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TVA ”I“; “w ‘ ‘ {3}“: v u. ‘5 .14. ~ .“ l . 1.1,.hj‘1chx.’ t 1 LIBRARY Michigan State University This is to certify that the dissertation entitled Low Tempeaaxune Efiectaonic ThanApont PROM/1119A 05 um AVE/60915 and Thin POW/Mum way.» presented by ling Zhao has been accepted towards fulfillment of the requirements for Ph.D. Physica degree in Major professor glee” Date AuguA/t 2, 1988 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 lllilllllllllllllllllli'llllllflllll L 31293 00658 0116 MSU LlBRARlES -;— 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. JV" ; a? £91935? ‘ LOW TEMPERATURE ELECTRONIC TRANSPORT PROPERTIES OF ALKALI ALLOYS AND THIN POTASSIUM WIRES BY Jing Zhao A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHLOSOPHY Department of Physics and Astronomy 1988 ABSTRACT LOW TEMPERATURE ELECTRONIC TRANSPORT PROPERTIES OF ALKALI ALLOYS AND THIN POTASSIUM WIRES BY Jing Zhao We have performed high-precision measurements of the electrical resis- tivity and the thermoelectric ratio of high concentration l_(_Rb alloys, and ENa and Eng alloys, and of thin potassium wires, over the temperature range from H.2K to 0.07K. The alkali alloy study reveals that the resistivity anomaly found in previous measurements of 53b alloys is a general phenomenon, and thus likely to be fundamental. The dependences of the anomaly upon temperature, residual resistivity, and magnetic field have been studied. For all three alloy systems with pO <10.7 Qm, the anomaly is found to be approximately linear in both temperature and residual resistivity. Localization, electrwnr-electrwni interaction, and Kondo effects appear to be ruled out as explanations for this regime. The anomaly is found to approach the electron-electron interaction limit when pO >1O—7 am. We also found the un- expectedly large magnitude of electron-electron scattering in pure Li two be due directly to the band structure of Li rather than indirectly to anisotropic scattering. The thermoelectric ratio G data suggest that adding impurities to K quenches phonon drag and many-body effects dominate the sign as well as the magnitudes of the thermal diffusion component of G. Previous measurements of the temperature dependent resistivities of thin potassiwniflJwfi have been extended to samples of different lengths, different bulk purities, and different surface conditions. The new data all show size—effect anomalies. The anomalies appear to be affected by sunrface corrosion, and do not generally depend upon sample length. For a given size of the anomaly the temperature dependence is independent of all parameters studied. The data do not appear to be explained by current theories of : interference between electron-electron scattering and surface scattering; reduction in surface scattering due to electron—phonon scattering; localiza- tion effects combined with Charge-Density-Waves (cow); or sinnile cow behavior. The low temperature limits of G for almost all of the samples are consistent with values of of; =- --o.6io.2v"1 and G; =- 0:0.2V-1. G: is the surface contribution, and G1 is the impurity contribution. The surface con— 0 tribution is compatible with completely diffuse surface scattering. ACKNOWLEDGMENTS I gratefully thank my advisor Professor Jack Bass, whose invaluable supports and criticisms guided me through every stage of this research. I would also like to thank Professors P.A. Schroeder and w.i%. Pratt 4h". for their precious advice, discussions, and help during this research. I wish to thank Professor Hide Sato for his helpful discussions and generous help in many stages of this study. I am also deeply indebted to Dr.Zhao—Zhi Yu, mainly from whom I learned to operate the dilution refrigerator and to make alkali metal samples. I thank everyone else for their aid and interesting discussions: Yao-Jin Qi an for his help in taking some data and for permission use some of his data, and Yi-Yun Huang for her help in using computers. Finally, the financial support of Michigan State University, the National Science Foundation, and the MSU Center for Fundamental Materials Research are acknowledged. ii TABLE OF CONTENTS Chapter Page LIST OF TABLES vii LIST OF FIGURES viii I. INTRODUCTION 01 1.1 Alkali Metals 03 1.2 Simple Transport Properties on 1.3 Previous Work 06 1.3.1 General Background About the Resistivities of Simple Metals 06 1.3.2 Previous High Precision Low Temperature Electrical Resistivity Measurements of Potassium 08 1.3.3 Previous Work on High Precision Low Temperature Electrical Resistivity Measurements of 5RD Alloys 12 1.3.4 Previous High Precision Low Temperature Electron Resistivity Measurements of Pure Li 1A 1.3.5 Previous Work on High Precision Low Temperature Electrical Resistivity Measurements of Lng Alloys 15 1.3.6 Previous Work on High Precision Low Temperature Electrical Resistivity Measurements of ENa Alloys 15 1.3.7 Previous Work on the Thermoelectric Ratio G of 5RD, ENa, and Lng Alloys 15 iii II. III. 1.” Present Thesis 1.A.1 Low Temperature Electrical Resistivity 1.4.2 Thermoelectric Ratio G THEORY 2.1 Inelastic Electron-Impurity Scattering 2.2 Temperature-Dependent Elastic Scattering Contribution to the Impurity Resistivity 2.3 Kondo Effect 2.A Two-Level Systems 2.5 Weak Localization and Interaction 2.6 Electron-Electron Scattering 2.7 Thermoelectric Power EXPERIMENTAL TECHNIQUES 3.1 3.2 3.3 Measuring System 3.1.1 Reaching Low Temperatures 3.1.2 High Sensitivity 3.1.3 High Precision Measuring Method 3.2.1 Resistivity 3.2.2 Temperature Derivative of Resistivity dp/dT 3.2.3 Thermoelectric Ratio G Sample Preparation 3.3.1 Glove Box 3.3.2 Sample Can 3.3. 3 Sample 3.3.“ Potential Leads of Sample iv 18 18 21 22 22 26 27 29 29 3a 36 no no Ln m m 1:2 143 an us as as A6 us 1:7 IV. RESISTIVITY ANOMALIES AND THERMOELECTRIC BEHAVIOR IN 5RD, ENa, AND LlMg ALLOYS AT LOW TEMPERATURES 49 4.1 Electrical Resistivity 50 4.1.1 Residual Resistivity per Atomic % Impurity 51 4.1.2 T2 Resistivity Component 54 4.1.3 High Temperature Term 62 4.1.4 Low Temperature Anomaly 68 4.1.4.a Temperature and p0 Dependences of the Anomaly 68 4.1.4.b Comparison with Theoretical Models 77 4.1.4.0 Conclusions Concerning the Low Temperature Resistivity Anomaly 85 4.2 Thermoelectric Ratio G 86 4.2.1 The Lorentz Ratio L 87 4.2.2 G Data 88 4.2.3 The Electron Diffusion Component G 93 4.2.4 The Phonon Drag Components 96 4.2.5 Conclusions 99 V. ELECTRON-ELECTRON SCATTERING 100 REFERENCES (Chapters 1 - 5) 106 VI. ELECTRONIC TRANSPORT PROPERTIES OF THIN POTASSIUM WIRES BELOW 1 K 111 6.1 Derivative of Electrical Resistivity, dp/dT 111 6.1.2 Theoretical Background 113 6.1.3 Samples Characterization and Experimental Procedure 116 6.1.4 Experimental Data and Analysis 123 6.1.5 Summary of Behavior and Comparison with Theories 133 6.1.6 Conclusions and Suggestions for Further Work 135 REFERENCES (Chapter 6.1 - 6.1.6) 137 6.2 Thermoelectrical Ratio G 152 6.2.1 Introduction 152 6.2.2 Review of Previous Work and Background Information 152 6.2.3 Experimental Data 156 6.2.4 Summary and Conclusions 160 REFERENCES (Chapter 6.2 - 6.2.4) APPENDIX 1 169 APPENDIX 2 173 vi Table (1): (2): (3): LIST OF TABLES Page Experimental and Theoretical Values of B for 5RD, ENa, and Lng Alloys 61 Experimental and Theoretical Values of G0 for 5RD, ENa, and Lng Alloys 95 Experimental and Theoretical Values of A for 5RD, ENa, and Lng Alloys 100 vii Chapter 3 (Figure 1.) 4 (Figure 1.) J: 2' (Figure (Figure (Figure (Figure (Figure (Figure 2.) 3.) 4.) 5.) 6.) 7.) LIST OF FIGURES Page The Low Temperature Circuit: The components inside the Broken Line are inside the sample can. 48 p0 versus atomic percent impurity concentration 0 for 5RD, ENa, and Lng alloys. The insert shows the detailed data for dilute alloys. The solid lines indicate the best experimental values from reference 59, and the broken line for 5RD alloys is fit to a formula p0 = 1.14 x 10'7 (1 - 0) am. The open squares are the data measured by Oomi et al. 53 dp/dT versus T for pure Rb, Li, K, and Na. The RRR for each pure alkali metal approximately has the following value: 400 for Rb, 1000 for Li, 5800 for K, and 4700 for Na respectively. 56 (1/T)dp/dT versus T for all §Rb alloy samples. 57 (1/T)dp/dT versus T for all ENa alloy samples. 58 (1/T)dp/dT versus T for all Lng alloy samples. 59 (1/T)dp/dT versus T for all concentrated Lng alloy samples. 60 The coefficients of the T2 term A + Bp0 versus p0 for 5RD, fiNa, and Lng alloy samples. The data were obtained by averaging over the flat regions on figures 3, 4, and 5. 64 viii 4 (Figure 8.) 4 (Figure 9.) 4 (Figure 10.) 4 (Figure 11.) 4 (Figure 12.) 4 (Figure 13.) 4 (Figure 14.) 4 (Figure 15.) 4 (Figure 16.) 11(Figure FL) The coefficients of the T2 term A + BpO versus pO for all 5N3 alloy samples. Notice the different pO values for each Na concentration. 65 X versus T for ERb and ENa alloys. The solid curves are fit to Eq. 4.10, and the broken lines are fit to Eq. 4.5. 66 (1/T)dp/dT versus T-1, T-2, and T-3/2 for the 23.6% ERb alloy. 71 X versus T for 23.6% 5RD. The long dashed, dashed, and dotted curves are fit to Eqs. 4.7, 4.6, and 4.8 respectively. The solid curve is fit to Eq. 4.6 along with an additional T5 term. 72 dp/dT versus for §Rb and ENa alloys. 74 dp/dT versus T for Lng alloys. 75 Intercepts of the data in Figures 11. and 12. versus po. 76 -1 -3/2 (1/T) dp/dT versus T and T for 1%, 10%, and 32% Lng alloys. 78 The coefficients of the corresponding T—3/2 term in (1/T)dp/dT versus pO for all alloy samples. The dashed line is a po dependence. The solid and long dashed lines are 93/2 dependences for Li and K based alloys, respectively, from electron-electron interaction theory. 80 (1/T)d1np/dT versus T for 9.75Rb and 1% Lng alloys for magnetic fields of B = 0 (open symbols) and B = 0.2T (filled symbols). 83 4 (Figure 4 (Figure 4 (Figure 4 (Figure 4 (Figure 4 (Figure 4 (Figure 11(Figure 4 (Figure 6 - dp/dT 18.) 19.) 20.) (1/T)dp/dT versus T for 5RD alloys with the solid lines fit to the ineffectiveness of electron-phonon scattering model by Kaveeh and Wiser. The dashed lines indicate the behavior expected from the standard theory. This figure is taken from reference 45. 84 (1/T)dp/dT versus T for dilute Lng alloys with the solid lines fit to the ineffectiveness of electron- phonon scattering model. 89 The Lorentz ratio L versus T for 32% Lyhg and 2.2% KRb alloys, and pure K. 90 G versus T for all KRb alloys, and pure K and Rb. 91 G versus T for all KNa alloys, and pure K. 92 G versus T for all Lng alloys, and pure Li. 97 GO versus 90-1 for all alloy samples. 98 0 versus T2 for KRb alloys. 104 The values of A + BpO determined from Figure 7. against p0 for Lng alloys (open triangles) and pure Li (full triangle). The broken curve repre- sents the expected behavior for Lng with anisotropic scattering, using the parameters suggests by Sinvani et al. Data for KRb alloys (full circles) and for pure K (open circle) are given in the same units in the inset. Note that both the Lng and KRb data extra— polate linearly to their respective pure metal data points. 105 dp/dT versus T for the K(7300) samples, which were (Figure 1.a) 6 - dp/dT (Figure 1.b) 6 - dp/dT (Figure 2.) 6 - dp/dT prepared and cooled in a He atmosphere. This figure is taken from Reference 1, but the data have renormal- ized as described in the text, a few plotting errors have been corrected, some of the nominal sample diameters have been revised as described in the text, and some additional samples from Reference 3 have been added. Two nearly identical samples were always pre- pared and measured together; for the samples in Figure 1., the data for both wires in a pair was always fair- ly close. For simplicity, we omit the pairs of the thicker samples; paired samples are indicated by brackets. Two pairs of samples were annealed at room temperature to thin them further after their initial measurements; the arrows indicate the changes which occurred due to these annealings. 139 p(T) versus T for selected data from Figure 1.a. were integrated by hand. Note that the integrated data have qualitatively similar form to the data of Figure 1.a. 140 dp/dT versus T for thin K(7300) wires cooled in an Ar atmosphere or in partial vacuum. This figure is taken from Reference 2, but the data have been re- normalized as described in the text. The straight line indicating bulk behavior is the same line as in Figure 1. The samples connected by brackets were prepared and measured together. 141 The T2 coefficient A + Bpi for "bulk" (d 3 1 mm) xi (Figure 3.) 6 - dp/dT (Figure 4.) 6 - dp/dT (Figure 5.) 6 - dp/dT (Figure 6.) samples of K as a funciton of residual resistivity p1. The filled symbols indicate the data used to determine "bulk" behavior for K(7300). K(4800), and K(1700) (for definitions see text) in Figures 1., 2., 5., and 6. respectively. The K(7300) wires had d 1.5 mm, the K(4800) and K(1700) wires had d = 1.0 mm. The straight line is a best fit to all of the data. 5 2 Its slope of 2.5 x 10- K- is somewhat larger than those for Rb (1.3 x 10’5K'2) and Na (0.8 x 10'5K'2) impurities in K 142 p versus 1/d for K wires of different diameter and different bulk purity. The solid lines have a slope taken from Reference 16. The letter "3" indicates samples with shiny surfaces 143 dp/dT versus T for the K(1700) samples. Pairs of samples prepared together are designated by identi- cal symbols, with the open symbol designating the sample prepared first. Note that in each pair the anomaly is always larger for the sample prepared first, and the anomaly is independent of sample length. After their initial cooling, some of the samples were given room temperature anneals and then cooled and measured again. The progres- sion of behavior after such anneals is indicated by the arrows. 144 dp/dT versus T for the K(4800) samples. Pairs of samples prepared together are designated by identi- xii 6 - dp/dT (Figure 7.) 6 - dp/dT (Figure 8.) 6 - dp/dT (Figure 9.) cal symbols, with the open symbol designating the sample prepared first. Note that in each pair the anomaly is always larger for the sample prepared first, and the anomaly is independent of sample length, except for the sample pair denoted by diamonds. This pair was the only one that showed a length dependence approximately proportional to L2. After their initial cooling, some of the samples were given room temperature anneals and then cooled and measured again. The progression of behavior after such anneals is indicated by the arrows 145 Intercomparison of data sets with four different size anomalies involving wires having different thicknesses, different bulk RRRs, and/or cooled in different gases. The curves through the low tem- perature data are fits up to 1.2K to an equation of the form p(T) = (A + Bp1)T2 - CT:{/3 where (A + 891) is determined by the behavior of thick K wires 146 A versus 1/d for the data of Figures 1., 2., 5., and 6. A is the deviation at 1.0K of the anomalous values of dp/dT in Figures 1., 2., 5. and 6. from the bulk behavior shown in each figure. The letter "3" indicates samples with shiny surfaces. 147 A versus (RRR/d) for the data of Figures 1., 2., 5., and 6. The scale for A versus R/d is given at the top of the graph. The letter "s" indicates samples with shiny surfaces. 148 xiii 6 - dp/dT (Figure 10.) 6 - dp/dT (Figure 11.a) 6 — dp/dT (Figure‘H.b) 6 - 6 (Figure 1.) 6 - G (Figure 2.) 6 - 6 (Figure 3.) 6 - 6 (Figure 4.) 6 - 6 (Figure 5.) 6 - 6 (Figure 6.) 6 - 6 (Figure 7.) APPENDIX 1: Figure A.1 A versus pO - pi for the data of Figures 1., 2., 5. and 6. The letter "3" indicates samples with shiny surfaces. 149 Figure 8. with arrows indicating changes in A which occur upon corrosion. 150 Figure 8 with arrows indicating changes in A which occur upon corrosion. 151 G versus T for He cooled K(7300) 162 G versus T for Ar and Vac cooled K. For the vacuum cooled samples, about 10qu of residual He gas was left in the sample can at room temperature to ensure that the samples would cool properly. This gas affected the G data above 1K, which is thus not as reliable as the remainder of the data in this paper. 163 0 versus T for He cooled K(4800) 164 G versus T for He cooled K(1700) 165 GO versus 1/pO for the data of Figures 1. - 4. 166 Test of G(T) = G0 + DT2 below 1K for selected data from Figure 4. 167 The T2 coefficient D from Figure 6., versus p;1, for the data of Figures 1. — 4. 168 K2 /(2kf)2 versus a and Y 172 xiv APPENDIX 2: 3/2 Figure A.2 (1/T)dp/dT versus T- for all 32% Lng alloys 175 Figure 1.3 G versus T for all 32% Lng alloys 176 XV Chapter 1 INT RODUCT I ON Electrical transport measurements are an importtunz‘tool for inves- tigating electrons and phonons in metals. Historically such measurements have played an essential role in our understanding of metals. A special class of metals, called simple metals, have distinguished themselves be- cause of their neanly free-electron like Fermi surfaces, for which the theoretical situation is enormously simplified. Among these "simple" met- als, the alkali metals are believed to have the simplest electronic structure. Transport measurements on these metals have proved to be the best testing ground for theories concerning scattering of electrons by phonons in a metal.1 The most recent revival of interest in electrical transport of simple metals started about a decade ago with the advent of high-precision low temperature electrical resistivity measurement tech- niques which made feasible investigations of some not well explored electron scattering processes, such as electron-electron scattering and in- elastic electron-impurity scattering. Over the past decade coupled advances iJl theories and in experiments on simple metals have greatly en- hanced our quantitative understanding of some fundamental electronic transport properties of metals.2’3’u’5 However, anomalous behavior of the resistivity below 1K, and of a variety of other properties of alkali met- als, have been discovered,6 which are inconsistent with free—electron behavior in a defect free sample. These anomalies have raised the question as to whether a "simple" metal is really simple. The underlying motivation for this dissertation is to try to answer this question through exploring ultra low temperature resistivity anomalies. If the alkali metals do be- have as expected for free-electron metals, we wish to understand anomalies which they display, and these simplest metals should provide a best testing ground to quantitatively elucidate important phenomena like localization7 and interaction7 effects which have substantial current interest. On the other hand, if the hypothesis of an intrinsic Charge Density Wave ground state6 applies to the alkali metals, we hope through thhsvmwk to find evidence for the CDW. This dissertation is a report of experimental studies of low tempera- ture transport properties of some alkali metals and their alloys, mainly istudies of the effects of adding various impurities to the alkali metals K and Li and of reducing the geometric sizes of K samples. In these studies ‘we have performed a series of ultra-high precision (two parts in 108 ) and -15 V) electrical resistivity measurements on alkali ultra-small signal (10 metals and their alloys at temperature from 4.2K to 80mK , using a SQUID and a high precision current comparator with the aid of a computer averag— ing technique. Also, thermoelectric ratios of all samples were measured which provided information additional to the electrical resistivity measurements. At temperatures below about 1K, the electrical resistivity and the thermoelectric ratio contain all the information about the macrosc- opic transport properties of our samples. The main topics to be presented are: resistivity anomalies and ther- moelectric behavior in KRb, KNa, and Lng alloys below 1K (ch. 4); electron-electron scattering in Li (ch.5); and electronic transport properties of thin potassium wires below 1K (ch.6). Ch.6 is composed of two papers which will appear in Physics Revew B. 1.1 Alkali Metals In the past few years the alkali metals have attracted renewed theoretical and experimental interest from two contrasting points of view: 1) 1TB monovalent metals Lithium, Sodium, Potassium, Rubidium and Cesilnn are generally considered to have the simplest electronic structures of all metals, with nearly spherical Fermi surfaces lying entirely inside the first Brillouin zone, and with no unfilled d or f shells to complicate calculations. Indeed, de Haas-Van Alphn measurements.cn? their Fermi sur- faces show the free electron picture to be valid to a remarkable degree,8 with maximum deviations from sphericity ranging from 0.14% in K to 3.5-5.5% in Li where the Fermi surface comes closest to the Brillouin zone bound- aries. Many properties of the alkali metals behave exactly as expected fromfree-electron-theory.9 Furthermore, they all have a bcc structure at room temperature, and no superconducting phenomena have been seen in the alkali metals at any temperature yet reached. The alkali metals should serve as ideal testing grounds for theories of very low temperature transport properties, since for free-electron metals the Fermi surface in— tegrals involved in calculating transport properties become tractable. 