Photoemission of quantum materials

The emergent phenomena that characterize quantum materials have received prominent exposure thanks to experimental techniques based on photoemission. In turn, the challenges and opportunities presented by quantum materials have driven improvements in the photoemission technology itself.

Angle-resolved photoemission spectroscopy (ARPES) is an essential tool for learning about the electronic structure of quantum materials. The technique is based on the photoelectric effect and, by measuring the kinetic energy and emission angle of the electrons ejected from a solid, it is used to establish the relationship between their crystal momentum and binding energy inside the solid. More specifically, ARPES measures the occupied part of the single-particle spectral function, which contains information both about band dispersions and many-electron interactions^1,2.

Over the years, ARPES has become a widespread method to probe the inner properties of quantum materials, and it has played an important role in a number of landmark discoveries in the field. Here we survey the impact of these experiments on three important classes of quantum materials: cuprate superconductors, iron-based superconductors, and topological insulators.

Cuprates

Although ARPES experiments date back to the 1970s, it was with the advent of the cuprate high-temperature superconductors in the late 1980s that the technique found its ideal use. Famously, these materials display superconductivity at temperatures in excess of the boiling point of liquid nitrogen, yet the precise mechanism through which this superconductivity comes about remains an enigma. Nevertheless, at each turn of the high-temperature superconductivity saga, ARPES experiments have played a central role.

One of the very first challenges in cuprates was to identify the symmetry and momentum-dependence of the superconducting gap — the lowest-lying quasiparticle excitation energy. The gap turned out to be anisotropic around the Fermi surface³ and, in what was an early triumph for the technique, ARPES measurements provided evidence for a $d_{x^2-y^2}$ superconducting pairing symmetry⁴ (Fig. 1). This was initially shown through an isotropic closing of the gap across the superconducting transition temperature, T_c, at the positions on the Fermi surface now known to be the node, where the superconducting gap is zero, and the antinode, where the superconducting gap reaches its maximum⁴. Concurrently, the sign-changing properties of the superconducting gap were demonstrated through phase-sensitive measurements⁵, and penetration-depth experiments gave further support for nodes in the gap function⁶; together, all of these experiments helped confirm $d_{x^2-y^2}$ pairing in cuprates.

Figure 1: Photoemission experiments on cuprates.

a, Initial ARPES support for d-wave superconducting gap (right), revealed by gap opening at the antinode (B) but not the node (A) below T_c = 78 K. Lower panel shows schematic of d-wave sign-changing superconducting gap along a circular Fermi surface. b, Pseudogap in slightly underdoped cuprate with T_c = 90 K, which manifests itself as spectral gap that persists well above T_c, up until T*. Panel shows spectra at antinodal k_F, which are symmetrized to visualize depletion of electronic density of states near the Fermi level in the pseudogap regime. Adapted from ref. 3, Macmillan Publishers Ltd (a); and ref. 10, Macmillan Publishers Ltd (b).

Just when it seemed that the ARPES picture of spectral gaps in cuprates was clear, the mysterious pseudogap above T_c became apparent⁷. Like d-wave superconductivity, the pseudogap is characterized by a gap that is anisotropic around the Fermi surface with maximum energy at the antinodal momentum^8,9 (Fig. 1b). Much of the debate around the origin of the pseudogap focuses on whether it is a precursor of superconductivity or a distinct electronic state. Experimental data have been interpreted to support both points of view, as well as hybrid scenarios¹⁰, and most of these arguments are made on phenomenological grounds. For example, a precursor scenario seems reasonable if one considers that the magnitude and momentum dependence of the pseudogap can be consistent with a thermally smeared superconductor¹¹, but a 'two-gap' scenario is supported by different doping and temperature dependence in different regions of the Fermi surface^12,13. The mystery of the pseudogap is not resolved to this day, but detailed momentum-dependent spectroscopic information has narrowed down the field of possibilities.

A major step-change in ARPES technology also took place in the midst of the cuprates mania: the development and adoption of two-dimensional position-sensitive electron detectors that could simultaneously measure energy against momentum along a one-dimensional trajectory in momentum space. This allowed for much more efficient data collection and more direct visualization and mapping of band dispersions.

