Graef M. Introduction to conventional transmission electron microscopy

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598 Phase contrast microscopy

(recall that the interaction constant σ is in the range 0.005–0.009 V

−1

). This

means that we can expand the wave function in powers of the exponential argument,

and only keep terms of ﬁrst order in V

φ(r) = 1 + iσ V

(r) +···≈1 + iσ V

(r). (10.17)

This is known as the weak phase object approximation or WPOA.

In the back focal plane of the objective lens, the wave function becomes

φ(q) = F[1 + iσ V

(r)] = δ(q) + iσ V

(q). (10.18)

In the absence of an aperture, the microscope transfer function consists only of the

phase shift factor, so that the aberrated wave function is given by

(q) = [δ(q) + iσ V

(q)]e

−iχ(q)

. (10.19)

Working out the product we ﬁnd:

(q) = [cos χ(q) − i sin χ (q)]δ(q) + iσ V

(q) cos χ (q) + σ V

(q) sin χ (q).

(10.20)

The ﬁrst term only contributes for q = 0 and is equal to unity. For the remaining

terms we must ﬁrst consider the behavior of the aberration function. Fig. 10.6 shows

-1

0123 123

D=0.5

D=1.5

D=2.5

D=0.0

D=1.0

D=2.0

Fig. 10.6. The function sin χ (Q) for several of the defocus values D of Fig. 10.4.

10.2 The microscope as an information channel 599

the function sin χ (q) for the same dimensionless defocus values as Fig. 10.4. The

function oscillates wildly for large values of q, but there is a range of defocus

values D for which the function changes relatively slowly for smaller values of q.

In particular, around D = 1 the function sin χ(q) is close to −1 for a reasonably

large range of q. This range of q is known as a passband, since the value of sin χ(q)

is nearly constant and equal to −1 in this range. The value of cos χ (q) is then nearly

zero. Let us now assume that we can tune the microscope defocus value so that

sin χ(q) =−1 for all values of q; this is obviously another approximation, since

the q-range for which this can be done is ﬁnite, not inﬁnite. The aberrated wave

function under this assumption becomes

(q) ≈ 1 − σ V

(q). (10.21)

In the image plane we ﬁnd

(r) = F

−1

[

(q)

]

= 1 − σ V

(r). (10.22)

The image intensity is then given by

I (r) =|φ

(r)|

= 1 − 2σ V

(r) + σ

(r) ≈ 1 − 2σ V

(r), (10.23)

where the quadratic term in σ V

is neglected, since we only keep terms linear

in V

Equation (10.23) can be interpreted as follows: we know from Section 2.6.7

that the interatomic potential V (r) has sharp peaks at the atom positions (see also

Fig. 2.12). The projected potential along a given zone axis direction will also show

large positive peaks at the projected atom positions. Therefore, the intensity I (r)

will be less than unity at the projected atom positions. In other words, if we can ﬁnd

a microscope setting for which sin χ(q) =−1 over a signiﬁcant range of q, then

the corresponding image will show dark regions at the projected atom positions.

This is known as dark atom contrast. In the following section, we will explore

how closely we can actually approximate this idealized microscope setting in a real

microscope.

10.2.1.4 The microscope point resolution

The function sin χ(Q) is shown as a grayscale plot in Fig. 10.7. White (black) areas

correspond to positive (negative) values. It is clear that this function oscillates

rapidly for large spatial frequencies Q, as is also obvious from Fig. 10.6; for small

spatial frequencies, the behavior for under-focus and over-focus conditions is dif-

ferent. For under-focus (D > 0) conditions, the ﬁrst minimum (dark band) appears

for D ≈ 1. At this point the function sin χ(Q) reaches its minimum of −1, which

600 Phase contrast microscopy

-101234

Dimensionless Frequency Q

Dimensionless Defocus D

Fig. 10.7. The function sin χ (Q) displayed as a grayscale plot; vertical lines indicate integer

values of the dimensionless defocus D and correspond to the left-hand column of Fig. 10.6.

can be expressed by the following two conditions:



− DQ



=−

(4n + 1) (n ≥ 0); (10.24)

∂χ

∂ Q

= 2π Q(Q

− D) = 0. (10.25)

The ﬁrst relation expresses that sin χ (Q) =−1, the second that χ (Q) reaches

a minimum. It is easy to see that minima are located at Q = 0 and Q =

√

Substituting this in the ﬁrst relation results in

8n + 2

, (10.26)

10.2 The microscope as an information channel 601

which, using equation (10.11), translates to

 f



8n + 2



. (10.27)

The ﬁrst minimum lies at the position (D, Q) = (1, 1), as indicated by the ﬁlled

white circle in Fig. 10.7. If we move to a slightly larger defocus D, then the

minimum splits into two minima with a small bump in between. It is customary to

allow this bump to increase to a value of sin χ ≈−0.7, which can be accomplished

by replacing the factor 8n + 2 in equation (10.27) by 8n + 3. As a consequence

of this change, the ﬁrst zero crossing of sin χ (black ﬁlled circle) moves towards a

larger value of Q, as indicated by the white arrow in Fig. 10.7. The defocus value

D moves from 1 to



= 1.2247.

