2.2.1.3 Surface periodicity: notations for surface structures

This chapter discusses surface periodicity of reconstructed surfaces. Reconstruction of the clean surface usually results in a change of the surface periodicity, and the new lattice is called a superlattice. The new surface periodicity is usually deduced from the LEED pattern, which gives directly a map of the 2D reciprocal lattice. To express the relationship between the ideal surface lattice and the superlattice, two systems are most commonly used, denoted by a and b the basis vectors of the primitive lattice, and similarly by A and B the basis vectors of the superlattice. A superlattice is said to be commensurate when all matrix elements M_ij are integers. If at least one matrix element M_ij is an irrational number, then the superstructure is said to be incommensurate. Alternatively to the matrix method of denoting surface structures, another system, originally proposed by Wood is more commonly used. Whereas the matrix notation can be applied to any system, Wood's notation can only be used when the angle between the superlattice vectors A and B is equal to the angle between the primitive lattice vectors a and b. Thus the label p(1 x 1) or simply 1 x 1 means that the surface has the same periodicity of the ideal bulk termination. For surfaces with more than one atom per unit cell, this does not necessarily imply that the surface is unreconstructed, as it is the case, for instance, for the (110) surfaces of III-V semiconductors.