Conductivity scaling and the effects of symmetry-breaking terms in bilayer graphene Hamiltonian

Abstract

We study the ballistic conductivity of bilayer graphene in the presence of symmetry-breaking terms in effective Hamiltonian for low-energy excitations, such as the trigonal-warping term ($\gamma_3$), the electron-hole symmetry breaking interlayer hopping ($\gamma_4$), and the staggered potential ($\delta_{AB}$). Earlier, it was shown that for $\gamma_3\neq{}0$, in the absence of remaining symmetry-breaking terms (i.e., $\gamma_4=\delta_{AB}=0$), the conductivity ($\sigma$) approaches the value of $3\sigma_0$ for the system size $L\rightarrow{}\infty$ (with $\sigma_0=8e^2/(\pi{}h)$ being the result in the absence of trigonal warping, $\gamma_3=0$). We demonstrate that $\gamma_4\neq{}0$ leads to the divergent conductivity if $\gamma_3\neq{}0$, or to the vanishing conductivity if $\gamma_3=0$. For realistic values of the tight-binding model parameters, $\gamma_3=0.3\,$eV, $\gamma_4=0.15\,$eV (and $\delta_{AB}=0$), the conductivity values are in the range of $\sigma/\sigma_0\approx{}4-5$ for $100\,$nm$\ <L<1\,\mu$m, in agreement with existing experimental results. The staggered potential ($\delta_{AB}\neq{}0$) suppresses zero-temperature transport, leading to $\sigma\rightarrow{}0$ for $L\rightarrow{}\infty$. Although $\sigma=\sigma(L)$ is no longer universal, the Fano factor approaches the pseudodiffusive value ($F\rightarrow{}1/3$ for $L\rightarrow{}\infty$) in any case with non-vanishing $\sigma$ (otherwise, $F\rightarrow{}1$) signaling the transport is ruled by evanescent waves. Temperature effects are briefly discussed in terms of a phenomenological model for staggered potential $\delta_{AB}=\delta_{AB}(T)$ showing that, for $0<T\leqslant{}T_c\approx{}12\,$K and $\delta_{AB}(0)=1.5\,$meV, $\sigma(L)$ is noticeably affected by $T$ for $L\gtrsim{}100\,$nm.