TRANSMISSION ELECTRON MICROSCOPY STUDIES OF

THE SPECIFIC STRUCTURE OF CRYSTALS FORMED BY

PHASE TRANSITION IN IRON OXIDE AMORPHOUS

FILMS

V. Yu. KOLOSOV 1 and A. R. THOÈ LEÂ N 2{1Engineering Department, Ural State Economic University, 8th March Street, 620026 Ekaterinburg,

Russia and 2Department of Experimental Physics, Chalmers University of Technology,41296 GoÈ teborg, Sweden

(Received 16 August 1999; accepted 9 December 1999)

AbstractÐThe crystal growth and the structure of crystallised areas in amorphous iron oxide ®lms pre-pared by pyrolysis is studied (in particular, in situ ) by means of transmission electron microscopy. Thebend-contour (BC) technique reveals an unusual phenomenon of regular internal bending of the lattice planesassociated with a phase transition also observed earlier in some other amorphous ®lms. The peculiarity ofthe crystals studied (transrotational crystals) is a very complex texture usually formed with a permanent ro-tation of [001] at the front of the growing crystals resulting in a variable change of imperfection andgrowth rate attributed to crystal anisotropy. 7 2000 Acta Metallurgica Inc. Published by Elsevier ScienceLtd. All rights reserved.

Keywords: Transmission electron microscopy (TEM); Amorphous materials; Nucleation & growth; Thin®lms; Internal lattice bending

1. INTRODUCTION

Thin layers of iron oxide produced by chemical

vapour deposition as semitransparent masks [1, 2]

are easily obtained in an amorphous state.

Crystallisation of amorphous ®lms is of interest

both from a practical and a scienti®c point of view.

For practical applications of amorphous layers, it is

important to understand the mechanisms of crystal-

lisation to prevent the transition to the more stable

crystalline state. From a scienti®c point of view

crystallisation from the amorphous solid state is of

special interest, since it results in imperfect crystal

growth and formation of exotic structures inherent,

in particular, in ``explosive'' crystallisation [3, 4]

and ``spherulite'' crystallisation [5, 6]. Signi®cantly

more complex microstructures, unusual in solid

state physics, and with a strong internal bending of

the crystal lattice planes, can be formed during

amorphous±crystalline transformation in thin ®lms.

Initially, they were discovered for crystal growth in

chalcogenide amorphous ®lms [7, 8]. Later this unu-

sual phenomenon was observed for many other

substances [9] of dierent chemical nature and

therefore it deserves much more attention. In earlierstudies [10, 11] the general features of the structureof pyrolytic iron oxide amorphous ®lms and crys-tals formed after crystallisation were described.

Here we have performed a detailed electron mi-croscopy study of the speci®c structure of the crys-tallised areas. Though when they have grown large

enough (2 mm in diameter or larger) in some areasthey acquire in addition a polycrystalline structure,where the general feature is a regular, complex tex-

ture caused by internal bending of lattice planesformed in the course of crystal growth from an iso-lated single crystalline nucleus (surrounded byamorphous material). Keeping in mind this last

essential fact, we preferably use the term crystal forthe crystallised areas we study, but now modify itto transrotational crystal, considering other import-

ant features mentioned above.

2. CRYSTAL STRUCTURE OF aa-Fe2O3

The crystallised phase a-Fe2O3 (hematite) we stu-died has a rhombohedral (trigonal) structure andbelongs to the space group R�3c: It is similar to thecrystal structure of sapphire (corundum) described

elsewhere [12]. Although sapphire is rhombohedral,it is commonly described in terms of a hexagonal

Acta mater. 48 (2000) 1829±1840

1359-6454/00/$20.00 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.PII: S1359 -6454 (99 )00471 -1

www.elsevier.com/locate/actamat

{ To whom all correspondence should be addressed.

structure with a larger, hexagonal unit cell [12]. Weshall follow the same description for our phase,

using the lattice parameters a = 0.503 nm, c =1.374 nm [13].

3. EXPERIMENTAL

3.1. Thin ®lm preparation

Amorphous iron oxide ®lms were produced bythermal oxidation of iron pentacarbonyl Fe(CO)5 as

described earlier [2] on freshly cleaved (in air)sodium chloride crystals as a substrate. The free-standing ®lms studied were separated from the sub-

strate in distilled water and placed on electronmicroscope grids (mesh 100±400).

