In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, screening is the damping of
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
s caused by the presence of mobile
charge carriers. It is an important part of the behavior of charge-carrying
fluids, such as ionized gases (classical
plasmas),
electrolyte
An electrolyte is a medium containing ions that is electrically conducting through the movement of those ions, but not conducting electrons. This includes most soluble salts, acids, and bases dissolved in a polar solvent, such as water. Upon ...
s, and
charge carriers in electronic conductors (
semiconductor
A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way ...
s,
metal
A metal (from ancient Greek, Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electrical resistivity and conductivity, e ...
s).
In a fluid, with a given
permittivity , composed of electrically charged constituent particles, each pair of particles (with charges and ) interact through the
Coulomb force as
where the vector is the relative position between the charges. This interaction complicates the theoretical treatment of the fluid. For example, a naive quantum mechanical calculation of the ground-state energy density yields infinity, which is unreasonable. The difficulty lies in the fact that even though the Coulomb force diminishes with distance as , the average number of particles at each distance is proportional to , assuming the fluid is fairly
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...
. As a result, a charge fluctuation at any one point has non-negligible effects at large distances.
In reality, these long-range effects are suppressed by the flow of particles in response to electric fields. This flow reduces the ''effective'' interaction between particles to a short-range "screened" Coulomb interaction. This system corresponds to the simplest example of a renormalized interaction (see sections 1.2.1 and 3.2 of).
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 th ...
, especially for
metals
A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typical ...
and
semiconductor
A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way ...
s, the screening effect describes the
electrostatic field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
and Coulomb potential of an
ion inside the solid. Like the electric field of the
nucleus
Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to:
* Atomic nucleus, the very dense central region of an atom
*Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA
Nucl ...
is reduced inside an atom or ion due to the
shielding effect, the electric fields of ions in conducting solids are further reduced by the cloud of
conduction electrons.
Description
Consider a fluid composed of electrons moving in a uniform background of positive charge (one-component plasma). Each electron possesses a negative charge. According to Coulomb's interaction, negative charges repel each other. Consequently, this electron will repel other electrons creating a small region around itself in which there are fewer electrons. This region can be treated as a positively charged "screening hole". Viewed from a large distance, this screening hole has the effect of an overlaid positive charge which cancels the electric field produced by the electron. Only at short distances, inside the hole region, can the electron's field be detected. For a plasma, this effect can be made explicit by an
-body calculation (see section 5 of
). If the background is made up of positive ions, their attraction by the electron of interest reinforces the above screening mechanism. In atomic physics, a germane effect exists for atoms with more than one electron shell: the
shielding effect. In plasma physics, electric-field screening is also called Debye screening or shielding. It manifests itself on macroscopic scales by a sheath (
Debye sheath) next to a material with which the plasma is in contact.
The screened potential determines the inter atomic force and the
phonon
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechani ...
dispersion relation in metals. The screened potential is used to calculate the
electronic 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 ...
of a large variety of materials, often in combination with
pseudopotential models. The screening effect leads to the
independent electron approximation, which explains the predictive power of introductory models of solids like the
Drude model, the
free electron model and the
nearly free electron model.
Theory and models
The first theoretical treatment of electrostatic screening, due to
Peter Debye
Peter Joseph William Debye (; ; March 24, 1884 – November 2, 1966) was a Dutch-American physicist and physical chemist, and Nobel laureate in Chemistry.
Biography
Early life
Born Petrus Josephus Wilhelmus Debije in Maastricht, Netherland ...
and
Erich Hückel,
dealt with a stationary point charge embedded in a fluid.
Consider a fluid of electrons in a background of heavy, positively charged ions. For simplicity, we ignore the motion and spatial distribution of the ions, approximating them as a uniform background charge. This simplification is permissible since the electrons are lighter and more mobile than the ions, provided we consider distances much larger than the ionic separation. In
condensed matter physics, this model is referred to as
jellium.
Screened Coulomb interactions
Let ''ρ'' denote the
number density
The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric num ...
of electrons, and ''φ'' the
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
. At first, the electrons are evenly distributed so that there is zero net charge at every point. Therefore, ''φ'' is initially a constant as well.
We now introduce a fixed point charge ''Q'' at the origin. The associated
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system i ...
is ''Qδ''(''r''), where ''δ''(''r'') is the
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
. After the system has returned to equilibrium, let the change in the electron density and electric potential be ''Δρ''(''r'') and ''Δφ''(''r'') respectively. The charge density and electric potential are related by
Poisson's equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with ...
, which gives
: