I wanted to calculate the change in distance "d" between the two layers in bilayer graphene for a given pressure "p" perpendicular to the surface. For continuous materials the relation between stress tensor \sigma and straintensor \epsilon is described by a compliance tensor s, so that \epsilon = - s \sigma. Maybe one can use the compliance tensor for graphite, and then use \epsilon_zz, but I doubt it works on the atomic level. On the atomic level one probably has to determine the reaction of bilayer graphene to pressure from Raman spectroscopy. However, I could not find such studies yet, where the perpendicular pressure has been related to the layer distance (i.e. strain). Has anyone got a reference or some clarifications how to estimate d from p? Thank you all!
A simple estimate is to just look at a calculated binding energy curve for the layers in graphite. The authoritative reference there is Lebegue et al., Phys. Rev. Lett. 105, 196401 (2010), where they used the direct random-phase approximation, which is the best method currently available (or at least feasible to use) for this problem. The force you just get as dE/dd (not so good notation there, but I assume that you know how to get a force from an energy as function of a spatial variable...) normalize per area. They give the lattice constant in the paper. This works assuming that the in-plane lattice constant does not change, which should work for not-so-large pressures at least.
I am not aware of any RPA calculation that gives the binding curve for bilayer graphene, but there is probably tons of data using less sophisticated methods that don't work properly in van der Waals systems, if you feel adventurous you can try them out. Otherwise it is easily doable by other DFT-based methods.
A simple estimate is to just look at a calculated binding energy curve for the layers in graphite. The authoritative reference there is Lebegue et al., Phys. Rev. Lett. 105, 196401 (2010), where they used the direct random-phase approximation, which is the best method currently available (or at least feasible to use) for this problem. The force you just get as dE/dd (not so good notation there, but I assume that you know how to get a force from an energy as function of a spatial variable...) normalize per area. They give the lattice constant in the paper. This works assuming that the in-plane lattice constant does not change, which should work for not-so-large pressures at least.
I am not aware of any RPA calculation that gives the binding curve for bilayer graphene, but there is probably tons of data using less sophisticated methods that don't work properly in van der Waals systems, if you feel adventurous you can try them out. Otherwise it is easily doable by other DFT-based methods.
Have a look at the following paper:
B. Verberck, B. Partoens, F.M. Peeters, and B. Trauzettel: Strain-induced band
gaps in bilayer graphene, Phys. Rev. B 85, 125403 (2012)
I very much appreciate the two answers, thanks! Regarding the Paper on Strain induced gaps, I have some trouble relating the strain \epsilon_zz to stress, i.e. the force per unit area to be applied to achieve this strain. For the sample I have in mind - bilayer embedded between two films producing a perpendicular force - I can measure or simulate stress but the strain would be very difficult determine for the embedded bilayer graphene. I am not sure one can apply the relation cited in the paper p = \epsilon_zz c_44 , where p would be the perpendicular force per unit area and c_44 relates in this case perpendicular strain to shear forces.
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