Nonergodicity and microscopic symmetry breaking of the conductance fluctuations in disordered mesoscopic graphene

We show a dramatic deviation from ergodicity for the conductance fluctuations in graphene. In marked contrast to the ergodicity of dirty metals, fluctuations generated by varying magnetic field are shown to be much smaller than those obtained when sweeping Fermi energy. They also exhibit a strongly anisotropic response to the symmetry-breaking effects of a magnetic field, when applied perpendicular or parallel to the graphene plane. These results reveal a complex picture of quantum interference in graphene, whose description appears more challenging than for conventional mesoscopic systems.

Figure 1

(a) Optical micrograph of the monolayer-graphene device. Measurement probes are indicated and the graphene flake can be seen in the inset. (b), (c) Resistance versus back-gate voltage for the bilayer and monolayer devices, respectively, at several temperatures. The arrow in each plot identifies the Dirac point. (d) CF in the monolayer device at different temperatures. (e) Magneto-CF of the monolayer device at several gate voltages (12–54 V) at 4.2 K. Each curve has had a smooth background subtracted.

Figure 2

(a) Comparison of CF obtained by sweeping $B_{\perp }$ (upper curves, upper horizontal axis) and back-gate voltage (lower curves, lower horizontal axis) at a cryostat temperature of 0.04 K. Fixed gate-voltage and magnetic-field values are indicated on the plot. Colored arrows indicate gate voltages at which the magneto-conductance measurements were made. (b) Blue data points show $\delta g_{\mathrm{rms}}$ determined from gate-voltage induced CF at different $B_{\perp }$ (open symbols) and $B_{\parallel }$ (filled symbols). Red data points show $\delta g_{\mathrm{rms}}$ from the magnetoconductance at various gate voltages. To allow the dependence of $\delta g_{\mathrm{rms}}$ on $B_{\perp }$ to be clearly seen, we have slightly displaced the other two data sets by the indicated increments. Dotted lines indicate the range of $B_{\perp }$ for which $\delta g_{\mathrm{rms}}$ drops to $1/\sqrt{2}$ and 1/2 of its zero-field value. Cryostat temperature was 0.04 K.

Figure 3

(a) In this contour, conductance was measured as a function of gate voltage, following which CF were obtained by background subtraction. The evolution of these CF as $B_{\perp }$ was incremented from $-$1 to 6 T was then plotted in the contour. (b) Variation of resistance with gate voltage for three representative values of $B_{\perp }$ (indicated). (c) As in (a) except gate-voltage induced CF are plotted as a function of $B_{\parallel }$. (d) As in (b) except now for three values of $B_{\parallel }$. $V_D$ in (a) and (c) denotes the Dirac point. Cryostat temperature was 0.04 K in panels (a)–(d). Curves in (b) and (d) are shifted vertically by the offsets (in kΩ) indicated in the panels.