Thermodynamic Properties From Static-Lattice Calculations with Soft Modes
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Thermodynamic properties from static-lattice calculations with soft modes Graeme J. Ackland and Neil D. Drummond Department of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 2LZ ABSTRACT While phase transition pressures and zero temperature thermodynamic properties can be accurately determined from electronic structure calculation, transition temperatures are more problematic because of the need to sample phase space and quantise vibrations. The quasiharmonic method has proved extremely accurate for calculating thermodynamic properties up to 90% of the melting point without explicit phase space sampling for most materials, but has low temperature divergences for soft mode materials such as perovskites. Here we present a modified quasiharmonic method which avoids these difficulties. Using a simple model system, we demonstrate trends of behaviour in both the classical and quantum limits showing the importance of phonon quantisation in soft-mode behavior. INTRODUCTION Accurate electronic structure calculation has revolutionized the description of high-pressure, zero temperature crystal structure in perovskite materials, however at finite temperatures a number of difficulties remain. The difficulty arises from the need to evaluate both energy and entropy, the latter requiring sampling an ensemble rather than making a single calculation[1,2]. For many materials the quasiharmonic approach [3] has enabled finite temperature free energy to be calculated, but in the case of perovskites the high temperature phases are typically dynamically stabilised, with imaginary phonon frequencies in their zero-temperature limit, and hence the quasiharmonic free energy diverges. This is unfortunate, since measured perovskite phonons are typically well suited to quasiharmonic treatment being harmonic away from the phase transition. The classic theory of critical phenomena in soft mode materials is to expand the free energy in terms of some order parameter. For a real system, the difficulty lies in defining the order parameter microscopically: typically it will be some combination of coupled phonons and strains, and in actually calculating the entropic contribution from the other degrees of freedom. In practice this type of appoach becomes equivalent to defining a free energy barrier at the symmetric structure and relating that to a temperature. An alternate approach is offered by the “effective hamiltonian” method[4]. Here the number of degrees of freedom in the system is reduced to a number of onsite terms (e.g. a quartic potential) and some interaction terms (e.g. elastic or electrostatic). This model is then parameterised from zero temperature calculations, and a Monte Carlo approach used to sample the surface. A great advantage is the ease of application of external fields in ferroelectric materials. Recently, the effective Hamiltonian approach has been expanded to incorporate quantum Monte Carlo causing a huge revision in the predicted phase diagram of BaTiO3[5]. However, the problem still remains of mapping the vibrati
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