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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 POSITIVE MUON IN ANTIMONY: EVIDENCE FOR AN ISOLATED KONDO IMPURITY By Thomas Michael Shaun Johnston B.Sc.E. Queen’s University, 1991  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  ©  Thomas Michael Shaun Johnston, 1993  ______________  In presenting this thesis  in  partial fulfilment of the requirements  for degree at the University of British Columbia, I agree that the Library freely available for reference and study. I further agree that permission copying of this thesis for scholarly purposes may be granted by the department  or  by  his  or  her  representatives.  It  is  understood  that  an  advanced  shall make it for extensive head of my copying  or  publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  The University of British Columbia Vancouver, Canada Date  DE-6 (2/88)  Abstract  The anomalous electronic structure of a positive muon in the semimetal antimony was investigated by the SR (Muon Spin Rotation/Relaxation) technique. Precise mea surements of the giant muon Knight shift (K=+1.4%) were made as a function of temperature (2-20 K) and magnetic field (18-21 kG) with the applied magnetic field parallel to the ê-axis of the antimony single crystal. No de Haas-van Alphen oscillations were observed at a temperature of 3 K, which indicates the electron spin density on the muon 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-van Alphen oscillations. The measured K, varies weakly with temperature below 10 K. The data were fit in three different ways. The best fit is to the form K  =  1/(a + bT 2 + cT ) 4  with parameters a=7.167(3) x i0, b=6.4(7) x 1O K , and c=1.8(2) x 2  10  . A 4 K  fit to an Arrhenius Law of the form 1— Ae/T yields a pre-exponential of 0.47(4) and an activation temperature Ea/kB of 40(2) K. If one fits to an expression for the Kondo susceptibility of the form K  =  a(1/2IrTK  —  /) at low temperatures (2-10 2 0.433T  K) 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 ob serve a crossover from a weakly temperature dependent Pauli paramagnetic behaviour below 10 K to a Curie-like behaviour above 90 K. This crossover is characteristic of a Kondo impurity indicating the anomalous electronic structure of a positive muon in antimony may be due to muonium formation, in which a local moment is centered on or near the muon. Within context of this model estimates of the hyperfine parameters =259,7(4) MHz and A±=129.9(2) MHz) for muonium in antimony are similar to 11 (A 11  those for Mu* in covalent semiconductor. The large anisotropy suggests a large spin density resides on the nearest neighbour antimony atom(s). Since the SR technique only permits one muon in the sample at a time, the anomalous electronic structure of a muon in antimony may be a unique example of a truly isolated Kondo impurity in so far as other magnetic impurities in the sample can be neglected. A large Korringa-like relaxation of the muon in antimony was observed in longitudinal field implying the spin dynamics of a muon in antimony are also anomalous when compared to normal metals. The Korringa constant S/(K,T T) is temperature independent as expected for 1 a Kondo impurity for T << TK. Above a temperature of 100 K, which is comparable to TK, a breakdown of the Korringa law is seen. In particular a peak in T’ at a temperature of 75 K is observed.  111  Table of Contents  Abstract  ii  List of Tables  vii  List of Figures  viii  Acknowledgements  x  1  Introduction  1  2  Crystal Structure and Fermi Surface of Antimony  7  3  The Knight Shift  11  3.1  Theory  11  3.2  The Muon Knight Shift in Sb: Review of Previous Experimental Results  14  3.2.1  First Discovery  15  3.2.2  Self-Consistent Molecular Cluster Calculations  18  3.2.3  Crystal Backscattering Resonance  20  3.2.4  SbBi and SbSn Alloys  20  3.2.5  De Haas-van Alphen Oscillations in Sn 119 and Cd” Knight Shift  21  4  The Kondo Effect  25  4.1  Introduction  25  4.2  Dilute Alloys  27  4.3  Some Models  28 iv  5  6  4.3.1  Moment Formation and the Anderson Model  4.3.2  The Spin-Fluctuation Model vs. the Spin-Compensation Model  28  for the Kondo Effect  32  4.3.3  The Resonant-Level Model  34  4.3.4  Exact Solutions of the Kondo Susceptibility  35  4.3.5  The Korringa Relation  35  tSR 1  38  5.1  Muons and Muon Beams  38  5.2  TF-SR Geometry and Technique  41  5.3  LF-SR Geometry and Technique  44  5.4  Experimental  45  5.5  TF-tSR Data  47  5.6  LF-,uSR Data  49  Experimental and Results  51  6.1  Shubnikov-de Haas Oscillations  51  6.2  De Haas-van Aiphen Oscillations of the Magnetization in Sb  52  6.3  jISR in Sb  54  6.3.1  Sample Characteristics  54  6.3.2  Measurement of Muon Knight Shift with TF-[LSR  54  6.3.3  Magnetic Field Dependence of the Muon Knight Shift  56  6.3.4  Temperature Dependence of the Muon Knight Shift below 20 K  61  6.3.5  Muon Knight Shift at High Temperatures  62  6.3.6  Discussion of TF-tSR Results  69  6.3.7  LF-tSR Experimental Procedure  73  6.3.8  tSR Temperature Scan 1 LF-  73  V  6.3.9  LF-tSR Field Scan  .  6.3.10 Discussion of LF-SR Results 7  Conclusion  75 75 82  Bibliography  84  A The de Haas-van Aiphen Effect  87  A.1 Quantization of Electron Orbits in an Applied Field  87  A.2 Landau Levels  88  A.3 Extremal Orbits of the Fermi Surface  93  B Experimental Data  95  B.1 TFSR Data: Knight Shift vs. Temperature 1  95  B.2 LFiSR Data: Relaxation Rate vs. Temperature 1  95  vi  List of Tables  2.1  Sb Lattice Parameters  2.2  Experimental de Haas-van Alphen Oscillation Periods  10  6.1  Sb Level Crossing Resonance Parameters  80  B.1 TF-iSR Data: Knight Shift vs. Temperature  95  B.2 LF-[LSR Data at 1 kG: T’ Relaxation Rate vs. Temperature  96  B.3 LF-uSR Data at 650 G: T’ Relaxation Rate vs. Temperature  96  9  vii  List of Figures  2.1  The Rhombohedral Crystal Structure and Brioullin Zone  8  2.2  Sb Electron Pocket on the Fermi Surface  9  3.1  Volume in k Space  13  3.2  Virtual Bound State  16  3.3  Sb Electronic Density of States and Spectral Density on the Interstitial Muon from Molecular Cluster Calculations  19  3.4  Temperature Scan: K,, in Pure Sb as well as in a 6.3 at.-% Bi Alloy.  3.5  De Haas-van Alphen Oscillations in Cd  23  4.1  The Virtual Bound States of the Anderson Model  29  4.2  Phase Diagram of Magnetic and Nonmagnetic Behaviour for the Ander  .  22  son Model  30  4.3  Exact Result for Kondo Susceptibility  36  5.1  Decay Positron Emission Probability  40  5.2  TF-RSR Experimental Geometry  42  5.3  tSR Geometry 1 LF-  44  5.4  TF-,uSR Experimental Apparatus  46  5.5  TF-iSR Time Spectrum, Asymmetry Histogram, and FFT  48  5.6  iSR Asymmetry Histogram 1 LF-  50  6.1  De Haas-van Aiphen Oscillations of Sb Magnetic Susceptibility  53  6.2  Cross-section of the Sb Single Crystal Sample  54  viii  6.3  Rotating Reference Frame Fit and FFT of Typical Asymmetry Histogram 57  6.4  TF-tSR Linewidth as a Function of Magnetic Field  58  6.5  Frequency Shift and Muon Knight Shift: 2-21 kG  59  6.6  Muon Knight Shift Field Scans: Sb  Bile T=2.7K  60  6.7  Four Fits to Low Temperature K in Sb Data  63  6.8  Temperature Scan: K, in Sb at B=4 and 15 kG  64  6.9  Fitting Curie and Curie-Weiss Behaviour to High Temperature K in Sb 66  6.10 Typical LFtSR Asymmetry Histograms in Sb 1  74  6.11 Predicted and Experimental Temperature Dependence of Longitudinal Relaxation Rate, T’  76  6.12 Field Dependence of Longitudinal Relaxation Rate, T’  77  6.13 Temperature Dependence of 0 TK 1 S/(T )  81  A.1 Explanation of dHvA Effect for Free Electrons in Two Dimensions in a Magnetic Field  89  A.2 Allowed Electron Orbitals and dHvA Oscillations  ix  90  Acknowledgements  The patience and interest of my advisor, Rob Kiefi, are much appreciated. I am also grateful for the assistance provided by Dave Williams. For the seemingly thankless task of sitting shifts I would like to say “Thanks!” to Kim Chow, Sarah Dunsiger, Tim Duty, Bassam Hitti, Evert Koster, Andrew MacFarlane, Jeff Sonier and Jürg Schneider. Also Evert Koster deserves another thank you for performing the Shubnikov-de Haas and magnetization studies.  The guidance of Curtis Ballard and Keith Hoyle with  experimental set up was also very much appreciated. Last but definitely not least I am thankful to my family, who always provide perspective and support for all my undertakings.  x  Chapter 1  Introduction  A positive muon  ()  implanted into a crystalline solid almost always occupies an  interstitial site. In normal metals the conduction electrons act to screen the positive charge of the muon. The strong Coulomb potential of this single positively charged impurity is so reduced by the electronic screening, that no bound electront+ state in a metal has been identified.’ With an applied magnetic field the conduction electrons may be slightly polarized causing a non-zero electronic spin density at the  .  This  contact interaction between the muon and the net electron spin density at the muon site [ri()  —  n(i)] determines the contact or isotropic hyperfine field experienced  by the muon: STr  -.  Bhf(r)  =  ——[n (rn)  —  n (r,L)],  (1.1)  where n() is the spin up electron density and n(i) is the spin down electron density at the  +  site. In an applied field B 0 the Knight shift constant K, is defined  as: Bhf()  zB K1L  =  B()  -  0 B  =  0 KB  (1.2)  0 B(i)—B 0  where B(i) is the total field experienced by the  +. [2]  The Knight shift is directly pro  portional to the magnetic susceptibility of the conduction electrons, which in moderate fields is independent of B . 0 The only exception occurs at the low electron density endohedral site in the metal .[1] 1 60 C 3 Rb  1  Chapter 1. Introduction  2  In typical metals the Knight shift is small (r10 to 100 p.p.m.)[3] and weakly depen dent upon temperature (since the Pauli spin susceptibility of a degenerate electron gas is temperature independent to first order) and crystal orientation (due to the s char acter of the local electrons in most non-transition element metals). Also the Korringa relaxation rate for a stationary  in metals is normally immeasurably small because  of the short muon lifetime and small contact hyperfine interaction. Intrinsic semiconductors and insulators have much lower carrier concentrations than metals do. In these materials the muon can be screened efficiently if it captures an elec tron during the slowing down process to form a hydrogen-like atom called muonium or Mu). In vacuum the p+e  contact interaction produces an energy splitting  between 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 ionic insulators. In most wide band gap materials, such as alkali halides, a single type of muonium is observed with a large hyperfine parameter close to that of muonium in vacuum. In covalent semiconductors, such as Si or GaAs, two paramagnetic muonium states are observed. One center in Si called normal muonium (Mu) is characterized by an 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 Si than in vacuum leading to a smaller electron density on the  +.  The other center called  anomalous muonium (Mu*) is characterized by a very much smaller hyperfine interac tion, 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 the bond centered positions along the [111] directions with the majority of the unpaired electronic spin density on the two nearest neighbour atoms along the [111] direction. At temperatures above 100 and 300 K respectively Mu and Mu* are unobservable due to very rapid decay of the TFiSR precession signals caused by charge changing reactions 1  Chapter 1. Introduction  3  to a diamagnetic center- either Mu+ or Mu [4] .  The small electron densities in the Group Vb semimetals (antimony: , arsenic: ‘-2 x 1020 cm 3 cm , and bismuth: 3  x 1019  x 10” cm ) are intermediate be 3  tween an intrinsic semiconductor and a normal metal. Consequently it may be possible that the presence of the muon results in the formation of a paramagnetic complex, in which the unpaired electron is centered on or near the muon. This system of a single magnetic impurity in a metal (i.e. a Kondo impurity), which at first glance seems to be a trivial problem, is in fact of great historical significance in condensed matter physics. Kondo systems have been subjected to numerous theoretical and experimental investigations. [6,7,8] An exact solution to model Kondo systems has eluded researchers until recently.[9,10,11] Due to the long range of the RKKY impurity-impurity interac tion in dilute magnetic alloys even the most dilute alloys (i.e. p.p.m. levels) have exper imental impurity concentrations, which are often too great to avoid impurity-impurity interaction. [2,6] The isolated nature of a local moment of a muonium impurity in a semimetal or metal would provide an ideal testing ground for these theories, which have modelled the magnetic susceptibility and other bulk properties of single Kondo impurity systems. Previous 1 tSR 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 these semimetals.[12,13] Also Al, Zn, Cu, and C have been examined with SR for signs of 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 a temperature dependence, which is also unusual for normal metals. Within experimen tal uncertainty at all temperatures K with B 0 perpendicular to ê is half the size of K,, with B 0 parallel to ê (K-  =  K,’j). In order to gain insight into the origins of the  anomalous K,, alloys of Sb were studied. This work with SbSn and SbBi alloys shows,  Chapter 1. Introduction  4  K, is sensitive to impurity concentration. In general as the number of electrons on the Fermi 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 mea  surements of K,, in Sb with B 0 parallel to the ê-axis were made as a function of tem perature (2-20 K) and magnetic field (2-21 kG). At a temperature of 3 K de Haas-van Alphen oscillations of K were sought, but not found, which indicates the electron spin density does not simply scale with the magnetic susceptibility of the conduction elec trons. An upper limit of 0.2% of K, can be placed on the oscillation amplitude. The temperature dependence of K , below 20 K is best fit to a power law of the form: 1 1, K —  a+2 bT + cT 4  (1.4)  with parameters a=7.167(3) x iO, b=6.4(7) x 1O K . 