Conductance fluctuations in chaotic bilayer graphene quantum dots

Previous studies of quantum chaotic scattering established a connection between classical dynamics and quantum transport properties: Integrable or mixed classical dynamics can lead to sharp conductance fluctuations but chaos is capable of smoothing out the conductance variations. Relativistic quantum transport through single-layer graphene systems, for which the quasiparticles are massless Dirac fermions, exhibits, due to scarring, this classical-quantum correspondence, but sharp conductance fluctuations persist to a certain extent even when the classical system is fully chaotic. There is an open issue regarding the effect of finite mass on relativistic quantum transport. To address this issue, we study quantum transport in chaotic bilayer graphene quantum dots for which the quasiparticles have a finite mass. An interesting phenomenon is that, when traveling along the classical ballistic orbit, the quasiparticle tends to hop back and forth between the two layers, exhibiting a Zitterbewegung-like effect. We find signatures of abrupt conductance variations, indicating that the mass has little effect on relativistic quantum transport. In solid-state electronic devices based on Dirac materials, sharp conductance fluctuations are thus expected, regardless of whether the quasiparticle is massless or massive and whether there is chaos in the classical limit.

Figure 1

Transmission versus energy for the cosine billiard quantum dot in both relativistic and nonrelativistic regimes: (a) bilayer graphene (BGQD); (b) 2DEG quantum dot (NRQD). The blue, green, and red curves from top to bottom show the mixed, intermediate, and chaotic cases: $W/L=0.36$ and $M/L=0.22,W/L=0.27$ and $M/L=0.165$, and $W/L=0.18$ and $M/L=0.11$, respectively. The circles in (a) indicate the energy values used in Fig. 4. For all cases, we use $N_{\mathrm{mode}}=96$ for BGQDs and $N_{\mathrm{mode}}=48$ for 2DEG quantum dots.

Figure 2

Resonance width profile ${\gamma }_{\alpha }$ for relativistic quantum dots and their nonrelativistic quantum counterparts for different types of classical dynamics: (a) BGQD, mixed (b) BGQD, intermediate (c) BGQD, chaotic, (d) NRQD, mixed, (e) NRQD, intermediate, and (f) NRQD, chaotic. The energy values at which the self-energy is evaluated are $E_0=0.2t$ for (a)–(c) and $E_0=t$ for (d)–(f).

Figure 3

The mean value of ${log}_{10}({\gamma }_{\alpha }/t)$ in a moving window with the window size $0.04t$ and the step size $0.004t$. The upper line in different panels in Fig. 2 are excluded in the average as it accounts for the smooth variation rather than sharp conductance resonances. Panel (a) is for BGQD (left panels in Fig. 2) and panel (b) is for NRQD (right panels in Fig. 2). The blue circles, green asterisks, and red plus symbols show the mixed, intermediate, and chaotic cases, respectively.

Figure 4

Representative local density of states (LDS, the first row) and the corresponding $x$ component of the electron flow (the second row) for BGQDs with mixed (a), intermediate (b), and chaotic (c) classical dynamics. For the LDS patterns, the dark (red) region denotes higher values, and the color scale is normalized for each panel for better visualization. The left, middle, and right panels are for the first layer, the second layer, and combined layers, respectively. The Fermi energies are $0.3137t$ for (a), $0.4187t$ for (b), and $0.3784t$ for (c), with the respective maximum and minimum values of the LDS patterns as $\left(3.0016,1.2\times {10}^{-3}\right),\left(0.4987,7.7\times {10}^{-3}\right)$, and $\left(0.4823,5.6\times {10}^{-3}\right)$.

Figure 5

Ratio of the maximum to the minimum values of the LDS versus energy: (a) BGQD and (b) NRQD. From top to bottom the blue circles, green asterisks, and red plus symbols show the mixed, intermediate, and chaotic cases, respectively.