HW #3, begin immediately, due Tuesday if possible

As we have discussed, one can start with a depletion approximation, in which we "pretend" n(x) and p(x) are both zero in the "depletion region", then, starting with that, on can calculate V(x) and hence Ec(x) and Ev(x) and then use them to recalculate n(x) from n(x) = Nc exp{-(Ec(x)-mu)/kt}, and p(x) from an analogous relation.

1) Suppose Ec-mu = mu-Ev = 0.1 eV, and Egap=1. eV (and kT=.025 eV)
a) Graph Ec(x). What is total variation in Ec(x) from left to right in your graph.
b) where are the boundaries of the depletion region?

2) Suppose Ec-mu =0.15 ; mu-Ev = 0.05 eV, and Egap=1. eV (and kT=.025 eV)
(How does that change things?)
a) Graph Ec(x). What is total variation in Ec(x) from left to right in your graph.
b) where are the boundaries of the depletion region?

3) (re)Calculate n(x) for each case.

4) For a homogeneous doped semiconductor, given n=4x10^17, m*=.2me, and hbar/tau=3 meV:
a) what is tau in sec?
b) what is sigma in (Ohm-cm)-1 ?
c) what is mu in cm^2/V-sec ?
d) What is rho? and what is R (resistivity) in Ohms for a "strip" 1 mm wide, 100 nm thick and 2 cm long?
e) What is I for that strip for a voltage of 10 microvolts end to end? How much heat will that generate?

5) how come nobody uses the concept of mean-free-path for semiconductors? (but they do for metals)

6) In a p-n junction, one can write J=n(x) e mu Ex(x) - e D dn(x)/dx (eqn 1) (c.f. eq 5.3 or so in Streetman), where n(x) is the density of electrons in the CB, mu is the mobility, Ex(x) is the electric field in the x direction and D is the diffusion coefficient. Turns out D = mu KT/e (Einstein relation).
a) Show that we can rewrite eqn. 1 (above) as:
J=n(x) e mu Ex(x) - n(x) mu KT d(ln{n(x)}/dx

b) take the derivative using the expression way above for n(x) = ...

c) evaluate at a value of x such as x=0 for a nice symmetric junction (as in problem #1). Show that this is, or is not, zero. [side points: What is Ec(x)? How is it different from and related to Ex(x) in terms of definition, units, physical relationship, etc. ]

7) Consider two different junctions, one in which Ec-mu =mu-Ev = 0.1 eV, and another in which
Ec-mu =mu-Ev = 0.15 eV. What is the diffusion current across the junction interface (x=0) in each case? How does the diffusion current depend on "delta mu"? It is a weak or strong dependence?

8) Suppose you apply an external voltage of 0,3 Volts. You may represent that by a linear dependence of mu through the depletion region (but mu is independent of x outside of that range). Do it in a way the reduces the offset of the band edges (so that Ec and Ev have less x dependence.)

a) How does this effect the extent (length scale, x_0) of the depletion region?

b) Sketch the band edges and mu

c) (major extra credit) Starting with the expression for J (in #6, above) consider, analyze and discuss how this effects the current through the p-n junction. Big change, small change...???

Questions and discussion are welcome and encouraged.