Depletion Width Calculation for a Silicon P-N Junction

Given Data:
- ND (Donor concentration) = 10^16/cm^3
- NA (Acceptor concentration) = 10^18/cm^3
- Temperature (T) = 300 K
- Intrinsic carrier concentration (ni) = 10^10 cm^-3
- Permittivity of free space (ε0) = 8.854 x 10^-14 F/cm
- Relative permittivity of Silicon (εSi) = 11.7

Depletion Width (W) Formula:
The depletion width of a P-N junction can be calculated using the formula:

W = sqrt((2 * εs * (Vbi + VR)) / (e * (1/ND + 1/NA)))

Where:
- W is the depletion width
- εs is the effective permittivity of the semiconductor
- Vbi is the built-in potential
- VR is the applied reverse bias voltage
- e is the elementary charge (1.6 x 10^-19 C)

Calculation Steps:

Step 1: Calculate the Effective Permittivity (εs)
The effective permittivity (εs) is calculated by multiplying the permittivity of free space (ε0) by the relative permittivity of Silicon (εSi).

εs = ε0 x εSi
= (8.854 x 10^-14 F/cm) x (11.7)
= 1.034 x 10^-12 F/cm

Step 2: Calculate the Built-in Potential (Vbi)
The built-in potential (Vbi) is calculated using the following formula:

Vbi = (k * T / e) * ln(NA * ND / (ni^2))

Where:
- k is Boltzmann's constant (8.617 x 10^-5 eV/K)
- T is the temperature in Kelvin
- e is the elementary charge (1.6 x 10^-19 C)
- ln is the natural logarithm function

Vbi = (8.617 x 10^-5 eV/K * 300 K / 1.6 x 10^-19 C) * ln((10^18/cm^3 * 10^16/cm^3) / (10^10 cm^-3)^2)
= 0.026 V * ln(10^34 / 10^20)
= 0.026 V * ln(10^14)
= 0.026 V * 31.83
= 0.827 V

Step 3: Calculate the Depletion Width (W)
Substituting the values into the depletion width formula:

W = sqrt((2 * εs * (Vbi + VR)) / (e * (1/ND + 1/NA)))
= sqrt((2 * 1.034 x 10^-12 F/cm * (0.827 V + VR)) / (1.6 x 10^-19 C * (1/10^16/cm^3 + 1/10^18/cm^3)))

Let's assume VR = 0 V for simplicity:

W = sqrt((2 * 1.034