Diffusion of the electronic excitation in disordered systems

The kinetic equations describing the spreading ot the electronic excitation at zero temperature as a stochastic Brownian motion have been found. In the case of a structural disorder they imply the diffusion ot the excitation into the ranges of the maximum density of atoms. The relaxation time is also density-dependent having the minimum value in the ranges of maximum density of atoms and vice versa. In the case of a substitutional disorder the master equation in the “k”-space imply the spreading of excitation as a decaying wave packet. The relaxation lifetime in the first case of disorder and the lifetime of the coherence state in the second one are much smaller than the radiation lifetimes.     

Diffusion and hopping conductivity in disordered one-dimensional lattice systems

We investigate one-dimensional lattice systems with (symmetric) nearest neighbor transfer ratesW _{n, n+1} which are independently distributed according to a probability density(w). For two general classes of(w), we rigorously determine the asymptotic behavior of the relevant single site Green function ₀() near=0, and obtain exact results for the long time decay of the initial probability amplitude and for the low energy density of states. A scaling hypothesis, accurately confirmed by computer simulations, is used to relate the low frequency hopping conductivity() uniquely to ₀(–i), and we conjecture that the resulting asymptotic behavior for() is also exact. The critical exponents associated with the various asymptotic laws depend on(w) and show a crossover from universal to non-universal behavior. Comparison is made with the results of several approximate treatments.     

Diffusion in disordered media

Diffusion in disordered systems does not follow the classical laws which describe transport in ordered crystalline media, and this leads to many anomalous physical properties. Since the application of percolation theory, the main advances in the understanding of these processes have come from fractal theory. Scaling theories and numerical simulations are important tools to describe diffusion processes (random walks: the 'ant in the labyrinth') on percolation systems and fractals. Different types of disordered systems exhibiting anomalous diffusion are presented (the incipient infinite percolation cluster, diffusion-limited aggregation clusters, lattice animals, and random combs), and scaling theories as well as numerical simulations of greater sophistication are described. Also, diffusion in the presence of singular distributions of transition rates is discussed and related to anomalous diffusion on disordered structures.     

Diffusion in disordered media

Diffusion in disordered systems does not follow the classical laws which describe transport in ordered crystalline media, and this leads to many anomalous physical properties. Since the application of percolation theory, the main advances in the understanding of these processes have come from fractal theory. Scaling theories and numerical simulations are important tools to describe diffusion processes (random walks: the ‘ant in the labyrinth’) on percolation systems and fractals. Different types of disordered systems exhibiting anomalous diffusion are presented (the incipient infinite percolation cluster, diffusion-limited aggregation clusters, lattice animals, and random combs), and scaling theories as well as numerical simulations of greater sophistication are described. Also, diffusion in the presence of singular distributions of transition rates is discussed and related to anomalous diffusion on disordered structures.     

Lattice vibrations in ordered and disordered thiourea

The critical dynamics of the Syozi model for dilute ferromagnetism is considered by the use of master equations. The dynamics is soluble as it is assumed that the time scale of motion on the sublattice on which the impurities move is so much faster than on the other sublattice that fast relaxing variables may be adiabatically eliminated, leaving a new soluble master equation. It is found that the linear and non-linear relaxation of magnetization exponents ^(l) and ^(nl) increase on dilution to ^(l)/(1–) and ^(nl)/(1–) respectively ( is the specific heat exponent for the pure system, which itself changes on dilution to –/(1–)). Thus if the exponents for the pure system obey the scaling law of Rácz and Fisher ^(nl)= ^(l)– ( is the magnetization exponent which changes on dilution to /(1–)) then so do the exponents for the diluted system. Similarly the exponent for spin diffusion changes on dilution to /(1–).     

Lattice vibrations in ordered and disordered thiourea

Raman scattering and infrared reflection spectra of single crystals of thiourea were recorded at several temperatures between 2 K and 240 K in the spectral region (03500) cm^–1. The intention was to search for changes in frequency, linewidth, and intensity of optical phonon modes near the various phase transitions which are known to exist in this material. All observed phonon modes have been tabulated according to a group-theoretical enumeration, and they are attributed to internal motions of the molecular groups and external lattice vibrations. A survey of the temperature dependence of all observed lines is given.Some modes are in fact influenced by the phase transitions in a characteristic manner. The observed temperature dependences can be interpreted by a simple pseudospinphonon coupling model (following contribution II).     

Density of states of vibrations in a disordered chain

We study the local density of vibrational states of a disordered binary chain using a renormalization approach suitable for the alloy problem. We include isotopic and spring constant disorder. Short range order effects in the alloy are taken into account in the pair approximation. We present some numerical results for the case of spring constant disorder and discuss their relevance for the interpretation of neutron scattering experiments in some metallic amorphous systems.     

Discontinuities in disordered systems

The entropyS _T (j) of a two-dimensional Ising spin glass with an independent distribution of the random couplingp(J)=x·δ(J+1)+(1-x)δ(J-j) is discontinuous for temperatureT=0 and rationalj>0 and continuous elsewhere. The integrated density of frequenciesk _M (ω ²) of an one-dimensional chain of coupled oscillators with an independent distribution of the random massesp(m)=x·δ(m-1)+(1-x)δ(m-M) has the same behaviour, whereω ² corresponds toj andM to 1/T. The discontinuity points for infiniteM are, for sufficiently large but finiteM, special, frequencies, wherek _M (ω ²) has a Lifshitz singularity.     

