# Work function

Simple Band Diagram with denoted vacuum energy EVAC, conduction band edge EC, Fermi level EF, valence band edge EV, electron affinity Eea, work function Φ and band gap Eg

In solid-state physics, the work function (sometimes spelled workfunction) is the minimum thermodynamic work (i.e., energy) needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. Here "immediately" means that the final electron position is far from the surface on the atomic scale, but still too close to the solid to be influenced by ambient electrostatic potentials in the vacuum. The work function is not a characteristic of a bulk material, but rather a property of the surface of the material (depending on crystal face and contamination).

## Relationship between work function, vacuum electric potential, and Fermi level (voltage)

If we remove an electron from a solid, to a state of zero total energy, then the thermodynamic work required is given by -EF, where EF is the Fermi level (electrochemical potential) in that system. The work function is however defined by placing the electron into the vacuum nearby the surface, where there is an electrostatic potential ϕ; at this point, the electron's total energy is not zero but instead Evac=-eϕ. Thus, the work function W is defined by

$W \equiv -e\phi - E_{\rm F}.$

In other words, it is the sum of the work required to completely remove the electron (-EF), and the work required to place it back just outside the surface (-eϕ).

The work function is in fact a property of the surface material, and EF is fixed by the electrode that is attached to the material. Practically, this means that the material's work function acts to determine the value of ϕ (rather than the other way around). We can define the electrode's internal voltage V as V = -EF/e,[1] giving

$\phi = V - \frac{W}{e}$

In other words, if we use a battery to apply a voltage V to an electrode, then the actual ϕ produced in the vacuum will vary depending on the work function. W depends on what material the electrode is made from. The reason for this dependence of W on material can be attributed to a variety of effects (binding energy, surface dipoles, etc.), discussed below.

## Applications

An important implication of the work function is that there will be spatial variations in the vacuum electrostatic potential inside a vacuum chamber, if it is not lined with a uniform material. These equilibrium variations in ϕ (called Volta potentials or contact potential differences) can disrupt sensitive apparatus that rely on a perfectly uniform vacuum. For example, the Gravity Probe B experiment was significantly impacted by variations in W (and ϕ) over the surface of a free spinning gyroscope, as the resulting electric dipole torques resulted in precession and slowing by eddy currents.

In a thermionic electron gun, the work function (as well as temperature) is a critical parameter in determining the amount of current that is emitted from a hot filament. Although tungsten (the common choice for vacuum tube filaments) can survive to high temperatures, its work function is approximately 4.5 eV. Various oxide coatings can substantially reduce this and enhance the emission.

In electronics the work function is important for design of the metal–semiconductor junction in Schottky diodes and for design of vacuum tubes. The work function difference between metal and silicon in a MOS capacitor is related to the flat-band voltage (i.e. the voltage that induces zero net charge in the underlying semiconductor) and the equivalent oxide charge per unit area at the oxide-silicon interface by $V_{fb} = \phi_{ms} - {Q_{ox} \over C_{ox}}$.

## Measurement

Certain physical phenomena are highly sensitive to the value of the work function. The observed data from these effects can be fitted to simplified theoretical models, allowing one to extract a value of the work function. These phenomenologically extracted work functions may be slightly different from the thermodynamic definition given above.

Many techniques have been developed based on different physical effects to measure the electronic work function of a sample. One may distinguish between two groups of experimental methods for work function measurements: absolute and relative.

Absolute methods employ electron emission from the sample induced by photon absorption (photoemission), by high temperature (thermionic emission), due to an electric field (field electron emission), or using electron tunnelling.

All relative methods make use of the contact potential difference between the sample and a reference electrode. Experimentally, either an anode current of a diode is used or the displacement current between the sample and reference, created by an artificial change in the capacitance between the two, is measured (the Kelvin Probe method, Kelvin probe force microscope).

### Methods based on thermionic emission

The work function is important in the theory of thermionic emission, where thermal fluctuations provide enough energy to "evaporate" electrons out of a hot material (called the 'emitter') into the vacuum. If these electrons are absorbed by another, cooler material (called the collector) then a measurable electric current will be observed. Thermionic emission can be used to measure the work function of both the hot emitter and cold collector.

#### Work function of hot electron emitter

Energy level diagrams for thermionic diode in forward bias configuration, used to extract hot carriers coming from emitter. In order to escape, hot electrons must exceed the Fermi level EF,e by an energy We, the work function of the emitter.

