When a surface is exposed to electromagnetic radiation above a certain threshold frequency (typically visible light for alkali metals, near ultraviolet for other metals, and extreme ultraviolet for non-metals), the radiation is absorbed and electrons are emitted. This phenomenon was first observed by Heinrich Hertz in 1887. Johann Elster (1854-1920) and Hans Geistel (1855-1923), students in Heidelberg developed the first practical photoelectric cells that could be used to measure the intensity of light. In 1902, Philipp Eduard Anton von Lenard observed that the energy of individual emitted electrons increased with the frequency (which is related to the colour) of the light. This appeared to be at odds with James Clerk Maxwell's wave theory of light, which was thought to predict that the electron energy would be proportional to the intensity of the radiation. In 1905, Albert Einstein solved this apparent paradox by describing light as composed of discrete quanta, now called photons, rather than continuous waves. Based upon Max Planck's theory of black-body radiation, Einstein theorized that the energy in each quantum of light was equal to the frequency multiplied by a constant, later called Planck's constant. A photon above a threshold frequency has the required energy to eject a single electron, creating the observed effect. This discovery led to the quantum revolution in physics and earned Einstein the Nobel Prize in Physics in 1921.

It has been shown that it is not necessary for light to be "quantized" to explain the photoelectric effect. The most common method employed by physicists to calculate the probability of an atom ejecting an electron relies on "Fermi's golden rule". Although based upon quantum mechanics, the method treats the incident light as an electromagnetic wave that causes an atom and its constituent electrons to transition from one energy state ("eigenstate") to another.

While one can use the classical electromagnetic theory of light to describe the effect, one may also use the modern quantum theory of light to describe the photoelectric effect. However, the modern quantum theory of light is not a "particle model", as it does not always predict results which one would expect from a naïve "particle" interpretation. An example would be in the dependence on polarization with regard to the direction electrons are emitted, a phenomenon that has been considered useful in gathering polarization data from black holes and neutron stars.

Traditional explanation

The photons of a light beam have a characteristic energy determined by the frequency of the light. In the photoemission process, if an electron within some material absorbs the energy of one photon and thus has more energy than the work function (the electron binding energy) of the material, it is ejected. If the photon energy is too low, the electron is unable to escape the material. Increasing the intensity of the light beam increases the number of photons in the light beam, and thus increases the number of electrons emitted, but does not increase the energy that each electron possesses. Thus the energy of the emitted electrons does not depend on the intensity of the incoming light, but only on the energy of the individual photons. (This is true as long as the intensity is low enough for non-linear effects caused by multiphoton absorption or level shifts such as the AC Stark effect to be insignificant. This was a given in the age of Einstein, well before lasers had been invented.)

Electrons can absorb energy from photons when irradiated, but they usually follow an "all or nothing" principle. All of the energy from one photon must be absorbed and used to liberate one electron from atomic binding, or the energy is re-emitted. If the photon energy is absorbed, some of the energy liberates the electron from the atom, and the rest contributes to the electron's kinetic energy as a free particle.

Experimental results of the photoelectric emission

1. For a given metal and frequency of incident radiation, the rate at which photoelectrons are ejected is directly proportional to the intensity of the incident light.
2. For a given metal, there exists a certain minimum frequency of incident radiation below which no photoelectrons can be emitted. This frequency is called the threshold frequency.
3. For a given metal of particular work function, increase in frequency of incident beam increases the intensity of the photoelectric current.
4. Above the threshold frequency, the maximum kinetic energy of the emitted photoelectron depends on the frequency of the incident light, but is independent of the intensity of the incident light so long as the latter is not too high
5. The time lag between the incidence of radiation and the emission of a photoelectron is very small, less than 10⁻⁹ second.
6. The direction distribution of emitted electrons peaks in the direction of polarization (the direction of the electric field) of the incident light, if it is linearly polarized.

Mathematical description

The maximum kinetic energy K_max of an ejected electron is given by

where h is the Planck constant, f is the frequency of the incident photon, and φ = hf₀ is the work function (sometimes denoted W), which is the minimum energy required to remove a delocalised electron from the surface of any given metal. The work function, in turn, can be written as

where f₀ is called the threshold frequency for the metal. The maximum kinetic energy of an ejected electron is thus

Because the kinetic energy of the electron must be positive, it follows that the frequency f of the incident photon must be greater than f₀ in order for the photoelectric effect to occur.

