Universal description of granular metals at low temperatures: Granular Fermi liquid

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.

Figure 1

Schematic phase diagram for granular metals showing crossovers between three distinct phases. At small bare tunneling conductances, $g_T^{\left(0\right)}<g_T^C$, a granular metal is in an insulating phase at zero temperature. The metal-insulator transition in three dimensions at $T=0$ occurs at $g_T^C=(1∕6\pi )\ln (E_C∕\delta )$, where $E_C$ and $\delta$ are the charging energy and the mean energy level spacing in a single grain respectively. At large tunneling conductance, $g_T^{\left(0\right)}>g_T^C$, the two different types of conductivity behavior are possible: (i) the high-temperature phase, $T>g_T\delta$, is characterized by the logarithmic temperature dependence of the conductivity in all dimensions; (ii) the low-temperature phase, $T<g_T\delta$, is the universal granular Fermi-liquid phase described in terms of low-energy interacting diffusion modes.

Figure 2

Diagram (a) defines the bare propagator of fields $B$ given by Eq. (4). Diagram (b) represents the interaction vertex that appears from the expansion of terms $\mathrm{Tr}\left[V_i{Q}_i\right]$ in Eq. (2) to the first order in the field $B$. Diagrams (c) represent the Dyson equation that defines the screened Coulomb interaction (thick dashed line). Thin dashed lines denote the bare Coulomb interaction.

Figure 3

Diagrams (a) appear from the expansion of terms $\mathrm{Tr}\left[\epsilon Q_i\right]$ and $\mathrm{Tr}\left[Q_i{Q}_j\right]$ in Eq. (2) to the fourth order in the field $B$. Diagram (b) appears from the expansion of interaction term $\mathrm{Tr}\left[V_i{Q}_i\right]$ in Eq. (2) to the second order in the field $B$. Diagrams (c) represent the renormalization of the diffusion (Cooperon) self-energy due to Coulomb repulsion leading to Eq. (6a). The thick dashed lines denote the screened Coulomb interaction. The tunneling vertices are represented by circles.

Figure 4

Diagram (a) represents the renormalized polarization operator that includes (i) the renormalized expression for the diffusion propagator $D({\omega }_n,\mathbf{q})$ in Eq. (6a) (black rectangle) and (ii) the renormalized vertex (black triangle) shown in diagram (b). The thick dashed line denotes the screened Coulomb interaction.