QB_PH_SEC7_012: Band Theory of Solids & Semiconductors

Silicon at $T = 300\text{ K}$ has an intrinsic carrier concentration of $n_i = 1.5 \times 10^{10}\text{ cm}^{-3}$. If it is doped with phosphorus atoms at a donor concentration of $N_D = 1.0 \times 10^{16}\text{ cm}^{-3}$, what is the minority hole concentration $p$ in $\text{cm}^{-3}$? (Enter your answer in scientific notation as $A \times 10^B$, where $A = 2.25$. Enter only the exponent $B$)

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Law of Mass Action**: In thermal equilibrium: $n \cdot p = n_i^2$ 2. **Extrinsic Majority Carrier Balance**: For complete donor ionization in an $n$-type semiconductor: $n \approx N_D = 1.0 \times 10^{16}\text{ cm}^{-3}$ 3. **Minority Hole Density Calculation**: $p = \frac{n_i^2}{N_D} = \frac{(1.5 \times 10^{10}\text{ cm}^{-3})^2}{1.0 \times 10^{16}\text{ cm}^{-3}} = \frac{2.25 \times 10^{20}}{1.0 \times 10^{16}} = 2.25 \times 10^4\text{ cm}^{-3}$ 4. **Exponent Question**: $B = 4$.