2) On the otlmn~‘hand, a variety of properties6 of these metals show anomalous behavior that is inconsistent with free-electron Fermi surfaces. These anomalies have been attributed6 to the presence of a Charge-Density- Wave (CDW) ground state due to a Fermi-surface instability of simple metals--a theory proposed by Overhauser in 1968? The interest in this pos- sibility has recently been restimulated by the report of CDW satellites in potassium by Giebultowicz et al.10 from neutron diffraction experiments. However, an alternative interpretation of most of these anomalies involving possible martensitic embryos has also just been proposed.11 Among the alkali metals, Potassium is usually most favorable to study, since unlike Lithium and Sodium, which undergo martensitic phase transfor— mations at about 75K and 35K respectively, Potassium does not undergo such .a phase transformation. Moreover, the Debye temperature a of both Rubidium and Cesium is unusually low (55K and 40K, respectively) so that fkn~‘these metals, electron-phonon scattering cannot be neglected until very low temperatures. The presence of electron-phonon scattering complicates in- vestigations of less well understood scattering.(e.g. electron-electron scattering,and electron-dislomation scattering) Besides, Potassium has been claimed to be the best material to exhibit the existence of a CDW.6 1.2 Simple Transport Properties The linearized macroscopic transport equations which relate fluxes and forces are: 3. Z + J = o E - e VT (1a) + 3+ 3 q=TeE-KvT (1b) or,for experimental convenience, 9 E: Dink ..) 3 + S VT (2a) 1v 3— I VT (2b) UN: _) q = T Here, the Onsager Relations have been used to reduce the number of inde- pendent coefficients in Eqs.1 and 2 to three. 3 is the electrical current density, a the heat flow current density, E the electric field, and VT the 0+4: temperature gradient. For the tensor coefficients: is knownlas the .) + electrical conductivity and p, its inverse, as the electrical resistivity; + + + + -> E as the thermoelectric tensor; K the thermal conductivity; and S = the + .y 9 (0+4: .) thermopower. I is related to E by mid D++ m++ (3) At low temperatures, the second term is very small for metals. Due to the cubic symmetry of simple metals, and in the absence of a magnetic field, all the tensors reduce to scalars, which greatly simplifies the situation. Experimentally one can measure each of these coefficients directly by con- trolling the situation in such a way that either VT, 3 or E is made zero. The thermoelectric ratio, G defined by G=iatE=o, (A) Q is equal to S G _ ET , (5) where L = Kp/T. When elastic electron scattering is dominant, L becomes a constant: LO =n2/3(k/e)2—-the ideal Lorentz number. G is favorable to our study because no AT measurements are required to measure it. L0 = Kp/T further reduces the independent transport coefficients to two. This means, as long as L=LO, measuring p and G of a sample will complete the measure- ment of all macroscopic transport coefficents of the sample. In the following we primarily study electron scattering mechanisms by measuring the DC electrical resistivity, for which electrons are completely responsible, and the thermoelectric ratio,which gives additional informa- tion complementary to resistivity. 1.3 Previous Work 1.3.1 General Background about the Resistivities of Simple metals The resistivities of simple metals are expected to have the following contributions:12 o= 9. + is + via + 9222. m Here pO is the temperature independent residual resistivity caused by elas- tic scattering of electrons from impurities or lattice defects. At low temperatures, it‘s magnitude is much larger than the rest of the terms in Eq.6 . The temperature dependent resistivity is composed of an electron- electron component pe-e’ an inelastic electron-impurity component pe—i’ and an electron-phonon component pe-p' The existence of the second term pe_e,arising from electron-electron scattering and varying as T2, was first predicted half a century ago by 13. It‘s coefficient was predicted to be insensitive Landau and Pomeranchuk to small amounts of impurities or defects.12 Since this term is very small, generally more than four orders of magnitude smaller than the residual resistivity p0, it was not observed until about 1976 when an experimental breakthrough occurred in the measurement of low temperature electric resistivity. Use of a current comparator, and the applicaticuitflua different atmospheres. These measurements stimulated proposals of three alternative models for the anomalous behavior:localization effect327; reduction in electron-surface scattering due to electron-phonon scatteringea;anuiinterference between electron-surface scattering and electron-electron scatteringzg. The authors of these models all challenged the applicability of the Gurzhi effect to the data of tfluarneasurements. These models, especially the localization model, led to our new experimen- tal studies as we describe later. 1.3g3 Previous Work on High Precision Low Temperature Electrical Resistivity Measurements of KRb Alloys 30 In 1980 C.W. Lee et al. reported resistivity measurements on bare, free-hanging KRb alloy samples with nominal concentrations (H’IJ.05, 0.13, (3.32 and 2.24 At.% Rb from 180 mK to 4K. Below 1.3K the resistivities were found to have the expected form 2 p = Do + (A+BpO)T (10) where A was consistent with the A found in pure K, and B = (8.5:O.3)x10—6/K2 was comparable to the theoretical values of 13.7x10.6/K2 from P.L.Taylor16 and 12.5x10-6/K2 from Kus et al.3'. Lee et al. cilaimed 13 these results as evidence of the contribution to the resistivity from in— elastic electron-impurity scattering. With improved accuracy of temperature measurement, and improved resistivity measurement precision by nearly an order of magnitude, further measurements of KRb alloys up to concentrations of 9.4 at.%KRb were carried out by M.L. Haerle et al.32 in 1984. They reported data consistent with Eq.10 down to about 0.3K. However, below that temperature they observed divergences from the expected T2 behavior in the resistivities.(which could not be seen in Lee et al.‘s measurements due to their larger uncertainties). These divergences were especially surprising because KRb should be the "simplest" alloy one can imagine. This is due to a combina- tion of the simple electronic structure of K, the complete mutual solubility of K and Rb, and also their practically equal sizes. Haerle et al. characterized the data by using Eq.10 plus an additional term. The best form of this extra term was found to be —CpOT. It was also noted that tie anomaly in the KRb alloys could not be due to the localization or electron-electron interaction effects because the measured anomalies are about two orders of magnitude larger than the predicted by both theories, and because the anomalies had a different pO dependence from the predictions--the localization and interaction theories predicted that the g and pg/Z respectively. Since no extra term should be proportional to p Kondo like anomalies were seen in the thermoelectric properties of the samples, it was noted that the Kondo effect was also unlikely to be the cause . 14 1.3.4 Previous High Precision Low Temperature Electrical Resistivity Measurements of Pure Li The electrical resistivity of Li down to Helium temperature was first measured by G. Krill in 197133 . The sample was high-purity Li with p0=73pflhn. He reported that for the temperature range 10—40K the data for p varied as T” , and for the temperature range 4.5-10K the data for p ex- ‘hibited.a.T'2 temperature dependence with a prefactor A=33me/K2. A decade later, M. Sinvani et al.3u extended measurements to 1.2K. Their sample was high-purity Li with p0=121p0m. Their data could be well fit to a T2 de- pendence with prefactor A=3011me/K2. This prefactor is more than an order of magnitude larger than that predicted for the electron-electron scatter- ing contribution to the resistivity of Li by MacDonald et al.1u, and Sinvani et al. attributed this anomalously large value of A to anisotropy of the electron relaxation time of pure Li, in accord with the theory of Kaveh and Wiser. Later on, we will discuss our new experimental studies designed to test the validity of Sinvani et al.‘s attributitni. Recently, Yu et a123 . (1984) measured free hanging, bare, high-purity Li samples down to 100mK. The data further confirmed the T2 dependence for p down to 1.2K , and the coefficient A was found to be consistent with the previous measurements. At temperatures below 1.2K, however, the resistivities were found to deviate from a T2 dependence; a turn up in (1/T)dp/dT was seen. Yu. et a1. tentatively attributed this deviation to dislocatitni vibration effects similar to those seen in K as mentioned above. 15 1.3.5 Previous Work on High Precision Low Temperatu:e_§legt_ri_cal Resistivity Measurements of Lng Alloys 35 In 1985 G Oomi et al. reported measurements of the resistivity of three 11ng alloys with magnesium concentrations of 1.0, 5.0 and 10at.% from 2 to 60K. Below 10K a term proportional to T2 and to the residual resis- tivity was identified and associated with inelastic impurity scattering of -14 electrons. The data could be fit to Eq. 10 with A = (4.0:0.6)x10 ime”2 and B = (1.6:0.2)x10”6 K-Z. Within the uncertainties, the value of A was in agreement with that obtained for pure L1, as noted above, and the value of B was comparable to two theoretical predictions: one by P.L. Taylor16 B = 3.4x10—6K-2; and one by Kus et al.31 6KDZ, when these were B = 3.1x1o' evaluated using p0=1.0x10_80m for 1at.%L_i_Mg . However, due to large uncer- tainties of the data, the detailed picture of how A changed upon adding Mg to Li was still not completely clear. 1.3.6 Previous Work on High Precision Low Temperature Electrical Resistivity Measurements of KNa Alloys No measurements have been reported prior to ours. 1.3.7 Previous Work on the Thermoelectric Ratio G of K, KRb, KNa, Li and Lng More then twenty years ago, the Canadian group of D.K.C. MacDonald36 measured the thermopower S of pure K, pure Li, dilute KRb and KNa alloys, and dilute L_i_Mg alloys at temperatures from 3K down to 100mK. They ap- Proximated their thermopower data with the expression 3 * S = AT + BT + Cexp(-e /T) . (11) 16 where the first term is due to electron diffusion (S ); the second term diff is due to Normal phonon drag (SN ); and the last term is due to Umklapp ph phonon drag (83h)' This form for S was based upon the predicted low tem— perattuws behavior for a simple, free electron metal with a spherical Fermi surface that does not contact the Brillouin Zone boundary. Fkn*l( samples with RRRs ranging from 4000 to 10,000 and with diameters frmn(LCfl to 0.8mm, MacDonald et al. found the data could all be:fit to Eq.11, but the data showed a large variability between samples and also in the same sample measured at different times. They concluded that there was ru><3bvious de- pendence on the residual resistivity or on the diameter of the specimen. The thermopower of Li was found to be linear with T below 2 or 3K, which is what would be expected fCW’the electron diffusion component alone. MacDonald and coworkers reported that the effects of alloying on the phonon drag component of dilute K and Li alloys were to decrease the magnitude of both normal and Umklapp phonon drag. The diffusion thermopowers of all the dilute alloys were found to obey the Gorter-Nordheim rule:36 8 (p S + pisi)/(pp + pi). (12) diff= p p which distinguished two contributions to S : Sp due to the host metal diff and Sidue to the impurity. However, the signs and the magnitudes of the im- purity parts of S for KRb, KNa, and Lng alloys could not be understood diff by the simple model of the scattering by impurities of free electrons on a spherical Fermi surface. In 1974, Nielsen and Taylor37 investigated theoretically the many- body contributions tt>the electron diffusion thermopower. They found that some second-order corrections to the matrix elements for electron scatter- ing involving intermediate virtual phonon states could make a large contribution to the diffusion thermopower while leaving the conductivities 17 essentially unaltered. Based on this model, they explained the signs and magnitudes of the diffusion thermopowers of KRb and.KNa alloys. In 1980, C.W. Lee38 reported G measurements on pure K and KRb alloys from 4.2K down to 80mK. They fit their pure K data to the form * it as - G = GO ‘1‘ B T2 + (C /T)e 8 /T This is approximately derived from Eq.11 as G = SL/T and L a L0 at low tem- (13) peratures (for K, L=L B*T2 tflw Normal phonon drag term, and the exponential is the Umklapp 0 only below 1K). In Eq.13, G0 is the diffusion term, pnunuan drag term. For pure K, Lee et al. found G0 = -0.03:0.03V-1, B* = - 0.30:001V—1, K72, and 6* = 2322K. These results were consistent with those of MacDonald et al. For KRb alloys, Lee et al. found the same three terms as above, with the normal and Umklapp phonon drag terms quenched more and more as the impurity concentration increased. The diffusion term 0 was 0 found to approximately obey the Gorter—Nordheim rule with G1 +0.48:0.01V71, the Rb impurity scattering part of the diffusion ther- moelectric power. In 1983, 14. L. Haerlxs et al.25 reported G measurements on deformed pure K samples from 4.2 to 0.08K. They claimed that Eq.13 could fit their data for T20.2K, with 0*=18:2K, G <0, B*O. Below 0.2K, they 0 found G of the deformed samples to deviate from Eq.13. An additional term A*T, contributed by dislocation scattering, was needediJiEq.13tx)fit their data. Recently, Yu et al.23 measured G for thin K wires and KRb alloys up to 9.4at.%. The thin K data were reported fit to Eq.13 well. They found the magnitude of 00 became larger, and the magnitudes of 8* and C* became smaller, as the sample diameter decreased. They also reported that.(}.for 18 two 0.07at.% KRb samples were well fit by Eq.13 with G 1 2 o = 0.35:0.05v'1, B* - - - x = -0.20:0.01 V K , C* = 2570:270V 1K, and 6 = 23. For 9.4%KRb they found the simple Eq.13 could no longer be fit to the C data. No explana- tion was provided. 1.4 Present Thesis 1.4.1 Low Temperature Electrical Resistivity As noted about, it has been shown by Lee et a139 and Yu et al.23 that the behaviors of pure, bulk, free~hanging K and Li samples are as expected from standard theory (representative primarily of standard electron- electron scattering) down to at least 0.3 and 1K for K and Li respectively. It also has been shown that adding small amounts of Rb to K produces the behavior expected for inelastic electron-impurity scatteriru;.38 However, at lxnv enough temperatures, resistivity anomalies which deviate from con- ventional theories were discovered in pure K, in thin K wires, iriigi, and in KRb alloys by previous investigators in our laboratory. The sources of most of these anomalies are not yet understood. This dissertation is a report of three additional experimental studies designed to further extend our understanding of the ultra—low temperature resistivity of alkali metals and their alloys. The three specific topics which we focused on are the following: 1. Low temperature resistivities of different alkaliafljtws and con- centrated alkali alloys The resistivity anomaly found in KRb alloys below 0.5K by Haerle and 32 Yu et al. seems very interesting. The anomaly was found to be too big to be due to localization or interaction, and the G behavior did not support a 19 Kondo effect either. We pursued the sources of the anomaly in two directions: (a) To see whether the anomalous behavior is not unique to KRb al- loys, and thus likely to be fundamental, we measured KNa and 11ng alloys. KNa was measured because Na has a very different ion size from K. Lng was measured because Mg has, a different number of valence electrons from Li, and also because the alloys undergo a phase transition to a different crys- tal structure upon cooling. Li was chosen because it is the only alkali metal which has significant solubility for heterovalent solutes? 9 We also investigated the influence of a magnetic field on the anomaly to test for a Kondo effect. (b) To see whether the anomaly approaches the behavior predicted by Localization and Interaction as the impurity content increases,(i.e.higher po ), we extended our study to more concentrated 1_(_Rb and L__i_Mg alloys (i_(_Na alloys have a very limited solubility range). This extension is sig- nificant because these alloys consist of metals withtmxflisimpler electronic structures than any previously used to study localization and interaction effects, and because the dominant inelastic scattering mechanism in these alloys--inelastic electron-impurity scattering--differs from the inelastic electron-phonon scattering which dominated previous studies. The Lng alloys are of special interest since their p0 can be made comparable to those studied in other systems that show localization and interaction effects. 2. Electron-electron scattering in Li: explaining the unexpectedly large A and testing Kaveh and Wiser's theory As noted above, the unexpectedly large magnitude of A, the electron- electron scattering contribution to the resistivity in pure Li, was a 20 imazzle. As expected for electron-electron scattering, this A was found to be insensitive to small amounts of impurities or defects. However, it is about one order of magnitude larger than calculated for Li in the BCC structure with a free electron Fermi surface. In contrast, the is the average scattering vector. 2110 and 2W1 are typically of cumkn~ 10"2 and 10“5 respectively. The term p(T)=2W1pOT2 is the pé11=BT2 term in Eq.8 arising from incoherent electron scattering. Thus 2 2 2 B -2W1 - N h'K /2MkBOD The resistivity iriim1.2.1 can also be separated into two terms, elastic 3 (2.2) and inelastic: 2 el inel + 2 p = p0(1—2w —2W1T ) + 4pOW1T = pe_i pe_i . (2.3) 0 At low temperatures the Debye-Waller factor has the form e72w= exp{- T2+...)}, tluua the first term in Eq.2.3, due to elastic electron- impurity scattering, is just the expansion in powers of T2 of the low 2(WO+W1 temperature Debye-Waller factor . From Eq.2.3 we also see that the elastic electron-impurity scattering reduces the inelastic scattering effect by half, amuiireduces the residual resistivity pO by 2W at the zero of tem- 0 perature, which is a quantum zero-point motion effect. Recently several authors 38’35estimated the magnitude of B by taking Ks 2k which ap- f! proximately corresponds to isotropic scattering.(Ap.1) They found that the calculations gave magnitudes close to the experimental results for KRb and Lng alloys. In 1966 Kagan and Zhernov considered the more general case that electrons are not only scattered by the impurity ion itself, but also by a noticeable number of atoms that surround the impurity atom and whose oscil- lations are perturbed. In additional to the same incoherent terms obtained 24 by Taylor, they derived another term, due to coherent inelastic electron- impurity scattering, which they estimated might be of the same order as the incoherent term at temperatures near 1K. This term has the form Ap « pOSTS/Og (2.u) where S = (lip-MO)/MO + (Z — ZO)/ZO. Z and Z are the charge of the host 0 1 ion and the charge of the impurity ion, MO and M1 are the mass of the host 1 ion and the mass of the impurity ion. Thus, this term is positive for KRb and Lng alloys but negative for KNa alloys. 31 l inel _1and pe_1 (Eq.2.3)i%w KRb and Lng alloys. The derivation was made without the Kus et al. in 1979 made a more accurate calculation of p: Debye temperature approximation and contained the effect of the mass dif' ference between host and impurity ions. By using the phonon dispersion curve and the RPA screened Ashcroft pseudopotential of K together with a Rb Ashcroft pseudopotential in which the potential radius was determined frcm the experiummtal residual resistivity results, they obtained B=12.5><106K-2 for KRb alloys. For Lng alloys the valence difference means that time RPA screening is not very reliable. However, the same procedure was used to estimate B for Lng , and they obtained B= 3.1x1-06K-2 . Very recently in association with the calculation of a new resistivity term caused by a phonon vertex correction, see below, Hu and Overhouser re- derived Taylor‘s formula. However, they criticised the use of K = 2kf as an approximation. (2kf. is the maximum momentum transfer). By assuming a Gaussian pseudopotential for Rb in K (see Ap.1) with the scale factor a=1, they argued that the Koshino-Taylor term is too small.1uo account for the experimental results in KRb alloys. Their result is very sensitve to the choice of a (Ap.1), and thus without justification of a specific value of 0 their argument is weak. 25 Since the pseudopotential used in Kus‘s calculation of _i_.__i_Mg is not proper, we decided to try an alternative estimation by using a screened Coulomb potential which should be a reasonable approximatiq (2.5) min where C is a temperature-independent constant and w(q) is the frequency of the phonon of wave vector q. Assuming that the electron-phonon interaction is cut off at qmin -- n/l, Eq.2.5 modifies D to be a positive constant (4W1) at high temperatures and to smoothly change to a negative constant (- 4W1) at zero temperature. The C in Eq.2.5 was determined from experimental data at high temperatures, and l was obtained from pO measurements. Since 26 provides the value of the "transport" electron mean free path 1 they PO tr' used the Born-approximation expression for p0 and an Ashcroft pseudopoten- tial appropriate to homovalent impurities to estimate the ratio of l to ltr as l = O°Sltr (2.6) For heterovalent impurities, (Mg in Li),vuatme a screened Coulmnb potentialuu which yields 1 = 0.661tr (2.7) Kaveh and Wiser claimed this modification of the Koshino-Taylor term gave an excellent fit to the low temperature anomalous resistivities of KRb al- loys up to 23.6%. 2.2 Temperature-Dependent Elastic Scattering Contribution to the Impurity Resistivity A Very recently S. Hu and A.W. Overhauser proposed a model 6 in which the many-body electron—phonon vertex correction, together with a partial failure of Migdal's theorem, leads to a new temperature-dependent term in the electrical resistivity caused by elastic impurity scattering. Apaip (T/o )2 1n(o /T) (2.8) 0 D D where A is the electron-phonon coupling constant. As mentioned above, they argued that the magnitude of the Koshino-Taylor term should be smaller than the experimental results based on their evaluation of the average scatttnh- ing vector K by a Gaussian.potential with a=1. With this Gaussian potential (a=1) they calculated the magnitude of the term iriik1.2.8, and 2J2 || = (EN) where f()<") is the Fermi function for the intermediate state K". It is this f(K") (originating from the Pauli principle ) which produces the tem- perature dependent scattering. Evaluating the Fermi function near T=0 the E becomes f de"[2f(e") - 1} z _21n g5 ck... 8" lEfl (2.10) where we let the initial state energy differ frcm the Fermi energy by 62 (which is the order of kBT). Thus the additional resistivity from this ef- 48 fect has the form kT = Cpm[1 +£15- in(-E—B-—)J, (2.11) p K r f where c is the impurity concentration and pIn is the first Born scattering term. The contribution to the resistivity of Eq.2.11 is called the Kondo effect. This effect is expected to have a strong magnetic field depend- ence, since the application of a magnetic field can be thought“8 of as equivalent to displacing the initial state from the Fermi energy by the Zeeman energy, oe=-uOH. We see from Eq.2.10 that this reduces the diver- gence just as does a change in temperature. Further increasing an external field to guB sz will break up the spin correlated state of the conduction k electrons near the ion, removing the local spin degrees of freedom neces- sary for Kondo scattering, and thus destroying the Kondo effect. Estimation of other higher-order terms in the scattering led to an energy dependent resistivity and this was expected to give a giant thermoelectric power . 29 2.4 Two-level Systems In order tc»explain the logarithmic temperature dependent resistivity anomaly found in disordered metals which showed no magnetic field depend- ence, Cochrane et al.”9 proposed a "two level systems" model. (TLS) In analogy to the Kondo model, the TLS are internal degrees of freedom which result from atoms being free to tunnel between two alternative positions of local equilibrium. Such double wells are assumed to result from the disor- dered structure and to disappear on crystallization. In order to have a mechanism similar to the Kondo effect, Cochrane et al. further postulated that the electron--perhaps corresponding to d-like wave functions localized around the atom-~could distinguish between the two positions of the tunnel- ing atom, a degree of freedom analogous to the spin degree of freedom for the electrons in the Kondo model. A derivation similar to that of the Kondo effect gives the resistivity caused by TLS as A9 « intk§(12 + T§)/DZJ, (2.12) where TK is a temperature which is analogous to Kondo temperature, and D is the electron band width. More recently, it has come to be generally believed trunz‘the logarithmic behavior of the resistivities of disordered metals is donminated by quantum localization and interaction effects. “The applicability of the TLS thus remains cloudy. 2.5 Weak Localization and Interaction The periodicity of the crystal permits the classificationcfl? electronic wave functions as Bloch waves. However, the crystalline state is an ideal theoretical situation. Traditionally, the weak-disorder limit is described by the scattering of Bloch waves by impurities, and this leads to a Boltzmann transport equation for the quasiparticles. In the past few years there has been a growing realization that disordered materials cannot 30 be understood by forcing them into the mold of ordered systems. New con- cepts must be introduced which treat the disorder from the beginning. The new understanding is based on advances in two different areas of the problem. The first is the problem of Anderson localization. 'Nmatraditional view of the electronic wave function in a random potential had been that scattering causes the Bloch waves to lose phase coherence on the length scale of the mean free path 1. Nevertheless, the wave function remains ex- ‘tended throughout the sample. In 1958, Anderson50 pointed out that if the disorder is very strong, the wave function may become localized, in that the envelope of the wave function decays exponentially from some point in space. Thus the electrons are diffusively instead of freely propagating. Fkn'iveak disorder, i.e., for (kfl)-1<<1, it is possible to calculate cor- rections to the Baltmnann transport theory for a by using diagrammatic perturbation in the first principles calculation of a noninteracting electron gas which is weakly scattered by rigid random impurities. It turns out that to higher order in (krl)-1 there are significant scale- dependent corrections to conductivity arising from singular backscattering. Thus, the physics of the weak localization phenomenon can be interpreted as an echo of plane wave scattered by impurities. (kfl)-1<<1 can also be called the quantum interference regime. There are two different lifetimes cm the conduction electrons, the elastic lifetime I and the inelastic 0 lifetime Ti. To tum, whereas I is the lifetime of the electron in an eigenstate of momen- 1 is the lifetime in an eigenstate of energy. At low temperature the latter can exceed the former by several order of magnitude. 31 As a consequence, a conduction electron in state k can be scattered by inr- Innflities without losing its phase coherence. Since the electron has wave- like character, in reality one has to consider two partial waves of the electron which propagate on a closed loop in opposite directions. (This has been beautifully demonstrated in the experimental observation of Flux Quantization in Normal Rings.51) If one considers the two partial waves to interfere, this gives twice the probability to return to the origin as com- pared tx) classical diffusion (their amplitudes add instead of their intensities). This quantum peak in the diffusion profile can be considered as an ennui. The tendency of the electron to return to its starting point is called weak localization because it is thought of as a precursor of localizaticni. Therefore, there are corrections of quantum mechanical in— terference to the classical transport theory which assumes that the particle moves quasiclassically between the collisions. It can be shown theoretically, that the resulting conductivity is no longer a material con- stant run; a length-dependent conductivity. Thouless 52 (1977) pointed out that inelastic scattering introduces random fluctuations in the time evolu— tion of emu electronic state. Such fluctuations destroy the coherent phase and, thus, limit the quantum interference necessary for localizaticnn (flue scale-dependent localization effects are cut off beyond Lth (coherent length). This brings the temperature into play. 2 Lth = (011) (2.13) where D is the diffusion constant. Including only the singular back scat- tering process, and supposing Ti a: T-p, the correction to conductivity, arising from localization has the form 7 32 2 Tp/2 AolO(T) = :2 a (2.14) 11 where Lth = a T-p/Z. Thus, the correction to (l/T)dp/dT, which we are in- terested in, is _ =-__..._._ (2.15) where we take p a pO in the prefactor. The second aspect of the disordered problem is the interaction among electrons in the presence of a random potential. The fact that electrons are diffusively instead of freely propagating leads to a profound modifica- tion of the traditional view based on the Fermi-liquid theory of metals. 53 in 1979 that Coulomb in- It was first predicted by Altshuler and Aronov teractions in a disordered electron gas lead to another resistivity anomaly. The physical idea is that since the electron motion is diffusive, the electrons spend a longer time in a given region in space relative to the plane-wave state, and their interaction is enhanced. The effective electron-electron interaction is retarded. A sudden change of the charge distribution in a disordered metal cannot be screened immediately, since the electrons can only propagate by diffusion they need time to screen the charge distribution. The conductivity can be calculated using the Kubo formula to lowest order in the interaction, which gives the following cor- rections to the conductivity of three dimensional systems:7 2 1.3 oin‘hLI27 11' (2.16) wt: le rm v "\ x v—l \ o 33 where F = [32/3][1+3F/4 - (1 + F/2)3/2]F, F = (1/x)ln(x+1), x = (2kf/ko)2, and k0 is the Thomas-Fermi screening vector. This gives a correction to (1/T)dp/dT of the form 2 1:131“, -iiii.” (217) T dT T GT 3 In the alloy systems we studied, the dominant electron inelastic scat- tering term is pégil= ZBpOT? as indicated from Eq.2.2 and 2.3. Thus 1/11-(ne2/m)2BpOT2, where n is electron density. In Eq.2.16 D is related to the conductivity by the Einstein relation 0 =D(e2/h)(dn/def). By using the free electron value of the Fermi wave vector kf,‘we can calculate the inagnitudes (M? the quantum correction terms of Eq.2.14 and 2.16. Assuming that alloying does not change the host Fermi wave vector,uu we have for the K and Li based alloys 1 do _ x 7 3 -1 _ ~ 4 5/2 -3/2 -2 A T at = 9.6 10 p0 T 3.7 10 p0 T (an ) (2.19) 1 dp _ x 7 3 —1 _ ~ 4 5/2 —3/2 —2 A 7 87 = 7.8 10 p0 T _ 5.7 10 p0 T (QmT ) (2.20) respectively. At temperatures below about 0.6K, for all of our alloys with 8 < 17x10- 0m, the second term in Eq.2.19 and 2.20 , (arising from the in- po teraction effect) has a magnitude about one order bigger than the first term (arising from the localization). Thus the interaction effect is predicted to be dominant below that temperature. Considering the change of electron density from the host in the high concentration Lng alloys, Eq.2.20 can be represented as 34 %§)p5/2T-3/2 6 2/3 3 '1 4 1/6 4 n p 0 E ,.- - - _ - T 4.2x10 0 OT 2.610 no (3 rate 04o. 2)(2.21) swun~e nox1027 is the electron density of the Lng alloy with concentration (nm'r' 0. Assuming ncx1027= (1—c)nLi-+ O Ifevk¢k§£ dK)|2 Ask (2.22) p(T) = Ziman12gives the complete expression of <¢,P¢> for electron-electron scat- tering as 1 + + + + 2 <4>,P<1>> - WET fff{<1>(k1) + <1>(k2) (k3) 90%)} 3.1.1 -> -) -> -) xP1,2 dk1dk2dk3dku (2.23) Here P?’: is the transition probability of the two electrons being mutually + + k to state k scattered from states R1, 2 3. E“. This can be calculated once the scattering potential is known. In the relaxation time approximation it + + + + is easy to show from the BTE that <1) = r(k)v(k) E. This makes the factor from 0(R) in Eq.2.23 looks like: (“MENU-£1) + 1(E2)V(R2) - «1623)1/(12 ) - 1(Ru)V(Ru)}2 (2.211) 3 In the isotropic limit where 1(R) is a constant, and in the simple metals which have a free—electron spherical Fermi surface so that v(-k))ac R, this factor becomes: (2.25) We seetflum.Eq.2.25 vanishes for Normal scattering (momentum conservation), and thus from Eq.2.23 and 2.22 pe-e = 0. On the other hand, for Umklapp scattering, Eq.2.25 = 32 (g is a reciprocal lattice vector). and thus in an isotropic system only U-scattering gives a contribution pe—e to the resistivity . 36 Assuming that the electron-electron interaction can be approximated by a screened Coulomb potential, Eq.2.22 can be calculated13, with trm31°esult P 2 2 e_e « g (kBT) . Using different pseudopotentials, screening lengths, etc. other es- timates of the magnitudes of this T2 term for simple metals have Meai published. Lawrence and Wilkinssu developed a theory of the screened Coulomb irnnecaction between electrons and calculated the screened Coulomb interaction contribution in K, Na, and Rb. MacDonald et al.1n refined this theory, and found that in alkali metals the contribution of the screened Coulomb interaction was much smaller than calculated by Lawrence and Wilkins. MacDonald et al. proposed that the magnitude for U-electron— electron scattering in the alkali metals is dominated by a phomonsexchange contribution. 2.7 Thermoelectric Power It is believed that there are two principal contributions to the ther- moelectric power of a metal; a contribution from thermal diffusion of conduction electrons though the metal, S and the phonon-drag ther- d! moelectric power, Sp, arising from interaction between the electrons and the non—equilitndhun phonon system of the lattice. Sp can be further con- sidered.as composed by two parts; 3: , arising from N-electron-phonon scattering processes, and s: arising from U-electron-phonon scattering. Generally, Sd can be written as37 2 2 n k T Sd - _3§_— E (2.26) where the dimensionless parameter E is given by the Mott rule 61np(ef) dlnef g g .. (2.27) 37 and for free-electron like metals it can be written as 61np(rf) -3. ____.._.. a - 2 + ( élnef ). (2.27) Applying Matthiessen's rule p = p1 + p2 to Eq.2.26 and Eq.2.27, one gets the Gorter-Nordheim relation S = (9181 + p282)/p (2.28) where the subscripts 1 and 2 represent the two different electron scatter- ing processes. The mechanism of the thermoelectric power due to phonon-drag can be understood as follows: For a pure unstrained sample at low temperatures, phonon-electron scattering is the dominant scattering process for the phonons. As a result of the phonon current generated by the temperature gradient, an electron within the metal will be "dragged" alonglnrthe phonon current as in viscous flow. Consequently, electrons tend to pile up at the cold end of the sample. This charge imbalance generates an internal electric field which exerts a retarding force on the streaming electrons, and ultimately a steady state is attained in which the total electric cur- rent vanishes. The resulting thermoelectric voltage divided by the temperature gradient at the two ends of the sample is SP' For the normal phonon-drag contribution, Klemens55 and others gave the following theoretical expressions valid for free electrons and the Debye approximation. O/T 4 -x N k T 3 x e SP e(e) f f(x)dx, (2.29) o (1 - e"‘)2 38 where: x = hm/kT and O is the Debye temperature; f(x) is the fractional probability of a phonon being scattered by an electron rather than by any- thing else; i.e. 1/le ’ 1/1 + 1/1 ’ i , e (2.30) where 11 and 1e are phonon mean free paths for impurity and electron scat- tering, respectively. The Umklapp scattering contribution s: in alkali metals was predicted 36 by Guenault and MacDonald to be (2.31) The exponential temperature dependence is due to the fact that iriaainetal with a spherical Fermi surface which does not touch the Brillouin zone,a minimum phonon wave vector (qmin) is needed for a U-phonon-electron scat- tering to take place . In Eq.2.31 0* a qmin' Thus at low temperatures, one expects the thermoelectric power S of alkali metals to have the general form 3 s = AT + BT + C(T)exp(-0*/T). (2.32) A theory of S which considers the electron scattering as a many-body problem was developed by Nielsen and Taylor.” They considered a model in d which free electrons were scattered by phonons or impurities, and calcu- lated the second—order corrections to the scattering probabilities for these processes. They found the second-order electron-phonon interactions are strongly energy dependent near the Fermi level and, thus, contribute significantly to the thermopower. For pure metals they proposed a correction to Eq.2.27 39 efNVm T A51 = m 1111(6) (2.33) where V is the q=2/3kf component of the Fourier-transformed screened ionic pseudopotential, N is the number of ions, M and m are the mass of ion and electron respectively, and $1 is a temperature dependent function. At low temperatures 111 a: T3lnT. Since such an effect is close to the T3 depend- ence of the simple phonon-drag theory of SE (Eq.2.29), and yet involves no phonon flux, the A51 component is often referred to as "phony phonon drag". Due to the attraction of the ion potential, Ag1 is inherently negative. For dilute alloys at low temperatures, Nielsen and Taylor obtained two second-order corrections to Eq.2.27 efNUm nO AE =6 ——- (2.34) B k202M N e NVm AEC =- 1; 2 (539/3 (2.35) k 0 M 0 where U is the q=2/3kf component of the Fourier-transformed immudty- scattering potential, nO/N is the valence. While AEC is irflun~ently negative, the sign of A58 will be the same as the net scattering potential of the impurity, and may be positive or negative. 40 CHAPTER 3 EXPERIMENTAL TECHNIQUES ExperiJnental techniques are very essential to this study. Typically, in the metal systems we studied the temperature dependent resistivity is inuch smallem than the residual resistivity; the ratio of p(T)/p0 is on the u order of 10- to 10-6. Thus, if one wants to measure the resistivity change as a function of temperatures with 1% accuracy, one has to resolve 6 to 10—8 . Furthermore, the resistances of our samples are in the range of 10"6 to 10-“ 9. In order to avoid self magnetoresistance one part in 10- and heating, the current passing through the sample should be kept as small as possible. For a current of 100mA, to obtain the above high resolution, one has to be able to detect a signal as small as 10_15 V . Recent develop- ments in high-precision and small signal low temperature measurement techniques at MSU allow us to reach temperature below 100mK, with a voltage sensitivity about 10.15 V (limited by Johnson noise), and a precision in resistance measurement approaching two parts in 108. The system is unique. Moreover, the alkali metals are very reactive, so inert gas glove boxes and well sealed sample cans are crucial to preparing reliable samples. In this chapter, the main equipment used in the experiments is briefly described; details were given in ref.38 and 55. Sample preparation and some improvements in equipment are also described. 3.1 Measuring System 3.1.1 Reaching Low Temperatures Continuously variable temperatures from 4.2K down to below 100mK were obtained with a locally built dilution refrigerator. The system could be cooled down to liquid nitrogen temperature (~77.4K) in about 20 hours by 41 adding liquid nitrogen to the outer dewar of the refrigerator, and then cooled further to liquid helium temperature (4.2K) by transferring liquid helium to the inner dewar. The latter transfer took about:21mnms and about 25 liters of liquid He. 3.1.2. High Sensitivity High voltage sensitivity (3 10-15V) was obtained by using a SQUID null detector with several noise shielding facilities. It is essential to screen out radio-frequency noise, magnetic fields, and vibrations that can affect t1ua4.2K) and the thermal links are made of silver and copper. RL1 and RL2 were 0.1% AgAu alloys used for weak thermal links between the mixing chamber and the samples. U1, L , and R or U2, L , and 1 L1 2 RL2 were arranged so that ATxT measurements could be checked in-situ to tnake sure that no systematic errors appeared, for details see Ref.38. The master current was normally preset: for G measurements Im generally was set 43 to 2 or 5mA and for dp/dT measurements 20 or 50mA. The current dependences were checked; no current dependences were ever seen in this study. 3.2.1 Resistivity R(4.2K) could be obtained by comparing with R8 while the refrigerator was at Liquid He temperature. Rs, which was in series with the SQUID, was designed to superconduct when the dilution refrigeratcn‘1vas in Operation so that Rs would not affect the sensitivity of other measurements. Shortly after each run, circulation was stopped, and the refrigeratcw temperature rose to near 1K where it remained for a while due to the liquidation of the mixture helium. By warming up RS untill it be— came normal, R(1K) could be measured by comparing it with Rs‘ RRR is defined by RRR=R(295)/R(1K), where R(295) was measured at roan temperature by a Keithley digital nanovoltmeter with an error < 1%. Sample diameters(i2%) could be estimated from the diameters of the dies and lengths(:10%) were generally measured at room temperature so that p(1K) could calculated from them. The change in p(1K) caused by the thermal con- traction was estimated at less than 0.1%, which is smaller than other sources of uncertainty. For pure K and pure Li samples, the resistivities could also be obtained by p(1)=RRRp(295), where p(295) for K and Li are 71.9n0m and 93.2n0m, respectively. In this study for pure samples the dif- ference between pO(OK) and p(1K) was less than 0.1%, and for alloy samples the difference between p0(0K) and p(4.2K) was negligible. Hence, we take 90 = p(1K) for pure samples, pO p(4.2K) for alloy samples. The total errors in determining p(4.2K) and p(1K) were about 15%, if the samples were not severely corroded during the cooling, otherwise, a special method had to be used to estimate p0.