TRANSMISSION ELECTRON MICROSCOPE OBSERVATIONS OF FERROMAGNETIC DOMAIN STRUCTURES

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TRANSMISSION ELECTRON MICROSCOPE OBSERVATIONS OF FERROMAGNETIC DOMAIN

STRUCTURES

R. Wade

To cite this version:

R. Wade. TRANSMISSION ELECTRON MICROSCOPE OBSERVATIONS OF FERROMAG- NETIC DOMAIN STRUCTURES. Journal de Physique Colloques, 1968, 29 (C2), pp.C2-95-C2-109.

�10.1051/jphyscol:1968216�. �jpa-00213531�

JOURNAL DE PHYSIQUE Colloque C 2, supplément au no 2-3, Tome 29, Février-Mars 1968, page C 2 - 95

TRANSMISSION ELECTRON MICRO SCOPE OBSERVATIONS OF FERROMAGNETIC DOMAIN STRUCTURES

Laboratoire de Physique du Solide, C. E. N. G., Grcnoble.

Résumé. - Les champs magnétiques jouent le rôle d'objets de phase pour les ondes électroniques.

Les images obtenues en microscopie électronique des domaines ferromagnétiques dans les lames minces sont ainsi interprétées et les résultats comparés aux calculs classiques d'optique géométrique.

Abstract. - Magnetic fields act as phase objects for electron waves. The imaging of domain structures in thin ferromagnetic films by transmission electron microscopy is discussed in this light and compared to the geometrical optical approach based on the calculation of classical tra- jectories.

1. Introduction. - Several methods have been used to obtain electron images associated with magne- tic structures in thin ferromagnetic films. The aim of the present article is to present an account of the image formation in these different methods.

In thc off-focus method of Fuller and Hale [ l ] the images have usually been accounted for by geomctrical optics using classical electron trajectories. Recently, Wohlleben [2] has emphasised the limitations of the geometrical approach and showed that wave optical treatments are preferable [3, 41. Such treatments are based on the Aharanov and Bohm [5] calculations of phase changes accompanying the motion of electron waves through magnetic field ~egions. Viewed in tliis way the magnetic layer becomes a phase object and the observational inethods become means of produ- cing amplitude variations in the images.

We show that the geometrical approach can account for the qualitative aspects of many diffraction (Fraun-

film which we treat as a region of uniform magnetic flux (B) of thickness D. We write the non-relativistic Schrodinger equation for an electron in the magnetic field of the film as

where e , m and E arc respectivcly the electron charge, mass and kinetic cnergy, A is Plank's constant, c is the velocity of light, A is the vector potential of the magne- tic field, I/I is the electron wave function.

We can write the solution of the wave function immediately below the object in the form :

where S i s thc power series

hofer) images from magnetic structures, but that

certain experiments require explanation by the wave the values Sj are supposed to be real. A classical

optical approach. approximation of S includes only the first term S,

In single crvstal toils the local diffraction conditions and on substituting for I/I in equation (1) we find

- -

can be changed by the flux configurations within 1 2

and near the foi1 enabling the magnetic structure to be ^---(VSo 2 n? + ;r ^{4 ) = E}

revealed in a manner completely different from the

direct imaging of the fields themselves [6, 71. which corresponds to the Hamilton - Jacobi equa- The initial discussion of the action of magnetic tion of classical mechanics. S, is the action given by fields as phase objects, largely due t o Guigay [8], is

S, = p.dr where p is the canonical momentum and similar to a W. K. B. or eikonal approximation. S

the integral along the classical trajectory is approxi- II. The phase approximation of wave optics. - mated by an integral along the z axis which is justified We ignore -the atomic structure of a ferromagnetic for small deflections about an initial trajectory along z.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1968216

C 2 - 9 6 R. H. WADE To first order in h equation ( 2 ) becomes

where A(r) ⁼ exp S, is the amplitude of the wave function whilst the exp~nential term describes the phase. On the basis of thc experimental observation that a focussed image of a ferromagnetic film shows no amplitude variations we suppose that A(r) is a constant.

In many problems we are only interested in the flux component along one direction in aplane perpendicular to the direction of motion of thc electron beam.

Taking the z axis to coincide with the direction of the clectron motion and assuming the flux within the film to be Bx = Bz = O, B, = B,(x) given by A, = A, = O B,(x) d x , whilst above and below the layer A, = A, = A, ⁼ O, we find

above the film So ⁼ pz _I

within the film So = z ( p +

.

where k = ( 2 m ~ ) i / h .

TheIterm xeBo Dlclz in the phase is equivalent to a rotation of the wave front through the anglc

showing that the magnetic vector potential A has a physical significance had previously been obtained by Ehrenberg and Siday [9] and Franz [IO]. The wave function of equation (5) represents a first order appro- ximation in h as is pointed out by Franz.

III. Off focus images. - (i) GEOMETRICAL ^OPTICS.

- The magnetic field region is assumed to be illumi- nated by a completely incoherent electron beam and the local deflections are calculated using the Lorentz force equation

Variations in the direction of the component of magnetic flux perpendicular to the electron beam produces intensity fluctuations in the image formed at an off-focus plane. A domain wall in a ferromagne- tic films yields an intensity distribution which can be related to the distribution of the magnetisation vector M within the wall [ l , 4 , 111. The wall is found to be imaged in dark or bright contrast depending on the relative directions of M within the domains on either side of the wall, the wall image contrast is reversed in the virtual image formed above the specimen plane, the in-focus image shows no contrast, as shown sche- matically in figure la. Electrons passing outside the

The direction of the classical motion of an electron is obtained from the relation p ⁼ (VS,) which applicd to equation ( 4 ) above and below the film shows the particle to be deflected through the angle cp given by the equation (6). It is natural that the classical, geometrical optics, calculation should give the same result as the wave optical treatment in t h e absence of diffraction

o r interference effects. _{OFF FCCUI !WU}

Equations ( 4 ) and (5) show that the effect of the magnetic flux is to change the phase of thc wave bet- ween trajectories separated by distance x by an amount

A L

FIG. 1. - ( a ) At distance Z off-focus dornain walls are equal to the flux enclosed between the tiajectories irnaged in dark or bright contrast.

In an off-focus image the distortion of the edgc can reveal which do net themselves necessarily pass through the stray fields; (h) betwcen antiparallel dornains; ( c ) in the rcgion field region. This result of Aharanov and Bohm ofdornainwalls.

TRANSMISSION ELECTRON MICROSCOPE C 2 - 9 7

specimen yield shadow images of an edge which are related to the local stray field distribution [12, 131. The stray fields between antiparallel domains and near Néel walls produce the edge distortions shown respec- tively in figure l b and c. Domain wall images at A and wall stray field images at B are shown in figure 2a whilst the stray fields between antiparallel domains in coba1,t ptoduce edge images of the type shown in figure 2b.

FIG. 2. - (a) Domain wall images (A) in a permalloy film.

The dark band along th: cdge is produccd by the component of M parallel to the edgc whilst the bumps at (B) arc due to thc stray field of the Néel walls. ( b ) Slightly and strongly defo- cussed images of a n cdgc of a cobalt foi1 the distortcd edge profile is produced by the stray fields b2tween the domains.

However Wohlleben [2, 141 has pointed out that the calculated image profiles of domain walls are based on the supposition that an electron passing through

an element of width dx at the position x in the wall contributes to the image intensity within dX,, at the position X in the image, as shown on the figure 3a, with the geometrical relations,

I I (

'Y' ;@&

1-1

FIG. 3 . - (a) Illustrating domain wall imaging in the geome- trical optical theory. (b) At large distances off-focus a n angular dispersion cx = 2 l i d x is associated with the aperture of width dx.

It is as if the image were recorded by placing a sinall aperture of width dx successively at each position in the wall. Such an aperture will yield an image of width dX, = ^za where a = 2 Â/dx, at the plane z, figure 3b. With dx = cm. a = rad whilst with dx ⁼ IO-' cm, a = rad. Since a is of the same order of magnitude as the magnetic deflection cp, dX, $ dX,. It is unlikely that in general a detailed wall magnetization distribution can be obtained from this geometrical approach.

Wohlleben gives the condition A@ > h/2 e as a criterion of the limit of applicability of the geometrical optical approach, where A@ = (AX.AB.D) is a measure of the difference of flux between two points separated by the distance Ax.

C 2 - ^{98 R. H. WADE} (ii) WAVE ^OPTICS. - When the magnetic object is

illuminated coherently the off focus images can be calculated using the wave-optical approach of Boersch et al. [3]. In this treatment, the image is calculated in the following twa stages :

(i) The phase term in the object function f (x) is calculated by the enclosed flux rule,

f ( x ) = Aexp Ry(x)dx ]

where the amplitude A is assumed constant, f (x) is equivalent to the wave function $(x) leaving the object.

(ii) The Kirchoff integral incorporating f (x) is used to calculate the intensity distribution I ( X ) at a distance Z below the specimen.

' ^+cc _{x x !}

I ( X ) = constant 1 1-, ^{exp[ik/($ + $):---2-)}

-

*

where 7, is the distance between the source and the object.

We note that the object function f (x) describes a pure phase object.

Such calculations show that a converging domain wall acts as a magnetic biprism in that interference fringes are produced in the region where the beams from either side of the domain wall overlap. Such fringes are sliown in the figure 4 due to Wohlleben.

The biprism fringes are modified by the wall structure in a manner which must be calculated using specific wall models.

The separation of the fringes given by the usual biprism formula

determines as a condition necessary for the observa- tion of fringes that the illumination angle

which for 100 k V electrons A. - 4 x 10-'O cm ; Zo = 10 cm ; Z = 2 cm, cp - ^{5 x} IO-' rad yiclds

p = rad corresponding to an electron source 1 000 A wide. Since p is about an order of magnitude smaller than the usual values obtained by overfocus- sing both condenser lenses of the electron microscope interference fringes are not generally observable under normal operating conditions.

In practicc images are obtained under conditions of partial coherence, an investigation of the degradation of the images under the effect of increasing source size is being carried out at present by the author in collaboration with Guigay.

In principle a wave optical approach which compares the experimental intensity profiles of domain wall images with those calculated for specific wall models can yield precise information on domain wall struc- tures.

IV. Diffraction images. - (i) GEOMETRICAL OPTICS.

- An electron beam passing through a region of uniform magnetic flux is deviated through the angle c p = - eDBxDv, where &,, is the flux component per-

cmo

pendicular to v . The beam is deviated in opposite senses either side of a domain wall in a ferromagnetic film yielding in a Fraunhofer diffraction image two spots separated by the angle cp, = cp cos x where % is

FIG. 4. - Interference fringes in a converging domain wall in a pernialloy film with off-focus distance 6.5 cm, electron accelerating potential 45 kV.

(From Wohlleben, unpublished.)

TRANSMISSION ELECTRON MICROSCOPE C 2 - ⁹⁹ the angle between M x , and the domain wall. The wall

itself will contribute intensity between the spots from the domains.

The magnetisation and stray field distributions of Bloch and Néel domain walls are shown schemati- cally in figure 5. The electron being sensitive to the flux B = 4 nM + H, obligcs us to consider the field distribution over the entire trajectory. T o obtain the form of the intensity distribution due to the wall in the diffraction images we need only consider the deflection ^q, parallel to the wall direction [15, 16, 171.

In the case of an infinitely long wall of arbitrary angle we find at any position x , in the domain wall

this general result is obtained by applying the Gauss relation

V . B d v = B . n ds

suriace

and the Maxwell equation V.B = O

to the volume dY, X, Z of the figure 5a where n is a unit vector perpendicular to the surface and X, Z ⁺ and d Y + O [13].

Equation (7) shows that a t a large distance bAow the specimen the interna1 deflections along y a t any position x , within a Neel wall are exactly compensa- ted by the stray field deflections. In the case of thc Bloch wall the compensation is between the stray fields above and below the foil. The results hold to within a few percent for wall lengths greater than about 10 (2 6), where 2 6 is the wall width. We are lead to expect that diffraction images from both types of wall will show a straight streak between the main spots, figure 5c.

Iinages obtained from Neel, and cross-tie walls, which consist of antiparallel Neel segments, are shown in figure 6. We find that in al1 cases the wall contributes a straight streak between the diffraction spots from the domains. In figure 7 we show images

(cl

- -

tlic two walls. 2.5 x I O - ' rad.

FIG. 7. - Th: region ( a ) of a cobalt foi1 containing rnany antiparallel domains separated by Bloch walls yields the diffrac- tion images (h) and (c) which differ only in the better angular resolution of the image (c). The scale lines in (6) and ( c ) repre- sent respectivety and 3.3 x IO-' rad.

from domains in cobalt which are separated by Bloch walls, again we find a straight streak but using the highest resolution, 7c, the streak decomposes into discrcte maxima. Since we cannot account for this fine structure with the geometrical approach we are led to consider the wave optics of Fraunhofer diffrac- tion.

(ii) WAVE ^OPTICS. - We seek to describe the object by an appropriate function f (x) and to perform the Fourier transformation o f f (x) to obtain the ampli- tude distribution A(s) in the diffraction image,

f (x) exp(- 2 nisx) dx (8) s is a reciprocal lattice distance.

In several cases of interest the analysis can be carried out analytically in other cases we have recourse to numerical methods [18, 191.

FIG. 8 . - ( a ) The magnetic structures 1 and II with (b) their corresponding phase forms and (c) their diffraction images.

Figure 8a shows two simple antiparallel domain structures, My = M,, which abut directly in I a (wall width zero), and in IIa across a wall of width 26, in which Mx = M,. These magnetic structures corres- .pond to the phase structures of 86. As an example we write the phase structure 1

where 2 a is the width of the specimen or more probably the lateral coherence distance of the electron beam at the specimen. We consider only the one dimen- sional problem. Substituting f (x) = exp[iA(x)] in equation (8) we obtain the amplitude for the structure 1

1 ^{sin *(do - 2 nsa)}

A(r) a = ( ^exp [- ^{f (A, - 2 nsn) . % ( A , - 2 nsa)}

1 ^{sin 4(Ao + 2 nsa)}

+ enp [- ^{( A , + 2 nsa)} _{$ ( A , + 2 nsa)}

The corresponding intensity distribution given by I(s) = A(s). A*(s) is shown in figure 8c together with that obtained by a similar calculation for structure II.

We find that the structure between the main diffraction spots oscillates with the period l/a and has an ampli-

TRANSMISSION ELECTRON MICROSCOPE C 2 - 101

tude dependent on the wall width. The principal maxima fa11 at s = rt A , / 2 na which correspond to the classical Lorentz deflection positions. Since the aperture width 2 a is usually greater than about 5 p the inten- sity fluctuations between the main maxima have not yet been resolved experimentally.

Regardless of the wall structure the streak again runs straight between the principle maxima since it is easily shown that the geometrical proof that q, = O holds in the wave optical treatment because the enclosed flux between trajectories separated by any distance y at a position x, in the wall is necessarily zero if the trajectories are parallel to one another which is true when the source distance is sufficiently great.

The system of regularly spaced antiparallel domains in the cobalt foil, shown in figure 7, constitutes a phase grating of the form of figure 9a, which can be expressed for an object of infinite extent as the function g(x) given by the convolution,

where f (x) is the object function over a unit cell. The diffraction image amplitude G(s) is given by

1 I 1 rt-t atb -;

I I

-a-b 4

rd, ,l'

I I

FIG. 9 . - ( a ) The periodic magnetic structure on the left constitues the phase grating shown on the riglit.

When the enclosed flux is not exactly zero over one period there is a continuous phase change which occurs in (b) due to the non-equal domain widths and in (c) due to possibility of different wall structures when there is a magne- tisation component + ^Mz in the domains. ( d ) The diffraction image from (a), the open arrowheads mark the shift of the maxima due to the structures (b) and (c).

C 2 - ¹⁰² R. H. WADE with F(s) the Fourier transform of f ( x ) . The diffrac-

tion image then shows a structure of period 112 a within the intensity enveloppe shown in the figure 9d.

The unit ce11 function f ( x ) can be chosen in many ways to produce different intensity enveloppes which, however, al1 predict identical intensities at equal order diffraction peaks, in common with the example in light optics of a square wave phase object. The principal maxima are again at the Lorentz deflection positions and the intensity between these maxima is sufficient to allow al1 the peaks to be distinguished whilst for

1 s 1 > A0/2 na there is a sharp cut-off in intensity.

The image of figure 7c shows al1 these features.

which has the Fourier transform

showing al1 the peaks to be shifted by

with respect t o the normal positions depending on whether the phase is advanced or retarded. Either of the structures 9(b) or (c) could lead to the observed spot splitting and we are at present unable to choose between these possibilities.

A phase calculation allows us to account for the experimental result that the deflections produced by vacuum condensed ferromagnetic films are often lower than the theoretical values [20]. If the films grow as separate columnar crystals [21] schematised in figure l l a a geometrical approach would show that part of the beam is undeflected and part deflected through the angle q, = Bedlmcv. The phase structure of the film is schematised in figure 1 l b and a simplified

FIG. 10. - The region (a) of a cobalt foi1 with the diffraction images (b) and (c) in which the maxima appear as doublets.

In fig. 10 b, the scale line on the right represrnts 3 . 2 5 x 10-5

rad., and in fig. 10 c, 1 . 6 x rad. , t ^\

Sometimes we find that the maxima are split as shown in figure lob and c. This occurs because there

- ^I

/ 1 1

⁺

is a continous phase change A(2 a) from one unit ce11 _CC) ^{/ / > . . s} to the next so that we rewrite the 6 function of equa- _{-A+Acri -w&-}

tion (10) : a IAn

&a t +"

FIG. 11. - (a) Schematic structure of a discontinuous magnetic film, (6) The associated phase structure, (c) the diffraction image.

TRANSMISSION ELECTRON MICROSCOPE analysis shows the diffraction irnage to have the form

shown in figure l l c . The central spot is displaced by ds ⁼ A(a)/2 na which corresponds to the deflection - b, showing that <p2 depends on the

<p2 = 7 muc

film density in agreement with the experimental results of ~ e r r i e r - a n d ~ a d e [20]. We noce that figure I l c shows that only the zero order maxima is retained in the diffraction image.

V. Diffraction contrast. - The diffraction con- trast method constitutes the basic means of investiga- ting crystal defects by transmission electron microsco- py. When a single crystal is set in a position of strong Bragg reflection the local diffraction conditions are changed by the presence of a defect. The magnetic fields associated with ferromagnetic specimens by changing the local diffraction conditions play the role of defects leading to contrast effects in direct images 16, 22, 231.

One way of approaching the problem is to include the crystal lattice potential V(r) in the Schrodinger equation (1). However, we choose to follow the approach of Jakabovics [6] in which the Howie and Whelan two beam dynamical equations modified to take account of the phase effects due to the magnetic field of the specimen become,

T and S are respectively the amplitudes of the transmit- ted and diffracted intensities; ^,&, &; and EL are the real and imaginary parts of the extinction distances ; s, is the deviation from the Bragg position in the absence of a magnetic field ; g is the diffraction vector and @ is a vector perpendicular to B giving the electron beam deflection as a function of depth in the magnetic field. Numerical integration of the equatinns (12) yields the rocking curves with the form of figure 12 for the transmitted and diffracted intensities. Thecurves predict the intensity as s varies about the exact Bragg position as at an extinction contour*. The separation of the curves ds is proportional to g @ cos a where a is the angle between @ and g. ds is maximal for g//@ and minimal for g I m. As a reference we take ^ci,

* In this section s denotes the deviation of the diffraction vector g from the exact Bragg position. In figure 12 W = s e,.

Elsewhere s is an ordinary reciprocal lattice distance.

FIG. 12. - Bright and dark field rocking curves calculated for the (2020) reflection in cobalt with foil thickness = 3 q.

The intensity scale is arbitrary, 8; and eh are taken as 10 8,.

(From Jakabovics [6 (b)], by courtesy of the Philosophical Magazine.)

as the angle between g and the component of @ per- pendicular to the length of the domains.

We consider the case of a cobalt foil shown schema- tically in figure 13a, the stray field and internal magne- tisation cmfigurations being mutually perpendicular distort an extinction contour in characteristic fashions.

The step effect of figure 13b and the zig-zag effect of figure 13c are due respectively to the internal flux and the stray field, which effect predominates depends on the angle a,. In figure 14a we show an example of the step effect in cobalt whilst in figure 14b we see the difference in intensity between domains in a region where s varies slowly. In figure 1% and b we show respectively the zig-zag effect in cobalt and in stripe domains in an iron foil. The step effect reveals domain structures in a focussed image whilst the zig-zag effect reveals flux configurations of the form of figure 13c.

In cobalt such a structure corresponds to the stray fields whilst in stripe domains, where Bourret and Kleman 1231 have shown the stray fields to be negligible, it corresponds to an oscillation of Mx of the form of figure 13c.

The curves of figure 12 show that for a given value of s there is an intensity difference between B = + ^My,

an effect shown in figure 14b for the transmitted beam, whilst the dark field micrograph of figure 16

R. H. WADE

FIG. 13. - ( a ) Schematised flux distri- bution in and around a cobalt foil. The step and zig-zag effects are produced in a n extinction contour by the different flux distributions shown. Which effect predominates depends on the angle ai.

FIG. 14. - Bright field images from a cobalt foil showing (a) the step effect at sn extinction contour, (b) the difference in transmitted intensity brtween antiparallel domains in the region A at an extinction contour where s varies slowly. In both cases

a1 = o.

TRANSMISSION ELECTRON MICROSCOPE C 2 - 105

FIG. 15. -Bright field images showing extinction contours with ai ^E 90° (a) in cobalt where the zig-zag effect is due to the stray field. (b) In a (1 11) iron foi1 containing stripe domains where the zig-zag effect is produced by an oscillation of the interna1 magnetisation.

((6) is from Bourret and Kléman, unpublished.)

FIG. 16. - Dark field micrograph of a cobalt foi1 showing the effect of magnetisation direction on the diffracted intensity. The operating reflection is (20%).

(From Jakabovics [ 6 (b)] by courtesy of the Philosophical Magazine.)

C 2 - 106 R. H. WADE

shows the effect in the diffracted beam. The asymme- images for a given value of s to study stripe domain try in the diffracted intensities, having no apparent structures in iron foils. In such domains My oscillates physical explanation, can be exploited to obtain measu- with a period D and Mx with a period 2 D, so that rements of the anomalous absorption parameter eL[6]. one or other periodicity is shown depending on the Bourret and Dautreppe [7], Bourret and Kleman [23] angle a,. For g parallel to the domain directions the have used the difference in intensity of the bright field periodicity is 2 D whilst for g perpendicular it, it is D.

In addition for other angles the periodicity varies with

FIG. 17. - Bright field images of stripe domains in iron foils, ( a ) in the region of the extinction contour A with xi = 60° the observed doinain periodicity is a function of ^S. The contour at B shows the zig-zzg effect. (b) The period D is revealed for a i = 90°

and 2 D for al = O0 at the regions E and F respectively.

(From Bourret and Kleman, unpublished.)

the value of s, as is show; in figure 17a where g rnakes an angle of 600 to the domains. Figure 17b shows for a given value s that the observed periodicity depends on the angle between g and the domain directions, at F we observe the period 2 D whilst at E the period is D.

FIG. 18. - Diffraction contrast at domain walls in cobalt a,= O, (a) focussed image, (b) defocussed image of the same area.

(From Bourret, unpublished.)

TRANSMISSION ELECTRON MICROSCOPE C 2 - 107

We show finally in the micrograph of a cobalt foi1 figure 18, that domain walls themselves can be revealed by the method of diffraction contrast, since a compa- rison of the in-focus and off-focus micrographs of figure 18a and b, show that the focussed image reveal al1 the domain walls in the same contrast. Although it is undoubtedly the stray field associated with the wall which is responsible for the contrast a detailed analysis of this case has not yet been made.

VI. Other methods. - There are several classical methods in light optics which enable amplitude images to be obtained from phase objects. It is naturally interesting to know whether these methods are appli- cable to the observation of ferromagnetic films in the electron microscope. We have already dealt with the defocussed mode which yields shadow images related to phase changes produced in the object. Two other methods use in-focus images and are based on the Abbe lheory of image formation which shows that the imaging of an object by a lens can be regarded as occuring in the stages : object + Fraunhofer diffrac- tion image + final image. We have already indicated that the diffraction image amplitude is given by the Fourier transform of the object function. The final image amplitude is related to the diffraction ampli- tude by a further Fourier transform. If we allow al1 the information in the diffraction image to pass unmo- lested into the final image we recuperate the object.

The amplitude distribution A,(s) in the diffraction image of a zero width converging wall, corresponding to equation (9) and figure 8 1, is plotted in figure 19a.

The diverging wall diffraction amplitude A,(s) is shown in figure 19b. The diverging and converging wall diffraction amplitudes are related by A,(s) ⁼ AC(s).

(i) PHASE CONTRAST. - In the case of weak phase oscillations the object function of a grating is written as

f (x) = et"(") = 1 + icp(x)

for an object of infinite extent. The diffraction anpli- tude contains a real zero order maximum with the rest of the spectrum pure imaginary. A quarter wave plate placed over the zero order maximum renders this maximum imaginary also. A second Fourier transformation now yields an image showing intensity fluctuations corresponding to the original phase structure of the object.

It is clear from the amplitude distributions of figure 19 that with a domain wall as object it is not possible to modify independently the phase of the real and imaginary parts in the diffraction amplitude.

Consequently the c!assical phase contrast method will not be applicable to domain wall observations.

(ii) FOCAULT. - Part of thv diffraction image cut off by an aperture does not contribute to the image.

Placing an opaque screen in the region O < s < ^co of figure 19 cuts out the intensity in the corresponding domain in the direct image. Antiparallel domains in cobalt are imaged in this way in the figure 20.

WohIleben [14] has recently calculated the intensity distributions in Focault images from domains separated by zero width walls. His results show that :

FIG. 19. - Diffraction imagr amplitude distributions for (a) converging, (6) diverging walls-of zero width. The real and iml-

ginary parts are shown respectively as the full and dotted lines. FIG. 20. - Domains in cobalt revealed by the Focault mode.

C 2 - 1 0 8 R. _{H. WADE} 1. Fringes appear parallel to the walls and there is a

broad transition from the bright to shadow part of the image.

2. The form of the image is sensitive to the position of the screen a t the diffraction plane.

3. The images of convergent and divergent wall regions show mirror symmetry about the classical intensity step when the aperture edge is at s = 0.

The first two points show that the classical determi- nation of the wall magnetisation distribution [25] is not possible in this mode which had at first sight appeared to offer the advantage of the good resolution associated with in-focussed images.

These results can be extended and generalised since the symmetry in the phase forms of diverging and converging walls of identical but arbitrary width and structure shows that the diffraction amplitudes are necessarily related by :

These conditions are readily confirmed for the simplest case of the amplitudes from zero width walls shown in figure 19. Denoting the arbitrary position of thr knife aperture edge by s, the rela- tions (13) yield the entirely general results, in the one dimensional case considered, that for apertures spanning the region s, < ^s ,< co in the two diffrac- tion images the final images of the two walls show mirror symmetry. Apertures disposed to obscure the region ^s, < ^s ,< co in one diffraction image and the symmetrical region - co ,< s < - sl in the other cause the final images of the two walls to become identical.

