Supercell lattice

Minimize the Gibbs energy Gk of each supercell at temperature T with respect to all the lattice vectors and atom positions. 

Such an approximation of periodicity was made for the calculations discussed in the next section (section 4). The supercells for these calculations were composed of either 12 or 32 primitive Li M02 unit cells (M = 3d TM ion 0 < X < 1) that contained various M defects. The lattice parameters of the supercells were kept constant at the parameters for the undetected structure, while the ionic coordinates were allowed to relax. A 2 x 2 x 2 Tr-point mesh was used for the calculations on the 12-unit supercells and a 1x1x1 Tr-point mesh for the 32-unit supercells. The primitive LiJV[02 unit cells used to construct the super cells had previously been calculated with full relaxation of lattice parameters as well as ionic coordinates. 

You now know how to define a supercell for a DFT calculation for a material with the simple cubic crystal structure. We also said at the outset that we assume for the purposes of this chapter that we have a DFT code that can give us the total energy of some collection of atoms. How can we use calculations of this type to determine the lattice constant of our simple cubic metal that would be observed in nature The sensible approach would be to calculate the total energy of our material as a function of the lattice constant, that is, tot(a). A typical result from doing this type of calculation is shown in Fig. 2.1. The details of how these calculations (and the other calculations described in the rest of the chapter) were done are listed in the Appendix at the end of the chapter. 

Because we can always choose each atom so it lies within the supercell, 0 < j), < 1 for all i and j. These coefficients are called the fractional coordinates of the atoms in the supercell. The fractional coordinates are often written in terms of a vector for each distinct atom. In the hep structure defined above, for example, the two atoms lie at fractional coordinates (0,0,0) and (5,5, 5). Notice that with this definition the only place that the lattice parameters appear in the definition of the supercell is in the lattice vectors. The definition of a supercell with a set of lattice vectors and a set of fractional coordinates is by far the most convenient way to describe an arbitrary supercell, and it is the notation we will use throughout the remainder of this book. Most, if not all, popular DFT packages allow or require you to define supercells using this notation. 

In Chapter 2 we mentioned that a simple cubic supercell can be defined with lattice vectors a, = a or alternatively with lattice vectors a = 2a. The first choice uses one atom per supercell and is the primitive cell for the simple cubic material, while the second choice uses eight atoms per supercell. Both choices define the same material. If we made the second choice, then... 

The three-dimensional shape defined by the reciprocal lattice vectors is not always the same as the shape of the supercell in real space. For the fee primitive cell, we showed in Chapter 2 that... 

There are many examples where it is useful to use supercells that do not have the same length along each lattice vector. As a somewhat artificial example, imagine we wanted to perform our calculations for bulk Cu using a supercell that had lattice vectors... 

The calculations above allowed the positions of atoms to change within a supercell while holding the size and shape of the supercell constant. But in the calculations we introduced in Chapter 2, we varied the size of the supercell to determine the lattice constant of several bulk solids. Hopefully you can see that the numerical optimization methods that allow us to optimize atomic positions can also be extended to optimize the size of a supercell. We will not delve into the details of these calculations—you should read the documentation of the DFT package you are using to find out how to use your package to do these types of calculations accurately. Instead, we will give an example. In Chapter 2 we attempted to find the lattice constant of Cu in the hep crystal structure by doing individual calculations for many different values of the lattice parameters a and c (you should look back at Fig. 2.4). A much easier way to tackle this task is to create an initial supercell of hep Cu with plausible values of a and c and to optimize the supercell volume and shape to minimize... 

In the exercises for Chapter 2, we suggested you compute the lattice constants, a and c, for hexagonal Hf. Repeat this calculation using an approach that optimizes the supercell volume and shape within your calculation. Is your result consistent with the result obtained more laboriously in Chapter 2 How large is the distortion of c/a away from the ideal spherical packing value ... 

Suppose we would like to carry out calculations on a surface of an fee metal such as copper. How might we construct a slab model such as that depicted in Fig. 4.1 It is convenient to design a supercell using vectors coincident with the Cartesian x, y, and z axes with the z axis of the supercell coincident with the surface normal. Recall that for fee metals, the lattice constant is equal to the length of the side of the cube of the conventional cell. The supercell vectors might then be... 

All calculations in this chapter used the PBE GGA functional. For calculations related to Cu surfaces, a cutoff energy of 380 eV and the Methfessel-Paxton scheme was used with a smearing width of 0.1 eV. For calculations related to Si surfaces, the cutoff energy was 380 eV and Gaussian smearing with a width of 0.1 eV was used. The k points were placed in reciprocal space using the Monkhorst-Pack scheme. For all surface calculations, the supercell dimensions in the plane of the surface were defined using the DFT-optimized bulk lattice parameter. 

All calculations in this chapter used the PBE GGA functional and a plane-wave basis set including waves with an energy cutoff of 380 eV. For all surface calculations, the supercell dimensions in the plane of the surface were defined using the DFT-optimized bulk lattice parameter. 

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