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'I '1111 I 11‘I 1'1I"I:I'I',I'11" III"'I1II'II II I' 'IIII1' 'J LIBRARY Michigan State University This is to certify that the thesis entitled DE HAAS—VAN ALPHFN STUDY OF THE AuGaZ ALLOY PHASE presented by John Jesse Higgins has been accepted towards fulfillment of the requirements for Ph.D. degreein Physics Physics flflfiw/Ax. /,;(/ mt professor [hue 18 November 1°77 0-7 639 "V "I’ '5 yo '5 Po - . DE HAAS-VAN ALPHEN STUDY OF AuGa2 ALLOY PHASE By John J. Higgins A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1977 . ..‘4 _V ..L. tincture e Q U -.’v-' \ .~V .. L. ._ .c ABSTRACT DE HAAS-VAN ALPHEN STUDY OF THE AuGa2 ALLOY PHASE BY John J. Higgins The de Haas-van Alphen frequency (F) of the third zone neck orbit (C5), the Dingle temperature (TD), and the residual resistance ratio (RRR) of the intermetallic compound AuGa2 were measured on samples cut from the top, middle, and bottom of three single crystals to see if these parameters vary over the range of composition in which the fluorite structure exists. The three crystals, 4 to 6 cm long, were grown by the Bridgman method from melts that deviated from stoichiometry by having excess Au (.287 at.% from stoichiometry) and excess Ga (.203 and .549 at.%). F ranged from 3388 RC to 3384.2 kG to 3405.7 kG. Assumption of the rigid band model implies for these samples a concentration range about stoichiometry less than .06 at.% Ga. But analogy to the case of Pd impurity associates a value between .22 and .43 at.% Ga with the range of F. An independent analysis of uncertain reproducibility indi- cates a concentration difference of .351510 at.% between two groups of samples. Stoichiometry does not coincide exactly with the congruent point. The deA phase constant was found to be .46 :_.09. RRR and TD varied from 55 to 1800 and from 5.0 K to 1.1 K, respectively. ‘ ‘ ' .1 2:: 7.15 CC-unber .5 :“ . C..d A.i{\6d 1n T‘a ,. ‘12? CA . L'dr the in a is ACKNOWLEDGMENTS I wish to thank Professor Peter A. Schroeder for his guidance and assistance during the course of this research; and Professor F. J. Blatt, for his counsel in preparing a preliminary report. I am grateful to Brent Blumenstock for his diligent assistance in measuring the residual resistance ratios, and to Boyd Shumaker, for growing the crystals and preparing samples for analysis. I wish to express my appreciation to the physics department machinists, who manufactured much of the cryogenic apparatus and to the pe0ple in the electronics shop, especially Dan Edmunds, who donated several overtime evenings towards improving the NMR system. Vivian Shull, MSU micrOprobe specialist, and Professor W. C. Bigelow of the University of Michigan, and his students Jerry Hoffman and Sumio Ono were generous with their time in guiding and running the micro- probe experiments. I am indebted to my wife, Debbie, who typed most of the first draft and inked in many of the drawings, and for whom, in partial indemnifica- tion for the intrusion of this work into nearly every aspect of her life, a special copy has been reserved for her to tear apart. The support of the National Science Foundation is gratefully acknowl- edged. ii "if OF TABLES . "v 1.: r:.~":':< ,. cots UL v~.\¢.uo 1 1:22": u... .mk Vfl'YY'V'T "i" .M. .A.:“..VI‘ 1.2 P 1.3 Pr LI. an s “r; 2-1 Pr 2-2 Fe: 2.3 CI“ 2.4 F 2.5 (J. ~11 I ' BASIC Iii-3F: TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . vi LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . . . . . . . viii CHAPTER I. MOTIVATION AND PURPOSE. . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . DeHaas-van Alphen effect in studies of metals . A metallurgical question. . . . . . . . . . . . 1.2 Hypotheses and Objectives. . . . . . . . deA effect and concentration . . . . . . . . . Phase diagrams and crystal growth . . . . . . . 1.3 Proposed Program and Summary of Results. . . . . . OOOOCDNHt—I II. FERMI SURFACE OF AuGaz. . . . . . . . . . . . . . . . . . . 11 2.1 Preliminary Definitions. . . . . . . . . . . . . . 11 Ideal crystal . . . . . . . . . . . . . . . . . 11 Perfect crystal . . . . . . . . . . . . . . . . 11 Static crystal. . . . . . . . . . . . . . . . . 12 Real crystal. . . . . . . . . . . . . . . . . . 12 Free electron model . . . . . . . . . . . . . . 12 Coordinate systems. . . . . . . . . . . . . . . 13 Magnetic field nomenclature . . . . . . . . . . 15 Sign of charge carriers . . . . . . . . . . . 17 2.2 Fermi Surface of an Ideal Crystal. . . . . . . . . 17 Empty lattice model . . . . . . . . . . . . . 17 Limitations of the empty lattice model. . . . . 18 Methods to include the ionic potentials and spin-orbit interactions. . . . . . . . . . . 19 2.3 Crystalline Structure of AuGa2 . . . . . . . . . 20 2.4 Fermi Surface of AuGa . . . . . . . . . . . . 21 Empty lattice Fermi surface . . . . . . . . . . 21 Band structure. . . . . . . . . . . . . . . . . 26 2.5 Orbits on the Fermi Surface. . . . . . . . . . . . 26 III. BASIC THEORY OF THE DE HAAS-VAN ALPHEN EFFECT . . . . . . . 30 3.1 Two Aspects of Electronic Magnetization. . . . . . 31 Moving charges. . . . . . . . . . . . . . . . . 31 Thermodynamic approach. . . . . . . . . . . 33 Implications for the deA effect. . . . . . . . 34 iii .u A 2G E“ .L 3 3 at .VL ..¢ MUM n\. T. YJ Pkg 6 .I GO 9 o o o 3 3 3 3 h s 7.. /% a he .1“ \ y y a .1 .. T. F .. a P .l . 9 I a m. \u . 1 NIL S 1 y L . . t o \. .xJ .3 n. s.“ V 6 ms . a; Mu. Vi .. CHAPTER 2 History of the de Haas-van Alphen Effect . . . . .3 Landau Levels. . . . . . . . . . . . . . . . . . . Free electron model . . . . . . . . . . . . . . Degeneracy. . . . . . . . . . Periodicity . . . . . . . . . Real crystals . . . . . . . . . . . . . . . . . 3.4 Quantization of Orbits . . . . . . . . . . . . . . Derivation. . . . . . . . . Cylinders of orbits . . . . . . . . Periodicity and deA frequency. . . . . . Degeneracy. . . . . . . . . . . . . 3.5 Temperature and Scattering Effects . Nonzero temperature . . . . . . Scattering. . . . . . . . . . . Spin-splitting . . . . . . . . . . . . . . . Lifshitz-Kosevich Theory . . . . . . . . . . . . Justification of Semiclassical Theory. . . . . . . Corrections to Semiclassical Theory. . . . . . . Magnetic interaction (Many-body effects). Demagnetization . . . . . . . . . . . . . . . Magnetic breakdown. . . . . . . . . . . . . wwww \OmNG IV. DE HAAS-VAN ALPHEN EFFECT AS A PROBE OF CRYSTAL COMPOSITION 4.1 De Haas-van Alphen Frequency . Rigid band model. . . . . . . . . Relation to AuGa2 . . . . . . . . . . . . 4.2 Dingle Temperature . . . . . . . . . . . . . 4.3 Considerations for Precision . . . V. EXPERIMENTAL TECHNIQUE FOR THE DE HAAS—VAN ALPHEN EFFECT. 5.1 Techniques Available . . . . . . . . . . . . . . . Torque. . . . . . . . . . . . . . . . . . . . . Pulsed fields . . . . . . . . . . . . . . . . . Field modulation. . . . . . . . . . . . . . . . Other techniques. . . . . . . . . . . . . . . 5. 2 Field Modulation Technique . . . . . . . . . . . . Magnetic field variable . . . . . . . . . . . . Voltage . . . . . . . . . . . . . . . . . . . Spectral analysis . . . . . . . . . . Data reduction. . . . . . . . . . . . . . . . . VI. DE HAAS-VAN ALPHEN APPARATUS. . . . . . . . . . . . . . . . 6.1 Probe. . . . . . . . . . . . . . . . . . . . . . Flange and support tube . . . . . . . . . . . . Mobile base . . . . . . . . . . . . . . . . . . Control tubes and motor . . . . . . . . . . . . Sample holder assembly. . . . . . . . . . . . . deA coils. . . . . . . . . Coil winding and specifications . . . . . . . Electrical lines; Heat Leaks. . . . . . . . . iv 34 37 37 38 40 42 42 42 43 44 47 50 50 51 53 54 59 62 62 64 64 69 69 71 72 74 76 78 78 80 80 82 82 83 83 90 94 97 98 102 103 104 106 108 110 118 a can; w". gun p.) to IC LI’ {'1 O‘O‘O‘ 1.‘ v.7 to?:?v\,r‘u¢; ‘ALO ho u:\&...—.\.o 7.1 C: 7.2 0: 7.3 SR 7.4 D;- TIZI. GE‘JXTPX A‘JD 8.1 T 8.3 V 8.3 yL Ty . “‘° CPRACTERI" 9.2 Ci 9-3 R t-—-‘ O H ('3 ru (L (L CHAPTER 6. 2 Electronics. . . . . . . . 6. 3 Superconducting Solenoid and Dewar . . . . . . . 6. 4 Magnetic Field Measurement . Nuclear magnetic resonance. Calibrated resistors. VII. EXPERIMENTAL PROCEDURE AND DE HAAS-VAN ALPHEN DATA. 7.1 General Procedure. . . . . 7.2 Orientation. . . . . . 7.3 Skin Effect. . . . . . 7.4 Data Reduction . VIII. GROWTH AND PREPARATION OF SAMPLES . 8.1 Theory of Freezing for Solid Solutions . . . . . Phase equilibria. . . . Homogeneity . . . Growth kenetics . Hybrid approach . 8.2 Methods. . . . . . Growing . . . . . . Cutting . . . . . . . . . . X-Raying. . . 8.3 Melt Concentrations. IX. CHARACTERIZATION OF SAMPLES . 9.1 Anticipated Difficulties . 9.2 Direct Analysis. . . . . Chemical analysis . Microprobe. . . . . . Other methods considered. 9.3 Residual Resistance Ratio. . Relation to concentration and ture . . . . . . . . AC method . DC method . X. DISCUSSION OF THE RESULTS . 10.1 Concentration. . . . . . . . . . 10.2 deA Phase Constant. . . . . . . 10.3 deA Frequency . . . . 10.4 Possible Phase Width . . . LIST OF REFERENCES AND NOTES . . . . . . . Dingle tempera— 119 122 124 124 125 127 127 128 133 134 142 142 142 145 148 150 157 157 160 162 165 168 168 170 170 175 204 206 206 210 216 218 219 219 220 223 231 DlRO' I b . “J‘- Structure Fr: Sn '5' ' "“ fini‘“-‘ Ur‘CkLsALG5AK KER Medul “1‘3 On the Hetica 10. 11. 12. 13. 14. 15. 16. 17. LIST OF TABLES RRR and Impurity Concentration of AuX2 Crystals . . . . . . Comparison of Vacancy Concentrations and RRR in AuAl2 . Structure Factors of AuGa2 . . . . . . . . . . . Specifications for de Haas-van Alphen Coils N5 and N6 and NMR.Modulation Coil NMR—M. . . . . . . . . . . . . . Thermal Leaks: Conductive Heat Flow from Top of Probe (Room T) to LHe Level (10' Above Sample) . . . . Superconducting Solenoid. . . . . . . . . . . . . . . . Characteristics of the Standard Resistors for the Super— conducting Solenoid. . . . . . . . . . . . . . . . . Sample Characteristics by Position and Parent Crystal's Melt Concentration . . . . . . . . . . . . . . . . . . Data on the Three AuGa2 Crystals Grown by the Bridgman Me thOd O O O O I C O O O O O O O O O O O O O O O I O O 0 Dimensions of deA and RRR Samples. . . . . . . . . . . . Chemical Analysis of Slug A (AuGa +.204 at.% Ga). . . . . 2 Microprobe Analysis of Slugs A and B. . . . . . . . . . Typical Statistics Used to Compute Table 12 . . . . . . Reproducibility and Resolution Tests of the MSU Microprobe. Tentative Conclusions for the Relative Change in the Gallium Concentration with Respect to the Bottom of Slug Error in Concentration Differences Ac Due to Hypothetical Deviations from Linearity in the X-ray Count versus Concentration. . . . . . . . . . . . . . . . . . . . . Comparing Variances in Mocroprobe Data. . . . . . . . . . . vi 22 112 120 123 126 140 158 162 174 183 184 188 191 195 201 - —. u- n .- “an- av. TABLE 18. Scattering Rates from deA and RRR in Cu and Au. . . . . . . 210 19. Results of Comparing RRR from DC and AC Methods. . . . . . . 215 vii pvt-cc“!- I #- ..Jt.\~ 3 “h” :I“ a _ n ~CLkLr "C orientati ~. rystal Str' w - , ' J. Hc.es 1n tn: I. :ourtn zone A ”A,“ a “body 111 .‘.. LIST OF FIGURES FIGURE 1. RRR of AuGa2 crystals versus excess Ca in the melt . . . . 2. Coordinate systems in real and reciprocal spaces . . . . . . 3. The four axes in kfspace: kx, ky, kz, and kg, and their orientation with respect to the magnetic field. . . . . . 4. Crystal structure of AuGa2 . . . . . . . . . . . . . . . . . 5. Holes in the third zone of the empty lattice model of AuGa2 in the reduced zone scheme. . . . . . . . . . . . . . . . 6. Holes in the third zone of the empty lattice model of AuGa2 in the repeated zone scheme . . . . . . . . . . . . . . . 7. Fourth zone sheet of the empty lattice Fermi surface of AuGa2 in the repeated zone scheme . . . . . . . . . . . . 8. Some extremal cross-sections of the empty lattice Fermi (surface of AuGa when the magnetic field is scanned in a {110} plane a o o o o o o o I o c o o o o o o o o o o o o 9. Energy bands of AuGa2 by a non-relativistic APW calculation. 10. Intersection of the path Q with the orbits C3 (hexagon) and C4 (star) . . . . . . . . . . . . . . . . . . . . . . . . 11. Orbits on a hypothetical Fermi surface . . . . . . . . . . . 12. The relation between the Landau levels and the Fermi level EF 0 I O O O O O O O O O O O O O O O O O O O I I O O O 13. A slice in kfspace showing quantized orbits superposed on a Fermi surface of arbitrary shape, at some value of kH . . 14. Changes in the quantization scheme as the principal axes of an ellipsoidal Fermi surface vary their orientation in B = E O I O O O I O O O O O O O O O O O C O O O O I O O O 15. Former kfstates lying between quantized orbits are swept onto the orbits . . . . . . . . . . . . . . . . . . . . . viii 14 16 23 23 24 24 25 27 29 41 45 45 49 . 1., Ho ... \o 0 1a 1.5. 3‘1 'I Orbits cent» W‘ ' 1 A apical Cut, ’7". A A -. .u \v‘ln: +h(t). FIR AL; Escalation har:c:1c tion V .‘TIC FIGURE 16. 17. 18. 19. 20. 21. 22. 23. 240 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. Theoretical dependences of magnetization and free energy upon the inverse field, l/B for T = TD = 0. Hypothetical orbits to illustrate magnetic breakdown . Cross-section of the enpty lattice surfaces in the third (clear) and fourth (black) zones at the kz values given for B || <111>. Orbits centered on point L on the Brillouin zone . Typical data for the C3 (Third zone neck) orbit of AuGa +h(t). 2 . The construction of M(Ho ,t) from M(H) and h(t), with H = H Modulation of the steady field Ho by h cos wmt causes each harmonic component“!r (H) of the oscillatory magnetiza- tion.MOSC of equation (71) to induce an alternating voltage v(t) a er(Ho,t)/dt . Plan view of the de Haas-van Alphen probe (top). Plan view of the de Haas-van Alphen probe. Exploded view of the probe's coil former and sample holder assembly. . . . . . . Angle map of mobile base tilt angle. Dimensions of coil formers . Coil winding contour map of coil strength in Gauss across a coil on half of Former CF2. T= 200 C. Coil winding contour map of coil strength in Gauss across a coil on half of Former CF2. T = 4.2K. Block diagram of deA and NMR electronics. . Theoretical curves of constant phase and deA rotation graphs for orbit C3 . . Typical experimental rotation graphs . Identification of positive peaks in a deA rotation graph by (l) rotation, (2) reset field, Skin effect test on Sample 12. ix (3) rotation. 0 per volt per volt 58 66 66 68 87 89 92 99 100 101 107 111 116 117 121 130 131 132 135 31 LA.) \‘7 Skin effect Selected da: Buzzary of : Hypothetical nonzero ; Gradient for in the :. Concentrati. fractions De Haas-‘15:] A FIGURE 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. Skin effect test on Sample 30. . . . . . . . . Selected data for AuGaz. . . . . . . . . . . . . . . . . Summary of deA and RRR data . . . . . . . . . . . . Simplified equilibrium phase diagram for Au-Ga . Hypothetical intermetallic compound (solid phase S) with a nonzero phase width and cm not at stoichiometry . . Gradient for a dilute concentration of an alloying element in the melt . . . . . . . . . . . . . . . . . . . . . . Gallium concentration of the melt at the interface of solid and liquid is lower than for an equilibrium system. Concentration of melt at solid-liquid interface versus fractional distance along crystal . . . . . . . . . . . De Haas-van Alphen and RRR samples (circled numbers) and samples for microprobe analysis (not circled) . . . . . Location of samples cut from Slug A. . . . . . . . . Error introduced into measurements of concentration differ- ences by deviations from linearity in x-ray count . . . Display of the data: 10 second x-ray count N for the Ga K line versus trial number in Run 2 . . . . . . . . . . Display of the data: 10 second x-ray count N for the Ga L line versus trial number in Run 2 . . . . . . . . . . . . Display of the data: 10 second x-ray count N for the Au M line versus trial number in Run 2 . . . . . . . . . . . . Theoretical frequency distributions of x-ray count N . Histograms of x-ray count N for three x-ray lines of AuGaz, Run 2 on 8 lug A O O O O O I O O O O O O O O O I O I O 0 Apparatus for residual resistance ratio measurements . . . . Possible AuGaz phase boundary. Based on assuming TD varies continuously with position. . . . . . . . . . . . . . . . Hypothetical phase boundaries for AuGaz consistent with continuous Dingle temperature and the prepared melt concentrations. . . . . . . . . . . . . . . . . . . . . An expansion of the first phase boundary in Figure 53, dia— grammed in the manner of Figure 52. . . . . . . . 136 139 141 143 147 147 153 156 161 171 194 196 197 198 202 203 211 225 227 228 o A H Lh“” R" N I :1 Intro ‘ Q 1 . A“ - ' 6.3:. 958 1?." 132'81' ’ .811 t bLA L11: .« .4 "LI . ..ov a» ‘ 5. HO COTE-379." .. .. #5.. aka: I N. nr' #4; .7 k f‘ O h! r »8 {1 tea u~ CHAPTER I MOTIVATION AND PURPOSE 1.1 Introduction De Haas-van Alphen effect in studies of metals. The magnetization of a nonferromagnetic substance arises from various physical phenomena. The local atomic or ionic magnetic moments, the exchange interaction between these moments and the electrons, and the preferential occupation of the lower energy, spin-up electronic states (Pauli spin magnetism) may contribute to the total paramagnetism. The diamagnetic component of the susceptibility arises from the quantum mechanical response of electrons in motion to an applied magnetic field. Under certain experi- mental conditions this response of the conduction electrons in metals adds two components to the magnetization, one of which is a steadily increasing diamagnetism, while the other oscillates, becoming alternately parallel and anti—parallel to the applied field. The last phenomenon, the presence of oscillation in the magnetic susceptibility with increasing magnetic field, is called the de Haas-van Alphen (deA) effect, after W. J. de Haas and P. M. van Alphen,l who observed it in 1930 when measuring the magnetic susceptibility of a single crystal of bismuth at liquid hydrogen temperatures. At low temperatures and big fields it is easily observed in most metals. The deA effect has been used since the middle of the 1950's to obtain much information about the shapes and sizes of Fermi surfaces. In the past few years quantitative research into the effect of the impurity concentration on the deA measurements has led to the investigation of the :13va ette . . _ . . 2321f CCEtEHLId.. T215 cevelogrent :C EYES: 123:8 S :......: ' ' fl‘ 5 O1 ‘h'kvsALakLan C: ., b Liu0r156 81 A 28.x“ n, . u“- k‘snnln S. a:~ N‘I -;:‘V:H ‘ ‘.V“$ .1 ' hv “Cu 1'] F- A \a A .._‘ i'. \-V I .;_1‘ “£37 n A . h, U a; ~ 1". of the deA effect as a probe of scattering potentials of impurities, their concentrations, and their effect on the Fermi surface of the host. This development combined with speculations and questions that arose in the course of a study of the galvanomagnetic properties of an homologous series of intermetallic compounds to suggest the use of the deA effect to investigate some phenomena that apparently occur during the growth, by the Bridgman technique, of the single crystal AuGaz, one of the compounds used .in the galvanomagnetic study. The next two subsections tell how these questions arose and describe the hypotheses suggested by them. The last section of this chapter outlines the program for the investigation of the hypotheses. A metallurgical question. AuGa2 is an example of an intermetallic compound, also called an ordered alloy. Its crystal structure is that of fluorite and is shown in Figure 4. It is grown by mixing gold and gallium within several atomic percent of stoichiometry (one Au atom for each two Ga atoms), melting this charge, and slowly cooling it, as described in Chapter VIII. According to the equilibrium phase diagram Figure 38, the precipitate, at least initially, will be of fixed compo- sition AuGa2 independently of the prepared melt concentration, so long as it lies within a rather broad range about stoichiometry. In particular, the long range order should be the same. The long range order relates to the tendency of the Au—sites of the lattice to be occupied preferen- tially by Au atoms rather than by Ca atoms, impurities, or vacancies, and the tendency of the Ga-sites to be occupied by Ca atoms.2 The lattice of an ordered alloy is divided into two or more sublattices extending throughout the whole crystal, and each preferentially occupied by one species of atom. The residual resistance ratio (RRR) is an experimental measure of 3.: tezeeraturt— .;-"" ”’1‘ (1,. . "‘r‘trGLvre ..C :1215 in tee 15:: long range order. Because the Au and Ca ions in the crystal have dif- ferent scattering potentials, randomly intermixing them destroys crystalline symmetry. The Bloch wave of the conduction electron thus sees the site of a sublattice containing anything other than its correct species as a scatterer. At low temperatures (normally in liquid He at one atmosphere: 4.2 K) such scattering dominates the crystal's resistiv- ity, which consequently varies with the amount of long range order. At room temperature the phonons dominate the resistivity. Because at room temperature the phonon spectrum is much less sensitive to small varia— tions in the long range order the resistivity is, too. Thus the residual resistance ratio, RRR = = RRT/R4.2’ fixed geometry [1] CRT/04.2 increases monotonically with long range order. From the above discussion one would expect a single crystal of AuGa2 to have a uniform RRR, and crystals grown from different melts, all near stoichiometry, to have the same RRR. But it is not unusual for 3RRR.to vary widely in crystals grown from melts of nearly the same con— <2entration and the same purities. Specifically, in 1968 J. Longo3 pre- Pared AuGa and AuAl 2 2 Perties as AuGaz) from melts that were within 10.5 th Al or Ga of being (with the same fluorite structure and similar pro— SStoichiometric. He found (see Table l and Figure 1) for both alloys, (1) Crystals grown from melts prepared rich in the Group III element had much higher RRR than crystals grown from melts prepared stoichiometric; and for AuGaZ, (2) a maximum in the RRR of the most pure samples taken from different AuGa2 crystals plotted against the prepared melt Concéfli ‘. I A” range 3.0 to .. U 9" 3k; .1 n'. v An. mu.» aralte want 1'." ‘1': y an ‘1 L 'lfi-c ‘Vdefig' k A Ck". ue .omstent wi: D‘N-‘y{,‘ u . “Neal rEIati C3‘A‘n ~qtratlon o C‘; “:CCUI concentration); (3) within a given AuGa2 crystal, a big spread of RRR about the average value. The average RRR versus melt concentration also exhibits a clear maximum; (4) traces of Ga on the surface of crystals grown from melts of concentration > 0.5 th Ga. Longo concluded that these data are consistent with a report by M. E. Straumanis and K. S. ChopraA that AuAl2 has a homogeneity range from 78.18 to 78.94 th Au (32.92 to 33.92 at.% Au) in the temperature range 300 to 400 °C, with the phase being fcc, fluorite structure. Straumanis and Chopra also said that at stoichiometry there are 0.152 .Al-site vacancies per cell and 0.076 Au-site vacancies per cell, giving a vacancy concentration of 1.9 at.% on each sublattice. They further maintained that at the Al—rich end of the homogeneity range the Al-sites are completely filled, resulting in a crystal grown from an .Al-rich melt having a higher RRR. However, Longo reported that his values for RRR seemed too high to ‘be consistent with Straumanis and Chopra's vacancy concentration. The empirical relation (RRR)(I)=1. [2] Vflnere I = impurity (or vacancy) concentration in percent, mamam>uu /\ Le 45¢; -Ln Noosq JbN N m05< mo mam «H: .H munmwm [room lunYOnu_ 82:11:! Table 1. RRR ar (Adapt 0 DJ wr< r.) In Table l. RRR and Impurity Concentration of AuX 2 Crystals. (Adapted from Longo, Ref. 3) RRR of AuX AuAl2 Melt Prepared exact Al-rich 1408 550 Crystals AuGa2 Melt Prepared exact Ga-rich 1. 250 2. 190b 904 950b a. Longo (Ref. 3) cites JPSST (Ref. 10). b. Taken from graph in Ref. 3. Table 2. Comparison of Vacancy Concentrations and RRR in AuAl2 Straumani54 (RRR)(I)=1 (RRR)(I)=10 and Chopra Stoichiometry 1.9 at.% 0.0071 at.% 0.071 at.% Al-rich end of homogeneity range (0.56 at.% excess Al) 0.634 at.% 0.0018 at.% 0.018 at.% :av coalesce to He states that t nether pcssibi- RRR valt 1"? ('1. (7‘ r 4- 0") 0‘ stick were not 0 :E:«vin9{ n O; ’1 5. ~~A‘..~‘, n . may coalesce to form small voids dispersed throughout a macroscopic region. He states that the variation of RRR over one crystal supports this belief.3 Another possibility consistent with a high concentration of vacancies and big RRR values is that the vacancies are ordered, as in Al N12 and in Fe0.8758. However,such crystals would exhibit x-ray superlattice lines, which were not observed by Straumanis and Chopra.7 Impurities, dislocations, crystallites, and deviations from stoichiometry could account for LongissRRR observations. His description of the care taken in the growing of the crystals, the good x-ray pictures, and the regular pattern of the RRR values suggest deviations from stoichiometry may be more important than the other mechanisms. Some binary alloys have equilibrium phases with a wide variation in the range of homogeneities (in atomic percent of one of the components). The range of concentration for Longfsssamples corresponding to a range of RRR from 190 to 950, obtained from the relation (3), is .10 to .053 at.% deviation from stoichiometry, a range difficult to detect by mmethods of compositional analysis. Consequently, a phase this narrow could be reported as nominally fixed composition. The fact that AuGa2 crystals with the biggest RRR must be grown from melts prqaared off stoichiometry is not unusual.8 This phenomenon may atrise from other phases existing near the phase of interest, from the (nongruent point (the maximum in the liquidus curve) lying slightly off stoichiometry, and other conditions. An example of another condition is Mgsz, which has a peritectic point that looks very much like a congruent point.9 See Figure 39 and discussion there for a hypothetical case of the congruent point lying off stoichiometry. H u .> =9 "9 'J "J I V r 7‘ (It ’I) (1; (IA The hypctne ofthe use of ti this. They are exactl I.‘ (J) V The as CORCET on he; ;:V| “Mi“ W ' signals, .‘.1 «UK '1 55.15 L ’ u: 811;}. tl, i‘ S:‘-\ «L195 15 the '1" utLEUtrat‘0 {I‘EIUatl 1.2 Hypotheses and Objectives The hypotheses of this thesis are suggesnxiby the above discussions of the use of the deA effect in the study of metals and the RRR varia- tions. They are: (1) When a AuGa2 crystal is grown from a melt of given initial composition by the Bridgman technique the crystal's composition varies along the direction of solidification. (2) The AuGa2 line of the equilibrium phase diagram is not exactly vertical, but slanted, and crosses stoichiometry. (3) The deA effect is capable of detecting the above range of concentration. Furthermore, deA studies may yield information on how Au, Ga, and vacancy impurities affect the Fermi surface and scattering of electrons. If these hypotheses are correct for AuGaZ, they may extend to other intermetallic compounds which are of nominally fixed composition. deA effect and concentration. AuGa2 yields moderately strong deA signals, which have provided Fermi surface information since 1965.10 Our primary objective is to see if there is a correlation between the «deA signal and RRR of samples all nominally AuGaZ, and thus extend the study of Fermi surface dependence upon impurity concentration to an analo- gous, but slightly different condition. Some method of characterizing the samples is desirable, and much work.went toward a direct analysis of the concentration. The results were only partially intelligible, so the evaluation of the data relies mostly on indirect methods of characteri- zation. These results are discussed in Chapters IX and X. Further discussion of the deA effect is deferred until Chapter III. sa:;les are c' 1 , vs «.3 “r A“ .§ 9"; “,‘II 21" 'A \- Vts t 16 . “1.2 Phase diagrams and crystal growth. A secondary objective is eluci- dation of the growth pattern of the AuGa crystal. The concentration of 2 an alloy may depend upon both the equilibrium phases and the kinetics of growth. The data presented later indicate that the variations among the samples are due to both effects. Having considered the puzzle (the range of RRR values in AuGaz), the suitability of this alloy for an extension of the study by means of the deA effect of the dependence of the Fermi surface upon concentra— tion, and the influence of the method of growing the crystal, it is appropriate to outline the experimental program. 1.3 Proposed Program and Summary of Results Grow single crystals of AuGa by the Bridgman method, each from a 2 melt with initial concentration slightly (less than one atomic percent) off stoichiometry. Cut samples from along the axis of each crystal and measure deA frequency, Dingle temperature, residual resistance ratio, and concentration. Chapter VII contains the deA and Dingle temperature data; Chapter 'VIII gives the melt concentrations for growing the crystals and the .locations of the samples cut from the crystals; Chapter IX gives both ‘the data for the direct analysis for concentration in the crystals and the residual resistance ratios for the samples. Chapter X discusses these data and draws conclusions, an epitome of which follows: (1) The samples exhibit a definite and non-random variation in deA frequency (F), Dingle temperature (TD), and residual resistance ratio (RRR). (2) The variation in RRR correlates with that in T according to simple models. That is, samples ordered by increasing RRR are consequently ordered by decreasing T , both corresponding to increasing long range crystalline order. (31 The se: tne CI' C8131”; stelcn (3) (4) 10 The samples of lowest T and highest RRR came from regions of the crystals which soliHified at a time when the average con- centration of the remaining melt was gallium rich (compared to stoichiometry). Direct analysis of the concentration of the crystals was only partially successful due to the differences in concentration being of the same order as the resolution of the available methods of analysis. Some trends were evident and are dis— cussed in Chapter IX. u 1" Y" ‘ ‘* -A.'.ai are 119 P11 onOHS; CHAPTER II FERMI SURFACE OF AuGaz The Fermi surface is a mathematical construction related to the dynamical properties of the conduction electrons of a crystal.11 Before mentioning some of the methods used to calculate the Fermi surface it is useful to list some important, well-known terms from the physics of crystals. Their enumeration here will highlight several distinctions often ignored in general use but important in deA theory. 2.1 Preliminary Definitions Discussion of the theory refers from time to time to the following physical models of a real, crystalline solid: Ideal crystal. A perfectly periodic and static repetition of an atomic basis throughout all space. From the formal relation, crystal structure = lattice + basis, [4] it is seen that the cluster of ions and atoms forming each basis makes tip the content of each primitive cell. A static repetition means there Eire no phonons; each basis is rigidly fixed with respect to its lattice point. Perfect crystal. A perfectly periodic repetitionof an atomic basis throughout a finite region of space. As with the ideal crystal there are no impurities, vacancies, cracks, or other imperfections. The presence of phonons gives this model different properties from those of the ideal crystal. 11 arranzezent or 31233035 and 11;. M- 1L $638308 0? _ Free elect“ R.“ D (J :J 0 7‘9‘ (D ’L’ OJ () (I) erg-5e meal or 59:33 free to .7. LY?" ,. “D' SUrIaCeS wnére . ~::‘ c.1eCtl‘Je lass agij 2 3 hPOI‘r- .1; deriVEd Q 6:31 _ “ ma‘e 1 \J ..‘50n COnStr 3:;‘v, 12 Static crystal. An imperfect, but approximately periodic and static repetition of an atomic basis throughout a possibly finite region of space. This is a model for calculations of the effects of alloying or of imperfection on physical properties when phonons are ignored. Real cgystal. The physically obtainable, approximately periodic arrangement of atoms assumed by numerous elements and compounds. Both phonons and imperfections are present to a degree that varies widely and depends on the crystal's history and environment. Free electron model. Non-interacting electrons confined to finite region of space throughout which the potential energy is constant. This crude model of a crystal includes the idea of the conduction electrons being free to move throughout the crystal, but ignores the crystalline structure, whose potentialvariestxnfiodically. It is useful for rough calculations of physical properties because it does lead to discrete krstates when periodic boundary conditions are imposed and, in conjunction with the Pauli exclusion principle, to the idea of a Fermi surface. The energy surfaces in kfspace are given by E = m [5] where m0 is the free electron mass. Often this model is extended to real materials with ellipsoidal Fermi surfaces by replacing mo with an effective mass m* or with an effective mass tensor with elements m1, m2, and m3. Important concepts of the deA theory for real metals were first derived and are most easily understood using this model. Extension is then made to more complicated cases. Such is the case for the Harrison construction for the Fermi surface (see the description of the empty lattice model in the next section) and the Landau levels (Chapter III). Coordinate ~"_ systens need to ': Figure 2 illusrrn :ns thesis has n Latzice. Hence : 33, also called : oblique. o: the corres;onc nv:-{'.‘u ' ' Lathe oasrs R - 4".“ 7"“ :.~cao 111E 2011C annoum zone 11‘ .czeede ‘- ' L .5" «Led to as it.» ‘ v. ~: . " 5! 5:16 7. ' L 01 lentat‘. \‘L‘.’ A IJAS S: stezs a . ‘ 'o - -- ze.o field h.- a. .. ~ {sate becaUSe ..~-¢» planes (31" A k. 25:; n . chosen for 1 13 Coordinate systems. Some conventions of notation for coordinate systems need to be established for later use in presenting the theory. Figure 2 illustrates these. The intermetallic compound AuGa2 studied in this thesis has a cubic conventional cell and a face centered cubic lattice. Hence the conventional basis is the orthogonal triad 31, §_, 33, also called the crystalline axes. The primitive basis 21’ 22, 23, is oblique. The axes along 31,.§2,.§3 are also denoted by [100], [010], [001], etc., concisely represented by <100>, called a form. The lattice of the corresponding reciprocal space is body centered cubic, and its primitive basis is Al, A2, A3. The bisector planes of the vectors §_= nlél + ... yield the zone planes, which are parallel to the atomic planes. The zone planes closest to the origin define the first Brillouin zone in reciprocal space. Axes in reciprocal space are usually labeled k k k3 in the literature, so reciprocal space is often 1’ 2’ referred to as krspace. For a given orientation of the g1 and hence the .gi, the orientations of the kfaxes may be chosen for convenience. In cubic systems a triad that coincides with the crystalline axes is used. In zero field this is the most convenient choice for coordinates in krspace because spatial directions in the crystal are defined by the atomic planes and their intersections. Spatial axes x, y, and z are chosen for convenience in representing the geometry of the eXperi- mental apparatus, and convenience may require that their unit vectors 11 not be parallel to the 51’ as for a crystal at arbitrary orientation 14 3 ’? n 1 1 I I. / ._ 2 1 A =—%> Conventional -—>Primmve (a) (b) kz 1, ix -Q ky (c) kx (d) Figure 2. Coordinate systems in real and reciprocal spaces. (a) Basis vectorsgi (conventional) and Bi (primitive) of the fcc lattice. (b) The primitive reciprocal basis vectors A: and their relation to the Brillouin zone. The reciprocal lattice is body centered cubic. (c) The perpendicular axes in reciprocal space (also kfspace) and their orientation with the Brillouin zone. (d) External conditions (e.g. field direction) may be used to define spatial axes x, y, and 2 not necessarily parallel to the}Li axes. C, n: F! ’1‘ , 01 1.18 me :1“; ii), the Dingle £173.) of the int: froztne top, :1; partnerers vary .: st nature exists. .r. .1- . . 2.15.43 netnoo 1.) ' » ABSTRACT DE HAAS-VAN ALPHEN STUDY OF THE AuGa2 ALLOY PHASE By John J. Higgins The de Haas-van Alphen frequency (F) of the third zone neck orbit (C3), the Dingle temperature (TD), and the residual resistance ratio (RRR) of the intermetallic compound AuGa2 were measured on samples cut from the tap, middle, and bottom of three single crystals to see if these parameters vary over the range of composition in which the fluorite structure exists. The three crystals, 4 to 6 cm long, were grown by the Bridgman method from melts that deviated from stoichiometry by having excess Au (.287 at.% from stoichiometry) and excess Ga (.203 and .549 at.%). F ranged from 3388 kG to 3384.2 kG to 3405.7 kG. Assumption of the rigid band model implies for these samples a concentration range about stoichiometry less than .06 at.% Ga. But analogy to the case of Pd impurity associates a value between .22 and .43 at.% Ga with the range of F. An independent analysis of uncertain reproducibility indi- cates a concentration difference of .35:.10 at.% between two groups of samples. Stoichiometry does not coincide exactly with the congruent point. The deA phase constant was found to be .46 i;.09. RRR and TD varied from 55 to 1800 and from 5.0 K to 1.1 K, respectively. q I! so“ also calla v e we . . ~1,.,A L‘L‘u in ‘M .6. C intensity Di r O u... an 3. .4 4L :2 .7. ., h.“ -AC J. .51 ;L 15 in a uniform field. Magnetic field nomenclature. The vector B_is the magnetic induction, also called the flux density. H_is the magnetic intensity and does not include the magnetic contributions of the material medium. M_is the intensity of the magnetization, or simply the magnetization. The term "magnetic field" and its abbreviation, "field," are used variously to mean H_or B, depending on the context, when distinction is not important. Correct interpretation of deA data can require one to distinguish H and B, even in nonmagnetic media, when magnetic interaction is significant. (See section 3.9) The orientation of the magnetic field with respect to the Fermi surface is important because the topology of the Fermi surface enters the expressions in the deA theory. Thus two coordinate systems in kfspace are maintained: the former k1, k2, k3, which are parallel to the crystalline axes for the cubic structure, and a new direction kH parallel to the field, plus two axes normal to it (Figure 3). In this thesis the spatial axes are always chosen so that §_is parallel to the z-axis. (Normally H_is also considered parallel.) General crystal orientation will thus render the crystalline axes noncongruent with the spatial axes, x, y, and z. The orientation of the field with respect to the crystalline axes and hence with respect to the Fermi surface is given by the polar and azimuthal angles, a anddg and a gradient operator is defined: a a . 1 a v= _._+1 ._+ ———. 6 H 1Hon 9 so 1¢ Hsine so [ 1 and Figure 3 illustrates the relations among the kfaxes, kH, and the angles of the field direction. 16 II Figure 3. The four axes in krspace: kx, ky, kz, and kg, and their orientation with respect to the magnetic field. 90. Most eqa holes by Chang qis used for negative. R'. 1 . ul.€~tlons i:- ..1e energy: f 1 cm metal de. “hi Surfan thus . prw‘a ' \a‘ r: i. Q Used A ‘ePe sxsal regui a at . cse-‘ "wk-3.11 . 111% E‘eg 17 Sign of charge carriers. The charge of the electron is -e, where e>O. Most equations are written for electrons and can be changed for holes by changing the sign. (When e appears as a universal constant, as in the flux quantum hc/e, no sign is implied.) When necessary for distinction the positive electronic charge is written p or Iel. Sometimes q is used for general charge, and may be implicitly positive or negative. 2.2 Fermi Surface of an Ideal Crystal The eigenfunctions of the conduction electrons in an ideal crystal for the single-particle approximation are Bloch waves,wk, indexed by the quantum numbers_k, called the wave—vector. The quasi-cdntinuous distribution of eigenenergies Eb(k) in k-space gives rise to constant energy surfaces in each Brillouin zone b of kfspace. Along given directions in k—space the functions Eb(k) correspond to the energy bands. The energy functions and the number n of electrons per unit volume of the metal determine the maximum energy of occupied states at 0 K (the Fermi energy EF). The continuum of states k_ belonging to EF form the Fermi surface in k-space. The problem of calculating the Fermi surface is thus primarily the evaluation of EF(k). Which of various methods available is used depends on the properties of the metal (e.g., a transition metal requires more sophisticated methods than does a simple metal) and the accuracy required. Some methods of calculating the Fermi surfaces are given below. Results of applying two of these to AuGa2 then follow. Empty lattice model. This combines zone theory, which uses the crystalline structure to derive the Brillouin zone, and the free electron model. The spherical energy surfaces E = fi2k2/2mo are divided t: "in to 51183 are exa. ‘n ‘5‘ use th V.‘ .C D. :18, my 7. r‘ 1'8: .1 :eat' necel usua' structicn. eg I a 1 a a: are is: 3 st Cr 55 ..t 18 into sheets by the zone boundaries. The sheets of the Fermi surface are examined in various ways, depending on the problem at hand. Thus one may use the extended, reduced, or repeated zone schemes. The reduced zone, empty lattice model is obtained most easily by the Harrison con- 12 which is extensively used to explain and predict gross struction, features of the Fermi surfaces of metals and properties related to it. This model is also frequently referred to as l—OPW and free electron. The Harrison construction of the Fermi surface of AuGa2 is discussed in Section 2.4. Limitations of the empty lattice model. Although the empty lattice model usually gives reasonable accurate first order results (considering the free electron sphere as the zeroth approximation), second order corrections are almost always needed to obtain good qualitative‘ agreement with experiment, and occasionally first order predictions have signifi— cant error. This is because, although the model reflects the primitive space lattice of the crystal, the influence of the atomic basis (the ionic potentials) enters the model only through the effective valence. The ionic potential usually causes changes in the :topology of the empty lattice Fermi surface where it approaches zone boundaries, and these changes can be significant with regard to some of the metal's properties. The well known necks of the noble metals, absent in the empty lattice model, are examples of this effect. Further, if the atoms forming the metal have occupied d orbitals in- the free state whose energies are close to those of the valence states after crystallization, both the band structure and the effective valence of the crystal may differ from those of this model. Spin-orbit interactions, an important cause of the lifting of a reference sore of the centioned b (2} , jh‘n‘ “4L Ln ls 1h“ 2‘31 C:«€Ct Cf ._ 3“?- :1 Cf 9.. E 7‘1 ‘r‘e CIrD‘ sue 1: 19 degeneracies in crystals of heavy atoms, are not included in this model. In spite of these limitations, the empty lattice Fermi surface is adequate for this study of AuGa But for completeness, and also because 2. a reference to a band structure calculation for‘AuGa2 will be given, some of the standard methods of band structure calculations are briefly mentioned below. Methods to include the ionic_potentials and spin-orbit interactions. In the nearly free electron (NFE) model Bloch waves are treated as an expansion in plane waves. The expansion coefficients and the eigen- energies for given k_are obtained from perturbation theory. At first thought, the NFE method appears to have fundamental defects. J. M. Zimanl3 (this reference is hereafter referred to as Ziman) lists the following points: (1) The deep potential well at each ion means the Fourier components Vg of very short wavelength are important, so that the series would be expected to converge slowly. (2) Rapid oscillation of wk near the ions also requires short wavelengths and implies slow convergence. (3) Simple perturbation theory effectively uses the Born approximation, which is invalid for deep atomic potentials. These arguments led to the neglect of this method as a practical scheme for band structure calculations. But it now appears that it can be made formally valid by the introduction of the pseudopotential (Ziman, p. 76). The NFE model is not used directly for band structure calculation, but remains useful as a means of illustrating the periodicity of E(k) and its gaps at zone boundaries. The orthogonalized plane waves (OPW), or pseudopotential method is due to Herring,14 and is important in the calculation of band structures. It can be used when the ions lack spherical symmetry, a requirement for the augmented plane waves method (see below). After the ionic potentials , . saee ortncgo: :unctlo-ns, V: were f on: t it: ‘31.; . rvLc‘ltlal 1: 3'6 “I h MA “as cute are ~—. to k- L" 393116. 7 i ' Cr"'5ta1 111C, . £32 in 20 are replaced by pseudopotentials an iterative process approximates the Bloch functions by summing orthogonalized plane waves (plane waves made orthogonal to the core states). The first step yields the l-OPW functions, which are often quite sufficient to represent the electronic wave functions over large regions of k-space; only 3 or 4 terms may be needed even in the corners of the Brillouin zone (Ziman, p. 94). The Fermi surfaces given by the free electron, the empty lattice, and the l-OPW models are often equal or nearly equal, so that these names are used somewhat interchangeably. OPW assumes a distinction between core and conduction electrons.15 This condition may not be well satisfied for transition metals and other elements with high level d orbitals, such as Au. The method of augmented plane waves (APW) was suggested by J. C. Eflater.16 The space of the crystal is divided into spheres centered on the ions and interstitial regions of constant potential. The ionic potential is assumed spherical. Plane waves in the interstitial regions are augmented by spherical harmonic waves in the cores, and the two waves are matched at the surface of the sphere. Assuming a valence l for Au leaves its outer subshell 5d10 filled and hence spherically symmetrical. Similarly, assuming a valence 3 for Ca leaves outer subshell 3d10. These valences are supported by experiment and theory, and allow the APW method to be applied to AuGaz, with results to be given in Section 2.4. 2.3 Crystalline Structure of AuGa2 AuGa2 has the fluorite structure.10 The face centered cubic (fcc) Bravais lattice has an atomic basis of one Au and two Ga atoms (Figure 4). The structure factor is obtained '0': positions anc tor-R :act‘ scatterina ncu. direction (1.13. i 1 G 3180 21 S(hk£) = E f1 exp(-2ni(hxi+kyi+izi)) [7] is obtained by using the cubic conventional cell and summing over the positions a_ [81 r°= 13 + a + z —1 yi—Z x,a r—l and form factors fi of the ions in the cell, expressed with respect to the conventional cubic axes 31' This structure factor predicts constructive interference of diffracted x-rays in the same directions as for a monatomic fcc crystal, but with different intensities due to the unequal scattering power of Au and Ca. Structure factors associated with the direction (hki), that is, diffraction from planes (hki), are given in Table 3. Because the primitive Bravais lattice is fcc, the primitive reciprocal lattice is body centered cubic (bcc), and the first (and each reduced) Brillouin zone is the polyhedron of Figure 5. 2.4 Fermi Surface of AuGa2 Empty lattice Fermi surface. Jan, Pearson, Saito, Springford,and Templeton10 determined the empty lattice Fermi surface of AuGa2 by the method of Harrison12 in 1965, assuming a valence of seven electrons (one for Au and three for each Ga). It is in fair agreement with most of the deA data, reported first by them10 and confirmed and extended by others.17"20 The first zone is full. The most notable failure of the model is the prediction of a big octahedron of holes in the second zone, whereas extensive experiments have not shown a signal that could belong only to that zone. The conclusion that the second zone is full is 21,22 supported also by a band structure calculation. The third and Z R 1: .c a t t .H r. i i. I S .1 E r .l 4.. u. ..u S e e C e v . v .. V A D . a n. E E E C ..T. E A +1 .T. E at .C u‘.‘ :1 .... A» «C 22 TABLE 3. Structure Factors of AuGa 2 Character of Diffraction Structure Factor Plane, (hkfi) S(hk£) h, k, and l are all even 4f +8f (_1)(h+k+£)/2 Au Ga h, k, and R are all odd 4f Au h, k, 2 mixed even and odd zero fourth zone sheets have been generally verified by the deA, magneto- 23 24 Tiny pockets resistance, and a few other types of measurements. predicted in zones five and six by the model are expected to be prevented by the effect of the crystal potential's rounding off corners of the empty lattice Fermi surface. No experiments give definite evidence of occupation of zone five.19 Zone six is certainly empty.19’22 The third zone sheet, the fourth zone sheet, and the extremal cross- sectional areas for all sheets for directions <100> are shown in Figures 5, 6, 7, and 8, all for the empty lattice model. (The figures are from Ref. 10; its notation for cross sections is used in the litera— ture. A, B, and C refer to directions <100>, <110>, and <111>.) The deA data of this thesis all pertains to the necks of the third zone, about the point L on the zone boundary, where the cross-sectional area is a minimum and the charge carriers are holes. The measured cross—sectional area here is .32 times that given by Figure 5, so that the necks are more "pinched off" than the figure suggests. igdre HOIE: in t? Bril; 23 I/ Figure 4. Crystal structure of AuGa . Open circles are Au and filled circles are Ga. (Ref. 10 Holes in the third zone of the empty lattice model of AuGaz in the reduced zone scheme. The polyhedron is the Brillouin zone. (Ref. 10) Figure 5. F‘lgure 6. H. In Figure 6. Holes in the third zone of the empty lattice model of AuGaz in the repeated zone scheme. Extremal cross sections are shown at right. (Ref. 10) Figure 7. Fourth zone sheet of the empty lattice Fermi surface of AuGa2 in the repeated zone scheme. It has both electron and hole character. Extremal cross sections are shown at right. (Ref. 10) 7' (1’ 1541) (H HHHIALI IN HNII'; IIHMI (H 3 ( HF ( IN‘N I X II?! M“! 25 .1 r O U (‘I / )u —-1 \ 1 1 1 1 O ( FiI \ /// . R l EXTREMAL SECTION OF FERMI SURFACE IN UNITS OF (Zn/012 0 OO ' 1 ' ' 1 1 11 1 1 1 I -10° 0 IO 20 3O 40 5O 60 7O 80 90 100° GOO) Gm) ORIENTATION OF MAGNETIC FIELD IN {no} PLANE Figure 8. Some extremal cross-sections of the empty lattice Fermi surface of AuGa when the magnetic field is scanned in a {110} plane. Sections correSponding to the 5th and 6th bands are shown by dotted lines. The top line framing the graph (7.02) refers to a great circle of the Fermi sphere. (Ref. 10) ..C a «D structure F; A l ure 10 S the star f I m u Can an 3 Vi .1 E .1 t C i «k E ‘1‘ .8 .\ \\ As \ 2 is 26 Band structure. Switendick21’22 calculated the nonrelativistic band structure (Figure 9) by the method of augmented plane waves (APW). Fig- ure 10 shows the path Q, between points L and W, intersecting the hexa- gon formed by the third zone neck sheet (labeled '+' in Figure 10) and the star formed by the fourth zone sheet (labeled '-'). 2.5 Orbits on the Fermi Surface Conduction electrons subjected to a magnetic field and the forces generated by the crystalline potential move in real space on complicated paths. Semi-classical theory relates this to the motion of gestates on the Fermi surface. The Lorentz force is _ _ .1. E-B-q(§t+cz><§t), [9] wheregt andfit are the total electric and magnetic fields seen by signed charge q of velocity v and canonical momentum 2, Assume (9) holds for the conduction electrons, and assume that replacement of p, g, and g by crystal momentum hg, and applied fields Ea andfia respectively leaves a valid dynamical equation, ' l E-‘hk —q(§a+c_vx§a). [10] Then it is easy to show that the application of a constant, uniform magnetic field causes electrons to undergo a continuous change of state while remaining at constant energy: the electron moves along the inter- section of its energy surface and a plane normal to g, Such a path is called an orbit, whether or not it is closed. For a given Fermi surface and field direction there may be a continuum of orbits. The deA effect is dominated by closed orbits having extremal cross-sectional (3,1 (HEY (NY I figure 9. Ener (Ref CI ref Pa~ f“ 27 X5' X4' 1.1?- x5 F15 ['15 1'25' X1 {'2' ENERGYIRYI 'er' ‘ GouyCAUJUM X A I" A L U W K Z r Figure 9. Energy bands of AuGaz by a non-relativistic APW calculation. (Ref. 22) F"""" ‘7\w / \ / Q \ / \\ ( 0 > ——Orbit cf, / \\ / \ / / \..._....._....._./ - - - -2 one Boundary Orbit 0'3 Figure 10. Intersection of the path Q with the orbits C' (hexagon) and C' (star). The empty lattice orbits have been modified to reflect the gap shown in Figure 9. Plane {111} lies in the paper. areas. Figun surface at t'.‘ galong the 2- surface of Au The appli to move along area, in a“.CCI' is the origin 28 areas. Figure 11 illustrates such orbits on a hypothetical Fermi surface at two different orientations with respect to the applied field §_along the z-axis, and Figuras6 and 7 show extremal orbits on the Fermi surface of AuGa2 when the field is along symmetry directions. The applied magnetic field does more than cause the electron states to move along orbits. It turns out that closed orbits are quantized in area, in accordance with the Bohr-Sommerfeld quantum condition, and this is the origin of the deA effect. The theory is described in Section 3.4. I 29 B-field Figure 11. Orbits on a hypothetical Fermi surface. (a) All extremal orbits (M = maximal area, m = minimal area) and one general orbit (G) for §_along a symmetry direction. (b) The same types of orbits for the Fermi surface at an arbitrary orientation with field. CHAPTER III BASIC THEORY OF THE DE HAAS-VAN ALPHEN EFFECT The de Haas-van Alphen effect is the oscillatory magnetization (or equivalently, susceptibility) of crystals with free charge carriers manifested at low temperatures (usually below 20 K) and big fields (usually above 1 RC). The oscillations are periodic in reciprocal field l/B with a period that is directly related to the extremal cross- sectional areas of the Fermi surface. Their amplitudes are dependent upon the curvature of the Fermi surface, the cyclotron effective mass m*, and scattering. The amplitudes yield density of states information and a scattering parameter, called the Dingle temperature and denoted either TD or x, related to crystal imperfections. The deA effect arises as part of the response of band state electrons and holes to an applied magnetic field. The theory requires a quantum mechanical approach, as can be seen from calculations25 of zero diamagnetic suscepti- bility for a classical gas of free electrons. Paraphrasing Dingle,35 in classical theory the electrons pursue any path consistent with Maxwell's equations. Under such lax conditions the average electronic current at any point vanishes. Quantum theory restricts the orbits; the current at each point no longer averages out, and magnetic behavior becomes possible. Fortunately the gross features of the deA effect are predicted by semi- classical theory, and the important deviations can be expressed in phenom- enological relations. Calculations using the full panoply of quantum mechanics and many body theory justify these results and generally give 26 only small corrections, if any. The basic semiclassical theory will 30 ed SO; '11 e Lt Tnen are 'NO 1C E .I 31 be given in this chapter, with mention when appropriate of features pre— dicted solely by more complete methods. The plan of this chapter is first to compare briefly two ways of looking at the magnetization of the conduction electrons in a magnetic field: the first involves the dynamics of electrons moving in a field, and the second is a thermodynamic approach. These give equivalent results. Then are presented a brief history of the deA effect, basic concepts, a sketch of the standard semiclassical theory, and finally some of the corrections to the semiclassical theory that are relevant to this study. 3.1 Two Aspects of Electronic Magnetization 26 Movinggcharges. This discussion follows Pippard. The effective Lorentz force (10), with no applied electric field and with the magnetic field along the z-axis, can be integrated to yield 15-30 = g x (1:10). [11] where kc and £0 are constants of integration (which can be written as one constant), and §Dis a scaling factor, .2 . §'- hc/e [12] Thus closed orbits in kfspace are related to the path in real space as follows: Project the real space path onto a plane normal to_§; it will form a closed path. Rotate this path by a positive angle fl/Z about §_and multiply its area by the scaling factor 32. The result is congruent with the kfspace orbit. (This is Onsager's theorem.) Equation (10) is sometimes called the dynamical expression for charge carriers in crystals. The velocity !_it contains is given by the kinematical expression, where defines the E, area of the The real essarily C10$1 fitha Pitch I :Otina ted b‘.’ \ 32 _ l l - E Vk RTE) . [13] d42 These lea to the following expression for the cyclotron frequency of a charge carrier on a closed orbit on the Fermi surface: B (0* = fit-C- , [14] where 2 3A h k * = 751? BE [15] defines the effective mass for the orbit, and AR is the cross—sectional area of the orbit in.kfspace. The real space path corresponding to an orbit in kfspace is not nec- essarily closed. But the field does render the path periodic along g, with a pitch 2 in real space. An analysis of the electron's corresponding 26 motion in kfspace, as was done for (15), leads to sZ = BAk/akz . [16] For reasons that will become apparent in Section 3.4, the deA signal is dominated by charge carriers on extremal orbits, that is, closed orbits whose areas Ak in kfspace are extremal. For these the pitch is zero. Pippard26 shows that the magnetization due to electrons on extremal orbits is the same as for the equivalent current loop: g_= ewéij , [17] where glis the vector area of the closed, real space path, with components 3A Ax =.l§.__§ , [18a] s as y 1 3Ak A = _._, 18b y 37an [ l wherei re 3' to rotat x tion of er}: the extreme shows .V. a: _ 7.: u‘ a IR?