While 2-D materials certainly have elastic constants, the reason for this issue when doing calculations relates to the "standard' definitions of the elastic constants and how they are normalized.

While the VASP tags for obtaining elastic constants are quite convenient, especially for materials whose lattice vectors aren't 90 degrees, we can gain a little insight to what I mention above by considering a simple cubic example. For a cubic system, the elastic energy density takes the form (essentially the generalized Hooke's law):

U = U0 + (1/2)* C11 (dxx2 + dyy2 +  dzz2) + C12* (dxx*dyy + dxx*dzz + dyy*dzz) + (1/2)*C44 *(txy2 + tyz2 + txz2)

Here U is energy density (so E / V where E would be the total energy calculated and V the volume). U0 is the equilibrium energy density. The dij are relative displacements along various directions (i.e. the percentages stretched). Therefore, we could obtain C11 by applying unaxial strain along one axis and keeping the other two fixed. This would allow us to extract C11. Likewise, we could apply biaxial strain and also a form of strain where we increase one axis by an amount and decrease the other by the same magnitude to obtain C11+C12 and C11-C12.

So from the above, we see that traditionally, the elastic constants in their quoted units are pressure (e.g. often Pa or kB) which is a force/area.  Force/area is the same as energy/volume (i.e. 1 Pa = 1 N/m2 = 1 J/m3).

The issue in 2-D materials is that when we do a calculation, there is technically no truly defined value for the volume since we use a supercell with as much vacuum as is necessary to minimize layer to layer interactions. To get around this, there are essentially two ways:

(1) Redefine elastic constants to be in terms of Force/length (e.g. Pa*length). In this case, you would take the VASP value's an multiple by whatever length your cell is in the perpendicular direction to the graphene surface.

(2) Define an effective thickness of the material.  Doing this is somewhat arbitraryl; however, various authors have defined this effective thickness to be the planar seperation between graphene layers in bulk graphite  (or MoS2 layers in bulk MoS2, etc). One can also use charge density confinement to define an effective thickness too.  For the effective thickness method, you would first do the same as in (1) and then divide by your effective thickness to get elastic constants in the usual units.

Thank you Mr. Andrew O'Hara. But this didn't work for me. The tags in my INCAR are as follows :

System = Graphene

IALGO = 38

ISMEAR = 0

SIGMA = 0.05

ICHARG = 2

ISTART = 0

ENCUT = 900

PREC = Accurate

EDIFF = 1E-8

NELM = 100

IBRION = 6

ISIF = 3

EDIFFG = -1e-3

PSTRESS = 0.0

POTIM = 0.02

I have calculated using the above tags in INCAR and the value that I get after calculation is C11 = 2340.5 kbar and C12 = 412 kbar (in vasp units). Could you please help me to find my mistake.

Dear Manju M S , could you please help me to grep C11, C22, C33 etc value from OUTCAR file ? where is it ?

I am attaching my OUTCAR file here.

Thank You

Shivam Kansara Please look at the end of your OUTCAR file (shift+g) and then slightly move up. TOTAL ELASTIC MODULI is what you need if you are looking for the elastic moduli of your system. To grep it, grep -A1 -A10 "TOTAL ELASTIC MODULI" OUTCAR. This will grep the total elastic moduli matrix.