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On R.E Size Effect Measurements Fermi V "" ON R.E SIZE EFFECT MEASUREMENTS FERMI SURFACE IN INDIUM D.G.deGROOT V-' VRIJE UNIVERSITEIT TE AMSTERDAM ON R.F. SIZE EFFECT MEASUREMENTS AND THE FERMI SURFACE IN INDIUM ACADEMISCH PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE WISKUNDE EN NATUURWETENSCHAPPEN AAN DE VRIJE UNIVERSITEIT TE AMSTERDAM, OP GEZAG VAN DE RECTOR MAGNIFICUS MR.I.A.DIEPENHORST, HOOGLERAAR IN DE FACULTEIT DER RECHTSGELEERDHEID, IN HET OPENBAAR TE VERDEDIGEN OP DONDERDAG 2 MEI 1974 TE 13.30 UUR IN HET HOOFDGEBOUW DER UNIVERSITEIT, DE BOELELAAN 1105 DOOR DIRK GEERT DE GROOT. GEBOREN TE EINDHOVEN *hk r r PROMOTOR: DR. A. LODDER COREFERENT: DR. J.H.P. VAN WEEREN This investigation was part of the research program of the "Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)", which is financially supported by the "Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (Z.W.O.)". STELLINGEN 1. Het is gewenst in de berekening van de laag freguente ruis in de verzadigingsstroom in avalanche diodes naast de schrootruis in de injectiestroom andere ruiscomponenten te betrekken. 2. Bij het onderzoek van anomalieen in de transmissie van electromagnetische golven door metaalkristallen vormt de ondergrond in de transmissie in vele gevallen een niet te scheiden component van het signaal. 3. De benaming O.P.W. berekening voor een pseudopotentiaal interpolatie schema, waarbij de Fouriercomponenten van de pseudopotentiaal als aan te passsen parameters worden ge- hanteerd, wekt ten onrechte de indruk dat de gebruikte golffuncties orthogonaal staan op de atomaire pit golf- functies. 4. Het is gewenst dat de Nederlandse regering een prejudiciele beschikking uitlokt, die tot gevolg zal hebben dat dienst- merken onder de beschermende werking van de Eenvormige Beneluxwet op de Warenmerken worden gebracht. 5. Om het atherosclerose probleem met enige kans op succes het hoofd te bieden dienen reeds bij kinderen preventieve maat- regelen genomen te worden., in de vorm van onderwijs in de voedingsleer aan kinderen. 6. Het door Foudraine beschreven leefklimaat in psychiatrische inrichtingen is kenmerkend voor veel inrichtingen van ver- pleging of verzorging. J.Foudraine, Wie is van hout,(AMBO),p.256 7. De huidige lange leveringstermijnen bij de aankoop van meubelen en de daaraan aangepaste verkoopcondities maken het gewenst dat een onderzoek wordt ingesteld naar een betere produktie-proces besturing bij de meubelfabricage. D.G. de Groot aan mijn ouders voor Suse Marjon Voorwoord Ret onderzoek, dat in dit proefsohvift is besohreven, kon slechte worden verwezenlig'kt, dankaig de steun en inzet van velen, Graag wil ik hen alien mig'n dank betuigen* Zeergeleerde promotort beste Adri3 de vrig'heid die g'e mis bio de keuze van het ondevzoek hebt gegevent de hulp die g'e mig' bood in wiskundige problemen3 en de plezierige en nauwgezette manier van werken tiqdens de besprekingen, waardeer ik big'zonder. Zeevgeleevde oovefevent,, beste Jaapa g'e vooptdurende belangstel- ling voor het ondex>zoeka g'e enorme inzet en geduld big de disaue- sies3 g'e aanmoedigingen en adviezen zig'n van onsohatbave waavde voov mig geweest, Voor deze plezievige samenwevking ben ik g'e zeev dankbaav. De diveotie van het Natuuvkundig Labovatorium dank ik voov de mig geboden gelegenheid dit ondevzoek te verriahten, Jacob Batema3 Ton Hoff en Jaap Maana dankzig g'ullie beveidwil- lige medewerking soms big' naaht en ontig is dit wevk voltooid* Voor de hig'dragen die gullie aan het onderzoek hebt geleverd en welke in dit proefsohrift zig'n terug te vinden, dank ik g'ullie hartelijk, Daan Rig'senbrig, de door g'ou ontwikkelde computer programma's zig'n van gvoot nut gebleken, De vele hulp die g'e ons daarmee ge- boden hebtt vermeld ik met dankbaarheid. Voor de big'dragen van Hans Fondse* Kees Goudswaardt Jan van Ree en Hans Radder ben ik zeer erkentelig'k. Menco Danet Jan Koolstra en Koos v.d. Maas g'ullie big'dragen in de versahillende stadia van het onderzoek heb ik zeer op prig's gesteld. Kees Graswinckelt de tig'dens het maken van plakg'es groeiende sa- menwerking heb ik altig'd zeer gewaardeerd. De prettige samenwerking met velen van het personeel wil ik graag memoreren, De her en Jongsma, Reig'nen, Knpl en Ras en hun medewerkers met name Henk Schwarz, alsmede de heer Petiet ben ik zeer erkentelijk voor hun aandeel in da tot&tdndkoming van de apparatuur. De her en Pomper en van Sig'.pveld dank ik voor de goede verzorging van de vele figuven* CONTENTS INTRODUCTION CHAPTER I THE R.F. SIZE EFFECT AND THE K.K.R.Z. METHOD 1 Introduction 4 1.1 The R.F. Size Effect 5 1.2 Line shape calculation 11 1.3 The K.K.R.Z. formalism 15 CHAPTER II EXPERIMENTAL