a question about temperature in two probe calculation

[benhuzhou]

Dear Nordland and everyone:
I have a question about temperature in two probe calculation, I calculate the transport energy spectrum of two probe graphene system by using theATK, and I have written a paper and submitted to APL, I received the reviewer comments today. One quesiton which I do not know how to answer, the quesiton is "The temperature at which the transmission probability and I-V curves are obtained is not stated". I do not know how to answer? what is the temperture for two-probe in ATK? is it 0K, 300K or other? 300K is a default value for two leads when I calculate the transport property of two probe system.
Thank you for your attention!!!

[Anders Blom]

If we understand the temperature parameters as a way to specify the width of the Fermi distribution used in the calculation (and this is, indeed, its only role), then you are correct: if you ran the calculation with default parameters, it corresponds to 300 K.

As we all know, this temperature has nothing to do with phonons etc, and in that respect the calculations correspond to the 0 K limit. But that's probably not the point of the referee's question, so stick to the answer "300 K". That's the temperature used in the calculation to set the width of the Fermi distribution, and that's the only role this parameter plays.

[benhuzhou]

Dear Anders Blom:
    Thank you very much!
    But I am still confused according to Nordland's explanation. The follows are his explanation:"
The temperature in ATK is only used for determining the shape of the fermi distribution,
and hence it is only used for calculating the density matrix and for evaluating the current.
The manual explains that the user should not think of this temperature as an experimental, physical temperature.
The two-probe calculation is using the temperature defined in the electrodes, but only for smearing of the fermi distribution.
If you force me to say what temperature ( as I would rather not bring physical temperature into the picture)  the two-probe calculation is done
under, I would say it was zero temperature, but the electrons feels a finite temperature"
and accordong to his explanation, I think it was zero temperature, but I am still not sure. By the way, the principle of calculation transport property for the two probe system is basisd on the Green's function, according to my knowledge, the Green's function works around 0K. I don not sure whether I have understood or not, Please give me a detail explanation.Thank you!

[Anders Blom]

Well, his answer is actually just a longer version of mine

It is in principle not appropriate to discuss temperature in this at all. We have no effects of real temperature, like phonons etc, and I think we can all agree on this.

What we do have is a Fermi distribution, and you are right, in principle we should use a step function and compute everything at 0 K to be on solid theoretical soil regarding the NEGF formalism. However, with a step-like Fermi distribution, the calculation will most likely not converge for numerical reasons. Therefore we make it have finite width, say 0.025 eV. Now, using E=kT we can express this energy as a temperature, which is 300 K. Some calculations might not even converge until we increase the broadening to 100 meV, or 1200 K.

If you want to be really thorough, you could in principle try to reduce the temperature, instead of increasing it, if the convergence is stable. Or, after you reach convergence, you can restart from the converged result and "anneal" the system down towards zero temperature. Only in this limit is, for example, the zero-bias conductance equal to the conductance quantum times the Fermi-level transmission coefficient.

This approach would be more relevant if you performed the calculation at a very high temperature, for convergence reasons, to anneal down to 300 K. Going down from 300 K to, say, 100 K, most likely the influence of the finite Fermi distribution width (a.k.a. finite electron temperature) on the observables like current etc will be negligible, and also it will probably be smaller than other inaccuracies due to k-point sampling, basis set, mesh cut-off, exchange-correlation functional, etc.

[serhan]

Hi,

The current saturates in a two-probe system as the bias voltage increases in ATK. What is the reason of this? As I understand the reason is not phonons, from this thread.

Cheers,
Serhan

[Anders Blom]

It is not a general feature that the current saturates. Different systems have different behavior; saturation, increase, or even NDR (an important application area!). It's simply a matter of how the energy bands in the electrode line up compared to the Fermi level. When you sweep the bias a certain number of bands (and a certain fraction of these bands) will be inside the bias window, and this determines the transmission coefficient.

Thus, if the lowest conduction band is narrow (less than 1 eV, say), and above it there is a new gap, then you will see a saturation at 1 V. If, on the other hand, the conduction band is a pure continuum (in DOS), you should see a steady increase.

Of course, we also have to take into account how these bands align with e.g. molecular levels of the scattering region.

And so on...

Physics is fun because the answers are seldom simple!

[Anders Blom]

We should perhaps revise the statements above a little, little bit...

The "temperature" actually enters the calculations in ATK twice. The first time is for the SCF loop, where we have one Fermi distribution, and the second time is in the evaluation of the transmission spectrum, which also contains a Fermi function.

As we have discussed above, the "temperature" for the SCF is really not physical at all, but just a numerical help for convergence.

