Aberration Correction and Possible Structural Tomography in Complex TEM

Fu-Rong Chen(1) and Ji-Jung Kai(1)

Dept. of Engineering and System Science, National Tsing-Hua University, Hsin Chu, Taiwan

-8.23.2003 Beijing TEM conference

It is very easy to answer many of these fundamental biological questions; you just look at the thing! Make the microscope one hundred times more powerful, and many problems of biology would be made very much easier.

It would be very easy to make an analysis of any complicated chemical substance; all one would have to do would be to look at it and see where the atoms are. The only trouble is that the electron microscope is one hundred times too poor.

Better electron microscopes

-Richard F. Feynman-12.29.1959 American Physics Society, CIT-(There’s plenty of room at the bottom)

Develop new tool to discover new science !

=/2(Cs34+2f2)

ElectronsSpecimen

g 2go-g-2g

Diffraction Plane

Image Plane

o

Iii *

Microscope transfer function (Cs ,Cc,Δf…)

The total phase shift due to spherical aberration and defocus is

)](exp[ iAe

)()exp(1 HPiei

Exit wave is blurred by the function exp(-iχ)

f

gg*=Ig

(phase lost)

(phase lost)

Phase

Reciprocal Space Real Space

=C∮V(r,z)dz-e/h∫∫B⊥(r).dA

V1(r,z) V2(r,z)

V2(r,z)

V1(r,z)

(h,k,l): atomic positions

Phase Retrieval

a) Transport Intensity Equation(TIE)

Kinematic Diffraction

Electron Dynamic Diffraction ?(oversampling, phase extension)

Resolution extension and exit wave reconstruction in complex TEMF.-R. Chen et. al., Ultramicroscopy (2003), PRL(submitted) JEM(1999), JEM(2001)

Non-interferometric Phase Retrieval MethodNon-interferometric Phase Retrieval Method

b) Electro-Static Phase Plate(Zernike was awarded Nobel prize at 1953)

I1 I2 I3Initialwave

Propagating wave

f

(I(r,0) )=-kzI(r,z)

Amplitude

phase

I=1+[AF-1(cosP(H))+ F-1(sinP(H))]

Reciprocal Space Real Space

Process Flow (Images 3D Structure)

Exit Wave Reconstruction(aberration correction)

Exit Wave->Structure(quantification of EW)

Structural Tomography ?

Process Flow (Images 3D Structure)

Exit Wave Reconstruction(aberration correction)

Exit Wave->Structure(quantification of EW)

Structural Tomography ?

CTF

PhaseModulus

PhaseModulus

Initial Phase of Imageusing TIE

under-focusunder-focus over-focusover-focusf = f = ±±60 nm60 nm

zI(r,z) Reconstructed Phase

Phase objectPhase object (I(r,0) )=-kzI(r,z)

Process Flow (Images 3D Structure)

Exit Wave Reconstruction(aberration correction)

Exit Wave->Structure(quantification of EW)

Structural Tomography ?

Refine the phase bySelf-consistent propagation

(Gerchberg-Saxton algorithm)

CTF

PhaseModulus

PhaseModulus

1i

2i

.

.

ni

e= (ni)N1

1i

2i

.

.

ni

√ Ini

Initial Phase of Imageusing TIE

)()exp(1 HPiei

Process Flow (Images 3D Structure)

Exit Wave Reconstruction(aberration correction)

Exit Wave->Structure(quantification of EW)

Structural Tomography ?

Refine the phase bySelf-consistent propagation

(Gerchberg-Saxton algorithm)

CTF

PhaseModulus

PhaseModulus

Initial Phase of Imageusing TIE

Aberration correction(linear imaging)e=Aoexp(io)

1i

2i

.

.

ni

W

exp((f1))P1(H)

exp((f2))P2(H)

.

. exp((f2))P2(H)

e

e=F-1( )W*

W* W

1i

Process Flow (Images 3D Structure)

Exit Wave Reconstruction(aberration correction)

Exit Wave->Structure(quantification of EW)

Structural Tomography ?

Refine the phase bySelf-consistent propagation

(Gerchberg-Saxton algorithm)

CTF

PhaseModulus

PhaseModulus

Initial Phase of Imageusing TIE

Aberration correction(linear imaging)e=Aoexp(io)

Refinement byNon-Linear imaging

A=Ao+dA=o+d

I(g)=∫e(g+g’) e*(g’)T(g+g’,g’)dg’

I(r)=ii*+()2n/n!(2ni2ni*)

2=fi2=(Ii(r)-I(r)i

exp)2

dA (FA)2 FAF FAFo

d FAF F)2 FFo

FA=fi

F=

fi

GGG

ee

ee

ee

dGGGGTGGG

GTG

GTG

GI

',0'

*

*

**

')','()'()'(

),0()()0(

),0()()0(

)(

Linear image

Non-linearimage

G2

G1

0

Part 1:Part 1:Consider Non-linear Contributions to Consider Non-linear Contributions to

the Imagethe Image

)','()','()'()'()','( * GGGEGGGEGpGGpGGGT s

GG

ee dGGGGTGGGGI'

* ')','()'()'()(

Temporal coherence spatial coherence

))(exp()( GiGp

})]'()'([)/(exp{)','( 22 GGGGGGEs

Pure phase transfer function

22 ))'(()'()'(2))'(( GGGGGG

For FEG TEM α<<1, this term can be ignored

]')'()exp[()',0()0,'()','( 222 GGGGEGGEGGGE

i *

i

Represent the diffraction wave in the regular Fourier optics approach

'')')('()'()'()'()!2

(

'')')('()'()'()'()(

')'()'()'()'()(

44**4

22**2

**

dGGGGGtGGtGGG

dGGGGGtGGtGGG

dGGtGGtGGGGI

ee

ee

ee

'

** )'()'()'()'()(G

ee GtGGGtGGGI

For FEG TEM, ~1

]')'()exp[()]'()'()(

2exp[ 2222

GGGGGG

Taylor expansion

)]()([)!2

(

)]()([)()()()(

*444

*222*

GGGG

GGGGGGGI

dd

dddd

)]()([)!

()()()( *222

1

* GGGGn

GGGI ddnn

ndd

)]()([)!

()()()( *222

1

* rrn

rrrI in

inn

nii

sei EEir )~exp()(

Fourier Transform

Linear image Non-linear image

where

Higher resolution information

)exp( iAe

These images were recorded using a JEOL 3000F FEGTEM, the lens aberration parameters are Cs=0.6mm at 300kV, focal spread f=4 nm, divergent angle =0.15 mrad and three-fold astigmatism a3=855.8 nm and a3=117.11 mrad.

f ( 30 im

ages)

Complex Oxide

(NbW)O3

Reconstructed exit wave

Amplitude

Phase

ie Ae

Cation: (Nb, W)

Anion: (O)

The Nb16W18O94 has a structure of M30-3xM4xO90+x with lattice constants a= 1.2251 nm, b= 3.6621 nm, and c=0.394 nm.

Amplitudea

b

Phase

Super High Resolution

• How to achieve Super How to achieve Super High ResolutionHigh Resolution phase extensionphase extension Exit wave → structureExit wave → structure

Information limit

Reconstructed Exit wave

The phase had been corrected inside The phase had been corrected inside the information limitthe information limit

Fourier Transform

1) The retrieved Exit Wave contains structural information up to

“information limit”

2) To extend the structural information beyond “information limit”,

we need diffraction intensities for “resolution extension”.

Phase extensionPhase extensionExit wave reconstructionExit wave reconstruction

For 200 kV FEG TEM phase transfer function (in Scherzer defocus)

PhasesPhases

AmplitudesAmplitudes

(1(1Å)Å)-1-1

(2(2Å)Å)-1-1

From images → From images → phases & amplitudesphases & amplitudes inside the small cycle inside the small cycleFrom diffraction intensities→ From diffraction intensities→ amplitudesamplitudes inside the big cycle inside the big cycle

For HRTEMFor HRTEMIt is possible to enhance the resolution by It is possible to enhance the resolution by phase extensionphase extension

