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X-Ray Diffraction from Crystals with Defects

  • Andrei BenediktovitchEmail author
  • Ilya Feranchuk
  • Alexander Ulyanenkov
Chapter
Part of the Springer Series in Materials Science book series (SSMATERIALS, volume 183)

Abstract

In the Chap.4, the X-ray diffraction from perfect crystals has been considered in details, however, the real crystals always posses the defects.

References

  1. 1.
    I. Robinson, R. Harder, Coherent x-ray diffraction imaging of strain at the nanoscale. Nat. Mater. 8, 291–298 (2009)Google Scholar
  2. 2.
    B. Jakobsen, H.F. Poulsen, U. Lienert, J. Almer, S.D. Shastri, H.O. Sorensen, C. Gundlach, W. Pantleon, Formation and subdivision of deformation structures during plastic deformation. Sci. 312(5775), 889–892 (2006)Google Scholar
  3. 3.
    Vladimir M. Kaganer, Karl K. Sabelfeld, X-ray diffraction peaks from partially ordered misfit dislocations. Phys. Rev. B 80, 184105 (Nov 2009)ADSCrossRefGoogle Scholar
  4. 4.
    V. Holy, T. Baumbach, D. Lubbert, L. Helfen, M. Ellyan, P. Mikulik, S. Keller, S.P. DenBaars, J. Speck, Diffuse x-ray scattering from statistically inhomogeneous distributions of threading dislocations beyond the ergodic hypothesis. Phys. Rev. B 77, 094102 (2008)ADSCrossRefGoogle Scholar
  5. 5.
    L.D. Landau, E.M. Lifshitz, Theory of Elasticity, vol 7, 3-rd edn, (Butterworth-Heinemann, Oxford, UK, 1986)Google Scholar
  6. 6.
    S. Timoshenko, J.N. Goodier, Theory of elasticity, 3-rd edn. (McGraw-Hill, New York, 1951)Google Scholar
  7. 7.
    M.A. Krivoglaz. X-ray and neutron diffraction in nonideal crystals, (Springer, Berlin, 1996)Google Scholar
  8. 8.
    V. Holy, J.H. Li, G. Bauer, F. Schaffler, H.-J. Herzog, Diffuse X-ray scattering from misfit dislocations in SiGe epitaxial layers with graded Ge content. J. Appl. Phys. 78(8), 5013–5021 (1995)Google Scholar
  9. 9.
    B.E. Warren, X-ray studies of deformed metals. Prog. Met. Phys. 8, 147–202 (1959)CrossRefGoogle Scholar
  10. 10.
    Ryogo Kubo, Statistical mechanics: an advanced course with problems and solutions (Elsevier, Amsterdam, 1965)Google Scholar
  11. 11.
    R.P. Feynman, Statistical Mechanics: A Set Of Lectures, (Westview Press, 1998)Google Scholar
  12. 12.
    S. Takagi, J. Phys. Soc. Japan 26, 1239 (1969)ADSCrossRefGoogle Scholar
  13. 13.
    A. Authier, Dynamical Theory of X-ray Diffraction (Oxford University Press, New York, 2001)Google Scholar
  14. 14.
    V. Holy, K.T. Gabrielyan, Dyson and Bethe-Salpeter equations for dynamical X-ray diffraction in crystals with randomly placed defects. Phys. Stat Solidi. 140(1), 39–50 (1987)Google Scholar
  15. 15.
    L.A. Apresjan, J.A. Kravcov, Teorija perenosa izlučenija: Statističeskie i volnovye aspekty. Nauka, 1983.Google Scholar
  16. 16.
    A.N. Polyakov, F.N. Chukhovskii, D.I. Piskunov, Dynamic scattering of X-rays in disordered crystals: statistical theory. Zh. Eksp. Theor. Phys. 99, 589–609 (1991)Google Scholar
  17. 17.
    Z.G. Pinsker Dynamical Scattering of X-Rays in Crystals (Springer, New York,1978)Google Scholar
  18. 18.
    M.A. Naimark, Linear Differential Operators (F. Ungar Pub. Co, New York, 1968)Google Scholar
  19. 19.