3.2. Electron beam annealing

Crystallisation was initiated by the in¯uence of

the electron beam inside the electron microscopeused (Tesla BS-540, JEM 200CX, JEM 2000FX andPhilips CM200FEG). This made it possible to trace

the phase transformation in situ. To prevent furthercrystal growth in the crystallised areas, the structure

studies were performed with the lowest possibleintensity of the beam. Increased intensity wasobtained by focusing or changing the beam current

(for some of the most stable amorphous ®lms eitherthe largest condenser aperture or none was used)with the purpose of crystallising new ®lm areas. As

was shown earlier using Se±Te ®lms as an example,the structure of the crystallised areas after electronbeam irradiation and after an ordinary heat treat-

ment carries the same main features [7, 8]. Besides,using an electron beam focused to a diameter of 1±100 mm, one can also perform and trace localannealing for dierent heating conditions (resulting

in dierent crystallisation rates) at any desiredregion of the sample [Fig. 1(a)]. Thus the outer areais left untreated and is suitable for further studies,

unlike after an ordinary annealing.

3.3. In situ studies

Many free-standing amorphous ®lms are suitablefor in situ studies of crystal growth in the trans-mission electron microscope. During uniform ®lm

heating (irradiation) crystallisation starts from iso-lated nuclei embedded in an amorphous matrix,where one can follow the growth of individual crys-

tallites before they start to touch [Fig. 1(b)]. Forcrystal growth rates in the range 0.001±10 mm/s, theobservations are inherently made in situ. Besides, inthe range 1±100 mm/s the growth is suitable forframe-by-frame analysis using video or CCDrecording. The usual presence of prominent extinc-tion bend contours (which actually visualise the

®eld of lattice orientation and dierent imperfec-tions, as well as the crystal phases) enables a pre-liminary analysis of the process. Though the ®lms

we study usually demonstrate a distinct dynamicbehavior of the crystal front propagation [11] (seealso below), we shall not go into further detail thanis necessary for a structure characterisation of the

crystallised areas.

3.4. Structure studies

Our transmission electron microscopy (TEM) stu-dies include conventional microscopy: selected areadiraction (SAD), bright and dark ®eld images, in

the last case in conical mode (using the re¯ectionsincluded in a whole diraction ring of spherulitecrystal), was found to be very useful [Figs 5(b) and

(d)]. Furthermore, convergent beam electron dirac-tion (CBED), energy dispersion analysis of X-rays(EDX), electron energy loss spectroscopy (EELS)

and high resolution electron microscopy (HREM)were performed with the help of CM200 FEGwith a slow scan CCD camera and a Gatan imaging®lter.

Fig. 1. Crystallised spots in amorphous iron oxide ®lm: (a)series of areas, crystallised subsequently by short pulses(fraction of a second) of focused (about 1 mm) electronbeam of increasing (from left to right) intensity: (b) crys-tals growing with a small growth rate 0.05±0.10 mm/swhile the ®lm is irradiated by a defocused (diameter

>100 mm) electron beam.

1830 KOLOSOV and THOÈ LEÂ N: AMORPHOUS FILMS

3.5. Bend-contour technique

Extinction bend contours (contrast lines with

dierent intensities, various shapes and pro®les)

depicting the regions where Bragg's law is satis®ed,

are often inherent in TEM micrographs of crystals

and crystallised areas in amorphous ®lms (Figs 1±

6). Due to the small Bragg angles for high energy

electrons (usually

approach suitable for both crystal and lattice bend-ing [7, 15].

3.6. Thickness measurements

Most strong BCs corresponding to diraction

from densely packed crystal lattice planes demon-

strate a ®ne structure of diraction maxima, best

resolved in dark ®eld. Measuring the distances l

between subsidiary maxima (in dark ®eld) and L

between the contours hkl and hkl (in bright ®eld),

which usually appear close together for the strong

lattice curvature considered here (Fig. 2), one can

determine the crystal thickness t from the equation

Fig. 4. Central area of the transrotational crystallite (a) and corresponding SADP (b), CBED (c),HREM (d). The arrows in (a) point at splitting and ``multiplication'' of bend contours.

1832 KOLOSOV and THOÈ LEÂ N: AMORPHOUS FILMS

t d 2L=ll where d is the interplanar distance and lis the electron wavelength [21]. The other method

we used for estimating the ®lm thickness was

measuring the thickness of non-transparent (parallel

to electron beam) regions in folded parts of the

foils (Fig. 3). The crystallised ®lms were 15±35 nm

thick.

4. RESULTS

4.1. Initial stages

Electron irradiated areas of an amorphous ®lmusually crystallise after some latent period of time(0.1±1 min, for intermediate intensities and corre-

sponding growth rates near 1 mm/s). Crystallisation

Fig. 5. Annular alternating zones of transrotational crystallised area with the texture T2 (a, b) and their®ne structure (c, d) in bright ®eld (a, c) and dark ®eld (b, d) obtained in a conical mode from thewhole ring, at the pointer in the SADP (inset). The pointer in the image (c, d) points at more perfect

zones Z+ with [001] near normal to the ®lm plane.

