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Anomalous electronic structure of the positive muon in antimony : evidence for an isolated kondo impurity Johnston, Thomas Michael Shaun 1993

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ANOMALOUS ELECTRONIC STRUCTURE OF THE POSITIVEMUON IN ANTIMONY: EVIDENCE FOR AN ISOLATED KONDOIMPURITYByThomas Michael Shaun JohnstonB.Sc.E. Queen’s University, 1991A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIA© Thomas Michael Shaun Johnston, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)__________________________________Department of_________________The University of British ColumbiaVancouver, CanadaDateDE-6 (2/88)AbstractThe anomalous electronic structure of a positive muon in the semimetal antimony wasinvestigated by the SR (Muon Spin Rotation/Relaxation) technique. Precise measurements of the giant muon Knight shift (K=+1.4%) were made as a function oftemperature (2-20 K) and magnetic field (18-21 kG) with the applied magnetic fieldparallel to the ê-axis of the antimony single crystal. No de Haas-van Alphen oscillationswere observed at a temperature of 3 K, which indicates the electron spin density on themuon does not scale simply with the magnetic susceptibility of the conduction electrons.An upper limit of 0.2% of K, can be placed on the amplitude of the K, de Haas-vanAlphen oscillations. The measured K, varies weakly with temperature below 10 K. Thedata were fit in three different ways. The best fit is to the form K = 1/(a + bT2 + cT4)with parameters a=7.167(3) x i0, b=6.4(7) x 1O K2, and c=1.8(2) x 10 K4. Afit to an Arrhenius Law of the form 1— Ae/T yields a pre-exponential of 0.47(4) andan activation temperature Ea/kB of 40(2) K. If one fits to an expression for the Kondosusceptibility of the form K = a(1/2IrTK — 0.433T2/) at low temperatures (2-10K) one obtains a parameter a=14.0(0.3) K and a Kondo temperature TK=160(3) K.Together with existing measurements of K at higher temperatures (90-180 K) we observe a crossover from a weakly temperature dependent Pauli paramagnetic behaviourbelow 10 K to a Curie-like behaviour above 90 K. This crossover is characteristic ofa Kondo impurity indicating the anomalous electronic structure of a positive muon inantimony may be due to muonium formation, in which a local moment is centered onor near the muon. Within context of this model estimates of the hyperfine parameters(A11=259,7(4) MHz and A±=129.9(2) MHz) for muonium in antimony are similar to11those for Mu* in covalent semiconductor. The large anisotropy suggests a large spindensity resides on the nearest neighbour antimony atom(s). Since the SR techniqueonly permits one muon in the sample at a time, the anomalous electronic structure ofa muon in antimony may be a unique example of a truly isolated Kondo impurity in sofar as other magnetic impurities in the sample can be neglected. A large Korringa-likerelaxation of the muon in antimony was observed in longitudinal field implying thespin dynamics of a muon in antimony are also anomalous when compared to normalmetals. The Korringa constant S/(K,T1T)is temperature independent as expected fora Kondo impurity for T << TK. Above a temperature of 100 K, which is comparableto TK, a breakdown of the Korringa law is seen. In particular a peak in T’ at atemperature of 75 K is observed.111Table of ContentsAbstract iiList of Tables viiList of Figures viiiAcknowledgements x1 Introduction 12 Crystal Structure and Fermi Surface of Antimony 73 The Knight Shift 113.1 Theory 113.2 The Muon Knight Shift in Sb: Review of Previous Experimental Results 143.2.1 First Discovery 153.2.2 Self-Consistent Molecular Cluster Calculations 183.2.3 Crystal Backscattering Resonance 203.2.4 SbBi and SbSn Alloys 203.2.5 De Haas-van Alphen Oscillations in Sn119 and Cd” Knight Shift 214 The Kondo Effect 254.1 Introduction 254.2 Dilute Alloys 274.3 Some Models 28iv4.3.1 Moment Formation and the Anderson Model 284.3.2 The Spin-Fluctuation Model vs. the Spin-Compensation Modelfor the Kondo Effect 324.3.3 The Resonant-Level Model 344.3.4 Exact Solutions of the Kondo Susceptibility 354.3.5 The Korringa Relation 355 1tSR 385.1 Muons and Muon Beams 385.2 TF-SR Geometry and Technique 415.3 LF-SR Geometry and Technique 445.4 Experimental 455.5 TF-tSR Data 475.6 LF-,uSR Data 496 Experimental and Results 516.1 Shubnikov-de Haas Oscillations 516.2 De Haas-van Aiphen Oscillations of the Magnetization in Sb 526.3 jISR in Sb 546.3.1 Sample Characteristics 546.3.2 Measurement of Muon Knight Shift with TF-[LSR 546.3.3 Magnetic Field Dependence of the Muon Knight Shift 566.3.4 Temperature Dependence of the Muon Knight Shift below 20 K 616.3.5 Muon Knight Shift at High Temperatures 626.3.6 Discussion of TF-tSR Results 696.3.7 LF-tSR Experimental Procedure 736.3.8 LF-1tSR Temperature Scan 73V6.3.9 LF-tSR Field Scan.756.3.10 Discussion of LF-SR Results 757 Conclusion 82Bibliography 84A The de Haas-van Aiphen Effect 87A.1 Quantization of Electron Orbits in an Applied Field 87A.2 Landau Levels 88A.3 Extremal Orbits of the Fermi Surface 93B Experimental Data 95B.1 TF-1SR Data: Knight Shift vs. Temperature 95B.2 LF-1iSR Data: Relaxation Rate vs. Temperature 95viList of Tables2.1 Sb Lattice Parameters 92.2 Experimental de Haas-van Alphen Oscillation Periods 106.1 Sb Level Crossing Resonance Parameters 80B.1 TF-iSR Data: Knight Shift vs. Temperature 95B.2 LF-[LSR Data at 1 kG: T’ Relaxation Rate vs. Temperature 96B.3 LF-uSR Data at 650 G: T’ Relaxation Rate vs. Temperature 96viiList of Figures2.1 The Rhombohedral Crystal Structure and Brioullin Zone 82.2 Sb Electron Pocket on the Fermi Surface 93.1 Volume in k Space 133.2 Virtual Bound State 163.3 Sb Electronic Density of States and Spectral Density on the InterstitialMuon from Molecular Cluster Calculations 193.4 Temperature Scan: K,, in Pure Sb as well as in a 6.3 at.-% Bi Alloy. . 223.5 De Haas-van Alphen Oscillations in Cd 234.1 The Virtual Bound States of the Anderson Model 294.2 Phase Diagram of Magnetic and Nonmagnetic Behaviour for the Anderson Model 304.3 Exact Result for Kondo Susceptibility 365.1 Decay Positron Emission Probability 405.2 TF-RSR Experimental Geometry 425.3 LF-1tSR Geometry 445.4 TF-,uSR Experimental Apparatus 465.5 TF-iSR Time Spectrum, Asymmetry Histogram, and FFT 485.6 LF-1iSR Asymmetry Histogram 506.1 De Haas-van Aiphen Oscillations of Sb Magnetic Susceptibility 536.2 Cross-section of the Sb Single Crystal Sample 54viii6.3 Rotating Reference Frame Fit and FFT of Typical Asymmetry Histogram 576.4 TF-tSR Linewidth as a Function of Magnetic Field 586.5 Frequency Shift and Muon Knight Shift: 2-21 kG 596.6 Muon Knight Shift Field Scans: Sb Bile T=2.7K 606.7 Four Fits to Low Temperature K in Sb Data 636.8 Temperature Scan: K, in Sb at B=4 and 15 kG 646.9 Fitting Curie and Curie-Weiss Behaviour to High Temperature K in Sb 666.10 Typical LF-1tSR Asymmetry Histograms in Sb 746.11 Predicted and Experimental Temperature Dependence of LongitudinalRelaxation Rate, T’ 766.12 Field Dependence of Longitudinal Relaxation Rate, T’ 776.13 Temperature Dependence of S/(T1TK0) 81A.1 Explanation of dHvA Effect for Free Electrons in Two Dimensions in aMagnetic Field 89A.2 Allowed Electron Orbitals and dHvA Oscillations 90ixAcknowledgementsThe patience and interest of my advisor, Rob Kiefi, are much appreciated. I am alsograteful for the assistance provided by Dave Williams. For the seemingly thanklesstask of sitting shifts I would like to say “Thanks!” to Kim Chow, Sarah Dunsiger, TimDuty, Bassam Hitti, Evert Koster, Andrew MacFarlane, Jeff Sonier and Jürg Schneider.Also Evert Koster deserves another thank you for performing the Shubnikov-de Haasand magnetization studies. The guidance of Curtis Ballard and Keith Hoyle withexperimental set up was also very much appreciated. Last but definitely not leastI am thankful to my family, who always provide perspective and support for all myundertakings.xChapter 1IntroductionA positive muon () implanted into a crystalline solid almost always occupies aninterstitial site. In normal metals the conduction electrons act to screen the positivecharge of the muon. The strong Coulomb potential of this single positively chargedimpurity is so reduced by the electronic screening, that no bound electront+ state ina metal has been identified.’ With an applied magnetic field the conduction electronsmay be slightly polarized causing a non-zero electronic spin density at the . Thiscontact interaction between the muon and the net electron spin density at the muonsite [ri()— n(i)] determines the contact or isotropic hyperfine field experiencedby the muon:-. STrBhf(r) = ——[n (rn) — n (r,L)], (1.1)where n() is the spin up electron density and n(i) is the spin down electrondensity at the + site. In an applied field B0 the Knight shift constant K, is definedas:Bhf() zB = B() - B0 = KB0 (1.2)KB(i)—B01L0where B(i) is the total field experienced by the +. [2] The Knight shift is directly proportional to the magnetic susceptibility of the conduction electrons, which in moderatefields is independent of B0.1The only exception occurs at the low electron density endohedral site in the metal Rb3C60.[1]1Chapter 1. Introduction 2In typical metals the Knight shift is small (r10 to 100 p.p.m.)[3] and weakly dependent upon temperature (since the Pauli spin susceptibility of a degenerate electron gasis temperature independent to first order) and crystal orientation (due to the s character of the local electrons in most non-transition element metals). Also the Korringarelaxation rate for a stationary in metals is normally immeasurably small becauseof the short muon lifetime and small contact hyperfine interaction.Intrinsic semiconductors and insulators have much lower carrier concentrations thanmetals do. In these materials the muon can be screened efficiently if it captures an electron during the slowing down process to form a hydrogen-like atom called muoniumor Mu). In vacuum the p+e contact interaction produces an energy splittingbetween the triplet and singlet states: LE = hAvac where Avac/2ir = 4463.30288(16)MHz. A wide variety of muonium centers are observed in semiconductors and ionicinsulators. In most wide band gap materials, such as alkali halides, a single type ofmuonium is observed with a large hyperfine parameter close to that of muonium invacuum. In covalent semiconductors, such as Si or GaAs, two paramagnetic muoniumstates are observed. One center in Si called normal muonium (Mu) is characterized byan isotropic hyperfine parameter reduced to about half that of muonium in vacuum. [4]The smaller hyperfine parameter indicates that the Mu electron is less localized in Sithan in vacuum leading to a smaller electron density on the +. The other center calledanomalous muonium (Mu*) is characterized by a very much smaller hyperfine interaction, which is a few percent of Avac and axially symmetric about the [111] directions.[5]The small highly anisotropic hyperfine interaction arises because Mu* sits near thebond centered positions along the [111] directions with the majority of the unpairedelectronic spin density on the two nearest neighbour atoms along the [111] direction. Attemperatures above 100 and 300 K respectively Mu and Mu* are unobservable due tovery rapid decay of the TF-1iSR precession signals caused by charge changing reactionsChapter 1. Introduction 3to a diamagnetic center- either Mu+ or Mu . [4]The small electron densities in the Group Vb semimetals (antimony: x 1019cm3, arsenic: ‘-2 x 1020 cm3, and bismuth: x 10” cm3) are intermediate between an intrinsic semiconductor and a normal metal. Consequently it may be possiblethat the presence of the muon results in the formation of a paramagnetic complex, inwhich the unpaired electron is centered on or near the muon. This system of a singlemagnetic impurity in a metal (i.e. a Kondo impurity), which at first glance seemsto be a trivial problem, is in fact of great historical significance in condensed matterphysics. Kondo systems have been subjected to numerous theoretical and experimentalinvestigations. [6,7,8] An exact solution to model Kondo systems has eluded researchersuntil recently.[9,10,11] Due to the long range of the RKKY impurity-impurity interaction in dilute magnetic alloys even the most dilute alloys (i.e. p.p.m. levels) have experimental impurity concentrations, which are often too great to avoid impurity-impurityinteraction. [2,6] The isolated nature of a local moment of a muonium impurity in asemimetal or metal would provide an ideal testing ground for these theories, whichhave modelled the magnetic susceptibility and other bulk properties of single Kondoimpurity systems.Previous 1tSR studies of the semimetals As and Bi found only small shifts (K-0.01%)at all experimental temperatures indicating no local moments were formed in thesesemimetals.[12,13] Also Al, Zn, Cu, and C have been examined with SR for signsof local moment formation without success.[14,15] However, in Sb K,, is about 1.4%,which is two to three orders of magnitude larger than in normal metals. K exhibits atemperature dependence, which is also unusual for normal metals. Within experimental uncertainty at all temperatures K with B0 perpendicular to ê is half the size ofK,, with B0 parallel to ê (K- = K,’j). In order to gain insight into the origins of theanomalous K,, alloys of Sb were studied. This work with SbSn and SbBi alloys shows,Chapter 1. Introduction 4K, is sensitive to impurity concentration. In general as the number of electrons on theFermi surface of the alloy decreases, K,, decreases.This thesis extends the work of previous experiments to higher fields (‘-.- 20 kG),lower temperatures (‘ 2 K), and higher statistics (‘-S-’ 1O’’ events/point). Precise measurements of K,, in Sb with B0 parallel to the ê-axis were made as a function of temperature (2-20 K) and magnetic field (2-21 kG). At a temperature of 3 K de Haas-vanAlphen oscillations of K were sought, but not found, which indicates the electron spindensity does not simply scale with the magnetic susceptibility of the conduction electrons. An upper limit of 0.2% of K, can be placed on the oscillation amplitude. Thetemperature dependence of K1, below 20 K is best fit to a power law of the form:K1,— a + bT2 + cT4(1.4)with parameters a=7.167(3) x iO, b=6.4(7) x 1O K2, and c=1.8(2) x 10—10 K—4.Good fits were also obtained from an Arrhenius Law with a pre-exponential of 0.47(4)and an activation energy of Ea/kB=40(2) K and from a theoretical expression for thelow temperature Kondo susceptibility:= a(1/2IrTK — 0.433T2/). (1.5)At low temperatures (2-10 K) this fit yielded a parameter a=14.0(0.3) K and a Kondotemperature TK=160(3) K, which leads to a Kondo exchange parameter J 3 eV. Together with existing measurements of K1, at higher temperatures we observe a crossoverfrom Curie-like behaviour above 90 K to weakly temperature dependent Pauli paramagnetic behaviour below 10 K. This crossover is characteristic of a Kondo impurityindicating that the anomalous electronic structure of a positive muon in antimony maybe due to local moment formation (i.e. muonium), in which the positive muon binds anelectron, which in turn is strongly coupled to the conduction electrons by an exchangeChapter 1. Introduction 5interaction. From the fit to Curie behaviour, estimates of the muonium hyperfineparameter can be made:A11 = 259.7(4) MHz = 0.05A (L6)A Aii, (1.7)which are similar to those of the anomalous muonium center Mu* in covalent semiconductors such as GaAs.Since the tSR technique only permits one ,u in the Sb sample at a time, impurity-impurity interactions are not present. If indeed muonium is formed in antimony, thissystem would be a unique example of a truly isolated Kondo impurity. Furthermore itwould provide information on the transition from the diamagnetic çj+ state in metalsto the paramagnetic muonium state characteristic of semiconductors and insulators.Large Korringa-like relaxation in longitudinal field was also observed implying thatthe spin dynamics of a muon in Sb are also anomalous when compared to normalmetals. A peak at a temperature of 75 K in the T’ relaxation rate also provides anestimate of TK. Below a temperature of 100 K the Korringa constant S/(KT1T) istemperature independent as is expected for a Kondo impurity. As the temperatureapproaches TK the breakdown of the Korringa law is seen.Chapter 2 discusses the band structure, crystal structure, and the Fermi surfaceof antimony. This is followed by a discussion of the standard theory of the Knightshift as well as previous SR investigations of K,,, and other relevant papers in Chapter3. Then Chapter 4 offers a glimpse at some of the theory behind the Kondo Effectand then gives some models for both the T’ relaxation rate and the Kondo magneticsusceptibility, which we assume is proportional to the Knight shift. An overview of the1tSR techniques and the experimental apparatus needed to investigate K,,, and T’ in Sbcan be found in Chapter 5. Chapter 6 deals with the specific experimental proceduresChapter 1. Introduction 6followed by a discussion of the results. The Conclusion provides a short summary ofthis thesis and raises several questions, some of which were answered and some of whichstill await answers. Appendix A outlines the de Haas-van Alphen effect in general. TheK and T’ data is tabulated in the Appendix B.Chapter 2Crystal Structure and Fermi Surface of AntimonyAntimony (Sb) is a Group Vb semimetal with the same A7 crystal structure as theother semimetals bismuth and arsenic, Figure 2.1 a). The rhombohedral primitive unitcell is similar to the fcc primitive cell, which has e=0:= a0(E,1,1) (2.1)= a0(1,e,1) (2.2)= a0(1,1,e) (2.3)(2.4)where e is a small parameter related to the angle c between any two primitive latticevectors by:= [1 — (1 + coscx — 2cos2a)hh’2j/cos. (2.5)These Group V atoms tend to associate in pairs in the lattice and so each unit celllattice point has a basis of 2 atoms at (0,0,0) and (2u,2u,2u). This structure can bevisualized as follows: stretch a NaC1 unit cell along the body diagonal very slightly,displace the Cl sites a small amount also along the body diagonal, and place Sb atomsat all the Na and Cl sites.[16]The fifth and sixth bands overlap slightly yielding a conduction band electron density and a valence band hole density of about 5.5 x 1019 carriers cm3 each.[17] Thelocation of the electrons and holes in the Brioullin zone found from band structure7Chapter 2. Crystal Structure and Fermi Surface of Antimony 8Figure 2.1: a) Rhombohedral crystal structure and primitive unit cell: Sb and otherGroup V elements have a basis of 2 atoms at (0,0,0) and (2u,2u,2u) at each point of thespace lattice. [16] b) Brioullin zone of the A7 crystal structure. [16] The mirror plane cdefined by the points FTXL and the point H on c are omitted for clarity.• = space latticepointc)b)Chapter 2. Crystal Structure and Fermi Surface of Antimony 9Structure u INaC1 0.250 60° 0Sb 0.234 57° 14’ 0.0416Table 2.1: Sb Lattice Parameters.[16]tarTrxI.FLbis.BsectrixElectrons4D5au-tFigure 2.2: Sb electron pocket on the Fermi surface. [17]Chapter 2. Crystal Structure and Fermi Surface of Antimony 10Extremal Period I’T-TU planein iO G1Holes 16.33 5.06Electrons 14.66 2.30Table 2.2: Experimental de Haas-van Alphen oscillation periods for electrons andholes. [16]calculations compare favourably with experimental results. The conduction electronsare in three equivalent pockets centered at the point L in the sixth Brioullin zone.The three FL directions correspond to the cubic [111] directions, which are not alongthe “stretch” direction, which is described in the previous paragraph. The “stretch”direction is I’T, the fourth [111] direction. The holes are in six closed pockets centeredon the point H in the mirror plane a defined by the points FTXL. The FX directionscorrespond to the three cubic [100] directions. The shape of these electron and holepockets is roughly ellipsoidal (see Figure 2.2). The extremal areas of these electronand hole pockets, which make up the Fermi surface, have distinct de Haas-van Aiphenoscillation periods shown in Table 2.2. See Appendix A for an elementary discussionof the de Haas-van Alphen effect.Chapter 3The Knight Shift3.1 TheoryThe Knight shift was first observed by Walter Knight in a Nuclear Magnetic Resonance(NMR) study of Cu63. He observed, that the resonance frequency in metallic copperwas 0.23% higher than in a diamagnetic copper compound, such as CuC1. In fact thisfrequency shift is ubiquitous in metals and it is defined by:‘mWd+1 (3.1)where L.Ym is the metallic resonance frequency and wj is the resonance frequency ofthe same nucleus in a diamagnetic reference material. In general there are severalcontributing factors to the Knight shift. However, the dominant contribution for themost commonly observed cases arises from the Fermi contact interaction between theconduction electrons and the nucleus. In this case there are four common experimentalobservations: (1) /w is positive, (2) the fractional shift (/c’.’/c4.d) is field independent,(3) the fractional shift is temperature independent, and (4) the fractional shift increaseswith nuclear charge Z due to the increased contact interaction. [18]In a metal a nucleus experiences magnetic couplings with the conduction electrons.In the zero field case the electrons have no net spin polarization and thus on average the nucleus experiences zero contact interaction. However, in the presence of astatic, external magnetic field H0 the electron spins have a preferred orientation antiparallel to H0 and the magnetic coupling will be non-zero. The nucleus experiences11Chapter 3. The Knight Shift 12a larger effective field due to the s-state coupling with polarized conduction electrons.Therefore, the shift L is positive. The shift is proportional to the net electron spinpolarization, which in turn is proportional to H0 or wd. Therefore, the fractional shiftis independent of magnetic field. Since the Pauli spin susceptibility (and hence also themagnetization) of a Fermi gas of highly degenerate conduction electrons is temperatureindependent for temperatures below the Fermi temperature, the Knight shift shouldalso be temperature independent.For a system with weakly interacting electrons the Hamiltonian is:Ee+’11n+’flen, (3.2)where fle is the weakly interacting electron Hamiltonian, ?-t. is the nuclear spin Hamiltonian, and ?len is the magnetic interaction of the electron spins and the nuclear spins.The main term of interest is the contact interaction 7-i. of the l electron at withthe j nucleus atflen 7enh2.S(i— ). (3.3)In H0 the polarized electronic spins will produce an average magnetic moment:>= H0. (3.4)The total spin susceptibility of the electrons is:= Jx8(E)g(E,A) dE dA, (3.5)where8(E) is nonzero over a region of width kBT near EF, since only electron statesnear EF have empty electron states, into which they can be excited as the spin flipsover. We define g(E, A) dE dA as the number of electron levels or allowed k valuesin a certain volume of space, which lies between two constant energy surfaces, EChapter 3. The Knight Shift 13and E + dE, and a small area element dA on one of the constant energy surfaces, asseen in Figure 3.1. The energy and the particular coordinate on the energy surface aredenoted E and A respectively.The contact interaction with the jh1 nuclear spin is:en = Yn1’zj [(Iuk(o)2)EFx:iIo], (3.6)where the average value of the probability of finding a Fermi surface electron at theposition of the j nucleus is denoted: (u(O)2)EF. u() is a Bloch function with theperiodicity of the lattice.The contact interaction can also be written asEk±dEk AreadAFigure 3.1: Volume in k space associated with dE dA.[18]LtjH0. (3.7)Chapter 3. The Knight Shift 14The magnetic interaction of the electron spins with a nuclear spin is equivalent to achange Lt1 in the moment of the nucleus, which produces a change in the resonancefrequency and hence a larger effective field at the nucleus of H0 + LH. The Knightshift is:K== -lUk(O)2)EFX. (3.8)As discussed previously, in most cases the Knight shift is independent of temperature and magnetic field. Since electronic wave functions are temperature and fieldindependent, the anomalous dependence of the muon Knight shift in Sb on temperature might reflect the behaviour of the total spin susceptibility, which is proportional tothe magnetization. As seen in Appendix A, the number of electrons at the Fermi surface and thus the magnetization exhibit de Haas-van Alphen oscillations at high fieldsand low temperatures. So theoretically it is possible to observe Knight shift dHvAoscillations, which are ultimately due to oscillations of the number of electrons at theFermi surface.3.2 The Muon Knight Shift in Sb: Review of Previous Experimental ResultsK in Sb is totally unlike the nuclear Knight shift normally observed in metals. Theanomalously large K in Sb is temperature and orientation dependent. This much isknown from previous SR studies,[13,19,20,21,22] which are briefly reviewed below.Nuclear magnetic resonance (NMR) has not yet observed a Knight shift on the Sbnucleus in a solid due to the low crystal symmetry, high atomic number, and a largequadrupole moment.[23,24] De Haas-van Alphen (dHvA) oscillations in the anisotropicand temperature dependent Knight shift of cadmium have been observed by NMR. [25]Cadmium is also an unusual system because the Knight shift in cadmium is anisotropicChapter 3. The Knight Shift 15and temperature dependent.One of the purposes of this thesis is to determine if the conduction electrons in Sbare responsible for the giant K. Since the Knight shift is proportional to the totalspin susceptibility of the electrons, which in turn is proportional to the magnetization,dHvA oscillations of K14 should be seen at the dHvA frequency or frequencies of thoseelectrons, which contribute to the Knight shift. In this way the relative contributionsof the electron states on the Fermi surface to 1(14 could be identified.On the other hand if the local electronic structure of the muon plays a dominant rolerather than the extended band electrons on the Fermi surface,then dHvA oscillationswould not be seen.3.2.1 First DiscoveryThe first observation of K,, in Sb showed that K,, was anomalously large, temperature, and orientation dependent.[13,19] Hartmann et al. concluded, that the temperature dependence does not exactly follow Curie behaviour as is expected for a paramagnetic moment, but the anisotropy of the frequency shift reflects the hyperfine structureof local moment ions. There had been no previous observation of local electronic moments on the site in a metal. It was suggested the large anisotropic 1(14 could beexplained by a muonium-like state, in which the unpaired electron was in a p-like orbitalon the muon.The authors suggested that the electron bound to the muon was in a virtual boundstate (VBS), as seen in Figure 3.2. Varying the electron concentration by varyingthe host metal changes the Fermi energy and the width of the virtual level. A localmoment occurs if the Fermi level is near a virtual level and if the virtual level issufficiently narrow in order to maintain the spin splitting.[26] The state is ‘virtual’Chapter 3. The Knight Shift 16Figure 3.2: Two virtual bound states (VBS) with opposite spins. The spin up VBS ispartly filled causing the formation of a localized moment. [26]Chapter3. TheKnight Shift 17because the diamagnetic + precession is still present and ‘bound’ because of evidenceof local moment formation. The hyperfine field associated with this VBS at the lowesttemperatures was estimated to be on the order of 25 G:Hhf = (0.4 MHz)(13.55 MHz/kG) 25 G (3.9)or about four orders magnitude lower than the vacuum value. [13] The authors suggestedthe giant K, seen in Sb has not been observed in Bi or As because perhaps the positionof the VBS relative to high and low density of states levels plays a key role.At high temperatures the thermal energy kBT is large compared to the energy ofthe contact interaction between the unpaired electron and the muon and, therefore,the local moment orientation fluctuates. However, at low temperatures, the smallhyperfine field is able to fully align the electron moment on the muon and saturationof the frequency shift is expected. For temperatures less than 20 K the frequency shiftshould be saturated. The temperature dependence fit well to a Brioullin function withJ=1/2 and 3/2:[19]Bj(x) = 2J+ ctni ((2J+1)x) — ctnh (3.10)where the total angular momentum quantum number is J = L+S and x =gJ1UBB/kBT.However, at low temperatures, the frequency shift showed no sign of saturating withapplied field, as one would expect from Equation 3.10 in the limit of high field and lowtemperature. The authors speculated that perhaps the temperature saturation was dueto muon diffusion and motional averaging or perhaps the localized wavefunction itselfis field dependent.