This work supported by the Director, Office of Science, Office of Basic Energy Sciences,

This work supported by the Director, Office of Science, Office of Basic Energy Sciences, Materials Science Division, DOE under contract DE-AC03-76SF00098,

and Asst. Sec. for EERE, Office of FreedomCAR and Vehicle Tech. for the HTML User Program, ORNL, managed by UT-Battelle, LLC for DOE under contract DE-AC05-00OR22725.

Sub-Ångstrom Electron Microscopy

for Materials Science

NNI Interagency Workshop January 27-29, 2004 Instrumentation and Metrology for Nanotechnology Grand Challenge Workshop

National Institute of Standards and Technology, Gaithersburg, MD

Michael A. O'KeefeMaterials Sciences Division

Lawrence Berkeley National Laboratory, Berkeley, CA 94720

Lawrence F. AllardHigh-Temperature Materials Laboratory

Oak Ridge National Laboratory, Oak Ridge, TN 37831

and

Track 1- Instrumentation and Metrology for Nanocharacterization

Breakout Session: Current State of the Art



The high-resolution electron microscope can provide essential feedback in the nano- theory/construction/measurement loop.

The Role of Measurement

Rose (1994)

Measurement with the electron microscope

• Better microscope resolution leads to less de-localization of higher spatial frequencies, so better precision in measurement of atomic coordinates.OÅM -- 0.78Å (2001)

TEAM -- 0.5Å (2006?)

[1] “Correction of aberrations, a promising means for improving the spatial and energy resolution of energy-filtering electron microscopes” H. Rose, Ultramicroscopy 56 (1994) 11-25.[2] “Sub-Ångstrom resolution of atomistic structures below 0.8Å”, M.A. O’Keefe, E.C. Nelson, Y.C. Wang and A. Thust, Phil. Mag. B 81 (2001) 11, 1861-1878.[3] “HRTEM at Half-Ångstrom Resolution: from OÅM to TEAM”, M.A. O’Keefe, Microscopy & Microanalysis 9 (2003) 2: 936-937.

• Better resolution allows characterization in more viewing directions, leading to atomic-resolution 3-D images -- locate every atom in place!

• The OÅM demonstrated sub-Angstrom microscopy to 0.78Å resolution in 2001 [2], using hardware correction of three-fold astigmatism and software correction of spherical aberration.

• The next-generation TEAM is designed for sub-0.5Å resolution [3], using hardware correction with lens current stability of 0.1ppm (rms) and a mono-chromator to reduce FWHH beam-energy spread below 0.35eV at 300keV or 0.18eV at 200keV.

• In 1994, in a paper on aberration correction [1], Harald Rose showed resolution over time. He predicted 0.5Å resolution by 2015.

1.4Å simulation 1.4Å reconstructionfrom 5 images

1.6Å Scherzer-focus image

Model

"Resolution of oxygen atoms in staurolite by three-dimensional transmission electron microscopy", Kenneth H. Downing, Hu Meisheng, Hans-Rudolf Wenk, Michael A. O'Keefe, Nature 348 (1990) 525.

1990: resolution extension by focal series reconstruction.Images of oxygen atoms on JEOL-ARM 1000

O

1.51.00 Spatial Frequency (Å-1)


151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

+1

-1

0

OÅM = 20Å

0.78Å

151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

CM300FEG/UT = 36Å+1

-1

0

Resolution, information limit, and focal series - CTFs show transfer of spatial frequencies.

151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

= 0.25 millirad

+1

-1

0

1.1Å

151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

n = 2 +1

-1

0

1.03Å

151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

n = 36 +1

-1

0

0.89Å

1.7Å resolution

1.07Å info limit

Resolution (Å) 1.0

151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

OÅM with CS of 0.6mm and Delta of 20Å

Info Limit (0.78Å)

CS corrected OÅM with CS at 0.02mm and Delta of 20Å

Info Limit (0.78Å)

With CS corrected, phase reversals are gone. Better mid-range transfer

Compare OÅM (CS = 0.6mm) with CS-corrected (0.02mm)

Resolution (Å) 1.0

What does aberration-correction (CS-correction) do?