(see Ch.6). 44 3.2.2 Temperature derivative of resistivity dp/dT. dp/dT was obtained by the resistance bridge in the following way: If R2 was to be measured, then R1 was kept at a constant temperature T1 by regulating the mixing chamber temperature with a commercial temperature controller. When the system reached equilibrium, R2 stayed at a constant temperature T (T2zT1). Comparing R with R2 would give 2 C(T2)=R2(T2)/R1(T1). The temperature of R 1 2 was then warmed up to (T2+AT) by using heater L2. Comparing R1 with R2 again gave C(T2+AT)=R2(T2 +AT)/R1(T1). Hence, AC _ C(T2+AT) - C(T2) g AR C C(T2) R AT was typically 0.1K and the change of L/A in AT was less than 10-7%.23 (where L is the length of the sample and A is the cross section of the sample). Hence AC = AR2 3 Ap2 ; 1 dp2 ATC RZAT p2AT dT p2 Since in the temperature range considered in this study (p-p0)/p0< 0.1%. We have do a pOAC dT ATC Now we summarize the uncertainties in the dp/dT measurement. The thermometers GRT1 and GRT2 were carefully calibrated and tested. Between 0.1 and 2K, the accuracy of the AT measurements was estimated to be 2.6%?5 In the calculation of the quantity pOAC/(CAT), C had precision better than 0.1ppm, which was the distinguishable magnitude from temperature dependence 45 as mentioned earlier. The error in pO would affect the quantity of inter- est dp/dT only by multipling dp/dT by a constant close to 1. Except at the lowest temperatures, typically the uncertainty in AC was less than a few percent. The uncertainty at the lowest temperatures could be as high as 25-100%. Hence, the main error in dp/dT at high temperatures came from measuring AT and came from AC at low temperatures. 3.2.3 Thermoelectric ratio G. As defined Ch.1, G = §,at E=0, i.e. G = % , at E=0. G was measured by heating the end of the sample with G1 or 62 and sending the slave current IS of the current comparator through the sample to cancel out the resulting thermally generated voltage, which was detected by the SQUID. Since IS was always equal to CIm and the heating power was equal to 2 RGIG , (RG was resistance of the G heater and IG was the heating current), 2 G - CIm / RGIG The currents in G measurements were determined with an accuracy of less (Juan 0.5%. Due to the thermal EMF noise and Johnson noise, the major source of uncertainty in G was in determination of C. At the lowest tem- peratures, the uncertainty in C could be 5% or more. 3.3 Sample Preparation 3.3.1 Glove box Two commercial (Vacuum Atmospheres Company) glove boxes were used to prepare the highly reactive alkali samples. One was filled with He and maintained an Oxygen level less than 0.5 ppm indicated by an Oxygen analyzer. The He gas was purified by a Dri-train Mo 40-1 purifier. The other one was filled with Ar and purified with a locally built Ar purifier. 46 Fresh potassium inside these two glove boxes stayed shiny for at least 2 hours. 3.3.2 Sample can Specially designed sample cans were used in this study. These were sealed with indium o-rings and equipped with superconducting leads and sil- 38 ver thermal link wires. After samples were sealed in a can inside the glove boxes, the can was transferred to the dilution refrigerator where the measurements were made. For cans filled with He gas, molecular sieve was used to absorb the residual He gas in the cans at low temperatures. The samples studied in chapter 4 and 5 were all made and kept under Ar. The original Niomax superconducting wires of the sample cans were later re- placed by copper clad superconducting wires with about 2 inches of copper cladding etched off in the middle of the wires to isolate them thermally. Since copper is much easy to solder, this switch solved the problem of stability of superconducting wire solder joints. 3.3.3 Sample The samples were fabricated from 99.95% pure K, Rb and Na obtained from Callery Chemical Division of Mine Safety Inc., and 99.99% pure Li ob- tained from Atomergic Chemetals Corp. All the samples in this study were wires prepared by extrusion.from stainless steel presses through stainless steel dies. The thin K wires were made from a special die with a 0.1mm stainless steel capillary. K, Rb and Na melt not far above room temperature. Inside the Ar filled glove box, the alloy constituents for the KRb and KNa alloys were weighed out, melted and mixed together inside a glass container on a hot plate. The liquid alloys were poured into the stainless presses. Because of the high melting point of Mg, an initial master Li(1%) Mg alloy was made inside a stainless steel crucible under Ar 47 atmosphere in an induction furnace, and then diluted to make less con- centrated alloys by adding Li and melting on a hot plate inside the glove box. We later found that mixing Mg into molten L1 in a stainless crucible on a hot plate inside the glove box also gave satisfactory alloys, and this procedure was used for 1%L_i_Mg, 10%Lng, 20%Lng, and 30%Lng alloys. The two 1%L_i_Mg alloys prepared in different ways gave the same results on all the measurements. A 49%L_i_Mg sample was made in the induction furnace. The high concentration 30% Lng and 49 1:ng samples were annealed at 70% of their melting temperatures under vacuum for 2 days. 3.3.4 Potential leads of sample Our measurements are four robe measurements. Most of the two potential leads were made of the same material as the samples, which are, generally, soft and sticky. For higher than 10% Lng samples, potential leads made from the same alloys were directly soldered with a soldering iron inside the glove box, because the alloys were hard and unsticky. This method sometimes gave unexpected dp/dT results which might be due to the metals on the iron tip diffusing into the samples or some compounds formed during the relatively high temperature soldering of the high concentration 11ng samples. We later developed a cold welding technique which used potassium as the low temperature thermal and electrical connector. Potassium is insoluble in Li, Mg and Cu, and is sticky at room temperature which makes it easy to cold weld onto samples and copper connectors. 48 Mixing Chamber . z.¢///.U////A////////////////// //////////////////// 4227; 2 U (I? 2L an m e U nu nu mm 8 nuts A 0m MO- SC 1 L an 1 $2. “a tnnn 1 .1“ U umn a :..m mhm Who wa 62 Sample Can J The low temperature circuit. The components inside Fig.1 the broken line are inside the sample can, 49 Chapter 4 RESISTIVITY ANOMALIES AND THERMOELECTRIC BEHAVIOR IN 5R1), _K_Na, AND L_i_Mg ALLOYS BELOW 1K In order to understand the low temperature resistivity anomaly found 32 in KRb alloys, we further studied the low temperature resistivities and thermoelectric ratios of high concentration KRb alloys, and KNa and EMg alloys. The anomaly seems to be a very general effect, and thus important 56’57’58 have shown that to elucidate. Very recently a number of workers as the electronrmeNIfree path becomes short the quantum corrections of localizaticniand interaction become important in analysing the resistivity of various disordered metals for which p0 >5x1077f2m. Our KRb, KNa, and Lng alloy samples have relatively simpler electronic structures than other systems studied,euuithis permits us to hope that the theoretical prediction based on a free-electron model will give quantitative comparison with the experimental data. Moreover, our samples cover the residual resistivity range of 10-109m to 1.7X10—7nm, which allows us to study how the quantum corrections become important as impurities are introduced into the crystalline metals. The study of bulk samples has an advantage in solving the relative importance and interplay of the interaction and the localization effects, since, in 20 samples both of the effects, though having large magnitudes, have very similar formulae. 50 4.1 Electrical Resistivity As noted before, at low temperatures where the contribution of electron-phonon scattering to the resistivity can be neglected, the resistivity p of dilute KRb and Lng alloys could be understood in terms of Eq.4.1, 2 2 p=p0+AT +pOBT . (4.1) except at the very lowest temperatures. The term AT2 is attributed to electron-electron scattering, and the term BpOT2 is attributed to inelastic electron—inunnfiity scattering. At the lowest temperatures, both previous 32 and high-precision measurements of the electrical resistivities of KRb our new meastuwmnents of KRb, KNa, and Lng alloys reveal anomalous departures from the expected T2 dependence at low tenperatures, and these departures grow with increasing p0. We assume that the resistivities of these alloy systems can be described as 2 2 p = pO + f(pO,T) + AT + Bp T + g(pO.T) (4.2) 0 where f is a low temperature anomalous contribution to p and g is a high temperature term which only becomes significant at T >1.2N(. The quantity' we studied can be expressed as D. .2 _ £' 5' T—T +2A+ZBpO+T (4.3) raLa where f' is the derivative of the low temperature anomaly and g' is the derivative of the high temperature term. To examine if f and g arealiinear with p0, we can normalize the data as ' ' X=—1—-T-%r%—g—&=£—T+2B+§—T (4.4) Do Do po po 51 If both f and g are proportional to p0, then X should be independent Of the concentrations of the alloy samples. Thus a plot Of X vs T for all samples should fall on a single curve. In Eq.4, 2A/pO is small for the concentrated alloys, so that we will make little error ir1)( if we take A to have the value derived from the low concentration data. (A=2.4me/K2 for K) In this section we present our experimental results, together with several dilute KRb data measured by previous investigatorszg The data are analysed in the following terms:(1) Residual Resistivity per Atomic % Impurity; (2) T2 resistivity component; (3) High tenperature components ; and (4) The low temperature anomaly. 4.1.1 Residual Resistivity per Atomic % Impurity Fig. 1 shows the residual resistivities Of our samples as functions of the nominal impurity concentrations 0. As shown in the insert to Fig.1, all of our dilute alloy data fall well on the best values of de/dc-- straight lines--Obtained by other people.59 K and Rb are mutually soluble at all concentrations. In contrast, K and Na do not tend to alloy with each other, and the maximum solubility of Na in K is only about 1at.% even with fast cooling. Mg has a maximum solubility of 70at% in Li. Fig.1 also shows that our concentrated Lng alloys (filled squares) are consistent, tO within experimental uncertainties, with the measurements of Oomi et al 35 (Open squares). These results give us confidence that the impurities in our samples were in solution. The pO for all KRb alloys can be fit into the formula p0= Dc(1-c) with D = 1.14x10-70m, as the dotted line shown in Fig.1. This implies that the solutes occupy random lattice sites in KRb alloys. Some segregated KNa alloy samples with pO much smaller than that 52 expected from their concentrations (0 20.1) were also made by keeping the samples at room temperature for an extended time. 53 /l ,. deC 1n _ / \ 10"ilm/at% ) \ /' \ “ 1-d1?/c1c=1.3 / / l 12 -- I 3 E x ‘70 ,_l \ I: 1 1 . 1 \ Q. l \ I \ h 1 " '— \‘ t-KFlb / \ / ‘ k I ‘ / \ fKNa \ ’ 1 1 1 1 o 20 40 so so mo C(Atomic Per Cent) 1'-‘igure.1:pO versus atomic percent impurity concentration c for KRb, KNa, and Lng alloys. The insert shows the detailed data for dilute alloys. The solid lines indicate the best experimental values from ref.59. and the broken line for KRb alloys is fit to a formula po - 7 1.14x10‘ C(1-c) 0111. The open squares are the data measured by Oomi et al. 54 4.1.2 T2 Resistivity Component We now turn to the temperature dependent resistivity, choosing first a method of presentation which displays the T2 component as a horizontal line, and also displays the low temperature anomaly and high temperature behavior most clearly. We start with the data of pure alkali metals depicted in Fig.2. The turn up of the data at high temperatures is due to electron-phonon scattering processes. Notice that the turn up points of each metals are in the order of their Debye temperatures [0(Li) > 9(Na) > 0(K) > 0(Rb)], which can be explained based on the Umklapp electron-phonon 71 The scattering and the presence of phonon drag in the alkali metals. turn up at the lowest temperatures are probably due to electron-dislocation scattering.6O From Fig.2, we see in spite of a turn up at both low and at high temperatures the data for pure Li, Na, and K all show a region of flat behavior expected from simple electron-electron scattering theory. As impurities are added, we would expect that the inelastic electron-impurity scattering becomes dominant at low temperatures (Eq.4.1). Fig.3 .4, 5, 6 show plots of 1/T (dp/dT) versus T for KRb, KNa, and Lng alloy systems respectively. In the absence of an anomaly, each of these plots would be a horizontal straight line as given by Eq.4.1. This is generally true for very dilute (0.38%) KRb alloys as shown in Fig.4, for pure K up to 1 K (since electron—phonon scattering becomes important about 1K) as shown in Fig.4, and for pure Li up to high temperatures (since OD(Li) is higher) as shown in Fig.5. As the impurity concentrations are increased we see that an anomalous turn-down begins to develop at the lowest temperatures in all three alloy systems, and at temperatures above about 1.2K a turn up also develops for KRb and KNa alloys. Notice that both the T2 term and the 55 anomalies grow in size with increasing impurity content. FigJSshows concentrated Lng data tO illustrate how large the low temperature anomaly grows as pO becomes large. An interesting feature of the Lng data in Fig.5 is the smooth transition from an upward turning anomaly in pure L1 to a downward turning anomaly in concentrated Lng. This seems to be a detailed display of the competition between two different mechanisms. Fig.7 shows the coefficients of the T2 terms in KRb, KNa, and Lng as a function of the residual resistivity p0. We see that the coefficients are consistent with straight lines for each alloy, as would be expected from Eq.4.1. The large uncertainties for the more concentrated alloys result from the fact that the T2 terms can no longer be well determined due to the large anomalies. Besides, the Koshino-Taylor theory, which predicted a T2 dependence, is appropriate only for low impurity concentration alloys; as the impurity concentration increases the coherent scattering of electrons by impurities and by the deformed phonon spectrum may not be neglegible. lmm values of A found by extrapolating the data back to p0=0 agree well with values measured on the pure host metals. The detailed discussions of these values of A are the topic of ch.5. 56 N< Rb 1:: - E o ' c 9 3° "' —-—17o 1" a U , . a a L1 -130 U ”A, k 1- ‘~ ’ 2 " ~90 V 0 'v' v I it: ‘ v .4. + J- — “ - — L “3: *‘h'r';_6'+'-T+‘f+ 7+1 , 9.0—, 0 —l$O Kit-w ° l T(K) Figure.2:dp/dT versus T for pure Rb, Li, K, and Na. The RRR for each pure alkali metal approximately has the following value: 400 for Rb. 1000 for Li, 5800 for K, and 4700 for Na respectively. 57 6001 O O 00— a. 5 o i i *7, * * D " 400 ° 7 N L— t , Cl 52 1 1 E C] s * 1 . T's: 300- * 1: A . Q CL v .E ' ‘ <1 _ v i ' ‘ r: 200(— X f ‘ ‘ p. ho'mnm) +- ' A038% 4.76 \N i, . '1-3 % 15.1 g: V _ 19.4 % 101 V i K Rb ”-4 7. 104 _ 023.6% 181 10% 238.6% 261 059.57. 264 to... 9 09991999 0 l. 0 61'5— 1.0 WS— T (K) a Figure.3:(1/T)dp/dT versus T for all KRb alloy samples. (iQmK'zl ( 4.2/TH Aln p/AT) 58 120 - 80- KNa. 9000(1) m) . . 1% 73 ” :1: 2.4% 39 0 2.4% 26 . .A 136 :12 I — 0% 0.41 ' ' I I I ' =1: ' 1 >1: * O * * O C A O o O. 0 ‘ ‘ A o A A A A Figure.4:(1/T)dp/dT versus T for all _K_Na alloy samples. 59 1... o I 120— g. I . I 7“ I I I E O O O . - " '- . I : E 80"00 (a)... _. I o. A i A Ag.$8; A A g AAA A 6‘9’69913‘00 “ 0.1 Q A I A _=_ 40— <1 A :: PA 5... \ “it as ' C(AT.%) 1% (norm 0 -A - 1 14 . A 0.35 5.2 __A L' Mge 0.1 2.2 O 0.03 0.38 -40_ Pure Li I 0.14 L l ' L l i o 1 2 3 T(K) Figure.5:(1/T)dp/dT versus ‘1‘ for dilute Eng alloy samples. 60 o OO z‘if‘H'" _*. 4.4 .3 ._ _. _ _. (34'* .. .PDC' (:2313‘? A; “ ‘4 A1 11! . 0“ o OQDA A -21— 0‘ O ‘ A . 9A A A ‘4” 1x 0 E o‘ A «c “6" To 3 o p. U -8- a: .A .A E L" : -101—O Lng P(10°ilm) —- 1% 1.4 >1< 3% 3.2 _12_ O 10% 9.5 ‘ 020% 15.0 A32% 16.5 0 A49% 7.4 -14t- A ._l 1 l 1 O 1 2 3 4 T(K) Figure.6:(1/T)dp/dT versus T for all concentrated Lng alloy samples. 61 Table 1 8(10'6K'2) sample B(a) B(b) B(c) B(ex) l_(_Rb 12.5 3.6 12.3i0.5 .1913 7.5:0.5 £3113 3.1 2.2 1.5:0.1 a--Kus et al. b--Screened Coulomb potential c--Hu Gaussian Table 1 lists the values of the slopes (i.e. B) Of the alloys, and the values of theoretical calculations for comparison. B(a) was obtained by Kus et al, and is the best calculated value for K as discussed in chapter 2. The B(a) were calculated without the Debye temperature approximation and with a quite good pseudopotential for Rb in K. B(a) also took into account tluarnass difference between host and impurity ions. The screened Coulomb pseudopotential is considered“ a reasonable approximation for heterovalent impurity scattering. B(b) is evaluated from Taylor‘s expression for such a potential for Mg in Li with the screening length determined from de/dc data. Considering the approximations involved in the Koshino-Taylor theory, the agreement between the experiments and the theories is good. 62 As mentioned in Chapter 2, very recently S. Hu and A.W. Overhauser proposed a model“6 in which the many-body electron-phonon vertex correction, together with a partial failure of Migdal‘s theorem, leads to a new temperature-dependent term in the electrical resistivity caused by elastic impurity scattering. Based on a Gaussian potential for Rb in K they calculated the magnitude of B (B(c) in table 1) from Taylor‘s expression and the magnitude of the electron—phonon vertex correction term, and they argued that the magnitude Of the Koshino-Taylor term should be smaller than the experimental results and thus the many body effects Of the electron-phonon interaction should account for the T2 term in the KRb alloy resistivities. However, in their evaluations, they do not justify the determination of the parameter a, which is essential to the magnitude of their calculations. Considering the agreement of experiment results with the better calculation results B(a) and B(b) in table 1, we think it is unlikely that the electron-phonon vertex correction is the dominant contribution to the T2 behavior in the resistivities of these three alloy systems. Fig.8 shows a detailed (A + pOB) versus pO plot for all the KNa data. The fact that all of the data fall on a single line, suggests that the magnitude of T2 term is not sensitive to segregation in KNa alloys. 4.1.3 High Temperature Term The Debye temperature Of Li (OD=369K) is so high that electron- phonon scattering is unimportant in our current temperature range. As shown in Figs.5 and 6, for the samples containing up to 10 at.% Mg (OD-300K) no significant high temperature deviations are observed. (For the samples containing higher than 10% Mg, the data are dominated by the low temperature anomaly). 63 As discussed in Ch.1, for pure K (OD=90K) electron—phonon scattering can be neglected below 1K, since phonon-drag greatly reduces the Bloch T5 component of the N-electron-phonon resistivity and the exponential U- electron-phonon component becomes very small below 1K. When Rb or Na is added into K, the resistivities at high temperatures begin to increase faster than that Of pure K, as depicted in Figs.3 and 4. It seems that the data can be scaled by p0, as illustrated in Fig.10. We see there that the values of X for KNa all fall on a single curve, to within the uncertainties, as also do the values of KRb alloys up to 23.6%. The 1.3% KRb data which were measured by M.L. Haerle in the early days have a large uncertainty, and the deviation from the curve for 38.6% KRb may be due to the decreasing of the Debye temperature by Rb as will be discussed below. 64 b 7 (MBA ) (ms/.1 ) (10’13flm/K2) 0) l 1-KFlb 2!“- 1 _. 1-KNa ‘ . 4-Lng r 1 1 L 0 1 2 3 s (10'8ilm) Figure.7:The coefficients of the T2 term A + BpO versus pO for KRb, KNa, and L_ng alloy samples. The data were obtained by averaging over the flat regions on figs.3,4,5. 65 KNa (mag) (16'?) m/KZ) ( “p119 K I i I l 0 20 40 60 80 no (10"“0 m) Figure.8:The coefficients of the T2 term A + Boo versus pO {kw all KNa alloy samples. Notice the different pO values for each Na concentration. 