Moreover, the improvements in energy resolution have been dramatic, going from several tens of meV down to sub-meV within 25 years¹⁴. Improvements in experimental throughput achieved thanks to brighter light sources have played an equally important role. As an illustration, successive generations of ARPES experiments looking at ostensibly the same structure — the d-wave superconducting gap, for example — provided quantitatively new information, such as fine structure in the momentum-dependence of the superconducting gap that was connected to a coexisting pseudogap^3,15.

Iron-based superconductors

The discovery of the iron-based superconductors (FeSCs) in 2008 presented an opportunity for a high-T_c do-over. Physicists could address a problem of similar importance, but with the advantage of all of the synthesis, characterization, experimental and theoretical tools that had been developed in the preceding two decades.

When a new superconductor is discovered, it is important to map out the electronic band structure in order to narrow the field of potential mechanisms giving rise to its behaviour. How many Fermi surfaces there are, their size and location within the Brillouin zone, as well as the bandwidth and band topology, can all circumscribe plausible explanations for the superconducting state. The early studies of FeSCs sought to clarify this information in collaboration with theory, which simultaneously yielded information about electronic correlations in these materials. It quickly became clear that ARPES would be a crucial tool for these materials because they have multiple Fermi surfaces originating from the d-orbitals of the iron atoms, potentially with different superconducting gap structure and different relevance to the pairing mechanism¹⁶. In an earlier era, direct experimental evidence for this complex fermiology would have come primarily from quantum oscillations, and the modern capabilities of ARPES ensure that multiple techniques can be used to ascertain this information that is critical to pinning down the mechanism of emergent phenomena.

The fermiology observed in early ARPES data showed hole pockets at the Brillouin zone centre, Γ, and electron pockets at the Brillouin zone corner, which sometimes have similar size and are separated by a (π, π) nesting vector shared by the spin density wave that defines many parent compounds in FeSCs¹⁷ (Fig. 2). This gave early support for interband pairing, possibly mediated by spin fluctuations. However, FeSCs represent a diverse set of materials, some of which lack hole pockets at Γ altogether^18,19. These outliers can imply different mechanisms, such as orbital fluctuations²⁰ or nematic fluctuations²¹.

Figure 2: Photoemission experiments on iron-based superconductors.

Complex Fermi surface structure, with multiple pockets at both Γ and M (bottom) and superconducting gap (top) in Ba_0.6K_0.4Fe₂As₂. This work reported nodeless superconducting gaps. Top-left inset shows gap closing at T_c on three bands. Adapted from ref. 22, IOP.

The superconducting gap function on all Fermi surfaces, and, specifically, the presence or absence of nodes, is something that ARPES is uniquely suited to directly measure in this multiband system, but the picture is still mixed^22,23. Some studies observe nodes, while others do not (Fig. 2), and others only observe nodes as a function of k_z. It should be noted that some of this discrepancy is consistent with an s_± pairing symmetry, which can exhibit accidental nodes from materials-dependent details of the pairing interaction²³. This variety of results is not surprising given the variation in chemistry and band structure among FeSCs, and it highlights how open questions still remain in many well-studied quantum materials.

One system that has captivated the field in particular is monolayer FeSe deposited on strontium titanate²⁴ — a clear example of the widely adopted strategy of searching for new and higher-temperature superconductors at the interface of two materials²⁵. This system has been reported to remain superconducting up to temperatures as high as 109 K, and although most experiments place T_c at ∼60 K, this is still considerably higher than the bulk T_c of 9 K at ambient pressure²⁶.

Perhaps the biggest contribution of monolayer FeSe to ARPES technology is accelerating the trend of incorporating thin film growth capability into a single instrument, thus integrating synthesis and electronic structure measurements into a tight feedback loop²⁷. One such set-up yielded ARPES data showing peculiar band replicas, which were interpreted in terms of interfacial mode coupling being responsible for enhanced T_c in the monolayer system²⁸. This potential mechanism, gleaned from measurements of momentum-space electronic structure, represents an actionable approach for engineering higher-T_c superconductors — interfacing a material with high-frequency phonons with a materials with an existing tendency towards superconductivity.

Topological insulators

Although they gained notoriety around the same time as FeSCs, topological insulators (TIs) — and their intellectual descendants that constitute the wider family of so-called Dirac materials — represent a different approach to ARPES experiments.

