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 find the donor concentration in the bulk germanium, we can use Fick's second law of diffusion, which describes the diffusion of dopant atoms into a material.
Fick's second law of diffusion is given as:
dC/dt = D * (d^2C/dx^2)
Where:
- dC/dt is the rate of change of dopant concentration with time,
- D is the diffusivity of the dopant atoms in the material, and
- d^2C/dx^2 is the second derivative of concentration with respect to position x.
In this case, we know the surface concentration of antimony (Sb) is 8 x 10^18 cm^-3 and that the diffusion is carried out for three hours at 790°C. We are asked to find the donor concentration in the bulk germanium.
To solve this problem, we need to integrate Fick's second law of diffusion with respect to time (t) and position (x), and then use the given values to solve for the donor concentration.
Let's break down the problem step by step:
Step 1: Calculate the diffusion length (L):
The diffusion length (L) can be determined using the following equation:
L = sqrt(4 * D * t)
Where D is the diffusivity of Sb atoms in germanium and t is the diffusion time.
Substituting the given values:
D = 4.8 x 10^-11 cm^2/s
t = 3 hours = 3 * 60 * 60 seconds
L = sqrt(4 * 4.8 x 10^-11 cm^2/s * 3 * 60 * 60 seconds)
L = sqrt(2.304 x 10^-8 cm^2) ≈ 4.8 x 10^-4 cm
Step 2: Calculate the effective surface concentration (Ceff):
The effective surface concentration (Ceff) is the difference between the original surface concentration (Cs) and the concentration at the diffusion length (L):
Ceff = Cs - (Cs - Cbulk) * exp(-L/LD)
Where Cs is the surface concentration, Cbulk is the bulk concentration, and LD is a length constant equal to sqrt(D * t).
Substituting the given values:
Cs = 8 x 10^18 cm^-3
Ceff = 8 x 10^18 cm^-3 - (8 x 10^18 cm^-3 - Cbulk) * exp(-(4.8 x 10^-4 cm) / sqrt(4.8 x 10^-11 cm^2/s * 3 * 60 * 60 seconds))
Step 3: Calculate the bulk concentration (Cbulk):
Since the question asks for the donor concentration in terms of number of donor atoms per cubic centimeter, we can express Cbulk as atoms/cm^3 by multiplying it by the Avogadro's number (NA):
Cbulk (atoms/cm^3) = Cbulk (cm^-3) * NA
Substituting the given values:
Cbulk (atoms/cm^3) = Cbulk (cm^-3) * 6.022 x 10^23 atoms/mol
Step 4: Solve for the bulk concentration (Cbulk):
To solve for Cbulk, we need one more equation. This equation relates the diffusion length (L) and the bulk concentration (Cbulk):
L = sqrt(D * t) * sqrt(Ceff / Cbulk - 1)
Substituting the given values:
(4.8 x 10^-4 cm) = sqrt(4.8 x 10^-11 cm^2/s * 3 * 60 * 60 seconds) * sqrt((8 x 10^18 cm^-3 - Cbulk) / Cbulk - 1)
Now, we have two equations. Let's combine them:
First equation:
Ceff = 8 x 10^18 cm^-3 - (8 x 10^18 cm^-3 - Cbulk) * exp(-(4.8 x 10^-4 cm) / sqrt(4.8 x 10^-11 cm^2/s * 3 * 60 * 60 seconds))
Second equation:
(4.8 x 10^-4 cm) = sqrt(4.8 x 10^-11 cm^2/s * 3 * 60 * 60 seconds) * sqrt((8 x 10^18 cm^-3 - Cbulk) / Cbulk - 1)
We need to solve these equations simultaneously to find the value of Cbulk. However, this requires iterative numerical methods or specialized software.