Superconducting, structural, and magnetic properties of lithium and lithium-rich compounds

Lithium is generally considered to be a simple metal, given its simple electronic structure with one valence electron. It is considered to follow a nearly free electron model and have a nearly spherical Fermi surface. However, away from ambient conditions, the behavior of lithium become much less simple. Under high pressures, lithium undergoes a series of symmetry-breaking phase transitions, even a metal to insulator transition; at low temperatures, lithium also undergoes a temperature-driven martensitic transformation. In this work, these deviations from simple models in lithium are investigated, both at ambient pressure and under high pressures, from the relative high temperature phenomenon of melting to low temperature measurements of superconductivity.

SUPERCONDUCTING, STRUCTURAL, AND MAGNETIC PROPERTIES OF LITHIUM AND LITHIUM-RICH COMPOUNDS by Anne Marie Schaeffer Richards A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Department of Physics and Astronomy The University of Utah May 2016 Copyright © Anne Marie Schaeffer Richards 2016 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Anne Marie Schaeffer Richards has been approved by the following supervisory committee members: Stephan LeBohec Chair 11/24/2015 Date Approved Jordan Gerton Member 11/24/2015 Date Approved Eugene Mishchenko Member 11/24/2015 Date Approved Lowell Miyagi Member 11/24/2015 Date Approved Clayton Williams Member 11/24/2015 Date Approved and by Carleton DeTar Chair/Dean of the Department/College/School o f _____________Physics and Astronomy and by David B. Kieda, Dean of The Graduate School. ABSTRACT Lithium is generally considered to be a simple metal, given its simple electronic structure with one valence electron. It is considered to follow a nearly free electron model and have a nearly spherical Fermi surface. However, away from ambient conditions, the behavior of lithium becomes much less simple. Under high pressures, lithium undergoes a series of sym­metry breaking phase transitions, even a metal to insulator transition; at low temperatures, lithium also undergoes a temperature-driven martensitic transformation. In this work, these deviations from simple models in lithium are investigated, both at ambient pressure and under high pressures, from the relative high temperature phenomenon of melting, to low temperature measurements of superconductivity. CONTENTS A B S T R A C T ............................................................................................................................... iii LIST OF FIGURES ................................................................................................................ vii LIST OF T A B L E S .................................................................................................................... xvi A C K N O W L E D G M E N T S .................................................................................................... xvii CH A P T E R S 1......IN T R O D U C T IO N ........................................................................................................... 1 1 . 1 Alkali Metals.................................................................................................................. 1 1.2 Lithium.......................................................................................................................... 2 1.3 Quantum Solids.............................................................................................................. 2 1.3.1 Nearly Free Electron M o d e l ............................................................................. 3 1.4 Extreme Conditions....................................................................................................... 4 1.4.1 Pressure.................................................................................................................. 4 1.4.2 Temperature......................................................................................................... 5 1.5 High Pressure Melting.................................................................................................. 5 1.5.1 Quantum Melting............................................................................................... 7 1.6 Superconductivity......................................................................................................... 8 1.7 Crystal Structure........................................................................................................... 10 1.8 Fermi Surface ................................................................................................................ 14 2. E X PE R IM EN TA L T E C H N IQ U E S ........................................................................ 18 2.1 Pressure ............................................................................................................................ 18 2 .1 . 1 Diamond Anvil Cell ........................................................................................... 18 2.1.2 Paris-Edinburgh Press......................................................................................... 21 2.1.3 Pressure Measurement ...................................................................................... 22 2.1.3.1 Ruby Fluorescence .................................................................................... 22 2.1.3.2 Equation of State ...................................................................................... 26 2.2 Temperature .................................................................................................................. 27 2.2.1 Low Temperature ................................................................................................ 27 2.2.1.1 Cryostat ....................................................................................................... 27 2.2.2 High Temperature ................................................................................................ 28 2.2.2.1 Laser Heating............................................................................................. 29 2.2.2.2 Resistive Heating...................................................................................... 29 2.3 High Pressure Resistance Measurements................................................................. 30 2.3.1 Resistance for Detecting Melting of Metals ................................................. 36 2.3.2 Resistance for Detecting Superconductivity................................................. 38 2.4 AC Magnetic Susceptibility ......................................................................................... 40 2.5 de Haas van Alphen Effect ........................................................................................ ...45 2.5.1 Torque Magnetometry........................................................................................ ...45 2.6 Diffraction...................................................................................................................... ...46 2.6.1 X-Ray Diffraction...................................................................................................46 2.6.2 Neutron Diffraction............................................................................................. ...50 2.6.3 Analysis.................................................................................................................. ...51 3. HIGH PR E S SUR E MELTING OF L IT H IU M ................................................. ...54 3.1 Abstract ......................................................................................................................... ...54 3.2 Introduction .................................................................................................................. ...55 3.3 Methods ......................................................................................................................... ...56 3.4 Experiment.................................................................................................................... ...59 3.5 Results .............................................................................................................................. ...61 3.6 Conclusion ....................................................................................................................... ...63 3.7 Notes on Laser Heating............................................................................................... ...63 4. TW IN SAMP LE C H AM B E R FO R SIMULTANEOUS E LE C TRON IC T R A N S PO R T M E A SU R EM EN T IN A D IAM O N D AN V IL CELL . . . 65 4.1 Abstract ......................................................................................................................... ...65 4.2 Introduction .................................................................................................................. ...6 6 4.3 Experimental Setup ..........................................................................................................67 4.4 Conclusion ....................................................................................................................... ...71 4.5 Notes ............................................................................................................................... ...72 5. H IGH -PR E S SU R E SU P E R C O N D U C T IN G PH ASE D IA G R A M OF 6LI: ISO TO PE EFFECTS IN DENSE L ITH IUM .......................................... ...73 5.1 Abstract ......................................................................................................................... ...73 5.2 Introduction .................................................................................................................. ...74 5.3 Experiment and Results ............................................................................................. ...75 5.4 Discussion...................................................................................................................... ...81 5.5 Notes on Pressure Distribution.....................................................................................84 6 . N EW BOU N D A R IE S FO R M A R TE N S IT IC T R A N S IT IO N OF 7LI U N D E R PRES SUR E .................................................................................................... ...87 6 . 1 Abstract ......................................................................................................................... ...8 8 6 . 2 Introduction .................................................................................................................. ...8 8 6.3 Results .............................................................................................................................. ...90 6.3.1 Structural Analysis of Martensitic Transition..................................................90 6.3.2 Mixed Phases under Pressure..............................................................................90 6.3.3 Complementary X-Ray Diffraction................................................................. ...92 6.4 Discussion ....................................................................................................................... ...93 6.5 Summary ......................................................................................................................... ...96 6 . 6 Methods ......................................................................................................................... ...96 6.6.1 High Pressure Neutron Scattering Experiments.......................................... ...96 6.6.2 High Pressure X-Ray Diffraction Experiments............................................ ...97 6.6.3 Structural Analysis............................................................................................. ...97 v 7. F E RM I SURFACE OF L ITH IUM IS O T O P E S ................................................. 98 7.1 Abstract ......................................................................................................................... 99 7.2 Introduction .................................................................................................................. 99 7.3 M e th o d ........................................................................................................................... 100 7.3.1 Single Crystal Growth........................................................................................ 100 7.3.2 Data Collection.................................................................................................... 103 7.4 Dingle Temperature.......................................................................................................104 7.5 Effective Mass................................................................................................................104 7.6 Discussion......................................................................................................................105 7.7 Conclusion......................................................................................................................107 8 . SU P E R C O N D U C T IV IT Y OF B A L I 4 U N D E R P R E S SU R E .....................109 8 . 1 Abstract ......................................................................................................................... 109 8.2 Introduction .................................................................................................................. 110 8.3 Experiment.................................................................................................................... 112 8.4 Results.............................................................................................................................113 8.5 Discussion ....................................................................................................................... 115 8 . 6 Conclusion......................................................................................................................118 9. S U M M A R Y ......................................................................................................................... 119 9.1 Results and Discussions............................................................................................... 119 9.2 Conclusion ....................................................................................................................... 124 A P P EN D IX : COLLECTED N EU TRON S P E C T R A ............................................ 126 REFERENCES ......................................................................................................................... 138 vi LIST OF FIGURES 1 . 1 An example of different symmetry operations. Panel A shows translational symmetry in one dimension, a motif copied at a certain distance away. Panel B shows 2 -fold rotational symmetry. The center of rotation is the origin of the two-dimensional coordinate system drawn in the panel. Panel C shows a reflection operation. The mirror plane is drawn along the vertical axis. Panel D shows an inversion operation. The inversion plane, for convenience, is drawn diagonally across the panel. The opposite side of the motif is drawn in red to illustrate the operation. The underside of the motif in the top left quadrant of the panel is red, after the inversion operation, the underside of the motif is now face up......................................................................................................................... 1 1 1.2 An example of the reduced zone scheme presenting a Fermi surface.................... 16 2 . 1 Two different cuts of diamonds commercially available from Almax-easylab. On the left is a conventional cut of diamond and seat, on the right is a Boehler cut seat and seat designed to fit the cut [17]. Images are courtesy of Alamx-easylab.................................................................................................................. .............. 19 2 . 2 Cross section diagram of a DAC. This figure was inspired by the diagram presented in http://pubs.rsc.org/en/content/articlelanding/2013......................... 20 2.3 A plate DAC from Almax-easylab; inside a home-made oven it is capable of reaching temperatures of & 600 K. Pressure inside the DAC is applied by tightening the three inner screw simultaneously by means of a customized gear box. The diamond seat fits into the bottom plate secured by three set screws, which allow for lateral adjustment of the bottom diamond. The top seat is pressed into top plate and not mobile. The three outside screws can be adjusted to align the tilt of the top diamond. This DAC allows for accurate alignment of diamonds and is capable of pressures of ~ 50 GPa with 500 fim diameter culet diamonds and higher pressures with smaller culets. It has little change in pressure ( ± 2 GPa) with the application of low temperature ( K) to moderately high temperature (~ 500 K). Picture courtesy of Alamx-easylab. 21 2.4 Another style of plate DAC; this is a miniature model manufactured by HPDO (hpdo.com). The pressure is applied by means of the three pressure screws. The posts attached to the bottom plate fit snugly into the top plate and preserve the diamond alignment as pressure is applied by gently tightening each screw separately. Both seats are attached to the plate by means of set screws, allowing for lateral adjustments. The only mechanism to adjust the tilt is by properly epoxying the diamonds; see section 2 .1 . 1 ................................... 2 2 2.5 A piston cylinder style DAC designed for resistive heating, capable of reaching temperatures of & 1 0 0 0 K. Pressure is applied by means of four pressure screws which engage the piston via through-holes in the cylinder, or through the use of a steel membrane which attached to the cylinder directly and applied pressure when inflated with gas. This figure shows the Helios DAC, which is designed for high temperature. Many measurements in the works were also taken using the Diacell Bragg DAC. The styles and alignment of the two are very similar. The bottom seat is secured to the cylinder with four set screws, allowing for lateral adjustment. The top seat is affixed to the cylinder by means of three set screws on a rocker, which allows for very fast and accurate tilt adjustment. Both DACs provide pressures >100 GPa using 200 fim diameter culet; pressures of ~ 50 GPa are routinely achieved. The pressure drift with temperature in the piston-cylinder type DAC is generally not great ± 2 GPa, except in the case when the helium membrane is employed when cooling to low temperatures. In this case, the pressure tends to increase by very noticeable amounts of >5 GPa. Picture courtesy of Alamx-easylab. . . 23 2.6 Diagram of PE Press. In the configuration shown, the two detectors would be parallel to the page in front and behind the page. Image is the PE Press from SNAP at ORNL: https://neutrons.ornl.gov/snap/sample...................................... 24 2.7 This is a typical nonhydrostatic ruby under moderate pressure. The y-axis represents the intensity of the spectrum in arbitrary units. The doublet has already begun to merge, and with increasing pressure, the doublet will continue to broaden and merge as the nonhydrostaticity increases. Finding the middle of the peak in such a situation does not take into account the broadened doublet. The pressure is determined by recording the value of the middle of the peak and the value of two-thirds to the higher wavelength, where the R1 peak would be visible under hydrostatic conditions................................................. 25 2 . 8 A cartoon of the relation between resistance and resistivity................................... 33 2.9 A diagram of the circuit used to measure resistance inside of the DAC. The function generator of a SR830 Lock Amplifier was set to values near 0.1 V and < 15 Hz to provide a nearly constant current through the sample. Since the Lock In Amplifier provides a constant voltage source, not a constant current source (though the use of a resistor with a resistance » than the sample's resistance serves partially regulate the current), the resistance of the sample and any leads inside of the voltage probe can have an effect on the current of the overall circuit. For metals, this correction is smaller than the noise in the measurement itself............................................................................................................. 34 2.10 Methods for measuring the voltage drop, which can be related to resistance, across a sample. A) a cartoon of the 4-point probe arrangement. B) a cartoon of the quasi-4-point arrangement................................................................................... 35 2 . 1 1 A cartoon of a typical electric probe built on a DAC. When size constraints allow, a four-point probe is built (culets of & 350 - 500 im ) ; when smaller culets are used, a quasi-4-point probe is built. When building the quasi-4-point probe, all the leads are placed as close to the sample as possible to have a minimal signal from the leads. Figure by William B. Talmadge........................... 36 viii 2.12 The melting transition of natural lithium measured by means of electrical resistance. The resistance was measured via a four-point probe. The lithium was kept inside of a argon-atmosphere cell to prevent reaction with moisture or air. Melting was confirmed both against the literature values for the melting temperature of lithium as well as confirmed visually (video of the sample was taken under a metallurical microscope during heating and cooling, the sample was large enough to confirm change of shape upon melting). The red curve shows the cooling of the sample and a significant hysteresis consistent with a first-order phase transition.............................................................................................. 39 2.13 An example of a superconducting transition of BaLi4 under pressure at ~ 22 GPa (see Chapter 8 ). This transition is very sharp despite of the high pressure applied due to the moderate change of dp in this pressure region....................... 40 2.14 An example of a superconducting transition of Li at about 30 GPa. The transition here is very broad and does not actually complete all the way. The lines demonstrate the onset of the superconducting transition and the possible double transition................................................................................................................ 41 2.15 A typical set of coils placed on a D'Anvils nonmagnetic DAC.............................. 42 2.16 The wiring configuration of the coil system. The magnetic fields of the two field coils (in black) are in the same direction. The magnetic fields of the two pick-up coils (shown in red) are in opposite directions............................................ 43 2.17 AC magnetic susceptibility signals of Li at 18 and 25 GPa (Pressures 2 and 3, respectively). A sample of Nb of comparable size to the sample was placed in the compensating coil, creating a signal of roughly equal magnitude, yet opposite direction, of the signal of the sample under pressure. The x-axis shows the temperature and the y-axis is the amplitude of the magnetic sus­ceptibility signal; here, the raw measured voltage is shown. The arrows show the superconducting transitions..................................................................................... 44 2.18 An example of quantum oscillations in lithium with natural isotopic composi­tion. The x-axis is the magnetic field in Telsa and the y-axis is the intensity of the signal (measured in voltage from the torque magnetometry). The frequency of the change in the inverse field, shown as F in the figure, can be related to Fermi surface............................................................................................. 46 2.19 A Wheatstone bridge circuit for measuring quantum oscillations by means of a piezoresistive cantilever. R1 and R2 represent two outside dedicated boxes. The resistances of the cantilever arms, Rs is the arm that holds the sample, and Rr is the reference arm, are shorted together at point A. AC voltage is applied from points B to C, and the voltage across points A and B is measured with a Lock In Amplifier (LIA in the figure). This figure is adapted from [64]. 47 2.20 Example of Thomson scattering. The concept of this figure was inspired by Pecharsky et al. [139]. This shows a two-dimensional cartoon of the concept behind Thomson Scattering............................................................................................ 49 2 . 2 1 An example of simple diffraction of light by a grating. Constructive interfer­ence only occurs at the condition CD-HG = nA (A is the wavelength of the light). This idea for this schematic comes from chemwiki.ucdavis.edu................ 49 ix 2.22 An example of simple Bragg's law. The concept of this figure comes from chemwiki.ucdavis.edu........................................................................................................ 50 3.1 Aspects of melting lithium under pressure. a) Micrographs of sample in reflected light. Red dotted lines represent the contour of the sample. Dark spots are regions of sample or electrodes deformed under pressure and are shiny if viewed from a different angle. b) Schematic drawing of the quasi-four-probe resistance. c) The arrangement of electrodes prior to loading the lithium sample. The square region marks the approximate area of the sample in the measurement. d) Large hysteresis between melting and recrystallization temperatures of a lithium sample at ambient pressure due to rapid cooling. . . 57 3.2 Jump in resistivity at a) 19 GPa, well within the boundaries of the fcc phase, and b) 40 GPa, which is at the boundary of fcc and CI16. The broadening of the melting signal in the latter was reproducible, indicating the existence of mixed solid phases. The higher temperature arrow defines the completion of the melting transition and the lower temperature arrow gives the lower limit of melting. The resistances are estimated after subtraction of the lead contribution and would approximate L40GPa ~ 3 kJ per mol. and L igGPa ~ 5 kJ per mol., using Mott's equation............................................................................... 58 3.3 The melting curve of lithium [70, 100, 110]. Solid lines represent the bound­aries of solid structures determined by X-ray diffraction [75, 70]. The dotted line is the interpolated bcc - fcc phase boundary. The dashed line is to guide the eye along the melting curve of lithium. The shaded area is the region below the melting curve in which X-ray diffraction lines disappear in condensed lithium. Pressure uncertainties are ± 1 GPa. Slope changes at 9 and 35 GPa represent small but clear change in the resistance versus temperature (+) . . . . 60 3.4 The resistance as a function of temperature. a) The resistance of lithium at different pressures. The sample size was smaller in the run shown by the black curve marked with a star in which the jump in the resistivity is proportionally smaller. b) The resistance versus temperature from room temperature to 77 K at 35, 41, and 64 GPa. Melting at 64 GPa was observed above room temperature inside the cryostat (see arrow); melting at 35 and 41 GPa was measured in the homemade oven outside of the cryostat (see text). Red dotted lines in the subset graph are linear guidelines. A change in slope is observed at 35 GPa. Arrow in the subset graph points to the change in slope at 35 GPa near the boundary of fcc - CI16/hR1 phases.............................................................. 62 4.1 The correlation between the pressures in the two chambers up to 11 GPa. Three rubies across each sample chamber are measured for each pressure. The blue line is a guide to the eye, having a slope of 1. Red is a linear fit for PA vs. PB........................................................................................................................... 6 8 4.2 Evolution of gasket under pressure. The pressures shown are the pressure in either sample chamber. The probes are mounted on the opposite side............... 69 4.3 Diagram of the electrical circuit used to measure the two electrically isolated samples. The inset shows the quasi-four-point probes built on the twin cham­ber gasket........................................................................................................................... 71 x 4.4 The superconducting phase diagram of two YBa2Cu3O7-x (0 < x < 0.65) samples with slightly different initial critical temperatures were measured as a function of pressure to test the feasibility of simultaneous transport measurements at high pressure in twin chamber design. Open symbols are plotted from previous high pressure studies for comparison [191]......................... 72 5.1 A picture of a twin chamber gasket drilled with alumina pressure medium and rubies. The symmetry of the two holes is key to having an even pressure pressure distribution across both holes. Rubies are loaded into each sample chamber, and three rubies in each pressure chamber are measured...................... 76 5.2 Experimental set up for simultaneous measurements. a) Twin chamber gasket built on a 500 fim culet diamond which is used in the present experiments in a DAC for simultaneous measurements of superconductivity. Each pressure chamber has a pair of extra Pt leads and contains several pieces of ruby for accurate determination of pressure gradient within each pressure chamber. The insets show the gasket and samples under reflected light only, at 21.3 GPa, demonstrating the metallic appearance of both samples and map of ruby pieces inside each pressure chamber in the same run. b) Schematic drawing of the twin chamber design used in the experiments. Small portions of the platinum leads in the path contribute to the total resistance measured for each sample. The electrodes for measuring the resistance of 6Li and 7Li are shown in different colors............................................................................................................... 78 5.3 Electrical signals from lithiums samples. a) Superconducting transitions as determined by electrical resistivity. All black lines show transitions for 7Li, red lines are the transitions for 6Li. Each pair shows the simultaneous measure­ment. The double step in 18.1 GPa transition of 6Li may be related to presence of mixed phases with different T'cs at a structural phase boundary. The samples' resistances in a normal state above their superconducting transitions is 0.5-10 m Q varying by the sample size and geometry. These values would give an estimate of p &0.5 - 1^Qcm, at room temperature, for a typical sample size of 50 x 50 x 10^m3. An RRR value of & 75 is estimated from ambient pressure measurements on the samples used here [162]. The transitions above are scaled for ease of comparison. b,c) The shift of Tc with an applied external magnetic field of B & 100 Oe for 6Li at 23.3 and 26.6 GPa. B & 100 Oe for green lines and B=0 for red lines.................................................................................. 79 xi 5.4 Results of the measurements of the superconducting T'cs of lithium isotopes. a) Superconducting phase diagram of lithium isotopes. Open shapes repre­sent 6 Li, solid shapes represent 7 Li. The different shapes designate separate loadings. The pink shaded region from 18 GPa to 21.5 GPa shows the direct isotope effect with a large difference in the T'cs for 6Li and 7 Li. The grey shaded region shows the inverse isotope effect from 21.5 GPa to 26 GPa. The pressure error bars represent the maximum pressure difference between all the rubies shown in Figure 1 in the two chambers (the value is generally is equal to the difference between the pressure from the ruby in the smallest radii of one chamber and the ruby in the largest radii of the opposite chamber). The Tc onset error is determined from the resistance signal, the upper error marking the first drop in resistance from a normal metallic state, the lower error is determined by the completion of the change in slope of resistance. b) Comparison of the various superconducting phase diagram of natural lithium measured by various techniques (Open triangles: Deemayd and Schilling [42], Open squares: Struzhkin et al. [172], diamonds: Shimizu et al. [160]. and circles: this study). The solid lines are guide to eye. Dashed line is the speculated boundary between hR3 and fcc at low temperature. c)The isotope coefficient, a, as a function of pressure. The dashed line at a = 0.5 shows the expected value for a conventional isotope effect........................................................ 80 5.5 A diagram of the placement of rubies inside both twin chambers. The labeled radii refer to the distance of the rubies from the center of the culet.................... 84 5.6 Using the radii measured and shown in Figure 5.5, a plot of the pressures for both samples as a function of radius is plotted. The black symbols and lines represent the pressures of the rubies in chamber A, and the red symbols and lines represent the pressure measured in Chamber B. Whilst the pressure error inside the entire cell could be ± 1 GPa, both samples appear to experience the same pressure gradient.............................................................................................. 85 5.7 The pressures of the two samples chambers plotted as the measured pressure in chamber B as a function of the pressure in chamber A. The black line has a slope of 1 , and serves as a guide to the eye to determine the drift of pressure between the sample chambers. Whilst there is some deviation from the line, the correlation between the pressures in each sample chamber is indeed rather close to the slope of 1. Comparing to the results from Figure 4.1, the improvement in the technique is quite evident........................................... 8 6 6.1 Phase boundaries obtained from data collected while cooling. Background shades are used to highlight the regions of the P-T phase diagram of lithium in which different structures are observed. Symbols designate different struc­tures. Symbol overlap indicates the presence of mixed phases. The blue region here indicates two possibilities for the fcc/hR3 phase boundary, one boundary in which a transition from fcc to hR3 is not supported and one boundary drawing the upper limit. The inset shows the proposed structural boundaries of martensitic phase transition in lithium overlaid with previous reports. Orange region represents the superconducting region [160, 172, 152, 42]. 91 xii 6.2 Equation of state for 7Li measured at room temperature. Squares and trian­gles represent the experimental data points by neutron scattering and x-ray diffraction in bcc and fcc phase respectively. Open symbols are x-ray data. Dashed line is previously reported eos for natural lithium. Solid lines are Vinet fits to neutron and x-ray data........................................................................................ 92 6.3 Equation of state and bulk modulus of lithium at 85 K. Equation of state fits and calculated bulk modulus (K = jpsT) for 7Li at base temperature (~80 - 85 K), in which bcc and hR3 phases coexist, form neutron data. Solid squares and circles are experimental data points of the present study in bcc and hR3 phase respectively. ......................................................................................................... 93 6.4 Diffraction signals from isotopically enriched 7 Li. A) and B) Neutron diffrac­tion pattern of lithium at 2 GPa for cooling and heating respectively. The calculated peaks for the structures for this pressure are plotted below the spectra. The conversion upon cooling to hR3 is evident, and hysteresis is observed. The refined a lattice parameter for the bcc phase at room temper­ature is 3.323 ± 0 . 0 0 2 A, and at base temperature is 3.322 ± 0 . 0 0 2 A. For the hR3 phase the calculated lattice constants at base temperature are a = 2.959 ± 0.004 A and c = 21.726 ± 0 . 0 0 1 A. C) The neutron diffraction at 5 GPa shows no indication of hR3 phase down to the lowest temperature. The lattice parameter a for the bcc phase at room temperature is 3.196 ± 0.001 A , and for the fcc phase at base temperature a = 4.002 ± 0.003 A . D) Structures of lithium at 3 and 7 GPa measured by x-ray in a DAC shows the presence of fcc phase to 7 K. Intensities are scaled for each data set. The dotted and solid lines show the location of the peaks at 3 and 7 GPa respectively......................... 94 6.5 Phase fractions are presented in order during each isobaric cooling and sub­sequent heating. The gray area reported for 2.75 and 3.25 GPa are represen­tative of areas with peaks of hR3 phase which appear as weak shoulders on characteristic bcc and fcc peaks. This could be indicative of a co-existence of bcc/fcc/hR3, or other phases. The slope of the hR3-fcc boundary in the region between 3.5 and 7 GPa below 80 K cannot be well defined by the current data. It should be noted, however, that at these pressures, once some percentage of the sample undergoes the transition to fcc, that percentage does not decrease upon further cooling to 80 K. This raises the possibility that bcc/fcc/hR3 co-exist in these pressure-temperature regions, and it is only the bcc that transforms to hR3, and the fcc percentage is stable or increasing upon cooling. It remains an open question if the transformation of fcc to hR3 upon cooling is possible.............................................................................................................. 95 7.1 Design by Nash and Smith [131]. A stainless steel crucible is heated with a hot plate and it is lined with petroleum jelly to prevent reaction between the lithium ad the steel. The lithium melt forms an ingot in the crucible and the plunger is applied to the melt. The nucleation occurs at the top from the temperature difference of the cooling wings................................................................1 0 1 7.2 Modified crucible used to grow single crystals of lithium. Crucible and plunger are machined from Mo. The hole in the plunger was made using an EDM; the hole is & 500 im and tapers to & 200 im hole at the top. The size of the crucible is & 0.75 cm........................................................................................................ 102 xiii 7.3 X-ray diffraction pattern from a Bruker system of single crystal lithium. The diffraction image shows clear single crystal spots......................................................103 7.4 Lithium crystal on cantilever. The crystal is noticeably quite large (>400 /im) in order to compensate for the low mass of lithium..................................................104 7.5 The two ambient pressure phases of lithium. A) The unit cell of the low temperature hR3 (or 9R) phase. B) The first Brillouin Zone of this phase. . . . 108 8.1 Unit cell of BaLi4, P6 3/mmc (194)hexagonal, a = 9:1875 A, c = 9.1875 A, a = 1 :0 0 0 0 , V = 671.62 A3, Z = 2 [164]......................................................................... 1 1 1 8.2 AC resistivity measurement setup. Inset shows the arrangement of the leads in quasi-four probe measurement.................................................................................. 113 8.3 Resistivity as a function of temperature for selected data points. The size of the transition in different runs depends on sample size. In the graphs above, all data have been scaled to show the same size transition for comparison. Arrows indicate the onset of Tc. Resistivity curves a), b) are for BaLi4, the asterisk near the 2 GPa curve is a possible onset of a superconducting transition near 2.5 to 2 K. The resistivity curves in c) are for barium at 35 and 43 GPa. Red and blue bars indicate the upper or lower bounds of Tc as defined by [47]. The curve in d) is the data point taken at 12 GPa by the AC magnetic susceptibility method. The ratio of the residual resistivities at ambient pressure is RR(5KK) ~ 14 and samples were showing metallic behavior throughout all pressure runs...........................................................................................114 8.4 Superconducting phase diagram of BaLi4. The data represent the Tc onset, de­termined by the change in slope of the resistance as a function of temperature. The symbols show the first instant of of the change in slope and the lower error bar marks the end of the change in the slope. The pressures are determined from measuring 3 ruby pressure markers at each pressure. The x-axis error, which are not clearly visible, are determined by the standard diavation of the measured pressure markers. The data point at 53 GPa, marked by a star, remains unconfirmed due to the diamond failing before the pressure was confirmed after heating. The data point in run 6 (12 GPa, 6.9 K) is taken by the AC magnetic susceptibility method.......................................................................116 8.5 Superconducting phase diagram of BaLi4 (this study) in comparison to ele­mental lithium [42] and barium [47, 125]. The dashed lines mark the structural phase boundaries of barium [133]. Black squares show the superconductivity of barium that was measured in the current study for better comparison of the superconductivity of barium and BaLi4 ............................................................... 117 A.1 The ambient pressure measurement of 7Li with neutron diffraction. Clearly in the bcc phase.................................................................................................................126 A.2 7Li at ~0.5 GPa. The same was not cooled sufficiently to see the transition to the hR3 phase............................................................................................................... 127 A.3 7Li at ~1.7 GPa. Clearly in bcc phase at high temperature and the emer­gence of a new phase in the measurement at ~94 K can be seen. This low temperature is consistent with peaks from the hR3 phase (though that phase does have several peaks in common with the fcc phase)..........................................128 x iv A.4 7Li at ~2 GPa. The evidence of the hR3 phase over the the fcc phase is much more clear at this pressure.............................................................................................. 129 A.5 7Li at ~2.75 GPa. At this pressure, the hR3 phase is present at the lowest temperatures; however, upon heating, we see the loss the hR3 peaks and the strengthening of the fcc peaks up to ~ 163 K , then a return to the bcc phase above 200 K ....................................................................................................................... 130 A . 6 7Li at ~3.3 GPa. At this pressure, the hR3 phase has very little evidence. Though it is still possible, the only peaks that would be exclusively hR3 are very broad and weak, whilst the other peaks, which are shared with the fcc phase, are strong and distinct. Again, we see the strengthening of the fcc upon warming to the sample to ~ 153 K but then a return to the bcc phase above 200 K .................................................................................................................................. 131 A.7 7Li at ~3.8 GPa. There does not appear to be any evidence for the hR3 phase at the low temperature in this pressure. At the base temperature, not only do we only see the hR3 peaks that overlap with fcc peaks, but we also see a distinct lessening of the bcc 1 1 0 peak (which would be shared with the hR3 phase). This appears to indicate that the sample is transforming into fcc phase. However, the fcc 200 peak appears somewhat broad, unlike the other peaks of the phase.............................................................................................................132 A . 8 7Li at ~4.3 GPa. This pressure very closely resembles 3.8 GPa, including the broad fcc 200 peak. The main difference is that the fcc phase clearly comes at a higher temperature...................................................................................................133 A.9 7Li at ~5 GPa. This pressure show some mixed fcc and bcc phase even at room temperature. Upon cooling, the fcc 200 peak becomes quite strong and distinct, becoming more prominent than the fcc 1 1 1 peak, showing that there is some texturing of the sample..................................................................................... 134 A.10 7Li at 5.6 GPa. This pressure shows the fcc phase becoming more dominant at room temperature. The fcc 200 peak appears alongside the bcc peaks; only at low temperature (~85K) does the fcc 111 peak begin to appear.....................135 A.11 7Li at ~ 6 GPa. The fcc phase is dominant at room temperature; however, there are still peaks from the bcc phase, and the fcc 1 1 1 peak is conspicuously absent until the sample is cooled to ~ 210 K . It appears that this peak grows only at the expense of the bcc 110 peak......................................................................136 A.12 7Li at 1,5 GPa. The pressure of the sample was released to look for evidence of whether the hR3 phase at lower pressure could be obtained again. We do indeed see some evidence for the hR3 below 100 K , though the exclusive hR3 are very weak......................................................................................................................137 x v LIST OF TABLES 1.1 Crystal Systems................................................................................................................. 13 1.2 Lattice Centering............................................................................................................... 14 ACKNOWLEDGMENTS I would like to acknowledge the mentorship of Professor Shanti Deemyad. All experi­mental skills and scientific expertise were learned under the guidance of Professor Deemyad. All of the experiments presented in this dissertation were either carried out in the Deemyad Lab at the University of Utah or at User Facilities with access granted as a result of Professor Deemyad's approved research proposals. This dissertation would not have been possible without the help from Professors LeBohec and Mischenko in allowing my defense to go forward. Also, the support from Vice President of Faculty Amy Wildermuth and Dean of Students Lori McDonald were instrumental to the presentation of this work. Professor Raikh provided insightful feedback regarding this work as well as the presen­tation of the thesis defense. Also, this work benefited as a result of the help and support from the Physics and Astronomy GSAC. Chapter 3: The authors would like to acknowledge the generous donation of some of the major equipment used in this study by Linda and Robert Grow from UPRI Co. The authors would like to thank Dr. Eugene Mishchenko for insightful discussions on the topic of melting and reviewing this manuscript, Matthew DeLong for providing technical and intellectual support in design of the experiment, and the University of Utah UROP fund for partial financial support. Chapter 5: Authors are grateful for experimental assistance from R. McLaughlin, D. Sun, H. Malissa, Z. Jiang, F. Doval, and A. Friedman. This work is supported by NSF-DMR grant num. 1351986. JKB acknowledges the financial support from the University of Utah UROP fund. Chapter 8 : The authors thank Dr. Richard Hennig for insightful discussions on electronic properties of BaLi4. Experimental support by S.R. Temple, J. Jue, Z. Xu, and J. Bishop in sample preparation and low temperature studies and analysis of XRD spectrum by M. Sygnatowicz in Materials Characterizations Lab of University of Utah, Material Science Department is acknowledged. One of the authors, W. Talmadge, would like to acknowledge the financial support from University of Utah UROP program. Initial sample synthesis was done with great help from the late W. Wingert. x vii i CHAPTER 1 INTRODUCTION This work is mainly concerned with the properties of low Z materials under extreme conditions, in particular, lithium and lithium-rich materials. This dissertation serves to give some theoretical background of alkali metals, lithium in particular, and to describe experimental studies. There are three main parts: an introduction and overview of the experimental techniques used, Chapters 1 and 2 ; results and/or discussions of experimental work, Chapters 3 through 8 ; and a summary of findings, Chapter 9. 1.1 Alkali Metals The alkali metals are characterized by one valence electron (s-electron) and tightly bound core electrons. Intuitively, this simple electronic structure should lead to simple behaviour. Indeed, under ambient conditions, the alkali metals behave in manners described well by the nearly free electron model. However, with the application of high pressures, the alkali metals display nonintuitive trends: lithium, sodium, potassium, and rubidium all display maximums and subsequent minimums [65, 70, 100, 110, 155, 130] in their melting curves as a function of pressure (the measured melting temperatures at differing pressures), as well as a series of symmetry-breaking structural phase transitions, going from the highly symmetric bcc to much lower symmetry phases [132, 75, 65, 130, 4]. Lithium and sodium also undergo a martensitic phase transformation at low temperature from bcc to hR3 at ambient pressure [13]. Lithium is the lightest of the alkalis that is a metal under ambient conditions. Though hydrogen has the same electronic configuration of the other elements in the alkali group, it is yet to be confirmed as metallic in laboratory conditions [189, 109, 129, 49]. However, the emergence of a metallic phase (even a nearly room temperature superconducting phase) is predicted to occur under extreme pressures [190, 5, 18, 115] and the strong magnetic field of Jupiter indicates that hydrogen is conductive at least within the core ( ~ 40 million 2 atmospheres). Due to the difficulty in reaching metallization pressures of hydrogen, the would-be lightest alkali metal, this work is focused on the next lightest alkali, lithium. 1.2 Lithium Lithium is not only the lightest metal under ambient conditions, but it is also the lightest known superconductor [177, 42, 160, 172]. While under ambient pressure, the superconducting critical temperature (Tc) is rather low at & 100 fjK, with the application of extreme pressure, Tc increases to above 3.5 K near 15 GPa, and then begins to increase sharply at about 20 GPa, reaching a maximum near 30 GPa [42, 160, 172, 152]. This trend in the superconductivity of lithium is most likely related to the series of structural phase transitions that it undergoes as a function of pressure [70, 75]. Lithium has also been predicted and observed to undergo a metal to insulator transition (near 75 GPa) [119, 132]. 1.3 Quantum Solids A quantum solid is a material that is dominated by zero point energy (e = 1 hw) and is intrinsically restless. A visualization of a quantum solid would be to imagine a solid with lattice spacing r. The lattice ions form a periodic structure, they are confined by fixed boundaries. If these lattice ions have a wave function with the width of the de Broglie wavelength, A = , comparable to r, A > 0 .1 r, corrections to the energy due to quantum mechanic effects must be considered. Since the thermal de Broglie wavelength, A = ^2JhkBT, is inversely proportional to temperature, T , lowering the temperature appears to increase the width of the de Broglie wavelength. Thus at low temperatures, we may expect the wavefunctions of the lattice ions to spread in this manner, leading to significant ground state or zero point motion in the solid. A useful parameter to qualify the extent of dominance of quantum effects on a system is the de Boer parameter (A). The de Boer parameter is the ratio of the de Broglie wavelength and the minimum distance separating atoms in a crystal, A = = ----- h,- . The de Boer ° rmin rmin\/ 2mt parameter shows the amount of delocalization of a system compared to its size [43]. The higher the de Boer parameter, the higher the 'quantumness' of the system. This appears very similar to the Lindemann criterion of melting, discussed in section 1.5. Writing the de Boer parameter in terms of the potential energy i yields: hr 2 - 1 A = h - = (1 .1 ) V 2m which is very reminiscent of the criterion for quantum melting, further explored in section 1.5.1. 3 An example of such a material is 4He, which does not even solidify under ambient pressure [113, 197]. Other low Z elements, such as lithium, may display quantum solid behaviour. There is even a small but measurable difference in the lattice constants of lithium isotopes, showing that the lattice constant of the 6Li, the lowest mass isotope, is larger than that of natural lithium (~ 92 % 7Li), indicating an expanded lattice for the lower mass isotope [35]. Lithium is an intriguing candidate for quantum solidity. Unlike solid 3He, H2 or D2, which would be characterized by van der Waals interactions, a ^6 , a quantum solid lithium (at least at pressures <~70 GPa) would be in a metallic phase. Whilst insulators form interatomic bonds characterised by a short-range potential, a ^6 , metals are characterised by long-range Coulomb interactions, a 1. (The full term for Coulomb potential is V (r) = ^ e~kDr , with e~k° r , kD is the inverse of the Debye length, describing Debye electron screening. This term shows that the electric field of the positively charged ions in an electron gas diminishes more rapidly than 1 as the electron gas tends to gather around the positive charge. As the main difference between metals and insulators in this work is concerned with the comparative range of the potentials, Coulomb potential is estimated to be a 1 as in [170].) This type of metallic quantum solid opens the possibility of a material that could exhibit an increase in quantum solid characteristics, and even perhaps quantum melting, with the application of pressure [170](see section 1.5.1 and Chapter 3). 1.3.1 Nearly Free Electron Model The alkali metals are generally considered to be simple metals, following a nearly free electron model. According to this model, the conduction electrons are considered to be free in the lattice. The conduction electrons in a periodic potential behave as unbound electrons. This is described by the Bloch Theorem, in which the wave function of the free electron experiences periodic potential wells determined by the lattice: t k (r) = ek - ^ t k (r) = eikrUk(r) (1 .2 ) where uk (r) is a function with the periodicity of the crystal lattice. In this model, the lattice is assumed to be static. It is assumed that the energy contribution of the motion of lattice ions does not significantly contribute to the overall energy of the system. This means that this model could only be valid at zero temperature, given the always present motion of the lattice ions from thermal energy, a k sT . However, considering a solid even at zero temperature, there is still the contribution from zero-point energy. In the case of a quantum solid, the zero-point energy dominates the total energy of the system. 4 1.4 Extreme Conditions Ambient conditions are quite unusual when considering the universe as a whole. Com­plex life on the surface on the earth exists in a remarkably narrow range of pressures and temperatures. Even when considering 'extremophiles' , the range of the pressures and temperatures, ~2x104 Pa (~ 0.2 bar) to ~108 Pa (^1000 bar) and ~ 250 K to ~ 400 K; whereas elsewhere in the universe, pressure and temperature range from vacuum ~ 1 0 " 17 Pa to ~ 1034 Pa inside a neutron star, and temperatures from 3 K to 1012 K. What we may think of as extreme, in many ways, can be the more typical state of a material. Though extreme conditions are difficult and generally expensive to achieve, it is important to explore such extreme conditions in a laboratory setting. Only viewing a material in our narrow range of ambient conditions does not allow sufficient range on any thermodynamic axis to properly gauge the behaviour of that material. Studying materials in extreme conditions allows for further characterization of said material, and for more accurate predications of its behaviour. 1.4.1 Pressure Pressure is the thermodynamic variable with the largest range in the universe. Even in laboratory conditions, the range of achievable pressures is impressive, from ~ 10" 7 Pa (~ 10" 12 bar) in a vacuum to ~ 7x 1011 Pa or ~ 700 GPa (~ 7x 106 bar) static pressure and >1012 Pa or 1 TPa (107 bar) dynamic pressure. This makes pressure one of the most important accessible thermodynamic variables and invaluable for the characterization of materials. In this work, only static pressure techniques were applied. Static pressure generally consists of two types, hydrostatic and nonhydrostatic. Hydrostatic pressures are achieved by using a liquid pressure transmitting medium, allowing the pressure to be isotropic across the sample, transmitted equally from all directions. In practice however, all pressure transmitting media solidify under sufficient pressure. Materials become stiffer as they are compressed, and different materials have varying hydrostatic limits under pressure (the pressures at which they become solid) [92, 3]. Hydrostatic and nonhydrostatic pressures may be applied in several types of pressure cells. In this work, pressure was applied in a Diamond Anvil Cell (DAC) or Paris-Edinburgh (PE) Press (see Chapter 2, sections 2.1.1 and 2.1.2). In the present work, generally either a solid pressure medium or no pressure medium was used (liquid argon was used in the visual observations detailed in Chapter 3; however, such hydrostatic measurements in this work are the minority). 5 1.4.2 Temperature A wide range of temperatures is accessible under laboratory conditions and even inside a DAC. High power lasers can be focused through the diamond onto the sample or even a portion of the sample, and temperatures >1000 K can be achieved. Conversely, a DAC can be made to fit inside a cryostat or dilution refrigerator and <100 mK temperatures can be reached. For the characterization of materials, temperature, like pressure, is a very useful thermodynamic variable. Temperature governs many important aspects of a given material; the superconducting Tc, the melting temperature, Tm, and in the case of lithium, there is the temperature-driven martensitic transition at Ms & 80 K at ambient pressure [14, 13, 167, 151]. Used together, simultaneously, temperature and pressure allow one to map many interesting regions of the phase diagram: the superconducting phase diagram, the structural phase diagram, and melting curve to list a few. 1.5 High Pressure Melting Melting is the transition of a material from a solid phase to a liquid phase as a function of either temperature or pressure. A useful concept for melting is the Lindemann Criterion, in which melting is described as the vibrational motion of the lattice ions exceeding a certain proportion of the lattice spacing (& 10 %) [106]. If we consider a solid with a lattice with an interatomic distance of r, the rms (root mean squared) amplitude of atomic vibration, {dr‘2ms) 2 , and a constant fraction, 5, (generally assumed to be & 0.1), the melting according to the Lindemann Criterion occurs when: l d r " » 1 = 5 (1.3) r This criterion was inspired by the observation of the relation between the thermal expansion coefficient, a, and melting temperature, Tm: aTm = e (1.4) in which e is a constant. Intuitively, if we imagine the application of pressure as decreasing the interatomic spacing, r, then {drlms) 2 must decrease proportionally in order to preserve the ratio. However, the interaction potential of the lattice ions also depends upon the interatomic distance r . In the case of a van der Waals type solid, the interaction potential is a ^6 , so when r is reduced with the application of pressure, the potential energy increases, resulting in more energy required to affect {dr1ms) 2 . In short, as the interatomic spacing decreases, the amount of energy necessary to displace the lattice ions through vibrations increases. This results in an increase of Tm as a function of pressure. This is true even 6 in the case of metals, in which the interaction potential is of the form a 1. The intuitive expectation is that materials become more classical with the application of high pressure. A thermodynamics approach to melting allows us to see effects of pressure upon the melting temperatures (Tm) of materials (in terms of classical mechanics). Starting with the relation of Tm and pressure (P ) from the Clausius-Clayperon relationship [32] which appears in many standard thermodynamic textbooks: 1 dTm A V Tm Up = ~l (L5) where A V is the change in volume and L is latent heat of melting, both are in terms of unit mass. However, from [69, 169] it appears that the entropy of melting, the change in disorder from the first-order phase transition: AS = L (1 .6 ) T m is independent of pressure. We can introduce the Gruneisen parameter, aK ft -7\ Y = P a : (L7) in which a is the thermal expansion coefficient, K is the bulk modulus, either isothermal or adiabatic depending upon the path, p is density, and Cv is the specific heat at constant volume. We can insert the Gruneisen parameter into the Clauius-Clayperon relation [168]: aK Y = pCv (1 .8 ) {d T X= a K = YpCv dP = J dTYpCv ~ YP^ A E ----- = 2L - > dP = 2ypCv m A V dP = K I ^ dP = K A V p for unit cell volume change 1 dTm A V 2y Tm ~dF = ~L = K Here, we can see that the slope of the melting curve as a function of pressure is inversely proportional to the incompressibility (bulk modulus K ), showing that the expectation is that the slope j p is positive as the material is compressed. However, this relation is based upon the assumption that the material does not undergo any reordering as it is compressed, that there are no pressure induced structural phase transitions. In section 1.1, it is noted that the alkali metals, lithium, sodium, potassium, 7 and rubidium all have a maximum in their melting curves. This type of anomalous melting curve [169], occurs when the liquid phase is more dense than the solid phase. Since these materials do show symmetry-breaking phase transitions with the application of pressure, the maxima in the melting curves can be understood as a classical effect, and it is expected that if the material is compressed to a sufficiently close-packed structure, the equation 1 . 