a question about temperature in two probe calculation

[Nordland]

There are two fundamental assumptions in almost all DFT codes.

1) The first is called the Born-Oppenheimer approximation.
I recommend reading the wiki-link here, but in short the BO-approximation states that the electrons has a adiabatic behavior
in relation to the quantum mechanics of the atoms. Therefore we can solve the electronic Schrödinger equation separate from
nucleus Schrödinger equation.

2) The movement of the nucleus are treaded as a classical particles, and furthermore it is assumed that nucleus are frozen in their
position with out any movement of all. This means that the potential from nucleus does not depend on time, and hence we can solve
the time-independent electronic Schrödinger equation instead of the time-dependent. However if the nucleus is not allowed to move, we know from thermodynamics, that the temperature must be zero. (Phonon is a quantized mode of vibration, but with no movement of the nucleus, there are no
vibration modes, and hence there are no phonons at T=0)

This two assumption is foundation for most DFT calculations, and therefore it is basic assumption that the calculation is performed at zero temperature.

However, if you want perform a very crude approximation how your system depends on the temperature you can adjust the fermi temperature, however
the reason for this being a crude approximation is that the second approximation is still being imposed despite it being wrong.

If your referee ask for a temperature, then your reply should be: "The calculation is performed at T=0, but with a fermi temperature of 300 K. And there is no phonon-electron interaction included in the simulation."

[Quantamania]

Related to the electron temperature for the Fermi-Dirac distribution function in calculations, is there an explanation for why an electron temperature set to zero K causes failure of the calculation?  In this instance, the distribution is discontinuous at the Fermi energy level, but at other values, it is continuous.  This was one of the challenges I received from my mentor while discussing the methods I used in the thesis.  We can easily generate results using temperatures of 5 K, but never with zero K.

Is there a reason why the discontinuous distribution at 0 K cannot be used, while a continuous distribution at 300 K leads to successful calculations, aside from the difficulty of derivatives being taken from it (such a distribution is well treated in integral calculus)?

[Anders Blom]

The reason is indeed purely numerical, but perhaps a bit different than you guessed.

The problem is most noticeable for metallic systems (as the electrodes should be for a two-probe system to make sense); for semi-conductors the Fermi level is in the band gap, and changing it even by quite much, inside the gap, has no effect on the occupations.

Moreover, the problem is related to how the surface density of states looks around the Fermi level. If the DOS contains many sharp peaks, then if you change the Fermi level by a small amount, and if the distribution function is very sharp (low temperature), you change the occupancies (related to the DOS) by a large amount. With a finite k-point sampling (as we must have to avoid infinite matrices...), there will be a limited number of states which can account for the new occupancies, and this can reshuffle the density matrix quite a lot. So, a small change in one place gives a large change in another place, and this is not what you want for successful self-consistent convergence.

If you had an infinite k-point sampling, then it wouldn't be such a problem, because you would then have energy points infinitely close to each other around the Fermi level, to account for the changes in occupancies etc.

This is why you can often cure convergence problems either by increasing the temperature or the k-point sampling. Increasing the temperature is cheaper, since more k-points takes more time to compute, but you are also, perhaps, sacrificing the accuracy a bit. So often a combination is best; a bit more k-points and a bit higher temperature.

Someone else can perhaps provide a more stringent description, but this is how I picture it.