Graef M. Introduction to conventional transmission electron microscopy

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198 The transmission electron microscope

(a)

(b)

Fig. 3.27. (a) Ray diagram for a strongly excited C

lens. Electrons leave the C

aperture

at large angles with respect to the optical axis. (b) Ray diagram with a second condensor

lens C

. The lens has the ﬁrst condensor cross-over as the object plane, and the specimen

as the image plane. The diameter of the C

cross-over on the specimen is known as the spot

size. The variable diameter aperture C

limits the angular range of electron trajectories at

the expense of beam current.

under-focused; if the image cross-over falls before the sample, then the lens is said

to be over-focused. The lens is in-focus when the smallest possible spot diameter

is observed on the sample. Although over- and under-focused conditions may give

rise to a spot with as identical diameter on the sample, one must be aware of the fact

that the beam divergence angle θ

is different in the two cases. Observation modes

that require a nearly parallel incident beam (such as high-resolution observations,

see Chapter 10) must hence be carried out with as underf-ocused C

lens (unless

additional condensor lenses are present). On microscopes with rotary lens control

knobs, the under-focus condition usually corresponds to a counterclockwise rotation

(i.e. less current through the lens).

Using the two-lens condensor system, we can routinely obtain 100 nm spot

sizes. The ﬁrst condensor lens control is often known on the microscope console

as the spot size; the spot size can only be varied in discrete steps. The C

current

is often labeled as intensity or brightness on the microscope console; it can be

varied continuously. Since the C

cross-over changes location whenever the spot

size is changed it is necessary to change the C

current to obtain the smallest beam

diameter on the sample. The second condensor aperture C

can be set to one of

3.9 The specimen stage 199

three or four different diameters. It can be seen from Fig. 3.27(b) that a change in

diameter does not change the size of the illuminated area on the specimen, only

the beam intensity changes (less beam current for a smaller aperture diameter). A

smaller aperture will also change the maximum beam divergence angle by limiting

the solid angle subtended by the C

cross-over at the sample. This improves the

coherence of the incident beam at the expense of lower image intensity. Modern

microscopes often have a third so-called mini-condensor lens, located right above

the objective lens upper pole piece. Since this lens is an integral part of the objective

lens stage, we will discuss it in the next section.

3.9 The specimen stage

3.9.1 Types of objective lenses

We have seen that the objective lens of a TEM is a type of immersion lens, since the

sample is located well within the magnetic ﬁeld of the lens. There are two major

types of objective lenses: the high-contrast lenses, also known as asymmetric

lenses,

and the twin or symmetric lenses. In an asymmetric objective lens the sample is

located at one-third of the distance between the upper and lower pole pieces, as

shown in Fig. 3.28. The objective or diffraction aperture is located two-thirds of

the way down and coincides with the lens back focal plane. The upper pole piece

usually has a narrower bore diameter than the lower pole piece, hence the name

“asymmetric”. In a symmetric lens design, the sample is located in the center

between the two pole pieces, as indicated in Fig. 3.29. Both bore diameters are the

same in the symmetric lens.

1/3

B(z)

Back focal plane

Specimen

Objective aperture

Fig. 3.28. Schematic diagram of a high-contrast objective lens. The specimen and back

focal plane are located at one-third and two-thirds the distance along the axis of the lens.

The angles between the rays and the optical axis are highly exaggerated and are actually

only a few tens of milliradians. The lens forms a ﬁrst intermediate image.

200 The transmission electron microscope

B(z)

Back focal plane

Specimen

Objective aperture

Objective

condensor lens

Objective

imaging lens

Condensor

mini lens

Microprobe mode

B(z)

Back focal plane

Specimen

Objective aperture

Objective

condensor lens

Objective

imaging lens

Condensor

mini lens (off )

Nanoprobe mode

(a)

(b)

Fig. 3.29. Schematic diagram of a symmetric objective lens. With the condensor mini-lens

switched off (a) a ﬁne probe is obtained (nanoprobe mode). When the condensor mini-lens

is switched on (b) a parallel beam is obtained (microprobe mode).

Asymmetric lenses have a much longer focal length than symmetric lenses (often

in the range 5–10 mm), and hence a smaller lateral magniﬁcation M or, equivalently,

a larger angular magniﬁcation M

, which makes them well suited for large camera

length diffraction observations. Since the incident electrons experience only a frac-

tion of the objective lens magnetic ﬁeld before reaching the sample (as indicated

by the B(z) proﬁle in Fig. 3.28), this preﬁeld acts only as a weak third condensor

lens. Recalling that the primary task of the condensor lenses is to demagnify the gun

cross-over, it is clear that the asymmetric objective lens cannot be used to obtain

nanometer probe sizes.