[19] The K,, temperature dependence might also reflect a broadeningof some critical feature of the density of states.Chapter 3. The Knight Shift 183.2.2 Self-Consistent Molecular Cluster CalculationsMore recently the giant K, in Sb has been interpreted as an effect of the electronicdensity of states. [22] Self-consistent molecular cluster calculations suggest the positiveK, could be due to a large peak in the density of states near the Fermi energy anda high electron density on the muon (or spectral density), as shown in Figure 3.3.Unfortunately the + site in Sb is not known. In the calculations the muon wasassumed to be at the octahedral site in the center of a cluster of 26 Sb atoms. Thiscould have a dramatic influence on the result.The width of the peak in the finite cluster broadens in the bulk semimetal to0.2 eV. In order for Zeeman splitting to be of a comparable width, a magnetic field(B = /.E/gB) of roughly 2 to 15 MG are needed. (In Sb the electron g factor isapproximately 15 for fields applied parallel to the ê axis, but if the is a strongperturbation to the Fermi surface, then the free electron g factor of 2 might be moresuitable. [28]) Zeeman splitting separates the peaks in the density of states for spin-upand spin-down electrons. If the spin-up peak is shifted enough by an applied field tolie above the Fermi level, then a local moment will arise due to the spin-down electronsand saturation will occur. In the case of moderate magnetic fields the frequency shiftwill increase linearly with field.Thus this model qualitatively explains the lack of field saturation, but does not provide a natural explanation of the temperature dependence. Although the exact featuresof this large peak in the density of states are not known, the temperature dependence ofI(. was attributed in a somewhat unsatisfactory way to thermal repopulation of levelswithin this peak. [22]Chapter 3. The Knight Shift 1910eVII ‘ IIEFFigure 3.3: Sb electronic density of states and spectral density on the interstitial muonfrom molecular cluster calculations. [22]I I I?Chapter 3. The Knight Shift 203.2.3 Crystal Backscattering ResonanceThe giant has also been interpreted as a crystal backscattering resonance (CBR),which arises from the combined resonance from electron scattering by an impurity orother defect and backscattering by the host lattice.[29] The CBR occurs at isolateddefects, when the host density of states is low. Bound states may arise through theCBR mechanism because at the resonance it is possible, that the density of statesis increased. The authors of Reference [29] suggest, that in the case of Sb the giantenhancement of KL reflects the increase of the density of states near the Fermi leveldue to an assumed CBR near the Fermi level. The broadening of this large hostinterstitial density of states on the due to either thermal broadening from theFermi-Dirac distribution or phonon-electron interaction, is consistent qualitatively withthe temperature dependence of K in Sb. Thermal motion or diffusion of the mayalso be contributing factors.3.2.4 SbBi and SbSn AlloysBrewer et al. have measured K as a function of temperature in pure Sb and in Sballoys with homovalent bismuth (Bi) and with heterovalent tin (Sn).[20,21] Data weretaken with the ê axis parallel and perpendicular to the applied field of 4 kG, Figure 3.4.In pure Sb it was found, that I Kj/2 in the temperature range from about10 to 100 K. Also the frequency shift showed no saturation in applied field up to9 kG in agreement with the 0.5-2.0 kG data of Hartmann et al. Alloying with Bidecreases both electron and hole concentrations. (At about 70% Bi the alloy becomes asemiconductor. [20]) As seen in Figure 3.4, at 6.3 at-% Bi K,. is reduced and K decreasesmore rapidly with temperature compared to K, in pure Sb. At 12.5 at.-% Bi, a K oforder 0.1% has been observed with stroboscopic SR.[20] However, working with suchChapter 3. The Knight Shift 21concentrated alloys produced results which depended on the local Bi configuration.Alloying with heterovalent Sn decreases the number of conduction electrons whileincreasing the number of holes.[21J For Sn concentrations less than 0.1 at.-% a largerK1. than in in pure Sb was seen, whereas for Sn concentrations greater than 0.1 at.-%K1. was dramatically reduced. Since the electron pockets at the Fermi surface do notvanish until a concentration of 0.63 at-% Sn, it seems a more subtle effect is at work.It was speculated, that the anomalous K1. in Sb involves some feature of the densityof states and that the temperature dependence was due to thermal broadening of thisfeature.At temperatures below 50 K the observed K1. appears to be due to muons, whichare isolated from the Sn impurities, whereas at higher temperatures a second signalattributed to a muon trapped at a Sn impurity appears. Zero field (ZF) tSR measurements on pure Sb confirmed, that the hop rate was relatively low at temperaturesbelow 50 K and that the muon hop rate rises very rapidly at higher temperatures.Although this conclusion is probably still valid, it may have to be reinterpreted in lightof results in this thesis. (See Chapter 6.) The high sensitivity of K1. to alloying withBi and Sn suggests that K1. is quite sensitive to the Fermi level.3.2.5 De Haas-van Aiphen Oscillations in Sn”9 and Cd” Knight ShiftThe effects of Landau quantization of electrons were searched for in NMR studies ofboth Sn1’9 and cadmium (Cd”).[25,30] Reproducible dHvA oscillations of the Knightshift in Sn were not observed and an upper limit on their amplitude A was established:A/H=1 x i0, where H is the magnetic field. In Cd the temperature dependence ofthe Knight shift and the dHvA oscillations of the Knight shift were clearly observed,as shown in Figure 3.5. In order to explain these effects, details of the Fermi surface,the scattering mechanisms, and the temperature dependence of the band structure dueChapter 3. The Knight Shift 22Figure 3.4: K14 as a function of temperature in pure Sb (solid lines) and in a 6.3 at.-%Bi alloy (dashed lines) with ê parallel and perpendicular to the applied field of 4 kG.[20j0.0 0 20 30 40 50 60 80 90 100 120T (K)70Chapter 3. The Knight Shift 230353--0.351—-—0.349— H 19724G H 9569I I I I-0.508 05100.336 HI6025G HI5873G -I I t I I0.625 0 627 0 629(I_4G )Figure 3.5: De Haas-van Aiphen oscillations in Cd.[25]to the lattice potential must all be understood. Since the amplitude and period of thedHvA oscillations strongly depend on the shape of the Fermi surface, single crystalsamples were used.A more general form of Equation 3.8 is the following sum over single particle states:K 16t IUk,fl(O)26(EF — E(k)), (3.11)k,nwhere n is the band index. [25] As seen in Appendix A Landau quantization of theelectronic states at sufficiently high magnetic fields and low temperatures gives rise toan oscillatory magnetization and thus an oscillatory Knight shift. The Knight shiftmay be written as the sum of an oscillatory K and a non-oscillatory component K:[25}K + K T Uk(O)I2)EF + uk(O)2)Q EF, (3.12)Chapter 3. The Knight Shift 24where (Iuk(O)2)o EF is the wave function averaged over a certain portion of the Fermisurface. The ratio of oscillatory to non-oscillatory Knight shift is given by:K = (Uk(O)I2)Q EF (3 13)K X (IUk(O)12)EFand for Cd1.5!, (3.14)where the Knight shift is measured by NMR and the susceptibility by a force magnetometer.Chapter 4The Kondo Effect4.1 IntroductionThe appearance of a resistivity minimum at a temperature on the order of 10 K inmetals with small concentrations of magnetic impurities was first observed in the 1930s.Prior to 1964 only the leading term of perturbation theory for impurity scattering hadbeen examined. [31] In this analysis the behaviour of a magnetic impurity was found tobe no different from that of a nonmagnetic impurity. In 1964 this resistivity minimumwas explained by Kondo’s analysis of higher order terms of perturbation theory, whichshowed some unexpected behaviour arising from impurities with magnetic moments.[32]The origin of the Kondo effect is the exchange interaction of the conduction electronsspins with the local moment spin S of the impurity ion:(4.1)where J is the exchange coupling parameter and is typically about 0.1 eV.[33] Thisinteraction leads to spin exchange scattering, in which the spins of the bound electronsand the conduction electrons near the Fermi energy flip flop. The Kondo Effect arisesfrom high order terms in perturbation theory, which yield a magnetic scattering crosssection, which increases as the temperature decreases. This process is characterized bya so-called Kondo temperature:TK TFe_hhIJIM) (4.2)25Chapter 4. The Kondo Effect 26where TF is the Fermi temperature and N(O) is the density of states at the Fermilevel.[33] The combination of the phonon contribution to resistivity, which decreasesas temperature decreases, and the spin flip scattering terms yield a resistance minimum around TK. However, since this exchange model assumes the existence of a localmagnetic moment, the model does not explain the stability of the magnetic momentin the presence of conduction electrons. Although the problem of a single local moment in a metal crystal seems very simple at first glance, an exact solution of the modelHamiltonian involving Equation 4.1 has eluded researchers until recently.[9,1O,11] SinceKondo’s perturbation analysis of the exchange Hamiltonian, hundreds of papers andnumerous review articles have been published on a wide variety of Kondo phenomenaarising from flev.[6,7,8]In dilute solid solutions of a magnetic ion in a nonmagnetic metal crystal the magnetic ion can be dramatically altered by the metal. If the partially occupied ioniclevel responsible for the magnetic moment lies slightly below the Fermi energy, thenconduction electrons could fill some ionic levels and thus the magnetic properties ofthe impurity ion will change. Also since the ionic energy level in question lies in thecontinuum of conduction electron levels, there is a strong mixing of localized ionicand conduction electron levels. The ionic levels become partly delocalized and theconduction band wave functions are altered near the ion.[31] The nature of this exchange interaction between the impurity and the host electrons and its consequenceson the bulk effects, such as the susceptibility, have been investigated both experimentally and theoretically. At high temperatures (T>> TK) the Kondo impurity displaysCurie-like behaviour, whereas at low temperatures (T << TK) Pauli paramagnetismis evident.[7,9,1O,34,35,36] The physical interpretation of these phenomena is that athigh temperatures the moment behaves as though it were isolated, since the conductionelectrons produce no fundamental change in magnetic properties. On the other handChapter 4. The Kondo Effect 27at low temperatures the influence of the conduction electrons is profound, since themagnetic behaviour is determined by the electrons and the moment.4.2 Dilute AlloysSo far NMR experiments to study the Kondo effect have had to take into account thepossibility of interactions between impurities, which are always present even at lowimpurity concentrations. The long range of the RKKY interaction, which is the oscillatory polarization of the conduction electrons by dilute local moments that decreaseswith distance as 1/,’, makes the study of isolated Kondo impurities difficult. pSR experiments are ideal in this respect, since only one p+ is in the sample at a time andso there can be no interactions between muons. In order to avoid impurity-impurityinteraction, it has been found empirically, that the following relation must hold:T0/TF > c, (4.3)where T is the Curie-Weiss temperature, Tp is the Fermi temperature, and c is theimpurity concentration. [6] For many alloys even p.p.m. concentrations are too high.The RKKY interaction arises from a first order perturbation estimate involvingthe exchange Hamiltonian of two impurities and leads to an oscillatory polarization ofthe conduction electron spin density proportional to cos(2kp’r)/r3.The Kondo effectarises from higher order perturbation terms. This strong RKKY interaction is whatmakes the Kondo effect for an isolated impurity so difficult to detect in all but the mostdilute alloys. RKKY interactions must be avoided for experimental measurements torepresent accurately the behaviour of an isolated impurity.Chapter 4. The Kondo Effect 284.3 Some ModelsIn this section we briefly review a few models of an isolated Kondo impurity. Theirresults for the bulk magnetic susceptibility will be compared with experimental resultson K, in Sb, which is thought to be proportional to the magnetic susceptibility.4.3.1 Moment Formation and the Anderson ModelThe development of the theory of dilute magnetic alloys evolved from two seeminglyseparate investigations: one on the interactions of local moments and conduction electrons and the other on the formation of a local moment in a metal. The former topicis addressed by the exchange Hamiltonian, Equation 4.1, and the latter by the morefundamental Anderson Hamiltonian.The Anderson model was developed in 1961, before Kondo’s analysis of the higherorder perturbation terms, to describe the scattering of conduction electrons by magneticiron-group ions dissolved in nonmagnetic metals:[37]= ?Iof + ?loci + 7Icorr + ?Iex. (4.4)The unperturbed free-electron Hamiltonian is:‘Hof = Ekflk,0-, (4.5)where ek is the free electron energy for a state with momentum k and is the numberoperator for electrons with momentum k and spin a. The energy of the unperturbed dstate of the impurity atom is given by:‘Hod = Ed(nd+ + rid_), (4.6)where Ed is the unperturbed d state energy level and d+ is the number operator forelectrons with spin up. The eigenfunctions for the localized impurity d state are theChapter 4. The Kondo Effect 29EEd+UEF—Ed_____Edg(E)Figure 4.1: The position of the virtual bound states in the Anderson model. U is theCoulomb repulsion energy, Li is the width of the virtual d state, EF is the Fermi energy,and Ed is the energy of the d state.[38]Chapter 4. The Kondo Effect 301.002.07v/Y=7vA/UFigure 4.2: Phase diagram of magnetic and nonmagnetic behaviour for the AndersonModel. U is the Coulomb repulsion energy, L is the width of the virtual d state, EF isthe Fermi energy, and Ed is the energy the unperturbed d state. When U/ is large,localization is easier.[7] In the following two cases localization is impossible. When(EF— Ed)/U is less than zero, the d state is empty. When (EF — Ed)/U is one, thelevel is doubly occupied. [37]0 1.0Chapter 4. TheKondoEffect 31wavefunctions . The Coulomb repulsion between these d states is described by:7tcorr = Und+nd_, (4.7)whereU= f Id(r1)I2k ( 2cr — i-’ dr1T2. (4.8)(ev is the exchange interaction, Equation 4.1. The Anderson model describes a highlysimplified quantum state of an impurity in a metal, in which a moment may or maynot exist. The essence of the problem is that the one electron bound states in metals,which are responsible for the local moment, almost certainly lie in the continuum ofunbound or free electron states because of the conduction band’s large width. Thesefree electron states cannot be localized completely and so the concept of a virtual levelof width L\ is used.[37] The d electrons of an impurity atom may retain much of theirlocal character, but the conduction electrons can broaden and shift the unperturbed denergy level giving rise to a virtual bound state. This d state is a good eigenstate ofonly 7- and not of ?-t. Therefore the d state conduction electron in this virtual boundstate eventually returns to a conduction band state of similar energy. [39]When both the Coulomb repulsion energy and the distance of the d state belowthe Fermi level are much greater than , the Anderson Hamiltonian reduces to theexchange Hamiltonian.