For values of n larger than 0, the same situation occurs: the function sin χ shows

two minima separated by a bump at which sin χ ≈−0.7. The general expression

for the microscope defocus values at which this bump occurs is given by

 f



8n + 3



. (10.28)

Of particular importance is the value for n = 0(D =

√

1.5):

 f

(

λ, (10.29)

which is known as the Scherzer defocus. The corresponding sin χ function is shown

in Fig. 10.8. The ﬁrst zero crossing of sin χ at Scherzer defocus is determined from



−

(



= 0, (10.30)

which is solved easily to result in Q = 6

1/4

. Using equation (10.10) we ﬁnd

≡ 6

1/4





−1/4

. (10.31)

In real space, the frequency q

corresponds to the distance

≡ 6

−1/4





1/4

≈ 0.64





1/4

= 0.64 Gl. (10.32)

The number ρ

is known as the point resolution of the microscope

†

and it is the

number stated by microscope manufacturers as one of the microscope parameters.

†

It is also possible to take the high-frequency edge of the ﬁrst passband and deﬁne it to be the Scherzer frequency;

the point resolution then becomes ρ

= 0.67 Gl.

602 Phase contrast microscopy

-1

0123

1/4

-0.7

sinχ

First

Passband

400 kV, C

= 1.0 mm

300 kV, C

= 1.2 mm

111

200

220

Fig. 10.8. sin χ(Q) at the Scherzer defocus D =

√

1.5. The {111}, {200}, and {220} recip-

rocal lattice spacings of aluminum are superimposed on the drawing for 300 and 400 kV

microscopes with C

= 1.2 and 1.0 mm, respectively.

Example 10.3 Consider a 400 kV microscope with a spherical aberration of

1.1 mm. What is the Scherzer defocus  f

? What is the defocus value for the

third passband?

Answer: From Table 2.2 we have for the electron wavelength λ = 0.001 644 nm.

The Scherzer defocus is given by D =

√

1.5 which translates to

 f



1.5C

λ = 52.08 nm.

For the third passband we have n = 2 or D =

√

9.5 which translates to  f

131.07 nm.

Example 10.4 Consider a 300 kV microscope with C

= 1.2 mm, and a 400 kV

microscope with C

= 1.0 mm. Determine for each microscope which reﬂections

of the [110] zone axis pattern of aluminum will fall within the ﬁrst passband.

Answer: The spatial frequencies corresponding to the ﬁrst three reﬂections in the

[110] zone are given by: q

111

= 4.2777,q

200

= 4.9395, and q

220

= 6.9855 nm

−1

The conversion factors from q to Q are 0.309 37 Gl for the 300 kV machine, and

0.258 18 Gl for the 400 kV machine. Therefore, the reﬂections at 300 kV correspond

to Q

111

= 1.323,Q

200

= 1.528 and Q

220

= 2.161;at400 kV we ﬁnd Q

111

= 1.104,

200

= 1.275, and Q

220

= 1.803. Since the Scherzer frequency is equal to 6

1/4

1.565, the {111} reﬂections fall within the ﬁrst passband for both microscopes,

the {200} reﬂections are near the edge of the passband at 300 kV, and the {220}

10.2 The microscope as an information channel 603

reﬂections are well beyond the ﬁrst passband, as indicated by the open circles and

open squares in Fig. 10.8.

The preceding example shows that the microscope parameters determine which

spatial frequencies of a crystalline sample (in this case aluminum) will be trans-

mitted with the same phase shift and therefore similar contrast. For the 400 kV

microscope, the 111 and 200 reﬂections are transmitted with nearly identical phase

shifts, whereas the 300 kV microscope would have nearly zero transmission for

the 200 reﬂections. Reﬂections with Q > Q

are also transmitted (for coherent

illumination), but the phase shifts vary rapidly with defocus and spatial frequency,

resulting in images which are difﬁcult to interpret.