“ a- 50'- ~V‘ 5‘1 ‘Q‘LHH‘ \4 a... . «q ‘ u 4. i ‘43. s SI. “fire 11 i _ C ‘7». rue \ 92‘s 33 1 A2 = ;2'Ak , [18c] where 6y refers to rotation of B about the y-axis (coincident with ky), 6x to rotation about the x-axis, and equation (18c) follows from integra— tion of equation (11) about the orbit. The derivatives are evaluated at the extremal cross sections for the given field direction. Equation (18) shows 5 and_§ are not generally parallel. However, from (18) one expects g parallel to §_when along symmetry directions of the Fermi surface, and this is usually the case. Thermodynamic approach. The magnetization of a system is related to the dependence of its Gibbs free energy G on the magnetic intensity H through the thermodynamic relation This can be demonstrated by considering the work and heat energy changes of a magnetic system. It is analogous to V = —(3G/3P), where volume V corresponds to magnetization.M, and pressure P corresponds to magnetic intensity H.27 Frequently no distinction is made between the Gibbs and Helmholtz free energies in a solid system, since the effects of volume changes are negligible and reference is made simply to the free energy 9. Special cases are considered separately. For example, magnetostriction calculations require that the free energy explicitly include the dependence on strain. The general expression for the free energy per unit volume of an electron gas 1328 $2 = nEF + kBT Z 2n[1—£0(E)] , [20] states where n is the total number of conduction electrons per unit volume, fo is the Fermi-Dirac function, and the summation is over all conduction states. Equation (19 sufficient 13' this region E can make sev. be carried D-Q) with equal OCCupied or éi’u’En CrOSS den mics, I an aEO’Jnt f are relate; A bri 34 Equation (19) then is rewritten as 11 = -(VHO)T. [21] Implications for the deA effect. When a region of the sample is sufficiently extensive that it can contain the electronic motion and sufficiently crystalline, pure, and cold that there is associated with this region a well defined Fermi surface on which the electronic states can make several orbits before scattering, then both approaches above can be carried further to yield the deA effect. From the point of view of moving charges and orbits on the Fermi surface, the orbits become quantized, with equal increments of area Ak between orbits and some-finite number of occupied orbits (i.e. the number of quantized orbits that fit within the given cross section of Fermi surface). From the point of view of thermo- dynamics, the conduction electrons contribute to the total free energy an amount Qosc’ which oscillates with field. Of course, these two approaches are related, as demonstrated at the end of Section 3.7. A brief history of the deA effect is followed by definitions of these basic concepts: Landau levels, quantization of orbits, periodicity in inverse field, and deA frequency. 3.2 History of the de Haas-van Alphen Effect The deA effect is named after W. J. de Haas and P. M. van Alphen,l who observed it in 1930 in a single crystal of bismuth. Later that same year another quantum oscillation effect, in magnetoresistance, was observed by de Haas and Shubnikov.29 Independently of de Haas and van Alphen, but in the same year, 1930, Landau30 considered the quantum mechanical problem of free electrons in a steady magnetic field and remarked that the magnetization of a metal would the ort «L in. ..O Q 1C 828 have C was Pe Lari-1 3' 7 IESZT A... ‘~.d I A9 9?. . t S ”1.x. 1L 5 C 35 be expected to exhibit periodic variations because of the quantization of the orbits of the conduction electrons. The energy levels of this system, called Landau levels, enter directly into the calculation for the deA effect. Although calculations from first principles are limited to quadrat- ic energy surfaces, the concept is extended to real metals, most of which have complicated Fermi surfaces not resembling ellipsoids. Thus Landau's name is associated with the early development of the deA theory. But it was Peierls31 who in 1933 first addressed the deA effect, by showing that Landau's quantization extends to conduction electrons in systems with peri- odic potentials. Between the years 1933 and 1939 Landau,30 Peierls,31 Blackman,32 and Shoenberg33 developed the theory of Landau levels, still restricted to ellipsoidal energy surfaces. Not until the years 1952-56 6 7 35’3 and Lifshitz and Kosevich3 develop the theory did Onsager,34 Dingle for Fermi surfaces of general shape, usually referred to as the Lifshitz- Kosevich (LK) semiclassical theory. As the name suggests, this was not a calculation directly from the Schrodinger equation, but concepts from the earlier theory were incorporated. This raised the question of whether all the results of the LK semiclassical theory were generally applicable. In recent years full quantum mechanical calculations have given this theory a formal foundation and extended many of the important results to the general case. (See Ref. 38, hereafter called Gold, for examples and references to these calculations.) The deA effect was thought to be a peculiarity of bismuth until ob- served in zinc in 1947 and shortly afterwards in a number of other met-. als.39’31A Its importance as a tool to study the Fermi surface was perhaps not fully appreciated until Onsager34 in 1952 showed that the frequencies of the oscillations (with respect to inverse field) are directly proportional to the extremal cross-sectional areas of the Fermi surface perpendicular to the field. crystals. Within were done or. early work a characterize day. For a izpurities 1‘. of theories ' much more on Chapter IV. 36 the field. It is now extensively used to study the Fermi surface of crystals. Within a few years of the discovery of the deA effect, experiments were done on dilute alloys of bismuth.69 In fact, some feel that all the early work was on samples that were effectively alloys, albeit poorly characterized ones, when compared with the highly purified metals used to- day. For a number of years the goal of deA studies on samples to which impurities had been added was the study of the scattering and the testing of theories predicting amplitude dependence. These studies have now become much more quantitative and have increased their scope, as mentioned in Chapter IV. Meanwhile, experimentalists were looking for deA signals in other types of crystals. In 1961 they were first observed in a semiconductor40 (PbTe) and an intermetallic compound41 (InBi). Intermetallic compounds possess both long range order and a high density of charge carriers, so their Fermi surfaces have been studied in nearly as much detail as those of pure metals. In recent years other investigations have profited by the direct access the deA effect gives to the Fermi surface. One example is pressure derivatives, the effect of pressure on the band structure. The relation of the free energy to the deA effect (see Section 3.1) and the fundamental importance of the electronic energy suggest that phenom- ena other than magnetic susceptibility will exhibit oscillations as the field is changed, and these are observed in abundance, in both equilibrium properties (magnetic susceptibility, quasi-adiabatic temperature, heat capacity, magnetostriction, and contact potential) and transport properties (thermal conductivity, magnetoresistance, thermo-electric power and Hall constant). The generic term for these phenomena is quantum oscillations. Because OH a Vb". 84.. m {.1 (1' ’1 “here s 31.11 l/ t: (D H- ' f c‘ 37 Because quantum oscillations involve orbits on the Fermi surface they yield information about its topology and various parameters, such as extremal orbit areas, energy levels, level widths (related to crystalline perfection), and cyclotron effective masses. The deA oscillations are usually easier to detect than the other oscillations, and have made the biggest contribu- tion to the measurement of the cross-sectional areas of Fermi surfaces. Measurements of the other quantum oscillations sometimes give more accurate or precise values for the other parameters, depending on the substance and conditions. 3.3 Landau Levels Free electron model. Free, non—interacting electrons constrained to y’ 22 satisfying periodic (Born-von Karmén) boundary conditions have single particle energy levels given by Equation a volume V of dimensions 1x, R (5). Application of a magnetic induction _B_ = B2 [22] changes the quantization scheme: new wave functions replace the plane waves, kx and ky are no longer good quantum indices, and two new quantum numbers arise, one of which is n = O, l, 2, ... and appears in the new energy, 2k2 z . [23] En(kz) = hwo(n + 1/2) + h Ino where m0 is the free electron mass, the cyclotron frequency is a = LB [24] and 1/2 is a phase constant. These levels are called Landau levels, after their first calculation by Landau,3O who started from the SchrOdinger equation for magnetic field, where the kin He used the L for flux dens E is substitu given S i:- p 1 Iv because one 1 Of Energy wit The abse the Landau 1e CUStalllne P 0fthe Landau D936ne K113 7M .14) ShOWS (7 38 (ErEA)2 w 2mo = Ev . [25a] where the kinetic momentum operator is V [25b] He used the Landau (also, linear) gauge for the vector magnetic potential A = B(O,x,0) [25c] for flux density B parallel to the z-axis. (Frequently magnetic intensity ‘H is substituted for B in the literature.) Usually the Landau level is given simply as En = hwo(n + 1/2) , [26] because one is usually interested in a fixed value of kz, so the variation of energy with kz can be ignored. The absence of the second new quantum number in (23) and (26) renders the Landau levels of higher degeneracy than the energy levels of the crystalline potential, a characteristic which is intimately related to the existence of the deA effect. In view of this, a sketch of the derivation of the Landau levels and their degeneracies is warranted. Degeneracy. A standard derivation (see, for example, Ziman, pp. 269- 274) shows (25) is satisfied by 2 2 E = a' +h—E5 [27] 2mo and v(X.y,Z) = U(X) exP(i(By+kZZ)). [28] where u(x) satisfies the one-dimensional Schrhdinger equation for a simple harmonic oscillator, of frequency A The eigenvalue Equations (27) quantizes k 2 0f k2 is unres those orbits 1 then for nacrc 313' ing T ““5 the numbe mung each 1e 39 2 2 m _ _E_.§_E£§l.+ —9-(w x4fi§)2u(x) = E'u(x) , [29] 2mo dx2 2 o m of frequency “0 and centered at the point x =_1_f1§ [30] 0 mo mO The eigenvalue problem (29) has eigenvalues E' = hwo(n + 1/2) . [31] Equations (27) and (31) give the final result (23). Single-valuedness of w quantizes kz and B in units of 2n/zz and 2n/RY respectively. The range of k2 is unrestricted, as in zero field, but the range of B is restricted by (30) if one considers only those states corresponding to electrons whose orbits lie completely within volume V (which is practically all of them for macroscopic V). Thus 0 é=xo é=lx , [32] giving me. 0:8;—-—°fi°x=§32x- [33] m w po = °h° sixty , [34] making each level (23) p-fold degenerate. Note that = lezy = total flux [35] p0 hc/e flux quantum ’ so that the degeneracy of each level equals the number of flux quanta threading the specimen. All these derivations are for a right parallelpiped and for B112, but are directly extendible to a specimen of general shape and for general ,1. field or ien‘ the field, . the levels occupied le' (E.g., nO = at which le 33) Bl be Yielding mch is it. 1/39 with a 40 field orientation, kz becoming kH for the latter case. Periodicity. Figure 12 showsthe relation between the Landau levels, the field, and the Fermi level. Equation (26) shows the spacing between the levels increases in proportion to the field so that the highest occupied level no suddenly depopulates as its energy value rises above EF. (E.g., no = 7 at B = B1 and no = 6 over Bl < B ;=B2.) Let Bl be the field at which level En = EF for some specific quantum number n = no and let B2 > Bl be the field at which level no-l equals EF. From (26) this gives EF W = B1(nol + 1/2) — B2(n0 - 1/2) $ [36] yielding EL. QL.= 1 =‘he/moc [37] Bl B2 B1(no + 1/2) EF which is independent of the quantum numbers at the Fermi level; depopula- tion of the levels occurs at equally spaced intervals in inverse field, l/B, with a period 1 = (he/moo) l P=__.— Bn Bn-l EF [38] Each depopulation means a sudden change in the energy (23), En(kH), where kH replaces kz to allow for a general orientation of the crystal in the field, in line with the definitions in Section 2.1. This sudden energy change implies, by Section 3.1, a consequent pulse in magnetization, with pulses from different kH generally out of phase because En(kH) = EF is satisfied at different fields B (through mo). Near extremal values of En(kH), with respect to kH, the pulses are in phase, and these dominate the signal. Energy Figure 12. The 1 an a: highe 41 \\ \\\\Z\ \\\\\\ B a. (no=7) thno=6) Figure 12. The relation between the Landau levels and the Fermi level EF. At any field B the Landau levels are equally spaced by an amount that is proportional to B. As B increases the highest occupied level n suddenly depopulates as its energy value rises above EF. 0 depopulat i' electron :I of state 01 holes as t? the Fermi l with quadre replaced by one masses plicated Fe talline in; hhitsimil tion for the lan iSnot nece. Vet? near 1, .I. 3" Quanti: Der ivat \ A T. he ddas~van Sha . pe 13} Con c “Cimerf eld Q} The “yr 0 eq ls {17 42 Real crystals. The existence of Landau levels and the periodic depopulation occur in systems far more complicated than the simple free electron model. All quantum oscillations arise from the sudden change of state of a non-negligible fraction of the conduction electrons and holes as the magnetic field causes successive Landau levels to exceed the Fermi level EF' The quantitative theory can be extended to crystals with quadratic energy surfaces by generalizing to the effective mass, mO replaced by m* = (m1m2)1/2 in equation (23), where mi are the band struc- ture masses (elements of the effective mass tensor). But even for com- plicated Fermi surfaces in the presence of finite temperatures and crys- talline imperfections the properties of the band states continue to ex- hibit similar periodicities, governed by the slightly more general equa- tion En = fiw*(n + y) [39] for the Landau levels, where w* is given by (14) and the phase constant y is not necessarily 1/2. But full quantum theory and experiments give y very near 1/2 for most metals (Gold, pp. 45 and 85). 3.4 Quantization of Orbits Derivation. In 1952 Onsager34 was able to extend the theory of the de Haas-van Alphen effect in crystals with Fermi surfaces of arbitrary shape by considering the effect of the applied magnetic field on the elec- tronic orbits, rather than the electronic energy levels. In a field the electrons assume orbital motion, which is quantized according to the Bohr- Sommerfeld quantum condition §PC dq = h(n + y) , [40] where q is the generalized coordinate, p is the canonical momentum c A is the v of the ele force (10) orbits are where n an Landau lev to be unif Retur to Obtain From the d the path 0 SubstitutiI 43 _ .9 RC - TIE + CA 9 [41] Alis the vector potential, and y is a phase constant. This quantization of the electron motion in real space leads, by the semiclassical Lorentz force (10), to the quantization of orbits in kfspace, so that only those orbits are allowed whose area is given by An = 2ns(n + Y) , [42] where n and y are the same quantum number and phase constant as in the Landau levels (39), s is the scaling factor (12), and B (in s) is assumed to be uniform over the electron's orbit. For quadratic energy surfaces Y =1/2- Returning to (40), combine it with equations (11), (12), and (41) to obtain §f§£_x d£_- SAde = 2ns(n + y). [43] From the definition of flux 0 through the area AH defined by projecting the path onto a plane normal to B, Bf§£_x d3 = 2® . [44] By Stokes's theorem §Agdr_= fcurloéfda = ¢ . [45] Substitution into (43) gives 21rs(n + y) = ct = BA , [46] H showing that the flux in the projected real space area AH of the orbit is quantized. Onsager's theorem transforms (46) directly into (42). Cylinders of orbits. Figure 13 shows orbits at kH on a Fermi surface of arbitrary shape. This is referred to as a slice of krspace. AF is the cross-sectional area at the Fermi surface. Because kH is a good quantum number the orbits are quasi-continuous along the field direction and form tubes of I parallel ‘ tube is nt quadratic, in which c As th face in Pi, depopulace level (39). causes a .3‘» 0f 1'10?ng C Slices alOr HEXt SUbSeC a . . .erl _.. \& bet"i‘ien the stilt“ to 1%) be t1 nom- ~°1 to g \ Q“ C vurlace Com: 5 o 44 tubes of constant cross-sectional area with a common axis, not necessarily parallel to_§, unless B_is parallel to a symmetry axis. The shape of a tube is not necessarily constant along kH unless the energy surfaces are quadratic, f3 INK) = mk' 'k 9 [47] in which case they are elliptic cylinders (Gold, p. 44). See Figure 14. As the field increases and a given orbit passes beyond the Fermi sur— face in Figure 13 the Fermi-Dirac distribution requires that it suddenly depopulate. Each orbit (42) corresponds to a highly degenerate Landau level (39), and the sudden change in the distribution of orbit states causes a pulse in the magnetization, as expected from the point of view of moving charges, Section 3.1. The way in which pulses from different slices along kH contribute to the total deA signal is discussed in the next subsection. Periodicity and deA frequency. The period in inverse field l/B between the pulses coming from a given slice can be found by an analysis similar to that made for Landau levels. For a fixed field direction let AF(kH) be the area of the intersection of the Fermi surface with the plane, normal to B_at kH, that defines the slice of kfspace. (Note that the Fermi surface contains an orbit only for discrete values of B.) As suggested by the discussions of pulses in the magnetization, the conditions En(kH) = EF [48] and An = AF(kH) [49] are equivalent. Comparison of equations (39) and (42) gives immediately the period in inverse field for the orbits breaking through the Fermi surface: 45 Slice 1 To kH Figure 13. A slice in kfspace showing quantized orbits superposed on a Fermi surface of arbitrary shape, at some value of kH. This is not necessarily an extremal cross—section. .B is parallel to kH' Figure 14. Changes in the quantization scheme as the principal axes of an ellipsoidal Fermi surface vary their orientation in §_= H. The number of cylinders (levels) increases as the maximal cross-sectional area of the ellipsoid increases. (From Gold, p. 44) n: ..c {I ~ -\_ a ‘ fir su. T ,7 i. J . a“. 46 l 2ne l 1 P=___=__——_—.' 50 En Bn_l hc AF(k.H) 1 1 43 It can be shown that magnetization from each slice versus l/B is a saw- tooth wave (Figure l6(a)), with a period that varies with kH, so that sig- nals tend to cancel except those from that part of the Fermi surface for which AF(kH) is stationary with respect to kH: the extremal orbits. Figures l6(c) and l6(d) show the net magnetization after summing over slices in the vicinity of a maximum and minimum in cross-sectional area. (One figure is for no spin, and the other for spin 1/2 electrons.) The inverse of the grouping in (50) occurs often enough that it is convenient to define the deA frequency _ he F(O,¢) I 2ne AF,ext ° [51] where the deA frequency F depends on the field direction (O,¢) through the extremal cross—sectional area AF ext of the Fermi surface, hereafter 3 denoted AF. The deA signals (Figures l6(c) and (d)) can be resolved into harmonic components, each having a distinct period when plotted against l/B: Mosc’r on sin[21rr(—§ — y) i %] , [52] for the r—th deA harmonic, where the positive sign in the second phase constant is for orbits of minimal cross-sectional area, and the negative sign, for maximal. Figure l6(e) shows the fundamental harmonic for both maxima and minima. There may be more than one value of AF for a given field orientation; then there are terms like (52) for each deA frequency. The total magnetization will also have a non-oscillatory term, due to additional magnetism from the electrons and from the lattice. Equation (52) shows the deA signal is directly related to the Fermi surface 0 D) :3 00 H- ‘J m 2.1 (L H a {3 47 surface's extremal cross-section in each direction, information that can give good clues about the shape and dimensions of the Fermi surface, and in simple cases allows direct calculation. As outlined above, the sum of pulses in magnetization from all orbits passing through the Fermi surface generates the oscillations in the net deA signal. Equivalently, the oscillations can be imagined as occurring when each cylinder (Figure 13) or tube of orbits bursts the Fermi surface, which can occur only at extremal cross-sections. Degeneracy. As mentioned in the subsection on cylinders of orbits, each orbit represents a degenerate energy level. It turns out that the degeneracy is the same for each level and equals the numbercfifformer firstates that lie between the orbits, a result known as sweeping out the area between orbits. Although a rigorous proof of degeneracy is beyond the‘sem'iclaa-ss'ical arguments presented so far in this section, we present here a calculation Showing the validity of this heuristic concept for the free°electron model. In zero magnetic field the free electron model has discrete eigen— energies E = h2k2/2mo. In magnetic fields the Schrodinger equation be- comes H = E' , where H is given in (25a), with discrete eigenenergies E' given by (23). The energy difference Un = E' - E is the change in energy of the state 5 as it is transformed into one of the states of the set (n,kH). By the correspondence principle, in the limit of big quantum numbers we expect both systems to have the same energy, since classical charge carriers do not gain energy in a steady magnetic field. Assuming this carries over to each slice of kfspace, the question is, in what region can an orbit "sweep out" the kfstates so that U = O for any field? This is answered by summing over states ‘93 orbit (I? where and Ysing the ifSPace in films THE limits 48 states within a slice of unit thickness over a trial region for the n-th orbit (Figure 15). Consider fi2k2 U = (E' -E) n =‘hw (n + 1/2)-+-——JB [53a] n age 0 mo 2 = ;§_.[23(n + 1/2) - k2 ) [53b] 2m P o _ n2 _ - 2nmo (An Ak) , [53c] where k% = 18% + k; [54] and AR = “kg . [55] Using the density of states per unit volume of sample per unit area of kfspace l 2 Wk = __§_X [56] (Zn)2 in Ak,2 Uslice = I Un(Ak) wk dAk : [57] Ak,l gives A k, 2 ex 2 n.2 2 Uslice = 1 [i1 (An ' "kpi kp dkp - [58] Ak, 1 4U2 mo The limits shown in Figure 15(a), A = (An - AA/2)o.5 [59a] k,l n Assi of 5- 10 1e 10) Assignment Of h-states to level ..-- _ (a) n-1 n-1 Assignment of it; states tO level -...-- (b) Figure 15. Former k—states onto the orbits. circular orbits 49 n n+l \ \ \ \ \ AA = quantum 1 \ 1 of orbit area. 1 | 1 —— orbit. 1 I 1 -—-- boundary 1 / / Of regions / / / defined by / / / having area 1 I/2 AA. In n+l n n+l n+2 I n- n n+ 1 I l 1 I 1 lying between quantized orbits are swept Alternative schemes (a) and (b) for in a slice from a spherical Fermi surface. and where is the ce: a result c in Figure and give the IE 50 and A + AA/2 Ak,2 = 1—E---do°5 . [59b] where AA 2ns [59c] is the center area associated with the phase constant 1/2, give Uslice = 0 a [60] a result consistent with the correspondence principle. The limits shown in Figure 15(b), 0.5 Ak,l = Lia [613] and An+l 0-5 Ak,2 = ( 11 1 . [61b] give the result ezlxlX 2 Uslice = - 1 2) B ’ [62] 4nmoC which is not consistent with the correspondence principle. Thus the scheme of Figure 15(a) is the one to use. 3.5 Temperature and Scattering Effects The effect of bath temperature T and scattering on the Fermi surface, the latter represented by the scattering parameter X, or equivalently, Dingle temperature TD, is to reduce the amplitudes of the deA signal (52). The higher harmonics of MOSC are reduced the most. Nonzero temperature. The equations of Sections 3.3 and 3.4 assume the sharp Fermi surface and Fermi energy obtaining at zero absolute (I: ,I are are :llr 51 temperature. At temperatures above zero both of these quantities are "smeared out," the Fermi level EF by an amount kBT and the Fermi surface by an equivalent amount in k§Space. Hence from the points of view of either Landau levels passing through EF or orbits passing through the Fermi surface, depopulation of states occurs over a broader field range, with a consequent attenuation of magnetization. The existence of quantum oscillations requires hw* .2... kBT , [63] where hm* is the quantum of energy for the system with magnetic field. In Gaussian units this is equivalent to B(kG) ;_ 7.45 :1 T(K). [64] 0 Another consequence of finite temperature is in increased scattering of the electrons by phonons. This increases the width of the energy levels, with an effect on MOSC amplitude that goes approximately as exp(-const. T/B). Scatteringio Impurities and strains in the crystal also attenuate the deA signal. A nonrigorous approach is to argue that the Landau levels are given a finite width as the electron lifetimes in the orbital states are reduced from infinity to finite values by scattering, and that this finite width further decreases the rate of level depletion as the level passes EF. Dingle36 first modified the deA theory for the effect of scat— tering, expressing it in terms of an increase TD in the effective temper- ature, which appears in an attenuation factor added to (40): M or e‘mTD/B sin[21rr(%— y) 1‘ .2], [6S] OSC where and k1 lifet: these 52 a = 2n2kBm*c/he , [66] and kB is Boltzmann's constant. Dingle also investigated the relations among TD, the mean inverse lifetime l/r on the orbit, and the level width P. The present forms for these relations are exp(—n/w*r) = exp(—2nF/hw*) = exp(-2n2kBTD/hw*) . [67] These relations assume a Lorentzian line shape for the energy levels (Gold, pp. 57-58). Dingle assumed the free electron model, but the field dependence of the amplitude of MOSC as given in (65) is observed to hold very well for real metals, and a Green's function calculation by Brailsford44 only slightly modified Dingle's results, with a factor 2 in lifetime, giving (67). Others45 extended these expressions to general Fermi surfaces. However, accurate numbers for TD must be found experimentally, and the theory is not clear on just how disordered a crystal may be before the deA effect can no longer exist. It used to be thoughtl‘é,47 that if point and line defects approached a density of one per electron orbit in real space that no deA oscillations would be observed, but recent work48 shows that even in this limit the signal is strong enough for measurement and analysis, and the theory is still under development, as described in Chapter IV. Two important characteristics of TD are its independence of the mag- nitude of magnetic field in nonmagnetic media,60 and its noticeable depen— dence on field orientation. The latter effect arises because scatter on the Fermi surface is generally anisotropic, and as the orientation of the field is changed different regions of the Fermi surface make the dominant COHCI and sh S=+l transt 53 contribution to the deA signal. 3.6 Spin Splitting Dingle35 first considered the effect on electronic magnetization of the interaction of the electron's spin with the magnetic field. Lif- shitz and Kosevich37 extended his result to arbitrary effective mass. The effect of Zeeman splitting of each Landau level is to give, for any slice through the Fermi surface, two sets of levels with the same spacing but shifted in phase, giving rise to two signals of half the original amplitude and shifted in phase. The field's interactions with the electron spin S =‘il/2 perturbs the spinless Landau levels (39) by the amount SguBB, transforming (39) into ['11 ll hw*(n.+ y):_%guBB [68a] hm*(n + y : gm*/4mo) [68b] upon removing common factors. Thus the phase constants in the two deA signals Mé;%,r analogous to (52) should be replaced by Y :_gm*/4mo, so that the total signal is _ (‘1') Mosc,r ' Mosc,r osc,r [69a] = Ar siannr(F/B - 'y - gm*/41110) :1: n/4l + Ar sin[21rr(F/B - y + gm*/41110) I n/a] [69b] = Ar sin[2nr(F/B -y) ;:u/4] cos[r %.g 2E3, [69c] m0 a trigonometric relation giving the Dingle cosine factor in (69c), which shows that some values of gm* can give zero signal, called a spin- splitting zero. By multiplying the phase shift gm*/4m0 by the deA period it can also be expressed in terms of l/B, giving the total splitting bet LA) ‘1 1-. under Part 54 between peaks (see Figure l6(d)): O[l/B] = gm*/ZmOF. [69d] In magnetic media the signals from the two spin systems generally have different amplitudes, so this simple relation no longer holds because the Agt) are not equal and the energy of perturbation is no longer direct— ly proportional to field.60 3.7 Lifshitz-Kosevich Theory In 1956 Lifshitz and Kosevich37 started with the expression (20) for the free energy of a gas of nearly free electrons, assumed that the elec- trons do not interact (i.e., B_= ED, and showed that summing over states under the condition of the Onsager relation (42) yields an oscillatory part for the free energy, from which the oscillatory magnetization and sus- ceptibility are easily obtained by differentiation. In succeeding years the important results of this semiclassical theory have been validated by full quantum theory. The free energy summation in (20) can be evaluated by expressing the . density of states in terms of the coordinates (E, kH, H), and using the Poisson summation formula. The perfectly sharp Landau levels are then assumed to be replaced in the real crystal by a series of Lorentzian curves of half-width F to reflect the effects of crystal impurities. The derivation also includes non-zero temperature in the Fermi-Dirac distri- bution. After including the Zeeman splitting the result for the oscilla- tory part of the free energy is (see Gold, pp. 55-67 for a derivation): [70a] where where a and E‘Jaluat Th Cosine field, Where 55 where 3/2 kBT exp(—raTD/H) 1 He 2' m* 9 -__ r /§'(rhc) [Aextnlk sinh(raT/H) OS[2 r8 m0) Tl 41. [7013} x cos[2nr(%~- y):¥ where a is from (66), H is the magnetic intensity, F is the deA frequency, and 2 A .. ___ 3 AF 9 akHZ ext [70c] evaluated at the kH corresponding to the extremal cross-sectional area. The magnetization is given by (21), taking the derivative of only the cosine term since the other factors are comparatively independent of field. Using (6) gives r=°° r=°° so... = 2 so.” =2 use) “we , 171a] r=l r=l where wr = Zurl—L—F'eH ‘1’) - v] 47% , [71b] 2 lg 8 3/2 kBT exp(—raTD/H) 1T m*(e’¢) 3:11. = -1—1 9. fig “>317 r8 T") "r 'fii sinh(raT/H) o F e x ——£—J%?:', [71c] IAext 5 _ _ ~ 13 - ___1____§E . 