Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Amin, Mohammad H. Sharifzadeh
Doctor of Philosophy - PhD
University of British Columbia
This thesis is organized in two independent parts, in which I study two different aspects of high Tc superconductivity. The first part begins with an introduction aimed to briefly introduce some relevant experimental and theoretical works performed in recent years, that have helped us to think about cuprates the way we do now. Afterwards, I introduce Landau Fermi liquid theory in a standard text book way. The question of validity of Fermi liquid theory in 2-d is then raised and investigated by searching for singularities in Landau's /-function. I show that the interaction function between two quasiparticles whose momenta approach each other near a curved point of the Fermi surface, contains a 1-d singularity not strong enough to change the Fermi liquid behavior. On the other hand, inflection points provide 2-d singularities that have to be taken seriously in Fermi liquid considerations. I then introduce nearly antiferromagnetic Fermi liquid theory (NAFL), which is a phenomenological theory proposed to describe high Tc systems. I mainly focus on the self-consistency of the theory in calculations. I criticize the theory on the basis of overlooking the vertex corrections in the strong coupling calculations of the transition temperature Tc. I calculate the first vertex correction for an optimally doped system and show that it is of the same order of magnitude as the bare vertex. Migdal's theorem is therefore not valid and Eliashberg formalism is not applicable to this situation. The same conclusion is obtained even after inclusion of the quasiparticle residue Z to the calculation. The sign of the vertex correction is then considered. I show that the positive sign of the vertex correction for the optimally doped system requires a phase transition of some sort as the doping is decreased. Part II of the thesis is devoted to the vortex lattice properties of high Tc superconductors. I establish a method to study vortex lattice properties of d-wave superconductors based on a generalization of the London model. The method has the advantage of simplicity as well as having very few free parameters (one at most) compared to other methods. The generalized London free energy is obtained from an s-d mixing Ginsburg-Landau free energy and also from the microscopic theory of Gorkov. The generalized London equation is found to be analytic at high temperatures. At very low temperatures however nonanalyticities arise as a result of the nodes on the superconducting gap. I then present the results of our calculations of some measurable quantities, such as the vortex lattice geometry and the effective penetration depth (as denned in μSR experiments). Comparison between our results and different experimental data is then performed. Especially our prediction for the magnetic field dependence of the effective penetration depth at T = 0, which is recently observed in //SR experiments with excellent agreement, is discussed.
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VERTICES AND VORTICES IN HIGH Tc SUPERCONDUCTORS By Mohammad H. Sharifzadeh Amin B. Sc. (Electrical Engineering) Shiraz University, Shiraz, Iran. M . Sc. (Physics) Sharif University of Technology, Tehran, Iran. A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E O F D O C T O R OF PHILOSOPHY in T H E FACULTY OF GRADUATE STUDIES PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA September 1999 @ Mohammad H. Sharifzadeh Amin , 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada Department Date Oct. Ilj, DE-6 (2/88) Abstract This thesis is organized in two independent parts, in which I study two different aspects of high Tc superconductivity. The first part begins with an introduction aimed to briefly introduce some relevant experimental and theoretical works performed in recent years, that have helped us to think about cuprates the way we do now. Afterwards, I introduce Landau Fermi liquid theory in a standard text book way. The question of validity of Fermi liquid theory in 2-d is then raised and investigated by searching for singularities in Landau's /-function. I show that the interaction function between two quasiparticles whose momenta approach each other near a curved point of the Fermi surface, contains a 1-d singularity not strong enough to change the Fermi liquid behavior. On the other hand, inflection points provide 2-d singularities that have to be taken seriously in Fermi liquid considerations. I then introduce nearly antiferromagnetic Fermi liquid theory (NAFL) , which is a phenomeno-logical theory proposed to describe high T c systems. I mainly focus on the self-consistency of the theory in calculations. I criticize the theory on the basis of overlooking the ver-tex corrections in the strong coupling calculations of the transition temperature Tc. I calculate the first vertex correction for an optimally doped system and show that it is of the same order of magnitude as the bare vertex. Migdal's theorem is therefore not valid and Eliashberg formalism is not applicable to this situation. The same conclusion is obtained even after inclusion of the quasiparticle residue Z to the calculation. The sign of the vertex correction is then considered. I show that the positive sign of the vertex correction for the optimally doped system requires a phase transition of some sort as the doping is decreased. ii Part II of the thesis is devoted to the vortex lattice properties of high Tc superconduc-tors. I establish a method to study vortex lattice properties of d-wave superconductors based on a generalization of the London model. The method has the advantage of simplic-ity as well as having very few free parameters (one at most) compared to other methods. The generalized London free energy is obtained from an s-d mixing Ginsburg-Landau free energy and also from the microscopic theory of Gorkov. The generalized London equation is found to be analytic at high temperatures. At very low temperatures however non-analyticities arise as a result of the nodes on the superconducting gap. I then present the results of our calculations of some measurable quantities, such as the vortex lattice geom-etry and the effective penetration depth (as denned in fiSR experiments). Comparison between our results and different experimental data is then performed. Especially our prediction for the magnetic field dependence of the effective penetration depth at T = 0, which is recently observed in //SR experiments with excellent agreement, is discussed. i i i To my parents iv Table of Contents Abstract ii List of Tables viii List of Tables viii List of Figures ix List of Figures x Acknowledgement xi I Fermi Liquids, Non-Fermi Liquids and Nearly Antiferromagnetic Fermi Liquids xiii 1 Introduction 1 2 Fermi Liquid Theory 11 2.1 Landau Theory of Fermi Liquids 11 2.2 Microscopic Foundation of Fermi Liquid Theory 14 2.2.1 Calculation of fpp/ for a Dilute Fermi Gas 15 2.2.2 Fermi Liquid Theory and Many-Particle Theory . 17 2.3 Fermi Liquid Theory in 2-d 20 2.3.1 Calculation of fpp' for a Curved Fermi Surface 22 2.3.2 Calculation of f n D ' near an Inflection Point 25 2.4 Calculation of Self Energy 28 2.4.1 Non-Singular Interaction 29 2.4.2 Singular Interaction 31 2.5 Summary 35 3 Nearly Antiferromagnetic Fermi Liquid Theory 37 3.1 Paramagnon Theory 38 3.2 Antiferromagnetic Spin Fluctuations 41 3.3 d-wave Superconductivity 43 3.4 Self-Consistency of N A F L 45 3.5 Calculation of Vertex Correction 48 3.5.1 The Effect of Quasiparticle Renormalization Z 53 3.5.2 Sign of the Vertex Corrections 55 3.6 Concluding Remarks 56 II Vortices in d-wave Superconductors 59 4 Introduction 60 5 Phenomenological Model 66 5.1 Ginsburg-Landau Theory 66 5.2 London Theory in Vortex State 67 5.2.1 Derivation of the London Equation from G L Theory 68 5.2.2 The Effect of the Vortex Core 70 5.2.3 Solution for a Vortex Lattice 73 5.3 Mixing of d-wave with s-wave 75 5.4 Numerical Results and Experimental Realizations 80 vi 6 Microscopic Model 87 6.1 Gorkov Theory and Quasi-Classical Approximation 87 6.1.1 Local Approximation 88 6.1.2 Calculation of Current in a Superconductor 92 6.2 Generalized London Equation 94 6.2.1 Nonlinear Corrections 95 6.2.2 Nonlocal Corrections 98 6.2.3 Combining Nonlinear and Nonlocal Corrections . . . 104 6.3 Numerical Calculations 105 6.4 Conclusion I l l Bibliography 118 vii List of Tables Calculated values of the vertex correction 8A^ for two different wave-vectors ki and kh on the Fermi surface . . . viii List of Figures 1.1 Crystal structure of Y B C O 2 1.2 Phase diagram of high temperature superconductors. 