Extensive Study of Crystal Vibrations

Extensive Study of Crystal Vibrations
Comprehensive Study of Crystal Vibrations - J. Berges/University of Bremen

A new set of equations that underpin computer research captures the dynamic interaction of electrons and vibrations in crystals.

Although a crystal is a highly ordered structure, it is never at rest because its atoms are constantly bouncing around their equilibrium positions, even at absolute zero temperature. These vibrations, known as phonons, interact with the electrons that hold the crystal together and contribute to its optical qualities, its capacity to conduct heat or electricity, and even to its vanishing electrical resistance if it is a superconductor. It is these interactions that give crystals their unique optical properties. An accurate description of the interaction between electrons and phonons is necessary to predict or at least understand such properties.

This effort is enormous, given the difficulty of the electrical problem, which assumes that atomic nuclei remain stationary, and the lack of an exact solution. By building on a long list of past achievements, researchers have managed to take an important step towards a comprehensive theory of electrons and phonons.

The electron-phonon dilemma is formulated simply at a basic theoretical level. First, one begins by considering a configuration of enormous point charges that stand in for electrons and atomic nuclei. Secondly, the Schrödinger equation and Coulomb's law are allowed to govern how these charges evolve, with the possibility of intermittently adding a perturbation. The Hamiltonian of such a system is the mathematical description of its energy, consisting of kinetic and interaction factors.

However, understanding the exact theory is insufficient, as the equations involved are only formally simple. In practice, the approximations are necessary because they are so complex – not least because so many particles are involved. Therefore, a high-level, practical theory must provide the means to produce approximations that are plausible and result in equations that can be solved by modern computers.

Removing the focus from the image of individual particles in favor of an effective quasiparticle specific to the system at hand may help simplify the problem. A classic example of a quasiparticle is the phonon, which considers the collective vibrations of its constituent atomic nuclei about their location in a predetermined crystal structure, rather than the individual vibrations of the nuclei that could theoretically be located anywhere in space. For almost a century, researchers have studied these “elastic waves,” often using two well-known approaches: the Born-Oppenheimer approach, which assumes that electrons respond instantaneously to displacements of nuclei, and the harmonic approach, which assumes that this response produces restoring forces proportional to the displacements.

Research conducted in the mid-20th century using the methods of quantum field theory to study the interaction of quasiparticles forms the basis of the work of Stefanucci and his colleagues. Gordon Baym published a theory about electrons and phonons in 1961; Accordingly, the phonon field gave a displacement to places in space and time. Feynman diagrams, which graphically depict interaction processes and can be converted into mathematical formulas using simple rules, are one of the tools discussed above. All potential processes taking place in physical reality can be described by combining such diagrams into recursively interconnected sets of equations. Such equations that fully describe systems of interacting electrons were first demonstrated by Lars Hedin in 1965.

In a study he conducted in 2017, Feliciano Giustino combined these methods and created the Hedin-Baym equations in the context of state-of-the-art material simulations and answered many unanswered questions.

Many of the ongoing issues have been addressed by Stefanucci et al. First, they placed constraints on the electron-phonon Hamiltonian to avoid the mistake of trying to solve a problem that had never been properly addressed. They emphasized that the construction and evaluation of the Hamiltonian is an iterative process because the equilibrium state in which the theory is developed is not known in advance. They also emphasized that, contrary to popular belief, this Hamiltonian cannot typically be expressed in terms of physical phonons. Second, the researchers extended Giustino's work to systems that are forced out of equilibrium at any temperature. This was an important development because it reflected both technological and experimental conditions. This generalization allows time to take on mathematically complex values.

Third, researchers produced the first comprehensive collection of diagrammatic Hedin-Baym equations and associated rules for Feynman diagrams. These equations form the basis for systematic approaches that neglect some diagrams, and also provide a test for the adherence of the resulting dynamics to fundamental conservation rules. Although the effects of electrons on phonons and vice versa have been extensively investigated separately, in this case it is essential that both occur simultaneously.

Today, Born-Oppenheimer and harmonic approaches, which form the basis of the theory called density functional perturbation theory, are extensively used in parameter-free simulations of electrons and phonons. On the other hand, parameterized model Hamiltonians are often—but not always—used in conjunction with diagrammatic approaches. Currently available doubling-down methods for the electron–phonon problem are the result of attempts to combine both approaches.

The information collected by Stefanucci and colleagues will undoubtedly help build further bridges between the various approaches. Advances beyond thermal stability will also be crucial because they are necessary to elucidate cutting-edge time-resolved spectroscopic research and develop more effective photovoltaics. Finally, since the team's findings are applicable to any fermion-boson system, such as an interacting light-matter system, many industries will benefit from this groundbreaking discovery.

Source: physics aps.org/


📩 12/09/2023 15:02