Fermi Energy - Andrew Nordmeier, Milad Hashemi, Sirikasame ...

Fermi Energy - Andrew Nordmeier, Milad Hashemi, Sirikasame Aromsook, Meredith Brown
The Fermi energy, of a substance is a simple concept with far reaching results. It is a result of the
Pauli exclusion principle when the temperature of a material is lowered to absolute zero. By this principle,
only one electron can inhabit a given energy state at a given time. When the temperature is lowered to
absolute zero all of the electrons in the solid attempt to get into the lowest available energy level. As a
result, they form a sea of energy states known as the Fermi Sea. The highest energy level of this sea is
called the Fermi energy or Fermi level. At absolute zero no electrons have enough energy to occupy any
energy levels above the Fermi level. In metals the Fermi level sits between the valence and conduction
bands. The size of the so called band gap between the Fermi level and the conduction band determines if
the metal is a conductor, insulator or semiconductor.
Once the temperature of the material is raised above absolute zero the Fermi energy can be used
to determine the probability of an electron having a particular energy level. As figure 1 below shows, the
probability that an energy state is filled by an electron depends on the Fermi energy.
Figure 1: Fermi Function. [1]
In this equation Ef is the Fermi energy, T is the absolute temperature, and k is Boltzmann’s constant. kT is
also proportional to the thermal energy of the particles. Figure 2 below illustrates the Fermi level of a
semiconductor. As the temperature rises the probability of electrons with energy greater than the Fermi
energy increases. These electrons take part in electric conduction which is an important application of the
Fermi energy.
Figure 2: Fermi energy of a semiconductor. [2]

As Figure 2 shows, there is a finite probability for electrons to exist in the band gap, but there are no
energy states in the band gap because the density of states is zero.
The Fermi energy is important to both electrical and thermal conductivities. As stated above,
increased temperature leads to increased numbers of electrons jumping into the conduction band from the
Fermi level. The electrons in the conduction band bump into each other causing electrical current, and
heat from the collisions. With more electrons available, more current, and heat, can flow. Hence, the
Fermi energy is intrinsic to both of these properties.
For metals, the Fermi energy actually lies within the conduction band; hence, it takes relatively
little energy to bump these electrons into the band gap, which is why metals are typically good conductors
of both electricity and heat. For semiconductors, the Fermi energy lies between the valence band and
conduction band, and more energy is required to excited the electrons sufficiently, and thus both electrical
and thermal conductivity are lower than for metals. For insulators, the Fermi energy is considerably
below the conduction band. Which means that it takes an incredible amount of energy to raise an electron
to the conduction band, and thus few electrons do; therefore, heat flow and electrical current are reduced
significantly, in some cases to near zero. Note, however, that the correlation between thermal and
electrical conductivity is only valid for certain substances, specifically metals and metalloids. Diamond,
for instance, has an incredibly high thermal conductivity but very low electrical conductivity. These
differences are due to differences in structure, which makes the Fermi energy less important than other
properties [3].
Fermi energy also answers a perplexing question; why electrons do not contribute to the specific
heat of a solid. After all, if electrons are crucial to thermal and electrical conductivity, it seems logical
that specific heat would also be affected. However, this is not the case and the Fermi energy provides the
reason. Recall that the Fermi energy is the highest level an electron can occupy at 0K. It may be expected
that this energy is significantly lower than the energy of an electron and ordinary physical reactions with
its environment, but that is not actually the case. Take Copper for instance, the Fermi energy of copper is
7 eV. The thermal energy of an electron at 300K is about .026 eV, based on 3/2 kT, which tells us that
the thermal energy can only interact with the electrons on the very top level of the Fermi sea. Since so
few electrons are excited by thermal energy their presence is trivial and does not contribute to the specific
heat.
Interestingly enough, the Fermi sea is why white dwarf stars don't collapse in upon themselves
due to gravitational effects. The Fermi energy represents a certain minimum "zero-point" energy (the
absolute minimum that can be reached) and a "zero-point" pressure. This pressure is due to the electrons

