Numerical Evidence of an Electronic Localization Transition in a Disordered Layer of Metal Atoms

Recent observations of transitions in the non-linear polarizability of submonolayers of metal atoms on silicon, and between insulating and metallic behavior for electrons in silicon inversion layers, indicate the need for further study of the electronic states in such systems. In this work, the projected density of states and normalizations for individual states are used to characterize the localization of independent electrons in a model for a layer of metal atoms randomly deposited at low density on a crystalline surface of silicon. We find a localization transition, whose sharpness in energy is limited only by the sample size of over 80,000 sites for a model of the Si(001)2x1 surface. This is consistent with previous analytic and numerical calculations, by similar methods, of the localization properties of other models for electronic states in weakly disordered layers, and with the observed transition in the non-linear polarizability of submonolayers of metal atoms on silicon. 1. Electronic states at Disordered Interfaces The nature of electronic states at disordered interfaces is a long standing problem. In the 1970s it was thought that the independent electronic spectrum of a disordered layer had mobility edges which separated energies where the states were exponentially localized from energies where the states were plane-wave-like. Many theoretical and numerical attempts were made to determine the location of these edges for the Anderson [1] and related models, but the mobility edges were elusive. During the 1980's, many opinions changed to the view that all states were exponentially localized and that the localization length varied smoothly with energy, becoming very large at energies where the states were previously thought to have been delocalized. Recent observations by Kravchenko and others [2] of an apparent metal-insulator transition in silicon inversion layers has raised doubts about this view. A variety of theoretical and numerical approaches have been applied to the behavior of electrons at disordered interfaces. The scaling theory of Abrahams and others [3] persuaded many that the electronic states in such systems must all be exponentially localized, and this has been supported by field theoretic approaches[4] (for a review see Lee and Ramakrishna[5]) as well as the extensive numerical work of McKinnon, Kramer, and collaborators, see Ref. [6] for a review. On the other hand, Haydock[7] presented analytical arguments, and Godin and Haydock (GH) [8] produced numerical evidence that there is a sharp transition between exponential and power-law localized states in disordered two-dimensional systems. Starting from high magnetic fields where individual Landau levels are known to carry currents, Azbel [9] argued that there is a metal-insulator transition at zero field. In addition to the work on independent electrons, there have been efforts to take electron-electron interactions into account, and it has been suggested [10] that interactions can delocalize electrons in circumstances where independent electrons would be localized. Most of the experimental information about electron localization at interfaces has come from silicon inversion layers, devices in which electrons are trapped by an electric field at the (001) interface between p-type silicon and a layer of silicon oxide. Uren and others [11] found two distinct kinds of localization, and Davies and Pepper [12] extended this work to show evidence for a transition between exponential and power-law localization. Kravchenko and others [2] saw evidence similar to that of Ref. [11] of two kinds of localization in low mobility devices, and found a much more dramatic transition in high mobility devices. While electron-electron interactions clearly play an important role in the Kravchenko devices, as long as the electronic excitations close to the Fermi level can be expanded in localized excitations, no matter how complicated, the electronic Hamiltonian expanded in these excitations is of the same form as the one used below. In a different experimental approach Arakat, Kevan, and Richmond (AKR) [13] deposited alkali metal atoms on a 2x1 reconstructed Si(001) surface, and measured the second harmonic produced by an optical, in-plane electric field. The advantages of this approach are that the distribution of metal atoms, the disorder, can be observed directly, and that an electric field is applied directly to the electrons rather than a potential difference being applied between contacts to a tenuous electron gas. The disadvantage of this approach is that the measurements are at optical frequencies rather than DC. The second harmonic showed a sharp threshold at about 1/6th of a monolayer for each of three different alkali metals. Although the AKR suggested that this threshold is due to a metal-insulator transition induced by electron-electron interactions, we were intrigued to see if a localization transition could be excluded. In the first of the following four Secs., we present an independent particle model for the electrons of metal atoms deposited on the 2x1 reconstructed surface of (001) silicon. Section 3 contains a description of the calculations for different coverages, of normalizations for states at the Fermi level, and of the projected densities of states at various atoms. In the subsequent Sec. the results of these calculation are compared with those obtained by GH using the Anderson Model to simulate an inversion layers complete with contacts. In the last Sec we discuss the relation between the results of these calculations and the metallization of overlayers. 2. A model for Metal Atoms on Si (001) 2x1 Our model for the metallic electrons is similar to that of Debney[14] with a basis of a single s-orbital centered on each metal atom. We assume that at low coverages only the most binding site in each 2x1 surface cell is occupied. We take the energies of these orbitals to be independent of whether nearby surface cells are occupied, and allow the electrons to hop from one metal atom to another with matrix elements decreasing exponentially with the distance between cells. In our calculations, metal atoms occupy each surface cell randomly, independent of the occupation of any other cells. The bulk terminated Si(001) surface has two dangling silicon bonds for each surface atom. For each surface silicon atom, one of the dangling bonds dimerizes with that of a neighboring atom to produce a 2x1 reconstruction which has two dangling bonds per surface cell . The remaining dangling bonds further hybridize and transfer electrons to produce one band of doubly occupied surface states, and one band of unoccupied surface states separated in energy by about 1 eV. See Ref. [13] for further discussion of the Si(001) surface. The valence s-orbital of the metal adatom hybridizes most strongly with bands of surface states to produce a singly occupied orbital whose energy must consequently lie in the gap between the occupied and unoccupied surface bands of the silicon. The energy of the orbital is determined by the chemical bond formed between the metal and silicon, and this in turn fixes the exponent with which the orbital decays with distance outside the cell in which it is centered. It is this hybridized orbital of the isolated metal adatom which forms our basis for the electronic states of the metal overlayer, and because the spatial decay of the orbital outside its central cell is determined by the chemistry of the metal-silicon bond, we expect the different alkalis to behave similarly. In the electronic Hamiltonian for this model, the energies of each orbital are the same for the reasons described in the previous paragraph. Outside the central cell, each orbital decays exponentially, and we take this to be independent of direction so that the matrix element of the Hamiltonian between metal orbitals in cells i and j takes the Hückel form: