Angle-resolved photoemission spectroscopy (ARPES) is an experimental technique used in condensed matter physics to probe the allowed energies and momenta of the electrons in a material, usually a crystalline solid. It is based on the photoelectric effect, in which an incoming photon of sufficient energy ejects an electron from the surface of a material. By directly measuring the kinetic energy and emission angle distributions of the emitted photoelectrons, the technique can map the electronic band structure and Fermi surfaces. ARPES is best suited for the study of one- or two-dimensional materials. It has been used by physicists to investigate high-temperature superconductors, graphene, topological materials, quantum well states, and materials exhibiting charge density waves.
ARPES systems consist of a monochromatic light source to deliver a narrow beam of photons, a sample holder connected to a manipulator used to position the sample of a material, and an electron spectrometer. The equipment is contained within an ultra-high vacuum (UHV) environment, which protects the sample and prevents scattering of the emitted electrons. After being dispersed along two perpendicular directions with respect to kinetic energy and emission angle, the electrons are directed to a detector and counted to provide ARPES spectra—slices of the band structure along one momentum direction. Some ARPES instruments can extract a portion of the electrons alongside the detector to measure the polarization of their spin.
Principle Electrons in crystalline solids can only populate states of certain energies and momenta, others being forbidden by quantum mechanics. They form a continuum of states known as the band structure of the solid. The band structure determines if a material is an insulator, a semiconductor, or a metal, how it conducts electricity and in which directions it conducts best, or how it behaves in a magnetic field.
Angle-resolved photoemission spectroscopy determines the band structure and helps understand the scattering processes and interactions of electrons with other constituents of a material. It does so by observing the electrons ejected by photons from their initial energy and momentum state into the state whose energy is by the energy of the photon higher than the initial energy, and higher than the binding energy of the electron in the solid. In the process, the electron's momentum remains virtually intact, except for its component perpendicular to the material's surface. The band structure is thus translated from energies at which the electrons are bound within the material, to energies that free them from the crystal binding and enable their detection outside of the material.
By measuring the freed electron's kinetic energy, its velocity and absolute momentum can be calculated. By measuring the emission angle with respect to the surface normal, ARPES can also determine the two in-plane components of momentum that are in the photoemission process preserved. In many cases, if needed, the third component can be reconstructed as well.
Instrumentation A typical instrument for angle-resolved photoemission consists of a light source, a sample holder attached to a manipulator, and an electron spectrometer. These are all part of an ultra-high vacuum system that provides the necessary protection from adsorbates for the sample surface and eliminates scattering of the electrons on their way to the analyzer.
The light source delivers to the sample a monochromatic, usually polarized, focused, high-intensity beam of ~1012 photons/s with a few meV energy spread. Light sources range from compact noble-gas discharge UV lamps and radio-frequency plasma sources (10–40 eV), ultraviolet lasers (5–11 eV) to synchrotron insertion devices that are optimized for different parts of the electromagnetic spectrum (from 10 eV in the ultraviolet to 1000 eV X-rays).
The sample holder accommodates samples of crystalline materials, the electronic properties of which are to be investigated. It facilitates their insertion into the vacuum, cleavage to expose clean surfaces, and precise positioning. The holder works as the extension of a manipulator that makes translations along three axes, and rotations to adjust the sample's polar, azimuth and tilt angles possible. The holder has sensors or thermocouples for precise temperature measurement and control. Cooling to temperatures as low as 1 kelvin is provided by cryogenic liquefied gases, cryocoolers, and dilution refrigerators. Resistive heaters attached to the holder provide heating up to a few hundred °C, whereas miniature backside electron-beam bombardment devices can yield sample temperatures as high as 2000 °C. Some holders can also have attachments for light beam focusing and calibration.
The electron spectrometer disperses the electrons along two spatial directions in accordance with their kinetic energy and their emission angle when exiting the sample; in other words, it provides mapping of different energies and emission angles to different positions on the detector. In the type most commonly used, the hemispherical electron energy analyzer, the electrons first pass through an electrostatic lens. The lens has a narrow focal spot that is located some 40 mm from the entrance to the lens. It further enhances the angular spread of the electron plume, and serves it with adjusted energy to the narrow entrance slit of the energy dispersing part.