Sub-Ångstrom Resolution

by Image Reconstruction

Principal Investigator: Michael A. O’Keefe 1992 -- 2002

OÅM team: J.-O. Malm 1992 -- 1993

E.C. Nelson 1995 -- 2002

C.J.D. Hetherington 1995 -- 1997

Y.C. Wang 1997 -- 1998

C. Kisielowski 1998 -- 2000

Aim: to produce sub-Ångstrom resolution for NCEM users.

*Supported by DOE/SC/BES/DMS

1992-2002: the LBNL One Ångstrom Microscope ProjectMaterials Sciences Division

NCEM

OÅM image taken close to alpha-null defocus shows pairs of C atoms separated by 0.89Å in the diamond structure.

Model of diamond structure in [110] orientation. Pairs of C atoms are separated by 0.89Å to form the ‘dumbbells’.

OÅM image shows 0.89Å spacings in test specimen of diamond

Y.C. Wang, A. Fitzgerald, E.C. Nelson, C. Song, M.A. O’Keefe et al, Microscopy and Microanalysis 5 (1999) 2: 822-823.

1998: first sub-Ångstrom result from OÅM

0.89Å

(b)

|A2| = 2.46m

(a)

OÅM image averaged

004

simulated

004

Before correction, diamond image shows effect of 3-fold astigmatism

After correction, diamond image shows 0.89Å atom pairs in “dumbbells”

OÅM image averagedImages -- Wang & O’Keefe, 1998

|A2| < 0.05m

1998: aberration correction -- three-fold astigmatism

Zemlin tableaux -- O’Keefe, Wang & Pan, 1998

Si444 (0.78Å) Si622 (0.82Å)

Si531 (0.92Å)

Image taken near alpha-null defocus shows pairs of Si atoms separated by 0.78Å.

Silicon structure model in [112] orientation. Pairs of Si atoms are separated by 0.78Å in ‘dumbbells’.

Experimental 0.78Å Transfer at 3kV Electron Gun Extraction Voltage

0.78Å

M.A. O’Keefe, E.C. Nelson, Y.C. Wang and A. Thust, Philosophical Magazine B 81 (2001) 11: 1861-1878.

Diffractogram confirms transfer of spacings to 0.78Å.

“Last-Century” Cutting-Edge Resolution [112] Si images from STEM and TEM

Best possible STEM- HB603U -

Best possible TEM- OÅM -

0.78Å

[112]

0.78Å

“Sub-Ångstrom resolution of atomistic structures below 0.8Å”, M.A. O’Keefe, E.C. Nelson, Y.C. Wang and A. Thust, Philosophical Magazine B 81 (2001) 11, 1861-1878.

“Quantitative interpretation and information limits in annular dark-field STEM images”, P.D. Nellist & S.J. Pennycook,

Microscopy and Microanalysis 6, 2: (2000) 104-105.

[112] Si has become the “de facto” test specimen

Atom-atom spacings for diamond-cubic test specimens from 1.62Å to 0.51ÅD

um

bb

ell

Sp

acin

g (

Å) [110] series

[112] series

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

3.0

0.89Å

[110] diamond [112] silicon

0.78Å

Testing Microscope Resolution (the A-OK test series)

3.0 4.0 5.0 6.0 6.55.54.5 7.03.5

1.4

1.2

1.0

0.8

0.6

0.4

1.6

Lattice Parameter (Å)

diamond

-SiC

-InN

SiGe

AlSbCdTe

0.51Å

0.64Å

0.72Å0.78Å

0.82Å0.89Å

0.94Ådiamond

-SiC

-InN

SiGe

AlSb

CdTe

0.89Å

1.11Å

1.24Å

1.36Å

1.41Å

1.53Å

1.62Å

OÅM images reconstructed from focal series of 20 component images

• LiCoO2 is the most commonly used positive electrode materials for lithium rechargeable batteries

– Energy storage lithium insertion into and extraction from LixCoO2

• Ultra high resolution is needed to resolve light elements in a heavy matrix

– Conventional HRTEMs with resolutions to 1.6Å can routinely image the heavier metal atoms in structures such as oxides.

