Reflection from a multilayer

For measurements far above the critical angle, where multiple refraction of the transmitted waves is negligible, the reflectivity can be calculated by kinematical approximation. However in this work only measurements near the critical angle are described and one has to use Parratt's exact recursive method [24] for reflectivity calculations of multilayers.

Figure: Schematic drawing of a typical multilayer consisting of N bilayers. One of the bilayers with the thickness $\Lambda$ is shown at the right side. The ratio of the individual layer thicknesses in the bilayer is defined by $\Gamma$.

The most usual structure of a multilayer is a stack of N bilayers, consisting of two layers with different thicknesses. The schematic picture of such a multilayer is shown in figure . In general each multilayer consists of a certain number of layers on a substrate. For every layer $j$ the refractive index $n_{j}$, the thickness $\Delta_{j}$ and the phase factor $p_{j}$ can be defined. Following the equations and for each layer the wave vector transfer $Q_{j}$ and the reflectivity $r^{'}_{j,j+1}$ can be calculated

$$n_{j}=1-\delta_{j}+i\beta_{j}$$

$$Q_{j}=\sqrt{Q^{2}-8k^{2}\delta_{j}+i8k^{2}\beta_{j}}$$

$$p_{j}=e^{i\Delta_{j}Q_{j}}$$

$$r^{'}_{j,j+1}=\frac{Q_{j}-Q_{j+1}}{Q_{j}+Q_{j+1}}$$ (28)

Figure: Calculation of the reflectivity of a multilayer structure with the bilayer thickness 20 Å and 10 repetitions according to Parratt's exact recursive method. The superstructure peak at about 21 $mrad$ corresponding to the bilayer thickness is clearly visible. The additional Kiessig oscillations correspond to the total thickness of the sample.

The substrate is treated as infinitely thick. According to equation the reflectivity at the layer-substrate interface can be written as

$$r^{'}_{N,\infty}=\frac{Q_{N}-Q_{\infty}}{Q_{N}+Q_{\infty}}$$ (29)

The reflectivity on the top of the Nth layer is, according to equation and ,

$$r_{N-1,N}=\frac{r^{'}_{N-1,N}+r^{'}_{N,\infty}p^{2}_{N}}{1+r^{'}_{N-1,N}r^{'}_{N,\infty}p^{2}_{N}}.$$ (30)

It is straightforward that the reflectivity for the (N-1)th layer is

$$r_{N-2,N-1}=\frac{r^{'}_{N-2,N-1}+r_{N-1,N}p^{2}_{N-1}}{1+r^{'}_{N-2,N-1}r_{N-1,N}p^{2}_{N-1}}$$ (31)

Please note that the reflectivities without prime include the multiple refractions shown in figure . This recursive method can be continued until the total reflectivity $r_{01}$, from the interface between the multilayer and vacuum, is reached. The reflectivity from a multilayer consisting of (10x(10Å+10Å)) bilayers is shown in figure . It is clear that Kiessig oscillations corresponding to the total thickness of the multilayer appear. Another important feature is the superstructure peak appearing at the position corresponding to the thickness of the bilayer. The position can be calculated simply using the equation . Looking carefully at equation one can recognise its similarity with the well known Bragg equation (for small angles $\sin \alpha\approx\alpha$). The analogy is that the superstructure peak is the Bragg peak of a crystal with very small number of lattice planes (bilayers) with the "lattice constant" $\Lambda$.
At this point one has to emphasise that no detailed assumptions have been made about the crystalline structure of the layers or multilayers. The only material parameter which enters the calculation is the index of refraction . It means they are applicable to amorphous and polycrystalline samples (using the proper index of refraction).