Advanced Transmission Electron Microscopy Lecture 2: Electron Holography by James Loudon

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Advanced Transmission Electron MicroscopyLecture
2 Electron Holographyby James Loudon
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The Transmission Electron Microscope
electron gun
specimen (thinner than 200 nm)
electromagnetic lens
viewing screen
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Electron Holography

• When an electron wave passes through a specimen,
  its intensity and phase change.
• An image records only the intensity and not the
  phase.
• This is unfortunate as the phase contains
  valuable information about the electric and
  magnetic fields in the specimen.
• The term holography is used to describe an
  imaging technique which encodes the phase
  information in an image.
• There are several methods to produce images which
  contain the phase information and the main ones
  will be covered in this lecture.

y
x
z
wavefronts
specimen
phase shift between the two rays
electron wave which went through the specimen
electron wave which went through vacuum
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Electron Holography
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Electron Holography
A conventional image measures the intensity
I(x,y) y(x,y)2 a2(x,y). The phase, f(x,y),
is lost. How can we recover it?
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Examples of E and B-fields Measured using
Electron Holography

1. Semiconductor physics built-in voltage across a
   p-n junction.
2. Nanotechnology Upper panel remnant magnetic
   state in exchange-biased CoFe elements. Lower
   panel micromagnetic simulation of the same
   elements.
3. Field Emission Electrostatic potential from a
   biased carbon nanotube.
4. Geophysics Exolved magnetite elements in the
   titanomagnetite system.
5. Biophysics Chains of magnetite crystals which
   grow in magnetotactic bacteria and are used for
   navigation

Refs (a) Twitchett et al., J. Microscopy 214,
287, 2003. (b) Dunin-Borkowski R.E. et al., J.
Appl. Phys. 90, 6, 2899, 2001. (c) Cumings J. et
al., Phys. Rev. Lett. 88, 5, 056804, 2002., (d)
Harrison R.J. et al., Proc. Nat. Acad. Sci. 99,
26, 16557, 2002. (e) Simpson E.T. et al., J.
Phys. Conf. Ser. 17, 108, 2005.
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Magnetic Imaging

• In normal operation, the main objective lens of
  the microscope applies a vertical field of 2T to
  the sample.
• This is obviously undesirable for magnetic
  imaging and so the objective lens is usually
  turned off and the diffraction lens which is
  lower down the column (and is normally used to
  produce diffraction patterns) is used as an
  objective lens.
• Some microscopes like the Cambridge CM300 and
  Titan TEMs are equipped with a Lorentz lens
  which has a higher acceptance angle and lower
  aberrations than the diffraction lens whilst
  still keeping the sample in a low field.
• With judicious fiddling, the specimen can be in a
  field of lt5G.

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Obtaining Information from the Phase
Geometry for the integrals
x
electron beam
z
B
Constant determined by acceleration voltage
Magnetic flux density
specimen
S
Electrostatic potential
f(0) 0
f(x)
Origin of the Mean-Inner Potential
Note that the electrostatic potential can either
come from specimen charging (not usually what is
wanted) or from the mean inner potential, V0,
which accounts for the fact that electrons travel
faster through material than vacuum.
Electric field (or force or acceleration)
Electric potential, V
Electron beam
Electron accelerates
Specimen
Electron decelerates, returning to its original
speed
V0
Atomic nuclei
z
z
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Electrostatic Contribution to Phase Shift
Calculation the same as for a Potential Step
Schrodinger equation (E is the energy of the
electrons)
This is the equation of simple harmonic motion
Solution is
with
So the phase shift is
This can be Taylor expanded as E/e 300kV, V0
10V which gives
or, if V is not constant
m electron mass, c speed of light in a
vacuum, l electron wavelength in the vacuum.
The calculation SHOULD BE DONE RELATIVISTICALLY
this changes CE to
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Magnetic Contribution to Phase Shift
F evB ma so a evB/m and vxatime
For small deflections, q vx/v
t
B
q
v
In general for non-constant B
e-
SHOULD ALSO BE DONE RELATIVISTICALLY (but in fact
all the relativistic bits cancel)
Based on Hirsch, Howie, Nicholson, Pashley,
Whelan Electron Microscopy of Thin Crystals
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To Reiterate
Geometry for the integrals
x
electron beam
z
B
Constant determined by acceleration voltage
Magnetic flux density
specimen
S
Electrostatic potential
f(0) 0
f(x)
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Phase Recovery Method 1 Off-Axis Electron
Holography
Note many people (including me) use the term
holography to refer to off-axis holography
rather than a collective term for methods to
recover the phase.
The electron biprism is a positively charged wire
placed in the column to interfere electrons which
went through vacuum with electrons which went
through specimen.
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How Does Off-Axis Holography Work?

