In
solid state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...
the electronic specific heat, sometimes called the electron heat capacity, is the
specific heat
In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
of an
electron gas
An ideal Fermi gas is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. T ...
. Heat is transported by
phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
s and by free electrons in solids. For pure metals, however, the electronic contributions dominate in the
thermal conductivity
The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
. In impure metals, the electron mean free path is reduced by collisions with impurities, and the phonon contribution may be comparable with the electronic contribution.
Introduction
Although the
Drude model
The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials (especially metals). Basically, Ohm's law was well established and stated that the current ''J'' and voltage ...
was fairly successful in describing the electron motion within metals, it has some erroneous aspects: it predicts the
Hall coefficient
The Hall effect is the production of a voltage difference (the Hall voltage) across an electrical conductor that is transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. It was discov ...
with the wrong sign compared to experimental measurements, the assumed additional electronic heat capacity to the lattice
heat capacity
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity i ...
, namely
per electron at elevated temperatures, is also inconsistent with experimental values, since measurements of metals show no deviation from the
Dulong–Petit law
The Dulong–Petit law, a thermodynamic law proposed by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states that the classical expression for the molar specific heat capacity of certain chemical elements is constant for tempe ...
. The observed electronic contribution of electrons to the heat capacity is usually less than one percent of
. This problem seemed insoluble prior to the development of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. This paradox was solved by
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
after the discovery of the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
, who recognised that the replacement of the
Boltzmann distribution
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability t ...
with the
Fermi–Dirac distribution Fermi–Dirac may refer to:
* Fermi–Dirac statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pa ...
was required and incorporated it in the
free electron model
In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quant ...
.
Derivation within the free electron model
Internal energy
When a metallic system is heated from absolute zero, not every electron gains an energy
as
equipartition dictates. Only those electrons in
atomic orbitals
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any spe ...
within an energy range of
of the
Fermi level
The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by ''µ'' or ''E''F
for brevity. The Fermi level does not include the work required to remove ...
are thermally excited. Electrons, in contrast to a classical gas, can only move into free states in their energetic neighbourhood.
The one-electron energy levels are specified by the
wave vector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
through the relation
with
the electron mass. This relation separates the occupied energy states from the unoccupied ones and corresponds to the spherical surface in
k-space. As
the
ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
distribution becomes:
:
where
*
is the
Fermi–Dirac distribution Fermi–Dirac may refer to:
* Fermi–Dirac statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pa ...
*
is the energy of the energy level corresponding to the
ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
*
is the ground state energy in the limit
, which thus still deviates from the true
ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
energy.
This implies that the ground state is the only occupied state for electrons in the limit
, the
takes the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
into account. The
internal energy
The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
of a system within the free electron model is given by the sum over one-electron levels times the mean number of electrons in that level:
:
where the factor of 2 accounts for the spin up and spin down states of the electron.
Reduced internal energy and electron density
Using the approximation that for a sum over a smooth function
over all allowed values of
for finite large system is given by:
:
where
is the volume of the system.
For the reduced internal energy
the expression for
can be rewritten as:
:
and the expression for the electron density
can be written as:
:
The integrals above can be evaluated using the fact that the dependence of the integrals on
can be changed to dependence on
through the relation for the electronic energy when described as
free particle
In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. I ...
s,
, which yields for an arbitrary function
:
:
with
which is known as the density of levels or density of states per unit volume such that
is the total number of states between
and
. Using the expressions above the integrals can be rewritten as:
:
These integrals can be evaluated for temperatures that are small compared to the
Fermi temperature
The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature.
In a Fermi ga ...
by applying the
Sommerfeld expansion
A Sommerfeld expansion is an approximation method developed by Arnold Sommerfeld for a certain class of integrals which are common in condensed matter and statistical physics. Physically, the integrals represent statistical averages using the Fe ...
and using the approximation that
differs from
for
by terms of order
. The expressions become:
:
For the ground state configuration the first terms (the integrals) of the expressions above yield the internal energy and electron density of the ground state. The expression for the electron density reduces to
. Substituting this into the expression for the internal energy, one finds the following expression:
:
Final expression
The contributions of electrons within the free electron model is given by:
:
, for free electrons :
Compared to the classical result (
), it can be concluded that this result is depressed by a factor of
which is at room temperature of order of magnitude
. This explains the absence of an electronic contribution to the heat capacity as measured experimentally.
Note that in this derivation
is often denoted by
which is known as the
Fermi energy
The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature.
In a Fermi ga ...
. In this notation, the electron heat capacity becomes:
:
and for free electrons :
using the definition for the
Fermi energy
The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature.
In a Fermi ga ...
with
the
Fermi temperature
The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature.
In a Fermi ga ...
.
