Fractal conductance fluctuations of classical origin

In mesoscopic systems, conductance fluctuations are a sensitive probe of electron dynamics and chaotic phenomena. We show that the conductance of a purely classical chaotic system, with either fully chaotic or mixed phase space, generically exhibits fractal conductance fluctuations unrelated to quantum interference. This might explain the unexpected dependence of the fractal dimension of the conductance curves on the (quantum) phase breaking length observed in experiments on semiconductor quantum dots.

Figure 1

Classical conductance $g\left(B\right)$ through a stadium (left, geometry as in Ref. 5) and a square billiard (right, geometry as in Ref. 11) versus magnetic field $B$. Both fluctuating conductance curves are fractals (see insets and text). Their dimensions are $D\approx 1.28$ for the stadium and $D\approx 1.25$ for the square billiard. The fractal dimensions are in good agreement with experimental measurements [5, 11].

Figure 2

(Color) How lobes translate into fluctuations. In the bottom row, the entry set of the standard map with absorbing boundary conditions at $\pm 3\pi$ for $K=7.5$ and 7.6, respectively, can be seen. The three pictures in the center row show the magnification of the central sections of the entry set for three different values of $K=7.5$, 7.55, and 7.6. The transmission $T\left(K\right)$ for $K=7.5–7.6$ [20] is shown in the top left picture. Note that a small change in $K$ shifts the lobes vertically, but conserves the overall phase space structure, and that the largest fluctuations are caused by intersection with the apex of lobes. Starting from $K=7.5$, a large transmission lobe is cut by the horizontal line (see text), i.e., the transmission increases with $K$. In the same way, e.g., the fluctuations of $T\left(K\right)$ near $K=7.55$ can be understood. The box-counting analysis reveals a fractal structure (top right).

Figure 3

(Color) Transmission $T\left(X\right)$ for lobes (red upper curve, shifted along the $y$ axis for clarity) and stripes (black lower curve) for one and the same random distribution with $\alpha =1.9$, $\beta =0.5$. The inset shows the box-counting analysis for the upper (red triangles) and lower (black squares) transmission curves. The regression line is drawn for the upper curve, whose fractal dimension is 1.41.

Figure 4

(Color) (A) Schematic transmission according to Fig. 3 (bottom), covered with boxes of size $s$. There are three contributions marked (a)–(c). (B) Total number $N_{\mathit{int}}\left(w\right)={\int }_w^{\infty }n\left(w^{\prime }\right)d{w}^{\prime }$ of lobes (for the open standard map with $\mid p\mid <4\pi$) of width larger than $w$ on a double-logarithmic scale. The four curves show estimates for increasing resolution $w_{\min }={10}^{-5}\left(\text{pink}\right)–{10}^{-8}\left(\text{black}\right)$. The curves clearly approach a power law corresponding to $n\left(w\right)\propto w^{-1.9}$. The insets show the transmission curve $T\left(K\right)$ for values $K=8.0–8.1$ calculated from $2\times {10}^{13}$ trajectories, and its box-counting dimension.