Conductivity, scattering and transport in granular materials.

phenomena, random processes, noise, and Brownian motion.
The electronic transport in granular conductors is governed by the nontrivial interplay the diffusive intra-grain
electron motion and grain-to-grain tunneling which is accompanied by sequential charging of the grains involved in
the particular electron transfer process [3]. Transport properties are controlled by competition between the intergrain
coupling and electron-electron Coulomb interaction. The basic parameter that characterizes transport
properties is the dimensionless tunneling conductance, gт. Depending on the bare tunneling conductance gт (0), the
conductivity can demonstrate either (i) exponential (insulating)-, at gт (0)<< g c
т=(1/2 π d) ln (E c / δ ), where E c and
δ are the charging energy and the mean energy level spacing in a single grain respectively, or (ii) logarithmic
(metallic), at gт(0)>> g c
т, temperature dependencies, experiencing metal-insulator transition at gт(0)= g c
We investigate transport in a granular metallic system at both limiting cases. We show that in the metallic region,
gт>>1, and low temperatures, T ≤ gт δ , where δ is the single mean energy level spacing in a grain, the coherent
electron motion at large distances dominates the physics, contrary to the high temperature (T> gт δ ) behavior where
conductivity is controlled by the scales of the order of the grain size. The conductivity of one and two dimensional
granular metals, in the low temperature regime, decays with decreasing temperature in the same manner as that in
homogeneous disordered metals, indicating thus as insulating behavior. However, even in this temperature regime
the granular structure remains important and there is an additional contribution to conductivity coming from short
distances. Due to this contribution the metal-insulator transition in three dimensions occurs at the value of tunnel
conductance g c
т = (1/6 π ) ln (E c / δ ), where E c is the charging energy of an isolated grain, and not at the
generally expected g c
т ∝ 1. Correction to the density of granular metals due to the electron-electron interaction are
calculated. Our results compare favorably with logarithmic dependence of resistivity in the high-T c cuprate
superconductors indicating that these materials may have a granular structure.
We investigate the effect of Coulomb interactions on the tunneling density of states (DOS) of granular metallic
systems at the onset of Coulomb blockade regime in two and three dimensions. We derive the analytical expressions
for the DOS as a function of temperature T and energy є. We show that samples with the bare intergranular
tunneling conductance g т
0 less than the critical value g c
т =(1/2 π d) ln (Ec/ δ ), where Ec and δ are the charging
energy and the mean energy level spacing in a single grain respectively, are insulators with a hard gap in the DOS at
temperatures T → 0.
In 3d systems the critical conductance g c
т separates insulating and metallic phases at zero temperature, whereas
in the granular films g c
т separates insulating states with hard (at g т
0 <g c
т) and soft (at g т
0 >g c
т) gaps. The gap in
the DOS begins develop at temperature T*~Ec g т
0 exp (-2 π d g т
0 ) and reaches the value Δ ~T* at T → 0.
We further study the electron thermal transport in granular metals at large tunnel conductance between the
grains, gт>1 and not too low a temperature T>gт δ , where δ is the mean energy level spacing for a single grain.
Taking into account the electron-electron interaction effect we calculate the thermal conductivity and show that the
Wiedemann-Franz law is violated for granular metals. We find that interaction effect suppress the conductivity less
than the electrical conductivity.
We present a unified description of the low temperature phase of granular metals that reveals a striking generality
of the low temperature behaviors. Our model explains the universality of the low-temperature conductivity that
coincides exactly with that of the homogeneously disordered systems and enables a straightforward derivation of
low temperature characteristics of disordered conductors.
We investigate the suppression of superconducting transition temperature in granular metallic systems due to: 1.
fluctuation of the order parameter (bosonic mechanism); 2. Coulomb repulsion (fermionic mechanism) assuming
large tunneling conductance between the grains gт>>1. We find the correction to the superconducting transition
temperature for 3d granular samples and films. We demonstrate that if the critical temperature T c > gт δ , where δ is
the mean level spacing in a single grain the bosonic mechanism is the dominant mechanism of the superconductivity
suppression, while for critical temperatures T c > gт δ the suppression of superconductivity is due to the fermionic
Turning to insulating regime, we develop a theory of a variable range hopping transport in granular conductors
based on the sequential electron tunneling through many grains in the presence of the strong Coulomb interaction.
The processes of quantum tunneling of real electrons are represented as trajectories (world lines) of charged classical
particles in d+ 1 dimensions. We apply the developed technique to investigate the hopping conductivity of granular
systems in the regime of small tunneling conductances between the grains gт<<1 and derive the Efros-Shklovskii
law of Coulomb-interaction controlled transport σ ∝ exp [-(T 0 /T1/ 2 )].
Further analysis will focused on the analysis of the Coulomb blocade effect in granular conductivity within the
framework of effective single junction approach [4].
We wish to thank N.Vandewalle for stimulating discussion of the reported results.
1. A. Georges, G. Kotliar, W.Krauth, M. J. Rosenberg Rev. Mod. Phys. 68, 13 (1993).
2. D.D.Arovas, F. Guinea, C. D. Herrero, D.San Jose Jouru. Of Phys. Cond. Matt. 29, 021168 (2002).
3. O. I. Gerasimov, G. Mondio. Electrical conductivity of granular matter. Progress in Condensed Matter. Physics, v.84( 2004).
YPS, Bologna, Italy.
4. J. S. Herman, B. Abeles. Phys. Rev. Lett. 37, 1429 (1976).

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