9— ‘ 1H ‘e F as 'o F sine 23¢ ’ [71d] an an so :18 th te Q 5... 56 and a = 2n2kBm*c/eh, (146.9 kG/K for m* = mo). [7le] The term y is generally ignored or taken to be a constant in both direction and magnitude of field; hence it is called the phase constant here. Mea- surement of its angular variation requires a series of precision measure- ments of phase, and apparently has not been made for any metals, although the values of y at symmetry directions has been reported for the noble metals.49 The total free energy (70) and the magnetization (71) are best illustrated graphically by first taking the limit as T and TD (i.e. scat- tering) go to zero (the ideal crystal condition) and then setting g = 0, to give spinless states. The resulting waveforms are shown in Figure l6(b) (free energy) and Figure l6(c) (the component of the magnetization along iH). In Figure l6(c) the solid line is for a maximal cross-sectional area (of either an electron or a hole sheet of the Fermi surface), and the dOtted line is for a minimum (again either electron or hole). Now "turn on" spin. As discussed in Section 3.6 the signal arising from each spin sheet will be identical in nonferromagnetic crystals except for a phase shift downward in l/B for the spin-up electrons and an equal shift upward for the spin-down electrons, giving a waveform like that in Figure l6(d). The above ideal crystal condition gives the maximum relative ampli- tudes for the harmonics in 903 and Mosc‘ As evident from (65) and 7l(c), c finite T and TD tend to wash out the oscillations in each harmonic, with the higher harmonics being attenuated the most. Because of this rapid attenuation of the harmonics, their detection beyond the third normally requires very pure samples and very low temperatures, and for many experi- ments only the fundamental harmonic, illustrated in Figure l6(e), is observed. For such a case Zeeman Splitting of the levels attenuates the Figure 16. Theoretical dependences of magnetization and free energy upon 57 the inverse field, 1/B for T = TD = 0. (Figures (a), (c), and (e) are from Cold, p. 48) (a) (b) (C) (d) (e) Magnetization of a single slice of the Fermi surface. Oscillatory part of the free energy contributed by spinless electrons near an extremal section of the Fermi surface of an ideal crystal. All harmonics are present. Magnetization, summed over slices, conditions as in (b), for a maximal cross—section (solid) and a minimal one (dotted). All harmonics are present. Magnetization from a Fermi surface maximum, conditions as in (c) except electrons have spin % with consequent level splitting. Fundamental component of each of the two curves in (c). Figure (e) could also represent the two fundamentals of each of the two spin signals of (d), coming from a single extremal orbit, in which case the amplitude of the sum is reduced by the factor cos(ngm*/2mo). (0) lb) (e) fl+2 / / / / / {1+1 58 / / / / / (a) Figure 16 1.1 tr (.0 59 signal amplitude by the cosine factor in 7l(c), but does not change the shape of the wave from a sinusoid. Important properties of crystals can be inferred from measurements of the fundamental (or from any one harmonic). These are l) the extremal cross-sectional areas of the Fermi surface (given by deA F), 2) cyclotron mass m* (obtained from the dependence of amplitude Mr(H) upon the tempera- ture), 3) TD (from the dependence of Mr upon field), and 4) the effective g-factor (from extrapolation to the infinite field phase). The g-factor is more commonly measured directly from a quantum oscillation waveform rich in harmonics, as shown in Figure l6(d). 3.8 Justification of Semiclassical Theory Early theory rigorously developed applied only to single—particle states of independent electrons, and most of the present theory easily com- pared with experimental data is either semiclassical or has been extended by analogy from its free electron origins to real crystals. The concept most important to Fermiology, quantization of orbits, has two vulnerable areas, as pointed out by its originator, L. Onsager,34 in 1952. These are the ambiguity of the connection between E(k) and the Bohr—Sommerfeld condition due to the fact that the components of the kinetic momentum do not commute, and the implicit assumption of wave packets. Onsager specu- lated that it was "reasonable to hope that neither previous theories of diamagnetism nor the present generalization will be invalidated by the error involved...or at least that the error in the computed susceptibility will not vary rapidly with the field intensity.”34 So far his hope has been upheld, with subsequent rigorous calculations tending to support various elements of the theory. Let us consider the theoretical support 60 for three important semi—classical features: firstly, the concept of wave packets representing the orbiting electron; secondly, the relation between width of the energy levels and electronic lifetime; and thirdly, the cone cept of independent electrons. (See References 26 and 38 for more refer- ences and details than are given below.) Regarding wavepackets and the use of the dynamical relation (10) and the kinematical relation (13), when the field can be considered a perturba— tion of the total crystal potential one may write the wave function of an orbiting electron as (Ziman, pp. 147—157) (_r_,t) = E Z f(i,t) Ting—g). [72] n 2 The Wn(£_- g) are Wannier functions, obtained from a unitary transforma- tion of the Bloch waves and so are a suitable set of basis function for a representation of the electron wave functions. The f(£,t) is an envelope function obtained by solving an equation with the equivalent Hamiltonian and very similar to the Schrbdinger time dependent wave equation, and the sum is over all energy bands n and sites £_in the sample. As time passes, f(§,t) peaks at successive sites along the real space path of the orbiting electron, picking up wave functions centered On the sites. The various eigenfunctions that have been calculated this way suggest the idea of a time-dependent mixing of states: the uniform, static perturbation asso- ciated with the magnetic field generates a mixing of kfstates. As time passes the kfstate that most nearly characterizes the oscillatory wave function moves around the orbit. Increasing the field intensity draws more kfstates into the mixing, corresponding to an increased increment of area, AAk = 2ns. It is to be noted that as B increases so that orbit radius decreases, AAk increases, so that k at any instant of time is less well defined, illustrating the well known relationship between quantization 61 and the uncertainty principle. Regarding the width of the energy levels, considerations by Falicov and Stachowiak5 present the wave packet as a superposition of many orbital functions which spread in all directions, coming together periodi— cally after one cyclotron period, thus forming a series of pulses. They show that the density of states about the center of a level is a Fourier transform of the total, time dependent wave packet, energy being conjugate to time in quantum mechanics. So long as the pulses continue unchanged indefinitely in time they add to give a sharp energy level. Scattering of individual orbital functions attenuates the pulse more and more as time passes, with a consequent increase in the width of the energy level. This is the theoretical foundation of the relation (67) between level width F and mean inverse lifetime l/r. The problem of the electron-electron interaction has been approached from many directions. This sketch follows Cracknell and Wong (pp. 416-420 of Ref. 15, hereafter called CW). It is assumed that the behavior of the electrons lies between the two extremes of completely independent and completely correlated (plasma wave) motion. To a first approximation the interactions can be put into the crystal potential through a screening parameter A: V(£) = 2 Sign 3‘113711 . [73] 2 -—- . This still assumes the system can be described by single-particle states. Landau's51 phenomenological theory of the Fermi liquid, originated for liquid 3He, can apply to the conduction electrons as the interaction is turned on, and the Coulomb repulsion between electrons becomes partially balanced by the attractive force due to the exchange of virtual phonons, thereby generating the eigenstates of the interacting system by a continuous 62 transformation from the old eigenstates. The entities associated with the new eigenstates are called quasi-particles. They are independent modes that can replace the electrons, a replacement justified by perturbation theory.52 A finite discontinuity in momentum space at zero temperature remains, leaving the concept of the Fermi surface intact. Implications of the theory for TD and deA F are not clear. Experiment shows that deviations from predictions of the semiclassical theory are negligible, if observable at all (Gold, pp. 96-97). The LK amplitude 7l(c) agrees (within an experimental uncertainty of 5% to 20%) with the data on the noble metals.53 Also, because the measured Fermi volume of copper lies within 3% of that predicted by the single-particle LK theory,61 effects due to electron-electron interaction should change deA F less than 2% (Gold, p. 97). 3.9 Corrections to Semiclassical Theory As stated above, full quantum calculations have supported most results in the semiclassical theory, with minor adjustments of energy levels or deA frequency usually smaller than the experimental resolu- tions. But two intrinsic effects not predicted by semiclassical theory, magnetic interaction and magnetic breakdown, are easily observed. A third observed deviation is due to the demagnetization in a sample of finite size. Magnetic interaction (Manyébodyyeffects). Each electron sees the flux density B_and not just the applied magnetic intensity H, which Lifshitz and Kosevich assumed in their derivation. Replacement of H_by B_in (70) and (71) was first suggested by Shoenberg,54 in order to explain an abnormally rich harmonic content in certain samples giving a strong 63 55’56 He referred to deA signal and subsequently justified by theory. this as the B-H effect. The reason that the very small magnetization of non-ferromagnetic samples can alter the deA signals is that the field appears in the phase wr of Mr' This gives r=m Mose = X ELIE) sinwr(§) , [74a] r=l where org) = 2nr(§£%_¢l - y) 1% , [74b] and H is retained in Mr because these functions are not measurably changed by magnetic interaction. The flux density is given by p = g + (my , [75a] with the total magnetization M_of the electrons the sum of the oscilla- tory part due to quantization of the orbits and a steady part, E-= Ilisteady +L"lose ' [75b] Because Mose is the dominant part of M'at the fields and temperatures used to study the deA effect, one normally approximates M_in 75(a) by M . Equation (74) is thus an implicit equation for M . A calcula- ‘osc —osc tion by Gold (pp. 69-78) shows that 74(a) has additional, nonharmonic frequencies arising from extremal orbits labeled a and b, truncating 74(a) to the fundamental term for both orbits and adding the two signals gives, after solving the implicit equation to first order in the harmonics, the following frequencies: Fa and F normally the strongest, 2Fa, 2Fb, Fa - Fb, b and Fa + F the next strongest, and additional frequencies of smaller ampli- b tudes. The magnetic interaction effect is strongly field dependent. Thus the character of the oscillations may rapidly change with field to a more complex pattern, sometimes exhibiting beats. 64 Demagnetization. When the deA amplitudes and the magnetic suscep- tibility are big enough that magnetic interaction is observable, then demagnetization produced by the shape of the sample may also be important, entering the deA phase through -§internal = £1applied +fldemag + ATE ' [76] This can increase the difficulty of calculating the relative amplitudes of the harmonics in l/H. If Mosc H2/8n2F IIV 1 . [77] then the magnetic response for a sample of arbitrary shape will be exceed- ingly complicated, and the only simple fact is that it will oscillate with the same period as for the LK theory (Gold, p. 75). 57 Magnetic breakdown. Cohen and Falicov were the first to point out the possibility of an electron's jumping between adjacent semiclassical orbits on the Fermi surface (an interband transition by the magnetic potential). Under the influence of a moderate magnetic field states will move along arc C to arc D of the hypothetical orbits of Figure 17 to form orbit A(2). A stronger field may cause a transition from one zone to the other, so that the state moves from arc C to arc B. By Onsager's theorem, such orbits correspond to real space orbits, with resultant quantization. The sign of the area was ignored in the derivation of Onsager's rela- tion (42), with no distinction between hole and electron orbits. Breakr down orbits require explicit consideration of the sign which arises in step (43), where_£ x d£_may be negative on some arcs and positive on other arcs. It is obvious that the integral of £_x d£_about an orbit that intersects itself is equal to the algebraic sum of areas from each simple loop of the orbit if one assigns negative 65 value to hole orbits and positive value to electron orbits. Magnetic breakdown thus generates new deA frequencies which are sums and differences of the old ones, and proportional through (51) to such areas as A(3) and A(4) of Figure 17. 58 Blount showed that the condition for significant probability of magnetic breakdown is (F.F fiuu*);§ i E [78] g 9 where Eg is the energy gap between bands at the point in kfspace where breakdown occurs. The breakdown parameter BO gives the probability of a , 59 jump as P = exp (-Bo/B) , [79a] where E2 B = , ° ZfiszzK [79b] where G is the reciprocal basis length, and K is the distance in kfspace from the center of the orbit to the breakdown point. These conditions are so strong that almost all experimentally ob- served breakdown occurs across the small energy gaps due solely to the spin-orbit interaction, which may lift degeneracies allowed by the crystal field. Breakdown across the basal planes of the hexagonal close packed structures is the most common example. However, breakdown across energy gaps generated by the crystal field does occur, although rarely, with the needles of Zn being the oft-quoted example. So investigation of the possibility of breakdown on orbit C5 is justified. Consider the only two zones containing sheets of the Fermi surfaceanAuGaz, namely the third and fourth zones, and superpose them, with B_11(lll). All the extremal orbits labeled C shown in Figures 6 and l (l) (21 .43 fl-fl Figure 17. Hypothetical orbits to illustrate magnetic breakdown. Original orbits and new orbits created by magnetic breakdown (arrows MB). The areas of electron and hole orbits have different signs. §_into the paper. .._ _/ .52() Figure 18. Cross—section of the empty lattice surfaces in the third (clear) and fourth (black) zones at the kz values given for p H <111>. Height of the unit cell is 2.0. (From Refs. 3 and 23) n: H n 67 7 can exist on the Fermi surface. Orbits C' and CA have the same kH. A 3 3,23 . (given here as Figure 18) clearly shows thlS construction by Longo property (as well as the origin of some of the other extremal orbits that lie in the {111} planes). The last diagram of Figure 18 is in the plane of the hexagonal zone boundary of Figure 5. (Half the distance from a corner to the center of a cube of edge 2 is .866). This is redrawn in Figure 19(a) and (b), showing the two orbits concerned. Figures 19(c) and (d) show hypothetical breakdown orbits, with therxints of breakdown circled. Orbit (d) is self-intersecting. To calculate the magnetic fields required for significant breakdown probability, take Eg = 0.54 eV from Figure 9, and m* = .175 m0 from Ref. 10. Condition (78) requires B 3 479 kG . [so] This value is only an estimate because a bandstructure calculation is not absolutely accurate. The parameter Bo can be calculated if one knows G and K. The lattice constant10 a = 6.055 X at 4.2 K gives G, and an estimate of K can be obtained from either the empty lattice diagram Figure 5 or the band structure of Figure 9, by calculating ratios of distances on the Brillouin zone. The empty lattice Fermi surface figure gives K = .llG and the band structure graph gives K = .O6lG. These give Bo as 1.6 x 106 Gauss and 2.8 x 106 Gauss, respectively, with corresponding transition probabil- 14 and 10-25. Even ities at the highest field used (5 x lo4 Gauss) of 10' assuming the true value of K is twice that of the empty lattice calcula— tion (an unlikely event since the lattice potential tends to reduce small cross sections near zones), P = 10- . The "significant" P associated with the field (80) is between .2% (for K = .061) and 4% (for K = .11). These models show magnetic breakdown should have negligible effect on the deA oscillations of the C5 orbit of AuGaZ. 68 A A (III) zone VQV boundary lies AVA in the paper. (a) Orbits in zones 3 and 4. The crystal field smooths the sharp corners of (b) the free electron orbits. /'\7 Zone 4 Zone 3 (c) (d) Orbits possible by magnetic breakdown. O= breakdown point. Orbit (d) self - intersects. Figure 19. Orbits centered on point L on the Brillouin zone. (a) Intersection of <111> plane with Fermi spheres. (b) Neighboring hole orbits: C3 (hexagon) and C4 (star). (c) and (d) TWO examples of many possible hypothetical break- down orbits. However, all have essentially zero probability. CHAPTER IV DE HAAS—VAN ALPHEN EFFECT AS A PROBE OF CRYSTAL COMPOSITION Within a few years of the discovery of the deA effect, experi— ments were done on dilute alloys of Bi.33’69 For a number of years the goal of deA studies on samples to which impurities had been added was to study the scattering effect and test theories predicting amplitude dependence. But quantitative measurements on even dilute alloys were hindered by the strong dependence of the signal's amplitude on crys- talline purity. Typically, the amplitude is attenuated by factors of 100 to 1000 per atomic percent of concentration of impurity. The de- 49,62—68 has velopment of sensitive and precise measurement techniques made possible recent and continuing use of the deA effect to study the effect of composition on the tapology of the Fermi surface, the scatter— ing of conduction electrons, and the contribution of the impurity atoms to the density of conduction electrons. Such studies are still limited to dilute alloys, however. 4.1 De Haas-van Alphen Frequency The introduction of impurities changes the topology of the Fermi surface of a metal. In the case of impurities having a valence differ— ent from that of the host, the consequent change in the electron to 69 70 atom ratio causes coarse changes in the Fermi surface. Finer changes are caused by the alteration of the band structure, due to the differ- ent atomic potential of the impurity. The biggest change in deA F occur usually for big valence difference Z, small pockets of the Fermi surface, and the highest concentrations of impurity allowing observable deA signals. The maximum concentrations of impurity useable have been about one atomic percent. The changes in deA F are typically less than 0.1%. Although early work on alloys, summarized by Heine,70 was valuable in showing alloy experiments to be feasible, it did not allow significant comparison with the theory. As referenced above, the development of methods for precise measurements of deA F allowed more quantitative ex- periments. One result was the determination of the average number of conduction electrons contributed by each impurity atom in some alloys of 71,72 the noble metals. Various other alloy systems have yielded a range of results, from no change in F (ZnGe in A173) to a modification in the Fermi surface sufficient to produce new deA F (In in Pt74). Derivation of a theory began in 1956 with Heine's7O interpretation of much of the published data using the theory of primary solid solu- tions and Friedel's75’76 rigid band structure model. This model assumes that the band structure of the host does not change upon adding impu— rities, the only effect being a change in EF. It was later extensively deveIOped by Stern.77 A theory to allow band structure and density of states to change upon alloying was put forward in 1958 by Cohen and Heine78 for monovalent metals. Coleridge79 has introduced partial wave analysis into the theory. Quantitative correlation of the data with the more complete theory requires accurate knowledge of composition. Such knowledge is not ob— tainable for our samples, as will be described in Chapter IX. 71 Furthermore, it turns out that the compositional variations in our sam- ples have only a slight effect on deA F. Thus only a general, qual- itative description of the simpler theory, i.e. the rigid band model, is pertinent. Rigid band model. Following Stern77 and Coleridge and Templeton,71 the addition of impurities is assumed to have no effect on the electronic band structure of the host, and the only basic change is in the density of conduction electrons, i.e. the number per unit sample volume, if the relative valence of impurity to host is not zero. Secondary changes follow in the density of states at the Fermi level and in the topology of the Fermi surface, specifically in cross sectional areas. The rigid band (RB) model can be used for cubic systems even when alloying changes the lattice parameter because the relative dimensions in reciprocal space remain unchanged. Let N be the density (number/volume) of atoms in the alloy, N the 1 density of impurity atoms, ai their fractional concentration, Ni 3 = F, [81] and Z their valence difference relative to the valence of the host atoms. Then the increase in the density of conduction electrons is dn = ZaiN. [82] For small ai the change in Fermi level is Za N [83] where D(EF) is the density of states evaluated at the Fermi level. The RB model then predicts a relative change in the deA F; (f) 72 _dF ='_l(dAF) dE = m*cNZ a F A dE F fieF D(E ) 1' F EF F [84] Thus changes in F are directly proportional to both impurity concentra- 71’72 observed tion and to valence difference. Coleridge and Templeton this behavior for Cu alloys with 121 §_2. But Si and Ge did not fit the RB predictions. The behavior of the transition metal impurities varied. For example Ni fit the model approximately,72 by others showed no simple correlation with Z. Furthermore, the rate of change of F with ai did not agree quantitatively with the rate measured for Cu necks. A para- meter sometimes used to characterize the change in the Fermi surface is _ dF/F S — dn/n [85a] _ dF/F " NZ a /n [85b] 1 _ m*cn _ heF D(EF)' [85°] The empty lattice model gives S = 2/3. The measured71 values for Cu are S = 0.69 for the belly orbit and S = 6.2 for the neck orbit. RB theory has limitations when applied to metals with more compli- cated Fermi surfaces and structures than those of the noble metals. For example, RB depends on an unambiguous valence assignment, not possible when d bands lie close to the Fermi level. Experiment also shows that the alloy band structure is not really independent of the lattice para— meter, even for a cubic lattice. For a discussion of RB failure, see cw, pp. 499-501. Relation to AuGa2. The observation by Coleridge and Templeton71 that RB fails for 121 > 2 in noble metal hosts suggests that a reliable 73 prediction for the response of the C5 (neck) orbit of AuGa2 to a varia- tion in vacancy concentration requires either theoretical calculation more sophisticated than RB or experimental evidence. (For Ga-site vacancies, Z = 3.) The calculated S for this orbit is 27.33, and by equation (85c) is independent of the relative valence Z and concentration Both E = 9.19 eV and D(EF) = 2.06 x 1022 eV-1 were calculated from 31' F the free electron equations. The lattice constant and the assumed 23 valences of Au and Ga give n = 1.26 x 10 cm—3. Two data points for dF of C5 versus concentration of Pd, taken from work of Schirber,20 give dF/F = 7.14 a1 [86] (with large uncertainty) near a .005 atomic fraction of Pd). The Pd 1 substitutes for Au in Au dexGa Substitution of (86) and (82) into 1— 2' (85a), using the lattice constant to get N = 5.41 x 1022, and assuming the valence of Pd is zero (i.e., Z = —1) give S = 3.8, a much slower change in F than the RB model predicts. This discrepancy could arise three ways: (1) the use of a free electron value for D(EF), (2) Z is not -1 for Pd, and (3) the RB model fails, so that dF/F is not due entirely to changes in n. It is diffi- cult even to estimate the effect of (1). As for (2), taking Pd valence to be zero may be unreliable for quantitative work, but probably gives a good idea of the direction of changes, and has been found acceptable in the interpretation of deA data on some intermetallic compounds con- taining Pd. Finally, part of the discrepancy is certainly due to (3), the inappropriateness of applying pure RB theory to the C3 orbit, which is clearly demonstrated by measurement of the pressure derivative: d(ln F)/dP. Schirber and Switendick18 and Schirber20 measured 74 d(ln F)/dP = (~13i4) x 10-4 kbar-l. (There was a rapid decrease in the derivative above 6 kbar.) The pressures used correspond to volume changes about 1%.20 The result of APW band structure calculations18 using a lattice parameter reduced 1% below that used previously (Refer- ence 22 and Figure 9) agree with the sign of the pressure derivative. These show that the second zone begins to empty and the third zone neck region begins to £111.20 Both these investigations show RB fails to the extent that band structure depends on the average lattice parameter a. It is not clear how much change in a is caused by the Pd impurity, but the direction of the discrepancy in S (i.e. the fact that measured dF/F is smaller than that predicted) requires a decrease of a in view of the reports of Schirber and Switendick. A decrease in a is suggested by the smaller a of pure Pd (3.88 A versus 4.07 A in pure Au; Reference 42, p. 29), while both elements have the fcc structure. Also, the ionic radius of Pd (.80 A) is smaller than for Au (1.37 A) (Reference 156, p. F-ll7). Vacancies would give a smaller a. They also have zero valence. If the deviation of the composition of AuGa from stoichiometry is due to 2 vacant lattice sites, the opposing tendencies for increasing and decreas— ing F could give a very small change. 4.2 Dingle Temperature In contrast to deA frequency F, the Dingle temperature TD, also called the scattering temperature, or scattering parameter, is unaf- fected by ordered changes of composition. However, disorder strongly increases TD, and the most important cause is lattice dislocations, due 75 to the fact that small angle scatter (any angle greater than l/n, where n is the deA phase) is sufficient to eliminate the contribution of the 48,80,81,82 scattered electron. Dislocations have strain fields that fall off very slowly (l/r) in comparison to those of impurities (l/r3).80’81 The scattering of an electron through the strain field can be treated equivalently as successive small angle scattering and as dis- persion of a wave by a medium of variable refractive index.48 Impurities act almost as paint scatterers, and the strain field is 80,81,83,84 usually ignored. Most of the measurements of scattering anisotropy, i.e. obtaining r by inverting T have been done for impu— k D’ rity scattering.85’86’87’88 Such inversion requires thorough knowledge of both the Fermi surface topology and the electron velocities.87 There is appreciable anisotrOpy in the noble metals, depending upon the impu- rity.87 See Section 9.3 and Table 18 for further comparison of the types of scattering. Scattering by vacancies has received little attention. Lengeler and Uelhoff89 reported TD/ai for various orbits in Au, obtained by quench— ing and extrapolating the vacancy concentration from its value (720 ppm) at the melting temperature. The range of TD/ai was 35 to 51 K/at.%, and 40 K/at.% on the neck orbit. For comparison, Lowndes g£_§1.87 found TD/ai = 9.1 K/at.% for Au(Ag). They also verified the linear dependence on concentration. The theory has been develOped by Soven90 and Coleridge g; 31.91 Using multiple scattering theory, Soven90 computed the oscillatory den- sity of states when a dilute, random distribution of atomic scattering potentials is put into a free electron gas in a uniform magnetic field. The shifts in deA F can then be calculated. Coleridge g£_§1.91 based 76 their theory on partial wave analysis, extracting from anisotropy measurements the relative amount of interaction with s, p, and d waves. 