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{{Short description|Physical model of solid metals as electron gases}}
{{Use American English|date = February 2019}}

'''Jellium''', also known as the '''uniform electron gas''' ('''UEG''') or '''homogeneous electron gas''' ('''HEG'''), is a [[quantum mechanical]] model of interacting [[electron]]s in a solid where the [[positive charge]]s (i.e. atomic nuclei) are assumed to be uniformly distributed in space; 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 [[Crystal lattice|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 [[Electric-field screening|screening]], [[plasmon]]s, [[Wigner crystal]]lization and [[Friedel oscillation]]s.

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.<ref>{{cite journal|doi=10.1162/posc.2006.14.4.457|last=Hughes|first=R. I. G. |year=2006|title=Theoretical Practice: the Bohm-Pines Quartet|journal=Perspectives on Science|volume=14|issue=4|pages=457–524|s2cid=57569991|url=http://muse.jhu.edu/journals/perspectives_on_science/v014/14.4hughes.pdf}}</ref>

==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 (quantum mechanics)|Hamiltonian]] for ''N'' electrons confined within a volume of space Ω, and with [[electronic density]] ''ρ''('''r''') and (constant) background charge density ''n''('''R''')&nbsp;=&nbsp;''N''/Ω is<ref>{{cite book|last=Gross| first=E. K. U. |author2=Runge, E. |author3=Heinonen, O. |title=Many-Particle Theory| publisher=Verlag Adam Hilger | location=Bristol| year=1991|pages=79–80 |isbn=978-0-7503-0155-8}}</ref><ref>{{cite book|last=Giuliani| first=Gabriele |author2=Vignale |author3=Giovanni |title=Quantum Theory of the Electron Liquid| url=https://archive.org/details/quantumtheoryofe0000giul| url-access=registration |publisher=Cambridge University Press|year=2005 |pages=[https://archive.org/details/quantumtheoryofe0000giul/page/13 13–16] |isbn=978-0-521-82112-4}}</ref>

<math display="block"> \hat{H}=\hat{H}_{\mathrm{el}}+\hat{H}_{\mathrm{back}}+\hat{H}_{\mathrm{el-back}},</math>

where

*''H''<sub>el</sub> is the electronic Hamiltonian consisting of the kinetic and electron-electron repulsion terms:<math display="block"> \hat{H}_{\mathrm{el}}=\sum_{i=1}^N\frac{p_{i}^2}{2m}+\sum_{i<j}^N\frac{e^2}{|\mathbf{r}_i-\mathbf{r}_j|}</math>
*''H''<sub>back</sub> is the Hamiltonian of the positive background charge interacting [[electrostatic]]ally with itself:<math display="block"> \hat{H}_{\mathrm{back}}=\frac{e^2}{2}\int_{\Omega}\mathrm{d}\mathbf{R}\int_{\Omega}\mathrm{d}\mathbf{R}'\ \frac{n(\mathbf{R})n(\mathbf{R}')}{|\mathbf{R}-\mathbf{R}'|} = \frac{e^{2}}{2}\left(\frac{N}{\Omega}\right)^{2}\int_{\Omega}\mathrm{d}\mathbf{R}\int_{\Omega}\mathrm{d}\mathbf{R}'\ \frac{1}{|\mathbf{R}-\mathbf{R}'|}</math>
*''H''<sub>el-back</sub> is the electron-background interaction Hamiltonian, again an electrostatic interaction:<math display="block"> \hat{H}_{\mathrm{el-back}}=\int_{\Omega}\mathrm{d}\mathbf{r}\int_{\Omega}\mathrm{d}\mathbf{R}\ \frac{\rho(\mathbf{r})n(\mathbf{R})}{|\mathbf{r}-\mathbf{R}|} = -e^{2}\frac{N}{\Omega}\sum_{i=1}^{N}\int_{\Omega}\mathrm{d}\mathbf{R}\ \frac{1}{|\mathbf{r}_{i}-\mathbf{R}|}</math>

