# Angle-resolved Photoemission Spectroscopy

Many-body quantum physics lies at the very heart of modern condensed matter. Every undergraduate is familiar with the semiclassical Bloch-wave description, which treats electrons as *independent* and subject to a lattice potential in which they move. It no longer suffices here. Instead, non-negligible **interactions*** *involving electrons in quantum materials yield a bountiful cornucopia of physics.

To capture this physics, it is useful to introduce such many-body objects as fermionic quasiparticles and bosonic elementary excitations. For example, electrons in materials are described as **quasiparticles** when interactions do not change their qualitative behavior, but merely adjust their parameters, such as introducing an *effective* mass different from its *real* mass (an action known as renormalization). The **single-particle spectral function** *A*(** k**,

*ω*), a multi-dimensional function, is a crucial quantity that extensively describes the behaviour of interacting electrons. It is also experimentally accessible by angle-resolved photoemission spectroscopy (ARPES). This direct link between theory and experiment has enabled ARPES to become one of the most mainstream, potent, and consequential experimental techniques over the past three decades. It has made — and will continue to make — many landmark contributions to our understanding of quantum many-body physics.

This page summarizes how the single-particle spectral function is obtained using ARPES, and use it to illustrate two works by former graduate students of our group. We also introduce two key experimental components of ARPES: the light source, and the photoelectron spectrometer.

### Contents

- Photoemission
- Spectroscopy
- Many-Body Physics
- Modern Light Sources
- Photoelectron Spectrometer
- Scientific Impact
- Further Readings

## Photoemission

ARPES is based on the photoelectric effect. A photon impinges on a material and is absorbed by an electron, which then escapes from its surface. The utility of ARPES as a spectroscopic tool derives from the fact that the **kinematics of the photoemission process** can be exploited to deduce the binding energy *E*_{B} and crystal momentum *h*** k** of an electron before emission. A generic ARPES measurement involves a single crystalline sample irradiated by monochromatic light of energy

*hν*, resulting in photoemission of electrons in all directions. A fraction of these electrons is collected by a photoemission spectrometer and analyzed.

How do we translate data collected by the spectrometer back into photoemission kinematics? The spectrometer records the kinetic energy *E*_{kin} and emission angle (*ϑ*, *φ*) of each detected electron. (*ϑ* is the electron's polar angle with respect to the surface normal, and *φ* is its azimuthal angle typically defined with respect to the experimental geometry or crystal axis.) By imposing energy and momentum conservation, electronic states before (*k*_{//}) and after emission (*E*_{kin}, *ϑ*) can be related,

where *k*_{//} is the crystal momentum of an electron parallel to the surface, *ϕ* the sample surface work function (not to be confused with angular *φ*), *E*_{kin} the photoelectron kinetic energy, and *E*_{B} its binding energy prior to emission. In-plane periodicity of the crystal structure is always maintained, thus *k*_{//} is conserved throughout the photoemission process (modulo an in-plane reciprocal lattice vector *G*_{//}). While the orthogonal component *k*_{⊥} is not conserved during transmission through the surface, but can be deduced if certain assumptions hold.

The energetics of the photoemission process are depicted in the figure below. A technical note: *E*_{kin} is defined with respect to the sample vacuum level, *E*_{vac}. Instead of directly using *E*_{kin}, which depends on *hν*, energy *E* is typically referenced with the Fermi level *E*_{F}, where *E*_{B} = *E*_{F} − *E*. In practice, the value of *E*_{kin} corresponding to *E*_{F} is calibrated by fitting a Fermi-Dirac distribution to the spectrum of a polycrystalline metal placed in electrical contact with the sample so their Fermi levels align.

## Spectroscopy

Having found a way to retrieve photoemission kinematics from the spectrometer, we now ask: how can we use this to obtain information about the quantum many-body state of the material? We begin by writing many-body quantum states as ket-vectors of the form |Ψ〉. The photoemission process can be described by the **transition probability** *w _{fi}* of an

*N*-electron

*initial*state |Ψ

*〉to an*

_{i}^{N}*N*-electron excited

*final*state |Ψ

*〉under a perturbation*

_{f}^{N}*H*

_{int}. This is approximated by Fermi’s golden rule,

where *E _{N}^{i}* and

*E*are initial and final state energies. The perturbative Hamiltonian

_{N}^{f}*H*

_{int}arises from the interaction of the material with an incoming photon,

and takes after the dipole approximation. It is simply the inner product (overlap) between electron momentum * p* and the electromagnetic gauge potential

*of the incident photon. At this point, the astute reader may spy scenarios where this overlap is diminished or forbidden entirely — this leads to*

**A****matrix element effects**which have to be accounted for. This effect can instead be exploited, such as in multi-orbital iron-based superconductors or topological insulators, to selectively photoemit from orbitals or bands with particular electron momenta

*.*

**p**The transition probability *w _{fi}* is expressed using

*N*-electron states, yet photoemission clearly removes electrons from the sample, which are later detected by the spectrometer. How, then, shall we express the final

*N*– 1 electron state of the material after electron removal?

