Session V38: Disordered Electronic Systems

Power-law temperature dependence of the conductivity of disorder-tuned magnetic thin films

Using a specialized high vacuum deposition/characterization chamber, we study the \textit{in situ} temperature-dependent conductivity \textit{$\sigma $}($T$,$R_{0})$~of thin magnetic films (Fe, Gd and Cr) prepared at different stages of disorder where disorder is characterized by the sheet resistance $R_{0}$ measured at $T$~=~5~K. Our observation of power-law dependences of the form, $\sigma $(T,R$_{0})$~=~A~+~BT$^{P}$ , has also been noted in numerous experiments on other materials including indium oxide, 2D Si inversion layers and amorphous NiSi alloys. For our samples, the fitting parameters $A$, $B$ and $P$ vary systematically with increasing disorder. We discuss two regimes of behavior: In the first weak disorder regime, the power-law dependence represents a quantum correction to the classical Boltzmann conductivity due to spin-wave scattering. This contribution can appear either alone (Cr) or simultaneously (Gd) with the well-known logarithmic corrections due to weak localization. In the second regime with greater disorder, we utilize $T$-dependent reduced activation energy plots of \textit{$\sigma $}($T)$ for sets of films spanning a large range of disorder strengths to show that there are narrow regions of disorder strength where pure power-law behavior (A~=~0) dominates.   

The Kubo-Greenwood expression and 2d MIT transport

The 2d MIT in GaAs heterostructures (p- and n-type)features a mobility that drops continuously as the reduced density x= n/n$_{c}$-1 is decreased. The Kubo-Greenwood result [1] predicts $\mu$ = (e$\epsilon$$_{h}$/hn$_{c}$)$\alpha$$^{2}$(x) where $\alpha$ is a normalized DOS. $\alpha$(x)is obtained from the data [p-type, Gao et al. [2]; n-type Lilly et al. [3]]. Interact -ion corrections yield a Fermi energy E$_{F}$=E$_{c}$[x-A$_{HF}$ x$^{1/2}$/$\gamma$+A$_{CT,ic}$x$^{1/4}$/$\gamma$$^{3/2}$] where E$_{c}$ = n$_{c}$/a$_{2}$ with a$_{2}$ the noninteracting DOS and $\gamma$ = (2$\pi$n$_{c}$)$^{1/2}$a*, A$_{HF}$ is a Hartree -Fock term, while A$_{CT,ic}$ is a repulsive interaction term between the charged traps (CT) and the itinerant carriers (ic) that are confined in conducting filaments. N(E$_{F}$,x)is determined using N(E)dE=2kdk/2$\pi$ where k$_{F}$=(2$\pi$n$_{c}$ x)$^{1/2}$. The data determines the ``self-energy'' terms A$_{HF}$ and A$_{CT,ic}$. The effective mass m*(x) is given by m*(x)/m$_{b}$* = 1/$\alpha$(x) and diverges as x$^{-3/4}$ as x goes to zero at T=0. $\alpha$(x) goes to 1 for large x. The interaction approach will be compared with the percolation approach. [1] Mott and Davis, Elect. Prop. in Noncryst. Mat. [Clarendon Press 1971]; [2) Gao et al. PRL93,256402 (2004);[3] Lilly et al.PRL90,056806 (2003)   

Anderson localization for weak disorder

More than 25 years ago the correctness of Anderson's original argument for localization of states of noninteracting particles in a strongly disordered lattice potential was proved, but arguments for diffusive states in weak disorder are less compelling. A barrier to treating the disorder as a perturbation is the extreme sensitivity of extended states to boundary conditions in the absence of disorder. We find that a promising approach, which avoids boundary conditions, is to consider an infinite, homogeneously but weakly disordered, lattice. The Hamiltonian on the lattice is converted from configuration space to tridiagonal form by using the Lanczos transformation, and the disorder is treated perturbatively. We argue that the effect of disordering a two dimensional lattice may, in the tridiagonal representation, fall off like the inverse square root of the distance from the origin, and this can lead to an initial power law localization. In three dimensions the effect of disorder falls off like the inverse distance from the origin of the tridiagonal representation, and that this should lead to diffusive behavior except near the band edges.   

