where every odd term is zero (x,x^3 ,x^5 ) and the other ones are just numbers. If you put in these
terms you get
n=∫∞−∞H(E)f(E)dE=K(μ) +
π^2
6(kBT)^2
dH(E)
dE E=μ+
7 π^4
360(kBT)^4
d^3 H(E)
dE^3 E=μ+...
Sommerfeld Expansion: Chemical Potential 3D
It is possible to get the chemical potential out of the Sommerfeld expansion. IfH(E) =D(E), which
you get out of the dispersion relationship, you can getK(E)which is just the integral ofH(E)over
all energies. You get the non-linear relationship of theμand theT (EF as the fermi energy):
n=
nμ^1 /^2
E^3 F/^2+
π^2
8
(kBT)^2
nμ−^1 /^2
E^3 F/^2+...
Dividing both sides byngives:
1 =
μ^1 /^2
EF^3 /^2+
π^2
8(kBT)^2
μ−^1 /^2
EF^3 /^2+...
Sommerfeld Expansion: Internal energy
In this caseH(E) =D(E)E. So we get the expression for the internal energy:
u=∫∞−∞H(E)f(E)dE=
3 n
5 EF^3 /^2μ^3 /^2 +
3 π^2
8(kBT)^2
n
E^3 F/^2μ^1 /^2Sommerfeld Expansion: Electronic specific heat
If you differentiate the internal energy once you get the specific heatcV. So you see that the specific
heat is linear with the temperature.
cV=
du
dT=
3 π^2
4(kBT)
n
EF^3 /^2μ^1 /^2 +....If you take a metal and measure the specific heat at a certain temperatureT, some of the energy goes
into the electrons (∝T) and some into the phonons (∝T^3 ), because both are present in a metal:
cV=γT+AT^3
C
T is plotted versusT(^2) to get a line (see fig. 21). The slope of the line is the phonon constantAand
the interception with the y-axis givesγ, the constant for the electrons.
Typically metals are given aneffective massm∗, which was reasonable when comparing the calcu-
lated free electron case (massme,γ) with the results measured (γobserved) (The effective mass is a way
to parameterize a system - interaction was neglected). What was really done with this was specifying
a derivative of the Sommerfeld expansion. So because of history we are talking about effective mass,
but we mean this derivative in the Sommerfeld expansion.
m∗
me
=
γobserved
γand m∗=2
9
dH
dEE=μ·me