Phase Phase ExtensionExtension

real spaceComplex Maximum Entropy(Complex Exit wave)

reciprocal spacereciprocal spaceGerchberg-Saxton algorithm(Electron diffraction)

Corrected Data

Resolution enhancement by phase extension

Maximum entropy image processing can extrapolatemissing information to yield a most-probable answer

I(blurred)=I(good image) O(blurring function)’

L=-I(good)log( )+(I(good) O-I(blurring))I(good)I(initial)

PhasePhaseextensionextension

““Correct” phase information was extrapolated to next Correct” phase information was extrapolated to next higher frequency after MEM deconvolutionhigher frequency after MEM deconvolution

Phase information before MEM deconvolutionPhase information before MEM deconvolution

The thickness of this particle is 6 nm. The lens aberration parameters used were Cs=0.6mmfocal spread, f =3 nm and divergence angle, =0.2 mrad.

-260 nm -268 nm -276 nm

-284 nm -294 nm -298 nm

CdSe nanoparticle viewed along the [112] directionCdSe nanoparticle viewed along the [112] direction

Information limit

Extrapolated information from MEM {444}

Phase of exit wavePhase of exit wave

After phase extensionAfter phase extension

{444} →0.87Å

Process Flow (Images Structure (positions and atomic type)

Exit Wave Reconstruction(aberration correction)

Structural reversionExit Wave->Structure(quantification of EW)

Structural Tomography ?

S-State model

n(R,z)≈{1+P3exp(-|P1|(R-Ri)2)[exp{-kP1P2/2Eo}-1]

P1=Eoo (energy of s-state)P2=tP3=Cj (excitation parameter)Eo=voltage of TEMk=1/

Analytic Multi-Slice

=qo=exp(iV) z

t=n*z

1=(qoPz)qo

n(R,z)≈qon-(iz/4)[qo

i(2qon-i)]

n(R,z)=Aexp[i(nV+)]

=tan-1[C1(2V)+iC2((V)2]A= |1+iC1(2V)+iC2((V)2|

C1=t(n-1/8C2=t(n-1)(2n-1)/24V=projected potential of a unit cell

Exit wave can be related tophase grating of one unit cell

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

100

200

300

400

500

600

AuPt

AgSr

Ni

Ca

Si

Al

foca

l len

gth

(A)

1/Z

Optical lens Atomic lens

Al Au

Classical Approach: Classical Approach: Multislice MMultislice Methodethod

q1(x,y)

q2(x,y)q3(x,y)

qn(x,y)

qn(x,y)=exp{iσVn(x,y)}

),(])],()],(),([),([[),(),( 2112 yxpyxpyxpyxqyxqyxqyx nne

specimen can be subdivided into thin slice The potential of each slice projected into a plane

From Exit wave to StructureFrom Exit wave to Structure

),(10 yxViq 01 q

][ 002 zpqq )exp( 2gcipz c

02

00 4qq

icq

)(4

)(4

02

02

0

2

2200

2233 000

qqqic

qqqqic

q

20

1

4

42

c

c

......)](

)()([4

)(4

220

20

0222

002

022

2

320

2220

2344

0

00

00000

qqq

qqqqqqic

qqqqqqic

q

1 unit cell

n

i

ininn qq

icq

1

2

000 4

Analytic Solution

Exit wave from analytical solution and Multislice (10nm)

MultiSlice

Modulus

Modulus

Modulus

Modulus

Phase Phase

PhasePhase

Au Si

Quantification of Exit Wave- (S-state Wave) Structure

Modulus

Phase

Cu segregates to Al grain boundary

Modulus Phase

Exit Wave of CdSe nano-particle

(reconstructed by Complex TEM)

Structural Tomography (Discrete)?

(112)

(111)

(110)

(112)(111)

(c)

f

1. Projection-Slice theoremBracewell, Aust. J. 198 (1956)2. Convolution-Backprojection AlgorithmProc. Nat. Acad. Sci. 68,2236-2240 (1971)

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祝 郭先生有八十歲的智慧 和青春的健康

Conclusions

1.一流的設備≠一流的研究2.不要忽略人類的智慧才能讓發

揮功能