Lf§WJ M31) 1 RETURNING MATERIALS: Place in book drop to [JBRARJES remove this checkout from .—_—. your record. FINES will , be charged if book is returned after the date stamped below. i :- 1 ABC 2. 4.." l m I W n I it!" ‘ SPIN-GLASS BEHAVIOR IN DILUTE ALLOYS WITH ATOMIC ORDER-DISORDER TRANSITIONS By James Stephen Shell A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1982 :.A ‘." - J (I so. G//738? ABSTRACT SPINeGLASS BEHAVIOR IN DILUTE ALLOYS WITH ATOMIC ORDER-DISORDER TRANSITIONS By James Stephen Shell An experimental investigation of the influence of binary host atomic order-disorder transitions on the spin-glass behavior of dilute alloys has been conducted. The investigation was carried out with the use of a homebuilt vibrating sample magnetometer (VSM), a commercial squid based susceptometer, and an AC mutual inductance bridge apparatus. These instruments were used to study the magnetic susceptibility of three host alloy systems with the emphasis on (Cu3Pt)]_anx. An overview-of general spin-glass theory and a discussion of ex- perimental considerations including the design, calibration, and operation of the VSM are presented. ‘ The nature and reliability of the samples is discussed prior to the presentation of experimental results. The observed experimental behavior is presented and dis- cussed in light of mean field theories, free electron RKKY couplings, mean free paths, Brillouin zones, and Fermi momentum effects. None of the above approaches can explain the results in even a qualitative James Stephen Shell fashion. Finally, preliminary investigations were started on Cu3AuMn alloys and the hysteresis behavior of Cu3PtMn alloys. Perhaps this work will help make clear to them why I chose to do physics. In any case, I would like to dedicate this dissertation to my parents. ii ACKNOWLEDGMENTS I would like to thank Professor Carl L. Foiles for suggesting the thesis topic and providing expert guidance throughout the course of the research. His clarity of thought and patience made this re- search most enjoyable. I would also like to express my appreciation to Professor Jerry Cowen for many fruitful discussions and for introducing me to the S.H.E. susceptometer. The AC susceptibility data in this thesis is due largely to the efforts of Professors Foiles and Cowen. I would like to thank Professor Gerald Pollack for constant encourage- ment during the course of this work, especially as my outside interests multiplied. I am also deeply grateful to various members of the tech- nical staff for their support. In particular, I would like to thank Mr. Daniel Edmunds, without whose help much of the electronics would lay idle or never have been built. Thanks are also due to Mr. John Yurkon for assistance in troubleshooting the magnet power supply, Mr. Donn Schull for general electronics assistance, Mr. Boyd Shumaker for sample preparation assistance and Mr. Bob Cochrane, Mr. Dick Hoskins, and Mr. Floyd Wagner for their assistance in machine shop matters. I am also grateful to Mrs. Peri-Anne Warstler for her ex- cellent typing of this thesis especially under such short notice and to Mrs. Carol Edmunds for many a fine home-cooked meal. I would also like to thank the Chemistry Department for the use of their Squid susceptometer. iii In: 1.15 (Fan's vi 01b. The financial support of the Physics Department and the National Science Foundation has been appreciated. iv yv ‘6' In t“. (D (“5 Ln (D (Jj 4 TABLE OF CONTENTS Chapter LIST OF TABLES ........................ LIST OF FIGURES ....................... I. INTRODUCTION ....................... A. Why Study Magnetism? ................. B. Preface of Thesis Content and Introduction to Spin-Glasses . .. ................. C. Magnetic Behavior in Order-Disorder Alloys ...... II. GENERAL SPIN-GLASS THEORY ................ A. Mean Field Theories ................. B. Virial and High Temperature Expansions ........ III. INSTRUMENTATION . . . . ................ A General Magnetometer Systems ............. B. General VSM Considerations ............. C Signal Detection and Modulation ........... D Signal Detection Coils ................ E Field Noise ..................... F Holder Design .................... G. Cryostat Design . ._ ................. H Temperature Control and Measurement ......... I. Typical Run ..................... J. Calibration ..................... Page viii ix l l 4 9 I6 16 38 46 46 57 61 65 66 69 72 75 77 81 .. r3"- .xu-r' Chapter Page K. Further Improvements ................. 87 IV. SAMPLE PREPARATION AND RELIABILITY ........... 90 A. CgMn ......................... 90 l. Preparation ................... 9O 2. Magnetization vs. Field ............. 91 3. Annealed vs. Quenched Behavior .......... 92 B. Ternary Systems ................... 99 1. Cu3Pt ...................... 99 2. CuPd(l7) ..................... 101 3. Cu3Au ...................... 102 V. Experimental Results ................... 103 A. QuMn ......................... 103 B. Cu3PtMn ....................... 110 C. CuPd(l7)Mn ...................... 116 D. Hysteresis Study ................... 120 VI. INTERPRETATION ..................... 129 A. Binary Host Alloys .................. 129 B. Effective Mn Moments ................. 134 C. Mean Field Interpretation .............. 138 1. SK Phase Diagram ................. 138 2. Mean Random Field ................ 141 D. Free Electron RKKY Coupling ............. 147 E. Mean Free Path Effects ................ 150 F. Brillouin Zone and Fermi Momentum Effects ...... 158' Quantum Effects ................... 161 vi .I«_fi‘ r J!” vE‘ ‘91 N 1.3! J! |QIAP‘IFY- '11."..1“. ( c L.) m I! v . w 3;- Chapter Page VII. GENERAL CONCLUSIONS AND THE FUTURE ........... 163 APPENDICES A. Signal Detection Coil Considerations ......... 167 B. Schematic Diagrams .................. 174 C. Magnet and Power Supplies .............. 179 D. Grounding and Shielding Considerations ........ 183 E. Phase Shift Considerations .............. 187 F. Hysteresis and Anisotropy .............. 191 G. Anisotropic Exchange Interations ........... 196 H. Sherrington-Kirkpatrick Phase Diagram ........ 200 I. Mean Random Field .................. 208 Larsen Integral ................... 211 K. guaguMn Data ..................... 214 LIST OF REFERENCES ...................... 217 vii Table 10 11 LIST OF TABLES Page Cluster Probabilities for Specific Dilute Concentrations .................... 8 Summary of Experimental Results Relating to the Freezing Temperature .............. 8 Quadrature and In-Phase Components of Clipped Sine Waves .................. 68 Comparison of Annealed and Quenched CuMn ....... 98 Curie-Weiss Parameters for guMn ............ 106 Experimentally Determined Parameters for Ordered and Disordered CuafltMn Alloys ......... 115 Experimentally Determined Parameters for Ordered and Disordered CuPd(17)Mn Alloys ....... 121 Selected Properties of Binary Hosts and Pure Copper ................... A. . . 130 Effective Manganese Moments .............. 137 Resistivity of guafltMn (0.81%) and CuPd(l7)Mn (1%) .................... 152 CalcUlated SpinéGlass Freezing Temperatures . for CgaPtMn (0.81%) and CuPd(l7)Mn (1%) ........ 156 viii Figure 0101450)“) 10 11 12 13 14 15 16 17 18 LIST OF FIGURES Concentration dependence of magnetic phases in metallic alloys .................. Mean field theory development flowchart ....... P(H) for different models .............. EA order parameter .................. ng vs a for guagtMn ................. Basic techniques for measuring bulk sus- ceptibility ..................... Squid based magnetometer ............... Conventional Foner style VSM ............. VSM used in this study ................ Holder design .................... Basic cryostat layout ................ Palladium calibration run .............. Palladium calibration run .............. Possible future VSM ................. CuMn (3%) - annealed and QgMn (2%) - M vs. H ....................... Brillouin function for spin S = 2 .......... CuMn (1%) quenched vs. annealed behavior ....... CuMn (2%) quenched vs. annealed behavior ....... ix Page 5 21 25 3O 45 49 55 58 6O 71 73 85 B6 88 93 94 95 96 Figure 19 20 21 22 23 24 _ 25 26 27 28 29 30 31 32 33 34 35 Page CuMn (3%) - quenched vs. annealed behavior ...... 97 QuafltMn (3%) OS and DOS - M vs H ........... 100 CuMn - inverse susceptibility vs. temperature . . . . 105 Curie temperature vs. concentration (selected data) ........................ 108 QuagtMn (OS) - inverse susceptibility vs. temperature ..................... 111 CgafltMn (DOS) - inverse susceptibility vs. temperature ..................... 112 'Qg3_tMn — AC susceptibility vs temperature ...... 113 QuafltMn (3% DOS) - typical squid ng data ...... 114 ng (c) and e (c) for QuagtMn ............ 117 CuPd(l7)Mn - inverse susceptibility vs temperature - VSM .................. 118 CuPd(l7)Mn - inverse susceptibility temperature - Squid . . ._ .............. 119 CuPd(l7)Mn - susceptibility vs tempera- ture - Squid ..................... 122 ng (c) and 0 (c) for CuPd(l7)Mn ........... 123 CuMn (2%) - hysteresis curve ............. 124 QuafltMn (3%) OS - hysteresis curve .......... 125 ggagtMn (3% DOS - hysteresis curve .......... 126 Brillouin zone structure for our binary alloys ........................ 133 Figure 36 37 38 39 4o 41 42 43 44 45 45 47 48 49 50 51 52 53 54 55 56 Electron/atom ratio vs. domain size ........ CuafltMn-SK phase diagram .............. CuPd(l7)Mn - SK phase diagram ........... ggaggnn (2%, 3%) - Klein diagram .......... QuagtMn (1%, 1/2% 005) - Klein diagram ....... Brillouin zone effects on the Fermi surface . . . . Optimum pick-up coil geometry ........... Tuned amplifier .................. "Interrupter" ................... Temperature controller ............... Current source ................... Grounding scheme (low level signal circuitry) . . . Grounding scheme (overall) ............. Quadrature sensitivity and homogeneity curves . . . Soft spin-glass hysteresis curve .......... DM Interaction vectors ............... SK phase diagram - BPW approximation ........ SK phase diagram - spherical model ......... SK phase diagram - rescaling calculation ...... CgafiuMn (1%) - inverse susceptibility vs temperature - (Squid) ............... CgaguMn (1%) - susceptibility vs. temperature - (Squid) ..................... xi Page 135 142 143 145 146 160 171 175 176 177 178 184 185 189 192 198 201 203 206 215 216 In an .n no i4 ‘H L: I. INTRODUCTION A. Why Study Magnetism? To the uninitiated, the topic of this thesis and the field of magnetism in general may seem either esoteric and without applica- tion, or an old field of study whose prime is long past. In fact, just the opposite is true and it is the purpose of this section to discuss the role of magnetism and dilute alloys in the field of condensed matter physics. Historically, many important developments in condensed matter physics and many-body physics first appeared in the theory of magnetism. The reason for this may lie with the rela- tive ease with which magnetic systems can be visualized, as compared with other interacting many-body systems. The flexibility of being able to work both classically and quantum mechanically, as well as in spaces (spin or real) of arbitrary dimension is most certainly an important aid. In fact, one might argue that the modern era of many-body theory actually began in the 1950's when the theory of spin waves in metallic ferromagnetic materials was developed. Even the Kondo problem has provided an excellent many-body system which can be checked experi- mentally. It was also one of the first problems to be attacked using the powerful, newly developed theory of the renormalization group.(]) It has recently been solved exactly.(2) Another example of an important concept whose origin lay in the theory of magnetism is that of broken symmetry. This idea arose from early attempts to understand anti-ferromagnetism.(3) The problem was this. The conventional alignment of spins is an accept- able idea within mean field theory. However, such a state is not an eigenstate of the system and therefore cannot be the true ground state. More serious is the problem that such a state does not satis- fy the symmetry requirements of quantum mechanics. A ground state of a quantum mechanical system must be an eigenstate of any symmetry which the Hamiltonian possesses. The Heisenberg Hamiltonian has rotational symmetry in spin space. A single Neel state does not. The true ground state must therefore be a symmetric linear combination of all the orientationally degenerate Neel states. This symmetry has definite consequences for the energy level spectrum. It requires the existence of anti-ferromagnetic spin waves, which are probably the first example of the use of Goldstone bosons in theoretical physics. This introduction of Goldstone bosons is currently being used in certainLagrangian 'field theories of elementary particle physics. Another area where magnetism has made important inroads is in the field of phase transitions and critical phenomena. The first ex- ample of an interacting many-body system which is completely solv- able and undergoes a phase transition was probably the 2-0 Ising ( model in zero field solved by Onsager in 1944. 4) Magnetic systems were natural systems to consider because the order parameter is generally quite transparent and very physical. Most of the phase transitions to date have dealt with systems in which the ordered phase has simple long range order. These include ferromagnets, antiferromagnets, ferrimagnets, helical and canted structures. The dilute magnetic alloys studied in this thesis do not order into such simple geometrical arrangements. Rather, the spins appear to freeze in random directions. This may be an example of a new, more subtle, form of phase transition to an ordered phase where the spins are strongly correlated but exhibit no simple long range order. A currently investigated system, the 2-D XY model, shows some promise towards explaining the spin-glass transition, the helium superfluid transition, and a possible mechanism for quark confinement.(5) It is also possible that the new field of dynamic critical phenomena will enter the picture. The dynamic properties of a system (transport coefficients, relaxation rates, etc.) all depend on the equations of motion and are not simply determined by the equilibrium distribution of the particles at a given instant of time. The static properties of a system (thermodynamic coefficients, linear response to time independent phenomena) are determined by the equilibrium distribution of particles at a single time. The modern field of dynamic critical phenomena may be necessary to explain spin-glass behavior since the spin-glass transition does appear to be coopera- (6) tive and yet time dependent. Finally, condensed matter physics has entered the new field of disordered systems, and magnetic systems are once again playing an important role, and will certainly aid in establishing fundamental concepts about these systems. It is likely that other areas of physics will draw on this knowledge, because often the magnetic analogs are the simplest systems that can be checked experimentally. ........ 8. Preface of Thesis Content The measurements described in this thesis are an attempt to shed further light on the nature of impurity-impurity interactions in dilute magnetic alloys. Our studies have dealt with metallic spin-glass systems in general, and the low temperature behavior of the magnetic moments in order-disorder binary alloys. By controlling the degree of order of the host we have a nice way of modifying the interactions between impurities in a systematic fashion. The goal of the research was to determine the effect of the host condition on the paramagnetic and spin-glass properties of the alloys. Since the alloys under study exhibit spin-glass behavior at sufficiently low temperatures, a general introduction to these systems is in order. In the context of this thesis, the following definition of a spin-glass can be given. A spin-glass is a random, metallic, magnetic system characterized by a random freezing of the magnetic moments without long range order. Whether or not this freezing occurs gradually or suddenly is a question not easily answered, but certain thermodynamic measurements do show a sharp behavior. The discontinuity in the slope of X versus T is often associated with this transition, and defines what we shall call the freezing temperature Tf (or the spin-glass temperature T59). The most common metallic spin-glasses are typically transition metal impurities in non-magnetic hosts. Usually the spin-glass regime exists for only a limited concentration range of the impurity. We can picture the situation roughly as follows. See Figure 1. O - In the very dilute regime the isolated impurity-conduction electron .zroaoomzmocm _ MET mE:.. xosao 05mm 03mm Egoamozm" roan mmaom 360.36 _ : Oqamq )v 56le 00:06:- Sam».o: u mo 003 u in 4.0 “530520: mfg m7$ :21" msocxm #. noznmanxmadoz amumsamanm 04 amazmflmn u:mmmm A: swam..¢n m._o (deg/dc)os? . (4) There are also weak indications that 6(c) may change sign for the QuagtMn alloys for sufficiently dilute amounts of manganese. Is this sign reversal real and how does it relate to an analogous sign reversal in CuMn? II. GENERAL SPIN-GLASS THEORY The purpose of this section is to give a brief introduction to the general theory of spin-glasses. It should be noted that there does not exist any generally accepted theoretical understanding of these systems. There has been a great deal of theoretical ac- tivity in recent times, but no unifying thread has emerged thus far. The discussion will be presented in two parts. First, a discussion of mean field theories will be.presented. The utility of mean field theories resides primarily in their simple physical interpretation and ease with which one can calculate physical properties. Thus it is logical that this be the first step in probing a theoretically com- plex topic. The second part of the discussion will touch on some of the other non-mean-field approaches,especially virial expansions and high temperature expansions. The discussion is intended to present the topic from a somewhat historical/logical approach and is certainly a reflection of the author's personal viewpoint. This will also serve as a brief introduction to some of the ideas and models that will be used to interpret the experimental results. Before I begin the discussion of mean-field theories proper, let me comment on why the spin-glass problem has been one of such great difficulty. It is the goal of equilibrium statistical mech- anics to calculate the free energy FN(B) of a system of N particles. Then the thermodynamic limit (N +1m) is taken to find the free 16 17 limit energy per particle f(8) = N +,m FN(B)/N. Now the thermodynamic limit is often needed in the evaluation of the partition function ZN before the logarithm is taken. This allows one to neglect fac- tors in ZN of lower than exponential order in N. This particular convenience is upset by the introduction of randomness. A random system is described by a Hamiltonian which contains random param- eters which usually represent the interactions in the random quenched configuration. To calculate the physical properties we must average over all possible configurations. This is achieved by averaging FN(8) over some probability distribution for the random parameters. This average must be computed after taking the logarithm, but before taking the thermodynamic limit. That is, we must calculate limit 1 ‘Bf(B) = N +,m N <1" ZN(B)> (2-1) randomness Notice that the thermodynamic limit can no longer be used directly in the evaluation of ZN- In addition, averaging over the random parameters is difficult because the logarithm prevents any useful fac- torization into a product of one parameter averages. In short, the primary difficulty in these theories stems from the quenched nature of the disorder. Two methods of attack have been formulated within mean-field theory. One is the n + 0 replica method of Edwards and Anderson. They remove the disorder at the outset and are left with performing a thermal trace over the exponential of some effective Hamiltonian. 18 The second approach is typified by the work of Thouless-Anderson and Palmer. They perform the thermal averages at the outset and are left with averaging over the disorder. These theories will be considered in a bit more detail shortly. Neither of these methods allows us to use the thermodynamic limit in the evaluation of Z. In the case of the replica method, statistical mechanics demands that we calculate limit 1111111; -1 N +4w n + O n ' but most theoreticians would prefer to interchange the limits. The legality of this interchange is by no means obvious. Because the non-equilibrium nature of this problem is so important, I would like to say a few more words about it. As shown in Equation (2.1), the desired free energy involves the calculation of <£n S>c when < >s denotes a thermal trace over the spins and < >c denotes a configuration average. In fact, if we were to do the simpler calculation in s,c’ what would we have? We must have in that case a system where the positions of the particles are in thermal equilibrium as well as the spins. That is, we have described a gas of paramagnetic particles interacting with spin-dependent ex- change forces. And in this case there is a real correlation between the relative spins of the particles and their positions. But in the spin-glass systems of interest, the atoms have condensed into the solid state and are no longer free to move. The former systems are referred to as annealed, the latter are called quenched. In 19 the spin-glass case the randomness is immobile. Therefore, the averaging must be carried out over a physical observable. The free energy is such an observable, the partition function is not. There is undoubtedly still some correlation between the spin and spatial configurations in the condensed systems, but it is certainly different from the case of the paramagnetic gas. Rather than go through a detailed discussion of any one par- ticular mean-field theory, a general overview of the work to date will be given. This overview is certainly not complete, but is designed to show how the various models fit into the overall scheme of things. See the flowchart in Figure 2. Before the first actual development of mean field theories for random systems, the problem was being attacked with diagrammatic expansions. The formalism of cumulants (or semi-invariants) was already in place due to work by J. G. Kirkwood in 1938. He used the theory of cumulants to expand the free energy in power of kT. i.e.) F(B) = Mn (2.2) 3 ID ' 3 IIMB n l The primary contribution of Brout [B] was to rearrange the terms in the Kirkwood expansion. He abandoned the expansion in powers of l/kT in favor of expansion in powers of l/Z. Here 2 is the effec- tive number of spins interacting with a given spin. This is useful because in the limit 2 +1w the mean field theory becomes exact. Therefore, the mean field theory results should coincide with the leading order term in Brout's expansion. Horwitz and Callen [HC] 20 Abbreviation Reference B 15 HC 16 E 17 RMF 18 PP 19 MRF 20 EA 21 SK 22 TAP 23 VP 24 BGG 25 S 26 KSS 27 AT. 28 BM 29 P 30 1959 B RMF m 1971 Canelja _& 5431903941.... 1975 E A .3 K‘ r. BGG 1 KSS rd A ‘1“ <= '0 Figure 2. Mean field theory development flowchart. 22 extended Brout's method by regrouping the cumulants in such a way that certain restrictions in the summation of indices could be lifted. This, plus an expansion in terms of spin-deviation opera- tors Oi = S - Siz led to the development of vertex renormalization. With vertex renormalization one can eliminate all reducible diagrams. (Reducible diagrams are linked diagrams which can be separated into unlinked parts by a single cut.) This renormalization procedure led to an irreducible linked diagram expansion for the Ising model. Englert [E] extended these ideas further by exploiting the similari- ties between the semi-invariant expansion of the partition function and the methods of propagators in quantum field theory. This opened the door to the use of some of the more powerful many-body theory techniques. These l/Z expansions are useful as a means of verifying mean-field theory and suggesting higher order corrections to them. About the same time as Brout‘s work, the first mean-field theory was being developed by Klein and Brout. The statistical problem that needed to be solved was the calculation of z = 2 e'BH all states where H = Z v..S.S. i a where

= ZAa/n

= 2Aa3/3n. At low temperatures Marshall argued that P(H,T) should approach the form shown in Figure 3a. At low temperatures the correlation between spins is larger and the total energy of the system must go down. But this can only be done by arranging for the spins to sit in fields of larger magnitude. Thus the mean values of H must get pushed out to larger values. Klein and Brout find more or less agreement with Marshall. They find P(H) to be a Lorentzian with width proportional to the impurity concentration. If they properly exclude large fields from impurities close to the origin, the dis- tribution changes from Lorentzian to Gaussian. They also tackle the problem of including spin-spin correlations. They express the 2-particle correlation function g(r]2) as a power series in cz(r) where c is the impurity concentration and z(r) is the number of sites included within some radius r = Rc' For r > Rc the cluster expansion breaks down. Thus they conclude that the impurities are fully or partially correlated to the spin at the origin if they are located within some correlation radius Rc and are approximately P(H) 11 (a) ‘3 H P(H) .4 (b) h) H P(H) A (C) 11 Figure 3. P(H) for different models. 26 randomly oriented if located outside Rc' So P(H) is the sum of 2 contributions. For r < Rc’ P(H) is the field obtained from the cor- related spins and for r > Rc the field distribution is essentially Gaussian. The next development was the mean-random field approxima- tion [MRF]. The idea here is straight forward. When calculating the probability distribution of the internal field at a particular site, replace all functions of the internal field at the other sites by their mean values. Continue to neglect spin-spin correla- tions and use the random molecular field approximation. In principle P(Hj) could be different at each site. If one imposes the self- consistency requirement which says that each P(Hj) has the same functional form, then an integral equation for the probability distribution is generated. This model was reasonably successful in explaining experimental low temperature properties as long as one used Ising spins. The Heisenberg mean random field predictions dis- agree with experiment. This is unfortunate since the actual spin systems studied experimentally were Heisenbergflike. The probability distribution P(H) in the_Sherrington:Kirkpatrjck_“ _ model (to be discussed shortly) is shown in Figure 3b and 3c for one and two dimensional spins respectively as computed using num- erical simulations by Palmer and Pond [PP]. They find P(H) to be linear in h for small h for Ising spins, and find a "hole" in the 2-dimensional spin case. Muon Spin depolarization or Mossbauer studies may be able to choose among these models. In 1971 Canella and Mydosh published their now famous low field ac susceptibility measurements and this caused theoretical activity 27 to greatly increase. In 1975 a new kind of spin-glass theory was developed by Edwards and Anderson [EA]. Their model consists of a set of spin variables on a periodic lattice interacting via ex- change bonds that are randomly ferromagnetic and anti-ferromagnetic. They obtain some of the experimentally observed behavior of real spin-glasses (like a cusp in the susceptibility) and some behavior which isn't observed experimentally. It is not clear how well the [EA] approach is withstanding the test of time. Some high tempera- ture series expansion work of Fisch and Harris leads to the conclu- sion that the spin-glass phase cannot exist below 5 dimensions. Bray, Moore and Reed have performed numerical simulations which they interpret as evidence that the spin-glass phase has no frozen in order but only very long relaxation times. They argue that the frozen in magnetization results from the mean field approximation, and that in systems with fewer than 4 spatial dimensions, this state does not persist. Nevertheless, its useful to examine the 3 crucial elements on which the EA model is based. (1) They perform an average over configurations of random bonds in such a way that all spatial inhomogeneities are averaged out. Recall therefore 28 e'BnF = (Tre'BH)n = Zn for any n 50 _ ’BZLJ-- Z 5o(a)S-(q) (Tre 8H)n = True 13 13 a=1 1 J and 2 2 2 $3 §_9ij( cg] 51(G)Sj(a)) or where 08 ij lj 51(0)SJ(G)Sj(B)Si(B) j has been eliminated and replaced by its mean which is translationally invariant. Anderson argues that the The random interaction Ji only limit in which this average is safe is the limit n.+ 0 when it becomes Jij. So the trick is to do the problem for all integer n to get an expression for F(n) and take the limit n1+ 0 when finished. This formal structure is a coupling between dif- ferent replicas (2) The second element is the characterization of the spin- glass phase by the order parameter q = <->J where §(x) 29 is the spin variable at the lattice site x and the inner brackets correspond to a spin-average for a fixed configuration of bonds. The outer brackets < >J denote an average over the bonds. This order parameter was introduced because many people interpreted the cusp in X as evidence for a phase transition. Edwards and Anderson thought about the transition in the following way. Since there is not thought to be any long range order in space, perhaps there is still long range order in time. This idea led them to suggest the follow- ing order parameter. q = tJTEM» <>. The connec- tion with the replica method is that the replica-replica correlation an = seems to behave in exactly the same manner as the long time correlations . Apparently 2 replicas a,8 can be considered as the same system at two times t1 and t2 with |t1 - t2| +-m. Using a variational method, EA were able to show the existence of a critical temperature below which q is non-zero. See Figure 4. The parameter q is then given by a self-consistent equation and one can deduce the thermodynamic and magnetic properties of the system. (3) The third crucial element is their use of replication techniques to allow the bond and spin averages to be performed interchangeably. A bit more on this shortly. The appealing aspect of the EA model is the following. If eij is the probability of finding a pair of spins at sites i and j, and ZJijEij f 0 then one can have ferromagnetic or antiferromagnetic ordering at sufficiently low temperatures. In their model, zJ..e can equal zero on any 13 1'3“ scale, and the mere existence of a ground state is sufficient to cause a transition and a consequent cusp in the susceptibility. Ttime Figure 4. EA order parameter. 31 Because the replica trick has enjoyed widespread use, let's discuss it briefly. The goal of the replica trick is straight- forward. It allows one to interchange the bond and spin averages. Recall that when the randomness is quenched, the averaging must be carried out over the free energy and not the partition function. The replica method essentially transforms the free energy average to a partition function average in the following way. . n Recall, 2nx= 11m 111—"D- n + O n n n for integral n, Z = H Z“ a=1 therefore 2nZ = lim Zn'l n + 0 n lim -1 n implies <2nZ> n + 0 Notice this transformation was not achieved without sacrifice. We have, as pointed out earlier, increased the effective spin dimen- sionability of the problem and have been forced to introduce limit- procedures. Van Hemmen and Palmer [VP] present a very thorough analysis of the replica trick using a slightly different form -_d_ n <£nZ> — dn £n n = 0 and conclude that the fatal error in the method is the analytic continuation from integer n to real n. There is not universal 32 agreement on this point. An infinite-ranged model for which the mean-field approximation of EA should be exact was formulated by Sherrington and Kirkpatrick [SK]. Their solution agreed with that of EA but also contained a pathology. The entropy becomes negative at sufficiently low tem- peratures. This is not possible for a discrete Ising—type model. Since they used the replica method, this may be due to passage from integer n to n + 0 as pointed out by van Hemman & Palmer. However, there are other possibilities: (l) Reversal of the limits N + m and n + 0. (2) The steepest descent method. (3) The stability of the SK stationery point in the steepest descent calculation. The third possibility has led to the development of several new theories. It begins with the work of deAlmeida and Thouless [AT]. They found the stationery point chosen by SK to be a maximum of the integrand at high temperatures, but not at low temperatures in the spin-glass and ferromagnetic phases. They consider the SK spin-glass and ferromagnetic phases to be unstable. They restore the stability using the concept of replica symmetry breaking. This comes about in the following way. SK derive the following result for the configura- tion averaged n-replica partition function -—- 292 Zn = egnNB .1 fags mg e-Nf{q} (2.3) M217 where 33 2~2 B J 2 q 2 - 2n Tr e “<3 68 a8 a B _ 2~2 S S f{q} - 58 J 2 q a<8 Because of the factor N in the exponent in Equation (2.3), the integral is dominated in the thermodynamic limit by that set {an} which give the absolute minimum of f{q}. This leads to the mean field equations .21 aqua O B J (an ) SK assumed the absolute minimum of f{q} occurs when an = q for all pairs (6,8), but AT showed this was not the case. The idea behind replica symmetry breaking is just to divide the replicas in different groups and allow the order parameter to be different for each group. A great deal of work in this area has been done by Bray and Moore [BM]. Parisi [P] takes the idea one step further. He argues that the order parameter for spin-glass is a zero by zero matrix having the diagonal elements equal to zero. This can be parameterized by a function on the interval [0,1]. This would seem to imply the need for an infinite number of order parameters to characterize the spin-glass transition. Parisi also finds that there are an infinite number of eigenvalues which accumulate toward zero. He speculates that this infinite set of first order transitions is likely connected to remanence and is related to a breakdown of linear response theory. Bray and Moore also suspect that linear response theory breaks down. 34 Because of the subtle difficulties involved using replication theory, Thouless, Anderson, and Palmer [TAP] looked for a solution that did not require its use. They found a solution of the SK model which did behave reasonably at low temperatures (no negative entropies), and also confirmed the SK solution at and above the critical temperature. Above TC they use a high temperature expan- sion. Below TC they use a mean field theory which takes into account not only the average spin at each site, but also the fluctuations from this average. They performed a diagrammatic sum where the diagrams were classified according to powers of l/Z. The largest contributions (single chains) give the SK high temperature result. The ring diagrams of order N/Z are negligible if they converge, but TAP found they diverge at Tc 2 5. Below Tc’ the only way to make the diagram series converge was to introduce a random mean spin at every site which satisfied the following mean field equations. 5 ll tanh hi/T (2.4) 111. .- 71—; 3%. (1 - m?) (2.5) .1 :- I M c. 3 Those can in fact be derived from the following free energy, 2 2 2 F X J..m.m. - $58 2 Jij(l-m1.)(l-mj) + 3 (1.1) '3 ' J (1.1) 4. 1 grumpmnmp + (1-m,)2n5(1-m,)1 1 35 The first term is the internal energy of a frozen lattice. The second term is the correlation energy of the fluctuations. The last term is the entropy of a set of Ising spins constrained to have means mi. Equations 2.4 and 2.5 can be rewritten as _ 2 2 _ 2 mi - tanh(8 g Jijmj - 8 § Jij(1 mj)mi) The meaning of this equation is fairly transparent. The term 8 2 Jijmj is the total mean field hi experienced by the spin Si. 3 From this is subtracted the contribution which is induced by the spin Si itself. This can easily be seen. A mean moment mi at site i . at site j. This induces a mean moment produces a mean field Jijm1 J .m. at site j (here 13x33 1 XJJ mean field afijxjjmi at site 1. This is the field which must be subtracted from the total mean field at site i, since the field = amj/ahj). This in turn produces a which orients a given spin is that due to only the other moments in the system. The final step involves the use of linear response theory to derive ij = 8(l-mg). This may be a weak link in the theoretical development. This added term can be thought of as a kind of feedback term. It may be able to explain blocking or remanent phenomena. Since the TAP equations involve N simultaneous mean field equations for N spins, they can only be solved numerically, and then only near T m 0 and T m Tc. Their theory leads to the startling prediction that not only at the transition temperature T , but all temperatures T < Tc have some of the fluctuation c properties of a critical point. 36 This completes my brief introduction to the mean field theories of spin-glasses except for a brief mention of a few other approaches which lend support to some of the previously noted theories. For example, Sommers [S] solves the EA model by summing a renormalized diagrammatic expansion. His high temperature phase is identical with the SK solution, and his low temperature phase is similar to a particular solution of the TAP equations. Klein, Schowalter, and Shukla [KSS] study Ising spins interacting via random potentials in the Bethe-Peierls-Weiss approximation. They claim as the ef- fective number of neighbors approaches infinity, the magnetic properties arising from the BPW approximation, the mean random field (MRF) and the SK replica treatment are identical. They are also able to obtain the microscopic free energy derived by TAP. Blandin, Gabay and Garel [BGG] derive a mean field theory using a Landau theory approach. To third order they obtain a solution similar to TAP (saddle point in the free energy). In fourth order this saddle point breaks into a maximum (which they identify as the SK result) and a minimum. Kosterlitz, Thouless and Jones solve the spherical model of a spin-glass in the limit of infinite ranged interactions. They use the properties of large, random matrices and show that the results are identical to those obtained by the n + 0 replica trick. Finally, as we close this section on mean field theories let's briefly examine the predictions they make for the magnetic sus- ceptibility. In the Edwards-Anderson model for example, one finds x = xc(l-q) where Xc = %- is the usual Curie Law and q 37 is the previously mentioned EA order parameter. Thus X = Xc above T = Tc' Below Tc, T q = 4.1147912) therefore, T -T T -T c c c . 2 x =-— [l-(--)]{l + ( ) + o(T -T) } Tc Tc . Tc c or -g. _ 2 x - T - o(Tc T) C Thus, the cusp is linear on the high temperature side of Tc and quadratic on the low temperature side. This is an artifact of the molecular field approximation since experimentally the cusp is symmetric. ' If one allows a non-zero mean for the Gaussian distribution of exchange interactions, (such as the Sherrington-Kirkpatrick model)’ then one finds - [Hum 11“” T _ = X( ) {kT-Jo(l-q(T))} 1-30x 0 (0) where X .js the result for J0 = 0. Therefore, above the ordering 38 temperature (when q = 0) we have a Curie-Weiss Law. In the spin-glass phase fluctuations decrease X(O) and X giving rise to a cusp. Positive 30 enhances X at all temperatures. More conventional mean field theories, such as the mean random field predict the high temperature behavior to be almost Curie-Weiss like. They predict "ocpszug 2 3 9(1) xm =3—kafmt1-1-E) kBT] (See Appendix 1). Typically, one does begin to see deviations away from Curie-Weiss behavior some distance above TS , in contradiction with the EA mean field theories. Mydosh,(3]) for example, would argue that significant deviations occur as high as 5 159. B. Virial and High Temperature Expansions Most of the theories presented in the previous pages are based on the concept of a mean field. Approaches which do not use this concept generally take the form of expansions (density or tempera- ture) or computer simulations. For completeness, we will briefly examine two expansions. In 1970 Larkin and Khmelnitskii(49) in- troduced a virial expansion which is valid at high temperatures and low concentrations. In order to obtain a virial expansion they introduced the recurrence relations: 39 A N V m ll 2 f + 2: f + 2 'f + f.. k k kr kr kr...m kr...m 1j...n where the summations are carried out over different sets of the indices. The nature of the series becomes clear if a few of the terms are worked out. fijk ‘ F11k ' (F11 + Fik T ij) + (F1 + F1 1 F k) A simpler way to write it might be _ F(1) ‘03 I f = F(z) - ZF(]) _ F(3) _ 2F(2) + 2F(1) "h 1 “+9 I - F(4) - 2F(3) + ZF(2) - 2F(1) where the superscript refers to free energies of clusters of that size. The terms in the expression for fij alternate in sign with the ..n first term always positive. This leads to cancellation of terms in the proper fashion if you simply sum the fij n' Larkin and Khmelnitskii assume a Hamiltonian of the form + H - :3 Vij§1§j - pH g S, 40 where H is the external magnetic field and vij is the asyomptotic 2k R form of the RKKY interaction (i.e., V(R) = VO cos g ). Thus, R indirect interactions only are considered. Since the distribution of impurities doesn't depend on temperature, one must calculate the thermodynamic functions for a given configuration of impurities, and than average over all configurations. The first term is just the free energy of non-interacting spins given by F(]) = -NT 2n sinh(pH(S+%)/T)/sinh(pH/2T) The second term in the expansion takes the form V n 0 + 42 (1%)] F(2) . - i unv01s12s+mn 3 T(a n) 3 The physical quantities do not depend on the cut-off parameter a. The general form of the m-cluster free energy is nV m-l 170“) = -NT (‘TQJ (D Summing all such terms, LK argue that the free energy is of the form nV F = -NT<1>(—TQ- . #1). So in the range of temperatures larger than the Kondo temperature but smaller than V0, the free energy does not depend on the three parameters n, H, T, but only on their ratios. This is essentially 41 (95) what Souletie and Tourner argue using only RKKY interactions. The important point to be made here is that for H >>T, the series represents an expansion in terms of the parameter nvo/ H. Thus in very strong magnetic fields and low temperatures we have a way of estimating the magnitude of V0. Using the expression for F(z) and making the assumption H >> T one can derive v . 2 -92-(zs+1)] . m '-’ 11$N[]"§ pH This expression is applicable when the second term in parenthesis is small. This approach allows us to try and estimate the magnitude of V0 from magnetization data at high fields. Larkin and Khmelintskii also derive an expression for the magnetic susceptibility at high temperatures in weak fields. They find(33) X_] - 3(1+csnv0) - Nu25(5+1) gy_ _ M y2 [1 Z y 1 n 11 you: 0.. 25 - k z J(J+l)(2J+l) J=O ye :z E: II ZS -yeJ S(S+l) Z (2J+l)e and ed = J(J+l)/2. J=O N A ‘< v II 42 The numerical values of CS are equal to: 0.984 C(Z) 1.31 C(4) C(1/2) c(5/2) 0.667 C(l) = 0.85 C(3/2) 1.17 C(3) = 1.25 C(7/2) 1.09 .36 II _3 Thus, for spin 1/2 T* = Can = 0.667 nV . 0 0 For large S, T* = 8/95an0 nS. This work is in contrast to Klein's mean field approach where the Curie-Weiss constant is put in by hand. Thus LK find a to be a linear function of the concentration. (5]) which is It is also worth mentioning the work of K. Matho in some ways an extension of the LK approach. For example, LK evaluate 62 only in the limit uH >> T, except for S = 1/2 when they do the calculation analytically. Matho extends this work by evaluat- ing F(z) in the limit T << T1 but for arbitrary quantum spin S and magnetic field H. According to Matho the second virial contribution is given by Bf(2) = - gidWD(W)£n{Z1z(BN.z)/A§(Z)} when Z12 is the partition function of isolated pairs coupled by an exchange energy W. The distribution of couplings is given by D(W) and is assumed to be symmetric. Matho assumes D(W) = 111/W2 for |W| 5_kT and D = 0 outside. The high energy cut-off is related to the second moment of the full RKKY distribution by 0(2)/W1 = 2T1. Matho argues that the exact formula for the Curie-Weiss temperature 1; of LK is given by kT: = cW105 where eS is given in terms of the 43 trigamma function _ -1 1 A O 1 5 0 BS - EFTSTSFTT'jZA{w( )(% + -l§;§-l) ‘ ¢( )(5 ‘l-%§—29 where yi.A = 1" ('Xj.x) A (z) 1/j ‘-l ' 2 -l ' . . xj’A(Z) = (-%;T§7-9 e1"( A )/J for j = 1, ...ZS, A = l, ... where the Aj(z) are the coefficients of the polynomial P(x,z) given by 25 . . P(X,Z) = Z Ao> Tg for large spin values. This is $9 (34) in agreement with other authors who claim that in order for the spin glass state to be stable, T must be greater than 9. If one 59 looks ahead briefly at our QuagtMn data, one finds this to be nearly the case for the disordered state alloys but is certainly not the case for the ordered state alloys if we interpret Tg = leNl. (See 44 Figure 5). One might argue that those results are based on an expansion of the free energy which up to now only includes pair contributions. How good an approximation this is depends to some extent on how well the near neighbor spins effectively screen out the more distant spins. Finally, Matho(35) points out the relationship between virial and high temperature expansions. He argues that the high tempera- ture expansion is valid only for temperatures higher than all the bond energies. For T >> T], the second virial corrections reduce . to the UT expansion with the moments 0(1) and 0(2) as coefficients. Here T1 is the high energy cut-off in the exchange distribution. For T << T], the corrections depend only on the central amplitude W1 through the parameter CW1/kT1. 45 _ u _ - >— _ d _ ...N I. D CCU—Hung: ow >.nvnvmm o OcflaegaOm ‘0 00w .3 l x: 0 mm «.1. b g D e. T m I. my D a..1 b 00 D N .1 p D U _ _ _ _ _ p _ _ b n e a m .m. :2 3 3 a 3 29:6 m. «we . a1 FZ - -X'VO1'H 3'2- It is advantageous to measure the magnetization of a material in a uniform field because such a measurement makes no requirement on the field dependence of the magnetization. Force measurements which require a field gradient are limited to situations where the magnetization either varies linearly with the field or is field independent. If one has the situation m = x(H)H, then the force on the sample will also include contributions from the derivatives ofxwith respect to H. In the GuOy method the magnetic field is uniform. The spatial inhomogeneity arises because one uses a long sample which is only partially immersed in the magnetic field. If the sample has its extent primarily in the z-direction then one can easily show that _ . 2 2 F2 - - 2XA(H2-H") 48 Figure 6. Basic Techniques for Bulk Susceptibility Measurements. AFM = Alternating Force Magnetometer NCPM = Null Coil Pendulum Magnetometer VSM = Vibrating Sample Magnetometer VCM = Vibrating Coil Magnetometer MIB = Mutual Inductance Bridge FM = Fluxmetric Magnetometer SQUID = Superconducting Quantum Interference Device 49 Force Methods Induction Methods (DC) (AC) VCM ES G 0 Eq F U yd VSM gu ara ay Ml B E I 5 (1 AFM NCPM. PM + Figure 6. Basic techniques for measuring bulk susceptibility. 50 where A is the cross-sectional area of the sample. The three techniques in the middle box straddle the ground be- tween force and induction techniques. The fluxmetric method is usually an induction approach but doesn't quite use all the power of other AC techniques. The idea is to immerse the sample and a pick- up coil in the magnetizing field so that the dipole field from the sample is coupled to the coil. Then the sample is gradually removed from the vicinity of the pick-up coils and the change in flux in- duces a voltage in the pick-up coil. This hduced voltage can be integrated by using a Ballistic galvanometer or an electronic integrator. Modern systems measure the flux change itself with unprecedented accuracy by using SQUID electronics. More on this later. The next two methods to be described are fundamentally force methods, but show some AC characteristics and also demonstrate null techniques. The first is the null-coil pendulum magnetometer used by C. A. Domenicali. In this apparatus, the sample is placed in an inhomogeneous magnetic field inside a “null coil" of known geometry. The current through the coil is adjusted so as to make the net magnetic moment of coil plus sample equal to zero. The coil is rigidly attached to the lower end of a sensitive pendulum and the null position of the pendulum is observed with the aid of a projec- tion microscope. This approach has two distinct advantages over conventional force methods, (1) the necessity for measuring the force on the sample is eliminated and (2) knowledge of the magnetic field gradient is not required. If the effective field gradients 51 acting on the coil and sample are equal, then the return to equilib- rium implies m = iAn, where i is the current through sample = mcoil the coil, A is its effective cross-sectional area, and n is the number of turns on the coil. If the effective gradients acting on the coil and sample are not equal, then even if the forces are equal, the magnetic moments are not necessarily equal and opposite. There are two ways around this problem. (1) the pendulum magnetometer may be calibrated with a standard of known magnetic properties and the same size and shape as the samples to be studied. (2) One may try and establish a magnetic field geometry in which the effective gradient is the same for sample and coil. The first method is in general much simpler. In the case of our VSM, both the coil and sample were in a uni- form magnetic field, so this problem did not arise. Nevertheless, we chose to calibrate the system by using a similar standard specimen. One can also worry about the small magnetic field generated by the null coil itself. This field will be negligible except for measure- ments in very small fields. This was not a problem in our measure- ments since the VSM typically operated in the 5 KOe to 9 KOe range. One can, however, make this correction arbitrarily small by using two concentric null-coils connected in series opposition. A more complete discussion of null determinations of magnetization and its implications for cryogenic measurements can be found in Reference 36. In general, the introduction of the sample into a uniform field will produce a nonuniformity in the field. This difficulty can be overcome if the sample is an ellipsoid of 52 revolution,for then the demagnetizing field is uniform. Even in this case there will be images induced if the magnet is an electro- magnet, and the image will change with changes in the permeability of the pole faces. Since our samples were essentially cyclindrical with a diameter of approximately 2 mm and a length of about 14 mm, ’there will exist a distortion of the field due to both the images and the non-uniformity of the demagnetizing field. However, if a Domenicali coil is wound around the sample, it is easy to show that the B field at every point in space will be unchanged and the mag- netization will be uniform throughout the sample and directly pro- portional to the current per unit length in the Domenicali coil. Another advantage of null coils not directly related to the measure- ment process is that it gives the experimenter a way to create a variable magnetic moment at the sample location which can be very useful for checking general system operation and for fine tuning or alignment just before a series of measurements is to be taken. Another variation of the force method is the so-called alter- nating force magnetometer.(37) The principle behind this instrument is to use AC techniques to improve the sensitivity beyond conventional force methods. The idea is to subject the sample to a small oscil- lating magnetic field gradient in addition to the steady uniform field which magnetizes the sample. The sample is held in position on the end of a rod which moves back and forth horizontally because of the alternating force on the sample. This motion is detected and is the desired signal. This instrument can also be designed to operate in a null fashion by putting a current carrying coil on the other end of the rod immersed in its own field gradient. The 53 direct current in the coil is adjusted until the vibration detector indicates zero. Let's turn now to a discussion of some of the induction methods. An interesting example is the vibrating coil magnetometer.(38) Here the sample is fixed in a uniform magnetic field and some signal detection coils are vibrated back and forth with respect to the sample. This is in contrast to the VSM where the sample is vib- rated and the coils are held rigidly. The advantage of the VCM is its greater flexibility in the setting of sample environment condi- tions. Temperature variation and control are somewhat easier, as compared to a VSM, and measurements under high pressure for example, are possible only in a VCM arrangement. The principle difficulty in producing a practical VCM is the elimination of the signal in- duced by curvature in the magnetizing field. Since our pressures were never to exceed one atmosphere absolute, the VSM was the logical choice. At this stage I would like to briefly discuss the mutual induc- tance bridge<39) technique since it was used to experimentally determine some of the spin-glass freezing temperatures. The basic idea behind the mutual inductance technique rests on the fact that the mutual inductance of two concentric solenoids depends on the magnetic characteristics of the material within the solenoids. The change in mutual inductance upon introduction of a sample of mass m is given by A(mutual inductance) = K m X 54 where K is a constant characteristic of the coils. In this particular case, the susceptibility coils used consisted of a primary and two oppositely wound secondaries. These were wound so as to make the total mutual inductance approximately zero. The change in mutual inductance produced by inserting the sample into one of the secon- daries was measured with a Cryotronics Model 17B mutual inductance bridge operating at 17 Hertz. Since the coils were not exactly balanced without a sample, and the balance was temperature and field dependent, readings with the sample in and out of the coils were taken and subtracted. Since the operating temperature range of this system was limited to the interval 1.2°K to 4.2°K, a way to measure ng for those samples where ng > 4.2°K had to be found. One approach would have been to measure hysteresis 100ps as a function of temperature and look for the onset of irreversibility. This is a long, inaccurate and tedious process. The best solution was to gain access to an S.H.E. Corporation VTS 800 Series Susceptometer. Since this is a squid-based device, accurate data could be taken in low dc fields (=25 gauss). These fields were low enough that the spin-glass transi- tion was not severely rounded out. Nogata and Keesom(4o) demon- strated that low field dc susceptibility was a practical way to determine T5 The instrument developed by S.H.E. Corporation is g' essentially a state of the art version of the fluxmetric magnetometer. Instead of using a ballistic galvanometer or electronic integrator to integrate the voltage induced in a pick-up coil, 3 squid sensor is used. A rough block diagram of the system is shown in Figure 7. 55 m.” Omo — “if! u - - - -.. _ ... a: 0—1 ..--1 mincxm V. _ n “ two T1..-1...........i...!...:.-.. 11111 ......11 L 1 (HI — mwcafi— 1%. k. Mazda ummma Becamnoamams. L: < a] 56 The sample and its holder are suspended on the end of a long titanium ribbon and they are cycled back and forth through two pairs of Helmholtz coils. The flux change caused by the movement of the sample induces an EMF in the coil pairs. This causes a current to flow through the signal coil of the squid. It is the job of the squid and associated electronics to detect this current and, if operated in a feedback mode, to keep it approximately constant. This is known as flux-locked loop operation. The current induced in the signal coil causes a flux change in one part of the squid cylinder. It is this flux change which is detected in the follow- ing manner. (For a good discussion of squid devices see Reference 41.) The RF oscillator supplies an RF signal to the squid tank circuit. This generates an RF supercurrent through the squid point contact. If the RF amplitude is great enough, the critical current of the weak link will be exceeded and it will go normal. This allows a readjustment of the flux between the two holes in the squid such that the point contact again goes superconducting. It should be noted that this RF induced transition takes place about the average DC flux in the squid cylinder which is constant since the rest of the squid cylinder is always in a superconducting state. Thus, if the average dc flux within one hole of the squid body should change due to the presence of current in the signal coil, a flux change will also appear in the hole containing the RF coil due to the constant shuffling back and forth of flux. Now the average dc flux in the RF hole is slowly modulated (at approximately 1000 Hz) by an audio signal introduced via the same RF coil. Thus the ampli- tude of the l KHz frequency component of the RF detector is a ' 57 measure of the average dc flux in the RF hole of the squid which is constantly communicating with the average flux in the signal hole of the squid. Thus the lock-in output is a dc voltage directly proportional to any small dc flux change in the squid body. To ensure that the change experienced by the RF coil is small, a dc current is fed back to the squid signal coil via a mutual induc- tance. The voltage caused by this current flowing through a standard resistance is the actual measured quantity. It varies linearly with the flux change in the signal coil. Flux changes as small as 2 x 10'7 gauss-cm2 can be detected. The dotted line indi- cates that many times this feedback current is fed back directly to the RF coil. 8. General VSM Considerations If one "inverts" the alternating force magnetometer, one ar- rives at essentially the vibrating sample magnetometer. That is, rather than driving the sample with an alternating magnetic field gradient and measuring the vibration of the rod, one could drive the rod (with say a loudspeaker) and measure the magnetization of the sample by measuring the induced voltage in a system of pick-up coils mounted in the vicinity of the sample. That is precisely what is done and Figure 8 shows the basic idea behind the vsn.(42) Here the system is designed to operate in a null comparison mode. The signal derived from the oscillation of the reference sample in the upper set of coils is first phase shifted so as to be 180° out of phase with respect to the sample signal. Then it 58 'Power Flef Amp 030 " Spkr .. --------------- cc----¢-1 ...-- ....... . ..... dun-q -------- [Dividerj ‘— - . Recorder A Null Detector _O-Scope Figure 8. Conventional Foner style VSM. 59 is attenuated with a precision divider and mixed with the sample signal. The combined signal is fed to the primary of a well shielded audio transformer. At balance, there is no current flow in the transformer primary. Any inbalance is amplified by a high- gain narrow bandpass amplifier and fed to the input of a phase sensitive detector. The reference signal for the phase sensitive detector is derived from the oscillator used to drive the loud- speaker. Balance is achieved when the lock-in output reads zero. The magnetic moment of the sample is proportional to the reference voltage divider setting. Because it's a null measurement, the divider settings are independent of vibration amplitude, frequency and the electronic circuitry following the mixer. One could also implement an automatic null balancing approach by taking the dc output from the phase sensitive detector and using it to drive a coil which replaces the permanent magnet (see dotted line in Figure 8). The machine described above is essentially the design of the magnetometer used in the MF study. The magnetometer used by the author was similar, but significant modifications were made (see Figure 9). Again, a null measurement was made, but in this case a Domenicali type coil was used and the signal was nulled out at the sample. This eliminates the need for phase shifters, dividers, and mixers. It also greatly reduces image effects. Of course, this approach has some drawbacks. Often times fairly large currents (mlOO mA) would be needed to null the sample moment and this large current caused problems in two ways. Since the Domenicali coil has a finite resistance even at 4.2°K, the 6O 0oémq m_© >30 .90: 0:3 m0rq 0 m W117 ( 1+ 03V V r\\\\ Y\\\11 1; 00300 WWW I <04 . _. 4 dream: mm_m< [F .30 m. mzfimq I Do" - Ul at the detector output due to Vn(t) at the detector input is T/2 T [f Vn(t)dt -/ Vn(t)dt] o T/2 an r TT/(D . 211/u) . = ‘2? Lo 5171(an + on)dt -11 s1n(nwt + 1tn)dt] /w -u—4 = n 63 The integral is written in two parts because of the sign reversal introduced by the demodulator during the interval T/2 < t < T. Therefore, A V eJl -.." n“ [1 ( 1) ] cosmn Since the filter is linear, superposition applies, and the average of the combination of N signals harmonically related is (-1)n] n = l,2,3,....N Thus, for n = 2, = O regardless of the phase 32. Once the signal frequency is chosen, in a null measurement it is the job of the electronics to detect amidst all the noise back- ground, the presence or absence of the known signal. In a system like the one used, the ac signal from the detection coils is first applied by a high-gain, narrow band-pass low noise amplifier. The amplifier gain is of the order of 106. The amplified ac signal is converted to a dc signal by the synchronous demodulator. The demodulator is driven by a reference voltage which maintains a constant phase relationship with the input signal voltage. Typi- cally, the dc signal is then fed directly to a low pass filter. In our case, provision was made to break that connection at certain times. This will be discussed in more detail later. The output of the filter is the average value of the signal which appeared at the output of the demodulation. It is the average 64 value that is of interest, not the peak or RMS value. For random noise signals the peak and RMS values depend on the amplitude of the noise, while the average value tends to zero as the averaging time increases. This affords another means of distinguishing between signal and noise. The detection coil signal, which may be only a few nanovolts, must first be sufficiently amplified in order to drive the input amplifier of the lock-in. I will not give a detailed description of the electronics, but will mention some general practices relating to low level signal amplification. It is generally known that for an amplifier con- sisting of several stages in cascade, the overall noise figure (NF = 10 log 2%;SE) is determined primarily by the noise figure of the first stage, provided this stage has a reasonable power gain. If the signal can be sufficiently amplified, then the noise generated in succeeding stages will have a negligible effect. Therefore, in the input stage, center frequency gain and low noise are more im- portant than narrow bandwidth. In our particular application, the detection coils are elec- trically floating and connected to the input terminals of a Triad GlO transformer. The transformer serves as a differential input. Any common mode voltage between the coils and the copper shield which encloses the input stage is not seen by the transformer. Problems can arise if the two input lines are not equally balanced with respect to ground. In that case, some fraction of the AC voltage between ground and the coils would be seen by the trans- former. 65 If we assume that the characteristics of the signal source (i.e., the pick-up coils) are fixed and not alterable, then the design problem is to choose the parameters of the amplifier so that the output is an amplified replica of the input, and the output signal to noise ratio is optimized. If we desire that the amplifier respond to the signal voltage generator in such a way that it is independent of the source impedance, then we must require the ampli- fier input impedance to be much larger than the source impedance. The optimum signal to noise ratio is achieved by properly matching source-amplifier impedances with the Triad transformer. The transformer secondary is connected to the control grid of a high-gain grounded cathode pentode. The plate voltage output is passed through a 60 Hz notch filter and further amplified by two triode stages. The narrow band section of the amplifier is simply an operational amplifier (White Model 212 A) with a 33 cycle Twin- T rejection filter in the feedback loop. This is followed by a cathode follower stage to lower the output impedance. The final stage is a low pass filter with a frequency cut-off of ~100 Hz. This signal is now appropriate to send to the phase-lock-amplifier. This same signal is mointored on an oscilloscope to assist in rapidly finding the proper value of null current and to keep watch on the signal from the magnetometer. 0. Signal Detection Coils The detection coils employed in our VSM were #986 Miller air core RF chokes configured in a standard four coil Mallinson array. 66 That is, all coil axes are parallel to the field and each pair is connected in series opposition and both pairs are connected in series opposition. The question of which coil size and configuration is optimal is very complex. Some degree of experimentation in this area was carried out by the author. A description of that work and a derivation of the induced EMF for a given coil-sample configuration is presented in Appendix A. E. Field Noise One of the major experimental problems facing users of VSM's is the ambient magnetic field noise. This problem was also given serious attention by the author. When the ambient magnetic field suddenly changed value, some voltage was induced in the pick-up coils because they are not perfectly balanced. This voltage impulse would eventually work its way to the tuned circuitry in the pre- amplifier. ’ The amplifier would ring with a sizeable component at the tuned frequency. Even though its phase might be different from the signal of interest, and it presumably decayed away experi- mentally, it would often overload the lock-in amplifier. The original idea was to build a precision clipper and insert it just before the tuned circuitry. If the signal of interest was 6 say 5 nV, one would have to amplify it by 10 to get into the 5-10 mV level before you could reliably clip it. There are commercially available voltage comparators with 1 mV sensitivity and repeatability.1 1Calex Model 540 Voltsensor (Calex Mfg. Co., Inc. Pleasant Hill, CA). 67 The next step was to investigate the effect of the clipping on the frequency spectrum of the signal. This was done by assuming the signal of interest to be c05wt and the added signal due to the ringing dt to be of the form ce' cos(mt+w). In particular, if one assumes a signal of the form 0’7tcos(10t) V(t) = colet + Se' so that the ringing is in-phase, and the burst is initially 5 times the signal amplitude and has a time constant of 1.4 seconds, the .frequency spectrum is as shown in Table 3. Also shown are the same values if the signal is clipped with an amplitude of (1.2). Also -.— ...... shown are the results for the same signal with the noise pulse shifted by l radium relative to the signal. One can see that for the in-phase noise signal, it appears as though the true measurement signal has more than doubled in ampli- tude. If the waveform is clipped with an amplitude of 1.2, the in- phase signal appears to have grown by only 17%. In both cases the quadrature component is very small. From Table 2 the in-phase com- ponent of the “true“ signal seems to grow by 59%. The clipped version of this shows only a 2% increase. However, in both cases, there is a sizeable quadrature component. This isn't a problem if the lock-in phase is properly adjusted (see Appendix E for some dis- cussion of this point). Overall it appears that some use might arise from such a precision clipper. However, about this time, a simpler and better idea was devised. It really doesn't matter Table 3. 68 Quadrature and In-Phase Components of Clipped Sine Waves. V(t) = cos (lOt) + te‘0-7t cos (lOt) vclipped cosine sine cosine sine Freq. coeff. coeff. coeff. coeff. 0 0.018 0.0 -0.0003 0.00 10 2.15 0.038 1.17 0.0020 20 0.031 0.102 0.0004 0.009 30 0.026 0.055 -O.20 0.0013 V(t) = cos (lOt) + 5e'0°7t cos (lOt + l) Vclipped cosine sine cosine sine Freq. coeff. coeff. coeff. coeff. O -0.056 0.0 -0.013 0.0 10 1.59 -O.92 1.02 -O.44 20 0.061 0.051 0.014 -0.009 30 0.032 0.029 0.043 0.14 69 if the tuned amplifier rings (provided it doesn't ring too long) so long as the final data readings don't reflect this fact. So the idea was to monitor the output of the tuned amplifier with comparators. If a noise burst suddenly appeared, then we would race ahead of the signal and disconnect the demodulator output from the integrator-low pass filter input. The capacitors in the low- pass filter are quite capable of holding their potential until the system reconnects. This scheme allowed one to use the full ampli- fication and narrow bandpass characteristics of the tuned amp, and did not require any modification of the incoming signal. This idea was considered feasible and the circuitry was constructed. A schematic diagram of the device (called the "interrupter") appears in the Appendix B. The operation of the circuit is straightforward. The signal passes through a non-inverting variable gain follower and is ex- amined by two comparators in parallel. If the absolute value of the signal exceeds a set amount, a 555 timer configured as a one-shot is triggered. The positive output pulse from the 555 eventually turns off the transitor which is keeping the reed relay closed, and the relay opens breaking the connection between the demodulator and the integrator. F. Holder Design Next to the detection coil design, the construction of the sample holder is probably the most crucial design element. Many holders 70 were built and tested during the course of this work, but we shall discuss only those three which were used to take most of the data in the thesis. (The oldest of the three incorporated the following features: (See Figure 10) (1) A bifilar wound platinum resistance thermometer with a room temperature resistance of approximately 10 ohms. This was wound on a thin aluminum cylinder. Aluminum was chosen because many magnetic impurities (especially iron) do not retain their magnetic moments in aluminum. Aluminum is also easily machined and etched. The platinum wires were insulated from the aluminum (with GE varnish and cigarette paper. (2) Next a Domenicali coil consisting of approximately 250 turns of #38 copper wire was wound. (3) Then approximately one inch of Evanohm wire (resistance =30 ohms) was twisted together and varnished to the holder and con- nected to a twisted pair set of leads. (4) Finally a AuFe (7%) versus Chromel thermocouple was at- tached directly to the holder. This thermocouple was quite long as it had to run the full length of the cryostat and exit through a wax seal and then move on to a liquid nitrogen reference bath. The other two holders were slight modifications of this design. The second holder was developed to measure hysteresis. Because the field strength was only several hundred gauss, the contribution from the holder had to be made as small as possible. This was accomplished by winding a single turn of wire on a plastic form, Nun CoH Current Source 71 AuFe ' Chromel C 1 Evanohm HM: I PRT__ 'T‘HLf __~ I [1 Ratio 11 D\/M RS .;3 JB Temp Control ...—J NanovoH Source Figure 10. Holder design. ——V. 72 varnishing the coil, and then softening the plastic and removing it. There was no aluminum form. The structural integrity was provided completely by the hardened GE varnish. In addition, there was no heater wire or thermocouple. AuFe (7%) is also a spin-glass and can show hysteretic effects. The thermocouple was left in place and taped higher up on the cryostat so as to still be useful in precooling. All the hysteresis data was taken at 4.2°K so the heater was not necessary. Of course, this restricts you to samples whose spin-glass temperatures are somewhat greater than 4.2°K. The last holder to be built was basically the same as the first except the platinum resistance thermometer was left off. It was decided that most of the work would be done in the 4.2°K to 77°K temperature range, and that the platinum thermometer would not be required. The vycor rod which transmitted the speaker vibration was re- duced in diameter under heat and given a flat end surface. The holders were then simply varnished onto the end. In addition, pre- vious holders employed a set screw to hold the sample in place. This meant making the holder wall thick enough to support at least several threads of the 0-80 variety. By utilizing silicon grease and eliminating the set screw, the metal form could be made arbi- trarily thin. G. Cryostat Design The basic mechanical layout of the cryostat is shown in Figure 11. One of the fundamental mechanical concerns in designing a VSM . 73 BaHast Leads ~\ U‘ \x F“ coHs 1r 0 Transfer Tube |‘\ 1“ Magnet Fl'Slure 11. Basic cryostat layout. 74 is to insure that no mechanical coupling exists between the detec- tion coils and either the cryostat or the pumping system. The vacuum pump was crudely shock mounted and the line going to the mag- netometer was at one point firmly anchored to the laboratory wall. All of the final vacuum connections were made with thick walled rubber vacuum hose flexed in such a way as to damp out vibration. Lead bricks were placed on top of the magnetometer to minimize any possible recoil. One should always check to see that the dewar does not contact the magnet pole face. Another point of general construction interest includes the use of stainless steels in cryostats. Their usefulness stems from their high strength and toughness as well as thermal insulating properties even at 4.2°K. The magnetic properties of stainless steel are probably less well-known. The austenitic phase of the A151 300 stainless steels is non-ferromagnetic. However, these steels are metastable at temperatures below room temperature and can be trans- formed to ferromagnetic martensite either by cooling or cold work- ing. A study of the martensite transformation due to cooling alone was done for a wide selection of the A151 300 stainless steels in Reference 43. They found that of the grades 302, 303, 304, 308, 310, 316, 321, and 347, only the 303 and 304 steels showed any transformation from the austenstic to the martensitic phase after repeated thermal cycling between room temperature and 77°K. Further work was done by Larbaleistur and King.