Artificial superlattices

m'ü

•>

l

l

$ i<<

r

-%

(v

«Uv

Ui

ARTIFICIAL SUPERLATTICES

Proefschrift

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus, prof.drs. P.A.Schenck,

in het openbaar te verdedigen ten overstaan van een commissie

door het College van Dekanen daartoe aangewezen,

op dinsdag 20 september 1988 te 14.00 uur

door

Roland Henri Marie van de Leur,

geboren te Roermond,

natuurkundig ingenieur.

Dit proefschrift is goedgekeurd door de promotoren

prof.dr.ir. F.Tuinstra en prof.dr.ir. J.E.Mooij.

Het in dit proefschrift beschreven onderzoek is mogelijk gemaakt door

financiële steun van de stichting voor Fundamenteel Onderzoek der

VOORWOORD

De mens is altijd al op zoek geweest naar materialen die niet in zijn natuurlijke omgeving voorkomen en die eigenschappen hebben waarmee werktuigen verbeterd kunnen worden. Waarschijnlijk zullen velen bij kunstmatige materialen in eerste instantie denken aan kunststoffen. Echter ook de meeste metalen (denk aan de velen soorten staal) zijn kunstmatig; ze zijn door de mens gemaakt.

Het eerste succes van de mensheid op het gebied van kunstmatige materialen is wellicht de fabrikage van de legering brons, bestaande uit ongeveer 90% koper en 10% tin. Voor de bronstijd werden allerlei eenvoudige wertuigen vervaardigd uit koper welke verkregen werd door verhitting uit het in de natuur voorkomende kopererts. Koper is vrij zacht en daardoor kon vrij gemakkelijk enige vorm gegeven worden aan een koperen voorwerp. In de bronstijd konden sterk verbeterde en veelsoortiger werktuigen vervaardigd worden door de ontwikkeling van de giet-techniek. Brons heeft een lager smeltpunt dan koper waardoor gieten mogelijk werd. Omdat brons veel harder is dan koper zullen bronzen bijlen ook heel wat efficiënter geweest zijn dan koperen.

In dezelfde trant is er een zeer recent succes van de speurtocht naar nieuwe materialen; de ontdekking in 1987 van de keramische materialen die, tot relatief hoge temperaturen, een elektrische stroom zonder weerstand kunnen geleiden. Ondanks het feit dat duizenden jaren zijn verstreken sinds de ontdekking van brons, is de methode waarmee deze nieuwe materialen worden verkregen in wezen niet ver­ anderd. Het materiaal ontstaat door de samenstellende elementen in de juiste ver­ houding en onder de juiste omstandigheden bij elkaar te brengen. De natuur bepaalt welk materiaal uit dit proces onstaat en de menselijke inbreng bestaat slechts uit het creëren van de juiste omstandigheden.

Recente technologische ontwikkelingen openen nu de weg voor een geheel nieuw soort materialen. Tegenwoordig is het in principe mogelijk laagjes van slechts één atoom dik op elkaar te leggen. Het zal duidelijk zijn dat deze technieken, waarbij men op atomaire schaal de samenstelling in de hand heeft, een onuitputtelijke bron van nieuwe materialen zijn. Het aantal nieuwe materialen wordt hierdoor nu slechts beperkt door de creativiteit van de mens. Een van de eenvoudigste voorbeelden van dit nieuwe type materialen is een zogenaamde multilaag. Hierbij worden

afwisselen-de lagen van twee typen atomen op elkaar gelegd, waarbij elke laag afzonafwisselen-derlijk slechts enkele atomen dik is. In de onderstaande figuur is zo'n multilaag, waarbij de afzonderlijke atoomlagen afwisselend twee en drie atomen dik genomen zijn, schematisch weergegeven.

Dit proefschrift handelt over een dergelijk soort materialen. Deze zijn vervaardigd door het verdampen van de samenstellende elementen en het vervolgens laten condenseren op bijvoor­ beeld een glasplaatje. Het verschil tussen de in dit proefschrift beschreven kunstmatige super-roosters en de bovengenoemde multilagen is dat in de onderhavige superroosters de concentratie van beide elementen sinusoidaal veranderd. De in dit proefschrift beschreven superroosters hebben dus geen scherpe grensvlakken. Aange­ zien de laagjes, zoals gezegd, slechts een paar atomen dik zijn is het niet mogelijk met een microscoop te kijken hoe het materiaal eruit ziet. De techniek die hiervoor wordt gebruikt is ver­ strooiing van röntenstraling aan het materiaal. Röntgenstraling die op het materiaal valt zal in slechts een paar richtingen, welke bepaald worden door de posities van de atomen in het materiaal, verstrooid worden. Het meten van zowel de intensiteit als de ruimtelijke verdeling van de verstrooide stralen levert informatie over de atomaire opbouw van het materiaal.

Stellingen behorende bij het proefschrift "Artificial Superlattices" van Roland H.M. van de Leur.

I

De oplossing van een stelsel van drie lineaire vergelijkingen met drie onbekenden waarvan de coëfficiënten experimenteel verkregen zijn is, in tegenstelling tot hetgeen Lutts beweert, niet vrij van (systematische) meetfouten.

A. Lutts, Z. Kristallogr. 164, 31 (1983).

n

Om, uitgaande van de samenstellende metalen in een atmosfeer van ongeïoniseerd moleculair zuurstof, de keramische supergeleider YiBa2Cu307-y te vervaardigen

d.m.v. elektronenstraalverdamping zonder uitstookprocedure achteraf, zal men eerst het technologische probleem van de realisatie van de daarvoor noodzakelijke gradiënt in de zuurstofdruk tussen substraat en elektronenkanon moeten oplossen.

AJ.G. Schellingerhout e.a., Z. Phys. B 71, 1 (1988).

III

Of monokristallijne groei van een multilaag bestaande uit afwisselende lagen van elementen met een verschillende roosterkonstante mogelijk is, wordt mede bepaald door de groeirichting van deze gelaagde struktuur.

rv

De meting van transporteigenschappen van een twee-dimensionaal elektronengas in een magnetisch veld wordt sterk beïnvloed door de macroscopische afmetingen van het preparaat parallel aan het twee-dimensionale gas.

R.H.M. van de Leur, doctoraalverslag, T.H.Twente (1983) en W. van der Wel, proefschrift, T.U.Delft (1987).

V

De veel gehoorde mededeling dat de tegenvallende bedrijfsresultaten veroorzaakt zijn door een te hoge of een te lage dollarkoers dient men te verstaan als het niet adequaat

VI

Om het dynamische proces van de generatie van misfit-dislocaties op het grensvlak tussen substraat en epitaxiale film te kunnen begrijpen zal men eerst de elastische eigenschappen van de betreffende materialen bij de tijdens de groei van de film heersende temperatuur moeten onderzoeken.

vn

Een autoriteit die (het lezen van) bepaalde boeken verbiedt geeft daarmee uiting aan haar minachting voor de intellectuele vermogens van de lezer; dit verbod dient derhalve genegeerd te worden.

vm

De volgorde van auteurs, zoals vermeld boven een wetenschappelijke publikatie, geeft geen informatie over de daadwerkelijke bijdrage van elk der auteurs afzonder­ lijk.

IX

De opmerking van People en Bean dat hun model overeenstemt met de experimentele gegevens is misleidend omdat het betreffende model ongefundeerde (numerieke) aannamen bevat.

R. People en J.C. Bean, Appl. Phys. Lett. 47, 322 (1985) en 49, 229 (1986). X

Voor de emancipatie van wielrensters is het aan te bevelen te spreken over de "Tour masculin" in plaats van "Tour de France" als men van het betreffende evenement slechts de wedstrijd tussen mannen bedoelt.

XI

Een promovendus is een Boorman die aan zichzelf het Algemeen Wereldtijdschrift verkoopt.

Het in dit proefschrift beschreven onderzoek was een gezamelijk onderzoek van de werkgroepen Fysische Kristallografie (FK) en Supergeleiding (SG) aan de faculteit der Technische Natuurkunde. In dit onderzoek zou de expertise op het gebied van de fabrikage en de elektrische eigenschappen van dunne metallische films, aanwezig in de groep SG, gecombineerd worden met de kennis op het gebied van natuurlijk gemoduleerde strukturen, aanwezig in de groep FK, om zo elektrische eigenschappen van metallische superroosters, in het bijzonder de elektron-fonon wisselwerking, te correleren aan de kristalstruktuur.

Ondanks de inzet van de studenten GJ.A. Adang en M.T. van Wees hebben de elektrische metingen niet tot publiceerbare resultaten geleid. De bijdrage van de student J. te Nijenhuis aan het struktuur onderzoek is beschreven in paragraaf 3.2.

Gedurende de afgelopen jaren heb ik nauw samengewerkt met collega promovendus ir. A.J.G. Schellingerhout. Gezamelijk hebben wij met en aan de opdampinstallatie gewerkt. Het was zijn voorstel om, na alle verbeteringen die we hadden aangebracht, halfgeleider moleculaire bundel epitaxie (MBE) te gaan bedrijven met de installatie. Mede daarom, maar ook gezien de inhoudelijke discussies die we gevoerd hebben met betrekking tot dit onderwerp, moeten vooral de hoofdstukken 5 en 6 van dit proefschrift gezien worden als het resultaat van onze samenwerking.

Ook op deze plaats wil ik dr.ir. A.J.G. Schellingerhout bedanken voor de prettige samenwerking gedurende de afgelopen jaren.

TABLE OF CONTENTS

VOORWOORD in

TABLE OF CONTENTS ix

GENERAL INTRODUCTION 1

CHAPTER 1 CRYSTALLOGRAPHIC STRUCTURE 13

Introduction 14 1 Sinusoidal deposition rates 15

2 Substitutional modulation 25 3 Displacive modulation 31 4 Combined substitutional and displacive modulation 38

5 Nonperiodical distortions in alloys 42

Introduction 42 5.1 Fluctuation in deposition rates 43

5.2 Drift in deposition rates 50 5.3 Finite crystal size 55 6 Nonperiodical distortions in superlattices 56

7 Substrate roughness 64

References 69

CHAPTER 2 FABRICATION OF METALLIC SUPERLATTICES . . . 71

Introduction 72 1 Deposition apparatus 73

2 Epitaxial growth of V, Nb, Ta and their binary alloys 79

3 Epitaxial growth of superlattices 84

CHAPTER 3 STRUCTURE ANALYSIS OF

METALLIC SUPERLATTICES 91

Introduction 92 1 Experimental x-ray methods 92

2 A new four circle diffractometer scanning routine 100

3 Structure of a Nb/V superlattice 105

References 109

CHAPTER 4 CRITICAL MODULATION AMPLITUDE IN

NB/V AND TA/V SUPERLATTICES I l l

Introduction 112 1 Stiffness constants 113 2 Elastic energy 117 3 Results 121 4 Discussion 123 References 125

CHAPTER 5 MASS SPECTROMETER CONTROLLED

FABRICATION OF Sl/GE SUPERLATTICES . . . . 127

Introduction 128 1 Fabrication 129 2 Silicon deposition control 130

3 X-ray diffraction 131

4 Results 132 5 Conclusion 134

CHAPTER 6 CRITICAL THICKNESS FOR PSEUDOMORFIC GROWTH

OF Sl/GE ALLOYS AND SUPERLATTICES 137

Introduction 138 1 Critical thickness 140 2 Superlattices 148 3 Energy barrier and strain 152

4 Discussion 155 5 Conclusions 157

References 158

APPENDIX A STIFFNES S CONSTANTS FOR CUBIC CRYSTALS 161

APPENDIX B ANALYSIS OF Y1BA2CU3O7.Y THIN FILMS ON SAPPHIRE SUBSTRATES MADE BY ELECTRON

BEAM EVAPORATION 167

Introduction 168 1 Fabrication 168 2 Results before annealing 170

3 Results after annealing 172

4 Conclusions 176 References 178

SUMMARY 179

SAMENVATTING 181

Despite its very general title which in fact includes a very broad field in solid state physics and material science, this thesis only contains a study of growth and structural characterization of artificial superlattices. Besides the concept "artificial superlattice" a large number of terms appears in literature for similar types of structures. In table 1 the terms which occur most frequently in literature, and their abbreviation, are listed. We define artificial superlattices as crystalline alloys with a modulated concentration of the constituents in one spatial direction.

Table 1. Synonyms for artifical superlattices

superlattice multilayer film

coherently modulated structure compositionally modulated structure compositionally modulated alloy artificial modulated metal periodically modulated structure artificial superstructure film layered composite

layered synthetic microstructure synthetic modulated material layered ultrathin coherent structure artificial metallic superlattice

macroperiodic electronic super structure synthetic modulated structure

CMS CMS CMA AMM PMS ASF LC LSM SMM LUCS AMS MESS SMS

We fabricated artificial superlattices of random binary metallic and semi­ conducting alloys in which no phase separation occurs and which are homogeneous for all proportions of the constituents. In figure 1 such a superlattice, with a modulation period A, is shown schematically. In the atomic layers perpendicular to the modulation direction the constituents are distributed at random on the lattice sites in a proportion determined by the modulation function. The modulation function

defines the concentration of the constituents in each atomic layer. Such a modulation in the concentration results in a modulation in both the electron density (substitutional modulation) and the displacement of each atomic layer from the average position (displacive modulation). In figure 2 both modulations are represented separately.

h—

A

- H

FIG. 1. A two dimensional model of a superlattice with an average concentration

of 50% of one of the constituents and a modulation period A which is chosen to be commensurate with the average lattice.

Our definition of artificial superlattice also includes the type of superlattices that has attracted most attention in literature, namely those consisting of alternating thin layers of the pure components. The modulation function of these superlatticess has a rectangular shape. In these superlattices the degree of structural order is higher than in the superlattice shown in figure 1 because the random distribution which is essential to obtain a non-rectangular modulation is absent if the layers consist of one element. Several physical properties such as superconductivity,1-4 electronic

transport in the normal state5-7 and magnetism8-10 in metallic superlattices, and

photoluminescence11 and magnetoresistance12 in semiconductor superlattices have

been measured as a function of the modulation period and the thickness of the individual layers forming a modulation period. Typical modulation periods in these superlattices are 1-10 nm. These multilayers have found application already as lasers, neutron monochromators and x-ray mirrors. Many of these multilayers reported in

literature, however, do not fulfil the condition of crystallinity in our definition of artificial superiattices. A typical x-ray mirror13 e.g. consists of polycrystalline

tungsten layers spaced by an amorphous carbon layer. It has to be kept in mind that these types of structures are called superiattices in literature frequently.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

o O O O o o .

o O O O o o .

o O O O P o .

o O O O o o .

o

o

o

o

O O O o o

O O O o o

0 0 O o o

O O O o o

FIG. 2. A sinusoidal modulation function (a) leads to a periodical displacement

(chosen to be logitunal) of the atoms from the average position (b) and to a modulation in the electron density (c) symbolically represented by the size of the circles.

The semiconductor superiattices which we will discuss have a sinusoidal concentration modulation and are monocrystalline. The metallic superiattices which we will discuss are, however, not monocrystalline in the strict sence of the word but they give a diffraction pattern that hardly can be distinquished from monocrystalline material. For this reason we wish to speak of monocrystalline superiattices to distinquish them from polycrystalline superiattices where the crystallites have a large variation in orientation. In the latter superiattices the direction of modulation is not

confined to one crystallographic direction of the average lattice and thus quite different from the superlattices which we will discuss, where the modulation direction varies only a little arround one crystallographic direction.

The modulation in an artificial superlattice is commensurate or incommensurate with the average lattice. From a crystallographic viewpoint only the commensurately modulated structures are superlattices in the strict meaning of the word. In this type of structures translation symmetry exists and a crystallographic supercell can be defined. In an incommensurately modulated structure there is in principle no translation symmetry in the modulation direction. In an incommensurate superlattice the crystallographic unit cell has an infinite dimension in one direction. In spite of the objection in using the word superlattice for the incommensurately modulated structures we use it because also in incommensurate superlattices there is strict long range order and the term superlattice is widely spread under workers in this field. If confusion is possible we will call the superlattices incommensurate or commensurate as the case may be.

Artificial superlattices are not in thermodynamical equilibrium. Often there is a tendency to form an homogeneous alloy of the constituents or at least fading of the sharp interfaces in multilayers.14"16 There exist, however, superlattices in nature,

and many minerals and alloys have a modulation period which is long in comparison with the average unit cell of the structure. As a function of temperature a modulation may appear or disappear in these structures. In natural superlattices this proces is reversible while in artificial superlattices the applied modulation only disappears. In alloys where spinoidal decomposition occurs, the modulation appears at a certain temperature17 and can not changed at will. This modulated phase is the energetically

most favourable state of the alloy and despite its analogy with artificial superlattices these structures will not be discussed. Also of great interest is the transition from a commensurate phase to an incommensurate phase in certain natural superlattices as has been observed in for instance Na2C03. A considerable amount of theoretical work has been done in this field of natural superlattices both with respect to symmetry aspects and to energy considerations calculations to explain the origin of the modulation in these structures.18 In artificial superlattices, however, the

modulation is applied during the growth of the structure and the resulting modulated structure is not the energetically most favourable state for these alloys. Energy

considerations with respect to the structure of artificial superlattices are mainly restricted to the elastic energy corresponding with the concentration modulation in superlattices and the lattice mismatch between superlattice and substrate.

Artificial superlattices are grown on substrates and thus the growth depends on the properties of the interface between the substrate and the superlattice such as substrate symmetry, temperature and the presence of impurities. In multilayers also the interface between the constituent layers is important to obtain continued growth of such a superlattice. In superlattices where the modulation has no large gradients there is not an interface within the superlattice but nevertheless continued monocrystalline growth may be impossible due to the large strains. There is an enormous amount of literature on epitaxial growth of semiconductors and metals starting with the work of Van der Merwe and coworkers about four decades ago. For a review the reader is refered to the work of Matthews.*9 Except for for the semiconductor systems, most attempts to grow superlattices have been based on an emperical approach rather than on controlling composition, temperature, substrate structure and other variables to obtain a good superlattice in terms of crystallinity. Systematic studies of these variables have been reported only recently for metallic superlattices2^ and some 10 years earlier for semiconductor superlattices.

DuMond and Youtz14 were, in 1940, the first to fabricate an artificially

modulated thin film and their problems are strikingly similar to the problems we met in making artificial superlattices. Their aim was to fabricate artificially modulated thin films to calibrate the x-ray wavelengths which were not well known at that time. In contrast, we use nowadays well known x-ray wavelengths to determine the modulation period in the superlattices. DuMond and Youtz grew a Au/Cu superlattice on a glass substrate. In using a glass substrate, which indeed has been a very common substrate for thin film deposition until recently, they could of course not obtain epitaxial growth. In contrast, our aim was to obtain epitaxial growth and thus a proper choise of substrate material was very important. DuMond and Youtz obtained the concentration modulation in the film by keeping the copper flux on the substrate constant while modulating the gold flux. We applied the same procedure by keeping the Si flux constant and modulating the Ge flux in our Si/Ge superlattices. Our metallic superlattices, however, were fabricated by modulating both fluxes of the constituents in opposite phase. The latter method has the advantage that a larger

modulation amplitude can be obtained. To modulate the gold flux DuMond and Youtz used a pendulum clock and a relay to modulate the heating current trough the gold source ( see figure 3 ). The copper flux was kept constant by a constant current through the source.

FIG. 3. Control circuits, relays

and ratchet by means of which a -o pendulum clock was made to

o control the heating and cooling of

~~° the trough containing the gold (DuMond and Youtz, ref. 14).

Our deposition apparatus, however, has a feedback loop which keeps the fluxes at the desired value. This method of course allows a much better deposition control. Despite this more sophisticated control, our conclusion (before modifications were made on the system) was the same as that of DuMond and Youtz, namely that the deposition process was not accurate enough to fabricate well defined superlattices. Nevertheless DuMond and Youtz observed that the intensity of the x-ray reflection corresponding with the modulation decreased with time to about half the initial value after the film was kept at room temperature for two days. This decay was due to interdiffusion in the modulated film and provided a method to measure much lower diffusion rates than any other available method. Only in the late 1960's this diffusion work was picked up by Hilliard et al?-l& and became an important application23

mainly for metallic but also for the semiconductor layered structures.

In the early 1970's Esaki et a/.20 introduced a new type of electronic structures

composed of alternating semiconductor layers with sharp interfaces between these layers. If the thickness of one of the layers composing the modulation period is less than the electron mean free path in that layer, the entire superlattice enters into a quantum state. The modulation leads to subbands in the conduction band similar to the energy bands obtained by the Kronig-Penney model. In a superlattice with an, in one spatial direction, varying energy gap between the valence and the conduction

band as illustrated in figure 4, the first Bnllouin zone is devided in a series of so called minizones. en 0) C c o n d u c t i o n b a n d

y/// y/// yy//

WÊÊÈÊËÊ

_{yyy yyy yy//}

v a l e n c e band

1 1 » >

FIG. 4. A spatial modulation with

sharp interfaces of the energy gap in a semiconductor superlattice with a modulation period A in the z direction.

2A 3A

These minizones lead to narrow subbands separated by forbidden regions in the conduction band of the semiconductor with the smallest energy gap as shown in figure 5. A proper choise of the amplitude and the period of the modulation gives rise to a two-dimensional density of states which was earlier obtained in Si-inversion layers.

FIG. 5. Subbands separated

by a forbidden region in the conduction band of a semi-conductor superlattice with a periodical distortion of the

average potential.

mi, minizone; sb, subband; A, modulation period; E, energy.

The growth of superlattices with the very good crystal quality which is essential for this type of electronic structures became technically possible in the early 1970's. Since that time this field of solid state physics has been expanding rapidly. The theoretical work, concerning e.g. bandstructure calculations, the experimental work on dislocation-free growth of these superlattices which we will discuss in the chapters 5 and 6 and the device applications are nowadays such a large field that an international conference series with a few hundred participants is devoted to superlattices only.

The great interest in metallic superlattices24'25 dates from the early 1980's, a

few years after the observation of an enhanced biaxial modulus in these films by Hilliard, Tsakalakos et a/..26"28 This enhanced modulus was observed in films

originally produced for diffusion measurements. Although the mechanism leading to this so called supermodulus effect is not totally clear by now, the origin is thought to be a modification of the Fermi surface of the metal. The promising results in semiconductor superlattices and the supermodulus effect in metallic superlattices gave rise to the assumption that in these films interesting phenomena may occur with respect to electronic transport in the normal state, superconductiviy and magnetism. Superlattices consisting of a superconducting and a normal metal,2 9-3 0 a

ferromagnetic and a non-ferromagnetic metal,31'32 a ferromagnetic and a super­

conducting metal33 or a metal and a semiconductor34 attracted much attention. Often

only a little attention was payed to the crystalline quality of these structures. The Nb/Ta superlattice was the first metallic superlattice for which epitaxial growth conditions were investigated systematically in relation to die crystalline quality.35'36

Because Nb, Ta and their binary alloys have an identical lattice their are no strains in these superlattices. In this thesis we report detailed studies of the metallic superlattice structures where, in contrast to the Nb/Ta superlattices, both substitutional and displacive modulation are present.

The scheme of this thesis is as follows. The first four chapters concern metallic superlattices and in the remaining two chapters the semiconductor Si/Ge superlattices are discussed. In the first chapter we discuss the origin of the modulation as well as the effect of this modulation on the diffraction pattern. Because in real superlattices several, nonperiodic, distortions of the ideal structure influence the diffraction pattern

we pay attention to distortions which are likely to occur in metallic superlattices. Chapter 2 concerns the fabrication method and the growth of metallic superlattices. Because several x-ray methods have been used to determine the epitaxial orientation and the structure of metallic superlattices, these techniques are discussed in chapter 3 briefly. Also in chapter 3, the structure of one particular Nb/V superlattice is discussed in more detail. In chapter 4 we will show that there exists a maximum strain for "monocrystalline" growth of sinusoidally modulated Nb/V and Ta/V superlattices which sets an upper limit to the modulation amplitude. Superlattices, such as Si/Ge, of a very good crystalline quality can be used to determine the performances of the deposition control during the growth of these structures as we will illustrate in chapter 5. The last chapter of this thesis concerns the growth of dislocation-free Si/Ge superlattices. A model will be presented that describes the relation between the thickness of Si/Ge films grown on a Si substrate and the substrate imposed strain in these lattice mismatched films.

In May 1988, when (almost) the whole physical community was under the spell of the new high Tc superconductors we started a project, which took a few

weeks, on the fabrication of YiBa2Cu307-y thin films. In appendix B our efforts to

REFERENCES

1. MJalochowski, Z. Phys. B- Condensed Matter 56, 21 (1984). 2. M.G. Karkut, D. Ariosa, J.-M. Triscone, and 0 . Fischer,

Phys. Rev. B 32, 4800 (1985).

3. I. Banerjee, Q.S. Yang, CM. Falco, and I.K. Schuller, Solid State Commun. 41, 805 (1982).

4. W.P. Lowe and T.H. Geballe, Phys. Rev. B 29, 4961 (1984). 5. P.F. Carcia and A. Suna, J. Appl. Phys. 54, 2000 (1983). 6. H. Hoffmann and P.Kücher, Thin Solid Films 146, 155 (1987). 7. M. Gurvitch, Phys. Rev. B 34, 540 (1986).

8. H.J.G. Draaisma, FJ.A. den Broeder and WJ.M. de Jonge, J. Magn. Mat. 66, 351 (1987).

9. E.M. Gyorgy, D.B. McWhan, J.F. Dillon Jr., L.R. Walker, and J.V.Waszczak, Phys. Rev. B 25, 6739 (1982).

10. I.K. Schuller and M. Grimsditch, J. Appl. Phys. 55, 2491 (1984).

11. E.E. Mendez, G.Bastard, L.L. Chang, L.Esaki, H.Morkoc, and R.Fischer, Physica 117B/118B, 711 (1983).

12. L.L. Chang, H.Sakaki, C.-A. Chang, and L. Esaki, Hys. Rev. Lett. 38, 1489 (1977).

13. T.W. Barbee Jr, Proc. Topical Conf. on Low Energy X-ray Diagnostics, A.I.P. Conf. Proc. 75, p.131, (Amer. Inst. Phys., New York, 1981). 14. D.W.M. DuMond and J.P. Youtz, J. Appl. Phys. 11, 357 (1940). 15. R.M. Fleming, D.B. McWhan, A.C. Gossard, W. Wiegmann, and

R.A. Logan, J. Appl.Phys. 51, 357 (1980).

16. P.M. Petroff, J. Vac. Sci. Technol. 14, 973 (1977). 17. J.W. Cahn, Acta Metall. 9, 795 (1961).

18. A.C.R. Hogervorst, thesis Technical University Delft, 1986.

19. J.W. Matthews, ed. Epitaxial Growth, (Academic Press, New York, 1975). 20. L. Esaki, L.L. Chang, and R. Tsu,

Proc. 12th Int. Conf. Low Temp. Phys.,Kyoto, Japan p.551 (1970). 21. H.E. Cook and J.E. Hilliard, J. Appl. Phys. 40, 2191 (1969). 22. E.M. Philofsky and J.E. Hilliard, J. Appl. Phys. 40, 2198 (1969).

23. A.L. Greer and F. Spaepen in ref.6 p.419.

24. J.M. Cowley, J.B. Cohen, M.B. Salamon, and B.J. Wuensch, eds. A.I.P.Conf. Proc. 53, (Amer. Inst. Phys., New York, 1979). 25. L.L. Chang and B.C. Giessen, eds., Synthetic Modulated Structures,

(Academie Press, Orlando, 1985).

26. W.M.C. Yang, T. Tsakalakos, and J.E. Hilliard, J.Appl.Phys. 48, 876 (1977).

27. J.E. Hilliard in ref.2 p.407. 28. T. Tsakalakos in ref.2 p.422.

29. I.K. Schuller, Phys. Rev. Lett. 44, 1597 (1980).

30. W.P. Lowe, T.W. Barbee. T.H. Geballe, and D.B. McWhan, Phys.Rev.B 24, 6193 (1981).

31. B.J. Thaler, J.B. Ketterson, and J.E. Hilliard, Phys. Rev. Lett. 41,336 (1978).

32. I.K. Schuller, CM. Falco, J. Hilliard, J. Ketterson, B. Thaler, R. Lacoe, and R. Dee in ref.2.

33. H.K. Wong, H.Q. Yang, B.Y. Jin, Y.H. Shen, W.Z. Cao, and J.B. Ketterson, J. Appl. Phys. 55, 2492 (1984).

34. S.T. Ruggiero, T.W. Barbee Jr, and M.R. Beasley, Phys. Rev. B 26, 4894 (1982).

35. S.M. Durbin, J.E. Cunningham, and C.P. Flynn, J. Phys. F: Metal Phys. 12, L75 (1982).

36. S.M. Durbin, J.E. Cunningham, J.E. Mochel, and C.P. Flynn, J. Phys. F: Metal Phys. 11, L223 (1981).

CHAPTER 1

INTRODUCTION

For the investigation of the physical properties such as superconductivity, magnetism or elasticity of artificially fabricated superlattices, knowledge of the crystallographic structure of these superlattices is a prerequisite. In contrast with naturally grown crystals and alloys, artificial superlattices, which are usually grown by deposition techniques such as sputtering and electron-beam-evaporation, are not in thermodynamical equilibrium and so the structure of such a superlattice depends heavily on several parameters controlling the growth. In addition to the deposition rates of the constituents, the vacuum in the apparatus and the crystallographic orientation and temperature of the substrate strongly influence the structure of the superlattice. After deposition interdiffusion may result in interfaces of the alloy composition in multilayers.1"3 It is also possible that one of the constituents of a

multilayer imposes its crystal structure on the other, as has been observed in Nb/Zr multilayers.4 In these multilayers the Zr undergoes a structural phase transition from

the hep to a bec structure at small modulation periods. Measurements of the physical properties are not meaningful without having detailed knowledge of the crystal structure in advance. It is not only the modulation period or the thickness of the separate layers in a multilayer structure and the average crystal structure being important, also the crystalline quality in terms of defects and variations in lattice constant and composition are essential. Extended defects and other distortions of the ideal lattice are of fundamental importance for the consideration of physical properties of these solids.

