X-ray Reflectivity (XRR)

X-ray reflectivity (XRR) is a non-destructive, non-contacting method to measure film thickness, interface and surface roughness and density of films ranging from 20 � to 1 mm total thickness. Films can be single or multilayer structures, and the thickness of individual layers can be determined with no optical constant corrections required. The films can be epitaxial, polycrystalline or amorphous. The thickness of the film is measured from the periodicity of fringes, the density from the angle at which the intensity begins to drop and the roughness from the damping of the thickness fringes and rate of intensity decrease with angle. These are not affected by the crystallinity of the film. XRR is basically a grazing incidence scattering technique, with the incident and scattered beams at equal angles to the surface of the sample. In XRR, an X-ray-beam strikes a solid-surface at a small angle (0-2^�) and is totally reflected. Above the critical angle of total reflectance q_c beam penetrates the sample. Measurement of the critical angle provides the determination of the density of the material. If the sample contains a thin layer, X-rays are reflected from the air/layer as well as from the layer/substrate interfaces. This leads to the interference fringes. In this case the position of the mth order fringe maximum q_m can be shown to be related to the layer thickness t and the critical angle q_c by the following simple equation.

where n_s and n_l are the refractive indices of the substrate and the layer for X-ray of wavelength l. From this formula, thickness of the layer can be derived.
 

Reflection from ideal layered structures  : The X-ray reflectivity from a layer structure can be calculated by applying the recursive theory of Parratt [1], a generalization of the Fresnel reflectivity theory, to a system of flat, ideal layers, each with a constant electron density. The refractive index n of a material, is given by:

where d+ib is related to the atomic scattering factors of a particular type of atom. Thus, the refractive index of matter for X-rays is less than unity. Due to this reason, at a grazing angle below a certain critical angle q_c a beam of X-rays incident on a surface will be totally reflected. Applying Snell's law in the small angle approximation one can show that q_c=�{2d}.

Figure 0.1: Multilayer representation for calculation of X-ray specular reflectivity. Each layer j is defined by its thickness t_j, refraction index n_j, and roughness of the bottom interface s_j. Total of N layers are considered plus the incident space vacuum and the substrate.

Let us consider the specular reflection of X-rays with an angle of incidence q from a system of N smooth homogenous layers as illustrated in Fig. 0.1. The refractive index of the jth layer is denoted by n_j=1-d_j-ib_j, the reflection amplitude (fresnel amplitude) of the interface between the j-1^th and j^th layers is denoted by r_j-1,j and the electric vectors of the incident and reflected waves at the interface between the j-1^th and j^th layers are denoted by E_j-1,j and E^r_j-1,j, respectively. Maxwell's equations require that the tangential component of the total electric field be continuous at each of the j^th interfaces. The solution to this problem leads to a recursive formula for the reflection amplitude R_j-1,j of the stack of layers from the substrate to the interface between the j^th and j-1^thlayers:

where
 

and a_j is the phase factor for half the thickness t_j

The recursion formula is solved by starting at the substrate (layer N) whose thickness is assumed to be infinite and therefore there is no reflected wave in it; hence R_{N, N+1} is zero.

One then solves the equation with j=N and continues recursively until the top of the layer of the material is reached (j=1), where a₁ is unity, giving the reflectivity
 

The ratio of the reflected to the incident X-ray intensities is given by the product of R_0,1 with its complex conjugate
 

In the above analysis, we assumed perfectly smooth surfaces. However, the real surfaces and interfaces are not infinitely sharp as assumed above. In general we have some roughness which can be taken into account by considering the so called Born Approximation (BA) or Distorted Wave Born Approximation (DWBA) [2]. The BA is valid when reflectivity is small, whereas DWBA is valid when reflectivity approaches unity. In general, the reflection amplitude r which is valid for both the BA and DWBA, is written in the following form:

where r_F is the reflection amplitude for a single ideal surface, also called as Fresnel amplitude. s_{j-1, j} is the root-mean-square roughness between the j-1^th and j^th layers. S_j and S_j-1 are the scattering vectors for the j-1^th and j^th layers, respectively. This can be substituted in the reflection amplitude from the interface between j-1^th and j^th layers.
 

This modified Parratt formalism will be used in this work for XRR simulations to evaluate the layer thickness and interface/surface roughness.

Bibliography

[1]

L. G. Parratt, Phys. Rev. 95, 359 (1954).

[2]

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, Phys. Rev. B 38, 2297 (1988).

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On 26 Oct 2005, 14:56.

Last Updated Wednesday October 26, 2005