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. Generally, these measurements involve fitting to Richardson's law, and so they must be carried out in a low temperature and low current regime where space charge effects are absent.

Energy level diagrams for thermionic diode in forward bias configuration, used to extract all hot electrons coming out from the emitter's surface. The barrier is the vacuum near emitter surface.

In order to move from the hot emitter to the vacuum, an electron's energy must exceed the emitter Fermi level by an amount

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

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 (per unit area of emitter), J_e (A/m²), is related to the absolute temperature T_e of the emitter by the equation:

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

where k is the Boltzmann constant and the proportionality constant A_e is the Richardson's constant 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

Energy level diagrams for thermionic diode in retarding potential configuration. The barrier is the vacuum near collector surface.

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:^[11]

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

where W_c is the collector's thermionic work function, ΔV_ce is the applied collector–emitter voltage, and ΔV_S is the Seebeck voltage in the hot emitter (the influence of ΔV_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, except now

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

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 ΔV_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

Photoelectric diode in forward bias configuration, used for measuring the work function W_e 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 ${\displaystyle \hbar \omega }$ required to liberate an electron (and generate a current) is

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

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.^[9] 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.^[12]

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 ΔV_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

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

Since the experimenter controls and knows ΔV_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 ΔV_sp 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.

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.^[10] 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).

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.^[7] 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)

Band diagram of semiconductor-vacuum interface showing electron affinity E_EA, defined as the difference between near-surface vacuum energy E_vac, and near-surface conduction band edge E_C. Also shown: Fermi level E_F, valence band edge E_V, work function W.

In a semiconductor, 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 (note that this has a different meaning than the electron affinity of chemistry); in silicon for example the electron affinity is 4.05 eV.^[16] 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

${\displaystyle W=E_{\rm {EA}}+E_{\rm {C}}-E_{\rm {F}}}$

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.^[17] 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.^[18]

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; 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,^[19] which allowed for oscillations in electronic density nearby the abrupt surface (these are similar to Friedel oscillations) 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_s) 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).^[7][20]

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. ^[21] 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 ${\textstyle \varphi (T)=\varphi _{0}-\gamma {\frac {(k_{\text{B}}T)^{2}}{\varphi _{0}}}}$ and ${\displaystyle \gamma }$ is a calculable material property which is dependent on the crystal structure (for example, BCC, FCC). ${\displaystyle \varphi _{0}}$ is the electron work function at T=0 and ${\displaystyle \beta }$ is constant throughout the change.