Angle-Resolved Photoemission Spectroscopy (ARPES) is an experimental technique based on several refinements of the photoelectric effect initially observed by Heinrich Hertz in 1887. When a monochromatic beam of photons of energy $h\nu$ are incident upon a sample, measurement of the electron's kinetic energy and exit angle gives information about the momentum and energy ("band structure") of the electron state in the studied material. Directly, ARPES gives the binding energy ($\text{E}_\text{b}$) of the emitted electrons and the components of momentum parallel to the sample surface ($\textbf{k}_\parallel$). Crystalline translational symmetry is broken by the vacuum interface, so less information is available for the out of plane momentum ($\textbf{k}_\text{z}$) without varying the incident photon energy.

Experiment

From simple energy and momentum conservation arguments, we can explain the process by which ARPES makes available the electronic band structure. In typical ARPES experiments, a hemispherical energy analyzer simultaneously measures the intensity distribution of photoelectrons along a one dimensional linecut of angle—defined by an entrance slit on the analyzer—and resolved by the photoelectron kinetic energy. Briefly, photoelectrons emitted from different angles (green, blue, and orange in the above depiction) are incident at different locations on the entrance slit and follow different circular trajectories between the analyzer plates. Meanwhile, more and less energetic electrons follow longer and shorter paths (depicted as a spread in blue, green, and orange curves) due to their different bend radii in the constant E-field produced by the analyzer. The analyzer therefore spatially filters the electrons to produce an image of the photoemitted beam. A final electron sensitive detector, typically a channel plate paired with a phosphor screen + CCD or delay line + timing hardware, turns the spatially filtered electron signal into a digital one. If a photoemitted electron leaves the sample with an energy $\text{E}_\text{kin}$ at an angle $(\theta, \phi)$ to the sample normal, $\hat{\text{\textbf{z}}}$, the binding energy and $\textbf{k}_\parallel$ are given by conservation: $\text{E}_{\text{b}} = \underbrace{h\nu}_\text{known} - \varphi - \underbrace{\text{E}_\text{kin}}_\text{measured}$ $\textbf{p}_\parallel = \sqrt{2 m \text{E}_\text{kin}}\sin{\theta} \left(\cos \phi \hat{\mathbf{x}} + \sin \phi \hat{\mathbf{y}} \right).$ The sample workfunction $\varphi$, giving the different between the Fermi and vacuum levels, may or may not be known. The Fermi edge of a metallic sample (actual or reference), nevertheless links the kinetic energy to the electron binding energy. By turning the sample, $\hat{\text{\textbf{z}}}$ can be rastered perpendicular to the analyzer entrance, making available the photoemission intensity over all values of $\textbf{k}_\parallel$. Consideration must be given to the difference between experimentally measured angles and the spherical polar angles relative to $\hat{\text{\textbf{z}}}$. To obtain high-quality data, ARPES experiments are conducted in an ultra-high vacuum chamber which minimizes surface contamination and interactions between the photoemitted electrons and any potential interference between the emission and detection processes. Additionally, ARPES experiments are often performed at cryogenic temperatures to minimize thermal broadening of the data. This capability also allows for the study of high-temperature superconductors below their critical temperatures, where the electrons take on a fundamentally different structure. ## Single + Many Body Physics ARPES provides a direct measurement of the angle resolved photocurrent $I(\textbf{k}, \omega)$. This can be expressed in terms of a dipole interaction matrix element $M$, the Fermi-Dirac distribution for state occupancy, and the single-particle spectral function $A(\mathbf{k}, \omega)$: $I(\textbf{k}, \omega) = M\left(\textbf{k}, \omega, \mathbf{k}\cdot\mathbf{A}\right)f(\omega)A(\mathbf{k}, \omega).$ Although matrix element contribution can in general be difficult to disentangle, it is typically slowly varying in momentum and photon energy, and therefore ARPES can be thought of as measuring also $A(\textbf{k}, \omega)$. $\textbf{A}(\mathbf{k}, \omega) = -\frac{1}{\pi} \frac{\textcolor{blue}{\Sigma''(\textcolor{black}{\mathbf{k}, \omega})}} {[\omega - \hspace{-0.25em}\underbrace{\textcolor{green}{\epsilon_0(\textcolor{black}{\mathbf{k}})}}_{\textcolor{green}{\text{Bare band}}}\hspace{-0.25em} - \hspace{-0.4em}\underbrace{\textcolor{red}{\Sigma'\left(\textcolor{black}{\mathbf{k},\omega}\right)}}_{\textcolor{red}{\text{E-renormalization}}}\hspace{-0.9em}]^2 + [\underbrace{\textcolor{blue}{\Sigma''\left(\textcolor{black}{\mathbf{k},\omega}\right)}}_{ \textcolor{blue}{\text{Lifetimes}} }]^2}$ From this quantity much can be extracted, including single-particle band structure, energy renormalization by interaction, and measurement of the quasiparticle lifetimes. ARPES therefore makes available momentum resolved information on electron-electron and electron-boson interaction in real material systems. ## Laser ARPES From the conservation equations, we can see that given a fixed angular resolution of $\Delta \theta$ of our analyzer, we may obtain a higher momentum resolution for smaller photoelectron kinetic energy, $\Delta\mathbf{k}\propto\sqrt{\text{E}_\text{kin}}\Delta\theta$. UV and VUV lasers based on frequency doubling in BBO or KBBF and operating at very high photon fluxes with narrow spectral linewidths of a few meV allow ARPES studies of materials at a much higher resolution than is possible at synchrotron sources. Pulsed laser sources also enable [ARPES studies of materials perturbed out of equilibrium](/tr-arpes) by a strong visible to IR pulse, at the penalty of larger bandwidth $\Delta \nu$ and therefore worse energy resolution.