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Suppose there's a metal in which electrons interact with themselves and with the phonons. The hamiltonian might look like this

\begin{equation} H= \sum_{k}\epsilon_k c^\dagger_k c_k + \sum_{k}\omega_ka^\dagger_ka_k+H_{e-e}+H_{e-p} \end{equation}

where $\epsilon_k$ is the energy of the electrons, $\omega_k$ is the energy of the phonons and the other two terms are interaction terms. Without going into the details, let's say that I calculate the resistivity only due to e-e scattering and only due to e-p scattering and I get

\begin{equation} \rho_{ee}=a T^2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \rho_{ep}=b T \end{equation}

where $a$ and $b$ are just constants. The question is: How should I add these resistivities? Is it reasonable to just say $\rho_{total}=\rho_{ee}+\rho_{ep}$? I'm trying to avoid a calculation where I need to treat both scattering processes simultaneously, of course.

Some extra information: The e-e resistivity is calculated in the context of Umklapp scattering at low temperatures when the Fermi surface has a similar size as the Brillouin zone. The e-p resistivity is the tipical calculation at high temperatures when the phonons become classical. There's a temperature range where both effects coincide and that's where I'm trying to get the total resistivity.

P.S.: I read about Matthiesen rule and also, in a different place, about the "parallel resistor" formula but I'm not sure how to apply those concepts here. (Just naming those terms in case they ring a bell).

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I would recommend reading more about Matthiessen's rule, preferably the section on it in Ashcroft and Mermin's near-indispensable textbook. In short the idea is that scattering rates from distinguishable scattering mechanisms (as you seem to have here) are additive. In the relaxation-time approximation that implies that the total relaxation time $\tau$ satisfies $$\frac{1}{\tau}=\sum_i \frac{1}{\tau_i}.$$ I.e. the relaxation times add like parallel resistances. If none of the $\tau_i$ depends on the momentum $k$ we obtain Matthiessen's rule for the resistivity: $$\rho=\sum_i \rho_i.$$ In other words, physical resistivities add like serial resistors, just like you proposed. If you think about it, this is quite natural. A bunch of electron going from point A to point B in a slab of the material will experience both scattering mechanisms - it can't split up and have a portion of it undergo only electron-phonon scattering.

However, there's a big assumption made above: that $\tau_i$ is $k$-independent for each $i$. When this doesn't hold Matthiessen's rule becomes an inequality $$\rho \ge \sum_i \rho_i.$$


We can obtain also obtain a formal upper bound. There's a variational result due to Kohler [cited as Z. Phys. 126, 495 (1949)], described in Ziman's book Electrons and Phonons (1960). It's an interesting result for metals, but I don't think it's particularly useful for microscopic descriptions/lattice system.

Introduce a function $\Phi_\mathbf{k}$ such that the electron distribution becomes $$f_\mathbf{k}\equiv f_\mathbf{k}^0 - \Phi_\mathbf{k} \frac{\partial f_\mathbf{k}^0}{\partial \epsilon_\mathbf{k}}.$$ We can view $\Phi_\mathbf{k}$ as the deviation from equilibrium, or the extra energy due to the transport process. For two given scattering mechanisms $P_1$, $P_2$ we can associate two exact solutions $\Phi_1$, $\Phi_2$. The resistivity of each mechanism in isolation can be written as an inner product, $\rho_i=\langle \Phi_i | P_1 \Phi_i\rangle$. Forming the linear combination $$\Phi' = \eta \Phi_1 + \left( 1-\eta\right) \Phi_2$$ we have that $$\rho \le \langle \Phi' | \left( P_1 + P_2 \right) \Phi'\rangle.$$

Minimizing with respect to $\eta$ we obtain the bound $$\rho - \left( \rho_1 + \rho_2 \right) \le \beta_1 \beta_2 \frac{\rho_1 \rho_2}{\beta_1 \rho_1 + \beta_2 \rho_2},$$ where $\langle \Phi_j|P_i\Phi_j\rangle = \left( 1+ \beta_i\right)\rho_i$.

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  • $\begingroup$ Great Answer! Is there a way to get an upper limit on the resistivity as well? Or adding them up gives a lower boundary on resistivity and that’s as far as we can go? $\endgroup$ Commented Aug 29, 2019 at 13:37
  • $\begingroup$ @P.C.Spaniel I'm not aware of any general results along those lines. The upside is that the inequality actually doesn't rely on the relaxation-time approximation - it can be derived under more general conditions than the equality. If you need a precise value for the resistivity (rather than rough temperature scaling trends) it'd likely be better to use something like Kubo's formula. $\endgroup$
    – Anyon
    Commented Aug 29, 2019 at 13:59
  • $\begingroup$ @P.C.Spaniel I did find an upper bound in Ziman's old book. I think it's mostly useful in analytically tractable cases, but if you have an idea about how far your system is from equilibrium it does let you go slightly further. $\endgroup$
    – Anyon
    Commented Sep 2, 2019 at 18:21

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