Work function
In solid-state physics, the work function is the minimum thermodynamic work 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 electric fields in the vacuum.
The work function is not a characteristic of a bulk material, but rather a property of the surface of the material.
Definition
The work function for a given surface is defined by the differencewhere is the charge of an electron, is the electrostatic potential in the vacuum nearby the surface, and is the Fermi level inside the material. The term is the energy of an electron at rest in the vacuum nearby the surface.
In practice, one directly controls by the voltage applied to the material through electrodes, and the work function is generally a fixed characteristic of the surface material. Consequently, this means that when a voltage is applied to a material, the electrostatic potential produced in the vacuum will be somewhat lower than the applied voltage, the difference depending on the work function of the material surface. Rearranging the above equation, one has
where is the voltage of the material, relative to an electrical ground that is defined as having zero Fermi level. The fact that depends on the material surface means that the space between two dissimilar conductors will have a built-in electric field, when those conductors are in total equilibrium with each other.
The work function refers to removal of an electron to a position that is far enough from the surface that the force between the electron and its image charge in the surface can be neglected. The electron must also be close to the surface compared to the nearest edge of a crystal facet, or to any other change in the surface structure, such as a change in the material composition, surface coating or reconstruction. The built-in electric field that results from these structures, and any other ambient electric field present in the vacuum are excluded in defining the work function.
Applications
;Thermionic emission: In thermionic electron guns, the work function and temperature of the hot cathode are critical parameters in determining the amount of current that can be emitted. Tungsten, the common choice for vacuum tube filaments, can survive to high temperatures but its emission is somewhat limited due to its relatively high work function. By coating the tungsten with a substance of lower work function, the emission can be greatly increased. This prolongs the lifetime of the filament by allowing operation at lower temperatures.;Band bending models in solid-state electronics: The behavior of a solid-state device is strongly dependent on the size of various Schottky barriers and band offsets in the junctions of differing materials, such as metals, semiconductors, and insulators. Some commonly used heuristic approaches to predict the band alignment between materials, such as Anderson's rule and the Schottky–Mott rule, are based on the thought experiment of two materials coming together in vacuum, such that the surfaces charge up and adjust their work functions to become equal just before contact. In reality these work function heuristics are inaccurate due to their neglect of numerous microscopic effects. However, they provide a convenient estimate until the true value can be determined by experiment.
;Equilibrium electric fields in vacuum chambers: Variation in work function between different surfaces causes a non-uniform electrostatic potential in the vacuum. Even on an ostensibly uniform surface, variations in known as patch potentials are always present due to microscopic inhomogeneities. Patch potentials have disrupted sensitive apparatus that rely on a perfectly uniform vacuum, such as Casimir force experiments and the Gravity Probe B experiment. Critical apparatus may have surfaces covered with molybdenum, which shows low variations in work function between different crystal faces.
;Contact electrification: If two conducting surfaces are moved relative to each other, and there is potential difference in the space between them, then an electric current will be driven. This is because the surface charge on a conductor depends on the magnitude of the electric field, which in turn depends on the distance between the surfaces. The externally observed electrical effects are largest when the conductors are separated by the smallest distance without touching. Since two conductors in equilibrium can have a built-in potential difference due to work function differences, this means that bringing dissimilar conductors into contact, or pulling them apart, will drive electric currents. These contact currents can damage sensitive microelectronic circuitry and occur even when the conductors would be grounded in the absence of motion.
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. For inhomogeneous surfaces, the work function varies from place to place, and different methods will yield different values of the typical "work function" as they average or select differently among the microscopic work functions.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, by high temperature, due to an electric field, or using electron tunnelling.
- 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 ). However, absolute work function values can be obtained if the tip is first calibrated against a reference sample.
Methods based on thermionic emission
In order to move from the hot emitter to the vacuum, an electron's energy must exceed the emitter Fermi level by an amount
determined simply by the thermionic work function of the emitter.
If an electric field is applied towards the surface of the emitter, then all of the escaping electrons will 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, Je, is related to the absolute temperature Te of the emitter by the equation:
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
The same setup can be used to instead measure the work function in the collector, simply by adjusting the applied voltage.If an electric field is applied away from the emitter instead, then most of the electrons coming from the emitter will simply be reflected back to the emitter. Only the highest energy electrons will have enough energy to reach the collector, and the height of the potential barrier in this case depends on the collector's work function, rather than the emitter's.
The current is still governed by Richardson's law. However, in this case the barrier height does not depend on We. The barrier height now depends on the work function of the collector, as well as any additional applied voltages:
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 resulting current density Jc through the collector is again given by Richardson's Law, except now
where A is a Richardson-type constant that depends on the collector material but may also depend on the emitter material, and the diode geometry.
In this case, the dependence of Jc on Te, or on ΔVce, can be fitted to yield Wc.
This retarding potential method is one of the simplest and oldest methods of measuring work functions, and is advantageous since the measured material is not required to survive high temperatures.
Methods based on photoemission
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 required to liberate an electron is
where We is the work function of the emitter.
Photoelectric measurements require a great deal of care, as an incorrectly designed experimental geometry can result in an erroneous measurement of work function. This may be responsible for the large variation in work function values in scientific literature.
Moreover, the 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.
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.