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

What’s a Modulated Structure ?What’s a Modulated Structure ?

tT

T = 0 (mod t) or MOD (T, t) Commensurate modulation

superstructures

TT

T 0 (mod t) or MOD (T, t) Incommensurate modulation

incommensurate structures

T

T

a*a*

b*b*qq

Schematic diffraction pattern

of an incommensurate modulated structure

Schematic diffraction pattern

of an incommensurate modulated structure

1 2 3* * *q q q q a b c1 1 2 2 3 4 43h h h h h bb b b

1 2 3 4

1 2 3 4

, , ,

( *,0), ( *,0), ( *,0), ( , )

h h h k h l h m b a b b b c b q d

* * *+h k l m h a b c q

ConclusionConclusionIn the reciprocal space:In the reciprocal space:

The diffraction pattern of an incommen-surate modulated crystal is the projection of a 4- or higher-dimensional weighted lattice

The diffraction pattern of an incommen-surate modulated crystal is the projection of a 4- or higher-dimensional weighted lattice

In the direct space: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

An incommensurate modulated structure is the “hypersection” of a 4- or higher-dimensional periodic structure cut with the 3-dimensional physical space

Representation of one-dimensionally modulated incommensurate structuresRepresentation of one-dimensionally

modulated incommensurate structures

1 1 2 2 3 3 4 4h h h h h b b b b

1 1 2 2 3 3 4 4x x x x x a a a a

, ( , 1,2,3,4)i j ij i j a b

1 1

2 2

3 3

4 (0, )

q

q

q

a a d

a b d

a c d

a d

1

2

3

4

( *,0)

( *,0)

( *,0)

( , )

b a

b b

b c

b q d

1 2 3* * *q q q q a b c

Lattice vectors in real- and reciprocal- space

1 1 2 2 3 31

( ) ( )exp[ 2 ( )]N

j j jjjF if h x h x h x

h h

Structure-factor formulaStructure-factor formula

1 1

4 3 4 3

0 0

1 1 2 2 3 3

4 4 3 (3 )

( ) ( ) ( , , )

exp 2 ( )

( )

on njjj

j j j

j n j n

f h d df x x x xP

i h U h U h U

h x h x

h

Modulated atoms

Modulated atoms

situated at theiraverage positionssituated at theiraverage positions

Modified Sayre Equations in multi-dimensional space

Modified Sayre Equations in multi-dimensional space

( ) ( ) ( )F F FV

h'

h h' h h'm m

m s

ss

( ) ( ) ( )

2 ( ) ( )

( ) ( )

F F FV

F F

F F

h'

h'

h'

h h' h h'

h' h h'

h' h h'

m m

m

s

s

( ) ( ) ( )

2 ( ) ( )

F F FV

F FV

h'

h'

h h' h h'

h' h h'm mm ( ) ( ) ( )F F FV

h'

h h' h h'

m m

m s

ss

( ) ( ) ( )

2 ( ) ( )

( ) ( )

F F FV

F F

F F

h'

h'

h'

h h' h h'

h' h h'

h' h h'ms s2

( ) ( ) ( )F F FV

h'

h h' h h'ms( ) ( ) ( )mF F FV

h'

h h' h h'

Strategy of solvingincommensurate modulated structures

Strategy of solvingincommensurate modulated structures

i) Derive phases of main reflectionsii) Derive phases of satellite reflectionsiii) Calculate the multi-dimensional Fourier mapiv) 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 ms( ) ( ) ( )mF F F

V

h'

h h' h h'

ms s2

( ) ( ) ( )F F FV

h'

h h' h h'

using

m mm ( ) ( ) ( )F F FV

h'

h h' h h'

using

Electron Crystallographic Study

of

Bi-based Superconductors

using

Multi-dimensional Direct Methods

Electron Crystallographic Study

of

Bi-based Superconductors

using

Multi-dimensional Direct Methods

Why Electrons ?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

Scattering of X-rays and Electronsby Different Elements

Scattering of X-rays and Electronsby Different Elements

Relative scattering power

Relative scattering power

OO OO

Sin/Sin/

BiBi

SrSr

CaCaCuCu

X-raysX-rays

ElectronsElectrons

Bismuthbi-layer

Perovskitelayer

Bismuthbi-layer

Bi-based SuperconductorsBi-based Superconductors

n = 1 n = 2 n = 3Bi2201 Bi2212 Bi2223

Bi-OBi-O

Sr-OCu-OSr-O

Sr-OCu-OCa-OCu-OSr-O

Sr-OCu-OCa-OCu-OCa-OCu-OSr-O

Bi-OBi-O

Bi-OBi-O

Bi-OBi-O Bi-O

Bi-O Bi-OBi-O

c

Bi2Sr2Can1CunO2n+4+xBi2Sr2Can1CunO2n+4+x

Electron diffraction analysis ofthe Bi-2223 superconductor

Electron diffraction analysis ofthe Bi-2223 superconductor

Space group: P [Bbmb] 1 -1 1a = 5.49, b = 5.41, c = 37.1Å; q = 0.117b*

*The average structure is known*

Bi-2223 [100] projected potentialBi-2223 [100] projected potentialSpace group: P [Bbmb] 1 -1 1a = 5.49, b = 5.41, c = 37.1Å; q = 0.117b*Space group: P [Bbmb] 1 -1 1a = 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

a3

a4

Bi-2223Bi-2223

cut at a2 = 0 and projected down the a1 axis

cut at a2 = 0 and projected down the a1 axis

Space group: P [Bbmb] 1 -1 1a = 5.49, b = 5.41, c = 37.1Å; q = 0.117b*

a1 = a, a2 = b 0.117d,a3 = c, a4 = d

4-dimensional metal atoms4-dimensional metal atoms

Image Processing of Bi-2212Image Processing of Bi-2212Space 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

FTFT

FT-1FT-1Phase extension

Phase extension

Enhanced image

Original image

Image Processing of Bi-2212 (continued)Image Processing of Bi-2212 (continued)

BiBiSrSr

CuCuCaCa

CuCuSrSrBiBi

bb

cc

22

8844

11

Oxygenin Cu-O layerOxygenin Cu-O layer

b

c

O atoms on the Cu-O layer

O atoms on the Cu-O layer

Bi-OBi-OSr-OSr-O

Sr-OSr-OBi-OBi-O

Cu-OCu-O

O (extra)O (extra)

Electron diffraction analysis of Bi-2201Electron diffraction analysis of Bi-2201

RT = 0.32 Rm = 0.29 RS1 = 0.29 RS2 = 0.36 RS3 = 0.52

RT = 0.32 Rm = 0.29 RS1 = 0.29 RS2 = 0.36 RS3 = 0.52

Space group: P[B 2/b] -1]; a = 5.41, b = 5.43, c = 24.6Å,= 90o; q = 0.217b* + 0.62c*Space group: P[B 2/b] -1]; a = 5.41, b = 5.43, c = 24.6Å,= 90o; q = 0.217b* + 0.62c*

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

Sample thickness: ~5Å Bi-2201Bi-2201

The effect of sample

thickness

Bi-O

Bi-O

Sr-O

Sr-OCu-O

~100Å~200Å~300Å

Extra oxygenOxygen inCu-O layer