KOLOSOV and THOÈ LEÂ N: AMORPHOUS FILMS 1833

usually starts with the formation of a crystallinenucleus with the a-Fe2O3 structure and with the[001] orientation approximately normal to the ®lmplane, unlike in some other substances studied ear-lier [22] which showed several preferential orien-

tations. The reproducibility of preferred latticeorientation in a-Fe2O3 nuclei is easily revealed fromthe patterns of crossed BCs, having a six fold sym-

metry and corresponding to the [001] zone axis[Fig. 1(b)]. This orientation is actually deducedfrom the corresponding diraction pattern

[Fig. 4(b)], which allowed (using dark ®eld imagesof the diraction spots) to assign indices to theBCs. The nuclei always show a strong curvature ofthe lattice, R11±2 mm, as is indicated by the small

intercontour distances between the BCs hkl and hkl,always coming in pairs. For a-Fe2O3 crystal nuclei,the two most prominent re¯ections correspond to{110} and {300} [Figs. 4(a) and (b)]. The mostintense BCs from the pair {110} come close

together, indiscernible from one another in bright®eld due to dynamical diraction, resulting in acomplex ®ne structure of the contours which can

only be distinguished in dark ®eld. The geometry oflattice bending in small crystals (about 1 mm in di-ameter) corresponds to that of a cup-shaped bent

crystal with [001] oriented strictly parallel to theincident electron beam (normal to the ®lm plane)only at the centre. Because of the strong internallattice bending (around an axis lying in the ®lm

plane), such crystals do not actually correspond tothe term ``single crystal'' (we use the term ``transro-tational crystal'' instead), even though they nor-

mally produce point diraction patterns andsometimes carry sixfold morphology [Fig. 6(a)] still,though less, preserved at subsequent stages of

growth.

4.2. Large transrotational crystals (crystalline areas)

During further radial growth of a transrotational

crystal up to several mm the SADP transforms froma spot pattern to a pattern with diraction circles(with or without texture) as for a polycrystalline

®lm [Fig. 5 (inset)]. However, in our case such anSADP normally corresponds to an individual trans-rotational crystal (crystalline aggregate) with a com-

plex texture of two dierent kinds, T2 (we usesubscripts ``+'' and ``ÿ'' to describe the orientationof [001] close to the ®lm normal and parallel to the®lm plane respectively, and ``2'' is used to describethe orientation range from T+ to Tÿ).(T+) Azimuthal misorientations of the lattice

around the ®lm normal [001] prevail. It is visually

indicated by splitting and ``multiplication'' ofradially oriented, mostly bright BCs {110} and{300}, distinguished at the initial stages of growth

[Fig. 4(a)]. Spherulite structure with a tendency for[001] to be slightly inclined from the ®lm normal[Fig. 6(b)], is formed as a consequence. The crystal-lisation front is moving radially at a more or less

constant rate leading to a uniform microstructureof circular symmetry, with BC patterns resemblingdendrites coming preferably from one centre.

(T2) Internal lattice bending (around an axislying in the ®lm plane) prevails. As a result, duringradial growth the orientation of [001] at the crystal-

lisation front, together with crystal imperfections, ischanged periodically, synchronously with a changein growth rate, and a crystallised area resembling a

target board with dark and light regularly alternat-ing annular zones is formed (Fig. 5). Thesecomplicated features of a generally observed micro-structure need to be characterised in detail.

Fig. 6. Crystalline areas with a tendency for orientation of[001] normal to the ®lm plane: crystals grown with a largegrowth rate, about 50 mm/s (a), and small growth rates,

below 1 mm/s (b).

1834 KOLOSOV and THOÈ LEÂ N: AMORPHOUS FILMS

4.3. Main regularities of the unusual transrotationalmicrostructure

Figure 7 schematically illustrates the main fea-tures of complex regularities of lattice orientationsacross the crystal. It is shown in Fig. 7 that a pro-

gressively increased de¯ection of [001] from its nor-mal position at the crystal centre (perpendicular tothe ®lm plane) is realised at the circular crystallisa-

tion front moving outwards. Most easily it can beseen from the sketch of [001] orientations outlinedin the crystal cross-section in the same ®gure. In

other words, the permanent rotation of the unit cell(if traced along any radial outward direction ofcrystal growth) around an axis lying in the ®lmplane is observed as far as the crystal growth takes

place. It is very important to underline here thatthe rotation of the unit cell is caused not by thebending of the specimen, but by internal bending of

the lattice itself (more accurately, bending of thecrystal lattice planes) around an axis lying in the®lm plane. It is the same unusual phenomenon as