8 will be valid [168]. If a material undergoes a phase transition to a structure with relatively low symmetry, then it is possible for a liquid to be more efficiently packed than a solid. At a constant temperature, the result is a pressure-induced melting transition; meaning the slope of dTm becomes negative, leading to a maximum in the melting curve as a function of pressure. Chapter 3 explores the melting curve of lithium under high pressures in more detail and results of experiment are presented therein. Although it is understood that the maxima (and even subsequent minima) in the melting curves of the alkali metals correspond to the series of symmetry-breaking phase transitions in these materials, i.e., there is a classical explanation; the concept of quantum melting is an intriguing possibility for low Z metals, such as lithium (or even possibly dense hydrogen). This is the possibility that purely quantum effects can serve to expand the lattice even as pressure is applied to compress it [170]. We start with restating the Lindemann criterion This comes from replacing the momentum term p in the uncertainty principle with energy, and equating that energy with the potential in the lattice (since it is at equilibrium). Finally 1.5.1 Quantum Melting (1.9) r If we rewrite the Heisenberg uncertainty principle: ArAp > h p = \/2 me 1 ( 1 .1 0 ) resulting in the relation: ( 1 .1 1 ) 8 for simplicity, we use a proportionality to a constant. We can then rearrange this into the familiar format of equation 1.9: i {dr2rms) 2 a r 2 - 1 ( 1 .1 2 ) r From this equation (1.12), we can see that if n = 1, as is the case for a metal with Coulomb type interactions ( 1 ), as the interatomic distance r decreases, the fraction sS 2 increases. Meaning the 5 in equation 1.9 increases, possibly to a value over the critical fraction to cause the lattice to melt. Whilst the results presented in Chapter 3 do not provide sufficient evidence to conclude that the minimum in the melting curve of lithium is due to quantum melting, it remains an open question to investigate whether the phenomenon may occur at higher densities. 1.6 Superconductivity Superconductivity as a physical phenomenon is important to this body of work, as it is a low temperature phase transition as well as a quantum mechanical effect. When searching for the possibilities of quantum solid behaviour in metallic, dense lithium, studies of the behaviour of the superconducting phase as a function pressure may provide insights into the physics governing this material. This is one of the main motivations for the investigation into the superconducting phase of dense lithium and its isotope effects (see Chapter 5 for experimental results). Superconductivity was first reported in mercury in 1911 by H.K. Onnes [136]. It was first thought that the electrical resistivity of a metal would decrease as a function of temperature continuously down to zero temperature. However, Onnes' laboratory measured the resistivity of mercury as a function of temperature down to liquid helium temperatures, and they observed an abrupt drop in the resistivity near 4.2 K. Soon after, it was found that not only was there a critical temperature (Tc), above which superconductivity could not exist, but there was also a relatively weak critical magnetic field (Hcm) that could also destroy the superconducting state. The Tc and Hcm of conventional BCS or type I superconductors can be described by the empirical formula [157]: Hcm(T) = Hcm(0) 1 - [=■ © ' (1.13) After the initial report, many other metals were found to demonstrate the same abrupt drop in electrical resistivity. This was assumed to be ideal conductance (p = 0) for ~ 22 years. In 1933, Meissner and Ochsenfeld reported that materials in the superconducting state also demonstrated zero internal magnetic field regardless of the material's history [124], known now as the Meissner-Ochsenfeld effect. This showed that superconductivity 9 was not simply ideal conductance, but rather a phase transition (characterized by electrical resistivity p = 0 and magnetic field B = 0), allowing for a thermodynamic approach to the study of the superconducting phase. After 1935, superconductivity was also described quite well in terms of electrodynamics by the London theory [108]. This approach treated a superconductor as having 2 types of electrons, superconducting and nonsuperconducting. However, a description of the behaviour of the superconducting electrons themselves was lacking (both in London theory and in the later Ginzburg-Landau theory [97]). It was not until 1957 that a microscopic theory of the phenomenon had been proposed by Bardeen, Cooper, and Schrieffer (BCS theory) [9]. A key feature of the BCS theory comes from the idea of bound pairs of electrons from [34]. One starts by considering a normal metal in the ground state. In momentum space (or k space) all the states for the noninteracting electrons are occupied inside the Fermi Sphere (see section 1.8), and the states outside the Fermi sphere are not occupied. If an extra pair of electrons with equal momentum but with opposite sign, one spin up (+k t ) and one spin down (-k ^) are introduced near to the Fermi surface, these electrons become attracted to each other by means of phonon interaction. One electron deforms the lattice, which in turn affects the other electron, allowing them to form a pair even though they repel each other. One consequence of this type of phonon-mediated BCS type superconductivity is that there exists an isotope effect for Tc. The lighter isotope of a given superconducting element (either in elemental form or as a part of a compound) has a well-described shift to a higher Tc (see equation 1.14). The mass of the lattice ions alone has a distinct and predictable effect on the Tc. This is strong evidence for the effect of phonons on superconductivity. The two attracted electrons become bound as a Cooper Pair, the total spin of the paired electrons is zero, meaning they now no longer obey Fermi Statistics and now obey Bose-Einstein Statistics. Now, at T<Tc, the Cooper Pairs are all able to exist in the lowest energy level, as in Bose-Einstein condensation. In the Bose-Einstein condensate, all the particles have the same wave-function and are in a superfluid state. This makes it impossible for particles to be scattered separately by impurities or defects in the lattice from the remainder of the condensate without destroying the superconducting state itself. The energy it would take to scatter the particles separately would have to exceed the energy needed to break the Cooper pairs, thus destroying the supconducting state. According to BCS theory, the Tc can be expressed as: v ) 10 Fermi level, and Vef f is the attractive potential between electrons [121]. From this equation 1.14, the isotope effect is apparent: Lighter isotopes are expected to show a proportionally higher T'cs than the heavier isotopes The structure of solids is one of the most important features of a material, not only affect­ing the macroscopic appearance of a material but also governing many physical properties. For a material to be considered a crystal, it must have a periodic repetition of a base unit cell in three dimensions. Lithium and indeed the remainder of the alkali metals discussed thus far have been assumed to be crystals, and the presence of the symmetry-breaking phase transitions has been referenced (see section 1.1). In light of the importance of this symmetry-breaking phase transitions in lithium to the melting curve and phase transitions in general to the superconducting phase diagram, a discussion of the basic principles of crystal symmetry is warranted. Structures of crystals are defined in terms of the symmetry present in the solid and dates to the 18th and 19th centuries. These crystal symmetries were first determined by analytical geometry, before the development of diffraction techniques necessary to experimental ob­serve the structures of crystals [59]. There are 4 possible symmetries in a three-dimensional model: translational, rotational, reflection, and inversion. All crystals by definition have translational symmetry; there must be a base unit cell that can be repeated overall space; in other words, the definition of a crystal is a periodic structure. This restriction in turn leads to some limitations in other symmetry elements that are possible for crystals. For example, a shape that has five-fold rotational symmetry (such as star fish with five legs) cannot exist in a crystal since five-legged base unit cell cannot be repeated over all space without gaps. Although such symmetry does exist in nature (e.g., star fish, flower petals), it is not crystalline; such solids with five-fold symmetry are aperiodic and referred to as quasi-crystals [139, 159, 186]. For each symmetry element, we can consider an associated symmetry operation; exam­ples are shown in Figure 1.1. We can consider a translational symmetry operation acting (1.15) (light^ ). The results from Chapter 5 show the test of this relation under compression. 1.7 Crystal Structure 11 Figure 1.1. An example of different symmetry operations. Panel A shows translational symmetry in one dimension, a motif copied at a certain distance away. Panel B shows 2-fold rotational symmetry. The center of rotation is the origin of the two-dimensional coordinate system drawn in the panel. Panel C shows a reflection operation. The mirror plane is drawn along the vertical axis. Panel D shows an inversion operation. The inversion plane, for convenience, is drawn diagonally across the panel. The opposite side of the motif is drawn in red to illustrate the operation. The underside of the motif in the top left quadrant of the panel is red, after the inversion operation, the underside of the motif is now face up. upon a basis, or to be more general, a motif, to copy the motif at some vector, Ti, away from the original. For example, if a motif is at the origin of a three-dimensional Cartesian coordinate system and a translation operation is carried out in the x-direction, the result is a motif at x = 0, T 1, 2T1, etc. A rotational operation simply rotates the motif about a center of rotation. This is limited by the necessity of translational symmetry, meaning only some rotations are allowed: 1-fold, 2-fold, 3-fold, 4-fold, and 6 -fold rotational symmetries. For example, 1-fold rotational symmetry centered about the origin of a coordinate system for a motif at a position of x = 1 , y = 1 , z= 0 rotates the motif by 2 n in x and y and the result is a copy of the motif at the exact same position as the original. As another example, 2 -fold rotational symmetry on the original motif at x = 1 , y = 1 , z= 0 would result in a copy of the 12 motif at x=-1, y=-1, z=0. For a reflection operation, we can consider drawing a mirror plane a distance d from the motif, then a reflection of the motif on the other side of the mirror plane at a distance of d. The result is two motifs of opposite handedness or chirality a distance of 2d from each other. If a motif were at position x=1, y=1, z=1, and a mirror plane were drawn on the x-axis, the result would be a motif with opposite relative chirality at position x=-1, y=1, z = 1 . An inversion symmetry operation is only possible in at least three dimensions; it is very similar to reflection, however, it simply takes into account the three-dimensional shape of the motif. An example would be a three-dimensional motif of a pyramid with the apex pointing out of the page; the result of an inversion operation with the center of inversion in the center of the motif would be a motif with the apex pointing into the page. It is interesting to note that in a two-dimensional space, inversion is not distinguishable from reflection, and only 3 symmetries are possible. Following this trend, in a four-dimensional space, there are five possible symmetries, and so on. At times, when considering other such factors as magnetism, electron spin can be treated as another dimension and another symmetry operation may be added. However, in this work, only three-dimensional solids are considered. There are also symmetry operations which involve the combination of two symmetry operations: glide planes and screw-axes. A glide plane is a reflection in a plane followed by a translation parallel to that plane. A nice example of this is footprints; starting with a left foot print at x = - 1 , y = 0 , then a reflection of that foot print makes a right foot print at x= 1 but the y position is y = 2 instead of y=0. A screw-axis is a rotation about a center followed by a translation. These symmetry operations are related to one other in some very specific manners, which can be explained in terms of group theory. The four symmetry properties from a finite group and demonstrate the four properties of a group: Closure, Associativity, Identity, and Inversion. Closure (G1 *G r = G3) requires that any combination of two elements result in a third element. It should be noted that * represents a generic binary operation. For example, if we start with a motif x = 1 , y = 1 , z= 0 , and apply 2 -fold rotation, then apply a mirror plane on the y-axis, the result is the same as if we drew a mirror plane on the x-axis. The application of two symmetry operations has the same result as the application of a third. Associativity of symmetry operations requires that the grouping of the symmetry operations can be changed and not affect the result, ((G1 * Gr) * G3 = G1 * (Gr * G3)). Referring to the previous example, we could apply two of the symmetry operations on a motif; draw the mirror plane on the x-axis first, then draw another mirror plane on the y-axis. We can 13 then applying a 2 -fold rotational operation on the resultant motif to return to the original motif. We can also take the original motif, draw a mirror plane on the y-axis, then apply a 2-fold rotational operation. We then draw a mirror plane on the x-axis and apply it to the resultant motif to return to the original motif. Identity (E * G1 = G1 * E = G\) requires that there be a symmetry operation that serves as unity. The 1-fold rotational operation serves this purpose. Inversion (G- 1 * G1 = Gi * G- 1 = E) requires that each symmetry operation has an inverse. Each nonrotational or translational symmetry operation serves as its own inverse. For rotational and translational symmetry, the direction of rotation and translation is reversed for the inverse. This finite number of crystal symmetries means these elements can only interact with each other forming a limited number of symmetry groups, shown in Tables 1.1 and 1.2. It is the combination of these symmetry elements in a group that define the crystallographic axes, lattice parameters a,b,c (which are measurable distances between lattice sites), and angles between the lattice parameters a, 3, 7 . All possible groups have been divided into 7 crystal systems, within which are 14 Bravais lattices. The Bravais lattices are generated from the combination the seven crystal systems with the seven types of lattices centering Table 1.1. Crystal Systems. Crystal System Symmetry Lattice parameters Bravais Lattices Triclinic no axes other than 1 ­fold rotation or inver­sion, no mirror plane a = b = c; a = 3 = 7 = 90° Primitive Mon o c l in ic Unique 2-fold and (or) single mirror plane a = b = c; a = 7 = 90°, 3 = 90° Primitive, Base- Centered (C) Orthorhombic 3 mutually perpendicu­lar 2 -fold axes, rotation or inversion a = b = c; a = 3 = 7 = 90° Primitive, Base- Centered (C), Body-Centered, Face-Centered Trigonal Unique 3-fold axis, rota­tion or inversion = 7 3 ;ac; b = ° 0 a 9 Primitive Tetragonal Unique 4-fold axis, rota­tion or inversion a = b,a = c; a = 3 = 7 = 90° Primitive, Body- Centered Hexagonal Unique 6 -fold axis, rota­tion or inversion 3 ;a c; ° 0 = 12 a = b, ,7 = °, 0 a 9 Primitive Cubic 4 3-fold axes, rota­tion or inversion along 4 body diagonals of a cube 7= 3 ;ac; b = ° 0 a 9 Primitive, Body-Centered, Face-Centered 14 Table 1.2. Lattice Centering. Lattice Centering T yp e Primitive (P ) lattice sites on each corner of the unit cell B od y -C en te red (I) lattice sites on each corner of the unit cell plus one in the center Face-Centered (F) lattice sites on each corner of the unit cell plus lattice site in center of each face of the unit cell Base-Centered (3 types: A ,B , or C) lattice site on each corner of the unit cell plus one additional lattice site on the face of each pair of parallel faces Rhomb ohedra l (R ) lattice site only on the corners of the unit cell where a = b = c and a = 3 = y = 90° which are outlined in Table 1.2. Using this understanding of symmetry and the constraint of periodicity for a crystal, the number of possible space groups for a crystal is limited to 230 distinct types. Metals tend to have very simple crystal structures, with the alkali metals at ambient pressure all in a body centered cubic phase, bcc. All the alkali metals undergo phase transitions to a face centered cubic phase, fcc, with the application of pressure. In the case of lithium (and to a more limited extent, Na and K), there is also a temperature-driven martensitic phase transformation to a hexagonal symmetry with a basis of 3 atoms, hR3 (space group 166) [13]. A martensitic phase transformation is a type of diffusion-less phase transformation that forms from crystal planes slipping along faults. Details of the phase transitions of the bcc, fcc, and hR3 phases of lithium are described in detail in Chapter 6 . At higher pressures, the akali metals all undergo phase transitions to crystal structures with lower symmetries than the cubic phases bcc and fcc. This change in the structure also causes other changes in the physical properties of the crystals; in the case of Na and Li, there is even a change from the metallic phase to an insulator phase [111, 119, 132]. 