Accurate derivation of electron-phonon scattering contribution to metal resistivity

Question

My lecture derived the expression for this contribtuin using the collision integral approach but I missed lot of details. He considers the lowest order correction to distribution function $f=f_0+\delta f_p$, where $f_0$ is the equilibrium function and "guesses" the expression for $\delta f=(-\partial f_0/\partial\epsilon)T\chi$. Then he writes down the expression of collision integrals for phonon-electron interaction. After several approximations, he represents collision integral as $-\hat{\Omega}\chi_p$ where $\hat{\Omega}$ plays the role of operator in "Hilbert space". After that, he expands $\chi_p$ by a basis $\lbrace\phi_l(p)\rbrace$ and then consider only the one function $\phi(p)$ in a basis.

Of course, I lost lot of details but the key idea is to reduce the problem to "eigenvalue" problem and solve it. For me, it seems very unnatural.

Hence, I will be very grateful for how to obtain the expression for $\rho$ wich has the correct behavior $\sim T$ for high temperatures and $\sim T^5$ for low temperatures.

Answer 1

I can maybe give a quick summary of the logic behind these calculations. The Boltzmann equation is, schematically, $$ lhs=rhs $$ where $lhs=\partial_0f_p+\ldots$ are the streaming terms, and $rhs=C[f_p]$ is the collision term.

1) We are seeking a solution of the form $f_p=f^0_p+\delta f_p$. What is the small parameter in this expansion? There are two length scales in the Boltzmann equation. On the $lhs$ we have $\vec{E}=\vec\nabla\Phi\sim\Phi/L$. On the $rhs$ we have a collision length $l$ (roughly, $C[f_p]\sim \delta f_p/\tau\sim v_p\delta f_p/l$). The expansion parameter is the ration of these two numbers $$ \delta f_p/f_p\sim Kn \sim l/L $$ where $Kn$ is the Knudsen number.

2) On the $lhs$ we have the Lorentz force acting on particles (this is a force term $\sim E\cdot\nabla_p f_p$). We expand and get something proportional to $$ v_p\cdot E (\partial f^0_p)/(\partial \epsilon_p) $$
3) On the $rhs$ we expand the collision integral $$ C[f^0_p+\delta f_p] \sim C_L[\delta f_p] $$ where we have used that $C[f^0_p]$ vanishes (equilibrium), and $C_L$ is a linearized collision operator.

4) The amusing observation about $C_L$ is that it is an anti-hermitean operator with respect to an inner product of the form $$ \langle\chi|\phi\rangle \sim \int f^0_p\chi_p\phi_p $$ This makes a certain amount of sense. Hermitean operators (in QM) give time evolution $\sim \exp(-iE_nt)$. The collision operator accounts for equilibration $\sim \exp(-t/\tau_n)$.

5) To study transport phenomena we have to look for the largest $t_n$ (just like in QM we look for the ground state, the lowest $E_n$), because this mode will dominate the relaxation process at large $t$. The corresponding eigenvector $\delta f_p\sim f^0_p\chi_p$ (factoring out $f^0$ or $(\partial f^0_p/\partial \epsilon_p)$ is a convenient normalization) gives $$ rhs = f^0_p\frac{\chi_p}{\tau} $$ and now I can just solve for $\chi_p$ (and then go ahead and compute the electric current). Clearly, $\chi_p\sim \tau/L\sim l/L$ as advertised.