4.3 Considerations for Precision In their discussion of experimental technique, Coleridge_g£ogl.68 and Coleridge and Templeton49 emphasize the need for highly controlled experimental conditions. Coleridge and Templeton49 used magnetic fields with homogeneities of 10 ppm over the sample diameter (a few millimeters), angular resolution of .03° (corresponding to a resolution in deA F of 5 in 107) by using the deA signal's symmetry to align the probe, and deA signals with little noise, yielding a maximal resolution in phase of .002 cycles and an assured resolution of .01 cycles. Their field was measured with an uncertainty of less than one Gauss by the use of in §i£p_NMR. Our superconductive solenoid had a resolution of about .02% over 4 cm between 30 and 50 k6, or 5 ppm over 1 mm. Our NMR probe could measure the field with 5 significant digits easily, but stability of the oscillator was not quite good enough for 6 digits. Thus the resolution was between 0.1 and l Gauss. Angular setting was achieved by a sample holder copied after that of Coleridge and Templeton,49 that would be tilted with the probe in place and recording a deA signal. The experimental apparatus being of suitable sensitivity, it re- mained to choose one orbit to examine. Since our intent was to see the relation between the deA effect and the composition of our samples, rather than a general study of the Fermi surface, our experiments measure the deA signal from only one extremal orbit, that of the third zone neck, C5. The reasons for choosing this orbit are as follows. Previous 77 3 on AuGa2 showed that the signal from C', the third zone neck orbit, w is a strong one. It has even symmetry about , and 1 experiments varies fast enough with angle to allow precise orientation of the probe. As a small orbit, its greater sensitivity to impurities is suspected, an assumption consistent with the more rapid increase (by a factor of two to three) of T of Cu neck with respect to Cu belly orbits when dis- 48,82 D locations are introduced. And equation (84) shows the relative change may be greater for small F. The sensitivity of the neck region of AuGa2 to structural changes was also implied by Schirber's20 work: V 3 biggest. Some of the other orbits with comparable pressure derivatives the magnitude of the pressure derivative of orbit C was among the had weaker signals. In the direction and at 4.2 K, the easiest 3 This frequency is also isolated, of help in obtaining accurate phase temperature to maintain, the signal from C was completely dominant. measurements. On the other hand, after much of this thesis data was collected, Templeton and Coleridge72 reported dF/ai to be three to six times greater for Cu belly than for Cu neck when Ni and A1 are the impurities. However, due to the plotting methods of obtaining pre- cision F, it isn't clear whether greater precision is obtained from an orbit with bigger absolute change in F, or bigger relative change. Detecting the signal from a AuGa orbit of greater area would have 2 been difficult in the presence of the dominant neck signal. CHAPTER V EXPERIMENTAL TECHNIQUE FOR THE DE HAAS-VAN ALPHEN EFFECT 5.1 Techniqpes Available Measurement of the frequency and amplitude of the oscillatory magne- tization (equation 71) of the deA effect yields extremal cross-sectional areas, cyclotron masses, lifetimes of the charge carriers, effective g-factors, and other properties of electrons on the Fermi surface. By recording the magnetization as a function of the magnitude of the field and of the temperature, with angle of orientation as a parameter, the angular dependence of the quantities can be mapped, and also the shape of the Fermi surface determined. Three techniques are commonly employed to measure Mose. One measures M, the total magnetization, directly by its torque in a uniform field, and the other two measure differential magnetization through the voltage induced in a coil by the response of M_to a time dependent field. Torque. The original observations made by de Haas and van Alphenl used Faraday's method of measuring magnetic susceptibility,92 which requires a non—uniform field. Shoenberg69 devised a method specifically for the deA effect: mechanical linkage to the torque couple of samples suspended in a uniform magnetic field. The couple is a measure of the differences of magnetic susceptibilities along the principal axes. A uniform field increases resolution and simplifies analysis of the effect. 78 79 A major advantage of the torque method over the induction methods is that it is a more direct measure of the amplitudes of the deA harmonics of 71(3), allowing comparison with LK theory. (See D. Shoenberg and 53 and references therein.) As will be seen, the induction J. Vanderkooy methods further resolve each deA harmonic into time harmonics, with each amplitude weighted by additional functions. Summing the time harmonics to obtain the deA harmonic is uncertain because the gains and phase shifts imposed by the apparatus vary with each harmonic and are usually difficult to determine. Two majcn: disadvantages of the torque method are that M_x B is zero when B_is parallel to a symmetry axis of the sheet of the Fermi surface being measured, since 3F/36 and 3F/8¢ in 7l(d) are then zero, and that the apparatus requires more access room and less freedom ofcnientation. This latter difficulty has been ameliorated by new designs, including measure- ment of the torque by the counter torque provided by a small coil about the sample.:53 A non-ferromagnetic sample may experience a torque because M_and B are not parallel for a general direction in a metal with a non-spherical Fermi surface. Whether the field is swept or rotated, the sample must not be permitted to move in response to the torque Mox B_acting on it if relatively uncomplicated signals are desired. Otherwise F(6,¢) will oscil- late about its mean with a frequency F and amplitude proportional to M, giving rise to sum and difference frequencies. (Sum and difference fre- quencies also arise from magnetic interaction. The cause is the same: oscillations in the phase (74b),ralthough in one case F oscillates and in the other case B oscillates.) For this reason counter torques are applied during the experiment. The induced voltage methods are also subject to this distortion, although it is not as inherently as big a problem since 80 with them the sample is rigidly mounted. The mounting of easily deformed crystals for both low stress and low compliance presents some difficulties at times, but these difficulties are small for AuGaz, which is a hard, brittle alloy. Pulsed fields. Banks of capacitors repeatedly drive big currents through solenoids, each pulse enduring for times of the order of micro- seconds. Fields to 300 kG may be generated easily. An oscilloscope displays the voltage v(t) = §2_m §§-+ 4n%%-. [87] The second term is isolated and recorded by photography or digital storage. The pulsed field technique was also originated by D. Shoenberg and extensively used by him and his collaborators to first observe many of the high deA frequencies.93 It remains the only method for observing the deA signals of very low amplitude (usually those with m* > mo, although recently the use of very strong solenoids permits the measurement of deA signals corresponding to m* two or three times mo). Limitations of the pulsed field technique include the limited resolu- tion, noise, and eddy currents. The last both complicate the analysis of the signal and create temperature instabilities. High speed collection and analysis of the data by digital electronics has recently improved both sensitivity and signal-to-noise ratio, but for most metals the field modula- tion technique, to be discussed next, remains the best method. Field modulation. The field modulation technique was frequently called the Shoenberg-Stiles method, after its inventors, D. Shoenberg and 94 P. J. Stiles, but the basic idea has been so extensively modified by so :many others 95-101 that it is now usually known by its acronym, FMT. 81 The field modulation technique induces v(t) in. (87) by superposing an alternating magnetic field h(t) on the steady field HO. The use of a steady field gives better control of the field and more time for removal of the noise in the deA signal. (Here, steady means the HO is considered fixed in time for the purposes of calculation. To analyze the deA signal obtained during a field sweep, the sweep rate must be slow enough that the time dependence of Ho affects the results by a negligible amount. This condition is easily satisfied in practice.) The modulation field h(t) is usually sinusoidal. Shoenberg and Stiles used modulation frequencies in the megahertz range, perhaps a carry-over from the microsecond electronics used in the pulsed field work previously. Skin effect in normal metals and alloys is significant at these frequencies. A. B. Pippard analyzed the extreme case (anomalous skin effect), and much Fermi surface information was obtained on various metals};3 However, the intermediate regime of strong, but not anomalous, skin effect is exceedingly difficult to analyze, and even in the anomalous regime the complications caused by the eddy currents prevent the extraction of the full wealth and precision of information about the Fermi surface contained in the deA signal.94’102 Shoenberg and Stiles also used small amplitude modulation, so their version of FMT could be described as high frequency, small amplitude. The low frequency, large amplitude version developed subsequently 95-101 is the one most often used today. It offers numerous advantages: a more straight—forward analysis of the data, higher resolution, more convenient signal processing, less noise in the signals, and the availability of an internal spectrometer action of the sample to discriminate against strongly interfering deA signals and to enhance weak ones. The theory of FMT will be further developed in the next section. 82 Other techniqyes. The three techniques described above are the most useful ones. For special conditions other techniques may be better, or they may be usable when none of the primary three is. One of these is a vibrating sample method, which is useful for semi-metals and semiconduc- tors, which have low conductivity and small cross-sections. It is rarely used for studies of the high frequency oscillations one ordinarily finds in metals and alloys. There are also methods which combine or modify the primary three. Field modulation with two modulation frequencies, one high and one low (the deA signal appearing as a pair of side bands of the modulation frequency), field modulation in combination with torque, and field modulation in combination with pulsed field (to utilize the maximum in field, where the induced voltage would otherwise be zero) have all been used. See Gold, pages 112-120, for references. 5.2 Field Modulation Techniqpe The deA data of this thesis was obtained with the low frequency, large amplitude FMT because it is the most convenient, an electromagnet and solenoids to provide steady fields were readily available, and the types of measurements desired were suitable for FMT. Because the deA signal from the C5 orbit is strong and m* = .l75mo the main advantage of the pulsed field, its ability to measure small signals, was not needed. Rather, since high precision was sought, it is doubtful that the pulsed field method could have been used; the development of high precision measurements has been done using the low frequency, large amplitude FMT. Furthermore, the torque method could not have been used because the field was in the symmetry direction . The remainder of this chapter gives the theory of FMT and shows how to determine the quantities of interest from the theoretical expressions 83 describing the sample's interaction with the detection system. Chapter VI examines our apparatus in detail. Chapter VII outlines the experimental procedure, and comments on some problems to be aware of when applying this theory to real measurements. Magnetic field variable. The fundamental field quantity is flux den- sity (or magnetic induction)_H, both in its microscopic (i.e., at points within a crystalline cell) and macroscopic (i.e., averaged over a small number of cells) forms. But as usual when dealing with macroscopic media, the magnetic intensity, defined as H_= H_- 4nH, is more convenient to use in measurement and calculation. In this chapter manipulations of the LK equations (71) and (74) for H_use H because LK theory was first derived assuming H'= H, the calculations in the literature use H, and it is easier to compare fflflds measured with different devices, since H discounts varia- tions due to different media. Clearly there is no difference between using H and using H when magnetic interaction is negligible. LTh‘e data of this thesis were taken under this condition. (Recordings in which magnetic interaction is visible were not analyzed.) Voltage. Let the steady field HO be modulated by a small, parallel, atlternating field H(t) so that the total field magnitude is H(t) = Ho + h(t) , [88a] With modulation field h(t) = h cos(wmt) , [88b] WI"Hare mm = 21rfm [88C] 313 the modulation frequency, and h is the amplitude of the modulation 84 field. An oblique geometry (Ho not parallel toIQ can be used to enhance different deA frequencies of the spectrum,98-101 but the data of this thesis were taken using condition (88). The only significant changes in the spectrum MOSc (71a) arising from the field modulation come through the phase (71b), which now depends explicitly on time as well as on the steady field H : —o 1pr“) = 21Tr[HO + h cos(wmt) - Y) + [89] £4: 0 where F is F(6,¢) evaluated for the direction (6,¢) of HO (see Figure 3) and is a constant in time due to.H H'flo° The skin effect is negligible because 2 w < C 2 9 [90] 2nod permitting H to uniformly penetrate the sample. For h << Ho an approximate treatment97 obviates the work of Fourier analyzing M(Ho,t): F ,3; h H + h cos(m t) 7 H [1 H OS(wth [91] o m o 0 30 that _ E;._ Fh _ ‘21 sin wr(t) - sin[2nr[H -§cos(wmt y] + 4). [92] o H 0 Let i=2n—F-'21. [93] H o Trigonometric relations transform (92) into sinwr(t) = Sim]:r cos[Ar cos(wmt)] + costr sin[Ar cos(wmt)], [94] .\'. fi 0u is 85 where wr is now (and for the rest of this chapter) the phase (71b) eval— uated at the steady field Ho, and is to be distinguished from the time dependent wr(t) of (89). The Fourier series for the time dependent func- tion involves the Bessel functions of the first kind: cos[Ar cos(wmt)] = 2 n2; (—l)n J2n(lr) cos(2nwmt), [95a] (D sintir cos(wmt)] = 2 néo (-l)n J2n+1(lr) cos[(2n+l)wmt], [95b] where the first term of the primed sum has a factor %. Substitution of (94) and (95) into (713) yields the following expression for the time de- pendent magnetization. It has been generalized for the case of more than one deA signal (i.e., more than one extremal orbit on the Fermi surface for the given field orientation) by summing over orbit index i. 5...“) % 2 1.21 Ff.” an x {sinw:l) (%JO(A:1)) + n20 (-l)n J2n(1:1)) cos(2nwmt)) -cos (i) r m n (i) 1r {O (-1) J2n+l(lr ) cos[(2n+l)wmt])}. [96] n: Note: that H6 has been substituted for the correct value H ianr. This aPPIroximation introduces less error than the previous approximation (91), 0f the order of 0 to 13% compared with O to 4% of the amplitude for the £181xi and temperature ranges we used. To particularize this equation to our exxperiment, putHo H [111] so that only the first component of (71d) 19 That zero. The fundamental (r = 1) signal from orbit C5 completely dominates, so only that term survives. The orbit area is a minimum, so the phase constant is + 1. Finally, detection is at the second time 4 liaruunnic, so only the J2 (n = 1) term, in the first Bessel series of (96) 86 is recorded by the signal processing equipment. Thus _Hosc(t) =-2iHM1(Ho) (sinwl) J2(A) cos(2wmt). [97] This quantity is measured through the voltage induced in a small pick—up coil surrounding the sample and in close flux linkage with it. From (87), and (97) we get v(t) = V2 cos(2wmt), [98] where V2 is the amplitude - 2 L- .1. V2 - anli(Ho) J2(2th/HO) sin[2'rr(Ho y + 8)], [99] where n > 0 includes various numerical constants and the amount of coupl- ing between the pick—up coil and the sample. Voltage polarity is important only for evaluating the phase constant y. Figure 20 shows a typical recording of the r.m.s. value of v(t) = VZI/E (versus field rather than inverse field). The envelope function is M1(Ho). Note that it is essentially a single sinusoid, in accordance with (99). Equation (93) shows that the amplitudes in (96) depend upon the ratio h/Hi in the argument of the Bessel functions. The optimum modula- tion amplitude for the production of the second time harmonic when the deA magnetization is dominated by its fundamental component is given by Ztho t to = ———2—L = 3.05, [100a] H O for which J2(Ao) = First Maximum = 0.486. [lOOb] 87 5.0 , , ,. I‘ 1‘111 1"] '1T"1""""1-n‘ 1 I “11111111“I l I _ E 1 11111111 _1" .....__ _ ._ _ __ _— Figure 20. Typical data for the C5 (Third zone neck) orbit of AuGaZ. m* = .175mo, T = 4.2, TD ~ 1 to 2 K. Plot of r.m.s. second time harmonic of v(t) induced by the sample versus field. 88 The Bessel function also strongly affects the time harmonics of the induced voltage v a: dM/dt. Figure 21 illustrates this graphically by showing the theoretical result of modulating the steady field Ho with h(t) in (88b). At a fixed HO the instantaneous magnetization M(Ho,t) may be a complicated function of time, even though the field dependence of the deA magnetization M(H) is sinusoidal, as shown in Figure 21. Further, the form of M(Ho,t), and hence its harmonic content in time, varies with HD and h, with h measured as a proportion of the hO corresponding to M0' pt Because the magnetization is shown over only a few oscillations, the amplitude is taken as constant in Figure 21. Note the distinction in this figure and in Figure 3 between the general variable H (a "dummy" variable for the function Mose) and Ho’ the steady field to which the modulation field is added. Because it is easier to visualize changes with field rather than inverse field, the concept of the field period is useful. Mosc,r is not strictly periodic in field, but over four or five oscillations it is nearly sinusoidal. Let the phase wr differ by 2n radians at H and H' > H. Then the field period is 2 H HH' H avg = '— =———=——é PH H H F F-H . F , [1013] where the second and third steps follow directly from equations (50) and (51), and the fourth step is an approximation, with Havg = %(H+H'). For AuGaZ, 50 RG 25 RC. 739 Gauss at H0 185 Gauss at H PH(C8) = [101b] 0 Note that for C5 (F = 3385 kG) and when HO = 50.18 kG, equation (100a) 3 f\ \2927k6 seine/KW fl fl 8 I \ c'{%‘ 1 : H t \ -..J I AZ I } 1 I I —‘—-- _____ __ 4 -.'-'— I l _Qaly Gauss I O f :/ .J ’ I M(t) displaced : i upward to show ~f\\ c‘z:::> graphical _ l \ h=h construction. h - ghopt v/I °pc c t = time 3 \/ < E \ It H = Magnetic Field h>hopt ‘ Identification _/,// of Graphs < w A /\\ . '3 I . . M(Ho,t) M(H) M(Ho,t) \ g,“ / \ \I. h(t) h(t) nonm HHoo mm manamm Exploded view of the probe's coil former and sample holder assembly. Figure 25. 102 precise measurement of the magnetic field, and the associated coils, one for modulating the field for NMR and a radiofrequency (RF) coil. The holder is not attached rigidly to the tube, but is suspended from the bottom of it and pulled upward by one or more springs, to be held firmly against three pointed rods. Two of these are threaded and may be rotated to adjust their lengths and thus tilt the holder to allow a well controlled orientation of the sample inside. Discussion of the probe falls naturally into a number of assemblies: flange and housing tube; mobile base, to which the holder is attached and which is able to tilt; control tubes and motor for tilting the mobile base and hence orientating the deA sample; a sample holder assembly consisting of the coil formers with their coils, places for the samples to be seated, and a spindle for attaching the holder to the mobile base; and lastly, electrical lines and feedthroughs for both deA and NMR. Letter codes in the following description correspond with those on the figures. Flange and support tube. The main housing tube (HT) is stainless steel (3.8.), .020" wall, l-3/8" 0D, and 153 cm in length, with an over- all length for the probe of 170 cm. A Quick-Seal (QS) soldered to a brass flange (SF) permits both a seal for the dewar of the cryostat sufficient for the bath vapor pressures we used and easy adjustment of the vertical position of the probe. Such adjustment allowed a check by NMR of the axial homogeneity of the solenoid, and a subsequent position- ing of the sample in the flatest part of the field profile. The flange is bolted to a c0pper tee of 2" bore that provides access to the dewar. HT is sealed at the top by a lift ring (LR) and feedthrough box (FB). 103 On top of FB four 3/16" Quick seals (08), of which the back two are hidden behind the front two, permit entry for two control tubes (CT) and the NMR RF coaxial line (RFL). The fourth seal was either blanked off or used for a tension wire (not shown; see below). The RFL is vacuum tight and terminates in a BNC connector on a copper elbow (E). The feedthrough box has four ports with o-ring seals for hermetic, electrical feedthroughs (EF), which have nine pins each. The sides of two are shown in Figure 23. Three threaded tripod posts (TP) hold the probe at the desired height and also support the motor at the very top (not shown). The housing tube also has a lower flange (LF) at the bottom, for guiding and attaching the lower assemblies. Mobile base. A brass disk (MB) has a small flange beneath for attaching the holder assembly and three smaller stainless steel disks (SS) seated on t0p to provide hard contact points for the three threaded, pointed, equidistant tripod legs (TL). One or more springs (SG) attached to the lower flange by a cross bar on two threaded spring posts (SP), hold the mobile base firmly against the tripod. A phosphor bronze wire leads from each spring, running close by TL and through small holes in LF and MB to beneath MB, where it is soldered to a small pivot cone, applying nearly vertical tension for the range of tilting angles used. One leg is a stud; the other two are rotated by means of control tubes (CT) extending beyond the top of the housing tube to tilt the mobile base and holder assembly. Threads 4-48 provided fine control. The arrangement of the springs changed from one central spring passing through center holes in the lower flange and in the mobile base (as in 104 Ref. 49), to the addition of three helper springs located symmetrically on the circumference of the lower flange, to two symmetrically posi- tioned, off-axis springs (SC in Figure 24) near and parallel to the rotating tripod legs. This last provided greater and more uniform ten- sion. Deterioration in the signal-to—noise (S/N) ratio when the axis of the modulation coils (coaxial with the mobile base) was more than about 2° from the direction of-B of the solenoid (vertical) prompted the addition of a strong central wire (not shown) running from the center of the mobile base through the feedthrough. After the crystal was oriented much more tension could be applied through the wire than through the springs. Although the tension wire returned S/N to almost the value prevailing when the axes of the coils were parallel to B, it was mar- ginally beneficial because the deA electrical signals obtained off- axis were used only for orienting the sample, and did not have to be especially clean. Clean signals were needed only from H_g, and except for Sample 6 this put the coil axis within 2° of B, Springs were of phosphor-bronze. Some were purchased from McMaster Supply; some were wound in the machine shop from wire of .020". Springs of (un- stretched) lengths ranging 2 cm to 4 cm and of two diameters, .18 cm and .22 cm, were used. All had spring constants 2.65:.06 Nt/m. Control tubes and motor. The control tubes (CT) are two 3/16" 3.5. tubes, each soldered to a tripod leg at the bottom and to a shaft at the top. Each shaft has two thick spur gears (G) spaced slightly apart, but with teeth aligned, to enable continuous engagement with a spur gear (MG) on the shaft of a synchronous motor as CT turns and moves vertically over a range of about 2 cm. The motor is pivoted manually to engage one 105 CT at a time. Because the deA signal depends upon the vector B, con- tinuous change of the crystal's orientation in a static field gives a deA rotation signal (a different signal from that obtained by a field sweep, but also oscillatory). The smooth change of orientation given by the synchronous motor (6 revolutions per minute) gives a signal much easier to interpret than those obtained from a hand rotation. However, unless S/N is very poor, the motor is a convenience, not a necessity. It is desirable to know the relation between the number of turns Ci of CTi and the polar angle and aximuthal angle of the mobile base. (The Euler angles are not appropriate because the mobile base has only two degrees of freedom, being unable independently to rotate about its axis. For the sample to realize a third degree of freedom it would have to be removed from the holder assembly and rotated about its long axis.) Knowledge of this relation permits a calculation of the angular resolu- tion for the crystal's orientation and from F(e,¢) calculation of the resolution in F. It is also a check on how closely aligned are and the axis of the cylindrical sample. Finally, the angular variation can be converted to translation of the NMR sample to obtain information about the radial homogeneity of the field. The equations are C p / tan 6 = [C—%—)2 + (gfi)2 (Cl — 2C2)2flI12 [108a] (2C - C )3 tan q; = 2:! c 1 [108b] 1 where Cl is the number of turns of the first control tube, CTl, attached to tripod leg TLl; and similarly for C2. The polar angles (e,¢) are with respect to the axes defined by the probe axis (z-axis) and the line (x-axis) connecting the tips of TL fixed and TLl, where TLl is chosen so 106 TL2 lies in quadrant I of the xy-plane. (Axis z is vertical and plane xy is horizontal.) The TL form an equilateral triangle in or nearly parallel to the xy-plane. The parameters are triangle side 5, height h, and pitch p of the 4-48 threads: 3 = .9544", h = .8265", and p = .0283". Figure 26 is a contour map of polar angle. The range of turns is roughly :8 from the level, corresponding to maximal 6 of about 25°. The resolution in setting 9 at a value near zero is 0.4°, corresponding to a resolution in deA frequency F of 28 ppm when orienting the sample by ' orbit of AuGa . (This value 3 2 was obtained from the equation of Ref. 10.) The orientation procedure the symmetry of the deA signal of the C is iterative: a theoretically infinite series of alternate adjustments of CT. (See Chapter VII for details.) It was normally carried through three steps (one step being an adjustment of each CT) to give an accuracy of orientation corresponding to an error of less than 1 kG in F, which is about the standard deviation of F for any one run. Sample holder assembly. This consisted of the coil formers with their coils, the samples, and a spindle (SL) and cap (CP) that held the assembly together and affixed it to the mobile base. There were two types of coil formers, one (CF2) for the split NMR modulation coils (NMR-M) another (CFl) for all three FMT coils, with numerous editions of each. The three FMT coils are solenoids and may be grouped into a long modulation (M) coil (which could be considered two coils in series) and a signal (S) coil consisting of a pick-up (P) coil and a balance (B) coil in series opposition. With the exception of one of the formers for NMR, all formers and the spindle are plastic, laminated epoxy with paper filler, EP-22 manufactured by the Synthane Corp. This was found to 2.59 2.15 1.2: 2.1., 2.29 /‘\ J: 1.'8 (1 C3 .75 OJ 1 #4 L4 ‘,94 ~ Bl Q, X.§0 .4 d ‘ m [0” i) 1.25 O U) A ‘olu’ (U \u’ l.32 0% a) "’ J) :1 .73 H .4 .91 C) $4 u OSC C '\ ' .95 U '54 .P) C) U) .IJ (3 H Is 13 LJ ..IJ U4 CD .02, LI 4 g .0). '_a I: L- 53 ..91 14 -.n) -.I: .o" ‘|.Ov 'l.lJ Figure 26. Number of turns of Control Tube .‘I) e.. 0.; I.L,l.0 (.3 dod:)o’ 7.9 ’0, ’00 I r '3" In; 7., Y.‘ ’.. / ’05 'o‘ ,IJ '0‘ i. / I 7 3 . . 5: .fi . out, o ,0! «y’ 1.2 7.! 3.0 0.! I / / (a: 5.3 007 90% 0.: 0.7 u.‘ 3.1 308 007 5.‘ 5" 5o. n.‘ ),Q I / - I :.J 09' ya ’0? 30’ ‘0‘ 50° 1‘.) 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I.s CO6 20“ Z.) 2.1/1.9l.‘l.h|.§l.ll.2 3'6 20é/510 l-Q la? l-S lob 1.2 lol 1’ 1.0' o/ .3 .V f: 1.9 3.6 3.6 3.5 3.6 J.J 3.1 3.2 1 3.2 J. 1‘ "1- 20° 2.9 2.6 2., 2.7 2.6 2.i 2.5 2.5 2.- 2.] (.2 2.2 2.I [.9 1.9 1.7 l.b l.b l.$ 1.6 l.) I.) 1 :.J l.|/.