Non UBC
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TRIUMF
Uemura, Yasutomo J.
TfdtJ~~F REPORT LU~RA Y Thesis for a Degree of Doctor of Science . ., ' " . ,lL DYNAMICAL PROPERTIES OF SPIN GLASSES STUDIED BY MUON SPIN RELAXATION BY YASUTOMO J, UEMURA Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-Ku, Tokyo, Japan. Submitted to University of Tokyo. December, 1981 . PREFACE When we visit a new city in our travelings, after a driving on a winding road, we feel a little pleasure and excitement since we may find a new friend, a different life of people ·, a small restaurant, a beautiful site, etc. In a similar way, an exciting moment comes when we obtain a new and "hot" experimental data, after some technical efforts for the measurement. Though may it be a tiny bit, we may find an unknown aspect of physical properties of substances, or confirm some other findings or principles. It is especially fascinating to study an interesting and open subject with a unique and developing experimental method, since we can contribute to many stages of such "travelings". A surprisingly sharp cusp of ac-susceptibility was f ound about ten years ago by Cannella and Mydosh in AuFe, a system of dilute magnetic moments distributed randomly in the host material. This discovery suggested a possible "phase transition", different from usual ferro- or antifer romagnetic orderings, and led to a collection of experimental and theoretical investigations of random magnetic systems, now called "spin glasses". The mechanism of the spin glass behavior is, however, not yet fully understood at present, and experimental studies of its spin dynamics have only just marked the first step. Here we have an attractive subject, related to a basic problem of "randomness" in the modern solid state physics. Thanks to the development of high intensity muon beams in the recent decade, muon spin rotation and relaxation (µSR) measurement has extensively been applied to the research of solids. After stopping polarized positive muons in the specimen, we can observe the time evolution of the direction and polarization of muon spins by detectfng decay positrons emitted fromµ+. Depolarization of muon spins reflects amplitude, randomness, and dynamical - 1 -fluctuations of local magnetic fields at muon sites in the specimen, and thus gives us important microscopic informations on magnetic materials. µSR is a very young member of magnetic probes in solids, and its experimental technique and b·asic theoretical backgrounds are in the stage of development. We have here a unique and powerful experimental method. This thesis will report experimental investigations of typical dilute-alloy spin glasses CuMn and AuFe by zero- and longitudinal-field µSR methods, which I performed at Vancouver, Canada, using a high intensity polarized muon beams provided at TRIUMF, one of the largest meson factories producing muons with a 500 MeV proton accelerator. In my graduate studies, as a member of a visiting group from University of Tokyo headed by Prof. T. Yamazaki, I had a chance to stay at TRIUMF since 1978 for about two and a half years in total, where I finished the present experiments with many collaborators. In 1978 at TRIUMF, we developed a new technique and succeeded in observing µSR in longitudinal magnetic fields or even without applying external field. Chapter I will introduce this experimental method and will compare it to the conventional µSR method in transverse external magnetic fields. As we shall see in this thesis, the zero- and longitudinal-field µSR measurement is particu larly suitable to detect dynamical spin fluctuations in spin glasses. I described a stochastic theory of spin rela xation of a magnetic probe, in nuclear dipolar system in Chap. 11-A, and in spin glasses in Chap. 11-B. Such a theory is used for interpret~tions of observed µSR spectra, and applied to deduce static and dynamical quantities of fluctuating local fields, as in the line-shape discussions of magnetic resonance. We shall see some characteristic features associated with the zero-field measurement, and also a new t reatment for a random dilute spin sy·stem. Introduced line shapes in static random local fields are given by Kubo and Toyabe, while the dynamical - 2 -treatments are mostly my "hand made" theories which I developed from my own ideas during the interval of experimental activities. Experimental results of zero-field µSR in AuFe and CuMn are surveyed in Chap. Ill-A, while Chap. 111-B introduces the observation of µSR in CuMn performed in finite longitudinal magnetic fields. In the analysis of Chap. Ill-A, we will assume a single value for correlation time Tc of all the Mn or Fe moments, and will deduce absolute values of T as a mean correlation ti'me of C spin fluctuations. We will check the static and dynamica l contributions to the muon spin depolarization, in Chap. 111-B, via the decoupling effects in longitudinal fields. As an attempt beyond the simplest case with a sing l e correlation time, wil 1 extend the stochastic theory and the analysis of observed data to include a spatial distribution of Tc as well as an effect of order parameter of impurity spins below the cusp temperature in Chap. IV-A. I will revisit reported results of transverse-field µSR experiments in Chap. IV-B, to study the possible change of impurity spin systems caused by the applied external fields, and wil 1 discuss the relation to the well-known reduction of ac-susceptibility of spin glasses in finite fields. In Chap. V-A, I wi 11 compare the present results of µSR experiments to the findings of other dynamical experimental methods, such as neutron scattering, ac-susceptibility, Mossbauer effect, ESR and NMR in dilute-alloy spin glasses. We shall see that dynamical and static quantities in different experiments can be compared in a normalized plot, thanks to the scaling law of dilute-alloy spin glasses. Chap. V-B will introduce another zero-field µSR experiment in an amorphous insulator spin glass (Coo) 40 (A12o3) 10(Si0 2) 50 which we performed at TRIUMF. A marked difference from the case of dilute alloys shall be noticed. Finally in Chap. VI, I will summarize the results of the present work, and discuss the physical meanings with possible future investigations. Study - 3 -of random systems is not simple. One first considers the averaged quantity, and then discusses its distribution. We shall further compare different physical situations to find out the origin of the distribution. It is a pleasure of the author if such attempts in this ~hesis, peformed step by step, can contribute to understand a complicated nature of the sp in glass behavior. - 4 -Some of the main results of the pr e sent thesis have been published in inv ited tal ks I pres ented at th e 2nd International Conference on Muon Spin Rotation µSR2, Vancouver, August, 1980 = Principal pub] ication ·A, and at the 16th International Conference on Low Temperature Physics LT- 16 , Los Angeles, August 1981 = Principal publication B. Different topics are also discussed in related pub] ications (r.p.) as Chap. I and 11-A in r.p. A; Chap. 11-A in r .p. B; Chap. 11-B and Appendix I in r.p. C; Chap. Ill-A in r.p. D; Chap. 111-B in r.p. E; and Chap. V-B in r.p. F. Principal publications attached for the degree requirement : A. "Probing Spin Glasses with Zero-Field µSR" Y.J. Uemura, Hyperfine Interactions 8 (1981) 739-748. (I nvited Paper at µSR2) B. "Dynamics of Spin Glasses Detected by Muon Spin Relaxation" Y.J. Uemura and T. Yamazak i, to be published in Physica B+C. (Invited Paper at LT-1 6) Related publications: A. "Spin Relaxation of Positive Muon in ZrH2 and MnO" Y.J. Uemura, Master Thesis, Univ. of Tokyo, 1979-- 5 -B. "Non-Secular Part of Nuclear Dipolar Broadening Detected by Zero-Field Spin Relaxation of Posit ive Muon" Y.J. Uemura, R.S. Hayano, Y. lmazato, N. Nishida and T. Yamazaki, Solid State Communications 31 (1979) 731-734. C. "Zero-Field Muon Spin Relaxation Reflecting Dynamics of Spin Glass" D. Y. J. Uemura , Solid State Communications 36 (1980) 369-372. + "Zero-Field Spin Relaxation ofµ as a Probe of the Spin Dynamics of AuFe and ~Mn Spin Glasses" Y.J. Uemura, T. Yamazaki, R.S. Hayano, R. Nakai and C.Y. Huang, Physical Review Letters 45 (1980) 583-587. E. "Muon Spin Relaxation in a Spin Glass CuMn Observed in Finite Longitudinal Magnetic Fields" Y.J. Uemura, K. Nishiyama, T. Yamazaki and R. Nakai, Solid State Communications 39 (1981) 461-465. Y.J. Uemura, C. Y. Huang, C.W. Clawson, J.H. Brewer, R.F. Kiefl, D.P. Spencer and A.M. de Graaf, Hyperfine Interactions 8 (1980) 757-762. - 6 -would like to acknowledge my thesis supervisor Prof. T. Yamazaki of Univ. of Tokyo for introducing me to the field of µSR, for giving me a chance to stay at TRIUMF and perform this work, and for valuable discussions and encouragements during the graduate studies. The application fo the zero-field µSR experiment to spin glasses was first proposed by Yamazaki. To develop zero-field µSR method in 1978 , I worked with R.S. Hayano, J. lmazato, N. Nishida and T. Yamazaki. Since these members returned to Japan in the end of 1978 to construct a new meson facility at KEK, I mainly stayed at TRIUMF as a single representative from the Yamazaki group afterwords, working for the preparation and analyses of the present experiments in spin glasses which we started in the spring of 1979. To perform µSR experiments, we are usually assigned a block of "beam time", continuing about a week throughout days and nights. We have to set up spectrometers, adjust counters electronics, maintain the stability of temperature, etc., during the beam time. I performed such activities with short term visitors from Japan and U.S., some members from the group of Prof . M. Doyama, and with some collaborators at TRIUMF, by helping one another on several different subjects, such as spin glasses, itinerant weak-ferreomagnets, diffusion problems, etc. They are: M. Tak igawa , R. Nakai, T. Yamazaki, H. Nakayama, et al . in March 1979 (slowly cooled CuMn 0.8 %) ; C.Y. Huang, D.E. Maclaughlin in May 1979 (slo wly cooled ~Mn 0.5 %); Y. Kuno, S. Takagi, R. Nakai, C.Y. Huang in September 1979 (slowly cooled ~n 2.4 %, rapidly cooled AuFe 1.0 %); R.S. Hayano , R. Nakai in October 1979 (AuFe 1.0 %, 1.4 %, slowly cooled ~Mn); C.Y. Huang, C.W. Clawson, J.H. Brewer, R.F. Kiefl, D.P. Spencer in November 1979 (an insulator spin glass) ; R.S. Hayano , R. Nakai in March 1980 (AuFe 1.0 %, quenced CuMn 1.1 %); K. Nishiyama and R. Nakai in September 1980 (LF-µSR in - ? -quenched ~Mn 1. 1%). Without their help, the present work would have never been completed. Specimens of CuMn and AuFe were prepared by H. Miyajima, and (Coo) 40 (Al 2o3) 10(Si0 2) 50 by A.M. de Graaf. Ac-susceptibility of AuFe and CuMn was measured by S. Chikazawa and Y. Miyake. D.E. Maclaughlin also supported to measure the susceptibility in the early stage. H. Yasuoka cared the chemical analyses of the specimens. these scientists. would like to express my sincere gratitude to I would like to thank M. Doyama for metallurgical suggestions and also for approving and supporting my stay at TRIUMF after 1979 under his grant for a J.S.P.S. international project; R. Kubo and S. Kobayashi for valuable discussions; J.H. Brewer, D.G. Fleming, A.S. Arrott, K.M. Crowe and all other µSR scientists at TRIUMF for technical collaborations and heartful encouragements; J.T. Sample and other staffs of TRIUMF for the operation of cyclotron and the hospitality at Vancouver; and Japan Society for the Promotion of Science Foundation, Grant-in-Aid of the Japanese Ministry of Education, Culture and Science, Mitsubishi Foundation, Toray Science Foundation, and NSERC of Canada for financial supports. I also acknowledge a financial support of D.G. Fleming in 1979 for my stay at TRIUMF. It has been my great pleasure to have many opportunities for discussions with scientists working on spin glasses. I thank A.T. Fiory, R.H. Heffner, D.E. Maclaughl in and A. Schenck for the discussions on their µSR investigations of spin glasses and comments on the present work; A.M. de Graaf, J.l. Tholence, R. Tournier, A.P. Murani and J.A. Mydosh for discussions and comments; I.A. Campbell for sending a preprint related to the present work; and many other scientists for useful conversations. It is my pleasure to acknowledge heartful encouragements of my parents. - 8 -INDEX Chapter I. Chapter I I -A. 11-B. Chapter Ill-A. lli-B. Chapter IV-A. IV-B. Chapter V-A. 1/-B. Chapter Ill. Experimental Procedures of µSR Measurements Muon Spin Relaxation in Nuclear Dipolar Systems A Stochastic Theory of Spin Relaxation in a Spin Glass Zero-Field µSR in AuFe and CuMn Longitudinal-Field µSR in CuMn Effects of an Order Parameter and Distribution of Correlation Titnes Field-Induced Effects Reflected in Transverse-Field µSR Dynamical Properties of Dilute-Alloy Spin Glasses Measured by Different Experimental Methods Zero-Field µSR in an Insulator Spin Glass (Coo)40 (Ai2o3) 10(Si0 2) 50 Discussions and Conclusions - 9 -Chapter I. EXPERIMENTAL PROCEDURES OF µSR MEASUREMENTS To begin with, let us see a brief sketch of "how" and "what" we measure in µSR experiments [1]. In major "meson factories" as SIN (Zurich), TRIUMF (Vancouver), LAMPF (Los Alamos) and CERN (Geneva), a high intensity proton beam of about 500-800 MeV is extracted to hit a meson-production target, where pions are produced as the primary part icle . With a sho r t life ti me t = 2.6 x 10- 8 1T sec, pion decays into a muon and a muon-neutrino during the transportation through t he beam channel . Due to the parity violation of weak interaction, muon spin is 100% polarized, by nature, along the direction of motion in the center of mass system of a pion decay. Then by choosing muons of a proper momentum with bending magnets of the beam channel, we obtain a muon beam whose spin is polarized (70~100%) along the beam direction. It is again the effect of parity violation which makes it possible for us to detect the time evolution of muon spins. After stopped in a specimen, positive muon (µ+) decays into a positron (e+) with the mean life time + + t = 2.2 µsec by a three body decayµ + e + v + v . Positrons are emitted µ µ e with a spatial asymmetry referred to muon spin direction, i.e., practicallv with about 2~30 % higher probability to the spin direction ofµ+ compared to the opposite side, thus telling us the direction and polarization of muon spins. Conventional way to measure the depolarization of muon spins, which has been used in majority of µSR experiments so far, is the transverse-field µSR (TF-µSR) method where we observe the damping of a coherent precession of muon spins by applying a transverse external ma~netic field HT perpendicularly to the initial direction of muon spins. As shown in Fig. 1-1-a, we stop polarized positive muons in a specimen, identifying an incident muon by a counter M, detect its decay positron by a counter E, and accumulate such an - 10 -event one by one. Since positrons are emitted preferentially to the muon spin direction, the time spectrum of decay positrons E(t), plotted versus residence -time t ofµ+ in the specimen, exhibits muon spin precession around HT E(t) N•exp(-t/T ){1 + A•G (t)cos(y HTt + $)} , µ X µ ( 1-1 ) where A is the initial asymmetry of muon decay (usually ~0.2), y is the µ gryomagnetic ratio 2n x 1.35 x 104 /sec/0e ofµ+, and G (t), the transverse X spin relaxation function , represents the depolarization of muon spins in the specimen . The spectrum is usually observed in a time ranqe t = 0 10 µsec, with a specimen of a size typically about 1 inch in diameter, where we stop 103 104 muons each second. As only one muon exists in the specimen when a positron is + detected,µ is a completely dilute magnetic probe without any probe-probe interaction. In transverse-field µSR, which exactly corresponds to the "T2" measurement of magnetic resonance, spin depolarization is caused both by static inhomogeneous random local fields and by fluctuating random fields, thus it is difficult to distinguish the static and dynamical effects in this experiment. Furthermore, it is sometimes difficult to deduce detailed shape of relaxation functions in this method, since G (t) is observed only upon the X sinusoidal modulation. In 1978 at TRIUMF, we established [2,3] a completely different method to observe muon spin depolarization. As shown in Fig. 1-1-b, we measure the decay positrons by forward (EF) and backward (EB) counters, applying a longitudinal magnetic field HL or even without applying any field. As long as muon spins keep the polarization, the asymmetry EF:EB is about 1.2:0.8 (after some corrections for the instrumental solid angle factors), but it becomes 1:1 when muon spins are depolarized. - 11 -EF(t) N•exp(-t/T ) {1 + A•G (t) } µ z (1-2) Thus the longitudinal muon spin relaxation function Gz(t) can directly be observed from the ratio (EF-EB)/(EF+EB). It is this simple technique that allowed us to start zero-field µSR (ZF-µSR) and longitudinal-field µSR (LF-µSR) exper iments . Zero-field µSR is particularly suitable for systems sens i tive to external fie l ds, such as spin glasses or superconductors , whereas the stat ic and dynamical effects of random local fields on muon spin depolarizat ion can clearly be d i stinguished in LF-µSR, as in the 11T111 measurement of magnetic resonance. We first applied this method to an antiferromagnet MnO [2 , 4], and then to a study of critical phenomena of a weak itinerant helimagnet MnSi [5,6]. ZF- and LF-µSR also opened ·a way for experimental investigatio ns of zero- and low-field spin relaxation phenomena, as we shall see in the following chapters. See related publication A for details of experimental ·techniques. - 12 -Chapter I I -A. MUON SPIN RELAXATION IN NUCLEAR DIPOLAR SYSTEMS In this chapter, we shall survey low-field spin relaxation phenomena in nuclear dipolar systems studied by ZF- and LF-µSR, which is deeply related to the µSR investigations in spin glasses. More than ten years ago, Kubo and Toyabe [8,9) theoretically studied low-field spin re l axation in random local field H with Gaussian distribution (width /',/ y ) µ and derived y 3 (-µ-) ili• 6 exp(-G 1 2 22 122 g (t) = - + -(1- 6 t )exp(- - 6 t) z 3 3 2 i=x,y,z (2-1) (2-2) (2-3) as the zero-field spin relaxation function of a probe in the static local fields. We have actually observed this "Kubo-Toyabe function" by ZF-µSR in + nuclear dipolar systems (10,11), as shown in Fig. 2-1 for an example ofµ in MnSi. Here the 1/3 component represents muons stopped at a site where local field is rather parallel to the initial polarization direction (i.e., the beam direction, z-axis), while the damped oscillation of the 2/3 component is caused by x- and y-components of random fields. By applying a small longitudinal field HL along z-axis, such static random fields can be "decoupled", and the observed soectra in Fig. 2-1 followed a theoretical curve 262 1 2 2 - - {1 - exp(- - 6 t )cos(w t)} 2 2 · L WL (2-4) - 13 -calculated by the same authors [8,9]. In static random fields from surrounding nuclear moments, zero-field µSR exhibited a faster depolarization of muon spins compared to the measurement in a high transverse field HT [1 1], as demonstrated in Fig. 2-2 where we found approximately five times greater rate cr2 for initial Gaussian-like damping 2 2 exp(- cr t) of G (t) compared to G {t). In TF-µSR, only one component of static Z X random fields parallel to HT contributes to depolarization, whereas the zero-field relaxation is caused by H and H, and this makes a difference of X y a factor two. Another factor 2.5 comes from the following effect. Surrounding nuclear moment I pre .cesses around HT in a high transverse external field, averaging out the components of I perpendicular to HT. In zero-field where these moments are completely randomly oriented and static, such an averaging does not take place and all the non-secular terms of dipolar interaction contribute to random fields, yielding the di polar width 2 2 2 2 2 - 6 A = - I(I+l)ij y y · E r. Z 3 I lJ i I ' (2-5) which is 5/2 times greater than the well-known Van Vleck ~alue in TF-vSR. The simplest way to take into account the effect of dynamical modulation of local fields would be the "strong collision model " [12] which assumes: 1) The local field onµ+ is suddenly changed by a "collision", and the local field after this collision is distributed over P(H. ) without any correlation I to the field before the collision. 2) The collision takes place with the rate v as a Markovian process <H(t)H(O) > / <H(0) 2> exp( -v t) . (2-6) The strong collision model was f i rst adopted by Kehr et a l. [13] for transverse-field muon spin relaxation. In t his model, the dynamical' zero-field spin rela xation function G (t, v ) is derived from the static function g (t) as z z - 14 -G (t,v) z 2 ftft + v • 0 o2 + .... } (2-7) I have illustrated this situation in Fig. 2-3. G (t) is given as an envelope z function made by the convolution of static functions g (t). Based on this z picture, I independently reached eq. 2-7 and proposed to apply it to ZF-and LF-µSR, since I found [14] a slight difficulty in earlier dynamical treatments of Kubo and Toyabe [8]. As in Fig. 2-3-a, fast modulations v/ 6 » 1 will make the Gaussian-like static function g (t) into the exponential shape z G 2 G (t,v) = exp(-2 6 t/v) , z (2-8) i.e., the narrowing effect in the words of magnetic resonance. A characteristic feature of ZF-uSR is its response to the slow modulation of local fields (v/ 6 < 1) via the decay of the 1/3 asymmetry. From the intuitive picture shown in Fig. 2-3-b, I derived [14] G2(t) =½exp(- fvt) (2-9) expected in the limit v / 6 « 1 and t• 6 » 1. We performed a numerical calculation of eq. 2-7, by using a Laplace transformation [10], and obtained the line shapes for different values of v shown in Fig. 2-4 [10,11]. Relaxation functions for LF- and TF-µSR can also be calculated in a similar way. - 15 -I Chapter 11-B. A STOCHASTIC THEORY OF SPIN RELAXATION IN A SPIN GLASS What kind of line shapes we would then expect in spin glasses? In this chapter, we shall develop a stochastic theory of spin relaxation of a magnetic probe in a random dilute spin system. To realizeµ+ in dilute-alloy spin glasses, the following information is helpful. (a) No effect of muon diffusion:µ+ is known to rest stationarily at an octahedral interstitial site of pure Cu between T = 2 70K (15], and similar situations can be expected in other fee noble metals Au or Ag (16]. lb) Random and uniform occupation of interstitial sites by muons: This assumption well explained the earlier results of TF-µSR in AuFe and ~Mn [17], and will be confirmed also in ZF-µSR. (c) Dominant atomic dipolar fields onµ+: Because of the weak coupling between conduction electrons and muon spins (Knight shift ~ 50 ppm forµ+ in Cu [18] is more than 40 times smaller than ~ 2300 ppm for the host Cu nuclei (19]), averaged RKKY field onµ+ in 1 at. % ~Mn is only around 10 G. Atomic di polar field is around 100 G, and nuclear dipolar field from Cu nuclei is around 4 G. Thus the random fields at muon sites in spin glasses are mainly originated from atomic dipolar moments of surrounding impurities, and the dynamical modulation of local f1elds is caused only by the impurity spin fluctuations. Static random local field on a probe from dilute moments in such a system has a Lorentzian distribution (H.W.H.M. a /y) for dilute limit [20,21] µ (effective for Mn or Fe concentrations c s 3 at. %)t PLlH.) = YJJ a x,y,z, I 1T 2 2 2 a + y H. µ I (2-10) PL ( 1 'HI) y 3 a 2 = _µ_ X 411IHI ' 2 (a 2+ y 2IHJ2)2 1T µ (2-11) and this will be a good approximation even for muons surrounded by the "randomly - 16 -ordered" moments (22]. When local field is static, the zero-field muon spin relaxation function has a simple form gzL(t) = _1_ + ~(1- a t)exp(- a t) 3 3 (2-12) calculated by Kubo (23]. As shown in Fig. 2-5, gL{t) has the characterist i c z ' 113 component similar to the Kubo-Toyabe function gG(t), but the damped z oscillation of the 213 component decays faster than gG{t), reflecting a larger z randomness of Lorentzian local field compared to Gaussian. When we apply the strong collision model to such a system, we have t o take note on the following feature: GL{t) calculated directly from the static z function gL(t) in eq. 2-7 does not exhibit the narrowing effect even for very z fast modulation v i a» 1, since the envelope function made by the exponentially decaying gL(t) damps as gL{t) itself. Figure 2-6 shows the relaxation z z functions calculated in this way. GL(t) = exp(-4 a tl3) for v i a» 1 without the z change of the initial decay rate, while a form similar to eq. 2-9 can be expected for slow modulations v i a« 1 and t• a » 1. In random dilute systems, however, there should be many different muon s i tes characterized by different surr ounding impurity con f igurations as illustrated in Fig. 2-7. When surrounding moments fluctuate randomly, the local field at each site will be modulated, but with different variable ranges for different sites. Then the assumption 1) of the strong coll is ion model should be applied not with the total Lorentzian distribution, but with the sub-distribution P.(H) at each j-th site. For simplicity, I assumed the J Gaussian shape for P.{H) with the width ~ - corresponding to the variable range J J of random fields at the j-th site. Then muon spin at the j-th site would exhibit the relaxation function GG(t, ~ .,v) in a fluctuating local field with the rate v. The possibility to z J find a muon site with ~ - can be given as J - 17 -p(ll) a 2 12hr - exp(- ...E_) , l u 2 (2-13) so that the Lorentzian local field can be expected in the bulk specimen: (2-14) Thus I obtained [24] the zero-field muon spin relaxation function in spin glasses SG Gz (t,a,v) ("" G J G (t,ll,v) p(ll) dll 0 z (2-15) shown in Fig. 2-8. In contrast to GL(t) in Fig. 2-7, G5G(t) really exhibits z z the narrowing effect. This is a simple example indicating an important role different operations (static+ dynamic; spatial averaging) in random systems. (c. f., GL(t) in Fig. 2-7 wi 11 describe the case of fast muon diffusion in z random dilute spin systems, where the local field onµ+ will be modulated with the full range of PL(H.).) I SG The static limit of G2 (t ,a,v), shown by the broken line of Fig. 2-8, L becomes gz(t) of eq. 2-12. The slow modulation via « 1 is reflected as 1 2 2 Gz(t) = J exp(- rt)+ 3(1-at)exp(-at) (2-16) as expected from eq. 2-9, since eq. 2-9 does not depend on the value of LI. of The narrowing effet, starting at vi a~ 2, reduces the decay rate of G5G(t), but z this takes place very slowly with increasing via, due to a wide variety of local fields at different sites . In the limit of fast modulation vi a » 1, G5G(t) becomes z G (t) = exp(-/ 4a2tlv ). z (2-17) This decay with a "root exponential" shape was noticed in NMR measurenets in random dilute systems [25], to which the present theory has provided a - 18 -microscopic theoretical basis with eqs. 2-8 and 2-15. The 1 imiting cases eqs. 2-16 and 2-17 have recently been confirmed by a computer simulation of Fiory [26]. Since magnetic moment ofµ+ is much smaller than the impurity spin, we can neglect perturbations on the spin glass systems caused by the existence ofµ+. Neglecting the spatial correlation of neighbouring impurity spins, we will expect the same auto-correlation function for an impurity spin Sand for local field Honµ+ as <S(t)S(0) > / <S(0) 2> = exp(- vt) (2-18) Hence we can deduce the correlation time, =1/ v of impurity moments (Mn or Fe) C by comparing the observed spectra of ZF-µSR with G;G(t) of Fig. 2-8. In 1 at. % ~Mn, where the averaged local field coefficient a is roughly 107 /sec, we can observe 105 /sec < v < 1010 /sec by ZF-µSR within the time window 30 nsec < t < 10 µsec of muon measurements. Relaxation functions in spin glasses expected for transverse-field µSR can also be calculated in a similar way. They are shown in Fig. 2-9. When we compare relaxation functions Fig. 2-8 and Fig. 2-9 of different µSR techniques in spin glasses, we find the following advantages of the zei0-field measurement! - a) No disturbances on the system caused by the external fields, b) Clear distinction of the static 1 ine shape from the dynamical case in ZF-µSR thanks to the characteristic 1/3 asymmetry of the static function, c) Response to the slow modulation of local fields via the 1/3 asymmetry in ZF-µSR, d) Possibi l ity to decouple small inhomogeneous fields (such as nuclear dipolar fields) by applying a small longitudinal external field. Figure 2-10 compares the detectable region of correlation time 'c of spin glasses for several experimental methods . ZF-µSR covers a wide region between neutron scattering and ac-susceptibility. - 19 -To calculate the static width a of random fields, I derived a convenient equation by using p (6) of eq. 2-13. Let us define 62 as the dipolar width Max expected when all the lattice points are filled up with impurity spins (c=100 %). Since 62 is an additive quantity for each impurity spin, the averaged dipolar field onµ+ in a dilute alloy can be expressed as (2-19) (2-20) where c is the impurity concentration. The right hand side of eq. 2-20 becomes 2 12/Tr a •6 M ax a 2 exp(- --- ) - /1i7"ir a26 2 a erfc(-- ). fi 6M (2-21) Max ax In the dilute limit 6M »a , the second term of this equation can be neglected ax compared to the first term, and we obtain a quite simple relation for the random fields in spin glasses a = ./rr/2 c· t:. Max (2-22) This treatment can be called a "sum rule", which automatically gives the static width a from the lattice sum and the impurity concentration. In spin glasses, the full contribution of non-secular terms of the dipolar interaction in zero-field , described in the previous chapter, will be somewhat reduced by the fast precession of the impurity spin S around its internal field. For sufficiently high internal fields on S, we obtain (2-23) where m denotes the magnetic quantum number of the spin Sand P(m) gives its occupation probability. ' 2 At higher temperatures, P(m)=l/(25+1), whereas 6 is proportional to s2 at T=0. - 20 -Chapter Ill -A ZERO-FIELD µSR IN AuFe and CuMn + The first experiment to probe spin glasses withµ was performed by Murn,ick et al. [17] in AuFe (1.5 %) and CuMn (0. 7%) using the transverse-field µSR method. They measured the muon spin precession and found a rapid change _of the rela xation rate around T g' as shown in Fig. 3-1, but the dynamical effect was not distinguished from the static 1 ine broadening in this experiment. At TRIUMF, we observed [27] the spin relaxation of + in zero-field (< µ in AuFe (1.0 at. %, T =9.lK deduced from the cusp of ac-susceptibility observed g by Chikazawa and Miyake; 1.4 at .%, T =12.8K: both are made by the rapid g cooling down after arc melti ng) and ~Mn (1. 1 at. %, T =10.8K: quenched from g 930 C) prepared by Miyajima. The specimens were checked by chemical analysis, ac-susceptibility (very sharp cusp exhibited in the following figures), and x-ray micro analysis (homogeneous impurity concentration within ± 5% of each concentration among several different spots of 10 µmin diameter). The temperature was stabilized within ± O.lK in a gas-flow type liquid He cryostat throughout the measurement. Since nuclear dipolar moments of Au is negligibly small, local fields + onµ are originated only from Fe moments in AuFe. Figure 3-2 - a shows the 1 observed zero-field muon spin relaxation function, which fitted well to the theoretical lines representing GSG(t) of eq. 2-15. An almost static relaxatio n z function like gl(t) of eq. 2-12 was observed at T 'v 0.5T, and a slow z g depolarization of the 1/3 asymmetry was foun d at T 'v 0.7T. Around T, the g g depolarization rate rapidly decreased with increasi ng temperature, reflecti ng t he narrowing effect due to the dynamical fluctuations of Fe moments . At higher temperatures T 1.3T, the fast spin fluctuation aver aged out the g . random fields onµ+ and held. G (t) 'vl, so muon spins exhibi t ed almost no z depolarization. - 21 -Oe) In ~Mn, nuclear dipolar fields Heu from Cu neclei als9 contribute to muon spin depolarization. At higher temperatures T 1.3T , where the effect g of local fields from Mn moments are eliminated by the fast spin fluctuations, we observed a typical Kubo-Toyabe function eq. 2-3, expected forµ+ in static nuclear dipolar fields, with the width~= 0.36 /µsec exactly correponding to the case ofµ+ in pure Cu. Thi s static line shape confirmed that muons are not diffusing in the crystal. When we cooled down towards T, relaxation g rate rapidly increased reflecting the local fields from Mn moments, and the observed spectra were fitted well by the solid lines of Fig. 3-2-b () SG G +. G t = G (t) x {g (t) fo r µ 1n pure Cu} z z z (3-1) as expected for coexisting effe c ts of fluctuating dynamical fields and small static random fields. The depolarization rate continues to increase with decreasing temperature below T. g SG At lower temperatures T 0.5T, G (t) g z was observed with the character i stic 1/3 component, as indicated by the broken line in Fig. 3-2-b, since the effect of nuclear dipolar fields is eliminated by the almost static and dominant atomic dipolar fields in this temperature region. The static width a of random fields, determined from observed spectra at T 0.5T, was a = 5.2 ± 0.3 /µsec (a/y = 61 0e) in 1.0 % AuFe and g µ a = 10.2 ± 0.8 /µsec (a/y = 120 0e) in 1. 1 at .% CuMn. By assuming S = 5/2 µ for the moment of Mn impurity in Cu, eqs. 2-22 and 2-23 give a theoretical estimate a = 11.4 /µsec for P(m) 1/(2S+l). A similar calculation for 1.0 % AuFe yields a = 3 . 5 / µsec f or S=l and a = 4.8 / µsec for S = 3/2. In this way, the observed dipolar width agrees fairly well with the theoretical estimates, supporting the assumption of uniform occupation of interstitial sites by muons. The observed static line shape indicated the Lorentzian local field, also su pporting this assumption. - 22 -By assuming these experimental values of a , and by comparing the SG observed data with Gz (t) of eq. 2-15 (Fig . 2-8) for AuFe, and with eq. 3-1 for CuMn at T 0.9T, we deduced the correlation time, = 1/v of Fe or Mn g C moments as shown in Fig. 3-3. , changes rapidly (about two orders of C magnitude) in the temperature region 0.8Tg $ T $ 1.2Tg' and the sharpness of this change roughly corresponds to a: {T/(T-T ) }2 'c g (3-2) above T 1.03T when we assume the power law behavior of the critical slowing g down in the paramagnetic region. When we assume the Arrhenius law 'c T exp(E /kT) , o a the slope around T corresponds to the activation energy E g a pre-exponential factor ,0 = 10- 15 sec. (3-3) 20kT and the g When we plot 'c of different specimens versus inverse temperature Tg/T normalized by T of each specimen, as in Fig. 3-3-b, all the data points follow g a unique curvature, indicating a "scaling property" existing for the dynamical quantity. The absolute values of 'cat T = T9 are roughly 'c -8 -9 10 'v 10 sec (3-4) for both AuFe and CuMn alloys. This must be a characteristic number for the dynamical spin fluctuations of spin glasses. From the slow decay of the characteristic 1/3 component of zero-field spectra, we found the correlation times in microsecond region below T $ 0.8T. g Another important bit of information ·provided by this experiment -is the systematic dependence of the experimental results on the heat treatment of the specimen. We measured ZF-µSR also in ~Mn alloys made by a slow cooling after induction melting. In the slowly-cooled specimens, the observed - 23 -spectra deviated considerably from eq. 3-1, and the change of T around T=T C g was slow and continuous (see Fig. 3-3). The ac-susceptibility of slowly-cooled alloys exhibited a much broader maximum compared with the very sharp cusp observed in rapidly-cooled samples. These features are explained by the inhomogeneous impurity concentration of slowly-cooled samples, found by an X-ray micro analyzer, as in Fig. 3-4. In such inhomogeneous slowly-cooled samples, the fit of G (t) was z significantly improved by taking into account a large distribution of Tc +2 (x 10- ), as shown by the broken line of Fig. 3-5. In contrast, the best fit of the spectra was obtained for eq. 3-1 with the single value of Tc in the rapidly-cooled samples at T Tg. In the temperature region 0.8Tg $ T $ 0.9Tg, we noticed a slight deviation of ZF-µSR spectra from the theoretical curve even in the homogeneous rapidly-cooled samples (see Fig. 3-2-b at T=lOK). We shall come to this point later in Chaps. 111-B and V-A. In this chapter, to grasp a general feature of the "mean correlation time", we started with the simplest model function which assumes a single value of Tc in the system and an exponential auto-correlation function eq. 2-18 for Mn (or Fe) moments. Yet within such limitations, the present zero-field µSR experiment has determined the absolute values of Tc' for the first time, in a wide temperature region around T. A similar measurement has recently g been performed at LAMPF in ~Mn (1.6 at. %, T =7.6K) by Brown et al. [28], g . and the values of Tc derived from a similar simple model function [29] agreed very well with our data, as shown in Fig. 3-6, confirmi .ng the "scaling of Tc by Tg" holding in another RKKY spin glass system. - 24 -Chapter 111-B. LONGITUDINAL-FIELD µSR IN A SPIN GLASS CuMn To investigate further details of dynamical aspects , we have performed the µSR measurement in ~Mn by applying finite longitudinal external magnetic fieids HL [30]. When random fields Hi are static, the applied longitudinal field HL > Hi will "decouple" the random fields, then help keep the muon spin polarization and change the shape of Gz(t), as can be seen in the static relaxation function g;(t,HL) = 1 - : L j 1(wLt)exp(- a t) - (£_)2 {j (wLt)exp(- a t) - 1} WL 0 - { 1 + (£_) 2} a ft j (wL·r)exp(- a ·r) d, WL O 0 (3-5) of Fig. 3-7 derived by Kubo [23], where j0 and j 1 denote spherical Bessel functions and wL = yµHLr In contrast, in the local fields fluctuating fairly fast, the depolarization rate r of muon spins is propo r tional to [31] V r a: --2--2-wL + v (3-6) so the observed spectra will exhibit the "root exponential" shape of eq. 2-17 and will not depend on HL for the fast modulation v » wl. In this way, as illustrated in Fig. 3-8, we can clearly distinguish between the dynamical and static effects in LF-µSR, as in the 11T111 measurement of magnetic resonance. At TRIUMF, by using the longitudinal-field µSR technique, we observed the muon spin relaxation function G (t) in CuMn (1.1 at.%, T =10.BK, quenched from z - g 930 C, the same specimen used in the previous ZF-µSR experiment) in longitudinal external fields HL = 0 640 Oe. Temperature was set to each point in zero-field, and HL was increased isothermally at each temperature. In this measurement, we monitored the temperature by two different resistors fixed at different points of the specimen, which exhibited a good coincidence of readings within the fluctuation ± 0. lK of the temperature. The experimental - 25 -data in zero-field well reproduced the results introduced in the previous chapter with the deduced correlation times shown in Fig. 3-9, and with the averaged random local field a/y = 120 0e estimated at T 5K. )J Figure 3-10 shows the observed LF-µSR spectra fort < µsec. The effect of nuclear dipolar field Heu is not dominant for this time region, and Heu is completely decoupled in a finite field HL > 40 0e since Heu ~ 4 0e. Together with the experimental data, we have attached the theoretical function SG Gz (t,a,v,HL) which I numerically calculated in the same way as the zero-field function eq. 2-15, by taking into account the effect of HL and by assuming an exponential auto-correlation f unction for random fields H. as in eq. 2-6. I Sol id I ines in Fig. 3-10 represent such functions of "single Tc" model drawn with the best fit values of Tc= 1/vZF in Fig. 3-9 determined from the zero-field measurements. See Fig. 3-10. At T = 20K, muon spin is hardly depolarized because of the fast fluctuation of Mn moments. The data in HL = 40 and 640 0e confirm no instrumental effects on G (t) caused by the applied field. z At T = 12K (T~1.1T) G (t) did not depend on HL; except for the effect of g z Heu in zero field, and this tendency holds in longer time region t < 5 µsec as shown in Fig. 3-11, being consistent with the "single Tc" model which predicts no field dependence of the spectra for v2F » wl. At T = 11K, just above T, G (t) depended on HL' but only slightly. A little static contribution g z to the muon spin relaxation was suggested, but this is not a dominant effect. In contrast, we found a remarkable field dependence of G (t) when z we cooled down below T, as demonstrated in Fig . 3-10. At T = l0K (T~0.9T ) , g . g the very beginning part (t ::; 0.3 µsec-) of . the zero-field data ex.hibits somewhat slower damping compared to G (t) at T = 5K, which indicates a fairly z fast modulation rate v2F = 8 x 107 /sec when eq. 2-18 is assumed. We - 26 -calculated the line shape for the highest field HL = 640 Oe expected in the "single Tc" model with this value of vZF' and obtained the upper line in Fig. 3-10 at T = 10K. The observed field dependence is much greater than the prediction of the simplest model, suggesting the existence of much slower spin fluctuation and an important role of the decoupling effect by the external fields. The decoupling effect becomes clearer at T = 8K. At T = SK (T~O.ST ), g the observed spectrum is well explained by an almost static relaxation function for a slow modulation v2F = 1.5 x 105 /sec, which is quite close in shape to the static function eq. 3-5. Another interesting feature observed below T g is the amplitude of longitudinal field HL required to cause a certain decoupling effect. HL effect is caused by HL 160 Oe held G (t) close to 0.5 at 10K, but a similar z 320 Oe at SK. In a certainly high longitudinal field, where the static effects are appreciably eliminated, the depolarization rate becomes maximum for v = wl, so we can select the dynamical contribution by "gating" with HL' as illustrated in Fig. 3-12. In this figure, we compared Gz(t) observed in HL = 640 De, where the dynamical spin relaxation is caused by the Fourier component of fluctuating random fields with the frequency wl = 5.7 x 107 /sec. The damping rate of G (t) z increases with decreasing temperature above Tg' corresponding to the slowing down of Mn moments. The fast depolarization observed at T = 10K indicates an important role of the dynamical effect at this temperature, coexisting with the static effect we found in Fig. 3-10. When we further cooled down, the damping rate of G (t) starts to decrease, showing the decoupling effect. z The observed data at SK exhibits a slow relaxation, expected for a slow fluctuation of local fields. We can conclude the following features from the analysis described above. - 27 -(a) In the "paramagnetic region" T > T , muon spins are mainly depolarized g by the fluctuating random fields. This confirms the absolute values of T C in Fig. 3-3, deduced in ZF-µSR, as a good measure of correlation times in this temperature region. (b) + At T O.ST , local fields onµ can be regarded to be almost static, and g a possible deviation from the "single Tc" model does not play an important role. This may partly be due to the 1 imited sensitivity of µSR measurement, where fluctuations slower than v 105 /sec looks "static". The amplitude a of averaged static random field, estimated in ZF-µSR, is clearly re-confirmed SG by the good agreement of G2 (t, a ,v,HL) to the observed decoupling effect at T = SK. (c) Just below T , a fast dynamical component of local field is suggested to g be coexisting with a static component, and the "single T 11 model does not work C to explain the observed data. The static component is somewhat smaller than the full amplitude a / y of local fields, as it is decoupled by a smaller µ longitudinal field. - 28 -Chapter IV-A. EFFECTS OF AN ORDER PARAMETER AND DISTRIBUTION OF CORRELATION TIMES In Chap. Ill-A, we performed an analysis of zero-field µSR by assuming a single correlation time T for all the impurity moments in the specimen. C The model function GSG(t), calculated for this simplest case, explained the . z observed data fairly well, but some deviation was not iced in the spectra around T 0.8 ~0.9T . This deviation was further clearly exhibited in LF-µSR, g as shown in Chap. 111-B, where we found an unexpectedly large decoupling effect in CuMn at T 0.9T in longitudinal fields. As each impurity moment has a - g different internal field from surrounding impurities in spin glasses, it may not be trivial, indeed, to expect a unique value of T in the system. We C restricted ourselves so far to such a simplest model, however, in order · to grasp a general feature with "zeroth order approximation". In this chapter, we shall consider more complicated situations, beyond the simplest "single Tc" model. We will survey a model with a distribution of Tc' as well as an effect of order parameter below T. g Let us consider, first, the case when a system is devided into some clusters or domains, each of which having its own correlation time Tel' but Tel changes from cluster to cluster. In such a cluster which contains more than about 20 30 impurity spins ; a dominant local field at most of the muon sites is originated from the impurities within the cluster. The spin relaxation SG function of muons in the cluster is then given by Gz (t,a,vcl'HL) with the modulation rate vc 1=1/Tcl of the cluster. If the system is an assembly of h l h 1 . GSG (" ) b b. l . sue c usters, we ave on y to integrate z vcl over a pro a 1 1ty distribution P(vc 1) of vcl' proportional to the number of muon sites with vcl' in order to calculate the muon spin relaxation function expected in the entire - 29 -specimen. For convenience and simplicity, we. chose a Gaussian distribution for (ln "cl) [see foot note 1], and obtained G (t) as z G (t) = I SG P(ln v) d(ln v) (4-1) z Gz (t,a,v) 2 P( ln v) 1 exp{ - [(ln v)-(ln v0)] } (4-2) 2 ffi ·(In w) 2(ln w) where v0 corresponds to the averaged modulation rate and w gives its width. When we fit the zero-field spectra observed in homogeneous rapidly-cooled specimens of CuMn and AuFe with the above equation, the width w is estimated to be smaller than"' 3 at T = T, as mentioned in Chap. Ill-A, and no significant g improvement of the fit is found compared to the simplest "single," model C above T, indicating that the effect of different clusters is not obvious in g this temperature region. Since the simplest "single," model was not C successful in explaining LF-µSR data just below T , we attempted to reproduce g the field dependence observed in CuMn at T = 0.9T with longitudinal-field g relaxation functions given by eq. 4-1. As shown in Fig. 4-1 with an example of v0 = 5 x 107 /sec and w = 100, such a cluster model provides a better fit compared to the single, model, but there is still an obvious deviation C from the observed data. Below the ordering temperature, iri u.sual ferro- or antiferrornagnetic systems, fluctuation of magnetic moments is not described ·by a simple exponential auto-correlation function eq. 2-18, since each spin starts to have an averaged polarization, i.e., the staggared magnet'ization. Even in spin glasses, without long range spatial ordering of a.ny Fourier component of impurity moments, such a polarization may be realize 'd as a finite probability for each spin to be oriented towards its "preferred direction" during the fluctuation. This picture can be described by a non-vanishing part Q of auto-correlation function of impurity spins - 30 -<S(t)S(O) > I <S(0) 2> (1-Q)exp(-vt) + Q ( 4-3) where < > denotes the thermal average for one impurity moment and represents the spatial average over different impurities. Such an '~rdering in time" is suggested by Edwards and Anderson [32] based on the replica theory, and Q is called an "order parameter". A simplest way to simulate eq. 4-3 is to assume each spin as a vector sum of a static component IQ•S and a dynamical component 11-Q•S randomly fluctuating with the rate v. Then the local field onµ+ can also be described by the same vector sum. When the fluctuation is appreciably fast (v/a » 1), the spin relaxation function at each muon site GG(t,6,v,Q) is given by a product z of static and dynamical relaxation functions as 6 IQ•6 , 6 2 Q•62 , 2 = 62 - 6 2 s s . 6d s 2 G { G of eq. 2-4 } x exp( -26d V t) G2(t,6,v,HL,Q) g (t,6 ,HL) 2 2 , z s (4-4) V +wL where 65 and 6d correspond to width ' s of Gaussian local fields representing static and dynamical components at each muon site. Then by taking the spatial average over p(6) of eq. 2-13 as in eq. 2-15, we obtain a model function for the entire system (4-5) which includes the effect of Q. For the limit of extremely fast fluctuation which eliminates the contribution of dynamical part, this function becomes the static function gL{t, a ,HL) of eq. 3-5 with a reduced width a = IQ•a of static z s s random fields, which exhibits IQ times slower initial decay of the zero-field function as well as the decoupling effect caused by IQ times smaller longitudinal field in LF-µSR, compared to the case with the full amplitude a. When we tried to reproduce the spectra of LF-µSR observed at T 10K, - 31 -this model of "coexisting static and dynamical fields" with eq. 4-5 has provided a further better fit to the experimental data, as demonstrated in Fig. 4-1, compared to the simplest "single, "model and to the cluster model. Sol id C lines in Fig. 4-1 represent the line shape which I numerically calculated for 40% of static part (IQ= 0.4) and the rest of dynamical component fluctuating 8 with v = 7 x 19 /sec. Though the agreement to the experimental data is still not complete, Fig. 4-1 supports our arguments of coexisting static and dynamical local fields in Chap. 111-B. The above treatment with eqs. 4-4 and 4-5 is very cl9se to the "mean field app roximation" in random systems [33), where we assume a unique polarization for all the surrounding moments . but take into account the difference of internal fields at different impurity site. It is interesting to note that the estimated static component 30 40% corresponds to the mean field order parameter Q = 0.09 0.16, which looks reasonable at T 0.9Tg [33]. In zero-field · where the static function in eq. 4-4 becomes eq. 2-3, the integration of eq. 4-5 can be performed analytically, and obtained 2 2 1 / 2 Gz(t) = 3 exp(- 4ad t/v 2 . + - {1 3 a t ----8----} exp(-/ 4a/t!v + a/t 2 / 4a/t!v + a/t2 a s IQ•a ' 2 Qa ' = a for a model function in the entire system. 2 - a s 2 (4-6) In the static limit Q + 1, a + a, . s L . the above equation becomes g (t) of eq. 2-12, whereas the "root exponential" z shape of eq. 2-17 can be expected for the dynamical I imit Q + 0, ad+ a , via» 1. This model function also indicates that the fast dynamical component can cause the decay of the 1/3 asymmetry in zero field, when it coexists with a static component. With the analytic form of eq. 4-6, I fitted the spectra of zero-field µSR - 32 -observed in AuFe (1 . 0 at. %) below Tg, by varying the static width as and the 2 decay rate A = 4ad Iv as free parameters. As illustrated in Fig. 4-2, this funct ion (sol i d 1 ine) exhibits better agreement to the data between 0.7T < T < 0.9T compared to the "single T "model with a value of a fixed by g g C t he results at T 0.STg. Temperatu re dependence of the width a of the s static component, thus obtained assuming eq. 4-6, is plott_ed in Fig . 4-3. also attached a theoretical value a of the full amplitude of local fie l ds at T 0 0 ' calculated by using eqs. 2-22 and 2-23 with P(m) = 1 form= Sand 0 form~ S, and by using the effective Bohr magneton number peff = ge l S(S+l) = 3.4 determ ined for another AuFe of 1 at. % by Cannella and Mydosh [34] in the susceptibility measu rement. Though t he value of a i s based upon several 0 assumptions and thus should not be taken too seriously, the experimental points of as exhibit a tendericy to increase towards a0, as in the magnetization curves of usual ferro- or antiferromagnetic systems . The decay rate A of the 1/3 asymmet ry was around A~ 0.4 /µsec at T 0.7Tg, and decreases monotonically with decreasing temperature towards A~ 0 . 1 /µsec at T 0.ST . By using the deduced values of A and by assuming g ad2(T) = a 2 - a 2 (T) for the amplitude of dynamical component, we can obtain 0 S the correlation time Tc of the fluctuating component as Tc= \/ 4ad2, which I plotted in Fig. 4-4 with open circles . These points should be regarded as a rough measure, since Tc is related to the uncertainty of a0, and also to a possible instrumental error for the sol id angle factor which prevents to determine precise values of \ . Figure 4-4 clearly indicates, however, the dynamical component keeps a fairly fast fluctuation below T , when we adopt g present model of coexisting static and dynamical local fields. The correlation time T = 1.4 x 10- 9 sec, estimated in Fig. 4-1 from LF-µSR in CuMn at T 0.9T, C g agrees well with the values of AuFe in Fig. 4-4. As described in Chap. Ill-A, the depolarization rate of muon spins in - 33 -zero field keeps increasing when we cooled down through T. The change above g T is clearly due to the dynamical fluctuating fields, since the spectrum was g not changed by a sma 11 1 ong i tud i na 1 fie 1 d. Below T , however, two basically g different interpretations could explain the observed temperature dependence. The picture A) of "single Tc" model ascribes it to further slowing down of impurity spins and regards the decay of the 1/3 component as the result of the slow modulation of impurity spins. An alternative view 8), the "static+ dynamical" model assumed in the analysis of this chapter, is to attribute the observed change below T mainly to the onset of a static component of local g fields, and explains the decay of the 1/3 component to be caused by a fairy fast fluctuation of dynamical component. Our attempt to reproduce the results of LF-µSR (Fig. 4-1), together with a comparison of the fit of ZF-µSR (Fig. 4-2), support the picture 8), i.e., the onset of the static component below T, even g though the agreement to the observed data is not complete and the real system may be somewhat more complicated. We have seen that the macroscopic cluster modal was not very successful in explaining the observed data. The observed co-existence of static and dynamical local fields, however, does not necessarily provide a direct evidence for the existence of the "mean-field order parameter", since the similar local field can be embodied also in another physical situation where impurity spins of different correlation times are distributed microscopically in the system. We need, then, further studies to determine the detailed mechanism which produces the static component below T . In this chapter, we neglected a possible change g of the spin dynamics of the system induced by the external fields in LF-µSR. This point will be discussed in the next chapter. I proposed another stochastic theory to simulate the effect of Q on G (t) in a time-dependent treatment [24], z details of which will be introduced in Appendix I. - 34 -Chapter I V-B. FIELD-INDUCED EFFECTS REFLECTED IN TRANSVERSE-FIELD µSR In the arguments of Chaps. 111-B and IV-A, we have neglected a possible change of spin dynamics of the spin glass system itself in external fields, i.e., the field-induced effects. The sharp cusp of ac-susceptibility of spin glasses is known to be appreciably rounded by a small external magnetic field such as 300 500 0e [34 , 35] as shown in Fig. 4-5 [35] . Then there may be a considerable field-indeced effects in µSR results. In the followings, to investigate this feature, I will revisit transverse-field µSR measurements, performed by Murnich et al . [17] in CuMn and AuFe, by Brown et al. [36] in ~Mn, and by Emmerich et al. [37] in ~Mn . G ( t) X All the TF-µSR spectra were analyzed by assuming the simplest form + exp(-t/T 2) for the transverse spin relaxation function ofµ . Murnick et al. [17] calculated the width W of random fields as 1/T2 y w, µ and plotted its temperature dependence in Fig. 3-1. The relaxation rate A 1/T2 for the results at LAMPF [36] and SIN [37] are compared in Fig. 4-6 [36], normalized by the value AF observed at lower temperatures (T s 0.5Tg) for the frozen spins. We should fisrt notice that larger relaxation rate is observed for higher transverse field HT at each temperature point. This tendency is especially clear in the paramagnetic region above T , as shown g in Fig. 4-6. Since averaged static random local field onµ+ is around 100 0e in these specimens, there may be some contribution of non-secular terms of dipolar interaction [11] at lower external fields HT s 200 0e. This e f fect, however , increases the relaxation rate for lower HT' opposite to the observed tendency, thus cannot explain the field dependence of A. Spin glasses exhibit a Curie-1 ike susceptibility above T , which is embodied by a small polarization g of impurity moments along the external field. Sol id 1 ines in Fig. 4-6 - 35 -indicate the expected relaxation rate for a small static field onµ+ originated from polar ·ized moments in each external field fol lowing Curie law. The data points in Fig. 4-6 exhibit much higher relaxation rate compared to the sol id 1 ines, suggesting the Curie-I ike polarization not to be a main origin of the observed field dependence of T2. Two possible explanations: 1) Dominant frequency spectrum of the fluctuating spins is shifted towards lower frequency by the applied external field, and Tc becomes longer for higher HT; 2) Some small static component is induced for fluctuating moments by HT' and thus increased the muon depolarization rate in TF-µSR. We can distinguish between (1) and (2) by comparing the field dependence of TF-µSR and LF-µSR. Since the system is in the region of very fast fluctuation (narrowing 1 imit) above T 1.1T , g similar increase of muon depolarization rate for higher HL should be observed if the case (1) is applied, then A(HT;TF-µSR) - A(HL=HT;LF-µSR). In contrast, in the case (2), LF-µSR data will not exhibit a large field dependence, since a small static component can be decoupled by HL. Then A(HT;TF-µSR) > A(HL=HT;LF-µSR). As shown in Chap. 111-B, LF-µSR in CuMn did not exhibit a large field dependence above T, and the depolaiization rate tends to be slightly decreased g . . in higher longitudinal field, opposite to TF-µSR, suggesting the case (2). Heffner et al. [38] recently extended their measuements in ·AgMn to cover LF-µSR, yielding negligibly small depolaiization rate above T 1.2Tg for HL = 0.15, 1.0, 5.0 k0e without any field dependence, which clearly indicates the case (2) with A(TF-µSR) » A(LF-µSR). The reported exponential shape of G (t) (36], corresponding to the static 1 ine shape in spin glasses, is also X supporting the picture (2). Thus we can conclude that the muon depolarization in high transverse -fields is caused by a field-induced static component of local fields. - 36 -Since negligibly small muon depolarization is observed in ZF- and LF-µSR above T 1.3T, there should be almost no contribution of fluctuating g dynamical fields on the relaxation rate A of TF-µSR observed in AgMn (Fig. 4-6) with HT= 1 and 5 k0e at T 1.6Tg' 2.2Tg, and 3.3Tg. Then we can estimate the field-induced static component of impurity moments as A(HT,T)/AF. Such a static component will not contribute to the ac-susceptibility x , if its ac modulation is slower than the measuring frequency of x , causing a possible ac reduction of xac from the Curie-like susceptibility above Tg as 2 X (H=0,T) - X (H=HT,T) = X (H=0,T) x {A(HT,T)/AF} ac ac ac (4-7) since A is proportional to the induced static width « IQ, while the removed susceptibility to Q. Mulder et al. [35] recently reported a precise result of ac-susceptibil ity measured in CuMn (2.0 at. %, T =15.5K) with external fields H = 0 10 k0e, g shown in Fig. 4-7 where they plotted the reduction of x (H,T) from the zero-field susceptibility x (T) versus external field H. As can be expeceted in 0 Fig. 4-1, the effect of external fields becomes greater when temperature approaches closer to Tg, and the largest reduction of xac was observed at T=T9• After sealing the external field by the cusp temperatures of different specimens according to the common scaling law in RKKY spin glasses [39], I plotted in Fig. 4-7 the reduction of Xac which l ~stimated with the TF-µSR results of AgMn by assuming eq. 4-7. The points of TF-µSR exhibit a good agreement with the susceptibility experiment, when we normalize _ the temperature with T , g suggesting that the well-known reduction of x in finite fields can be ac explained by the field-induced static component of impurity moments found in TF-µSR. It will really be interesting to investigate such an effect of finite external fields by using different probes with different time windows, such as x of different frequencies, µSR, and neutrons, in the same specimen. ac - 3? -What wi 11 then be the effect of external fields below T ? Even without g external field, x does not follow the Curie curve below T and exhibits a ac g sharp reduction, indicating the spontaneous onset of static component of impurity spins. Such an "original" static component is represented by the difference between Curie curve and Xac at H=0. A small static component will be induced by external fields, in addition to the "original" static part . We may be able to estimate the field-induced static component below T from the corresponding g curves in Fig ~ 4-7, which indicates the effect less than 5% at T 0.9T for g HL = 640 0e in 1 at. % ~Mn and smaller values for lower temperatures. Then we can assume that the observed decoupling effects in LF-µSR are mainly reflecting the "original" static component. We started the simplest analysis of LF-µSR in the previous chapters without taking into account the possi b le field-induced effects. As can be expected in the present discussions, general features of the observed spectra of LF-µSR would mainly be reflecting the "original unperturbed" system, since we performed the experiment in fairly low fields HL s 640 0e. We need, however, further systematic theoretical and experimental studies to fully understand the field-induced effects on the spin dynamics of spin glasses. - 38 -Chapter V-A. DYNAMICAL PROPERTIES OF DILUTE-ALLOY SPIN GLASSES MEASURED BY DIFFERENT EXPERIMENTAL METHODS In this chapter, we shall compare the present results of µSR measurements with dynamical experiments of dilute-alloy spin glasses performed by other experimental methods [39,40]. Recently, neutron spin-echo technique was applied by Mezzei and Murani [41] in ~Mn (5 at. %, T =27K), to obtain a time-correlation function of impurity g spins shown in Fig. 5-1. Though informations from neutons (closed circles) are 1 imitted to T $ 10- 9 sec, Fig. 5-1 demonstrates a rapid slowing down of C spin fluctuations above T, and an appearanee of long-1 ived spin fluctuations g below Tg. If we plot the time tl/e where t(q,t)/t(q,O) decays into 1/e at each temperature, as a rough measure of "mean correlation time", tl/e exhibits a _temperature dependence which agrees well with the µSR results at T Tg, as shown in Fig. 5-2 where we also normalized the value of Tc with Tg of different specimens. Ac-susceptibility experiments with different measuring frequencies [42] also provide dynamical informations. In the mesurement with a small ac-field of frequency v , the real part of ac-susce~tib i l ity x' is proportional to ,m x' a: 1 / { 1 + (2TI·v !vl } m (5-1) for the spin fluctuations with the rate v, so only the dynamical part with v > vm will respond to the external force. The remaining amplitude of the spin correlation function can then be estimated as the difference between the extrapolated Curie curve Xe . (T) and observed dynamical susceptibiliby ur,e below T g Q = { X (T) - X (T v ) . } / X (T) Curie ac ' m Curie ' (5-2) - 39 -which represents the portion of "static component" (v < v ) at the time t = 1/v . m m In this way, Murani et al. [43) plotted the t ime evolution of the "static" part , by changing the measuring frequency v , as open ~ ircles in Fig. 5-1 m dete rmined fr om x measured in the same spec imen used in the neutron experiment. ac Since x exhibits almost no frequency dependence below T = 0.9Tg [44), points ac for T 20K (T 0 . 7 T) keep almost constant values for di ffe rent time g ranges. As described by Murani et al. [43), it is interesting that the neutron data can be extrapolated smoothly towards the points of Xac· From static components of local fields a8 of Fig. 4-3 and the correlation time 'c of the dynamical component (Fig. 4-4) estimated in Chap. IV-A, we can calculate the auto correlation funct ion eq. 4-3 based on the ZF-µSR experiment in AuFe (1.0 at. %, T =9.1 K), which g (TIT 0.8). g I plotted with triangles in Fig. 5-1 for an example at T = 7K also indicated the "static " amplitude (a /a )2 by arrows at S 0 the edge of Fig. 5-1. When we scale the temperature by cusp temperatures of different specimens, neutron spin-echo, µSR, and susceptibility results are fairly consistent, supporting the picture of coexisting static and dynamical components below T , described in Chap. IV-A. This agreement sho~ld be very g important since the three measurements are performed in zero field with different time windows . Neutron results in Fig. 5-1 suggests a deviation of correlation functions from the simple exponential form starting even above T , as pointed out by g Murani [45). In contrast, we could not find a remarkable effect of distribution of T in µSR above T. Neutrons have a better resolution for fast spin C g -10 + fluctuations (, < 10 sec) compared to muons. Local fields onµ is C -originated from a few neighbouring impurity moments, thus there may be an averaging effect for, in µSR experiments. These factors would explain the C - 40 -difference between the two probes concerning the possible d i stribution of Tc' though a more precise comparison with the same spe cimen is required. From the l ine width 1/T2 and g-shift ~g/g measured by ESR in ~Mn(4, 25 at. %) , Salamon [46] estimated the correlation time of Mn moments as TC with T 0 T ( 1 - T /T) - l ( 5- 3) 0 g 7 x 10- 12 sec for 4% CuMn. This experiment was performed with an external fi eld H t 3 k0e in rather concentrated systems , and the observed ex variation of T2 was only around one order of magnitude with a few temperature points at T > 1.2T . These aspects would explain a dif f erent sharpness of g the correlation time compared to µSR. Field dependence of the line width was clearly exhibited in another ESR measurement of Dahlberg et al. · [47] in rather dilute AgMn (c = 0.2 0.5 at. %) performed with Hext = 600 0e and 3 k0e, as shown in Fig. 5-3. These authors found both a deviation of x and a ac positive g-shift of ESR, caused by the external field H t' appearing at the ex same temperature Td above Tg' and (Td-Tg) is larger for larger external field. This field dependence of g-shift indicates the field-induced effect of static component of spin fluctuations, similar to that observed by TF-µSR discussed in the previous chapter. An elaborate study of 57Fe Mossbauer effect in AuFe (0.8 ~8.0 at ,%) was performed by Violet and Borg [48]. From the spectra shown in Fig. 5-4, they deduced the temperature dependence of hyperfine field, plotted in Fig. 5-5. In 1965, when this experiment was reported, the "sharp cusp" of susceptibility was not known yet, so the "transition temperature" T0 was determined self-consistently from the Mossbauer results. As can be seen in Fig. 5-4, the splitting of characteristic six peaks is not clear above T 0.7T . Though g with such limitations, this was the first observation of "static" component of impurity spins in spin glasses, with the time window io- 7 sec of 57Fe. - 41 -To compare µSR measurement, I plotted the static component a /a estimated by S 0 ZF-µSR in AuFe (1.0 at. %) versus T/T in Fig. 5-5, as in the "magnetization" g curve. Results of both methods agree fairly well, providing another support to the picture of Chap. IV-A of coexisting static and dynamical effects below Tg. The transition temperature T0, found with their time window 10- 7 sec, is very close to the cusp temperature AuFe alloys. Spin fluctuation should Tg of xac measured later in different 8 then be faster than 10 /sec above T, g and a slower component starts to show up below T. This is consistent with g the absolute value T 10-S 10- 9 sec at T = T we deduced in ZF-µSR. It C g would be very fruitful to obtain Mossbauer spectra with much better statistics, and analyze it by taking into account the possible distribution of hyperfine fields as well as the dynamical effects [49]. Field-induced static component of impurity spins, discussed in Chap. IV-B, would be another interesting subject for future studies of Mossbauer effect. Host Cu NMR in CuMn spin glasses was carried out by Maclaughlin and Alloul [50,51] and Levitt and Walstedt [52]. They found a rapid increase of the line width when cooled down through T , as shown in Fig. 5-6, which g indicates the slowing down of Mn moments above T , and a change of static width g at lower temperatures T $ 0.5Tg. A ramarkable feature found in this experiment is a rapid reduction of the intensity of spin echo signals between T 0.5T g and T T , as demonstrated in Fig. 5-7 [53]. As an analogue to our µSR g measurement in HL = 640 0e shown in Fig. 3-12, this experiment is "gating" the dynamical effect with Hext = 1.6 k0e, wl = 1 x 107 /sec, since the loss of spin-echo signals reflects nuclear spin relaxation caused by dynamical effects within the deadtime 50 µsec of the spectrometer. Figure 5-7 roughly agrees with the µSR results, but the fastest relaxation was found at somewhat lower temperatures T 0.7Tg for a slower frequency wl in NMR, suggesting an existence of "slow modu.lation" v 107 /sec below T and favoring ou r analysis g - 42 -of µSR data with the simplest model in Chap. Ill-A. This loss of spin-echo signal is a major difficulty in applying NMR to spin glasses, together with a possible perturba t ion caused by external fields. In these aspects, µSR is an ideal method, since we can detect almost 100% of signals within the available time window of µSR spectrometer, even in zero field. Another important result of NMR experiment is the field dependence of the line width exhibited in Fig. 5-6 above T. This exactly corresponds to the field-induced static g effect found in the relaxation rate of transverse-field µSR measure ments , discussed in Chap. IV-B. - 43 -Chapter V-B. ZERO-FIELD µSR IN AN INSULATOR SPIN GLASS Amorphous alumino-sil icate glasses containing Co or Mn moments also exhibit the spin glass behavior [54]. Biemann et al. [55] applied transverse-field µSR to (Mno)40 (Al2o3)20(Si0 2)40 (Tg=2.8 ~3 . 0K), and observed a continuous cahnge of the relaxation rate ovser a wide temperature region 0.9Tg < T < 6Tg with an Arrhenius type dependence on temperature. E /k ST and a g ,0 = 6 x 10- 13 sec was deduced for the correlation time of Mn moment in eq. 3-3. We have tried the zero-field µSR measurement in (Coo)40 (Al203)10(Si0 2) 50 (14.3 at. % Co, T =6.4K) [56]. Fig. 5-8 shows zero-field spin relaxation g functions observed in this system. In contrast to the dilute alloys, the spectrum starts to change from very high temperatures (T 10T ). The relaxation g rate increases continuously down to T 2T, but we could not find a pronounced g difference of the spectrum below this temperature. This is due to the fast reduction of G (t) occurring within the deadtime t $ 30 nsec of the spectrometer, z since this system contains larger amount of magnetic impurities compared to dilute alloys. Though we do not know the muon site in this material, and though the local fields on µ+ may be different from Loren'tzian distribution in such a SG concentrated system, we anal~zed the data with 'z (t) of eq. 2-15 by assuming a random spatial distributions of muons and Co moments and a dominant dipolar fields from Co moments, in order to grasp a general feature of the mean correlation time. We also took into account an effect of nuclear dipolar fields [56] in a similar way as in CuMn. At higher temperatures, the obser'(ed spectra exhibited the "root exponential" shape in the beginning part, and fitted wel 1 to the theoretical curve of "single 'c" model, as illustrated by the sol id lines in Fig. 5-8. In contrast, at lower temperatures T $ 20K, the - 44 -best fit of the spectrum (broken 1 ines in Fig. 5-8) was obtained by taking into account a large distribution of correlation times ' c' w 100 in eq. 4-2, as illustrated in Fig. 5-9. Figure 5-10 shows the deduced correlation time, of Co moments, C corresponding to 1/v of eq. 2-15 for T 30K and 1/v of eq. 4-1 for T 20K. 0 One sees that, changes gradually over a fairly wide range of temperature, C following an Arrhenius law eq. 3-3 with an activation energy E /k = 80K and a h . 1 f -- 10- 12 sec. t e pre-exponent1a actor ,0 In a clear contrast to the case of dilute alloys, we thus found a continuous change of, starting from very C high temperatures (T 10Tg) and a large distribution of 'c in this insulator amorphous spin glass. The deduced values of 'c genera l ly agrees with the TF-µSR measurement in (Mno)40 (Al2o3)10(Si0 2) 50 when scaled with Tg, though E /k 13T for the Co glass is slightly larger than E /k ST for the a g a g Mn glass. At higher temperatures in these glasses, the correlation time is rather close to the exchange frequency of CoO or MnO, suggesting an important role of the short range exchange coupling of neighbour ing magnetic moments in amorphous insulating systems. - 45 -Chapter VI. DISCUSSIONS AND CONCLUSIONS In Chap. I 1-B, I developed a stochastic theory of spin relaxation of a magnetic probe in a random dilute spin system. The essential key idea of my treatment is first to calculate the relaxation function at each site by assuming an appropriate sub-distribution of random fields, and then to take the spatial average by summing up the responses from different sites. We have seen that this simple principle for the order of two operations, i.e., time dependent treatment and spatial averaging, plays a very important role in the calculation of dynamical spin relaxation functions. It will be helpful to remind this feature, and to take the spatial average not in the early stage but in the final stage of any calculation, when we deal with dynamical properties of random systems. Dynamical relaxation functions I calculated upon this principle were actually observed by zero-field µSR in dilute-alloy spin glasses. By using the fuction p(~) of eq. 2-13 connecting between the Lorentzian local field of the entire system and different Gaussian sub-distributions, I also derived a "sum rule" eq. 2-22 which gives the static width of random fields in dilute magnetic systems in terms of the lattice sum and the impurity concentration. In Chap. IV-A and Appendix I, I extended the dynamical treatment to cover the effect of an order parameter in spin glasses. All of the present theories can directly be applied to the calculation of line shapes of NMR and ESR in random systems. We established the zero-field µSR method at TRIUMF, and applied it to spin glasses. From the observed depolarization of muon spins in ~Mn and AuFe, we deduced absolute values of correlation time T of Mn (or Fe) moments shown C in Fig. 3-3 , for the first time in ze~o field. In the temperature region above the cusp temperature Tg, we fo und a rapid slowing down of impurity spins - 46 --10 -8 -9 from Tc~ 10 sec at T 1.2Tg towards Tc~ 10 10 sec at T = Tg, corresponding to the critical index N 2 or to the slope E /k 20T of the a g Arrhenius law. We also found a dynamical scaling Jaw existing for correlation times of different noble-metal based dilute-alloy spin glasses AuFe, CuMn and AgMn. By the µSR measurements in CuMn with finite longitudinal external fields, where we observed only little field dependence of the spectrum above T, we g found a dominant contribution of fluctuating dynamical fields on muon spin depolarization and confirmed the correlation times determined by ZF-µSR as a good measure in the "paramagnetic" region above T. g In the paramagnetic phase of usual ferro- or antiferromagnetic insulators, such as MnF2 , magnetic moments are randomly flipping one another via exchange coup! ing among neighbouring spins with exchange frequency w ex nw I zis(S+l) kT ex c (6-1) where z is the number of nearest neighbour spins, and a fast spin fluctuation with a temperature independent flipping rate of eq. 6-1, observed above T 1.2Tc' is known to slow down when the temperature approaches closer to T C expect a similar behavior also in spin glasses with nw kT ex g We can (6-2) -11 -12 corresponding to Tc~ 10 10 sec in ~Mn or AuFe for the RKKY exchange coup! ing among impurity moments. In magnetic insulators . as MnF2 , spin fl:rpping via spin-lattice interaction takes place with much smaller rate compared tow . ex After the thermal equilibrium of the spin system and the heat bath (lattice) via the spin-lattice interaction, the exchange fluctuations in insulators can then be reg arded as an internal excitation of magnetic spin system itself, which can be fully described when a temperature T is given. There exists, however, another process which flips impurity spins in - 47 -dilute-alloy spin glasses, i.e., sp in flip scatterings of conductions electrons by localized d-moments via s-d coup l ings. Bernier and Alloul (57] actually observed this effect by 63cu NMR in d i lute (50 1000 ppm) ~Mn between T = 0.3 20K, where they determined the correlation time T of Mn moments from C the relaxation rate of Cu nuclei as -11 T ·T 10 C sec•K (6-3) which exhibits a "Korringa" type temperature dependence, and the absolute value of Tc was consistent with an estimate from s-d coupling constant measured in other exper iments. Thus, in the dilute limit where Mn moment behaves as a free spin, a fairly fast spin fluctuation (e.g . T 10- 12 sec at T = 10K) is C caused by conduction electrons, with the rate independent of impurity concentration. In ESR measurements which detect the total magnetization M of coupled z d-spins ands-spins, relaxation rate (or I ine width) is governed by a rather weak interaction betweens-electrons and lattice, since a spin flipping via s-d interaction does not change the total magnetization. This effect is called "Bottleneck effect" (58,59] and is observed in ESR of ~Mn (60]. In contrast, the bottleneck effect does not take place in µSR, since muons probe the dominant dipolar field from d-moments as a point-like probe (see r.p. A for details), similarly to the case of NMR. Thus the depolarization rate of muon spins should directly reflect the spin flipping via s-d scattering, if it ever occurs. In spin glasses, spin flip scatterings of conduction electrons are suppressed by internal (or external) magnetic fields H. ton d-moments, as is 1n reflected also in the temperature dependent term p (T) of resistivity at r lower temperatures (61,62]. Yoshida (63] has shown that, for the limit gµbHint « kT the scattering occurs with the rate expected for free spins, while no spin flipping take place for '.'frozen" spins gµbHint » kT · - 48 -Since both RKKY exchange coupling and s-d scattering give a s imi Jar rate T C sec (6-4) at higher temperatures, competing and co-operative effects of these two mechanisms should be governing the spin dynamics of dilute-alloy spin glasses. Unfortunately, it is not possible to separate these effects in µSR experiments, where the correlation time reflects the additive effects as (6-5) Thus, from the present µSR experiments, we can conclude that both processes slow down rapidly above Tg' and that the upper limit has been determined by the present work for 1/TRKKY and 1/Tsd at each temperature. In regular ferro- or antiferro magnetic systems, where different muon sites have a translational symmetry, ii.SR spectra exhibit coherent precession when static local field shows up in . the ordered state, and we can easily measure the (sub-lattice) magneiization from observed precession frequency [3,7]. In contrast, it is not easy in random systems to distinguish the effect of such a static component from a simple slowing down of spin fluctuations, because random fields in both cases contribute to the depolar'ization of m.u.on spins. So we started the analysis in Chap Ill-A with the simplest model of "randomly fluctuating paramagnetic . spins" even below T, and deduced a slow flu .ctuation g of Mn or Fe moments with T in microsecond region as a rough measure of the C "mean correlation time". Then we observed the decoupling effect in finite longitudinal magnetic fields, and found that the experimental results are better explained by the model based on coexisting static and dynamical local fields. A precise 1 ine shape of zero-field µSR spectra, observed in AuFe around T 0.8T, - . g - 49 -also supports this picture. So I f i tted the spectra of ZF-µSR in AuFe with a model functfon calculated for the vector sum of static and dynamical local fields and deduced the static component a , shown in Fig. 4-3, which corresponds 8 to IQ of the order parameter Q. According to this model, the dynamical component keeps a fairly fast fluctuation below T as shown in Fig. 4-4, in g contrast to the monotonic temperature variation of 'c deduced from the simplest "single," model. C Though there still remain some ambiguities concerning the field-induced effects on spin dynamics, we can regard this onset of static component as a characteristic feature of spin glasses below T. For µSR measurements, g fluctuations slower than v < 106 /sec look "static" and exhibit the 1/3 asymmetry, so we extended the results of Mossbauer effect in Fig. 5-5 to a longer time region. In the analysis of µSR experiments, we also took into account the effect of dynamical components, so the arguments should be more persuasive compared to the case of Mossbauer effect analyzed by assuming the static effect a priori. As shown in Fig. 5-1, the present work has provided dynamical informations below T in the time region between neutron and g susceptibility measurements. Since we used the theoretical value a0, deduced upon several assumptions, to normal ize the static component, detailed absolute values of µSR results in Figs. 5-1 and 5-5 should not be taken too seriously. Good agreements of µSR results with other methods, however, support the picture with long-] ived static component of spin fluctuations below T. Yet, the g real system may be somewhat more comp] icated, since the observed LF-µSR spectra still exhibit s·ome deviation from the model functions, and also since the "slow spin fluctuation" is suggested by NMR experiments below T. g The inhomogeneous impurity concentration, found in slowly-cooled CuMn by an X-ray microanalyzer, is a warning to experimenters on dilute-alloy spin - 50 -glasses. A systematic dependence of ac-susceptibil ity experiments on the heat treatment of the specimen was reported in an elaborate work of Mulder et al. [35], who also found a sharper cusp for rapidly cooled specimens. In Chap. IV-B, I also pointed out the role of f i eld-induced static component of impurity spin fluctuations based on the reported results of transverse-field µSR experiments. The amplitude of static component, estimated from TF-µSR, was consistent with the reduction of ac-susceptibil ity caused by finite external fields, thus providing an explanation to the well-known field sensitivity of Xac· This feature is also observed in ESR [47] and NMR [51] measurements. In contrast, as we have seen in LF-µSR experiments, the dynamical component is not much changed by the external fields above T. This field-induced onset of static component should be g another important information for studies of spin glass behavior. We found a qualitative difference of µSR results between dilute-alloy spin glasses and amorphous insulator spin glasses. Dilute alloys exhibit a drastic change of correlation time T at T T without a large distribution C g of T , while the change of T was very continuous and T exhibited a large C C C spatial distribution in the insulator spin glass, as we have seen in Chap. V-B. Such a different sharpness of dynamics in different spin glasses was also po~nted out by Tholence [42] based on ac-susceptibil ity measurements with different measuring frequencies v • x of CuMn and AuFe depends only little m ac - -on v , while a larger frequency dependence of x was found in other systems m ac with weaker RKKY coup! ings, in insulator spin glasses, and also in the same specimen of (Coo) 40 (A12o3)10(Si0 2) 50 [64], as illustrated in Fig. 6-1. As expected from eqs. 5-1 and 5-2, x will exhibit a large frequency dependence ac in inhomogeneous systems with a large distribution of Tc· In contrast, a sudden onset of a very long-lived component of spin fluctuations can be expected - 51 -below T when x does not depend on v • Thus the sharp and homogeneous g ac m dynamical behavior was found both by µSR and x in CuMn and AuFe. This would ac be suggesting an important role of long ranged RKKY interaction for the co-operative spin freezing in dilute-alloy spin glasses, i n contrast to more "local" and inhomogeneous dynamics of insulator spin glasses governed by the short range exchange couplings. To explain the sudden reduction of susceptib i lity in spin glasses below Tg' we have two different pictures . In "polariza t ion picture", one assumes the onset of static component for each and every spin below T , as in g regular antiferromagnets. Another picture may be called "population picture", since a certain portion of impurity moments (e.g . , 10 spins out of 100) becomes "bloc ked" o r "in mobile" , thus stops responding to ex ternal forces in this picture. The "polarization picture" is representing the view point of mean-field treatments and co-operative phase transition at T, while the "population g picture" stresses the effect of randomness in the system and i s close to percolation or blocking models of spin glasses. It is not easy to distinguish between these two pictures in experiments, since both effects cause a reduction of responses to external forces. There is, however, a hope to study this problem. In the polarization picture, ac-susceptibil ity x and auto-correlation function ~(q,t)/ ~(q,0) ac are quadratic quantities of impurity spin S, while the local field a (T) of s ZF-µSR, relaxation rate A(T) of TF-µSR, and the hyperfine fields in Mossbauer experiments are linear quantities of the static spin polarization. So we 2 have to calculate the square term (a8/ a0) when we compare i t with xac in the polarization .picture, as I have done in Chaps. IV-Band V-A. For the population picture, the situation is a little bit different. Local field measured by - 52 -muons is linear to the population of static spins, since the static amp! itude a is proportional to the impurity concentration. Mossbauer spectra would exhibit a full amplitude of hyperfine field for static spins, while a single peak for fluctuating moments, so a superposit ion of different spectra would show up. The auto-correlation function ~ (q,t)/ ~(q,O) and the reduction of Xac are proportional to the number of static spins, so now becomes linear terms in the population picture. Even in the polarization picture, the static component of each spin, given by the hyparbolic-tangent term in mean field treatments, is different from one spin to another, reflecting different internal fields on impurity moments . So the reality would be somewhat inbetween these two pictures. Roughly speaking, the present results of µSR look favoring the "polar -ization picture" since we obtained a good agreement of µSR results calculated upon this picture with neutrons and x i n Fig. 5-1. We need, however, further detailed inve~tigations, ac since the absolute values of d /a are not very precise, and also since the 8 0 field-induced static component_ J\(T) measured in TF-µSR may have a possible dependence on measuring times. In this context, it should be very import .ant to extend ZF-µSR experiments to lower temperatures, and to determine a 0 experimentally. Also required are more detailed studies of TF-µSR and the comparison to experiments with other methods, such as x , neutrons, Mossbauer, ac etc., performed in finite external fields. Different experimental methods have enlightened different aspects of spin glass behavior. Probes with very narrow time windows, . such as Xac with fixed frequency or Mossbauer effect, distinguish static and dynamical effects as in a "all or nothing" way, thus found a drastic change at T ,= T. Neutron g scattering has a fairly wide time window, so it followed the continuous change of the system from very high temperatures. Specific heat 165] and - 53 -resistivity [66], which reflect det a iled change of microscopic clustering of the system, exhibited very conti nuous behavior at T = T without pronounced g anomaly. As we have seen in this thesis, muon spin depolarization is sensitive both to static local fields and to dynamical spin fluctuations, and covers an important region between x and neutrons with its time window particularly ac suited to T T. In the present work, however, we could not clarify detailed g aspects of possible . spatial correlation or clustering effects [foot note 2] of spin fluctuations above T , suggested in the resistivity and calorimetric . g measurements. Future theoretical studies of critical slowing down, taking into account the different strength of exchange couplings for different bonds and also the consequent topological effects on spin fluctuations, may be helpful to consider this aspect, and may provide an explanation for the rapid change of correlation times observed in random systems. It should also be important to include the effect of spin-flip s-d scatterings in dilute alloy spin glasses [67], as a term of fluctuating forces in kinetic treatments of spin dynamics above T, and as an extended basis of coupled impurity spins and g conduction electrons in the calculation of elementary excitations [68] at lower temp·eratures. In spite of many experimental and theoretical efforts, the nature of spin glass behavior is still not fully understood, and is open to future investigations. It is my pleasure if the present thesis work could contribute to a small progress in this field with some new informations, and could open a way to future experimental studies. To finish this thesis, I would 1 ike to emphasize the importance of systematic studies of spin glasses with different experimental methods in· zero and finite fields , by using the same specimen. - 54 -REFERENCES [1] Proceedings of the first and second internationa l conference on muon spin toataion are published in Hyperfine Interac t ions.§_ (1979) and !!_ (1981) respectively. [2] R.S. Hayano, Y.J. Uemura, J. lmazato, N. Nishida , K. Nagamine, T. Yamazaki and H. Yasuoka, Phys. Rev. Lett. 41 (1978) 421. [3) T. Yamazaki, Hyperfine Interactions 6 (1979) 115. [4] Y.J. Uemura, N. Nishida, J. lmazato, R.S. Hayano , M. Takigawa and T. 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Dodds, T.L. Estle, R.H. Heffner, M. Leon, D. E. MacLaughl in, C.E. Olsen and M.E. Schillaci, Hyperfine Interactions 8 (1981) 763. [29] M. Leon, Hyperfine Interactions 8 (1981) 781. [30] Y.J. Uemura, K. Nishiyama, T. Yamazaki and R. Nakai, Solid State Comm. 29 ( 198 1 ) 461. [31] See for example, C.P. Slichter , Prin ci ples of Magnet ic Respnance , (Harper & Row, New York, 1963). [32] S.F. Edwards and P.W. Anderson , J. Phys. F5 (1975) 965, F6 (1976) 1927. [33] For a review of theoris in spin glasses , see A. Blandin, J. de Phys. (Paris) 39 (1978) C6-1499. - 56 -[34] V. Cannella and J.A. Mydosh, Phys. Rev. B6 (1972) 4220. [35] C;A.M. Mulder, A.J. van Duyneveldt and J.A. Mydosh, Phys. Rev. B23 ( 1981) 1384. [36] J.A. Brown, R.H. Heffner, T.A. Kitchens, M. Leon, C.E. Olsen, M •. E. Schillaci, S.A. Dodds and D.E. Maclaughlin, J. Appl. Phys. 52 ( 1981) 1 766. [37] C. Emmerich and Ch. Schwink, Hyperfine Interactions 8 (1981) 767. [38] R.H. Heffner, M. Leon, M.E. Schillaci, D.E. Maclaughlin and S.A. Dodds, Proceedings of MMM conference 1981, Atranta, to be pub! ished in J. Appl. Phys. (1982). [39] . For a review of experiments in spin glasses, see J.A. Mydosh, J. Mag. Mag. Matrs. 7 (1978) 237, and also J.A. Mydosh, Proc. of Int. Con·f. on Disordered systems and Localzation, Rome, 1981, to be publised by Springer-Verlag. [40] See also A.P. Murani, J. de Phys. (Paris) 39 :C1978) C6-1517. [41] F. Mezei and A.P. Murani, J. Mag. Mag. Matrs. 14 (1979) 211. [42] J.L. Tholence, Solid State Comm. 35 (1980) 113, and references therein. [43] A.P. Murani, J. Mag. Mag. Matrs. 22 (1981) 271, A.P. Murani, F. Mezei and J.L. Tholence, Physica 108B (1981) 1283. [44] J.L. Tholence ·, Physica 108B (1981) 1287 and to be published. [45] A.P. Murani, Sol id State Comm. 33 (1980) 433. [46] M.B. Salamon and R.M. Herman, Phys. Rev. Lett. 41 (1978) 1506, M.B. Salamon, Sol id State Comm. 31 (1979) 781. [47] E.D. Dahlberg, M. Hardiman, R. Orbach and J. Souletie, Phys. Rev. Lett. 42 (1979) 401. [48] C.E. Violet and R.J. Borg, Phys. Rev. 149 (1965) 540. - 57 -[49] Such an attempt has been made by F.J. Litterst, J . M. Friedt, J.L. Tholence and F. Holzberg (submitted to J. Phys. C) in an insulator spin glass. [50] D. E. Maclaughlin and H. Alloul, Phys. Rev. Lett. 36 (1976) 1158. [51] D.E. Maclaughlin and H. Alloul, Phys. Rev. Lett. 38 (1977) J81. [52] D. A. Levitt and R.E. li,lalstedt, Phys. Rev. Lett. 38 (1977) 178. [53] H. Al loul, J. Appl. Phys. 50 (1979) 7330. [54] R.A. Verherlst, W. Klein, A.M. de Graaf and H.0. Hooper, Phys. Rev. 811 (1975) 4427. [55] L.H. Bieman, A.M. de Graaf, W.J. Kossler, A.T. Fiory, C.E. Stronach, W.L. Lankford and K.G. Lynn, Hyperfine Interactions 4 (1978) 861. [56] Y.J. Uemura, C.Y. Huang, C.W. Clawson, J.H. Brewer, R.F. Kiefl, D.P. Spencer and A.M. deGraa f , Hyperfine Interactions 8 (1981) 757, [57] P. Bernier and H. Alloul, J . Phys . F3 (1973) 869, H. Alloul and P. Bernier, J. Phys. F4 (1974) 870. [58] H. Hasegawa, Prog. Theo. Phys. (Kyoto) 21 (1959) 483. [59] R. 0rbach, J. Mag. Mag. Matrs. 15- 18 (1980) 706. [60] A.C. Gossard, J. Appl. Phys . 38 (1967) 1251. [61] M.T. Beal-Monod, Sol id State Comm. 9 (1971) 1443. [62] I.A. Campbell, Phys . Rev. Lett. 47 (1981) 1473 . [63] K. Yoshida, Phys. Rev. 107 (1957) 396. [64] L.E. Wenger , C.A. M. Mulder, A. J . van Deyneveldt and J.A. Mydosh, Phys . Lett. 77A, (1980) 378. [65] L.E. Wenger and P. H. Keesom, Phys. Rev. 811 (1975) 3497 , 813 (1976) 4053 . [66] J.A. Mydosh, P.J . Ford, M.P. Kawatra and T.E. Whal!, Phys. Rev. 810 (1974) 2845 . [67] An attempt assuming dominant s-d scatterings i s made in ref. [62]. [68] In ref. 22, excitations a r e calculated ford-moments only. - 58 -Foot Notes [1] This assumption corresponds to the Gaussian dist r ibution of the activation energy E of different clusters . a [2] See for example M. Suzuki and S. Miyashlta, Physica 106A (1981) 344, S. Miyashita and M. Suzuki, J . Phys. Soc. Japan 50 (1981) 1840. - 59 -TF-pSK I Hr (/) Beam r "Cµ. = z ' ' 2.2µ.sec w > /L w !HT TIME ZF-pSR LF- EB EF (/) r EF Beam z ' ' ' w " " > EB IL w TIME ( HL ) Fig. 1-1 Schematic view of experimental methods of (a) transverse-field µSR and (b) zero- and longitudinal - field µSR. An incoming polarized muon is identified by M counter, and stopped in a specimen (dotted piece). Decay positrons are recorded by counters E, EF, and EB, and the measured time spectra reflect muon spin polarization .in the way as shown in the figure. - 60 -1.0 >-a::: r w >-(/) <! 0 µ+ in MnSi (T = 285 K) 2 30 Oe t . Q~~-£;-l-t~ 10 0e 4 6 TIME (µ.sec) 8 10 Fig . . 2-1. Zero- and longitudi nal - field muon spin relaxation function observed in MnSi at room temperature. Ef f ects of hyperfine f ields fr om itinerant electrons are negligibly small at this temperature , and muon spins are depolarized by nuclear dipolar fields . The ze r o- field relaxation shows a characte r istic recovery of the asymmetry to 1/3 , and experimental data well followed the solid curves representing eqs . 2- 3 and 2- 4. The width t /y =3. 6 Oe of random z µ fields , deduced from the observed data , agrees with the order of magnitude of nuclear dipolar field from 55Mn nuclei . For the case of HL=lO and 30 Oe, longitudinal field HL acts as a holding field to preserve the pola r ization of muon spins. - 61 -z 0 u z ::> LL. z 0 1.0 ~0.5 X <{ ....J w a:: 0 f • Gx (t) o G2 (t) 2 Gx (t) and G2 (t) of µ.+ in ZrH2 3 TIME 5 6 Fig. 2-2. Comparison of transverse -f ield spin relaxation function G (t) and X zero - field spin relaxation funciton G (t) measured byµ+ in ZrH2. G (t) was Z X observed by applying a transverse external magnetic field of 5 kOe. ' In, this system, dominant nucl ear dipolar fields are made by the spin 1/2 proton moments free from the effect of quadrupolar interaction. Solid lines represent the best fit curves with 6 =0. 33 /µsec for G (t) corresponding to the Van Vleck value, X X while with 6 =0. 51 /µsec for G (tJ of eq . 2- 3. As explained in the text, the z z zero - field function exhibits five times faster damping compared to Gx(t) , reflecting the recovery of non- secular terms of dipolar interaction in zero field. - 62 -Fig. 2-3 . I 1.0 ,_ 0.2 1.0 G2 (t) 0.6 0.2 0. 0 STRONG v/6 = 5 v/t. =0.1 ',, ', ',, 10 COLLISION MODEL NARROWING 2 3 SLOW MODULATION G2{t) - 1 ·exp(-~ v t J -----30 Sahematia view of the dynamiaal spin relaxation funations aalaulated from the statia relaxation funation in a strong aollision model eq. 2~?. Dynamiaal funation is given as an envelope funation of statia funations. An example of the Gaussian loaal field PG(H.) of eq. 2- 1 with the statia funation eq. 2-3 is 'l, presented. In the aase of fast modulation v/11 > 1 s_hown in (a), Gz(t) beaomes an exponential shape aorresponding to the narrowing effeat. Slow modulation in (b) aauses a damping of the aharaateristia 1/3 asymmetry at t• /1 »l . . - 63 -1. 0 a.a 0-6 --N (!)0.4 0.2 o.o 0 1 2 t·6 3 4 5 Fig . 2-4. Zero- field muon spin relaxation functions GG(t) in Gaussian random z local fields calculated numerically for several modulation rates v with the use of st rong collision model eq. 2- ?. The characteristic recovery of asyrronetry of the static case to 1/3 is shown to be very sensitive to a slow modulation , while the initial Gaussian shape is modified by v/~ > 1 via the narrowing effect for a fast modulation of random local fields . - 64 -P(H;) P(IHI) 0 LQt:eatz:iaa PL(H.) y =1-I 1.0 L gz ( t) 0.5 I 3 1T i=x,y,z H, 0 IHI I a (WALSTEDT &WALKER, 1974) 2 2 2 (a +y H. ) )J I STATIC CASE L l 2 g2(t) = 3 + 3(1-at)exp(-at) ( KUBO 1979) 0----~----~----~----~--2 f • a 3 4 Fig . 2-5. Zero-field muon sp-in relaxation function in static random fields with Lorentzian distribution (H.W.H.M. a/y ). The characteristic 1/3 asymmetry µ is associated with the 2/3 component which exhibits an exponential initial damping and faster decay of the oscillation compared to the Gaussian case. The Lorentzian local field was suggested by Wa 1 stedt and Walker [20] for a magnetic probe in dilute-alloy spin glasses, and the static spin relaxation function was given by Kubo [23]. - 65 -1. e, e,. 8 - c,. 6 ._. -N !) c,. 4 11/0 =o c,. 2 0.2 0.5 Cl. Cl 0 1 2 3 ta 4 5 6 7 8 Fig. 2-6 . Dynamical spin rela:xation function rf(t) calculated directly from the z static function g~(t} of Fig. 2- 5 in Lorentzian local fields with the use of strong collision model eq. 2-?. Even for the fast modulation of local fields, the initial decay rate exp(-4at/3) does not exhibit a narrowing effect , since the envelope function of exponentially decaying gL(t) damps as gL(t) itself . J, . z z This function Cr{t) does not describe the case ofµ+ in spin glasses, but will z correspond to the case with a fast muon diffusion in random dilute spin systems. - 66 -Au 0 0 P(H) fl+* • 0 Fe • 0 0 0 0 0 0 0 P(H) H 0 0 * 0 QH Fig. 2-7. Schematic view of different amplitude and variable ranges of local fields onµ+ in different sites of dilute - alloy spin glasses . When the Fe moment changes its orientation, local fields on / will be modulated within each sub- distribution of random fields associated to different sites . - 67 -MUON RELAXATION FUNCTION G~G (t) IN SPIN GLASS BEGINNING PART 5000 0.5 0 ~-------;!;---~--' ta 2 3 4 10 30 ta 50 70 30 ta 5o 70 Fig. 2-8. Zero-field spin relaxation random loaal field with H.W.H.M. a/y is µ ofµ+ in spin glas ses. Lorentzian assumed, and different sub-distributions of random fields at different muon sites are taken into aaaount. The broken line shows the static ease, whiah corresponds to gL(t) of eq. 2-12, and the z dynamical relaxation functions are attached with the normalized modulation rate v/a. Typically, the width a is around 107 /sea in 1 at .% CuMn, so the unit 1 in the time saase corresponds to ~100 nsea. The slow modulation of loaal fields is reflected as the slow deaay of the 1/3 asymmetry, and cfG(t) is z approximated by eq . 2- 16 .. for v/ a $ 0. 2. The fast fluctuations of loaal fields r>eduae the initi,al deaay mte of G.:c(t) via the narrowing effect for v/a 2, and c!G(t) exhibits a "root exponential " shape of eq. 2-17 for the limit of fast modulation of loaal fields. - 68 -1.0 ,..-.... +-' ...__.. (!) (J)X (.9 0.5 0 BEGINNING PART via= 500 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ' ' ', ', ,._ 0 .......... "'-0.5 ---------I ta 2 3 4 10 30 ta 50 70 Fig. 2-9. Transverse-field spin relaxation .function of -i/ in a spin glass. calculated in a similar way as the zero -fi eld function in Fig . 2-8. The static case , indicated by the broken line , exhibits an exponential shape exp(-at). Slow modulations of local fields v/a $ 0. 5 can not be distinguished from the static case in the transverse -f ield measurement. c!/cG(t) exactly corresponds to the Fourier transforms of the resonance line shapes in NMR or ESR. - 69 -OBSERVABLE RANGE OF SPIN CORRELATION TIME T OF SPIN GLASS C 10-10 T (sec) C Neutron Moessbauer TF-µSR ZF-)JSR ac Susceptibility Remanence Fig. 2-10. Comparison of the observable time range of the eorrelation time T of spin glasses in different experimental methods. Zero-field µSR e eovers an important region between neutron seattering and ae-suseeptibility measurements. - 70 -.-1 395 Oe 8.5 X :,-.o . ic:ri"½-· ~ · . .. /Vy <.-1 0 I r----.-----r----,,----~ ,-I :,-.o k .., cu • {o ~'; 0 395 Oe 7.3 K I +----,-----.-----,---,..J 0 100 200 300 400 Time Channel (2.2 ns/chan.) ...... ----:------------, ,-I 0 0 ,-I 0 co N 00 4 8 CuMn 0.7% o 85 Oe re • 330 Oe fc 0 1890 Oe fc • 1890 Oe zrc 12 TEMPERATURE 16 (K) 20 Fig. 3-1. (a) Muon spin precession signal in a spin glass CuMn (0.7%) observed in transverse-external magnetic field 330 Oe. (b) Width W of random fields estimated from depolarization rate 1/T2 of muon spins as yµW = 1/T2. Both (a) and (b) from Murnick, Fiory and Kossler [17]. The observed relaxation rate exhibits a rapid change around T, but static and dynamical effects of g random local fields are not distinguished in this TF-µSR measurement. The width Wat lower temperatures corresponds well to a theoretical estimate assuming a random selection of interstitial sites by muons. Some dependence of 1/T2 on the transverse-field HT is noticed. - 71 -1.0 Gz(t) 0.5 o. LO 0.5 o. µ+ in Au Fe (I at.%) r9 = 9.1 K Hext=0 9K 4.5K 6 K 0.2 0.4 0£ 08 1.0 T IME (µ sec) + µ in Cu Mn ( I.I at.%) Tg=l0.8 K 2 Hext= O • 30 K 0 12 K • 11 K "' 10 K ..,. 6 K 3 4 5 6 7 T IME (µsec) Fig. 3-2. Zero- field muon spi n r elaxation functions observed i n spin glasses (a) AuFe (1 . 0 at . %, T =9. lK) , and (b) Cu.Mn (1. 1 at . %, T =10.BK) . Observed spectra - g - g fitted well to the model function cfG(t ) of eq . 2- 15 (Fig. 2- 8) . Solid lines in . 2 AuFe represent G$G(t) for a = 5.2 /µsec (a/y = 61 Oe) and the modulation rate - 2 µ v = 1/ T shown in Fig . 3- 3-a . In Cu.Mn, which exhibits an effect of nuclear dipolar C -field from Cu nuclei , observed spectra followed GSG(t)x{gG(t) in pure Cu} indicated 2 2 by solid lines at higher terrperatures T 0.9T , while the broken line GSG(t) g 2 was found at lower terrperatures T. ,n, 0. 5T . The static width a/y = 120 Oe was g µ estimated at T=5K for random fields from Mn moments, and the correlation times in Fig . 3-3 -b were deduced from the observed spectra of Cu.Mn. - 72 -I Au Fe ( I at.%) Hext = O 0 • • ,. 10 Irr "' *"' ,. * ' 9 ::, (.) i Q) I /\ ,. (I) Tg Au Fe ./ '\ 'o 1--..° •' \ , _ >- l I at.% ,. II ···, .. , 7 !::: ,. ...J · 1.4 II * LLl ··. iii ,. Ir i== a. CuMn LIJ 10-8 II (.) z 5 If) ::, I.I at.% • 0 If) (quenched) _J I1 0.8 II 0 w et: ( slow cooling) et: l I I 0 T u le a: (T-Tg) Tg T 2 10 * re er (T-Tg) • 4 6 8 10 12 14 ().6 1.0 1.4 1.8 TEMPERATURE (K) Tg/T Fig. 3-3. (a) Correlation time T of Fe moments in AuFe (1.0 at . %) estimated C -from zero-field µSR. The error bar indludes the systematic error and the scattering of data among a few measurements. The ac-sus ceptibility was measured by Chikazawa and Miyako. (b) Correlation time T plotted versus C inverse reduced temperature T /T normalized by the cusp temperature of g different specimens of spin glasses. Statistical error is roughly the size of each point. The sharpness of the change of T roughly corresponds to the C ·index 2 of critical slowing down above T, while the activation energy E =20kT g a g is obtained from the slope of T around T. Dynamical scaling law is found C g for correlation times of different specimens of RKKY spin glasses . - ?3 -- CuMn 2:-0 '-/ 1.1 at% quenched -1l '-0 >-I-(/) z w I-z +-----+ >. 20µm 0.8 at% 0 slow cooling '-I X C :E X-ray MICROANA LYSIS SCANNING ·-··-- - ~ Fig . 3-4. X- ray micro analysis of .Cu.Mn alloys made with dif f eren t heat treatments. The intensity of characteristic X-ray is proportional to the concentration of Mn impurity. I n slowly cooled specimen, a large inhomogeneity is found in the scale of around 100 µm, while the quenched-made specimen exhibited a homogeneous consentration within a statistical error. - ?4 -1.0 µ+ in CJd Mn ( 1.1 at % ) quenched G2(t) 0.6 T= 11 K 0.2 o. o. 0.2 0.6 1.0 TIME (µsec) 1.0 µ+ in Cu Mn (0.8 at%) slow cooling G2(t ) 0.6 T = 7.9 K 0.2 o. o. 0.2 0.6 1.0 TIME (µse c ) Fig. 3-5. Corrrparison of the fit of zero - field µSR spectra with a single value of v (solid line) and the fit allowing a distribution of v (broken line) in GSG(t). A Gaussian distribution for (log v) is assumed: the estimated width z 6(log 10v) at T~Tg was around 2 for slowly - cooled 0. 8% Cu.Mn, while less than 0. 4 for quenched Cu.Mn. In this way, the observed ZF- µSR spectra in the ·homogeneous rapidly-cooled specimens above T > T were fitted well with the sirrrplest g model function eq. 2- 15 without a large distribution of v. - 75 -Hext = O • • • T • - 6 *T T * - 10 (.) • Q) T (/) Tg Au Fe rrR1uMFJ -1--.,0 ! I at.% T T w 1.4 II * T * I-T ... *. • Cu Mn rrR1uM~ 168 z I. I II • 0 Ag_Mn rLAMPFJ _J w 1.6 II • n:: n:: 0 T 0 r0 oc (T-Tg) ··-·-·-· 10-IO T 2 re a: (T-Tg) • 0.6 1.0 1.4 1.8 Tg/T Fi g. 3-6. Corrrpar>ison of the correlation time 'e of Mn (or Fe) moments observed in the present experiment at TRIUMF and in another ZF-µSR experiment performed at LAMPF in ~Mn (1 . 6 at . %, Tg=?. 6K) by Brown et al . [28]. All the data points show a unique curve , indicating that the absolute values of correlation time of these RKKY spin glasses can be sealed with Tg. - ?6 -•' . 1.0 wL/0 = 20 10 5 --------N (.9 2 0.5 0 I ta 2 3 4 Fig . 3-7 . Longitudinal - field muon spin rela:ration function eq. 3- 5 in static random local fields with Lorentzian distribution (H. W.H.M. a/y ) derived µ by Kubo [23]. Static random fields are decoupled by the longit udianl field HL° The normalized amplitudes wL/a of external fields are attached . - ?? -DECOUPLE STATIC RANDOM FIELDS NO CHANGE FAST DYNAMIC FIELDS \ \ \ \ ' ', LF , , ZF , , ___________ . time rcx: 11 112+w2 L time Fi g . 3-8 . Sahematic Viel.J of the muon_ spin r e laxation funations for statia and dynamiaal loaal fields expeated in longitudinal - field µSR measurements . Statia random fields will be deaoupled and G (t) is held to a aonstant value z by the appliaation of a longitudinal field , while no field dependenae is expeated for fluatuating dynamiadl ·loaal fields with the rate v » wL. - ?8 -w -I-• • • 1. 1 at % T =I0.8K g • • • 6 8 10 12 14 TEMPERATURE CK) ... C. Q) u "' ::, "' Fig . 3-9. Correlation time, of Mn moments determined by zero - field µSR C in CuMn (1 . 1 at.%) . Ac- susceptibility data (measured by Chikazawa and Miyako) is also attached . - 79 -C.F., An expanded graph of this figure is attached in the nest page. Q.6 Q,2 I.O G20) Q.6 Q,2 G,(tl Q.6 Q,2 ~Mn (I.I 01%) Tg • 10.8 K HL (Oe) • • 0 ° 40 • 80 • 160 • 320 o 640 T• 20 K T • 12 K T• I0K T• 8 K T• 5K o. ~--------0. Cl.2 · 0.4 0.6 Cl.8 LO 0. Cl.2 0.4 0.6 Cl.8 LO TIME (µsec) TIME (µsec) Fig. 3-10. Longitudinal-field µSR spectra observed in CuMn (1.1 at.%, T =10.BK). - g Solid lines represent theoretical model functions cf/<t,a,v 2F,HL) for "single 'c" model with vZF of Fig. 3-9 determined in zero-field measurements. At T=12 and 11K, no field dependence is expected for cfG, and two lines at T=l0K denote cfG for z z HL=0 and 640 Oe. Six lines at T=B and 5K represent ~G for each different HL. The observed data does not depend much of HL at T Tg, while a remarkable decoupling effect is exhibited below Tg. - 80 -CuMn (I.I at%) Tg= 100 8K HL (0e) = • 0 ° 40 • 80 ,. 160 • 320 a 640 0.6 T = 20 K T= 12 K 0.2 Tc IOK T• 8 K T• 5K 0. Q2 · 0.4 0.6 O.S LO 0. Q2 0.4 0.6 O.S LO TIMF (µsec) TIME (µsec) - 81 -1.0 G2 (tl 0.6 0.2 o. • • 0 HL (0e) • • 0 T = 12 K 0 0 • • • • • • • • 2 3 4 T IME (µsec) 40 • 80 0 • • 5 • 160 • 320 . 0 • 0 0 • .. 0 • •a• o a 640 T = II K 0 t:.• aJ 0 •o• '°,/" ., • r/J •o 0 0 • • •• 2 3 4 TIME (µsec) 5 Fig. 3-11. Longitudinal - field µSR spectra obs~rved in Cu.Mn above Tg, plot t ed with a longer time range t $ 5 µsea . Almost no field dependence is observed at T=12K, except for the effect of nuclear dipolar fields in · .zero - field , while Gz(t) slightly depends on HL at r=llK , just above Tg=10. 8K. - 82 -Fig. 3-12. o.6 ~2 o. 0 m=+l/2 \ 7 UJL = 5 X 10 -1/2 .. J I sec ,,/ CuMn (I.lat%) Tg=IO.BK HL = 640 0e 2 3 4 5 TIME (µsec) T= = 20 • I 2 • I I = 10 0 8 , 5 K Muon spin relaxation function in ·cw'efn (1.1 at.%) observed in HL = 640 Oe, plotted with a longer time range t $ 5 µsea. In a considerably high ongitudinal fields, the effect of static random fields is appreciably eliminated, and G (t) selectively reflects the dynamical component of random z fields fluctuating with the rate v "' wL' as in the "T1" measurement of magnetic resonance. - 83 -C.F., An expanded graph of thes figure is attached in the next page. at T • 10 K HL (Oe) • • 0 • 40 • 80 • 160 • 320 o 640 single Tc claste rs Stilt.+ /Jct. if I le ~v ~~ran)flct V _ ;x/-I.O I 110 1cf 108 to"" [/sec) Hr~ ... I 10 10 ( /sec > Hex! ;stat Gz (I) O .o.o 0 0 0.15 ... .. 640 Q.2 320 0 0 0. o. Q.2 0.4 o.s a.a I.O o. Q.2 0.4 Q,6 a.a I.O o. Q.2 0,4 Q.6 a.a I.O TIME (µsec) TIME (µsec) TIME (µsec) Fig. 4-1. Comparison of some attmpts to reproduce the field dependence of LF-µSR observed in CuMn (1.1 at.%, T =10.8K) at T=lOK (~0.9T) by different - g g model functions representing different physical situations. The "single T 11 C model with an exponential auto - correlation function of impurity spins can not explain the large decoupling effect . The cluster model eq . 4- 1, which takes into account a loarge macroscopic spatial distribution of Tc' improves the fit considerably . The best agreement to the data is obtained by the model function eq. 4- 5 assuming coexisting static and dynamical local field as a vector sum, with 40% of static component and the rest of dynamical component fluctuating with v =? x 108 /sec. - 84 -1.0 Gz (f) 0.6 0.2 o. o. CuMn (I.lat%) (Tg=IO.BK) at T = 10 K HL (0e) = • 0 single Tc J V l06 108 '10 10 (/sec) D D D 0 ••• a a D O O O 0.2 0.4 TIME . . . . . 6 6 • • 0.6 0.8 (µsec) 0 0 • • 0 1.0 o. 0 40 • 80 6 160 cluster s ~i I I • 320 V I I I I I I 106 108 1010 (/sec) 0.2 0.4 TIME • • 0.6 0.8 (µsec) • 1.0 640 320 0 0 640 o. stat.+ /let. "~ra'!}f let ____ ,;..,½;~~-Hext ;stat 0 0 ~-------- 0 640 -- ~320 160 -so 0. • 0 o• 0~ 40 _-., o. • • 0 • 0.2 0.4 TIME 0.6 o.s (µsec) 1.0 v:, co I 1.0 0.6 0.2 0. 0 Fig . 4-2. Au Fe I at.% Tg = 9.1 K T = 7.5 K ---·--· single Tc stat.+ /let. 2 TIME (µ. sec) 3 Corrrparison of the fits of zero-field µSR speeutum in AuFe (1. 0 at.%, T =9. lK} at T=?. SK ('v 0. BT } by "single , " model (broken line} g g C and by "static+ dynamical " model (solid line} . Single, model assumes C eq . 2-15 with a = s.2 · /µsec and v = 32 /µsec, while the coexistence of static corrrponent as = 2 . 8 /µsec and the dynamical corrrponent with v =? x 108 /sec in eq . 4- 6 gives the soliri line. A significantly better fi .t is obtained . + with the coex isting static and dynamical local field s onµ - 86 -6 .._a 0 - 5 u Q) + + t + (/) 4 + + -(/) 0 3 + + + 2 Au Fe ( I at.%) Tg 00 • 2 3 4 5 6 7 8 9 10 TEMPERATURE ( K) Fig . 4-3 . Amplitude a of t he static component of random f ields estimated s fo r zero -field µSR spectra in AuFe (1 . 0 at . %, T =9. l K) by assuming coexisting - g static and dynamical local fields on µ+ eq . 4- 6. The arrow of a0 indicates a theoretical estimate for the full amplitude of random fields onµ +. As in the magnetization curves of usual ferro - or antifer r omagnetic materials , as exhibits a tendency to increase towards a0• - 8 7 -I AuFe { I at.%} -6 Irr Cl 10 ' - 9 ::, (.) E Q,) /\ a, (/) ... ·, U) .. ' - , .. .. \ 'o .. '· '•, .....,o ·, . '· >-II ····,. 7 r-Ll.J ·, •.. ...J ffi Ir j:: a. w 168 II 0 z 0 5 (/) 0 00 ::::> 0 (/) 0 _J 0 II w 00 et: 0 et: l I I 0 0 10-IO Tg 4 6 8 10 12 14 TEMPERATURE {K) Fig. 4-4 . Correlation time t of Fe moments in AuFe (1 . 0 at . %, T =9.lK) C - g estimated for the fluctuating dynamical dOmponent of impurity moments (open circles) by assuming the coexisting static and dynamical local fields and by fitting the zero - field µSR spectra with eq . 4- 6. When we assume such "static+ dynamical" model, the fluctuating component keeps rather fast fluctuation below T, in ' g contrast to the monotonic increase ·of , found in the simplest model of single . C correlaiton time. - 88 -0 - ,---- ,--- , . ·-- ,-- ·. .IIb t O.t. oi :; 0 ,), ,,, -i ·., H=lOO Oe O,~, ~ ---,/ ',~ ,' • \ , H=SOO Oe "---.--,./ b C L . 0 •• 0 • 00 - 0.3f - . 0 0 0 0 -~,- \ • H=SOO Oe ., . ';---,--0! ., f' ' ;:'. -.. 0 Ila 'o (bl I I l TIT f - 02 2 --:, E QJ >< Fig. 4-5. Ac-susc eptibility of spin glasses Cu.Mn (IIA = 0.57 at.%, T =6.0K; - g IIB = 0.70 at.%, T =7.65K) measured by Mulder et al. [35]. The sharp cusp g at T=Tg (=Tf in the figure) exhibits some reduction in finite external fields. - 89 --< ' < w I-C[ It: z 0 1- 0.1 C[ N It: C[ ..J 0 a. w 0 + ::I.. 0 'j 0 .0 1 C[ u (/) 0 .1 £.!!.Mn e • 0 .87 ot . % (0) 200 0e cJ (X ) 1000 0 e A_q Mn 1.6 ol.% (Cl) I 50 0e (+ ) 100 0 0e (. ) 5000 Oe o0x~ • + ) ,XX·"" ' I~+" \ x~ 101 o~+ '0,x( I 10 SCALED TEMPERATURE T/ T9 Fig . 4-6. Muon spin depolarization rate A measured in transverse - field µSR in ~Mn (1.6%, Tg=?. 6K) by Brown et al. [36], and in CuMn (0 . 8? at . %, Tg=9.5K) by Emmerich et al. [37]. Transverse spin relaxation function Gx(t) = exp( - At) is assumed, and the depolarization rates are normalized by the value AF expected for the random frozen configuration of impurity spins . Solid lines . 1 •• indicate a theoretical value of k ;e~timated for a static field from Curie - Zike polarization of impurity moments in external fields. A remarkable field - dependence of the depolarization rate is observed . Taken from Brown et al. [36]. - 90 -ac-suscept ibility i n..~Mn .(2 at. %) 10-1 10-2 A I ' 0 I ,' I ?<- t:,. ...._ fa 'o ' ?<- I ' I "'' l! ~ 10-3 I I 0 r,. i ti ,. I .... • 0 0 T = 15. SK = T g 17K = 1. 1 OT · g 14K = 0.90T g 12K = 0. 77T g 31 K = 2T g TF- µSR i n ~ Mn I ,' fq I I I ' 6 ' ( 1. 6 at .. %, T g =7. 6 K) + T = 12K = 1;57r I I ' I ' I I 10- 4 102 103 ·104 10 MAGNETIC . FIELD (Oel g 0 17K = 2.24T g ,os Fi g . 4-7. Reduction of ac- susceptibility measur ed in finite fields x(H) from the zero-field susceptibility x measured in ·cuMn (2.0 at . %, T =15. SK) by 0 - g Mulder et al. [35]. I also plotted the r eduction of x expected from the ac field - induced static component of impurity spins estimated in transverse - field µSR of ~Mn , by normalizing the external fields with the values of Tg. When we scale the temperature with T , the results of µSR roughly agrees with the g . susceptibility measurement. TF-µSR experiment is performed by Brown et al. [36] . - 91 -C\:, 0 cr' w cr' ---w I I I I I I . 1.0 I ----::::::::::: ' .._!_ "" • 1 f • . •• SK (a /a ) 2 in S O . ZF-µSR of AuFe 0.5 - ______ -0-----0--o---o- , ._ T = 6K l:>. l:>. l:>. I- T = ?K o_). (q)a,0093.8.~ 1 ~~lOK, ' '--~-~~-.1-T = BR -12 -10 -8 -6 -4 -2 Log{Tim .e) • Neut ron spin echo o ac-susceptibility £>. · zero-field µSR l in Cu.Mn (5 at . %, T =27K) - g in AuFe (1.0at.% , T =9.1K) g Fig. 5-1. Spin correlation function of irrrpurity moments in Cu.Mn (5 at.%, T =27K) determined by - g neutron spin echo experiment (closed circle) of Mezzei and Murani [41], and also by ac-susceptibility with different measuring frequencies in the same specimen (open circles) , taken from Murani [43]. We also plotted the auto-correlation function eq. 4-3 determined by zero-field µSR in AuFe (1.0 at . %, T =9.1K) at T=?K with triangles, and the static arrrplitude (a /a ) 2 for a few terrrperature points g S O . at the edge of the figure. When we scale the terrrperature with T of different specimens, neutron, g µSR, and susceptibility experiments exhibit a good agreement . u Q) "' en 7 10-Q) ...__ Oo • • ••• o •• • • • o'b • ZF-µSR ~Mn(l.6ot.%) • !_!!Fe(l.4ot . •t.) • Cu Mn ( I.I at. 0/o) o o• neutron Cu.Mn(5at.%) D • ~. • • 0 ,:i • Fig. 5-2. Corrrparison of correlation time Tc of'CuMn (1.1 at.%); AuFe ( 1. 4 at.%) and ~Mn ( 1. 6 at.%) determined by zero-field µSR to the decay time tl/e where the spin correfotion function measured by neutron spin echo measurement in Fig. 5-1 decays into 1/e in ·cuMn (5 at.%). Neutron experiment was performed by Mezei and Murani [41]. Correlation times and tl/e for dirrerent specimens are also scaled by T in this figure. µSR data in ~Mn g and original figure for the corrrparison of µSR results are taken from Heffner et a 1. [38). - 93 -0 0 q, 0 •• 0 0.,, a • 0 . 0 0 . • too ; ,o ... ~., . . . . ... "'".. . ••••• .. 8: ., • • •••• '~ ....... I 10 CONCENTRATION U.1%1 T/ClA ( C ~-UNITS If 100 :. l 5.0 ; - 4.0 + t• ft I- 9 i 3.0 • e fQ f 8. E 2.0 t Ag:Mn 1,5 1000 2000 3000 4000 5000 Mn Concentration ppm 4 Fig. 5-3. (a) ESR line width in ~Mn (0 . 2~0. 5 at . %) with Hext = 600 Oe (open circles) and with Hext = 3 KOe (closed circles) . - A clea r dependence on external field is exhibited. (b) Temperature Td(ESR) • D where a positive g- shift shows up in ESR; Td(x ) + (> where x starts to deviate from Curie ac ac law in finite fields . D and (> are measured in H ext = 600 Oe , while • and + in 3 kOe. Positive g-shift and ~-he,_ deviation of Xac from Curie law both reflect the static component of impurity moments in finite fields , being consistent with TF- µSR. Both (a) and (b) from Dahlberg-et al. [47] . - 94 -~~·- .:~ .... ...................... -·- ..... ·-.-... _._ .. ___ ..-3 -2 -, 0 r 5t -.. t .'!·6·\·-···-····· ........ -···· -···-·········---··-- .·- ·······--·---...:.:.:~:-~ . ......... -, .. ····-·· ... - ·····-·.-- ......... _ .. ,._.. b- l0 ,5•K , .•. -. - ... -· •••.• _ 0•-., ..... _ . ...... . . ,,:• ·.·- . · • • •••••••••• • ,.· • ... -....... .. .. ... ...... 04•',°"' ·-o- 4 .&•K ............ .-··' · - •..•. ""··~ _ • .-·· ••. _ . .,,. . ........... , ., .-. _ _ . ___ .. _.. ,.,.. ··-... -· · ~ '·----- 6 -4 -2 O . 2 4 6 VELOCITY (mm/sec) Fig. 5-4. Mossbauer spectra of 57Fe in AuFe (1. ? at.%, T0=14.BK) measured by Violet and Borg [48]. Characteristic six peaks clearly appears below T $ O. ?T , and increase of hyperfine fi elds is suggested for lower temperatures. 0 - 95 -x 1.0 -- .... ... --r-"' oa Mossbauer ~o 0.6 -:: r, CONC£\ITRATION (oL •1,,) ... 0.4 0 .64 0 1.7 .. 0 .2 . 1.9 0 2 .9 • 4 .4 • 6.7 • 7.4 • A ZF-µSR 0~0-'---=-o"°.2-'---,-0_~4-'---"o .~s--"'---,Jo.""e--"--'1.-o __, T/T 0 Fig. 5-5. Temperature dependence of hyperfine field on Fe nuclei in AuFe (O.B'v8.0 at.%) measured by Violet and Borg [48]. Temperature is normalized by the "transition temperature" T deduced self-consistently 0 from the Mossbauer results. I also plotted the static part a /a of s 0 impurity spins estimated by zero-field µSR in AuFe (1.0 at.%), normalized by the cusp temperature Tg. Results of Mossbauer effect and µSR exhibit a good agreement. - 96 -., 2 :,: <1 :,: I >-. c j "' K)2 ::E :c 3 :,: 10 T1IJ.i.sec -4-- flHsot Ho(kOe l 300 "9 Sulk 63cu NMR I/Io --- --'7 1.3 Cu-Mn 1 ¾ ---v 3 .5 " --Y 8.9 200 ~. W = 10 MHz T2 '~ 1i iOO \ ._\¾ 10 0.5 -T1(msec) 5 Tl I Cu Mn ; 1.01 at¾ I, 0.2 I 2 \ I 0.1 10 ' TEMPERA TURE (K) .0.5 1 2 5 10 20 50 T(-K) Figs. 5-6 and 5-7. Line width (5-6) and the spi n- echo intensity and rela.xation times (5-7) measured by NMR in Cu.Mn (1 at . %) by Maclaughlin an·d Alloul [51 , 53). Line width above T exhibits a field dependence similar to g . TF- µSR. The loss of spin echo intensitoy reflects the dynamical spin rela.xation of Cu nuclei within the dead time of spectrometer . Fig. 5-? can be corrpared to Fig. 3- 12 of LF- µSR, since both experiments are "gating " dynamical corrponent of spin fluctuations . For wL = 107 /sec in NMR, the fastest rela.xation is observed at T 0.?T , suggesting a slow spin fluctuation v 107 /sec below T . g g - 97 -Fig. 5-8 . 1.0 G2(t) 0.6 0.2 o. o. 02 µ + in (CoO¼o<A~~),o(SiO;t50 Tg= 6.4K -rH-L"t+W 4-_ 60 K 0.6 TIME (µsec) 40 K 30K 20 K 10 K 1.0 Zero-field µSR spectra observed in an amorphous insulator spin glass (Co0)40(Al 203) 10(Si0 2) 50 (14.3 at . % Co, Tg=6.4K). Depolarization rate starts to increase from very high temperature T 10 T . Solid lines represent the "single T " model GSG (t ) , while a large distr~bution of T C Z C is suggested at lower temperatures as indicated by broken lines. The effect of nuclear dipolar fields is rz,~gligible in this time region. The observed , I• • spectra does not depend much on. temperature below T = 10K, due to a rapid reduction of asymmetry with a dead time~ 30 nsec of the spectrometer . - 98 -1.0 + in (Coo40 (A~03)10 (Si0~50 µ Gz(t) Tg= 6.4K I I T =15 K 0.6 I \ \+1 Hext = 0 ', ........ ; 0.2 0. o. 0.2 0.6 1.0 TIME (µsec) Fi g . 5-9 . Comparison of the fit of zero -field µSR spectT'Um in an insulator spin glass (CoOJ4/Al 2o3) 10(Si0 2) 50 observed at T=15K (T=2.3Tg) by "single 'c " model (solfd line) and by the model function eq . 4-1 with a large distribution of ,: (broken line). The observed spectT'Um fitted better to the broken line , C suggesting a large spatial distribution of correlation times in the specimen . - 99 --(.) Q) (/) -1-..iu Lu z 0 _j w 0::: 0::: 0 u ,o-r o. 100 TEMPERATURE (K) 20 (Coo4 0 (A~03)10 (Si0~ 50 Tg= 6.4 K Q05 1/T (1/K} 10 o., Fig. 5-10. Correlation time Tc of Co moments in (Co0)40(Al 203) 10(Si0 2) 50 estimated by zero - field µSR. Values at T 30K represent the best fit for "single T " model eq . 2-15, while the averaged value v of eq. 4-1 in the C 0 fit with a large distribution of Tc is plotted at T $ 20K. The observed correlation times exhibit an Arrhenius law with the activation energey · - 12 E /k 13T and the pre - exponential factor T = 10 sec . a g o - 100 -90 80 70 60 50 i.o Xa.c . (a .u.) o 17.3 Hz A 197.6 a 2102 CuMn 1.,. T(K) L. 6 · 8 . 10 12 14 16 18 20 B .··• (ol 0 .65 752 .-:-~. (bl 10.4 746 6 (o!•.-:.•::..-11 • le) 234 724 . ·:···· . Id) 1330 67.6 "' 1,1,•.•.•1dl .. .... : :\:; ::, )4 b 2 X 2 0 5 10 T(K) 15 20 Fi g. 6-1. Fr equency dependence of ac- susceptibility measur ed in Cu.Mn (1 at.%) (a) by Tholence [42], and i n t he same specime n of (CoOJ40(Al 2o3J10(Si0 2J50 (b) by Wenger et al. [64]. The dilute - allo y spin glass exhivits alomst no dependence on the measuring f requency of x , while a large change was obse rv ed in the ac amorphous insulator spin glass , suggesting inhomogeneous distribution of cor r elation times in the insulator spin glass. - 101 -Appendix I. A TIME-DEPENDENT TREATMENT TO CALCULATE RELAXATION FUNCTIONS IN THE EXISTENCE OF ORDER PARAMETER In Chap. IV-A, we considered the spin fluctuations with an order parameter <S(t)S(O)> I <S(o) 2> (1-Q)ex p (- vt) + Q (A-1) by assuming each sp in Sas a vector sum of static and dynamical components. We shall see an alternative model in this appendix , where the non-vanishing part Q of the above equation is embodied by a tendency for the spin S to be oriented towards "preferred directio n" during dynamical fluctuations . Suppose two situations A and B for an impurity spin S. In A, the spin is in its preferred direction. In B, we find this spin in any direc t ion at random. We can then simulate eq . A-1 in the following model. 1) S i s in its "preferred" direction (state A) with probability p = IQ. 2) Sis fluctuating without any preference of direction (state B, B', B", •.. ) with probability (1- p ). 3) The change A+ B occurs a s a sp in fluctuation with the rate (1-p)v, while B + A with the rate pv (i.e., t he detailed balance between A and B). 4) Modulation among states B, i.e., B + B', takes place with the rate (1- p )v. Let us adopt this model also t o describe the time evolution of local field H. Muons will feel the field of A as a static field , and the modulation among the states B can be treated via the strong collision approximation. Then the relaxation function at a muon site becomes 8 1 - a , . G + q• exp(- pv t)•G (t, 6 ,qv ) z - 102 -(A-2) PA(t;a;v;p) = pqv•exp(-vt+u+w)•{2I0(z) + (/ulw + lwlu)I 1(z)} u = pavt, w = qBvt, where I0 and I 1 are the Modified Bessel functions 00 I (z) = E (f)2k I (k!)2, 0 k=O I1(z) =; (f)Zk+l I {k!(k+l)!} . k=O (A-3) (A-4) The first term of eq. A-2 corresponds to the case where the local field + onµ is in its preferred direction throughout the time interval t, while the second term describes muons which only feel the field of B, B', B", PA in . the integrand of eq. A-2 gives the probability thatµ+ stays in the field A for time ta in total out of the interval t. The term of PA with k=O part of I, I [k=O], corresponds to the fluctuation A+B and B+A, I 1[k=O] to A+B+A 0 0 and B+A+B (or B'), I [k=l] to A+B+A+B(B') and B+A+B(B')+A, and so on. 0 We can finally obtain the model function for the entire specimen by G averaging G2(t,6,v,p) over the probability p(6) as in eq. 2-15. Figure A-1 shows an example of this model function for via= 50. By the onset of order parameter Q = p2 , the line shape is drastically changed. The 113 asymmetry exhibits a decay due to the dynamical fluctuation. In actual systems, random local field onµ+ is originated from a few impurity moments, so it may not be appropriate to assume the same time evolution f.or an impurity spin Sand for a random field H. So employed the simpler "vector sum" model for the analyses of experimental data in the text. The present time-dependent treatment, however, covers any combination of Q and v, and has an advantage over the "vector sum" model in chap. IV-A which is limited only to the case of fast fluctuation via »1. Some view points of this time-dependent treatment may be applied also to stochastic calculations of other physical systems, e.g., ferro- or antiferromagnetic materials. - 103 -' BEGINNING PART 1.0 N (.!) 0.5 0 0.1 0 .2 0.9 0.6 0.4 0 I ta 2 3 4 30 ta 50 70 Fi g. A-1 . Zero- field muon spin relaxa t ion function G (t) in a spin glass wi t h order paramete r Q = p2 calcu l ated according to the !ime - dependent model of Appendix I fo r an example of t he modulation rate vi a = 50. A dras t ic change of line shape is caused by the onset of order par amet er Q. - 104 -
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Dynamical Properties of Spin Glasses Studied by Muon Spin Relaxation
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