For high-temperature superconductors and, more broadly, correlated-electron systems, emergent phenomena are typically first identified by means of transport or thermodynamic experiments, and if these phenomena manifest themselves through changes to the electronic structure, ARPES experiments can provide clues that, in turn, inform theory. For TIs, on the other hand, theory has very much led the way, informing ARPES experiments performed soon after the initial materials' synthesis is achieved. This is partly because of the surface-sensitive nature of ARPES, which, in this case, is a big advantage that enables direct probing of the topological surface states. Indeed, an entire lineage of topological Dirac materials has been discovered since the TIs came onto the scene²⁹, all generally following the paradigm of theory driving ARPES experiments.

Bi₂Se₃, Bi₂Te₃ and related Bi-chalcogenide materials have long been known to be very good thermoelectric materials, and since 2009 they have enjoyed a new career as three-dimensional (3D) TIs. Bi₂Se₃ was shown to have the electronic structure characteristic of a topological insulator in May 2009³⁰ (Fig. 3a), followed by Bi₂Te₃ two months later³¹. Materials that are predicted to be 3D TIs have several characteristic observables in an ARPES experiment: small bulk bandgap, robust surface state with a Dirac dispersion, and spin–momentum locking. Most of these properties were reported in the initial 2009 experiments^30,31 (Fig. 3a).

Figure 3: Photoemission experiments on topological insulators.

a, ARPES spectra of Bi₂Se₃, along two high-symmetry cuts in the Brillouin zone. Sharp V-shaped dispersion is Dirac surface state (SS) and regions of broader intensity at Γ are the bulk bands. b, Time- and angle-resolved photoemission snapshots of the band structure of the topological insulator Bi₂Se₃ as it is radiated by a 160-meV, 300-fs light pulse. The coherent interaction between the time-periodic potential of the light pulse and electrons in the Dirac cone results in Floquet–Bloch states that appear as replicas of the original Dirac cone. Dynamics gaps in the electronic structure open up at positions where the replica cones intersect the original cone. The replica bands disappear once the driving field shuts off. Adapted from ref. 30, Macmillan Publishers Ltd (a); and ref. 42, Macmillan Publishers Ltd (b).

The bandgap was shown to correspond to the bulk because of its k_z dispersion, which can be accessed by varying the photon energy, while the surface state lacks this k_z dispersion. The surface state displays a linear relationship between its energy and momentum, particularly very close to the Γ point of the Brillouin zone, identifying the surface Dirac cone and Dirac point. One inconvenient truth about these bismuth chalcogenides is that they are not actually insulators — the volatility of the chalcogen makes them naturally n-type, such that the Fermi level lies above the bandgap. Still, the Dirac surface state seems to be impervious to this, and is clearly observed in this doped system. Considerable efforts have to be made to dope Bi-chalcogenides back to the charge-neutrality point, and the surface state appears largely unchanged when this is done^30,31, unless the impurities are magnetic. The fact that the most commonly studied TIs are not really bulk insulators amplified the role of ARPES in this field, as other experimental tools had more difficulty extracting quantitative information about the topological surface states from early samples.

The second characteristic of 3D TIs' electronic structure is spin–momentum locking, which manifests itself as a spin texture that winds in a circle around a constant-energy cut of the Dirac cone surface state. Spin-resolved ARPES experiments based on Mott detectors confirmed the expected spin texture of Bi₂Te₃ soon after initial band structure measurements³². Another way of discerning spin texture is to use the light helicity dependence of the matrix elements in highly spin–orbit-coupled systems. Studies using laser-based ARPES looked at the difference in photoemission intensity between right and left circularly polarized light in TIs, and found results consistent with theoretically predicted spin texture, including 3D warping³³. Finally, spin-ARPES has also been deployed using low-energy exchange scattering from a thin film ferromagnetic target, which allows direct spin detection to be integrated with a laser photoemission lightsource³⁴. Bi-chalcogenide TIs provide an excellent test bed for ongoing development of spin-resolved ARPES because of their characteristic spin texture and the large distance in momentum space between bands of opposite polarization.

Time resolution

Another important development in twenty-first-century ARPES has been the addition of time resolution. In these time-resolved ARPES experiments, a femtosecond laser pulse (the pump) is used to perturb a material from its equilibrium state, and another time-delayed ultraviolet pulse (the probe) is used to study how the material's electrons respond or decay back to equilibrium. Besides providing energy- and momentum-resolved electronic relaxation dynamics, time-resolved ARPES allows the probing of unoccupied states above the Fermi level by occupying them through photo-excitation.

