x-ray analysis of thin films and multilayers_2.pdf

Rep. Prog. Phys.59 (1996) 1339–1407. Printed in the UK

X-ray analysis of thin films and multilayers

Paul F FewsterPhilips Research Laboratories, Cross Oak Lane, Redhill RH1 5HA, UK

Received 11 June 1996

Abstract

X-ray diffraction is sensitive to thin films of atomic dimensions to thicknesses of manytens of microns, by virtue of the x-ray wavelengths employed and the very high diffraction-space resolutions attainable. X-ray methods are generally non-destructive, in that samplepreparation is not required, and they can provide a very appropriate route to obtain structuralinformation on thin films and multilayers. Analysis can be performed across the wholespectrum of material types from perfect single crystals to amorphous materials. The choiceof the x-ray diffraction analysis procedure depends on the quality of the structural formand therefore this review has been organized to reflect this. Following a description ofthe various material types, some typically important material parameters have been givenarising from the application areas. These material parameters relating to the structure arethen categorized into macroscopic and microscopic properties which can then be furthersubdivided and correlated to the most appropriate analysis method. It is clear from this thatx-ray analysis covers the whole range and from this the recent developments will becomeclear. It is intended therefore that the reader is not compelled to read from beginning toend, but rather be able to find the structural parameter of interest to him/her in a simpleway and from that discover the various approaches to its determination.

0034-4885/96/111339+69$59.50c© 1996 IOP Publishing Ltd 1339

1340 P F Fewster

Contents

Page1. Introduction 13412. The physical properties of thin films 13413. Definition of structural types 13424. Material application 1344

4.1. Electronics 13444.2. Optical materials 13464.3. Organic polymer thin films 13484.4. Magnetic multilayers 13484.5. Multilayer mirrors 13494.6. Protective coatings 13494.7. Active coatings 13494.8. Conducting layers 1350

5. The sensitivities of x-rays to structural properties 13505.1. Macroscopic shape 13505.2. Microscopic shape 13655.3. Macroscopic composition 13705.4. Microscopic composition 13795.5. Macroscopic form 13815.6. Microscopic form 13815.7. Macroscopic orientation 13825.8. Microscopic orientation 13855.9. Macroscopic distortion 13875.10.Microscopic distortion 13925.11.Macroscopic homogeneity 13945.12.Microscopic homogeneity 13945.13.Macroscopic interfaces 13965.14.Microscopic interfaces 13985.15.Macroscopic density 1400

6. Diffraction equipment 14017. The future 1402

References 1403

Analysis of thin films and multilayers 1341

1. Introduction

Much of the technology available today arises from the specific physical properties ofmaterials, for example magnets, semiconductors and abrasive surfaces. Many of thesematerials exist in bulk form but additional physical properties can be accessed by reducingthe dimensions of the material and also by combining many thin layers to create furtherproperties. These possibilities in tailoring the material to create the desired physicalproperties has led to large research activities in many fields. The consequence of this is thatthe material shape, especially film thickness, is a very important property since it influencesthe electronic and magnetic properties, etc, and has necessitated the requirement foraccurate characterization. Various analytical methods have been used; electron microscopy,ellipsometry, surfometers, x-ray scattering and more indirect approaches, for examplephotoluminescence for quantum-well structures, etc. All techniques have a part to playin determining the structural details of thin films and multilayers and these should beconsidered remembering that each method will determine something different. As withall fields of science new methods are continually being developed, not only because of thenew challenges in the understanding of materials, but also because of the availability ofnew x-ray sources, improved understanding in physics and advances in instrument design.This review will endeavour to encompass the advances in analysis of thin films with x-raysacross a wide spectrum of materials, although the methods are not specific to the problemstudied, but to include all developments and an even bias is not possible.

2. The physical properties of thin films

The physical properties of thin films must relate to their structure and it is the structurethat can be studied by x-ray diffraction. The diffraction effect in some cases is very clearwhereas with others it can be very subtle. This whole field of understanding physicalproperties from the interaction of x-rays with matter is a very large field so I will confinethis review to the structural properties. X-rays are electromagnetic with wavelengths closeto the interatomic distances and it is usually considered that the electric-field displacementhas the strongest interaction with matter by being scattered from the electrons. The majorityof scattering is from x-rays scattered by the tightly bound electrons, but the outer electronscan have a measurable effect so that electron-charge studies are possible. Also the magneticmoment of atoms can also be investigated by x-ray scattering because of the magnetic-field displacement of the x-ray beam although this is a very weak interaction. There arealso significant advantages with investigating the scattering with x-ray wavelengths closeto a resonance absorption, for one or more of the atoms, to enhance some of these moresubtle effects or to make the analysis more element specific. This latter point is especiallyrelevant to synchrotron radiation studies. The fundamental equation relating the strengthof scattering (the structure factor amplitude,FS) to the electron densityρ(r) at positionrover all of the sample volume, or because we have a periodic unit of volumeV is givenby:

FS =∫

V

ρ(r) exp(2π iS · r) dV

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where|S| = 2 sinω′/λ (ω′ is half the scattering angle andλ is the x-ray wavelength). Ingeneral we make the approximation that the electron density is concentrated at the atomicsites as discussed above hence this can be approximated to a summation. Since we canmake this a discrete function we can consider the structure factor amplitude as a series ofsummations over the individual atomic sites:

FH =N∑j

fj (S) exp(−2π iH · rj )

where fi is the scattering factor of theith atom in the repeat unit and includes atomicvibration effects, a reduction with scattering angle (ω′ = θ ) and resonance scattering(complex) components.H is the vector form of the Miller indiceshkl.

To prevent this review from becoming an extremely long catalogue of possibilitiesI shall restrict it to the more generally available techniques that can be practised usinglaboratory equipment that is commercially available with the occasional referral to moreexotic techniques during the discussion. In general I will restrict this review to elasticscattering apart from a small deviation into fluorescence techniques when they complement,or are used in conjunction with, scattering methods.

Perhaps at this stage we should define what we mean by a thin film. A thin film to asurface scientist is a few atomic layers whereas a ‘thin’ protective coating could be several100 microns. In general though the definition of a thin layer depends on the physicalproperty of interest, since if the physical property of interest is modified by the reduction inthe thickness then as far as this specific physical property is concerned this could be classifiedas a thin layer. Likewise x-ray diffraction will have its own definition of a thin layer whenit reaches a thickness that cannot distinguish it from the bulk, this could be related to theabsorption depth or extinction length. These factors depend on the material being studied(the density, the elements and the degree of perfection), also the x-ray wavelengths willinfluence both of these factors.

3. Definition of structural types

This section is intended to define the structural types so that it becomes more relevant tomake an appropriate x-ray diffraction experiment. Basically x-ray diffraction experimentsare all similar in that they produce a three-dimensional distribution of intensities that reflectsthe three-dimensional nature of the material under investigation. This leads to diffractionmaxima whose position relate to certain length scales in the sample, its height relates to thescattering power and the relative arrangement of the atoms contributing to this peak andfinally a peak shape that relates to a distribution of length scales or a correlation length, e.g.from finite long-range order. Superimposed on this are the more subtle diffraction effectsthat will be discussed later.

I will define the basic units as far as the x-rays are concerned for completeness, althoughthey are well known.

Atom: x-rays interact most strongly with the tightly bound electrons and weakly withthe electrons defined by the extended wave function. The scattering strength is representedby the scattering factor amplitude that is a function of the scattering angle and wavelength.The scattering factor is derived from quantum mechanical considerations of the electronorbitals and exist in tabulated form, Macgillavry and Riek (1962).

Molecule: this is composed of atoms forming a representative group for the materialand defines the chemical nature of the material.


Phase: is composed of a collection of molecules that can be arranged to form anamorphous or crystalline solid. A phase can be characterized by a series of characteristiclength-scales: if it is amorphous then this could be purely the interatomic distances, whereasin the crystalline form this will be represented by the interatomic plane separations.

Materials can be multiphase, with mixtures of crystalline and amorphous forms leadingto very complex materials for structural studies.

Below I have taken commonly used names and given them definitions since this leads tocommon x-ray analysis techniques. Generally the x-ray technique is defined by the material;here I have attempted to reverse this trend by considering the structural property of interestand then assigning the appropriate method. As with any technique a bewildering arrayof instrument names and methods arise from specific procedures, for example ‘powderdiffractometry’ that really examines polycrystalline material in general (powders being aspecial form of polycrystalline material) yet the same instrument can be used for the analysisof perfect semiconductor materials. ‘High resolution’ suggests something that might bedesirable for certain analyses, but ‘high resolution’ is really high angular resolution whichrefers to its diffraction space performance not its ability to examine small ‘real’ spacedimensions, although it can.‘Perfect’ epitaxial: this is defined as a single extended crystal having perfect registry withthe orientation of the underlayer, which in itself would have to be perfect.Textured epitaxial: in this case the registration of the orientation is close to, but variesfrom, the underlying substrate both within the surface plane and parallel to the surfacenormal. The orientation of mosaic crystal components are determined by the underlayer.The crystals in general extend through the layer depth but will have limited dimensionslaterally.Textured polycrystalline: these crystals are preferentially orientated along a certain direction.The orientation is determined more by growth than the underlayer and the crystals aregenerally smaller than the layer thickness. Also the crystals are generally randomlyorientated parallel to the surface plane.‘Perfect’ polycrystalline: these crystals are randomly orientated to give no preferredalignment. The crystals would almost certainly be smaller than the layer thickness, allof a similar size and isotropic in their dimensions. Polycrystalline powders can be preparedto produce near ‘perfect’ polycrystalline samples.Amorphous extended lattice structures: these materials will have similar strength bondingbetween all atoms in the solid, yet there is no long-range periodicity. The definition of the‘x-ray’ amorphous state is defined as material whose bond lengths are the only correlatedlengths; this gives rise to broad diffraction features characteristic of the distribution of bondlengths in the structure.Aligned molecules: here each molecule behaves rather like a repeat unit by forming weakinterlinking between neighbouring molecules. These molecules are generally long and sothis ordering is dependent on the molecular shape.Random molecules: these materials are essentially amorphous with weak interlinkingbetween molecules and no long-range order.

These definitions in themselves are functions of the analysis method. For examplestructurally a semiconductor multilayer material may be clearly periodic yet its opticalproperties may give the same response as though it is a random alloy structure, etc. Hencewhen relating the results of any two analytical techniques it should be realized that almostalways they are measuring something different. Examination of any x-ray diffraction patternat the highest angular resolution will reveal imperfection in the most ‘perfect’ of materials,yet with a more imperfect instrument ‘probe’ it will appear perfect (Fewster and Andrew

1344 P F Fewster

(1993a)). It is fair to say that x-ray methods are presently at a stage of detecting deviationsfrom perfection in the most perfect material even to the level of point defects, etc. Thismust be borne in mind in all analyses and therefore the frequently used statement in manypublications ‘the material was of very high quality’ only reflects the x-ray method used orthe requirements of the physical property to be studied.

The advancement in the developments in x-ray diffractometer design has changedsignificantly over the last 10 years especially in the area of analysing thin films andmultilayers. There have also been significant advances in the analytical methods. I shallendeavour to keep the arguments general and not too material dependent. Having said thatI will briefly review the link from material applications to structural features accessible tox-ray diffraction methods.

4. Material application

In this section I shall briefly review the application areas of thin films to link the importantparameters that are accessible to x-ray scattering methods. The overlap of the applicationareas is obvious in the sense that each area uses a range of materials and so some cancome under several headings. This aspect is important since the device physicist, polymerchemist, etc, is more interested in the most appropriate method to characterize the structureand relate to some physical or chemical property. Whereas the x-ray diffraction physicistwill choose an appropriate method depending on the definition of material type given above.The definition of the material type defines the technique, not the application area.

4.1. Electronics

Modern electronic devices rely heavily on the modification of the electronic conductivityof semiconductors by impurity doping and energy-band structure modification. Theintroduction of impurities at levels in excess of∼ 1018 cm−3 can create measurable changesin the average lattice parameter over and above that due to changes in stoichiometry orfrom inbuilt stresses during crystal growth (Fewster and Willoughby 1980, Dobsonet al1979). By measuring this difference and armed with a theoretical model of defect sizes,shapes and positions in the crystal lattice it is possible to contribute to the physics ofthese systems (Lesznzski 1993). Concentrations below this are also accessible by studyingthe diffuse scattering or using sensitive quasi-forbidden reflections (Fuijmoto 1984), andrelies on good defect modelling and very careful measurements. Generally a tremendousamount is assumed about structures grown by molecular-beam epitaxy, vapour-phase epitaxyand chemical-beam epitaxy because of the apparent control of layer-by-layer growth;occasionally though too heavy reliance on the growth parameters is assumed. Becauseit has been shown that monolayers of these semiconductor materials have been grown itis often assumed that the growth methods have control at the monolayer level. X-raydiffraction offers a good characterization tool for checking the control of the thickness andcomposition in these structures.

Much of the work in x-ray diffraction of electronic materials has concentrated on defectanalysis by topography, the determination of composition and, to a lesser extent, thicknessmeasurements of semiconductor materials. X-ray diffraction is more than a characterizationtool and has been used very successfully to obtain detailed analysis of dislocation generation,dopant segregation, layer thickness fluctuations, layer tilting and distortion and interfacialextent and roughness, etc. Examples of all these are included in this review.

X-ray diffraction from semiconductors is the ideal case for detailed studies since in high


resolution the scattering is sensitive to the deviation from perfection. Perfection allows theuse of dynamical diffraction theory and therefore the diffraction profiles can be modelledprecisely provided that the correct model of the sample can be given. This diffraction theorytreats the whole process of diffraction as the creation of a wavefield within the crystalsample, and, therefore, as yet, there is no way of deriving the model from the profile, butonly by calculating a diffraction pattern from a proposed structure that matches the observeddiffraction pattern. Although automatic fitting methods have been used to help the analyst.Diffraction from semiconductors is also very strong because of the generally high level ofperfection giving rise to fairly rapid data collection times for basic characterization. Fordetailed analysis the collection times increase considerably.

The details obtainable by undertaking a careful analysis include: strain relaxation inthe layers, dislocation observation, thin-layer tilting, lattice distortion, mosaicity, in-depthstrain variation, interfacial roughness, etc. All this information can help in ensuring the mostfundamental understanding of the structure of thin layers and is not easy or is impossibleto achieve by other analytical techniques. In time these x-ray methods may become moreroutine since the instrumentation to obtain the data has become commercially available inthe last 10 years.

On the processing of integrated circuits there are some very specific requirements forthe electrical interconnections where the sheet resistance needs to be controlled to withina few per cent. This is achievable by controlling the sputtering process by ensuring thatthe aluminium target is planar, has a well-defined orientation and a specific grain size. For1 µm deposited thin films thickness uniformities of better than 1.5% can be achieved. Theorientation distribution in the target and in the deposited thin film can be analysed by x-raymethods although the grain size can be determined by conventional metallurgical techniques,since it is the large dimensions parallel to the surface that most strongly influence the sheetresistance and these can be observed optically.

Extending quantum-size effects from two-dimensional confinement, in lasers forexample, to one or zero dimensions opens up a whole series of possibilities. These arequantum-wire and quantum-dot devices which are often arranged in periodic structuralarrangements on a substrate. These are very amenable to x-ray analysis. These structuresjust use the electric field–electronic interaction, but there are whole new areas of possibilitywith periodic magnetic fields confining the electrons, etc. These new areas of research usedysprosium for the magnetic contact or in some cases highTc superconductors and thesewill create whole new challenges to structural characterization methods.

Many of the devices above rely on single-crystal and epitaxial growth but there aremany devices now that rely on amorphous and polycrystalline structural forms especiallyin the area of liquid crystal display (LCD) drivers. The switch electronics can be addressedby amorphous thin-film transistors or diodes in a matrix array yet the electronics for drivingthe matrix requires higher electron mobilities. This can be achieved by recrystallizing theamorphous Si. The degree of recrystallization, the crystallite size and shape and film porosityall influence the electron mobility and device performance: these structural parameters areall accessible to x-ray analysis.

Insulating layers form another important group of materials for capacitor dielectricsand with the demands for faster device action, capacitor dimensions have had to decreaserequiring very effective insulators. This has led to important applications of ferroelectricmaterials. Another important group of materials for display electronics are the transparentconducting layers, the actual composition and microstructure and thickness can have abearing on their proper function. A commonly used material is polycrystalline indium tinoxide.

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Table 1. Some device properties and the relevant structural properties used in the electronicsindustry.

Device Physical property Structural properties and type

LASER, tunnel diode, Quantum size, barrier height, Thickness, composition,high electron/hole interface extent, defects interfaces, defectsmobility transistors ‘Perfect’ epitaxial

Delta-doped structures Thickness, segregation, Thickness, composition,composition, site occupancy, local structurediffusion ‘Perfect’ epitaxial

Porous silicon Nature of porosity, column/hole size Defect size, average strainoptoelectronic devices ‘Perfect’ epitaxial

Relaxed buffer layers Growth model dislocation Layer tilts, tilt distribution,distribution, surface defects, mosaic block size,unit-cell dimension unit-cell size

Textured epitaxial

Quantum wire and dots Dimensions, shape, composition Lateral period, thickness, shape‘Perfect’ epitaxial

Thin film transistors, diodes Amorphous/crystalline ratio, Amorphous/crystalline ratio, thicknessPolycrystalline Si devices thickness, crystallite shape and size Amorphous extended lattice structure,

textured polycrystallinePolymers and biological Alignment, thickness, chemistry Thickness, periodicity, compositionsensors Aligned molecules/random molecules

Deposition conditions can also influence the stress levels in thin films and multilayers—this can influence the bowing and warping of the substrate and result in poor film adhesion,peeling and cracking. Engineering these structures is now an important part of devicedesign. Also these stress levels can induce hillock formation, etc.

SiC has a high thermal conductivity and is compatible with Si technology and so couldbecome very important for high-power microwave devices. Questions of structural qualityare arising since ultimately, as with all device materials, this will influence the performanceor the property that the device is trying to harness. Ferroelectrics also have a range ofimportant application areas as non-volatile semiconductor memories, etc. The importanceof the structural parameters will gradually become apparent in time.

Polymers are also gradually finding application in electronic devices for conducting thinfilms and light emitters and polyacetylene has been proven to act as a pure semiconductingmaterial in field-effect transistors. The latter has potential in the field of large-areaelectronics. Also at the research stage are biological materials with the possibilities ofvery low-resistance one-dimensional conductors. The characterization of these materialsrelate to the molecular alignment, twisting, long-range order, thickness, etc.

Some of the devices, the physical property of interest and the relevant structural propertyfor x-ray analysis are given in table 1.

4.2. Optical materials

Optical materials cover many types and forms of material including: metals, polymers,glasses, ceramics, semiconductors, composites, diamond, carbides, nitrides and dielectric


multilayers. There is a clear requirement here for establishing structure–propertyrelationships for example surfaces and interfaces, and local chemistry. Changes at the atomiclevel can dramatically alter the complex refractive index and hence the optical response andRisser and Ferris (1992) have related aspects of the optical response to the microstructureusing finite element analysis calculations. It has also been known for some time that theperformance of semiconductor laser diodes is degraded by the presence of dislocations inthe active region. The emission wavelength of quantum-well lasers is very dependent on theactive layer thickness and composition. In multiple quantum-well structures the thicknessesand compositions of the individual barrier layers begin to become important especially if thedevice requires coupling between wells, i.e. a superlattice, since the optical properties relyon the structure of the whole periodic structure. The quality of the interfaces will modifythe energy levels and hence the optical performance. Any defects will create recombinationcentres that may wholly or partly result in non-radiative emissions and dramatically influencethe efficiency of an emitter for example. In dielectric multi-layer structures variations inthe individual layer compositions will alter the performance of anti-reflection coatings andhighly reflective mirrors.

One significant disadvantage of Si has been its indirect band gap, thus competingprocesses always swamp any photon emission. This has given rise to the importance ofGaAs and other direct band-gap structures that are suitable for light emitting diodes andlasers. However, developments in creating quantum size effects in Si by etching has openednew possibilities. The etching produces an array of quantum wires in this porous Si andcould open the opportunities for optical interconnects (removing the cross-talk) and alsoopto-electronic devices integrated with Si technology. The porosity, wire dimensions andstrains are accessible to x-ray diffraction methods.

The optical properties are not only dependent on the precise control in achieving thecomposition and thickness required to satisfy the device design, but also evaluate themicrostructure and stress induced during preparation. For example sputter deposited materialproduces a columnar grain structure, also a porous microstructure can influence the waterabsorption and change the local chemistry, etc. The stress in thin films can result fromdifferences in thermal expansion coefficient when the lattice matching is close at sometemperature value during processing and on cooling this difference increases creating elasticdistortions (stress) in the thin film. Stresses can also exist within the crystallites, which isa function of the crystallite size, homogeneity of the phase, variation in stoichiometry andimpurities. All these properties, stress, homogeneity, impurities, microstructure and filmthickness are very important if the thin layer will be pumped by an intense laser source.

Non-resonant Raman scattering also has an overlap of capabilities with that of x-raydiffraction methods and the relative merits of the two techniques is wholly dependent onhow well the method helps untangle the structure–property relationship. The most relevanttechnique should be chosen on whether it answers the question posed and also whetherthe characterization technique provides a complete understanding of the material properties.Often a combination of techniques is required.

The demand for short wavelength lasers has led to the exploitation of more exoticmaterials including II–VI materials (ZnSe based), GaN and possibly SiC. The quest forthe ‘blue’ or ‘green’ laser will lead to increased information packing for optical discsand lessen the optical tolerances for reading that is necessary for equipment based on theconventional ‘red’ GaAs laser diode. Defects will shorten the operating lifetime and theepitaxial crystalline quality will determine whether the material will lase at all. Alternativeapproaches use frequency doubling methods, much of this work has been based on bulksamples but surface laterally periodic structures of reversed structural polarity provide

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another route to achieve this by interference of the reflected and incident wave. This lateralperiodicity is observable with x-ray topographic methods to image the inverted structurecreated for reversing the polarity.

Materials like diamond are important for transparent windows for x-rays to microwavefrequencies. These transparent thin films (∼ 25 µm) are created by removing the substrateafter growth. The crystallite size should be controlled to be below 1000A to reduce lightscattering effects, this is just the range for x-ray crystallite size measurements.

4.3. Organic polymer thin films

Organic polymer thin films cover a wide range of applications from biological and chemicalsensors, corrosion- and wear-resistant coatings, adhesives and by altering the functionalgroups they can be used for lubricants, plasticizers, anti-oxidant coatings, etc. Thestructurally important parameters mainly relate to conformational arrangements and thecomposition of these functional groups. These parameters are influenced by heating,chemical attack and through mechanical distortions. Other parameters include molecularorientation, porosity, crystallinity and interfacial roughness, depending on the application.The film thicknesses involve depends strongly on the application and can range from 100µmfor dielectrics in microelectronic applications to< 100 nm for low friction coatings formagnetic discs.

Biotechnological interest in thin films ranges from determining the adsorption of proteinmolecules onto polymer implants, that can promote clotting, to sensing protein adsorption.These thin films are amenable to x-ray scattering studies. Langmuir–Blodgett films havea range of applications from biotechnology (membranes and biosensors), gas sensors toelectronic devices and lithography. Chemical and biological sensors are deposited on piezo-electric substrates and corrosion- and wear-resistant layers rely on self-assembled moleculesby virtue of their dense stable structure. The structurally important parameters in these thinfilms include lateral order (crystallinity) the arrangement of the head and tail groups, etc.The latter will relate to the variation of density as a function of depth, which is accessibleto x-ray methods.

Another important application area of polymers is for adhesive films. These depend ontheir orientation (texture), crystallinity and surface roughness, etc. The latter will obviouslyinfluence the strength of the adhesion whereas the former properties relate to the toughnessof the adhesive. Again estimates of these parameters are available through x-ray scatteringmethods.

4.4. Magnetic multilayers

The demand for high-density recording systems presents a challenge to the physicist,material scientist and for structural analysis techniques. There are two main areas; magneto-optical recording and magneto-resistive random access memories. The magneto-opticalrecording relies on the Faraday effect and reading relies on the Kerr effect and the limit onthe packing density of the information depends on the diffraction limit of the wavelengthused and the structural imperfections that limit the optical depolarization and hence theread out capability. Reducing the wavelength resolves the diffraction limit but requires ahigh magneto-optic effect at this wavelength, a preferred magnetization normal to the film, aCurie temperature of a few hundred degrees Celsius and a ‘good’ microstructure. Multilayerstructures of Cu/Ni and Co/Pt can satisfy these requirements. X-ray diffraction is a valuabletool here for characterizing the microstructure and providing a feedback for the thicknesses


obtained from growth.The discovery of the giant magneto resistive effect in ferromagnetic/non-ferromagnetic

superlattices offers a route to small bit size random access memories. The strength of theantiferromagnetic coupling between the layers depends on the spacer thickness and can bedisrupted by significant interface roughness.

4.5. Multilayer mirrors

Monochromators of hard radiation, extreme ultra-violet (15–100 eV) and x-rays (100 eV–3 keV) covering the 800–4A wavelength range have important applications in astronomy,x-ray fluorescence and x-ray photoelectron spectroscopy studies at synchrotrons. The basicstructures consist of alternating high- and low-density materials to enhance the reflectivityand by varying the incidence angle of the incoming beam the appropriate wavelength can beselected. The individual thicknesses of the layers, the periodicity and the interface qualityall influence the reflectivity at a given wavelength.

4.6. Protective coatings

There is a demand for corrosion-resistant coatings for metallic layers in corrosiveatmospheres at elevated temperatures, especially in the energy industries when thetemperatures can be as high as 600◦C in a low-oxygen high-sulphur atmosphere. Thisis a significant challenge and requires understanding of the chemistry (composition—thisgives the chemical barrier), the micro-structure (crystallite size, porosity and stress—thisrelates to the penetration of corrosive chemicals and the adhesion of the layer).

The use of multiple metallic thin films can create new physical properties over those ofbulk samples. Metallic multilayers have been used to create structures that have a hardnesscoefficient in excess of the individual layers in bulk form and this hardness depends onthe wavelength of the multilayer periodicity Shinnet al (1992). Yanget al (1977) hasalso observed an enhanced elastic modulus for Au/Ni and Cu/Pd multilayers. The realunderstanding of these phenomena is unknown but do depend strongly on the parametersthat are easily accessible to x-ray diffraction techniques.

Abrasive or very hard wearing coatings have been produced by creating thinpolycrystalline diamond films, cubic boron nitride andβ-SiC. The crystalline size, filmdensity and film thickness all affect the wear properties. The adhesion of the film is clearlyvery important but the microstructure will determine the performance of the film. Theobvious microstructural properties are the crystallite size and distribution and the orientationdistribution can influence the effectiveness of the wear.

4.7. Active coatings

Thin films of zinc oxide deposited with a specific crystallite orientation can exhibit peizo-electric properties, other suitable materials are aluminium nitride and Pb(Zr, Ti)O3 (PZT).The layers are in the micron range in thickness and have applications as acoustic wavedevices, crystal resonators and Lamb filters (Lamb waves are at least an order of magnitudeslower than surface acoustic wave devices) and will be important for non-volatile RAMand microswitches. These ferroelectrics also allow a higher degree of miniaturization andintegration with Si technology. These layers are usually sputtered, often by magnetronsputtering, and adjusting the growth conditions to obtain the required orientation is animportant parameter. X-ray diffraction is ideally suited to determining the preferred

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orientation. Other structural parameters of importance are the crystallite size, strain anddislocation density. The domain formation and therefore the domain switching in these thinlayers is heavily influenced by the grain size, which should be> 1 µm (Tuttle et al 1994).All ferro-electric materials are pyro-electric and therefore thin-film detectors can be made,but again the orientation dependence of the film will have a bearing on the response.

4.8. Conducting layers

The general importance of conducting layers for interconnecting devices in electroniccircuitry has led to a demand for an improved understanding of deposition of thin metalliclayers. Clearly to obtain good conductivity the chemistry has to be well understood:oxidation states, etc, as well as other microstructural parameters that can influence theconductivity, porosity or film coverage and crystallite size. These are all accessibleparameters to x-ray analysis. HighTc superconductors have also been considered forinterconnects and switching elements, based on the Josephson effect, and because oftheir chemical complexity growth of the correct phase requires diffraction methods forconfirmation as well as a detailed understanding of the microstructure. A common substratefor these superconducting thin layers is strontium titanate (SrTiO3) because of the smalldifference in lattice parameter of this cubic substrate and those of the basal plane ofthe YBaCuO unit cell. The small differences of the two lattice parameters in the basalplane also results in two possible orientations of the YBaCuO unit cell, the proportion ofthese orientation ‘twins’ can be determined by x-ray diffraction. HighTc superconductormultilayers also have interesting physical properties where of course the whole structuralquality is important.

5. The sensitivities of x-rays to structural properties

We have defined the basic structural types and these are crucial to deciding on the appropriatex-ray analysis method. I have then tried to relate the materials within various applicationareas to the type of parameters of importance and in this section I shall discuss the individualparameters in turn and relate them to the various methods available. These methods dependon the structural type described earlier. The structural properties have also been separatedinto macroscopic and microscopic properties. Macroscopic properties I define as beingrelated to the sample as a whole, for example external dimensions and layer thicknesses,whereas microscopic properties are concerned with crystallite size dimensions, defects, etc.This categorization has been listed in table 2.

5.1. Macroscopic shape

The layer thickness is the fundamental parameter for thin layers since this is the property thateither creates the desired physical property on its own or in combination with other layersand the underlying substrate. The first obvious effect on the diffraction pattern is the changein the magnitude of diffracted intensity with thickness; the thinner the layer the lower thescattering volume. This is the only general statement for the full range of material types.This effect has been used with reasonable success on a limited range of material types by anumber of authors and the most appropriate method depends on the material type since thisdefines the simplicity or complexity of the diffraction model to apply. Another feature ofscattering from thin layers is that the scattering will exhibit interference fringes whose periodis related to the layer thicknesses; these interference fringes rely on a constant thickness


Table 2. Structural properties accessible to x-ray methods.

Type of property General property Specific property

Macroscopic Shape Layer thicknessesComposition Structural phase

Elements presentPhase extent

Form AmorphousPolycrystallineSingle crystal

Orientation General preferred textureLayer tilt

Distortion Layer strain tensorWarping

Homogeneity Between analysed regionsInterfaces Interface spreadingDensity Porosity

Coverage

Microscopic Shape Average crystallite sizeCrystallite size distribution

Composition Local chemistryOrientation Crystallite tilt distributionDistortion Crystallite lattice strain

Crystallite strain distributionDislocation strain fieldsPoint defectsCracksStrain from precipitates

Interfaces RoughnessHomogeneity Distribution within region of sample studied

over the diffracting volume of the sample. For ‘perfect’ epitaxial and reasonable qualitytextured epitaxial material the diffraction profile can be seen to broaden with decreasinglayer thickness and also these interference fringes can be seen. Hence by modelling thestrength of the scattering, the broadening and the fringing an accurate measurement ofthickness can be achieved.

Clearly since the thickness of a layer is defined as a change in composition as afunction of depth, the link between thickness and composition is not always distinct andtherefore many of the methods described below will yield both or require knowledge of thecomposition.

5.1.1. Scattering strength method of thickness determination.The fundamental equationapplied to this method relates the linear absorption coefficientµi for material of thicknessti to the intensity ratioIi/Io of the diffracted intensity from the material compared with thatfrom an infinitely thick sample of the same material:

Ii

Io

= 1 − exp

(−µiti

[1

sinω+ 1

sin(2θ − ω)

])where 2θ is the scattering angle andω the angle subtended by the incident beam on thesample. This equation is valid for the polycrystalline, amorphous and molecular structuresdescribed above provided that the long-range order or the perfect crystalline regions within

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these materials is less than the extinction length. The extinction length is a dynamicaldiffraction effect that is defined as the depth at which the energy flow of the transmittedbeam swaps into the diffracted beam and represents the coherently diffracting depth:

ξ = πV

reCλ|FH | [| sinω| sin(2θ − ω)]1/2

whereFH is the structure factor andC is the polarization factor (1 or cos 2θ for the twopossible polarization states),V is the volume of the unit cell,re is the electron radius andλ is the x-ray wavelength. Layers that are thicker than their extinction depth cannot usethis simple intensity ratio because the incident beam has been diminished and the diffractedintensity can no longer increase significantly with increased material thickness.

In the most simple case when the material is amorphous then the extinction lengthis infinite but the scattering is weak; so in general a long experiment would have to beperformed to capture sufficiently reliable data. For polycrystalline material the importantparameter is the size of the crystallites and these have to be less than the extinction length,unless the crystallites have significant defect densities that reduces the coherently diffractingdimensions. Another aspect that is important in determining the intensity of the scattering issecondary extinction; this arises from crystallites diffracting the incident beam thus reducingthe flux available to crystallites deeper into the sample that are similarly orientated. Thethickness can be determined by several methods based solely on the scattering strength.

(i) The layer can be determined by measuring the scattered intensity from the layerwith that from an equivalent bulk sample, Brandt and van der Vliet (1986). Thelinear absorption coefficients for various materials are derived from the mass absorptioncoefficients, Macgilavry and Riek (1962), and the density of the material. So an estimate ofthe structural form is required and knowledge of the elements and their relative proportionsis necessary, unless the material of interest exists in known thicknesses to create a calibrationof the intensity ratio versus thickness. This approach is most suitable for polycrystallinelayers on non-crystalline substrates provided that the layer and the bulk reference materialare both similar structurally, i.e. ideally ‘perfect’ polycrystalline or predictable texturedpolycrystalline samples. If the layer is amorphous as well as the substrate it is unlikely thatthe scattering from the layer can be separated and x-ray reflectometry methods should beused, this is discussed later. If on the other hand the substrate is crystalline and the layer isamorphous then method (ii) would be preferred. The above equation can be simplified foramorphous and polycrystalline samples when used with conventional symmetrical diffractiongeometry, that is the diffraction planes are parallel to the sample surface:

Ilayer = Ibulk

(1 − exp

(−2µlayertlayer

sinθlayer

))whereµ is the linear absorption coefficient andtlayer is the thickness of the layer,Ilayer isthe intensity scattered from a reflection from the layer andIbulk is the intensity scatteredfrom the same reflection from a bulk sample of the same material as the layer,θ is theBragg angle.

(ii) This method is similar but relies on the comparison of the scattering from thesubstrate with and without a thin layer (i.e. layer absorption). The appropriate equation forsymmetric diffraction geometry is given by:

Isubstrate= Isubstrate with no layerexp

(−2µlayertlayer

sinθsubstrate

).

This method has greatest precision for small thicknesses compared with monitoring thelayer diffraction which is best for larger thicknesses. The absolute thicknesses for these


different ranges is a function of the absorption coefficients for the x-radiation used. Butin general these methods are suitable for thicknesses of the order of microns. Kolerovetal (1984) have used this idea by monitoring the reduction of the substrate diffraction untilit disappeared for determining the thickness of metal foils mounted on a polycrystallinesubstrate. The diffracted intensity of the substrate is reduced by decreasing the angle ofincidence until the substrate reflection is unobservable. The precision of this method is8–20% and is suitable for amorphous layer materials on ‘perfect’ polycrystalline substrates.

(iii) If both layer and substrate are crystalline then the precision for intermediatethickness measurement can be used very effectively by combining the two methods; thishas been used successfully by Anderson and Thomson (1989). The appropriate equationthen becomes:

Ilayer

Isubstrate= Ibulk layer[1 − exp(−2µlayertlayer/ sinθlayer)]

Ibulk substrateexp(−2µlayertlayer/ sinθsubstrate).

To determine the thickness from this expression Anderson and Thomson (1989) used aniterative approach and have derived a series of curves for the variance:thickness ratio asa function thickness for nickel and zirconium oxides for all three methods. Clearly as theIlayer/Isubstrateratio changes to the extremes, i.e. when the layer is very thin or very thick,all three methods start to become quite unreliable.

The importance of having both reference bulk and layer materials of the same structuralform for method (i), will become more apparent later with the discussion on orientation.Unfortunately the likelihood of having randomly orientated crystallites in bulk materialsis rare and hence thin-layer polycrystalline samples with a natural macroscopic anisotropywill almost certainly create some orientation texture, resulting in a non-random intensitydistribution. Method (ii) from a materials point of view, may well be more reliable sincethe same reflection from the substrate can be measured before and after deposition of thelayer, and does not depend on the crystal form of the layer. The layer may well have apreferred orientation at the early nucleation stage before the growth establishes, thus thelayer may not be homogeneous.

For ‘perfect’ and textured epitaxial layers, that are homogeneous in depth, all thesemethods can be applied in principle since all the intensity from the layer and the substratereflections can be measured. Chandhuri and Shah (1991) have used this approach butthe problems of extinction become very important since, in general, the crystal quality isgood and dynamical effects apply. Chandhuri and Shah (1991), however, worked fromthe kinematic theory and have modified the equations to include primary and secondaryextinction effects so that the theoretical intensity becomes:

Isubstrate

Io

=(

e2

mc2

)f (A)|Fsubstrate|2λ3

sinωVsubstrate

1 + cos2 2θ

2 sin 2θ

exp(−µlayertlayer

[1

sinω+ 1

sin(2θ−ω)

])µsubstrate

[1

sinω+ 1

sin(2θ−ω)

]wheref (A) is the influence of the primary extinction. Primary extinction is a dynamicaleffect that results in a reduction of the intensity of the incident beam through diffractionwithin a distinct crystalline region.

f (A) = tanhA + | cos 2θ | tanh|A cos 2θ |A(1 + cos2 2θ)

A =(

e2

mc2

) |F |λtcryst

V sinθ

1 + cos 2θ

2

wheretcryst is the thickness of the individual crystallites or mosaic blocks within the layeror substrate. If the layer is thin thentcryst cannot be large and the primary extinction will be

1354 P F Fewster

small, although it could be very significant for substrate material. Also the linear absorptioncoefficient,µ, in this case is modified by secondary-extinction effects. Secondary extinctionis the reduction of the incident-beam intensity available to crystallites due to diffraction fromthose above. The modified absorption coefficient is:

µ = µi + 1√2πη

exp

(− 12

2η2

) ∣∣∣∣ e2Fi

mc2V

∣∣∣∣2

λ3 1 + cos2 2θ

2 sin 2θ

where1 represents the angular deviation of the crystallite block from the mean orientationandη is the standard deviation of the crystallite orientation, Zachariasen (1945). The indexi

refers to the substrate or layer and in this particular case assumes that the orientation ofthe crystallites takes on an error-function distribution. Therefore, it is clear to see thatsecondary extinction is negligible for weak reflections (i.e.F is small) as in the case ofthin layers, but this correction must be made for reasonable-quality single-crystal substratematerial.

Chandhuri and Shah (1991) have used these equations to determine the thicknessesof AlGaAs semiconductor thin layers on GaAs substrates by estimating the mosaic-blockorientation spread from the rocking curve width and measuring the integrated intensityfrom the substrate. The thicknesses derived from this method are within about 5% of thoseobtained by reflection high energy electron diffraction (RHEED). However, this study hasindicated the limit of kinematic diffraction modelling, since without correction for extinctionthe errors in the thickness determination increase to 130%! However weak are reflections(small F ) or short wavelengths (smallλ) the errors decrease by reducing the extinctioneffects, however, the measurement becomes more difficult. Ideally dynamical diffractionshould be used as discussed later.

5.1.2. Thickness measurement based on interference fringes.This method no longer relieson the uncertainties of measuring the scattered intensity from one of the components butmakes use of the interference of the diffracted beam from the changes in x-ray refractiveindex. This method can be used in any material form, crystalline or amorphous but requiresa flat sample over the region studied. There are two general approaches to this type ofanalysis; reflectometry and ‘high’ angle diffraction. Basically reflectometry is concernedwith diffraction close to the 000 maximum (i.e. the incident beam direction) whereas ‘high’angle diffraction is concerned with diffraction close to all other accessiblehkl reflections.The existence ofhkl reflections is dependent on a regular lattice periodicity whereas the 000maximum is always present hence diffraction features close to the latter are independent ofthe crystal form.