– The OÅM (One-Ångstrom Microscope) at the NCEM has achieved resolutions to 0.8Å and, in addition to heavy atoms, has previously imaged columns of lighter atoms, including O, N, and C.

– In this work, we have used the OÅM to image all the component atoms, including columns of Li atoms in a matrix of CoO2.

Resolution of light atoms -- imaging lithium

Schematic of Layered LiCoO2 Structure

Li atoms

CoO6 octahedra

Single unit cell projected in the [110] orientation

Co atoms

O atoms

Reconstructed Exit-Surface Wave of LiCoO2

Comparison of simulated and experimental ESWs shows that Li atom columns are visible at 0.9Å resolution in the OÅM.

The reconstructed exit-surface wave shows that the specimen is tilted away from exact [110] zone axis orientation and also reveals buckling and possible electron beam damage.

CoO

OLi

Experimental

Co is “fuzzy” O is strong

Li is weak

Simulation

Model

ESW phase (peak height) is proportional to the number of atoms in the column producing the peak. Line trace shows the one-atom difference between

adjacent columns.

Simulated Pd cube-octahedron analysis -- Line trace shows peaks in ESW phase --

ESW phaseb

aa

b

0.286 radian

6 atom column

11 atom column

# atoms in columns

6 7 8 9 10 11 10 9 8 7 6a b

Analysis of experimental image of 70Å Au nanoparticle

Single image at -2600A underfocus Phase shows white atom columns

FSR of particle

Twinning in ESW phase becomes clearer after application of a high-pass filter

Particle image High-pass image

Analysis of 70Å gold nanoparticle by peak profile

Line trace of ESW phase shows initial increase from outer edge, followed by groups of peaks with very similar heights.

Edge Center

“Quantization” of ESW phase peak steps suggests that height differences may be due to different integral numbers of atoms.

Zero?

57 7

9

The technique of profile tracing of phase to measure peak heights suffers from the lack of a well-defined zero level, especially for supported nanoparticles.

Z-Contrast Microscopy

• Atomic structure

Detector

0.2 nm

Sr Sr

Ti Ti

Spectrometer

1

54

6

2

3

and electronic structure

550 600 6500

2

4

6

8

10

12

14

Energy Loss (eV)

Mn L II/IIIO-K

1

2

3

45

6

Courtesy of S. Pennycook

STEM Probe Size is Limited by Spherical Aberration

No spherical aberration

FWHM ~ 0.8 Å

Current density is concentrated into central maximum

FWHM ~2 Å

Significant current is

lost in probe

“tails”

Aberration limited

Aberration correction can achieve the smaller brighter probe

VG Microscope’s HB501UX, 100 kV


Electron Microscopy in 2003 -- aberration-corrected STEM

Single Atom Spectroscopy

5 Å Spectroscopic identification of a single atom within a bulk material.

8% collection

efficiency

820 850 880

Inte

nsi

ty

Energy (eV)

La M4/5

La in CaTiO3 grown by MBE


Linetrace of STEM Intensities

Au to Au spacing 2.88 Å

Single Au

Single Au

First Column

Carbon film background


Measurement of gate-oxide width

“Thin Dielectric Film Thickness Determination by Advanced Transmission Electron Microscopy”, A.C. Diebold et al., Microscopy & Microanalysis 9 (2003) 493–508.

Electron Microscopy in 2003

Diebold et al. (2003), compared measurements of gate-oxide width using TEM and STEM.

(a)TEM shows silicon [110] dumbbells (left) up to nitrided gate oxide, then oxide, then poly silicon.

(b)STEM (HAADF) with 10 millirad aperture agrees with TEM

(c)STEM with 13 millirad aperture shows oxide as wider

(d)STEM with larger aperture shows even “wider” oxide

Advanced TEM

Diebold et al. (2003).

