Graef M. Introduction to conventional transmission electron microscopy

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288 Getting started

types it is necessary to obtain a series of two-beam bright ﬁeld–dark ﬁeld image

pairs for three or more different reciprocal lattice vectors g. It is not sufﬁcient to

simply know the family {g} of the reﬂection; we must determine which member

of the family corresponds to the active reﬂection, so that all g-vectors are known

unambiguously. The reader is encouraged to practice the procedure outlined in this

section; ﬁrst, start on a large-grained material, or even on the single-crystal GaAs

study material, so that you do not have to worry about losing the region of interest

when tilting the sample. Then, when you have mastered the procedure, you can

work on more difﬁcult situations, for instance a small-grained material, such as the

one used for Fig. 4.21.

4.6.4 Convergent beam electron diffraction patterns

All images and diffraction patterns obtained so far used a parallel incident beam,

which means that the second condensor lens was used in an under-focus condition.

It is also possible to use a small spot size

†

and a large condensor aperture to obtain

a conical incident beam. Observations made with such a conical beam are known

as convergent beam observations.

We will describe the full geometry and symmetry of the convergent beam electron

diffraction (CBED) pattern in Chapters 6 and 7. For now, we will simply illustrate

how CBED patterns can be obtained for systematic row and zone axis orientations.

We will assume that you have inserted one of the study materials in your microscope,

and that you have obtained either a systematic row orientation or a zone axis

orientation. Most likely you obtained this pattern using a large spot size (i.e. a

small spot size number), and an under-focused second condensor lens, to generate

a parallel incident beam. Next, carry out the following steps:

in image mode, set the ﬁrst condensor lens to one of the smallest spot sizes, and correct

the condensor astigmatism;

focus the image using the objective focus control (you may have to use the binoculars to

do this, since the image might be quite dim);

focus the second condensor lens to cross-over (i.e. the smallest possible illumination

spot), and center the spot on the region of interest;

switch to diffraction mode and remove the diffraction aperture.

The resulting pattern is a convergent beam diffraction pattern. Next, try each of the

following modiﬁcations.

Change the diameter of the condensor aperture; the radius of the diffraction disks

will

change, because you are changing the range of incident beam angles (see Fig. 3.27b). If

†

On the microscope console this usually means a high spot size number. On a JEOL 2000EX microscope for

instance, spot size 8 results in a much narrower beam than spot size 2, as illustrated in Fig. 4.8(c).

4.6 Basic CTEM observations 289

000 002 004

(a)

(b)

(c)

(d)

Fig. 4.24. g

002

systematic row CBED pattern for GaAs, obtained on a Philips EM420

microscope operated at 120 kV. The selected condensor aperture corresponds to a beam

convergence angle of θ

= 6.6 mrad, slightly larger than the Bragg angle θ

002

= 5.92 mrad.

The foil thickness increases from (a) to (d).

you completely remove the condensor aperture from the beam path,

†

you will observ

a complex pattern of lines and intensity variations inside a large-diameter central disk.

You may need to reduce the image intensity to observe this pattern, in particular if you

use a digital camera to record the image, but you should

not use the second condensor

lens control to do this! Why not? In CBED mode, image intensity should be changed by

means of the ﬁlament emission control.

Move the condensor aperture using the aperture translation controls; note that the location

of the diffraction disks changes, but the intensity distribution remains in the same location.

If you have a zone axis diffraction pattern you can use this method to precisely position

the central disk at the zone center.

Change the beam tilt (you may want to do this using the dark ﬁeld memory setting); both

the location of the diffraction disks and the intensity distribution inside them changes.

You can use this to precisely align the beam along a given direction, and then return the

central beam to the center of the screen using the projector alignment.

Change the sample orientation using the primary tilt axis; this has the same result as

changing the beam tilt, except that the diffraction disks remain in the same location

during tilting.

Change the second condensor lens current or the objective lens current and observe how

that changes the CBED pattern. There is only one current setting for which the resulting

pattern is a true CBED pattern, and you should experiment with both lens currents to

discover for which setting the pattern is obtained.