[7,9,37] In other words the addition of one electron to the d stateoccurs at energy Ed, as shown in Figure 4.1. The addition of another electron will beat the energy Ed + U due to Coulomb repulsion. If the energy difference U exceeds thewidth of the virtual level and (Ed + U) > EF, then the d state is singly occupied andthe impurity possesses a magnetic moment.[9] However, when the Coulomb repulsionis small compared to either the width of the virtual level or the distance (EF — Ed) ofthe d state below the Fermi level, the population of the impurity level is the same forboth spin states and no magnetic moment exists.Chapter 4. The Kondo Effect 32Assuming a local moment exists means the d state of the impurity with spin up isfull or partially filled and with spin down is empty. When there is equal occupationof spin up and spin down states, the local moment disappears. When the AndersonHamiltonian is analyzed under the Hartree-Fock approximation, it yields a phase diagram, which is sharply split into regions of magnetic and nonmagnetic behaviour, asseen in Figure 4.2. Although this sharp transition is unphysical for a local system,the phase diagram does provide some insight into the properties of dilute alloys basedupon the location of the ci state relative to the Fermi level and the ratio of the Coulombrepulsion of the d electrons to the width of the virtual state.[7,37] When U/ is large,localization of the d state in a level below EF is easier because the virtual state has alonger lifetime (.-h/LS. In other words when U is large compared to L, screening isineffective and the magnetic moment persists.[7] In the following two cases the d statehas no local moment. When (EF — Ed)/U is less than zero, the d state is above theFermi level and the d state is empty. When (EF — Ed)/U is greater than one, the levelis doubly occupied. [37]4.3.2 The Spin-Fluctuation Model vs. the Spin-Compensation Model forthe Kondo EffectTwo seemingly contradictory explanations of the Kondo effect have emerged: the localmoment spin flip model, as originally proposed by Kondo, and the spin fluctuationmodel.[8,33,34,40] The dual explanation originates from the two ways one can lookat the typical magnetic properties of dilute alloys (enhanced Pauli paramagnetism atlow temperatures and Curie-Weiss behaviour at high temperatures): one way is todetermine how surrounding electrons interact with a well-defined spin to create a nonmagnetic system at low temperatures and the other way is to find how a nonmagneticalloy with spin fluctuations diplays magnetic properties at high temperatures. [34]Chapter 4. The Kondo Effect 33In the spin-compensation picture the impurity electron is surrounded by an increasing number of opposite spin host electrons, as the temperature decreases. The spincloud compensates the impurity spin and the paramagnetism is degraded. This spincompensation can be destroyed with sufficiently high temperatures or applied fields.The binding energy of this quasibound spin cloud to the impurity is of the order ofkBTK. The details of the smooth transition from the spin correlated to the magneticstate, which occurs around TK, are unknown. More information about the low-lyingexcited states of the Kondo system would be needed. [33]The spin-fluctuation picture assumes a nonmagnetic state: a single virtual level inthe Anderson model at Ed. However such an impurity, which is nonmagnetic in theHartree-Fock approximation of the Anderson Model, does in fact exhibit magnetic behaviour when localized spin fluctuations are taken into account. A conduction electroncan occupy the virtual level at Ed (described by the Anderson Model) of width L fora time proportional to h//I The electron in question retains memory of its spin for alonger time than h//i because of the Coulomb repulsion U between d state electronsof opposite spin. This so called memory time is the lifetime Tsf of the spin fluctuation.This lifetime is in fact infinite and a static local moment appears, if U is large comparedto Li. The local moment is a dynamic effect of the localized spin fluctuations. [40]When the localized spin fluctuation rate is much greater than the thermal fluctuation rate, the impurity spin amplitude averages to zero. The impurity becomesnonmagnetic and shows a Pauli susceptibility. The slow transition between magneticand nonmagnetic behaviour of a dilute alloy occurs at temperatures on the order ofthe spin fluctuation characteristic temperature T3f = h/kBr8f.When the localized spinfluctuation rate is less than the thermal fluctuation rate, then the local susceptibility decreases and shows a Curie-Weiss dependence. The Curie-Weiss temperature isproportional toT3f.[33,40]Chapter 4. The Kondo Effect 34Although the spin-compensation and the spin-fluctuation models appear very different mathematically, it is not clear whether the spin-compensated and the spin-fluctuation states apply to physically distinct systems. A superficial link between thesetwo models can be made by equating T3f and TK.[33]4.3.3 The Resonant-Level ModelAs previously mentioned the magnetic behaviour of the dilute alloy is either a welldefined spin at high temperatures which is compensated by the host spin cloud at lowtemperatures or a nonmagnetic spin fluctuation at low temperatures which becomes alocal moment at high temperatures. These two models can be shown to be equivalent(T8f = TK) in a special case (U = 0 and Ed = 0) of the Anderson model, the resonant-level (RL) model.[35] The RL Hamiltonian is:(4.9)The authors of Reference [35] compared the thermodynamics of the RL model to thecase of a one dimensional classic Coulomb gas; positive and negative charge is analogousto spin up flips and spin down flips. There is a resonance level at the Fermi energywith a width , which is roughly kBTK. The static susceptibility obtained from thismodel[35] at low temperatures (T << TK) is:X+bT2. (4.10)In a later paper the authors of Reference [35] provide another expression for the lowtemperature susceptibility: [36]2S I 2ir /kT’\2”\x = (gi)- 3(2S +1) k-K)) (4.11)Chapter4. The Kondo Effect 35This model[36] predicts behaviour of Curie-Weiss form at high temperature (T>> TK):2S(S+1) 1x=(gIB). ,,(4.12)01GB I +Uwhere is related to /. by:k8 = 0 233S + 3S + 3/2 (4.13)s+1The authors of References [35,36] remark that, although the RL model describes experimental behaviour at a finite field for an arbitrary spin at most temperatures, themodel is not exact. Rather its usefulness lies with its simplicity.4.3.4 Exact Solutions of the Kondo SusceptibilityBoth the Kondo and Anderson models possess integrable Hamiltonians, which can besolved exactly by the Bethe-Ansatz technique.[9] Numerical results[1O] for the Kondomodel obtained by the Bethe-Ansatz technique yielded the magnetic susceptibility fora spin 1/2 impurity in weak applied magnetic fields and at low temperatures (,UBH <<kBT << kBTK):1 T2x =____— O.433—-. (4.14)2TrTK TKThe full temperature dependence of the zero field susceptibility for spin 1/2, 1, and3/2 for values of T/TK from about 0.001 to 735 is tabulated in Reference [9].Numerical results[11] for the infinite- U Anderson model obtained by the BetheAnsatz technique yielded the exact magnetic susceptibility for a Kondo impurity, asseen in Figure 4.3. However, no closed form expression was obtained.4.3.5 The Korringa RelationAn impurity nucleus will experience relaxation due to field inhomogeneity arising fromthe interaction of the nuclear and the conduction electron spins. (This concept isChapter 4. The Kondo Effect 36I IIIIII I I 111119 I I 111111$ I I 111111to —- GIN t112JI 1111111 I I 1111111 1 1111111 1111110 1 10 100 1000(2j + 1 )T/T0Figure 4.3: Exact result for Kondo susceptibility, where N=2 is the 2-degenerate level(i.e. spin up and spin down), which corresponds to a Kondo impurity.Chapter4. The Kondo Effect 37discussed in Sections 5.2 to 5.6 and in Section 6.3.10.) By examining the local spinsusceptibility the Korringa relation(K2T1=constant) relating the spin relaxation timeT1 and the Knight shift K was shown to be valid for the Anderson model at lowtemperatures (T << TK) and low fields (gBB << kBTK).[41,42] For U/Li 0.5 therelaxation rate T1 has a peak around T TK.[42]Chapter 5SRProbing physical and chemical properties of matter with positive muons is a technique,whose versatility is reflected in the acronym ,uSR, which stands for Muon Spin Rotation/Relaxation/Resonance. Detailed discussions of iSR are found in References [2,46].This chapter discusses transverse field (TF) and longitudinal field (LF) tSR, which areused to study K1 and T’ in Sb respectively.5.1 Muons and Muon BeamsPositive muons are obtained from the decay of positive pions:(5.1)which have an average lifetime (Tn) of 26 ns. The positive pions are produced atmedium energy meson factories like TRIUMF by colliding intense proton beams ofenergy 500-800 MeV with suitable production targets (usually carbon or beryllium).The elementary pion production reactions are:p+p—7r+p+n (5.2)p+n+n+n (5.3)Low energy pions, which stop at or near the surface of the production target, are usedto produce spin polarized muon beams. Since the pion decay is a two body problemand the muon neutrino spin is antiparallel to its momentum, the muon spin must also38Chapter 5. SR 39be antiparallel to its momentum in order to conserve angular momentum. In the restframe of the pion the muon is emitted with a momentum of 29.7894 MeV/c. If muonsemitted within a small element of solid angle are used to make a beam, then this socalled “surface beam” will be very nearly 100% spin polarized.The muon is a spin 1/2 lepton with a mass of 105.6595 MeV/c2 or 0.1126096 m,where m is the proton mass. The mean life of the muon (rn) is 2.19714(7) R5 in freespace. The positive muon decays into a high energy positron, an electron neutrino, anda muon antineutrino:e + “e +i• (5.4)In this three body decay the emitted positron energy varies continuously from zero (ifthe two neutrinos travel antiparallel and carry away all the kinetic energy) to some Emax(if the two neutrinos travel together and antiparallel to the positron). After integratingover the neutrino momenta, the probability per unit time for a positron emission at anangle 6 to the spin is:dW(e, 6) = [1 + a(e)cos(6)]n(e) de dcos(6), (5.5)TIAwhere a(e) = (2c— 1)/(3 — 2e) and n(e) = 2E(3 — 2c). The asymmetry parameter a is1 for maximum positron energy and when averaged over all energies. The reducedpositron energy is e = E/Emaz, and the maximum e+ energy is Emax = 52.8 MeV orabout half the muon rest mass energy. The radial plot of the probability Win Figure 5.1is given for energies between the maximum e+ energy e = 1 and e = 0.5. The averagee energy, , equals 0.682. The anisotropic distribution of decay positrons is the key to[tSR. The asymmetry arises from parity violating terms of the weak decay Hamiltonian.Positron contamination of the beam is effectively eliminated by a Wien filter consisting of crossed electric and magnetic fields, which are both perpendicular to the muonbeam. In addition the Wien filter acts to rotate the muon spin away from the directionChapter 5. 1SR 40I +/7//i’ll//If/Ill / \\\\\\\\\\\\\\ \ +I / - \\\\\\\\\\\ \ \ \ e26 MeV 52 MeVI 0muon spinFigure 5.1: Decay positron emission probability for angle 6 with respect to the spinfor reduced energies from e = 0.5 (26 MeV) to 1.0 (52 MeV).Chapter 5. ,uSR 41of the muon momentum due to precession in the magnetic field. With this spin rotatorthe spin can be aligned either parallel to the applied magnetic field at the targetfor LF-1tSR or perpendicular for TF-SR. The beam passes through a thin vacuumwindow in the beam snout, before entering the experimental area. At the TRIUMFM20 beamline the surface beam has a maximum intensity at a muon momentum of 28MeV/c which corresponds to a range of about 120 mg cm2 in carbon. Just beforeentering the cryostat the muon beam is collimated to a spot size of about 1 cm, whichis smaller than the typical sample.5.2 TF-1tSR Geometry and TechniqueIn TF—1SR the muon spin polarization is perpendicular to the applied field, as shownin Figure 5.2. In high magnetic fields it is necessary, that the muon momentum isalong the field direction in order to avoid deflection of the beam. The superconductingsolenoid HELlOS used in this experiment is capable of magnetic fields up to about 70kG, but normally fields above 20 kG are not used because conventional scintillationcounters have a timing resolution of about 2 ns, which limits the frequency to less thanabout 300 MHz. At higher frequencies there is a precipitous drop in the precessionamplitude. Also at 20 kG the beam luminosity at the sample position is maximum dueto the behaviour of the beam in a large solenoidal field. [47]The precession of the muon in the applied field is detected by measuring the timeevolution of the anisotropic positron decay distribution. Each detector in Figure 5.2consists of a plastic scintillator connected to a photomultiplier tube (PMT) via a lucitelight guide. When a charged particle such as a or a e+ passes through and depositssome energy in the scintillator, it emits a light pulse, which travels down the light guideto the PMT. The light guide is needed because the PMTs, which do not function inChapter 5. SR 42Figure 5.2: TF-tSR experimental geometry. A schematic showing four positron counters (Up, Down, Left, Right), the thin muon (TM) counter, and the initial muonpolarization (Pu), which is perpendicular to both the beam momentum (pa) and theapplied field (B).BChapter 5. 