Jellium, also known as the uniform electron gas (UEG) or homogeneous electron gas (HEG), is a
quantum mechanical
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, qua ...
model of interacting
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
s in a solid where the
positive charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectiv ...
s (i.e. atomic nuclei) are assumed to be uniformly distributed in space; the
electron density
In quantum chemistry, electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial va ...
is a uniform quantity as well in space. This model allows one to focus on the effects in solids that occur due to the quantum nature of electrons and their mutual repulsive interactions (due to like charge) without explicit introduction of the
atomic lattice and structure making up a real material. Jellium is often used 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 ...
as a simple model of delocalized electrons in a metal, where it can qualitatively reproduce features of real metals such as
screening
Screening may refer to:
* Screening cultures, a type a medical test that is done to find an infection
* Screening (economics), a strategy of combating adverse selection (includes sorting resumes to select employees)
* Screening (environmental), a ...
,
plasmon
In physics, a plasmon is a quantum of plasma oscillation. Just as light (an optical oscillation) consists of photons, the plasma oscillation consists of plasmons. The plasmon can be considered as a quasiparticle since it arises from the quantiz ...
s,
Wigner crystal
A Wigner crystal is the solid (crystalline) phase of electrons first predicted by Eugene Wigner in 1934. A gas of electrons moving in a uniform, inert, neutralizing background (i.e. Jellium Model) will crystallize and form a lattice if the electr ...
lization and
Friedel oscillation
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 or Fermi liquid. Friedel oscillations are a quantum mechanical analog ...
s.
At
zero temperature, the properties of jellium depend solely upon the constant
electronic density
In quantum chemistry, electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial va ...
. This property lends it to a treatment within
density functional theory
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
; the formalism itself provides the basis for the
local-density approximation Local-density approximations (LDA) are a class of approximations to the exchange–correlation (XC) energy functional in density functional theory (DFT) that depend solely upon the value of the electronic density at each point in space (and not, ...
to the exchange-correlation energy density functional.
The term ''jellium'' was coined by
Conyers Herring
William Conyers Herring (November 15, 1914 – July 23, 2009) was an American physicist. He was a Professor of Applied Physics at Stanford University and the Wolf Prize in Physics recipient in 1984/5.
Academic career
Conyers Herring completed ...
in 1952, alluding to the "positive jelly" background, and the typical metallic behavior it displays.
Hamiltonian
The jellium model treats the electron-electron coupling rigorously. The artificial and structureless background charge interacts electrostatically with itself and the electrons. The jellium
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
for ''N'' electrons confined within a volume of space Ω, and with
electronic density
In quantum chemistry, electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial va ...
''ρ''(r) and (constant) background charge density ''n''(R) = ''N''/Ω is
where
*''H''
el is the electronic Hamiltonian consisting of the kinetic and electron-electron repulsion terms:
*''H''
back is the Hamiltonian of the positive background charge interacting
electrostatic
Electrostatics is a branch of physics that studies electric charges at rest (static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...
ally with itself:
*''H''
el-back is the electron-background interaction Hamiltonian, again an electrostatic interaction:
''H''
back is a constant and, in the limit of an infinite volume, divergent along with ''H''
el-back. The divergence is canceled by a term from the electron-electron coupling: the background interactions cancel and the system is dominated by the kinetic energy and coupling of the electrons. Such analysis is done in Fourier space; the interaction terms of the Hamiltonian which remain correspond to the Fourier expansion of the electron coupling for which q ≠ 0.
Contributions to the total energy
The traditional way to study the electron gas is to start with non-interacting electrons which are governed only by the kinetic energy part of the Hamiltonian, also called a
Fermi 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. ...
. The kinetic energy per electron is given by
:
where
is the Fermi energy,
is the Fermi wave vector, and the last expression shows the dependence on the
Wigner–Seitz radius The Wigner–Seitz radius r_, named after Eugene Wigner and Frederick Seitz, is the radius of a sphere whose volume is equal to the mean volume per atom in a solid (for first group metals). In the more general case of metals having more valence ...
where energy is measured in
rydbergs.
Without doing much work, one can guess that the electron-electron interactions will scale like the inverse of the average electron-electron separation and hence as
(since the Coulomb interaction goes like one over distance between charges) so that if we view the interactions as a small correction to the kinetic energy, we are describing the limit of small
(i.e.
being larger than
) and hence high electron density. Unfortunately, real metals typically have
between 2-5 which means this picture needs serious revision.
The first correction to 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 ...
for jellium is from the
Fock exchange contribution to electron-electron interactions. Adding this in, one has a total energy of
:
where the negative term is due to exchange: exchange interactions lower the total energy. Higher order corrections to the total energy are due to
electron correlation
Electronic correlation is the interaction between electrons in the electronic structure of a quantum system. The correlation energy is a measure of how much the movement of one electron is influenced by the presence of all other electrons.