Diffusion and electromigration on disordered surfaces

We study a one-dimensional disordered solid-on-solid model in which neighboring columns are shifted by quenched random phases. The static height-difference correlation function displays a minimum at a nonzero temperature. The model is equipped with volume-conserving surface diffusion dynamics, including a possible bias due to an electromigration force. In the case of Arrhenius jump rates a continuum equation for the evolution of macroscopic profiles is derived and confirmed by direct simulation. Dynamic surface fluctuations are investigated using simulations and phenomenological Langevin equations. In these equations the quenched disorder appears in the form of time-independent random forces. The disorder does not qualitatively change the roughening dynamics of near-equilibrium surfaces, but in the case of biased surface diffusion with Metropolis rates it induces a new roughening mechanism, which leads to an increase of the surface width as . Received 7 February 2000     

Diffusion in disordered lattices and related heisenberg ferromagnets

We study the diffusion of classical hard-core particles in disordered lattices within the formalism of a quantum spin representation. This analogy enables an exact treatment of noninstantaneous correlation functions at finite particle densities in terms of single spin excitations in disordered ferromagnetic backgrounds. Applications to diluted chains and percolation clusters are discussed. It is found that density fluctuations in the former exhibit a stretched exponential decay while an anomalous power law asymptotic decay is conjectured for the latter.     

Off-perturbative states in disordered systems

The systematic approach for the off-perturbative calculations in disordered systems is developed. The proposed scheme is applied for the random temperature and the random field ferromagnetic Ising models. It is shown that away from the critical point, in the paramagnetic phase of the random temperature model, and in the ferromagnetic phase of the random field one, the free energy contains non-analytic contributions which have the form of essential singularities. It is demonstrated that these contributions appear due to localized in space instanton-like excitations.     

Electronic localization in disordered systems

A brief review is given of the current understanding of the electronic structure, transport properties and the nature of the electronic states in disordered systems. A simple explanation for the observed exponential behaviour in the density of states (Urbach tails) based on short-range Gaussian fluctuations is presented. The theory of Anderson localization in a disordered system is reviewed. Basic concepts, and the physics underlying the effects of weak localization, are discussed. The scaling as well as the self-consistent theory of localization are briefly reviewed. It is then argued that the problem of localization in a random potential within the so-called ladder approximation is formally equivalent to the problem of finding a bound state in a shallow potential well. Therefore all states are exponentially localized in d=1 and d=2. The fractal nature of the states is also discussed. Scaling properties in highly anisotropic systems are also discussed. A brief presentation of the recently observed metal-to-insulator transition in dequals;2 is given and, finally, a few remarks about interaction effects in disordered systems are presented.     

Stark resonances in disordered systems

We study further the metastable behavior of Metropolis dynamics for the two-dimensional nearest neighbor ferromagnetic Ising model, with positive and small external field, in the limit as the temperature vanishes (see [NS]). We focus on the typical features of the escape (nucleation) from the (metastable) configuration with all spins –1, to the (stable) configuration with all spins +1. Using the reversibility of the process as the main tool, we prove (for the discrete time version of the model) that the first step of a typical escaping path is the time reverse of a typical time evolution of a shrinking subcritical rectangular droplet, which is one slice smaller than a critical droplet. This subcritical droplet then evolves in a time of order 1 to a critical droplet, which finally grows with features described in [NS].Work partially supported by the Brazilian CNPq and by the American NSF, under grant DMS91-00725     

Wave functions in disordered systems

Particular solutions of the stationary Schrödinger equation for ad-dimensional disordered tight binding model are found. The particular solution is defined by boundary conditions on one face of the system. The determination of the rate of growth of the mean square wave function leads to an exactly soluble eigenvalue problem ind – 1 dimensions. Ford 2 there are three types of particular wave functions in which the mean square amplitude (a) grows exponentially (b) decays exponentially (c) does not grow or decay but oscillates.Supported in part by the National Science Foundation under grant No. DMR 78-10276.     

Systematic approximations in disordered systems

We discuss two systematic approximation schemes for disordered systems (random lattices) by extending the single site CPA. In the first scheme all self energy diagrams up to those includingν scattering centers are summed up. Thisν-center approximation has been discussed before and is here derived using a straightforward projection operator formalism. The second scheme is a “true” cluster-CPA in which the scattering from an arbitrarily large cluster of neighbouring atoms (ν-cluster) embedded in a random crystal is treated just in the same way as in the single site CPA. In contrast to other cluster formulations our theory preserves translational invariance and determines the self energy in a natural way. The two schemes are compared with each other concerning their practical applicability.     

Instanton in disordered Peierls systems

We study disordered Peierls systems described by the fluctuating gap model. We show that the typical electron states with energies lying deep inside the pseudogap are localized near large disorder fluctuations (instantons), which have the form of a soliton-antisoliton pair. Using the “saddle-point” method we obtain the average density of states and the average optical absorption coefficient at small energy.