Surface State Wave Function Research Articles

Quantum transport of topological surface states in presence of circular n-p junctions

The surface states of topological insulators are characterized by spin-momentum locking, where the electron spin is locked perpendicular to its momentum. This property is preserved under gate potentials which can be used to manipulate spin currents without an external magnetic field enhancing the prospects for spintronic applications. Here, we investigate the scattering of the wave functions of topological surface states within the circular n-p junction positioned on the upper surface of the quintuple-layered three-dimensional topological insulator. With the analytical calculations we explain the scattering of these surface states in such systems.

Influence of surface optical phonon on the electronic surface states in wurtzite group-III nitride ternary mixed crystals

An intermediate-coupling variational method is presented to investigate the surface electron states in wurtzite AxB1−xN (A, B=Al, Ga and In) ternary mixed crystals (TMCs). Corresponding effective Hamiltonian are derived by considering the surface-optical-phonon (SO-phonon) influence and anisotropic structural effect. The surface-state energies of electron, the coupling constants and the average penetrating depths of the electronic surface-state wave functions have been numerical computed as a function of the composition x and the surface potential V0 for the wurtzite AlxGa1−xN, AlxIn1−xN and InxGa1−xN, respectively. The results show that the surface-state levels of electron are reduced with the increasing of the composition x in wurtzite AxB1−xN. It is also found that the electron-surface-optical-phonon (e-SO-p) coupling lowers the surface-state energies of electron and the shifts of the electronic surface-state energy level in the wurtzite AlxGa1−xN and AlxIn1−xN increase with the increasing of the composition x. However, in the wurtzite InxGa1−xN, the case is contrary. The influence of the e-SO-p interaction on the surface electron states can not be neglected in wurtzite AxB1−xN.

Bulk versus surface contributions to the Rashba spin splitting of Shockley surface states

Shockley surface states of a bulk crystal with both time reversal and space inversion symmetries exhibit the Rashba spin splitting due to the broken space inversion symmetry at the surface. Since the evanescent states in the bulk region are doubly degenerate with respect to spin degrees of freedom even in the presence of spin-orbit interaction (SOI), one might think that the entire spin splitting occurs via the SOI in the surface region where the potential energy deviates from the bulk one. In the present work, we elucidate why this is not the case. Namely, in the presence of SOI, the complex energy bands are modified such that a pair of evanescent states having the same energy and the same complex wave number become two distinct solutions of the Schr\odinger equation that do not satisfy the same boundary condition. Since the tail of the surface-state wave function is expressed as a superposition of these evanescent waves, the bulk region also contributes to the Rashba spin splitting. We illustrate this effect by a first-principles calculation for Au(111) and by a simplified $sp$-band model Hamiltonian on a square lattice.

Rashba spin splitting of L -gap surface states on Ag(111) and Cu(111)

Spin-resolved band structures of L-gap surface states on Ag(111) and Cu(111) are investigated by spin- and angle-resolved photoelectron spectroscopy (SARPES) with a vacuum-ultra-violet laser. The observed spin textures of the Ag(111) and Cu(111) surface states agree with that expected by the conventional Rashba effect. The Rashba parameter of the Ag(111) surface state is estimated quantitatively and is 80% of that of Cu(111). The surface-state wave function is found to be predominantly of even mirror-symmetry with negligible odd contribution by SARPES using a linearly polarized light. The results are consistent with our theoretical calculations for the orbital-resolved surface state.

Generating surface states in a Weyl semimetal by applying electromagnetic radiation

We show that the application of circularly polarized electromagnetic radiation on the surface of a Weyl semi-metal can generate states at that surface. These states can be characterized by their surface momentum. The Floquet eigenvalues of these states come in complex conjugate pairs rather than being equal to $\pm 1$. If the amplitude of the radiation is small, we find some unusual bulk-boundary relations: the Floquet eigenvalues of the surface states lie at the extrema of the Floquet eigenvalues of the bulk system, and the peaks of the Fourier transforms of the surface state wave functions lie at the momenta where the bulk Floquet eigenvalues have extrema. For the case of zero surface momentum, we can analytically derive scaling relations between the decay length of the surface states and the amplitude and penetration length of the radiation. For topological insulators, we again find that circularly polarized radiation can generate states on the surfaces; these states have much larger decay lengths than the surface states which are present even in the absence of radiation. Finally, we show that radiation can generate surface states for trivial insulators also.