66 .26 ') K") x 110'“ Figure.9:X versus 1' for KRb and KNa alloys. The solid curves are fit to Eq.4.10, and the broken lines are fit to Eq.4.5. 67 The different horizontal positions of the two curves are just the difference between the two 83 as indicated in Eq.4.4. The fact that such normalization brings the data for each alloy onto a single curve means that f and g are both approximately proportional to p0 for not too high concentration alloys. Fits of X=28+DT3 for the high temperatures are shown in Fig.9 as the solid lines, and this suggests that n s 0‘ poT . n~5 (4.5) Due to the experimental uncertainties n only can be determined to within 42n26. As mentioned in Ch.2, there is a theoretical prediction for a term similar to Eq.4.5 by Y. Kagan and A.P. Zhernov61 . In their calculation of coherent electron-impurity scattering, they obtained a term proportional to pOT5 due to scattering by the deformed phonon spectrum. However, the leading term, apOTS, was predicted to be positive for KRb but negative for KNa; this contradicts the experimental results. Alternatively, quenching of phonon-drag might provide an explanation for a T5 term. The effects of impurities quenching phonon-drag can also be seen in the thermoelectric ratio data of the alloys, as we will present 3 later. We noted above that phonon-drag effects largely reduce the N- electron—phonon Bloch T5 term in pure K. As impurities are added to K, the T5 N-electron-phonon component might become visible if the impurities quench phonon-drag. In such a case, we would expect the magnitude Of T5 3 component (T in 1/po/dT) to grow as p0 increases. To compare the magnitude with such a theory, we plot the Ekin and Maxfield theoretical calculation of the Bloch T5 term of pure K in the limit of no phonon-drag -16T5 as the dotted line in Fig.4. ( = 3.SX10 Q m/TS). We see that N Pe_p adding the Bloch Tl5 term (dotted line) to the pure K data (solid line) _.1l 68 could give comparable magnitudes to account for the increases of the dilute KNa and KRb data at high temperatures (see Fig.3). For more concentrated alloys we would then have to invoke decreasing of 0D due to alloying to further increase the magnitudes of electron—phonon scattering c0 +(1— of both Normal and Umklapp scattering processes. ealloy‘ imp c)0 host: W410“ will decrease Upon adding Rb (o = 55K) impurity to K. The magnitude of the Bloch T5 component (Normal part) is predicted to be proportional 1x3 1/08 , and the magnitude of the Umklapp part of electron- phonon scattering should grow fast in alloys with very high Rb concentration, as illustrated in the extreme case of the data of relatively pure Rb in Fig.2 where the turn up at high temperature is believed to be dominated by U-electron-phonon scattering process. Therefore, as the Rb concentration further increases, we would expected the magnitudes of both the normal T5 term and the Umklapp component tc>increase in high concentration KRb alloys; the U-electron—phonon component dominates iJT‘the very high concentration KRb alloys; and the behavior finally approaches that of Rb. In fact a rough estimation show that both the Normal T5 term and the Umklapp term are needed to account for the big magnitudes of the high concentration KRb alloys data. Hence the high temperature resistivity behavior shown in Fig.3 can be attributed to the decreases Of the Debye temperatures Of the alloys. 4.1.4 Low Temperature Anomaly 4.1.4.a Temperature and p9 Dependences Of the Anomaly As illustrated in Figs. 3, 4, 5, and 6, a low temperature resistivity anomaly begins to develop as pO increases in all three alloy systems. In particular, Fig.6 illustrates how in Lng this anomaly---plotted in the form of (1/T)dp/dT--crosses the zero line (indicating a resistivity 69 minimum) and turns down enormously as p0 is further increased. The L_i_Mg alloy system was chosen because it gives such a high p0 and yet has a relatively simple electronic structure. As displayed in Fig.9 the low temperature resistivity anomaly f(pO,T) for all KNa and KRb alloys is proportional to p0. The large turn up at low temperatures in dilute EMg alloy data complicates a similar analysis, and we will demonstrate below in Fig.13 the linear p0 dependence of dilute L_i_Mg data by a different plot. TO analyze in detail how the low temperature anomaly varies with T, we first consider the three temperature dependences expected for localization,7 electron-electron interaction7 , and the Kondo effectu7. f a -T (localization) (4.6) 1/2 . f a -T (electron interaction) (4.7) f a -lnT (Kondo effect) (4.8) -1 '3/2 -2 for which (1/T)dp/dT should be proportional to T , T‘ , and T respectively. When we plot (1/T)dp/dT as a function of these three powers of T for various samples, we find that for all the KRb and KNa samples and for dilute Lng the graphs with the T—1 abscissa give slightly better fits than those with T"3/2 , and that the T”2 graphs give the worst fits. (Caution should be exercised in concluding that a specific form is appropriate.) Examples of such fits are shown in Fig.10 for a 23.6% KRb alloy. The dashed, long dashed, and dotted curves in Fig.11 illustrate how the three fits to the 23.6% KRb alloy data in Fig.10 look when converted into a plot of (1/T)dp/dT versus T. We note that all three curves fall below the data at sufficiently high temperatures. This is due to the onset of the extra term discussed in the previous section. The solid curves in Fig.11 and in Fig.9 illustrate how a resistivity term linear in T (the 70 .long—dashed line) along with an additional T5 term fit the 23.6% KRb data. A similar fit for the KNa alloys is illustrated by the solid line in Fig.9. 71 40(3 g, dlng (mm/k2) 10C) 40C) 1()0 *2 3/2 -1 - Figure.10:(I/T)dp/dT versus T , T , and T for the 23.6% KRb alloy. 72 250e- 200-— "x 150—- ;\, is X 100— . 23.6% KFlb 50-5 1 l L 0 0.5 1 1.5 T(K) Figure.11:X versus T (Ewe 23.6% KRb. The long dashed, dashed, and dotted curves are fit to Eqs.4.7, 4.6, and 4.8 respectively. The solid curve is fit to Eq.4.6 along with an additional TD term. 73 If we choose two high-temperature terms (i.e: T5 and an exponential) to adjust, we can fit the entire range of data for each alloy--to within experimental uncertainty—-using each Of the three forms given in Eq.4.6, 4.7, 4.8. We thus cannot absolutely rule out any one of these three alternatives. In the absence of detailed knowledge of the magnitudes of these two high temperature terms, we choose tO parameterize our results using the low temperature form which, by itself, fits the data over the widest temperature range. This is f or T, Combining this best fit with the fact that fa pO we offer as an approximate empirical equation for our data at low temperature p=pO+AT2+BpOT2-CpOT (1.9) If Eq.4.9 is correct, then plots of dp/dT versus T for the data should give straight lines which do not pass through the origin. The intercept with the abscissa should give the coefficient Cpo. Such plots are given in Figs.12 and.13. We see that for KRb, KNa, and dilute Lng the data appear to be consistent with straight lines, corresponding to a resistivity anomaly of the form f m T. Fig.13 also shows that the data begin to deviate from a straight line for Lng alloys with Mg concentration higher than 10%. Fig.14 contains the coefficient CpO versus p0. We see that for 7 all alloys with p0<1O’ am, the data are consistent with a po dependence to within the uncertainties. On the other hand, when the Lng alloy data are extended to po>1o"7 0m, the data increase faster than p0. Therefore, both the temperature dependences and the residual resistivity dependences Of the low temperature resistivity anomaly of Lng alloy system seems to be 74 A p / AT ( 10'13nm/K) (a I 9 59.5% ‘ . * 38.6% K-Rb' 23.6% . 9.4% . 1.3% K-Nao 1% 0 1% Figure.12:dp/dT versus T for KRb and KNa alloys. 22" ,o 1.. 7°?” " ‘ 7"” L £112.. . ' 1 dT 2(A+BPO)T C 1 l l L l P 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ' T(K) (10"3Om/k) Ap/AT 75 ’ f as" 0 20% ' 10% * 3% o 1% 10.35% l i 3 Figure.l3:do/dT versus 1' for £113 alloys. Intercept C (10“ (Tm/K) 0.1 76 l lilJ l [LIL I [ll 1 10 100 1000 p. (nilm) Figure.14:Intercepts of the data in Fig.11 and Fig.12 versus p0. 77 separable into two categories by p0 = 10—7Qm. To seelxfifiithe T and p0 dependences of the anomaly for high concentration Lng alloys, we plot the -3/2 —1 '3/2 (1/T)dp/dT data versus T and T in Fig.15. This shows that the T form becomes a better fit as the Mg concentration increases. Fig.16 is a plot cfl’ the coefficients of the T-3/2 7 term in (1/T)dp/dT versus pO for all Gm are forced to this form). We see how 5 samples (i.e. the data for p0<10- the data change from a pO dependence (dashed line) to pO /2 (solid line) as p0 inceases. To summarize the experimental data, we rewrite the best fit form for the low temperature resistivity anomaly as ’7 pa=f— CpOT for p0<10 pm, (1.10) pairs-ng/ZT'3/2 for pO>1o'7nm (4.11) where the power of the temperature dependence in Eq.4.10 can only be determined to between 1/2 and 2, due to the uncertainties. For the 32% Lng samples we found that heat treatment begins to affect tie resistivity anonaly; annealing always reduces the magnitude of the anomaly. Also a K cold-welding technique was necessary for making the potential leads of the 32% Lng samples, since the soldering technique was found to sometimes cause anomalous behavior. Thus we only consider the 32% Lng sample which was well annealed and with good potential leads, a detailed discuss of the other 32% Lng data is in Ap.2. 4.1.4.b Comparison with Theoretical Models As discussed above, the anomaly contributes a negative resistivity term at low temperatures, and its magnitnuhs grows with increasing p0. In the following we compare the data with models which might explain such behavior. These models are (a) the Kondo effect, (1/T)dP/dT (10”om ‘2) 78 -3/2 '3/2 T 0 10 20 30 l l I 1% Lng i <<————() "-————>- 10% Lng "‘ \ \3Q%Lng ._ I I.— — .4 i\ — l l l l l l | O 2 4 6 8 1 12 14 T“ (K") Figure.15:(l/T)dp/dT versus T“1 and T73/2 for 1%, alloys. 10%, and 32% Lng 79 (b) two level systems, (0) ineffectiveness condition for electron—phonon scattering, (d) localization and interaction. The detailed descriptions of these models are given in Ch.2. (a) The Kondo effect 'To test if the anomaly is caused by the Kondo effect, we63 measured the magnetic field dependence of the anomaly. In Fig.17 the open symbols are with no external magnetic field and the filled ones are with a 0.2 Tesla field. Both the 9.7% KRb and 1% Lng alloy data show essentially no 1nagnetic field dependence. Applying the theory mentioned in Ch.2, a field of 0.2 T should decrease the Kondo temperature by about 0.3Ku8, which means approximately the patterns of filled symbol should move towards low temperature by about 0.3K. Therefore, the Kondo effect seems to be ruled (out. Moreover, as we will describe later, we find no Kondo-type anomalies in the thermoelectric properties. (b) Two level system Two level systems were, for a while, proposed to explain the negative logariiflunic temperature dependent resistivities in structurally disordered metals.“9 The anomaly for TLS is expected to be independent of magnetic field. Moreover, the extra parameter T in the TLS logarithmic temperature K dependent form permits an improvement in the logarithmic fit to the p(T) of 1O-7Qm. To compare the predicted magnitude of interaction effects with the experimental data, the coefficients of the T—3/2 term have been estimated for our KRb, KNa, and Lng alloys from plots such as the one shown in Fig.10. We plot these coefficients versus pO on a log-log plot in Fig.16. The solid line and the broken solid line in Fig.16 are the predicted behavior an° interactitni effects in Li and K based alloys respectively. For small values of p0 the data in Fig.16 are consistent with a linear dependence or1 p0, as indicated by the dashed line, and the data fall well above the solid and brohailines. This shows that the quantum effects are too small to account for the anomaly in these samples. As po increases above about 10..7 0m, the data break away from the dashed line, and appear to approach the solid line, although the high concentration alloys show large variations from sample to sample for a given Mg concentration. These variations are associated with different heat treatments, which suggests that microscopic differences within these alloys affect the data. The 49% Lng alloy has a much bigger magnitude than the prediction. Changing the electron density due to alloying can also affect the magnitude of the interaction effect, however, as estimated in Ch.2 such density changes should only turn the solid line up a little at large pO in Fig.16. 83 0 II ° 8 1.5- O i \ K—Hb _ TpdT! (10'7K’) 1.0 - .. U-Mg \ A. 05 — I _ 0 AB=0.2T A o A B=0 . l O 1 1:3 0.5 T(K)(l Figure.17:(1/T)dlnp/dT versus T for 9.75m: and 1% L_i_Mg alloys for magnetic fields of B - 0 (open symbols) and B . 0.2T (filled symbols) . 84 __ ——— -—- ——— -—- ——— -—— l—— .——— ——4 14C)” 30 _ AT 1 l _. St E o 20 - C3 A CL :: A *— 1: \\ 13‘ 10 - J L- “— — O c . . . , c eipilcnh O 0307. 00470 O 1.3 'I. 0.131 V 9.4 VI 1.01 5 9.47. 1.04 ('0 1- I 23.6% 1.81 1 . 1 i 0.0 ' 0.5 1.0 T(K) Figure.18:(1/T)dp/dT versus T for KRb alloys with the solid lines fit to the ineffectiveness of electron-phonon scattering model by Kaveeh and Wiser. The dashed lines indicate the behavior expected from the standard theory. This figure is taken from ref.45. 85 Other possibilities for increasing the anomaly magnitudes in the high concentration Lng alloys are: a) since annealed samples always show smallen‘rnagnitudes than do unannealed samples, while their pO's differ not much, (Ap.2) the disorder formed in the very high concentration alloys :3eems to enhance the electron-electron interaction over the simple theory. b) the solid line in Fig.16 completely neglects the localization effect. However, localization might become important in very high concentration Lng alloys, since the argument that the localization effect in Lng alloys is smaller than the interaction effect is based on the assumption that the "nonmav'rmsistivity is dominated by the Koshino-Taylor T2 term. This assumption may no longer be true in the very high concentration Lng alloy, especially in the sample with 49% Mg. 4.1.4.0 Conclusions Concerning the Low Temperature Resistivity Anomaly We conclude that a change in the low temperature resistivity Of our data takes place at p0: 10—7flm. (A) For KRb, KNa, and 1.1113 samples with pO < 10'7om, the resistivity anomaly is approximately proportional to pOT, and is not sensitive to an applied magnetic field, nor to the spatial distribution of the impurities. The anomaly issruM; due to the Kondo effect, or to localization and interaction effects. The anomaly can be fit by thelKaveh and Wiser ineffectiveness electron—phonon scattering model, if substantial variation in the cutoff phonon wave vector is allowed. However, it is not yet clear whether the anomaly might be caused by two-level effects or by other mechanisms. (B) For LIMg samples with p0 > 10—7Qm, the anomaly approximately . 5/2 '3/2 varies as the form predicted for the interaction effect, (or p0 T ), and its magnitude is about what would be expected from electron-electron 86 interaction theory. For very high concentration Lng alloys the details of the microscopic structure of the samples seem to affect the anomaly. This behavior is not understood. 4.2 Thermoelectric ratio G In addition to the resistivity, we also always measured the thermoelectric ratio G. As described in the above section, when the (L = 2.44x10’8V2/K2). and measuring p temperature is low enough, L = L 0 0 and G provide complete information about electron transport properties. This is true for most of our samples at temperatures below 1K. As discussed in ch.2, the thermoelectric ratio arises fundamentally in two different ways when a temperature gradient is applied to a metal: either ‘because of tuna thermal "diffusion of the conduction electrons, or because of direct transfer of momentum from excited lattice wave (phonons) to the conduction electrons, the latter being known as "phonon-drag". The free electron theory predicted that 0 = (LO/L)[GO — bT2 + o(1/T)exp(—o*/T)] (4.12) where G0 2 -4YT is due to normal phonon drag, and the last term is an Umklapp phonon is a contribution from thermal diffusion of conduction electrons, drag term appropriate to the alkali metals with bcc lattice structure. At low temperatures where L=L and the exponential term becomes negligible, 0 Eq.4.12 reduces to a simple form 0 = 00 — 0T2. (4°13) 87 In the following section we present our experimental data.cfi'1nn~e alkali metals and their alloys, and discuss the results in terms of Eq.4.12 and Eq.4.13. 4.2.1 The Lorentz ratio L As shown in Fig.20, several Ls Of different samples were measured to asertairitnua general temperature dependence of L(T). The values of L are determined from the Wiedemann-Franz law QR L g TAT 3; (1.111) where K and o are the thermal and electrical conductivities respectively, Q is the heating power, R is the sample resistance, and AT is the temperature gradient caused by Q. From Fig.20 we see that for pure K and KRb samples at low temperatures below about 1.1K, L(T) is very close toiL 1.03s 0’ L(T)/LO $0.97. For quantitative analysis of G of K and K based alloys we thus concentrwune upon TS 1.1K. Above 1K the values of L(T) for K and KRb 70 as due to samples turn down, as shown in Fig.20 . This can be explained the fact that.aM;'these intermediate temperatures, electron-phonon interactions produce changes in electron energy which are large compared to kT, and but changes in electron momentum that are relatively small; thus tflmethermal conductivity is reduced much more than the electrical conductivity. The L(T) of the 32%LMg alloy approaches L at very low 0 temperatures and has a different behavior at high tenperatures then expected from the above simple electron-phonon scattering consideration. Apparently the electron behavior in very concentrated L_i_Mg alloys is more cmnnplex. However, as we will display later, for not very high concentration Lng alloys the electron behavior seems simple. 88 4.2.2 G Data Figs. 21, 22, 23 illustrate the C data for KRb, KNa, and L_i_Mg alloys. Several points are immediately obvious. For Pure K and Rb samples, the data behave as expected from Eq.4.12. The solid lines are fit to Eq.4.12. with L=LO. The deduced parameters from the fit are 00 = - 0.08v", b = 0.48v'1k‘2, c = 6200 V-1K, and 0* = 23K fcw~1< and 0 = 0 1, b =0.21v'1K72, c = 561 v'lk, and o* = 9.7K for Rb. For pure Li 1.06v' and Lng samples with Mg concentration up to 20 at%, tHDldithln the uncertainty one hardly sees phonon drag effects, due to the high Debye temperatures of the samples. The flat data means that the electron diffusion effectG(3 in Eq.4.12 is dominant in all of those samples. For samples with Mg concentrations up to 32% in Li we begin to see deviation from horizontal line at low temperatures, and the deviation becomes prominent in 49% Lng sample. The deviation at low temperatures indicates that the electron behavior no longer can be described by simple free- electron theories, and such behavior also show in L(T) and the1~esistivity behavior as discussed in the above section. 89 1. Cu 1. 120— 0, 1. __ 4.5... ' C) O E ‘55 (3C1"Cf) 9%.. v - i. -— ‘8 33 ‘A A A _. A :1 8 Q 6 0 0‘. O ‘0 elf a 5 40- Q E. \ .— 9? C(AT.%) 8 (nflm) 0 - 1 14 . A 0.35 5.2 L' M94 0.1 2.2 O 0.03 0.38 -40_ pure L1. ' 0.14 1 L L 1 l 1 o 2 3 T(K) Figure.l9:(1/T)dp/dT versus T for dilute Lng alloys BH.Cn the soil 1 lines fit to the ineffectiveness of electron-phonon scattering model. 90 ~32%L1Mg 1.2 '- / r— .0 O) I Figure.20:The Lorentz ratio L versus T for 32% Lng and 2.2%KRb1alloys, and pure K. 91 <— Pure Rb v V o v (M, vvv ' o 8 ‘3‘63‘ O‘AOA A ‘A I I . . I ' O O OO O O O O I O o 4—PUI'9K (01— 1'00 Figure.21:G versus T for all KRb alloys, and pure K and Rb . 92 0.5 o W") K-Na 901108) m) c1 f%> '73 __ . 2.4% 39 2'0 . 24% 26 A 1% 22 1 l 1 L / 0 1 2 3 4 T (K) Figure.22:G versus T for all KNa alloys. and pure K . 93 4.2.3 The electron diffusion component GO by extrapolating the data in Figs.21, 22, and 23 to We determined GO zero temperature. Generally, 00 can be written as G0 = (e/ef)E. (4.14) Here e is the electron charge. (which give a negative sign in Eq.4.14) The dimensionless parameter for free-electron like metal (essentially spherical Fermi surface) has the form t = 1 + dln i/dlnel8 + At (4.15) f where 2 is the electron mean free path and A5 is the manny-body contrilmxtions. Thus G is determined essentially by the properties of the 0 Fermi surface of the conduction electrons and the electron scattering processes. The electron diffusion component Go of free-electron like metals enltxeseparated into different scattering contributions by the Gorter-Nordheim relation 00 = (08G: + ppGg)/po (4.16) where the subscripts a and p represent the two different scatterers e.g. the alloy hmnndty'"a" and the impurities and defects in the relatively + p . Due to the lack of detail knowledge of a P the residual impurities and defects inside the relatively pure K, Rb, and pure samples "p". p0 = p Li, here we only focus on G arising from electron scattering by the O deliberately added impurities. Eq.4.16 can be rewriten to eliminate the variable pa. _ a p _ a G0 - G0 + (pp/p0)(GO GO) (4.17) 94 Since G3, same purity host metal, a plot of G GS, and pp are assumed constant for a alloy system made from the 0 versus (1/p0) should yield a straight a line with intercept GO and slope pp(Gg - 0:), and this is displayed in Fig.24. For dilute alloys the data show a linear behavior. It is easy to see from the above equations that a straight line relationship suggests the Fermi surface of the host metal has not been affected by the amount of alloying solute involved. Hence, from Fig.21 we can conclude that the K Fermi surface appreciably begins to change at Rb concentrations about 39 at% , and from Fig.23, Li Fermi surface changes take plaeseu.Mg concentratiomsiulto about 3%. It is interesting to note that no significant Fermi surface changes even up to this high concentration of Rb in K. Rb appears to fit very well into the K lattice; for the atomic volumes of the two metals do not differ much and the two metals form a continuous range of solid solution with a very narrow liquid + solid region in the phase diagram. The G0 due to KRb, KNa, and Lng alloying are listed in table 2, which are Obtained from the intercepts in Fig.24. 95 Table 2 00(v’1) Alloy G0(exp.) £(exp.) E(N&T) ka +0.50:0.05 '1.1i0.1 -O.3iZO.5 KNa +0.07i0.05 '0.2i0.1 +0.9i20.5 lgMg -O.2:0.1 +1.0:O.5 37 N&T--Nielsen and Taylor From Eq.4.15 we see that without the many-body effect correction E is explicitly dependent on the scattering, namely (dlnl/dlne)e . For elastic f scattering one would expect that the electron mean free path 2 would increase with e. For example, with unscreened Coulonb scattering, 9. is proportional to 22, and for perfect screening or hard sphere scattering, 2 is independent of 6. Thus the value of 6 should be in the range of 1 ~ 3. From the experiment results and Eq.4.15, we see that the effects of the inany-body domiruflmithe sign as well as the magnitude of the thermal diffusion effects. The signs of A6 were estimated by Nielsen and Taylor37 to be negative, which is in agreement with the experimental results. However, the predicted magnitudes are not quite agree, which be may due to, as emphasized by Nielsen and Taylor, the large uncertainty arising from the choice of potential, since no accurate pseudopotentials for Rb and Na 96 impurities inside K seem available. It should be pointed out that the listed theoretical values were fit to the experimental results of the thermoelectric power of KRb and KNa alloys at 3K measured by Guenault and 36 MacDonald . From Figs.21, 22, we see that G obtained by the data at 3K 0 are a very rough approximation, which neglect the effect of phonon-drag below 0.3K. Such an approximation decreases the values of GO -1 0.3V for _K_Rb and KNa alloys, and this will makes 6 close to the listed by about theoretical values. 4.2.4 The Phonon Drag Components The effects of alloying on the phonon drag components of KRb and KNa alloys can be seen from Figs.21 and 22. Both Normal and Umklapp phonon drag (in terms of Eq.4.12) are quenched more and more as the impurity concentration increases, i.e., the magnitudes of these two terms decrease as p0 increases (this is also true for the segregated KNa samples). A more detailed illustration of the Normal phonon drag component of pure K and KRb is shown Fig.25, where we plot G versus T2 for the low temperature data. We see that for both pure K and dilute _K_Rb alloys the data fall nicely on straight lines as expected from Eq.4.13 ,and the slope decreases as the impurities concentration increases as expected from impurity quenching phonon drag. Fig.25 also illustrates that the Normal phonon drag component deviates from the simple T2 behavior by the 9.4% KRb sample. The observation of impurity quenching phonon drag is consistent with the resistivity measurements as discussed in section 1. (SN) 97 1.‘ . . . . o - PUO U U‘MQ “' o 0.03% ‘ 0.1% . 0.35% F - 1% ‘ 3% __ . 10% + 20% ' 32% - 49% _O O O O O O O ._ + + + _.A :: A4. A f. . ..A ..5 . r . xaa; A o xA A A 5 - ; ' x "v ' ' l 1 i 0 1 2 3 T (K) Figure.23:G versus T for all png alloys. and pure Li. 6.1V‘) 98 r“ A ._.g 1- UMg .1 KFib — Figure.24:G 0 , KNa p. “’7' -1 versus pO for all alloy samples. 99 4.2.5 Conclusions (1) We obtairuxiireliable values of electron thermal diffusion components GO for KRb, KNa, and Lng alloys. (2) The thermoelectric properties of dilute KRb, KNa, and Lng alloys are simple, as expected from the free-electron theories. For L_ng samples with Mg concentrations higher than about 30%, the thermoelectric properties become much more complicated. (3).Alloying dose not significantly change the Fermi surfaces of the host for:hmnmity concentrations up to about 39% in KRb and 3% in Lng alloys. (4) The many-body effects dominate the sign as well as the magnitudes of the thermal diffusion component. (5) Adding impurities to K quenches both N-phonon-drag and U-phonon- drag. 100 Chapter 5 ELECTRON-ELECTRON SCATTERING There has been considerable interest in the past few years in electron-electron scattering in very pure metals at low temperatures. As discussed in Ch.2, the electron-electron scattering contribution to the resistivity pe_e(T) in alkali metals is predicted to be dominated by U- scattering and to have the form: 2 pe_gT) = AT (5.1) The experimental data and the best theoretical calculations are listed in table.3, where the experimental values of A for K and Li are obtained frmmi Fig.8 and A for Na from Fig.2. Table 3 A (meK_2) metal A(MD) A(exp.) (bcc alkali) fcc Al Li 2.1 27:2 Na 1.4 1.7:0.3 K 1 7 2.4:0.4 Al 4.3 2.7(Rt) MD --MacDonald et al. 198167 Rt --Ribot et al. 198165 101 In K, Na, and Al, the values of A for freely hanging bulk samples are in satisfactoy agreement with the latest calculations and are insensitive to small amounts of impurities or defectsgu’23’65 The unexpectedly large value of A in very pure Li, on theeother hand, is a puzzle. Although this A appears to be insensitive to small amounts of impurities or defects::’u"23 it is about an order Of magnitude larger than the predicted value. As we mentioned in Ch.1 Li has the most distorted spherical Fermi surface among the alkali metals, and it undergoes a 75%‘ complete martensitic transformation at about 75K to a more complex structure?7 for which the Fermi surface either contacts or nearly contacts several Brillouin zone boundaries that have substantial energy gaps. The discrepancy between experiment and theory is surely related to the fact that the prediction was made for Li in BCC crystal structure with a nearly spherical Fermi surface which does not contact Brillouin zone boundaries. There are two very different ways in which the phase transformation might produce a value of A larger than predicted: (A) indirectly or (B) directly. (A) The argument for an indirect mechanism was made by Sinvani et al,3 4 as follows. As derived in Ch.2, for a metal with a spherical Fermi surface which does not contact Brillouin zone boundaries and in which the dominant (e.g. impurity) scattering mechanism is isotropic in k-space, rununal electron-electron scattering does not contribute to p(T). The only cxnmribution is due to Umklapp electron—electron scattering which, as listed in table 3, is estimated for BCC Li at 2.1 f9 mK—z. If, instead, the dominant scattering mechanism is anisotropic in k-space, then normal electron-electron scattering can contribute to A, (i.e. Eq.2.24 does not vanish), and A is the sum of Umklapp and Normal terms, A = A + A . Sinvani et al. argued that there should be two sources of anisotropy in L1; 102 1) Due to the large deviation of the Fermi surface of Li from sphericity in the 110 direction, it seems necessary to use a multiple plane-wave pseudo- wavefunction, which leads to a mildly anisotropic T(R) even for the scattering cd‘anl electron by an isotropic impurity. They calculated this 2 "impurity-scattering dominated" AN to be about 0.8 f0 mK- . 2) A large 1 "effective dislocation density" might be introduced intoimichwing its transformation, andinight produce highly anisotropic scattering. This would put AN into its anisotropic limit AZ, they estimated A: to lie between 10 and 40f0 mK72. Therefore, they argued that A: is the explanation of the large experimental value of 27f0 mK72. (Hung their estimates, the impurity-scattering dominated A = AU + A? would be about 3f0 mK—g (B) The argument for a direct mechanism is that the large value of 27f!) mKw‘2 for AU arises from greatly increased Umklapp electron-electron scattering associated with the complex Fermi surface of Li at low temperatures. That is, that the AU calculated for bcc lattice is just not relevent to real Li at low temperatures. To discriminate between these two mechanisms, we have measured the temperature-dependent resistivities of pure Li and several dilute Lng alloys from 4K down to 0.1K. Lng alloys with concentrations up to 20 at.% undergo martensitic transformations upon cooling69 similar to that for pure Li. As noted before, except at the lowest temperatures, addition of Mg to L1 should add to equation 5.1 an inelastic impurity scattering term of the form BpOT2 to give p(T) = AT2 + BpOT2 (5.2) 103 As is true f1n~1nost impurities in most host metals, we would expect Mg to scatter electrons nearly isotropically in Li, consistent with the estimate) of Sinvani et al. for impurity-dominated scattering. Addition of Mg to Li should then lead to one of two possible outcomes: (1) Indirect. An earlier paper by Kaveh and Wiser68 (1982) indicated that AN should decrease rapidly as isotropic impurity scattering increases. Usiru; information given by Sinvani et al (1981), we estimate that addition N N N of only about 0.05% of Mg to Li would reduce A from ”Aa to “A1, and that extrapolation of A+Bp0 for more concentrated Lng alloys back to the Mg- free limit should yield A=AU + AN“4me K72. i (ii) Direct. If the effects of anisotropic scattering are small, then A=AU will be large, and will hardly change upon addition of a small.emnount of Mg to Li. Ihl this case, a linear extraplation should yield a coefficient A of about 27 fnm K72. In figure 26 we replot, from figure 7, the T2 coefficient<fl’p(T) against pO for pure Li and the Lng alloys, along with the broken curve which represents (nu: estimate for the indirect mechanism using the parameters of Sinvani et al. The alloy data are plotted as open triangles and the pure Li as a full triangle. We see that the alloy data extrapolate linearly back to the pure Li value of A=27f2mK-2. The data thus follow the form of equation (5.2) with a constant value of A, and not the broken curve estimated for the indirect mechanism of Sinvani et al. ‘We conclude that the unexpectedly large A thA.is due not to dominance of anisotropic.scattering, but rather to the change in Fermi surface which accompanies the transformation that Li undergoes at about 75K. We note that the T2 coefficients for dilute KRb and dilute KNa alloys (see figure 7) also extrapolate directly to the coefficient for pure bulk 104 K. It thus appears that anisotropic scattering is.rKM: needed to explain the behavicnn' of the T2 components of the resistivities of either L1 or K in the vicinity of 1K. Figure.25:G versus T2 for KRb alloys. 105 80 O‘ 0 #890110 1111(2) 4“ O 20 L l l 0 10 20 30 9,, (nil m1 Figure.26:The values of A + BpO determined from figure 7 against p0 for Lng alloys (open triangles) and pure Li (fWflJ. triangle). ‘The broken curve represents the expected behavior for Lng with anisotropic scattering, using the parameters suggests by Sinvani et al. Data for KRb alloys (full circles) and for pure K (Open circle) are given in the same units in the inset. Note that both the Lng and KRb data extrapolate linearly to their respective pure metal data points. 10. 11. 12. 13. 14. 15. 106 REFERENCES (Chapters 1 - 5) R. Taylor, C.R. Leavens, and R.C. Shukla, Solid State Comm.15, 809 (1976); C.R. Leavens and R. Taylor, J.Physis. F.5, 1969 (1978). H. van Kempen, J.S. Lass, J.H. Ribot, and P. Wyder, Phys. Rev. Lett. 31. 1574 (1976). J.A. Rowlands, (3.1)uvvury, and 3.8. Woods, Phys. Rev. Lett. 59, 1201 (1978) C.W. Lee, W.P Pratt Jr., J.A. Rowlands, and P.A. Schroeder, Phys. Rev. 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Overhauser,Phys.Rev.Lett.53,64 (1984). M.Kaveh and N.Wiser,J.Phys.F 12. 935 (1982). 0.00mi and S.B.woods, Solid State Ccmmun.53,223 (1985). 110 70. George Terence Meaden, "Electrical Resistance of Metals", Plenum Press,1965.71. N. Wiser, Contemp.Phys. 25, 211 (1984). 111 Chapter 6 ELECTRONIC TRANSPORT PROPERTIES OF THIN POTASSIUM WIRES BELOW 1K 6.1. DERIVATIVE OF ELECTRICAL RESISTIVITY, dp/dT 6.1.1 Introduction. We recently published1 measurements of the temperature dependent electrical resistivities p(T) below 1K of thin, high purity potassium (K) wires prepared and cooled to low temperatures in the presence of He gas. As ShOWTlIJlI?ig. 1, wires thinner than diameter d a 1 mm showed anomalous deviations from the simple T2 variation expected for electron-electron scattering. Wires thinner than the elastic bulk mean-free-path for impurity scattering (9. = 0.2 mm) displayed negative values of dp/dT, the tnanperature derivative of the resistivity. When Rb was added to the K to reduce 9. to ~ 0.02 mm, a d = 0.25 mm wire of the dilute alloy showed a much smaller anomaly, as we discuss below. We concluded that we were seeing a size-effect, and tentatively attributed the anomalous behavior to the Gurzhi effect? a reduction in electron-surface scattering due to normal electron—electron scattering. It was noted1 that wires prepared and cooled in Ar gas, or prepared in He but cooled in vacuum, displayed anomalies havingiflmesame form as those prepared and cooled in He, but with magnitudes generally representative of thicker wires and with more variation from wire to wire of a given thickness. This somewhat different behavior, illustrated in Fig. 2, was rationalized3 as being due to the different atmospheres. Indeed, part of the stimulus for the investigation in ref. 1 of the behavior of thin K wires cooled in He gas was the apparent 112 contradiction between observations by Rowlands et al.uof anomalous behavior in d = 0.8 mm K wires cooled in He, and those by Lee et al. of little or no such anomalies in d = 0.9 mm K wires cooled in Ar? The measurements of ref. 1 stimulated proposals of three alternative models for the anomalous behavior: (1) localization effects;6 (2) reduction in electron-surface scattering due to electron-phonon scattering;7 and (3) interference between electron-surface scattering and electron-electron scattering.8 The authors of these models, as well those of a more recent paper? all challenged the applicability of the Gurzhi effect to the data of ref. 1. Of particular interest was the localization model? which predicted that vauni dp/dT < 0, the anomalous behavior should be proportional to L2, the square of the length of the wire, provided that a parameter v, defined as the rnunber (n? distinct conduction channels in the thin wire, is constant. This model stimulated us to make new measurements on thin wires with diameter d g 0.1 mm, but with different lengths. The new samples had bulk resistance ratios RRR = R(295K)/R(OK) which differed from that for the old samples. We were thus able to investigate the effects of different bulk electron mean-free-paths on the behavior of thin K wires. This paper is a report of the results of our measurements. In addition to our measurements of dp/dT, we also routinely measured the thermoelectric ratio G of our samples, which provides information that complements dp/dT. The behavior of G is generally compatible with the patterns described in the 113 current paper. Details of the behavior of G are given in the following paper]0 This paper is organized as follows. In section II we briefly review the various models proposed to explain the original size-effect data on dp/dT. In section III we discuss experimental details, focussing upon sample characterization and modifications of previous procedures. In section IV we present our data and some data analysis. Section V contains a summary of the most important features of the data and a more detailed analysis, as vmfll as comparisons with various theories. Section VI contains our conclusions. 6.1.2 Theoretical Background. For a simple nearly-free-electron metal such as K (neglecting, for the moment, the possibility of a Charge-Density-Wave (CDW) ground state11). we would expect1 the very low temperature electrical resistivity of a thick sample to have the form: . (1) 2 2 p - 91 + p(T) - pi + AT + BpiT + pe-ph Here pi is the temperature independent residual resistivity due to impurities in the sample and p(T) is composed of an electron-electron component AT? an inelastic electron-impurity component BpiTz, and an electron-munmnicomponent p that drops off exponentially with e-ph 114 decreasing temperaturel2 In bulk K, p is negligibly small below about e-ph 1K. We can eliminate the unknown constant term p1 by measuring dp/dT, the temperature derivative of p. If we neglect p ‘then from Eqn. 1 we e-ph’ expect dp/dT to have the form dp/dT = 2(A + Bpi)T, (2) so that a plot of dp/dT versus T should yield a straight line passing through the origin. Indeed, as illustrated by the +-symbols in Fig. 1, Ekui. 2 is closely obeyed for bulk samples of high purity K. However, for thin wires we find large deviations from this equation. It is these deviations which we call the "anomaly". The model originally proposed to explain the data in Fig. 1 is the (Surzhi effect? In this effect, normal electron-electron scattering (NEES) (whid1,tw'itself, cannot contribute to the electrical resistivity in a rnetal \vith a spherical Fermi surface and isotropic scattering) reduces the probability that a typical electron will reach the sample surface where it is diffusely scattered, compared to this probability in the absence of NEES. In ref. 1, we noted that our data were not in the regime where the Gurzmm. effect predominates, but we suggested that the effect might persist down to where our data lay. Subsequent theoretical work, both 9 13 numerical, and analytical, has shown that this suggestion is incorrect. 115 The next model, proposed by de Gennero and Rettori? involved an interference between NEES and surface scattering. This model predicted an anomaly in p(T) that was proportional to T? and that passed through a maximum value as a function of l/d at 51./d z 1. The anomaly was a complex function of l/d, the form of which had be to determined numerically for the range of values of Q/d of interest. 11m predicted T2 temperature dependence disagreed with the data of Fig. 1. In additicni,x"ecent theoretical analysis9 calls into question the validity of this model. Kaveh and Wiser7 proposed an alternative explanation for the anomaly in terms of a reduction in the effectiveness of electron-surface scattering due to electron-phonon scattering. This model predicted the form: p (T) « -3(p2 5 anom o/pspi)(2R'/d)T (3) where pS is the residual resistivity due to surface scattering, and p0 = pi + pS is the total residual resistivity of the wire. Ihitflme"thin wire" limit we expect pS a p1(l/d), and this model predicts panom a (1 -+ l/d)22 provickui that no contaminants are introduced into the sample during thinning. The predicted temperature dependence of T5 was much too rapid to explain the data of Fig. 1. Finally, Farrell et al.6 proposed that the data of ref. 1 were due to a combination of Charge-Density-Wave (CDW) effects for thicker wireslu plus localization effects for sufficiently thin, high purity wires. The CDW 116 model was used to describe the data when dp/dT > 0, and the localization effects were assumed to take over for dp/dT < 0. This model predicted that localization would predominate for 9./d > 1, provided that certain other conditions were also satisfied. The model also predicted a variation of dp/dT with L2 in the localization regime, provided that the number of parallel conducting channels remained constant. Our new data allow us to investigate whether two different phenomena are present, but we were not able to obtain enough data on samples with different lengths and negative values of dp/dT to convincingly test for an L2 variation of the anomaly under the conditions predicted. 