Superdiffusive Conduction: AC Conductivity with Correlated Noise

We present evidence of the existence of a superdiffusive regime in systems with correlated disorder for which localization is suppressed. An expression for anomalous electrical conductivity at low frequencies is found by using a generalized Langevin equation whose memory function accounts for the interactions between the carriers. New mechanisms inducing a superdiffusive conductivity are discussed and experimental possibilities for observing that phenomenon in nanotubes and superlattices are presented. One of the most fundamental quantities in transport theory is electrical conductivity whose understanding has permanently represented a challenge for most of the last century. Discoveries of new aspects of this property have always been accompanied by technological innovation. Since finite conductivity is the consequence of the presence of one or more scattering mechanisms, most of the observed conductivity phenomena have always been obtained for diffusive or subdiffusive propagation of the charge carriers. The subdiffusive regime occurs in many different disordered materials such as ion conducting materials or glassy plastics. Measurements of the AC conductivity as a function of the frequency reveal the existence of universal behavior of this quantity whose origin seems to be the only feature shared by those systems: the presence of disorder [1]. The presence of correlated disorder may introduce drastic changes in the transport properties of a system by originating a superdiffusive behavior of the Preprint submitted to Elsevier Science 2 February 2008 charge carriers. This behavior can be observed e.g. in the quantum Heisenberg chain [2] where one can find the existence of both a superdiffusive regime emerging at weak correlations and a ballistic regime resulting from strong correlations. A similar result was obtained for a disordered chain [3]. It has been proved as well that the Anderson model with long-range correlated diagonal disorder displays a finite phase of extended states in the middle of the band of allowed energies [4,5]. Moreover, the suppression of Anderson localization has recently been confirmed experimentally in semiconductor super-lattices with intentional correlated disorder [6]. Quasi-ballistic and ballistic conduction at room temperature has also recently been observed in carbon nanotubes [7]. The aim of this Letter is to propose a general scenario to describe the conduction mechanism in disordered materials valid for both correlated and uncorrelated disorder. The result can be expressed by a simple equation for the variation of the conductivity as function of the frequency at low frequencies. We show that subdiffusive conductivity increases with the frequency while superdiffusive conductivity decreases. The result is general and independent from a particular microscopic mechanism. In the linear regime, transport properties can be determined through the knowledge of AC conductivity σ(ω). The first proposed model for AC conductivity, the Drude model, goes back more than a century. In spite of great progress in the field, and the discovery of many forms of universal behavior [1] no general model accounting for the main characteristics of complex disordered systems has been set forth. Although a general relation for σ(ω) is not available, some basic relations involving that quantity established long ago still remain valid. The frequency-dependent diffusion coefficient D(ω) is related to the conductivity through the expression σ(ω) = ne kBT D(ω) , (1) where, e is carrier charge, n the carrier density and T is temperature and kB is the Boltzmann constant. The above equation is a form of the fluctuationdissipation theorem. The frequency dependendent diffusion coefficient is defined as D(ω) = ∫ Cv(t) exp(iωt) dt , (2) where Cv(t) = 〈v(0)v(t)〉 is the velocity correlation function of the carriers and the brackets 〈· · · 〉 indicate ensemble average. Our aim here is to describe the interactions responsible for this behavior using a generalized Langevin equation (GLE) which makes the dependence more explicit and may display superdiffusion as well as normal diffusion and subdiffusion. Our starting point is that it does not matter which way Cv(t) is determined once established, the diffusion and conductivity can be found. In general, complex materials present a non-Ohmic behavior in the conductivity, and this effect is equivalent to the