magnetic field. The spin-orbit splitting of the energy spectrum of 2D states, due to inversion asymmetry of the surface quantum well, and the broadening of 2D states at the energies, when they are in resonance with the heavy hole valence band, are investigated in structures with different strength of the surface quantum well. A quantitative analysis is carried out within the framework of the Kane model of the energy spectrum. The theoretical results are in good agreement with the tunnelling spectroscopy data.Comment: 29 pages, RevTeX, submitted in Phys.Rev.B. Figures available on request from [email protected]

The classification of irreducible admissible mod p representations of a p-adic GL_n

Let F be a finite extension of Q_p. Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over \bar F_p to be supersingular. We then give the classification of irreducible admissible smooth GL_n(F)-representations over \bar F_p in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel-Livne for n = 2. For general split reductive groups we obtain similar results under stronger hypotheses.Comment: 55 pages, to appear in Inventiones Mathematica

Elliptic Curves over Real Quadratic Fields are Modular

We prove that all elliptic curves defined over real quadratic fields are modular.Comment: 38 pages. Magma scripts available as ancillary files with this arXiv versio

Development of an eight-band theory for quantum-dot heterostructures

We derive a nonsymmetrized 8-band effective-mass Hamiltonian for quantum-dot heterostructures (QDHs) in Burt's envelope-function representation. The 8x8 radial Hamiltonian and the boundary conditions for the Schroedinger equation are obtained for spherical QDHs. Boundary conditions for symmetrized and nonsymmetrized radial Hamiltonians are compared with each other and with connection rules that are commonly used to match the wave functions found from the bulk kp Hamiltonians of two adjacent materials. Electron and hole energy spectra in three spherical QDHs: HgS/CdS, InAs/GaAs, and GaAs/AlAs are calculated as a function of the quantum dot radius within the approximate symmetrized and exact nonsymmetrized 8x8 models. The parameters of dissymmetry are shown to influence the energy levels and the wave functions of an electron and a hole and, consequently, the energies of both intraband and interband transitions.Comment: 36 pages, 10 figures, E-mail addresses: [email protected], [email protected]

General boundary conditions for the envelope function in multiband k.p model

We have derived general boundary conditions (BC) for the multiband envelope functions (which do not contain spurious solutions) in semiconductor heterostructures with abrupt heterointerfaces. These BC require the conservation of the probability flux density normal to the interface and guarantee that the multiband Hamiltonian be self--adjoint. The BC are energy independent and are characteristic properties of the interface. Calculations have been performed of the effect of the general BC on the electron energy levels in a potential well with infinite potential barriers using a coupled two band model. The connection with other approaches to determining BC for the envelope function and to the spurious solution problem in the multiband k.p model are discussed.Comment: 15 pages, 2 figures; to be published in Phys. Rev. B 65, March 15 issue 200

Interface electronic states and boundary conditions for envelope functions

The envelope-function method with generalized boundary conditions is applied to the description of localized and resonant interface states. A complete set of phenomenological conditions which restrict the form of connection rules for envelope functions is derived using the Hermiticity and symmetry requirements. Empirical coefficients in the connection rules play role of material parameters which characterize an internal structure of every particular heterointerface. As an illustration we present the derivation of the most general connection rules for the one-band effective mass and 4-band Kane models. The conditions for the existence of Tamm-like localized interface states are established. It is shown that a nontrivial form of the connection rules can also result in the formation of resonant states. The most transparent manifestation of such states is the resonant tunneling through a single-barrier heterostructure.Comment: RevTeX4, 11 pages, 5 eps figures, submitted to Phys.Rev.