Table of Contents

Analyzing defects in graphene

In this exercise we follow-up on what was started previously with defects in silicon, but this time you will have to figure out the setup as well.

When comparing scaled coordinates between papers and code input scripts, always make sure that they use the same coordinate systems and definitions for a unit cell (both real and reciprocal space). For example while many sources (like the paper of Curtarolo, Setyawan) assume a 120° degree angle between $a$ and $b$ for a hexagonal cell, you can also define it to be a 60° angle (like the default in CP2K).
Once you have verified that your calculation setup works, use nohup mpirun -np 8 cp2k.popt … & to run the calculations in parallel and in the background since they may take longer to complete than before.

Vacancy in graphene

Comparing energies

Use the template and initial geometry provided when calculating the projected density of states for graphene to setup a single point energy calculation for a 6x6x1 supercell of graphene.

Create a vacancy by removing one carbon atom from this supercell and perform the energy calculation again.

Quick question: Does it matter which carbon atom you remove? (hint: what kind of boundary conditions do we impose?)

Calculate the energy of the vacancy formation, that is $E_v = E_2 - \frac{N-1}{N} \cdot E_1$ where $E_1$ is the energy of the complete system, $E_2$ that of the system with a vacancy and $N$ the number of atoms.

Analyze the PDOS

Would you expect the vacancy to haven any influence on the projected density of states? Check whether your assumption was right by visualizing the PDOS.

Replacement with oxygen

Now, instead of removing one carbon atom from the 6x6x1 supercell, simply replace it with an oxygen atom. Perform first a single point calculation and second a geometry optimization and compare the energy of adsorption for both cases.

Oxygen atom adsorbed on graphene

Now we are going to investigate the effect an adsorbent has on graphene.

Change in energy

In order to adsorb an oxygen atom on top of a graphene layer, modify the coordinate section by adding one oxygen atom which has them same coordinates as a carbon (except the z-component of course). Check whether the addition of an oxygen atom has an effect on the structure of graphene by optimizing its geometry and calculate the adsorption energy.

Use $E_\text{ad} = E_3 – (E_1 + \frac{1}{2}E_2)$, with:

$E_\text{ad}$
energy of adsorption
$E_1$
energy of graphene
$E_2$
energy of molecular oxygen (-31.929714235694995 a.u.)
$E_3$
energy of oxygen adsorbed on graphene
The formation of oxygen is more difficult to obtain, here we use $\frac{1}{2}$ of the total energy of molecular oxygen. This approximation introduces a large uncertainty to our calculations because the calculation of dioxygen is difficult using KS-DFT.

Try to explain based on the lecture what might be the problem.

Displacements

We are furthermore interested in the change of structure this adsorbent causes. Try to visualize which atoms have to assume a new position in order to minimize the total energy. That is: plot $\sqrt{(x^i-x^i_0)^2 + (y^i-y^i_0)^2 + (z^i-z^i_0)^2}$ in a sensible manner (one which also retains the geometry of graphene).

Analyzing defects in hexagonal Boron-Nitride

Repeat the calculations of the vacancy formation, defect formation and adsorption for the h-BN-layer structure, taking into account that now both the N and the B can be replaced.

Compare the energies for the two cases, where is a vacancy more likely to be and on top of which atom does an oxygen atom preferably adsorb.

Use the total energy of a B or N atom when calculating the vacancy formation energy.

Since N and B are radicals, you have to include the following keywords/options in the right places (use the CP2K manual):