Longitudinal conductivity from density of states (DOS)

Question

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.

Answer 1

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.