In order to move from the hot emitter to the vacuum, the electrons must overcome an energy barrier

$E_{\rm barrier} = W_{\rm e}$

determined simply by the thermionic work function of the emitter. If an electric field is applied into the surface of the emitter, then the escaping electrons will all be accelerated away from the emitter and absorbed into whichever material is applying the electric field. According to Richardson's law the emitted current density (per unit area of emitter), Je (A/m2), is related to the absolute temperature Te of the emitter by the equation:

$J_{\rm e} = -A_{\rm e} T_{\rm e}^2 e^{-E_{\rm barrier} / k T_{\rm e}}$

where k is the Boltzmann constant and the proportionality constant Ae is the Richardson's constant of the emitter. In this case, the dependence of Je on Te can be fitted to yield We.

#### Work function of cold electron collector

Energy level diagrams for thermionic diode in retarding potential configuration. Electrons must gain an even higher energy to reach the collector. This configuration is used to determine the work function Wc of the cold collector.

The same construction can be used to instead measure the work function in the collector. If an electric field is applied out of the emitter instead, the electrons must overcome an additional barrier before reaching their destination (the collector). This collector barrier depends on the work function of the collector, as well as any additional applied voltages:[2]

$E_{\rm barrier} = W_{\rm c} - e (\Delta V_{\rm ce} - \Delta V_{\rm S})$

where Wc is the collector's thermionic work function, ΔVce is the applied collector–emitter voltage, and ΔVS is the Seebeck voltage in the hot emitter (the influence of ΔVS is often omitted). The resulting current density Jc through the collector (per unit of collector area) is again given by Richardson's Law, except now

$J_{\rm c} = A_{\rm c} T_{\rm e}^2 e^{-E_{\rm barrier}/kT_{\rm e}}$

where Ac is the Richardson constant of the collector. In this case, the dependence of Jc on Te can be fitted to yield Wc.

This retarding potential (or retarding diode) method is one of the simplest and oldest methods of measuring work functions, and is advantageous since the collector need not be heated.

### Methods based on photoemission

Photoelectric diode in forward bias configuration, used for measuring the work function We of the illuminated emitter.

The photoelectric work function is the minimum photon energy required to liberate an electron from a substance, in the photoelectric effect. If the photon's energy is greater than the substance's work function, photoelectric emission occurs and the electron is liberated from the surface. Similar to the thermionic case described above, the liberated electrons can be extracted into a collector and produce a detectable current, if an electric field is applied into the surface of the emitter. Excess photon energy results in a liberated electron with non-zero kinetic energy. It is expected that the minimum photon energy $\hbar \omega$ required to liberate an electron (and generate a current) is

$\hbar \omega = W_{\rm e}$

where We is the work function of the emitter. Note that this minimum energy can be misleading in materials where there are no actual electron states at the Fermi level that are available for excitation. For example, in a semiconductor the minimum photon energy would actually correspond to the valence band edge rather than work function.[3]

Of course, the photoelectric effect may be used in the retarding mode, as with the thermionic apparatus described above. In the retarding case, the dark collector's work function is measured instead.

### Kelvin probe method

Kelvin probe energy diagram at flat vacuum configuration, used for measuring work function difference between sample and probe.

The Kelvin probe technique relies on the detection of an electric field (gradient in ϕ) between a sample material and probe material. The electric field can be varied by the voltage ΔVsp that is applied to the sample relative to the probe. If the voltage is chosen such that the electric field is eliminated (the flat vacuum condition), then

$e\Delta V_{\rm sp,flatband} = W_{\rm s} - W_{\rm p}, \quad \text{when}~\phi~\text{is flat}.$

Since the experimenter controls and knows ΔVsp, this directly gives the work function difference between the two materials. Although the Kelvin probe technique only measures a work function difference, it is possible to obtain an absolute work function by first calibrating the probe against a reference material (with known work function) and then using the same probe to measure a desired sample.

Typically, the electric field is detected by varying the distance between the sample and probe. When the distance is changed but ΔVsp is held constant, a current will flow due to the change in capacitance. This current is proportional to the vacuum electric field, and so when the electric field is neutralized no current will flow.

The Kelvin probe technique can be used to obtain work function maps of a surface with extremely high spatial resolution, by using a sharp tip (see Kelvin probe force microscope).

## Electron work functions of elements[4]

Note: The work function can change for crystalline elements based upon the surface orientation. For example Ag:4.26, Ag(100):4.64, Ag(110):4.52, Ag(111):4.74. This is due to differences in the surface dipole of the different crystal faces. Ranges for typical surfaces are shown in the table below.

