Trigonal warping, pseudodiffusive transport, and finite-system version of the Lifshitz transition in magnetoconductance of bilayer graphene Corbino disks

Using the transfer matrix in the angular-momentum space we investigate the impact of trigonal warping on magnetotransport and scaling properties of a ballistic bilayer graphene in the Corbino geometry. Although the conductivity at the charge-neutrality point and zero magnetic field exhibits a one-parameter scaling, the shot-noise characteristics, quantified by the Fano factor $\mathcal{F}$ and the third charge-transfer cumulant $\mathcal{R}$, remain pseudodiffusive. This shows that the pseudodiffusive transport regime in bilayer graphene is not related to the universal value of the conductivity but can be identified by higher charge-transfer cumulants. For Corbino disks with larger radii ratios, the conductivity is suppressed by the trigonal warping, mainly because the symmetry reduction amplifies backscattering for normal modes corresponding to angular-momentum eigenvalues $\pm 2\hslash$. Weak magnetic fields enhance the conductivity, reaching the maximal value near the crossover field $B_L=\frac{4}{3}\sqrt{3}(\hslash /e)t^{\prime }{t}_{\perp }{\left[t_0^{2}a(R_{\mathrm{o}}-R_{\mathrm{i}})\right]}^{-1}$, where $t_0$ ($t_{\perp }$) is the nearest-neighbor intralayer (interlayer) hopping integral, $t^{\prime }$ is the skew-interlayer hopping integral, and $R_{\mathrm{o}}$ ($R_{\mathrm{i}}$) is the outer (inner) disk radius. For magnetic fields $B\gtrsim B_L$ we observe quasiperiodic conductance oscillations characterized by the decreasing mean value $\langle \sigma \rangle -{\sigma }_0\propto B_L/B$, where ${\sigma }_0=(8/\pi )e^2/h$. The conductivity, as well as higher charge-transfer cumulants, show beating patterns with an envelope period proportional to $\sqrt{B/B_L}$. This constitutes a qualitative difference between the high-field ($B\gg B_L$) magnetotransport in the $t^{\prime }=0$ case [earlier discussed in Rut and Rycerz, J. Phys.: Condens. Matter 26, 485301 (2014)] and in the $t^{\prime }\ne 0$ case, providing a finite-system analog of the Lifshitz transition.

Figure 1

Systems discussed in the paper (schematic). (a) The Corbino disk in Bernal-stacked bilayer graphene. The current flows through the disk-shaped area with the inner radius $R_{\mathrm{i}}$ and the outer radius $R_{\mathrm{o}}$ in a perpendicular magnetic field $\mathbf{B}=(0,0,B)$. The coordinate system and the electrostatic potential $U\left(r\right)$ (with $r=\sqrt{x^2+y^2}$) are also shown. The leads (white areas) are modeled as infinitely doped graphene regions ($|U_{\infty }|\to \infty$). (b) An artificial (nanotubelike) system formed of a BLG strip of the width $W$, contacted by two electrodes at a distance $L$ in uniform field $B$, upon applying periodic boundary conditions.

Figure 2

Minimal conductivity of unbiased BLG Corbino disk with the radii ratio $R_{\mathrm{o}}/R_{\mathrm{i}}=2$ as a function of the radii difference $R_{\mathrm{o}}-R_{\mathrm{i}}$, specified in the units of $\ell$ [see Eqs. (17) and (18) in the main text], for different values of $t^{\prime }$. Triangles, squares, and circles represent the data obtained numerically for $t^{\prime }=0.3, 0.2$, and $0.1\mathrm{eV}$ (respectively), with the lines (dotted-dashed, solid, and dotted) depicting the approximating Eq. (15) with best-fitted parameters listed in Table 1. The inset in the top panel presents the scaling function $\beta \left(\sigma \right)$ [see Eq. (11)], with $L\equiv R_{\mathrm{o}}-R_{\mathrm{i}}$, extracted numerically from $\sigma \left(R_i\right)$ data sets. (Notice that the best-fitted lines almost overlap each other.) The bottom panel is a zoom-in, for smaller radii differences, which allows to depict the region where the actual value of the conductivity may deviate from Eq. (15). We further notice that only selected data points from the bottom panel are shown in the top panel for clarity.

Figure 3

Same as Fig. 2 but for the radii ratio $R_{\mathrm{o}}/R_{\mathrm{i}}=1.5$. For the parameters in Eq. (15) corresponding to the lines depicted, see Table 2.