Distribution of conductance for Anderson insulators: A theory with a single parameter

We obtain an analytic expression for the full distribution of conductance for a strongly disordered two and three-dimensional conductor within a perturbative approach based on the transfer-matrix formulation. Our results confirm the numerical evidence that the log-normal limit of the distribution is not reached even in the deeply insulating regime. We show that the variance of the logarithm of the conductance scales as a fractional power of the mean, while the skewness changes sign as one approaches the Anderson metal-insulator transition from the deeply insulating limit, all described as a function of a single parameter. The approach suggests a possible single parameter description of the Anderson transition that takes into account the full nontrivial distribution of conductance.   

One Parameter Scaling Theory for Stationary States of Disordered Nonlinear Systems

We show that the normalized average participation number of the stationary solutions of disordered nonlinear lattices obeys a one-parameter scaling law. Our approach opens a new way to investigate the interplay of Anderson localization and nonlinearity based on the powerful ideas of scaling theory.   

Critical exponents of the three-dimensional Anderson transition from multifractal analysis

We use high-precision, large system-size wave function data to analyse the scaling properties of the multifractal spectra around the disorder-induced three-dimensional Anderson transtion in order to extract the critical exponent $\nu$ of the localisation length. We study the scaling law around the critical point of the generalized inverse participation ratios $P_q=\langle |\Psi_i|^2\rangle$ and the singularity exponent $\alpha_0$, defined as the position of the maximum of the multifractal spectra, as functions of the degree of disorder $W$, the system size $L$ and the box-size $\ell$ used to coarse-grained the wave function amplitudes. The values of $\alpha_0$ are calculated using a new method entirely based on the statistics of the wave function intensities [Phys.~Rev.~Lett.~102, 106406 (2009)]. Using finite size scaling analysis we find agreement with the values of $\nu$ obtained from transfer matrix calculations.   

Perturbative approach to the RG $\beta $-function for the 3-d Anderson localization

The $\beta $-function of the conductance for Anderson Metal-Insulator transition in 2+ $\varepsilon $ dimensions is known from the non-linear sigma model. However, the result is valid for small $\varepsilon $ only. Recently, the $\beta $-function for the two-dimensional unitary case up to two-loop order was reproduced within a standard diagrammatic perturbation theory by including contributions from the ballistic regime in a consistent way [P. Ostrovskii (2009), unpublished]. An extension of the method to three dimensions will be discussed. The result in leading order in 1/g (g=dimensionless conductance) is $\beta $(g)=1-a/g, where a is a constant.   

Quantum $k$-core Percolation on the Bethe Lattice

Quantum percolation is the study of hopping transport of a quantum particle on randomly diluted percolation clusters. We investigate the Landauer conductance through the dilute Bethe lattice. We show that (1) $p_q$, the quantum percolation critical probability, is greater than $p_c$, the geometric percolation critical probability, and (2) for $p_q<1$ that the quantum conductance transition is continuous with a quantum conductance exponent of 2, as in the classical case. We also study the Landauer conductance through a dilute Bethe lattice where the dilution is subject to the condition that each occupied bond/site must have at least $k$ occupied neighboring bonds/sites. This geometric constraint defines $k$-core percolation. We find, again, that $p_q>p_c$ and, for $p_q<1$, we calculate a quantum conductance exponent of 2 for $k=3$ and a coordination number of four.   

The Mott-Hubbard Insulator: localization and topological quantum order

An insulating state of condensed matter is characterized by localization of the center of mass of the electrons. This criterion can be addressed in terms of the ground state on a torus with boundary conditions $\Psi_{K}(\{x_{1}+L,x_{2}, \ldots\}) = exp( i K L) \Psi_{K}(\{x_{1},x_{2}, \ldots\})$. As shown by Kohn[1], in an insulator the energy is insensitive to $K$ as $L \rightarrow \infty$, whereas in an ideal metal it increases as $K^{2}$. In addition, Souza, et al. derived expressions for the localization length in terms of the wavefunction as a function of $K$. The present work generalizes the arguments to provide a fundamental distinction between ``band'' and ``Mott-Hubbard'' insulators. The criteria involve only counting of electrons and experimentally measurable quantities independent of models, and they lead to the requirement that a Mott-Hubbard insulator with no broken local symmetry must have topological quantum order.\\[4pt] [1] W. Kohn, Phys. Rev. 133, A171 (1964)\\[0pt] [2] I. Souza, et al., Phys. Rev. B 62, 1666 (2000).   

