A granular metal is an array of metallic nano-particles imbedded into an insulating matrix. Tuning the intergranular coupling strength a granular system can be transformed into either a good metal or an insulator and, in case of superconducting particles, experience superconductor-insulator transition. The ease of adjusting electronic properties of granular metals makes them most suitable for fundamental studies of disordered solids and assures them a fundamental role for nanotechnological applications. This Review discusses recent important theoretical advances in the study of granular metals, emphasizing on the interplay of disorder, quantum effects, fluctuations and effects of confinement in formation of electronic transport and thermodynamic properties of granular materials.

We investigate transport in a granular metallic system at large tunneling conductance between the grains, gT ≫ 1. We show that at low temperatures, T ≤ gT 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 > gT 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 an 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 T = (1/6π) ln(EC/δ), where EC is the charging energy of an isolated grain, and not at the generally expected g C T ∝ 1. Corrections to the density of states of granular metals due to the electron-electron interaction are calculated. Our results compare favorably with the logarithmic dependence of resistivity in the high-Tc cuprate superconductors indicating that these materials may have a granular structure. A great deal of research in the current mesoscopic physics focuses on understanding properties of granular metals(see [1,2,3]). The interest is motivated by the fact that while their properties are generic for a wealth of strongly correlated systems with disorder, granular metals offer a unique experimentally accessible tunable system where both the interaction strength and degree of disorder can be controlled.The key phenomenon revealing the most of the underlying physics is transport, where the effects of interactions play a crucial role. The processes of electron tunneling from grain to grain that govern electron transfer, are accompanied by charging the grains involved after each electron hop to another grain. This may lead to a Coulomb blockade, and one justly expects this effect to be of the prime importance at least in the limit of weak coupling. It makes it thus clear, on a qualitative level, that it is the interplay between the the grain-to-grain coupling and the electron-electron Coulomb interaction that controls transport properties of granular metals; yet, despite the significant efforts expended, a quantitative theory of transport in metallic granular systems is still lacking.A step towards formulation such a theory was made recently in [ 3]. It was shown that depending on the dimensionless tunneling conductance g T one observes either exponential-, at g T ≪ 1, or logarithmic, at g T ≫ 1 temperature dependence of conductivity. The consideration in [ 3] was based on the approach developed by Ambegaokar, Eckern and Schön (AES) [5] for tunnel junctions. This technique however, as shown in [4], applies only at temperatures T > g T δ, where δ is the mean energy level spacing in a ...

We generalize the Beliaev-Popov diagrammatic technique for the problem of interacting dilute Bose gas with weak disorder. Averaging over disorder is implemented by the replica method. Low energy asymptotic form of the Green function confirms that the low energy excitations of the superfluid dirty Boson system are sound waves with velocity renormalized by the disorder and additional dissipation due to the impurity scattering. We find the thermodynamic potential and the superfluid density at any temperature below the superfluid transition temperature and derive the phase diagram in temperature vs. disorder plane.Superfluidity in random environments enjoys a long standing yet intense attention. The effect of disorder on the behavior of systems possessing long-range correlations is central to contemporary condensed matter physics, and superfluid Bose gas offers an exemplarily unique and accessible tool for both experimental and theoretical researches. One of the fascinating properties of such systems is their ability to maintain superfluidity (i.e. long range correlations) even in the strongly disordered environment. He 4 , for example, remains superfluid when absorbed in porous media [1]. The problem of influence of disorder on superfluidity (and on its close analogsuperconductivity) has been under extensive theoretical attack (see seminal works [2,3]) and remarkable progress in qualitative understanding of disordered Bose systems was achieved. Recent papers [4,5] discussed a continuum model of the dilute interacting Bose gas in a random potential. The advantages of this model are that (i) it is microscopically related to the original problem and (ii) it is very well understood in the clean limit. The proposed model describes, in particular, the quasiparticle dissipation and depletion of superfluidity at zero temperature and marked an important step towards quantitative description of disordered Bose systems.In this Letter, building on the model of Refs. [4,5], we develop a systematic diagrammatic perturbation theory for the dilute Bose gas with weak disorder at finite temperatures below the superfluid transition temperature T s . We obtain disorder corrections to the thermodynamic potential which completely determine thermodynamic properties of the superfluid system. We derive for the first time the disorder-induced shift of T s resulting from disorder scattering of quasiparticles with energy ǫ ∼ T . We find that the superfluid density decreases monotonically with the temperature. This completely agrees with the experimental data, while being in some contradiction with the theoretical result of Ref.[4] where a non-monotonic temperature dependence of superfluid density dependence was claimed. In the limit T → 0 our theory reproduces all the results of Refs. [4,5].The model. The starting point of our model is the Lagrangian density:where ϕ = ϕ(r, τ ) is the field representing Bose particles, r is the real space coordinate, τ is the Matsubara time and u(r) is the disorder potential. As usual we consider a soft intera...

We investigate effects of Coulomb interaction and hopping transport in the insulator phase of granular metals and quantum dot arrays. We consider a spatially periodic as well as an irregular array, including disorder in a form of a random on-site electrostatic potential. We study the Mott transition between the insulating and metallic states in the regular system and find the dependence of the Mott gap upon the intergranular coupling. The conductivity of a strictly periodic array has an activation form with the Mott gap as an activation energy. Considering irregular systems we concentrate on the transport properties in the dielectric, low coupling limit and derive the EfrosShklovskii law for hopping conductivity. In the irregular arrays electrostatic disorder results in the finite density of states on the Fermi level giving rise to the variable range hopping mechanism. We develop a theory of tunneling through a chain of grains and discuss in detail both elastic and inelastic cotunneling mechanisms; the former dominates at very low temperatures and/or very low applied electric fields, while the inelastic mechanism controls tunneling at high temperature/fields. Our results are obtained within the framework of the new technique based on the mapping of quantum electronic problem onto the classical gas of Coulomb charges. The processes of quantum tunnelling of real electrons are represented in this technique as trajectories (world lines) of charged classical particles in d + 1 dimensions. The Mott gap is related to the dielectric susceptibility of the Coulomb gas in the direction of the imaginary time axis.

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. Using the renormalization group technique we derive the analytical expressions for the DOS as a function of temperature $T$ and energy $\epsilon$. We show that samples with the bare intergranular tunneling conductance $g^0_{\scriptscriptstyle T}$ less than the critical value $g_{\scriptscriptstyle T}^{\scriptscriptstyle C}=(1/2\pi d) \ln(E_{\scriptscriptstyle C}/\delta)$, where $E_{\scriptscriptstyle C}$ and $\delta$ are the charging energy and the mean energy level spacing in a single grain respectively, are insulators with a {\it hard gap} in the DOS at temperatures $T\to 0$. In 3d systems the critical conductance $g_{\scriptscriptstyle T}^{\scriptscriptstyle C}$ separates insulating and metallic phases at zero temperature, whereas in the granular films $g_{\scriptscriptstyle T}^{\scriptscriptstyle C}$ separates insulating states with the hard (at $g^0_{\scriptscriptstyle T}g_{\scriptscriptstyle T}^{\scriptscriptstyle C}$) gaps. The gap in the DOS begins to develop at temperatures $ T^* \sim E_{\scriptscriptstyle C} g_{\scriptscriptstyle T}^{\scriptscriptstyle 0} \exp (-2\pi d g_{\scriptscriptstyle T}^{\scriptscriptstyle 0})$ and reaches the value $\Delta \sim T^*$ at $T\to 0$.Comment: 4 pages, 1 figur