4 , and c=1.8(2) x 10—10 K— 2 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 the low temperature Kondo susceptibility: =  a(1/2IrTK  —  0.433T / 2 ).  (1.5)  At low temperatures (2-10 K) this fit yielded a parameter a=14.0(0.3) K and a Kondo temperature TK=160(3) K, which leads to a Kondo exchange parameter J  3 eV. To  gether with existing measurements of K 1, at higher temperatures we observe a crossover from Curie-like behaviour above 90 K to weakly temperature dependent Pauli para magnetic behaviour below 10 K. This crossover is characteristic of a Kondo impurity indicating that the anomalous electronic structure of a positive muon in antimony may be due to local moment formation (i.e. muonium), in which the positive muon binds an electron, which in turn is strongly coupled to the conduction electrons by an exchange  Chapter 1. Introduction  interaction.  5  From the fit to Curie behaviour, estimates of the muonium hyperfine  parameter can be made: 11 A  =  259.7(4) MHz A  =  (L6)  0.05A  (1.7)  Aii,  which are similar to those of the anomalous muonium center Mu* in covalent semicon ductors such as GaAs. Since the tSR technique only permits one ,u in the Sb sample at a time, impurityimpurity interactions are not present. If indeed muonium is formed in antimony, this system would be a unique example of a truly isolated Kondo impurity. Furthermore it would provide information on the transition from the diamagnetic  çj+  state in metals  to the paramagnetic muonium state characteristic of semiconductors and insulators. Large Korringa-like relaxation in longitudinal field was also observed implying that the spin dynamics of a muon in Sb are also anomalous when compared to normal metals. A peak at a temperature of 75 K in the T’ relaxation rate also provides an estimate of TK. Below a temperature of 100 K the Korringa constant S/(KT T) is 1 temperature independent as is expected for a Kondo impurity. As the temperature approaches TK the breakdown of the Korringa law is seen. Chapter 2 discusses the band structure, crystal structure, and the Fermi surface of antimony. This is followed by a discussion of the standard theory of the Knight shift as well as previous SR investigations of K,,, and other relevant papers in Chapter 3. Then Chapter 4 offers a glimpse at some of the theory behind the Kondo Effect and then gives some models for both the T’ relaxation rate and the Kondo magnetic susceptibility, which we assume is proportional to the Knight shift. An overview of the tSR techniques and the experimental apparatus needed to investigate K,,, and T’ in Sb 1 can be found in Chapter 5. Chapter 6 deals with the specific experimental procedures  Chapter 1. Introduction  6  followed by a discussion of the results. The Conclusion provides a short summary of this thesis and raises several questions, some of which were answered and some of which still await answers. Appendix A outlines the de Haas-van Alphen effect in general. The K and T’ data is tabulated in the Appendix B.  Chapter 2  Crystal Structure and Fermi Surface of Antimony  Antimony (Sb) is a Group Vb semimetal with the same A7 crystal structure as the other semimetals bismuth and arsenic, Figure 2.1 a). The rhombohedral primitive unit cell is similar to the fcc primitive cell, which has e=0:  =  (E,1,1) 0 a  (2.1)  =  (1,e,1) 0 a  (2.2)  =  (1,1,e) 0 a  (2.3) (2.4)  where e is a small parameter related to the angle c between any two primitive lattice vectors 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 cell lattice point has a basis of 2 atoms at (0,0,0) and (2u,2u,2u). This structure can be visualized 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 atoms at all the Na and Cl sites.[16] The fifth and sixth bands overlap slightly yielding a conduction band electron den sity and a valence band hole density of about 5.5 x 1019 carriers cm 3 each.[17] The location of the electrons and holes in the Brioullin zone found from band structure 7  Chapter 2. Crystal Structure and Fermi Surface of Antimony  • =  8  space lattice  point  c)  b)  Figure 2.1: a) Rhombohedral crystal structure and primitive unit cell: Sb and other Group V elements have a basis of 2 atoms at (0,0,0) and (2u,2u,2u) at each point of the space lattice. [16] b) Brioullin zone of the A7 crystal structure. [16] The mirror plane c defined by the points FTXL and the point H on c are omitted for clarity.  Chapter 2.  Crystal Structure and Fermi Surface of Antimony  Structure NaC1 Sb  u 0.250 0.234  I 60° 57° 14’  0 0.0416  Table 2.1: Sb Lattice Parameters.[16]  tarT rx I.FL bis.Bsectrix  Electrons 4D5au  -t  Figure 2.2: Sb electron pocket on the Fermi surface. [17]  9  Chapter 2. Crystal Structure and Fermi Surface of Antimony  Extremal Period in iO G 1 Holes Electrons  10  I’T-TU plane 16.33 14.66  5.06 2.30  Table 2.2: Experimental de Haas-van Alphen oscillation periods for electrons and holes. [16]  calculations compare favourably with experimental results. The conduction electrons are 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 along the “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 centered on the point H in the mirror plane a defined by the points FTXL. The FX directions correspond to the three cubic [100] directions. The shape of these electron and hole pockets is roughly ellipsoidal (see Figure 2.2). The extremal areas of these electron and hole pockets, which make up the Fermi surface, have distinct de Haas-van Aiphen oscillation periods shown in Table 2.2. See Appendix A for an elementary discussion of the de Haas-van Alphen effect.  Chapter 3  The Knight Shift  3.1  Theory  The Knight shift was first observed by Walter Knight in a Nuclear Magnetic Resonance (NMR) study of Cu . He observed, that the resonance frequency in metallic copper 63 was 0.23% higher than in a diamagnetic copper compound, such as CuC1. In fact this frequency shift is ubiquitous in metals and it is defined by: (3.1)  ‘mWd+1  where  L.Ym  is the metallic resonance frequency and  wj  is the resonance frequency of  the same nucleus in a diamagnetic reference material. In general there are several contributing factors to the Knight shift. However, the dominant contribution for the most commonly observed cases arises from the Fermi contact interaction between the conduction electrons and the nucleus. In this case there are four common experimental observations: (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 increases with 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 aver age the nucleus experiences zero contact interaction. However, in the presence of a static, external magnetic field H 0 the electron spins have a preferred orientation an tiparallel to H 0 and the magnetic coupling will be non-zero. The nucleus experiences 11  Chapter 3.  12  The Knight Shift  a 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 spin polarization, which in turn is proportional to H 0 or  wd.  Therefore, the fractional shift  is independent of magnetic field. Since the Pauli spin susceptibility (and hence also the magnetization) of a Fermi gas of highly degenerate conduction electrons is temperature independent for temperatures below the Fermi temperature, the Knight shift should also be temperature independent. For a system with weakly interacting electrons the Hamiltonian is: (3.2)  Ee+’11n+’flen,  where fle is the weakly interacting electron Hamiltonian, ?-t. is the nuclear spin Hamil tonian, and ?len is the magnetic interaction of the electron spins and the nuclear spins. The main term of interest is the contact interaction the  j  -i. 7  of the l electron at  with  nucleus at nh2 e 7  flen  .  S(i  —  ).  (3.3)  In H 0 the polarized electronic spins will produce an average magnetic moment: . 0 >= H  (3.4)  The total spin susceptibility of the electrons is: =  J  (E)g(E,A) dE dA, 8 x  (3.5)  where 8 (E) is nonzero over a region of width kBT near EF, since only electron states near EF have empty electron states, into which they can be excited as the spin flips over. We define g(E, A) dE dA as the number of electron levels or allowed k values in a certain volume of  space, which lies between two constant energy surfaces, E  Chapter 3.  13  The Knight Shift  Ek±dEk  Figure 3.1: Volume in  k  Area dA  space associated with dE dA.[18]  and E + dE, and a small area element dA on one of the constant energy surfaces, as seen in Figure 3.1. The energy and the particular coordinate on the energy surface are denoted E and A respectively. The contact interaction with the en =  jh1  nuclear spin is:  ’zj [(Iuk(o)2)EFx:iIo], 1 Yn  (3.6)  where the average value of the probability of finding a Fermi surface electron at the position of the  j  nucleus is denoted:  )EF. u() 2 (u(O)  is a Bloch function with the  periodicity of the lattice. The contact interaction can also be written as . 0 LtjH  (3.7)  Chapter 3.  The Knight Shift  14  The magnetic interaction of the electron spins with a nuclear spin is equivalent to a change  1 Lt  in the moment of the nucleus, which produces a change in the resonance  frequency and hence a larger effective field at the nucleus of H 0 + LH. The Knight shift is: K  =  )EFX. 2 -lUk(O)  (3.8)  =  As discussed previously, in most cases the Knight shift is independent of temper ature and magnetic field. Since electronic wave functions are temperature and field independent, the anomalous dependence of the muon Knight shift in Sb on tempera ture might reflect the behaviour of the total spin susceptibility, which is proportional to the magnetization. As seen in Appendix A, the number of electrons at the Fermi sur face and thus the magnetization exhibit de Haas-van Alphen oscillations at high fields and low temperatures. So theoretically it is possible to observe Knight shift dHvA oscillations, which are ultimately due to oscillations of the number of electrons at the Fermi surface.  3.2  The Muon Knight Shift in Sb: Review of Previous Experimental Re sults  K in Sb is totally unlike the nuclear Knight shift normally observed in metals. The anomalously large K in Sb is temperature and orientation dependent. This much is known 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 Sb nucleus in a solid due to the low crystal symmetry, high atomic number, and a large quadrupole moment.[23,24] De Haas-van Alphen (dHvA) oscillations in the anisotropic and 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 anisotropic  Chapter 3.  The Knight Shift  15  and temperature dependent. One of the purposes of this thesis is to determine if the conduction electrons in Sb are responsible for the giant K. Since the Knight shift is proportional to the total spin susceptibility of the electrons, which in turn is proportional to the magnetization, dHvA oscillations of K 14 should be seen at the dHvA frequency or frequencies of those electrons, which contribute to the Knight shift. In this way the relative contributions of 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 role rather than the extended band electrons on the Fermi surface,then dHvA oscillations would not be seen. 3.2.1  First Discovery  The first observation of K,, in Sb showed that K,, was anomalously large, tempera ture, and orientation dependent.[13,19] Hartmann et al. concluded, that the tempera ture dependence does not exactly follow Curie behaviour as is expected for a paramag netic moment, but the anisotropy of the frequency shift reflects the hyperfine structure of local moment ions. There had been no previous observation of local electronic mo ments on the  site in a metal. It was suggested the large anisotropic 1(14 could be  explained by a muonium-like state, in which the unpaired electron was in a p-like orbital on the muon. The authors suggested that the electron bound to the muon was in a virtual bound state (VBS), as seen in Figure 3.2. Varying the electron concentration by varying the host metal changes the Fermi energy and the width of the virtual level. A local moment occurs if the Fermi level is near a virtual level and if the virtual level is sufficiently narrow in order to maintain the spin splitting.[26] The state is ‘virtual’  Chapter 3.  The Knight Shift  16  Figure 3.2: Two virtual bound states (VBS) with opposite spins. The spin up VBS is partly filled causing the formation of a localized moment. [26]  Chapter3.  17  TheKnight Shift  because the diamagnetic  +  precession is still present and ‘bound’ because of evidence  of local moment formation. The hyperfine field associated with this VBS at the lowest temperatures 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 suggested the giant K, seen in Sb has not been observed in Bi or As because perhaps the position of 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 of the contact interaction between the unpaired electron and the muon and, therefore, the local moment orientation fluctuates.  