° .e .7 l.S 5.7 I.) 1.5 [.6 1.7 l.a The numbers in the Rough contour lines The numbers along the axes are 108 have excellent mechanical preperties both for machining and for use at LHe temperatures. Plastic was used for lightness, to increase the force of contact between the mobile base and tripod. It was also felt that if the spindle (SL) passing through the center of the NMR sample and RF coil were metal, it would attenuate the NMR signal. In order to make an accurate measurement of field, the deA sample lies in the center of the NMR sample. The cap has a center access hole through which the leads exit. They are taped to the outside of the probe nearly its whole length from the cap to just under the flange (SF), where they pass into the interior of HT to the electrical feedthroughs (EF) in the feed— through box (FB). deA coils. Reference 49 used a coil similar to NMR—M but with half the diameter, for both NMR and deA modulation, but our larger ratio of magnetic moment to center modulation field h decreased our S/N by a factor of 40 when NMR-M was used to modulate the field for detec- tion of the deA signal. Bigger h and S/N were possible from winding M directly over S. The deA magnetization induces a voltage directly into the pick-up coil (P). The balance coil (B) has two jobs: to reduce the noise volt- age and to reduce the voltage induced in the P coil directly by the M coil (transformer action). A large contribution to the noise in the final signal could come from vibration of the P coil in the slightly inhomogeneous magnetic field of the solenoid. (The vibration is due to the alternating magnetic moments of the M coil and the sample.) Cancelation of vibrational noise will occur if coils P and B have the same flux linkage to the steady field. This is approximated by having them physically close and with the same area-turns. The B coil sees 109 roughly the same field variation with time, and so tends to cancel the voltage induced in the S coil by this vibration. Although vibrational noise is greatly attenuated, as shown by comparing v(t) with and with- out the B coil at typical, fixed values of H and h, it remains the over- whelming contribution to the total noise in the final signal, as is shown by the fact that v(t) from a well balanced signal coil in high, constant field with typical modulation current is much noiser than v(t) recorded under the following test conditions (except for the indicated change conditions are as for recording deA signals): (1) H = 0, (2) h = 0, and (3) the input to the detector shunted by an impedance similar to that of the S coil. Weak flux linkage between the B coil and the sample is required to prevent serious attenuation of the desired signal. The design of coils N5 and N6 used in these experiments satisfies the above requirements: P and B have the same dimensions, the same number of turns of wire, and they are close together. An alternative design is to wind the B coil over the P coil; this decreases their separation but also decreases the net signal from the sample. The pr0per choice probably depends upon the homogeneity of the field and the intensity of magnetization. The first design was found to have significantly lower S/N for our experi- mental conditions. The second job of the B coil is to cancel the strong voltage in- duced directly in the P coil by the M coil, a voltage which is typi- cally 102 to 106 times greater than the deA voltages.101 When measuring the fundamental time harmonic of M(t) it is essential to reduce this overriding background signal, coming at the same frequency, to a level that enables the electronic amplifiers to properly process the small 110 oscillatory component. Even when detecting at overtones, which is usually the case, filtering out such a strongly interfering signal electronically is impractical and leads to deterioration of S/N. A much improved signal results from reducing this interference at its source by having the P and B coils see the same modulation field. This is accomplished by the twin M coils shown. The balance ratio (the ratio of voltages from the S and P coils) depends upon the characteristics of the sample and coils; typical values101 can range from 10.2 to 10—3 and the required effectiveness of balance depends upon the strength of the deA and the harmonic of detection. Most detection equipment has filters becasue the deA signal has strong time harmonics, so that reducing the balance ratio below the value necessary to allow Optimal electronic fil- tering does not improve S/N. Very low ratios of the order of 10.4 to 10'.5 cannot be maintained over wide field ranges because of magneto— resistance and diamagnetism of the sample and wire.97 Coil winding and specifications. The dimensions of the formers for coils NMR-M, M, P, and B are shown in Figure 27. Coil specifica— tions are given in Table 4. The coil strength of the modulation coils was calculated from -1 -l -—; -?;;:;ISE-{22 (sinh (r2/|22|)-sinh (rl/IZZI) - zl [sinh—l(r2/|zl|)-sinh-1(rl/Izll)]} , [109] where the units are mks, with the final answer for Sc in Gauss/Ampere- 111 - ' , l L . 1 I- -..—...}. ..r.. . --.-_.....-. ......_-..-_.. ,_ .- '..-...—..- 1 I \ V 1060 - . o — ’ —-—-p—-« - w- ~——?-—.—.—- .... _. . . . . . ‘ E, . .‘l , ; --... _: _ . . --. ‘ I . I y . . . . I -. 47-.-; . '. -... --. . I .- , ,_ . , .--. . I . ' I I Ir-T—I I g .' l : 1 . g I . 25 4\ . . . . . . . I ' I ' “‘130 ~-I¢8 I ~I30 - ; -5 ~-- I 089 228 L...I._ .44."... Jan. (O)- (b) Figure 27. (a) (b) __-__,- ' U—‘L-W ' I: . :L I +139 ""9" T 090 I . ’ "gfitflé3’u‘ -: i . .6501; w" ; '7 ' Thaw» *' - . 1., -.- .- L . bore: '.. - '. .. i, :zA ‘ A .../7". 15 : '.mv”' _ fl - . W- I . IE! .I7 7. . . . ..775V, . . €-f-T-33?0"r—+--+J, Dimensions of coil formers. L L..- Q- _c’ Former CR1 for deA coils N5 and N6. Each has windings P (pick-up), B (balance) and M (dH A modulation). Dimensions in mils (thousandths of an inch). Former CF2 for coil NMRAM, for NMR modulation. Dimensions in inches. 112 TABLE 4. Specifications for de Haas-van Alphen Coils N5 and N6 and NMR Modulation Coil NMR—M WINDINGS Wireb Number DC Resistance (0) Size of 4.2 K Coil Windinga (AWG) Turns RTC zero kG N5 P 41 250 14 - B 41 250 14 - M 43 1326 152 2.6 N6 P 50 2300 .99 k 12 B 50 2300 (note d) 12 M 50 6936 3.88 k 48 NMReM 42 6240 2.66 k 33 COMPLETE COILS e Input Impedancef Coil Strength S Balance at 100 Hz, 4.2 K, (Gauss/milliAmp) Coil VS/VP and zero kG Theoryg Experiment -5 0 N5 10 148 Q at 1 1.25 1.28 (note i) N6 0.5% 63 II at 47° 6.42 5.8-7.0 (note d) a P stands for pickrup; B for balance; M for modulation. bLeads were usually AWG 36. cRoom temperature. dAs wound, balance was 0.5% and B resistance was .99 k0. Later the balance became 2.4% and resistance .96 k0, probably due to a few turns shorted on one winding. eV is voltage induced in the signal windings (P and B in series opposition). VP is for pick-up winding alone. At 4.2 K in zero field. fThe presence of the deA sample made no difference. gFrom Equation (109). hFrom fitting data to Bessel function of Equation (99). 1Additional data for room temperature and zero field: measured inductance of 4.3 mH and Q of .18, both at 1 kHz. 113 turn equal to 10.48c in units of Tesla/Ampere—turn. L, r1, and r2 are the solenoid's length and inner and outer radii (meters), uO/4fl = 10-7(mks), 21 and z are signed distances from the field point to the 2 left and right ends, respectively, of the solenoid (note the absolute values in the arguments of the inverse hyperbolic sine functions), and the field points lies on the long symmetry axis. Equation (109) comes from integrating the Biot-Savart expression, and its accuracy is about 2%. This was determined for deA formers N5 and N6 from the Bessel zeros of the deA signal and the roots of J2 in (99), and for NMR for- mer M by direct measurement with a Hewlett—Packard field/current meter. For coil strength at the center (109) reduces to -7 D D _ B = 4x10 . -1 2 _ . -1 1 Sc - NI -B;:BI—(31nh (L ) Slnh (L )), [110] where D is diameter. It also reduces to the usual thin solenoid for— mula p B__0 NI — 2L (cos a1 + cos a2), [111] where 31 are the angles subtended at the axial field point by the basal radii. Estimates for the Optimal values of the size of wire and number of turns were made as follows. Matching the impedance of the oscillator (500) permits the maximal power and the maximal modulation field. The desired maximum for second harmonic detection (see Figure 45) is one— half (amplitude 367 G) the field period (735 G) at the maximal field (50 RC). The available power of the oscillator is 0.5 W, which is lost to Joule heat and flux leakage in both the coils and the impedance matching transformer at the front end of the detector. Room temperature 114 DC resistance Of the M coil is 2 k0, and the RRR Of fine (AWG 42 to 50) Cu magnet wire is 95 to 100 (as measured). Assuming all power goes to the Joule heat, the maximal current into the cold coil is 35 mArmS. The maximal coil strength for fine wire on formers of the given dimensions is around 7 G/mA (1.25 G/mA for coil NS and 6.42 G/mA for coil N6), giving a maximum amplitude Of 330 Gauss. This shows maximizing the power into the M coil is important. Flux leakage, less than Optimal matching, and magnetoresistance Of the Cu reduced the maximal modula- tion field so that equation (100a) could be satisfied only for H near 40 kG. TO wind the M coil to match impedance assume that the coil and in- put transformer truly reflect the infinite input impedance Of the amplifier, ignore magnetoresistance, and wind the coil to have resis- tance Rc = R8 of the source (oscillator) when the coil is at liquid helium (LHe) temperature. For a constant voltage supply the useful para- meter field per volt, and the equations are SCNV B = S , [112a] 2 2%. ((RS+RC) +X) where VS and RS are the source voltage and resistance; Rc and X are the coil's resistance and impedance at frequency w = Zflf: X = IIISCCN2 [112b] 2 where C is the cross-sectional area (m ) (r 2 + r 2). [112C] TI C ‘ 2 1 2 The packing densities Of fine wires Of diameter dw were both calculated and measured. Using them and (112) the field B(N,dw) per 115 volt VS was calculated for N windings of copper magnet wire of diameter dw on one recess Of former CF2 with our oscillator (R = 500, f = 100 Hz) as the source. lfimrfield point is on the coil's axis and at the center Ofthe single recess. (Note that the two recesses Of NMR-M are a modified Helmholtz pair.) Sketches Of contours for B/VS are shown in Figure 28 for 20°C and in Figure 29 for 4.2 K. Because fringing effects reduce the accuracy of (lle), the frequency was adjusted by a factor found experimentally. Spot checks of these calculations against measured values yielded 4% accuracy for frequencies near 100 Hz. The contours Of Figures 28 and 29 assume a length L and inner coil diameter ID. The third constraint, outer diameter, determines another boundary on the map: an OD line (dashed), the locus of the points (dw’ Nd), where Nd is the maximal numbercflfturns Of wire Of diameter dw that the former has room for. Maximal field per volt for the former is given by the maximum within the accessible region defined by the two axes and the OD line. In Figure 29 the OD line happens to pass close by the absolute maximum. During collection Of the data an unexpected problem arose that later appeared to be temperature rise Of the sample, caused by the heating in the M coil Of CFl. Thus choosing the best coil specifications is not as simple as described above. The maximal modula- tion field h that will be required, the combination Of turns N and modulation current Im necessary to establish it, the resulting coil resistance Rc and Joule heat, the rate Of heat dissipation in the coil and holder, and the maximum temperature rise Of the sample all are interrelated parameters that must be adjusted for Optimum detection. The coils were all wound on a Meteor coil winding machine. The wires had solderable insulation; it vaporized from the heat Of soldering 116 .aaonm mum mmoamumammu on meow .0 om u H .Nmo “wagon mo mama so HHOO m mwouow uao> you mmnmu OH nuwcmmum HHOO mo ama usouaoo wcfiwcfia HHOU .wm musmam w .35.}. v o "I III 7 _ . . _ o vw 0v 3.0m 0950 922 06 ”2.2.100"— CZ m; I o; m6 mmouo. ace :0 meta. z oovw OoON . .P . 2-x22 .o to; .3 23> \333 36.25 coon 117 .M ~.q u H .Nmo Heston we mama so Haou .3 Haoo .mN musmam m mmouom uao> Hon mmsmu CH nuwcwuum Hfiou mo mos Hsouaoo wcwv: a .1252... v IIIII o wIIIIII _ _ _ . _ cc. IlllIIl. 2. 3. 2. 2.3 / / ha I!!! oaaow 03< \/ I\ a V 2.: co / o— : / a / can. // z coma 33.: 23 co 2:3 2 x N... u ... ZImzz .o to; 3. 2.6>\:=63 36.23 cavn 118 with ordinary PbSn solder. The tips of heavy leads (AWG 36) were cleaned, tinned and anchored to the coil former by threading through small (drill NO. 72) holes on the shank, with 1 mm left protruding. The former was put onto a mandrel with a drum on which the heavy leads were secured. The fine wire was wrapped about the tip, soldered, wound onto the former, cut, and that end soldered to the second heavy lead. Each layer was glued with very thin DuCO cement. Electrical lines; Heat Leaks. The heavy (AWG 36 or 38) leads from CFl led through the hole in the cap (GP) to above LF, where each was soldered to a separate probe lead (AWG 32) running to electrical feed— throughs (EF) on FB. Likewise for CF2 leads. All leads were rigidly taped, except for the last few inches at the tOp, either to the ex- terior side of HT or to a narrow aluminum bridge (not shown) to facilitate taping on the holder assembly. Coil leads were twisted pairs; probe leads ran parallel. Experience showed several centimeters Of loosely dangling (but twisted) coil lead did not have noticeable effect on S/N. Taping also protected against mechanical damage. Three NMR RF lines were used, one at a time. RFLO is a 3/16" 3.3. tube (the outer conductor) with a 1/32" 3.3. tube as the center con— ductor, spaced by three or four Teflon triangles and tension at each end. These sizes give an impedance close to that Of the R658 cables used (549). The impedance changed slightly with the level of LHe. The major design consideration was low thermal leakage. An effort was made to improve NMR S/N in RFLl by using as the center conductor a Cu magnet wire, size AWG l8 chosen for current coaxial impedance matching, spaced by Teflon triangles every 2 or 3 inches. S/N improvement was small but 119 noticeable. Later RELZ was made vacuum tight by a hermeticlflfllcon- nector on a COpper elbow because running procedure required frequent removal and reinsertion of the probe. Water Of condensation collected inside the tube and absorbed the FR power, killing the signal. RFL2 is another 3/16" tube enclosing an RG58 cable with the insulation and ground braid removed, leaving the AWGZO Cu center plus foam insulation. S/N was about the same, perhaps lower. This RFL was used only when the probe was left in LHe during the entire run, so it wasruflzvacuum tight. RFLl was used most. Because the sample and coils were in the LHe bath, thermal leaks were a concern only in conserving LHe. Heat flux down the probe, with the top at room temperature and the bottom in LHe to about 10" above the sample was calculated from tables of integrated thermal conduc- tivities.103 Results are given in Table 5. These are upper limits be- cause the loss of heat to the rising cold He gas carries away much heat before it reaches the bottom of the probe. 6.2 Electronics The electronics were standard for the field modulation technique (FMT). See References 95 through 101 for typical circuits. Lower case v and i are time dependent variables, capital V and I are amplitudes or phasors. In Figure 30 oscillator number 1(OSCl) with voltage Vm generates a sinusoidal current Im of frequency fm and distortion less than .02%. Im generates the deA modulation field h. A digital volt meter (DVM) measures Im but sometimes an oscilloscope (S) was used. The deA 120 TABLE 5. Thermal leaks: Conductive heat flow from top Of probe (room T) to LHe level (10" above sample). Path Heat Flow (Watts) -3 1 wire, 32 AWG 2.88 x 10 All probe leads except RFL (18 wires, 32 AWG each) .052 RFLO (s.s. center) .02 RFLl (18 AWG Cu center + air space) .073 Probe housing tube (HT) (3.3.) .166 Control tube (CT) (3.5.) (each) .025 Solenoid's (SCS) 3/8" suspension tubes (all three) 0.44 Total (Probe with RFLl + SCS suspension; exclude dewar conduction, radiation leaks, gas thermal exchange) .39 a. Assumes no heat given up to rising cold He gas (no exchange). voltage vS = VP + VB from the deA coils enters a lock-in amplifier (LA) via a switch box (SB), where connections are shielded. In addition to the B coil and electrical filters, a null control is frequently employed. A phase coherent voltage, Obtained from either the B coil Or from a third coil in the probe, and available for ex— ternal phase and amplitude control, is added in series to null the fundamental voltage from the S coil. During development of the appa- ratus a phase shifter and a voltage attenuator modified vS before it entered LIA. But data presented in this thesis was taken either with VS going directly to LIA or with a commercial ratio transformer (RT) improving the balance first. The inset at the top left Of Figure 30 121 I CR PM 3 D it .001 I ’ Rw or RLN PROBE NJ i E Figure 30. Block diagram of deA and NMR electronics. 122 shows the two cases. RT enables a variable, precise fraction of VB to be added in Opposition to v but no phase change is possible. Coil B, N5 was balanced well enough that it needed no help; coil N6 had about 2%% imbalance which was reduced by RT (no phase shifting) enough that the bandpass filter Of the LIA could handle it. The switch box (SB) had internal shields to prevent cross—over of the modulation and signal voltages. The LIA input was a transformer, found to give much improved S/N compared to the other Option, transistor input. The LIA output Vo was recorded against steady field HO Of the superconducting solenoid (SCS) on a chart recorder (CR). Vo was proportional to the n-th harmonic component (in time) of v The component has amplitude Vn and fre- S. quency fn = (the detection frequency). Except for special measure- fd ments, n = 2 for the data. Thus the LIA effectively performed a Fourier analysis to v , recording V cos(6 - ¢). where 6 is the phase S 2 angle of V2 relative tO V1 and d is the LIA phase. The x—axis of CR was the steady magnetic field Ho. Field measure- ment is described in Section 6.4. 6.3 Superconducting Solenoid and Dewar A cryostat was built to suspend the SCS. The superconducting sole- noid (SCS) was a NbTi, filaments in a Cu matrix, 50 kG at 60 A and 4.2K, manufactured by Oxford Instruments, Cambridge. Hysteresis appeared to be negligible. The NMR field measurements are unaffected by hysteresis; when the calibrated resistors were used, field points HO for deA fre- quency data were approached by one or two decreasing oscillations. Some of the deA phase plots show a small hysteresis effect. See Table 6. 123 TABLE 6. Superconducting Solenoid Design Hysteresis (from deA phase separation between up-field and down—field sweeps) Maximum error in deA frequency F phase phase constant Homogeneity (Specification) (Axial) (NMR) (Leads to negligible phase smearing) Calibration (field measured by NMR.versus voltage VW across standard resistor as measured by a 5 digit, 1 microvolt resolution DVM. Average and Standard Deviation Of 22 points over 29 - 49 kG) Maximum error in deA frequency F phase phase constant Oxford Instruments, 50 RC, NbTi, 850 Gauss/Amp (Spec.) .0222 lag at 37 kc .037 31 .11 21 (Normally avoided by cycling into the field point.) i 7.5 kG j: .04 cycles at 31 kG j: .17 cycles .1% over a 1" diameter volume, which gives an average of 40 ppm/mm. However, the profile may be jagged. Five or six points over 4 cm gave jagged profiles. Either in field or poor NMR precision in reading line center. Average Typical Field over 4 cm Gradient Avg . (RC) (1%) (imam/mm) 4cm 9.82 .1 92 25 26.76 .019 37 4.7 30.53 .029 28 7.4 35.42 .008 2.8 2.1 Across RW (water-cooled tube): (141.433 :_.055) kG/V :_.001 kG Q: .044 accuracy) (as given by above and RLanw): (850.825 1 .331) kG/V i .001 kG Across RLN ;: 3.4 RC 11.074 cycles at 20 kG j; . 096 cycles 124 6.4 Magnetic Field Measurement The field was measured by NMR and a standard resistance Rw or RLN in series with the solenoid's current. Later a precision calibra— tion of these resistors by NMR while measuring SCS current I with a DVM (not shown) enabled their use for field measurements sufficiently pre- cise and reproducible for our deA measurements Of F, given the complica- tions presented by the crystal itself. Calibration is given in Table 6. Nuclear magnetic resonance. Precise measurements of magnetic field were made by NMR of 27A1. A marginal oscillator (M0) put RF power into the sample through the electrical feedthrough (NMR-RF) on the probe. A detector in MO demodulated the RF signal, whose amplitude decreased at resonance, giving a dip in the demodulated voltage. This was seen on 8 after being cleaned by filters F. Filters were 60 Hz twin-T rejection plus sometimes a bandpass set close to the modulation frequency, which was 70 to several hundred Hz. For more precise field measurements, or for weak signals, another LIA replaced S. Frequency was measured by a Hewlett-Packard Frequency Counter (FC). The method of tuning was by changing the lengths of RG58 coaxial cables, as described in Ref. 104, which also gives the MO circuit diagram (ours was slightly modified). The sample was made by mixing filings of shop grade A1 alloy with epoxy. This paste was packed into a cylindrical shell into which the RF coil has previously been inserted. After the epoxy had set a trans— verse hOle was drilled tO allow insertion Of the deA coil former and sample holder. The S/N for the NMR signal was only fair. Below 30 kG the decreas- ing NMR signal amplitude and the difficulty in making the M0 work at 125 the required lower frequencies combined to make field measurements very difficult. Various sizes Of wire and numbers of turns were tried for this type Of 27A1 sample in order to improve S/N. Solid A1 metal and ruby were tried out Of curiosity. Niobium in NbSn superconducting wire was also tried. The best NMR sample was 27A1 with RF being about six turns of wire in the Al-epoxy. The size of the sample was 1.5 cm x 1.0 cm diameter, with a hole of diameter 0.57 cm at the center. The wire was AWG20 but size is apparently unimportant. Calibrated resistors. Two low resistance, high current resistors were used, one at a time, to provide both a voltage for the Ho scale Of the chart recorder and a precise measurement of field, the latter in conjunction with a 5% digit DVM, with a sensitivity of l microvolt. Both resistors were calibrated and checked occasionally with NMR. See Table 6 for calibration data and Table 7 for characteristics. Tempera- ture was monitored with a thermocouple meter. Note that RLN may be more constant over 0 - 60A than RW' The calibration ratio that was used was obtained by dividing that for RW by the RLN/RW ratio at 60 Amperes. 126 TABLE 7. Characteristics Of the Standard Resistors for the Superconducting Solenoid Resistor Rw Design Water-cooled 3/8" 8.8. tube. Distance between potential leads = 5 cm. Resistance (By RLN/RW) (6.017 r .002) x 10'.3 Ohms Temperature coefficient 1.77 x 10.4 (0C)..1 at 11 OC (AR/R/AT) Joule heat: AT(I) AT = (.00145)Il°94 OC/A (at 200C) T rise: At 60 A (Amp) AT 2.70C (By thermocouple on outer surgace. But AT Of cooling water = 0.2 C +T gradient in 8.8. shell.) I-Vw linearity over 60 A (from T coef. & T rise) .07% if uniform T in shell .035% if linear T grad in shell |+ H- N Error by above effect in .49 kG error in field measurement extrapolating the NMR calibration to 20 kG .021 cycle error in deA phase at 20 kG field. The error in deA F and phase constant is variable. That in F would show as a curve in the phase plot. Resistor RLN Design 500 Ampere standard resistor by Leeds-Northrup. Air—cooled. Very big (about 40 pounds). Resistance (specification) 1 x 10-3 i .04% Temperature coefficient Given as very low RLN/RW (by DVM) .16617 at 0 Amp .16624 at 60 Amp (.04% increase) B—VLN CALIBRATION (by RLN/Rw ratio)=(850.825 i .331) kG/V i .008 kG - CHAPTER VII EXPERIMENTAL PROCEDURE AND DE HAAS-VAN ALPHEN DATA 7.1 General Procedure The general procedure was to cool the superconducting solenoid (SCS) and probe, record the deA oscillations on a long field sweep to see their general form, coax the marginal oscillator into working, and re- cord the data. The deA frequency F and phase constant y were Obtained by measur— ing the deA phase as a function Of a steady field H. The Dingle tem- perature T is obtained from the measured variation Of the signal D amplitude with field H. Theoretically, the amplitude is M1(H) Of (71a). The recorded voltage V is really the r.m.s. value Of (98), uniformly 2 prOportional to its amplitude (99). Phase data was Obtained by recording V2 while sweeping down in field, stOpping every 5 to 10 oscillations to note the phase and measure the field with either NMR or by recording the voltage Vw across Rw’ the solenoid's standard resistor. (W stands for both the standard resistors W and LN unless distinction is necessary.) In the first runs NMR was used mostly, but as it became evident that the physics Of the crystal was complicating the signal so that the slightly lower preci— sion Of H measured by Vw across Rw was irrelevant, RW was used ex- clusively tO greatly speed up the collection Of data. 127 128 Dingle data was recorded separately for convenience. At spaced field values the modulation current Im was adjusted for constant Bessel factor, the amplitude Of V was recorded, and H measured. Both Vw.and 2 the calibrated current dial of the SCS power supply were accurate and precise enough for Dingle data. 7.2 Orientation The crystal was oriented by turning the control tubes (CT) while Observing the symmetry in V2. The concept Of surfaces Of constant deA phase101 is useful in this method Of orientation. Let the direction of .H be fixed in real space (xyz) while both the crystalline and kraxes may undergo general angular displacement. Given a Fermi surface (FS) with cross—sectional area A(0,¢) corresponding to F(0,¢), the field's magnitude is varied to keep the deA phase Of (71b) constant as the crystal undergoes general displacement in real space and the direction ofig undergoes a corresponding displacement in krspace (see Figure 3). The constant phase surface is the locus of H_vectors, and the family of surfaces is generated by varying the phase. A spherical FS gives a spherical surface of constant phase; a prolate ellipsoidal FS gives an oblate ellipsoid. Thus there are two types of deA signals: field sweep and rotation. In the latter, H remains constant in.magnitude as its direction in the crystal changes, giving oscillations as it crosses the surfaces of constant phase. Ref. 10 has shown that an hyperboloid of one sheet models the third zone neck Of AuGa2 very well, giving the extremal cross-sectional area as a function Of the poldr angle 0: 129 A(0) A(9) = 2 1, [(l+b) cos 0 -b]’5 [113] where b = mt/m£ = .16, the ratio of the transverse and longitudinal masses. H a A gives bowl-shaped surfaces whose axis is parallel to <111>. Figure 31(a) shows their intersection with a plane containing <111>. The solid lines are quarter phase and the dashed lines are three—quarter. The other two arcs are the paths in this plane of two .H vectors as the crystal undergoes a small angular displacement about . The arcs are distorted from being circular due to the unequal scales Of the figure. The corresponding rotation signals are shown in Figure 31(b). One peak does not correspond to quarter or three-quarter phase; it is a turning point (TP), where the field and are exactly parallel, and may be a positive or negative peak, as shown in Figure 31(b). Typical rotation diagrams from our runs are shown in Figure 32. If the phase wTP at which the turning point occurs is close to W1/4’3/4 (as is most likely for arbitrary field), the two will be confused, and the rotation graph will show no obvious TP. The fact that the envelope function of the rotation graph can change greatly with angle and its being generally different from the sweep envelope further confuse the identification Of TP. SO frequently a rotation graph at fixed H does not identify TP clearly and each peak must be examined to see if it is due to wTP or 01/4’3/4. An example is shown in Figure 32(c). Figure 33 shows how to identify a positive rotation peak as a turning point, a peak preceding TP, or a peak following TP. This allows one to check only positive peaks (3 similar analysis applies to negative peaks) and thus move over angle more rapidly. The method is as follows: the 130 .munufiamma tamam x unnumaoo um A m mumum> .hauawam>w=vm HOV oawcm msmum> o can mafivcoamouuou .mOHmOm Hmnvm How umaooufio on casoa muu< .Awfixm m>fiuwmoov H9 manna unnumnou mo mm>uao uummumuna ouaufiswma camam usmumcoo mo Amwxm o>fiumwoav mou< .m All 0 9:20.05 m.w+ .05.... md: =H 19 41038 9'0? =H Ia 4Iuw 60v 2 a 2: ”ON .0 ufinuo mom magnum coaumuou <>mw van manna unnumnou mo mmphnu Hwoauouoona x: Anv Amv .Hm mummfim Nduu o 0.0% 041 0: c. I =< Figure 32. (a) (b) (C) (d) 131 ' . 4.4.4-.. ”.474 4.4 4-4. ‘ ‘ ’ 4 , ! 4 5 f I i I 1' I .. +1-1-.. 1.41-, . r‘ -..—4—4-4- - “4.1-+4 4;- £11.?!" .. 4 4 ,. -L: . ,3 i ‘- ‘ I 4- 44’ ; In ! . ' | -.4 — ' l .. _ ; . i T I ' 1. 4 . V. .,... ,. - 4 I + . -... Y z 4 ..4 -. .4..,._ ...-,7- . I . '3‘ "’ ‘ 1 “ ‘4‘ r“ 4 ‘ 4 . .6 -4-.. . .... 4.. .L I 4 4 -.. >-x——h—l>! Lr -I__.I4. :§:- . . I. .. ‘ -‘ ' ' ‘ ..‘ .4..,.+ -. t... .. .-.—4. p. l, 4 .4w 4. -,4. 1 4. 41-4.4 .4. ,4 ..4. _ ' 4 I ' _ 1. IL‘T'Tl'H -- l ‘ ‘ .‘il iii! -I .u ...) "...—'- ' ~—-—.- ”...—.... Horizontal segments Typical experimental rotation graphs. denote the limits of rotation. Positive turning point (TP). Negative turning point (TP). The same sample and angular range as in (a). A small change in H obscures the turning point. Another example of an obscured TP. Numbers are revolutions of one control tube. TP identified by subsequent data. 132 {—— Rotation (c) Figure 33. Identification of positive peaks in a deA rotation graph by (1) rotation, (2) reset field, (3) rotation. (a) Peak is a turning point. (b) Peak precedes a turning point. (c) Peak follows a turning point. 