If you started from a systematic row orientation, then your CBED pattern may be

similar to the one shown in Fig. 4.24, which shows a section of the g

002

systematic

row of GaAs for four different thicknesses. The patterns were obtained on a Philips

EM420, operated at 120 kV, equipped with a Gatan DualView camera system.

The Bragg condition is satisﬁed for the g

004

reﬂection, and the g

002

reﬂection is

†

It is not always possible to completely remove the condensor aperture from the column. This depends on the

microscope model. Without a condensor aperture, a large number of x-ray photons are generated in the column.

290 Getting started

barely visible. From Fig. 4.5 (part 2) we ﬁnd that this is in agreement with the

relative magnitudes of the corresponding Fourier coefﬁcients V

002

004

= 0.127.

The number of vertical fringes in the 004 diffraction disks increases with increasing

sample thickness. We also note that the intensity distribution in the central disk is

asymmetric: the left half is signiﬁcantly brighter than the right half, in particular for

the thickest foil. Chapters 6 and 7 will explain the details of the contrast variations

in these CBED patterns. For now, you should familiarize yourself with obtaining

and recording these patterns.

If you started from a zone axis orientation, then your pattern might be simi-

lar to those shown in Fig. 4.25. The patterns correspond to a [110] zone axis of

000 002 004002

111

004

111

113113

113 113

220

220 222222

222222

000

mm2

(a) (b)

Fig. 4.25. (a) and (b) Zone axis CBED patterns of the [110] orientation of GaAs, recorded

at 120 kV on a Philips EM420, for two different foil thicknesses. Part (c) shows a schematic

of the patterns with Miller indices labeling each disk; (d) shows the internal symmetry of

the central disk.

4.7 Lorentz microscopy: observations on magnetic thin foils 291

GaAs, again obtained at 120 kV in a Philips EM420, for two different sample thick-

nesses. The beam convergence angle is 5.1 mrad, just below the Bragg angle θ

5.13 mrad for the g

111

reﬂections. With increasing foil thickness, more details be-

come visible within the disks. The disks are indexed in Fig. 4.25(c). From the

pattern in Fig. 4.25(b) we see that the overall pattern has a horizontal mirror plane,

and therefore has planar point group symmetry m. The central disk itself has an

additional vertical mirror plane, as shown schematically in Fig. 4.25(d), and hence

has planar point group symmetry mm2. The intensity distributions inside the 1

and 1

11 disks are not identical and this difference is due to the non-centrosymmetric

nature of the GaAs crystal structure.

4.7 Lorentz microscopy: observations on magnetic thin foils

4.7.1 Basic Lorentz microscopy (classical approach)

As described in equation (3.7) on page 146, an electron moving through a region

of space with an electrostatic ﬁeld E and a magnetic ﬁeld B experiences a velocity-

dependent force, commonly known as the Lorentz force F

=−e

(

E + v × B

)

. (4.2)

We will assume that all ﬁelds are static. Whereas Chapter 3 dealt with the behavior

of electrons in the magnetic ﬁeld of round lenses, deﬂectors, and stigmators, in this

section we will describe how the electron interacts with the magnetic ﬁeld caused

by the sample itself.

Since the magnetic component of the Lorentz force acts normal to the travel

direction of the electron, a deﬂection will occur. Only the component of B normal

to v will contribute to the deﬂection, and we will refer to this component as the

in-plane magnetic induction B

⊥

, i.e.

B = B

⊥

+ B

n, (4.3)

where n is a unit vector parallel to the beam direction. Since only two out of the

three components of B act on the electron trajectory, it is clear a priori that a

complete determination of all three components will require the use of at least two

independent incident beam directions n.

While it is possible to mount the sample in a ﬁeld-free region (see Section 4.7.2)

it is in general not possible to completely remove the electrostatic ﬁeld E, since

the crystal structure itself generates an electrostatic lattice potential. The mag-

netic deﬂection caused by the Lorentz force does not change the electron energy,

whereas the (positive) electrostatic component causes the electron to accelerate

which changes its kinetic energy (refraction).

292 Getting started

Fig. 4.26. Schematic of a magnetic thin foil and the resulting deﬂection of an incident

electron

beam.