3-D STEM

Work by

P.A. Midgley and M. Weyland Cambridge U.

Electron Microscopy in 2003

Fig. 3a. Result of adding successively more projections to the reconstruction, using direct (left) and weighted (right) back-projection over a tilt range of 90.

Fig. 2. Non-uniform sampling of Fourier space over-emphasizes lower frequencies, giving a blurred reconstruction. The greater density of low-frequency data is compensated by using weightedback-projection reconstruction.

2-D test object for simulation

P.A. Midgley and M. Weyland, Cambridge U.

Fig. 3b. Effect of tilt range. Limited tilt produces a missing wedge in Fourier space. Missing data limit the reconstruction resolution in the vertical direction, causing streaking. Figure shows tilt ranges from 10 to 60. Tilt axis is into the plane of the figure.

Object Reconstruction

WeightedDirect

Recent advances in tomographic specimen holders allow tilts to 70 around two axes within the 2.2mm polepiece gap of modern ultra-high-resolution electron microscopes. With a tilt series in x and one in y, the “missing wedge” becomes a 20 “missing pyramid”.



An individual nanoparticle in the reconstructed data set can be isolated to show that it is anchored to the wall of a 3nm-diameter mesopore. The particle is about 1nm in diameter.

3-D image of nanoparticles. Reconstructed using weighted back projection from 55 STEM HAADF images of Pd6Ru6–MCM 41 catalysts. Tilts from +60 to -48 in 2 steps at 300kV. Metal particles have been colored red for clarity.



The electron microscope will continue to evolve and provide essential feedback in the nano- theory/construction/measurement loop.

Conclusion

(b)

(a) One Ångstrom Microscope • spread of focus = 20Å • information limit = 0.78Å

(a)

(a) Standard CM300FEG/UT* • spread of focus = 35Å • information limit = 1.05Å

OÅM information limit is at sub-Ångstrom level

*Hans Bakker, Arno Bleeker, and Peter Mul, Ultramicroscopy 64 (1996) 17-34.

1.05Å 0.8Å 0.8Å1.05Å

Compare standard CM300FEG/UT with OÅM-spec CM300

1 2 3 0

1.01Å

d A2

0.68Å

0.80Å

0.1 0.10.050.03A2 (m)A2 (m)

a b

Before correction:mean = 2.46m

1Å limit

1998: correction of OÅM three-fold astigmatism

1 2 3 0

1.01Å

d A2

0.68Å

0.80Å

0.1 0.10.050.03A2 (m)A2 (m)

a b

1Å limit

Correction method uses 2-fold stigmators to provide an approximation to a 3-fold field (D. Typke & K. Dierksen, Optik 99, 4: (1995) 155-166)

After correction:means = 0.03m

O

LiCo

• Sub-Angstrom image of LiCoO2 battery material shows all atom species.

• Superimposed model identifies the strong white peaks with the positions of oxygen atom columns, the strong fuzzy peaks with cobalt sites, and the weak white peaks at lithium positions.

LiCoO2 Exit-Surface Wave with less Smoothing

Exit-Surface Wave Simulation of LiCoO2 - the [110] Zone

Resolution0.8Å 1.0Å0.9Å

39.4Å

45.1Å

50.7Å

Th

ickn

ess

Li Li Li

ESW simulations suggest that Li atom columns should be clearly visible for resolutions of 0.8 to 1.0 Ångstrom at specimen thickness of 40 to 50 Ångstrom.

EpicierCo50-Pt50_20-zoomed-3M6.jpg five-fold Pt-Co particle with an 'artistic' schema... Not as well-oriented, not as well-resolved, but significantly smaller (2.5 - 3 nm...; the scale is missing) http://cecm.insa-lyon.fr/people/people.php?name=epicier

Simulation study of Pd cube-octahedron

Model

Exit-surface wave

Scherzer image

"Deceptive "Lattice Spacings" in High-Resolution Micrographs of Metal Nanoparticles", J.-O. Malm & M.A. O'Keefe, Ultramicroscopy 68 (1997)13-23.