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How Does Off-Axis Holography Work?
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Separating the Phase
To get the phase, we Fourier transform the
intensity and use
wrapped phase
Inverse Transform
SrTiO3
gives amplitude and phase
Fourier transform
vacuum
Original Image (called the hologram)
100 nm
Extract sideband and put origin at centre
glue
SrRuO3
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Extracting the Phase cont.
Select sideband ?
Inverse transform sideband ?
The original wavefunction!
The spatial resolution of the technique is
determined by the size of the mask placed around
the sideband.
Minor difficulty the image you recover is the
real and imaginary part of the wavefunction. To
calculate the phase, you take the inverse tangent
(actually arctan2) of the imaginary part upon the
real part which gives the phase modulo 2p. So the
phase image contains phase wraps which must be
removed by adding 2p to selected areas of the
image. This can be difficult if there are many
phase wraps.
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Methods for Separating B and V
Constant determined by acceleration voltage
Magnetic flux density
Electrostatic potential
In a magnetic sample, the phase will be a sum of
electrostatic and magnetic contributions. How can
you separate B and V? Method 1 If the specimen
has a uniform thickness (t) and composition, the
electrostatic term will just be constant any
changes in the phase will be the result of B only.
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Separating B and V (cont.)
If B is confined to the sample and constant
throughout the sample thickness, we can get the
component of B normal to the electron beam
explicitly as
Method 2 If the sample can be heated above its
Curie point so that it is no longer magnetic we
have
See Loudon J.C. et al. Nature 420, 797, 2002.
The magnetic contribution to the phase is then
the difference of these two.
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Separating B and V (cont.)
Method 3 If the magnetisation of the sample can
be reversed by tilting the specimen and applying
a B-field (usually done using the objective lens
which can apply a vertical field of up to 2T), we
have
The magnetic contribution to the phase is then
This, of course, relies on being able to exactly
reverse the magnetisation. See R.E.
Dunin-Borkowski et al., Microscopy Research and
Technique, 64, 390, 2004 and refs. therein.
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Separating B and V (cont.)
Method 4 If the magnet is hard so that it tends
to stay in a fixed magnetic state, holograms can
be taken then the sample removed from the
microscope and turned upside down when holograms
are taken, remarkably, the magnetic contribution
to the phase is reversed but the electrostatic
contribution remains the same.
Thin
Thick
Thick
Thin
Turn over
F -ev B
F -ev B
Electron beam
v
v
See R.E. Dunin-Borkowski et al., Microscopy
Research and Technique, 64, 390, 2004 and refs.
therein.
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Phase Recovery Method 2 Out-of-Focus Imaging
This technique is also known as Fresnel imaging
or in-line holography. Unlike off-axis
holography, where electrons which pass through
the specimen are interfered with those which pass
through vacuum, different regions of specimen are
interfered by the simple method of taking an
out-of-focus image. This is easier than off-axis
holography as no biprism is required and the
specimen area of interest does not need to be
close to the vacuum so the field of view can be
much larger - the field of view achievable by
electron holography is 1 mm. The disadvantage is
that getting the phase is difficult. It is good
for a semi-quantitative overview of the specimen.
Example a specimen with three magnetic domains.
In-focus image (blank)
Intensity
Displacement
Out-of-focus image
The distance telling you how far out of focus you
are is called the defect-of-focus or defocus, Df
and is usually measured in mm.
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Method 2 Out-of-Focus Imaging
(c)
(a) Magnetic domain walls in a magnetic thin film
(of La0.7Ca0.3MnO3) at a defocus of 1.4 mm and
(b) a montage of images at different defoci (Df)
(c) Magnetic domain walls in Nd2Fe14B. Taken from
S. J. Lloyd et al., Phys. Rev. B 64, 172407, 2001
and J. Microscopy, 207, 118, 2002.
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The Transport of Intensity Equation
There is a method of obtaining the phase using
out-of-focus imaging. It requires two images
equally disposed either side of focus and an
in-focus image.
Combining Schrodingers equation
with the condition for a steady electron current
and re-expressing the answer in terms of the
intensity I and phase f gives the Transport of
Intensity Equation
This is a non-linear equation and so difficult
(but by no means impossible) to solve in the
general case. If, however, the in-focus image has
a constant intensity, I0 (this depends on the
specimen), the equation simplifies to Poissons
equation which can be solved by Fourier methods.
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Simplifying the Transport of Intensity Equation
The Transport of Intensity Equation (TIE)
If the in-focus intensity is constant I0, we have
Taking the 2D Fourier transform gives
(q is the Fourier space coordinate)
So
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Using the Transport of Intensity Equation
x
z
z -Df
To obtain the phase, take one image at positive
defocus, another image at negative defocus and
subtract. Fourier transform, divide the answer by
q2 and multiply by all the constants. Inverse
transform and you have the phase.
z 0
z Df
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How Well Does TIE Work?
Flat, circularly magnetised permalloy elements
(J.C. Loudon, P. Chen et al. in
preparation.) Note phase images are often
displayed as the cosine of the phase this gives
a contour map where there is a phase shift of 2p
between adjacent dark lines. The contour maps
also resemble magnetic field lines.
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Phase Recovery Method 3 Foucault or Phase Plate
Imaging
When a electron is deflected by a magnetic field,
the scattering angle is
This is much smaller than the scattering angles
for Bragg scattering from a crystal which are
several mrad.
The effect of a magnetic field is to shift the
diffraction pattern.
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Phase Recovery Method 3(a) Foucault Imaging
If several magnetic domains are present, the
spots in the diffraction pattern are split.
If an aperture is used to block one of the split
beams, only one set of domains appear bright.
This form of dark-field imaging is called
Foucault imaging.
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Phase Recovery Method 3(b) Phase Plate Imaging
Instead of blocking one set of beams, a thin
sheet of carbon can be used to induce a phase
shift in one of the split beams. For magnetic
imaging a phase shift of p should be used. The
optimal phase shift depends on the object. For
weak phase objects, p/2 is best. After some
maths, it can be shown that the resulting image
should have black field lines with a phase
shift of 2p between each.
Image has field lines
Carbon sheet giving p phase shift.
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Method 3 Phase Plate Imaging
J.C. Loudon, P. Chen et al. in preparation.
cos(phase) recovered using TIE.
Phase plate image of circular permalloy elements.
This method was suggested by A.B. Johnson and
J.N. Chapman (J. Microscopy, 179, 119, 1995) for
visualising magnetic fields and it is very rarely
used. It is also not very clear how to obtain the
phase itself from a phase-plate image. Phase
plates are more commonly used to enhance the
contrast from biological specimens.