The strong interference between the different boundaries of refractive index give rise tofringing, figure 1, from which the thickness can be estimated using

t = (i − j)λ

2(sinω′i − ω′

j )

where i and j are the fringe orders andω′ is half the scattering angle. This is simplyderived from Bragg’s law and was first used for thickness determination with reflectometryby Keissing (1931), in a more appropriate form for small scattering angles and by Bartelsand Nijman (1979) for ‘high’ angle diffraction studies. However, the observation of thesefringes depend on the smearing effect of the diffractometer probe, the wavelength used,the bowing of the sample and the thickness of the layer. Clearly if the thickness variesover the studied region this will also smear the fringes quite considerably and this ismore pronounced at small angles close to the 000 reflection when the projected beam


Figure 1. The fringing arising from a thin 0.5 µm layer of AlAs on GaAs calculated usingdynamical-diffraction theory. Note also the width of the ‘layer’ peak compared with that of thesubstrate.

size on the sample is large. The bowing effect of a sample effectively requires a differentangle of incidence to satisfy the scattering conditions for different positions on the sample,although this can be overcome by triple-crystal methods in reflectometry Cowley and Ryan(1986) and in diffractometry Fewster (1989a). The presence of a bowed sample makesno difference to the scattering angle (2ω′) but rather the angle of incidence (ω) hence bycollecting data at several 2ω′ angles with a range ofω settings the influence of the variableincidence angle is eliminated. The probe of a triple axis diffractometer allows this separation.Cowley and Ryan (1986) used a standard triple-crystal diffractometer which requires closematching of the first and third crystal with that of the sample to obtain good resolution:and they were able to analyse the oxide on a silicon wafer to determine the thickness androughening of the interfaces. Fewster showed with a versatile multiple-crystal multiple-reflection diffractometer that the fringe visibility can be recovered for allhkl reflectionsand for reflectometry. The requirement for changing the crystal optics depending on theproblem no longer arises.

Macrander et al (1988a) considered a method to make the analysis of thicknessmeasurement from the fringe spacing more automatic by directly Fourier transforming thediffraction pattern. The original intensity-angle curve is converted into a Fourier transform-thickness curve that contains a series of peaks corresponding to the thickness within thestructure. The method is not without problems however, since for structures with morethan three layers the combinations become too complex to interpret. Basically, all possiblecombinations of thickness can arise taking away the advantage of the method. Armin andHalliwell (1991) have extended the work of Macranderet al (1988a) to extract thicknessvalues from the Fourier transform of the diffraction curves. In this case they recommendtaking the first differential of the intensity-angle curve before transformation to enhance

1356 P F Fewster

the Fourier components to clarify the important length scales in the structure. However, itcannot be extended beyond a three-layer structure. A similar approach has been applied toreflectometry profiles by Bridou and Pardo (1994), who make use of the extracted data asa starting model for their subsequent trial-and-error modelling.

The major drawback with the simple relationship of Bartels and Nijman (1979) andFourier transformation methods of Macranderet al (1988a), Armin and Halliwell (1991)has been illustrated by Fewster (1993a) and that is that these are only an approximationthat can give large differences in determined thicknesses extracted from a calculated patternbased on dynamical theory. These methods use simple conceptual understandings thatcannot take into effect the complex dynamical diffraction effects. Therefore, these methodsshould only be used as a very first approximation to a subsequent simulation.

As with all methods the temptation is to find rapid analysis methods by extractingreal information directly from the reflectometry or diffraction pattern, as described abovebut problems of complex diffraction effects as described by Fewster (1993a) lead touncertainties. The limitations of these methods also arise from the fringe spacing resolution,i.e. the observation of distinct oscillations, that is dependent on the coherence and the layerthickness fluctuations over the probed area.

5.1.3. Thickness measurement by fringe spacing and scattering strength.The measurementsof thickness in multilayers can be approached in a very different manner, since the multilayerintroduces an extra periodicity in the structure. The periodicity produces satellite intensities,figure 2, and, using the relationship given above, the wavelength of the multi-layer is givenby:

3 = (i − j)λ

2(sinω′i − sinω′

j )

where i and j are the satellite orders andω′ is half the scattering angle for that satellite,Fewster (1987). The period can be determined to high accuracy, to within anangstrom for

Figure 2. The theoretical diffraction profile from a{[AlAs] 50 A [GaAs] 50 A}× 50 multilayeron (001) GaAs, CuKα radiation and 002 reflection.


periods close to 100A whereas below this the accuracy improves still further, since it isdetermined by the relative separation of the satellites. However, the simple angular spacingcan be subject to errors of several per cent due to complex diffraction effects, Fewster(1996a). If the average composition,xave of the multilayer is known then we have twobasic equations for a two-layer repeat structure:

3 = D1 + D2

and

D2x2 + D1x1 = xave3

for a simple structure with two layers of the formAxB1−xC with x1 and x2 being thecompositions in layer 1 and layer 2. Hence combining these we have:

D2 = (xave − x1)3

(x2 − x1).

Hence, if we know the individual compositions and the ‘average’ composition,xave we canvery easily determine the thicknessD2 and D1. Now for a multilayer composed of layerthicknessesD1 and D2 below a thickness of∼ 400 A, we can consider the multilayer todiffract as a structure with an average compositionxave with a perturbation giving rise tothe satellite peaks. For thicknesses much above these values the concept of an average peakno longer exists, Fewster (1993a, 1996a). The determination of the ‘average’ compositionis explained in more detail in the following section, but at this stage we can consider it tobe derived from the measured lattice parameter. Clearly if the compositions are unknownthen we have too many variables and we must include extra information.

Additional information is contained in the satellite intensities. The intensity of thesatellite peaks in the kinematical theory approximation is given by:

IH = |FH |2 =∣∣∣∣ N∑

j

fj exp(−2π iH · rj )

∣∣∣∣2

where H represents the Miller indices of the reflection of interest andrj is the atomiccoordinate of thej th atom in the unit-cell repeat that has a scattering factorfj . The unitcell in this case can be considered as the whole repeat unit of the compounds composingthe multilayer structure. Provided that the reflectivity of the satellite intensities are lessthan ∼ 7% then the kinematic approximation is applicable. The intensity calculated withthis formula is based on an infinite structure and is used on a relative basis, hence a scalefactor needs to be applied during the fitting. Also since the satellites exist over a significantrange of scattering-angle geometrical factors (the length of time the diffraction conditionis satisfied—Lorentz factor) and physical parameters (modification of the polarization stateand variations in the absorption) will modify the intensity for comparison. The measuredintensity should therefore be divided by:

X = 1 + cos 2θm cos2 2θ

sin 2θ(1 + cos2 2θ)

(1 − exp

−2µt

sinθ

)5 exp

−2µiti

sinθ.

The first term relates to the Lorentz and polarization factors, withθ being the Bragg anglefor the sample andθm that for the monochromator or analyser crystal. The second termaccounts for the absorption in the multilayer that has a linear absorption coefficientµ, andthicknesst , and the third term is the product of the absorption terms for the layers abovethe multilayer.

The intensity obtained is the integrated intensity. Therefore, combining theserelationships betweenDi , xi and 3, etc, to reduce the number of independent variables

1358 P F Fewster

and matching the measured integrated intensity of the satellites with those calculated, theunknown parameters can be extracted. Kervarecet al (1984) have applied this method tothe analysis of an AlxGa1−xAs/GaAs multilayer structure to extract the thicknessesD1 andD2 and the composition,x. They obtained two possible solutions, given in monolayers,that straddled the true values.

Since these are real materials they do not exist as idealized structures but will haveinterfacial mixing of the components. These satellites are the Fourier coefficients of thisperiodic function and the interface shape influences the high-order coefficients or satelliteintensities, i.e. those most remote from the ‘average composition’ peak. This will introducetwo more parameters, providing the interface shape is assumed to follow a well-knownfunction. However, if the grading is a small proportion of the overall thickness then theeffect can be ignored. What is far more important though is the fact that the periodicityof the multilayer may not coincide, and is very unlikely to do so, with an integer numberof the ‘average’ unit-cell periodicity. Likewise this can happen for the individual layerthicknesses comprising the multilayer. The structure will then be incommensurate and thismust be taken into account, Fewster (1986, 1987). This incommensurate nature of mostmultilayer structures can be accommodated by adding the amplitudes of the two diffractionpatterns that straddle the correctD1 : D2 ratio for each period. Also the amplitude of thisdiffraction pattern should be added to that of the similar profile for the alternative nearestcommensurate period. The various combinations should be added according to the ratio ofthe difference from the measured period value. This incommensurate nature must be takeninto account if accurate evaluation of the intensities is required for measuring interfaces,etc. In fact it is always far better to include a best estimate of the interface intermixing fordetermining individual thicknesses.

As with methods in reflectivity the Fourier transform approach has been used forextracting layer thicknesses from periodic multilayer structures. This was first performedby DuMond and Youtz (1940) using a gold–copper multilayer where they modelled thecomposition profile to obtain the interdiffusion coefficient. For Bragg reflections thecomposition modulation is also accompanied by a strain modulation and this gives riseto an added complexity in the satellite intensities and must be modelled as well. Fleminget al (1980) have used this method to determine the thicknesses of AlAs/GaAs periodicmultilayer structures using the approach suggested by Guinier (1963), where the intensityof the satellites is given by:

FH±m

FH≈ am

2

[(H

d± m

3

)3ε ∓ η

].

The scattering factor between successive layers then oscillates betweenf (1+η) andf (1−η)

and the interplanar spacing byd(l +ε) andd(1−ε). am is themth order Fourier coefficientor the amplitude of the satellite reflection arising from the multilayer periodicity3 andHis the reflection order for the ‘average’ lattice periodicity.

Clearly since we are unable to measure the structure factor amplitudes, only the intensityratio is obtained, the phase information of the complex Fourier coefficients is lost. However,if the problem is simplified by including knowledge of the expected thicknesses andassuming the interface between the two components to be equal then the phases are simply 0or π and only the cosine components of the Fourier expansion exist andam can be obtained.Having obtained the Fourier coefficients the composition profile can be determined.

In general the results are very close to those obtained by modelling and fitting asdiscussed above, but problems can arise when there are very few satellites measured.This becomes obvious at the limit when only the first-order satellites are measurable then


this approach will result in a sine wave, i.e. the Fourier transform approach in the limitwill produce equal layer thicknesses. Fewster (1988) has compared the two methods andhas shown that for straightforward thickness measurement, provided that several orders ofsatellites are observed then the agreement is good. The situation for investigating interfaces,however, can cause problems, see later.

5.1.4. Thickness measurement by profile simulation.It is clear from the above that themore direct measurements of the scattering strength and the fringe periodicity can lead toserious errors in analysis, whereas, combining fringe periodicity and integrated intensityfor periodic multilayers produces a much improved estimate, but again it is limited to verydefinite applications. A good theoretical model of the diffraction process, however, canyield a far more reliable value of thickness, but leads to an iterative process. Basicallythere are three main diffraction models; dynamical, kinematical and optical theories. Thesecan be categorized into ranges of applicability. The dynamical theory is the most exactingand is a direct derivation from Maxwell’s equations, there are many variants from this thatgive differences in detail, but more significantly can accommodate multilayer structures,crystal distortions and defects. This is the only model that correctly calculates the observedscattering from thick near-perfect materials. The kinematical theory is an approximationof the dynamical theory and is applicable to weak scattering events whereas the opticaltheory has been used very successfully for reflectometry profile calculations when the latticeperiodicity effects are remote from the observed region. Relating the diffraction processto the structural feature is much more transparent in the latter theories since the aspectsof extinction and the concept of wavefield generation are excluded, although extinctioncorrections as mentioned earlier can be applied but can never be exact. There are modelsof semi-kinematical theory that have some dynamical aspects included: but Shufan andZhenhong (1990) have found that the semi-kinematical model of Tapfer and Ploog (1986)is inadequate for structures of any complexity (e.g. a three-layer structure) when comparedwith the full dynamical theory. However, qualitative results can be obtained from theformer.

Very simply we could categorize dynamical theory to have the widest applicationin semiconductor diffraction, kinematical theory for polycrystalline diffraction and theoptical theory for reflectometry. Since in this section we are concerned with themeasurement of thickness we are in general considering high-quality material that can usedynamical diffraction and polycrystalline and amorphous materials that are best analysedby reflectometry with optical theory.

The basic equations of dynamical theory relate the electric-field displacements as afunction depth into the sample. Although there are many slightly varying expressions, themost useful for our purpose are those proposed by Takagi (1962, 1969) and presented alsoby Taupin (1964):

iλγH

π

∂DH

∂z= 90DH + C9HD0 − αH (ω)DH

and

iλγ0

π

∂D0

∂z= 90D0 + C9HDH

where γH is the direction cosine for the wave with an electric-field displacementDH

and Miller index H . z is the spatial coordinate normal to the crystal surface andC isa polarization term that is either 1 or cos 2θ for theσ or π states. The symbol9 is related

1360 P F Fewster

to the structure factor amplitude by:

9H = −λ2re

πVFH

where these parameters have their normal meaning. We can see from these equations that theelectric-field displacements for the incident or refracted beamD0 is coupled toDH and thesolution depends on the structure factor amplitudes of the 000,hkl and−h−k−l reflections.Because the final diffraction profile is dependent on the wavefield generation within thecrystal we cannot simply work from the profile and arrive at the structure factor amplitudevariation with depth. Also the deviation parameter (α(ω) ≈ 2 sin[(ω − ω0)/2] sin 2θto a good approximation, Fewster, (1996a) contains strain information and lattice planeorientation effects:ω is the diffracted-beam direction with respect to the crystal surface andω0 refers to the Bragg condition.

This dynamical model now becomes a very powerful tool since we can extend ourqualitative approach to the interpretation of a diffraction profile and change our model andfit the diffraction profile exactly. Using dynamical theory Fewster (1993a) has shown thatin fact a simple qualitative approach for the simplest single thin-layer structures can giveerrors in the determination of thicknesses from fringe spacing. An attractive aspect of themodelling with dynamical theory is that the intensity of the layer scattering can be relatedto the underlying substrate and in combination with the shape and fringe spacing a veryexact value can be obtained. A few examples are illustrated in figure 3.

Numerous authors use these methods and dynamical diffraction software is commerciallyavailable. Because of the power of these methods they have been used for a variety ofcomplex structures especially to model superlattice and multiple quantum-well structures.Since this readily accommodates the incommensurability discussed here, and, above, thevarious thicknesses can be obtained relatively quickly with practice.

As shown in figure 3 the measurement of thin layers is perfectly possible, in factthin buried layers can be observed to show very large diffraction effects by creating aninterference pattern between the thicker surrounding regions. There does not have to be anindividual layer peak. However, the sensitivity of this interference permits the investigationof very thin (submonolayer) layer structures. This is very well illustrated by the work ofmany authors (Wieet al 1989, Ferrariet al 1994, Fewster 1986, 1989b, Chu and Tanner1986). One problem though is that the thickness is strongly correlated to the compositionand cannot easily be resolved from a single-diffraction profile, especially for small mismatchvalues as in the AlGaAs system. Wieet al (1989) have given a relationship for this,∑ 1d

di

ti ∼ constant for a given structure

for thin layers of a few 1000A. The composition is derived from the mismatch and thereforethis correlation is very evident. Also Tapfer and Ploog (1989) have shown that the fringeintensity is proportional to the lattice mismatch for small mismatches (1d/d < 1%).

However, this problem can be solved by measuring the profiles of several reflectionsto separate these two parameters for large mismatch structures. This has been reported byFerrari et al (1994) and Bocchi and Ferrari (1995) for a 1A InAs layer buried in GaAs.The two reflections chosen were the 004 and 224 reflections, in grazing exit geometry, onan 001 substrate and layer, with CuKα radiation. The 004 reflection was used to derivethe (1d/d)t ratio and this appears to follow the proportionality relationship given above,whereas this relationship breaks down for the 224 reflection and positive mismatches. Inthis very special case Ferrariet al (1994) found that the fringe intensity is almost entirelydetermined by the thickness.


(b)

(a)

Figure 3. (a) The diffraction profile from a single 23A layer of AlAs on a GaAs substrateincluding the calculated profile and (b) the profile from a complex laser structure showing howthe individual thicknesses, composition and In segregation could be obtained by careful matchingof the simulated-to-experimental profile.

Gianniniet al (1993) have tackled this problem, but have had to resort to a synchrotronby combining rocking-curve analysis as above with standing-wave analysis. The presenceof a periodic set of atomic planes generates a standing wave in the crystal that has nodes andantinodes that can be interchanged when the crystal is rocked from one side of the intrinsicdiffraction profile to the other. The subsequent fluorescence or photoemission when theantinode corresponds to a specific atomic plane will indicate the local composition andresolve the thickness–composition correlation.

The analysis of multilayers formed withδ-doped layers periodically deposited in asemiconductor matrix can create some very impressive diffraction effects. Aδ-doped layercomprises of an atomic or a subatomic layer of the dopant atoms. The periodicity givesrise to satellite peaks as described previously because of the strain and scattering factor

1362 P F Fewster

modulation. Gillespieet al (1993) have measured the individual layer thicknesses ofSi δ-doped layers in a GaAs matrix. With Si layer thicknesses of a fewangstroms thesensitivity of the simulation method indicated variations in thickness of 0.1A. Likewise, asshown in the last section and by diffraction profile modelling the period can be determinedto the Angstroms level. A study of the bounds of the InGaAs and InAlAs systems onInP has been determined by Bennet and del Alamo (1993). The bounds of the rangeof applicability of various thicknesses, compositions and diffraction conditions can bedetermined by simulation.

Aleksandrov and Afanas’ev (1985) have applied dynamical theory to determine thethickness of an amorphous layer. Basically it has required the solution of all the wavefieldinteractions created in the underlying perfect crystal as well as those generated in theamorphous top layer and vacuum above. There are four wavefields created within theamorphous film and substrate and three above the sample surface in the grazing incidencegeometry used. By solving these coupled equations they were able to obtain an expressionfor the variation of intensity from planes nearly parallel to the surface normal directionwithin the substrate and the influence of the amorphous layer above. The former gives afast variation because of the crystalline quality, whilst the amorphous layer gives slowlyvarying intensity. The diffraction geometry is based on that first employed by Marraet al(1979), where the diffraction plane is almost perpendicular to the surface and the incidentbeam is very close to the condition of total external reflection. The total external reflectionarises from the subunity refractive index of x-rays. Aleksandrov and Afanas’ev (1985)were able to achieve a 10% reliability in the thickness for a 250A amorphous layer createdby disturbing the surface of perfect Si. They have found this technique is sensitive toamorphous layers down to 5A. This uses a laboratory (1.2 kW) source in a specializeddouble-crystal arrangement.

Kinematic. Since the scattering for multilayer structures in normal diffraction mode isgenerally weak the simpler and more transparent kinematical theory can be applied. A fewgroups have used this approach very successfully; Speriosu (1981), Vreeland and Paine(1986), Vandenberget al (1988). But now that the computation speed has increased themore exacting and commercially available dynamical theory simulation is preferred. Thesegroups have used kinematical modelling to successfully determine the period and individuallayer thicknesses in semiconductor multilayers and the intermixing of the constituents at theinterfaces.

One area where the transparency of the kinematic model has been useful is in theinterpretation of laterally periodic structures since the diffraction processes require thecombination of several modulations. Macrander and Slusky (1990) tackled this problemand derived an expression for the periodicities in a multilayer with a laterally periodicstructure. The periodicities lie in perpendicular directions to each other in diffraction spaceand so when they collected the data with a standard double-crystal arrangement (with anopen detector) the various satellites all lay on the one profile. De Caroet al (1994) useda combination of kinematic theory and a Fraunhofer ‘envelop’ slit function to derive anexpression for the reflectivity as a function of the deviation from the Bragg angle and fromthis he could determine the composition and thickness of the AlxGa1−xAs/GaAs multiplequantum-well structure, the quantum-wire period and width.

Gailhanouet al (1993) have also combined kinematical theory and the Fraunhoferapproximation for optical gratings. The difference they have made is to use a differentdata collection method: reciprocal space mapping. This has allowed them to separate the


periodicities from a line profile to a two-dimensional intensity distribution. The diffractionis complex since the intensity distribution cannot be explained assuming simple Braggdiffraction by the surface grating since there is some Laue transmission through the grating.Therefore, the scattering is complicated such that the ratio of the diffracted to incidentamplitudes is given by:

X(kh) = Xgrating(kh − k0) +∫

T (kt − k0)X0(kht − kt )T (kh − kht ) dkt

where k0, kh, kt , kht are respectively the incident, diffracted, transmitted and diffractedtransmitted beam vectors,X0 is the amplitude ratio from the substrate andT is the amplituderatio of the transmitted beam through the grating. All the lateral periodic satellites withtheir individual crystal truncation rods can therefore satisfy the diffraction condition andcreate transmitted beams and these can reach the substrate. Subsequently these beams arediffracted by the substrate and on passing out through the grating they interfere with thediffracted beam from the grating thus creating resonance peaks. This complex interferencegives rise to diffraction space streaking that is characteristic of the grating shape, figure 4.

