Graphene Ripples as a Realization of a Two-Dimensional Ising Model: A Scanning Tunneling Microscope Study

Abstract

Ripples in pristine freestanding graphene naturally orient themselves in an array that is alternately curved-up and curved-down; maintaining an average height of zero. Using scanning tunneling microscopy (STM) to apply a local force, the graphene sheet will reversibly rise and fall in height until the height reaches 60-70 percent of its maximum at which point a sudden, permanent jump occurs. We successfully model the ripples as a spin-half Ising magnetic system, where the height of the graphene is the spin. The permanent jump in height, controlled by the tunneling current, is found to be equivalent to an antiferromagnetic-to-ferromagnetic phase transition. The thermal load underneath the STM tip alters the local tension and is identified as the responsible mechanism for the phase transition. Four universal critical exponents are measured from our STM data, and the model provides insight into the statistical role of graphenes unusual negative thermal expansion coefficient.

12 pages, 5 figures, 1 table

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Cited by in corpus (5)

• Anomalous Dynamical Behavior of Freestanding Graphene Membranes
• Thermal control of nucleation and propagation transition stresses in discrete lattices with non-local interactions and non-convex energy
• Bifurcation analysis and phase diagram of a spin-string model with buckled states
• STM driven transition from rippled to buckled graphene in a spin-membrane model
• Buckling in a rotationally invariant spin-elastic model