1.tSR 43a strong magnetic field, must be placed well outside the magnet. The PMT convertsthe light pulse into an electronic signal, which is carried via coaxial cable to the fastelectronics for processing. The thin muon (TM) counter between the beam snoutand the sample detects incoming muons and, with much less efficiency, high energypositrons and gamma rays. In order to minimize multiple scattering of the incomingmuons, which will cause the muons to miss the sample and add to the backgroundsignal, the muon counter is about 250 ,um thick. In TF-uSR the four positron counterssurround the target, as shown in Figure 5.2. They detect the decay positrons frommuons implanted in the sample and are labelled, as if you were riding the muon: Up(U), Down (D), Left (L), and Right (R). With these counters the anisotropic decaypositrons are detected.The beam rate is reduced so that only one muon at a time is in the sample duringthe typical observation period of 10 s. The signal from the TM counter starts a clockand defines time zero. A signal from either the U,D,L, or R positron counter stopsthe clock and then the time bin in the appropriate decay positron counter histogramis incremented by one event. A good event is one, in which there is precisely one muonand one positron in the 10 1us data gate. Otherwise the event is discarded becausethe decay positron cannot be unambiguously related to the parent muon. Typicallyabout iO events are recorded. The resulting time histograms may be regarded as theensemble average of such decays.The muon spin precesses in an internal magnetic field consisting of the applied fieldplus local fields. Consequently the anisotropic positron decay distribution also precessesat a frequency = yB1 where ‘y = 2ir x 0.01355342 MHz/G. Inhomogeneitiesor fluctuations in the internal field will result in dephasing or spin relaxation of theprecession amplitude. Typically a gaussian relaxation function is used todescribe the dephasing due to inhomogeneity from nuclear dipolar fields or the magnet.Chapter 5. ,uSR 44FTMFigure 5.3: LF-tSR geometry showing forward (F) and backward (B) counters, thethin muon (TM) counter, the polarization (Pu) and the beam momentum (pp).Both P/A and p/A are parallel to the applied field direction.5.3 LF-SR Geometry and TechniqueIn longitudinal field muon spin relaxation (LF-/USR) the muon polarization is parallelto the applied field. In the absence of spin relaxation the implanted would preserveits initial polarization because its spin would be locked along the applied field direction.However, dynamic processes in the sample, such as fluctuating local moments, createmagnetic field fluctuations at the site causing muon spin relaxation. The decaypositron asymmetry is measured using backward and forward counters, as shown inFigure 5.3. Typically the B counter has a hole in the middle in order to avoid the pathof the muon beam and to accommodate cryostats or other geometrical constraints.Chapter 5. uSR 455.4 ExperimentalThe actual experimental apparatus is shown in Figure 5.4. In this experiment a singlecrystal of 99.9999% Sb was studied. The single crystal Sb sample, shown in Figure 6.2,was oriented with the ê axis parallel to the applied field. A thin strip of high purity99.9% Ag was mounted on top of the Sb, in order to provide a reference + precessionfrequency. Ag was chosen because it has few nuclear moments to cause broadeningand the Knight shift (K80=94.0(3.5) p.p.m.) is known to be small and very weaklytemperature dependent. [2,23]The Sb sample and the Ag reference were mounted inside a helium gas flow cryostatshown in Figure 5.4. The temperature was monitored with two carbon glass resistorson the helium diffuser and on the sample holder. The fluctuation in the temperaturewas less than 0.1 K and the accuracy is estimated to be better than 0.5 K. De Haas-vanAlphen oscillations require high magnetic fields and low temperatures, as is explainedin Appendix A. Consequently in this part of the experiment the magnet was variedin the maximum range possible to avoid severe loss of precession amplitude and thecryostat was operated close to its base temperature of 3 K.The four positron counters are arranged cylindrically because of geometrical restrictions imposed by the magnet’s cylindrical bore. The counters are labelled Forward TopRight (FTR), Forward Top Left (FTL), Forward Bottom Right (FBR), and ForwardBottom Left (FBL). In the positions shown in Figure 5.4 these four counters functionas U,D,L, and R described earlier during the discussion of TF-SR geometry. “Forward” is specified for these counters because for LF-jtSR these counters are logicallyconnected (“or”ed) using coincidence units to form a single forward counter. Foursimilar counters upstream of the target are “or”ed to form a backward telescope.Chapter 5. uSR 46Sample RodHe SpaceVacuumCryostatTM Counter+/.L spin,ubeamSb SampleBeamVacuumHELlOSMagnetBore0150mmApprox. Scale:0 75 150mmFT RCounterFBRCounterFigure 5.4: A cutaway showing the central portion of TF-SR experimental apparatus.Chapter 5. 1uSR 475.5 TF-1tSR DataA typical histogram from one of the positron counters, such as FTR, displays theoscillatory pattern of the precession in a transverse field of 100G superimposed onthe exponential decay of the muon, as seen in Figure 5.5a):HFTR(t) = N(0)e_t’T[l + aP(t)] + b, (5.6)where a is the asymmetry parameter defined in Section 5.1, P(t) describes the timeevolution of the x component of the spin polarization, and b is the random background from uncorrelated +e+ events. Asymmetry histograms, such as the one inFigure 5.5b), are formed after removing the random background b. If the muon decayed isotropically, then each decay positron histogram would be identical at all timeswith no asymmetry except for trivial normalization effects. However due to muon decayasymmetry histograms corresponding to detectors, which are 180 degrees out of phase,will display a time dependent asymmetry due to the precession of the anisotropic decayof the muon. The asymmetry of two matched counters is the main observable quantityin TF-tSR:A(t) — HFTR(t) — HFBL(t) (5 7)HFTR(t)--HFBL(t)’where the only remnants of the muon’s decay are the error bars, which are proportionalto the square root of the number of events in the time bin and which grow exponentiallywith time.In TF-SR with an applied field B = B, which is perpendicular to the axis joiningthe two counters, the asymmetry as seen in Figure 5.5b) is given by:A(t) = A0P(t) =A0G(t)cos(wt + ), (5.8)where g is the phase. The amplitude of precession, A0, is about 25% because ofthe counters’ solid angle, positron absorption, and muon decay kinematics. As notedChapter 5. 1iSR 48a) 800b)E1______________________Time (,us)2.0 I Ic)L 1.5001.0ci)S.00.5I I0 1.0 2.0 3.0 4.0Frequency (MHz)Figure 5.5: a) Decay positron time spectrum showing muon precession and exponentialdecay. b) Asymmetry histogram showing muon precession. c) Fast Fourier Transform.Chapter 5. 1uSR 49previously the maximum observable frequency is limited by the timing resolution of thescintillation counters. The resolution of a matched pair of counters can be estimatedby looking at the full width half maximum of a time histogram of pulses, which aregenerated simultaneously in each counter with back to back y rays. Typical timingresolution is on the order of 2 ns. A gaussian distribution of internal fields leads to agaussian relaxation function:G(t) =e22. (5.9)Static nuclear dipolar fields are one source of such line broadening effectsOften the signal is analyzed online with fast Fourier transforms in order to obtainan initial estimate of the precession frequency, as shown in Figure 5.5c). In order todetermine more accurately the experimental parameters (asymmetry, phase, Gaussianrelaxation rate, and precession frequency), x2 minimization is used to fit the timehistograms. If the muon precession frequency is very high the data are often analysedby first transforming to a reference frame rotating at a frequency slightly less than w.In this way the data can be packed heavily thereby increasing the speed of the fittingprocedure.5.6 LF-tSR DataIn this type of measurement the muon polarization seen in the forward and backwardtelescopes is given by:F(t) = N(O)e_t’T[1 + aP(t)] + b (5.10)B(t) = N(0)ethfT1[1— aP(t)] + b, (5.11)where the asymmetry parameter a is defined in Section 5.1, P(t) describes the timeevolution of the component of the muon spin polarization along the field direction, andChapter 5. ,uSR 500.030.02—‘ 001N0— 00—0.01—0.028Time (ts)Figure 5.6: An example of LF-pSR asymmetry histogram showing an exponentiallyrelaxing muon spin polarization.b is the random background. After removing the background terms b the time evolutionof the asymmetry is found, as shown in Figure 5.6:aP(t) = (5.12)In general relaxation of the muon polarization will occur as a result of the interactionbetween the spin and other magnetic moments in the material. Typically thisrelaxation can be fitted with an exponential function. LF-SR is particularly sensitiveto the temporal fluctuations of these moments. Often the temperature has a dramaticinfluence on the longitudinal relaxation rate T1.0 2 4 6Chapter 6Experimental and ResultsMost of the data for this thesis were obtained through TF and LF-SR experiments.De Haas-van Aiphen oscillations of K1. were sought for, but not found. The temperature dependence of 1(1. was examined from 2-20 K. Previously published data[20J from90-180 K were combined with our low temperature data. A crossover from a weaktemperature dependence from 2-6 K to a Curie-like behaviour from 90-180 K is seen.With LF-1tSR the Tj1 relaxation rate showed only a weak dependence on field, but theKorringa constant(S/TiTI0)showed an anomalous temperature dependence. Twoadditional experiments were performed on the same Sb crystal: Shubnikov-de Haasoscillations of the conductivity were examined to determine if there was a strong orientation dependence. Finally de Haas-van Alphen oscillations of the magnetization,which were expected to be proportional to K1., were measured, in order to comparewith measurements of K1..6.1 Shubnikov-de Haas OscillationsA steady state NMR spectrometer was used and the applied field was scanned from 18to 20 kG and the Sb sample was held at a temperature of about 3 K. The ê axis ofthe Sb single crystal was aligned at angles from -11 to +20 degrees from the appliedfield. This experiment confirmed the presence of Shubnikov-de Haas oscillations inthe conductivity of Sb. In addition to the main field a small modulation field wasused. Using phase-sensitive detection techniques the first derivative of the conductivity51Chapter 6. Experimental and Results 52oscillations was observed by measuring the Q of the pickup coil, which is a function ofthe surface impedance of the sample in the coil. It was found that the orientation ofthe sample does not strongly affect the amplitude of the oscillations over the range ofangles examined. Any misalignment of the sample in the pSR experiments would notexceed more than one or two degrees. Thus before the present SR study was done, itwas known through these measurements, that small misalignments of the Sb sample’sê axis away from its parallel orientation to the applied field would not strongly affecteither the period or the amplitude of the oscillations.6.2 De Haas-van Aiphen Oscillations of the Magnetization in SbIn a force magnetometer at a temperature of 2 K strong susceptibility oscillations, asseen in Figure 6.1, were detected in a 0.5 mm thick triangular piece of the Sb crystalwith the sides measuring 3 mm, 3 mm, and 5mm. The peak-to-peak amplitude of theseoscillations was about 15% of the total magnetization. The period was 353(9) G in Bor about (9.8+0.2) x i0 G1 in 1/B, which is close to the expected dHvA period inTable 2.2.Although dHvA oscillations are oscillations in the diamagnetic susceptibility, thedensity of states at the Fermi surface also oscillates and therefore the paramagneticspin susceptibility shows the same variation with magnetic field. It is this effect whichproduced the oscillations in the Cd Knight shift discussed in Section 3.2.5. If electronstates at the Fermi surface are responsible for the large K in Sb, then one might expectto see K osillations of similar amplitude and periodicity in the 1tSR study.Chapter 6. Experimental and Results 532.75+ +2.7 ++o ++ ++2.65 +++—‘__________________________________ZcO ++ +o LU + + +2.55 + + +25 +I— E + + + +E-++••++i++++:+++++18.000 18.400 18.800 19.200 19.600 20.00018.200 18.600 19.000 19.400 19.800FIELD (GAUSS)(Thousands)Figure 6.1: De Haas-van Aiphen oscillations of Sb magnetic susceptibility at a temperature of 2 K showing a period of 353(9) G and a peak-to-peak amplitude of 15% of themagnetization.Chapter 6. Experimental and Results 54cm C-CXIS0.75 cm____U_(2cm 0.5cmFigure 6.2: Cross-section of the Sb single crystal sample.6.3 SR in Sb6.3.1 Sample CharacteristicsIn all the pSR experiments in this thesis a piece of the same single crystal of 99.9999%Sb used in the magnetization experiment was studied. The Sb sample is sketched inFigure 6.2. For this thesis the ê axis crystal was always oriented parallel to the appliedmagnetic field. A thin strip of high purity 99.9% Ag from Aldrich was mounted on topof the Sb, in order to provide a reference precession frequency, which is unaffectedby nuclear moments or large Knight shifts.The experiment was performed on the M20 beamline at TRIUMF. The surfacemuon beam has a maximum intensity at a muon momentum of 28 MeV/c. Withsurface beams, targets with masses per square centimeter as low as 10 mg cm2 can bestudied.[2] Our 0.5 cm thick antimony (pSb=6.691 g cm3) sample was clearly thickenough to stop all the muons.6.3.2 Measurement of Muon Knight Shift with TF-1iSRIn these TF-.tSR experiments, clear identification of the precession frequency inboth the Sb sample and the Ag reference, a thin strip of 99.9% pure Ag mounted onChapter 6. Experimental and Results 55top of the Sb sample, was sought. Since K in Sb is so large, the Ag reference frequencyand the Knight shifted Sb frequency were easily resolved in fast Fourier transforms, asseen in Figure 6.3. Ag was chosen as a reference due to its small muon Knight shift,K=94.0(3.5) p.p.m., and the lack of nuclear dipolar relaxation. [2]About ten million events per asymmetry histogram were recorded. However strongrelaxation in the Sb signal, as seen in Figure 6.4, limited our resolution of the precession frequency to about 0.001 MHz, which ultimately limited the precision of ourobservations of the muon Knight shift in Sb to about 0.1% of K. The relaxation inthe Sb is attributed primarily to nuclear dipolar interactions. The field dependence ofT’ in Figure 6.4 may be due to magnetic field inhomogeneity. Relaxation due to themagnetic field’s spatial inhomogeneity would show a linear increase with field. However, in high solenoidal fields the muon beam spot size on the sample decreases andso the spatial inhomogeneity becomes less important.[47] This could explain the shapeof the curves in Figure 6.4. The stronger field dependence in Sb likely arises from thelarger size of the Sb compared with Ag. The relaxation in the Ag at low fields may bedue to trapping at impurity sites.The decay e+ asymmetry histograms, such as the one in Figure 6.3, were analyzedwith fast Fourier transforms to estimate the frequencies and then fitted in time byx2 minimization, in order to obtain accurate values for the experimental parameters:asymmetries (a), phases (), Gaussian relaxation rates (T’), and + precession frequencies (w) in both the Ag reference and the Sb sample. The time evolution of themuon spin polarization was assumed to be of the following form:P(t) =aA9Re{exp[_(t/Tj)2— Q’Agt + bAg)] } + asbRe{exp[—(t/Tj)2— i(wSbt + bsb)] }.