A Microscopic View of Ohm's Law — Question 3

Question 3

Independent scattering mechanisms add rates rather than relaxation times

Two independent scattering mechanisms have times $\tau_1=30$ fs and $\tau_2=60$ fs. Use $1/\tau=1/\tau_1+1/\tau_2$ and $\rho=m/(ne^2\tau)$, with n and m unchanged.

1. Find the combined time and express total resistivity in terms of the two individual resistivities.

2. If the second mechanism is completely suppressed, find the conductivity increase factor.

Question 3 - Solution

Independent scattering mechanisms add rates rather than relaxation times

(a) Method and calculation

$$\tau=(1/30+1/60)^{-1}=20.0\,\mathrm{fs}.$$$$\rho=\frac{m}{ne^2}(\tau_1^{-1}+\tau_2^{-1})=\rho_1+\rho_2.$$

(b) Interpretation and further calculation

$$\tau_{\mathrm{new}}=30.0\,\mathrm{fs},\quad\sigma\propto\tau.$$$$\sigma_{\mathrm{new}}/\sigma_{\mathrm{old}}=30/20=1.50.$$

Physical check

Adding times would predict weaker scattering when a new scattering channel is introduced, the wrong trend.