was discovered earlier for crystal growth in amor-phous ®lms of dierent substances [7±9, 22]. Thetransrotational crystals and the transrotational crys-

talline areas themselves are not bent and the surfaceremains ¯at after the transformation. In the presentcase, this can easily be seen at the folded parts ofthe ®lm (Fig. 3; cracking and folding of transrota-

tional crystallised area is sometimes observed aftera long period of time). But it can be understoodalso a priori, without any additional experimental

evidence, since in crystals large enough the wholerotation of [001] across the crystal exceeds one com-plete turn, i.e. 3608 [e.g. 7208 for the transrotational

crystallites shown in Fig. 5 and 14408 for the crystalin Fig. 3(a)]. This obviously cannot be realised in

pure bending of the crystal as a whole body.As a result, circular zones with a de®nite orien-

tation of [001] with respect to the ®lm plane are

observed. We shall conditionally classify them astwo kinds of alternating zones, Z+ and Zÿ. Nearthe centre line of odd zones Z+, including the cen-

tral point of the ®rst central zone, [001] is almostnormal to the ®lm plane and the zones look lighterin a bright ®eld image (Fig. 5). Near the centre line

of even zones Zÿ, [001] is nearly parallel to the ®lmplane and the zones appear to be darker in theimage. One of the main reasons for this apparentdierence in darkness of these two zones is that

they dier drastically in the amount of imperfec-tions. While the central parts of zones Z+ arealmost perfect (typically single crystalline) the cen-

tral parts of zones Zÿ are most imperfect. Actually,the change from one zone to another is gradual andtherefore a division of the transrotational crystal

into two zones is schematic.

4.4. Dynamics of crystal growth

The intensity of the beam (or rather its heatingeect) determines the rate of crystal growth: the

higher the intensity, the higher the crystal growthrate. In turn the crystal growth rate, in the range 1±100 mm/s, determines the width of the alternatingzones: the larger the growth rate, the larger the dis-

tance between the centre lines of the zones. Thismeans that the degree of internal bending decreases.For a mean growth rate of about 1 mm/s the widthof both zones is about 1 mm (corresponding to aninternal lattice bending of about 908/mm) andbecomes larger (internal bending, smaller) for a lar-

ger growth rate. Thus the diameter of the centralzone can increase by an order of magnitude forgrowth rates above 50 mm/s [Fig. 6(a)]. It is worth

Fig. 8. Variation of crystal growth rate of an Fe2O3 trans-rotational crystallite with time, revealed from an analysis

of successive frames of video records.

Fig. 7. Sketches of lattice orientations. Transrotationalcrystallised area having zonal structure. (a) [001] is indi-cated by arrows in the crystal cross-section and by corre-sponding projections (dashes and dots) in plane view. Thealternating annular zones Z+ and Zÿ are indicated byshadowing of less perfect crystalline zones Zÿ. Atomisticmodel of the transrotational crystal structure (internal lat-tice bending). (b) (100) and (001) planes are schematicallyindicated (about 1 line per 10 atomic planes with orien-

tation gradients exaggerated about 10 times).

KOLOSOV and THOÈ LEÂ N: AMORPHOUS FILMS 1835

adding that for small growth rates (much below1 mm/s), the width of zone Z+ also becomes larger,even the whole crystal can correspond to one zoneof such a kind [texture T+, Fig. 6(b)].Last but not least, the crystal growth rate (being

large enough) is not constant and depends upon theorientation of the lattice in the growing transrotato-nal crystals, roughly upon the kind of zone that is

currently formed. The growth rate is higher forzones Z+ and decreases by a factor of 2±100 duringthe growth of zones Zÿ (Fig. 8). Therefore, oneshould keep in mind that it is an average growthrate that we usually refer to here (for the growth oftransrotational crystals with texture T2, see Fig. 5).

4.5. EDX and EELS studies

EDX spectra taken from an area 300 nm across,either from the centre or from dierent pointsinside both the ``perfect'' and ``imperfect'' zones, aswell as spectra taken from the whole transrotational

crystal, are identical with peaks corresponding toFe and O present.EELS spectra taken in the image mode from

dierent transrotational crystalline areas (in particu-lar from the single crystalline parts free from extinc-tion BCs) and from amorphous areas are also the

same, with the same M2,3 maximum for Fe, near60 eV. The spectra are similar to the publishedspectrum of iron oxide ®lms [23].