Consider a planar thin foil of thickness t, normal to the incident beam. If the

in-plane magnetic induction is directed along the x-axis (see Fig. 4.26) with mag-

nitude B

⊥

, then the momentum transferred to the electron is given by

dτ, (4.4)

with τ the time it takes for the electron to traverse the sample. In the absence of elec-

trostatic ﬁelds, the magnitude of the Lorentz force along the y direction is given by

= ev

⊥

. (4.5)

The integral can be rewritten in terms of the sample thickness t = vτ as

= e

⊥

dz = eB

⊥

t. (4.6)

The deﬂection angle θ

is then to a good approximation (Fig. 4.26) given by the

ratio of the y and z momentum components:

⊥

, (4.7)

where m is the relativistic electron mass. This equation can be rewritten in a more

useful form by means of the de Broglie relation between particle momentum and

wavelength ( p = h/λ), and we ﬁnd for the Lorentz deﬂection angle θ

eλ

⊥

t = C

(E)B

⊥

t. (4.8)

The constant C

(E) is determined by the acceleration voltage E of the microscope,

and is given by

(E) =

9.377 83

√

E + 0.974 85 × 10

−3

µrad T

−1

;

4.7 Lorentz microscopy: observations on magnetic thin foils 293

(to obtain the Lorentz angle in µrad, B

⊥

must be stated in tesla, t in nm and E

in kV). For the commonly used acceleration voltages we have: C

(100) =

0.895 018, C

(200) = 0.606 426, C

(300) = 0.476 050, C

(400) = 0.397 511,

and C

(1000) = 0.210 834 µrad T

−1

. A 100 nm thin foil with an in-plane

magnetic induction of B

⊥

= 1 T, will give rise to a beam deﬂection of θ

39.75

µrad at 400 kV. For comparison, typical Bragg angles for electron diffraction

are in the range of a few milliradians, i.e. two to three orders of magnitude larger than

Lorentz deﬂection angles. Note that the sample thickness and the in-plane induction

component both appear in the expression for the deﬂection angle. Since the sample

thickness is notoriously difﬁcult to quantify experimentally, it should be clear that

any measurement based on the Lorentz deﬂection angle will at best be qualitative

and provide only the product B

⊥

t. Localized thickness variations and local

out-of-plane excursions of the magnetic induction will produce identical changes

in the Lorentz deﬂection angle θ

. Only when an independent thickness mea-

surement is available can Lorentz methods provide a direct map of the in-plane

induction component.

4.7.2 Experimental methods

Direct observations of the magnetic substructure of a material in a TEM require

that a thin foil be made, using appropriate thinning procedures. One must always

be aware that the magnetic structure of a thin foil may be different from the bulk

magnetic structure. In particular, in soft magnetic materials, the introduction of

two free surfaces may completely change the energetics of magnetic domains, and,

indeed, the very nature of the magnetic domain walls.

In order to preserve the magnetic microstructure of the thin foil, the sample must

be mounted in a ﬁeld-free region in the microscope column. Since the objective lens

in a standard TEM is an immersion-type lens, the requirement of a ﬁeld-free region

has profound consequences on the electron optical properties of the microscope.

The low-ﬁeld sample environment and consequent increase in the focal length of the

objective lens result in a reduced ﬁnal image magniﬁcation compared with that of

conventional transmission electron microscopy. This represents a serious limitation

in the quantitative study of nanoscale magnetic structures. Furthermore, inelastic

scattering in the sample contributes noise to the images, which further limits the

attainable resolution of the standard Lorentz modes.

A number of different sample +lens conﬁgurations can be used.

Mount the sample above the main objective lens. This generally requires the introduction

of a second goniometer stage in between the condensor and objective lenses, and is

therefore only attempted in dedicated instruments.

Use a dedicated low-ﬁeld Lorentz pole piece. This is perhaps the most efﬁcient and

reliable method, but may require a pole piece change, which is not always an easy thing

294 Getting started

to do. Furthermore, if the microscope is used for many different types of materials,

the loss of magniﬁcation due to the reduced ﬁeld may be unacceptable to other users. A

permanent Lorentz pole piece downstream from the sample holder is therefore preferable,

assuming it does not deteriorate the electron optical properties of the column when used

in conventional or high-resolution observation modes.

