In
solid-state physicsSolid-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 large-scale properties of solid materials result from...
, the
free electron model is a simple model for the behaviour of
valence electronIn chemistry, valence electrons are the electrons of an atom that can participate in the formation of chemical bonds with other atoms. Valence electrons are the "own" electrons, present in the free neutral atom, that combine with valence electrons of other atoms to form chemical bonds. In a single...
s in a
crystal structureIn mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry...
of a metallic
solidSolid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
. It was developed principally by
Arnold SommerfeldArnold Johannes Wilhelm Sommerfeld was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics...
who combined the
classicalWhat "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...
Drude modelThe Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials...
with
quantum mechanicalQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
Fermi-Dirac statisticsFermi–Dirac statistics is a part of the science of physics that describes the energies of single particles in a system comprising many identical particles that obey the Pauli Exclusion Principle...
and hence it is also known as the
Drude–Sommerfeld model. The free electron
Empty Lattice ApproximationThe Empty Lattice Approximation is a theoretical electronic band structure model in which the potential is defined not more precisely than periodic and weak. The Empty Lattice Approximation describes a number of properties of energy dispersion relations of non-interacting free electrons that move...
forms the basis of the
band structureIn solid-state physics, the electronic band structure of a solid describes those ranges of energy an electron is "forbidden" or "allowed" to have. Band structure derives from the diffraction of the quantum mechanical electron waves in a periodic crystal lattice with a specific crystal system and...
model known as
nearly-free electron modelIn solid-state physics, the nearly-free electron model is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model is closely related to the more conceptual Empty Lattice Approximation...
. Given its simplicity, it is surprisingly successful in explaining many experimental phenomena, especially
- the Wiedemann-Franz law which relates electrical conductivity and thermal conductivity
In physics, thermal conductivity, k, is the property of a material's ability to conduct heat. It appears primarily in Fourier's Law for heat conduction....
;
- the temperature dependence of the heat capacity
Heat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...
;
- the shape of the electronic density of states
In solid-state and condensed matter physics, the density of states of a system describes the number of states per interval of energy at each energy level that are available to be occupied by electrons. Unlike isolated systems, like atoms or molecules in gas phase, the density distributions are not...
;
- the range of binding energy values;
- electrical conductivities;
- thermal electron emission
Thermionic emission is the heat-induced flow of charge carriers from a surface or over a potential-energy barrier. This occurs because the thermal energy given to the carrier overcomes the binding potential, also known as work function of the metal. The charge carriers can be electrons or ions, and...
and field electron emission from bulk metals.
Ideas and assumptions
As in the Drude model,
valence electronIn chemistry, valence electrons are the electrons of an atom that can participate in the formation of chemical bonds with other atoms. Valence electrons are the "own" electrons, present in the free neutral atom, that combine with valence electrons of other atoms to form chemical bonds. In a single...
s are assumed to be completely detached from their
ionAn ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...
s (forming an
electron gas). As in an
ideal gasAn ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
, electron-electron interactions are completely neglected. The electrostatic fields in metals are weak because of the
screening effectIn solids, especially in metals and semiconductors, the electrostatic screening or screening effect reduces the electrostatic field and Coulomb potential of an ion inside the solid...
.
The crystal lattice is not explicitly taken into account. A quantum-mechanical justification is given by
Bloch's Theorem* For the theorem named after Felix Bloch on wave functions of a particle in a periodic potential, see Bloch wave.* For the theorem in complex variables named after André Bloch, see Bloch's theorem ....
: an unbound electron moves in a periodic potential as a free electron in vacuum, except for the electron mass
m becoming an effective mass
m* which may deviate considerably from
m (one can even use negative effective mass to describe conduction by
electron holeAn electron hole is the conceptual and mathematical opposite of an electron, useful in the study of physics, chemistry, and electrical engineering. The concept describes the lack of an electron at a position where one could exist in an atom or atomic lattice...
s). Effective masses can be derived from band structure computations. While the static lattice does not hinder the motion of the electrons, they can well be scattered by impurities and by
phononIn physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...
s; these two interactions determine electrical and thermal conductivity (
superconductivitySuperconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...
requires a more refined theory than the free electron model).
According to the
Pauli exclusion principleThe Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
, each
phase spaceIn mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
element (Δk)
3(Δx)
3 can be occupied only by two electrons (one per
spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
quantum numberQuantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...
). This restriction of available electron states is taken into account by
Fermi-Dirac statisticsFermi–Dirac statistics is a part of the science of physics that describes the energies of single particles in a system comprising many identical particles that obey the Pauli Exclusion Principle...
(see also
Fermi gasA Fermi gas is an ensemble of a large number of fermions. Fermions, named after Enrico Fermi, are particles that obey Fermi–Dirac statistics. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density,...
). Main predictions of the free-electron model are derived by the
Sommerfeld expansionA 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...
of the Fermi-Dirac occupancy for energies around the
Fermi levelThe Fermi level is a hypothetical level of potential energy for an electron inside a crystalline solid. Occupying such a level would give an electron a potential energy \epsilon equal to its chemical potential \mu as they both appear in the Fermi-Dirac distribution function,which...
.
Energy and wave function of a free electron
For a free particle the potential is
. The Schrödinger equation for such a particle, like the free electron, is
The wave function
can be split into a solution of a time dependent and a solution of a time independent equation. The solution of the time dependent equation is
with energy
The solution of the time independent equation is
with a
wave vectorIn physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave , and its direction is ordinarily the direction of wave propagation In...
.
is the volume of space where the electron can be found.
The electron has a kinetic energy
The plane wave solution of this Schrödinger equation is
For
solid stateSolid-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 large-scale properties of solid materials result from...
and
condensed matterCondensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
physics the time independent solution
is of major interest. It is the basis of
electronic band structureIn solid-state physics, the electronic band structure of a solid describes those ranges of energy an electron is "forbidden" or "allowed" to have. Band structure derives from the diffraction of the quantum mechanical electron waves in a periodic crystal lattice with a specific crystal system and...
models that are widely used in
solid-state physicsSolid-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 large-scale properties of solid materials result from...
for model
HamiltoniansIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
like the
nearly-free electron modelIn solid-state physics, the nearly-free electron model is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model is closely related to the more conceptual Empty Lattice Approximation...
and the Tight binding model and different models that use a
Muffin-tin approximationThe muffin-tin approximation is a shape approximation of the potential field in an atomistic environment. It is most commonly employed in quantum mechanical simulations of electronic band structure in solids. The approximation was proposed by John C. Slater. Many modern electronic structure methods...
. The
eigenfunctionsThe eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix...
of these Hamiltonians are
Bloch waveA Bloch wave or Bloch state, named after Felix Bloch, is the wavefunction of a particle placed in a periodic potential...
s which are modulated plane waves.
Dielectric function of the electron gas
On a scale much larger than the inter atomic distance a solid can be viewed as an aggregate of a negatively charged plasma of the free electron gas and a positively charged background of atomic cores. The background is the rather stiff and massive background of atomic nuclei and core electrons which we will consider to be infinitely massive and fixed in space. The negatively charged plasma is formed by the valence electrons of the free electron model that are uniformly distributed over the interior of the solid. If an oscillating electric field is applied to the solid, the negatively charged plasma tends to move a distance
x apart from the positively charged background. As a result the sample is polarized and there will be an excess charge at the opposite surfaces of the sample.
The surface charge density is
which produces a restoring
electric fieldIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
in the sample
The dielectric function of the sample is expressed as
where
is the
electric displacement and
is the
polarization densityIn classical electromagnetism, polarization density is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is...
.
The electric field and polarization densities are
and the polarization per atom with
n electrons is
The force
F of the oscillating electric field causes the electrons with charge
e and mass
m to accelerate with an acceleration
a
which, after substitution of
E,
P and
x, yields an harmonic oscillator equation.
