Download Chapter 13 Ideal Fermi gas

Chapter 13
Ideal Fermi gas
The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We
shall consider the limit:
µ >> kB T ⇔ βµ >> 1,
which defines the degenerate Fermi gas. In this limit, the quantum mechanical nature of
the system becomes especially important, and the system has little to do with the classical
ideal gas.
Since this chapter is devoted to fermions, we shall omit in the following the subscript (−)
that we used for the fermionic statistical quantities in the previous chapter.
13.1
Equation of state
Consider a gas of N non-interacting fermions, e.g., electrons, whose one-particle wavefunctions ϕr (~r) are plane-waves. In this case, a complete set of quantum numbers r is
given, for instance, by the 3 cartesian components of the wave vector ~k and the z spin
projection ms of an electron:
r ≡ (kx , ky , kz , ms ).
Since the Hamiltonian operator of the system,
Ĥ =
N
X
H(i) =
i=1
N
X
p~ 2i
+ V (~ri ),
2m
i=1
is spin independent, the summation over the states r (whenever it has to be performed)
can be reduced to the summation over states with different ~k (~p = h̄~k) as
X
r
...
⇒
(2s + 1)
X
...,
~k
where we have already summed over ms = −s, −s + 1, . . . , s to get the prefactor (2s + 1).
163
164
CHAPTER 13. IDEAL FERMI GAS
We assume that the electrons are in a volume defined by a cube with sides Lx , Ly , Lz
and volume V = Lx Ly Lz . This imposes on the one-particle wavefunction
1 ~
ϕk (~r) = √ eik·~r
V
the following periodicity condition at the cube’s walls:
eikx x = eikx x+ikx Lx
⇒
eikx Lx = ei2πnx ,
which is then translated into a condition for the allowed k-values:
kx,y,z =
2π
nx,y,z ,
Lx,y,z
nx,y,z ∈ Z.
Each state has in k-space an average volume of
∆k =
(2π)3
(2π)3
=
.
Lx Ly Lz
V
In the thermodynamical limit, i.e., when
V → ∞, N → ∞ and n =
∆k → 0 so that the sum
X
r
→
P
N
= const,
V
can be substituted by an integral over k-space as
Z
Z
1
V
3
(2s + 1)
d k = (2s + 1)
d3 k
3
∆k
(2π)
Z
V
= (2s + 1) 3 d3 p,
h
r
(13.1)
where the equality k = p/h̄ has been used.
1
⋆ Note that the factor h̄1 ( h3N
for N particles), which we had to introduce ”ad hoc”
in classical statistical physics, in quantum statistical physics appears naturally.
Grand canonical potential
We consider now the expression for the grand canonical potential Ω(T, V, µ) that we
derived in Chapter 12:
X ln 1 + e−β(εr −µ) .
Ω(T, V, µ) = −kB T
r
Through the substitution (13.1), it can be rewritten as
Z ∞
h
2 k2 i
V
2
−β h̄2m
,
−β Ω(T, V, µ) = (2s + 1)
dk
k
ln
1
+
z
e
4π
(2π)3
0
(13.2)
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13.1. EQUATION OF STATE
where we also introduced z = eβµ and used an explicit expression for the one-particle
energies for free electrons εr given by
εr
⇒
h̄2 k 2
.
ε(~k) =
2m
Expression (13.2) is then transformed further by introducing a dimensionless variable x,
x = h̄k
r
β
2m
⇒
2
k dk =
into
4V
−β Ω(T, V, µ) = (2s + 1) √
π
m
2πβh̄2
3/2
2m
βh̄2
3/2 Z
∞
0
x2 dx,
2
x2 dx ln 1 + z e−x .
By making use of the definition of the thermal de Broglie wavelength λ,
s
2πβh̄2
,
λ=
m
we get
(2s + 1) 4V
√
−β Ω(T, V, µ) =
λ3
π
Z
∞
0
2
dx x2 ln 1 + z e−x .
In order to calculate the integral, we perform a Taylor expansion of the logarithm:
ln(1 + y) =
∞
X
(−1)n+1
n=1
yn
n
for |y| ≤ 1.
