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Theory of tamm surface states in approximation higher than tight-binding approximation
Abstract
The influence of the non-zero value of exchange integrals between Wannier functions, localized in non-neighbouring elementary cells (higher approximation than tight-binding), on the conditions of existence of Tamm surface states and the position of the energy level corresponding to the surface state is shown.
A very simple crystal model of a semi‐infinite heteroatomic planar lattice like boronitride is investigated for the energy, wave functions and existence conditions of surface states, using the one‐electron Green function method. σ and π electrons are treated separately. For π electrons, we find two surface state bands, both for the electropositive and the electronegative surface. The consequence is the existence of Shockley “subsurface” states. For σ electrons, similar results were found as for the sphalerite‐type lattice.
The investigated model can also be used to draw qualitative conclusions about the effect of electronic correlation on surface states of planar graphite. The possibility of finding a certain type of “antiferromagnetism” with π electrons localized on the planar graphite boundary is suggested.
On étudie un modèle très simple d'un cristal avec un réseau sémi‐infini, héteroatomique, comme le boronitride. L'énergie, les fonctions d'onde et des conditions d'existence d'états de surface ont été obtenus par la méthode de la fonction de Green à un électron. Les électrons σ et π ont été traités séparément. Pour les électrons π on trouve deux bandes d'états de surface pour la surface électropositive ainsi que pour la surface électronégative, ce qui implique l'existence d'états de “sous‐surface” de Shockley. Pour les électrons σ on trouve des résultats semblables aux ceux qu'on a obtenu pour le réseau de type sphalérite.
Le modèle étudié ici peut aussi être employé pour tirer des conclusions qualitatives sur l'effet de la corrélation électronique sur les états de surface du graphite plan. On propose qu'il serait possible de trouver un certain type de “antiferromagnétisme” avec les électrons π localisés sur le bord du graphite plan.
Es wurde ein einfaches Modell für ein halb‐unendliches, heteroatomares Kristallgitter, wie Boronitrid untersucht. Die Energie, die Wellenfunktionen und Existenzbedingungen für Oberflächenzustände wurden mit der Einelektron‐Green‐Funktionsmethode erhalten. Die σ ‐ und π ‐Electronen wurden getrennt behandelt. Für die π ‐Elektronen finden wir zwei Bände für Oberflächenzustände, sowohl für die elektropositive als für die elektronegative Oberfläche. Als Folgerung erhalten wir die Shockleysche unteroberflächenzustände. Für die σ ‐Elektronen wurden ähnliche Resultate als für das Sphaleritgitter gefunden.
Das untersuchte Modell kann auch dafür angewendet werden, urn den Effekt der elektronischen Korrelation auf die Oberflächenzustände planares Graphits in qualitativer Weise zu diskutieren. Es wurde vorgeschlagen, dass es möglich wäre, eine gewisse Art von “Antiferromagnetismus” der π ‐Elektronen die auf der Grenze planares Graphits lokalisiert sind, zu finden.
A new method is presented for the calculation of surface and adsorption
effects on one-electron states in crystals. Conceptually, this method is
similar to the Koster-Slater resolvent method for impurity states, which
has often been applied to surface states in semi-infinite crystals. The
elaboration is very different, however. The proposed scheme works for
finite crystals, the resolvent matrix is calculated numerically instead
of analytically, and the applicability of the method depends on a
suitable algorthm for the numerical solution of the Koster-Slater
equations. Such an algorithm is described. In comparison with the
resolvent method for semi-infinite crystals, this method permits a more
quantitative treatment of real crystals, such as transition metals or
semiconductors. On the other hand, compared with standard
molecular-orbital methods on finite clusters, it can handle much larger
crystals.
Working within the framework of finite crystal models, a comparative study is made of the various shortcomings of the molecular orbital (MO) theory of surface states based on semi-infinite models. Particular attention is paid to the removal of the degeneracy of surface states and the inconsistency of the boundary conditions encountered with semi-infinite models. The main conclusion drawn is that a physically more realistic MO theory of surface states is obtained by using finite models.
