As per my understanding, the electrical conductivity of a material can be expected to increase with the increased carrier concentration. However, beyond a certain (extreme) point, the electrical conductivity should peak and begin to decrease with increasing carrier concentration due to the increased scattering.

The problem is that, as reported in multiple articles (for example, Figure 2 of Reference 1 and Figure 7 of Reference 2), as well as my own personal experience, BoltzTraP2 (which operates under the constant relaxation time approximation) is capable of capturing a similar effect.

Given that the program solves the Boltzmann's transport equation at a constant relaxation time, this effect of non-monotonic variation in electrical conductivity with carrier concentration can be concluded to occur even without a change in electron scattering rates.

Therefore, is there any other explanation to the non-monotonic trend in electrical conductivity with increasing carrier concentration?

Note: I am aware that the chemical potential and carrier concentration aren't the same thing and that they are related by the Fermi-Dirac distribution. However for a given material, an increase in chemical potential would occur with an increase in carrier concentration right?

I have also contacted the BoltzTraP2 developers on this issue, however I am still unable to understand the exact reasoning behind this.


[1] Tang, C.; Huang, Z.; Pei, J.; Zhang, B.-P.; Shang, P.-P.; Shan, Z.; Zhang, Z.; Gu, H.; Wen, K. Bi2Te3 Single Crystals with High Room-Temperature Thermoelectric Performance Enhanced by Manipulating Point Defects Based on First-Principles Calculation. RSC Advances, 2019, 9, 14422–14431. DOI

[2] Pandit, A.; Haleoot, R.; Hamad, B. Thermal Conductivity and Enhanced Thermoelectric Performance of SnTe Bilayer. Journal of Materials Science, 2021, 56, 10424–10437. DOI

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    $\begingroup$ It has been few months now, I would like to know whether you got any answers for this question? or new insights that you could share. Thank you $\endgroup$ Dec 9, 2023 at 9:40

1 Answer 1


In case someone else starts wondering on the same phenomenon, what I figured out is as follows.

When observing the band structure of the material, it should be noted that only electronic states with energies close to the chemical potential contribute to electrical conductivity. The "filled" states significantly below the chemical potential have practically zero values of the derivative of the Fermi-Dirac distribution entering the integrals and are therefore suppressed (refer to equations 8-12 in ref [1]). If this were not the case, the core states of electrons would hold a contribution to the electron transport. At moderate carrier concentrations, the high group velocity of the bands coupled with the increased amount of charged carriers would result in a higher conductivity. However, as the carrier concentration increases further, the high carrier concentration bands which show a complex nature (entanglement, multiband, non-parabolic band etc.) could result in a lower group velocity. This would therefore reduce the overall electrical conductivity of the material.

[1] Georg K. H. Madsen, Jesús Carrete, Matthieu J. Verstraete, BoltzTraP2, a program for interpolating band structures and calculating semi-classical transport coefficients, arXiv:1712.07946. DOI

  • $\begingroup$ Thank you for answering this question, with your explanation, I would like to ask whether the carrier type (p or n) would have any effect? $\endgroup$ Dec 13, 2023 at 4:12
  • $\begingroup$ @JaafarMehrez, no. There is no effect based on the carrier type. $\endgroup$
    – PBH
    Dec 17, 2023 at 2:45

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