Nearly localized states in weakly disordered conductors

Abstract

The time dispersion of the averaged conductance G(t) of a mesoscopic sample is calculated in the long-time limit when t is much larger than the diffusion traveling time t~. In this case the functional integral in the effective supersymmetric field theory is determined by the saddle-point contribution.If t is shorter than the inverse level spacing A (At/h « 1), then G(t) decays as exp[t/t o]. In the ultra-long-time limit (At/h )) 1) the conductance G(t) is determined by the electron states that are poorly connected with the outside leads. The probability to find such a state decreases more slowly than any exponential funcion as t tends to infinity. It is worth mentioning that the saddle-point equation looks very similar to the well known Eilenberger equation in the theory of dirty superconductors.We consider long-time relaxation phenomena in a disordered conductor that is attached to ideal leads. For simplicity we assume that the electrons in this conductor do not interact with each other, the temperature is zero (T = 0) and there are no inelastic processes. The total current I(t) at time t depends upon the voltage according to the Ohm law,We are interested in the asymptotic form of the conductance G(t) as t M oo.The same problem has been considered earlier by Altshuler, Kravtsov, and Lerner (AKL). Our initial goal was to obtain their results by means of a more direct calculation. At present, we can neither confirm nor disprove the AKL results. We have found an intermediate range of times, where the conductance G(t) decays more slowly than it was predicted in Ref. 1. The AKL asymptote could be valid at longer times (see discussion below).There are three time scales in the problem.(1) The mean-Bee time w = l/v~, where v~is the Fermi velocity and l is the mean-&ee path. This time scale determines the dispersion of the Drude conductivitỹ t/7-(2) The time of difFusion through the sample tD L2/D, where D = l2/3r is the difFusion coefficient, and L is the sample size.(3) The inverse mean level spacing 5/b, = hvV, where v is the density of states and V is the volume of the sample.In a macroscopic sample the inequality r « tD « h/A is valid, provided that L » l and the disorder is weak. Indeed, the product Lt D is connected with the dimensionless conductance of the sainple g = 2vrh/(tLib, ), which is large for a weak disorder. The times t~and h/4 enter into time dispersion only due to quantum corrections to conductivity. At times t « fi/b, an electron can be considered a wave packet of many superimposed states propagating G(t) = G, e '~+ e ' 'f VQ(r)p(Q)exp f -4], d(dt 2' A = dr Str(D(V'Q) + 2i~AQ). 8 (2) We vary the action A with respect to Q, taking into account the constraint Q2 = 1, and obtain the saddlepoint condition which recalls the diffusion limit of the Eilenberger equation 2DV'(QV'Q) +ice [A, Q] = 0.(2) semiclassically.Therefore, it is natural to assume that the conductance G(t) is proportional to the probability of Gnding a Brownian trajectory that remains in the sample for the time t. For t » to such a probab...

Relaxation process in a regime of quantum chaos

“…At time t H , ln P q (t H )/ ln P (t H ) ∼ 2. As it was pointed out recently [5], less strong deviations (ln(P q (t q )/P (t q )) ∼ 1) should take place at a shorter time t q ∼ t c √ g due to weak localization corrections according to equations obtained in [4]. Up to now these theoretical predictions for open systems have not been checked neither by numerical computations nor by laboratory experiments.…”

“…The scale t q can be also explained in a more standard way based on weak localization corrections [5,4]. Indeed, the quantum interference gives a decrease of the diffusion rate 1/t c ∝ D → D(1 − at/t H ) where a is some constant (diffusion stops at time t H ).…”

“…For quasi one-dimensional metallic samples the results of [3,4] predict that the current in the sample, being proportional to the probability P (t) to stay inside the sample, will decay, up to a very long time, in an exponential way according to the classical solution of diffusive equation which describes the electron dynamics in disordered metallic samples: P (t) ∼ exp (−t/t c ). Here t c ∼ t D = N 2 /D is the diffusion time for a system of size N with diffusion coefficient D.According to [3,4] the strong deviation of quantum probability P q from its classical value P takes place only for t > t H where the quantum probability decays as P q (t) ∼ exp (−g ln 2 (t/t H )). Here, t H = 1/∆(h = 1) is the Heisenberg time, ∆ is the level spacing inside the sample and g = t H /t c = E c /∆ is the conductance of the sample with Thouless energy E c = 1/t c .…”

“…Recently the interest to this problem was renewed and new effective methods based on the supersymmetry approach have been developed to study the problem in more detail [4]. For quasi one-dimensional metallic samples the results of [3,4] predict that the current in the sample, being proportional to the probability P (t) to stay inside the sample, will decay, up to a very long time, in an exponential way according to the classical solution of diffusive equation which describes the electron dynamics in disordered metallic samples: P (t) ∼ exp (−t/t c ). Here t c ∼ t D = N 2 /D is the diffusion time for a system of size N with diffusion coefficient D.…”

“…According to [3,4] the strong deviation of quantum probability P q from its classical value P takes place only for t > t H where the quantum probability decays as P q (t) ∼ exp (−g ln 2 (t/t H )). Here, t H = 1/∆(h = 1) is the Heisenberg time, ∆ is the level spacing inside the sample and g = t H /t c = E c /∆ is the conductance of the sample with Thouless energy E c = 1/t c .…”