[edit] Three-step model

In the X-ray regime, the photoelectric effect in crystalline material is often decomposed into three steps:

1. Inner photoelectric effect (see photodiode below). The hole left behind can give rise to auger effect, which is visible even when the electron does not leave the material. In molecular solids phonons are excited in this step and may be visible as lines in the final electron energy. The inner photoeffect has to be dipole allowed. The transition rules for atoms translate via the tight-binding model onto the crystal. They are similar in geometry to plasma oscillations in that they have to be transversal.
2. Ballistic transport of half of the electrons to the surface. Some electrons are scattered.
3. Electrons escape from the material at the surface.

In the three-step model, an electron can take multiple paths through these three steps. All paths can interfere in the sense of the path integral formulation. For surface states and molecules the three-step model does still make some sense as even most atoms have multiple electrons which can scatter the one electron leaving.

Early observations

In 1839, Alexandre Edmond Becquerel discovered the photovoltaic effect while studying the effect of light on electrolytic cells. Though not equivalent to the photoelectric effect, his work on photovoltaics was instrumental in showing a strong relationship between light and electronic properties of materials. In 1873, Willoughby Smith discovered photoconductivity in selenium while testing the metal for its high resistance properties in conjunction with his work involving submarine telegraph cables.

Hertz's spark gaps

In 1887, Heinrich Hertz observed the photoelectric effect and the production and reception of electromagnetic waves. He published these observations in the journal Annalen der Physik. His receiver consisted of a coil with a spark gap, where a spark would be seen upon detection of electromagnetic waves. He placed the apparatus in a darkened box to see the spark better. However, he noticed that the maximum spark length was reduced when in the box. A glass panel placed between the source of electromagnetic waves and the receiver absorbed ultraviolet radiation that assisted the electrons in jumping across the gap. When removed, the spark length would increase. He observed no decrease in spark length when he substituted quartz for glass, as quartz does not absorb UV radiation. Hertz concluded his months of investigation and reported the results obtained. He did not further pursue investigation of this effect, nor did he make any attempt at explaining how this phenomenon was brought about

Stoletov: the first law of photoeffect

In the period from February 1888 and until 1891, a detailed analysis of photoeffect was performed by Aleksandr Stoletov with results published in 6 works; four of them in Comptes Rendus, one review in Physikalische Revue (translated from Russian), and the last work in Journal de Physique. First, in these works Stoletov invented a new experimental setup which was more suitable for a quantitative analysis of photoeffect. Using this setup, he discovered the direct proportionality between the intensity of light and the induced photo electric current (the first law of photoeffect or Stoletov's law). One of his other findings resulted from measurements of the dependence of the intensity of the electric photo current on the gas pressure, where he found the existence of an optimal gas pressure P_m corresponding to a maximum photocurrent; this property was used for a creation of solar cells.

In 1899, J. J. Thomson investigated ultraviolet light in Crookes tubes. Influenced by the work of James Clerk Maxwell, Thomson deduced that cathode rays consisted of negatively charged particles, later called electrons, which he called "corpuscles". In the research, Thomson enclosed a metal plate (a cathode) in a vacuum tube, and exposed it to high frequency radiation. It was thought that the oscillating electromagnetic fields caused the atoms' field to resonate and, after reaching a certain amplitude, caused a subatomic "corpuscle" to be emitted, and current to be detected. The amount of this current varied with the intensity and colour of the radiation. Larger radiation intensity or frequency would produce more current.

Nikola Tesla described the photoelectric effect in 1901. He described such radiation as vibrations of aether of small wavelengths which ionized the atmosphere. On November 5, 1901, he received the patent US685957, Apparatus for the Utilization of Radiant Energy, that describes radiation charging and discharging conductors. This was done by using a metal plate or piece of mica exposed to "radiant energy". Tesla used this effect to charge a capacitor with energy by means of a conductive plate, making a solar cell precursor. The radiant energy threw off with great velocity minute particles (i.e., electrons) which were strongly electrified. The patent specified that the radiation (or radiant energy) included many different forms. These devices have been referred to as "Photoelectric alternating current stepping motors".

