I , t '_ . .. b D ’0'! ‘U a u .. ‘C‘ c .0 L. u '0 . v .. nugunuhfll..2.‘v¢lo. .. c I‘ -v. v . I! ‘lvlflllbvoz .3 £63 I. I. . .iummlu- wan”. 2. I. l A v ..vb..z.. . v. lunf v....2'......o 23“." Q. 2 I'll . a o 4..”vi I i I. .133Pfishvlv23‘.. . . 7 to l 0 I A .1 “till; i“)!' , ‘1'... I .00 . fl ‘ I 1 I 7 0v. '5! x u v n. I: u: n a I 001 10 If u 3 . I... c O .0 ‘ a ‘ . t «11. .980 9.1. n v r..- .a .04 .o A u ‘ ‘ ‘ 'x ' “1"; o t . '2 .;' ”Ana: 1' ‘ n '1'9H‘ ': ' v 12: . . . an . . .- .. U: ism... “i' Michigan Siesta .-wer5211 This is to certify that the dissertation entitled NERNST-ETTINGSHAUSEN MEASUREMENTS 0N ALUMINUM BELOW 1K presented by Ahmad Amjadi has been accepted towards fulfillment of the requirements for PA Do «grain—M lac/M (J Major professor D... E— l 8- 85 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES —_ RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. NERNST-ETTINGSHAUSEN MEASUREMENTS ON ALUMINUM BELOW 1K By Ahmad Amjadi A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1986 ABSTRACT NERNST- ETT INGSHAUSEN MEASUREMENTS ON ALUMINUM BELOW 1K By Amjadi Ahmad We have constructed a system for transport measurements below 1K in magnetic fields up to 30kG, and used it to measure the high magnetic field limit of the Nernst-Ettingshausen coefficient for a pure polycrystalline Aluminum sample at 210mK, where we expect phonon-drag contributions to be negligible. Previous measurements on Al were limited to above 1.8K. Our data indicate that the electron-phonon mass enhancement continues to appear in the off- diagonal component of the thermoelectric tensor coefficient at temperatures well below 1K. To within our measuring uncertainty, the enhancement hicnu'data between 10 k6 and 20 kG is consistent with (1410), in agreement with the low-temperature, high-field NE coefficient measurements of Al from 1.8K to 5K by Thaler, Fletcher and Bass (ref.3 ). The data are less consistent with an alternative prediction of (1 + 2/3AO). Ahmad Amjadi As shown in Figure (“2), above 20 k0 the measured values of the NE coefficient start to drop from the saturated value at lower fields (i.e. between 10 kc and 20 kG). Thisieffect is most likely due to magnetic breakdown in Al above 20 k0. By taking closely spaced data points from 20 kc to 30 kc, we looked for evidence of the quantum oscillations which accompany such breakdown in the thermopower of single crystal samples (ref.8 ). However, no convincing evidence of such oscillations was found. We assume that their absence is due to the polycrystalline nature of our sample. TO MY WIFE ACKNOWLEDGEMENTS It is a great pleasure to acknowledge my thesis advisor Professor Jack Bass, whose suggestions and criticisms throughout the course of this research were invaluable. I would also like to thank Professor Pratt, Professor Schroeder and Professor Spence for their precious advice, discussions, and help at many stages of this study. Specific thanks go to Professor Foiles for his suggestions on Thermometer preparation. I am indebted to Dr. Mark Haerle and Dr. Vernon Heinen for their unselfish help on different stages of this research. I would like to extend thanks to the good humored fellows in the machine shop for their help in constructing the precise centering device, sample can, high vacuum can, heat exchangers, and all the mechanical supports for. the superconducting magnet, the Liquid Nitrogen and Helium dewars, and the sample. Finally, the financial support of the Natnnmu.Science Foundation is gratefully acknowledged. TABLE OF CONTENTS Chapter LIST OF TABLES LIST OF FIGURES I. II. INTRODUCTION 1.1 Introduction 1.1.1 Transport Equations 1.1.2 Compensated Metals 1.1.3 Uncompensated Metals 1.2 Previous Work 1.3 Present Thesis 1.3.1 Construction 1.3.2 Thermoelectric Measurements of Aluminum EXPERIMENTAL TECHNIQUES 2.1 Introduction 2.2 Dilution Refrigerator 2.3 High Precision (0.1 PPM) Resistance Bridge 2.A SQUID 2.5 Vibration Isolation of the Cryostat 2.5.1 Magnet Supports 2.5.2 Centering Device a. Stage 1, Thermal isolation between the mixing chamber and the 1.2K copper band heat exchanger iv Page viii ix 01 02 O7 08 12 2H 2H 25 28 28 31 31 32 32 33 36 Chapter b. Stage 2, Thermal isolation between the copper band and the surrounding vacuum can 2.6 Thermometry in High Magnetic Field at Low Temperature 2.6.1 2.6.2 2.6.3 2.6.N 2.7 Sample 2.7.1 2.7.2 2.7.3 2.7.4 2.7.5 2.7.6 Different Thermometers Thermometer Preparation Thermometer Calibration Temperature Regulation Sample Size Sample Preparation Annealing Sample Connections a. Electrical connections b. Thermal connections 0. Mechanical connections Electrical contacts to the sample a. Spotwelding b. Soldering Indium c. Plating Magnetic Field and Current Density Dependence of Different Solders 2.7.7 Sample Can 2.8 Superconductive Magnet 2.8.1 Introduction Page 37 37 39 N1 U6 N8 "9 51 51 53 53 53 5M 56 56 57 58 61 6” 65 65 Chapter 2.8.2 Electrical connections to the Super- Conducting magnet 2.8.3 Pressure Relief Valves 2.9 Reference Resistor 2.10 SQUID's in a Feedback Loop III. THEORY 3.1 Electron-Phonon Mass Enhancement 3.2 Fundamental Transport Theory 3.3 Transport in High Magnetic Field 3.” The High Magnetic Field Nernst-Ettingshausen Coefficient IV. THE EXPERIMENTS AND RESULTS “.1 Temperature Range 1K to 4.2K u.1.1 The Hall Coefficient (RH) above 1K H.1.2 The NE Coefficient above 1K u.2 Measurements at Temperatures below 1K u.2.1 The Hall Coefficient (RH) below 1K N.2.2 The NE Coefficient measurements below 1K u.2.3 Correction of QX by RL Measurements u.3 Conclusion REFERENCES APPENDIX A vi Page 66 67 70 71 75 75 83 88 95 101 101 101 102 103 103 107 108 113 115 118 Table (1.1): (“.1): (“.2): LIST OF TABLES Page Comparison of Calculated and Experimental 13 Effective Masses in Na, A1, Pb. RH for Pure Aluminum Sample at Different 105 Magnetic Fields and a Constant Temperature 137 mK. Nernst-Ettingshausen Coefficient Measure- 110 ments for Different Values and Direction ("z"-direction and "-z"-direction) of the Magnetic Field from Direct Measurements and Corrected Values using R Coefficient L Measurements. Figure (1.1) (1.2.a,b) (2.1) (2.2) LIST OF FIGURES Page a EE-as a Function of Magnetic Field B for 20 Temperatures 2.3K and H.6K. Pa 2 3 -—- as a Function of T (a) and T (b) for 20 B Magnetic Fields of 1.5 T(A), 1.8 T(o), 2.0 T(o) and 2.1 T( ). The broken lines indicate the value predicted from the electronic specific heat Yc, which contains the enhancement factor (1 + A). Diagram of the Sample, inside the super- 30 conducting Magnet, with the connections between the mixing chamber and the vacuum can through the centering device. Cross section view of the centering device. 35 A-Brass Band. B-Copper Band Heat Exchanger. C-The Upper Side of the Mixing Chamber. D-Vespel Cylinder (stage 1 of the thermal isolation). E-Vespel Cylinder (stage 2 of the thermal isolation). F-Stainless Steel Rod. G-Adjusting Screw. H-Locking Nut. I,J-Attaching Screws. \Iiii Figure (2.3) (2.”) (2.5) (2.5.b) (2.6) Page Block diagram of the two stages of thermal 38 isolation of the centering device. Stage 1-The thermal isolation between the mixing chamber and the copper band connected to the 1.2K heat exchanger. Stage 2-The thermal isolation between the copper band and the surrounding vacuum can. Speer Carbon Thermometer: A-Speer Carbon Resistor N3 Body. B-Speer Graphite. C-Superconducting NbTi Leads. D-Copper Clad is removed from the super- conducting wire. E-Silver paint for electrical contacts between the leads and the speer carbon thermometer. F-Superconducting wire with the electrical insulation removed from it. Sample Holder: A-Main copper support. IB-Pb M5 shield for sample transverse voltage wires. C-Heater leads. D-Heater. E-Heater stand. F-Aluminum sample. G-Vespel substrate. H- Thermometer. I-Thermometer holding-spring. J-Current leads to the sample. Admitance of Tc as a Function of Temperature D7 in Magnetic Fields; 15, 20, 25. k0. Brass sample cutter cross section. 52 it Figure (2.7) (2.8.a) (2.8.b) (2.9) (2.10) (2.11) Page Schematic diagram of the electrical circuit 55 containing the Al sample for Nernst Ettingshausen and Hall coefficient measurements. Schematic diagram of the set up for magnetic 62 field and current density dependance studies of different solders. (A)-Electrical connections out of the magnetic field. (B)-Solder under test inside the magnetic field. M-Voltage and current leads. N-Electrical contacts. 0-Superconducting leads. P-Pb tube. U-Sample can. Q-Superconducting magnet. R-Solder under test. S-Helium dewar. T-Liquid Helium. Magnetic Field Distribution. 62 A-The magnetic field dependence of the contact 63 resistance. B-The current density dependence of the contact resistance a) Woods metal, b) Sn, c) Roses Alloy, d) In, e) Pb. Helium pressure relief valve: A-Weight holder 69 screw. B-Extra weights. C-Top flange. D-Top valve holder. E-O-ring seal. F-Elbow. G-Brass pipe. H-Copper pipe casing. Simplified block diagram of the SQUID circuit 73 in a feedback loop, for measuring i-switch position (A), Hall coefficient ii-switch position (B), Nernst Ettingshausen coefficient. Figure (3.1) (3.2) (H.1) (u.2) Electronic density of states as a function of energy, where NC(E) and NCO(E) are the enhanced and unenhanced electronic density of states. The dispersion relation, E(k) for enhanced and E°(k) for unenhanced electrons. The enhancement has the effect of producing a different slope in the vicinity of Ef. Hall coefficient of Al as a function of magnetic field B at temperature 0.1N7K. The broken line is the value predicted from the known electronic structure of Al. Nernst-Ettingshausen coefficient of Al as a function of magnetic field at temperature 0.1N7 K. The broken line indicates the value predicted from the electronic specific heat YC. vi Page 81 106 112 CHAPTER I I. INTRODUCTION (1.1 Introduction). Transport properties of metals have been studied over the years by numerous investigators, because they provide information about the physics of the materials. In recent years, the introduction of SQUIDs (superconducting quantum interference devices (Ref.1)) has greatly enhanced the sensitivity of low temperature transport measurements and has allowed considerable improvements in our understanding of different scattering processes, such as electron- electron and electron-phonon scattering in metals. Magnetic fields have also been employed extensively to investigate the magneto- transport properties in metals. However, due tn: the extreme sensitivity of SQUIDs to magnetic fields, it is difficult to use the powerful combination of SQUIDs plus magnetic field, particularly at ultra-low temperatures (T<1K) where the complexity of the low temperature apparatus inhibits proper shielding of the SQUID. 'This dissertation is a report of the construction of a general purpose experimental system for carrying out high-magnetic field transport coefficient measurements using SQUIDs at ultra-low temperatures, and of the results of measurements of the off-diagonal diffusion component of the thermoelectric tensor hfl' ) of yx d’ Aluminum below 1K. These measurements were used to test the system; they are also of interest because this component is expected to be subject to a many-body renormalization involving the electron-phonon mass enhancement (A). In this introduction we start with a background on Transport equations and define the thermoelectric tensor and its components. Then we define the appropriate Nernst-Ettingshausen (NE) coefficients for compensated and uncompensated metals, and other coefficients such as the Hall coefficient and Righi-Leduc (woefficient which are necessary for our experiment. We then review previous work on NE measurements, and finally, describe the present thesixs and the reasons for measuring these coefficients on Aluminum at very low temperatures. (1.1.1)TRANSPORT EQUATIONS: The electrical and thermal current densities 3 and U are related to the electric field E and the temperature gradient VT by the fundamental transport equations (Ref. 2). ‘ ”E+E"(vr) (11) J-LH 12 ° + ++ E+++ 6T (12) U - L21 L22 ( ) - . o = The electrical conductivity tensor ++ ++ L12 = e" = The thermoelectric tensor ++ ++ ++ L = + e = T . e" (1.3.a) 21 ++ ++ ++ _ +3 + 8" L22 = + A - A T '3 (1030b) 0 where ++ A" = The thermal conductivity tensor. All of these tensor coefficients are functions of magnetic field E and temperature T. We focus primarily on the high-field limit of Enyx’ tune off-diagonal element of the thermoelectric tensor. This element is the sum of two terms, a diffusion component (e"yx)d and a phonon-drag component (5"yx)g (ref. 3) n g n n e yx (e yx)d + (e yx)g (1.