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 as a simple model of delocalized electrons in a metal, where it can qualitatively reproduce features of real metals such as screening, plasmons, Wigner crystallization and Friedel oscillations.
At zero temperature, the properties of jellium depend solely upon the constant electronic density.
This property lends it to a treatment within density functional theory; the formalism itself provides the basis for the local-density approximation to the exchange-correlation energy density functional.
The term jellium was coined by Conyers Herring in 1952, alluding to the "positive jelly" background, and the typical metallic behavior it displays.
[1] The jellium model treats the electron-electron coupling rigorously.
The artificial and structureless background charge interacts electrostatically with itself and the electrons.
where Hback is a constant and, in the limit of an infinite volume, divergent along with Hel-back.
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.
is the Fermi wave vector, and the last expression shows the dependence on the Wigner–Seitz radius
The first correction to the free electron model for jellium is from the Fock exchange contribution to electron-electron interactions.
Higher order corrections to the total energy are due to electron correlation and if one decides to work in a series for small
, Chachiyo's correlation energy density can be used as the higher order correction.
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.
Here the lowest-momentum plane-wave states are doubly occupied by spin-up and spin-down electrons, giving a paramagnetic 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 Fermi fluid.
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.
As a result, jellium at zero temperature at a sufficiently low density will form a so-called Wigner crystal, in which the single-particle orbitals are of approximately Gaussian form centered on crystal lattice sites.
When Wigner crystallization occurs, jellium acquires a band gap.
[6] Furthermore, Hartree–Fock theory predicts exotic magnetic behavior, with the paramagnetic fluid being unstable to the formation of a spiral spin-density wave.
[7][8] 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 (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 method was Ceperley and Alder's famous 1980 calculation of the zero-temperature phase diagram of 3D jellium.
[12][13] The most recent QMC calculations indicate that there is no region of stability for a ferromagnetic fluid.
[15] Experimental results for a 2D hole gas in a GaAs/AlGaAs heterostructure (which, despite being clean, may not correspond exactly to the idealized jellium model) indicate a Wigner crystallization density of
It is employed in the calculation of properties of metals, where the core electrons and the nuclei are modeled as the uniform positive background and the valence electrons are treated with full rigor.
From quantum Monte Carlo calculations of jellium, accurate values of the correlation energy density have been obtained for several values of the electronic density,[9] which have been used to construct semi-empirical correlation functionals.
[20] The jellium model has been applied to superatoms, metal clusters, octacarbonyl complexes, and used in nuclear physics.