Dimensional Crossover in Quantum Networks: From Macroscopic to Mesoscopic Physics

We report on magnetoconductance measurements of metallic networks of various sizes ranging from 10 to ${10}^6$ plaquettes, with an anisotropic aspect ratio. Both Altshuler-Aronov-Spivak $h/2e$ periodic oscillations and Aharonov-Bohm $h/e$ periodic oscillations are observed for all networks. For large samples, the amplitude of both oscillations results from the incoherent superposition of contributions of phase coherent regions. When the transverse size becomes smaller than the phase coherent length $L_{\varphi }$, one enters a new regime which is phase coherent (mesoscopic) along one direction and macroscopic along the other, leading to a new size dependence of the quantum oscillations.

Figure 1

Scanning electron micrograph of several samples of various sizes; the two contacts are visible for the small sample.

Figure 2

Magnetoresistance of a square network containing 3000 plaquettes: (a) low field data; (b) high field data; (c),(d) Fourier (FFT) amplitudes of (a),(b), respectively. Data are taken at 400 mK.

Figure 3

AAS amplitude $\Delta g_{\mathrm{AAS}}$ and AB amplitude $\delta g_{\mathrm{AB}}$ as a function of the number $N$ of plaquettes for different networks of various sizes $L_x\times L_y\propto N$, for two topologies, square (squares) and ${\mathcal{T}}_3$ (diamonds) [17].

Figure 4

$\Delta g_{\mathrm{AAS}}/(\delta g_{\mathrm{AB}}{)}^2$ as a function of the number of plaquettes $N$ for square networks (squares) and diamonds networks (diamonds).