Weak Disorder Limit Research Articles

Non-equilibrium noise in low-dimensional systems

The transmission coefficient of a thin disordered wire varies dramatically with the energy of electrons attempting to diffuse along the wire. Hence on a microscopic scale the relationship between current and voltage is a highly nonlinear one. This results in noise, in the form of harmonic and intermodulation distortion of the input signal, which has a l/f2 behaviour up to frequencies where hf is comparable with kBT. Analytic expressions are derived for the noise in the limit of weak disorder.

Power-law localization in two-dimensional systems Theory and experiment

Abstract It is shown that all the non-metallic transport phenomena in the weak-disorder limit for small length scales result from a quasi-extended wavefunction φE=ψ(1) ext+ ψ(2) ext/r. For large length scales, φE changes into pure power-law states, φE = C/r S. For S > 1, diffusive transport is predicted with σdif ∼T S/(S+1). For stronger disorder, S increases and for S < 1, hopping conduction is predicted. In this region the negative magnetoresistance should disappear. Experimental evidence is given for all the above predictions. We also construct a two-parameter scaling function, appropriate to power-law localization, which is in agreement with experiment. We show that all the available data support the existence of power-law localized states which are separated by a mobility edge from exponential localized states.

Diffusion on a disordered Cayley tree

The model proposed recently by Brezini (1983) to calculate the average probability and the average size of the localisation domain for an electron being localised at a given site in a disordered Cayley tree is extended to the case of a uniform distribution for site energies. Thus, numerical results are presented in the limit of weak disorder. Particular attention is paid to the states near the mobility edge and critical exponents are deduced for the localisation length.

Quantum diffusion and localization in disordered systems

Abstract We describe quantum diffusion of the electrons in a disordered system by requiring that ψ(r, t)|2 obeys a diffusion equation, where ψ(r, t) is the time-dependent wavefunction. It is found that, regardless of the weakness of the disorder, this requirement leads to electron eigenstates which consist of a power-law component for dimepsion d1 and a logarithmic correction for d=1, in addition to an extended function. For d ≤ 2, this is correct only below a certain length scale. As a result, even for kFl≫l, the conductivity σ is reduced from the Boltzmann conductivity σB in agreement with diagrammatic calculations. By expanding the eigenstate in terms of 1/rn, it is shown that the 1/rd-1 term is responsible for the reduction of σ in the weak-disorder limit. It is demonstrated that in three dimensions one can extrapolate the formula for the conductivity down to the Anderson transition to obtain σ = g2σB[1 - (C/g2k2 Fl2) (1-l/L)] where g is the reduction in the density of states due to disorder and C is ...

Scaling behavior of one-dimensional weakly disordered models

The density of states and various characteristic lengths of one-dimensional tight-binding models and disordered harmonic chains are calculated in the limit of weak disorder at the band edge of the ordered system. The density of states and a localization length of the one-dimensional Anderson model were already calculated by Derrida and Gardner; we recover their results. For the tight-binding models with off-diagonal disorder our results are in agreement with numerical calculations of Krey.

A microscopic approach to the statistical fracture analysis of disordered brittle solids

The static fracture properties of disordered brittle solids, having ordered crystalline structure and a random distribution of flaws (broken bonds), are studied here from a microscopic point of view. In the weak disorder limit, starting from McClintock's description, we have obtained the empirical Weibull formula for the average fracture strength σ c , giving all its fitting parameters. The result of a molecular dynamic simulation of fracture in a square Lennard-Jones system, containing 400 atoms and randomly broken bonds, are found to compare very well with the theory.

Logarithmic frequency dependence of the dynamical conductivity of one-dimensional disordered systems at low frequencies: a computer simulation for a very long chain

The low-frequency behaviour of the dynamical conductivity sigma ( omega ) for electrons on very long (106 sites) one-dimensional disordered chains is investigated by computer simulation. Berezinskii's result (1974) which is asymptotically exact in the limit of weak disorder and shows the behaviour sigma ( omega )/ sigma 0 to 32( omega tau )2(ln2(4 omega tau )- pi 2/4+(2C-3)ln(4 omega tau )+ ...) at low frequencies ( sigma 0, tau and C denote the Drude value of the conductivity, the relaxation time of the electrons and Euler's constant, respectively), is established for the first time by a direct numerical method. A discussion is given of a simple derivation of the coefficient 32. It is also found that this low-frequency behaviour remains almost when disorder is introduced, unchanged even when the disorder is strong.