1.8 Fermi Surface The location and topology of the Fermi surface describes the low temperature properties of metals. This is due to the fact that the current that flows through a metal is due to the changes in the occupancy of states near the Fermi surface [91]. The Fermi surface is the direct result of the Pauli exclusion principle, which only allows two electrons (one spin up, t, one spin down, I) per orbital. The Fermi surface is the abstract boundary in momentum space (k-space) which separates the filled orbitals from the unfilled orbitals at absolute 15 zero. This boundary constructs a surface of constant energy in the k-space known as the Fermi-energy, ep. A free electron Fermi surface is perfectly spherical with a radius kF, the Fermi vector, which is determined from valence electron concentration. We can define the Fermi energy by finding the solution for the energy levels from the Schrodinger equation [91]: hr f dr dr d2 \ - ^ . { x r + d p + sT r ) 0k(r) + 0kV (r) = ek 0k (r) (1.16) V (r) - > 0 if we consider an electron in a cube with sides of length L, and boundary condition, 0 k(x, y, z) = 0 at x ,y ,z = 0, L 1.16 has the solution: 0 n(x, y, z) = A sin ( ) sin ( ) sin ( ) ( 1 .1 7 ) For a three-dimensional box, the wavefunction is a standing wave, confined to space of length L. For a solid with a lattice, however, the wavefunction is required to be periodic in all three dimensions. Considering a cubic lattice with a lattice constant of length a, we require that the wavefunction, in all dimensions, have a period a: 0 k(r) = elhr0n(x, y, z) = 0n(x + a, y, z) = 0n(x, y + a, z) = 0n(x, y ,z + a) (1.18) The wavevectors are defined as kx ,ky,kz = ± , n = 0,1, 2, 3.... Then, solving for the energy, we have: hr ek = ---- (kr) (1.19) k 2m ! v ! Now consider nF to be the topmost filled energy level in a free electron gas (nF is the number of filled orbitals), and represented by points inside a sphere in momentum space. This volume of this sphere is defined by the Fermi vector kF. The volume (in real space is a3) can be expressed in terms of the Fermi vector: . ( 3nrnF ) 3 , , k 1 = ( - ^ ) ( i -2°) and the constant energy at the surface of that sphere is the Fermi energy eF: hr ?r hr ( 3nrnp ) 2 ^f = 2m |kr 1 = 2m ( ^ J (L21) The total volume of the Fermi surface, the overall size, depends only upon electron con­centration [91]. Since lithium, like all alkali metals, has only one valence electron, it has 16 a low electron concentration and the calcultaed volume enclosed by the Fermi surface is significantly smaller than the volume of the first Brillouin Zone (BZ). The actual shape of the Fermi surface in real metals can be much more complex than the surface of a sphere. Even in the alkali metals, the shape is only nearly spherical. The shape is distorted by interactions with the lattice. In the case of the alkali metals (at least in the bcc phase), the Fermi surface is far enough from the boundaries of the first BZ, since the alkali metals have one conduction electron and a low electronic density of states, and thus have minimal lattice interaction and distortion [91, 102]. In the case of metals in which the Fermi Surface is close to the boundaries of the BZ, or it crosses the first BZ, the shape of the Fermi surface becomes very far from a sphere. The Fermi surface is generally presented in the reduced zone scheme, much like band structure, in which all bands are shown folded in on the first BZ; see Figure 1.2. Constructions of the Fermi surface can be made with measurements of quantum oscil­lations in resistance (Shuninkov de Haas Effect [161]) or magnetic moment (de Haas van Alphen Effect [161], see section 2.5). These measurements are often accompanied by detailed information regarding the crystal structure of the metal (generally obtained through x-ray or neutron crystallography) and the band structure. However, some general qualitative rules outlined by Kittel [91] can be followed to estimate the shape of the Fermi surface: The interaction of the electron with the lattice creates energy gaps in the zone boundaries of the periodic lattice. In all cases (unless time-reversal symmetry is violated), the Fermi surface 1st Zone 2nd Zone 3rd Zone Figure 1.2. An example of the reduced zone scheme presenting a Fermi surface. 17 perpendicularly intersects in the zone boundaries from the condition dE = hm (k - 1 G), where G represents the reciprocal lattice vector. The lattice potential serves to " round" any sharp corners of the Fermi surface. Applying these constraints to the shape of the Fermi surface and producing them in the reduced zone scheme can lead to many exotic shapes [91]. CHAPTER 2 EXPERIMENTAL TECHNIQUES This chapter presents the experimental techniques used to obtain the results presented in Chapters 3, 4, 5, 6 , and 8 . These include the techniques for reaching high pressures, high or low temperatures, as well as measuring physical properties such as resistivity and magnetic susceptibility. 2.1 Pressure In section 1.4.1, the pressure as a thermodynamic variable is explained in detail. This section is concerned with practical application of pressure. In this work, these pressures were achieved most commonly in a Diamond Anvil Cell (DAC) (pressures ranging from 0 to & 65 GPa), also using a Paris-Edinburgh (PE) Press (0 to & 7 GPa). 2.1.1 Diamond Anvil Cell In a DAC, high pressure is applied by squeezing the sample between two diamond tips (culets). Typical sizes of the culets in this work range from 200 to 500 fim in diameter; different cuts of diamonds are shown in Figure 2.1. The small size of the culet and the hardness of the diamond are the two main factors that allow for very high pressures. There are many different designs of DACs; the common elements are shown in Figure 2 .2 . Two designs used in this work are plate DACs and piston cylinder DACs shown in Figures 2.3, 2.4, and 2.5. The basic designs of these DACs have mostly the same components: two diamonds, positioned opposite of each other with the culets facing each other; the diamonds are placed on seats (generally something hard such as tungsten carbide); one or both seats may be positionable by means of lateral screws that hold the seat in the body of the DAC; one seat may also be attached to a rocker, held in the body of the DAC with screws, the positioning of which allows for relative tilt adjustment. The elegantly simple idea of applying very high pressures between two diamonds is complicated by the degree of precision required to machine and manufacture bodies of DACs capable of securing the seats and diamonds 19 Figure 2.1. Two different cuts of diamonds commercially available from Almax-easylab. On the left is a conventional cut of diamond and seat, on the right is a Boehler cut seat and seat designed to fit the cut [17]. Images are courtesy of Alamx-easylab. in a stable yet adjustable manner (the ability of the DAC to be adjusted is necessary for the alignment of the culets, since if the culets are not aligned with respect to each other, pressure will not be possible). Before any measurement is possible, the two opposing diamonds must first be aligned with some precision to guarantee that the two culets are as parallel as possible. The amount of alignment possible depends upon the design of the DAC. Some DACs allow only lateral alignment, others have means of adjusting the relative tilt. The technique used in this work is as follows: First the diamonds are epoxied on the seats (typically, these are made of tungsten carbide, for its hardness). The interface of the diamonds and seats must be completely cleaned prior to the application of epoxy, otherwise the subsequent alignment may be affected. After the diamonds and seats are completely clean, the diamonds are aligned relative to each the seat according to Figure 2.1 such that the culet of each diamond and the opening in each seat form concentric circles. When using Boehler cut diamonds [17], see the right side of Figure 2.1, this step is much easier. Each diamond is then pressed tightly against the seat and minimal amounts of epoxy are applied, generally in three small dots spaced evenly around the diamond. After the epoxy has set enough to prevent the diamonds from falling out of the seats, the DAC can be preliminarily aligned. After aligning the lateral and the tilt (to some extent) if possible, a flattened piece of metal (generally a 250 fim stainless foil) is placed between the diamonds, and epoxy is applied to the remainder diamond-seat interface. Enough pressure is applied to slightly deform the metal, making imprints of the diamond culets in the metal. The epoxy is then allowed to cure with the pressure applied. This helps to align the tilt, and in the cases where the DAC does not have a tilt adjustment, it may be the only way to align the tilt. After the epoxy has cured, the alignment may be completed. All alignments pertain to the relevant position of the culets to each other. Looking from the back of a diamond through the culet, one should see the two culets as concentric circles, if the diamonds are brought close enough to barely 20 Figure 2 .2 . Cross section diagram of a DAC. This figure was inspired by the diagram presented in http://pubs.rsc.org/en/content/articlelanding/2013. touch each other, the culets ought to lay on top of each other. The amount of the tilt misalignment between the culets can be seen by means of interference fringes between the two diamond surfaces. Depending of the style of DAC, the tilt can be adjusted to have very few interference fringes (such as two or three colors, or a partial fringe) or even one solid color. The tilt can also be limited by the size of the culet, with smaller culets having more ease of alignment. If the diamonds are not aligned, then when pressure is applied to the sample, the diamonds will be in danger of breaking. If the tilt is far off and the diamonds touch, even with very little pressure, they can break each other. Another flattened metal foil (which is now the 'gasket') is then placed between the diamonds and an indent is made. This deforms the gasket, making an impression of the diamonds in the metal. The indentation process allows one to drill a small hole (^ | to 3 the culet size) in the center of culet indent, it also precompresses the gasket material so that the sample chamber hole will be stable (it will be harder and less liable to collapse as excess material flows into the hole when pressed, nor expand as the material flows away from the culets, and when centered properly, it ought not drift far from the center). If too much pressure is applied, the gasket will fail by becoming too thin, then the diamonds will 21 I Figure 2.3. A plate DAC from Almax-easylab; inside a home-made oven it is capable of reaching temperatures of & 600 K. Pressure inside the DAC is applied by tightening the three inner screw simultaneously by means of a customized gear box. The diamond seat fits into the bottom plate secured by three set screws, which allow for lateral adjustment of the bottom diamond. The top seat is pressed into top plate and not mobile. The three outside screws can be adjusted to align the tilt of the top diamond. This DAC allows for accurate alignment of diamonds and is capable of pressures of ~ 50 GPa with 500 fim diameter culet diamonds and higher pressures with smaller culets. It has little change in pressure (±2 GPa) with the application of low temperature (~4 K) to moderately high temperature (~ 500 K). Picture courtesy of Alamx-easylab. punch through the gasket, touch each other, and break. Also, if there is misalignment of the diamonds, the gasket hole will drift. The danger to the diamonds is if the hole drifts off of the edge of the culet. At this point, the diamonds would be pushed through the hole, touch each other, and break. 2.1.2 Paris-Edinburgh Press Experiments performed at Oak Ridge National Laboratory used a Paris-Edinburgh (PE) Press to apply pressures from ambient pressure to & 7 GPa. A PE Press consists of two opposing pistons which can be driven together by means of hydraulics (chambers can be filled with oil to displace the pistons, thus pushing them together), shown in Figure 2.6. Anvils typically made of a hard material are placed on the opposing pistons. For these experiments, the anvils were made of cubic boron nitride. The anvils are designed with an indentation that fits a gasket (in this case, a single toroidal gasket was used). The two 22 Figure 2.4. Another style of plate DAC; this is a miniature model manufactured by HPDO (hpdo.com). The pressure is applied by means of the three pressure screws. The posts attached to the bottom plate fit snugly into the top plate and preserve the diamond alignment as pressure is applied by gently tightening each screw separately. Both seats are attached to the plate by means of set screws, allowing for lateral adjustments. The only mechanism to adjust the tilt is by properly epoxying the diamonds; see section 2 .1 .1 . anvils, much like in the case of the DAC, must be aligned in order to assure that the two gasket indentations are directly oppose each other. Also as in the case of a DAC, if the anvils are misaligned, then the entire pressure apparatus, would not only fail to provide pressure to the sample but could also fail catastrophically. 2.1.3 Pressure Measurement The pressure can be measured using various pressure markers, ruby fluorescence being one of the most common. Another means of measuring pressure is using a known equation of state (eos) of the material, the change in unit cell volume as a function of pressure. 