Graef M. Introduction to conventional transmission electron microscopy

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448 Systematic row and zone axis orientations

0.4 nm

(a) (b) (c)

(d)

102030405060708090100

Fig. 7.30. (a) Hypothetical

crystal structure consisting of two W atoms; (b) projected

potential for one slice of thickness 0.05 nm; (c) real part of the propagator (200 kV);

(d) thickness series of amplitude (top row) and sin (phase) (bottom row).

also affected by the neighboring atom. The central feature corresponds to a tightly

bound 1s state, the ring is a 2s state and the more diffuse feature is a combination of

the 3s state and other Bloch waves. At a particular depth in the crystal each of the

Bloch waves has a different excitation amplitude so that the intensity of the various

features oscillates with depth in the crystal, similar to the intensity oscillations in

the two-beam bright ﬁeld–dark ﬁeld case.

The probability for a beam electron to be in a bound 1s state oscillates with

depth in the crystal. The oscillation period depends on the electron energy, the

type of atoms in the column, and the distance between the atoms along the col-

umn. Figure 7.31 shows this oscillatory behavior for 12 selected elements. The

computations were carried out with a modiﬁed

ms.pro program for the following

simulation parameters: one atom in the center of unit cell, 200 kV, g

max

= 80 nm

−1

slice thickness 0.05 nm, 2000 slices, tetragonal axis c = 0.2 nm, and no absorption.

The thickness increases from left to right, and each vertical intensity proﬁle corre-

sponds to the line PQ in Fig. 7.30(a). The bound 1s state is centered on the atom

itself and the intensity oscillates with a periodicity which generally increases with

increasing atomic number. With increasing atomic number the 2s state becomes

occupied and oscillates with a different period. For the heaviest elements the 3s

bound state is occupied as well. The oscillation periods are not simply related to

the atomic number, but instead vary in a way similar to the variations in electronic

structure across the periodic table of the elements.

7.4 Computation of the exit plane wave function 449

Be (4)

Na (11)

Cl (17)

Ti (22)

Zn (30)

Sr (38)

Ru (44)

Te (52)

Nd (60)

Er (68)

Au (79)

At (85)

050100

Thickness [nm]

Fig. 7.31. Radial intensity proﬁle as a function of foil thickness (indicated at the top)

for 12 selected elements (multi-slice computation, slice thickness 0.1 nm, 200 kV, 3205

beams or g

max

= 80 nm

−1

, Doyle–Turner scattering factors, no absorption). The oscillatory

character of the bound Bloch states of 1s,2s, and 3s type is clearly visible.

What we learn from this type of simulation is that the propagation of a plane

incident wave along a zone axis orientation gives rise to bound states at each

atom column. If the columns are sufﬁciently far away from each other, then the

oscillatory behavior of the symmetrical lowest-order Bloch waves dominates the

scattering behavior and the wave function will be peaked at the column positions.

If columns are close together, as they are in many projected crystal structures, then

there will be a ﬁnite probability for electrons from one column to scatter into a

neighboring column, which means that the non-symmetrical Bloch waves will be

excited (see Fig. 7.30d). Which particular waves are excited will depend on the

conﬁguration of columns. For many crystal structures we can therefore describe

the zone axis scattering process as essentially a collection of bound s-type states;

the interaction between the bound states could then be viewed as a perturbation. It

is possible to work out analytical solutions for scattering from an isolated column;

this lies at the basis of the atomic column approximation described in [VDOdB96].

An extensive study of the channeling behavior in terms of Bloch waves can be

found in [BLS78].

The reader is encouraged to experiment with the

ms.pro routine, in particular to

study what happens when the illumination is tilted slightly away from the zone axis

orientation. The routine returns an animated GIF ﬁle for up to 400 slices.

450 Systematic row and zone axis orientations

7.5 Electron exit wave for a magnetic thin foil

7.5.1 The Aharonov–Bohm phase shift

In 1959 Aharonov and Bohm published a revolutionary paper on the importance of

the phase of the electron wave function in the presence of electrostatic and magnetic

potentials [AB59]. While the existence of such phase shifts was known to others at

that time, their paper provided the ﬁrst detailed discussion of the effect and drew

a lot of attention from the physics community. They found that even in regions

of space where all the ﬁelds vanish, the wave function of a charged particle could

still experience changes due to the corresponding electromagnetic potentials. It has

taken several decades for the scientiﬁc community to accept the notion that the

wave function of a particle can be affected by something other than a force, and

an extensive review of the quantum effects of electromagnetic ﬂuxes can be found

in [OP85]. There is now ample evidence that the Aharonov–Bohm (or A–B) effect

is indeed real, and a direct experimental proof based on electron holography was

provided by A. Tonomura and coworkers [TOM

86] in 1986. The existence of the

A–B phase shift is related to the fact that the electric and magnetic ﬁelds do not

appear directly in the Schr¨odinger equation, but only in the form of the potentials;

while the electromagnetic potentials are not observables in the classical theory, they

do become the fundamental quantities in the quantum mechanical framework.

The A–B phase shift imparted on an electron wave with relativistic wavelength

λ by the presence of electromagnetic potentials V and A is given by [AB59]:

φ(r

⊥

) =

λE

V (r

⊥

, z)dz −

A(r

⊥

, z) · dr (7.35)

where E

is the total energy of the beam electron and the integrals are carried

out along a straight line L parallel to the incident beam direction (i.e. crossing

the plane of the sample at the point (x, y, 0)). It is left as an exercise for the

reader to show that the prefactor of the electrostatic component of the phase shift

is equal to the interaction constant σ introduced in Chapter 2. A 100 nm foil with

a mean inner potential V

of 30 V will hence give rise to an electrostatic phase

shift of 18.36 radians at 400 kV (using Table 2.2 on page 93). The prefactor of the

magnetic component of the A–B phase shift equals e/h

= 0.001 519 27 T

−1

−2

and is independent of the electron energy.

Since it is rather difﬁcult to measure absolute phases, we usually work in terms

of the phase difference φ(r

⊥

) between a given electron trajectory crossing the

sample at the location r

⊥

and a reference trajectory which is conveniently chosen

to coincide with the optical axis of the microscope, as schematically indicated in

Fig. 7.32. In the absence of an electrostatic potential, the magnetic component of

7.5 Electron exit wave for a magnetic thin foil 451

(x, y, 0)

∞

Fig. 7.32. Schematic of the two trajectories involved in the formulation

of the Aharonov

–

Bohm phase shift.

the phase difference φ

− φ

between the trajectories 1 and 2 is equal to

φ

⊥

) =



+∞

−∞

A · dr

−

+∞

−∞

A · dr



A · dr.

This last line integral can be converted into a double integral using Stokes’ theorem

(e.g. [Jac75]):

φ

⊥

) =

∇×A · dS, (7.36)

with S a unit vector normal to the integration surface. This integral is equal to the

magnetic ﬂux 

enclosed between the two trajectories:

φ

⊥

) =



⊥

) = π



⊥

)



, (7.37)

where 

= h/2e is the ﬂux quantum. The phase shift is therefore proportional to

the enclosed ﬂux, measured in units of the ﬂux quantum. If we take trajectory 1

to be the reference phase (i.e. “zero phase”), then the phase at any other point r

⊥

can be calculated if either the vector potential A or the magnetic induction B are

known. In Section 7.5.3 we will discuss an example of the use of this relation.

Consider a thin foil of thickness t with parallel top and bottom surfaces. The foil

contains three magnetic domains, as illustrated in Fig. 7.33, separated from each

other by 180

◦

domain walls. The origin of the reference frame is taken to be in the

center of the foil, with a right-handed Cartesian reference frame oriented as shown

in the ﬁgure. The z-axis coincides with the optical axis of the electron microscope,

with positive z in the direction of the beam. The magnetic induction B

⊥

in the

domains is uniform and parallel to the x-direction, so that the Lorentz deﬂection

will be in the ±y-direction. In the absence of fringe ﬁelds we can calculate the phase

452 Systematic row and zone axis orientations

φ(y)

(a) (b)

Fig. 7.33. Schematic illustration of the phase computation for a set of three alternating

magnetic domains (magnetization normal to the plane of the drawing). The domain wall

proﬁle is taken to be discontinuous (black line) and with a ﬁnite domain wall width δ (gray

line).

shift of the electron wave function along the line 0–y from the A–B equation (7.36):

φ(y) =

+t/2

−t/2

⊥

dy dz =

⊥

ty. (7.38)

If the domain wall is inﬁnitely narrow, then the slope of the phase changes dis-

continuously across the domain wall, as indicated in Fig. 7.33. For a more realistic

domain wall proﬁle, the slope change is continuous. The range over which the

slope changes corresponds to the domain wall width. The phase function is a linear

function inside magnetic domains with a uniform magnetization and a constant foil

thickness. We leave it to the reader to show that the phase becomes a quadratic

function of y for a linear wedge-shaped foil. Small local thickness variations will

cause non-linearities in the phase as well, but generally do not change the sign of

the slope. The same is true for magnetization ripple, which causes ﬂuctuations in

the orientation of the gradient vector of the phase function.

The Lorentz deﬂection angle, equation (4.8), can be converted into a phase

gradient:

∇φ =

2π

e, (7.39)

where e is a suitably oriented unit vector. For a uniformly magnetized foil of constant

thickness t this results in a phase gradient of

∇φ =

⊥

te =−

(

B × n

)

t. (7.40)

This expression is independent of the electron energy. Since the electrostatic com-

ponent of the phase is proportional to the interaction constant σ , and σ decreases

with increasing microscope acceleration voltage E, it follows that magnetic con-

tributions to the phase shift are larger relative to the electrostatic contribution for

higher-voltage microscopes. Since the Lorentz deﬂection angle is proportional to the

7.5 Electron exit wave for a magnetic thin foil 453

electron wavelength, it is clear that intermediate voltages in the range 200–400 kV

provide the best compromise between a strong relative magnetic contribution and

reasonably large Lorentz deﬂection angles.

7.5.2 Direct observation of quantum mechanical effects

The most readily observable quantum effect in Lorentz microscopy is the ap-

pearance of interference fringes at convergent wall images in Fresnel mode (see

Fig. 4.29). We will show that there is one fringe per ﬂuxon h/2e in the image,

independent of defocus, magniﬁcation, and electron energy [Woh71].

Consider again a ﬁlm of thickness t, with an inﬁnitely narrow domain wall

centered at y = 0 (Fig. 7.34, based on Fig. 5 in [Woh71]). A point source S at

a distance s from the sample illuminates the area containing the domain wall. In

Fresnel mode the image plane is located at a distance  f below the sample. The

Lorentz deﬂection of electrons passing through the domain on the left is −θ

and at

the image plane the electrons appear to originate from an effective source S



, located

at a distance sθ

from the actual source (measured at the source plane). Similarly,

electrons deﬂected by the angle +θ

on the right-hand side appear to originate

from the point source S”. The two sources will give rise to a region containing

interference fringes with width δ

. The angle subtended by the two sources at the

∆f

SS' S"

Fig. 7.34. Schematic illustration of the relation between a single fringe period in a Fresnel

image (defocus  f ), and the corresponding region in the sample (gray rectangle). The ﬂux

through this region is equal to one ﬂux quantum (based on Fig. 5 in [Woh71]).

454 Systematic row and zone axis orientations

image plane is equal to γ =

2sθ

s+ f

. The width of a single fringe in the image plane

is then given by

;

h(s +  f )

2ets B

⊥

where we have made use of equation (4.8). Next, we refer this fringe width back to

the width δ

in the sample plane:

= δ

s +  f

;

2et B

⊥

;



⊥

We ﬁnd that δ

⊥

= 

, or, that the ﬂux contained in a region in the sample

corresponding to a single projected fringe width in the image is equal to one ﬂux

quantum. The fringe width depends inversely on both ﬁlm thickness and magnetic

induction.

7.5.3 Numerical computation of the magnetic phase shift

Magnetization conﬁgurations in thin foils rarely have an analytical representation

of either the vector potential or the magnetic induction. For such cases the mag-

netic phase shift imparted on the beam electron may be calculated by assuming that

the magnetization has in-plane periodicity. It is usually not too difﬁcult to create

a periodic continuation of any given magnetization pattern; one can either pad the

region of interest with a zero magnetization edge (making sure that the magnetiza-

tion goes to zero smoothly as the edge is approached), or, if the spatial derivatives

of the magnetization vanish at the edge, then one can readily use the pattern as a

single unit cell. We will hence assume in this section that a two-dimensional (2D)

magnetization unit cell can be found, and to each pixel (i, j) in the cell we assign

a three-dimensional (3D) magnetization vector M(i, j). The magnetization is as-

sumed to be constant along the beam direction and we ignore the demagnetization

or fringe ﬁeld outside the foil. An example of such a magnetization conﬁguration

is shown in Fig. 7.35(a). This conﬁguration commonly occurs in materials which

have the 111directions as the so-called “easy axes”, i.e. low-energy directions for

the magnetization vector. The incident beam is normal to the plane of the drawing

along the [110] direction.

7.5 Electron exit wave for a magnetic thin foil 455

500

400

300

200

100

0 100 200 300 400 500

[001]

[110]

(a)

(b)

(c)

|F[e

iφ

Fig. 7.35. (a) Periodic 2D magnetization conﬁguration with four different domain ori-

entations; magnetization vectors are along 111-type directions. 71

◦

(vertical) and 180

◦

(zig-zag) domain walls with widths δ

= 24 nm and δ

180

= 48 nm are present. (b) The

calculated Aharonov–Bohm phase shift φ for the conﬁguration in (a); (c) is the power spec-

trum of a wave e

iφ

; the central spot is split into four components, one for each magnetic

domain.

456 Systematic row and zone axis orientations

The angle between two 111directions is either 70.53

◦

, 109.47

◦

,or180

◦

. There-

fore the possible magnetic domain walls are known as 71

◦

, 109

◦

and 180

◦

domain

walls. The ﬁrst set of domain walls is normal to the [

110] direction and is of the 71

◦

type, i.e. the magnetization vector rotates by 71

◦

upon crossing the domain wall.

The magnetization vectors are at an angle of 35.3

◦

with respect to the plane of the

drawing. The second set of domain walls is parallel to the magnetization vectors

and lie along {11

2}-type planes. These walls are 180

◦

walls. Each wall has an asso-

ciated domain wall width δ, typically a few tens to a few hundreds of nanometers.

The magnetization changes direction smoothly across a domain wall. The domain

wall widths in Fig. 7.35(a) are δ

= 24 nm and δ

180

= 48 nm. This magnetiza-

tion conﬁguration is periodic in both x and y. The conﬁguration in Fig. 7.35(a)

is deﬁned for an array of P × Q = 512 × 512 pixels, and the pixel size is equal

to D = 1 nm. The routine

PhaseExample in the lorentz.f90 module can be used to

create the magnetization vectors corresponding to Fig. 7.35(a).

It has been shown by Mansuripur [Man91] that for such a periodic conﬁguration

the magnetic A–B phase shift may be written as a 2D Fourier series. We refer

the reader to the original paper for the lengthy proof of this statement, and we will

continue by stating the explicit result. We represent the discrete Fourier components

of the magnetization over a unit cell of P × Q pixels with pixel spacing D by M

with m = 1,... ,P and n = 1,... ,Q; in other words, we have



M(i, j)exp



−2πi





. (7.41)

The A–B phase shift is then given by the discrete 2D Fourier transform:

φ(r) =



m=0



n=0



|q|

(t|q|)(

q × e

) ·

[

p × (p × M

)

]

2πir·q

, (7.42)

where the prime on the summation indicates that the term (m, n) = (0, 0) does not

contribute to the summation, q =

∗

is the frequency vector, e

∗

and e

∗

are the reciprocal unit vectors, p is the beam direction expressed in the orthonormal

reference frame, t is the sample thickness, a hat (ˆ) indicates a unit vector, and the

function G

(t|q|)isgivenby

(t|q|) =

(p ·

+ p

sinc



πt|q|

p ·



, (7.43)

where sinc(x) ≡ sin(x)/x. For normal beam incidence (p  e

) the function

(t|q|) takes on the value 1. Numerical implementation of this equation is straight-

forward and the corresponding source code can be found on the

website as the

PhaseMap routine in the lorentz.f90 module.

7.5 Electron exit wave for a magnetic thin foil 457

φ (rad)

Fig. 7.36. Shaded surface representation of the calculated phase shift of Fig. 7.35(b).

The magnetic phase shift φ for the magnetization conﬁguration of Fig. 7.35(a)

(with t = 1 and |M(i, j )|=1 for all i, j ) is shown in Fig. 7.35(b) as a grayscale

image. The minimum and maximum phase shifts are ±0.325 rad. This phase shift

must be multiplied by the sample thickness (t, in nm) and the saturation magneti-

zation (M

, in tesla). An alternative representation as a shaded surface (Fig. 7.36,

t = 100) clearly shows that the magnetic phase shift inside a domain is a linear

(planar) function of position, whereas the slope of the phase changes upon crossing

a magnetic domain wall.

A plane wave which travels through a thin foil with the magnetization conﬁg-

uration of Fig. 7.35(a) will experience a position-dependent phase shift such that

the exit wave has the phase φ. In the back focal plane of the objective lens we

can observe the modulus-squared of the Fourier transform of the wave function

|F[e

tφ

.ForM

t = 100 nm T the resulting diffraction pattern is shown in

Fig. 7.35(c). The central beam, represented by a small circle, is split into four

separate beams, one for each magnetic domain.

Note that the magnetic phase shift does not affect the amplitude of the wave.

Since the phase shift can be large, several radians, a magnetic thin foil is known as

a so-called strong phase object. If there are thickness variations across the foil, then

the phase computation becomes much more involved, since we can no longer simply