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.

specimen

wavefronts

x

z

y

electron wave which went through the specimen

electron wave which went through vacuum

phase shift between the two rays

Electron Holography

specimen

wavefronts

x

z

y

Electron Holography

specimen

wavefronts

x

z

y

The specimen exit plane

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.)

The phase changes if the electrons move at a different speed or direction through the specimen than through vacuum.

A conventional image measures the intensity I(x,y) = |(x,y)|2 = a2(x,y). The phase, (x,y), is lost. How can we recover it?

yxikzftiyxa ,2exp, kzfti 2exp0

Symbols: f = frequency, t = time, k = wavenumber = 1/.

Examples of E and B-fields Measured using Electron Holography

a

200 nm

b 200 nmc

200 nm

d

200 nm

e

(a) Semiconductor physics: built-in voltage across a p-n junction.(b) Nanotechnology: Upper panel: remnant magnetic state in exchange-

biased 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

Bz

electron beam

specimen

(0) ≡ 0 (x)

x

SConstant determined by acceleration voltage

Electrostatic potential

Magnetic flux density

SB dzyxh

edzzyxVCyx E .,,

2,,),(

Geometry for the integrals

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.

+ + +++ + ++

+ + ++ + +

Electron beam

Electron accelerates

Specimen

Electron decelerates, returning to its original speed

Electric field (or force or acceleration)

z z

Electric potential, V

V0

Origin of the Mean-Inner Potential

Atomic nuclei

Electrostatic Contribution to Phase Shift – Calculation the same as for a Potential Step

t

z

VV0

This is the equation of simple harmonic motion

2 2

022

deV E

m dz

2

02 2

2d mE eV

dz

This can be Taylor expanded as E/e = 300kV, V0 ~ 10V which gives:

The calculation SHOULD BE DONE RELATIVISTICALLY: this changes CE to

t

z e-

specimen

Schrodinger equation (E is the energy of the electrons)

Solution is with

So the phase shift is

ikzAe 2 02

22 eVE

mk

tEm

eVEm

tkkV

202free

222

2

2

2

2

mcEE

mcEeCE

tVCteVE

mE 002

2

2

1

dzzyxVCyx E ,,, or, if V is not constant

m = electron mass, c = speed of light in a vacuum, = electron wavelength in the vacuum.

Magnetic Contribution to Phase Shift

For small deflections, = vx/v

Based on Hirsch, Howie, Nicholson, Pashley, Whelan: Electron Microscopy of Thin Crystals

t B

x

z

v

vx

e-v

xx

In general for non-constant B

SHOULD ALSO BE DONE RELATIVISTICALLY (but in fact all the relativistic bits cancel)

F = ev×B = ma so a = evB/m and vx=a×time

hk

eBt

mv

eBtt

vtmevB

//

h

eBtxxkx

22

SB.dh

eyx

2,

To Reiterate:

Bz

electron beam

specimen

(0) ≡ 0 (x)

x

SConstant determined by acceleration voltage

Electrostatic potential

Magnetic flux density

SB dzyxh

edzzyxVCyx E .,,

2,,),(

Geometry for the integrals

Phase Recovery Method 1: Off-Axis Electron Holography

+

electron gun

specimen

Lorentz lens

electron biprism

viewing plane

interference region

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.

200nm

Holographic fringes

How Does Off-Axis Holography Work?

+

z

x kzfti 2exp

kzftiyxi eeyxa 2,,

iQxkzftiyxi eeeyxa 22,,

kzftie 20

iQxkzfti ee 220

kzftiiQxiQxi eeeae 2220total

The waves interfere

How Does Off-Axis Holography Work? iQxiiQxiikzftiiQxiQxikzfti eeeaeeeeeaee 22/22/2/2222

total

iQxiiQxiiQxiiQxi eeeaeeeeaeI 22/22/22/22/totaltotal

*2

totaltotal

yxQxyxayxaI ,4cos,2,1 2total

cosine fringes

Latex spheres (image from Lai et al. in Tonomura et al. (Eds.), Electron Holography, North Holland, Elsevier, Amsterdam 1995.)

fringes bend on moving from vacuum into the specimen

The phase information now appears in the image

…but how do you separate it?

