Jellium, also known as the uniform electron gas (UEG) or homogeneous electron gas (HEG), is a quantum mechanical model of interacting electrons in a solid where the positive charges (i.e. atomic nuclei) are assumed to be uniformly distributed in space whence the electron density 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 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.
Contents
- Hamiltonian
- Contributions to the total energy
- Zero temperature phase diagram of jellium in three and two dimensions
- Applications
- References
At zero temperature, the properties of jellium depend solely upon the constant electronic density. This 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, 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 for N-electrons confined within a volume of space Ω, and with electronic density ρ(r) and (constant) background charge density n(R) = N/Ω is
where
Hback is a constant and, in the limit of an infinite volume, divergent along with Hel-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, which yields the free electron gas model. The kinetic energy per electron is given by
where
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
The first correction to the free-electron model for jelium 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 and if one decides to work in a series for small
The series is quite accurate for small
For the full range of
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
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 plane waves, with the lowest-momentum plane-wave states being 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. 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, 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.
Within Hartree-Fock theory, the ferromagnetic fluid abruptly becomes more stable than the paramagnetic fluid at a density parameter of
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. They calculated the paramagnetic-ferromagnetic fluid transition to occur 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
Applications
Jellium is the simplest model of interacting electrons. 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. Semi-infinite jellium slabs are used to investigate surface properties such as work function and surface effects such as adsorption; near surfaces the electronic density varies in an oscillatory manner, decaying to a constant value in the bulk.
Within density functional theory, jellium is used in the construction of the local-density approximation, which in turn is a component of more sophisticated exchange-correlation energy functionals. From quantum Monte Carlo calculations of jellium, accurate values of the correlation energy density have been obtained for several values of the electronic density, which have been used to construct semi-empirical correlation functionals.
The jellium model has been applied to superatoms, and used in nuclear physics.