HW 4 and HW 4 solutions

Here is a solutions link for HW 4. Please feel free to comment.

https://drive.google.com/file/d/0B_GIlXrjJVn4eGRGLWNJR1g3TXc/edit?usp=sharing


I was thinking that, now that we have the motivation of wanting to understand a p-n junction, maybe we should back up a little and review the different kinds of current (drift and diffusion). How does that sound? Also, perhaps you would like to read about the two kinds of current:
1) drift current (current directly associated with the force that an Electric field exerts on an electron.
2) diffusion current (current associated with a density gradient!).
Also, you can read about the continuity equation for electrons in a semiconductor, and the recombination time.  I think that is related to the life-time of an excited electron (and there is one for holes as well). Here is a guiding question for that: suppose you excited some electrons into the conduction band (with photons) and thus created a state not in (thermal) equilibrium. What would happen over time after that? What mathematical description might be suitable to describe the return to equilibrium? What would the characteristic time scale be? What is it called?

For the following problems let's use: \( D_2 = 0.6 \times 10^{22} \, states/(eV-cm^3)\) and \( D_3 = 0.3 \times 10^{22} \, states/(eV-cm^3)\). (Or would you rather do them with D2 and D3 equal? comment here if you have an opinion on that. (simplicity vs generality))

Warm-up problems related to \mu.
1. Calculate the intrinsic carrier concentrations (\(n_i\) and \(p_i\)) for a semiconductor:
a) with a gap of 1 eV at room temperature.
b) with a gap of 0.25 eV at room temperature.
c) with a gap of 0.25 eV at 60 Kelvin.



2. For a semi-conductor with a 1.0 eV gap, let's set our zero of energy at the top of the valence band and calcuate:
a) n and p for \(\mu =  0.7 \, eV\)
b) n and p for \(\mu =  0.8 \, eV\)
c) n and p for \(\mu =  0.9 \, eV\)
d) how does the product np depend on \(\mu\)?

Doping:
3. Doping with donors adds electrons to a semiconductor. Why do we assume that they all go into the conduction band? Don't some of them end up filling holes (empty states) in the valence band? Discuss. Why do people usually ignore that? Why (when) would that be okay or not okay?

4. Here is a bigger doping problem. Imagine doping an Si crystal with some P (phosphorus) to a level of \(N_d\) (number of donors). That means there are  \(N_d\) extra electrons on the crystal, right? But I have heard that there are also \(N_d\) local trap states (bound states localized in the vicinity of each P atom due to its net + charge). Why don't the electrons just go into those (shallow) trap states instead of into the conduction band? (Let's suppose those bound states have an energy about 10 or 15 meV below the conduction band edge.) Would your opinion be different if those states were 200 meV below the band edge?

Transport:
5. A semiconductor is doped to a level of \(5 \times 10^{17}\, cm^{-3}\). It has a transport scattering time of \(\tau = 10^{-14}\, seconds\) and an effective mass of 0.2. Suppose its shape is a rectangle 1 mm x 1 mm x 1 cm.
a) How much current flow would there be if you applied 0.05 Volts along its long direction?
b) What is the current density?
c) What is the electric field in the semiconductor?

6. a) Calculate the conduction band (electron) diffusion current in a semiconductor with a 1x1 mm cross-section where \(D_n = 100 \, cm^2/sec\) and \(n(x) = 10^{17} e^{-x/L}\,cm^{-3}\) from x = 0 to 20 L.
b) Plot this current as a function of x.
(This problem is mostly a math exercise and won't really make intuitive sense because the current depends on x.)
c) What is the current density (as a function of x)?

Electrons out of equilibrium:
7. Consider a p-doped semiconductor (e.g., Boron doped Si) with \( D_2 = 0.6 \times 10^{22} \, states/(eV-cm^3)\) and \( D_3 = 0.3 \times 10^{22} \, states/(eV-cm^3)\) doped at a level of \(10^{17}\, cm^{-3}\) holes in the valence band.
a) What is the concentration of electrons in the conduction band, n, for this hole-doped material?
b) Does it depend on the energy gap? If so, use 1 eV.
c) Suppose that at t =0 a flash of light is used to excite carriers into the conduction band. Suppose n increases to 100 times what you got in part a). What is your guess as to what will happen after that (regarding the time dependence of n).
d) Use the equation \(\frac {dn(t)}{dt} = -(n(t) - n_{eq})/\tau_r \) to solve for n(t) after t = 0. Here \(\tau_r\) is called the recombination time. It is completely different from the transport scattering time in problem 5.
e) To what sort of event does  \(\tau_r\) refer?

8. Study the equation:  \(\frac{\partial n(x,t)}{\partial t} = \frac{\partial J_n(x,t)}{\partial x} - (n(x,t) - n_{eq})/\tau_r \).  This is a very important equation.
a) Describe what each term in this equation means. Explain this equation in words.
b) Suppose a laser pulse is used to excite carriers, as in the previous problem, but instead of a uniform excitation the same everywhere in the semiconductor, suppose there is a spatially somewhat localzed excitation. Describe the time dependence of n(x,t). Related your description to this equation.