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solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as solid-state chemistry, quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state phy ...
, the work function (sometimes spelled workfunction) is the minimum thermodynamic work (i.e., energy) needed to remove an
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
from a solid to a point in the
vacuum A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
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 (depending on crystal face and contamination).


Definition

The work function for a given surface is defined by the difference :W = -e\phi - E_, where is the charge of an
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
, is the
electrostatic potential Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work needed ...
in the vacuum nearby the surface, and is the
Fermi level The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by ''μ'' or ''E''F for brevity. The Fermi level does not include the work required to re ...
( electrochemical potential of electrons) 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 Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), ...
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 :\phi = V - \frac where is the voltage of the material (as measured by a
voltmeter A voltmeter is an instrument used for measuring electric potential difference between two points in an electric circuit. It is connected in parallel. It usually has a high resistance so that it takes negligible current from the circuit. A ...
, through an attached electrode), 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 An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
, when those conductors are in total equilibrium with each other (electrically shorted to each other, and with equal temperatures). The work function refers to removal of an electron to a position that is far enough from the surface (many nm) 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 Thermionic emission is the liberation of charged particles from a hot electrode whose thermal energy gives some particles enough kinetic energy to escape the material's surface. The particles, sometimes called ''thermions'' in early literature, a ...
: In thermionic
electron gun file:Egun.jpg, Electron gun from a cathode-ray tube file:Vidicon Electron Gun.jpg, The electron gun from an RCA Vidicon video camera tube An electron gun (also called electron emitter) is an electrical component in some vacuum tubes that produc ...
s, the work function and temperature of the
hot cathode In vacuum tubes and gas-filled tubes, a hot cathode or thermionic cathode is a cathode electrode which is heated to make it emit electrons due to thermionic emission. This is in contrast to a cold cathode, which does not have a heating element ...
are critical parameters in determining the amount of current that can be emitted.
Tungsten Tungsten (also called wolfram) is a chemical element; it has symbol W and atomic number 74. It is a metal found naturally on Earth almost exclusively in compounds with other elements. It was identified as a distinct element in 1781 and first ...
, 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 (approximately 4.5 eV). By coating the tungsten with a substance of lower work function (e.g.,
thorium Thorium is a chemical element; it has symbol Th and atomic number 90. Thorium is a weakly radioactive light silver metal which tarnishes olive grey when it is exposed to air, forming thorium dioxide; it is moderately soft, malleable, and ha ...
or
barium oxide Barium oxide, also known as baria, is a white hygroscopic non-flammable chemical compound, compound with the formula BaO. It has a Cubic crystal system, cubic structure and is used in cathode-ray tubes, crown glass, and Catalysis, catalysts. It ...
), the emission can be greatly increased. This prolongs the lifetime of the filament by allowing operation at lower temperatures (for more information, see
hot cathode In vacuum tubes and gas-filled tubes, a hot cathode or thermionic cathode is a cathode electrode which is heated to make it emit electrons due to thermionic emission. This is in contrast to a cold cathode, which does not have a heating element ...
). ;
Band bending In solid-state physics, band bending refers to the process in which the electronic band structure in a material curves up or down near a junction or interface. It does not involve any physical (spatial) bending. When the electrochemical potential ...
models in solid-state electronics: The behavior of a solid-state device is strongly dependent on the size of various
Schottky barrier A Schottky barrier, named after Walter H. Schottky, is a potential energy barrier for electrons formed at a metal–semiconductor junction. Schottky barriers have rectifier, rectifying characteristics, suitable for use as a diode. One of the p ...
s 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 The triboelectric effect (also known as triboelectricity, triboelectric charging, triboelectrification, or tribocharging) describes electric charge transfer between two objects when they contact or slide against each other. It can occur with d ...
: 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 A surface charge is an electric charge present on a two-dimensional surface. These electric charges are constrained on this 2-D surface, and surface charge density, measured in coulombs per square meter (C•m−2), is used to describe the 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 (once brought into contact, the charge will instead flow internally through the junction between the conductors). 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 (photoemission), by high temperature (thermionic emission), due to an electric field (
field electron emission Field electron emission, also known as field-induced electron emission, field emission (FE) and electron field emission, is the emission of electrons from a material placed in an electrostatic field. The most common context is field emission from ...
), 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 (the Kelvin Probe method,
Kelvin probe force microscope Kelvin probe force microscopy (KPFM), also known as surface potential microscopy, is a noncontact variant of atomic force microscopy (AFM). By raster scanning in the x,y plane the work function of the sample can be locally mapped for correlati ...
). However, absolute work function values can be obtained if the tip is first calibrated against a reference sample.


