To find the boundary of the depletion layer in the n-region (Xn) of a pn junction at room temperature, we can use the following formula:
Xn = sqrt((2 * ε_si * (Vbi + VR))/(q * (1/N_A + 1/N_D)))
Where:
- Xn is the boundary of the depletion layer in the n-region
- ε_si is the permittivity of silicon (11.8 in this case)
- Vbi is the built-in voltage
- VR is the reverse bias voltage
- q is the charge of an electron (1.60E-19 Coulombs)
- N_A is the acceptor concentration
- N_D is the donor concentration
Now, let's calculate the values needed to find Xn:
Step 1: Calculate the built-in voltage (Vbi)
Vbi = (k * T/q) * ln(N_A * N_D / ni^2)
Where:
- k is the Boltzmann constant (8.62E-05 eV/K)
- T is the temperature in Kelvin (300 K)
- ni is the intrinsic carrier concentration
Step 2: Calculate the intrinsic carrier concentration (ni)
ni = sqrt(N_C * N_V) * exp(-Eg / (2 * k * T))
Where:
- N_C is the effective density of states in the conduction band
- N_V is the effective density of states in the valence band
- Eg is the energy gap of silicon (1.12 eV)
Step 3: Calculate the reverse bias voltage (VR)
VR = 0 (since we are considering no reverse bias)
Step 4: Substitute the values into the formula for Xn
Xn = sqrt((2 * 11.8 * (Vbi + VR))/(1.60E-19 * (1/2.5E+18 + 1/7.0E+16)))
Now, let's calculate each step in detail:
Step 1: Calculate Vbi
Vbi = (8.62E-05 * 300)/1.60E-19 * ln((2.5E+18 * 7.0E+16)/(1.5E10)^2)
Vbi = 0.714 V
Step 2: Calculate ni
ni = sqrt(2.8E19 * 1.04E19) * exp(-1.12/(2 * 8.62E-05 * 300))
ni = 9.65E9/cm³
Step 3: Calculate VR
VR = 0
Step 4: Calculate Xn
Xn = sqrt((2 * 11.8 * (0.714 + 0))/(1.60E-19 * (1/2.5E+18 + 1/7.0E+16)))
Xn = 4.77E-5 cm
Therefore, the boundary of the depletion layer in the n-region (Xn) is approximately 4.77E-5 cm.
Main answer: The boundary of the depletion layer in the n-region (Xn) is approximately 4.77E-5 cm.
Explanation: To find Xn, we use the formula Xn = sqrt((2 * ε_si * (Vbi + VR))/(q * (1/N_A + 1/N_D))). We first calculate the built-in voltage (Vbi) using the formula Vbi = (k * T/q) * ln(N_A * N_D / ni^2). Then, we calculate the intrinsic carrier concentration (ni) using the formula ni = sqrt(N_C * N_V) * exp(-Eg / (2 * k * T)). Finally, we substitute the values into the formula for Xn and calculate it.