Intermediate stages of surface state formation and collapse of topological protection to transport in Bi2Se3

Surface states consisting of helical Dirac fermions have been extensively studied in three-dimensional topological insulators. Yet, experiments to date have only investigated fully formed topological surface states (TSS) and it is not known whether preformed or partially formed surface states can exist or what properties they could potentially host. Here, by decorating thin films of Bi2Se3 with nanosized islands of the same material, we show for the first time that not only can surface states exist in various intermediate stages of formation but they exhibit unique properties not accessible in fully formed TSS. These include tunability of the Dirac cone mass, vertical migration of the surface state wave-function and the appearance of mid-gap Rashba-like states as exemplified by our theoretical model for decorated TIs. Our experiments show that an interplay of Rashba and Dirac fermions on the surface leads to an intriguing multi-channel weak anti-localization effect concomitant with an unprecedented tuning of the topological protection to transport. Our work offers a new route to engineer topological surface states involving Dirac–Rashba coupling by nano-scale decoration of TI thin films, at the same time shedding light on the real-space mechanism of surface state formation in general.

Two-band superconductivity of bulk and surface states in Ag thin films on Nb

We use epitaxial strain to shift the energy of the two-dimensional Ag(111) surface states of Ag islands on Nb(110) substrates, allowing to spatially tune the bottom of the surface-state band $E_{\rm SS}$ through the Fermi level $E_{\rm F}$. Bulk and surface-state contributions to the Ag(111) local density of states (LDOS) can be separated with scanning tunneling spectroscopy. For thick islands ($\approx$\, 20 nm), the Ag surface states are decoupled from the Ag bulk states via orthogonality, and the superconductive gap induced by proximity to Nb is due to bulk states only. However, for thin islands (3-4 nm), surface-state electrons develop superconducting correlations as identified by a complete energy gap in the LDOS when $E_{\rm F}$ is smaller than but close to $E_{\rm F}$. The induced superconductivity in this case is of two-band nature and appears to occur when the surface-state wave function reaches down to the Ag/Nb interface.

Decay length of surface-state wave functions on Bi(1 1 1)

We calculate the decay length in surface normal direction of the surface-state wave functions on a clean Bi(1 1 1) surface as a function of two-dimensional (2D) wave vector along the line. For this purpose, we perform a first-principles density functional theory (DFT) calculation for semi-infinite Bi(1 1 1) by employing the surface embedded Green’s function technique. The decay length of the two surface bands is found to be ∼24 Bi bilayers at , while it remains less than 5 Bi bilayers when is away from and . At , the degenerate surface bands are split from the upper boundary energy of the projected bulk valence bands only by 5 meV. In spite of this, the decay length of these bands at is less than 10 Bi bilayers due to the large effective mass (small curvature) of the highest valence band in the surface normal direction.

Helical Majorana surface states of strongly disordered topological superconductors with time-reversal symmetry

Noncentrosymmetric superconductors with strong spin-orbit coupling and the B phase of $^{3}\mathrm{He}$ are possible realizations of topological superconductors with time-reversal symmetry. The nontrivial topology of these time-reversal invariant superconductors manifests itself at the material surface in the form of helical Majorana modes. In this paper, using extensive numerical simulations, we investigate the stability and properties of these Majorana states under strong surface disorder, which influences both bulk and surface states. To characterize the effects of strong disorder, we compute the level spacing statistics and the local density of states of both two- and three-dimensional topological superconductors. The Majorana surface states, which are located in the outermost layers of the superconductor, are protected against weak disorder due to their topological characteristic. Sufficiently strong disorder, on the other hand, partially localizes the surface layers, with a more pronounced effect on states with energies close to the gap than on those with energies close to zero. In particular, we observe that for all disorder strengths and configurations there always exist two extended states at zero energy that can carry thermal current. At the crossover from weak to strong disorder the surface state wave functions and the local density of states show signs of critical delocalization. We find that at this crossover the edge density of states of two-dimensional topological superconductors exhibits a zero-energy divergence, reminiscent of the Dyson singularity of quasi-one-dimensional dirty superconductors.