''H''<sub>back</sub> is a constant and, in the limit of an infinite volume, divergent along with ''H''<sub>el-back</sub>. 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'''&nbsp;≠&nbsp;'''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]].  The kinetic energy per electron is given by
: <math> K = \frac{3}{5} E_{\rm F} = \frac{3}{5}\frac{\hbar ^2k_{\rm F}^2}{2m_{\rm e}} = \frac{3}{5}\biggl(\frac{9\pi}{4}\biggr)^{\frac{2}{3}} \frac{1}{(r'_{\rm s} / a_0)^2} \textrm{Ry} \approx \frac{2.21}{(r'_{\rm s} / a_0)^2} \textrm{Ry} </math>
where <math> E_{\rm F}</math> is the Fermi energy, <math> k_{\rm F} </math> is the Fermi wave vector, and the last expression shows the dependence on the [[Wigner–Seitz radius]] <math> r'_{\rm s} </math> where energy is measured in [[Rydberg constant|rydbergs]]. <math> a_0 </math> is the [[Bohr radius]]. In what follows <math> r_{\rm s} </math> is the normalized value <math> r_{\rm s} = r'_{\rm s}/a_0 </math>

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 <math> 1/r_{12} </math> (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 <math> r_{\rm s} </math> (i.e. <math>1/r_{\rm s}^2</math> being larger than <math>1/r_{\rm s}</math>) and hence high electron density.  Unfortunately, real metals typically have <math> r_{\rm s} </math> between 2-5 which means this picture needs serious revision.

The first correction to the [[free electron model]] for jellium is from the [[Hartree–Fock method|Fock exchange]] contribution to electron-electron interactions.  Adding this in, one has a total energy of
: <math> E = \frac{2.21}{r_{\rm s}^2} - \frac{0.916}{r_{\rm s}}</math>

where the negative term is due to exchange:  exchange interactions lower the total energy.  Higher order corrections to the total energy are due to [[Electronic correlation|electron correlation]] and if one decides to work in a series for small <math>r_s</math>, one finds
: <math> E = \frac{2.21}{r_{\rm s}^2} - \frac{0.916}{r_{\rm s}}  + 0.0622 \ln (r_{\rm s}) - 0.096 + O(r_{\rm s})</math>
The series is quite accurate for small <math>r_{\rm s}</math> but of dubious value for <math>r_{\rm s}</math> values found in actual metals.

For the full range of <math>r_{\rm s}</math>, Chachiyo's correlation energy density can be used as the higher order correction. In this case,
: <math> E = \frac{2.21}{r_{\rm s}^2} - \frac{0.916}{r_{\rm s}}  +  a \ln \left( 1  + \frac{b}{r_{\rm s}} + \frac{b}{r_{\rm s}^2} \right)</math>,<ref>{{cite journal | title = Simple and accurate uniform electron gas correlation energy for the full range of densities | author = Teepanis Chachiyo | journal = J. Chem. Phys. | volume = 145 | pages = 021101 | year = 2016 | doi = 10.1063/1.4958669 | pmid = 27421388 | issue = 2| bibcode = 2016JChPh.145b1101C | doi-access = free }}</ref> which agrees quite well (on the order of milli-Hartree) with the [[quantum Monte Carlo]] 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 <math>1/r_{\rm s}^2</math>, where <math>r_{\rm s}</math> is the [[Wigner–Seitz radius]], whereas the interaction energy operator scales as <math>1/r_{\rm s}</math>.  Hence the kinetic energy dominates at high density (small <math>r_{\rm s}</math>), while the interaction energy dominates at low density (large <math>r_{\rm s}</math>).