The energy dispersion is carried out for a narrow range of energies around the so-called pass energy in the direction perpendicular to the direction of angular dispersion, that is perpendicular to the cut of a ~25 mm long and ⪆0.1 mm wide slit. The angular dispersion previously achieved around the axis of the cylindrical lens is only preserved along the slit, and depending on the lens mode and the desired angular resolution is usually set to amount to ±3°, ±7° or ±15°. The hemispheres of the energy analyzer are kept at constant voltages so that the central trajectory is followed by electrons that have the kinetic energy equal to the set pass energy; those with higher or lower energies end up closer to the outer or the inner hemisphere at the other end of the analyzer. This is where an electron detector is mounted, usually in the form of a 40 mm microchannel plate paired with a fluorescent screen. Electron detection events are recorded using an outside camera and are counted in hundreds of thousands of separate angle vs. kinetic energy channels. Some instruments are additionally equipped with an electron extraction tube at one side of the detector to enable the measurement of the electrons' spin polarization.
Modern analyzers are capable of resolving the electron emission angles as low as 0.1°. Energy resolution is pass-energy and slit-width dependent so the operator chooses between measurements with ultrahigh resolution and low intensity (< 1 meV at 1 eV pass energy) or poorer energy resolutions of 10 meV or more at higher pass energies and with wider slits resulting in higher signal intensity. The instrument's resolution shows up as artificial broadening of the spectral features: a Fermi energy cutoff wider than expected from the sample's temperature alone, and the theoretical electron's spectral function convolved with the instrument's resolution function in both energy and momentum/angle.
Sometimes, instead of hemispherical analyzers, time-of-flight analyzers are used. These, however, require pulsed photon sources and are most common in laser-based ARPES labs.
Basic relations
Angle-resolved photoemission spectroscopy is an angle-selected refinement of ordinary photoemission spectroscopy. Light of frequency made up of photons of energy , where is the Planck constant, is used to stimulate the transitions of electrons from occupied to unoccupied electronic state of the solid. If a photon's energy is greater than the binding energy of an electron , the electron will eventually leave the solid without being scattered, and be observed with kinetic energy
at angle relative to the surface normal, both characteristic of the studied material.
Electron emission intensity maps measured by ARPES as a function of and are representative of the intrinsic distribution of electrons in the solid expressed in terms of their binding energy and the Bloch wave vector , which is related to the electrons' crystal momentum and group velocity. In the photoemission process, the Bloch wave vector is linked to the measured electron's momentum , where the magnitude of the momentum is given by the equation .
As the electron crosses the surface barrier, losing part of its energy due to the surface work function,For simplicity reasons, the work function has been included in the expression for as part of (true meaning of the binding energy). In practice, however, the binding energy is expressed relative to a material's Fermi level, which can be read off of an ARPES spectrum. The work function is the difference between the Fermi level and the vacuum level where electrons are free. only the component of that is parallel to the surface, , is preserved. From ARPES, therefore, only is known for certain and its magnitude is given by . Here, is the reduced Planck constant.
Because of incomplete determination of the three-dimensional wave vector, and the pronounced surface sensitivity of the elastic photoemission process, ARPES is best suited to the complete characterization of the band structure in ordered low-dimensional systems such as two-dimensional materials, ultrathin films, and nanowires. When it is used for three-dimensional materials, the perpendicular component of the wave vector is usually approximated, with the assumption of a parabolic, free-electron-like final state with the bottom at energy . This gives: . The inner potential is an unknown parameter a priori. For d-electron systems, experiment suggest that . In general, the inner potential is estimated through a series of photon energy-dependent experiments, especially in photoemission band mapping experiments.
Fermi surface mapping
Electron analyzers that use a slit to prevent the mixing of momentum and energy channels are only capable of taking angular maps along one direction. To take maps over energy and two-dimensional momentum space, either the sample is rotated in the proper direction so that the slit receives electrons from adjacent emission angles, or the electron plume is steered inside the electrostatic lens with the sample fixed. The slit width will determine the step size of the angular scans. For example, when a ±15° plume dispersed around the axis of the lens is served to a 30 mm long and 1 mm wide slit, each millimeter of the slit receives a 1° portion—in both directions; but at the detector the other direction is interpreted as the electron's kinetic energy and the emission angle information is lost. This averaging determines the maximal angular resolution of the scan in the direction perpendicular to the slit: with a 1 mm slit, steps coarser than 1° lead to missing data, and finer steps to overlaps. Modern analyzers have slits as narrow as 0.05 mm. The energy–angle–angle maps are usually further processed to give energy–kx–ky maps, and sliced in such a way to display constant energy surfaces in the band structure and, most importantly, the Fermi surface map when they are cut near the Fermi level.
Emission angle to momentum conversion