Electrons can be emitted from solids under irradiation with photons of sufficiently low wavelength. This process, called photoelectron emission, has played a key role as experimental support for quantum mechanics. Einstein got the Noble prize in physics for explaining why photoelectrons cannot be emitted from a metal by red light, no matter how high the intensity (they did not have intense lasers at that time). Photoelectrons are emitted when a single photon (quanta) of energy hn is absorbed by the solid, where h is Planck's constant and n the frequency of the light used. The energy of the photon must be larger than the energy separation between the top of the valence band and the vacuum level. At low temperatures and in metals, this energy is the work function. For non metals, electrons may exist in the band gap (electronic states due to the existence of the surface (surface states), impurities or defects). Thus, in semiconductors, the photoelectric threshold (minimum photon energy) is usually larger than the work function. In insulators, the Fermi level is not defined in practice, so the photoelectron threshold is just the energy between the most weakly bound electron state (usually a band-gap state) and the vacuum level.
For metals at any finite temperature, there will be electrons in the conduction band above the Fermi level, as a results from the Fermi-Dirac distribution. Thus, a sharp photoelectron threshold does not exist.
Because the photon carries essentially no momentum, photoelectron emission cannot result in the "free-electron" model. This is because when a free electron absorbs energy from the photon, its momentum also increases. For the process to be possible, a third body must absorb the momentum. The most likely participant is the lattice.
Retarding potential measurements
Photoelectron emission is measured by applying a voltage between the specimen (cathode) and another electrode, called the collector or anode, and measuring the current. If the anode is positive, the photoelectron current saturates (except for the Schottky effect, see notes on thermionic emission). If the anode is negative, electrons will be repelled and not collected, except those of sufficient energy to overcome the potential difference (applied voltage plus Df = fA - fc, the work function difference between anode and cathode). A cut-off anode voltage Vc is defined as the minimum negative voltage at which the photoelectron current is zero. For V < Vc all electrons are returned to the cathode. The value of Vc is then that at which even the highest energy electron cannot be collected. The maximum photoelectron energy results when the photon is absorbed by an electron at the Fermi level, and is equal to hn - fc when referred to the vacuum level, or hn - EF when referred to the Fermi level. The electron energy at the anode is hn - EF - eVc (or hn - fA - eVc with respect of the vacuum level of the sample. At the cutoff, the electron can barely make it above the vacuum level of the anode, so hn - fA - eVc = 0. Thus, the measurement of the cutoff voltage for a particular photon energy can be used to measure the work function of the anode. If one wants to determine the work function of the specimen, then one needs to shine light on the anode and bias the specimen to a negative voltage with respect to the anode.
Photoelectric threshold and Fowler's law
The photoelectric thrshold occurs when hn equals the work function of the sample, f. In this case, only electrons at the Fermi level can be excited outside the solid (electrons below the Fermi level can also absorb energy in a metal, but they will end up in excited states below the vacuum level). As the photon energy is increased over f there is an increase in the fraction of valence electrons of energy below the Fermi level that can be excited above the vacuum level. Also, as the electron energy above the vacuum level increases, the range of angles that can result in escape increases (increase in escape cone, see class on thermionic emission). The effect of these two factors make the photoelectric current i to be:
i = c (hn - f)n
where c is a constant. For most metals, n = 2; other exponents apply to semiconductors. This is called Fowler's law. This equation can be used to determine the work function of the sample by measuring i(hn) and fitting it to a power law. The intercept on the hn axis (threshold) gives the work function f, or more correctly, the photoelectron threshold. Thermal effects cause some electrons to exist above the Fermi level, so the threshold becomes rounded, but this effect can be taken into account with a corrected Fowler expression. If the temperature is very high, thermoionic emission will occur in addition to photoelectron emission.
The photoelectric effect is used to detect light. The most sensitive light detectors are the photomultipliers, where the current of electrons from the cathode (photocathode) is amplified by secondary electron emission (see class on this topic). The fraction of photoelectrons that can be detected is called the quantum efficiency. A low work function will ensure a high quantum efficiency and also the ability to detect photons of low energy (high wavelength l). The minimum work function of pure materials is that of Cs, around 2 eV, which allows detection of photons in the red of the spectrum (l ~ 620 nm). However, Cs is very unstable because it oxidizes readily. Typical metals have f ~ 4 eV, so that only UV light of l < 310 nm can produce photoelectrons. A semiconductor like CscSb has a band-gap of 1.6 eV and an intrinsic photoelectron thresold of 2.05 eV, due to a low electron affinity. Doping this material with electrons that go to states close to the Fermi level lead to a photoelectric threshold close to the work function which is 1.25 eV. This corresponds to a photon wavelength in the near infrared (about 1 mm).
The design of photocathodes is based on the 3-step model. In this approximation, the photoelectron emission is divided in three separate steps: (1) photoabsorption or excitation, (2) transport of excited electrons to the surface, and (3) passage through the surface and escape. To maximize quantum efficiency, attention is given to each one of the steps. Large excitation results from materials with large density of electrons states close to the vacuum level, e.g., high electron density and low photoelectron threshold. The excited electrons may lose energy during transport to the surface and so their flux is attenuated. The attenuation is less for materials where inelastic electron scattering is hindered, like insulators or wide band-gap semiconductors. Finally, a low surface barrier is required for maximum escape.
The best photocathodes are those with negative electron affinity (NEA) cathodes, which mainly affect step (3). These ae made by applying a monolayer of Cs plus controlled amounts of O to a GaAs substrate. NEA photocathodes are used for night-vision applications.
The measurement of the energy distribution of photoelectrons produced by a photon of a single energy (monochromatic) is called photoelectron spectroscopy. The electron energy at the anode, above the vacuum level, is just the photon energy hn minus the binding energy of the electron. Thus different electron energies correspond to electrons in states with different binding energy. Thus, these measurements give information of the density of states of the electrons prior to excitation for those electron levels bound by less than hn - fA. If hn is in the UV, the technique is called UPS, or ultraviolet photoelectron spectroscopy. If X-rays are used, the technique is called XPS, or x-ray photoelectron spectroscopy. Both techniques are very powerful for analyzing surfaces and will be covered in detail later.
For drawings, graphs, see photoelectron.PDF
Copyright 2002, by Raśl Baragiola, University of Virginia. All rights reserved.