In
solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as solid-state chemistry, quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state phy ...
, Friedel oscillations,
[
] named after French physicist
Jacques Friedel, arise from localized perturbations in a metallic or semiconductor system caused by a defect in the
Fermi gas
A Fermi gas is an idealized model, 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. These statis ...
or
Fermi liquid
Fermi liquid theory (also known as Landau's Fermi-liquid theory) is a theoretical model of interacting fermions that describes the normal state of the conduction electrons in most metals at sufficiently low temperatures. The theory describes the ...
.
Friedel oscillations are a quantum mechanical analog to
electric charge screening of charged species in a pool of ions. Whereas electrical charge screening utilizes a point entity treatment to describe the make-up of the ion pool, Friedel oscillations describing fermions in a Fermi fluid or Fermi gas require a quasi-particle or a scattering treatment. Such oscillations depict a characteristic exponential decay in the fermionic density near the perturbation followed by an ongoing sinusoidal decay resembling
sinc function
In mathematics, physics and engineering, the sinc function ( ), denoted by , has two forms, normalized and unnormalized..
In mathematics, the historical unnormalized sinc function is defined for by
\operatorname(x) = \frac.
Alternatively, ...
. In 2020, magnetic Friedel oscillations were observed on a metal surface.
One-dimensional electron gas
As a simple model, consider one-dimensional electron gas in a half-space
. The electrons do not penetrate into the half-space
, so that the boundary condition for the electron wave function is
. The oscillating wave functions that satisfy this condition are
:
,
where
is the electron wave vector, and
is the length of the one-dimensional box (we use the 'box" normalization here). We consider degenerate electron gas, so that the electrons fill states with energies less than the Fermi energy
. Then, the electron density
is calculated as
:
,
where summation is taken over all wave vectors less than the
Fermi wave vector , the factor 2 accounts for the spin degeneracy. By transforming the sum over
into the integral we obtain
:
.
We see that the boundary perturbs the electron density leading to its spatial oscillations with the period
near the boundary. These oscillations decay into the bulk with the decay length also given by
. At
the electron density equals to the unperturbed density of the one-dimensional electron gas
.
Scattering description
The electrons that move through a
metal
A metal () is a material that, when polished or fractured, shows a lustrous appearance, and conducts electrical resistivity and conductivity, electricity and thermal conductivity, heat relatively well. These properties are all associated wit ...
or
semiconductor
A semiconductor is a material with electrical conductivity between that of a conductor and an insulator. Its conductivity can be modified by adding impurities (" doping") to its crystal structure. When two regions with different doping level ...
behave like
free electrons of a
Fermi gas
A Fermi gas is an idealized model, 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. These statis ...
with a
plane wave
In physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of ...
-like
wave function
In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
, that is
:
.
Electrons in a metal behave differently than particles in a normal gas because electrons are
fermion
In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s and they obey
Fermi–Dirac statistics
Fermi–Dirac 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 Pauli exclusion principle. A result is the Fermi–Dirac distribution of part ...
. This behaviour means that every k-state in the gas can only be occupied by two electrons with opposite
spin
Spin or spinning most often refers to:
* Spin (physics) or particle spin, a fundamental property of elementary particles
* Spin quantum number, a number which defines the value of a particle's spin
* Spinning (textiles), the creation of yarn or thr ...
. The occupied states fill a sphere in the
band structure
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
k-space, up to a fixed energy level, the so-called
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 radius of the sphere in k-space, ''k''
F, is called the
Fermi wave vector.
If there is a foreign atom embedded in the metal or semiconductor, a so-called
impurity
In chemistry and materials science, impurities are chemical substances inside a confined amount of liquid, gas, or solid. They differ from the chemical composition of the material or compound. Firstly, a pure chemical should appear in at least on ...
, the electrons that move freely through the solid are scattered by the deviating potential of the impurity. During the scattering process the initial state wave vector k
i of the electron wave function is scattered to a final state wave vector k
f. Because the electron gas is a
Fermi gas
A Fermi gas is an idealized model, 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. These statis ...
only electrons with energies near the Fermi level can participate in the scattering process because there must be empty final states for the scattered states to jump to. Electrons that are too far below the Fermi energy ''E''
F can't jump to unoccupied states. The states around the Fermi level that can be scattered occupy a limited range of k-values or wavelengths. So only electrons within a limited wavelength range near the Fermi energy are scattered resulting in a density modulation around the impurity of the form
:
.
Qualitative description

In the classic scenario of electric charge screening, a dampening in the electric field is observed in a mobile charge-carrying fluid upon the presence of a charged object. Since electric charge screening considers the mobile charges in the fluid as point entities, the concentration of these charges with respect to distance away from the point decreases exponentially. This phenomenon is governed by
Poisson–Boltzmann equation
The Poisson–Boltzmann equation describes the distribution of the electric potential in solution in the direction normal to a charged surface. This distribution is important to determine how the electrostatic interactions will affect the molecules ...
. The quantum mechanical description of a perturbation in a one-dimensional Fermi fluid is modelled by the
Tomonaga-Luttinger liquid.
[D. Vieira ''et al''., “Friedel oscillations in one-dimensional metals: From Luttinger’s theorem to the Luttinger liquid”, ''Journal of Magnetism and Magnetic Materials'', vol. 320, pp. 418-420, 2008.]
(arXiv Submission) The fermions in the fluid that take part in the screening cannot be considered as a point entity but a wave-vector is required to describe them. Charge density away from the perturbation is not a continuum but fermions arrange themselves at discrete spaces away from the perturbation. This effect is the cause of the circular ripples around the impurity.
''N.B. Where classically near the charged perturbation an overwhelming number of oppositely charged particles can be observed, in the quantum mechanical scenario of Friedel oscillations periodic arrangements of oppositely charged fermions followed by spaces with same charged regions.''
In the figure to the right, a 2-dimensional Friedel oscillations has been illustrated with an
Scanning tunneling microscope, STM image of a clean surface. As the image is taken on a surface, the regions of low electron density leave the atomic nuclei 'exposed' which result in a net positive charge.
See also
*
Lindhard theory
In condensed matter physics, Lindhard theoryN. W. Ashcroft and N. D. Mermin, ''Solid State Physics'' (Thomson Learning, Toronto, 1976) is a method of calculating the effects of electric field screening by electrons in a solid. It is based on qu ...
References
External links
* http://gravityandlevity.wordpress.com/2009/06/02/friedel-oscillations-wherein-we-learn-that-the-electron-has-a-size/ - a simple explanation of the phenomenon
{{DEFAULTSORT:Friedel Oscillations
Condensed matter physics