moving around, even at 0K. At high temperatures and low densities, classical pressure dominates. But at
low temperatures and incredibly high densities, such as those found in the white dwarf, the Fermi energy
and pressure become radically more important. This is due to Heisenberg's Uncertainty Principle: as the
electrons become more and more confined, the uncertainty in their momentum gets increasingly large,
and the electrons begin to go increasingly faster. As such, the Fermi energy is principally responsible for
keeping white dwarf stars from becoming black holes. Of course, eventually the star becomes so
compressed that the Fermi energy is not sufficient, and the star collapses. [4]
For a sample material with high critical temperature (Tc), the Fermi energy can be estimated by
forcing the superconducting component (SNS) in one part of a C-shaped sample to undergo pair-breaking
events while moving under the application of a temperature gradient. [5]
The thermal voltage resulting from the same temperature gradient acting on both the continuous
and Indium-filled halves of the sample were detected as a function of temperature around the
superconducting transition temperature. By comparing the difference between the thermopowers of two
halves,
Fig. 2: The difference between the thermopower between each half
the Fermi energy of the superconducting material can be calculated using the equations
*
S S A(
Tco
T ) − − ≈ ∆ 2
A ≅ ( 7.
04 / 2)(
k / )
B eE f
where S is the peak of the graph shown in Fig.2, A is the constant that is given by the best linear fit to the
data point in Fig. 2, Tco is the temperature that corresponds to the peak.

The angular correlation of positron-annihilation radiation from a polycrystalline metal can also
provide a value for the Fermi energy of the conduction electrons.
Because a positron entering a metal is usually thermalized before annihilation, the momentum of the
emitted gamma-ray pair is approximately equal to that of the electron which is annihilated with the
position. The angel between the emitted gamma rays is equal to π − θ where θ ≈ θ ≈ p / mc , and
sin 1
p1 is the component of the electron momentum perpendicular to the direction of emission of gamma
rays.[6]
The goal of the experiment is to find the angular correlation of the gamma rays which can be
achieved by counting coincidences between pulses in two detectors placed on opposite sides of the metal
sample in which the annihilation take place. From the correlation, the Fermi Energy can be calculated by
finding the value of α that yields the maximum momentum (pm) of the conduction electrons. After that,
the Fermi energy can be calculated by using the maximum momentum (pm) with an appropriate equation.
Photoluminescence experiments can be used to measure various energies, including the Fermi
energy, by increasing the excitation density until electron-hole systems undergo a phase transition, and
measuring the luminescence signal. The Fermi energy can then be calculated by EF(T)=EF(0)−βT 2
where β is a material specific constant. [7]
Around the 1920s Enrico Fermi, a professor in Rome, began work on the statistical study of
discrete energy states.[8] So called Fermi energy, the highest energy level of electrons at 0K, is an
important concept in the study of metals, insulators, semiconductors, and white dwarf stars, and in
understanding other material properties such as thermal conductivity. Knowledge of the Fermi energy of a
material has allowed deeper study in many areas of science, such as the thermal and electrical properties
of non conductive materials, like diamonds, electron tunneling, and the kinetics of free-electrons. The
concept of Fermi energy has allowed us to understand more about the interactions of electrons, and the
correlation between energy states and physical properties.

References
[1] (http://hyperphysics.phy-astr.gsu.edu/hbase/solids/fermi.html)
[2] (http://hyperphysics.phy-astr.gsu.edu/hbase/solids/fermi.html)
[3] (http://hyperphysics.phy-astr.gsu.edu/hbase/solids/fermi.html#c1)
[4] (http://www.sciencebits.com/StellarEquipartition)
[5] Richard A. Doyle, Owen L. de Lange, and Vladimir V. Girdin, “Estimation of Fermi energy from
thermoelectric-power measurement on a C-shaped polycrystalline sample of Bi1.6Pb0.4Sr2Ca2Cu2Oy”,
1991.
[6] John D. McGervey, “Student Measurement of Fermi Energy by Positron Annihilation”, 1963.
[7] (http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TWV-485V1CD-
5&_user=582538&_coverDate=07%2F31%2F2003&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&
_acct=C000029718&_version=1&_urlVersion=0&_userid=582538&md5=16cc69a689ec57c16d1314372
37813e6#sec3), 2003
[8] (http://www.britannica.com/ebc/article-9364347)