Methods based on thermionic emission

The work function is important in the theory of
thermionic emission Thermionic emission is the liberation of charged particles from a hot electrode whose thermal energy gives some particles enough kinetic energy to escape the material's surface. The particles, sometimes called ''thermions'' in early literature, a ...
, 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 An electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. The moving particles are called charge c ...
will be observed. Thermionic emission can be used to measure the work function of both the hot emitter and cold collector. Generally, these measurements involve fitting to
Richardson's law Thermionic emission is the liberation of charged particles from a hot electrode whose thermal energy gives some particles enough kinetic energy to escape the material's surface. The particles, sometimes called ''thermions'' in early literature, a ...
, and so they must be carried out in a low temperature and low current regime where
space charge Space charge is an interpretation of a collection of electric charges in which excess electric charge is treated as a continuum of charge distributed over a region of space (either a volume or an area) rather than distinct point-like charges. Thi ...
effects are absent. In order to move from the hot emitter to the vacuum, an electron's energy must exceed the emitter Fermi level by an amount :E_ = W_ 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 Thermionic emission is the liberation of charged particles from a hot electrode whose thermal energy gives some particles enough kinetic energy to escape the material's surface. The particles, sometimes called ''thermions'' in early literature, a ...
the emitted
current density In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional ...
(per unit area of emitter), ''J''e (A/m2), is related to the absolute
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
''T''e of the emitter by the equation: :J_ = -A_ T_^2 e^ where ''k'' is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
and the proportionality constant ''A''e is the
Richardson's constant Thermionic emission is the liberation of charged particles from a hot electrode whose thermal energy gives some particles enough kinetic energy to escape the material's surface. The particles, sometimes called ''thermions'' in early literature, ...
of the emitter. In this case, the dependence of ''J''e on ''T''e can be fitted to yield ''W''e.


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 ''W''e. The barrier height now depends on the work function of the collector, as well as any additional applied voltages: :E_ = W_ - e (\Delta V_ - \Delta V_) where ''W''c is the collector's thermionic work function, Δ''V''ce is the applied collector–emitter voltage, and Δ''V''S is the Seebeck voltage in the hot emitter (the influence of Δ''V''S is often omitted, as it is a small contribution of order 10 mV). The resulting current density ''J''c through the collector (per unit of collector area) is again given by
Richardson's Law Thermionic emission is the liberation of charged particles from a hot electrode whose thermal energy gives some particles enough kinetic energy to escape the material's surface. The particles, sometimes called ''thermions'' in early literature, a ...
, except now :J_ = A T_^2 e^ 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 ''J''c on ''T''e, or on Δ''V''ce, can be fitted to yield ''W''c. This retarding potential method is one of the simplest and oldest methods of measuring work functions, and is advantageous since the measured material (collector) is not required to survive high temperatures.


Methods based on photoemission

The photoelectric work function is the minimum
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
energy required to liberate an electron from a substance, in the
photoelectric effect The photoelectric effect is the emission of electrons from a material caused by electromagnetic radiation such as ultraviolet light. Electrons emitted in this manner are called photoelectrons. The phenomenon is studied in condensed matter physi ...
. If the photon's energy is greater than the substance's work function,
photoelectric emission The photoelectric effect is the emission of electrons from a material caused by electromagnetic radiation such as ultraviolet light. Electrons emitted in this manner are called photoelectrons. The phenomenon is studied in condensed matter physic ...
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 Photon energy is the energy carried by a single photon. The amount of energy is directly proportional to the photon's electromagnetic frequency and thus, equivalently, is inversely proportional to the wavelength. The higher the photon's frequenc ...
\hbar \omega required to liberate an electron (and generate a current) is :\hbar \omega = W_ where ''W''e 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 In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in ...
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.