133 first peak reached in a rotation is examined by reducing H to put V2 at a positive going zero crossing and rotating slightly about the new position. The signal recorded is then compared against theoretical diagrams like those of Figure 33. If the peak precedes a TP, continue to rotate in the same direction; if it follows a TP rotate the other direction. Continue in this manner until either TP is found or the next peak examined is one that follows TP, in which case TP is the previously recorded negative peak. Resolution in orientation and resulting error in F and was mentioned in Chapter VI. 7.3 Skin Effect When the absolute value of the volumetric magnetic susceptibility of the sample is less than 100 and the sample's conductivity and the modulation frequency are such that the skin depth 6, where 5 = ___—c ir’ [114] (21mm) is several times the sample's greatest axial diameter D, then the com- bination of electrical filters and a moderately balanced S coil gives good S/N. The condition 6 '1')- > 1 [115] for most metal samples requires a modulation frequency less than 10 Hz if the sample diameter is one or two millimeters, RRR is greater than 500, and magnetoresistance is less than 100. Atrade—off is made between increasing frequency and field inhomogeneity. Increased frequency has 134 the advantages of increased voltage (until the skin effect becomes severe), improved electrical performance, and ease of adjusting the filters and detectors, decreased time constant on the final DC filter of the detector, and generally increased S/N and ease of operation. Regarding improved performance of the filters, at very low frequencies finding the resonant frequency is difficult and time consuming. Voltage transients arise from turning the deA signal off to make an NMR measurement and then on, and from other adjustments. Each transient caused the filters to ring many minutes when a 10 Hz modulation frequency was attempted. The modulation frequency of 100 Hz we used permitted the slight eddy current arising from the skin effect to induce a fundamental component of several milliVolts. A null control (ratio transformer) divided this by more than 100 to give a level which the filters could easily reject, leaving S/N as high as when no sample was in the coil. The residual eddy current does not affect the results. Figure 34 and 35 show the results of two tests to check that the deA amplitudezis not seriously affected by any residual skin effect at lOOHz. Figure 34 shows the amplitude is at least 96% of its zero frequency limit for Sample 12, with RRR = 205. Figure 35 shows a test for a sample of presumably lower RRR. Voltage has become independent of frequency at 100 Hz. The dashed line shows the phase of VS' The theoretical value for a uniform field is zero. 7.4 Data Reduction The deA frequency F and phase constant Y were obtained by plotting the number of recorded oscillations (starting with the highest field OSCillation) against inverse field, a technique described in References 49 135 .0. do. no. we. .NH waaamm do ummu uommmm awxm .qm muswfim 3...... I A>3u> .< q— ...E . ...s. .... 136 [0001600] 9 so. 3do¢ 000! .om maaemm so “mm“ “gamma aaxm .mm muswam 3.. E. o mm ON fix W\ nu 093% MM 0.0 137 and 68. In accordance with equations (71a) and (87), a positive-going zero crossing of voltage with respect to decreasing field denoted the start of another cycle. Let m+r denote the relative phase (m whole cycles plus remainder r) as measured from the high field reference phase N, chosen to be integral (i.e., a positive-going zero crossing). Then m+r = wlm) - N. [lle] From (71), (87), and (lle), m+r clearly has slope F and intercept INF=—Y+l/8-N+€ [115c] where INF stands for the infinite field intercept (i.e., at 1/H = 0) and E = 0 if the Dingle factor is negative and the number of coil- detector polarity reversals is even, and E = :5 if either the Dingle factor is positive or there is a polarity reversal. For our system v a +dM/dt. N is determined from the known field and the value of F from the slope. For evaluation of (llSc), see Section 10.2. The trick is to isolate the fundamental component of the extremal orbit desired. Ordinarily digital Fourier analysis, electronic filtering of a sweep signal made with time t a l/H, and the sample's own signal discrimination, all discussed in Chapter V, are used to separate fre- quencies differing by a big factor. Very close frequencies that must be precisely determined require graphical analysis and consideration of the phasors involved. The latter approach was made for this data, in which the C5 signal dominated. The error introduced by residual oscillations in both the phase plots and Dingle plots is the same order as the effect of uncertainty in measurements from the instruments, the 138 temperature rise in the sample, magnetic interaction, and the occasional presence of crystallites. Very small signals from crystallites could be tolerated, but several samples were discarded because of too much inter- ference. Figure 36 shows typical data. Table 8 and Figure 37 summarize the measurements of the deA frequencies and phase constants, Dingle temperatures, and residual resistance ratios. These results are discussed in Chapter X, after the presentation in the next two chapters of some data and theory on the growth and analysis of the samples. 139 SELECTED DATA lo: AU602 deA Phase Plot I deA Sweepl 0 1 aucocotva Ea; notxu I l. lul'lllll It it'llllll'. 40" \ll ll||ll|1 I“ I. II 44-44 IIHII’UIIII‘I 1 "IIIIIuI 4!- .IIII.'II|| 4"" 4 I 11”.... .Illllllllllil 4'! III! Inllllllltlll llltl I I. l -.IJ' | I] illill'! Fixed modulation field i l l l l l l l l 10 E .06 l/H the") H (kG) 35 Residual Resistance Ratio -RRR l Dingle Temperature . ‘20 A15 lo'2 , . . Modulation low and at H‘ Exes-...... {.35 .-a 5500 1500 25 6‘0; 4‘0 Sample is Sample l2 l/H (kG l .02 (psec) Time l. H ll . Unfiltered deA signal. Modulation at lOO Hz. Selected data for AuGaz. Figure 36. 140 TABLE 8. Melt Concentration deA Sample Freq. AcSnOte a No. INFb (kG) Melt C 4 30 -.90 3388 i4 +.549 at.% 27 26 l a\ 6 Melt A. -‘\\\\ +.204 at.% F}‘\\ 16 15 -.82 f ~\\\\‘ _ 94 3384.2:0.6 5 LQSE/ 10 “'90 3385.2:1.2f —.75 f -.83 3389.3i0.5 -.92 3396.4:0.3 Melt B f -.288 at.% -.79 3391.8i1.4f -.65 3402.4:0.4 -.75 _.88 3405.7i0.8 aDeviation of Ga concentration from stoichiometry: Acs means Ga excess (deficit). b Sample Characteristics by Position and Parent Crystal's Dingle Temp. (K) RRR o :.5 .75 .88i.12 c 287C .09d 3l800e _>_I_17ooe .34 31500e .47 31000e .91 .3 500e .02:.1o 205:20 .55:.15 157 .00i.l4 56 55 positive (negative) INF = infinite field phase = -Y-3/8 cycles for orbit C' of AuGa , where Y = phase constant of LK theory. (INF includes net -% voltage reversals.) Average INF = -.83i.09, giving Y = .46i.09. CAxis ||<111>. dAxis 230 off <111>. eLower bound. {In situ NMR.measurements for field. 141 AuGa2 Sample characteristics by position and crystal's melt concentration parent «.22 ...... c... 2.: :o a: 2.4. A2: 4 3x(.. 00R..o i:e.co_£u_o.. So.— 60:02.00. 72.50.: a Susan—um: ca:a_< anger-moo: on 0.3.0.2392“... «_mfo vocation .0333“ SAMPLE NUMBER 2(\)\ j & Melt ill-ill --- - l r .. ... I- - l llle .2\-\AV m m .l [ll-l... lllli l m m lll llivll ll llllll C u M L ll l _n l.lllllllyll.llllA'. \\\)//J I A e o a e .ii-liizz-Ii-eia- l-i1.-.m ...- ..I l .. -. l--- -ll-ii .l./..m\ .... .3...- ,. {.---z-- --.- m /5/ .. e ..--Ill 4 I il l i ..... i ...... . é ... .------- -- -- - -..-l. K/ . / m Dill lAll l - illlu l- ll/b @ Z I .I .m u O-.l- m ...-m _ llll.l-l - llllll. E “law I. Illil ill. I | lIIl ll ll ll b”\\A/I\ ...H ...--. .ll.+.%--- - - ...-l ..- - l..- --.---.i-l .... _ _ _ k L _ _ _ _ 5 o 5 0 5 5 4 3 2 1| 0 0 0 7 2 1. 9 87 O 0 2 4. 4 3 33 O 0 z 3 3 3 33 2 1 o .. ... 00 3: p atom B A Melt +0.20 at ‘7. C Melt ’O.29 (ll-70 EXCESS ‘05 5 at.’/. exc ess Ga Ga EXCESS Summary of deA and RR data. Figure 37. CHAPTER VIII GROWTH AND PREPARATION OF SAMPLES This chapter considers enough of the theory of growing alloys to give an idea of what to expect to find in an alloy grown from the melt, although reliable quantitative answers are difficult to obtain because of the scarcity of data and because making measurements on a freezing metal is a full project in itself. Next are given some of the details of growing, cutting, and x-raying our crystals, all of which used standard techniques. These crystals presented no special diffi- culties other than the usual uncertainties involved in growing single crystals. Questions regarding some of the samples are discussed in Chapter IX. The last section is an exposition of the melt concentra- tions we chose and why. In this and the following chapters the big crystal taken from the crucible after growing is sometimes called a slug to distinguish it from the small crystals cut from the slug and used as samples in the deA and RRR experiments. Both are single crystals (unless something went wrong). 8.1 Theory of Freezinggfor Solid Solutions Phase equilibria. Figure 38(a) shows a simplified equilibrium phase diagram for the Au-Ga system, omitting some details not 142 143 L+ S 4 9 2 I- ::,=\;_('_ II A 3 I 9 / ~ V AuGa AuGaz l— 74 s E 46% 0 l i g l O 20 4O 60 (a) C (01% Ga) c Til ‘ "‘ T4 T2 a b KLS A T 3') ’ “II h—*—k-—-§ l" c i d— (o AuG a (qu‘le 12 _ l L I I J 1 Ce C2 C3! C4 c5 (b) c (at.% Go) Figure 38. Simplified equilibrium phase diagram for Au-Ga. (After Ref. 105) (a) Full range of concentration. (b) Sketch of the region near AuGa stoichiometry expanded, illustrating reversible processes. 144 pertinent to AuGaz. Figure 38(b) shows the phases near AuGa2 stoichiometry. The relevant phase boundaries are Ls=liquidus and the AuGa, AuGa2 and Ca vertical boundaries. I=isothermal lines, and the phase regions are L=liquid, S=solid, S+L=solid + liquid, and E=eutectic. The concentration at the congruent point (the relative maximum in melting temperature) is cm, a value near, but not necessarily exactly at, stoichiometry. A.melt of concentration cl < cm and at temperature T1 may cool in a reversible manner (i.e., the system remains close to ithequilibrium states) until at T a phase mixture starts to form: a 2 nucleus of precipitate (solid AuGa2 crystal) in the bulk of liquid melt remaining. After additional slow cooling the system, represented by the three points (c1,T3), (c2,T3), and (AuGa2,T3), still has two phases: liquid melt of concentration c constituting the molar fraction 2 b/(a+b) of the system (lever rule), and solid AuGa of concentration 2 66.7 at.% Ga and constituting molar fraction afl(a+b) of the system.- The average concentration of the whole system is c1. Phase separation continues with further cooling until the melt reaches ce at Te=451°C, the eutectic point. The temperature remains at Te upon further slow removal of heat, and the remaining melt freezes to form a eutectic alloy, an intimate mixture of AuGa and AuGa2 in the proportion d/c. The eutectic forms inclusions in the pure AuGaz. As another example, if a melt of concentration c4 > c is cooled reversibly to T phase m 4 separation into solid AuGa2 and liquid melt again commences, but now the melt becomes richer in Ca as cooling continues, with the melt concentration following the liquidus line to the right, passing through c5, and the system finally reaching a different isothermal line 12, corresponding to the solidification of pure Ga. Again, the average concentration of the total system is constantly c4. 145 Homogeneity range. In Figure 38(a) the phases Au, AuGa, AuGaz, and Ca are represented by vertical lines, indicating that these phases are of fixed composition, at least to the accuracy of measurement for the diagram. The phases labeled 8 and Y have a non-zero range of homogeneity, or width in concentration, even though they are single- phase systems (unlike the extended eutectic region E) and have been assigned specific formulas:105 Au3Ga for the 8 phase, of maximum width 2.7 at.%, and Au7Ga3 for the Y phase, of approximate width 1.0 at.%. An ordered alloy of fixed composition can be described in accordance with equation (4) by alloy structure = § (sublatticer + basisr), [116] where, except for the most complicated structures, the basis is monatomic. A range of concentration for a fixed crystal structure implies some disorder in the occupation of the sites of each sublattice: substitu— tions, interstitials, or vacancies. In some crystal structures one sublattice represents ordered vacancies. Two examples are6 A13Ni2 and FeO.87SS° One way of characterizing the amount of order in an alloy system with ordered phases is by order parameters, a simple set being f1, the fraction of atoms of species A on the i-th sublattice.2 Thermal and x-ray measurements show the f1 change continuously with the temperature and concentration of the samples, corresponding to continuous transformations among states of relatively high order and states of relatively low order (with respect to separation of the atomic species onto their own sublattices). Furthermore, these transformations (i.e., changes of crystal structure) usually extend over non-zero ranges of concentration before arriving at the next 146 well—defined phase, whether the transformations be homologous (a continuous transformation from one single-phase system to another) or heterogenous (passing through a two—phase stage). Both of these observations lead one to expect that most of the distinct phase boundaries of a phase diagram are in reality blurred, and that inter— mediate phases of nominally unique concentration (i.e., intermettalic compounds) will have some width. Additionally, for intermetallic compounds having a congruent melting point, there is no §_priori reason that it must lie at the point of stoichiometry for the compound, as attested by numerous phase diagrams for binary allows. Figure 39 shows a simple scheme, devised to resemble the AuGa system of Figure 38(a), which could describe a 2 real alloy with a homogeneous range of concentration and with the congruent point not at stoichiometry. Suppose that the solid is an intermetallic compound. Consider this figure in light of the quasi- static (reversible) process discussed earlier, so that the system is always in equilibrium. As it is cooled from T1 to T2 it is a two phase system: the melt (the saturated solution) and the solid (com- pound, or precipitate). The concentration of the solid ranges from 11°C c during cooling. Imposition of equilibrium conditions means 1 2 that at any TszfT the entire solid phase of the system is uniformly l at the corresponding concentration given by the solidus line, and at Ts the solid is at the stoichiometric concentration of the intermetallic compound. As before, average concentration of the system is constant, since it is closed, permitting no change in the amount of each com- ponent. A number of variations can be envisioned, such as solidus lines of different shape and stoichiometry on the other side of the congruent point. crn 147 L ‘\\\\\= Ls ' Ss L+S ‘ 3: L-I—S - l stoichiomit ry —L I l / \ CI Ca C (CLO/o) Figure 39. Hypothetical intermetallic compound (solid phase S) with a nonzero phase width Direction of G rowth C? .2 G >.# - c ': a, 3% Skflid h48"' 32(3) 93 3 C) :3 CL E . +93 Bottom Top Figure 40. Gradient for a dilute concentration of an alloying element in the melt. (After Ref. 106) Temperature and cm not at stoichiometry. 148 So far we have discussed the behavior of the alloy system in equilibrium or in quasi-static processes. But in the Bridgman crystal grower the system, which is the whole charge in the crucible, is certainly not in equilibrium for three reasons: (1) There is always a temperature gradient. (2) The process for growing most metal crystals reduces the temperature of any portion of the solid shortly after its solidification to an extent that diffusion is too slow for the solid to approach new equilibria in practical times. (3) Some degree of mass flux, thermal fluctuations, and other transient phenomena are normally present during crystal growth. What predictions can the equilibrium phase diagram make about the concentrations of samples grown by the Bridgman method? That phase diagrams have been very useful for many years of crystal growing is convincing evidence that they do contain useful information, even for this nonequlibrium system. For slow cooling rates the region in the vicinity of the interface is near equilibrium. As the interface velocity (rate of solidification) increases one would expect the transient conditions to change the composition of the solid that freezes. This is the regime of growth kinetics, which is discussed next. Growth kinetics. This is the study of the time dependent, microscopic process of solidification at the interface of solid and liquid.106 Not only are such considerations necessary for calculating the behavior of systems not in equilibrium but they also give additional information on what happens during solidification under quasi—static conditions. Growth kinetics depend on such things as diffusion, con- centration and thermal gradients at the interface, the amount of stirring of the melt, the free energies of the components of the 149 system, and the mass of the system. For example, the equilibrium phase diagram shows simply that AuGa will precipitate; but the solid 2 can be dispersed as crystallites or concentrated in single crystals of macroscopic size, depending upon conditions such as the presence of a temperature gradient (reversible process assumes uniform temperature throughout the system), the rate of cooling, and the shape of the crucible. Regarding deviation (3) in the subsection above, even an unstirred melt has at least three fluxes:152 diffusion flux, arising from the concentration gradient; a convection flux arising from the temperature gradient; and a third flux due to the precipitation of atoms from the melt at the interface. All of this is referred to as solute redistri- bution. Calculations are complicated and depend on such details of the system as various thermodynamic quantities (specific heats, heats of fusion, distribution coefficients), kinetic properties (mass, velocity) and atomic and ionic characteristics. Figure 40 shows a concentration gradient typical of those calcu- 106’107 for the melt of‘a dilute, binary alloy lated or measured undergoing solidification in a closed system, such as a crucible sealed in a pyrex tube, as is done in the Bridgman method. The gradient for the solid is not shown because it varies with the equilibrium con- centration (represented by the solidus line of the solid phase) and with the ability of the solid to come to equilibrium under the given conditions. For most intermetallic compounds it would be nearly a horizontal line on the scale of Figure 40. Naturally the concentration gradient in the melt tends to disappear under the effect of stirring. Our Bridgman system may be considered partially stirred: no mechanical 150 stirring, but the radiofrequency furnace we used induces convection currents. Most of the analysis in the literature is for small concentrations of an alloying element (which can also be considered an impurity), but the concepts may be appropriate for our system: stoichiometric AuAg2 is considered the host phase, and the excess Au or Ga is considered the impurity. It is unlikely that the equations can be used directly, because here the "impurity" has the same diffusion coefficient and free energy considerations as one of the components of the host phase. Hybrid approach. Experience shows that theoretical considerations of the freezing of metals into crystals must include elements of both the phase equilibria and growth kinetics. The composition of grown crystals approaches what the equilibrium phase diagram shows, and the conditions of slow growth are an attempt to obtain a quasi-static process. But as discussed above, nonequilibrium conditions exist, and may have a significant effect on the grown product. In fact, a true quasi—static growing process would nullify hypothesis (1) of Chapter I on the variation of concentration along the growth axis of the crystal, and an explanation for the range in RRR could not invoke such a variation. The main question in the hybrid approach is to what extent each side should be included. The inability of the solid phase to equilibrate in the time over which the charge is solidified seems, from common experience, to be one of the deviations from conditions of equilibrium that will have as big an effect as any other deviation. Consider Figure 38(b). If the precipitate is exactly AuGa as indicated by the 2 vertical line, then that phase is already in thermal equilibrium for 151 all lower temperatures. But postulate for AuGa2 a solid phase of variable concentration, near stoichiometry, and of finite width (Figure 39). As the temperature is lowered from T1 to T2 the point representing the solid phase of the system changes from c1 to c2. The solid formed at T can be in equilibrium at T only by changing 1 2 its concentration. The required mass transport is a slow process in a solid and is strongly temperature dependent. Attainment of thermal equilibrium in a binary alloy is possible inthe times normally used to grow crystals, but only for temperatures in the upper third of the range from room temperature to melting point. Consequently, one suspects the whole solid will have a concentration which varies between c1 and c over the length of the solid. 2 Assuming a phase width such as Figure 39 for AuGa2 and assuming that the preceding argument applies to its growth, how is the concentra- tion range C1 to c2 of the solid AuGa2 distributed over the physical length of the crystal? Define cS to be the concentration of the solid at the fractional distance 2 from the end of the solid that froze first. Knowledge of cs(z) is of obvious importance for the cutting of samples frounthe crystal in order to measure physical prOperties dependent on cs. First a qualitative answer. For slow cooling rates the region in the vicinity of the interface is assumed to be near equilibrium because for it Tsz and the equilibrium redistribution in that part of the solid can occur before the temperature at the region changes much, and before distant changes in the whole system can propagate to this "interface system." But this interface system is not closed to mass transport, so that as solidification proceeds in the presence of a temperature gradient, the concentration 152 at the interface changes more slowly than it would were the system closed. This effect is illustrated in Figure 41. The details depend upon the temperature profile, how it changes in time, and diffusion coefficients. But the general effect is that most of the freezing occurs over a concentration range in the melt much smaller than would be expected from the equilibrium phase diagram. For example, in Figure 38(b) most of the charge solidifies while the concentration of the melt goes from c4 to c5; only a small fraction of melt remains beyond c5, and only a very thin layer remains by the time the isothermal line, 12, is reached. This follows from the lever rule. It should be repeated that this subsection has considered two different phenomena causing a variation of concentration in the solid. The first arises from a solid phase boundary that is slanted rather than vertical, in conjunction with the inability of the solid's con- centration to rapidly respond to temperature changes. The second effect, discussed immediately above, arises from the melt concentration at the interface changing during crystal growth in a manner different from that predicted by the phase diagram. We continue to consider the second effect. Some simplifying assumptions about the system are made preparatory to making a calculation for AuGaz. Calculations giving one of cs, 2, and T as functions of the other two are made by assuming or deriving functional forms for the liquidus and solidus phase boundaries, and are common in the literature.107’108 But they either assume phase boundaries or conditions inappropriate to our case, or require the knowledge of thermodynamic or kinetic variables unavailable for our system. Furthermore, the accuracy of such calcula- tions is low, so one would not have confidence in even qualitative results for the extremely narrow AuGa phase. 2 Figure 41. (a) (b) (C) (d) 153 (O) (b) Ch0'96 Charge 1 '. ' -' - ' .' ‘ 1- -:-I-':.=2'-.'.-'M;i'r.'-:--'.=:'-i1 . . - Molt .- ;::,-,v..;.L.,.,. .339. . .. .. c:> um; T T T Tl 1" T! n- —————— J l T2 72 Ga conc. Bottom Top Bottom Top Position Position h Bottom Top Bottom Top Bottom Top PosHion Fashion Fashion Gallium concentration of the melt at the interface of solid and liquid is lower than for an equilibrium system. Temperature step T1 to T2 shown on the AuGa2 phase diagram. Reduction of T in the equilibrium system and consequent precipitation along the whole charge leaves excess Ga atoms in the remaining melt. (The dots represent Ga atoms in excess of stoichiometry.) Upon lowering the temperature profile in the crystal grower, VT allows only the melt near the interface to solidify. The excess Ga atoms left in the melt rapidly diffuse into the rest of the melt, lowering the concentration. 154 We therefore make a different calculation specifically for AuGaz, abandoning hope of making even a rough calculation of cS(z) and looking instead to c£(z), the concentration of the melt at the interface between solid and liquid when the interface is located at a fraction 2 of the length of the charge. This approach takes advantage of the fact that the AuGa2 phase boundary (unlike those generally assumed in the calcula- tions in the literature) is either a vertical line or close to being one. Thus knowledge of c£(z), while not giving the details of cs(z), will at least give a general idea of its behavior. For example, if c£(z) changes slowly about some value of 2, then cs(z) will, too. For use in the calculation a numerical function relating c1 and T for 2/3fiumaou «N.N HH.N oo.q Aggy nuwcma ow.o ow.o mw.o AsuV Ages EOHuom unmoxmv mommamflo omNo.oH mmqm.oH ommm.q~ Amamuwv ommmz Hmuoa Awsamv Hmumauo amemaaam wosumz dewwfiHm mnu %n c30uu mamummnu .AHHHV mam mfixm mafia omo3uon .am Hooo.OH N.um sooo.on emSmEaumm .musaomnm N.um moo.on .Nu3 Nooo.0H m.H N.um men.o+ sumo.~e m.z N.um wm~.o: Noma.aq H.~ N.um som.o+ ammo.ae Aue\aav AN.umv ANusv Huou am we aAm\~v .sum on amusaoma< emmam “was m>aumamm uamz waumaoum onu mo deflumuuaoocoo aofiaamw V mHmG< HouHmo Hounm HOHH Wm Hmong mmmoonm mafiaouw Nm05< omusfi map so mama .a mam¢e 159 concentration. The charge was mixed by melting several times in one induction (radiofrequency- RF) furnace (Lepel 2% kW) and grown in another (Stanelco, 30 kW, 380 kHz). To start, the charge was put into a vitreous (pyrolytic) graphite crucible about 3" long, which acts as susceptor in the induction furnace. The crucible with its charge was then put into a combination vacuum chamber and RF coil and held in a vertical position by a short vycor tube. The chamber was then evacuated and filled with one-third of an atmosphere of a mixture of argon plus hydrogen (10%). The furnace was turned on and the charge heated to approximately 1100 to 1200°C, as measured by a pyrometer, and then held about 30° above the melting temperature for about one-half hour. After the water-cooled RF coils brought the chamber to room temperature (about one-half hour after the furnace was shut off) the chamber was opened, and the charge removed. It was then weighed as a check that no evapora- tion of metal out of the crucible occurred to change the concentration. After being washed with methanol, the charge was placed upside-down on the vertical crucible: the end of the charge formerly up being now downward, to increase the amount of mixing beyond the normal stirring generated by the induction furnace. The chamber was once again evacuated, and the whole procedure repeated, at least four times for each crystal, following which the charge was once again weighed, washed, and reversed in its crucible in preparation for the final sealing and the growing. This was done by inserting the crucible and charge into a vycor tube 2% feet long by 19 mm o.d. and closed at one end. The tube was flushed with argon, and then sealed closed by flame while maintaining one-third of an atmosphere of the argon-hydrogen mixture in it. The closed tube was placed vertically within the RF coil of the Stanelco induction furnace. 160 The axial length of the coil was only three or four times that of the charge, and the temperature gradient was changed by slowly raising the coils (see Table 9). Rough estimates for temperature gradients are 100 to 200°C over the length of the crucible. The roughly equal vapor pressures109 of Au (1 mm at l867°C) and Ga (1 mm at 1349°C) ensured that any mass loss during the heating cycles would be small, and any difference in mass loss would be even smaller. Cutting. The crystals were cut by spark erosion. A .002" molybdenum wire cutter at about 300 volts was used to cut off sections of the right thickness and face angle, preparatory to cutting cylindrical samples .4 cm to .5 cm by .1 cm in diameter with a Servomet Spark cutter. This used a small tube of the proper diameter as the cutting electrode. After a (111) axis of the slug was identified sections about .5 cm in thickness (and about the diameter of the slug) were cut by the wire. These were glued face down to a brass mounting block on a goniometer and oriented so that an x-ray beam paralleled a (111) direction. The crystal, still on the goniometer, was transferred to the spark cutter, which was aligned so that its axis was automatically in the (111) direction of the crystal. The spark cutter then lowered onto the crystal to cut out the sample. The plane had been previously cut so that it was close to being perpendicular to (111); thus the samples were nearly right cylinders. The accuracy of orientation of the crystal in the probe and cryostat needed to be only within several degrees, because final orientation was obtained from crystalline symmetry as observed in the deA signal. Figure 43 sketches the locations of the deA samples. This figure also shows from where were cut the samples used in the analysis of composition by a microprobe (Chapter IX). The dimension of the deA samples are 161 CRYSTAL A CRYSTAL B FNRR T—©49 H } 4., (outline) 3 V (outline) ®_\ @— L‘@ ”i ‘Microprobe Mount B Microprobe Mount A Figure 43. De Haas-van Alphen and RRR samples (circled numbers) and samples for microprobe analysis (not circled). Cut from the crystals of AuGa2 grown by the Bridgman method. Sample 19 is for RRR only. 162 TABLE 10. Dimensions of deA and RRR Samples Sample Type of Length L Diametera D Ratio No. Measurement (cm) (cm) L/D 5 deA and RRR .60 approx. .13 square (b) 6 deA only .38 approx. .218 . (c) 10 deA and RRR .518 .218 2.38 12 " " .483 .221 2.19 13 " " .163 .216 0.75 14 " " .259 .211 1.23 15 " " .231 .218 1.06 16 " " .203 .211 0.96 19 RRR only .902 .150 x .145 rect. 6.12 20 deA and RRR .602 .216 2.79 26 deA only .445 .218 2.05 27 " .287 .224 1.28 30 " .290 .224 1.29 aDiameters are maxima. Some of the "cylinders" were oblate at one end, due to spark erosion in the tube cutter, with a minor axis as much as 15% smaller than the diameter of the bigger end (the one entered). Some samples have different shapes, as noted. bVariable cross-section. L/D roughly 3.3. cVariable cross-section. L/D roughly 1.6. 