Granular electronic systems

Granular metals are arrays of metallic particles of a size ranging usually from a few to hundreds of nanometers embedded into an insulating matrix. Metallic granules are often viewed as artificial atoms. Accordingly, granular arrays can be treated as artificial solids with programmable electronic properties. The ease of adjusting electronic properties of granular metals assures them an important role for nanotechnological applications and makes them most suitable for fundamental studies of disordered solids. This review discusses recent theoretical advances in the study of granular metals, emphasizing the interplay of disorder, quantum effects, fluctuations, and effects of confinement. These key elements are quantified by the tunneling conductance between granules $g$, the charging energy of a single granule $E_c$, the mean level spacing within a granule $\delta$, and the mean electronic lifetime within the granule $\hslash ∕g\delta$. By tuning the coupling between granules the system can be made either a good metal for $g>g_c=(1∕2\pi d)\ln (E_c∕\delta )$ ($d$ is the system dimensionality), or an insulator for $g<g_c$. The metallic phase in its turn is governed by the characteristic energy $\Gamma =g\delta$: at high temperatures $T>\Gamma$ the resistivity exhibits universal logarithmic temperature behavior specific to granular materials, while at $T<\Gamma$ the transport properties are those generic for all disordered metals. In the insulator phase the transport exhibits a variety of activation behaviors including the long-puzzling $\sigma \sim \exp [-(T_0∕T{)}^{1∕2}]$ hopping conductivity. Superconductivity adds to the richness of the observed phases via one more energy parameter $\Delta$. Using a wide range of recently developed theoretical approaches, it is possible to obtain a detailed understanding of the electronic transport and thermodynamic properties of granular materials, as is required for their applications.

Figure 1

Scanning electron microscope photographs of indium evaporated onto ${\mathrm{SiO}}_2$ at room temperature. From 169.

Figure 2

Transmission electron micrographs showing periodic granular (a) bilayers, (b) trilayers, (c) tetralayers, and (d) thick films. The insets on the left sides of panels (a) and (c) are the zoomed-in images. The scale bars correspond to (a)–(c) $200\mathrm{nm}$ and (d) $40\mathrm{nm}$. From 159.

Figure 3

Transmission electron micrograph image of nanocrystals (a) monolayer and in-plane electrodes, (b) highly ordered superlattice between electrodes visible at the upper left and lower right. Adapted from 137.

Figure 4

Schematic phase diagram for granular superconductors at temperature $T=0$. Symbols $S$ and $I$ stand for superconducting and insulating phases, respectively.

Figure 5

Resistance of $3\mathrm{D}$ granular $\mathrm{Al}-\mathrm{Ge}$ sample as a function of temperature on a log-log scale, as measured at (zero) $(\times )$ and $100\mathrm{kOe}$ field (open circles). Sample room-temperature resistance is $500\mathrm{\Omega }$. From 86.

Figure 6

Conductance $g_0$ vs inverse temperature $T^{-1∕2}$ for periodic granular multilayer and thick film shown in Fig. 2. Inset: for the high-temperature range $g_0$ has been replotted as a function of $T^{-1}$, indicating Arrhenius behavior from 100 to $160\mathrm{K}$. From 159.

Figure 7

Self-energy of the electron Green’s function averaged over the impurity potential inside the grains and over tunneling elements between the grains. Averaging over the impurity potential is represented by (a) the dotted line while tunneling elements are represented by (b) crossed circles.

Figure 8

Diagrams represent the Dyson equation (a) for a single-grain diffusion propagator Eq. (2.20) and (b) for the whole granular system Eq. (2.29). Dotted lines represent the impurity scattering while crossed circles stand for intergranular tunneling elements.

Figure 9

The diagram representing the conductivity to leading order in $1∕g$. The crossed circles represent the tunneling matrix elements $t_{k{k}^{\prime }}^{ij}{e}^{i{\phi }_{ij}\left(\tau \right)}$ where phase factors appear from the gauge transformation.

Figure 10

Diagrams representing (a) the vertex correction and (b) the renormalized Coulomb interaction.

Figure 11

Diagrams describing the conductivity of granular metals: diagram (a) corresponds to ${\sigma }_0$ in Eq. (2.5) and it is the analog of the Drude conductivity. Diagrams (b)–(e) describe first-order corrections to the conductivity of granular metals due to the electron-electron interaction. The solid lines denote the propagators of electrons and dashed lines describe effective screened electron-electron propagators. The sum of the diagrams (b) and (c) results in the conductivity correction $\delta {\sigma }_1$ in Eq. (2.5). The other two diagrams, (d) and (e), result in the correction $\delta {\sigma }_2$. From 33.

