From Exit Wave to Structure: Is the Phase Object Approximation Useless? ° University of Antwerp, Department of Physics, B-2020 Antwerp, Belgium °°NCEM,

From Exit Wave to Structure: Is the Phase Object Approximation Useless?

° University of Antwerp, Department of Physics, B-2020 Antwerp, Belgium°°NCEM, Lawrence Berkeley Laboratory, U.S.A.

D. Van Dyck°, P. Geuens°, C. Kisielowski°°, J.R. Jinschek°°

Cairns, Australia

July 2, 2003

Evolution in Science

describe understand design

macro micro nano

Evolution in theory

• Prediction of properties (materials, molecules from “first principles”

• Ingredients: atom positions with high precision (0.01 Å)

experiment theory

strong interaction nanostructures

sub surface information

easy to detect

use of lenses (real space Fourier space)

bright sources “A synchrotron in the electron

microscope”[1]

less radiation damage than X-rays[2]

sensitive to ionization of atoms[3].

[1] M. Brown[2] R. Henderson[3] J. Spence

Advantages of electrons:

Radiation Source Brightness Elastic Mean-Free Absorption Length Minimum Probe Size (particles/cm2 / Path (nm) (nm) (nm)

eV/steradian)

Neutrons 1024 107 108 106

X rays 1026 103 105 102

Electrons 1029 101 102 10-1

1

Source: NTEAM Project

Electron microscope

Electron microscope = coherentimaging

Image wave = object wave * impuls response

Deblurring (deconvolution) of the electron microscope1) retrieve image phase: holography2) deconvolute the impulse response function3) reconstruct exit (object) wave

OB*P

IIM = |IM|2

Focus variation method

transport of intensity equation

Phase of total exit wave5 Al: Cu

Courtesy C. Kisielowski (NCEM,Berkeley)

Phase of total exit waveAu [110] wedge

Meyer R.R. et al., Science 289 (2000), 1324-1326.

meE

h

2

),,(2),,('

zyxVEme

hzyx

d(x, y,z)2dz

' 2

dz

2

dz

EV (x,y,z)

E 1

The phase object approximation

Wavelength of the electron

Wavelenght inside the object

 Relative phase shift

Total phase shift

),(),,(),( yxVdzzyxVyx p

Transmission function:

(x,y) = exp iVp (x,y)

d(x, y,z)

V (x,y,z)dz

E /

Weak object

With

Zone axis orientation: channelling

• Atoms superimpose along beam direction

• Strong scattering

• Plane wave methods not appropriate

• Atom column as a new basis

From exit wave to structure: channelling theory

light atoms heavy atoms light atoms heavy atoms

High energy equation:

e- feels the mean potential of the atom column:

Expansion in eigenfunctions of the Hamiltonian:

1),()0,,(0,

yxcyxEnmnm

nmnm

1exp),(1),,(00,

zkE

Eiyxczyx z

nm

Enmnmnmnm

with

Energy

Delocalized states

Localized 1s state

U(x,y)

< 0.1 nm

S-state model

S-state

parameterization of the

analytic expression of the wave function:

• fast calculation

• analytic derivatives

S-state model

multislice

phaseamplitude

GaN [110] thickness 8 nm 300 keV

[001]

[110]

Exit wave of column

Amplitude peaked at the atom column position

Phase constant over the atom column

Van Dyck D., Op de Beeck M., UM 64 (1996), 99-107.

Amplitude of 1 Phase of 1

Cu

Au

Phase of total exit wave 5 Al: Cu

Amplitude of

Phase of

Courtesy C. Kisielowski, J.R. Jinschek (NCEM, Berkeley)

5 Al + Cu

Phase of

Im ()

1

Re ()

0

0

Au [110] – Vacuum wave

Courtesy C. Kisielowski, J.R. Jinschek (NCEM, Berkeley)

Re ()

= exit

wave

Im (

)

exit wave - vacuum

vacuum

=

Re ()

Im (

)

layer 1

layer 2

layer 10layer 9

Courtesy C. Kisielowski, J.R. Jinschek (NCEM, Berkeley)

Au [110] – Vacuum wave

Courtesy C. Kisielowski, J.R. Jinschek (NCEM, Berkeley)

EW phase image

EW amplitude image

exit wave - vacuum

vacuum

=

“vacuum” measured in hole

Courtesy C. Kisielowski, J.R. Jinschek (NCEM, Berkeley)

Au [110] hole (300 keV)

exit wave - vacuum

vacuum

=

Im (

)

Re ()

Courtesy C. Kisielowski, J.R. Jinschek (NCEM, Berkeley)

counts

phase [rad]

Im (

)

Re ()

amplit

ud

e

Gauss fitting: sigma 0.1 rad

Radial data distributionAveraged amplitude

Courtesy C. Kisielowski, J.R. Jinschek (NCEM, Berkeley)

Ultimate resolution = atom

Transfer functions

0 1 20,0

0,5

1,0

1/A

1/A

1/A

1/A

detector

0 1 2-1

0

1electron microscope

0 1 20,0

0,5

1,0

thermal motion

0 1 20,0

0,5

1,0

Si atom

Resolving atoms = new situation

Model based fitting (quantitative)

resolution precision

resolving refining

resolution precision

1 Å 0.01 Å

CRN

resolution

dose

ρ = 1 ÅN= 10000σCR= 0.01 Å

Å

ρ

σCR

resolution versus precision

Precision (error bar)

Is HREM able to resolve amorphous structures?

1/a

1/ρ

REC IPRO C AL SPAC E

2

2

a2N

2 2

N 1.5

a pRequirement: or

parameters data

3D HR Electron Tomography (HRET)

3

3

14

3N13a

parameters data

Amorphous structures never resolvable in 2D

N/a3 < 1.5/

2 Ångstrom resolution sufficient in 3D

Conclusions

• All object information can be obtained from the exit wave

• Single atom sensitivity

• The phase object approximation is not appropriate

• The channelling wave should be used instead

Scanning Electron Microscopy & HREM & Spectroscopy

A STEM / HRTEM : Tecnai G2

Scanning coils

Sample

Focused e-beam

HAADF Detector

Image Filter

Upgrade to HRTEM/STEM @ NCEM in 2002 First instrument of this kind in the US

•Probe size 0.13 nm (currently at NCEM: ~1 nm)•Energy resolution: 200 - 300 meV (currently: ~1eV)•Information Limit : < 0.1 nm @ 200 kV•Phase Contrast & Z-Contrast & Spectroscopy on identical areas

0

1000

2000

3000

4000

5000

6000

390 400 410 420 430 440

On core

Off core

Co

un

ts (

arb

. un

it)

Energy Loss (eV)

Current technology: HAADF-image Local energy spectrum

Dislocation core in GaN [0001] 0.2 nm

N. Browning, C. Kisielowski, LDRD, 2002-2003

Courtesy: L.M. Brown, Inst. Phys. Conf. Ser. 153 (1997), p. 17-22.

Courtesy: L.M. Brown, Inst. Phys. Conf. Ser. 153 (1997), p. 17-22.

Experiment design

Intuition is misleading

“Ideal” HREM: Cs = 0f = 0

“Ideal object”:phase object

we need a strategy

no image contrast

• Spherical aberration corrector?improves the point resolution

• Chromatic aberration corrector?improves the information limit

• Monochromator?improves the information limit reduction of electrons

Ultramicroscopy 89(2001), 275-290

Do these correctors improve the precision as well?

The electron microscope of the future

• Quantitative 3D structure determination on atomic scale

• Spectroscopy on atomic scale

• Flexibility, experiment design

• Nanolab

The ideal instrument for the characterisation of nanostructures