163 given in Table 10. All of the deA and RRR samples were numbered in one series (circled in the figures) and all other pieces cut from a slug were numbered in a separate series for each slug. Some samples cut for deA runs were not used because they appeared not to be single crystals. Some of them are shown in the figures by unlabeled, dotted lines. Sample 19 was cut specifically for a RRR measurement. The (111) axis is at an angle to the long axes of the slugs. Note that Sample 6 is not (111); it was used for a measurement of T only. Figure 43 also D shows the microprobe samples, discussed in Chapter IX. X—Raying. Both orientation and checking for single crystals were done on a Phillips x-ray machine, with a copper target, without filters, by the Laue method (back-scattered x-rays). The cubic crystalline structure made orientation relatively easy using the standard techniques: Grenninger chart, measurement of angles between x—ray spots that were the intersections of zones, and identification of symmetry. With the crystalline surface 3.0 cm from the film plane it was possible to have two of the <100>, <110>, and axes on the emulsion and the third not far off, so that the arcs formed by spots from zone planes could easily be extended to intersection. (The objective was always to put the axis at film center.) In general, though, only one axis might be in the picture. A good picture was usually required in order to have enough spots to identify the correct zone arcs. Even with a good picture, an arc passing through the (burned out) center of the picture was difficult to identify. The exposure with the copper target and Polaroid type 57 B&W film was 3/4 to 1 hour at 30 kV and 22 to 24 mA. 164 The side surface of the slug as taken from the crucible gave good x—ray pictures without further preparation. Surfaces freshly (that day) spark-eroded by the wire cutter generally gave good pictures; sometimes a very light cleaning with acetone and a toothbrush was needed to remove the carbon breakdown products of the kerosene. A soft toothbrush left no surface damage. Two exceptions were encountered. Worn guide tips for the wire permitted slight motion with resulting surface striations when the cutting head was fixed to move vertically. Visible striations frequently gave bad pictures, almost certainly from the striations and not a crystalline defect because each time it happened spark planing (using a spinning disk as the electrode or the Spark cutter) to just barely remove the striations resulted in good pictures. It was dis— covered that this nuisance could be obviated by setting the cutter to draw the cutting head horizontally. The other exception was a mysterious "aged cut" effect: a surface resulting from a fresh cut by the wire would give very good pictures. From it a cylindrical sample would be cut and the remaining crystal left on the mounting block shut in a drawer. Sometimes nothing would be done to a piece between the initial cut and x-ray and the storing of it. A week or more later the piece would be removed and x-rayed again, perhaps to check the orientation, and the picture would be very bad, or unusable. Acetone did not improve it; only a light spark planing did. Before cutting the smaller pieces and the samples, the pieces of parent crystal were checked to see that they were single crystals. Each sample was also checked for the same purpose, and those samples which gave unusual de Haas-van Alphen signals received repeated scrutiny. All of the final samples were determined by numerous checks to be Single crystals within the limits prescribed by the resolution of 165 angular measurement with a Grenninger chart Q:O.5 degrees), and by the resolution in comparing spot patterns using tracings (£0.25 degrees). However, despite the thoroughness of the x-ray check, there is always the possibility of inclusions of other crystallites in the parent crystal, which remain undetected in the sample. Two types of single crystal checks were used. The first was a series of x—rays over one face or side of a crystal, with transport between pictures provided by a track and screw mounted on the x-ray machine to keep the crystal's orientation in the beam as constant as possible. No attempt was made to identify spots; the pictures were simply compared to see that the spot patterns for each picture were a single crystal pattern and that the patterns were the same on each picture. This method was also used on the sides of the samples. But the beam diameter was only a little smaller (between -05 and .10 cm) than that of the samples, so that a good picture from a basal surface was taken to indicate a single crystal, at least on that end. The second method of checking for single crystals was used only for the samples, and had a higher confidence level, because it involved ascertaining that both ends had the same orientation. The probability of another crystallite being included between the ends was low (but not zero) because samples were smaller size than the parent pieces. A 180° rotation about the vertidal axis, with the x-ray beam horizontal and the film plane vertical, transforms a spot into its mirror image across the horizontal axis of the film plane. 8.3 Melt Concentrations Slug A (see Table 9) was grown from a melt prepared 0.20 at.% Ga-rich, since Longo found that such a starting point gave crystals 166 with (RRR) residual resistance ratios (a measure of crystalline per- fection) among the highest of the crystals that he grew. Samples cut from the bottom, middle, and top of this slug were judged from x-ray pictures to be single crystals and were measured for deA frequency F, Dingle temperature TD, and RRR. Only RRR showed a variation definitely above uncertainty in measurement, and that increased monotonically from the bottom to the top of the slug. It was apparent that unless the measurable range of variation of AuGa was bigger than the range over 2 this slug, our deA measurements would not be able to distinguish the samples. Furthermore, the monotonic increase in RRR suggested that all portions of slug A lay to one side of stoichiometry. The growing of another crystal, from a melt of different concentration, seemed in order. Interested in seeing what would happen if the melt were slightly Au-rich, we grew a second sample (B) from a melt prepared .29 at.% on the Au-rich side of stoichiometry. (This is expressed in the tables and figures as a deviation from stoichiometry of —.29 at.% in Ga con- centration.) This time lower RRR were obtained, as well as a definite variation in deA F and TD. However, there were no extrema, as one would expect in going through stoichiometry. Upon reconsidering the behavior of RRR in slug A we realized that moving more toward the Ga- rich side would approach and perhaps pass through the concentrations of high RRR (and presumably stoichiometry). The distortion of con- centration versus distance discussed in the first section made it purely a guess as to what the concentration of the tOp samples of slug A may have been, but taking a clue from Longo's data (Figure l), we grew a final slug (C) from a melt +.55 at.% Ga-rich. The values for TD passed through a definite minimum, signaling the passage of our 167 samples through stoichiometry (assuming the stoichiometry corresponds to the most order, the least scatter, the lowest Dingle temperature). A possible minimum in deA F was observed. RRR has not yet been measured on these samples. CHAPTER IX CHARACTERIZATION OF SAMPLES A variation of the de Haas-van Alphen frequency of the C5 orbit with the position of the sample in the Bridgman crystal could arise from a number of mechanisms, some of which relate an increase of deA fre- quency to an increasing proportion of Ga while others, to a decreasing proportion. Therefore, an independent measure of composition would be useful. This chapter reports the results for the following methods of analysis performed on samples from the single crystals grown by the Bridgman method. For brevity, the entire single crystal taken from the crucible is termed a slug. Slug A......Chemical, Microprobe, RRR Slug B................Microprobe, RRR Slug C......None All of the samples were cut from regions that were either adjacent or close to the corresponding deA samples. 9.1 Anticipated Difficulties It was suspected that a method of high precision relative to that of the x—ray, thermal, electrical, and microscopic methods commonly employed to determine phase relationships would be needed to characterize 168 169 the samples. Phase equilbria studies of the Au-Ga system report AuGa2 105 to be of apparently fixed composition. Furthermore, AuGa2 has been grown and studied for years by many metals physicists, who have not reported any prOperties which indicated to them the existence of a phase width. One way to achieve high resolution in the determination of composi- tion (of the order of .01 wt.%) is by accurate prior weighing of the components, followed by melting and annealing for a sufficient time at the desired temperature. The structure of the resulting equilibrium system is then analyzed, usually by x—ray methods. Obviously the composition of our samples cannot be predetermined, since one of our hypotheses is that the composition of a crystal grown by the Bridgman technique from a melt of prepared composition varies along the direction of solidification in a manner determined by the equilibrium phases and the growth kinetics. Annealing would render the composition uniform; samples from different positions along the slug would no longer show the variation of composition resulting from the growth process. Of course, one could study the phase equilibria about stoichiometry by annealing crystals grown from.melts with compositions in a small range about stoichiometry, but the method chosen, i.e., measurements on samples taken from the unannealed crystal as grown by the Bridgman technique, permits both the investigation of the growth pattern and the possibility of obtaining information on the existence of a solid, equilibrium phase about stoichiometry for AuGaz. 170 9.2 Direct Analysis For the purpose of characterizing the samples, a direct measurement of the composition is the most satisfactory. This was attempted by means of wet chemical analysis and xeray analysis by electron microprobe. No conclusion could be drawn from the chemical analysis because the variation was less than the quoted error of the method. The method is discussed briefly, for the data gives an upper limit on the range of concentration. The differences in concentration were not much greater than the uncertainties of measurement by the microprobe, either. However, the data taken on four samples from Slug A were rather con- sistent, with eight of nine partially or completely independent results showing the top of the slug to have a lower concentration of Ga than the bottom (see Table 15). Therefore, despite the relatively high uncertainty in the data, some conclusions were drawn about the existence and direction of a variation of concentration cs(z) of the solid with position in Slug A. Four of six microprobe measurements on Slug B gave the same result: Ga concentration decreases in the distance from the bottom of the slug. The other two results would be quantitatively close to the first four if one changed their signs; there may have been a mistake in recording relative positions. Chemical analysis. Wet chemical analysis was performed on samples taken from the t0p and bottom of Slug A. Figure 44 shows both the samples for chemical analysis (numbered) and the de Haas—van Alphen samples (numbered in a separate series and circled). The samples were sent for analysis to laboratories in groups of two: one sample from the top and one from the bottom of Slug A, except that one "group" 171 <1”) glug Axis *\ 23° 7“\ .59 < t .553 53'119J I: Grown from melt prepared * +.20 at.Z Ga-rich 5| Figure 44. Location of samples cut from Slug A. Numbers are order of cutting, in two series: deA samples (circled) and samples for chemical analysis (not circled). Not drawn to scale. 172 had only one sample in it. Each group was sent to a different laboratory or in a different week. The samples are identified by group number. Gold content was measured two ways: by titration with hydroquinone of a solution containing the dissolved sample and by gravimetry. Gallium was measured by gravimetric precipitation of gallium-oximate from an ammonical solution. This method could have a bias of l to 2% of Ga concentration due to a failure to recover all of the precipitate. Much effort was made to obtain the most precise commercially available methods, and consultations with other analytical chemists suggested these errors are as small as could be expected short of undertaking a special project. The atomic (or molar) fractional Ga concentration is a = nGa , [118a] nGa + “ Au where n is the atomic density (atoms cm-3). The atomic percent of Ca is c = 100 a. [ll8b] The absolute change Ac in concentration in going from the bottom of the slug to the t0p is the ultimate quantity sought. It is Ac = ctop - cbottom' [119a] A related variable is the percentage change XAC = 100 (Ac/c ). [119b] bottom Frequently Ga concentration is reported with respect to stoichiometry: — -.E Acs - c 3 100. [120] 173 The "unit" for c, Ac, and AcS is atomic percent Ga (at.% Ga). It will not always be written out. This percentage is distinct from other percentages in which the data is occasionally expressed, such as per- centage changes. Table 11 shows that the chemical analysis does not appear to distinguish between the concentrations at the top and bottom of Slug A. The absolute errors in the analysis for gold at the two laboratories were 0.5wt% (Schwarszpf) and 0.25 wt% (National). By taking the differential of the formula for converting weight fraction w to atomic fraction a, M 1 1 1 —= 1+fi—(r- 1), [121] l 2 1 m where Mi are the atomic masses, these errors become 0.46 at.% Ga and 0.23 at.% Ga, respectively. It is interesting to note that all five values of wt% Au for the five samples in Groups I, II, and III lie within 0.11 wt% Au of their average; and if all four groups are averaged, the reported values lie within 0.39 wt% of their average. Thus, either Group IV was a blunder and the resolution of the chemical test for Au is better than estimated, or Groups I, II, and III just happen to cluster together, and a bigger sample would exhibit a more normal distribution. It should be mentioned that Schwartzkopf said the basic error in the test for Au is only 0.1 wt% Au, but that other considerations increase it to 0.5 wt%. The total assay for Group IV was 2% deficient, whereas the other complete analysis for Au and Ca are less than 0.2 % deficient. But even when Group IV is dropped from the average, the standard error in Ac is as big as Ac itself. The distribution of the values of c for 174 .Hm>uouafi moaowwmaoo Now onu o>fiw Ou woumsnwm own new .uouuo cmumafiumo huoumuonma scum amnu Honumu moon mo mamas: mam owsmu scum woawmuno mum mmaucwmuuoocan .mu N.um aw .ASOuuonvo I Aaouvo ma oH asouu noun .mo N.um ea .huuoaofinoaoum scum coaumfi>ow we a .casaou :uunom man scum mo Nos mchDw m o H a HHH masons you no N.um as.“ cam .HH a H masons now am N.um em.“ mo muouum ousaomnm woumsHuwo .Aqlmv coaumaom nwsousu.mmpfiw o ouo: ca muouum mo aowmuo>aoo w .>H w HHH museum you :< Nua m.“ use .HH a H wmoouw How s< NuB mm.“ uouum ousaomnm wmumEHumm huoumuonon an >H a HHH masons HNH.HqH.I Awe o< N nmo.Hoo.| .zoeaom fiNO.HmO.I . fimO.Hmfi.l . mOH woo N.um woo N.um u.emeam mo< mo 04 nowmuo>< Ammo N.um som.+ wmu N.um m mo.+ wo.l qo.t oq.+ NN.I wd.l no.1 moHaamm to mo< .w.z .ovawwoo3 .huoumuonmg ofiuhamamouofiz maoxnumanom mm.mm oq.~q om.mm mm.H¢ mo Nuz ovouuooom q~.m¢ om.~¢ HQ.H¢ mm.H¢ HN.H¢ mN.Hq oq.~q s< NIOOH am nome mu NuB musmao .mownoumuonma ofinamnwouuooam HoaOHumz hp vou%am:m HH a H wmsouu n .N.um «ON. >n nofiu abfiaamw woumaoua uHmE m Baum asouom >H HHH HH >H HHH HH H anon n u 175 Groups I, II, and III suggests some separation, but the sample of five is so small that no conclusion can be drawn. Microprobe. Samples from Slugs A and B were analyzed on two electronprobe microanalyzers. Slug A was analyzed on an Applied Research Lab (ARL) model EMX-SM microprobe at MSU by the technician, V. Shull, with the assistance of the author, and samples from both slugs were analyzed at the University of Michigan under the direction of Prof. W. C. Bigelow, of the Dept. of Materials and Metallurgical Engineering. The most reliable measurements correspond to a difference of Ga concentration of —.35 1;.10 (std.dev.) at.% Ga between the tap and bottom of Slug A. The range in Slug B may be -.24 i .10 at.% Ga. Again, the differences were not much greater than the resolution of the measuring technique. The micrOprobe samples were cut from the top, middle, and bottom of the slugs A and B, from regions adjacent to the deA samples, and mounted in plastic (Figure 43). Samples from each slug were mounted and tested separately. Mount A contains two pairs of samples from slug A: one pair from each end. Within each pair one sample exhibits a face approximately parallel to a {111} plane and the other sample, perpendicular to {111}, a precaution against the dependence of the mircoprobe response an orientation. However, no such dependence was subsequently found. Mount B contains samples from the top, middle, and bottom of Slug B, and has a test for reproducibility of the microprobe data: samples 8 and 16 exhibit faces that were originally congruent. The data from them was more nearly the same than was data from the other samples. The exposed faces were polished ultimately with six micron alumina after mounting. Early mounted samples were lightly etched with dilute aqua regia and examined under a metallurgical microscOpe. Occasionally 176 regions of slightly different, homogeneous appearance were seen. They appeared to be neither surface residues nor crystallites of different orientation; they may have been different phases. But for the most part the observed surfaces were without structure. The later samples, for which data is presented, were not etched because the slight gold color upon etching indicated the probability of removing a higher percentage of gallium atoms from the surface, thus potentially biasing the data. Possible inclusions were avoided by examining the samples when they were in the microprobe, using both an optical microscope and the image current of the microprobe (described below). A description of the operation of the microprobe at MSU follows. The essential features were the same for both machines (UM and MSU). The sample mount was inserted into a vacuum chamber in the microprobe and the chamber was evacuated. An electron beam, usually about one nanoAmpere and with a diameter 1 or 2 microns, was directed onto the target area on the polished face of the sample, and conducted out of the sample by conducting paint previously applied. X-rays resulting from the collision of the electron beam with the sample were analyzed by three LiF x-ray photometers set to detect the Au M, the Ga L, and the Ga k lines. The electron beam was swept over a small surface area, usually 80 x 64 microns. Sometimes the beam was swept along a line on the face of the sample while the intensities were recorded against position by a chart recorder. Repeated sweeps over the same line averaged out the noise. The latter mode was used to check for regions of different composition. A11 quantitative comparisons of samples were made with data taken by the first mode. In addition to.the quantitative information obtained from the intensities of the x-ray lines, the microprobe can give an image of the 177 surface, showing its general quality: regions of different appearance, pits, fissures. The electrons of the beam which penetrate the sample's surface and are conducted out of the sample constitute the image current. Electrons ejected from atoms in the x—ray process contribute to a lesser extent. Surface conditions modulate the image current, which in turn controls the intensity on a c.r.t. screen. This image helped in selecting a homogeneous region for measurement of the composition. The Ca K and Ca L photometers used electronic filters to prevent interference from neighboring x-ray lines, but the Au line was suffi- ciently isolated that no electronic filter was required. During the first runs sporaddc deflections of the beam indicated charge build-up occurring in the sample and holder region for some unknown reason. Coating the faces of the plastic holder and sample with about one micron of carbon greatly reduced this tendency. The beam had to be swept over a small area (80 x 64 microns) to obtain Ga x-ray intensities that did not decrease with time. Presumably the higher energy flux preferentially boiled off the Ga. What is the relation of the x—ray count N to the desired information, namely Ac or %Ac? To answer this question requires both a consideration of the physical principles of the microprobe and statistical analysis. The discussion below summarizes the relevant information on the operation of the microprobe from References 110 through 112. Basic principles of the statistical analysis are given in Young,113 with a more complete discussion in Mandel.114 Reference 110 also contains a discussion of statistical analysis tailored specifically for the microprobe. Regarding the physical principles of operation, the LiF detectors count only a small fraction (about .01), called here the x—ray count N, of the total x-ray intensity generated by the electron beam. The total 178 intensity is composed of both bremsstrahlung and characteristic radiation. The detectors are set to detect a specific x-ray line of a specific element, whose x-ray intensity I is roughly proportional to both the mass concentration of that element and the beam current Ib. For our thick samples I is independent of the absolute mass density of the element,110 and depends nonlinearly on the voltage. Both I and N depend on a multitude of other factors, most of which have either overwhelming or significant effect on the precise results. Most of these factors are common to all x-ray spectrographic techniques, but some are unique to the microprobe. The most important ones are mentioned here. Sta— tistical errors are discussed below. Machine parameters must be held constant or changed in a controlled manner. Those to which the count N is most sensitive are take-off angle (the angle between the sample surface and the small element of solid angle subtended by the detector), focus of the detector, and beam current I and voltage V. Sample preparation b is also important. Its most important factor is that the surface is smooth and that each surface be presented at the same angle to the beam (usually perpendicular). For example, a groove half a micron deep or a 2 degree tilt can both give a possible 10% error for N when the beam is focused to a spot.112 A swept beam is not so sensitive, resulting in a .2% to 1% error.115 The samples should not be etched, and must be thoroughly cleansed of the polishing slurry if it contains elements with interfering lines. Obviously inhomogeneities in the sample could give unexpected results. Finally, there are various inherent errors (in addition to statistical variation) common to x-ray analysis, of which the largest are absorption and enhancement (related to fluorescence). Thus the first answer to the question above is that the x-ray count N of a specific x-ray line is roughly proportional to the weight 179 concentration of the corresponding element, but that many complicated corrections are necessary for accuracy of the order of l to 5 absolute wt.% determinations in the absence of standards.111’112 However, the precision of measurements is an order of magnitude better. The ultimate in reliability and reproducibility comes with the use of a standard. One has two samples, one the standard, both being uniform and nearly identical in composition, with smooth surfaces prepared in the same manner. In this case the measurement of the relative weight fraction is very precise and straightforward:llo’111 Nu = '13—: [12.2] S S Slat! where u is the unknown, 3 is the standard (nearly identical in both physical and chemical aspects with the unknown), and N is the x-ray count when the instrument is set to detect the same characteristic radiation from both samples. The weight fraction w is, w = ————- [123] where m1 is the mass, or mass density, of element 1 of a binary sample. Our samples meet the rigid requirements above because they are essentially the same composition (in fact, the question is will the microprobe be able to detect any difference in composition), are homogeneous, and are held in the same mount, where they are polished simultaneously. The bottom samples are considered the standards when comparisons between the top and bottom (or middle and bottom) of a slug are made. Sometimes the distribution of counts from each sample is considered separately. 180 Regarding the statistical analysis, if one makes a number of trails (or, replications) of the x-ray count on a sample under fixed conditions, one obtains a sample of counts Nt’ t = 1, 2, 3, ..., whose population is governed by two distributions. One of these arises from the small fluctuations in the parameters mentioned above, unavoidable even though the conditions may be controlled well enough to prevent bias or major error. This distribution is usually taken to be Gaussian, although for any specific method of measurement this is always open to question.110 The other distribution is that of Poisson, and applies to any phenomena for which the population is huge and the probability of an event from any one member is infinitesimal,110:113 in 1917.110 The shape of the Poisson distribution is markedly different as first considered by Einstein from the normal curve only if a count of zero has a nonrnegligible probability. As the average of the distribution increases so that a zero result means only that the machine broke, the distribution becomes a special Gaussian whose standard deviation is the square root of the mean.llo’113 The fact that x—ray emission has Poisson's distribution provides a check on the reliability of data from x-ray spectrography. This comes from considering means and standard deviations as follows. The definition of the sample variance SV is independent of the distribution function of the population. In symbols appropriate to the x—ray count it is one run 2 (Nt4fi)2, [124a] trial t 1 SV = (T:I) where T is the total number of trials in the run and N is the mean of Nt over the run. The sample standard deviation SSD is 181 ;, SSD = (3V) 2. [1241:] Suppose that the variation in N is entirely due to the x-ray mechanism, t with no error contributed by the measuring technique. Then the parent distribution is that of Poission, and for a large sample, for which the statistics approach the population statistics, the relation mentioned above holds: SSD = N%, [125a] sv = 1T1. [125b] For all of the data of this chapter N is greater than 5000, large enough that the x-ray distribution has the shape of the normal (i.e., the Gaussian) curve. If the distribution of the errors introduced by the machine and sample are also normal, then the total distribution is also normal and the great body of statistical theory for such a distribution may be used. Equation (125) is a theoretical minimum for SV and SSD in (124), and a substantial increase in either signals a deterioration of precision by random errors in the process of measurement. This analysis contains no safeguard against bias or blunder. Also, because the sample is finite (125) does not hold exactly, and when machine error is very small, it may happen that SSD is slightly (1% to 2%) smaller than N%. This occurs occasionally in the data presented in the tables and in examples in References 110 and 112. An 80% confidence interval,113 computed from N, SSD, and Student's values, implies that there is an 80% chance that if the experiment were repeated under the same conditions and for the same number of trials, -' - the new average N would lie within one confidence interval of N. The 182 relative 80% confidence interval RCI for a run is the confidence interval CI divided by N: , [126a] where T is the number of trials in the run and t is the corresponding Student's t for an 80% interval. An interval could have been computed for any percentage of confidence, but the value 80% was chosen as perhaps being sufficient to distinguish real effects without demanding resolution not available from the method of Measurement. Table 12 summarizes the microprobe data and Table 13 shows typical run data used to compute Table 12. It turns out that although the un- certainty can be reduced to the statistical minimum (125) for a count taken with most of the microprobe controls left unchanged, the error in the reproducibility is several times greater. Thus the data is grouped and compared by run. A run (column 1) is all the data on samples in one count and one setting of the machine, with only the precision controls being varied. Each trial (column 6) holds all variables fixed while counting for a precise time, usually 10 seconds, to get the number N, proportional to the characteristic intensities detected (referred to below as an x-ray line). Using N, the mean from all of the trials in a run, improves the reliability of the results. Columns 7 through 9 of Table 12 give a slight modification of the relative 80% confidence interval: it is expressed as a percentage, %CI = 100 RCI, [126b] in order to more easily compare it with the results, which are very small fractions. 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