Separating the Phase

2

2 ( ) 4 ( ) 4

1 ( ) 2 ( )cos 4 ( )

1 ( ) ( ) ( )i x iQx i x iQx

I x a x a x Qx x

a x a x e e a x e e

2 ( ) 4 ( ) 4F.T. ( ) ( ) F.T. ( ) * ( ) F.T. ( ) *F.T. F.T. ( ) *F.T.i x iQx i x iQxI x q a x q a x e e a x e e

To get the phase, we Fourier transform the intensity and use

2 ( ) ( )F.T. ( ) ( ) F.T. ( ) * ( ) F.T. ( ) * ( 2 ) F.T. ( ) * ( 2 )i x i xI x q a x q a x e q Q a x e q Q

Inverse Transform

Original Image (called ‘the hologram’)

Fourier transform

Extract sideband and put origin at centre

wrapped phase

vacuum

glue100 nm

gives amplitude and phase

SrTiO3

SrRuO3

2 ( )( ) i q Q xq Q e dx

Extracting the Phase cont. 2 ( ) ( )F.T. ( ) (0) F.T. ( ) * (0) F.T. ( ) * ( 2 ) F.T. ( ) * ( 2 )i x i xI x a x a x e q Q a x e q Q

Select sideband ( )F.T. ( ) i xa x e

Inverse transform sideband ( )( ) i xa x e 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.

real

x

2

0

wrapped

x

2

0

Methods for Separating B and V

Constant determined by acceleration voltage

Electrostatic potential

Magnetic flux density

SB dzyxh

edzzyxVCyx E .,,

2,,),(

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.

SB dzyxh

etVCyx E .,,

2),( 0

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

yxx

y

et

hyx ,

/

/

2,

B

Method 2: If the sample can be heated above its Curie point so that it is no longer magnetic we have

SB dzyxh

edzzyxVCyx E .,,

2,,),(cold

dzzyxVCyx E ,,),(hot

yxyxdzyxh

eyx ,,,,

2),( hotcoldmagnetic S.B

The magnetic contribution to the phase is then the difference of these two.

See: Loudon J.C. et al. Nature 420, 797, 2002.

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:

SB dzyxh

edzzyxVCyx E .,,

2,,),(

yxyxdzyxh

eyx ,,

2

1,,

2),( -magnetic S.B

The magnetic contribution to the phase is then:

SB dzyxh

edzzyxVCyx E .,,

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.

yxyxdzyxh

eyx ,,

2

1,,

2),( down upsideupright way magnetic S.B

See R.E. Dunin-Borkowski et al., Microscopy Research and Technique, 64, 390, 2004 and refs. therein.

B B

vElectron beam

F = -ev × BTurn over

v

Thin Thick ThinThick

F = -ev × B

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 m. 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.

Intensity

Displacement

In-focus image (blank)

Out-of-focus image

The distance telling you how far out of focus you are is called the defect-of-focus or defocus, f and is usually measured in mm.

Method 2: Out-of-Focus Imaging

(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 (f) (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.

(c)

The Transport of Intensity EquationThere 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 EVm

22

2

with the condition for a steady electron current 0*Im ..J

and re-expressing the answer in terms of the intensity I and phase gives the Transport of Intensity Equation:

z

II xyxy

2

.

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

z

II xyxy

2

.

z

II xy

22

0

z

I

Iq F.T.

2~4

0

22

f

fIfI

Iq 2F.T.

2

1~

02

If the in-focus intensity is constant I0, we have:

The Transport of Intensity Equation (TIE):

Taking the 2D Fourier transform gives:

(q is the Fourier space coordinate)

So

Using the Transport of Intensity Equation

f

fIfI

Iq 2F.T.

2

1~

02

x

z

z = f

z = -f

z = 0To 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.

How Well Does TIE Work?f = 3mm f = 0 f = -3mm

Recovered phase cos Simulated Phase (and cos)

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

t B

x

ze-v

rad 50 h

eBt

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.

mrad

rad

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.

Phase Recovery Method 3(b): Phase Plate ImagingInstead 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.

Carbon sheet giving phase shift.

Image has ‘field lines’

Method 3: Phase Plate Imaging

Phase plate image of circular permalloy elements.

cos(phase) recovered using TIE.

J.C. Loudon, P. Chen et al. in preparation.

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.