Chemical Sciences: A Manual for CSIR-UGC National Eligibility Test for Lectureship and JRF/Photoelectric effect

Photoelectric effect
A diagram illustrating the emission of electrons from a metal plate, requiring energy gained from an incoming photon to be more than the work function of the material.
Light-matter interaction
Low energy phenomena	Photoelectric effect
Mid-energy phenomena	Compton scattering
High energy phenomena	Pair production

The photoelectric effect is a phenomenon in which electrons are emitted from matter (metals and non-metallic solids, liquids or gases) as a consequence of their absorption of energy from electromagnetic radiation of very short wavelength, such as visible or ultraviolet light. Electrons emitted in this manner may be referred to as "photoelectrons". As it was first observed by Heinrich Hertz in 1887, the phenomenon is also known as the "Hertz effect", although the latter term has fallen out of general use. Hertz observed and then showed that electrodes illuminated with ultraviolet light create electric sparks more easily.

The photoelectric effect takes place with photons with energies from about a few electronvolts to, in high atomic number elements, over 1 MeV. At the high photon energies comparable to the electron rest energy of 511 keV, Compton scattering, another process, may take place, and above twice this (1.022 MeV) pair production may take place.

Study of the photoelectric effect led to important steps in understanding the quantum nature of light and electrons and influenced the formation of the concept of wave–particle duality.

The term may also, but incorrectly, refer to related phenomena such as the photoconductive effect (also known as photoconductivity or photoresistivity), the photovoltaic effect, or the photoelectrochemical effect which are, in fact, distinctly different.

Introduction and early historical view

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 color) 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.

Modern view

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

$K_{\mathrm {max} }=hf-\varphi$

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

$\varphi =hf_{0},$

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

$K_{\mathrm {max} }=h\left(f-f_{0}\right)$

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.

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.

History

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.

JJ Thomson: electrons

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.

Radiant energy

Photoelectric motor. Rays falling on insulated conductor connected to a capacitor: the capacitor charges electrically.

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.

Von Lenard's observations

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.

Einstein: light quanta

Albert Einstein's mathematical description of how the photoelectric effect was caused by absorption of quanta of light (now called photons), was in one of his 1905 papers, 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.

Effect on wave–particle question

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.

Uses and effects

Photodiodes and phototransistors

Solar cells (used in solar power) and light-sensitive diodes use a variant of the photoelectric effect, but not ejecting electrons out of the material. In semiconductors, light of even relatively low energy, such as visible photons, can kick electrons out of the valence band and into the higher-energy conduction band, where they can be harnessed, creating electric current at a voltage related to the bandgap energy.

Photomultipliers

These are extremely light-sensitive vacuum tubes with a photocathode coated onto part (an end or side) of the inside of the envelope. The photocathode contains combinations of materials such as caesium, rubidium and antimony specially selected to provide a low work function, so when illuminated even by very low levels of light, the photocathode readily releases electrons. By means of a series of electrodes (dynodes) at ever-higher potentials, these electrons are accelerated and substantially increased in number through secondary emission to provide a readily-detectable output current. Photomultipliers are still commonly used wherever low levels of light must be detected.

Image sensors

Video camera tubes in the early days of television used the photoelectric effect; newer variants used photoconductive rather than photoemissive materials.

Silicon image sensors, such as charge-coupled devices, widely used for photographic imaging, are based on a variant of the photoelectric effect, in which photons knock electrons out of the valence band of energy states in a semiconductor, but not out of the solid itself.

The gold-leaf electroscope

The gold leaf electroscope.

Gold-leaf electroscopes are designed to detect static electricity. Charge placed on the metal cap spreads to the stem and the gold leaf of the electroscope. Because they then have the same charge, the stem and leaf repel each other. This will cause the leaf to bend away from the stem. The electroscope is an important tool in illustrating the photoelectric effect. Let us say that the scope is negatively charged throughout. There is an excess of electrons and the leaf is separated from the stem. But if we then shine high-frequency light onto the cap, the scope discharges and the leaf will fall limp. This is because the frequency of the light shining on the cap is above the cap's threshold frequency. The photons in the light have enough energy to liberate electrons from the cap, reducing its negative charge. This will discharge a negatively charged electroscope and further charge a positive electroscope. However, if the electromagnetic radiation hitting the metal cap does not have a high enough frequency (its frequency is below the threshold value for the cap), then the leaf will never discharge, no matter how long one shines the low-frequency light at the cap.

Photoelectron spectroscopy

Since the energy of the photoelectrons emitted is exactly the energy of the incident photon minus the material's work function or binding energy, the work function of a sample can be determined by bombarding it with a monochromatic X-ray source or UV source, and measuring the kinetic energy distribution of the electrons emitted.

Photoelectron spectroscopy is done in a high-vacuum environment, since the electrons would be scattered by gas molecules if they were present. The light source can be a laser, a discharge tube, or a synchrotron radiation source.

The concentric hemispherical analyser (CHA) is a typical electron energy analyzer, and uses an electric field to change the directions of incident electrons, depending on their kinetic energies. For every element and core (atomic orbital) there will be a different binding energy. The many electrons created from each of these combinations will show up as spikes in the analyzer output, and these can be used to determine the elemental composition of the sample.

Spacecraft

The photoelectric effect will cause spacecraft exposed to sunlight to develop a positive charge. This can get up to the tens of volts. This can be a major problem, as other parts of the spacecraft in shadow develop a negative charge (up to several kilovolts) from nearby plasma, and the imbalance can discharge through delicate electrical components. The static charge created by the photoelectric effect is self-limiting, though, because a more highly-charged object gives up its electrons less easily.

Moon dust

Light from the sun hitting lunar dust causes it to become charged through the photoelectric effect. The charged dust then repels itself and lifts off the surface of the Moon by electrostatic levitation. This manifests itself almost like an "atmosphere of dust", visible as a thin haze and blurring of distant features, and visible as a dim glow after the sun has set. This was first photographed by the Surveyor program probes in the 1960s. It is thought that the smallest particles are repelled up to kilometers high, and that the particles move in "fountains" as they charge and discharge.

Night vision devices

Photons hitting a gallium arsenide plate in night vision devices cause the ejection of photoelectrons due to the photoelectric effect. These are then amplified into a cascade of electrons that light up a phosphor screen.

Cross section

The photoelectric effect is simply an interaction mechanism conducted between photons and atoms. However, this mechanism does not have exclusivity in interactions of this nature and is one of 12 theoretically possible interactions . As noted in the prologue; Compton scattering and pair production are an example of two other competing mechanisms. Indeed, even if the photoelectric effect is the favoured reaction for a particular single-photon bound-electron interaction, the result is also subject to statistical processes and is not guaranteed, albeit the photon has certainly disappeared and a bound electron has been excited (usually K or L shell electrons at nuclear (gamma ray) energies). The probability of the photoelectric effect occurring is measured by the cross section of interaction, σ. This has been found to be a function of the atomic number of the target atom and photon energy. A crude approximation, for photon energies above the highest atomic binding energy, is given by :

$\sigma =\mathrm {constant} \cdot {\frac {Z^{n}}{E^{3}}}$

Here Z is atomic number and n is a number which varies between 4 and 5. (At lower photon energies a characteristic structure with edges appears, K edge, L edges, M edges, etc.) The obvious interpretation follows that the photoelectric effect rapidly decreases in significance, in the gamma ray region of the spectrum, with increasing photon energy, and that photoelectric effect is directly proportional to atomic number. The corollary is that high-Z materials make good gamma-ray shields, which is the principal reason that lead (Z = 82) is a preferred and ubiquitous gamma radiation shield.