In the symmetric lens design (the so-called Riecke–Ruska lens) the electron

experiences nearly half of the objective lens magnetic ﬁeld before reaching the

sample, and this upper half of the ﬁeld is known as the objective condensor lens. It

can be used to further demagnify the incident probe, down to the single nanometer

3.9 The specimen stage 201

size range. This observation mode is thus known as nanoprobe mode (Fig. 3.29a).

When an additional mini-condensor lens is present, often as part of the objective

lens circuit, then this lens can be used to create another cross-over, this time right

above the objective condensor lens (Fig. 3.29b). The electrons leaving this cross-

over are then focused in a parallel but very narrow beam and after passing through

the sample and the objective imaging lens, they go through a symmetric cross-

over. This mode is known as microprobe mode. In microprobe mode, the electrons

travel in a parallel beam through the sample, whereas in nanoprobe mode they are

focused into a small probe. Microprobe mode is therefore useful as a diffraction

technique, since the parallel beam can be focused readily into diffraction spots,

while nanoprobe mode is more useful for analytical observation techniques. The

focal length of the symmetric lens is much smaller than that of the asymmetric

lens, and is usually in the range 1–3 mm. This means that the objective aperture is

located very close to the sample, which imposes limitations on the tilt range of the

specimen holder. The lens gap of a symmetric lens (the distance between the upper

and the lower pole pieces) is of the order of a few millimetres; the asymmetric lens

has a much larger lens gap, in the range of 10–20 mm.

While the primary function of the illumination stage is to create an electron

beam in a well-deﬁned reference state, the primary function of the objective lens

is to bring the various diffracted electron beams to a cross-over while introducing

minimal lens aberrations (see also Section 3.5). When we discussed the Fraunhofer

diffraction conditions we saw that we do not need to have a lens in order to have

a diffraction pattern; indeed, simply projecting a laser beam through a ﬁne optical

grating onto a wall on the other side of a room will produce a diffraction pattern

without using a lens. The main purpose of the objective lens is to reduce the inﬁnite

“focal length” of the lensless system to a ﬁnite length, so that both diffraction

pattern and image can then be magniﬁed by other lenses. It is, in principle, possible

to create microscopes without lenses, since all of the relevant physics occurs in the

interaction of the electron beam with the specimen. One such microscope is the

point projection microscope [SQM93], which uses no lenses at all, and hence does

not suffer from lens aberrations.

3.9.2 Side-entry, top-entry and special purpose stages

There are two different types of specimen stages: those where the sample enters

the objective lens from the side (side-entry) and those where the sample is lowered

into the lens from the top (top-entry).

The side-entry stage is by far the most common stage and is used for both

conventional and analytical electron microscopy and for most in situ experiments.

As shown in the schematic drawing in Fig. 3.30, the sample is mounted at the end

202 The transmission electron microscope

primary

tilt axis

secondary

tilt axis

Fig. 3.30. Side-entry double-tilt holder; the primary and secondary tilt axes are indicated.

(Photograph reproduced with permission of B. Armbruster, Gatan, Inc.)

of a rod-like specimen holder. The sample ﬁts into a 3.05 mm diameter cup, which

can either pivot around one axis in the plane of the sample, or rotate around an

axis perpendicular to the sample. The rod-axis represents the primary rotation axis.

These holders are referred to as double-tilt and rotation-tilt, respectively. Depending

on the objective lens gap the maximum tilt angle may be as high as 60

◦

or as low

as 15

◦

. Note that most specimen holders are made out of a soft aluminum alloy

(or pure Be for some analytical holders), so utmost care is necessary whenever a

holder is manipulated. Depending on the diameter of the microscope column, the

specimen rod is anywhere from about 25–60 cm long. This rod is entered into the

column through a vacuum airlock. Refer to the microscope manuals for the correct

procedure to insert a sample holder into the column.

For most microscopes the sample has ﬁve degrees of freedom as illustrated in

Fig. 3.31: two lateral movements in the horizontal plane (x and y, controlled by the

specimen translation wheels or a joystick/trackball system), one vertical movement

(z, specimen height) and two rotation angles α and β (either double-tilt or rotation-

tilt). The ﬁve numbers {x, y, z,α,β} describe the specimen attitude. The rotation

angles are deﬁned as positive for counterclockwise rotations, when looking from

the positive side of the x, y,orz axes towards the specimen. The reader should

carefully study the geometry of the holder(s) he/she will be using, in particular

how the holder is oriented inside the microscope column; for several microscopes,

the holder is actually placed upside down in the column.

