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Low temperature specific heat of LixNbS2 intercalation compounds Dahn, Douglas Charles 1985

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LOW TEMPERATURE SPECIFIC HEAT OF Li NbS, x INTERCALATION COMPOUNDS by DOUGLAS C. DAHN B.Sc, Dalhousie University, 1977 M.Sc, Dalhousie University, 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1985 © Douglas C. Dahn, 1985 In presenting t h i s thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the The University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It i s understood that copying or publication of th i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permi ssio n . Department of Physics The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: July 1985 A b s t r a c t T h i s t h e s i s d e s c r i b e s a study of the low temperature s p e c i f i c heat of L i NbS 2, where x i s between 0 and 1. Samples were prepared by i n t e r c a l a t i n g l i t h i u m i n t o niobium d i s u l f i d e in e l e c t r o c h e m i c a l c e l l s . S t r u c t u r a l data obtained by x-ray d i f f r a c t i o n are presented. These, together with e l e c t r o c h e m i c a l measurements, show that staged phases e x i s t f o r some values of x. The e l e c t r o n i c s p e c i f i c heat of L i NbS 2 i s c o n s i s t e n t with complete charge t r a n s f e r from the i n t e r c a l a t e d l i t h i u m to the bands of the NbS 2 host. The l a t t i c e s p e c i f i c heat a l s o shows l a r g e changes as a f u n c t i o n of x. A d i s c u s s i o n of the data i n terms of continuum e l a s t i c i t y theory suggests that i n t e r c a l a t i o n produces l a r g e changes i n the shear e l a s t i c constant c 4 „ . A b r i e f d i s c u s s i o n of s u p e r c o n d u c t i v i t y i n L i NbS 2 i s a l s o i n c l u d e d . i i Table of Contents Abstract i i List of Tables vi List of Figures v i i Acknowledgements x 1. INTRODUCTION 1 1.1 Intercalation 1 1.2 Niobium Disulfide 5 1.3 Band Structure and the Rigid Band Charge Transfer Model 6. 1.4 Contributions of this Thesis 10 2. PREPARATION AND STRUCTURE OF INTERCALATED NIOBIUM DISULFIDE 13 2.1 Preparation of NbS2 13 2.2 Preparation of Li NbS2 19 2.3 The Structure of Li NbS2 24 X 3. THE LOW TEMPERATURE SPECIFIC HEAT EXPERIMENT 39 3.1 Introduction 39 3.2 Techniques for Low Temperature Specific Heat Measurements 39 3.2.1 Adiabatic Calorimeters 39 3.2.2 The AC Temperature Method 42 3.2.3 The Relaxation Time Method 43 3.2.4 Discussion 46 3.3 The Specific Heat Cryostat 48 3.3.1 General Features 48 3.3.2 Measurement and Control of the Reference Block Temperature 55 3.3.3 Sample Thermometry 57 3.3.4 Computer Data Aquisition System 63 3.4 Measurement Procedure and Data Analysis 66 3.4.1 Measurement Procedure 66 3.4.2 Computation of the Specific Heat 70 3.4.3 Accuracy of the Specific Heat 75 3.4.4 Determination of the Linear and Cubic Terms in the Specific Heat 82 4. RESULTS OF THE SPECIFIC HEAT MEASUREMENTS 86 4.1 Introduction 86 4.2 Sample Platform Heat Capacity 86 4.3 The Specific Heat of NbS2 89 4.4 The Specific Heat of Li NbS2 94 X 5. THE ELECTRONIC SPECIFIC HEAT 116 5.1 Introduction 116 5.2 The Electronic Specific Heat of a Metal 117 5.3 The Electronic Specific Heat in the Rigid Band Charge Transfer Model 118 5 . 4 Discussion 124 6. THE LATTICE SPECIFIC HEAT 128 6.1 Introduction 128 6.2 Mobility of Intercalated Lithium 129 6.3 Phonon Specific Heat 135 6.4 The Phonon Specific Heat of Li NbS2 in the Elastic Continuum (T3) Limit 140 6.5 Beyond the Elastic Continuum Limit 152 6.6 Simple Microscopic Models 159 6.6.1 Vibrational Motion of a Single Intercalated Lithium Ion 160 6.6.2 One Dimensional Models 163 iv 6.7 Discussion 170 7. SUPERCONDUCTIVITY 174 7.1 Introduction 174 7.2 Meissner Effect Measurement 176 7.3 Comparison With Previous Work 179 7. 4 Discussion 181 8. CONCLUSION 188 8.1 Summary of this Thesis 188 8.2 Suggestions for Future Work 190 BIBLIOGRAPHY 194 APPENDIX 1: INTENSITIES OF X-RAY BRAGG PEAKS FOR STAGE TWO Li NbS2 200 x z APPENDIX 2: SOLUTION OF THE HEAT FLOW EQUATIONS 210 v L i s t of Tables Table Page I. P r o p e r t i e s of 2H-NbS2 16 I I . S p e c i f i c heat r e s u l t s f o r L i x N b S 2 samples......... 113 I I I . Hopping time as a f u n c t i o n of temperature 131 IV. Decoupling temperatures f o r var i o u s c o o l i n g rates.135 V. E l a s t i c c o n s t a n t s f o r some lay e r e d t r a n s i t i o n metal d i c h a l c o g e n i d e s with the 2H s t r u c t u r e 143 VI. j3 and T 1 Q % f o r the L i x N b S 2 samples 158 VI I . Superconducting t r a n s i t i o n temperatures of the L i x N b S 2 samples 175 A1-I. C a l c u l a t e d i n t e n s i t i e s of Bragg peaks f o r stage 2 L i NbS 2 207 A1-II. I n t e g r a t e d (00/) i n t e n s i t i e s for stage 2 L i NbS, 209 v i L i s t of F i g u r e s F i g u r e Pag 1. Schematic diagram of an i n t e r c a l a t i o n c e l l 2. The s t r u c t u r e of 2H-NbS2  3. D e f i n i t i o n of the l e t t e r n o t a t i o n f o r close-packed atomic planes 4. Schematic band s t r u c t u r e of 2H-NbS2 5. A t y p i c a l flange c e l l 2 6. The (008) region d u r i n g the discharge of a L i / L i x N b S 2 x-ray c e l l to 2.670V 7. Hexagonal l a t t i c e parameters a and c f o r L i x N b S 2  8. C e l l v o l t a g e as a f u n c t i o n of x f o r L i NbS 2 9. -dx/dV data f o r L i / L i NbS 2 c e l l DD45, showing the s t a g i n g t r a n s i t i o n s 10. -dx/dV data showing the aging e f f e c t 11. Schematic diagram of an apparatus for r e l a x a t i o n time heat c a p a c i t y measurements 12. S i m p l i f i e d diagram of the low temperature s p e c i f i c heat c r y o s t a t 13. D e t a i l of the vacuum can i n t e r i o r 14. D e t a i l of the top s i d e of the sample p l a t f o r m 15. C i r c u i t diagram of the AC bridge used to measure R s 16. Sample thermometer r e s i s t a n c e as a f u n c t i o n of temperature 17. The s p e c i f i c heat measurement c y c l e 18. Data f o r a t y p i c a l thermal decay 19. S p e c i f i c heat of L i 3gNbS 2 as a f u n c t i o n of temperature '. 20. Sample p l a t f o r m heat c a p a c i t y 21. S p e c i f i c heat of NbS 2  v i i 22. Specific heat of Li l3NbS2 95 23. Specific heat of Li 16NbS2 97 24. Specific heat of Li 2 3 N b S 2 9 8 25. Specific heat of Li 25 N b S 2 1 0 0 26. Specific heat of Li 30NbS2 101 27. Specific heat of Li 32NbS2 103 28. Specific heat of Li 35NbS2 104 29. Specific heat of Li 41NbS2 106 30. Specific heat of Li 5 0 N b s 2 1 0 7 31. Specific heat of Li 68NbS2 108 32. Specific heat o f • L i1 00N b S 2 1 0 9 33. Thermal decay of the L i1 0 o N b S 2 s a mPl e a t 2.73K 110 34. The linear specific heat coefficient 7 as a function of x for the Li NbS2 samples 114 X 35. The cubic specific heat coefficient (3 as a function of x for the Li NbS2 samples 115 X 36. Tight-binding f i t (Doran et a l . 1978) to. the band structure of NbS2 120 37. Brillouin zone and Fermi surface of NbS2 121 38. The dz2 band density of states for NbS2 122 39. The density of states at the Fermi level of Li NbS2..125 40. The electronic specific heat coefficient 7 as a function of x in Li NbS2 126 X 41. Staging phase diagram for a typical intercalation compound 132 42. 'Average' sound velocity v for the LixNbS2 samples 141 43. Debye temperatures for the LixNbS2 samples 142 44. Polar plot of the inverse sound velocities calculated for NbS2 145 45. Acoustic mode 1 at 8=n/2 147 v i i i 46. P as a function of c,, 150 47. Phonon dispersion curves from inelastic neutron scattering data on NbSe2 154 48. Definition of terms used in the one-dimensional 'spring and plate' model 164 49. Model dispersion relations for the longitudinal mode propagating along the c-axis in Li NbS2 169 50. Meissner effect data on a Li 25NbS2 sample 178 51. T as a function of x for a series of Li- NbS2 samples prepared at high temperatures (McEwan 1983)..180 52. T as a function of x, using the rigid band cHarge transfer model and the BCS equation for T . ...183 A1-1. Unit cells of NbS2 and stage 2 Li NbS2 202 A1-2. The Bragg-Brentano focusing geometry 204 A2-1. Model system for heat flow calculations 211 ix Acknowledgements It i s a pleasure to thank Jim Carolan and Rudi Haering f o r t h e i r guidance and encouragement throughout t h i s p r o j e c t . I have a l s o b e n e f i t e d from working with numerous other people both in the low temperature p h y s i c s l a b and i n Rudi's i n t e r c a l a t i o n r e s e a r c h group. My co-workers have been a constant source of support, and I wish t o thank a l l of them. Several people have made important d i r e c t c o n t r i b u t i o n s to the work d e s c r i b e d i n t h i s t h e s i s , and should be mentioned by name. Joos Perenboom and Barbara F r i s k e n helped with the design of the c r y o s t a t . Beat Meyer's expert machine work and h e l p f u l suggestions are a p p r e c i a t e d . Summer students Ravi Menon and J e f f B e i s a s s i s t e d with the s p e c i f i c heat experiment on many o c c a s i o n s . Steven S t e e l and Walter Hardy c a l i b r a t e d one of the thermometers used i n the c r y o s t a t . Walter and h i s students a l s o were always an in v a l u a b l e source of advice: on low temperature techniques. Ed S t e r n i n made the NMR measurements mentioned on page 13. I a l s o wish to thank B i r g e r Bergersen, J e f f Dahn, and John B e r l i n s k y f o r t h e i r i n t e r e s t and f o r u s e f u l d i s c u s s i o n s . F i n a n c i a l support from the F a c u l t y of Graduate Stud i e s i n the form of a U n i v e r s i t y Graduate F e l l o w s h i p i s g r a t e f u l l y acknowledged. F i n a l l y , I would l i k e to thank my wife J u l i a and our c h i l d r e n f o r t h e i r support and encouragement. x 1. INTRODUCTION 1.1 INTERCALATION Intercalation, as the term is used here, is the reversible insertion of 'guest' atoms or molecules into a 'host' solid, in such a way that the crystal structure of the host is not drastically altered. A necessary condition for intercalation is the existence of sites in the host which are available for occupation by the guest. These sites must be accessible from the surface of the host, and the guest, or intercalant, must be mobile in the host. Perhaps the best known intercalation host is graphite, in which a wide variety of atoms and molecules can be inserted between the carbon layers. Graphite intercalation compounds have been comprehensively reviewed by Dresselhaus and Dresselhaus (1981). Other layered materials which have been shown to be intercalation hosts include some of the layered s i l i c a t e s , and the layered transition metal dichalcogenides (LTMDs) such as NbS2. For reviews of intercalation in the LTMDs see Whittingham and Jacobson (1982), Whittingham (1978), Levy (1979), or Marseglia (1983). Intercalation has also been observed in some non-layered hosts; for example lithium intercalates into V205 (Murphy et a l . 1979) and some of the other transition metal oxides, and into Mo6SB (Schollhorn and Kumpers 1977, Mulhern 1982). Electrochemical cells based on intercalation (Whittingham 1976) have received a great deal of attention 1 2 in recent years, s i n c e they can form the b a s i s of long l i f e , h igh energy d e n s i t y rechargable b a t t e r i e s . The most promising systems from a p r a c t i c a l p o i n t of view are those i n v o l v i n g l i t h i u m i n t e r c a l a t i o n i n t o the LTMDs. C e l l s based on l i t h i u m i n t e r c a l a t i o n i n t o T i S 2 have been s t u d i e d e x t e n s i v e l y (J.R.Dahn 1982, Whittingham 1979) and Li/MoS 2 c e l l s (Py and Haering 1983) are now being produced c o m m e r c i a l l y 1 . T h i s t h e s i s i s concerned with the i n t e r c a l a t i o n compound L i NbS 2 r which can a l s o be prepared and s t u d i e d by means of e l e c t r o c h e m i c a l i n t e r c a l a t i o n . A schematic diagram of a L i / L i NbS 2 c e l l i s shown i n f i g u r e 1, to i l l u s t r a t e the o p e r a t i o n of an i n t e r c a l a t i o n c e l l . The h a l f c e l l r e a c t i o n s are xL i x L i + + xe" (1-1) • at the l i t h i u m metal anode and x L i + + xe" + NbS 2 *-^ > L i NbS 2 (1-2) X at the L i NbS 2 cathode. The e l e c t r o n s move from the anode to x * the cathode through an e x t e r n a l c i r c u i t , and the L i + ion moves through the e l e c t r o l y t e . In a d d i t i o n to t h e i r p r a c t i c a l a p p l i c a t i o n s , i n t e r c a l a t i o n c e l l s can be used to obt a i n a great deal of thermodynamic i n f o r m a t i o n about the i n t e r c a l a t i o n compound. T h i s i s because the open c i r c u i t v o l t a g e of a l i t h i u m i n t e r c a l a t i o n c e l l i s given by (McKinnon and Haering 1983) 'Moli Energy L t d . , Burnaby, B.C. 3 • V v V Li LixNbS2 Electrolyte F i g u r e 1: Schematic diagram of an i n t e r c a l a t i o n c e l l . The e l e c t r o l y t e c o n s i s t s of a l i t h i u m s a l t d i s s o l v e d i n an o r g a n i c s o l v e n t . 4 V(x)=(u -u )/e (1-3) a C where M= and n are the chemical potentials of lithium atoms in the anode and cathode, respectively, and e is the electronic charge. A complete review of the literature of intercalation cells would be beyond the scope of this thesis, so only a few examples will be given. High resolution electrochemical measurements of V(x) and 9x/9V (J.R.Dahn and McKinnon 1984a) can be compared with theoretical models of the intercalation compound (Dahn, Dahn, and Haering 1982, McKinnon and Haering 1983). These and related electrochemical techniques have been used to study a number of phenomena which occur in layered intercalation compounds. One of these is staging. An intercalation compound of stage n is one in which every n interlayer gap contains a higher concentration of guest than the intervening n-1 gaps. Staging is best known in the graphite intercalation compounds, and also occurs in some of the intercalated LTMDs, such as AgxTaS2 (Scholtz and Frindt 1980), Li NbSe2 (D.C.Dahn and Haering 1982) and Li NbS2. X X LixTaS2 (J.R.Dahn and McKinnon 1984b) exhibits staging as well as two-dimensional lithium ordering in the interlayer gaps. In addition to V(x) measurements, the temperature coefficient 3V/9T of c e l l voltage has been used to obtain the entropy of intercalation compounds (J.R.Dahn and Haering 1983) . 5 Intercalation c e l l s can also be used as a convenient sample preparation technique, and i t i s primarily in thi s role that they appear in thi s t h e s i s . By preparing a Li/NbS2 c e l l and allowing i t to discharge, the cathode material i s converted to L i NbS2, where x can be accurately determined A. by time-integrating the c e l l current. As shown in Chapter 2 for LixNbS2, materials prepared t h i s way may be in a different c r y s t a l phase than materials of the same composition prepared by other methods such as direct high temperature reaction of the elements. 1.2 NIOBIUM DISULFIDE NbS2 is one of the layered t r a n s i t i o n metal dichalcogenides. These are compounds of the form MX2, where M i s a group IV, V, or VI t r a n s i t i o n metal and X i s s u l f u r , selenium, or tellurium. The LTMDs consist of strong covalently bonded MX2 layers separated by so-called Van der Waals gaps. The interlayer bonding i s r e l a t i v e l y weak, although i t i s no longer believed to be due e n t i r e l y to the Van der Waals interaction (Umrigar et a l . 1983, Hibma 1982). A number of review a r t i c l e s dealing with the LTMDs (Wilson and Yoffe 1969, Yoffe 1973, Hullinger 1976, Lieth and T e r h i l l 1977, Vandenberg-Voorhoeve 1976), their charge density waves (Wilson et a l . 1975, Williams 1976), and superconductivity (Frindt and Huntley 1976), are a v a i l a b l e , and I w i l l therefore r e s t r i c t the following discussion to NbS2 as much as possible. 6 NbS2 i s found in two p o l y t y p e s ; one ( r e f e r r e d to as 2H) with a two l a y e r high hexagonal u n i t c e l l and one (3R) with a three l a y e r rhombohedral s t r u c t u r e . The NbS2 used i n t h i s work was 2H. These s t r u c t u r e s are shown in f i g u r e 2. I f we c o n s i d e r the s t r u c t u r e s as s t a c k s of two dimensional c l o s e packed p l a n e s , the s t a c k i n g sequences are BaB-CaC f o r 2H-NbS2 and BaB-CbOAcA f o r 3R-NbS2. The l e t t e r s r e f e r to the three i n e q u i v a l e n t p o s i t i o n s marked in f i g u r e s 2 and 3. C a p i t a l l e t t e r s r e f e r to S and small l e t t e r s to Nb. By analogy with L i T i S2 (J.R.Dahn et a l . 1980), i n t e r c a l a t e d l i t h i u m atoms are b e l i e v e d to l i e i n the o c t a h e d r a l l y c o o r d i n a t e d s i t e s i n the i n t e r l a y e r gaps. .1 .3 BAND STRUCTURE AND THE RIGID BAND CHARGE TRANSFER MODEL The e l e c t r o n i c energy band schemes f o r NbS2 and the other LTMDs f i r s t proposed by Wilson and Y o f f e (1969) have s i n c e been confirmed by a number of experiments, as w e l l as by d e t a i l e d band s t r u c t u r e c a l c u l a t i o n s (Mattheis 1973, Wexler and Wooley 1976). The q u a l i t a t i v e f e a t u r e s of the 2H-NbS2 bands are shown in f i g u r e 4. The valence bands are predominantly of s u l f u r 3s and 3p c h a r a c t e r , and the p a r t i a l l y f i l l e d conduction band i s d e r i v e d p r i m a r i l y from niobium 4d s t a t e s . The 3R p o l y t y p e has an almost i d e n t i c a l d e n s i t y of s t a t e s , at l e a s t f o r the d s t a t e s , s i n c e the c o o r d i n a t i o n of S around Nb i s t r i g o n a l p r i s m a t i c i n both p o l y t y p e s , and t h i s i s the most important f a c t o r i n determining the d s t a t e s p l i t t i n g . NbS2 i s m e t a l l i c , s i n c e Figure 2 : The structure of 2H-NbS2• Black c i r c l e s indicate niobium atoms and open c i c l e s are s u l f u r . Intercalated lithium i s believed to l i e i n the octahedral s i t e halfway between the two niobium atoms. 3R-NbS2 has layers of the same type, but a different i n t e r l a y e r stacking sequence. Figure 3: De f i n i t i o n of the l e t t e r notation close-packed atomic planes. for F i g u r e 4 : Schematic band s t r u c t u r e of NbS 9 the Fermi level lies in the middle of the half f i l l e d dz2 band. The properties of LTMDs and their intercalation compounds have frequently been discussed in terms of the rigid band charge transfer model (RBCT). This model assumes that: 1. The d bands at least are not affected very much by intercalation (rigid bands). 2. On intercalation of an alkali metal atom, its valence electron is donated to the lowest unoccupied state in the host bands (complete charge transfer). Other intercalated electron donors such as certain organic molecules exhibit incomplete charge transfer. RBCT has been used extensively to explain the optical and electrical properties of intercalated LTMDs. The reviews by Marseglia (1983) and by Yoffe (1982), for example, discuss a number of experimental results from this point of view. Occasionally the predictions of RBCT f a i l ; this is o.ften due to intercalation induced changes in the s and p bands which can lead to overlap with the d band. Theoretical insight into the RBCT model has come from the calculations of McCanny (1979), and of Umrigar et a l . (1983). In both these works, f i r s t principles band structure calculations for TiS2 and Li TiS2 are performed and compared. The calculated d bands are not significantly altered by intercalation, except by f i l l i n g due to charge transfer. The sulfur 3s and 3p bands are, however, modified considerably, as is a 10 h i g h - l y i n g T i 4 s ~ l i k e band. Although the t i g h t - b i n d i n g model of band s t r u c t u r e i s not accurate enough f o r c a l c u l a t i o n s i n the LTMDs, one can d i s c u s s the r e s u l t s i n a t i g h t - b i n d i n g framework. From t h i s p o i n t of view, the s and p bands are more s t r o n g l y a f f e c t e d by i n t e r c a l a t i o n because the atomic s and p s t a t e s on which they are based are la r g e i n s i z e , and extend s i g n i f i c a n t l y i n t o the i n t e r l a y e r gap. (The s u l f u r p s t a t e s extend f a r t h e s t i n t o the gap, and i t i s t h e i r o v e r l a p which i s p r i m a r i l y r e s p o n s i b l e f o r i n t e r l a y e r bonding i n the u n i n t e r c a l a t e d LTMDs.) On i n t e r c a l a t i o n , there i s a l a r g e o v e r l a p of the s and p s t a t e s with the L i 2s s t a t e . The metal atom d s t a t e s , because of t h e i r smaller s i z e , do not extend i n t o the gap and are not a f f e c t e d . From the above d i s c u s s i o n , there i s good reason to expect RBCT to be a v a l i d way of understanding those e l e c t r o n i c p r o p e r t i e s of i n t e r c a l a t e d LTMDs which depend p r i m a r i l y on e l e c t r o n s at or near the Fermi l e v e l (that i s , in the d bands). One such property i s the e l e c t r o n i c c o n t r i b u t i o n to the low temperature s p e c i f i c heat, and i t was the p o s s i b i l i t y of making a q u a n t i t a t i v e t e s t of the RBCT model i n t h i s way which motivated the work d e s c r i b e d in t h i s t h e s i s . 1.4 CONTRIBUTIONS OF THIS THESIS Chapter 2 of t h i s t h e s i s i s concerned with the p r e p a r a t i o n and c h a r a c t e r i z a t i o n of NbS 2 and L i NbS 2. I t w i l l be shown X that staged phases e x i s t i n L i NbS 2, as in L i NbSe 2 11 (D.C.Dahn and Haering 1982). This has not been realized by previous authors (McEwan and Sienko 1982, McEwan 1983). Chapter 3 describes the low temperature heat capacity cryostat built for this work, and the procedures used for measurements and data analysis. The results for eleven Li NbS2 samples covering the range 0 to 1 in x are given in chapter 4. The specific heat of a normal metal at sufficiently low temperatures has the form (Ashcroft and Mermin 1976, for example) c = 7T+j3T3 (1-4) where 7 and 0 are constants2. The f i r s t term in (1-4) is due to the electrons and is proportional to the density of states at the Fermi level. It can be separated from the cubic term by f i t t i n g the data to (1-4). In the RBCT model, as x is increased by intercalation, the Fermi level moves up through the host bands, and the variation of 7 with x directly maps out the d band density of states. The interpretation of the electronic specific heat data in this fashion is discussed in chapter 5 . 2 Note that it has not been specified whether this is the specific heat at constant volume or constant pressure. Experimental evidence and thermodynamic arguments show that the difference between these is insignificant in solids, especially at low temperatures (Ashcroft and Mermin 1976, p427). Although we can ignore the difference between the two, the experimental data in this thesis were measured at constant pressure, while the theoretical expressions in chapters 5 and 6 are, s t r i c t l y speaking, valid for constant volume. 12 The c u b i c term i n the low temperature s p e c i f i c heat i s due to l a t t i c e v i b r a t i o n s . The c o e f f i c i e n t 0 of t h i s phonon term a l s o changes as a f u n c t i o n of x, and some p o s s i b l e reasons f o r t h i s are presented i n chapter 6 . Chapter 7 i s a b r i e f d i s c u s s i o n of s u p e r c o n d u c t i v i t y i n L i NbS 2 and chapter 8 summarizes the t h e s i s . Suggestions f o r f u t u r e work are a l s o g i v e n . 2. PREPARATION AND STRUCTURE OF INTERCALATED NIOBIUM DISULFIDE 2.1 PREPARATION OF NbS, The NbS2 used in this work was prepared by reaction of the elements in evacuated quartz ampoules. The starting materials were 99.9% pure niobium powder and 99.9999% pure sulfur powder3. The niobium powder was reduced by heating it to 500°C in hydrogen." After reduction the niobium was handled only in an argon atmosphere. Weighed amounts of niobium and sulfur were placed in quartz ampoules which were then evacuated using a diffusion pump, and sealed. Enough excess sulfur was added to produce approximately 6 atmospheres of sulfur gas pressure at the annealing temperature of 750°C5. The excess sulfur is required in order to get a stoichiometric product with the 2H structure 3 Both from SPEX Industries, Metuchen, N.J. o Hydrogen can be absorbed into Niobium to form a metal hydride. It has recently been learned that the reduction procedure that was used leaves a significant amount of hydrogen in the niobium metal. X-ray diffraction measurements of the niobium that was used to prepare NbS2 batch DD12 indicated that it contained about 20 atomic percent hydrogen. (The lattice parameters of H Nb are known as a function of x; see Schober and Wenzl 1978X) However, proton NMR measurements failed to detect any hydrogen in the NbS2 that was produced from this material. When NbS2 is prepared by high temperature reaction, the hydrogen may react with some of the excess sulfur present to form H2S. 5 The thermodynamic properties of sulfur vapour have been measured by Rau et a l . (1973) 13 14 (Fisher and Sienko 1980). Typical ampoules had an interior volume of about 12cm3, and contained about a 6g total charge, of which about .2g was excess sulfur. The ampoules were heated slowly to the reaction temperature of 950°C, left there for 2 to 3 days, then annealed at 750°C for one day. This was followed by a quench into cold water. The quench is necessary in order to obtain the 2H phase, and also neatly separates the excess sulfur from the NbS2, since a l l the sulfur vapour condenses out on the cold walls of the ampoule during the quench. The product is a free flowing powder. Since the excess sulfur adheres to the walls, only the material which can be poured freely out of the ampoule when i t is cracked open is used. X-ray powder diffraction measurements were made on the NbS2 powders. For the three batches of NbS2 used in this work, the dimensions of the hexagonal unit c e l l are listed in table I. A l l these batches are pure 2H phase. No Bragg peaks corresponding to the- 3R- structure were seen. The lattice parameters were determined by a least squares f i t to the positions of at least 8 Bragg peaks6. The three batches of NbS2 have the same lattice parameters, within the accuracy of the measurements. The values depend to some extent on the details of the f i t t i n g procedure and the methods used to correct for diffractometer errors such as 6Using computer programs written by J.R.Dahn, P.Mulhern and the author. 15 the off-axis effect (J.R.Dahn et a l . 1982). Estimates of these possible systematic errors have been included in the errors quoted in the table, and this is why the differences between the three a values, in particular, are less than the uncertainty in each one. Lattice parameters from the previous literature are also included in the table, and agree reasonably well with the present values. As noted by previous authors (Jellinek et a l . 1960, Revelli 1973, Fisher and Sienko 1980), some of the Bragg peaks are quite broad. For example, the (104) peak of DD9 NbS2 has a width at half maximum of 1.3° in 29, where 6 is the Bragg angle7. The broad peaks indicate some disorder in .the crystals. The type of disorder can be deduced from the fact that a l l the lines with Miller indices (hkl)=(0Ql) or (11/) are sharp. It can be shown (Revelli 1973, for example) that these lines are not broadened if the disorder is due only to stacking faults. Stacking faults occur frequently in layered transition metal dichalcogenide crystals, and.are, in this case, errors in the registry between adjacent S-Nb-S sandwiches. In terms of the letter notation of section 1.2, the sequence . . .BaB-CaC-BaB-AbA-CbC... has a stacking fault between the third and fourth layers. 7DiffTactometer r e s u l t s are u s u a l l y given i n terms of 2d rather than 6 s i n c e the instrument a c t u a l l y measures 2d. Table I Properties of 2H-NbSp Included are hexagonal lattice parameters a and c, superconducting transition temperature, T and specific»heat coefficients y and 3 . Unless indicated otherwise, T„ was measurea by a magnetic susceptibility method, and Tc taken as the temperature when the transition was 50% complete. Reference a (A) c (A) T. C(K) range of Tc (K) Y 2 (mJ/mole-K ) 6 (mJ/mole-K ) This work: DD6 DD9 DD12 3.325 3.323 3.324 (±.005) 11.96 11.96 11.95 (±.01) 5 .7* 5.5-6.0 19.3±1.5 0.31±0.04 Fisher and Sienko (1980) McEwan (1983) 3.324 (±.003) 11.95 (±•02) 6 6 .33 .46 6.25-6.41 0.4IK wide Aoki et al (1983) Nakamura and Aoki (1983) 6 6 . lf .06 + 18.2 0. 33 Revelli (1973) 3.31 11.90 6 .1 5.5-6.6 Van Maaren and Harland (1969) 5 .5* 10. 7 0.31 Van Marren and Schaeffer (1966) powders single crystals 5.8-6.2 6.1-6.3 Molinie et al (1974) 6 .23 + f - onset temperature * - measured calorimetrically 1 7 Superconducting transition temperatures for NbS2 from the literature are also given in table I. There is considerable disagreement on the superconducting transition. The transitions are a l l rather broad. This has been explained by some authors (Van Maaren and Schaeffer 1966, Revelli 1973) in terms of variations in stoichiometry within the sample. Many LTMDs are known to have metal rich phases, which are essentially the stoichiometric phases with some excess metal intercalated between the layers. The assumption made is that the samples were actually 2H-Nb1+yS2, with slight variations y within each sample and between samples. The transition temperature is said to drop very rapidly with increasing y, in agreement with the rigid band charge transfer model, since intercalated excess Nb should donate electrons to the dz2 band and hence lower the density of states at the Fermi le v e l . Revelli (1973) stated that the transition drops by about 1.5K for each change in y of .01. Non-stoichiometry might be a satisfactory- explanation of the variations of the transition temperature, except for the results of Fisher and Sienko (1980), which indicate that Nb,, S2 exists only in the 3R structure, and that 3R-Nb,, S2 1 +y 2 1 ' 1 +y is not a superconductor. Fisher and Sienko did not, however, offer an alternate explanation of the broad and variable transitions. The superconducting properties of stoichiometric 3R-NbS2 are also unclear. Jones et a l . (1972) reported a transition at 5.9K, while Van Maaren and Schaeffer(1966) saw 18 a transition extending from 5.0 to 5.5K. Fisher and Sienko were not able to prepare stoichiometric 3R-NbS2 and suggested that the earlier samples were a l l non-stoichiometric, and that the observed superconductivity was due to small amounts of 2H present as impurities. However, further work by the same group confirms that stoichiometric 3R-NbS2 can be prepared, provided the reaction temperature is sufficiently low (McEwan 1983). McEwan found Tc=4.67K for the stoichiometric 3R phase. Previous specific heat results for the 2H phase (table I) are also in disagreement. Although the values for 0 in c=7T+0T3 (1-4) agree, the value for 7 given by Van Maaren and Harland (1969) is much smaller than that reported by Aoki et al.(l983). The specific heat of NbS2 was measured during the course of the work leading to this thesis, and the results (section 4.3) are in agreement with those of Aoki et a l . The differences in 7 values may be due to the methods used to f i t the data to equation (1-4). Equation (1-4) is only valid in the normal state. As will be discussed in section 4.3, however, the normal state data alone are not sufficient to determine 7 accurately. It is possible to use the specific heat data in the superconducting state to derive an extra constraint on the f i t , and only i f this is done can the parameters in (1-4) be determined accurately. 19 2.2 PREPARATION OF Li^NbS; To prepare Li NbS2, lithium was intercalated into NbS2 in electrochemical c e l l s . The c e l l cathodes (positive electrodes) were prepared by fixing NbS2 powder to a nickel f o i l substrate using the following procedure. F i r s t , the nickel f o i l substrates were etched in nit r i c acid to clean and roughen them. They were then thoroughly rinsed and dryed. NbS2 was ground using a mortar and pestle, until i t passed through a 400 mesh (38Mm) sieve. The powder was then mixed with cyclohexane to form a thick slurry, which was spread evenly over the substrates. After the cyclohexane evaporated, the cathodes were passed between two steel rollers, which compact the NbS2 layer, thereby improving the electrical contact between the NbS2 grains, and between the NbS2 and the substrate. Inserting a sheet of weighing paper between the cathode and the upper roller helps to prevent the NbS2 from sticking to the r o l l e r . The mass of NbS2 on the cathodes was established by weighing the- bare substrates and the finished cathodes. Cathodes used for preparation of low temperature specific heat samples were 1.75 inches in diameter, and contained typically 0.3g of NbS2. Similar but smaller cathodes containing 10 to 20 mg of NbS2 were used in cells intended for the electrochemical measurements to be described later in this chapter. Anodes for the cells were lithium metal f o i l8. The From Foote Minerals, Exton, Pa. 20 c e l l s were assembled in an argon f i l l e d glovebox. They were of the flange c e l l type (figure 5). Cells were assembled by placing a porous polypropylene film separator9 between the anode and cathode. The separators were wet with an electrolyte consisting of a 1 molar solution of LiAsF61 0 in propylene carbonate. The active components of the cells were sandwiched between stainless steel flanges, which were separated by Viton rubber o-rings. The o-rings served to provide airtight seals for the c e l l s , as well as electrically isolating the flanges. The anode and cathode are each in electrical contact with one of the flanges, so that electrical connection to the c e l l is accomplished by simply connecting a lead to each flange. Li NbS2 samples for low temperature specific heat measurements were made by discharging flange cells to a preset voltage using a Princeton Applied Research model 173 Potentiostat/Galvanostat. The current which passed through the c e l l during discharge was- integrated by a PAR model 179 dig i t a l coulometer. Since for each L i+ ion which moves from anode to cathode during the discharge, one electron moves through the external c i r c u i t , the value of x in the LixNbS2 samples could be calculated using QM x = (2-1) m (96,500 Coul/mole) 9Celanese Plastics Celgard #2500 or 3501 10U.S.Steel Agrichemicals, 'Lectrosalt' brand. 21 F i g u r e 5: A t y p i c a l f l a n g e c e l l . 2 2 where Q i s the charge which has passed through the c e l l , M i s the molecular weight of NbS 2 (1 5 7.Og/mole), and m i s the mass of NbS 2 on the cathode. In order to have a u n i f o r m l y i n t e r c a l a t e d sample, a c e l l must be allowed to f u l l y e q u i l i b r a t e . When a f r e s h l y prepared c e l l ( x = 0 , open c i r c u i t v o l t a g e = 3 . 2 V ) i s connected to a f i x e d v o l t a g e V 0 such as that p r o v i d e d by the PAR 1 7 3 , c u r r e n t should flow u n t i l the cathode m a t e r i a l i s uniformly i n t e r c a l a t e d to a composition given by V 0 = V ( X ) = (M ~*x (x) )/e ( 2 - 2 ) 3 C (see equation 1 - 3 ) . The e q u i l i b r a t i o n of a c t u a l c e l l s i s not q u i t e t h i s simple, s i n c e the approach to e q u i l i b r i u m can be rather slow, and at some p o i n t as the i n t e r c a l a t i o n c u r r e n t slowly d i e s away, spurious changes i n Q due to coulometer d r i f t and c e l l leakage c u r r e n t s may become s i g n i f i c a n t . A u s e f u l way of monitoring an e q u i l i b r a t i o n i s to make a p l o t of c u r r e n t as a f u n c t i o n of Q, or e q u i v a l e n t l y , x as determined from ( 2 - 1 ) . T h i s was done d u r i n g the e q u i l i b r a t i o n s of the sample p r e p a r a t i o n c e l l s . The p l o t s of c u r r e n t a g a i n s t x e x h i b i t e d an almost l i n e a r appearance as the c u r r e n t approached z e r o . T h i s s o r t of behavior i s not s u r p r i s i n g , s i n c e at l e a s t near e q u i l i b r i u m i t i s reasonable to expect the c u r r e n t to be p r o p o r t i o n a l to the d e v i a t i o n from e q u i l i b r i u m . Discharges were stopped by o b s e r v i n g t h i s l i m i t i n g behavior and d i s c o n n e c t i n g the c e l l only when x had come w i t h i n about 1% of the apparent l i m i t i n g v a l u e . The times r e q u i r e d f o r t h i s were u s u a l l y 5 to 1 0 days, and the 23 f i n a l c u r r e n t s were always l e s s than 5uA. C e l l s were g e n e r a l l y not l e f t on the PAR 173 f o r more than 10 days, i n order to minimize the e f f e c t s of d r i f t and leakage c u r r e n t s . One of the problems that has been a s s o c i a t e d with e l e c t r o i n t e r c a l a t i o n i n the past i s cathode u t i l i z a t i o n . Often some of the cathode p a r t i c l e s are not i n good e l e c t r i c a l c ontact with the s u b s t r a t e , and consequently cannot be i n t e r c a l a t e d . T h i s was seen, f o r example, in the author's p r e v i o u s work on L i NbSe 2 (D.C.Dahn and Haering 1982) The cathodes used i n that study were not r o l l e d , however, and the a d d i t i o n of the r o l l i n g s tep i n the cathode p r e p a r a t i o n procedure appears to have e l i m i n a t e d a l l problems with cathode u t i l i z a t i o n . X-ray d i f f r a c t i o n on L i NbS 2 prepared using r o l l e d cathodes c o n s i s t e n t l y shows no t r a c e of Bragg peaks due to u n i n t e r c a l a t e d m a t e r i a l . Because of t h i s , the u n c e r t a i n t y i n x i s determined p r i m a r i l y by c e l l leakage c u r r e n t s , coulometer d r i f t , and p o s s i b l y by s i d e r e a c t i o n s i n the c e l l . The magnitude of these e f f e c t s i s not easy to estimate a c c u r a t e l y , but i s b e l i e v e d to be a few percent of x. The next step i n the p r e p a r a t i o n of a s p e c i f i c heat sample was to take an e q u i l i b r a t e d c e l l back i n t o the argon glovebox, open i t , and scrape the i n t e r c a l a t e d cathode m a t e r i a l o f f the s u b s t r a t e . To remove the e l e c t r o l y t e which remained on the s u r f a c e s of the L i NbS 2 g r a i n s , they were X r i n s e d w i t h pure propylene carbonate and d r i e d i n vacuum. The propylene carbonate used was s p e c i a l l y d i s t i l l e d and 24 co n t a i n e d about 10 to 20 p a r t s per m i l l i o n of water. T h i s was the same m a t e r i a l used i n the p r e p a r a t i o n of e l e c t r o l y t e . About 1ml was used f o r the r i n s e . A f t e r d r y i n g , some of the L i NbS 2 powder was pressed i n t o a p e l l e t f o r use as a s p e c i f i c heat sample. The p r e s s i n g was done i n a s t e e l p i s t o n d i e which forms a p e l l e t 6mm i n diameter and a few mm hi g h . The f o r c e r e q u i r e d to form a p e l l e t i s of order 1000N and i s a p p l i e d with a c-clamp. The p e l l e t was then weighed and mounted in the c r y o s t a t as d e s c r i b e d in chapter 3. The remaining L i NbS 2 powder c o u l d be used f o r x-ray d i f f r a c t i o n . A l l h a n d l i n g of the L i NbS 2 samples took p l a c e i n an argon atmosphere. X 2.3 THE STRUCTURE OF L i NbS, x <L X-ray d i f f r a c t i o n and e l e c t r o c h e m i c a l measurements have been used to determine the c r y s t a l s t r u c t u r e s and approximate phase boundaries of the L i x N b S 2 phases formed by i n t e r c a l a t i o n at room temperature. For x>.23, L i NbS 2 has a stage 1 s t r u c t u r e , that i s , there i s an equal c o n c e n t r a t i o n of l i t h i u m i n each i n t e r l a y e r gap. For x between (roughly) .11 and .19 there i s a w e l l - o r d e r e d stage 2 s t r u c t u r e . In stage 2, every second gap c o n t a i n s l i t h i u m , and the i n t e r v e n i n g gaps are e i t h e r empty or n e a r l y so. There i s evidence f o r a d i s o r d e r e d stage 3 phase ( l i t h i u m i n every t h i r d l a y e r on average but with no long range order i n the s t a g i n g sequence), which e x i s t s f o r compositions near x=.08. Samples with average compositions between those of the 25 staged phases are phase mixtures. The staging behavior is similar to that observed in Li NbSe2 (D.C.Dahn and Haering 1982). An especially powerful way of observing the staging phase transitions is the use of electrochemical cells with beryllium x-ray windows (J.R.Dahn et a l . 1982). These cells are similar in construction to the flange cells described in the previous section. The NbS2 powder, instead of being fixed to a nickel substrate, is fixed directly to the inner surface of a .25mm thick beryllium f o i l window which is set into one of the flanges. To keep preferred orientation of the cathode powder to a minimum, x-ray c e l l cathodes are not rolled. Ideally, it would be best to have completely random orientation of the cathode particles, since this simplifies interpretation of powder diffraction measurements. However the NbS2 particles are thin platelets with their crystallographic c-axis normal to the flat faces. Because of their shape, they tend to settle with their c-axis normal to. the substrate to which they are attached. With r o l l i n g , this orientation is enhanced to the point where only (00/) Bragg peaks can be seen. Figure 6 shows portions of diffTactometer scans made while an x-ray c e l l was slowly equilibrating to a final voltage of 2.760V. The region around the (008) Bragg peak is shown in the figure, although complete scans from 10 to 90° 2d were made in each case. Only NbS2 peaks were seen in the fi r s t scan, made before the discharge started. As gure 6: The (008) region during the discharge of a Li/LixNbS2 x-ray c e l l to 2.760V. a- Before discharge (x=0) b- After 3 days (x=.06) c- After 6 days (x=.08) d- After 15 days (x=.14). At this point i n t e r c a l a t i o n was e s s e n t i a l l y complete. The remaining int e n s i t y in the NbS2 peak i s due to material which was not e l e c t r i c a l l y connected to the substrate. 27 intercalation proceeded, these peaks shrank. They did not disappear completely, because of incomplete cathode utilization* At the end of the equilibration the cathode (except for the non-utilized fraction) was in the stage 2 phase. A third set of peaks, corresponding to what is believed to be a disordered stage 3 phase, was seen during the intercalation process. This occured because while going from NbS2 to stage 2, it is necessary to pass through stage 3 as an intermediate state. During the intercalation process in this c e l l , each cathode particle had NbS2 at the center, surrounded by a region of stage 3, surrounded in turn by a region of stage 2 at the surface. Intercalation apparently proceeded by both phase boundaries propagating into the center. That the phase between x^.11 and .19 is truly stage 2 can be seen from the presence of a (009) Bragg peak. In pristine 2H-NbS2 and in the stage 1 intercalation compound, (00/) Bragg peaks with / odd are a l l extinct. This happens because of a symmetry of the two layer high unit c e l l which causes the geometrical structure factor for these lines to be zero. In the stage 2 compound, only one of the two interlayer gaps in the unit c e l l contains lithium and is expanded. The two layers in the unit c e l l are no longer equally spaced along the c-axis, and (00/) peaks with / odd are allowed. Note that staging is observed primarily through the distortion of the host lattice due to the fact that intercalated gaps expand, as the scattering power of lithium 28 is very small. Although of the (00/) peaks with / odd, only (009) is observed, intensity calculations (Appendix 1) show that the (003), (005), and (00JJ_) peaks are weak. The (001) peak has a scattering angle which is too small for it to be detectable in the x-ray c e l l configuration. The (007) peak should be observable, but unfortunately a beryllium Bragg peak originating in the window interferes with i t . The stage 1 and 2 phases both appear to have the same structure as 2H-NbS2, except for the addition of the lithium and the resultant expansion of the interlayer gaps. This is an important point, since McEwan (1983) and McEwan and Sienko (1982) report that LixNbS2 prepared by direct high temperature reaction of the elements forms in the 3R phase or in a 2H-3R phase mixture for x between .01 and .13. Apparently, doing the intercalation at room temperature avoids this. Another comment on McEwan and Sienko's work should be made here. Although they state that a l l their samples were stage 1, they report the presence of (007) and (009) Bragg peaks in 2H-LixNbS2 for x between .13 and .17. McEwan (1983) surmises that the (007) peak may be due to a superlattice of period 7c0 along the c-axis. This is clearly an incorrect explanationsince such a superlattice would produce (0,0,1/7) and related peaks, rather than a (007) peak. The most likely explanation is that McEwan and Sienko's samples were stage 2 between X=.13 and .17. The dimensions a and c of the unit c e l l are shown as a function of x in figure 7. Accurate values of c and a could 3.36 3.35 h 3.34 h 3.33 h 3.32 0.0 gure 7: 0.4 0.6 0.8 x in LixNbS2 Hexagonal l a t t i c e parameters a and c f o r L i NbS2- The crosses r e p r e s e n t data from t n i s work, and the p o i n t s are from McEwan (1933). 30 not be obtained for the' stage 3 phase, since it has so far only been observed as an intermediate state during intercalation from NbS2 to stage 2. An approximate value for c in the stage 3 phase is 12.2A. Also shown in figure 7 are the lattice parameters of McEwan and Sienko's high temperature prepared Li NbS2. The X precision of the data is good because Li NbS2 prepared at high temperature has very sharp Bragg peaks, indicating fewer stacking faults than in Li NbS2 prepared by room temperature intercalation. Only those samples which were 2H, or where 2H was the major component of a 2H-3R phase mixture, are included. Plateaus at the stage 3 and stage 2 compositions can be seen in the c-axis data. Note that the c values for x>.2 are lower than the results from this work. McEwan and Sienko prepared their samples by reaction in evacuated quartz tubes, and some of the lithium was lost due to reaction with the quartz.1 1 The amount of tube attack increased as a function of lithium concentration (McEwan 1983). As a result, the x values quoted by McEwan and Sienko are too high, and the error in x increases as a function of x. Comparison of the c data shows, for example, that McEwan and Sienko's 'x=.33' sample actually had a composition near x=.25. i 1 Similar tube attack by lithium has been observed in the course of high temerature compound preparation in this laboratory (J.R.Dahn and P.J.Mulhern, unpublished). 31 F u r t h e r i n f o r m a t i o n on the phase t r a n s i t i o n s i n L i x N b S 2 was o b t a i n e d from e l e c t r o c h e m i c a l measurements. As mentioned in chapter 1, the v o l t a g e of a L i / L i NbS 2 c e l l can be used to study the thermodynamics of the i n t e r c a l a t i o n compound, si n c e V(x)=(u -M ( x ) ) / e a C where u and u„ are the chemical p o t e n t i a l s of l i t h i u m i n l i t h i u m metal and i n the i n t e r c a l a t i o n compound, r e s p e c t i v e l y . Only a b r i e f d i s c u s s i o n of the i n t e r p r e t a t i o n of e l e c t r o c h e m i c a l measurements w i l l be r e q u i r e d here; f o r more complete d i s c u s s i o n s see Johnson (1982) or J.R.Dahn and McKinnon (1984a). As the value of x i n a c e l l cathode i s i n c r e a s e d by i n t e r c a l a t i o n , the cathode i s sometimes observed to undergo a f i r s t order phase t r a n s i t i o n between two compositions, say x, and x 2 . During the t r a n s i t i o n the average composition x i s given by x = f 1 x 1 + ( 1 - f , ) x 2 (2-3) where f^ and (1-f,) are. the f r a c t i o n s of the sample i n the x, and x 2 phases, r e s p e c t i v e l y . Regions of constant v o l t a g e V i n a c e l l ' s V(x) curve are the s i g n a t u r e s of such phase t r a n s i t i o n s , s i n c e as long as the sample i s a phase mixture M i s constant and t h e r e f o r e V i s c o n s t a n t . These f e a t u r e s c in V(x) curves may be d e t e c t e d more e a s i l y by n u m e r i c a l l y c a l c u l a t i n g the i n v e r s e time d e r i v a t i v e dt/dV d u r i n g a slow constant c u r r e n t charge or d i s c h a r g e of the c e l l . T h i s i s because ( i g n o r i n g k i n e t i c e f f e c t s due to l i t h i u m d i f f u s i o n g r a d i e n t s i n the cathode and the i n t e r n a l impedence of the 32 c e l l ) dt/dV = (Q,/i)(dx/dV) (2-4) where i i s the c e l l current (positive on discharge) and is the amount of charge corresponding to a change in x of 1. Phase transitions w i l l therefore produce sharp peaks in dt/dV. The peaks are not i n f i n i t e l y high because of the lithium d i f f u s i o n gradients mentioned above. V(x) data for L i / L i NbS2 are shown in figure 8. Because X of incomplete cathode u t i l i z a t i o n (91% u t i l i z a t i o n ) in the small flange c e l l used, the data have been scaled so that x=1 occurs at 1.90V. Also included are the V(x) data from s p e c i f i c heat sample preparations. The results are in agreement with less accurate previous measurements (Holleck et a l . 1975, DiPietro et a l . 1982). Using a microcomputer based instrument which calculates dt/dV during a constant current discharge or charge, dt/dV measurements were made on several c e l l s . Typical results are shown in figure 9. The data are shown in terms of -dx/dV (equation 2-4) The staging phase transitions are c l e a r l y seen. The higher peak at about 2.74V i s due to the stage 2 to stage 1 t r a n s i t i o n . A smaller broader peak at about 2.78V is seen on the f i r s t recharge and second discharge. This i s believed to be due to the stage 2 to stage 3 t r a n s i t i o n . As can be seen from figure 8, L i / L i NbS2 c e l l s cannot be recharged a l l the way back to x=0 X in the L i NbS2 cathode. This is the reason that the f i r s t x discharge curve in figure 9 is d i f f e r e n t than the other two. The area under the -dx/dV curve at voltages above the stage 33 3 -co 2 -o > > 1 -0 0.0 0.2 0.4 0.6 0.8 x in L i x N b S 2 1.0 Figure 8 Cell voltage as a function of x for L i NbS2- The lines show the f i r s t discharge anct the subsequent f i r s t charge of c e l l DD65. The discharge and charge were both at a rate of Ax=l in 60 hours. Also shown (x) are the V(x) values of the specific heat samples. 34 2.6 2.7 2.8 2.9 V (volts) F i g u r e 9 : -dx/dV data f o r L i / L i NbS ? c e l l D D 4 5 , showing the s t a g i n g t r a n s i t i o n s . a- F i r s t d i s c harge b- F i r s t recharge c- Second d i s c h a r g e . 35 2 - stage 1 peak is larger for the f i r s t discharge than for subsequent charges or discharges. (The area under a -dx/dV curve between two voltages is the change in x between those voltages.) This presumably happens because of some kinetic barrier which prevents de-intercalation of the stage 3 phase. Similar behavior has been observed in the Li NbSe2 X system, where the presence of residual lithium in fully charged cathodes has also been verified directly by x-ray diffraction (D.C.Dahn and Haering 1982), as well as in the intercalated graphite 'residue compounds'. There is also apparently some kinetic or nucleation barrier at the very beginning of the intercalation process. Since the x-ray results clearly show a succession of three transitions (NbS2 to stage 3, stage 3 to stage 2, and stage 2 to stage 1), there should, in principle, be three peaks in -dx/dV on the f i r s t discharge. There are, however, only two. For some reason, the f i r s t intercalation can only proceed (at a rate of Ax=1 in about 200 hours- in this- case) when the c e l l voltage has already dropped into the stage 2 region. The kinetics of the staging transitions in Li NbS2, Li NbSe2, and related materials might be an interesting topic for more detailed study in the future. An additional feature in -dx/dV was observed in two cells made using freshly prepared NbS2 from batch DD12. Two small peaks near 2.67V (x^.3) could be seen (figure 10). The cells which showed this feature were assembled 14 days and 28 days after batch DD12 was prepared and ground. Also shown 0 2.68 V L_ 2.72 (volts) 2.76 Figure 10: -dx/dV data on L i / L i NbS2 c e l l s , showing the aging effect. Thi solid line is data from a c e l l made with freshly prepared NbS2 from batch DD12, and the dashed line with aged NbS2 from the same batch. The dashed line has been displaced downward by .5 V" for c l a r i t y . The arrows indicate the extra peaks mentioned in the text. 37 in the figure is data from a c e l l after i t was made from DD12 NbS2 which had been stored in a closed v i a l for 177 days. The extra peaks are not seen in this data. The extra peaks in -dx/dV were never observed in the other batches of NbS2; however, dx/dV runs were not made on these batches until they were a few months old. The extra peaks are similar to those due to lithium ordering on a /3a triangular superlattice at x=1/3 in Li TaS2 (J.R.Dahn and McKinnon 1984). The same type of lithium ordering may be involved here. A possible reason for the disappearance of the extra peaks in 'old' material is loss of sulfur. There is evidence that NbS2 slowly gives off sulfur; i t smells faintly of H2S when it is in a i r , and smells very strongly when being ground1 2. The smell presumably comes from sulfur lost by the NbS2, which then reacts with moisture in the a i r . Aging effects have also been observed by Dutcher(1985) in TaS2. He found that the intensities of x-ray Bragg peaks due to the charge density wave changed as a function of time and storage conditions, and also believes that sulfur loss is the cause. If sulfur is lost from NbS2, we are left with excess Nb, which would intercalate into the interlayer gaps. The presence of randomly placed niobium in the gaps could serve 1 2 Loss of sulfur during grinding is not the explanation for the extra peaks in -dx/dV, since a l l the cells were made using ground material. 38 to i n h i b i t l i t h i u m o r d e r i n g . The amount of s u l f u r l o s t i s not known, but i t must be r a t h e r small s i n c e no s i g n i f i c a n t changes i n the l a t t i c e parameters were observed. C r y s t a l l o g r a p h i c data on 2H-Nb 1 +^S 2 f o r small y are not a v a i l a b l e , but Huisman et a l . (1970) have measured the l a t t i c e parameters of 2H-Nb 1 +ySe 2. If the behavior of 2H-Nb 1 +yS 2 i s s i m i l a r , the f a c t t h a t no changes i n the l a t t i c e parameters were observed means that changes i n y were about .01 or l e s s . 3. THE LOW TEMPERATURE SPECIFIC HEAT EXPERIMENT 3.1 INTRODUCTION This chapter begins with a brief review of the various experimental methods used for s p e c i f i c heat measurements at low temperatures. The reasons for choosing the relaxation time method for t h i s work are given. There follows a discussion of the cryostat used, the measurement and control of the reference block temperature, and the sample temperature. The measurement cycle and data analysis are then discussed in d e t a i l . 