The limit of high density is where jellium most resembles a [[Fermi gas|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]] 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 transition]]s 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 <math>r_{\rm s}=5.45</math> in three dimensions (3D) and <math>2.01</math> in two dimensions (2D).<ref>{{cite book|last=Giuliani|first=Gabriele|author2=Vignale |author3=Giovanni |title=Quantum Theory of the Electron Liquid|url=https://archive.org/details/quantumtheoryofe0000giul|url-access=registration|publisher=Cambridge University Press|year=2005|isbn=978-0-521-82112-4}}</ref> However, according to Hartree–Fock theory, Wigner crystallization occurs at <math>r_{\rm s}=4.5</math> in 3D and <math>1.44</math> in 2D, so that jellium would crystallise before itinerant ferromagnetism occurs.<ref>{{cite journal|author1=J. R. Trail |author2=M. D. Towler |author3=R. J. Needs |title= Unrestricted Hartree-Fock theory of Wigner crystals|journal= Phys. Rev. B |volume= 68 |issue=4 |pages= 045107 |year=2003|doi= 10.1103/PhysRevB.68.045107|arxiv = 0909.5498 |bibcode = 2003PhRvB..68d5107T |s2cid=8932393 }}</ref> Furthermore, Hartree–Fock theory predicts exotic magnetic behavior, with the paramagnetic fluid being unstable to the formation of a spiral spin-density wave.<ref>{{cite journal|author= A. W. Overhauser|title= Giant Spin Density Waves|journal= Phys. Rev. Lett. |volume= 4 |issue= 9|pages= 462–465 |year=1960|doi= 10.1103/PhysRevLett.4.462|bibcode = 1960PhRvL...4..462O }}</ref><ref>{{cite journal|author= A. W. Overhauser|title=Spin Density Waves in an Electron Gas |journal= Phys. Rev. |volume= 128 |issue=3 |pages= 1437–1452 |year=1962|doi= 10.1103/PhysRev.128.1437|bibcode = 1962PhRv..128.1437O }}</ref> 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.<ref name=Ceperley>{{cite journal|author1=D. M. Ceperley |author2=B. J. Alder |title= Ground State of the Electron Gas by a Stochastic Method|journal= Phys. Rev. Lett. |volume= 45 |issue=7 |pages= 566–569 |year=1980|doi= 10.1103/PhysRevLett.45.566|bibcode = 1980PhRvL..45..566C |s2cid=55620379 |url=https://digital.library.unt.edu/ark:/67531/metadc1059358/ |type=Submitted manuscript }}</ref> They calculated the paramagnetic-ferromagnetic fluid transition to occur at <math>r_s=75(5)</math> and Wigner crystallization (to a body-centered cubic crystal) to occur at <math>r_{\rm s}=100(20)</math>.  Subsequent QMC calculations<ref>{{cite journal|author1=F. H. Zong |author2=C. Lin |author3=D. M. Ceperley |title= Spin polarization of the low-density three-dimensional electron gas|journal= Phys. Rev. E |volume= 66 |issue=3 |pages= 1–7 |year=2002|doi= 10.1103/PhysRevE.66.036703|pmid=12366294 |arxiv = cond-mat/0205339 |bibcode = 2002PhRvE..66c6703Z |s2cid=11606173 }}</ref><ref>{{cite journal|author1=N. D. Drummond |author2=Z. Radnai |author3=J. R. Trail |author4=M. D. Towler |author5=R. J. Needs |title= Diffusion quantum Monte Carlo study of three-dimensional Wigner crystals|journal= Phys. Rev. B |volume= 69 |issue=8 |pages= 085116 |year=2004|doi= 10.1103/PhysRevB.69.085116|arxiv = 0801.0377 |bibcode = 2004PhRvB..69h5116D |s2cid=18176116 }}</ref> have refined their phase diagram: there is a second-order transition from a paramagnetic fluid state to a partially spin-polarized fluid from <math>r_{\rm s}=50(2)</math> to about <math>100</math>; and Wigner crystallization occurs at <math>r_{\rm s}=106(1)</math>.

