Conductance anomaly near the Lifshitz transition in strained bilayer graphene

Strain qualitatively changes the low-energy band structure of bilayer graphene, leading to the appearance of a pair of low-energy Dirac cones near each corner of the Brillouin zone, and a Lifshitz transition (a saddle point in the dispersion relation) at an energy proportional to the strain [Mucha-Kruczynski, Aleiner, and Fal'ko, Phys. Rev. B 84, 041404 (2011)]. Here, we show that in the vicinity of the Lifshitz transition, the conductance of a ballistic $n$-$p$ and $n$-$p$-$n$ junction exhibits an anomaly: a nonmonotonic temperature and chemical potential dependence, with the size depending on the crystallographic orientation of the principal axis of the strain tensor. This effect is characteristic for junctions between regions of different polarity ($n$-$p$ and $n$-$p$-$n$ junctions), while there is no anomaly in junctions between regions of the same polarity ($n$-$n$${}^{\prime }$ and $n$-$n$${}^{\prime }$-$n$ junctions).

Figure 1

Left: Top view of an unperturbed (top panel) and a strained (bottom panel) bilayer graphene (BLG) crystal. The top and bottom layers are shown in yellow and red, respectively. Strain modifies the intralayer nearest-neighbors coupling ${\gamma }_0$, as well as the interlayer coupling ${\gamma }_3$ between atoms at the center of the other layer's hexagons. Right: Electronic band structure in the vicinity of the Brillouin zone corners K and K${}^{\prime }$, with focus on the low-energy dispersion near the K point for unperturbed and strained BLG.

Figure 2

Schematic representation of a suspended BLG device with strain axis oriented along the $x$ direction (as defined in Fig. 1). The sketch illustrates the example of an $n$-$p$-$n$ configuration of such a device ($\mu <0$). In the highly doped contact regions, the Fermi level (dotted line) lies high up in the conduction band (yellow), where the dispersion is parabolic. In the central region, the Fermi level lies in the valence band (red), and is close to the charge neutrality point, where the dispersion is modified due to the two Dirac mini-cones and the saddle point associated with the Lifshitz transition. Shading indicates occupied states.

Figure 3

Left: isoenergetic lines at $\epsilon =-2$, $-5$, $-6$, and $-8$ meV for strained bilayer graphene, with $w=5$ meV. Center: transmission probability $\mathcal{T}(\epsilon ,\theta )$ across a single potential step ($n$-$p$ or $n$-$n$${}^{\prime }$ junction), from a highly doped region to a barely doped region, as a function of energy and incidence angle of incoming electrons. Right: linear response conductance of the junction as a function of chemical potential $\mu$ and temperature $T$. Results are shown for unstrained bilayer graphene (a), as well as uniaxially strained bilayer graphene for various orientations of the strain axis with respect to the crystallographic axis $x$ in Fig. 1: $\phi =0$ (b), $\phi =\pi /4$ (c), and $\phi =\pi /2$ (d).

Figure 4

Transmission coefficient and conductance of $n$-$p$-$n$ and $n$-$n$${}^{\prime }$-$n$ junctions with nonstrained bilayer graphene (a), as well as strained bilayer graphene with the uniaxial strain axis at an angle for $\phi =0$ (b), $\phi =\pi /4$ (c), and $\phi =\pi /2$ (d) from crystallographic axis $x$. Left: transmission probability $\mathcal{T}(\epsilon ,\theta )$ obtained in an exact calculation. Center: transmission probability obtained by averaging over fast oscillations after the contribution of evanescent waves is neglected. Right: linear response conductance as a function of chemical potential and temperature. All calculations are performed for experimentally accessible values $w=5$ meV, $V_0=50$ meV, and $L_x=1\mu$m.