However, at low temperatures, the small  hyperfine field is able to fully align the electron moment on the muon and saturation of the frequency shift is expected. For temperatures less than 20 K the frequency shift should be saturated. The temperature dependence fit well to a Brioullin function with J=1/2 and 3/2:[19] Bj(x)  =  2J+  ctni  ((2J+1)x) —  ctnh  where the total angular momentum quantum number is J  =  L+S and x  (3.10) =  UBB/kBT. 1 gJ  However, at low temperatures, the frequency shift showed no sign of saturating with applied field, as one would expect from Equation 3.10 in the limit of high field and low temperature. The authors speculated that perhaps the temperature saturation was due to muon diffusion and motional averaging or perhaps the localized wavefunction itself is field dependent.[19] The K,, temperature dependence might also reflect a broadening of some critical feature of the density of states.  Chapter 3.  3.2.2  18  The Knight Shift  Self-Consistent Molecular Cluster Calculations  More recently the giant K, in Sb has been interpreted as an effect of the electronic density of states. [22] Self-consistent molecular cluster calculations suggest the positive K, could be due to a large peak in the density of states near the Fermi energy and a 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 was  assumed to be at the octahedral site in the center of a cluster of 26 Sb atoms. This could have a dramatic influence on the result. The width of the peak in the finite cluster broadens in the bulk semimetal to 0.2 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 is  approximately 15 for fields applied parallel to the ê axis, but if the  is a strong  perturbation to the Fermi surface, then the free electron g factor of 2 might be more suitable. [28]) Zeeman splitting separates the peaks in the density of states for spin-up and spin-down electrons. If the spin-up peak is shifted enough by an applied field to lie above the Fermi level, then a local moment will arise due to the spin-down electrons and saturation will occur. In the case of moderate magnetic fields the frequency shift will increase linearly with field. Thus this model qualitatively explains the lack of field saturation, but does not pro vide a natural explanation of the temperature dependence. Although the exact features of this large peak in the density of states are not known, the temperature dependence of I(. was attributed in a somewhat unsatisfactory way to thermal repopulation of levels within this peak. [22]  19  Chapter 3. The Knight Shift  I I  I  I  ‘  I? II  10eV EF  Figure 3.3: Sb electronic density of states and spectral density on the interstitial muon from molecular cluster calculations. [22]  Chapter 3.  3.2.3  The Knight Shift  20  Crystal Backscattering Resonance  The giant  has also been interpreted as a crystal backscattering resonance (CBR),  which arises from the combined resonance from electron scattering by an impurity or other defect and backscattering by the host lattice.[29] The CBR occurs at isolated defects, when the host density of states is low. Bound states may arise through the CBR mechanism because at the resonance it is possible, that the density of states is increased. The authors of Reference [29] suggest, that in the case of Sb the giant enhancement of KL reflects the increase of the density of states near the Fermi level due to an assumed CBR near the Fermi level. interstitial density of states on the  The broadening of this large host  due to either thermal broadening from the  Fermi-Dirac distribution or phonon-electron interaction, is consistent qualitatively with the temperature dependence of K in Sb. Thermal motion or diffusion of the  may  also be contributing factors. 3.2.4  SbBi and SbSn Alloys  Brewer et al. have measured K as a function of temperature in pure Sb and in Sb alloys with homovalent bismuth (Bi) and with heterovalent tin (Sn).[20,21] Data were taken 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 about  10 to 100 K. Also the frequency shift showed no saturation in applied field up to 9 kG in agreement with the 0.5-2.0 kG data of Hartmann et al.  Alloying with Bi  decreases both electron and hole concentrations. (At about 70% Bi the alloy becomes a semiconductor. [20]) As seen in Figure 3.4, at 6.3 at-% Bi K,. is reduced and K decreases more rapidly with temperature compared to K, in pure Sb. At 12.5 at.-% Bi, a K of order 0.1% has been observed with stroboscopic SR.[20] However, working with such  Chapter 3. The Knight Shift  21  concentrated alloys produced results which depended on the local Bi configuration. Alloying with heterovalent Sn decreases the number of conduction electrons while increasing the number of holes.[21J For Sn concentrations less than 0.1 at.-% a larger . than in in pure Sb was seen, whereas for Sn concentrations greater than 0.1 at.-% 1 K 1. was dramatically reduced. Since the electron pockets at the Fermi surface do not K vanish until a concentration of 0.63 at-% Sn, it seems a more subtle effect is at work. It was speculated, that the anomalous K 1. in Sb involves some feature of the density of states and that the temperature dependence was due to thermal broadening of this feature. At temperatures below 50 K the observed K 1. appears to be due to muons, which are isolated from the Sn impurities, whereas at higher temperatures a second signal attributed to a muon trapped at a Sn impurity appears. Zero field (ZF) tSR measure ments on pure Sb confirmed, that the  hop rate was relatively low at temperatures  below 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 light of results in this thesis. (See Chapter 6.) The high sensitivity of K 1. to alloying with Bi and Sn suggests that K 1. is quite sensitive to the Fermi level. 3.2.5  De Haas-van Aiphen Oscillations in Sn” 9 and Cd” Knight Shift  The effects of Landau quantization of electrons were searched for in NMR studies of both Sn 9 and cadmium (Cd”).[25,30] Reproducible dHvA oscillations of the Knight ’ 1 shift 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 of the 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 due  Chapter 3.  The Knight Shift  22  0.  0  0  20  30  40  50  60  70  80  90  100  120  T (K)  Figure 3.4: K 14 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.[20j  Chapter 3. The Knight Shift  0353  0.351  23  -  -  —  0.349 —  -  H  H 19724G I  I  I  -  0510  HI6025G I  9569  I  I  0.508  0.336  —  HI5873G I  I  0 627  0 629  t  0.625 (I  _4  G  -  )  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 the dHvA oscillations strongly depend on the shape of the Fermi surface, single crystal samples were used. A more general form of Equation 3.8 is the following sum over single particle states: K  16t  IUk,fl(O) 6 2 (EF  —  E(k)),  (3.11)  k,n  where n is the band index. [25] As seen in Appendix A Landau quantization of the electronic states at sufficiently high magnetic fields and low temperatures gives rise to an oscillatory magnetization and thus an oscillatory Knight shift. The Knight shift may 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.  where  The Knight Shift  (Iuk(O) ) 2 o  EF  24  is the wave function averaged over a certain portion of the Fermi  surface. The ratio of oscillatory to non-oscillatory Knight shift is given by: K K  (Uk(O)I2)Q  =  X  EF  (IUk(O)12)EF  (3 13)  and for Cd 1.5!,  (3.14)  where the Knight shift is measured by NMR and the susceptibility by a force magne tometer.  Chapter 4  The Kondo Effect  4.1  Introduction  The appearance of a resistivity minimum at a temperature on the order of 10 K in metals 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 had been examined. [31] In this analysis the behaviour of a magnetic impurity was found to be no different from that of a nonmagnetic impurity. In 1964 this resistivity minimum was explained by Kondo’s analysis of higher order terms of perturbation theory, which showed some unexpected behaviour arising from impurities with magnetic moments.[32] The origin of the Kondo effect is the exchange interaction of the conduction electrons spins  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] This interaction leads to spin exchange scattering, in which the spins of the bound electrons and the conduction electrons near the Fermi energy flip flop. The Kondo Effect arises from high order terms in perturbation theory, which yield a magnetic scattering cross section, which increases as the temperature decreases. This process is characterized by a so-called Kondo temperature: TK  TFe_hhIJIM)  25  (4.2)  Chapter 4.  The Kondo Effect  26  where TF is the Fermi temperature and N(O) is the density of states at the Fermi level.[33] The combination of the phonon contribution to resistivity, which decreases as temperature decreases, and the spin flip scattering terms yield a resistance mini mum around TK. However, since this exchange model assumes the existence of a local magnetic moment, the model does not explain the stability of the magnetic moment in the presence of conduction electrons. Although the problem of a single local mo ment in a metal crystal seems very simple at first glance, an exact solution of the model Hamiltonian involving Equation 4.1 has eluded researchers until recently.[9,1O,11] Since Kondo’s perturbation analysis of the exchange Hamiltonian, hundreds of papers and numerous review articles have been published on a wide variety of Kondo phenomena  arising from flev.[6,7,8] In dilute solid solutions of a magnetic ion in a nonmagnetic metal crystal the mag netic ion can be dramatically altered by the metal. If the partially occupied ionic level responsible for the magnetic moment lies slightly below the Fermi energy, then conduction electrons could fill some ionic levels and thus the magnetic properties of the impurity ion will change. Also since the ionic energy level in question lies in the continuum of conduction electron levels, there is a strong mixing of localized ionic and conduction electron levels. The ionic levels become partly delocalized and the conduction band wave functions are altered near the ion.[31] The nature of this ex change interaction between the impurity and the host electrons and its consequences on the bulk effects, such as the susceptibility, have been investigated both experimen tally and theoretically. At high temperatures (T>> TK) the Kondo impurity displays Curie-like behaviour, whereas at low temperatures (T << TK) Pauli paramagnetism is evident.[7,9,1O,34,35,36] The physical interpretation of these phenomena is that at high temperatures the moment behaves as though it were isolated, since the conduction electrons produce no fundamental change in magnetic properties. On the other hand  Chapter 4. The Kondo Effect  27  at low temperatures the influence of the conduction electrons is profound, since the magnetic behaviour is determined by the electrons and the moment.  4.2  Dilute Alloys  So far NMR experiments to study the Kondo effect have had to take into account the possibility of interactions between impurities, which are always present even at low impurity concentrations. The long range of the RKKY interaction, which is the oscil latory polarization of the conduction electrons by dilute local moments that decreases with distance as 1/,’, makes the study of isolated Kondo impurities difficult. pSR ex periments are ideal in this respect, since only one p+ is in the sample at a time and so there can be no interactions between muons. In order to avoid impurity-impurity interaction, it has been found empirically, that the following relation must hold: /TF > c, 0 T  (4.3)  where T is the Curie-Weiss temperature, Tp is the Fermi temperature, and c is the impurity concentration. [6] For many alloys even p.p.m. concentrations are too high. The RKKY interaction arises from a first order perturbation estimate involving the exchange Hamiltonian of two impurities and leads to an oscillatory polarization of the conduction electron spin density proportional to cos(2kp’r)/r . The Kondo effect 3 arises from higher order perturbation terms. This strong RKKY interaction is what makes the Kondo effect for an isolated impurity so difficult to detect in all but the most dilute alloys. RKKY interactions must be avoided for experimental measurements to represent accurately the behaviour of an isolated impurity.  Chapter 4. The Kondo Effect  4.3  28  Some Models  In this section we briefly review a few models of an isolated Kondo impurity. Their results for the bulk magnetic susceptibility will be compared with experimental results on K, in Sb, which is thought to be proportional to the magnetic susceptibility. 4.3.1  Moment Formation and the Anderson Model  The development of the theory of dilute magnetic alloys evolved from two seemingly separate investigations: one on the interactions of local moments and conduction elec trons and the other on the formation of a local moment in a metal. The former topic is addressed by the exchange Hamiltonian, Equation 4.1, and the latter by the more fundamental Anderson Hamiltonian. The Anderson model was developed in 1961, before Kondo’s analysis of the higher order perturbation terms, to describe the scattering of conduction electrons by magnetic iron-group ions dissolved in nonmagnetic metals:[37] = ?Iof  +  ?loci  +  7Icorr  (4.4)  + ?Iex.  The unperturbed free-electron Hamiltonian is: ‘Hof  where  ek  =  (4.5)  -, 0 Ekflk,  is the free electron energy for a state with momentum k and  is the number  operator for electrons with momentum k and spin a. The energy of the unperturbed d state of the impurity atom is given by: ‘Hod =  (4.6)  Ed(nd+ + rid_),  where Ed is the unperturbed d state energy level and  d+  is the number operator for  electrons with spin up. The eigenfunctions for the localized impurity d state are the  Chapter 4.  The Kondo Effect  29  E  Ed+U  EF—Ed  Ed  g(E) Figure 4.1: The position of the virtual bound states in the Anderson model. U is the Coulomb 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  30  1.0  0 0  1.0  2.0  7v/Y=7vA/U Figure 4.2: Phase diagram of magnetic and nonmagnetic behaviour for the Anderson Model. U is the Coulomb repulsion energy, L is the width of the virtual d state, EF is the 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, the level is doubly occupied. [37] —  —  Chapter 4. TheKondoEffect  wavefunctions  .  31  The Coulomb repulsion between these d states is described by: 7tcorr =  (4.7)  Und+nd_,  where U = (ev  f  kd(r2)I Id(r1)I r c 2 1  —  i-’  (4.8)  dT 1 dr . 2  is the exchange interaction, Equation 4.1. The Anderson model describes a highly  simplified quantum state of an impurity in a metal, in which a moment may or may not 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 of unbound or free electron states because of the conduction band’s large width. These free electron states cannot be localized completely and so the concept of a virtual level of width L\ is used.[37] The d electrons of an impurity atom may retain much of their local character, but the conduction electrons can broaden and shift the unperturbed d energy level giving rise to a virtual bound state. This d state is a good eigenstate of only 7- and not of ?-t. Therefore the d state conduction electron in this virtual bound state eventually returns to a conduction band state of similar energy. [39] When both the Coulomb repulsion energy and the distance of the d state below the Fermi level are much greater than  ,  the Anderson Hamiltonian reduces to the  exchange Hamiltonian.[7,9,37] In other words the addition of one electron to the d state occurs at energy Ed, as shown in Figure 4.1. The addition of another electron will be at the energy Ed + U due to Coulomb repulsion. If the energy difference U exceeds the width of the virtual level and (Ed + U) > EF, then the d state is singly occupied and the impurity possesses a magnetic moment.[9] However, when the Coulomb repulsion is small compared to either the width of the virtual level or the distance (EF  —  Ed) of  the d state below the Fermi level, the population of the impurity level is the same for both spin states and no magnetic moment exists.  Chapter 4.  The Kondo Effect  32  Assuming a local moment exists means the d state of the impurity with spin up is full or partially filled and with spin down is empty. When there is equal occupation of spin up and spin down states, the local moment disappears. When the Anderson Hamiltonian is analyzed under the Hartree-Fock approximation, it yields a phase dia gram, which is sharply split into regions of magnetic and nonmagnetic behaviour, as seen 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 based upon the location of the ci state relative to the Fermi level and the ratio of the Coulomb repulsion 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 a longer lifetime (.-h/LS. In other words when U is large compared to L, screening is ineffective and the magnetic moment persists.[7] In the following two cases the d state has no local moment. When (EF  —  Ed)/U is less than zero, the d state is above the  Fermi level and the d state is empty. When (EF  —  Ed)/U is greater than one, the level  is doubly occupied. [37] 4.3.2  The Spin-Fluctuation Model vs. the Spin-Compensation Model for the Kondo Effect  Two seemingly contradictory explanations of the Kondo effect have emerged: the local moment spin flip model, as originally proposed by Kondo, and the spin fluctuation model.[8,33,34,40] The dual explanation originates from the two ways one can look at the typical magnetic properties of dilute alloys (enhanced Pauli paramagnetism at low temperatures and Curie-Weiss behaviour at high temperatures): one way is to determine how surrounding electrons interact with a well-defined spin to create a non magnetic system at low temperatures and the other way is to find how a nonmagnetic alloy with spin fluctuations diplays magnetic properties at high temperatures. [34]  Chapter 4. The Kondo Effect  33  In the spin-compensation picture the impurity electron is surrounded by an increas ing number of opposite spin host electrons, as the temperature decreases. The spin cloud compensates the impurity spin and the paramagnetism is degraded. This spin compensation 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 of kBTK.  The details of the smooth transition from the spin correlated to the magnetic  state, which occurs around TK, are unknown. More information about the low-lying excited states of the Kondo system would be needed. [33] The spin-fluctuation picture assumes a nonmagnetic state: a single virtual level in the Anderson model at Ed. However such an impurity, which is nonmagnetic in the Hartree-Fock approximation of the Anderson Model, does in fact exhibit magnetic be haviour when localized spin fluctuations are taken into account. A conduction electron can occupy the virtual level at Ed (described by the Anderson Model) of width L for a time proportional to h//I The electron in question retains memory of its spin for a longer time than h//i because of the Coulomb repulsion U between d state electrons of 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 compared to 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 fluc tuation rate, the impurity spin amplitude averages to zero. The impurity becomes nonmagnetic and shows a Pauli susceptibility. The slow transition between magnetic and nonmagnetic behaviour of a dilute alloy occurs at temperatures on the order of the spin fluctuation characteristic temperature T f 3  =  f. 8 h/kBr  When the localized spin  fluctuation rate is less than the thermal fluctuation rate, then the local susceptibil ity decreases and shows a Curie-Weiss dependence. The Curie-Weiss temperature is proportional to T f.[33,40] 3  Chapter 4.  The Kondo Effect  34  Although the spin-compensation and the spin-fluctuation models appear very dif ferent mathematically, it is not clear whether the spin-compensated and the spinfluctuation states apply to physically distinct systems. A superficial link between these two models can be made by equating T f and TK.[33] 3 4.3.3  The Resonant-Level Model  As previously mentioned the magnetic behaviour of the dilute alloy is either a well defined spin at high temperatures which is compensated by the host spin cloud at low temperatures or a nonmagnetic spin fluctuation at low temperatures which becomes a local moment at high temperatures. These two models can be shown to be equivalent f 8 (T  =  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 the case of a one dimensional classic Coulomb gas; positive and negative charge is analogous to spin up flips and spin down flips. There is a resonance level at the Fermi energy with a width  ,  which is roughly kBTK. The static susceptibility obtained from this  model[35] at low temperatures (T << TK) is: (4.10)  X+bT2.  In a later paper the authors of Reference [35] provide another expression for the low temperature susceptibility: [36]  x  2S 2 =  (gi)  I -  2ir 2 ”\ 2 /kT’\ 3(2S +1)  k-K))  (4.11)  Chapter4.  The Kondo Effect  35  This model[36] predicts behaviour of Curie-Weiss form at high temperature (T>> TK): 1  S(S+1) 2 x=(gIB)  .  01GB  where  I  (4.12)  ,,  +U  is related to /. by: k8  0 233 2 + 3S + 3/2 2S s+1  =  (4.13)  The authors of References [35,36] remark that, although the RL model describes ex perimental behaviour at a finite field for an arbitrary spin at most temperatures, the model is not exact. Rather its usefulness lies with its simplicity. 4.3.4  Exact Solutions of the Kondo Susceptibility  Both the Kondo and Anderson models possess integrable Hamiltonians, which can be solved exactly by the Bethe-Ansatz technique.[9] Numerical results[1O] for the Kondo model obtained by the Bethe-Ansatz technique yielded the magnetic susceptibility for a spin 1/2 impurity in weak applied magnetic fields and at low temperatures (,UBH << kBT << kBTK): x  =  1 2 T — 433 O. —-. 2TrTK TK  (4.14)  The full temperature dependence of the zero field susceptibility for spin 1/2, 1, and 3/2 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 Bethe Ansatz technique yielded the exact magnetic susceptibility for a Kondo impurity, as seen in Figure 4.3. However, no closed form expression was obtained. 4.3.5  The Korringa Relation  An impurity nucleus will experience relaxation due to field inhomogeneity arising from the interaction of the nuclear and the conduction electron spins.  (This concept is  Chapter 4.  The Kondo Effect  I  to  IIIIII  36  I  I  111119  I  I  I  111111$  I  111111  —  GIN t112  -  J I  0  1111111  1  I  I  1111111  1  1111111  10 (2j + 1 )T/T 0  100  111111  1000  Figure 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  37  discussed in Sections 5.2 to 5.6 and in Section 6.3.10.) By examining the local spin susceptibility the Korringa relation (K T=constant) relating the spin relaxation time 1 T 2 1 and the Knight shift K was shown to be valid for the Anderson model at low T temperatures (T << TK) and low fields (gBB << kBTK).[41,42] For U/Li relaxation rate T 1 has a peak around T  TK.[42]  0.5 the  Chapter 5  SR  Probing 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 Rota tion/Relaxation/Resonance. Detailed discussions of iSR are found in References [2,46]. This chapter discusses transverse field (TF) and longitudinal field (LF) tSR, which are used to study K 1 and T’ in Sb respectively.  5.1  Muons and Muon Beams  Positive 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 at  medium energy meson factories like TRIUMF by colliding intense proton beams of energy 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 used to produce spin polarized muon beams. Since the pion decay is a two body problem and the muon neutrino spin is antiparallel to its momentum, the muon spin must also 38  Chapter 5. SR  39  be antiparallel to its momentum in order to conserve angular momentum. In the rest frame of the pion the muon is emitted with a momentum of 29.7894 MeV/c. If muons emitted within a small element of solid angle are used to make a beam, then this so called “surface  beam” will be very nearly 100% spin polarized.  The muon is a spin 1/2 lepton with a mass of 105.6595 MeV/c 2 or 0.1126096 m, where m is the proton mass. The mean life of the muon (rn) is 2.19714(7) R 5 in free space. The positive muon decays into a high energy positron, an electron neutrino, and a muon antineutrino: e + “e  +i•  (5.4)  In this three body decay the emitted positron energy varies continuously from zero (if the 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 integrating over the neutrino momenta, the probability per unit time for a positron emission at an angle 6 to the  spin is: dW(e, 6) = [1 + a(e)cos(6)]n(e) de dcos(6),  (5.5)  TIA  where a(e) = (2c  —  1)/(3  —  2e) and n(e) = 2E (3 2  1 for maximum positron energy and  —  2c). The asymmetry parameter a is  when averaged over all energies. The reduced  positron energy is e = E/Emaz, and the maximum e+ energy is Emax = 52.8 MeV or about half the muon rest mass energy. The radial plot of the probability Win Figure 5.1 is given for energies between the maximum e+ energy e = 1 and e = 0.5. The average e 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 consist ing of crossed electric and magnetic fields, which are both perpendicular to the muon beam. In addition the Wien filter acts to rotate the muon spin away from the direction  Chapter 5. 1 SR  40  /7//i’ll//If/Ill  I I  I  /+  \\\\\\\\\\\\\\ \  / \\\\\\\\\\\ \ \ \ 26 MeV -  e  +  52 MeV  0 muon spin  Figure 5.1: Decay positron emission probability for angle 6 with respect to the for reduced energies from e = 0.5 (26 MeV) to 1.0 (52 MeV).  spin  Chapter 5. ,uSR  41  of the muon momentum due to precession in the magnetic field. With this spin rotator the  spin can be aligned either parallel to the applied magnetic field at the target  for LFtSR or perpendicular for TF-SR. The beam passes through a thin vacuum 1 window in the beam snout, before entering the experimental area. At the TRIUMF M20 beamline the surface beam has a maximum intensity at a muon momentum of 28 MeV/c which corresponds to a range of about 120 mg cm 2 in carbon. Just before entering the cryostat the muon beam is collimated to a spot size of about 1 cm, which is smaller than the typical sample.  5.2  tSR Geometry and Technique 1 TF-  In TF— SR the muon spin polarization is perpendicular to the applied field, as shown 1 in Figure 5.2. In high magnetic fields it is necessary, that the muon momentum is along the field direction in order to avoid deflection of the beam. The superconducting solenoid HELlOS used in this experiment is capable of magnetic fields up to about 70 kG, but normally fields above 20 kG are not used because conventional scintillation counters have a timing resolution of about 2 ns, which limits the frequency to less than about 300 MHz. At higher frequencies there is a precipitous drop in the precession amplitude. Also at 20 kG the beam luminosity at the sample position is maximum due to 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 time evolution of the anisotropic positron decay distribution. Each detector in Figure 5.2 consists of a plastic scintillator connected to a photomultiplier tube (PMT) via a lucite light guide. When a charged particle such as a  or a e+ passes through and deposits  some energy in the scintillator, it emits a light pulse, which travels down the light guide to the PMT. The light guide is needed because the PMTs, which do not function in  Chapter 5. SR  42  B  Figure 5.2: TF-tSR experimental geometry. A schematic showing four positron coun ters (Up, Down, Left, Right), the thin muon (TM) counter, and the initial muon polarization (Pu), which is perpendicular to both the beam momentum (pa) and the applied field (B).  Chapter 5. 