Transport signatures of relativistic quantum scars in a graphene cavity

Wave function scars refer to localized complex patterns of enhanced wave function probability distributions in a quantum system. Existing experimental studies of wave function scars concentrate nearly exclusively on nonrelativistic quantum systems. Here we present a combined experimental and theoretical study of a relativistic quantum cavity system realized by etching a graphene sheet. The conductance of the graphene cavity has been measured as a function of the back gate voltage (or the Fermi energy) and the magnetic field applied perpendicularly to the graphene sheet, and characteristic conductance contour patterns are observed at low temperatures. In particular, two types of high-conductance contour lines, i.e., straight and paraboliclike high-conductance contour lines, are found in the measurements. The theoretical calculations are performed within the framework of the tight-binding approach and Green's function formalism. Characteristic high-conductance contour features similar to those in the experiments are found in the calculations. Specifically, the equally spaced, parallel, straight high-conductance contour lines signify the persistence of relativistic quantum scars. The wave functions calculated at points selected along such a straight conductance contour line are found to be dominated by a chain of scars of high-probability distributions arranged as a necklace following the shape of the cavity, and the current density distributions calculated at these point are dominated by an overall vortex in the cavity. These characteristics are found to be insensitive to increasing magnetic field. However, the wave function probability distributions and the current density distributions calculated at points selected along a paraboliclike contour line show a clear dependence on increasing magnetic field, and the current density distributions at these points are characterized by the complex formation of several localized vortices in the cavity. Our work brings insight into quantum chaos in relativistic particle systems and should greatly stimulate experimental and theoretical efforts in this still emerging field.

Figure 1

Graphene cavity structure. (a) False-colored atomic force microscope (AFM) image of the graphene device and schematics for the device structure and measurement setup. Yellow parts are electrodes. The red region is graphene. Green highlighted regions are graphene pieces isolated from the cavity by etched trenches, which could be used as side gates but are not used in this work. The device is made on a ${\mathrm{Si}/\mathrm{SiO}}_2$ substrate which is used as a back gate. $V_1$ and $V_2$ denote the voltage drops over the cavity and a graphene bulk region, respectively, and $V_H$ is the Hall voltage generated in the graphene bulk region. (b) Zoom-in AFM image of the cavity structure in (a). (c) Hall resistance $R_{xy}$ and longitudinal resistance $R_{xx}$ of the graphene bulk region measured at perpendicularly applied magnetic field $B=5\mathrm{T}$ and temperature $T=60\mathrm{mK}$. $V_{BG}$ is the applied back gate voltage, and $V_{\text{Drift}}$ is the drifting gate voltage of the Dirac point.

Figure 2

Low-temperature magnetotransport in the graphene cavity structure. (a) Longitudinal conductance of the graphene bulk region measured as a function of the magnetic field $B$ at different back gate voltages $V_{BG}$ and temperature $T=60\mathrm{mK}$. Curves are successively vertically offset by $1e^2/h$ for clarity. (b) Conductance of the graphene cavity measured as a function of $B$ at different back gate voltages $V_{BG}$ and temperature $T=60\mathrm{mK}$. Curves are successively vertically offset by $0.15e^2/h$. Black arrows denote the positions of conductance peaks which evolute with increasing back gate voltage. (c) Conductance map of the bulk graphene region, i.e., longitudinal conductance measured for the bulk graphene region as a function of $V_{BG}$ and $B$ at $T=60\mathrm{mK}$. (d) Conductance map of the graphene cavity at $T=60\mathrm{mK}$.

Figure 3

Characteristic conductance patterns. (a) Conductance map of the graphene cavity measured at temperature $T=60\mathrm{mK}$ over a large range of $V_{BG}$. (b) Simulated conductance map of the graphene cavity. Here, similar characteristic conductance patterns are found in the measurements and the calculations. The red arrows point to the direction towards the Dirac point.

Figure 4

Conductance patterns in region I near the Dirac point. (a) and (b) Zoom-in plot of the measurements shown in rectangular region I of Fig. 3 and zoom-in plot of the calculations in the corresponding region shown in Fig. 3. Straight (paraboliclike) high-conductance contour lines are highlighted with yellow (green) dashed lines. (c) and (d) Calculated wave function probability distributions and current density distributions at points $a_n$, where $n=1$, 2, 3, 4, and 5, selected along straight line A. Here, it is seen that the scar pattern does not show a significant change with increasing magnetic field, and the current density distribution in each panel shows only one clockwise current vortex. (e) and (f) Calculated wave function probability distributions and current density distributions at points $b_m$, where $m=1$, 2, 3, and 4, selected along straight contour line B. Characteristic features in the wave function probability distributions and the current density distributions similar to those in (c) and (d) are observed, except that the current vortex seen in each current density distribution is counterclockwise.

Figure 5

(a) and (b) Calculated wave function probability distributions and current density distributions at points $c_k$, where $k=1$, 2, 3, and 4, selected along paraboliclike contour line C in Fig. 4. Here, the scar distribution pattern shows change sensitive to change in magnetic field, and the current density distribution exhibits the formation of a complex structure consisting of several localized clockwise and counterclockwise current vortices.

Figure 6

Conductance patterns in region II. (a) and (b) Zoom-in plot of the measurements shown in rectangular region II in Fig. 3 and zoom-in plot of the calculations in the corresponding region shown in Fig. 3. Note that region II is far from the Dirac point compared with region I. Yellow dashed lines highlight straight high-conductance contour lines observable in the measurements and the calculations. (c) and (d) Calculated wave function probability distributions and current density distributions at points $d_l$, where $l=1$, 2, 3, and 4, selected along straight line D in (b). Characteristic features in the charge density distributions and the current density distributions similar to those in Fig. 4 are observed, except that a well-defined additional current path along the edge of the cavity is observable in each current density distribution panel.