3.2 TECHNIQUES FOR LOW TEMPERATURE SPECIFIC HEAT  MEASUREMENTS • 3.2.1 ADIABATIC CALORIMETERS Although some low temperature calorimetry on solids was done late in the last century, i t was not u n t i l the work of Nernst, Eucken, and their collaborators beginning in 1909 that s a t i s f a c t o r y results over a wide range of low temperatures were obtained. (For a review of early work see Partington 1952.) The 'adiabatic calorimeter' f i r s t used by Nernst i s , with improvements, s t i l l in wide use today. In i t s simplest form, an abiabatic calorimeter consists of a vacuum chamber immersed in a low temperature bath and containing the sample, which i s suspended on supports having very low 39 40 thermal conductance . A thermometer and r e s i s t a n c e heater are mounted on the sample; here again care i s taken to minimize the thermal l i n k between the sample and the bath through the l e a d s . To make a measurement the sample i s f i r s t c o o l e d by the i n t r o d u c t i o n of exchange gas. T h i s i s then pumped out, l e a v i n g the sample (approximately) thermally i s o l a t e d . A pulse of heat Q i s then a p p l i e d to the heater, causing the temperature of the sample to r i s e by an amount AT. The heat c a p a c i t y of the sample, thermometer, and heater assembly i s then given by C=Q/AT (3-1) The p u l s e d heating i s then repeated, p r o v i d i n g measurements of C at s u c c e s s i v e l y higher temperatures. Improvements s i n c e Nernst's day (Gmelin 1979, f o r example) include b e t t e r thermometry, the i n t r o d u c t i o n of a temperature-regulated r a d i a t i o n s h i e l d around the sample, and computer data a q u i s i t i o n . Since l a r g e amounts of exchange gas can be adsorbed on the sample, e s p e c i a l l y i f i t i s powdered or porous, and s i n c e the removal of exchange gas t y p i c a l l y r e q u i r e s s e v e r a l hours of pumping, many modern a d i a b a t i c c a l o r i m e t e r s are equipped with a mechanical heat switch f o r c o o l i n g the sample. A b i a b a t i c and r e l a t e d methods s t i l l p rovide the most a c c u r a t e r e s u l t s on l a r g e samples (mass a few grams) at temperatures above 1K. The a b s o l u t e accuracy 41 can be b e t t e r than 0.5% (Gmelin 1979). The l a r g e samples are r e q u i r e d so that the i n e v i t a b l e heat leaks along thermometer l e a d s , e t c , have an ac c e p t a b l y small e f f e c t on the sample temperature. Below 1K, f r i c t i o n a l h eating by the heat switch i s a s e r i o u s problem. A v a r i a t i o n on the a d i a b a t i c method i s the q u a s i - a d i a b a t i c heat p u l s e method. Here the thermal l i n k between the sample and the regulated' r a d i a t i o n s h i e l d or r e f e r e n c e block i s made l a r g e enough so th a t a f t e r a heat pulse the sample c o o l s again i n a reasonable time, t y p i c a l l y a few minutes. T h i s e l i m i n a t e s the need for a thermal switch or exchange gas. As long as the c o o l i n g time i s very long compared to the d u r a t i o n of the heat p u l s e , the maximum AT a f t e r a heat pulse w i l l s t i l l be give n to high accuracy by (3-1). In some cases, c o r r e c t i o n s must be made to account f o r the heat l o s t down the thermal l i n k d u r i n g the hea t i n g p u l s e ( S e l l e r s and Anderson 1974, Fagaly and Bohn 1977). When the sample's thermal c o n d u c t i v i t y i s low, the sample may not be isothermal immediately a f t e r the p u l s e . As long as the i n t e r n a l r e l a x a t i o n time of the sample i s short compared to the sample to r e f e r e n c e block c o o l i n g time, i n t e r n a l r e l a x a t i o n e f f e c t s may a l s o be c o r r e c t e d f o r ( L a s j a u n i a s et a l . 1977). 42 3.2.2 THE AC TEMPERATURE METHOD Sullivan and Seidel (1968) introduced a new method of low temperature heat capacity measurement. In this method, the sample is connected to a temperature regulated block by a thermal link of thermal conductance k. A heater and thermometer are attached to the sample. An AC heater current at frequency co/2 is applied to the heater, and produces an AC temperature in the sample at frequency w. In the simplest case, the AC temperature is given by AT =P /2uC ( ac ac where P is the heater power and C is the total heat capacity of the sample and its addenda (heater, thermometer, supports, etc.) For (3-2) to be valid, the internal thermal response times r^nt of the thermometer and heater must be very short compared to 1/a>, the sample-to-block thermal relaxation time r=C/k must be much longer than 1/CJ, and the sample's thermal conductivity must be sufficiently high. If these conditions are not met, the analysis becomes more complicated (Sullivan and Seidel 1968), but in principle the method can s t i l l be applied. The greatest d i f f i c u l t i e s arise when the sample has a low thermal conductivity. In this case (assuming again that r. <<1/CJ<<T), we have, for heater and thermometer on 43 opposite sides of the sample P AT = (l+2k/3k_) (3-3) a C 2coC where kg is the thermal conductance across the sample. The frequency independent correction factor in (3-3) means that in order to make measurements of high absolute accuracy, we need either to have k/kg small, or to make an accurate independent measurement of k and kg. It is therefore d i f f i c u l t to make accurate measurements using the AC method, on samples with low thermal conductivity. The LixNbS2 samples used in this work were pellets of compacted powder and were very poor conductors of heat. The main advantages of the AC method are that it can be used with very small samples (<1mg mass) and that it can be used to give a continuous readout of C (via a lock-in amplifier) as the temperature and other parameters such as magnetic f i e l d are varied. The continuous nature of the measurement makes i t the method of choice in studies were high precision but not necessarily high absolute accuracy is needed. 3.2.3 THE RELAXATION TIME METHOD The relaxation time method was introduced by Bachmann et a l . (1973), and is also discussed in the review by Stewart (1983). A simplified apparatus is shown in figure 11. The sample is attached to a platform 44 Schematic diagram of an apparatus for relaxation time heat capacity measurements. a- Sample b- Sample platform c- Heater d- Thermometer e- Temperature regulated block. The platform is supported by the heater and thermometer wires, which make thermal contact to the block. 45 such as a sapphire s l i d e , to which are connected a heater and thermometer. The sample p l a t f o r m i s suspended from a temperature r e g u l a t e d r e f e r e n c e block by means of the heater and thermometer l e a d s . These wires are t h e r m a l l y anchored to the block, and p r o v i d e a thermal conductance k w between the p l a t f o r m and the block. A measurement begins with the sample and p l a t f o r m at the block temperature T 0 . A DC c u r r e n t i s then a p p l i e d to the heater, producing power P. The temperature d i f f e r e n c e 6 between the sample and block then r i s e s , e v e n t u a l l y reaching a constant maximum value 60 given by 0 o = P A w (3-4) The heater c u r r e n t i s then turned o f f . Assuming the sample's thermal c o n d u c t i v i t y i s high and there i s no thermal r e s i s t a n c e at the boundary between the sample and the p l a t f o r m , the sample and p l a t f o r m w i l l have a uniform temperature T g as they c o o l . T h i s w i l l r e l a x back to T 0 a c c o r d i n g to k w ( T s - T 0 ) = - C ( d T s / d t ) (3-5) or; T s - T 0 = t 9 0 e " t / T (3-6) where C i s the t o t a l heat c a p a c i t y of the sample and p l a t f o r m and T=C/k w i s the r e l a x a t i o n time. Measurement of T,60i and P i s s u f f i c i e n t to determine C, s i n c e C=rk w=TP/0 o (3-7) 46 When the sample's thermal conductance kg is not in f i n i t e , the sample is not isothermal during cooling and (3-5) to (3-7) no longer hold. It i s , however, possible to determine kg from the decay data. A fai r l y straightforward calculation then determines C. The data analysis in the presence of a finite kg is discussed in Bachmann et a l . (1973), and later in this thesis (section 3.4 and Appendix 2). The specific heat can be determined with reasonable accuracy when kg is of order kw or greater. This condition is much less restrictive than the corresponding one for the AC temperature method, making the relaxation time method more suitable for samples of low thermal conductivity. To avoid confusion, i t should be pointed out that some authors (for example Fagaly and Bohn 1977) use the term 'relaxation time method' to describe a form of the quasi-adiabatic method discussed in section 3.2.1. A measurement of the exponential relaxation time after a short heat pulse can be used to extrapolate the sample temperature back to the time of the pulse. A relaxation time measurement, as the term is used by Bachmann et a l . and in this thesis, should involve a measurement of both T and k . w 3.2.4 DISCUSSION For the specific heat measurements described in this thesis, the relaxation time method was used, since 47 i t a l l o w s accurate measurements on small samples of low thermal c o n d u c t i v i t y . Since the L i NbS 2 samples were p e l l e t s of pressed powder, t h e i r thermal c o n d u c t i v i t y was q u i t e small (see s e c t i o n 3.4.2). As we have seen, the AC temperature method i s not s u i t a b l e f o r such samples. Samples of about 150mg mass can e a s i l y be prepared i n e l e c t r o c h e m i c a l c e l l s . Although samples of t h i s s i z e are q u i t e adequate f o r a r e l a x a t i o n time measurement, i t would be d i f f i c u l t to reduce the heat l e a k s through the heater and thermometer leads enough to allow a d i a b a t i c or heat-pulse method measurements, at l e a s t at temperatures below about 10K. At higher temperatures, such as 15K and above, the r e l a x a t i o n times become much lo n g e r , t y p i c a l l y of order 100s with the present c r y o s t a t and sample s i z e s . At 4K, decay times are t y p i c a l l y 10s. T h i s happens because the sample's heat c a p a c i t y i n c r e a s e s r a p i d l y as the temperature i s r a i s e d , as does k g, the, sample's thermal c o n d u c t i v i t y . Because of t h i s , the present system c o u l d be run i n a q u a s i - a d i a b a t i c heat pulse mode at higher temperatures. I f i t i s ever d e s i r a b l e to make extensive measurements at temperatures above 15K, a s l i g h t m o d i f i c a t i o n of the computer software c o n t r o l l i n g the experiment would allow t h i s , and permit measurements to be made more q u i c k l y than with the present r e l a x a t i o n time method. The present method i s rather slow at high T, s i n c e one must wait s e v e r a l r e l a x a t i o n times with the 48 sample heater on i n order to get a s t a b l e maximum sample temperature. 3.3 THE SPECIFIC HEAT CRYOSTAT 3.3 .1 GENERAL FEATURES The s p e c i f i c heat c r y o s t a t i s of standard d e s i g n , except f o r p r o v i s i o n s to allow the mounting of a i r - s e n s i t i v e samples. F i g u r e 12 i s a s i m p l i f i e d drawing of the c r y o s t a t , showing i t s important f e a t u r e s . The vacuum can i s supported i n s i d e a "He bath by i t s pumping l i n e . I nside the can i s a copper temperature r e g u l a t e d block which supports the sample p l a t f o r m . The c r y o s t a t i s designed to allow samples to be mounted i n s i d e an argon f i l l e d glovebox. Since the a i r l o c k used to t r a n f e r a r t i c l e s i n and out of the glovebox cannot accomodate the e n t i r e c r y o s t a t , the vacuum can may be detached from the pumping l i n e . Samples are mounted i n the glovebox, and the vacuum can i s c l o s e d . A brass plug with two o - r i n g s i s used to s e a l the port in the top f l a n g e of the can. At t h i s p o i n t the vacuum can, f u l l of argon and containimg a sample, can be removed from the glovebox and j o i n e d to the r e s t of the c r y o s t a t by means of a indium s e a l . The e l e c t r i c a l leads pass out of the vacuum can i n t o the l i q u i d helium space through epoxy feedthroughs and are j o i n e d to w i r i n g l e a d i n g t o the top f l a n g e of the c r y o s t a t by means of Amphenol m u l t i - p i n 49 Figure 12: (facing page) Simplified diagram of the low temperature specific heat cryostat (not to scale). a- Pumping line for vacuum can. b- Liquid helium f i l l port. c- Glass liquid helium Dewar. d- Pumping line (3/4 inch thin wall stainless steel tubing). e- Joint with indium seal. f- Electrical feedthrough. g- Vacuum can. h- Temperature regulated block and sample holder assembly. For details of this area see figures 13 and 14. i - Socket for pumping line plug. j - Pumping line plug. k- Control rod. 1- Radiation shields. m- Pumping port for Dewar. n- Control rod feedthrough. The distance from the top of the Dewar to the bottom of the vacuum can is 35 inches. 51 connectors. Once the cryostat has been assembled and installed in a liquid heliun dewar, the pumping line is evacuated. A rod is then lowered down the center of the pumping line and threaded into the brass plug. By raising the rod, the plug is removed to a chamber at the top of the cryostat, and the argon is pumped out of the vacuum can. The sample is never exposed to a i r . After the can is evacuated, the system is precooled to liquid nitrogen temperature, and then liquid helium is transferred into the dewar in the usual way. If necessary, at the end of a measurement when the system is again at room temperature the plug may be replaced, sealing the sample in vacuum. The vacuum can may then be removed from the cryostat and brought back into the glovebox for inspection or removal of the sample. The copper reference block inside the vacuum can is shown in detail in figure- 13. It is supported on three thin wall stainless steel tubes. To allow cooling of the block, a thermal link to the bath is made by means of a brass rod and a length of copper braid. This arrangement gives a block to bath thermal conductance of 0.8mW/K at 4.2K, which allows the block to be regulated above the bath temperature without excessive amounts of power. The block cools from 80K to 4.2K in about 90 minutes. The sample platform (figure 14) is a sapphire chip 6mm square and ,1mm thick. On the side facing the copper 52 Figure 13: Detail of the vacuum can interior, a- Copper plug b- Brass rod (3/16 inch solid) c- Copper braid.'This and the brass rod form the block to bath heat link, d- Support used in sample mounting, e- Sample (see figure 14). f- Radiation shield. g- Germanium resistance thermometer, h- Copper reference block, i - Block heater. j - Support (thin wall stainless steel tubing-one of three). k- Top flange of vacuum can (stainless steel). 1- 8-lead electrical feedthrough (one of three). 53 Figure 14: De t a i l of the top side of the sample platform (the side away from the sample). a- Gold contact pad. b- Nichrome f i l m heater. c- Au+.07%Fe vs chrome! thermocouple. d- Carbon r e s i s t o r s l i c e . This i s glued on top of the heater. 54 block (the upper side) there i s an evaporated thin film nichrome heater. The two contact pads were made f i r s t . They consist of about 1 Mm of gold, underlaid by a very thin layer of chromium to provide good adhesion. The chromium and gold were evaporated in a standard vacuum deposition unit, using e l e c t r i c a l l y heated 'boat' sources and a copper f o i l mask. A nichrome layer about 15nm thick was then deposited through a d i f f e r e n t mask using an electron beam gun source. The resistance of this f i l m is about 6000 and is nearly temperature independent. The leads to the heater are .003 inch diameter brass wires and are soldered to the gold contact pads with about 1mg of pure indium. Two sample thermometers were used. The f i r s t was a Au+.07%Fe vs chromel thermocouple (Sparks and Powell 1972, Rosenbaum 1968) in a d i f f e r e n t i a l configuration with on junction on the block and the other on the sample platform. The second sample thermometer is a small s l i c e of an Allen-Bradley carbon composition r e s i s t o r which is glued on top of the sample heater. Both the re s i s t o r and thermocouple were bonded to the sample platform using Emmerson and Cummings Stycast 2850 high thermal conductivity epoxy. Sample thermometry w i l l be discussed in more d e t a i l in section 3.3.3. 55 Samples are attached to the platform using a few milligrams of Cry-Con13, a high thermal conductivity grease. Sample mounting is facilitated by a small foam cushioned support which holds the platform while the sample is pressed onto i t . The specific heat of Cry-Con has been measured (Torikachvili et a l . 1983) and the grease heat capacity can therefore be subtracted from the data, as will be discussed in section 3.4.2. Samples can be removed by very carefully sliding them sideways off the platform, or by dissolving the grease in cyclohexane. 3.3.2 MEASUREMENT AND CONTROL OF THE REFERENCE BLOCK  TEMPERATURE The temperature of the reference block is measured by means of two encapsulated doped germanium resistance thermometers. These were purchased from Lakeshore Cryotronics1 *. Lakeshore also supplied calibrations covering the temperature range 1.4 to 100K. One thermometer is used in a feedback loop to control the block temperature, the other is used as a backup and a check on the stability of the calibration. The resistances of the thermometers are measured with a low 1 3 Air Products and Chemicals, Inc., Allentown, PA 1fl Westerville, Ohio, models GR-200B-1500 and GR-200B-1000 56 power automatic resistance bridge.1 5 This instrument makes a four terminal AC resistance measurement, using 30Hz excitation, phase sensitive detection, and a typical power dissipation in the thermometer of 1nW at 4K. The temperatures derived from the two thermometers agreed within about ±3mK below 10K and ±l0mK above. The differences were random, and correspond roughly to the accuracy with which the resistances were measured. Lakeshore's calibration of one of the thermometers was checked between 2K and 20K by S. Steel and W.N.Hardy. The standards used were "He and H2 vapour pressure and the susceptibility of the paramagnetic salt Gd2(SO,)3-8H20. The salt was used to interpolate the temperature scale between the regions where vapour pressure could be used. The Lakeshore calibration gave .temperatures which were consistently about 10 to 15mK higher than the vapour pressure-salt temperature. Unfortunately, the existence of temperature gradients of this magnitude between the vapour pressure c e l l and the Ge thermometer in the calibration cryostat could not be ruled out. The Lakeshore calibration was therefore used without any adjustments. The accuracy of the block thermometry is now summarized. Block temperatures were measured with a 15AVS-45 from R.V.Elektroniika, Finland, or similar. 57 p r e c i s i o n of ±3mK below 10K and about 10mK above 10K. The two thermometers agree with each o t h e r , i f the manufacturer's c a l i b r a t i o n s are used. Any systematic e r r o r s i n the temperature s c a l e are b e l i e v e d to be l e s s than about 15mK, because of the c a l i b r a t i o n check. Thermometry of t h i s accuracy i s more than adequate f o r t h i s experiment. As p r e v i o u s l y mentioned, one of the Ge r e s i s t o r s was used i n a feedback loop to c o n t r o l the block temperature. The automatic r e s i s t a n c e bridge has an a d j u s t a b l e i n t e r n a l r e f e r e n c e r e s i s t a n c e , and a voltage output which i s p r o p o r t i o n a l to the d i f f e r e n c e i n r e s i s t a n c e between the thermometer being measured and the r e f e r e n c e . Using t h i s as an e r r o r s i g n a l , i t i s a simple matter to r e g u l a t e the block temperature. The s t a b i l i t y of the block temperature during the time r e q u i r e d f o r a s p e c i f i c heat measurement was ±0.1mK or b e t t e r , which i s s u f f i c i e n t . 3.3.3 SAMPLE THERMOMETRY The f i r s t sample thermometer i n s t a l l e d was a d i f f e r e n t i a l thermocouple with one j u n c t i o n on the sample p l a t f o r m and the other on the re f e r e n c e b l o c k . The m a t e r i a l s were Au+.07%Fe f o r the block to p l a t f o r m l e g and chromel ( a l s o known as KP) for the o t h e r s . T h i s thermocouple p a i r was chosen because of i t s u n u s u a l l y high s e n s i t i v i t y at low temperatures, f o r example 13MV/K 58 at 4.2K and 9/zV/K at 2K (Sparks and Powell 1972, Rosenbaum 1968). The thermocouple voltage was measured with a Keithley 148 nanovoltmeter. The advantages of the thermocouple include its small size and low heat capacity, and that it directly measures the sample to block temperature difference 8. However, the thermocouple proved unsatisfactory for several reasons. F i r s t , the total noise was about ±.02MV at the nanovoltmeter input, corresponding to a temperature noise of ±2mK at 4K. This should be compared to a typical value of 80, the maximum temperature difference during a measurement, which is 50mK. (One tries to keep 0O<.O2T - see section 3.4) The noise situation is even worse at lower temperatures, since smaller 80s must be used and the sensitivity of the thermocouple decreases. Another problem was the lack of an absolute calibration of the thermocouple; thermocouples made from different samples of Au-Fe wire may have sensitivities which differ by 5% or more. Standard tables can only be used for these thermocouples if one is prepared to accept these possible systematic errors. Although the problems with the thermocouple could, in principle, have been solved by calibrating our batch of wire and signal averaging to reduce the noise, a simpler solution was to add a small resistance thermometer to the sample platform. This is a .5mm thick 59 slice cut from the centre of an Allen-Bradley 15J2, .1W carbon resistor. Its mass is lOmg. Small portions of the original resistor leads remain, and two .003 inch diameter brass wires were soldered to each of these to allow for a four terminal resistance measurement. Brass was chosen because its thermal conductivity is low enough that wires of reasonable mechanical strength can be used without creating too large a sample to block heat link. One side of the resistor slice was covered with a 6nm mylar sheet. This was/then glued to the sample platform, directly on top of the thin film heater. The mylar served to prevent electrical contact. The adhesive was Stycast 2850 epoxy. Since the response time of the automatic resistance bridge used for the block thermometers is too long to allow i t to follow the thermal decays of the sample, the sample resistor was measured with the AC 'bridge' circuit shown in figure 15. It is similar to one described by White (1979). In the diagram, A1 and A2 are OP-07 low noise operational amplifiers, Rg is the sample thermometer, and r represents the resistances of the leads (the lead resistances need not be equal). R^  is a resistance box, and the other resistors are 1% metal film types. The c i r c u i t makes a four terminal AC measurement. A1 is a non-inverting follower, which forces the voltage at point P to be equal to . A2 is a constant current 60 lOOkfl ' O S C + I5V (offset null) 100 kQ AAA Rc 100 kQ lOOkfl r •WH ref. lock-in F i g u r e 15: C i r c u i t diagram of the AC b r i d g e used to measure R . I t s o p e r a t i o n i s d e s c r i b e d i n the t e x t . 61 source which delivers an AC current i=V /lOOkfl through osc 3 Rg. The detector is a Princeton Applied Research HR-8 lock-in amplifier referenced to the oscillator which drives the c i r c u i t . The bridge is balanced by adjusting R^  to give zero signal at the lock-in. In this case we have iR =V R /l00kR=V1=V R. / ( lOOkG+R. ) s osc s' 1 osc b' b Rs=Rb/(1+Rb/I00kfi) (3 Since Rg<1400fl in the temperature range used, this effectively is Rs=£b- When the bridge is out of balance, the detector voltage is (R,-R )V /lOOkfi. This linear JJ *D VJ O w response is one advantage of using a bridge with active components. When using the bridge, the oscillator level must be kept low enough to avoid significant heating of the sample platform. The levels used were 800mV rms above 8K, 400mV from 4 to 8K, and 200mV below 4K. The power dissipation in the resistor i s , for example, 5nW at 4.2K. Considering that the thermal conductance between the sample platform and the block is about 6MW/K at this temperature, this is acceptably small. At 4.2K, dRg/dT= 108J2/K, so the temperature sensitivity of the bridge is 430MV/K. The lock-in is normally operated on a 20MV f u l l scale range. The frequency used was 394Hz. Quadrature signals were 5% of f u l l scale or less when the bridge output in phase with V was zeroed. Total noise at the lock-in input is 62 about ±1iriV peak-to-peak, although of course the lock-in is insensitive to most of t h i s . With a time constant of .1s, the noise at the lock-in output corresponds to ±0.2MV at its input, or ±0.5mK at 4.2K. This is much better than the thermocouple. Although the noise can be reduced further by increasing the lock-in time constant, this was not done because i t is also important to keep the time constant much shorter than the sample's thermal relaxation time. Using the calibrated germanium thermometers, it is easy to make an accurate calibration of the sample thermometer. With the sample heater off, the sample is at the same temperature as the reference block. During each experimental run, values of the sample thermometer resistance and the germanium thermometer temperature T were recorded. About 25 temperatures were used in a f i t to an equation of the form (White 1979) N lnT=Z P InR (3 n=l nThe f i t s were made using an orthogonal polynomial least squares method16. The program determines the best polynomial order N by increasing i t u n t i l l the improvement in the f i t is no longer s t a t i s t i c a l l y significant. N was always 5,6, or 7. Residual deviations 1 6 UBC subroutine DOLSF (Moore 1981) 63 between the data and the f i t were typically ±2mK. A new calibration had to be made for each run. The calibration changed each time the cryostat was warmed up to room temperature for sample replacement. Typical calibration changes corresponded to temperature errors of about 50mK. It is possible the. calibration changes are related to the procedure used to remove most of the samples. This involves immersion of the entire sample platform, including the resistor, in cyclohexane, which dissolves the grease holding the sample on. As an example of one of these calibrations, figure 16 shows the data and f i t obtained during the run on a Li 2 QN d S2 sample. The data are shown as small crosses. Note that, as might be expected, the f i t breaks down rapidly outside the temperature range covered by the calibration points. For this reason, the derivative dRg/dT of the f i t is not expected to be valid at the endpoint temperatures.. Since an accurate dRg/dT is required for specific heat data analysis, specific heat data were not taken at the endpoints. The other calibration points are, for the most part, the same temperatures at which the specific heat was measured. 3.3.4 COMPUTER DATA AQUISITION SYSTEM The thermal decays used to determine the heat capacity are recorded using a microcomputer, which also controls the sample heater. The computer is based on a Figure 16: Sample thermometer resistance as a function of temperature. The line is a f i t using equation (3-9). 