Boron Diffusion in n-type Silicon: Impurity Concentration & Time

[kidia]

I have a question here,
A uniform doped n-type silicon substrate of 0.1ohm-cm resistivity is to be subjected to a boron diffusion with constant surface concetration of 4.8*10^17/cm cubic.The desired junction depth is 2.8*10^-6m .Calculate the impurity concetration for the boron diffusion as a function of distance from the surface and how long will it take to cover the distance if the temperature at which this diffusion is conducted is a 1100Cetigrade

 

[Jhero]

Hello,

Thank you for your question. To calculate the impurity concentration for the boron diffusion as a function of distance from the surface, we can use the following equation:

N(x) = N0*exp(-x/δ)

Where N(x) is the impurity concentration at a distance x from the surface, N0 is the initial impurity concentration (4.8*10^17/cm^3 in this case), and δ is the junction depth (2.8*10^-6m in this case).

Plugging in the values, we get:

N(x) = 4.8*10^17*exp(-x/2.8*10^-6)

To calculate the time it will take to cover the desired junction depth, we can use Fick's Second Law:

δ = √(Dt)

Where δ is the junction depth, D is the diffusion coefficient, and t is the time.

We can find the diffusion coefficient using the following equation:

D = D0*exp(-Ea/kT)

Where D0 is the pre-exponential factor, Ea is the activation energy, k is the Boltzmann constant, and T is the temperature in Kelvin.

Assuming a pre-exponential factor of 10^3 cm^2/s and an activation energy of 3.5 eV, we can calculate the diffusion coefficient at 1100 degrees Celsius (1373 Kelvin):

D = 10^3*exp(-3.5/8.62*10^-5*1373) = 8.54*10^-5 cm^2/s

Now, plugging in the values for δ and D, we can solve for t:

2.8*10^-6 = √(8.54*10^-5*t)

t = (2.8*10^-6)^2/(8.54*10^-5) = 9.17*10^-10 seconds

Therefore, it will take approximately 9.17*10^-10 seconds or 0.917 nanoseconds to cover the desired junction depth at a temperature of 1100 degrees Celsius.

I hope this helps answer your question. Let me know if you have any further inquiries.