Advanced Transmission Electron
Microscopy
Lecture 2: Electron Holography
by James Loudon
The Transmission Electron Microscope
electron gun
specimen (thinner
than 200 nm)
electromagnetic lens
viewing screen
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
wavefronts
z
specimen
phase shift
between
the two rays
electron wave
which went
through the
specimen
electron wave
which went
through
vacuum
Electron Holography
y
x
wavefronts
specimen
z
Electron Holography
The amplitude changes if electrons
are ‘absorbed’ i.e. if the number
coming out the specimen is fewer
than the number that went in.
(‘Absorbed’ is used to refer to all
the electrons which do not
contribute to the image: high-angle
scattering also counts as
absorption.)
y
x
wavefronts
z
The
specimen
exit plane
specimen
y  ax, y exp2i ft  kz  if x, y 
The phase changes if the electrons
move at a different speed or
direction through the specimen
than through vacuum.
y 0  exp2i ft  kz
Symbols:
f = frequency,
t = time,
k = wavenumber = 1/l.
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?
Examples of E and B-fields Measured
using Electron Holography
a
b
200 nm
c
200 nm
d
200 nm
e
200 nm
(a) Semiconductor physics: built-in voltage across a p-n junction.
(b) Nanotechnology: Upper panel: remnant magnetic state in exchangebiased CoFe elements. Lower panel: micromagnetic simulation of the
same elements.
(c) Field Emission: Electrostatic potential from a biased carbon nanotube.
(d) Geophysics: Exolved magnetite elements in the titanomagnetite
system.
(e) 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.
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
<5G.
Obtaining Information from the Phase
f ( x, y )  CE  V x, y, z dz 
Geometry for the integrals
2e
Bx, y, z .dS
h 
x
electron beam
z
Constant determined
by acceleration
voltage
Electrostatic
potential
Magnetic
flux density
B
specimen
S
f(0) ≡ 0
f(x)
Origin of the Mean-Inner Potential
Note that the electrostatic
Electric field
potential can either come
Electric
(or force or
from specimen charging (not
Electron beam
potential, V
acceleration)
usually what is wanted) or
Electron accelerates
from the mean inner
potential, V0, which accounts
+ + + + + + +
Specimen
for the fact that electrons
+ + + + + + +
travel faster through material
Electron decelerates,
than vacuum.
V0
returning to its original
Atomic
z
z
speed
nuclei
Electrostatic Contribution to Phase Shift –
Calculation the same as for a Potential Step
2
d
y
Schrodinger equation 
 eV0y  Ey
2
(E is the energy of the
2m dz
electrons)
d 2y
2m


E  eV0 y
2
2 
2
specimen
dz
t
This is the equation of simple harmonic motion
Solution is

So the phase
f  2 kV  kfree t  
shift is
et
z
V

V0
z
2m
E  eV0 
2
2m
2m 


E

eV

E t
0
2
2



y  Ae2ikz with 2k 
f
This can be Taylor expanded as E/e = 300kV, V0 ~ 10V
which gives:
f  x, y   C E  V  x, y, z dz
1 2m
eV0t  CEV0t or, if V is not constant
2 2 E
2e  E  m c2
The calculation SHOULD BE DONE

CE 
2
l
RELATIVISTICALLY: this changes CE to
 E E  2m c





m = electron mass, c = speed of
light in a vacuum, l = electron
wavelength in the vacuum.
Magnetic Contribution to Phase Shift
F = ev×B = ma so a = evB/m and vx=a×time
For small deflections, q = vx/v
q
t
q
vx
f x   2kqx  
x
z
evB / m t / v 
t
eBt eBt


m v hk
B
v
e-
q
v
2eBtx
h
qx
x
q
In general for non-constant B
f  x, y   
2e
B.dS

h
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
To Reiterate:
f ( x, y )  CE  V x, y, z dz 
Geometry for the integrals
2e
Bx, y, z .dS
h 
x
electron beam
z
Constant determined
by acceleration
voltage
Electrostatic
potential
Magnetic
flux density
B
specimen
S
f(0) ≡ 0
f(x)
Phase Recovery Method 1: Off-Axis Electron
Holography
electron gun
200nm
specimen
Lorentz lens
electron biprism
interference
region
+
Holographic fringes
viewing plane
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.
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.
How Does Off-Axis Holography Work?
x
y  exp2i ft  kz
z
y  ax, y eif  x, y e2i  ftkz 
y 0  e2i  ftkz
+
y  ax, y eif  x, y e2i  ftkz e2iQx
y 0  e2i  ftkze2iQx
The waves interfere
y total y y 0  aeif e2iQx  e2iQx e2i  ftkz
How Does Off-Axis Holography Work?
y total  e2i ftkz aeif e2iQx  e2iQx   e2i  ftkzeif / 2 aeif / 2e2iQx  eif / 2e2iQx 


I total  y total  y * totaly total  ae if / 2e 2iQx  eif / 2e 2iQx ae if / 2e 2iQx  e if / 2e 2iQx
2
I total  1  a2 x, y   2ax, y cos4Qx  f x, y 

The phase information now
appears in the image
…but how do you separate it?
Latex spheres (image from
Lai et al. in Tonomura et al.
(Eds.), Electron
Holography, North Holland,
Elsevier, Amsterdam 1995.)
cosine fringes
fringes bend on moving from
vacuum into the specimen
Separating the Phase
I  x   1  a 2 ( x)  2a( x) cos  4 Qx  f ( x) 
 1  a 2 ( x)   a( x)eif ( x ) e4 iQx  a( x)eif ( x ) e4 iQx 
To get the phase, we Fourier transform the intensity and use  (q  Q) 

e
2 i ( q Q ) x
dx

F.T. I ( x)    (q)  F.T.  a 2 ( x)  *  (q)  F.T.  a( x)eif ( x )  * F.T. e 4 iQx   F.T.  a( x)e if ( x )  * F.T. e 4 iQx 
F.T. I ( x)    (q)  F.T.  a 2 ( x)  *  (q )  F.T.  a ( x)eif ( x )  *  (q  2Q)  F.T.  a ( x)e if ( x )  *  (q  2Q)
wrapped phase
Inverse
Transform
gives
amplitude
and phase
Fourier
transform
Original Image (called
‘the hologram’)
Extract sideband and
put origin at centre
SrTiO3
vacuum
100 nm
glue
SrRuO3
Extracting the Phase cont.
F.T. I ( x)    (0)  F.T.  a 2 ( x)  *  (0)  F.T.  a( x)eif ( x )  *  (q  2Q)  F.T.  a( x)e if ( x )  *  (q  2Q)
Select sideband 
F.T.  a ( x)eif ( x ) 
Inverse transform sideband 
a( x)eif ( x )
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 2. So the phase image contains ‘phase wraps’ which must be removed by
adding 2 to selected areas of the image. This can be difficult if there are many phase wraps.
fwrapped
freal
2
2
0
x
0
x
Methods for Separating B and V
f ( x, y )  CE  V x, y, z dz 
Constant determined
by acceleration
voltage
2e
Bx, y, z .dS

h
Electrostatic
potential
Magnetic
flux density
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.
f ( x, y )  C EV0t 
2e
B x, y, z .dS
h 
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
h    / y 

f x, y 
B  x, y  
2et   / x 
Method 2: If the sample can be heated above its Curie point so that it is no
longer magnetic we have
fcold ( x, y )  C E  V x, y, z dz 
fhot ( x, y )  C E  V x, y, z dz
2e
Bx, y, z .dS

h
See: Loudon J.C. et al.
Nature 420, 797, 2002.
The magnetic contribution to the phase is then the difference of these two.
fmagnetic( x, y )  
2e
Bx, y, z .dS  fcold x, y   fhot x, y 

h
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:
2e
B x, y, z .dS

h
2e
f ( x, y )  C E  V x, y, z dz 
Bx, y, z .dS

h
f ( x, y )  C E  V x, y, z dz 
The magnetic contribution to the phase is then:
fmagnetic( x, y )  
2e
1


f x, y   f- x, y 
B
x
,
y
,
z
.
d
S


h
2
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.
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
B
Electron
beam v
fmagnetic( x, y )  
F = -ev × B
Thin
B
Turn over
F = -ev × B
v
2e
1
fright way up x, y   fupside down x, y 


B
x
,
y
,
z
.
d
S

h 
2
See R.E. Dunin-Borkowski et al., Microscopy Research and Technique, 64, 390,
2004 and refs. therein.
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 outof-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.
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.
Intensity
In-focus
image (blank)
Displacement
Out-of-focus
image
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.
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 Schrodinger’s equation
2 2

 y  Vy  Ey
2m
with the condition for a steady electron current
.J  . Imy * y   0
and re-expressing the answer in terms of the intensity I and phase f gives the
Transport of Intensity Equation:
2 I
 xy .I xyf   
l z
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 Poisson’s
equation which can be solved by Fourier methods.
Simplifying the Transport of Intensity
Equation
The Transport of Intensity Equation (TIE):
 xy .I xyf   
If the in-focus intensity is constant I0, we have: I 0 2 xyf  
2 I
l z
2 I
l z
2
 I 
2 2~
F.T . 
Taking the 2D Fourier transform gives:  4 q f  
lI 0
 z 
(q is the Fourier space coordinate)
So
~
f
 I Df   I  Df 
1
F.T
.


2q 2lI 0
2
D
f


Using the Transport of Intensity Equation
~
f
 I Df   I  Df 
1
F.T
.


2q 2lI 0
2Df


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.
x
z
z = -Df
z=0
z = Df
How Well Does TIE Work?
Df = 3mm
Recovered phase
Df = 0
cosf
Df = -3mm
Simulated Phase
(and cosf)
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 2 between adjacent dark lines. The contour maps also resemble magnetic field lines.
Phase Recovery Method 3: Foucault or
Phase Plate Imaging
When a electron is deflected by a
magnetic field, the scattering angle is
q
t
B
h
 50 mrad
This is much smaller than the
scattering angles for Bragg scattering
from a crystal which are several
mrad.
q
x
z
leBt
v
eThe effect of a magnetic field is to
shift the diffraction pattern.
mrad
mrad
Phase Recovery Method 3(a): Foucault
Imaging
If several magnetic domains are present, the spots in the diffraction pattern
are split.
2q
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.
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  should be used. The optimal phase shift depends on the object. For
weak phase objects, /2 is best. After some maths, it can be shown that the
resulting image should have black ‘field lines’ with a phase shift of 2 between
each.
2q
Image has
‘field lines’
Carbon sheet
giving 
phase shift.
Method 3: Phase Plate Imaging
J.C. Loudon, P. Chen et
al. in preparation.
Phase plate image
of circular
permalloy
elements.
cos(phase)
recovered using
TIE.
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.