65 Z80 microprocessor. It was assembled by the Physics Dept. Electronics Shop, using, for the most part, commercially available S100 bus circuit boards. Two 8-inch floppy disk drives and 64 kbytes of memory are used. A department standard interface system (PHYS44), designed and built by the Physics Electronics Shop, is used for data aquisition and control of the experiment. A number of input and output devices are available in the PHYS44 system. Those used for the specific heat measurements are 8, 12, and 16 bit analog to digital converters, and a computer controlled relay used to switch the sample heater on and off. The 16 bit converter is a dual slope device1 7 with a conversion time of about one second. This was used when the sample's relaxation time was greater than about 30s. For typical samples, this occurred at temperatures above about 8K. At lower temperatures, the relaxation time is shorter, and a shorter conversion time is required in order to get a sufficient number of data points during a thermal decay. For the f i r s t few runs of the system, the only fast A/D converter available was an 8 bit device1 8. This did not have sufficient resolution, and was replaced by 1 7 Intersil ICL 8068/ICL 7104 1 8National ADC 0801 66 a 12 b i t s u c c e s s i v e approximation c o n v e r t e r 1 9 . A m u l t i p l e x e r i n the PHYS44 system a l l o w s up to 8 d i f f e r e n t analog v o l t a g e inputs to be connected to e i t h e r of the A/D c o n v e r t e r s . Inputs from the s p e c i f i c heat experiment were the sample heater v o l t a g e , and the recorder outputs of two automatic r e s i s t a n c e b ridges (Ge thermometers), the nanovoltmeter (sample thermocouple), and the l o c k - i n a m p l i f i e r i n the sample r e s i s t a n c e thermometer b r i d g e . 3.4 MEASUREMENT PROCEDURE AND DATA ANALYSIS 3.4.1 MEASUREMENT PROCEDURE The general p r i n c i p l e s of the r e l a x a t i o n time method have been d i s c u s s e d i n s e c t i o n 3.1. The experimental procedure used w i l l now be d e s c r i b e d i n more d e t a i l . The measurement c y c l e i s shown i n f i g u r e 17. I t begins with the r e f e r e n c e block and sample at a common temperature T set by the block temperature r e g u l a t o r . The c a l i b r a t e d r e s i s t a n c e thermometers on the block are read at t h i s time, and the AC br i d g e measuring R g i s balanced. The microcomputer then switches on the sample heater. The sample heater c u r r e n t comes from a re g u l a t e d DC power supply and passes through a r e l a y c o n t r o l l e d by the computer. The v o l t a g e a c r o s s the 1 9 A n a l o g Devices AD572 67 Figure 17: Specific heat measurement cycle. The major steps in the cycle are: a- Balance sample thermometer bridge b- Sample heater on c- Balance sample thermometer bridge to measure 6 0 -d- Reset bridge. e- Sample heater off f- Computer monitors the thermal decay. 68 sample heater is read using a different pair of leads and a Keithley 177 digital multimeter. After the heater is turned on, the temperature difference 9 between the sample and block rises, eventually reaching a maximum value 0O whick is typically about 50mK. This causes the R bridge to go out of balance. The R bridge is nulled again, providing a measurement of 6Q which is independent of the gain of the bridge c i c u i t . The resistance box is then set back to its original value. The next step is to prepare the microcomputer to collect data. While the heater is s t i l l on, the experimenter gives it a sampling rate and a sampling duration. The duration of the measurement was usually chosen to be about four times the expected thermal relaxation time T. The sampling rate was then chosen to give 90 to 100 samples during the measurement. The operator also chooses which analog to dig i t a l converter is to be used. In early runs, the 8 bit converter was used for sampling rates greater than 1 sample/s, otherwise the the 16 bit converter was used. After the 12 bit converter was installed, i t was used in place of the 8 bit device. Once a l l this information has been entered, the microcomputer reads the sample thermometer voltage, which at this time corresponds to t?0. It then shuts off the sample heater by opening the relay, and repeats the thermometer reading at the .preset sampling rate. The 69 timing is done with a quartz oscillator counter/timer which is part of the microcomputer system. At the end of the measurement, the microcomputer stores the values of sample thermometer voltage and time on a floppy disk for later analysis. If desired, several thermal decays can be made at the same block temperature T, and the results averaged to reduce the noise. Signal averaging was usually performed when the thermocouple sample thermometer was being used, but was not normally necessary when using the carbon resistor sample thermometer. The thermal relaxation time r is a function of temperature. It is therefore important to have 80 small enough that r does not change significantly during the thermal decay. Sample heater powers were chosen so that 0O/T<O.O2, which ensures r is constant to about 2% or better for typical samples. To get the specific heat as a function of temperature, the thermal relaxation measurements are repeated for different block temperatures. Typically about 25 temperatures covering the range 2.6 to 25K were used. Data analysis is done on the UBC Computing Centre's Amdahl V/8, since the calculation speed of the microcomputer is too low. The data are transferred to the Computing Centre over a high speed data l i n e . 70 3.4.2 COMPUTATION OF THE SPECIFIC HEAT The analysis begins with the data in the Amdahl in the form tN, V ( tn) (n = 1 to N), where V(tn) is the sample thermometer voltage at time tn, N is the number of samples, and, since the sampling was carried out at a fixed rate 1/ ts, tn= n ts« Since the maximum temperature difference 80 is small, V is proportional to 8 even for the resistor sample thermometer, and V i t s e l f can be f i t to an exponential decay without converting each voltage to a temperature. So, V is f i t to an equation of the form V ( t ) = VI E"T / T ,+ V (3 os where V" is a possible small voltage offset. (V=V os c 3 os when 69=0.) V,,!,, and Vq s are a l l parameters determined by the f i t . The f i t t i n g i s done by minimizing the reduced chi-squared parameter x2 = ? (V(t )-(Vi e-t / T l+V ) ) 2 (3-r (N-3)6V2 n=l n 1 O S where N-3 is the number of degrees of freedom, and 5V is the standard deviation of each of the V(t ), or, -n equivalently, the noise at the sample thermometer voltage output. To illustrate the data f i t t i n g and analysis, the data on a Li 3gNbS2 sample at T=4.60K will be used (figure 18). The resistor sample thermometer was used for this measurement. The noise at the lock-in amplifier 71 Figure 18: Data for a typical thermal decay. The figure shows the voltage at the lock-in output as a function of time, as recorded by the microcomputer. The line is a f i t to equation (3-10). The arrow marks the beginning of the data used in the f i t . 72 output was about ±.002V, so for the f i t 6V was taken to be .002V. It should be pointed out that the parameter values which minimize xr2 are independent of 8V, so that only a rough estimate is needed. When a l l of the data are used in the f i t , the best f i t value of xr2 is 17.2. This is unacceptably high, since for a good f i t we expect xr2-1. The reason for the large value is clear on examination of figure 18; the thermal decay was not truly a single exponential. This happened because the sample's thermal conductivity was low enough that i t was not isothermal during the decay. A solution of the heat flow problem for this case (Appendix 2) shows that the temperature at the sample platform decays according to the infinite sum of exponentials 0(t) = I e „ e ~t / Tn (3-12) where n=l (3-13) 2k T 6 i = w I  0 0 m K c o t y i l + tan y1l ) ( k w x 1 - Cpl) + kwx 1 + Cp]_ and C s = ( k w T i " C p l ) P l 1 ' t a n ( 3 _ 1 4 ) Here Cg and C ^ are the heat capacities of the sample and sample platform, respectively, kw is the platform to block thermal conductance, and /u,l is an eigenvalue defined in appendix 2. One finds that Tn« r , for n>1, so 73 that after dropping the f i r s t few data points the rest of the data f i t a single exponential very well, allowing us to determine 8, and r, from the f i t . In the example, the f i r s t eight points are dropped, resulting in a best f i t with xr2=.83. xr2 does not ' decrease significantly if more points are dropped. The f i t determines values of T , , V, (the voltage corresponding to 0, ) , and Vq s. In the example, these are 21.36s, 1.588V, and .005V, respectively. The ratio 8y/80 is given by ei/0o=V,/(V(t=O)-Vos)=1.588/(1.771-.005)=.899 To determine Cg, we also need kw. With the sample heater off, the thermometer .resistance Rg was 285.39J2, corresponding to T=4.5981K using the fi t t i n g function (3-9). At the maximum sample temperature, R was s 281.08O, corresponding to T+0O = 4. 651 2K, or c90 = 53. 1mK. The sample heater power P was .371uW, giving kw=P/0o=6.98MW/K. The sample platform heat capacity had been measured previously (see section 4.2), and at this temperature was 29.6MJ/K. TO this should be added the heat capacity of the 6mg of Cry-con grease used to mount the sample, (the grease is considered part of the platform rather than part of the sample.) Torikachvili et a l . (1983), have measured the specific heat of Cry-con, and give a polynomial f i t to their data which is valid between .56 and 20K. Using this f i t , the heat capacity of 6mg of 74 grease at 4.60K is 4.93MJ/K (±7%), which when added to the heat capacity of the bare platform gives a total platform heat capacity of 34.5MJ/K. The values of 9\/dQ, r,, kg, and C ^  are now inserted into equation (3-13) which is solved numerically to get M,1=.609. This value is inserted into (3-14) to get Cs= ( kwr 1 "Cp l * M 1 1/t a n*i 11 = •1 00mJ/K. Since the sample mass was 117mg and the molecular weight of NbS2 is 157.1g/mole, the molar specific heat c is . 136J/mole-K. Note that if the non-exponential nature of the decay is not taken into account, we would use simply W ^ p l ( 3"1 5 ) which would result in a 13% error. (In some of the other samples the effect is larger; the worst case was the L i ^ b S j sample at 2.73K, which had M , 1/tanju , 1= . 1 5 . ) The value of Mil is related to the thermal conductivity Kg of the sample. From (A2-10), U i D ^ C g l / K g A r , (3-16) where 1 is the thickness of the sample and A its cross-sectional area. Taking approximate values l=3mm and A=7r(3mm)2 gives Ks=.029W/m-K for the Li 30NbS2 sample at 4.6K. For comparison, commercial copper has a thermal conductivity of about 500W/m-K at the same 75 temperature, and nylon has about .OlW/m-K (White 1979). Since the sample is a pressed powder, the value for Kg is not characteristic of undivided Li 2 QN b s2 f which should have a considerably higher thermal conductivity. 3.4.3 ACCURACY OF THE SPECIFIC HEAT Most of the specific heat data is estimated to be accurate to a few percent. In cases where the 8 bit analog to digital converter and thermocouple sample thermometer were used, or where the sample thermal conductivity was unusually low (Li,NbS2), the accuracy is poorer. These cases are pointed out in section 4.4 where the data are presented. In this section, an error estimate of a typical measurement is carried out, in order to show the factors which limit the accuracy of the measurement. The example used is Li 3gN b^ 2 a t 4.60K, as in the previous section. Errors which contribute to the scatter in a plot of c as a function of T for a particular sample will be called random errors and are treated f i r s t . Errors which do not contribute to the scatter in a c(T) plot will be called systematic errors. Examples of systematic errors are errors in the temperature scale or the mass of a sample. The reason for making the distinction between random and systematic errors in this way is to allow an understanding of the scatter in the c(T) plots. This understanding aids in distinguishing noise from genuine 76 fine structure in the specific heat. The final step in data analysis is the calulation of C starting from k , C T , , V,, and V . The s w pi 1 1 os uncertainties in T , , V,, and V „ are correlated, since os a l l these are determined from the same f i t . The uncertainty in VQ g is so small that i t makes an insignificant contribution to the uncertainty in the final result. Using this fact, and treating only random errors at this point, we have '_S-j 2_|_ SR^ ^1S\2_|_ f x\T S\ 2 2 dcs 3c 3c («0 - (6k — S ) 2 + ( S T — S ) Z + ( 6 VX—f a) S W3 kw 3T l 3VX 6c 6c + 2 aT v (6TX — S) ( 6 V ! —S) T i V i (3-17) 6TI 6V: where 6 in front of a quantity indicates the uncertainty in that quantity, and v is the correlation coefficient between 6T, and 5V,. The correlation coefficient is determined by the f i t t i n g program. The uncertainties S T , and can also be determined from the f i t , but not without some ambiguity. According to standard s t a t i s t i c a l theory (Bevington 1969, for example), if the errors in each data point are random, then the uncertainty in a parameter determined from a non-linear least squares f i t is the amount by which that parameter must be changed in order to increase x2 by one from it s minimum (best f i t ) value. As the parameter in question is varied, x2 must remain minimized with respect to a l l the other parameters. Here 77 (cf. 3 -11) , X2 = TW2 * ^ ( ^ - ( V ^ ^ - V o s ) )2 = (N-3)X r2 (3-18) V • n=l If the errors in the different samples v( tn) are correlated, for example i f there is a significant amount of low frequency noise in the data, then the criterion above can lead to parameter errors that are too small. Roughly speaking, some of the 'averaging out' which one has expected has not taken place. Some low frequency noise is present in the thermal relaxation data, and i t is therefore safer to determine the uncertainties in the fitted parameters using a different criterion. This is that the uncertainty in a parameter is the amount by which i t has to be changed in order to double x2. When this is done for our example, the results are 6 T , = . 4 S , 6 V 1 = . 0 2 V , and 6 V Q S = . 0 0 5 V . A comparison of the last two of these justifies the earlier statement that 6 Vq s is not important. The correlation coefficient between 7 , and V , is - . 8 0 . Percentage errors are 2% in T, and 1% in V , . The thermal conductance of the wire is given by kw=P/0O. P is known with an uncertainty of 0 . 1 % , which is insignificant. 80 is determined from two resistance values, say R, and R2r both of which are determined with a precision of 0 . 0 1 Q , . Assuming no significant error in the calibration, the result for the example is 78 6kw/kw=6t90/c90 = 6(R1-R2 )/(R1-R2)=v/2( .01fi)/4.31fi=0.3% The various terms in (3-17) can be evaluated using finite differences, giving (recall Cs=.l00mJ/K) 5kw(9Cs/3kw)=+.0003mJ/K 5 r , O C /3T,)=+.0023mJ/K 8V,OC /3V,)=+.00l3mJ/K Combining these using (3-17) gives 6CS=.0015mJ/K or a percentage error in Cg of 1.4%. Total random errors of about this size (1 or 2 percent) are typical, and are consistent with the scatter in the c(T) plots for most of the samples. In most cases, the largest contribution to the uncertainty in the specific heat comes from 6T, and 6V,. Measurements made with the thermocouple sample thermometer have a larger uncertainty, because of the noisier temperature signal. It is also necessary to consider systematic errors in the results. As discussed in section 3.3.2, systematic errors in the temperature scale derived from the germanium thermometer are believed to be 15mK or less and are insignificant Since the resistor sample thermometer is calibrated against the germanium thermometer, it too should contribute l i t t l e error (section 3.3.3). The thermocouple sample thermometer will be assumed to measure temperature differences with an accuracy of ±5%. The sample masses were measured with 79 an accuracy of ±1mg or about ±1%. As discussed in section 4.2, the sample platform heat capacity Cp-^  was measured with an accuracy of about ±3%. It is typically about 20% of the total heat capacity. Since the effect of the data analysis is (approximately) to subtract Cp-^  from the total to get the sample heat capacity C , we can take dCs/dCp^=-1. This implies that the uncertainty in C i contributes about ±1% to the error in C . The pi s grease heat capacity is known with an accuracy of about ±15%, due mostly to uncertainties in the grease mass. Since the grease heat capacity is typically only about 4% of the total, this is insignificant. Combining the errors mentioned so far, yields a total possible systematic error of about ±2% or less (roughly ±6% with the thermocouple). Another possible source of error is residual gas in the vacuum can. An upper limit on the magnitude of this effect will now be estimated. During measurements, the can was pumped continually with a diffusion pump. The pressure measured at room temperature with an ionization gauge near the pump was about 2' 10"7 torr in most cases, which is the lowest pressure obtainable with the pumping system used. In some of the experimental runs, however a small superleak was present somewhere in the vacuum can. No leak could be detected as long as the temperature of the helium bath was above its superfluid transition. When the bath 80 was cooled below the transition, the pressure inside the cryostat rose. By comparing two heat capacity measurements made at the same reference block temperature, one with the superleak present and the other with the bath above the superfluid transition, we can determine how sensitive Cg is to the residual gas pressure. Such a pair of measurements was made on the L iQ 16NbS2 sample at a block temperature of 2.73K. The f i r s t was made with the bath at 2.3K and not superfluid, the second with a superfluid bath at 1.4K. The vacuum system pressures at the ionization gauge were 1.7-10~7 and 4.2•10"7 torr, respectively. Thermal relaxation measurements gave a sample to block thermal conductance k =3.89yW/K and heat capacity C =47.6uJ/K with the w S normal bath, and k =6.80MJ/K and C =52.0MJ/K with the w s superfluid bath. The large difference in thermal conductance; is, due to conduction of heat through the gas. At these low pressures, the thermal conduction through the gas is proportional to the pressure in the cryostat, which in equilibrium is proportional to the pressure at the gauge20 (White 1979, p130). The relative difference in C between the two s measurements is much smaller than the difference in k . w 2 o The pressures in the gauge and the cryostat are not equal even in equilibrium, because of thermomolecular effects. 81 This is because of the way the measured thermal conductance is used in the calculation of the specific heat (section 3.4.2). The only reason that C is s different for the two measurements is that the residual gas distorts the heat flow pattern in the sample. The equations used for data analysis assume that a l l the heat leaves the sample through the platform. When there is a significant amount of gas around the sample there is also heat loss through its free face and sides. Most data were taken with the bath above the superfluid point, and the error due to residual gas should be small. We can roughly estimate the size of this error by assuming measured Cg values deviate from the true value by an amount which is proportional to the pressure at the ionization gauge. The error to be obtained from this assumption is really an upper li m i t , since high pressures mean there is flow out of the cryostat. From the two measurements described above, the proportionality constant can be obtained, and is dCs/dP=l8 J/K-torr. At the lower pressure of 1.7«l0~7torr this gives a deviation from the true Cg of 3juJ/K or 6%. Measurements at higher temperatures are affected less by residual gas, since both kw and the sample thermal conductivity increase with increasing temperature. The Li 1gNbS2 sample discussed above had a fairly typical thermal conductivity and size, so the 82 effect residual gas had on it should be typical of most samples. In summary, then, systematic errors due to causes other than residual gas are believed to be about ±2% for typical measurements (±6% with the thermocouple sample thermometer). The error due to residual gas is d i f f i c u l t to estimate accurately, but is believed to be about +6% or less near 3K for typical samples. The effect of residual gas should decrease rapidly with increasing temperature. 3.4.4 DETERMINATION OF THE LINEAR AND CUBIC TERMS IN THE  SPECIFIC HEAT Specific heat data are often presented in the form of plots of c/T against T2. This- is because the specific heat of a normal metal at sufficiently low temperatures has the form (see equation 1-4) c = yr+0T3 where 7 and 0 are constants. The linear and cubic terms are due to electrons and phonons, respectively. When c/T is plotted as a function of T2, data satisfying this equation l i e on a straight line with slope /3 and intercept 7 . Such a plot for the Li 2QNhS2 sample between 1.8 and 10K is shown in figure 19. The data l i e on a straight line for temperatures between 2.8 and 6K. Above 6K, the phonon specific heat begins to deviate from 83 Figure 19: Specific heat of L i 3QNbS2 as a function of temperature. The" line is the f i t described in the text. 84 cubic behavior. (This is to be expected - see section 6.5.) To reach temperatures below 2.8K, the bath temperature had to be taken below the superfluid point. The small superleak in the vacuum can caused a high helium gas pressure in the can and a large positive systematic error in the specific heat (see the previous section). Because of this, the deviation from the line of the points between 1.8 and 2.8K is not significant. Error bars are shown on some of the points. These are the results of error calculations similar to those in the preceding section, and include random errors only. The scatter of the points around the line is consistent with the calculated random error. The values of the constants 7 and 0 were determined using a standard linear least squares f i t to the straight line region between 2.8 and 6K. The results are 7=9.4±.6mJ/mole-K2 and 0=.96±.03mJ/mole-K4. The errors quoted here are due to the random errors in the data only. In addition, we saw in the previous section that there was a possible residual gas effect of 6% or less near 3K, as well as possible systematic errors due to other causes of about 2%. Since the effect of residual gas should decrease rapidly with temperature, the maximum possible effect of residual gas on 7 and /3 can be estimated by assuming +6% error at 3K and no error at 6K. Deviations in 7 and p° due to this can be easily estimated graphically, and when these are combined with 85 the other errors the error bounds are 7=9. 4(+!M) mJ/mole-K2' _ 1 . D / 3 = . 9 6 ( * , Q J ) mJ/mole-K" Analyses similar to this were performed on a l l the Li NbS2 samples. 4. RESULTS OF THE SPECIFIC HEAT MEASUREMENTS 4.1 INTRODUCTION In this chapter, the results of the low temperature specific heat measurements are presented. Section 4.2 describes a measurement of the heat capacity of the sample platform. Section 4.3 contains results on NbS2, and section 4.4 on LixNbS2 samples. A table at the end of the chapter summarizes the data. The interpretation of these results is discussed in chapters 5, 6, and 7. 4.2 SAMPLE PLATFORM HEAT CAPACITY In order to extract the heat capacity of.a sample from the raw data, the heat capacity of the sample platform must be known. This was measured as a function of temperature on two occasions, once before the carbon resistor sample thermometer was installed, and again after. Since most of the measurements on Li NbS2 were performed with the carbon A. resistor in place, the sample platform results with the resistor will be presented f i r s t and in more d e t a i l . The measurements were made as described in section 3.4.1. Recall that in the case where a sample was present, the thermal decays could not be f i t with a single exponential (section 3.4.2). With no sample attached, the decays were single exponentials and the heat capacity C ^ could be calculated directly from Cp l = Tkw ( 4" ° 86 87 where T is the relaxation time and k is the thermal w conductance of the wires (see equation 3-7). The results are shown in figure 20 in the form of C ^/T as a function of T2. To avoid introducing extraneous scatter into the LixNbS2 data, the sample platform heat capacity data were smoothed. The data below 5.5K were f i t to an equation of the form Cpl=aT+bT3 (4-2) using the linear least squares method. Since normal solids are expected to obey an equation of this form at sufficiently low temperatures, (4-2) is the best equation to use for the extrapolation of the platform heat capacity to temperatures below 2.74K, the lowest temperature at which i t was measured. The f i t parameters are a=.42357MJ/K2 and b=.28365MJ/K4. At higher temperatures, the data deviate from (4-2), and so the smoothing was done using a least squares f i t to a fourth order polynomial of the form 4 C_, = aT + bT3 + I pn(T-5.5K)n (4-3) p i n=2 n In (4-3), a and b have the same values as in (4-2). This ensures that at 5.5K, where the transition from (4-2) to (4-3) is made, both Cp^(T) and its derivative are continuous. The parameters pn are determined from a f i t to a l l data between 5.5 and 20K, and are p2=-.31569MJ/K3, p3=1.1635nJ/Kft, and p„=-.13494nJ/K5. When the platform heat 88 Figure 20: Sample platform heat capacity. Shown is the heat capacity of the sample platform with the resistance thermometer (x), and without i t ( A ) . The line is the f i t described in the text. 89 capacity is needed for analysis of data on LixNbS2 samples, (4-2) is used below 5.5K and (4-3) from 5.5 to 20K. The data deviate from the smooth curve of (4-2) and (4-3) by ±3% or less in a l l but 2 cases, and this is believed to be approximately the accuracy with which the smoothed curve represents the actual platform heat capacity. The possible systematic error due to residual gas, discussed in section 3.4.3, should not be present in this measurement, because the thermal conductivities of the platform components are much higher than those of the samples. For the f i r s t several runs of the specific heat cryostat, the carbon resistor was not in place. The heat capacity of the sample platform in this condition was also measured and, is shown as the triangular points in figure 20. Because of the poor accuracy of both Cp^ and Cg in these early runs where the thermocouple sample thermometer and 8-bit D/A converter were used, i t was not considered worthwhile to smooth the data. Instead, C n values for data p i analysis were obtained simply by linear interpolation in Cp^ against T2 between the measured points. Below 4.4K, the straight line extrapolation shown in figure 20 was used. 4.3 THE SPECIFIC HEAT OF NbS2 Figure 21 shows the specific heat of a sample of NbS2 from batch DD9. The most significant feature is the specific heat anomaly at the superconducting transition temperature T . The transition takes place between 5.5 and 6.OK, and is 50% Figure 21: a- The specific heat of NbS2. b- Detail below 7.1 K. The line is the f i t to the normal state specific heat described in the text. 91 complete at 5.7K. In the normal state, at low enough temperatures, the specific heat is expected to s a t i s f y2 1 cN=7T+/3T3 (4-4) It is d i f f i c u l t to assign accurate values to 7 and 0 using the normal state data alone, because of the very limited temperature range (about 6 to 8K) over which (4-4) is valid. It is possible, however, to derive an additional constraint on the f i t from the superconducting data. Because the superconducting transition is second order, the entropy in the superconducting and normal states must be equal at T . Thus, T T c c J(cs/T)dT = J(cN/T)dT (4-5) 0 0 By substituting (4-4) in (4-5), we get T c /(cs/T)dT = 7Tc+(1/3)0TC3 (4-6) 0 By evaluating the left-hand side of (4-5) a constraint on 7 and j3 can be obtained (Schwall, et al 1 976). To do this, i t is necessary to extrapolate the superconducting state data to T=0. Since the electronic specific heat of a 21 This equation is essentially (1-4). In this section cN is used for the normal state specific heat, Cg for the superconducting state specific heat, and c for the measured specific heat. 92 superconductor is appoximately exponential well below T , (Tinkham 1975, p8), the extrapolation is done by assuming c=cs=Ae"b/T+/3T3 (4-7) at temperatures below the range covered by the data. The constants A and b in (4-7) were determined by plotting the measured specific heat c in the form of ln(c-/3T3) against T. On such plots, the four data points below 3.2K l i e on a reasonably straight l i n e . For several values of 0 between .3 and .4mj/mole-K", plots were made, and A and b were determined. The lowest temperature at which data was taken was 2.645K. The integral in (4-6) was therefore split into two parts, covering the temperature ranges 0 to 2.645K and 2.645K to T , respectively. The low temperature part is determined from the extrapolation (4-7) and is described by 2.645K / (cg/T)dT = 14.0mJ/mole-K + (1.5K3)|3 (4-8) 0 The weak dependence on /3 comes about because the values of A and b determined from the plot depend on the value of /3 used for the plot. The other part of the integral was determined by numerical integration of the data. Because of the broad transition, i t was decided to take T to be 5.7K (the c midpoint of the transition), but to include in the integral 93 the excess specific heat between 5.7 and 6.OK. That i s , T 6.OK 6.OK / (cs/T)dT = / (c/T)dT - / ((TT+6T3)/T)dT (4-9) 2.645K 2.645K 5.7K When this is calculated and added to the low temperature part of the integral, the result is Tc J (cQ/T)dT = (l37±5)mJ/mole-K - (,3K)7 + (8.8K3)0 (4-10) 0 b Using (4-10) and Tc=5.7K in (4-6) gives (6.0K)7+(70.5K3)/3 = (1 37±5 )mJ/mole-K (4-11) Requiring y and 0 to satisfy (4-11) and to f i t the data between 6 and 7K yields 7=19.3±1.5 mJ/mole-K2 /3 = 0 . 3 1 ± . 04 mJ/mole-Kft. These values are in agreement with those of Aoki, et al (1983), which are y-18.2mJ/mole-K2 and 0=0.33mJ/mole-K". Another quantity of interest is the size of the specific heat jump Ac at Tc. For NbS2, the ratio bc/yTc is 1.3, compared to 2.14 in NbSe2 (Schwall, et al 1976), and 1.43 in the BCS theory (Tinkham 1975, p36). It is also possible to use the data to calculate the thermodynamic c r i t i c a l f i e l d H (0) from c Tc JC(cs-cN)dT = (V/8TT) Hc2(0) (cgs units) (4-12) 0 94 (Zemansky 1957) where V is the molar volume. Evaluating the integral using the exponential extrapolation (4-7) gives H (0)=1.0kG, compared with 1.28kG in NbSe2 (Schwall, et al 1976). 4.4 THE SPECIFIC HEAT OF Li NbS, x 2-This section presents specific heat data for eleven Li NbS2 samples. The data are given in order'of increasing x. Unless stated otherwise, the data were measured using the resistor sample thermometer, 12 bit A/D converter, and the method of section 3.4. The specific heat of each sample is plotted as c/T as a function of T2 in figures 22 to 33. The error bars on some of the. data represent error calculations similar to those of section 3.4.3, and include random errors only. The coefficients of the linear ( 7 ) and cubic (j3) terms in (1-4) were determined by least squares f i t s as described in section 3.4.4. The results are summarized at the end of the section in table II. Comments on the individual samples. x=. 1 3 This was a stage 2 sample. There is a slightly larger than usual uncertainty in the value of x for this sample (X=.13±.01). This is because of a problem during the discharge of the c e l l . For a time of about 1 day near the end of the discharge, the coulometer was not connected and the c e l l current was not integrated. The x 95 (a) 100 200 T2 (K2) 300 400 ( b ) 50 i 0 10 20 30 40 50 T2 (K2) Figure 22: a- Specific heat of L i 13NbS2. b- Detail below 7.1 X. The line is the f i t described in the text. 96 value was obtained, with'the accuracy stated above, by estimating the lost charge. This same x value is obtained in other c e l l s discharged to the same voltage. The specific heat (figure 22) was measured using the thermocouple sample thermometer and 8 bit A/D, which is the reason for the large scatter in the data. The least squares f i t to determine y and /3 uses the data between 2.67 and 7K. x=. 16 This sample was also stage 2. Note the large slope of the c/T vs T2 plot, and the relatively low temperature (=*7K) at which the specific heat begins to deviate from its cubic behavior (figure 23). The small superleak mentioned in section 3.4.3 was present in this run. The four points at lowest temperature were taken with the bath superfluid, and therefore with a high residual gas pressure in the cryostat. Because of this, they contain a large positive systematic error> and their- deviation from the line is not considered significant. They were not used in the f i t made to determine y and j3, which extends from 2.73 to 7K. x=. 23 This and a l l remaining samples are stage 1. The specific heat jump at 3.1K is due to a superconducting transition (figure 24). A f i t from 3.2 to 7K was used to determine 7 and j3. CM I V o E \ E 240 200 h 160 h 120 h (a) 50 100 150 200 250 T 2 ( K 2 ) 300 CM I o E \ —> E o 20 h Figure 23 a- Specific heat of L i ^gNb^ b- Detail below 7.1 I o E o 10 h Figure 24: T< (K 2) a- Specific heat of L i b- Detail below 7.1 K. A superconducting transition is seen at 3.1 K .23NbS2 99 x= .25 This sample was measured using the thermocouple and 8 bit converter, and the data is therefore of rather poor quality (figure 25). There is a superconducting transition at 3.1K. That the specific heat anomaly is truly due to superconductivity was confirmed by a magnetic susceptibility measurement to be described in chapter 7. This was performed on a small piece of the specific heat sample, which was broken off (in the glovebox) after the specific heat measurement. x=.30 Here again the 4 lowest temperature points were taken with the superleak present. The apparent rise in specific heat below 2.8K (figure 26) is not significant. 7 and 0 were determined from a f i t between 2.8 and 6K. This sample has an unusually high 0 value. The discharge of the c e l l used to prepare this sample was unusual. It was held at 2.730V (in the stage 2 voltage range) for 3 days, then the voltage was lowered to 2.670V and fi n a l equilibration took place. x= .32 This was intended as a repeat of the x=.30 run. The x=.30 and .32 samples were both prepared by discharging cells to the same voltage (2.670V). The reason for the 8% difference in x is not completely clear. It is possible that i t is related to the unusual mode of discharge of the x=.30 c e l l . If part of the cathode 100 CM I o E 240 200 h 160 h 120 h 80 h 40 200 (o) 400 600 T 2 (K 2) 800 1000 1 02 material became disconnected while the c e l l was at 2.730V, i t would have been 'left behind' in the stage 2 phase, resulting in the lower overall x value. X-ray diffraction measurements were performed on both the x=.30 and .32 samples, however, and neither one shows any stage 2 Li NbS2, or NbS2 Bragg peaks. If the x=.30 sample contains more than a few percent of stage 2, it must be either highly disordered or in the form of very fine particles (=100A or less). Another possible reason for the different x values is the difference in electrochemical behavior in this region between 'fresh and 'aged' NbS2 (section 2.3). The specific heat results for the x=.32 sample are shown in figure 27. A f i t from 2.8 to 7K was used to determine 7 and /3.. x=. 35 This sample was prepared and measured by D.Li using the author's apparatus. Good specific heat data are available only between 2.8 and 7K (figure 28). A l l of the data were f i t to determine 7 and 0. x=.41 This was the f i r s t sample measured using the resistor sample thermometer. Rather than the 4 terminal bridge of section 3.3.3, a simple 2 terminal AC Wheatstone bridge was used to measure the thermometer resistance. The detector was a PAR 129 lockin amplifier. This arrangement was noisier than the 4 terminal bridge used (a) 100 50 100 150 200 250 300 T 2 ( K 2 ) CM I o E 40 30 \ 20 h o 10 h 10 (b) 20 30 T 2 ( K 2 ) 40 Figure 27: a- Specific heat of L i „„NbS b- Detail below 7.1 K. 32r""2 50 Figure 28: Specific heat of L i .,NbS 1 05 in later measurements, and this is reflected in the scatter in the data (figure 29). A f i t from 2.6 to 7.5K was used to determine 7 and )3. .50 The specific heat data are shown in figure 30. A f i t between 2.8 and 8K was used to determine 7 and $. .68 These data were taken using the thermocouple sample thermometer and 8 bit converter (figure 31). 7 and 0 were found using a f i t between 2.8 and 8.2K. 1 .00 The results are shown in figure 32. The interpretation of these data is complicated by the fact that the sample had a very low thermal conductivity, which resulted in large systematic errors below about 6K. The low conductivity caused the thermal decays to be highly non-exponential, as shown in figure 33 for the 2.73K measurement. As discussed in section 3.4.2, the difference e?(t) between the sample and reference block temperatures should decay according to 6(t) = Z 6 e_ t / Tn (4-13) n=l n Values of T, and 0,/0o determined from a f i t to the data are used to calculate the specific heat. Since T 1<<T1, r n> 1 simply dropping the f i r s t few data points is sufficient to isolate the n=1 term in (3-12). For this measurement, however, 0,/0o=.13, and by the time the n=1 term is CM I o E o 20 10 10 20 30 40 Figure 29: a-T2 (K2) Specific heat of L i ^NbS, b- Detail below 7.1 K. 107 (b) 20 , 0 10 20 30 40 50 T 2 ( K 2 ) Figure 31: a- Specific heat of L i 6gNbS2. b- Detail below 7.1 K. 109 Figure 3 2 : The specific heat of L i ^ Q Q J ^ ^ . The solid line is the result of a least squares f i t between 7 and 10 K. The dashed line shows that the data are also consistent with y=0 (see text). A large, positive systematic error is present at the lower temperatures, due to the unusually low thermal conductivity of this sample. Figure 33: Thermal decay of the L i , nnNbS0 sample at 2.73K (see text). 1 > U 0 2 The maximum thermometer signal of 1.8V corresponds to a temperature difference 90=19mK. 111 isolated the signal level is only about 5 times the noise. (Measurements on a l l other samples had . 5^0,/0O< 1 . ) As a result, the uncertainties in the fitted parameters 0, and r, are unusually large for this sample, resulting in a large random error in the specific heat (±40%). In addition, a low thermal conductivity sample will enhance the systematic error due to residual gas. Because of this, the apparent upturn in the specific heat below 6K is not believed to be significant. At 7K and above, the sample's thermal conductivity was high enough that accurate specific heat measurements could be made. (For example, at 8K, 0i/0o=.9) The coefficients 7 and /3as determined from a f i t to the data between 7 and 10K, are 1.3mJ/mole~K2 and .18mJ/mole-K", respectively. This f i t is shown as the solid line in figure 32. The data are also consistent with 7 = 0 , and 0=.19mJ/mole-K* (the dashed li n e ) . The rigid band charge transfer model (section 1.3) predicts 7=0 for this sample, since the dz2 band should be f u l l , and the sample should be a semiconductor. The low, strongly temperature dependent thermal conductivity is also consistent with a semiconducting sample. No superconducting transition or other specific heat anomalies were observed in this sample, although because of the poor quality of the data below 6K, i t would have been d i f f i c u l t to observe such an anomaly 1 12 even if it did occur. There is no reason to expect Li,NbS2 to be a superconductor. If more accurate measurements of the specific heat of Li,NbS2 are ever required, they could perhaps be made by mixing the powder sample with a known amount of a thermal contact agent such as vacuum grease. Because of the high reactivity of the samples, the chemical compatibility of the contact agent with lithium would have to be checked. The values of 7 and /3 for a l l the samples are listed in table I I . The table also contains details of the sample preparation conditions, the sample masses, and the masses of Cry-Con grease used to attach the samples. 7 and 0 are shown as functions of x in figures 34 and 35, respectively. The interpretation of these results is the subject of the following three chapters. Table II Specific Heat results for Li y NbS2 samples The table includes x, the equilibrium voltages of the electrochemical cells used to prepare the samples, the masses of the samples and the grease used to mount them, the specific heat coefficients y and 6, and the superconducting transition temperature, T Where no Tc was observed, Tc is shown as being lower than the lowest temperature at which measurements were made. x stage ce l l voltage (V) NbSo batch 0 - - DD9 . 1 3 ± . 0 1 2 2. 760 DD9 .16 2 2. 755 DD12 .23 1 2.710 DD12 .25 1 2. 707 DD9 .30 1 2.670 DD12 .32 1 2.670 DD12 .35 1 2.640 DD12 .41 1 2.600 DD9 .50 1 2.500 DD12 .68 1 2.350 DD9 1.00 1 1.900 DD9 (mg) mass (mg) Y(-mJ mole-K r) B(-mJ mole-K (K) 231 124 111 117 106 117 98 129 143 103 201 155 4.8 3 5 3 1.5 6 7 7 6 2 2.8 4 1 9 . 3 ± 1 . 5 f+1.0) 10.9 13.1 11.6 10.3 9.4 11.4 6 10 6 5 4 1 1.4 +0.5 •1.5 +0.4 - 1 . 0 +1.0 - 1 . 4 +0.6 - 1 . 6 +0.3 1.0 +0.3 - 1 . 0 0+0.3 { . 8 ± 0 . 2 .8+0.3 •3(ti:S] 0.31+0.04 0 . 7 0 ± 0 . 0 6 + .05 - .02 + .03 - . 0 1 + .05 - . 0 3 + .06 - . 0 4 + .03 - . 0 1 + .03 - . 0 1 0 . 3 6 ± 0 . 0 1 1. 32 0.55 0.66 0.96 0.36 0.29 0.187+.005 0 . 2 4 ± . 0 1 0 . 1 8 ± . 0 1 5 .7 <2. 7 <2.0 3 .1 3 .1 <1.8 <2. 8 <2.8 <2.6 <2.8 <2.8 none observed 114 20 Figure 34: The l i n e a r s p e c i f i c heat c o e f f i c i e n t y as a function of x for the LixNbS2 samples. The dashed l i n e i s intended only as a guide for the eye. 115 0.4 0.6 x in L i x N b S 2 F i g u r e 35: The c u b i c s p e c i f i c heat c o e f f i c i e n t 6 as a f u n c t i o n of x f o r the L i NbS ? samples. The dashed l i n e i s p r o v i d e d only to guide the eye. ° 5. THE ELECTRONIC SPECIFIC HEAT 5.1 INTRODUCTION This chapter begins with a short introduction to the theory of the electronic specific heat of metals. In section 5.3, the effect of intercalation on the electronic specific heat is discussed in terms of the rigid band charge transfer model. The electronic density of states calculated by Doran et a l . (1978) is used to predict the electronic specific heat of Li NbS, as a function of x. The theoretical x i predictions are compared with the data in section 5.4. In disordered systems, i t is possible for there to be a linear term in the low temperature specific heat which is not of electronic origin (Anderson et a l . 1972). In a disordered solid such as a glass, some atoms or groups of atoms may be in positions were they have access to two closely separated potential energy minima. A distribution of these 'tunneling states' is responsible for the linear specific heat. The magnitude of this linear term is generally rather small. Typical glasses have linear specific heats which are of order 100 times smaller than the electronic specific heats of metals such as LixNbS2 (Hunklinger et a l . 1975, for example). There is disorder in Li NbS2, most significantly that due to the arrangement of the lithium atoms in the interlayer sites. As is discussed further in sections 6.2 and 6.6.1, the lithium is believed to l i e on the octahedral sites in the interlayer gaps, and 1 16 1 17 for most values of x i s di s t r i b u t e d at random among these s i t e s . The lithium-lithium interaction i s believed to be small compared to the energy required for a lithium atom to hop from one s i t e to the next. Therefore, in spite of the disorder, lithium atoms (except perhaps a few near defects, or possibly at phase boundaries or domain walls i f these exist) should be in deep, single potential energy minima. The linear s p e c i f i c heat due to l a t t i c e disorder i s probably much smaller than in glasses, and we can be reasonably sure that the measured lin e a r s p e c i f i c heat i s electronic in o r i g i n . 5.2 THE ELECTRONIC SPECIFIC HEAT OF A METAL Consider a system of independent electrons with a density of states in energy given by N(e). At temperature T, the occupation of these states i s given by the Fermi-Dirac d i s t r i b u t i o n where n i s the chemical potential of the electrons. The to t a l electronic energy at temperature T i s therefore given 1 f ( e ) = e ( e - y ) / kBT _ 1 (5-1) by CO U = J e N ( e ) f ( e ) d e (5-2) — oo From t h i s , i t is possible to derive the well known expression for the electronic s p e c i f i c heat of a metal (see, 1 18 for example, Ashcroft and Mermin 1976, chapter 2). cel= ( 9U/9T) N= ( TT2/3 ) kB2TN ( eF ) (5-3) where ep is the Fermi energy. In a real metal, this must be corrected for the electron-phonon interaction. Instead of (5-3) we have cel=( 1+X) (7r2/3)kB2TN(eF) (5-4) where X is the electron-phonon coupling constant2 2. The linear specific heat coefficient 7 (see equation 1-3) is 5.3 THE ELECTRONIC SPECIFIC HEAT IN THE RIGID BAND CHARGE  TRANSFER MODEL An introductory discussion of the rigid band charge transfer (RBCT) model has been given in section 1-3. As was mentioned there, the Fermi level in NbS2 l i e s in the center of the h a l f - f i l l e d dz2 band. According to RBCT, as x is increased in LixNbS2, the electrons donated by the intercalated lithium progressively f i l l the band, until i t is completely f u l l at x=1. As x increases, ep moves up through the dz2 band, and N(eF) changes, eventually falling to zero at x=1. Although X may also change on intercalation, the dominant factor affecting the electronic specific heat is expected to be N(eF). The behavior of the electronic specific heat coefficient 7 should therefore directly reflect the 7=(1+X) ( 7 r2/3)kB2N(eF) (5-5) 2 2 This is also called the mass enhancement factor. 119 structure of the dz2 band density of states. In this section, a calculated density of states is used with RBCT to predict j ( x ) . Wexler and Wooley (1976) made band structure calculations for NbS2 and several other layered compounds using the layer method. The results are in reasonable agreement with those calculated for some of the layered compounds (but not for NbS2) by Mattheis (1973), who used an augmented plane wave method. Doran et a l . (1978) made a tight-binding f i t to Wexler and Wooley's bands and used this to calculate the density of states of NbS2. The f i t to the bands is shown in figure 36. The labels on the horizontal axis of figure. 36 refer to points of high symmetry in the hexagonal Brillouin zone (figure 37). Figure 38 shows the calculated density of states for the dz2 band. The density of states given by Doran et a l . , has been multiplied by two to account for electron spin, and the energies have been converted from Rydberg units to electron volts. (1 Ryd=l3.6eV). Some comments should be made at this point. The weak splitting of the two dz2 sub-bands disappears at the top face of the Brillouin zone (the plane containing A,H,and L in figure 37). This is due to the same symmetry of the unit c e l l which causes the geometrical structure factor of (00/) Bragg peaks to be zero for / odd (section 2.3, appendix 1). Because of this, the Fermi surface can be 'unfolded' into a doubled zone, as is done in figure 37. In stage two Li NbS2 1 2 0 Figure 36: Tight-binding f i t (Doran et al.1978) to the layer method band structure of NbS2- The small crosses represent the layer method results of VJexler and Wooley (1976). A separate f i t was used 2 to determine the dz band density of states. 121 K Figure 37: Brillouin zone and Fermi surface of NbS The Fermi surface has been 'unfolded' into a double zone (both r and r 1 are the zone center). After Wexler and Wooley (1976). 122 e-eF (eV) Figure 38: The dz band density of states for NbS (Doran, et a l , 1978) 1 23 this symmetry disappears, and some sub-band splitting will occur even at the zone boundary. This is the only topological change in the bands to be expected in any of the intercalation compounds, unless lithium ordering occurs. Ordering in the interlayer gaps would result in a larger unit c e l l and further band s p l i t t i n g , although this splitting would probably be weak. The splitting due to staging is also expected to be weak, since it is due to interlayer interactions, and i t should not have a large effect on the density of states. Another point concerns the shape of the dz2 band density of states. The Fermi level lies on the side of an extremely sharp peak in the density of states. Because of this , the calculated N(eF) should not be considered to be very precise. A slight change in e „ would change N ( e „ ) drastically. There is also a 'shoulder' (van Hove singularity) in the density of states at e-eF=.27eV. This shoulder is due to the saddle point in the lower sub-band at T. Although the exact size and location of the shoulder depend on the details of the calculation, its existence does not. The next step is to treat Li NbS2 in the RBCT model by adding electrons to the dz2 band. Consider the Fermi energy to be a function of x. With N(e) expressed in units of states per unit energy per formula unit, the assumption of complete charge transfer means that 124 e F(x) x = / N(e) de (5-6) This integral equation has been used to calculate N(ep(x)) as a function of x. The results are shown in figure 39. The electronic specific heat coefficient 7 is given simply by 7(x) = (l+X)(7T2/3)kB2N(eF(x)) (5-7) 5.4 DISCUSSION To use (5-7) to f i t the 7(x) data we need, in principle, to know X as a function of x. Given the lack of any information which- would allow the independent determination of X, we will proceed by making the assumption that i t is constant. One way of assigning a numerical value to X is to use the values of 7 and N(ep) at x=0. These are 19.3±1.5 mJ/mole-K2 and 2.94 states/eV-formula unit, respectively, giving X=1.8 using (5-7). This is in reasonable agreement with the value of 1.94±(10 to 20%) which Aoki et a l . (1983) have calculated from NMR relaxation time and specific heat measurements. The curve obtained by the use of (5-7) and X=1.8 is shown in figure 40, along with the data. A curve for X=1.2 is also included. The general features of the data and the calculated curves agree reasonably well. Because of the approximate nature of the band calculations, exact agreement should not be expected anyway. Variations in X as a function 125 C o E *- 2 -I > CO Q) -f-» D CO 1 -X <0 0 J I I I I I I I L 0.0 0.2 0.4 0.6 x in L i x N b S 2 0.8 1.0 Figure 39: The density of states at the Fermi level of LixNbS2, plotted as a function of x. This curve was calculated using the rig i d band charge transfer model and the density of states of Doran et a l . (1978). 1 2 6 Figure 40: The electronic specific heat coefficient Y as a function of x in L i NbS0. The data x 2 are shown (chapter 4), together with the predictions of the ri g i d band charge transfer calculation for two values of the electron-phonon coupling constant X (see text). 1 27 of x are also possible, and could contribute to the differences between the data and the calculated curves. (As we shal l see in Chapter 6, the s p e c i f i c heat data also imply s i g n i f i c a n t changes in the phonon spectra as a function of x.) Several conclusions can be drawn from the data: 1 . 7 tends to a value near zero at x= 1 . This supports the hypothesis of complete charge t r a n s f e r , that i s , each intercalated lithium atom donates one electron to the NbS2 bands. 2. Use of the dz2 bands of the NbS2 host gives reasonable agreement with the LixNbS2 data. There is no evidence that i n t e r c a l a t i o n leads to either major changes in the dz2 band, or to band overlap in t h i s material. 3. The shoulder in the s p e c i f i c heat data at x=.4 indicates that t h i s i s the value of x for which the Fermi l e v e l of Li NbS2 crosses the saddle point in the L i NbS2 dz2 A X band. If we assume a completely r i g i d band (that i s , the Li NbS2 dz2 band i s exactly the same as the NbS2 dz2 X band), t h i s puts the following constraint on the density of states of NbS2 eS J N(e) de = .4 states/formula u n i t . (5 e F Here ep and eg are the Fermi energy of NbS2 and the energy of the saddle point, r e s p e c t i v e l y . Note that t h i s result does not depend on any assumptions concerning X. 6. THE LATTICE SPECIFIC HEAT 6.1 INTRODUCTION This chapter is concerned with the specific heat due to lattice vibrations (phonons). The well known fact that the phonon specific heat of a three dimensional solid is proportional to T3 at sufficiently low temperatures has already been mentioned. The theory which explains this will be briefly reviewed in section 6.3. The T3 behavior occurs at temperatures so low that only long wavelength acoustic phonons can be thermally excited. Because these are ordinary sound waves, the specific heat can be calculated using continuum elasticity theory. This allows a discussion of the specific heat data in terms of the macroscopic elastic constants of LixNbS2 (section 6.4). In section 6.5, the question of deviations from T3 is addressed. The mechanisms which cause these deviations as the temperature is raised are discussed. It is possible to make a rough theoretical estimate of the temperature above which significant deviations from T3 begin to occur, and this is found to be in agreement with the data. In section 6.6, we take a different approach which sheds further light on the problem. Simple one dimensional lattice-dynamical models are used to investigate possible effects of lithium intercalation on the elastic behavior. These models are also used to describe the effects of staging. Section 6.7 concludes the chapter. 128 1 29 Before beginning a discussion of the specific heat which includes only vibrational motion, we recall that one of the most important features of Li NbS2 is that, at least near room temperature, the intercalated lithium is highly mobile in the host. The internal energy of the system depends on how the lithium is distributed among the sites in the interlayer gaps, because of lithium-lithium interactions and elastic energy effects (J.R.Dahn, D.C.Dahn, and Haering 1982). The equilibrium configuration of the lithium atoms i s , in general, a function of temperature. We might therefore expect a term in the specific heat which is related to the changes in lithium configuration as a function of temperature. The following section (6.2) wil l show, however, that at the temperatures used in this study, motion of lithium between sites will have essentially stopped. The specific heat at these temperatures is therefore due only to the thermally excited vibrational motion of a crystal in which each lithium is 'frozen' on one particular s i t e . 6.2 MOBILITY OF INTERCALATED LITHIUM It is clear that lithium is free to move about in the interlayer gaps of intercalated layered compounds at room temperature, since intercalation is not possible otherwise. Neutron diffraction studies of Li TiS2 (J.R.Dahn et a l . X 1980) show that intercalated lithium spends most of its time localized in the octahedral sites in the interlayer gaps. 1 30 Because the local environment of the lithium is very similar, this is also expected to be true in LixNbS2. The accepted point of view of lithium motion between sites involves occasional thermally activated 'hops' over a potential energy barrier. As is well known, the time between hops in such a situation is given by T h = A eE/kBT (6-1) where E is the energy barrier and A is a constant. Direct evidence for this picture comes from the NMR measurements on Li TiS2 made by Kleinberg (1982) and Silbernagel (1975). From the temperature dependence of the 7L i linewidth, Kleinberg was able to evaluate the constants A and E and found A=1.9-10'11S and E=.29leV (E/kB=3370K). Similar values of A and E should occur in LixNbS2. Using for convenience the approximate values E=.3eV and A=10"11 allows a calculation of as a function of temperature, the results of which are shown in table III. Since the temperature corresponding to a given value of depends only logarithmically on A, it need not be accurately known. The temperatures do depend strongly on E, however, and since E was measured on L ixT i S2, the values in the table must be considered to be only approximate. It is clear that at' the temperatures of the specific heat measurements, lithium hopping will be frozen out, and the configurational degrees of freedom will not contribute to the specific heat. So far i t has been shown that at low temperatures the intercalated lithium will be fixed in one particular 131 Table III Hopping time as a function of temperature. !h<s> T(K) 10"6 302 10- 3 188 1 137 103 101 10s 89 109 76 configuration; nothing has been said about the nature of that configuration. This is an important point, since phase transitions may occur on cooling. As an example, the phase diagram of LixNbS2 is expected to qualitatively resemble one calculated using the 'spring and plate' model (J.R.Dahn, D.C.Dahn, and Haering 1982; see also Safran 1980, and Millman and Kirczenow 1983). This is a lattice gas model which includes in the Hamiltonian the elastic energy associated with the expansion of the host lattice on intercalation. The phase diagram for staging is shown in figure 41. Clearly, i f the system remains in thermodynamic equilibrium, there is a possibility that some samples will change stage on cooling. Another possibility is that phase 132 2.0 0 0.25 0.50 0.75 1.0 x Figure 41: Staging phase diagram for a typical intercalation compound according to the model of Dahn, Dahn, and Haering (1982). The integers represent regions of pure stages; the shaded areas are phase mixtures. The model allows for phases of higher stage, although these are not included in the diagram. 1 33 transitions involving in-plane lithium ordering may occur. The question of what configuration is frozen in as the temperature is lowered can be addressed using a simple theory f i r s t used by Bragg and Williams (1934) in their classic paper on atomic ordering in quenched alloys. Consider an intercalation compound at temperature T, in which the lithium atoms happen to be in a configuration which is the equilibrium state at another temperature 8. We assume that the system relaxes to equilibrium at T with a time constant T , which we assume is of order r^. That i s , we write d0/dt = -(0-T ) /T~-(0-T)Ah (6-2) If we now assume the intercalation compound is being cooled at a constant rate dT/dt=-r, d0/dT^(0-T)/rhr (6-3) An analytic expression for the solution of this equation can be easily obtained, but it is rather unwieldy. As pointed out by Bragg and Williams there is a simple approximate solution which is good enough for our purposes. As long as r^r<<1, 0-T will be small, and the solution is essentially 0=T. As T is lowered, begins to increase rapidly. At f i r s t , we s t i l l have d0/dT=*1 , and the solution i s 0^T+rrh (6-4) At s t i l l lower temperatures, r r ^ becomes large enough that the approximation of (6-4) breaks down, and after this 8 134 remains essentially constant as T is decreased further. This point where 6 'decouples' from T occurs roughly at the point where d0/dT=O in (6-4), that i s , when dx rAE -r — " = —2eE / kBT = 1 (6-5) n k T: dT KBX Temperatures calculated from (6-5) for various cooling rates are shown in table IV. The precooling from room temperature to liquid nitrogen temperature before a specific heat experiment takes 3 to 4 hours, so that the cooling rate is about 0.1 K/s. The Li NbS2 samples should therefore end up in a state X characteristic of equilibrium at about 120K (assuming E=.3eV). Note that the cooling rate does not have a large effect on this temperature. This calculation shows that the low temperature staging and ordering phase transitions predicted by equilibrium lattice gas models of intercalation compounds (J.R.Dahn, D.C.Dahn, and Haering 1982, and references therein) will not actually occur at temperatures below about 100K, no matter how slow a cooling rate is used. Since the exact phase diagram and E value for Li NbS2 are not known, these calculations do not provide an unambiguous answer to the question of whether or not some LixNbS2 samples changed stage on cooling. If the samples remained in thermal equilibrium down to T=0, the spring and plate model indicates that a l l the samples would be phase mixtures of the staged structures at X=1/3,1/2, etc.(figure 41). The 1 35 Table IV Decoupling temperature for various cooling rates. Cooling Rate (K/s) Decoupling Temperature (K) 100 1 54 1 29 0.1 1 20 0.001 1 04 arguments of this section do make it clear that this extreme type of phase separation wi l l not occur. 6.3 PHONON SPECIFIC HEAT In this section the theory of low temperature lattice specific heat will be briefly reviewed. This theory is based on the well known quantum theory of the harmonic lattice (see, for example, Ashcroft and Mermin 1976, chapter 23). The lattice specific heat depends on the dispersion relations "S(K) of phonon modes. (Here s is a branch index and k~ is the phonon wave vector.) Most rigorously, the specific heat c is given by a sum over the discrete set of phonon modes. In a macroscopic crystal, however, the allowed phonon wavevectors are very close together, and the sum may be replaced by an integral in the usual way. This leads to 1 36 the following expression for the specific heat per mole; ° = V W s B ^ Z T7^3 e* u8< R W kBT _ x ( 6"6 ) where V is the molar volume, and the integral is taken over the Brillouin zone. This general relation can be simplified for low temperatures, since the Bose-Einstein occupation factor ( e * i o ) s ( k ) / k B T _ 1 } - 1 becomes vanishingly small for fta>( k~)>>kgT. At low enough temperatures in a three dimensional material only the three acoustic phonon branches, for which w(k~)-»0 as k"-K), make a significant contribution to c. The following simplifications in (6-6) may then be made: 1. Optical phonon modes may be ignored. 2. At very low temperatures, only the very long wavelength portion of the acoustic phonon dispersion curves will be important. In this region, the phonon dispersion relation is linear, and we can use a»e(Tc)=ve(R)k (6-7) s s where k is the magnitude of k~, R is a unit vector in the direction of k\ and v (R) is the velocity of sound in s that direction. 3. Since the integrand is vanishingly small except near k=0, we may take the integral in (6-6) to be over a l l of 1 37 k-space. These simplifications lead to 3 dit frvs(k)k The integral will be rewritten in spherical coordinates. Taking dk~=k2dkdS2 (0 is the solid angle); 3 dk fivs(k)kd c = V ^  | / dfi / ^3 ( e f t V s ( R ) k / k B T _ x ) (6-9) Making the change of variables y=fivg(R)k/kgT in the k integration gives 3T s ( T i l ( t V s( f i ) ) 3 o e ^ - l Since the definite integral over y is just 7r/l5, carrying out the derivative with respect to T yields 7TkBVT3 dfi c = — ^ i f (6-11) 30ft3 s (vs(£)) If 1/v3 is defined to be the average over mode index s and solid angle of the inverse cubed sound velocity,that is 1 _ 1 y r 1 (6-12) v3 " 3 I 1 5? v7(kT3 1 38 then c is given by 3 c = (6-13) The phonon specific heat coefficient 0 is given by c 2TT2 1 - 1, (6-14) These relations hold at temperatures low enough that the only phonons which have a significant probability of being thermally excited are the long wavelength acoustic phonons which satisfy the linear dipersion relation (6-7). The well-known Debye formula was devised to describe the specific heat over the entire temperature range from T=0 up. It is often used to f i t data at intermediate temperatures where (6-13) f a i l s . However, i t f i t s the data for layered compounds very poorly, primarily because, as we shall see, their phonon dispersion relations are highly anisotropic. In spite of this, specific heat data even on layered compounds are often presented in terms of an effective Debye temperature, and in order to define this i t will be useful to briefly review Debye theory. The Debye formula is derived by setting o> (k~)=vk in (6-6) and taking the integration not over the Brillouin zone, but over a sphere in k-space which contains the same number of allowed 1 39 phonon wave vectors2 3. This condition on the size of the sphere specifies its radius, the Debye wavenumber kp. The Debye frequency wQ and temperature e?D are given by kB0D=fiwD-nvkD (6-15) The result for the molar specific heat is /T\3 D y*ey dy C = 9 NA M ! J <6-1 6 ) ;ey - i )2 0 where NA is Avogadro's number. In the limit T-?0, this becomes 1 2 - " l 3 C = — F ~ NAkB (Ij «"17> and the coefficient /3 is 1 2 ^ NAkB . 6 - (6-18) Since the definition (6-15) of t?D in terms of the Debye sphere does not make sense for layered compounds, effective Debye temperatures for these materials are defined in terms of the low temperature limits (6-17) and (6-18). 2 3 Note that both the sound velocity and the distance from the origin to the zone boundary are assumed to be isotropic. 140 6.4 THE PHONON SPECIFIC HEAT OF Li NbS, IN THE ELASTIC x_ 4  CONTINUUM (T3) LIMIT As we have just seen, the cubic specific heat coefficient 0 can be related to an 'average' sound velocity v and a Debye temperature #D by means of equations (6-14) and (6-18), respectively. Using the measured 0 values (figure 35), v and 6n have been calculated for the Li NbS2 samples and are i-) X shown in figures 42 and 43, respectively. Clearly, there are large changes in the lattice dynamics as a function of x. In this section, an elastic continuum model wil l be used to discuss the specific heat in terms of the elastic constants of Li NbS2. To show how 0 depends on the elastic constants, a value for NbS2 will be calculated. Materials of hexagonal symmetry have 5 independent elastic stiffness constants, which are, in the standard abbreviated subscript notation, c1 1 f c3 3, ca f t, c6 6 = ( cn- c ,2) / 2 , and c1 3 (Auld 1973). To the author's knowledge, the elastic constants of NbS2 have not been measured. McMullen and Irwin (1984) have recently f i t Raman spectra for NbS2 with a simple 4-parameter valence force model. In principle, elastic constants can be calculated from these inter-atomic forces., but because of the simplicity of the model and the limited amount of data used in the f i t i t is not clear that the values would be reliable. More extensive data are available for several related compounds, however, and these are listed in table V. The values for a l l three compounds in the table are similar, and 141 3.0 2.6 E o 2.2 1.8 1.4 1.0 \_ \ \ x / Xj X */ ,x - I — I — I I I I I I I 0.0 0.2 0.4 0.6 0.8 x in Li x NbS 2 1.0 Figure 42.- 'Average' sound velocity v (equation 6-14) for the L i NbS2 samples. The dashed line is intendea only as a guide for the eye. 142 220 180 Q C O 140 h 100 0.0 0.2 0.4 0.6 0.8 x in L i x N b S 2 1.0 Figure 43: Debye temperatures for the Li NbS2 samples. The dashed line is a guide fo? the eye. 1 43 Table V Elastic constants for some layered transition metal dichalcogenides with the 2H structure. The values in the table are in units of l01oN/m2. Elastic Compound ic 4* ic ic Constant TaSe2 NbSe2 MoS2 c,, 19.6 to 12.4 10.8 17.4 c3 3 5.3 4.6 5.2 Cnu 1 .74 1.9 1.9 c6 6 5.5 to 5.4 4.6 7.3 cn 1 .34 to . 76 =0 . 2.3 * Feldman (1982) t 76, 1982); Jericho et a l . (1980) ** Feldman (1976, 1982) the same may be expected of NbS2. For the purposes of this illustration the elastic, constants of NbSe2 will be used. Most of the NbSe2 constants in the table were deduced from inelastic neutron scattering data by Feldman (1976). Jericho et a l . have measured c,, ultrasonically. In Feldman's 1976 work c1 3 could not be specified very accurately; i t was given as between +3.1 and -0.2 in units of 101oN/m2. A more sophisticated analysis involving an atomic force model (Feldman 1982) gives c1 3 between 1.34 and 0.76 in the same units, for TaSe2. This result should be approximately 1 4 4 applicable for NbSe2 as well, and so for the following calculations c , 3 = 1 . 0 • 1 01°N/ m2 w i l l be used. To calculate 0 from the elastic constants we f i r s t find the sound velocity as a function of the direction of propagation, then calculate the average of the inverse cube of the velocity (equation 6 - 1 2 ) . The sound velocities in a hexagonal crystal are functions only of the angle 6 between the c-axis and the direction of propagation. The velocities of the three acoustic modes are (Auld 1 9 7 3 ) ' V i ( e ) = { C i i s i n26 + c3 3cos29 + C i ^ - ( ( c1 1- C M ) s i n28 + ( c ^ - c 3 3 ) c o s28 ) 2 + ( c13 + c ^ ) 2 s i n 2 6 } ' * / ( 2 p ) v 2 ( e ) = { ( c 6 6 s i n2 e + c<4 4cos 2 e) /p}^ (6-19) v 3 ( 9 ) = { C i i s i n29 + c3 3cos26 + + ( ( c n - c . O s i n 2 e + ( c 1 ) , - c 3 3 ) c o s 2 e ) H ( C l 3 + c ^ ) 2 s i n 2 6}'V(2p) In these equations, p is the density (4.6g/cm3 for NbS2). Polar plots of the inverse sound velocities as a function of 6 are given in figure 44. Mode 1 is a quasi-shear wave. At 0=0 i t becomes a pure shear (transverse) wave propagating along the c-axis, with atomic displacements in the basal plane. This is an example of what is called a 'rigid layer' shear mode. Since intralayer bonding forces are much 145 In plane component of 1/v (1 unit = 10~4s/m) s Figure 44: Polar plot of the inverse sound velocities calculated for NbS? (see text). The curves are labelled with the mode index s. Modes 1 and 2 intersect the vertical axis at ( p/c. '44 axis at . Mode ( p / c3 3 3 intersects the vertical )2. Modes 1, 2, and 3 inter-sect the horizontal axis at ( p / c ^ ) ( e /c6 6 >% and (P/C-Q)2, respectively. 1 46 stronger than the interlayer ones, this acoustic wave involves essentially rigid layers vibrating as units. At 0=TT/2, mode 1 again is a pure shear wave, now propagating in the basal plane with the atomic displacements along the c-axis. That this is also essentially a rigid layer shear mode (at least for long wavelength) is clear from figure 45. The fact that mode 1 is a rigid layer shear wave at both 0=0 and tt/2, results in its sound velocity being /c« 8/p in both those directions. c4, is the elastic constant associated with 'rigid layer' shear. Since mode 1 is the mode with the lowest sound velocity i t makes the largest contribution to the specific heat. Mode 2 is a pure shear wave for a l l 0, polarized in the direction normal to both the c-axis and the direction of propagation. It is a rigid layer shear wave at 0=0, but at 0=7r/2 i t involves shearing of the layers themselves (elastic constant c6 6) . Mode 3 is a quasi-longitudinal wave, which becomes pure longitudinal at 0=0 and TT/2. Because the sound velocity is a function only of 0, equation (6-12) for the average inverse cube velocity becomes i i -\ 7 1 / 2 _ = 2 i f s i n e d e (6 -20) v3 3 s=l 0 v3 (6) The integral for the pure shear wave (mode 2) can be 147 4* c Figure 45: Acoustic mode 1 at 6=TT/2. The curved lines represent the NbS2 layers. The parallelogram would be a rectangle in the undistorted material, and indicates the nature of the strains associated with the wave. 1 48 evaluated easily and is •n/2 S sine de 1/2 = 4.9*10 - I i s 3 / m 3 (6-21) 0 v 3 ( 9 ) c 6 6  *- k it The numerical value was calculated using NbSe2 elastic constants and the density of NbS2. The integrals for the other two modes were calculated numerically, and are 9.21 • 10" 1 'sVm3 for mode 1 and 1 . 50 • 1 0"1 1 s3/m3 for mode 3. The average of these is 1/v3=5.21•10"11s3/m3, which, when used in (6-14) gives /3=. 22mJ/mole-Kfl. The experimental value for NbS2 is . Sli^ma/mole-H*, which is 30% higher. This is reasonable agreement, considering the calculations were made using the elastic constants of NbSe2, a closely related, but different, material. The difference in 0 indicates that NbS2 has somewhat softer elastic constants than NbSe22fl. The corresponding calculated and experimental Debye temperatures are 206K and 187±7K, respectively. Now the effect of intercalation on the specific heat can be considered in this elastic continuum l i m i t . Of the five elastic constants, there are two, c3 3 and cfl(1, which depend primarily on interlayer forces. c3 3 is associated with compression along the c-axis, and c«„ is associated McMullen and Irwin's (1984) f i t to the Raman spectra does not seem to agree with this conclusion. For example, the f i t implies c«4^2.5•101°N/m2, which is s t i f f e r than NbSe2. However, this discrepancy may not be significant, since the elastic constants depend on the model used to f i t the Raman data. 2 a 1 49 with r i g i d layer shear. These two elastic constants are expected to change significantly as x in LixNbS2 is varied. Of these two interlayer elastic constants, cu u plays a much larger role in determining the low temperature specific heat. In the calculation above the contribution to 0 from mode 1, which depends mainly on c4 a, is much larger than the contributions from the other modes. Another illustration of the importance of c^, comes from a numerical calculation of the derivatives of /3 with respect to the c's. These are, in units of 1 0- 1 ^ J/N-mole-K", dp/dc, ! = - . 4 2 a/3/9c 3 3 = - 1 . 5 2 3 / 3 / 3 C , I , = - 7 . 9 1 30/9c66=-1.47 9j3/9c , 3=+1 . 0 5 9/3/9c«a is much larger than the others. To roughly estimate the size of the changes in cafl which are required to explain the data, /3 was calculated for several values of c,,4. The results are shown in figure 4 6 . A similar plot was also made for c3 3. As can be seen from the plot, the range of 0 values covered by the Li NbS2 data ( . 1 8 to 1 . 3 2 mJ/mole-K") corresponds to changes of cfttt of about a factor of 1 0 , provided the other elastic constants are fixed at their NbSe2 values. Much larger fractional changes in c3 3 (about a factor of 20 ) would be required. In addition, the simple 'spring and plate' elastic model to be presented later in this chapter (section 6 . 6 . 2 ) Figure 46: 3 as a function of c,,. The other elastic constants arg fixed at their NbSe2 values (see text). 151 shows that c3 3 should increase monotonically as a function of x in LixNbS2. Basically, the argument is that since the interlayer gaps expand on intercalation, the lithium must be pushing the layers apart. Intercalated lithium atoms can be thought of as compressed springs which act against the original NbS2 interlayer forces to separate the layers. Since we are adding more interlayer springs as x increases, c3 3 must also increase. The Li NbS2 samples with x between .12 and .5, however, have j3 values higher than at x=0. This indicates an intercalation induced reduction in whichever elastic constant is primarily responsible for the variations in 0 . The elastic constant primarily responsible for the changes in 0 is therefore almost certainly ca„ . Another point concerning c3 3 can be made here, although i t is not essential to the argument that the data reflect the behavior of c „ . Elastic stability conditions may be derived from the requirement that the elastic energy must be a positive definite function of the strains. If it is not, there will be some strain for which the elastic energy is negative, and the crystal w i l l spontaneously distort. This argument leads to the conditions (Born and Huang 1954, Feldman 1976) c3 3>c23/c,, (6-22) c33>2c23/(c1,+c,2) (6-23) There are no restrictions on C i , , , except, of course, that i t must be positive. The stability conditions can be evaluated using the NbSe2 elastic constants. In units of l01oN/m2, 1 52 using c13=1.0 as in the specific heat calculations yields c33>.09 and c33>.16, respectively. If the maximum c1 3 (3.1) consistent with Feldman's (1976) estimates is used we get c33>.89 and c33>1.5, respectively. To explain the data in terms of c3 3' alone would require i t to take on a value of about .3 in the x=.16 sample, which might result in a violation of the stability c r i t e r i a , depending on the actual value of c1 3. The lattice distortions that would result from this have not been observed in either LixNbS2 (chapter 2), or in LixNbSe2 (D.C.Dahn and Haering 1982). 6.5 BEYOND THE ELASTIC CONTINUUM LIMIT This section will show how, as the temperature rises, the phonon specific heat begins to deviate from its low temperature T3 behavior. The reasons for this will be explained, and approximate calculations of the temperature at which significant deviations set in will be made. An understanding of the deviations from T3 is important, since it 1. allows us to be confident that the experiments have truly found the low temperature l i m i t , and 2. provides some additional insight into the mechanism for changes in the specific heat on intercalation, supporting the conclusion that c«4 is primarily responsible. Deviations from T3 behavior at higher T come about because the phonon dispersion curves are linear only in the 153 very long wavelength limit, and because they are truncated at the Brillouin zone boundaries. (At higher T we can no longer make the approximation that the integral in (6-6) extends over a l l of k-space.) Phonon dispersion curves for 2H-NbSe2 have been measured at room temperature by inelastic neutron scattering (Moncton et a l . 1977) and are shown in figure 47. The dispersion curves for NbS2 and i t s intercalation compounds should be similar. The labelling of the different phonon branches is that of Moncton et a l . The long wavelength parts of branches L3 and A6 correspond to acoustic mode 1 of the previous section.(the rigid layer shear mode), with propagation along the a and c axes, respectively. Note that at point A on the zone boundary there is no splitting between the A6 acoustic and A5 optical shear branches, or between the A, and A2 longitudinal branches. This is because of the symmetry of the two layer high unit c e l l . We could think of the A6 and A5 branches as an acoustic branch in the double zone, which has simply been folded over. In a stage 2 intercalation compound the symmetry is broken and a small gap should appear at the zone boundary. Looking further at the dispersion curves of figure 47, we notice two features that may cause the f i r s t deviations from T3 as the temperature is raised. One of these is the relatively low energy of the top of the A6,A5 branch. The other is the anomalous upward curvature of the E3 branch. 154 i I i 1 r f 2H-NbSe2 (300 K) -[ooc] [COO] -<2«-/c) WAVE VECTOR (4»/VTo> Figure 47: Phonon dispersion curves from inelastic neutron scattering measurements on NbSe After Moncton et a l . (1977). 155 This upward curvature can be explained in terms of forces associated with bending of the layers, as was f i r s t demonstrated in theoretical studies of the specific heat of graphite (Komatsu 1955, Bowman and Krumhansl 1958). Acoustic waves propagating in the basal plane with atomic displacements along the c-axis, although they are basically shear waves, also involve bending of the layers (see figure 45 in the previous section). Since the layers are s t i f f and cfl„ is small, the energy due to bending the layers can be significant, especially at short wavelengths. The layer bending energy density is proportional to the square of the layer curvature, that i s , i t is proportional to (92u/9y2)2, where u is the atomic displacement associated with the wave, and y is a coordinate along the direction of propagation. The shear elastic energy is c«f l(9u/9y)2. It can be shown (Komatsu 1955) that when the potential energy is a sum of layer bending and shear terms, the dispersion relation for waves is pcj2=cI1(tk2+bk'' (6-24) where b is a positive constant. The second term is due to the layer bending forces. This explains the upward curvature of the E3 phonon branch. The value b can be estimated graphically from, figure 47, and is approximately 4- 1.0- 1 " r n V s 2 . To estimate the temperatures at which the bkft term and the truncation of the A6,A5 mode will cause deviations from T3 in the specific heat, we f i r s t recall (6-6), the general 1 56 expression for the phonon specific heat. This involved integrals of the form /dit 1 (6-25) efia) s(it)/k BT _ x If we assume for the moment an isotropic elastic material (as in the Debye model), this becomes 2 irk 2fivk / dk (6-26) e*vk/kBT _ 1 I f we define z=fivk/k_T, then the integrand is proportional a to z3/(e - 1 ) . This function is the' same as the black body radiation spectrum (Kittel 1969, p256, for example) and has its maximum value at z=*3. It drops to zero at large z, and has half i t s peak value at z^5.5. What this means is that the specific heat is quite sensitive to phonons with energies up to about 5 times kgT. The general conclusion that the specific heat is sensitive to phonon energies up to several times kgT is expected to be true even for anisotropic materials. It is now possible to produce rough estimates of the temperatures at which the two different effects being considered will cause deviations from T3 in the specific heat. The top of the A 6 , A 5 branch occurs at a phonon energy fuj of order fi/c,„/p(2n/c), where c is the height of the two layer unit c e l l . As we have just seen, deviations from T3 will occur when this phonon energy becomes less than about 5k_T. If T . is defined as the temperature above which this 1 57 c-axis" truncation causes deviations from T3, we have fi 2ir Similarly, we can define as the temperature above which the bk" term in the dispersion relation for the layer bending (£3) branch causes deviations from T3. This can be estimated by setting 5k^T^ equal to the energy at which the two terms cattk2 and bk" in the dispersion relation (6-24) are equal. This gives For c«„=1.9«101°N/m2, the value used for the calculation of /3 for NbS2, we get Tfc=16K and Tb=30K. The estimates of of T and T^ are very approximate, but we can see that both mechanisms are probably important. This is unlike the case of graphite, where T^ appears to be significantly lower than T (Komatsu 1955). To compare these estimates with the data, the quantity T^Q^ wil l be used. This is defined as the temperature at which the lattice specific heat data deviate by 10% from 0 T3. The results are given in table VI. The value for NbS2 (16+1K) is reasonably close to the estimated temperatures Tfc and T^. This suggests that the explanation of the deviations from T3 given above is correct. The T^Q^ data also show that there is a strong correlation between high 0 values (which we believe to be 158 Table VI P a n d T10% for the Li NbS2 x 2 samples. X 0(mJ/mole-Ka) T10%( K ) 0 .31 1 6±1 .13 .7 9.5±1 . 1 6 1 .32 9.5±.5 .23 .55 13±.5 .25 .66 9±2 .30 .96 9±.5 .32 .36 1 6±2 .35 .29 Insufficient data .41 .36 >9 .50 .19 >1 5 .68 .24 13±1 .5 1 .00 .18 1 5±2 due to low ca„'s), and low T^^. This is what is expected on the basis of the equations (6-27 and 6-28) for Tfc and T^. Taking another numerical example, the value of /3 for the x=.16 sample implies a c„„ value of about 0.2•101°N/m2. This yields Tfc = 5K and Tb=3K. T1 Q 5. was 9.5±.5K. Again, this is reasonable agreement, considering the roughness of the theoretical estimates. 