†

For top-entry stages, the

reader should consult the microscope manual to determine the relative orientation

of the specimen holder and rotation axes with respect to the microscope column.

In modern microscopes, some or all of the ﬁve coordinates may be computer

controlled, while in older instruments x, y, and z motion are manually controlled.

The primary tilt angle α can be changed manually or via a motor, and the secondary

tilt/rotation angle β is always controlled by a motor. Both motors can be actuated

†

Upside down with respect to the orientation of the specimen cup during loading.

3.9 The specimen stage 203

-z

-y

-x

+β

+α

+β

(a)

(b)

Fig. 3.31. Degrees of freedom for a typical specimen holder: (a) translation axes (positive

x points along

the length axis of the holder, positive

y points to

wards the front of the

microscope column); (b) primary tilt angle α; (c) secondary tilt angle β for a double-tilt

holder; (d) secondary angle β for a rotation-tilt holder.

by ﬂoor-mounted pedals, or, in some microscope models, by rotary or push but-

tons on the microscope console near the column. More recent computer-controlled

microscopes are equipped with a joystick-like mechanism to control the specimen

attitude.

We have seen in Section 3.5 that lens aberrations increase rapidly for electrons

traveling at large angles with respect to the optical axis of the microscope. This

axis is deﬁned as the line connecting the ﬁlament tip to the center of the viewing

screen, and for a well-aligned microscope column the center line of each lens

should coincide with the optical axis. The size of the electron transparent region in

a thin foil can range from a few to several hundred micrometers, and the specimen

translation controls x and y are used to bring a region of interest in the sample onto

the optical axis of the column and therefore in the electron beam.

For both top- and side-entry stages, the specimen is located inside the objective

lens magnetic ﬁeld. Surrounding the specimen holder is the so-called cold-ﬁnger or

anti-contamination device (ACD). It is a piece of metal, typically copper, linked to

a liquid nitrogen dewar. By keeping the cold ﬁnger at liquid nitrogen temperature,

residual gases in the microscope vacuum (or petroleum derivatives from an oil

diffusion pump) are trapped onto this cold surface instead of on the sample. The

ACD keeps the sample clean so that prolonged observations are possible. On some

microscope models, one can insert an ACD-heater into the dewar at the end of a

microscope session so that the remaining nitrogen is boiled off.

204 The transmission electron microscope

A transmission electron microscope is an ideal instrument for in situ experiments.

A variety of special purpose stages have been designed:

heating and cooling;

magnetic ﬁeld (one must use special low-ﬁeld pole pieces in order to apply a controlled

magnetic ﬁeld on the sample);

electric ﬁeld, e.g. to study ferroelectric domain walls;

straining stages, mostly tensile, some torsion or mode I deformation.

Combinations of two or more externally applied ﬁelds (e.g. electric ﬁeld and defor-

mation) are also possible. We refer to [Val79] for an extensive overview of different

stages. Each stage has its own peculiarities and one should always be aware of the

fact that performing an experiment on a thin-foil sample may not give rise to the

same sample response as that expected for a bulk sample. The recent advances in

spherical aberration correctors will undoubtedly rekindle interest in in situ obser-

vations, since the lens gap was kept small to reduce the spherical aberration. If C

can be varied at will, then there is no longer a need for a small lens gap.

3.9.3 The objective lens and electron diffraction geometry

The objective lens is used with a high excitation current which means that a high

lateral magniﬁcation is obtained. From our discussion on electron optics we know

that the angular magniﬁcation M

is the inverse of the lateral magniﬁcation M,

which means that rays leaving the objective lens will travel at very small an-

gles to the optical axis in all subsequent lenses – this is why most aberrations

of the subsequent lenses are unimportant and can be ignored. The aberrations

of the objective lens cannot be ignored since in this lens electrons travel at the

largest angles with respect to the optical axis. The spherical aberration C

and

the chromatic aberration C

determine the ﬁnal image quality for conventional

TEM work (assuming correctable aberrations, such as astigmatism, have been taken

care of).

A cross-sectional view of the goniometer area of the JEOL 120CX microscope

is shown in Fig. 3.32(a). The specimen rod enters from the right and is ﬁrmly

seated in a metal cup which is linked to one of the specimen translation controls.

The specimen holder sits right above the bottom plate of the anti-contamination

device (cold ﬁnger) which, in turn, sits above the objective aperture. The objective

stigmator follows a few millimeters below the lower objective pole piece. The OL

pole pieces are located above the lens coil which is visible in the lower left-hand

corner. Figure 3.32(b) shows the entire imaging stage of the microscope, from

the objective lens area down to the projector lens; the bottom three lenses will be

discussed in more detail in the following section.