Optical. The modelling of the scattering close to grazing incidence remote from Braggpeaks was pioneered by Parrat (1954) and is still the most popular algorithm. It is a verypowerful and general method that essentially models the scattered intensity with sample

Figure 4. The distribution of intensity in diffraction space created by periodic multilayer witha laterally periodic optical grating (produced by kind permission from Marc Gailhanou). Thisform of diffraction is best modelled by a combination of kinematical diffraction theory and theFraunhofer approximation.

1364 P F Fewster

Figure 5. The change in fringe separation with thickness for a SiO2 layer on Si in a reflectometryprofile simulated for various layer thicknesses, CuKα radiation.

information in terms of the x-ray density as a function of depth,t :

Xj =(

exp−iπRj+1tj+1

λ

)4Xj−1 + Rj+1

1 + Xj−1Rj+1

where

Rj = rj − rj+1

rj + rj+1exp(−2rj rj+1σj+1)

and

rj =√

ω2 − 2δj − i2βj

whereβ is the imaginary component of the refractive indexδ to account for absorptioneffects andj is the layer index. The parameterσ can be used to introduce interfacialsmearing or by including additional layers in this recursion formula. As in the previoussimulation methods a model of the sample is simulated and the comparison is made withthe experimental profile. The process is iterative although as with the other methods variousautomatic processes can be applied. In figure 5 it is possible to see how the thickness fringevaries for different thicknesses for a crystalline SiO2 layer on Si close to the origin 000(reflectometry condition).

The basic problem in fitting this reflectivity profile is the fact that it is a nonlineartransform of density profile to the scattering profile. Consequently similar structures canappear almost indistinguishable. Siviaet al (1991) have used a route to help solve thisproblem by using a method based on a speckle imaging technique developed for astrophysics.The process is to make a Fourier approximation to the reflectivity data by considering therate of change of the density profile, this relates to the data through a phaseless Fouriertransform above some critical value forQ (= 4π sinθ/λ), therefore if this relationship holdsthen the autocorrelation function or Patterson function (i.e. how the density relates to itself)is related toQ4 times the reflectivity data. Hence the autocorrelation function of the rate ofchange of density with depth represents the real-space information in the reflectivity data.The results of this are not simple to interpret but by introducing a reference density (at


an interface for example) then the pattern can be changed so dramatically that the densityinformation could be understood. Siviaet al (1991) have applied this approach to theanalysis of titanium on sapphire.

Bartelset al (1986) have modelled carbon tungsten multilayers with a period of 60Awith both dynamical theory and optical theories and the results are almost identical. Thereason for this agreement is that since both approaches effectively consider the interaction aswavefields, thus including multiple internal scattering events, both are similar. However theTakagi (1962, 1969)–Taupin (1964) dynamical model used is based on just a strong incidentand diffracted beam and is not valid in the specular reflection region below the critical angle.This is where the difference exists between the two models. Although specular reflectivityis a dynamical diffraction effect in the absence of Bragg scattering.

It must be born in mind, however, that the presence of diffuse scattering arising fromimperfect interfaces is very strong in the reflectometry region and this can boost the intensityin certain regions of the diffraction profile and possibly lead to uncertainties in the detailsof the interpretation, de Boeret al (1995b). The influence of roughness on these profileswill be discussed later.

5.2. Microscopic shape

The microscopic shape is defined as those structural features that are smaller than theincident-beam dimensions and are therefore observed by the modification of the scatteringas opposed to being observed as an isolated feature. For example, the individual crystallitesize in a polycrystalline sample, the mosaic-block size in a textured epitaxial sample andtheir distributions. The microscopic shape determination in polycrystalline materials wasfirst proposed by Scherrer (1918) and has been developed by a number of researchers usinga variety of methods. The approaches used are more reliable the smaller the crystallitesize, below∼ 500 A the method can be very reproducible. Examination of mosaic-blockdimensions has been approached in different ways either by topography or by the simulationof reciprocal-space maps.

5.2.1. Crystallite size determination.The crystallite size,L, is generally simply basedon the expression derived by Scherrer (1918), that is simply derived from the diffractionbroadening effects due to finite number of crystal planes based on the kinematicapproximation.

L = Kλ

βtruecosθ

whereβtrue is the profile breadth on the angular scale andK is a geometrical factor relatedto the shape of the crystallites (for a sphereK ∼ 1.07, whereas for other shapesK can varydepending on the reflection order and shape, Wilson (1963)). This method will producesome average of the crystallite sizes, to obtain a distribution the situation becomes verymuch more complex and uncertain unless there are a few distinct average sizes.

Stokes and Wilson (1942) have defined an integral breadth to determine the effectivecrystallite dimensionLeffective, which is the volume average of the crystallite dimensionLV .Since the volume can be expressed as the product of this volume average〈LV 〉 and theprojected cross-sectional area the effective crystallite dimension is given by:

Leffective = 〈L2V 〉

LV

.

1366 P F Fewster

The integral breadth is defined as the integrated intensity divided by the peak intensityand is a very convenient form. This approach of Stokes and Wilson (1942) is much moregeneralized and removes the dependency on the shape distribution of the crystallites andhenceK = 1 in the Scherrer equation.

The strain variation in a thin polycrystalline-layer sample either grain to grain(distributed macrostrain) or within individual grains (microstrain) is given by the followingexpression:

e = 1d

d= βstrain

4cotanθ

wheree is the upper limit of the lattice distortions and is equal to 1.25〈ε2〉1/2 (1.25× theroot mean square strain) andβstrain is the profile integral breadth contribution due to thisstrain.

The influence of size and strain on the width of the diffraction profile fortunately havedifferent dependences on the scattering angle 2θ and this has been exploited by Warren andAverbach (1950) and Williamson and Hall (1953). The first is based on determining theFourier coefficients of the diffraction profile and extracting those that depend on strain andsize components. The latter is more graphically based. But both require several reflectionorders to separate these effects. Basically these approaches are still used today althoughsome groups have extended these basic methods. The elegance of the Warren and Averbach(1950) approach is that it can be extended to cope with stacking faults, defects, etc, althoughthe number of parameters creates significant correlation problems. This is especially the casewith thin layers when the diffraction peaks are usually broader and fewer, and frequentlyno higher-order diffraction peaks are observable.

Single-line methods rely on the profile shape that can vary from Gaussian for straineffects to Lorentzian for size effects. Very good statistics and high intensities are requiredfor this form of analysis which is rarely the case for thin layers. All the above methodsalso depend strongly on the knowledge of the instrument response. The necessity for highintensity requires a broad instrument function that results in broad profiles or a very intensesource. However, the above approaches can produce reliable results but in general thebest results are obtained for crystallite size measurements below 1000A and small strains(< 0.0005) or for strain measurements above∼ 0.0005 with crystallite sizes in excess of2000A, using standard laboratory powder diffractometers.

An indication of the limits of this technique using standing Bragg–Brentano geometrycan be illustrated in the analysis of nanostructures. Fewster and Andrew (1993c) haveanalysed a series of thin-layer (300–2000A) small-diameter silver spheres imbedded in aSiO2 amorphous matrix, that has applications in zero dimension quantization devices. Thespheres occupied approximately 30% of the layer volume and had a random orientationdistribution and were approximately 70A in diameter. The scattering was very weak and inthe limit this amounted to< 1 photon per second detected on a background of∼ 2 photonsper second, but with count times of 60 seconds the sphere diameters could be determinedand agreed with the lattice images from electron microscopy. The method of analysis usedwas to fit the background very precisely and model the diffraction shape using a standardshape, the Scherrer equation, and to include the noise. The latter is very important to obtainrealistic bounds of the accuracy, which was close to±3 A for these dimensions. Clearlyas discussed above for these dimensions the ‘micro-strain’ would have to be very large toupset these results. The measured lattice parameter was found to be identical to that of bulksilver.


The crystallite size determination using the Scherrer equation and the apparent strainexpression are widely used in the analysis of thin films because of their simplicity. Senet al (1982) have performed a careful analysis of ZnO thin films using an asymmetricalfocusing geometry of the Seeman–Bohlin x-ray diffractometer (Seeman 1919, Bohlin 1920).They used an expression that is derived on the assumption that the grains are small and anon-uniform strain exists within each grain to determine the mean grain size:

β2trueLeffectivecosθ = λβtrue + 16e2Leffectivesinθ tanθ

from Halder and Wagner (1966). In this case they combine the information from the 0002and 0004 reflection to obtainLeffective and e. Senet al (1982) were then able to extractthe variation in grain size and lattice strain with film thickness by studying a series of ZnOlayers. The crystallite size increased dramatically with film thickness, whereas the strainlevels reduced and similarly this response was observed for increasing substrate temperature.This is a good example of the use of these methods for controlling the microstructure bythe growth conditions and for observing the structure evolution.

Of course the shape of crystallites is not often simple and Langfordet al (1986) havereconstructed crystallite shapes by detailed analysis of the apparent crystallite size alongcertain crystallographic directions for a ZnO bulk specimen. For thin-layer analysis this ispossible provided all geometrical effects of the diffractometer are accounted for. The majorproblem being the non-parafocusing conditions with conventional geometries for layershaving orientation texture. For applications of total pattern-fitting methods based on theRietveld method the reader is referred to the cautions and procedures expounded by Delhezet al (1993). These basically ensure that a detailed analysis of those procedures that arewell known, size-strain analysis, instrumental aberrations, etc, are determined initially. Thisensures that the correlation of an essentially least-squares method is minimized.

5.2.2. Determination of mosaic-block dimensions.A mosaic block is a discrete diffractingentity similar to a crystallite discussed above except that its relationship to the matrix ofthe sample is different in that it is connected via small-angle grain boundaries. This linkleaves no voids but a series of dislocations that accommodate the misorientation betweenneighbouring blocks. The analysis of the mosaic structure, dimensions and orientationdistribution, is often what people mean when they are interested in the crystal ‘quality’, forexample Koschinskiet al (1992) and Capperet al (1993). The analysis method applicablecan be defined by the size and the instrument resolution. Generally, since the misorientationangles are small, double-crystal diffractometers or triple-crystal diffractometers are oftenrequired. The presence of mosaic blocks in some systems simplify the analysis if theirdimensions are less than the extinction distance since the kinematical diffraction modelcan be applied. The presence of mosaic blocks in ‘ideally imperfect’ crystals allows thedetermination of molecular structures without the complications of dynamical theory.

I shall discuss two approaches to this problem and the interpretation of diffractionspace to isolate size effects from other effects and a route to obtaining the full three-dimensional shape. Basically a mosaic crystal will consist of finite dimensions or correlationlengths perpendicular and parallel to the surface and each mosaic block will have a differentscattering vector direction to its neighbour. Also in general the relative lattice parameterbetween mosaic blocks is small and so the scattering angle will be essentially unchangedbut the angle of incidence at which each will diffract will differ according to their relativemisorientation. Clearly isolating an individual block with a very small probe will implythat the probe is smaller than the block or the interference of other blocks will influence

1368 P F Fewster

the observation. Hence, either the diffracted response has to be modelled or we have toemploy special experimental techniques.

The simulation method requires a method of reproducing the reciprocal-space map ofthat observed from a triple-crystal or double-crystal diffractometer, Holy et al (1993). Holyet al (1993) have applied kinematical diffraction theory to this problem, and therefore theapproach is only valid for layers much less than the extinction depth, and the substratediffraction is ignored. The approximation made is that the disruption of the lattice dueto the finite block size will only influence the phase term in the wave equation, i.e. theperiodicity of the lattice is much smaller than the dimensions of the distortion, Takagi(1969), which is the same as that made in dynamical diffraction theory of distorted structures.From this they were able to derive the correlation function (that consists of the coherentlydiffracted wave and the partially coherent diffuse-scattering contribution) that includes theshape of the blocks and their misorientation. For nearly all practical examples for the blockdimensions and misorientations the coherent part of the scattering is negligible and thereforefor various values of these two parameters the experimental double-crystal profiles or triple-crystal diffraction-space maps can be modelled. They have used this method to successfullymodel the diffraction from 6.2 and 1.4 µm layers of ZnTe on GaAs. The root-mean-squaremosaic-block radii were determined to be 0.2 µm.

Holy et al (1994b) have taken the argument further by showing how this mutualcorrelation function can be derived directly from Fourier transforming the diffuse-scatteringdistribution. Holy et al (1994b) have also developed two mosaic-block models with differentcharacteristic diffuse-scattering distributions that correspond to the limiting cases and haveused a combination of these to create a combined model. However, as they state to obtainan unambiguous interpretation it should be supported by another independent method.

With the development of the high resolution multiple crystal multiple-reflectiondiffractometer (HRMCMRD: Fewster 1989a) it has been possible to obtain diffraction-space maps with the minimal artifacts and therefore the diffuse scattering can be observed,this was the configuration used by Holy et al (1994b) for their studies. The shape of thediffraction peaks are sensitive to microscopic tilts and lateral correlation lengths and thesecan be separated using a very pragmatic approach and hence the mosaic-block dimensionsare accessible almost directly from these maps, Fewster (1996b). If the block sizes approacha few microns then an imaging method can be used, Fewster (1991a). Because of thevery high angular resolution of the instrument individual mosaic blocks can be isolated indiffraction space and imaged. Consider the diffraction-space map of a mosaic GaAs crystalwith a AlGaAs thin layer, figure 6, then the instrument can be set to record the diffractedintensity for any position on this map. By simply placing a high resolution (submicron grainsize) photographic plate in front of the detector the dimension of the region giving rise tothis scattering can be obtained. If the misorientation is greater than the angular resolution(∼ 10′′ arc) and the blocks are significantly larger than the film resolution then this providesa very quick method for observing these dimensions.

For a highly distorted layer a multiple-crystal topograph can reveal the break-up of thelayer into regions of extended ‘blocks’, figure 7 (Fewster 1991b), and the dimensions,position on the sample, etc, can be retrieved easily. This multiple-crystal topographytechnique has been used extensively and will be discussed later for finding informationon interface structures and studies in microscopic homogeneity, etc.

However, the development of a further technique that crosses the boundaries ofmicroscopic shape, orientation and distortion is that of three-dimensional reciprocal-spacemapping, Fewster and Andrew (1995a). The results of this type of measurement are bestillustrated in figure 8, where it is possible to observe the shape of the diffraction of four


Figure 6. The distribution of intensity around the 004 reflection of a thin AlGaAs layeron a GaAs substrate, showing the misorientation approximately represented by differences inincidence angleω.

Figure 7. A multiple-crystal topograph of a ZnSe layer. The dimensions of the mosaic regionscan be determined as well as the position of the regions on the sample.

Figure 8. A three-dimensional reciprocal-space map of four mosaic blocks of GaAs and thecorresponding AlGaAs blocks in the thin layer.

GaAs substrate mosaic blocks with their corresponding AlGaAs layers. In fact it is thethree-dimensional image of figure 6. From these maps we have the possibility to extract thefull three-dimensional shape of these mosaic blocks as well as their relative misorientation,absolute lattice parameter, etc.

1370 P F Fewster

Figure 9. The distribution of diffuse scattering resulting from the presence of Si pillars inporous Si, where the porosity is≈ 60%, 004 reflection, CuKα radiation.

5.2.3. Quantum wires arranged in a random way approximately normal to the surface.Thepossibilities of porous Si creating an opportunity for optical devices and interconnects in Sihas prompted work on extracting correlation lengths that relate to the Si wire dimensions.Bellet et al (1992) have used double-crystal methods to determine these characteristic lengthscales in a variety of structures. The sub-50% porous structures consist of ‘defect voids’whereas above 50% the structure consists of Si pillars. The diffuse scattering is veryextensive and can easily be measured by double-crystal methods, but to have a deeperunderstanding of the strains and distributions reciprocal-space mapping reveals more detail(figure 9). The width of the diffuse-scattering profile directly relates to a correlation length,as for crystallite size, to give an average quantum wire width. The distribution of intensityin figure 9 will relate to the wire direction, etc. Diffuse scattering topography, Fewster(1993b) of this weak scattering suggested that this diffuse scattering arises from distributedsmall scatterers as was assumed.

To obtain information concerning the size of the wires after oxidation then the alterationsto the diffuse scattering can be rather subtle, but still measurable using synchrotron sources.Naudon et al (1994) have applied the theory of Porod (1982) to interpret the diffusescattering from small-angle scattering experiments. The total integrated intensity is afunction of the electron-density variation and the voids and from this relationship of Porodand assuming that the deviation from abruptness of the Si–void interface is due to the oxidethickness, the latter could be determined.

5.3. Macroscopic composition

Determining the composition of a completely unknown thin film depends first on whetherit is composed of phases that already exist in a data base or is an unknown compound.Phase identification is a standard x-ray diffraction method for determining the compositionof polycrystalline materials. The difficulties arise with thin films because the scattering isweak and the higher probability of non-randomness in the crystallite orientation as wellas the other common problems with the method (e.g. material not in database, database


value incorrect, etc), Jenkins and Smith (1981). Enhancing the scattering from a thin filmcan be achieved with an appropriate choice of x-ray scattering geometry. For completelyunknown materials a full structure analysis is required that would give the molecularstructure. This type of analysis is carried out on bulk samples in general, and becomesvery difficult on thin layers because of the low number of accessible reflections, the weakscattering and preferred orientation effects. All these create additional corrections to theraw data and creates real limits to the applicability although careful matching of a fewlines can be successful. For structures that have a significant phase extent, where theproportion of elements exist over a significant composition range, an understanding of thesolid solubility of the constituent elements is required and is used extensively for determiningthe composition in semiconductors. On the other hand this can be troublesome for phaseidentification as described below.

The determination of the macroscopic composition in thin layers just brings in a levelof complexity in terms of a lack of intensity and possibly more pronounced orientation andcrystallite shape effects. Thin-film growth rarely creates a ‘perfectly random’ distributionof crystallites and in general is more likely to have preferred orientation that can alsovary as a function of depth. A common mode of growth exhibits a random distributionof small crystallites at the initial stages of nucleation of the film until the more rapidlygrowing planes cover the surface laterally and a slower growth plane normal to the surfacedominates the orientation. This results in large crystallites towards the surface with well-defined orientation whereas at the substrate interface the crystallites would be small withrandom orientations. These effects have a significant implication on the determination ofthe molecular structure and the identification of the phase by comparison with those in adata base.

5.3.1. The determination of the molecular structure.The molecular structure determinationmethods for single crystals is very well established and has been applied from complexbiological molecules of very large molecular weights (Kendrewet al 1960) and samplesof submillimetre dimensions to relatively simple inorganic single crystals of sub-10 microndimensions (Harding 1995). These methods rely on the determination of the intensities thatare related via kinematical theory to the structure factor amplitude,FH , as shown previouslythat in turn is related to the electron density at a positionr within the periodic unit:

ρ(r) = 1

V

∑H

FH exp[−2π iH · r]

whereV is the volume of the periodic unit,H is the reflection order and the integratedintensity∼ |FH |2.

However, a thin layer on a substrate poses an altogether different challenge, becauseexcept in the case of a nearly ‘perfectly random’ polycrystalline layer not all the reflections toaffect a structure determination will be accessible. Hence the simplest method could well beto remove the layer and prepare the sample for a more conventional polycrystalline-structureanalysis method. First an approximate model has to be found to determine the phasesof the structure factor amplitudes by one of several methods; maximum entropy (Gilmore1993), simulated annealing (Su 1995), direct methods (Giacovazzo 1993), Patterson methods(Patterson 1934) or trial and error methods (when considerable knowledge of the structureis known). Once a good first estimation of the molecular structure has been establishedthen the comparison of the data to the calculated values can be carried out using eitherFourier difference methods (Woolfson 1970) or total pattern-fitting methods using Rietveldrefinement (Cheetham 1993). From here the details of the structures can be refined. The

1372 P F Fewster

popularity of the Rietveld method for polycrystalline material is that microstructural effects(orientation, crystallite size and strain can simply be added as extra parameters in the fit).Rietveld has proved very successful for structure determination or for microstructural studiesof known structures, but when all these parameters are refined the correlations can lead touncertainties in the derived parameters. The Rietveld method is basically a least-squaresmethod that associates the intensity at any point in the diffraction pattern,I (2ω′), to thestructural parameters of the sample through the relation:

Icalc(2ω′) = S∑H

LHCHPHA8(2ω′ − 2θH ) + Ibkgd(2ω′)

whereS is a scale factor,L the geometrical Lorentz factor,C the polarization factor,Ais an absorption factor,P is a preferred orientation factor due to non-randomness of thepolycrystalline sample and8(2ω′ − 2θ) is the profile shape that is often approximatedto Gaussian, Lorentzian, pseudo-Voigt or Pearson VII form. The background is usuallysubtracted from the original measured data or fitted to a polynomial but is rarely calculatedas a scattering contribution. The summation is then over all reflections and this calculatedintensity at the scattering angle 2ω′ can be compared with the measured intensity at thatpoint. Therefore, since the method basically relies on a least-squares analysis a good startingmodel for the parameters is very important. However, a total pattern-fitting method like thisdoes indicate that a good fit is indicative of a possible solution but if this is not the propersolution then the data is inadequate or the relationship between intensity and the parameteris inadequate.