A well suited technique to investigate the structure of artificially fabricated superlattices is x-ray diffraction. It is non-destructive which allows measurement of the physical properties after the structure has been determined. It provides us with information of the structure on atomic scale. X-ray diffraction also makes it possible to recognize several types of distortions in a crystal. To apply this technique effectively, it is important to have a model for several kinds of distortions to be expected in order to see what the effect of a certain type of distortion is on the diffraction pattern. X-ray diffraction gives us the structure of a crystal in relation with the average structure of the area of the crystal illuminated by the x-ray beam, i.e. the average over a macroscopic area. Typical areas are 0.4 mm2 for a crystal if

mounted on a four circle diffractometer and about 10 mm2 for a sample in a powder

diffractometer. The thickness of the samples, typically 0.5 |im, permits diffraction from the total thickness of the sample at not too low angles of diffraction. X-rays are diffracted by the electrons of an atom. To describe the diffraction pattern we therefore start with the classical distribution p(r) of electrons around the atom. The Fourier transform of this distribution gives the atomic form factor ƒ(&). In the quantum mechanical description/(fc) is the Fourier transform of the probability of finding an electron in a given area. The numerical values of f(k) used in this thesis are the quantum mechanical ones as given in the International Tables.5

Superlattices have an extra periodicity with respect to the average lattice. This extra periodicity results in extra peaks (satellites) in the diffraction pattern. The number of extra peaks and their intensity strongly depends on the type of modulation and the exact shape of the modulation function which describes the period, amplitude and phase of the modulation. The quality of the heteroepitaxially grown metallic superlattices allows us to use the kinematic theory of x-ray diffraction instead of the more complicated dynamical theory which has successfully been applied in the field of the ideal semiconductor superlattices and multilayers.6"8 In the latter case the

superlattices are dislocation-free or nearly so. For metallic superlattices the epitaxial growth conditions never led to dislocation free growth.

During the last ten to fifteen years many analyses have been published in the field of diffraction on natural incommensurate superstructures. For this purpose symmetry and diffraction theories in more than three dimensions have been developed.9 These theories seem unnecessary complicated for the,

crystallo-graphically simple, artificial superlattices. The essential elements and the terminology of these theories are nevertheless used here.

1. SINUSOIDAL DEPOSITION RATES

The non-square wave modulated superlattices reported in literature are obtained by interdiffusion of the constituent elements after deposition of a layered structure of the pure elements. It has first been shown by DuMont and Youtz10 that in such layered

solid solutions for every concentration. The arguments of DuMont and Youtz are based on the diffussion equation

dc(z,t) _ d2c(z,t)

dt ~U dz2 ' ( L 1 )

were D is the diffusion constant, c(z,t) the concentration of one of the constituents, t the time and z the spatial co-ordinate in die direction of the modulation. Describing the periodic composition modulation by a Fourier series, higher order harmonics in that series will, as a function of time, decay quadratically with their order according to equation (1.1). The consequence is that after some time of interdiffusion the composition modulation will become sinusoidal or nearly so before ending up as a homogeneous alloy.

Our purpose was to fabricate sinusoidally modulated alloys during the growth of the superlattice instead of achieving this result by interdiffusion. Modulation during growth presents better perspectives to obtain a larger variety of modulated monocrystalline alloys. The reason for this is that it is not necessary to grow a layered structure of the constituents first. In the case where monocrystalline growth of a layered structure of the pure components is not possible, a monocrystalline modulated alloy can not be obtained by die diffusion method. This situation occurs if the difference in the stress-free lattice parameters (misfit) of the constituents is too large to obtain monocrystalline growth of a layer with a thickness equal to half the modulation period of one constituent on top of the other. Modulation during growth, however, allows us to limit the amplitude of the concentration modulation and consequently to limit the maximum strain in the crystal. Even in the case were the amplitude is chosen such as to obtain one atomic layer of the pure elements in each modulation period, the total elastic energy in the crystal is lower than in a layered structure. This originates from the fact that the elastic energy depends quadratically on die strain.

In this section we will demonstrate, using a simple model for the growth of a superlattice, in which way sinusoidal deposition rates lead to a sinusoidally modulated superlattice. The derived expression for the substitutional and displacive modulation will form a basis for the discussion in subsequent sections. This expression is different from those reported in literature11-12 because we start with

modulated deposition rates leading to modulated structures while others use the sinusoidal modulation to fit their observed diffraction results from multilayers in which interdiffusion is assumed to occur. For the time being we assume in our model absence of interdiffusion. Because interdiffusion only decreases the amplitude of modulation and not the period, it is very easy to incorporate interdiffusion in our model as we will see.

OOO Q

msm

(222822282

©6>

O O

oo ooo

m* OOO OOO

tm 00000000

®#®ÉI@OOOOOO

0 # ® ® ® © 0 0 0 0 0

0 0 ® ® 0 ® © 0 © 0 0

ooo@©@®©o ■

~QQQQ§f@®P

000<§>®@©0

FIG. 1. Artificial superlattices can only be obtained if the growth

mechanism is layer by layer growth (a). If the growth mode is island growth (b), clusters of the constituents occur.

To obtain superlattices by a deposition technique it is important that the growth mechanism of the superlattice is planar growth in contrast with island growth. Only in the case of planar growth the modulation in the deposition rates is reflected in the crystal structure. This can easily be demonstrated for a layered structure but the same arguments apply to other modulated structures. To illustrate the effect of the growth mechanism we deposit the number of atoms corresponding to two atomic layers of each element alternately. In the case of planar growth we obtain a superlattice as indicated in figure la. If the growth mode is island growth a structure as indicated in figure lb occurs and no superlattice is obtained. In these two examples, the substrate is assumed to be flat on atomic scale. In the case where surface steps are present a more complicated situation occurs. But also in this case layer by layer growth (on each plateau) is essential to obtain a superlattice. The effect of the surface roughness

on the structure will be discussed in section 7.

A sinusoidal concentration variation in the superlattice during growth can only be achieved when the deposition rates of the constituent elements meet two requirements concerning the phase and amplitude of modulation. First, the deposition rates have to be modulated at opposite phase. This means that the rate of one element is maximal when the other reaches its lowest value. The other requirement is that the sum of both deposition rates has to be constant in time or, what is equivalent, the amplitude of modulation has to be the same for both elements. We restrict ourselfs to superlattices with an average concentration of 50% of the elements A and B and express the deposition rates R(i) in numbers of atoms per unit area and unit time. With the requirements mentioned before we obtain the following expression for the deposition rates of the elements A and B :

R A ( 0 = j R t o t [ 1 + Kcos(cot + (p) ] and

R B ( 0 = \ Rtort 1 - K-cos(cot + (p) ], (1.2)

with R^ the total depostion rate, K the amplitude and O) the angular frequency of modulation, t the time and <p the phase. There are several ways to obtain a description of the resulting superlattices. In a "gedanken experiment" we assume initially no displacive modulation and the concentration of element A is given by

1 r 2rtZ -i

C A ( Z ) = £ [ 1 + K C O S ( — + 9 ) ] , (1.3)

z A

where the time and angular frequency in equation (1.2) are replaced by the spatial co­ ordinate z in the modulation direction and the modulation period A. In a second step, relaxation due to the concentration variation is allowed by replacing z by

z' = z + P-cos( + (p), (1.4) A

where J3 is the amplitude of the displacive modulation. The combination of equations (1.3) and (1.4) leads to an expression for the concentration of element A as a

function of the spatial coordinate z' in the superlattice. This concentration can also be obtained from the deposition rates R(t) and the atomic volume of the constituents. In that case the spatial co-ordinate in the growth direction of the last deposited material is given by the sum of the deposition rates, integrated over the time T of deposition, i.e.

