Non-zero value of Born effective charge in metallic system

[raihan_ahammed]

Dear Admin and VASP users,

I have been trying to calculate the Born effective charge tensors (Z*) of a bulk metallic system using LEPSILON, with different k-point meshes and smearing settings. Surprisingly, I consistently obtain non-zero (Z*) tensors.

Based on my understanding of this previous discussion (viewtopic.php?t=20517&utm_source=chatgpt.com), I expected (Z*) to vanish for a metallic system, since the non-adiabatic method is not implemented in VASP. Is this interpretation correct, and how should I interpret the non-zero values?

Any clarification or suggestions would be greatly appreciated.

Kind regards,
Raihan

[martin.schlipf]

Your understanding is correct: There should be no Born effective charges in a metallic system. The way this shows up in practice is that the dielectric constant is huge and that your Born effective charges show an erratic convergence behavior. For an insulator you would expect to see a rapid convergence instead. Fundamentally it comes down to a linear response where you have the change of the Fermi occupancies in the numerator and the energy eigenvalues in the denominator. For an insulator the numerator is 1 (occupied - empty) and the denominator is the gap. For a metal both numerator and denominator approach zero and the results become numerically unstable.

As a side note, there are a few potential issues in your INCAR file. Please review that this is what you intended

• you set both ALGO and IALGO
• you set LREAL twice
• you calculate without spin polarization, are you sure Cr does not have a magnetic moment in this compound

[sophie_weber1]

Hello Martin,

I am also involved in this project. Thanks a lot for your quick response.
Your physical explanation makes sense, but then I am a little uncertain how the results in VASP should be interpreted; you seem to be saying that even if your BEC should be zero, you will by default get nonzero values when calculating using LEPSILON, but they will converge erratically. Can you define what you mean (w.r.t which convergence parameters, and what you mean by "erratic") more explicitly? for example, in this system to me the change in BEC with k-mesh did not seem so bizarre or erratic (the number changed appreciable only when the k-mesh was increased dramatically). Or is the printed dielectric constant a better indicator (and what constitutes "huge"?)

If one can't expect to actually get BECs approaching zero in the calculation, even when they should be, it would be useful if some more detailed information regarding how to tell when BEC values are or are not unphysical in the vasp wiki would be very helpful.

More generally, I would expect that if you had a semimetallic, anisotropic system where at least one direction is insulating, once could still get physically meaningful, finite BECs for electric fields along this direction. Can you please clarify whether this is indeed the case, or whether any metallic directions negates the possibilities of BECs?

[martin.schlipf]

I was only partially correct in my last post. While the argument is correct that you need to compute (f_nk - f_mk)/(ε_nk - ε_mk), VASP skips over the critical n = m term instead. Therefore you do not encounter the divergence and the value of Z* actually converges albeit noisy and slowly. Keep in mind that by skipping the n = m term, we do not compute the real Born effective charge and therefore should not expect to reach the value 0. For the dielectric constant the convergence is indeed erratic and small changes to the setup have large changes to the dielectric constant.

As a test, I computed TiN and MgO (both cubic rocksalt structure). You can see that in the metallic system the dielectric constant is erratic, and the Born effective charges converge much slower and do not approach the correct limit.

bec_convergence.png

[sophie_weber1]

Hi Martin,

Ok, thanks a lot for this. I think this would still be a very good clarification to make in the wiki, as one would assume from the way it is written that you should get zero calculated BEC in a metal.

Would it be possible to answer my question regarding semimetallic systems? I would guess that if both components of a BEC are along insulating directions, one could still have a physically meaningful nonzero value even if other directions have occupied states at the fermi level. I guess given that VASP is not printing the "true" BEC this would be hard to verify, but if certain components converged quickly whereas others didn't, I guess this could be an indicator?

[martin.schlipf]

I did not know the answer to this question so I tested it. I will describe the process since it is illustrative how to think about these problems: If we assume that BEC are correct for slabs that are insulating along z, then a the BEC of a MgO/TiN interface, should converge to bulk BEC in the MgO. So I simulated a single layer of TiN separated by m layers of MgO. You can see in the blue lines, that Z*_zz and ε_zz converge slowly with the film thickness towards the bulk value.

sl_vs_m.png

There is a physical reason for this: Within the TiN the field is screened (E = 0), so the field in the MgO is larger than it would be in bulk. You need to consider the electric displacement field D instead of the electric field E. The displacement field D is independent of the layer thickness. The corresponding Z*(D) is obtained by dividing Z*(E) that VASP reports by ε_zz. To compare Z*(D) to the bulk Z*(E), we need to multiply with ε_MgO. The orange lines show that this approaches the bulk value quickly. You can also fit ε_zz to a capacitor model with a width of 0.86 layers.