This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
To determine the density of donor ions (N_D), we use the formula for the barrier potential (V_0) of a P-N junction: V_0 = V_T ((N_A N_D)/(n_i^2)) where V_T is the thermal voltage, N_A is the acceptor concentration, and n_i is the intrinsic carrier concentration. Given values from the problem: • Absolute temperature, T = 350 \, K • Barrier potential, V_0 = 0.7 \, V • Concentration of acceptor ions, N_A = 7.5 × 10^15 \, m^-3 • Intrinsic concentration, n_i = 5 × 10^13 \, m^-3 Constants (from question 4b): • Boltzmann constant, k = 1.38 × 10^-23 \, J/K • Electronic charge, q = 1.6 × 10^-19 \, C Step 1: Calculate the thermal voltage (V_T). The thermal voltage is given by V_T = (kT)/(q). V_T = (1.38 × 10^-23 \, J/K) × (350 \, K)1.6 × 10^-19 \, C V_T = 4.83 × 10^-21 \, J1.6 × 10^-19 \, C V_T = 0.0301875 \, V Step 2: Rearrange the barrier potential formula to solve for N_D. V_0 = V_T ((N_A N_D)/(n_i^2)) (V_0)/(V_T) = ((N_A N_D)/(n_i^2)) e^(V_0)/(V_T) = (N_A N_D)/(n_i^2) N_D = (n_i^2)/(N_A) e^(V_0)/(V_T) Step 3: Substitute the values and calculate N_D. First, calculate the exponent term: (V_0)/(V_T) = 0.7 \, V0.0301875 \, V ≈ 23.186087 e^(V_0)/(V_T) = e^23.186087 ≈ 1.1706 × 10^10 Next, calculate n_i^2: n_i^2 = (5 × 10^13 \, m^-3)^2 = 25 × 10^26 \, m^-6 = 2.5 × 10^27 \, m^-6 Now, substitute all values into the equation for N_D: N_D = 2.5 × 10^27 \, m^-67.5 × 10^15 \, m^-3 × (1.1706 × 10^10) N_D = (0.33333 × 10^12 \, m^-3) × (1.1706 × 10^10) N_D = 3.902 × 10^21 \, m^-3 Rounding to two significant figures: N_D ≈ 3.9 × 10^21 \, m^-3 3 done, 2 left today. You're making progress.