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1. Sommerfeld Expansion in 2D
a. Show that every term in the Sommerfeld expansion for density n vanishes
except for the
T
=0 term. Also show that this procedure (erroneously) predicts
the exact relationship
μ
=E
F
.
b. By performing the exact integration analytically, derive the following result:
μ
+k
B
T*ln(1+exp(
μ
/k
B
T))=E
F
c.
Using a suitable expansion of the natural log, show that the correction from
μ
=E
F
is negligible.
d. The conclusion in (a) included all orders of the expansion – so why is there a
discrepancy with the exact solution?
2.
Free electron gas with an anisotropic dispersion
a. Suppose one has a free electron gas in 3D where the dispersion relation is
given by
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 Fall '11
 Appelbaum
 Solid State Physics

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