Comparison of Surface Roughness Estimations by X-ray Reflectivity Measurements and TEM observations

Estimations of surface and interfacial roughness by x-ray reflectivity measurements were compared with those from TEM and AFM observations. The x-ray reflectivity was calculated based on the Parratt formalism, accounting for the effect of roughness by the theory of Nevot-Croce. Estimated surface and interface roughnesses from the x-ray reflectivity measurements did not correspond to the TEM image observation results. This disagreement suggests that the modified Fresnel reflectance coefficient introduced by Nevot and Croce cannot be applied for a surface and interface having a large degree of roughness. In addition, the calculated result showed a strange phenomenon in that the amplitude of the oscillation due to interference effects increases in the case of a specific roughness of the surface. This strange result suggests that there is a serious problem in how the effect of the roughness is added to the Parratt formalism. The modification of Nevot and Croce for Fresnel coefficients has been used for only surface and interface reflection. However, the modification of the transmission coefficients has an important role when the roughness of the surface or interface is large, and the effect of diffuse scattering due to the roughness should not be ignored in the calculation. This is an essential problem regardless of the size of the roughness. An inaccurately addition of the Fresnel reflectance coefficient to the Parratt formalism causes this disagreement and strange result, and therefore the effect of roughness was properly incorporated into the theoretical formulation of the x-ray reflectivity. In this paper, we only describe these problems. Further investigation of this problem will be reported elsewhere where we propose a new formalism to derive the x-ray reflectivity for systems having surface and interfacial roughness.