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It is well-known that using the so-called Streda formula, the transversal conductivity $\sigma_{xy}$ and thus the Hall conductivity in a two-dimensional material is given as the derivative of the integrated density of states up to fermi energy with respect to a magnetic field

$$\sigma_{xy} = \sigma_{\text{Hall}} = \frac{\partial \rho(E_F)}{\partial B}$$ where $E_F$ is the Fermi energy.

Does there exist a similar formula for the longitudinal conductivity $\sigma_{xx}$ that in terms of the density of states?

Remark: I should say that looking at Ando's original paper, here, the answer to my question should be yes, since he has an expression relating the DOS with his quantity X in (2.5) and an expression for the longitudinal conductivity and X in (2.6). However, his formula is for a very special model, so there should be a generalization of that.

Disclaimer: Of course, I know that there exist the Kubo formulas from linear response theory that yield expressions for $\sigma_{xx}$, but I am looking for an expression in terms of the density of states.

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  • $\begingroup$ It appears Streda formula is valid only when the Fermi level is in the gap: canvas.harvard.edu/courses/5194/files/1278866/… . Check last paragraph of the very first section $\endgroup$
    – Pavlo. B.
    Commented Apr 14, 2021 at 20:38
  • $\begingroup$ "It should be emphasized that Streda’s formula applies only when the Fermi-level is in a true energy gap. It does not apply when the Fermi level is a region of localized states, because the derivative of the electron density with respect to the magnetic field in the limit of small but finite frequency does not then coincide with the derivative of the density with respect to the magnetic field in an equilibrium state." $\endgroup$
    – Pavlo. B.
    Commented Apr 14, 2021 at 20:41
  • $\begingroup$ I would think one can obtain a similar result for the longitudinal conductivity, but it will yield zero if the Fermi level is in the energy gap. $\endgroup$
    – Pavlo. B.
    Commented Apr 14, 2021 at 20:44
  • $\begingroup$ @Pavlo.B. you are correct about the applicability of said formula. It would be interesting to know for sure whether such a formula exists... $\endgroup$
    – Sascha
    Commented Apr 14, 2021 at 22:45
  • $\begingroup$ You mean a version of Streda formula that is valid for any material? I would be delighted if it does, but is there any reason to believe that there may be one? $\endgroup$
    – Pavlo. B.
    Commented Apr 15, 2021 at 3:15

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I suspect that there is no single formula with a well-known name. The formula cited in the OP is valid for a particular geometry: bulk material with constant electric field, constant magnetic field, and no interactions (which means one can use one-particle DOS). When studying conductivity one is usually interested in more complex situations: fields varying in time and space, heterostructures, etc.

The closes one comes to using the logic of the OP is when describing the ballistic conductance in one-dimensional structures, which is proportional to the electron group velocity $v(E)$ times the density of states, $\rho(E)$. Since the density of states is one dimension is inversely proportional to the group velocity, the two cancel out, resulting in conductance quantization: $$ v(E)\times \rho(E) = const $$

Another common case where the conductance is reduced to a density of states is the well-known Meir-Wingreen formula (also here) for transport through nanostructures. Its particularity is that in some cases it can be applied even when the interactions are present, as they did for Kondo effect.

Judging by this applications, expressing conductance in terms of the DOS in bulk materials is judged "obvious", even though it is embarrassing that one cannot readily come up with a reference/name for widely used formula.

Remark
Another area where one heavily focuses on the density of states in relation to conductance is studying the conductance through disordered materials, Anderson localization, weak localization, etc.

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  • $\begingroup$ I am not sure, I communicated my question properly. What I was wondering about is if there exists a formula for $\sigma_{xx}$ for bulk material with constant electric field, constant magnetic field and no interactions. $\endgroup$
    – Sascha
    Commented Apr 11, 2021 at 12:14
  • $\begingroup$ @Sascha yes, I understood your question, and I understand that my answer is not exactly what you are looking for. But it gives you directions where to look. Have you consulted the books by Imry or Datta on transport in semiconductors? If I have time to look, I will reedit my answer to be more to the point. $\endgroup$
    – Roger V.
    Commented Apr 11, 2021 at 12:19
  • $\begingroup$ @Sascha note that the ballistic conductance described in my answer is exactly what happens in IQHE - the conductance is quantized, because the strong magnetic field makes electrons effectively one-dimensional. My understanding is that Streda formula applies to QHE, even though it is not specified in the OP. $\endgroup$
    – Roger V.
    Commented Apr 12, 2021 at 11:47
  • $\begingroup$ sorry, I am asking only about the $\sigma_{xx}$ conductivity and I would prefer if your answer would address that quantity. $\endgroup$
    – Sascha
    Commented Apr 13, 2021 at 7:28

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