ESW phase is proportional to the specimen potential projected through thickness H in the direction of the

incident electron beam.

In the image, large phase changes have produced white peaks in atom columns near the particle center.

Delocalization has produced strong Fresnel fringes, masquerading as “white atoms”, near the particle edges.

The nanoparticle model used in the test simulations has 561 atoms of palladium

arranged as a cube-octahedron.

}),,(exp{),(0H

dzzyxiAyx

Ewald sphere

specimen shape function

g

2. The (complex) exit-surface wave is transferred to the image plane by the objective lens, forming a (complex) image amplitude.

Central maximum in shape function falls to zero atsin (gt)/(gt) 0then gt and t 1/ g 2/(u2) or u2 2/(t)and d2 t/2

Image formation and exit-wave reconstruction

Incident electron beam

Specimen

Exit-surface wave

Objective lens

Diffraction amplitudes

Image

Image formation to microscope information limit

Image formation at the microscope information limit

1. The incident electron beam passes through the specimen to produce the specimen exit-surface wave.


Exit-surface wave resolution is limited only by the electron scattering described by the interaction of the Ewald sphere with the specimen shape function.


Exit-surface wave resolution is limited only by the electron scattering described by the interaction of the Ewald sphere with the specimen shape function.

For electron wavelength and specimen thickness t, the scattering resolution is given by

dscatt = ( t / 2)

At 300keV, = 0.02Å, and values of thickness of 100Å and 65Å give dscatt = 1Å and 0.8Å. 2. The (complex) exit-surface wave is transferred to the image plane by the objective lens, forming a (complex) image amplitude.

During transfer, the objective lens imposes phase changes on the components of the exit-surface wave due to the lens defocus.

Lens transfer blocks the exit-surface components that describe specimen spacings finer than the information limit of the microscope.





Lens transfer blocks the exit-surface components that describe specimen spacings finer than the information limit of the microscope.

For electron wavelength and microscope spread of focus of the information limit is given by

d = (/2)

At 300keV, = 0.02Å, and values of of 35Å and 20Å give d = 1Å and 0.8Å.

Image formation and exit-wave reconstruction

Reconstruction of exit-wave to microscope information limit

Incident electron beam

Specimen

Exit-surface wave

Objective lens

Diffraction amplitudes

Image

Image formation to microscope information limit

Focal series of images

x (Å)

(Å)

3-D Fourier transform

u (Å-1)

Stack of image diffractograms

(Å-1)

Locus of linear image contributions

Project paraboloid to zero-focus plane

Fourier components of exit-surface wave

Estimate of exit-surface wave to information limit

Inverse 2-D Fourier transform

”Direct Structural Retrieval from high-resolution electron micrographs", D. Van Dyck and M. Op de Beeck, in Computer Simulation of

Electron Microscope Diffraction and Images, A TMS Publication, William Krakow and Michael A. O'Keefe (eds) (1989) 265-271.

1 3 5 7 9 11 9 7 5 3 1

0.57 radian

ModelESW phase

ab

a b

ESW phase (peak height) is proportional to the number of atoms in the column producing the peak. Line trace shows the two-atom difference between

adjacent columns. # atoms in columnsa b

11 atom column

1 atom column

Simulated Pd cube-octahedron analysis -- Line trace shows peaks in ESW phase --

Experimental Au nanoparticle analysis Color coding shows phase of normalized ESW ((x,y) - (1+0i)).

Complex pixel maps of (x,y) - 1

(a) Wide view

(b) Particle

(c) SupportComplex pixel map of large area (a) shows a pink peak near 3pi/4 phase due to the nanoparticle. Amorphous support (c) contributes random phase to particle pixel map (b).

