A p-n junction is obtained at a depth of 3 x 10-3 cm below the surface by diffusion of antimony (Sb) into p-type germanium. What is the donor concentration in the bulk germanium if the diffusion is carried out for three hours at 790 C? Express your answer in number of donor atoms per cubic centimeter. Assume the surface concentration of antimony is held constant at 8 x 1018 cm-3 and that the diffusivity at 790 C is 4.8 x 10-11 cm2/s.
To calculate the donor concentration in the bulk germanium, we need to use Fick's Second Law of Diffusion, which relates the diffusivity, time, depth, and concentration gradient.
The formula is:
D = (2 * ΔC * δ) / (t * π)
Where:
D = Diffusivity
ΔC = Change in concentration
δ = Depth of diffusion
t = Time
π = Pi (approximately 3.14159)
To solve for the change in concentration (ΔC), we rearrange the formula as follows:
ΔC = (D * t * π) / (2 * δ)
Now let's substitute the given values into the formula:
D = 4.8 x 10^-11 cm^2/s (Given)
t = 3 hours = 3 * 60 * 60 seconds = 10800 seconds
δ = 3 x 10^-3 cm (Given)
ΔC = (4.8 x 10^-11 cm^2/s * 10800 s * π) / (2 * 3 x 10^-3 cm)
= (1.6464 x 10^-6 * π) / 6 x 10^-3
Now we need to calculate the change in concentration (ΔC) in atoms/cm^3, using Avogadro's number (6.022 x 10^23 atoms/mol) and the molecular weight of antimony (Sb). The atomic weight of antimony is 121.76 g/mol.
Surface concentration of antimony = 8 x 10^18 cm^-3
ΔC (atoms/cm^3) = ΔC (cm^-3) * (1 mol / Molecular weight of Sb) * (Atoms / mol)
Now let's calculate ΔC (atoms/cm^3):
ΔC (cm^-3) = ΔC * 8 x 10^18 cm^-3 (Surface concentration of antimony)
ΔC (cm^-3) = (1.6464 x 10^-6 * π) / 6 x 10^-3 * 8 x 10^18 cm^-3
Finally, we need to calculate ΔC (atoms/cm^3):
ΔC (atoms/cm^3) = ΔC (cm^-3) * (1 mol / Molecular weight of Sb) * (Atoms / mol)
= ΔC (cm^-3) * 1 / (121.76 g/mol) * (6.022 x 10^23 atoms/mol)
Now we can calculate ΔC (atoms/cm^3) with the given values.
After substituting the given values and evaluating the expression, we get the donor concentration in the bulk germanium as the final result.