Two-scale localization in disordered wires in a magnetic field

Calculating the density-density correlation function for disordered wires, we study localization properties of wave functions in a magnetic field. The supersymmetry technique combined with the transfer matrix method is used. It is demonstrated that at arbitrarily weak magnetic field the far tail of the wave functions decays with the length $L_{{\rm cu}}=2L_{{\rm co}}$, where $L_{{\rm co}}$ and $L_{{\rm cu}}$ are the localization lengths in the absence of a magnetic field and in a strong magnetic field, respectively. At shorter distances, the decay of the wave functions is characterized by the length $L_{{\rm co}}$. Increasing the magnetic field broadens the region of the decay with the length $L_{{\rm cu}}$, leading finally to the decay with $L_{{\rm cu}}$ at all distances. In other words, the crossover between the orthogonal and unitary ensembles in disordered wires is characterized by two localization lengths. This peculiar behavior must result in two different temperature regimes in the hopping conductivity with the boundary between them depending on the magnetic field.Comment: 4 page

Symmetry Dependence of Localization in Quasi- 1- dimensional Disordered Wires

The crossover in energy level statistics of a quasi-1-dimensional disordered wire as a function of its length L is used, in order to derive its averaged localization length, without magnetic field, in a magnetic field and for moderate spin orbit scattering strength. An analytical function of the magnetic field for the local level spacing is obtained, and found to be in excellent agreement with the magnetic field dependent activation energy, recently measured in low-mobility quasi-one-dimensional wires\cite{khavin}. This formula can be used to extract directly and accurately the localization length from magnetoresistance experiments. In general, the local level spacing is shown to be proportional to the excitation gap of a virtual particle, moving on a compact symmetric space.Comment: 4 pages, 2 Eqs. added, Eperimental Data included in Fig.

Strong localization of electrons in quasi-one-dimensional conductors

We report on the experimental study of electron transport in sub-micron-wide ''wires'' fabricated from Si $\delta$-doped GaAs. These quasi-one-dimensional (Q1D) conductors demonstrate the crossover from weak to strong localization with decreasing the temperature. On the insulating side of the crossover, the resistance has been measured as a function of temperature, magnetic field, and applied voltage for different values of the electron concentration, which was varied by applying the gate voltage. The activation temperature dependence of the resistance has been observed with the activation energy close to the mean energy spacing of electron states within the localization domain. The study of non-linearity of the current-voltage characteristics provides information on the distance between the critical hops which govern the resistance of Q1D conductors in the strong localization (SL) regime. We observe the exponentially strong negative magnetoresistance; this orbital magnetoresistance is due to the universal magnetic-field dependence of the localization length in Q1D conductors. The method of measuring of the single-particle density of states (DoS) in the SL regime has been suggested. Our data indicate that there is a minimum of DoS at the Fermi level due to the long-range Coulomb interaction.Comment: 12 pages, 11 figures; the final version to appear in Phys. Rev.

Electron-Assisted Hopping in Two Dimensions

We have studied the non-ohmic effects in the conductivity of a two-dimensional system which undergoes the crossover from weak to strong localization with decreasing electron concentration. When the electrons are removed from equilibrium with phonons, the hopping conductivity depends only on the electron temperature. This indicates that the hopping transport in a system with a large localization length is assisted by electron-electron interactions rather than by the phonons.Comment: 5 pages, 4 figure

Low-Temperature Dephasing in Disordered Conductors: the Effect of ``1/f'' Fluctuations

Electronic quantum effects in disordered conductors are controlled by the dephasing rate of conduction electrons. This rate is expected to vanish with the temperature. We consider the very intriguing recently reported apparent saturation of this dephasing rate in several systems at very low temperatures. We show that the ``standard model'' of a conductor with static defects can {\em not} have such an effect. However, allowing some dynamics of the defects may produce it.Comment: 6page