The present molecular orbital theory of Tamm surface states, which is based on the tight-binding approximation (TBA), has been extended to include the next-nearest-neighbour (NNN) interaction. The effect of the NNN interaction on the width of the volume band and the density of states has been investigated. Contrary to the TBA, this more elaborate method has shown that, under certain conditions, it is possible for the energies of the surface states to lie within the volume band. Furthermore, depending on the various existence conditions, the number of surface states is two, one or zero.
The usual tight-binding (TB) approximation, employed in electronic structure calculations, is extended to include the more-distant neighbor sites than the first. In doing so, a simple power law is adopted to describe the bond and overlap contributions, which enables the energy dispersion relation to be obtained in a closed analytic form. The resulting energy band structure has markedly different features from its TB counterpart to which it reduces in the appropriate limits. In particular, local extrema always occur at the TB stationary values, although these may not both be band edges. Another intermediate critical point may arise, which is one of the band edges, the other being located at one of the TB extrema. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 341–346, 2003
The energy spectrum of localized states in a simple cubic crystal is analysed, using the next-nearest neighbor approximation. The main distinguishing feature between the nearest and next-nearest neighbor approximation is the energy distribution of the localized states. Contrary to the tight-binding approximation, a non-symmetric distribution is found for clean surfaces, and split energy bands are found for surfaces completely covered by adsorbed atoms.
A unified picture of the electronic and vibrational spectra of solids is presented and discussed. with particular reference to the role of the surface.
Using the direct crystal orbital approach, a study is made of the effect on the crystal energy band edges of including more distant neighbour interactions than the first. Various ways of incorporating these higher-order interactions are discussed.
The semi-infinite chain of identical atoms is discussed using molecular orbital theory including next-nearest-neighbour interactions. It is shown that localized electronic surface states can exist in this model, the maximum number possible being two. The number which occur depends on the relative magnitudes of the various interactions. In all cases the energy of any surface state which exists is found to lie outwith the band of energy for the volume (i.e. non-localized) electronic states.
IntroductionLocalized Surface and Chemisorption StatesCalculation of Chemisorption EnergyConcluding Remarks
Beim eindimensionalen Gitter wird systematisch zwischen zwei Arten von Oberflädienzustän-den (O.Z.) unterschieden, je nachdem sie aus einem oder aus zwei Bändern entstammen. Die O.Z. erster Art werden aus einem neuen, einfachen Gesichtspunkt heraus, ohne jede Rechnung anschaulich verständlich gemacht. Bei den O.Z. zweiter Art wird nicht mehrdas Energie-Spektrum, wie das bisher üblich war, als Funktion der Gitterkonstanten a. sondern als Funktion eines Parameters a, wie er In der Theorie der Hillschen Differentialgleichungen verwendet wird, aufgetragen. Indem man auch negative Werte von a heranzieht, lassen sich die bisher anschau-lich nicht zu erfassenden O.Z. und der ganze Mechanismus des Austretens aus den Bändern durch ein einfaches Abzählvcrfahren mit Hilfe von Überlegungen, wie sie bei den O.Z. erster Art angestellt wurden, direkt verstehen. In diesem Schema zeigt sich dann auch in besonders deutlitber Weise, wie die Arbeit von Maue⁴ mit der von Shockley⁸ zusammenhängt, Unsere Betrachtungsweise gestattet noch einige wichtige Schlüsse über die Energiespektren der Elektrnnen in Kristallen. Es läßt sich die Erwartung