Rashba spin splitting of Shockley surface states on semi-infinite crystals

We study the Rashba spin splitting of the $L$-gap surface state on the (111) surfaces of Au, Ag, and Cu, and that of the topologically protected surface state on Sb(111) by a first-principles calculation based on the embedded Green's function technique. By taking advantage of semi-infinite surface geometry, we address several issues that are difficult to handle with the finite slab calculation. For Au(111), we investigate the nonlinear dependence on wave vector $\mathbf{k}$ of the spin-splitting energy, and also at what wave number each dispersion curve of the two spin-split bands is merged into a surface-projected bulk band. Then, we discuss why Cu(111) exhibits a larger Rashba effect than Ag(111) despite the fact that the atomic orbitals of Ag have larger spin-orbit matrix elements than those of Cu. We reveal that, while the wave function of the $L$-gap surface state has strong ${p}_{z}$ character (the $z$ axis is surface normal), the Rashba spin splitting occurs through the spin-orbit matrix element between the ${d}_{{z}^{2}}$ and ${d}_{uz}$ orbital components hybridized with ${p}_{z}$, where ${d}_{uz}$ denotes a linear combination of ${d}_{xz}$ and ${d}_{yz}$ with its orbital lobes pointing to the $\mathbf{k}$ direction (defined as the $u$ axis). Cu(111) exhibits a larger spin splitting than Ag(111), since the amount of both the ${d}_{{z}^{2}}$ and ${d}_{uz}$ components mixed in its surface-state wave function is much larger than that of Ag(111). In contrast to the (111) surfaces of noble metals, the spin splitting of the topologically protected surface state on Sb(111) arises from the spin-orbit matrix element between the ${p}_{z}$ and ${p}_{u}$ orbital components. We also discuss under what condition the energy dispersion curve of a surface state intersects the boundary line between a projected bulk band and a projected bulk band gap at a large angle, instead of gradually approaching the boundary line with increasing $|\mathbf{k}|$. Finally, we comment on a recent theoretical work of Krasovskii [Phys. Rev. B 90, 115434 (2014)] on the microscopic origin of the Rashba spin splitting.

Topological protection, disorder, and interactions: Survival at the surface of three-dimensional topological superconductors

We consider the interplay of disorder and interactions upon the gapless surface states of 3D topological superconductors. The combination of topology and superconducting order inverts the action of time-reversal symmetry, so that extrinsic time-reversal invariant surface perturbations appear only as "pseudomagnetic" fields (abelian and non-abelian vector potentials, which couple to spin and valley currents). The main effect of disorder is to induce multifractal scaling in surface state wavefunctions. These critically delocalized, yet strongly inhomogeneous states renormalize interaction matrix elements relative to the clean system. We compute the enhancement or suppression of interaction scaling dimensions due to the disorder exactly, using conformal field theory. We determine the conditions under which interactions remain irrelevant in the presence of disorder for symmetry classes AIII and DIII. In the limit of large topological winding numbers (many surface valleys), we show that the effective field theory takes the form of a Finkel'stein non-linear sigma model, augmented by the Wess-Zumino-Novikov-Witten term. The sigma model incorporates interaction effects to all orders, and provides a framework for a controlled perturbative expansion; the inverse spin or thermal conductance is the small parameter. For class DIII we show that interactions are always irrelevant, while in class AIII there is a finite window of stability, controlled by the disorder. Outside of this window we identify new interaction-stabilized fixed points.

