Multifractal magnetoconductance fluctuations in mesoscopic systems

We perform a multifractal detrended fluctuation analysis of the magnetoconductance data of two standard types of mesoscopic systems: a disordered nanowire and a ballistic chaotic billiard, with two different lattice structures. We observe in all cases that multifractality is generally present and that it becomes stronger in the quantum regime of conduction, i.e., when the number of open scattering channels is small. We argue that this behavior originates from correlations induced by the magnetic field, which can be characterized through the distribution of conductance increments in the corresponding “stochastic time series,” with the magnetic field playing the role of a fictitious time. More specifically, we show that the distributions of conductance increments are well fitted by $q$ Gaussians and that the value of the parameter $q$ is a useful quantitative measure of multifractality in magnetoconductance fluctuations.

Figure 1

Schematics of disordered graphene nanowires connected to leads (red) with two boundary configurations: (a) armchair and (b) zigzag, and of (c) a graphene chaotic billiard.

Figure 2

Single realizations of the magnetic field-induced conductance fluctuations of graphene disordered wires with armchair boundaries with (a) $N=1$, (b) $N=2$, (c) $N=5$, (d) $N=10$ propagating channels are shown on the left panel and with zigzag boundaries with (e) $N=1$, (f) $N=3$, (g) $N=5$, (h) $N=11$ propagating channels are shown on the right panel. The conductance is given in units of $2e^2/h$ and the magnetic flux in units of $B{a}^2/(h/e)$.

Figure 3

Left panel shows the mean generalized Hurst exponent $H\left(p\right)$ of the conductance fluctuations of (a) graphene disordered wires with armchair boundaries with $N=1,2,5,10$ propagating channels, (c) graphene disordered wires with zigzag boundaries with $N=1,3,5,11$ propagating channels, and (e) two-dimensional electron gases with a square lattice with $N=1,2,5,10$ propagating channels. The right panel shows the multifractal singularity spectrum of the conductance fluctuations of (b) graphene disordered wires with armchair boundaries with $N=1,2,5,10$ propagating channels, (d) graphene disordered wires with zigzag boundaries with $N=1,3,5,11$ propagating channels, and (f) two-dimensional electron gases with a square lattice with $N=1,2,5,10$ propagating channels.

Figure 4

Single realizations of the magnetic field-induced conductance fluctuations of graphene chaotic billiards with (a) $N=1$, (b) $N=2$, (c) $N=5$, (d) $N=10$ propagating channels. The conductance is given in units of $2e^2/h$ and the magnetic flux in units of $B{a}^2/(h/e)$.

Figure 5

Left panel shows the mean generalized Hurst exponent $H\left(p\right)$ of conductance fluctuations of (a) graphene chaotic billiards with $N=1,2,5,10$ propagating channels and (c) chaotic billiards with a square lattice with $N=1,2,5,10$ propagating channels. The right panel shows the multifractal singularity spectrum of conductance fluctuations of (b) graphene chaotic billiards with $N=1,2,5,10$ propagating channels and (d) chaotic billiards with a square lattice with $N=1,2,5,10$ propagating channels.

Figure 6

NCI histograms of MCF data of graphene chaotic billiards divided by the flux step $\mathrm{\Delta }\varphi$ with $N=1$ (blue squares), $N=2$ (red squares), $N=5$ (green squares), $N=10$ (maroon squares) propagating channels, which were obtained from the same series we have used to perform the multifractal analysis in Sec. 3. The solid lines show the best fits with a $q$-Gaussian function. Notice that the peak of the distribution becomes narrower as $N$ decreases. The parameter values of the $q$-Gaussian functions which best fit the points are $q=2.09\pm 0.05,2.09\pm 0.06,1.73\pm 0.05,1.34\pm 0.07, \beta =2375\pm 335,4022\pm 643,952\pm 79,364\pm 31$ and $x_0=0.011\pm 0.001,0.009\pm 0.001,0.013\pm 0.001,0.016\pm 0.001$, for $N=1,2,5,10$, respectively. The curves have been arbitrarily shifted in the vertical direction for clarity.

Figure 7

Dependence of (a) the width of the multifractal singularity spectrum $f\left(\alpha \right)$ with $N$ and of (b) the value of $q$ that best fits the data of the distribution of conductance increments divided by $\mathrm{\Delta }\varphi$. The black squares, blue circles, red diamonds, green up triangles, and violet down triangles represent the results for two-dimensional chaotic billiards with a square lattice (2DCB), two-dimensional electron gases with a square lattice (2DEG), graphene chaotic billiards (GCB), armchair graphene nanoribbons (AGNR), and zigzag graphene nanoribbons (ZGNR), respectively.