Historical research guide · 0835 · 2015
Graphene ripples as a realization of a two-dimensional Ising model: A scanning tunneling microscope study
J. K. Schoelz · P. Xu · V. Meunier · P. Kumar · M. Neek-Amal · P. M. Thibado · F. M. Peeters
In one minute
A sheet of graphene one atom thick cannot lie flat. It holds itself up with ripples that curve alternately up and down and average out to no height at all. Paul Thibado’s group at Arkansas, with collaborators at Maryland, Rensselaer and Antwerp, put a scanning tunnelling microscope tip over a freestanding sheet and pulled on it. The sheet rose smoothly and reversibly by tens of nanometres, and then, once it had reached 60 to 70 per cent of its full travel, jumped about thirty nanometres more and stayed there. The team models each ripple as one spin in a two-dimensional Ising magnet — curved up or curved down — and the jump becomes an antiferromagnetic-to-ferromagnetic phase transition. Four of the six universal critical exponents come straight out of the measured data and land on the two-dimensional Ising values. The strangest part is that the transition happens on heating, the reverse of an ordinary magnet, because graphene contracts when warmed and that added tension rewrites the coupling between neighbouring ripples.
Connection
Chapter 6 asks what can be drawn out of the ambient fluctuation field, and this is the paper that turns graphene’s ripples from mere roughness into a countable, controllable two-state system with a measured phase transition. Chapter 2 gains a room-temperature laboratory system whose spontaneous motion is described by exact statistical mechanics rather than by analogy.
From the source
Ripples in pristine freestanding graphene orient themselves naturally in an array that is alternately curved up and curved down, maintaining an average height of zero. Applying a local force with a scanning tunnelling microscope tip makes the sheet rise and fall reversibly in height until it reaches 60 to 70 per cent of its maximum, at which point a sudden permanent jump occurs.Abstract; Sect. III, Experimental Results, Fig. 1
The ripples are successfully modelled as a spin-half Ising system in which the height, or curvature sign, of the graphene is the spin. One spin represents an entire ripple about 10 nm across containing roughly a thousand carbon atoms, and the coupling energy between neighbouring ripples is made a function of the total magnetisation. The permanent jump in height, controlled by the tunnelling current, is equivalent to an antiferromagnetic-to-ferromagnetic phase transition.Abstract; Sect. IV, Eqs. (1) and (2); Sect. V, Fig. 3(a)
Four of the six static two-dimensional universal critical exponents were measured from the STM data — the pair correlation exponent near one quarter from the decay of the height-height autocorrelation function, the critical isotherm exponent near fifteen, the spontaneous polarisation exponent near one eighth and the susceptibility exponent near seven quarters — and all fall within the two-dimensional Ising universality class.Sect. VI, Discussion; Table 1; Fig. 5(a) to 5(c)
The transition from the flexible state to the rigid state occurs with increasing tunnelling current, that is on heating the sample, which is opposite to ordinary two-dimensional Ising magnet behaviour. The authors attribute this to graphene’s negative thermal expansion: heating raises the internal tension, which changes the coupling between nearest-neighbour ripples and shifts the lowest-energy configuration from the alternating state to a single bulged-out state.Sect. VI, penultimate paragraph; Sect. VII, Summary
The thermal load underneath the tip alters the local tension and is identified as the mechanism responsible for the transition. Across many data sets and samples the current and voltage pairs at which the sheet switches follow an inverse-square-of-current trend, mirroring the critical field versus temperature curve of some ferromagnetic systems.Abstract; Sect. VI, Fig. 4
The result supplies a two-dimensional Ising framework for graphene’s ripples in which a ripple’s curvature is a genuine two-state variable that a modest local field and a little heat can flip, and the authors present it as a new statistical footing for understanding the role those ripples play.Sect. I, final paragraph; Sect. VII, Summary
The way in
https://doi.org/10.1103/physrevb.91.045413How to cite it
J. K. Schoelz, P. Xu, V. Meunier, P. Kumar, M. Neek-Amal, P. M. Thibado, F. M. Peeters (2015) Graphene ripples as a realization of a two-dimensional Ising model: A scanning tunneling microscope study doi:10.1103/physrevb.91.045413