Universal critical conductivity in the metal to Anderson insulator transition in the two dimensional Anderson-Hubbard model

We demonstrate, through extensive quantum Monte Carlo simulations, the existence of a universal critical conductivity in an Anderson insulator to metal transition in two dimensions. The universality of the critical conductivity across various models of disorder is presented, thus pointing to the existence of a quantum critical point with universal properties. We also present the behavior of the compressibility and magnetic susceptibilities across the phase transition and compare them to experimental data and analytical renormalization group investigations.   

Origin of the zero-bias anomaly in the Anderson-Hubbard model

The combination of disorder and electron-electron interactions is known to suppress the density of states at the Fermi level, with two important examples being the Altshuler-Aronov anomaly in metallic systems and the Efros-Shklovskii Coulomb gap in the atomic limit. In both these cases the interactions are nonlocal. Recent Monte Carlo and exact diagonalization studies of the Anderson-Hubbard model, in which the interaction is purely local but strong, obtain a zero-bias anomaly with a curious linear dependence on hopping. Here we demonstrate that this anomaly arises not from diagonal terms in the self energy but from renormalization of the hybridization function, and that it is specifically associated with neighboring sites for which the lower Hubbard orbital on one site is near in energy to the upper Hubbard orbital on the other. Based on these two points, we construct an approximate analytic expression for the anomaly and physical understanding of the linear dependence on hopping.   

Disorder-induced pseudogap in the two-site Anderson-Hubbard model

Several recent exact diagonalization calculations have established that the disordered Hubbard (Anderson-Hubbard) model has a disorder-induced pseudogap or zero bias anomaly (ZBA) in the density of states. Motivated by these numerical results, we have studied the density of states of the two-site Anderson-Hubbard model, for which analytical results are possible. We find that, while strong correlations generally suppress valence fluctuations and lead to $t^2/U$-type corrections to the density of states, large valence fluctuations occur when the lower and upper Hubbard orbitals of neighboring sites are nearly degenerate. For these configurations, the level repulsion between many-body states, and therefore the width of the ZBA, is of order $t$.   

Metal-Insulator and Magnetic Transitions in Strongly Correlated and Disordered Systems

We present a theoretical study for the effects of potential fluctuation on the spin and transport properties of correlated electron systems. Our study is based on the Fermion Hubbard model with on-site disorder. Our main results are: (1) The local potential fluctuation induces variations in the local carrier density and spin polarization, leading to the changes in collective optical and transport properties. (2) The Mott gap in the strongly correlated system closes with increasing disorder but leaves a distinctive pseudogap in the density of states. (3) The combined analysis of density of states, optical conductivity, and spin susceptibility reveals a phase diagram that features the Mott insulator to metal to Anderson insulator phase transition.   

Geometrical frustration, heavy fermions, and lattice disorder in uranium and cerium intermetallics

Geometrical frustration may lead to a variety of interesting states of matter such as spin super-solids, spin-ice, or spin-liquids. While frustration has been widely studied in oxides such as the pyrochlores or Mott insulators, the effect of geometrical frustration on the development of the heavy-fermion state or quantum criticality in intermetallic compounds has received much less attention. Samples from two classes of geometrically frustrated heavy fermion materials based on the hexagonal CaCu5 and cubic AuBe5 have been synthesized: CeCu$_{4-x}$Al$_x$, UAuPt$_4$, UAuCu$_4$, and USnCo$_4$. Magnetic data will be presented to try and quantify the degree of frustration. In addition, since lattice disorder can play a large role in defining magnetic properties in frustrated systems and because of the known Pd/Cu site/anti-site disorder in UPdCu4, extended x-ray absorption fine-structure (EXAFS) data have also been obtained. The local structure results will be discussed and related to the magnetic properties.   

Effect of Kondo Defects in Heavy Fermion Systems

The effects of impurities in heavy-fermion materials, in which the competition between Kondo screening and antiferromagnetic ordering is likely the cause for the experimentally observed non-Fermi-liquid behavior, are poorly understood. We present a newly developed real-space large-N theory to demonstrate that defects, by inducing significant perturbations in the local electron and magnetic correlations of heavy-fermion systems, provide a new approach to understanding their complex properties. The real-space form of the local density of states around defects, which can take the form of missing Kondo atoms, i.e., Kondo holes, or non-magnetic atoms, reveals insight into the heavy or light character of the perturbed states, and the strong correlations between them. The strongly correlated nature of these materials leads to highly non-linear quantum interference effects between defects that can drive the system through a phase transition to a novel inhomogeneous ground state.