Phys. Rev. Lett. Phys. Rev. A Phys. Rev. B Phys. Rev. C Phys. Rev. D Phys. Rev. E Phys. Rev. ST AB Rev. Mod. Phys. Phys. Rev. (Series I) Phys. Rev. Volume: Page/Article:

Phys. Rev. B 51, 2311–2321 (1995)

[Issue 4 – 15 January 1995 ]

[ Previous article | Next article | Issue 4 contents ]

X-ray diffraction from Si/Ge layers: Diffuse scattering in the region of total external reflection

J.–P. Schlomka, M. Tolan, L. Schwalowsky, O. H. Seeck, J. Stettner, and W. Press

Institut für Experimentalphysik, Christian-Albrechts-Universität Kiel, Olshausenstrasse 40-60, 24098 Kiel, Germany

In this paper it is shown that diffuse-scattering experiments within the region of total external reflection can be explained quantitatively using the distorted-wave Born approximation for layer systems. Three Si/Ge samples with different degrees of complexity were investigated. The simultaneous analysis of the specular reflected intensity and the diffuse scattering leads to one consistent set of interface and layer parameters, which is able to fit both the shapes and the locations of all dynamic peaks in the off-specular scans and the characteristics of the reflected intensity. Therefore the distorted-wave Born approximation seems to give a correct and complete description of the diffuse scattering in the region of total external reflection.

©1995 The American Physical Society

URL: http://link.aps.org/abstract/PRB/v51/p2311
DOI: 10.1103/PhysRevB.51.2311
PACS: 61.10.Lx, 61.10.Dp, 68.55.Jk

[ Previous article | Next article | Issue 4 contents ]

