An elegant solution to a quantum problem
The theory of electromagnetic polarisation has been under development for more than 150 years since British scientist Michael Faraday provided the first experimental evidence of the interaction between electricity, magnetism and light. An EU-funded project is now helping to fit together the missing pieces of the puzzle.
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Led by Marie Curie fellow Irina Lebedeva at the Universidad del País Vasco in Spain, the RESPSPATDISP initiative has conducted cutting-edge research to develop a theoretical framework for defining and predicting how solid materials respond to electromagnetic perturbations at the quantum scale. The work not only contributes to efforts to complete the modern theory of electromagnetic polarisation, but will stimulate further research in fields such as materials science, condensed matter physics, nanotechnology and bioscience.
Recent developments in materials science, for example, have led to new materials with unusual properties, such as graphene, which consists of a two-dimensional monolayer of carbon atoms.
When a group of isolated atoms form a solid, some of the electrons move to a different quantum state, which allows the solid to hold together. These electrons are central to the properties of the solid and so determining this electronic structure is key to understanding a materials properties and these properties can have unusual and useful effects when exposed to light, electricity or magnetism.
Graphene and other new materials such as metal oxide nano-sheets have been shown experimentally to demonstrate extreme Faraday Rotation, a magneto-optical phenomenon that rotates the polarisation of light and which can be exploited for advanced optical communications and ultra-efficient memory technologies.
These can be, used in an array of ever-shrinking electronic devices found in our homes, vehicles and pockets. To understand this behaviour and to be able to predict new magneto-optical phenomena that can occur in these and other materials a tool for their theoretical description is needed.
For single molecules, i.e. finite systems, there is already a well-established theory of responses to magneto-electric fields based on perturbation theory, which enables complex systems to be studied based on knowledge of simpler ones by gradually adding electromagnetic perturbations and measuring the effects, Lebedeva explains. However, the problem arises when we try to consider systems composed of a large number of atoms, for example materials such as graphene, under periodic boundary conditions: imagine a crystal as an infinite number of equal unit cells multiplied in all directions and electronic states as periodic waves and it becomes unclear how to define the positions of the electrons.
One solution to this problem would be to consider the crystal as a very big but finite system, calculate its surface and use localised quantum wave functions of electrons to describe its electronic states, essentially extending perturbation theory from a finite to a periodic system.
A tool to predict magneto-optical phenomena
Lebedeva, however, opted for a more elegant solution that allows the material to be characterised purely within its existing periodic state. Instead of looking at the wave functions of single electrons, she applied a density matrix, a mathematical function to describe a quantum system in a mixed state. Characterising the system under external electromagnetic fields therefore does not depend on the position of a single electron, but rather on the differences in positions between electrons. This also overcomes the problem of wave functions changing dramatically over time in the presence of a varying magnetic field due to rapid alterations in the trajectory of electrons.
We have demonstrated that the density matrix approach, which has previously been applied to describe responses to static electric and magnetic fields, can be naturally extended to varying time-dependent fields. In fact, it turns out that this approach can serve as a unified framework for assessing responses to arbitrary electromagnetic fields for any type of system, both periodic and finite, potentially providing the basis for a full theory of material response to electromagnetic fields, Lebedeva says.
The RESPSPATDISP procedures were extensively tested on molecules and solids under periodic boundary conditions, including widely used semiconductors such as silicon and germanium, and the results were compared to both available literature and standard characterisation processes.
With the reliability of the procedures validated, the equation has been used to develop a code for the calculation of the magneto-optical response of both periodic and finite systems. The code has been incorporated into the open-source software package Octopus, which provides scientists with essential tools to perform density functional theory calculations.
Octopus is an ideal framework for implementing our developments, making them available to the wider scientific community and ensuring ongoing research, according to Lebedeva, who is currently working on refining the code to reduce computational requirements.
Lebedeva says that receiving the Marie Curie fellowship greatly advanced her career; she is already applying knowledge gained in RESPSPATDISP to practical applications. These include software to model the stability of organic phosphorescent light-emitting diodes which can greatly improve the energy efficiency of electronic devices and lighting systems. This work is also the subject of a new EU-funded project called MOSTOPHOS.
Magneto-optical responses are already widely used to characterise materials. Novel materials, such as topological insulators and carbon nanostructures, that could lead to the discovery of unexpected phenomena, have become available recently, Lebedeva notes. In turn, improving our understanding of magneto-optical responses will contribute to the development of more new materials and quantum information processes, including fast and efficient light-driven technologies for the future.