1. Electron Crystallographic Study of Incommensurate Modulated Structures Fan, Hai-fu Institute of Physics, Chinese Academy of Sciences, Beijing, China Fan, Hai-fu Institute of Physics, Chinese Academy of Sciences, Beijing, China

2. What’s a modulated structure ? Muti-dimensional direct methods of solving modulated structures Incommesurate modulation in Bi-based supercondutors from electron crystallography What’s a modulated structure ? Muti-dimensional direct methods of solving modulated structures Incommesurate modulation in Bi-based supercondutors from electron crystallography

3. What’s a Modulated Structure ? t T T = 0 (mod t) or MOD (T, t)  Commensurate modulation  superstructures T T  0 (mod t) or MOD (T, t)  Incommensurate modulation  incommensurate structures T T

4. Schematic diffraction pattern of an incommensurate modulated structure Schematic diffraction pattern of an incommensurate modulated structure a* b* q q

5. Conclusion In the reciprocal space: The diffraction pattern of an incommen- surate modulated crystal is the projection of a 4- or higher-dimensional weighted lattice In the direct space: An incommensurate modulated structure is the “hypersection” of a 4- or higher- dimensional periodic structure cut with the 3-dimensional physical space

6. Representation of one-dimensionally modulated incommensurate structures Lattice vectors in real- and reciprocal- space

7. Structure-factor formula Modulated atoms Modulated atoms situated at their average positions situated at their average positions

8. Modified Sayre Equations in multi-dimensional space Modified Sayre Equations in multi-dimensional space

9. Strategy of solving incommensurate modulated structures Strategy of solving incommensurate modulated structures i) Derive phases of main reflections ii) Derive phases of satellite reflections iii) Calculate the multi-dimensional Fourier map iv) Cut the resulting Fourier map with the 3-D ‘hyperplane’ (3-D physical space) v) Parameters of the modulation functions are measured directly on the multi-dimensional Fourier map using

10. Electron Crystallographic Study of Bi-based Superconductors using Multi-dimensional Direct Methods Electron Crystallographic Study of Bi-based Superconductors using Multi-dimensional Direct Methods

11. Why Electrons ? 1. Electrons are better for studying minute and imperfect crystalline samples 2. Electron microscopes are the only instrument that can produce simultaneously EM’s and ED’s for the same crystalline sample at atomic resolution 3. Electrons are better for revealing light atoms in the presence of heavy atoms 1. Electrons are better for studying minute and imperfect crystalline samples 2. Electron microscopes are the only instrument that can produce simultaneously EM’s and ED’s for the same crystalline sample at atomic resolution 3. Electrons are better for revealing light atoms in the presence of heavy atoms

12. Scattering of X-rays and Electrons by Different Elements Scattering of X-rays and Electrons by Different Elements Relative scattering power O O O O Sin  / Bi Sr Ca Cu X-rays Electrons

13. Bismuth bi-layer Perovskite layer Bismuth bi-layer Bi-based Superconductors n = 1 n = 2 n = 3 Bi  2201 Bi  2212 Bi  2223 Bi-O Sr-O Cu-O Sr-O Cu-O Ca-O Cu-O Sr-O Cu-O Ca-O Cu-O Ca-O Cu-O Sr-O Bi-O c Bi 2 Sr 2 Ca n  1 Cu n O 2n+4+x

14. Electron diffraction analysis of the Bi-2223 superconductor Electron diffraction analysis of the Bi-2223 superconductor Space group: P [Bbmb] 1 -1 1 a = 5.49, b = 5.41, c = 37.1Å; q = 0.117b* *The average structure is known*

15. Bi-2223 [100] projected potential Space group: P [Bbmb] 1 -1 1 a = 5.49, b = 5.41, c = 37.1Å; q = 0.117b* Space group: P [Bbmb] 1 -1 1 a = 5.49, b = 5.41, c = 37.1Å; q = 0.117b* R symM = 0.12 (Nref. =42) R symS = 0.13 (Nref. = 70) R m = 0.16 R s = 0.17

16. a3a3 a4a4 Bi-2223 cut at a 2 = 0 and projected down the a 1 axis cut at a 2 = 0 and projected down the a 1 axis Space group: P [Bbmb] 1 -1 1 a = 5.49, b = 5.41, c = 37.1Å; q = 0.117b* a 1 = a, a 2 = b  0.117d, a 3 = c, a 4 = d 4-dimensional metal atoms

17. Image Processing of Bi-2212 Image Processing of Bi-2212 EM image from Dr. S. Horiuchi Space group: N [Bbmb] 1 -1 1 a = 5.42, b = 5.44, c = 30.5Å; q = 0.21b* + c* Deconvolution Phase extension Phase extension FT  1 c b Bi Sr Cu Ca Cu Sr Bi Oxygen in Cu  O layer

18. b c O atoms on the Cu-O layer Bi-O Sr-O Bi-O Cu-O O (extra) Electron diffraction analysis of Bi-2201 R T = 0.32 R m = 0.29 R S1 = 0.29 R S2 = 0.36 R S3 = 0.52 Space group: P[B 2/b] -1]; a = 5.41, b = 5.43, c = 24.6Å,  = 90 o ; q = 0.217b* + 0.62c* Space group: P[B 2/b] -1]; a = 5.41, b = 5.43, c = 24.6Å,  = 90 o ; q = 0.217b* + 0.62c*

19. Bi-2201 Influence of thermal motion (B) and Modulation (M) to the dynamical diffraction Bi-2201 Influence of thermal motion (B) and Modulation (M) to the dynamical diffraction Experimental B and M B set to zero M set to zero B,M set to zero

20. Sample thickness: ~5Å Bi-2201 The effect of sample thickness Bi-O Sr-O Cu-O ~100Å ~200Å ~300Å Extra oxygen Oxygen in Cu-O layer

22. Acknowledgements Li Yang, Wan Zheng-hua, Fu Zheng-qing, Mo You-de, Cheng Ting-zhu Li Fang-hua Li Yang, Wan Zheng-hua, Fu Zheng-qing, Mo You-de, Cheng Ting-zhu Li Fang-hua

23. Thank you! E-mail: fan@aphy.iphy.ac.cn URL: http://cryst.iphy.ac.cn E-mail: fan@aphy.iphy.ac.cn URL: http://cryst.iphy.ac.cn