7
$\begingroup$

I am trying to calculate 2D elastic constants of a hexagonal boron nitride (hBN) monolayer. The unit cell is the primitive cell and the Cartesian coordinate is set as follows.

enter image description here

The elastic constants $C_{j} \:\: (j=x, y)$ have relations with the corresponding strain, $$ \frac{(E-E_{0})}{A_{0}} = \frac{1}{2}C_{j}\varepsilon_{j}^{2}, $$ where $E_{0}$ and $A_{0}$ is the energy and area at the equilibrium.

To find $C_{j}$, I made four deformed structures of the strain $\varepsilon_{j} = \Delta a_{j}/a_{0,j} = \pm 0.005,\; \pm 0.01$. The schematic diagrams for the deformation is the following.

enter image description here

POSCARs of deformed structures have the same fractional coordinates written in the original POSCAR of the optimized geometry, but the lattice vectors are changed according to the above strain. I conducted single-point calculations with deformed structures using VASP and fitted the above equation with a quadratic function.

However, the problem is that $C_{x}$ and $C_{y}$ is not the same (I obtained $C_{x}=326\; \mathrm{Nm^{-1}}$ and $C_{y}=300\; \mathrm{Nm^{-1}}$). Those two values are to be the same due to the hexagonal symmetry [Mouhat, F.; Coudert, F.-X. Necessary and Sufficient Elastic Stability Conditions in Various Crystal Systems. Phys. Rev. B 2014, 90, 224104.].

I suspect the deformed structures I made are wrong somehow, but I am not sure how to define the deformation correctly.

EDIT: I added more data related to the fitting process.

A. The POSCAR lattice vector matrices in the x-direction

$\varepsilon = 0$ (the optimized structure):

 2.5122293853799835   -0.0000000000000046   -0.0000000000000000
-1.2561146926399998    2.1756544679020906    0.0000000000000000
 0.0000000000000000   -0.0000000000000000   15.0000000000000000

$\varepsilon = 0.005$:

 2.52479053    -0.00000000    -0.00000000
-1.25611469     2.17565447     0.00000000
 0.00000000    -0.00000000    15.00000000
 

$\varepsilon = 0.01$:

 2.53735168    -0.00000000    -0.00000000
-1.25611469     2.17565447     0.00000000
 0.00000000    -0.00000000    15.00000000

$\varepsilon = -0.005$:

 2.49966824    -0.00000000     0.00000000
-1.25611469     2.17565447     0.00000000
 0.00000000    -0.00000000    15.00000000

$\varepsilon = -0.01$:

 2.48710709    -0.00000000     0.00000000
-1.25611469     2.17565447     0.00000000
 0.00000000    -0.00000000    15.00000000

B. The POSCAR lattice vector matrices in the y-direction

$\varepsilon = 0.005$:

 2.51222939    -0.00000000    -0.00000000
-1.25611469     2.18653274     0.00000000
 0.00000000    -0.00000000    15.00000000

$\varepsilon = 0.01$:

 2.51222939    -0.00000000    -0.00000000
-1.25611469     2.19741101     0.00000000
 0.00000000    -0.00000000    15.00000000

$\varepsilon = -0.005$:

 2.51222939    -0.00000000    -0.00000000
-1.25611469     2.16477620     0.00000000
 0.00000000    -0.00000000    15.00000000

$\varepsilon = -0.01$:

 2.51222939    -0.00000000    -0.00000000
-1.25611469     2.15389792     0.00000000
 0.00000000    -0.00000000    15.00000000

C. The quadratic fittings

The x axis is dimension-less and the unit of y axis is $\mathrm{eV}/\text{Å}^{2}$. Thus, the elastic constants in $\mathrm{Nm}^{-1}$ is given by factoring $\sim 16 \times 2 = 32$ to the 2nd order coefficient. enter image description here

$\endgroup$
6
  • $\begingroup$ Maybe it would be helpful for diagnosing if we could see explicitly how the 3 by 3 lattice vector matrix in the POSCAR file is changed for the deformed structures. Perhaps the plots of the energy against strain with your obtained data points and the fitted quadratic curve might be helpful too. $\endgroup$
    – CW Tan
    Commented Apr 30, 2023 at 13:46
  • $\begingroup$ @CWTan Thank you for advice. I add some details related to the fitting process. $\endgroup$
    – Patche
    Commented Apr 30, 2023 at 14:21
  • $\begingroup$ Thanks, I didn't spot anything immediately off with the way you're deforming the structures, but I'm suspecting a sensitivity issue. Since you're using energies and the strain is small, and your trying to extract a second derivative, even small numerical errors in the energy might lead to a bad fit. I gather from your tag that you're using VASP - what EDIFF are you using? And how tightly did you do the convergence tests for ENCUT and KPOINTS? You could compare that to the energy differences of your deformed structures. $\endgroup$
    – CW Tan
    Commented Apr 30, 2023 at 14:38
  • 1
    $\begingroup$ @CWTan I used EDIFF = 1E-8 and ENCUT = 600 eV. The kpoint sampling is a 60 * 60 * 1 Gamma-centered Monkhorst-Pack k-mesh. I'll check the calculation condition again. Thanks. $\endgroup$
    – Patche
    Commented Apr 30, 2023 at 15:07
  • 1
    $\begingroup$ If the conversation gets too long here, I recommend the VASP chat room: chat.stackexchange.com/rooms/109983/vasp $\endgroup$ Commented Apr 30, 2023 at 19:00

1 Answer 1

2
$\begingroup$

I think you should take a rectangular unit cell instead of the hexagonal unit cell depending on this article (The response of mechanical and electronic properties of graphene to the elastic strain M. Topsakal, S. Cahangirov, and S. Ciraci)
( https://doi.org/10.1063/1.3353968 )..... can I get your email please, I have a question

$\endgroup$
6
  • $\begingroup$ Hi Mr. Patche How can I contact with you $\endgroup$ Commented Jun 4, 2023 at 5:39
  • $\begingroup$ Please add a hyperlink to the referenced article. $\endgroup$ Commented Jun 4, 2023 at 9:19
  • $\begingroup$ This is link doi.org/10.1063/1.3353968 $\endgroup$ Commented Jun 4, 2023 at 9:38
  • 2
    $\begingroup$ Your answer could be improved with additional supporting information. Please edit to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center. $\endgroup$
    – Community Bot
    Commented Jun 4, 2023 at 12:54
  • $\begingroup$ @ahmedhassan Sorry. I didn't check SE for a while. If you need some help, could you use the chat room? I will check the question later. chat.stackexchange.com/rooms/109983/vasp $\endgroup$
    – Patche
    Commented Jun 7, 2023 at 8:42

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .