in Fermionic Systems with Multiple Internal Degrees of Freedom

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2 Condensation of Cooper Pairs and Cooper Quartets in Fermionic Systems with Multiple Internal Degrees of Freedom A dissertation submitted to the Division of Research and Advanced Studies of the University of Cincinnati in the partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY (Ph.D.) in the Department of Physics of the College of the Arts and Sciences 008 by Aseem Talukdar M.S. University of Cincinnati, USA, 003. Committee chair: Michael Ma ii

3 Abstract Condensation of Cooper pairs and Cooper quartets in fermionic systems with multiple internal degrees of freedom is studied. In this thesis work, I work on two major projects. On the first project I discuss Cooper pair condensation and while on the second I discuss Cooper quartet condensation. Due to the restrictions imposed by Pauli s principle, no two identical fermions can occupy a single quantum state. Therefore for electronic systems with two internal states, the maximum number of electrons that could be bound together is two. However; for a fermionic system having more than two internal states, it is possible that the bound state structure could be quite different. On my thesis I focus on systems that have four internal states. On one hand it is possible that the system will still undergo some types of pairing condensation, but there is also a possibility that the fermions will form a more complex structure where four fermions are bound together which we call a quartet. Physical systems where fermions can have four internal states include a system of spin- 3 fermionic atoms and a two band electronic system. I look at possible two and four particle bound state structures in such systems. First I discuss pairing condensation in the system. I extend the original Cooper problem to the pairing of two quasiparticles excited out of two decoupled superconductors. I show that two quasiparticles can form a bound state but can t destabilize the underlying system. I derive the Landau Ginzberg free energy for the system and use it to describe the pairing structure that will exist under different limits of the interaction among the fermions. In the second work, I discuss quartet condensation in the system. I modify the Landau Ginzberg approach to include fluctuations in the order parameters and to allow for a quartet order parameter. We show that under the special SU(4) symmetric limit of interaction, the system has a tendency to undergo a quartet instability which will suppress the pair instability. More importantly, the same tendency can be seen iii

4 even if the interaction is tuned away from the SU(4) limit. iv

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6 Acknowledgements Here with great pleasure I want to take the opportunity to express my sincere gratitude to everyone who have contributed in direct and indirect ways to my doctoral work. Although it is really difficult to acknowledge everyone; first of all I would like to thank my thesis advisor Dr. Michael Ma for his intensive supervision, endless patience and for being always there for me throughout the course of my thesis work. It has been real pleasure for me to work under him as a graduate student. I would like to thank Dr. Philip Argyres, Dr. Young Kim and Dr. Slava Serota for serving on my thesis committee. Special thanks to Prof Fu-Chun Zhang for stimulating my interest towards this field and to Dr. B Goodman for helping me with numerous discussions. I am especially grateful to Dr. Rohana Wijewardhana for his support and for his invaluable help at various stages of my graduate career. Again I would like to express my strong appreciation to all the faculty members, staff and students in the physics department for their wonderful help and support throughout. I take this opportunity to thank my previous academic institutions, Ramjas College and Delhi University for their active research oriented environment which played an important role in my development as a physicist. My special admirations to Prof. S K Muthu and Prof. P Dasgupta for their encouraging words during my stay at Delhi University. I also would like to extend my special thanks to my close friends Rupak and Chiranjib who have made my life enjoyable, and have supplied me with endless support and also for bearing with me during the down and depression stages of my graduate v

7 student life. Also I am rather fortunate to have Param and Sherry who have always been there for me. I will always cherish their constant love and support. Again I must thank all my friends in the physics department for making my stay at Cincinnati more enjoyable. Last but not the least, I would like to thank everyone in my family for bestowing their blessings good wishes on me. Nothing would have been possible without the constant love and support of my Maa, Deuta, Dada-Bou, Bhanti, and Abhi-Ani-Geet. And finally my special appreciation for Monmee for everything. vi

8 Table of Contents List of Figures..... ix Chapter 1 Introduction Physical System: Two Band System Cold Atoms Outline of the Thesis: Chapter Superconductivity: A Brief Introduction Properties: Electromagnetic Properties: London Equations: Thermodynamic Properties: The BCS Theory: Attractive Interaction of Electrons: Cooper problem: BCS solution: Calculation of thermodynamic functions: Chapter 3 Landau-Ginzberg Theory of Superconductivity LG Theory for Homogeneous Systems: LG Theory for Inhomogeneous Systems: LG Theory in a Magnetic Field: Analysis of LG Equations: Coherence Length: Phase Stiffness in LG Theory: Flux Quantization: Fluctuation in LG theory: Chapter 4 Pairing Condensation Cooper Problem for Quasiparticles: Diagonalization of the Effective Hamiltonian: Landau-Ginzberg Free Energy: Analysis SU(4) symmetric case: SU(3) symmetric case: vii

9 4.4.3 Non Symmetric Case: Non Diagonal Situations: Pairing in Spin- 3 Atoms with Spin-Spin Interaction: Presence of an External Zeeman Field: Zero Temperature Case Finite Temperature Case Four species case: Spin-Spin and Zeeman Field : Summary and Results Chapter 5 Quartet Condensation of Fermions Introduction Effective Hamiltonian SU(4) Symmetric Case: Energy Functional Self Consistent Equations: Pair Susceptibility Function: Non SU(4) Symmetric Case: Energy Functional Self Consistent Equations Pair Susceptibility: Critical Curve in the Non SU(4) Case: Quartet Amplitude: Analysis of the Free Energy: Quartet Instability in Spin- 3 System Free Energy Functional: Summary and Results Chapter 6 References Chapter 7 Appendix Appendix Invariance of the Free Energy: Appendix LG Free Energy for Two Band Model: Appendix Instability of the Normal State Appendix Free Energy: SU(4) Case Appendix Free Energy Functional: Appendix SC Equation and Minimization Approach viii

10 List of Figures 1.1 Intraband and interband interactions Free energy at different temperatures Phase diagram in presence of external field Phase diagram in presence of external field, Two decoupled Cooper pairs and a Cooper quartet Attractive and repulsive interaction in the Hamiltonian Free energy for T > T c and T T c Critical curve in the SU(4) limit Non interacting susceptibility Diagonal element of the susceptibility matrix Off diagonal element of the susceptibility matrix Non SU(4) phase diagram Non SU(4) phase diagram Q and m r vs m(= T Tc T c ) Free energy for h h f Free energy for h h f Free energy for h > h f Free energy for h = h f Free energy phase diagram Critical curve for spin-3/ quartet ix

11 Chapter 1 Introduction Superconductivity and superfluidity are among the most fascinating phenomena observed in condensed matter systems. In addition to its potential for applications, superconductivity is intriguing to physicists as a macroscopic manifestation of quantum mechanics. From the perspective of applications, most of the present day research in superconductivity is driven by the quest of designing a room temperature superconductor. At the same time from the perspective of basic science, it also inspires people to seek for new unconventional types of superconductors, to seek for new phases of matter and to understand the physics of complex physical systems. In my thesis, I study condensation in a complex physical system where an unconventional superconductivity may arise. Superconductivity involves condensation of bosonic Cooper pairs (CP) at very low temperature. In this thesis I will discuss condensation in fermionic systems having multiple internal degrees of freedom. Fermions obey the Pauli Exclusion Principle, which forbids more than one identical fermion from occupying a single quantum state. Therefore for electronic systems with two internal states, namely spin up and down; the maximum number of electrons that could be bound together is two. They are the Cooper pairs in conventional superconductivity. However; for a fermionic system having more than two internal states, it is possible that the pairing structure could 1

12 be quite different. In my thesis we focus on systems that have four internal degrees of freedom. On one hand it is quite possible that the system will still undergo some types of pairing condensation, but there is also a possibility that the fermions will form a more complex structure where four fermions are bound together which we call a Cooper quartet (CQ). CQ s are also bosonic in nature and can condense at low temperature giving rise to an unconventional type of superfluid. Pairing and quartetting of fermions will in general be a competing feature for fermionic systems with multiple internal states. Therefore it will be quite interesting to investigate the pairing and quartetting phenomenon in a suitable system and this is the main objective of my thesis work. Some physical systems where fermions can have four-fold internal states include two band electronic systems, spin- 3 cold fermionic atoms and bi-layered systems with electrons and holes. In this work I will concentrate specifically on the first two. An appropriate Hamiltonian describing such a system can be written as H = H s + H i In above H s is the represents single particle Hamiltonian and H i includes interactions among the fermions. In momentum space H s can be written in the second quantized notation as H s = kα ǫ kα c kα c kα where α corresponds to the internal indices of the fermions. We assume that fermions interact with each other only with two body point interactions. Thus H i contains a pairwise interaction term which can be written as H i = λ lmnj drψ l (r)ψ m (r)ψ n(r)ψ j (r) lmnj In the sum l,m etc again refer to the internal indices.

13 1.1 Physical System: Physical systems where the discussions presented in this thesis will be applicable include a two band electronic system and a system of spin- 3 alkali atoms. Here I will discuss them briefly Two Band System A two band electronic system can be observed in transition metals like niobium where fermi surface of s and d bands overlap. In case of such systems, fermions could be electrons from s and d bands. Four internal indices for the electrons are due to band indices and electron spins; one of them could be s band with up spin (s ). Similarly other three internal indices can be identified as s band down spin, d band up and down spins respectively; that is (s ), (d ) and (d ). For simplicity we will use the following short hand notations (s ) 1, (s ), (d ) 3, and (d ) 4. Consider the following form of interaction H i = λ drψ (r)ψ (r)ψ (r)ψ (r) It is convenient to carry out calculations in momentum space. Taking Fourier transformation of the field operators ψ (r) = kn c kn φ kn (r) where n is the band index and the volume has been taken as unity. In momentum space the interaction term takes the following form H i = {k i } V c k 1 l c k m c k 3 nc k4 j λ lmnj drφ k 1 l (r)φ k m (r)φ k 3 n(r)φ k4 j(r) lmnj } {{ } lmnj (1.1) 3

14 In general the sign of V lmnj will be arbitrary depending on (lmnj). For formation of pairs two fermions must have attractive interaction between them. So we take V lmnj as positive. Our assumption is that λ lmnj is positive. Again to make sure that H i include attractive interactions, the integral in it must be positive. The space integral imposes momentum conservations. To ensure that the integrand will be positive definite we take φ k3 n(r) = φ k4 j(r) (1.) where the bar refers to time reversal. Therefore we consider pairing between time reversed states. Finally this will lead to several possible interaction structures. Because of the restrictions imposed by (1.), we can make the following choice in (1.1) k 1 = k = k; (l, m) {(1, ), (3, 4)} k 3 = k 4 = p; (n, j) {(1, ), (3, 4)} Depending on the choice of (l, m) and (n, j); the interaction may be intraband or interband. Intraband channel will be the one when (l, m) = (n, j). Some possible types of such interactions are shown in fig-1.1. Here it can be noticed that for interband interaction, the sign of V lmnj is not important. We can make a U(1) gauge transformation and change the sign. In general the interaction parameter V lmnj will be different for different channels. Physically several limits of the interaction can be identified. One is the intraband limit when dominating interaction is in the intraband channels. Another limit would be interband limit when the dominating interaction is the in the interband channel. We will investigate the pairing structure for each of these limits. Two electrons will experience strong Coulomb repulsion in space. But in order to form a bound state structure, there must be a net attractive force between them. They can overcome the Coulomb repulsive force with the aid of electron phonon 4

15 Figure 1.1: Intraband and interband interactions interaction and that way can attract each other. Once there is a net attraction between two electrons, they can form a bound state called a Cooper pair Cold Atoms The other physical system we will concentrate on is a system of cold spin- 3 alkali atoms in optical traps. With the rapid development of trapping and cooling techniques people have been able to study and realize new and exotic physical phenomenon in such systems. In fact the first direct experimental realization of Bose-Einstein condensation was done in ultra cold Rb atoms 1. This discovery has triggered open many new directions for research. Among them fermion pairing and superfluidity have attracted considerable attention. 5

16 Alkali atoms have one single electron in the outermost s-orbital. Other electrons are in completely full shells of quantum states; giving rise to a net zero angular momentum and a zero spin. The other contribution to the total spin comes from nuclear spin. If the nuclear isotope is one with even number of nucleon, then nuclear spin (I), combined with electron spin (σ) will give a net half integer hyperspin (f = I ± σ). For example 13 Cs has a hyperspin 3/ in its lowest hyperfine manifold. Here four internal states can be identified as the z components of the hyperspin which we are going to identify by numbers 1,,3 and 4. For the spin- 3 system in addition to the pairwise interaction as described in H i, we also include spin-spin interaction. I assume an effective interaction with s-wave phase shift only. It leads to a delta function interaction. In that case the spin channel wave function of two fermions needs to be antisymmetric. Therefore only channels considered here are with total spin 0 and. Since there are only two independent parameters, in the expansion (s i.s j ) n, we take n = 0, 1. Thus the spin-spin interaction term has the following form. H ss = J ij s i.s j δ(r i r j ) In cold atom system the strength of the pairing interaction can be controlled externally by manipulating the Feshbach resonance. And a wide range of interaction strength can be explored. Spin- 3 atoms are neutral. Their pairing and quartetting condensation will form a neutral superfluid unlike tn case of a two band system, where the condensates will carry a net charge resulting in a charged superfluid or a superconductor. 6

17 1. Outline of the Thesis: The organization of the thesis is as follows. It begins with a review of the relevant textbook materials on the necessary theoretical background for understanding my thesis work. In the second chapter I briefly discuss about some of the fundamental properties of a conventional superconductor and the Bardeen Cooper Schrieffer (BCS) theory of superconductivity. In the third chapter, basics of Landau Ginzberg (LG) theory of superconductivity is discussed. I discuss LG theory for both homogeneous and inhomogeneous situation and also in presence of magnetic field. Later we discuss inclusion of fluctuations in the theory and its importance. The fourth and the fifth chapters are devoted to discuss my research work. In the fourth chapter pairing condensation is discussed in the fermionic system. As mentioned earlier, in cold atoms the strength of the interaction can be tuned externally. First for certain specific types of interaction; I extend the Bogoliubov diagonalization and the BCS analysis for the four species case. However, my main objective is to use the LG approach to address pairing condensation. Therefore I derive the free energy for the problem under consideration and minimize it to address the pairing structure. I also discuss the limits of interactions mentioned earlier for the two band system. Later an extension of the original CP problem is presented for the case of quasiparticles in order to discuss the stability of the solutions obtained from LG analysis. Next, I consider presence of spin-spin interactions among the fermions and discuss the pairing structure and map the results to the existing models. After that introduction of an external Zeeman field to the system is studied. Finally the modification in the pairing structure is studied in the presence of spin-spin interaction and Zeeman field together. Condensation of CQ s is discussed in the fifth chapter. In the strong coupling limit; because of the higher energy gain the system will prefer undergoing quartet condensation rather than pair condensation. On the other hand if the coupling is 7

18 weak, it is not so clear whether pairing instability will dominate over quartet instability or vice versa. We are interested if there could be a situation where the system can undergo quartet instability at the cost of pair instability. LG free energy is used to study quartets. Minimization of LG free energy gives condensation of CP s below the transition temperature. Here we are interested in the Bose condensation of quartets; so I generalize the approach to include fluctuations in CP order parameter and include a CQ order parameter. Rather than minimizing, I consider LG free energy as an effective Hamiltonian. As an approximate calculation the Bogoliubov inequality is used. The free energy functional so obtained is minimized with respect to the variational parameters. At first I consider the very special limit of the interaction when it is equal among all the species which is the so called SU(4) symmetric limit of interaction. After that deviation from the SU(4) limit is studied. For both the cases I discuss the instability of the system with respect to the formation of quartets. Finally quartet instability in presence of spin-spin interaction is discussed. 8

19 Chapter Superconductivity: A Brief Introduction Superconductivity in a material refers to the phenomenon of vanishing of its electrical resistance together with perfect diamagnetism below certain temperature called the critical temperature T c. It was first observed in mercury by K. Onnes in Mercury becomes superconducting at a temperature around 4. K. Superconductivity has been observed in many metals and alloys. A number of elements superconduct at low temperature, out of them niobium (Nb) has the highest T c of 9. K at atmospheric pressure. On one hand good conductors such as aluminium (1. K) and Tin (3.7 K) do become superconductors at low temperature; while equally good conductors such as gold, silver and copper do not show any evidence for superconductivity at all. Most of the superconductivity research is driven by the quest for a room temperature superconductor. So far highest T c has been obtained in YBCO compounds. According to superconductor.org, the first ambient superconductor has been observed in Antarctica with a T c of 185 K in an optimized 13/11 intergrowth. Superconducting state is fundamentally different from the normal metallic state. In fact this is a different phase separated by a phase transition from the normal state. In a superconducting material a finite fraction of the electrons condense to form a 9

20 superfluid which extend over the entire volume of the system. In this chapter we first review and discuss the physical properties of superconductors. We then review the microscopic theory of superconductivity developed by Bardeen Cooper and Schrieffer (BCS) 3 and expanded on by Bogoliubov. Extension of this theory will be discussed in the next chapter..1 Properties: Here we briefly review some of the experimentally observed phenonemena for a superconductor and discuss their significance 4,5, Electromagnetic Properties: Superconducting state is characterized by zero or extremely close to zero dc resistivity that persistent electric currents have been observed in superconducting rings to flow without decaying for more than a year. Experiments on decay of currents on super conducting solenoids indicated that decay time of supercurrents is not less than years. So for all practical purposes we can assume an infinite electrical conductivity for a superconductor. But superconductivity is more than just an ideal conductor. An ideal conductor is a normal conductor with zero resistivity. The fundamental property that distinguishes a superconductor from an ideal conductor is the Meissner effect shown by the former. A superconductor expels a weak external magnetic field and B = 0 inside it. This particular property can not be realized if we consider a superconductor as an ideal conductor. From Ohm s law we know that E = ρj, where E is the electric field and j is the current density. Now if the resistivity ρ goes to zero and current is kept finite, then E must be zero. From Maxwell s equation we know that E = B. Thus if E is zero, then B should not change. This in turn t will imply that presence of magnetic field inside the specimen will depend on history. 10

21 If the specimen is cooled from a higher temperature to a temperature below T c in presence of an external weak magnetic field; the field lines will continue to penetrate it even after the transition occurs. But for a superconductor, the quite opposite is observed experimentally. Once the specimen is cooled the field lines are pushed out as the transition occurs. Meissner effect is accepted to be the more definitive proof of superconductivity. Among several reasons, one is that it is an equilibrium property, whereas resistivity is a non equilibrium transport effect. Moreover it is probably easier to demonstrate flux expulsion than zero resistivity. F. and H. London proposed a phenomenological theory in 1934 to explain the observed electromagnetic behavior in a superconductor. We discuss that in the next section..1. London Equations: London brothers pointed out the underlying physics of the Meissner effect. They postulated two equations that are known as London equations. If m is the mass, e is the charge of electron and n s is the superconducting electron density then assuming the motion of such electrons through the lattice is frictionless we get Also since mẍ = ee j t = en s ẍ We can eliminate ẍ from the above two equations and get j t = e n s m E (.1) Equation (.1) is the London s first equation. Taking curl of it j t = e n s m E = n s B e m t 11

22 In above we have used Maxwell s 3 rd equation E = B. Upon simplifying t t (B + m ne j) = 0 B + m ne j = B 0 Here B 0 is the constant of integration. To describe Meissner effect, B 0 must be taken as zero. Using the result for j obtained by taking curl of Maxwell s fourth equation B = µ 0 j; we finally get B = 1 B (.) λ L where λ L has the dimension of length and called the penetration depth of a superconductor. It is given by m λ L = µ 0 n s e Equation (.) is the second London equation. It accounts for Meissner effect because it does not allow a solution uniform in space. So a uniform magnetic field can not exist in a superconductor unless it is identically zero. In the pure superconducting state the only field allowed is exponentially decayed as we go from the surface inside to the specimen. If a semi-infinite superconductor occupy the space on the positive x-axis, then the solution of the second London equation can be written as B = B 0 e x/λ L B 0 is the field on the boundary at x = 0. Thus λ L physically signifies the distance inside the surface over which external magnetic field is screened. Typical value of penetration depth in a metal is around 500Å. But the result derived above and also the London equation itself are not entirely correct because the mean free path of electrons and also the restrictions imposed by uncertainty principle on localization are totally neglected. A modified form of London equations was proposed by Pippard. 1

23 It generalizes the London equation by relating current a point j(r), to the vector potential at other points. j(r) = 3n se /m 4πξ 0 d 3 r R(R.A(r )) R 4 e R/r 0 (.3) where R = r r. The parameter r 0 is the distance over which the points contribute to the integral and is defined as 1 r 0 = 1 ξ l (.4) Here l is the mean free path of the electrons at the Fermi surface and l = v F τ, τ being the scattering time from Drude conductivity formula. Also v F is the velocity of the electrons at the Fermi surface. ξ 0 is called the coherence length. The coherence length is connected to the energy gap, ξ 0 = v F π Physically coherence length signifies the size of the Cooper pair bound state..1.3 Thermodynamic Properties: One of the experimentally observed phenomenon which provided ample insight to the development of microscopic theory of superconductivity is the isotope effect. It is found that the critical temperature of supeconductors varies with the isotopic mass. Transition temperature changes smoothly once different isotopes of the same material is mixed. If the isotopic mass is M and the transition temperature is T c, they can be fitted to a relation of the form M α T c = constant From the dependence of T c on isotopic mass, it is clear that lattice vibrations and therefore electron phonon interaction have to play an important role in the occurence 13

24 of superconductivity. Original BCS model says that α is 1 which is approximately correct with experimental data for many superconductors. There is an energy gap in the elementary excitation spectrum for a superconductor. This gap has an origin entirely different from the energy gap in insulators. The observed heat capacity for many material can be fitted to a relation of form C e /kbt. Here is called the energy gap parameter. This gap parameter decreases continuously to zero as temperature is raised towards T c. In zero magnetic field the transition into normal state from superconducting state is is observed to be a second order one. Such a transition is characterized by zero latent heat but with a discontinuity in the heat capacity within the mean field theory. In the next section we will discuss the fundamentals of the BCS theory.. The BCS Theory: The BCS theory 3 was the first complete microscopic theory to explain superconductivity. Before that several theories were proposed including the London theory discussed earlier and the Landau Ginzberg theory; but none of them identify the mechanism for superconductivity and did not explain what gives rise to superconductivity. We will discuss Landau Ginzberg theory in a subsequent chapter. Although Landau Ginzberg theory could explain several experimentally observed phenomenon, nevertheless it was reliable only at temperatures close to transition temperature. On the other hand BCS theory which is regarded as one of the most spectacular successes of many body theory, could explain a wide range of experimentally observed phenomenon for most common superconductors. For example it could explain the isotope effect, it also could explain the existence of an energy gap ( ) at the Fermi level. Later Gor kov was able to derive Landau Ginzberg equations from BCS theory at a suitable limit 8. It provided a simple microscopic explanation of the order param- 14

25 eter Ψ in Landau Ginzberg theory. He found that Ψ is directly related to the wave function of the Cooper pair and more importantly to the gap parameter. The BCS theory is built upon three major insights. 1) Firstly, effective forces between two electrons could become attractive under suitable conditions with the aid of electron phonon interactions. ) Attractive interaction between two electrons excited above the Fermi sea can form a bound state and can destabilize the underlying system. More importantly it can happen irrespective of the strength of the attraction. 3) Schrieffer proposed a variational wave function in which all the electrons near to the Fermi sea are paired up. The BCS energy gap comes out of this analysis as corresponds to the energy for breaking up a pair of two electrons. We will briefly discuss above three aspects below...1 Attractive Interaction of Electrons: The first key idea in BCS theory that there is an attractive interaction among electrons near the Fermi surface was first formulated by Frölich in Two bare electrons will strongly repel each other with Coulomb repulsion. But with the help of electron phonon interaction, they can overcome the repulsion and attract each other! This attraction is viewed as an exchange of phonons, phonon being the quanta of lattice vibrational energy. As one electron moves through the lattice, it slowly deforms the lattice and induces a positively charged polarization into it. Maximum deformation occurs at a time τ π ω d s after the electron had passed. During this time it travels a distance of about v F τ 1000Å. The positive charge of the deformation can attract another electron without feeling the repulsion from the first electron. This way they can overcome the Coulomb repulsion and attract each other. The net effect on the two electrons is then to create an attractive interaction which tend to pair time reversed quasi-particle states. They form a spin singlet so that the spatial part of the 15

26 wave function could be symmetric and node less so as to take maximum advantage of the attractive interaction... Cooper problem: In the previous section we have seen that two electron can attract each other under suitable conditions. Cooper considered two such electrons above the Fermi surface and looked at their pairing. He found a startling result that they can in fact form a bound state and destabilize the underlying system. This is a crucial step towards the BCS theory. In the literature this problem is referred to as the Cooper Problem. In Cooper model, a spherical Fermi surface is assumed at zero temperature where all the states with k < k F are occupied, k F being the Fermi momentum. Consider two electrons excited above the Fermi surface where they have an attractive interaction between them but at the same time are inert to the electrons inside the Fermi sea. Again assume that the two electrons are located at positions r 1 and r and their spins are σ 1 and σ. Consider a wave function of the following separable form Ψ(r 1,r ; σ 1, σ ) = φ(r 1,r )χ(σ 1, σ ) For the BCS case, the pairing is in the time reversed state. The spin part of the wave function is singlet; so that the space part in the wave function is symmetric one. Going to the center of mass frame of reference and assuming that the center of mass is at rest, the wave function above can be written as Ψ(r 1,r ; σ 1, σ ) = φ(r 1 r )χ(σ 1, σ ) We now solve the Schrödinger equation for the two electron system. Since the gradient operator does not act on the spin part of the wave function we get [ m ( 1 + ) + V (r 1 r )]φ(r 1 r ) = Eφ(r 1 r ) (.5) 16

27 Taking Fourier transformation of φ φ(r 1 r ) = k φ k e ik.(r 1 r ) and substituting it in (.5) we get, k k m φ ke ik.r + (V (r) E) k φ k e ik.r = 0, here (r = r 1 r ) Multiplying both sides with e ip.r and integrating over volume we arrive at the following result. (ǫ p E)φ p = k V kp φ k (.6) Here ǫ p and V kp are defined as ǫ p = p m and V kp = d 3 re ip.r V (r)e ik.r Equation (.6) is called the Bethe Goldstone equation. We assume a simple form for the kernel V kp such that it is attractive in a thin shell around the Fermi surface, otherwise it is zero, ie V if ǫ k, ǫ p < ω d, V kp = 0 otherwise ω d is the Debye frequency Again here V > 0 so that the interaction is attractive. With the simplified form of the kernel we get (ǫ p E)φ p = V k φ k 1 V = k 1 ǫ k E Converting the sum over k into an integral gives 1 V 0 = 1 (π) 4π dkk 1 3 k>k F ǫ k E 17

28 Assuming a constant density of states N 0 in the thin shell of width ω d around the fermi level, the expression can be simplified to 1 ωd = N 0 V 0 o which for small coupling (N 0 V 0 1) is given by 1 dǫ ǫ E E = ω d e N 0 V 0 (.7) If E < 0, then the eigen state is a bound state. It is found that a solution with E < 0 always exists provided V o > 0. Thus a bound state exists, irrespective of how small the coupling is. Therefore the normal state is unstable with respect to formation of bound pairs. These bound pairs of electrons are called Cooper pairs (CP). Again because of the exponential factor, the bound state energy will be very small for small values of coupling. As in the full BCS solution the energy scale for superconductivity is set by the Debye energy; but multiplied by a very small factor. This explains why the transition temperature T c should be very small as compared to other energy scales in solids. For most materials, the Debye energy corresponds to typically energy scales of order K. And it leads to T c 1 K for most metallic superconductor. Because the binding energy is small, the size of the bound state is large. We can make an estimate of the size of such a pair, let it be ξ 0. Only electrons near the Fermi surface form pairs and their energy must be of order of k B T c. Also the momentum range is δp k BT c v F. Then from uncertainty relation δxδp, it gives = δ( p m ) p Fδp m ξ 0 δx = δp p F m = E F k F Typical size of a Cooper pair is around 1000Å 18

29 ..3 BCS solution: Equipped with the insights about the origin of attractive interaction between two electrons and from solution to the Cooper problem, BCS proposed a variational wave function and obtained the essential physics of superconductivity. The relevant Hamiltonian for the system involving electron phonon interaction is given by Ĥ = (ǫ k µ)c k,σ c k,σ + V (k,k,q)c k+q,σ c k q,σ c k,σ c k,σ k,σ k,k,q σ,σ Using BCS approximation the retarded nature of the interaction is modelled by restricting the interaction to only values of k such that ǫ k is within ± w d of Fermi energy. Again we assume that the most important interactions are those involving Cooper pairs (k, ) and (-k, ), then dropping all other interactions, the effective Hamiltonian is written as Ĥ = k,σ (ǫ k µ)c k,σ c k,σ V k,k c k c k c k c k (.8) The wave function BCS wrote down is a coherent state of Cooper pairs given by Ψ BCS = k (u k + v k c k c k ) 0 > (.9) where 0 > is the vacuum state. The normalization condition on Ψ requires u k + v k = 1. u k and v k are the variational parameters of the theory which are to be adjusted to minimize the total energy < Ψ BCS Ĥ Ψ BCS >. The BCS wavefunction is an example of correlated many particle ground state. It is a mean field ground state and not an exact solution to the many body problem under consideration. But for certain class of superconductors it becomes nearly exact for most purposes. For BCS theory we will discuss here we will assume that the coupling is small. In that case the gap parameter becomes much smaller than other relevant energy scales of the system. To solve the complete problem we use the BCS wave function as a reference state and 19

30 consider low lying excitations in the system. To find the low lying excitations we decouple the interaction term using Wick s theorem as follows c k c k c k c k < c k c k > c k c k + c k c k < c k c k > Here expectation values denote thermal averages. Although from Wick s theorem we should also consider averages such as < c k c k >, but they are essentially same in the normal state as in the superconducting state. So they can be absorbed in the definition of single particle energy ǫ k. Therefore we neglect them. Also assuming that fluctuations are small, the interaction term in the Hamiltonian in (.8) can be rewritten as V k,k c k c k c k c k V k,k (< c k c k > c k c k + c k c k < c k c k > < c k c k >< c k c k >) Introduce an order parameter = V k < c k c k > (.10) Using (.10), the effective Hamiltonian can then be written as Ĥ = k,σ (ǫ k µ)c k,σ c k,σ k ( c k c k + c k c k ) + V (.11) It can be diagonalized by using the following canonical transformation c k = u k v k c k v k u k γ k γ k The requirement for the transformation for being canonical implies that the variational parameters should obey u k + u k = 1. After diagonalization the Hamiltonian takes the following form 0

31 Ĥ = k ǫ k + k E k (γ k γ k γ k γ k ) + V = E o + k E k (γ k γ k + γ k γ k ) (.1) Here E o is the ground state energy and is given by E o = (ǫ k E k ) + V k with E k = ǫ k + (.13) The Hamiltonian is now expressed in terms of new operators γ and γ called quasiparticle annihilation and creation operators. They are fermion operators and obey standard anti-commutation rules for fermions. The BCS ground state is the vacuum for these quasiparticle operators in the sense that γ annihilates it. γ k Ψ BCS> = 0 γ k Ψ BCS> = 0 The excited states then corresponds to adding one or more quasiparticles to the ground state. The excitation energy to do this is given by E k. At a finite temperature (T), the quasiparticles will have occupations given by the Fermi-Dirac distribution. < γ k γ k >= f(e k ) = Imposing self consistency in (.10), we get 1 e βe k + 1 = V k E k [1 f(e k )] = V k tanh βe k E k As earlier introducing a constant density of states N(ǫ) = N o in a thin shell of width ω d around the Fermi surface we convert the sum over k in the above expression into an integral. Again noting that the integral is even in ǫ, we finally obtain = N o V ωd 0 dǫ ǫ + tanh β ǫ + 1 (.14)

32 This is the so called BCS gap equation. It allows us to calculate the gap parameter as a function of temperature. For a general solution of (T) one needs to carry out numerical calculations; nevertheless certain results can be extracted analytically. We will consider two such cases. First, we will calculate the gap parameter at zero temperature and later we will evaluate T c. At T = 0 the gap equation becomes = N o V ωd 0 dǫ ǫ + It gives a trivial solution = 0 which is the normal state. We also get a nontrivial solution for given by T=0 = 0 = ω d exp( 1 N o V ) (.15) Gap equation also enables us to calculate value of T c. Assuming a second order transition, T c is the temperature at which order parameter goes to zero. In that case (.14) takes the following form 1 = N o V ωd 0 dǫ ǫ tanh[ ǫ ] k B T c Upon solving we get transition temperature as a function of dimensionless coupling k B T c = 1.13 ω d exp( 1 N o V ) (.16) Now comparing (.15) and (.16), we see that the gap parameter at zero temperature and the transition temperature is related to each other by an expression of the form 0 k B T c = 3.53 Thus this ratio is a universal constant for superconductors within weak coupling BCS theory! Experimentally it has been found that this is obeyed by a wide range of superconductors. We can also calculate the condensation energy for the system. Condensation energy is defined as the difference of superconducting ground state energy and normal

33 state energy. We obtained the former as E o = k (ǫ k E k ) + V Also the normal state energy in which = 0, is given by E n = k (ǫ k ǫ k ) = k<k F ǫ k Therefore the condensation energy is E o E n = k = k (ǫ k E k ) + V ǫ k k<k F (ǫ k E k ) + 1 ǫ k E k k k<k F ωd = N o dǫ[ǫ ǫ + + ǫ + ] 0 where we have used the self consistent equation at zero temperature to substitute for 1. In the weak coupling approximation, it gives V E o E n = N o (.17) Thus we see that superconducting state has lower energy as compared to the normal state. However this energy gain is extremely small, of the order of /E F per particle. For = 10K, E F = 10 4 K, this value is approximately 10 K...4 Calculation of thermodynamic functions: Here we calculate some thermodynamic quantities of the system from the BCS theory. Earlier we had seen that the effective Hamiltonian for the system can be rewritten in terms of quasiparticle operators as Ĥ = k [ǫ k E k + E k (1 f(e k ))] + k E k (γ k γ k + γ k γ k ) = E 0 + k E k (γ k γ k + γ k γ k ) 3

34 Calculating the grand partition function, Z = Tr[e β(ĥ) ] = k exp[ β(ǫ k E k + E k (1 f(e k )))] Tr[exp( βe k γ k γ k )] Tr[exp( βe k γ k γ k )] γ γ is the number operator for excitation spectrum. We use number representation to calculate the trace. Since any operator is diagonal in its own representation, it gives Then the partition function is Tr[e βe kγ k γ k ] = 1 + e βe k Z = k [1 + e βe k ] exp[ β(ǫ k E k + E k (1 f(e k )))] So that the thermodynamic potential is Ω = 1 β ln Z = ln[1 + e βe k ] + β k k = Ω 1 + Ω (say) [ǫ k E k + E k (1 f(e k ))] Ω 1 and Ω are the first and the second term in the sum. Once Ω is known, entropy can be calculated as S = ( Ω T ) µ,v = k B β Ω β (.18) If we calculate the normal state entropy, Ω will not contribute towards it. Also in that case E is to be replaced with ǫ in Ω 1. This way for normal state entropy we get, S n = k B β Ω 1 β = k B k [ln(1 + e βǫ k ) + βǫ k e βǫ k + 1 ] = k B [f(ǫ k ) ln f(ǫ k ) + {1 f(ǫ k )} ln{1 f(ǫ k )}] (.19) k 4

35 Now consider the superconducting case. The result derived in (.19) will still hold but ǫ is to be replaced by E k = ǫ k +. But here is a function of temperature, so there will be additional contribution coming from Ω 1. So that k B β Ω 1 β = k B [f(e k ) ln f(e k ) + {1 f(e k )} ln{1 f(e k )}] k +k B β k f(e k ) E β For evaluating the contribution from Ω, we first note that from gap equation Therefore contribution from Ω 1 f(e k ) = 0 β E k k k B β Ω β = k B β k [ E k β + 1 f(e k) E k β ] Again since E k = ǫ k + E k β = 1 E k β Thus the entropy in the superconducting state is S c = k B [f(e k ) ln f(e k ) + {1 f(e k )} ln{1 f(e k )}] (.0) k If we compare expression for normal state and superconducting state entropy we see that they are continuous at the transition point. Once we have entropy, we can immediately calculate the specific heat. Specific heat is given by C = T S T = β S β = k B β k = k B β k = k B β k f β ln f 1 f βe k e βe k (e βe k + 1) (E k + β E k β ) ( f )[Ek E + β k β ] 5

36 Converting the sum over k into an integral over ǫ we rewrite the result for specific heat as C = k B βn 0 dǫ( f E )[E + β β ] (.1) We will evaluate the explicit expression for specific heat for the following two special cases. 1) Firstly at a temperature close to zero is finite and 0. In this case k B T, so that E = also f(e) = ǫ ǫ 1 e βe + 1 e βe 0 f E = βe βe (.) Since is almost constant, the second term inside the integral is very small. We will neglect that contribution. So that the specific heat is C = k B βn 0 dǫ( f E ) 0 = k B β N 0 0 dǫe β( 0+ ǫ ) 0 π = N 0 k B T 3 e k B T (.3) We can compare this result to the specific heat of the normal Fermi system. Specific heat for the normal system is given as c n = γt = π 3 N okb T (γ = Sommerfield Constant) It is straightforward to show that c = 3.13 T γt c 3 c T 3 e 1.76Tc T 6

37 In above we have used the fact that 0 k B T c = We can rewrite the above result as ln( c ) = ln γt c ln T c T 1.76T c T (.4) ) Now we consider the other extreme case when T T c. Close to the transition temperature, is very small. In this case we can safely put E ǫ. Therefore C = k B βn 0 dǫ( f ǫ )[ǫ + β β ] = I 1 + I (say) (.5) First consider I 1 ; using f = βeβǫ we get ǫ (e βǫ +1) βe βǫ I 1 = k B βn 0 dǫ (e βǫ + 1) ǫ The integrand is even in ǫ. Carrying out the integration one obtains, I 1 = 4N 0 Tβ 0 dx x e x (e x + 1) = π 3 N okb T = γt (.6) Which is the normal specific heat. Therefore the contribution from the first term is the normal state contribution, c n. It is clear that there is a discontinuity in the specific heat near T c given by the term I. c s c n = k B β N 0 = k B β N 0 ( β ) = k B N 0 ( d dt ) dǫ( f ǫ ) β Therefore the jump in specific heat at T c is given by c s c n = k B N 0 d βe βǫ dǫ (e βǫ + 1) }{{ } 1 dt T=T c (.7) 7

38 The factor d dt specific heat at T c is can be evaluated numerically5. The final result for the jump in c s c n = 10.k B N 0 The finite jump in specific heat indicates that within mean field theory superconducting transition is indeed a second order phase transition. In the next section we will discuss Landau Ginzberg theory of superconductivity which is based on the theory of second order phase transition. 8

39 Chapter 3 Landau-Ginzberg Theory of Superconductivity The superconducting state and the normal metallic state are two separate thermodynamic phases of matter in just the same way as liquid He 4 and superfluid He-II are separated by thermodynamic phase transition. Such phase transitions can be characterized by the nature of the singularities in specific heat and other thermodynamic variables at the transition point. The theory of superconductivity introduced by Landau-Ginzberg(LG) in 1950 describes superconductivity from thermodynamic point of view. Originally it was introduced as a phenomenological theory. Later Go rkov showed that it can be derived from Bardeen Cooper Schrieffer (BCS) theory in a certain limit. One important aspect of LG theory is that it can be used to solve many difficult problems of superconductivity without solving microscopic BCS theory. The LG theory as applied originally is a mean field theory of the thermodynamic state. However one of the most important aspect of it is that it can be extended to go to beyond mean field limit so as to include the effects of thermal fluctuations. Effects of thermal fluctuations are negligible in case of conventional superconductors making the mean field approximation essentially exact except when the temperature is so close to T c to be experimentally accessible. Since I will use the LG theory and 9

40 its extension to include fluctuation in my work, I present here a review of the LG theory. The LG theory of superconductivity is built upon the general approach to the theory of second order phase transitions 7 which Landau developed in 1930s. Landau noticed that second order phase transitions involve some changes in the symmetry of the system. For example above the Curie temperature a magnet does not have any magnet moment; while below it a spontaneous magnetic moment is developed. In Landau theory such a phase transition is characterized by an order parameter. Above a particular temperature called the transition temperature, the system is disordered and the order parameter is zero. Below the transition temperature the system is ordered and the order parameter is non zero. In case of ferromagnetic transition, the magnetization m below the Courie temperature is an example of order parameter. In LG theory of superconductivity, existence of an order parameter Ψ(r,T) is postulated which characterizes the superconducting phase of the material below the transition temperature (T c ). Above T c which is the normal state, it is zero while below T c it continuously rises from zero. LG assumed that the free energy of the system is analytic and can be expanded in a power series in Ψ. Ψ is in general a complex function. Free energy is real and it can depend only on Ψ. Close to the transition temperature the Taylor expansion of the free energy can be truncated at the quartic order in Ψ. 3.1 LG Theory for Homogeneous Systems: First assuming Ψ is spatially constant, the free energy density is written as f s (T) = f n (T) + a(t) Ψ + 1 b(t) Ψ 4 (3.1) In above f s and f n are respectively superconducting and normal state free energy densities. The free energy written in (3.1) is the only possible function at quartic 30