Editing Work function (section)

== Physical factors that determine the work function ==
Due to the complications described in the modelling section below, it is difficult to theoretically predict the work function with accuracy. However, various trends have been identified. The work function tends to be smaller for metals with an open lattice,{{what|date=October 2020|reason=open not defined}} and larger for metals in which the atoms are closely packed. It is somewhat higher on dense crystal faces than open crystal faces, also depending on [[surface reconstruction]]s for the given crystal face.

=== Surface dipole ===
The work function is not simply dependent on the "internal vacuum level" inside the material (i.e., its average electrostatic potential), because of the formation of an atomic-scale [[Double layer (interfacial)|electric double layer]] at the surface.<ref name="venables"/> This surface electric dipole gives a jump in the electrostatic potential between the material and the vacuum.

A variety of factors are responsible for the surface electric dipole. Even with a completely clean surface, the electrons can spread slightly into the vacuum, leaving behind a slightly positively charged layer of material. This primarily occurs in metals, where the bound electrons do not encounter a hard wall potential at the surface but rather a gradual ramping potential due to [[image charge]] attraction. The amount of surface dipole depends on the detailed layout of the atoms at the surface of the material, leading to the variation in work function for different crystal faces.

=== Doping and electric field effect (semiconductors) ===
[[File:Semiconductor vacuum junction.svg|thumb|[[Band diagram]] of semiconductor-vacuum interface showing [[electron affinity]] ''E''<sub>EA</sub>, defined as the difference between near-surface vacuum energy ''E''<sub>vac</sub>, and near-surface [[conduction band]] edge ''E''<sub>C</sub>. Also shown: [[Fermi level]] ''E''<sub>F</sub>, [[valence band]] edge ''E''<sub>V</sub>, work function ''W''.]]

In a [[semiconductor]], the work function is sensitive to the [[doping (semiconductor)|doping level]] at the surface of the semiconductor. Since the doping near the surface can also be [[field effect (semiconductor)|controlled by electric fields]], the work function of a semiconductor is also sensitive to the electric field in the vacuum.

The reason for the dependence is that, typically, the vacuum level and the conduction band edge retain a fixed spacing independent of doping. This spacing is called the [[electron affinity]] (note that this has a different meaning than the electron affinity of chemistry); in silicon for example the electron affinity is 4.05 eV.<ref>{{cite web|url=http://www.virginiasemi.com/pdf/generalpropertiessi62002.pdf|title=The General Properties of Si, Ge, SiGe, SiO2 and Si3N4 |author=Virginia Semiconductor|date=June 2002|access-date=6 Jan 2019}}</ref> If the electron affinity ''E''<sub>EA</sub> and the surface's band-referenced Fermi level ''E''<sub>F</sub>-''E''<sub>C</sub> are known, then the work function is given by
:<math> W = E_{\rm EA} + E_{\rm C} - E_{\rm F}</math>
where ''E''<sub>C</sub> is taken at the surface.

From this one might expect that by doping the bulk of the semiconductor, the work function can be tuned. In reality, however, the energies of the bands near the surface are often pinned to the Fermi level, due to the influence of [[surface state]]s.<ref>{{cite web|url=http://academic.brooklyn.cuny.edu/physics/tung/Schottky/surface.htm|title=Semiconductor Free Surfaces|website=academic.brooklyn.cuny.edu|access-date=11 April 2018}}</ref> If there is a large density of surface states, then the work function of the semiconductor will show a very weak dependence on doping or electric field.<ref>{{Cite journal | last1 = Bardeen | first1 = J. | title = Surface States and Rectification at a Metal Semi-Conductor Contact | doi = 10.1103/PhysRev.71.717 | journal = Physical Review | volume = 71 | issue = 10 | pages = 717–727 | year = 1947 |bibcode = 1947PhRv...71..717B }}</ref>

=== Theoretical models of metal work functions ===
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Theoretical modeling of the work function is difficult, as an accurate model requires a careful treatment of both electronic [[many-body problem|many body effects]] and [[surface chemistry]]; both of these topics are already complex in their own right.

One of the earliest successful models for metal work function trends was the [[jellium]] model,<ref>{{Cite journal | last1 = Lang | first1 = N. | last2 = Kohn | first2 = W. | doi = 10.1103/PhysRevB.3.1215 | title = Theory of Metal Surfaces: Work Function | journal = Physical Review B | volume = 3 | issue = 4 | pages = 1215 | year = 1971 |bibcode = 1971PhRvB...3.1215L }}</ref> which allowed for oscillations in electronic density nearby the abrupt surface (these are similar to [[Friedel oscillation]]s) as well as the tail of electron density extending outside the surface. This model showed why the density of conduction electrons (as represented by the [[Wigner–Seitz radius]] ''r<sub>s</sub>'') is an important parameter in determining work function.

The jellium model is only a partial explanation, as its predictions still show significant deviation from real work functions. More recent models have focused on including more accurate forms of [[electron exchange]] and correlation effects, as well as including the crystal face dependence (this requires the inclusion of the actual atomic lattice, something that is neglected in the jellium model).<ref name="venables"/><ref>{{cite book | isbn = 9780080536347 | title = Metal Surface Electron Physics | last1 = Kiejna | first1 = A. | last2 = Wojciechowski | first2 = K.F. | date = 1996 | publisher = [[Elsevier]]  }}</ref>

=== Temperature dependence of the electron work function ===
The electron behavior in metals varies with temperature and is largely reflected by the electron work function. A theoretical model for predicting the temperature dependence of the electron work function, developed by Rahemi et al. <ref>{{cite journal|last1=Rahemi|first1=Reza|last2=Li|first2=Dongyang | title=Variation in electron work function with temperature and its effect on Young's modulus of metals| journal=Scripta Materialia| date=April 2015|volume=99|issue=2015|pages=41–44 | doi=10.1016/j.scriptamat.2014.11.022 | arxiv=1503.08250|s2cid=118420968 }}</ref> explains the underlying mechanism and predicts this temperature dependence for various crystal structures via calculable and measurable parameters.  In general, as the temperature increases, the EWF decreases via <math display="inline">\varphi(T)=\varphi_0-\gamma\frac{(k_\text{B}T)^2}{\varphi_0}</math> and <math>\gamma</math> is a calculable material property which is dependent on the crystal structure (for example, BCC, FCC). <math>\varphi_0</math> is the electron work function at T=0 and <math>k_\text{B}</math> is constant throughout the change.