4.6. Grains, grain boundaries and grain boundarystatistics

It is well known from TEM that displacement ofBCs is one of the best indications of grain bound-aries. Owing to the presence of numerous BCs and

circular symmetry of the complex orientational tex-ture of the transrotational crystallised areas, conicaldark ®eld images make it possible to visualise the

grain boundaries (more than 1000 within a crystal-lised spot with a diameter of 25 mm, shown inFig. 5) in the imperfect zones Zÿ. The positions ofthe visible grain boundaries in the crystallitesshown in Fig. 5 were translated to transparent foilsby hand from magni®ed negatives to obtain a largemap of the grain boundaries. All the grains were

enumerated and their width and grain boundarylength were measured. The results are summarisedin Fig. 9. The histograms illustrate the distribution

of the lengths of grain boundaries, and of thewidths of crescent-like and ribbon-like grains. Inthe most disturbed areas of imperfect crystalline

zones, the grains are usually very narrow andelongated (very roughly, elongated parallel to thebasal planes). Grain size and shape vary from zone

to zone in a regular manner from equiaxed in zoneZ+ to crescent- and ribbon-like (up to 20 nm inwidth and 1000 nm in length) in zone Zÿ, thus theaspect ratio is changing by almost two orders of

magnitude.

4.7. HREM studies

The centre of the transrotational crystal (where itwas nucleated) with [001] strictly normal to the ®lm

plane is highly perfect (Fig. 4), as well as the wholeneighbouring area around the crystal centre corre-sponding to a zone Z+.

At the grain boundaries, which are very commonin regions Zÿ, no mis®t or dislocation-like contrasthas been observed. In some of the places examined

in more detail, lattice fringes are passing continu-ously from one grain to the other and with no verysharp osets at the boundary [Fig. 10(a)], which isexactly what would be expected from the geometry

of internal lattice bending (see Section 5.2).Neither inclusions of amorphous material nor

dislocations have been observed inside the trans-

rotational crystalline regions.The interface between amorphous material and

the crystallised region Z+ is rather sharp

[Fig. 10(b)], with the transition region about 1 nmcharacterised by intermediate ordering and a changeof diraction contrast.

5. DISCUSSION

5.1. Internal lattice bending

The main striking peculiarity observed for theFig. 9. Histograms of distribution of grain width (a) and

grain boundary length (b) in zones Zÿ.

1836 KOLOSOV and THOÈ LEÂ N: AMORPHOUS FILMS

crystal growth is a regular permanent internal lattice

bending (transrotational atom periodicity) inherent

in the crystallisation process.

It is worth emphasising that internal lattice bend-

ing cannot be attributed to the peculiarities of chemi-

cal bonding, since it was revealed earlier for

substances of dierent chemical nature: Se, Te,

CuSe, Cu2±xTe, a-Fe2O3, CoPd, Re [24] and the fulllist of examples that have been observed is much

longer [25]. No special features are observed by

HREM, either at the centre of the transrotational

crystalline nuclei, where the unusual phenomenon

starts, or at the crystallisation front at the periphery

of the growing crystal. Neither EDX nor EELS

spectra reveal any dierence in composition which

can be associated with the unusual lattice orien-

tation rotation. Internal lattice bending is dislo-

cation independent, since it can also be observed in

perfect dislocation-free transrotational crystals, e.g.

Se [15] and Cu2±xTe [26].

To perform a simple estimate of the upper value

of possible elastic strain, let us consider convention-

al cylindrical bending of the crystal (as a whole

body), though it is not realised in our experiments.

In this case the maximum strain e located in thesurface layer is e gradj� t=2, where gradj is the

Fig. 10. HREM of grain boundaries in the most imperfect regions of zones Zÿ (a) and of amorphouscrystalline interface (b) of the crystal shown in Fig. 6(a), tilted 188.

KOLOSOV and THOÈ LEÂ N: AMORPHOUS FILMS 1837

gradient of lattice orientation associated with lattice

bending and t is the crystal thickness. For examinedcrystals with a mean lattice bending of 908 per mm(gradj 1 1.5 rad/mm) and for a thickness of 20 nm,we obtain e=1.5%. Though large and near thetheoretical limit, such high strains have earlier beenexperimentally proved to be possible for perfect

thin crystals and whiskers [27]. Actually the highelastic strains accomodated at the surface layers of

the single crystalline zones Z+ should be even smal-ler, since in the zones Zÿ plastic deformation takesplace as discussed above and therefore part of the

total lattice rotation (in this case, a sum of discretemisorientations) is ``taken up'' by the grain bound-aries.

To give the reader some insight into how atomscan be arranged in the unusual transrotational crys-

tals to ®t the experimental data, let us go into somedetail of hypothetical atomistic models and mechan-isms of the phenomenon. We shall keep the full set

of experimental facts and arguments aside, since itneeds special consideration which is out of thescope of the present paper.

From a general point of view, a similar rotationalmisorientation of the crystal lattice (as a whole) can

be built by placing wedge disclinations [28] into aperfect crystal. They are common in liquid crystalsand can be considered as rotational analogues to

the well-known translational line defects, dislo-cations. Our ®rst model for the case of cylindricalinternal bending was based on this idea [29].