Power law localisation in disordered systems

A power law term in the eigenfunctions Psi of electrons in weakly disordered systems (for which kFl>>1, where l is the elastic mean free path) is derived from first-order perturbation theory. It is shown that psi =A psi ext(1)+(B/Rd-1) psi ext(2) for d>1 where d is the dimensionality, A is a constant, B is a function of the angle Theta between K and r, and psi ext(1), psi ext(2) are extended wavefunctions with different phases. This result confirms the recent conjecture of M. Kaveh and N.F. Mott (1983 Phil. Mag. B47 L9) which was based on a semiclassical diffusion model. The authors conclude that for any number of dimensions the corrections to the Boltzmann formula for the metallic conductivity in the weak disorder limit, which for d=3 can in some cases lead to a value of the conductivity tending continuously to zero, are due to this power law component of the wavefunction.

Eigenfunction in disordered systems near the mobility edge

A model is proposed to calculate the average probability and the average size of the localisation domain for an electron being localised at a given site in a Cayley tree lattice. The numerical results are presented in the limit of weak disorder in the case of the Cauchy distribution for site energies. Attention is paid to the states near the mobility edge in the localised regime. In particular, features exhibited in the linear-chain case are observed for the first time for higher dimensions.

Frequency dependence of the conductivity in presence of an electric field in one dimension: Weak-disorder limit

Frequency dependence of the conductivity in presence of an electric field in one dimension: Weak-disorder limit

Interface energy in random systems

We study the interface energy $\ensuremath{\sigma}$ as a function of disorder in two-dimensional Ising-type systems at $T=0$ for the percolation and frustration models. Our approach consists in calculating this energy for long strips of varying widths by a random sampling method, then extrapolating the results. In the weak-disorder limit, very wide strips may be studied and accurate values are obtained for the first-order correction to the interface energy in both models. Only moderate widths (up to $N=9\ensuremath{-}10$) can be studied in the general situation and we use finite-size scaling to analyze the data in the region of the threshold, where $\ensuremath{\sigma}$ vanishes with a critical exponent $v$. For percolation, we obtain $\frac{v}{\ensuremath{\nu}}=0.98\ifmmode\pm\else\textpm\fi{}0.05$, where $\ensuremath{\nu}$ is the correlation-length exponent, in agreement with Deutscher and Rappaport's proposal that $v=\ensuremath{\nu}$ exactly. The analysis of the results is more difficult in the case of frustration, because size effects are important and the scaling region is not reached for $N=9$. Our data show that Monte Carlo results for $\ensuremath{\sigma}$ are unreliable and that much care is necessary to reach firm conclusions on the frustration threshold ${x}_{c}$ or the exponents $\ensuremath{\nu}$ and $v$.

Anderson localization in different one-dimensional systems with off-diagonal disorder and spin-dependence

The localization properties of certain spin-dependent, one-dimensional electronic systems with only off-diagonal disorder are studied. In higher dimensions (d=2,3) the models considered would correspond to different universality classes, whereas ford=1 no qualitative difference is found: ForE=0, all eigenstates are exponentially localized, whereas forE→0 the localization length diverges logarithmically, such that exactly atE=0 the geometric average of the transmission coefficient would decay with increasing chain lengthL as exp (-const. ·L 1/2), instead of the usual, exponential decay. ForE=0, in the interior of the band, the localization lengthr 0 diverges ∼W 2 −2 in the limit of weak disorder (W 2→0), whereas just at the band edge one has roughlyr 0∼W 2 −2/3. A universal recursion relation, depending only on the energy and on certain randomly distributed determinants, determines the localization length and the density of states for all systems considered.

Phase randomness in the one-dimensional Anderson model

The probability distribution $P(\ensuremath{\nu})$ of the phase $\ensuremath{\nu}$ which determines the scaling properties of the dimensionless resistance $\frac{R}{T}$ is analyzed for the one-dimensional Anderson model with diagonal disorder. For weak disorder ($\frac{W}{V}\ensuremath{\ll}1$) and $E\ensuremath{\ne}0$ we find that $P(\ensuremath{\nu})$ is initially sharply peaked for short chains but becomes uniform over $2\ensuremath{\pi}$ as $N$ increases. Moreover, the length at which $P(\ensuremath{\nu})$ becomes uniform is found to be independent of $\frac{W}{V}$ (as long as $\frac{W}{V}\ensuremath{\ll}1$); instead it appears to depend only on the distance in energy from band center, i.e., on the extent to which the electron's wavelength is incommensurate with the lattice. Thus, unlike the disordered Kronig-Penny model, there is a phase randomness length, shorter than the localization length but longer than the lattice spacing. $P(\ensuremath{\nu})$ is also studied in the limit of large disorder and analytic expressions for the mean and variance of the inverse localization length are derived for weak and strong disorder. Our results suggest that uniform phase randomness is indeed a property of most disordered 10 systems in the limit of weak disorder.

