Resistance of Si cube doped with Sb

Assume: 1 cm cube ($L = 1$ cm, $A = 1$ cm$^2$), $n_i = 1.45\times10^{10}$ cm$^{-3}$, $\mu_e = 1350$, $\mu_h = 450$ cm$^2$V$^{-1}$s$^{-1}$ at 300 K.

Pure silicon

$$\begin{aligned} \sigma_i &= en_i(\mu_e+\mu_h) = 1.602\times10^{-19}\times1.45\times10^{10}\times(1350+450) \\ &= 4.18\times10^{-6}\ \Omega^{-1}\text{cm}^{-1} \\ R &= \frac{\rho L}{A} = \frac{1}{\sigma_i}\times\frac{1\ \text{cm}}{1\ \text{cm}^2} = 2.39\times10^{5}\ \Omega \end{aligned}$$

Doped with 1 Sb per $10^9$ Si atoms

Sb is a donor:

$$\begin{aligned} N_d &= \frac{5\times10^{22}}{10^9} = 5\times10^{13}\ \text{cm}^{-3}, \quad n \approx N_d \\ p &= \frac{n_i^2}{N_d} = 4.2\times10^{6}\ \text{cm}^{-3}\ (\text{negligible}) \\ \sigma &= eN_d\mu_e = 1.602\times10^{-19}\times5\times10^{13}\times1350 = 1.081\times10^{-2}\ \Omega^{-1}\text{cm}^{-1} \\ R' &= \frac{1}{\sigma} = 92.5\ \Omega \end{aligned}$$

Change in resistance

$$\Delta R = 2.39\times10^{5} - 92.5 \approx 2.39\times10^{5}\ \Omega \ (\text{decrease}), \qquad \frac{R}{R'} = 2586$$

Answer: Pure Si: $2.39\times10^{5}$ $\Omega$; doped: 92.5 $\Omega$. Resistance drops by about $2.39\times10^5$ $\Omega$ (about 2600 times), even though only one atom in a billion is replaced.