References

1. H. Dosch, in Critical Phenomena at Surfaces and Interfaces (Evanescent X-Ray and Neutron Scattering), edited by G. Höhler, Springer Tracts in Modern Physics Vol. 126 (Springer-Verlag, Berlin, 1992).
2. R. W. James, The Optical Principles of the Diffraction of X-Rays (Ox Bow, Woodbridge, 1982).
3. L. G. Parrat, Phys. Rev. 95, 359 (1954).
4. A. Abelés, Ann. Phys. (Paris) 5, 596 (1950).
5. J. Lekner, Theory of Reflection (Nijhoff, Dordrecht, 1987).
6. J. Lekner, Physica B 173, 99 (1991) [ INSPEC].
7. L. B. Lurio, T. A. Rabedeau, P. S. Pershan, I. F. Silvera, M. Deutsch, S. D. Kosowsky and B. M. Ocko, Phys. Rev. Lett. 68, 2628 (1992).
8. E. E. Fullerton, J. Pearson, C. H. Sowers, S. D. Bader, X. Z. Wu and S. K. Sinha, Phys. Rev. B 48, 17432 (1993).
9. M. K. Sanyal, S. K. Sinha, A. Gibaud, S. K. Satija, C. F. Majkrzak, and H. Homma, in Surface X-Ray and Neutron Scattering, edited by H. Zabel and I. K. Robinson, Springer Proceedings in Physics Vol. 61 (Springer-Verlag, Berlin, 1992), pp. 91–94.
10. S. K. Sinha, M. K. Sanyal, S. K. Satija, C. F. Majkrzak, D. A. Neumann, H. Homma, S. Szpala, A. Gibaud and H. Morkoc, Physica B 198, 72 (1994) [ INSPEC].
11. C. Thompson, G. Palasantzas, Y. P. Feng, S. K. Sinha and J. Krim, Phys. Rev. B 49, 4902 (1994).
12. D. E. Savage, N. Schimke, Y.-H. Phang and M. G. Lagally, J. Appl. Phys. 71, 3283 (1992) [ INSPEC].
13. Y.-H. Phang, R. Kariotis, D. E. Savage and M. G. Lagally, J. Appl. Phys. 72, 4627 (1992) [ INSPEC].
14. D. E. Savage, Y.-H. Phang, J. J. Rownd, J. F. MacKay and M. G. Lagally, J. Appl. Phys. 74, 6158 (1993) [ INSPEC].
15. M. R. Fitzsimmons and E. Burkel, Phys. Rev. B 47, 8436 (1993).
16. S. K. Sinha, E. B. Sirota, S. Garoff and H. B. Stanley, Phys. Rev. B 38, 2297 (1988).
17. G. H. Vineyard, Phys. Rev. B 26, 4146 (1982).
18. R. Pynn, Phys. Rev. B 45, 602 (1992).
19. D. K. G. de Boer, Phys. Rev. B 49, 5817 (1994).
20. D. K. G. de Boer, Phys. Rev. B (to be published).
21. Note that the specular reflected intensity calculated with the first-order DWBA causes this problem. In the present paper the specularly reflected intensity is always calculated using the exact Parrat formalism [Eq. (1)]. Fortunately, it turns out that the first-order diffuse scattering cross section as obtained by Sinha et al. (Ref. 16) is correct up to second order (Refs. 19 and 20).
22. V. Holý, J. Kubena, I. Ohlídal, K. Lischka and W. Plotz, Phys. Rev. B 47, 15896 (1993).
23. V. Holý and T. Baumbach, Phys. Rev. B 49, 10668 (1994).
24. A. V. Andreev, A. G. Michette and A. Renwick, J. Mod. Opt. 35, 1667 (1988) [ INSPEC].
25. A. P. Payne and B. M. Clemens, Phys. Rev. B 47, 2289 (1993).
26. J. Daillant and O. Bélorgey, J. Chem. Phys. 97, 5824 (1992) [ INSPEC].
27. D. Bahr, W. Press, R. Jebasinski and S. Mantl, Phys. Rev. B 47, 4385 (1993).
28. Y. Yoneda, Phys. Rev. 131, 2010 (1963).
29. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, 1964).
30. W. A. Hamilton and R. Pynn, Physica B 173, 71 (1991) [ INSPEC].
31. A. Messiah, Quantenmechanik (de Gruyter, Berlin, 1985), Band 2.
32. Similar expressions were also derived by S. K. Sinha, J. Phys. III 4, 1543 (1994) [ INSPEC]; D. K. G. de Boer (unpublished).
33. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982).
34. M. F. Barnsley, R. L. Devaney, B. B. Mandelbrot, H.-O. Peitgen, D. Saupe, and R. F. Voss, The Science of Fractal Images (Springer-Verlag, Berlin, 1988).
35. H. O. Peitgen, H. Jürgens, and D. Saupe, Bausteine des Chaos-Fraktale (Springer-Verlag, Berlin, 1992).
36. Although the correlation function given by Eq. (4) allows values h_j>1, it can be shown that for this particular C_j(R) with h_j>1, no real surface exists.
37. R. Chiarello, V. Panella, J. Krim and C. Thompson, Phys. Rev. Lett. 67, 3408 (1991).
38. G. Palasantzas, Phys. Rev. B 48, 14472 (1993).
39. G. Palasantzas, Phys. Rev. B 49, 10544 (1994).
40. Note that this is only true for small-q_z values. For large-q_z values the Born approximation yields that the diffuse scattering cross section is not proportional to the PSD, but to the Fourier transform of lcurl exp [q²_zC( R vec )]-1 rcurl /q²_z (Ref. 16).
41. H. Sirringhaus, C. Schwarz, and H. v. Känel (unpublished).
42. Z. H. Ming, A. Krol, Y. L. Soo, Y. H. Kao, J. S. Park and K. L. Wang, Phys. Rev. B 47, 16373 (1993).
43. A correlation function C_jk(R) between interfaces has to fulfill the additional conditions for all R vec: |C_jk( R vec )|²<= C_j(0) C_k(0) and |C_jk( R vec )|²<= 1/2 [C_j(0)+C_k(0)] [see, e.g., J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley-Interscience, New York, 1971)]. The particular function given by Eq. (5) always fulfills these conditions.
44. R. J. Temkin, G. A. N. Connell, and W. Paul, in Amorphous and Liquid Semiconductors, edited by J. Stuke and W. Brenig (Taylor & Francis, London, 1974).
45. L. Brügemann, R. Bloch, W. Press and M. Tolan, Acta Crystallogr. Sec. A 48, 688 (1992).
46. Numerical tests show that for small-h_j values (h_j<0.2) and rather large correlation lengths xi _j, the diffuse scattering shows a narrow peak at the specular condition (Refs. 16 and 18).
47. J. Daillant and O. Bélorgey, J. Chem. Phys. 97, 5837 (1992) [ INSPEC].
48. O. H. Seeck, P. Müller-Buschbaum, M. Tolan, and W. Press (unpublished).
49. A. Gibaud, G. Vignaud and S. K. Sinha, Acta Crystallogr. Sec. A 49, 642 (1993).
50. The specularly reflected beam was fitted assuming a Lorenzian line shape (see also Ref. 49) and the calculated intensities were convoluted with the resolution of the diffractometer. In addition, corrections which takes into account the narrow slit in front of the detector and the small incidence angles were done.
51. International Tables for Crystallography, edited by A. J. C. Wilson (Kluwer Academic, Dordrecht, 1992), Vol. C.
52. L. Brügemann, R. Bloch, W. Press and P. Gerlach, J. Phys. C 2, 8869 (1990).
53. M. Kardar, G. Parisi and Y. Zhang, Phys. Rev. Lett. 56, 889 (1986) [SPIRES].

[ Previous article | Next article | Issue 4 contents ]