Enhancing and reducing the Rashba-splitting at surfaces by adsorbates: Na and Xe on Bi/Cu(111)

The surface alloy Bi/Cu(111) shows a paradigmatic free-electron-like surface state with a very large Rashba-type spin–orbit splitting. Using angle-resolved photoemission we investigate how adsorbates of different chemical nature influence the size of the spin splitting in this system. We find that the adsorption of small amounts of monovalent Na atoms leads to an enhancement of the spin splitting while an overlayer of the closed-shell rare gas Xe causes a reduction. The latter result is in contrast to the Au(111) surface for which an increased splitting size after Xe-adsorption was observed. We discuss these experimental findings in terms of the characteristic differences of the surface state wave functions and their spatial deformation in the presence of different types of adsorbates. Our results provide insight into the complex interplay of atomic and interface potential gradients governing the Rashba effect.

Surface states and annihilation characteristics of positrons trapped at the oxidized Cu(100) surface

In this work we present the results of theoretical studies of positron surface and bulk states and annihilation probabilities of surface-trapped positrons with relevant core electrons at the oxidized Cu(100) surface under conditions of high oxygen coverage. Oxidation of the Cu(100) surface has been studied by performing an ab-initio investigation of the stability and electronic structure of the Cu(100) missing row reconstructed surface at various on-surface and subsurface oxygen coverages ranging from 0.5 to 1.5 monolayers using density functional theory (DFT). All studied structures have been found to be energetically more favorable as compared to structures formed by purely on-surface oxygen adsorption. The observed decrease in the positron work function when oxygen atoms occupy on-surface and subsurface sites has been attributed to a significant charge redistribution within the first two layers, buckling effects within each layer and an interlayer expansion. The computed positron binding energy, positron surface state wave function, and annihilation probabilities of the surface trapped positrons with relevant core electrons demonstrate their sensitivity to oxygen coverage, atomic structure of the topmost layers of surfaces, and charge transfer effects. Theoretical results are compared with experimental data obtained from studies of oxidation of the Cu(100) surface using positron annihilation induced Auger electron spectroscopy (PAES). The results presented provide an explanation for the changes observed in the probability of annihilation of surface trapped positrons with Cu 3p core-level electrons as a function of annealing temperature.

Effect of electron–phonon interaction on surface states in wurtzite nitride semiconductors under pressure

A variational theory is proposed to study the electronic surface states in semi-infinite wurtzite nitride semiconductors under the hydrostatic pressure. The electronic surface state energy level is calculated, by taking the effects of the electron–Surface–Optical–phonon interaction, structural anisotropy and the hydrostatic pressure into account. The numerical computation has been performed for the electronic surface state energy levels, coupling constants and the average penetrating depths of the electronic surface state wave functions under the hydrostatic pressure for wurtzite GaN, AlN and InN, respectively. The results show that electron–Surface–Optical–phonon interaction lowers the electronic surface state energy levels. It is also found that the electronic surface state energy levels decrease with the hydrostatic pressure in wurtzite GaN and AlN. But for wurtzite InN, the case is contrary. It is shown that the hydrostatic pressure raised the influence of electron–phonon interaction on the electronic surface states obviously. The effect of electron–Surface–Optical–phonon interaction under the hydrostatic pressure on the electronic surface states cannot be neglected, in specially, for materials with strong electron–phonon coupling and wide band gap.

Theoretical aspects of studies of oxide and semiconductor surfaces using low energy positrons

This paper presents the results of a theoretical study of positron surface and bulk states and annihilation characteristics of surface trapped positrons at the oxidized Cu(100) single crystal and at both As- and Ga-rich reconstructed GaAs(100) surfaces. The variations in atomic structure and chemical composition of the topmost layers of the surfaces associated with oxidation and reconstructions and the charge redistribution at the surfaces are found to affect localization and spatial extent of the positron surface-state wave functions. The computed positron binding energy, work function, and annihilation characteristics reveal their sensitivity to charge transfer effects, atomic structure and chemical composition of the topmost layers of the surfaces. Theoretical positron annihilation probabilities with relevant core electrons computed for the oxidized Cu(100) surface and the As- and Ga-rich reconstructed GaAs(100) surfaces are compared with experimental ones estimated from the positron annihilation induced Auger peak intensities measured from these surfaces.

