402 Problems d Solutions on Thermodynamics €4 Statistical Mechanic8
where no is the concentration of particles at z = 0, whence we get
(b) By definition,eE
kT= -D-noexp(eEz/kT).
(c) The particle flux along the applied electric field is(%).
J,' = n(z)?i = n(z)pE = pEno exp
(d) The total flux is zero at equilibrium. Hence Jo + J,, = 0, giving
eD
P=@'2206
Consider a system of degenerate electrons at a low temperature in
thermal equilibrium under the simultaneous influence of a density gradient
and an electric field.
(a) How is the chemical potential p related to the electrostatic potential
d(z) and the Fermi energy EF for such a system?
(b) How does EF depend on the electron density n?
(c) From the condition for p under thermal equilibrium and the consid-
erations in (a) and (b), derive a relation between the electrical conductivity
o, the diffusion coefficient D and the density of states at the Fermi surface
for such a system.
(SUNY, BUflUlO)
Solution:
obtain EF = p~g + ed(z), where po = p(T = 0).
(a) From the distribution n, = {exp[(e - p - e+(z))/kT] + l}-', we
v4 v4
(b) N = 2 --T& = 2. -. -~(2rnEF)~/~
h3 3 h3^3