After a little algebra the relation between polarization density and electric field can be expressed as
The frequency dependent dielectric function of the solid is
At a resonance frequency
, called the
plasma frequency, the dielectric function changes sign from negative to positive and real part of the dielectric function drops to zero.
This is a
plasma oscillationPlasma oscillations, also known as "Langmuir waves" , are rapid oscillations of the electron density in conducting media such as plasmas or metals. The oscillations can be described as an instability in the dielectric function of a free electron gas. The frequency only depends weakly on the...
resonance or
plasmonIn physics, a plasmon is a quantum of plasma oscillation. The plasmon is a quasiparticle resulting from the quantization of plasma oscillations just as photons and phonons are quantizations of light and mechanical vibrations, respectively...
. The plasma frequency is a direct measure of the square root of the density of valence electrons in a solid. Observed values are in reasonable agreement with this theoretical prediction for a large number of materials. Below the plasma frequency, the dielectric function is negative and the field cannot penetrate the sample. Light with angular frequency below the plasma frequency will be totally reflected. Above the plasma frequency the light waves can penetrate the sample.
The Schrödinger equation
For a free particle the potential is
, so the Schrödinger equation for the free electron is
This is a type of
wave equationThe wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
that has numerous kinds of solutions. One way of solving the equation is splitting it in a time-dependent oscillator equation and a space-dependent wave equation like
and
and substituting a product of solutions like
The Schrödinger equation can be split in a time dependent part and a time independent part.
Solution of the time dependent equation
The
peculiar time dependent part of the Schrödinger equation is,
unlike the Klein-Gordon equationThe Klein–Gordon equation is a relativistic version of the Schrödinger equation....
for
pionIn particle physics, a pion is any of three subatomic particles: , , and . Pions are the lightest mesons and they play an important role in explaining the low-energy properties of the strong nuclear force....
s and most of the other well known
wave equationThe wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
s, a
first order in time differential equation with a
90o out of phase driving mechanism, while most oscillator equations are
second order in time differential equations with
180o out of phase driving mechanisms.
The equation that has to be solved is
.
The complex (imaginary) exponent is proportional to the energy
The imaginary exponent can be transformed to an angular frequency
The wave function now has a stationary and an oscillating part
The stationary part is of major importance to the physical properties of the electronic structure of matter.
Solution of the time independent equation
The wave function of free electrons is in general described as the solution of the time independent Schrödinger equation for free electrons
The
Laplace operatorIn mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...
in Cartesian coordinates is
The wave function can be factorized for the three Cartesian directions
Now the time independent Schrödinger equation can be split in three independent parts for the three different Cartesian directions
As a solution an exponential function is substituted in the time independent Schrödinger equation
The solution of
gives the exponent
which yields the wave equation
and the energy
With the normalization
and the wave vector length
we arrive at the
plane waveIn the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....
solution with a wave function
for free electrons with a
wave vectorIn physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave , and its direction is ordinarily the direction of wave propagation In...
and a kinetic energy
in which
is the volume of space occupied by the electron.
The traveling plane wave solution
The product of the time independent stationary wave solution and time dependent oscillator solution
gives the traveling plane wave solution
which is the final solution for the free electron wave function.
External links
See also
- Fermi gas
A Fermi gas is an ensemble of a large number of fermions. Fermions, named after Enrico Fermi, are particles that obey Fermi–Dirac statistics. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density,...
- Bloch wave
A Bloch wave or Bloch state, named after Felix Bloch, is the wavefunction of a particle placed in a periodic potential...
- Nearly-free electron model
In solid-state physics, the nearly-free electron model is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model is closely related to the more conceptual Empty Lattice Approximation...
- Free electron laser
A free-electron laser, or FEL, is a laser that shares the same optical properties as conventional lasers such as emitting a beam consisting of coherent electromagnetic radiation which can reach high power, but which uses some very different operating principles to form the beam...
- Particle in a one-dimensional lattice
In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so electrons are subject to a regular potential inside...