(44) They found ferro- magnetic behavior in stainless steeptypes 321, 347 and 310. The ferromagnetism in the 321 and 347 steels increased in magnitude with 75 each additional cooling to low temperatures. Both types 321 and 304 were used in the construction of the original magnetometer. It was used to construct the inner dewar and the vibration guide tube. Thus, it became clear these were potential troublespots. Indeed, when samples of these steels were tested in the magnetometer, they were extremely magnetic at low temperatures. Thus, two modifications were made. The lower half of the guide tube was replaced by aluminum tubing and its length was reduced. This allowed the current carry- ing wires to the holder to be attached to the rod a greater distance from the detection coils. The second modification was to replace the lower half of the inner dewar by pyrex glass using a glass-metal seal. This also allowed visual observation of the sample which is a slight added bonus. Finally, the magnetometer was able to take data above 77°K. One could either transfer liquid nitrogen into the dewar as one did with liquid helium, or one could use a small dewar built by the MSU Scientific Glass Blowing Shop. It was supported by a wooden platform which rested on the NMR coils. H. Temperature Control and Measurement The necessity of temperature control is one of the other major differences between this magnetometer and the MF system. The MF system which operated in the 77-300°K range did not employ heaters. Their method involved pouring in the liquid nitrogen and by adjusting the nitrogen gas exchange pressure, control the rate of cooling of the sample. Data points were taken in a quasi-static fashion. As one proceeds to liquid helium temperatures, there are three factors 76 which make this technique more difficult. The first is that the specific heat of the cryostat is going to be on a steeper part of the Debye curve. Therefore, less energy needs to be removed to cool the sample. This makes the process of cooling much faster. The second point is that the latent heat of 4He is 2.6 J/ml and is 161 J/ml for N2. What this means effectively, is that in order for helium liquid to collect, the inner dewar space must be at 4.2°K. Unless the exchange gas pressure is very low, the sample and holder will rapidly approach 4.2°K because of their close proximity to the liquid. Finally, the magnetic susceptibility of most of the materials studied tended to follow a Curie-Law type of behavior. This means that the rate of change of x with T is much greater at low tempera- tures than at high temperatures. The mechanical forepump in our system is incapable of pulling a hard enough vacuum to use the exchange gas technique. Construc- tion of a diffusion pump system to do this was started. Some ideas of how to insert small calibrated amounts of helium gas were also studied, but in the end, it was decided to use electronic tem- perature control. This sytem operated on a null basis. The thermocouple output was approximately 1225 uV at 4.2°K and decreased to zero at 77°K (the reference bath was liquid nitrogen) and then switched sign at higher temperatures. The method was to take the thermocouple out- put and add in series a bucking voltage provided by a Keithley Model 260 Nanovolt source. This sum was fed into a Keithley Model 155 Null detector/microvoltmeter which in turn output a voltage 77 between 0 and 1 volts, depending on the null imablance. This vol- tage served as the input to a homemade controller whose job it was to output heater current to the Evanohm heater. Its schematic is included in the Appendix. A brief discussion of the design of temperature controllers can be found in Reference 45. In that article the basis of control theory is presented including the effects of time constants on stability, and how to alter the fre- quency dependence of g(w) (the gain and frequency response of the controller) using proportional, derivative, and integral control. Proportional control is when the control signal is simply propor- tional to the error signal. Derivative control is a control signal proportional to the time—derivative of the error signal, and integral control is when the control signal is proportional to the integral of the error signal. It is easy to construct such control signals with the aid of operational amplifiers. If one adds to that the unity gain followers and inverters, the design of the controller should be fairly obvious. The overall gain of the loop is control- led by the size of the resistor placed in series with the heater resistance. It's also a good idea to place a DC ammeter in series with the heater to monitor the current being delivered. I. Typical Run The following is intended to be a rough guide to operation of the system when taking low temperature data. The details of each step vary according to the taste of the experimentalist. (l) Periodically, one should check the detection coil balance by 78 applying a 33 Hz signal to the NMR coils. One should also occasion- ally check the phase relationship between the reference magnet and the rotating flip coil. The PFC-4 manual describes the sequence of events to minimize the quadrature signal. (2) The magnet is rolled into position and noted with the plumb , bob. The holder is centered in the magnet gap and vertically be: tween the detection coils. (3) Most of the samples are rolled and cut to size so that mounting of the sample consists of simply covering it with a light layer of silicon grease and inserting it into the holder. (4) Before the dewars are put into place it is generally worth the time to check lead continuity. This includes checking thermo- couple leads, null current leads, and heater current leads. (5) The inner dewar is fastened into place, usually using some shims to keep it parallel and centered with the magnetometer driving rod. (6) The outer dewar is added and rotated so that the un- silvered strip exposes the transfer tube. Fine positioning of this dewar can be accomplished by pumping on the outer dewar airspace and gently applying pressure to the inner walls of the dewar near the top. Any contact between the magnetometer drive and the inner dewar should be visible on the oscilloscope. The output from the permanent reference magnet gives a good indication of vibration - quality. (7) Most of the electronics is now turned on. The precooling time is approximately four hours, and this gives the equipment time 79 to stabilize. Generally the FFC-4 is turned on approximately one hour before the start of data taking, and the main current supply is energized but not delivering current about 15 minutes prior to data taking. (8) The vacuum jacket of the outer dewar should be pumped out, flushed with nitrogen and pumped out again. Then both dewar spaces are pumped out with a mechanical forepump and filled with gaseous nitrogen. (9) The precooling is done with liquid nitrogen in the dewar jacket and also in a separate dewar which surrounds the tail. (10) The sample is given a minimum amount of vibration to prevent the shaft from freezing up. (11) When the cryostat is sufficiently cold ( 100°K), the nitrogen gas is evacuated and replaced with helium gas. (The inside dewar space is filled to 1000 microns and the outer dewar space is filled to just above atmospheric pressure. (12) If the system is to be cooled to 4.2°K in zero field, then one can proceed with the separate dewar in place. If the system is to be cooled in some applied external field, then the external dewar must be removed and the magnet rolled into position and the proper magnetic field maintained. (13) One can proceed to transfer helium in the normal fashion. Usually it's a good idea to evacuate the vacuum jacket of both halves of the transfer tube prior to transferring helium. If the free half of the split transfer tube should go soft, a flexible transfer tube can be used. 80 (14) As the helium transfer commences one should monitor both the temperature as indicated by the thermocouple and the end of the transfer tube inside the dewar. A good transfer typically requires approximately 5 liters of helium. (15) After filling, but before taking data, one should check the exchange gas pressure. A check should also be made that the dewar is not touching either pole face of the magnet. The vibration coupled to the coils will make the data meaningless. (16) In addition, on humid days, water may condense on the out- side of the dewar. Usually a fan will reduce this greatly. I would not recommend taking data unless the exterior of the dewar is dry. (17) The sample drive.amplitude should be increased to approxi- mately 3 VAC and checked for quality with the oscilloscope. (18) The frequency of vibration should be checked occasionally to see that it is in the center of the narrow band pass of the tuned amplifier. The phase of the sample relative to the lock-in ampli- fier should now be adjusted using the phase adjust in the reference channel. (19) The interupt levels of the "Interrupter" should be adjusted according to the ambient magnetic field noise. (20) If everything checks out, data taking can commence. Generally for data at 4.2°K a relatively large exchange gas pres- sure is used, because the nulling currents at 4.2°K in reasonable fields can be large enough to cause the sample temperature to rise slightly. 81 (21) As the temperature is raised, the inner dewar space should be evacuated to about 25 microns so that excessive heater currents are not required. Since the volume of the inner dewar space is small, small changes in the amount of helium gas present cause relatively large changes in the exchange gas pressure. This makes temperature control more difficult. For that reason a ballast tank was added to the system, and should be opened to the inner dewar space at this time. If operating properly, the temperature con- troller should be able to keep the temperature constant to within ml uV on the thermocouple output. (22) A typical data point involves writing down the null cur- rent values required to generate a slightly positive and slightly negative phase-lock voltage. Perhaps a dozen values on each side of zero is reasonable. These values are separately averaged and then one interpolates to find the value of the current that nearly produces zero output voltage. (23) When the data taking is finished, its important to remember to close the ballast tank valve. It is also a good idea to pump on the inner dewar space. If the pressure within the inner dewar should rise because the gas has warmed up, it could conceivably come out of the quick-disconnect and cause damage. J. Calibration The calibration of vibrating sample magnetometers is a very interesting topic. The degree of complexity is a strong funtion of the type of calibration desired. Two approaches will be discussed 82 The first is an absolute calibration of the instrument based on the dipole approximation. The second is the safer but less flexible comparison method. The dipole approximation is merely a statement that the detection coils are sufficiently far away that the only relevant multipole componet of the field is the dipole component. In that case, the voltage induced in the detection coils due to the changing flux is given by d + v = 3% = [mlech = xlillvocscc where X is the volume susceptibility H is the applied external magnetic field V is the sample volume w is the frequency of vibration GS is some geometrical factor due to the sample geometry Gc is some geometrical factor due to the coil geometry. In this way, if one inserts a sufficiently small spherical sample which has a well known moment in a given field (typically the satura- tion moment of a small sphere of pure nickel), then one can num- erically determine G = GSGc‘ Then the machine is calibrated as long as any sample measured produces essentially only a dipole field. Complications arise when the samples do not meet this criterion. Foiles and McDaniel(46) investigated the case of' right circular cylinders having a diameter to length ratio of :1 to 5. 'This involved two basic measurements. One was the effect of fixing the sample length and varying the diameter. The second . 83 was varying the length and holding the diameter fixed. In the first case, they found the voltage output to be directly proportional to the sample area. (The coil diameter was never greater-than 1/8 that of any coil dimension. In the second case they found that the di- pole approximation was not valid and the voltage output behaved like V a (volume)-(l + BL). Thus, if one is going to use samples in this size and shape regime, more system parameters are going to have to be determined. The easiest way around this is to use the second calibration technique which is just the familiar comparison method. Here the idea is to use a standard sample with exactly the same geometry as the unknown samples. Since the samples studied by the author were essentially of the same geometry as that investi- gated by Foiles and McDaniel, their results could be employed. In particular, one could use as a reference standard, a sample of pal- ladium which had the same length as the unknown samples, but somewhat smaller diameter. This data will be presented shortly. There is another approach which can be used if the magnetometer employs a Domenicali coil. In this case, when the axis of the cylinder is parallel to the applied field, the induced dipole moment in the sample can be cancelled by an equivalent current shell pro- duced by the Domenicali coil. Thus, the absolute voltages produced in the former method are replaced by dc nulling currents. In principle one could absolutely calibrate this instrument because presumably one can calculate the magnetic moment of a finite solenoid. In practice, we also employed the comparison method. The coil to coil distance across the gap is approximately 66 mm in our VSM. 84 Since the sample length is about 14 mm, we are certainly not in the pure dipole regime. Thus one begins to wonder if the results for varying diameter samples have the same behavior as when they were nulled out electronically. For now one must ask the question; of what form is the field produced by two concentric solenoids of finite length but different diameter? And is the detection coil system such that it only detects dipolar fields and ignores quad- rupolar fields? The Foiles-McDaniel study indicates that the coils are detecting more than just simple dipolar fields. My guess is that they are detecting quadrupolar fields also and so if two con- centric solenoids produce a quadrupole field we are still 0K. The point I'm trying to bring home is that with the nulling current technique one must have some grasp of how the addition of the coil moment changes the effective field of the sample + Domenicali coil. In particular, if different multipole moments become dominant, is the detection coil configuration sensitive to these new components? If it is not, there exists the possibility for error. Figures 12 and 13 show the room temperature palladium calibration runs for the two holders used to take the data shown in this dissertation. The conversion of raw data to useable numbers proceeds as follows: m Ix A s x s n x H s X = I. o — o I — o X (30]) A mx I: 1is Hx A where ms, mx are the masses of the standard and unknown, respectively; I:, I: are the nulling currents; 85 1* l I I 6 a. Room Temp. n5— magnet and 2" voice coil capable of using 70 watts RMS was decided upon. Its frequency response was 60-9000 Hz. The plan was to drive this speaker at 2100 Hz with a Krohn-Hite Model UF-lOlA ultra low distortion power amplifier. Since the induced emf varies linearly with frequency, one might expect an increase of a factor of three from the frequency change. A sample to detection coil distance reduction of 33 mm to say 19 mm gives a signal VI'IIE' 1 '1 88 coils l vacuum Figure 14. Possible future VSM. #03100: 89 increase of (5.2). A reasonable noise reduction of (3/16)/(ll/l6) = 0.27 could reasonably be expected. This leads to an overall signal/noise ratio improvement of about 60. There may, however, be a serious problem with this design. A convincing technique for pre- venting detection coil motion was not available. Indeed, that may be a fatal flaw with this design. Another advantage to this design is the reduction in thermal noise due to the cooling of the detec- tion coils. It is also worth noting that other attempts to design a VSM which is insensitive to field noise and microphonics have been made. See for example Reference 47. IV. SAMPLE PREPARATION AND RELIABILITY A. QpMn 1. Preparation The preparation of the QpMn samples was accomplished in a straight forward fashion using a Lepel induction furnace. A graphite crucible served as the active heating element. Inside the graphite was an alumina crucible which contained the copper and manganese. Both crucibles were placed inside a vycor glass holder which fit into the induction coil windings. The crucible and glass were baked in an atmosphere of less than 10'4 mm Hg. Prior to melting, the copper was etched in 50% nitric acid and the manganese was etched in acetic acid followed by nitric acid. The appropriate masses were weighed on a Mettler balance. The copper and manganese was induction heated in approximately 1/3 atmosphere of argon to approximately 100°C above the melting tem- perature of the alloy. After being reasonably certain the mixture was homogeneous, it was swiftly poured into a copper chill cast mold. The copper mold was lightly etched in nitric acid before use. The melt is quenched very quickly due to the sizable mass of the copper mold. The alloy was remelted and repoured usually two more times to insure a good mixture. The resultant alloys are probably quite close to their nominal composition. 90 91 The major uncertainty arises from the copper which evaporated and traces were found deposited on the vycor glass holder. Some uncertainty may also be due to not all the contents of the crucible leaving the crucible. Typical numbers for a 5 gram alloy would be a loss of about 0.04 grams. If this is attributed to copper, the deviation of manganese concentration from nominal is probably less than 0.1 at %. . The quenched samples were then rolled into approximate cylinders with diameters of about 3 mm. They were also cut in 14 mm lengths. A final etch was performed before the samples were removed. In addition, some of the samples were annealed in order to ascertain whether or not cold working had any effect, and also to check for possible clustering effects. A typical anneal consisted of first sealing the samples in vapor tubes at approximately 1.5 x 10'4 torr. Then they were baked at approximately 600°C for approximately 30 hours and then cooled at 50°C per hour until the samples reached 450°C. Then they were extracted from the furnace. A check of sample masses before and after annealing showed virtually no change (<0.0005 gm). 2. Magnetization vs. Field After the CpMn samples were prepared, two basic tests were made to check the quality of the samples. Our principal concerns were whether or not the Mn atoms had gone into solution in a uniformly random fashion, and whether or not the cold working caused by the 92 rolling process had any effects. Two samples at each concentration were made. One would serve as a control sample and would not be further modified. The second would have its susceptibility measured as a function of temperature, both before and after an anneal. It would also have its magnetization measured as a function of magnetic field at 4.2°K. The magnetization of CuMn (2 at%) and CuMn (3 at%) - annealed samples vs. field are shown in Figure 15. Now the mag- netization of weakly coupled moments can be expected to exhibit a magnetization curve based on a Brillouin function: 1M5= NngB BJ gJu H . (-E—%—). By going to low temperatures and high magnetic fields, we 3 . can best look for evidence of possible cluster formation.. At 4°K and 9000 gauss,3%§%E-3 (0.60). Figure 16 shows a plot of 32(x). The linear portion of B2(x) extends to just above this value, so we would expect M to be approximately linear in H. The plots in Figure 15 are linear within experimental error. The plots shown, incidentally, are the "worst case" possibilities. One would expect clustering, if it were to occur, to be present in the most concen- trated, annealed samples. 3. Annealed vs. Quenched Behavior The second test was to examine the temperature dependence of X for annealed and cold worked samples. Figures 17-19 show plots for annealed and cold worked 1%, 2%, 3% Mn samples. The results are summarized in Table 4. The e-values are essentially independent of heat treatment within experimental error. There may be a slight hint of increasing values of A0 for increasing concentration. 93 l 1 11 l l l O ... oCuMn(3%ann.) . oCuMn(2%) ’o? . 'E 1? r- ‘3 3 o C .9. '- m . N O ‘5 C 01 m o z .. L- r ‘7 l I _l I l l O 2 4 6 8 10 Magnetic Field (KOe) Figure 15. CpMn (3%) - annealed and CpMn (2%) - M vs. H. 94 p- «macaw do. mad.do=¢= «cannaos «o1 m0dz m u N. ml— 95 l l 1 F 22 1- 20 __ a annealed o quenched 18 - 6 h 3‘. 16 - ..n to a \ O 2 CE) 14 1- \ o :3 S :1 12 l" a 10 - a a 8 n- O 6 1- O a 4 - a? 1 I L 1’ o 20 4o 60 Temperature (K) Figure 17. CuMn (1%) quenched vs. annealed behavior. 96 «macaw 0m. X'1 Cemu/mole/at.%) . _ _ _ 10 w a no I b mzamfima L o acmaosma wJ D O D 8 I o l D O in I. w an o m r. o a; P O . D o. D o ‘ T 1' _ _ _ _ _ b no ac mo 403003340 23 neg: Away acmznsma 0.5 at.%, the Curie-Weiss constant is positive and proportional to c. We would like to have answered the question, does the a dependence enjoy an interesting and perhaps drastic reversal of sign in the region 0.05 at.% < c < 0.5 at.%? 103 104 The data for several CpMn samples is shown in Figure 21. There are several features worth noting about this data. (1) the greater the concentration of Mn impurities, the higher the temperature at which the maximum in X occurs. Since this curvature is due to the rounding of a spin-glass cusp due to the large magnetic field, this is in accord with other experimental work. Incidentally, other workers report a dependence of the cusp position on the magnitude of the magnetic field. We did not look into this. (2) Some of the data lines are continuously changing smooth curves. The implica- tions here, of course, are that we have not gone to sufficiently high temperatures. Some workers would argue that one should be at best at 5 Tf (Tf is the freezing temperature) before one attempts to use a Curie-Weiss analysis. But work at high temperatures re- quires an extremely sensitive magnetometer, since the signal falls off as l/T. It is also advantageous to work at small fields, es- pecially if the spin-glass transition is being examined. The Curie-Weiss constants and effective moments are shown in Table 5. The parameters a and Peff are in the realm of expected values, but show clear difficulties. The more or less commonly accepted range of values of Peff for Mn in copper is 4.9 us to 5.1 “B' Our observed values are clustered around those values, but the devia- tion seems a little extreme. The average Peff for the five samples is 5.15 pa. The e-values are also perhaps a bit scattered. A comparison of our data with that of other workers is shown in Figure 22. One should keep in mind, however, that the Mn concentrations are estimates based on the amount of material used in preparing the 105 . . m . m u m NUT .‘\h_* 03* o o ..x I max 0 11. %~ M.u_l 0 our . o . o t D a I mm o m_ o b .o m ‘ml. . D o ' w e a a a. r. m 0 I p e o 0 r1. ..c I o I b ... . 4w . s x D o o u D D D N a... . . u o o .u em 0 O o e r . - p, . . . . .O NO we 5c ma GO NO .4...: 0.0..9 ...q.u mam— mdmcxm Nd. mm:: 1 *:[)C)53 o 3L4-96C355 8 2t!- 0 o r‘ a 5? #:24- A 0 (U D \ 2 O EE2CDF- 9 \ :3 a5, 2 Ujem O 'r a >< A o 12 r- 0 6 a 8.. a a I 4— A o '1 I L L 1 L O 20 4O 60 80 100 Temperature (K) Figure 29. CuPd§l7an - inverse susceptibility vs. temperature - Squid. 120 10.3 ”8 (for CuPdMn (l% 0S and 3.4% 05)) (see Table 7). Figure 30 shows some typical low field X vs T Squid data that was used to determine some of the freezing temperatures. The actual concentra- tion dependence of TS and e for the CuPd§l7)Mn alloys is shown in 9 Figure 3l. As one can see, the concentration dependencies are not so clearly established as in the QuagtMn case, but the two basic trends seem to reappear. That is, all e's are negative with [elos > lelDOS and (ng)Dos > (ng)05' These are the same trends displayed by the QuafitMn samples. 0. Hysteresis Study Only a very slight beginning into the study of the hysteresis behavior of our alloys was made, but the results are quite interest- ing. As a first check of our ability to make reliable low field 'hysteresis measurements, we examined one of our guMn (2 at %) samples. The hysteresis curve for a sample cooled in -4 kilogauss is shown in Figure 32. It agrees quite well with the general sort of behavior other investigators have found. It has a width of ap- proximately 320 gauss and a displacement of about l95 gauss. The magnetization reversal at approximately 350 gauss is quite sharp. Similar measurements for QuafltMn (3%) OS and DOS are shown in Figures 33 and 34 respectively. The ordered state data exhibits no sharp changes in magnetization and very little hysteresis. The width of the hysteresis loop is only about 40 gauss with a displacement of only 20 gauss. This is in many ways very similar to what one would see for CuMn if the hysteresis loop was measured at a relatively 121 Table 7. Experimentally Determined Parameters for Ordered and Dis- ordered CuPd§l7)Mn alloys. Alloy Peff (118) 6C (K) ng (K) Disordered 0.22% Mn 0(1) 21.25 0.7% Mn 5.23 -0.59 1.0% Mn 4.99(2) -3.03 5.5 Ordered 0.22% Mn 4.88 -0.065 <1.25 0.7% Mn 5.55 -7.5 3.3 1% Mn 5.02 -10.3 3.5 5.35(2) -7.3(2) 3.4% Mn 4.15 -14.3 8.7 5.52‘2) -15.9(2) 1T. w. McDaniel - Reference 9. 2Squid Data. 122 1. _ — _ — - q I. 0 IL no 0 nu o nu o nu o O \J .9 b ay _9 b s l t I. O nu “u > o b I o l r a r\ b a a m .. L m > ,5 ....I... II D ..l e n pdfiow > “w o d menunvmW M l owL$Om > h m — _ b _ n d n u h m a N .O dd «macaw mo. ...maumqmficqo CC ncva_dw~z= . mcmnmcescm.¢fik m> ++: +1 + m_u_z 0...me _ _ mmmmo. — b ..Nb ....O 0.0 .H\m ocuv~331mx wzemm mimosms. ..o No we . 143 N.O ._ xa\ql UP .r Eb ..0 (I44 2, mU_Z arbmm _ nub 1m.o mflocxm um. oceawdwyzzimx 33mm ....0 m afimmxma. 0.0 H\q _.0 Nb MO 144 In the temperature range Tmax < T < ZTmax’ A is only weakly temperature dependent and a plot of LI:§%XLIl-vs. %-should yield a straight line with a slope proportional to A(T 2 0). At sufficiently high tem- perature A approaches zero and therefore LliglK-should become a constant. When the figures are made for the Cu3PtMn system, several interesting features result. The 2% DOS and 3% DOS do appear to follow the predicted behavior with the intersection of the linear portion and constant portion in the interval (ng, ZTSg) (see Figure 39). The 2% OS and 3% 05 do not show this behavior at all. Rather than having a linear portion with a negative slope, there is a slop- ing curve with a positive slope. This behavior certainly seems contrary to MRF expectations since 2 2 c 3kB n kBT can never exhibit a positive slope. The 1% DOS and 1/2% 005 data are even more pathological. The slope in the temperature regime (2TS , T59) is positive and quite large and at temperatures less than 9 ng the curves become concave down. (See Figure 40.) The reasons for this unusual behavior are not known, but one possibility certainly exists. The early MRF calculations (including the one on which the data is analyzed) predict a probability distribution _ 1 AU) P(H,T) - T {MHZ + H2 which is non-zero as H + 0. More recent calculations by Held and 145 (T -9)X/C (emu-K/mole/at.%) mdmcsm we. w a w ~ 0 «so ”mom «mom ”*aom axaom D h b . p . 6.9.... one Ohm 9N6 900 d \ ... A X -3 mm 33: 3a. may .. .33: 3335. 146 _ q q a _ _ _ _ — mm a.» r. .1 t. 43 A o a i m. s.» 1 .m / u.“ u Ara.ii .1 m m C / s... 1 L x . e O . - . ~H L n we 1 . no ..x. 00m . ~800m . D |de/dc|Dos. (3) Why is (dTSg/dc)DOS > (dTSg/dc)os? It would be nice if one could find some common ground on which to analyze all the different kinds of behavior. The first to come to mind is the use of a free electron RKKY coupling between impurities. Simple explanations based on RKKY coupling are probably not possible. Either a more sophisticated RKKY treatment is needed, or additional interactions are required. Nevertheless, some general thoughts may be worthwhile. ' For example, the dependence of e on host condition may have some explanation in the work of Kok and Anderson.(13) They were able to explain the experimental observations that 8 2 0 for amorphous and liquid alloys, but is not zero for the same alloys in crystalline form. Their analysis is based in part on the performance of a lattice sum. It is generally accepted that e a It Jij and ng a ( II Jgj)1/2. Kok and Anderson argue that in an1gmorphous or liquid matiix, the lattice sum for e averages nearly to zero. However, the lattices for ordered and disordered hosts are quite similar, so ap- plication of Kok-Anderson ideas is not quite straightforward. Mattis<64) has tabulated FCC lattice sums as a function of the Fermi momentum. If one could predict the change in Fermi momentum upon ordering, one could make a definite prediction about the be- havior of e. A similar study has been carried out by A. Freudenhammer<65) 149 for the spin-glass freezing temperature. He calculates the depen- dence of TS on the Fermi momentum and also finds a strong quasi- 9 periodic dependence. Such an approach is nice conceptually, but runs into trouble when looked at in detail. First of all, CuMn also has an FCC lattice sum. Since e/a z 1 for both QgMn and ggafitMn the change in sign of de/dc is puzzling. Secondly, the strong quasi-periodic dependence of TS on kf is only true if the 9 nearest neighbor interaction is assumed to be zero and RKKY other- wise. The oscillations are all but wiped out if one assumes a nearest neighbor direct exchange 0 and RKKY for larger distances (where D/J0 = l). The original calculation was probably based on the idea that the direct Mn-Mn interaction would be anti-ferro- magnetic and might essentially cancel the ferromagnetic RKKY near— est neighbor exchange. One must certainly worry about suppression of the oscillations for a non-zero nearest neighbor interaction. Finally, some electron-positron annihilation studies done by I. Y. Dekhtyar et al. have indicated that the Fermi momentum doesn't change upon ordering for the Cu3Au alloys.(66) Cu3Pt may very likely exhibit similar behavior. It was originally thought that if the Brillouin zone splits in half due to ordering of the alloy, and the zone was originally half filled, the energy of the electrons (and Fermi energy) may decrease due to the compression of the energy levels near the zone boundary. It's more likely that the ordering transition takes place because a binary alloy can lower its energy by surrounding all A atoms by 8 atoms and vice versa. The greater the size difference in A and B, the stronger the tendency to order. 150 E. Mean Free Path Effects Another approach to the understanding of experimental results might be to focus on the role of the Pt, Pd, or An and their in- fluence in the copper matrix. One could argue that the role of the transition metal is that of an impurity scatterer which reduces the range of the RKKY interaction. This may be true, but a com- parison of (CuAl)Mn, (CuZn)Mn, and (CuPd)Mn alloys suggests there is more to the story. In addition, although the e-values decrease with increasing electron concentration, they never go negative except in Pd and Pt systems. It appears as though the Mn ions behave quite .. _——— --..- .—-— -..-.—_.-.——«.—-.' “...-Wa- .. differently in host matrices with electron concentration > 1 (CuAl, ‘~_.-...-—.—._—.. -W- _.—...——.— -w.—w _-— CuGc, CuZn) compared to their behavior in host matrices with electron concentration less than 1 (Cu3Pt, CuPd). This implies that changes in Fermi wavevector and density of states at the Fermi level may be important in addition to impurity scattering processes. Both aspects of this will be touched upon briefly. The binary host spin-glass alloys, which are the concern of this dissertation, are all of the form (A B , when A and 8 con- y l-y)l—xMx stitute the binary alloy and M is the magnetic impurity. The con- centration parameter y is held fixed and only the concentration of magnetic impurities X is allowed to vary. Since the disorder in the matrix is large, 0.17 §_(l-y) g_0.25, there is considerable change in resistivity depending on whether the host is ordered or dis- ordered. As one would expect, the total resistivity is smaller for the ordered host alloy, but perhaps surprisingly, the phonon mm—_—.___ - _-_...A ...._.,, 161 contribution can be larger for the ordered alloy (ex., Cu3Pt) (See Table 10 for some typical resistivities of our CuafltMn alloys). Several items should be noted. (1) There is little change in the resistivity of the Cu3Pt (DOS) upon addition of 1 atomic percent manganese (ml% change). (2) This is not the case for the ordered state alloys (m43%). This is a good indication that the ordered state alloys are indeed ordered, but also raises one subtle pos- sibility. Any attempts to understand changes in behavior of £9333M" (1%) based solely on mean free path effects, should bear in mind‘ that the ordered state resistivities have a large contribution from the magnetic impurities, whilst the disordered state alloys do not. In the discussion to follow, we will distinguish between these sources of disorder. As derived by Ruderman and Kittel, the response of an electron gas to a magnetic impurity consists of spin density, oscillations. One can think of these as arising from the inter- ference of the scattered outgoing spherical wave with the incident Bloch wave. If the translational symmetry of the lattice is dis- turbed by the addition of non-magnetic scattering centers, (i.e., Pt, Pd, Au), the exact states of the free electrons are no longer simple Bloch waves, but linear combinations of many such waves with different k values. This leads to a damping of the interference oscillations such that the RKKY spin density oscillations are of the cosZkfr 4% form o(r) m ——3—-— e r looked at this problem in somewhat more detail(]4) and found the when A is the mean free path. DeGennes magnitude was damped in approximately the expected manner, but in addition, there was also a shift in the phase of the oscillations. 152 Table 10. Resistivity of §u_PtMn (0.81%) and CuPd(l7)Mn (1%). 3— (77°K) Ap(77°K) Alloy uO-cm uO-cm Disordered Cu Pt 40.58 _3_ 0.42 Cu3PtMn (0.81% 41.0 CuPd(l7) 11.5 2.1 CuPd(l7)Mn (1%) 13.6 Ordered Cu Pt 7.81 ‘3— 3.35 ggagtMn (0.81%) 11.16 CuPd(l7) 6.5 1.3 CuPd(l7)Mn (1%) 7.8 153 It should also be noted, that there are arguments which contend that the concept of a mean free path does not really enter into these dis- cussions. P. F. deChatel‘67) argues that in disordered systems the non-local susceptibility of a homogeneous, isotropic conduction electron system x(Ri,Rj) = x(lRi-le) must be replaced by a suscept- ibility that depends on the particular distribution of atoms in the region between sites i and j. _ f(€ )-f(e 1) x(R..R.) - Z n n * * 1 J n’n. €n"€n wn(Ri)wn (Rj)wn.(Ri)wn.(Rj) If one chooses . 2 2 * 1kn'(Ri‘Rj) h kn wn(Ri)wn(Rj) = e and En = -§fi- one gets back the free electron RKKY interaction.) The point here is that the non-local susceptibility is determined by the energy eigenfunctions, whether they are wavelike or not. One does not need an undisturbed propagation of Bloch waves. DeChatel argues that x(Ri,Rj) will not be drastically cut off even if the eigenfunctions are far from Bloch waves. Even though he does argue that the mean free path cannot play the role of a damping length, he advocates its use as such since it is the best one can do when working within certain approximations, although these approximations may be fatal. In that spirit, we will briefly discuss a model developed by U. (95) Larsen to examine the effects of damping and fluctuations in nearest neighbor distance on the spin-glass temperature of RKKY 154 coupled alloys. In particular, the evidence to be presented will suggest that the changes in ng due to the order-disorder trans- formation cannot be attributed solely to mean free path effects. The formalism developed by Larsen is an RKKY version of the Sherrington and Southern model.(62) They obtain a freezing tempera- ture kng = bS{§A1j} ”I where Aij distribution ofJ the exchange coupling Jij = J(Rij)' Larsen assumes = A(Rij) is the width of a Gaussian that J(R) is the RKKY interaction and A(R) is its envelope, so that g 9nJE(2£+1)2 e-R/A A(R) 3 25F (ZkFR) If one temporarily omits the damping (assume pure RKKY) and sets all nearest neighbor distances equal to the average, then one finds kTSg= 4k3 2:93}1 R4} v'xa (4.1) 11“ where 2 A = bSJ (22+1) b = [(25+114'111/2 4EF ’ S 12 ’ )1/3 and <§> = a0 (l6nc is the radius of a sphere containing one spin. In the case of finite A, spins beyond the distance A are excluded and kng is reduced below the value given by (4.1). When the dominant scattering is due to magnetic impurities and not defects or phonons, this is called self-damping and A h.