The most common implementation of this experiment uses a variety of pump frequencies (anything from mid-infrared to visible) and a 6-eV probe energy for photoemission. The latter is chosen because it can be achieved through fairly routine frequency quadrupling of commercial Ti:sapphire lasers operating at 1.5 eV using nonlinear crystals. It provides excellent momentum resolution³⁵ and relatively high photon counts. The major disadvantage of 6 eV is that it has a poor cross-section for most materials and it cannot access electrons far from the Brillouin zone centre. To address this, higher harmonic generation in gases has been employed to obtain tens of eV light energies to do ARPES^36,37,38. The challenge here is to get enough photons in a small bandwidth of energy, and this can be optimized by using specialty monochromators to select and shape different harmonics of the laser. This is a rapidly evolving field that will likely shape the future of time-resolved ARPES.

Time resolution has been used to study many different systems including charge density wave materials, high-temperature superconductors and graphene³⁹. Recently, 3D TI materials have provided a particularly productive platform for these measurements by virtue of having a robust and sharp two-dimensional state very close to the Γ point that can be probed by 6-eV photons. These sorts of experiments in 3D TIs clarified dynamics of Dirac fermions including phonon-assisted surface-bulk coupling and persistent surface-state population⁴⁰.

Besides probing electronic dispersion and its dynamics, time-resolved ARPES can also be used to coherently control the electronic bands and potentially engineer new states of matter. This was recently demonstrated on the surface of a TI by using a pump pulse with energy less than the bulk bandgap^41,42. In this case, even though the pump does not have enough energy to make interband transitions, the electrons still feel a temporally periodic perturbation due to the electric field of the light. Since direct absorption and the resulting electronic scattering is suppressed, the coherent nature of the electric field of the laser pulse can be imprinted onto the electronic states. In analogy to Bloch's theorem, just as spatially periodic potential in a solid gives rise to replication of the dispersion in momentum, subjecting a TI to an oscillating electric field produces 'replicas' of the bands in energy separated by the pump photon energy (Fig. 3b). This is clearly visible when the bands in question are the sharp Dirac surface states. These bands are called Floquet–Bloch (FB) states and can be viewed as hybrid states between electrons and photons. For all practical purposes, FB states behave like real bands. Similar to gaps at the zone edges of Bloch states in solids, hybridization gaps open up at the crossing points of the FB states, indicating their fully coherent nature.

The FB bands offer a new and exciting way for engineering band structure of materials. Traditionally, one needs to play with the chemistry of the material to modify the electronic band structure. Since FB states are hybrid states of electrons and photons, one can simply change the properties of light (such as intensity, frequency or polarization) to change the band structure. In the case of the TI, for example, it was shown that by making the pump light circularly polarized, one can break the time-reversal symmetry and open up a gap at the Dirac point^41,42, making the massless Dirac fermions massive. In this case, circularly polarized light mimics the application of a magnetic field, which is normally not compatible with ARPES experiments. The ultimate goal here is to use similar approaches to realize new phases of materials or switch between existent ones, such as converting trivial insulators to TIs by simply shining light onto them⁴³.

Conclusion and outlook

ARPES has been a critical tool for learning about the electronic structure of quantum materials, even before the category was coined. At every stage, ARPES has permitted direct access to electronic information, with momentum-space specificity or with the ability to discern surface from bulk. At the same time, the challenges of every new prominent quantum material drove developments and augmentations to the technology that are subsequently applicable to myriad other materials, showing how a powerful experimental technique can connect many seemingly different materials under one intellectual umbrella. Cuprate high-temperature superconductors propelled ARPES to its mature, modern implementation, which allowed it to play a prominent role right from the start of FeSCs. Just as cuprates, and specifically Bi-based cuprates, served as the canonical systems for conventional ARPES experiments, 3D TIs later became the type-1a supernova of spin- and time-resolved ARPES. Taken as a whole, this ability to study, and increasingly manipulate, the electronic structure of a wide variety of materials, together with the support of theory and a variety of synthesis and characterization methods, pushes us closer to the goal of materials by design and — more exciting from a physics standpoint — the possibility of designing new emergent phenomena in quantum materials⁴⁴.