    H. Feshbach, P.M. Morse, Methods of Theoretical Physics (McGraw-Hill, New York, 1953)Google Scholar
  20. 20.
    T. Ungár, J. Gubicza, G. Ribárik, A. Borbély, Crystallite size distribution and dislocation structure determined by diffraction profile analysis: principles and practical application to cubic and hexagonal crystals. J. Appl. Crystallogr. 34(3), 298–310 (2001)CrossRefGoogle Scholar
  21. 21.
    M. Leoni, J. Martinez-Garcia, Paolo Scardi, Dislocation effects in powder diffraction. J. Appl. Crystallogr. 40(4), 719–724 (Aug 2007)Google Scholar
  22. 22.
    V.M. Kaganer, R. Köhler, M. Schmidbauer, R. Opitz, B. Jenichen, X-ray diffraction peaks due to misfit dislocations in heteroepitaxial structures. Phys. Rev. B 55, 1793–1810 (1997)ADSCrossRefGoogle Scholar
  23. 23.
    Jens Lothe, John Price Hirth, Theory of Dislocations (Krieger Pub. Co., New York, 1982)Google Scholar
  24. 24.
    S. Suresh L.B. Freund, Thin Film Materials: Stress Defect Formation and Surface Evolution (Cambridge University Press, Cambridge, 2004)Google Scholar
  25. 25.
    John E. Ayers, Heteroepitaxy of Semiconductors: Theory, Growth, and Characterization (Taylor and Francis, Boca Raton, FL, 2007)Google Scholar
  26. 26.
    K.P. Ryaboshapka, Physics of X-ray Scattering from Deformed Crystals (Navukova Dumka, Kiev, 1993)Google Scholar
  27. 27.
    J.D. Eshelby, W.T. Read, W. Shockley, Anisotropic elasticity with applications to dislocation theory. Acta Metall. 1(3), 251–259 (1953)CrossRefGoogle Scholar
  28. 28.
    A.N. Stroh, Dislocations and cracks in anisotropic elasticity. Phil. Mag. 3(30), 625–646 (1958)MathSciNetADSCrossRefzbMATHGoogle Scholar
  29. 29.
    J. Martinez-Garcia, M. Leoni, P. Scardi, A general approach for determining the diffraction contrast factor of straight-line dislocations. Acta Crystallogr. A 65(2), 109–119 (2009)Google Scholar
  30. 30.
    A.K. Head, Edge dislocations in inhomogeneous media. Proc. Phys. Soc. London, Sect. B 66(9), 793 (1953)ADSCrossRefzbMATHGoogle Scholar
  31. 31.
    B. Yu, Bolkhovityanov, L.V Sokolov, Ge-on-Si films obtained by epitaxial growing: edge dislocations and their participation in plastic relaxation. Semicond. Sci. Technol. 27(4), 043001 (2012)Google Scholar
  32. 32.
    A. Benediktovitch, F. Rinaldi, S. Menzel, K. Saito, T. Ulyanenkova, T. Baumbach, I.D. Feranchuk, A. Ulyanenkov, Lattice tilt, concentration, and relaxation degree of partly relaxed InGaAs/GaAs structures. Phys. Status Solidi (a). 208(11), 2539–2543 (2011)Google Scholar
  33. 33.
    Vladimir M. Kaganer, Karl K. Sabelfeld, Short range correlations of misfit dislocations in the X-ray diffraction peaks. Phys. Status Solidi (a). 208(11), 2563–2566 (2011)Google Scholar
  34. 34.
    Péter Dusán Ispánovity, István Groma, Géza Györgyi, Evolution of the correlation functions in two-dimensional dislocation systems. Phys. Rev. B, 78, 024119 (Jul 2008)Google Scholar
  35. 35.
    V.M. Kaganer, O. Brandt, H. Riechert, K.K. Sabelfeld, X-ray diffraction of epitaxial films with arbitrarily correlated dislocations: Monte Carlo calculation and experiment. Phys. Rev. B 80, 033306 (2009)Google Scholar
  36. 36.
    V.M. Kaganer, K.K. Sabelfeld, X-ray diffraction peaks from correlated dislocations: Monte Carlo study of dislocation screening. Acta Crystallogr. A 66(6), 703–716 (2010)Google Scholar
  37. 37.