Atom ...
and if one decides to work in a series for small
, one finds
:
The series is quite accurate for small
but of dubious value for
values found in actual metals.
For the full range of
, Chachiyo's correlation energy density can be used as the higher order correction. In this case,
:
, which agrees quite well (on the order of milli-Hartree) with the
quantum Monte Carlo
Quantum Monte Carlo encompasses a large family of computational methods whose common aim is the study of complex quantum systems. One of the major goals of these approaches is to provide a reliable solution (or an accurate approximation) of the ...
simulation.
Zero-temperature phase diagram of jellium in three and two dimensions
The physics of the zero-temperature phase behavior of jellium is driven by competition between the kinetic energy of the electrons and the electron-electron interaction energy. The kinetic-energy operator in the Hamiltonian scales as
, where
is the
Wigner–Seitz radius The Wigner–Seitz radius r_, named after Eugene Wigner and Frederick Seitz, is the radius of a sphere whose volume is equal to the mean volume per atom in a solid (for first group metals). In the more general case of metals having more valence ...
, whereas the interaction energy operator scales as
. Hence the kinetic energy dominates at high density (small
), while the interaction energy dominates at low density (large
).
The limit of high density is where jellium most resembles a
noninteracting free electron gas. To minimize the kinetic energy, the single-electron states are delocalized, in a state very close to the Slater determinant (non-interacting state) constructed from plane waves. Here the lowest-momentum plane-wave states are doubly occupied by spin-up and spin-down electrons, giving a
paramagnetic
Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, d ...
Fermi fluid.
At lower densities, where the interaction energy is more important, it is energetically advantageous for the electron gas to spin-polarize (i.e., to have an imbalance in the number of spin-up and spin-down electrons), resulting in a
ferromagnetic
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
Fermi fluid. This phenomenon is known as ''itinerant ferromagnetism''. At sufficiently low density, the kinetic-energy penalty resulting from the need to occupy higher-momentum plane-wave states is more than offset by the reduction in the interaction energy due to the fact that exchange effects keep indistinguishable electrons away from one another.
A further reduction in the interaction energy (at the expense of kinetic energy) can be achieved by localizing the electron orbitals. As a result, jellium at zero temperature at a sufficiently low density will form a so-called
Wigner crystal
A Wigner crystal is the solid (crystalline) phase of electrons first predicted by Eugene Wigner in 1934. A gas of electrons moving in a uniform, inert, neutralizing background (i.e. Jellium Model) will crystallize and form a lattice if the electr ...
, in which the single-particle orbitals are of approximately Gaussian form centered on crystal lattice sites. Once a Wigner crystal has formed, there may in principle be further phase transitions between different crystal structures and between different magnetic states for the Wigner crystals (e.g., antiferromagnetic to ferromagnetic spin configurations) as the density is lowered. When Wigner crystallization occurs, jellium acquires a
band gap
In solid-state physics, a band gap, also called an energy gap, is an energy range in a solid where no electronic states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference (in ...
.
Within
Hartree–Fock theory, the ferromagnetic fluid abruptly becomes more stable than the paramagnetic fluid at a density parameter of
in three dimensions (3D) and
in two dimensions (2D). However, according to Hartree–Fock theory, Wigner crystallization occurs at
in 3D and
in 2D, so that jellium would crystallise before itinerant ferromagnetism occurs. Furthermore, Hartree–Fock theory predicts exotic magnetic behavior, with the paramagnetic fluid being unstable to the formation of a spiral spin-density wave. Unfortunately, Hartree–Fock theory does not include any description of correlation effects, which are energetically important at all but the very highest densities, and so a more accurate level of theory is required to make quantitative statements about the phase diagram of jellium.
Quantum Monte Carlo
Quantum Monte Carlo encompasses a large family of computational methods whose common aim is the study of complex quantum systems. One of the major goals of these approaches is to provide a reliable solution (or an accurate approximation) of the ...
(QMC) methods, which provide an explicit treatment of electron correlation effects, are generally agreed to provide the most accurate quantitative approach for determining the zero-temperature phase diagram of jellium. The first application of the
diffusion Monte Carlo Diffusion Monte Carlo (DMC) or diffusion quantum Monte Carlo is a quantum Monte Carlo method that uses a Green's function to solve the Schrödinger equation. DMC is potentially numerically exact, meaning that it can find the exact ground state ener ...
method was Ceperley and Alder's famous 1980 calculation of the zero-temperature phase diagram of 3D jellium.
They calculated the paramagnetic-ferromagnetic fluid transition to occur at
and Wigner crystallization (to a body-centered cubic crystal) to occur at
. Subsequent QMC calculations have refined their phase diagram: there is a second-order transition from a paramagnetic fluid state to a partially spin-polarized fluid from
to about
; and Wigner crystallization occurs at
.
In 2D, QMC calculations indicate that the paramagnetic fluid to ferromagnetic fluid transition and Wigner crystallization occur at similar density parameters, in the range