The previous sections discussed the ground state of a system of fermions like electrons. The ground state corresponds to absolute zero temperature. This section has a look at what happens to the system when the temperature becomes greater than zero.
For nonzero temperature, the average number of fermions
The biggest difference is that
It reflects the exclusion principle: there cannot be more than one fermion in a given state, so the average per state cannot exceed one either. The Bose-Einstein distribution can have many bosons in a single state, especially in the presence of Bose-Einstein condensation.
Note incidentally that both the Fermi-Dirac and Bose-Einstein distributions count the different spin versions of a given spatial state as separate states. In particular for electrons, the spin-up and spin-down versions of a spatial state count as two separate states. Each can hold one electron.
Consider now the system ground state that is predicted by the
Fermi-Dirac distribution. In the limit that the temperature becomes
zero, single-particle states end up with either exactly one electron
or exactly zero electrons. The states that end up with one electron
are the ones with energies
To see why, note that for
The correct ground state, as pictured earlier in figure
6.11, has one electron per state below the Fermi energy
Next consider what happens if the absolute temperature is not zero but
a bit larger than that. The story given above for zero temperature
does not change significantly unless the value of
holes (states that have lost their electron)
immediately below the Fermi surface.
Put in physical terms, some electrons just below the Fermi energy pick
up some thermal energy, which gives them an energy just above the
Fermi energy. The affected energy range, and also the typical energy
that the electrons in this range pick up, is comparable to
You may at first hardly notice the effect in the wave number space
shown in figure 6.15. And that figure greatly exaggerates
the effect to ensure that it is visible at all. Recall the ballpark
Fermi energy given earlier for copper. It was equal to a
One of the mysteries of physics before quantum mechanics was why the
valence electrons in metals do not contribute to the heat capacity.
At room temperature, the atoms in typical metals were known to have
picked up an amount of thermal energy comparable to
The Fermi-Dirac distribution explains why: only the electrons within a
distance comparable to
To discourage the absence of confusion, some or all of the following
terms may or may not indicate the chemical potential
Fermi energy to absolute zero temperature, but to not
do the same for
Fermi level or “Fermi
brim.” In any case, do not count on it. This book will
occasionally use the term Fermi level for the chemical potential where
it is common to do so. In particular, a Fermi-level electron has an
energy equal to the chemical potential.
electrochemical potential needs some
additional comment. The surfaces of solids are characterized by
unavoidable layers of electric charge. These charge layers produce an
electrostatic potential inside the solid that shifts all energy
levels, including the chemical potential, by that amount. Since the
charge layers vary, so does the electrostatic potential and with it
the value of the chemical potential. It would therefore seem logical
to define some
intrinsic chemical potential, and add
to it the electrostatic potential to get the total, or
For example, you might consider defining the
In particular, the strengths of the double layers adjust so that in
thermal equilibrium, the electrochemical potentials
Unfortunately, the assumed
potential in the above description is a somewhat dubious concept.
Even if a solid is uncharged and isolated, its chemical potential is
not a material property. It still depends unavoidably on the surface
properties: their contamination, roughness, and angular orientation
relative to the atomic crystal structure. If you mentally take a
solid attached to other solids out to isolate it, then what are you to
make of the condition of the surfaces that were previously in contact
with other solids?
Because of such concerns, nowadays many physicists disdain the concept
of an intrinsic chemical potential and simply refer to
the chemical potential. Note that this means that the
actual value of the chemical potential depends on the detailed
conditions that the solid is in. But then, so do the electron energy
levels. The location of the chemical potential relative to the
spectrum is well defined regardless of the electrostatic potential.
And the chemical potentials of solids in contact and in thermal equilibrium still line up.
The Fermi-Dirac distribution is also known as the “Fermi factor.” Note that in proper quantum terms, it gives the probability that a state is occupied by an electron.
- The Fermi-Dirac distribution gives the number of electrons, or other fermions, per single-particle state for a macroscopic system at a nonzero temperature.
- Typically, the effects of nonzero temperature remain restricted to a, relatively speaking, small number of electrons near the Fermi energy.
- These electrons are within a distance comparable to
of the Fermi energy. They pick up a thermal energy that is also comparable to .
- Because of the small number of electrons involved, the effect on the heat capacity can usually be ignored.
- When solids are in electrical contact and in thermal equilibrium, their (electro)chemical potentials / Fermi levels / Fermi brims / whatever line up.