### Non-interacting case

Recall that the non-interacting limit corresponds to the simple behaviour of independent electrons treated by the venerable Bloch description. We are lucky here, for the *N*-electron initial and final states can both be trivially factorized into one-electron and *N *– 1 electron states,

where |Φ* _{i/f }^{k}*〉represents one-electron states, with a tensor product taken with |Ψ

_{i/}

*〉, which are*

_{f }^{N-1}*N*– 1 electron states. The antisymmetrizing operator (

*A*) imposes Pauli exclusion. The non-interacting limit allows for a dramatic simplification: |Ψ

*〉= |Ψ*

_{i}^{N-1}*〉. This is because the rest of the electrons, being non-interacting, do not respond to the removal of the single photoelectron. Denoting energies of initial and photoelectrons as*

_{f}^{N-1}*ϵ*and

_{k}*ϵ*, we can express photocurrent

_{f}*I*as a function of purely one-electron states |Φ

*〉.*

_{i/f }^{k}*M*are the matrix elements responsible for the overlap between these states and the electromagnetic perturbation

_{f,i}^{k}*H*

_{int}.

### Interacting case

All real materials, of course, have interacting electrons. Many-body initial and final states cannot be trivially factorized, as was done in the non-interacting state. The *N* – 1 electron system can no longer be regarded as unchanged under electron removal; that is, other electrons really do notice that the photoelectron has been removed. Nevertheless, to construct a relatively tractable formalism, we cautiously adopt the **sudden approximation,** where the photoelectron is removed sufficiently quickly and the *N* – 1 final electron state is left in a number of excited states. (One is reminded of the chemist's Frank-Condon principle, except we work with macroscopic materials rather than single molecules.) The total transition probability, and thus photocurrent *I*, is proportional to the sum over these excited states,

where the one-electron dipole matrix element *M _{f,i}^{k}* was defined in the non-interacting case. The

**Green’s function formalism**enables us to ascribe meaning to the photocurrent. It states that the total photocurrent

*I*, a function of momentum

**and frequency**

*k**ω*, is proportional to the Fermi-Dirac distribution

*f*(

**,**

*k**ω*), and — the linchpin of ARPES — the

**single-particle spectral function**

*A*(

**,**

*k**ω*):

## Many-Body Physics

What is the significance of the spectral function? It contains information not only on the material's bandstructure (the physics of which had propelled the semiconductor revolution of yesteryear), but also many-body effects, namely Coulomb and spin interactions of electrons with themselves and with the lattice. All interactions are taken into account via the **self-energy** Σ(** k**,

*ω*), which enters the spectral function:

In the non-interacting case Σ(** k**,

*ω*) = 0. The spectral function is merely a series of delta peaks that traces out the bandstructure

*ϵ*of a material. Once interactions are present, the real component Σ

_{k}*'*(

**,**

*k**ω*) of self-energy shifts band energies

*ϵ*, and its imaginary component Σ

_{k}*''*(

**,**

*k**ω*) broadens delta peaks into Lorentzian curves:

At this point one has waded ankle-deep into many-body physics, and stands by its shore with eyes set upon a vast ocean of unsolved problems. Let's introduce two of these problems.

The curious or interrogative reader asks: when the interaction strength is increased, is there a point beyond which the behavior of an electron in the material qualitatively changes, and it ceases to be a quasiparticle? The answer to this question is made apparent by re-expressing the spectral function in the approximate form, with *ϵ_{k}* + Σ

*'*(

**,**

*k**ω*) →

*ϵ*and Σ

_{k}*''*(

**,**

*k**ω*) → Γ

*,*

_{k}where the first term (the Lorentzian peak) is scaled down by a **coherence factor**, 0 < *Z*_{k} < 1. In order to conserve spectral weight (that is, the spectral function must integrate to unity), the rest of its weight is transferred to an incoherent component *A*_{inc} lacking peak structure. This is shown in the previous diagram. In the **strange metal** phase featured most famously in the cuprates, the spectral function is swamped entirely by its incoherent component, and any trace of a quasiparticle peak or bandstructure is lost. *The behavior of electrons in this phase is thus fundamentally different from those of non-interacting or weakly-interacting electrons*.

The microscopic origin of strange metallic behaviour is an open problem. This phase was previously associated with **quantum criticality**, where a change in a material parameter such as doping induces a phase transition, a sharp change in its properties, at zero temperature We have (Su-Di Chen *et al.*) recently performed a systematic ARPES study that suggested otherwise. Its existence was instead closely tied to another cuprate phase, the **pseudogap**.

What, then, gives the pseudogap its name? We turn to another longstanding challenge in many-body physics: understanding the origins and mechanisms of **unconventional superconductivity**. These superconductors, first discovered in 1986, are famous for having high transition temperatures in certain systems, promising a new revolution in energy transfer. Some superconductors also take on a topological character, potentially underpinning future implementations of quantum computing. *Elucidating the physics of unconventional superconductivity is a central research focus of the Shen group*.

