Theoretical spectroscopy: unravelling electronic correlations for materials design

Matteo Gatti

1 Spectroscopy and the “music” of materials

As our eyes need light to see shapes and colours, spectroscopy can tell us about the behaviour of electrons in materials, disclosing the extremely rich variety of their properties. In a spectroscopy experiment, a sample is perturbed by an external probe, such as a photon or an electron. In the standard setup, the sample is initially in its ground state and the perturbation is weak enough so that the sample is not altered by the experiment. As a result of the interaction with the probe, the sample gets excited, reacting to the external perturbation: its response can be measured, delivering precious information about its characteristic excitations. The outcome of the experiment usually has the form of a spectrum: a collection of peaks, encoding the possible excitation energies of the material, which defines its properties. Quantum mechanics gives us the keys to decode the measured spectra and to make the best use of the great deal of information that they carry. It allows us to explain existing materials properties and to predict new ones.

As we strike a bell to hear its sound, spectroscopy reveals the “music” of materials: a complex symphony where many different instruments are played at the same time. The goal of spectroscopy is to identify the instruments and the notes played by this orchestra. It allows us to transcribe the music into a score, and, possibly, to recombine different sounds into new symphonies that may be played again by nature.

2 Why do we care? The dream and the challenge

Understanding and mastering the interaction of light with matter is a crucial step in order to chase the dream of designing new materials with tailored properties. This holds true both for traditional spectroscopies, where new materials are interrogated by light beams on their properties, and for advanced experiments, where strong coupling with light is used to steer materials to provide properties on demand, or even to realise new states of matter.

The usefulness of the research domain on the electronic structure of materials can be hardly overestimated. New materials are more and more demanded to better perform specific functions in a very large variety of technological domains. For example, materials advances are urgently needed to obtain more efficient solar cells for energy conversion and storage, or to build faster and less consuming optoelectronic devices. Indeed, materials discovery plays a key role to attain the Sustainable Development Goals of the United Nations’ 2030 Agenda, and the International Year of Basic Sciences for Sustainable Development that we are celebrating precisely highlights the crucial importance of fundamental science to respond to humankind’s needs.

On the other hand, the theoretical description of excited states of matter, as those measured by spectroscopy experiments, is rooted into the search for effective solutions of a quantum many-electron problem, which poses a great intellectual challenge on its own. The electronic structure of materials is the result of the interactions among many particles. In principle, the origin of the extremely rich variety of materials properties is rather simple, and well known: essentially, just atoms from the periodic table, located at different positions, and the Coulomb interaction. Like the DNA for the human genome, those remarkably simple ingredients represent the cornerstone of the “materials genome”. However, even the idealised picture of solids as perfect crystals at 0 K (made of positively charged, massive ions at rest and interacting electrons wandering in the crystalline field) represents a quantum many-body problem: a prominent example of a complex system. Indeed, two important aspects of interacting many-electron systems are emergence and high sensitivity to control parameters, such as temperature, pressure, or doping. The fact that a water molecule is different from the O and H atoms it is built of, or that the interface between two insulators can be metallic, and that two sheets of graphene stacked together at a slightly twisted angle can be superconducting, are three simple illustrations of the basic principle that the whole is different from the sum of its parts. Collective behaviours, phase transitions with drastic changes, or spontaneous symmetry breakings, are all noteworthy examples of the manifestation of the complexity of interacting many-electron systems. This implies that, in order to describe, explain, and predict materials properties, very often the centre of interest needs to be shifted from the characteristics of the elementary constituents to the role of the inter-relations among them. One can even imagine to exploit the effect itself of the interaction between particles in order to devise new functional materials. For example, excitonics is a new research field that aims to replace electrons (or holes) with excitons, i.e. interacting electron-hole pairs, as more efficient carriers to realise faster and more compact optoelectronic devices.

Spectroscopy investigations of many-electron systems combine the question of unravelling the effects of interactions together with the need for an accurate description of the physical process at the basis of a specific experiment. The rest of this contribution will propose some thoughts on possible strategies to address the interacting many-electron problem (underlining concepts that can be shared also in other contexts), and will focus on the prototypical example of the signatures of electron correlation in photoemission spectra.