Figure 12

Resistance as a function of temperature of a granular sample from 147. In spite of the fact that the sample exhibits strong insulating behavior in the intermediate temperature range it finally turns to the superconducting state. From 147.

Figure 13

Temperature dependence of resistivity of granular (a) Ga and (b) Pb ultrathin films according to 95. Different curves correspond to films with different normal-state resistance which is tuned by the sample thickness. The low-temperature state is controlled by the film normal-state resistance. From 95.

Figure 14

Illustration of the renormalization of the Coulomb energy of a normal grain placed in contact with bulk superconductor. Due to virtual electron tunneling processes the Coulomb energy of a normal grain $E_c$ is renormalized down to the values ${\tilde{E}}_c=\Delta ∕2g$. The conductance of the contact is supposed to be large, $g\gg 1$, and the initial Coulomb energy is assumed to be larger than the superconducting gap $\Delta$.

Figure 15

Zero-temperature phase diagram of the granular superconducting array. The dimensionless parameter $\alpha$ is related to the conductance $g$ as $\alpha =\pi g$. From 55.

Figure 16

Low-temperature resistance of $3\mathrm{D}\mathrm{Al}$ granular samples as a function of magnetic field at low temperatures $T\ll T_c$. The curves are shown for two samples with different normal-temperature resistance. The inset shows a much less pronounced peak in the resistance at higher temperatures $T-T_c\ll T_c$. From 86.

Figure 17

A phase diagram sketch of a granular superconductor in the coordinates critical temperature vs tunneling conductance for two cases $E_c\ll {\Delta }_0$ and $E_c\gg {\Delta }_0$. At large tunneling conductances $g\gg 1$ the Coulomb interaction is screened due to electron intergrain tunneling and the transition temperature is approximately given by the single-grain BCS value. In the opposite case $g\ll 1$ the boundary between the insulating and superconducting states is obtained by comparing the Josephson $E_J=g\Delta$ and Coulomb $E_c$ energies.

Figure 18

Phase diagram of a superconducting grain at $T=0$ in coordinates of the Zeeman energy $h$ vs pair-breaking parameter $\alpha$. The dashed line separates the gapless (GS) and gapped (S) superconducting states. First- and second-order phase transition lines are shown by thick and thin solid lines, respectively. From 35.

Figure 19

Schematic representation of two mechanisms for the suppression of superconductivity. Diagrams (a)–(c) describe the correction to the superconducting transition temperature due to Coulomb repulsion. Diagram (d) describes the correction to the transition temperature due to superconducting fluctuations. All diagrams are shown before averaging over the impurities. The solid lines denote the electron propagators, the dashed lines describe screened Coulomb interaction, and the wavy lines describe the propagator of superconducting fluctuations. From 34.

Figure 20

Corrections to the classical conductivity and density of states (DOS). Diagram (a) describes the correction to the DOS; diagrams (b) and (c) describe corrections to the conductivity due to superconducting fluctuations. The wavy lines denote the propagators of the fluctuations; the dashed lines stand for the impurity scattering. From 30.

Figure 21

Schematic picture for resistivity behavior of granulated superconductors as a function of the magnetic field at low temperatures $T\ll T_c$. The resistivity at low temperatures grows monotonically on decreasing the magnetic field. It reaches asymptotically the value of the classical resistivity $R_0$ only at extremely strong magnetic fields. The transition into the superconducting state occurs at a field $H_{c_2}$. From 32.

Figure 22

Diagram representing the tunneling probability via elastic co-tunneling processes. The crossed circles represent the tunneling matrix elements $t_{k,k^{\prime }}^{ij}{e}^{i{\varphi }_i\left(\tau \right)}{e}^{-i{\varphi }_j\left(\tau \right)}$ where the phase factors appear from the gauge transformation. The wavy lines represent the average of the phase factors $⟨e^{i\varphi \left({\tau }_1\right)}{e}^{-i\varphi \left({\tau }_2\right)}⟩$ with respect to the Coulomb action.

Figure 23

Diagram representing the tunneling probability via inelastic co-tunneling processes. The crossed circles represent the tunneling matrix elements $t_{k,k^{\prime }}^{ij}{e}^{i{\varphi }_i\left(\tau \right)}{e}^{-i{\varphi }_j\left(\tau \right)}$ where the phase factors appear from the gauge transformation. The wavy lines represent the average of the phase factors $⟨e^{i\varphi \left({\tau }_1\right)}{e}^{-i\varphi \left({\tau }_2\right)}⟩$ with respect to the Coulomb action.