3 2.3 Crossed channel diagrams contributing in Ippi 21 2.4 Fermi surface near a curved point and an inflection point 23 2.5 Singular and non-singular regions near an inflection point 27 2.6 Self energy to second order in \ p , p - q • • 32 3.7 Fermi surface in the first Brillouin zone, with the value for t and t' given in the text 43 3.8 First correction SA^ to the bare vertex 49 5.9 Magnetic field distribution in a triangular lattice, using the ordinary Lon-don equation with Gaussian cut off 74 5.10 Distribution of magnetic field in a vortex lattice for e = 0.3 and H = 6.8T 81 5.11 a) Gibbs free energy as a function of p. b) Equilibrium angle /?MIN as a function of H 82 5.12 Magnetic field distribution function 85 6.13 Circular Fermi surface with a dx2_y2 gap 96 6.14 Two high symmetry orientations of centered rectangular unit cell. . . . . 103 6.15 Equilibrium angle ft as a function of reduced temperature t = T/Ad for various fields 106 6.16 Apex angle /? as a function of magnetic field B at T = 0 108 ix 6.17 The effective penetration depth as a function of the magnetic field. . . . 109 6.18 Magnetic field distribution obtained from the nonlinear-nonlocal London equation as compared with the ordinary London calculation I l l 6.19 Magnetic field distribution obtained from the nonlinear-nonlocal London equation and also from nonlocal London equation 112 6.20 Comparison between our nonlinear-nonlocal theory and a recent /iSR result. 114 6.21 Magnetic field as a function of the distance from the center of the vortex (a-direction) for an isolated vortex 115 Acknowledgement First, I would like to give my special thanks to my both supervisors Ian Affleck and Philip Stamp for teaching me the basics of physics and research and also for all their support, encouragement and patience. I especially would like to thank Philip Stamp for directing me to condensed matter physics and teaching me superconductivity and basics of high T c superconductivity, and also Ian Affleck for teaching me the concepts of condensed matter physics during several magnificent courses and informal discussions, and supporting me whenever I needed. I would like to thank both for being a friend and sometimes like a caring father to me. I also would like to take the opportunity to thank Victor Barzykin, Alexander Zagoskin and most of all Igor Herbut for very stimulating and encouraging discussions and for ex-plaining difficult materials to me in simple language. I am also grateful to Walter Hardy, and Doug Bonn and Rob Keifl for providing me up to date experimental information, Jeff Sonier for several useful discussions about aSR experiments and providing me his data, Marc Julien for teaching me the basics of N M R experiments and especially Saeid Kamal for helping me understand microwave experiments as well as being a supporting friend for me during these five years. I also would like to thank all my friends, specially Shahrokh Parvizy and A l i Sheik-holeslam for encouraging me to choose physics as my career, all my friends in Canada especially Siavash Jahromi, Mahan Movassaghi, A l i Mirabedini and Nima Ahmadvand for all good times I had with them during my years of Ph.D., and Andrea and Jeff for lending me their laptop with which I wrote most parts of this thesis. Outside the physics world, I would like to thank my parents and my sister Parvaneh xi for their loving support and encouragement and believing in me and providing me all the supports I needed all through the years of my study. I also want to thank my uncle A l i Joukar and his wife Roya Sadeghi for making Vancouver my home by all their love and support. Finally I would like to give my especial thank to my gorgeous wife Roxana Pardis for her love and support, for always being with me during difficult times of life, giving me lessons of living and making me believe in myself. xii Part I Fermi Liquids, Non-Fermi Liquids and Nearly Antiferromagnetic Fermi Liquids xiii Chapter 1 Introduction It has been more than a decade since the discovery of the first high temperature super-conductor [1], and still no consensus about the microscopic theory relevant to cuprates exists. New theories have been emerging one after another and disputes about the older theories are still unsettled. A general feature of all cuprates is their strong anisotropy which makes them effectively 2-d electronic systems. Fig. 1.1 shows the crystal structure of the Y B C O compound which is one of important high Tc materials. At the center there are two Cu02 planes separated with Y atoms in between. Between these two planes and another two planes above or below them, there are oxygen chains (that play crucial role in the doping of the system) and also Ba atoms. It is generally agreed that the C u 0 2 planes play an important role in determining the characteristics of cuprates at all dopings. Fig. 1.1 shows the crystal structure of Y B C O material In the early days of high Tc's, Anderson [2] proposed an effective Hamiltonian which describes a single band 2D Hubbard model with an infinite on site repulsion U. Soon after, Zhang and Rice [3] tried to justify the Anderson's suggested model (known as the t-J model) for cuprates. They proposed that the holes which primarily reside on oxygen sites would delocalize onto four O sites around C u + 2 ions. The combination of a delocalized 0 hole and a Cu+ 2 ion forms a singlet which would hop between C u + 2 sites in the same way as a hole does in a simple C u + 2 square lattice. In the absence of doping, C u 0 2 planes are half filled and form a quasi-2D antiferro-magnet with an insulating Neel ground state. Numerical simulations, experiments, and 1 Chapter 1. Introduction 2 Figure 1.1: Crystal structure of Y B C O scaling theory, all suggest that above the Neel temperature, the antiferromagnetic spin correlations are well described by the isotropic 2D Heisenberg model [4] with exchange coupling J ~ 1550K between nearest neighbors. However, according to the Hohenberg-Mermin-Wagner theorem [5], classical fluctuations destroy Neel ordering at an arbitrarily small nonzero temperature. The small (~ 10 _ 5 J ) , but finite correlations between the planes are believed to be responsible for the observed long-range order [4]. Chakravarty, Halperin, and Nelson [6] on the other hand, have employed quantum non-linear a model for the long-wavelength action of the 2D, s = 1/2 Heisenberg model. A scaling analysis of this model describes correctly both experimental and numerical data [4]. Introducing a small amount of holes into the C u 0 2 planes, the Neel temperature falls off rapidly from its undoped value. Increasing the doping level even further, the system moves into a state with a metallic normal state and a transition to a superconducting ground state at low temperatures (cf. Fig. 1.2). The superconducting transition temper-ature increases initially as doping is increased. After reaching its maximum at optimal Chapter 1. Introduction 3 X Figure 1.2: Phase diagram of high temperature superconductors, x represents doping, A F antiferromagnetically ordered phase and SC superconducting phase. Upper line is the crossover to pseudogap state. [12] doping, it starts to decrease and the superconductor enters a so called overdoped regime (Fig. 1.2). A l l over the metallic phase, the antiferromagnetic correlation is short ranged with £ between one and (up to) fifteen lattice spacings, depending on the material, doping and temperature. Experiments evidently indicate that the electronic properties of the underdoped and overdoped materials are different [7, 8, 9, 10, 11]; especially the pseudogap phase which exits only for underdoped compounds [12] (Fig. 1.2). Among experiments, inelastic neutron scattering (INS) and nuclear magnetic resonance (NMR) experiments provide important information about the magnetic properties of the system. INS directly probes the spin response function, x"(<f, In Lai. 8 5Sr 0.i5CuO4 the INS experiments [8] display four sharp magnetic peaks located at 5q ~ 7r/4 away from the antiferromagnetic wave vector. These incommensurate peaks were also observed recently in YBa2Cu 306.6- [13] Like many experiments, INS has some inherent problems. For example it is difficult to determine experimentally the absolute value of intensity. It is also not easy to separate the phonon spectrum from the magnetic excitations. N M R experiments [14, 15] on the other hand, are much more exact, and can detect Chapter 1. Introduction 4 much lower intensities. However, N M R is an indirect probe and its data analysis involves using a theoretical Hamiltonian connecting nuclear and electron spins, the hyperfme Hamiltonian. Moreover, since N M R experiments are done at very low frequencies (~ nuclear spin Zeeman splitting), no information about w-dependence of x(v,T) can be obtained. Early measurements of the Magnetic Hyperfme Shift (Knight shift) at the oxygen sites showed that most of the total spin polarization was carried by the copper sites [16, 17]. Alloul et al. [18] reported that in Y B a 2 C u 3 0 6 + i , the knight shift of Yttr ium scales linearly with the macroscopic susceptibility. This was interpreted as a proof of the existence of a single spin fluid in agreement with the t-J model, because yttrium is coupled to the magnetic susceptibility of the Cu02 planes mainly through the polarization of the oxygen orbitals. Further measurements of knight shift showed the same T-dependence on all nuclear sites [14]. Similar result was obtained for all other cuprates indicating that the single spin fluid nature is a common feature of all these materials. Assuming a single fluid picture, Shastry [19] and also Mila and Rice [20] (SMR) proposed a hyperfme Hamiltonian in which the only source of coupling is the localized electronic spins at the copper sites. While the magnetic hyperfme shift (Knight shift) gives a measure of the uniform magnetic susceptibility, the Nuclear Spin Lattice Relaxation Rate (1/Ti) provides infor-mation about the imaginary part of the dynamic spin susceptibility x{q, U)- A n important experiment by Takigawa [16] on Y B a 2 C u 3 0 7 , in the early days, displayed a significant difference between the temperature dependence of 1/Ti for the C u 0 2 plane copper and oxygen sites. The relaxation rate for oxygen sites (and also 8 9 Y sites) obeys Korringa relation 1 7 T i T « Const, while the relaxation rate for copper sites is much shorter than for 1 7 0 and 8 9 Y sites and shows non-Korringa behavior. Using SMR Hamiltonian, Moriya Chapter 1. Introduction 5 [21] showed that m ^ o c l i m E ^ ^ ^ (1-1) Q where x"(<f, w) is the imaginary part of the magnetic susceptibility and F(q) is a form factor which depends on the site and the direction of the magnetic field. For 1 7 0 sites, F(q) is peaked at q = 0 and vanishes at the antiferromagnetic (AF) wave vector Q = (7r/a, 7r/a). For 6 3 C u sites on the other hand, it is peaked at q = Q. Thus different nuclei probe different regions of the momentum space of x"((f>0). The significant difference between the 6 3 C u and 1 7 0 relaxation rates could then be understood as a result of an enhanced susceptibility at q = Q. Another important quantity measured by N M R experiments is the nuclear spin-spin relaxation rate ( 1 / T 2 G ) which is the rate of relaxation of the nuclear spins among them-selves and is usually much larger than 1/Ti. In most solids T2G is temperature in-dependent and is related to spin-spin dipolar coupling and therefore doesn't give any information about electron susceptibility. In cuprates however, because of the antiferro-magnetic enhancement of the spin susceptibility, indirect coupling of spins via nonlocal spin susceptibility dominates the nuclear spin-spin coupling. Thus T2Q can provide im-portant information about the antiferromagnetic exchange between electron spins as was pointed out first by Pennington et al. [22]. Measurements of 1/T 2 G in different materials with different dopings have found 1 / T 2 G to increase with decreasing temperature, with sometimes a flattening or small decrease close to Tc [14]. This is considered as a proof for the existence of A F correlations in all cuprates. To have a quantitative description, Millis, Monien and Pines (MMP) [23] proposed a phenomenological form for the magnetic susceptibility x(<l, w ) which has a peak near the Antiferromagnetic (AF) wave vector Q = [IT/a, ir/a) x & w ) = i i ti(-* %2—~i— + 1—^T7r~ ( L 2 ) i + £ (q - Q) - iu/u)Sf l - i w / i o Chapter 1. Introduction 6 where xo is the measured bulk spin susceptibility which in general is temperature depen-dent, XQ is the staggered susceptibility which is some orders of magnitude greater than the bulk susceptibility and To is an energy scale of order of quasiparticle bandwidth or Fermi energy. The first term in Eq.(1.2) produces a peak at A F wave vector Q and the second term is the ordinary Fermi liquid susceptibility. For the 1 7 0 sites, the effect of the first term is suppressed by the form factor F(q) (cf. Eq. (1.1)) and the second term (the Fermi liquid term) is the only term that remains. To the lowest order, Eq.(1.2) gives ( 1 7 T i T ) _ 1 oc Xo(T) which is consistent with the Korringa relation. For copper sites however, the contribution of the first term is dominant near Q and one can safely neglect the second term. Assuming large correlation length £, the form factor can be treated as a constant and the integration can be simply carried out to get ( ^ T ) " 1 cx = — (1.3) where a is defined by XQ — A £ 2 a n d assumed to be temperature independent. Thus the spin lattice relaxation rate provides information about the temperature dependence of usf in the M M P model ^TtTxUsf/a (1.4) This is completely different from the 1 7 0 case and has non-Korringa behavior. In the early days of high T c 's, there existed some speculations that antiferromagnetic correlation was responsible for Cooper pairing in high T c materials [24, 25, 26, 27, 28] (the idea of d-wave pairing already existed in the context of heavy fermions [29, 30]). Inspired by paramagnon theory, Pines and collaborators [31, 32] proposed a (phenomenological) nearly antiferromagnetic Fermi liquid (NAFL) theory to describe pairing process as well as normal state properties of high Tc superconductors. The theory assumes that the low lying excitations of a high Tc system in its normal state, are Fermi liquid quasiparticles interacting with each other via antiferromagnetic spin-fluctuations. The susceptibility Chapter 1. Introduction 7 (1.2) can be thought of as the propagator of bosonic particles that describe spin fluctu-ations. One of the first predictions of N A F L theory was dX2_^2-wave pairing which was then confirmed by various experiments [33, 34]. Transition temperature was calculated in weak [31] and strong [35, 36] coupling limits and a Tc as high as 90K was obtained using several fitting parameters. N A F L theory, as we mentioned above, is constructed based on the assumption that the excitations of the system are Fermi liquid quasiparticles. Almost none of the normal state properties of cuprates agree with Fermi liquid theory (at least in low to moder-ate doping regimes). The proximity to Mott-insulating antiferromagnetic phase might suggest that non-Fermi liquid properties are actually signatures of non-Fermi liquid exci-tations. N A F L theory on the other hand believes that the anomalous behavior is a result of strong interaction between the Fermi liquid quasiparticles due to antiferromagnetic spin fluctuations. Orthogonal to the above philosophy was Anderson's proposal in 1990 [37, 38] that 2-dimensional Fermi liquids are unstable to the formation of a 2-d Luttinger liquid. Since Anderson, theorists have been trying to understand if and when "non-Fermi liquid" (NFL) can appear in itinerant Fermi systems. There is strong evidence [39] for N F L behavior in the v = 1/2 quantum Hall liquid; theoretical understanding of this assumes a singular gauge interaction between the quasiparticles [40]. The high T c problem is more complex because the lattice introduces an on-site constraint for strong Hubbard repulsion- which means the lattice problem is fundamentally different from a continuum model. Perturbative investigations of the nature and existence of singular interactions in the low-density regime [41, 42, 43] have not shown N F L behavior, even in a finite box [43] (although they do reveal weak singular structure in the quasiparticle interactions [42, 43]). However, these results do not rule out non-perturbative singular behavior in the higher density lattice problem- no consensus yet exists on whether these exist in 2 Chapter 1. Introduction 8 dimensions. We stress that this question is quite different from that of the existence of instabilities to ordered phases (e.g., Superconducting or SDW instabilities), which can be understood using standard methods [44]. As discussed by various authors [37, 38, 43], N F L behavior is typically expected if the interaction between quasiparticles with momenta p and p* is singular when Q = p—p' —> 0. It is nevertheless essential to satisfy all relevant Ward identities and crossing symmetry in any calculation, given the way singular behavior in one channel can influence that in the others. A n approach which reveals the structure in the irreducible interactions as a function of Q and which satisfies the above requirements, involves calculating the irreducible 4-point vertex ffff perturbatively [45], and then using this to determine the full scattering function and quasiparticle dispersion. If f^j,' is not singular, then it is nothing but Landau's /-function. In the next chapter, after an overview on Landau Fermi liquid theory, we perform a calculation of along the line mentioned above for a regular Fermi surface and also near inflection points of the Fermi surface. Our calculation for a curved Fermi surface reproduces the previous results [43] existing in the literature. Near inflection points however, we find that f£g is singular in Q already at 2nd order in a short-ranged repulsive interaction. We stress that the N F L behavior found previously for Fermi surfaces having finite curvature everywhere appeared only at infinite order, and was too weak to destroy Fermi liquid behavior [42, 43]. Calculating the imaginary part of the self energy to second order in the singular interaction, we find a frequency dependence different from the expected Fermi liquid form. Besides the controversy about the Fermi liquid starting point, the validity of Eliash-berg formalism used in N A F L theory to calculate Tc is under question. Migdal's theorem is an essential ingredient of Eliashberg formalism which states that the vertex corrections to the bare coupling between electrons and the bosonic medium (here spin fluctuations) Chapter 1. Introduction 9 are small. In the case of phonons, the small ratio of Debye frequency and Fermi energy (ujr)/EF) makes the vertex corrections small [46]; despite the large bare coupling. In the case of spin-fermion coupling on the other hand, the only relevant energy scales are EF and the antiferromagnetic exchange coupling J. The ratio J/EF however, is of order one [47]. Thus it is not clear why vertex corrections should be small in the spin fermion interaction. Calculations of the vertex corrections close to the antiferromagnetic ordered state [47, 48, 49] showed that the vertex corrections are actually the same order as the bare vertex and therefore are not small. Schrieffer [47] showed that in the ordered state, the dressed vertex should vanish at the antiferromagnetic wave vector. The vertex corrections therefore should be equal to the bare vertex but with opposite sign. The magnitude of the vertex correction at higher dopings is more important because the superconducting transition is suppressed at low doping. In chapter 3, we first give a brief introduction to paramagnon theory, which is a theory for Fermi liquid systems near a ferromagnetic instability. We then turn to the nearly antiferromagnetic Fermi liquid theory pointing, out the similarities and differences between N A F L and paramagnon theory. The self-consistency of N A F L theory is then discussed. We calculate the first correction to the bare spin-fermion vertex. We find that the vertex correction is actually big and of the same order as the bare vertex. The self-consistency of the calculation including quasiparticle renormalization is discussed afterwards. The sign of the vertex corrections is another subject of dispute. As we mentioned above, the sign of the vertex corrections is negative near the ordered state. At optimal doping the sign of our vertex correction is also negative in disagreement with some other results [50]. We talk about the sign of the vertex corrections at the end of the 3rd chapter. Our discussion is based on the flow of the renormalized vertex versus change in doping. Chapter 1. Introduction 10 We argue that the positive sign of the vertex corrections at the optimal doping requires a phase transition at some intermediate doping. In fact, for other reasons, this phase transition has been suggested by Chubukov to be a Lifshitz transition in the topology of the Fermi surface [51]. However, there is no experimental evidence for this or any other phase transitions at intermediate doping at the time of writing [52]. Chapter 2 Fermi Liquid Theory 2.1 Landau Theory of Fermi Liquids Systems of strongly interacting electrons are very difficult, if not impossible, to solve. Perturbative methods fail to work as a result of the strong coupling between particles. Non-perturbative methods, also, are not quite well developed for systems of more than one dimension. It was Landau's ingenuity that realized the low energy excitations of a strongly interacting fermion system could still be fermionic quasiparticles similar to non-interacting electrons. Proposed in 1957 [53, 54], Landau's Fermi liquid theory has been extremely successful in describing low temperature properties of liquid 3 He [55, 56] as well as interacting electrons in metals [57]. It especially explains the success of non-interacting electron theory for low temperature properties of metals. In his theory, Landau assumes that if one adiabatically turns on the interaction be-tween electrons, the non-interacting ground state evolves into an interacting ground state with a one to one correspondence between the bare particle states and the quasiparticles in the interacting state. A necessary condition for this to happen is that no bound states should form during this process. A system that possesses this property is called "Normal Fermi Liquid". Superconductors therefore do not belong to this category. In Landau's picture, the number of quasiparticles, N , is the same as the number of bare electrons. They carry the same charge as electrons and obey Fermi statistics. One has to be careful not to go too far with this picture. For example, the ground state energy is not the same 11 Chapter 2. Fermi Liquid Theory 12 as the sum of the quasiparticles energies. As we will see later, the quasiparticle states are meaningful only near the Fermi surface. A better way, is to define the quasiparticles as excitations of the system above its ground state and also quasiholes as absence of particles in the ground state. More precisely, a quasiparticle state with momentum p (> pF, Fermi momentum) is defined by adiabatic evolution of a non-interacting Fermi sea with an extra electron at momentum p (and likewise a quasihole state). Yet another assumption is necessary to establish the Landau Fermi liquid theory. The assumption is that the quasiparticles interact with each other via an interaction function ff$m ("^or s m i P l i c i t y from now on we suppress the spin indices and denote the /-function by fPP'- We put them back when necessary later.) The scattering of quasiparticles due to this interaction causes the quasiparticles to decay. If the decay rate is large, then the quasiparticle lifetime, rp may be too short. The quasiparticle states therefore will be ill-defined. In other words, the quasiparticle states are not exact eigenstates of the interacting Hamiltonian, i.e. the energy levels have finite width (~ 1 / T P ) . In order to have distinguished energy levels, the width should be smaller than the energies, or The key point here is that the scattering of these quasiparticles occurs at a vanishingly small rate as the quasiparticles get close to the Fermi surface. The reason behind this lies in the Pauli exclusion principle which restricts the phase space available for quasiparticles to scatter to. At T = 0 this phase space is proportional to (p — pF)2 where p is the momentum of the scattered particle [58]. The scattering therefore, is very small when p « pF. At higher temperatures, the scattering rate increases like T 2 . As a result, the quasiparticle picture is meaningful only at low temperatures and for quasiparticles with momenta close to the Fermi surface. Thus Landau's picture, with N quasiparticles filling up the states up to the Fermi surface, should be considered only as a formal assumption. (2.5) v Chapter 2. Fermi Liquid Theory 13 Still this picture is useful especially in the context of Luttinger's theorem that allows to state that for a spherical Fermi surface, the value of Kp is the same in the interacting and non-interacting system. Let us apply a weak perturbation to the system to take it away from its ground state. The effect of this perturbation is to change the occupation number by 8np. Landau postulated that the change in the energy of the system is given by the expansion 6E = Y, + - L £ fpp,5n°6n}, (2.6) where is the quasiparticle energy and V is the volume of the system. In general, one can expand to higher powers of 5np. However, expansion to the second order is enough for most practical purposes. The quasiparticle energy affected by other quasiparticles is given by *=£>=4+TrY,f*>ty (2-7) p p Landau's /-function is therefore the second derivative of the energy with respect to 5np and <5np-, ^^MSnJ, ( 2 - 8 ) p p Since there is a one to one correspondence between quasiparticle states and non-interacting electron states, the quasiparticles should obey Fermi statistics (a more rigorous proof in-volves maximization of the entropy. [57]) NP = e/J(e?-/0 + i ( 2- 9) where / i is the chemical potential; which can be shown to be equal to the Fermi energy EF at T=0 [59]. For a SU(2) symmetric system, one can write (putting back spin indices again) Chapter 2. Fermi Liquid Theory 14 These functions are usually expressed in terms of spherical harmonics A = ( ^ ) / « f i ( « » ) f e . ] I W N P /; and 4>i are known as Landau parameters. In terms of these parameters, the quasipar-ticle's effective mass is given by (for all derivations and proofs see Ref. [60]) - = A + ft/i (2-12) m m* 37r Observable quantities are usually given in terms of dimensionless Landau parameters Fi = N(0)fi and Zi = TV (0)0/, with the density of states at the Fermi surface ^(0) = ^ E ^ - / ^ ) = ^ (2-13) In terms of F;'s and Z/ 's , we have [60] Effective Mass : — = (1 + F L Specific Heat : cy = \m*pFk2BT m V 3 1 3 1 Tip2 Compressibility : - = — ^ ( 1 4- FQ) K 3m* v2 Sound Velocity : c? = — ^ — ( 1 + F 0 ) 3mm* y2p m* Magnetic Susceptibility : \ M — F 4TT2(1 + Z 0 ) where n = N/V is the density of electrons and 7 = eh/mc is the electron's gyromagnetic ratio. 2.2 Microscop ic Foundat ion of Fermi L i q u i d Theory As emphasized in the previous section, the basic ingredient of Landau's Fermi liquid theory is the function f v p i ; which describes the interaction of low energy quasiparticles. Chapter 2. Fermi Liquid Theory 15 The simple form of the interaction term in (2.6), essential in Fermi liquid calculations, is far from obvious. Renormalization group theory [44] provides a plausible derivation for a low energy effective Hamiltonian in agreement with Fermi liquid theory. As we will see later however, renormalization group might miss some essential singularities in the quasiparticle interaction by putting all the momenta on the Fermi surface. Moreover, although renormalization group is a conserving approximation [61, 62], it does not satisfy the Pauli principle [63, 64, 65]. Another way, is to find a relation between fppi and the dressed two-particle interaction or 4-point vertex function in many-particle theory. What we do in this section is to first derive fppi for a low density electron gas by variation of the ground state energy using second order perturbation theory. We then discuss the connection between many-particle theory and Fermi liquid theory. In particular, we find a relation between Landau's /-function and 4-point vertex function in many-particle theory. 2.2.1 Calculation of fpp' for a Dilute Fermi Gas In a dilute Fermi gas, one can calculate / p p / using Abrikosov-Khalatnikov's formalism [45] (for more detailed derivation, see also Ref. [66], or Ref. [43, 67] for derivation in 2-d). In the dilute limit one can assume short range interaction between electrons. The interaction Hamiltonian therefore can be approximated by where U is the coupling constant and V is the volume of the system. We do our expansion in powers of the s-wave scattering amplitude, a, which is given to the first order in U by cp>lcpt (2.14) U = (2.15) m Chapter 2. Fermi Liquid Theory 16 Applying perturbation theory to Hint, the first order correction to the ground state energy is given by Here the index (1) represents both momentum and spin, is the occupation number and Qij requires spins to be antiparallel Qii = ^ ( l - d W i ) (2.17). where CT'S are the Pauli matrices. To second order in perturbation theory we have „(2) _ V - l(#int)mo|2 _ ^ V - r Tl^jl - W 3)(l - n 4 )Qi 2 Q 3 4 (t) - v Eq. (2.18) is divergent at large momenta. To get away from this divergence we have to expand everything in terms of a instead of U. First notice that (2.15) is not exact. To second order in U, (2.15) can be written as 4 ™ T-T TT , U2 „ r Q, m = U = U+yE *Pi+P2,P3+P4 e + "3 4e e (2-19) V 1,2,3,4 4 ei + e2 — e3 — e4 Here, U is the renormalized coupling. Inverting (2.19), U can be written in terms of a 4 ™ ( 4 7 r a ) 2 ^ Q 3 4 r g 9 m " W ^ ^ ^ . - e i - e , ( 2 - 2 0 ) substituting this into (2.16), we get a term second order in a which should be combined with (2.18) to get the second order correction to the ground state energy. Combining all, after some manipulations, we get ( 2 ) _ 327r2a2 ^ nin2n3QL2Q34 ( 2 ^ m 2 y 2 i , w P l + P 2 ' P 3 + P 4 e i + e 2 - e 3 - e 4 l ' ' This now converges at large momenta, since all momenta are limited because of the raj's. Taking the derivatives with respect to npa and n p v , we get fac> _ f ™ r ( 4 7 r a ) 2 v 2^ ,_^ • V " m ^ m2V \ U % + V - es -+ — — — J — + * _ f ] ; (2.22) eP V Ck+p-p' ep' eP~i~€k €k+p'-p Chapter 2. Fermi Liquid Theory 17 The first term in the bracket is the "Cooper" channel and the last two terms belong to the "crossed" channels. The denominators of the last two terms vanish when p = p'. The singularities however get canceled between the two terms except for special cases (we will return to this point later). 2.2.2 F e r m i L i q u i d Theory and M a n y - P a r t i c l e Theory In many-particle theory [58, 60, 66, 68], the excitations of the system above its ground state are given by the poles of the single particle Green's function G(uj,k) [58]. In the absence of any phase transition, G(oo, k) has a form very similar to non-interacting Green's function. One can write [58] G(u, k) = — r r + $(w, k) (2.23) u - ek 4- %rk where Zk and rk are quasiparticle residue and quasiparticle lifetime respectively. The first term in the right hand side of (2.23) is similar to the non-interacting electron Green's function G<°>M) = (2-24) where 8k = S sgn(efe - p) with 0 < <5 •< 1. The second term, $(u,k), represents the incoherent part of the Green's function. For most calculations, the first term in (2.23) is sufficient to give correct physical behavior up to the desired accuracy. This explains why Fermi liquid theory, which is very similar to non-interacting electron theory, works so well for complex systems like metals. In any calculation however, one has to make sure that the incoherent part of the Green's function does not make significant contribution. In terms of quasiparticle self-energy T,%(u), Zk and rk are give by [68] TK~L = -Zk ImEfc-(efc) (2.25) du d Zk=[l- —Re^k(uj) (2.26) u=ek Chapter 2. Fermi Liquid Theory 18 We can see from (2.24) that the quasiparticle lifetime in non-interacting systems is infinite (Tfc —> oo). In interacting systems however, rk is finite. In fact, one can show that [58] for a 3-d Fermi liquid rkx oc (e^ ~ A*)2, and therefore the lifetime diverges as the particle gets close the Fermi surface. The closer to the Fermi surface, the more stable the quasiparticle state is. For states with momenta far away from the Fermi surface, the quasiparticle concept is not meaningful as emphasized before. For small u, the self energy behaves like ImZPF (u) ~ OJ2 and R e £ P F ( o ; ) ~ u ; (2.27) Substituting into (2.26) we find Z (= Z^=PF)) a nonzero positive constant (always < 1). Z measures the amount of the spectral weight accumulated in the quasiparticle peak [68]. It has other meanings as well. It can be shown [69, 59] that the occupation number rik is always discontinuous at the Fermi surface with a jump exactly equal to Z. Thus the Fermi surface concept survives even after turning on the interaction. The existence of the Fermi surface, in fact, is equivalent to the validity of Fermi liquid theory. When Z = 0, as it is in 1-d Luttinger liquids, the occupation number doesn't have a jump at pF. Consequently, there is no well defined Fermi surface; although higher order derivatives of n/t still are singular right at the position of the Fermi surface. Fermi liquid theory therefore breaks down in such cases. The dressed interaction between two quasiparticles is given by 4-point vertex function T^(Pi, P 2 ; P 3 , Pi) which is related to the 2-particle Green's function G ( 2 ) by G<2>(Pi, P 2 ; P 3 , PA) = (2n)8G(Pl)G(P2) [6(Pi - P3)5(P2 - P 4 ) - 5(P1 - P 4 )5(P 2 - P 3)] + i(27r)4 c7(P1 + P 2 - P 3 - P 4 ) ^ (2.28) with P = (v,p). Since in Fermi liquid theory, interaction between two quasiparticles is described by fppi, it is natural to expect connection between /pp< and T ^ . To see the Chapter 2. Fermi Liquid Theory 19 connection, let us define T(P, P'; K) by T(P, P'; K) = r<2> (P, P'; P + X , P ' - K) (2.29) where i f = (u,k). It has been shown that the following integral equation holds under certain assumptions [60] Z2r? r h -k T(P,P';K) = r^)(p,p') + — % / dnrM(p,Q)—±—^F(Q,P'-K) (2.30) (27r)d J co- q- k where Z is the quasiparticle residue, Q = (^,PFq), with /z being the chemical potential, and the function is defined by The order of the limits plays important role here (we will see this shortly). The integration in (2.30) is over all directions of q. The function is closely related to the Landau's /-function [66]. More precisely, at low energies putting all the momenta on the Fermi surface we have fa = Z2T^\p,p') . (2.32) The notation means that P = (p,pFp) and P' = (p,pFp'). fpp' does not have direct physical meaning by itself [66]. However, one can change the order of limits in (2.31) to define r(*)( P ) p/) = l i m l i m p ( P j P ' ) K } (2.33) Z2T^ is actually the physical forward scattering interaction between two quasiparticles [66]. Taking the limit k —>• 0 and co/k —> 0 in (2.30), one can find the relation between Z2TW and fpp/ Z2rW (p, #) = fpp, --^fdSl f„ Z2F^ p>) (2.34) Chapter 2. Fermi Liquid Theory 20 The quantity Z2T^ is usually called scattering amplitude or T-matrix [56, 43]. More precisely the T-matrix tppi (q, ui) is defined by tppl(q, u) = ZPZP>T(P, P'; Q) (2.35) A modified version of (2.30) gives tpp'(q,uj) in terms of fppi [56] —* —* v (£ w) = fw' + 2 — 4 0 ) v ( 9 . • (2-3 6) - q • vk — UJ -j- %o This equation is usually known as Bethe-Salpeter equation. One can again write t°< = tU + 4a • a't% (2.37) PP pp pp v ' In the limit q —> 0, uj/q —> 0, and for p and p' on the Fermi surface, Eq. (2.36) gives Fi ts — 1 + Ft/(21 4-1) Z t t = l + Zt/(2l + 1) ( 2 " 3 8 ) where and are spherical harmonics. 