Coulomb cutoff and vacuum size

[Christian Koenig]

Hi all,

I try to converge the energy gaps of a thin film/slab with respect to the vacuum size (G0W0 approximation).

The Coulomb cutoff technique is used as described in Phys. Rev. B 73, 205119 (2006) (https://journals.aps.org/prb/abstract/1 ... .73.205119) and I would expect a behaviour similar to Fig. 6.

My film is 10 Angstrom thick, so - assuming that the DFT calculation includes enough vacuum - I would expect that once the cutoff is a bit larger than 0.5 times the thickness of the slab the gap energy converges to a constant value. The reason for the factor of 0.5 is the definition of the cutoff region which is a slab with thickness 2*cutoff according to the reference above. In the attached plots I use the effective value of 2*cutoff as the x axis. Different lines correspond to different vacuum sizes in the DFT calculation.

Indeed, the calculations converge to a constant value between about 16 and 24 Angstrom. However, once the cutoff is increased too much so that the images of the slab can interact again, there should be a deviation from the converged value. In the reference this deviation is obvious but not out of control. As you can see in the attached figures, in my calculations the oscillations become much stronger than I would expect (order of eV) and the onset happens always at around 24 Angstrom, independent of the vacuum size (which I want to converge along with the cutoff). In principle as the distance between the images of the slabs grows, the oscillations should start later.

Could someone please explain the onset of the oscillations and why this onset is independent of vacuum size? Is something wrong in the procedure that I use to converge the gap energies?

I also attached the input file for one of the calculations. Note that the parameters are not fully converged to make the calculations feasible.


Best,

Christian

[Daniele Varsano]

Dear Christian,
as documented in various posts, the cutoff length of the box should be set slightly smaller than the size of the unit cell in the finite direction.
Maybe this is anti-intuitive, but it is how at the moment the cutoff-box is coded. To be clear, for a supercell of (Lz=40Bohr side size), you should set the cutoff length something like 39Bohr.

Having said that, my strong suggestion, in case you have an orthorhombic cell, is not to use a box geometry, but the Wigner Seitze cutoff.
You can activate it as:
CUTGeo= "ws z"
CUTwsGvec= 0.700000

There is a kinetic energy cutoff to be controlled, but the default value should be ok.
Best,
Daniele

[Christian Koenig]

Dear Daniele,

I have a hexagonal cell with the vacuum in the c direction (z in the cartesian coordinate system).
The total length of the cell in this direction would be

30 Angstrom for 20 Angstrom vacuum
40 Angstrom for 30 Angstrom vacuum and
50 Angstrom for 40 Angstrom vacuum, respectively.

In that sense I hit the oscillations well before I come close to the cell size, at least for the case of 30 and 40 Angstrom vacuum.
It is clear that the calculations are not meaningful if the cutoff is increased too much, but the threshold should depend on the vacuum size.


Best,

Christian

[Daniele Varsano]

Dear Christian,
that sounds a bit strange, anyway I'm a bit confused about the plot notation (2*cutoff).
Anyway, it is clear that there are numerical instabilities here (that's why I was suggesting to use ws cutoff that you can give a try, but I'm afraid yambo will complain).
Having very anisotropic cells I suggest you build the cutoff potential using RandGvec on the order of 100 instead of that 1 (only first BZ).
Next, you are using terminators but I can't see GTermEn and XTermEn in your input, be sure at the end of the report that they are set correctly at the default value.
Can you post also a report file of one of the runs, indicating which point is in the plot?
Best,
Daniele

[Christian Koenig]

Dear Daniele,

The notation refers to the reference given above:

A region D is defined outside of which there is no Coulomb interaction. On page 3: "The choice of the region D for step 1 is suggested by symmetry considerations, and it is [...] an infinite slab of thickness 2R for 2D-periodic systems." Thus, at least if my interpretation is correct here, a cutoff of 20 Bohr as provided in the example input file corresponds to a slab thickness of 40 Bohr or 21.2 Angstrom. This probably means that for a point somewhere in the slab, interaction with all points within a range of +-10.6 Angstrom deviation in the z-direction is allowed.


The output file for the first point of the 40 Angstrom calculation (blue line at 21.2 Angstrom) is attached.
I could find the value of GTermEn (Sc Terminator pole : 40.81708) but not the value of XTermEn.


Best,

Christian

[Daniele Varsano]

Dear Christian,
there is a bit of confusion, probably there is a quite missing documentation for this issue.
The cutoff size provides you with the interaction range. A cutoff of 20 Bohr in the input file builds a coulomb potential extending up to 10 Bohr (because of a misleading factor 2), i.e. about 5.3 Angstrom. This is not correct as you avoid interaction inside your slab.
Now, one wants that all the points in the slab do interact with each other, avoiding interaction with the replica. This is obtained. for a slab thickness of 10 Ang (~19 bohr ) setting a potential extending up to 25 bohr (considering some spill-out of the charge) and it is obtained setting a box size of about 50 bohr, accordingly, also your box size should be large enough (more than 50 bohr). Finally, you should consider a cutoff size a little smaller than your box size, provided that the cell size is large enough to include interaction in your slab but not between replica.
I hope now this is more clear, maybe could be useful also looking to this thread:
viewtopic.php?p=1793#p1793

Best,
Daniele

[Christian Koenig]

Dear Daniele,

So apparently I misunderstood the meaning of the parameter, thanks very much for the explanation. I attached a plot with new calculations including a random integration over 100 RL and explicitly including the terminator energies. The gap energies stabilize above 45 Bohr cutoff box size which agrees well with your estimate.

Thanks again, this was really helpful and everything seems to be clear now.


Best,

Christian

[Jing Liu]

Dear Christian,
as documented in various posts, the cutoff length of the box should be set slightly smaller than the size of the unit cell in the finite direction.
Maybe this is anti-intuitive, but it is how at the moment the cutoff-box is coded. To be clear, for a supercell of (Lz=40Bohr side size), you should set the cutoff length something like 39Bohr.

Having said that, my strong suggestion, in case you have an orthorhombic cell, is not to use a box geometry, but the Wigner Seitze cutoff.
You can activate it as:
CUTGeo= "ws z"
CUTwsGvec= 0.700000

There is a kinetic energy cutoff to be controlled, but the default value should be ok.
Best,
Daniele

Dear Daniele,
I want to do a G0W0 gap convergence test with respect to the vacuum spacing. According to the tutorial, the cutoff length should be set slightly smaller than the size of the unit cell. For example, the side size is 28.34589 bohr (15 angstrom), so the cutoff length I chose is 27.
My question is that if I increase my side size to 16, 17, 18 or 19 angstrom to do the gap convergence test, should I fix the cutoff length as 27, or increase it as my unitcell increased?
Thanks so much!

Best,
Jing Liu

[Daniele Varsano]

Dear Jing,

you should set it always just a bit smaller of your cell.
If your cell is orthorhombic I suggest you to use the CUTGeo= "ws z" , in this case you do not need to take care of the box size.
Very soon we are going to release a new shape cutoff that avoids all these settings.

Best,
Daniele

[Jing Liu]

Dear Daniele,
Thanks so much for your quick reply!
I got it!

Best,
Jing Liu