151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1


+1

-1

0

CM300FEG/UT

OÅM

= 0.25 millirad

n = 2

n = 36

= 36Å

= 20Å

0.78Å

1.07Å

+1

-1

0

+1

-1

0

+1

-1

0

+1

-1

0

OÅM CTF shows transfer of 0.89Å spacings from diamond

1.1Å

1.03Å

0.89Å



151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

+1

-1

0

OÅM = 20Å

0.78Å

151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

CM300FEG/UT = 36Å+1

-1

0

CTFs show transfer of spatial frequencies, resolution, information limit

151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

= 0.25 millirad

+1

-1

0

1.1Å

151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

n = 2 +1

-1

0

1.03Å

151413121110987654321k,(nm-1)

0.0670.0710.0770.0830.0910.1000.1110.1250.1430.1670.2000.2500.3330.5001.000d,(nm)

1

0

-1

1

0

-1

n = 36 +1

-1

0

0.89Å

1.7Å resolution

1.07Å info limit

Reconstructed Exit-Surface Wave of LiCoO2

Comparison of simulated and experimental ESWs shows that Li atom columns are visible at 0.9Å resolution in the OÅM.

The reconstructed exit-surface wave shows that the specimen is tilted away from exact [110] zone axis orientation and also reveals buckling and possible electron beam damage.

CoO

OLi

Experimental

Co is “fuzzy” O is strong

Li is weak

Simulation

TEAM STEM/TEM building blocks:• Probe CS corrector is required for sub-Å probe to allow sub-Å

Z-contrast and accurate spectroscopic imaging.

• Monochromator for high energy resolution (better than 0.05eV) to provide high energy resolution for chemical characterization and improved information limit for high-resolution sub-Å microscopy.

• Biprism for holographic studies of phase at high resolution.

• Energy filter (in-column or post-column) with better than 0.05eV resolution (hi-res GIF already there). Post-column filter also provides extra magnification for holography and sub-Å imaging (>5Mx).

• Objective-lens CS corrector to extend microscope resolution to the information limit.

• High-stability lens and HT power supplies. Lens to 0.1ppm (FEI UT already at 0.3ppm). HT to 0.25ppm (FEI now at 0.25ppm at 200keV).

• Large CCD camera for sufficient field of view at high magnification and holographic reconstruction (Gatan UltraScan has 4k by 4k now).

• Low drift stage with sub-Å piezo-electric control (JEOL has 0.05Å).

• Automated (computerized) procedures for alignment, aberration correction, image acquisition, and focal-series reconstruction.

Transmission Electron Achromatic Microscope >>> the TEAM project <<<

Information limit is set by temporal coherence damping function:

E(u) = exp{-½222u4}

Push TEAM information limit to 0.5Å level

E(u) imposes an information limit for the microscope of

d = 1/|u| = () at a level of exp(-2) or 13.5%

where = CC{( 2(Ebeam)/E2 + 4 2(I)/I2}

• CC is the chromatic aberration coefficient for the objective lens

• (Ebeam)/E is the rms energy spread in the electron beam as a fraction of total beam energy over the time of image acquisition.

• (I)/I is the fractional rms ripple in lens current.

d = 0.8Å requires 20Åand d = 0.5Å requires 8Å

OÅ TEA

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3

Allowed incident beam energy spread (FWHH) for 0.5Å resolution

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3

Objective Lens Current Stability (I)/I (rms ppm)0 0.1 0.2 0.3

E (eV)

0

0.1

0.2

0.3

0.4

300keV

200keV

120keV80keV

Improved lens current stability allows greater energy spread

Greater energy spread allows more incident beam current

If an objective lens current stability of 0.1 (rms) ppm can be achieved, an information limit of 0.5Å can be achieved with an incident beam energy spread (FWHH) of up to 0.35eV at 300keV and up to 0.18eV at 200keV, allowing reasonable incident beam currents for HREM imaging.

Series of Simulated “Weak-Phase-Object” Images of LiCoO2

1.8Å 1.6Å 1.2Å1.4Å

1.0Å 0.8Å 0.6Å 0.4Å

• Positions of Co atom columns should be seen clearly at resolutions as poor as 1.8Å.

