For the calculation of the bulk value of the Young

For the calculation of the bulk value of the Young׳s modulus the simplest choice is a 2-atom tetragonal unit cell with cell vectors along [], [110], and [001]:This cell, however, does not allow to introduce twin boundaries. Therefore, the calculation of was repeated for a second, 6-atom orthorhombic unit cell with cell vectors along [], [] and [111]:The cell contains three atomic layers with ABC stacking along [111], with 2 atoms per plane in the unit cell. This cell can also be used for calculating Young׳s modulus . In a cubic crystal, however, and are equal due to symmetry. It is important to note that for the orthorhombic unit cell the Poisson contraction gives rise to a tilt of the basis vectors and , if a strain is applied in the direction of . Thus, the angle between and has to be relaxed in order to get the correct value for the Young׳s modulus . Unit Ion Channel Compound Library for the bulk crystal with twin boundaries are derived from the orthorhombic cell, but with a different number of atomic planes in the [111] direction ( axis). Periodic boundary conditions require that always 2 twin boundaries are included in one unit cell. We used cells with 6, 8 and 10 atomic layers (thus containing 12, 16 and 20 atoms) with stacking sequences of ABC BAC, ABCA CBAC and ABCAB ACBAC, respectively, in which the twin boundaries are separated by , , Å, 9.57Å, 11.96Å. In all cases, the Young׳s modulus was calculated by changing the length of between −0.024 and +0.030 in steps of 0.06 in units of . This corresponds to applied strains between −3.4% and +4.2%. A polynomial of degree 6 was fitted to the 10 total energy values.
Results of the DFT benchmark calculations Since elastic constants are second derivatives of the total energy, DFT calculations have to be very well converged in order to be able to extract elastic constants in a reliable way (within the limits of the accuracy of the chosen functional). First, we thoroughly tested the influence of the plane wave basis set and density cut-off energies. For our choice of these cut-off energies (30Ry and 120Ry, respectively), the elastic constants are well converged within ±0.1GPa. Much more crucial is the convergence with respect to k-point density and Gaussian smearing parameter σ. An additional problem arises from the fact that different unit cells have to be used for the calculation of the bulk value of the Young׳s modulus and of structures containing twin boundaries, since k-point meshes will not be equivalent. Therefore, to be able to identify changes in elastic properties due to twin boundaries, absolute values of elastic constants have to be converged as good as possible. To have an estimate on how well our calculations are converged and what accuracy for changes in the elastic constants can be expected, we applied the following 3 step strategy: The results for step (1) are shown in Fig. 1 and Table 1. k-point grids were always of the Monkhorst–Pack type with divisions of (n, n, n) of the three basis vectors of the reciprocal lattice. Fig. 1 shows the typical behavior for the convergence of properties with k-point density n: the larger the Gaussian smearing parameter σ, the faster the k-point convergence, but the k-point-converged result depends on the value of σ. This is most obvious for the elastic constant C12. However, for our choices of σ of 0.005, 0.010 and 0.015Ry the deviation from the σ=0 limit is less than 0.1GPa for all elastic constants. Table 1 summarizes the results of the elastic constants for the different smearing values σ together with the k-point density, which is required for obtaining convergence within an accuracy of ±0.1GPa. Obviously, for elastic properties a much denser k-point mesh than for the calculation of energy differences (for example, surface and interface energies) are required. In Table 1 also the experimental values for the SOEC at 0K and 300K are given. The DFT calculations underestimate the elastic constant by up to 20%. This is typical for the PBE functional and other functionals based on the generalized-gradient approximation (GGA). In the local density approximation (LDA), on the other hand, elastic constants are overestimated by up to 20% [11]. However, this uncertainty of DFT calculations (the dependence of the results for SOEC on the choice of functional) is not crucial for our aim: we are not interested in the absolute values of elastic properties, but only in the change of the Young׳s modulus after the introduction of twin boundaries.