A Simple Closed-form Expression for the X-ray Reflectivity from Multilayers with Cumulative Roughness

We present a simple closed-form expression for the reflectivity horn a multilayer which includes the effects of absorption, refraction, surface and substrate reflections but neglects dynamical effects. This expression reproduces the exact dynamical calculation except for the regions near the critical angle and for intense Bragg reflections. The expression is generalized to include cumulative interface roughness which follows a t”* power-law growth. X-ray reflectivity is commonly used to study the interface structure and morphology of metallic superlattices.( l-7) The specular reflectivity is a measure of the average electron density profile of the superlattice and in principle can be used to determine the interdiffusion and average roughness of the interfaces. The most common approach to calculate the low angle profile is the recursive application of optical theories where the layers are assumed to be continuous media of constant electron density. (I-3) These theories are equivalent to dynamical calculations and include effects of multiple scattering, absorption, refraction, and surface and substrate reflections. (8) Although it is straight forward to calculate the reflectivity using this technique, the calculations are time consuming when the superlattice contains a large number of layers. In this paper, we present a simple closed-form expression for the reflectivity of a superlattice which includes the effects of absorption, refraction, and substrate reflections but neglects dynamical effects. The expression is then generalized to include cumulative interface roughness which follows a t’” power-law growth. The kinematic calculation of the x-ray reflectivity from a general A/B superlattice is shown schematically in Fig. 1. The reflectivity of the superlattice is given by summin g the reflectivity from each interface ignoring multiple reflections. The surface and substrate reflectivities are given by r, and r&, and the A-on-B and B-onA reflectivities are given by r, and reA respectively. Each layer is characterized by a thickness t and index of refraction n given by: (1) n=lPNd 2x (& + Af' + iAf “> + Permanent address: Universidad Complutense, Madrid, Spain.