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% 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 graphene's unusual negative thermal expansion coefficient.

STM-driven transition from rippled to buckled graphene in a spin-membrane model

“…The correlation length ξ may also change discontinuously and, counterintuitively, the temperature decreases as the tunneling current increases [15]. The M versus h curve of this model is as follows [15]. At zero field, M = 0 and thus J = −1.…”

“…The spin-spin interaction is antiferromagnetic (J = −1) for 0 < M < M 0 and ferromagnetic (J = 2) for M > M 0 , M 0 is 60%-70% of the maximum possible value of the magnetization. The correlation length ξ may also change discontinuously and, counterintuitively, the temperature decreases as the tunneling current increases [15]. The M versus h curve of this model is as follows [15].…”

“…Thus, the out-of-plane displacements would make it possible to stabilize the graphene sample. The understanding of this rippling has triggered a great amount of theoretical work, both starting from first principles [3][4][5][6][7][8][9][10] and using simple statistical mechanics models [11][12][13][14][15].…”

“…Schoelz et al proposed a phenomenological Ising model to explain their experimental results [15]. In their model, each local spin σ ij represents one ripple composed of ∼ 1000 carbon atoms and the value of the spin indicates the curvature of the ripple.…”

“…To further mimic experimental results, a smaller jump in the magnetization for h 0 < 2.5 is induced by an increase in the correlation length ξ; see Figure 3 in ref. [15].…”