Turn the objective lens off and use an objective minilens (if present) to obtain a back

focal plane at the location of the selected area aperture. If a post-column energy ﬁlter is

available, the loss of image magniﬁcation may be partially compensated by the internal

magniﬁcation of the ﬁlter [DDG97].

All images shown in this section were obtained on a JEOL 4000EX top-entry

high-resolution TEM, operated at 400 kV in Low Mag mode (with the main objective

lens turned off). The objective minilens provides a cross-over near the selected

area aperture plane, at a maximum magniﬁcation of 3000×. A Gatan imaging ﬁlter

(GIF) is then used to remove most inelastically scattered electrons (using an energy

selecting slit width of 20 eV), and to magnify the image by an additional factor

of about 20×. All images were acquired on a 1K×1K CCD camera, and are gain

normalized and background subtracted. Unless mentioned otherwise, no additional

image processing was carried out on the experimental images.

4.7.2.1 Fresnel mode

The Fresnel or out-of-focus Lorentz mode is perhaps the easiest observation method

to carry out experimentally, since it only requires a change in the lens current for the

main imaging lens (either a Lorentz lens or an objective minilens). The method is

schematically shown in Fig. 4.27. Consider a sample with three magnetic domains,

separated from each other by 180

◦

magnetic domain walls. The character of the

walls is unimportant at this point. The outer domains have their magnetization point

convergent

wall image

divergent

wall image

∆f

underfocus

overfocus

Fig. 4.27. Schematic illustration of the Fresnel or out-of-focus imaging mode.

4.7 Lorentz microscopy: observations on magnetic thin foils 295

into the plane of the drawing, and the central domain has the opposite magnetization

direction. The resulting trajectory deﬂections are towards the positive y-direction

for the outer domains.

A standard bright ﬁeld image, obtained by selecting only the transmitted beam

with a sufﬁciently large aperture in the back focal plane of the imaging lens, does

not show any contrast (other than diffraction or absorption contrast) for an in-focus

condition. In other words, when the object plane of the imaging lens is located

at the sample position, no magnetic contrast is observed. If the current through

the imaging lens is reduced, the focal length increases (causing a reduction in the

lateral image magniﬁcation), and the object plane is located at a distance  f below

the sample. Because of the small Lorentz deﬂection angles, a rather large defocus

 f is needed to obtain an overlap between the electrons deﬂected on either side

of the left domain wall in Fig. 4.27. In the overlap region, an excess of electrons

will be detected and this will show up in the image as a bright line feature at the

projected position of the domain wall. The domain wall on the right will show up

as a dark feature, since electrons are deﬂected away from the domain wall position.

The bright domain wall image is known as the convergent wall, the dark wall is the

divergent wall.

If the imaging lens current is increased, so that the focal length is reduced and the

image magniﬁcation increased, then an over-focus image is obtained. The image

contrast of convergent and divergent domain walls reverses, as can be seen from

the extended electron trajectories in Fig. 4.27.

Figure 4.28 shows a set of experimental Fresnel images for a cross-tie wall

arrangement in a 50 nm thick Ni-20 wt% Fe Permalloy ﬁlm. The in-focus image is

shown in (a), while the under-focus and over-focus images are shown in (c) and

(d), respectively. A schematic of the magnetization conﬁguration around the cross-

tie wall is shown in (b). Note that this conﬁguration can not be deduced from the

Fresnel images; the Foucault images shown in the next section (Fig. 4.32) were

used to derive the magnetization arrangement shown in the schematic. The bright

points along the main domain wall (Fig. 4.28c) correspond to the centers of regions

around which the magnetization has a circular pattern (vortices); this arrangement

acts as a lens which focuses the electrons into a single spot. The magnetization at

the vortex is predominantly out-of-plane (see e.g. [Chi78, pages 402–408]), and

opposite to the magnetization at the intersection of the main wall and the divergent

walls in Fig. 4.28(c). This divergent wall is known as a cross-tie wall. Such walls

gradually disappear with increasing distance from the main 180

◦

domain wall.