Later, we proposed [30, 37] an alternative atomis-tic model and a sketch of internal bending of two

main sets of crystallographic planes correspondingto the crystals studied is shown in Fig. 7(b). Thelast model complies with further experimental facts

and also allows one to propose an appropriatemechanism for the formation of the unusual in-ternal lattice bending of a growing crystal. This

hypothetical mechanism is based on surface nuclea-tion and subsequent relaxation of initial clusters

formed with lattice distortions, which are caused bychanges of the equilibrium interatomic distanceswhen the crystallisation front is moving from the

®lm surface towards the bulk of the material. Inorder not to return to this question again, we shalladd here that the reproducibility of the orientation

of a nucleus with its basal, most densely packed lat-tice planes lying parallel to the surface ®ts the con-

cept of surface nucleation.The basic idea while constructing the model of

the ®nal structure with a distorted lattice [illustrated

schematically and with a large exaggeration inFig. 7(b)] was to conserve symmetry of the latticewhen going down in dimensions to the unit cell (to

allow the same electron diraction) preferably keep-ing the angles constant and thus incidentally mini-

mising the energy of the deformed crystal structure.Atom ordering in the model corresponds to a com-bination of translation and slight rotation (inevita-

bly accompanied by slight tension or compression),

which is why we prefer the term ``transrotational''.Among possible analytical descriptions of atomic

positions in the crystal cross-section [Fig. 7(b)] the

mathematical tools of conformal transformation asapplied to a two-dimensional (2D) set of regularpoints (corresponding to the unusual crystal lattice)

seems particularly intriguing, though it usually uti-lises complex variables and thus is limited only to a

2D case. Conformal transformations arose fromphysical concepts and the methods of conformalbodies in mathematics in its turn found numerous

applications in dierent ®elds of physics.Conformal transformations preserve angles (andalso produce constant tension [31]), that makes the

choice of a corresponding model of lattice distor-tion reasonable from a physical point of view.

There are dierent bi-unique analytical functionsproducing conformal transformations (for the pur-pose of depicting internally bent lattice planes one

can deal with conformal mapping). Thus the 2D setof points {W }, W=U+iV described by the confor-mal transformation W ÿi lnkZ =k (k is a par-ameter) [32] of the 2D translational lattice with theset of points {Z }, Z X iY is one of the possibleanalytical descriptions of atom positions in the ®lmcross-section (plane U, V) of Fig. 7(b) [33]. Wetook this appropriate form of a complex logarithm

as it was given in the description of a 2D ``gravityrainbow structure'' of small magnetised iron spheresin a magnetic ®eld (slightly tilted with respect to the

gravity ®eld) [32]. The authors used the term ``con-formal crystals'' for such a real physical system con-sisting of interacting particles located at the sites

{W }. Its cross-section in a recent paper [34] iscalled a ``conformal lattice'' and the corresponding

idealised physical system is called a ``strictly confor-mal crystal''.In conclusion, let us return to the terms de®ning

the crystals with an internal lattice bending (trans-rotational atom periodicity) we observed. In thephysics of condensed matter, the term ``crystal'' is

normally reserved for solid bodies with a 3D peri-odic atomic structure. That is why we do not sup-

port the idea to use it for the 2D array of particlescited above. At the same time, we do not ®nd itappropriate to follow the intuitive idea to use it to

de®ne our crystals, not only because it is alreadyused but also since for dimensions higher than 2D,the class of conformal transformations is very slen-

der (restricted to bilinear, MoÈ bius, transformationand inversion [35]). In particular, the logarithmic

conformal transformations used above cannot begeneralised for 3D space. For the last case thereexists a much broader class of quasi-conformal

transformations [35] with limited distortion (dilata-tion) which from the physical point of view seemsmore appropriate for constructing models of a crys-

tal lattice with internal bending. Though specialconsideration should be given primarily to dierent

1838 KOLOSOV and THOÈ LEÂ N: AMORPHOUS FILMS

geometries of internal lattices observed earlier [36],one can use the term ``quasi-conformal'' crystal for

this purpose in a mathematical sense. In fact, themathematical restrictions of quasi-conformal trans-formations of translational 3D lattice for the

description of internal crystal lattice bending, canappear to be too strong or too weak, dependingupon the complex physicochemical nature of atom

bonding in the crystals.

