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According to Peierls and Landau, 2D crystals were thermodynamically unstable. They can't exist! Of course, this theory was disapproved in 2004 (example: graphene).

What is the general definition of stability of a general system?

What is the thermodynamics' stability?

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Well, when we talk of stability of systems, at least for equilibrium systems, we require the free energy to be bounded below and be convex. As the free energy is obtained by a Legendre transformation (which preserves convexity), the energy functional is required to be convex. This essentially allows us to minimize energies to find ground states. Within the realm of regular equilibrium thermodynamics, higher order quantities (depending on fluctuations) like the heat capacity or susceptibility/compressibility are put in by hand to be finite quantities.

Now, coming to your point about stability of ordered states, it is precisely these fluctuations (that in thermodynamics you assume are finite, but in statistical mechanics, are calculable) that diverge for arbitrarily large systems for some dimensions and/or at some critical points corresponding to phase transitions. So, the so called argument against the stability of 2D crystals with short ranged interactions is the Mermin-Wagner theorem, which basically shows that in 2 dimensions, fluctuations about the ordered state (in this case the lattice structure) de-correlate over large distances, thereby destroying any large scale order in the system (again only in the thermodynamic limit of large system sizes).
Coming to the specific case of graphene as a stable 2D crystal, the loophole it exploits to "violate" the Mermin-Wagner theorem is rather subtle. Mind you that, just because it fluctuates as a sheet embedded in a higher dimensional space does not make it a non-2D structure. As an aside, fluid membranes that can self-intersect (this means that all the interactions are only local) are equivalent 2D sheets embedded in a higher dimensional space, and they do not have a flat phase (ordering of normals) and are always crumpled. So just by allowing fluctuations into another dimension is not enough to have ordering in graphene.

What instead happens is that the lattice structure of graphene is fixed, so it corresponds to what is called a tethered elastic membrane (which unlike the fluid membrane can stretch.) It is this in plane stretching that allows for phonon mode to propagate and couple in-plane degrees of freedom to transverse fluctuations out of the plane. This effectively mediates a long range interaction, thereby circumventing Mermin-Wagner. Another way to see this technically, is that the bending rigidity of the sheet gets renormalized such that it diverges at large length scales, effectively making the sheet stiffer on larger scales. So in short, the thermal undulations are essential for the stability of graphene as a 2D crystal.

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  • $\begingroup$ O, it's still complicated for me. I don't know any thing in statistical mechanics. Thank you. $\endgroup$ – aayyachi Sep 27 '14 at 19:13
  • $\begingroup$ Well, that is going to be hard then. The first part should be alright as it only deals with thermodynamics. With regard to the latter half on graphene, I just wanted to point out that the reason for existence of graphene as a 2D stable crystal is a subtle and non-trivial argument. $\endgroup$ – surajshankar Sep 29 '14 at 16:03
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Well, Landau's statements were not as definitive as you appear to think. His views are summarized in Statistical Physics (Landau and Lifshitz). From my copy of the 3rd edition, part one, they are found in sections 137 and 138. The discussion is on thermal fluctuations as a function of temperature and size of the 2D film. The following quotes will get you started. I've found that when I disagree with Landau, I'm always wrong.

"The result obtained, strictly speaking, means only that the fluctuational displacement becomes infinite when the size (are) of the two-dimensional system increases without limit (so that the wave number may be arbitrarily small). But, because of the slow (logarithmic) divergence of the integral, the size of the film for which the fluctuations are still small may be very great." (section 137)

"Let us note first of all that, when T=0, a two-dimensional lattice of any size could exist..." (section 138).

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  • $\begingroup$ If you could give some explanation, that would be helpful. $\endgroup$ – aayyachi Apr 30 '14 at 8:00
  • $\begingroup$ On actual stability: as T increases, so does the vibrational energy in the system. Some of that energy is in the out-of-plane direction. At some point the restraining force is overcome and atoms go flying off or the 2D structure buckles and isn't 2D anymore. $\endgroup$ – Jon Custer Apr 30 '14 at 18:52
  • $\begingroup$ Next, stable compared with what? Diamonds are forever, but are not the stable phase of carbon at human conditions. Similarly, graphene is not thermodynamically stable relative to graphite. If it were, than at some point (ignoring kinetics) graphite would float apart into a bazillion layers of graphene. Since kinetics keep diamonds around at surface conditions one can't rule them out completely, but Landau's arguments are a good starting place. $\endgroup$ – Jon Custer Apr 30 '14 at 18:59

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