Homework 4

Here is a video related to this homework.


For the following problems, consider a semiconductor for which  \(E_g = 1 eV, \quad kT=.025 eV\) and  
\(D_c = 12 \times 10^{21} \frac{states}{eV*cm^3}, \quad B_c = 3 eV\)
\(D_v = 12 \times 10^{21} \frac{states}{eV*cm^3}, \quad B_v = 3 eV\).
(Assume that within each band the density of states is independent of E.)
Let's set our zero of energy at the top of the valence band.
Please mention any possible typos, confusing things etc.

It will probably help to discuss a lot of this in the comments here. For example, how do the units work going from charge to electric field to potential? How does one end up with a potential in eV? (You may have to convert from cm to meters when you use surface charge to calculate electric field if epsilon_o is in Farads/meter.)

1. a) If this semiconductor is undoped, then what is the value of n at room temperature?  What is the relationship between n and p? What is \(E_f\) for this un-doped case?
b) Suppose that this semiconductor is doped with 10^17 donors/cm^3. In that case we like to assume that each donor contributes one electron to the conduction band. What is \(E_f\) in this case?
(What do you think might be the rationale behind choosing \(12 \times 10^{21} \frac{states}{eV*cm^3}\) for the density of states? Why that value? (post here))

2. a) Plot the density of states as a function of energy from E from E = -4 eV to +4 eV.
b) Calculate n and p for \(E_f = 0.5 eV\).
c) Calculate n and p for \(E_f = 0.2 eV\).
d) Calculate n and p for \(E_f = 0.8 eV\).
e) Do a semi-log graph of n and p as a function of \(E_f\) for \(E_f\) in the range 0.1 to 0.9 eV. Can you use the approximate form of the Fermi function for this calculation?

3. For the same semiconductor, suppose you dope it with 10^17 donor atoms per cm^3. Then you get 10^17 electrons/cm^3 in the conduction band and 10^17 positively charged donor atoms/cm^3 embedded in the lattice.
a) What is the charge density associated with just the electrons in the conduction band (in coulombs/cm^3)?
b) What is the charge density associated with the positive ions (in coulombs/cm^3)?
c) What is the net charge density?
d) What would the net charge density be if all the electrons magically disappeared?

2. Consider a semiconductor as above. Suppose that it is doped with 10^17 donors for the half to the left of the plane x=0, and doped with 10^17 acceptors/cm^3 to the right of x=0. (The plane x=0 defines an interface between to two differently doped regions.) Suppose that within a distance \(x_d\) of the interface all the electrons in the conduction band from the left side cross over to the other side and fill up previously empty valance band states (holes) there.
a) When \(E_f\) is independent of x, that represents an equilibrium state. What value of \(x_d\) would enable to bands to bend and shift by just the right amount to enable \(E_f\) to be independent of x?
b) Plot the conduction band edge and valence band edge as a function of x for this case.

6. Suppose the doping on each side is 10^18 cm^-3 instead of 10^17 cm^-3.
a) why might one guess that in that case the equilibrium value of \(x_d\) might be 10 times shorter than for the 10^17 case?
b) What is the actual equilibrium value of \(x_d\) for this case?  Why is it not exactly 10 times smaller? (What additional factor influences x_d?)

3  (optional extra problem). Suppose the doping on each side is 10^16 cm^-3.
What is the equilibrium value of \(x_d\) for this case?

4. a) For the n-p junctions you solved in problem 2 (doped with \(10^{17} 1/cm^3\)), use your calculated value for \(E_c (x)\) to calculate n(x) as a function of x.  You can use the equation \(n(x) = KT D_c e^{-(E_c (x) - E_f)/kT}\)?
b) What is \(n(-x_d)\)? What is \(n(x_d)\)? Sketch a rough graph n(x) as a function of x.
c) Graph the product of the conduction electron density times the electric fields a function of x. At what x does its maximum value occur?
d) Calculate the drift current as a function of x in the junction region. (You can just do this for x less than zero since that's easier and I think the maximum is in that range.)

5. Use your equation for n(x) from the previous problem to calculate the diffusion current in the junction (for x less than zero). This is an important problem. Please spend some time on this one.

6. Graph this two currents. Compare and discuss here.

Notes:  \(n(x) e^2 \tau/m^*m\) multiplied time electric field has units of current per unit area. I think that if you use your expression for n(x) (from the previous problem) and multiply it time the electric field in the junction, which varies linearly with x, that correspond to a current associated with electrons accelerated by the electric field.   

Special bonus: Calculate the peak value of that current in Coulombs/second (amperes) for a specific junction with an area of 1 cm^2, using m*=0.2 and \(\tau = 10^{-12}\) seconds. (Use the specific 10^17 doping case above. We want an actual number. Is it big, small, negligible, 10^-15 amps, 2 amps or what?
   Hints: One can separate out the term \( \mu= e \tau/m = e \tau c^2/(m^*m c^2)\). With mc^2 in eV and c in cm/s that can have units of \(cm^2/(Volt*seconds\). (The e turns eV into Volts...).
      Does the maximum in this product occur somewhere between -x_d to zero? Where is it?

PS. Here is how one might approach graphing current density.

Here is a video on the integration to get V(x):