Comparison with experimental results for the heat capacity of metals
For temperatures below both the
Debye temperature
In thermodynamics and solid-state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (Heat capacity) in a solid. It treats the vibrations of the atomic lattice (hea ...
and the
Fermi temperature
The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature.
In a Fermi ga ...
the heat capacity of metals can be written as a sum of electron and
phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
contributions that are linear and cubic respectively:
. The coefficient
can be calculated and determined experimentally. We report this value below:
The free electrons in a metal do not usually lead to a strong deviation from the
Dulong–Petit law
The Dulong–Petit law, a thermodynamic law proposed by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states that the classical expression for the molar specific heat capacity of certain chemical elements is constant for tempe ...
at high temperatures. Since
is linear in
and
is linear in
, at low temperatures the lattice contribution vanishes faster than the electronic contribution and the latter can be measured. The deviation of the approximated and experimentally determined electronic contribution to the heat capacity of a metal is not too large. A few metals deviate significantly from this approximated prediction. Measurements indicate that these errors are associated with the electron mass being somehow changed in the metal, for the calculation of the electron heat capacity the
effective mass of an electron should be considered instead. For Fe and Co the large deviations are attributed to the partially filled
d-shells of these
transition metals
In chemistry, a transition metal (or transition element) is a chemical element in the d-block of the periodic table (groups 3 to 12), though the elements of group 12 (and less often group 3) are sometimes excluded. They are the elements that can ...
, whose d-bands lie at the
Fermi energy
The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature.
In a Fermi ga ...
.
The
alkali metal
The alkali metals consist of the chemical elements lithium (Li), sodium (Na), potassium (K),The symbols Na and K for sodium and potassium are derived from their Latin names, ''natrium'' and ''kalium''; these are still the origins of the names ...
s are expected to have the best agreement with the free electron model since these metals only one s-electron outside a closed shell. However even sodium, which is considered to be the closest to a free electron metal, is determined to have a
more than 25 per cent higher than expected from the theory.
Certain effects influence the deviation from the approximation:
* The interaction of the conduction electrons with the periodic potential of the rigid crystal lattice is neglected.
* The interaction of the conduction electrons with phonons is also neglected. This interaction causes changes in the effective mass of the electron and therefore it affects the electron energy.
* The interaction of the conduction electrons with themselves is also ignored. A moving electron causes an inertial reaction in the surrounding electron gas.
Superconductors
Superconductivity
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
occurs in many metallic elements of the periodic system and also in alloys,
intermetallic
An intermetallic (also called an intermetallic compound, intermetallic alloy, ordered intermetallic alloy, and a long-range-ordered alloy) is a type of metallic alloy that forms an ordered solid-state compound between two or more metallic elemen ...
compounds, and doped
semiconductors
A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...
. This effect occurs upon cooling the material. The
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
decreases on cooling below the critical temperature
for
superconductivity
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
which indicates that the superconducting state is more ordered than the normal state. The entropy change is small, this must mean that only a very small fraction of electrons participate in the transition to the superconducting state but, the electronic contribution to the heat capacity changes drastically. There is a sharp jump of the heat capacity at the critical temperature while for the temperatures above the critical temperature the heat capacity is linear with temperature.
Derivation
The calculation of the electron heat capacity for super conductors can be done in the
BCS theory
BCS theory or Bardeen–Cooper–Schrieffer theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) is the first microscopic theory of superconductivity since Heike Kamerlingh Onnes's 1911 discovery. The theory describes sup ...
. The entropy of a system of
fermionic
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
quasiparticles
In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum.
For exam ...
, in this case
Cooper pairs
In condensed matter physics, a Cooper pair or BCS pair (Bardeen–Cooper–Schrieffer pair) is a pair of electrons (or other fermions) bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Coope ...
, is:
:
where
is the
Fermi–Dirac distribution Fermi–Dirac may refer to:
* Fermi–Dirac statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pa ...
with
and
*
is the particle energy with respect to the
Fermi energy
The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature.
In a Fermi ga ...
*
the energy gap parameter where
and
represents the probability that a
Cooper pair
In condensed matter physics, a Cooper pair or BCS pair (Bardeen–Cooper–Schrieffer pair) is a pair of electrons (or other fermions) bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper ...
is occupied or unoccupied respectively.
The heat capacity is given by
.
The last two terms can be calculated:
:
Substituting this in the expression for the heat capacity and again applying that the sum over
in the reciprocal space can be replaced by an integral in
multiplied by the
density of states
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
this yields:
:
Characteristic behaviour for superconductors
To examine the typical behaviour of the electron heat capacity for species that can transition to the superconducting state, three regions must be defined:
# Above the critical temperature
# At the critical temperature
# Below the critical temperature