3.3.1 The Miller Index Notation

The Miller indices can be used to specify directions and planes in a crystal [Ashcroft76,Kittel96]. The Miller indices of a plane are defined in the following way: First, three lattice vectors have to be defined. For cubic crystal systems, the lattice vectors are chosen along the edges of the crystallographic unit cell (unit cube). Any crystal plane intercepts the axes in certain points. The Miller indices are the ratios of these points and are given as a triplet of integer values $(hk\ell)$. A Miller index 0 means that the plane is parallel to the respective axis. Negative indices are indicated with a bar written over the number.

In the notation of [Ashcroft76], $[hk\ell]$ with square brackets instead of round brackets, denotes a direction in the basis of the lattice vectors. The notation $\{hk\ell\}$ denotes all planes that are equivalent to $(hk\ell)$ by the symmetry of the crystal. Similarly, the notation $\langle hk\ell\rangle$ denotes all directions that are equivalent to $[hk\ell]$ by symmetry.

In cubic crystal systems the Miller indices of a plane are the same as those of the direction perpendicular to the plane.

Figure 3.3: Three planes in the cubic system along with their Miller indices.

E. Ungersboeck: Advanced Modelling Aspects of Modern Strained CMOS Technology