T

z(x) = ƒ VARA(t) + VBRB(t) dt

0

= ?Rtot { x [VA + VB] + -[sin(cox + 9) - (VA - VB)sincp ] }, (1.5)

z CO

with VA and V& the volume occupied by an atom A or B, assuming that the volume

per atom of an alloy A0.5B0.5 is the average of the volume of the constituents. At moment T the concentration of element A arriving at the substrate is

CAOO =2 t1 + Kcos(cox + (p)]. (1.6)

The combination of equations (1.5) and (1.6) leads to an expression for the concentration cA as a function of the z co-ordinate. It is evident that in the case of

monocrystalline growth, the direction of modulation z is in principle independent of the crystallographic orientation of the lattice. The growth planes of crystals grown from a melt or in a saturated solution are usually the most densely packed planes. For epitaxial growth by a deposition technique on a substrate, however, the growth direction is more likely to be perpendicular to the optical surface while the crystallographic orientation of the deposited material is determined by the crystallographic orientation of the substrate surface.

The preceeding two descriptions of the compositional modulation assume a continuous distribution of the concentration and disregard the corpuscular character of the solid. This description will be valid as long as the modulation period is much larger than the crystallographic cell dimensions. We will now take into account the time of formation of an atomic layer and assume that each atomic layer can be characterized by a single well defined average concentration. For each atomic layer the deposition rates are integrated over the formation time of a monolayer and so a

discrete concentration distribution in the modulation direction will occur. To simplify the resulting expressions, the deposition rates as given by equation (1.2) are now the deposition rates in monolayers per second. To obtain these rates the actual deposition rates in atoms per second per unit area have to be devided by the number of atoms per unit area in the alloy AQ.SBQ 5. The concentration of element A in the n-th atomic layer is obtained by integrating the deposition rate over l/Rlot seconds, the

deposition time of one monolayer, and is

n/Rtot 2 K R Q) 9 1

cA(n)= f RA ( 0 d t = U 1+ ^Isinfej—)-cos(~£-co +cp)].

(n-l)TRtot tó töt t0t

(1.7) The angular frequency co of the modulation in the deposition rate can be expressed in the average atomic layer spacing dav in the direction of modulation, and the

superlattice period A, i.e.

co = 27tdavRtot ( 1 8 )

A

With this expression for co, the concentration of element A in the n-th atomic layer can be written as

cA( n ) = 4 [ W p - c o s ^ " - 1 1 ^ ) ] . (1-9)

L A

with P = [KA/7cdav]sm(ndav/A). Thus, the amplitude of the substitutional modulation

in the superlatice is reduced by a factor [A/ndav]sin(ndaw/A) with respect to the

amplitude K of the deposition rates. This is a consequence of the assumed discrete character of layer by layer growth, equation (1.9) provides the average concentration in the n-th atomic layer, which is independent of the distribution of the atoms in this plane. In the following discussion we will assume that the atoms A and B are distributed at random in this plane. Only in this case it is possible to assign a uniform layer thickness to this atomic layer and a uniform modulation period throughout the whole crystal. The short range disorder due to the different atomic sizes of the constituents will not be discussed here. In the subsequent sections where we introduce a more general expression for the modulation, deviations from this ideal

situation will be discussed.

If we assume the validity of Vegard's law for the alloys Ai_xBx .which means

that the lattice constant of the alloy varies linearly with the concentration x, then the layer spacing can be expressed in the spacing in the same crystallographic direction of the constituents, i.e.

d = CA-dA + CB-ds, (1.10)

with dA and ^B t n e layer spacings in the elements A and B. A more exact discussion

requires that the contraction and stretching in the modulation direction due to the matching to the average lattice in the plane perpendicular to the modulation direction is taken into account also. In a cubic lattice with modulation along a principal axis these elastic effects only lead to a larger amplitude of the displacive modulation than deduced from the concentration in the way mentioned above, as long as Hooke's law is valid. We will discuss this effect at the end of this section.

The thickness of the superlattice after n atomic layers have been deposited since the starting point t = 0 is the product of the layer spacing and the deposition rate in layers per second, integrated over the time of deposition:

n/Rtot

D(n) = J [dA-RA(t) + dB-RB(t)] dt = n-dav + ^[sin(27tn ^-+ (p) - sin <p ],

(1.11) with rfav the average layer spacing and t, = KA-[d\-d^\IAKdaw.Ths. position of the

n-th layer is given by

z(n) =^[D(n) + D(n-l)]

= (n - i-)-dav + £■ { cos(7t^)-sin([2n-l]7t^+ (p) - sin (p } (1.12)

1 A A

and its thickness is (see Fig. 2 ) :

d(n) = D(n) - D(n-l) = dav + 2^-[dA - dB]-sin( -&-)-cos(n[2n-\]-®-+ q>).

A A (1.13)

The amplitude ^cos(JirfavM) of the displacive modulation can be larger than the layer

distances due to the cumulative character of the displacement. The cosine term in equation (1.12), which does not appear in continuum models, clearly illustrates that for small modulation periods the discrete character has to be taken into account. This can easily be demonstrated for a superlattice with A = 2dav. In that case the phase of

the modulation can be chosen such as to obtain alternating layers with different composition. This superlattice has only substitutional modulation and no displacive modulation. For superlattices with a long modulation period (A » dav) equation

(1.13) reduces to

d(n) = dav + hdA - dB]-cos(7t[2n-l]d-^+ 9). (1.14)

z A

OOOO

O O O O

I FIG. 2. To obtain an expression for displacive I and substitutional modulation in a superlattice I each atomic layer n is assumed to have a thickness

d(n) and an electron density p(n) which are both

OOOO-I

determined by the concentration of the

O O O O •»

OOOO-

1

OOOO

OOOO

The electron density p(«) in the n-xh layer is given by

in electrons per atom, where PA and PB the number of electrons per atom A and B. With equation (1.9) for c^(n), and Cg(«) = 1 - C/^n) we obtain

p(n) = pa v + TVcos(Ji[2n-l]-a?-+q>), (1.16)

A

with 77 = [KA/27tdav][pA-pB]sin(7tdav/A) and pa v = [p^ + P B ] / 2 . The condition A »

dw is satisfied if the modulation period A is much larger than the atomic distances in

the crystal but also if the modulation direction is in a crystallographic direction with high indices. The latter is the case in Nb/Ta13 and Nb/V14 superlattices grown on

(012) (X-AI2O3 substrates where the modulation vector is inclined at about 3 degrees with respect to the (002) direction in the superlattice. For superlattices with A » dav

the expression for rj reduces to 77 = K-[pA - P R ] / 2 . The expression for the layer thickness d{n) and the electron density p(n) in the n-th atomic layer indicate that in a superlattice of two elements with different lattice constants, there is in general substitutional as well as displacive modulation. Both types of modulation are in phase when the atom with the largest number of electrons also has the largest lattice constant. Usually this condition is fulfilled but there are exceptions like the transition metals Nb and W with element numbers 41 and 74 respectively. The lattice constants of these metals, both with a bcc structure, are 0.330 and 0.316 nm for Nb and W.

