THE APPLICATION OF TWO FLUID MODEL TO IR SPECTRA OF HEAVY FERMIONS A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Prabuddha Madusanka Hathurusinghe Deawage November, 2018 THE APPLICATION OF TWO FLUID MODEL TO IR SPECTRA OF HEAVY FERMIONS Prabuddha Madusanka Hathurusinghe Deawage Thesis Approved: Accepted: Advisor Dean of the College Dr. Sasa V. Dordevic Dr. Linda M. Subich Faculty Reader Dean of the Graduate School Dr. Ben Yu-Kuang Hu Dr. Chand Midha Faculty Reader Date Dr. Sergei F. Lyuksyutov Department Chair Dr. Christopher J. Ziegler ii ABSTRACT This work studies conductivity data experimentally determined using IR spectroscopy for three heavy fermion materials: YbFe4Sb12, CeRu4Sb12 and CeCoIn5. We describe them using the two fluid model of electrons and mathematically analyze the materials using the Drude-Lorentz model. In particular, using the program RefFit, the optical data in the frequency range of 4-25,000 cm−1 for temperatures ranging from 8-300 K was considered. We found the plasma frequency and scattering rate and observed how temperature effected their magnitudes and the distributions of electrons in our system. Through analysis of these Drude-Lorentz model parameters, the character- istic temperature - the point below which large conductivity exists - of all three materials was found to be in the range of 70-80 K. Further, using these characteristic temperatures, we showed that the optical data was described by a modified form of the hybridization order parameter equation. This demonstrates that the IR spectra of these materials can be described using the two fluid model. iii ACKNOWLEDGEMENTS I would like to express my sincere thanks to my advisor Dr. Sasa V. Dordevic for his continuous support of my research and his patience, motivation, and immense knowledge. His guidance helped me in all throughout my research and writing of this thesis. I would also like to extend my thanks to Dr. Ben Yu-Kuang Hu and Dr. Sergei F. Lyuksyutov for being on my thesis committee and for their insightful comments and encouragement. I would also like to thank the rest Physics Department’s academic staff for their support. I warmly thank my friends for sharing their knowledge and for their invaluable help. Finally, last but by no means least, my parents for their unconditional love and support in a number of ways throughout my time here. iv TABLE OF CONTENTS Page LISTOFTABLES................................. vii LISTOFFIGURES ................................ viii CHAPTER I. INTRODUCTION............................... 1 II. LITERATUREREVIEW........................... 3 2.1 Electrodynamic Solids . 3 2.2 Maxwell’s Equations . 4 2.3 Conductivity ............................... 5 2.4 DrudeandLorentzmodel ........................ 6 III.HEAVYFERMIONS ............................. 12 3.1 YbFe4Sb12, CeRu4Sb12, CeCoIn5 .................... 13 3.2 Two-fluid Model and Hybridization Order Parameter . 23 IV.REFFIT .................................... 25 4.1 UsingRefFIT............................... 26 V. RESULTANDDISCUSSION ........................ 30 5.1 YbFe4Sb12 ................................ 34 5.2 CeRu4Sb12 ................................ 47 v 5.3 CeCoIn5 .................................. 59 5.4 HybridizationOrderParameter . 71 VI.CONCLUSION ................................ 76 BIBLIOGRAPHY ................................. 78 APPENDICES ................................... 81 APPENDIX A. APPENDIX TITLE GOES HERE . 82 APPENDIX B. SECOND APPENDIX: THE TWO DIMENSIONAL WAVEEQUATION .................... 83 APPENDIX C. EXAMPLE OF A TABLE AND A FIGURE . 84 vi LIST OF TABLES Table Page 3.1 Properties of YbFe4Sb12........................... 13 5.1 Temperaturesforthreematerials. 33 5.2 Fitting parameters for real and imaginary component of YbFe4Sb12 at10K. ................................... 36 5.3 Drude model fitting parameters for YbFe4Sb12 at various temperatures. 38 5.4 Lorentz models fitting parameters for YbFe4Sb12 at various temperatures. 40 5.5 γ fitting parameters of first three modes for YbFe4Sb12 with temperature. 45 5.6 Fitting parameters for real and imaginary componant of CeRu4Sb12 at10K. ................................... 49 5.7 Drude models fitting parameters for CeRu4Sb12 at various temperatures. 51 5.8 Lorentz models fitting parameters for CeRu4Sb12 at various temperatures. 53 5.9 γ fitting parameters of first three modes for CeRu4Sb12 with temperature. 57 5.10 Fitting parameters for real and imaginary componant of CeCoIn5 at8K..................................... 61 5.11 Drude models fitting parameters for CeCoIn5 at various temperatures. 63 5.12 Lorentz models fitting parameters for CeCoIn5 at various temperatures. 65 5.13 γ fitting parameters of first three modes for CeCoIn5 with temperature. 