1 .tSR  43  a strong magnetic field, must be placed well outside the magnet. The PMT converts the light pulse into an electronic signal, which is carried via coaxial cable to the fast electronics for processing.  The thin muon (TM) counter between the beam snout  and the sample detects incoming muons and, with much less efficiency, high energy positrons and gamma rays. In order to minimize multiple scattering of the incoming muons, which will cause the muons to miss the sample and add to the background signal, the muon counter is about 250 ,um thick. In TF-uSR the four positron counters surround the target, as shown in Figure 5.2. They detect the decay positrons from muons 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 decay positrons are detected. The beam rate is reduced so that only one muon at a time is in the sample during the typical observation period of 10 s. The signal from the TM counter starts a clock and defines time zero. A signal from either the U,D,L, or R positron counter stops the clock and then the time bin in the appropriate decay positron counter histogram is incremented by one event. A good event is one, in which there is precisely one muon and one positron in the 10 1 us data gate. Otherwise the event is discarded because the decay positron cannot be unambiguously related to the parent muon. Typically about iO events are recorded. The resulting time histograms may be regarded as the ensemble average of such decays. The muon spin precesses in an internal magnetic field consisting of the applied field plus local fields. Consequently the anisotropic positron decay distribution also precesses at a frequency  =  1 where ‘y yB  =  2ir x 0.01355342 MHz/G. Inhomogeneities  or fluctuations in the internal field will result in dephasing or spin relaxation of the precession amplitude. Typically a gaussian relaxation function  is used to  describe the dephasing due to inhomogeneity from nuclear dipolar fields or the magnet.  Chapter 5. ,uSR  44  F  TM  Figure 5.3: LF-tSR geometry showing forward (F) and backward (B) counters, the thin 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 Technique  In longitudinal field muon spin relaxation (LF-/USR) the muon polarization is parallel to the applied field. In the absence of spin relaxation the implanted  would preserve  its initial polarization because its spin would be locked along the applied field direction. However, dynamic processes in the sample, such as fluctuating local moments, create magnetic field fluctuations at the  site causing muon spin relaxation. The decay  positron asymmetry is measured using backward and forward counters, as shown in Figure 5.3. Typically the B counter has a hole in the middle in order to avoid the path of the muon beam and to accommodate cryostats or other geometrical constraints.  Chapter 5. uSR  5.4  45  Experimental  The actual experimental apparatus is shown in Figure 5.4. In this experiment a single crystal 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 purity 99.9% Ag was mounted on top of the Sb, in order to provide a reference  +  precession  frequency. Ag was chosen because it has few nuclear moments to cause broadening and the Knight shift (K =94.0(3.5) p.p.m.) is known to be small and very weakly 80 temperature dependent. [2,23] The Sb sample and the Ag reference were mounted inside a helium gas flow cryostat shown in Figure 5.4. The temperature was monitored with two carbon glass resistors on the helium diffuser and on the sample holder. The fluctuation in the temperature was less than 0.1 K and the accuracy is estimated to be better than 0.5 K. De Haas-van Alphen oscillations require high magnetic fields and low temperatures, as is explained in Appendix A. Consequently in this part of the experiment the magnet was varied in the maximum range possible to avoid severe loss of precession amplitude and the cryostat was operated close to its base temperature of 3 K. The four positron counters are arranged cylindrically because of geometrical restric tions imposed by the magnet’s cylindrical bore. The counters are labelled Forward Top Right (FTR), Forward Top Left (FTL), Forward Bottom Right (FBR), and Forward Bottom Left (FBL). In the positions shown in Figure 5.4 these four counters function as U,D,L, and R described earlier during the discussion of TF-SR geometry. “For ward” is specified for these counters because for LF-jtSR these counters are logically connected (“or”ed) using coincidence units to form a single forward counter. Four similar counters upstream of the target are “or”ed to form a backward telescope.  Chapter 5. uSR  46  Sample Rod He Space Vacuum Cryostat TM Counter  FT R Counter  +  /.L  spin  ,ubeam Sb Sample Beam Vacuum HELlOS Magnet Bore 0150mm  FBR Counter Approx. Scale: 0  75  150mm  Figure 5.4: A cutaway showing the central portion of TF-SR experimental apparatus.  Chapter 5. u 1 SR  5.5  47  TFt 1 SR Data  A typical histogram from one of the positron counters, such as FTR, displays the oscillatory pattern of the  precession in a transverse field of 100G superimposed on  the 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 time evolution of the x component of the  spin polarization, and b is the random back  ground from uncorrelated +e+ events. Asymmetry histograms, such as the one in Figure 5.5b), are formed after removing the random background b. If the muon de cayed isotropically, then each decay positron histogram would be identical at all times with no asymmetry except for trivial normalization effects. However due to muon decay asymmetry histograms corresponding to detectors, which are 180 degrees out of phase, will display a time dependent asymmetry due to the precession of the anisotropic decay of the muon. The asymmetry of two matched counters is the main observable quantity in TF-tSR: A(t)  —  HFTR(t) HFBL(t) HFTR(t)--HFBL(t)’ —  (5 7)  where the only remnants of the muon’s decay are the error bars, which are proportional to the square root of the number of events in the time bin and which grow exponentially with time. In TF-SR with an applied field B  = B, which is perpendicular to the axis joining  the two counters, the asymmetry as seen in Figure 5.5b) is given by: A(t) where g is the phase.  =A G(t)cos(wt + ), 0 P(t) = A 0  (5.8)  The amplitude of precession, A , is about 25% because of 0  the counters’ solid angle, positron absorption, and muon decay kinematics. As noted  Chapter 5. 1 iSR  48  a)  800  b)  E1 Time (,us) 2.0  I  I  c) L  1.5  0 0  ci)  1.0  S.  0  0.5 I  0  1.0  I  2.0  3.0  4.0  Frequency (MHz)  Figure 5.5: a) Decay positron time spectrum showing muon precession and exponential decay. b) Asymmetry histogram showing muon precession. c) Fast Fourier Transform.  Chapter 5. 1 uSR  49  previously the maximum observable frequency is limited by the timing resolution of the scintillation counters. The resolution of a matched pair of counters can be estimated by looking at the full width half maximum of a time histogram of pulses, which are generated simultaneously in each counter with back to back y rays. Typical timing resolution is on the order of 2 ns. A gaussian distribution of internal fields leads to a gaussian relaxation function: G(t)  =  22 e .  (5.9)  Static nuclear dipolar fields are one source of such line broadening effects Often the signal is analyzed online with fast Fourier transforms in order to obtain an initial estimate of the precession frequency, as shown in Figure 5.5c). In order to determine more accurately the experimental parameters (asymmetry, phase, Gaussian relaxation rate, and precession frequency), x 2 minimization is used to fit the time histograms. If the muon precession frequency is very high the data are often analysed by 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 fitting procedure.  5.6  LF-tSR Data  In this type of measurement the muon polarization seen in the forward and backward telescopes is given by: F(t)  =  N(O)e_t’T[1  B(t)  =  N(0)ethfT1[1  + aP(t)] + b  (5.10)  aP(t)] + b,  (5.11)  —  where the asymmetry parameter a is defined in Section 5.1, P(t) describes the time evolution of the component of the muon spin polarization along the field direction, and  Chapter 5. ,uSR  50  0.03 0.02 —‘  001  N  0— 0  0 —0.01 —0.02 0  2  4  6  8  Time (ts) Figure 5.6: An example of LF-pSR asymmetry histogram showing an exponentially relaxing muon spin polarization.  b is the random background. After removing the background terms b the time evolution of 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 interaction between the  spin and other magnetic moments in the material. Typically this  relaxation can be fitted with an exponential function. LF-SR is particularly sensitive to the temporal fluctuations of these moments. Often the temperature has a dramatic influence on the longitudinal relaxation rate T . 1  Chapter 6  Experimental and Results  Most of the data for this thesis were obtained through TF and LF-SR experiments. De Haas-van Aiphen oscillations of K . were sought for, but not found. The tempera 1 ture dependence of 1(1. was examined from 2-20 K. Previously published data[20J from 90-180 K were combined with our low temperature data. A crossover from a weak temperature dependence from 2-6 K to a Curie-like behaviour from 90-180 K is seen. With LFtSR the Tj 1 1 relaxation rate showed only a weak dependence on field, but the Korringa constant(S/TiTI ) showed an anomalous temperature dependence. Two 0 additional experiments were performed on the same Sb crystal: Shubnikov-de Haas oscillations of the conductivity were examined to determine if there was a strong ori entation dependence. Finally de Haas-van Alphen oscillations of the magnetization, which were expected to be proportional to , 1. were measured, in order to compare K with measurements of . 1. K  6.1  Shubnikov-de Haas Oscillations  A steady state NMR spectrometer was used and the applied field was scanned from 18 to 20 kG and the Sb sample was held at a temperature of about 3 K. The ê axis of the Sb single crystal was aligned at angles from -11 to +20 degrees from the applied field. This experiment confirmed the presence of Shubnikov-de Haas oscillations in the conductivity of Sb. In addition to the main field a small modulation field was used. Using phase-sensitive detection techniques the first derivative of the conductivity 51  Chapter 6. Experimental and Results  oscillations was observed by measuring the  52  Q of the  pickup coil, which is a function of  the surface impedance of the sample in the coil. It was found that the orientation of the sample does not strongly affect the amplitude of the oscillations over the range of angles examined. Any misalignment of the sample in the pSR experiments would not exceed more than one or two degrees. Thus before the present SR study was done, it was known through these measurements, that small misalignments of the Sb sample’s ê axis away from its parallel orientation to the applied field would not strongly affect either the period or the amplitude of the oscillations.  6.2  De Haas-van Aiphen Oscillations of the Magnetization in Sb  In a force magnetometer at a temperature of 2 K strong susceptibility oscillations, as seen in Figure 6.1, were detected in a 0.5 mm thick triangular piece of the Sb crystal with the sides measuring 3 mm, 3 mm, and 5mm. The peak-to-peak amplitude of these oscillations was about 15% of the total magnetization. The period was 353(9) G in B or about (9.8+0.2) x i0 G 1 in 1/B, which is close to the expected dHvA period in Table 2.2. Although dHvA oscillations are oscillations in the diamagnetic susceptibility, the density of states at the Fermi surface also oscillates and therefore the paramagnetic spin susceptibility shows the same variation with magnetic field. It is this effect which produced the oscillations in the Cd Knight shift discussed in Section 3.2.5. If electron states at the Fermi surface are responsible for the large K in Sb, then one might expect to see K osillations of similar amplitude and periodicity in the 1 tSR study.  Chapter 6. Experimental and Results  53  2.75  +  +  2.7  ++ +  ++  o  2.65  +  ++ +  ZcO  —‘  LU o 2.55 I—  E  E-  25  +  ++ + +  +  +  + +  +  +  +  +  +  +  ++  ++++  ••++ + i  +++:  20.000 19.600 19.200 18.800 18.400 18.000 19.800 19.400 19.000 18.600 18.200 FIELD (GAUSS) (Thousands)  6.1: De Haas-van Aiphen oscillations of Sb magnetic susceptibility at a temper of 2 K showing a period of 353(9) G and a peak-to-peak amplitude of 15% of the  Figure ature  magnetization.  Chapter 6. Experimental and Results  54  C-CXIS  cm 0.75 cm___ (  2cm  0.5cm  Figure 6.2: Cross-section of the Sb single crystal sample. 6.3  SR in Sb  6.3.1  Sample Characteristics  In 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 in Figure 6.2. For this thesis the ê axis crystal was always oriented parallel to the applied magnetic field. A thin strip of high purity 99.9% Ag from Aldrich was mounted on top of the Sb, in order to provide a reference  precession frequency, which is unaffected  by nuclear moments or large Knight shifts. The experiment was performed on the M20 beamline at TRIUMF. The surface muon beam has a maximum intensity at a muon momentum of 28 MeV/c.  With  surface beams, targets with masses per square centimeter as low as 10 mg cm 2 can be studied.[2] Our 0.5 cm thick antimony (pSb=6.691 g cm ) sample was clearly thick 3 enough to stop all the muons. 6.3.2  Measurement of Muon Knight Shift with TFiSR 1  In these TF-.tSR experiments, clear identification of the  precession frequency in  both the Sb sample and the Ag reference, a thin strip of 99.9% pure Ag mounted on  Chapter 6. Experimental and Results  55  top of the Sb sample, was sought. Since K in Sb is so large, the Ag reference frequency and the Knight shifted Sb frequency were easily resolved in fast Fourier transforms, as seen 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 strong relaxation in the Sb signal, as seen in Figure 6.4, limited our resolution of the pre cession frequency to about 0.001 MHz, which ultimately limited the precision of our observations of the muon Knight shift in Sb to about 0.1% of K. The relaxation in the Sb is attributed primarily to nuclear dipolar interactions. The field dependence of T’ in Figure 6.4 may be due to magnetic field inhomogeneity. Relaxation due to the  magnetic field’s spatial inhomogeneity would show a linear increase with field. How ever, in high solenoidal fields the muon beam spot size on the sample decreases and so the spatial inhomogeneity becomes less important.[47] This could explain the shape of the curves in Figure 6.4. The stronger field dependence in Sb likely arises from the larger size of the Sb compared with Ag. The relaxation in the Ag at low fields may be due to trapping at impurity sites. The decay e+ asymmetry histograms, such as the one in Figure 6.3, were analyzed with fast Fourier transforms to estimate the frequencies and then fitted in time by 2 minimization, in order to obtain accurate values for the experimental parameters: x asymmetries (a), phases  (),  Gaussian relaxation rates (T’), and  +  precession fre  quencies (w) in both the Ag reference and the Sb sample. The time evolution of the muon spin polarization was assumed to be of the following form: P(t)  =  Re{exp[_(t/Tj) 9 aA 2  —  Q’Agt + bAg)]  2 } + asbRe{exp[—(t/Tj)  —  i(wSbt  + bsb)] }.  (6.1) As explained previously the signal was fitted in a reference frame rotating about 0.5  Chapter 6. Experimental and Results  MHz less than the Ag reference  56  precession frequency. The muon Knight shift relative  to Ag was then calculated: K  —  fsb  —  Hsb  —  —  HAg  —  —  JAg  K  —  —  Ag  206.699(1) MHz 203.859(2) MHz  1  —  0 01393 ( 1  63  —  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 uncer tainty of 0.1% is an improvement by a factor of about five over previous JLSR investi gations. This is due to our increased statistics, the high magnetic field, which permits clear distinction of the Ag reference frequency and the shifted Sb frequency, and the simultaneous measurement of both  6.3.3  precession frequencies.  Magnetic Field Dependence of the Muon Knight Shift  At a temperature of 2.7 K a preliminary investigation was conducted: a field scan from about 2.3 to 21.1 kG with B parallel to the ê axis of Sb, as seen in Figure 6.5. In Figure 6.6a) there was a hint of an oscillation in K with a period of about 0.5 kG in B or about 14 x iO G’ in 1/B, which is close to the expected dHvA period for Sb given in Table 2.2. However the observed variation in K was not much larger than the statistical error bars. Thus even from this preliminary study it is clear that the amplitude of any oscillations is less than or equal to about 0.2% of the total Knight shift. The magnetization oscillations had an amplitude about 40 times greater under similar 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  57  a)  0.10 0  x  0  o.10 Time (ps) 0  x  5 b)  4 a) 0  3 2  a)  1  0  0  244 248 252 Frequency (MHz)  Figure 6.3: a) Rotating reference frame fit to asymmetry histogram showing beating of the Ag reference and the Sb sample precession frequencies. b) FFT showing + precession frequency in Ag reference and Sb sample. The Sb frequency Knight shift can be clearly seen. Data from a run at T=2.2 K and B=18.05 kG.  Chapter 6. Experimental and Results  58  0.45 Cl)  ci)  04  Sb  0.35  ---J  0  c  0.25  C  .2  0.2  -H  0  x  Ag  0.15  0 -  0.1 0.05  cs  F—  0 0  2  4  6  8  10  12  14  16  18  20 22  Magnetic Eied ‘(kG) Figure 6.4: TF-SR linewidth or Gaussian relaxation rate as a function of magnetic field at T=2.7 K.  Chapter 6. Experimental and Results  I.. .%1  I  EE  I  I  59  I  I  I  I  4.0  4  c)  :3.87 -  _c  cz5,  -)  <  (f)  >N (_)  L)  -  3.6  19.25  21.25  Cl) 4—  1  CG)  -  I  I  I  1.42  I  I  I  I  I  b)  1.41 Cf)  1.40  •c  1.39  1..38  + I  0  I  I  I  I  I  I  16 20 4 8 12 Magnetic Fied (kG)  Figure 6.5: a) Frequency shift in Sb relative to Ag reference as a function of magnetic field and b) the Knight shift as a function of field at T=2.7 K.  Chapter 6. Experimental and Results  60  a)  1.395  4-, 9-  1.390  I I  20.5 21.0 20.0 Magnetic Field (kG)  19.5  1.400  b)  +  1.395 C’)  1.390  4 4+  .++  +++++ 19.2  18.4 18.8 Magnetic Field (kG)  18.0  1.398  ‘1.394  1.390  1.386 18.0  I  iIfi I  I  ‘f  c)  hi  I  18.4 18.8 Magnetic Field (kG)  I 19.2  Figure 6.6: a) Muon Knight Shift: Sb BHê T=2.7K Preliminary Investigation b) Muon Knight Shift: Sb BIe Scan Up T=2.7K c) Muon Knight Shift: Sb BIe Scan Down T=2.7K  61  Chapter 6. Experimental and Results  The Knight shift as a function of field is shown in Figure 6.5b). This frequency shift Lf  indicates the extra field /.H experienced by the  due to the interaction with the  Sb conduction electrons: (6.4)  H=K,H=Hhf, where Hhf is the hyperfine field. [2] 6.3.4  Temperature Dependence of the Muon Knight Shift below 20 K  The precise low temperature behaviour of K,, was investigated in a magnetic field of 15 kG in the temperature range of 2 to 19 K, as seen in Figure 6.7. The data were fit in 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  2 x  was 2.9  for six degrees of freedom. In Figure 6.7b) we show a fit to Equation 4.14, an exact result for the susceptibility of a Kondo impurity in weak fields and at temperatures much less than the Kondo temperature: K  =  a  —  O.433)  ,  (6.6)  (2rK  In this case there were only two fitted parameters, the constant a=O.1150(9) K and TK=131(1) K. The x 2 was 9.1 for seven degrees of freedom. When the temperature range of the fit is from 2 to 10 K as seen in the inset of Figure 6.7b), a better fit with a x 2 of 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  =  2 a+bT  (6.7)  Chapter 6. Experimental and Results  62  . The 2 The fitted parameters were a=7.151(3) x i0 and b=1.19(2) x 10—6 K  2 x  was  11 for seven degrees of freedom. Another polynomial function with one more parameter was used to obtain a better fit, as shown in Figure 6.7d):  =  (6.8)  a+2 bT + cT 4  , and c=1.8(2) x 2 The fitted parameters were a=7.167(3) x iO, b=6.4(7) x i0 K— 1O K— . The 4  2 x  1 at low temper was 1.5 for six degrees of freedom. A constant K  atures, which is expected for a Kondo impurity, is reproduced best by a polynomial with even powers.  6.3.5  Muon Knight Shift at High Temperatures  Unfortunately the full temperature dependence has not yet been measured with the same precision as shown in Figure 6.7. In the absence of such data we used previous data (T=90 to 180 K) on this sample from Reference [21], in order to compare with the Kondo model. It should be noted, that the data in Figure 6.9 (which are the squares in Figure 6.8) were taken in a lower magnetic field of 4 kG. The data were reanalyzed using the same procedure as in the present experiment to eliminate other possible systematic effects. to exhibit Curie-Weiss behaviour for T > TK.  For a Kondo impurity we expect  Figure 6.9a) shows a fit of the high temperature data to: K,  =  (6.9)  . 1 a(T+&)  The fitted parameters were a=0.0170(2) K and &=-10(1) K. The  2 x  was 32 for five  degrees of freedom. For comparison we show a fit to straight Curie law behaviour: =  a/T,  (6.10)  Chapter 6. Experimental and Results  63  1.38 1.36\\ 1.34 1.32  cn -c  0-i  N  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  12  16 20  1.381.36 1.39  1.34 1.32  1.38 5 I  0  4  I  I  8  10 I  I  I  12  I  16  20 0  4  8  Temperature (K) Figure 6.7: Four fits to K 1 data at temperatures below 20 K. The magnetic field was nominally 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  64  1.50 1.25 -C  0.75 _c  c  0  r  fEC  U.DU  a  0.25  I  410  80 ‘120’160 Temperature (K) I  I  111111  I  I  I  I  11111  60  b)  40 0  r;,n  0 0 0 0 0  0  0 11111,  2  I  I  4 102040 100 Temperature (K)  Figure 6.8: a) Temperature Scan: K, in Sb showing Pauli paramagnetism at low tem peratures and Curie-Weiss behaviour at high temperatures. Circles are data obtained at 15 kG for this thesis and squares are obtained from data taken at 4 kG in Refer ence [21]. The two low temperature “squares” are not used in fits. b) This semilog plot of a) can be compared qualitatively to Figure 4.3. Quantitative comparison is not possible because neither a closed form expression nor a table of values was provided. [11]  Chapter 6. Experimental and Results  65  where the constant a was 0.644(1) K%. In this case the  2 x  was 37 for six degrees of  freedom. Note that the poor fit is caused by a single point at T=180 K. Clearly further measurements are required to determine the high temperature behaviour. From the constant a of the fit to Curie behaviour, Equation 6.10, the hyperfine parameter for muonium in Sb can be estimated as follows. Consider the spin Hamil tonian for the muon-electron system, which includes terms for the electron Zeeman interaction, the muon Zeeman interaction, and the muon-electron contact interaction: S•B—hI•B+AS•I, 7 (=h where  ‘ye/  (6.11)  is the gyromagnetic ratio of the electron/muon, S is the electron spin op  erator, I is the muon spin operator, and A is the hyperfine parameter.[5j In high magnetic fields where S is a good quantum number the off diagonal elements may be neglected. In this case one can decompose ?-( into two effective muon Hamiltonians: one for S=+1/2 and the other for S=-1/2. For an electron spin parallel to the magnetic field direction (S=+) the spin Hamiltonian becomes: =  <1171:11>  (6.12)  —h ,BI + 7  =  4iIz  (6.13)  For an electron spin antiparallel to the field direction (S=-) the spin Hamiltonian becomes: 7-C-  =  71:  =  <4 17-tI 1> hWe  —  (6.14)  h’yBI  —  (6.15)  In this approximation it is clear, that the effective magnetic field seen by the muon is: B  =  B  2h-y,  .  (6.16)  Chapter 6. Experimental and Results  66  a)  0.8 0.6 0.4  b)  0.4 80  120 160 200 Temperature (K)  Figure 6.9: a) Fit to Curie-Weiss behaviour to high temperature K, in Sb yielding a x 2 of 32 for five degrees of freedom and a Curie temperature of -10(1) K. b) Fit to Curie behaviour yielding a x 2 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  67  Thus the muon will precess at two frequencies depending upon the orientation of the electron spin: = 71 B + A/2. 4 .  (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 two equilibrium 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 probability the 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 high temperature the above equation reduces to: e, where  E  (6.20)  = ‘yeBh/4kBT. Substituting Equation 6.20 into Equation 6.18 leads to: (6.21)  such that the muon Knight shift is given by: ‘yB  = L 1 e 7 h A 4kBT-y,.  (6.22)  Equating the coefficient of 1/T to fitted parameter a in the fit to Curie law leads to A=259.7(0.4) MHz. A 11 is the component of the hyperfine tensor parallel to the crystalline ê-axis. Comparing with the hyperfine parameter of muonium in a vacuum: Avac Au  4463.30288(16) MHz  (6.23)  0.0582(1)Avac.  (6.24)  Chapter 6. Experimental and Results  68  Although no data above 80 K with H I ê-axis is available yet, we know that at lower temperatures: [21] K.  (6.25)  Thus it is reasonable to assume: A  11 A /2  129.9(2) MHz.  (6.26)  From these we can estimate the isotropic and anisotropic hyperfine parameters: 8 A A  11 +2i1± A  =  =  Au  —  A  1 A  173 MHz  (6.27)  Aii  43 MHz.  (6.28)  The large anisotropic hyperfine parameter and small isotropic hyperfine parameter suggest, that the majority of the spin density is near but not on the muon. A similar situation occurs in covalent semiconductors. For example the hyperfine parameters of the Mu* center in GaAs are 218.5 and 87.87 MHz. In this case the unpaired electron is known to be on the two nearest neighbour nuclei on the [111] axis. It is interesting to compare A with what one would expect from an electron in a hydrogenic 2p orbital: [49]  4  =  12.531-y, <r > MHz. 3  (6.29)  The muon gyromagnetic ratio 7/21r is 135.53 MHz/T and the expectation value of r 3 for a hydrogenic 2p orbital (n=2, 1=1) is:[50] 3 > <r  1(l +1)(21 + 1) n 3 a  ’ 3 24a  (6.30)  where a  =  meao/Z,  (6.31)  Chapter 6. Experimental and Results  where me is the electron mass,  t  69  is the reduced electron mass for the muon-electron  system, a 0 is the Bohr radius, and Z is the nuclear charge. In units where both me and 0 are one, a a  (1 + m)/m . 1  =  1.005  (6.32)  and 175.3 MHz.  (6.33)  This result is about four times larger than the experimental value of 43 MHz. On the other hand we do not expect the electron to be in a 2p orbital on the muon, but rather on the valence orbital of the neighbouring Sb atom(s). However this result does emphasize that the deduced hyperfine parameter can only be explained, if there is a large spin density near but not on the muon. 6.3.6  Discussion of TF-iSR Results  In this section the results are discussed in terms of earlier models based on local moment formation and peaks in the density of states. Also new insights on K 1 in Sb are offered by 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 is consistent with earlier work, which explained this behaviour as an effect of a peak in the density of states, as was discussed in Section 3.2.2.[20,22] In Figure 6.6 the field scans show, that the amplitude of the dHvA oscillations of  in Sb under these experimental  conditions is less than the statistical error of 0.2% of K, which is about 40 times smaller than the amplitude expected from the dHvA oscillations of the magnetization- about 8% of the magnetization. Observations in Cd show, that the amplitude of magnetization and Knight shift oscillations were within a factor of about two. It was thought K might be proportional to the spin susceptibility and hence the magnetization.  