### 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.

`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

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.

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

`UKS = .TRUE.`

`MULITPLCITY = …`