Interaction effects and phase relaxation in disordered systems

This paper is intended to demonstrate that there is no need to revise the existing theory of the transport properties of disordered conductors in the so-called weak localization regime. In particular, we demonstrate explicitly that recent attempts to justify theoretically that the dephasing rate (extracted from the magnetoresistance) remains finite at zero temperature are based on the profoundly incorrect calculation. This demonstration is based on a straightforward evaluation of the effect of the electron-electron interaction on the weak localization correction to the conductivity of disordered metals. Using well-controlled perturbation theory with the inverse conductance $g$ as the small parameter, we show that this effect consists of two contributions. First contribution comes from the processes with energy transfer smaller than the temperature. This contribution is responsible for setting the energy scale for the magnetoresistance. The second contribution originates from the virtual processes with energy transfer larger than the temperature. It is shown that the latter processes have nothing to do with the dephasing, but rather manifest the second order (in $1/g$) correction to the conductance. This correction is calculated for the first time. The paper also contains a brief review of the existing experiments on the dephasing of electrons in disordered conductors and an extended qualitative discussion of the quantum corrections to the conductivity and to the density of electronic states in the weak localization regime.Comment: 34 pages, 13 .eps figure

Geometry dependent dephasing in small metallic wires

Temperature dependent weak localization is measured in metallic nanowires in a previously unexplored size regime down to width $w=5$ nm. The dephasing time, $\tau_{\phi}$, shows a low temperature $T$ dependence close to quasi-1D theoretical expectations ($\tau_{\phi} \sim T^{-2/3}$) in the narrowest wires, but exhibits a relative saturation as $T \to 0$ for wide samples of the same material, as observed previously. As only sample geometry is varied to exhibit both suppression and divergence of $\tau_{\phi}$, this finding provides a new constraint on models of dephasing phenomena.Comment: 6 pages, 3 figure

Decoherence in Disordered Conductors at Low Temperatures, the effect of Soft Local Excitations

The conduction electrons' dephasing rate, $\tau_{\phi}^{-1}$, is expected to vanish with the temperature. A very intriguing apparent saturation of this dephasing rate in several systems was recently reported at very low temperatures. The suggestion that this represents dephasing by zero-point fluctuations has generated both theoretical and experimental controversies. We start by proving that the dephasing rate must vanish at the $T\to 0$ limit, unless a large ground state degeneracy exists. This thermodynamic proof includes most systems of relevance and it is valid for any determination of $\tau_{\phi}$ from {\em linear} transport measurements. In fact, our experiments demonstrate unequivocally that indeed when strictly linear transport is used, the apparent low-temperature saturation of $\tau_{\phi}$ is eliminated. However, the conditions to be in the linear transport regime are more strict than hitherto expected. Another novel result of the experiments is that introducing heavy nonmagnetic impurities (gold) in our samples produces, even in linear transport, a shoulder in the dephasing rate at very low temperatures. We then show theoretically that low-lying local defects may produce a relatively large dephasing rate at low temperatures. However, as expected, this rate in fact vanishes when $T \to 0$, in agreement with our experimental observations.Comment: To appear in the proceedings of the Euresco Conference on Fundamental Problems of Mesoscopic Physics, Granada, September 2003, Kluwe

Non-linear effects and dephasing in disordered electron systems

The calculation of the dephasing time in electron systems is presented. By means of the Keldysh formalism we discuss in a unifying way both weak localization and interaction effects in disordered systems. This allows us to show how dephasing arises both in the particle-particle channel (weak localization) and in the particle-hole channel (interaction effect). First we discuss dephasing by an external field. Besides reviewing previous work on how an external oscillating field suppresses the weak localization correction, we derive a new expression for the effect of a field on the interaction correction. We find that the latter may be suppressed by a static electric field, in contrast to weak localization. We then consider dephasing due to inelastic scattering. The ambiguities involved in the definition of the dephasing time are clarified by directly comparing the diagrammatic approach with the path-integral approach. We show that different dephasing times appear in the particle-particle and particle-hole channels. Finally we comment on recent experiments.Comment: 28 pages, 6 figures (14ps-files