PHYSICS 617 – Jan. 22, 2016
Properties of Fermi Gases
(1) Introduction:
These notes are to accompany chapter 2 of the text, which covers noninteracting electron gases,
or “Fermi gases”. Some background & terminology: “Fermi gas” refers to a non- or weaklyinteracting gas of Fermions, which when condensed will exhibit a Fermi surface due to Pauli
exclusion. The Fermi gas idea applies surprisingly well for treating simple metals, such as
sodium or copper. The term “Fermi liquid” generally refers to a more strongly interacting
system; Landau Fermi liquid behavior is discussed in chapter 17 of the text. Fermi liquids
occupy a regime for which interactions between electrons are strong, yet even for such a case it
can be shown that many of the characteristics of non-interacting Fermi gases persist, for example
the existence of a Fermi surface. In the last 10-20 years there has been considerable interest in
understanding systems for which the electron-electron interactions increase to the extent that the
Fermi liquid characteristics break down: there is a class of properties referred to as “non-Fermi
liquid” behavior, and a particular model sometimes applied to such systems is the “Luttinger
liquid”. The metals exhibiting high-temperature superconductivity may fall into these categories.
Measurement techniques to analyze Fermi surfaces in metals are described in chapter 14 of the
text. One important addition is based on photoemission; with the development of high-intensity
synchrotron light sources in recent years, analysis of the ejected electrons now provides provides
an extremely powerful analysis tool for the electron states in solids. Angle Resolved
PhotoEmission Spectroscopy (ARPES) is the term for the technique as often applied to solids.
Briefly, with the incoming photon energy and momentum known, by detecting the momentum of
the outgoing electron, by conservation principles one can extract considerable information about
the electrons in the valence band. In many cases this allows mapping of the Fermi surface, and
the energy vs. momentum character of the valence band. The references at the end include a
recent review of general photoemission techniques1 and applications to high-temperature
superconductors2. Also see reference 3 for a description of Fermi surface and related behavior in
condensed cold atoms and optical lattices.
(2) Fermi Gas low-T behavior, Fermi energy and state counting:
For non-interacting Fermions, the thermal-equilibrium occupation function is given by:
f (ε ) = [exp(ε − µ ) / k B T + 1] ,
[1]
which is the “Fermi function,” expressing the probability of a Fermion particle occupying a state
at a given energy, ε . The parameter µ is the chemical potential, which depends on the density of
the Fermi gas, as analyzed below. When µ is much smaller than kB T , the distribution [1] goes
over to the smoothly-varying Boltzmann distribution, with occupation proportional to
exp(−ε / k B T ) , as in a classical gas. When µ is large (low T or high density of particles), Pauli
exclusion becomes important, and the distribution [1] is essentially 1 or 0 except for a transition
region near ε = µ . The transition region has a width of approximately 4 kB T , as in the sketch.
−1
At vanishingly low temperatures (or very high densities), the Fermi distribution becomes a step
function. The zero-temperature limit of µ is the position of the step, defined as εF , the Fermi
energy. For electron densities corresponding to typical metallic materials, εF has a
corresponding temperature ( kTF ≡ ε F defining the Fermi temperature) on order of 10,000 K, as
spelled out in section 2 below. For such materials the step-function picture can be an excellent
approximation. For this case the surface surrounding occupied states in momentum space is
called the Fermi surface.
To understand the Fermi surface picture, we consider plane-wave states for free electrons of the
 
form, ψ = s exp(ik ⋅ r ) / V . For simplicity we consider a box of dimensions L × L × L to
represent the crystal, and we assume periodic boundary conditions: the wavefunction is assumed
to be periodic with respect to translation by L in any direction x, y, or z. That
 is,




ˆ
ˆ
ˆ
ψ ( r ) = ψ( r + Li ) = ψ ( r + Lj ) = ψ ( r + Lk ) . This restricts the wavevector k to discrete values,
[2]
k = (2π / L)(n1iˆ + n2 ˆj + n3 kˆ ) ,
forming
a dense array of allowed values when plotted in k-space, the three-dimensional space of

k vectors. Periodic boundary conditions are not particularly realistic when applied to the surface
of a real crystal, but as we are concerned for now with the behavior of an essentially infinite
crystal, the boundary details are not important, and we can verify in the end that the measurable
quantities do not depend on L.
The energy of free, non-interacting electrons is limited to thekinetic energy, which is
ε = 2 k 2 2m . Since this is independent of the orientation of k , in the T = 0 ground state when N
electrons occupy the N/2 lowest wavevectors (factor 2 for spin), the filled states make up a
sphere confined to the Fermi surface as defined above, in this case corresponding to the Fermi
sphere, as in the sketch above, pictured with the discrete states for a sample of size L.