• Resolution of 1.4Å is required to make the O atom columns visible.

• Li atoms can be seen at 1.0Å and become more visible at a resolution of 0.84Å.

• In this approximation, atom columns appear black and proportional to atom mass (scattering cross section).

• Resolutions of 0.6Å and 0.4Å are not attainable experimentally (yet).

Co O

LiLiO

Phase of ESW, (x,y) is color coded from light blue (zero phase) to red (pi radian).

Simulated Pd cube-octahedron analysis -- Argand pixel map shows ESW amplitude and phase for every pixel --

11 1

2

3

4

5

67

8

9

10

1+0i Re

Im

ESW, displayed in the form of an Argand plot with the same color coding, shows 11 atom columns.

ESW analysis program by T. Tomaszewicz (2003, to be published).

Argand plots can be used to show the trajectory of a complex function as specimen thickness is increased (O’Keefe, Ph.D. thesis, 1975). In this case the Argand plot of the ESW (Tomaszewicz, 2003), taken over every pixel in the (marked) frame containing the particle, shows 11 quantized traces (numbered) corresponding to the 11 different column heights making up the particle. The light blue area near the 1+0i point on the Argand plot contains the background (“vacuum”) pixels (since exp {i0} = 1+0i). The 11 traces contain all the pixels making up the ESW phase peaks at the positions of the atom columns. The highest phase change of each trace corresponds to the central pixel of each phase peak. Because only one atom column contains 11 atoms, trace number 11 is much weaker.

Phase of ESW, (x,y) Complex pixel map of ESW, (x,y)

2-2

-2i

2i

Phase angle advances in “ticks” of 0.282 radian-i

i

-3 -1

Phase of (ESW minus 1) is color coded from dark blue (zero phase) to red (pi radian).

Simulated Pd cube-octahedron analysis -- Argand pixel map shows ESW amplitude and phase for every pixel --

11 1

2

3

4

5

67

8

9

10

0+0i Re

Im

ESW, displayed in the form of an Argand plot with the same color coding, shows 11 atom columns.


The constant complex “vacuum” amplitude can be subtracted out of the ESW and the resulting function ((x,y) -1) plotted in the Argand plane. These plots, and the resultant shifted trajectory have been used by Sinkler and Marks (1999) in an implementation of a direct-methods structure refinement. With the constant complex “vacuum” amplitude subtracted out, the background pixels become black, and the Argand plot of the “normalized” ESW ((x,y) -1) now shows the 11 column traces emanating from the origin (Tomaszewicz, 2003).

Phase of ((x,y) - (1+0i)) Complex pixel map of (x,y) - (1+0i)

2-2

-2i

2i

"Dynamical Direct Methods for Everyone", W. Sinkler & L.D. Marks, Ultramicroscopy 75 (1999)251-268.

1-1

-i

i

Experimental Au nanoparticle analysis -- Argand pixel map shows ESW amplitude and phase for each pixel --

Phase of experimental ESW, (x,y) is color coded from light blue (zero phase) to dark blue (pi/4 radian).

Experimental ESW, as an Argand plot with the same color coding, shows elongation but no individual atom columns.


The Argand plot of the ESW is taken over every pixel in the (marked) frame containing the particle (Tomaszewicz, 2003).

Re

Im

-1

-i

i

Conclusions

• Profile traces of the ESW phases from experimental nanoparticles may show quantization of peak heights, but there is no reliable zero line. Contributions from the amorphous support material blur the pixel map traces for columns in nanoparticles. Experimental phases have less sensitivity, due to damping by the Stobbs factor.

• Simulations of nanoparticle exit-surface wave images demonstrate that atom columns produce peaks with proportional phase heights. This result can be confirmed by the “peacock” traces seen in pixel maps of the complex ESW

1 3 5 7 9 11 9 7 5 3 1

0.57 radian

Documents

This work supported by the Director, Office of Science, Office of Basic Energy Sciences,