Stress and Strain in Solids

The geometry optimization of crystal structures within first-principles calculations are on the way to be a standard tool. This possibility of finding the energetically most favorable structure has been opened by the steadily increasing computer power, as well as by more efficient computer programs. For many applications this procedure is a way of systematically improving theoretical results, which is desirable in the spirit of parameter-free calculations. Moreover, for some problems, it is even an imperative requirement: e.g. studies of crystal growth, surface reconstructions or the behavior of solids under pressure have thus become possible making first-principles calculations a powerful predicting tool. For a rigorous structure optimization, not only the internal parameters - the atomic positions - have to be relaxed, but also the lattice constants and angles between the crystal axis. In principle, all the information can be obtained by total energy calculations. In practice, however, this procedure is time consuming and can hardly be performed for a complex crystal structure. In order to achieve an optimized structure within reasonable time, two quantities are necessary: the atomic forces and the stress-tensor. While the former allow for a more efficient relaxation of the internal structural parameters, the latter quantitatively determines the stress as the reaction of a strain induced into a lattice. The resulting tensor - essentially containing the gradient of the energy with respect to the lattice parameters - is a guideline towards the equilibrium lattice constants. While the atomic forces have become a more or less standard tool for all methods within density functional theory, the stress-tensor is yet not as widely used. It was e.g. worked out for the pseudopotential method (with plane-waves as a basis set) where it has proven its success for linear and non-linear elastic constants, strain-induced phonon splitting, or surface stress. To our knowledge, there is no general implementation of the stress tensor within the linearized augmented plane-wave (LAPW) method so far. Within this project, the stress-tensor shall be developed for the LAPW method and applied to two different topics: the investigation of elastic properties of intermetallic compounds on the one hand, and high Tc materials on the other hand.

The geometry optimization of crystal structures within first-principles calculations is on the way to become a standard tool. For many applications this procedure is a way of systematically improving theoretical results, which is desirable in the spirit of parameter-free calculations. Moreover, for some problems, it is even an imperative requirement: e.g. studies of crystal growth, surface reconstructions or the behavior of solids under pressure have thus become possible making first-principles calculations a powerful predicting tool. The procedure is to determine the lattice parameters as well as the atomic sites inside the unit cell on searching for the energetically most favorable configuration. The methods based on density functional theory have been proven their success in predicting structural and elastic properties as well as lattice vibrations with high accuracy. For solids with 10-100 atoms, however, such investigations becomes an obstacle in terms of computer power. Within this project, a formalism has been developed to describe the reaction of a crystalline solid to strain on a quantum-mechanical basis. To this extent not only the total energy but also its derivative with respect to the strain tensor is calculated. This method allows for a dramatic reduction of computing time compared to the usual procedure and makes the investigation of complex crystals as a function of pressure or strain feasible. At the same time elastic constants can be computed with a relatively small effort. In parallel to this quite complicated methodical work and the programming effort, some representatives of high temperature superconductors were studied as a function of pressure. These materials, some of which exhibit a pronounced pressure dependence of the superconducting transition temperature, represent an example for future applications of the above described program package. In this context, the changes of the structural properties - the lattice parameters and the corresponding relaxations of the atomic positions - were a focal point. But also the resulting influence on the electronic properties were at the center of the investigations. The most intensive studies were performed for the mercury based high Tc family, where an increase of the critical temperature was observed not only as a function of pressure but also with oxygen doping. At the same time it strongly depends on the composition, i.e. the number of copper-oxygen layers within the unit cell, where the charge carriers (holes) form the Cooper pairs in the superconducting phase. In order to find the link between these three phenomena detailed investigations have been performed. A comparison reveals that and how the hole concentration in the Cu-O planes is increased as a function of pressure, doping and composition. In case of oxygen doping it could even be shown what the limiting effect is.