Coherent transport in disordered metals: zero dimensional limit

Abstract

We consider non-equilibrium transport in disordered conductors. We calculate the interaction correction to the current for a short wire connected to electron reservoirs by resistive interfaces. In the absence of charging effects we find a universal current-voltage-characteristics. The relevance of our calculation for existing experiments is discussed as well as the connection with alternative theoretical approaches. In recent years considerable attention has been devoted to the effects of the Coulomb interaction on the transport properties of small structures, like thin diffusive films and wires [1,2,3,4], tunnel junctions [5,6,7], and quantum dots [8]. One interesting issue concerns the way an applied bias voltage affects the interaction corrections to the electrical conductivity. In diffusive metals these corrections arise from the combination of the electron-electron and impurity scattering and yield well known singularities at low temperature [9]. It has been shown that a finite voltage or, more in general, a non-equilibrium situation leads to a suppression of these singularities [10,11,12,13,14]. In particular, in [10,14] non-equilibrium transport in a short wire connected to electrical reservoirs by ideal interfaces has been considered. However, in actual experiments the interfaces need not be ideal. Recently Weber et al. [15] investigated experimentally the non-equilibrium transport through a metallic nano-scale bridge. Both in [15] and in [16] it has been suggested that the Coulomb interaction effects are responsible for the observed temperature dependence of the conductance and the current-voltage-characteristics. Whereas [15] found an agreement between theory and experiment starting from a tunneling Hamiltonian, [16] pointed out that the experimental data agree with what they expect for a diffusive conductor. In this paper we develop a formalism in which both the resistive behavior due to the interfaces and due to the diffusive wire region are treated on the same footing. From our results we conclude that the main resistive behavior in [15] occurs at the interfaces.To begin with we recall the classical description of electrical transport through structures consisting of both interface barriers and diffusive regions. To be definite we consider a system made by a diffusive wire of length L which is attached to the reservoirs by two interface barriers. We study the system in a non-equilibrium situation with an applied voltage V l − V r = V where the subscripts l and r indicate the left and right reservoirs, respectively. The classical resistance of the structure is the sum of the wire resistance and the interface resistances R tot = R wire + R l + R r , so that the current as a function of voltage is

Quasiclassical theory of charge transport in disordered interacting electron systems

“…506 P. Schwab and R. Raimondi: Charge transport in disordered interacting electron systems Finally, as far as the experiments are concerned, a ln T behavior in the linear resistivity of short metallic bridges, together with an I-V -characteristic which agrees well with the universal function (190) has been observed in [113]. The bridges have been about L = 80 nm long, 80 nm wide and 10 nm thick; the temperature region where the log T in the linear resistivity has been seen is between T ≈ 100 mK and T ≈ 2 K. In [113] the Coulomb correction to the tunneling conductance has been suggested as the explanation; the considerations presented here and in [112] show that, also in the intermediate regime with both diffusive and interface resistivity the predicted I-V -characteristic does not change and thus agrees with the experimentally observed one. In [113] the Thouless energy, which sets the scale for the lowest diffusive mode in an open system and therefore for the low temperature saturation of the conductance, has been estimated to be of the order of several Kelvin, whereas the ln T is observed down to 100 mK.…”

“…2.4 we demonstrated how to obtain the classical resistance of a system which is composed of diffusive pieces and resistive interfaces in the framework of the quasiclassical description. In this subsection, following closely [112], we include the Coulomb interaction, and in particular we will concentrate on structures of a size comparable to the thermal diffusion length L T . In Sect.…”

“…In all curves the linear conductance has been subtracted. The figure is taken from[112]. energy scales related to the temperature and to the voltage remain below the energy of the second lowest diffusive mode.…”

Transport of interacting electrons in arrays of quantum dots and diffusive wires

“…Although the formula (9) has been derived under the assumption that the distribution function does not depend on time, we will show later that it remains valid also in non-stationary situations. In the latter case the function G n (t 1 , t 2 ) can be understood as the Keldysh component of the quasiclassical Usadel Green function.…”

“…Eqs. (9,15,(17)(18)(19)(20) form a complete set of equations, which allow to find the first order interaction correction to the I − V characteristics for an array of quantum dots. These equations represent a straightforward generalization of the Langevin approach employed in Ref.…”