(iii) INTERFERENCE MICROSCOPY. - Another method used in light optics is interference microscopy in which the beam transmitted by the object is divided and subsequently caused to overlap so as to produce a t once parallel interference fringes and a superposi- tion of object points A, B previously separated by a distance x perpendicular to the fringe direction. The overlap distance x and the fringe separation are depen- dent. A difference in phase between the two points A and B appears as a distortion of the fringe system.

In the case of a magnetic sample the electron optical phase difference is given by the enclosed flux rule.

In an idealised example fringes formed perpendicular to a 1800 Néel wall are displaced within the wall region in a way directly related to the component of magnetisa- tion perpendicular to the wall direction. Unfortunately in this case the treatment of section IV(i) shows that

the wall stray field will undoubtedly destroy the effect.

However other orientations of the fringe system with respect to the wall may enable the variation of the magnetisation component parallel to the wall to be related to the fringe dispIacements.

VIT. Summary. - The aim of the article has been to present as coherent an account as in my power of the problem of imaging magnetic strurtures. The principle conclusion is that the application of a wave optical approach to different observational modes should provide a sound basis for the detailed study of domain wall structures. The geometrical optical approach provides a qualitative description of the imaging processe;.

ACKNOWLEDGEMENTS. - 1 am grateful to D. Wohlle- ben for the micrograph of figure 4 and for sending me a copy of the article ⁽⁽ Diffraction effects in Lorentz Microscopy » before publication ; to Dr J. Jakubovics for the micrograph of figure 16 and t o Phil.',Mag. for allowing the reproduction of this figure and figure 12 ; to A. Bourret and M. Kleman for the micrographs of figure 15b, and figure 17 ; and to A. Bourret for the micrographs of figure 18. 5. P. Guigay kindly allowed me to use his unpublished treatment of the phase approximation of section II.

1 have lately received a copy of a report by M. S. Cohen 1241 which deals at much greater length than 1 have done with the application of classical optical methods to the observation of the phase objects formed by magnetic structures. However Our approaches are rather different.

References

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[2] WOHLLEBEN (D.), Phys. Letters, 1966,22, 564.

[3] BOERSCH (H.), HAMISCH (H.), WOHLLEBEN (D.) and GROHMANN (K.), (a) Z. Physik, 1960, 159, 397.

(b) Z. Physik, 1962, 167, 72 ; (c) Z. Physik, 1961, 164, 55.

[4] WARRINGTON (D.), Phil. Mag., 1964,7,261.

[5] AHARANOV (Y.), BOHM (D.), Phys. Rev., 1959, 115, 485.

[6] JAKUBOVICS (J.), (a) Phil. Mag., 1964,10, 277; (b) Phil.

Mag., 1966, 13, 85.

[7] BOURRET (A.), DAUTREPPE (D.), Phys. Sfaf. Sol., 1966, 13, 559.

[8] GUIGAY (J. P.), private communication.

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(London), 1949, B 62, 8.

[IO] FRANZ (W.), (a) Physik. Ber., 1940, 21, 686 ; (b) Z.

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[Il] FUCHS (E.), Z. angew. Physik, 1962,14,203.

TRANSMISSION ELECTRON MICROSCOPE [12] JAKUBOVICS (J.), Phil. Mag., 1964, 10, 675.

[13] WADE (R. H.), Bull. Soc. Française Min et Crist., to be published.

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DISCUSSION

M. VAUTIER. - Quelles sont actuellement les épais- seurs les plus fortes que l'on peut étudier par ces techniques de microscopie électronique ?

M. WADE. - Elles sont de l'ordre de 1 000 à 2 000 A

pour les électrons de 100 keV.

M. PIRCHER. - Y a-t-il moyen d'observer les propriétés magnétiques des lames minces en transpo- sant à l'optique électronique les techniques, de l'ho- lographie optique ?

M. WADE. - Je crois qu'en principe oui. Je vous signale que Cohen a abordé cette question dans l'article récent : (( Wave optical aspects of Lorentz Micros- copy », Report of Lincoln Lab., M.I.T.