Quantum many-body effect locates at the heart of modern condensed matter physics, where the interactions between electrons become non-negligible in materials, leading to the failure of classical Bloch-wave description. To capture the rich physics in those quantum materials, it is instructive to introduce fermionic quasiparticles and bosonic elementary excitations. For fermionic quasiparticles, one of the most descriptive quantities is the single-particle spectral function, which is experimentally accessible by angle-resolved photoemission spectroscopy (ARPES). Therefore, over the past two decades, ARPES has become one of the mainstream experimental techniques, making many landmark contributions to our understanding of quantum many-body effect.

### Contents

## General Description

ARPES is based on the photoelectric effect, in which a photo impinges on a material and is absorbed by an electron, which then escapes from the material. The utility of ARPES as an spectroscopic tool derives from the fact that one can exploit the kinematics of the photoemission process to deduce the binding energy E_{B} and crystal momentum hk of the electron before it was emitted from the from the material. A generic ARPES measurement consists of a single crystalline sample irradiated by monochromatic light of energy hν, resulting in photoemission of electrons in all possible directions. A fraction of these electrons are collected by a photoemission spectrometer, which records the kinetic energy E_{kin} and emission angles (ϑ, ϕ) of each detected electron. Here ϑ is the polar angle with respect to the surface normal, and ϕ is the azimuthal angle typically defined with respect to the experimental geometry or crystal axis. Based on energy and momentum conservation, one can then derive the following relationships between the pre- and post-emission electronic states:

Due to the discrete in-plane periodicity of the crystal structure, k_{//} is conserved throughout the photoemission process (modulo an in-plane reciprocal lattice vector G_{//}). The orthogonal component k_{⊥} is not conserved during transmission through the surface but can be deduced under certain assumptions. The energetics of the photoemission process are depicted in figure below. Note that E_{kin} is defined with respect to the sample’s vacuum level E_{vac}. Rather than directly report E_{kin}, which is dependent on hν, the energy E is typically referenced to the Fermi level E_{F} (where E_{B} = E_{F} − E). In practice, the E_{kin} corresponding to E_{F} is calibrated by fitting a Fermi-Dirac distribution to the spectrum of a polycrystalline metal which is placed in electrical contact with the sample to ensure that their Fermi levels align.

## Formal Description

Formally, the photoemission process can be described by the transition probability w_{fi} of an N-electron initial state |Ψ_{i}^{N}> to an excited final state |Ψ_{f}^{N}>, which can be approximated by Fermi’s golden rule:

### Non-interacting case

In the non-interacting case, the N-electron initial and final states can both be trivially factorized:

We denote energies of photoelectrons as ε

_{K}and ε

_{f}. |Ψ

_{i}

^{N-1}> and |Ψ

_{f}

^{N-1}> are the initial and final state of N-1 electron system. The non-interacting limit allows for a dramatic simplification: |Ψ

_{i}

^{N-1}> = |Ψ

_{f}

^{N-1}>, since the (N-1) electron system is not affected by the removal of one electron.

Therefore, we can calculate the total photocurrent as:

### Interacting case

For an interacting system, the many-body final and initial states cannot be trivially factorized as in non-interacting state. Nevertheless, for a more tractable formalism, we can cautiously adopt these forms as approximations under sudden approximation and main field approximation. For strongly interacting electron systems, the (N-1) electron systems can no longer be regarded as unchanged due to electron removal. Instead, under sudden approximation, the N-1 final electron state can be left in any number of excited states. Therefore, the total transition probability is then a sum over excited states:

Green’s function formalism enables us to further understand the structure of the formula, where interactions are taken into account via the proper self-energy, in terms of which the spectral function is given by

Therefore, we are finally motivated to write the photoemission intensity as:

## Modern Light Source

What light source is most suitable for ARPES experiments? From the basic conservation rule and its derivative of energy and momentum in ARPES experiment, we learn that ideally, we would like high photon energy to access large momentum space and bands with high biding energy, while contradictively, low photon energy for high momentum resolution. In addition, the intensity, flux, and polarization all play important roles in the quality of the final ARPES spectrum if more experimental details are considered. Given the above considerations, we summarize the following guiding rules for an ideally optimized light source for ARPES study of complex materials:

(1) Sufficiently high photon energy ( > 10 eV) to access the first Brillouin zone and probe valence bands;

(2) Sufficiently low photon energy ( < 20 eV) to facilitate high momentum resolution;

(3) Sufficiently high repetition rate ( >> 100 kHz) to limit the loss of energy and momentum resolutions due to space-charge effect while maintaining sufficiently high signal-to-noise ratio;

(4) Sub-meV bandwidth, high flux, and long term stability;

(5) Variable polarization for control over photoemission elements;

(6) Additional requirements may include a small ( < 1nm) beamspot and short ( < 100 ps) pulse duration for time-of-flight detection schemes.

Unfortunately, some of the aforementioned requirements are intrinsically incompatible with each other. Therefore, we have to design different light sources to enhance some of the requirements at the necessary sacrifice of others. On one extreme, we introduce modern laser technique into ARPES to achieve high energy and momentum resolution (refer to laser ARPES section). One the other extreme, we adopt synchrotron photons to significantly expand the momentum space ARPES can probe (refer to synchrotron ARPES section).

## Photoelectron Spectrometor

A photoelectron spectrometer uses electrostatic elements to manipulate the trajectory and energy of elec- trons and impinge them onto a detector. Modern spectrometers feature lensing elements that can be operated to record either the angular or spatial distribution of electrons, where the angular mode is used for ARPES measurements. The detector records the energetic and angular distributions of the photoelectrons which can be traced back to the electron single-particle spectral function prior to emission. The detector typically consists of a multichannel plate (MCP), which amplifies the signal by converting a single electron into a cloud of ∼ 106 electrons while maintaining the spatial distribution of the incident electrons. Most commonly, the MCP output is impinged onto a phosphor from which the resulting luminescence can be read into a computer using a CCD camera.

There are many commonly used photoelectron spectrometers and we mainly adopt hemisphere analyzers in our group. The hemisphere analyzer consists of two concentric hemispherical electrodes with different electrostatic potentials, resulting in a radial electric field that causes the electrons to undergo elliptical orbits. Thus, electrons with different kinetic energies are dispersed along the radial dimension onto the detector. At the same time, the electron position orthogonal to this axis is determined by its emission angle within the window accepted by the slit. The detector therefore records the two-dimensional photocurrent dis- tribution with respect to (Ekin,ϑy). Energy resolutions of order 1 meV are routinely obtained, though under pristine conditions sub-100 μeV. Typical acceptance angles are 15 degree with resolutions down to 0.1 degree.

## Scientific Impacts

The past two decades have witnessed an explosive growth of research on quantum materials, with ARPES playing the central role as a direct probe of electronic structure. In parallel, theoretical calculations from first principles and model Hamiltonians have developed rapidly, greatly adding value to the scientific output of ARPES experiments. In strongly correlated states of matter (especially the Cuperate superconductors and Iron-based superconductors), ARPES has allowed for detailed investigation of Fermi surfaces, order parameters, and mode-couplings, in phases ranging from conventional Fermi liquid to spin- ordered, charge-ordered, or superconducting states (For more details, please refer to Quantum Material Section). In topological materials, it has directly visualized the bulk electronic structure responsible for non-trivial topology, as well as the associated boundary states, particularly in cases in which they are inaccessible to transport probes. Advanced synthesis techniques have made it possible to explore these materials in lower dimensions, unlocking interactions and phenomena that are non-existent in their three-dimensional forms. These scientific pursuits have been accelerated by increasingly sophisticated ARPES measurement capabilities with space, spin, and time resolutions, all while pushing to ever lower temperatures, thus resolving all the quantum numbers of the photoemit- ted electrons in the relevant regions of the phase diagram.

## Further Readings

[1] A. Damascelli *et al.* Angle-resolved photoemission studies of the cuprate superconductors. *Rev. Mod. Phys.* **75**, 473 (2003)

[2] 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)

[3]J.A. Sobota, Y. He, Z.-X. Shen, *Reviews of Modern Physics* **2**, 93 (2021)