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High Resolution Transmission Electron Microscopy (HRTEM

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Title: High Resolution Transmission Electron Microscopy (HRTEM


1
High Resolution Transmission Electron Microscopy
(HRTEM)
  • Resolution?
  • Minimum resolving power?
  • High resolution?

2
Resolution Criteria
Rayleighs description
Abbes description
Aberration free systems
3
The Role of an Optical System
means convolution which Indicates the two
functions are folded together
h(r) is the point spread function
4
  • In reciprocal space

The factors contributing to H(u)
include Aperture ? The aperture function
A(u) Attenuation of the wave ? The envelope
function E(u) Aberration of the lens ? The
aberration funciton B(u)
B(u) exp -i c(u)
  • Each point in the specimen plane is transformed
    into an extended region in the final image
  • Each point in the specimen has contributions from
    many points in the specimen

This limits the resolution!
5
Weak-Phase-Object Approximation (WPOA)
  • Model f(r) exp-i sVt(r)
  • s is the interaction constant (It tends to be a
    constant when energy of electron increases)
  • Vt is the potential an electron sees when passing
    the specimen
  • Treating the specimen as a phase object no
    attenuation of the intensity
  • WPOA very thin sample so Vt ltlt 1
  • Then f(r) 1-i sVt(r)
  • The wavefunction seen in the image is given by
  • 1-i sVt(r) h(r)

Exit wavefunction (or specimen wavefunction)
6
Applying WPOA to TEM
  • Apply WPOA f(r) 1-i sVt(r)
  • to
  • B(u) exp -i c(u)
  • The observed intensity is I 1 2sVt(r)
    sin(r)
  • We can then set B(u) 2 sinc(u)
  • The Objective lens transfer function is defined
    as
  • T(u) A(u)E(u) 2 sin c(u)

Why only see sin(r) ???
Pay attention to the 2, which is always ignored!
Also called contrast transfer function (CTF) in
the WPOA
7
The Phase Distortion Function
  • Phase distortion function c(u)
  • has the form of a phase shift expressed as
  • 2p/l times the path differences travelled by
    those waves affected by
  • spherical aberration (Cs), defocus (?f), and
    astigmation (Ca).

stigmated
Real space and inverse space is connected from
the Braggs law 2d sinq n l ? 2q lu for
small angle q
8
The Contrast Transfer Function
The effect of different Cs and ?f on CTF
9
  • Important points to notice
  • CTF is oscillatory there are "passbands" where
    it is NOT equal to zero (good "transmittance")
    and there are "gaps" where it IS equal (or very
    close to) zero (no "transmittance").
  • When it is negative, positive phase contrast
    occurs, meaning that atoms will appear dark on a
    bright background.
  • When it is positive, negative phase contrast
    occurs, meaning that atoms will appear bright on
    a dark background.
  • When it is equal to zero, there is no contrast
    (information transfer) for this spatial frequency.

10
  • Other important features
  • CTF starts at 0 and decreases, then
  • CTF stays almost constant and close to -1
    (providing a broad band of good transmittance),
    then
  • CTF starts to increase, and
  • CTF crosses the u-axis, and then
  • CTF repeatedly crosses the u-axis as u
    increases.
  • CTF can continue forever but, in reality, it is
    modified by envelope functions and eventually
    dies off. Effect of the envelope functions can be
    represented as

11
Scherzer Defocus
  • In Scherzer defocus, one aims to counter the term
    in u4 with the parabolic term ?fu2 of c(u). Thus
    by choosing the right defocus value ?f one
    flattens c(u) and creates a wide band where low
    spatial frequencies u are transferred into image
    intensity with a similar phase.
  • In 1949, Scherzer found that the optimum defocus
    (extended Scherzer defocus) is

The similar phase or flat responding is the
underlying principle governing phase-contrast
imaging in HRTEM
The minimum contrast is very useful since it is
the dark-field focus condition in STEM
The resolution at the Scherzer defocus is
obtained when Sinc(u) first crosses the
axis rsch 0.66(Csl3)1/4
For a 200 KeV microscope
The yellow curve is the damping envelope function
12
The Envelope function
The resolution is also limited by the spatial
coherence of the source and by chromatic effect
Teff T(u)EcEa
The envelope function imposes a virtual
aperture in the back focal plane of the
objective lens
13
  • Information limit goes well beyond point
    resolution limit for FEG microscopes (due to high
    spatial and temporal coherency).
  • For the microscopes with thermionic electron
    sources (LaB6 and W), the info limit usually
    coincides with the point resolution.
  • Phase contrast images are directly interpretable
    only up to the point resolution (Scherzer
    resolution limit).
  • If the information limit is beyond the point
    resolution limit, one needs to use image
    simulation software to interpret any detail
    beyond point resolution limit.

14
Spherical Aberration Correction
A post-objective corrector proposed by Rose H. in
1990 and 1991
The hexapoles only affect non-paraxial rays (the
dashed line)
15
FEG on Information Limit
Other factors affect the envelope function Es
spread angle of the source Es specimen drift Es
specimen vibration ED detector (requires very
high resolution CCD camera, ok for film)
16
Focus Variation Technique
Simulation of exit wave from a stack of recorded
images for GaN0001
For studying atomic structure and the quantum
mechanical interaction, the information is
contained in the exit wave function. We are not
able to directly derive it from the Scherzer
defocus method produced images
17
The Negative Spherical-aberration Imaging (NCSI)
Cs -40 mm
Philips FEG CM 200
Image of the atomic structure of BaTiO3 011
The information limit is about 0.125 nm
The best current resolution is about 0.08 nm at
NCEM 300 KeV FEG
Jia, C.L. and Urban, K. Science, 303, 2001 (2004)
18
The Negative Spherical-aberration Imaging (NCSI)
Atomic structure of ferroelectric PZT
Local displacement and polarization
The precision of aberration-corrected TEM allows
values for physical parameters to be derived
directly on the atomic scale
Urban, K. Science, 321, 506 (2008)
19
Summary
  • Resolution of HRTEM lt 1 nm
  • HRTEM is useful for direct atomic level study
    like interface, dislocation, defects, etc.
  • Point-spread function
  • Weak-phase-object approximation (WPOA)
  • Contrast transfer function (CTF)
  • Point resolution and information limit
  • Image reconstruction
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