Since the Fresnel image mode highlights the domain walls in the thin foil, it

should be possible to derive from the images an estimate of the domain wall width δ.

This parameter plays a central role in the energetics of magnetic materials, and a

direct experimental measurement method would be an extremely valuable tool.

296 Getting started

5 µm

(a)

(b)

Fig. 4.28. (a) In-focus, (c) under-focus, and (d) over-focus images (zero-loss ﬁltered with

20 eV slit width) of cross-tie

domain walls in a 50 nm thick Ni-20 wt% Fe Permalloy

ﬁlm.

Note that the image magniﬁcation changes slightly with defocus.

The magnetization pattern

in (b) is derived from the Foucault images in Fig. 4.31. (Sample courtesy of Chang-Min

Park.)

One can use the divergent wall image to obtain an estimate of the domain wall

width δ; a reasonable estimate of δ is obtained when the measured width of a

divergent wall is plotted against the defocus value and extrapolated to zero defocus

(e.g. [SWP68, SW69, Woh71]). This extrapolation method works well for large

domain wall widths, but overestimates δ for the narrow walls found in hard magnetic

materials. A discussion of the accuracy of the extrapolation method can be found

in [GC87].

The classical Lorentz deﬂection theory provides a good ﬁrst approximation for

the understanding of image contrast in the Fresnel imaging mode. However, when

a more coherent electron beam is used (for instance, by under-focusing the second

condensor lens), then the convergent wall image reveals the presence of interference

fringes which cannot be explained by the classical model. Figure 4.29 shows a

sequence of under-focus images of a 180

◦

domain wall (upper right-hand corner)

in a permalloy thin ﬁlm at increasing defocus  f ; all images were taken with

an under-focused second condensor lens, or, equivalently, a reasonably coherent

electron beam. The traces below the micrographs are averages over 40 pixels along

4.7 Lorentz microscopy: observations on magnetic thin foils 297

∆f 2∆f 3∆f 4∆f

1 µm

Fig. 4.29. Under-focus images of a 180

◦

domain wall (upper right) which splits into two

◦

domain walls in a 50 nm Ni-20 wt% Fe Permalloy thin ﬁlm. The images were obtained

with a strongly under-focused second condensor lens, for increasing under-focus value  f .

The fringe contrast is clearly visible; intensity proﬁles across the 180

◦

domain wall are

shown at the bottom.

the wall proﬁle. While the number of fringes increases with increasing defocus, the

spacing between the fringes appears to be constant. The classical model for the

Fresnel observation mode fails to explain the presence of such fringes. We will

return to the origin of the fringes in Section 7.5.2; image simulations for coherent

Fresnel images will be discussed in Section 10.4.1.

While the Fresnel imaging mode is easy to use and provides a direct, qualitative

image of the domain wall conﬁgurations in the sample, its quantitative usefulness is

rather limited because of the sometimes large magniﬁcation changes caused by the

defocus. However, in Chapter 10 we will introduce a method, based on the Fresnel

mode, that can be used to determine quantitatively the magnetic microstructure.

4.7.2.2 Foucault mode

A second commonly used observation mode employs an in-focus image and is

somewhat similar to the dark ﬁeld imaging mode. As illustrated in Fig. 4.30 an

aperture is introduced in the back focal plane of the imaging lens (an objective

minilens or a dedicated Lorentz lens), and by translating the aperture normal to

the beam a section of the split central beam is cut off. The regions in the sample

that give rise to deﬂected electrons, which are passed by the aperture, will appear

bright in the image. This observation mode produces high-contrast in-focus images

of magnetic domain conﬁgurations, which are known as Foucault images. Since a

typical aperture stage has two translation controls which are normal to each other,

one would acquire typically either two or four images, one or two for each main

aperture translation direction. An example of Foucault images for the same sample

region as shown in Fig. 4.28 is shown in Fig. 4.31; the approximate aperture shift

directions are indicated by white arrows in the upper left-hand image. Since we