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