Periodic Boundary Conditions and Slab Models

If our goal is to study a surface, our ideal model would be a slice of material that is infinite in two dimensions, but finite along the surface normal. In order to accomplish this, it may seem natural to take advantage of periodic boundary conditions in two dimensions, but not the third. There are codes in which this technique is implemented, but it is more common to study a surface using a code that applies periodic boundary conditions in all three dimensions, and it is this approach we will discuss. The basic idea is illustrated in Fig. 4.1, where the supercell contains atoms along only a fraction of the vertical 

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 

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