Then,
Z
∞
0
2
x dx ln 1 + z e
−x2
=
=
=
∞
X
n=1
∞
X
n=1
∞
X
n=1
=
(−1)
(−1)
(−1)
n+1 z
n
n
n+1 z
n
n
n+1 z
n
n
Z
∞
dx x2 e−nx
2
0
∞
d
−
dn
d 1√ 1
−
π√
dn 2
n
Z
√ X
∞
π
zn
(−1)n+1 5/2 .
4 n=1
n
dx e
−nx2
0
We define the following functions:
4
f5/2 (z) = √
π
Z
∞
0
∞
X
zn
2
dx x2 ln 1 + z e−x =
(−1)n+1 5/2
n
n=1
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CHAPTER 13. IDEAL FERMI GAS
and
∞
X
d
zn
f3/2 (z) = z f5/2 (z) =
(−1)n+1 3/2
dz
n
n=1
so that the grand canonical potential for an ideal Fermi gas can now be written as
β Ω(T, V, µ) = −
Since Ω = −P V ,
βP =
2s + 1
V f5/2 (z).
λ3
2s + 1
f5/2 (z)
λ3
with λ = λ(T ).
(13.3)
In order to find the thermal equation of state, we substitute the fugacity z by the particle
density n =< N̂ > /V . Thus,
< N̂ >= z
∂
ln Z
∂z
n=
⇒
=Vz
T,V
∂
βP
∂z
T,V
2s + 1
< N̂ >
= 3
f3/2 (z) .
V
λ (T )
(13.4)
Substituting eq. (13.4) into eq. (13.3) one can obtain, in principle, the thermal equation
of state. The caloric equation of state can be obtained from
U =−
∂
ln Z + µ < N̂ > .
∂β
One first finds µ < N̂ >:
2s + 1 d
z f5/2 (z)
λ3
dz
λ3
2s + 1 ∂
ln Z
= V
λ3 ∂β V (2s + 1)
∂
31
∂
3
=
ln Z +
ln Z =
ln Z + P V
∂β
2β
∂β
2
µ < N̂ > = µ V
and then, using eq. (13.3), U :
3
2s + 1
3
U = kB T V
f5/2 (z) = P V.
3
2
λ
2
The equation U = 23 P V is also valid for the classical ideal gas, but it is not anymore valid
for relativistic fermions.
167
13.2. CLASSICAL LIMIT
13.2
Classical limit
Starting from the general formulae (13.3) and (13.4), we first investigate the classical limit
(non-degenerate Fermi gas) corresponding to
z = eβµ << 1.
Under this condition, the Fermi-Dirac distribution function reduces to the MaxwellBoltzmann distribution function:
1
< n̂r >= −1 βεr
≈ ze−βεr .
z e +1
Also, in the series expansion for f5/2 (z) and f3/2 (z) only the first two terms can be retained,
i.e.,
z2
,
25/2
z2
f3/2 (z) ≈ z − 3/2 .
2
f5/2 (z) ≈ z −
Then, eq. (13.3) and (13.4) reduce to
and
z βP λ3 ≈ (2s + 1)z 1 − 5/2
2
z nλ3 ≈ (2s + 1)z 1 − 3/2 .
2
The simplest approximation in this limit is
z (0) ≈
nλ3
,
2s + 1
which is equivalent to assuming that nλ3 << 1, i.e., that the particle density and the
thermal de Broglie wavelength are small and, since λ ∼ √1T , high temperatures.
In the next approximation step we have:
−1
z (0)
z (0)
(1)
(0)
(0)
z ≈z
1 + 3/2 = z
≈ z (0) 1 + z (0) 2−3/2 ,
1 − 3/2
2
2
where
1
≈ 1 + x, x << 1
1−x
was used. The equation of state for pressure is then
βP λ3 ≈ (2s + 1) z (0) 1 + z (0) 2−3/2 − 2−5/2 (z (0) )2
3
3
−5/2 nλ
= nλ 1 + 2
2s + 1
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CHAPTER 13. IDEAL FERMI GAS
or
P V =< N̂ > kB T
nλ3
1+ √
4 2(2s + 1)
.
