Which energy should I consider while trying to analyse the variation in transport coefficients with temperature?

Question

I am using the BolzTraP code [I'm completely new to this code] to calculate the thermo-electric coefficients as per this tutorial. And the output file obtained from BoltzTraP is given below. The column labels and what they stand for are given below :

• Ef[Ry] : Fermi Energy level

• T [K] : Temperature

• N: Number of Carriers

• DOS(Ef) : Density of States

• S: Seebeck Coefficient

• s/t : conductivity/ electron relaxation time

• R_H: Hall Coefficient

• kappa0: Thermal Conductivity

• c: Electronic Specific Heat

• chi: Pauli Magnetic Susceptibility

    Ef[Ry]     T [K]            N         DOS(Ef)           S             s/t               R_H        kappa0         c                 chi
   0.39136    1.0000     14.59160339  0.27749010E+04  0.00000000E+00  0.15404697E+22 -0.16852985E-11 -0.00000000E+00  0.00000000E+00  0.41426895E-07
   0.39136    2.0000     14.59160339  0.13874505E+04 -0.12814074E-20  0.77023484E+21 -0.33705969E-11  0.10277212E+01  0.65901673E-14  0.20713447E-07
   0.39136    3.0000     14.59160339  0.92496701E+03 -0.44231269E-15  0.51348989E+21 -0.50558954E-11  0.23649732E+06  0.15165172E-08  0.13808965E-07
   0.39136    4.0000     14.59160339  0.69372528E+03 -0.23870272E-12  0.38511743E+21 -0.67411938E-11  0.95722814E+08  0.61381369E-06  0.10356724E-07
   0.39136    5.0000     14.59160340  0.55498083E+03 -0.98897905E-11  0.30809428E+21 -0.84264870E-11  0.31727480E+10  0.20344954E-04  0.82853882E-08
   0.39136    6.0000     14.59160341  0.46249068E+03 -0.11450615E-09  0.25674891E+21 -0.10111701E-10  0.30612680E+11  0.19630110E-03  0.69045896E-08
   0.39136    7.0000     14.59160348  0.39645474E+03 -0.64293135E-09  0.22008938E+21 -0.11796403E-10  0.14734045E+12  0.94481076E-03  0.59187295E-08
   0.39136    8.0000     14.59160376  0.34700703E+03 -0.23031393E-08  0.19263855E+21 -0.13479177E-10  0.46195667E+12  0.29623023E-02  0.51805175E-08
   0.39136    9.0000     14.59160447  0.30870652E+03 -0.61253650E-08  0.17137573E+21 -0.15156879E-10  0.10928617E+13  0.70082134E-02  0.46087238E-08
  

Here you can see that the coefficients from column 4 to 10 vary with temperature, for a given Ef or N.

I've got the BoltzTraP user guide but it doesn't mention as to which energy I should study the variation at. Should it be analysed at the Fermi Level which is obtained during the SCF calculation? or at the experimentally determined carrier concentration?

Answer 1

The general answer to your question is that it depends on what scientific question you are wanting to answer. However, since you're interested in thermoelectric properties, the material you're studying is probably a semiconductor and has a band-gap. This means that the conductivity is essentially zero unless you dope it, so using the undoped Fermi energy from the SCF will not give you anything interesting.

If you want to compare to an experiment, you will probably want to look at the temperature variation at the experimental doping concentration; however, be aware that in practice the material may have intrinsic defects or other impurities which may act as additional dopants or compensate the doping, and you may need to account for these when you set the dopant concentration in BoltzTraP.

It may be more instructive to look at S vs T and doping concentration. I suggest you plot S vs doping concentration and overlay the curves for a series of temperatures, e.g. T = 300 K, 400 K, 500 K etc.

Be aware that BoltzTraP assumes that the material's underlying electronic structure is unchanged by doping. In other words, it assumes that, when you dope the material, all that happens is conduction states are occupied (n-dopants) or valence states are unoccupied (p-dopants). This is perfectly reasonable for small doping concentrations, but in thermoelectric applications it is common to dope the material very heavily, sometimes almost to the solubility limit, and BoltzTraP's assumptions could break down. I recommend you benchmark the results against a separate calculation with some explicit dopants in the model (i.e. actually make the substitution in your DFT program, or whatever you're using to compute the electronic structure).

Finally, I recommend BoltzTraP2 in general, as a superior tool. Not only does it have all the functionality of BoltzTraP, but it has some additional functionality such as being able to plot the interpolated bands, which is extremely useful for checking that the interpolation is sensible.