The deﬁnition of microscope resolution is not a straightforward one in the case of

transmission electron microscopy. For an optical microscope, resolution is deﬁned

by the Rayleigh criterion [Str99, BW75], which states that two intensity distribu-

tions (originally described by

(

sin x/x

)

-type functions) can be distinguished from

each other if the minimum intensity at the center is smaller than 0.81 times the

maximum intensity. As shown in Fig. 10.9, this corresponds to the second function

maximum being located at the ﬁrst zero of the ﬁrst function. If the two functions

are closer together then they can no longer be resolved. This criterion can be re-

formulated for more general intensity distributions, in particular for distributions

which do not have zeros anywhere [Pap68].

The attentive reader may notice that, if we know a priori the analytical shape

of the functions, then a simple curve ﬁt will allow separation of the two functions

even when they are spaced more closely than shown in Fig. 10.9 [van92]. This

type of curve ﬁtting is commonly used in many branches of science. Since careful

measurements of intensity distributions along with curve ﬁtting (taking into account

1.0

0.8

0.6

0.4

0.2

0.0

-10 -50510

Fig. 10.9. Illustration of the Rayleigh resolution criterion for optical observations (based

on Fig. 7.62 in [BW75]).

604 Phase contrast microscopy

all measurement errors) are currently possible, the Rayleigh resolution criterion is

not a useful criterion for electron optical observations.

To conclude these comments on resolution it is useful to consider the in-

ﬂuence of other aberrations, such as two-fold, three-fold, and four-fold astig-

matism. Following Krivanek [Kri94] we write the aberration function χ(q)as

follows:

χ(q)

2π

=−

λq

[

 f + C

cos 2(φ − φ

)

]

−

[

G cos 3(φ − φ

) + 3K cos(φ − φ

)

]

−

[

H cos 4(φ − φ

) − C

]

, (10.33)

where (H,φ

) represent the four-fold astigmatism. The maximum possible value

for this function would occur when all cosines are simultaneously equal to +1,

assuming that the phase angles are all equal to each other. The maximum phase

difference between a perfect microscope (with zero phase shift) and the phase shift

of equation (10.33) is then given by

max



χ(q)

2π



≡ δ =

λq

(

 f + C

)

(

G + 3K

)

(

H − C

)

(10.34)

Let us consider each aberration separately. If only intrinsic two-fold astigmatism

is present, i.e. the stigmator coils are turned off, then the maximum phase shift δ is

equal to

δ =

λq

This function is shown as a log–log plot in Fig. 10.10 for a typical value of C

µm and λ = 2.508 pm (200 kV). The function appears as a straight line with

slope 2. If we consider a phase shift of π/2 to be the maximum tolerable phase shift

due to aberrations (indicated by the horizontal dashed line at 0.25 in Fig. 10.10),

then only spatial frequencies smaller than about 0.45 nm

−1

will experience small

phase shifts. This will limit the point resolution of the microscope to about 2.2 nm.

Before the invention of quadrupole stigmators, this was indeed the best attainable

resolution in any TEM. The quadrupole stigmator allows us to correct the two-fold

astigmatism so that it is no longer the resolution-limiting aberration.

The next aberration in equation (10.34) is spherical aberration. The maximum

associated phase shift is given by

δ =

10.2 The microscope as an information channel 605

100

0.1

0.01

0.001

10 5 2 1 0.5 0.2 0.1 0.05 0.02 0.01

0.1 0.2 0.5 1 2 5 10 20 50 100

= 1

= 0.5mm

G = 1

H = 1

Resolution (nm)

Spatial frequency (nm

-1

)

max (χ(q)/2π)

Fig. 10.10. Maximum phase shifts as a function of spatial frequency for two-fold,

three-

fold, and four-fold

astigmatism, and for spherical aberration. This

ﬁgure is based on Fig.

in [Kri94].

which, for C

= 0.5 mm, results in the leftmost straight line with slope 4 in

Fig. 10.10. This line can be moved sideways by selecting the Scherzer defocus,

for which the effect of spherical aberration is partially canceled. This shift is in-

dicated by arrows pointing to the dashed line with slope 4. The resulting point

resolution is slightly better than 0.2 nm, which is what many current microscopes

are capable of.

The next aberration is three-fold astigmatism G, along with axial coma K which

has the same functional dependence. The maximum associated phase shift is

δ =

(G + 3K );

taking G + 3K = 1

µm, we have the third line with slope 3 in Fig. 10.10. This line

intersects the horizontal line at a point corresponding to a point resolution of about

0.2 nm, slightly larger than the Scherzer point resolution! This means that the point

resolution is limited by three-fold astigmatism and/or axial coma, and that these

aberrations must be corrected along with the spherical aberration.

