Non uniformly distributed p-n junction related problem

[THE HARLEQUIN]

Hello guys, I stumbled upon this problem while studying non uniformly distributed pn junctions and finding difficulty solving this. Any help will be greatly appreciated.

A diffused silicon p-n junction has a linearly graded junction on the p side with a = 2 x10^19 cm-4, and a uniform doping of 10^15 cm-3 on the n side. If the depletion width on the p side is 0.7 micro meter at zero bias, find the total depletion width, built-in potential, and maximum electric field at zero bias.

 

[eq1]

This is a fairly standard homework question. What have you tried? Did you find this in a textbook? Was there a figure that gave the slope of the gradient? Usually to solve it one needs the slope of the gradient like in this question.
https://www.chegg.com/homework-help/questions-and-answers/3-15p--diffused-silicon-p-n-junction-linearly-graded-junction-p-side-na-ax-x-distance-10-1-q20562763

 

[THE HARLEQUIN]

This is a fairly standard homework question. What have you tried? Did you find this in a textbook? Was there a figure that gave the slope of the gradient? Usually to solve it one needs the slope of the gradient like in this question.
https://www.chegg.com/homework-help/questions-and-answers/3-15p--diffused-silicon-p-n-junction-linearly-graded-junction-p-side-na-ax-x-distance-10-1-q20562763

Thanks eq1 for your reply. I tried to solve the built in potential part, though not sure if it's the correct approach. The gradient is given in the question as the parameter 'a' . the problem is from Neamen's semiconductor physics and devices exercise. I attached the photos of my solution. Please share your valuable thoughts on this.

 

[eq1]

I didn't understand the first two images, but the third looks like it has the correct approach; rho (charge density) is modified to include the ax term (it has a linear dependency and is not just a constant), then integrate charge to get the E-field and integrate that to get the built-in potential. I haven't checked the math or the constants you plugged in though. (The obvious thing is there needs to be an x^2 term in E-field which I saw, so I suspect it's ok)