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 nontrivial function of the applied stress σ and of the system size L, i.e., ν=νL(σ). We consider a generic two-dimensional membrane embedded into space of dimensionality 2+dc. (The physical situation corresponds to dc=1.) A particularly important application of our results is to freestanding graphene. We find that at a very low stress, when the membrane exhibits linear response, the PR νL(0) decreases with increasing system size L and saturates for L→∞ 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-consistent scaling analysis. By increasing σ, one drives a sufficiently large membrane (with the length L much larger than the Ginzburg length) into a nonlinear regime characterized by a universal value of PR that depends solely on dc, in close connection with the critical index η controlling the renormalization of bending rigidity. This universal nonlinear PR acquires its minimum value νmin=−1 in the limit dc→∞, when η→0. With the further increase of σ, the PR changes sign and finally saturates at a positive nonuniversal value prescribed by the conventional elasticity theory. We also show that one should distinguish between the absolute and differential PR (ν and νdiff, respectively). While coinciding in the limits of very low and very high stress, they differ in general: ν≠νdiff. In particular, in the nonlinear universal regime, νdiff takes a universal value which, similarly to the absolute PR, is a function solely of dc (or, equivalently, of η) but is different from the universal value of ν. In the limit of infinite dimensionality of the embedding space, dc→∞(i.e., η→0), the universal value of νdiff 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.