“) As discussed in chapter III below, when E is directed along an axis of at least 3-fold symmetry, the high-field limit of the diffusion component of the thermoelectric coefficient (6"yx)dlforeanmtal which has no open orbits in the plane perpendicular to g is just, n _ 2 2 I_ (E ) - 1! k Nt(Ef) 38 W d (1.5) where k = Boltzmann's constant Nt(Ef) = The transport Electronic density of states at the Fermi energy Ef T = Temperature (K) (IN = Magnetic field (Tesla) As we will see later in this introduction, Nt(Ef) is expected to be enhanced by the electron-phonon mass enhancement (x). The high-field limit of the Phonon-drag component of the thermoelectric coefficient is To (€"yx)8 = C §_ (1.6) where:(3 is a constant which depends upon the metal of interest, and a ~ 3, Opsal (Ref. A). 1311968 Blewer et al (Ref. 5) argued that for a metal with a Debye Phonon spectrum, and neglecting electron- phonon-Umklapp scattering, the high-field limit of (HE/X)8 should 3. In 1977 Thaler et al (Ref. 3) suggested vary with temperature as T that even if the phonon spectrum were not Debye like, and electron- phonon Umklapp scattering were not negligible, one might hope that in equation (1.6) a is still much greater than one, so that the phonon-drag component (e"yx)g varies much more rapidly anklthe electron-diffusion component (e"yx)d as a function of temperaturen Ihi such a case (6" ) can be separated from (€" ) experimentally yx d YX 8 by means of their different temperature dependences as follows. Substitute equations (1.5) and (1.6) nnx>(1.u) andrmfltiply this equation by g. This gives: 5" N (E ) yx B _ n2k2 t f + (a-l) T 3 CT (1.7) (a-1) A plot of equation (1.7) against T where (a~3) should 2 2 Nt(Ef) yield a straight line with intercept 1r k T° In theory, this value of Nt(Ef) could be compared with a calculated value to see whether ii; is enhanced. 1H1 practice, the enhancement for a given metal is rarely known well enough for accurate comparison. It is thus better to compare Nt(Ef) with another experimental quantity which is known to be enhanced. This comparison is made by means of the electronic specific heat YO, which is enhanced (Ref. 6 ) and which can also be written in terms of a density of states Y = n k -————— (1.8) Here Nc(Ef) is the electronic density of states determined from c measurements of Y . For simplicity, let us call the intercept of equation (1.7) Yt Y=11k———— (1.9) where Yt stands for "transport" specific heat. Then we can write the high-field limit of the diffusion component of the thermoelectric coefficient as, n _ I— (e ) — Y B ( C) (1.10) Equation 1.10 tells us that measurement of (5" yx) provides a d direct determination of the ratio Yt/YC. Since Ye is known to be enhanced, this ratio determines whether or not Yt is enhanced. For example, the electron-phonon enhancement A in Al is about o.u5. If Ytis enhanced, the ratio Yt/YC should be 1.0. If Yt is not enhanced, the ratio should be 0.7. With this background, let us see what are the best quantities to be measured in compensated and uncompensated metals in order to determine the diffusion component of the thermelectric coefficient (6"yx)d. For simplicity, let us call the intercept of equation (1.7) Yt Y=11k——— (1.9) where Yt stands for "transport" specific heat. Then we can write the high-field limit of the diffusion component of the thermoelectric coefficient as, (6" ) = Y - (—) (1.10) Equation 1.10 tells us that measurement of (5" y ) provides a x d direct determination of the ratio Yt/Yc. Since Y0 is known to be enhanced, this ratio determines whether or not Yt is enhanced. For example, the electron-phonon enhancement A in Al is about 0.115. If Ytis enhanced, the ratio Yt/Yc should be 1.0. If Yt is not enhanced, the ratio should be 0.7. With this background, let us see what are the best quantities to be measured in compensated and uncompensated metals in order to determine the diffusion component of the thermelectric coefficient '1 (e yx)d’ (1.1.2)Compensated Metals: For a compensated metal the adiabatic Nernst-Ettingshausen (NE) coefficient 03 is the most convenient experimental quantity from which to evaluate Enyx' Qa is defined as -E ‘(AV /W) Y a = y : ._______ Q aT/ax (AT/Ax) (1'11) where W is the sample width, Ey is the electric field produced in the y-direction when we apply the temperature gradient 21 = AT 3x A? in the )rwiirection and a constant magnetic field B in the z-direction. AVy = Ey-W is the voltage across the sample in the y-direction. The boundary conditions are 3 = O and Uy= Uz= 0. (Uy and U2 are the heat flows per unit area in the y and 2 directions respectively). In chapter III we will see that as B + on the diffusion component of Qa, call it Qad, reduces to; Q: = e"yx pyy (1.12) where pyy is the electrical resistivity. We see from Equation 1.12 a that measurements of and ield e" . Qd pyy Y yx For a compensated metal, the transverse electric field Ey is relatively large compared to an uncompensated metal. This makes AVy fairly easy to measure. But on the other hand from equation (1.11) the NE (naefficient measurement for a compensated metal requires the measurement of g; which is limited by accuracy in thermometery. Because accurate thermometry in high magnetic field and low temperature is not easy, if one has enough voltage sensitivity, inunmnpensated metals provide scope for more accurate determinations of Yt, as we will see next. (1.1.3)Uncompensated Metals: In an uncompensated metal the most convenient parameter for determining e"yx is the adiabatic NE coefficient defined as, Pa=—y- (1.13) Here UK is the applied heat flow per unit area in the x-direction,§ is in the z-direction, and the boundary conditions are again 3’, 0 and U = U = 0. y z In chapter III we will observe that as B + w the diffusion component of Pa, let us call it P: approaches the value P = ——-—————— = -Y (1.1M) where L0 is the Sommerfeld-Lorenz number and ne and11 are h respectively, the number of electrons and holes per Luiit volume in the metal. Now from equation (1.13) (AV /W) AV ° t pa.___¥____.______ (Qx/t°W) Qx 1< (1.15) where AVy is the transverse voltage difference, Qx is the total heat flow along the sample in the x-direction, and t and W are, respectively, the thickness and the width of the sample. To discuss limitations, let us look at equatnni(1.15). At very low temperature Pa ~ Pa , where Pa is the diffusion component d d of the NE coefficient which has the high field limit shown in eq (1.1M). Qx is the total heat flow along the sample in the x- direction, which is limited by the power of the dilution refrigerator. Because the maximum power of the dilution refrigeratcm at very low temperature is limited, in order to have a significant transverse voltage AVy, we need a very thin sample. But 10 a thin sample needs a substrate to hold it in the cryostat. Under these conditions two problems arise. The first problem is the measurement of the thickness (t) of a very thin sample at the cross section where Qx is measured, and the second one is the measurement of Qx (again at the same cross section where AVy is measured) along a sample which is attached to the substrate. Fortunately, there are cross-check measurements which allow these problems to be addressed. i) The experimental check for the thickness t is the Hall coefficient p E (AV/W) AV t 32.31."... Y=___L_=__Z_ (116) H B J-B I -B I-B ' X __)-(_______ X w-t where pyx is the transverse component of the electrical resistivity, Jx is the current density in the x~direction, Ix is the current in the x-direction. Since the Hall coefficient has the high-field limit (Ref. 2) 1 R =- _ (1.17) H (ne nh)ec which can be calculated from properties of the metal of interest, the thickness t can be calculated from equation (1.16) 11 I . B x tg— _ (1.18) Avy (ne nh)ec ii) The experimental check for the thermal current along the sample Qx is the Righi-Leduc RL coefficient (ref. 7) which is defined as: = (aT/ay) (ATy/W) R =-——-—-—- (1.19) L Qx B Qx B where BT/By is the temperature gradient transverse to both the heat current 0x and the magnetic-field B, and AT is the transverse temperature difference across the sample. RL has a high field limit of, 1 H ‘ R = _ = — ——— (1.20) L Lo T (ne nh)ec LOT where Lo is the Lorenz number and T is the temperature. Measurement of RL and ATy thus provides a cross-check on Qx' 12 (1.2)Previous Work: From 1956 to 1958, Landau (Ref. 9) shed a great deal of light on many-body interacting systems. In 196%, Prange and Kadanoff (Ref. 10) extended the theory for electron-phonon interactions in metals to the nonequilibrium case, and asserted that the effect of electron-phonon mass enhancement was unobservable in dc electronic transport in metals. The above statement was proved microscopically for electrical conductivity in the same year by Holstein (Ref. 11), but not for thermal conductivity or thermoelectric effects. In 1965, Ashcroft and Wilkins (Ref.12) reported that the low temperature electronic specific heat [equation (1.8)] is enhanced by electron-phonon and electron-electron interactions. To show this, they chose simple metals Na, Al and Pb because of their known Fermi :nn~faces and band structures. They compared experimental values of * %— --extracted from low temperature electronic specific heat * measurements in the literature --with their calculated £7- taking into account band structure effects, electron-electron enhancements, and electron-phonon enhancement. 13 Na Al Pb m'l' (7111—)133 1.00 1.06 ~1.12 6m (‘31—)e1-el 0.06 -0.01 0.00 (5“) 8 11 .5 el-ph 0.1 0. 9 1.05 m* 6m 6m (F)Bs[1*("6)ei-e1 (3)el_ph] 1.211 1.57 2.30 m‘! (Tn-)exp 1.25 1.115 2.00 (Table 1) Comparison of calculated and experimental effective masses in Na, Al, Pb. Both the electron-electron enhancement and band structure effects were small compared to the electron-phonon enhancement. 1! I: The experimental term %— for Aluminum was %— ~ 1.M5, from which m* = m (1 + lo) (1.21) gives a mass enhancement for A1 of A0 - .us. In 1966, Grenier et al (Ref. 13) measured e"yx of Cd in the Hell range. They found a value in the vicinity of the free electron value, but various uncertainties made them unable to determine the presence or absence of mass enhancement. 