Kelvin probe method

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 Δ''V''sp that is applied to the probe relative to the sample. If the voltage is chosen such that the electric field is eliminated (the flat vacuum condition), then :e\Delta V_ = W_ - W_, \quad \text~\phi~\text. Since the experimenter controls and knows Δ''V''sp, then finding the flat vacuum condition gives directly the work function difference between the two materials. The only question is, how to detect the flat vacuum condition? Typically, the electric field is detected by varying the distance between the sample and probe. When the distance is changed but Δ''V''sp is held constant, a current will flow due to the change in
capacitance Capacitance is the ability of an object to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are two closely related ...
. This current is proportional to the vacuum electric field, and so when the electric field is neutralized no current will flow. 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. 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 for the probe (see
Kelvin probe force microscope Kelvin probe force microscopy (KPFM), also known as surface potential microscopy, is a noncontact variant of atomic force microscopy (AFM). By raster scanning in the x,y plane the work function of the sample can be locally mapped for correlati ...
).


Work functions of elements

The work function depends on the configurations of atoms at the surface of the material. For example, on polycrystalline silver the work function is 4.26 eV, but on silver crystals it varies for different crystal faces as (100) face: 4.64 eV, (110) face: 4.52 eV, (111) face: 4.74 eV. Ranges for typical surfaces are shown in the table below.


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, 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 reconstructions 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 electric double layer at the surface. 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)

In a
semiconductor A semiconductor is a material with electrical conductivity between that of a conductor and an insulator. Its conductivity can be modified by adding impurities (" doping") to its crystal structure. When two regions with different doping level ...
, the work function is sensitive to the doping level at the surface of the semiconductor. Since the doping near the surface can also be 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 The electron affinity (''E''ea) of an atom or molecule is defined as the amount of energy released when an electron attaches to a neutral atom or molecule in the gaseous state to form an anion. ::X(g) + e− → X−(g) + energy This differs by si ...
(note that this has a different meaning than the electron affinity of chemistry); in silicon for example the electron affinity is 4.05 eV. If the electron affinity ''E''EA and the surface's band-referenced Fermi level ''E''F-''E''C are known, then the work function is given by : W = E_ + E_ - E_ where ''E''C 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 states. 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.


Theoretical models of metal work functions

Theoretical modeling of the work function is difficult, as an accurate model requires a careful treatment of both electronic many body effects and
surface chemistry Surface science is the study of physics, physical and chemistry, chemical phenomena that occur at the interface (chemistry), interface of two phase (matter), phases, including solid–liquid interfaces, solid–gas interfaces, solid–vacuum int ...
; 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, which allowed for oscillations in electronic density nearby the abrupt surface (these are similar to
Friedel oscillation In solid-state physics, Friedel oscillations, named after French physicist Jacques Friedel, arise from localized perturbations in a metallic or semiconductor system caused by a defect in the Fermi gas or Fermi liquid. Friedel oscillations are a ...
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 ''rs'') 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).


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. 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 \varphi(T)=\varphi_0-\gamma\frac and \gamma is a calculable material property which is dependent on the crystal structure (for example, BCC, FCC). \varphi_0 is the electron work function at T=0 and k_\text is constant throughout the change.


References


Further reading

* * For a quick reference to values of work function of the elements: *


External links


Work function of polymeric insulators (Table 2.1)

Work function of diamond and doped carbon




* ttp://academic.brooklyn.cuny.edu/physics/tung/Schottky/surface.htm Physics of free surfaces of semiconductors {{Authority control Condensed matter physics Physical quantities Vacuum Vacuum tubes