Temperature and magnetic-field dependence of the quantum corrections to the conductance of a network of quantum dots

We calculate the magnetic-field and temperature dependence of all quantum corrections to the ensemble-averaged conductance of a network of quantum dots. We consider the limit that the dimensionless conductance of the network is large, so that the quantum corrections are small in comparison to the leading, classical contribution to the conductance. For a quantum dot network the conductance and its quantum corrections can be expressed solely in terms of the conductances and form factors of the contacts and the capacitances of the quantum dots. In particular, we calculate the temperature dependence of the weak localization correction and show that it is described by an effective dephasing rate proportional to temperature.

Figure 1

An example of a quantum dot network with ${\mathcal{N}}_D=3$ quantum dots. The conductance of the network is dominated by the conductances of the contacts between the dots. We assume that all dots in the network are open, i.e., all contact conductances are much larger than the conductance quantum $e^2/h$.

Figure 2

(Color online) Schematic drawing of a trajectory $\alpha$ and its time-reversed $\overline{\alpha }$ that contribute to the cooperon propagator $\tilde{c}$.

Figure 3

(Color online) Calculation of the cooperon propagator for a network of quantum dots. A trajectory $\alpha$ originating in dot $j$ and ending in dot $i$ and duration $t$ is separated into two segments of duration $d\tau$ and a remaining segment of duration $t-2d\tau$ if $2d\tau <t$. A self-consistent equation for ${\tilde{c}}_{ij}$ is obtained by considering the combined effect of escape, the magnetic field, and the fluctuating potential to first order in $d\tau$.

Figure 4

(a) Diagrammatic rules for the ensemble average using random matrix theory. The weight factors depend on the symmetry present: ${\lambda }^{\prime }=\lambda$ in the presence of time-reversal symmetry, while ${\lambda }^{\prime }$ is reduced in the presence of a weak magnetic field and ${\lambda }^{\prime }=0$ where time-reversal symmetry is fully broken. (b) Expansion of the full matrix propagator in terms of single propagators $1/\left(\epsilon +i\pi \nu W{W}^{†}\right)$, depicted by single lines, and the matrix elements $H_{\alpha \beta }$, depicted by two open circles. (c) Dyson equation for the self-energy $\Sigma$.

Figure 5

(a) Diffuson ladder and (b) cooperon ladder.

Figure 6

(a) Diagrammatic representation of the leading contribution $g^{\text{cl}}$ to the ensemble-averaged conductance $⟨g⟩$. [(b)–(e)] Diagrams contributing to the weak localization correction $\delta g^{\text{WL}}$. (f) Definition of the Hikami-box used in (c)– (e).

Figure 7

Dyson equation for corrections to $⟨G_{ii}⟩$ due to the possibility of cooperon like ladders in the time-reversal symmetric case. Double hatching indicates a retarded-retarded or advanced-advanced pairing. These ladders are parametrically small and for that reason can also not extend across multiple dots.

Figure 8

(Color online) Diagrams for the first-order dephasing correction. Diagrams depicted in (b) and (c) as well as in (e) and (f) are weighed with a factor 1/2, in line with Eq. (57). Together (a)–(c) constitute the correction to the diffuson propagator, which cancels to leading order. Hence the only relevant contributions are the corrections to the cooperon in (d)–(f). In both cases, complex-conjugate contributions exist which are obtained by placing the vertices on the opposite matrix propagation lines.

Figure 9

(Color online) Renormalization of the interaction vertex by ladder diagrams involving Green’s functions of the same type (retarded-retarded or advanced-advanced).

Figure 10

(Color online) Dyson equation for the cooperon obtained by perturbation theory in the high-temperature limit. The hatched boxes indicate noninteracting cooperon ladders, while gray shading indicates that interactions are taken into account. Wiggly lines indicate the equal-time interaction propagator, which can either connect back to the same propagation line or to the opposite time-reversed one.

Figure 11

(Color online) Diagrams contributing to $\delta g^{\text{int}}$. The Hikami box is defined in Fig. 6.

Figure 12

Schematic drawings of two double quantum dots. Panel (a) shows a linear configuration; panel (b) shows a side-coupled configuration.

Figure 13

Diagrammatic depiction of Hikami boxes. Different diagrams contribute depending on where the cooperon and diffuson like ladders end and begin.

Figure 14

Diagrammatic depiction of contribution from Hikami boxes placed adjacent to leads.