aussprechen, daß für „richitig zwischen zwei Atomen abgeschnittene“ Kristalle im ersten verbotenen Gebiet, wegen des Fehlens von Kreuzungsstellen, keine O.Z. zweiter Art auftreten. Bei mehrdimensionalen Kristallen wird darauf hingewiesen, daß die von anderer Seite versuchte Übertragung des Ergebnisses bezüglich der O.Z. zweiter Art vom eindimensionalen auf mehrdimensionale Gitter ein ganz irreales Potential zugrundelegt. Damit wird die auch schon vom Experiment widerlegte Behauptung, daß der Diamant eine Oberflächenleitfähigkeit zeigen sollte, entkräftet. Nur im Falle fast freier Elektronen ist die eindimensionale Rechnung übertragbar. Es laßt sich aber dann meist für mehrere Brillouitisdte Zonen zeigen, daß für sie keine O.Z. und damit keine leitenden Oberflächenbänder auftreten können. Unter gewissen Annahmen ist es erlaubt, diesen Zonen dieinteressierenden Valenzbänder zuzuordnen. Um die Kreuzungsstellen von Bändern mehrdimensionaler Probleme weiter zu untersuchen, wird zum erstenmal die Zellenmethode zur Berechnung der O.Z. benützt. An dem durchgerechneten Gitter, das als zweidimensionales Analogonzum Diamanten angesprochen werden kann, treten die O.Z. an der Kreuzungsstelle nur aus dem oberen leeren Band aus, führen also zu keiner Oberflächenleitfähigkeit. Die Zuverlässigkeit der Zellenmethode läßt sich hierbei noch nicht völlig sicher beurteilen.
The existence or non-existence of surface states on the boundary of a monovalent metal is investigated fully for simple cubic models in 1, 2 and 3 dimensions, using the molecular modification of Bloch wave functions. Limiting values are found for the necessary changes required at the surface atoms, both for their Coulombic (electronegativity) terms α and their resonance integrals β. It is shown that surface states may be induced by the perturbation caused by the approach of a polar, or ionic, group; and that such surface states are more easily produced close together than separately. In certain non-cubic systems, e.g. graphite, surface states exist even without any necessary changes in α or β on the boundary. Finally, a series of recent theorems on the charge distribution in large molecules is applied to a study of the alternating charge distribution sometimes associated with surface states.
A formal treatment has been given of the surface states associated with a semi-infinite crystal. The treatment assumes that the surface states are produced by a potential which is a function only of the distance from the surface but is otherwise unspecified. A formal expansion is made in a complete set of Wannier orbitals, but the conclusions of the paper are based on the assumptions that (a) it is sufficient to use the orbitals of a single band, and (b) the tight-binding approximation is adequate for that band. Some theorems are given on the localization of Wannier orbitals, the localization of surface states, and the number of surface states for a given penetration of the surface potential.
It is shown under what conditions there exist solutions to the Hartree-Fock equation, which are localized in the neighbourhood of the surface of the crystal. The results obtained earlier as to the number and character of the localized solutions of the single electron Schrödinger equation of electrons in a limited crystal, can be applied to the localized solutions of the Hartree-Fock equation after accepting plausible assumptions. A linear combination, having analogical properties to the Wannier function, can be formed from the solution of the Hartree-Fock equation for an infinite crystal.
Zusammenfassung Es wird die Aufgabe der Auffindung des energetischen Spektrums Es und der Eigenfunktionen? des Elektronenzustandes in einer beschränkten ebenen dünnen Metallschicht behandelt. Zur Betrachtung dieser Probleme gelangte der Autor durch die Untersuchung des photoelektrischen Effektes an einer dünnen Metallschicht auf metallischer Unterlage oder auf einer solchen aus einem Dielektrikum.
