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Van Hove SINGULARITIES in BCS THEORY Van Hove SINGULARITIES in BCS THEORY Van Hove SINGULARITIES IN BCS THEORY Van Hove SINGULARITIES IN BCS THEORY By Armando Ga.ma Goicochea, B.Sc. A Thesis Submitted to the Faculty of Graduate Studies in Partial Fulfilment of the Requirements for the Degree Master of Science McMaster University August 1992 MASTER OF SCIENCE{1992) McMASTER UNIVERSITY (Physics) Hamilton, Ontario TITLE: Van Hove Singularities in BCS Theory AUTHOR: Armando Gama Goicochea, B.Sc. (Universidad Nacional Autonoma de Mexico) SUPERVISOR: Dr. Jules P. Carbotte NUMBER OF PAGES: vii, 115 ii ABSTRACT The influence of a logarithmically dependent (van Hove singularity) electronic density of states is studied in the weak-coupling limit. Through analytic and numerical analysis it is found that the model can give rise to temperatures in the 100 K range, and that universal BCS ratios such as 2tfa./k8 Tc and tfa.C/'rTc do not change essentially from their constant BCS values. The consequences of this model on the calculation of the isotope effect and specific heat are discussed in detail and compared to recent experimental results. iii To my parents, Eva Goicochea and Armando Gama ACKNOWLEDGEMENTS I wish to thank my supervisor, Dr. Jules P. Carbotte for his advice and encouragement. His devotion for his work makes of him an excellent role model. This is perhaps the most trascendentallesson one can learn from him. I am also indebted to Drs. David Goodings and Yuki Nogami for their careful reading of this thesis, and for the very useful comments they had. For the multiple enlightening discussions we had, his interest in this work, and his willingness to offer help, I would like to thank Mohamed Man­ sor. I have also benefitted from discussions with Phil Fischer. My gratitude goes to all those who have made these two years a very enjoyable learning experience, in particular Peter Arberg, Anthony Boey, Alex Bunker, Charles Curry, Mohamed Mansor, Peter Mason, R.achid Ouyed, Harald Schwichow, and Mark Walker. For sharing their enormous talent and friendship with me I thank them all. It is an honour to count them as friends. The generous hospitality of Harald Schwichow is also acknowledged. The staff of the Department of Physics and Astronomy has always been very helpful. Their efficiency and friendliness are exceptional. They all certainly contribute to enhance the reputation of the Department. For financial assistance, thanks are due to the Department of Physics and Astronomy of McMaster University. Finally, I would like to thank the Universidad Nacional Aut6noma de Mexico for taking care of my graduate education through a D.G.A.P.A. scholarship. v Table of Contents Chapter 1 Introduction ............................................... 1 Chapter 2 The BCS Theory . 5 2.1 Historical Introduction ........................................... 5 2.2 The Ground State . 8 2.3 Zero-Temperature Energy Gap .................................. 10 2.4 Temperature Dependent Energy Gap ........................... 12 2.5 The Critical Temperature. 13 2.6 The Isotope Effect . 15 2.7 Thermodynamic Properties ..................................... 16 2.8 Concluding Remarks. 18 Chapter 3 IDgh Tc Superconductivity . 21 3.1 Historical Introduction .......................................... 21 3.2 Structural Properties . 23 3.3 Physical Properties . 24 3.4 Concluding Remarks. 28 Chapter 4 Van Hove Singularities in BCS Theory ................ 31 vi Table of Contents 4.1 Introduction .................................................... 31 4.2 Energy Dependence in the DOS. 32 4.3 The Model . 34 4.4 The Critical Temperature. 36 4.5 Zero Temperature Energy Gap. 47 4.6 The Isotope Effect . 56 Chapter 5 Thermodynamic Properties . 43 5.1 Zero Temperature Condensation Energy . 43 5.2 :Free Energy . 66 5.3 Specific Heat . 76 5.4 Concluding Remarks. 94 Chapter 6 Conclusions . 97 Appendices. 101 A The Density of States. 101 B A Useful Integral ............................................... 105 C Weak-Coupling Constant . 107 D The Energy Gap near Tc . 109 References .......................................................... 