2.3 Fermi Liquid Theory in 2-d In the last section, we introduced Landau's Fermi liquid theory as a powerful tool to describe fermionic systems in 3-d. It is well known, on the other hand, that 1-d electron systems are not Fermi liquids but Luttinger liquids [70]. The question for 2-d systems is more subtle and has been a subject of controversy for almost a decade. Anderson in 1990 [37, 38] suggested that 2-d interacting electron systems are also Luttinger liquids and not Fermi liquids. He conjectured that Landau's quasiparticle interaction function would become singular in 2-d. The singularity he suggested had the form A ^ - T ^ (2-39) Chapter 2. Fermi Liquid Theory 21 P P P P P' P P' Figure 2.3: Crossed channel diagrams contributing in Ipp>. This singular interaction will result in vanishing quasiparticle residue and break down of Fermi liquid theory [43]. One immediately notices that the numerator of (2.39) vanishes (as p —>• p') when both particles are located on the Fermi surface. Thus the singularity can be detected only when the particles are away from the Fermi surface. Methods such as Renormalization Group, which put their initial momenta on the Fermi surface, are unable to detect this singularity. Since Anderson, there have been several attempts to prove or disprove Fermi liquid theory in 2-d [43, 41, 71, 42, 72]; all of them in agreement with Fermi liquid theory (at least for systems with isotropic Fermi surface). On the other hand, it is well known that Van Hove singularities [73] or nested Fermi surfaces [74, 44] do not preserve Fermi liquid theory. Therefore the geometry of the Fermi surface plays an important role in the Fermi liquid behavior of the system. In this section, we shall study the effect of local geometry of the Fermi surface on the interaction function fppi. We especially find a singularity at points where the Fermi surface has zero curvature. This singularity is shown, in the next section, to change the Fermi liquid behavior of the quasiparticles near the inflection points. Chapter 2. Fermi Liquid Theory 22 2.3.1 Calculation of fpp' for a Curved Fermi Surface Let us start with a 2-d Hubbard Hamiltonian H = E e A * c f r + ^ E 4+Q-t cp-gi cp'i cPt (2-4°) P'a <f,p,p' where €p~ defines the "bare" band structure and Q, is the area of the sample. We wish to find fpp,' up to second order in small coupling U, or more precisely, the renormalized coupling U, the same way as we did in Sec. (2.2.1). Note that by small coupling we mean U/t <C 1 (this is not necessarily a low-density limit, but it is a weak-interaction limit). Following the method we described in Sec. (2.2.1), we find faPp' = (h29PP>/m)5a>-a, - (h2gppl/m)2(Ippl + C$) (2.41) where the dimensionless coupling gpp> is given by [45, 43] mil m 9PP> = ~ff=tf U2 2 q 6P + EP> 6P'+q (2.42) we also identify the "Cooper" channel and "crossed" channel by (cf. Fig. 2.3) - \tp-!-tp, tk t~+pl_kJ Ipp, = E f , Z _f (2-44) k W V k efe+Q / The Cooper term, , is singular when p — (p+p')/2 —> 0, but regular near Q = p—p1 = 0 (as is gpp>). We thus drop C ^ ' and approximate gpp> « g0. Ippi, on the other hand, may become singular when Q —> 0. This can be seen by the fact that the denominators in (2.44) vanish at Q = 0. We therefore have to examine Ippi more carefully. Let us define Chapter 2. Fermi Liquid Theory 23 - A A - A 2 ^ - A k, _L A - A 3 Figure 2.4: The two situations considered in the text. In (a) we assume the Fermi surface has finite curvature at k = ko; in (b) the curvature is zero when k = ko (inflection point). Hatched lines show the integration regions discussed in the text. Expanding the denominators to second order in Q, we have IPP' r d2k (_ nr. nk -vk)-Q-\Q-Mf-Q (vp -vk)-Q + \Q • Ml1 • Qj (27T) [(vP - vn) • QY Al k,Q oc h(Q) where The "gap" A ^ is defined by Ak,Q d2k-. [(vp - vk) • Q]2 - A f Q 2\Q\ (2.46) (2.47) (2.48) and Q = Q/\Q\. For a quadratic dispersion relation = k2/2m, M^1 (— l/m) is —* —* independent of k. In more general cases however, it has ^-dependence but this does not affect our calculation (we will consider the ^-dependence of M^1 when appropriate later). Thus for the moment, we can safely neglect the ^-dependence and write A ^ = A ^ . Our goal is now to understand the behavior of Ip-(Q) as Q —>• 0. Infrared singularity occurs when (vp — Vk) • Q = 0. This condition is-usually satisfied on a line in /c-space Chapter 2. Fermi Liquid Theory 24 inside the Fermi surface (corresponding to a straight line in v-space). Let's confine our integration to a region within a cutoff A around this line, where \Q\ <C A <C pF, and pF is the Fermi momentum. The integration inside the Fermi surface then has the form r J-A du 1 , J A - A 3 I 1 2 A Q N 2 _ ^ l n ( ^ _ ^1) „ = (2.49) A U 2 - A | A Q K \ A + A $ \ J AQ A A v ; Thus the integration inside the Fermi surface contains no singularity as AQ —> 0. Singu-larity, if any, should therefore come from the integration near the Fermi surface. Let k0 be a point on the Fermi surface, satisfying (vp — Vk0) • Q = 0. We first consider points where Fermi surface has finite curvature (Fig. 2.4 (a)). Writing k = ko + k±e± + k\\e\\, where ey (e±) are unit vectors parallel (perpendicular) to the Fermi surface at ko, we then define by the expansion e£ = EF +vkok± + ak2± + bk2 + ck±kl{ (2.50) —* Near ko therefore, vk can be written as Vk = vkoe± + 7j|fc|| + ;y±k± . (2.51) where 71 = 2ae± + ci\\ , 7 | | = cej. + 2be\\. (2.52) Fermi surface also is defined by Vk0k± + ak\ + bk2 + ck±k\\ = 0 (2.53) —# Since the Fermi surface has finite curvature at ko, j\\ and b are non-zero. Rescaling the integral in (2.46) so that vko = b = 1, we have a Fermi surface near A;0 obeying A:2 pa.—k± —* —* (cf. (2.53)). It is not difficult to see that singularities in Ipp>, for p,p' close to ko, arise only when 7|| -Q^O (2.54) Chapter 2. Fermi Liquid Theory 25 In that case: /o rV—kZ dkw 7T -W - ^ i ^ ik~ ioF ( 2'5 5 ) To get (2.55) we have made use of the following integral y/u 1 , 1 , \y/u-VA\ Fu d - ^ = ^ G > ^ 7 ^ + « a n - \ x ) . (2-56) u This singularity is just that found already for a circular Fermi surface [42, 43]. It only —* occurs if p and p' approach each other along the direction orthogonal to 7||. For a circular Fermi surface, this direction is orthogonal to the Fermi surface. For non-circular Fermi surfaces (with no inflection point) on the other hand, 7|| • Q = 0 for directions which are not necessarily normal to the Fermi surface; but the singularity is still along lines. This case was also studied away from the Fermi surface by Fukuyama and Ogata [72]. A closer look at the singular region in fc-space for the case of a circular Fermi surface shows that it is bounded by circular arcs of opposite curvature which touch at the Fermi surface [43]. This singularity does not change the Fermi liquid behavior [43]. A simple way to see this, is to look at the quasiparticle energy ^=4+Hfpp ,Snp (2-57) f Singularity in fpp> is summed over in this equation. In our case here, the \Q\1^2 singularity is always along a line. Integration over this term removes the singularity and gives Fermi liquid behavior as one approaches the Fermi surface [42, 43]. 2.3.2 Calculation of fpp' near an Inflection Point At points where the Fermi surface has zero curvature, -fy —¥ 0 and consequently, condition (2.54) is always satisfied. It is therefore natural to expect a singularity in any direction Chapter 2. Fermi Liquid Theory 26 of Q; which may now cause break down of Fermi liquid theory. We wish to study this singularity near an inflection point. We must therefore extend (2.53) to higher order, and also include the /c-dependence of Ai^1 (which now vanishes in certain directions). Expanding about an inflection point at k = ko, we write e£ = vkok± + ak]_ + gk\ (cf. Fig.l(b)). Substituting directly into (2.44) we get <-9kfi/vko I dk± (2.58) Ip(Q) qQw J-A. J-, + [Q -> -Q) (2.59) 3gQ{lJ-A ""J-gA*/vkQ \ct.k± + kfl+Qfy+T(K,Q)i We have defined K = p — k0 (i.e., the center of mass momentum relative to k0), cQ = 2aQ±/3gQ\\ and T(K,Q) = 4-^- + CQ(K± + ^) (2.60) Integration over K±_ is straightforward. One finds dk\\ In fcjj + Qfy - (cQg/vko)kf{ + T(K, Q) k2 + Qfy-(cQg/vko)A* + T(K,Q) + (Q -> -Q) (2.61) The fty term in the numerator is small compared to the other terms unless if Q\\ —> 0 (or cQ —> co). We therefore neglect this term except when Q\\ <C Qx (we will come back to this case later). To calculate the above integrals we should use J du In\u2 + (3u + 7| = [u + (3/2) In\u2 + (3u + 7| - 2u + d{W) tan 9(-W)-\n. . , 2 \u + p/2-J\¥\) (2.62) where W = 7 — f32/4. However , it turns out that all the terms in the above integral are regular as Q —»• 0 except for the " tan - 1 " term. Keeping only the singular terms we find W)~°(w^)l^f ^ f A + J £ 2 ^ + t a n _ j A ^ 2 + (Q -> -Q) (2.63) Chapter 2. Fermi Liquid Theory 27 I: f , o c ^ Figure 2.5: Three different regions in fc-space around the inflection points with different singular behavior. We have put p = k0 therefore the regions shown are actually the regions where p' can be. Between I and II, there is a crossover whereas between II and III there is a sharp edge. where WRA = T(K, Q) + ^ = -K\ - CQ(K± + QJ2) (2.64) In the limit where K, Q <C A we approximately get ip(Q) = J^-l\w+\1?2e(w+) - \w-\^9(w-)} +.. (2.65) where (...) represents non-singular terms and W± = W-±Q. This behavior of Ipp> is rather complicated. We can get some feeling for it if we first take the case where p = k0, i.e., where one quasiparticle