Universal features of localized eigenstates in disordered systems

Localization–delocalization transitions occur in problems ranging from semiconductor-devicephysics to propagation of disease in plants and viruses on the internet. Here, we report calculations of localized electronic and vibrational eigenstates for remarkably different, mostly realistic, disordered systems and point out similar characteristics in the cases studied. We show in each case that the eigenstates may be decomposed into exponentially localized islands which may appear in many different eigenstates. In all cases, the decay length of the islands increases only modestly near the localization–delocalization transition; the eigenstates become extended primarily by proliferation (growth in number) of islands near the transition. Recently, microphotoluminescence experiments (Guillet et al 2003 Phys. Rev. B 68 045319) have imaged exciton states in disordered quantum wires, and these bear a strong qualitative resemblance to the island structure of eigenstates that we have studied theoretically. In crystalline materials, electronic (and vibrational) eigenstates are extended (or delocalized) throughout the material. Since the 1950s [2], it has been recognized that disorder can cause some electron states to decay exponentially in space; such spatially confined states are called ‘localized’. The earliest and simplest model of disorder for electrons is the celebrated ‘Anderson model’, in which the disorder is usually introduced in the electronic Hamiltonian by choosing the diagonal matrix elements (on-site energy terms) of the Hamiltonian from a uniform distribution. While important insights have accrued from this work, the relevance of these calculations to physical systems has not been certain, as Mott emphasized in his Nobel Lecture [3]. For this reason, our work employs disorder extracted from models which accurately reproduce experiments sensitive to the disorder. For example, the electronic case is best described by constructing the Hamiltonian matrix from a realistic topologically disordered model of a material. Structural disorder also leads to localization of vibrational normal modes [4], which we similarly explore with realistic models of two different amorphous materials. To enable comparison with conventional calculations, we analysed a simple three0953-8984/05/300321+07$30.00 © 2005 IOP Publishing Ltd Printed in the UK L321 L322 Letter to the Editor dimensional Anderson model [2]. Our work is also relevant to macroscopic vibrations, and the stability of structures [5]. For a particular disordered system, the character of vibrational or electronic states is determined by the corresponding eigenvalues (eigenenergy or eigenfrequency of the corresponding Hamiltonian or dynamical matrix, respectively), with a critical point, the ‘mobility edge’, separating extended from localized states. This localization–delocalization (LD) second-order phase transition has been the subject of intense research on toy models, and much is now known about the statistical properties of the states at, and around, the critical eigenvalue [6]. Localized eigenstates are characterized by an exponential spatial decay of the magnitude of the envelope of the wavefunction. The actual charge density or vibrational displacement at a particular point is understood to be a product of this exponentially decaying envelope with a rapidly spatially varying ‘filling’ function that ensures that the eigenstates remain orthogonal. The behaviour of the envelope function has been previously studied: it is now known that it changes from being exponentially decaying for localized eigenstates to being multifractal at the LD transition, and is spatially extended for delocalized states according to analytical considerations [6], accurate numerical analysis for simple lattice models [7] and numerical calculations on small realistic systems [8]. It has been established that the filling function is highly inhomogeneous and consists of ‘lumps’ of probability density, and Dong and Drabold introduced a ‘resonant cluster proliferation model’ to characterize the evolution of the states from localized to extended [9]. However, not much is known about the nature of such lumps, and possible spatial correlations between them, for different eigenstates. Our aim is to reveal such correlations and shed light on the origin of the localized states. To achieve this aim, we use a numerical approach and investigate the spatial structure of localized states of different type (electronic and vibrational excitations) in a diverse range of models, starting from lattice models exemplified by the Anderson Hamiltonian and concluding with realistic models of amorphous Si and vitreous silica with long-range Coulomb interactions between atoms. Technically, we solve the eigenproblem for large random matrices (Hamiltonian and dynamical matrices for electronic and vibrational problems, respectively) using either direct diagonalization (for moderate sizes) or a Lanczos method for larger systems. For structurally ordered systems (e.g. crystals), all the states (electronic, vibrational, spin) are spatially extended and their energies form bands, separated by gaps. For systems with disorder, ‘band tails’ appear at the edges of the bands, with possibly also states deep in a gap. First, we confirm that the localized (electronic) band-tail states consist of clumps or ‘islands’ (of charge density). This can be seen from figure 1, where a band-tail part of the calculated electronic density of states for a realistic model of amorphous (a-)Si is shown, together with representative eigenstates for selected eigenvalues in the tail. The number of such islands in a localized eigenstate increases on approaching the LD transition. In order to demonstrate the strong spatial correlations between islands belonging to different eigenstates, we show, in figure 2(a), the charge density of three adjacent-energy localized electron eigenstates for a-Si, with each of the states depicted by a different colour (red, green and blue). Any departure of colour in the figure from these fundamental colours implies spatial overlap of states: the inset shows the secondary colours resulting from overlap of two islands, and regions of white indicate overlap of three islands. It is clear that the island structures in such localized eigenstates overlap spatially, very strongly so in certain regions, meaning that the same islands can participate in different eigenstates. This fact leads us to conjecture on the existence of a bare-state (island) basis from which all the localized states can be constructed. Indeed, we were able to demonstrate numerically that the localized states in all the models studied can be decomposed into overlapping ‘bare’ Letter to the Editor L323 –18 –17 –16 –15 –14 E / eV 0 0.0002 0.0004 0.0006 0.0008