6.1.3 Sample Characterization and Experimental Procedure. A) Samples and sample preparation The samples were prepared from high purity (99.95%) K purchased from Mine Safety Appliances (MSA) Division of Gallard Schlesinger Corp. The K was supplied in 5 gm or 20 gm glass ampoules filled with argon gas. Different batches of K from MSA contain different amounts of trace impurities, as shown by the different impurity tables supplied by the manufacturer. For example, the Na content of the K studied in ref. 1 and the purer K in the present study were both listed as containing 15 ppm Na, while the less pure K in the present study was listed as 48 ppm Na. The typical bulk RRR = R(295K)/R(0K) (or bulk residual resistivities pi) for the three different batches of K were also different. To account for these differences, we compare the data for thin samples against data for thick ("bulk") samples prepared from the same batch. This procedure is not 117 perfect, since thick samples prepared from different ampoules can show different values of pi, as can, in some cases, even samples prepared from the same ampoule. Variations from within a single ampoule are normally only 10—20‘% (see table 3.1 in ref. 3), but variations between ampoules can be much larger. These variations can produce significant scatter in the inferred thin sample behavior. The samples in ref. 1 were made from material taken from several different 5 gm ampoules purchased at the same time. The data of ref. 3 show that the bulk RRR varied somewhat from ampoule to ampoule. Based upon all of the available data, we assume a bulk RRR = 7300 for both the samples prepared in He and those prepared in Ar. The samples used in the current study were made from two different sets of K. The first set of new samples was fabricated from a single 20 gm ampoule. The data from this set are shown below in Fig. 6. Measurements on one thick sample from this ampoule gave RRR = 4800, which we take as the bulk RRR. The second set of new samples was fabricated from several 5 gm ampoules purchased together at a later time. The data from this set are shown in Fig. 5 below. From measurements on two different samples which were in close agreement, the bulk RRR for the set is taken as RRR = 1700. We show in ref. 10 that the behavior of G supports this division of our samples into batches having three different bulk purities. For simplicity, we refer to samples as being from batches labelled by K(7300), K(4800), and K(1700). 2 From Eqn. 2, we would expect the T coefficient of p(T) for bulk samples to vary linearly with pi (i.e. inversely with RRR). Such behavior is illustrated in Fig. 3 for samples prepared in both He and Ar. Note 118 especially the substantial variations in p1 (and the associated values of A + Bpi) for the samples of K(7300) from different ampoules. The filled symbols in Fig. 3 indicate the values of A + Bpi used as the reference "bulk" behavior in Figs. 1, 2, 5, and 6. The sample wires were extruded at room temperature from stainless steel presses through stainless steel dies. As in ref. 1, two wires were extruded and mounted together in a sample can inside a He filled glove box. The sample can was sealed with an In o-ring and transferred to a dilution refrigerator, with which the samples were cooled. When the sample can was filled with He gas at atmospheric pressure, molecular sieve was used to absorb the residual He gas in the can at low temperatures. In the current measurements, the sieve was freshly baked in vacuum at 3000C for each sample pair, except in one case noted below. To better clean the sieve, this temperature was 1000C higher than that used in ref. 1. Although the surfaces of the samples were still shiny when the sample can was sealed, by the time the thin samples had been measured and brought back up to room temperature their surfaces were normally covered with a thin film of white material. To check whether this white material was essential to the anomalous behavior, for the last few thin samples studied we further cleaned the atmosphere in the sample can by wrapping a thin copper foil freshly coated with K around the inside surface of the can. This foil was inserted into the can along with the freshly baked sieve and the can was initially sealed a day before the samples were extruded and mounted. The large area of fresh K cleaned the He atmosphere in the sample can both before and after the samples were mounted. For the first time, 119 this procedure yielded surfaces of thin wires (d‘g 0.1 mm) that were still shiny when the samples were warmed back to room temperature after being measured. The lengths of the thinnest wires in ref. 1 were about 10 mm, although length measurements were not generally recorded. The carefully measured lengths of the new wires varied from 2 mm to 14 mm. For wires of diameter d 3 0.25mm, the sample length was determined by two K potential leads of the same diameter as the sample. Thinner wires were connected between 1 mm diam. wires as described in ref. 1. B) Measuring System and Procedures. The basic measuring system and procedures are described elsewhere!’15 We note here only one important improvement upon previous procedure, namely that we were able to extend our experimental resolution from a part in 107 to a few parts in 108 using computer averaging of the last digit of our current comparator. C) Sample Thickness Determination. It is important for this study to know the diameters of the samples. For samples with d 3 0.25 mm, measurements of sample length L (with an uncertainty of < 10%) and room temperature resistance R(295K) (with an uncertainty of < 1%), combined with the known p(295K) for K, showed that the diameters of the sample wires were closely equal to the diameters of the dies through which the samples were extruded. For samples with d4: 0.1 mm, however, surface corrosion usually caused the effective diameter of the 120 sample to 12 the K(7300) As approach the others. The remainder 131 of the Ar and Vacuum cooled data fall among the overlapping K(4800) and K(1700) data sets. If each set of data is separately fit to the form (1/d)? then the K(7300) data suggest n a 1/2, while the K(4800) and K(1700) data seem to give 1 _<_ n i 2. The As for d = 0.25 mm K(0.1%Rb) and d a 0.1 mm K(0.08%Rb) are the smallest in Fig. 8, and both are consistent with no anomaly at all. Aside from its slightly higher purity, the only treatment of the K(7300) sample set which we know differed from that for the K(4800) and K(1700) sets was the 1000C lower temperature at which the molecular sieve was baked. Perhaps this lower baking temperature led to greater surface contamination of the K(7300) samples which, in turn, led to larger anomalies in thick samples. The data of Fig. 4 are consistent with somewhat greater surface corrosion in the K(7300) samples. Since Kaveh and Wiser7 have predicted that A should vary with l/d, we 1~eplot A versus l/d in Fig. 9. This plot rescales each data set along the abscissa, but leaves the ordinate of each point unchanged. Fig. 9 brings t1m21((7300) and K(4800) data closer together than in Fig. 8, but separates the K(4800) and K(1700) data. Because the residual resistivities of the samples in each set are not simple functions of 1/d (see Fig. 4), we examine in Fig. 10 how A varies with "corrected" sample diameter, pO - p1 or (1/d'). If the increase in pO is due mainly to non-uniform thinning of the samples, then the data should vary more nearly as (1/d') than as (1/d). The data for the different samples in Fig. 10 overlap rather like those in Fig. 8. However, the data for the thinner samples clearly lie more to the right in Fig. 10 than in Fig. 8 relative to the thicker sample data, and the data for the purer 132 samples--K(7300) and K(4800)—~lie to the right of the data for the least pure samples--K(1700). There are two alternative interpretations of the data in Figs. 8 and 10. 1) The higher than expected values of pO in Fig. 4 are due primarily to non-uniform diameters, in which case Fig. 10 indicates that A increases less rapidly than linearly with (LA?) and may even be "saturating" for the thinnest wires. 2) The higher than expected values of pO for the thinnest samples are due in large measure to internal contamination of the thinner wires, leading to less rapid than expected irun°eases 1J1 A witli (1/d'). To test this latter alternative, we compared the increases in A for wires thinned by extrusion through smaller dies with the increases due to explicit surface corrosion--as indicated by the arrows in Figs. 11a and 11b. We conclude that changes in A due to corrosion are generally'iJuiistinguishable from those due to thinning. The one obvious exception is the d = 0.8 mm sample, for which the effect of corrosion is unusually large in the plot versus 1/d, but unusually small in the plot versus pO - pi. Both behaviors are compatible with highly non-uniform thinning, leading to an effective average diameter much smaller than the nominal one. In the following paper}0 we examine the behavior of the thermoelectric ratio 0 of these same samples. We see from Fig. 5 in ref. 10 that unmet of our data points fall on three separate lines on a Gorter— Nordheim plot, in a manner consistent with the separation into three different sample sets that we have been assuming in the current paper. Only three data points deviate significantly from the three expected lines in Fig. 5 of ref. 10. Two are for the K(1700) samples which have unusually 133 low values of pO in Fig. 4. The deviation in Fig. 5 of ref. 10 is due primarily to these unusual values of po. The third is the corroded d = 0.8 mm sample of K(1700)-"the six-pointed star in Fig. 5 of the current paper. Its Go is consistent with a much smaller effective diameter, as we have just argued. 6.1.5 Summary of Behavior and Comparison with Theories. From the foregoing experimental data, we summarize the behavior of dp/dT as follows: (1) All of our pure K samples thinner than d - 1 mm show anomalies. (2) The anomalies are generally larger the thinner the wires and the purer the host material, but there are some large variations for thin wires of a given diameter. (3) Both white and shiny surfaces yield anomalies, but anomalies associated with the former are usually larger. (4) If 9. is greatly reduced by adding small amounts of Rb, the anomalies become tcm>small to reliably isolate. (5) The anomalies all have the same temperature dependence for a given anomaly size, independent of wire thickness, tnxlk resistivity, or the gas in which the wire is prepared and cooled. (6) Surface corrosion both thins the wires and increases the sizes of the auuxnalies--as a corollary, the anomaly is invariably larger in the sample mounted first in a given sample pair. (7) There is no systematic length dependence of anomalies for which dp/dT remains positive. We do not have enough data to draw conclusions about length dependence for auuxnalies for which dp/dT becomes negative. (8) The variation of the anomaly size with 1/d apparently has a different form for the K(7300) samples from that for the K(4800) and K(1700) samples. (9) The data are not unique functions 134 of any of the following variables: 1/d, i/d, or pO - 91' (10) Below about 1.2 K, where electron-phonon scattering is small, the anomaly can be Parameterized as p(T) a T7/3 independent of the anomaly size. This form seems to require an additional negative contribution to dp/dT for data in the vicinity of 1.2K. We now consider whether any of the proposed models can describe these results. We omit the Gurzhi effect, which can be ruled out on theoretical grounds.6”9’13 A combination of CDW effects with localization seems unlikely' because, as illustrated in Fig. 7. the temperature dependence of the anomaly follows a single pattern. We thus do not seem to have a combination of two quite different phenomena-~a CDW effect for thicker samples and localization for thinner ones-~which just accidentally smoothly match together. The model of interference effects between normal electron-electron scattering and surface scattering makes two predictions: (1) a T2 variation of the anomaly; and (2) that the magnitude of the anomaly passes through a rnaximum value at 24Ri==1-2. The first prediction is at variance with the temperature dependent behavior shown in Fig. 7. The second prediction could be compatible with the behavior shown in Fig. 10 if the maximwn occurs for the largest values of 9./d we have been able to produce, i.e. i/d a 2.5.. We note that this model rms recently been criticized theoretically.9 Reduction of the effectiveness of surface scattering by normal electron-phonon scattering can be ruled out as the primary source of the 135 anomaly by the temperature dependence of the data illustrated in Fig. 7. The anomaly varies approximately as TY/3 rather than the T5 variation predicted by this model. If this T7/3 term is taken as reliable, then there i_s_ need for a small additional negative contribution to dp/dT in the vicinity of 1.2K, which could be due to this T5 term. Finally, we consider what alternatives exist to these models. If we combine the separate contributions of interference effects plus reduction of surface scattering by normal electron—phonon scattering, we can generate a temperature dependence of about T.2'33 But we have the claim9 that the interference effect model is incorrect. The possibility of a complete explanation in terms of a CDW state coupled with scattering from the sample surface cannot be ruled out. A CDW-based model has the advantages that: (1) it provides for unexpected variations from sample to sample due to differences in orientation of the CDW domains, and (2) it provides for deviations from simple T2 behavior. It has the disadvantages that: (1) no CDW—based model without localization yet predicts a negative dp/dT; (2) it is hard to see why the data for a given bulk purity fall into such generally nice patterns for a given sample purity if the anomaly is sensitively dependent upon the CDW domain structure, and (3) there appears to be no need for a CDW model to explain the behavior of either bulk K or dilute K-based alloys.“5 6.1.6 Conclusions and Suggestions for Further Work. We have summarized the results of our measurements in the numbered items in the previous section. We conclude that there does appear to be an anomalous "size-effect" in thin K wires, the nature of which is not yet 136 clear. Especially unclear is the contribution and role of surface corrosion. We know of no theory that seems able to adequately describe all of our data. Additional measurements which might help clarify the situation include the following. (1) Measurements on still thinner wires with both :shiny and corroded surfaces. Such measurements are not easy, because very thin wires take heavy pressure and long times to extrude, so that extrusion is difficult and the wire surfaces will invariably corroma. (2) Measurements with still higher purity bulk K. This will require additional purification of commercial K by vacuum distillation. (3) Incorporating white, corroded material into the body of K wires prepared and cooled in Re, to clearly establish whether or not introducing such material into K produces an anomaly. This work was supported in part by the NSF Divisioncfi'Matafieus Research through Low Temperature Physics grants DMR-83-03206, DMR-83-05289, and DMR-8700900. The authors would like to thank Z.-Z. Yu for allowing us to include in this paper some of his unpublished data. 10. 11. 12. 13. 137 REFERENCES (Chapter 6.1 — 6.16) Z.-Z.Yu, M.Haerle, J.W.Zwart, J.Bass, W.P.Pratt,Jr., and P.A.Schroeder, Phys. Rev. Lett. 52, 368 (1984). R.N.Gurzhi, Zh. Eksp. Teor. Fiz. 44, 771 (1963). Z.-Z.Yu, The Electrical Resistivity and The Thermoelectricliatio of K, Na, Li, Rb, and K-Rb alloys from 0.07K to 4.2K. Ph.D. Thesis, Michigan State University, 1984 (Unpublished). J.A.Rowlands, C.Duvvury, and 3.8. Woods, Phys. Rev.1mmt. 49,12m1 (1978). C.W.Lee, M.l“}hnm"le, V.Heinen, J.Bass, W.P.Pratt Jr., and P.A. Schroeder, Phys. Rev. 525, 1411 (1982). M.E.Farrell, M.F.Bishop, N. Kumar, and W.E. Lawrence, Phys.Rev.Lett.55, 626 (1985). M.Kaveh and N.Wiser, J. Phys. F. 15, L195 (1985). S.DeGennaro and A.Rettori, J. Phys. F.14, L237 (1984); ibid. 15, 2177 (1985). D.Movshovitz and N.Wiser, J. Phys. F. 17. 985 (1987). J.Zhao, Z.-Z.Yu, W.P.Pratt Jr., P.A.Schroeder, and J.Bass, Phys. Rev. 5 (1988), Following Paper. A.W.Overhauser, Adv. in Phys. 21, 343 (1978). H.van Kempen, J.S.Lass, J.H.Ribot, and P. Wyder, Phys. fhsv. Lett. _31, 1574 (1976). D.A.Stump (Unpublished). 14 15. 16. 17. 138 M.F.Bishop and W.E.Lawrence, Phys. Rev. B32, 7009 (1985). D.L.Edmunds, W.P.Pratt Jr., and J.A.Rowlands, Rev. Sci. Inst. 51, 1516 (1980). J. Bass, Landolt—Bornstein New Series, Group III, Vol. 155, "Metals: Electronic Transport Phenomena", Springer Verlag, Berlin, 1982. The two exceptions to the pattern of increasing A with increasing (1/d) in Fig. 8 are the corroded d = 0.8 mm K(1700) wire and one shiny K(4800) wire. As noted above, we believe that the d = 0.8 mm K(1700) wire has portions much thinner than its average diameter. The reason for the unexpectedly small A for the one shiny K(4800) wire is less clear, but may involve accidental internal contamination. 139 K(7300) (In mIK) dDIdT Q -41- (p0 m) (mm 5 in. W m H w 20 21 29 30 1m 99 1” ms - -11¢ Corr. 7 Days 0 I a: 1 .QI’CCXIV‘f I m I 10) Corroded 5 Days 0.08et.%KRb l l L i l l 0 0.4 0.8 1.2 1.6 T(K) 0‘ Noobo‘-memo“ uomuommuuo _000000090090r' _ - Figure 1: a) dp/dT versus 1' for the K(7300) samples. ’which were prepared and cooled in a He atmosphere. This figure is taken from Ref. 1, but the data have been renormalized as described in the text, a few plotting errors have been corrected, some of the nominal sample diameters have been revised as described in the text, and some additional samples from ref. 3 have been added. Two nearly iden- tical samples were always prepared and measured together; for the samples in Fig. 1, the data for both wires in a pair was always fairly close. For simplicity, we omit the pairs of the thicker samples; paired samples are indicated by brackets. Two pairs of sales were annealed at room temperature to thin them further after their initial measurements; the arrows indicate the changes which occurred due to these annealings. P(19(f{lnfl Figure 1: b) p(T) versus T for selected data from Fig. 1a. 1a were integrated by hand. qualitatively similar form to the data of Fig. 1a. T 1 ‘ K(7300) 21— + .4... 1.— 1” I r- . X. 1. X . +I§Po °o -1... -2. ‘2 d (pflm) (mm) L. + 10 1.5 I 16 0.50 X 20 0.25 -3- o 29 049 9112 0.09 .. 0125 0.08 -41- i 0 (14 0.8 The data of Fig. Note that the integrated data have 141 K(7300) A X \ E c: H 1.. “a \ Q Q d L Gas U 2 (p0 m) (mm) (mm) V V V {Y 56 0.10 Ar v 76 0,10 Ar -3- {O 71 0.09 V O 76 0.09 V {A 30 0.11 7 Ar '4r A 43 0.11 8 Ar ('1' 19 0.25 V _5_ x 20 0.25 v {t 75 0.09 v f! 67 0.09 V l i 1 l l L l l o 0.4 0.8 1.2 1.6 T(K) Figure 2: dp/dT versus ‘1' for thin K(7300) wires cooled in an Ar atmosphere or in partial vacuum. This figure is taken from ref. 2, but the data have been renormalized as described in the text. The straight line indicating bulk behavior is the same line as in Fig. 1. The samples connected by brackets were prepared and measured together. 142 .. 31- N x E .. 9 ~ + (5.2; v + K(7300) :2 Ar 1 K(7300) AAK17300) " Ha I K(4800) eoK11700) 1 L I 1 l 0 10 20 30 40 p, (pOm) Figure 3: The T2 coefficient A + Bpi for "bulk" (d 1 1 mm) samples of K as a function of residual resistivity pi. The filled symbols indicate the data used to determine "bulk" behavior for K(7300), K(4800), and K(1700) (for definitions see text) hiflgs. 1, 2, 5, and 6, respectively. The K(7300) wires had d = 1.5 mm, the K(4800) and K(1700) wires had d : 1.0 m. The straight line is a best fit to all of the data. Its slope of 2.5 x 10.5152 is somewhat larger than those for Rb (1.3 x 10-5K-2) and Na (0.8 x 10'5K'2) im- purities in K. 143 0 (mm) 0.4 0.2 0.1 0.08 0.06 150 , , , r 1 HeoK11700) He IK14800) HeAK17300) v +K17300) “ + Ar xK(7300) s-Shiny ‘A A. I . 1001" Oo-Corroded in Air I I o 5 O ‘ II 3 w. ~1<117001 Q. (K(4800) 50 -K(7300) 1 LA, 1 1 L l I All 1 0 4 8 12 16 20 1/d (mrfi‘) Figure 4: p versus lid for K wires of different diameter and different bulk purity. The solid lines have a slope taken from ref. 16. The letter "3" indicates samples with shiny surfaces. 144 , o K(1700) ' .g o x \ E C t h —1 3 (p0 mHmrn) (mm) Q -1... (4- 42 1.0 11 U x 42 1.0 12 * 94 0.82 11 Conodedin Ah -2- {s 34 0.10 a 412 0.10 9 I 63 0.11 12 _3__ {c3 66 0.10 7 1 Shiny Surface (A. 63 0.11 2 A 65 0.11 5 '4- e 74 0.08 “a 84 0.08 g 1 Conoded 22 Days 0 75 0.08 2 5 (o 90 0.07 5 1 Corroded 35 Days ( V 67 009 12 - -192 0.09 12 O.1at.%KRb l l L l l 4 I l 0 0.4 0.8 1.2 1.6 T(K) Figure 5: dp/dT versus T for the K(1700) samples. Pairs of samples prepared together are designated by identical symbols, with the open symbol designating the sample prepared first. Note that in each pair the anomaly is always larger for the sample prepared first, and the anomaly is independent of sample length. After their initial cooling, some of the samples were given room temperature anneals and then cooled and measured again. The progression of behavior after such anneals is indicated by the arrows. Figure 6: 145 / v 50 K(4800) V o V V O I t t t .. y 3 it 0 \, a .7 . 