Weak localization in arrays of metallic quantum dots: Combined scattering matrix formalism and nonlinear $\sigma$ model

Combining scattering-matrix formalism with nonlinear $\sigma$-model technique we analyze weak localization effects in arrays of chaotic quantum dots connected via barriers with arbitrary distribution of channel transmissions. With the aid of our approach we evaluate magnetoconductance of two arbitrarily connected quantum dots as well as of $N\times M$ arrays of identical quantum dots. Our results can be directly used for a quantitative decription of magnetoconductance measurements in structures containing metallic quantum dots.

Figure 1

1D array of $N-1$ quantum dots coupled by $N$ barriers. Each quantum dot is characterized by mean level spacing ${\delta }_n$. Each barrier is characterized by a set of transmissions of its conducting channels $T_k^{\left(n\right)}$.

Figure 2

Single quantum dot connected to the leads via two barriers.

Figure 3

Most general system with two quantum dots.

Figure 4

The magnetoconductance of two dots of Fig. 3 for $d_1,d_2⪢l_e,$ $d_1∕d_2=5$, $g_{ij}=g_0$, ${\beta }_{ij}=0$, ${\beta }_y=0$, ${\tau }_{\phi 1}={\tau }_{\phi 2}=\infty$. Here $H_1=1∕4\sqrt{{\alpha }_1{\tau }_{D1}}$ is the field at which weak localization is effectively suppressed in the first dot. For $g_y=0$ the magnetoconductance is given by superposition of two Lorentzians with different widths (decoupled dots), while for large $g_y$ only one Lorentzian survives corresponding to the contribution of a single “composite dot.”

Figure 5

Two quantum dots in series.

Figure 6

The system of two connected quantum dots, only one of which is in turn connected to the leads.

Figure 7

Magnetoconductance of a 1D array of $N-1$ identical open $(\beta =0)$ quantum dots in the absence of interactions $({\tau }_{\phi }\to \infty )$. The field $H_N$ is defined in Eq. (58).

Figure 8

2D array of identical quantum dots. Here the number of barriers in the $x$ direction is chosen to be $N=4$, and the number of quantum dots in the $y$ direction is $M=2$. The barriers are characterized by the dimensionless conductances $g_x$ and $g_y$ and the Fano factor $\beta$.