(6.1)As explained previously the signal was fitted in a reference frame rotating about 0.5Chapter 6. Experimental and Results 56MHz less than the Ag reference precession frequency. The muon Knight shift relativeto Ag was then calculated:K — fsb — Hsb — HAg —— —JAg AgK— 206.699(1) MHz1 — 0 01393 1 6 3— 203.859(2) MHz — (At a temperature of 2.3 K and in a nominal field of 15 kG, K, is 0.01393(1) or 1.393(1)%.Typically the value of the statistical error was about 0.1% of the total K. This uncertainty of 0.1% is an improvement by a factor of about five over previous JLSR investigations. This is due to our increased statistics, the high magnetic field, which permitsclear distinction of the Ag reference frequency and the shifted Sb frequency, and thesimultaneous measurement of both precession frequencies.6.3.3 Magnetic Field Dependence of the Muon Knight ShiftAt a temperature of 2.7 K a preliminary investigation was conducted: a field scan fromabout 2.3 to 21.1 kG with B parallel to the ê axis of Sb, as seen in Figure 6.5. InFigure 6.6a) there was a hint of an oscillation in K with a period of about 0.5 kG inB or about 14 x iO G’ in 1/B, which is close to the expected dHvA period for Sbgiven in Table 2.2. However the observed variation in K was not much larger thanthe statistical error bars. Thus even from this preliminary study it is clear that theamplitude of any oscillations is less than or equal to about 0.2% of the total Knightshift. The magnetization oscillations had an amplitude about 40 times greater undersimilar conditions. Clearly K,, in Sb does not scale simply with the magnetization.Two additional scans were performed with higher statistics as seen in Figures 6.6b)and c). We concluded that the amplitude of dHvA oscillations is less than 0.1% of K.The frequency shift (Lf= fs,— f) increases linearly up to 21 kG, Figure 6.5a).Chapter 6. Experimental and Results 57Figure 6.3: a) Rotating reference frame fit to asymmetry histogram showing beatingof the Ag reference and the Sb sample precession frequencies. b) FFT showing+ precession frequency in Ag reference and Sb sample. The Sb frequency Knight shiftcan be clearly seen. Data from a run at T=2.2 K and B=18.05 kG.0.100x0o.10a)b)Time (ps)0xa)0a)0543210244 248Frequency252(MHz)Chapter 6. Experimental and Results 580.45Cl) 040.35ci)---J0c0.25C.2 0.2-H0x 0.150- 0.10.05csF— 00 2 4 6 8 10 12 14 16 18 20 22Magnetic Eied ‘(kG)Figure 6.4: TF-SR linewidth or Gaussian relaxation rate as a function of magneticfield at T=2.7 K.SbAgChapter 6. Experimental and Results 59I....%1 I I I I I I IEE______4 4.0 c):3.87-cz5, -)_c <(f) - 3.6 19.25 21.25>N Cl)(_) 4—L) 1C- -G)I I I I I I I1.42 I1.41 b)+Cf) 1.40•c 1.391..38 I I I I I I I0 4 8 12 16 20Magnetic Fied (kG)Figure 6.5: a) Frequency shift in Sb relative to Ag reference as a function of magneticfield and b) the Knight shift as a function of field at T=2.7 K.Chapter 6. Experimental and Results 601.395 a)1.3904-,9-II19.5 20.0 20.5 21.0Magnetic Field (kG)1.400 b)+1.395C’)1.390 .++ 4 4+ +++++18.0 18.4 18.8 19.2Magnetic Field (kG)1.398I iIfic)‘1.394 I II1.390 ‘f hi I1.38618.0 18.4 18.8 19.2Magnetic Field (kG)Figure 6.6: a) Muon Knight Shift: Sb BHê T=2.7K Preliminary Investigation b) MuonKnight Shift: Sb BIe Scan Up T=2.7K c) Muon Knight Shift: Sb BIe Scan DownT=2.7KChapter 6. Experimental and Results 61The Knight shift as a function of field is shown in Figure 6.5b). This frequency shiftLf indicates the extra field /.H experienced by the due to the interaction with theSb conduction electrons:H=K,H=Hhf, (6.4)where Hhf is the hyperfine field. [2]6.3.4 Temperature Dependence of the Muon Knight Shift below 20 KThe precise low temperature behaviour of K,, was investigated in a magnetic field of15 kG in the temperature range of 2 to 19 K, as seen in Figure 6.7. The data were fitin three separate ways. Figure 6.7a) shows the fit to an Arrhenius law of the form:= a(1 — be_’T), (6.5)The fitted parameters are a=O.0139(5), b=O.47(4), and E/kB=40(2) K. The x2 was 2.9for six degrees of freedom. In Figure 6.7b) we show a fit to Equation 4.14, an exactresult for the susceptibility of a Kondo impurity in weak fields and at temperaturesmuch less than the Kondo temperature:K = a(2rK— O.433) , (6.6)In this case there were only two fitted parameters, the constant a=O.1150(9) K andTK=131(1) K. The x2 was 9.1 for seven degrees of freedom. When the temperaturerange of the fit is from 2 to 10 K as seen in the inset of Figure 6.7b), a better fit with a x2of 1.4 for four degrees of freedom was obtained. The fitted parameters were a=14.0(0.3)K and TK=160(3) K. Using Equation 4.11, which is similar in form to Equation 6.6,we find L1/kB=144(1) K. Figure 6.7c) shows a fit to a polynomial function:K= a+bT2(6.7)Chapter 6. Experimental and Results 62The fitted parameters were a=7.151(3) x i0 and b=1.19(2) x 10—6 K2. The x2 was11 for seven degrees of freedom. Another polynomial function with one more parameterwas used to obtain a better fit, as shown in Figure 6.7d):= a + bT2 + cT4(6.8)The fitted parameters were a=7.167(3) x iO, b=6.4(7) x i0 K—2, and c=1.8(2) x1O K—4. The x2 was 1.5 for six degrees of freedom. A constant K1 at low temperatures, which is expected for a Kondo impurity, is reproduced best by a polynomialwith even powers.6.3.5 Muon Knight Shift at High TemperaturesUnfortunately the full temperature dependence has not yet been measured with thesame precision as shown in Figure 6.7. In the absence of such data we used previousdata (T=90 to 180 K) on this sample from Reference [21], in order to compare with theKondo model. It should be noted, that the data in Figure 6.9 (which are the squares inFigure 6.8) were taken in a lower magnetic field of 4 kG. The data were reanalyzed usingthe same procedure as in the present experiment to eliminate other possible systematiceffects.For a Kondo impurity we expect to exhibit Curie-Weiss behaviour for T > TK.Figure 6.9a) shows a fit of the high temperature data to:K, = a(T+&)1. (6.9)The fitted parameters were a=0.0170(2) K and &=-10(1) K. The x2 was 32 for fivedegrees of freedom. For comparison we show a fit to straight Curie law behaviour:= a/T, (6.10)Chapter 6. Experimental and Results 631.381.36\\1.341.32I I I I I I I I I I I I I I Icn______________ ______________-c0-i 1.38-N 1.361.341.391.32 1.38_ __ ___5 10I I I I I I I__ ____ __ ____ __ __ ____ __ ____ __0 4 8 12 16 20 0 4 8 12 16 20Temperature (K)Figure 6.7: Four fits to K1 data at temperatures below 20 K. The magnetic field wasnominally 15 kG. a) Arrhenius fit, Equation 6.5. b) Fit to exact result, Equation 6.6.c) Fit to polynomial, Equation 6.7. d) Fit to polynomial, Equation 6.8.Chapter 6. Experimental and Results 641.501.25-C0.75_c0r fECc U.DU a0.25 I 410 80 ‘120’160Temperature (K)I I 111111 I I I I 1111160 b)400r;,n 0000000 11111, I I2 4 102040 100Temperature (K)Figure 6.8: a) Temperature Scan: K, in Sb showing Pauli paramagnetism at low temperatures and Curie-Weiss behaviour at high temperatures. Circles are data obtainedat 15 kG for this thesis and squares are obtained from data taken at 4 kG in Reference [21]. The two low temperature “squares” are not used in fits. b) This semilogplot of a) can be compared qualitatively to Figure 4.3. Quantitative comparison is notpossible because neither a closed form expression nor a table of values was provided. [11]Chapter 6. Experimental and Results 65where the constant a was 0.644(1) K%. In this case the x2 was 37 for six degrees offreedom. Note that the poor fit is caused by a single point at T=180 K. Clearly furthermeasurements are required to determine the high temperature behaviour.From the constant a of the fit to Curie behaviour, Equation 6.10, the hyperfineparameter for muonium in Sb can be estimated as follows. Consider the spin Hamiltonian for the muon-electron system, which includes terms for the electron Zeemaninteraction, the muon Zeeman interaction, and the muon-electron contact interaction:(=h7S•B—hI•B+AS•I, (6.11)where‘ye/ is the gyromagnetic ratio of the electron/muon, S is the electron spin operator, I is the muon spin operator, and A is the hyperfine parameter.[5j In highmagnetic fields where S is a good quantum number the off diagonal elements may beneglected. In this case one can decompose ?-( into two effective muon Hamiltonians: onefor S=+1/2 and the other for S=-1/2. For an electron spin parallel to the magneticfield direction (S=+) the spin Hamiltonian becomes:= <1171:11> (6.12)= —h7,BI + 4iIz (6.13)For an electron spin antiparallel to the field direction (S=-) the spin Hamiltonianbecomes:7-C- = <4 17-tI 1> (6.14)71: = hWe — h’yBI — (6.15)In this approximation it is clear, that the effective magnetic field seen by the muon is:B = B . (6.16)2h-y,Chapter 6. Experimental and Results 660.8 a)0.60.4b)0.480 120 160 200Temperature (K)Figure 6.9: a) Fit to Curie-Weiss behaviour to high temperature K, in Sb yielding a x2of 32 for five degrees of freedom and a Curie temperature of -10(1) K. b) Fit to Curiebehaviour yielding a x2 of 37 for six degrees of freedom and a constant a of 0.644(1)K%. Data were taken in a magnetic field of 4 kG.Chapter 6. Experimental and Results 67Thus the muon will precess at two frequencies depending upon the orientation of theelectron spin:= 71.4B + A/2. (6.17)If we assume, that the muonium is in thermal contact with a heat bath (i.e. a metal),then the observed /LSR precession frequency will be a Boltzmann average of the twoequilibrium states of the electron: [48]17(T) = pv +p_v_, (6.18)where p is the probability the electron is in state S = + and p... is the probabilitythe electron is in state S = —:= eeBh/2kBT + e+eBh/2kBT’(6.19)where the electron gyromagnetic ratio ‘y is about 2.8 MHz/G. In the limit of hightemperature the above equation reduces to:e, (6.20)where E = ‘yeBh/4kBT. Substituting Equation 6.20 into Equation 6.18 leads to:(6.21)such that the muon Knight shift is given by:= h7eA1L (6.22)‘yB 4kBT-y,.Equating the coefficient of 1/T to fitted parameter a in the fit to Curie law leadsto A=259.7(0.4) MHz. A11 is the component of the hyperfine tensor parallel to thecrystalline ê-axis. Comparing with the hyperfine parameter of muonium in a vacuum:Avac 4463.30288(16) MHz (6.23)Au 0.0582(1)Avac. (6.24)Chapter 6. Experimental and Results 68Although no data above 80 K with H I ê-axis is available yet, we know that at lowertemperatures: [21]K. (6.25)Thus it is reasonable to assume:A A11/2 129.9(2) MHz. (6.26)From these we can estimate the isotropic and anisotropic hyperfine parameters:A8= A11 +2i1± A1 173 MHz (6.27)A = Au— AAii 43 MHz. (6.28)The large anisotropic hyperfine parameter and small isotropic hyperfine parametersuggest, that the majority of the spin density is near but not on the muon. A similarsituation occurs in covalent semiconductors. For example the hyperfine parameters ofthe Mu* center in GaAs are 218.5 and 87.87 MHz. In this case the unpaired electron isknown to be on the two nearest neighbour nuclei on the [111] axis. It is interesting tocompare A with what one would expect from an electron in a hydrogenic 2p orbital: [49]4 = 12.531-y, <r3> MHz. (6.29)The muon gyromagnetic ratio 7/21r is 135.53 MHz/T and the expectation value of r3for a hydrogenic 2p orbital (n=2, 1=1) is:[50]<r3 >a3n1(l +1)(21 + 1) 24a3’ (6.30)wherea = meao/Z, (6.31)Chapter 6. Experimental and Results 69where me is the electron mass, t is the reduced electron mass for the muon-electronsystem, a0 is the Bohr radius, and Z is the nuclear charge. In units where both me anda0 are one,a (1 + m)/m1.= 1.005 (6.32)and175.3 MHz. (6.33)This result is about four times larger than the experimental value of 43 MHz. Onthe other hand we do not expect the electron to be in a 2p orbital on the muon, butrather on the valence orbital of the neighbouring Sb atom(s). However this result doesemphasize that the deduced hyperfine parameter can only be explained, if there is alarge spin density near but not on the muon.6.3.6 Discussion of TF-iSR ResultsIn this section the results are discussed in terms of earlier models based on local momentformation and peaks in the density of states. Also new insights on K1 in Sb are offeredby examining the possibility that muons in Sb form a Kondo impurity.In Figure 6.5a) the linear increase of the frequency shift with magnetic field isconsistent with earlier work, which explained this behaviour as an effect of a peak in thedensity of states, as was discussed in Section 3.2.2.[20,22] In Figure 6.6 the field scansshow, that the amplitude of the dHvA oscillations of in Sb under these experimentalconditions is less than the statistical error of 0.2% of K, which is about 40 times smallerthan the amplitude expected from the dHvA oscillations of the magnetization- about8% of the magnetization. Observations in Cd show, that the amplitude of magnetizationand Knight shift oscillations were within a factor of about two. It was thought K mightbe proportional to the spin susceptibility and hence the magnetization. The clearChapter 6. Experimental and Results 70absence of dHvA oscillations is evidence, that the electron(s) responsible for the giantdo not have the character of the conduction electrons. This fact suggests, that thelocal electronic structure around the muon plays a dominant role.The anomalously large K, indicates perhaps muonium or more specifically a paramagnetic Sb complex is formed in Sb, The conduction electron density in Sb is threeto four orders of magnitude less than in normal metals. The smaller electron densitiesin the Group Vb semimetals (Bi: ‘-‘-‘3 x 1017 cm3, Sb: ‘--‘5 x 1019 cm3, and As: ‘—2 x102cm3)provide ineffective screening and therefore the chance exists for an electronto bind weakly to the iz or a tSb complex. Although the small carrier density mayplay a role, it is clearly not the sole actor, since the giant K,4 is seen only in Sb and notin As or Bi. This may be taken as further evidence for the importance of the electronicstructure near the muon in Sb. For example it is possible that the muon chemicallybinds to one or more Sb’s in such a way that a virtual bound state is formed withparameters in the Anderson model which favour a local moment.The proposed muonium-like center in Sb has a large anisotropic hyperfine parameter (A=43 MHz) and a small isotropic hyperfine parameter (A= 173 MHz), whichexplains the anisotropic K. The hyperfine parameters suggest, that the majority of theelectronic spin density is near but not on the muon. A similar situation occurs for theanomalous Mu* center in covalent semiconductors such as GaAs, where the unpairedelectron is known to be on the two nearest neighbours on the [111] axis. This largespin density on or near the muon and the rapid decrease of the susceptibility (or theKnight shift, which is thought to scale with the susceptibility) as temperature increasesare characteristic of a Kondo impurity.Below 6 K K, and presumably the susceptibility are constant or at least only weaklytemperature dependent, which indicates the local moment is zero or near zero. In termsof the Kondo effect at low temperatures the local moment of an unpaired electron onChapter 6. Experimental and Results 71the muon disappears because the conduction electrons have a strong influence on thebehaviour of the system at low temperatures. This can be understood in two differentways depending on the choice of model. In the spin-compensation model the electronscompensate the muon spin by forming an oppositely polarized spin cloud around themuon. Alternatively in the spin-fluctuation model the local moment fluctuation rateis greater than the thermal fluctuation rate at these low temperatures and the spinmagnetic moment averages to zero. As the temperature increases, the compensationand/or the fluctuation rate decreases and so local moment behaviour becomes increasingly important. As seen in Figure 6.8 K1 decreases with temperature, as would beexpected from a local moment. Thus the Kondo effect provides a natural explanationfor the large and anomalous K, in Sb.From temperatures of 2 to 19 K the temperature dependence is best fit to a powerlaw, Equation 6.8. A good fit can also be made to an Arrhenius Law, Equation 6.5, withan activation temperature (E/kB) of 40(2) K, which might reflect the electron-muonbinding or perhaps the binding of the spin-compensation cloud to the muon. Anotherfit to a scaled universal Kondo susceptibility, Equation 6.6, with a TK of 131(1) K canalso be done, as seen in Figure 6.7. The fit to this model is clearly not as good, butit nevertheless provides some support for the idea of local moment formation in Sb.Equation 6.6 is valid for T << TK, which is not satisfied in the latter fit. When thetemperature range is restricted from 2 to 10 K, this fit to Equation 6.6 yields a smallerreduced x2 and a 7’K of 160(3) K.At much higher temperatures (80 to 180 K) K has a Curie-like behaviour (see Figure 6.9), which is characteristic of a Kondo impurity. The interpretation in terms of theKondo effect is, that at these high temperatures either the local moment spin fluctuation rate is less than the thermal fluctuation rate and the muonium impurity appearedas a well-defined local moment. An alternative interpretation of the Kondo effect isChapter 6. Experimental and Results 72that the spin compensation cloud is disrupted or shaken off by thermal excitation.The gradual change from the Curie-like to Pauli paramagnetic behaviour predictedby Kondo theory is characterized by either T8f or TK. The TK of 160(3) K obtained fromthe low temperature data and the activation temperature of 40(2) K from the Arrheniusfit do not agree. However it should be noted, that the Kondo impurity models weredeveloped for magnetic impurities with d state valence electrons in normal free electron-like metals. Sb is a semimetal with a relatively small Fermi energy and has both holeand electron carriers. The valence orbital is composed primarily of p character. Thesedifferences might alter the details of the temperature dependent screening. Qualitativeagreement with the crossover from Pauli paramagnetic to Curie-like behaviour is good,but quantitative consistency for the parameter TK is lacking.In order to get an estimate of the Kondo exchange parameter J from Equation 4.2,the following values were used: the TK of 160(3) K is from the fit to Equation 6.6, theSb free electron Fermi temperature TF is calculated to be 1.3 x i0 K measured relativeto the bottom of the valence band,[17] and the Sb density of states at the Fermi levelof 4.7 x 10-2 (atom eV)*[51] The result was a J of about 3 eV, which is an order ofmagnitude larger than typical values of 0.1 eV.[33] In the case of a semimetal it maybe appropriate to use:TF = ±(3fl2)2/3 6100 K, (6.34)kB 2mwhere m* is the effective mass of carriers ( 0.lme) and the carrier concentration n is5.5 x 1019 electrons/cm3.This yielded a J of about 6 eV.In summary the Kondo effect provides a new explanation of the giant Knight shiftin Sb and its anomalous temperature dependence and anisotropy. Muons in Sb maytherefore be a unique example of an isolated Kondo impurity. Since only one isin the sample at a given time, 1iSR observations of and its interactions with theChapter 6. Experimental and Results 73Sb conduction electrons is a system free of impurity-impurity interactions, such as thelong range RKKY interaction, and therefore a test of theoretical solutions of the singleKondo impurity problem.6.3.7 LF-1tSR Experimental ProcedureWith LF-1tSR two field scans and one temperature scan were performed. In this experiment two temperature scans, one at field of 650 G and the other at 1 kG, in therange from 2 to 300 K were done. One field scan at a temperature of 75 K was donefrom 0.2 to 1.8 kG.In these LF-jSR experiments the behaviour of the longitudinal relaxation rateT’ as a function of temperature and field was sought. Prior to the LF-tSR measurements in Sb, the T’ in Ag alone was determined to be 1.3(5.6) x 10 ts1. Thisshowed, that the relaxation rate in Ag was effectively zero. The smallest nonzero relaxation rate, which could be measured under these conditions, was on the order ofi0 jts. Also prior to the LF-SR measurements the asymmetry of both the Sband Ag signals were determined by transverse field measurements. The zero T’ inAg as well as both the Ag and Sb asymmetries were then fixed for the analysis of theLF-uSR asymmetry histograms. In other words the decay positron asymmetry histograms were fitted with two fixed amplitude signals: a nonrelaxing reference Ag andthe exponentially relaxing Sb signal. T1 in Sb was obtained from these fits by x2minimization. One asymmetry histogram showing a large T’ and another showing asmaller T’ can be seen in Figure LF-1tSR Temperature ScanIn Figure 6.lla) the LF-SR study of relaxation rate showed three interesting features:a sharp rise until about 40-50 K, a peak at about 75 K, and an abrupt cutoff at 200 K.Chapter 6. Experimental and Results 740.180.14I I I I I I I0 2 4 6 8 10Time (ps)Figure 6.10: Typical LF-1tSR asymmetry histograms taken in a field of 1 kG showing ina) fast relaxation (large T’) at T=75 K and in b) slow relaxation (small T1) at T=250K. Each histogram was fitted with two fixed amplitude components: a nonrelaxing Agand an exponentially relaxing Sb signal.Chapter 6. Experimental and Results 75Presumably the relaxation rate reflects the spin exchange rate of muonium with the Sbconduction electrons. For the 1 kG data there appears to be a constant “background”relaxation rate, which is discussed below.6.3.9 LF-uSR Field ScanA coarse field scan was conducted from 0.2 to 1.8 kG at a temperature of 75 K. InFigure 6.12 there is a peak in the relaxation rate at about 0.7 kG, but as can be seenfrom the relative size of the error bars, this is a small effect at this particular temperature. The weak temperature dependence of the relaxation is expected for Korringarelaxation.6.3.10 Discussion of LF-SR ResultsThe coupling of the to the spin magnetic moments of the Sb conduction electronsleads to T1 relaxation. In other words when the applied field is parallel to the muonspin, the + can either absorb or release energy to the electrons within kBT of theFermi surface, which leads to depolarization of the muon and relaxation in the decaypositron asymmetry. The muon transition is accompanied by an electron transitionfrom initial state Iks) to final state k’s’). The muon goes from initial state rn) to finalstate n). In the standard theory of Korringa relaxation of nuclear spins in a metal, ascattering process with a transition rate W is described by Fermi’s Golden Rule:Wms,nis = I <msVn’s’> 28(Em + E8 — E—E,), (6.35)where V is the s-state coupling of the muon and electron, which is responsible forthe scattering. From this independent electron model, the Korringa relation wasdeveloped.[18,52] As mentioned in Section 4.3.5 the Korringa relation is valid for aKondo impurity provided T << TK. The Korringa relation relates the relaxation rate(I)H—100 200 300Temperature (K)Figure 6.11: a) Experimental temperature dependence of longitudinal relaxation rate,T’, at 650 G and 1 kG. b) Predicted T’ using K from TF-1SR data with theKorringa relation. The electron g factor used was 2. c) Predicted T1 with g=15.Chapter 6. Experimental and Results 760 ‘ Ui=650 G0 0=1 kG00000000•-aI I I I I0.060.040.0200.080.060.04U)H- 0.020Cl)0.5H-0mg=2: UII I IUImm g=150a00•0UIUIaaI I0Chapter 6. Experimental and Results 770.0560.054- 0.052F-0.0500 0.4 0.8 1.2Magnetic Field (kG)Figure 6.12: Field dependence of longitudinal relaxation rate, T’, at a temperatureof 75 K.T’ to the temperature and the isotropic component of the Knight shift:T’ = TK0/S, (6.36)wheres = 47rkB2(6.37)andI K(6) = K80 + Kaniso(3O2— 1). (6.38)For temperatures up to 80 K it is known, that K11 = 2K within experimental error. [20]Therefore the isotropic component of the Knight shift is: KAll the available K1 data from temperatures of 2 to 180 K was used in the Korringarelation, in order to make a prediction of the longitudinal relaxation rate, T1 for twodifferent electron g factors, as seen in Figures 6.llb) and c). For electrons on the FermiChapter 6. Experimental and Results 78surface in Sb and for applied fields parallel to the ê axis, the electron g factor is 15 and50 7e = g/Lb/h 15/Lb/h.[28] In this case the predicted Tj’ is in very poor agreementwith the data as seen in Figure 6.llc). However with g=2 the agreement is muchbetter. This is a further indication of the importance of the local electronic structurerelative to the band structure.Ideally, in the absence of electron-electron interactions, the Korringa ratioT1K0/Sis unity, but Korringa ratios as high as 29 and as low as 0.16 have been reported.[23]These large deviations arise from known factors, such as core polarization and orbitalspin paramagnetism. However, since the muon has no core electrons, core polarizationis not a factor. Also Sb does not display orbital spin paramagnetism. Small deviationsfrom unity arise, when the independent electron model breaks down or in other wordswhen electron-electron interactions become important. For the muon in Sb problemwe expect the “normal” Korringa relation to hold for T << TK or in other words theKorringa ratio should be temperature independent and of order unity at temperatureswell below TK.[41] From K1 data TK is estimated to be in the temperature range from40 to 160 K. The peak in T’ occurs at T TK 75 K in Figure 6.lla).[42]The experimentally determined Tj’ and K values were used to calculate the quantity S/(T1TI0),which is shown in Figure 6.13a). The T1 values were taken fromthe LF-SR data and K1 was estimated by interpolating the TF-pSR data. This wasnecessary, since the K,, data points were not taken at the same temperature as theT data points. The Arrhenius fit was used to interpolate K,L for 50 K and below,while the Curie-Weiss fit was used for 75 K and above. In Figure 6.13a) the quantityS/(T1TK0)appears to diverge as the temperature approaches zero. This is due to thefact that the low temperature data extrapolated to a non-zero value at zero temperature. The origin of this temperature independent contribution to T’ is unknown.However in the event it arises from something other than an electronic mechanism, weChapter 6. Experimental and Results 79subtracted this “background” from the T’ data and calculated the “corrected” valueof S/(T1TK0)seen in Figure 6.13b). The “corrected” temperature independence ofS/(T1TI0)below 100 K is expected from the Korringa relation, which is only validfor a Kondo impurity at temperatures well below TK. At temperatures above TK wesee the breakdown of the Korringa relation in Figure 6.13.A possible origin for the “background” or non-electronic relaxation follows. At mostmagnetic fields the muon energy level splitting is different from the nuclear splitting.In this case no spin flip interactions between a muon and a nucleus are possible, sinceenergy must be conserved. However at certain fields, where there is a level crossing resonance (LCR), relaxation is possible. If the muon Zeeman splitting matches the nuclearsplitting due to quadrupolar and Zeeman interactions, then spin exchange between themuon and the nucleus is possible, since energy is conserved:7i = 71nuc (6.39)—g4H I = —gyH . J + Q[3J2 — J(J + 1)], (6.40)where I is the muon spin operator, J is the nuclear spin operator, and Q is thequadrupole parameter. If the T1 data was taken at such a field, then the observed“background” may be relaxation due to spin exchange between the nucleus and themuon, which would be temperature independent. In Figure 6.lla) the “background”relaxation seems to be temperature independent, since T’ at T=0 and above 200 Kare similar. At fields further away from the LCR this relaxation should decrease. Inmost cases for the 650 G data T’ is indeed smaller than for the 1 kG data.An estimate of the LCR field for the (1/2—*-1/2) nuclear transition accompaniedby a muon spinflip transition can be made:111cr = , (6.41)7I — 7nChapter 6. Experimental and Results 80where is the muon/nuclear gyromagnetic ratio. The field gradient introduced bythe positive muon will doubtlessly change Q, but since this parameter is unknown, theQ values for pure nuclear quadrupole resonance in Sb from Table 6.1 are used. For‘215b the expected 111Cr is about 6 kG. Although this is substantially larger than thefield at which the T’ data were taken, it should be noted that the muon can causelarge changes in Q by its presence.J ‘y/2ir (MHz/kG) Abundance (%) Q/h (MHz)‘215b 5/2 1.0189 57.3 76.867(1)‘23Sb 7/2 0.5518 42.7 97.999(1)Table 6.1: Sb LCR parameters.[23]Chapter 6. Experimental and Results 811.2a)0.80.400h-0.80.6 b)0.40.200Temperature (K)Figure 6.13: These two diagrams were produced with the K, data from Figure 6.8.Since there is no K, data above 200 K, K, was extrapolated to higher temperatureswith the Curie-Weiss fit to produce the figures seen here. a) S/(T1TI0evaluatedwith experimentally determined Tf’ and fitted K14. b) S/(T1TK?33 evaluated with azero Tj’ at zero temperature and a “corrected” Tf1 at all temperatures.r rii=1 kG- cP=650 CLEli-wuIRww w-ø UiUiI I I _.I. I100 200 300Chapter 7ConclusionIn this thesis we presented additional data on the anomalously large, anisotropic, andtemperature dependent K1 in Sb. We propose a model to explain the results in whicheither muonium or a paramagnetic Sb complex is formed. In this case Sb wouldbe the only metal or semimetal, in which muonium has been observed. Hyperfineparameters for muonium in Sb have been estimated from the temperature dependence ofK, and are similar to those of Mu* in GaAs indicating, that a large spin density resideson the nearest neighbour Sb. This single paramagnetic muonium impurity in Sb isinteresting, since it would represent a truly isolated Kondo impurity. 