This is a developing field with many groups working on the Rietveld method sorefinements in these relationships are emerging. For detailed understanding of this methodthe reader is referred to Young (1993).

Usually if the layers are ‘perfect’ epitaxial then this indicates that the material is knownor partially known and theseab initio analyses are not relevant. However, for the analysisof surface reconstruction, when the atom positions and bonding arrangements are unknown,then these methods are very useful. Carvalhoet al (1995b) and Carvalhoet al (1995a) haveused maximum entropy methods to determine the two-dimensional structure of the 7× 7silicon (111) surface where they were able to observe all the 102 atoms of the Takayanagidimer-adatom-stacking fault model. In this case the reflections are weak so the kinematicaldiffraction model is valid although, because of the difficult geometry and weak scattering,synchrotron radiation was used. This examines diffraction planes normal to the surface ofa sample in a vacuum chamber thus reducing the number of high order (H) reflectionsaccessible. Consequently Fourier methods are less reliable and the advantages of maximumentropy methods is to effectively extend the resolution beyond the measurement limit.

Methods relying on the fitting of a calculated model to the measured data and themeaning of ‘goodness’ of fit (as in this case the residual factor or discrepancy index) hasbeen the subject of many debates and have been discussed fully in the crystallographicliterature. Fortunately though, in almost all cases, there is always additional informationthat increases or decreases the confidence in the determined molecular structure, e.g. thebond lengths, the chemistry and energy calculations, etc.

5.3.2. Identification of a known structural phase.There are in access of 70 000 compoundsthat have been identified and kept on the International Crystal Diffraction Database whichis used extensively for identification. The methods relying on this database, or a calculateddatabase based on the scattering angles or atomic plane spacings only, assume the materialis close to or is a ‘perfect’ polycrystalline sample. Since most of this work is with


powdered materials this is not a serious problem, except that careful preparation techniquesare required. However, polycrystalline thin layers for identification are rarely ‘perfectly’orientated and the diffraction is weak. This means that the conventional Bragg–Brentanomethod or single profile-scan methods can give rise to fewer peaks than expected and theintensities can be unreliable. The signal can be enhanced by grazing-incidence geometry toreduce the depth penetration and scanning the detector (scattering angle) only, whereas theorientation distribution can be partially compensated by several scans at different incidenceangles and rotating the sample during data collection, etc. Of course the intensities are notreliable but this is less important than the actual scattering angle of the diffraction peaks.Capturing several reflections at once with area detectors (linear position sensitive detectors,two-dimensional detectors) can speed up the process but the principle is the same. Of courseall early work was done with x-ray sensitive film and this has been used very successfullyby Wallace and Ward (1975) to identify polycrystalline thin layers. The method dependson a grazing-incident beam impinging on a rotating sample with a cylindrical film wrappedaround the sample. The distorted positions of the peaks and the interpretation relies on verysimple geometrical relationships.

Figure 10. The diffraction profile from In2−δSnδO3 where the Sn content is such that the alloycould not be found in the data base and a better-quality lower-Sn content alloy was found bymatching. Note the very different intensity ratios indicative of significant orientation texture.

An example of the diffraction pattern from a thin layer of essentially similar materialsis shown in figure 10. In this case the In2−δSnδO could not be identified from searching thedatabase because only In2O3 is listed, for a very good reason. The range of stoichiometryof In2−δSnδO is large and the structure is essentially In2O3 with Sn in solid solution. Thepresence of Sn is to change the lattice parameter of the In2O3 and shift the peaks and ifa suitable model was available or a sample of In2O3 and In2−δSnδO (where the Sn wasjust beginning to precipitate) existed and Vegard’s law was obeyed then the Sn compositioncan be determined. In figure 10 we see another In2−δSnδO sample prepared in a differentmanner and the basics of the pattern are present but the influence of significant orientationtexture changes the detected intensity. These two patterns arise from the differences incrystallite size and depend on the growth kinetics and this is where a simple scan like thiscan prove to be a very quick characterization tool of the growth method.

The presence of multiple phases just confuses the issue but standard methods thathave been developed for bulk samples are applicable. Although there are many methodsto explore including data collection with several wavelengths to increase or decrease the

1374 P F Fewster

influence of different phases, etc, Fiala (1989). Methods of quantifying the proportions ofdifferent phases in a sample rely very heavily on the accurate determination of the intensitiesfrom reflections from the various phases. As discussed above this problem is accentuatedin the case for thin layers. Again there is a significant choice of methods primarily basedon bulk samples from calibration methods, averaging the intensities to total pattern-fittingmethods. The former relies on having similarly prepared samples of various compositionsand obtaining a relationship between intensity and composition (Klug and Alexander 1974),the intensity-averaging methods (Dickson 1969) require averaging several reflections of thesame phase with their calculated intensities and comparing between phases whereas thetotal pattern-fitting methods rely on modelling and comparison based on Rietveld methods(Madsen and Hill 1990). The latter two methods clearly do depend on knowledge of themolecular structure whereas the calibration methods do not.

If the deposited layer is very thin then grazing-incidence geometry, where the x-raypenetration is exceedingly small (incidence angles close to the total external reflectioncondition∼ 0.2◦ typically) can significantly enhance the layer scattering. Horiiet al (1995)have exploited this to distinguish between the C49 and C54 structures of TiS2, the formerbeing metastable and high resistance and the latter low resistance and stable as required forVLSI technology. The deposits were 300A thick and could easily be distinguished by thismethod. Although this work was carried out on a synchrotron it is not generally requiredwhen using this method.

5.3.3. Determining the phase extent in nearly perfect crystals.As indicated above if thematerial is an alloy of two phases and their relative proportions are required, then this isvery possible by x-ray methods provided that the relationship between lattice parametersand relative proportions is known. This approach is used extensively in the study ofsemiconductor composition (SiGe; Dismukeset al 1964: AlGaAs; Estopet al 1976, Bartelsand Nijman 1978: InGaAs/InP; Halliwell 1981: InGaAs/GaAs; Woodbridgeet al 1991).All but Dismukeset al (1964) found that the relationship between the composition and‘effective’ lattice parameter change is linear. Whereas Dismukeset al (1964) found avery small discrepancy that is important to include for accurate determination of the Geconcentration in SiGe multilayer structures. The basis for all these works is to assume thatthe thin layer has perfect registry of the surface-plane atoms of the layer and substrate andthe layer is distorted in a well-defined way (tetragonally for [001] surface normals). Themeasured difference in the incident angles for the layer and substrate to diffract,1ω, isthen related to the perpendicular strain by(

1a

a

)⊥

≈ −cotanθ1ω

cos2 φ

whereθ is half the scattering angle. This basically is an approximation for closely separatedpeaks using Bragg’s equation with an additional term inφ to account for diffraction planesinclined by this angle to the surface plane, Fewster (1986). The ‘effective’ lattice parameter(i.e. that which would be obtained if the layer was able to relax to its natural cubic shape)is then related to the vertical strained value by the elastic moduli, Hornstra and Bartels(1978). This ‘effective’ lattice parameter is then related directly to the composition byassuming a linear relationship with the lattice parameter (Vegard’s law) or some other form.The elastic moduli also varies with composition and is usually assumed to have a similarrelationship. There have been various comments on this relationship that are covered inthe review by Fewster (1993a) and will not be repeated here except to say that in generalthe linear relationship will give the composition to within 1% in ‘x’ (on an absolute scale


and to within 0.1% on a relative scale) for a ternary AlxGa1−xAs which is better than canbe determined by other analytical methods. For larger mismatch structures the accuracyimproves and is therefore perfectly adequate for relating the composition to other physicalproperties.

The above arguments all assume that the Bragg equation can be applied to this situation.Fewster (1987) and Fewster and Curling (1987) have shown that dynamical-diffractioneffects can be highly significant to invalidate the simple Bragg equation for single layersbelow 0.5 µm thickness and buried layers up to 2µm in thickness. This arises from the factthat the diffraction of the layer and substrate cannot be considered in isolation and the peakscan have very large shifts in their angular position that can lead to errors in compositionup to about 15% in ‘x’ depending on the geometry. Fortunately the correct value can beobtained by simulation with dynamical-diffraction theory.

The use of simulation for extracting the composition is now the most reliable way andhas the significant advantage that it can be applied to very complex multilayer structuresfrom graded composition layers (Hillet al 1985) to sophisticated laser structures (Fewsterand Andrew 1995a). In the latter case segregation of In from the quantum well as well asall the compositions and thicknesses could be determined.

The assumptions made in all these methods is that the substrate is a reliable internallattice parameter standard, i.e. unstrained and of known composition, and that the layerlattice parameter in the plane of the interface matches that of the substrate. In general withinthe accuracy required (< 1% in ‘x’) it is fair to assume that variations in stoichiometry aresmall for binary components (GaAs, InP, etc) whereas for ternary compounds (CdSeTe, In-doped GaAs, etc) significant uncertainty will exist. However, the absolute lattice parameterfor these materials can be obtained by several methods and will be discussed later in thesection on macroscopic distortion. Likewise, if the atomic planes of the layer are notcoherent with those of the substrate then it is not possible to obtain the ‘effective’ latticeparameter of the layer from one reflection. In fact eight double-crystal diffractometer profilesare required, ‘rocking-curves’, Halliwell (1990) or four diffraction-space maps, Fewster andAndrew (1993b). This is to account for the relative tilts between the layer and substrateand a way of obtaining a component of the lattice parameter parallel to the substrate layerinterface for two orthogonal directions as well as obtaining the lattice parameter normalto the interface. The ‘rocking-curve’ method is less direct and relies on comparing allthe 1ω which contain the lattice parameter difference, relative tilts and differences indiffraction plane inclination for the planes not parallel to the interface. The advantage ofthe diffraction-space map method is that the strain and tilts are clearly distinguished andmuch more additional information can be obtained and will be discussed under the sectionson distortion.

The use of simulation methods for the case when defects are present at the interfacealso can be misleading because of the breakdown of the coherence of the wavefield in thesample. This aspect will be discussed under the section on interfaces.

5.3.4. The determination of the composition from the x-ray density.Another approach toobtaining the composition is by determining the x-ray density of a layer. This methodrelies on determining the position of the critical angle of total external reflection and in thesimple case this requires the accurate location of the direct beam as a reference. Ideally theinstrument should be angularly selective for the accuracies necessary (i.e. defined by crystaloptics and not slits). The critical angle is simply related to the refractive index and hencethe possible composition (ωc = √{2δ}).

1376 P F Fewster

For a periodic multilayer structure an alternative method can be applied as demonstratedby Miceli et al (1986). A periodic structure produces a series of satellite reflections andall these will be displaced by virtue of the refractive index,δ, which can therefore bedetermined by obtaining the position of the zeroth order fringe. Miceliet al (1986) havederived the following relationship:

sin2 ωj =(

λ

23

)2

j2 + 2δ

where 3 is the modulation wavelength of the periodic multilayer andj is the satelliteorder. This method relies on the high accuracy of determining the angular positions ofthe satellites (they use a triple-crystal arrangement) and is only valid for low-absorptionmaterials when the refractive index becomes almost wholly real (i.e.β = 0 in therelation for the refractive indexη = 1 − δ + iβ). Hence by plotting sin2 ω versusj2 the average density is determined from the intercect. The accuracy of this methoddepends on an accurate determination ofω and therefore the direct-beam (incident-beam)direction has to be determined accurately. However, it is a very simple method andapplicable to all low-absorbing materials independent of the crystal form. The compositionderivation from the average x-ray density is less straightforward and is therefore lesshelpful for complex materials than those whose chemistry is totally unknown. Miceliet al (1986) used this method for determining the composition in GaAlAs/GaAs andNb/Ta structures, which they could do to a certainty of∼ 4% absolute in the alloyfraction.

The reflectometry can also be simulated and this is a very powerful method since thethickness and composition can be obtained at the same time. As discussed earlier forthe evaluation of thickness the Parrat (1954) formulism is very suitable in most cases,however the question of uniqueness is always a concern with any multiple-parameterfitting method. Unlike diffraction methods that can compare several reflections withdiffering sensitivities to different parameters, reflectometry does not have this flexibilityunless the material is crystalline. However, the use of a second wavelength can resolveambiguities. Ohkawaet al (1995) have used this effect to determine the compositionas a function of depth of an Al/C multilayer on Ge when the expected structure wasquite different from the actual. Ohkawaet al (1995) measured the reflectometry profileat the GeK absorption edge and at a point well away from it using a synchrotronsource.

5.3.5. The determination of the composition by x-ray fluorescence methods.Thecombination of measuring the x-ray fluorescence yield with grazing-incidence reflectometrygives an approach for obtaining elemental analysis of thin-layer structures. There isa significant enhancement of the fluorescence yield when the penetration depth is verysmall. These methods were first employed by Yoneda and Horiuchi (1971) followed byWobrauschek and Aiginger (1975) and Schwenke and Knoth (1982). Beckeret al (1993)then showed that the technique could provide depth information by varying the angle ofincidence and this promoted the development of composition determination in thin layersby fluorescence methods. De Boer and Hoogenhoff (1991) and Wiesbrodet al (1991)then showed how the data could be analysed to reconstruct the composition profile. Thegeometry of the diffractometer for this form of analysis is the same as that for normalreflectometry with an energy-sensitive detector above the sample to detect the fluorescentradiation. The angle of incidence is varied and the penetration depth is changed as shownin figure 11.


Figure 11. The variation in penetration depth as the incident angle is varied for GaAs close tothe region of the critical angle for total external reflection, CuKα radiation. The reflectometrycurve is also given.

At very low angles of incidence an evanescent wave is created that decays exponentiallywith depth; as the angle is increased the critical angle is reached when the electric-fieldpenetration increases rapidly. If an interface exists where the refractive index changes thenthe diffracted wave will interfere with the incident wave creating the nodes and antinodesof a standing wave parallel to the surface. The depth profile information (compositioninformation at a specific depth) is then obtained by varying the angle of incidence whichin turn alters the wavelength of this x-ray standing wave. Because of the very low anglesof incidence the peak-to-trough depth distance can be large and comparable to thin-filmthicknesses. Clearly as the angle of incidence increases the peak-to-trough depth decreasesrapidly and to be sure of the interpretation of the fluorescence yield this should be calculated,de Boeret al (1995a). The ratio of the fluorescent radiation from element ‘a’ in layer j tothe incident intensity is given by:

I (a)j

I0= C(a)j τ (a, λ)J (a, λ)ω(a)g(a) exp

(−

j−1∑n=1

µ(a)nρndn

sinθx

)

×|X↓

j |21 − exp

[−

(µ(λ)jsinθj

+ µ(a)jsinθx

)ρjdj

]µ(λ)jsinθj

+ µ(a)jsinθx

+|X↑j |2

1 − exp[−

(−µ(λ)jsinθj

+ µ(a)j

sinθx

)ρjdj

]−µ(λ)jsinθj

+ µ(a)jsinθx

+ 2 ReE

↓j ∗ E

↑j

−4πi sinθjλρj

+ µ(a)jsinθx

(1 − exp

[−

(−4πi sinθj

λρj

+ µ(a)j

sinθx

)ρjdj

])whereC(a) is the mass fraction,τ(a, λ) is the mass absorption coefficient of the element ‘a’at the wavelengthλ, J (a, λ) is the absorption jump factor,ω(a) is the fluorescent yield,g(a) is the emission rate,µ(λ) andµ(a) are the mass absorption coefficients for the layer

1378 P F Fewster

under consideration at the incident wavelength and the fluorescing wavelength, andX↑ andX↓ are the electric-field ratios for the outward and inward waves.

Clearly this approach of obtaining depth sensitivity relies on the establishment of anx-ray standing wave which requires interfering electric fields. To achieve this a fairlyabrupt change in the refractive index is required thus requiring smooth interfaces. It is alsoclear that the standing wave antinodes will become close together as the angle of incidenceincreases and therefore the fluorescence response will depend on the matching of the periodof these standing waves to the structure. Clearly multilayer structures can be analysed inthis way. Interfacial roughness is a significant complicating effect on the fluorescent yieldsince it will change the transmission at the interface and the created diffuse scattering canproduce fluorescence excitation.

Van den Hoogenhof and de Boer (1993) have used this fluorescence method to resolveambiguities in the layer order of a metallic Fe9Cr/TiN structure for which the reflectometrycurve could be fitted approximately in either order. In fact with the layer order reversedthe fit appeared more convincing, but a combination of the fluorescence method and theaddition of interface grading a perfect fit was obtained. They have also used the sensitivityof this fluorescence method to detect the presence and depth of contaminates of Fe and Cuin a Mo/Si multilayers and the presence of unexpected layers.

Zheludevaet al (1993) have used this method of depth-sensitive fluorescence to examinethe location of heavy Pb2+ ions in deposited PbSt2 Langmuir–Blodgett thin films on a Sisubstrate. The layer–substrate interface creates a standing wave and by measuring thefluorescence yield as a function of glancing angle they were able to show that the heavyions were located at the head of these organic bilayers. They have also investigated thepenetration of Pb2+ ions into manganese stearate and from this deduce the alignment of theorganic molecules from the distances of these ions, which could be determined to within±10 A from the substrate interface. These experiments can be undertaken on a synchrotronor using conventional sealed sources.

Zheludevaet al (1993) have taken the method further by creating a synthetic multilayerof Rh (46A), C (246 A) and Ni (40A) on a glass substrate and with the organic layer ofinterest on top. This multilayer structure is used to create an x-ray standing wave which notonly exists within the sample but also extends into the region above and can therefore beused to probe the organic layer. Again by determining the fluorescence yield of the heavyions the tail lengths could be determined.

Of course these techniques examine large areas (millimetres), because of the grazing-incidence beam. However, Noma and Iida (1994) have used grazing-exit geometryand a nearly normal incident beam confined to micron dimensions. This geometry isbased on the optical reciprocity theorem and is, therefore, surface sensitive although thefluorescence yield is low and special geometrical focusing with a synchrotron was requiredfor these experiments. Noma and Iida (1994) were able to chemically map Au/Cr/SiO2

patterned structures used in microelectronics with a spatial resolution of 4µm althoughcomplications of surface morphology are problematic with this grazing-exit condition.A further development by Tsujiet al (1995) is to combine the incident- and exit-beamtechniques to enhance the depth sensitivity. Essentially the variation with incident anglefor a detector at 90◦, 2ω′, the sensitivity is moderate; whereas for the detector in the planeparallel to the surface the fluorescence yield is very sensitive to differences in depth. Theangle that the detected fluorescence makes with surface can be tuned to enhance the signal,and when combined with the variation in the angle of incidence the possibilities may besignificant. Tsujiet al (1995) have studied the diffusion of Au in Si to obtain the variationin the Au concentration (10s of %) over depths of∼ 50 A.


5.4. Microscopic composition

The measurement of the ‘microscopic composition’ is defined as that which cannot bedetermined by spatially defining the beam and extracting the ‘macroscopic composition’locally. The methods used are necessarily more complex since the experiment or analysismethod has to cope with the surrounding matrix. I shall deal with two methods here, onethat is generally applicable and the other that applies to periodic multilayers, however bothmethods are only applicable to ‘ideally epitaxial’ layer structures since the effects are subtle.

