Scattering from the substrate



After analyzing the scattering from a monolayer, the question arises whether there is also diffuse surface scattering from the substrate. Let us consider a semi-infinite crystal with a simple cubic lattice. For the x and y directions parallel to the surface we get the same result as in 3D. If the interface is located at z=0 and the substrate occupies the negative half-space, the summation reads:

S(qz) = Sn=-¥0 exp(i n a qz) = Sn=0¥ exp(-i n a qz)
This is a simple geometric series, and we get, as before
S(qz) = 1 / {1 - exp(-i a qz)} = exp(-i a qz /2) / [ 2i sin(a qz /2) ]
Again we find that there is diffuse scattering perpendicular to the surface, the so-called crystal truncation rods (CTR). With a bit of effort, we can redo the calculation for a general atom distribution within the unit cell and arbitrary lattices. We will call this structure factor SCTR(q).
 

Complete surface model

We have now the two basic ingredients of a surface model, the monolayer scattering and the diffuse scattering from the substrate. In general, the surface layer has a relative shift D parallel to the substrate and is at a distance d from the top layer of the substrate. Hence we can write:

S(q) = SCTR(q) + exp( i [QD + qzd] ) Sfilm(q)


Fractional-order and integer-order reflections

Because of the diffuse scattering from the substrate we have now two types of surface reflections: fractional order reflections (relative to the substrate unit cell) only contain information about the monolayer, whereas the integer-order reflections contain information both about the substrate and the surface layer. If there are multiple domains of the surface structure, there may be furthermore overlap of the reflections from the various domains at the integer-order positions.

If the substrate has a Bragg reflection in the surface plane, the intensity is too large to be able to gain information about the surface: interference only works in a useful way, if the two contribution signals are not more different than 1 or 2 orders of magnitude. However, common substrates of fcc or bcc materials have index rules that result in that the surface plane ends up at an anti-Bragg position. In this case the CTR intensity is of the same order of magnitude as the scattering from a monolayer, and the interference between the two signals contains the information about the registry D of the monolayer on the substrate. Additionally to an in-plane data set, various CTR will also be measured. The integer-order CTR's yield the spacing d between the surface layer and the substrate.

The picture below shows the diffraction from a simple (2x1) structure on a square lattice, taken to be a bcc(001) surface and illustrates the various possibilities.