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ON R.E SIZE EFFECT MEASUREMENTS

FERMI SURFACE IN INDIUM

D.G.deGROOT

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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 golffuncties.

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 geboden 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 samenwerking 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 preparation of materials of high-purity both have stimulated this research. 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 magnetic resonance. Since the measuring technique used can affect the line shape we decided to compare the applicability of several techniques. We studied line shapes obtained with two different 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 impedance have to be separated completely. We paid much attention to this separation. Although the choice of a metal is limited due to the high demands 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 obtained 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 determination 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 surface there are some discrepancies. From the galvanomagnetic experiments it is concluded that the Fermi surface of indium is closed. Van Weeren et al.{1973) concluded from their calculations that the second zone hole surface may be open. Hughes e.a. (1969) assuming that the Fermi surface should be closed decided Note: We use the abbreviation e.a. for et al. • to introduce spin-orbi-t coupling in their calculations. Achiev- ing in this way a good agreement between calculated and experimental results, they found an "ondulating" structure in the connections of the g-arms. Concerning the structure in these connections it can be remarked that in the free electron model of indium so-called oc-arms, lying along the <101> edges of the Brillouin zone, are connected to the 8-arms. Therefore, the detailed structure of the connections of the $-arms may show remnants of the a-arms. The experimental results of the investigations mentioned above are used only in pseudopotential (ip.P.W.) calculations. Gantmakher e.a. (1966) compared their results with a free electron model of the Fermi surface. We used the more refined K.K.R.Z. method. Considering the fact that in the fitting of model parameters to the experimental results the use of linear dimensions of the Fermi surface instead of cross-sectional areas will be more accurate, and that we were able to do more accurate measurements of linear dimensions than Gantmakher e.a. (1966), we expect that we can obtain an improved model of the Fermi surface of indium. In chapter 1 we give the theoretical framework, which consists of the R.F.S.E., the line shape calculations of Juras, and the K.K.R.Z. method. In chapter 2 the experimental arrangements are described. In chapter 3 the line shapes observed with two different measuring techniques and the temperature dependence of the amplitude of some R.F.S.E. lines are given as well as the peak near zero field, connected with magnetic field induced surface states. In chapter 4 the determination of the critical field values from which linear dimensions of the Fermi surface are obtained are given as well as the Fermi surface of indium calculated with a K.K.R.Z. interpolation scheme. ; : / CHAPTER I

THE R.F. SIZE EFFECT AND THE K.K.R.Z. METHOD

1 Introduction In this introduction we will describe for the simple case of free electrons the phenomenon of the r.f. size effect (R.F.S.E.) \ A thin (150-400ym) plate of a very pure metal is placed within \ a coil at liquid helium temperatures: 4.2 K (see section 2.5). An external static magnetic field is applied parallel to the surface of the plate. In the measurement of the resistance of ; the coil at radio-frequencies: 1-30 MHz (see section 2.1 and j 2.3), are observed well-defined changes in this resistance (see section 3.1) at definite field values. This can be explained as follows: : The electrons, involved in the transport properties of metals, ] are not bound to the atoms. These electrons are moving in the potential due to all ions of the metal. However, we consider the simple model of completely free electrons. The movement of an \ electron in external fields may be considered as classical, although we need sometimes a quantum mechanical approach. Each e- < lectron in a metal has its distinctive velocity v and energy e. According to the Fermi-Dirac distribution function not all e- \ lectrons are relevant in the transport properties of metals. J

Only electrons with an energy e=eF, where ep is the Fermi energy { take part in conduction processes, since they solely can absorb j energy from external electromagnetic fields. So, we consider e- .j lectrons with energies eF and corresponding velocities vp. Ap- plying a magnetic field S the energy of the electrons does not j change. Due to the Lorentz force the velocity changes according i to ij d^F e - i m —^-£ = — vF x H (m and e are the electronic mass and charge ) resp., c is the velocity of light). The component of the velocity ; parallel to H does not change. The component of the velocity ] CHAPTER I 5

perpendicular to H changes. Consequently, the electron describes a helical trajectory of which the projection on a plane normal to § is a circle. The radius of this circle is given by

m vp sin a ^

r = c g , where a is the angle between vp and H. In consequence -of the free electron model the distribution of the elec-

tron velocity vp is uniform. Therefore, a large number of equal- sized trajectories occur if

m vp ^ a = 90 : r = R = c —-=-. Changing the magnitude of the field H eii the size of these trajectories vary inversely proportional to H. At a definite value H the diameter of the trajectory 2R equals the size d of the plate. A r.f. electromagnetic (e.m.) field applied by means of the coil penetrates only a few microns into a very pure metal at liquid helium temperatures, which is called the skin effect. Electrons passing through the skin layers can absorb much energy from the e.m. field, which can be observed in the resistance of the coil. The resistance as function of the magnetic field exhibits a pronounced change if a large number of electrons pass through both skin layers, this occurs when 2R = d. This is called the R.F • S. £<.. In section 1.1 we will deal with the R.F.S.E. more explicitly. We will give a review of the historical development and describe the size effect in terms of the ineffectiveness concept. In section 1.2 we will give a more quantitative description of the R.F.S.E. with regard to the calculation of the R.F.S.E. line shapes. The K.K.R.Z. formalism used in Fermi surface calculations is outlined in section 1.3.

1.1 The R.F. Size Effect Size effects, which can be observed in the d.c. conductivity of metals having a long mean free path compared to characteristic dimensions of the sample, were discovered a long time ago (first discovery in 1898, pee Sondheimer 1952). The discovery of the R.F.S.E. by Gantmakher (1962,1963) has shown that size J 6 CHAPTER I

effects can also be observed under anomalous skin effect (A.S.E.) conditions. By application of a magnetic field the R.F. currents can penetrate deeply into the metal. The current carriers are the electrons travelling along cyclotron trajectories. The anomalous skin effect. An e.m. wave incident on a surface of a conductor is nearly totally reflected. The effective pene- 1/2 —4

tration depth 6 = c/(2Truao) ' is very small: "-10 cm (oQ is the conductivity of the metal). However, in very pure metals at liquid-helium temperatures the carrier mean free path 1 may be ^10 cm, so 1>>S . Reuter and Sondheimer (1948) have shown that, if 1»SQ, the penetration of the e.m. wave can not be described with an exponential decay as in the case that 1 is small. The e.m. field distribution at the metal surface can be observed in the surface impedance defined as ,1.x, where E and E' are evaluated at the metal surface: E. is the tangential component of the e.m. field and E* is its derivative with respect to the z-coordinate, which is directed along the normal to the surface (inwards). Reuter and Sondheimer calculated that under A.S.E. conditions the surface impedance

/3 21 1 3 z « ( ™ j / (1 + /3 i) (1.2) is nearly independent of the character of the scattering (diffuse or specular) of the carriers at the metal surface. Utilizing the expression 6 = Z (c2/47riw) which is valid if 1 is small relative to 5, we can define an effective penetration depth in the case of the A.S.E.

Seff " < 2f% )1/3 with » * 7

It follows that the effective skin depth. 6 -.. is independent of 1 and dependent on the frequency according to u ' . Under A.S.E. conditions the surface currents are carried mainly CHAPTER I 7

by those carriers which are moving parallel to the surface within the skin depth. The rest of the electrons are more or less ineffective dependent on the part of their free paths which extend in the field-free interior of the metal. This approach can be used in a qualitative description of the A.S.E.. It is known as the "ineffectiveness concept" of Pippard (1947,1954). The effective regions, where v =0, can be located on the Fermi surface (see also form. 1.6). It is a region having a width 6/1 and located along the lines on the Fermi surface where v . n = 0 (1.4) (v is the carrier velocity; n the normal to the sample surface) Cyclotron resonance. Azbel1 and Kaner (1957) have shown in their theory of cyclotron resonance that under A.S.E. conditions carriers submitted to an external magnetic field parallel to the metal surface, are accelerated by the e.m. field at the metal surface, causing high surface currents if to = nw , UT>>1 (W is the cyclotron frequency, T the relaxation time). This is a temporal resonance. It is observed in the surface impedance of the metal. Kaner (1958) pointed out that at sufficiently low fields the cyclotron resonances will disappear, since the trajectory dimensions exceed the finite sample thickness (see also Khaikin 1962). A well-defined cut-off is observed if w T»1. The R.F.S.E.. Gantmakher (1962,1963) suggested that a similar phenomenon as described by Kaner (1958) could be observed if the frequency of the e.m. field is in the r.f. range i.e. u«u and if u T>>1 i.e.the charge carriers can interact several times with the e.m. field at the sample surface (A.S.E. conditions are valid), but the phase of the e.m. field is effectively constant (b)T<D, where D is the characteristic trajectory dimension, the e.m. field penetrates into the metal according to (K3), but significant changes in the surface impedance (1.2) can occur, connected with high surface currents, when the

J 8 CHAPTER I

trajectories span the sample thickness. This is a spatial resonance. An additional condition for observing this spatial resonance is that the spread AD in the trajectory dimensions D is small compared to D:AD/D<<1. This means that only extremal trajectories are observable in the R.F.S.E. Relation trajectory-orbit. The trajectories in the real space (the sample) are related to orbits on the Fermi surface in the }c-space. (To avoid misunderstanding we will use in this chapter the term orbit for an electron movement in the k-space and the term trajectory for the movement in the real space). The Fermi

surface is defined by en(£) = eF, where ic characterizes the state of each electron by the three quantum numbers k-^x'kyfkg)» n is the Brillouin zone number and e-, is the Fermi energy (see

section 1.3). All states below the Fermi energy ep are occupied. Only the electrons at the Fermi surface take part in conduction J processes, since they can make transitions to unoccupied states. -1 Therefore, studying the movements of electrons in external fields 1 we have to consider only electrons at the Fermi surface. Apply- J ing a magnetic field S the rate of change of the vector & due to "i the Lorentz-force is given by (Gauss units) j *£•§*«* »•» i The velocity of the electron with wavevector k is given by j v=f% (1.6) f h die ,j From the integrals of motion: k" S kj. = —==— = const, and '£= const. it is clear that the movement of an electron on the Fermi surface is defined by the line of intersection of this constant en- \ ergy surface e(ic)=ep with a plane normal to the direction of the magnetic field H. From (1.5) it follows that the orbit in ic-space and the projection of the trajectory in the real space on a plane normal to S are similar with the similarity factor (eH/hc) and are rotated relative to each other through an angle ir/2. It is seen by integration of (1.5) that the size of a tra- i CHAPTER I 9 •|

•\ ] ] jectory varies inversely proportional to H, since the size of | the corresponding orbit remains constant. At a definite value I of the field, the critical field H , the trajectory size equals ] the sample thickness d. By integration of (1.5) it follows then: ; n Ak = (e/c) d H (1.7)

I The location of Ak on the Fermi surface can be determined as | follows: Since we measure in real space a distance along the ] direction n, and only effective points are relevant according i to the ineffectiveness concept (see equation 1.4), we project ) the effective points of the Fermi surface on a plane normal to | n. Then, we take the distances between these projected effective | points measured in a direction perpendicular to S (H.n=0). ;] Since, in the experiment extremal dimensions are observed the ;| measured Ak is equal to the extremal value of these distances. However, apart from extremal dimensions an anomaly in the surface impedance occurs also in other cases (see below). Types of trajectories/orbits. Having defined orbits in k-space, we can distinguish in the case of an arbitrary Fermi surface several types of orbits and by means of the relation (1.5) the corresponding types of trajectories. We can have closed, helical and open trajectories. Closed trajectories correspond to cross-sections of the Fermi surface containing a centre of symmetry. The velocity component

of an electron vfl = \v.§)/H averaged over the orbit: is equal to zero. The helical trajectories correspond to non-central orbits i.e.

cross-sections containing no points of symmetry and f 0. Open trajectories exist only in metals with open Fermi surfaces. The main feature of this trajectory, is that the electron motion is infinite in a plane normal to a well-defined direction of §. Some special types of orbits (trajectories), which do not correspond to extremal cases, give also rise to surface impedance anomalies, (for example in indium)? ^ ^

(i) A cut-off orbit, defined by a certaia value k* (~p = kR)

which determines a plane normal to H, such that if &H

10 CHAPTER I

orbits exist and if ktI>k., similar orbits do not exist and vice

versa (see Matthey, 1969). We call kH a critical point, (ii) Trajectories containing a break point. A break point of a trajectory (or an orbit) is a point at which the velocity of an electron shows a sharp change in its direction. The size effect connected with breaks is explained by Gantmakher (1967) using only the sharp change in the direction of the electron velocity. Matthey (1969) takes into account possible reflections of the electrons at the metal surfaces. (iii) A chain of trajectories. Azbel1 (1961) pointed out that the group of charge carriers, which pass through the skin layer, carry currents into the bulk of the metal. They create an image of the surface currents layer in the effective points of the trajectories at depths into the metal equal to multiples of the trajectory diameter D. The electric fields associated with these image currents are called field splashes. This fields can in turn excite charge carriers* This leads to a chain of trajectories. This so-called anomalous penetration of the e.m. field.can be observed in the surface impedance if the chain of trajectories spans the sample thickness. Tilt effect. In the analysis of experimental results we have to distinguish central orbits from non-central orbits. This can be done by means of the tilt effect, where the magnetic field is tilted over a small angle from the sample surface. Gantmakher e.a. (1965) describe the effect of an inclined field on the R.F.S.E. lines. In case of a central orbit the line shifts to higher or lower fields, depending on the shape of the Fermi surface. The-non-central orbits are connected with helical trajectories. These helical trajectories contain several effective points. In an inclined magnetic field the effective points can be located subsequently within the skin layer by varying the magnitude of the field. This causes a splitting of the R.F.S.E. line as a function of the tilt angle. Op till now we considered the R.F.S.E. only in terms of trajac- { tories, which are fitting to the sample thickness. We will describe now the R.F.S.E. more quantitatively. I JI CHAPTER I 11

First of all we consider the amplitude of the R.F.S.E line. The amplitude is mainly determined by the relaxation time T. Using the ineffectiveness concept Chambers (1969) has shown that in the case of electrons which pass repeatedly through both skin layers (double-sided excitation) the amplitude is proportional to

-Try ———• , Y = i(u>/w ) + 1/u) T (1.8)

The denominator represents the effect of the repeated passages through the skin layers. Since w<1 the expression c c (1.8) can be approximated by -s/1

Where s is the path travelled between the skin layers and 1 the mean free path averaged over an orbit. This expression is used in section 3.3. In the next section we give an outline of the calculation of the R.F.S.E. line, including its shape and amplitude.

1.2 Line shape calculation The R.F.S.E. is observed experimentally in the derivative of the surface impedance Z, given in eg.(1.1), with respect to the magnetic field. Consequently, we have to calculate the electric field distribution in the sample in the case of a thin plate submitted to an e.m. field and an external magnetic field parallel to the sample surface. This calculation has been performed by Juras (X969,1970,1970') in the case of a cylindrical Fermi surface with its axis parallel to the sample surface and of a spherical Fermi surface. We will describe the main aspects of this calculation. j A thin metal plate at liquid-helium temperatures is placed in a J e.m. field and a static magnetic field parallel to the metal I surface. Assume that the A.S.E. conditions are valid i.e. t 6<<1, 5<>D, where D is the characteristic dimension of an j T

12 CHAPTER I

electron trajectory in the magnetic field. The electric field distribution in the metal can be evaluated from the Maxwell e- quations together with the continuity equation and the kinetic (Boltzmann) equation for the electron distribution function f(z,p), (p=hic) , where the z coordinate is directed along the normal to the metal surface (inwards). The resulting equations for the configuration of fig. 1.1 are: (E" is the electric field, ]" the current density) d2E (2) S= i (} j dz2 c2 a

2 3 j (Z) = - H / d p v (p) f(z,p) ; j =0 (1.10) a (2Trh)3 a z where f(z,p) is the Chambers (1952) solution of the kinetic e- quation. This solution is written as an integral over the travelled path, which is related to orbits in the Jc-space, defined by eqs. (1.5) and (i.6) and reads as

f f(Zrp-)=M __£ A(2f*}/* d«« exp[-Y(*-* il v

(1.11) where (ji^w t, is connected with the time of electron motion in the orbit, w c is the cyclotron frequency, £ o the equilibrium distribution function, y see (1.8), e is the electron energy and p = 2i_ defines an orbit in k"-space. The exponential function expresses the scattering probability of an electron travelling from to '. In the electric field £ is involved the penetration of electrons deep into the metal due to the large 1 i] -*• I (non-local case). The X(z,p) is the "time" preceding that an | electron collided with the metal surface (see fig.1.1 and below). |

It is clear from the given formulae that the R.F.S.E. line | shape is dependent on: (.•i) The dispersion law e(x")*eF. The dispersion law determines the shape of the orbit defined by (<{>,Px,e) in (1.11). The in- T

CHAPTER I 13

A -i ~w \ Z=0

Fig.l.I Free electron trajectories in the presence of an external magnetic field H. a)specularly reflected surface trajectories, colliding with the z-d surface at the instants A.,X2,... and with the scattering probability S(6). b)specularly reflected surface trajectories colliding with both surfaces at the instants *]**2'*3'**° an(* w*-t*1 fcê scattering probabilities 8.(8.) at z*0 and S2^92^ at z*^* câ ^ûse^y scattered surface trajectory. d)an ineffective bulk trajectory. e)an effective bulk trajectory.

tegrations over ' and " is along the orbit. Since the shape of the orbit and the magnetic field dependence is involved in v.E in (1.11), the interaction depends on the orbit shape and the magnetic field. (ii) The electric field distribution in the sample. From v.^ in equation (1.11) it follows that an optimum coupling between the e.m. field and the electrons is obtained in the effective regions of the trajectories and if the electron velocity and the e.m. field are in phase (spatially). This coupling depends also on the magnetic field, since this field determines the position of the effective regions in the sample. However, the e.m. field depends self-consistently on the number of effective current carriers. (iii) The scattering of the electrons at the sample surfaces, which affect the effectiveness of the electrons. The scattering appears in (l'.ll) by means of X(z,p). If the electron collides never with any surfaces of the metal, then is A(z,p)» -». If the electron colliding with a metal surface is always scattered

^ T

14 CHAPTER I

diffusely, then \(0,p)-=4>, v >0 and A(d,p)=4>, v <0 and generally A(z,p) satisfies the equation

<|) z = (l/toc)A/ d" vz(0 . In this case we can evaluate (1.11) by splitting (1.11) in two terms: the first term represents the path from the last scattering centre ^ to the point z; in the second term all trajectories colliding with both surfaces are involved (see fig.1.1), introducing a product of scattering probabilities, (iv) The mode of excitation: bilateral symmetric or antisymmetric excitation and unilateral excitation. This are boundary conditions for equation (1.9). (v) The measuring technique. Instead of the surface impedance Z (see 1.1) the first of second derivative of Z with respect to the magnetic field is measured. Besides, the line shapes of the real or imaginary part of dZ/dH are different. Assuming an isotropic relaxation time T. Juras calculated numer- ically the line shape of a R.F.S.E. line and the influence of the mean free path and the specularity function on the line shape. We refer to the original articles for details. An example is given in fig.1.2, where the calculated derivatives of the real and imaginary part of the surface impedance with respect to the field S is plotted as function of H/H in the case of the cylindrical Fermi surface and completely diffuse scattering. Some special features are:

The first dip at H/HQ=a, disappears with decreasing 1. The occurrence of such a dip depends strongly on the shape of the orbit. The width of a peak is about H/Ho=86/d, The peak at a field H/Ho=2 is reversed in sign with respect to the peak at

H/HQ=1. Juras showed that the signal for fields H/HO<1 is strongly dependent on the contribution of specular scattering at1 the sample surfaces; the first dip disappears in case of snecular reflection. Likewise the dependence on 1 is strongly CHAPTER I 15

Fig.1.2 The derivative of the surface resistance dR/dH and of the surface reactance dX/dH as functions of the magnetic field for antisymmetric excitation and a cylindrical Fermi surface as calculated by Juras (1969).

affected by the specular scattering at the sample surface. In order to be able to distinguish between several affects on the line shape it is necessary to measure the dR/dH and dX/dH signals correctly i.e. they must be separated completely.

1.3 The K.K.R.Z, formal-Cam In order to understand the behaviour of the electrons in the R.F.S.E. we need the knowledge of the Fermi surface i.e. we wish to know the wavevectors % of the electrons which satisfy

the relation e(k*)=£F. Therefore, we are faced with the problem of finding e(Jc) from the Schrodinger equation for a crystal

h £2 .1.. /•£» • TT/"£» „«;_._ /~i.\ _ - /'i\ ,t,->. ii\ (1.12) where V(r) is the potential, which has the periodicity of the lattice, V(r+l)=V(r), for any translation vector t. 16 CHAPTER I

The equation (1.12) is a one-electron problem. The potential V(r) is supposed to include the electron-electron interaction in a self-consistent way. Several methods are developed to solve equation (1.12). In this section we give an outline of the Green's function method developed by Korringa (1947), Kohn and Rostoker (1954) and Zi.man (1965). Utilizing the Green's function or propagator of the free par- ticle the Schrodinger equation (1.12) can be transformed into a homogeneous integral equation. As a result the Bloch function i/»j£(r) is composed of wavelets scattered from all different points in the lattice. Suppose, we have a muffin-tin potential, that is a crystal potential, which is spherically symmetrical within a sphere of radius R centred around each atom, inscrib- ed in the Wigner-Seitz cell. Outside the spheres the potential is defined to be constant and to be the zero of the energy scale. Writing this muffin-tin potential as a sum of contribu- tions of separate unit cells, the integral equation can be transformed into an integral equation within one unit cell. As a result the Bloch function ^g(r) can be considered as composed of wavelets scattered from all points within a unit cell, but by means of a structural Green's function the whole crystal takes part in it. Because of the spherically symmetrical potential we can expand the unknown wave functions ito(r) in spherical harmonics. Like- wise the Green's function can be expanded in spherical harmonics. These functions have to satisfy the boundary conditions i.e the scattered wave functions and the radial wave functions inside the sphere must have the desired continuity and continuous gradient at the surface r=R_ of the sphere. We obtain a system s of linear, homogeneous equations in the expansion coefficients, which equations can be solved if the secular determinant is e- qual to zero <3et II A, • ,, . + /(e) cot n, <5,,,

The so-called structure constants A, „ ,, . are determined from i,m;x ,m the structural Green's function and the boundary conditions at T

CHAPTER I 17

the faces of the unit cell. They represent the crystal structure. The crystal potential is represented in the phase shifts n., which are energy dependent (see below). Ziman (1965) considered equation (1.13) for the case of nearly free electrons. Since indium is more or less free-electron like we will use the formalism of Ziman. He transformed the secular equation (1.13) which is given in the angular momentum representation into a secular determinant in the reciprocal lattice representation:

2 det || (|£ + g-| - e) 6ggl + Tgg, (t,e) | | - 0 (1.14)

where r ,= ? j, (|k*-g| R ) j, (lic-g'l R) P, (cos 6^ ,) ETA (1) gg 1=o 1 si si gg and ETA(l) = -.51 i. (21+1) — V K r • cot n1-n1(KRs)/J1(KRg)

x 1 s 1 s cot nx - p- ° **

R is the Slater radius, V the volume of the unit cell, j, and n^ are spherical Bessel and Neumann functions, R^ is the solution of the radial Schrodinger equation for angular momentum 1, the primes denote the derivatives of the mentioned functions with respect to the argument,K2=e, and g is a reciprocal lattice vector. The terms V , have the function of a pseudopotential, which are energy and ic dependent. The summation in F i may be trun- cated when ETA(l) becomes small enough, which usually occurs for 1>2. This feature is very important since the number of parameters in a K.K.R.Z. interpolation scheme can be limited because of physical reasons. It can be seen from (1.14) that the K.K.R.Z. formalism satisfies the empty lattice test. In practice, the ETA(l) are the adjustable parameters, which are chosen in such a way that for a fixed energy e^Ep equation (1.14)

generates a Fermi surface E(JC)=EF in agreement with the experi-

^ 18 CHAPTER I mental Fermi surface. In the case of indium we use only three parameters 1=0,1,2 (see section 4.2).

2 Intvoduetion In this chaptez used for the r. The preparation for the flatnes measurements wi will describe t pecially to the part of the sur ing the transmi 2.3. A comparist S.E. measuremen will give some

2.1 Sample prepi The preparation on the R.F.S.E. (Sharvin and GaJ Zaitsev 1970, G| consist of grow] demountable raolJ low melting poij softness of indj methods to prej failure of the to prepare thin The ingots of i were 99,9999% p Arnhem,Holland. • about 10 mm werj The seed crystaj in the air. Prc T

CHAPTER II

EXPERIMENTAL ARRANGEMENT

2 Introduction In this chapter we will give a survey of the experimental methods used for the r.f.size effect (R.F.S.E.) measurements. The preparation of thin indium plates which meet the conditions for the flatness and uniformity of thickness required for R.F.S.E. measurements will be described in section 2.1. In section 2.2 we will describe the twin-T bridge method. We will pay attention es- pecially to the problem of separation of the real and imaginary part of the surface impedance. This separation is impossible using the transmission technique which will be described in section 2.3. A comparison of several experimental techniques used in R.F. S.E. measurements will be given in section 2.4. In section 2.5 we will give some details of the complete experimental set-up.

2.1 Sample preparation The preparation of monocrystals of indium suitable for experiments on the R.F.S.E. is described extensively in the literature (Sharvin and Gantmakher 1963, Yaqub and Cochran 1965, Revenko and Zaitsev 1970, Gregory and Superata 1970). The quoted methods consist of growing a single crystal plate from melts in demountable molds. The use of molds is possible because of the low melting point of indium. This property together with the softness of indium material are the reasons why the usual cutting methods to prepare thin plates are failing. First we discuss the failure of the usual cutting method and then report our technique to prepare thin plates of monocrystalline indium. The ingots of indium material we used to prepare single crystals were 99,9999% pure metal supplied by the Kawecki-Billiton Co., Arnhem,Holland.. Rods of monocrystalline indium with a diameter about 10 mm were grown from the melts by the Czochralski method. The seed crystals to be used ; were grown on a Kapitza furnace in the air. From the single crystal rods thin plates were 20 CHAPTER II

cutted by means of a spark-erosion machine, model STM 1 from Agietron, Switzerland. This apparatus is a so-called relaxation type. In practice the operation conditions of this spark-erosion machine can be varied in a broad range. The most important factors are the cutting rate, the operating energy and the cleaning effect of the used dielectric (Shell Sol K). i) To avoid any damage of the crystal and short-circuiting the cutting rate has to be low. The cutting rate amounts to a 1 mm/10 hours, ii) The operating energy determines to a high degree the flatness of the crystal plates. The sparks produce a number of craters on the plate. Lowering the operation energy the depth :, of the crates can be minimized. The used energy was as low as 1 possible (<10~ Joule). iii) Furthermore any stagnation in the '] spark process will produce an unevenness in the surface of the ;1 crystal plates. Hence, the refuse produced during the sparking .) process has to be removed as soon as possible, since the '! produced indium grains can cause short-circuitings. A flow of ] dielectric fluid was directed along the sloping-mounted :j tungsten cutting-wire. •") Taking the precautions mentioned above good results were not -A guaranteed. During the spark-erosion process the cutted indium 4 plate becomes warped. In order to prevent this the indium rods I have to be kitted on a carrier in such a way that the cutted !'j indium plate is fully supported by the carrier. However, since 4 .f the indium material is very soft a profile of the used kit J ' tit appeared at the other side of the cutted plate dependent on the plate thickness. The resulted unflatness can be removed by electro-polishing both sides of the indium slice. In order to produce perfect flat surfaces a prolonged electro-polishing procedure should be needed; in practice, however, such a prolonged procedure causes rounding of the edges. In order to obtain R.F.S.E. measurements of high accuracy the sample thickness have to be constant within a 1%. This high demands cannot be satisfied using the spark-erosion technique. A suitable alter- native method is the one in which a demountable mold is used. A schematic diagram of the mold and the filling arrangement is given in figure 2.o. The mold consists of an upper and lower "I CHAPTER II 21

>— GLASS SYRINGE

-. HEATING ELEMENT c < <

i GLASS SPACER GLASS MOLD HEATING ELEMENT

BRASS THERMOCOUPLE HOLES COPPER

KAPITZA FURNACE

HEATING ELEMENT INJECTION HOLE q

SPACER SEEN THROUGH THE UPPERGLASS

Fig.2.0 Arrangement of the Kapitza furnace and the mold used to grow thin monocrystalline;indium plates. Upper parts the heated syringe used to fill the mold with molten indium. Lower part: the construction of the mold. The dimensions are given in millimeters. -

• 22 CHAPTER II

3 polished glass plate with dimensions 25 x 70 x 1.5 mm and a shaped glass spacer. The used spacers have a thickness of about 160y and 320u. The circular part of the spacer was used for pre- paring samples suited for use in the double-sided excitation equipment (see sections 2.2 and 2.5). The rectangular part of the spacer shapes samples suitable for the transmission arrangement (see section 2.3 and 2.5). The molten indium was injected into the small mold cavity by means of a syringe shown in figure 2.0. The process of producing single crystal specimens with this mold and syringe is now described sequentially. (1). The mold and the syringe both made from (pyrex) glass were thoroughly cleaned. The mold and syringe were cleaned first in a solution of potassium bichromate in sulphuric acid during about half an hour. Then the mold and syringe were rinsed in distilled water and subsequently cleaned in aqua regia. The last cleaning process is used to eliminate traces of metals. The --1 cleaned mold and syringe were finally rinsed in bidistilled j water at 100° C and dried with a high-purity low residue organic J solvent such as ethyl alcohol or acetone. :j (2). A second cleaned syringe was filled with indium ingots of ! 99.9999% purity. The two syringes were placed in a vacuum better '-•% than 10 mm Hg. The molten indium (we used induction-heating) ?| was injected into the first syringe and outgassed for about one '?\ hour. The indium was then cooled down to room temperature. "i: (3). In order to prevent the metal from adhering to the mold its fl surfaces were treated. The upper and lower glass plates of the f| mold were blackened with soot from a high-purity ethyl alcohol i| flame. The soot produced by a natural gas-flame can also be used, but the soot from alcohol adhere more firmly to the glass. Besides the soot grains arising from alcohol are much smaller than those from natural gas. (resp. < 0.5y, * 2u) The edges of the spacer were always blackened with soot arising from the gas flame. (4). The mounted mold was placed on the Kapitza furnace! The syringe provided with a heating wire was placed perpendicular to the mold in the injection hole of the upper glas plate, (fig.2.0) The mold and syringe were kept at a temperature just above the melting point of indium (156 C). The assembly was r CHAPTER II 23

placed in the open air. The molten indium was injected into the mold by applying a pressure at P by means of a balloon (5). Subsequently the syringe was replaced by a seed crystal. The seed crystals grown from the melts in vacuum were oriented by means of X-ray diffraction. Before placing the seed crystal in the injection hole the seed and the indium bud protruding from the mold were first cleaned with soldering-liquid. When ; the indium liquid and the seed are fused the mold is cooled ; down slowly by reducing the temperature of the Kapitza furnace "\ gradually. The solidification started at the seed and the - liquid-solid boundary is shifting to the right (see fig. 2.0). | Reaching room temperature the seed was cutted with a razor 1 blade.

;; (6) . The desired crystal orientation being the direction of J the normal to the grown indium plate was checked with an i X-ray back reflection Laue-photograph. The samples used in the I experiments are oriented with their normals along the [lOoJ - ;< and [poij directions to within one degree. 'i The samples prepared in a mold blackened with soot from the 1 gas-flame have a higher reflectivity than those blackened with '! soot from burning alcohol. We expect that some oxidation can ^ have occurred. \1 The roughness of the surfaces were checked with a Mirau-inter- 1 ference assembly (Leitz made). The unevennesses on microscopic ;!f scale were within one wavelength of the sodium d-line. The || unflatness on macroscopic scale or the variations in thickness i| were within 2-3 wavelengths of the sodium d-line. The thicknesses of the samples were measured by means of a Leitz metal-microscope focussing successively at the sample surface and the carrier of the sample. The accuracy of this method amounts to ly. Occasionally a measuring ocular (Swift made, England) has been used (accuracy « 5y). After excecuting our R.F.S.E. measurements the thicknesses were determined using an electronic micrometer (the Micro-Comparator "Microtonic", Rank Taylor Hobson Division, England) with an accuracy of ly. The orientations and thicknesses of the samples are given in table 2.1. 24 CHAPTER II

Table 2.1 Data of the samples

sample orientation thickness thickness of the normal at R.T. at 4.2 K t (n) (y)

2 [ioo] 161 ±0.6% 159.5 ±0.6% 3 [100] 158.5 ±0.6% 157 ±0.6% 4 [001] 151.5 ±0.6% 151 ±0.6% 5 [001] 312.5 ±0.5% 311.5 ±0.5% 6 [IOO] 285 ±0.5% 282.5 ±0.5%

(7). The grown single crystals are not suited for mounting in the experimental apparatus because of the appendages and different dimensions. The single crystal plates were therefore shaped with the spark-erosion machine. The essential operation conditions are described above. Finally it will be noticed that we don't use a mold with a filling channel and a seeding channel and that the mold has an open end at the right (fig. 2.0). Using one channel we are secured that upon cooling down the sample will be strain free I-..H on account of the absence of a second fixed point. On demounting the mold the appendages in bhe two channels should make it difficult to remove the sample free from damage. The open end of the mold facilitates the demounting of the mold. Besides, upon freezing there will be a contraction of the indium material so that excess material will be needed. In molds with a closed end we have often observed pinholes or defects at arbitrary places in the crystal.

2.2 Twin-T bridge method

In this section we discuss the experimental set-up with a twin-T bridge. The use of this method is based on the concept that with the twin-T bridge it will be possible to measure the field L derivative of the real and imaginary part of the surface CHAPTER II 25

impedance Z=R+iX simultaneously. Actually in this technique we are detecting the change of the high-frequency resistance and inductance of- a coil wound around the indium slab. First, we will describe the relation between the surface impedance and the resistance and inductance of the coil. Then we will deal with the twin-T bridge and the way of automatic control of the bridge. The advantages and disadvantages of the bridge method with respect to other methods will be discussed in section 2.4. 2.2.1 Relation coil-sample The relations between the resistance r and the inductance L of the coil and the real part R and imaginary part X of the surface impedance of the sample have been derived by Kittel (1967) and quoted by Pukumoto and Strandberg (1967) and Carolan and Koch (1973). Due to the special definitions Kittel is using, coil quantities do not appear in his formulae. Therefore, we will shortly give here a derivation of the mentioned relations. Consider a flat solenoid with a flat slab of conductor with thickness d inside. Although usually the coil has been wounded tightly around the sample, which is the best experimental situation, we consider the case that the sample does not fill up the coil volume completely. There are several experimental reasons why we have chosen for this configuration. First of all indium is a very soft material not easy to handle with. So, in order to prevent deformation and unflattness of the indium plate we did not wind a copper wire around the sample directly. In most cases the samples were mounted on a glass carrier which was placed within the coil in a loose way. Secondly, the shrinkage of the two materials, the indium plate and the copper wire, is different on lowering the temperature. This fact can cause deformation of the indium plate with the coil fitting close to the sample on cooling down the system to helium temperatures. So, consider a flat solenoid with length 1, width w and heigth h and let the sample have a thickness d (d <. h), width w and '<§ length 1 or larger. A r.f. current through the solenoid excites an electro-magnetic field in the coil. When the characteristic 26 CHAPTER II

dimensions of the sample are small compared to the wavelength the external field, say H may be regarded as homogeneous. The electric field connected to it is in the y-direction. The electro-magnetic field in the sample depends on the penetration depth z, so H =H (z) and E =E (z). The radiation energy falling xx y y on the sample per second and per unit area normal to the z- direction is given by the vector of Poynting (in Gauss units): S — ^ v w i o i ^ "~ A XT * Combined with the definition of the surface impedance:

Z = R+iX = 2£ H* (2.2)

this leads to

S = (~) Z Hx (0) (2.3) where E (0) and H (0) are the r.f. electric and magnetic field at the sample surface. The time average of the real part of (2.3) is the energy dissipated in the sample per unit area:

< Re{S} > = 4 (-£") Hv(0) Re{Z} (2.4) The boundary conditions at the sample surface are: (IWint= ^tanW (2.5)

supposing that there are no surface currents. Together with the formula

„ _ 4ir N . /o cx Hext " ~ T xc (2*6> where N is the number of turns of the coil and i the r.f. current through the coilf equation (2.4) becomes. < Re{S} > s L L i Re{z} (2.7) I ^2 G This dissipation of energy in the sample can be observed as an increase in resistance of the coil Ly an amount r1 given s through ^J i^ (2.8) CHAPTER II 27

where O is the total area of the coil being 2*(1 * w). Together with the a.c resistance this leads to the following

relation between the resistance rs of the coil and the real part R of the surface impedance Z:

r *r-+— OR (2.9) s a.c. ,2 We now calculate the contribution of the* sample surface to the inductance. First we consider the flux through the slab only. We define the inductance in Gauss units: L = Re {c ^lux} (2.10) c d h = Re {£- N wQ/ yHx(z)dz} (2.11) ° dEv i Using the second Maxwell equation -**- = -~ H and the antisymmetric excitation condition E (d) = -E (0) (2.11) becomes L = Re {|£i 4-2 E (0)} (2.12) with (2.5) and (2.6) and (2.2)

L = Re w The part of the volume of the coil outside the sample gives a contribution to the^inductance of

L = 4ir ~ w(h-d) (2.14) The total inductance is

Kt2 Or.2 »j2 L = 4TT ~ w(h-d) + — V w X (2.15) Si. b) X Consider a change of the surface impedance due to, for example, a change of the external static magnetic field. The relative change in the inductance is L (X+AX)-L_(X) AV -——— = (2.1b; hs[X) X + 4TT -Si- h{l-d/h) 2c2 As mentioned above, we conclude that the experimental set-up with the coil fitting close to the sample (*v*d) is the best one. This concerns the dependence of the amplitude of the measured signal AI*s on the ratio of the thickness of the sample and the Ls 28 CHAPTER II

height of the coil. Concerning the dependence of the signal amplitude on the ratio of the thickness of two different samples we have to take into account that the signal amplitude of the R.F.S.E. is proportional to exp(-s/l) where s is the length of the path of the electron in the sample and 1 is the mean free path. This means that in the case we are doing measurements on two different samples with the same coil going from a thin to a thick sample the advantage of having a better filling competes with the disadvantage of having a more unfavourable f ctor exp(-s/l). Notice that the relative change in the inductance is not proportional to the filling factor as is the case of nuclear magnetic resonance.

When the coil forms a part of the twin-T bridge it is appropriate to consider it as an inductance L parallel with a resistance R . The relative changes of L and R in the parallel configuration are:

AL 2 AL 2 Ar P _ 1-U/Q ) s . 0 (1/Q ) s p 1+(1/Q2) s 1+(1/Q2) s (2.17)

AR . ft2 Ar n2 AIi Rp 1+Q2 rs 1+Q2 LS

uLs Rb where Q = —— = —rp- . When Q is not too small, for example in s W1

AL AtoL , .;; Writing --— = -—— pr and assuming that Ar =AwL we see that —! x.s rs u s s *

ALS Ar -r— is about an order smaller than -rr— . Actually the Ls rs situation is more favourable, however. In practice we found that the "R-signal M being proportional to AR/R (see form. 2.35) is about 100 times larger than the "ID-signal" being proportional to AL /L (see form. 2.35). From the

L CHAPTER II 29

relation (ARp/Rp) / = -(Ars/rg) / UL8/LS) + 2 we conclude therefore that Ar /r is much larger than &L /L . s s s s Therefore the relative change in the parallel resistance can be taken in good approximation equal to the relative change in the series resistance. An additional argument can be found in (2.16) which shows that the smaller the filling of the coil the smaller AL /L . 2.2.2 Twin-T bridge In this section we discuss the twin-T bridge itself. The characteristic lines such as the central grounding of the system and the independent regulation of each T-branch has been described by Tuttle (1940) for a bridge in his general form. The earliest applications of the twin-T bridge to r.f. techniques have been done by Anderson (1949) for nuclear magnetic resonance, by Taylor (1965) for Doppler-shifted cyclotron resonance with helicon waves and by Matthey (1969) for R.F.S.E. Although the twin-T bridge is described sufficiently by Grivet, Soutif and Gabillard (1951) and by Losche (1957) the importance of the bridge for the separation of the real and imaginary part of the surface impedance legitimates a review of the main aspects.

The twin-T bridge consists of two T-networks in parallel. A T-network is shown in fig. 2.1a. In order to simplify the

Fig.2.1 Transformation of a T-network (a) to a n-network (b). clscussion of the twin-T bridge we> transform each T-network to a n-network with the transformation formulae (see fig.2.la and b) (and cyclic permutations) (2.18) 30 CHAPTER II

After transformation we have to deal with two n-networks in parallel. We will denote the second n-network with primes. The output voltage V relative to the input voltage e^^ is given by U " if =K (ZB+ZB> where (2.19)

From (2.19) it follows that for Zg+Z^ = o the output will be zero, so the output of one n-branch is in balance with that of the other U-branch. The circuit diagram of the twin-T bridge is shown in fig. 2.2. In (2.19) Z +Z'=o leads for the

-=J2

7 D

Fig.2.2 Circuit diagram of the twin-T bridge. mentioned circuit to the following balance conditions: a = •) - 1 = o (2.20) P = -) - 1 = o

The "a" is connected with the resistance R of the coil, the "p" is connected with the inductance L of the coil. As R P P primary determines the amplitude of the signal, while L leads to a phase shift, we call "a" the amplitude signal (or R -signal and "p" the phase signal (or L -signal). Under the conditions CHAPTER II 31

(2.20) the output and input impedance become C, R {1+^} J Z.+Z' out A A

(2.21)

•§

In general R is large, while C. ,C9,Co are of the same order C P /-* 3 1 or =— is larger' than ?r~ and 1. Therefore, the real part of C2 C2 of Z . is of the order R, the imaginary part about

u(2C2+C1+C3)*

We return to equation (2.19) and consider the bridge transfer function U near the point of balance. A small change of the coil impedance Z will produce an output voltage which in first approximation is proportional to

AU = {(Z +Z ) dK -} (2.22) B B K dZ

So when the bridge is competely balanced (ZB+Z^=o) and a small detuning occurs, then the output voltage is in first order proportional to the change in Zg+Zi. The ratio K has to be taken in the balance point. Consequently the factor K in (2.19) can be approximated in first order by a constant i.e. the value of K in the point of balance. As a result the bridge transfer function

U = K (a-iQp) (2.23) which after complete bridge adjustment will be zero, deviates from zero, ddui e to a change of R with AR and of L with &L according to 32 CHAPTER II

K AL AU = (2.24)

From (2.20) we conclude that the variation AR produces only an "amplitude" deviation Aa and analogue the variation AL "•:$. produces only a "phase" deviation Ap. Also, we can readjust

the bridge in such a way that we reduce with the adjustable •A capacitor C, the "amplitude" deviation only and with the 1 adjustable capacitor C the "phase" deviation. This important property forms the basis for the separation of AR AL -=£ and -=•*- . Moreover, with the capacitors C~ and C we can RP LP 3 simulate the changes of the coil quantities R and L P P respectively. Finally, we consider the noise characteristics of the twin-T bridge. We will show that the bridge produces a small amount of

additional noise, which depends on C~/C2. A comparison of the used bridges will be given. Let us define first the voltage across the coil:

Z3ei Z V l o (2.25)

Therefore the output voltage can be written as:

V7 (ZR+Z«) (2.26) Zf ZB+Z1+Z3+ zf Z3

Introducing a slightly detuning of the bridge out of the balance point the output voltage will be according to (2.24 and 2.25)

(2.27) 'p S C2 C2 where AR_ and AL are the applied changes which produce the CHAPTER II 33

detuning. The thermal noise voltage generated by the bridge in thermal e- quilibrium at the temperature T is given by the Nyquist foriuula:

2 V » 4 kBTR Af (2.28) R is the real part of the output impedance of the bridge and af is the frequency band-width within which the voltage fluctua- tions are measured. Prom equation (2.21) we can deduce an expression for the output resistance, which holds for balance conditions:

R = (2.29) C-, C. (2+-1+-V +

From equations (2.27), (2.28) and (2.29) we obtain the signal- to-noise ratio for a twin-T bridge under balance conditions /R. AR (2.30) V /(4KBTAf)

where the voltage Vw across the coil has been expressed in Z2 the current through the coil I_ producing the r.f. field AL within the coil. Considering that is '2 Q2 the voltage across the coil arising from a change of R and L with AR and AL respectively and that R /{4KnTAf (-*)} is the thermal noise generated in the series B Q2 resistance of the coil we can write the noise figure F as C F = /(2-Hri) (2.31)

The described twin-T bridge has been build for three frequencies namely 1,2 and 5 MHz. In table 2.2 values of the J 34 CHAPTER II bridge components for the three operating frequencies are given.

Table 2.2 Values of bridge components

frequency capacitance resistance

f (MHz) Cx (pF) C2 (pF) C3 (pF) C (pF) R (n)

1 120 12 200-670 6200-7260 220 2 .120 120 30-490 1180-1880 220 5 33 15 95-215 70- 220 33

With these values and chosen values of the adjustable capacitance C3 and C we calculate by means of the equations (2.20), 1 (2.21) and (2.31) the quantities as mentioned in table 2.3.

Table 2.3 Characteristic quantities of the bridge

frequency (MHz) 1 2 5 'An C3 (pF) 624 450 140 chosen capacitance 1 C (pF) 6500 1600 115

inductance L (vH) ~3.3 -A

resistance R (fl) ~149 "390 "2700

quality Q ~8 ~9 ~21

output impedance Z (fi) 151-228i 126-1251 21.1-1555i

noise figure F 7.34 2.40 3.36 CHAPTER II 35

We see from table 2.3 that the series resistance of the coil being (R /Q2) increases with increasing frequency due to the skin effect. The increase of quality Q with the frequency is not only due to the increase of the frequency but also due to the use of a coax cable with which the coil forms a part of the twin-T bridge. The- influence of a 50-Ohm coax cable on the actual observed resistance R and inductance L will be discussed, (section 2.2.6). The output impedance appears to be a resistance of small value in series with a capacitance of the order of 700 pF for 1 and 2 MHz and of 200 pF for 5 MHz. The "signal-

to-noise ratio" calculated from /R / {Q/(2+(C3/C2))} appears to be the best for the 2MHz bridge. The 5MHz bridge has about the same "signal-to-noise ratio", but for the 1MHz bridge it is about a factor 4 lower.

2.2.3 The tuned amplifier The output signal defined in equation 2.24 being a r.f.signal was amplified by a tuned amplifier. For each of the three different operating frequencies we have a separate tuned amplifier. The two-stage r.f. amplifiers were build of the CA 3005 integrated circuit operating in the so-called B- £j mode. For each used frequency the twin-T bridge and the tuned jj amplifier belonging to it were placed as near to each other | as possible and both the amplifier as well as the bridge were | housed in a metal box, while adequate shielding of the jj several components and supply decoupling are provided. A |j schematic diagram of the twin-T bridge and the two-stage ^ r.f. amplifier is shown in figure 2.3. The values of the amplifier components are listed in table 2.4. The CA 3005 integrated circuits are placed in the differential-amplifier configuration and operate from dual supplies requiring the least number of external components. We use the CA 3005 in his B-mode because of the small value of the total d.c.current about 0.9 m& with good noise performance and his low power dissipation 15.0 mW. The input resistance of the CA 3005 is about several kilo ohms in parallel with a capacitance of about 10 pF. The output resistance, however, is about 105 ohm in parallel J 36 CHAPTER II

_0.0J w _

w A i_j I L '( **«/«.. i \ootu I onin 0-12V

Fig.2.3 Schematic diagram of the twin-T bridge and the tuned amplifier. The component values are given in the tables 2.2 and 2.4. with a capacitance of 4 pF. The advantage of the use of an integrated circuit is the low feedback of this circuits provided that adequate shielding and layout has been taken care of. As the noise figure of the CA 3005 increases both with an increase in current and with an increase in frequency we have ft chosen for low frequencies and as mentioned above for a low operating current. The noise figure is nominal 3 a" 4 dB. For good noise performance the source resistance for d.c operating conditions should be low. An inductance at the input of the CA 3005 is used to satisfy this requirement. The gain of one stage of the r.f. tuned amplifier is about a 100 times. We now pay our attention to the coupling of- the twin-T bridge to the r.f. tuned amplifier. In the preceding section we have seen that the bridge has an output impedance consisting of a resistance of a hundred Ohms in series with a capacitance of several hundreds pF. Besides, a load resistance will not change the balance conditions. From figure 2.3 w

CHAPTER II 37

Table 2.4 Bridge and amplifier components (fig.2.3)

frequency

components 1 Mhz 2 MHz 5 MHz

C/Cj t*~2'3 see table 2.2 C4 l-400pF l-400pF 22-52pF C5 2-430pF 2-430pF 27-75pF 22-52pF C6 2-430pF 2-430pF lx l-300pF 4x l-30pF CD 3* l-300pF 3* 1- 30pF 3* l-30pF CD3 3x 1- 30pF R see table 2.2 lOOyH 30yH Ll 300wH 4.7mH lOOyH L2 4.7mH 20pH 10pH L3 lOOyH : i 20yH lOyH L4 lOOuH Remarks: the capacitors are sintered silver mica capacitors, tol. 0.5%, T.C. 30 ppm/°C and the tuning capacitors I-30pF in 20 turns and J-400pF in 180°, T.C. 50 ppm/°C; the varicaps are 1-30pF, BB 105G, Philips and .a l-300pF, BB 113, Siemens; the resistors are metal film • 'I resistors, tol. 0.5%, T.C. 50 ppm/°C; the inductors are h.f. inductors Q « 40, T.C. < 20 ppm/°C.

" I we can see that the coupling between the bridge and the ampli- i fier occurs by means of a tuned circuit. The equivalent circuit of the twin-T bridge and the tuned circuit is given in figure 2.4? The output voltage of the bridge is represented by

V and the output impedance by R and CQ. Because of the capa- citive character of the output impedance of the bridge we prefer the given configuration to a transformer coupling. Evaluating the output voltage V across the inductor we obtain CHAPTER II

Fig.2.4a Equivalent circuit of the twin-T bridge together with the tuned entrance circuit of the r.f. amplifier.

(2.32) H As we have seen K given in equation (2.19) is for Zfi+Z^ = 0 in first order approximation a constant. At the same time the output impedance defined in equation (2.21) may be considered as a constant during the measurement of a r.f. size effect line. However/ in order to diminish any influence of the bridge output impedance on the tuning we require that C^ *10 - C . This leads to an inductance of a 300uH for 1 MHz, lOOpH for 2 MHz and about 30pH for 5 MHz. In resonance the transfer function will be H

Rs+ia)Ls VRo (2.33) The advantage of the low output resistance R of the twin-T bridge is now clear. Putting R_ = 0 the voltage at the input of the CA 3005 is larger than the bridge output voltage V by a factor of about 10. In practice the factor may be smaller due to the lower quality of the circuit. Prom (2.33) we see that the tuned circuit introduces a change of phase of about CHAPTER II jy

90 degrees. More important than a phase rotation by the tuned amplifier is the constancy of the phase. In order to minimize the influence of external effects such as temperature drift and microphony we choose components (see table 2.4) with a temperature drift of about a ppm/°C, while the components were firmly^soldered in the metal box. A typical plot of the bandwidth of the tuned amplifier is given in figure 2.4.

1.0 S 0.5

£ 0.2 01 UJ 0.05

UJ cc. 0.02 Q «7S 0.92 0.96 1.04 1.08

Fig.2.4b Frequency response of the 2MHz and 5MHz amplifiers.

2.2.4 Mixing procedure In this section we deal with the method of separating the real and imaginary part of surface impedance changes i.e. AR /R and 1 Pi AL /L in equation (2.24). In the next section we will describe the steps to be taken in realizing the separation. We start with the situation that the twin-T bridge is not completely balanced. Then the output of the bridge is given by equation (2.23). Equation (2.23) may also be written as

exp (a-iQp) EQ exp (2.34)

where E exp.(in t) is the input voltage of the bridge and

KQ exp.(i<|>o) the factor K defined in (2.19). Let a and p be so small that the first order approxation given by (2.22)

Note: in this chapter we often omit the subscript o in WQ, except in exp(iw t). 40 CHAPTER II

still holds. So K and <{> are the amplitude and phase of K taken in the balance point. Now, we introduce a variation of "a" and "p" by changing the resistance R and inductance L of the coil with an indium plate inside. These variations can be realized by modulating the external magnetic field in which the coil is placed. If the modulation field is small enough the periodical variation of AR may in first order approximation be replaced by (dR /dB)b COS(ID t+if ), where B w t+< tne is the magnetic field of induction and b cos ( m f>m) modulation field.Similarly, AL may be written as (dL /dB)b cos (cu t+i|i ) . Substituting this in equation (2.24) the additional voltage induced by the modulation field can be expressed as

K exp (14) AVQ = — ~{M}Eoexp. (io)Qt) with 1 p H2.35) dR dL 1 t> b 1 n b M IT 3T b m eost^t+^-iQ IT dT bm eos(«omt+«.n) * IT 3T m eost^t+^-iQ IT dT m eos(«omt+«.n)

In equation (2.35) we have introduced a phase m and i|»m. The phase <|> describes the phase-lag of the dR /dB-signal with respect to the applied modulation field b cosa>mt, while dR /dB is taken in a fixed point B of the external static magnetic field. Similarly, the phase \pm describes the phase-lag of the dL /dB-signal with respect to the applied modulation field, where dL /dB has to be taken in the fixed point B . Gen- erally and \\) are different. We may expect a phase difference of 90 degree according to the character of a resistance :• and inductance. However, due to the field distribution in the V) sample (see Awater (1973)) the phase difference can deviate I) from 90 degree. Also, on varying slowly the static magnetic •'{ field the dR /dB and dL /dB signals have different phases >^ with respect to the external field B i.e. dR /dB and dL /dB | are not at their maxima simultaneously. The dL /dB signal will be about 90 degree in advance of the dR /dB signal. Returning to equations (2.34)'and (2.35) we can obtain the total output voltage of the bridge, when the bridge is near LJ CHAPTER II 41

his balance position and a small modulation field is applied by adding equation (2.35) to equation (2.34).

K exp (iA) a Vt - -{A-iP}EQ exp (iuQt) with (2.36 )

(2.36b) Let us consider the expression (2.36 ). We can represent this expression by a vector in the complex plane (fig.2.5). The coordinate axes are referred to the input signal E exp (iw t)

The vector OA rotating with the frequency wo indicates the signal (^- - i-E-)exp.(iw t) so a signal when no modulation P p field is applied. Applying a modulation field and choosing the

phases <|>m and \pm arbitrary the vector OB representing the signal: dL.. cos (um t+\bm )}exp(iow t)

(2.37)

'it

Fig.2.5 Vector representation of formula (2.37). (see text). It is assumed that the real and imaginary part of the surface impedance change harmonically with the field B.

1 i 42 CHAPTER II i 1 describes a locus denoted with 1 in figure 2.5. The derivatives "\ dR /dB and dL /dB were assumed to be taken in a fixed point B "^ of the static magnetic field. When, however, the magnetic field i B varies with time dR /dB and dL /dB change in magnitude. If ! • i for instance the variation of R and L with B can be repre- (| sented by a bubble superimposed on a flat background the vector j OB describes the locus 2 in figure 2.5, supposing that just be- j fore and after the bubble a/R^ and p/wL are not changed. -,] Connected with each point of the locus 2 there will be a locus .;i 1 due to the applied modulation field. | Now, we wish to separate the amplitude signal defined as j dR A = =— + —1 -sD^ b cos((D t+d> ) (2.38) M^ Rp RadB m « m ^ from the phase signal defined as $1 dL ^g2 b^ cos(o) t+i// ) (2.39) u)Lp

It may be remarked here that when we are able to separate the f| amplitude signal and phase signal a/R and p/wL respectively, r °;| i.e. the amplitude and phase signals without applied modula- f| tion field we also have separated the signals I

(1/Rp) (dRp/dB)bm cos(wmt+m)and(l/o)Li) (dL /dB)bm cos (^mt+^>m) • 4] This property will lead to a method of separation as we shall :| see below. :| The signal given in expression(2.37) is not the output voltage J of the twin-T bridge as defined in (2.36). Along with an ampli- J 2 fication of the signal with KQE /w C* there is also a phase | rotation of $ -. Moreover the tuned amplifier gives an addi- M tional amplification G and phase rotation a. VThen we write the I / A\ . J1 vector description of (2.37) as ( ) and the phase rotation \-P/ as a matrix we can express the output voltage of the tuned amplifier by

^cosa -sina\ v _ /cos<|>_ -sin(j>_\ / A

cosa/ "* 1 \^sin(j)o cos$oJ \-P CHAPTER II 4 3

A cos( +a)+P sin( or V = O O ° w > +a)-P cos(4> J In consequence of the several phase rotations the output signal '{ has the components A cos ( +a)+P sin( +a) and i A sin(^) +a)-P cos( +a) referred to the coordinate system of s fig.2.5 corresponding to the input voltage of the bridge. This :,\ means that the amplitude signal (2.38) and the phase signal (2.39) 4 " j are mixed up. In order to obtain the components A and -P sep- | arately we have to rotate our coordinate system over an angle 1 -(+a). In the procedure of separating the amplitude and phase '•i signals we use the fact that they are perpendicular to each I other. ^j For the separation we used a double balanced mixer which to- -] gether with a phase shifter meets the needs of rotating our I coordinate system and cancelling one of the two perpendicular ":-j signals. /I This mixer consists of a ring of four matched diodes in con- ?•"! junction with broadband transformers. The two pairs of diodes j are placed in a balance circuit. The diode can be regarded as : 1 a switch. Whether a diode is open or closed depends on the in- ;)] stantaneous value of the applied voltage. We give here a short ;lj outline of the operation of the mixer. The output voltage is ! proportional to the current through the output transformer f when two voltages e,, - and e, _ are applied across the diodes. v,i r • x • x * o • •* i The current through the transformer is equal to the difference ;| of currents through the diodes of one pair of diodes. The jf voltage across one diode of the pair is equal to the sum I er f +el o an<^ t*ie v°lta9e across the other diode of the pair C| is equal to the difference e, -e - . Expanding the current i through a diode in a series according to where e is the voltage across the diode, we obtain the output H5 voltage e± f . ei.f^al(2^r!f.)+a2(4el.o.er.f.)+a3<6ei.o.er.f.+2er.f.) + Due to the presence of a second pair of diodes in balance with 44 CHAPTER II

the first one the odd powers in this series disappear. In practice the term a«(4e, e -) is dominating. This term leads to two sidebands whose frequencies are equal to the sum and the difference between the L.O. and R.F. frequencies. The upper- sideband is usually suppressed by filtering. We use the lower sideband. When signals are applied simultaneously to the so- called L.O. and R.F. ports and both input frequencies are i- dentical an output appears at the so-called I.F. port which is D.C. and proportional to the cosine of the phase difference of the input signals and proportional to the input voltage at the ,| R.F. port. In practice the output voltage is independent of the l.o. voltage drive when this voltage is high enough (7dBm). On account of the given explanation it is allowed to describe the operation of the mixer with signals with identical frequencies at the L.O. and R.F. port applied as a scalar product of two vectors representing both input voltages in a complex plane taking the L.O. vector of unit length. For the output voltage is proportional to the projection of the R.F. signal on the L.O. signal. Let the output voltage given in formula (2.40) be the input voltage at the R.F. port of the mixer. At the L.O. port a signal of the signal generator is applied 7dBm in magnitude and with -of course- a frequency a) , while the phase with respect to the input voltage of the bridge is 8. Then, the output voltage at the I.F. port being the scalar product of e,. ~ and e, „ can be obtained by r.r. l.o. •* GE^K^ /A COS( +a)+P sin(* +a) (cosB sing) I I r A •1 which becomes: }(2.41) GE K {A cos( +a-S)+P sin(<)> +a-B)}

So, by using a mixer together with a phase shifter we first rotate our coordinate system over an angle 6 and then we project the obtained output voltage on the real axis of the coordinate system in a complex plane. CHAPTER II 45

We may consider two different cases. First, when g=<|> +a then the output of the mixer is proportional to the amplitude signal A. Secondly, when 6+ir/2=* +a then the output of the mixer is proportional to the phase signal +P. In our experiment we have used a quadrature hybrid which produces from one signal two signals equal in magnitude but 90 degree out of phase. The splitting of the output signal and the application of two mixers and phase shifters enables us | to separate the amplitude and phase signals at the same time. 2.2.5 Set-up of the twin-T bridge measuring system The detailed description of the mixing-procedure in the foregoing section has been given in order to show the importance of the phase shifting i.e. the choice of our coordinate system in a complex plane for the separation of the amplitude and phase signals. In the present section we will describe the separation procedure. A Block-diagram of the experimental set-up is given in figure 2.6. In order to satisfy the relations (2.20) at any time changing none of the bridge components a crystal-controlled digital signal generator with a frequency stability of 3 x10/24 hours was used. The power divided signal of the generator is led on the one side to the twin-T bridge, on the other side after amplification to the phase shifters. When the twin-T bridge is near his balance position the r.f. signal of the generator is for the most part suppressed. A small signal defined in (2.36) still remains. In accordance with formula (2.40) the output voltage of the tuned amplifier, provided the assumptions made in the foregoing sections still hold, is proportional to

{ (Acos(o+a) +Psin(o^a)) +i(Asin(<|>o+a)-Pcos(o+a)) }exp (2.42) The quadrature hybrid Q.H. in figure 2.6 is capable of split-' ! ting this signal to two isolated quadrature phased outputs. jj There may also be a phase shifting between the input and the outputs of the quadrature hybrid. For simplicity we define contrary to the definition in section 2.2.4, the phase shift a r

46 CHAPTER II

PSi

INP

PD « - 1 PD2 RF A

PS2

Fig.2.6 Block diagram of the experimental set-up with the 3 twin-T bridge Schlumberger FS 30/7,digital signal generator Schlucberger TP "3/10,broadband power amplifier Merrimac model PD 20-17, two-way power divider Twin-T bridge and tuned amplifier,see fig.2.3 Merrimac model QH-7-17,broadband quadrature hybrid Merrimac model DM-1-250,broadband balanced mixer Low-pass filter PAR model 211,broadband preamplifier .PAR model 210A,selective amplifier PAR model 220,phase-sensitive detector Moseley recorder 2 FAM, xy recorder Audio power amplifier DC amplifier,see fig. 2.. 10 Varian V-3900,twelve-inch electromagnet Varian V-2803,10 kW regulated power supply Varian V-FR 2803,fieldial regulator MARK II Switch box for applying reference signals across varicaps 1 Tektronix 581A, cathode ray oscilloscope CHAPTER II 47

in (2.42) as the phase shift of the tuned amplifier including the quadrature hybrid. Then the output voltage of the quadrature hybrid at channel 1 is proportional to the expression given in equation (2.42). The output voltage at channel 2 can be achieved by replacing a by a+Tr/2 in (2.42). The output voltage at channel 2 is proportional to

(-Asin((J>o+a)+Pcos(o+a) }exp (ia> t) (2.43) The mixers MX 1 and 2 in figure 2.6 driven at the L.O. ports by the phase shifted signals of the digital signal generator are detecting only a component of the signals given in (2.42) and (2.43) respectively. As we have seen in (2.41) this component is the projection of the signal on the real axis of an arbitrary coordinate system in a complex plane defined by the phase 3 (see figure 2.7). dm)' Im

Asin («po+a )-

Fig.2.7 Vector representation of formula (2.40). The projection of the vector OB on the Re'-axis represents the signal (2.41). (see text). Amplitude factors are excluded.