5.4.1. Standing wave methods.As described in a previous section an incident wave caninterfere with its diffracted wave and produce a standing wave. However, in the case ofdiffraction the periodicity is locked into the atomic lattice thus creating a very fine probethat can investigate the composition variations at the atomic level. The method depends onthe layer being ‘ideally’ perfect and produces fluorescent x-rays or emits photo-electronsfrom planes perpendicular to the diffraction vector. The position of the anti-nodes variesfrom mid-way between the atomic planes to the atomic planes as the crystal is ‘rocked’from one side of the diffraction profile to the other. The high interaction of the x-rayswith the lattice at this point leads to increased inelastic scattering processes. Again anexpression can be derived for the fluorescent yield as a function of angle of incidence,Koval’chuk and Kohn (1986). Since these authors have extensively reviewed this workthere is little point in repetition. However, for the analysis of impurities (type and position)in structures that may occupy interstitial or substitutional sites then this type of probe isexceedingly powerful. The combination of diffraction and photo-electron spectroscopy isalso a very promising technique where the observable emission depth is< 1 µm comparedwith fluorescent radiation which is several microns, Koval’chuk and Mukhamedzhanov(1983). These diffraction experiments are possible with sealed x-ray sources although muchof the work now is carried out on synchrotron sources with various collimator arrangements,Kazimirov et al (1992).

To obtain the positions of atoms on the surface of a crystal, that is their lateral positionthen creating a lateral standing wave is possible by examining planes perpendicular to thesurface plane. This is a non-trivial experiment, however, Sakata and Hashizume (1995)have carried out such an experiment on a Si (111) surface. This is work for the synchrotronand specialized vacuum equipment. Sakata and Hashizume (1995) have carried out detaileddynamical theory calculations that required consideration of Laue (transmission) and Bragg(reflection) case diffraction to determine the electric field strength at the sample surface.The position of the adsorbed As atoms above a Si surface could then be determined frommonitoring their fluorescent yield with incident-wave direction.

5.4.2. Lattice parameter effects: amphoteric impurities.Impurities or dopants will alterthe lattice parameter in solids since in general they will have a different covalent radii tothe matrix. This effect is used to determine the phase extent in polycrystalline and perfectcrystal layers. Some dopants in semiconductor materials have a choice of atomic sites andfor the electrical conductivity it is very important to know this distribution. Si is a knownamphoteric dopant in GaAs and under certain conditions of growth it will preferentiallyoccupy Ga sites or As sites. The difference in the site occupancies will give the conductiontype but the number of charge carriers will not match the Si concentration. To increase thecharge carriers before the solubility limit of the dopant is reached the idea of creating layersof the dopant within the semiconductor matrix has a lot of attractions (δ doping). Hartet al(1993) have studied Siδ-doped GaAs multilayers to try and extract the site occupancy. The

1380 P F Fewster

Figure 12. The experimental and simulated profiles of a 60 period GaAs/Si multilayer, 004reflection and CuKα radiation.

Figure 13. The 002 reflection from the same sample as figure 12, illustrating the excellent fitsobtainable with dynamical diffraction theory to extract the site occupancy of Si.

overall Si concentration in the GaAs(500A)/δ-Si periodic multilayer relates to the positionof the ‘average’ or zeroth order satellite peak, whereas the satellites relate to the thicknessesof the GaAs and Si layers, figure 12. However, this single profile, although producing anexcellent fit, does not uniquely determine the site occupancy, so the 002 reflection wasmeasured and modelled. The 002 reflection relies on the difference of the scattering fromthe Ga and As sites and this revealed that Si only existed in significant numbers on the Gasites. The 002 reflection and fit to the satellites is given in figure 13.

These results are consistent with other analysis techniques (SIMS and LVM infraredabsorption). Additional information that naturally results from fitting were that the periodvariations and delta dope spreading were determined as 3% and at the 2-monolayer levelrespectively. At these levels this method is purely dependent on the knowledge of thestraining effects of Si. The covalent radii for tetrahedrally bonded atoms have been


determined by a number of authors, including Pauling (1960) and Phillips (1973), but tobe absolutely sure detailed calculations of the actual environment of the atoms is required(Jones 1995).

5.5. Macroscopic form

Determining the crystallographic form of a thin layer does not largely differ from that ofbulk material except that the intensity can be very weak. Clearly the difference betweenepitaxial (essentially perfect crystal) and polycrystalline is relatively trivial to determinewhereas whether a layer is amorphous or crystalline is less clear.

In a ‘perfect’ epitaxial layer the correlation lengths are well defined and create a discreteset of diffraction peaks and similarly for polycrystalline material, whereas in amorphousmaterial the range of correlation lengths is ill defined. The calculation above is basedon nearest neighbours whereas additional diffraction maxima may exist from next-nearestneighbours, etc. However, the formula of Debye (1930), that will produce these secondarymaxima, is only appropriate to within a few per cent for structures of the extended latticetype, since many have a highly disorderedfcc or hexagonal form by virtue of their bonding,Guinier (1963). In general though the position of the maxima give a correlation length fromthe Bragg equation.

5.6. Microscopic form

Determining the proportion of amorphous and crystalline forms in a thin layer is noteasy because of the difficulty in observing the amorphous peak, although it is possible bycalculation as given above. An example is given in figure 14 where the different proportionsof crystal material with different degrees of long-range order have been modelled, butthis does require very careful measurements and very careful background determination.The importance of understanding the energy for recrystallizing amorphous Si with a laser,for integrating crystalline and amorphous components, requires some indication of theseproportions. Mathe et al (1992) have approached this problem by determining the ratio

Figure 14. A thin layer of recrystallized amorphous Si showing the different long-range orderspresent in the sample. The correlation lengths were determined by modelling using the Scherrerequation (11% of the sample had lengths of 500A and 89% consisted of 45A lengths). Theaddition of calculated noise and careful background fitting assists the analyst in obtaining reliableresults.

1382 P F Fewster

of the intensity of the 111 reflection form of Si to that of the immediate background andplotting this against the laser energy. The extrapolation toI111/Ibackground→ 0 then gavethem the threshold energy for crystallization.

Other subtle effects of recrystallized material, in this case a thermal SiO2, have beenstudied by Takahashiet al (1993). They have observed small crystalline regions in thisSiO2 amorphous matrix that have an epitaxial relationship and occupy a few per cent of thelayer volume. They detected a weak diffraction peak on the tail of the substrate that has awidth that corresponds to the layer thickness and from modelling this profile they concludedthat these SiO2 micro-crystallites nucleate at the interface and are of the order of a unit cellwide. This data was obtained with a high-resolution diffractometer on a synchrotron source.

5.7. Macroscopic orientation

There are several aspects to the determination of macroscopic orientation and these areapproximately classified below. Basically the general orientation of the film perpendicularto the surface can be determined rather easily with a simple scan, whereas to determinethe full orientation texture requires a very detailed analysis which is outside the scope ofthis review. However, some technique developments will be included that aid the analysisfor thin layers, which basically only differs from bulk in that the scattering is weaker.Non-centrosymmetric crystals are also very important for consideration since their physicalproperties are often determined by these polar directions.

5.7.1. Basic analysis of orientation texture.The orientation of ZnO thin layers for peizo-electric thin films can be analysed in a crude routine way by undertaking a symmetric(ω = 0.5 (scattering angle)) scan on a slit-based diffractometer, McCullochet al (1985).It is assumed that for a randomly orientated ZnO film that the ratio of the 100 and 002reflected intensities give a calculated integrated intensity ratio ofIr100/Ir002 then

p = 2

( Ir100Ir002

I002

I100 + Ir100Ir002

I002− 1

2

)whereIhkl are the measured intensities of the reflections. This expression essentially onlygives a quantitative estimate of the orientation texture normal to the surface plane and solid-angle spread crystal-plane normals is defined by the horizontal and vertical divergences ofthe diffractometer used.

To extend this to obtain the orientation in the plane requires scanning along directionswhereω 6= ω′ (half the scattering angle) or using film methods. The camera of Wallaceand Ward (1975), discussed earlier, has been used very successfully for determining thegeneral orientation of thin layers. The positions of the peaks on the photographic film havea relatively straightforward geometrical relationship to the scattering angle and the relativeorientation of the scattering planes in the sample.

5.7.2. Pole figure and orientation distribution methods.Experimentally the determinationof the distribution of the orientation of crystallites involves the mapping of the intensityfrom a series of diffraction planes that are sufficiently representative of the space-groupsymmetry. The detector is set to the scattering angle for each of the chosen reflections andthe sample is then tilted and rotated to create a map of the intensity and hence ‘probability’function of the crystal orientations or pole figure. To create the orientation distribution ofthe crystallites requires involved mathematical methods of combining the results of the pole


figures, Dahms and Bunge (1989); Jarvinen (1993); Yuet al (1995). The general methodof determining the orientation distribution function has been reviewed recently by Bunge(1992). These methods are used for bulk structures but can equally well be applied to thinlayers.

Morawiec (1991) has derived a series of relationships that relate the intensity measuredat any given point in the pole figure to have a depth dependence and by combining these theorientation distribution as a function of depth can be determined. Bonarski and Morawiec(1992) have applied this method with a standard texture goniometer to study Al foils packedtogether, each having a different orientation texture. This procedure works well for systemswhere the change in texture is abrupt. Nikolayav and Ullemeyer (1994) have shown thatreducing the data collection time and smoothing the resultant pole figure maps of intensitydoes not alter the conclusions. Having a thin layer is very similar and therefore this could bea route to determining the orientation distribution of thin layers provided that the interferenceof the substrate reflections is considered.

Jarvinen (1995) has also considered the case for an in-depth variation of orientationtexture, since different reflections will be influenced in very different ways depending ontheir penetration and hence change the intensity distribution. In thin layers this could beimportant. If a standard parafocusing diffractometer is used for undertaking the textureanalysis of thin films then measuring intensities away from the symmetrical (i.e. forω 6= ω′

(half the scattering angle)) then the focusing condition is compromised and a correction to theintensities is required. The influence of these effects for the analysis of thin YBa2Cu3O7

layers on MgO has been demonstrated by Chateigneret al (1994) and depend on thepenetration depth, film thickness and the tilt used during the measurement. The degreeof defocusing and the broadening of the intensity profiles is more pronounced for thickerlayers and the intensity is reduced whereas for the substrate the effect is reversed.

However, many workers concerned with thin films are mainly concerned with themagnitude of the preferred orientation and can work quite well with just a suitable polefigure. But if the molecular structure is required then a good estimate of the intensities isneeded and consequently some form of texture correction will have to be performed beyondthat of a pole figure.

Energy dispersive methods create greater possibilities in geometry by introducinganother variable and Playeret al (1992) have used this to good effect in collecting pole-figure data. They have studied the orientation of 6000A erbium thin films on molybdenumusing a synchrotron source. Scanning the energy effectively collects different reflections soseveral pole figures can be collected simultaneously, however, the correction to the measuredintensity becomes quite complex (e.g. fluorescent radiation and film thickness). Playeretal (1992) kept the incident-beam direction constant by rotation of the sample about thisdirection to effectively vary the sample tilt and rotated about the sample normal with thescattering angle fixed to produce a series of pole figures.

The presence of a wavelength distribution can also bring complications to studies witha conventional sealed source. This arises from the relatively poor energy discrimination ofconventional detectors so that different crystallographic planes from the one of interestcan contribute to the measured intensity of the crystallographic planes diffracted withthe principal wavelength. Wenk (1992) has shown the advantages of using a graphitemonochromator between the source and sample for studying superconducting thin layers.The contribution of scattering from the substrate and the uncertainties introduced to thebackground estimation make integrated intensity measurements for evaluating texture lessreliable. This is especially important for these highly textured thin films. For weak texturethese effects are less significant.

1384 P F Fewster

Figure 15. A reciprocal-space map of YBa2Cu3O7 on MgO. The substrate peak is the 004reflection from MgO and the layer peak is the 0011 reflection. The angular displacement of thelayer peak (vertically) corresponds to the misorientation at this azimuthal position. Obtainedwith the low-resolution diffraction-space mapper, Fewster and Andrew (1993c).

5.7.3. Layer tilts with respect to underlying material.The principle behind this method isquite simple and is best illustrated by the reciprocal-space map given in figure 15 for a highTc superconductor grown on SrTiO2. This diffraction-space map is collected so that the longedge points to the origin of reciprocal space and therefore we can see that a line from thesubstrate peak to the origin does not intercept the layer peak. The angular displacement ofthis layer peak to this line is the misorientation of the layer with respect to the substrate. Thismap only represents one component of the misorientation and an orthogonal component willalso be required of the sample through its surface normal. For a well-aligned diffractometer,as in this case, the parallelism of the surface planes to the diffraction planes can be checkedby the deviation of the crystallographic planes from the central line of this map for whichω = ω′ (whereω′ is half the scattering angle), i.e. the symmetric diffraction condition.

The accuracy of this method is very dependent on the accuracy of locating the centreof the peaks, but for a good epitaxial layer this can be determined within a few secondsof arc using high-resolution methods, Fewster (1989a). Usually if the peaks are broad thenthere will be a distribution of misorientations (see next section). A reciprocal-space mapof a structure deposited on a misorientated substrate (i.e. the crystallographic planes areinclined to the surface plane) will yield this inclination angle. The dynamical streak fromthe surface plane will be normal to this surface and, inclined with respect to the radialdirection, form the reciprocal lattice origin. For a superlattice structure this effect can bevery pronounced since the satellites lie on a direction perpendicular to the surface plane,Neumanet al (1983).

This analysis can also be performed with more conventional diffraction equipmentwhere the detector window is large, e.g. a double-crystal diffractometer. With this typeof instrument the sample is rotated and the effective incidence angle for diffracting the twopeaks for two azimuths 180◦ apart will yield the relative angular misorientation,

1φ = 12(1ωπ − 1ω0)

where 1ω is the difference in the incidence angle for the layer and substrate planes todiffract. Again the accuracy of this method is high depending on the intrinsic diffractionwidth of the layer and substrate. Problems of a spread in mosaic-block orientationsignificantly broadens the peaks and the accuracy declines.

5.7.4. Polarity determination. The differences in scattering from two opposite faces ofa crystal in a polar direction are very small and are often enhanced by choosing an x-ray wavelength close to the resonance absorption edge of one of the atoms. Fewster andWhiffin (1983) have used such an approach to ascertain the orientation of a CdTe substrateand correlating these to various polarity-sensitive etches determined the polar direction of


a subsequently grown CdHgTe epitaxial layer. However, the method used opposite sidesof the sample to measure the intensity ofhkl and −h − k − l reflections. Stevenson andWilkins (1989) have made use of Bijovet pairs accessible from the top surface in CdTe toessentially undertake the same experiment. This method can, therefore, be applied aftergrowth but does rely on the sample appearing to be infinitely thick for simple analysis.However, the diffraction profiles can be determined with dynamical theory for these Bijovetpairs to determine the intensities from thin layers, although Stevenson and Pain (1990) haveextended the kinematic approximation to micron-layer thicknesses.

5.8. Microscopic orientation

Orientation at microscopic level, as defined previously, is concerned with the spread oforientations about some mean macroscopic value. Clearly in the extremes a ‘perfect’epitaxial layer will have no orientation distribution and a ‘perfect’ polycrystalline layer willhave no orientation texture, whereas all real-layer structures are between these extremes.We can, therefore, consider the problem as deviations from these two extremes and themethods are all similar in that they determine the shape of the diffraction peaks or the rangeof diffraction.

5.8.1. Mosaic-block orientation distribution.The distribution of tilts of mosaic blocks ina layer will exhibit itself as an extended region of scattering as we rock the sample in thex-ray beam. In the limit of a perfect distribution of crystallites we will have a continuousunvarying intensity ignoring instrumental aberrations. The simplest experiment is, therefore,to simply rotate the crystal about its axisω; this is used as a routine rapid analysis ofcrystalline quality in metallic multilayer systems, since the angular spread directly relatesto the distribution. When the effects are smaller then high angular resolution diffractionmethods are required which can be correlated with x-ray topographic methods. Gaoetal (1991) have studied Cd1−xMnxTe/Cd1−yMnyTe using double-crystal rocking curves incombination with Lang reflection topography. They found that the rocking curves weresensitive to tilts in the [110] direction and were very complex and broad, whereas thosealong [−110] were narrower. These results on their own could be misleading but bycombining this analysis with the x-ray topographs they were able to correlate the featuresobserved. They used the 004 reflection with CuKα1 radiation for both experiments.

Suzuki and Chikawa (1991) have taken a different approach to investigatingmisorientated mosaic blocks that they found applicable to GaAs on Si. The tilts werelarge (several minutes of arc). The instrument is composed of a finely collimated beam anda sample scanning and rotation mechanism so that the scattered intensity at 50µm areason the sample can be collected on a detector that selects the scattering angle through a slit.A whole series of these measurements are taken at different scattering angles to locate thepeak intensity and an image is constructed by plotting the peak position versus position ofthe sample. The angular sensitivity is of the order of a few minutes of arc and is sufficientfor reconstructing an orientation topograph. The orientation distributions in the GaAs werefound to be approximately 6 minutes of arc whereas the Si was a factor of 2–3 smaller.

A mosaic crystal will exhibit an extended intensity distribution perpendicular to thediffraction vector (diffraction plane normal) and this is most clearly observed in a reciprocal-space map, Fewster (1991a, b). From a reciprocal-space map the individual orientationscan be determined and in combination with topography the dimensions of the individualmosaic regions can be determined. The reciprocal-space map also separates strain effectsfrom differences in mosaic block tilts. An example of this is given for a ZnSe layer grown

1386 P F Fewster

Figure 16. The distribution of tilts evident in a ZnSe layer on GaAs. Figure 7 is a topographat the centre of the layer peak, 004 reflection CuKα radiation.

on a GaAs substrate, figure 16, where we can see that the layer is tilted with respect tothe substrate, within the layer there is a distribution of tilts and with topography we candetermine the dimensions of these tilted regions, figure 7.

Koppensteineret al (1993) have defined a correlated function which directly relatesto the Fourier transform of the reciprocal-space distribution of intensity multiplied by theFourier transform of the perfect reflectivity curve. If the root-mean-square distribution ofmosaic blocks orientation is1 then this correlation function is

G(r − r′) = P(|r − r′|) exp

[−2

3(πhρ1)2

]whereP(|r − r′|) is the probability of finding positionsr andr′ in the same block and istherefore shape dependent, andρ is the component ofr−r′ perpendicular to the diffractionvectorh. When this randomly misorientated disc-shaped mosaic-block model is comparedwith the random elastic-deformation-field model (Holy et al 1994a), Koppensteineret al(1993) could distinguish between distortions and misorientations by fitting the model shapesto the distribution of scattering in the reciprocal-space maps for Si–Ge superlattice structures.

Of course the contribution due to the mosaic-block orientation and correlation lengths areconvoluted and a simple approach (Fewster 1996b) to separate these two components is toexamine reflections from diffraction planes inclined to the surface. The spread in orientationsof the mosaic blocks creates a broadening perpendicular to the diffraction vector, whereasmosaic-block size components are not confined to this direction. This can be observed bythe characteristic elliptical shape having an inclination to the diffraction vector, figure 17. Ifthe mosaic blocks extend through the depth of the layer yet have limited lateral dimensionsthen the components of the microscopic tilt is given by:

1ωL2 = cos(φ + ξ)

sinφL3

where the various parameters are described in figure 17.

5.8.2. Three-dimension diffraction-space mapping.The concept behind this method isthat a three-dimensional sample will create a three-dimensional diffraction pattern not onlyin its distribution of diffraction maxima but also in the shape of these maxima. This is avery high angular resolution technique and has been applied to semiconductors, Fewster andAndrew (1995a), and polycrystalline bulk samples, Fewster and Andrew (1996). The generalprinciple behind this method is to create an effective three-dimensional ‘δ-function’ that is


Figure 17. The various components that contribute to the shape of the reciprocal-space map ofan imperfect epitaxial layer.

scanned through diffraction space and thus builds up the distribution of intensity. Fewsterand Andrew (1995a) have used this method to examine the mosaic structure of a AlGaAsepitaxial layer on a GaAs substrate and were able to measure the relative misorientations ofthe epitaxial blocks of the layer with respect to each other and with respect to the underlyingsubstrate. This method is of such high resolution that it was possible to detect the deviationsfrom perfect epitaxy within each mosaic block that is continuous from the substrate throughto the layer. This is a time-consuming approach but reveals a wealth of information that isinaccessible to other methods. An example of a three-dimensional diffraction-space map isgiven in figure 8.

