Start up VNL and create a new project for this tutorial. You will use VNL for the calculation, and it is recommended that you go through the VNL tutorial to be familiar with the basic work flow. SETTING UP THE CALCULATION You will in this section set up a DFT calculation using the local density approximation (LDA) for the BaTiO3 crystal and calculate the polarization.
# Set up configuration bulk_configuration = BulkConfiguration( bravais_lattice=lattice, elements=elements, fractional_coordinates=fractional_coordinates ) The structure is given in the ATK format below # Set up lattice lattice = SimpleTetragonal(3.9945*Angstrom, 4.0335*Angstrom) # Define elements elements = # Define coordinates fractional_coordinates = [[ [ [ [ [Ġ.
In this tutorial we use the experimental lattice constants and coordinates as obtained from the Inorganic Crystal Structure Database (ICSD). An internal stress further shifts the fractional coordinates in the c-direction away from their high symmetry positions.
THE BATIO3 CRYSTAL STRUCTURE Barium titanate (BaTiO3) has a tetragonal crystal structure at room temperature, where the unit cell is slightly elongated in the c-direction. SPONTANEOUS POLARIZATION OF FERROELECTRIC BATIO3 There is no explicit check for this in the code, but the results cannot be expected to be correct if a non-orthogonal cell is used.ĬHAPTER 2. Important In the current implementation of polarization in ATK, the unit cell must be orthogonal (simple cubic, tetragonal, or orthorhombic). ces in polarization, ATK computes and reports the electronic and ionic contributions separately, and also reports the polarization quantum. Given the multivalued nature of the polarization, it is perhaps not surprising that only differen, between two different structures is a well-defined property. The polarization is thus a periodic function, and the period is called the polarization quantum,, where Likewise, the ionic contribution would attain a different value if all ionic positions were displaced by a lattice constant in either direction. The reason is that the electronic polarization phase, which is only defined modulo. was that the polarization is a multivalued quantity, and in fact is determined by the Berry forms a lattice. The total polarization is simply the sum of the electronic and ionic contributions,Īn important finding in Ref. The number of k-points in the parallel direction should be larger, however. The integral over the perpendicular directions can easily be converged with a few number of k-points. The last integral is known as the Berry phase. Is a reciprocal lattice vector in the same direction. Is parallel to the direction of polariza. Where the sum runs over occupied bands, and where tion, and The electronic contribution to the polarization is obtained as Where and are the valence charge and position vector of atom, volume, and the sum runs over all ions in the unit cell. The latter is calculated using a simple classical electrostatic sum of point charges It is common to divide the polarization of a material into electronic and ionic parts. MODERN THEORY OF POLARIZATION The theoretical understanding of FE materials is described by the so-called modern theory of polarization. Before continuing with the calculations, let us briefly summarize some central theoretical concepts first. One of the most studied FE materials is barium titanate (BaTiO3), which is the topic for this tutorial. FE materials find applications in capacitors, ferroelectric random access memory (RAM), and more recently in ferroelectric tunnel junction (FTJ) displaying giant electroresistance effects. 13įerroelectric (FE) materials have a spontaneous electric polarization that can be reversed by the application of an external electric field. Spontaneous polarization of ferroelectric BaTiO3.
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Spontaneous polarization of ferroelectric BaTiO3: Tutorial on how to use ATK for polarization calculations Version 2015.2 Copyright © 2008–2015 QuantumWise A/S Atomistix ToolKit Copyright Notice All rights reserved. Spontaneous polarization of ferroelectric BaTiO3 Tutorial on how to use ATK for polarization calculations Version 2015.2