Insulators — Question 1

Question 1

A wide gap suppresses intrinsic carriers exponentially

Two ideal intrinsic solids have equal density-of-states prefactor C and gaps $E_{gA}=1.0\,\mathrm{eV}$ and $E_{gB}=3.0\,\mathrm{eV}$. At a chosen temperature $k_BT=.0250\,\mathrm{eV}$, use $n_i=C e^{-E_g/(2k_BT)}$.

1. Find their intrinsic-carrier density ratio.

2. If their electron and hole mobilities are also equal, find the resistivity ratio.

Question 1 - Solution

A wide gap suppresses intrinsic carriers exponentially

(a) Method and calculation

$$\frac{n_{iB}}{n_{iA}}=\exp\left[-\frac{E_{gB}-E_{gA}}{2k_BT}\right]=e^{-40}.$$$$\boxed{n_{iB}/n_{iA}=4.248\times10^{-18}}.$$

(b) Interpretation and further calculation

$$\sigma_i=en_i(\mu_n+\mu_p),\quad\rho_i=1/\sigma_i.$$$$\boxed{\rho_B/\rho_A=e^{40}=2.354\times10^{17}}.$$

Physical check

The comparison isolates band-gap effects. Defects, impurities, surfaces, and other conduction mechanisms can dominate measured insulating behavior.