The temperature in the transmission spectrum is, however, more physical, and by changing it we can actually determine how the temperature influences the quantum ballistic tunneling probability, via the broadening of the electrode Fermi distributions it introduces. In fact, you can even drive a current at zero bias by applying a different temperature to the two electrodes.

Now, the ability to do this was not included in ATK 2008.10 and earlier versions, but it is included in ATK 2009.06 (the semi-empirical edition) and will also be supported in the next DFT released.

For a very interesting example on this, see http://www.quantumwise.com/documents/manuals/GrapheneDevice/index.html, specifically the section called Extracting information from the Transmission Spectrum and the two subsection there called Calculating the temperature dependent conductance and Electron thermal transport!

[serhan]

Thank you very much Dr. Blom. So, is it possible to include phonon effects in the next versions of ATK?

Cheers,
Serhan

[Anders Blom]

No, no, as I wrote many times this has nothing to do with phonons

It is well-known that a difference in temperature can drive a current without applied bias (thermoelectric or Peltier-Seebeck effect). In a simplified picture it's quite obvious from the expression for the current (simplified notation):

[tex]I = \frac{2e}{h} \int_{-\infty}^{\infty} T(E) [f_L(E)-f_R(E)] dE [/tex]

As long as the two terminals have the same Fermi energy (no bias) and the same temperature, [tex]f_L(E)=f_R(E)[/tex] for all energies and the current becomes zero. But if the temperature is different, we can get [tex]I \neq 0[/tex] even at zero bias.

[serhan]

Thanks Dr. Blom. Yes, I understand where the Landauer formula comes from and usage from some textbooks. However, I tried to ask that apart from calculating the current with the formula you've mentioned; would it be possible to include phonon effects by considering the average number of phonons in a crystal and relating its effects on the transmission of the system, T(E). Then, after modifying T(E), the Landauer formula could be again used to calculate the current. I just thought loudly, these may or may not be possible mathematically.

Or can the effects of phonons be included in a semi - emprical way

Thanks very much for your kind response again.

Cheers,
Serhan

[Anders Blom]

You are in fact not the first one to come up with this idea, or something similar at least. Have a look at this reference for a study with ATK about "Electron- and phonon-derived thermal conductances in carbon nanotubes"!

[benhuzhou]

Dear Anders Blom:
    Thank you very much!
    I understand what you said, according to your explanation, I think my results are obtained under 300K (room temperature). So the referee may be ask  another question. As we all know, as  the temperature is 300K, the interaction between electrons and phonons should be considered, and he will ank me whether the interaction is considered or not, according to your explanation, the temperature has nothing to do with phonons. So I think he will not believe what I said. How to further explanation this question? If my results are abtained under 0K, or around 0K, Is the answer which the referee really want? I really get confused. By the way, I read a lot of papers, and the papers are not referred to the temperature, so l think it is a difficult question to answer.

[zh]

As we all know, as  the temperature is 300K, the interaction between electrons and phonons should be considered, and he will ank me whether the interaction is considered or not, according to your explanation, the temperature has nothing to do with phonons. So I think he will not believe what I said. How to further explanation this question? If my results are abtained under 0K, or around 0K, Is the answer which the referee really want? I really get confused. By the way, I read a lot of papers, and the papers are not referred to the temperature, so l think it is a difficult question to answer.

It is a quite simple question. Just tell him what you know and what is the meaning of temperature in your calculations. It is useless to guess what the referee will ask further. The DFT calculations are carried out at 0 K, it is a well known fact in the quantum chemistry community.  Strictly speaking, only in the finite temperature DFT where the universal energy function is based on the free energy rather than the total energy, the temperature in the Fermi-Dirac function has the physical meaning, i.e, it is related to the temperature of electrons.

Recommend you to read the following references in order to deeply understand the meaning of "T" parameter in the Fermi-Dirac function and the purpose of other broadening scheme in the band structure calculations:
1) N. David Mermin, Thermal Properties of the Inhomogeneous Electron Gas, Phys. Rev. 137, A1441 - A1443 (1965), http://link.aps.org/doi/10.1103/PhysRev.137.A1441
2) Matthieu Verstraete and Xavier Gonze, Metals at finite temperature: a modified smearing scheme, Computational Materials Science, Volume 30, Issues 1-2, May 2004, Pages 27-33, http://dx.doi.org/10.1016/j.commatsci.2004.01.006
3) M. Methfessel and A. T. Paxton, High-precision sampling for Brillouin-zone integration in metals, Phys. Rev. B 40, 3616 - 3621 (1989), http://link.aps.org/doi/10.1103/PhysRevB.40.3616

[benhuzhou]

Dear zh
Thank you very much, now I know how to do.
Best regards.

[serhan]

Thanks for the reference.

Cheers,
Serhan