The large p-type surface conductivity and field-induced surface conductivity of the cleaned germanium surface have been measured over the temperature range 77°-300°K. It is found that the surface conductivit and the field-induced surface conductivity are almost independent of temperature, varying by a factor of two over the temperature range investigated. The effect of oxygen, atomic hydrogen, and water vapor on the surface conductivity has also been observed. A qualitative two-dimensional band model is presented which correlates most of the experimental results. In this model, which is somewhat similar to the three-dimensional band model of graphite, there is a two-dimensional surface state band at the surface which overlaps in energy a two-dimensional valence band just beneath the surface. The states which form the surface state band are assumed to be perturbed out of the conduction band. The filling of these surface states with electrons out of the valence band gives rise to the observed p-type conductivity associated with the cleaned germanium surface. The states perturbed out of the conduction band are shown to be associated with the unfilled orbitals of the germanium surface atoms. A second system, where a similar two-dimensional matrix of unfilled orbitals is found, is in the dislocations associated with medium-angle grain boundaries. It is shown that the transport properties of these grain boundaries are similar in magnitude and temperature dependence to that observed for the clean germanium surface. The same model presented for the surface can be used to explain the transport properties of grain boundaries.
Zusammenfassung Mit Hilfe der gewöhnlichen Methode der Wellenmechanik (Randbedingungen?=0 fürx=± 8 und Stetigkeitsbedingungen von ? und ??/?x) sind alle möglichen Eigenwerte und Eigenfunktionen des in einem endlichen eindimen-sionalen Kristallmodelle bewegten Elektrons, sowohl wie im idealen Kristalle (das Gittermodell von Penney), als auch mit einer Lockerstelle, ermittelt worden, wobei zur Bestimmung willkürlicher Konstanten Differenzengleichungen verwendet wurden. Die erhaltenen Ergebnisse sind zur Erklärung einiger Erscheinungen aus dem Gebiete der lichtelektrischen Leitung benutzt worden.
The conditions under which bound states associated with atoms in the surface of a metal may exist are investigated, using the tight-binding approximation. These states arise as a result of modifications in the parameters of certain atoms. The modifications required to produce (a) bound states associated with all the atoms in the surface (surface states) and (b) bound states associated with particular small groups of atoms are found for the simple cubic lattice. It is also shown that most of the simpler crystal structures do not exhibit surface states without such modifications; in the graphite and diamond lattices, however, surface states exist solely by virtue of the lattice geometry.
DieGoodwinschen Rechnungen (Lsung derSchrdinger-Gleichung fr denbegrenzten Kristall mittels der Methode des gebundenen Elektrons) werden ausgebaut, indem erstens dieRaumgitter-Eigenschwingungen ausfhrlicher behandelt werden. Zweitens werden fr dieOberflchenzustnde im entarteten Fall dadurch bedeutend bersehbarere Ergebnisse gewonnen, da\ vereinfachende Annahmen ber gewisse Austausch- (bzw.Coulombsche) Integrale gemacht werden. Auf diese Weise wird der Zusammenhang mit einer Arbeit vonShockley hergestellt.
Germanium surface behavior has become of great interest recently, chiefly for its importance in the understanding of diode and transistor technology. In general, the surface may be treated as an assemblage of allowed electron states occurring in the normally forbidden energy range. A review of the measurements of the electrical properties suggests that there are two distinct types of state. The ``fast'' state has a hole or electron capture time not greater than a microsecond and is chiefly involved in the recombination process. The ``slow'' state has capture times from a millisecond to several minutes and determines the density and type of carrier at the surface. ``Fast'' states are believed to occur at the interface between the germanium and the oxide layer, and their density of about 1011 cm-2 is determined by the initial surface treatment. ``Slow'' states are associated with the structure of the oxide layer and the gaseous ambient, and have a density greater than 1013 cm-2. Since these states determine the conductivity type at the surface, they contribute to surface ``leakage'' in diodes and transistors and, because of their long equilibrium times, to low‐frequency noise. The adsorption of gases such as water vapor, not only controls the density and energy of the ``slow'' states but also leads to possible electrolytic conduction along the surface, in addition to the normal electron flow in the bulk semiconductor.
Phys. Zs. d. Sowjetunion1 (1932), 733; Zs
- I. Tamm
- E T Goodwin
- E. T. Goodwin