What’s a modulated structure ? Muti-dimensional direct methods of

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

2. T T T T T t T = 0 (mod t) or MOD (T, t) = 0 Commensurate modulation Þ superstructures T¹ 0 (mod t) or MOD (T, t) ¹ 0 Incommensurate modulation Þ incommensurate structures What’s a Modulated Structure ?

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

4. 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

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

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

7. Modified Sayre Equations in multi-dimensional space

8. using using 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

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

10. 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

11. Bi X-rays Electrons Sr Cu Ca Scattering of X-rays and Electrons by Different Elements Relative scattering power Sinq /l ~ 0.31 O O

12. Bismuth bi-layer Perovskite layer Bismuth bi-layer Bi-based Superconductors Bi2Sr2Can-1CunO2n+4+x n = 1 n = 2 n = 3 Bi-2201 Bi-2212 Bi-2223 Bi-O Bi-O Bi-O Bi-O Bi-O Bi-O Sr-O Cu-O Ca-O Cu-O Ca-O Cu-O Sr-O c Sr-O Cu-O Ca-O Cu-O Sr-O Sr-O Cu-O Sr-O Bi-O Bi-O Bi-O Bi-O Bi-O Bi-O

13. 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*

14. Bi-2223 [100]projected potential Space group: P [Bbmb] 1 -1 1 a = 5.49, b = 5.41, c = 37.1Å; q = 0.117b* RsymM = 0.12 (Nref. =42) RsymS = 0.13 (Nref. = 70) Rm = 0.16 Rs = 0.17

15. a3 a4 Bi-2223 4-dimensional metal atoms cut at a2 = 0 and projected down the a1 axis Space group: P [Bbmb] 1 -1 1 a = 5.49, b = 5.41, c = 37.1Å; q = 0.117b* a1 = a, a2 = b -0.117d, a3 = c, a4 = d

16. FT FT-1 Image Processing of Bi-2212 Space group: N [Bbmb] 1 -1 1 a = 5.42, b = 5.44, c = 30.5Å; q = 0.22b* + c* EM image from Dr. S. Horiuchi Phase extension

17. 2 Bi Sr Cu Ca Cu Sr Bi 1 8 Oxygen in Cu-O layer 4 Image Processing ofBi-2212(continued) Original image Enhanced image c b

18. O atoms on the Cu-O layer Bi-O c Sr-O Cu-O Sr-O Bi-O b O (extra) Electron diffraction analysis of Bi-2201 Space group: P[B 2/b] -1]; a = 5.41, b = 5.43, c = 24.6Å, b = 90o; q = 0.217b* + 0.62c* RT = 0.32 Rm = 0.29 RS1 = 0.29 RS2 = 0.36 RS3 = 0.52

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

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