3.9 The specimen stage 205

diffraction

aperture plate

ACD

scan coil

sample holder axis

"cold finger"

objective lens coil

specimen rod

support

objective

stigmator

optical axis

diffraction

aperture

selected area

aperture

objective

lens coils

cooling water

diffraction

lens

intermediate

lens

projector lens

objective stigmator

(a)

(b)

Fig. 3.32. (a) Cross-sectional view of the goniometer area of the JEOL 120CX microscope;

(b) cross-sectional view of the objective lens and magnifying lenses.

206 The transmission electron microscope

2θ

λfg

2θ

r ~ 2θL

(a)

(b)

thin

foil

bfp

screen

Fig. 3.33. (a) Schematic representation of the diffraction geometry; (b) deﬁnition of the

diffraction camera length L.

The OL is characterized by two different planes: the back focal plane, in which

the diffraction pattern is formed, and the image or selected area plane. In the

back focal plane we can introduce an aperture, typically known as the diffraction

aperture. Similarly, in the image plane we can introduce the so-called selected area

aperture (see Fig. 3.32b). Since electrons leaving the OL form an image at a large

distance compared to the sample dimensions, Fraunhofer diffraction conditions are

satisﬁed and hence the back focal plane contains the Fourier transform of the object

function. The reciprocal lattice points in the back focal plane are at a distance λ f |g|

from the origin of reciprocal space, as shown in Fig. 3.33(a) (this follows from

equation 3.41). The BFP hence contains an “image” of reciprocal space, magniﬁed

by a factor of λ f . Only those reciprocal lattice points that lie near the Ewald sphere

and satisfy the Bragg condition will give rise to a diffracted beam.

Figure 3.33(b) deﬁnes the camera length L of the microscope in diffraction mode.

The camera length is a measure for the magniﬁcation of the diffraction pattern. The

relation between the camera length L, the electron wavelength λ, the lattice spacing

d and the measured spacing r on an electron micrograph can be derived easily as

follows: from the Bragg equation we know that 2d sin θ = λ. Since Bragg angles

for high-energy electrons are small, it is a good approximation to replace sin θ

by θ, and we have 2dθ = λ. From Fig. 3.33(b) we have r = L tan(2θ) ≈ 2Lθ .

Combining the two relations we ﬁnd

r =

λL

= λLg, (3.71)

where g = 1/d. The constant L is known as the camera length, and the product

λL as the camera constant; it has dimensions of [nm mm]. For a given lattice

spacing d, an increase in the acceleration voltage will cause a corresponding de-

crease in the electron wavelength and therefore a decrease of the distance r in the

3.9 The specimen stage 207

diffraction pattern. The camera length L converts the diffraction angle 2θ into the

distance r .

For the aluminum example in Table 2.3 on page 102, 2θ = 12.38 mrad for the

(200) planes (at 200 kV), and for a typical camera length (the distance between

the sample and the observation screen) of L = 1000 mm we ﬁnd r = 12.38 mm.

Rather than changing the magnitude of L by physically changing the location of

the observation screen (a bit impractical), we can use the imaging lenses to change

the effective value of L, while keeping the object and image planes constant. All

microscopes have a predeﬁned set of camera lengths, and the microscope operator

can select the value of L best suited for the observation at hand. This value would

typically be shown on the microscope console and on the diffraction micrograph.

Care must be taken to accurately calibrate the camera length; this will be discussed

in Section 4.5 in the following chapter.

From the preceding paragraphs we can deduce the following geometrical in-

terpretation of electron diffraction (see Fig. 3.34a): the incident electron beam is

represented by the wave vector k, which is parallel to the optical axis of the ob-

jective lens. According to the Ewald sphere construction, this wave vector ends in

the origin O of reciprocal space, which is located at the intersection of the optical

Ewald

sphere

Incident

beam

Laue

circle

1015

(a)

(b)

(c)

(d)

Fig. 3.34. Perspective representation of the spatial relation between the Ewald sphere and

a plane in reciprocal space. In (a) the plane is tangential to the sphere, and the relrods cor-

responding to individual reciprocal lattice points intersect the sphere. When the reciprocal

lattice plane is tilted around an axis in the plane, the intersection of this plane and the Ewald

sphere becomes a circle through the origin, the Laue circle, indicated by a white arc. With

increasing tilt angle (b, c, and d) the circle becomes larger.