2.1.3.1 Ruby Fluorescence In this work, ruby (Al2O3 :Cr, alumina with chromium doping) fluorescence was the main source of pressure measurements, typically under nonhydrostatic conditions using the standard calibration [114, 45, 54, 195, 28]. Ruby fluoresces when excited by laser as shown in Figure 2.7. The wavelength of this excitation has a pressure dependence. To calibrate the ruby pressure, the fluorescence of the ruby is measured along with a pressure marker that has a well-defined eos (equation of state). The pressure can be determined from measurements of the pressure marker's unit cell (the compression of the unit cell with increasing pressure) and the wavelength of the ruby's fluorescence is recorded at each pressure. The dependence 23 Figure 2.5. A piston cylinder style DAC designed for resistive heating, capable of reaching temperatures of & 1000 K. Pressure is applied by means of four pressure screws which engage the piston via through-holes in the cylinder, or through the use of a steel membrane which attached to the cylinder directly and applied pressure when inflated with gas. This figure shows the Helios DAC, which is designed for high temperature. Many measurements in the works were also taken using the Diacell Bragg DAC. The styles and alignment of the two are very similar. The bottom seat is secured to the cylinder with four set screws, allowing for lateral adjustment. The top seat is affixed to the cylinder by means of three set screws on a rocker, which allows for very fast and accurate tilt adjustment. Both DACs provide pressures >100 GPa using 200 im diameter culet; pressures of ~ 50 GPa are routinely achieved. The pressure drift with temperature in the piston-cylinder type DAC is generally not great ± 2 GPa, except in the case when the helium membrane is employed when cooling to low temperatures. In this case, the pressure tends to increase by very noticeable amounts of >5 GPa. Picture courtesy of Alamx-easylab. of the ruby fluorescence as a function of pressure has the form: where A and B are constants, X0 is the ruby fluorescence wavelength (m) measured at ambient pressure, A is the ruby fluorescence wavelength under pressure, and the constant A has units of pressure (GPa). The values of the constants depend upon the specific calibration being employed. For the works in this dissertation, the nonhydrostatic pressures were determined from the calibration of Mao and Bell, later extended by Mao et al. [116, 114], in which A = 1904 GPa and B = 7.665; the authors did not report error bars. This was later refined by Dorogokupets and Oganov, with A = 1871 GPa and B = 10.06; no specific error bars are presented [45]. The calibrations performed by Mao et al. were performed using Ar as a pressure medium. Even though this is the ruby calibration typically used for (2 .1 ) 24 Figure 2.6. Diagram of PE Press. In the configuration shown, the two detectors would be parallel to the page in front and behind the page. Image is the PE Press from SNAP at ORNL: https://neutrons.ornl.gov/snap/sample. nonhydrostatic measurements, the conditions of the electrical resistance measurements in the following chapters were far more nonhydrostatic than the conditions of the calibration. This leads to a greater systemic error in the pressure measurements. For the AC magnetic susceptibility or XRD measurements, in which a pressure medium can be used, a different calibration was used. For hydrostatic measurements, the calibration of Chijioke et al. was used, A = 1879 ±6.7 GPa and B = 10.71 ±0.14 [28]. The typical spectrum of ruby fluorescence appears as a doublet with distinct R1 and R2 peaks. For pressure determination, the R1 peak is measured. However, under nonhydro­static pressure conditions, the shape of the doublet can change noticeably. As described by [23], the shape of the doublet can change depending upon the type of strain experienced by the ruby crystals (trigonal hR30, R-3c). Broadening of the ruby peaks measured under pressure is often attributed to be a sign of nonhydrostatic conditions [141]; however, it has since been shown to be mostly the result of " nonuniform strain" on the ruby crystals themselves [23]. When strained along the a-axis, the distance between the R1 and R2 ruby peaks increases with increasing pressure; the distance decreases with increasing pressure if the ruby experiences strain along the c-axis [23]. For the data presented in this work, the nonhydrostatic rubies showed a distinct broaden­ing and smearing of the R1 and R2 peaks with increasing pressure. The majority of electrical measurements presented in subsequent chapters relied upon alumina as a layer of electrical insulation between the lithium samples and the metal gasket. Alumina (Al2O3 ) is a hard and relatively incompressible material [123, 83]. With a bulk modulus of Ko ~ 254 GPa [83], alumina inside the sample chamber will result in highly nonhydrostatic conditions and 25 Figure 2.7. This is a typical nonhydrostatic ruby under moderate pressure. The y-axis represents the intensity of the spectrum in arbitrary units. The doublet has already begun to merge, and with increasing pressure, the doublet will continue to broaden and merge as the nonhydrostaticity increases. Finding the middle of the peak in such a situation does not take into account the broadened doublet. The pressure is determined by recording the value of the middle of the peak and the value of two-thirds to the higher wavelength, where the R1 peak would be visible under hydrostatic conditions. anisotropic pressure. Alumina is used in these experiments due to the paucity of materials that are nonreactive with lithium. In this work, the width of R1 and R2 spectra lines tends to increase with pressure; this indicates that the rubies, embedded in the alumina, appear to experience such nonuniform strains as described in [23]. Also, the distance between the R 1 and R2 peaks in the nonhydrostatic pressures presented in this work decrease with increasing pressure. This is a consistent trend present throughout different samples and experimental runs. It is unlikely that such a majority of ruby pressure markers would be loaded in a manner that they would experience strain along the c-axis; however, most of the nonhydrostatic measurements were made with alumina insulation inside the sample chamber. The method of producing the alumina insulation is described in section 2.3. This method involves forming a sheet of alumina by pressurizing 0.05 fim3 particle alumina (7 -Alr O3) powder in the sample chamber prior to loading of the sample. 7 alumina differs from bulk (a) alumina in its structure 26 and compressibility (K0 = 144 ± 21 GPa (using a first-order Bridgeman Equation to fit the Equation of State)) [25, 60]. Ruby powder was used in the pressure measurements as well as ruby spheres; however, the spectrum from the particular ruby spheres used was often too weak of a signal to be reliable. Thus, ruby powder was more common for the pressure determination. It is plausible that the alumina powder pressurizes into a sheet with a preferred orientation (the c-axis perpendicular to the diamond culets), causing the distinct decrease in the R1 and R2 distance with increasing pressure. Considering this effect, pressure determinations will benefit from using the R2 peak, as the R1 peak appears underestimate the pressure of the sample [23]. The ruby fluorescence also has a measured temperature dependence; the wavelength of the fluorescence decreases as a function of temperature as measured against pressure markers with known eos as well as thermal expansion coefficients [54, 195]. Thus, the temperature at which the ruby fluorescence is measured must also be recorded. Also, the R2 line disappears at low temperatures, leaving a sharper single R1 line. For low temperatures (<10 K ), a different calibration is necessary [54, 195]: P = Ao l n ( aA^ (2.2) the constant A0 has units of GPa, and A0 = 1762 ±13 GPa. 2.1.3.2 Equation of State The eos of lithium or NaCl, [40, 75], were also used in Chapter 6 of this work to measure the pressure inside the cell. The equation of state describes the variation of a solid's volume as a function of pressure and temperature. Measurements of X-ray or neutron diffraction give the spacings (d) of the lattice planes, which in turn give the unit cell volume. Measuring the equation of state of a material requires determination of the unit cell at different pressures. This allows for the determination of the bulk modulus and its derivative: k = -v d P , K (2.3) K = -dP This parameter describes how much the material resists compression. A lower value for K indicates a more compressible material, a material which will display a rather steep decrease in volume as a function of pressure. The equation of state of many materials, including NaCl and Li, have been previously measured [182, 40, 75, 74]. Thus, using these data as a calibration, we are able to measure the unit cell volume of the Li or NaCl and 27 determine the pressure of the sample. For more sensitive pressure markers, compressible materials with small bulk moduli are preferred. There are several functions which can be used to fit the volume versus pressure or temperature data [1]. In the results presented in Chapter 6 , the Vinet fit was used to fit the data for lithium, as this particular fit is derived from a general interatomic potential and is appropriate for simple solids [1, 181, 33] and was also used in previous works [74]. The Vinet fit: In this work, low temperature and moderately high temperature techniques were em­ployed. 2.2.1 Low Temperature At low temperatures, the energy becomes less and less dominated by kBT , allowing more subtle energies, such as zero point motion, to become detectable. 2.2.1.1 Cryostat Low temperatures, <2 K, for table top experiments at the University of Utah were achieved by means of cooling inside a liquid 4He continuous flow cryostat (Janis Research Co. Model SVT-200-5), and a cryogen-free closed cycle cryostat (Janis Research Co. Model SHI-950-15). A continuous flow cryostat typically consists of several different sections or layers. The innermost section is the sample tube, in which a rod with the sample (in our case a DAC) attached at the bottom is inserted. The next layer out can be filled with a cryogen, either liquid helium or liquid nitrogen. In the case of liquid helium use, there is another layer which is filled with liquid nitrogen to prevent excess boil off of liquid helium. The cryostat is thermally insulated with a high vacuum jacket. This type of cryostat achieves low temperatures by transferring the cryogen, typically liquid helium, from a storage dewar to inner cryogen chamber of the cryostat and then controlling the rate of the flow of the cryogen to the sample tube. Due to inconsistent rates of cooling, most notably in the liquid 4He system, heating curves were analyzed for temperature effects by letting a small pool of where the subscript 0 denotes the values at zero pressure (or ambient pressure). An example of the Vinet fit used for lithium can be seen in Chapter 6 . 2.2 Temperature 28 liquid helium into the bottom of the cryostat sample tube, applying a vacuum, and allowing the system to heat passively. A closed cycle cryostat has a chamber in which a similar rod with the sample (in our case a DAC) attached at the bottom is inserted. Cold helium gas is pumped through the chamber and an external mechanical pump extracts the warmer helium gas to cool and recycle. The DAC on the rod is cooled by being in thermal contact with a metal cold point of contact with the sample chamber walls. Once the system reaches the temperature of liquid helium, the sample chamber can be filled with helium gas which will liquefy upon contact with the metal cold chamber walls. In this manner, a small amount of liquid helium can pool into the bottom of the cryostat. A steady, slow heating can be achieved by forming a small pool of helium applying a vacuum and allowing the system to passively heat (in the same manner as in the continuous flow cryostat). This also allows the measurement to be taken without the vibrations of the chiller affecting the data. For the experiments performed with the High Pressure Collaborative Access Team from the Carnegie Institute of Science in the Advance Photon Source at Argonne National Laboratory, a home-made continuous flow cryostat was used. The cryostat was made to be of a small enough size to fit into the synchrotron X-ray beam, which limited the base temperature to & 10 K. As with the Janis systems, the cryostat is insulated with a vacuum jacket. A liquid helium dewar is connected to the cryostat and an open flow cools the system to base temperature by flow pipe of liquid helium in contact with the metal (Cu, due to it thermal conductivity) sample holder. A series of heaters near the DAC and Cu DAC holder serve to balance the temperature if higher than base temperature is desired. For experiments performed in the National High Magnetic Field Laboratory, low tem­peratures of 0.3 K were achieved using an Oxford systems top loading 3He cryostat. For experiments performed at Oak Ridge National Laboratory, low temperatures were limited to about the boiling point of nitrogen (& 77 K). A home-made system of circulating LN2 was used to keep the pressure cell (a PE press) cold and a series of heaters were employed to raise the temperature above the base temperature, and balancing the heaters versus the circulating LN2 could keep a mostly constant temperature (± 5 K). Due to the large mass of the pressure cell, using liquid helium was prohibitively expensive. 2.2.2 High Temperature While low temperatures are key for the search of quantum solids, high temperature experiments also play a role. There are two popular strategies for applying heat to a sample inside a DAC: to apply heat to the sample only, keeping diamonds, gasket, and 29 body of the DAC at a colder and more constant temperature; or to heat the entire DAC evenly (the same strategy as cooling the entire DAC in the cryostat), slowly, and steadily to avoid thermal lag issues. 2.2.2.1 Laser Heating Laser heating is a common technique used when one simply wants to heat the sample locally (not the entire DAC), or even to heat only a small portion of the sample. There are many reasons why one would desire to do this, most notably when wishing to heat the sample to temperatures that would damage, weaken, or even melt the gasket material or the other materials of the DAC. For instance, attempting to recreate conditions to melt iron under the extreme pressures (conditions near the molten/solid mantle interface) would require temperatures certainly high enough to melt an entire DAC (if it was made of a material such as hardened steel). In order to heat only the sample to these >1000 K temperatures, a high power laser (generally in the near IR range), is focused on the sample, or a portion of the sample. Increasing the laser power increases the temperature of the sample. For some samples, such as hydrogen, an absorber is placed in the sample chamber of the DAC with the sample to absorb the laser power and heat the sample [41]. In this type of laser heating experiment, the temperature must be measured remotely. A popular technique used to measure temperature is the collection of the black body radiation emitted from the sample [41, 16]. The error in temperature can be determined though the fit to a calculated black body spectrum, which can lead to large error bars. The emitted radiation is collected from the sample and split into two portions; one portion goes to the detector (in order to fit into the black body curve), and the other portion is generally used for a visual image of the sample which is in turn used to confirm that the laser is striking the appropriate piece of the sample and also to possibly monitor the laser reflections from the sample (for example, speckle motion can be used to determine if the laser is reflected from a solid or a liquid, see section 3.3 [16]). 