Bulk Wave Functions Research Articles

Electronic structure of semiconductor quantum films.

The electronic structure of thin (\ensuremath{\le}30 A\r{}) free-standing ideal films of Si(001), Si(110), and GaAs(110) is calculated using a plane-wave pseudopotential description. Unlike the expectation based on the simple effective-mass model, we find the following. (i) The band gaps of (001) quantum films exhibit even-odd oscillation as a function of the number N of monolayers. (ii) In addition to sine-type envelope functions which vanish at the film boundaries, some states have cosine envelope functions with extrema at boundaries. (iii) Even-layer Si(001) films exhibit at the valence-band maximum a state whose energy does not vary with the film thickness. Such zero confinement states have constant envelope throughout the film. (iv) Optical transitions in films exhibit boundary-imposed selection rules. Furthermore, oscillator strengths for pseudodirect transitions in the vicinity of forbidden direct transitions can be enhanced by several orders of magnitude. These findings, obtained in direct supercell calculations, can be explained in terms of a truncated crystal (TC) analysis. In this approach the film's wave functions are expanded in terms of pairs of bulk wave functions exhibiting a destructive interference at the boundaries. This maps the eigenvalue spectra of a film onto the bulk band structure evaluated at special k points which satisfy the boundary conditions. We find that the TC representation reproduces accurately the above-mentioned results of direct diagonalization of the film's Hamiltonian. This provides a simple alternative to the effective-mass model and relates the properties of quantum structures to those of the bulk material.

Theory of semiconductor superlattice electronic structure

The authors review the theory of semiconductor superlattice electronic structure. First a survey of theoretical methods is presented. These methods can be divided into two general classes: the supercell approach in which the superlattice is viewed as a material with a large unit cell, and the boundary-condition approach in which bulk wave functions in the constituent semiconductors are matched at the superlattice interfaces. Supercell approaches are essentially the same as conventional band-structure methods. They can only be applied to thin-layer superlattices because of numerical cost. The authors discuss problems of interface matching that occur in various boundary-condition methods and relate these methods to each other. A particular boundary-condition method is used to discuss the electronic structure of various III-V semiconductor superlattices. Emphasis is placed on discussing the qualitatively different behavior that can arise because of different energy-band lineups, strain conditions, and growth orientations. The authors compare the results of three commonly used boundary-condition methods and find generally good agreement.

Theory of Auger core-valence-valence processes in simple metals. II. Dynamical and surface effects on Auger line shapes.

Auger CVV spectra of simple metals are generally believed to be well described by one-electron-like theories in the bulk which account for matrix elements and, in some cases, also static core-hole screening effects. We present here detailed calculations on Li, Be, Na, Mg, and Al using self-consistent bulk wave functions and proper matrix elements. The resulting spectra differ markedly from experiment and peak at too low energies. To explain this discrepancy we investigate effects of the surface and dynamical effects of the sudden disappearance of the core hole in the final state. To study core-hole effects we solve Mahan--Nozi\`eres--De Dominicis (MND) model numerically over the entire band. The core-hole potential and other parameters in the MND model are determined by self-consistent calculations of the core-hole impurity. The results are compared with simpler approximations based on the final-state rule due to von Barth and Grossmann. To study surface and mean-free-path effects we perform slab calculations for Al but use a simpler infinite-barrier model in the remaining cases. The model reproduces the slab spectra for Al with very good accuracy. In all cases investigated either the effects of the surface or the effects of the core hole give important modifications and a much improved agreement with experiment.

Surface contribution of bulk states to the optical properties of metals

The influence on the optical properties of the tails of the bulk wavefunctions near the surface is investigated. It is shown that this surface contribution to the effective dielectric function is strongly anisotropic (in the author's model it is uni-axial with the optical axis perpendicular to the surface) and of the same order of magnitude as the intraband contribution observed experimentally. For the imaginary part of the surface contribution to the effective dielectric function in the intraband region and in the limit omega <<kF2 the authors obtain epsilon 2eff=(16kF4 omega p/ pi c omega 2) cos2 theta , where theta is the angle between the polarisation vector and the surface normal. If, in addition 1/ tau << omega , this leads to a surface contribution to the intraband relaxation time of 1/ tau ys=(3kF omega p/c) cos2 theta .

Spin Polarization of Electrons Field Emitted from Single-Crystal Iron Surfaces

The spin polarization of electrons field emitted from single-crystal iron surfaces was observed to be (+25\ifmmode\pm\else\textpm\fi{}5)% for emission along [100], (+25\ifmmode\pm\else\textpm\fi{}5)% along [111], and (-5\ifmmode\pm\else\textpm\fi{}10)% along [110] (the polarization is positive for majority-spin electrons). These results correlate qualitatively with expectations from the bulk band model. However, quantitative calculations with bulk wave functions gave strongly parameter-dependent results. For meaningful calculations the effect of the surface on the wave functions must be included.

Abstract: Electronic structure of Si and Ge surfaces—clean and with chemisorbed layers

Practical techniques have been developed for performing self−consistent calculations of the electronic structure of solid surfaces. The method exploits the fact that the disturbance produced by the surface is significant only in the first few atomic layers, and that beyond this the potential may be assumed to be identical to that of the bulk solid. The charge density in the surface region is calculated by numerically solving Schroedinger’s equation for the surface region tails of the bulk wavefunctions and for bound surface states. The self−consistent surface region potential is constructed using a fitted model potential for the ion cores, the exact Hartree potential, and a local approximation for the exchange and correlation potential. The calculations outlined above have been carried out for the Si(lll) surface. An inward relaxation of the surface atom plane of 0.34 Å is suggested by empirical chemical considerations, and the consequences of this and several other amounts of relaxation were explored. In all cases, a band of surface states was found within the absolute energy gap whose charge has a clear ’’dangling bond’’ character. Its shape was dependent on relaxation, going from sp3 to pz with increasing relaxation. A major qualitative difference found in going from 0.17 to 0.34 Å was the appearance of two additional bands of surface states below the valence bands and in internal gaps. These states are clearly identified with the strengthened bonds between the first and second layers, and contribute structure to the surface density of states, which has recently been verified in spectroscopic measurements. The calculated workfunction and surface Fermi level pinning are in excellent agreement with experiment. Similar calculations have been carried out for Ge(lll). The main difference from the Si results is a change in the nature of the energy vs parallel momentum relation for the dangling bond surface state. A second energy minimum appears which has a charge distribution that is not dangling bond−like, and that could participate in chemisorption bonds to second layer Ge atoms. The theoretical methods employed are also suitable for treating ordered adsorbed overlayers. H chemisorbed on Si(lll) has been studied, and a single fully occupied band of surface states has been found to lie in internal energy gaps of the valence band, 4−7 eV below the valence band maximum. The charge density contributed by this band is concentrated in the Si−H bond. Spectroscopic structure associated with this band of surface states has been seen experimentally. Another chemisorption system studied was Al substitutionally adsorbed on Si(lll), one of several structures which has been observed for this system. Since Al in this position has its valence requirements satisfied, a filled−band (saturated bond) picture might have been anticipated. Instead, three partially occupied bands of surface states have been found, indicating a metallic surface. This is especially remarkable considering that the Al atoms are 3.84 Å apart, compared to 2.86 Å in Al metal. A consequence of this result is the possibility that adsorbed metal films only a few layers thick may provide an adequate model of semiconductor metal interfaces which is suitable for study on an atomic scale by our methods.