    D.K. Satapathy, V.M. Kaganer, B. Jenichen, W. Braun, L. Daweritz, K.H. Ploog, Periodic array of misfit dislocations at the \({\rm {MnAs}}{\rm {GaAs}}\) interface studied by synchrotron X-ray diffraction. Phys. Rev. B 72, 155303 (Oct 2005)Google Scholar
  38. 38.
    B. Yu. Bolkhovityanov, O.P. Pchelyakov, S.I. Chikichev, Silicon germanium epilayers: physical fundamentals of growing strained and fully relaxed heterostructures. Phys. Usp. 44(7), 655–680 (2001)Google Scholar
  39. 39.
    S. Danis, V. Holy, J. Stangl, G. Bauer, Diffuse X-ray scattering from graded sige/si layers. Europhys. Lett. 82(6), 66004 (2008)Google Scholar
  40. 40.
    V.M. Kaganer, K.K. Sabelfeld, X-ray diffraction peaks from correlated dislocations: Monte Carlo study of dislocation screening. Acta Crystallogr. A 66, 703–716 (2010)Google Scholar
  41. 41.
    A. Benediktovich, A. Ulyanenkov, F. Rinaldi, K. Saito, V. Kaganer, Concentration and relaxation depth profiles of In\(_x\)Ga\(_{1-x}\)As/GaAs and GaAs\(_{1-x}\)P\(_x\)/GaAs graded epitaxial films studied by x-ray diffraction. Phys. Rev. B 84, 035302 (2011)ADSCrossRefGoogle Scholar
  42. 42.
    A. Zhylik, A. Benediktovich, A. Ulyanenkov, H. Guerault, M. Myronov, A. Dobbie, D.R. Leadley, T. Ulyanenkova, High-resolution X-ray diffraction investigation of relaxation and dislocations in SiGe layers grown on (001), (011) and (111) Si substrates. J. Appl. Phys. 109, 123714 (2011)ADSCrossRefGoogle Scholar
  43. 43.
    A. Zhylik, F. Rinaldi, M. Myronov, K. Saito, S. Menzel, A. Dobbie, D.R. Leadley, T. Ulyanenkova, I.D. Feranchuk, A. Ulyanenkov, High-resolution reciprocal space mapping of distributed bragg reflectors and virtual substrates. Phys. Status Solidi a. 208, 2582–2586 (2011)Google Scholar
  44. 44.
    J. Tersoff, Dislocations and strain relief in compositionally graded layers. Appl. Phys. Lett. 62(7), 693–695 (1993)ADSCrossRefGoogle Scholar
  45. 45.
    J. Tersoff, Erratum: Dislocations and strain relief in compositionally graded layers [appl. phys. lett. 62, 693 (1993)]. Appl. Phys. Lett. 64(20), 2748–2748 (1994)ADSCrossRefGoogle Scholar
  46. 46.
    V.M. Kaganer, O. Brandt, A. Trampert, K.H. Ploog, X-ray diffraction peak profiles from threading dislocations in GaN epitaxial films. Phys. Rev. B 72, 045423 (Jul 2005)Google Scholar
  47. 47.
    M.A. Moram, M.E. Vickers, X-ray diffraction of iii-nitrides. Rep. Prog. Phys. 72(3), 036502 (2009)ADSCrossRefGoogle Scholar
  48. 48.
    Ferenc F. Csikor, István Groma, Probability distribution of internal stress in relaxed dislocation systems. Phys. Rev. B 70, 064106 (Aug 2004)ADSCrossRefGoogle Scholar
  49. 49.
    M. Wilkens, The determination of density and distribution of dislocations in deformed single crystals from broadened X-ray diffraction profiles. Phys. Status Solidi (a), 2(2), 359–370 (1970)Google Scholar
  50. 50.
    A.J.C. Wilson, X-ray diffraction by random layers: ideal line profiles and determination of structure amplitudes from observed line profiles. Acta Crystallogr. 2(4), 245–251 (Aug 1949)CrossRefGoogle Scholar
  51. 51.
    M. Wilkens. In Fundamental aspects of dislocation theory, pages 1195–1221. National Bureau of Standards, Institute for Materials Research, U.S. Govt. Print, April 1969.Google Scholar
  52. 52.