5.9. Macroscopic distortion

Stress levels can have a significant impact on adhesion of thin films and the disruption ofthe interface by the formation of dislocations and cracks. To measure these stress levels werequire a good value for the elastic-strain tensor and good values of the interplanar spacingsof the material concerned. X-ray diffraction is a very powerful tool for determining theinterplanar spacing. An important point to be mentioned is that x-rays in general measurethe elastic stresses whereas strain gauges, etc, do not distinguish between elastic and plasticstrains.

5.9.1. The evaluation of stress in polycrystalline materials.If the strength of scatteringfrom the thin layer is sufficient and the material is polycrystalline with a reasonably randomorientation distribution to examine a high Bragg-angle reflection at several tilt angles thenthe strain distribution as a function of crystal-plane to surface-plane angle,9, can bedetermined. Barret (1952) has shown for an isotropic sample that deforms homogeneouslywith a uniaxial stress ofσ‖ parallel to the surface then the associated strainε‖(9) will be

1388 P F Fewster

given by:

ε‖(9) = σ1

E[(1 + ν) sin2 9 − ν]

where E is Young’s modulus andν is Poisson’s ratio. Hence plottingε‖ as a functionsin2 9 for a given reflection will yield the stress if the elastic parameters are known or theelastic parameters of the stress are known. This can be generalized for anisotropic strainvariations in the surface by resolvingσ‖ into (σx cos2 8 + σy sin2 8) where8 is the anglebetween the projected direction andx axis.

The strain measurements require high-precision lattice-parameter measurements and so avery good understanding of the capabilities of the instrument is required. For polycrystallinethin films and laboratory sealed x-ray sources this requires high intensity and good angularresolution either with instruments having parafocusing geometries over large angular offsetsin 9 or pin-hole collimator arrangements. If theε‖(9) versus sin2 9 plot is not linear thena probable cause is that the stress varies as a function of depth. This nonlinearity has beenstudied in detail by Vinket al (1991) on columnar Mo thin layers. The variation in stress isdiscussed in terms of the detailed microstructure and interaction between the columns andvoids, with the degree of texture increasing towards the surface. They also found that thepeaks were highly asymmetric and attributed this to stress variations along the columns andclose to the voids through relaxation.

To obtain in-depth variations in stress and especially subtleties in the surface thenthe complexity increases and the most reliable approach using conventional diffractiongeometries is based on using a stable inverse formulism with trigonometric basis functions,Wern (1995). This method, as that above, is applicable to isotropic materials and can beapplied to conventional texture goniometers Wern and Suominen (1994, 1996). It is basedon assuming that the stress state can be represented by a Fourier series that can have an‘effective’ period greater than the depth of interest. This will then allow the stress gradientto be determined up to the surface of the sample and also by judicious choice of integrationlimits allow thin layers to be studied in this way, Noyanet al (1995).

Because of the complexities of the orientation texture on this simple linear relationshipvan Acker et al (1994) have devised a constant-penetration-depth method of analysingstresses in thin films. The problems of nonlinearities due to orientation texture and samplingdifferent volumes is reduced. The incident beam is kept small and constant at about 5◦

and by measuring different Bragg reflections this effectively gives the lattice parametersat different χ values. Van Ackeret al (1994) have also used the same principle fordetermining the orientation texture in thin films although this analysis is always incompleteespecially with directions close to the surface normal. However, this approach is really anecessity to determine the stress values from the strain analysis measurements of the regionmeasured, since the x-ray elastic constants are derived from all the crystal orientations andthis distribution function will weight the transformation.

Doerner and Brennan (1988) have approached the problem in a very different wayby using the ‘non-coplanar’ geometry first proposed by Marraet al (1979). The incidentx-rays impinge on the surface at grazing incidence and the scattering planes studied areperpendicular to the surface plane. The depth penetration can vary from a few atom planesto the whole-layer thickness by varying this angle of incidence close to the total externalreflection critical angle, see figure 11. Since the sensitivity to depth is very sensitiveto incident angle accurate calibration or goniometry is necessary. Doerner and Brennan(1988) carried out a study of 2600A and 6000A Al films deposited on Si wafers using asynchrotron source and measuring the (220) interplanar spacing perpendicular to the surfaceas a function of depth. The thinner film was found to be uniform in strain with incidence


angle whereas the 6000A film showed a significant variation of the order of 0.1% suggestingthat relaxation does occur close to the surface of thicker films.

5.9.2. Evaluation of strains in non-epitaxial layers with single-crystal substrates.Thestresses in thin films deposited on crystalline substrates can be inferred from the warpingof the underlying substrate. This requires the elastic parameters of the two materials andtheir relative thicknesses to be known. The principle is quite simple in that it dependson the change in incidence angle of a well-collimated beam between two positions on thesample. The radius of curvature,R, is then related to the change in rotation of the sampleto maintain the same incident angle,1ω, for scattering to occur from different regions onthe sample separated by a distance1x by the expression:

R = 1x

1ω.

To obtain sufficient precision the separation should be as large as possible, the angularresolution as high as possible and the translation mechanism should be perfect! One of thefavoured goniometers for this work is with a Lang camera since this has a very reliabletranslation stage with minimal wobble, Fewster (1991c). The accuracy ranges from 1% at10 m radius of curvature to 4% at 100 m for a 4-inch diameter sample for example. Fromthe radius of curvatureR we can simply determine the stress,

σ = E

6(1 − ν)

t2substrate

tlayerR

whereE is Young’s modulus,ν is Poisson’s ratio andt are the thickness of the variouscomponents. Liet al (1989) have used this method to study the stresses in GaAs on Siunder various conditions and came to the conclusion that as the epitaxial layer thicknessincreased the curvature increased yet the stress was unchanged.

5.9.3. The evaluation of strain in perfect and inhomogeneous epitaxial thin films.Theevaluation of strain in epitaxial thin films is approached with high angular resolutiondiffractometers and can be determined very precisely on an absolute or relative scale.The section on composition measurement using simulation methods where we rely onthe relationship between the differences in lattice parameter between the substrate andlayer works well for perfectly crystalline samples. However, if there is uncertainty in thesubstrate lattice parameter then we require a more absolute method that relates the interplanarspacing to some external standard. Often the most convenient internal standard is the x-raywavelength since this exists on all laboratory sources with characteristic radiation, Bond(1960). However, comparative methods can be used provided a good reliable standardis obtained, Bowen and Tanner (1995). The method of Bond (1960) has been exploitedby Bartels (1984) in combination with a low divergence dispersive monochromator and byFewster (1982) in a ‘reverse’ double-crystal diffractometer to resolve the layer and substratereflections. This relies on the determination of the difference in the two angular positionsof the sample for diffraction from opposite ray paths from the same set of diffractionplanes. The resulting angle is purely a function of the Bragg angle. After correctionfor the refractive index this will give a lattice-parameter value within 1 part per million.Care must be taken though over the interpretation since the peak positions of single layers< 0.5 µm and buried layers< 2 µm are very sensitive to diffraction effects and ideallythe profile should be simulated to allow for this, Fewster and Curling (1987). It is alsoimportant to remember that for highly asymmetric geometries (incident angles< 5◦) the

1390 P F Fewster

refractive index differences become apparent, Pietsch and Bochard (1987) and so any simplepeak comparison should account for this or the profile should be simulated using dynamicaltheory. The scattering volume for these two angular determinations is different and thereforerelies on homogeneous samples. To overcome this Fewster and Andrew (1995b) have takena very different approach, that is a single measurement, and rely on high angular resolution ofa multiple-crystal diffractometer that angularly selects the scattered beam. The precision isthe same as that of the Bond method since sample centring and incident-beam misalignmentsare eliminated. Fewster and Andrew (1995b) have compared the two methods and appliedtheir method to inhomogeneous and polycrystalline materials.

Fatemi et al (1993) have carried out a detailed study of the shapes of various GaAsand related heterostructure compounds on Si using a double-crystal diffractometer. Themismatch in all cases is large. By combining the peak separations from a whole seriesof reflections; 004, 026, 224, 115 and 135 types they concluded that the layers have atriclinic unit cell with deviations from orthogonality close to 0.005◦. The layers weregrown on samples of varying offcuts from the (001) planes. This work illustrates that tofully characterize a thin layer to the highest resolution will result in a triclinic structureand careful measurements of a large unique set of reflections is required to determine theincreased number of parameters.

5.9.4. Evaluating strains in relaxing epitaxial layers.There has been a significant numberof publications on the relaxation of epitaxial layers because of the importance of being ableto know the limits of combining different material systems with significant differences inlattice parameters (Orders and Usher, 1987; Wie, 1989), the results have been comparedwith Raman spectroscopy (Macranderet al 1988; Halliwell et al 1989). Dunstanet al(1991) have combined a whole series of results to show that a logarithmic plot of thestrain versus thickness produces a linear curve above the critical layer thickness until verylarge layer thicknesses are considered. All these works are obtained using a double-crystaldiffractometer which requires a higher level of assumptions than using reciprocal-spacemapping where the tilts and strains are naturally separated.

The strains in relaxing epitaxial layers is best done by reciprocal-space mapping, Fewsterand Andrew (1993b). The alternative method involves taking a series of eight double-crystalrocking curves and leads to lower precision, Halliwell (1994). However, this may beadequate but requires more knowledge of the structure since there can be considerable peakoverlap from the consequential peak broadening, van der Sluis (1991). The basis of bothmethods is to measure the tilt component between the layer and the substrate and componentsof the lattice parameters both perpendicular and parallel to the surface plane, figure 17.The calculations are relatively straightforward and just rely on the determination of peakseparations and in the case of reciprocal-space mapping can result in the determination ofdegree of relaxation of less than 1%, which is defined as

R = a‖ − asubstrate

a − asubstrate

wherea‖ is the lattice parameter parallel to the interface anda is the average lattice parameterof the layer in bulk form. Below these levels it is possible to see individual dislocations bytopography to estimate the relaxation. A series of diffraction-space maps for a structure thathas a relaxed interface between the substrate and a thick buffer layer is given in figure 18.The misalignment of the buffer layer and superlattice peaks and the substrate peak for the004 reflection indicate a tilt between the layer and the substrate. The 115 reflection comesfrom planes inclined to the surface and therefore has a component of the lattice parameter


Figure 18. A series of diffraction-space maps from a partially relaxed 6µm layer of In0.1Ga0.9Ason GaAs. This thick buffer layer supports a superlattice structure of{[In0.5Ga0.95As] 100 A +[GaAs] 100A} × 10 with a 200A GaAs top layer.

parallel to the surface; after accounting for this effect this can be determined. Fewsterand Andrew (1993b) have analysed a series of these structures of different buffer-layerthicknesses in combination with topography to postulate a mosaic-grain-growth model athigh levels of relaxation.

The lattice parameter from planes parallel to the growth direction can be examined bymeasuring the diffraction from the side of the wafer if it has a good cleaved surface. Thediffraction from the layer will be weak compared with the substrate because of the relativethicknesses but has been used on a ZnTe single layer and a ZnSe/ZnTe superlattice onGaAs, Yanget al (1990). These results were combined with those obtained from planesperpendicular to the growth direction to obtain the lattice distortions. The complicationsof mosaicity and anisotropic relaxation can make this interpretation more complex. Thismethod would have to be weighed against using inclined planes with a significant lattice-parameter component perpendicular to the growth direction measured with very highresolution as described above.

Another approach to directly analysing the lattice parameters, and hence relaxation paral-lel to the interface plane, is to use incidence angles close to the critical angle for total externalreflection and investigate planes perpendicular to the surface. This gives a way of investigat-ing the degree of relaxation as a function of depth (Munekataet al 1987; Pietschet al 1993).Pietschet al (1993) have modelled the diffraction kinematically but had difficulties resolvinga unique model of the relaxing structure but they did conclude that the relaxation did takepart in domains with a distribution of in-plane lattice parameters for their InGaAs layers onGaAs. The information depth was considerably less than the structure thickness and the con-clusion was that this method should be complemented with more conventional geometries.Similarly Munekataet al (1987) also detected a domain structure with different strains.

Evaluating the strain in multilayers requires a combination of methods if we wish toknow the individual lattice parameters that compose a modulation period for example. In theprevious example for determining the thicknesses and compositions in multilayer structureswe have assumed that the lattice parameter parallel to the interfaces is constant. Becauseof this we could use a few variable parameters and calculate and compare the intensities

1392 P F Fewster

from different models. Since the scattering from a superlattice creates an ‘average’ peakand satellites due to the modulation we cannot easily obtain the actual lattice parametersperpendicular and parallel to the interfaces. To overcome this problem Birchet al (1995)have combined the results of reflectometry and reciprocal-space mapping. Reflectometrygives the individual layer thicknesses by simulation and comparison and a combinationof the average lattice parameters reduces the number of variable parameters to produce aunique solution. This was applied to a Mo/V superlattice and it was possible to derive thereduction of the coherency strain, which in this case corresponded to a relaxation of 33%.

5.10. Microscopic distortion

The distortion of crystallites in a polycrystalline sample is often referred to as microstrainand is determined in the same way as first proposed by Warren and Averbach (1950). Thebasic method has been refined and extended to give more detailed information but theprinciples are unchanged, this has been covered in an earlier review by Langford and Louer(1996). There are special problems with the analysis of thin layers, weaker scattering,smaller crystallite sizes, etc, but essentially the same physical principles apply.

5.10.1. Dislocation analysis in polycrystalline thin films.To obtain more detailedinformation about dislocations, Burgers vector directions, and stress levels in polycrystallinethin films detailed analysis is required on the breadth of the diffraction profiles as wellas their position. Vermeulenet al (1995) have derived some analytical expressions todetermine the dislocation configuration in Al layers on Si. The principle is based on thevariation of line broadening in different directions and relating these to the contributionof dislocation Burgers vectors and slip planes. Influences due to instrumental smearingeffects and crystallite size have to be subtracted. From this study they were able to showthat the density of dislocations with Burgers vectors inclined to the surface was two ordersof magnitude greater than those with Burgers vectors parallel to the surface. Also theydetermined that the dominant dislocations were screw in nature and that these dislocationsdecreased linearly in density as the stress relaxed at room temperature.

5.10.2. Dislocations in nearly perfectly crystalline material.The general presenceof dislocations and other microdefects will create diffuse scattering that has a shapecharacteristic of the reciprocal-space transform. The diffuse scattering though can be dueto several origins and this can be revealed by diffuse-scattering topography, Fewster andAndrew (1993a). However, once the predominant or proportion of defects is known thenthe diffuse scattering can yield useful information on the details of the distortions due tomicrodefects. The overall shape and position can be related to the local distortion andthe defect shape. The influence of broadening of the diffuse scattering due to misfitdislocations has been discussed by Fewster (1992) in which the strain fields from thedisrupted interface creates diffuse scattering extending normal to the diffraction vector witha parallel component that relates to the local relaxation. By modelling this diffuse scatteringthe strain distribution close to the interface was determined, the association of this diffusescattering with the presence of dislocations was determined by topography. The influenceof threading dislocations is less well defined because they are not linear and confined to theinterfaces. Kyuttet al (1995) have concluded from their study of highly mismatched GaSblayers on GaAs that threading dislocations were randomly distributed. Their reasoning wasthat the diffuse scattering broadening along the diffraction vector was linear with tanθ ,Ryaboshapka (1981), and did not conform to a cosθ broadening with a sinθ broadening


Figure 19. The interpretation of the diffuse scattering in the reciprocal-space map. This gavean estimate of the size of the dislocation strain fields at the early stages of relaxation in anIn0.09Ga0.91As layer on GaAs.

perpendicular to the diffraction vector. The latter components would be characteristic ofsize effects. Because the density of defects was high this could be confirmed with electronmicroscopy.

Reciprocal-space mapping offers the most sensitive approach to examining the diffusescattering and has been used for very low misfit dislocation densities by Kidd and Fewster(1994). They detected very weak diffuse scattering close to the layer peak that has beeninterpreted as illustrated in figure 19. The transformation from reciprocal space to real spaceclearly reveals two regions, those that are perfect and large and those that are small anddefective. This gave the extent of the dislocation-strain fields laterally which correspondedclosely to the layer thickness. This model fitted well with the concept that the strain fieldwill extend until it reaches some strain-relieving boundary. These results were combinedwith simulation of the diffuse scattering projected onto the diffraction vector and otherstructures of varying thickness to determine a very detailed understanding of the evolutionof the structure close to the critical-layer thickness. The nature of the diffuse-scatteringdistribution has been modelled using the kinematical theory based on the work of Krivoglaz(1995) by Kaganer and Kohler (1996) and qualitative agreement has been obtained.

5.10.3. Interparticle separation determination.Belish et al (1992) have measured theseparation between C60 spheres in a thin film on float glass and single-crystal Si substrates.The films were spin cast from solution and studied with reflectometry by analysing theintensity variation along the diffraction vector normal to the surface and perpendicular tothe diffraction vector. Determining the critical angle gave the electron density of the filmand from the transverse scans the ‘Yoneda’ peaks stayed constant for the bare glass substratebefore and after deposition suggesting that these correspond to the ‘critical angle’ for thesubstrate. Hence the electron density of the glass substrate could also be determined withinthe structure. The interface and surface roughnesses were found to be small and by plottingthe measured film thickness against the expected number of monolayers deposited they

1394 P F Fewster

derived the monolayer thickness as 10.9 (1.0)A (assuming the structure to befcc). Fromthe electron density they obtained 10.6A which is larger than the bulk value of 10.0A.

5.11. Macroscopic homogeneity

Some thin layers depend on structural features smaller than the probe size of thediffractometer and this gives us a series of choices, either we can reduce our probe sizeto match the feature or analyse the scattered information using a large probe. The formermethods are labelled macroscopic homogeneity and the latter microscopic homogeneity.These are clearly x-ray definitions and both methods have their place.

A common application of studying the macroscopic homogeneity is wafer mapping ofsemiconductor layers to assess the ‘quality’ and composition by monitoring the diffractionprofile half height width and the substrate–layer peak separation. Clearly these methodsassume that both assumptions in the interpretation are correct, see sections on compositionand orientation, and of course that the variation varies slowly on the millimetre scale.An example of this was undertaken and reported in Woodbridgeet al (1991) which wasconcerned with the In desorption across an InGaAs layer on a GaAs substrate. The majordifficulty with In is that for growth temperatures above 560◦C the In re-evaporates andshows up as significant lateral inhomogeneities in the composition.

Of course if the features are on a smaller scale∼ 100–30µm, then microdiffractiontechniques are possible using a microfocus x-ray source and total reflecting glass capillaries.Pin-hole collimators reduce the intensity by about an order of magnitude and so are lessfavourable. Eatoughet al (1995) have applied these methods to study the domain structurein Pb(Zr, Ti)O3 under electric-field bias. By collecting a large-angle diffraction profile froma device size region (0.7 mm diameter) with and without an applied bias they could observesubtle differences in the diffraction patterns indicative of 90◦ domain wall formation thatwas a function of grain size. The conclusion was that the 90◦ domain wall formation andswitching is very limited in sub-0.5 µm grain size material.

5.12. Microscopic homogeneity

Microscopic inhomogeneities are those that exist as variations within the probing-beamdimensions and exhibit themselves as changes in shape of the profile. Many ofthese influences have already been discussed; microscopic distortions and local chemicalinformation with standing-wave analysis, etc. These are mainly associated with variationsin the average signal from the detector, however, examining this distribution of intensity inthe scattered beam can be very revealing. By placing a high spatial resolution area detectorin the scattered beam (usually photographic emulsion) we can obtain a topograph that givesthis distribution of intensity across the probed area. This technique is generally only used onnear-perfect materials and is a very powerful method of detecting dislocations, misorientatedgrains, twins, etc. This is a well-established technique although applications on thin layersare less widespread because the contrast is generally weak for thin layers because ofdiffraction effects. Examples of mosaic block sizes has been described and similar methods,using multiple-crystal topography (Fewster 1991a, b), have been demonstrated by Huet al(1995) for examining thin surface regions of laterally periodic (∼ 20 µm period) domaininverted nonlinear optical materials. Huet al (1995) were also able to study the defectcontrast differences in the inverted and non-inverted regions. This contrast arises from thetilts from strain relaxation compared with the surrounding material. The depth of inversionwas approximately 1 mm.