3 Strategies to find our way out of the many-body labyrinth

Quantum mechanics tells us that a system of $N$ electrons (embedded in the static potential due to the positive ions, at 0 K) is described by the wavefunction $\Psi ( x_{1}, ..., x_{N} )$, solution of an $N$-body Schrödinger equation. $\Psi$ depends conjointly on the position and spin coordinates $x_{i} = ( r_{i}, \sigma_{i} )$ of each, indistinguishable, electron: this dependence cannot be factorised in such a way that $\Psi$ would depend separately on each of them ($\Psi$ could thus take the form of a Slater determinant, as in the Hartree-Fock approximation). In other words, as a result of the Coulomb interaction, electrons are correlated. In an extended system when $N$ is large, the direct calculation and the storage of the whole $\Psi$ is impractical (or even impossible). However, most importantly, at the end this is not even needed. Indeed, we are usually interested in observables, which are obtained as expectation values of specific operators. Mathematically, observables are the result of integration over most of the degrees of freedom of $\Psi$: its detailed information over the $N$ electrons is traced out. Powerful Quantum Monte Carlo (QMC) methods show that a stochastic evaluation of these complex integrals can be indeed performed with great accuracy.

Much effort in modern theory of many-electron systems is devoted to rewriting basic quantum mechanics equations in such a way that one can calculate only what is needed, allowing at the same time for a much clearer interpretation of the results. A successful strategy followed by many approaches is to introduce alternative (often called auxiliary) $N$-body systems, with effective particles and effective interactions between them (and sometimes with no interaction at all). This choice also represents an important paradigm shift. We move from the difficult search of approximate solutions of a known, very complicated, Schrödinger equation to the (not always easy) development of approximations of (in practice) unknown equations that are simpler and can be exactly (or almost exactly) solved.

The massive success of density functional theory (DFT), with thousands of applications in very different domains, is a clear demonstration that this is indeed a rewarding strategy. DFT is an exact reformulation of quantum mechanics, and another many-body theory of a collective variable: the ground-state electron density $n ( r )$. DFT takes a synthetic point of view and can be also called a minimum information theory, as it identifies $n ( r )$ as the simplest quantity that in principle one needs to know in order to calculate observables. The big advantage is that, in contrast to $\Psi$, the number of arguments of the function $n ( r )$ is always only one, independently of the number $N$ of electrons. The density is therefore also called a “reduced quantity”.

In this context, the genius idea of Kohn and Sham was twofold (see fig. 1). First, they introduced an auxiliary system of non-interacting particles that, by construction, has the same density as the original interacting many-electron system. In this way, they realised a mean-field scheme that is exact for one selected observable: the density (but not for anything else). Second, they showed that the unknown effective potential in which the auxiliary Kohn-Sham electrons live can be approximated by introducing another, simpler, model system (in their case it was the homogeneous electron gas). The unknown potential is imported, thanks to a simple prescription (now we call it a “connector”), from the model where it can be evaluated much more easily. The local density approximation (LDA) of DFT was born, which has been later refined (notably by making use of known exact constraints on the objects to approximate). The great popularity of the method is due to the fact that the difficult portion of the interaction contributions has been calculated, once for all, in the model (for the LDA, by accurate QMC methods), and the model results can be reused with a simple recipe, over and over again, by any researcher, in principle for any material.

We can certainly learn from this very successful strategy, and also try to generalise it along several directions. However, despite the ingenious conceptual leap, and the spectacular progress that has been made possible, Kohn-Sham DFT also has intrinsic limitations (as anything else). Mathematically, in traditional quantum mechanics observables are (known) functionals of $\Psi$: within DFT they become different functionals in terms of $n$. However, for most observables these density functionals are not known or cannot be easily approximated. In particular, the Kohn-Sham system itself does not have a direct connection with the elementary excitations of the real many-electron system, which are the focus of theoretical spectroscopy. Therefore, very often we are still obliged to look for alternatives.