These projections of the signals (2.42) and (2.43) can be recognized as amplitude modulated radio-frequency signals with u) as the frequency of the carrier and u the frequency of the modulation signal. The output signal at the I.P. port of the mixer MX 1 because of the identical frequencies at the L.O. and R.F. ports is a demodulated low-frequency signal superimposed on a d.c. level. This can be explained by substituting (2.38) and (2.39) in (2.41). The obtained output voltage for the signal (2.42) has a d.c. level proportional to 48 CHAPTER II

cos ( +a-8) + ~j - sin(4>o+a-8) (2.44) RP and a low-frequency signal given by (except for a factor) dL l +<|> )-t— bmccs(Va-S)c dBbmsin (*o+a-8)cos(w L % P (2.44) Similarly, at the mixer MX 2 we have a d.c. level proportional to

- ^- sin(o+a-6) + ^ cos (o+a-6) P and a signal given by (except for a factor) }(2.45) 1 dRn - rr d^Vin(Va-3)cos(%t+V RP From this we conclude that when a=o and p=o no d.c. level is present at both I.F. ports. Furthermore when 8= +a, as described above, the mixer MX 1 has an output proportional only to dR /dB and the mixer MX 2 an output proportional only to dL /dB. Simultaneously the d.c. level is independent of p or a respectively.

The separation procedure is as follows: In order to satisfy the relations (2.20) i.e. a=o and p=o we have to adjust the capacitors Cg and C (see figure 2.3). The d.c. output of both mixers will be zero. Next we adjust 8 by means of the phase shifters PS 1 and 2. When 8 is not equal to $o+a the d.c. level is dependent on both a and p. As the capacitor C, according to (2.20), determines only p and the expression for a is independent of C, we can change C starting from the balance condition. We readjust 8 by means of phase shifter PS 1 so that the d.c. level given in (2.44) does not alter on changing C. Likewise we readjust the phase shifter PS 2 so that the d.c. level given in (2.45) does not alter on changing C3, as p is independent of Cj. It will be noticed that two phase shifters were used, although in both cases (2.44) and (2.45) 8 has to be equal to o+a. However, using two phase shifters we are able CHAPTER II 49

to correct for different phase shifting in both channels and for the inaccuracy in the phase quadrature of the quadrature hybrid. As mentioned above (section 2.2.4), and in accordance with the expressions in (2.44) and (2.45),when a/R and p/wL are separated the same holds for dR /dB and dL /dB. •r P P The described method is rather rough. On changing C or C, one disturbs the balance too much so that even K can not be regarded as a constant. (Note: (2.44) and (2.45) have to be mul- h tiplied with a factor 6K E /u2C?). Therefore we used in addition j an other method for separation. First, the twin-T bridge was automatically controlled in a way to be described below to satisfy at any time the relations (2.20). Secondly, a modulation technique was used which will be presented now. We start from the situation that K is equal to the balance value K . From the expression (2.34), in analogy with the derivation of (2.36), we can evaluate without modulating the magnetic field the output voltage of the twin-T bridge in the case that the capacitors C and C_ are changed by means of a small low-frequent modulation signal e cosa/t applied across the variable capaci-

tance diodes CD and CD- (see figure 2.3) by way of the inputs E and F of the bridge (see figure 2.6):

^^ >Boe*p(i»ooot> P (2.46) We neglect phase shifting in the modulation signal. Consid- ering the similarity of the formulae (2.36) and (2.46) it is clear that we obtain similar formulae for (2.44) and (2.45).

We only imitate the changes in R and L with C3 and C respectively. Taking a-o and p»o and modulating either C- or C we can separate the real and imaginary part of the expression between braces in (2.46) by adjusting the phase shifters PS 2 and PS 1 respectively in order to obtain no modulation signal at either the I^F. port of mixer MX 2 or the I.F. port of mixer MX 1. Suppose we have carried out this separation procedure, then two further tests are available. 50 CHAPTER II

We return to the expressions (2.44) and (2.45). The output of the njixers MX 1 and MX 2 is, after correct separation, i.e. cos u t+< ) and 8=o+a proportional to(l/R*)(dR /dB) ( m l ra) (1/wL2)(dL /dB) cos((D t+* ), where R and L are taken in the balance point. These signals were phase sensitively detected by two lock-ins (see figure 2.6). The phase adjustment of the lock-in has been executed by studying the out-of-phase signal during a sweep of the external (static)magnetic field. As dR /dB and dL /dB are functions of the magnetic? field the recorded signal will deviate from a straight line when a small in-phase component is present. A correct phase adjustment is achieved when a straight line is obtained for the out-of-phase signal. Let us assume that the separation is incorrect. Then, according to (2.44) and (2.45) the dR /dB is mixed up with the dL /dB signal and reversed. Because of the dependence of dR /dB and dL /dB on the(static)magnetic field, the added signal given in

(2.44) has no unique pfhase during a field sweep. The out-of- phase signal detected by the lock-in will deviate from the straight line. Simultaneously the signal given in (2.45) has no unique phase. Correct separation is only achieved when the out- of-phase detected amplitude and phase signals give a straight line during a field sweep. In fact, a suppression of the in- phase peak signals to a magnitude less than 1% of the peak value were achieved. A second test for the correct separation of the real and imaginary part of the derivative of the surface impedance Z is the study of a nuclear magnetic resonance line shape. The n.m.r. line is originating from the "epibond" material in which the measuring coil is embedded. The line shapes of the, absorption and dispersion signals of a nuclear magnetic resonance peak are well known from calculations of the real and imaginary part of the complex susceptibility. Using the field modulation technique we are not measuring the real and imaginary part of the susceptibility but its derivatives,- with respect to the magnetic field. In figure 2.8a is shown the absorption signal of potassium as given by Halbach (1954) while in figure 2.9a the dis- CHAPTER II 51 persion signal of potassium is given (Halbach (1954)). Figures 2.8b and 2.9b show the corresponding signals of a proton resonance originated from the epibond material. /I c 3 / 0 ) HP , / r . IG

« a H (orb. units) Htarfcunits)

Fig.2.8 The differential absorption signal due to nuclear magnetic resonance of potassium (a; Halbach 1954) and of protons in epibond (b; present work). A in In uni t uni t / \ 1 . I 1 x~ "\ / 1 r~ T3 w w \l \ / « • o " b H(arb. units) H(arb.units)

Fig.2.9 Tha differential dispersion signal due to nuclear magnetic resonance of potassium (a; Halbach 1954) and of protons in epibond (b; present work).

The measurements of proton resonance have been performed at a temperature of about 1.7 K. The extremes in the absorption signal appear at magnetic fields of 1167.7 G and 1179.5 G. The successive extremes in the dispersion signal appear at 1161.4 G, 1174.3 G and 1185.7 G. Taking the value 1174.3 G as the peak position and using the operating frequency of 5 MHz we obtain the frequency-field relation of 4.25785 JeHz/G. Com- paring this value with the proton resonance value of 4.25776 kHz/G and taking in account an accuracy of 0.1°/oo we conclude that proton resonance in epibond occurs. The line J 52 CHAPTER II

shapes of the two corresponding signals are identical. Which is connected with an exact separation. A condition for the correct operation of the twin-T bridge including the separation is that the bridge is in or close to the balance point. An electronic control device is designed which allows the amplitude and phase of the twin-T bridge to be tuned independently. When correct separation of the dR /dB and dL /dB signals is achieved the d.c. level defined in (2.44) is proportional to a/R , while at the I.P. port of mixer MX 2 the d.c. level is proportional to p/coL , according to (2.45). We use the d.c. levels in a feedback system as given in the Block diagram of figure 2.6 to readjust the twin-T bridge continu- ously to its balance point a=o and p=o. An automatic control of the twin-T bridge is described by Mehring and Kanert (1965). Therefore, we should like to confine ourselves to some remarks about the used feedback system. From figure 2.6 we see that the mentioned d.c. voltages after filtering are amplified by a d.c. amplifier and then fed-back to the variable capacitance diodes. The dynamic behaviour of both control systems is dependent on the dominant time constants of the control system and the closed loop gain. There are a number of three significant time constants/ which are of the same value for both systems. The twin-T bridge embodies a time constant of about —4 i 10 sec. Both filter-networks have time constants of -4 5.10 sec. The control systems are designed in. such a way that the frequency response of the open-loop transfer function is essentially determined by the third time constant incorperated in the d.c. amplifier. A wire-diagram of the d.c. amplifier is, given in figure 2.10. The d.c. amplifier has a variable gain (0-8000*) and a variable time-constant (0.2-200 sec). Besides with an incorporated compensation network we are able to compensate a possible d.c. level present when the tt-rin-T bridge is balanced. Generally, in our experiments we have used a time-constant of the d.c. amplifier of 50 sec. The critical gain, defined as the gain for which the response of the system shows an aperiodic behaviour and oscillations are just absent, can be calculated from L 'MC-^'&s »

+ 15V II 10 987654321 O 0 0 O 0 O O

OUTPUT n

H H

Fig.2.10 Schematic diagram of the d.c. amplifier. The gain of the amplifier can be varied from 0 to 8000. The time constant can be varied from 0.1 sec. to 200 sec. in a 1-2-5 sequence. 54 CHAPTER II

4 (g+D ^ ^ = (^ + £•>* (2.47)

where g is the gain and T. and T2 the two largest time con- j stants. Prom(2.47) it follows that the critical gain will be '' about 2.5X101*. However, with a large gain the effective time constant of the d.c. amplifier being T./g+l is very small. In consequence the low-frequent measuring signals (dR /dB) cos(w_t+<|>_) and (dLVdB) cos(w_t+ijj ) can be sup- p m HI p mm pressed partly by the feedback control system. In order to

prevent this suppression the following condition has to be t .

satisfied: (g+l)<2irfmT1, where f is the modulation frequency, 1 being 7.8 Hz. Hence, the gain has to be smaller than 2450 . As the gain g and time constant T. are adjustable (see fig. 2.10} the gain and time constant can be adjusted for optimum operating conditions. It will be noticed that for maximum available gain and T.=50 sec. the peak amplitude is reduced to about 1/3 of the original value. The signals to be measured were detected with two lock-in amplifiers. The used time constants of the lock-in amplifiers were 1-3 sec. In order to prevent shifting of the peak with respect to the magnetic field we had to sweep slowly. The sweep rate was 0.1-2.5 G/sec. The sensitivity of the described system defined as the relative change of R or L that produces a signal-to-noise ratio of 1 is about 10~8. 2.2.6 The use of a coax cable At the end of section 2.2.2 we mentioned an influence of a 50ft coax cable on the resistance R and inductance L as seen P P by the twin-T bridge. In this section we will discuss this problem more extensively. Coax cables or tubes are commonly used to connect the coil to the twin-T bridge, (see for example Taylor (1965);. Collins (1957) and Hui (1967)). The coil consists of 50 windings of 50 micron thick copper wire. The d.c. resistance .is about 10 Ohm and the inductance, as we can see from tvle 2.3 is about 1 CHAPTER II 55 .'i j 3yH. Generally, this coil is not matched to the coax cable for } all operating frequencies and temperatures. This feature has i important consequences. j Let us consider the case of a transmission line of length 1

|j with a characteristic impedance ZQ and an arbitrary termination tJ Z. It is well known that the input impedance of the terminated line can be expressed by

(Z+Zo)+(Z-Zo)exp(-2yl) Z = Z in o (Z+Zo)-(Z-Zo)exp(-2yl) (2.48)

vrhere y = /(R+iwL) (G+iwC) with R, L, G and C as the characteristic resistance, inductance, conductance and capacitance respectively. Writing

y = «+iko (GZ/ 2 _ R/2Z }2 2 + GZ /2) + iu>/(LC() ) {{1 + ^ 0 Z 0 Z U)2LC equation (2.48) can be transformed to

^ {ch(al)c(kol)+ish(al)s(kol)}Z+{sh(al)c(kol)+ich(al)s(kol)}Zo = ^ {sh(al)c(kol)+ich(al)s(kol)}Z+{ch{al)c(kol)+ish(al)s(kol)>Zo (where s,c,sh,ch stand for sin/COS/Sinhjeosh resp.) ^ * When the termination Z is a coil with the impedance R +iuL c s s

and z « /(L/C) is real and equal to RQ then we can cal-

culate the real and imaginary part of the impedance Zin. We confine ourselves to a special case. Let us consider the commonly used transmission line RG 58 A (see for example White (1967)). For the operating frequencies 1/2 and 5 MHz the attenuation coefficient a amounts to 3 2 (1.7 - 3.7)10" /m, while the wavenumber kQ is (3-15)10" /m. As

a result the influence of the phase shift kQl in the expression (2.49) is more important than the attenuation. Neglecting the the attenuation we obtain from (2.49) the real part Rs and imaginary part wLs of the impedance Z. .

_ i , !S_ (2.50) (~)2 sin2k l+Ccosk^l-f^- Ko ° ° • o 56 CHAPTER II

cos2k 1{U>L +R (1 - -)tan2k 1} S a>L = - -2—_° (2.50) (=^)2 sin2k l+{cosk l- (^) sink 1} 2 Ro ° °0 Ro We consider a 50& coax cable, so R =50ft, with a length l=lm.

Taking RS«RQ (Rg/Ro<0.2)f kQl equal to 1°48', 3°36' and 9°, while wL is 18.8, 37.7 and 94.2 for the operating frequencies 1,2 and 5 MHz respectively, it follows from (2.50) that the denominator of the expression (2.50) decreases on increasing the frequency. In consequence RS>R and Rs increases with in- s s creasing frequency. Likewise OJL >O>L and wL increases with increasing frequency although the term (wL /R_)2 in the numer- ator of (2.50) will flatten this increase. Further the quality factor Q becomes Q= u>Ls/Rs* VRs{(wLs/Ro)+ (1"(Rs/Ro)"(aj2Ls/Ro))ko1} From this formula it follows, that for small wL /R_ and R_/R_, Q^wL/s R s. However, with increasing frequency the term 2 s/ R o) ko 1 will reduce the increase of Q with frequency* , Suppose that the preceding description represents the experimental situation, being a 50ft coax tube at liquid helium temperatures (the dielectric is helium). In other words, we ig- nore the influence of the attenuation and consider only a phase shift k 1, which we assume to be small (k 1< 0.15 for all frequencies used).In that case we may conclude that the increase of R , coL and Q shown in table 2.3 are in accordance P P with the described features of R wL_ and Q. f s However, from (2.50) it follows that the resistance R and inductance wLs as seen by the bridge are dependent on the resistance R and inductance wL of the coil both. In consequence of this dependency there may be a mixing of the amplitude and phase signals as defined in (2.38) and (2.39). As (R^/R..)2 = 0.007 and sink_l * k_l«l we neglect the terms 2 2 (RS/RQ) sinkQl and (RS/RQ) in the equations (2.50). The influence of R on wLs is therefore negligible in contrary to the influence of o>L on R . Therefore, let us consider a small s deviation AR., from R,, and AwLB from wL . The relative change S 5 S S CHAPTER II 57

in Rs can be evaluated toL 1 1 Aps AR R" ^o AcoL if—ir2*2 9-^L sr* (2-51) K s (cosk 1- =-S sink 1) s O K_ vJ

" R + B TZLT^ (2.51) s s In (2.51) the factor B is for all frequencies used

(Hansen, Grimes and Libchaber (1967)) and studies of the elec- trodynamic properties of superconductors in the mixed state (Bianconi and Iannuzzi (1971)). Secondly, the hollow box configuration where the homogeneous e.m. field is excited by a coil wound around a metal box of which a part of one side is the sample to be studied with a pick-up coil placed inside the box. This method has been developed independently by Druyvesteyn* and Smets (1969) and Gabel (1970). In figure 2.11 the configuration of the box with primary and rotating secondary coil is given.

SOLDERED CONTACT

SOLDEREO CONTACT

PRIMARY COIL -»

SECONDARY COIL TABLE

Fig.2.1] Hollow box arrangement with a rotating secondary coil as designed by Druyvesteyn. (see text).

The metal box was made of industrial polycrystalline copper. The copper frame has a thickness d of about 1 mm. and is used as screening material. The box dimensions l,a,b, are respectively 15,3.5 and 6 mm. An indium crystal of dimensions 8><5 mm. was soldered to the copper only with the short ends. A primary coil consisting of about 32 windings of 150u copper wire was placed around the box. A secondary coil consisting of 42 windings of 50y copper wire was mounted on a rotating We are grateful to Dr. W.F. Druyvesteyn for the discussions and the hollow box arrangement. i CHAPTER II 59

• i j table and was placed inside the metal box as close as possible ; to the indium sample. The dimensions of the secondary coil are small compared to those of the indium sample. The transmission of the e.m. field through the sample into the box in the presence of a uni orm magnetic field has been calculated by Cochran (197Q). It is shown that the transmission coefficient T_c can be written as

j Tc E ir * iriFi Gx <2-52>

where H is the alternating magnetic field outside the cavity, H. is the uniform magnetic field inside the cavity, H and H^ are in the y-direction, "a" is the height of the cavity, A

is the free space wavelength of the r.f. field, Gv(z) is the component of the electric field in the slab along the x-direction, generated by a r.f. magnetic field of unit strength at the surface z=0 in the y-dxrection while d is the thickness of the specimen in the z-direction. It has been assumed that 6/X«iand 6/d<

have no influence on the electric field Ev(d) at z=d. If T <<1 it follows that H inside the cavity has only a negligible .. effect on the electric field distribution in the specimen. The transmission through a slab into a box is enhanced over the transmission through a single slab by a factor X/a2iT. This is due to the fact that the impedance of the cavity is better matched to the specimen than free space will be.(Panofsky (1964)). The factor will be of the order of 103-10"for the operating frequencies. The change in the transmission coefficient T due to the occurrence of the R.F.S.E. may be related to the change of the surface impedance Z. We define a skin depth 5 (all time dependent quantities vary as exp(-iut)). 60 CHAPTER II

(2'53)

where c is the velocity of light and u> the radio-frequency. Using the field distribution function derived by Cochran(1970). (2.53) becomes

6 w = TTT (Gw(d)-Gv(0)) (2.54)

Generally, for(6/d)<

Gx(0)

« - IS G (0) . ic V". {2 55) O

The surface impedance can be defined as

4TT Ex(0> O and X2.56)

_ 4iT h) - u — ""^c™ x. c 0y In relation (2.52) the transmission coefficient is related to the electric field at z=d, while according to (2.56) and '2.55) the surface impedance depends on the field at z=0. Cochran (1970) made the hypothesis that in case of a size effect

AG (0) = -AGv(d). Then from (2.54)

AGx (2.57) with (2 .52) we have

AT = -

Combining the expression (2.56) and.(2.58) we obtain

AT = AT^1' + iAT^2) = - -jr^- (iAR-AX) (2.59)

If relation (2.57) holds then the real part of AT_ is proportional to the imaginary part of the surface impedance and the imaginary part of ATC is proportional to the real part of the CHAPTER II 61

surface impedance. The validity of relation (2.57) has been investigated By Gabel (1970) for gallium. He concluded that relation (2.57) is certainly a good first approximation. Gabel's pick-up signal was amplified, filtered and then rectified. The use of a rectifier means that one measures the modulus of the pick-up signal. However the background signal is about 100 to 1000 times larger than the transmitted signal. In consequence of this the background signal had to be bucked out. Gabel shows that on account of this compensation he measured the transmitted signal in phase with the background signal. As the background signal according to Gabel's investigations is in phase with the r.f. magnetic field he concluded that he has measured the imaginary part of the surface impedance. (The phase shift between the pick-up signal and the signal generator voltage appeared to be 77°+ 15°. Signal generator voltage and current through the excitation coil are 90° out of phase. 2 1 (1) AT =AH./H = T^'+iAT* * , AT* * in phase with En and AT is equal to (c2/8-rraw)AX ). We will show that the given reasoning is rather doubtful. A Block diagram of the transmission arrangement is shown in fig. 2.12

OUTPUT for unmodulated signals

from ficldial

modulation coil*

OUTPUT for unmodulated signals Fig.2.12 Block diagram of the transmission arrangement. S: sample; see fig.2,11. ATT: Merrimac model AR-2, broadband attenuator. TA: tuned amplifier. The other symbols are given in fig.2.6. 62 CHAPTER II

The excitation circuit is tuned by means of the variable capacitor C, so that a high current flows through the coil L. and there is no phase shift between the current and generator

voltage. The pick-up coil is tuned by means of C2- Due to the quality factor Q of the coil we obtain a gain of about 50 in signal-to-noise ratio of the transmitted signal. Considering for the moment the coil configuration as an ordinary transformer there will be no phase shift between the alternating magnetic field at the excitation side and the input voltage of the tuned amplifier T.A. (it has been assumed that the input resistance of T.A. is very large). Experimentally because of the phase shifts in the sample the excitation field and the input voltage of T.A. will be out of phase. There will be also a background signal. By adding to the transmitted signal a part of the generator signal of equal magnitude but I80°out of phase by means of the quadrature hybrid QH1 it is possible to compensate the background signal. We can bypass the compensation network by connecting the cable denoted by U.C. in fig. 2.12. The quadrature hybrid QH2 produces two signals 90° out of phase. The demodulated signal (we use the modulation technique as described in section 2.2) , using a mixer, can be detected phase-sensitively by means of a lock-in (see fig.2.12 and 2.6) and plotted on a two-pens recorder R. With the aid of the d.c. amplifiers DCA1 and DCA2 it is possible to record the transmitted signals when no modulation field is applied. It may be expected that the output voltage of QH2 has a form similar to (2.36)

exp(ia) •&- (T* +iTv '+M ) E exp(ico t) Rl c c ° ° }(2.60)

* ^c1} where M = —3=— b cos (u> t+d> )+i aB m in in where A is the amplification factor,a the total phase shift of the circuit, Q the quality factor of the secundary coil, Rj the a.c. resistance of the primary coil, w the operating frequency, bmcos wmt the alternating modulation field with fre- r

CHAPTER II 63

quency wm and m, \\>m possible phase shifts; EQ exp(iu)Qt) is the input voltage of the excitation circuit. As shown in section 2.2.4 the use of a mixer means a projection of the in (2.60) defined vector voltage on the real axis of a coordinate system in a complex plane defined by phase angle 8 adjusted by PS1 (fig. 2.12). However we don't have a criterion how to adjust the phase B in such a way that we separate the real and imaginary part of the expression between parenthesis in (2.60). Adjusting the phase 8 so that the d.c. level at the IP port of the mixer is extremal we measure the modulus of (2.60). On sweeping the external static magnetic field the unmodulated part of (2.60)MT* +iT*7') may change. Apart from the background signal due to a leakage there may be a contribution to the total output signal to be detected originating fromT^ViT^,2*. From our experiments it follows that the background signal (d.c. level of the mixer output) is dependent on the static magnetic field (fig. 2.13). It is clear that the shown field dependency is also determined by the phase B of the signal at the LO port of the mixer. When the background signal is not constant in magnitude and phase we cannot adjust 8 for an extremal d.c. level as this level depends on the magnetic field. Otherwise choosing a fixed 8 we measure a signal proportional to (Re)cos(a-8)+(Im)sin(a-8) where (Re) is the real and (Im) the imaginary part of the expression between parenthesis in (2.60). Let us say that the measured quantity is in first approximation the real part (see Gabel (1970)). Then a-8=0 or a-3=0+6, where 6<

In view of the above mentioned problem it is clear that n6 decisive answer on the Cochran hypothesis has been given. Finally it is remarked that perhaps an other phenomenon may be used for adjusting the phase 0. As indium has a 64 CHAPTER II

£

Fig.2.13 Two components, being in quadrature, of the amplitude of the transmission signal as a function of the magnetic field. superconducting critical temperature T of about 3.4K in our experiments carried out at 1 ..5K we mostly observe a peak due to the fact that indium goes from the superconductive state to the normal state. Suppose that the signal in the superconductive state originates from the leakage and is in phase with the ex- citing magnetic field. We can adjust 3 in such a way that we are measuring in quadrature with the "superconductive" signal. The observed signal is then proportional to the real part of the surface impedance. The latter method is rather disputable. The origin of the transmitted signal when the sample is in the superconducting state is not clear. Beside that, the background signal in the normal state depends on the magnetic field as shown in fig.2.13.

2.4 Comparison of several experimental techniques In practice several experimental techniques are used in R.F.S.E. measurements. In the two preceding sections we have described the twin-T bridge method and the transmission method. Two other methods are commonly used in R.F.S.E. studies i.e. the marginal oscillator and oscillators operating well above the oscillation margin. Most of these techniques are originated from n.m.r. experiments. Although the R.F.S.E. and the n.m.r. measuring techniques have much in common there are some differences. In the

j CHAPTER II 65

measurements of the R.F.S.E. the surface impedance or the coil impedance varies strongly with the external magnetic field so the base line varies strongly in contrary to the n.m.r. base line. Besides the resistance of the r.f. coil made of 50 micron copper wire is larger than the resistance of the coil used in n.m.r*« studies. Furthermore saturation of the sample does not occur. We will present the essential points of the techniques mentioned above. The advantage of the bridge method and the transmission arrangement over the marginal oscillator is that the r.f. power can be considerably larger than the r.f. level of the oscillator operating in the margin. This means that the sensitivity defined as the magnitude of the signal due to a change of the coil impedance devided by the magnitude of the change itself can be higher for the methods described in sections 2.2 and 2.3 than for the marginal oscillator (Penz and Kushida (1968)). Further, in the bridge method only passive linear components are used and the operating frequency is constant during the occurence of the R.F.S.E. peaks which can be of advantage in questions of linearity, stability and noise. The noise figure of the twin-T bridge given in (2.31) can be op- timized. (An improvement of this noise figure can be achieved by modifying the twin-T bridge as described by Alderman (1970) and Miyoshi and Cotts (1968))). As shown above the twin-T bridge can be adjusted for a complete separation of the real part R and imaginary part X of the surface impedance provided that the twin-T bridge is automatically controlled (Mehring and Kanert (1965) and Collins (1957) . If desired the phase shifters P.S 1 and 2 (fig.2.6) can be adjusted automatically for maintaining the complete separation when the mode adjustment appears to be unstable. The advantage of the twin-T bridge over the transmission method is that the twin-T bridge is essentially a nul3-indicator and that a sensitive r.f. amplifier can be used. A disadvantage of the transmission method is the absence of a criteron for separating the R and X mode. With the experimental arrangement pre- 66 CHAPTER II

sented in fig. 2.12 it will be possible to obtain a phase-amplitude plot (see Hansen, Grimes and Libchaber (1967)) i.e. the Gantmakher signal is obtained in a vector representation, where the amplitude corresponds with a radial distance from an origin and the phase is a polar angle. However an inherent problem is the background signal, (see section 2.3) Using a marginal oscillator provided it has been made sufficiently critical we do not have to deal with the separating problem as only the R mode is measured. A comparative research of the several oscillators used in n.m.r. studies is presented by Wind (1970). The marginal oscillator behaves as a L.C. circuit with an actual bandwidth Aco^ which depends on the oscillation level and the nonlinearity of the transistor. The magnitude of the signal |AU| using field-modulation with a mod-

ulation frequency flm can be given by

|AU| = 2—-_ (2.61) 2C(Au>J + )l/2 where |u| is the oscillation level and AG_ the change in the

conductance GQ of the L.C. circuit. Prom this formula it fol-

lows that the sensitivity |AU|/AGQ will be high for small bandwidths Aco^. This means that the marginal oscillator has to be adjusted very critical. At the same time the oscillator level is very low corresponding with a small signal-to-noise

ratio. The signal defined in (2.61) depends on flm. The voltage

AU will be higher for small Slm as used in R.F.S.E. measurement. One can calculate the condition for maximum sensitivity

|AU|/AGQ (see Bruin and Schimmel (1955)): AUj = 8m where Awj =tro+Po^ ^Lo' ro' po' Lo are the res*stance °f tne coil, the negative resistance of the oscillator element and the inductance of the coil in the operating point of the oscillator. 1 6 Therefore fimLo • rQ + pQ. For Qm • 50 sec." , LQ * 3.10" H we find r + p « 1.5 x 10 fl.The resistance of the coil r^ as mentioned above, is » 1 Ohm and a rather delicate setting of

pQ will be needed. This requires an oscillator level which is automatically controlled. This will also be needed when the CHAPTER II 67

signal to be measured is superimposed on a background which varies rapidly with the external magnetic field (see Hui (1967)). In consequence, however, the signals depend rather arbitrarily on the oscillation level |u| (see Wind (1970)). This can effect the shape of the peaks. The automatic mode selection of the marginal oscillator will be an advantage. However when both modes R and X are desired the marginal oscillator technique fails. The change of the oscillator frequency which is correlated to the X mode appears to be (Bruin (1961))

A.. Ato, uo 4uo where w* = 1/LC. Since jAo)1|<

2.5 Dewar system, sample holders and magnetic field calibration 2.5.1 Dewar system A conventional glass double dewar system was used. The nitrogen liquid in the outer dewar was kept at a constant level with a mechanical controlled filling system. The helium dewar was connected with a 30 litre helium vessel by way of an evacuated helium cooled siphon. With this closed helium system five days of uninterrupted operation can be achieved. Temperatures between 4.2K and 1.2K were obtained by pumping on the helium bath by means of a Leybold mechanical pump. The temperature was measured by means of the vapour pressure of the bath. The pressure was monitored with two Wallace & Tiernan manometers for pressures drvn to about 0.1 Torr. The vapour pressures were converted to temperatures using the He" scale of temperatures. Temperatures could be held constant within + 0.01K by adjusting the pumping value. This was done by means of "Leybold's Kryokonstanter B" electronic control system. An accurate temperature regulation was not needed since the Gantmakher peak position does not depend on temperature. The amplitudes however are dependent on the temperature. 2.5.2 Sample holders Two different sample holders were designed, one for each of the measuring methods described in sections 2.2 and 2.3. The sample holder for the twin-T bridge arrangement (fig.2.14) consists of a copper frame which can be oriented with respect i to the magnetic field by means of three tubes of german silver attached to the head of the copper frame. By rotating the tubes they can be translated in vertical direction. This arrangement was used only to align the magnetic field parallel to the sample surface. The alignment of the magnetic field parallel to the sample surface was executed by studying the shift or splitting of a Gantmakher resonance peak (see Gantmakher and Krylov (1965)). The sample was placed on a glass table which was attached to th« bottom of the copper body. The coil consisting of 50 CHAPTER II 69

CROSS SECTION A-B

Fig.2.14 Sample holder for the twin- Fig.2.15 Sample holder for the T bridge arrangement. 1. transmission arrange- copper frame. 2.sample- ment. 1.frame made of table holder. 3.teflon top delring. 2.copper box of the copper frame. 4.rods with the sample. 3. (see top view) to adjust primary coil. 4.second- the sample parallel to the ary coil. 5.worm gear to magnetic field. 5.coil rotate the copper box table. 6.rod to rotate the with respect to the coil. 7.lever to adjust magnetic field. 7.GE the sample-coil distance. 7031 varnish. 8.sample table. 9.coil embedded in epibond. 10. GE 7031 varnish. 70 CHAPTER II

windings of 50y copper wire was embedded in epibond in such a way that inside the coil no epibond was present. The sample mounted on the table was placed inside the coil, while the coil body was attached to the rotating table at the top of the copper frame. This arrangement enabled us to rotate the coil with respect to the sample orientation. In this way only samples with thicknesses of about 200y or less could be used. The samples with thickness larger than 200p were placed directly inside the coil body. All samples were fixed at one point with Nonaq Stopcock Grease. In the last configurations the direction of the magnetic field with respect to the crystallographic orientations could only be varied by rotating the Varian 12-inch iron magnet. The direction of the magnetic field in relation to the sample was determined from the measurements using symmetry properties and could be found within 0.5 degree.

The sample holder used for transmission measurements (fig.2.15) has facilities for rotating the secondary coil inside the metal box and for rotating the metal box. A sample with dimensions 5x7.8 iron was soldered with Woods- metal to one side of the metal box. Care was taken to avoid high soldering temperatures (<90°C) and to prevent the sample from touching the soldering-bolt or the excitation and pick-up coil during the mounting. In this case the alignment of the magnetic field parallel to the sample surface was achieved by rotating the magnet. The rotation of the magnetic field with respect to the crystallographic orientations of the sample could be obtained by rotating the me'tal box. The secondary coil could be rotated simultaneously in the same direction. Also we could rotate the secundary coil for maximum signal. 2.5.3 Magnetic field calibration The used magnet was a Varian 12-inch iron magnet which was field regulated. The homogeneity of the field within 1 cm3 in the center of the magnet amounts to 1 part in 105 for

II CHAPTER II 71

most field ranges. The pole distance is 7 cm with a maximum field value of 16 kG. Field sweeps could be executed starting from -300 G to 15 kG in different sweep ranges and with different sweep rates. The linearity of field sweeps was within 0.5%. The calibration of the fieldial was done with proton resonance. The, accuracy of the fieldial calibration in the field range of interest (<5 kG) for the Gantmakher measurements is only within several Gauss over long terms. Accurate field measurements executed during the Gantmakher resonance measurements could be done by means of a Rawson- Lush rotating coil Gaussmeter type 924-944 with an accuracy <0.01%. Determining an extremum of a Gantmakher resonance peak the field dials were adjusted to the peak position. Subse- quently the sweep range was decreased so far that the extremum could be just observed. Then the field dials were adjusted to the extremum. The dewar was moved out of the magnet and the rotating coil was placed at the center. The accuracy of the determined peak position is dependent on the peak amplitude and width. Mostly an accuracy better than l°/oo could be a- chieved. This cumbersome method was only done for a few samples and a few crystallographic orientations. Mostly the fieldial was calibrated with the rotating coil Gaussmeter in the field range of interest. This calibration was checked many times during the measurements. The external magnetic field could be modulated by a small modulation field. The modulation frequency was about 7.8 Hz in almost all cases. Some experiments were carried out at modulation frequencies of 5 Hz and 2 Hz. The skin depth 6 defined as

6 - w" Jd H(z) dz Ho ° where HQ is the external modulation field and H(z) the field distribution in the sample, may be of the order of the thickness d of the sample (say [6[=ad, l£a<5)(see Lyall and Cochran (1967)). In consequence the field distribution in the 72 CHAPTER II sample is not uniform causing phase shifts in the signals to be measured (see sections 2.2.4 and 2.3). The magnitude of the modulation field could be changed from 0.1 Oe-25 Oe dependent on the r.f. size peak width and amplitude under measurement. The modulation field used as a rule amounted about 1 Oe. 73

CHAPTER III

LINE SHAPES AND MEAN FREE PATHS

3 Introduction * In this chapter we make a comparison of the measured line shapes with the results of theoretical calculations of Juras. The agreement between the theoretical line shapes and our experimental results described in section 3.1 is quite reasonable. In section 3.2 we pay attention to the peak near zero field and connect it with the magnetic field induced surface states. For some of our samples we obtain from the peak near zero field values of the mean free paths in a special point of the Fermi surface. Values of the mean free path averaged over the orbit can be obtained from the temperature dependence of the amplitude of the peak in some particular cases as shown in section 3.3. In section 3.4 some plots are given which expose some special features of observed lines e.g. the background signal and change of the line shape due to the adjustments of the transmission set-up.

3.1 Parallel-field R.F,S.E, line shapes In this section we will pay attention to the R.F.S.E. line shapes measured in indium under conditions of a magnetic field parallel to the sample surface and either bilateral antisymmetric excitation or unilateral excitation. Theoretical work has been done by Juras (1969,1970,1970') for the case of bilateral antisymmetric excitation and by Kaner e.a. (1967) for the case of unilateral excitation. Experimental studies on the line shape has been carried out by Koch (1966) and Wagner (1968) in potassium, by Tsoi e.a. (1969) in potassium and by Boudreau e.a. (1971) in cadmium. These R.F.S.E. lines originate all from electron orbits. In this section we study line shapes due to electron as well as hole orbits of indium and make a comparison with the theory of Juras* A 74 CHAPTER III

For reasons of convenience we mark the lines in conformity with the notation of Gantmakher {1966). In fig. 3.1 is given the dR/dH signal of the a- and a+g- peak in a (001) sample with S parallel to [lOO^ direction versus the magnetic field. We also observe weak lines in double and triple field connected with the size effect due to a chain of trajectories (2a and 3a). The a-peak is due to a hole-orbit lying in a plane normal to ll in the jc-epace. The distance between opposite square "cups" spans the sample. This distance is nearly constant over the whole cup. Due to this fact we expect that the line shape calculated by Juras for a cylindrical Fermi Surface (F.S.) will be in accordance with the experimental situation mostly. The agreement in the line shapes is strikingly good (see chapter I). Gen- erally, the calculations of Juras show that the R.F.S.E. line in double field is reversed with respect to the line in single field. This is independent of the F.S. model and the excitation

SAMPLE 5 T =1.86K fo= 5MHz fm= 78Hz it// [100]

in •*-» c 2a

jQ a+g %- a x TO •u

400 1200 1800 H(Oe) Fig.3.1 The dR/dH signal as a function of the magnetic field H. The line shapes of the a, 2a and 3a peak are similar. The line a+g is due to a chain of trajectories. CHAPTER III 75

mode. Experiments in potassium (Koch 1966, Peercy e.a. 1968), cadmium (Goodrich e.a. 1967, Jones e.a. 1968), gallium (Fukumoto e.a. 1967) and magnesium (Roach 1971) confirm this theoretical result. However, the 2a-peak shown in fig. 3.1 is very similar to the a-peak. This is in accordance with the results of Gantmakher (1967). Boiko e.a. (1969) studying the R.F.S.E. in molybdenum have also observed a double field peak similar to the single field peak. This peak is due to a hole orbit in contrary to the peaks observed in the metals mentioned above where the double and single field peaks are reversed. They are all due to electron orbits. More important than the difference in type of the orbit (electron or hole) may be the form of the orbit. The hole orbits in question spanning the sample in the R.F.S.E. measurement have such a form that the holes move nearly parallel to the sample surface in the main part of the orbit. In the double field configuration this nearly straight part of the orbit passing through the skin layer forms a well defined current

SAMPLE 3 «C(H,[OO1]]«3O° T s 2.2K

f0 i 5 MHz c dX fmx 78 Hz 3 X) o *—' tu a

600 1A00 2200 2400 H(0e)

Fig.3.2 The dR/dH and dX/dH signals as a function of the magnetic field H. The peaks at about 600, 1200 and 1800 Oe have nearly similar line shapes. The small peak at 117A Oe is due to n.m.r. T