As already mentioned briefly, elastic effects can influence the positions of the atomic layers in a superlattice as derived before. In the atomic planes parallel to the substrate, matching of the superlattice to the substrate might impose the lattice constant of the substrate on the superlattice. This occurs e.g. in Si/Ge alloys and superlattices grown on Si substrates and will be discussed in the chapters 5 and 6. But also in situations where the substrate does not impose an inplane lattice constant, the lattice constant parallel to the substrate has to be constant (apart from a component of the modulation in this plane) throughout the whole superlattice to get monocrystalline growth. This implies that the atomic planes parallel to the substrate are subject to stress. If we take a cubic alloy modulated in the crystallographic in­ direction according to the expressions derived before, the effect of this stress can easily be illustrated. The strain in the (001) or xy-plane is

exx(n) = ^ k (1.17)

where a is the actual lattice constant in the xy-plane and ac(n) is the lattice constant of

the rt-th atomic layer in absence of stresses and only determined by the concentration of the constituent elements in an alloy with the same composition as this layer. This imposed strain in the xy-plane results in a strain perpendicular to the substrate of

ezz(n) = -2(Ci2/Cn)exx(n) = d ( n^( n ) , (1.18)

where d(n) is the actual thickness of the n-th atomic plane in the c-direction and Cyi and C\\ are stiffness constants. With the expression for exx and e2Z, and ac(n) =

c\(n)a\ + CB(n)-flB. the layer thickness can be written as

d(n) = [l+2(Ci2/Cn)HcA(n)-aA + cB(n)-aB] - 2(Ci2/Cn)-a . (1.19)

The total thickness of the superlattice after deposition of n atomic layers is the sum of the layer thicknesses, i.e.

n

D(n) = £ d(m) = n- { kl+2(Ci2/Cn)]-[aA + aB] - 2(Ci2/Cn)a}

m=l

+ [1 + 2(Ci2/Cii)K-[sin(2jui(^-+ cp) - sin cp ], (1.20)

A

with <jj = KA(d\ - dB)/2ndav and dav the average layer thickness. In a superlattice

where the lattice constant parallel to the substrate is not equal to the average of the lattice constants of the constituent elements, the superlattice is tetragonally distorted. For a superlattice with an average cubic lattice, a = dav = (aA + ag)/2 and the layer

positions are

z(n) = (n - ë da v + [1 + 2(Ci2/Cn)]-^{cos(7td-^-)-sin([2n-l]ji^v.+ cp) - sin (p }.

2 A A

(1.21)

The amplitude of the displacive modulation is enhanced by a factor [1 + 2(Ci2/Ci i)] due to the stress parallel to the substrate. The quotient C12/C11 is of the order of 0.5

and thus the amplitude of the displacive modulation is enhanced by a factor of about 2. This shows that the elastic behaviour of the superlattice cannot be neglected.

In the next two sections we will deal with substitutional and displacive modulation separately. This separate treatment allows us to illustrate the effects of both types of modulation on the x-ray diffraction pattern and so, to understand the complicated pattern of a superlattice with both types of modulation which will be discussed in section 4.

2. SUBSTTTUTIONAL MODULATION

A superlattice with substitutional modulation only, is a crystal where the contents of the unit cell varies according to a function periodic in the spatial direction of the modulation vector. Although it is possible to have a superlattice with more than one modulation vector, we will limit the discussion to superlattices with only one modulation vector because the modulation in the artificial superlattices under consideration is one-dimensional. The atomic sites in a superlattice with substitutional modulation are the sites of a corresponding crystal without modulation. By substituting periodically different atoms at these sites a modulated structure is obtained. In a commensurate superlattice the crystalline period in the direction of modulation is a rational fraction of the modulation period. In this case it is possible to define a so called supercell to obtain translation symmetry in the superlattice. If the modulation period is incommensurate, three dimensional translation symmetry in the crystal is absent.

The metallic superlattices we discuss in this thesis are based on the bcc type of structure these alloys have. The conventional unit cell contains two atoms. To simplify the expressions we prefer the description based on the primitive unit cell with a basis of one atom only. It is not possible to chose a primitive cell with only one atom for the description of the diamond structure on which the Si/Ge superlattices are based. Here the conventional cube contains eight atoms and the primitive cell two atoms. In the following discussion the expressions are based on the metallic superlattices but can easily be extended to the Si/Ge superlattices by addition of a second atom to the basis.

The superlattices we discuss in this thesis form solid solutions for every concentration of the constituents. No spinoidal decomposition occurs and the distribution of the constituent elements is at random over the lattice sites. In contrast with ordered alloys, which can be described exactly, the disordered alloys have to be described by an average structure. This structure is defined as the electron density of the unit cell averaged over all cells in the crystal. The diffuse scattering of x-rays arising from this disorder has been treated in literature.15'1^ The electron distribution

in a crystal with one atom per unit cell can be described by a convolution of the electron distribution p(r) around the nucleus and the lattice. To obtain the electron distribution in a binary alloy the lattice has to be multiplied by a probability function

P(rj) which determines the probability of finding the atom on the lattice site rj for

each of the constituents. This gives the following expression for the electron distribution:

[ PA(r) ® £ PA(r)-5(r-rj) ] + [ pB(r) ®XPB(r)-8(r-rj) ] (2.1)

j j

where ry = ua + vb + wc, with a, b, and c the lattice vectors and u,v, and w running through all integer numbers. The number j symbolically replaces the triplet u,v and

w. The electron distribution p(r) and the probability function P(r) of the two

constituents are indicated by the subscripts A and B. In a superlattice with a sinusoidal substitutional modulation, the probability functions are given by

P A ( r ) = j [ l + a-sin(r-Q + (p)],

and PB(r) =\ [ 1 - a sin(r-Q + <p) ] , (2.2)

if the average concentration of the constituents is 50%. The modulation is determined by the modulation vector Q and the amplitude a. The scattering amplitude is the Fourier transform of the electron distribution and is in such a superlattice a convolution in reciprocal space of the transform of the sinusoidal modulation and the transform of the lattice:

fB(k)-{ 156(k) -|oi-[exp(-i<p)-8(k + Q) - exp(+icp)-8(k - Q)] } ]

®[ (27t)3/a-bxc] £ 5(k - kj) , (2.3)

where kj = ha* + kb* + lc*, with a*, b* and c* the reciprocal lattice vectors of the primitive cell. Here we assumed the crystal to be infinitely large. Introduction of a k-dependent modulation amplitude a(k)= a-\f\(k) - fB(k)]/fav(k), with/av(fc) =

[/A(*)+/B(*)]/2 leads to

A(k) = (27t)3-fav(k> { 5(k) + |a(k)i-[exp(-i9)-5(k + Q) - exp(+icp)-5(k - Q)] }

® [ (27t)3/a-bxc] 2 5(k - kj) . (2.4)

ÜJ

The intensity distribution in reciprocal space is proportional to A*A, i.e.

I(k).-. fa2v(k) £ 8 ( k - kj) + [|a(k)]2 £ [ 8(k - kj + Q) + 8(k - kj - Q) ]

j j

-2[ia(k)]2cos(2(p) £ 5(ki - kj + 2Q)5(k - kj + Q) . (2.5) j i

The third term in equation (2.5) corresponds to the overlap of two satellites. In the case under consideration, where each main reflection has only one satellite at each side in the modulation direction and the reflections are represented by 5 functions, this overlap occurs only if the modulation period is half the atomic layer distance in that direction. Apart from this special situation the third term in equation (2.5) disappears and the intensity distribution in reciprocal space is

I(k) /. fa2y(k)X {5(k - kj) + [ Ja(k)]2-[ 6(k - kj - Q) + 8(k - kj + Q) ] }. (2.6)

j

The intensity of the satellite reflections is determined by the modulation amplitude

cc(k) only and is given by

The intensity of the satellites thus is independent of the direction of the modulation. The number of satellites in a superlattice with a sinusoidal substitutional modulation is two; one at each side of the main reflection, as is indicated in figure 3 for a superlattice with an average bcc structure and Q = [c* - a*]/4 + (3/8)6*. Here a*, b* and c* are the primitive translation vectors in reciprocal space.

O O

o

o

FIG. 3. The (111) plane (conventional notation: (100)) of a bcc superlattice with

a sinusoidal substitutional modulation and modulation vector Q - (c* - a*)/4 + (3/8)b*. The positions of main and satellite reflections are indicated by large and small dots.