(3) Summations and averaging in k-space and energy – Density of states: According to the
expression [2] for allowed k-values, the volume of a state in k-space is (2π / L)3 . So, if the crystal
contains N electrons with spin, the occupied volume is ( N / 2)(2π / L)3 , or per electron the kspace volume is 4π 3 /V , with L3 replaced by V = crystal volume. Defining n = N/V as the
electron density, the occupied k-space volume can be written 4π 3 n , and the number of states in a
given region of k-space can be found by dividing the volume by 4π 3 /V , which is to say,
[3]
Dk = V /(4π 3 ) ,
as sometimes defined for the density of states in k-space (sometimes given per real-space volume
of the crystal, in which case V does not appear). Notice that this result corresponds to a uniform
density in k-space, and that the details of the chosen volume and its dimensions do not appear in
the final expression.
From [3] it is easy to find the Fermi surface dimensions: setting 4π 3 n = 4 3 πkF3 , where the latter is
the Fermi surface volume, solving yields
[4]
kF = 3 3π 2 n .
2 2
Correspondingly the Fermi energy is εF =  kF
, and the Fermi temperature TF = εF .
kB
2m
A typical metal has a unit cell dimension of 1 nm or smaller, and if (as in the case of Cu) we
have 4 atoms/cubic cell each with one valence electron, a cell of this size will correspond to n =
4 × 1021 cm-3, a lower bound for the electron density expected for metals. Using [4] this gives kF
= 4.9 × 107 cm-1, for which we can calculate εF = 0.9 eV. This corresponds to TF equal to
approximately 10,000 K, similar to what was quoted above. Another measure is the Fermi
velocity, vF = kF / m , which is roughly 108 cm/s for this case. Typically a metal will melt or
decompose well below TF, and many of the properties can be obtained from T = 0 results by a
perturbation treatment in terms of T/TF (see the Sommerfeld expansion discussion below).
Instead of summing over k-space, we often deal with quantities that depend only on the energy,
independent of direction in k-space. In this case, the density used for summation is the energy
density of states [ g(ε ) ], defined so that g(ε )dε is the number of electron states per volume in the
energy range ( ε to ε + dε ). Considering the Fermi sphere, the region between ( ε and ε + dε ) is a
2
spherical shell, volume 4πk 2 dk . For free electrons we have dε = (  k )dk , and so the volume
m
2
in k-space for this shell is 4πk(m /  )dε . Using [3] the number of states in this volume is
Vk(m / π 2 2 )dε , and converting k to terms of energy we obtain,
2⎛ m⎞
g(ε ) = 2 ⎜ 2 ⎟
ε.
[5]
π ⎝ ⎠
The form [5] is the free-electron form for g(ε ) . We will also use g(ε ) later in the course for
more general situations, where the energy vs. k has a more complicated structure in real crystals,
but the summation procedures such as outlined below remain valid.
3/2
(4) Series analysis of temperature dependence; Sommerfeld expansion.
We start with the average value, summed over all electrons, of some function H which depends
on the energy of an individual particle:
H =V
∞
∫ H (ε )g(ε ) f (ε )dε .
[6]
−∞
Note that this is N times the average value of H per particle; V is included since g is defined per
volume. Integrating by parts:
∞ ⎡ε
∞ ⎡ε
⎤ df
⎤ df
[7]
H = −V ∫ ⎢ ∫ H (ε′)g(ε′)dε′⎥ dε = +V ∫ ⎢ ∫ H (ε′)g(ε′)dε′⎥ dε .
⎦ dε
⎦ dµ
−∞ ⎣−∞
−∞ ⎣−∞
For the last term in [7], since f (ε − µ ) = [exp(ε − µ ) / k B T + 1] ,
−1
df
df
=−
. However, note that
dε
dµ
df
df
is still a function of the energy ε, and
is zero except for a delta-function-like peak in ε
dµ
dµ
near the Fermi energy, so only energies very close to µ will contribute in the integration. The
Sommerfeld expansion is a Taylor expansion in (ε – µ) of the square bracket in [7], starting from
the case ε = µ. Proceeding we have:
∞ ⎡µ
⎤ df
H = +V ∫ ⎢ ∫ H (ε′)g(ε′)dε′⎥ dε
⎥⎦ dµ
⎣−∞
−∞ ⎢
+V
∞
df
∫ [ H (µ)g(µ)](ε − µ) dµ dε
−∞
.
[8]
∞⎡
⎤ (ε − µ )2 df
d
+ V ∫ ⎢ ( H ( µ )g( µ ))⎥
dε
dµ
⎦ 2
−∞ ⎣ dµ
+ ...
