"FLUCTUATIONS IN THE EXTRINSIC CONDUCTIVITY OF DISORDERED CONDUCTORS" by ALTSHULER B.L. (1985)

The conductivity of the metal is determined by the scattering processes without the conservation of the momentum of the electron system. At low temperatures, the residual resistance is caused by scattering on randomly placed impurities. In the frames of the standard quasi-classical approach, the only characteristic of the disorder, a term of the Drude formula for conductivity is transport scattering time. How fair is such approach?

To give answer on this question it is necessary to consider the system of finite size. The conductivity is not convenient for description of electron transport in the systems. More appropriate is the description in terms of the conductance $ G $: the reverse resistance of the sample (with a certain way of connecting external circuit). In the quasi-classical approximation, the average conductance of $ d $ -dimensional cube with a side of $ L $, measured between opposite faces, is $ G_0 (L) = \sigma_0L ^ {d-2} $, where $ \sigma_0 $ is Drude conductivity.

The conductance of the sample with randomly distributed impurities is also a random variable. Different samples of the same form with the same nominal average defect density will be showing slightly different conductances. The magnitude of the fluctuations is characterized by a standard deviation of $ \langle \delta G ^ 2 \rangle = \langle (G-G_0)^ 2 \rangle $. In the framework of classical physics it is natural to expect the self-averaging $ G(L) $ with an increase of the size of the system. Indeed, with the growth of $ L $ the number of scattering centers $ N_{\text {imp}} $ grows as $ L ^ d $, and the relative fluctuation of $ N _ {\text {imp}} $ decreases as $ [\delta N_ { \text {imp}} / \langle N _ {\text {imp}}\rangle] ^ 2 \propto L ^ {- d} $. Thus, fluctuations in the conductance of $ \delta G ^ 2 (L) \propto L ^ {d-4} $ should decrease with increasing $ L $ (in spaces of dimension $ d <4 $), confirming the hypothesis of self-averaging.

In the quantum world, however, situation is different. A mechanism, leading to much greater fluctuations of the conductance was predicted theoretically in the pioneering work of B.L. Altshuler [1] (and independently published a few months later by P.A. Lee and A.D. Stone [2]). The mechanism is based on the phenomenon of quantum interference in the scattering of electrons on impurities.

Interference manifests itself most clearly in mesoscopic samples where the inelastic processes do not break the coherence of the electron, i.e., under the condition $ L_ \phi \gg L $ (where $ L_ \phi $ is dephasing length). In this limit, $ \langle \delta G ^ 2 \rangle \sim (e ^ 2 / h) ^ 2 $ a factor of order unity, depending only on the shape of the sample. The fact that this ratio is independent of the degree of disorder and the size of the sample, allowed the authors \cite {LeeStone} to call such fluctuations universal conductance fluctuations.

Thus, the conductance of mesoscopic sample is not a self-averaging quantity: regardless of size $ L $, at sufficiently low temperatures $ G (L) $ fluctuates on the order of the quantum conductance of $ e ^ 2 / h $. For good metal relative magnitude of fluctuations is small, but near the transition to an insulator when $ G_0 \sim e ^ 2 / h $, fluctuations become strong. In this case, to describe the electron transport one should take into account the distribution function of the conductance $ P (G) $.

Universal conductance fluctuations are caused by diffusion modes (diffuson and cooperons) moving in mesoscopic system. This long-range interaction is responsible for the violation of the classical scaling $ \delta G ^ 2 (L) \propto L ^ {d-4} $. Classical scaling restores in the limit of $ L_ \phi \ll L $. Under this condition the sample actually splits into incoherent subsystems having the size $ L_ \phi $.

Another consequence of the universal conductance fluctuations determined by long-range diffusion modes is their sensitivity to the symmetry of the problem. Spin symmetry or time reversal symmetry breaking reduces the number of long-range diffuson and cooperons which leads to a change in the expression for prefactor $ \langle \delta G ^ 2 \rangle $. Accurate analysis of the dependence of the conductance fluctuation on the symmetry of the system and the temperature can be found in [3].

The theory of universal conductance fluctuations has been confirmed in many experiments. Normally, investigations are not carried out in a variety of different samples.The properties of a single sample are changed by applying a weak magnetic field or changing the chemical potential of the electrons due to the field-effect..

The universal fluctuations manifest themselves in experiment as reproducible conductance irregularities when changing the control parameter.

The idea suggested in [1], proved to be extremely fruitful. It has changed the language for electrical transport description by switching attention from size independent conductivity to the conductance of the finite-size sample and led eventually to the emergence of a new field of research - mesoscopic physics, where quantum properties significantly affect the electron transport on a large scale.

[1] Altshuler B.L. JETP LETTERS **41**, 648 (1985)

[2] P. A. Lee and A. D. Stone, Phys. Rev. Lett. **55**, 1622 (1985).

[3] Altshuler, Shklovskii B.I., JETP 91, 220 (1986).