69 5.14 Parameter values for YbFe4Sb12, CeRu4Sb12 and CeCoIn5. ....... 72 vii LIST OF FIGURES Figure Page 3.1 YbFe4Sb12 has a cubic structure with the space group Im3. The color representaion is, Blue - Sb, Red - Fe, Yellow - Yb (https: www.researchgate.net/ publication/ 255251265 Lattice dynamics and anomalous softening in the YbFe4Sb12 skutterudite). 14 3.2 This graph from Dordevic S.V. et.al. [1] shows the resistivity of YbFe4Sb12 asafunctionoftemperature. 15 3.3 CeRu4Sb12 crystal has body centered cubic (BCC) structure (https : //www.sciencedirect.com/science/article/pii/S0022459616302079). 16 3.4 This graph from Dordevic S.V. et.al. [1] shows the resistivity of YbFe4Sb12 dependenceoftemperature. 18 3.5 This graph from Yuji Matsuda [2] shows CeCoIn5 crystal structure. 19 3.6 This graph from Singley E. J. et.al. [3] shows the conductivity of CeCoIn5 dependenceoftemperature. 21 3.7 This graph from Dordevic S.V. et.al. [1] shows band structure of conduction electrons (ǫk) and localized f-electrons (ǫf ) at high tem- perature (dashed lines) and low temperature (solid line). 22 4.1 Firstinterface................................. 26 4.2 DatasetManagerwindow. ......................... 26 4.3 GraphwindowandContentswindow. 27 4.4 Model window and Parameter control window with the Graph con- tentwindow.................................. 28 4.5 Changing the parameters of the Drude and Lorentz model. ... 29 4.6 Changing the parameters of Drude and Lorentz model. .. 29 viii 5.1 (a) Simple Drude mode, (b) Three Drude modes with different pa- rameters for conduction and heavy electrons, (c) Two Drude modes for conduction electrons and heavy electrons, (d) Two Drude modes for conduction and heavy electrons with a Lorentz mode for inter- bandtransition................................ 32 5.2 This graph from Dordevic S.V. et.al. [1] shows the real (top) and imaginary (bottom) parts of the conductivity of YbFe4Sb12 as a function of wavenumber for different temperatures. .. 35 5.3 Graph showing experimentaly determined conductivity (black) with the real part of the fitted conductivity curve (red) for YbFe4Sb12 as a function of frequency at 10 K. The modes represent the individual components(blue)oftheoverallcurvefit. 37 5.4 Graphs showing ω of mode 1 (red), mode 2 (blue) and total ω p − − p (black) for YbFe4Sb12 asafunctionoftemperature. 39 5.5 Graphs showing ω of mode 3 (red), mode 4 (blue) and total ω p − − p (black) for YbFe4Sb12 asafunctionoftemperature. 41 5.6 Graphs showing normalized total ωp for both Drude (black) and Lorentz (red) models of YbFe4Sb12 as a function of temperature. 43 5.7 Graph showing total ωp for both Drude and Lorenz models of YbFe4Sb12 asafunctionoftemperature. 44 5.8 Graphs showing γ of mode 1 (Drude), mode 2 (Drude) and mode 3 (Lorentz) for YbFe−Sb as a function of− temperature. 46 − 4 12 5.9 This graph from Dordevic S.V. et.al. [1] shows the real (top) and imaginary (bottom) parts of the conductivity of CeRu4Sb12 as a function of wavenumber for different temperatures. .. 48 5.10 Graph showing experimentaly determined conductivity (black) with the real part of the fitted conductivity curve (red) for CeRu4Sb12 as a function of frequency at 10 K. The modes represent the individual components(blue)oftheoverallcurvefit. 50 5.11 Graphs showing ω of mode 1 (red), mode 2 (blue) and total ω p − − p (black) for CeRu4Sb12 asafunctionoftemperature. 52 5.12 Graphs showing ω of mode 3 (red), mode 6 (blue) and total ω p − − p (black) for CeRu4Sb12 asafunctionoftemperature. 54 ix 5.13 Graphs showing normalized total ωp for both Drude (black) and Lorentz (red) models of CeRu4Sb12 as a function of temperature. 55 5.14 Graph showing total ωp for both Drude and Lorenz models of CeRu4Sb12 asafunctionoftemperature. 56 5.15 Graphs showing γ of mode 1 (Drude), mode 2 (Drude), mode 3 − − − (Lorentz) and mode 6 (Lorentz) for CeRu4Sb12 as a function of temperature..................................− 58 5.16 This graph from Singley E. J. et.al. [3] shows the real part of the conductivity of CeCoIn5 as a function of wavenumber for different temperatures. ................................ 60 5.17 Graph showing experimentaly determined conductivity (black) with the fitted conductivity curve (red) for CeCoIn5 as a function of frequency at 8 K. The modes represent the individual components (blue)oftheoverallcurvefit. 62 5.18 Graphs showing ω of mode 1 (red), mode 2 (blue) and total ω p − − p (black) for CeCoIn5 asafunctionoftemperature. 64 5.19 Graphs showing ω of mode 3 (red), mode 4 (blue) and total ω p − − p (black) for CeCoIn5 asafunctionoftemperature. 66 5.20 Graphs showing normalized total ωp for both Drude (black) and Lorentz (red) models of CeCoIn5 as afunction oftemperature.