5.2. Variation of imperfection and growth rate

Qualitatively, the dependence of grain boundary

origin, number and nonequiaxial features uponchange in lattice orientation for crystallised areaswith texture T2 can be explained in the following

way. Because of the anisotropy of elastic propertiesof the crystal, the elastic limit diers for dierentorientations (for aFe2O3 the most signi®cant dier-ence should be between crystalline areas where thehexagonal axis [001] is perpendicular or parallel tothe ®lm plane, zones Z+ and Zÿ, respectively).Therefore only for some lattice orientations (in our

case zone Zÿ) large strains (caused by internal lat-tice bending) exceed the elastic limit and plastic de-formation with grain boundary formation starts.

High resolution electron microscopy studies of thegrain boundaries support this consideration, reveal-ing tilt grain boundaries with a tilt axis lying in the

®lm plane, as do the axes of internal lattice bend-ing. We can make rough estimates of the misorien-tations across the grain boundaries by dividing the

global (total) misorientation at zone Zÿ, e.g. 458,by the number of grain boundaries (ratio of zonewidth, 1 mm, and minimum grain width, 20 nm)which gives a misorientation of about 18.The above consideration also coincides with the

variation of grain shapes in dierent zones. Thetendency of single crystal growth (followed by large

equiaxial grain growth) is observed in the centre ofzones Z+, where [001] is normal to the ®lm planeor close to it. Also, it is well known that the hexa-

gonal structure can be considered elastically isotro-pic in the plane (001). At the same time, the largestaspect ratio is observed in zones Zÿ, where [001]has a tendency to be parallel (while {100} is nor-

mal) to the ®lm plane, thus making thin crystalsanisotropic for internal bending around an axislying in the ®lm plane. The variation in grain width

and grain boundary length can be explained by afull set of intermediate orientations for [001] passingfrom zone Z+ to Zÿ.Moreover, for the texture T+, [001] is oriented in

the same way across a crystallised area (but isslightly inclined with respect to the ®lm plane) and

as a result the imperfection is not very large andrelatively uniform as well as the growth rate.The anisotropy of the hexagonal lattice also

makes evident the experimentally observed aniso-

tropy of the crystal growth rate for crystallisedareas with the texture T2, in which the growth rate

varies synchronously with the change of [001] orien-tation at the front of the growing crystal.

5.3. Complex texture

All the transrotational crystals we studied above

a size of several mm in diameter (practically roundcrystalline spots) carry the main feature of thespherulite [5, 6] with one of the main crystallo-

graphic directions being oriented radially (actuallyspherulites formed in thin ®lms have a cylindricalsymmetry). The texture of these crystals is compli-

cated by strong internal lattice bending. For themost complex case of crystal texture (T2, Fig. 7) itis characterised (if traced along any radial direction,

i.e. the direction of crystal growth) by a constantrotation of [001] around the axes oriented tangen-tially in the ®lm plane.It is worth noting that the origin of the circular

zones of the crystallised areas studied has a naturecompletely dierent from that known from ``explo-sive'' crystallisation of amorphous Ge or Si ®lms

[3], formed as a result of spiral tangential growth.

6. CONCLUSION

An unusual phenomenon of strong regular in-ternal lattice bending as revealed by the TEM bend-contour technique can govern crystal growth in thin

amorphous ®lms of iron oxide (as it does for someother substances) and, in particular, can result in acomplex texture. The crystallised area, formed in

the course of the crystal growth from a single crys-talline nucleus under certain conditions, demon-strates a regular, synchronous variation of crystalgrowth rate (rhythmic crystallisation) and imperfec-

tions (formation of concentric alternating zones). Itcan be attributed to the anisotropy of the crystallattice exhibited as a result of the permanent change

of lattice orientation at the front of a growing trigo-nal transrotational crystal (quasi-conformal crystal).

AcknowledgementsÐThis work was partially supported bythe Royal Swedish Academy of Sciences, by the SwedishResearch Council for Engineering Sciences and by theRussian Basic Research Foundation through grant 97-01-17784 allocated for V.Yu.K. V.Yu.K. wishes to acknowl-edge A. L. Tolstikhina (Institute of Crystallography,Moscow) for the collaboration at the initial studies andproviding amorphous samples and N. Rivier for his inter-est in the studies and sending a reprint [32].

REFERENCES

1. MacChesney, J. B., O'Connor, P. B. and Sullivan,M. V., J. Electrochem. Soc., 1971, 118, 776.

2. Razuvaev, G. A. (ed.), Deposition of Films andCoatings by the Decomposition of MetalloorganicCompounds. Nauka, Moscow, 1981, p. 293, (inRussian).

KOLOSOV and THOÈ LEÂ N: AMORPHOUS FILMS 1839

3. KoÈ ster, U., Physica status solidi (a), 1978, 48, 313.4. Kolosov, V. Yu., The 9th Int. Conf. Crystal Growth,

Sendai, Japan, p. 322, 1989.5. Palatnik, L. S. and Kosevich, V. M., Growth of

Crystals, Vol. 3, Consultants Bureau, New York,1962, 121.