In the special commensurate structure were

Q=£<i, (2.8)

I(k) .-. fa2v(k)X { 5(k - kj) + 2^x(k)]2[l-cos(2(p)]5(k - kj +iki) } . (2.9)

j

It is not possible to assign a satellite to one particular main reflection because the satellite lies in the middle between two main reflections. In contrast to the previous case which led to equation (2.7), the intensities of the satellite reflections vary from total extinction to a\k) times the intensity of the main reflections, depending on the phase (p of the modulation function. To illustrate the effect of the phase we take a superlattice with a simple cubic structure and a reciprocal modulation vector Q equal to the vector c*/2. If (p = 0, the diffracted intensity at the reciprocal lattice points ha*

+ kb* + Ic* is equal to 1 for the points with h, k and / running through all integer

numbers and thus the superlattice obeys the diffraction condition of the average lattice. This is expected because there is no modulation at all in the crystal (see Fig. 4a). When (p = TC/2, (Fig. 4b) the atomic layers in the c direction have alternating an average electron distribution of [1 + oc]p(r) and [1 - a]p(r). This gives a diffracted intensity equal to 1 for u + v + w equal to an integer number, and o?(k) at reciprocal points with u and v any integer number and 2w odd. If we had chosen a modulation vector in the c-direction which was not equal to 2c, the number of different atomic planes perpendicular to the c direction would be infinite because each layer would have a different average electron distribution between [1 + cc]p(r) and [1 - tt]p(r). This is equivalent with chosing a different phase (p in each unit cell of the lattice. The phase cp therefore is of no significance in superlattices with substitutional modulation only, apart from the special commensurate case treated here. The diffracted intensity per unit volume of the crystal for an incommensurate superlattice as discussed in this section is proportional to [/av(fc)]2-[l + ce2(fc)/2]. For a commensurate superlattice

this intensity varies between [fav(fc)]2 and \fa\(k)]2[l + o?(k)], depending on the

phase of the modulation. The diffracted intensity in a superlattice with substitutional modulation thus is larger than the squared average scattering factor due to the fact that the intensity is determined by the sum of the squared atomic scattering factors.

A more general discussion of substitutional modulation in a superlattice with two constituent atoms is possible by introducing Fourier series for the electron distribution. In general, the electron distribution in a superlattice with substitutional modulation only, is given by equation (2.1) with

W¥

FIG. 4. In a commensurate superlattice with a sinusoidal substitutional

modulation and a modulation vector Q = 2c*, the crystal consists of atomic layers with an alternating concentration. The difference in concentration is determined by the phase (pof the modulation; (a) (p = 0; (b) <p = n/2.

N

PA(T) = CA [ 1 + E {an-sin(nr-Q) + pVcos(nr-Q)} ]

n=l

PB(r) = 1 - PA(r) (2.10)

where CA is the average concentration of atoms A in the crystal and an and p\ are

the Fourier coefficients of the modulation function. Introduction of fc-dependent Fourier coefficients On(Jt) and p\{k) for the modulation amplitude and an average atomic scattering factor/av(&) on a similar manner as in the sinusoidally modulated

superlattice gives an intensity distribution in reciprocal space of

I(k).'. f^v(k)5(k - kj) +

N

R

(k) X [«m

(k) + P

(k)][8(k - kj + mQ) + 5(k - kj - mQ)], (2.11)

provided that no reflections overlap. The satellite reflections at both sides of the main reflection again have almost the same intensity |/av(*:)/2]2[am2(*:) + Pm2(k)] and this

intensity depends, via the Fourier coefficients, on the order m of the satellites. This indicates that the accuracy in which the modulation function can be determined depends on the highest order of satellite reflections observable. We also derived here an important property of superlattices with substitutional modulation only, namely that the intensity of the satellite reflections is symmetric around the main reflection apart from the different atomic scattering factors/(fc).

In equation (2.10) the electron distribution is described by the convolution of an average distribution around an atom and an analytical periodical function, represented by Fourier series, sampled at equidistant (lattice) points. This sampling proces sets an upper limit N to the order of the Fourier coefficients given by NQ =

kjl, with ki the smallest lattice vector in reciprocal space in the modulation direction.

Because nQ always lies in the first Brillouin zone no overlap of satellites occurs apart from the special case where the N-th order Fourier term fulfils the condition NQ =

kill.

3. DISPLACIVE MODULATION

In a superlattice with displacive modulation only, the position of the atoms in the unit cell varies according to a function which is periodic in the direction of the modulation vector. Such a superlattice can be obtained by a statical periodical displacement of the atoms. We characterize the displacive modulation in a sinusoidally modulated superlattice by two vectors. First, the modulation vector Q which represents the modulation period and the direction of modulation. The polarization vector, a, defines the direction and the amplitude of the displacement of the atoms from their average position. The electron distribution in a superlattice with displacive modulation is a convolution of the electron distribution p(r) around the atomic nucleus and the periodically distorted lattice. There are two ways to describe this periodic distortion in a crystal with one atom per unit cell. In a superlattice where the atoms are displaced sinusoidally from their average position, the electron distribution is given by

p(r) ® £ 6(r - rj - asin(rj-Q + <p)) , (3.1) j

with (p a phase fractor and ry the vectors of the average lattice. It is also possible to define a sinusoidal displacive modulation where the displacement is ruled by the final position r of the atoms. For these superlattices the electron distribution is

p(r) ® X 5(r " rJ " a s i n ( r - Q + 9)) . (3.2)

j

The difference between these two superlattices is of a subtile nature but nevertheless important as we will see when we discuss the diffraction pattern. The meaning of both a and Q is illustrated in figure 5 where the situations with a parallel to Q (longitunal modulation) and a perpendicular to Q (transversal modulation) are shown. The Fourier transform of the electron distributions, given by the equations (3.1) and (3.2), provides the scattering amplitudes of both types of superlattices with displacive modulation only.

A(k) = [f(k)-(2jt)3/a-bxc ] £ Jm(-k-<x)-exp(imq>)-8(k - kj - mQ) ,

j m

(3.3) is the Fourier transform of the electron distribution given by equation (3.1), and equation (3.2) gives17

A(k) = [f(k)-(2K)3/a-bxc ] £ Jm(-kj-a)-exp(im<p)-5(k - kj - mQ) (3.4)

j m

where Jm is a Besselfunction of the first kind and the m-th order and f(k) the atomic

scattering factor. The vectors £j are the reciprocal lattice vectors of the average lattice. For the derivation of equations (3.3) and (3.4) we used the relation18

+00

exp(-ik-ocsin(r-Q + <p)) = X Jm(-k-a)exp(im(27cr-Q + (p)) .

m = -00

The intensity distributions in reciprocal space in the case no satellites or satellite and main reflections coincide are

I(k) .-. ! % ) £ J

(-ka)-5(k - kj - mQ) and (3.5)

j m

I(k) .-. f2(k)J Jm2(-[k - mQ]a)-5(k - kj - mQ), (3.6)

j m

for the superlattices given in equations (3.1) and (3.2) respectively.

- i

A

FIG. 5. In a superlattice

with a longitunal displacive modulation (a) the modulation amplitude a is parallel to the modulation vector A.

If a is perpendicular to A the modulation is transversal (b). The positions of the satellite reflections in reciprocal space are the same for both superlattices.

The difference between both expressions for the intensity is that in the second superlattice the argument of the Bessel function only depends on the reciprocal lattice vectors of the average lattice and not on the modulation vector. This can be seen by writing k = kj + mQ. One of the important consequences is that the second superlattice has no satellite reflections near the origin in reciprocal space; for cubic superlattices with a longitunal modulation satellites are absent for all (hkO) main reflections. We will return to this kind of superlattice in the next section and proceed here with the first one which corresponds to the deposition model discussed in the