1 59 An interesting feature to note is that because of the different ways that Tfc and depend on c „a, the plate bending mechanism will become relatively more important as C m is lowered. This suggests that the specific heat vs temperature curves of different Li NbS2 samples will have different shapes. This is unlike simple Debye theory, where the specific heats of different materials a l l f a l l on one universal curve i f they are plotted as a function of T/0D. The present Li NbS2 data for higher temperatures (above 10K) are not, unfortunately, of sufficient quantity or quality to allow a satisfactory test of this result. A more complete discussion of the specific heat above the T3 regime would best be based on a detailed atomic force, constant model such as those used by Wakabayashi and Nicklow (1979), or Feldman (1982). Such models attempt to f i t the entire phonon spectrum, and allow calculation of the specific heat at arbitrary temperatures directly from the general expression (6-6). At present, because of the limited high temperature specific heat data, and the lack of other measurements of the phonon spectra, the effort involved in such an approach would not appear to be j u s t i f i e d . 6.6 SIMPLE MICROSCOPIC MODELS So far in this chapter, the approach to lattice dynamics has been primarily through an elastic continuum approximation. To complement this view, and to gain some further insight into the specific heat of Li NbS2, i t is useful to consider 1 60 some simple microscopic dynamical models. 6.6.1 VIBRATIONAL MOTION OF A SINGLE INTERCALATED  LITHIUM ION When one lithium ion is inserted into a site between the layers of a NbS2 crystal, we expect the appearance of three new vibrational modes, because of the three new degrees of freedom associated with motion of the lithium ion about the center of its three dimensional s i t e . These new modes are localized vibrational modes involving the lithium as well as neighboring sulfur and niobium atoms (see Pryce 1969, for example). However, because the mass of lithium is so small compared to the other atoms in the compound, the amplitude of vibration of the lithium w i l l be very much larger than that of the surrounding heavy atoms. The approximate frequency of the localized modes can therefore be calculated assuming the surrounding NbS2 layers are fixed. To model the forces on the lithium, we assume i t is connected to the fixed rigid layers by springs. A reasonable value for the spring constant Gc associated with lithium motion in the c-axis direction is 160 N/m. This will come out of the 'spring and plate' model calculations in the next section. The vibrational frequency of the mode where the lithium motion is along the c-axis is then a>c=j/Gc/m (6 161 where m is the mass of a lithium atom. Numerical values are UQ= 1. • 1 0 1 •s- 1 and ficjc/kfi=900K, so that this vibrational mode will clearly not contribute to the low temperature specific heat. The spring constant G associated with motion parallel to the layers can be estimated from the fact that there is a potential energy barrier of order ,3eV high between adjacent lithium sites (Kleinberg 1982). Assuming the lithium atom sits in a harmonic potential well of depth .3eV and width equal to half the distance between sites yields G »4N/m2. This yields an in-plane vibrational frequency CJ =/G /m=2- 101 3 S " 1 (6-3 a. which is equivalent to a temperature of 140K. This is also much too high to be seen in the low temperature specific heat. In the discussion above, i t was assumed that the lithium atom was in a 'typical' s i t e , and that the only forces on i t were those associated with localizing i t on its site (that i s , those due to interaction with the host layers). The situation may be somewhat more complicated. Staged intercalation compounds are generally believed to posess a domain or island structure as proposed by Daumas and Herold (1969). Within each domain there is a well defined staging sequence, but globally there are guest atoms in every interlayer gap. As an 1 62 example, a stage 2 Li NbS2 crystal would have lithium in gaps 1,3,5, etc. in some domains and in gaps 2,4,6, etc. in the others. Kaluarachchi and Frindt (1983) have found that the domain size is of order 130A in Ag Ti S2. At the boundaries between the staging domains, the host layers must bend, and this raises the possibility that some of the lithium sites are significantly distorted. Lithium atoms in these sites would have different vibrational frequencies than the others, and we cannot rule out the possibility that they would contribute to the low temperature specific heat. Another complication arises from the fact that intercalated lithium atoms interact with each other. There is some evidence that this interaction is relatively weak. Lattice gas model f i t s to electrochemical data on L ixT i S2 (J.R.Dahn, D.C.Dahn, and Haering 1982) used a repulsive nearest-neighbor lithium-lithium interaction of 50meV, which is small compared to the .3eV barrier between sites. Because of this, it may be that the effect of lithium-lithium interaction on the vibrational frequencies is small. On the other hand, the lithium-lithium interaction is the driving force for the lithium ordering transitions which occur in Li TaS2 (J.R.Dahn and McKinnon 1984), and X probably in 'fresh' Li NbS2 (section 2.3), and it cannot be completely ignored. A wide range of unusual elastic behavior is possible in systems where there is a 163 competition between a periodic background potential (in this case due to the NbS2 layers) and an interparticle interaction. (For discussions of one dimensional systems of this type see, for example, Von Hohneyen et a l . 1981, Sharma and Bergerson 1984, and references therein.) One possibility in the case of Li NbS2 is that for lithium concentrations near but not equal to values such as X=1/3 where ordering occurs, the lithium configuration in each gap may consist of ordered two dimensional regions separated by discommensurations (domain walls). There could be soft modes associated with these domain walls. Further theoretical and experimental work is needed to determine if any such soft modes actually exist in Li NbS2. x 2 6.6.2 ONE DIMENSIONAL MODELS The basic model to be used in this section is a one dimensional infinite chain of masses M separated by springs of spring constant K. The model will be used to describe rigid layer longitudinal modes, so each mass can be thought of as representing an entire NbS2 layer. K represents the interlayer forces (figure 48). Since both M and K are both proportional to the area of the layer and because the vibrational frequencies depend only on their ratio K/M, the layer area is arbitrary. For convenience, we will take the area to be the base of a unit c e l l , so that M is just the mass of one NbS2 164 t Figure 48: Definition of terms used in the one-dimensional 'spring and plate' model (see text). 165 unit. We also take the system to l i e along the z axis. The distance between masses is c/2, because the unit c e l l is two layers high. The dispersion relation for waves in this system is well known, and is where k and co are the wavenumber and frequency, respectively. The spring constant K is related to c3 3 by where c and a are the lattice parameters. The NbSe2 value c33=4.6•101°N/m2 gives K=7.4 N/m. The same equations apply for vibrations normal to the z-axis (rigid layer shear waves), except that K is then To deal with the longitudinal rigid layer waves of intercalated material, i t is possible to use the 'spring and plate' model of intercalation (J.R.Dahn 1982; J.R.Dahn, D.C.Dahn, and Haering 1982) In this model the host material is again considered to be a system of rigid plates joined by springs of strength K. The equilibrium length of these 'host springs' is taken to be the host layer spacing c0/2. Intercalation is modelled by the insertion of 'lithium springs' of strength G and length cT/2>c0/2. In Li NbS2, there are x Li X lithium springs for each host spring. Balancing the forces of the springs yields an equation for the c-axis (6-31) K=/3c33a2/c (6-32) /3c a ua2/c. 166 of the intercalation compound as a function of x. c(x) - c0 . ( 6_3 3 ) cL - c0 x+K/G With appropriate values of cL and K/G, this equation gives a c-axis expansion in approximate agreement with experimental results for many intercalation compounds. A rough f i t to the Li NbS2 lattice expansion data requires X K/G=*.2. The most striking success of the spring and plate model has been its use in s t a t i s t i c a l mechanical lattice gas models of intercalation compounds. These models can be used to calculate the voltage V(x) of intercalation c e l l s , but f i t the data for systems such as Li TiS2, Li NbSe2, and Li NbS2 very poorly unless the X X X elastic energy associated with the lattice expansion (6-33) is included in the Hamiltonian. The elastic energy also provides a mechanism which produces staging. To use the spring and plate model to discuss vibrational modes, consider f i r s t the case of Li,NbS2. According to the model, each pair of adjacent layers is now separated by two springs in p a r a l l e l , with spring constants K and G. This is equivalent to one spring with spring constant Kej^=K+G. The lithium ion of mass m is placed in the center of this effective spring, dividing i t into two springs each of strength2 5 The spring constant G of the previous section is 4K or about 160 N/m. c 1 67 ^Ke f f * T^e °^^sPe r s^o n relation for this system has two branches, corresponding to the + and - signs in 2K 2 e f f to = m^ + M ±y/m2+M2+2mM cos(kc(l)/2) ) (6-34) mM where c(1) is the c-axis of Li,NbS2. The - branch is an acoustic branch. Using the fact that m/M=.045 is small, we can justify the use of the approximate relation tO M . kc(l) (6-35) in which the mass of the lithium has been ignored. For the optical (+) branch, the same approximation results in cj=2v/Keff/m (6-36) As might be expected, the frequency is the same as that of a single lithium vibrating between stationary layers. The new modes associated with the lithium degrees of freedom are contained in this optical branch. The optical branch is at too high a frequency to contribute to the low temperature specific heat, and it wil l therefore be ignored in the rest of this discussion. Since only the acoustic mode is important, we can, as we have just seen, ignore the lithium mass and say that the only effects of intercalation are to alter the spring constant from K to and to expand the c-axis. (Of these, the f i r s t is far more important.) This approach wil l now be used to discuss Li NbS2 with x<1. 1 68 For stage one compounds, the spring constant between each layer is Ke^^=K+xG, and the dispersion relation is just R ? " kc(x) - - if*" s i n -(6-37) ffA2+K'eff+2KKeff cos kc(x) (6-38) M where c(x) is the height of the two layer unit c e l l of LixNbS2. A stage two compound may be modelled by alternating springs of strength K (empty gaps) and Kg f£=K+2xG ( f u l l gaps), where x is the overall lithium concentration. The dispersion relations are K+K f f. 2 e r f , w = ± Mi M 1 Here again there is an optical and an acoustic branch. In figure 49, dispersion relations representing NbS2, stage 2 Li 1gNbS2, and stage 1 Li 2N^S2 are shown. The dispersion relations for the stage 1 compounds have been 'folded over' into the smaller one dimensional Brillouin zone of the stage 2 compound. We see that the i n i t i a l slope of the stage two curve lies between that of the x=0 and stage 1 curves, in spite of the lowering of part of the acoustic branch due to the gap at the zone boundary. This is a general result, and indicates that in the spring and plate model the contribution to the specific heat from the rigid layer longitudinal mode (elastic constant c3 3) will be a monotonically decreasing function of x, even when staging is taken 169. Figure 49: Model dispersion relations for the longitudinal mode propagating along the c-axis in L i NbS2• Included are curves representing NbS2 (lower solid lines), stage 2 L i 1 5 ^ 8 2 (dotted lines) , and stage 1 L i -MbS,, (upper solid l i n e s ) . 170 into account. 6.7 DISCUSSION The simple theoretical arguments put forward in this chapter indicate that the elastic constant co u and the rigid layer shear modes associated with it are primarily responsible for the differences in the specific heat coefficients 0 of the different LixNbS2 samples. The extra vibrational modes due to the addition of lithium are at frequencies too high to contribute to the specific heat. Looking at the data again in this lig h t , i t is possible to draw the following conclusions: 1. In the samples with x<.3, c fl „ was significantly smaller than in pure NbS2. Most of the c-axis expansion which . takes place on intercalation happens at low x (figure 7). By x=.3, the expansion is almost complete, and if we assume the expansion is a l l in the interlayer gaps, the gaps have expanded by about 15%. Intercalation and gap expansion appear to greatly reduce the interlayer shear forces that were present in the pure host. It is interesting to note that the sample with the highest 0 (lowest c„„) was stage 2 (x=.16). This is surprising, since only half of the interlayer gaps contain lithium and are expanded. If the shear forces between two layers depend only on the lithium concentration in the gap between them, we would expect 171 Ci,„ for stage 2 to be between the values for x = 0 and stage 1. That this is not so seems to imply that the shear forces across an interlayer gap are sensitive not only to the lithium in that gap, but also to the lithium in neighboring gaps. It is possible that charge transfer has something to do with this. Another possibility has been mentioned in section 6.6.1; there may be soft modes associated with Daumas-Herold staging domain boundaries. As x approaches 1, the lattice stiffens up again. At x=1, p is actually smaller than x=0, indicating ca„ is greater than in the pure host. A rough calculation shows that i t may be possible to explain the stiffening at large x by assuming shear stresses are transmitted from one NbS2 layer to the next through the intervening layer of lithium. From the activation energy for lithium hopping between sites i t was estimated (section 6.6.1) that an effective spring constant for in-plane motion of a lithium atom near the center of its site is 4 N/m. If we consider this as being due to two 'springs', one connecting the lithium ion to each of the two adjacent layers, the spring constant of each is 2 N/m. These springs are joined end to end at the lithium atom, and are therefore equivalent to a single spring of strength 1 N/m connecting the layers. If we assume these springs are the only interlayer shear forces, the elastic constant cfl„ is given by c,,=2cGx//3a2 (6-1 72 where c and a are the dimensions of the unit c e l l and G is the spring constant (1 N/m) per lithium. For Li,NbS2 this yields c„„=1.3•101°N/m2. Although this is smaller than the value of about 2.5«l01 oN/m2 implied by the specific heat data, i t is at least of the right order of magnitude, and indicates that lithium contributes to the interlayer shear forces. 3. /3, and therefore ca^, do not appear to be smoothly varying functions of x for the set of samples studied. In particular, the samples at x=.16 and .30 have higher |3 values than the samples near them in x. These two samples were both prepared from NbS2 from batch DD12, shortly after that batch was grown. As mentioned in chapter 2, freshly prepared and aged DD12 material behaved differently electrochemically. Cells made from fresh material showed two small peaks in -dx/dV near 2.67V, which may be due to lithium ordering. Recall that 2.67V is also the voltage used for preparation of the x=.30 (fresh) and x=.32 (aged) specific heat samples. Since a l l of the specific heat samples except for x=.16 and .30 (which were anomalous) and x=.50 (which was prepared at a voltage far from the extra dx/dV peaks) were made from relatively old NbS2, i t seems likely that whatever aging effect caused changes in the electrochemistry also caused changes in the specific heat. If the aging effect is due to loss of sulfur and subsequent intercalation of excess niobium into the 1 73 interlayer gaps, this might serve to help bind the layers together and reduce the specific heat. The presence of lithium ordering might itself influence the specific heat. It should be noted that both the x=.3 and x=.16 samples had compositions near values where ordering might be expected, since the x=.16 sample was stage 2 and therefore had a lithium concentration near X=1/3 in the f i l l e d gaps. The effects of aging obviously need to be investigated further. 7. SUPERCONDUCTIVITY 7.1 INTRODUCTION Specific heat anomalies due to superconductivity were observed in only three of the samples, NbS2 (Tc=5.7K), Li 23N b sz (3.1K), and Li 25N b S2 (3.1K). The data are reproduced here (table VII), together with the electronic specific heat coefficient 7 . It i s , of course, likely that at least some of the other samples were superconductors, but with Tc's below the temperatures at which the measurements were made. These minimum temperatures are also listed in the table. Some results related to superconductivity in the NbS2 sample have already been presented in section 4.3. Because of the reduction in N(ep), the density of electron states at the Fermi level, which is due to charge transfer into the dz2 band, the general result that superconductivity is eventually destroyed at large x is to be expected. Some of the results, however, clearly cannot be explained on the basis of rigid band charge transfer alone. The stage 2 samples at x=.12 and .16 did not exhibit superconductivity, even though the .16 sample was measured down to 2.OK. This is surprising, since the 7 values for the stage two samples are roughly equal to those of the superconducting samples, indicating that they have about the same N ( e „ ) . A similar situation occurs for the stage one samples at x=.30, .32 and .35, These a l l had 7 values comparable to the superconducting samples, but did not have 174 175 Table VII Superconducting transition temperatures and electronic specific heat coefficients for the Li NbS2 samples. Where no transition was observed, T is listed as being lower than the lowest temperature at which measurements were made. X Tc(K) 7(mJ/mole-K4) 0 5.7 19.3 .13 <2.7 10.9 .16 <2.0 13.1 .23 3.1 11.6 .25 3.1 10.3 .30 <1 .8 9.4 .32 <2.8 11.4 .35 <2.8 10.6 .41 <2.6 6.0 .50 <2.8 5.8 .68 <2.8 4.8 1 .00 None observed 1 .3 superconducting transitions. This puzzling state of affairs will be discussed further at the end of this chapter. In the next section, a Meissner effect measurement on one of the Li NbS2 samples is described. This verifies that the specific heat anomalies observed were truly due to 176 superconductivity. In section 7.3, the present results will be compared with previous data (McEwan 1983, McEwan and Sienko 1982). 7.2 MEISSNER EFFECT MEASUREMENT A magnetic measurement was made on a piece of the Li 25NbS2 sample26. The cryostat used (R.H.Dee and J.F.Carolan, unpublished) contains two identical, oppositely wound, superconducting coils in series. The sample was placed in the center of one of these pickup c o i l s , and a piece of pure indium for calibration purposes in the other. The coils are connected to an RF-SQUID, which, together with i t s associated electronics, produces an output signal proportional to the total magnetic flux through the pickup c i r c u i t . The arrangement is insensitive to uniform magnetic fi e l d s , because the coils are oppositely wound. Whenever either the sample or the indium standard becomes superconducting, i t expells magnetic flux (Meissner effect), and produces a signal. Transitions in the sample and indium can be distinguished because the flux change signals are of opposite sign. Thermometry for the experiment was provided by an Allen-Bradley carbon resistor which had previously been calibrated in the specific heat system by comparison with the germanium thermometers. The magnetic f i e l d in the vicinity of the sample could be varied by means of a 26The experiment was performed by J.Beis and the author 177 solenoid surrounding the pickup c o i l assembly. The fields used were of order a few gauss or less. Data taken during a temperature sweep are shown in figure 50. The transitions due to both the sample and indium can be seen. There is also a slowly varying background, presumably due to paramagnetism in some of the components of the pickup c o i l - sample holder assembly. The indium transition is measured at 3.403K. The accepted value is 3.404K (Weast 1970), which provides a check on the thermometer calibration. The transition of the Li 25N dS2 sample occurred over the range 3.31 to 2.93K, with 50% of maximum flux expulsion at 3.20K. The calorimetrically measured transition was centered at about 3.1K. The siightly'higher magnetic transition is not surprizing, if the width of the transitions is due to inhomogenieties in the sample. This is because a reasonably complete Meissner effect can be seen, even i f a significant part of the sample is s t i l l normal. A l l that is necessary to block magnetic flux is that superconducting regions somewhere in the sample extend a l l the way across a cross-section'perpendicular to the f i e l d . The calorimetrically measured transition, on the other hand, is sensitive to the bulk of the sample. Similar temperature scans were made in several different low magnetic fie l d s , going both up and down in temperature. The transition temperature and width were both independent of the sweep direction and the f i e l d . (The 178 Thermometer Res is tance ( i l ) Figure 50: Meissner effect data on a L i 2 5 ^ ^ ? samP^-e-The horizontal axis is the resistance of the carbon resistor used as a sample thermometer. The temperatures of important features have been calculated: a- Indium transition at 3.403 K. b- 3.31 K: transition in sample 10% complete, c- 3.20 K: transition in sample 50% complete, d- 2.94 K: transition in sample 90% complete. The dashed line is the estimated background. 1 79 solenoid was not calibrated, but the fields used were between zero and a few gauss. The small applied magnetic fiel d was used only to provide some flux to be expelled at the transition. To avoid depressing the transition temperature, the fields used must be much less than the c r i t i c a l f i e l d . Fields of a few gauss are expected to satisfy this condition, and the fact that the transition was independent of f i e l d , for fields of this magnitude, shows that this is indeed the case.) 7.3 COMPARISON WITH PREVIOUS WORK The superconducting transition temperature of LixNbS2 has been measured by McEwan and Sienko (McEwan 1983, McEwan and Sienko 1982). The results are shown in figure 51. The samples were prepared by high temperature reaction, and for X<.13 are phase mixtures of the 2H and 3R crystal types. Samples which were phase mixtures sometimes showed two separate transitions, and this is why the figure has two T 's for some values of x. Li NbS2 prepared by room c X temperature intercalation is 2H at a l l x, and so it is not possible to directly compare the results for x<.13. For x>.13, however, McEwan and Sienko's samples were pure 2H, and in principle should have had the same properties as room temperature prepared material. For x between .13 and .17, McEwan and Sienko's samples exhibited (007) Bragg peaks in x-ray diffaction. Although they did not realize this, this line indicates that these samples were 180 0.0 0.1 0.2 0.3 x in L i x N b S 2 0.4 0.5 Figure 51 T as a c function of x for a series of L i NbS9 samples prepared at high temperatures (McEwan 1983). For samples where no Tc was observed, the symbol T indicates the lowest temperature measured. 181 stage 2, or stage 2-stage 1 phase mixtures (section 2.3). Their result that Tc in the stage 2 region is lower than in the stage 1 region near x=.25 agrees with the results of the present study. As McEwan and Sienko increased x above .17, they f i r s t passed through a region where the presence of two Tc's probably indicates a stage 2-stage 1 phase mixture. Near x=.25, close to the low x limit of the stage 1 phase, they observed one transition at about 3.2K, also in agreement with the present work. Between x=.30 and .35, however, McEwan and Sienko's T values remain in the 3.2 to 3.5K c range, while in this work, the x=.30, .32 and .35 samples showed no superconductivity. This disagreement is not significant, since as mentioned in chapter 2 in relation to lattice parameter data, McEwan "and Sienko's x values appear to be too high (by about .05 or more) in this region. 7.4 DISCUSSION In the BCS theory of superconductivity, (Bardeen, Cooper, and Schrieffer 1957), Tc is given by T c - *£S> exp C Kfi N(eF)V (7-1) where <CJ> is an average phonon frequency, V is the strength of the phonon mediated effective interaction, and N(ep) is the density of states at the Fermi le v e l . Based on this equation, McEwan argued (qualitatively), that i f <o> and V remain constant on intercalation, the rigid band charge 182 transfer model implies that Tc should be a monotonically decreasing function of x. This argument is correct as far as it goes. Although the BCS equation for Tc does not give accurate numerical values, it does correctly identify the general trends. (For a recent review of the theory of T , see Allen and Mitrovic 1982.) As N(ep) decreases due to charge transfer and band f i l l i n g , Tc should drop. Unlike more accurate equations for T , the BCS equation displays the dependence of Tc on N(ep) e x p l i c i t l y . It is therefore possible to use i t , together with rigid band charge transfer, to calculate numerical values for Tc in Li NbS2. Values for V and <u> can be estimated from data on x 2 NbS2. Using N(ec,) from the calculation of Doran, et al r (1976), and setting •n<co>/kB=0D, allows V to be determined by solving (7-1) with Tc equal to the observed value of 5.7K. In chapter 5, values of N(ep) as a function of x were obtained using the rigid band charge transfer assumption. Putting these values into the BCS equation (7-1 ) gives the results shown in figure 52. The results calculated using rigid band charge transfer and the BCS equation do not agree with any of the data. The lack of precise numerical agreement is not serious, since there is no reason to expect it anyway. What is significant is that although the calculation predicts a Tc which decreases almost monotonically as a function of x, McEwan and Sienko's data show a rapid drop at low x, followed by a recovery between x=.1 and .3. McEwan (1983) has explained 183 Figure 52 T as a function of x, using the r i g i d band charge transfer model and the BCS equation for T£ (7-1). 184 the' depression of Tc between x^.02 and .3 in terms of a hypothetical charge density wave which is present only in this composition range. Near X=1/7, some form of lithium ordering along the c-axis with period 7c is supposed to play a role, causing, for example, the lack of any Tc in their x=.15 sample. There are serious problems with this explanation, however. First of a l l , McEwan searched for evidence of the CDW"in the resi s t i v i t y and magnetic susceptibility, and found none. McEwan also argues that Thompson's empirical relation for CDW transition temperatures in layered transition metal dichalcogenides (Thompson 1975) predicts that a CDW should occur. The Thompson relation is based on the c/a ratio of the crystallographic unit c e l l . It was not originally intended for use in intercalation compounds. Considering the drastic changes in the Fermi surface that will be produced by intercalation and charge transfer, there is no real reason to expect that the Thompson relation will apply to intercalation compounds. CDW's are now generally believed to be related to Fermi surface nesting (Wilson, et al 1975, Friend and Jerome 1979), and although there may well be a simple relation between Fermi surface geometry and unit c e l l geometry within a group of materials with similar structures, this will probably break down as soon as intercalation raises the Fermi level. Even if a CDW does occur, i t must be a relatively low amplitude, low temperature one, similar to that in NbSe2. Otherwise it 185 should have been easily observable. In NbSe2, the charge density wave transition temperature T0 is 33K and T"c is 7.2K. Data on the pressure dependence of T"c (Berthier, et al 1976) gives an idea of the magnitude of the effect that the CDW has on T . The application of pressure decreases T0 and increases T , until at 36kbar T0 disappears and Tc is 8.2K. Further increase in the pressure has l i t t l e effect on T . This can be understood by saying that the CDW opens gaps on the Fermi surface, reducing N(ep) and depressing Tc (from 8.2 to 7.2K). Destroying the CDW with pressure then raises T . The fractional depression of Tc (=*12%) by the CDW in NbSe2 is much smaller than the 50% differences between T 's ' c in Li NbS2 near x=.13 and .25. It may not be possible for a very weak CDW to produce the Tc variations observed. McEwan's evidence for the c-axis lithium ordering is the presence of the (007) Bragg peak, and, as we have seen in section 2.3, this is actually due to a simple stage 2 structure. Ordering along the c-axis with period 7c would give rise to (0,0,1/7) and related peaks, not (007) ones. A correct and complete explanation of the behavior of Tc as a function of x is not available at present. As we have seen, the rigid band charge transfer model alone cannot explain the data. The behavior of Tc as a function of x should be influenced as well by the large changes in the phonon modes which are caused by intercalation (chapter 6). The BCS equation tends to overemphasize the importance of the density of states and in any case is valid only in 186 the limit of weak electron phonon coupling ( X « 1 ) . Since NbS2 and its intercalation compounds have the coupling constant X approximately equal to 1.8 (chapter 5), we are clearly in the regime of 'strong coupling' superconductivity and should really be using a different equation for T . The most widely used such equation is the 'modified McMillan equation' (McMillan 1968, Allen and Mitrovic 1982).' This is •ft GO J . O 0 -exp T = l°K -1.04(1+A) 1.2kfi [_ (1+0.62 ) (7-2) where ^ *s a l° 9a rit n mic average phonon frequency, and u is an adjustable parameter of order .1 which represents Coulomb repulsion. This equation was used by Aoki, et a l , (1983) to discuss superconductivity in NbS2 intercalated with organic molecules. Another approach which works well * for many materials with 1.2<X<2.4 and . 1£/i £.15 (the usual range) is the empirical relation of Leavens and Carbotte (1974) T =0.1477A (7-3) c where A is the area under the electron-phonon coupling spectrum a2F(u), that is oo A = /a2F(co) dco (7"4> 0 The interaction spectrum a2F(co) is a dimensionless measure of the effectiveness of phonons of frequency u in scattering electrons between different points on the Fermi surface. It can be obtained from tunneling experiments. The structure in 1 87 a2F ( c o ) generally bears a close resemblance to the phonon density of states. The coupling constant X (also known as the mass enhancement factor), can be related to the interaction spectrum by x = 2 / ^ a2F(u)) ( 7 _ 5 ) 0 w for isotropic materials (Allen and Mitrovic 1982). Although there is insufficient information to actually use the Tc equations (7-2) or (7-3), they do indicate that the phonons and electron-phonon coupling are extremely important in determining T . It is clear from the lattice specific heat data that intercalation produces large changes in the phonon spectra, although the low freqency acoustic phonons important in the specific heat are not necessarily the most important in determining T . The mechanism for the Tc variations in LixNbS2 might be revealed by measurements of phonon spectra by tunneling or inelastic neutron scattering. It is not clear, however, that samples of suffient quality to allow these measurements could be prepared using the present methods. 8. CONCLUSION 8.1 SUMMARY OF THIS THESIS Most of this thesis is concerned with measurements of the low temperature specific heat of LixNbS2. This is the f i r s t low temperature specific heat study of lithium intercalation in a layered transition metal dichalcogenide. Sample preparation was carried out by intercalating lithium into NbS2 in electrochemical c e l l s . Electrochemical and x-ray diffraction measurements were used to study the structure of LixNbS2. Stage 2 and stage 3 phases were identified for the f i r s t time. In addition, there is some preliminary electrochemical evidence for in-plane lithium ordering near X=1/3. The extra peaks in -dx/dV which suggest ordering were seen only in electrochemical cells made from freshly prepared NbS2. The changes in electrochemical behavior may happen because of sulfur loss during storage. A cryostat suitable for specific heat measurements on small samples of air sensitive compounds was b u i l t . I t , and the experimental procedure, were described in chapter 3. Measurements were made on NbS2 and eleven Li NbS2 samples, covering the range 0<x^1. The original reason for doing this work was to test the rigid band charge transfer model of the electronic properties of intercalation compounds. The results for the electronic specific heat are consistent with complete charge transfer from the intercalated lithium atoms to the bands of 188 189 the NbS2 host. Because the electronic specific heat of Li NbS2 is determined by the f i l l i n g of the original NbS2 bands, the data provide information on the electronic density of states of NbS2. In particular, a shoulder in the density of states predicted by earlier band structure calculations was reflected in the data, and its position was determined. There were also large changes in the phonon specific heat as a function of x. In chapter 6, we showed that the configurational degrees of freedom of the lithium will not contribute to the low temperature specific heat, since lithium motion will be 'frozen out' at temperatures below about 100 K. Simple models of the vibrational motion of intercalated lithium show that the new vibrational modes due to the addition of lithium are at high frequencies, and will not be seen in the specific heat. Because of this, the data could be discussed in terms of an elastic continuum model of lattice vibrations. The results suggest that intercalation induced changes in the elastic constant c4fl associated with rigid layer shear are primarily responsible for the changes in phonon specific heat as a function of x. For x less than about .3, Cnn is significantly lower than in pure NbS2, indicating that small lithium concentrations between the layers weaken the interlayer shear forces. At higher x, c«ft increases again, and by x=1 is larger than in NbS2. This suggests that the bonding in Li1NbS2 is more three dimensional than in NbS2. 190 Superconductivity in Li NbS2 is discussed briefly in" chapter 7. It is shown that the variations in N(ep) due to rigid band charge transfer are not sufficient to explain the available data. Intercalation induced changes in the phonon spectrum and the electron-phonon interaction must also be involved. 8.2 SUGGESTIONS FOR FUTURE WORK The electronic specific heat of LixNbS2 is now reasonably well understood in terms of rigid band charge transfer. The electronic specific heat of other intercalation systems might be of interest. For example Ti S2, which has an empty dz2 band and is either a semiconductor or semimetal, becomes metallic on intercalation. Li TiS2 would show an electronic x ^ specific heat which would increase with x. This might be a good system in which to make detailed comparisons between the data and the predictions of band theorists, since very detailed and supposedly accurate calculations are available for both TiS2 and Li,TiS2 (Umrigar, et al 1983, McCanny 1979). A class of intercalation hosts for which simple rigid band charge transfer will not work at a l l are MoS2 and NbyMo.]_yS2 (0<y<l). These compounds undergo structural phase transitions when lithium is added. Py and Haering (1983) suggest that the transition is driven by the electronic energy. Electronic specific heat measurements on samples with lithium concentrations near the value at which the 191 transition occurs could improve our understanding of this process. The lattice specific heat data show that there is probably a softening of the shear elastic constant cqfl in samples with 0<x<.3. It is of interest to know whether this is typical of a l l lithium intercalated transition metal dichalcogenides, or is peculiar to LixNbS2. As seen in chapters 2 and 6, aging the NbS2 seems to have had an effect on the electrochemical properties and lattice specific heat of Li NbS2 prepared from i t . We have suggested that this may be due to loss of sulfur. It may be possible to test this hypothesis directly by preparing the non-stoichiometric compound Nb1+yS2, with a small well controlled amount y of excess niobium, and then intercalating this with lithium. (It may not, however, be possible to get this to grow in the 2H phase - see Fisher and Sienko 1980.) Another, rather time consuming, approach would be to start with the stoichiometric compound, and make a systematic study of the effects of aging and storage conditions on the electrochemical properties. A series of specific heat measurements on samples prepared from 'fresh' NbS2 would be of interest. By 'fresh' NbS2, we mean material which shows the extra peaks in -dx/dV, and which yields specific heat samples which behave like the x=.30 and .16 samples in the present study. It is not yet clear whether the high /3 values of the x=.30 and .16 samples are related to lithium ordering which was not 192 present in the other samples, or to the presence of interlayer excess niobium i t s e l f . 'Fresh' samples might, in the f i r s t case, show peaks in (J at x values corresponding to the ordered states, or, in the second case, they might have higher /3 values throughout the low x range. The interpretation of the phonon specific heat data in terms of intercalation induced changes in the elastic constant cM suggests a possible application for Li NbS2 or related intercalation compounds. Layered materials, chiefly graphite and MoS2, are widely used as solid lubricants. Their lubricating ability is related to the fact that because of the relatively weak interlayer interactions, the layers can slide over one another in response to a mechanical force. When some powdered MoS2, for example, is placed between two sliding surfaces, layer slipping allows it to spread into a smooth lubricating film. A low value of c„„ means that the interlayer shear forces are weak, and that layer slipping can occur easily. If we assume that the other elastic constants have values close to the ones we used for NbS2, the specific heat results for Li 1gNbS2 imply that i t had c42 • 1 01° N / m2. For comparison, graphite and MoS2 have C„ 4 • 1 0 1 °N/m2 and 1 . 9 • 1 0 1 °N/m2 , respectively (Wakabayashi and Nicklow 1979, Feldman 1976). It is possible, therefore, that Li NbS2 with 0<x<.4 may be a superior solid lubricant for c r i t i c a l applications, and its lubricating properties should be investigated. Although the presence of reactive lithium might seem to rule out the 193 practical use of Li NbS2, solid lubricants are often used mixed with grease or o i l , and this might be sufficient to protect the intercalation compound from a i r . It may also be that layer compounds intercalated with other, less reactive, species w i l l show similar behavior. As mentioned in chapter 7, the superconducting properties of Li NbS2 remain something of an enigma. It should be pointed out that the superconducting transition temperature of Li NbSe2 varies as a function of x in a way that is very different than for LixNbS2 (McEwan 1983). McEwan's measurements on LixNbSe2 were made on samples prepared at high temperatures. The samples had the 2H structure, and the presence of (007) and (009) Bragg peaks in the x-ray diffraction patterns for some of the samples indicates the presence of a stage 2 phase similar to that in Li NbS2, and in Li NbSe2 prepared by electrochemical X X intercalation (D.C.Dahn and R.R.Haering 1982). The band structure of NbSe2 is very similar to that of NbS2 (Wexler and Wooley 1976). 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Wilson, J.A., and Yoffe, A.D. 1969. Adv. Phys. ^8, 193. Yoffe, A.D. 1973. Ann. Rev. Mat. Sci. 3, 147. 199 Yoffe, A.D. 1982. Ann. Chim. Fr. ser.15, 7, 215. Zemansky, M.W. 1957. Heat and Thermodynamics, McGraw-Hill, N.Y. p394. APPENDIX 1; INTENSITIES OF X-RAY BRAGG PEAKS FOR STAGE TWO Li NbS2 —— The purpose of this appendix is to briefly outline a calculation of the Bragg peak intensities for stage 2 LixNbS2, to present the results, and to compare them with measurements on Li ^NbS2 made using an electrochemical c e l l with a beryllium x-ray window. Portions of this data have already been presented in figure 6(d). The hexagonal lattice parameters were c=12.35 A and a=3.330 A. To calculate the intensities we need to know the positions of the atoms in the unit c e l l . Atom positions will be given in terms of the three basis vectors c, a,, and a2. c has length c and l i e s in the direction normal to the layers, a, and a2 both have length a, l i e in a plane parallel to the layers, and make an angle of 120° with each other. In the notation to be used (p,q,r) indicates a position pa1+qa2+rc. Picking the origin to l i e halfway between the two niobium atoms in the unit c e l l (that i s , at one of the octahedral sites in the gap), the atom positions for the 2H-NbS2 structure are (see figure 2): Nb at ±(0,0,-1/4) S at ±(1/3,2/3,z), ±(1/3,2/3,1/2-z) where z is defined in figure A1 — 1. These atom positions lead to a structure factor of zero for (00/) Bragg reflections with / odd. The exact value of z for NbS2 is not known, but, as in most of the 2H transition metal dichalcogenides, it is approximately 1/8 (Hulliger 1976). The extinction of (00/) 200 201 peaks with / odd does not depend on the value of z. To get atom positions for stage 2 Li NbS2, we start by taking Z=1/8 in pure NbS2. We then make the assumption that the change in the c-axis in going from NbS2 to stage 2 Li NbS2 is due entirely to the expansion of only one of the two interlayer gaps in the unit c e l l (we choose the one at the origin). This is shown schematically in figure A1 — 1. Since the scattering factor of lithium is very low, the inclusion of the lithium in the calculation makes l i t t l e difference in the results. The most important factor determining the intensities is the gap expansion. For completeness, however, we will include the intercalated lithium, and will assume it lies in the octahedral sites in the expanded gap. There is not a lithium atom in every unit c e l l , so for the purposes of the intensity calculation we take the scattering factor at the lithium site to be 2x times the scattering factor of a lithium atom. Here x is the overall value in Li NbS2. Since- every second gap is empty, the concentration in the f i l l e d gaps is 2x. In terms of the expanded unit c e l l , the atom positions in stage 2 Li NbS2 are: Nb at ±(0,0,-.257) S at ±(1/3,2/3,.136), ±(1/3,2/3,.379) 2x Li at (0,0,0) Starting with these atom positions, the integrated intensities of the peaks were calculated using a computer 202 b • - Nb O - S 0 - L i Figure Al-1: Projections onto a (110) plane of the unit cells of a- NbS2 b- stage 2 L i NbS9. (not to scale) 203 program written by R.Marselais2 7. The program begins by calculating the structure factor F in the usual way. Atomic scattering factors from Ibers and Hamilton (1974) were used. The measurements were made with a Philips PW1050/70 vertical goniometer, which uses the Bragg-Brentano pseudofocusing geometry shown in figure A1-2. For a goniometer with an angular divergence 6 of the incident beam, and a diffracted beam monochromator, the integrated intensity of a Bragg peak centered at angle 28 is (for a thick sample) l+cos22 8cos22(^ I = I0m|F|26 _ sin6sin2cj) (Al-1) where I0 is a constant, m is the multiplicity of the reflection, and the angles 8 and <t> are defined in the figure. The results of calculations using (Al-1) are listed in table A1-I for peaks at angles 2 0 < 9 O°. They are in the column labelled 'standard intensity'. In our diftactometer, 8 is not constant because of a Philips PW1386/50 automatic divergence s l i t , which instead of providing a beam of constant divergence as would a fixed s l i t , keeps the illuminated area of the sample approximately constant. In addition, when using an x-ray c e l l , the intensities must be corrected for absorption in the beryllium window. Corrections for these effects have been discussed by Py and Haering (1983) and J.R.Dahn (1982), and are included in the computer program. The corrected 27XBAT:SPECTRUM.S 204 Figure Al-2: The Bragg-Brentano focusing geometry. In most diffractometers the sample is f l a t , not curved. Because the sample dimensions are much less than R, the focusing condition is s t i l l approximately satisfied, hence pseudofocusing. 205 intensities are also included in the table, as are the diffaction angles 26 calculated for copper Ka radiation. Detailed comparison of a l l the intensities with experiment was not done. This would be d i f f i c u l t because of preferred orientation effects (the calculation assumes random orientation in the powder sample), and in any case we are primarily interested in the (00/) peaks, especially those with / odd. Calculated and observed relative integrated intensities for the (00/) lines are given in table A1-II. Preferrred orientation should have an equal effect on a l l of the lines in this group. The agreement between calculated and experimental values is reasonably good. The calculation shows why, of a l l the (00/) peaks with / odd, only the (007) and (009) peaks have been observed in stage 2 Li NbS2 and stage 2 Li NbSe2 (this work, McEwan X X 1983, D.C.Dahn and Haering 1982). The (009) peak was considerably wider than the other observed (00/) peaks. The f u l l width at half maximum of the (009) peak was ^.7° in 26, while the (008) peak, for example, was .3° wide. This indicates some disorder in the staging sequence. It can be shown (J.R.Dahn 1982), that staging disorder broadens the (00/) lines with / odd much more than those with / even. The experimental (10/) lines had relative intensities in qualitative agreement with the calculation. Since these lines depend strongly on the stacking sequence of the NbS2 layers, this agreement indicates that the host stacking 206 sequence is not affected by intercalation (that i s , it is s t i l l BaB-CaC in the notation of section 1.2). 207 Table A1-I Calculated integrated intensities and angles for stage 2 Li NbS2 (see text). 29 h k 1 (degrees) 0 0 1 7.15 0 0 2 14.34 0 0 3 21 .57 0 0 4 28.90 1 0 0 30.98 1 0 1 31 .84 1 0 -1 31.84 1 0 2 34.29 1 0 -2 34.29 0 5 36.35 1 0 3 38.07 1 0 -3 38.07 1 0 4 42.88 1 0 -4 42.88 0 6 43.97 1 0 5 48.49 1 0 -5 48.49 0 7 51 .79 1 0 6 54.76 1 0 -6 54.76 1 1 0 55. 1 1 1 1 1 55.65 1 1 2 57.26 1 1 3 59.87 0 8 59.88 1 0 -7 61 .58 1 1 4 63.42 2 0 0 64.58 2 0 1 65.07 2 0 -1 65.07 2 0 2 66.54 2 0 -2 66.54 1 1 5 67.83 0 0 9 68.32 2 0 3 68.95 2 0 -3 68.95 1 0 8 68.96 1 0 -8 68.96 Standard Corrected Intensity Intensity (relat ive) (relat ive) 0.74 0.07 100.00 48.08 0.40 0.39 1 .78 2.71 20.76 34.85 7.87 1 3.73 8.54 14.90 31 .58 61 .04 31.21 60.32 0.07 0.14 8.89 1 9.78 0.82 1 .82 27.49 71 .52 38.44 100.00 4.00 10.75 6.42 1 9.56 0.33 1 .00 0.82 2.71 7.95 28.16 8.75 31.01 24.21 86.46 0.03 0.12 15.23 57.00 0.16 0.63 5.27 20.82 2.21 9.04 1 .55 6.58 2.59 11.21 0.89 3.90 0.81 3.54 4.34 1 9.42 4.38 1 9.64 0.07 0.33 0.58 2.69 0.11 0.53 1 .39 6.48 3.25 15.17 0.86 4.04 208 Table A1-I (Continued) Standard Corrected 29 Intensity Intensity h k 1 (degrees) (relat ive) (relative) 2 0 4 72.26 7.77 38.24 2 0 -4 72.26 5.59 27.53 1 1 6 73.04 5.18 25.81 2 0 5 76.43 0.08 0.42 2 0 -5 76.43 1 .62 8.48 1 0 9 76.89 0.07 0.40 1 0 -9 76.89 1 .61 8.47 0 0 10 77.20 1 .04 5.53 1 1 7 79.02 1 .40 7.58 2 0 6 , 81.43 2.77 15.50 2 0 -6 81 .43 2.51 14.07 1 0 1 0 85.48 1.81 1 0.63 1 0 -10 85.48 1.31 7.70 1 1 8 85.77 11.81 69.62 0 0 1 1 86.67 0.11 0.64 2 0 7 87.26 0.87 5.21 2 1 0 89.93 2.19 13.51 209 Table A1-II Integrated (00/) intensities for stage 2 LixNbS2. The (007) peak was not observed because i t s position coincided with.a beryllium Bragg peak originating in the c e l l window. The (002) peak may have been partially obstructed by the c e l l case. Calculated Corrected Intensity Observed Intensity Peak (relative to (008)) (relative to (008) ) (001 ) 0.35 out of range (002) 230 150 ' (003) 1 .9 not observed (004) 13. 1 2 (005) 0.67 not observed (006) 51 70 (007) 1 3 obscured by Be (008) 100 1 00 (009) 1 3 1 5 (00JJ0) 27 20 (001 1 ) 3.0 not observed APPENDIX 2; SOLUTION OF THE HEAT FLOW EQUATIONS This Appendix contains the solution of the heat flow problem for a relaxation time heat capacity measurement in which the sample's thermal conductivity is f i n i t e . Some of the results are given in Bachmann, et al (1972), but it will be useful to outline the derivation here. Also, in the cryostat used in this work, the heat capacity of the wires which support the sample platform is very small compared to the heat capacities of the sample and the platform. This leads to a useful simplification of the resulting equations, which is also discussed. Consider the system of figure A2-1. If the thermal conductivity of the platform is high enough that it is always essentially isothermal, and if there is no heat loss from the sample's edges, the temperature in the sample will be a function only of z, the coordinate normal to the plane of the platform. In this case, we have a one-dimensional heat flow problem. It is convenient to work in terms of a relative temperature where T0 is the temperature of the block. The heat equation for an inhomogeneous system is then (Carslaw and Jaeger where s(z) is the heat capacity per unit length and K(z) is 0(z,t)=T(z,t)-To (A2-1) 1959) (A2-2) 210 211 R e f e r e n c e Block (TQ) Figure A2-1: Model system for heat flow calculations. 212 the (1D) thermal conductivity. The thermal diffusivity K(z)/s(z) is constant in each of the three parts of the system and can be described by s(z)/K(z)=Cw/kwL2 0<z<L s(z)/K(z)=Cp l/kp l/2 L<z<z0 (A2-3) s(z)/K(z)=Cs/ksl2 z0<z<z, where the C's and k's are, respectively, the appropriate heat capacities and thermal conductances of the parts. At z=0, the temperature is fixed at the block temperature, and at z=z, there is no heat flow, so the boundary conditions are 0(O,t)=O (A2-4) 90(z,,t)/9z=0 (A2-5) At the internal boundaries z=L,z0 we require that the heat flow be continuous: K(z) 90(z,t)/9z continuous at z=z0,L (A2-6) If there is no thermal contact resistance at either of the boundaries, we also have 0(z,t) continuous at z=z0,L (A2-7) The case of a non-zero thermal boundary resistance (e.g. a poor grease joint) has been treated by Bachmann, et al (1972). To solve (A2-2), we begin by separating variables 0(z,t)=0(z)<//(t) (A2-8) 213 Then, defining the separation constant as -1/T, s(z) dz L_ dzj 1_ 1_ [ ^( z )d i l dt 1 (A2-9) T which gives </>(t)=e -t/r (A2-10) and dtj> dz + s(z) T 0 (A2-11) Equation (A2-11) and the boundary conditions define a Sturm-Liouville system. The eigenfunction solutions will be called 0n and the corresponding eigenvalues T . Using the homogeniety of the boundary conditions, i t can be shown that the. eigenfunctions are orthogonal with respect to s(z), that is , (Strictly speaking, we must also require s(z) and K(z) to be continuous; for this reason (A2-3) should be considered only an approximation to the actual s(z) and K(z) which change rapidly but continuously at z=l and z=z0.) The next step is to find the <t> . The following form z fs('z)tf>n(z)4>m(z)dz=0; 0 m*n (A2-12) 214 satisfies (A2-11) and the boundary conditions; <t> (z) = sinX z 0<z<L n n <p (z) = a COS7 z + b cosy L<z<z0 (A2-13) n n n V>n(z) = d cos(in(z,-z) z0<z<z, where (XnL)2=Vkw7n ( V ) 2 = C p l / k p l T n ( A 2 ~ 1 4 ) and a, b, and d are constants to be determined from the internal matching conditions (A2-6) and (A2-7). These conditions yield the equations sinX L = a sin7 L + b C O S 7 L n n n d cosjunl = a sin7n(L+/) + b cos7n(L+/) (A2-15) X k L cosX L = a7 k ,/ C O S 7 L - b7 k , / sin7 L n w n n pi n n pi n dM k 1 sinji 1 = a7 k , / C O S 7 (L+/) n s n 'n pi 'n - b7 k ,/ sin yn(L+l ) ' n pi n We wil l be interested in the case kpi*0 0 (or 7n-*0)» so the problem can be simplified by expanding the right hand sides of these equations to order 7n2. Doing this and solving the f i r s t three equations for a, b, and d gives a - 7nL sinXnL + (VVrn/Cpl) cosXnL b - sinXnL(1-72L/2) - (V^V T n / C p l > 7 n c o s X n L (A2-16) d sinXnL/cosjunl 215 Substituting these into the fourth equation of (A2-15) and dropping terms of order 7 * gives the eigenvalue equation -(Cw/XnD cotXnL + (Cs/MnD tanM nl + cp l = 0 (A2-17). which is valid in the limit k (This equation can also be derived by taking kp^=°° at the start of the problem, that i s , taking the sample platform to be isothermal. In this case, the internal matching condition (A2-6) is meaningless and must be replaced by a heat balance equation for the platform, which then reduces directly to (A2-17).) Therefore, the eigenfunctions 4>n are given by <t> = sinX z 0^z<L n n <t> = sinX L L<z<z0 (A2-18) n n <j> = sinX L cosu (z ,-z) /cosu 1 z0^z<z^ n n n n where u and X are defined in terms of T by (A2-14) and T n n n •* n is the nfc^ solution of the eigenvalue equation (A2-17). The solution to the entire time-dependent heat flow problem is then 0(z,t) = I An«n( z ) e Z/Tn (A2-19) n=l where the coefficients A are determined by the i n i t i a l n conditions. In the case of a relaxation time heat capacity measurement, we take t=0 to be the time the sample heater is switched off. Heat has been supplied to the sample platform at a rate P for t<0. Thus the i n i t i a l condition is 216 0(z,O)=0o=P/kw L<z<z, (A2-20) 0(z,O)=0z/L 0^z<L So, since the #n form an orthogonal basis, / dz s(z)9(z,0)ct)n(z) A = 0 . (A2-21) / dz s(z){cb (z)}2 0 Evaluating the integrals and using (A2-14) and (A2-17) gives 2e0c . c _ c , - l A = (l+tan2ynl + —(1+cot 2AnL) + ) n C A2L2sinX L C " C s n n s s (A2-22) which, together with (A2-14), (A2-17), (A2-18), and (A2-19), completely specifies the solution. Bachmann, et al (1973) outline a method of data analysis which is based on a numerical solution of equations (A2-17) and(A2-22) with n=1. It uses as input data the values of Cw, ky, C Ti» a n^ h)<t> ^  (L)/60. and Cw must be known beforehand; the other values can be determined from a f i t to the thermal decay data. (h^<p^{L) is available from the data because the temperature is measured at the platform, that i s , at z=L.) In the cryostat used for this work, however, the wire heat capacity Cw is not accurately known. It i s , however, much smaller than either the platform heat capacity Cp^ or the sample heat capacity Cg. Rough estimates also show that C /k r,=(X,L)2 is typically about .02, and that C /C is of W W s w 217 order 100. It is therefore useful to examine the equations in the limit where Cw and X,L are small. Setting tanX^-X^ in (A2-22), and using the fact that C « C , gives AiSinAiL 2k n (A2-23) 0o Cg(l+tan2yil) + y i + Cp l Treating the eigenvalue equation (A2-17) the same way and solving for Cg gives Cs=(kwT1-Cp l)^il/tanM,l (A2-24) Substituting this into (A2-23) gives (A2-25) AiSinXiL 2k T i w 0o yxKcotyil + tanu a ) (kWT ! - Cp]_) + kWT x + Cp l Section 3.4.2 describes how (A2-24) and (A2-25) were used in analysis of LixNbS2 specific heat data. In the notation used there, A,sinX,L is 0,. 

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