If three-fold astigmatism and axial coma are corrected, and the spherical aber-

ration is reduced using a C

-corrector, then the next resolution-limiting aberration

606 Phase contrast microscopy

will be four-fold astigmatism, which has a maximum phase shift of

δ =

shown as the rightmost line with slope 4 in Fig. 10.10. Four-fold astigmatism limits

the resolution to about 0.06 nm. The precise positions of these lines depend of course

on the actual values of the aberration coefﬁcients, including the phase angles of

the aberrations. The main point of this comment is that resolution improvements are

only possible if the next level of aberrations can be corrected. To bring the resolution

from 2–3 nm down to 0.2 nm required the removal of two-fold astigmatism by

means of quadrupole stigmators. Improvements down to 0.1 nm require correction

of C

, three-fold astigmatism and axial coma, whereas 0.05 nm resolution requires

correction of (among others) four-fold astigmatism.

10.2.2 The inﬂuence of beam coherence

In Chapter 3, we introduced the concept of coherence and we distinguished between

temporal and spatial coherence. Perfect coherence is only possible if the electron

gun produces perfectly monochromatic electrons which all travel precisely along

the same direction, in other words, the electron source produces the ideal plane

wave. In reality, each electron gun has an associated energy spread which is small

compared with the beam energy but is not zero. The electrons travel with a range

of wave vectors that is sharply peaked along the optical axis, but we cannot ig-

nore the fact that the beam has a ﬁnite (non-zero) divergence angle θ

. Both of

these factors will contribute to the fact that the electron beam is not perfectly

coherent.

The double-slit thought experiment described in Chapter 2 can be used to intro-

duce the concept of partial coherence, which is crucial to a realistic description of

the information transfer in the TEM. In the incoherent case, electrons which travel

through either slit will not interfere with each other and the intensity pattern at a

point Q on the screen is simply the sum of the intensity patterns due to each of the

slits:

inc

(Q) =|

(Q)|

+|

(Q)|

= I

(Q) + I

(Q). (10.35)

In the coherent case, amplitudes are superimposed instead of intensities, and we

have

coh

(Q) =|

(Q) + 

(Q)|

= I

(Q) + I

(Q) + 2



(Q)



(Q)γ

cos 2πβ,

(10.36)

where the factor γ

is a slowly varying function describing the amplitude of the

intensity oscillations (fringes) and β is a function which depends on the geometry

10.2 The microscope as an information channel 607

and determines the frequency and position of the fringes (see Fig. 2.1). If there is

no interference (incoherent case), then we have γ

= 0. Hence, we can combine

the coherent and incoherent cases into a single equation:

I (Q) = γ

(Q) + I

(Q) + 2



(Q)



(Q) cos 2πβ

+ (1 − γ

)

[

(Q) + I

(Q)

]

. (10.37)

The factor γ

is known as the degree of coherence.

†

Partial coherence simply

means that 0 <γ

< 1. The degree of coherence refers to the amplitude of the

interference fringes and it is easy to see that γ

is related to the interference term



∗

+ 

∗



Consider an exit wave ψ(r) which, in the perfectly coherent case, is modiﬁed by

the point spread function T (r). The image intensity is then given by

I (r) =|ψ(r) ⊗ T (r)|

;

[

ψ(r) ⊗ T (r)

]

[ψ

∗

(r) ⊗ T

∗

(r)].

Taking the Fourier transform of this expression and using the convolution and

multiplication theorems we ﬁnd [Kir98]:

I (q) =

[

ψ(q)T (q)

]

⊗ [ψ

∗

(q)T

∗

(q)];

ψ(q



)T (q



)ψ

∗



+ q)T

∗



+ q)dq



;

≡



, q



+ q)ψ(q



)ψ

∗



+ q)dq



. (10.38)

The new function T

is known as the transmission cross coefﬁcient and it depends

on two spatial frequency vectors q



and q



+ q.

The transmission cross coefﬁcient (or TCC for short) has a simple interpretation:

equation (10.38) expresses the fact that each spatial frequency q



in the exit wave

function will interfere with every other spatial frequency q



+ q to produce the ﬁnal

image intensity. The TCC describes the strength of this pairwise interference. For

the case of the linear image theory discussed in Section 10.2.1, only interference

effects between the forward scattered beam and the diffracted beams are considered.

The more general case is described by equation (10.38). In the following section,

we will take a closer look at the nature of the TCC and its consequences on image

formation under partial coherence conditions.

†

A more complete and rigorous analysis reveals that γ

is actually the modulus of a complex number, which

is known as the complex degree of coherence. We refer the reader to Chapter 10 in [BW75] for more

details.