The clear  Chapter 6. Experimental and Results  70  absence of dHvA oscillations is evidence, that the electron(s) responsible for the giant do not have the character of the conduction electrons. This fact suggests, that the local electronic structure around the muon plays a dominant role. The anomalously large K, indicates perhaps muonium or more specifically a para magnetic Sb complex is formed in Sb, The conduction electron density in Sb is three to four orders of magnitude less than in normal metals. The smaller electron densities , and As: ‘—2 x 3 in the Group Vb semimetals (Bi: ‘-‘-‘3 x 1017 cm , Sb: ‘--‘5 x 1019 cm 3 ) provide ineffective screening and therefore the chance exists for an electron 3 cm 20 10 to bind weakly to the iz or a tSb complex. Although the small carrier density may 4 is seen only in Sb and not play a role, it is clearly not the sole actor, since the giant K, in As or Bi. This may be taken as further evidence for the importance of the electronic structure near the muon in Sb. For example it is possible that the muon chemically binds to one or more Sb’s in such a way that a virtual bound state is formed with parameters in the Anderson model which favour a local moment. The proposed muonium-like center in Sb has a large anisotropic hyperfine param eter (A=43 MHz) and a small isotropic hyperfine parameter (A= 173 MHz), which explains the anisotropic K. The hyperfine parameters suggest, that the majority of the electronic spin density is near but not on the muon. A similar situation occurs for the anomalous Mu* center in covalent semiconductors such as GaAs, where the unpaired electron is known to be on the two nearest neighbours on the [111] axis. This large spin density on or near the muon and the rapid decrease of the susceptibility (or the Knight shift, which is thought to scale with the susceptibility) as temperature increases are characteristic of a Kondo impurity. Below 6 K K, and presumably the susceptibility are constant or at least only weakly temperature dependent, which indicates the local moment is zero or near zero. In terms of the Kondo effect at low temperatures the local moment of an unpaired electron on  Chapter 6. Experimental and Results  71  the muon disappears because the conduction electrons have a strong influence on the behaviour of the system at low temperatures. This can be understood in two different ways depending on the choice of model. In the spin-compensation model the electrons compensate the muon spin by forming an oppositely polarized spin cloud around the muon. Alternatively in the spin-fluctuation model the local moment fluctuation rate is greater than the thermal fluctuation rate at these low temperatures and the spin magnetic moment averages to zero. As the temperature increases, the compensation and/or the fluctuation rate decreases and so local moment behaviour becomes increas ingly important. As seen in Figure 6.8 K 1 decreases with temperature, as would be expected from a local moment. Thus the Kondo effect provides a natural explanation for the large and anomalous K, in Sb. From temperatures of 2 to 19 K the temperature dependence is best fit to a power law, Equation 6.8. A good fit can also be made to an Arrhenius Law, Equation 6.5, with an activation temperature (E/kB) of 40(2) K, which might reflect the electron-muon binding or perhaps the binding of the spin-compensation cloud to the muon. Another fit to a scaled universal Kondo susceptibility, Equation 6.6, with a TK of 131(1) K can also be done, as seen in Figure 6.7. The fit to this model is clearly not as good, but it 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 the temperature range is restricted from 2 to 10 K, this fit to Equation 6.6 yields a smaller reduced  2 x  and a  ’K 7  of 160(3) K.  At much higher temperatures (80 to 180 K) K has a Curie-like behaviour (see Fig ure 6.9), which is characteristic of a Kondo impurity. The interpretation in terms of the Kondo effect is, that at these high temperatures either the local moment spin fluctua tion rate is less than the thermal fluctuation rate and the muonium impurity appeared as a well-defined local moment. An alternative interpretation of the Kondo effect is  Chapter 6. Experimental and Results  72  that the spin compensation cloud is disrupted or shaken off by thermal excitation. The gradual change from the Curie-like to Pauli paramagnetic behaviour predicted by Kondo theory is characterized by either T f or TK. The TK of 160(3) K obtained from 8  the low temperature data and the activation temperature of 40(2) K from the Arrhenius fit do not agree. However it should be noted, that the Kondo impurity models were developed for magnetic impurities with d state valence electrons in normal free electronlike metals. Sb is a semimetal with a relatively small Fermi energy and has both hole and electron carriers. The valence orbital is composed primarily of p character. These differences might alter the details of the temperature dependent screening. Qualitative agreement 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, the Sb free electron Fermi temperature TF is calculated to be 1.3 x i0 K measured relative to the bottom of the valence band,[17] and the Sb density of states at the Fermi level of 4.7 x 10-2 (atom eV)*[51] The result was a J of about 3 eV, which is an order of magnitude larger than typical values of 0.1 eV.[33] In the case of a semimetal it may be appropriate to use: TF  = ±(3fl2)2/3  kB 2m  where m* is the effective mass of carriers  6100 K,  ( 0.lme)  (6.34)  and the carrier concentration n is  . This yielded a J of about 6 eV. 3 5.5 x 1019 electrons/cm In summary the Kondo effect provides a new explanation of the giant Knight shift in Sb and its anomalous temperature dependence and anisotropy. Muons in Sb may therefore be a unique example of an isolated Kondo impurity. Since only one in the sample at a given time, 1 iSR observations of  is  and its interactions with the  Chapter 6. Experimental and Results  73  Sb conduction electrons is a system free of impurity-impurity interactions, such as the long range RKKY interaction, and therefore a test of theoretical solutions of the single Kondo impurity problem. 6.3.7  tSR Experimental Procedure 1 LF-  With LFtSR two field scans and one temperature scan were performed. In this ex 1 periment two temperature scans, one at field of 650 G and the other at 1 kG, in the range from 2 to 300 K were done. One field scan at a temperature of 75 K was done from 0.2 to 1.8 kG. In these LF-jSR experiments the behaviour of the longitudinal relaxation rate T’ as a function of temperature and field was sought. Prior to the LF-tSR mea . This 1 surements in Sb, the T’ in Ag alone was determined to be 1.3(5.6) x 10 ts showed, that the relaxation rate in Ag was effectively zero. The smallest nonzero re laxation rate, which could be measured under these conditions, was on the order of i0  jts.  Also prior to the LF-SR measurements the asymmetry of both the Sb  and Ag signals were determined by transverse field measurements. The zero T’ in Ag as well as both the Ag and Sb asymmetries were then fixed for the analysis of the LF-uSR asymmetry histograms. In other words the decay positron asymmetry his tograms were fitted with two fixed amplitude signals: a nonrelaxing reference Ag and the exponentially relaxing Sb signal. T 1 in Sb was obtained from these fits by  2 x  minimization. One asymmetry histogram showing a large T’ and another showing a smaller T’ can be seen in Figure 6.10. 6.3.8  tSR Temperature Scan 1 LF-  In 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  74  0.18  I 0.14  0  2  I  I  I  I  I  4 6 8 Time (ps)  I  10  Figure 6.10: Typical LFtSR asymmetry histograms taken in a field of 1 kG showing in 1 a) fast relaxation (large T’) at T=75 K and in b) slow relaxation (small T ) at T=250 1 K. Each histogram was fitted with two fixed amplitude components: a nonrelaxing Ag and an exponentially relaxing Sb signal.  Chapter 6. Experimental and Results  75  Presumably the relaxation rate reflects the spin exchange rate of muonium with the Sb conduction 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 Scan  A coarse field scan was conducted from 0.2 to 1.8 kG at a temperature of 75 K. In Figure 6.12 there is a peak in the relaxation rate at about 0.7 kG, but as can be seen from the relative size of the error bars, this is a small effect at this particular temper ature. The weak temperature dependence of the relaxation is expected for Korringa relaxation. 6.3.10  Discussion of LF-SR Results  The coupling of the  to the spin magnetic moments of the Sb conduction electrons  leads to T 1 relaxation. In other words when the applied field is parallel to the muon spin, the  +  can either absorb or release energy to the electrons within kBT of the  Fermi surface, which leads to depolarization of the muon and relaxation in the decay positron asymmetry. The muon transition is accompanied by an electron transition from initial state  Iks)  to final state k’s’). The muon goes from initial state rn) to final  state n). In the standard theory of Korringa relaxation of nuclear spins in a metal, a scattering process with a transition rate W is described by Fermi’s Golden Rule: Wms,nis  =  I  <msVn’s’>  8(Em 2  8 +E  —  E  —  E,),  (6.35)  where V is the s-state coupling of the muon and electron, which is responsible for the scattering.  From this independent electron model, the Korringa relation was  developed.[18,52] As mentioned in Section 4.3.5 the Korringa relation is valid for a Kondo impurity provided T << TK. The Korringa relation relates the relaxation rate  Chapter 6. Experimental and Results  76  0.06 0  (I)  0.04  Ui=650 G 0=1 kG  ‘  0  0  0  0  0.02  0  H—  0000  0  •  -a I  0.08 U)  I  I  I  m  0.06  I  g=2  :  UI  0.04 H-  0.02 0  I  I  I  UI  g=15  mm 0  a  Cl)  0  0  •0 UI  0.5 H-  0  UI  a a  I  0  I  200 100 300 Temperature (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 TFSR data with the 1 Korringa relation. The electron g factor used was 2. c) Predicted T 1 with g=15.  Chapter 6. Experimental and Results  77  0.056 0.054 -  0.052  F-  0.050 0  0.4 0.8 1.2 Magnetic Field (kG)  Figure 6.12: Field dependence of longitudinal relaxation rate, T’, at a temperature of 75 K.  T’ to the temperature and the isotropic component of the Knight shift: T’ = TK /S, 0  (6.36)  where  s  = 47rkB  2  (6.37)  and I  K(6) = K 80 + Kaniso(3O 2  —  1).  (6.38)  For temperatures up to 80 K it is known, that K 11 = 2K within experimental error. [20] Therefore the isotropic component of the Knight shift is: K All the available K 1 data from temperatures of 2 to 180 K was used in the Korringa relation, in order to make a prediction of the longitudinal relaxation rate, T 1 for two different electron g factors, as seen in Figures 6.llb) and c). For electrons on the Fermi  Chapter 6. Experimental and Results  78  surface in Sb and for applied fields parallel to the ê axis, the electron g factor is 15 and 50 7e = g/Lb/h  15/Lb/h.[28] In this case the predicted Tj’ is in very poor agreement  with the data as seen in Figure 6.llc). However with g=2 the agreement is much better. This is a further indication of the importance of the local electronic structure relative to the band structure. Ideally, in the absence of electron-electron interactions, the Korringa ratio 0 T 1 T / S K is 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 orbital spin paramagnetism. However, since the muon has no core electrons, core polarization is not a factor. Also Sb does not display orbital spin paramagnetism. Small deviations from unity arise, when the independent electron model breaks down or in other words when electron-electron interactions become important. For the muon in Sb problem we expect the “normal” Korringa relation to hold for T << TK or in other words the Korringa ratio should be temperature independent and of order unity at temperatures well below TK.[41] From K 1 data TK is estimated to be in the temperature range from 40 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 quan tity 0 T, 1 S/(T ) I which is shown in Figure 6.13a). The T 1 values were taken from the LF-SR data and K 1 was estimated by interpolating the TF-pSR data. This was necessary, since the K,, data points were not taken at the same temperature as the T 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 quantity TK appears to diverge as the temperature approaches zero. This is due to the 1 S/(T ) 0 fact that the low temperature  data extrapolated to a non-zero value at zero tem  perature. The origin of this temperature independent contribution to T’ is unknown. However in the event it arises from something other than an electronic mechanism, we  Chapter 6. Experimental and Results  79  subtracted this “background” from the T’ data and calculated the “corrected” value of 0 TK seen in Figure 6.13b). The “corrected” temperature independence of 1 S/(T ) TI below 100 K is expected from the Korringa relation, which is only valid 1 S/(T ) 0 for a Kondo impurity at temperatures well below TK. At temperatures above TK we see the breakdown of the Korringa relation in Figure 6.13. A possible origin for the “background” or non-electronic relaxation follows. At most magnetic 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, since energy must be conserved. However at certain fields, where there is a level crossing res onance (LCR), relaxation is possible. If the muon Zeeman splitting matches the nuclear splitting due to quadrupolar and Zeeman interactions, then spin exchange between the muon and the nucleus is possible, since energy is conserved: 7i = 71nuc —g4H I =  (6.39)  —gyH J + Q[3J 2 .  —  J(J + 1)],  where I is the muon spin operator, J is the nuclear spin operator, and  (6.40)  Q  is the  quadrupole parameter. If the T 1 data was taken at such a field, then the observed “background” may be relaxation due to spin exchange between the nucleus and the muon, 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 K are similar. At fields further away from the LCR this relaxation should decrease. In most 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 accompanied by a muon spinflip transition can be made: 111cr =  ,  7I  —  7n  (6.41)  Chapter 6. Experimental and Results  where  is the muon/nuclear gyromagnetic ratio. The field gradient introduced by  the positive muon will doubtlessly change  Q  80  Q,  but since this parameter is unknown, the  values for pure nuclear quadrupole resonance in Sb from Table 6.1 are used. For  21 the expected 111Cr is about 6 kG. Although this is substantially larger than the ‘ 5b field at which the T’ data were taken, it should be noted that the muon can cause large changes in  5b 2 ‘ 1 23 ‘ Sb  Q  by its presence. J 5/2 7/2  ir (MHz/kG) 2 ‘y/  1.0189 0.5518  Abundance 57.3 42.7  (%) Q/h  Table 6.1: Sb LCR parameters.[23]  (MHz) 76.867(1) 97.999(1)  Chapter 6. Experimental and Results  1.2  81  r  rii=1 kG cP=650 C  -  0.8 0.4  a)  Eli L  -w uIRww -ø  w Ui  Ui  0  0  I  I  I  _.I.  I  8 . 0 hb)  0.6 0.4 0.2 00  100 200 300 Temperature (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 temperatures with the Curie-Weiss fit to produce the figures seen here. a) 0 TI evaluated 1 S/(T ) with experimentally determined Tf’ and fitted . 14 b) 33 K TK? evaluated with a 1 S/(T ) zero Tj’ at zero temperature and a “corrected” Tf 1 at all temperatures.  Chapter 7  Conclusion  In this thesis we presented additional data on the anomalously large, anisotropic, and temperature dependent K 1 in Sb. We propose a model to explain the results in which either muonium or a paramagnetic Sb complex is formed. In this case Sb would be the only metal or semimetal, in which muonium has been observed. Hyperfine parameters for muonium in Sb have been estimated from the temperature dependence of K, and are similar to those of Mu* in GaAs indicating, that a large spin density resides on the nearest neighbour Sb. This single paramagnetic muonium impurity in Sb is interesting, since it would represent a truly isolated Kondo impurity. 1 tSR experiments in Sb are unique in this respect because only one muon at a time is in the sample and RKKY interactions between two or more impurities are avoided. The other interesting feature about this system is the fact that Sb is a semimetal. Consequently the theory for a Kondo impurity may have to be modified to take into account the much smaller carrier densities. The qualitative agreement with typical Kondo behaviour is good; the K, data displays a crossover from the weakly temperature dependent Pauli paramagnetism of the conduction electrons at low temperatures to Curie-like local moment behaviour at high temperatures. Quantitative support for Kondo behaviour comes from a good fit of K data at temperatures from 2 to 10 K to a scaled equation for the universal Kondo susceptibility. The absence of dHvA oscillations in K, which was expected to scale with the  82  Chapter 7.  83  Conclusion  magnetization, confirms that the extended electronic band structure has been perturbed dramatically by the positive muon and that the local electronic structure of the muon plays a dominant role in determining K. The T’ relaxation rate displays a temperature dependence, which presumably reflects the anomalous spin dynamics of the muonium atom and the Sb conduction electrons.  The temperature independent Korringa ratio 0 TI seen below a 1 S/(T )  temperature of 100 K is expected for a Kondo impurity. The experimental data for K, and T 1 are tabulated in Appendix B, in order to provide a test for theoretical models. 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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 A  The de Haas-van Aiphen Effect  In 1930 de Haas and van Aiphen discovered the magnetization of bismuth oscillated as a function of magnetic field in the range 5-20 kG at a temperature of 14 K. The usefulness of these oscillations in Fermi surface studies of metals was not fully appre ciated until 1952, even though the quantization of electron orbits in an applied field and the possibility of directly observing the macroscopic oscillations due to this purely quantum 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 Field  In an applied magnetic field the orbit of an electron is quantized. The area of the semiclassical orbit for free electrons in k space is given by:  =  We see, that the area in  1  (n, + 1/2)-H.  (A.1)  space of two successive orbits is equal for equal increments  of 1/H:  1  1  Si\Tln+1  1’\ —  -fl-LZnJ  =  lrr-. 2  (A.2)  itC  If the orbital areas are the same, then we expect properties dependent on the Fermi surface to be the same.  This quantization of conduction electron orbits in metals  87  Appendix A. The de Haas-van Aiphen Effect  88  leads to oscillations of macroscopic properties such as resistivity, heat capacity, and susceptibility as a function of 1/H with period: (12e \.H) hcS  A3  —  The de Haas-van Aiphen effect is the oscillation in 1/B of the magnetic moment of a metal. Low temperatures and high magnetic fields are required to prevent thermal population of adjacent orbits and to keep the orbital period much less than the collision time. [48]  A.2  Landau Levels  For 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 are quantized 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 space . In the presence of a magnetic field a total of D previously free electrons 2 is (2ir/L) coalesce into a single Landau level, where 2 = pH D = (2lreH/hc)(L/27r)  (A.5)  and p = eL /27rhc. (If electron spin is taken into account, then D = 2pH.) As H 2 increases, the degeneracy D of a Landau level also increases, Figure A.2b). As long as there are electrons in a given Landau level, labelled by quantum number s+1, the Fermi level will remain in level s+1. However, as H and D increase, the electrons move into  Appendix A. The de Haas-van Aiphen Effect  89  <H 2 H,<H 3 H=O  1 H  2 H  H=O  3 H  /E  K  0  0  a)  b) c)  d)  e)  f)  g)  h)  Figure A.1: For spinless, free electrons in two dimensions at absolute zero, the electron orbitals below EF are filled. In a) and f) the applied field is zero. In b), e), and g) the free electrons coalesce into magnetic energy (Landau) levels of degeneracy D = pH, = ehH/m*c. In b) and which are split by /.E = g) the total energy of the system is 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 been raised in energy. If spin is taken into account the degeneracy increases by a factor of two to D = 2pH and the “spinless” Landau levels split by the Zeeman energy gpBH into a spin-up level and a spin-down level, as seen in c), d), and h). In d) the lower energy spin-down levels will lie below the Fermi energy, whereas the spin-up levels will be vacant leading to a net magnetic moment.[48]  Appendix A. The de Haas-van Aiphen Effect  k  a) .... •... •••• I...  .... .... .... ....  b)  •• .... .... •I••  .... I...  .... ....  s=1  c) ,  -o E  90  50,  s=2  1  s=3 I  11  I  2  3  l  I  4  100/H Figure A.2: a) Allowed electron orbitals without an applied field occupy an area of (27r/k) 2 in k space. With an applied field in the z direction the electrons are restricted to circles in the x-y plane. In k space the area between circles is: = 2irk(/k) )E = 2irm/h = 2-ireH/hc. b) The periodicity of 2 (2irm/h dHvA oscillations with 1/H is seen here for a two dimensional electron gas of 50 elec trons with p=O.5.[48] The completely filled magnetic levels are denoted by s. For example at 11=40 or 100/11=2.5 the levels s = 0,1 are fully occupied and there are ten states in s=2. At temperature above absolute zero thermal smearing of the magnetic levels lead to oscillations, which are more sinusoidal than sawtooth-shaped. [48]  Appendix A. The de Haas-van Aiphen Effect  91  lower Landau levels and eventually there are no more electrons in level .s+1. Now the Fermi level lies in level s. This movement of the highest occupied Landau level relative to 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  E  =  0, is:  (n + )hw,  (A.6)  eH/m*c  (A.7)  where =  is the cyclotron frequency.  The electrons in a Landau level can have an arbitrary  value of k. Consider now only the electrons in a volume of thickness 6k at k. The degeneracy per unit volume is: 6 eH  =  (If electron spin is taken into account, then  (A.8)  increases by a factor of two.) The Landau  levels up to and including level s will be completely filled and the total energy of an electron 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 the slice Sk is N So as H and  =  (s + 1).  (A.10)  increase, N increases linearly with H until level s reaches the Fermi  energy. As H increases further, electrons leave level s for level s-i and in general this situation will occur whenever 1  E  m*cE 1 he  ,  (A.11)  Appendix A. The de Haas-van Aiphen Effect  92  So the population of the slice N oscillates with a period in 1/H of m*cE/he and with an amplitude of +/2 around the value N , the total number of electrons in 6’k in the 0 absence of a magnetic field. [26] When the population of the slice is N , the energy of the electrons in a magnetic 0 field is: (A.12)  Uo 0 = hw(s + 1)2/2 + N U 0  2rn  but N 0 = (s + 1) and, therefore, = hwN . 0 +N 2 2m  (A.14)  For only a small change in magnetic field, Landau level s will still be the topmost filled state and = 0 + (N  +  -  The last term in the above equation represents the transfer of N Fermi level and (N 0  —  (A.15)  N)EF. —  0 electrons to the N  N)EF is the energy change associated with the rest of the Fermi  sea. So the change U  —  0 represents the total energy change of the Fermi surface U  electrons for a small change in magnetic field: =E(N2_N)+(N SU=U_U _ 0 N)E,  (A.16)  but E = hN / and, therefore, 0 = (N  —  . 2 ) 0 N  (A.17)  The magnetization of the slice 6k at absolute zero is given by: 6M =  —  = -(N  —  ), 0 N  (A.18)  Appendix A. The de Haas-van Aiphen Effect  93  but dN  (s + 1)/H  (A.19)  and, therefore, SM  (N  —  ) 0 N  N  —  (A.20)  . 0 N  Since the electron population of the slice Sk varies periodically with 1/H, the magne tization will also oscillate as a function of 1/H. This oscillation of the magnetization is the de Haas-van Alphen effect. In the slice Sk the change of susceptibility is given by: =  SM/H  =  (N 1  —  ) cc N 0 N  —  . 0 N  (A.21)  The bulk susceptibility of the electrons depends upon the oscillatory quantity N, which in 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 this  thermal energy is comparable to the spacing hw between Landau levels, then the oscillations in 1/H will be washed out. From the inequality ehH 1 m*c —>1 kBT  (A.22)  T/H < 0.134 K/kG.  (A.23)  1tw kBT we see  In order to avoid thermal smearing of the oscillations, operating with a field of order 20 kG requires temperatures of order 2 K.[48J  A.3  Extremal Orbits of the Fermi Surface  By measuring the period of dHvA oscillations, the areas S of a Fermi surface, which are perpendicular to B, can be calculated. In principle we might expect to see dHvA  Appendix A. The de Haas-van Aiphen Effect  94  periods contributed by all the orbits, which are perpendicular to the applied field and which have been quantized by the magnetic field. However, the dominant contribution comes 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. 247 of 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 Fermi surface, the area of the orbit changes as  changes. Therefore, clear resonances can  be obtained in dHvA measurements of very complicated Fermi surfaces because only the extremal orbits have slowly varying areas. A mathematical discussion of extremal orbits and observed dHvA oscillations can be found in Reference [26]. Thus by analyzing dHvA oscillations the particular electrons or holes on the Fermi surface contributing to particular phenomena, such as the magnetic susceptibility and the Knight shift, can be determined. [48]  Appendix B  Experimental Data  B.1  TF-tSR Data: Knight Shift vs. Temperature  All data was taken on M20 beamline at TRIUMF from November 25 to December 1, 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 is summarized below in Table B.1. Temperature (K) 2.298(1) 2.888(1) 4.18(2) 5.9(1) 8.00(5) 10.17(5) 13.05(.10) 15.98(7) 18.77(.27)  Knight Shift 1.393(1) 1.396(1) 1.393(1) 1.392(1) 1.386(2) 1.380(1) 1.362(2) 1.343(1) 1.314(1)  (%)  Table B.1: TFiSR Data: Knight Shift vs. Temperature taken in a magnetic field of 1 15 kG.  B.2  tSR Data: Relaxation Rate vs. Temperature 1 LF-  All data was taken on M20 beamline at TRIUMF in July 1993. The temperature scans at 1 kG and 650 G as well as the field scan at 75 K can be found in runs 674 to 709. 95  Appendix B. Experimental Data  96  The temperature and field scan data are summarized below in Tables B.2 and B.3. Temperature (K) 2 5 10 15 20 35 50 75 100 100 125 150 175 187 187 200 225 250 275 300  T’ Relaxation Rate (x 10 jis’) 2 0.79(.11) 1.21(.12) 1.862(.044) 2.44(.11) 3.00(.17) 4.105(.091) 4.95(.12) 4.95(.15) 4.59(.11) 4.85(.15) 4.52(.14) 3.936(.075) 2.91(.11) 1.14(.13) 1.40(.11) 0.78(.11) 0.68(.10) 0.92(.11) 0.79(.12) 0.58(.11)  Table B.2: LF-tSR Data at 1 kG: T 1 Relaxation Rate vs. Temperature  Temperature (K) 20 75 175 200  ts 2 10 ) Tj Relaxation Rate (x 1 1 2.423(.071) 5.59(.15) 2.54(.14) 0.000(.008)  Table B.3: LFtSR Data at 650 G: T’ Relaxation Rate vs. Temperature 1  

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