%u Thus the number of shells of width <;> which contribute in the sum for kTSg behaves 155 as - 1 1 - A/ - E/Ufi - c'z/3 . Thus self-damping is increasingly important at high concentrations. If A is due to a fixed number of non-magnetic impurities (such as Pd, Pt, or Au), the number of shells within A is A/<;> m c1/3, and is important at low concentrations. Larsen's final result, in- cluding fluctuation and self-damping effects is given by . _ 3 kT = Adifi’i e'rcx [1 - ec (1 x )111/2 (4.2) $9 c 1 x4 9&3; where r = a hc . Here p is assumed to be the total resistivity. 0 One can see already that there are major problems between theory and experiment. Larsen's theory would predict that the disordered state alloys have a lower spin-glass temperature than the ordered state alloys because their total resistivity is higher. In fact just the opposite is true. (The calculated spin-glass temperatures are shown in Table 11 based on the resistivities shown in Table 10.) If one assumes that for some reason the magnetic resistivity is dominant in these alloys (for the purposes of determining ng) then the trend is in the correct direction for the QuagtMn (0.81%) sample but still in the wrong direction for the CuPd(l7)Mn (1%) sample. In addition, the calculated difference in ng between ordered and disordered hosts is generally considerably smaller than observed experimentally. 156 Table 11. .Calculated Spin-Glass Freezing Temperatures for guafitMn (0.81%) and CuPd(l7)Mn (1%). Spin-Glass Temperature (K) Material Calculated(3) Calculated(4) Experimental (])Cu3PtMn (0.81%) 05 3.33 3.10 2.3 (])Cu3PtMn (0.81%) 005 3.59 1.79 3.3 (2)CuPd(17)Mn (1%) 05 4.89 _ 4.18 3.5 (2)CuPd(l7)Mn (1%) DOS 4.79 ‘ 3.7 5.5 1J = 0.1 ev, EF = 7 eV, 5 = 2, d = 3.70 3, 8 = 3.5. 23 = 0.1 ev, EF = 7 eV, 5 = 2.3, d = 3.70 A, 8 = 3.5. 3ng calculated using magnetic resistivity. 4T calculated using total resistivity. 59 157 However, the Larsen theory is not at a complete loss. The gen- eral behavior of TS as one adds Pt to the copper is to decrease the 9 spin-glass temperature. This is in accord with Larsen's theory. In addition, resistivity studies have been done(68) to check the mean free path dependence of the RKKY coupling in (Au]_xCux)Mny alloys and reasonable agreement was found. His theory also seems to work well for dilute spin-glasses where the host is a single noble metal. This suggests to the author that changes occur during the order- disorder transformation that are not accounted for by A. It is quite possible that Jsd or EF change as a result of the transformation. The effects could be sizeable since kng m Jz/EF. This possibility will be discussed briefly in the next section. 158 F. Brillouin Zone and Fermi Momentum Effects By way of introducing the suggestion that Fermi momentum depen- dencies may be a useful approach to sorting out T59 and a behavior in our spin-glass alloys, I would like to briefly discuss the possibility of Brillouin zone effects. Admittedly, these arguments are handwaving, but in the mind of the author, they are strongly suggestive. First of all, if one examines the magnetic susceptibility of Cu3Au and Cu3Pt, the ordered state is considerably more diamagnetic than the disordered state. See Table 8. Forgetting about the ion core con- tribution (since it doesn‘t involve conduction elections, it is not susceptible to Brillouin zone effects) one can write x = Xs + Xd’ where X5 and Xd are the Pauli paramagnetic and orbital diamagnetic contributions. Now, the spin paramagnetic contribution, even if significant in magnitude, is considered to be insensitive to the orderedisorder transformation since electronic specific heat measure- 'ments show only a few percent change. This follows because the Pauli susceptibility and the electronic specific heat are both strongly dependent on the density of states at the Fermi surface. Therefore, one's attention turns to the orbital diamagnetic term given by 3:E 82E 3E2 2 5E2' (3k xak —) } d5 y Y _2 Xd = 34nech If |gradEi{ 020) .33.! when the integral is over the Fermi surface. Such a term is there- fore highly dependent on the curvature of the energy surfaces. Therefore, 159 changes in Fermi surface geometry caused by Brillouin zones could have a significant effect on Xd‘ To explain such strong diamagnetic ef- fects, one might be tempted to adopt ideas out f0rward by Mott and ( Jones. 69) They examine the case of small overlap between the Bril- louin zone and the Fermi surface (See Figure 41). Here the surfaces of constant energy form approximately a family of ellipsoids. h2 2 2 3 E = 25'{“1kx + a2ky + a3k2} Electrons near states A and B may have an a'which is 30 times the value of 02 or a3 and makes large contributions to the diamagnetism in the y-axis direction. Now, one might ask if there is any reason to believe the BZ- Fermi surface overlap to be small. Generally speaking the answer is yes because we know from the Hume-Rothery rules that other things being equal, that structure is preferred in which the overlap out of the first Brillouin zone is as small as possible. To be more specific, one need only examine studies of the long-period superlattice alloy phase of Cu3Pt.(7O) This phase has been well documented by x-ray studies. These systems are superlattices with stable antiphase domains of definite size. A detailed study of the origin of the long-period superlattice has been carried out by Sato and Toth.(71) They find a definite relation between the electron-atom ratio and the ddmain size M. They develop a model based upon the stabilization of alloy phases using the Brillion zone theory which fits the experi- mental data very well. In addition, they study A38 type alloys 160 truncation factor= b/a Figure 41. Brillouin zone effects on the Fermi surface. 161 including the Cu-Pt system and find they can fit the experimental data (M vs e/a) very well if they use a truncation factor t = 0.958, (See Figure 41) which seems to agree with the general ideas of Mott (72) and Jones. Furthermore arguments by J. C. Slater and J. F. (73) claim that the order-disorder transformation in CuPt, Nicholas CuAu, and AgBMg are driven by Brillouin zone effects and do not simply follow from lattice strain relief. Because the local environ- ment of Cu and Pt in Cu3Pt (ordered vs. disordered) is considerably different than that for CuPt (ordered vs. disordered) a straightfor- ward generalization is not possible, but only suggestive. Finally, it should be remarked that determining changes in the Fermi surface due to an order-disorder transformation based on non-equilibrium (transport) properties is difficult at best. The principle dif- ficulty is that the Boltzmann transport equation can be solved in a simple way only for the unrealistic case of a spherical Fermi sur- face and for scattering elements that depend only on 191 where a is the scattering vector. Changes in resistivity are difficult to sort out because of dependencies on periodicity, and number and kind of carriers. In the case of thermopower,(49) individual contributions do show systematic changes with atomic ordering, but they can be opposite in presumably similar alloys. G. Quantum Effects It has been suggested (see Reference 75) that subtle effects, such as the change in sign of 6 with concentration are due to quantum 162 mechanical effects. Matho argues that the distribution of exchange energies D(w) = 26(w-w(R)) summed over all lattice positions R f O can be represented by a continuous function with high energy cut-offs. In fact, if D(w) = w1/w2 ang)the high energy cut-offs are chosen according to T1f + Tla = E—qufll- and‘ (l) (Tlf ‘ 719/011 ” Tla) ‘ tan“ (077111141) then the continuous distribution reproduces the moments 0(1) and 0(2). This leads to asymmetric saturation values for the spin-spin correlation function: + + 2 That ‘15, = +5 1 2 SW)” -S(S+l) -Bw >> 1 This appears to manifest itself as a distribution in favor of anti- ferromagnetic couplings. A real asymmetry toward ferromagnetism can cancel this effect and lead to the speculated sign change 0f 8. This brings up the point that most of the work dealing with spin- glasses to date has employed classical spins. Recent work by (98) indicates that there may not exist an Edwards-Anderson spin Klemm glass phase for S = 1/2 Heisenberg Spins. More quantum mechanical calculations are needed. VII. GENERAL CONCLUSIONS AND THE FUTURE The principle experimental claim of this thesis is that spin- glass behavior does occur in dilute magnetic alloys when the average interaction is antiferromagnetic. More specifically, unlike the ferromagnetic case, the condition lac] = ng need not be associated with the onset of magnetic order. Present data for two different alloys systems (ggagtMn and CuPd(l7)Mn) suggest that lecl nearly equal to 3TS does not cause the loss of spin-glass behavior. 9 The majority of the experimental results presented are an attempt to systematically explore the effect on the magnetic properties of host atomic order-disorder transitions. It is the author's view that this novel approach to the study of spin-glass behavior will enjoy a bright future. This approach is similar to the crystalline versus amorphous host studies commonly performed, but may be superior be- cause the alteration in host structure is not so drastic. It's one of the few methods available to alter the impurity-impurity interaction without changing the concentration. In addition, because the basic lattice structure is always present, the more powerful, conventional solid state theory techniques can be brought to bear on this problem. The basic experimental trends which have been established by this research include: 163 164 (l) The paramagnetic Curie temperature is a strong function of host order with le'ord > 191005‘ The concentration dependence is approximately linear with some possible interesting subtleties at low concentration. (2) The spin-glass temperatures are also a strong function of host order with (ng)DOS > (159)05. The spin-glass temperatures are nearly proportional to the concentration. (3) The spin-glass temperature ng appears to be a stronger func- tion of concentration than the Curie temperature for the disordered host samples. The opposite seems to be true for the ordered state alloys. (4) The beginnings of some very interesting hysteresis work have been presented. Earlier studies by other workers with dilute amounts of Pt find the width and the displacement field to have the same linear Pt concentration dependence as the reversal field. Our con- centrated (25%) Pt samples are certainly not an extension of this behavior. The value of the reversal field is difficult to ascertain, the width is quite small (~40 gauss) and the displacement field is non-negligible only for the disordered host data. (5). Finally, some preliminary results for 993393" (1%) indicate that this is an alloy system where the spin-glass temperatures are greater for the ordered host samples than for the disordered host samples. The reader also found presented in this thesis a few possible approaches one might take to try and understand the experimental trends in a qualitative fashion. An attempt to use the Mean Random ..— _.- ——— #, ‘---_.-.. .e -..— -—.-_4 .e—_ ..- .— -fi--— .—..... 165 Field approximation to learn something from the high temperature susceptibility proved essentially impossible. Nevertheless, the fact that the ggafltMn (2% DOS and 3% DOS) followed the expected behavior but the other £93330" samples did not may help clarify weaknesses in the MRF approximation. Conversely, it may indicate that there exists a fundamental difference between ordered and disordered host spin-glasses. Experimental trend #3 above already hints at this. Attempts to understand the data using a naive free electron RKKY interaction and involving Fermi momentum changes upon ordering look promising on the surface but fade somewhat when examined in detail. The periodic and quasi-periodic dependence of e and T59 on kF respectively looks hopeful at first, but is probably too simple-minded. One must keep in mind that even drastic changes in sign probably occur for e as a function of concentration in simple noble metal hosts. The situation is not likely to be any simpler for the binary hosts. Along these lines, considerable discussion was also given to the possible importance of Brillouin zone effects. The diamagnetism of the hosts, the formation of long-period superlattice structures, and studies of CuPt, CuAu, and Ag3Mg all suggest Brillouin . ...—......- -—-.- -——-———-—— - .‘ ”~— .. -... -..—---- _ mm---—n- We- ---~.-H._—.— .... ,- .3 zone effects may be important. However, there does not exist, to the best of the author' 5 knowledge, any theories relating changes in Fermi surface geometry to spin-glass properties. They will probably be needed. Finally, attempts to understand the spin-glass freezing temperature on the basis of mean free path effects also failed dismally. The qualitative behavior was not even correct. However, it's quite possible that the effect on mean free path due to ordering 166 at a fixed concentration of non-magnetic scatterers is qualitatively different from the addition or removal of further scatterers. In short, none of the theoretical attempts to understand the data were effective in the short run, but it is the author's view that new insights into the physics of spin-glasses will be achieved through the analysis of spin-glass behavior in order-disorder alloys. In closing, I would like to briefly discuss where one goes from here. Some future experiments which immediately come to mind in- clude: (i) diffuse x-ray scattering to establish the nature of the chemical short range order, (ii) extend the maganese concentration in the @3fit)]_anx and (CuPd(l7))]_anX alloys to see how far one can extend the SK phase diagram; (iii) initiate similar studies in alloys with different lattice structures.' For example (Cgflt)1_anx has a layered structure (al- ternating planes 0f Cu and Pt). (Quflg)l_anx has a body-centered cubic structure; (iv) the placement of Mn impurities on specific chemical sites. For example, Cu3_thMnx, where the Mn resides only on the copper sites or Cu3Pt Mnx where the Mn resides only on the Pt sites; l-x (v) Continue studies of FCC order-disorder hosts like (EEGAE)l-anx which was already begun; (vi) continue to study the hysteresis properties of these ternary alloys. A systematic study of (Cu1_thx)Mny for fixed y with variable x may lead to new physical insight. APPENDICES A. SIGNAL DETECTION COIL CONSIDERATIONS Probably the most complex and yet most crucial design aspect of VSM's is the size, shape, and relative position of the signal detec- tion coils. For a good discussion of this subject see References 76-78. I would like to discuss briefly the work of Mallinson(79) since a set of coils based on his calculations were made but never used to take actual measurements. His derivation is based on the principle of reciprocity which states that the mutual flux threading two coils is independent of which one carries a current 1. His final working result is <15 = “(Hy—W (Al) where he assumes the sample is magnetized in the x-direction and vibrates in the y-direction. A more general derivation of Mallinson's result is given by R. Reeves and J. Reeves.(8o) They consider the induced EMF in a filamentary circuit due to a magnetic dipole. Let d1 be an infinitesimal element of the loop, 5 be the dipole moment of the sample, and F the vector from 31 to m. + Then the vector potential at d2 due to m'is -: 1+ + "9"” 4n r3 .y. A: 167 168 + 31 .. + ., using E = - 55' the electric field produced at d2 due to m 15 given by 'E = 32.3L.(fi’x43) 4n at r3 Therefore, the EMF generated around the loop is given by 11 + "' + 0 3L_ m x r . V 411 ¢ 3t '3 d2 Interchanging dot and cross and integration and differentiation leads to < I :15 an: iv 5:1 wx + r + + = _ uo jL_ a , 2 x r 1'1? at ,3 41 + + where N: ,1de is a purely geometric function defined by the coil configurat mi (One can see reciprocity entering here since N is really the magnetic field per unit current which would be produced at'F due to a unit current flowing in the coil). Thus, - __+.‘+ V--T1Fatm N (A2) Now equation A2 is a very general result. For uexample, if the dm %N dt |ml is constant but its orientation changes with angular velocity 5 dipole is stationary then N is constant and V = - If 169 u then V = - -9- ° (m x N). If the dipole does not move or rotate but 4nn u .. changes in magnitude, then V = - Zg-N'° m §%'° Finally, if the dipole moves with velocity V without otherwise chang- ing then L1 + v=-—%fi-%§- (A3) This is the equation which applies to the VSM. Equation A3 can be rewritten as + + +a—N.v +§flv 11 + N v x x 3y y 32 z — - .2.+ ..i_ V - 4n m {3 or no 8"x x 3"x v=-fi{mx (Ti-vx+_2va+—3§vz) 3N 3N 3N + m (——x-v + —-x-v + —~X-v ) y 8x X By y 32 2 EN 3N 3N z z z + m2 ( ax vx + By vy + 32 v2)} (A4) This is the principle result of this appendix. If our sample vibrates only in the y-direction then equation A4 reduces to u 3N 3N 8N V = - o m -5-+ m __X.+ m -—Eqv n X By y 3y 2 32 y ° If the magnetic field is in the i direction, so that fi'is essentially a vector in the ; direction, then 170 V =x(1fi) vy But Nx can be identified with Mallinson's hx using the Biot-Savat Law. Now the idea is to maximize 5 by maximizing 75% . So let's con- sider the situation shown in Figure 43 where the sample is located at the origin, the distance of closest approach is X0: and the sample vibrates in the y-direction. The output voltage from this loop is dh dhx wu—d— ") Jib—)3] This implies that maximum output is achieved if conductor A is placed at the coordinates'(xo,0) and conductor 8 is placed anywhere along (X0, «5'x0). Thus Mallinson preducts the optimum coil configuration to be that shown in Figure 42. Here four identical N turn coils were used. By connecting them properly in series opposition the signal output was increased and some immunity to magnetic field fluc- tuations was achieved. Such a set of coils was constructed by the author. They were approximately 2.55 inches in diameter and consisted of 1700 turns of #36 HVY polythermaleze copper wire per coil. Each coil had a dc resistance of approximately 460 ohms. Their self- resonant frequency was measured to be close to 1400 Hz. Their BO return A. feed 0 III-IUIIIII...III.III.v.II.IIIIII..III.I.”x“ ... .MLV YA l7] 0 ulllnoouooul-ooo-o. X . . ...-II......ICIIIIIIIUIIIIII XXXX run-II. Optimum pick-up coil geometry. Figure 42. 172 inductance was l.l4 Henries. These coils were constructed in an attempt to increase the signal-to-noise ratio. The previous set of coils were #986 Miller air core RF chokes. Their mean diameter is approximately ll/lG inches and they have approximately 1300 turns of wire with a DC resistance of 95 ohms each. The signal was ex- pected to increase by a factor of (2.3) (%%%%9 z 3.0. The actual measured increase in signal was of the order of 2. However, the increase in noise due to magnetic field fluctuation was greater than 2 and the overall signal-to-noise ratio decreased. The sensitivity to applied field noise goes as er. Therefore 2 new gNr 2 . §i7oogéi.53; _ In fact, the only way to really increase the signal-to-noise ratio is to make the detection coils smaller and move them closer to the sample. One final attempt to reduce field noise effects was made by at- tempting to externally balance the Miller coils. By sweeping the magnetic field linearly, the output from the four pick-up coils was measured to be 10 uV/4000 gauss/min. This implies a total turn imbalance of approximately 15 turns. Coil turns were added and sub- tracted systematically in an attempt to balance the coils. Once the coils were reasonably well balanced with these quasi-static fields, an attempt was made to match reactive components at 33 Hz. This was done by driving the NMR coils at approximately 33 Hz and adjusting 173 the values of trimming capacitors. This seemed to meet with some success, but the reactive components were never completely balanced out. Overall, these measures probably increased the signal/noise ratio, but a quantitative estimate was never made. B . SCHEMATI C DIAGRAMS 174 175 A__A 39.2.. 3. :53 ~53 p.22. . ML '7' Eama 950:2? 176 I'll"'l‘| l'l'll"ll' 3693.. ‘0.» mar 0.2: tr / ‘33 mx wma .....m _ amt. MOO m: "2:33:03 ”295?. mdccxm a». =~=~mwxccfim1=. Qua do.» .Jawoqwctnaqg J a H 99.: adwal '99 aunfigj °uallou1uoa aunqeu 177 ... '< lfllllii 37} Li... W J> 178 05:23 4 < H ...m. +u. .1... won .10 - , 4, , . 0 p-I’1(((I— . a? X 1 $0.91? Eamongm mooL m - _ asst’mm‘fi fizz“ _ 3:3: «£20 £230: 3:: E. u . 1.. . ..H , ... 2:329 ..IM 33 as +W+WTW 0:26.: mo: «no :6: .3: 3:“ as.“ in. IR. miccsm am. newsman mocxnm. C. MAGNET AND POWER SUPPLIES Some discussion of low noise amplifiers and the like was pre- sented, but the major factor limiting the sensitivity of the system was ambient magnetic field fluctuations. For this reason I should like to briefly discuss the magnet and associated power supplies. The magnet is a Magnion lZ-inch diameter electromagnet with a 3- inch gap. It has an H-frame cast magnetic yoke, is indirectly water- cooled, and utilizes low impedence foil wound copper coils capable of continuous operation in excess of 65 amperes. The power supply is a model HS-1365/FFC-4 8.5 kilowatt magnetic field regulated power supply. It has a maximum output of 65 amperes at 130 VDC. The PFC-4 field regulator utilizes a rotating flip coil to sense the strength of the magnetic field directly. A permanent magnet in a controlled environment is mounted on the same shaft as the flip coil and provides a standard voltage output. A ratio transformer selects some fraction of the reference generator signal and this is compared with the flip coil output (which is about 50 mV/kilograms). The probe coil is turned at 1800 RPM by a synchronous motor and generates a 30 Hz signal. The difference between the flip coil and the precisely attenuated signal from the reference generator is amplified by a tuned amplifier. A demodulator selects that portion of the signal _ which is at the correct phase and converts it into a DC signal. This in-phase error signal is used in the field set or regulate 179 180 modes for control of the magnetic field. The regulator also employs a stationary field rate sensing coil mounted around one pole cap. It develops a voltage proportional to the ratio of change of the field. The in-phase error signal is combined with the signal from the rate coil and sent to an integrating amplifier. This causes the power supply to act as a current amplifier in controlling the current to the magnet. The power supply consists of basically 2 servo loops. The coarse or secondary loop consists of a motor-driven variac which is designed to control the amount of DC voltage supplied to the precision regulator. Its job is to maintain approximately 3 volts between the collector and emitter of the series bank transistors. The precision servo loop is essentially the PFC-4 field regulator. The integrating amplifier receives the demodulated chopper signal and the AC signal from the rate coil. It raises the power level sufficiently to drive the power transistor bank. The advertised stability of one part in 105 means $0.09 gauss at 9000 gauss. In fact, the field stability was often times more like :1 gauss. Sudden changes in the magnetic field would cause voltage surges which were large enough to overload the tuned amplifier and cause it to ring at =33 Hz. The phase-lock amplifier was only partially successful in discriminating against these unwanted signals. Attempts to deal with this problem began with an investigation of the magnet power supply itself. The most puzzling discovery was the presence of a 500 KHz signal at the output of the tuned amplifier, even when the input of the amplifier was grounded. It appeared as though some part of the system had begun to oscillate. Additional work suggested 181 problems with the Twin-T network. If certain capacitors were slightly increased in value the high frequency noise could be eliminated. Other games were played including an in situ tuning of the Twin-T. This was done by observing the field deviation meter which sees the dc output from the demodulator. By adjusting certain trimming re- sistors so that one acquired minimum oscillation of the field devia- tion meter around its correct value (as specified by the ratio transtrmer) a slight improvement of the field stabilization was effected. It is also possible that due to the age of ihe supply (approximately 14 years) and the germanium semiconductor technology employed, field stabilities of one part in 105 are unlikely to be re- gained. Some further testing yielded no new clues, it was decided to try and treat the symptons rather than the cause. More on this later. For those days when the ambient field noise was just too great to take data, we often switched power supplies. Most of the data at 5 kilograms utilized an Alpha Scientific Laboratories Model 3002 current-regulated power supply. It had output current and voltage capabilities of -0.3 to 30 amperes and -0.35 to 50 volts, respectively. Since the dc impedance of the magnet coils is approximately 2 ohms, supply currents as great as 25 amps were available. It was found necessary to plug this unit into the 235 volt single phase line to obtain 25 amps. In general, the field produced by this supply was quieter. Since this is a current regulated supply, the strength of the magnetic field had to be independently measured. This was gen- erally done with an LDJ Model 1018 Gaussmeter which utilizes a Hall probe. Finally, when the hysteresis measurements were being taken 182 in relatively low field, it was found that the Heathkit IP-2710 power supply was the quietest of them all. It was capable of delivering 30 V at 3 amps. D. GROUNDING AND SHIELDING CONSIDERATIONS Since this experiment involved the measurement of very low level signals in a somewhat noisy environment, some remarks about the grounding scheme would be in order. The principles of proper grounding are straightforward, but not so easily optimized in practice. Figure 43 shows the grounding scheme for the low level signal processing A circuitry. The power supply, tuned amplifier, phase-lock amplifier and digital multimeter are all isolated from the rack and the grounded neutral of the AC line. The signal ground is connected to the rack and the neutral line only at the oscilloscope. This means the link on the back of the phase-lock amplifier must be disconnected and the position of S1 doesn't really matter. The 31 Ohm resistor is designed to semi-float the amplifier input from common should the . possibility of ground loops exist. The connections as shown embody two rules of practical shielding.(81) Rule One requires that the shield must be tied to the zero-signal reference potential. Rule Two requires that the shield conductor should be connected to the zero signal reference potential at the signal earth connection. Since the oscilloscope is intimately tied to the rack but is not connected to neutral, everything was tied as close as possible to it. A view of the overall grounding scheme can be seen in Figure 44. When ground- ing instrumentation with widely varying power and noise levels, one should group the leads selectively. The signal ground for low-level 183 184 .no._m u .qu g.“— s: I. A menu: 1.. _ “Fan... - E 50..-... .3 e I ,L/ H r _ fl 1. oz: <90 a. _ IIIIJ, .— H 1; coon-00::- mu+l .llll. . - - w‘ m ma fl # m m .20.“ a .m.. _ floor mdccxm bu. awocaaszc mn3mam Ado: .m .. F> ‘ F - 1.0 p ‘ i . 1 1 ‘\ 1— ‘ —: so ‘ so \ 1 \ s \ L 1 ‘21 1 1 \ 1x1 Jo/J _’ Figure 52. SK phase diagram - BPW approximation. 202 both methods give the same results. The phase diagram obtained by Kosterlitz et al. is shown in Figure 49 and looks remarkably like the SK phase diagram. Now, the spherical model Hamiltonian has been shown to be equivalent to a standard Heisenberg Hamiltonian where the dimensionality of the spins approaches infinity. This would indicate some insensitivity of the phase diagram to spin dimensionality or quantum spin effects. The use of Ising model spins in the SK solution should also be addressed. Nature appears to be cooperating with theory in this respect. Even though the manganese impurities are believed to be ‘ correctly represented by a Heisenberg Hamiltonian, and the suscep- tibility is reasonably isotropic. they seem to exhibit Ising-like properties. For example, remanence is an effect which requires the existence of potential barriers in configuration space. Such poten- tial barriers may exist for Ising spins. but are normally not found in normal Heisenberg ferromagnets or antiferromagnets. Also, the linearity of the specific heat with temperature-at low temperatures implies Ising spins. Thereare at least two possible explanations. (l) There are anisotropic forces which act to reduce the free- dom of the spins. The experimental fact that the remanence obeys scaling laws characteristic of a l/r3 interaction suggest that it is due either to the RKKY exchange or dipole interactions. Probably the single-ion cubic anisotropy is most widely accepted. (2) The Heisenberg system may have potential barriers (or some other form of two level system) which make it appear like an Ising 203 Paramag net 1J3 Spin Glass Ferromagnet 0 / up 11) Figure 53. SK phase diagram - spherical model. 204 spin-glass.(88) One shouldn't concentrate on the energy levels of a spin in a fixed. random, magnetic field. Instead, one should look at the potential energy as a function of the orientation of all the spins. The meta stable states of the spin-glass correspond to local minima in the energy of the spin configurations. Tunnelling between local minima can occur if the separation in configuration space is not too great. Such a system can lead to a linear specific heat.(89) Another way to approach the problem is to look at the internal field distribution P(H) predicted by Ising and Heisenberg spins and measured by experiment. Ising model calculations typically give an internal field P(H=0)#0. J. Kopp<90) found a similar distribution of internal fields based on nearest neighbor RKKY couplings between spins. The aforementioned agreement between Ising model calculations and observed properties would suggest a non-vanishing P(H=0). However, experiments by (9]) et al. with polarized muons suggest P(H=0) + 0 which is Fiory consistent with a Heisenberg set of spins.(92) So the question is why the Ising model calculations give the correct properties of the system even though the fields are Heisenberg like distributed.(93) One possible hand-waving argument is the following. In the polarized "HKHI‘ experiment, the internal field distribution is measured relative to a fixed external direction (lab frame) and the randomness of the fields in amplitude and orientation give essen- tially a Heisenberg-like distribution. Globally there appear to be no constraints and all orientations are equally probable. When the system orders. the local spin at a particular site will not have an 205 arbitrary direction of orientation. The spin will tend to be frozen in the direction of the local order parameter. The volume element is no longer HZdH and is closer to an Ising distribution. There are, however, calculations which predict significant devia- tions in detail from the SK phase diagram. One of these is a real space rescaling study of spin-glass behavior in three dimensions.(94) They study an S = l/2 Ising Hamiltonian with nearest neighbor inter- actions only. The basic idea in rescaling studies is to progressively remove degrees of freedom from the problem and examine the new effective interactions between the remaining spins. In random systems, the interaction probability distribution changes upon re- scaling but retains its symmetry. The idea is to follow the distribu- tion through successive rescalings and monitor its width. (We shall characterize the width by J and the mean by Jo). In the paramagnetic phase the interactions decrease under repeated rescalings. This is because the effective interactions give information about correla- tions at larger and larger distances, but the correlation length is finite in the paramagnetic phase. If the width of the distribution continues to increase (but the mean stays small). one is believed to be in the spin-glass regime. If the exchange distribution has a finite mean ferromagnetic order is possible. In the case of ferro- magnetic order the interactions grow under rescaling and the distribu- tion is characterized by Jo +1m; J/Jo + 0. Using these definitions, the phase diagram derived by rescaling techniques is shown in Figure 50. In their model the spin-glass-ferromagnetic transition occurs at a considerably larger value of 36/3. They predict l.63 whereas the mean field value is between l.0 and 1.25. Also. the boundary 206 ._.\.._ pmqm. ‘.nvfl Ohuv .l \ qumfiqo. mumalmgmm arm. dkv dam M;u N.m mincxm ma. mx usmmm admaxma 1 1mmnmdm=m nmdncdmamoa. 207 between the paramagnetic and spin-glass phases occurs at a lower value of (T/3). They predict $0.5 whereas mean field theory pre- dicts l.O. I. MEAN RANDOM FIELD Earlier in the thesis (Section VI.C.2) an attempt was made to extract information from the high temperature magnetic susceptibility of dilute alloys using the mean-random-field approximation.(95) Some of the details leading to equation (6.l) are now presented. The mean random field has as its goal the determination of the temperature dependent probability distribution P(Ho) where no is the random variable corresponding to the effective field at site F . It is 0 given by HO = g vojuj where “j 15 the thermal average of uj. That is, - tre'BHp. lb: _ 9 J tre 8” and voj is the interaction between the moment at site j and the origin. Once the probability distribution function is known, the thermo- dynamic function can be obtained by integrating the thermodynamic variables for a single spin in a fixed internal field H over all fields. The thermal averages above are for a fixed set of position coordinates. The average over position coordinates must be performed last. The Hamiltonian describing the interaction between the im- purities H = .5. Vijuiuj can be rewritten in terms of the effective fields at the1sites as 208 209 where the thermal average uses the effective field +B§Hiui _ tre “j “j ‘ +B§Hiui tre 1 - For a fixed set of spatial positions, HO and DJ are constants. The distribution of the Ho arises from the requirement that the positions of the impurities are random variables. The distribution P(Ho) is calculated as follows. Pick an impurity “0 at site F6. Average over an ensemble of systems, each containing N spins, with fixed co- ordinates. but where each member of the ensemble has a different set of fixed coordinates. This effectively calculates the thermodynamic properties of a single spin in an internal field which is an average over all possible spatial configurations of all spins except that at .+ r . o The essential approximation (which in effect allows one to do the calculation) is that each P(Hj) has the same functional form (i.e., P(Hj) = P(Ho)). Also. when calculating the internal field at site 0, all functions of the internal field at sites other than 0 are replaced by their mean values. This allows one to derive an integral equation for P(H). One finds that 210 _ - _22 - ...--- where A(B) - ycllull, y - 3n lalnO and ||u|| = faDP(H)|u[dH. This theory is a molecular field theory which is one step better than the Weiss molecular field approximation. Instead of all fields being replaced by a mean field which is the same at all sites. they are replaced by a distribution of fields which is the same at all sites. J. LARSEN INTEGRAL 3 1/2 kTSg - Act; I} where A = st2(22+1)‘ b = [(2er1)4-111/2 4EF ’ s 12 J = effective s-d coupling constant 2 = angular momentum quantum number (for d electrons 1 = 2) s = the magnitude of the spins; and l 3 I = f gl-e‘rcx [1 - ey (1" )J 1 x4 where 2 ( ) 4BeQT o l y r = and y = _ aohy 1 Y .and p = total resistivity at the freezing temperature B = O/Otr (ratio of ordinary cross section/transport cross section) 3.5 to 4 for 3d impurities e = charge of the electron a0 = lattice spacing h = Planck's constant y c = impurity concentration Letting rc = w. 211 212 co co .. ] I=fe dx-e'yf e‘””" y I =£(2-+2)+£(057722+£n- +£3-93) 1 6 “W 6 ‘ 0"” 4 l8 1 l 3 2 f e'w/x + y /x xzdx is integrated using an IMSL library 0 routine called DCADRE (Cautious Adoptive Romberg Extrapolation). I 213 Program Temp (Input. Output) Integer IER Real DCADRE,F,A,B,AGRR,RERR,ERROR,C External F A 1.0 E-S B l.O REER = 0.0 AERR = 1.0 E-4 C = DCADRE (F,A,B,AGRR,RERR,ERROR,IER) Print lO,c FORMAT (Ell.5) END REAL FUNCTION F(X) REAL X,N,YP N = 6.l553 E-3 (or 4.9096 E-2) YP = 0.8166 E-2 F = X**2*EXP(-(N/X)-(YP/X**3)) RETURN End Output C = 0.31630 (or 0.29766) K. guafluMn Data We mention the preliminary results for Cu3AuMn (1%) data because we have data which suggests that the ordered state alloy has a large spin-glass temperature than the disordered state alloy. This is in direct contrast with QuagtMn and CuPd(l7)Mn alloys. Figure 51 shows high temperature x'1 vs T data and Figure 52 shows the X vs T data used to determine the spin-glass temperatures. However, a word of caution is in order here. It is a difficult task to obtain good quality ordered state samples of Cu3Au. The magnetic parameters for these samples are: 00$ 05 6 +3.31 +3.82 T 5.75 6.2 $9 214 215 - d -1 _ - _ - - mo I. o .. a. Om o_‘mw0nvw ) WM um I o a / m m mm Noii w m m e rs am I. a .... X a. dc I a m .1 o O o h _ .. _ — _ _ L _ NO #6 GO GO ...maumqmgqm CC 3.9.3 mm. Dumb; :3 1 mz