76 CHAPTER III

sheet which will be reproduced at the centre of the sample. We suggest that the field distribution at the centre of the sample may be more pronounced than the smouth variation calculated by Juras (1969), i.e. the first dip in R(H) and X(H) observed at single field may also be present at double field. An additional example is shown in fig. 3.2 where the line shapes of the double and triple field peaks are similar to the line shape of the single field peak both for the dR/dH and dX/dH signals. This peak is due to the so-called X-orbit. The orbit runs via the square "cups" of the second zone hole F.S. and has therefore also a straight part. The X-orbits have a smaller size than the second zone a-orbits. This is observable in the amplitude of the successive peaks. The amplitude of the double field line is about 1.5 smaller than the single field X-peak. The triple field peak is relative to the double field peak about a factor 3 weaker. For the a-peak it is about 4 and 10 times respectively. From fig. 3.2 it follows that the regions of sharp variation of

dX,

b in Q

a. < 2X

H (arb. units) Fig.3.3 Characteristic line shapes observed in the case of double- sided excitation. The peak a, b, X and 2X are' due to 2nd zone hole orbits, the peaks g are due to 3rd zone electron orbits (see fig.4.6). The sigh of all dX/dH-curves have to be reversed (see text). 1 CHAPTER III 77

the dR/dH curves coincide with the extrema of the dX/dH curves in accordance with theory. This property was observed in all samples and crystallographic directions. This may be considered as an evidence of correct separation of the dR/dH and dX/dH signal. The line shapes in dR/dH and dX/dH observed for several orbits in the second and third zone of indium are shown in fig. 3.3. These lines were observed in the (001) samples as well as in the (100) samples. Generally, the line shapes do not vary with the field direction i.e. as long as the special peak can be observed its line shape does not alter very much. Tsoi and Gantmakher have reported line shape variations in potassium as a function of both the mean free path and the sample thickness. Since in our case the mean free path and the sample thickness are not varied independently we cannot observe the influence of the sample thickness on the line shape. In connection with fig. 3.3 it has to be remarked that due to the phase-sensitive detection the sign of the signals has not been determined unambiguously. However, with respect to the sign of the signals all line shapes are brought in correspond- ence to each other. Comparing the line shape with the theoretical line shape calculated by Juras we come to the following conclusions. The line shape of the derivative of the real part of the surface impedance, dR/dH, for the a-peak resembles strikingly well the calculated shape obtained for a cylindrical F.S. As mentioned a- bove we expect that certainly for the case that the magnetic field is parallel to a symmetry direction the cylindrical model is a good approximation for the actual situation. The line shape of the derivative of the imaginary part of the surface impedance, dX/dH, for the a-peak is similar to the shape calculated for a cylindrical F.S. except as far as the sign of the signal is concerned. The b-lines connected with orbits across the hexagonal cups of the second zone F.S. and occurring for H" more than 30° from a symmetry direction may be described with a spherical F.S. model. 78 CHAPTER III

Indeed, the line shapes look like the calculated ones for a spherical F.S. except for the sign of the dX/dH signal. The third zone electron g-orbit observed in a (001) sample with 2 in a symmetry direction gives rise to the given g-lines. Comparing the g-line shapes with the line shapes calculated for a spherical F.S. the resemblance irrespective of the sign is reasonable. The shown X and 2X lines are similar to the calculated lines for a spherical F.S. except that in this case the dR/dH signal is reversed in sign. Let us consider further the question of the reversal of sign as mentioned above. We expect that the reversal of sign is connected with the special form of the Fermi surface. The electron orbits which give rise to the R.F.S.E. in the calculations of Juras for a cylindrical and spherical F.S. are both circular. The smaller number of orbits of equal size in the case of a spherical F.S. compared to a cylindrical F.S. causes not only less pronounced R(H) and X(H) curves but also a monotonic increasing background in the X(H)-signal* The first dip in the R(H) and X(H) peaks for a cylindrical F.S. is caused by the many orbits of equal size. Let us suppose that such a dip can be caused not only by orbits of equal size but also by special forms of the orbits. In particular let us suppose that a straight part of the orbit causes an intense dip. It may be expected that in that case the X(H) curve has a similar form as given by Juras for symmetric excitation and for instance for a spherical F.S. The observed line shape is reversed in sign with respect to the line shape in dX/dH obtained for antisymmetric excitation and a spherical F.S. The given explanation may be applied on the dX/dH line shapes for the X and 2X-peaks. Then the dX/dH line shapes given in fig. 3.3 are all reversed in sign. Assuming that this is the result of the phase-sensitive detection all dX/dH signals given in fig. 3.3 are 180° out of phase. An additional argument for the out-of phase detection of the dX/dH-signals illustrated in fig. 3.3 can be found in the i shape of the df/dH-signal obtained by Gantxnakher (1962) in tin for a fourth zone hole orbit. The given df/dH-signal corres- CHAPTER III 79

ponding with -(dX/dH) has a form similar to the a-peak given in fig. 3.3. The dR/dH line shape for the X and 2X peaks disagree with the calculated one for a spherical F.S. also due to the absence of a first dip in the calculated R(H)-curve. The ap- pearance of a first dip in the R(H)-curve for a spherical F.S. does shift the, dR/dH line shape for a spherical F.S. to the line shape for a cylindrical F.S. as shown in fig. 3.3. As a result we may conclude that the measured line shapes in the case of antisymmetric excitation qualitatively agree with those calculated by Juras. In addition to the line shapes for the double-sided excitation we will also consider the unilateral excitation. The line shape for the unilateral excitation cannot be constructed by averaging the solutions for symmetric and antisymmetric excitation (see Juras 1969). The first two lines illustrated in fig. 3.4 show the shape of the a-peak and X-peak in the transmitted signal.

o (A a Ul

2X H (orb.units) Fig.3.4 Line shapes of the transmitted signal (unilateral excitation). In the case of the second X peak and the 2X peak the electrically coupled signal are minimized, (see also fig.3.3). The lines are obtained under the condition that the d.c. output voltage of the mixer MX 1 or MX 2 (fig. 2.12) is extreme. Com- paring the line shapes with those in fig. 3.3 the similarity with the dX/dH signals of fig. 3.3 is striking. However, this resemblance is by accident, since the dX/dH signals shown in fig. 3.3 are for antisymmetric excitation. The third and fourth line given 80 CHAPTER III

in fig. 3.4 are an X and 2X peak. The line shape is similar to the dR/dH line shape for the X peak shown in fig. 3.3. The difference between the two X lines given in fig. 3.4 is due to the fact that in the case of the second X peak the position of the primary and secondary coils are chosen in such a way that the d.c. output voltage of the mixer is minimized. This lines are not very similar to those calculated by Kaner e.a. (1967). As discussed in section 2.2 and demonstrated in section 3.4 the line shape depends on the experimental adjustments (PS 1 in fig. 2.12). Finally, we will discuss experiments analogue to the work of Boudreau and Goodrich (1971) and in connection with the effect of the value of the mean free path and of surface scattering as described by Juras (1970).

&

I SAMPLE 5 1 7 »189K f0 x5MH2

. HII [100] | omplitudc of a ^ Q. I amplitude of b < \ NJ n

\ P

300 500 700 900 H{0e) Fig.3.5 The dR/dH and dX/dH signals as a function of the nagnetic field H. a) Line shapes at the beginning of the experiment, b) Line shapes of the same peak after 8 weaks of continuous operation, showing the deterioration of the sample. CHAPTER III 81

In fig. 3.5 recorder tracings of the dR/dH and dX/dH signals for the a-peak as function of the magnetic field are shown. The upper trace was obtained at the beginning of the experiment,The lower trace was recorded after about & weeks of continuous operation. The sample once placed in the dewar, was not removed for the duration and was kept at low temperatures (cycling to room temperature never occurred). The original condition of the sample surface was rather shiny, although not brightly. After eight weeks the sample surface was dirty but not damaged. These experimental conditions are similar to the descriptions of Boudreau e.a. (1971). After eight weeks the initial dip observed in the dR/dH and dX/dH peaks both have disappeared. In consequence the peak-to-dip amplitude ratio is increased as measured in Cd by Boudreau. This may be correlated to a decrease of the ineffectiveness parameter n as introduced by Juras (1970). A decrease of the ineffectiveness parameter produces a more pronounced peak with a sharp rise at the low field side and a smoothly varying signal at lower fields in disagreement with the experiment.

We observe that the amplitudes of the dR/dH and dX/dH peaks are reduced by a factor of about 20. Assuming a decrease of the mean free path, or of K=l/d as defined by Juras, we have to distinguish between dominantly specular scattering and dominantly diffuse scattering at the sample surface. For the case of dominantly specular scattering two properties are associated, i) on decreasing the mean free path the signal amplitude increases (see Boudreau e.a. 1971) (K = 4 relative to K - 1) ii) The temperature dependence of the amplitude is not monotonic (see Boudreau e.a. 1971) It will be shown in section 3.3 the temperature dependence of a peak observed in sample 5 after. 8 weeks behaves according to Tn (n > 2). Moreover, the peak amplitudes are reduced. Therefore, we suggest that the predominant scattering mech- anism is the diffuse scattering. However, in agreement with the results of section 3.2. •ome specular scattering may occur i.e J 82 CHAPTER III

the specularity parameter p may be 5*0 but <<1 or in the spec-

ularity function S(6) the critical scattering angle QQ, de-

fined by S(9) = 1 for 6 < QQ, S(6) =0 for 6 > 0Q, may be different from zero. In the upper trace of fig.3.5 the effect of surface reflection on the background signal is not observable. The large value of tc gives rise to extraordinary intense peaks so that no high sensitivity of the detection apparatus will be needed and the background signal looks nearly constant, -rhe deteriorated sample may be characterized by a diminished bulk mean free path. As shown by Juras the amplitude decreases and the positions of the extrema shift to higher fields in accordance with our experiment. An important feature of the diminished m.f.p. is the disappearance of the initial dip. On decreasing amplitude the sensitivity of the apparatus has to be increased and simultaneously the surface reflection can be observed in the rapidly varying background signal. The character of the curves shown in fig. 3.5 (lower traces) is similar to the curves given by Juras (1970) for the surface resistance R as function of the magnetic field in dependence on the critical scattering angle 9 . Boudreau and Goodrich (1971) present the derivatives of the curves for eQ = 0.2 and 6O = 0.0. The given dR/dH signal for

9O = 0.2 agrees strikingly well with the data of fig. 3.5. The sharp changes in the slope of R that appear between H = 0 and

H = Hd, the magnetic field of the onset of the R.F.S.E. signal, are defined in terms of 6Q and the skin depth 6. The field Hg where the skipping orbits are expelled from the skin layer is given by (see Juras 1970)

H6 = Hd 2T (1 " cos V t3'1*. Taking (2S/d)= (AH/H,) we obtain from fig. 3.5. 9 « 13.5°. a O The relation (3.1) is an approximation in the case of a real- istic F.S. (The equation (3.1) is derived for a cylindrical F.S).

The obtained value of 9Q is reasonable and is in agreement with the value used by Boudreau: 9 * 11.5° Consequently the experimental data of fig. 3.5 show the dirain- CHAPTER III 83

ishing of the bulk mean free path during the course of the experiment due to the recycling in temperature over a small temperature range near the helium boiling point. 3.2 The peak near zero field

On measuring the derivative with respect to the magnetic field H of the real and imaginary part of the surface impedance during a field sweep through zero field we observe a peak or jump at zero field, (fig. 3.6a,b). These distinctive signals are mostly superimposed on a gradual bump. This nonmonotonic dependence of dR/dH or dX/dH on the magnetic field exhibits a maximum at a field much higher than the zero field (100-200 Oe). The maxima in dR/dH and dX/dH occur at different fields, (fig. 3.7)

The peak near zero field is reported by many authors and for several metals: Cochran e.a. (1965) in Ga, Gantmakher e.a. (1969) in K, Cleveland e.a. (1971) in Mo, Sibbald e.a. (1971) in Cu, Boiko e.a. (1972) in W, Soffer e.a. (1973) in Cd and Awater e.a. in Sn. The observed peaks near zero field in the mentioned metals have different shapes. There are two types clearly distinguishable. First, a narrow and intense peak at the lowest fields (<10 Oe) and secondly a nonmonotonic background signal with a maximum at either low fields (<10 Oe) or higher fields (100-200 Oe), An example of the first type is given by

-200 -100 100 200 -200 -100 100 200 H(Oc) H(0c) Fig.3.6 The dR/dH signal (a) and the dX/dH signal (b) as a function of the magnetic field and the orientation of the field H relative to the {001] direction (1...6), observed in sample 3. 84 CHAPTER III

Boiko e.a. (1972) and an example of the second type is given by Sibbald. e.a. (1971) and Cleveland e.a. (1971). We notice also two different experimental findings about the dependence of the peak on external influences. The rapidly varying background observed by Cleveland e.a. in Mo is independent of the sample; the intense peak at zero field in W (Boiko e.a. (1972)) is equally well observed for samples with different surface qual- ities and its shape and intensity is independent of the field orientation relative to the crystallograph!c directions. How- ever, the line shape, intensity and position of the peak observed at zero field in K, Ga and Cu depend on the temperature (or mean free path), the surface quality, field direction and accidental strains.

From fig. 3.6 it follows that in our experiments for H parallel to surface of sample 3 ( ]_ [lOO] ) the line shape depends on the field orientation relative to the [OOl] direction. The line shape of curve 1 given in fig. 3.6 is quite different from that shown in fig. 3.8 for sample 2 under identical circumstances. The line shape and the dependence of the peak on the field direction observed in a (001)-sample is shown in fig. 3.9. The curve 1 shown in fig. 3.6a has a shape similar to the peak at zero field in tin reported by Awater (1973). The peak is attributed to the field induced surface states of electrons as described by Sibbald e.a. (1971), A detailed theory of the electronic surface quantum states in a low magnetic field as well as the surface impedance oscillations that result from microwave transitions between the surface quantum states is given by Prange e.a. (1968) and Nee e.a. (1968). The contribution of magnetic surface states to the radio-frequency impedance of metals is described by Sibbald e.a. (1971). It is expected that the peak at zero field in our experiments can be likewise ascribed to magnetic surface states. This means that we have to make the assumption that conduction electrons which hit the sample surface are specularly reflected. The gradual bump observed at higher fields (see fig. 3.7) may also be connected to the specular reflection of electrons at CHAPTER III 85

J2 dR/ '£ 'dH 3

0 3 SAMPLE 3 T « 4.2K fo*5MHz i fm* 78MHz H*// [001]

-250 250 500 -100 -50 50 100 H(Oe) H(0e)

Fig.3.7 The peak near zero field Fig.3.9 The dX/dH signal as a function and the rapidly varying of the magnetic field and the background signal at low orientation of the field rel- fields. ative to the JOIO] direction, observed in sample 4.

(arb . units ) h 0 ' 1 SAMPLE 2 / / T « 3.42K / fo.5MHz AMPLIT I

1 / HV[OOI]

•n — 1/ • ' • -150 -100 -50 50 100 150 H(Oe)

fig.3.8 Line shapes of the.peak near zero field observed in sample 2 under the same conditions as curve I in fig.3.6. J 86 CHAPTER III

the sample surface (see Juras (1970)). However, it has to be remarked that the condition of a sufficiently smooth sample surface connected with specular reflection seems not to be fulfilled, (see section 2.1). In a magnetic field the electrons which are reflected at the surface can become bound to the surface in quantized states. The electrons are captured in a potential well formed on one side by the vacuum-metal interface, on the other by the magnetic potential. The electrons moving in the skin layer only contribute to the skin layer current. The maximal penetration depth is given by (see Koch (1969)). = a n 4' H173 Kl/3 where K is the radius of curvature of the relevant part of the Fermi surface and n the quantum number. Only a small, well defined group of electrons at a single point of the Fermi surface are moving all the time in the skin layer (zn<6). For this group the electron velocity is essentially parallel to the surface and equal to the Fermi velocity Vp. On decreasing the magnetic field H the length of the skipping trajectories being 2(2Rz )1/2

50r • SAMPLE 2

• > 5 MHz •4-* * 7.8 Hz 3 ,[oot])- 24°

10 _ / aUJ 3 ... . 1

a. • < < • vz H(Oc)

1 1 • 50 55 T3(K3) Fig.3.10 Plot of the amplitude (see inset) of the peak near zero field as a function of T3. - CHAPTER III 87

where R is the cyclotron radius, increases and becomes equal to the mean free path 1. Sibbald,- Mears and Koch suggest that the extremum of the peak near zero field occurs at a field H where 1/2 ° 2(2Rz1) ' =1. The field at which the destruction of the surface current starts is given by (m.k.s. units)

From the given explanation it follows that the peak depends on the Fermi surface and the field direction relative to the crys- tallograhic orientation as seen in fig. 3.6 and 3.9. The peak position depends on K, what means that in indium H may occur ac fields slightly higher than in other materials (Ga,Cu). The amplitude of the peak depends on the temperature (see Sibbald e.a. (1971)). In fig. 3.10 is shown the temperature dependence of the amplitude A (see inset) of the peak observed in sample 2 over a small temperature range. We observe in this temperature range a T dependence of the amplitude on the temperature. It suggests that the amplitude is linked to the mean free path of the electrons. We don't observe a shift of the peak position H with temperature due to the small temperature range. The line shape depends on the special field distribution in the skin layer which in its turn depends on the special group of very shallow trajectories at a single point of the Fermi surface. Comparing fig, 3.6 curve land fig. 3.8 we may decide that the line shape also depends on the mean free path (m.f.p.) of the electrons, as the average m.f.p. of sample 2 is larger than that of sample 3 (see below). For some cases we evaluate the electron m.f.p, from the peak position. As the peak position shifts to higher fields with increasing modulation field we calculate from the peak position- modulation field dependency by means of extrapolation the peak position at low modulation field. i) sample 3 , S // [OOl] , HQ * 4.3G in the [oio] direction for a trajectory in a plane normal to S K « 3.19 x 1010m . Then 1 « 140y at T « 3.47K ii) sample 3 , § // [oio] , H * 2,5G; in the [ooi] direction 88 CHAPTER III

for a trajectory in a plane normal to H i:*1.93xiolom-1 Then 1 = 175p at T = 3.47K 10 1 iii) sample 2, 3 // [OOl] , HQ « 2.35G , K * 3.19 x io m" (see i)) Then 1 « 210y at T = 3.44K

iv) sample 4, H // [lOO] , HQ = 8.8G , in the [oio] direction for a trajectory in a plane normal to S K = 2,37 x 1010 m~" Then 1 a 80.5y at T « 4.2K. From the cases i) and ii) it follows that the m.f.p. can not cause the large amplitude of curve 1 in fig. 3.6a relative to the amplitude of curve 6. However in the CoioJ direction there are many trajectories with nearly the same curvature.

3.3 Temperature dependence of the R.F.S.E, amplitudes The amplitude of a R.F.S.E. peak depends on the shape of the orbit in question (which is connected to the special form of the F.S.), the skin depth 8 relative to the thickness of the plate d, the m.f.p. 1 and on the surface scattering. On studying the temperature dependence the amplitude of a peak changes mainly due to the change of the m.f.p. 1. In this section we will show that the amplitude-temperature dependence is also related to the scattering of the carriers at the sample surface. In fig. 3.11 is shown the temperature dependence of the dR/dH- signal in sample 3 with the magnetic field parallel to the sample surface and // [ooij. The shape of the peak is in accordance with the calculations of Juras (1970) as well as the increase of the initial dip and the shift of the positions of the extrema to lower fields on lowering the temperature. We compare this plot with the recorder traces obtained for sample 5 in the parallel field configuration and S // [lioj as given in fig. 3.12. The lower trace of fig. 3.5 and the traces shown in fig. 3.12 are recorded sequentially i.e. fig. 3.12 shows the results for a deteriorated sample (see section 3.1). According to the analysis given in section 3.1 the mean free path is reduced and the rapidly decreasing background is connected to the surface scat- I CHAPTER III 89

SAMPLE 5

£AMPLE 3 = SMHz = 7.8 Hz H" // [001] U> c 3 3.49 K 3.23 K X 2.69 K •o 2.30 K 1.85 K

800 1200 1600 300 500 700 900 1100 H(0e) H(0e) Fig. 3.11 Temperature dependence of Fig. 3.12 Temperature dependence of the dR/dH signal in sample 3 the dR/dH signal in sample 5 tering. Considering the shape of the first peak in fig. 3.12 and the peak of fig. 3.11 it will be noted that the shapes need not to be the same since different orbits play a part in it. The peaks in question are the a-peak (fig. 3.11) and the X-peak (fig. 3.12). The extrema of the X-peak shift also to lower fields with decreasing temperature and the amplitude increases monotonically. The increase of the amplitude with decreasing temperature would be nonmonotonically if specular reflection is more predominant. It is this argument we referred to in section 3.1 (see Boudreau e.a. 1971 and Haberland e.a. 1967). The temperature dependence of the R.F.S.E. peaks in indium has been measured by Krylov and Gantmakher (1967) and by Snyder (1971). They used the limiting point size effect to study the mean free path (m.f.p.) of the holes in a fixed direction of the ic-space. The variations of the amplitude of the peak on changing the temperature in the parallel field size effect is studied by 90 CHAPTER III

Snyder too. The relation between the amplitude A^ at a temperature T and the temperature dependent part of the mean free path 1 , averaged over the relevant orbit may be expressed by (see Druyvesteyn e.a. 19 71)

1 ln(AQ/AT) = sd^-i; ) (3.1) where A is the signal amplitude at T=0K, s the distance travelled by the charge carriers on going from one sample surface to the other and 1 the temperature independent part of the mean free path 1 . Multiple passages through the skin layers are ignored (see below). Considering the value ln(A /A_) as function of the temperature the authors mentioned above observed a temperature dependence of T3, connected with small angle scattering (see Soffer 1972). In fig. 3.13a is shown the logarithm of the amplitude of the dR/dH-signal as well as of two amplitudes of the dX/dH-signal (see inset) of the X-peak shown in fig. 3.12 as function of T3. The errors in the measured amplitudes are 3-7%. The measuring points are fitting fairly well to a straight line indicating a T3-dependency. By means of extrapolation of the straight lines to T=0K we obtain a value for A . Plotting ln(A /A_) as function of T3, as shown in fig. 3.13b, we obtain three straight lines with different slopes. Since the value ln(A /AT) is related to the temperature dependent reciprocal m.f.p. 1 averaged over one specific orbit (see also Snyder 1971) the slopes have to be the same. In fig. 3.12 we observe a rapidly varying background signal on which the X-peak is superimposed. We suggest that the different slopes are due to this background signal which is connected to the surface scattering (see section 3.1). The value of the slope depends on the field. At lower fields the slope will be larger since the background increases with decreasing field.

