Stress-controlled Poisson ratio of a crystalline membrane: Application to graphene
We demonstrate that a key elastic parameter of a suspended crystalline membrane---the Poisson ratio (PR) $ν$---is a non-trivial function of the applied stress $σ$ and of the system size $L$, i.e., $ν=ν_L(σ)$. We consider a generic 2D membrane embedded into space of dimensionality $2+d_c$. (The physical situation corresponds to $d_c=1$.) A particularly important application of our results is free-standing graphene. We find that at very low stress, where the membrane exhibits a linear response, the PR $ν_L(0)$ decreases with increasing $L$ and saturates for $ L\to \infty$ at a value which depends on the boundary conditions and is essentially different from the value $ν=-1/3$ previously predicted by the membrane theory within a self-consisted scaling analysis. By increasing $σ$, one drives a membrane into a non-linear regime characterized by a universal value of PR that depends solely on $d_c.$ This universal non-linear PR acquires its minimum value $ν_{\rm min}=-1$ in the limit $d_c\to \infty.$ With the further increase of $σ$, the PR changes sign and finally saturates at a positive non-universal value prescribed by the conventional elasticity theory. We also show that one should distinguish between the absolute and differential PR ($ν$ and $ν^{\rm diff}$, respectively). While coinciding in the limits of very low and very high stresses, they differ in general, $ν\neq ν^{\rm diff}$. In particular, in the non-linear universal regime, $ν^{\rm diff}$ takes a universal value which, similarly to absolute PR, is a function solely of $d_c$ but is different from the universal value of $ν$. In the limit $d_c\to \infty$, the universal value of $ν^{\rm diff}$ tends to $-1/3$, at variance with the limiting value $-1$ of $ν$. Finally, we briefly discuss generalization of these results to a disordered membrane.