X-ray scattering on rough and profiled surfaces: rigorous analysis and a non-linear model of film growth

 

 

Understanding of the evolution of surface profiles and roughness of gratings, random asperities, quantum wires and dots, nanowhiskers, etc., during growth is required for further technological improvement. X-ray scattering on surfaces with different types of nano-asperities (periodic, random, and self-organized ones, as well as their combinations) is considered based on the rigorous electromagnetic theory and a non-linear continuum model of surface growth.

It is well-known that linear continuum models of surface growth and evolution (see, for example, Ref. [1]) cannot reasonably reproduce the PSD spectra of the films deposited on a strongly profiled substrate or on a flat substrate with an rms roughness/correlation length of more than about a few nm. The linear continuum equation used in the models does not properly treat the complexity of the processes of island nucleation, growth and coalescence on profiled surfaces leading to a significant discrepancy of the predicted and measured PSD spectra [2]. The non-linear growth model proposed in the present work accounts for the non-linear growth effects and deals with a substrate that is a general rough grating with a groove spacing larger than the width/correlation length of the substrate’s asperities. The temporal evolution of the surface height distribution (h(r,t) (r is the radius-vector and t is time) is a function of its derivatives and a stochastic term η(r,t):

 

 

The function f is the sum of linear gradⁿh(r,t) and non-linear grad^l([gradⁿh(r,t)]^k), (l, k, n ∩ N), terms; the latter describes nonlinear effects and influence of the surface profile on the growth kinetics and resulting surface morphology. This model is capable simulating the growth of various crystalline, polycrystalline, and amorphous multilayers on planar and structured substrates. The model is used to predict the roughening and smoothing behaviors of Mo/Si, Al/Zr, and GaAs/AlGaAs films.

The boundary integral equation method (MIM, Ref. [3]), where the border structure is considered as a 2D grating with wavelength λ to period d ratios of much less than one, is used to obtain absolute specular and non-specular x-ray intensities. Note that even for 1-D surfaces and, especially, in the EUV and x-ray range, finding an exact solution to the problem of scattering of electromagnetic waves from a profiled surface is extremely difficult for any rigorous method. In spite of the convergence and accuracy problems, ensemble averaging via Monte Carlo simulations is required in order to obtain scattering intensities. A generalization of MIM has been recently proposed [4] to describe rough multilayer gratings that is suitable for present calculus (see Figure).

The authors are grateful to D. L. Voronov, Yu. V. Trushin, V. V. Yashchuk, and W. McKinney for useful discussions.

 

1. C. Herring, in The Physics of Powder Metallurgy, ed. by W. E. Kingston (McGraw-Hill, New York, 1951), 143.

2. D. L. Voronov, E. H. Anderson, R. Cambie, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, Proc. of SPIE (2011) 8139, 81390B.

3. L. I. Goray, J. Appl. Phys. (2010) 108, 033516.

4. L. I. Goray, Wav. Rand. Med. (2010) 20, 569.