In this expression, the first term corresponds to the equation of state for the classical ideal
gas, while the second term is the first quantum mechanical correction.
13.3
Fermi energy
In the low temperature limit, T → 0, the Fermi distribution function behaves like a step
function:
1
0 if εk > µ,
T →0
nk = β(ε −µ)
−−−→
1 if εk < µ,
e k
+1
i.e.,
nk
T →0
−−−→
θ(µ − εk ).
This means that all the states with energy below the Fermi energy εF ,
εF = µ(n, T = 0),
are occupied and all those above are empty. In momentum space the occupied states
lie within the Fermi sphere of radius pF (see Fig. 13.1). Such a situation is denoted
”quantum degeneracy” as we already mentioned at the beginning of this chapter.
Figure 13.1: Fermi surface in momentum space.
The Fermi energy can be determined from the following condition:
X
N=
1,
states r
with εr < εF
which, in the case of free fermions, can be written as
Z
X
(2s + 1)V
(2s + 1)V 4 3
N=
1 →
πk = N.
d3 k =
3
(2π)
(2π)3 3 F
|k|<|kF |
r
εr < ε F
169
13.4. GROUND STATE
Here, kF =
pF
h̄
is the Fermi wave number. Since
2s + 1 3
N
=
k ,
V
6π 2 F
2 1/3
6π n
kF =
2s + 1
n=
⇒
so that we can get the following expression for the Fermi energy:
h̄2
h̄2 kF2
=
εF =
2m
2m
13.4
6π 2 n
2s + 1
2/3
.
Ground state
At T = 0, the system is in its ground state, with the internal energy U0 given by
Z
2 2
X
V (2s + 1) kF
2 h̄ k
U0 =
ǫk =
dk
(4πk
)
(2π)3
2m
0
|~k|<kF
2
h̄
4π 5
V
k .
=
3
(2π) 2m 5 F
Using the expression for total particle number N ,
N=
V (2s + 1) 4π 3
k ,
(2π)3
3 F
one can show that the internal energy per particle at absolute zero is given as
3 h̄2 kF2
U0
3
= ǫF
=
N
5 2m
5
(independent of s).
Since P V = 23 U , one can derive now an expression for the zero-point pressure P0 :
2
P0 ≡ PT =0 = nǫF .
5
⋆ This zero-point pressure arises from the fact that there must be moving particles at
absolute zero since the zero-momentum state can hold only one particle of a given
spin state.
Let us use our model of a Fermi gas of electrons contained in a box in order to describe
electrons in a metal. For this case, since the density of electrons in a metal is n ≈ 1022
cm−3 , we find a zero-point pressure to be equal to:
P0 ≈ 1010 erg cm−3 .
170
13.5
CHAPTER 13. IDEAL FERMI GAS
Fermi temperature
At a finite temperature, the Fermi distribution function (occupation number) < nr >=
n(ε) has the shape as shown in Fig. 13.2. Such an evolution of n(ε) with increasing
Figure 13.2: Fermi distribution function.
temperature can be interpreted in terms of fermions within a layer beneath the Fermi
surface being excited to a layer above and leaving ”holes” beneath the Fermi surface (Fig.
13.3).
Figure 13.3: Momentum space.
We define the Fermi temperature TF as
εF = kB T F .
For low temperatures T << TF , the Fermi distribution deviates from that at T = 0 mainly
in the neighborhood of εF in a layer of thickness kB T , i.e., particles at energies of order
kB T below the Fermi energy are excited to energies of order kB T above the Fermi energy.
For T >> TF , the Fermi distribution approaches the Maxwell-Boltzmann distribution.
For electrons in a metal,
εF ≈ 2eV
⇒
TF ≈ 2 × 104 K,
which means that at room temperatures the electrons are frozen below the Fermi level,
except for a fraction TTF ≈ 0.015 of them. Since the average excitation energy per particle
171
13.6. LOW TEMPERATURE PROPERTIES
is kB T , the internal energy of the system is of order
T
N kB T
TF
and the specific heat capacity CV ,
CV
T
∼
,
N kB
TF
tends to zero so that the electronic specific heat contribution can be neglected at room
temperature.