Weak absence of diffusion in two dimensions in the weak-disorder limit

Working within the single-electron picture of two-dimensional disordered systems in the crucial weak-disorder regime, the authors calculate the average two-particle Green function by summing the maximally crossed diagrams occurring in the approximation of Neal and Langer (1966) as developed by Abrahams and Ramakrishnan (1980) to show that Anderson's criterion for location is not satisfied, while there is only a weak absence of diffusion in the sense of Ishii (1973). Summation of ladder diagrams appearing in the Langer-Neal scheme gives an identical result in the small-disorder regime. They reach the same conclusion upon using the CPA to perform the required averaging. Recent mathematical results associate such a weak absence of diffusion with a singularly continuous spectrum.

Scaling studies of the resistance of the one-dimensional Anderson model with general disorder

The resistance $\ensuremath{\rho}$ of a one-dimensional Anderson model with both diagonal and off-diagonal disorder is studied by analytic and numerical techniques. A recursive method is developed and used to derive an exact scaling law for the average resistance at $E=0$ for arbitrary disorder, and for $E\ensuremath{\ne}0$ in the limit of weak disorder. The average resistance grows exponentially with $L$, the length of the sample, in all cases. The typical resistance $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\rho}}=\mathrm{exp}[〈\mathrm{ln}(1+\ensuremath{\rho})〉]\ensuremath{-}1$ is also found to grow exponentially with $L$ in all cases, except for purely off-diagonal disorder at $E=0$, where $〈\mathrm{ln}(1+\ensuremath{\rho})〉\ensuremath{\propto}\sqrt{L}$. An explanation is given for the existence of this special case and it is shown that all our results are consistent with a lognormal probability distribution of the resistance for $\ensuremath{\rho}&gt;&gt;1$. Quantitative estimates are made of the reliability of numerically performed averages which show that a numerical average will converge only very slowly to the analytic result. This provides a qualitative explanation of the slower than linear growth of $\mathrm{ln}〈\ensuremath{\rho}〉$ with $L$ found in several numerical calculations; its consequences for experiment are also explored.

Anomaly in the band centre of the one-dimensional Anderson model

We calculate the density of states and various characteristic lengths of the one-dimensional Anderson model in the limit of weak disorder. All these quantities show anomalous fluctuations near the band centre. This has already been observed for the density of states in a different model by Gorkov and Dorokhov, and is in close agreement with a Monte-Carlo calculation for the localization length by Czycholl, Kramer and Mac-Kinnon.

Two-particle spectral function and ac conductivity of an amorphous system far below the mobility edge: A problem of interacting instantons

We use a variational approach to calculate the two-particle spectral function ${S}_{2}({x}_{1}, {x}_{1}, {x}_{2}, {x}_{2}, {E}_{1}{E}_{2})$ of a Gaussian-disordered electron system in the limit of deeply localized states and small energy difference $\ensuremath{\omega}={E}_{1}\ensuremath{-}{E}_{2}$. The solution of the variational equations yields a two-center potential, each center in lowest order being determined by the square of an instanton function. The two instantons interact via the constraint that the Hamiltonian has to have lowest eigenvalues ${E}_{1}$, ${E}_{2}$. As the two centers approach the minimum distance allowed for given $\ensuremath{\omega}$ by the tunnel effect, we are confronted with a problem of confluent saddle points, which forces us to introduce an additional constraint. Our method is rigorous in the limit of weak disorder $|{E}_{1}+{E}_{2}|\ensuremath{\rightarrow}\ensuremath{\infty}$, $\frac{\ensuremath{\omega}}{|{E}_{1}+{E}_{2}|}=\mathrm{const}\ensuremath{\ll}1$. We also apply it to the hydrodynamic limit $\frac{\ensuremath{\omega}}{|{E}_{1}+{E}_{2}|}\ensuremath{\rightarrow}0, |{E}_{1}+{E}_{2}|$ large. It is found that these limits cannot be interchanged. In both limits we evaluated the ac conductivity. The result $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{2}{(\mathrm{ln}\ensuremath{\omega})}^{d+1}$ is found in the hydrodynamic limit.

Localization length of one-dimensional random systems

Considering the behavior at infinity of the amplitude of wave functions in one-dimensional disordered systems, as effectively described by a wave transmission through an infinitely large cluster, we derive an exact expression for the decay length ${L}_{d}(E)$ of eigenstates around energy $E$. We obtain from it simple formulas for ${L}_{d}$ at $E=0$ and in the limit of weak and strong disorder, and verify them in the case of Lorentzian type of disorder where exact results are independently available. The empirical result of Bush concerning the behavior of the maximum ${L}_{d}$ in random binary alloys is found to be approximately correct, but only in the limit of weak disorder.

On the size of the localisation domain

We introduce and discuss a delocalisation criterion for electrons in disordered systems, based on the size of the localisation domain. Results are presented for weak disorder limit and a generalisation to arbitrary disorders is indicated.