Tunable Luttinger Liquid Physics in Biased Bilayer Graphene

Electronically gated bilayer graphene behaves as a tunable gap semiconductor under a uniform interlayer bias $V_g$. Imposing a spatially varying bias, which changes polarity from $-V_g$ to $+V_g$, leads to one dimensional (1D) chiral modes localized along the domain wall of the bias. Because of the broad transverse spread of their low-energy wave functions, we find that the dominant interaction between these 1D electrons is the forward scattering part of the Coulomb repulsion. Incorporating these interactions and the gate voltage dependence of the dispersion and wave functions, we find that these 1D modes behave as a strongly interacting Tomonaga-Luttinger liquid with three distinct mode velocities and a bias dependent Luttinger parameter, and discuss its experimental signatures.

Figure 1

(a) Schematic diagram of external gates. $V_T+V_B$ controls the bilayer doping while $V_T-V_B$ controls the gap via the depicted perpendicular electric field. (b) Structure of bilayer graphene with a “bias kink” in $V_g=V_T-V_B$ at $y=0$.

Figure 2

Dispersion about the $\pm \mathbf{K}$ points (where $\mathbf{K}=4\pi /3$) with (a) $V_g=0.02t$ and (b) $V_g=0.08t$. Edge-mode bands are indicated by the labeled arrows and bulk states by the hatched region.

Figure 3

The modulus square of the zero-energy wave function of (a) the 0 band and (b) the $\pi$ band on the dominate sites $a_2$ and $b_1$ at $V_g=0.02t$ and $V_g=0.08t$ (independent of being a right or left mover).

Figure 4

Examples of various scattering processes.

Figure 5

Fermi velocity $V_F$ and mode velocities $u_{c\pm }$ in the charge sector (in units of $td$), and the Luttinger parameter of the total charge sector as a function of $V_g$. We assume a screening length of $1000d\approx 0.3\mu \mathrm{m}$ for the Coulomb interaction and a short distance cutoff of 0.5 d.