Specific Heat of Sr 3Cupt

Specific Heat of Sr 3CuPt. 51r. 50 6 Below 1K by Adam D. Poleyn A.B. Physics Princeton University (1992) Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1999

Author ...... Department of Physics May 6, 1999

Certified by ...... /..-.-. . 1 ...... Thomas J. Greytak Professor of Physics Thesis Supervisor

Accepted by...... 7...f...... Asso.. Thomas J. Greytak Professor, Associate Dep rtment Head for Education

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Specific Heat of Sr 3CuPtO.Oro. 506 Below 1K by Adam D. Polcyn

Submitted to the Department of Physics on May 6, 1999, in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Abstract

The alloy Sr 3CuPtO.5 Iro.0 O6 was first fabricated by zur Loye and his collaborators at MIT in 1994. Lee and his collaborators have modeled the material as a spin-' chain with randomly distributed ferromagnetic (FM) and antiferromagnetic (AF) nearest neighbor bonds of equal strength. Magnetization measurements indicate that the material obeys the Curie Law to 4K, even though the AF and FM interactions have strengths of order 30K. Lee's model explains this unusual Curie Law behavior, and predicts that at temperatures well below 1K, the material should exhibit a scaling behavior in specific heat and susceptibility characteristic of a new universality class of disordered quantum spin systems, and contain an unusually large amount of spin entropy. If a large amount of spin entropy is available in the material below 1K, the material could be useful for magnetic refrigeration to temperatures as low as 100pK. Motivated by the work of zur Loye and Lee, I have constructed a new apparatus to measure specific heat u of Sr 3CuPtO.5 Iro.50 6 between 0.1K and 1K in fields to 7T. I have developed thermal characterization techniques to verify that the thermal properties of the calorimeter are appropriate for specific heat measurement below 1K, and used the apparatus to measure specific heat of potassium ferricyanide K 3 Fe(CN) 6 and Sr 3 CuPtO.5 1ro.5 0 6 using the AC and thermal relaxation methods. I find no evidence for a phase transition to a long-range ordered state between 0.1K and 2K in Sr 3 CuPtO.5 Iro.0 O 6 , and that u is consistent with the scaling law predicted by Lee's model between 0.1K and 0.4K at zero field. Application of fields below 10kG suppresses a, and at 10kG or obeys a T3/ 2 power law below 0.5K. I show that given my data, Sr 3 CuPtO.5 Iro. 5 0 6 is not superior to known paramagnetic salts for magnetic refrigeration to lmK, and suggest that a successful understanding of the physics of Sr 3 CuPtO.5 IrO.5 0 6 below 1K may require consideration of interactions between spins on different chains.

Thesis Supervisor: Thomas J. Greytak Title: Professor of Physics

Acknowledgements

The work described in this thesis is a testimony to the love, support, and patience of my wife, Amy Fronduti Polcyn. She has been a wonderful friend and helper to me throughout my years as a graduate student, and I am grateful and honored to be her husband. I would also like to thank my advisor, Professor Thomas Greytak, for his support and patience. The laboratory environment that he has created is an excellent one in which to obtain an education in experimental physics, and I am honored to have been his student and to have worked in this laboratory. Despite the inevitable frustrations and setbacks, I have very much enjoyed working on this project and learning about the physics of magnetic materials, and am grateful to him for proposing this project for my thesis work. I have been blessed with an excellent group of colleagues during my graduate career. I have learned a great deal from each of them, and wish I had taken even more advantage of the pool of talent that exists in this laboratory. Many of the key ideas in this thesis resulted from conversations with them. In addition to their talents as scientists, they are an excellent group of people, always ready to listen, teach, help, and support. My classmate Dale Fried has been a great friend, brother, and confidant to me. Tom Killian was an excellent office mate and colleague, and probably took as many phone messages for me as I did for him. I have had many enjoyable conversations with Lorenz Willmann. I am grateful to Stephen Moss for his friendship and support, and for many enjoyable hours on the tennis court. It has been a pleasure to work with David Landhuis, and I am grateful to him for taking over as safety officer. In addition to these current members of the group, I would like to acknowledge past members of the group, from whom I have learned most of what I know: Mike Yoo, Claudio Cesar, Albert Yu, Jon Sandberg, and John Doyle. Finally, I would like to acknowledge Professor Daniel Kleppner, with whom I worked early in my graduate career. It was always a pleasure to talk with him about physics and to listen to his stories. I have also had the pleasure of supervising several undergraduate projects during my graduate career. It was a pleasure to supervise Carlo Mattoni's thesis, and a lot of fun to work with him. I have also enjoyed working with Mihai Ibanescu, who I in many ways consider more of a colleague than a student. Paul Jackson has been an excellent friend and brother to me over the latter half of my years at MIT. His constant support, prayers, and concern have been a great source of encouragement to me, as have the many joyful times we have shared together. There are many other people who have made my years at MIT more enjoyable, and supported me along the way. Among them, I would like to mention Bryan Atchi- son, Jeff Niemann, Scott Socolofsky, members of the Eastgate Graduate Christian Fellowship Bible Study, and members of my St. Ignatius small group, especially Ed and Mary Dailey. I would also like to acknowledge the love and support of other members of my family, especially my parents, Dr. Daniel Polcyn and Elizabeth Polcyn. They have been a constant source of encouragement, comfort, and support. I am grateful to my son, Stephen, who always smiled for me when I came home after a long day at the lab and has brought both Amy and me so much joy. I would also like to acknowledge the support and love of Anne and Bob Fronduti, John and Meghan Fronduti, Karen Fronduti, and Sarah Polcyn. Finally, I would like to acknowledge and thank my Lord and Savior, Jesus Christ. He is the one who has put all these people in my life, Who has given me the op- portunity to study physics, and Who has created the amazing things I have had the pleasure to study as a graduate student. Psalm 116

I love the Lord, because He hears My voice and my supplications. Because He has inclined His ear to me, Therefore I shall call upon Him as long as I live. The cords of death encompassed me, And the terrors of Sheol came upon me; I found distress and sorrow. Then I called upon the name of the Lord: "0 Lord, I beseech Thee, save my life!" Gracious is the Lord, and righteous; Yes, our God is compassionate. The Lord preserves the simple; I was brought low, and He saved me. Return to your rest, 0 my soul, For the Lord has dealt bountifully with you. For Thou hast rescued my soul from death, My eyes from tears, My feet from stumbling. I shall walk before the Lord In the land of the living. I believed when I said, "I am greatly afflicted." I said in my alarm, "All men are liars." What shall I render to the Lord For all His benefits toward me? I shall lift up the cup of salvation, And call upon the name of the Lord, Oh may it be in the presence of all His people. Precious in the sight of the Lord Is the death of His godly ones. O Lord, surely I am Thy servant, I am Thy servant, the son of Thy handmaid, Thou hast loosed my bonds. To Thee I shall offer a sacrifice of thanksgiving, And call upon the name of the Lord. I shall pay my vows to the Lord, Oh may it be in the presence of all His people, In the courts of the Lord's house, In the midst of you, 0 Jerusalem. Praise the Lord! For Amy

Contents

1 Introduction 21 1.1 Properties of Spin Chains ...... 23 1.2 Sr 3CuPt1pIrpO6 ...... 26 1.3 Specific Heat of Sr 3CuPtO.5 1ro.5 0 6 below 1K ...... 32

2 Random Quantum Spin Chains and Sr 3CuPt 1 _pIrpO 6 35 2.1 Theoretical Work on RQSC ...... 35

2.2 Experimental Work on Sr 3CuPt1pIrpO6 ...... 43

3 Methods 51 3.1 Quasi-Adiabatic Calorimetry ...... 51 3.2 Thermal Relaxation Calorimetry ...... 52 3.3 AC Calorim etry ...... 54

4 Apparatus 59 4.1 Dilution Refrigerator and Magnet ...... 59 4.2 Heat Capacity Experiment ...... 59 4.2.1 Support Structure ...... 59 4.2.2 Calorim eter ...... 63 4.2.3 Calorimeter Models ...... 64 4.2.4 Thermometry ...... 67

5 Potassium Ferricyanide Experiment 69 5.1 Calorimeter Preparation ...... 69 5.2 AC Method Procedures ...... 71 5.3 Thermal Relaxation Method Procedures ...... 74 5.4 Empty Calorimeter Results ...... 77 5.5 K 3Fe(CN)6 Calorimeter Results ...... 79

6 Sr 3CuPtO.5 Iro.506 Experiment 87 6.1 Calorimeter Preparation ...... 87 6.2 AC Method Procedures ...... 88 6.3 Thermal Relaxation Method Procedures ...... 88 6.4 Empty Calorimeter Results ...... 93 6.4.1 AC Method Results ...... 93

11 6.4.2 Relaxation Method Results ...... 96 6.5 Sr 3CuPtO.5 IrO. 506 Calorimeter Results . .... 103 6.5.1 AC Method Results ...... 103 6.5.2 Relaxation Method Results ...... 107

6.6 Sr 3CuPtO.5 IrO. 5 0 6 Specific Heat Determination 111

7 Conclusions and Future Work 113

7.1 Sr 3CuPtO.5 IrO. 5 0 6 Specific Heat Results ...... 113 7.2 Discussion I: Entropy, Field Dependence ...... 116 7.3 Discussion II: Comparison with RQSC Theory ...... 119 7.4 Discussion III: Comparison with Beauchamp Results ...... 120 7.5 Discussion IV: Miscellaneous Interpretations ...... 121

7.6 Prospects for Adiabatic Demagnetization of Sr 3CuPtO.5 Iro. 5 0 6 ... . 122 7.7 Future W ork ...... 124

A Calorimeter Conductance from Power/Temperature Curves 127

B Exponential Fits with Instrumental Response 133

C Schwall Model 135

D Model for AC Transfer Function in Presence of - 2 Effect 139

Bibliography 143

12 List of Figures

1-1 Ordered Ising chain at T = 0 (top) and Ising chain with fluctuation (bottom )...... 22 1-2 Ordered Ising net at T = 0 (left) and Ising net with fluctuation (right). 22 1-3 Specific heat of various types of spin chains. The AF and FM chain results were taken from [5], extrapolated to zero temperature using simple spin wave theory, and the classical result from [4]. These results are plotted assuming the Furusaki Hamiltonian (1.2); the Bonner and Fisher Hamiltonian uses 2J rather than J...... 24 1-4 Representation of T = 0 FM spin wave. Rather than reducing the chain magnetization by breaking the chain at a single point, which costs energy 2J, the magnetization is reduced in a spin wave by tilting all spins slightly off the z axis, which costs much less energy. The spin wave is a traveling wave in which the difference in azimuthal angle between adjacent spin sites in the wave is constant...... 25

1-5 Crystal structure of Sr 3MM'0 6 . Left: view along the length of a chain, showing alternating MO 6 trigonal prisms (M represented by the small ball at center of prism) and M'O 6 octahedra (M' represented by the large ball at center of octahedron). Right: view down the c axis. Clusters of 3 Sr ions surround each chain ...... 27

1-6 M/H data for (top left) Sr 3CuPtO6 , (top right) Sr 3CuIrO6 , and Sr 3CuPt0 5. Ir 0 .5 0 6 (bottom)...... 29

1-7 1/x data for Sr 3CuPtI_,IrO 6 taken by Nguyen [15]...... 30 1-8 1/X theoretical prediction of Lee and collaborators [17]...... 31

2-1 Schematic view of spin wave confinement. Due to different dispersion relations for FM and AF segments, spin waves from a given segment do not propagate into adjacent segments. A mechanical analogy is a rope with discontinous changes in thickness...... 36 2-2 Missing entropy as a function of p, calculated with (line) Equation 2.10 and (open circles) HTE...... 38 2-3 Spin chain considered by Westerberg et al. (Equation 2.17). The FM segments form large effective spins, separated by AF segments of variable length...... 39

13 2-4 Schematic of the RSRG method. A renormalization step proceeds by first identifying Ao, the strongest bond in the chain. Then the two spins linked by Ao are frozen to form an effective spin SL + SR, and

the nearest neighbor bonds renormalized to A1 and A 2 . As the chain cools below kBT - Ao, it effectively carries out a step in the RSRG. . 40 2-5 M versus H data of Beauchamp and Rosenbaum on p = 0.5, 0.667 sam ples...... 45 2-6 AC susceptibility data of Beauchamp and Rosenbaum...... 46

2-7 Recent AC susceptibility data on Sr 3CuPt. 5 Iro.5O 6 taken by Beauchamp. 47

2-8 Magnetization versus field for Sr 3CuPtO.5 1r o.5 0 6 samples used in this thesis...... 48 2-9 Heat capacity data of Ramirez. In each case, the sample consisted of 0.2g of the Sr 3MM'0 6 material, and 0.2g of silver powder, which were compressed together to form a pellet. The pellet was then glued to the calorim eter [32]...... 49

3-1 Thermal circuit for Schwall model...... 53 3-2 Thermal circuit for Sullivan and Siedel model. The calorimeter and sample are assumed to be in excellent thermal contact, so that the slab with heat capacity Ct0, represents the entire sample/calorimeter assembly. Heat flux = -e"t (a is slab cross-sectional area) is applied at one end of the calorimeter by the heater, and temperature T = Td, + Tac is sensed at the other side of the calorimeter. Heat is dumped to thermal ground (the heat bath) through the weak link conductance K ...... 54

4-1 Overall view of apparatus. Only one calorimeter stage is shown. Note that devices (heater, thermometer) are not shown on the calorimeter. 61 4-2 Wireframe view of calorimeter stage. The calorimeter sandwich (sample dough between two quartz plates) is supported by vertical and horizontal vespel pegs. These pegs are glued into brass L brackets. .. 62 4-3 Top view of one calorimeter plate, showing configuration of heater, thermometer, and thermal link...... 62 4-4 Expected heat capacity of various calorimeter components. The N grease data below 0.4 K was obtained by linear extrapolation, which is reasonable for glassy materials [47]. The quartz data below 0.3 K is a T 3 extrapolation, which underestimates the true heat capacity [47]. Also shown are expected empty calorimeter heat capacity and measured empty calorimeter heat capacity from Sr 3CuPt. 5 Iro. 5O6 experiment, and Sr 3 CuPtO.5 1r o.5 0 6 heat capacity measured here (below 1K) and by Ramirez [15](above 1K)...... 65

14 4-5 Expected thermal conductance of various calorimeter components. The

PtW entry is for 16 1 mil diameter PtW wires, each 21 inch long (representing the heater and thermometer lead wires). The NbTi entry is for the same number and lengths of 5 mil diameter NbTiwires. The quartz/sample boundary resistance is taken to be the same as that between copper and glue...... 66

5-1 Monoclinic unit cell of K3Fe(CN)6 . Closed spheres represent Fe, open spheres K, open diamonds C, and closed squares N. 3, the angle between a and c, is approximately 107', and a = 7.04A, b = 10.44A, and c = 8.4A. Six cyanide groups surround each Fe, in approximately octahedral coordination. Here, only two Fe ions are shown with all cyanide groups. The closest cyanide groups on nearest neighbor sites are 2.74A apart on the chain axis a. For nearest neighbors along b, the closest cyanide groups are 6.14A apart...... 70

5-2 Apparatus used to measure AC heat capacity for Sr 3CuPtO.5 Iro. 5 0 6 and K3Fe(CN)6 experiment...... 72 5-3 LR-400 bridge transfer function measured with JFET ...... 74 5-4 Effect of LR-400 transfer function on empty calorimeter 102 mK thermal transfer function. The transfer function is normalized for power. 75 5-5 Typical power/temperature curve for K3Fe(CN)6 calorimeter, without baseline drift correction...... 76 5-6 Typical power/temperature curve for K3Fe(CN)6 calorimeter, with baseline drift correction...... 77 5-7 Empty calorimeter thermal transfer functions, measured and as calculated using the two-wire model and published data on calorimeter m aterials...... 78 5-8 Upper limit on AC empty calorimeter heat capacity...... 79 5-9 K3Fe(CN)6 calorimeter 108 mK thermal transfer function, with fit to two-wire model and resulting SSGI transfer function...... 80 5-10 K3Fe(CN)6 calorimeter 140 mK thermal transfer function, with fit to two-wire model and resulting SSGI transfer function...... 80 5-11 K3Fe(CN)6 calorimeter 303 mK thermal transfer function, with fit to two-wire model and resulting SSGI transfer function...... 81 5-12 K3Fe(CN)6 calorimeter zero field AC and relaxation heat capacities, with data for K3Fe(CN)6 single crystals published by Fritz...... 83 5-13 K3Fe(CN)6 calorimeter field-dependent AC heat capacity...... 84

6-1 Typical power/temperature curve for the Sr 3CuPtO.5 1ro. 5 0 6 experiment, with drift correction...... 90 6-2 Apparatus used to measure relaxation heat capacity for Sr 3CuPtO.5 1ro.5 0 6 experiment...... 91 6-3 Raw AT(t) (bottom curve) and AT(t) after averaging ten decays (top curve). Top curve is shifted up by 2 mK for comparison...... 92 6-4 Empty calorimeter 128 mK thermal transfer functions...... 93

15 6-5 Empty calorimeter 400 mK thermal transfer functions...... 94 6-6 Empty calorimeter AC heat capacity with off-plateau correction, and relaxation heat capacity...... 96 6-7 Empty calorimeter AC (no off-plateau correction) and relaxation heat capacity, 0 kG and 1 kG...... 97 6-8 Empty calorimeter AC (no off-plateau correction) and relaxation heat capacity, 3 kG and 5 kG...... 97 6-9 Empty calorimeter low temperature AT(t), showing best fit to sum of two exponentials...... 99 6-10 Empty calorimeter zero field thermal conductance to bath Kb, measured and predicted from published data on copper...... 99 6-11 Field dependence of Kb...... 100

6-12 Sr 3CuPtO.5 Iro.50 6 calorimeter 136 mK thermal transfer functions. . 103 6-13 Sr 3CuPtO.5 1r o.5 0 6 calorimeter 400 mK thermal transfer functions. . 104 6-14 Sr 3CuPtO.5 Iro. 5 0 6 calorimeter AC heat capacity with off-plateau correction, and relaxation heat capacity data...... 106

6-15 Sr 3 CuPtO.5 1ro. 5 0 6 calorimeter AC (no off-plateau correction) and relaxation heat capacity, 0 kG and 1 kG...... 106

6-16 Sr 3CuPtO.51rO. 5 0 6 calorimeter AC (no off-plateau correction) and relaxation heat capacity, 3 kG and 5 kG...... 107

6-17 Sr 3CuPtO.5 1ro. 5 0 6 calorimeter AC heat capacity data above 1 K. .. . 108 6-18 Excess Sr 3CuPtO.s5 ro.0 O 6 calorimeter AC heat capacity data above 1 K (see text)...... 108 6-19 Sr 3CuPtO.5 Ir o. 6 calorimeter zero field thermal conductance to bath Kb, measured and predicted from published data on copper...... 110

6-20 Field dependence of Kb, Sr 3CuPtO.5 1ro.5 0 6 calorimeter...... 110

6-21 High field heat capacity of Sr 3 CuPtO.5 Iro. 5 0 6 and empty calorimeters. 111

7-1 Zero field u/T for Sr 3CuPtO.51r o.5 0 6 , fit below 0.4K...... 114 7-2 Zero field u/T for Sr 3CuPtO.5 1r o.5 0 6 , fit to all data...... 114

7-3 Zero field specific heat for Sr 3CuPtO.5 Iro.50 6...... 115 7-4 Low field u/T for Sr 3CuPtO.5 IrO.5O6 ...... 115 2 7-5 o at 10kG for Sr 3CuPtO.5 Iro. 5 0 6 , and fit to a = AT3/ + B...... 116

7-6 Specific Heat of Sr 3CuPtO.5 IrO. 5 0 6 as a function of field at 130 mK. . 117 7-7 Comparison of various paramagnetic salt specific heats with

Sr 3CuPtO.s5 r o.5 O 6 . The dotted line is an extrapolation of the power law predicted by theory for CPI in the universal regime...... 123

A-i Thermal circuit for case 1. Temperature Tt is measured with the bottom calorimeter thermometer, referenced to the bath temperature. Power Qt0, is applied to top heater or Qbot is applied to bottom heater. 128 A-2 Thermal circuit for case 2. Temperatures is measured with the bottom (Tt) or top (Tt0p) calorimeter thermometers, referenced to the bath temperature. Power Q is applied with the bottom heater...... 130

16 C-1 Thermal circuit for two-link Schwall model. All temperatures are referenced to the bath temperature, and T2(t) is measured...... 136 D-1 Thermal circuit for calculation of Schwall model transfer function. The heater applies flux q at position 0. The thermometer is located at position m ...... 140

17 18 List of Tables

5.1 K3Fe(CN)6 calorimeter K, estimates. K, was determined from a fit of the transfer function to the two-wire model (2WTF Fit), and a lower limit on K, was determined by comparison of the data and calculated two-wire transfer functions with various K. (TF Low. Lim.)...... 82 5.2 Heat capacity data taken with relaxation method. Note that the empty calorimeter results ("Empty Cal. C") are an upper limit due to the low LR-400 bandwidth, and are for 68 mg of N grease. The K3Fe(CN)6 calorimeter results ("K 3Fe(CN)6 Cal. C") are for 6.44 mg K3Fe(CN)6 and 13.7 mg N grease. The "Fritz C" column is the heat capacity of 6.44 mg of K3Fe(CN) 6 from the Fritz paper. See text for explanation of the "errors" in the Fritz data...... 85

6.1 Estimates of empty calorimeter K.. K, was determined from a fit of the transfer function to the two-wire model (2WTF Fit), and from the slopes of the power/temperature curves and Equation (6.1) (PT Slope). Lower limits on K, were determined by comparison of the data and two-wire transfer functions with various K, (TF Low. Lim.), and by varying the power/temperature curve slopes appropriately by one standard deviation (PT Low. Lim.)...... 95 6.2 Relaxation heat capacity from various thermometer/heater combinations on the empty calorimeter. The notation Cth refers to heat capacity measured with thermometer t, heater h ...... 100 6.3 Parameters for Schwall model of empty calorimeter. The values with an "x" next to them are from fits of the measured AC transfer function to the Schwall model. The Cu Fing. column shows estimated heat capacity for two sets of copper fingers. The Vespel column shows estimated heat capacity and conductance for the Vespel pegs. . . . . 102

6.4 Sr 3CuPtO.5 1ro.5 0 6 calorimeter K,. K, was determined from a fit of the transfer function to the two-wire model (2WTF Fit), and from the slopes of the power/temperature curves and Equation (6.1) (PT Slope). Lower limits on K, were determined by comparison of the data and two-wire transfer functions with various KS (TF Low. Lim.), and by varying the power/temperature curve slopes appropriately by one standard deviation (PT Low. Lim.)...... 105

6.5 Parameters for Schwall model of Sr 3CuPtO.5 Iro.5 0 6 calorimeter. .. .. 109

19 6.6 Relaxation heat capacity from various thermometer/heater combinations on the Sr 3CuPtO.5 Iro.50 6 calorimeter. The notation Cth refers to heat capacity measured with thermometer t, heater h...... 109

20 Chapter 1

Introduction

One-dimensional spin systems ("spin chains") have been a subject of interest in condensed matter physics and statistical mechanics for many decades. Originally, such systems were of interest because theoretical problems in many-body physics, phase transitions, and critical phenomena were more tractable in one dimension than in three; hence, it was hoped that a greater understanding of critical phenomena in our three-dimensional world could be obtained by solving the corresponding one- dimensional problem [1],[2]. In the late 1960's and early 1970's., powerful theoretical methods such as the Renormalization Group were developed and applied to these many-body one-dimensional problems. It quickly became clear that the hope of "ex- trapolating" one-dimensional results to three-dimensional systems would not be re- alized, as the behavior of spin chains exhibited profound qualitative differences from that of three-dimensional systems. Around the same time, three-dimensional materials whose behavior approximated that of the one-dimensional systems were discovered in the laboratory, and the study of one-dimensional spin systems became a subject of interest in its own right. This study continues today, and has been characterized by a remarkable interplay between theory and experiment, due to the ability of the- orists to solve many one-dimensional problems on the one hand, and the ability of experimenters to produce materials to which these theories apply on the other. Why do one-dimensional spin systems differ from three-dimensional? Consider two spin systems, one one-dimensional and the other three-dimensional, in which nearest-neighbor spins interact with a coupling of strength J. This coupling could be produced by (for example) direct exchange, superexchange, or dipole interactions. For temperatures kBT > J, the interaction between the spins is unimportant; in this temperature range, the systems will behave identically. They will have zero magnetic heat capacity (in zero field), and their susceptibility X will obey the Curie Law x = c/T. At thermal energies kBT ~ J the systems will behave quite differently. The primary factor that distinguishes their behavior is the relative importance of thermal fluctuations of the spin degree of freedom. Consider a hypothetical Ising chain (that is, one in which the spin can have only two values, up or down) in which all spins are aligned (Figure 1-1). If a fluctuation occurs at a single site, the chain is split into two sections, one in which all spins are up, the other in which all are down. The energy cost of such a fluctuation is 2J, where J is the coupling between nearest

21 I\ lIT1 II l l I I

Figure 1-1: Ordered Ising chain at T = 0 (top) and Ising chain with fluctuation (bottom). neighbor spins. However, since the fluctuation can occur at any of the N sites in the chain, the entropy gain is k, In N. Hence the free energy change for this fluctuation is AF = 2J - kBT inN (1.1)

For the ordered state to be stable, AF > 0, that is N < e 2 J/kBT. For a macroscopic sample, this condition is likely not to hold. Hence it is .easy" to introduce thermal fluctuations into the chain and destroy the order [3].

Figure 1-2: Ordered Ising net at T = 0 (left) and Ising net with fluctuation (right).

The same is not true in higher dimensions. For example consider a two-dimensional net of aligned Ising spins (Figure 1-2) with a fluctuation analogous to the one-dimensional case. Here, the energy cost is 2Jv/N, and the entropy gain kB In A. The stability condition is then N < e4J /kBT, which holds for large N. Hence we see that it is more difficult to destroy the order in higher dimensions. In fact, due to these fluctuation effects, a one-dimensional system will not exhibit

22 long-range order down to T = 0. On the other hand, short range order does develop in the chain as it is cooled below kBT - J, and the entropy of the system is gradually reduced over a wide temperature range through this short-range ordering. Hence the correlation length (T), that is the length in the chain over which this short-range order is maintained, is a quantity of great interest in the theory of one-dimensional spin systems, and increases slowly as temperature is reduced. Within one dimension, there are other properties of the chain that are important for determining its behavior. One is the nature of the interaction J between the spins; in particular, whether it is ferromagnetic (FM) or antiferromagnetic (AF), whether it depends on the spin directions, and whether it involves only nearest neighbor spins. If the interaction depends on spin direction, we may have Ising (if J2, Jy = 0, J, # 0), XY (if J1 = Jy :L 0, JZ = 0), or Heisenberg (if Jx = Jy = J, = 0) spins, all of which show different behavior. Another important property is the magnitude of the spin. For example, a spin-J chain behaves quite differently from a spin- chain. In the next section, I will consider effects of the spin magnitude in particular. This will highlight the differences between quantum spin chains (with spin near 1) and the more intuitive classical chains (with S - oc) that are important for an understanding of this work.

1.1 Properties of Spin Chains

Because the theory of Sr 3CuPtjpIrpO6 assumes a nearest-neighbor Heisenberg description for the spin interactions, I will also assume that for the chains I describe below. In the Heisenberg model, the interaction between nearest-neighbor spins is isotropic. The Hamiltonian is written

L-1 L-1 H = JZ Si -Sii - pHz E Si (1.2) i=O i=O where the sum is taken over all sites in a chain of length L (that is, having L sites; length will always be measured in units of the lattice constant), and the Si are in general quantum spin operators. For J < 0, we have a Heisenberg ferromagnet; for J > 0, an antiferromagnet. For simplicity, I will take H, = 0. I will also focus primarily on the entropy and specific heat. (In this thesis, specific heat is taken to be n! dT'I, with units J/mol/K, and heat capacity Q/dT, with units of J/K.) I begin with the classical spin chain. In this case, the spin Si is treated as a classical vector. While this may seem like an unphysical idealization (real spins are quantum), the classical approximation already works quite well for spin- at temperatures not too close to T = 0 [2]. However, it is considered here because it follows intuition most closely, and so will help to highlight the non-intuitive properties of the quantum chains. In order to obtain a classical vector from the quantum operator Si, one has the sense that the limit S -+ oc should be taken, as this will lead to an infinite number of possible Sz, Sy, S,, as is the case for a classical vector. In order to take this limit, define the unit operators si = Si/S [4], and note that the commutation relations for

23 these unit operators are, for example,

ss - ss = (1/S)is (1.3)

Also, set JS2 = Jc; then the Hamiltonian (1.2) becomes

L-1 H = Jc E Si *si+1 (1.4) i=o Given the commutation relations (1.3), in the classical limit S -+ oo all spin operators commute. Hence for the FM case, the ground state (attained only at T = 0) will have all spins pointing in the same direction, and in the AF case nearest neighbor spins will point in opposite directions. Excited states can be formed by changing the relative orientations of nearest neighbors by an infinitesimal amount (in this sense, the T = 0 long-range order for the classical chain is even less stable than that of a quantum FM chain; i.e., the energy cost of excitation is infinitesimally small, whereas the entropy gain is still k, ln N). The classical nature of the spin leads to unphysical results for the entropy and heat capacity; at infinite temperature, the entropy is expected to be infinite; also, Fisher found for the specific heat [4] (Figure 1-3)

1.0

0.8 -Classical o FM * AF

0.6

0.4 - - - 0

0.2 00 00 0 0 0 CO 0 K I I I 0 0.2 0.4 0.6 0.8 1.0 kT/J

Figure 1-3: Specific heat of various types of spin chains. The AF and FM chain results were taken from [5], extrapolated to zero temperature using simple spin wave theory, and the classical result from [4]. These results are plotted assuming the Furusaki Hamiltonian (1.2); the Bonner and Fisher Hamiltonian uses 2J rather than J.

24 2

c =1 c =- 1 ( 2k,2 TB sinh T ~ hJ 2 ( 151.3 c(T) does not approach zero for T -+ 0, so the entropy S(T) = f6 c(T')/T'dT' approaches minus infinity as T -+ 0. Hence near zero temperature, the classical approximation will not describe any real system and an appropriate treatment will have to consider the spins to be quantum in some way. Another important observation about the entropy and specific heat in the classical chain is that they are the same for both the FM and AF chains. I turn now to the FM spin-} chain. In this case the Si do not commute, and it is not immediately clear that the classical ground state, with all spins pointing in the same direction, will also be the spin-! chain ground state. However, a straightforward analysis [6] shows that the ground state is indeed the same as in the classical case, and that the lowest-lying excited states at T = 0 can be described by introducing spin waves into the chain. A spin wave, or magnon, reduces the total chain magnetization by one unit, and is a traveling wave with dispersion hw = 4SJ(1-cos ka). A schematic representation of a FM spin wave is shown in Figure 1-4. For temperatures above T = 0, spin waves are still expected, although the approximation S_ ~ S on which the T = 0 analysis relies no longer holds [7]. Also, the absence of order in the id chain at finite temperature changes the spin wave spectrum. In particular, spin waves with wave vector k less than the correlation length ((T) will exhibit overdamped behavior, whereas spin waves with larger k will continue to exhibit oscillatory, traveling wave behavior [2]. These facts, along with the observation that spin waves do not obey the superposition principle [6], make it seem unlikely that it will be possible to compute the specific heat using a Debye-like model with the small wave-vector approximation w = 2SJk2 of the dispersion for the T = 0 spin wave. The Debye-like model predicts c oc v/Y; detailed numerical calculations by Bonner and Fisher [8] confirm that the specific heat has this temperature dependence at the lowest temperatures, although the amplitude of the v'T term is a factor of 1.3 smaller than is predicted by spin-wave theory. The Bonner and Fisher result is plotted in Figure 1-3.

T7TT7TVT

Figure 1-4: Representation of T = 0 FM spin wave. Rather than reducing the chain magnetization by breaking the chain at a single point, which costs energy 2J, the magnetization is reduced in a spin wave by tilting all spins slightly off the z axis, which costs much less energy. The spin wave is a traveling wave in which the difference in azimuthal angle between adjacent spin sites in the wave is constant.

25 For the AF spin-} chain, the classical ground state is not an eigenstate of H. In 1931, Bethe found the ground state eigenfunction of the spin-! chain, and showed that it was not ordered: that is. this chain does not exhibit long-range order even at T = 0 [9]. Given this, the physical nature of the T = 0 excited states is not clear. Nevertheless, des Cloizeaux and Pearson [10] have computed the dispersion for the low-lying excited states and found w o( Isin kal, which is the same k dependence found if one assumes the classical ground state for T = 0 and computes the spin wave spectrum [11]. However, there are some differences between the true excited states and the classical ones (for example, the true first excited state is a triplet, whereas the classical state is a doublet), and the physical nature of the excitations remains unclear. Again using a naive Debye-like model to obtain the temperature dependence of specific heat, one expects c cx T (for w c< k dispersion, valid for long wavelengths). Surprisingly, this temperature dependence is confirmed by the Bonner and Fisher results, although the amplitude of the linear term is a factor of three smaller than the spin wave result (experiment also supports the Bonner and Fisher results [1]). The Bonner and Fisher result is shown in Figure 1-3. To summarize, the 1D classical model yields the AF and FM ground states that we intuitively expect, and shows high-temperature behavior that approximates the behavior of chains with S = 5 and larger. However, its low-temperature behavior is unphysical, and its thermodynamics is the same for both AF and FM chains. The spin-- FM ground state is the same as the classical one, and the T = 0 excitations are spin waves, each of which lowers the total spin of the chain by one quantum. At temperatures much above T = 0. it is not clear that the simple spin wave picture is valid; however, it gives a qualitative account of the low-temperature specific heat. For the spin-j AF, the ground state is not the same as the classical one; moreover, it is disordered. However, assuming the classical ground state and computing the resulting spin wave spectrum again leads to an accurate qualitative account of low- temperature specific heat. For our purposes, the most critical feature of the quantum picture is the result that the excitations near T = 0 are spin waves, with differing dispersions for the FM and AF cases.

1.2 Sr 3 CuPt 1 _pIrpO 6

Sr 3 CuPt1 _pIrpO 6 is one member of a family of new one-dimensional magnetic materials with general formula Sr 3MM'O6 , where M and M' refer to sites in the crystal structure that will accept various magnetic and non-magnetic ions. Work on

Sr 3 MM'0 6 began in 1991 with the discovery of Sr 3CuPtO6 by Wilkinson et al. [12]. Soon after, Nguyen and zur Loye [13] began a systematic study of the synthesis and magnetic properties of several members of this family, including Sr 3CuPtO6 , Sr 3CuIrO6 , and their alloy Sr 3CuPtlpIrpO6 . The general crystal structure of Sr 3 MM'0 6 is shown in Figure 1-5. The structure consists of chains of alternating face-sharing MO 6 trigonal prisms and M'0 6 octahedra, with the chain axis parallel to the c-axis of the crystal. In the ab plane, one sees a hexagonal net of chains, with each chain surrounded by six clusters of

26 Q fiU: G * 0)

06. k C9 *

Figure 1-5: Crystal structure of Sr 3MM'0 6 . Left: view along the length of a chain, showing alternating MO 6 trigonal prisms (M represented by the small ball at center of prism) and M'0 6 octahedra (M' represented by the large ball at center of octahedron). Right: view down the c axis. Clusters of 3 Sr ions surround each chain.

27 three Sr+ ions. Typically, a = 9.6)1, while c = 11.2A (c = 6.7 A for Sr 3 CuPtO6 and

Sr 3CuIrO6 ) [13] [14]. c is four times the distance between M and M', due to the geometry of the trigonal prism and octahedron. Hence the magnetic ions on the M and M' sites are at most 2.8A apart, whereas sites on different chains are 5.5A apart. The separation between these two distance scales leads one to expect that the magnetism of these materials will be one-dimensional over some temperature range.

Interest in Sr 3 CuPti1 ,IrO 6 in particular was motivated by measurements of Ml/H at low fields (below 10kG) and at temperatures 2K < T < 300K for Sr 3CuPtO6 ,

Sr 3CuIrO6 , and Sr 3 CuPtjpIrpO6 . Nguyen [13] found that NI/H for Sr 3 CuPtO6 fit well to a one-dimensional AF Heisenberg model, with IJI/kB = 26.1K. For Sr 3CuIrO6 , Nguyen hypothesized that M/H showed one-dimensional FM behavior, with the peak in MT/H indicating a coupling on the order of J/kB ~ 30K. However, measurements of M/H on Sr 3CuPti_pIrpO6 showed Curie Law behavior down to 2K (Figure 1-6). This was quite surprising, given that J for the parent materials was around 30K. The chemistry of Sr 3 CuPtO6 indicates that a Cu2+ ion, with spin-), will be located in the M site. Cu 2+ generally behaves as a Heisenberg spin- [1]. Pt 4 + is located in the M' site, and has spin zero. Hence the AF behavior of Sr 3 CuPtO6 can be explained if two Cu 2+ on nearest M sites interact via superexchange through the Pt 4 + on the intervening M' site. Since superexchange is always an AF interaction [7], and since the M-M' distance is only 1.71 in Sr 3 CuPtO6 , this explanation is reasonable. 2 The chemistry of Sr 3 CuIrO6 leads to a spin-1 Cu + ion occupying the M site in this material as well. The M' site will be occupied by Ir4+, which also has spin-!. The FM 2 4 behavior of Sr 3CuIrO6 can be explained if the nearest-neighbor Cu + and Ir + ions interact ferromagnetically via direct exchange, which is reasonable given the Cu2+_ 4 Ir + distance of 1.7A. Magnetization versus field data at 5K for Sr 3 CuIrO 6 shows that only one-third of the full moment 2p, is obtained for fields up to 20T in powder samples [15]. If Sr 3 CuIrO6 were a Heisenberg FM, there would be no preferred direction and it would be possible to achieve the full moment in the powder samples at reasonably low field. This result suggests that there is some strong anisotropy in Sr 3 CuIrO6 that restricts the spins to align along a particular direction in the crystal. One possible reason for this anistropy is ferrimagnetic ordering of Ising-like spins on a hexagonal net. Such ordering could occur in Sr 3 CuIrO6 if a strong, AF, Ising coupling existed between short FM segments in different chains. Magnetization data on Ca 3 Co 2 0 6 [16], which is isostructural with Sr 3 CuIrO6 , has been interpreted in terms of this type of Ising ferrimagnetism. Of course, if ferrimagnetism is the correct explanation of the M versus H data, then Sr 3CuIrO6 is not one-dimensional below

5K. On the other hand, this does not mean that Sr 3 CuPt0 5. Ir 0 .5 0 6 could not be one-dimensional below 5K. In any case, possible consequences of such anisotropy for

Sr 3 CuPt0 .5 Ir 0 .5 O6 will be discussed further in Chapters 2 and 7. Given the magnetic structure of Sr 3CuPtO6 and Sr 3CuIrO 6 , we expect the alloy

Sr 3 CuPtj.pIrpO6 to consist of spin-! chains with a random distribution of nearest- neighbor FM and AF bonds, with probability p that a given bond is FM, 1 - p that it is AF. This model has been considered theoretically by P.A. Lee and his collaborators in a series of papers, and will be discussed in detail in Chapter 2. This theory was found to give a good account of the unusual Curie Law behavior (Figures 1-7 and

28 Sr Sr 3CuPtO6 3 CuIrO6 0.006.

I =:.Z1 0.6 - 0.005 - :u6.1 K K S 0.004 - 0 0.4 -

0.003 - S 0.2 - 0.002 - S I-D Halse::er3 Modei 0.001 0.0 Sp*------6 50 100 150 :00 250 300 0 5b 100 150 200 250 300 Temperaturz K Temeraure(Kj

Sr3Cu Pt0 .I r05 06 0.04 0- E 003 0 E 0.02 * 0.01 - *@*00... 0 0 10 20 30 40 50 Temperature [K]

Figure 1-6: M/H data for (top left) Sr 3CuPtO6 , (top right) Sr 3CuIrO6 , and Sr 3CuPtO.5 IrO.5 0 6 (bottom).

29 1-8), and made several predictions about the low-temperature (< 2K) properties of the alloy. In particular, the alloy was predicted to have an unusually large spin entropy content at low temperatures (10 - 30% of R ln 2, depending on p), and at the lowest temperatures to exhibit behavior characteristic of a new universality class of random quantum spin chains, with characteristic scaling laws in specific heat and susceptibility.

800 * Sr 3CuPtO6 A Sr 3CuPto.75Iro.2506 I

Tv Sr 3CuPto.soIro.5006 A 600 * A F + Sr3CuPto.:Iro.75Ose6 A * SrsCuIrO6 *V A V A A 400 I A V AA U A V 200 A* T : VT .MMMMU 0 0 50 100 150 200 250 300 Temperature (K)

Figure 1-7: 1/x data for Sr 3CuPtl-,IrO 6 taken by Nguyen [15].

30 20.0

15.0 h

10.0 -

5.0 p=1

0.0 0.0 4.0 8.0 4k BT/J

Figure 1-8: 1/x theoretical prediction of Lee and collaborators [17].

31 1.3 Specific Heat of Sr 3CuPt0 5.1r0.o50 6 below 1K

The goal of this work is to investigate the specific heat of one particular alloy, Sr 3 CuPt0 .5 Ir 0 .5 0 6 , at temperatures below 1K. Of course, one reason to measure specific heat of Sr 3CuPto.5 Iro.50 6 below 1K is to discover whether it is a material that obeys the theory. However, there is another reason for interest in Sr 3CuPto.5 Iro. 5 0 6 (and Sr 3 CuPtipIrpO6 in general). If it does obey the theory, or at least is found to have a substantial low-temperature entropy content, the material may be useful for refrigeration via adiabatic demagnetization. Adiabatic demagnetization of paramagnetic salts was the only technique available for refrigeration below 300mK until the early 1970's, when dilution refrigerators became widely available. However, most dilution refrigerators do not provide access to temperatures below 10mK, and for temperatures below 1mK the only known refrigeration technique is nuclear demagnetization [18]. The hope is that Sr 3CuPt 0 .3Ir0 .3O6 may prove superior in some way to dilution refrigerators or paramagnetic salts in the 10mK - 1K range, or that it may even provide competition for nuclear demagnetization in the low millikelvin to 100pK range.

Sr 3CuPt0 5 Ir 0.5 0 6 is focused on for two reasons: first, the most extensive experimental work thus far has been done on Sr 3CuPto.5 Iro.50 6 ; second, the distribution of FM and AF bonds in this particular alloy is (in principle) completely random. Specific heat is measured for several reasons. First, no other groups have yet reported specific heat measurements on Sr 3CuPt 0 .5 Ir 0 .50 6 below 1K. Second, specific heat gives the most direct access to entropy content. Third, if Sr 3CuPt0 .5 Ir 0 .5 0 6 undergoes a transition to long-range (two or three-dimensional) ordering at some temperature, specific heat provides a definitive signature for such a transition. Finally, if there are phases in addition to pure Sr 3CuPto.5 Iro. 5 0 6 or magnetic impurities of some kind in the measured Sr 3CuPto.5 Iro.05 6 sample, these are likely to have lower entropy content than Sr 3CuPt 0 5. Ir 0 .30 6 . Therefore the presence of such phases should not confuse the comparison of the specific heat data with theory.

In order to achieve this goal of specific heat measurements on Sr 3CuPt0 .5 Iro.5 0, it was necessary to design, build, and test a new experiment for specific heat measurements below 1K. The resulting apparatus allows for specific heat measurement from 100mK to 2K in magnetic fields to 7T using either AC or relaxation calorimetry. While based on earlier designs, the apparatus has a few unusual features. In particular, it allows for accurate AC calorimetry below 1K, incorporates a non-destructive sample mounting technique, and uses a new arrangement for relaxation calorimetry. The experiment was tested by measurements of the field-dependent specific heat of potassium ferricyanide K3Fe(CN)6 , a common material used in blueprinting and photography that behaves as a one-dimensional Ising AF below 1K and has a (three- dimensional) Nel transition at 130mK. This work is organized as follows. In Chapter 2, I discuss the theory of random quantum spin chains, developed by P.A. Lee and his collaborators, and experimental work to date on Sr 3CuPti-IrpO6 . In Chapter 3, I describe in detail the two methods used to measure specific heat, and mathematical models for these methods. In Chapter 4, the design and construction of the apparatus is described. In Chapter

32 5, measurements and results on K3Fe(CN)6 are described, and in Chapter 6, measurements and results on Sr 3CuPtO.*5 ro. 5 0 6 . Finally, in Chapter 7 I conclude with a presentation of the final results for specific heat of Sr 3CuPtO.5 1ro.5 0 6 , discussion of the results, and recommendations for future work.

33 34 Chapter 2

Random Quantum Spin Chains and Sr 3 CuPtipIrpO6

2.1 Theoretical Work on RQSC

The statistical mechanics of random quantum spin chains has been studied by Lee and collaborators [19] [5] [20] [21] [22] in a series of papers. They modeled Sr 3 CuPtIIrpO6 as a spin-! Heisenberg chain with Hamiltonian

L-- L-1 '= JiSi - Si+ - MHz E SZ (2.1) i=O i=O The Ji are given by the probability distribution

P(Ji) = p3(Ji + J) + (1 - P) (Ji - J) (2.2) This distribution indicates that a given nearest-neighbor bond will be of strength J and have probability p of being ferromagnetic and probability 1 -p of being antiferromagnetic. Hence the chain can be pictured as a collection of connected ferromagnetic and antiferromagnetic chain segments, where the typical lengths of the FM and AF segments depend on p. Furusaki et al. applied high-temperature expansion [23] [24] and transfer matrix methods [25] [26] [27] to analyze this system, and postulated the following physical picture based on their numerical results. For kT > J, the spins are decoupled and the system behaves as an ordinary paramagnet. For kBT < J, FM or AF correlations grow within the segments. Excitations of very short segments (a couple spins in length) will have energies of order J, and so will not contribute significantly to the thermodynamics in this temperature range. Longer segments can be described in a spin wave picture, with a spin wave population in each segment determined by kBT, the length, and the type (FM or AF) of segment. Since FM and AF segments have different spin waves dispersions, a spin wave from a FM segment will not propagate into the adjacent AF segments (and vice vesa) (Figure 2-1). Hence the thermodynamics of the system in this temperature range is the same as that of a collection of relatively short, decoupled segments. Since the segments are relatively short, there

35 is a substantial energy gap between the ground (zero spin wave) state and the first excited (one spin wave) state. Hence for kBT < J, these decoupled segments will be found in their local ground (zero spin wave) state, which is the state of maximal (minimal) total spin for a FM (AF) segment. In this low temperature regime, a FM segment containing n sites (for example) therefore behaves like a single "large spin" of spin n/2.

Figure 2-1: Schematic view of spin wave confinement. Due to different dispersion relations for FM and AF segments, spin waves from a given segment do not propagate into adjacent segments. A mechanical analogy is a rope with discontinous changes in thickness.

One might wonder whether the presence of these decoupled. large spins could be detected via a change in. for example, the Curie constant c = xT. The argument is most easily made for classical spins of magnitude So at each site rather than quantum spins-! as is the real case for Sr 3CuPt 1 -IrpO 6 ; however, the argument holds for the quantum case as well [17]. For kBT > J, the susceptibility per site x/N is just

2S2 x/N = 2 (2.3) 3kBT

3kBT

For kBT < J, short range order forms, with neighboring spins locked either parallel or antiparallel to each other (depending on whether their bond is FM or AF) in clusters of length (the correlation length). Note that these clusters will contain both FM and AF bonds-in the classical case, the spin wave picture does not apply and so there is no decoupling of FM and AF segments. In fact, a simple change of variables maps the classical problem to a simple classical FM chain without changing the energy spectrum [19]. However, these clusters form an effective spin Seff. To compute Seff for p = 0.5, one moves from one site to the next in the cluster, adding the spin at each site to Seff. For p = 0.5, there is an equal probability of increasing or decreasing Seff by one unit at each site. Therefore (S'2f) can be computed in a random walk

36 picture, with the result (Sjff) = So. In this case the total susceptibility is

N /p (2n f) (2.5) x - 3kBTB(Sf Hence the susceptibility per site is

xN (BSo) 2 (2.6)

2 (IB3kBT So) (2.7) The Curie constant is the same for the two temperature regimes. For a quantum spin S, the Curie constants are slightly different- for kT > J, S2 is replaced by S(S +1) above, and for kBT < J So is replaced by S [17].

Furusaki et al. found that their numerical methods began to fail at an energy scale kBT - J/5, and that the entropy per site obtained by integrating C/T from infinite temperature down to J/5 was less than the total entropy per site kBln2. The amount of this "missing" entropy varied with p: 12%, 24%, and 36% for p = 0.25, 0.5., 0.75 respectively. As a test of their physical picture of decoupled segments, they assumed that this picture applied at J/5 and computed the amount of entropy in the decoupled segments. A match between this amount and the amount "missing" from the HTE and TM calculations would lend credence to the decoupled segments picture.

In order to compute the amount of entropy in the decoupled segments picture, Furusaki noted that a FM segment containing I bonds has entropy kB In 1, an AF segment with I odd has zero entropy, and an AF segment with 1 even has entropy k, ln 2. (Furusaki determined that the boundary spins, which are shared by adjacent FM and AF segments, must be assigned to the AF segment.) Then the entropy contribution per site is given by S = E (2.8) N

where the sum is over all segments, and N is the total number of sites in the chain. This can be rewritten

S AFsegs SAF + ZFMsegs SFM (2.9) ZAFsegs nAF + FMsegs nFM that is, the sums are taken over the AF and FM segments separately. Noting for example that EFMsegs nFM = (nAF)N,AF, where Ns,AF is the number of AF segments, and that the number of FM segments is equal to the number of AF segments (they may differ by 1, which can be neglected for a chain with a large number of sites),

S(SAF) + (SM)(2.10) (rAF)+ (rFM)

37 Given the probability densities for FM and AF segments with 1 bonds,

PF(1) = - P)P'1 (2.11) PA (l) = p(1 - p)' (2.12) the quantities in (2.10) are found to be

00 (SAF) ln(2S +1) A (2m) (2.13) m=1

(SFM) PF(l) ln[2S(l - 1) + 1] (2.14) 11 1 +p (nAF) (2.15) p (2.16) (nFM) i-p From these, s can be computed as a function of p. Good agreement is found between s(p) found in this way and the amount of missing entropy found by Furusaki et al. from their HTE calculations (Figure 2-2). Hence the decoupled segment picture seems to provide a reasonable model for this low-temperature (kBT < J/5) regime.

I I I I ' I I 0.4

U.3 Model o HTE

0.2

0.1

0 I I I I I I 0 0.2 0.4 0.6 0.8 1.0 P

Figure 2-2: Missing entropy as a function of p, calculated with (line) Equation 2.10 and (open circles) HTE.

Since there is no frustration in this disordered system, one expects all of the missing entropy to be removed from the system between kBT - J/5 and T = 0 .

38 Furusaki et al. proposed that the coupling between segments, while much smaller than J, is not zero. To test this proposal, they carried out exact diagonalization of Hamiltonians of finite length chains (around 10 spins) with various configurations of FM and AF bonds, and found that in the ground state adjacent segments were indeed coupled. Accordingly, the lowest-lying excited states were intersegment spin waves, that is spin waves in which total Sz, summed over all large spins in the chain, was reduced by 1 for each spin wave. These low-energy spin wave excitations were of course at much lower energy than spin wave excitations within an individual segment, where S, of an individual large spin is reduced. Hence the "missing" entropy remaining below J/5 would be removed from the system by correlations between the segments that arise at much lower energies (temperatures) than the correlations within the segments. The sign and strength of the couplings between segments depended on the ordering (FM or AF) and size of the adjacent segments. This work on exact diagonalization of finite chains motivated Westerberg et al. to consider the Hamiltonian '= JiSi (2.17)

where the magnitude and sign of the Ji and the size of the spins Si are random (Figure 2-3). Here, the Si represent the large spins discussed above. Westerberg et al. took this Hamiltonian to model the low-energy properties of the RQSC, and attacked it using a generalization of the real-space renormalization group (RSRG) method developed by Dasgupta and Ma [28] for the study of random AF spin-! Heisenberg chains. The physical rationale for the RSRG is that at a given temperature T, the spins coupled by J, > kBT will have frozen into their local ground state, and the thermodynamics will be determined by spins coupled by Ji < kBT. Hence as the chain cools, the spins remaining to participate in the thermodynamics will be coupled by progressively weaker Ji, and the distribution of spin sizes and bond strengths will change (see Figure 2-4).

1 2 5/2 6 1/2 0 0

Figure 2-3: Spin chain considered by Westerberg et al. (Equation 2.17). The FM segments form large effective spins, separated by AF segments of variable length.

39 S 1 SL SR S2

A1 A0 A 2

RENORMALIZE

SI SL+ SR S2

A1 A2

Figure 2-4: Schematic of the RSRG method. A renormalization step proceeds by first identifying \O, the strongest bond in the chain. Then the two spins linked by .\o are frozen to form an effective spin SL +SR, and the nearest neighbor bonds renormalized to i\1 and Z12 . As the chain cools below kBT - Ao, it effectively carries out a step in the RSRG.

40 They found that the distributions of spin sizes and bond strengths in the chain flow to fixed-point distributions as the RSRG proceeded. From these fixed-point distributions of spins and bonds, scaling laws governing the thermodynamic properties of the system in the universal regime can be computed. For finite temperature T, large spins coupled by bonds weaker than kBT can be considered to be free spins, since in the universal regime the bonds are typically much weaker than kBT. Hence the (zero-field) entropy per site will be given by

s(T, H = 0)/L oc keln(2(S) + 1)/n (2.18)

where (S) is the average spin size (computed from the distribution found via RSRG), and n the number of spins-! per large spin. In the universal regime, they found n ~0 2, where A0 is the strongest bond in the chain and a = 0.22 ± 0.01. Also, (S) ~ A . Hence s(T, H = 0)/L oc akBT 2, | InT (2.19) and for the specific heat -(T)/L oc| InT I T (2.20) The effect of an applied field H will depend on how the magnetic energy p(S)H compares with kBT. If p(S)H < kBT, the field will not be able to overcome the thermal fluctuations of the free spins, and the specific heat will still be given by (2.20). On the other hand if p(S)H > kBT, then the field will start to align spins with couplings of A0 ~ p(S)H or less. Using the scaling result for (S), the energy scale at which the field starts to align spins is thus AH ~ H 1/(1+"). Hence if one were to study, as a function of applied field, the temperature at which the entropy started to decline rapidly, one would find the relation T ~ H1/('+"). The work of Westerberg et al. provides at least two methods to search for evidence of universal behavior via specific heat. First, one could simply measure specific heat as a function of temperature in zero field and look for the expected dependence given by equation (2.20). Second, in an applied field one should find a peak-like structure in the specific heat at a temperature corresponding to the energy scale Ao' described above. By studying the way in which the temperature at which this peak occurred varied with field, one could attempt to verify the relation T ~ H1/(1+c). Further theoretical work on the RQSC was done by Frischmuth and Sigrist [22]. They applied the continuous time quantum Monte Carlo loop algorithm [29] to the Hamiltonian (2.17) and assumed an initial distribution

1 -J 0 < J < Jo P (J)= 2J' 0, otherwise

This initial distribution was chosen so that the algorithm would be accurate well into the universal regime and because Westerberg's theory should apply to it. Hence Frischmuth and Sigrist's results provide an independent check on Westerberg et al.'s work. Frischmuth and Sigrist found that their results agreed very well with those of Westerberg; in particular, they found a = 0.21 ± 0.02, whereas Westerberg found

41 a = 0.22 ± 0.01. The theory presented above provides a consistent description of systems with Hamiltonian given by (2.1). Experimentally, it is critical to know whether and how the theory applies to Sr 3CuPtiJIrpO6 in particular. Note that the Hamiltonian differs from Sr 3CuPt1_JIrpO 6 in at least four ways: first, the magnitude of the Ji are different for FM (50K) versus AF (26K) bonds; second, the FM bonds always occur in pairs; third, in light of the high-field magnetization data on Sr 3CuIrO 6 (Chapter 1), it is not clear that it is appropriate to treat the FM segments as Heisenberg; finally, couplings between the chains are neglected. Regarding the first two differences, it is thought [5] that bond randomness is the factor that will determine the physics of Sr 3CuPtjJIrpO6 . Hence, within certain limits, the quantitative details of the bond distributions will not change the basic physical description of the system as given by the model. Regarding the third difference, the theory requires that some kind of spin wave picture can be used to describe the FM segments. This will be the case if the FM interaction is Heisenberg or XY [30], but not if it is Ising. The fact that the theory gives an account of the high-temperature data (see Figures 1-7 and 1-8) suggests that a spin wave picture does apply, at least in that temperature range. More measurements may help to clarify the situation. For example, high-field measurements on Sr 3CuIrO6 well above 30K would be desirable as a control on the 5K result, as would high-field measurements at high and low temperature for Sr 3CuPtO.51r o.50 6 itself. As for the fourth difference, since the distance between chains is large (10A), wave functions of electrons on different chains will not overlap directly. Hence the chains will not interact via exchange interactions. A band structure calculation (using the extended Hiickel method) for an isolated NiPtO6- chain and for Sr 3NiPtO6 [31] shows that the band structure for an isolated chain is virtually identical to a chain in Sr 3NiPtO6 . It also shows that there are very few Sr states available at the energies occupied by the unpaired Ni electrons. The Sr is therefore unlikely to perturb the Ni electron wave function, so no superexchange coupling via the Sr is expected. Hence the strongest interchain interaction will be the magnetic dipole interaction. This can be estimated via [6] 1 a U ~ 13 7 ()(Ry (2.21) where r is the distance between moments and Ry the Rydberg constant. Naively plugging in the interchain distance for r, U ~ 1OmK. Hence it is at least plausible that Sr 3CuPtJIrpO6 will remain one-dimensional down to the lowest temperatures accessible by a dilution refrigerator. Pursuing the question of couplings between chains further, the temperature T, at which a system consisting of one-dimensional chains is expected to undergo a transition to long-range, three-dimensional order is given by [2]

kBT ~ J's2 d(T) (2.22)

where s is the spin at a site in the chain, J' the coupling between chains, and 1d the correlation length. This result is obtained from the simple argument that as correlations grow within the chain, the correlated sections can be treated as a single

42 large spin of magnitude s 1d, and that three-dimensional order will occur when the interaction between two such large spins in adjacent chains is comparable to the thermal energy. This "amplification" of T, by 1d should be quite large for p = 0, 1, whereas for intermediate values of p the bond disorder will limit 1d. Hence one might expect an interesting dependence of T, on p, which might give one some indication of the sizes of the large spins in the chains at the given p.

Ramirez has measured low-temperature specific heat on Sr 3CoPtO6 [15], in which cobalt has spin 2 and platinum spin zero or spin 1. This material was thought to be a possible example of a classical random spin chain, for which Lee and collaborators have also made predictions [19]. Ramirez found a long-range ordering transition at 1.4K [32]. Using the classical expression for 1d [2], and the measured Tc, I find J _ 4mK. This result lends further support to the conjecture that the coupling between the chains is very weak. I emphasize that for a classical chain, as discussed above, the disorder does not limit the size of (T), so one would expect a higher T, for the classical system than for Sr 3CuPt 1 _pIrpO 6 . However, even if there are no materials properties of Sr 3CuPt 1 _,IrpO 6 that obvi- ate its description by the theory, the energy scale at which the actual distributions of bonds and spin sizes for Sr 3 CuPt 1 _pIrpO 6 begin to match those of the universal distribution may be experimentally inaccesible. Westerberg et al. attempted to address this issue by beginning the RSRG procedure with a distribution simulating Sr 3CuPt1pIrpO6 for p = 0.8. They found that the approach to the universal distribution was very slow; indeed, (S) and (n) do not show pure scaling behavior until Ao/J ~ 101. For J ~ 40K, this leads to temperatures clearly below anything available experimentally for condensed matter systems. Nevertheless, measurements in the experimentally accessible temperature range are of value as a test of the theory (for example, what if scaling behavior was observed at an accessible temperature?), and to determine the true low-energy physics of Sr 3CuPt 1 _,IrpO6 (for example, it may undergo a phase transition to long-range, three-dimensional order).

2.2 Experimental Work on Sr 3 CuPt1_pIrpO 6 As mentioned in Chapter 1, DC magnetic susceptibility for temperatures 2K < T < 300K has been measured for Sr 3CuPt1_pIrpO6 with various p and compared with the theory of Furusaki et al.. Excellent qualitative agreement between the kBT > J/5 theory of Furusaki et al. and experiment was found [17]. (see Figures 1-7 and 1-8). These results encouraged further experimental study of Sr 3CuPt 1iIrpO 6 , particularly at lower temperatures where the decoupled segment picture and the theory of Wester- berg et al. might apply. AC susceptibility of p = 0.5, 0.667 samples has been measured on 50mK < T < 30K and in fields OkG < H < 2kG by Beauchamp and Rosenbaum. Also, specific heat of a p = 0.5 sample has been measured on 2.5K < T < 50K in zero field by Ramirez [15]. I discuss these measurements below. Beauchamp and Rosenbaum (BR) found a broad peak in x'(w) at 1.7 K for frequencies of 200Hz, 500Hz, and 3000Hz in both p = 0.5 and p = 0.667 samples (see Figure 2-6). The peak is suppressed with increasing frequency and field. BR also

43 measured magnetization as a function of field M(H) for both p = 0.5 and p = 0.667, and found results shown in Figure 2-5. These results indicate that M(H) begins to saturate at fields as low as 100 gauss at 5K. Assuming that M(H) can be described by a Brillouin function, and noting that for the Brillouin function saturation begins for pagH(2j+1)/2kBT ~ 1, 1 find j ~ 400. This result is inconsistent with Furusaki's theory, for which one would expect j of order 1. Further AC susceptibility measurements to 0.3 K were done very recently on new samples of p = 0.3, 0.5, 0.7 material by Beauchamp. For all three, a peak was observed in x' around 1.7 K, with the peak temperature increasing slightly with p. The peak is much broader than would be expected for a long-range ordering transition [34]. Moreover, the peak is strongly suppressed by low (0.5kG) fields, and the peak temperature does not appear to shift very much if at all (Figure 2-7). If this peak represented long-range ordering with a coupling J ~ 2K, one would expect to need fields of 10kG or more (kBT ~~J ~ pH) to alter the peak substantially, and would expect the peak to shift in temperature with increasing field. Overall, these measurements suggest some qualitative change in the behavior of the material, but this change is unlikely to be a transition to a three-dimensional, ordered ground state. Sigrist [33] has proposed a model to explain the results of Beauchamp and Rosen- baum. According to this model, there are (non-statistically) long FM sections (referred to by Sigrist as "FM clusters") in the chain, which may have arisen from inadequate mixing during sample fabrication. These clusters correlate to form very large effective spins at temperatures below the exchange energy of 40K. The presence of these large effective spins is the cause of the large j observed in M(H) at 5K. Around 1.7K, large effective spins on the same and different chains begin to correlate due to competing AF and FM interactions between them. A spin system with competing interactions (frustration) and some kind of disorder can form a spin-glass state [61]. Hence Sigrist attributed the broad peak at 1.7 K to a spin-glass transition at which these FM clusters correlate. The Sr 3CuPtl._IrpO6 samples (p = 0.5 and p = 0.8) used for this study were subjected to repeated grinding and firing cycles in hopes improving the randomness of the bond distribution. However, these samples showed the same M(H) behavior found by BR (Figure 2-8). Ramirez and collaborators [15] found no indications of long-range ordering in the specific heat of Sr 3CuPtO.51r o.5 O6down to 2.5K (Figure 2-9). Com- parisons between Sr 3CuPtO.5 1r o.5 0 6 and diamagnetic Sr 3ZnPtO6 indicate additional weight in the spectrum below 20K, possibly due to magnetism in Sr 3CuPtO.5 1r o.5 0 6 . However, it is clear that access to lower temperatures would be necessary to make meaningful comparisons with the theory. The goal of this thesis is to measure specific heat of Sr 3CuPtO.5 Iro.50 6 at these lower temperatures.

44 40 I I 300 7

.20

C- --- x=0.667 20'20 -1o0 1 10 -20 - -= -x=0.5 0 7

-U

10 0 K -U- ' T=5 K

0 I I -500 0 500 1000 1500 2000 2500 H(G)

Figure 2-5: 1I versus H data of Beauchamp and Rosenbaum on p = 0.5, 0.667 samples.

45 0.40*

0.30 - *o x=0.667

0.20 -. x=0.5 E

0.10 * L U -) 0.00C ) 5 10 15 20 25 30 T(K)

Figure 2-6: AC susceptibility data of Beauchamp and Rosenbaum.

46 0.08

Magnetic Field

-. OT 0.06 -o-- 0.025 T 6 0.05 T

-- +0.5 T 0.2 T 0.04 u -~--05T01 T ~'

000 0.02

0 5 10 15 20 25 30 T (K)

Figure 2-7: Recent AC susceptibility data on Sr 3CuPtO.5 Iro.5 0 6 taken by Beauchamp.

47 60

50 - A p=Q. S 20K 7 7 p=O.S 5 K7

- - 40 - -

CU 30 E -7 E a) 20 - -

10 'V

10 0 200 400 600 800 1000 Magnetic Field [Gauss]

Figure 2-8: Magnetization versus field for Sr 3 CuPto.5 Iro.0 O 6 samples used in this thesis.

48 0.8 T V SraZCPtO, '9 A sracopto, a a

V enpy A* 0.6 - SeCu(FPtyO, 4 le1

0.4 --

0.2

9 V V 7 7 777VV7 7 0.09 0.050 0 10 20 30 40 50 T (K)

Figure 2-9: Heat capacity data of Ramirez. In each case, the sample consisted of 0.2g compressed together of the Sr 3MM'0 6 material, and 0.2g of silver powder, which were to form a pellet. The pellet was then glued to the calorimeter [32].

49 50 Chapter 3

Methods

3.1 Quasi-Adiabatic Calorimetry

Heat capacity at constant field is defined as

C = (4Q/dT)H (3.1)

Operationally, C can be most simply measured by applying a known heat to a sample and measuring the resulting temperature change. This is known as adiabatic calorimetry, so-called because the sample must be thermally isolated from its surroundings in order to obtain an accurate measure of 4Q. In order to use this technique at low temperatures, some kind of thermal switch must be used to connect the sample to a source of refrigeration. The switch is closed while the sample is cooled to the desired temperature, then opened for adiabatic calorimetry measurements. Several types of switches have been used successfully, including gas, mechanical, and superconducting types [18]. When trying to measure the specific heat of a new material, one generally has only a small (less than 1g) quantity of material available. Adiabatic calorimetry becomes very difficult for such small samples at low temperatures. Since low-temperature specific heats are small, small samples have low heat capacity, and so are very sensitive to heat leaks (dTeak = 9Qleak/C). These heat leaks could be due to conduction down electrical wires connected to thermometers or heaters on the sample, to radiation from warmer surfaces near the sample, and/or to residual exchange gas. As a result, the sample may warm to unacceptably high temperatures, and could undergo large temperature fluctuations if the heat leaks vary in time. The sample can be cooled and its temperature stabilized against the effects of heat leaks by increasing its thermal contact with the refrigerator, but of course this makes an accurate determination of 4Q impossible. Various solutions have been proposed for this tradeoff between thermal isolation and the need to cool and temperature control the sample. The main strategy is to deliberately make a weak thermal connection between sample and refrigerator. This thermal connection is chosen to be strong enough to provide sufficient sample cooling and temperature control to overcome heat leaks, yet weak enough so that the

51 thermal conductance across the sample is still much greater than that between sample and refrigerator. Under these conditions, various quasi-adiabatic techniques may be applied to measure sample heat capacity. Below I describe the two quasi-adiabatic methods, thermal relaxation and AC, which were used to measure specific heat in this thesis. I also discuss mathematical models for these two methods. These will be used later to describe the behavior of the calorimeters used in this thesis.

3.2 Thermal Relaxation Calorimetry

Thermal relaxation calorimetry was first proposed by Bachmann et al. [35] in 1972, and has subsequently been adopted by several groups for low-temperature calorimetry of small samples. [36] [37] A constant power P is applied to the sample until a steady- state condition is achieved. Due to the weak link between sample and refrigerator (the latter kept at fixed temperature), a temperature difference AT(Po) arises between sample and refrigerator. If P is turned off suddenly, AT decays exponentially to zero, with the time constant -r1 for the decay given by

r1 = C/Kb (3.2) where Kb is the thermal conductance of the weak link. TF can be determined by recording AT(t) and fitting to an exponential. If one applies various powers P and records the steady-state temperature difference AT(P), one can obtain Kb as the slope of the power/temperature curve P versus AT(P). Given T and Kb, C can then be determined. The above method works well provided the minimum conductance in the calorimeter Kmin is much greater than Kb. Ideally, one designs a calorimeter in which all of parts of the calorimeter are in good thermal contact with each other and with the sample; in that case, Kmin could be the sample conductance K8, if the sample were a poor thermal conductor (as Sr 3CuPt1_pIrpO 6 is). However, for Kmin -~ Kb, parts of the sample that are further from the link will see a significantly smaller thermal conductance to the refrigerator than those closer to the link, so that AT(t) can no longer be characterized by a single time constant T1. In general, one would expect that under these circumstances AT(t) would be given by a sum over (infinitely many) exponentials. However, by solving the heat equation for their geometry, Bachmann et al. found that under certain conditions, only the first two terms in such a sum are needed to describe AT(t). The time constants of these two terms were r = C/Kb as before, and T2 oc C/Kmin. Hence AT(t) which were best described with two exponentials were said to exhibit "r 2 effect". The required conditions for the two-exponential description are that the thermal link and addenda heat capacities be only a few percent of the sample heat capacity, and that Kmin > Kb/2. In this case, it was possible to extract the heat capacity (and Kmin) given T(t) and Kb. Pursuing Bachmann's result further, Schwall et al. noted that a simple model (Figure 3-1) consisting of two discrete heat capacities with thermal links between them and to the bath could be solved exactly, and that the solution would be a sum

52 C1 AT 1

Kmin

C 2 AT 2

Figure 3-1: Thermal circuit for Schwall model.

53 T Kb

->' tot I I

Figure 3-2: Thermal circuit for Sullivan and Siedel model. The calorimeter and sample are assumed to be in excellent thermal contact, so that the slab with heat capacity Ctot represents the entire sample/calorimeter assembly. Heat flux j = (e") (a is slab cross-sectional area) is applied at one end of the calorimeter by the heater, and temperature T = Tdc + Tac is sensed at the other side of the calorimeter. Heat is dumped to thermal ground (the heat bath) through the weak link conductance Kb.

of two exponentials, such as was seen in Bachmann's model when the calorimeter thermal properties met the conditions mentioned above. In the Schwall model, the decay AT(t) = AT 2(t); that is, the thermometer is on lump 2. The heater location is irrelevant, since both lumps are at the same temperature at t = 0 when the heater is turned off. One then measures AT2(t) and fits it to the expression

AT2 (t) = A 2 exp(-t/TA) + B 2 exp(-t/TB) (3.3)

to find A 2 , TA, B 2 , and TB. Kb is again the slope of the power/temperature curve. Then solving the model yields

CAot = C1B+TC2 Kb (3.4)

That is, the total heat capacity Crt of the calorimeter (sample + addenda) is just Kb times the weighted average of the two time constants. Schwall et al. found that they could reliably extract the sample heat capacity by application of their model solution to determine the total heat capacity Ctot, and then subtracting from Cot the (separately measured) heat capacity of the addenda. Schwall's approach will be used in this thesis to extract heat capacities in situations where AT(t) is best described by a sum of two exponentials.

3.3 AC Calorimetry

Sullivan and Seidel first proposed AC calorimetry in 1968 [38]. The specific geometry they considered is shown and described in Figure 3-2. For their analysis, they assumed excellent thermal contact between all addenda and the sample, and that the dominant thermal resistance and heat capacity were due to the sample itself. In

54 this technique, the reference of a lockin amplifier applies an AC voltage at frequency f to a heater connected to the sample. The heater then produces an AC power at angular frequency w = 47rf (the extra factor of two is due to the fact that power is proportional to the square of the applied voltage), which in turn produces a zero- frequency rise in sample temperature (due to the fact that the applied power is always greater than zero) and a temperature oscillation of amplitude ITac| at frequency w. ITac is then measured by the lock-in. Due to the noise bandwidth narrowing offered by the lockin, ITac can be measured precisely. Assuming wT > 1 and wr 2 < 1, where 71 and 72 are as defined in Section 3.2, the heat capacity can be found from

P 1 2 22 2K]1/ C = 2wT [1 + W2T+ W7 + 3 ]-1/2 (3.5)

The prefactor 2w IP TaI can be written QT/27rITac II where QT is the total heat applied over one oscillation of the power. Hence this prefactor is the quotient of heat applied per radian and ITacd, which is the temperature change produced over one radian. This prefactor will then be an accurate measure of the heat capacity provided that two conditions hold. Firstly, the time per radian 1/w must be much greater than the response time of the sample so that the sample can follow the temperature oscillation;

i.e. wT2 < 1. Secondly, 1/w must be much less than the response time of the weak thermal link so that no heat is dissipated to the bath over the measurement time; i.e. wT > 1. These conditions are expressed in the second factor in (3.5). This second factor also contains a term 2. Since K, and Kb are finite, there will be temperature drops across both the sample and the thermal link, and a corresponding attenuation of ITacd; this attenuation will be worse if most of the temperature drop occurs across the sample, rather than across the link. (3.5) can be rewritten P C = D(w, T) (3.6) 2w IT,,, where the factor D(w, T) contains all dependence on the thermal properties of the calorimeter. Unless the experimental conditions are such that D = 1, the ac method will only provide a precise measurement of C/D, not of C. D 1 can occur for two reasons. First, one could have a calorimeter with excellent thermal properties, such that D = 1 over some range of w, but for some reason heat capacity data were taken at some Wmeas for which D < 1 (how this might come about will become clear in Chapter 5). In that case, if one knows ITac(wi)I at an wi for which D = 1, D = ITac(Wmeas)I/ITac(wi)I and an accurate measurement of C can be obtained. Second, D = 1 could occur due to poor calorimeter thermal properties, such that there would be no w for which D = 1. In that case, D must be determined by measuring the thermal properties of the calorimeter-including its heat capacity-using methods other than AC calorimetry, and having an accurate model for the calorimeter that relates those thermal properties to D. Hence, the amount of calorimeter characterization necessary to obtain good measurements of D(w, T) makes the AC method seem much less attractive than the relaxation method unless D = 1 can be achieved. Whether or not D = 1 can be achieved below 1K will depend strongly on the

55 material under study. Following Sullivan and Seidel, note that the ratio of frequency- independent to frequency-dependent terms in D(w) above is

(2LKb/3AK)/ (w 2 T22) = 6O/ (W2 L 2 Ti) (3.7) where q = K/pc is the diffusivity of the material. Also note that the frequency- independent term itself is given by

2 2LKb/(3AK) = 2L /(3rq1 ) (3.8) In order to obtain D = 1, we want the frequency-independent term (3.8) to be small and the ratio (3.7) to be large so that the frequency-dependent terms are even 2 smaller than the independent term. Hence 7r 1/L > 1 is desired. Metals typically 4 2 have q - 10 cm /s below 1K. So for a typical Tr of 1s, metals will easily satisfy this requirement with reasonable L. Electrically insulating magnetic materials such 2 as Sr 3CuPt1_pIrpO 6 , however, will typically have i1 < lcm /s, since they are poor thermal conductors with relatively large magnetic specific heats below 1K. Hence the condition can be satisfied only by making L very small, perhaps 1mm or less. If the sample allows D = 1 over some range of w, the AC method can provide specific heat data of comparable accuracy and superior precision to the relaxation method. In addition, the AC method requires less data processing and analysis, and provides a continuous readout of heat capacity via ITac I. This latter feature allows one to scan a thermodynamic variable such as T or H and so obtain the heat capacity as a function of one of these variables in a relatively short time. Even if D < 1, the AC method is still valuable to obtain a qualitative picture of a material's specific heat. In particular, if the material has a phase transition, there will be a peak in specific heat over a narrow temperature interval. From (3.5), this will lead to a dip in the prefactor P/2wC, and peaks in T1 and T2 over that temperature interval. The overall effect will be a dip in ITac. Hence when studying a new material it is useful to use the AC method as a first step, since it will allow one to quickly identify interesting qualitative features in the spectrum such as phase transitions. These features can then be studied more quantitatively with the relaxation method. Finally, one can go further if the frequency-independent term is the dominant contribution to D over the temperature range of the phase transition. In this case, D is just a constant scaling factor, which can be determined by measuring C at a couple of temperatures with the relaxation method and comparing with the AC result P/2WITac. Then the precise data taken via the ac method can be rescaled by D to give both precise and accurate specific heat data over the phase transition. (3.5) was derived by Sullivan and Seidel from an exact solution of the thermal model shown in Figure 3-2. The model was solved using the matrix method [39], in which the temperature and heat flux at z = L can be found by multiplying the temperature and heat flux at z = 0 by a 2x2 matrix describing the thermal properties of the material between the two points. The matrix method is easily extended to the calorimeter geometry used for this experiment. Matrices describing the thermal link, sample, calorimeter, and any boundary resistances between them can be easily written down and included in the matrix multiplication. The result of this matrix

56 calculation is the "thermal transfer function", LoTacl (w), for ITacI measured at a given position in the calorimeter.

As mentioned above, (3.5) holds provided wr 2 < 1 and wT > 1. The condition wT < 1 can also be written lo L, where L is the sample thickness and lo = L 2K§ 2 > VCW is the characteristic thermal length (length over which T can change appreciably). Seen in this way, this condition states that the calorimeter temperature does not vary appreciably over the sample thickness, so that the entire calorimeter can be considered a single thermal object with a single temperature. Under these "one-lump" conditions, the positions of thermometer, heater, and thermal links are not important. Hence (3.5) should apply to any calorimeter geometry, provided the one-lump criteria hold. Hence I will refer to the Sullivan and Seidel geometry-independent (SSGI) transfer function. One test of a well-designed calorimeter is how well it corresponds with the SSGI transfer function having the same thermal parameters; for such a calorimeter, the details of where and how the various elements of the calorimeter are placed are not important. In particular, the calorimeter geometry used in this thesis will be modeled as a single slab with two thermal link wires, one on either side of the slab. As will be seen in Chapters 5 and 6, the transfer functions resulting from exact solution of this two-wire model and those resulting from SSGI will differ very little. In practice, an AC specific heat measurement begins with measurements of the transfer function at a few points over the temperature and field range of interest, in order to determine an appropriate frequency range for collection of specific heat data., 2 Provided r/T1 /L > 1, the transfer function will have a "plateau", where the transfer function is independent of frequency. D = 1 will hold on this plateau, and specific heat can be measured accurately for frequencies on the plateau. A frequency in this range is chosen, and specific heat is measured as temperature or field is varied. Since the thermal transfer functions are determined by the thermal characteristics of the calorimeter, they can be used for thermal characterization of the calorimeter. Also, one can measure parameters such as calorimeter heat capacity or link wire conductance via relaxation calorimetry, and use a calorimeter model to fit the transfer function and extract other calorimeter parameters, such as calorimeter thermal conductance. Finally, if one can also estimate calorimeter thermal conductance via other methods, one can use the model to predict the transfer function, and compare with the measured function as a consistency check on the other measurements and on the model. Thermal transfer functions will be used for all these purposes in this thesis (see Chapters 5 and 6).

57 58 Chapter 4

Apparatus

4.1 Dilution Refrigerator and Magnet

Refrigeration was provided by an SHE Corporation Model 430 dilution refrigerator that had a measured cooling power of 210pW at 100mK. Temperatures as low as 13mK at the mixing chamber were achieved with the experiment attached. Magnetic fields were generated by an American Magnetics superconducting magnet, which provided fields up to 7T. Field uniformity was better than 4 parts in 104 over the entire experimental area.

4.2 Heat Capacity Experiment

Because the dilution refrigerator was not top-loading, cycling even a single calorimeter from room temperature to low temperature and back required at least two weeks. On the other hand, the fact that the refrigerator was not top-loading also meant that a large volume was available for experiments. This made it possible to examine both a calorimeter with a sample and a control calorimeter without sample in a single cooldown, and reliably separate effects due to the sample from effects due to the calorimeter itself or to the refrigerator environment. This large volume also made it possible to consider calorimeter designs in which the powder sample was thin but had a large surface area. Such a design is desirable because L < T /pc for good AC calorimetry, and K, = KA/L > Kb for good relaxation calorimetry.

Since K is very small for ceramics such as Sr 3CuPtO.5 1ro.5 0 6 and electrical insulators such as K3Fe(CN)6 , and since both are expected to have large heat capacity at low temperatures, one would like to maximize A and minimize L in order to obtain good conditions for calorimetry. These considerations provided the motivation for the design of this heat capacity experiment, which I describe in further detail below.

4.2.1 Support Structure The overall structure of the apparatus is shown in Figure 4-1. An OFHC copper plate was bolted to the mixing chamber, and silver soldered to a 6. inch long, 1 inch ID/I

59 inch OD copper tube that in turn was silver soldered to another OFHC copper plate, called the bath plate. Threaded brass rods were screwed into the bottom side of the bath plate until a short section of rod stuck out on the top of the plate. Brass nuts were threaded onto these short sections and tightened against the top of the plate. This prevented the rods from rotating when other nuts were threaded onto the rod from below, and improved thermal contact between the rods and the bath plate. A 2 inch diameter, -I inch thick copper plate was passed through the brass rods and held in place with brass nuts. This plate provided a black body shield against radiation from higher parts of the refrigerator. The calorimeter stages were attached below this black body plate. A single stage is shown in Figure 4-2. The base of each stage was a copper plate identical to the black body plate. Screwed into the plate were three brass L brackets, with a long horizontal vespel peg and a short vertical vespel peg glued with Stycast 2850 epoxy onto the appropriate face of the bracket. These vespel pegs were individually machined on the face of a vespel rod, cut off the rod, and filed to a sharp point. The calorimeter itself consists of a mixture of powder sample and Apiezon N grease [40] sandwiched between two 1.5 inch diameter, 2 mm thick quartz plates. The calorimeter rested on the vertical vespel pegs, and was confined in the horizontal plane by the horizontal pegs. Electrical leads for the calorimeter thermometers and heaters (see Section 4.2.2) were strung through holes in the copper plate and plugged into a Microtech [41] plugboard epoxied to the bottom of the plate. 2j mil manganin wires from this plugboard were threaded through the copper tube to the mixing chamber, where they were connected to another Microtech plugboard. From this plugboard, more 2- mil manganin wires, thermally anchored to copper bobbins with GE 7031 varnish at the mixing chamber, base plate, still, and cold plate, were used to carry electrical signals in and out of the vacuum can. Between calorimeter stages and between the black body plate and bath plate, copper blocks (typical dimensions 1 inch x inch x 1 inch) were positioned to increase thermal contact between the calorimeter stages. Once all the stages were loosely installed on the support structure, the brass nuts on the bath plate and below each calorimeter stage were tightened as much as possible to ensure good thermal contact between the copper blocks and the plates. After assembly, the entire experiment was wrapped in aluminized mylar sheet, which was affixed to the experiment using Apiezon N grease and dental floss [42]. This shielded the experiment from black body radation emitted from the 4 K vacuum can surfaces. It was desired to have two four-wire thermometers and two four-wire heaters available on each calorimeter, both for diagnostic purposes and as insurance against the likely loss of electrical contact to some devices during cooldown. Given the number of wires available, this limited the number of calorimeters that could be measured in a single cooldown to two (labeled a and 3). One calorimeter contained the sample, and the other was a control calorimeter used to correct the sample data for effects due to the addenda or the refrigerator environment. The same quantity of N grease used for the sample calorimeter was placed between the plates of the control calorimeter, which will be referred to hereafter as the empty calorimeter. The number of calorime-

60 Cu Tube

Bath Plate Cu Block BBR Plate

Calorimeter Stage

Threaded Brass Rod

Figure 4-1: Overall view of apparatus. Only one calorimeter stage is shown. Note that devices (heater, thermometer) are not shown on the calorimeter.

61 CaLorimeter

Cu Plate

Vespel Peg Frame

6-32 Th ru for Brass Rod

Figure 4-2: Wireframe view of calorimeter stage. The calorimeter sandwich (sample dough between two quartz plates) is supported by vertical and horizontal vespel pegs. These pegs are glued into brass L brackets.

Thermao Link Wire

Hleaiter Wire

ThermaL Link Fingers

Chip Resistor Thermometer

Figure 4-3: Top view of one calorimeter plate, showing configuration of heater, thermometer, and thermal link.

62 ters that could be measured during a single cooldown is limited to four (assuming one thermometer and one heater per calorimeter) by the number of electrical wires available. Also, magnetic field profile calculations indicated that with four calorimeter stages, magnetic field variations of roughly 1% from top to bottom stage would be expected.

4.2.2 Calorimeter As mentioned above, the calorimeter can be described as a sandwich, with a mixture of sample powder and Apiezon N grease sandwiched between two quartz plates. Heat is intended to flow primarily perpendicular to the plates, so that the heat flow problem can be considered to be one-dimensional. This design geometry takes advantage of the large (2- inch) diameter of the vacuum can tail piece (where the experiment is located) to minimize the sample thermal conductance. Since for this one-dimensional case the sample conductance K, = rKA/L, a larger A not only improves K, directly, but also makes it possible to make L smaller for a fixed sample mass. In this situation, KS Oc #4 , where 0 is the diameter of the calorimeter plate. Given the tail piece diameter, # can be made relatively large (1.5 inches in this case). The material for the plates had to be chosen with care. It was desirable for it to be an electrical insulator (since the heater and thermometer would be glued to it), have excellent thermal conductivity, and very low specific heat. The last of these requirements was especially important since the plate is large, and so potentially could contribute a great deal to the overall heat capacity of the calorimeter. Of course this is undesirable since for accurate sample heat capacity measurement, one wants the calorimeter heat capacity to be dominated by the sample, not the addenda. The material chosen for the plates was single crystal Z-cut quartz [43]. For temperatures below 1 K, its thermal conductivity is surpassed only by pure metals such as copper and silver, and by superfluid 4He. The Z-cut ensures that the 001 axis, which has a slightly higher thermal conductivity than the other crystallographic directions, is perpendicular to the face of the plate. More impressive is the specific heat of quartz, which below 1 K is among the lowest of commonly used low-temperature materials [42]. Finally, it is electrically insulating. One surface of each plate was roughened with 120 pm grit to increase the surface area of contact between sample and plate. This was done in hopes of decreasing thermal boundary resistance. Attached to the non-roughened face of each quartz plate were a thermal link, thermometer, and heater. This face is shown in Figure 4-3. The thermal link consisted of a set of copper fingers cut from a sheet of 1 mil thick copper shim stock and glued to the plate with GE 7031 varnish, and a 21 inch long, 36 gauge copper wire affixed to the copper fingers with silver epoxy [44]. The copper fingers were used to improve thermal contact between the quartz plate and copper wire, and to improve the thermal conductivity of the calorimeter in the plane of the plate below 200 mK (see Figure 4- 5). A Dale Electronics Model RCWP-575 chip resistor (1%, 1 kQ RuO 2 thick film)was used as a thermometer, and glued to the center of the plate with Stycast 1266 epoxy. The heater was an 18 inch length of 1 mil diameter 92% platinum/8% tungsten wire (called PtW wire henceforth). The wire was wound over the entire surface of the

63 plate and glued down with GE 7031 varnish. After the GE varnish dried, the thermal link and heater were coated with a thin layer of Stycast 1266 epoxy. This was done after it was found (in another experiment in this laboratory) that GE varnish bonds to sapphire plates did not reliably withstand repeated thermal cycling. The chip resistor was chosen for thermometry because of its small size, relatively low heat capacity [45], low magnetoresistance, relative insensitivity to thermal cycling, and its common use in our laboratory for secondary thermometry in the 0.1K to 1K temperature range. The PtW wire was chosen because it has the lowest specific heat of common resistance wires at low temperatures [46]. Copper was chosen for the thermal link due to its well-known thermal properties and availability. In retrospect, copper was not a good choice for the thermal link because of its large nuclear heat capacity at high (several tesla) fields and low temperatures. Also, the use of industrial copper shim, which likely has large quantities of unknown impurities, was not wise. High purity silver would have been a much better choice.

In the K 3Fe(CN)6 experiment and the initial Sr 3CuPt. 5 Iro. 5O6 experiment, 5 mil diameter NbTi wires were used as electrical leads connecting the thermometers and heaters to the Microtech plugboard. NbTi wires are superconducting in fields of 8T below 1K, and so have excellent electrical conductivity but very low thermal conductivity and specific heat. These are excellent properties for electrical leads for low- temperature calorimetry. On the other hand, these wires are not at all malleable and so often could not be plugged in without placing them under stress; furthermore, it is very difficult to make reliable solder joints with them. As a result, one of the four calorimeter thermometers was lost during cooldown of the K3Fe(CN)6 experiment, and three of four were lost during the first cooldown of the Sr 3CuPtO.5 Iro.0 O 6 experiment. For the second Sr 3 CuPt0 5. Ir0 .5 0 6 experiment, the NbTi wires were replaced with 1 mil PtW wires, which are thin enough to have thermal properties comparable to those of NbTi, but are quite malleable, solderable, and surprisingly robust (1 mil diameter manganin wires, for example, are much more fragile). With the calorimeters mounted on their respective stages, the copper thermal link wires were threaded through holes in the copper plates to the bath plate, where they were placed between one of the copper blocks and the bath plate before final tightening of the brass nuts. When the brass nuts were tightened, the thermal link wires were pressed firmly between blocks and plate, ensuring good thermal contact to the bath plate and hence to the mixing chamber of the refrigerator.

4.2.3 Calorimeter Models

Models for the expected behavior of the empty and K3Fe(CN)6 calorimeters were developed for comparison with experimental data. In these models it was always assumed that the calorimeter could be modeled as a one-dimensional system; that is, the calorimeter temperature T was a function of z only, where the z axis is perpendicular to the plane of the calorimeter plates. This assumption is reasonable because the heater wire and thermal link fingers are distributed uniformly over the entire surface of the plates, and because the thermal conductivity of quartz is excellent. The expected heat capacities of various components of the empty calorimeter are

64 shown in Figure 4-4. The expected thermal conductances of various components of the calorimeter, and thermal boundary resistances between various surfaces, are shown in Figure 4-5. These data are based on published specific heat and thermal conductivity data for the various materials used, and on measurements or estimates of the relevant geometrical factors. In Chapters 5 and 6, these expected values will be compared with values of link conductance and calorimeter heat capacity measured with the relaxation method.

10 3 OkG mt cal, nea'su'red' total calcd addenda *CPI zero field

N grease Quartz 104 Cu fingers - 46 l1d) Dale Chip O V 0 PtW htr/Iead wires 0 * * Cu wire

0 0 0 L~3 VO Cj 0- 6 dt y, b~ 0 0 0 0 0 0 0 0 a NU V A AA A A, A, A A 10~~7 V

10-8 I I 0.0 1 0.02 0.05 0.1 0.2 0.5 1 2 5 10 Temperature [K]

Figure 4-4: Expected heat capacity of various calorimeter components. The N grease data below 0.4 K was obtained by linear extrapolation, which is reasonable for glassy materials [47]. The quartz data below 0.3 K is a T 3 extrapolation, which underestimates the true heat capacity [47]. Also shown are expected empty calorimeter heat capacity and measured empty calorimeter heat capacity from Sr 3CuPtO.5 1ro. 5 0 6 experiment, and Sr 3CuPtO.5 Iro.50 6 heat capacity measured here (below 1K) and by Ramirez [15](above 1K).

Also, thermal transfer functions for the empty calorimeter were computed using these expected values. The calorimeter was modeled as a single slab of material with a thermal link wire on either side. I will refer to this as the two-wire model. The matrix method was then used to derive exact, analytical expressions for the thermal transfer functions. These transfer functions will be compared with experimental transfer functions in Chapters 5 and 6.

65 10 2 I I .. I , . I I I I .. I I I I . III T

VV VV 10 0 V V v Quartz normal V - Quartz/sample bdry ---Cu finger/quartz bdry- V 0 Quartz in plane A N grease 102 V 0000 4 Cu finger - C-U o Copper wire C * vespel pegs _ 0 Ile * PtW wires ~4-) * NbTi wires Ui -4 D~ 10 V do CdVO3 00 C 0 OC 0 0 000000 0 IR 10- 0 000000 L CU 0- *

00 WV

100 0. 01 0.02 0.05 0.1 0.2 0.5 1 2 5 10 Temperature [K]

Figure 4-5: Expected thermal conductance of various calorimeter components. The PtW entry is for 16 1 mil diameter PtW wires, each 1 inch long (representing the heater and thermometer lead wires). The NbTi entry is for the same number and lengths of 5 mil diameter NbTiwires. The quartz/sample boundary resistance is taken to be the same as that between copper and glue.

66 4.2.4 Thermometry

Primary thermometry for 0.1K < T < 0.3K was provided by a 3 He melting curve thermometer (MCT) of the type described by Greywall and Busch [48]. The MCT is thermally anchored to the mixing chamber, and its capacitance measured by a tun- nel diode oscillator located on the coldplate. The accuracy of temperatures measured with the MCT was typically better than +lmK. As mentioned above, secondary thermometry was provided by Dale chip resistor thermometers. For the K3 Fe(CN)6 experiment, the mixing chamber chip resistor was calibrated against the MCT, and the calorimeter chips calibrated against the mixing chamber chip with the mixing chamber temperature controlled. For the Sr 3 CuPtO.5 1ro.5 0 6 experiment, resistances of chip resistor thermometers on the calorimeters and mixing chamber were recorded along with the corresponding melting curve thermometer temperature as the refrigerator was slowly cooled. The two methods did not give significantly different results. For the /3 calorimeter top plate thermometer, the maximum difference between the two methods was 6mK at 120mK, decreasing to a negligible amount by 209mK; for the o calorimeter top plate thermometer, the maximum difference was 2 mK at 133 mK, and was less than lmK at lower and higher temperatures. Magnetoresistance was measured by temperature controlling the mixing chamber and measuring chip resistance as a function of magnetic field. The resulting chip resistance versus temperature and field data were fit to R(T) = (A + RHH) exp(B/Ti) [49]. For the four calorimeter thermometers, typical values are 1 A = 0.37kM, B 1.67K4, and RH = 2.6Q/T. Variations in these parameters between thermometers are typically 5%. Chip resistor temperatures are typically taken to have errors of ± 2mK, due to scatter about the fit. For temperatures above 0.3K, the R(T) fit was extrapolated. It was assumed that the chip resistors and MCT were in excellent thermal contact, and so had no appreciable temperature difference, provided that R(T) fit well to the expression given above for the chip resistor. It has been shown [49] that even under conditions of excellent thermal contact, the chip resistor begins to deviate from the exp(B/Ti) behavior below 50mK, and that the behavior below 50mK depends strongly on the device. However, it was found during the second Sr 3CuPtO.5 1ro.5 0 6 experiment that all of the chips began to deviate at higher temperatures. For the empty calorimeter top thermometer, deviation began around 60mK; for the empty calorime-

ter bottom thermometer, 80mK; and for both of the Sr 3CuPtO.5 Iro.5 0 6 calorimeter thermometers, around 120mK. Heat capacity data was not taken below the deviation temperatures, or was taken only to get a qualitative idea of the heat capacity below those temperatures. Since all of the thermometers were from the same batch, it is likely that this difference between thermometers is caused by loss of excellent thermal contact between each thermometer and the MCT. For the empty calorimeter thermometers, it is likely that the bottom thermometer is not as well contacted to the quartz plate as the

top thermometer. For the Sr 3CuPtO.5 1ro.5 0 6 calorimeter thermometers, it is possible that neither thermometer is as well contacted to the quartz plates as is the empty

67 calorimeter top thermometer. It is also possible that the thermal link wires for the

Sr 3CuPtO.5 1r o.5 0 6 calorimeter are not as well-connected to the bath plate as those of the empty calorimeter. Finally, it is important to address the question of the lowest possible temperature at which heat capacity could be measured with the current apparatus. The MCT has measured temperatures as low as 13mK at the mixing chamber. Also, the calorimeters did continue to cool below the deviation temperatures (as evidenced by continued increase in resistance). Assuming that these thermometers have similar calibrations to others used in our laboratory, it is found that the empty calorimeter top thermometer has measured temperatures as low as 20mK, while all other thermometers (including the chip on the mixing chamber!) stop cooling in the 40 to 50mK range. Hence with more careful attention to thermal contact issues, it seems possible to cool the calorimeters to 20mK. This does not necessarily imply that heat capacity could be measured to that temperature. For the AC method, there is always a DC component of the power, proportional to the peak AC power, which will warm the calorimeter above the bath temperature during measurements. This can be reduced, but only to the extent that the signal-to-noise of lTac is not too low. Also, the resolution of the relaxation method is limited by the size of the throw AT(0) required to achieve reasonable signal-to-noise. However, these issues of thermal contact and signal-to- noise are technical, and in principle solvable, problems. Therefore it seems feasible, with some effort, to extend the measurement range of this apparatus to at least 50mK, and perhaps lower.

68 Chapter 5

Potassium Ferricyanide Experiment

The specific heat of potassium ferricyanide K 3Fe(CN)6 was measured as a test of the apparatus. This material (Figure 5-1) contains chains of Fe3+ ions, with effective spin-half and nearest-neighbor coupling. The chain axis is parallel to a. The coupling between nearest neighbors in the chain contains an isotropic part J of strength J ~ 0.36K, and an anisotropic part, with J, ~ 0.1K, Jb Ja ~ -0.03K (as usual, J > 0 indicates AF coupling). The coupling Jd between chains is isotropic and much weaker, Jd ~ 0.01K [50]. When cooled below 1K in zero field, K3Fe(CN)6 develops short-range antiferromagnetic order along the chains, with the spins in a given chain aligning antiparallel to one another and pointing along the c axis. The short-range order is described approximately by the 1D Ising model [50], and leads to a broad plateau in the specific heat between 200mK and 500mK [51]. K3Fe(CN)6 undergoes a transition to antiferromagnetic long-range order at TN = 129 mK. The transition to long-range order is observed as a peak in specific heat at 129 mK. (Note that the transition to long-range order occurs at higher temperature than would be suggested by Jd; this is due to uncertainties in Jd and to short-range order in the chains above TN, as discussed in Section 2.1.) Both the short- and long-range order are strongly affected by magnetic fields as low as 100 gauss [52]. This material was chosen to test the apparatus for three reasons. First, the sharp feature at 129 mK has been observed by multiple groups and so provides a test of temperature and heat capacity measurements. Second, since the specific heat has a strong response to weak (hundreds of gauss) applied fields, it provides a test of heat capacity measurement in a field. Finally, K3Fe(CN)6 is electrically insulating and has a quasi-one-dimensional structure, so it should have similar thermal properties to Sr 3CuPtO.5 1ro.5 06 .

5.1 Calorimeter Preparation

For this experiment, calorimeter preparation began with two complete, previously unused calorimeter plates. 6.44mg of small K3Fe(CN)6 crystals (Fluka, > 99%,

69 I0 IO

I ' I O I bN

Figure 5-1: Monoclinic unit cell of K3Fe(CN)6. Closed spheres represent Fe, open spheres K, open diamonds C, and closed squares N. 3, the angle between a and c, is approximately 1070, and a = 7.04A, b = 10.44A, and c = 8.4A. Six cyanide groups surround each Fe, in approximately octahedral coordination. Here, only two Fe ions are shown with all cyanide groups. The closest cyanide groups on nearest neighbor sites are 2.74A apart on the chain axis a. For nearest neighbors along b, the closest cyanide groups are 6.14A apart.

70 stock no. 60300) were measured out using a Mettler balance. 13.7mg of Apiezon N grease, measured with the same balance, was placed in an agate mortar with the K3Fe(CN)6 crystals. The two components were ground and mixed with an agate pestle until a reddish-orange "dough" was obtained. The dough was carefully and thoroughly scraped out of the mortar and onto one of the calorimeter plates. The second plate was pressed against the dough to form the calorimeter sandwich. The sandwich was placed in a hobby vise and the dough compressed further, taking care not to lose any dough out the sides of the sandwich. The completed calorimeter was mounted on the vespel peg brackets, and the NbTi leads plugged into the microtech connector under the stage plate. The empty calorimeter was assembled similarly. Here, 68mg of Apiezon N grease was used. Some grease was lost out the sides of the calorimeter during compression; the final amount of grease in the calorimeter was determined by massing the calorimeter before and after assembly. The amount of grease on this calorimeter was originally intended to correspond with the amount used to make K3Fe(CN)6 dough on an earlier run, in which far too much K3Fe(CN)6 was used (leading to extremely long thermal time constants). When an appropriate amount of K3Fe(CN)6 (6.44 mg) was finally used, it was decided not to reduce the amount of N grease on the empty calorimeter to 13.7 mg due to the expectation that the addenda (calorimeter and grease) would have a negligible contribution to the total heat capacity of the K3Fe(CN)6 calorimeter, and the fragility of the calorimeter. The empty calorimeter stage was mounted in the top position of the support structure, with the K3Fe(CN)6 calorimeter stage below it. The entire experiment was then bolted to the mixing chamber of the refrigerator.

5.2 AC Method Procedures

Thermal transfer functions were measured at a few temperatures over the temperature range of interest, 100mK to 450mK. As discussed in Chapter 3, thermal transfer functions are needed to determine the range of frequencies that will afford D = 1 over the temperature or field range of interest, and are generally useful for thermal characterization and modeling of the calorimeter. Typically, the excitation power angular frequency (EPAF) range used was 0.126 rad/s to 16.08 rad/s. (EPAF is w from Section 3.3). The setup diagrammed in Figure 5-2 was used for transfer function and AC specific heat measurements. Mixing chamber temperature control was achieved with a homemade AC resistance bridge and homemade lockin operating at 1 kHz to measure mixing chamber chip thermometer resistance, and a homemade PID temperature control circuit to drive the mixing chamber heater. For the K3Fe(CN)6 (empty) calorimeter, an AC voltage was applied to the calorimeter bottom (top) heater with the reference of a Stanford Research Systems SR830 lockin amplifier, and the calorimeter top (bottom) thermometer resistance measured by a Linear Research LR-400 Four-Wire AC Resistance Bridge in lOAR mode. (For the K3Fe(CN) 6 calorimeter, the top thermometer was used because the bottom thermometer wiring failed during cooldown.

71 IMixing Chamber LR-400 AC Resistanc e Bridge x1O AR RB Hlomemade output Tiemperature Controller

Input Excitation Lockin

PC Linux Box

Figure 5-2: Apparatus used to measure AC heat capacity for Sr 3CuPtO.5 IrO.50 6 and K3Fe(CN) 6 experiment.

72 The bottom heater was used to to ensure that heat went through the sample before reaching the thermometer. For the empty calorimeter, the bottom heater wiring failed during cooldown, so the top heater and bottom thermometer were used.) The LR-400 reading was fed to the SR830, which detected the RMS amplitude IVUcJ and phase # (at twice the frequency of the excitation voltage). A PC running Linux directed the SR830 (via GPIB) to acquire 75 amplitude and phase readings on this signal over five to ten lockin time constants (30 seconds here), and to report them to the PC. The reported ITac was found by first taking the mean and standard deviation of the 75 IVaCI readings and converting them to temperatures using the MCT calibrations. At least two additional corrections were made. First, the lockin reports the RMS value of IVacI, while the Sullivan and Seidel formula uses the amplitude of ITacd; to account for this, all measured IVaci were multiplied by v/2. Second, the electronics used for ITac measurement have their own transfer functions. The heater, thermometer, and associated wiring did not produce any attenuation of signals over this frequency range. However, the LR-400 bridge significantly attenuated signals above 0.2 Hz. The bridge transfer function was measured with a JFET (Motorola 2N5486) operated as a variable resistor [53]. A small (6 mV) signal from the lockin reference was applied to the gate of the JFET, and the drain-source resistance of the device (which oscillated at the same frequency as the gate voltage) measured with the LR-400/SR830 setup described above. Figure 5-3 shows the measured LR-400 transfer function. Given this transfer function, IVac at a given w was corrected for LR-400 attenuation by multiplication by the appropriate factor. Note that for a given frequency f in the figure, the corresponding EPAF is 27rf (as opposed to 47rf), since the LR-400 detects ITac directly. Unfortunately, the effects of the LR-400 transfer function were not discovered until after the experiment was over. Hence the thermal transfer functions used to determine an appropriate frequency for specific heat measurements all had a false knee at high frequency due to the LR-400 response. To clarify this point, the uncorrected and corrected transfer function for the empty calorimeter at 102 mK are shown in Figure 5-4. For the uncorrected function, there is a narrow "plateau" centered at 1.89 rad/s, which was chosen as the EPAF for the heat capacity experiments. Re- call from Chapter 3 that in order to obtain accurate heat capacity data from the formula C = P/2wITacI, one must choose w on the "plateau" of the transfer function. Otherwise, one must know ITacI (Wpat), that is the value of ITacl for some Wplat that is on the plateau of the transfer function at the given temperature. This is needed in order to determine the off-plateau correction factor D(w), which will equal ITac (Wmeas) / Tac I(wpat) , where Wmeas is the EPAF at which heat capacity was actually measured. For the situation shown in Figure 5-4, the chosen EPAF of 1.89 rad/s apparently is below the plateau (not shown in Figure 5-4) of the actual (corrected) function. Hence to obtain accurate heat capacity data, Tac(wpiat) has to be known so that D(w) can be determined. For the empty calorimeter, the EPAFs chosen (1.76 rad/s or 1.89 rad/s) based on the uncorrected transfer functions were well below plateau; moreover, the plateau region indicated by the corrected thermal transfer functions began at frequencies above 12.57 rad/s, above the region of frequencies examined for

73 100

20 - gos 0 . 100 10 -0

E 5

2 -

0.5 0

0.2 0 0.1 0.01 0.02 0.05 0.1 0.2 0.5 1 2 5 10 Frequency [Hz]

Figure 5-3: LR-400 bridge transfer function measured with JFET. the transfer function and in a region where attenuation due to the LR-400 was severe (factor of 19 at 1.28 Hz, corresponding to EPAF of 8.04 rad/s). Hence Tac(wpiat) cannot be determined, and it is only possible to partially correct the empty calorimeter heat capacity data for this off-plateau choice of Wmeas (Section 5.4). In particular, only an upper limit can be placed on the empty calorimeter heat capacity. For the K3Fe(CN)6 calorimeter, no off-plateau correction had to be applied, since the EPAFs chosen (0.377 rad/s or 0.503 rad/s) were on plateau for both the uncorrected and corrected transfer functions. With an excitation voltage frequency determined, ITac data were acquired, using the procedure described above, as a function of temperature at fixed field, with the magnet running in persistent mode. For the K3Fe(CN)6 calorimeter, several fields from zero to 9.5 kG were examined. For the empty calorimeter, only zero field was examined.

5.3 Thermal Relaxation Method Procedures

In order to measure heat capacity via the relaxation method, it is necessary to measure Kb and the decay of calorimeter temperature with time AT(t) upon turning off the calorimeter heater (see Section 3.2). In addition to providing the heat capacity, these data are useful for thermal characterization of the calorimeter. Obviously, the direct measurement of Kb is useful for thermal characterization. Also, the number of exponential decays needed to describe AT(t), and the fit parameters of each, can be used to extract information about calorimeter thermal properties.

74 1 06

S 50

0 * Corrected o Uncorrected 0

2 C,)

0 L 5 105 0~0

35 0 0

10 4 0.1 0.2 0.5 1 2 5 10 20 50 100 o [rad/s]

Figure 5-4: Effect of LR-400 transfer function on empty calorimeter 102 mK thermal transfer function. The transfer function is normalized for power.

Excellent temperature control of the heat bath is essential. Several minutes are required to measure Kb and AT(t), and fluctuations in heat bath temperature over that time will produce errors in Kb and AT(t). In this experiment, the heat bath is the mixing chamber of the refrigerator. Bath temperature control was achieved in the following way. A voltage proportional to the mixing chamber chip resistance was measured with a homemade ac resistance bridge and lockin amplifier. This voltage was input to a Princeton Applied Research Corporation Model 113 pre-amplifier, where it was compared with a setpoint voltage. The difference between the two voltages was amplified and sent to a homemade PID temperature control circuit, which drove the mixing chamber heater. This setup provided ±10Q stability of the mixing chamber chip resistance at 4 kQ, or +2 mK at 239 mK. As will be seen, even this degree of stability led to substantial errors in C. With temperature control established, the power Po to be applied to the heater for the AT(t) measurement was determined. PO was chosen to be large enough so that the signal to noise in the AT(t) data was at least 10, but small enough so that the temperature resolution of the resulting heat capacity data was good. For the K3Fe(CN)6 experiment, this led to PO's that produced AT/T - 3%. Given PO, the next step was to measure Kb. Kb can be determined from

P = Kb AT(P) (5.1)

where P is power applied to the calorimeter bottom heater, and AT(P) the resulting

75 top thermometer temperature change. Various powers P less than or equal to P are applied to the heater using a homemade current source, and the resulting ATs recorded. Each data point was acquired after waiting 7 to 10 link time constants. Initially, the heater voltage was incremented in equal steps to a maximum voltage, then decremented back to zero voltage to check for non-equilibrium effects. It was found later that the baseline temperature drifted during the measurement. In sub- sequent measurements, the temperature baseline (zero power) was checked between every two or three power/temperature points in order to correct for this. The power- temperature curves are then fit to a line, and the slope is taken to be Kb. Repre- sentative power-temperature curves obtained for the link conductance measurements are shown in Figures 5-5 and 5-6.

4.5

3.0

01.5 * Data -- Fit

0 - 0

117.5 118.0 118.5 119.0 119.5 120.0 120.5 Temperature [x10 3 K]

Figure 5-5: Typical power/temperature curve for K3Fe(CN)6 calorimeter, without baseline drift correction.

Finally, AT(t) was measured. The low bandwidth of the LR-400 bridge (see Figure 5-3) would be expected to distort AT(t) with time constants of a few seconds or less. However, typical time constants found in AT(t) measurements were in the range 10 to 60 seconds for the K3Fe(CN)6 calorimeter, so the LR-400 was deemed adequate for measurements of AT(t) on this calorimeter. AT(t) was measured simply by recording the lOAR output of the LR-400 bridge as a function of time with a digital voltmeter (Hewlett Packard Model hp34401a). These data were then transferred to the PC for further analysis. This method did not give accurate results for the empty calorimeter, where time constants of less than 1 second were encountered when measured with this method. However, the method gives an upper limit on the time constants for the empty

76 II 'I 'I 4.5

-3.0

1.5 - Data -- Fit

0.238 0.240 0.242 0.244 0.246 Temperature [K]

Figure 5-6: Typical power/temperature curve for K 3Fe(CN)6 calorimeter, with baseline drift correction. calorimeter. The heat capacity of the empty calorimeter calculated from this upper limit was so small compared to that of the K3Fe(CN)6 calorimeter that the empty calorimeter had no effect on the K3Fe(CN)6 results, to within experimental error. Hence no further improvements to the method were necessary to determine the K3Fe(CN)6 heat capacity.

5.4 Empty Calorimeter Results

In Figure 5-7, thermal transfer functions for the empty calorimeter at 102 mK, 279 mK, and 430 mK are shown, normalized for the applied power. Clearly the chosen EPAFs of 1.76 rad/s and 1.88 Hz are below plateau for all temperatures. As mentioned in Section 5.2 above, this is because the effects of the LR-400 bridge transfer function were not recognized and corrected for until well after the data was taken. The thermal parameters of all materials in the empty calorimeter can be computed from published data, and plugged into a calorimeter model to obtain expected transfer functions for comparison with the measured data. The two-wire model introduced in Chapter 3 was used. The parameters for the model were determined as follows. Since the thermal conductivity of N grease is 9 orders of magnitude less than that of quartz, the effect of the quartz on slab conductance was ignored and slab conductance computed using interpolated (or for T < 0.4 K, extrapolated assuming C/T = constant) values of N grease thermal conductivity. The dough surface area

77 106

1: 5 0,

C'o,

_- 2 .0 102mK Measured -102mK Materials data E" 4 a 279mK Measured a 0-o.- 279mK Materials data * 430mK Measured 430mK Materials data

10 , , 0.1 0.2 0.5 1 2 5 10 20 50 100 w [rad/s]

Figure 5-7: Empty calorimeter thermal transfer functions, measured and as calculated using the two-wire model and published data on calorimeter materials. was taken to be the surface area of the calorimeter plate, and dough thickness was determined to be 5 - 10 mil from rough measurements using a light microscope and reticle. The thermal link wire conductance was determined using the commercial copper wire data of Suomi [54]. Slab heat capacity was the sum of that of the two quartz plates, 68 mg N grease, two copper fingers, two chip resistors, and two PtW wire heaters. The amplitude of the heater voltage was as measured with an hp34401a digital multimeter on the voltage leads of the heater. The heater resistance was as measured with the LR-400 bridge at 4 K. Given these input parameters, the expected transfer functions were computed (with no adjustable parameters) and compared with the measured functions for 102 mK, 279 mK, and 430 mK (Figure 5-7). Given that the properties of materials used here could differ from those used for the published data by as much as a factor of two, and the error in the dough thickness measurement, the agreement is relatively good. The fact that the modeled functions fall off faster than the measured ones as frequency decreases is due in part to the fact that the expected Kb is substantially higher than was measured. Since the plateau and high frequency knee are not present in the measured functions, it is only possible to put a (very weak) lower limit on the sample conductance. Comparing expected transfer functions for various K, with the data, it is found that K, > 5pW/K for all temperatures. AC C data for the empty calorimeter are shown in Figure 5-8. These data were multiplied by the factor ITacI(W = Wmax)/TacI(W = Wmeas), where Wmax was the max-

78 imum power angular frequency at which the transfer function was measured, and Wmeas was the angular frequency at which C/D was measured. From Figure 5-7, this factor was taken to be five. Hence these data represent an upper limit on the heat capacity of the empty calorimeter. The points are all at least a factor of thirty below the K3Fe(CN)6 data, which confirms that it is safe to ignore the addenda in computing the K3Fe(CN)6 heat capacity.

1.0 -

0.8 - * * AC top therm,bot htr _ 0 o AC top therm,top htr

* 0.6 - 0-

.0 ,X4 - E

0.2 H

0 0.05 0.20 0.35 0.50 Temperature [K]

Figure 5-8: Upper limit on AC empty calorimeter heat capacity.

5.5 K3Fe(CN)6 Calorimeter Results

For the K3Fe(CN) 6 calorimeter, thermal transfer functions were measured at 108mK, 139mK, and 250mK. These are shown in Figures 5-9, 5-10, and 5-11. Clearly the chosen EPAFs of 0.38 rad/s and 0.50 rad/s used for specific heat measurement are on plateau for all temperatures. The plateaus are quite broad, spanning at least one order of magnitude in w. Indeed, the low-frequency knee is below the measured range of frequencies for the two low temperature functions, and the high-frequency knee is above the measured range for the 250 mK function. Given this broad plateau, one might expect that D = 1 will hold for the K3Fe(CN)6 calorimeter, and that the AC specific heat data will be accurate as well as precise. The plateaus are at lower frequencies than those for the empty calorimeter. This is not surprising, since C is much larger for the K3Fe(CN)6 than for the empty calorimeter, since Kb and K, are expected to be comparable for the two, and since we have w, oc Kb/C and wh oc K/C for the locations w, and wh of the low- and high-frequency knees.

79 103 * Measured - 2 Wire Model, Exact SSGI

04 0 - 3

10-5 1 0.0 1 0.02 0.05 0.1 0.2 0.5 1 2 5 10 w [rad/s]

Figure 5-9: K3Fe(CN) 6 calorimeter 108 mK thermal transfer function, with fit to two-wire model and resulting SSGI transfer function.

10-4 9 8 7 6

En 4

-c__ 3

* Measured -2 Wire Model, Exact -- SSGI

10-5 1 0.0 1 0.02 0.05 0.1 0.2 0.5 1 2 5 10 o [rad/s]

Figure 5-10: K3Fe(CN)6 calorimeter 140 mK thermal transfer function, with fit to two-wire model and resulting SSGI transfer function.

80 10 1 1 9 Measured -2 Wire Model, Exact 5 ... SSGI

L-z 10-S4

10-5, 0.01 0.02 0.05 0.1 0.2 0.5 1 2 5 10 o [rad/s]

Figure 5-11: K3Fe(CN)6 calorimeter 303 mK thermal transfer function, with fit to two-wire model and resulting SSGI transfer function.

Since the thermal conductance of the K3Fe(CN)6 dough could not be found a priori from published data, no attempt was made to compare the measured transfer functions with what would be predicted given this calorimeter geometry and published data on the materials involved. However, the data were fit to the two-wire model with one adjustable parameter, the calorimeter thermal conductance. For these fits, the error bars on the points initially were taken to be the standard deviation of the JTad data, and generally were smaller than the plotting symbols. The scatter is generally greater than the error bars, particularly at high frequencies. The cause for this scatter in the transfer functions is not known. The effect of the scatter and small error bars was that fits of the data to calorimeter models gave unreasonably large x2 . Given this, estimates of calorimeter thermal conductance were extracted from the data in two ways. First, the data were fit to the model with equal weight assigned to each point, rather than weighting with 1/o.2 as is usual for x 2 fitting. Second, various two-wire transfer functions assuming specific values of K, were calculated and compared with the data. If the assumed K, was too small, the calculated curve showed a high-frequency rolloff not seen in the data. From this analysis, a (very weak) lower limit on K, was obtained. The parameters for the fit were determined as follows. The thermal conductance of a single link wire was taken to be half the slope of the power/temperature curve, interpolated linearly from power/temperature data taken at nearby temperatures for the relaxation method. The dough surface area, heater voltage, and heater resistance were measured as for the empty calorimeter functions. Finally, for the 108 mK and

81 2WTF 2WTF Kb/Ks Temp Fit Low. Lim. Range (mK) (pW/K) (pW/K) 108 40 t 8 20 0.02-0.05 140 55 t11 30 0.03-0.07 303 85 + 20 40 0.07-0.18

Table 5.1: K3Fe(CN)6 calorimeter K, estimates. K, was determined from a fit of the transfer function to the two-wire model (2WTF Fit), and a lower limit on K, was determined by comparison of the data and calculated two-wire transfer functions with various K, (TF Low. Lim.).

139 mK functions, the slab heat capacity was taken to be the value found via AC specific heat measurement, which corresponds with that published by Fritz et al. [52] for 6.44 mg K 3Fe(CN)6 single crystals (the correction for the 13.7 mg N grease present was three orders of magnitude below this and so was ignored). (Note that the data reported by Fritz was actually compiled from measurements by Rayl et al. [51] and by Domb and Miedema [55]. However, I will refer to this as the Fritz data.) For the 250 mK function, the heat capacity measured with the relaxation method was used. The heat capacity values were taken in this way because the heat capacity of K3Fe(CN)6 is a very steep function of temperature below 250 mK, so even slight differences (10 mK) in the temperatures at which relaxation and transfer function data were taken were enough to make the relaxation heat capacity data inappropriate for transfer function modeling. Given these input parameters, the functions were fit to the two-wire model as described above, with the calorimeter thermal conductance K, as the adjustable parameter. The resulting K, values determined for various temperatures are shown in Table 5.1. The table also shows the lower limits on K, obtained by comparing calculated two-wire transfer functions with the data. The thermal parameters obtained from the two-wire model were used to obtain the SSGI transfer function (see Chapter 3). The two-wire and SSGI model functions are shown in Figures 5-9, 5-10, and 5-11 above. The models correspond well with each other and with the measured data on the plateau region, as expected. Small deviations are observed at the high-frequency knee. At high frequencies, the internal structure of the calorimeter will be probed most, so one expects the exact locations of heater and thermometer to begin to play a role, and for deviations to therefore occur between the SSGI function and the exact model. The fitted values for K, confirm that Kb/K is small, and our expectation D = 1 on the plateau to within a few percent. Hence it is expected that the AC specific heat data will be accurate as well as precise. Finally, again since Kb/K, is small, and since the two-wire and SSGI models fit well, the relaxation AT(t) should fit to a single time constant. Zero field AC C/D(w) data for the K 3Fe(CN)6 calorimeter is shown in Figure 5-12. For comparison, the Fritz and relaxation data are also shown. The AC data

82 were collected at various times over the course of two months, during which time the refrigerator was warmed to 2K several times, and to 77K once. The data show excellent precision, notwithstanding the (gentle) thermal cycling to which the calorimeter was subjected. Precision on the order of 2% is typical. The relationship of these data to the Fritz data will be discussed further below. AC data was also taken at various fields below 1 T (Figure 5-13). As expected, the peak is suppressed by application of even a low field [52].

1.6 * AC Data, OkG o Relaxation data, OkG 1.4 o Fritz Data, 6.44mg+Ngrease

1.2

>1.0-

.* Z0.8 - x U 14 0.6 - 0 I - to0

0.4 -e

0.2 0 0.1 0.2 0.3 0.4 0.5 Temperature [K]

Figure 5-12: K3Fe(CN)6 calorimeter zero field AC and relaxation heat capacities, with data for K 3Fe(CN)6 single crystals published by Fritz.

The thermal relaxation method (cf. Section 3.2) was used to obtain calorimeter- to-bath link conductances and to measure zero-field heat capacity for K3Fe(CN)6 at four temperatures between lOOmK and 300mK. As discussed in Section 3.2, this data is useful to obtain accurate heat capacity measurements when the thermal properties of a calorimeter are not ideal for accurate AC heat capacity measurements, and to provide an accurate measure of the link conductance for modeling of calorimeter behavior. To check for possible T2 effect, the decay curves were fit to a sum of two exponentials. These fits gave equal time constants for the two exponentials, and split the throw evenly between them. This indicates that a single exponential is sufficient to describe the decay, and that there is no observable T2 effect. Since heater currents could be measured to ± 0.1 pA, and the heater resistance was known to better than ± 2 Q, yielding errors in power of typically ±0.4 nW at 100 mK and ±1 nW at 250 mK. With these errors, X2 near one were typically obtained for the linear fits

83 1.6 AC Data, OkG * AC Data, 0.8kG 1.4 A AC Data, 7kG AC Data, 9.5kG 1.2

,'1.0 - 0 e0 00 C) 0

.0.8 02 A

%.) A

0.6 - .{A0

0 *

0.4 * l

0.2 I 0.1 0.2 0.3 0.4 0.5 Temperature [K]

Figure 5-13: K3 Fe(CN)6 calorimeter field-dependent AC heat capacity. of the power/temperature curves. At a given temperature, AT(t) and Kb were measured multiple times, and the resulting T and Kb values averaged to yield final values T and Kb. The errors in i and Rb were taken to be the standard deviations of the distributions of measured T and Kb when these were measured more than twice, and taken to be half the difference between the measured values when measured twice. The calorimeter heat capacity was then calculated as C = Kbr. No correction was applied for the addenda heat capacity, since the addenda contribution was smaller than the error bars (see Table 5.2). The resulting heat capacity data are shown in Table 5.2, along with the Fritz data. The "errors" in the Fritz data are not due to errors in their data, but to my uncertainty in the temperature, which in turn is a result of the temperature resolution of the relaxation method used to take my data. Obviously, the errors in the relaxation data are large. The main sources of error are the consequence of inadequate temperature control over time scales of several minutes. One source of error was that all of the Kb points, with the exception of two taken for the 245 ± 5mK point, were taken without correcting for drift in the baseline temperature, which occurred over the Kb measurement time of several minutes. This led to errors in Kb in the range 10% to 20%. Another source of error was inadequate temperature control during AT(t) measurement, in which typical time constants were 10s above the peak and 30 - 50s below and near the peak, and a single AT(t) measurement included ten time constants or more. This led to T errors at the 10% level, although at higher temperatures (200 - 250mK) temperature stability was better and T errors were at

84 Empty Cal. K3Fe(CN) 6 Cal. Fritz Temp C C C (mK) (pJ/K) (pJ/K) (pJ/K) 95 ±5 1.3 0.3 32 4 46 ± 9 131± 4 - 96 20 131±17 210 5 - 56 10 47.1+ 0.3 245 + 5 4 0.3 60 2 45.3 ± 0.2

Table 5.2: Heat capacity data taken with relaxation method. Note that the empty calorimeter results ("Empty Cal. C") are an upper limit due to the low LR-400 bandwidth, and are for 68 mg of N grease. The K3 Fe(CN)6 calorimeter results ("K 3Fe(CN)6 Cal. C") are for 6.44 mg K3Fe(CN)6 and 13.7 mg N grease. The "Fritz C" column is the heat capacity of 6.44 mg of K3Fe(CN)6 from the Fritz paper. See text for explanation of the "errors" in the Fritz data.

the few percent level. These problems with temperature control were overcome in the Sr 3CuPtO.5 1ro. 5 0 6 experiment described in the next chapter, through the use of superior instrumentation and due to the fact that thermal time constants were 3s or less. Now I compare the AC and Fritz data at zero field. The agreement for temperatures near TN is quite good, but at higher temperatures the AC data are significantly higher than the Fritz data. Furthermore, this enhancement above TN is confirmed by the relaxation data. Given the results of the transfer function analysis and the agreement between AC and relaxation data, it seems reasonable to conclude that D = 1 over the range of temperatures measured. If that is the case, then the physics of the K3Fe(CN)6 sample measured here must be somehow different from that of the samples reported on by Fritz. The data reported by Fritz were taken on single crystals of K 3Fe(CN)6 grown from aqueous solutions of analytical reagent grade K3Fe(CN)6 . As noted in Section 5.1, the material used here was > 99% pure single crystals, crushed with mortar and pestle. It is well-known that air and light (particularly UV) will slowly convert 2 K3Fe(CN)6 to K4Fe(CN)6 .K 4Fe(CN)6 contains Fe +, which has spin two in many circumstances [1] [2]. The samples used here were protected from light by covering in blackcloth or storing in dark cabinets, and were exposed directly to air for only one or two days. Little conversion of K3Fe(CN)6 to K4 Fe(CN)6 is expected under these circumstances [56]. However, the conversion may have been enhanced since the K3Fe(CN)6 was in powder form after it was crushed and mixed with N grease. One hypothesis regarding the difference between my sample and that of Fritz is based on the idea that the heat capacity above TN is determined by short-range ordering within the chains, and that at TN by long-range ordering. Then one would say that the short-range ordering is somehow enhanced in my sample relative to that of Fritz, whereas the long-range ordering is affected very little. Suppose there are impurities within the chains, for example sites with Fe2+rather than Fe3+. The

85 relevant parameters here are the impurity concentration x and the correlation length (. For an Ising chain at low temperatures [2],

!e/kBT2~ (5.2)

given J ~ 0.36K for K 3Fe(CN) 6 , this leads to ( ~ 2 at 0.3K, and ( ~ 8 at 0.13K. Hence it seems that impurity concentrations x of 10 - 50% would be necessary to produce dramatic changes in the short-range order. It is difficult to see how these kinds of impurity concentrations could be present in my sample. Nor is it clear how such impurities would enhance the heat capacity due to short-range order. One might argue that if Fe 2 is indeed in a spin two configuration, it would contribute extra spin entropy to the system, which might be removed through short-range ordering. An- other argument might be to point out [1] that superexchange, which is the source of nearest neighbor coupling in K3Fe(CN) 6 , is very sensitive to the relative positions of surrounding atoms and to the non-magnetic species participating in the superexchange. Hence, if some of the CN groups were replaced by impurities or water, or if the crystal structure were somehow modified (say by grinding), the superexchange coupling might be made less anisotropic. This would lead to a more Heisenberg-like behavior, which is known to have an enhanced short-range order contribution to heat capacity relative to Ising behavior. In any case, studies of impurities in TMMC [57] show that even very small impurity concentrations, with 1/x ~3, can produce downward shifts in TN of 3%, and impurity concentrations such that 1/x ~ lead to pronounced broadening in the peak at TN in addition to the shift. Hence it would be surprising if impurities in the chains enhanced the short-range order without affecting the long-range order in some way. In the end, it seems difficult to produce a simple physical model to explain the differences between the AC and Fritz data at zero field. A definite disadvantage of the use of K3Fe(CN)6 as a standard for heat capacity measurement is the complexity and sensitivity of the superexchange interaction that dominates its low-temperature behavior. On the other hand, it is clear that this apparatus detected the transition to long-range order in K3Fe(CN)6 , that the AC and relaxation data agree reasonably well to within (admittedly large) error bars, and that the AC data correlates well with published data, particularly around TN. It is also apparent from the transfer functions and heat capacity data that the thermal properties of the calorimeter are sufficient for accurate AC and relaxation method measurements, and that field-dependent measurements are feasible. This measurement of heat capacity of K3Fe(CN) 6 gives confidence that the apparatus will be able to detect any interesting features in the

Sr 3CuPt. 5 Ir 0 .5 O6 spectrum, and suggests changes (such as improved temperature control) that will improve the quality of the relaxation data.

86 Chapter 6

Sr 3CuPtO.51ro. 5 0 6 Experiment

Given the favorable results of the K3Fe(CN)6 experiment, it was appropriate to move on to measurement of Sr 3CuPtO.5 IrO.5 0 6 heat capacity. There were three goals for the Sr 3CuPtO.5 1r o.5 0 6 experiment: first, in light of the susceptibility measurements of Beauchamp, to look for a transition to long-range order above 1 K ; second, to measure zero field specific heat for T < 1K if there was no clear evidence for a transition to long-range order; and third, to measure field-dependence of specific heat to 70 kG. The first run of the Sr 3 CuPtO.5 Iro. 5 0 6 experiment had serious technical problems that made it impossible to achieve these goals. Due to the NbTi lead wires, three of the four thermometers lost at least one wire during cooldown, which meant that one four-wire thermometer was available on the empty calorimeter and two three-wire thermometers were available on the Sr 3CuPtO.5 1r o.5 0 6 calorimeter. This made it impossible to measure temperature and ITac accurately. Also, there was sufficient K3Fe(CN)6 residue (less than a few pg!) in the "empty" calorimeter to lead to a K3Fe(CN) 6 signature at 130 mK. Due to these difficulties, all the data reported below were taken during the second run of the Sr 3CuPtO.5 1r o.506 experiment, in which these technical problems were solved as described in Section 6.1.

6.1 Calorimeter Preparation

As for the K 3Fe(CN)6 experiment, calorimeter preparation began with two complete calorimeter plates. These were the same plates used for the K3Fe(CN)6 experiment. The Sr 3 CuPtO.5 Iro.5 O6 sample was mounted on the two plates formerly used for the K3Fe(CN)6 experiment empty calorimeter, while the empty calorimeter for this experiment used the two plates formerly used for the K 3Fe(CN)6 sample. The K 3Fe(CN)6 residue was mechanically removed from the plates using 20 pm (Number 600) silicon carbide grit. It was assumed that this change in surface texturing did not change the thermal properties of the plates significantly. The NbTi wires of the K3Fe(CN)6 experiment were replaced with PtW wires, as discussed in Section 4.2.2. 105.6 mg of Sr 3CuPtO.51r o.5 06 were used during this experiment. 73.1 mg of this were applied using the method described in Section 5.1, and the rest using a new technique. The new technique was developed because it was tedious to scrape all of

87 the dough out of the mortar and pestle. Sr 3CuPtO.s5 ro.0 O6 and N grease were measured out on separate tares. The Sr 3CuPtO.5 Iro.5 was ground by itself in the mortar and pestle, then transferred onto the N grease tare and mixed with the grease. This transfer was easy because the powder and mortar were dry at this point. The amount of dough was measured, and the dough scraped from the tare to the calorimeter. Then the final mass of tare plus dough residue was measured to determine how much of the dough had actually been transferred to the calorimeter. A total of 238.5 mg N grease was used in making the Sr 3CuPt0 ro.5 0 6 dough. 200.0 mg N grease was used on the empty calorimeter.

6.2 AC Method Procedures

Two thermal transfer functions were measured for each calorimeter, one just above 100 mK and one at 400 mK, at zero field. The EPAF range chosen was 0.13 rad/s to 25.13 rad/s. The method used to acquire and correct ITac| data was exactly as described in Section 5.2. As for the K 3Fe(CN)6 experiment, the need to correct thermal transfer functions for the LR-400 bridge transfer function was not recognized until after the experiment was completed. Hence the bridge again distorted the transfer functions at high frequency, and the frequencies chosen for heat capacity measurements were below plateau for both calorimeters. For the empty calorimeter, 2.51 rad/s was chosen for heat capacity measurements (for all fields and temperatures). For the Sr 3CuPtO.5 IrO.5 O6 calorimeter, 1.26 rad/s was chosen (for all fields and temperatures). Fortunately, these choices were not very far below the plateau (of the corrected transfer function), and a plateau region is visible in most of the corrected transfer functions. Hence it is possible to correct the zero field heat capacity data for the poor frequency choice (see Section 6.5). Nevertheless, due to the uncertainties involved in making this correction, the heat capacity data taken with the AC method will only be used as a check on the relaxation data, and only data taken with this latter method will be analyzed to determine the specific heat of Sr 3 CuPtO.s5 ro.5 O6 . In addition to zero field, AC heat capacity was measured for both calorimeters in fields up to 5 kG.

6.3 Thermal Relaxation Method Procedures

As mentioned in Section 5.3, Kb and AT(t) alone are useful for calorimeter thermal characterization. For the Sr 3 CuPtO.5 Iro.50 6 experiment an additional thermal characterization experiment was possible. Since all calorimeter thermometers and heaters except for the Sr 3CuPtO.5 1ro.5 0 6 calorimeter top heater were functional for the entire experimental run, it was possible to estimate the thermal conductance across the dough by examining differences in power/temperature curve slopes for curves obtained with different heater/thermometer combinations. For the empty calorimeter, power/temperature curves were measured at 128 mK and 400 mK using all combinations of heaters and thermometers. For two curves measured with the same ther-

88 mometer but different heaters, the difference in the slopes of these curves is related to the dough thermal conductance via (see Appendix A)

Sbb (6.1) K7 mbt mbb where mbt is the slope of the bottom thermometer, top heater curve; and mbb the slope of the bottom thermometer, bottom heater curve. (A similar expression holds for the top thermometer curves, with mbt replaced by mtb and mbb replaced by mtt.) Here it is assumed that the thermal link wires have the same thermal conductance.

For the Sr 3CuPtO.5 1r o.5 0 6 calorimeter, power/temperature curves were measured at three temperatures using the bottom thermometer and bottom heater in one case and the top thermometer and bottom heater in the other (this was necessary because the top heater was not functioning). In this case, K, is found from (see Appendix A)

Ks= mtb(( mtb - mbb ) 2 + 2 mTn - Mnbb_ (6.2) 'mbb mbb mtb is the slope of the top thermometer, bottom heater curve; and mbb the slope of the bottom thermometer, bottom heater curve. Again, the thermal link wires are assumed to have the same thermal conductance. The values of K, obtained using, these equations will be discussed in Sections 6.4 and 6.5 below.

Since the Sr 3 CuPtO.5 1ro.5 0 6 calorimeter top heater was not functioning on this run, and since it was desired to take heat capacity data using heater and thermometer on opposite sides of the dough to detect possible problems with dough thermal resistance, the top thermometer and bottom heater were used for heat capacity data acquisition for the Sr 3CuPtO.5 IrO. 5 0 6 calorimeter. A few points with bottom thermometer and bottom heater were taken to check for differences (see Section 6.5). For consistency with the Sr 3 CuPtO.5 1r o.5 0 6 calorimeter, the top thermometer and bottom heater were also used for heat capacity data acquisition on the empty calorimeter. A few points with other thermometer/heater combinations were also taken on this calorimeter to check for differences (see Section 6.4). The method used for mixing chamber temperature control for this experiment differed from, and was superior to, that used for the K3Fe(CN)6 experiment. The LR-400 bridge was used to measure mixing chamber chip resistance. The desired mixing chamber chip resistance was set directly on the LR-400 bridge, and the 1OAR output of the bridge fed to the PARC Model 113 pre-amp. The pre-amp output was fed into a home made temperature controller, the output of which drove the mixing chamber heater. This setup differs from that of the K 3Fe(CN) 6 experiment in that the LR-400 provides an amplified voltage, proportional to the difference between the chip resistance and the setpoint resistance, directly to the PARC pre-amp. In the K 3Fe(CN) 6 experiment (Section 5.3), the homemade bridge and lockin provided a voltage proportional to the chip resistance, which then had to be subtracted from the setpoint and amplified by the PARC pre-amp. In the Sr 3 CuPtO.5 1ro. 5 0 6 experiment, once temperature control was established at a new temperature (which typically required 10 minutes for a 100 mK temperature step), the mixing chamber chip resis-

89 tance fluctuated by less than +2 ohms at 6kQ and less than +1 ohm at 3kQ. This corresponds to temperature fluctuations of 0.1mK at 130mK and 0.5mK at 400mK. The power PO to be applied to the heater for AzT(t) relaxation data was determined by the condition that the temperature change AT produced by PO be at least 10 times as large as the noise in the T measurement. With PO determined, Kb was then determined using the K 3Fe(CN)6 experiment procedure that allowed for drift correction; i.e. between every two or three points, the temperature at zero power was checked, rather than ramping the power up to P and then back down. The drift was corrected for in the analysis or, if it was too severe (total drift 10 Q or more during the measurement was considered severe), the measurement was redone with better settings on the PID controller. A typical power/temperature curve is shown below.

3.5 -

3.0 -

2.5 -

c 2.0 -

S1.5

1.0 -

0.5 - Data -Fit 0

0.191 0.193 0.195 0.197 0.199 Temperature [K]

Figure 6-1: Typical power/temperature curve for the Sr 3CuPtO.5 ro.0 O6 experiment, with drift correction.

As mentioned in Section 5.3, the LR-400 bridge is not adequate for A T(t) measurement for time constants of a few seconds or less. In this experiment, time constants of both the sample and empty calorimeters were less than 3 seconds over the entire temperature range examined. To measure AT(t) for short time constants, the arrangement shown in Figure 6-2 was used. The Stanford Research Systems SR830 lock-in amplifier supplied a 3 Volt, 100 Hz AC voltage to a home made circuit. This circuit supplied 100 Hz, 0.25pA AC current to the current leads of the thermometer via a transformer, and sensed (with amplification provided by another transformer) the resulting voltage on the thermometer voltage leads. This voltage was then measured with the SR830. 100 Hz was chosen because it maximized AT(t) signal-to-noise. A 30 ms time constant was used for the SR830. This time constant was chosen as the

90 Mixing Chamber Homemade Bridge + SR8 30 Lock-In

RB Home nade LeCroy Tempe rature Scope Contr )ller

O)ut Current Source trl In C

- PC Linux Box

Figure 6-2: Apparatus used to measure relaxation heat capacity for

Sr 3 CuPtO.5 1ro.5 0 6 experiment.

91 best tradeoff between minimum distortion of AT(t) (which argues for a short time constant) and noise in AT(t) (which argues for a long time constant). However, even this lockin time constant led to a fairly noisy AT(t). This problem was solved by taking many (typically 10) AT(t) measurements and averaging them with a LeCroy 9410 digital oscilloscope. A typical AT(t) before and after signal averaging is shown in Figure 6-3. The resulting averaged AT(t) was transferred to the PC for further analysis.

I * 0.260

0.258

0.256

5 0.254

E 9 0.252

A. 0.250

0.248

0 6 12 18 Time [s]

Figure 6-3: Raw AT(t) (bottom curve) and AT(t) after averaging ten decays (top curve). Top curve is shifted up by 2 mK for comparison.

To check the homemade circuit for self-heating, the current supplied by the homemade circuit was reduced until the SR830 reading was found to be independent of current. The 0.25 pA used was in this current-independent regime. Also, the LR- 400 was connected to a calorimeter thermometer, and no change was recorded on it when the homemade circuit was connected to and disconnected from the calorimeter thermometer on the opposite plate. The time response of this SR830 setup was also checked. A simple circuit that allowed for manual switching between two different resistances was constructed, and the response of the SR830 setup and the hp34401a to switching was measured. The hp34401a found a switching time of less than 10 ms. For the SR830 setup, the response was exponential with a time constant of 51 t 3 ms. The measured AT(t) must be corrected for the finite response time of the SR830 setup. This matter will be discussed in Sections 6.4 and 6.5 below. Note that the thermometer resistance is not measured directly in this measurement scheme. However, measurements show a linear relationship between lockin voltage and thermometer resistance. To convert lockin voltage to thermometer resistance,

92 the resistances at zero heater power and at Po, measured for the Kb determination, are identified with the corresponding lockin voltages in the AT(t) data. Voltages between these two are converted to resistance by linear interpolation. Then R(T), known from the 3He MCT calibration, is used to obtain AT(t) from AR(t).

6.4 Empty Calorimeter Results

6.4.1 AC Method Results

10-2

10-3 * Measured 2 Wire Model, Exc t: ---- Schwall Model, Ex Vc t- --- Published Material s Data- 3 !: (Zr1 V r K / , S= P111 / * / * / 2

10-4 0.1 0.2 0.5 1 2 5 10 20 50 100 co [rad/s]

Figure 6-4: Empty calorimeter 128 mK thermal transfer functions.

As usual, thermal transfer functions were measured on the empty calorimeter in order to determine a proper EPAF for heat capacity measurements and for thermal characterization of the calorimeter. 128 mK and 400 mK empty calorimeter thermal transfer functions, corrected for the effects of the LR-400, are shown in Figures 6-4 and 6-5. The low-frequency knees are visible on each transfer function, and the plateau is visible in the 400mK function. Since the scatter in the data is again large relative to the assumed error, the transfer functions were analyzed as described in Section 5.4 to obtain estimates of K,. Both transfer functions were fit to the exact two- wire model to obtain the calorimeter thermal conductance, and the resulting thermal parameters used to obtain the SSGI transfer function. The model parameters were determined as for the K 3Fe(CN)6 experiment, with the exception of the calorimeter heat capacity. This was taken to be the value measured with the relaxation method.

Due to observation of a T2 effect in the empty calorimeter relaxation data at low

93 0.1

0.05

0.02

S0.01

0.005 Measured -2 Wire Model, Exact -- Schwall Model, Exact Published Materials Data SSGI, K=100 uW/K 0.002 "/SSIK=10w/

0.001 0.1 0.2 0.5 1 2 5 10 20 50 100 w [rad/s]

Figure 6-5: Empty calorimeter 400 mK thermal transfer functions. temperatures, these transfer functions were also fit to a Schwall-type model, discussed in Appendix D. For this Schwall transfer function, there were two fit parameters: heat capacity of lump one C1 , and the thermal conductance of lump one K1 (this is not taken as the sample conductance K, for reasons discussed in Section 6.4.2). The heat capacity of the second lump C 2 was taken to be the value Ctrt measured with the relaxation method, minus C1. The conductance of the second lump was taken from the power/temperature curve results. As Figures 6-4 and 6-5 show, the agreement between the models and the measured functions is good. The measured function agrees better with the Schwall function at low temperatures, as expected from the relaxation results, but the Schwall and two-wire models both describe the data well at 400mK, also expected from the relaxation results (the T 2 effect vanishes above 350mK). The calorimeter sample conductance K, as determined by the two-wire transfer function is shown in Table 6.1 (the fitted C1, K1 from the Schwall transfer function will be shown in Table 6.5). The K, value determined by the transfer function fit is clearly unreasonable at 128mK. This is because the transfer function does not extend to sufficiently high frequency to show the high-frequency knee. Without this knee, it is only possible to put a lower limit on K. The empty calorimeter thermal conductance K, was also measured by comparing power/temperature curves taken with bottom thermometer, bottom heater on the one hand, and bottom thermometer, top heater on the other. This was done at 127 mK and 394 mK. The resulting power/temperature curve slopes, and the thermal conductance K, derived from them using equation (6.1), are shown in Table 6.1. Also shown is a lower limit on K, obtained by shifting the slopes appropriately by one standard deviation. Finally, a

94 2WTF TF PT PT Kb/K Temp Fit Low. Lim. Slope Low. Lim. Range (mK) (ptW/K) (pW/K) (pW/K) (pW/K) 128 > 1026 20 28 26 0.10 - 0.13 400 93 40 50 470 180 0.03-0.08

Table 6.1: Estimates of empty calorimeter K,. K, was determined from a fit of the transfer function to the two-wire model (2WTF Fit), and from the slopes of the power/temperature curves and Equation (6.1) (PT Slope). Lower limits on K, were determined by comparison of the data and two-wire transfer functions with various K, (TF Low. Lim.), and by varying the power/temperature curve slopes appropriately by one standard deviation (PT Low. Lim.).

range for Kb/K, is found from the largest and smallest possible K, shown in the table at the given temperature. It should be noted that power/temperature curves were also taken using top thermometer combinations, and the resulting slopes were the same to within the error bars. The top thermometer data would then suggest that the K, are much larger than any of those reported here; hence even the largest values reported in the table must be considered lower limits. In addition, transfer functions based on the two-wire model and published data on the calorimeter materials were computed and compared with measured functions for the empty calorimeter. The model parameters were determined as for the computed empty calorimeter functions in the K3Fe(CN)6 experiment. The dough thickness was assumed to be 10 t 5 mil. Figures 6-4 and 6-5 compare the computed with the measured functions. Again, the agreement is relatively good. Since the computed Kb is higher than measured, it is no surprise to see the computed function fall off faster with decreasing frequency. The computed plateau is also at higher wITct than the measured, due to the fact that the measured heat capacity is higher than the computed. This also is not too surprising, since the materials used for the calorimeter may have impurity heat capacity contributions not seen in the published results. AC heat capacity data were taken for 163 mK < T < 430 mK at zero field. In addition to the usual corrections for the lockin RMS reading and LR-400 transfer function, these data were corrected for the fact that the chosen EPAF was below the plateau of the corrected transfer function. To obtain the off-plateau correction factor, the ratios JTact(Wp)/ITac(Wmeas = 2.51 rad/s), for wp = 5.03 rad/s and wp = 25.13 rad/s, were computed. These wp were chosen because they describe the boundary within which the actual plateau would be expected to fall. Then the off- plateau correction factor A was taken to be the mean of these two values, and was assigned an error equal to half of the difference between the two values. Using this method, A = 1.5 ± 0.1 at 128 mK, and A = 1.8 t 0.2 at 400 mK. To account for the temperature dependence of A, a simple linear interpolation was done for temperatures between 128 mK and 400 mK. The resulting C/T is plotted in Figure 6-6, along with the measured relaxation heat capacity. The agreement between the AC

95 and relaxation results is quite good. This suggests that D = 1 on plateau, consistent with the conclusions of the transfer function analysis. Note that the error bars in the AC data are dominated by the uncertainty in my determination of A. AC heat

-- 2.00

.75 -

.50

.25 - x C) .00

o OkG oc 0.75 k * OkG relax

0.50 - -I

0 0.2 0.4 0.6 0.8 1.0 Temperature [K]

Figure 6-6: Empty calorimeter AC heat capacity with off-plateau correction, and relaxation heat capacity. capacity data were also taken in several fields up to 5 kG. Since thermal transfer functions were not available for these fields, it was not possible to correct these data for any off-plateau effect. Hence these data are plotted (along with the uncorrected zero field data) without any off-plateau correction in Figures 6-7 and 6-8, and are useful mainly for qualitative analysis.

6.4.2 Relaxation Method Results Recall that heat capacity is obtained in the relaxation method (provided that the thermal properties of the calorimeter are not too bad) by measurement of Kb and AT(t), and fitting the latter to one or a sum of two exponentials. In the Sr 3CuPtO.5 Iro.0 O6 experiment, the finite response time of the SR830 setup must be taken into account in order to fit AT(t) properly. As mentioned above, the SR830 setup had a step response given by S 0 t < 0 (6.3) 1 - exp(-t/Tr) t > 0 where Tr, = 51 ± 3ms. Hence AT(t) is the convolution of the thermal response of the calorimeter and this exponential response. In Appendix B, the expected output

96 3 0 * OkG relax _ * 1kG relax o OkG AC o 1kG AC U 2.5 -

0 X2.0 - -,j 11~ 0 0 0 0 0 1.5

0.5 EPI .I 0.05 0.20 0.35 0.50 Temperature [K]

Figure 6-7: Empty calorimeter AC (no off-plateau correction) and relaxation heat capacity, 0 kG and 1 kG.

3.0 - 3kG relax 5kG relax 3kG AC v 5kG AC 2.5 -A

x A - 2.0 A

V A 1.5

1.0

U0.5

0.05 0.20 0.35 0.50 Temperature [K]

Figure 6-8: Empty calorimeter AC (no off-plateau correction) and relaxation heat capacity, 3 kG and 5 kG.

97 signal AT(t) given s(t) and the actual thermal response ATi(t) is computed for A Ti(t) given by one and two exponentials. The measured AT(t) were fit to the appropriate expression for r, = 48, 51, 54 ms. From these fits, the exponential coefficients A, B and time constants TA, TB of the thermal response

AT(t) = Ae-'I'A + Be-''B (6.4) were obtained for each r. For the power/temperature curves, heater currents again could be measured to better than ± 0.1 pA, and the heater resistance was known to better than ± 2 Q, yielding errors in power of typically t0.4 nW at the lowest temperatures and ±10 nW at the highest temperatures (near 1 K). With these errors, x2s less than or near one were obtained for the linear fits of the power/temperature curves (less than because the current measurement is probably better than the indicated error). A typical power/temperature curve was shown in Figure 6-1 above. The Kb data in zero field are plotted as a function of temperature along with the expected Kb in Figure 6-10. The expected curve was computed based on Suomi's data [54] for commercial copper wire. In Figure 6-11, field dependence of Kb is shown. It is expected that Kb will be dominated by the electron channel in the copper link wire at these temperatures. Given that the wire used here was common commercial wire, prepared without any special attention to purity, it is likely that there are magnetic impurities in the wire that could lead to significant magnetoresistance and a reduction in Kb at high fields. Relaxation heat capacity was measured in zero field for 137 mK < T < 846 mK, and in fields up to 70 kG for 130 mK < T < 450 mK. Above 350 mK, AT(t) fit best to a single exponential. Below 350 mK, all AT(t) curves required two exponentials with different time constants for a good fit (Figure 6-9). For the data above 350 mK, heat capacity was computed as C = KbT. For the data below 350 mK, the Schwall formula AT A + BTB Cot = Kb A B (6.5) was used. Use of this formula is discussed further below. With typical scatter in AT(t) of ± 0.03 mK, x2s close to one were obtained for the fits. The mean heat capacity was taken from Kb and the T, = 51 ms fit results. The error in the heat capacity was found by propagating the errors in the Kb and r, = 51 ms AT(t) fit through the heat capacity formulae, and adding to this result half the difference between the heat capacities obtained for T, = 48 ms and Tr, = 54 ms. As the field was increased, especially above 10 kG, the larger time constant became still larger at low temperature, while the smaller did not change or even fell slightly. Relaxation heat capacity was measured at 127 mK and 394 mK at zero field with all thermometer/heater combinations. These data are shown in Table 6.2. The differences in the heat capacity results due to the use of different thermometer/heater combinations are clearly larger than the errors due to the fits. This is thought to be due to the asymmetry of the calorimeter design, in particular that the top plate is in contact with longer vespel pegs than is the bottom plate. Hence it is important that the same thermometer/heater combination be chosen for both the Sr 3 CuPtO.5 Iro.50 6 and

98 129.0

128.5

o 128.0 ~ Data - Two-exponential fit --- One-exponential fit 127.5

E g 127.0 - \-

126.5

126.0 0 3 6 9 Time [s]

Figure 6-9: Empty calorimeter low temperature AT(t), showing best fit to sum of two exponentials.

e -

-D 2 0

E - * Measured - Published Data

.*~

0 I- _ 0.1 0.3 0.5 0.7 0.9 Temperature [K]

Figure 6-10: Empty calorimeter zero field thermal conductance to bath Kb, measured and predicted from published data on copper.

99 I I I 2.0 * OkG o 5kG * 10kG o 30kG e 1.5 A 50kG . 70kG 00 x . -.

Cc1.0 0

UA 00 0 AA

0* ~0.5 AA AA

0.1 0.2 0.3 0.4 0.5 Temperature [K]

Figure 6-11: Field dependence of Kb.

Temp Ctt Ctb Cbb Cbt (mK) (pJ/K) (pJ/K) (pJ/K) (pJ/K) 127 ± 1 1.39 ± 0.03 1.60 ± 0.05 2.03 ± 0.06 2.16 ± 0.07 392 + 1 4.3 ± 0.3 3.6 ± 0.06 4.6 ± 0.5 3.91 ± 0.07

Table 6.2: Relaxation heat capacity from various thermometer/heater combinations on the empty calorimeter. The notation Cth refers to heat capacity measured with thermometer t, heater h.

100 empty calorimeters, in order to obtain the appropriate amount to subtract from the

Sr 3CuPtO.5 1r o.5 0 6 calorimeter result to obtain Sr 3 CuPtO.5 1ro. 5 0 6 specific heat. It is somewhat surprising that the empty calorimeter heat capacity shows field dependence at low fields. The materials chosen for the calorimeter were chosen in part because they were thought to be non-magnetic. There are two possible reasons for this magnetic response. First, some pure component in the calorimeter, thought to be non-magnetic, actually is magnetic. Second, there are magnetic impurities in some component(s) of the calorimeter. Given the size of the magnetic response (at the lowest temperatures, C/T doubles between OkG and 1kG), it seems unlikely that the PtW wires, chip resistors, or copper thermal link wire could be responsible for the effect (see Figure 4-4). This leaves the quartz plates, N grease, or copper fingers. Since N grease is composed of petroleum-based hydrocarbons [58], its specific heat is unlikely to have any dramatic field dependence. Pure quartz is also non-magnetic. As mentioned in Section 4.2.2, the copper fingers were made from commercial copper shim stock, and so may contain large amounts of magnetic impurities such as iron. It seems most likely that the copper fingers are responsible for this low-field heat capacity response. I return now to the question of whether it is appropriate to use the Schwall formula to compute heat capacity for two-exponential fits with this calorimeter. Since two exponentials fit the data best (both one and three exponentials give worse fits), it is reasonable to model the calorimeter with two lumps, having heat capacities C1 and C2, and connected by some thermal conductance K 1. The only remaining question for the model is then how to connect these lumps to thermal ground. It might seem most appropriate, given the geometry of the calorimeter, to connect each to ground with a thermal link Kb/2. Given this model, and measured values of Kb and the AT(t) fit parameters A, B, TA, and TB, it is possible to predict C1, C2, and K1 (see Appendix C). Applying this model to the data gave unreasonable results. In particular, two solutions for C1, C2, and K1 were obtained for each temperature examined. One solution predicted K1 < 0, and the other predicted C1 + C2 that did not extrapolate smoothly to the higher temperature, one-exponential data (e.g. this model predicts that C +C2= 5.5pJ/K at 322 mK, whereas at 351 mK, the lowest one-exponential fit temperature, C + C2 = 3.18pJ/K). On the other hand, the standard Schwall model, with only one of the lumps connected to ground, is found to give more reasonable results: the K1 are positive, and C1 + C2 is consistent with the high temperature data. Furthermore, the results obtained with the standard Schwall model agree with the corrected AC heat capacity(Figure 6-6). Another argument against use of the two-link Schwall model can be made. Pre- sumably, in the two-link model the two lumps most likely represent the two plates, with the N grease in between providing K1 = K,. However, in light of the other thermal characterization data, the source of this two-exponential behavior could not be poor thermal conductance across the dough. The power/temperature curves taken with different heater/thermometer combinations suggest that the thermal conductance K, between the given heater and thermometer is reasonably large. However a small, weak thermal link to some part of the calorimeter that is not between them would not be detected in these measurements, and such a link could be the source of

101 Field Temp C1 C1, K 1 C1 C2 K1 Cu Fing. Vespel (kG) (mK) (pJ/K) (pJ/K),(pW/K) (pJ/K) (pJ/K) (pW/K) 0 128 0.1 0.2,0.01 0.4 ± 0.lx 1.2 t 0.lx 1.0 ± 0.6x 0 128 0.1 0.2,0.01 0.56 0.99 0.61 0 221 0.2 0.5,0.04 0.64 1.56 1.15 0 322 0.3 0.8,0.07 0.78 2.17 1.13 0 400 0.3 0.8,0.07 1.3 ± 1.1x 2.3 t 1.1x 17 ± 19x 70 129 25.2 - 13.66 0.66 1.40 70 200 10.6 - 4.35 0.97 0.75 Table 6.3: Parameters for Schwall model of empty calorimeter. The values with an "x" next to them are from fits of the measured AC transfer function to the Schwall model. The Cu Fing. column shows estimated heat capacity for two sets of copper fingers. The Vespel column shows estimated heat capacity and conductance for the Vespel pegs. this two-exponential behavior. Hence it seems appropriate to use the standard one-link Schwall model to compute heat capacity of the empty calorimeter for the case where AT(t) fits two exponentials. It is desirable to go further and try to connect such a model with the actual calorimeter geometry. Since K1 is not between the calorimeter plates, lump 2 would be associated with the calorimeter plates and dough, and lump 1 with the weakly connected part of the calorimeter. This weakly connected part could be all or part of the copper fingers, the vespel pegs, or the PtW lead wires, since these parts of the calorimeter are not between the heaters and thermometers. At 130 mK, the PtW wires have an estimated total heat capacity of 7 nJ/K, orders of magnitude below the measured heat capacity (1.55 paJ/K) at this temperature and below the resolution of the experiment. The vespel peg heat capacity at 130 mK is difficult to estimate, since no measurement of vespel's heat capacity could be found in the literature; assuming it behaves like N grease, the 6 pegs (massing less than 30 mg each) would have a total heat capacity of 0.3 pJ/K. Finally, the two sets of copper fingers have a total heat capacity (estimated from the literature) of 0.12 pJ/K at 130 mK. These heat capacity estimates suggest that the vespel pegs or copper fingers are the most likely candidates for lump 1. The values of C1, C2, and K1 derived from the AT(t) fit parameters and Kb for the one- link Schwall model are shown in Table 6.3 for zero field and 70 kG data. Also shown (values are marked with an "x") are C1, C2, and K1 as determined from the Schwall transfer function introduced in Section 6.4.1. Neither the computed copper finger nor the computed vespel numbers agree very well with the numbers derived from the data. Given that the Schwall model is not very detailed and only gives reliable results for Ctot, this is not too surprising. However, for the zero field data, it seems more likely that the vespel pegs are responsible for the two-exponential behavior. The heat capacity estimates are closer, and it is difficult

102 to imagine how K1 could be so low for the copper fingers. For the 70 kG data, the copper fingers are most likely to be responsible for the two-exponential behavior, since copper is known to have a large nuclear contribution to its heat capacity at these temperatures and fields. As shown in the table, the measured C1's here compare reasonably well with the expected nuclear heat capacity of copper. The fit parameters for the Schwall AC transfer function compare reasonably well with those found from the relaxation data, particularly at 128mK. At 400mK, the Schwall AC fit parameters have large errors, indicating that that model does not work as well at high temperatures, which is consistent with the fact that no T 2 effect was observed at high temperatures. This vanishing of T2 effect at high temperatures is consistent with the Schwall relaxation model assuming that C2 > C1 at high temperatures. This is reasonable, as one would expect the heat capacity of the quartz plates to rise more quickly than that of Vespel as the temperature rises. Finally, C/T for this empty calorimeter is at least a factor of three higher than C/T of the empty calorimeter used for the K 3Fe(CN)6 experiment (recall that C/T measured for the latter experiment is an upper limit). This can be attributed to the fact that 200 mg of N grease were used for this calorimeter, while only 68 mg were used for the other.

6.5 Sr 3CuPt0 .Ir 0 .50 6 Calorimeter Results

6.5.1 AC Method Results

103 9 8 7 . Measured -2 Wire Model, Exact 6 SSGI, K,=20 AW/K 5 4 ......

104 1 0.1 0.2 0.5 1 2 5 10 20 sa 100 w [rad/s]

Figure 6-12: Sr 3CuPto.5 1r o.5 0 6 calorimeter 136 mK thermal transfer functions.

103 0.01 0.009 0.008 0.007 0.006......

0.005

0.004

0.003

0.002 - 2 Wire Model, Exact -.-.-.-SSGI, K,=37 AW/K

0.001 0.1 0.2 0.5 1 2 5 10 20 50 100 w [rod/s]

Figure 6-13: Sr 3CuPtO.5 IrO.50 6 calorimeter 400 mK thermal transfer functions.

For the Sr 3CuPtO.51ro.5 0 6 calorimeter, thermal transfer functions were measured at 136 mK and 400 mK. These are shown in Figures 6-12 and 6-13, corrected for the effects of the LR-400. In the frequency range examined, the low-frequency knee, plateau, and the beginnings of the high-frequency knee are visible in each transfer function. Again, the functions were compared to the exact two-wire model in the manner described in Section 5.5 to obtain calorimeter thermal conductance, and the resulting thermal parameters used to compute the SSGI transfer function. A fit to the Schwall transfer function was attempted, but the fit parameters for C1 and K1 were negative. Thus the Schwall model was excluded as a possible model for the Sr 3CuPtO.5 IrO.5O6 calorimeter in this temperature and field regime, which is consistent with the one-exponential behavior found with the relaxation method (see 6.5.2). Model parameters were determined as for the empty calorimeter, and all parameters for the best-fit transfer functions are shown in Table 6.4. Calorimeter thermal conductance was also measured by comparing power/temperature curve slopes measured with bottom thermometer and bottom heater on the one hand, and with top thermometer and bottom heater on the other. K, was then obtained using equation 6.2. The calculated K, along with the lower limit on K, obtained by shifting the slopes appropriately by one standard deviation, are shown in Table 6.4. Given the presence of the plateau, and the low Kb/K, indicated in the table, D = 1 would be expected for the AC heat capacity data. Zero field AC heat capacity data were taken for 0.100 K < T < 2.1 K. Some of thes data, including all of the data above 0.5 K, were taken during slow cooldowns of the refrigerator from - 2 K. The lockin RMS reading and LR-400 transfer function corrections were applied. As for the empty calorimeter, the exci-

104 2WTF TF PT PT Kb/K Temp Fit Low. Lim. Slope Low. Lim. Range (mK) (pW/K) ([LW/K) (pW/K) (pW/K) 136 19 6 15 - - 0.14-0.24 145 - 41 25 0.1-0.15 306 - - 370 130 0.03 - 0.08 400 37 4 30 - - 0.38-0.52 430 - - 1400 193 0.01 - 0.09

Table 6.4: Sr 3 CuPtO.5 Iro.5 0 6 calorimeter K,. K, was determined from a fit of the transfer function to the two-wire model (2WTF Fit), and from the slopes of the power/temperature curves and Equation (6.1) (PT Slope). Lower limits on K, were determined by comparison of the data and two-wire transfer functions with various K, (TF Low. Lim.), and by varying the power/temperature curve slopes appropriately by one standard deviation (PT Low. Lim.).

tation power frequencies chosen for heat capacity measurement were below plateau for the Sr 3CuPtO.5 1ro.5 0 6 calorimeter. However, given that the plateaus are visible in the measured transfer functions, it is again possible to correct for the poor choice of frequency. For the 136 mK transfer function, the method described for the empty calorimeter could not be used to determine the off-plateau correction factor A, due to the scatter in the data at high frequencies. However, the model functions as well as the measured function support the choice A = 1.0. For the 400 mK function, the plateaus shown by the two model functions were used to estimate the range over which the plateau could occur, giving A = 1.2 ± 0.1. Again, intermediate temperatures were corrected using linear interpolation. The resulting C/T is plotted in Figure 6-14, along with the measured relaxation heat capacity. The agreement between the AC and relaxation results is quite good below 400 mK. This suggests that D = 1 in this temperature range, again consistent with the transfer function analysis. The divergence between the AC and relaxation data at higher temperatures suggests that the growth of A with temperature is stronger than linear. AC heat capacity data were also taken in fields up to 5 kG, for temperatures 140 mK < T < 440 mK. As for the empty calorimeter, it was not possible to correct these data for off-plateau effects. These data are plotted, along with uncorrected zero field data and relaxation data, in Figures 6-15 and 6-16. Comparison of the AC data with the low-field relaxation data (Figure 6-15) shows good qualitative agreement, and that the AC data is slightly higher than the relaxation. This quantitative difference is expected due to the off-plateau effect. The zero field data above 1 K merits special attention, since this is the range in which Beauchamp has seen a peak in AC susceptibility. Figure 6-17 shows this data, taken on two cooldowns, with correction for the LR-400 transfer function applied. Note that no correction for the off-plateau effect is applied, so Figure 6-17 shows C/T/D, not C/T as is shown in Figure 6-14. This implies that the analysis of this

105 I I I 1.4

1.2 o OkG ac * OkG relax

1.0 0 0 0 I 0.8 F x

- 0.6 U- |-

0.4 I- ieTTTf -

0.2 0-

0 0.4 0.8 1.2 1.6 Temperature [K]

Figure 6-14: Sr 3CuPtO.5 Iro.0 O6 calorimeter AC heat capacity with off-plateau correction, and relaxation heat capacity data.

1.2

1.0 0 * OkG relax * 1kG relax 0o OkG AC 0 1kG AC 0.8 10

x U~ 00

0.6 r 0 o 0

0 -

o 0.4 OIQO U CbEm 0.2 F

0 0.1 0.3 0.5 0.7 0.9 Temperature [K]

Figure 6-15: Sr 3 CuPtO.5 Iro.5 O 6 calorimeter AC (no off-plateau correction) and relaxation heat capacity, 0 kG and 1 kG.

106 10 -- 3kG relax 5kG relax 9 -3kG AC LO v 5kG AC

-2 A

A A 8 - V 6VA A V c 6 A V A

V V V.AA 0 y A

0.05 0.20 0.35 0.50 Temperature [K]

Figure 6-16: Sr 3CuPtO.51ro. 50 6 calorimeter AC (no off-plateau correction) and relaxation heat capacity, 3 kG and 5 kG. data is mainly qualitative, which is sufficient for the purpose of detecting the presence or absence of a feature. It is apparent that if there is a feature in heat capacity, it is small. The region around 1.5 K appears most promising. To examine the excess heat capacity in this region, the 21 November data points on the wings of the region (0.7 K < T < 1.23 K and 1.78 K < T < 2.10 K) were first fit to an eighth order polynomial. This polynomial was then subtracted from the data in the central region, and the result is plotted in Figure 6-18. Error bars are just those from the C data. The result suggests that there is a peak here. This will be discussed further in the Section 7.1 below.

6.5.2 Relaxation Method Results Relaxation heat capacity was measured in zero field for 137 mK < T < 846 mK, and in fields up to 70 kG for 130 mK < T < 450 mK. AT(t) curves fit to a sum of two exponential decays (plus the SR830 setup response) always yielded a single time constant for H < 10 kG. Hence these curves were best fit to a single exponential. Typical scatter in the curves was ±0.03 mK, leading to x 2 near one for the fits. Due to the Sr 3CuPtO.Or o.50 6 , the heat capacity C 2 of the calorimeter dominated that of any second lump C1, so no two-exponential behavior was observed in this regime. To check this against the model, the values for C1 and K1 found at 130 mK in zero field for the empty calorimeter were assumed for the Sr 3 CuPtO.5 IrO. 5 0 6 calorimeter, and C2 was taken to be KbT for the Sr 3CuPtO.5Iro.50 6 calorimeter at 140 mK. The model

107 - - 1.50

1. 25

- -3 -1. 00

x 0. 75

0.50 - o " 11/21 AC data . 0 0 0 12/29 AC data C, 0.25 -00- I 0.6 1.0 1.4 1.8 2.2 2.6 Temperature [K]

Figure 6-17: Sr 3CuPtO.5Iro.0 O6 calorimeter AC heat capacity data above 1 K.

2.0 * 11/21 AC data o 12/29 AC data

1.5 -

1.0 -

IC:) 0.5 -

-0.5

1.2 1.3 1.4 1.5 1.6 1.7 1.8 Temperature [K]

Figure 6-18: Excess Sr 3CuPtO.5 ro.5 06 calorimeter AC heat capacity data above 1 K (see text).

108 Field Temp C1 Cu. Fingers C1 C2 K 1 (kG) (mK) (pJ/K) (pJ/K) (pJ/K) (pW/K) 70 133 25.2 16.74 1.28 1.60 70 201 10.6 8.26 1.27 1.097 70 408 2.6 1.248 3.032 0.424

Table 6.5: Parameters for Schwall model of Sr 3CuPtO.5 Iro.50 6 calorimeter.

Temp Ctb Cbb (mK) (pJ/K) (pJ/K) 144.9 - 11.0 0.1 148.3 10.7 ± 0.2 - 305.9 - 14.9 i 0.2 302.8 15.5 ± 0.3 - 431.8 - 16.8 + 0.3 429.0 17.1 + 0.4 -

Table 6.6: Relaxation heat capacity from various thermometer/heater combinations on the Sr 3CuPtO.5 IrO. 50 6 calorimeter. The notation Cth refers to heat capacity measured with thermometer t, heater h.

then predicted that at least 97 % of the throw should be in a single time constant, consistent with observation. However, above 10 kG, a long, second time constant appeared. As for the empty calorimeter, this second time constant is thought to be due to the nuclear heat capacity of the copper fingers. Values of C1, C2, and K1 for

the Sr 3CuPtO.5 IrO. 5 0 6 calorimeter at 70 kG are shown in Table 6.5. It is reassuring that the K 1 are comparable to those obtained for the empty calorimeter at 70 kG (Table 6.3), and that the C2 are comparable but larger. The error analysis for the power/temperature curves here is identical to that for the empty calorimeter, with the same typical errors in power measurement and x2 of one typical for the fits. The Kb data in zero field are plotted as a function of temperature along with the expected Kb in Figure 6-19. The expected curve was computed based on Suomi's [54] data for commercial copper wire. In Figure 6-20, the field dependence of Kb is shown. As a check, relaxation heat capacity was measured for a few points at zero field using the bottom thermometer and bottom heater. The results were consistent with the usual top thermometer, bottom heater data to within the error bars of the respective measurements, as shown in Table 6.6.

109 5

LOl

C-)

* Measured - Published Data

* 0 0.1 0.3 0.5 0.7 0.9 Temperature [K]

Figure 6-19: Sr 3CuPtO.5Iro.0 6calorimeter zero field thermal conductance to bath Kb, measured and predicted from published data on copper.

I I ' I 10l 2.0 * OkG o 5kG m 10kG . o 30kG U)1. 5 S A5OkG * 05- 70kG x CU

C-)

00. A U -3 080 _0 |

- -

0 S . I I . I 0.1 0.2 0.3 0.4 0.5 Temperature [K]

Figure 6-20: Field dependence of Kb, Sr 3CuPtO.5 1r o.50 6 calorimeter.

110 6.6 Sr 3 CuPtO.5 1ro.5 0 6 Specific Heat Determination

Given the heat capacities of the Sr 3 CuPtO.5 Iro.5 0 6 and empty calorimeters, and the mass of Sr 3CuPtO.51ro. 506 used, the specific heat a of Sr 3CuPtO.5 1r o.5 0 6 was computed as a function of temperature and field. For all temperatures and fields, only the relaxation data was considered for determination of Sr 3 CuPtO.5 Iro. 5 0 6 specific heat. For fields of 10 kG and below, the heat capacity of the Sr 3CuPtO.5 1ro.5 0 6 sample was computed by simple subtraction of the Sr 3CuPtO.5 1ro. 5 0 6 and empty calorimeter heat capacities, and the sum of the errors in the two calorimeter heat capacities taken as the error in the Sr 3 CuPtO.5 1r o.5 0 6 heat capacity. There was a slight (38.5 mg) difference in the amount of N grease in the two calorimeters, but this was smaller than the error bars and so was neglected. Above 10 kG, the low amplitude, long time constant component of AT(t) occasionally produced large errors in the AT(t) fit, rendering a simple subtraction meaningless in some cases. Hence these data were examined by plotting the empty and Sr 3CuPtO.5 1r o.5 0 6 calorimeter data on the same plot, Figure 6-21.

104 * Sample cal. 70kG : 0 Empty cal. 70kG : 5 M Sample cal. 50kG - o Empty cal. 50kG - 2 A Sample cal. 30kG . 0A Empty cal. 30kG S, 10-5 -1 Set Cu fingers,70kG _'1 e .-- 1 Set Cu fingers,50k! 1 Set Cu fingers,30kG

0 2

ci6

107 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Temperature [Kelvin]

Figure 6-21: High field heat capacity of Sr 3CuPtO.51ro. 506 and empty calorimeters. The main conclusion to be drawn from the high field data is that the heat capacities of the Sr 3CuPtO.5 Iro. 50 6 and empty calorimeters are essentially the same at these fields. The Sr 3CuPtO.5 r o.50 6 is slightly higher, which makes sense as a small phonon contribution is still expected from the Sr 3CuPtO.51ro. 50 6 . These data support the hypothesis that the two calorimeters, measured with the same heater/thermometer combination, are essentially the same modulo the magnetic heat capacity contribution of the Sr 3CuPto.51r o.50 6 . This hypothesis is essential in order to obtain

111 Sr 3CuPtO.5 I r o.50 6 heat capacity by simple subtraction of the two calorimeter heat capacities. For given mixing chamber temperatures, small differences were seen between the temperatures of the empty and Sr 3CuPtO.5 Iro.50 6 calorimeters. Below 300 mK, these differences were always less than 3 mK, increasing to 5 mK at 375 mK and to as much as 50 mK at Sr 3CuPtO.5 Ir o.50 6 calorimeter temperature of 850 mK. The Sr 3 CuPtO.5 IrO.5 0 6 calorimeter, in the position closest to the mixing chamber, was always at the higher temperature. Since heat is applied at the mixing chamber for temperature control, and the amount of heat needed increases with mixing chamber temperature, a temperature gradient is expected across the calorimeter stack, with higher temperatures closer to the mixing chamber. When there were differences between the two calorimeter temperatures, linear interpolation was used to determine the proper empty calorimeter heat capacity at a given Sr 3CuPtO.5 IrO.5 O6 calorimeter temperature before subtracting the two results. Errors in measured heat capacities were propagated through the interpolation formula to give the final error in the empty calorimeter heat capacity.

112 Chapter 7

Conclusions and Future Work

7.1 Sr 3 CuPt 0. 5Ir 0.50 6 Specific Heat Results

In this section, I will restrict myself to presentation and analysis of the data, and will only refer to specific interpretations of the data where necessary. Figure 7-1 shows the zero-field u/T of Sr 3 CuPt. 5 1ro. 5 0 6 . u/T is plotted rather than the specific heat a because the former is equal to the temperature derivative of entropy, ds/dT. The zero- field u/T data were integrated using a simple trapezoidal rule integration, and gave 0.12 J/mol/K for the change in entropy over the temperature range shown. One mole of Sr 3CuPtO.5 1r o.5 0 6 has 1.5 spins-1 per formula weight, or 1.5 R In 2 = 8.6J/mol/K entropy content at infinite temperature. Hence the u/T data shown here accounts for only 1.4 % of the total spin entropy.

In Figure 7-3, a for Sr 3CuPtO.5 1r o.5 0 6 is shown directly. The fact that a decreases monotonically while u/T increases with decreasing temperature below 0.4 K indicates the possibility that a ~T, where 0 < 6 < 1, below 0.4 K. The a/T data below 0.4 K were fit to the RQSC expression

a/T = AT5-1 ln T/To1 (7.1)

The fit is shown in Figure 7-1; 6 = 0.50 ± 0.07 is found, with X2 = 4.0. The data above 0.4K seem to have more scatter than that below. It was found experimentally that temperature control was more difficult at higher temperatures, and this is one cause of the increased scatter. However, a power law fit to all of the data (above 0.4K as well as below) led to a/T oc T-', where a = 0.76 ± 0.04, with X2 = 15. In Figure 7-2, I show the fit to u/T = OT 0 76 +B, where # = 0.100±0.005J/mol/K' 2 4 , and B = -0.02 ± 0.01. Figure 7-4 shows the field dependence of a/T to 10 kG. The 10 kG a data were fit to the power law a = ATT', with the result -y = 1.52 t 0.07, A = 0.22 ± 0.02, and X2 = 2.8. Figure 7-5 shows the 10 kG a data as a function of T 3/2 , along with the best fit to a = A T3/2+B. A = 0.22± 0.01 J/mol/K5 / 2, B = -0.0003±0.001 J/mol/K are found, and X2 = 3.0. In Figure 7-6 a(H) is shown at 130 mK. This does not fit to a power law (C oc H-) (0 = 0.6 ± 0.1 was the best fit, with X2 = 64!), but does fit a second-order polynomial fairly well. This is shown in Figure 7-6. With X2 = 7,

113 0.45 * Data -Fit <0.4K data RQSC

0.30

0. 15

00 0 - -

0.1 0.3 0.5 0.7 0.9 Temperature [K]

Figure 7-1: Zero field c-/T for Sr 3CuPtO.51ro. 50 6 , fit below 0.4K.

0.45 I I I

* Data -Power Law Fit

0.30

0 15

0 --

0.1 0.3 0.5 0.7 0.9 Temperature [K]

Figure 7-2: Zero field a/T for Sr 3CuPtO.51r o.5 0 6 , fit to all data.

114 0.09

0 0.06 H E

0- 0.03 - V)I

0 I - 0.1 0.3 0.5 0.7 0.9 Temperature [K]

Figure 7-3: Zero field specific heat for Sr 3CuPtO.5 1r o.5 0 6 .

0.45 I I I I I OkG 1kG 3kG 5kG T, * 7kG 0.30 F 10kG

0 I. b 0. 15 P- I 0 0 0

0 I I I I 0.1 0.3 0.5 0.7 0.9 Temperature [K]

Figure 7-4: Low field -/T for Sr 3CuPtO.5 IrO. 50 6 .

115 I I I I I I I I 0.07 - Relaxation Dato -Fit 0.06

0.05

0.04 0 -E 0.03

0.02

0.01

0 I I I I I I I I 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 T3/2 [K3/ 2]

3 Figure 7-5: a at 10kG for Sr 3CuPtO.51ro. 50 6 , and fit to a = AT /2 + B. the fit was a(H) = 0.05t0.002J/mol/K-(0.0063t0.0006J/mol/K/kG)H+(21.2x10~ 5J/mol/K/kG2)H 2 (7.2)

7.2 Discussion I: Entropy, Field Dependence

I mentioned in the last section that the total spin entropy of Sr 3CuPtO.5 IrO.5 O 6 should be 1.5 R In 2, or 8.6 J/mol/K. 0.12 J/mol/K, or 1.4%, is removed in the temperature region studied. If it is assumed that u/T continues to follow the T6- 1 I n TI fit to the T < 0.4K data down to T = 0, the total amount of entropy removed from the system below 1K would still be only 3.1% of 1.5Rln2. Since my relaxation data extends only to about 1K, and since the AC data taken at higher temperatures is only useful for qualitative analysis, it is difficult to determine the amount of entropy removed above 1K on the basis of my data alone. However, the AC data can be used to estimate an upper limit on the amount of spin entropy removed between 1K and 2K. If I take the AC data shown there to be entirely due to Sr 3CuPtO.5 IrO.5 O 6 , entirely magnetic, and ignore any off-plateau effect, I find 7% of R In 2 there. Also, as noted in Chapter 1, Ramirez has taken heat capacity data on Sr 3CuPtO.5 1r o.5 0 6 and on diamagnetic Sr 3ZnPtO6 at temperatures from 2.5K to 50K. From this data, it should be possible to determine roughly how much of the magnetic entropy of Sr 3CuPtO.5 1ro.5 0 6 is removed on that temperature range. Ramirez's data

116 0.06

0.05 - Relaxation data

0.04

0 E 0.03

0.02

0.01 -

0 0 2.5 5.0 7.5 10.0 12.5 15.0 Field [kG]

Figure 7-6: Specific Heat of Sr 3CuPtO.51ro. 50 6 as a function of field at 130 mK.

were taken on 0.2g Sr 3CuPtO.5IrO. 5 0 6 and 0.2g Sr 3ZnPtO6 samples [32]. Each sample was mixed with the same mass of silver powder and compressed to improve sample thermal conductance. To estimate the magnetic entropy removed between 2.5K and 50K, note first that the lattice contribution dominates Sr 3CuPtO.5 Iro.5 0 6 and Sr 3ZnPtO6 heat capacities over most of the temperature range, and that the heat capacity of Sr 3ZnPtO6 is slightly higher than that of Sr 3CuPtO.51r o.50 6 at the highest temperatures (see Figure 2-9). The former observation indicates that the lattice heat capacity of Sr 3CuPtO.5 1ro.50 6 must be determined and subtracted from the total to determine the magnetic heat capacity. The latter observation indicates that the lattice heat capacity of Sr 3CuPtO.51ro.50 6 is lower than that of Sr 3ZnPtO6 (since Sr 3 ZnPtO6 is a diamagnetic insulator, all of its heat capacity is due to the lattice). Hence a simple subtraction of the integrals of -/T for the two materials, which would assume equal lattice heat capacities, will underestimate the magnetic entropy removed. A better estimate can be obtained using the method of Stout and Catalano [59]. Given that the crystal structures of Sr 3CuPtO.5 IrO.5 0 6 and Sr 3ZnPtO6 are very similar, the law of corresponding states would imply that the two lattice contributions can be written

Czpt = f(T/lOz) (7.3) Clcp = f(T/lOpi) (7.4) that is, the two heat capacities are identical modulo a scaling of the temperature axis. At sufficiently high temperatures, the total heat capacity will be just Cat

117 for both materials. Therefore the appropriate temperature scaling factor can be determined at high temperatures, and the Sr 3ZnPtO6 data multiplied by this factor to determine Ct. If the law of corresponding states is applicable, this scaling factor should not change with temperature (at high temperature). Assuming that 50K is sufficiently high temperature for Sr 3CuPt0 .5 Iro.5O 6 and Sr 3ZnPtO6 , it is found that a scaling factor of 1.07 overestimates the 50K Sr 3CuPt. 5 Iro*O 6 heat capacity, while a scaling factor of 1.1 underestimates it. Applying these scaling factors to the Sr 3ZnPtO6 data and integrating under the resulting c-/T curves, and subtracting the results from the integral under the Sr 3CuPt. 5 Iro.5O 6 curve, I obtain 5 J/mol/K as a lower limit, and 8 J/mol/K as an upper limit for the magnetic entropy removed from Sr 3CuPtO.51ro. 50 6 between 2.5K and 50K. These limits correspond to 58% and 93% of the total magnetic entropy, respectively. It should be pointed out that even these are only rough bounds. One reason for this is that the magnetic entropy is only about 10% of the total Sr 3CuPtO.5Iro.50 6 or Sr 3ZnPtO6 entropies, so the determination of magnetic entropy involves the subtraction of two large numbers. Due to the sparse- ness of the data set and use of the trapezoidal rule, it is not unreasonable to suppose errors in the integrals of at least a few percent. Another reason is that 50K is not a very high temperature compared to J/kB, so the applicability of this method is not clear. Finally, Ramirez made measurements on a different Sr 3 CuPtO.5 1ro.5 0 6 sample than was measured here. With these qualifications on the accuracy of this calculation in mind, I conclude that the calculation is enough evidence to cast doubt on the possibility that a significant amount of the spin entropy of Sr 3CuPtO.5Iro.0 O6 is removed at temperatures where specific heat has not already been measured. However, assume for a moment that the correct value for magnetic entropy removed between 2.5K and 50K is on the low end of the range given, so that a good deal of the spin entropy remains unaccounted for. In this case, and in light of Sigrist's glass model for Sr 3CuPt0 .5Iro.5O 6 below 1.7K (see Section 2.2), it is possible that some of the spin entropy is "frozen in" below 1.7K, and this is the reason that the amount of entropy removed below 1K is so small. If this hypothesis were correct, hysteresis might be observed in the field-cooled versus zero-field-cooled heat capacity. However, some of the heat capacity data points at 5kG were taken after cooling from 400mK in a 7kG field, while others were taken after zero-field cooling. The points from the two data sets showed no systematic differences. Similarly, some of the 10kG points were taken after field-cooling at 1kG, while others were taken after zero-field cooling. Again, both sets of points fall on the same curve. The suppression of -/T by magnetic field confirms that the rise in U/T is associated with the spin degrees of freedom in Sr 3CuPtO.51ro. 50 6 . Also, the fact that the temperature dependence changes from roughly T 1/2 at 0 kG to T 3/2 at 10kG is interesting. The T 3/2 dependence is expected for excitations with dispersion w = Ak 2 , such as occur for spin waves in a three-dimensional ferromagnet. However, in 3 2 Sr 3CuPtO.5 1ro.50 6 the T / dependence occurs only in a 10 kG field. For ferromagnetic spin waves in a field w = poH + Ak 2 [60], leading to gap behavior in specific heat for thermal energies below poH (po = (g/2)pIB). If po corresponds to one electron spin, then gap behavior should be observed below 1K for a 10kG field. If /-o corresponds to a cluster involving many electron spins, then gap behavior should be observed at

118 3 2 even higher temperatures. As it is, the data for Sr 3CuPtO.5 Iro.0 O 6 follow T / below 2 0.5K. If there were even a small gap in Sr 3CuPtO.5 Iro.0 O 6 , the fit to a = AT3/ + B would lead to a non-zero B. As it is, B is zero (to within experimental error), indicating that there is no evidence for the existence of a gap in Sr 3 CuPtO.5 Iro5 O 6 specific heat at 10 kG. This suggests that Sr 3CuPt 5 IrO. 5 0 6 is not a simple three-dimensional ferromagnet at these temperatures and fields. However, it does suggest that some spin-wave-like excitations, propagating in three dimensions, are present. This in turn suggests that interactions between spins on different chains may be important at these temperatures and fields. This would not be too surprising, given the interpretation of Beauchamp's data by Sigrist in terms of a glassy state involving interactions between clusters of spins on different chains (see Section 2.2), and given the possible importance of interchain interactions in isostructural Ca 3Co 2 0 6 and Sr 3CuIrO6 (see Chapter 1).

7.3 Discussion II: Comparison with RQSC Theory

There are two similarities between RQSC theory and the specific heat data. First, there does appear to be some spin entropy still present in Sr 3CuPtO.5 IrO. 50 6 below 1K. Second, RQSC theory predicts 6 = 0.44 ± 0.02 in the universal regime. The data are fairly well described between 0.1K and 0.4K by the functional form predicted by theory, with 6 = 0.50 ± 0.07. However, there are two important differences. First, the QMC results of Frisch- muth and Sigrist predict 25 % of the total entropy lies below temperature T* = 0.03JO, while Furusaki et al. predict that 24 % of the total entropy is missing from their high- temperature expansion results, valid to T 0.1Jo. This discrepancy between my data and theory could be explained if only some fraction of the sample actually exhibits RQSC behavior, say 10%. The fact that Beauchamp observes behavior quite different from what is expected for RQSC supports this hypothesis. Another possible reason for the discrepancy is that there may be anisotropy in the FM bonds, as suggested by the high-field M versus H data on Sr 3CuIrO6 at 5K (see Section 1.2). As a worst-case scenario, the missing entropy can be recalculated (using the results of Section 2.1) with (SFM) = 0- In this case, the missing entropy is 8%, still a factor of four larger than the observed value. Second, theory did not anticipate that universal or nearly-universal behavior could be seen at such a high temperature. Frischmuth and Sigrist [22] suggest that the

scaling regime of Sr 3CuPtO.5 1ro.5 0 6 occurs only at temperatures well below JO/1000, which would be roughly 40 mK here. Westerberg et al. [21] also question whether it would be experimentally possible to observe scaling in specific heat. These conclusions are based on the observation that certain choices for the initial distribution of coupling constants and spins converge to the fixed point at higher temperatures than others.

The distribution that seems to describe Sr 3 CuPtO.5 1r o.5 0 6 , namely delta functions at JO and -JO, converges very slowly. The measured field dependence could give insight into the actual distribution of J in this temperature range, which could be useful in addressing this discrepancy.

119 7.4 Discussion III: Comparison with Beauchamp Results

Perhaps the strongest evidence against the applicability of RQSC theory to the measured Sr 3CuPt*5 Ir0 .5O6 heat capacity data is the AC susceptibility data of Beau- champ. While those data do not seem to be a signature of a transition to a three- dimensional, fully ordered ground state (see Section 2.2), they do indicate some kind of qualitative change in the magnetic behavior of the material. As discussed in Sec- tion 6.5, qualitative AC heat capacity data above 1 K suggest the possible presence of a small peak in heat capacity centered at 1.5 K. The width is typical of a three- dimensional ordering transition, ±10 %T, [2]. The amount of entropy contained in the peak cannot be calculated directly from the data, since it was taken off-plateau. How- ever, over this narrow range one would assume that A is temperature-independent, so the data will equal the actual heat capacity times a scaling factor greater than one. Hence integration of this data will place an upper limit on the entropy associated with the peak. Again using the simple trapezoidal rule, the maximum entropy associated with the peak is 0.01 J/mol/K, or 0.1% of 1.5R In 2. The amount of entropy found in three-dimensional ordering transitions of systems that exhibit one-dimensional behavior at higher temperatures is expected to be a fairly small fraction of the total spin entropy. As an extreme example, only 1% of the total spin entropy is removed in the ordering transition of TMMC [2], with the rest lost to short-range order at higher temperatures. Other materials lose around 10% in the ordering transition. Hence, if the feature in Sr 3CuPtO.5 ro.5 0 6 is real, it seems unlikely that it is associated with a transition to a three-dimensional, fully ordered ground state. Are there any models that would be consistent with the susceptibility and specific heat data? Recall the model proposed by Sigrist (Section 2.2), who postulated that the distribution of bonds in the Sr 3CuPt. 5 Ir 0 .5 O 6 chains is not truly random. This led to large FM "clusters" at low temperatures, which then correlated to form a glassy state at 1.7K, where the peak in X is observed. The absence of a pronounced feature in specific heat around 1.7K also supports this idea of a transition to a glassy state. On the other hand, the formation of such a state is typically evidenced in specific heat as a broad maximum 20% above the "freezing temperature" Tf (Tf = 1.7K here), and a nearly linear dependence of specific heat on T well below Tf [61]. Pursuing this, I found that the low-temperature specific heat (below 0.4K) could be fit to a-= aT+3To2 , with x2 = 4.4, a = 0.060 ± 0.009 J/mol/K 2 , = 0.068 t 0.003 J/mol/K 2 . 2 The To term is consistent with a random exchange Heisenberg AF chain model [62], but I see no clear physical motivation for such a term in Sr 3CuPtO.5 ro.5 0 6 . Moreover, there is no evidence for a broad maximum in the AC data above 1.7K, although a broad feature may be difficult to detect given that some of the change in the AC data with temperature is due to changes in D(w, T). Perhaps a consistent model for the susceptibility and specific heat data is that there are some parts of the sample in which the bonds are distributed randomly, and other parts in which they are not. Hence it could be that part of the sample shows RQSC or RQSC-like behavior, which is detected in the heat capacity, while the rest

120 contains long FM chain sections and forms a spin glass state at 1.7K. Given the M versus H data shown in Figure 2-8, I estimate that the saturation M will not be much higher than 100 emu/mol. Furthermore, if all spins in FM segments are aligned, and if a powder sample for which only } of the spins align with the field is assumed (as is the case for Sr 3 CuIrO6 ), a saturation M of roughly 3000 emu/mol would be expected for a powder sample. Hence it seems that these long FM chain sections make up only a small part (a few percent) of the sample, whereas they would have to make up 90% of the sample if the measured entropy is to be reconciled with the "missing entropy" predicted by theory (see Section 7.3). Therefore it seems that this model is not consistent with the two data sets. Other susceptibility measurements will help to clarify the situation further, as would better heat capacity measurements above 1K. In particular, for a spin glass Tf generally shows a dependence on the frequency at which AC susceptibility is measured [34]. Also, the static (zero frequency) susceptibility should show characteristic hysteresis behavior when the sample is cooled in zero field versus in a finite field.

7.5 Discussion IV: Miscellaneous Interpretations

One other possible explanation for the data is that what is observed on the Sr 3CuPt. 5 Ir0 .5O 6 calorimeter is not due to the Sr 3CuPtO.5 Iro. 5O6sample. This is highly unlikely. The empty and Sr 3 CuPtO.5 Iro. 5 0 6 calorimeters are constructed to be identical with the exception of the presence or absence of Sr 3CuPtO.5 1r o.5 0 6 , and the heat capacities of both calorimeters were measured during the same cooldown. For the relaxation data, empty calorimeter data was taken just before or after Sr 3CuPt. 5 Ir. 5 O6 calorimeter data at a given temperature. The quantitative. agreement between the AC and relaxation results at zero field shows that the Sr 3CuPtO.5 1r o.5 0 6 results can not be explained by some problem with one of the measurement techniques. The calorimeter thermal parameters given by the transfer functions and power/temperature curves suggest that the thermal properties of the calorimeters are suitable for reliable heat capacity measurement. Also, the field dependence of empty and Sr 3CuPtO.5 Iro.5 O 6 calorimeter heat capacities are different, suggesting a spin contribution to the heat capacity that must be associated with the Sr 3CuPtO.5 1r o.5 0 6 sample. Finally, the fact that the two calorimeters have comparable heat capacities at the highest fields indicates that magnetism in the Sr 3CuPtO.5 1ro.0 6 sample is the most likely source of any differences between them at lower fields. Another possible explanation is that the low-temperature rise in ds/dT is due to

Sr 3CuPtO.5 Iro.0 O6 , but that some physics other than RQSC is responsible for it. For example, the rise could be due to a Schottky anomaly. The latter could in principle be caused by some crystal field splitting in the environment of the copper or iridium ions, or by nuclear hyperfine splitting (HFS) in the copper (nuclear spin-!) or iridium (nuclear spin 1). The crystal field produces a term DS2 in the Hamiltonian, and hence produces no splitting for spin-- ions such as copper and iridium in Sr 3 CuPt. 5 Ir 0.5 O6 . The nuclear hyperfine contribution is more worrisome, since the strength of such

121 interactions are typically 10 - 100 mK [63]. However, C(T) oc T- for the Schottky anomaly at temperatures well above the peak [63]. Since this dependence clearly does not describe the zero field data, HFS can not explain the low-temperature rise in ds/dT. To summarize this discussion, my specific heat data show that there is still spin entropy present in Sr 3CuPtO.5 Iro.50 6 below 1K. Furthermore, the zero-field data below 0.4K fits reasonably well to the scaling law predicted by RQSC theory. However, there are many inconsistencies between the behavior of Sr 3CuPtO.ro.5 0 6 and the theory. First, the theory does not predict the scaling law to be observed in specific heat at such a high temperature, given the initial bond distribution for Sr 3CuPtO.5 Iro.50 6 . Second, even if the scaling law continues to T = 0, the amount of entropy accounted for by the data is much less than predicted by the theory. Moreover, given the data of Ramirez at higher temperatures, it is at least conceivable that most or even all of the spin entropy is already accounted for at temperatures above 0.1K. The specific heat data show no transition to a long-range ordered state between 0.1K and 2K at zero field, and at 10kG show behavior characteristic of spin-waves in a three-dimensional FM. The AC susceptibility results of Beauchamp show that there is physics other than RQSC, perhaps some sort of spin-glass transition involving interactions between different chains, present in Sr 3CuPtO.5 1ro*5 0 6 as well. The possible presence of three- dimensional spin waves at 10kG, in combination with the possible glass transition indicated by the AC susceptibility at 1.7K, and the importance of interchain interactions in isostructural Ca 3Co 2 0 6 and possibly Sr 3 CuIrO6 (see Chapter 1), all suggest that a full understanding of the physics of Sr 3CuPtO.5 Iro.50 6 may require consideration of interactions between spins on different chains.

7.6 Prospects for Adiabatic Demagnetization of

Sr 3 CuPt0 5. Ir 0 5 O 6

As mentioned in Chapter 1, interest in Sr 3CuPtO.5 1ro.5 0 6 has been motivated in part by the possibility, suggested by the theory, that the material may be useful for adiabatic demagnetization. The specific heat data shows that the ordering temperature for Sr 3CuPtO. 5 IrO.5 0 6 is below 100 mK, and that there is spin entropy below 1 K that can be suppressed by application of relatively low (10 kG) fields. Hence it is worthwhile to compare Sr 3CuPtO.5 Iro.5 with paramagnetic salts traditionally used for adiabatic demagnetization, to see if Sr 3CuPtO.5Iro.0 O6 might offer some advantages over these other materials. There are several materials properties that are desirable for adiabatic demagnetization [42]. First, it should be possible to remove a significant fraction of the spin entropy with a modest field, and to do so starting at a temperature of 1 K. In this case, a pumped 4He cryostat will provide sufficient precooling, and the required field can be obtained with an electromagnet or even a (movable) permanent magnet. Sec- ond, the material should have an ordering temperature T, below the temperature range of interest. Third, the material should have a large zero-field specific heat at

122 the low end of the temperature range of interest, so that it will warm up slowly after demagnetization. Traditionally, the first criterion is tested with measurements of (OM/OT)H. Spe- cific heat gives no insight into this thermodynamic quantity; however, the fact that C/T rises with decreasing temperature indicates that there is some spin entropy available at 1 K. As for the second criterion, T, is below 100 mK, so the material could be used at least down to this temperature. If the dominant interaction between the chains is dipolar, then as discussed in Section 2.1 an ordering temperature of 1 mK is conceivable. This would be competitive with cerium magnesium nitrate (CMN), a well-studied material with T, ~ 2 mK. Finally, in Figure 7-7 the zero- field specific heats of several common paramagnetic salts is compared with that of

Sr 3 CuPt. 5 Ir0 .5 O6 . Apparently Sr 3CuPt. 5 Iro.5 O6 is inferior to these materials in this respect.

0 0

0 0 0 0 - 0 0 E 0

0 0.1

CL C-)

0.01 - Theory a CPA UM

no. 0.001 0. 01 0.02 0.05 0.1 0.2 0.5 1 Temperature [K]

Figure 7-7: Comparison of various paramagnetic salt specific heats with Sr 3CuPt0 5. Iro.5O6 . The dotted line is an extrapolation of the power law predicted by theory for CPI in the universal regime.

These considerations suggest that Sr 3CuPtO.5 1ro. 5 0 6 does not provide any significant advantage over traditional materials used for adiabatic demagnetization to 1 mK. Of course, these materials have not been in general use for refrigeration for more than two decades, since dilution refrigerators-which offer access to the same temperature range and provide continuous refrigeration-have become commercially available. However, the temperature range below 1 mK is currently only accessible

via nuclear demagnetization of copper or PrNi5 . If Sr 3CuPtO.5 1ro.5 0 6 has T, < 1 mK,

123 and obeys RQSC theory to those temperatures, it could be quite valuable as a re- frigerant. The main advantage it would have over nuclear demagnetization is that the electron spins, rather than the nuclear spins, provide the cooling. The electron spins are thermally well-connected to the phonons, while the nuclear spins are only weakly coupled (through the electron spins) [18]. Hence the time required to cool the Sr 3CuPt. 5 Iro.5O 6 lattice after demagnetization would be dramatically shorter than that required to cool the lattice with nuclear demagnetization. Also, the fields required for nuclear demagnetization are large, typically many tesla. For Sr 3CuPt. 5 Ir0 .5O6 , the behavior of C at 130 mK suggests that much lower fields would be required, 10 kG or less. (This difference in field scale is due to the relative moment of nuclear and electron spins). One severe disadvantage of Sr 3CuPtO.5 1ro. 5 0 6 for nuclear demagnetization, as compared with metals such as copper or PrNi5 , is its low thermal conductivity.

The best way to test whether Sr 3CuPt. 5 Ir. 5 O6 could compete with nuclear demagnetization would be to do the adiabatic demagnetization experiment directly. This will be discussed further in Section 7.7 below.

7.7 Future Work

It is apparent from the data taken in this thesis that the basic design principles for the calorimeter are sound, and that it is possible to measure specific heat on 100 mK < T < 1.0 K and in fields to 70 kG. However, a couple of small changes would improve the calorimeter further. First, the copper shim thermal link fingers and link wire should be removed and replaced with high-purity silver. A single silver wire could be wound on the surface of the plate in the same way that the PtW heater was, and the same wire used for the thermal link to the bath. Second, it would be desirable to further sharpen and narrow the vespel needles, as these may contribute to the two-exponential behavior seen in the empty calorimeter at the lowest temperatures. Also, a peg arrangement that would be symmetric with respect to top and bottom plates would be desirable, as the asymmetry in the current design may be responsible for the asymmetry seen in empty calorimeter heat capacity measured with different heater/thermometer combinations. Since the AC method is more precise than the relaxation method, and since data acquisition and analysis is easier, it would also be desirable for AC calorimetry to be the primary measurement method, with relaxation calorimetry used only as a check on the AC results. At present, the reverse is true. I have shown that it is possible to obtain accurate heat capacity data with the AC method, and future measurements should be approached with the goal of establishing the AC method as primary. This would have the additional advantage that, to my knowledge, there are no specific heat experiments in operation today that are designed to measure below 1K and in fields to 7T using the AC method.

As for Sr 3 CuPt0 .5 Ir 0 .5 O6 in particular, it would be desirable to do more careful, field-dependent, measurements of heat capacity on 1.0K < T < 4.0 K. It is not clear whether this apparatus is appropriate for such measurements-the addenda heat

124 capacity is larger than that of 105mg Sr 3CuPtO.5 1r o.5 0 6 at those high temperatures (see Figure 4-4). More AC measurements could be undertaken, in hopes of verifying or refuting the presence of the small feature around 1.5 K that can not be ruled out by the data taken thus far. Also, neutron diffraction below 2K in fields to 10kG could give valuable insights into the nature of the spin ordering in the material. From the perspective of RQSC theory, another important study would be the p dependence of Sr 3CuPt 1 _JIrpO6 specific heat. This would be useful for comparison with Beauchamp's data, and to see if the amount of missing entropy accounted for below 1 K changes with p. Also, it is important to resolve the issue of anisotropy in Sr 3CuPtjp.IrO 6 , perhaps through a study of M versus H to high (20T) fields at low (5K) and high (100K) temperatures. From the perspective of adiabatic demagnetization, the data presented in this thesis provide some motivation to proceed with a refrigeration experiment involving

Sr 3CuPt*5 Ir 0 .5 O6 . Of course, it would be prudent to complete studies of p-dependent specific heat and of anisotropy before proceeding with refrigeration. Initially, such an experiment would be performed in a dilution refrigerator with precooling to 100 mK, and would require a magnet capable of producing at least 10 kG fields. A large quantity of Sr 3 CuPtO.5 Iro. 506 would be necessary, at least several grams. Chip resistors would no longer be adequate for thermometry-magnetic thermometry based on CMN would be a relatively straightforward alternative for initial experiments. Also, a superconducting heat switch would have to be developed. If cooling to 1 mK or below were possible, careful consideration would have to be taken of heating due to electrical noise, RF, and mechanical vibration if it were desired to actually observe temperatures of 1 mK and below. Rough experiments on the facility here at MIT may be able to confirm T, < 30 mK. Given the amount of development necessary for even an initial experiment, and the limitations of the available apparatus, it may be advisable to make use of an existing demagnetization refrigeration facility for such experiments.

125 126 Appendix A

Calorimeter Conductance from Power/Temperature Curves

I derive equations for the thermal conductance K, of the sample dough given the slopes m of power/temperature curves measured with various heater/thermometer combinations. There are two cases relevant to the data taken in this experiment: first, for two curves measured with the same thermometer but different heaters; second, for two curves measured with the same heater but different thermometers. The thermal circuit for the first case is shown in Figure A-1. Power is applied to either the top or bottom plate, and temperature always measured at the bottom thermometer. The thermal circuit is simply analyzed by repeated application of energy conservation and of Ohm's Law for a thermal circuit. The latter is

Q= KAT (A.1) = AT/R (A.2) where Q is the power through the thermal resistance, AT the temperature change across the resistance, R the thermal resistance, and K its inverse, the thermal conductance. For power QLp applied to the top heater, TO TO Qto = + " (A.3) Rb Rb+ R( and Tto - Tot = RR (A.4) sRb + Rs

(A.3) and (A.4) are solved for top in terms of Tbot and the thermal resistances. Then the slope mbt of the power temperature curve is given by mbt = dQLp/dTot. In this case,

mt = Rb Rb+RS (A.5) R,+Rb Here, I assume that the two thermal links have the same thermal resistance.

127 Rs

Tbot TOP

Rb Rb

Figure A-1: Thermal circuit for case 1. Temperature Tbot is measured with the bottom calorimeter thermometer, referenced to the bath temperature. Power Q0t , is applied to top heater or Qbot is applied to bottom heater.

128 Similarly, for power bot applied to the bottom thermometer, Ohm's Law yields Tb Tb Qbot =T + T (A.6) Rb Rb + R, Hence mTbb + (A.7) Rb R, + Rb

Combining (A.5) and (A.7) and solving for R8,

RS 1 t - mbb (A.8) mbt Mbb

This result was used to determine R, from mbt and mbb measurements on the empty calorimeter. Note that one can also solve these equations for Rb. The result is

Rb = 1 (1 + mbt) (A.9) mbt mbb

Plugging in measured values for the empty calorimeter shows that Rb is larger than 2/mbb or 2/mbt by 5% at 127 mK and by 1% at 400 mK. Given that the approximation that the two thermal links have the same conductance may not be good to better than 5%, I did not correct the empty calorimeter relaxation heat capacity data for this effect. Now I turn to the second case, two power/temperature curves measured with the same heater but different thermometers. The situation is shown in Figure A- 2. Power is always applied to the bottom heater, and power/temperature curves measured with either the bottom or top thermometer. For the bottom thermometer, Ohm's Law yields Q = KT + K,(Tb - T) (A.10) But also 1 K(Tb - Tt)= 1 1 Tb (A.11) KS+

Combining (A.10) and (A.11) and again noting mTbb = do/dT 1 mnb=K+ K (A.12)

Similar analysis applies for power/temperature curve measurement with the top thermometer. Here, (A.11) is used along with

Ks(Tb - T) = KT (A.13)

Tt = Ks Tb (A.14) Ks + K

129 Ks

Tbot TOP

K K

Figure A-2: Thermal circuit for case 2. Temperatures is measured with the bottom (Tbot) or top (Top) calorimeter thermometers, referenced to the bath temperature. Power Q is applied with the bottom heater.

130 Hence mb = K + K(K + K) (A.15) Ks Combining (A.12) and (A.15) yields

Ks = mntb [( mtb - mbb ) 2 + 2 mtb - mbb_ (A.16) mbb mbb

K= mbb (A.17) 1+ Mb Mtb These results were used to determine Ks from mtb and mbb measurements on the

Sr 3 CuPtO.5 IrO.5 0 6 calorimeter. The K equation indicates that at 145 mK 2K is 2% lower than 2 mtb, and at 306 mK 2K is less than 1% lower than 2 mtb. Again, no correction was applied to the K data for this effect due to the fact that the two thermal link wire thermal conductances may in fact differ by a few percent.

131 132 Appendix B

Exponential Fits with Instrumental Response

In this portion of the Appendix, I compute the expected output signal AT(t) from the SR830 setup used for relaxation heat capacity measurements in the

Sr 3 CuPtO.5 1r o .5 0 6 experiment. For the one-exponential case, the input signal AT (t) is given by

T < 0 ZAT (T) = A7T(O) (B.1) AT (0)e- /71 T > 0

The normalized step response of the SR830 setup is

TO0

The impulse response h(r) is easily obtained from s(T) via h(T) = is(T) [64]:

h(t) = (B.3) T>r

Then AT(t) is obtained via the convolution

AT(t) = JM ATi(T)h(t - T)dT (B.4)

Here, the integrand is non-zero only for T < t. Considering only the case t > 0, I have Jt AT(0) e-T/T AT(t) = AT (0) e_(tT) dT + e(t_)/r dT (B.5) -- 00 r 0 Tr The final result is

- , Tr AT(t) T7(0) t/7] I (B.6) 1 -- L

133 A similar calculation holds for the two-exponential case, where

T/ T < 0 A T) { AT() (B.7) Ae-7*1^+ Be--Tl'rB T>0

Here,

AT(t) A etlA+ B etT-B _ [A TITA +BTr IB -/r (B.8) TA TB TA TB Equations (B.6) and (B.8) were used to fit all A T(t) measured with the SR830 setup.

134 Appendix C

Schwall Model

The calorimeter model used by Schwall et al. is shown in Figure 3-1. The two lumps

are assumed to have different temperatures T and T2 in general, and the bath is taken to be at zero temperature. The heat equations for this one-dimensional system are

C1iT = -Ks(T 1 - T2 ) (C.1)

C 2t 2 = Ks(T 1 - T2)-KT 2 (C.2)

The initial condition, appropriate for the relaxation method, is T1 (0) = T2 (0) = Ti. This system of equations can be solved for T (t) and T2 (t) using standard methods of linear algebra. In the Schwall model, T2 (t) is recorded during a relaxation experiment. Hence in the following all results are based on the T2 (t) solution.

Recall that T2 (t) satisfies

T2 (t) = Ae-'I'A + Be-t'TB (C-3)

The fit parameters A, B, TA, and TB are related to the thermal parameters of the model by the equations

-2 K + K, A = i 1 + C1 (C.4) 2 4 2 (C+Z + -K _F K C1 K 1 K (1+ + )2 B =Ti-A (C.5) 202 K/Ks (C.6) TA ~~ K 4I

2C K/KK 2 (C.7) TB K

+ 0 C- 1+ +E) 1 K 1C2KL2

135 Ks

C1 C2

K K

Figure C-1: Thermal circuit for two-link Schwall model. All temperatures are referenced to the bath temperature, and T2 (t) is measured.

Since it was non-trivial to obtain Equations (8) in Schwall's paper from the solution for T2 (t), and since these equations led directly to the final result 3.4, I mention here the tricks I used to obtain them. Equations (8) were

C = KbT1(1- KbT2/C 2 ) - C2±+ KT 2 (C.8)

T2= BT (C.9) [TKb(A + B)2 - AC2]

The first equation can be obtained by examining 1 and - -+ ) and eliminating TB K, between them. Examination of A/B and eliminating'rA theTBTA square root terms in favor of 1/TA and 1/TB leads to the second equation. C1, C2, and Kf were extracted from the equations for the fit parameters by first plugging the fit parameters into three equations. The first was the equation give above for A. The second was C1 + C2 = Cmeas, where Cmeas was the total heat capacity measured using the Schwall model. The third was

0102 Kf = (C.10) K TATB which is found by combining the equations for TA and TB above. Mathematica was used to extract C1,C2, and Kf, which were given implicitly by these three equations.

136 The Schwall model with two separate thermal links of equal conductance can be solved similarly. The model is shown in Figure C-1. The heat equations here are

Citi = -Ks(T 1 - T 2) - KT1 (C.11)

C 2t 2 Ks(T 1 - T2) - KT2 (C.12)

Here, the initial condition is T1 (O) = Tio, T2 (0) = T20 . Note that T10 # T20, although they become equal in the limit K/K, -+ 0. However, the solution for the fit parameters is (again considering only T 2 (t))

A = 20 _+C1 K. C1 (1 1() 2 (1++ C)11 ') _K A, _ 1 I)2)]

B = T 2 0 - A (C.14)

K _C 2

K (1+ 2)(1 + ) 1)- 1±2) _ TA 2C, KKC,C (C.15)

KB (i+2)(i+) [1+11 _ 2 _F]

11 Equations similar to Equations (8) of Schwall's paper can be obtained for this model, using the same algebraic tricks mentioned above. The results are

2 C (2K)(TA - TB) K,--TATBK C1, (C.1)

BC2T2 TB = KTA A+ B) - AC 2 These can be combined to yield

A TA±BTB ATB+BTA K 2 C 1 + C 2 = 2K K K T+AB (C.19) A+B A+B C1 Since the standard Schwall model describes the relaxation data better than this two-wire model (see Section 6.4.2), this equation was not used to analyze data. It is presented here for completeness. Finally, Mathematica was again used to determine C1, C2, and Kf given the fit

parameters. First C1 + C2 was determined using (C.19). Then this result for C1 + C 2 , along with the equation for A and the result (again found by combining the TA and TB results)

K 1 CC 2 - K (C.20) 2 KACB were used to obtain C1, C2, and Kf.

137 138 Appendix D

Model for AC Transfer Function in Presence of T2 Effect

In this appendix, I present a model consistent with that suggested in the text (Sec- tion 6.4) for calorimeters exhibiting T2 effect. Then I compute the expected transfer function for such a model. The model from the text was that the two quartz plates and sample dough formed effectively a single thermal lump, and that there was a weak connection to some other part of the calorimeter (copper fingers or vespel pegs) not between the calorimeter heater and thermometer. The two thermal link wires were treated effectively as a single wire of twice the conductivity connected to the calorimeter. This model led to two-exponential AT(t) behavior according to the Schwall model. The thermal circuit for this model is shown in Figure D-1. This differs from the Schwall model in that the two lumps are treated as continuous slabs, with temperature varying over the thickness of the slab. The transfer function was computed for this geometry, and for the same geometry with heater and thermometer reversed. This reversal led to a transfer function that did not improve the agreement between the measured function and the standard two-wire model. On the other hand, the function calculated from the exact geometry of Figure D-1 did improve the agreement; hence, this geometry is more representative of the actual experimental situation. The first matrix equation describing this situation applies to heat flow through the slab representing the calorimeter (two quartz plates plus dough):

Tm A B To q.) C D q a TO ) where

2 K(D.1)

A = cosh 0 (D.2) sinh 0 B =- (D.3) a

139 TM C, K

Figure D-1: Thermal circuit for calculation of Schwall model transfer function. The heater applies flux j at position 0. The thermometer is located at position m.

140 KO C==- sinh 0 (D.4) a D =cosh 0 (D.5) a is the cross-sectional area of the slab, and the other parameters are as shown in Figure D-1. The second equation, representing the weakly connected part of the calorimeter (probably vespel pegs or copper fingers) is

F )(Tm ( H q;n' where wCw (D.6) V 2K, E = cosh 6 (D.7) ,9 F = sinh (D.8) a = G - K sinhO0 (D.9) a H = cosh Ow (D.10)

and the cross-sectional area a is assumed to be the same as for the calorimeter lump. Solving the system for Tm, I find

B -AD G A-B -B Tm = - A Ib+ H C D a (D.11) Ca The measured transfer function was then fit to wTm. Note that as lump one disap-

pears, i.e. Cw + 0, Kw -+ 1, Tm reduces to Sullivan and Seidel's exact solution for their geometry [38].

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