Density of states and conductivity of a granular metal or an array of quantum dots

The conductivity of a granular metal or an array of quantum dots usually has the temperature dependence associated with variable range hopping within the soft Coulomb gap of density of states. This is difficult to explain because neutral dots have a hard charging gap at the Fermi level. We show that uncontrolled or intentional doping of the insulator around dots by donors leads to random charging of dots and finite bare density of states at the Fermi level. Then Coulomb interactions between electrons of distant dots results in a soft Coulomb gap. We show that in a sparse array of dots the bare density of states oscillates as a function of concentration of donors and causes periodic changes in the temperature dependence of conductivity. In a dense array of dots the bare density of states is totally smeared if there are several donors per dot in the insulator.

Figure 1

The shape of the Coulomb gap in the vicinity of the Fermi level. Bare density of states in the absence of long range Coulomb interaction is shown by the dashed line. Occupied states are shaded.

Figure 2

A sparse array of same size neutral metallic dots surrounded by an insulator.

Figure 3

The bare density of ground states (BDOGS) of a clean system of indentical dots consists of equidistant peaks. Occupied states are shaded.

Figure 4

A “super-dense” array of dots. Only one donor is shown by + and electron donated by it is shown by −.

Figure 5

BDOGS $g_0\left(\epsilon \right)$ at a certain filling factor. Occupied states are shaded.

Figure 6

BDOGS at the Fermi level $g_0\left(\mu \right)$ of a sparse 3D array as a function of filling factor (solid line). The reference to the equation appropriate for a part of the curve is shown next to it. Dashed lines describe locations of minima and maxima of the oscillating part.

Figure 7

Ranges of ES and Mott’s laws alternating with growth of filling factor $\nu$ at a given low temperature.

Figure 8

BDOGS at the Fermi level $g_0\left(\mu \right)$ of a “super-dense” array as a function of filling factor $\nu$.

Figure 9

The origin of $g_{\max }$. The solid horizontal line is the Fermi level, the dashed line is the fluctuating potential energy. Two peaks of BDOGS of the array are shown, as well as two corresponding levels of four dots.

Figure 10

The origin of $g_{\min }$. The meandering line represents the fluctuating potential energy as a function of coordinate. The regions of the shortest size $\sim l$ where dots are completely filled with electrons or holes are shown with thicker lines. The solid horizontal line in the middle is the Fermi level, the other two horizontal lines correspond to the nearest peaks of an isolated dot. Two peaks of BDOGS of the array are shown, as well as two corresponding levels of seven dots.

Figure 11

Two-dimensional cross section through a donor (with charge $+e$) located between dots A and B for the system shown in Fig. 4. The group within the dashed circle is neutral altogether and does not affect the energy levels of dots A and B. Dot A gets an electron donated by the donor because the donor is closer to its border.

Figure 12

BDOGS of isolated donors at $\nu ⪡1$. Energy $\epsilon$ is measured in units of $\alpha P_{ii}{e}^2∕2$. There is a hard gap of the width $\alpha P_{ii}{e}^2$ around the Fermi level.

Figure 13

An example of rare configurations leading to the finite BDOGS at the Fermi level at $\nu ⪡1$. The cluster of five dots and four donors is effectively positively charged in the ground state. Charges of five dots are shown in units of $e$. All the other dots in the vicinity of the cluster remain neutral.

Figure 14

An almost densely packed array of metallic spheres $(d⪡R)$. A donor is shown by + and electron donated by it is shown by −.

Figure 15

An array of coated dots. The width of coating insulator is $w$. A donor is shown by + and electron donated by it is shown by −.