For examining the twin structure in thin layers there are a choice of methods dependingon the type of twin, etc. Topography will give the dimensions of the twin regions and workson the principle of imaging one set of orientations at a time. This has been carried out toidentify the dimensions of twins in CdHgTe layers grown in CdTe. It has been knownfor some time that Cd0.2Hg0.8Te grown on (111) CdTe substrate will form 180◦ twinsquite readily and in fact the diffraction pattern will reveal an almost six-fold symmetryindicative of equal contributions of both twins (Horning and Staudenmann 1986). A seriesof topographs from the inclined 115 diffraction planes reveal one twin set whereas diffractingfrom the 115 planes of the other twin set after rotation of 60◦ about the surface normalinverted the image, Fewster (1991c). The twins were found to be of 20µm dimensionsand very irregular. The topography camera used was first proposed by Berg (1931) anddeveloped further by Barret (1945). This type of microstructure is very detrimental to deviceperformance (Capperet al (1989)).

Topography is not wholly confined to near perfect structures but has also been appliedto Langmuir–Blodgett films, Lider and Koval’chuk (1992), who used an analyser crystalto expand the width of the scattered beam from the sample. The reason for this is thatthe characteristic length scales in the lead stearate films was close to 50A and they wererestricted to very low scattering and incident angles giving rise to a very narrow projectionimage of the sample. By rocking the sample or sample and analyser they were able toisolate orientation effects from strain effects and observe 0.5 mm mosaic blocks with angularspreads less than 25′′ arc.

The high temperature phase of YBa2Cu3O7−δ has an orthorhombic unit cell with longc axis and similara andb axes. Growth of these thin layers on a substrate with a cubic-cell parameter close toa and b will create a finite probability of both orientations andmonitoring and controlling these orientations is important to obtain the desired physicalproperties. Measuring the individual proportions relies on topographic methods discussedabove or more simply by obtaining a digitized signal of both components. Low-resolutionreciprocal-space mapping, Fewster and Andrew (1993c), will give a more reliable method ofestimation if the twins are small, the material is poor quality and the scattering is weak. Bysuitable choice of reflections and measuring the intensity distributed in reciprocal space it ispossible to separate the scattering from the two orientations, figure 20. The important aspectto observed from this type of analysis is that there is no simple scan that will allow thetwo reflection profiles to be compared. However, the integrated area of the two reflectionsshowed that the ratio was 5:4.

Figure 20. The reciprocal-space map obtained with a slit-based diffractometer of aYBa2Cu3O7−δ layer on SrTiO2 for the 038 and 308 reflections. The existance of these tworeflections in the same region of reciprocal space indicates two orientations that are in proportionto their intensities and structure factors (CuKα radiation).

1396 P F Fewster

5.13. Macroscopic interfaces

The influence of rough interfaces on the diffraction pattern can be quite subtle and oftenrequire simulation and modelling methods to quantify the extent. First though we shalldistinguish between interface extent or smearing perpendicular to the plane of the interfaceand roughness that has a characteristic correlation length parallel to the plane of the interface.In general the interface extent will modify the scattering normal to the interface plane andtherefore can be observed from diffraction planes parallel to the surface with a very simplescan. Whereas interface roughness will result in diffuse scattering and requires examiningthe diffraction off from this direction and ideally should be collected by reciprocal-spacemapping.

We shall first examine the case of interface-extent measurement. Again severalapproaches will be discussed, modelling the integrated intensities and simulating thediffraction profile. Generally to obtain small-scale interface information will requiremeasurement of the intensity distribution remote from the average reflections of the layerof interest making multilayers an ideal structure for studying these effects.

5.13.1. Interface extent by modelling the integrated intensity.This can be a very convenientand fast method for evaluating interfaces especially for multilayers. The approach can bemost simply explained by taking the analysis discussed earlier where the square of thestructure factor (in the kinematical theory approximation) gives the scattered intensity. Verysimply then the Fourier transform of the structure factor amplitude represents the distributionof atoms (scattering factor) within the periodic repeat unit. So all that is required is theatom type (and its charge state to determine the scattering factor, Cromer and Waber (1974))and its fractional coordinates within the periodic unit. Hence the structure factor amplitudeis given by

Fhkl =N∑j

fj exp(−2π i[hxj + kyj + lzj ])

whereN is the number of atoms in the repeat unit. Hence we can very simply calculatethe structure factor amplitude of a repeat unit, which in this case is the multilayer periodincluding an interfacial grade. The scattering factor,f , will vary across an interface havingpartial occupancy of the intermixing atoms and an average interatomic distance should alsobe incorporated to reflect this change in composition.

This approach has been used by Fewster (1986, 1987, 1988) for semiconductor materialsand by Fullertonet al (1992) for metallic multilayer systems. Although the approach isvery general. A very important aspect of this type of analysis as discussed in the sectionon thickness measurement is that there will be some incommensurability in the period ofmodulation and the individual thicknesses of the layers and this must be taken into account.The general change in the diffraction profile from interfacial smearing is to reduce theintensity of the satellites, especially those of higher order, since these effectively representthe higher-order Fourier coefficients in the synthesis of the composition profile.

The method of determining the interface extent directly by using Fourier transforms asdescribed in the section on thickness has been used by Fleminget al (1980) to examine theinterdiffusion of Al in AlAs/GaAs superlattices on annealing. This method is very directbut does assume that both interfaces are identical because of the limited structure factorphase information. This assumption for interdiffusion is probably valid but for conditionsof segregation or natural roughness this may not be adequate. This is not a restrictionon the modelling and fitting method given above, Fewster (1988). Fewsteret al (1991)


have undertaken a detailed study of many structures based on the AlGaAs system and haveconcluded that the GaAs to AlAs and GaAs to AlGaAs interfaces are smoother that AlAsto GaAs which is smoother than AlGaAs to GaAs.

Growth of more complex semiconductor structures from the vapour phase can cause thecreation of different compounds at the interface and have dramatic effects on the satellites,arising from the gas sequencing. The diffraction from these structures can be modelledeither by using the modelling as above or by simulation and comparison (Vandenberget al1988; Vandenberget al 1988; Lyonset al 1989).

5.13.2. Interfacial extent characterization by direct determination.Manciu et al (1995)have tried to make reflectometry more transparent by taking a series of approximations tohelp relate structural parameters to the reflectrometry profile. They approximate the Abelesmatrices to create an analytical form that is valid for low reflectivities. The intentionhere is to develop a method suitable forin situ analysis. They have worked out theaverage individual thicknesses by looking at individual intensities by examining the envelopefunction of the periodic fringes in the pattern, where the intensity of thei the fringe,Ii is

Ii = R20 + 4R2

i N2 sin2

(D1

πD2

)+ (−1)Nl+i−14R0RiN sin2

(D1

πD2

)R

roughnessi = Ri exp

(−

(8π2

λ2

)S2 sin2 θ

)assuming that there is no absorption andN is the number of periods.R0 is the reflectivityfrom the surface andRi is the reflectivity at the top of the underlayer. The effect ofroughness was incorporated with a Debye–Waller factor and for the situation with manylayers 4N2R2

i � R20 then a plot of logI versusθ2 gave an estimate of the surface extent.

WhereI is the intensity at the nodes of the fringes since it is the rate of fall off of this‘truncation rod’ that is most affected. The internal interfacial smearing on the other handinfluences the fringe strength and hence measuring several successive fringe intensities andcomparison with the above equation will give an indication for this smearing. The approachassumes that all interfacial spreading are equivalent.

The presence of roughness can also have a significant effect on the satellites whenthe constituent layers have lattice parameters related by a simple ratio. Neerinck,et al(1995) have shown that for certain conditions of roughness some of the satellite peaksbecome unobservable in a Fe3O4/CoO multilayer when the CoO layer has a small thicknessfluctuation.

5.13.3. Diffraction profile simulation to determine interface extent.The structuresdiscussed in the previous section could equally well be studied by modelling the diffractionprofile and varying the parameters that are considered to be important. This has been usedquite extensively for multilayers but can be a lengthy process since it also requires a goodfit to the peak shapes as well as the intensities. This requires a very good experiment,preferably on a high resolution diffractometer with a small instrument function, that reducesthe intensity and therefore is less sensitive to interface smearing.

Clarke (1987) has approached the modelling of the diffraction profile of an Ir/Ru metallicmultilayer by effectively building the structure and carrying out a fast Fourier transformand comparing this output with the measured profile. This is effectively an extension ofthe method used by Fleminget al (1980). An intermixing ratio is included and the wholestructure can be included. Of course the whole approach becomes unwieldy with too many

1398 P F Fewster

parameters included. If the quality of the superlattice is good then the number of satellitescan be very high and the sensitivity to the roughness increases because of the increasednumber of satellites. Hartet al (1995) have used this to good effect for determining theroughness and period variations in an Alδ-doped GaAs superlattice.

Holy et al (1990) have determined the influence of different interface roughness types interms of their influence on the diffraction profiles. They used a semi-kinematical diffractionmodel and considered structural models with random thickness variations, as consideredin the approaches above, correlated roughness and lateral total thickness variations withroughness. By comparison with a GaAs/AlAs superlattice they were able to ascertainthat it corresponded to the latter configuration since all satellite intensities diminished andbroadened, whereas the second configuration would not effect the zeroth-order satellite andthe first only decreases the satellite heights. Therefore in any simulation approach it isimportant to have a good model or at least know the sensitivity of the model to the variousparameters. The influence of systematic- and random-period variation are also given byFewster (1996a), where the satellite shapes may be of similar width but totally differentshape. Fullertonet al (1992) have also considered this influence of random thickness andcumulative thickness variations in metallic multilayers by introducing a Gaussian distributionfor the interface distances.

Determining interface extent in non-periodic semiconductor multilayers can be carriedvery easily by simulation of a high-resolution diffraction profile, but not at the fine scaleas that for periodic structures. Bensoussanet al (1987) have shown that details in thefringe structure can be used to detect interfacial grades at the 0.02 µm level. They haveshown graphically how the ratio of the major layer fringes varies as a function of interfacialgrade assuming that it approximates to an erfc function as in diffusion processes. Of coursesome grades are less symmetrical, as in segregation, and these are best fitted by simulation.Provided that the data is of good quality then dopant segregation can be observed forinterface spreading that is about 10% of the layer thickness, which in absolute terms can bea very small width.

Reflectometry is very sensitive to interfacial roughness and interface extent and thelatter can be modelled by just including either a Debye–Waller factor or intermediate layersof varying composition (Boheret al 1990; Novet and Croce, 1980). Care has to be takenthough if the roughness is on a scale close to the lateral correlation length of the roughnesssince this will determine whether intensities or amplitudes are added. If the intensities areadded then this is analogous to introducing a Debye–Waller factor. If the lateral roughnessbecomes too great then the diffuse scattering created can produce enhanced apparent specularreflectivity that could also be misleading, de Boeret al (1995b). A reasonable estimate ofthe diffuse scattering on the specular reflectivity can be obtained by scanning radially justoff the specular (surface-normal) direction, although the diffuse scattering can vary rapidlydepending on the defects causing it.

5.14. Microscopic interfaces

The microscopic interfacial information refers to the actual roughness determination (lateralhillocks and undulations), compared with the more macroscopic interface smearing whichcan result from roughness when the roughness is averaged on a larger scale.

5.14.1. Interfacial roughness characterization by direct determination.Interfacialroughness introduces a lateral correlation length that will produce broadening for lengthsshorter than the coherence length of the experiment. The width of the profiles perpendicular


to the diffraction vector will now be a complex shape arising from the perfect region resultingin a width determined by the coherence length and other correlation lengths related toroughness hillocks, Fewster (1991b). For a superlattice the ‘average’ or zeroth-order peakwidth will be dominated by the coherence length and acts as a useful internal standard.The additional broadening of the satellites will arise from the roughness and this widthshould increase with satellite order as the correlation lengths are averaged over smallerthicknesses and because they are more sensitive to the detailed structure of the interfaces.The correlation lengths can, therefore, be determined by extracting the reciprocal latticecoordinates and transforming, Fewster (1996b). Chrzan and Dutta (1986) have deriveda two-dimensional structure factor for a superlattice that includes a variance in the layerthickness per unit distance parallel to the layers. This will produce a better estimate of themagnitude of the roughening whereas the simple transformation method also requires anestimate of the interface extent.

Aspects of the influence of dislocations from disrupted interfaces due to layer relaxationhas been covered in the section on microscopic distortion, where again lateral correlationlengths are extracted from the diffuse scattering.

5.14.2. Interfacial roughness characterization by simulation.The influence of interfacialroughness is most acute on the diffuse scattering and this is best analysed by reciprocal-spacemapping, as discussed previously. To model the diffuse scattering Holy (1994) has defineda mean lateral correlation length and taken all the interfaces to be represented by a fractalsurface. By taking the Fourier transform of the associated covariance the distribution ofthe diffused-scattering distribution in reciprocal space is determined using the kinematicaltheory. In this way Holy has been able to show theoretically that the diffuse scatteringshapes differ for mosaic structures, strain gradients and interfacial roughness.

The sensitivity to interfacial roughness was stated earlier to be very acute inreflectometry measurements. The resulting diffuse scattering has been largely analysedby conducting ‘rocking’ scans, i.e. examining the scattering perpendicular to the diffractionvector although more recently the modelling of reciprocal-space maps has been showinghigher prominence (Holy et al 1995; de Boeret al 1995b). The diffuse scattering can appearvery complex but is now largely understood. Much of this work on the simulation of thediffuse scattering (non-specular scattering) has been pioneered by Sinhaet al (1988). Theircalculations are based on the distorted-wave Born approximation which approximates to theoptical theory but will also naturally include the lateral roughness to create diffuse scattering.This is a perturbation method and therefore is only strictly valid for rms roughnesses,σ 6> 1/q whereq is the specular component of the wave vector. Whenever the incident orscattered rays make an angle equivalent to the critical angle of the surface then there is anenhancement of the diffuse scattering, Yoneda (1963), giving rise to characteristic ‘wings’in a scan perpendicular to the diffraction vector or ‘angles wings’ in a reciprocal-space map.If σ � 1/q then the Fourier transform of the diffuse-scattering profile along this scan willyield the height–height correlation function of the interface. Sinhaet al (1988) have useda self-affine random Gaussian model for the interfaces whereas other works have includedfractal models. Holy and Baumbach (1994) have extended this approach to the analysisof modelling the interfaces from periodic multilayers. De Boer (1996) has considered thevalidity of different approaches in extracting roughness and has proposed a formulation thatmay be valid for both small and large correlation lengths (the latter can be modelled wellusing the ideas of Strutt and Rayleigh (1896)). The validity of the various approaches havebeen considered in terms of analysing the total scattering, which in some models can exceed

1400 P F Fewster

that from the incident beam!The Sinhaet al (1988) model, however, does not work so well for modelling the off-

specular scattering from Langmuir–Blodgett multilayers, Stommeret al (1995). In this casethe multilayers were fatty-acid salts and the diffuse scattering was thought to arrive fromdomain structures of 100A dimensions and not be dominated by interfacial scattering. Soas with any modelling the dominant scattering process has to be considered.

5.15. Macroscopic density

The magnitude of the scattering is directly related to the electron density of the solid. Indiffraction the scattering includes strain and scattering strength effects but in reflectrometrythe x-rays are only sensitive to the latter. The position of the critical angle in thereflectrometry profile is very sensitive to the density of the material. To obtain this positionaccurately it is important to know the absolute angle and to model the diffraction profilebecause the critical angle is not as sharp because of absorption effects. With small samples(5 × 5 mm2) a combination of single-crystal monochromator and slits can be used andthis will give densities to within∼ 5%, de Boeret al (1965). For larger samples (a fewcm2) it is possible to increase the accuracy to< 1% by using crystal optics as demonstratedbelow by virtue of the high precision possible in the angular selectivity and absolute angularmeasurement, Fewster and Andrew (1995b). We have discussed already how Miceliet al(1986) have used the reflectometry to determine the composition in periodic multilayers bymeasuring the refractive index. Clearly we have to be aware that reflectometry, althoughbeing insensitive to crystalline quality, is sensitive to x-ray density that can vary fromamorphous to polycrystalline to single-crystalline material.

In this one example of the variation of critical angle with x-ray density for a∼ 300 ACr layer on glass, figure 21. It is possible to see that by determining the exact incident-beamdirection and determining the exact scattering angle (Fewster and Andrew, 1995b) this valuecan be determined very precisely. The scattering has been modelled using the Parrat (1954)formulism.

Figure 21. The variation in the position of the critical angle for two thin Cr layers depositedon glass. The density varies from 7190 to 6750 kg m−3, i.e. from the bulk value to densitieswell below this indicating significant porosity in these films. The film thickness is also obtainedfrom modelling the profile.


6. Diffraction equipment

The range of diffraction equipment is very extensive but again as we can see fromcharacterizing the various material types we can do likewise with x-ray diffractometersand cameras. Although now the differentiation between cameras (photographic film-basedmethods) and diffractometers (counter-based methods) is less distinct. This is because ofthe developments in area detectors based on new technologies.

Essentially x-ray diffraction equipment can be categorized into low- and high-angularresolution methods, the former diffractometers relying on slits to define the x-ray beamdirection and the latter relies on crystal reflections. There are also combinations of bothforms giving a significant diversity for the range of problems. Slit-based geometries definethe x-rays spatially, i.e. a small narrow beam has a higher angular resolution, whereas crystalreflections define the beam angularly and can therefore have large cross-sections. I shallbriefly mention two diffractometers that have specifically benefited the analysis of thin filmsin recent years, that also illustrate the points that have been raised.

For analysis of polycrystalline materials on a synchrotron Hart and Parrish (1986) madeuse of a parallel-plate collimator in the diffracted beam. The advantage of this geometryis that it can be used at grazing incidence to enhance the scattering from thin films. Theparallel-plate collimator allows angular selection of the expanded beam caused by the lowangle of incidence beam on the sample. The scattering is therefore dominated by that fromthe top thin layer. This is a very useful instrument for enhancing the scattering from thinpolycrystalline films, figure 22. There is very limited x-ray wavelength selection with thisinstrument making the overall resolution poor.

An instrument that defines the scattering by crystal optics and reduces the wavelengthdispersion and the horizontal divergence is illustrated in figure 23, (1λ/λ ∼ 7.8 × 10−5

and horizontal divergence∼ 6.7′′ arc for an intensity of> 106 photons s−1 and background< 10−1 photons s−1, using a 2kW CuKα sealed source). This is the high-resolution multiple-crystal multiple-reflection diffractometer (HRMCMRD, Fewster 1989a) that creates a nearδ-function-shaped probe to explore reciprocal space. The instrument-smearing effects arevery small and has been used extensively for reciprocal-space mapping described above.Reciprocal-space mapping at lower resolution can also be carried out by using slits insteadof crystals for weakly scattering materials, Fewster and Andrew (1993c).

Figure 22. The parallel-plate diffracted-beam collimator makes this geometry ideal for studyingpolycrystalline thin films by capturing the distributed scattering from a layer when using grazing-incidence geometry.

1402 P F Fewster

Figure 23. The high-resolution multiple-crystal multiple-reflection diffractometer that can havea large beam spatially but the divergence of the incident beam and scattered beam accepted aredefined by crystal reflections. The large spatial beam also permits topography to be performed.

The study of thin layers is not restricted to these arrangements and geometries andalthough the principle is similar in that we have to define our incident- and scattered-beamdirections, although in some cases the latter is poorly defined, with the resolution that suitsthe problem. For example both the diffractometers described can be used for reflectometry,etc. The geometry can appear complex although in general this just relates to the orientationof the sample with respect to the diffraction plane.

7. The future

The importance of thin-layer and multi-layer structures will continue to grow because theygive a greater range of structural, mechanical, electrical, magnetic and optical propertiescompared with those in the bulk. Analytical methods have been developing in tune withthese demands. X-ray methods have a special position in this respect because they interactwith structures at the atomic level and can investigate layer thicknesses in excess of thin-layer dimensions without the requirement to prepare or modify the sample. There is virtuallyno limit to the x-ray source flux that is accessible from a laboratory-sealed source thatmost of the reviewed work covers to third-generation synchrotron sources and beyond.The development in available instrumentation is rapid allowing complex geometries to beaccessible commercially and with easy-to-use software for experiment design and analysis.

The development in analysis methods for interpretation of the measured data willcontinue to increase primarily because of the increase in computer power. This will lead toincreasingly complex problems being solved and as yet there appears to be no limit to whatstructural problems can be solved by x-rays. The subtleties and scattering due to certainthin-layer structural parameters may be weak but with increasingly powerful sources (in thelaboratory as well as synchrotron sources) and advances in diffraction theory these shouldbe in our grasp. The limit is defined more by the imagination of the experimentalist thanthe problem itself.


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