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## How do you calculate defect formation energies?

Total energy, DFT+U, BigDFT,...

kalkm1
Posts: 6
Joined: Tue Jul 02, 2019 4:51 pm

### How do you calculate defect formation energies?

I am trying to calculate the defect formation energy for a Cd-vacancy (Vcd) and Te-antisite (Tecd) in CdTe, but I am struggling to get the expected formation energies in line with other works in the literature.

I want to ask what the fundamental steps are and the main points that must be considered in order to calculate the formation energy (as I feel that I must be missing something)?

What I am currently doing is as follows:
1. I assume a neutrally charged defect such that the formation energy (Ef) is given by: Ef = Etot(D) - Etot + u(Cd) - u(Te)
where Etot is the total energy of the whole bulk system, Etot(D) the energy of the total system that includes the single defect, and u(Cd) and u(Te) are the chemical potential energy of a single Cd atom and single Te atom respectively. u(Te) is only subtracted in the case of the Te-antisite calculation of course.

2. I calculate Ef using supercell sizes of 64, 216 and 512 atoms. Because of the spurious interaction between the defect in the repeating supercell due to the periodic boundary condition (PBC), the formation energy should change as a function of supercell size and reach its true value in the limit of an infinitely large supercell. Using these three points I should be able to through extrapolation approximate Ef at an infinitely large supercell. Even if extrapolation is not possible, the Ef obtained for a supercell of 512 atoms should be pretty close to its true value (within 0.1 eV).

3. For all total energy (Etot) calculations, I relax the supercell structure until the stresses are below 10^-4 Ha/Bohr. This has proved to be a considerable challenge for the supercells containing the defect, where the difference in energy between the SCF iterations begins to increase (instead of decrease!) and the relaxation abruptly stops. I've had the most luck when I fix all of the atom positions except for the defect and its nearest neighbours and use ionmov 2 and optcell 0 (i.e. fixing the volume to that of the relaxed bulk structure and only allowing atoms which are not fixed to move).

4 a). How do you get the chemical potentials of the atoms, u(Cd) and u(Te)? This part I'm very unsure about (yet its something papers in the literature seem to skip over as being trivial). I believe that the condition u(Cd) + u(Te) = Etot(CdTe), where Etot(CdTe) is the total energy of a CdTe pair, must be satisfied? Initially, for u(Cd) for example, I would place a single isolated Cd atom in the middle of the unit cell with a volume fixed to that of the relaxed corresponding bulk supercell size. When I did this for Cd and Te, the condition above was not satisfied. Furthermore, for CdTe, defect formation energies are often given in terms of Cd-rich or Te-rich conditions. If I use the isolated atom approach to calculate the chemical potential energies, do I get the Cd-rich or Te-rich formation energies? Or neither?

4 b). In order to calculate the formation energy of the defects in Te-rich conditions, I determine u(Te) from a calculation of bulk Te where I define a unit cell containing 3 Te atoms arranged in a trigonal lattice, and relax the structure. Then, u(Te) = Etot(Te)/3 and u(Cd) = Etot(CdTe) - u(Te). This of course satisfies the condition above.

Doing all this, the formation energies I obtain for Vcd and Tecd are still out by ~1-2 eV and neither is the relationship between Ef and supercell size (which I plot as 1/L where L is the cubic root of the number of atoms) linear - it appears more to plateau at the larger supercell sizes.

Some quick further questions: Am I correct in relaxing the supercell when I introduce the defect? Do I need to add a charge to the defect (and is this done using the charge variable)? Even in my current case of a neutrally charged defect am I correct in thinking there would still be spurious interaction between the supercells in the PBC due to valance electrons moving around the defect?

For context, I am using PAW datasets from the JTH table v1.1 with LDA.

Any help or tips from your own experience calculating formation energies would be greatly appreciated!

Cheers,
Kjell

ebousquet
Posts: 469
Joined: Tue Apr 19, 2011 11:13 am
Location: University of Liege, Belgium

### Re: How do you calculate defect formation energies?

Dear Kjell,
I cannot help you in calculating the formation energy but I would start to not relax the cell parameter as for a concentration going to zero the cell parameter should be the pure bulk one.
Additionally, if you have to add a charge to the supercell to get charge neutrality, then 100% sure you cannot relax the cell parameter as the stress is ill defined in this condition, see:
https://journals.aps.org/prb/abstract/1 ... .86.140103
https://journals.aps.org/prb/abstract/1 ... .91.024107

Hope this can help a bit.
Best wishes,
Eric

kalkm1
Posts: 6
Joined: Tue Jul 02, 2019 4:51 pm

### Re: How do you calculate defect formation energies?

Dear Eric,

Relaxing the structures containing the defects does seem to be the common approach in the literature when calculating formation energies and I too observe that it has a significant effect on the formation energies calculated, therefore I do believe I need to perform a relaxation.

I am a little confused about if my supercell is charged or not when I introduce the defects. If I remove a Cd atom to introduce the Cd Vacancy, will my supercell be charged by default (given that Cd is the cation in CdTe)? And I therefore need to include "charge +2" in my input file to achieve charge neutrality? Up until now I've been assuming that my supercell, even with a defect, is always neutral unless I explicitly tell abinit there is a net charge by using the "charge" input parameter.

Best regards and thank you in advance!

Kjell

gmatteo
Posts: 291
Joined: Sun Aug 16, 2009 5:40 pm

### Re: How do you calculate defect formation energies?

I am a little confused about if my supercell is charged or not when I introduce the defects. If I remove a Cd atom to introduce the Cd Vacancy, will my supercell be charged by default (given that Cd is the cation in CdTe)?
No, the unit cell is always charge-neutral unless you use the charge input variable
Up until now I've been assuming that my supercell, even with a defect, is always neutral unless I explicitly tell abinit there is a net charge by using the "charge" input parameter.
Right.