The onset of superconductivity below a material's transition temperature (*T*_{c}) produces an 'anomalous' counterpart Φ(** k**,

*ω*) to self-energy Σ(

**,**

*k**ω*), associated with the formation of the superconducting condensate. Its main effect is to force open a gap in the bandstructure across the Fermi level, the

**superconducting gap**. The superconducting gap closes above

*T*

_{c}, and this is visible in high-resolution ARPES measurements of electronic bandstructure. In the cuprates, a similar gap, which we call the

*pseudo*gap, does not close above

*T*

_{c}but remains open above it and in the absence of superconductivity. The origins of this immensely quixotic gap are unclear, but it is categorically not a superconducting gap — hence its moniker — and it is crucial to advancing our understanding of unconventional superconductivity in the cuprates.

The nature of unconventional superconductivity in the other great family of high-temperature superconductors, the iron-based superconductors (FeSCs), also remains unresolved. By looking at its *interaction with adjacent phases* in the phase diagram using ARPES, such as the spin-density wave, we have (M. Yi *et al*.) ruled out selected scenarios for the symmetry of the superconducting gap of a species of FeSC, and provided valuable insights about its origin. **Competition and cooperation ****between different phases** — superconductivity, the pseudogap, strange metallic behavior, and spin-density waves, for starters — is a recurring theme in our research. Investigating them, especially using ARPES, teaches us a great deal about the fundamental physics behind the behavior of quantum materials.

A great deal more can be said about the many systems probed by ARPES than can fit on this page, and the unsatiated reader is directed to our section on quantum materials. The two landmark studies mentioned above were conducted by former graduate students in our group. Perhaps you shall choose to join us, and make your own contribution to advancing our understanding of quantum materials.

## Modern Light Sources

What light sources are most suitable for ARPES experiments? We would ideally like high photon energy to access high momenta and bands with high binding energy. On the other hand, we may prefer low photon energy for high momentum resolution. A tradeoff between range and resolution has to be contended with. In addition, intensity, flux, and polarization all play important roles in the quality of the final ARPES spectrum.

These guidelines summarize an ideal light source for ARPES:

- Sufficiently high photon energy ( > 10 eV) to access the first Brillouin zone and probe valence bands;
- Sufficiently low photon energy ( < 20 eV) to facilitate high momentum resolution;
- Sufficiently high repetition rate ( >> 100 kHz) to limit the loss of energy and momentum resolutions due to the space-charge effect (repulsion between photoemitted electrons) while maintaining sufficiently high signal-to-noise ratio;
- Sub-meV bandwidth, high flux, and long term stability;
- Variable polarization for control over photoemission matrix elements;
- Additional requirements, such as a small beamspot ( < 1 nm) and short pulse duration ( < 100 ps) for time-of-flight detection schemes.

Some of these requirements are unfortunately intrinsically incompatible with each other. We must therefore use different light sources, each fulfilling a range of requirements at the expense of others. In the low-energy limit, we introduce modern laser technique to achieve high energy- and momentum-resolution. On the other limit, we may use high-energy synchrotron sources to significantly expand the momentum space ARPES can probe.

## Photoelectron Spectrometer

A photoelectron spectrometer uses electrostatic elements to manipulate the trajectory and energy of electrons, and impinge them upon a detector. Modern spectrometers feature **lensing elements** that can be operated to record either the angular or spatial distribution of electrons. 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 reconstruct the electron single-particle spectral function prior to emission. It typically features 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. The MCP output is usually impinged onto phosphor, from which the resulting luminescence can be read into a computer using a CCD camera.

There are many commonly used photoelectron spectrometers, but we mainly adopt **hemispherical analyzers** in our group. The hemisphere analyzer consists of two concentric hemispherical electrodes set at different electrostatic potentials, producing a radial electric field that forces incoming electrons into elliptical orbits. Electrons with different kinetic energies are thus dispersed along the radial dimension when detected. Energy resolutions at an order of 1 meV are routinely obtained, although under pristine conditions <100 μeV may be attainable. At the same time, their positions orthogonal to this axis are determined by its emission angle, within the window accepted by the slit. Typical acceptance angles are 15° with resolutions down to 0.1°. The detector therefore records the two-dimensional photocurrent distribution as a function of (*E*_{kin}, *ϑ*_{y}).

## Scientific Impact

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. At the same time, first-principles theoretical calculations and the analysis of model Hamiltonians have developed rapidly, greatly adding value to the scientific interpretation of ARPES experiments. In strongly correlated materials (especially cuprate and iron-based superconductors), ARPES has enabled detailed investigation of Fermi surfaces, order parameters, and mode-couplings, in phases ranging from conventional Fermi liquids to spin-ordered, charge-ordered, or superconducting states. In topological materials, it has directly visualized the bulk electronic structure responsible for non-trivial topology, as well as its associated boundary states, particularly in cases where they are inaccessible to transport probes. Advanced synthesis techniques have made it possible to explore these materials in lower dimensions, unlocking interactions and phenomena absent in their three-dimensional counterparts. These scientific pursuits have been accelerated by increasingly sophisticated ARPES measurement capabilities with spatial, spin-, and time-resolution, all the while pushing to access lower temperatures.

## 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)