A Microscopic View of Ohm's Law — Question 5

Question 5

A scattering law produces a residual resistivity and a temperature crossover

A model metal has fixed n and m, with $1/\tau(T)=a+bT^2$, where a,b>0. Define $K=m/(ne^2)$.

1. Find resistivity and its logarithmic temperature slope.

2. Find the temperature where residual and temperature-dependent contributions are equal, and the slope there.

Question 5 - Solution

A scattering law produces a residual resistivity and a temperature crossover

(a) Method and calculation

$$\rho(T)=K(a+bT^2),\quad\rho(0)=Ka.$$$$\frac{d\ln\rho}{d\ln T}=\frac{T}{\rho}\frac{d\rho}{dT}=\frac{2bT^2}{a+bT^2}.$$

(b) Interpretation and further calculation

$$a=bT_*^2\Rightarrow T_*=\sqrt{a/b}.$$$$\rho(T_*)=2Ka,\quad\left.\frac{d\ln\rho}{d\ln T}\right|_{T_*}=1.$$

Physical check

The logarithmic slope interpolates from zero to two; it is not fixed by the power of the temperature-dependent term alone.