1. **Problem Statement:** Design a p-type silicon semiconductor that maintains a constant conductivity of 100 $\Omega^{-1} \cdot \text{cm}^{-1}$ over a range of temperatures and comment on the purity level required. 2. **Relevant Formula:** The conductivity $\sigma$ of a semiconductor is given by: $$\sigma = q \cdot p \cdot \mu_p$$ where: - $q$ is the charge of an electron ($1.6 \times 10^{-19}$ C), - $p$ is the hole concentration (number of acceptor atoms for p-type), - $\mu_p$ is the hole mobility. 3. **Key Points:** - For p-type silicon, holes are majority carriers introduced by acceptor impurities. - Mobility $\mu_p$ decreases with increasing temperature due to increased phonon scattering. - Intrinsic carrier concentration $n_i$ increases with temperature, which can affect conductivity. - To maintain constant conductivity, the product $p \cdot \mu_p$ must remain roughly constant. 4. **Approach:** - Since $\mu_p$ decreases with temperature, $p$ (hole concentration) must increase to compensate. - However, $p$ is set by doping concentration (acceptor level $N_A$), which is fixed. - To keep conductivity constant, doping must be high enough so that intrinsic carriers are negligible compared to doped holes over the temperature range. 5. **Estimating Required Doping Level:** - Intrinsic carrier concentration $n_i$ in silicon at room temperature (~300 K) is about $1.5 \times 10^{10} \text{cm}^{-3}$ and increases exponentially with temperature. - To ensure $p \approx N_A$ dominates, $N_A$ must be much greater than $n_i$ at the highest temperature considered. 6. **Calculating Hole Concentration:** Given $\sigma = 100$ and assuming room temperature hole mobility $\mu_p \approx 450 \text{cm}^2/\text{V}\cdot\text{s}$, $$p = \frac{\sigma}{q \mu_p} = \frac{100}{(1.6 \times 10^{-19})(450)} \approx 1.39 \times 10^{18} \text{cm}^{-3}$$ 7. **Interpretation:** - The doping concentration $N_A$ should be about $1.4 \times 10^{18} \text{cm}^{-3}$ to achieve the target conductivity at room temperature. - This is a moderately high doping level, ensuring holes dominate over intrinsic carriers even at elevated temperatures. 8. **Purity Level Comment:** - To maintain constant conductivity, the silicon must be very pure except for the controlled acceptor doping. - Unintentional impurities or defects that introduce compensating donors or traps must be minimized. - High purity and precise doping control are essential to keep conductivity stable over temperature. **Final answer:** A p-type silicon semiconductor doped with acceptor concentration approximately $1.4 \times 10^{18} \text{cm}^{-3}$ will provide a conductivity near 100 $\Omega^{-1} \cdot \text{cm}^{-1}$ at room temperature and maintain it over a temperature range if the material purity is high and intrinsic carrier effects are minimized.