6. Wunderlich, B., Macromolecular Physics. Vol. 1,Crystal Structure, Morphology, Defects. AcademicPress, NY, 1973.

7. Kolosov, V. Yu., Electron microscopy study of imper-fections in crystals growing in amorphous ®lms.Thesis (in Russian) Inst. of Metal Physics Russian Ac.Sci, Ekaterinburg (Sverdlovsk), 1982.

8. Bolotov, I. E., Kolosov, V. Yu. and Kozhyn, A. V.,Physica Status Solidi (a), 1982, 72, 645.

9. Kolosov, V. Yu., Electron Microscopy 90, Proc.XIIth ICEM, San Francisco Press, Seattle, 1990, Vol.1, 574.

10. Kolosov, V. Yu. and Tolstikhina, L. A., Rasplavy (inRussian), N2, 20, English translationÐMelts, July1990, p. 95, 1989.

11. Kolosov, V. Yu., Electron Microscopy 92, Proc. 10thECEM, Vol. 2, Granada, Spain, p. 513, 1992.

12. Lee, W. E. and Lagerlof, K. P. D., J. ElectronMicrosc. Tech., 1985, 2, 247.

13. X-Ray diraction data cards, ASTM, Philadelphia,card no. 13-534.

14. Eades, J. A. and Steeds, J. W., Phys. Bull., 1975, 26,108.

15. Bolotov, I. E. and Kolosov, V. Yu., Physica statussolidi (a), 1982, 69, 85.

16. MoÈ llenstedt, G., Optik, 1953, 10, 72.17. Rang, O., Optik, 1953, 10, 90.18. Rozhankii, V. N. and Berezhkova, G. V., Physica sta-

tus solidi, 1964, 6, 185, (in Russian).19. Shannon, M. D. and Eades, J. A., Phil. Mag., 1979,

40, 125.20. ThoÈ le n, A. R. and Taftù, J., Ultramicroscopy, 1993,

48, 27.21. Delavignette, P. and Vook, P. W., Physica status

solidi, 1962, 3, 648.22. Kolosov, V. Yu., Veretennikov, L. M., Kuzmin, A.

A. and Mamaev, V. A., Electron Microscopy 1994,

Les Editions de Physique, Les Ulis, 1994, Vol. 2a,397.

23. Ahn, C. C. and Krivanek, O. L., EELS Atlas. Gatanand ASU, 1983.

24. Kolosov, V. Yu., Europhysics Conf. Abstracts, 8thGeneral Conference of the Cond. Matter Division,Vol. 12a, European Physics Society, Budapest, 1988,212.

25. Kolosov, V. Yu., unpublished.26. Kolosov, V. Yu. and Kozhyn, A. V., Phys.-Chem.

Studies of Metallurgical Processes, (in Russian), UPI,Sverdlovsk, 1988, N 16, 65.

27. Urusovskaya, A. A., in Modern Crystallography IV,ed. L. A. Shuvalov et al., Nauka, Moscow, 1981, p.47. (See also English translation: ModernCrystallography IV, Physical Properties of Crystals,ed. L. A. Shuvalov et al., Springer, Berlin).

28. Kleman, M., Dislocations in Solids, North Holland,Amsterdam, 1980, Vol. 5, 243.

29. Kolosov, V. Yu., Phys.-Chem. Studies of MetallurgicalProcesses, (in Russian), UPI, Sverdlovsk, 1987, N 15,48.

30. Kolosov, V. Yu., II All-Union Conference onSimulations of Crystal Growth, Abstracts, Vol. 1.,Latv. State University, Riga, p. 154, (in Russian),1987.

31. Lavrentiev, M. A. and Shabat, B. V., Methods ofTheory of Functions of Complex Variables. Nauka,Moscow, 1965, (in Russian).

32. Rothen, F., Pieranski, P., Rivier, N. and Joyet, A.,Eur. J. Phys., 1993, 14, 227.

33. Kolosov, V. Yu., in Aperiodic `94, ed. G. Chaupiusand W. Paciorek. World Scienti®c, Singapore, 1995,p. 126.

34. Rothen, F. and Pieranski, P., Phys. Rev. E, 1996, 53,2828.

35. Mathematical Encyclopedia. Soviet Encyclopedia,Moscow, (in Russian), 1972.

36. Kolosov, V. Yu., Z. Kryst., 1988, 185, 402.37. Kolosov, V. Yu., 7th All-Union Conference on

Crystal Growth, Extended Abstracts, (in Russian),Academy of Science USSR, Moscow, 1988, Vol. 1,13.

1840 KOLOSOV and THOÈ LEÂ N: AMORPHOUS FILMS