Shockley Surface State Research Articles

Calculation of Shockley and Tamm states for a one-dimensional lattice using Green's-function theory

The surface-state energy levels associated with a semi-infinite linear chain of atoms with two atomic orbitals have been investigated using the Green's-function formalism of Kalstein and Soven. We have made a systematic study of the dependence of the energies of both Shockley and Tamm surface states and their existence condition on parameters such as the hopping integral and the surface-potential perturbation.

Shockley Surface States on the (110) Plane of bcc Iron

Shockley surface states have been found on different faces of bcc iron. The (110) surface shows these states both below and above the Fermi level. Recent experiments on iron and other transition-metal surfaces are discussed tentatively in the light of the results obtained in this paper. The role of surface states in chemisorption and catalysis on transition metals is also mentioned.

On the extension of shockley surface states in metals

By considering a simple model for Shockley states in metals, it is shown that screening leads first, to an accumulation of the charge at the surface and second to a negligible shift of the energy of the localized state.

Stark Ladders for Finite, One-Dimensional Models of Crystals

Application of the methodology of analysis of Shockley surface states confirms the existence of Wannier's Stark ladders of energy levels in finite crystals for wave functions that decay exponentially towards both edges of the crystal and shows that the perfectly regular energy intervals will be minutely disturbed by tunneling effects. If phonon scattering is neglected, coherent oscillations of electrons associated with specified energy bands are predicted.

ПОВЕРХНОСТНЫЕ ЭЛЕКТРОННЫЕ СОСТОЯНИЯ МОДЕЛИ ОДНОМЕРНОГО ПОЛУБЕСКОНЕЧНОГО КРИСТАЛЛА ТИПА КРОНИГА-ПЕННИ

Using the MO LCAO formalism, exact energy spectrum and wave functions of electrons in the one-dimensional model of a semi-infinite crystal were obtained. The model can be interpreted as a semi-infinite chain composed of two kinds of atoms connected alternately by strong and weak bonds. In addition to Tamm and Shockley surface states, Shockley subsurface states were obtained. It was shown, that the character of the surface state can change depending on its position in the forbidden energy band. Reasons for the occurrence of Shockley surface states are discussed. Results of this paper are compared with those obtained within the framework of Huckel approximation.

Unrestricted Hartree—Fock Calculations on Linear-Atomic-Chain Models Yielding the ``Surface'' (End) States

Restricted and unrestricted Hartree—Fock methods have been applied to the study of the properties of the Tamm and Shockley surface states. Linear polyenic chains were chosen as models and the PPP-type semi-empirical method used. The SPO (spin-polarized orbital) results on Tamm surface states and their physical interpretations are discussed. Further, the radical—or polyradical—character of atomic chains yielding the Shockley surface states was obtained and is discussed. Possible implications for the surfaces of real semiconductor crystals and their geometry are suggested.

Simple model of surface and chemisorption states on the (111) plane of ZnS type crystals

The surface and chemisorption states of a simple model of a semi-infinite ZnS type (sphalerite) crystal, limited by the (111) plane were studied. The existence conditions of various states localised near the surface were studied and the classification of these states was carried out. The differences between characteristics of surfaces which are formed by more electronegative atoms and surfaces formed by more electropositive atoms were pointed out. In the case of more electronegative surfaces the energy band of Shockley surface states is placed closely above the top of the highest valence band, the formation of Tamm surface states with the energies below the valence band is facilitated and the formation of Tamm surface states with the energies above the conductivity bands becomes more difficult. As far as the more electropositive surface is concerned, the energy band of Shoekley states is placed near the bottom of the conductivity band, the formation of Tamm states with energies below the lowest valence band is made more difficult and the formation of Tamm states with the energies above the highest conductivity band facilitated. In the gaps between individual valence or conductivity energy bands of volume states, further energy bands of surface states of a new kind can originate. The earlier studied model of the diamond lattice is a special case of the model investigated in this work. From the studied model follows, that a semi-infinite crystal can become a degenerated p-or n-semi-conductor if, in the forbidden energy gap, Shockley surface states do not exist at least for one of the two possible (111) delimiting planes.

A simple model of hydrogen chemisorption on diamond-like crystals

A model for the chemisorption of hydrogen on the surface of a semi-infinite diamond-like crystal, assuming complete surface occupation, is examined by the simple semiempirical MO LCAO method. The main features of various types of localized surface states connected with the formation of the chemisorption bond and their relation to the Tamm and Shockley surface states for the same model are pointed out. An interpretation is given of surface states connected with bond character changes in surface layers of the crystal. It is demonstrated that the Shockley surface states existing on a free surface tend to disappear with increasing chemisorption.

