Homework 3. Due Feb 3

0. Watch the videos on pn junction dynamics. Post a question, comment or correction. Also, you can engage in discussion with other students in the comments.

For the rest of the problems in this assignment, let's imagine a semiconductor (with KT=0.025 eV) with the following density of states:
\(D(E) = 12*10^{22}\, states/(eV cm^{3})\)  for 1 eV < E < 5 eV ...  (conduction band)
Additionally in the range -4 eV to 0 eV,
\(D(E) = 12*10^{22}\, states/(eV cm^{3})\)  for -4 eV < E < 0 eV ...  (valence band).

1. Suppose this semiconductor is uniformly doped with 10^18 donors per cm^3 .
a) What would n have to be for that doping?
b) What would \(E_F\) have to be for that doping?
c) What would p, the minority carrier density, be for that doping?

2. Suppose, on the other hand, it is uniformly doped with 10^18 acceptors per cm^3 .
a) What would p be for that doping?
b) What would \(E_F\) be for that doping?
c) What would n, the minority carrier density, be for that doping?

3. (Some of these parts may take a long time to understand and answer)
Consider a p-n junction made using the parts as above (10^18 doping on each side).
a) What is the total amount of band bending you would get?
b) What is the depletion length? (email me this number as soon as you get it so I can gauge your progress. Feel free to include a discussion of unknown or ambiguous quantities.)
c) What is the value of the electric field at the interface? (x=0)
d) What is the value of n(x) at x=0?  (this is difficult. (maybe not as difficult as i thought...))
e) Sketch a rough graph of n(x). What is its value at \(x=-x_d\)? What is its value at \(x=+x_d\)?
f) What is the magnitude of the drift current at x=0?
g) What is the magnitude of the diffusion current at x=0?
h) Sketch a rough graph of the derivative of n(x) with respect to x (as a function of x).

4. extra credit. At what value of x does the maximum diffusion current occur.  Sketch a graph of the diffusion current as a function of x.

5. extra credit. (warning, this might be messy) In 3c you calculated the depletion length for this particular doping (an actual number in nm or cm). Additionally, so that you can understand the dependencies of \(x_d\) on various parameters, please obtain an expression for the depletion length as a function of like D(Ec), doping, kT and whatever else it depends on. To simplify the calculation a bit, please assume that the density of states is constant within each band, as above, and the D(Ec)=D(Ev);  also assume that the doping is equal and opposite on both sides (i.e, \(N_a = N_d\), where \(N_a\) refers to the doping on the left and \(N_d\)is the doping on the right side. Ask questions in the comments here or by email if anything, or everything, here is unclear.

Solution notes

1. a)\( n = 10^{18} electrons/cm^3\)
b) 0.8 eV
c) p=3.8 * 10^7 cm^-3

2. a)\( p = 10^{18} electrons/cm^3\)
b) 0.2 eV
c) n=3.8 * 10^7 cm^-3

3.
a) 0.6 eV
b) \((x_d)^2 = 0.3 volts * 2 \epsilon \epsilon_o/(e N_d) = 25 nm\)
c) Emax= \(2* 0.3 volts/x_d= 2.4 * 10^5 volts/cm\)
d) \( n(0) = 6.2 * 10^{12} electrons/cm^3\), because Ec-Ef=0.5 eV
e) 10^18 and at -xd n=3.8*10^7
h)dn(x)/dx, see plot here