Calculation of Donor Atom Concentration (ND) in a Silicon P-N Junction

Given:
NA = 10^18/cm^3 (Acceptor atom concentration)
ni = 10^10/cm^3 (Intrinsic carrier concentration)
Vbi = 0.77 V (Built-in potential)
Temperature (T) = 300 K

Explanation:
The built-in potential (Vbi) of a P-N junction is given by the equation:

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

Where,
k = Boltzmann constant (1.38 * 10^-23 J/K)
T = Temperature in Kelvin
q = Elementary charge (1.6 * 10^-19 C)
NA = Acceptor atom concentration
ND = Donor atom concentration
ni = Intrinsic carrier concentration

We can rearrange the equation to solve for ND:

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

Calculation:
Substituting the given values into the equation:

ND = (10^10/cm^3)^2 * e^(0.77 V * 1.6 * 10^-19 C / (1.38 * 10^-23 J/K * 300 K)) / (10^18/cm^3)

Simplifying the equation:

ND = (10^20/cm^6) * e^(3.2 * 10^19 / 4.14 * 10^-5) / (10^18/cm^3)

Using the fact that 1 cm^3 is equal to 10^6 nm^3:

ND = (10^20/nm^6) * e^(3.2 * 10^19 / 4.14 * 10^-5) / (10^12/nm^3)

ND = (10^8/nm^3) * e^(3.2 * 10^19 / 4.14 * 10^-5)

Calculating the exponential term:

e^(3.2 * 10^19 / 4.14 * 10^-5) ≈ 1.504 * 10^26

Substituting the exponential term back into the equation:

ND ≈ (10^8/nm^3) * 1.504 * 10^26

ND ≈ 1.504 * 10^34/nm^3

Conclusion:
The donor atom concentration (ND) in the silicon P-N junction to achieve a built-in potential of 0.77 V, with an acceptor atom concentration (NA) of 10^18/cm^3 at 300 K, is approximately 1.504 * 10^34/nm^3.