1tSR experimentsin Sb are unique in this respect because only one muon at a time is in the sample andRKKY interactions between two or more impurities are avoided. The other interestingfeature about this system is the fact that Sb is a semimetal. Consequently the theoryfor a Kondo impurity may have to be modified to take into account the much smallercarrier densities.The qualitative agreement with typical Kondo behaviour is good; the K, datadisplays a crossover from the weakly temperature dependent Pauli paramagnetism ofthe conduction electrons at low temperatures to Curie-like local moment behaviour athigh temperatures. Quantitative support for Kondo behaviour comes from a good fitof K data at temperatures from 2 to 10 K to a scaled equation for the universal Kondosusceptibility.The absence of dHvA oscillations in K, which was expected to scale with the82Chapter 7. Conclusion 83magnetization, confirms that the extended electronic band structure has been perturbeddramatically by the positive muon and that the local electronic structure of the muonplays a dominant role in determining K.The T’ relaxation rate displays a temperature dependence, which presumablyreflects the anomalous spin dynamics of the muonium atom and the Sb conductionelectrons. The temperature independent Korringa ratio S/(T1TI0)seen below atemperature of 100 K is expected for a Kondo impurity.The experimental data for K, and T1 are tabulated in Appendix B, in order toprovide a test for theoretical models.Equally accurate measurements of with the magnetic field both parallel andperpendicular to the crystal c-axis for all temperatures up to room temperature areneeded in order to gain more insight into the Kondo model as a possible explanation forthe anomalous K,. in Sb. Also a closer examination of the low temperature behaviour ofT1 is needed, in order to understand the so called “background” relaxation rate. Analloying experiment with small concentrations of Te might prove interesting, if indeedexpansion of the electron pockets leads to an enhanced K. Clearly something willhappen, since previous work and this thesis have shown K,. in Sb is extremely sensitiveto any change in either temperature, pressure,[22], or alloy concentration. [20,21]Bibliography[1] R.F. Kiefi, W.A. McFarlane, K.H. Chow, S. Dunsiger, T.L. Duty, T.M.S. Johnston,J.W. Schneider, J. Sonier, L. Brard, R.M. Strogin, J.E. Fischer, and A.B. Smith ITT,Phys. Rev. Lett. 70, 3987 (1993).[2] A. Schenck, Muon Spin Rotation Spectroscopy, Bristol: Adam Hilger Ltd., 1985.[3] A. Schenck, Hyp. Tnt. 8, 445 (1981).[4] R.F. Kiefi and T.L. Estle, Physica B 170, 33 (1991).[5] R.F. Kiefi and S.R. Kreitzman, Eds. T. Yamazaki, K. Nakai, and K. Nagamine,Perspectives of Meson Science, New York: North-Holland, 1992.[6] C. Rizzuto, Rep. Prog. Phys. 37, 147 (1974).[7] G. Grüner, Adv. Phys. 23, 941 (1974).[8] H. Suhi, ed., Magnetism: Magnetic Properties of Metallic Allo!,s, Vol. V, NewYork: Academic Press, 1973.[9] A.M. Tsvelick and P.B. Wiegmann, Adv. Phys. 32, 453 (1983).[10] V.1. Mel’nikov, JETP Lett. 35, 414 (1982).[11] P. Coleman and N. Andrei, J. Phys. C 19, 3211 (1986).[12] E. Lippelt, P. Birrer, F.N. Gygax, B. Hitti, A. Schenck and M. Weber, Hyp. 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Lindgren, 0. Hartman, E. Karisson, and R. Wppiing, Hyp. Tnt. 31, 439 (1986).[23] G.C. Carter, L.H. Bennett, D.J. Kahan, Metallic Shifts in NMR, Toronto: Pergamon Press, 1977.[24] R.L. Odle and C.P. Flynn, J. Phys. Chem. Sol. 26, 1685 (1965).[25] R.G. Goodrich et al., Phys. Rev. Lett. 183, 414 (1969).[26] C. Kittel, Quantum Theory of Solids, New York: John Wiley Sons, 1963.[27] D. Li. Williams, TRTUMF Research Proposal, Expt. 668, 1992 (unpublished).[28] Z. Altounian and W.R. Datars, J. Phys. F 6, 1297 (1976).[29] W.E. Pickett and B.M. Klein, Phys. Rev. B31, 6273 (1985).[30] E.P. Jones and D. Li. Williams, Can. J. Phys. 42, 1499 (1964).[31] N.W. Ashcroft and N.D. Mermin, Solid State Physics, Philadelphia: SaundersCollege, 1976.[32] J. Kondo, Progr. Th. Phys. 32, 37 (1964).[33] A. Narath and A.C. Gossard, Phys. Rev. 183, 391 (1969).[34] N. Rivier and M.J. Zuckerman, Phys. Rev. Lett. 21, 904 (1968).[35] K.D. Schotte and U. Schotte, Phys. Rev. B 4, 2228 (1971).[36] K.D. Schotte and U. Schotte, Phys. Lett. 55A, 38 (1975).[37] P.W. Anderson, Phys. Rev. 124, 41(1961).[38] G. Grüner and A. Zawadowski, Progress of Low Temperature Physics, Vol. VIIb,New York: North-Holland, 1978.Bibliography 86[39] M.D. Daybell and W.A. Steyert, Rev. Mod. Phys. 40, 380 (1968).[40] N. Rivier, M. Sunjic and M.J. Zuckerman, Phys. Lett. 28A, 492 (1969).[41] H. Shiba, Prog. Th. Phys. 54, 967 (1975).[42] H. Shiba, J. Low. Temp. Phys. 25, 587 (1976).[43] H.R. Krishna-murthy, J.W. Wilkins, and K.G. Wilson, Phys. Rev. B 21, 1003(1980).[44] W. Götze and P. Schlottmann, Solid State Comm. 13, 17 (1973).[45] T. Riseman, M.Sc. Thesis, University of British Columbia, 1989 (unpublished).[46] S.F.J. Cox, J. Phys. C 20, 3187 (1987).[47] V. Aseev and J. Beveridge, Surface Muons in the Solenoid HELlOS, TRIUMFDesign Note TRI-DN-92-1, January 1992 (unpublished).[48] C. Kittel, Introduction to Solid State Physics, Sixth Edition, Toronto: John Wiley& Sons, 1986.[49] J.R. Morton and K.F. Preston, J. Mag. Res. 30, 577 (1978).[50] J.J. Brehm and W.J. Mullin, Introduction to the Structure of Matter, Toronto:John Wiley 4z Sons, 1989.[51] W.R. Datars, Phys. Rev. B 7, 3435 (1973).[52] J. Korringa, Physica 16, 601 (1950).Appendix AThe de Haas-van Aiphen EffectIn 1930 de Haas and van Aiphen discovered the magnetization of bismuth oscillatedas a function of magnetic field in the range 5-20 kG at a temperature of 14 K. Theusefulness of these oscillations in Fermi surface studies of metals was not fully appreciated until 1952, even though the quantization of electron orbits in an applied fieldand the possibility of directly observing the macroscopic oscillations due to this purelyquantum effect had been predicted by Landau prior to the original de Haas-van Alphen(dHvA) experiment.[48]A.1 Quantization of Electron Orbits in an Applied FieldIn an applied magnetic field the orbit of an electron is quantized. The area of thesemiclassical orbit for free electrons in k space is given by:= (n, + 1/2)-H. (A.1)We see, that the area in 1 space of two successive orbits is equal for equal incrementsof 1/H:1 1 1’\Si — -fl-- =2lrr-. (A.2)\Tln+1 LZnJ itCIf the orbital areas are the same, then we expect properties dependent on the Fermisurface to be the same. This quantization of conduction electron orbits in metals87Appendix A. The de Haas-van Aiphen Effect 88leads to oscillations of macroscopic properties such as resistivity, heat capacity, andsusceptibility as a function of 1/H with period:(12e A3\.H)— hcSThe de Haas-van Aiphen effect is the oscillation in 1/B of the magnetic moment ofa metal. Low temperatures and high magnetic fields are required to prevent thermalpopulation of adjacent orbits and to keep the orbital period much less than the collisiontime. [48]A.2 Landau LevelsFor a two dimensional system of noninteracting electrons at absolute zero, Figure A.].shows the origin of the effect. [48]As seen in Figure A.2a), with an applied field H in the z direction, the orbits arequantized in k, k space and the area between adjacent orbits is:= S,,—= 2’ireH/hc. (A.4)Consider a square sample with side .L, such that the area an electron occupies in k spaceis (2ir/L). In the presence of a magnetic field a total of D previously free electronscoalesce into a single Landau level, whereD = (2lreH/hc)(L/27r) = pH (A.5)and p = eL2/27rhc. (If electron spin is taken into account, then D = 2pH.) As Hincreases, the degeneracy D of a Landau level also increases, Figure A.2b). As long asthere are electrons in a given Landau level, labelled by quantum number s+1, the Fermilevel will remain in level s+1. However, as H and D increase, the electrons move intoAppendix A. The de Haas-van Aiphen Effect 89H,<H2<H3H=O H1 H2 H=O H3/EK0 0a) b) c) d) e) f) g) h)Figure A.1: For spinless, free electrons in two dimensions at absolute zero, the electronorbitals below EF are filled. In a) and f) the applied field is zero. In b), e), and g) thefree electrons coalesce into magnetic energy (Landau) levels of degeneracy D = pH,which are split by /.E = = ehH/m*c. In b) and g) the total energy of the systemis at a minimum because as many electrons are raised in energy as are lowered. In e)the energy of the system is at a maximum because the topmost electrons have all beenraised in energy. If spin is taken into account the degeneracy increases by a factor oftwo to D = 2pH and the “spinless” Landau levels split by the Zeeman energy gpBHinto a spin-up level and a spin-down level, as seen in c), d), and h). In d) the lowerenergy spin-down levels will lie below the Fermi energy, whereas the spin-up levels willbe vacant leading to a net magnetic moment.[48]Appendix A. The de Haas-van Aiphen Effect 90ka)••b).... ....•... ....••••I... •I••.... ........ I....... ........ ....c) s=1 s=2 s=311 I l,50,-oE1 2100/HFigure A.2: a) Allowed electron orbitals without an applied field occupy an areaof (27r/k) in k space. With an applied field in the z direction the electronsare restricted to circles in the x-y plane. In k space the area between circles is:= 2irk(/k) (2irm/h)E = 2irm/h = 2-ireH/hc. b) The periodicity ofdHvA oscillations with 1/H is seen here for a two dimensional electron gas of 50 electrons with p=O.5.[48] The completely filled magnetic levels are denoted by s. Forexample at 11=40 or 100/11=2.5 the levels s = 0,1 are fully occupied and there are tenstates in s=2. At temperature above absolute zero thermal smearing of the magneticlevels lead to oscillations, which are more sinusoidal than sawtooth-shaped. [48]I I3 4Appendix A. The de Haas-van Aiphen Effect 91lower Landau levels and eventually there are no more electrons in level .s+1. Now theFermi level lies in level s. This movement of the highest occupied Landau level relativeto the Fermi level produces oscillations of transport properties or any other quantity,which depends on the density of states at the Fermi level, as we will see below.The energy of a Landau level n, where n 0, is:E = (n + )hw, (A.6)where= eH/m*c (A.7)is the cyclotron frequency. The electrons in a Landau level can have an arbitraryvalue of k. Consider now only the electrons in a volume of thickness 6k at k. Thedegeneracy per unit volume is:=eH6 (A.8)(If electron spin is taken into account, then increases by a factor of two.) The Landaulevels up to and including level s will be completely filled and the total energy of anelectron in any of these levels will be less than the Fermi energy:h(s + ) EF — E, (A.9)and above this energy all the levels are empty. The total number of electrons in theslice Sk isN=(s + 1). (A.10)So as H and increase, N increases linearly with H until level s reaches the Fermienergy. As H increases further, electrons leave level s for level s-i and in general thissituation will occur whenever1 E m*cE 1he , (A.11)Appendix A. The de Haas-van Aiphen Effect 92So the population of the slice N oscillates with a period in 1/H of m*cE/he and withan amplitude of +/2 around the value N0, the total number of electrons in 6’k in theabsence of a magnetic field. [26]When the population of the slice is N0, the energy of the electrons in a magneticfield is:Uo (A.12)U0 = hw(s + 1)2/2 + N0 2rnbut N0 = (s + 1) and, therefore,= hwN+N0. (A.14)2 2mFor only a small change in magnetic field, Landau level s will still be the topmostfilled state and=+ + (N0 - N)EF. (A.15)The last term in the above equation represents the transfer of N — N0 electrons to theFermi level and (N0— N)EF is the energy change associated with the rest of the Fermisea. So the change U — U0 represents the total energy change of the Fermi surfaceelectrons for a small change in magnetic field:SU=U_U=E(N2_N)+(N_ )E, (A.16)but E = hN0/and, therefore,= (N — N0)2. (A.17)The magnetization of the slice 6k at absolute zero is given by:6M = — = -(N — N0), (A.18)Appendix A. The de Haas-van Aiphen Effect 93butdN (s + 1)/H (A.19)and, therefore,SM (N — N0) N — N0. (A.20)Since the electron population of the slice Sk varies periodically with 1/H, the magnetization will also oscillate as a function of 1/H. This oscillation of the magnetization isthe de Haas-van Alphen effect.In the slice Sk the change of susceptibility is given by:= SM/H = 1(N — N0) cc N — N0. (A.21)The bulk susceptibility of the electrons depends upon the oscillatory quantity N, whichin this case is the number of electrons near the Fermi surface. At nonzero temperatures,levels within kB T of the Fermi level are populated by thermal excitations. If thisthermal energy is comparable to the spacing hw between Landau levels, then theoscillations in 1/H will be washed out. From the inequality1tw ehH 1—>1 (A.22)kBT m*c kBTwe seeT/H < 0.134 K/kG. (A.23)In order to avoid thermal smearing of the oscillations, operating with a field of order20 kG requires temperatures of order 2 K.[48JA.3 Extremal Orbits of the Fermi SurfaceBy measuring the period of dHvA oscillations, the areas S of a Fermi surface, whichare perpendicular to B, can be calculated. In principle we might expect to see dHvAAppendix A. The de Haas-van Aiphen Effect 94periods contributed by all the orbits, which are perpendicular to the applied field andwhich have been quantized by the magnetic field. However, the dominant contributioncomes from extremal- either maximal or minimal- orbits, such as the “neck” (minimal)or “belly” (maximal) orbit in the Fermi surface of copper, which is described on p. 247of Reference [48]. Extremal orbits or areas of the Fermi surface have periods in 1/H,which do not change for a small change in k. For non-extremal orbits on the Fermisurface, the area of the orbit changes as changes. Therefore, clear resonances canbe obtained in dHvA measurements of very complicated Fermi surfaces because onlythe extremal orbits have slowly varying areas. A mathematical discussion of extremalorbits and observed dHvA oscillations can be found in Reference [26]. Thus by analyzingdHvA oscillations the particular electrons or holes on the Fermi surface contributingto particular phenomena, such as the magnetic susceptibility and the Knight shift, canbe determined. [48]Appendix BExperimental DataB.1 TF-tSR Data: Knight Shift vs. TemperatureAll data was taken on M20 beamline at TRIUMF from November 25 to December1, 1992. The two field scans taken at T=2.7 K can be found in runs 1293 to 1345.The temperature scan taken at B=15 kG can be found in runs 1368 to 1377 and issummarized below in Table B.1.Temperature (K) Knight Shift (%)2.298(1) 1.393(1)2.888(1) 1.396(1)4.18(2) 1.393(1)5.9(1) 1.392(1)8.00(5) 1.386(2)10.17(5) 1.380(1)13.05(.10) 1.362(2)15.98(7) 1.343(1)18.77(.27) 1.314(1)Table B.1: TF-1iSR Data: Knight Shift vs. Temperature taken in a magnetic field of15 kG.B.2 LF-1tSR Data: Relaxation Rate vs. TemperatureAll data was taken on M20 beamline at TRIUMF in July 1993. The temperature scansat 1 kG and 650 G as well as the field scan at 75 K can be found in runs 674 to 709.95Appendix B. Experimental Data 96The temperature and field scan data are summarized below in Tables B.2 and B.3.Temperature (K) T’ Relaxation Rate (x 102jis’)2 0.79(.11)5 1.21(.12)10 1.862(.044)15 2.44(.11)20 3.00(.17)35 4.105(.091)50 4.95(.12)75 4.95(.15)100 4.59(.11)100 4.85(.15)125 4.52(.14)150 3.936(.075)175 2.91(.11)187 1.14(.13)187 1.40(.11)200 0.78(.11)225 0.68(.10)250 0.92(.11)275 0.79(.12)300 0.58(.11)Table B.2: LF-tSR Data at 1 kG: T1 Relaxation Rate vs. TemperatureTemperature (K) Tj1 Relaxation Rate (x 102ts)20 2.423(.071)75 5.59(.15)175 2.54(.14)200 0.000(.008)Table B.3: LF-1tSR Data at 650 G: T’ Relaxation Rate vs. Temperature


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