In 2D, QMC calculations indicate that the paramagnetic fluid to ferromagnetic fluid transition and Wigner crystallization occur at similar density parameters, in the range <math>30<r_{\rm s}<40</math>.<ref>{{cite journal|author1=B. Tanatar |author2=D. M. Ceperley |title= Ground state of the two-dimensional electron gas |journal= Phys. Rev. B |volume= 39 |issue=8 |pages= 5005–5016 |year=1989|doi= 10.1103/PhysRevB.39.5005|pmid=9948889 |bibcode = 1989PhRvB..39.5005T }}</ref><ref>{{cite journal|author1=F. Rapisarda |author2=G. Senatore |title= Diffusion Monte Carlo Study of Electrons in Two-dimensional Layers|journal= Aust. J. Phys. |volume= 49 |pages= 161 |year=1996|doi= 10.1071/PH960161 |bibcode = 1996AuJPh..49..161R |doi-access= free }}</ref> The most recent QMC calculations indicate that there is no region of stability for a ferromagnetic fluid.<ref>{{cite journal|author1=N. D. Drummond |author2=R. J. Needs |title= Phase Diagram of the Low-Density Two-Dimensional Homogeneous Electron Gas|journal= Phys. Rev. Lett. |volume= 102 |issue=12 |pages= 126402 |year=2009|doi= 10.1103/PhysRevLett.102.126402|arxiv = 1002.2101 |bibcode = 2009PhRvL.102l6402D |pmid=19392300|s2cid=35125378 }}</ref> Instead there is a transition from a paramagnetic fluid to a hexagonal Wigner crystal at <math>r_{\rm s}=31(1)</math>. There is possibly a small region of stability for a (frustrated) antiferromagnetic Wigner crystal, before a further transition to a ferromagnetic crystal.  The crystallization transition in 2D is not first order, so there must be a continuous series of transitions from fluid to crystal, perhaps involving striped crystal/fluid phases.<ref>{{cite journal|author1=B. Spivak |author2=S. A. Kivelson |title= Phases intermediate between a two-dimensional electron liquid and Wigner crystal|journal=Phys. Rev. B |volume= 70 |issue=15 |pages= 155114 |year=2004|doi=10.1103/PhysRevB.70.155114|bibcode = 2004PhRvB..70o5114S }}</ref> 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 <math>r_{\rm s}=35.1(9)</math>.<ref>{{cite journal|author1=J. Yoon |author2=C. C. Li |author3=D. Shahar |author4=D. C. Tsui |author5=M. Shayegan |title= Wigner Crystallization and Metal-Insulator Transition of Two-Dimensional Holes in GaAs at <math>B=0</math>|journal= Phys. Rev. Lett. |volume= 82 |issue=8 |pages= 1744 |year=1999|doi= 10.1103/PhysRevLett.82.1744|arxiv = cond-mat/9807235 |bibcode = 1999PhRvL..82.1744Y |s2cid=119371913 }}</ref>

==Applications==

Jellium is the simplest model of interacting electrons. It is employed in the calculation of properties of metals, where the [[core electron]]s and the nuclei are modeled as the uniform positive background and the [[valence electron]]s 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.<ref>{{cite journal|last=Lang|first=N. D.|year=1969|title=Self-consistent properties of the electron distribution at a metal surface|journal=Solid State Commun.|volume=7|issue=15|pages=1047–1050|doi=10.1016/0038-1098(69)90467-0|bibcode = 1969SSCom...7.1047L }}</ref><ref>{{cite journal|last=Lang|first=N. D.|author2=Kohn, W.|year=1970|title=Theory of Metal Surfaces: Work Function|journal=Phys. Rev. B|volume=3|issue=4|pages=1215–223|doi=10.1103/PhysRevB.3.1215|bibcode = 1971PhRvB...3.1215L }}</ref><ref>{{cite journal|last=Lang|first=N. D.|author2=Kohn, W.|year=1973|title=Surface-Dipole Barriers in Simple Metals|journal=Phys. Rev. B|volume=8|issue=12|pages=6010–6012|doi=10.1103/PhysRevB.8.6010|bibcode = 1973PhRvB...8.6010L }}</ref>

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,<ref name=Ceperley/> which have been used to construct semi-empirical correlation functionals.<ref>{{cite journal|last=Perdew|first=J. P.|author2=McMullen, E. R. |author3=Zunger, Alex |year=1981|title=Density-functional theory of the correlation energy in atoms and ions: A simple analytic model and a challenge|journal=Phys. Rev. A|volume=23|issue=6|pages=2785–2789|doi=10.1103/PhysRevA.23.2785|bibcode = 1981PhRvA..23.2785P }}</ref>

The jellium model has been applied to [[superatoms]], [[Metal cluster compound|metal clusters]], [[Alkaline earth octacarbonyl complex|octacarbonyl complexes]], and used in [[nuclear physics]].

==See also==
*[[Free electron model]] — a model electron gas where the electrons do not interact with anything.
*[[Nearly free electron model]] — a model electron gas where the electrons do not interact with each other, but do feel a (weak) potential from the atomic lattice.

==References==
{{reflist|2}}

[[Category:Condensed matter physics]]
[[Category:Density functional theory]]
[[Category:Nuclear physics]]