In the preceding description we made the assumption that the charge carriers are not able to complete more passages through the skin layers.. This may be justified for the X-peak, shown in fig. 3.12, since the hole orbit in question has sharp corners and diffuse scattering of the charge carriers at the sample CHAPTER III 91

Fig.3.13 a)Plot of the amplitudes (see inset)of the X-peak of fig.3.12 vs.T3. b)Plot of the corresponding relative amplitudes vs.T3.

50 r

SAMPLE 3 o= 5MHz HII [001]

28 36 T3(K3)

10 Fig.3". 14 a)Plot of the amplitudes

surface can occur. Gantmakher has shown the occurrence of this so-called breaks. Besides, because of the sample (see section 3.1) the m.f.p. will be low. However, in the case of a pure sample and a fitting orbit the charge carriers may complete many passages through the skin layers. For this situation results as 3 shown in fig. 3.14a, where In AT versus T is plotted, are obtained. The measuring points don't fit to a straight line. The expression (3.1) has to be changed for the case of multiple passages through the skin layers. We obtain (see Druyvesteyn e.a 1970)

In (A /A_) = stll1 - I"1) + ln{l+ £ ° (1~e 21} (3.2) -s/1 1-e s/1o Since the last term in equation (3.2) is =0 and increases with A increasing temperature the value of In(AO/ T) is at higher temperatures larger than the value which can be expected from the term s(l" - 1~ ) as shown in fig. 3.14a (the slope of the curve shown in fig. 3.14a is at high temperatures approximately equal to the slope of the line s(1~ - 1~ )).Consequently, the temperature dependence seems to be changed as given in fig. 3.14b where In In(A /AT) is plotted as function of In T. The obtained dependence is T2'1*6. Due to the scattering of the measuring points the accuracy is not high but the slope of the straight line deviates obviously from the slope of the dotted line which corresponds with a T3-dependency. Finally, we will calculate the m.f.p. 1 for the two samples 3 and 5. First, we consider sample 5; Via equation (3.1) we calculate the m.f.p. 1 from the tempe- rs rature dependence of the amplitude of the second peak (the "b"- peak) shown in fig. 3.12 (3//[lio]). Plotting lgAT as a function of T3 a straight line is fitting fairly well to the measuring points. Using this result and the value of s=485u we obtain for the mean free path I averaged over the. (110) orbit of the second zone hole Fermi surface, 1 (cm ) = (1.3+0.1)T3 From the assumption that the charge carriers complete one pass- age through the skin layer and using the sample thickness we can CHAPTER III 93

evaluate the ih.f.p. 1 1~ :> 30 cm" Snyder (1971) has calculated the m.f.p. 1 for a (101) orbit and obtained l~1(cm""1) = (1.1+O.DT3. P ~" It has to be remarked that the occurrence of the rapidly varying background in fig. 3.12 causes a slightly larger slope of the lg A.J, vs T curve. We will show now that also in the case of multiple passages through the skin layer (sample 3) it is possible to obtain the mean free paths 1 and 1 from the slope of the curves in fig. 3.14a at different temperatures. Let us suppose that the reciprocal m.f.p. I" varies according to I" =aTn, The slope of the P n P curve lg P^, versus T can be expressed by d In d(l/l ) -£- (3.3) n n d T d T l-exp(-s/lT)

and for T=0K

d In d(!/!„) - -s ; ) (3.4) d T d T l-exp(-s/lQ)

In fig.3.14a we made the assumption that n=3. If n^3 then the slopes at the temperature T=0 of the curves drawn in fig. 3.14a should be equal to zero for n>3, and infinite for n<3. Although the slopes of the curves are not well defined at very low temperatures we suggest that the two mentioned cases are not fulfilled. Therefore n is equal to 3. Applying the equations (3.3) and (3.4) to the case n=3 the term d (1/1 )/d T3 is inde- , P pendent of the temperature. Consequently we obtain from (3.2), (3.3) and (3.4)

d In ATj ln(Ao/V = O.5)

d T 9 4 CHAPTER III

For the lower curve of fig. 3.14a we obtain the reciprocal m.f.p. l"1 averaged over the (010) orbit (H//[oof] ) of the second zone hole Fermi surface using equation (3.5) at different points: I"1 (cm"1) = (1.4+0.15) T3 Using this result and taking the quotient of the expressions (3.3) and (3.4) we can calculate the temperature independent part of the m.f.p. 1T. We obtain 1Q = (545+ 10%)y. Comparing this result with the sample thickness (157y) it is clear that the charge carriers can complete many passages through the skin layers.

3.4 Special features of some lines In this section we give further some special features of lines recorded with the transmission set-up. As described in section 2.3 the main problem of the separation of the real and imaginary part of the R.F.S.E. signal in the case of the transmission set-up is connected with the presence of a background signal. In fig. 3.15 several recording traces are given for different adjustments and detection systems. In the cases I and II a mixer is used (see fig. 2.12). Case III gives the amplitude of the signal measured with a r.f.detector. A modulation field is applied in the cases I and III; in case II the d.c. output of the mixer is recorded when no modulation field is present. The shown peaks are the superconductive peak and the X-peak. The traces "a" show the signal in phase and 180 degrees out of phase with the electrically coupled signal. The traces b and c are recorded perpendicular to the electrically coupled signal and the superconductive signal respectively. The transition of indium from the superconductive to the normal state will be accompanied with an increase in the transmission. When this signal is in phase with the electrically coupled signal the cases b and c should be the same. As shown in section 2.3 the background depends on the magnetic field. In fig. 3.16 we further demonstrate the temperature dependence of the amplitude of the background signal. Using a r.f.detector the change of phase of the signal is not measured. The sharp increase of the transmitted signal H OQ * TRANSMITTED SIGNAL (arb.units) OQ TRANSMITTED SIGNAL (arb units)

Ul

(D to O n M CO Hi rt s-g. It d. rt (D (D rt O. O rt (t> O co rt fl» (D rt 3 P 5r* 3s o w fl> H O. co H rt ui (D IB rt co (a rt (D H (D CO 3 H (0 O H rt H- 6 (D O

(0 O CO i-h to rt I n> OQ rt 3 H. n> o ft 9 o H> rt e> (D M a.

UI 96 CHAPTER III

at high fields is probably due to the magneto-resistance of indium. Concluding we can say that the background signal is due to an electrically coupled signal on one hand and to a transmitted signal on the other, which depends on the magnetic field and the temperature. The superconductive peak has a line shape which depends on the adjustments (see fig. 3.15), the sample and the temperature. A detailed display of the peak is given in fig. 3.17. The height of the peak is comparable with an a- or X-peak.

SAMPLE A T = 2.1 K 5MHz

Fig.3.17 The superconductive peak 600 1000 observed in sample 4 at H(Oe) 2.1 K. 97

CHAPTER IV

CALCULATION OF THE FERMI SURFACE FROM CRITICAL FIELD VALUES

4 Introduction. In this chapter we deal with the determination of the Fermi surface of indium from the experimental results. In section 4.1 we discuss the method of determining the critical field values of the several R.F.S.E. peaks observed in our experiments. From these critical field values and the known sample thicknesses we obtain extremal dimensions of the Fermi surface as shown in section 4.2. Some of the extremal dimensions are used in a K.K.R.Z. interpolation scheme to calculate the Fermi surface. The remaining extremal dimensions are used to verify the calculations. These calculations along with a comparison with ty.P.W. calculations are described in section 4.2. In section 4.3 is shown that the dependence of the extremal dimensions on the orientation of the magnetic field in the plane of the sample agrees with the orientation dependence as deduced from the calculated Fermi surface.

4.1 Determination of the critical field In this section we will deal with the problem of the determination of the critical field i.e. the static magnetic field at which the relevant orbit spans exactly the sample thickness. Several methods to determine the critical field value H are o used, (i) Haberland e.a. (1969) put the critical field HQ at the first discernible departure of the line from the background slope and include a large error 5-10%. (ii) One identifies the critical field with the first extremum of the signal on the low-field side (Jones e.a (1968), Fukumoto e.a. (1966)). Some- times this value is corrected (Roach (1971)). (iii) The field value at which the first maximum rate of the deviation in R occurs is determined. (Fukumoto e.a. (1967), Boudreau e.a. (1971)). (iv) The second derivative d2R/dH2 is measured and 98 CHAPTER IV

the field value at which the first axtremum occurs is indicated as the critical field (Steenhout e.a. 1970, Herrod e.a. 1971, Boiko e.a. 1972). In one case we have measured the second derivative d2R/dH2. From fig. 4.1 we see that the first minimum in d2R/dHz indicates the point of sharp variation in dR/dH being the position of the critical field according to the calculations of Juras (1969). However, a disadvantage of the methods mentioned up till now is that, if the successive peaks have a complicated structure due to overlap or if due to the large widths of the successive peaks at high magnetic fields the variation of dR/dH with the field has an oscillating behaviour, it is very difficult to locate the critical field values precisely. In order to resolve the several overlapping peaks and to indicate the critical field a fifth method is commonly used, (v) Koch and Wagner (1966) and Krylov and Gantmakher (1967) have shown that the width of the R.F.S.E. peak decreases on increasing the operating frequency f , whereas the point which corresponds to the critical field value remains fixed. This decrease of the peak width with increasing frequency has been ascribed to the decrease of the penetration depth 6 of the electric field with frequency. Assuming that, since the anomalous skin effect conditions are satisfied, the skin depth 6 is proportional to f" ' each point -1/3 of the peak shifts to the fixed point according to f ' . In order to determine the critical field the magnetic field of each extremum is measured as function of the frequency f and plotted -1/3 ° as a function of f ' . Straight lines are fitted to the data and -1/3 r by means of extrapolation to fQ ' =0 the value H is obtained. This method has been used by Cleveland e.a.(1970,1971), Bradfield i e.a.(1973) and Matthij (1969) with success. I In our experiments we have chosen the last method. The measurements were carried out for sample 5 and 6 with H in the crystallographic symmetry directions [lOO] and £llO]. In fig. 4.2 we have shown the frequency dependence of the a-peak. The critical field value is indicated with a dot. In fig. 4.3 are plotted the magnetic field values for each extremum of the dR/dH-signal (crosses) as well as for'each extremum of the 1 3 dX/dH-signal (dots) as a function of f^ / (fQ=l,2 and 5 MHz), CHAPTER IV 99

. SAMPLE U 1 T * 145 K fos 5MHz

<(H,[iOO])x32° in \ 1 1 1 • b un i A I o y 1 I ' j \ UJ ** "'dHJ Q ^ A. \ 1 \ Am 'M 1— 1 a V

200 600 1000 1400 1800 H(0e) Fig.4.1 The dR/dH- and d2R/dH2-signals as a function of the magnetic field H.

•o

400 500 600 700 H(0e) Fig.4.2 Frequency dependence of the a-peak (see fig. 4.7b). The extremes shift to lower field with increasing frequency. The dots denote the fixed point. r

100 CHAPTER IV

460 -

Fig.4.3 Plot of the field positions of the extremes of the a-peak (a) and the b-peak (b) in saaple 5 (see fig.4.7b) at different -1/3 operating frequencies £ as a function of f . The dots (crosses) denote the extremes in the dR/dH (the dX/dH) signal. CHAPTER IV 101

Straight lines were fitted to the data by means of a least- squares fitting technique. The upper part of fig.4.3 gives the a-peak in sample 5 (H7/£lOCQ ). The measuring points fit fairly well to a straight line and all lines converge to approximately one field value. Six lines cross the ordinate within one Oersted at 522 Oe. This gives an error (standard deviation) in the mean value 5*22.1 Oe smaller than l°/oo. In the lower part of fig. 4.3 the situation is different. The measuring points fit not so well to the straight lines and the four lines do not converge to the same field. The mean value is 466 Oe whereas the standard deviation is smaller than 5°/oo. The convergency of the lines is certainly comparable with the results of Cleveland e.a. (1971) and Bradfield e.a. (1973). In most cases it is better. The two examples given in fig. 4.3 are examples of extreme cases. We have investigated closer the variation of the width of a peak with frequency. Let the magnetic field value of an extremum shifts with frequency according to H=H + A f~n. The difference AH in the position of two extremes of one R.F.S.E. peak varies consequently according to -A* ' } f~~n» In fig. 4.4 we

io2

50

5 20

10' mom olrcmum

12 5 10 f(MHz) Fig.4.4 Plot of the differences AH in the position of two extremes of an a-peak in sample- 5 (left part) and 6 (right part) as a function of the frequency. compare two cases: (i) in the left part is given the distance AH i of several extremes of the a-peak at 522*1 Oe (see fig. 4.3) a* function of the frequency. The slope of the straight lines, being n, is approximately the same and equal to 1/3. (ii) In the 102 CHAPTER IV

right part of fig. 4.4 the same method is employed for an a-peak in sample 6 (H//f010]). The points don't fit to a straight line for all values AH measured between two extremes. Besides, the slope of the line differs from n=l/3. This deviation from the generally expected AH^ f~ ' relationship has been observed earlier by Haberland e.a. (1967,1969) and Boudreau e.a. (1971). Fukumoto e.a. (1966) have observed that the low-field extremum in dR/dH has a weaker frequency dependence than the other, in accordance with our experience, (see fig. 4.4). It is known that the specular reflection of charge carriers at the sample surface effects the frequency dependence of the surface impedance (Dubovskii 1970). In the sections 3.1 and 3.2 we have shown that, although the diffuse scattering is predominant, specular reflection may occur dependent on the sample surface conditions and the orbit geometry. It is expected that in the case of small peaks and in the case of a deteriorated sample (see fig. 3.5) the influence of the specular reflection is clearly observable. In that case large deviations of the f~ ' rule can occur in the observed frequency dependence of the position of an extremum. As shown in fig. 4.4 (right side) the main extremum satisfies reasonably the f~ ' -rule. Therefore, -1/3 it is possible to use the AH=c f ' relationship for the determination of the critical field value, as described above, if convenient criteria are employed in order to discriminate the different critical field values H of one peak. The following procedure has been employed. (i) Straight lines according to a f~ ' dependency were fit to the data by means of a least squares fitting technique. In case that the differences in magnetic field values of the measuring points and the corresponding theoretical values for a specific extremum is too large (larger than 0.5% of the field value in question) this extremum has not been taken into account, (ii) The lines have to converge to within .75% of the possible critical field value. (iii) Additional arguments may be present to affirm the choice of the point of convergency, for instance the critical field CHAPTER IV 103 values obtained from peaks caused by a chain of orbits fitting to the sample have to be consistent with the values obtained for the single orbits. As a result the critical field values used in the calculations of the caliper values have in most cases an accuracy better than 0.5% (see next section).

4.2 Theoretical description of the Fermi Surface

The critical field values obtained by means of the extrapolation method described in the preceding section were used to calculate extremal dimensions k_v. of the Fermi surface according to eq. ( 1.7 ). The measured extremal dimensions are listed in table 4.1

Table 4.1 Extremal dimensions

Plane Direction of Zone Calculated Experimental value (CU) extremal value (CU) dimension Present itfork GK

(001) 100 2 1.7919 I.792 HI °.018* 1.838 + 0.046 (001) 100 2 0.7570 0.762 HL °.008 (001) 100 2 1.4526 1.459 *r 0.015 (001) 110 2 1.5947 1.599 i:.016 ° * 1.660 ± 0.064 (001) 110 2 1.2923 1.291 *:.01 °3 1.304 + 0.030 (001) 110 2 0.6264 0.622 4• 0.006 0 .688 + 0.076 (001) 110 2 0.4842 0.488 i:.00 °5 0 .469 1 °.055 (100) 001 2 1.4985 1.486 ->:.015 ° * 1 590 1 o.040 (001) 100 3 0.2769 0.275 + 0.003 0.281 + 0 .008 (001) no 3 0.2086 0.209 + 0.004* 0.210 + 0.008 (100) 001 3 0.1551 0.161 + 0.016 0.162 ± °-008 (001) 100 3 0.01049 0.0107+ 0.0035

The experimental values marked with are used in the fitting program 104 CHAPTER IV

We use the conventional computational units (CU) throughout i.e 2 27r/a=h /2m=l/ where m is the free electron mass. The conversion factor for indium is 1CU=7.246 eV. The accuracy of the experimental values of the extremal dimensions depending on the accuracy of the determination of the sample thickness and the critical field is about 1% (see sections 2.1 and 4.1). However, when the extremal dimension is obtained from critical fields connected with a R.F.S.E. due to combina- tions of orbits the experimental error is larger. In table 4.1 are also given the extremal dimensions as obtained from the plots given by Gantmakher and Krylov (1966), to be referred to as GK. As described by Krylov and Gantmakher (1967) GK have indicated the middles of the R.F.S.E. lines as to be the critical field. As a result their experimental values are a few percent too high. The measured extremal dimensions of the Fermi surface were used in a K.K.R.Z. interpolation scheme. They are appropriate for this purpose. But, not only the value of the extremal dimension, also the location of the extremal dimension on the Fermi surface must be known. Since the correct location is known beforehand only for a few dimensions we were able to use only four experimental values, marked with , in our fitting program. The remaining extremal dimensions were used to verify the calculations. Since the K.K.R.Z. method is rapidly convergent with respect to the angular momentum 1 (see Ziman 1971) we have chosen a description with three terms 1=0,1,2. As parameters in our fitting program we used the ETA(l) as defined in section 1.3 instead of the phase shifts n-j. We used the helium temperature lattice constant a=4.5557 8 and c/a=1.083 (Barrett 1962) and the Slater radius

Rg=2.22144 CU. As Fermi energy Ep we have chosen the free-electron value E°=1.205 CU. The position of the muffin-tin zero relative to the bottom of the band has been obtained from an ab initio calculation. We have used a value of 0.421 CU . With respect to.this it can be remarked that the choice of the muffin-tin zero is not very critical as shown by DeviHers and de Vroomen (1971) and by Coenen and de Vrooxnen (1972), t) Note: This calculation has been perforaed by D.B.B. Rijsenbrij and J.A. Radder with a K.K.R. program (Natuurkundig Laboratoriun, Vrije Universiteit, 1973) i CHAPTER IV 105

The three ETA-values are fitted to the four marked extremal dimensions by a weighted least squares fit procedure with the additional condition that the difference between calculated and measured value is smaller than the experimental error. The number of reciprocal lattice vectors needed in the calculations can be determined from fig.4.5, showing the dependence of the parameters ETA on the number of the reciprocal lattice vectors used in the fitting program.

3 1 5 10/ 30 50 number of r I. vectors

Fig.4.5 Dependence of the three parameters ETA (1) on the number of reciprocal lattice vectors 3 - used in the fitting program. '1=0

At least thirty reciprocal lattice vectors are needed. Using a smaller number the deviation of the calculated extremal dimension from the measured one increases and can exceed an experimental error. The three ETA-values obtained from the fitting program with thirty reciprocal lattice vectors are: -2.333 CU; -1.434 CU; +0.297 CU. We calculated with these parameters the remaining extremal dimensions, listed in table 4.1. The calculated extremal dimensions of the Fermi surface agree all within the experimental error with the measured values. In fig. 4.6 calculated cross-sections of the Fermi surface in several planes are shown. The extremal dimensions given in table 4.1 are designated in the cross-sections. In the case of the (001) and (100) central cross-sections of the second zone hole surface the projections of the square cups of the hole surface on the plane in question are also shown in fig. 4.6. 106 CHAPTER IV

(001) cross-section of the 3rd zone electron surface.

0.0 KU» [100]

(!10) cross-section of the 3rd zone electron surface.

Fig.4.6 Cross-sections of the Fermi surface of indium. The dimensions indicated in the cross-sections correspond with the calculated dimensions given in table 4.1 and 4.2.

i CHAPTER IV 107

[010] T

(001) central cross-section of the 2nd zone hole surface.

(100) central cross-section of the 2nd zone hole surface.

