The Electrical Properties of Solids — Question 8

Question 8

A temperature sweep separates activation from a geometrical resistance change

An insulating specimen has $\rho(T)=\rho_a e^{E_a/(k_BT)}$, with constant prefactor and negligible thermal expansion. Measured resistances satisfy $R(300\,\mathrm K)/R(400\,\mathrm K)=10$. Use $k_B=8.617\times10^{-5}\,\mathrm{eV/K}$.

1. Infer the activation energy.

2. Find the ratio $R(400\,\mathrm K)/R(600\,\mathrm K)$.

Question 8 - Solution

A temperature sweep separates activation from a geometrical resistance change

(a) Method and calculation

$$\ln(R_{300}/R_{400})=\frac{E_a}{k_B}(1/300-1/400).$$$$\boxed{E_a=1200k_B\ln10=.2381\,\mathrm{eV}}.$$

(b) Interpretation and further calculation

$$1/400-1/600=1/1200.$$$$\boxed{R_{400}/R_{600}=e^{E_a/(1200k_B)}=10}.$$

Physical check

The equal ratios follow equal differences in inverse temperature, not equal temperature increments. This activation model does not describe an ordinary metal over the same interval.