Figure 4

The Fano factor $\mathcal{F}$ (red solid lines) and the third charge-transfer cumulant $\mathcal{R}$ (blue dashed lines) for the Corbino disk with $R_{\mathrm{o}}/R_{\mathrm{i}}=2$ as functions of $R_{\mathrm{o}}-R_{\mathrm{i}}$. Skew-interlayer hopping $t^{\prime }$ is varied between the panels. For large system size, where the Fabry-Perot oscillations become negligible, the numerical results obtained in the presence of trigonal warping approach the pseudodiffusive values of $\mathcal{F}=\frac{1}{3}$ and $\mathcal{R}=\frac{1}{15}$ (horizontal lines).

Figure 5

Same as Fig. 4 but for $R_{\mathrm{o}}/R_{\mathrm{i}}=1.5$.

Figure 6

Radii-ratio dependence of the conductance $G=2\pi \sigma /ln(R_{\mathrm{o}}/R_{\mathrm{i}})$ (top panel) and the effective number of transmission channels (bottom panel) at a fixed $R_{\mathrm{i}}=15{ł}_{\perp }\approx 24\mathrm{nm}$ and varying $t^{\prime }$ (specified for each line). Vertical lines mark the values of $\mathcal{A}\approx 3$ (left), $\mathcal{A}\approx \frac{5}{2}$ (middle), and $\mathcal{A}\approx 2$ (right) following from Eq. (25). The corresponding results for two decoupled MLG disks ($t_{\perp }=t^{\prime }=0$) are also shown.

Figure 7

Conductivity of the rectangular BLG sample at the charge-neutrality point as a function of the magnetic field $B$. The sample length is fixed at $L=300l_{\perp }\simeq 480\mathrm{nm}$, the width is $W=10L$, and the skew-interlayer hopping integral $t^{\prime }$ is varied between the panels. Red solid lines: exact numerical results obtained from the mode-matching analysis via Eq. (12). Blue solid lines: results from Eqs. (35) and (36) for $L\gg l_B$. Dashed lines: results from Eq. (41) for $W\gg L\gg l_B$.

Figure 8

Magnetoconductance oscillations for the rectangular BLG sample of the length $L=45l_{\perp }\simeq 72\mathrm{nm}$ and the width $W=2L$ at the charge-neutrality point. Top panel: the conductivity as a function of magnetic field for three values of the skew-interlayer hopping integral $t^{\prime }$ (specified for each line). The lines for $t^{\prime }=0.3$ and $0.2\mathrm{eV}$ are shifted by 1 and $\frac{1}{2}$ for clarity. Bottom panel: consecutive periods of the beating envelope extracted from the data shown in the top panel, for $t^{\prime }=0.1\mathrm{eV}$ (circles), $t^{\prime }=0.2\mathrm{eV}$ (squares), and $t^{\prime }=0.3\mathrm{eV}$ (diamonds). Solid lines correspond to the approximating Eq. (44).

Figure 9

Conductivity $\sigma$ ($$specified in the units of $8e^2/h$), the Fano factor $\mathcal{F}$, and the $\mathcal{R}$ factor for BLG Corbino disk with $R_{\mathrm{i}}=300l_{\perp }, R_{\mathrm{o}}=2R_{\mathrm{i}}$, and different skew-interlayer hopping integrals $t^{\prime }$ (specified for each panel). Vertical lines correspond to $B=B_L$ [see Eq. (45)], the horizontal lines mark the pseudodiffusive values $\mathcal{F}=\frac{1}{3}$ and $\mathcal{R}=\frac{1}{15}$, the dashed lines represent the approximating Eq. (46).

Figure 10

Charge-transfer characteristics $\sigma , \mathcal{F}$, and $\mathcal{R}$ for BLG disk with $R_{\mathrm{i}}=15l_{\perp }, R_{\mathrm{o}}=4.84R_{\mathrm{i}}$. Each of the main panels compares the numerical results for $t^{\prime }=0$ (blue line) and $t^{\prime }=0.3\mathrm{eV}$ (red line). The vertical gray line in the top panel marks the value of $B_L$ obtained from Eq. (45) for $t^{\prime }=0.3\mathrm{eV}$. The inset: period of the beating envelope for three different values of $t^{\prime }=0.3\mathrm{eV}$ (circles), $0.2\mathrm{eV}$ (orange squares), $0.1\mathrm{eV}$ (triangles). Solid lines correspond to Eq. (47).