ARRANGEMENT 2 Introduction 19 2.1 Sample preparation 19 2.2 Twin-T bridge method 24 2.2.1 Relation coil-sample 25 2.2.2 Twin-T bridge 29 2.2.3 The tuned amplifier 35 2.2.4 Mixing procedure 39 2.2.5 Set-up of the twin-T bridge measuring system 45 2.2.6 The use of a coax cable 54 2.3 The transmission method 57 2.4 Comparison of several experimental techniques 64 2.5 Dewar system, sample holders and magnetic field calibration 2.5.1 Dewar system 68 2*5.2 Sample holders 68 2.5.3 Magnetic field calibration 70 CHAPTER III LINE SHAPES AND MEAN FREE PATHS 3 Introduction 73 3.1 Parallel-field R.F.S.E. line shapes 73 3.2 The peak near zero field 83 3.3 Temperature dependence of the R.F.S.E. amplitudes 88 3.4 Special features of some lines 94 CHAPTER IV CALCULATION OF THE FERMI SURFACE FROM CRITICAL FIELD VALUES 4 Introduction 97 4.1 Determination of the critical field 97 4.2 Tneoretical description of the Fermi Surface 103 4.3 Angular dependence of R.F.S.E. lines 111 SUMMARY 115 REFERENCES 117 INTRODUCTION In the past two decades elaborate work has been done in the field of £he electronic properties of metals. The development of refined measuring techniques in this field and the prepara- tion of materials of high-purity both have stimulated this re- search. It has induced numerous studies on the determination of Fermi surfaces of metals. The advent of fast electronic computers afforded theoretical physicists the opportunity to perform extensive calculations on the topology of the Fermi surface. This in turn required even more accurate experimental results. Beside that, since a few years the interest is also directed on the variation of the mean free paths of electrons over the Fermi surface. In view of these developments we looked for experimental methods appropriate to the study of the Fermi surface as well as the mean free paths. A suitable tool was presented in the radio-frequency size effect (R.F.S.E.) method. With this method accurate values of linear dimensions of the Fermi surface can be obtained. In studies of mean free paths by means of the R.F.S.E. the desired information can be obtained from the temperature dependence of the amplitude of the R.F.S.E. lines. The several techniques used in R.F.S.E. measurements, where the surface impedance of a metal as function of an applied magnetic field is measured, are all developed in studies of nuclear mag- netic resonance. Since the measuring technique used can affect the line shape we decided to compare the applicability of sev- eral techniques. We studied line shapes obtained with two dif- ferent techniques. Juras (1969,1970,1970') and Kaner and Fal'ko (1967) calculated for different experimental configurations and simple models of Fermi surface the line shapes of the real and imaginary part of the surface impedance. We chose therefore a measuring technique _, with which the real and imaginary part of the surface impedance can be measured simultaneously. In order to obtain the correct line shapes the real and imaginary component of the surface im- pedance have to be separated completely. We paid much attention to this separation. Although the choice of a metal is limited due to the high de- mands on the purity imposed by the R.F.S.E., many metals have been investigated yet. A metal which can be made sufficiently pure is indium. We are interested in indium for two reasons, (i) Gantmakher and Krylov (1966) have studied the R.F.S.E. in indium. Since they used another experimental technique the ob- tained line shapes and those studied in the present work are complementary. (ii) Since specific parts of the Fermi surface of indium are strongly dependent on the potential model accurate determina- tion of the Fermi surface will be of interest in studies on the potential. Many investigations of the Fermi surface of indium have been done. These investigations include cyclotron resonance(Bezuglyi 1960, Castle e.a.* 1961, Mina e.a. 1965,1966), magneto-acoustic effects (Rayne e.a. 1962, Rayne 1962), galvanomagnetic effects (Gaidukov 1966, Garland e.a. 1969, Trodahl 1971), the de Haas- van Alphen effect (Brandt e.a. 1963,1964, O1Sullivan e.a. 1968, Hughes e.a. 1969, Van Weeren 1973) and the R.F.S.E. (Gantmakher e.a. 1966, Krylov e.a. 1967). These studies have resulted to the following model of the Fermi surface of indium: (1) a large second zone hole surface consisting of 6 "square cups" connected through 8 "hexagonal cups"; (2) a third zone electron surface consisting of. 4 so-called 6-arms along the <110> edges of the Brillouin zone, which are joined together in a ring in the(001) Brillouin zone faces. However, in the details of the Fermi sur- face there are some discrepancies.
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