Thermal expansion in multiple layers of graphene

In this file, we apply the nonequilibrium Green’s function method to calculate the coefficient of thermal expansion (CTE) in multiple layers of graphene. We focus on the effect from different layer number $N$. Our main prediction is that: the increase of $N$ can either enhance or weaken CTE, depending on the strength of the substrate interaction $\gamma$. If $\gamma <ϵ$, where $ϵ$ is the inter-layer interaction, the CTE will increase with increasing $N$. Otherwise, if $\gamma >ϵ$, CTE will decrease with increasing $N$.