is at the inflection point (see Fig. 2.5). Then for most regions in fc-space, (2.65) gives 0(4cQi - Q j j M Q i i ) h0,ko+Q (2.66) I Q | | I 1 / 2 This is the same inverse square root as found for the circular Fermi surface (cf. (2.55)), but now extending over large "butterfly wing" regions in ft-space.) Fig. 2.5 shows different regions in fc-space with different singular behavior. The singularity in the left wing of the butterfly region comes from switching p and p1, i.e. putting;?' at the inflection point. Chapter 2. Fermi Liquid Theory 28 When Q\\ —> 0, the above approximation does not hold anymore. In that case (2.59) becomes r(vk0A/g)^3 1 rK /•(«*<, A / j Ip(Q) = 7TFT dk± 2aQ± J-A J-(vkQk± dk\\ = C ( ^ V 3 [(K± + QJ2)^ - (K± - QJ2)^} + ... (2.67) where C ~ f™oo dz/(z3 — 1) = — ir/y/3. If we again put p — hQ, we get Ip(Q)~QlV3 (Q\\<Q±) (2-68) The singularity here is stronger than what we found before (~ O^ 1 ^ 2 ) . There is a crossover between these two different behaviors (cf. Fig. 2.5). It is very important to note also the non-singular region (region III in Fig. 2.5) which lies close to the Fermi surface as Q approaches the origin. There is no singularity in the interaction between 2 quasiparticles if they are both on the Fermi surface. Thus any cal-culation (perturbative or renormalization group) of the scattering between quasiparticles confined to the Fermi surface will miss all the singular behavior in (2.66). In the more general case where neither particle is at A;0, the behavior of Ipp> is not so easily displayed. As the quasiparticles approach each other one finds the simple behavior 6{ — K\\ — CnK±) W^pT^-tp { Q < K ) (2'69) 2.4 Calculation of Self Energy Fermi or non-Fermi liquid nature of the system can be checked by calculating the self energy T,p(uj). We already have seen how the quasiparticle lifetime and quasiparticle residue are related to the real and imaginery parts of the self energy (cf. (2.25) and (2.26)). The imaginary part of is specially important because it can be calculated Chapter 2. Fermi Liquid Theory 29 easily using the Fermi golden rule. We saw that in a 3-d Fermi liquid, at low energies, we have ImEp F (LO) ~ LO 2 (2.70) In this section we first see how this relation is different for a 2-d Fermi liquid with non-singular interaction. We then find the effect of our singularities on ImE P f , (LO) and show the deviation from Fermi liquid behavior. 2.4.1 Non-Singular Interaction Let us first consider the non-singular case. Landau's /-function is therefore regular and given by the cylindrical harmonics // the same way as before. For the present calculation, it is enough to consider the isotropic harmonic and write UP' = fo (2-71) Having Landau's /-function, we can calculate the quasiparticle T-matrix using the Bethe-Salpeter equation (2.36). In the isotropic channel, which we consider here, the T-matrix is simply given by [56] V ( ^ ) = 4 W = 1 + / ^ f a > ) (2.72) where x 0^ 00) 1S the 2-dimensional Lindhard function [43] X ° ( ? » = iV(0) (2.73) r) = Lo/qvF and density of states at the Fermi surface iV(0) = pF/2nvF. A l l the physical quantities can then be found from the T-matrix. What we wish to do here however, is to calculate the quasiparticle self energy and compare it with the 3-d case discussed before (calculation we present here is based on Ref. [43]). Chapter 2. Fermi Liquid Theory 30 It is always easier to calculate the imaginary part of the self energy which is equivalent to the scattering rate and is given by Fermi golden rule [56] I m E p M = T T £ | V ( 9 > ) | V - , ( 1 - 4_f)(l - n°^)5(u + ep - ep^-ep+?) (2.74) We calculate the energy on shell (i.e. co = ep). Assuming q <C p F we have £p-_? =ep- vFqp = co - vFqp (2.75) where p = p • q. The 5-function in (2.74) can be written as /oo du5{u -co + ep^)S(u - ep - ep-,+-)) - O O /oo dv8(v — qvFp)5(v — qvFp!) (2.76) -oo with p! = p' • q = cos 9'. On the other hand (at T = 0) (1 - n°p+9) = 9(pF- p')9(p' + qp! - pF) (2.77) Eq. (2.77) requires p' < pF and p' > 0. Therefore vl& P P 9 J-«/2 2n JPF-W> 2TT (27r) 2y-./2 P Substituting all back into (2.74) we find I m £ » = ir£-T,f 1^0 J * (g . ^ 2 0 ( w - < ^ ) % ^ " " W " " (2-79) One can write Because of the ^-function, the ^-integral will require p = v/qvF =rj. As a result ImE p ( W ) = / / z , ^ ^ % 0 ( a , _ u)0(qvF - (2.81) Chapter 2. Fermi Liquid Theory 31 In the isotropic case, t(q,u) is only a function of rj (cf. (2.72)). For rj < 1, which is actually the case here, we have < ( 7 ? ) = 1 + / 0(1 - iJ/Vl - ^)/V(0) = N(0) 1 + iAov/Vl ~ V2 ^ where A0 = F0/(l + F0) with F0 = iV(0)/o dimensionless Landau parameter. In this case (2.81) becomes ./0 J W w c 7? 1 — 77^ (27rwF)3 7o y^/wc rj - n2 where LOc — AvF is the energy cutoff. Substituting t(rj) from (2.82) we get ImEpfw) = AT ^ - * ^ , « A" y 1 ^ (2.84) p Jo Ji>/uc r) 1 + (A0 - l)rj2 Jo Jv/uc n where K . = ^0/2^1;^^. In the last step we have dropped the ry2 term in the denominator. We therefore simply get 'lmZJw)~ ("dvv\n\ — | ~ u ; 2 ( l + 21n| — |) (2.85) Jo V UJ The leading u> dependence is therefore ImEp(w) ~ w 2 lnw (2.86) Although this is different from the 3-d case (2.70), it does not change Fermi liquid behavior [43]. Now let us see how our singular interaction will change this result. 2.4.2 Singular Interaction We now calculate the self energy for the singular interaction we found near inflection points. As in some previous studies of various kinds of singular interaction [43, 42, 40, Chapter 2. Fermi Liquid Theory 32 x(q) Figure 2.6: Self energy to second order in XPtP-q. 75, 76], it is convenient to begin with that part of T,p(u) which is 2nd-order in the singular interaction. Let us write fpp' = fpp^T + KP' (2.87) where fpp^T is the non-singular part of fpp> and Xpp> is the singular part. We calculate £p-(u;) to second order in Xpp>, but to all orders in fpp^T. Since we are not particularly interested in the angular dependence of the regular Fermi liquid contribution, we will simply let fpp^T —>• / 0 , a constant as before. We do our calculation for p — k0, i.e. at the inflection point. The singular term Xpp> therefore is given by (2.66) and (2.68). Since (2.68) is only valid in a region very close to the normal axis, we take Xpp> everywere to be V ~7S (2'88) We again use (2.74) as our starting point (note that (2.74) is valid for cu > 0, as can be immidiately seen by the form of the occupation numbers. A full version is given in Ref. [56]). In order to calculate to 2nd order in Xpp>, \tpp>\2 in (2.74) should be replaced by [43] .. ^ ( ^ ) | 2 ^ 1 1 + A>)|2 (2'89) Chapter 2. Fermi Liquid Theory 33 where X°{QIU) i s the generalization of (2.73) for a non-circular Fermi surface. This is equivalent to calculating the diagram presented in Fig. 2.6 [43], where x(<f, u) is the usual fluctuation propagator Since we are going to calculate everything near the inflection point, we can assume that X°(q,u) behaves regularly at the inflection point and treat it as a constant. At T = 0 we have I m £ f c » ~ £ ±e(-ep)9(ep^e(ep,+,)5(u + ep, - e ^ - - ep,+-) (2.91) —* We expand near the inflection point k0 by efc = «*o«J- + P « i ( 2 - 9 2 ) —* —# where K = k — ho- From now on, for simplisity, we take Vk0 ~ g — l.Thus 9 = -Q± - Ql tp-q = -QL — Q\\ epi+q- = q±-Q±. + (q\\-Q\\)3 (2.93) The self energy now becomes ImS f c 0 (w) ~ / ^ / DQ\\ fQ3 dQ± /7" ~ Q ± " Q 5 + q ± + * x 5 ( a ; - 3 Q f ^ + 3Q|| 9 | 2) (2.94) Performing the q± integral, we get I m E f e » ~ / ^ fdq\\ fk dQ±(u - Q± - Qp(uj - Q± - Qjj) y 7 ./-Qj} x % - 3 Q 2 ^ | | + 3 Q | | 9 | 2 ) (2,95) Chapter 2. Fermi Liquid Theory 34 Integrating over Q± is also trivial I m £ f e » ~ | * dq\\6(u - 3Qj 9 | | + 3Qlig2) (2.96) We now change the variables to Q\\ —>• a; 1/ 3^ and q\\ —» ui^y. Integration limits also change to i A / u ; 1 / 3 . For small a; (<§C A 3 , or more precisely <C #A3), the integration limits can be replaced by ±oo. However, one has to make sure that the integral does not diverge as the limits go to infinity. We finally get roo WT- roo Im£* 0(u;) - w 4 / 3 / — / dy5(l - 3x2y + 3xy2) (2.97) J—oo X J—oo The integral is regular at integration limits and also at x = 0. Therefore we can confi-dently write I m S J t o ( o ; ) ~ w 4 / 3 (2.98) This is clearly different from the Fermi liquid form (2.70) or the 2-d version of it (2.86). This is also different from the result obtianed by Fukuyama and Ogata [72] which they find the power to be (5/3) instead of (4/3) for inflection points. The reason for this descripency is that they miss the singular interaction in their calculation. Notice that if we drop the l/Q\\ factor (which comes from the singular interaction) in the integrand of (2.96), we get ImE ~ w 5 / 3 in agreement with Ref. [72]. We can repeat this calculation for a curved Fermi surface using ek = K± + /cjj (2.99) instead of (2.92). Eq. (2.96) then becomes /A rA dQ\\ / dq\\5(u + 2Q\\q\\ - 2q\) A | Q | | | uj2\nuo (2.100) Chapter 2. Fermi Liquid Theory 35 which agrees with (2.86). Although (2.98) is different from (2.100), the difference is not enough to break down Fermi liquid theory (in the sense that Z ^ 0 and Fermi surface exists) [72]. On the other hand, higher order calculations can change this result significantly. For example simple estimate indicates that the 4th order calculation gives marginal Fermi liquid behavior ImS ~ to. Systematic higher order calculation therefore is necessary to acheive the final conclusion [77]. 