(110) central cross-section of the 2nd zone hole surface*

Fig.4.6 Cross-sections of the Fermi surface of indiua. 108 CHAPTER IV

These projections are corresponding to the points of the square

cups where the derivative of the radial distance kp with respect to the polar angle measured in a planar section of the F.S that crosses the edge of a square cup, changes in sign. These projections are important in connection with possible "break"- orbits as described in chapter 1 (section 1.1). Up till now the geometry of the Fermi surface of indium is described by many authors with an i|/PW model. In order to compare our calculations with earlier work on this subject we give a survey of the literature. Ashcroft and Lawrence (1968), using a single-parameter model in a 6 ^PW description, compared their calculations with the experimental data for the third zone from the de Haas-van Alpen (dHvA) measurements of Brandt and Rayne (1963) and the R.F.S.E. measurement of GK (probably uncorrected) and with the cyclotron resonance measurements of Mina and Khaikin (1965,1966) of the second and third zone. Van Weeren and Anderson (1973) have fitted the Fourier coeffi-

cients V , VQ02 and V200 of the pseudopotential and the Fermi energy to the experimental data from dHvA measurements of the third as well as the second zone. In the earlier work of Feng-ling Cheng and Anderson (1967) this fitting procedure was executed with respect to the dHvA measurements of the third zone. Hughes and Shepherd (1969), using only four plane waves in contrary to the mentioned authors (6 t|>PW's), have included spin-orbit coupling in their calculation and used their low dHvA frequency results in order to determine the set of parameters. Arustamova, Mina and Pogosyan (1969), using three ^PW's, have calculated for a single parameter model the 6 tubes of the third zone electron F.S. They used the corrected R.F.S.E. measurements of GK (1966) and the dHvA effect measurement of Brandt and Rayne (1963) to determine the pseudopotential and the Fermi energy. Extensive calculations have been performed by Gaspari and Das (1968). The calculated energy values at symmetry points of the Brillouin zone obtained from an OPW calculation were used to Table 4.2 Comparison of theoretical extremal dimensions of the Fermi Surface (CU)

Zone Dimension Present work Ashcroft Van Weeren Feng-ling Cheng Hughes Gaspari Arustamova designation Lawrence Anderson Anderson Shepherd Das llina,Pogosyan i (1968) (1973) (1967) (1969) (1968) (1969)

K.K.R. Z. ip.p.w. tfi.P.W. tJj.P.W. ip.P.W. *.P.W. i|>.P.W.

2 a 0.8893 0.889 0.889 0.884 0.898 2 b 0.7974 0.803 0.783 0.780 0.811 2 a1 0.7420 0.746 0.747 0.747 2 X 1.2923 1.275 1.275 2 c 0.3132 0.303 0.316 2 c1 0.4842 0.500 0.467 3 W 3 f 0.2086 0.218 0.202 0.208 0.209 0.271 0.2084 H 3 8 0.1551 0.158 0.157 0.232 0.1503 3 f[l00] 0.2769 0.289 0.270 3 f24° 0.206 0.210 0.206 0.2026 3 f56° 0.170 0.180 0.171 0.1703

2 a_o 0.8987 0.895 2 a 0.7595 0.815

O vO 110 CHAPTER IV

determine the parameters for the pseudopotential interpolation scheme. The calculated F.S. of the second and third zone were compared with the dHvA measurement of Brandt and Rayne (1963) and with the magneto-acoustic measurements of Rayne and Chandrasekhar (1962) and Rayne (1963). We have performed i|»PW calculations with the parameters of Ashcroft and Lawrence (1968) and of Van Weeren and Anderson (1973) in order to obtain extremal dimensions of the Fermi surface. The numerical results along with the corresponding results gathered from the surveyed literature are listed in table 4.2. The extremal dimensions mentioned in table 4.2 are designated in fig. 4.6. We compare these results with the corresponding values calculated with the K.K.R.Z. interpolation scheme, which are also given in table 4.2. As far as the extremal dimensions of the third zone electron F.S is concerned the agreement of our results with the results of Hughes and Shepherd (1969) is extraordinarily good. We suggest that this agreement is due to the fitting of the pseudopotential parameters to the measurents of the third zone. The same argument will be valid too for the results of Arustamova, Mina and Pogosyan <1969) and the results of Feng-ling Cheng and Anderson (1967). In all calculations the extremal dimensions of the second zone hole F.S. agree fairly well as far as the a and a1 dimensions is concerned. The differences in the b dimensions don't exceed the 2.2%. The agreement of the results mentioned in the columns 4 and 5 and with our results in column 3 for the second zone is reasonably good. The third zone dimensions of the latter authors a- gree slightly better with our values. The discrepancy between the calculated third zone dimensions of Gaspari and Das and our corresponding dimensions is rather large. We suppose that the calculations of Gaspari and Das describe the F.S. poorly because of the fact that they have found the existence of a arms which is confirmed experimentally only by Brandt and Rayne (1963) and not by other investigators.

i CHAPTER IV 111

Finally, we can conclude that our calculations describe fairly well the experimental extremal dimensions and agree reasonably with the results obtained from pseudopotential calculations. Complemental results obtained from our calculations are i) At the connections of the £ arms remnants of the a arms are present in the LOOlj direction with a height of 0.0543 CU. m ii) The second zone hole surface is open near the point W in the Brillouin zone. The diameter of this connection is about 0.005 CU (see also Van Weeren and Anderson 1973).

4.3 Angular dependence of R.F.S.E. lines In this section we deal with the shift of the R.F.S.E. line position as a function of the direction of the external magnetic field in the plane of the sample. The direction of the magnetic field was varied by rotating the magnet. We have used the double- side excitation arrangement at one frequency, 5 MHz. The position of the R.F.S.E. line was determined by using the linearity of a field sweep range and several calibrated (see section 2.5) points in the sweep range. Since the location of the critical field in the line structure is in many cases difficult to distinguish, the accuracy of the measured extremal dimensions is smaller than the accuracy of the measurements in the preceding section, where the frequency dependence was used. The accuracy amounts about ± 2%. Some measurements are more inaccurate due to either the small amplitude of the peak or the overlap of successive peaks. We have confined ourselves to the angular dependence of the extremal dimensions of the central cross-sections of the second zone hole surface. The measured extremal dimensions are indicated in the calculated central cross-sections of the F.S., as shown in fig. 4.7. In the plots the a, a1 ,b, b' , X and d lines are designated. All the measured extremal dimensions fit fairly well to the calculated cross section. Consider the (100) cross section. The position of the a and a* peaks follows the calculated cross-sectional contour. The extremal dimensions parallel to the [OOl] direction or nearly parallel to the [ooi] direction seem- to be larger than the corres- i 112 CHAPTER IV

Fig.4.7 The (100) (a) and (001) (b) central cross-sections of the 2nd zone hole surface of indium. The measured extremal dinensi'ons (poss.error about 3Z) as a function of the magnetic field orientation are indicated in the calculated cross-sections with dots and straight lines. CHAPTER IV 113

ponding calculated extremal dimensions. However, the R.F.S.E. lines in question are not due to central orbits through the point r but due to an orbit slightly displaced from Y. This is in agreement with the fact that the R.F.S.E. lines split in two lines when the magnetic field is tilted over a small angle from the sampJLe surface. In the preceding section we took into account this fact in our fitting procedure (see fig. 4.6). Between the angles 20° and 60° GK don't give data for the so- called b1 orbits. We have observed very small peaks which shift on varying the magnetic field direction in accordance with the calculated angular dependence, as seen from fig. 4.7a. The X-lines are corresponding to orbits which on one side traverse a square cup and on the other side terminate at the sharp corner of the intersection of two hexagonal cups. It is remarkable that in the C0103 direction no X-line has been observed (see also GK). This is confirmed by the geometry of the calculated F.S.: A line drawn in the [oio] direction starting from the top in the [00l3 direction of the projected "central" square cup intersects the outer contour of the hole surface neither in a point of the square cup nor in a critical point (see for example the X-lines for 3° and 12°). It is clear that X-lines observed at angles between about 42°and 53° are due to orbits which are passing over two square cups (see GK). In the (001) cross section of the second zone F.S. the measured extremal dimensions are only plotted in a range of angles of 45° due to the symmetry of the indium lattice. The a-lines near the £l00]| direction are not originated from the central orbits through T, but from orbits slightly displaced from T. A particular feature of the X-lines is that for angles smaller than 14° relative to the [lOO] direction no X-lines are observed. It is clear from the plot that for smaller angles the shape of the orbit will change i.e. the orbit does not traverse the square cup. The X-peak observed when the magnetic field is 45° from the 114 CHAPTER IV

direction can be attributed unambiguously to a cut-off orbit. The square cups perpendicular to the £lO(T] and [010] directions projected on the (001) plane protrude from the central cross-section. The common tangent of two protrusions is parallel to the [lio] direction. The measured extremal dimension fits very well to the distance between the points of contact. The d-lines observed at angles from 16° to 30° are shown in the (001) cross-section. From the drawn lines it is clear that 16° and 30° are the extremal angles for observing the d-lines (see also GK). The measured extremal dimensions fit reasonably well to the calculated ones. From the given outline we can conclude that the observed orientation dependence of the extremal dimensions is in accordance with the geometry of the calculated Fermi surface. 115

SUMMARY

This thesis deals with the determination of the Fermi surface of indium and of mean free paths in indium by means of the R.F. size effect (R.F.S.E.). The effect of the experimental arrangement, the excitation mode, the temperature and the surface scattering of the charge carriers have been studied. The introduction elucidates the motivation of this work. The application of linear dimensions of the Fermi surface, obtained from accurate R.F.S.E. measurements, in a K.K.R.Z. interpolation scheme in order to calculate the Fermi surface may lead to an improved Fermi surface model of indium. The mean free path and the specular reflection at the sample surface of charge carriers will affect the R.F.S.E. line shape. In order to be able to distinguish between several affects on the line shape it is necessary to investigate the observed line shapes in connection with the actual surface impedance anomalies. After a theoretical chapter 1 in which the physical aspects of the R.F.S.E. are described, an extensive description is given in chapter 2 of the double-sided (twin-T bridge arrangement) and the single-sided (hollow box transmission arrangement) excitation set-ups in connection with the separation of the real and imaginary part of the surface impedance. Chapter 3 presents the line shapes obtained with the two experimental arrangements. The line shapes obtained with double-sided excitation agree roughly with those calculated by Juras. Mean free paths are calculated from the magnetic field induced surface states cut-off field and the temperature dependence of some R.F.S.E. peaks. The results are in accordance with earlier experiments . In chapter 4 we deal with the calculation of the Fermi surface of indium with a K.K.R.Z. interpolation scheme by fitting three parameters to four selected linear dimensions of the Fermi sur- 116

face, determined experimentally. The Fermi surface calculated with these parameters is in accordance with the experimental results for many orientations of the magnetic field within the experimental error. 117

REFERENCES

Alderman D.W. 1970 Rev.Sci.Instr. 4J[, 192 Anderson H.L. 1949 Phys.Rev. 7£, 1460 Arustamova M.E., Mina R.T. and Pogosyan V.S. 1969 Sov.Phys.-JETP 29_, 1065 Ashcroft N.W. and Lawrence W.E. 1968 Phys.Rev. 175, 938 Awater H.H.A. and Lass J.S. 1973 J.Phys.F: Metal Phys. 2» >H3 Azbel'M.Ya. and Kamer E.A. 1957 Sov.Phys,-JETP 5_, 730 Azbel'M.Ya. 1961 Sov.Phys.-JETP J_2, 283 Balibar S., Perrin B. and Libchaber A. 1972 J.Phys.F: Metal Phys. 2, 629 Barrett C.J. 1962 Advances in X ray Analysis 5_, 33-47 Bezuglyi P.A. 1960 Sov.Phys.-JETP J£, 1049 Bianconi A. and Iannuzzi M. 1971 Phys.Rev.B £, 3935 Boiko V.V., Gasparov V.A. and Gverdtsiteli I.G. 1966 Sov.Phys.-JETP 29, 267 Boiko V.V. and Gasparov V.A. 1972 Sov.Phys.-JETP 34, 1266 Boudreau D.A. and Goodrich R.G. 1971 Phys.Rev.B 3^. 3086 Bradfield J.E. and Coon J.B. 1973 Phys.Rev.B 7_, 5072 Brandt G.B. and Rayne J.A. 1963 Phys.Rev. J_32, 1512 Brandt G.B. and Rayne J.A. 1964 Phys.Letters ^2, 87 Bruin F. and Schimmel F.M. 1955 Physica XXI, 867 Bruin F. 1961 Adv. in Electronics and Electron Phys. J^5_, 327 Carolan J.F and Koch J.F. 1973 J.Phys. Chem.Solids 34, 1797 Castle J.G., Chandrasekhar B.S. and Rayne J.A. 1961 Phys.Rev.Lett. b> 409 Chambers R.G. 1952 Proc.Phys.Soc. A 65, 458 Chambers R.G. 1969 Phys.Kond.Mat. % 171 Cleveland J.R. and Stanford J.L. 1970 Phys.Rev.Lett. 24, 1482 Cleveland J.R. and Stanford J.L. 1971 Phys.Rev.B h_t 311 Cochran J.F. and Shiftman C.A. 1965 Phys.Rev. |40_, A 1678 Cochran J.F. 1970 Can.J.Phys. 48, 370 Coenen N.J. and de Vroomen A.R. 1972 J.Phys.F: Metal Phys. 2, 487 Collins R 1957 Rev.Sci.Instr. 28_, 502 Devillers M.H.C. and de Vroomen A.R. 1971 Sol.St.Comm. 9, 1939 Druyvesteyn W.F. and Smets A.J. 1969 Phys.Lett. 30A, 415* Druyvesteyn W.F. and Smets A.J. 1970 J.Low Temp.Phys. 2, 619 Dubovskii D.B. 1970 Sov Phys.-JETP 2i» 465 Feng-Ling Cheng and Anderson J.R. 1967 University of Maryland Tech.Rep. no. 711 Fukumoto A. and Strandberg M.W.P. 1966 Phys.Lett. 23, 200 Fukumoto A. and Strandberg M.W.P. 1967 Phys'.Rev. 155, 685 Gabel D.E. 1970 Thesis, Simon Fraser University Gaidukov Yu.P. 1966 Sov.Phys.-JETP 2j2, 730 Gantmakher V.F. 1962 JETP 43, 345 Gantmakher V.F. 1962 Sov.Phys.-JETP JJ5, 982 Gantmakher V.F. 1963 Sov.Phys.-JETP }J_, 549 Gantmakher V.F. and Krylov I.P. 1965 Sov.Phys.-JETP 20, 141b Gantmakher V.F. and Krylov I.P. 1966 Sov.Phys.-JETP 22, 734 Gantmakher V.F. 1967 Progr.Low Temp.Phys.V edited by C.J.Gorter (North- Holland, Publ.Co., Amsterdam 1967)p.l8l Gantmakher V.F., Fal'kovskii L.A. and Tsoi V.S. 1969 JETP Lett. £, 1** Garland J.C. and Bowers R. 1969 Phys.Rev. 188, 1121 Gaipari G.D. and Das T.P. 1968 Phys.Rev. 167, 660 Goodrich R.G. and Jones R.C. 1967 Phys.Rev. 156, 745 Gregory W.D. and Superata M.A. 1970 J.Cryst.Growth 7, 5 Grimes C.C. and Walsh W.H. 1964 Phys.Rev.Lett. J2» 523 I 118 I i Grivet P., Soutif M. and Gabillard R. 1951 Physica XVII, 420 \ Gutorsky H.S., Meyer L.H. and McClure R.E. 1953 Rev.Sci.Instr. 24_, 644 j Haberland P.H. and Shiffman C.A. 1967 Phys.Rev.Lett. \% 1337 : Haberland P.H., Cochran J.F. and Shiffman C.A. 1969 Phys.Rev. 184, 655 Haberland P.H., Cochran J.F. and Shiffman C.A. 1969 Phys.Lett. 30A, 467 \ Halbach K. 1954 Helv.Phys.Acta J27_, 259 • Hansen J.W., Grimes C.C. and Libchaber A. 1967 Rev.Sci.Instr. 313, 895 \ Herrod R.A., Gage C.A. and Goodrich R.G. 1971 Phys.Rev.B 4^ 1033 j Hughes A.J. and Shepherd J.P.G. 1969 J.Phys.C: Sol.St.Phys. £, 661 j Jones R.C., Goodrich R.G. and Falicov L.M. 1968 Phys.Rev. 174, 672 j Juras G.E. 1969 Phys.Rev. J87, 784 j Juras G.E. 1970 Pbys.Rev.Lett. J2£, 390 • Juras G.E. 1970' Phys.Rev.B 2, 2869 I Kaner E.A. 1958 Sov.Phys.-Doklady 3, 314 ! Kaner E.A. 1963 Sov.Phys.-JETP _I7_, 700 '• Kaner E.A. and Fal'ko V.L. 1967 Sov.Phys.-JETP 24, 392 i Khaikin M.S. 1962 Sov.Phys.-JETP _14-, 1269 Kittel C. 1967 Quantum Theory of Solids (Wiley, New York 4 printing)p.31O i Koch J.F. and Wagner T.K. 1966 Phys.Rev. j_51_, 467 Koch J.F. 1969 Phys.Kond.Mat. £» 14° i Kohn W. and Rostoker N. 1954 Phys.Rev. 94, 1111 Korringa J. 1947 Physica XIII, 392 • Krylov I.P. and Gantmakher V.F. 1967 Sov.Phys.-JETP 24_, 492 * Losche A. 1957 Kerninduktion (VEB Deutscher Verlag der Wissenschaften, 1 Berlin, 1957) p.147 ; Lyall K.R. and Cochran J.F. 1967 Phys.Rev. JL59_, 517 Matthey M.M.M.P. 1969 Thesis, Katholieke Universiteit, Nijmegen, Holland Mehring M. and Kanert 0. 1965 J.Sci.Instr. 42, 449 Mina R.T. and Khaikin M.S. 1965 Sov.Phys.-JETP 2J_, 75 Mina R.T. and Khaikin M.S. 1966 Sov.Phys.-JETP 24_, 42 Miyoshi D.S. and Cotts R.M. 1968 Rev.Sci.Instr. 39, 1881 Nee T.W., Koch J.F. and Prange R.E. 1968 Phys.Rev. VJ±, 758 * O'Sullivan W.J.O., Schirber J.E. and Anderson J.R. 1968 Phys.Lett.27A, 144 Panofsky W.K.H. and Phillips M. 1964 Classical electricity and magnetism (Addison-Wesley Publ.Co. London 2nd edition)p.219,226 Peercy P.S., Walsh W.H., Rupp L.W. and Schmidt P.H. 1968 Phys.Rev. 2ZJL» 713 Penz P.A. and Kushida F. 1968 Phys.Rev. 176, 804 Perrin B., Weisbuch G. and Libchaber A. 1970 Phys.Rev.B j^, 1501 ! Pippard A.B. 1947 Proc.Roy.Soc. A191, 385 ) Pippard A.B. 1954 Proc.Roy.Soc. A224, 273 ',' Prange R.E. and Nee T.W. 1968 Phys.Rev. J68_, 779 Rayne J.A. and Chandrasekhar B.S. 1962 Phys.Rev. 125, 1952 I Rayne J.A. 1962 Phys.Rev. 129, 652 | Revenko Yu.F. and Zaitsev G.A. 1970 Instr.S Exp.Techn.(USA) 6, 1799 i Reuter G.E.H. and Sondheimer E.H. 1948 Proc.Roy.Soc. A195, 336 I Roach P. 1971 Phys.Rev.B. £, 1748 j Sharvin Yu.V. and Gantmakher V.F. 1963 Priboryi Techn.Bksp. <>, 165 j Sibbald K.E., Mears A.L. and Koch J.F. 1971 Phys.Rev.Lett. 2£, 14 '! Snyder P.M. 1971 J.Phys.F: Metal Phys. J^ 363 ' I Soffer S. 1972 J.Low Temp.Phys. £, 551 i* Soffer S., Huguenin R. and Probst P.A. 1973 J.Low Temp.Phys. 11, 337 j Sondheimer E.H. 1952 Adv.Phys. 1,1 ! Steenhout O.L. and Goodrich R.GT 1970 Phys.Rev.B I, 4511 I Taylor M.T. J965 Phys.Rev. J_3£, A 1145 ~ j. 119

Trodahl H.J. 1971 J.Phys.C: Sol.St.Phys. £, 1764 Tsoi V.S. and Gantmakher V.F. 1969 Sov.Phys.-JETP 29_, 663 Tuttie W.N. 1940 Proc.Inst.Radio Eng. 28, 23 Wagner T.K. and Koch J.F. 1968 Phys.Rev. 165, 885 Weeren J.H.P.van and Anderson J.R. 1973 J.Phys.F: Metal Phys. 3_» 2!09 Wind R.A. 1970 J.Phys.E: Sci.Instr. 3, 31 Yaqub M. and Cochran J.F. 1965 Phys.Rev. J_3_7_, A 1182 Ziman J.M. 1965 Proc.Phys.Soc. 86_, 337 Ziman J.M*. 1971 Solid State Physics vol. 26, 90 Ziman J.M. 1965 Principles of the theory of solids (Cambridge University Press, 1964) p.90

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