1 111 In 1968, Blewer et al (Ref. 1A) extended the work done by Grenier and Long to lower temperature on Antimony for better understanding of the nature of the scattering mechanisms. They found Yt to be 10-20% larger than YO. Because of uncertainties in the measurements, and the fact that the value of A for Antimony was not known, this result also did not indicate whether there is an enhancement. In 1970, Long (Ref. 15) measured the NE coefficient for a pure tungsten crystal in the temperature range of 1.11 - 11.1K and argued that the density of states computed from the NE coefficient was consistent with the values of specific heat measurements; in other words Yt = YO. But the uncertainties in both values again made them unable to establish any enhancement. In 1971 , Averback and Bass (ref. 16) (see also Averback et al. ref. 17) measured the low-temperature magneto thermoelectric power of Al and a number of Al alloys and showed that the electron- diffusion component of 8 first became more positive as the magnetic (1 field increased, and then saturated to a high field limit. They showed that the difference AS between the high field (B = w) and (1 zero field (B = 0) values of S was independent of the type of d impurity in Al and had the value ASd= (2.2 i 0.2) T x 10-8 V/K. In 1972, Averback and Wagner (Ref. 18) showed that this value for ASd was larger than expected in the absence of electron-phonon 15 mass enhancement. However, their calculation was limited by several approximations and uncertainties. In 1975, Douglas and Fletcher (Ref. 19) measured the NE coefficients for Cd and W. Because the Debye temperature of Cd is low (~200K) the phonon-drag component of e"yx did not vary as T3 over the temperature range they studied, and since the diffusion part is separated from the phonon-drag component by extrapolation to T 3(OK, the uncertainty in their data did not able them to reach any conclusion concerning enhancement. For W, their data did not agree with Long's (Ref 15) and again the temperature dependence of the 3 phonon-drag component did not vary as T so that, again, no clear conclusion concerning any enhancement could be drawn. In March 1976, Opsal et a1 (Ref. 6)--for experimental and theoretical details see Refs. 20 and 21--for the first time demonstrated the presence of mass-enhancement using measurements of ASd for a dilute A_l_Ga alloy. Ca was chosen because it had been shown (Ref. 22) to be the nearly isotropic scatterer in Al needed for the model of Averback and Wagner to be applicable to the experimental data. Opsal et al. also extended the Averback and Wagner calculation beyond a free-electron model, and showed that band structure corrections could not explain the too-large value of ASd noted by Averback and Wagner. Having run out of ways for the experiment and the Averback and Wagner analysis to be wrong, Opsal 16 et al. then re-examined the question of whether mass-enhancement should be present in thermopower. They concluded that although Prange and Kadanoff (Ref 11) were correct that electron-phonon mass enhancement is not observable in electrical conductivity, they were not correct for thermopower. They showed that for elastic impurity scattering, if all of the properties of the electrons are consistently renormalized, then this renormalization cancels out of the electrical resistivity and the thermal conductivity, but does not cancel out of the diffusion component of the thermopower. Shortly thereafter, Opsal (Ref. 11) showed that the phonon-drag components of the thermoelectric tensor are not enhanced, and that in the high‘field limit, (e"yx)g has the same % variation with ° " magnetic field as does (6 yx)d' Although Opsal et al (Ref.6) had demonstrated both theoretically and experimentally the presence of electron-phonon mass enhancement in thermopower, the accuracy with which they could determinine the size of this enhancement was limited. The presence of both giant Quantum Oscillations (Ref.8 ) and small additional terms in the thermopower (Ref.6 ) make it unlikely that further measurements of AS for Al or any other metal would allow a more d accurate determination of the enhancement. 0n the other hand, Nernst-Ettingshausen (NE) coefficient measurements have two advantages over thermopower measurements 17 i) In the high field limit Enyx can be extracted directly from the NE coefficient, because it has (fine simple form of equation (1.1M) ii) Giant quantum oscillations do not seem to be as large in the NE coefficient as in the thermopower. In 1976, Fletcher (Ref. 22) measured the NE coefficient of a single crystal of Molybdenum and found the expected temperamuwe dependence of Uneeflectron-diffusion and phonon-drag components. From the data, he found Yt to be within 6% of Y0 and with this uncertainty he came out with the expected enhancement of A = 0.3 with uncertainty of 25%. In May 1977, Thaler et al (Ref. 3) measured the high-field Nerwust-Ettingshausen coefficient of polycrystalline Al from 1.8K to 5Kr and separated the coefficient into electron-diffusion and phonon-drag components as described earlier. From the diffusion component of the NE coefficient they found the transport heat capacity Yt to be identical to the experimental electronic heat capacity Yo for Al, within a probable uncertainty of 3%. Ewan the approximate value of )0 = 0.M5 for Al, this leads to agnmmable uncertainty of about 10% in the magnitude of many-body enhancement A0. The experimental difficulty in measuring Pa in an uncompensated 18 metal is the smallness of the NE voltage. Their Al sample, which was 0.25mm thick, produced only 5nV for 10mW of heat and a 2T a magnetic field. A plot of their data BE as a functicn1<3f magnetic field B for two different temperatures is shown in figure (1.1). At a both temperatures, g— is independent of field above 0.8T. Figure a (1.2.a) shows EB plotted against T2, which is expected to be a straight lhuein.the absence of Umklapp scattering, with an Pa intercept at T=0K of —§ (The diffusion component of NE). Since this a plot was not exactly a straight line, they also plotted -E§ against T3, 1diich gave an apparent straight line [Figure 1.2.b] that had no specific theoretical justification. As shown in Figures (1.2.a) and Pa (1.2.b) twua extrapolation of these two plots led to values for -§9 which differed by less then 0.5% from each other. Their best value a P — ‘- for —EE is 5.88 x 10 11 m3 j 1 with an uncertainty of 2%. This .YC value was compared with an expected value of ‘2 2 = 5.90 x Lo(ne-nh) e 10-11 153.1" calculated from the accepted electronic heat capacity ‘1'c of 1.360 mj mol_1 K.2 Dixon et al (Ref. 7 ) and the Aluminum 19 lattice parameter of 11.032AO Pearson (Ref. 23) The uncertainty in YT Yc was less than 1% and the ratio _5 from their measurement came out Y to be 1.00 i 0.03M. 20 I‘OA16 I I 1 I l I 1!}: T A: I, a / in" A“ 51' # J 4: _A a -1 5 °” 'Lsx ‘ m '2 4+ . 3 dz 1 53 ‘ 13 L’ 1% 20 MT) Pa \ . ngure (LA) —§-as a Function of Magnetic Field B for Temperatures 2.3K and H.6K. I10.“ 6' 5'6 1' I 52 1 L-B ‘FT-fifij' ) o 4 3 'P/B‘tn J - l X10116'Lh -< (b) ‘ rNKN a Figure (1.2.a,b)£§ as a Function of T2 (a) and T3(b)for Magnetic Fields of 1.5 T(A), 1.8 T(o), 2.0 T(o) and 2.1 T(o). The broken lines indicate the value predicted from the electronic specific heat Yo, which contains the enhancement factor (1 + A). 21 In 1977, Lyo S.K. (Ref. 211 ) studied the enhancement of the electron diffusion thermopower microscopically, and found that the thermopower is enhanced not only by the mass enhancement, but also by an additional electron-phonon modification of the quasiparticle velocity. In a model of weak s-wave scatterers this mechanism added an additional term A/2, so that the total enhancement was predicted to be 1 + 3/2 A. In May 1978 Vilenkin and Taylor (Ref. 25) found additional corrections to the thermopower which, for the case of weak s-wave scattering, increased the total enhancement of the thermopower to 1 +2A. A few months later in November 1978. Vilenkin and Taylor (Ref. 26) extended weak s-wave calculations still further, and found that the electron-phonon renormalization of the impurity scattering corrections to the low-temperature Seebeck coefficient of dilute alloys almost completely cancel corrections due to the electron- phonon renormalization of the electron energy, velocity, and relaxation time. They argued that "For normal valences the electron-phonon corrections are reduced by a partial cancellation to a few percent of some recently predicted values". In January 1980, Ono and Taylor (Ref. 27) investigated the Seebeck coefficient in dilute alloys at low temperature using a more realistic model than the weak s-wave model previously considered. They assumed that free electrons are scattered by a random array of 22 fixed impurities and interact with longitudinal Debye phonons through a Frohlich Hamiltonian. They found that for a screened Columb potential, the net electron-phonon enhancement of the Seebeck (usefficient is close to the mass-enhancement factor 1 + A suggested by Opsal et a1 (Ref. 6). In August 1980, Ono (Ref. 28) added two more impurity potentials to the previous investigation by Ono and Taylor (Ref. 27): i) The screened Coulomb potential with Friedel's extension (Ref. 29) of the Thomas-Fermi approximation: and ii) The square-well potential. He found that the electron-phonon enhancement of the Seeback coefficient again generally came out numerically close to the mass-enhancement factor 1+A. In 1982, Rammer and Smith (Ref. 30) noted that since 19611 the electron-phonon mass enhancement had been obtained only for free electron models and Debye-phonons with simple model impurity potentials. But they argued that for calculating thermoelectric properties even in the absence of renormalization, the free electron models are not adequate. They focussed on the high-field limit (new >>1 (1.22) where we is the cyclotron frequency and 1 is the collision time. To simplifying the problem, they neglected the effect of impurity and phonon scattering. To treat the effect of renormalization they 23 used the Keldysh method (Ref. 31). Their calculation gave a renormalization factor of m* = m (1 + A - caA) (1.32) where c is an energy-independent constant that is 1/2 for free electrons interacting with Debye-phonons and a is: E:F fog(E)dE a = (1.33) E(EF)°EF . . . 1/2 Here g(e) IS the electronic den31ty of states. For g(e) ~ e: as in case of free electrons, a = 2/3. This investigation yields an overall reduction in the electron-phonon enhancement factor from (1 +1) to(1+-23-A). In February 1983, Hansch and Mahan (Ref. 32) continued their previous work on the derivation of new transport equation for many- particle systems in do electric fields (Ref. 33). They derived the transport equation by both the Keldysh (Ref. 31) and the Kadanoff (Ref. 11) methods for nonequilibrium formulation of many-body systems, and calculated the low-temperature thermopower in high magnetic field. They essentially confirmed the statement made by Opsal, Thaler, and Bass that electron-phonon mass enhancement is 'V. 3 II) (I. .51: 5“." .”A '1‘: ea. UV)- 6. ‘d Q. ~- 211 present in the adiabatic thermopower in high magnetic field and that this enhancement is (1 + A). (1 .3)PRESENT THESIS: As noted in the introduction, this dissertation has two parts: 1. Construction of a general purpose system for measuring high-field transport coefficients using a SQUID at ultra-low temperature; and 2. Use of this system for measuring the off-diagonal diffusion component of the thermoelectric coefficient of Al below 1K. As also noted, this coefficient is expected to be subject to many body renormalization involving the electron-phonon mass enhancement. (1.3. 1) The Construction Problem: The main construction problem involved vibration isolation in a dilution refrigerator. To illustrate the problem, we use a simple examp l e . In the presence of a magnetic field, one of the most intensive sources of noise is due to mechanical vibration of the sample leads. With the refrigeration capacity of our refrigerator, a sample of thickl'iess a few tenths of a m, and a magnetic field of 10 k0 (1 T). we would expect a Nernst-Ettingshausen voltage across the sample of about 10-11V. Assume that the voltage leads and the sample make a 1009 Which has an effective change in area of 1 mm2 per vibration 25 and which vibrates with a mechanical frequency of 10 Hz . Then the noise voltage generated in the loop would be (103.11) _ VN - dt — 2Nf B A Cos (2wft). (2.1) 5 6 With the above assumptions, V ~ 6 x 10- volts, about 6 x 10 N times bigger than the assumed signal. This model illustrates the importance of vibration isolation of the cryostat, and explains the need frn~ the multiple-stage vibration isolation described in Chapt. II. Thermometry in high magnetic field at low temperature was another important subject in this study. For Righi-Leduc coefficient measurements, transverse temperature differences had to be measured very precisely, and part of this dissertation was to make and calibrate thermometers with high resolution at‘very low temperature in the presence of high magnetic field. The construction techniques and calibration of the thermometers are also described in Chapt. II. (1.3.2)Thermoelectric Measurements on Aluminum: For the following reasons we chose an Al foil for our experiments. 26 1. The most accurate measurements of anxy are possible on uncompensated metals like Al, as noted above. 2. Al has a large electron-phonon mass enhancement, A~0.M5. 