112 vii Chapter 1 Introduction A few years ago, the idea of superconductivity occurring at temper­ atures around 100 K would probably have seemed futile, or hopeless. And there was reason to believe so, since for decades scientists worked hard to raise the critical temperature, Tc (temperature at which a material undergoes the superconducting transition), only a few degrees above 20 K. This ideal became reality in 1987, when superconductivity was found to occur in copper-based oxides at temperatures close to 100 K. Through ingenious chemical substitutions, the highest Tc was later pushed up to 125 K, in 1988, and remains the record Tc known to date. These events constitute a major breakthrough that has profoundly revolutionized not only physics, but science in general. It has also produced a flood of multidisciplinary work hardly seen before. Soon after the discovery of high Tc oxides it became clear that the con­ ventional microscopic theory of superconductivity, formulated by J. Bardeen, 1 2 1 Introduction L. N. Cooper, and J. R. Schrieffer (BCS) in 1957 did not provide an ade­ quate description of these new materials. Experimental results such as a very small or near zero isotope effect, and unusual temperature dependence in the normal state resistivity led some authors to formulate models that were not based on a Fermi liquid description, and did not invoke a phonon-mediated pairing mechanism. By contrast, others believe that the Fermi liquid ap­ proach is valid, and that once the appropriate pairing mechanism is found, the BCS formalism will provide a good description of the oxide supercon­ ductors. A common feature found among these new materials is the presence of Cu. and 0 atoms occupying a nearly square planar arrangement. These Cu.02 planes, as they are usually referred to, are probably the most funda­ mental pieces, for they are believed to be responsible for superconductivity in the layered compounds. This is the reason why superconductivity in these compounds is generally considered quasi two-dimensional. On the other hand, it has been known for a long time that the density of states (DOS) for electrons in a two-dimensional periodic potential exhibits logarithmic van Hove singularities (vHs), which arise from saddle points in the energy dispersion relation. This motivated some theorists recently to introduce the concept of vHs in the DOS of the new Cu.-based superconductors, in an attempt to explain the strikingly high Tc's that they show. It has also been claimed that the small isotope effect can be understood with the help of this energy dependent DOS. 1 Introduction 3 We believe that this conclusion is of considerable importance. Moti­ vated by this interesting suggestion, we have introduced a vHs in the elec­ tronic DOS near the Fermi energy, within the framework of the weak-coupling BCS theory, and some properties have been calculated and compared to their constant-DOS counterparts. Our aim is to determine whether or not this model is able to account for some of the properties of the oxides, and to ob­ tain simple approximate analytic expressions for important superconducting parameters such as the critical temperature, and the specific heat jump. In Chapter 2 we outline briefly some of the important ideas involved in the BCS theory, as well as a few basic equations. No attempt is made to include all the relevant concepts of the theory, nor to provide a detailed derivation of the equations, for there is plenty of excellent literature on the subject accesible to those interested. Chapter 3 is a summary of some relevant characteristics of the high Tc cuprate superconductors. Because this is a rapidly evolving field, one cannot hope more than to merely describe a few common structural and physical properties, especially those that are important in the development of our model. The consequences of a vHs on the critical temperature are studied in Chapter 4, along with some other calculations regarding the zero temperature energy gap, and the isotope effect coefficient. The influence of the vHs in the thermodynamical behavior of weak­ coupling superconductors is the subject of Chapter 5. The free energy, tem­ perature dependent gap, and specific heat are investigated for this model and compared to the BCS results. 4 1 Introduction Fina.lly, the most important conclusions are gathered in Chapter 6. Additiona.lly, Appendices A, B, C, and D are included to provide some com­ pleteness to the work. Chapter 2 The BCS Theory 2.1 Historical introduction The phenomenon of superconductivity was discovered in 1911 by Kamerlingh Onnes1, who named it based on the striking electrical properties of this state, i.e. infinite conductivity below the transition temperature Te. He also discovered that superconductivity is destroyed if a strong enough magnetic field is applied , which is now called the critical field He . A superconductor is more than a perfect conductor, it is also a per­ fect diamagnet. This means that the magnetic field vanishes in the interior of a bulk specimen when cooled below its transition temperature Te. This phenomenon, known as the Meissner effect, was discovered in 1933 by W. Meissner and R. Ochsenfeld2• 5 6 2 The BCS Theory In 1934, C. J. Gorter and H. B. Casimir formulated3 a phenomeno­ logical theory that accounted for the thermodynamic properties of supercon­ ductors, based on a two fluid model. The following year, F. and H. London proposed4 an electromagnetic theory of superconductivity that was
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