Note that the terms written with square brackets in [8] do not contain the energy, and could be
placed outside the integrals. The middle term is odd in (ε – µ) and it integrates to zero. We keep
the first and third terms, neglecting higher-order terms. A more general form for the Sommerfeld
expansion can be seen in a Statistical Mechanics text such as Huang, and also in Ashcroft and
Mermin. The energy integrals for the remaining terms in [8] can be evaluated to yield,
µ
π2
d
H ≅ V ∫ H (ε )g(ε )dε + V (k B T )2
[9]
( H (µ)g(µ)) .
6
dµ
−∞
(5) T-dependence of chemical potential, and analysis of specific heat:
From the general definition of the density of states and the occupation probability, the total
number of particles must be N =
∞
∫ Vg(ε ) f (ε )dε . This is just [6] with H replaced by unity. So
−∞
using [9] we can write,
µ
π2
dg( µ )
(kB T )2
.
[10]
6
dµ
−∞
∂N ⎞ ∂µ ⎞ ∂T ⎞
There is a thermodynamic relation,
⎟
⎟
⎟ = −1 by which we can obtain the
∂µ ⎠T ∂T ⎠ N ∂N ⎠ µ
following, where the final form is obtained by directly evaluating derivatives of equation [10]:
∂N ⎞
2
π2
dg( µ )
⎟
V 2kB T
⎞
∂
T
⎠
∂µ
6
dµ
µ
=−
.
[11]
⎟ =−
⎞
∂T ⎠ N
Vg( µ ) + (small term with g ′′)
∂N
⎟
∂µ ⎠T
N ≅V
∫ g(ε )dε + V
Given that the T = 0 value is µ = εF, from [11] we obtain by a 1st order expansion,
⎧ dg( µ )
⎫
π2
µ ≅ εF − (k B T )2 ⎨
g( µ )⎬ .
6
⎩ dµ
⎭
[12]
This shows how µ changes with the temperature; for example for free electrons, where dg / dε is
positive (see [5]), µ decreases with increasing T. However near the top of a band, as we shall see
for example with a hole pocket, the opposite temperature dependence holds.
Note that the departure from µ = εF is small for a good metal with T<<TF: e.g. for free electrons
specifically, g(ε ) = (Const.) ⋅ ε , so that we find that g ′ / g = 1 , and [12] reduces to,
2ε
2
⎛
⎞
π
T
[13]
µ ≅ εF − kB T ⎜ ⎟ .
12
⎝ TF ⎠
The last term is indeed very small in a metal, and in very many cases one may simply use µ = εF .
We can similarly examine the average total energy, replacing H in our expansion by ε:
µ
d [µg( µ )]
π2
E ≅ V ∫ εg(ε )dε + V (kB T )2
,
[14]
6
dµ
−∞
I wrote E to emphasize that [14] yields the average total energy, which we could write as N
times a single-particle average, E ≡ N ε . With this replacement, equation [14] expands to
(dividing by V to get the energy density, using n = N/V, and keeping lowest-order terms),
εF
π2
π2
dg ⎞
2
u = n ε ≅ ∫ ε g(ε )d ε + ( µ − ε F )ε F g(ε F ) + (kBT ) g(ε F ) + (kBT )2
ε F . [15]
6
6
d µ ⎟⎠ ε F
−∞
The second and fourth term on the right are identical and cancel, which can be seen by using [12]
to simplify the fourth term. (Also making the approximation µ ≈ εF , which is true to lowest order
in T.) The first term on the right is the T = 0 energy density, which evaluates to 35 nε F for a freeelectron gas. The third term contains all the temperature dependence:
π2
u ≅ 35 nε F + (kBT )2 g(ε F ) ,
[16]
6
so that the volume heat capacity, which is the temperature derivative of the energy density, is:
π2
[17]
c = (k B )2 g(εF )T ≡ γT .
3
This is the characteristic linear-T heat capacity seen experimentally in metals at very low
temperatures. The T-linear coefficient is traditionally denoted γ, and its measurement is one way
of probing the electronic density of states in metals.
References:
1. Friedrich Reinert and Stefan Hüfner, “Photoemission spectroscopy - from early days to recent
applications” New J. Phys. 7 (2005) 97
2. W S Lee, I M Vishik, D H Lu and Z-X Shen, “A brief update of angle-resolved photoemission
spectroscopy on a correlated electron system”, J. Phys.: Condens. Matter 21 (2009) 164217.
3. Immanuel Bloch, “Ultracold quantum gases in optical lattices”, Nature Physics 1, 23 - 30
(2005).