T >> TF
→ classical limit
T << TF
→ degenerate limit
13.6
Low temperature properties
In section 13.1, we found that
nλ3 = f3/2 (z)(2s + 1),
with λ =
s
2πh̄2
.
kB T m
(13.5)
The low temperature limit, T << TF , at a fixed density corresponds to the condition
nλ3 >> 1.
In the low temperature limit, z is large and one cannot use power series expansion for
deriving an asymptotic formula for f3/2 (z). Instead, such a formula is obtained by the
Sommerfeld expansion:
4
π2 1
3/2
√
f3/2 (z) ≈ √ (ln z) +
+ ... .
(13.6)
8 ln z
3 π
We substitute eq. (13.6) into eq. (13.5) keeping only the first term:
ln z
(0)
2/3
√
εF
TF
µ
3 π nλ3
=
=
,
=
≈
4 (2s + 1)
kB T
kB T
T
which gives the correct limiting value for the chemical potential εF .
Including also the next order in the expression (13.6), we get
4
π2 1
3
3/2
√
nλ ≈ (2s + 1) √ (ln z) +
,
8 ln z
3 π
172
CHAPTER 13. IDEAL FERMI GAS
which is then rewritten as
(ln z)
3/2
√
3 π nλ3
π2 1
√
≈
−
.
4 2s + 1
8 ln z
We now substitute ln z on the right-hand side of this equation by ln z (0) = TF /T :
2 #
3/2 "
π2 T
TF
3/2
(ln z)
≈
1−
T
8 TF
"
2 #
TF
π2 T
⇒ ln z ≈
,
(13.7)
1−
T
12 TF
where we used
2
(1 − x)2/3 ≈ 1 − x + . . .
3
From (13.7) follows the asymptotic formula for the chemical potential:
"
2 #
π2 T
.
µ ≈ εF 1 −
12 TF
Figure 13.4: Chemical potential of the ideal Fewrmi gas at a fixed density.
In Fig. 13.4, the Maxwell-Boltzmann curve refers to the high-temperature limit
nλ3
µ = kB T ln
.
2s + 1
Now, we can calculate the internal energy U from
U=
X
k
V 4πh̄2
(2s + 1)
εk n k =
(2π)3 2m
Z
∞
0
dk k 4 nk
173
13.7. PARTICLES AND HOLES
by partial integration:
Z
k 5 ∂nk
4πh̄2 ∞
V
dk
(2s
+
1)
U = −
(2π)3
2m 0
5 ∂k
2 Z ∞
V
∂ε
4πh̄
k5
z −1 eβε
=
β
(2s
+
1)
dk
2
3
(2π)
2m
5 [z −1 eβε + 1] ∂k
0
Z ∞
V
k 6 eβ(ε−µ)
4
=
dk
βh̄
(2s
+
1)
2,
20π 2 m2
[eβ(ε−µ) + 1]
0
where
h̄2 k 2
.
ε=
2m
At low temperatures, the integrand is peaked at k = kF . By expanding k 6 around this
point and using the expression for µ obtained above, we get
#
"
2
3
5π 2 T
U =
+ ... ,
N εF 1 +
5
12 TF
#
"
2
2
2U
5π 2 T
+ ...
= mεF 1 +
P =
3V
5
12 TF
and the heat capacity
CV
N kB
=
π2 T
+ ...,
2 TF
with
N =< N̂ > .
13.7
Particles and holes
We can define that the absence of a fermion of energy ε, momentum p~ and charge e
corresponds to the presence of a hole with
174
CHAPTER 13. IDEAL FERMI GAS
energy
momentum
charge
=
=
=
−ε
−~p
−e
The average occupation number for a fermion is
nk =
1
eβ(εk −µ)
and for a hole
1 − nk =
+1
1
eβ(µ−εk )
with
εk =
+1
,
k 2 h̄2
.
2m
The concept of a hole is useful only at low temperatures T << TF , when there are few
holes below the Fermi surface. When T >> TF , the Fermi surface ”disappears” and the
system approaches the Maxwell-Boltzmann distribution function.
Only holes and excited particles participate in the conduction; the rest of fermions remain
inactive beneath the Fermi surface.