In order to keep a direct connection with excitation spectra measured by spectroscopy experiments, it is convenient to make use of dynamical (i.e., frequency-dependent) correlation functions, which are another kind of reduced quantities. Notable examples are response functions, which correlate the change of an observable with the perturbation that has caused it, and Green’s functions, which describe the propagation of additional particles interacting with all the others (Green’s functions are therefore also called propagators). In particular, the one-particle Green’s function $G$ is the propagation amplitude for one additional electron (or hole) between two space, time, and spin coordinates. In an orbital basis, it is the complex dynamical matrix $G ( \omega )$. It is a more expensive object to calculate than the density $n ( r )$. However, with respect to DFT, more observables are known as functionals of $G$.

Notably, the spectral function is simply $A ( \omega ) ( 1/\pi ) | \mathrm{Im} G ( \omega ) |$: its peaks correspond to the one-particle excitations such as electron addition and removal energies measured by photoemission spectroscopy.

As other correlation functions, the one-particle Green’s function can be obtained, in principle exactly, as a solution of a Dyson equation:

(1) $ G ( \omega ) = G_{0} ( \omega ) + G_{0} ( \omega ) \Sigma (\omega ) G ( \omega ) $

Dyson equations always share the same structure. The first term on the right-hand side is an initial approximation, which one can afford to calculate explicitly. In this case, $G_0$ can be the non-interacting Green’s function or the propagator of a quasiparticle, i.e., a particle that feels (or, as often said, “is dressed by”) the interactions with the other electrons. The kernel in the second term of the equation, in this case the complex and dynamical self-energy $\Sigma ( \omega )$, takes into account the remaining couplings with the rest of the many-electron system: it is generally unknown and has to be approximated. The kernel modifies the one-particle excitations in $G$ with respect to the initial $G_0$, and the non-linearity of the equation can also change their number. The important result is that the many-electron system is recast into a set of elementary excitations and couplings between them.

As in other computational schemes, differently known as embedding, partitioning or downfolding, where one is interested only in a part of a system and treats the remainder in an effective way, the frequency dependence of $\Sigma ( \omega ) $ replaces the need to deal explicitly with the many-electron excitations, when the target are only the one-electron excitations contained in $G$. Thus, by analogy with the Kohn-Sham scheme, the approximation strategy based on Dyson equations can be also understood to lead to auxiliary systems with dynamical and complex effective potentials (or kernels) describing couplings between effective particles or excitations.

An important advantage of the structure of Dyson equations is that eq. (1) can be equivalently rewritten as an infinite series:

(2) $ G ( \omega ) = G_{0} ( \omega ) + G_{0} ( \omega ) \Sigma ( \omega ) G_{0} ( \omega ) + G_{0} ( \omega ) \Sigma ( \omega ) G_{0} ( \omega ) \Sigma ( \omega ) G_{0} ( \omega ) +$ ...

This expansion shows that it is generally advantageous to approximate the kernel of the Dyson equation with respect to a direct approximation of the target correlation function itself: through the infinite resummation in eq. (2), a low-order approximation of the kernel can produce a higher-level correlation function. Alternative expansions to Dyson equations are also possible that share the same resummation spirit: for example, in the cumulant expansion approximation, $G ( \tau )$ (which is the Fourier transform from $\omega$ to time difference $\tau$) has an exponential representation $G ( \tau ) = G_{0} ( \tau ) e^{C ( \tau )}$, where approximations are then done on the cumulant function $C ( \tau )$.

Different approaches are followed to approximate the kernels of Dyson equations. For example, in dynamical mean field theory, similarly to the LDA, the self-energy is imported from a model (in this case, an Anderson impurity model). In approximations of many-body perturbation theory, instead, a selection of the most important couplings is made, on the basis of some physical criterion. In the following, we will see how the dielectric screening of the Coulomb interaction is the key physical mechanism to explain the excitation spectra of many extended materials, the simple metal aluminium will be their representative example.

4 A theorist’s view on photoemission

In a photoemission experiment (see fig. 2), one removes (or adds, in the case of inverse photoemission) one electron from a many-electron system and observes the reaction of all other electrons. The measurement of the electron removal (and addition) energies determines the electronic structure of materials and can reveal the effects of the Coulomb interaction between the electrons.