In practice, a polished insulated metal plate or other conducting-body in radiant energy (e.g. sunlight) will gain a positive charge as electrons are emitted by the plate. As the plate charges positively, electrons form an electrostatic force on the plate (because of surface emissions of the photoelectrons), and "drain" any negatively charged capacitors. In his patent application, Tesla noted that as the rays or radiation fall on the insulated conductor (which is connected to a capacitor), the capacitor will indefinitely charge electrically.

In 1902, Philipp Lenard observed the variation in electron energy with light frequency. He used a powerful electric arc lamp which enabled him to investigate large changes in intensity, and had sufficient power to enable him to investigate the variation of potential with light frequency. His experiment directly measured potentials, not electron kinetic energy: he found the electron energy by relating it to the maximum stopping potential (voltage) in a phototube. He found that the calculated maximum electron kinetic energy is determined by the frequency of the light. For example, an increase in frequency results in an increase in the maximum kinetic energy calculated for an electron upon liberation - ultraviolet radiation would require a higher applied stopping potential to stop current in a phototube than blue light. However Lenard's results were qualitative rather than quantitative because of the difficulty in performing the experiments: the experiments needed to be done on freshly cut metal so that the pure metal was observed, but it oxidised in a matter of minutes even in the partial vacuums he used. The current emitted by the surface was determined by the light's intensity, or brightness: doubling the intensity of the light doubled the number of electrons emitted from the surface. Lenard did not know of photons.^{[citation needed]}

Einstein: light quanta

Albert Einstein's mathematical description in 1905 of how the photoelectric effect was caused by absorption of quanta of light (now called photons), was in the paper named "On a Heuristic Viewpoint Concerning the Production and Transformation of Light". This paper proposed the simple description of "light quanta", or photons, and showed how they explained such phenomena as the photoelectric effect. His simple explanation in terms of absorption of discrete quanta of light explained the features of the phenomenon and the characteristic frequency. Einstein's explanation of the photoelectric effect won him the Nobel Prize in Physics in 1921.

The idea of light quanta began with Max Planck's published law of black-body radiation ("On the Law of Distribution of Energy in the Normal Spectrum". Annalen der Physik 4 (1901)) by assuming that Hertzian oscillators could only exist at energies E proportional to the frequency f of the oscillator by E = hf, where h is Planck's constant. By assuming that light actually consisted of discrete energy packets, Einstein wrote an equation for the photoelectric effect that fitted experiments. It explained why the energy of photoelectrons were dependent only on the frequency of the incident light and not on its intensity: a low-intensity, high-frequency source could supply a few high energy photons, whereas a high-intensity, low-frequency source would supply no photons of sufficient individual energy to dislodge any electrons. This was an enormous theoretical leap, but the concept was strongly resisted at first because it contradicted the wave theory of light that followed naturally from James Clerk Maxwell's equations for electromagnetic behavior, and more generally, the assumption of infinite divisibility of energy in physical systems. Even after experiments showed that Einstein's equations for the photoelectric effect were accurate, resistance to the idea of photons continued, since it appeared to contradict Maxwell's equations, which were well-understood and verified.

Einstein's work predicted that the energy of individual ejected electrons increases linearly with the frequency of the light. Perhaps surprisingly, the precise relationship had not at that time been tested. By 1905 it was known that the energy of photoelectrons increases with increasing frequency of incident light and is independent of the intensity of the light. However, the manner of the increase was not experimentally determined until 1915 when Robert Andrews Millikan showed that Einstein's prediction was correct.

The photoelectric effect helped propel the then-emerging concept of the dualistic nature of light, that light simultaneously possesses the characteristics of both waves and particles, each being manifested according to the circumstances. The effect was impossible to understand in terms of the classical wave description of light, as the energy of the emitted electrons did not depend on the intensity of the incident radiation. Classical theory predicted that the electrons would 'gather up' energy over a period of time, and then be emitted.

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