    Gabor Ribarik and Tamas Ungar. Characterization of the microstructure in random and textured polycrystals and single crystals by diffraction line profile analysis. Materials Science and Engineering: A, 528(1):112–121, 2010. Special Topic Section: Local and Near Surface Structure from Diffraction.Google Scholar
  53. 53.
    B.E. Warren, X-Ray Diffraction (Courier Dover Publications, New York, 1990)Google Scholar
  54. 54.
    Levente Balogh, Géza Tichy, Tamás Ungár, Twinning on pyramidal planes in hexagonal close packed crystals determined along with other defects by X-ray line profile analysis. J. Appl. Crystallogr. 42(4), 580–591 (Aug 2009)CrossRefGoogle Scholar
  55. 55.
    M. Barchuk, V. Holý, D. Kriegner, J. Stangl, S. Schwaiger, F. Scholz, Diffuse X-ray scattering from stacking faults in \(a\)-plane GaN epitaxial layers. Phys. Rev. B 84, 094113 (Sep 2011)ADSCrossRefGoogle Scholar
  56. 56.
    L.D. Landau, Phys. Z. Soviet. 12, 579 (1937)Google Scholar
  57. 57.
    Sterling Hendricks, Edward Teller, X-ray interference in partially ordered layer lattices. J. Chem. Phys. 10(3), 147–167 (1942)ADSCrossRefGoogle Scholar
  58. 58.
    H. Jagodzinski, Eindimensionale Fehlordnung in Kristallen und ihr Einfluss auf die Röntgeninterferenzen. I. Berechnung des Fehlordnungsgrades aus den Röntgenintensitäten. Acta Crystallogr. 2(4), 201–207 (Aug 1949)Google Scholar
  59. 59.
    H. Jagodzinski, Eindimensionale Fehlordnung in Kristallen und ihr Einfluss auf die Röntgeninterferenzen. II. Berechnung der Fehlgeordnetendichtesten Kugelpackungen mit Wechselwirkungen der Reichweite 3. Acta Crystallogr. 2(4), 208–214 (Aug 1949)Google Scholar
  60. 60.
    J. Kakinoki, Y. Komura, Diffraction by a one-dimensionally disordered crystal. I. The intensity equation. Acta Crystallogr. 19(1), 137–147 (Jul 1965)CrossRefGoogle Scholar
  61. 61.
    J. Kakinoki, Diffraction by a one-dimensionally disordered crystal. II. Close-packed structures. Acta Crystallogr. 23(6), 875–885 (Dec 1967)CrossRefGoogle Scholar
  62. 62.
    M.M.J. Treacy, J.M. Newsam, M.W. Deem, A general recursion method for calculating diffracted intensities from crystals containing planar faults. Proc. R. Soc. Lond. A 433(1889), 499–520 (1991)ADSCrossRefzbMATHGoogle Scholar
  63. 63.
    Matteo Leoni, Alessandro F. Gualtieri, Norberto Roveri, Simultaneous refinement of structure and microstructure of layered materials. J. Appl. Crystallogr. 37(1), 166–173 (Feb 2004)CrossRefGoogle Scholar
  64. 64.
    L. Velterop, R. Delhez, H. de Th, E.J. Keijser, Mittemeijer, D. Reefman, X-ray diffraction analysis of stacking and twin faults in f.c.c. metals: a revision and allowance for texture and non-uniform fault probabilities. J. Appl. Crystallogr. 33(2), 296–306 (2000)Google Scholar
  65. 65.
    V.S. Kopp, V.M. Kaganer, J. Schwarzkopf, F. Waidick, T. Remmele, A. Kwasniewski, M. Schmidbauer, X-ray diffraction from nonperiodic layered structures with correlations: analytical calculation and experiment on mixed Aurivillius films. Acta Crystallogr. A 68(1), 148–155 (2012)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Andrei Benediktovitch
    • 1
    Email author
  • Ilya Feranchuk
    • 1
  • Alexander Ulyanenkov
    • 2
  1. 1.Physics DepartmentBelarusian State UniversityMinskBelarus
  2. 2.Rigaku Europe SEEttlingenGermany

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