Oxidation and thermal reduction of the Cu(1 0 0) surface as studied using positron annihilation induced Auger electron spectroscopy (PAES)

Changes in the surface of an oxidized Cu(1 0 0) single crystal resulting from vacuum annealing have been investigated using positron annihilation induced Auger electron spectroscopy (PAES). PAES measurements show a large increase in the intensity of the annihilation induced Cu M 2,3VV Auger peak as the sample is subjected to a series of isochronal anneals in vacuum up to annealing temperature 300 °C. The intensity then decreases monotonically as the annealing temperature is increased to ∼600 °C. Experimental probabilities of annihilation of surface-trapped positrons with Cu 3p and O 1s core-level electrons are estimated from the measured intensities of the positron annihilation induced Cu M 2,3VV and O KLL Auger transitions. Experimental PAES results are analyzed by performing calculations of positron surface states and annihilation probabilities of surface-trapped positrons with relevant core electrons taking into account the charge redistribution at the surface, surface reconstructions, and electron–positron correlations effects. The effects of oxygen adsorption on localization of positron surface state wave function and annihilation characteristics are also analyzed. Possible explanation is proposed for the observed behavior of the intensity of positron annihilation induced Cu M 2,3VV and O KLL Auger peaks and probabilities of annihilation of surface-trapped positrons with Cu 3p and O 1s core-level electrons with changes of the annealing temperature.

Quantum Oscillations in Coupled Two-Dimensional Electron Systems

Quantum oscillations of the electron-phonon coupling of a Shockley surface state induced by the coupling to a metallic quantum well are observed experimentally as the thickness of the quantum well is increased. Microscopic calculations allow us to assign these oscillations to changes in the Eliashberg function, caused by a modulation in the electronic density overlap between the 2D surface state band and the 2D electron system of the quantum well and by the gradient of the one-electron potential in combination with the localized character of the Shockley-type surface state wave function. The findings are important in the context of the control of surface processes such as film growth and surface chemical reactions.

Calculational aspects of electron–phonon coupling at surfaces

We study the validity of two frequently used approximations in calculations ofelectron–phonon coupling at surfaces. The rigid-ion approximation is a standardapproximation used for the bulk metals. On the basis of density functional theorycalculations, we find that for Be this approximation is as valid for surface atoms as for bulkatoms. In addition, the slab method for calculations of a phonon induced surface statelifetime is examined. The convergence of the electron–phonon matrix element with respectto the thickness of the slab is studied for several systems. When the number of slab layersis increased, the net effect of decreasing overlap and increasing number of final statesdepends strongly on the decay length of the surface state wavefunction and the bandstructure.

Localized States at Zigzag Edges of Bilayer Graphene

We report the existence of zero-energy surface states localized at zigzag edges of bilayer graphene. Working within the tight-binding approximation we derive the analytic solution for the wave functions of these peculiar surface states. It is shown that zero-energy edge states in bilayer graphene can be divided into two families: (i) states living only on a single plane, equivalent to surface states in monolayer graphene and (ii) states with a finite amplitude over the two layers, with an enhanced penetration into the bulk. The bulk and surface (edge) electronic structure of bilayer graphene nanoribbons is also studied, both in the absence and in the presence of a bias voltage between planes.

Enhanced Rashba spin-orbit splitting inBi∕Ag(111)andPb∕Ag(111)surface alloys from first principles

We present first-principles calculations of a $(\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3})R30\ifmmode^\circ\else\textdegree\fi{}$ $\mathrm{Bi}∕\mathrm{Ag}(111)$-ordered surface alloy, which has recently been investigated experimentally using angle-resolved photoemission spectroscopy. The surface states in the $L$-projected bulk band gap show a Rashba-type spin-orbit splitting which is three times larger than what has been observed on a clean Bi(111) surface. This large enhancement can be explained by the strong distortion of the surface-state wave function which is caused by the substantial outward buckling of the Bi atom. Also, in a similar surface alloy, $\mathrm{Pb}∕\mathrm{Ag}(111)$, a strong Rashba-type splitting was found by our calculations. The comparison to the experimental data is more difficult due to the presence of a second, close-by surface state. We discuss the dependence of the two-dimensional band structure on the surface corrugation and compare to the experimental findings.