If one takes a book from a well stocked bookshelf, the other books would not move much. Similarly, without Coulomb interaction, the electron removal energy would directly correspond to the single-particle energy level that the removed electron was occupying in the many-electron system, with the other electrons not perceiving its absence (see fig. 2, middle panel). If, instead, one removes a book from the bottom of a disordered pile, the other books could fall down, and the pile could even collapse. Similarly, taking into account the Coulomb interaction, the removal of an electron creates a hole with a positive charge that induces a rearrangement of all other electrons (see fig. 2, right panel). As a result, the measured electron removal energy will be different with respect to the non-interacting case. This polarisation effect, which can be visualised as the creation of multiple electron-hole pairs (also called a “polarisation cloud”), suggests the definition of a quasiparticle: a hole together with the rearrangement of the surrounding electrons. Quasiparticles are said to be “renormalised”: their properties are different from “bare” particles and their interaction is also weaker, as the electron polarisation screens the Coulomb interaction.

The celebrated Hedin’s GW approximation (GWA) to the self-energy corrects the Hartree-Fock approximation (so it includes correlation effects) by precisely taking into account the dielectric screening in materials. The GWA replaces the Coulomb interaction $v$ by the dynamically screened Coulomb $W ( \omega ) = v+v\chi ( \omega ) v$ where $\chi ( \omega )$ is the density response function. The resulting self-energy is $\Sigma = iGW$, whence its name. For the rest, the GWA makes two main approximations: it assumes that the polarisation is made of non-interacting electron-hole pairs ($\chi ( \omega )$ is calculated at the level of the so-called random-phase approximation) and treats the interaction between the additional charge and the polarization charge at the classical (Hartree) level (so, for example, it neglects coupling with spin excitations).

Whereas Kohn-Sham particles are not individually measurable, quasiparticles have a direct link with experiments, which allows for physical insight and facilitates the development of approximations. The GWA is nowadays the state-of-the-art approximation to calculate quasiparticle properties in a wide range of materials, where dielectric screening is the key physical process. In particular, quasiparticle energies define the band structures of crystalline materials. They are the main peaks in angle-resolved photoemission (ARPES) spectra, where they are measured as a function of their wavevector. As an example, fig. 3 (left panel) shows the measured ARPES spectra of aluminium. The sharp peaks dispersing between ~ –11 eV binding energy and the Fermi energy, set at 0 eV, correspond to the parabolic valence band of aluminium.

Calculations in materials science are very often limited to these quasiparticle band structures. However, quasiparticles are not the only features in the one-particle excitation spectra. A hole in the many-electron system acts as a perturbation that can also induce additional excitations. Notable examples are plasmons (or magnons), collective excitations of the electronic charge (or spin). Therefore, photoemission spectroscopy is like watching a drop falling on a surface of water, when concentric ripples are formed on the surface around the falling point of the drop. Indeed, in addition to quasiparticle peaks, photoemission spectra in fig. 3 (left panel) also display satellites (or “side-bands”), which have an energy distance from the quasiparticle band equal to a multiple of ~15 eV, i.e. the energy needed to create a plasmon excitation in aluminium.

While plasmon satellites in core level spectra have been known for a long time, for the valence excitations measured by ARPES, satellites are much less explored. To describe these plasmon satellites in the spectral function $A ( \omega )$, including their shape dispersion, recently the GWA to the self-energy has been successfully combined with the cumulant expansion of the Green’s function in a wide range of materials, from simple metals to transition-metal oxides like SrVO3 or TiO2.

Plasmon satellites are genuine many-body effects that cannot, by definition, be interpreted within a quasiparticle picture and, therefore, carry information complementary to the band structure, featuring distinct length- and time-scales from the quasiparticles. They are the result of the dynamical screening of the Coulomb interaction and, in fact, a dynamical kernel is needed in order to produce satellites in the spectral function. Therefore, satellites reflect the strength of electronic correlation in materials. Here, we have reached the important conclusion that these dynamical correlation effects can be understood as a coupling of excitations, in this case the coupling between a hole and a collective plasmon excitation (similar photoemission satellites can be also due to coupling with other neutral excitations like excitons, magnons, phonons, photons, etc.).