Element eV Element eV Element eV
Ag 4.52 – 4.74 Al 4.06 – 4.26 As 3.75
Au 5.1 – 5.47 B ~4.45 Ba 2.52 – 2.7
Be 4.98 Bi 4.31 C ~5
Ca 2.87 Cd 4.08 Ce 2.9
Co 5 Cr 4.5 Cs 2.14
Cu 4.53 – 5.10 Eu 2.5 Fe: 4.67 – 4.81
Ga 4.32 Gd 2.90 Hf 3.9
Hg 4.475 In 4.09 Ir 5.00 – 5.67
K 2.29 La 3.5 Li 2.3
Lu ~3.3 Mg 3.66 Mn 4.1
Mo 4.36 – 4.95 Na 2.36 Nb 3.95 – 4.87
Nd 3.2 Ni 5.04 – 5.35 Os 5.93
Pb 4.25 Pd 5.22 – 5.6 Pt 5.12 – 5.93
Rb 2.261 Re 4.72 Rh 4.98
Ru 4.71 Sb 4.55 – 4.7 Sc 3.5
Se 5.9 Si 4.60 – 4.85 Sm 2.7
Sn 4.42 Sr ~2.59 Ta 4.00 – 4.80
Tb 3.00 Te 4.95 Th 3.4
Ti 4.33 Tl ~3.84 U 3.63 – 3.90
V 4.3 W 4.32 – 5.22 Y 3.1
Yb 2.60 [5] Zn 3.63 – 4.9 Zr 4.05

## Physical factors that determine the work function for a given material

### Free electron gas model

In the free-electron model non-interacting electrons bounce around inside a potential well of depth U. The Fermi Level is the highest energy level that is occupied by electrons. Here $E_F$ is defined relative to the bottom of the potential well, and the work function W is the energy required to eject the electron in the Fermi Level.

In the free electron model the valence electrons roam freely (zero force) inside the metal but find a confining potential step $U$ at the boundary of the metal. In the system's ground state, states with energy less than the Fermi Level are occupied, and states above the Fermi Level are not occupied. The energy required to liberate an electron in the Fermi Level is the work function. If, as in the diagram right, we define the Fermi Energy $E_F$ from the bottom of the well, the results reported in the Wiki page Fermi energy are applicable. However, usually the Fermi energy is referenced to energy zero: that of the lowest energy electron free of the metal. In that case the Fermi energy would have a negative value (i.e., the Fermi Level lies below those of escaped electrons) $E_F\approx -W$ (but see below).

### Work function and surface effect

The work function W of a metal is closely related to its Fermi energy $E_F \;$ (defined relative to the lowest energy free particle: zero in the above diagram) yet the two quantities are not exactly the same. This is due to the surface effect that exists even with a perfectly clean surface. The surface effect is a small buildup of charge at the surface (even in the absence of external fields), a natural result of the discontinuity between the crystal lattice and vacuum. Of course, the metal screens these charges but it does so imperfectly, leaving a layer of electric dipoles. The magnitude and sign of this surface dipole depends on the crystal face orientation, and this results in variability in the electrostatic potentials that exist just outside the surfaces. Since the work function is defined by moving a charge from the bulk to just outside surface, it is directly influenced by these surface potential variations.

In detail, a real-world solid is not infinitely extended with electrons and ions repeatedly filling every primitive cell over all Bravais lattice sites. Neither can one simply take a set of Bravais lattice sites $\{R\} \;$ inside the geometrical region V which the solid occupies and then fill undistorted charge distribution basis into all primitive cells of $\{R\} \;$. Indeed, the charge distribution in those cells near the surface will be distorted significantly from that in a cell of an ideal infinite solid, resulting in an effective surface dipole distribution, or, sometimes both a surface dipole distribution and a surface charge distribution.

It can be proven that if we define work function as the minimum energy needed to remove an electron to a point immediately out of the solid, the effect of the surface charge distribution can be neglected, leaving only the surface dipole distribution. Let the potential energy difference across the surface due to effective surface dipole be $W_S \;$. And let $E_F \;$ be the Fermi energy calculated for the finite solid without considering surface distortion effect, when taking the convention that the potential at $r \rightarrow \infty \;$ is zero. Then, the correct formula for work function is:

$W = - E_F +W_S,$

where $E_F \;$ is negative, which means that electrons are bound in the solid.

### Work function trends

The thermionic work function depends on the orientation of the crystal and will tend to be smaller for metals with an open lattice, larger for metals in which the atoms are closely packed. The range is about 1.5–6 eV. It is somewhat higher on dense crystal faces than open ones.

## References

1. ^ This is precisely the voltage whose differences are measured by a voltmeter; see Fermi level.
2. ^ "Thermionic Energy Conversion" [1]
3. ^ http://www.virginia.edu/ep/SurfaceScience/PEE.html
4. ^ CRC Handbook of Chemistry and Physics version 2008, p. 12–114.
5. ^ Nikolic, M. V.; Radic, S. M.; Minic, V.; Ristic, M. M. (1996-02). "The dependence of the work function of rare earth metals on their electron structure". Microelectronics Journal 27 (1): 93–96. doi:10.1016/0026-2692(95)00097-6. ISSN 0026-2692. Retrieved 2009-09-22.