3. Al has a simple, well understood electronic structure with a nearly spherical Fermi surface, and its Debye temperature is sufficiently high (about N30K) so that phonon-drag effects are manageably small. Ix.is not reactive, so that it is easy to work with. A. Previous measurements on Al extended down to only 1.8K. It is therefore of interest to go to lower temperatures to reduce uncertainties of extrapolation and to make sure that no unexpected behavior occurs. In this thesis, we extended measurements down to a refrigerator temperature of 0.1117K. From Fig. 1.2, we see that extrapolation to T = OK from such a low temperature, should introduce an uncertainty of much less than 1%. As noted above, in addition to measuring the Nernst- Ettingshausen coefficient, NE, we also measured the Hall Coefficient, R and the Righi-Leduc Coefficient, RL’ as cross- H’ checks on the sample thickness and the amount of heat flowing through the middle of the thin sample. These measurements helped us to reduce the uncertainties in the quantity of ultimate interest, 6" . YX The remainder of the thesis is organized as follows: 27 Chapter II provides a description of the experimental techniques. Chapter III provides the theoretical background for the calculathmucfi‘the high field limits of the Nernst-Ettingshausen (NE), Hall (RH) and Righi-Leduc (RL) coefficients. Chapter IV furnishes the experimental data and our conclusions R about the low temperature, high magnetic field NE, RH’ L coefficients of Al. CHAPTER II EXPERIMENTAL TECHNIQUE (2.1)Introduction: Since the dilution refrigerator, current comparator, SQUID, and other pieces of equipment have already been described elsewhere (Ref. 35, 36), in this chapter only some modifications are described in detail. After brief remarks concerning the refrigerator, bridge, and the SQUID, different techniques for vibration isolation of the system are described, especially the centering device . Then the low temperature high magnetic field thermometry is briefly reviewed, and the thermometer preparation techniques and calibration are discussed. Sample preparation and the sample contact techniques are next. The superconducting magnet and its connections to the cryos- tat are>described. Finally, the reference resistor is discussed, and the method of using the SQUID in a feedback loop is described. (2.2)Dilution Refrigerator: To obtain temperatures on the order of .1K continuously, a locally built dilution refrigerator was used. The details of this dilution refrigerator are described in Chi-Wai Lee's dissertation (Ref. 36). 'The refrigerator was capable of giving temperatures ranging from n.2x to 60mK. First the system was cooledckmn to gnu-4 "-5"- 3A... ,4. ‘— [l‘ n') ‘ ‘ a“. 'L‘. ‘- 29 liquhirfitrogen temperature (~77.NK) in about 12 hours using He exchange gas. Further cooling to liquid helium temperature (14.2K) was done by transferring liquid helium into the inner dewar (Figure 2.1).. Cooling to 1.3K was obtained by pumping on the liquid helium in a 1K pot. From this temperature the dilution refrigerator reached the lowest temperature (~60mK at the mixing chamber) after a few hours of circulation ofthe 3He/uHe mixture. ‘The power of the dilution refrigerator at very low temperatures was very important, because for NE measurements a thermal current is sent through the sample and the refrigerator must be able to absorb this heat at the temperature of interest. As the temperature decreases, the cooling power of the dilution refrigerator decreases also. At 150mK the cooling power of the dilution refrigerator is 100 uWatts; at 100mK, the power ‘40 Watts. At the lowest tempera- ture (60 mK), the cooling power drops to zero. As discussed below, the cooling power of the refrigerator plays an important role in determination of the sample thickness and the lowest temperature at which accurate NE measurements can be made. 30 MIXING CHAMBER ”A (L. MAW/WM); .1— 7“- L I Q U l D He J ".9033 a i Mggng' 1 SAMPLE ' 1 Sample CI" A) L Vac. Can Figure (2.1)Diagram of the Sample, inside the superconducting Magnet, with the connections between the mixing chamber and the vacuum can through the centering device. A - Centering Device. B - Magnet Support. C - Teflon rings for attachment of the magnet to the vacuum can. D - Vacuum Can. E - Sample Can. F - Sample Support. G - Magnet. H - Sample Can. I - Vacuum Can. J - Teflon rings for attachment of the magnet to the vacuum. L - Liquid He Dewar. M - Teflon ring for attachment of the sample can to thermixing chamber. Figure (2.1)Diagram of the Sample, inside the superconducting Magnet,vnth the connections between the mixing chamber and the vacuum can through the centering device. 31 (2.3)High Precision (0.1 PPM) Resistance Bridge: The resistance bridge consisted of a commercial direct current comparator (Ref. 37) and a SQUID (Superconducting Quantum Interference Device) (Ref. 36) which was used as a sensitive null- detector. The current comparator could read the ratio of two currents (Slave and Master) with a precision of better than 0.1 PPM using a set of eight decade-dials and the technique of averaging beyond the last dial. (2.“)SQUID: The SQUID was used as a high precision null-detector. It provides a very high current sensitivity, limited only by the ther- mal Johnson (voltage) noise in the SQUID circuit resistance R: Johnson Noise = VAkBTRAf (2.1) where kB is the Boltzmann constant and Af is the band width. The basic principle of the SQUID is based on the Josephson effect (Ref. 1). The SQUID used in our system was a symmetric point contact r.f. tuased device purchased from SHE Corp. (SHE Model RMPC with SHE Model 330 electronics). -. -i 32 (2.5)Vibration Isolation of the Cryostat: The first level of vibration isolation of the cryostat was made by using flexible bellows on pumping lines and an air mount for the cryostat. These are described in the thesis of Chi Wai Lee (Ref. 36). In addition, for standard measurements a 11 metal shield was wrapped around the whole cryostat to isolate it from the earths magnetic field which is ~0.5 G. In this study, where a magnetic field of 10-20 kG was to be applied, the noise due to vibration in the magnetic field could be very large, as explained in the Introduction. Therefore the vibra- tion of the system had to be minimized. This was done by reducing the relative vibration of the sample with respect to the supercon- ducting magnet in two steps. First, the magnet was rigidly attached to the vacuum can by two flanges at the ends of the magnet as described in the next section. Second, a centering device was used to produce a strong mechanical coupling between the sample and the vacuum can (Figure 2.1), together with poor thermal conduction. (2.5.1)Magnet Supports: The superconducting magnet was attached to the top of the vacuum can by three long 1/11" threaded brass bars, located inside the Helium bath. The magnet was rigidly attached to the vacuum can by two teflon rings, Figure (2.1). At room temperature the vacuum can had enough clearance to fit into the teflon rings, but as the .. 33 cryostat was cooled down, the teflon flanges shrank and caused the magnet to be rigidly attached to the vacuum can. (Since there should be no weight hanging on the vacuum can, the magnet support bars were made out of brass, such that in the process (sf cooling down the cryostat, the higher thermal contraction of the brass bars relative to the vacuum can (the top part of the vacuum can was made of stainless steel), causing the magnet to push up on the vacuum can. This pressure on the vacuum can was helpful in order to keep its connections leak tight. (2.5.2)Centering Device: Minimizing mechanical vibration, while providing accurate centering with very low heat loss, is done by clamping the mixing chamber of the dilution refrigerator into the surrounding H.2K vacuum can with a reentrant spider (Ref. 39) as shown in Figure (2.2). The thermal isolation of the system was done in tuna different stages. A 1" x 1/16" copper band was located between the center clamp Unmthe top of the mixing chamber) and the external brass cylinder» Twiis copper band was located between the two stages of thermal isolation Figure (2.3) and was thermally lagged to the continuous heat exchanger, which operated around 1.2K. The thermal isolation between the stages was provided by using Vespel (Sp-22), 311 purchased from DuPont. Vespel has a very low thermal conductivity. The heat flow was calculated by the general expression: é= -K (T) A (%) (2.2) 35 A G B H ck D— _ I / r J E F L.‘ Figure (2.2)Cross section view of the centering device. A-Brass Band. B-Copper Band Heat Exchanger. C-The Upper Side of the Mixing Chamber. D-Vespel Cylinder (stage 1 of the thermal isolation). E- Vespel Cylinder (stage 2 of the thermal isolation). F-Stainless Steel Rod. G-Adjusting Screw. H-Locking Nut. I,J-Attaching Screws. 1 36 where K(T) is the thermal conductivity, and A is the average cross section of the media where g;- is measured. If K (T) ~ Tn, and the heat influx is at the ends only, then the above equation may be simplified over the temperature range studied to: d: 1--'I3> [K(Th)/(n + 1)] Th (2.3) Th is the temperature of the heat path at the hot end, 1 is the length of the heat path, and n is a constant which depends on the material. Equation (2.3) was used to estimate the heat flows in the system which is described next. a)Stage 1. The thermal isolation between the mixing chamber and the 1.2K copper band heat exchanger: Three Vespel tubes were attached to the copper band (1.2K) from one end and fitted on to the three Vespel studs from the other end. The three Vespel studs were fixed on the mixing chamber clamp by three brass screws. The thickness of the Vespel tubes in this heat path was first estimated using Equation (2.3), and then corrected experimentally so that the noise of the vibration of the system was low enough to provide good voltage sensitivity for NE measurements. -1 Using equation (2,3) where T =1.2, Tc - 0.1K, K = 21-1 W K—1cm , h the tube radius R = 1/14", the tube thickness AR = 1/16", the length of the tube 1 = 1" and for Vespel n - 2. gives, 0(1) ~ 6 11 Watts. This is low enough to allow temperatures below 0.1K to be reached. 37 b)Stage 2, The thermal isolation between the copper band (1.2K heat exchanger) and the surrounding vacuum can (H.2K): Three Vespel tubes were inserted into the 3/8" holes on the copper band as shown on Figure (2.2). Then three 6-32 stainless steel screws were fitted inside the Vespel tubes so that they could be adjusted for centering the mixing chamber. For this adjustment three holes on the copper band provided access to the screws and their locking nuts. To fix the brass cylinder to the surrounding vacuum can, three screws with spherical shaped ends were attached to the brass cylinder. These screws could be tightened from inside by removing the lower part of the vacuum can. 'Ra allow access to the centering device from inside, the vacuum can was made out of two pieces. These two pieces were attached together just underneath the centering device. (2.6)Thermometry in High Magnetic Fields at Low Temperatures: Thermometry, even at zero field, becomes increasingly difficult at lower temperatures due to poor thermal conductivity of materials, thermal boundary resistance, subsequent long thermal relaxatnm1 times, and thermal gradients. The application of high magnetic fields adds to the complication because of the magnetic field de- pendence of the thermometers. In this thesis, high precision 38 To;1.2K O 6 - . 2..... Q{Q1 1» 5 2'. 4.2K . Q . e a, 1} ~100mN 2 9 9 9 A Vac.Can - A Capper gluing STAGE;2 3“ d STAG-6:1 ' ambe' 0230001116)I Figure (2.3)Block diagram of the two stages of thermal isolation of the centering device. Stage 1-The thermal isolation between the mixing chamber and the copper band connected to the 1.2K heat exchanger. Stage 2-The thermal insulation between the copper band and the surrounding vacuum can. .i‘ 1:.- tr .A 39 thermometry below 1K in the presence of high magnetic fields was reviewed and different kinds of thermometers such as capacitive thermometers, germanium, carbon and Speer carbon resistance ther- mometers were studied. Finally the Speer carbon resistance thermometers were chosen, calibrated in the presence of the magnetic field, and used for thermometry at low temperatures and high mag- netic field. In this sectnnn different kinds of thermometers are briefly reviewed, and then the Speer carbon thermometers are described in detail. (2.6.1)Different Thermometers: Capacitive thermometers are useful as transfer standards where magnetocapacitive effects are not a problem (Ref. N0, N1). But, for the following reasons they were not suitable for thermometry in this study. i)-their capacitance drifts slowly with time and oc- casionally shifts in value from run to run. 11)-the insensitive region of these capacitors is between 1K and 0.1K. (Ref. N2). Germanium resistance thermometers are very sensitive and stable thermometers for a wide range of temperatures, especially for tem- peratures below 1K. These thermometers are typically reproducable to better than .0005K at liquid Helium temperature when cycled to room temperature (Ref. N3). But, the problem of using germanium thermometers in our study was their magnetic field dependence. N0 Since in this study the thermometry is in the presence of a high magnetic field, the magnetic dependence of these thermometers was a fundamental problem. Different techniques for shielding these thermometers from the magnetic field were studied such as, (A)- superconductive shields and (B)-placing the thermometers out of the magnetic field with a thermal line connection to the sample. In technique (A), because of the non-uniform magnetic field produced by the magnetic shields around the sample and in technique (B), because of the large heat capacity of the thermal line which causes a very large time constant for thermal equilibrium, neither technique was practical for the primary thermometers. But the above techniques were used for thermometer calibration as described below. Carbon (Ref. NN, N5), carbon-glass composition (Ref N6), and Allen-Bradley (Ref N7) resistors are often used as thermometers in high magnetic fields above 1K. But these resistors are not suitable for thermometry below 1K (for Allen-Bradley below .SK), because of their very high resistances at lower temperatures. Speer Carbon resistors (Ref. N8, N9) were found to be the most appropriate sensors to be used as low temperature high magnetic field thermometers in this study. Matsushita carbon resistance thermometers (Ref. 50, 51) also seemed potentially suitable as thermometers. But, we could not find information, either in the U.S. or Japan, about the manufacturing company. Thus, Speer resis- tors were used as our thermometers. Speer resistors at zero -\~ .15 N1 magnetic field have a temperature dependence in the form of (Ref. N9) 1 /N R = R0 exp (AT_ /N) = R0 exp (AB1 ) (2.N) where R0 and A are constants and B = 71f“ The magnetic field dependence of the Speer resistors is defined 2% g [R(T,H) - R(T,0)]/R(T,0). In Ref. N9, it is shown that 9% as: is a linear function of log (H/T), i.e. IAEI ~ 108(H/T) (2.5) where the magnetoresistance is negative (i.e., application of a field decreases R at fixed temperature). From Equation 2.5, we see that each factor of 2 increase in H/T increments AB by a fixed R amount. For example, IAR/R| = 11% at H/T = 20 kG/K (H = 20 kG, T = 1K). 35%| increases to 8% at £13- - N0kG/K (H = 110 kG, T =1x or H = 20x0 T = .sx or ....), H and 9% increases to 12% at T = 80 kG/K (H = 20kg, T = ,.25K or H = 80KB, T=1K0r 0.00). (2 . 6 .2 )Thermometer Preparation: The thermometers were made from Speer carbon resistors which were purchased from the Speer carbon company. W.C. Black et a1 N2 (Ref. N8) found, that Speer resistors of grade 1002 in various nominal resistance values from 100 - 500 0 are the most suitable for very Ix»: temperature work. For this reason all of our thermometers were made from 100 0 1/N W speer carbon resistors. First the Speer resistors were cut in forms of discs with heights of about 3mm. Then one side of the discs were flattend so that the graphite appeared, Figure (2.N). (This is the side of the thermometer which sits on the sample, and it was made flat for better thermal connection with the sample.) 'Then the flat side was covered with cigarette paper which was soaked in GE 7031 varnish. The thermometer leads were superconducting NbTi (0.00N" diam) wires which were attached to the two sides of the Speer disk with silver paint. After a few hours, when the silver paint was completely dry, the thermometers were baked at 200°C for about ten hours. The temperature and the time of baking complete the diffusion process of the silver paint hfix>the Speer. (If the diffusion of the silver paint into the Speer is not complete, the resistance of the Speer thermometer changes as a function of time from run to run. 131 such a case the time or the temperature of baking must be increased). After baking, the thermometers were painted with Duco cement and then a little silicon glue was used on the leads at the ther- mometer ends to provide strength with flexibility. The leads Figure (2.N)Speer Carbon Thermometer: A-Speer Carbon Resistor Body. B-Speer Graphite. C-Superconducting NbTi Leads. D-Copper Clad is removed from the superconducting wire. E-Silver paint for electrical contacts between the leads and the speer carbon thermometer. F-Superconducting wire with the electrical isolation removed from it. NN were twisted together exactly the same way as the sample leads, and then were wetted with Apiezion N-Grease for better thermal conduc- tion to the cold line. Finally the Speer thermometers were mounted on the sample limbs with the help of a non-magnetic spring and some N-Grease (Fig 2.5), and its leads were glued to the sample support with GE 7031 varnish. As is shown in Figure (2.N), the superconducting wires have a copper clad. This copper clad was etched away over a short length (~3mm) , for better thermal isolation of the thermometers from the environment. Overall, about ten thermometers were made using this technique. Four of these were used for measuring the transverse and the longitudinal temperature gradients. These thermometers, desig- T T nated T T were mounted on the sample limbs. right’ left’ hot’ cold Two other themometers, used for calibrating the above four ther- mometers, were thermally connected together with an annealed pure polycrystalline silver wire. One of the calibration thermometers was mounted underneath the sample (in the magnetic field), while the other was mounted out of the magnetic field, by the mixing chamber. The Speer thermometer connected to the mixing chamber (Tmix) was isolated from the magnetic field at the mixing chamber with a NbTi box. N5 I: Figure (2.5)Sample Holder: A-Maine copper support. B-Pb shield for sample voltage wires. C-Heater leads. D-Heater. E- Heater stand. F-Aluminum sample. G-Vespel substrate. H- Thermometer. I- Thermometer holding-spring. J-Current leads to the sample. N6 (2.6.3)Thermometer Calibration: A germanium resistance thermometer, called GRTZ, was used as a reference thermometer for calibration of the Speer thermometers. GRT2 itself was calibrated in three steps. The details of the three step calibration are given in the Ph.D. thesis of Z.Z. Yu (Ref. 35). CRT, was mounted by the mixing chamber, inside a Niobium box which isolated GRT2 from the magnetic field. GRT2 was thermally anchored to the two Speer calibration thermometers, and the calibra- tion in the magnetic field was done in two stages as follows: First the Speer thermometer by the mixing chamber (Tmix) and the Speer thermometer by the sample (T3) were calibrated against GRT2 at zero magnetic field. Then the magnetic field was turned on and T}31was calibrated against Tmix at different magnetic fields (TS is in the magnetic field, while Tm is at zero field). After this ix calibration, T8 was our reference thermometer in the magnetic field, T T and T were calibrated L' C H against Ts at different magnetic fields,Figure (2.5.b). and all the other thermometers TR, For rough measureurets, the magnetic field dependence of the ther- mometers were fitted to equation (2.5). For very accurate measurements of T and T each thermometer was calibrated as fol- R L’ lows for every temperature reading at a given magnetic field. 117 D =15 (1(6) 3.54- o H= 20(kG) A H: 25 (k6) ILSO” 3u46" A 3042- e a E —s I“ 338- :L34" ' I l l l l I I l I l l l l 3.3970 .72 .74 .76 .78 .80 .82 T01) Figure (2.5.b) Admitance of To as a Function of Temperature in Magnetic Fields; 15, 20, 25, kG. N8 First, the magnetic field was ramped up to 80' Then the ther- mal current Qx was ramped up and the NE voltage (the transverse voltage) was measured. The conductances of TR and TL were measured with the SHE conductance bridge (Model PCB). Then for the calibra- tion of TR and TL’ QX was ramped down to zero while the magnetic field B was held constant. Consequently the temperature gradient (in all directions x and y of the sample) dropped to zero and, TR=TL=TC=TH=TS=TMIX=GRT2 For calibration of, for example T some heat was sent into the R! mixing chamber and the temperature of the whole sample was raised along with all of the thermometers until the conductance bridge read the same value for TR as was measured before (on RL measurements when Qx was flowing into the sample). Since all the thermometers were at the same temperature, T was calibrated directly from GRTz. R For the calibration of TL or TC exactly the same technique was used, but, obviously, different amounts of heat were needed to bring each thermometer to its appropriate temperature. (2.6.N)Temperature Regulation: SHE model PCB conductance bridges were used to measure the conductance of the thermometers in the N-terminal configuration with an accuracy of better than .5%. Using low excitation voltage (10 to 100 W), eliminated the problem of self heating. 100W excitation N9 Vtfltage could be used only at temperatures above .5K, while 10uV excitation voltage gave 1% accuracy with no self heating problems. The conductance bridges were self-balancing, so that in addition to their use in temperature measurements, their differential outputs were also used for temperature regulation(Ref.36). Because of their low noise, the bridges were ideal for use with a temperature con- troller to regulate the mixing chamber temperature. The temperature controller (Ref.36) had the usual differential and integral controls with adjustable time constants and proportional controls. Twelve output power levels were available for better temperature regulation at different temperature ranges. For regulating the temperature, T0 was used as the reference thermometer, and the mixing chamber heater was used for controlling the temperature. Using the conductance bridge, we could control the temperature with resolution of better A than 10- K. This resolution was required for RL measurements. (2.7)SAMPLE (2.7.1)Sample Size: The length and the width of the sample were limited by the dimensions of the superconducting magnet. The magnetic field was directed along the cylindrical axis of the magnet. To measure transverse Magneto-transport effects, the field had to be perpen- dicular to the sample. Since the inner diameter of the magnet was about 2", the length of the sample was limited to about 1-1/2". The widtr113f the sample was ~1/8" and the sample had six limbs, two for transverse and four for longitudinal measurements, (Fig. 2.5). 50 TTua thickness of the sample was also limited. Consider equa- tion (1.15) where Pa has its high magnetic field limit, and Ux = Qx/(W.t). The NE transverse voltage is; V=E °W=P°U°W=——-—-—- (2.8) Since Pa is a constant, and the heat flow Qx is limited by the power of the dilution refrigerator, Vy is proportional to 1/t--i.e. the thinner the sample, the larger Vy. On the other had, having a very thin sample brings problems. These include the difficulty of precise measurement of its thick- ness, the problems. associated with the handling a very thin foil vdJfli poor mechanical strenth, and the uncertainty in the magnitude of the heat flow density at the center of the sample due to heat flowing through the substrate. To ensure that the NE transverse voltage Vy was at least 10‘2 times the noise level in the SQUID circuit ~1O-13V, we needed to use a (2 x 10-3)" thick Aluminum sample. 51 (2.7.2)Sample Preparation: The Aluminum sample was purchased in form of a 1" wide and 3.. 2 x 10— thick Aluminum foil from Cominco American. We measured the RRR of this pure Aluminum foil and it was; RRR _ 9(300K) ‘ ‘p(‘—“u.21<) “ 230° The sample was cut with a spark cutter. The sample cutter was made out of Brass (Fig. 2.6). It had six limbs for transverse and longitudinal voltage and temperature measurements, and two wide ends for thermal and electrical current inputs. The limbs were elliptical pads (about 1/8" x 3/6N") which were connected to the sample by 1/32" wide and 1/16" long rectangular bridges (Fig. 2.6). The Aluminum foil was spark out under paraffin oil, while it was sandwiched between two thick Alumhumaplates. Without the Aluminum plates for holding the thin sample flat, the edges of the sample did not come out smooth. (2.7.3)Annealing: The sample was annealed at N20°C for twelve hours and the temperature was then gradually reduced to room temperatureeixi about five hours. During the annealing process, the sample was placed between two very clean pieces of Alumina to keep the sample flat. 52 --*1 1/8.1¢— 1 1 11:15? 5? fir. 1/8 1/2‘1/4" 1 , . _ 4 l .- . l’=1/16 1- 1425 was» 1 ,_ 64 , F _,_ F - .1. T E! 1 12 ' ' o l _’;1 if I 1 1 ' ' ”3'2?— l 1 I 1 ' 1 ' 1 ' | e- 1 L511." ~—43/1.‘ L—a/s' ——.' mal—.1 3/131«——snb—+1 1 1 1 1 I 1 ' I 1 1 1 I 1 1 1 .--. 1 . 7 I I T INBZ 1 1 1 1 1 1 1 _ __ ‘2 1 1 i :_ l 4 .r 1 , e » ‘ 3/3 / 4/», ‘ // Figure (2.6)Brass sample cutter cross section. 53 (2.7.N)Sample Connections: The sample was connected to the cryostat electrically, ther- mally, ENTd mechanically through three independent parallel lines as follows: a)Electrical Connection: Three pairs of NbTi (d = 0.00N") superconducting wires were connected to the sample; two for sending electrical current into the sample and four for measuring longitudinal and transverse voltages across the sample. Since the NE measurements were made through the two middle transverse leads, using the SQUID, this pair of NbTi wires was twisted together very carefully and then fed through a Pb tube to shield the wires from the magnetic field. It was essential that the electrical contacts of this pair of wires to the sample have very low contact resistance. Five paimws of NbTi (d = 0.00N") wires were also used for electrical connection of the thermometers TH, TC, TL’ TR' and TS to the conductance bridges. b)Thermal connection: The sample was thermally connected to the dilution refrigerator by a pure annealed polycrystalline wire of silver. For better 5N thermal. + + E(k) = Eo(k) + £(E(k) , k) (3.1) where 80(7) is the unperturbed band energy of the electron and Z(E(k’),k>) is the electron self energy due to electron-phonon interactions. We now consider the transport properties of these quasi- particles. The transport properties of quasi-particles which at low temperature are elastically scattered by a random distribution of impurities may be described in terms Of a mean free path I(R). This mean free path is related to the renormalized relaxation time T(k’), by I0?) = 1702’) - 10?) (3.2) where 170:) is the velocity of a quasi-particle in state E, and is defined as: 71—30?) (3.3) 71:11?) (3.11) From equation (3.1) we see that, 32+ 17 1502’) + -3—2 (3.5) 3E(k) 71:11?) = 7K1: (12’) + k 0 k OK or by rearrangement of terms and multiplying by %: 1 + + 1 32 — V E (k) - - 1 + + 'h k 0 h 3k — V E(k) = + (3.6) h k (1 _ 329 ) (1 _ 82+ ) 3E(k) 8E(k) Substituting equations (3.3) and (3.N)into eg (3.6) and defining the mass enhancement parameter A as 32113112) , 1?) 111?) = - , (3.7) 3E(k) We get + + 1 32 -> + V (k) ..— 11(1<) =—-—9——-+“3" (3.8) 1 + 11%) 1 + M1?) 78 1 3£(E(k),k) The term E 3k 18 defined as 1 82(E(k) . k) 3+ A KI v The quantity £(E(k), R) varies very slowly with respect to k’, and 130:) is much smaller than 70(R) (Ref. 55). and in most cases can be neglected. However, R(R) varies much more rapidly with energy near the Fermi energy than does 7011?) (ref. 55). Since, the diffusion thermopower depends upon energy derivatives (see the Mott Rule later 1J1 this chapter), R(E) can make a significant contribution to the thermopower as first pointed out by Lyo (Ref. 55). Since for a metal A(E) is positive at the Fermi energy BF (Ref. 56), the quasi-particles, as we expected from the elementary picture of the electron-phonon interaction, have lower velocities than non-interacting electrons. The density of states in the band is given by the general form of, OS 1 113 |$E(§)| N(E) = f (3.10) S(E) Where the integral is over the constant energy surface S(E). 7+E(§) is proportional to V(§) (eq. 3.3), and the Fermi Surface k is not affected by mass enhancement (Ref. 56). Thus, if A02) is independent Of R, then the electronic density Of states at Ef is 79 O NC(EF) = (1 + A) NC (E ) (3.11) F where NC(E) and NCO(E) are the enhanced and unenhanced electronic density Of states. The electronic density Of states as a function of energy is shown in Figure (3.1). In terms of the dispersion relation, this enhancement has the effect of producing a different slope in the vicinity of EF as shown in Figure (3.2). This slope is equal to g—E, and if we define m*, from equation (3.3) as (57) 1 + + h E 561(5)); = v<1<) E = F (3.12) F F then the slope at Ef is BE 1 + 5? BF 5; V(kF) — VO(kF)/(1+A) (3.13) which is (1 + A) times smaller than the slope for the non- interacting electron model. Using the same definition as above for the non-interacting electron model BE 1 EE— - ~ V (k ) (3.1N) F 7 m o F 80 N(E) .2 I: .1- v p-_-———_—--___-————_—_—_._ Figure (3.1)Electronic density Of states as a function of energy, where Nc(E) and Nc°(E) are the enhanced and unenhanced electronic density Of states. r- 81 Figure (3.2)The dispersion relation, E(k) for enhanced and E,(k) for unenhanced electrons. The enhancement has the effect of producing a different slope in the vicinity of Ef. 82 We conclude that m* =m(1+A) (3.15) Qualitatively, as noted above, one can think of the electron having to drag along a phonon cloud, which thereby slows its motion down and increases its mass. If, in addition, we think of the mean free path for scattering by impurities as ‘roughly' the distance between impurities, then the mean free path will be uneffected by the fact that tnua electron is slowed down; i.e. in?) - i’ (12’) (3.16) Then since 1 = VI, we must have .0?) = 1002’) (1 + 1) (3.17) Here, TH?) and IN?) are the perturbed (renormalized) and ICU?) and 10(7) are the unperturbed relaxation time and the mean free path between the scatterings, respectively. Finally, as mentioned in the introduction the electronic specific heat is linearly proportional to the electronic density of states [eq 1.8)], and the specific heat has been shown (Ref. 6) to be enhanced by the same factor as the density Of states. With this background, we now turn to the fundamental transport equations and define the proper coefficients for compensated and 83 umcompensated metals in order to look for the electron-phonon mass enhancement. (3.2)Fundamental Transport Theory: The electrical current density and the heat current density are defined as + + + J = fe v - f(k) - dk (3.18) + + + U = f(E-u) - V - f(k) - dk (3.19) where f(R) is the electron distribution function and u is the chemical potential. The electron distribution function f = f(Q) can be evaluated by solving the Boltzmann Transport Equatnn1(BTE), which in metals can be written as (Ref. 56) ; + + + 3f k ' Vk f V VP f - (5E)scatt (3.20) Here-ég) , is the rate of change Of the distribution function at scatt due to scattering, and R is the partial derivative of E with respect to time. Lf we take the solution f(k) of Eqn. (3.20) which is linear in E and VT, then by substituting this solution into equations (3.18) and (3.19) we Obtain the macroscopic transport equations. These 8N equations are written in two different forms with different tensor coefficients. Since in this study we are dealing with tensor coefficients from both forms of the transport equation, we explain each briefly. The first form of the transport equations were given in the introduction: 3=3“’-E+23- (1171) (3.21) + ++ + 4- -> U = e . E + A - (-VT) (3.22) where from the Onsager relations (Ref. 2) ..., e = T . E“. (3.23) The second form of transport equations are -> + E = p ' J + S - VT (3.2N) + 4n) + 4..) + U = n - J - A" . VT (3.25) p = The electrical resistivity tensor The thermopower tensor T = The Peltier tensor A" = The thermal conductivity tensor 85 «+ ++ Where from the Onsager relations fl = T S, and ++ C" 1+ 0 ++ ++ =A"+T (3.26) All Of the above tensor coefficients are functions Of temperature and magnetic field. We note that with either form of the transport equations, four coefficients would normally be needed to relate 3 and U to E and VT. However the Onsager relations reduce the number of independent coefficient tensors to three. The coefficient tensors of the two forms Of the transport equation are directly related together by ++ ++ ++ 0 ° 0 = 1 = Unit tensor (3.27) ++ ++ ++ 69 1 ++ e" = 0 - s or s = :3 . e" (3.28) 0 Now from the transport equation we return to the BTE for the ++ ++ ++ ++ evaluation Of the tensor coefficients a , p , E" and S . In a cubic metal with the magnetic field along the z-directior1 and also along a 3-fold symmetry direction, (Ref. 8), the tensor 0+ is - o o O 1 xx xy H o (3 29) 0 = -a 0 . YX YY O O 0 - zz. a is the ith, jth element of the conductivity tensor 0+. 13 86 ++ -> The tensor coefficient 0, which has a similar form as 3 , can be derived in terms Of 3+ tensor elements for a cubic metal as follows: First write, F pxx 0xy 0 . H (3 30) = - O . p pYX pYY . O O 922 Then from equation (3.20) with the boundary conditions Jy = J z = O and VT = O, we get 3 E+E (1) x - 0xx x Oxy y 3'3 a 3 13+): (10) y - oxy x cxx y - O 3.3 .p Jz = ozzEz = O (3.31c) Combixnhng equations (3.31a) and (3.31b), pxx and pyx can be deduced in terms of 0 and o as follows: xx xy Ey oxy 0xx 8 3_ 3 2 2 (3'32) x 0 + 0 xx xy E:x 0xx x o + o 87 These two equations show that in order tO determine the behaviors of pxx and pxy’ in general we need to know the behaviors of both a and o . xy xx The DC conductivity of a metal is simplified by Ashcroft and. Mermin (Ref. 2) as follows dk 3f 2 + + + + 0(8) - e T(E) f 1;? 5E V(k)V(k) (3.3N) where the integral is over the Fermi surface in k-space. The tensor coefficient 23 is related tO 3+ by the Mott Rule (Ref. 2) +9 d++(E) 0 '== ——_— E' eLOTE dE JE=EF (3.35) Then from equations (3.22) and (3.25) we get ++ 8L T ++ ++ e" 0 do 3 ' :2: = '7.— [53333.13 (3'36) 0 0 F Now that we have a general idea about the transport equations and the tensor coefficients, we focus our attention on the high magnetic field limit of such coefficients, and concentrate upon the Off diagonal component of the thermoelectric tensor 3 and, also upon the Nernst-Ettingshausen (NE), Righi-Leduc (RL) and Hall ‘1 an! 88 (RH) coefficients for compensated and uncompensated metals as defined in the introduction. (3.3)Transport in High Magnetic Field: We now show that for analysis of transport properties in magnetic field, the new variables E = the energy of the electron R = the component Of its crystal momentum parallel to the magnetic field 0 = an angular variable whose time derivative is the cyclotron frequency are more appropriate than the variables Rx, R , R2. After Y explaining the new variables, the Boltzmann Transport Equation is written in a new form in which f(kx,ky,kz) + 111(E,kz,¢). Then, in the high field limit by expanding the magnetic field dependence of w in a power series in %, (here we use the symbol H for magnetic field, so as to follow the notation in the references for the +9 ++ ++ following analysis) we will find for each tensor element 0 , p , e", the leading (non-zero) term for a metal with a given structure: compensated or uncompensated; Open or closed orbits. The semi-classical equation Of motion of a wave packet under an applied Electric and Magnetic field (Ref. 56) is n-—-= eE1-§ 07x R) GL3?) If the electric field E = o, and the magnetic field is in the z-direction then, + R -(h<“D =119— -> 9 e + + 2 E? (H . k)= 311- 01x H:)= o CL38) From 2_ (Hz - E) dt 0 we conccude that the 2 component Of Q constant). should be constant (Q2 Also for the energy E we can write 92.83.95 (3.39) dt k dt using equation (3.37), at zero electric field, we will find that the energy is conserved (E = constant). 