2.2.2.2 Resistive Heating For lower temperatures, temperatures which are not likely to damage the DAC, re­sistance heating techniques can be used rather than laser heating. In many cases, only relatively low increases in temperatures are desired, such temperatures would require low power lasers and would be so low that distinguishing the black body signal from the room temperature is problematic. In these cases, resistively heating the entire cell is a possible solution. In this work, two different resistive heating methods were used. The first used a 30 plate DAC made by Almax-easylab, and to measure temperature, a K-type thermocouple was anchored to the bottom side of the top plate, resting against the side of the diamond and insulated with Kapton. The DAC and a thermocouple were placed inside a clay cylinder which had a high resistance nichrome coil inside. This whole assembly was placed in a 4 inch diameter petri dish, with the thermocouple wire, two ends of the nichrome coil wires, as well as the four electrical probe wires leading to the outside. Either a whiteclay lid with a glass window or a standard petri dish cover was used to close the home-made oven, depending on the need for quality optical access. The oven was then mostly sealed with electrical tape; vents were left on purpose to prevent overpressure forming from expansion of hot gas. By applying voltage across the nichrome coils, the high resistance, R, of the nichrome generates heat, where the heat, P , is related to the voltage, V , by: P = R . This system was capable of reaching temperatures of & 600 K. Another system for resistively heating a DAC is using a DAC designed to reach high temperatures. The Almax-easylab Helios DAC is a piston-cylinder type diamond anvil cell that has a resistive heater built near the base of one diamond. A gasket may be placed in contact with the heater. The thermocouple may also be fed through a small opening to come into contact with the gasket. The heater is externally powered by leads reaching out of the cylinder. This setup is capable of reaching higher temperatures of & 1000 K. 2.3 High Pressure Resistance Measurements Resistance is an excellent material property to measure, since it is a relatively straight­forward measurement to perform and analyse. This makes resistance measurements at high pressure incredibly useful. Resistivity (the inverse of a material's conductivity) is an intensive material property; its value is independent of the amount of material present or the type of electrical measurement employed [6 6 ]. Resistivity is a function of temperature; the behaviour of the resistivity as temperature is varied reveals whether the material is a metal, or a semi-metal, or even if it is high quality metal or not. The resistivity of metal can be described by the Matthieson rule: p(T) = po + pi(T) (2.5) where p0 represents the resistivity, or impedance of current carrying electrons, from impuri­ties and pi is the intrinsic resistivity of the material [27, 120]. This work is chiefly concerned with resistivity of metals. If a material becomes less resistive (more able to carry current) as temperature is decreased, then it is exhibiting metallic behaviour [6 6 ]; the slope of that decrease gives information regarding the quality of the metal. 31 A cartoon picture to consider is the following: If one considers a metal as a material that has ' free' electrons, the electrons have the ability to carry electric current through a metal. A material at a nonzero temperature is not completely still. The ions forming the lattice will have motion due to the thermal energy, kBT (or in the case of a quantum solid, motion from zero-point energy). The motion of the lattice ions, expressed as phonons, can create random points of interaction for the electrons. Therefore, a portion of the current (consisting of current carrying electrons), instead of flowing through the lattice without impedance, actually experiences impedance through interaction with lattice phonons. Thus, the current is impeded by the thermal motion of lattice ions. A simple metal (with the Fermi surface contained entirely inside the first Brillouin Zone) has an intrinsic ability to conduct a current; in a completely defect-free crystal, the current would be infinite in the absence of lattice motion. Thus, at zero temperature, and in the absence of zero-point motion, a perfect simple metal crystal would be an ideal conductor, with zero resistivity (the inverse of conductivity). (For metals in which the Fermi surface crosses the Brillouin Zone boundaries, there is a nonzero resistivity due to umklapp processes.) The intrinsic resistivity of metal, pi, is due to electron-phonon interactions. This can be expressed as a function of temperature by the Gruneison-Bloch relation [138, 6 8 ]: rr © R C / T \ f r z5 Pi(T) = M b r ( o R j I dZ(ez - 1 ) ( 1 - e- z) (2.6) where C is a constant, M is the atomic weight, T is the absolute temperature, and 0 R is the empirical temperature characterizing the metal' s ideal resistivity, analogous to the Debye temperature, 0 D, which characterizes the specific heat of a sample [27]. At high temperature, T » 0 D, the number of phonons liable to interact with electrons is directly proportional to the temperature [6 ]. Thus, the resistivity becomes a linear function with respect to the temperature: Pi(T) - T ,T » 0 D (2.7) Below about 0.10R, 0 R & 0 D, the integral in 2.6 is a constant and the expression can be reduced to: Pi<T > & M e f (2'8) Cl T5 where is C1 = 124.4C [27]. The Gruneison-Bloch equation is relevant for simple metals characterized by nearly spherical Fermi surfaces, making it applicable to lithium and other alkali metals, though not over the entire temperature range. It is a good approximation for alkali metals, particularly at low and high temperatures. 32 Metals have a recognizable trend in resistivity as a function of temperature as seen in this equation. When the system is cooled, the thermal motion of the ions is reduced, thus decreasing the number of possible random electron-phonon interactions. Thus, the ability to carry electric current (the conductance) increases, reducing the resistivity as a function of temperature. If we see the resistivity decrease as a function of temperature, then it can be interpreted that the interactions between the phonons (produced by the thermal motion of the lattice ions) and electrons are decreasing. In other words, as the lattice ions move less and less due to the decreasing thermal energy, then the current flows through the lattice with less impedance, meaning the material has less resistivity. Impurities cause defects or distortions in the lattice, which serve as additional sources of impedance for the current (p0). The slope at which the resistivity decreases as a function of temperature can also be used to determine the quality of a metal sample. This is expressed by the RRR (residual-resistance ratio); this value of samples is sometimes reported to indicate how free a metal is from impurities. The RRR value is the resistance at room temperature divided by the resistance of the sample at low temperature, generally ~ 4 K; the higher the number (the greater the slope of the decrease of resistivity as a function of temperature) the more pure the metal is. If the slope of decrease is very rapid, then this may show that the majority of the resistance is due to intrinsic resistivity, pi(T) (as opposed to impurities or defects, po) [31]. The steeper the rate of descent of resistivity as function of temperature, the more pure the sample. The slope of the resistivity as a function of temperature may also show that the material is not a particularly good metal. The mean free path of the electrons through the materials may be short, and the slowing of the motion of the lattice ions may not strongly affect the short mean free path of electrons. Thus, we are left with the resistivity decreasing as a function of temperature, though perhaps with a shallow slope. The nature of the decrease of resistivity as a function of temperature may also change once lower temperatures have been reached. A common example is Pt, the slope of the resistivity is quite steep with the decrease in temperature until near 20 K [142]. At this point, magnetic impurities begin to dominate the trend, and the resistivity becomes nearly constant as a function of temperature (0 K < T < ~20 K). Whilst these magnetic impurities were not apparent at higher temperatures, once the intrinsic resistivity (pi) has diminished, they become the dominant source of the resistivity of the material. There is much information regarding the sample embedded in the resistivity as a function temperature, including information regarding the material's structure. Whilst resistivity is 33 an intensive material property, it must never be forgotten that what is actually measured is resistance. Resistance, as opposed to resistivity, is an extensive property; it does depend upon the amount of material measured and the configuration of the electrical probe em­ployed. These have a very simple relation: p = A R (drawn in Figure 2.8); where resistivity is p, resistance is R, cross-sectional area of the sample is shown by A, and length of the sample is l (for all practical purposes, this would be the length between the voltage probe). If we apply a known current through a material, we can measure the voltage difference at two points, shown in Figure 2.9. Using Ohm's Law V = R I , we control I , the current, and we measure V , the voltage, so we may solve for R, resistance, without much difficulty. However, the sample, as a part of the circuit (with a constant voltage supply), does affect the current. In the case of metals, the correction is too small to be significant. Though to be exact, we would have to include the effect that the sample itself has on the circuit, since the circuit used has a constant voltage source, not a constant current supply. This means the equation to solve for resistance becomes: V V ? Rc r = v s v (2 .9 ) ( 1 - # > where Vs is the voltage measured (the absolute value is considered as the sign of the voltage can be negative depending upon the phase), V is the constant voltage applied, and Rc is a resistor hard wired into the circuit to give nearly constant current with a constant voltage source. In all experimental setups, resistance is the parameter that is measured, then the resistivity is calculated from the geometric properties. However, since those geometric properties are present in the relation, p = R A, changes in the geometry of the sample can indeed change the signal. For example, if one measures the resistance and then applies an anisotropic pressure from say the top and bottom (as the case when driving the two diamond Figure 2 .8 . A cartoon of the relation between resistance and resistivity. 34 Figure 2.9. A diagram of the circuit used to measure resistance inside of the DAC. The function generator of a SR830 Lock Amplifier was set to values near 0.1 V and < 15 Hz to provide a nearly constant current through the sample. Since the Lock In Amplifier provides a constant voltage source, not a constant current source (though the use of a resistor with a resistance » than the sample's resistance serves partially regulate the current), the resistance of the sample and any leads inside of the voltage probe can have an effect on the current of the overall circuit. For metals, this correction is smaller than the noise in the measurement itself. anvils together with a nonhydrostatic pressure medium), the area, A , could decrease faster than the length, l, could either decrease or increase. Though most likely l would be constrained, then the resistance of sample will increase. In the case of pressure-induced structural phase transitions, these are certain to be reflected in the resistance data; however, they may be difficult to distinguish from all other sorts of geometric changes. For example, when applying nonhydrostatic pressure, the area, A , of the sample may be reduced, causing the measured resistance of the sample, R, to increase. To measure resistance under high pressures, an electrical four-point probe or quasi-four probe must be built to function inside the sample chamber of the DAC. For a four-point measurement, four separate wires electrically contact the sample inde­pendently. Two of the wires are used to induce a current through the sample and the other two wires are used to measure the voltage across the sample. Generally, this is considered to be the standard method of measuring voltage across a sample. The two voltage leads contact only the sample. However, even a true four probe arrangement, as seen in Figure 2 .1 0 , does require the resistivity of the sample itself to be taken into account if a constant 35 Figure 2.10. Methods for measuring the voltage drop, which can be related to resistance, across a sample. A) a cartoon of the 4-point probe arrangement. B) a cartoon of the quasi-4-point arrangement. voltage source is used in the measurement. This is because the portion of the sample between the current leads and the voltage leads determines the amount of current flowing through the sample (I = R where R = Rsample + Rleads). In the experimental setup, this effect is reduced by limiting the space between the current leads and voltage leads, and in the analysis by using equation 2.9. Due to size constraints present in a DAC, especially when attempting to reach higher pressures by using smaller culets, the quasi-four probe is often advantageous. The quasi-four probe works according to the same principles as the four-point probe, the only difference is that each current lead touches one voltage lead (as seen in Figure 2.11). The greatest consequence of this configuration is that a portion of the signal now comes from the leads themselves. Several of the experiments in this dissertation have used resistance under high pressure to measure either solid to liquid transitions or superconducting transitions under pressure. The technique is naturally modified according to the needs of the experiment; here, the basic outline of the process will be explained, though each chapter with results also contains some explanation that is specific to those results. To prepare a DAC for electrical resistivity measurements, first the DAC must be properly aligned and a gasket preindented and drilled. The gasket must then be electrically insulated. For these experiments, stycast BLU (2850 KT) epoxy is used to insulate the gasket. Inside the indented area of the gasket, alumina powder is pressurized to form a solid sheet. This acts as another layer of insulation as well as a pressure transmitting medium. A small hole is then poked in the center of the gasket chamber hole (which was filled with solid alumina); this is where the sample will be placed. Depending upon the material and the desired pressure range, other materials besides alumina may be used, such as NaCl or LiF for example. 36 Figure 2.11. A cartoon of a typical electric probe built on a DAC. When size constraints allow, a four-point probe is built (culets of & 350 - 500 ^m); when smaller culets are used, a quasi-4-point probe is built. When building the quasi-4-point probe, all the leads are placed as close to the sample as possible to have a minimal signal from the leads. Figure by William B. Talmadge. 37 The electrical probe itself consists of Cu wires (typically ~ 35 AWG) leading outside of the DAC (attached to wires eventually connecting to the Lock In Amplifier) and attached to smaller leads cut from very thin foil that eventually contact the sample. Pt is a good material for such leads; it cuts fairly cleanly allowing one to make very thin, sharp leads. However, at times other materials are desired such as Ni or Ta. The thin leads are glued to the gasket and arranged so that they will make a mechanical connection to the sample when under pressure. They are attached to the Cu leads by