Calculation of Surface Energies

TABLE 4.2 Surface Energies Calculated for Cu(100) and Cu(lll) from DFT as Function of Slab Thickness, in eV/A2 (J/m2)  

As an example, the DFT-calculated surface energies of copper surfaces are shown in Table 4.2 for the same set of slab calculations that was described in Table 4.1. The surface energy of Cu(l 11) is lower than for Cu(100), meaning that Cu(l 11) is more stable (or more bulklike ) than Cu(100). This is consistent with the comment we made in Section 4.4 that the most stable surfaces of simple materials are typically those with the highest density of surface atoms. We can compare our calculated surface energy with an experimental 

This definition neglects entropic contributions to the surface energy. 

An alternative is to describe the surface using a symmetric model. In the symmetric model, the center of the slab consists of a mirror plane. The atoms in the middle layers are typically fixed at bulk geometries and the layers above and below are allowed to relax. One advantage of a symmetric model is that any dipole generated by surface features will be automatically canceled. There is a cost involved, however, because it is typically necessary to include more layers in a symmetric slab than in an asymmetric slab. A symmetric slab with nine layers is depicted in Fig. 4.12. Note that, in this example, three layers are allowed to relax on each side of the slab. Recall that in our earlier model of an asymmetric slab, three layers were allowed to relax on one side of a five-layer slab. So in order to carry out calculations in which three layers are allowed to relax, one would need to employ nine layers in a symmetric model, compared to the five layers needed for an asymmetric one. 

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The question of how to terminate the box is fundamental to all the calculations of interfacial energy in compounds, including the calculation of surface energies. It has been addressed previously for particular cases by Chetty and Martin [11,12]. These authors pointed out that a suitable termination is one which is on a symmetry plane of the crystal, or which follows symmetry planes if it is not parallel to the boundary. However, it may not always be possible to find a symmetry plane. I offer a solution here which is more general. It reconciles the atomistic picture with the thermodynamic limit. 

For an interesting discussion of the calculation of surface energies, as well as a discussion of the accuracy of LDA, GGA, and several post-DFT methods, see D. Alfe and M. J. Gillan, J. Phys. Condens. Matter 18 (2006), L435. 

An approximate calculation of surface energy can be made by envisioning a surface being formed by mechanical forces across it and calculating the work required to separate the two halves of a crystal (see Figure 12.1). Here the surface... 

The minus sign is introduced because energy has to be consumed to create a surface. Calculations of surface energies based on Eq. (4.16) invariably yield values that are substantially greater than the measured ones (see Table 4.4). The reason for this discrepancy comes about because in the simple model, surface relaxation and rearrangement of the atoms upon the formation of the new surface were not allowed. When the surface is allowed to relax, much of the energy needed to form it is recovered, and the theoretical predictions do indeed approach the experimentally measured ones. 

T. Halicioglu, Calculation of surface energies for low index planes of diamond. Surf. Sci. 259(1-2), L714-L718 (1991)... 

Johnston K, Castell MR, Paxton AT, Finnis MW (2004) SrTi03 (001)(2 x 1) reconstructions first-principles calculations of surface energy and atomic structure compared with scanning tunneling microscopy images. Phys Rev B Condens Matter 70(8) 085415... 

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