'ETmmlE = constant and Ez = constant we reach the conclusion that in a magnetic field only ( i.e. no electric field), the electron moves in k-space along an orbit for which energy is constant at the Fermi energy, and the component Of 1: parallel to H is also constant. Of course when the electric field is not zero, the energy E and E2 are not precisely conserved any more, butizz will be a function only'cxf the electric field, and the energy will change little 90 9 compared to E IIIthiS case, kz will still be appropriate as one F. of the parameters, especially in the high-field limit when the second term on the right-hand-side of equation (3.37) is much larger than the first term. In such a case, the electronic trajecmmwr remains within a distance k T from the Fermi energy and can be B approximated as lying on EF. From equation (3.38) we also reach the conclusions that dkx e .....dt = (REWy Hz (3.110) .) dky e .EE_ = -(HE)VX Hz (3.N1) ‘The magnitIMe Of g; (s is the differential path length in the kx,ky plane) is then given by ds _ dkx 2 dky 2 1/2 _ _g 2 2 1/2 _ e a? 7 [(dt ) + (dt ) J -‘hc Hzwx + vy ) hc Hz V hc ds dt - (Efi-)‘V- (3.N2) 2 In general, (t), the differential time element for motion along the trajectory is used as the third variable, and 0 is an alternative to it in the specific case when we have a closed orbit. 0 and t are related together by the cyclotron frequency we, as 93.2.1... m (3.113) 91 and the cyclotron frequency is related to the cyclotron mass mC by (3.NN) We now rewrite the BTE equation in terms Of new variables 4), E and Rz under the conditions Of applied electric field E, magnetic field in the z-direction, and zero VT for a closed orbit. Setting 3% = ka = O the BTE becomes, -.> + 3f k ° Vk(f) - (8E)scatt (3.N5) In terms Of the new variables, Eqn. (3.N5) becomes 8f - 3f 0 3f - 3f (36) ¢ + HE E + 5?; k2 = (SE)scatt (3.N6) In the above equation E and k2 are unaffected by Hz because, from equations (3.3) and (3.37) E = Vk(E).E = e V - E (3.N7a) O e -) and kz a 7'1 Ez. (3.1170) However, 6 = we is proportional to H. (3.N7c) Substitute equation (3.N7a,b,c) into (3.N6) 3f 3f =1 (3;) scatt (3.N8) 92 o By change of f + fO BE 0 , we get 3f 3f 3f 8f 0 fl 0 +.+ _ 0 2+ 8111 _ _ O ( as )00 ¢ + (“as ) 13v E +( _—3E .h 1:“:z -——-akz -( _8E )IW) (3.119) Since 0 is linear in E, the third term on the left hand side of equaticnl (3.N9) is quadratic in E and is thus a higher order term. The linearized BTE then becomes, -I(w)=eV-E (3.50) E QJIQ) 06 TO derive the magnetic field dependences of the leading terms in the transport tensor elements for a general metal, we have to solve equation (3.50) in the high field limit. We begin by expanding 01(1 = x,y,z) in a power series in %. (0) (1) 1 2 (2) ...] + (1,911 + (E) 1i + 1 = 81, = 2111 (3.51) l i Note that because the linearity of the BTE, its solutions, 1px, (1y, wz are independent of each other. By substituting equation (3.51) into equation (3.50) and using the fact that 1(0) is a linear Operator, we derive general equations for the 1111“!) which is the coefficient of the term 6%)“ in the 1th component Of w. 93 Using the information we obtain, we get expansions of the transport tensor elements in powers Of %, and find, for each tensor element, the leading (non-zero) coefficient for any metal, either compensated or uncompensated, and for open or closed orbits. These «+4- coefficients determine the behavior of 0 , 0+ and :3 in the high magnetic field limit. From such calculations in the high-field limit we get the following results (Ref. 56) [In the foLknnng pages the ith, jth element of any tensor I’ is defined as xij = x' Oi), where fij(H) is tre magnetic field dependence Of the il'fil ith, jth element, and x'ij is its coefficient.]: The magnetic field dependence of 0+ in the high field limit in a compensated and uncompensated metal with no open orbits in the x-y plane and the magnetic field in z-direction is r01 1 o" l 0" .1.- xx H2 xy H xz H +7} m = I l I .1— 1 .1. ouncomg+ ) 0 yx H 0 yy H2 0 yz H (3.52a) 1 1 I __ I _ 9 _° zx H 0 zy H 9 22 d _ 1 1 1" 0' _ OI _ OI _ xx H2 xy H2 xz H *7 .. - 1 l 1 l 1 l “c0857 ) 0 yx H2 0 yy H2 0 yz H (3.52b) 1 1 I _ I _ I L o zx H 0 zy H 0 22 ‘ TO evaluate the structure of 23(H), since the Mott Rule applies to each element of 0+(H), the magnetic field dependence of E" is exactly the same as that for 0+, with the exception that in a 9N compensated metal the anyx term retains its % power. The magnetic field dependence of E" in the high field limit for a compensated or uncompensated metal with no open orbits in the plane perpendicular ..y to H is thus e' l e' l e' 1 xx H2 xy H xz H +7: m = ' l ' .1. ' .1. Ecomp or(H+ ) E yx H e yy H2 6 yz H (3.53) uncomop 1 1 I _ I _ I e zx H E zy H e zz Since the magnetic field dependences of the resistivity tensor elements for compensated metals are very different from those in ..., uncompensated metals, we list the tensor elements p for compensated and uncompensated metals as well. I I I p xx 9 xy 9 xz H 11+) ' 11 ' ' (511) puncorsip - p yx p yy p yz 3' a I I I L9 zx p zy p zz ' 2 2 " I I I p xxH p xy 9 xz *+ W = -—' 2 I 2 I u p comg+ ) p ny p yyH p yzH (3.5 b) I I I p zx p zy p 22 The Hall coefficient and the Righi-Leduce coefficient are defined by equations (1.19) and (1.20) and they can be written as 9 _ _X£ - 1 RH 7' B '7 0 .B (3056 yx _ (aT/BY) _ 1 RL — -5—7§—— - IF—_7§ (3.57) x yx In the high field limit, the Off-diagonal components of the ++ 4") electrical and thermal conductivity tensors 0 and A" should Obey the Wiedeman-Franz law at all temperatures, independent of the nature of the scattering integral (Ref. 2) . T (3.55) By applying the Weideman-Franz law to the Hall coefficient, the Righi-Leduc coefficient RLcan be evaluated. As described in the introducticwu R provides a means for evaluating the value Of Qx at L the center Of the sample, where the NE coefficient is measured. With this general background, we now turn to the high-field limit of the NE coefficient, which allows Enyx to be extracted directly. (3.N)The High Magnetic Field Nernst-Ettingshausen Coefficient: First we derive the high-field limit Of the NE coefficient for a compensated metal. Since for an uncompensated metal the derivation Of the high-field limit is similar to that for a compensated metal, we just write the result for a compensated metal. 96 The adiabatic NE coefficient for a compensated metal is defined as (Ref. 3) Q = 3’ (3.58) This coefficient can be written in terms of the transport coefficients as follows. First, multiply the first equation of the first form of transport equation (3.21), by the resistivity tensor coefficient 3+, 6+ + 6+ 6+ 6+ 6+ p ° J = 0 ° 0 - E + 0 ° e".(- 7T) (3.59) + + + Witfll'the experimental boundary conditions Jx = Jy = JZ = 0, and the 6+ 6+ fact that p - 0 = 1, we get + 6+ 6+ + E = -p - e" - VT (3.60) Comparing equation (3.58) and (3.60) a ———_EY (“ ‘7‘) ( 61) Q 7 BT/Bx = p E yx 3' The tensor element 3+ - E" is derived in Appendix A and its (yx) component is, 6+ a = - ‘n’ o 11 g 11 11 11 Q (p E )yx pyx 8 xx + pyy E yx + pyz 8 2X + ”-3 'v I.’ . A ' 1 u.- .- v" .“ 97 VTy 11 ... 11 ... E" . __ .62 (pyx xy pyy yy pyz zy 3T (3 ) x .) VT To derive J in terms of the transport coefficients, let us look at x the second equation of the second form of the transport equation (3.25), and write down its x, y, components with the boundary conditions Uy = U2 = O, J = O, and the fact that there is no temperature gradient in z-direction (the direction of the magnetic field) U = -(A"XXVTX + )"xery) (3.63a) CI =0 3.1" + 11 9 . Uy ( yxVTX + A yvay) (3 63b) From equation (3.63.b) we immediately get ——1= - ——¥-3‘- (3.611) Substituting this equation into equation (3.62), using equations (3.53) and (3.5Nb) for magnetic field dependences of a compensated metal in the high field limit, and using the fact that the tensor elements of A" and 3+ are related by the Weideman-Franz law, the leading term Of the NE coefficient in high-field limit for compensated metals is simply the second term in equation (3.62) 98 a a, 11 Qcomp(H+ ) +6 yxpyy (3.65) With a similar technique, and with the same boundary conditions, the high-field limit of the adiabatic NE coefficient for an uncompensated metal is (3.66) Notice that.ixlaa compensated metal, pyy is proportional to H2 and e"yx is proportional to H-1, consequently the NE coefficient Qa is proportional to H. Also in an uncompensated metal, since pyx is proportional to H, e"yx is proportional to H.1 and A"xy is proportional to H_1 , the NE coefficient Pa is again proportional to H. With these high-field limits of the NE coefficients for compensated and uncompensated metals, let us see how NE measurements at ultra low temperatures can be examined for presence of electron- phonon mass enhancement. The off diagonal component of the conductivity tensor, Oxy is well understood; and for no open orbits perpendicular to 13 it is found to be (Ref. 56) = e +-g+... (3.67) '{fiflfl fi-vv. U‘A l... hr.) 9:1 'O “J 99 ‘where ne and nh are the number Of electron and holes per unit volume, respectively. In an uncompensated metal where ne # nh, the first term in equation (3.67) is the dominant term. For a compensated metal, where ne = n the first term is zero, but its h! derivative has exactly the same form as we will find for the uncompensated metal (ref. 56 ). From equation (3.35) the off-diagonal component of the thermoelectric tensor is do n = __Z£ e yx eLOT( dE )E=EF (3.68) Bne 3n Since OE— and —3—E-:— have opposite signs, the densities of states due to the electrons and holes are simply additive, and n _ -1 '2 E yx - eLOT N(B ) + 0 (B ). (3.69) Here N(Ef) is the electronic density Of states at the Femmi energy, defined as One Bnh N(Ef) = fi— " O_E_ (3.70) Consequently, fran equations (3.65) and (3.66) we reach the conclusion that the NE coefficient for compensated and uncompensated metals is proportional to products Of terms involving pyy, pyx’ and 11 II .. II ' Axytimes e yx’ where from equations (3.35 3.69) e yx ls rO ortional to N(E . Since , , and A" are unenhanced p p F) pxy pYY XV 100 (Ref. 56), in each of these two cases any enhancement must enter through N(Ef). In the introduction, we designated N(Ef) by the symbol Nt(Ef) to distinguish it formally from the specific heat density of states Nc(Ef). Various theoretical estimates of Nt(Ef) have been reviewed in the introduction. The latest estimates predict a proportionality to (1 + A) (Ref. 32) or to (1 + 2/3A) (Ref. 30). Chapter N EXPERIMENTAL RESULTS High magnetic field limit (wc°T)>1) measurements of the Nernst- Ettingshausen (NE), Hall (RH), and Righi-Leduc (RL) coefficients were taken on pure polycrystalline Aluminum samples in two different temperature ranges. Rough measurements of NE and somewhat more careful measurements Of RH were made between 1K and N.2K, primarily to develop the measuring equipment and techniques needed to extend the experiments to lower temperatures. More careful measurements were taken below 1K, particularly at the single temperature 1N7mK, which is low enough so that phonon-drag effects on the NE coefficient should be small. 