Final Report Summary - RESPSPATDISP (First-principles theory of spatial dispersion in electromagnetic response of solids: Applications to natural optical activity and magnetoelectric effect)
The perturbation theory is a natural way to describe various types of responses to static and oscillating electromagnetic fields and has been successfully used for molecular systems for a long time. However, extension of this theory to solids is not straightforward. The difficulties come from the fact that potentials corresponding to uniform electric and magnetic fields break translational invariance of the Hamiltonian and dipole moments describing coupling to these fields and involving the position operator are ill defined under periodic boundary conditions. Incorporation of the uniform magnetic field is particularly challenging as it introduces vector coupling to electron dynamics and leads to non-perturbative changes in single-particle eigenstates. Some decisive steps in understanding of response of solids to static fields have been made recently and among others have given rise to the development of what is known as the modern theory of polarisation. The present project aimed at the extension of those theories to the time domain and development of a theoretical framework for first-principles description of the macroscopic electromagnetic response of solids with full account of spatial dispersion and memory effects. The objectives of the project include (1) development of complete perturbation theory for response of solids to electromagnetic fields in the framework of time-dependent density functional theory (TDDFT), (2) implementation of the developed formalism in the open-source electronic structure code Octopus, (3) application of the code for calculation of magneto-optical responses of technologically relevant materials.
The first stage of the project consisted in the analytical work on derivation of expression for magneto-optical response of periodic systems in terms of sums over states. Two approaches for description of uniform magnetic fields were compared. In the first approach, a long-wave limit of the current response to a slightly inhomogeneous vector potential was taken. In the second approach, the formalism of non-equilibrium Green functions was applied and perturbations to periodic counterparts of the Green functions were calculated. The both approaches were shown to recover the same final expression for the magneto-optical response. In addition, we showed that the Green function formalism can serve as a basis for development of a unified approach to calculation of all-order response to arbitrary electromagnetic fields both for periodic and molecular systems. The obtained expression of the magneto-optical response was analyzed and gauge-invariant analogues of the electric and magnetic dipole moment operators applicable both to finite and periodic systems were suggested. Symmetry restrictions for observation of magneto-optical response were identified.
Though the summation over Bloch states is useful to analyze different contributions to the response and demonstrate the similarity between periodic and molecular systems, it is clearly impractical for computational purposes as it requires calculation of many unoccupied states. Using the Sternheimer approach, in which the calculation of perturbations of states is reduced to a solution of a set of linear equations, allowed us to decrease substantially the computational cost. To preserve the gauge-invariant formulation of perturbations the density matrix-perturbation theory was adapted. The developed computational scheme involves direct calculation of corrections to the density matrix elements within the occupied and unoccupied subspaces on the basis of lower-order corrections and solution of the Liouville equation for corrections to the density matrix elements between the occupied and unoccupied subspaces. The differentiation of the density matrix with respect the wave vector is performed within k*p theory. The electron interaction is taken into account within TDDFT approach, which provides an optimal balance in accuracy and computational cost.
The developed formalism was implemented in Octopus code (http://www.tddft.org/programs/octopus) which aims to have a fully versatile free (GNU license) code able to describe the time-dependent electron-ion dynamics of finite and extended systems to arbitrary intense and time-dependent electromagnetic perturbations. The implemented procedures were extensively tested for molecules under periodic boundary conditions, both versus available literature data and results of calculations using existing routines treating the system as finite. The magneto-optical response of widely used semiconductors was investigated.
All project developments are accessible to the whole scientific community for free. Tutorials and other introductory material will be provided in due time under the tutorial webpage of Octopus. This part of the code is maintained and keep up-to date regularly by a set of Octopus developers.
Magneto-optical responses are routinely used to characterize materials and supplement optical adsorption data. There is no doubt that the main outcome of this project, the tool for calculation of magneto-optical response of both periodic and finite systems will stimulate research in condensed matter physics, materials science, nano and bioscience. The project is extremely timely as new materials, such as topological insulators and carbon nanostructures, have become available recently and can lead to discovery of new magneto-optical phenomena. The developed versatile tool for spectroscopic analysis can be widely used for interpretation and guidance of new experiments, including those at large facilities such as synchrotron. The methodology promoted in this project requires using top-level supercomputing centres (PRACE initiative) and will be applied in future scientific projects performed on the international computing facilities.
The first stage of the project consisted in the analytical work on derivation of expression for magneto-optical response of periodic systems in terms of sums over states. Two approaches for description of uniform magnetic fields were compared. In the first approach, a long-wave limit of the current response to a slightly inhomogeneous vector potential was taken. In the second approach, the formalism of non-equilibrium Green functions was applied and perturbations to periodic counterparts of the Green functions were calculated. The both approaches were shown to recover the same final expression for the magneto-optical response. In addition, we showed that the Green function formalism can serve as a basis for development of a unified approach to calculation of all-order response to arbitrary electromagnetic fields both for periodic and molecular systems. The obtained expression of the magneto-optical response was analyzed and gauge-invariant analogues of the electric and magnetic dipole moment operators applicable both to finite and periodic systems were suggested. Symmetry restrictions for observation of magneto-optical response were identified.
Though the summation over Bloch states is useful to analyze different contributions to the response and demonstrate the similarity between periodic and molecular systems, it is clearly impractical for computational purposes as it requires calculation of many unoccupied states. Using the Sternheimer approach, in which the calculation of perturbations of states is reduced to a solution of a set of linear equations, allowed us to decrease substantially the computational cost. To preserve the gauge-invariant formulation of perturbations the density matrix-perturbation theory was adapted. The developed computational scheme involves direct calculation of corrections to the density matrix elements within the occupied and unoccupied subspaces on the basis of lower-order corrections and solution of the Liouville equation for corrections to the density matrix elements between the occupied and unoccupied subspaces. The differentiation of the density matrix with respect the wave vector is performed within k*p theory. The electron interaction is taken into account within TDDFT approach, which provides an optimal balance in accuracy and computational cost.
The developed formalism was implemented in Octopus code (http://www.tddft.org/programs/octopus) which aims to have a fully versatile free (GNU license) code able to describe the time-dependent electron-ion dynamics of finite and extended systems to arbitrary intense and time-dependent electromagnetic perturbations. The implemented procedures were extensively tested for molecules under periodic boundary conditions, both versus available literature data and results of calculations using existing routines treating the system as finite. The magneto-optical response of widely used semiconductors was investigated.
All project developments are accessible to the whole scientific community for free. Tutorials and other introductory material will be provided in due time under the tutorial webpage of Octopus. This part of the code is maintained and keep up-to date regularly by a set of Octopus developers.
Magneto-optical responses are routinely used to characterize materials and supplement optical adsorption data. There is no doubt that the main outcome of this project, the tool for calculation of magneto-optical response of both periodic and finite systems will stimulate research in condensed matter physics, materials science, nano and bioscience. The project is extremely timely as new materials, such as topological insulators and carbon nanostructures, have become available recently and can lead to discovery of new magneto-optical phenomena. The developed versatile tool for spectroscopic analysis can be widely used for interpretation and guidance of new experiments, including those at large facilities such as synchrotron. The methodology promoted in this project requires using top-level supercomputing centres (PRACE initiative) and will be applied in future scientific projects performed on the international computing facilities.