Determine the amount (in grams) of boron that when substitutionally incorporated into of silicon will establish a charge carrier density of carriers/cm .
To determine the amount of boron (B) that, when substitutionally incorporated into a crystal of silicon (Si), will establish a desired charge carrier density (N), you need to use the equation related to the intrinsic carrier concentration:
n_i^2 = 2 * (N_d * N_a) * exp(-Eg / (2 * k * T))
Where:
- n_i is the intrinsic carrier concentration
- N_d is the donor concentration
- N_a is the acceptor concentration
- Eg is the energy bandgap of silicon
- k is the Boltzmann constant
- T is the absolute temperature in Kelvin
In this case, the desired charge carrier density is provided in "carriers/cm^3." To convert it to "carriers/m^3", you need to multiply by 1e6.
Now let's go step-by-step with the calculations:
1. Convert the charge carrier density from "carriers/cm^3" to "carriers/m^3":
- Multiply the charge carrier density by 1e6.
2. Calculate the intrinsic carrier concentration (n_i):
- Look up the value for n_i at the given temperature or interpolate if needed.
- Rearrange the equation to solve for n_i:
n_i = √(2 * (N_d * N_a) * exp(-Eg / (2 * k * T)))
3. Rearrange the equation to solve for the amount of boron (N_d):
- Rearrange the equation to solve for N_d:
N_d = n_i^2 / (2 * N_a * exp(-Eg / (2 * k * T)))
4. Calculate the amount of boron in grams:
- Use the molar mass of boron (B) to convert from moles to grams:
Mass (g) = (N_d * Molar mass (g/mol)) / Avogadro's number
By following these steps, you will be able to determine the amount of boron (in grams) needed to establish the desired charge carrier density. Remember to substitute the appropriate values for N, T, N_a, Eg, k, and the molar mass of boron.