3 £5 33 c: ~ ‘ O -2__ I p 0 'U - .. Q 1 d L ‘U 1. (PO 111) (mm) (mm) C)I I -4 x 15 1.0 O 8 O 52 0.09 8 . l ‘ _5_ ‘o 94 0.08 11’Drty 5"“ A (e 79 0.08 9 e 97 0.08 15 V 38 .11 -5" {v 55 3.10 ;)Sh1ny Surface 7 (:1 23 33; gicorroded 15 Days ‘ 1 A 99 0.06 9 Conoded 50 Days 0 49 0.08 15$h1ny Suriacs -8,- i I i l l l i 1 4 0 0u4 (L8 L2 L6 T(K) dp/dT versus T for the K(4800) samples. Pairs of samples prepared together are designated by identical symbols, with the open symbol designating the sample prepared first. Note that in each pair the anomaly is always larger for the sample prepared first, and the anomaly is independent of sample length, except for the sample pair denoted by diamonds. This pair was the only one that showed a length dependence approximately proportional to L? After their initial cooling, some of the samples were given room temperature anneals and then cooled and measured again. The progression of behavior after such anneals is indicated by the arrows. (film/K) dP/dT Figure 7: 146 o 6-- *v 5- =1: V 4.. 16 - V '0' . 3F 09‘ a I x+ . I 21- I ll ' A. 1+— / . - XAA ‘ a t . 14%;} ‘ ‘13 + 0 . .I E i 1 —4 I I. D D o _1.-. I -2... Q d K Gas ‘ O _3_ (p0m11mm) =1: 94 0.82 (17001He . v 56 0.07 Ar 0 o --4.- 049 0.08 (48001He U I 16 0.50 (73001He O c] _51 073 0.08 (170011-19 + 19 0.25 V X 20 0.25 (730011-18 _6_ A43 0.11 Ar 0112 0.09 (730011-10 C187 0.07 (48001He l 1 1 l ' ‘ l 1 0 0.4 0.8 1.2 1.6 T(K) Intercomparison of data sets with four different size anomalies involving wires having different thicknesses, different bulk RRRs, and/or cooled in different gases. The curves through the low temperature data are fits up to 1.2K to an equation of the form p(T) = (A +BpL)T2 - sz/B where (A + Bpl) is determined by the behavior of thick K wires. 147 HeOK11700) HeIK(4800) 3.. HeAK17300) l V +K17300) Ar xK173001 He*KRb s-Shiny F?— I (m le) l I O O 41- " xts . » Corroded in Air 0 + A Figure 8: A versus 1/d for the data of Figs. 1, 2, 5, and 6. A is the deviation at 1.0K of the anomalous values of dp/dT in Figs. 1, 2, 5. and 6 from the bulk behavior shown in each figure. The letter "3" indicates samples with shiny surfaces. 148 (Id 0 0.6 1.2 1.8 2.4 3.0 1* 1 1 1 I 1 r i r' '- HeoK11700) HelKi4800) 8F HeAK173001 I V +K173001 'AfXK17300) Ha*KRb _ S‘Shiny . . ‘ I 6- “ as o o E — o A ‘ A is o I I s—I c 4 1- : I x :3 0 + -oc-Corroded in Air I—s x 11- <3 A o 2- A Oo-—s ms 8:9 A T-1K L; L L 1 1 1 l l L l 0 2 4 6 8 10 Rid 110‘ mu?) Figure 9: A versus (RRR/d) for the data of Figs. 1, 2, 5, and 6. The scale for A versus i/d is given at the top of the graph. The letter ”3" indicates samples with shiny surfaces. A (10 lei 149 HGOK117001 HeIK14800) . BF' HGAK17300) V +K173001 Ar x517b3001 116* " s-Shiny ' ' ‘ I 6- A) O A O i— A‘ o O . I S x - 41- $ 0 x + ‘ oe-Corroded 1n Air 1— x-"S + + A O o o-s ‘ x ' T-1K * 1 i l l l l l l l l l oL—gL 20 40 50 go 100 120 po-pu (p0 m) Figure 10:A versus po - p1 for the data of Figs. 1, 2, 5, and 6. The letter "3" indicates samples with shiny surfaces. 150 HeoK(17001 HeIK14800) 8- HeAKi7300) V +K(7300) Ar xK173001 He*KRb _ “it 4— Corroded in Air (film/K) A 8 12 16 20 1/d (mn‘i‘) Figure 11:a) Figure 8 with arrows indicating changes in A which occur upon corrosion. (10 le1 A 151 HeoK117001 HeIK(48001 I 8- HeAK173001 v +Kf73001 Aer173001 HgakKRb I A I 4k/////’ )1 I A. + I .o 4-Corroded in Air + + T-1K 1 l 1 ‘ l ' Q‘pl (P0 m, Fig.11 b) corrosion. 120 Figure 10 with arrows indicating changes in A which occur upon 152 6.2 THERMOELECTRIC RATIO G. 6.2.1 Introduction. In the process of studying the low temperature electrical resistivity p(T) of thin Potassium (K) wires,“2 we also measured the thermoelectric ratio G of these same wires. In this paper we describe t1”: IWhNJltS we obtained, which shed additional light upon the behavior of thin K wires. As we discuss below, G provides information complementary to that obtained from p(T). We know of only one prior set of studies of the low temperature thermoelectric properties of thin K wires, the pioneering measurements by the Canadian group of D.K.C. MacDonald. MacDonald, Pearson, and Templeton3 measured the thermopower S of K wires with diameters ranging from d = 0.8 mm to 0.07 nun. These samples were not made from the same stock, and they found no systematic variation of properties of S with d. We will summarize their results below. This paper is organized as follows. In section II we review the results obtained by MacDonald et al. on the low temperature thermoelectric properties of high purity K and describe the expected behavior for G. In section III we present and analyze our data. Section IV contains a summary and conclusions. 6.2.2 Review of Previous Work and Background Information. 3 MacDonald et.al. fit their thermopower data for high purity K below about 3K to the equation: 153 s = AT + 8T3 + Cexp(-0/T). (1) The form of this equation was based upon the predicted behavior for a simple, free electron metal with a spherical Fermi surface that does not contact the Brillouin Zone boundary? The first term was attributed 1x3 the electron diffusion component of S, the second term to the Normal portion of the phonon-drag component, and the third term to the Umklapp portion of the phonon-drag component. They found values of A = (+ 0.5 + -1.0) x 10'8 V/K2 and B : (-0.15 + —0.30) x 10—8V/Ku for samples with RRRs ranging fwwnn 4000 1x3 10,000 and 0.07 mm i_d g_0.8 mm. Above about 1.5K, both the second and third terms in Eqn. 1 are significant, and they combine to produce a minimum in G at ~3.3K. Below 1K, where we will examine our data quantitatively, the third term in Eqn. 1 is completely negligible. The thermoelectric ratio G is related to S by the Eqn. G : S/LT (2) where L is the Lorenz ratio? Experimentally we have fimump that L is . . -8 2 2 equal to L0 (the Sommerf‘eld value of the Lorenz ratio, 2.44 x 10 V /K,) only at temperatures below 1K. For quantitative analysnscfl‘G we thus concentrate upon T 1 1K. From Eqns 1 and 2 we expect G to have the low temperature form: G:G +DT, (3) 154 where G0 = A/LO and Di: B/LO. The MacDonald et al. data described above gave GO z (+ 0.2 4 -0.4) v’1 and D = (- 0.06 + - 0.12) v‘1k‘? If two different scatterers are present, e.g. impurities "i" and the sample surface "s", then from the Gorter-Nordheim equation)4 we expect the value of Go to be: i 3 GO - (pi/po)Go + (pa/po)Go' (4a) Here 0; ammigai are, respectively, the bulk thermoelectric ratio and resistivity due to residual impurities, G: and pS are the thermoelectric rustic and resistivity due to surface scattering, and p0 : pi + p3. We can eliminate the variable pS from Eqn. (4a) and rewrite it as s i 3 G0 = C0 + (pi/p0)(GO - GO). (4b) Since 0:, 0;, and pi are assumed constant for a set of thin samplens from a given batmfli<3f K, a plot of GO versus (1/po) should yield a straight line . . s i s with intercept GO and slope pi(Go - GO). Eqns. 3 and 4b will tinnithe basis for analysis of our experimental data. Although Eqn. 3 was initially derived from a simple model ixnnalving ea free-electron diffusion term plus phonon-drag, we now know that its form is also appropriate to more modern analyses of low temperature thermopower 155 involving many-body contributions such as the Nielsen-Taylor effects“’6 and electron-phonon mass enhancement . 4 ’ 7 Nielsen and Taylor showed that a many-body contribution which they labelled "virtual recoil", provided a qualitative understanding of the nmnndty thermopowers produced by Na, Rb, and Cs inmnnrities in K. Another contribution which they labelled "phony- phonon-drag" gave a contribution to the diffusion thermopower in pure metals whicwirnhnicked the temperature dependence of phonon-drag. These effects complicate the analysis of experimental data, especially illii case such as ours where the total residual impurity content is small and its detailed composition is not known. We thus concentrate upon presenting our experimental fTfiMJltS, deriving values of the quantities in Eqn. 4b, and showing that the contribution of surface scattering to the thermopower is what would be expected if such scattering is completely diffuse. We continue norwfier to the two terms in Eqn. 3 as electron-diffusion and phonon-drag components, although the second term in Eqn. 3 may contain a contribution from "phony-phonon-drag". For a metal with an essentially spherical Fermi surface, like K, one should be able hovudte the low temperature diffusion term in the thermoelectric ratio to good approximation as:14 0 = (e/Ef)[(dln(nv)/dlnE)- (dan/dlnE]E_E. (5a) ‘ r Here ea is the electronic charge, n the electronic density of states, v the Fermi velocity, 2 the electron mean-free-path, and the logarithmic 156 iderivatives are to be evaluated at the the Fermi energy Ef. If the "nv" term is evaluated for a free-electron model, we obtain: 0 : -0.5[1 + d1nQ/d1nE]E v'1 (Sb) r We will use Eqns. 4b and 50 to find values of (dlni/dlnE) for both residual impurities in our samples and the sample surface. Eqn. 1 of ref. 1 and Eqn. 4 of the present paper demonstrate both the complementary nature of p and G, and an important fundamental difference between them. Eqn. 1 of ref. 1 shows that the contributions to p ffixnn two different sources of scattering are additive. For each scatterer, p increases linearly with the concentration of that scatterer, until time two concentrations become so large that the individual scatterers interact. In contrast, we see from Eqn. 4 above that, when one scatterer is strongly dominant, the value of 0 becomes independent of the concentration of both that scatterer and any minority scatterer. 6.2.3 Experimental Data. We have measured G on a variety of samples, as described in ref; 1. Following ref. 1, we categorize the samples as follows. Three sets were prepared and cooled in He gas; these sets are designated as K(7300), K(4800), auui K(1700), where the numbers refer to the measured residual resistance ratios (RRR : R(295K)/R(0K)) of bulk samples from each set. The fourth set contains samples of material from K(7300), but which were either prepared and cooled in Ar gas, or prepared in He and cooled in Vacuum. They will be designated by the symbols Ar or V, respectively. 157 Fig. 1 shows the C values for the K(7300) samples. The values cfl‘ dp/dT for these samples are given in Fig. 1 of ref. 1. If we examine first the data for the thickest--d = 1.5 mm-— samples (crosses), we find a relatively large, negative phonon-drag minimum with a magnitude > 3 V"1 at its lowest point at about 3.3K. Below 3.3K, the data rise smoothly to a low temperature limiting value which is close to zero and usually slightly negative. As the samples become thinner, G changes systematically; the phonon—drag minimum becomes shallower, and the low temperature limit G0 becomes more negative. According to the standard picture of phonon-drag in K, the phonon- drag minimum at ~ 3.3K involves a competition between a negatdAna "Normal" component and a positive "Umklapp" component].4 In this picture, the fact that the phonon-drag minimum becomes shallower with decreasing wire thickness and concurrent increase in pO indicates that the additional surface scattering in the thinner wires reduces either the Normal component alone, or both the Normal and Umklapp components together. As we noted above, many-body contributions to the temperature dependent porticniisf the thermopower may complicate this interpretation. The shift toward more negative values of G0 with decreasing sample thickness indicates: (a) that the dominant scatterer niiflmzsamples is changing from impurity scattering to surface scattering (possibly including effects of surface corrosion) as the sample thickness decreases, and (b) that surface scattering produces a negative contribution to 60' Similar behavior is shown in Figs. 2--Ar and Vac. cooled, 3--K(1700), and 4--K(4800). The behaviors of dp/dT for these same samples are shown in 158 Figs. 2, 5, sun: 6, respectively, in ref. 1. Comparison of Fig. 2 in the current paper with Figs. 1, 3, and 4, shows that the general behavitn" of G for samples prepared or cooled in Ar and Vacuum differs little from that for samples of the same diameter and purity prepared and cooled in He. To determine 0: and G6 as defined in Eqn. 4, we plot GO versus 1/po in Fig. 5. Aside from three data points? the data for the four sets of K fall on three straight lines, with most of the Ar and Vac data falling on the same line as the He data from the K(7300) batch of K. As expected from Eqn. 4, the slopes of these lines are in proportion to the values of pi for the different sample sets. Within experimental uncertainties, the data for the K(7300) and K(4800) samples determine the same values of G: and 6(1), namely G: z -0.55 1 0.1 V.1 and G; s -O.11 0.1 V11 The data for the K( 1700) samples determine the slightly different values 0: = - 0.65 1 0.1 V.1 and G; = + 0.1 1 0.1 11-] Within their mutual uncertainties, both sets 1 of data are consistent with the common values 0: = - 0.6 i 0.2 V- and G; = 0 + 0.2 VI‘ which we take as our best estimates for G: and 610. We note that this value of G; falls within the range of the values estimated from the data of MacDonald et 31.2 If we insert these estimates for G; and G: into Eqn. 5b, we find that 159 [dani/dlnE]E = - 1.: 0.3 (6a) r and [danS/dlnE]E = +0.2.: 0.3. (6b) r Here 21 and is are the electron mean-free-paths for scattering from impurities and the sample surface, respectively. The sign of'the impurity term (Eqn. 6a) is opposite to that expected for simple elastic scattering of electrons by impurities? This sign discrepancy argues for the presence of the many-body contributions of Nielsen and Taylor.6 To within experimental uncertainty, the surface term (Eqn. 6b) is consistent with the value zero. This is exactly what would be expected for completely diffuse surface scattering, in which an electron hitting the sample surface is scattered into a random direction independent of its incoming energy. According to Eqn. 3, we expect G(T) to vary as T2 for temperatures below the phonon-drag peak. This expectation is valid, to within experimental uncertainties, as illustrated in Fig. 6 with a subset of samples chosen to have minimal overlap of data. We have checked how the coefficient D varies with various parameters such as residual resistivity p0, nominal sample diameter d, etc. The best systematic variation was found to be with p81, as shown in Fig. 7. 160 6.2.4 Summary and Conclusions. From the foregoing data and analysis, we are able to drama the following conclusions. (1) The general form of G remains the same as samples are thinned, but the phonon-drag minimum generally becomes less negative and the low temperature electron diffusion limiting value Go becomes more negative. (2) The data for all four sets of samples are consistent with the Gorter-Nordheim rule, and mostly fall on three different straight lines, one for each bulk sample purity. These lines determine values of G; and G: which are the same for the K(4800) and K(7300) samples, but slightly different for the K(1700) samples. To within an experimental uncertainty of _+_ 0.2 VI‘ the data for all four sample sets 1 1 are consistent with the values: 0: z -0.6 1 0.2 V- and 0; z 0 _+ 0.2 V. This value for the surface term is compatible with completely diffuse surface scattering. (3) Below 1K, G varies approximately as DT? as expected from Eqn. 3, and the coefficient D varies approximately as pg] This work was supported in part by the NSF Division of Materials Research through Low Temperature Physics grants DMR-83—03206, DMR-83-05289, and DMR-87-00900. 161 REFERENCES (Chapter 6.2 - 6.24) J. Zhao, 1(J?. Pratt, Jr., H. Sato, P.A. Schroeder, and J. Bass, Phys. Phys. Rev. 5__, (1988) (preceding paper). Z.-Z. Yu, M. Haerle, J.W. Zwart, J. Bass, W.P. Pratt Jr., and P.A. Schroeder, Phys. Rev. Lett. 52, 368 (1984). D.K.C. MacDonald, W.E. Pearson, and I.M. Templeton, Proc. Roy. Soc. (London) 4255, 334 (1960). FU.J. Blatt, P.A. Schroeder, C.L. Foiles, and D. Greig, Thermo-electric Power of Metals, Plenum Press, N.Y., 1976. M.L. Haerle, W.P. Pratt Jr. and P.A. Schroeder, J. Phys. F. 13, L243 (1983). P.E. Nielsen and P.L. Taylor, Phys. Rev. 515,1HMH (1974) and references therein. J.L. Opsal, B.J. Thaler, and J. Bass, Phys. Rev. Lett. 35, 1211 (1976); B.J. Thaler, R. Fletcher, and J. Bass, J. Phys. F. 5, 131 (1978). [War the two data points for d : 0.10 mm samples of K(1700) -- see Fig. 6 of ref. 1, the deviation is due to the unusual values of p0 for these two samples (see Fig. 4 of ref. 1). Ekn'the datum point of the corroded d = 0.8 mm sample of K(1700)--six-pointed star in Fig. 5 of ref. 1--the deviation is probably related to the especially large amount of both bulk and surface corrosion in this sample. 0 ‘12}: Q d Gov-“23* ‘ (p0 ml (mm) todoao”.' + 10 1.5 32,o.* y 10 0.9 fit I A 11 0.9 -11 :3. I 16 0.50 e- (x 20 0.25 ,.. I 21 0.255 c C) 29 0.19 .. g 0 112 0.09 .* {A 111 0.08 o O 125 0.08 4- _ .0 -2... .v 0% A O 9 O I“ I . ° . m , o I O o " e v e _3_ a. O ' I ' " K(7300) ‘ . .m 4- + 1'“ 1 v 0 2 T(K) Figure 1: G versus T for He cooled K(7300) 163 0P 14! 4. + w *» K(7300) Wfiafl-fi' v 1++ { 11- 1| -11- A ‘7 f a ,7 '+ , A Q d Gas + 3 (pflmiimm) w ' (V 56 0.10 Ar l ‘9 -2- v 76 0.10 Ar + {I 71 0.09 V A v + + O 76 0.09‘V A 30 0.11 Ar A {A 43 0.11 Ar 4; 4 ,+ 19 0.253v X 20 0.25 V *3"{* 75 0.09 V A 67 0.09 V 1 I 1 ' o 1 2 3 4 T(K) Figure 2: G versus T for Ar and Vac cooled K. For the Vacuum cooled samples, about 10qu of residual He gas was left in the sample can at room temperature to ensure that the samples would cool properly. This gas affected the G data above 1K, which is thus not as reliable as the remainder of the data in this paper. G W") K(4800) 0" -—4 I ' ‘11:. Q d . , (pf) m1 (mm) ' 3 '1 x 15 1.0 1°." 839.. ,9 52 0.09 .d‘b O 534 ()JJB I (it 79 0.08 _,_ a 97 0.08 (V 38 0.11 17 (36 0.10 {I 58 0.08 1' o 87 0.07 °° A 99 0.06 ’ 9': o 49 0.08 3 _2__ ‘3 O o i O O 917 a :0. o v I 009 0' . O I v v e _3_ L l l 0 1 2 3 T(K) 164 Figure 3: G versus T for He cooled K(4800) 165 1—- K(1700) R d p d 0 (p0 m)(rnrn1 (p0 m) (mm) {1- 42 1.0 A 63 0-H x 42 1.0 A 65 0°” :1: 94 0.82 ' 74 0.08 (t 34 0.10 O 84 0.06 ‘1’ w 42 0.10 ‘9 75 0.03 0..s , {I 63 0.11 o 90 0.07 a». g o 86 0.10 v 67 0.09 it * 8v 0 1:" .3... . ~1L ‘ v v i t 8‘ v ' 8% “(p d' *1} i a: 3 ‘ f * 9 39 -2... e 9n- o "' I i l 1 0 1 2 3 T(K) Figure 4: G versus T for He cooled K(1700) 166 (Saw V + K(7300) Ar :1 K(7300) o K(1700) He{l K(4800) AK(7300) 1 1 L 1 1 ’1 1 1 0.04 0.06 0.08 0.100 Figure 5: 60 versus 1/po for the data of Figs. 1-4. 167 T (K) 0.4 0.6 0.8 ‘ 1.0 1.2 r; 11 (p0 111) (mm) 0.2- -O.8"' K(1700) -1.0 l l L l l l l 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 T2 (K2) Figure 6: Test of G(T) = Co 4» 0T2 below 1K for selected data from Fig. 1%. 168 HeoKi1700) He “((4800) He AK(7300) Ar XKWSOOD ’02 -0.3'- N... 'x 72 a -o.4- ‘0.5- 0 Figure 7: The T2 coefficient 0 from Fig. 6, versus p;: for the data of Figs. 1-4. 169 APPENDIX 1 Because it is important to the vertex correction for elastic scattering proposed by Hu and Overhauser, here we discuss the magnitude of B in Eq.2.3 iminnre detail. The magnitude of B dependents on two parameters: Debye temperature GD, (This is come from an approximation made for the density of phonon states at low temperatures), and the average momentum transfer per collision. 0D generally varys with temperature and the experimental measuring -3 D , this could give an error in methods. Since B is proportional to 0 evaluating B as high as 30%. From the experimental data we known GD of K anud Li have a smaller value at low temperatures than at high temperatures. The low temperature 0 for K and Li obtained from the specific heat D measurement are 90K and 350K respectively. The average momentum transfer per collision K is defined as P . K2(1-cosin9)dfl k k Pk.k(1-cosin9)dfl (A.1) where Pk‘k is the probability of electron with wave vector k scattering to state k‘, ( from Fermi‘s golden rule P kav2(K)), V(K) is the Fourier k‘ transform of the scattering potential, 9 is the angle between k‘ and i<, and K:|E-E‘l=2szin(G/2). Eq.A.1 can be written as 170 12kr K5V2(K)dK xiv = 3k 3 2 , (A.2) IO r K v (K)dK Since no reliable pseudopotentials of Rb impurity in K and Mg impurity in Li seem to be available, approximation has to be made about V(K) iJitarder to evaluate B. For a 8(r);xmential,\HK)=constant, corresponds to isotropic scattering, Eq.A.2 gives Kavz0.8(2kf)z2k This is a very rough f’ approximation, since 2k is the maximum momentum transfer, however, several f 30,35 auothor used this value to compare with their experimental results. An Gaussian potential V(r):V Oexp(-r2/azr:), where rS is the Wigner- Seitz radius was chose by S. Hu and 0verhouser661n their estimation of B. There are two independent parameters V0 and a. Kasa) changes significantly with a as showed in Fig.A.1, for examples when (1:1 Kav=0'52(2kf) and a:0.5 Kavz0.8(2kf), (this will result in Bs differ by a factor of two), thus without tim3_plstification of parameter a such a potential can not provide quantitatively calculation. A reasonable approximation of the scattering potential for heterovalent impurities Eng is the screened Coulomb potential Ze2 -ar V(P) = - (F—) e (A.3) In this potential the only parameter 1/a is the screening radius, which can be calculated, at least approximately, by the Fermi-Thomas method. This 171 gives a=1.5kf for Li, and with this value , KaV:O.77(2kf). The vaJima