Absence of rippling in graphene under biaxial tensile strain

Recent experiments [C. H. Lui, L. Liu, K. F. Mak, G. W. Flynn, and T. F. Heinz, Nature (London) 462, 339 (2009)] on graphene grown on ultraflat substrates have found no rippling in graphene when subject to temperature cycling. Unsupported/unstrained films of graphene as well as films grown on various substrates on the other hand have been found to show rippling effects. As graphene grown on a substrate is invariably strained, we examine the behavior of the out-of-plane acoustic-phonon mode with biaxial tensile strain. This mode is generally associated with the rippling of graphene. We find that it can be fit to a relation of the form w(2) = Ak(4)+Bk-2, where w and k are the frequency and wave vector, respectively. The coefficient A is found to show a weak dependence on strain while B is found to increase linearly with strain. The strain-induced hardening explains the absence of rippling in graphene subject to biaxial strain. In addition, we find that graphene when subject to a biaxial tensile strain is found to undergo a structural transition with the mode at K going soft at a strain percentage of 15%.