Moreover, fig. 4 shows that the calculated spectral function is actually only a small part of a photoemission spectrum. This clearly exemplifies the need to calculate what is really measured in experiments in order to obtain deeper understanding. Besides capturing correlation effects in the spectral function, an accurate description of the photoemission event itself is also necessary. This is, in part, still an open problem, and here we can delineate a few of its sides. First of all, while photoemission spectra strongly depend on the energy of the photon impinging on the sample, spectral functions, by definition, do not. In the latter, one missing piece of information is the probability for an electron (called the “photoelectron”) to escape from the sample (in practice, it is an optical transition that can be calculated as a matrix element of the photon-matter interaction). Moreover, the photoelectron can also inelastically scatter on its way out from the sample, losing a fraction of its energy. In its interactions with other electrons, it can induce additional plasmons, which also contribute to the plasmon satellites in the measured spectra. These additional contributions are called “extrinsic” effects, to distinguish them from the “intrinsic” plasmon satellites in the spectral function. And, since intrinsic and extrinsic processes are quantum phenomena, there are also “interference” effects between them. Another important aspect in the measured spectra in fig. 3 and fig. 4 is the thermal motion of the atoms (in the so-called Debye-Waller effect), which acts against the wavevector selectivity of ARPES, giving a contribution to the spectra that is integrated over all the wavevectors.

Only by taking into account the effects of the interactions, at the same time, in the spectral function and in the photoemission process, it has been possible to reach a quantitative agreement between experiment and theory (see fig. 3 right panel), even for the simple case of aluminium. The outcome has been the suggestion for a strategy to extract from the spectra precise information about correlation strength in materials and the detection of a new kind of satellite that is due both to electron-electron interaction and the atomic vibrations. This also shows, once more, the crucial importance of collaboration (in this case between theory and experiment), combining complementary competences in order to solve problems.

5 Conclusions and outlook

We have taken here a short exploration journey through the fascinating many-electron world. Electronic interactions make many-electron systems intriguing and challenging at the same time. They influence excitation spectra in quantitative and often even qualitative ways. In order to make full use of the information in the measured spectra, one needs both to accurately describe the interaction between the probe and the sample, and to be able to interpret signatures of electronic correlation in the spectra.

We have outlined some central concepts and strategies in many-body theories, and we have used a simple example to illustrate how a tight interplay between theory and experiment is essential to reach a deeper understanding. Those considerations can be similarly extended to other materials, spectroscopies, and contexts.

Theory has a very important role to play when it bridges different experiments and creates new synergies between experimentalists. For example, the view of dynamical correlation as the coupling between different excitations suggests the idea to study them separately. Referring again to our illustrative example, plasmon excitations that characterise the dielectric screening of materials can be accurately investigated by electron energy loss and inelastic X-ray scattering spectroscopies, which can be efficiently coupled with information from photoemission. Plasmons have their own interest as central objects in the research field of plasmonics. Moreover, as in a virtuous circle, energy loss spectra themselves feature dynamical correlation signatures such as double plasmons and double excitations (or multiplets) that pose new challenges to theory.

New discoveries and technological progress continuously open up new horizons, indicating new goals to reach and suggesting new roads to explore. The list of examples here can be endless: the development of extreme brilliant (fourth-generation) synchrotron sources is now allowing for improved spatial resolution and better exploitation of coherent phenomena; free-electron lasers can be now used for the investigation of ultrafast phenomena at the nanoscale; efficient artificial intelligence algorithms can be combined with physical insight for the design of better approximations and the discovery of novel materials; the advent of quantum computing will offer a revolutionary platform to benefit from the development of functional theories, and so on. All this just signifies that interacting many-electron systems will certainly continue to keep us busy for a very long time, in an intellectual endeavour that is constantly renewed.


The present article is based on a contribution presented at the session “Fisica della materia” organised by Matteo Paris at the 107th Congress of the Società Italiana di Fisica. Many stimulating discussions with members and friends of the Palaiseau Theoretical Spectroscopy Group are gratefully acknowledged.