Photoelectron spectroscopy is a general term which refers to all techniques based on the photoelectric effect originally observed by Hertz. This was later explained as a manifestation of the quantum nature of light by Einstein, who recognized that when light is incident on a sample, an electron can absorb a photon and escape from the material with a maximum kinetic energy, hν−φ, where hν is the light energy and φ is the work-function of the material. The energy of an electron inside a solid can be obtained using photoelectron spectroscopy — the core electrons will have a lower kinetic energy than the valence electrons when absorbing the same photon energy. In principle, the momentum of the electrons can also be obtained — different momentum electrons will escape at different angles from the surface of a material. However, since the electrons are being projected through the surface, the momentum perpendicular to the surface is not conserved. Therefore, angle-resolved photoemission is ideal for 2D materials where the principle momentum directions of interest are parallel to the surface. One of the primary materials of interest are the 2D copper-oxide perovskite family in which the undoped parent compounds are anti-ferromagnetic insulators and the doped compounds have the highest known superconducting transition temperatures, yet to be described by theory.
In practice, the electrons ejected from the material are collected using a hemispherical detector in which lens voltages direct the electron onto a two-dimensional (energy, momentum) multi-channel plate. The sample and the detector are kept in an ultra high vacuum (UHV) chamber in order to minimize surface contamination. Light sources are either synchrotron radiation at ~ 20-200 eV, plasma Helium discharge at ~ 20 eV, or more recently, modern-day lasers at 7 eV and 11 eV.
Since ARPES measures the energy and momentum of electrons inside a solid directly, it has a very natural theoretical description. The initial state of the electrons evolves, upon interaction with light, to a final state with a transition probability given by Fermi's golden rule:
where the interaction Hamiltonian is given by:
The initial state of this dipole-interaction Hamiltonian is a neutral solid, while the final state is a dipole—the hole in the material left by the ejected electron and the photoelectron itself, which is a combination of the light and the ejected electron. If one assumes that the solid does not relax in the time it takes to measure the photoelectron (the so-called sudden approximation), then the intensity I(k,ω) of the spectral weight can be thought of as:
where I0 describes a matrix element dependent on incident photon energy, and f(ω) is the Fermi function, signifying that ARPES measures the occupied density of states.
A(k,ω) is the single-particle spectral function that describes, theoretically, the energy and momentum of an electron in a solid. The bare-band electron dispersion is denoted by εk, while the complex quantity self-energy, Σ, describes all the interactions that an electron has with other electrons and the lattice in a solid. This description is based on Landau's Fermi liquid theory in which electrons, though being part of a solid, remain as independent quasi-particles near the Fermi level. They can therefore be understood as relatively free electrons, but with a renormalized dispersion due to interactions. Differences in how electrons interact lead solids to form metals, insulators, magnets, or superconductors. ARPES allows direct access to the electron spectral function through which these interactions take place.
ARPES, with its unique capability to directly resolve in energy-momentum space and hence image electronic structures of materials, has been demonstrated to be a powerful experimental probe to study materials of a wide range of interesting properties. A major area of research in condensed matter physics is the science of emergence - where complex quantum systems develop novel properties not pertinent to the individual constituencies on their own. An important example of such properties is the collective and nanoscale phenomena in quantum matter, such as high temperature superconductivity, colossal magnetoresistance and other unconventional phenomena in complex materials. Listed and described in many parts of the Quantum Materials section of the website are some examples of the impacts ARPES has had in various areas of condensed matter physics, such as elucidating the pairing symmetry of the cuprate high temperature superconductors and realizing the theoretically predicted novel materials of topological insulators. Over the last decade, the improvements in angular and momentum resolutions have been a key in advancing this technique into a sophisticated precision tool for the investigation of complex phenomena, as continuing efforts to push the limits and to improve capabilities such as spin resolution and time resolution will no doubt keep ARPES to be at the forefront of quantum condensed matter science contributing to the understanding of novel phenomena as they are discovered.
 A. Damascelli et al. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473 (2003)
 W. S. Lee et al. A brief update of angle-resolved photoemission spectroscopy on a correlated electron system. J. Phys.: Condens. Matter 21, 164217 (2009)