Reconciling two views of Matthiessen's rule for electron mobility

Question

My understanding of Matthiessen's rule is that when electron conduction in a solid is inhibited by more than one mechanism, that the overall resistivity $\rho$ is formed as if these resistance source were added "in series," $\rho = \sum_i \rho_i $, where $i$ enumerates the distinct resistive phenomena. This is easy for me to conceptualize by associating each resistivity with a scattering rate $\rho_i\propto\nu_i$. It seems reasonable then that the total resistivity is proportional to a total scattering rate $\rho\propto\nu$ formed from a weighted sum of the individual collision rates.

Now let's say I want to abstractly represent a solid as a circuit element, where each scattering mechanism is a resistor that contributes to the overall resistance of the solid: one for scattering off of ions, one for scattering off phonons, etc. The argument above says I should lay these resistors in series.

If we didn't have that knowledge, though, one might hypothesize that for a homogeneous solid, the electron current is split up equally amongst the different scattering mechanisms. This would lead to a circuit representation with the different scattering mechanisms' resistances added in parallel. Is there a way to deduce that this is wrong without appealing to kinetic theory ($\rho_i\propto\nu_i$)?

Answer 1

First, if we take the Matthiessen's rule formula, $\frac 1 {\mu}=\frac 1 {\mu_{impurities}}+\frac 1 {\mu_{lattice}}$ and take into account that the resistivity is inversely proportional to the mobility, we'll get $\rho=\rho_{impurities}+\rho_{lattice}$, which implies a series connection of resistors representing various scattering mechanisms.

We can also observe that, in the course of passing though a resistor, each electron will be subject to multiple scattering events associated with both (all) scattering mechanisms. Obviously, this would not be possible if the current was split into several branches, each representing a specific scattering mechanism.

We'd rather model a resistor as a chain of resistors, each representing a particular scattering mechanism, with the value of each resistor corresponding to the relative contribution of that mechanism to the total rate of scattering.