Contribution to the theory of surface and chemisorption electronic states on various crystal planes with special reference to the diamond - like lattice

A new variant of the one-electron Green function method is suggested, which enables an appreciable simplification of the calculation of surface and chemisorption electronic states on selected boundary planes of crystals. By this method, the (111) and (1̄1̄1) planes were investigated of a semi-infinite diamond-like crystal. New Shockley surface states (Λ 3 type) were found and conclusions were obtained concerning the existence and number of surface states on ideal surfaces of both kinds. Qualitative results are mentioned briefly of calculation of chemisorption of atoms (assuming complete surface occupation) on the (111) plane of the crystal considered. An LGP-30 “pocket” electronic computer was used to carry out numerical calculations.

The Shockley surface states of electrons in the Kronig-Penney model of a linear chain of atoms joined by alternately strong bonds

The allowed energies of the bond states of electrons in a one-dimensional potential, which is constant along the lines joining the atoms in a chain and has the form of the Diracδ-function in the neighbourhood of the atom, were calculated. The lengths of the bonds alternate. It is found that we get Shockley surface states in a semi-infinite chain, produced from an infinite chain by interrupting the shorter bond. This proves that the results for the given model are qualitatively in agreement with the conclusions reached by the MO LCAO method for an analogical model.

Shockley surface states for a graphite and diamond model with a delta function potential

The Shockley surface states problem was formulated for the Coulson graphite model and Hoerni diamond model with delta function potentials. It was shown that the solution of these problems can be reduced to the solution of analogous problems in the MO-LCAO method. A solution is given for one type of models. The analogy of both methods shows that the qualitative properties of Shockley surface states depend primarily on the geometry of the lattices.

An interpretation of the conditions for the existence of Shockley surface states

A study is made of the Shockley surface states in a linear chain of equal atoms, joined by alternately strong bonds. If the simple MO LCAO method is used, which considers the exchange integrals between nearest neighbours and next nearest neighbours, we get the surface states for a semi-infinite chain if the stronger bond is interrupted. The connection between Shockley and Tamm surface states is shown and the hypothesis is put forward that the condition for the existence of Shockley surface states has a generally simple physical interpretation, the validity of which is proved on the models of a semi-infinite crystal studied up to now.

Study of the Surface States of Diamond and Graphite by a Simple MO-LCAO Method

The surface states of diamond and two-dimensional graphite are investigated on the assumption that the delimitation of the crystal causes no perturbation within the elementary cells of the finite crystal. In both cases a plane perpendicular to the bond between a selected pair of carbon atoms is taken as the delimiting plane. The molecular orbitals are assumed in the form of a linear combination of $s{p}^{3}$ or $s{p}^{2}$ hybrids. In the case of graphite, molecular orbitals that are linear combinations of $2{p}_{z}$ orbitals, whose interaction with $s{p}^{2}$ hybrids is neglected, are further considered. It appears that in the case of diamond there exists a band of energies pertaining to Shockley surface states in the gap between the valence and conductivity bands. The number of atoms in this band equals the number of atoms in the surface. From a discussion of the pertinent wave functions, it follows that these states are an expression of unsaturated bonds of the surface carbon atoms. The electron density on the hybrids projecting from the surface is essentially greater than the density on the other hybrids of the surface atoms. Further bands of surface states exist in the region of energies allowed for the volume valence and the conductivity states. In graphite, a quite analogous behavior is shown by those surface states whose wave functions are linear combinations of $s{p}^{2}$ orbitals. Surface states whose wave functions are linear combinations of ${p}_{z}$ orbitals are a manifestation of unsaturated double bonds of surface atoms having only two neighbors. The analogy between these Shockley states and the nonbonding states of odd alternant aromatic hydrocarbons is pointed out.

Contribution to the theory of the Shockley surface states—I: General formulation and the case of zero deformation of the potential on the goodwin-artmann model

Within the framework of the tight-binding method, general equations have been set up for the calculation of orbital energies and wave functions of the electron in a crystal model in which an interaction between atomic s orbitals and p orbitals is assumed. Simple explicit expressions are derived for the case of zero deformation of the potential of the crystal surface. From a discussion of the hybridization of the wave functions of the volume and surface states the connection with the concept of free valence localized in the crystal surface follows.