1N7mK was chosen as the lowest temperature with enough cooling power with the present dilution refrigerator for measuring the NE and RL coefficients with a resolution Of 0.5%. In the last part Of this chapter we summarize the data and present our conclusion concerning electron-phonon mass enhanc ement . (N.1)Temperature range 1K tO N.2K: (N.1.1)The Hall coefficient (Ru) above 1 K: The Hall coefficient Of Aluminum was measured in a standard Helium cryostat, using a SQUID as shown in Figure (2.11), as a check on the thickness (t) of the thin Aluminum sample. The sample thickness was estimated independently by use of a caliper, and also 102 by weighing a piece of known length and width and determining it thickness from its known density. These measurements gave the following results. Technique Thickness (t) in Meter Caliper (5.1 i 0.5) x 10.5m Weighing (5.08 i 0.10) x 10.5m Hall coefficient (5.05 i 0.10) x 10'5m In later analysis we use the value t = 5.08 t 0.10 x 10-5m. From resistivity measurements, the RRR of the sample was evaluated, and it was R(3OOK) RRR = R(N.2K) = 958 i 10 (N.3) (N.1.2)The Nernst-Ettingshausen coefficient above 1K: The NE coefficient of Aluminum was also roughly measured in the He cryostat at about 3K and magnetic fields of 2-3T . These measurements were made to check our measuring technique and to establish the amount Of vibration isolation needed to achieve voltage noise in the 10—13 V range in the presence of a 3T field. We succeeded in achieving the necessary voltage sensitivity in the presence Of the field, and found NE values about 5% higher than those reported by Thaler, Fletcher and Bass (ref.3 ) in 1977. In view of the crudeness Of the measurements, this agreement was taken 103 as sufficiently good to justify going forward with a more precise measuring system on our dilution refrigerator. For checking our thermometers, at the same thmathe NE coefficient was measured, the gradient Of the temperature in the x- direction was also measured. Then by measuring the amount of heat applied to the sample,the thermal conductance in the x-direction was determined. From the Weidemann-Franz law, the resistance Rxx at Helium temperature was calculated. The value of this resistance was the same as its directly measured value to within a few percent. (N.2)Measurements at temperatures below 1K: As mentioned chapter 2, we reached temperatures below 1K by use of a dilution refrigerator. From the cooling power Of the dilution refrigerator as a function of temperature, and also the heat required to be sent through the sample for NE measurements, we found the temperature 1N7mK to be appropriate for our experiments. (N.2.1)The Hall coefficient (RH) below 1K 3" R measurements were made at 1N7mK for a 21x‘Hf' thick H Aluminum sample in different magnetic fields, with the magnetic field up (+z-direction) and down (-z-direction) in ten different runs. For these measurements, a typical current Of 0.5 mA was sent into the reference resistor and SOOuA through the sample. The SQUID 1ON noise, which was a function Of the magnetic field, was about : 1.N x -13 10 volts at 10 kG. The Hall voltage for this sample in 10 k0 magnetic field was on the order Of 10-6 volts which was 107 times larger thamlthe SQUID sensitivity. The Hall coefficient approached its saturated theoretical value of (1.025 t 0.005)x 10-1O m3c71, to within 0.5% above 10 kG . The latest and most reliable sets of data are listed in table (N.1) and a phat of the data as a function Of magnetic field is given in Figure (N.1). The thickness of the sample was taken as 5.08 x 10‘5m, as discussed above. In Figure -10m3C-1 (N.1) the broken line is the value Of RH = 1.023 x 10 justified in chapter.1. 105 Table (N.1) R for pure Aluminum sample at different magnetic H fields and a constant temperature 1N7 mK. \J'IAUICDGJU'IODUICDUICD U10 U10) R = 315 (m3C-1) Run # Magnetic field Field direction H B 1 (5.0)kG up (0.905 t 0.01)x10:18 2 (5.0)kG down (0.895 t 0.01)x10 6 (8.0)kG up (0.983 t 0.01)x10::8 6 (8.0)kG down (0.979 t 0.01)x10 (10.0)kG up (1.015 t 0.01)x10:18 (10.0)kG up (1.018 i 0.01)x10 (12.5)kG up (1.019 t 0.01)x10:}g (12.5)kG down (1.020 t 0.01)X1O (1N.0)kG up (1.018 1 0.01)x10::8 (15.0)kG down (1.017 t 0.01)X1O_1O (16.0)kG up (1.019 i 0.01)x10_1O (17.5)kG up (1.020 t 0.01)x10_1O (18.0)kG down (1.020 t 0.01)x10_10 (20.0)kG up (1.018 t 0.01)x10_1O (22.0)kG up (1.019 1 0.01)x10_1O (2N.0)kG down (1.017 t O.01)X10_1O (25.0)kG up (1.018 t 0.01)x10_1O 0 (26.0)kG up (1.019 t 0.01)X10_1O (27.5)kG Up (1.020 t 0.01)X10 10 (28.0)kG down (1.019 t 0.01)x10::g 1O (28.5)kG Up (1.020 t 0.01)x10 106 .2 wo 0.5333 3:09:36 859.2 Ono. Scum 85:06.8 Onam> one we OCHH cmxoue one .xmea.o .OESumEOQEOu um m EHme 030:me mo "830:3 m we 3. mo “COMOEMOOO Sm: :13 Pam?” 75m en 3 O." r... 3 no a q _ R _ 4 ad 1 36 c 1 35 o I use—On... m lodud / m r l w... 0. mm .11 T . i3 Ilzhuplmwhwuuyhv;11n~1.IE-111mwv1.mwhn1nwfuunr11hu lllllllll . _ _ _ _ p : 2x 0—1 107 (N.2.2)The NE coefficient below 1K: The NE coefficient of the Aluminum sample was measured with the refrigerator held at 1N7mK as a function of magnetic field in different runs. Because of the applied temperature gradient, the temperature at the center of the sample was about 210mK. An uncorrected value of the NE coefficent was first determined frOm equation (1.13). In this equation, Ux is the thermal current density: 2 U a Qx = Rheater ' Iheater (N 3) x, W- W-t ' cf Such uncorrected data from the last few runs, by which time the bugs had been worked out of both the equipment and our measuring procedure, are listed first in table (N.2). As mentioned in chapter 1, because the sample is very thin, a.significant portion of the heat sent into the sample might go through the substrate. In such a case, the Qx in equation (1.13) would not be exactly equal to RhIh2 as assumed in equation (N.3). A value of Qx corrected for heat flow through the substrate was estimated from RL measurements on the sample as discussed in chapter 1. The results are given next. 108 (N.2.3)Correction of 0x by RL measurements: The Righi-Leduc coefficient has a high-field limit shown in equation (1.20). From the known value of RL for Aluminum and the measured value Of VTy,the transverse temperature gradient, we calculated Qx as follows. Combining equations (1.19) and (1.20), QX could be deduced from: L T 3T 0 Qx - 3y R_B’ (”'14) H 3T AT where a—y— = W—is the transverse temperature gradient, and compared 2 with IxR' The ratio of the value determined from equation (N.N) to that determined from 13R determines the fraction of the heat passing through the center of the sample at a given magnetic field. These ratios were difficult to measure reliably, since the typical temperature differences across the sample were only 5-10 mK. Under the best conditions, we could resolve these differences to an uncertainty Of 5-10%. In the last run, when we had established our measuring procedures, we measured the heat ratios for an average sample temperature Of 0.215K at -16 kG, + 20 k0, + 20 k0 again, and - 20 kG. The values found were 10N% at -16 kG, 90% at +20 kG, 85% at + 20 kG, and 100% at -20 k0, each with an uncertainty Of 5-10%. We see that the negative field values were slightly more positive than the positive field values. If we linearly average these four values, so as to eliminate effects of + and - field, we find an 109 average of 95 1 5%, which we take as the "best" RL correction. Using this correction factor, we calculated the "corrected" values of NE listed second in table N.2. A plot of Pa/B against B in Figure N.2 shows that above 8 k0 the NE coefficient saturates to a value approximately independent Of the magnetic field, until about 21 kG, above which it drops Off in value with increasing field. We tentatively attribute this dropoff above 21 kG to the onset of magnetic breakdown, which is known to occur in Al (ref.8 ) Since the analysis for mass enhancement assumes both high field saturation of the NE coefficient, and no magnetic breakdown, we use the data between 10kG and 20kG for estimation of the presence and size Of any such enhancement. When we examine these data, we find that they clearly demonstrate the presence Of a mass enhancement, knit within are in slightly better agreement with an enhancement of Ao than with 2/3AO, but we cannot rule out 2/3AO to within our experimental uncertainty. 110 Table (N.2) Nernst-Ettingshausen Coefficient Measurements for Different Values and Direction ("z"-direction and." -z"-direction) Of the Magnetic Field from Direct Technique and its Corrected Values from RL coefficient Measurements. Magnetic Magnetic field NE (m3J'1) NE(m3J71) field direction direct measurement corrected by RL (5.0)kG up (N.23iO.20)x10-11 (11.111130.50)x10'1.1 (5.0)kG down (3.93+0.20)x10"11 (L1.13:0.50)x10‘11 (7.5)kG down (N.79iO.20)X10-11 (5.0NiO.50)x10-11 (8.0)kG down (5.52:0.15)x10'11 (5.79:0.115)x10".1 —11 -11 (10.0)kG up (5.00:0.15)x10 (5.25:0.N5)x10 (10.0)kG down (5.1123:0.15)x10'11 (5.69zt0.115)x10'11 (12.0)kG down (5.63:0.11)x10—11 (5.91:0.N0)x10-11 (1N.0)kG down (5.77:0.11)x10’11 (6.06:0.N0)x10-11 (15.0)kG down (5.62:0.11)x10"11 (5.90.+.0.110)x10'11 (15.0)kG up (5.22:0.11)x10-11 (5.118_+.0.110)x10"11 (16.0)kG up (5.22:0.11)x10~11 (5.N8iO.N0)x10-11 (16.0)kG down (5.85:0.11)x10'11 (6.111¢0.110)x10’11 -11 -11 (17.0)kG up (5.60:0.11)x10 (5.88:0.N0)x10 (17.5)kG down (5.38:0.11)x1O-11 (5.65:0.110)x10'11 (18.0)kG down (5.55.10.101110'11 (5.83:0.110)x10'11 (18.0)kG up (5.311;0.11)x10‘11 (5.11730.110)x10'11 (20.0)kG down (5.58:0.11)x10_11 (5.86::0.NO)x1O_11 (20.0)kG up (5.57:0.11)x10'11 (5.85:0.110)x10’11 (21.5)kG (21.0)kG (21.0)kG (21.0)kG (22.0)kG (22.0)kG (23.0)kG (2N.O)kG (2N.0)kG (25.0)kG (25.0)kG (25.5)kG (26.0)kG (26.5)kG (27.0)kG (27.5)kG down UP down up down UP UP up down UP down down UP down UP down 111 (5.12:0. (5.05:0 (5.02:0 (5.00:0 (5.52:0 (N.92:0 (N.86:0. (N.1N:0. (3.66:0. (2.86:0. (2.51:0. (2.53:0. (2.15:0. (2.08:0. (1.97:0. (1.65:0. 11)::10'11 11 11 11 10)x10- 10)x10- 10)x10- 11 11 10)x10— 10)x10‘ 20)::10'11 11 11 20)x10- 25)x10" 11 11 25)x10’ 25)x10_ 25)x10-11 25)::10’11 25)):10'11 30)x10'11 30)::10’11 (5.37:0. (5.30:0. (5.27:0. (5.25:0. (5.79:0. (5.17:0. (5.10:0. (N.35:0. (3.8N:0. (3.00:0. (2.63:0. (2.66:0. (2.26:0. (2.18:0. (2.07:0. N0)x10'-11 11 11 11 N0)x10- N0)x10- no)x1o' 11 11 N0)x10— N0)x10- NO)x10"11 11 11 N0)x10- 50)x10— 50)x10-11 50)x10-11 50)::10'11 50)x10-11 (115)1110'11 50)::10-11 (1.73:0.50)x10- o . .. a use: 338% 0295020 on... Scum 6338.5 03mg, 23 86.8205 0:: C935 05. Maze ORSHEOAEOH pm 303 032mm:— wo 530:3 m we ucmwowmmmoo m2 .313 0.53m :5 on ....u as m... o._ no a O . a. . _ . 0O " .. x 3.616... 3.. 6 m m :E.. O O m m Em N o o . . m m G.” . m m a _ . / A40; O u u . 8 o u . cc ) . I O m u _ 0 F8 _ " .... 7w: 0 o m . n ( C . O A . 1O 0. o O O O O. &+— T1 lllllll 111 " lbIIOIIIQIIIIIHIII.|1| IIIIIIII _ O O u u s . _ u 9x :I 113 We conclude with a brief demonstration that phonon-drag.h3 unimportant in the NE coefficient at 210mK. The value of the phonon-drag component of the NE coefficient can be estimated from Figure (1.2.b) as follows. In Figure (1.2.b) the intercept at TsO a 3 of 23 v.s. T is the electron-diffusion component of the NE coefficient. At higher temperatures i.e. 210mK, the phonon-drag component of NE coefficient is just a 02-) . 3(p /B) . 3 (T ) (u.6) B 3 3(T3) a From Figure (1.3). ASE—€13)- ~ 1.083 x 10-13 m3J'1K‘3 8(T ) phonon-drag component of NE coefficient at 210mK is . Therefore the a _P_)T=210mK ~10 x1015 m3J1 ( B g (N.7) This is about 5 x 10—3 times the electron-diffusion component of NE coefficient, and thus considerably smaller than our measuring uncertainty. (u.3)Conclusion: We have constructed a system for transport measurements below 1K in magnetic fields up to 30kG, and used it to measure the high magnetic field limit of the Nernst-Ettingshausen coefficient for a pure polycrystalline Aluminum sample at 210mK, where we expect 111-l phonon-drag contributions to be negligible. Previous measurements on Al were limited to above 1.8K. Our data indicate that the electron-phonon mass enhancement continues to appear in the off- diagonal component of the thermoelectric tensor coefficient at temperatures well below 1K. To within our measuring uncertainty, the enhancement in our data between 10 k0 and 20 kG is consistent with (1”,), in agreement with the low-temperature, high-field NE: coefficient measurements of Al from 1.8K to 5K by Thaler, Fletcher and Bass (ref.3 ). The data are less consistent with an alternative prediction of (1 + 2/3AO). 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Lett. 33, 363 (1977). 56. A.A.Abrikosov, "Introduction to the Theory of Normal Metals" Academic Press. New York and London. (1972). 57. P. L. Taylor "A. Quantumn Approach to the Solid State" P r e n t i c e H a l l , N e w J e r s e y ( 1 9 7 0 ) . APPENDIX 118 APPENDIX A 4-+ ++ From equation (3.60) the Tensor p - e" is 1 pxx pxy pxz 4n) 4-: 0 E a O D pyx pyy pyz pzx pzy pzz_ _ N " pxx E:10: + pxx exy + E" + E" .... pxy yx pXY YY " 11 pxz ezx pxz Ezy " " pyx Exx + pyx exy + = E" ... E" + pYY YX pYY YY " n pyz sz pyz ezy H " pzx 6xx + pzx exy + e" + e" + pzy yx pzy yy ' fl 1: p22 ezx pzz ezy 9 Consequently, Ey is g ( n n n ) 6T ... = + + . y pyx exx pyy ny pyz sz x < .n + + .m pyx xv pyy yy pyz 2y) N " (pyx Exz + pyy eyz yz zz + p a" ) - 9T 119 Since 9T2 =- 0. .) E J g p E" + p E" + p 81: 4. 9T YX XX YY Yx yz zx X (p E" + p 6" + p E" ) . _Z YX XY YY Y)’ Y2 Z)’ 6T X ATE ”11111111111111 |111||11111|1|111111|l1111111111“ 3 1213 03082 0942