2.5 Summary Our calculation of /pp< for a 2-d electron system reveals a 1-d singularity near curved points of the Fermi surface as p —> p1. This is in agreement with previous results and is shown to preserve Fermi liquid behavior [43]. Near inflection points of the Fermi surface, on the other hand, we find a 2-d singularity again with \p — p ' | - 1 / 2 behavior. To second order in the singular term, the imaginary part of the self energy shows an a>4/3-dependence in disagreement with the ordinary Fermi liquid result (i.e. ui2\nu>). This however does not invalidate Fermi liquid theory [72]. Inclusion of Higher order diagrams is necessary for any rigorous conclusion. We should point out that our result is different from Ref. [72] in a sense that they do not incorporate singular interaction at the inflection points in their calculation of the self energy. They report a;5'/3-dependence for the imaginary self energy near an inflection point, different from what we find (a;4/3). Ref. [78] has also investigated the effect of the inflection points. They however consider the scattering between two quasiparticles located on two inflection points opposite to each other with respect to the center of the Brillouin zone. This is clearly different from our calculation. Employing renormalization group technique they find that inflection points would increase the lower critical dimension for instability from Fermi liquid fixed point from d = l to d=3/2. Thus Chapter 2. Fermi Liquid Theory 2-d systems will still be stable under such an effect. Chapter 3 Nearly Antiferromagnetic Fermi Liquid Theory At zero doping, all cuprates are Mott insulators with antiferromagnetically ordered spins. On the other hand, the physical behavior of overdoped samples is well described by Fermi liquid theory. However, the most interesting region in the phase space of high T c 's is none of the above phases, but the intermediate doping regime. One can think about the intermediate doping as a continuation of the metallic phase or the extension of the insulating phase. Depending on which way to choose, one may end up with two different schools of thought based on Fermi liquid or non-Fermi liquid theories. At this time not only no consensus about the older theories of high Tc has been achieved, but also different new theories such as, SO(5) theory [79], theory of stripes [80], and more recently theory of nodal liquids [81], etc. have emerged. In this chapter we shall focus on one of the older theories which is based on the Fermi liquid assumption, and because of its simplicity in calculations, has attracted considerable attention especially among experimentalists. Nearly antiferromagnetic Fermi liquid theory (NAFL) is a phenomenological theory motivated by nearly ferromagnetic spin fluctuation theory or paramagnon theory, pro-posed in 1960's to describe metals near their ferromagnetic instability as well as liquid 3 He. Both theories are based on the assumption that the excitations of the electronic system are Fermi liquid quasiparticles, interacting among each other via the damped spin waves. Besides the similarities, there are fundamental differences between these two theories which have made N A F L not as widely accepted as paramagnon theory. In this chapter, after a brief review of paramagnon theory, we introduce N A F L emphasising on 37 Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 38 the similarities and differences between the two theories. 3.1 Paramagnon Theory Fermi liquid theory is primarily constructed to describe properties of low energy electron-hole excitations [68]. There also exist other excitations in a Fermi liquid, which are in-herently different from electron-hole excitations (although they can be views as collective resonances of them [68]), such as density oscillations and damped spin waves. These excitations are usually known as plasmons and paramagnons respectively. On the other hand, although thermodynamic and static measurements such as specific heat and spin susceptibility can be expressed in terms of the Fermi liquid parameters, the information contained in Landau parameters is not enough to describe some other measurements, e.g. transport properties and superconductivity (or superfluidity in 3 He). In particular, the interaction between the quasiparticles plays important role for these properties. We saw in the last chapter that the effective interaction between quasiparticles can be ob-tained from the Landau interaction function via Bethe-Salpeter equation (2.36). This is however, not always the easiest route to take. The other way is to assume a model Hamiltonian for interaction between quasiparticles and apply field theory techniques to calculate the physical properties. The reliability of these calculations should then be judged by experiments. One of the theories that follow this path is paramagnon theory which we describe briefly here. Magnetic susceptibility of a Fermi liquid is given by [60] *» = 5 $ £ w (3'101) If the Landau parameter Z 0 —> —1, the denominator of (3.101) vanishes and therefore magnetic susceptibility diverges. The system in that case undergoes a phase transition to a ferromagnetically ordered state. When the denominator in (3.101) is not zero but Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 39 very small, the magnetic susceptibility is still large and the neighboring spins still have the tendency to align, although the long range order no longer exists. On the other hand, the interaction between the quasiparticles is given by the T-matrix which can be found by Bethe-Salpeter equation (2.36). In particular (2.38) gives tg = V ( l + Z0) (3.102) which also diverges as Z0 —> —1. Thus strong magnetic susceptibility results in strong interaction between quasiparticles; which tends to align the neighboring spins. The dynamic susceptibility of a Fermi liquid was suggested by Leggett [55] to be (~ \ X ( 0 ) ( g » ^ Q i n o \ x M = TTz^K^yW) ( 3 ' 1 0 3 ) where x ^ is the susceptibility of the non-interacting electron system (particle-hole bub-ble). In 3-d isotropic systems, x ^ is given by Lindhard function [68]. Eq. (3.103) was proposed by Leggett using deductive reasoning. An attempt to derive it was first made by paramagnon theory [82]. The theory assumes that the (effec-tive) interaction occurs only between quasiparticles with the opposite spins. The model Hamiltonian for this interaction was first proposed by Berk and Schrieffer [83] as Hint = ljd3r nt(r)ni(r) = L £ 4+^_^ (3.104) k,k',q where ct (c^) is an operator that creates (annihilates) a quasiparticle with momentum k and / > 0 is the coupling constant. Performing Eliashberg type calculations, Berk and Schrieffer showed that in nearly ferromagnetic metals, such as Pd, superconductivity is suppressed as a result of ferromagnetically enhenced coupling between the quasiparticles. This model was then adopted by Doniach and Engelsberg [82] and Rice [84] for liquid 3 He. Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 40 Spin susceptibility can be calculated from (3.104) using Kubo formula [68] and R P A approximation [82] =i-iXl) (3-i05) Comparing (3.103) and (3.105) we find Z0 = -N(0)I (3.106) Thus the interaction / is closely related to Fermi liquid interaction function fppi. In low energy and momentum, one can expand (3.105) to get *(<f' U J ) = 1 - I + (1/12)?- i(,/A)Ioo/q ( 3 - 1 0 7 ) where I = IN(0) and q = q/pF. The susceptibility X(QIU) m (3.107) is peaked at q = 0, reflecting the proximity to ferromagnetism. Near the antiferromagnetic instability, one expects the susceptibility to be peaked near q = ( ± 7 r / a , ±7r/a, ± 7 r / a ) where a is the lattice spacing. x(q, to) can be viewed as the propagator of a bosonic field that mediates the interaction between the fermionic quasiparticles. The corresponding bosons are called paramagnons which are critically damped spin waves. Eq. (3.107) can be written as W) = 2° • , (3-108) • 1 + £ V - tUj/oJq where Xo — SN(0) is the enhanced static susceptibility, £ = ^JsI/12pF is the ferro-magnetic correlation length and toq = Aq/itlpp is an energy scale that characterizes the damping of the bosonic particles. Here, S = 1/(1 — /) is the Stoner enhancement factor which measures the proximity to the ferromagnetic instability. Although paramagnon theory gives the right form for the spin susceptibility, it has some internal inconsistencies [63]. For example if we calculate the density-density corre-lation, instead of (3.106) we get [68] F0 = N(0)I (3.109) Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 41 (3.106) and (3.109) can be satisfied simultaneously only if FQ = —Z0. This is obviously not always the case. For example for liquid 3 He [68], Fo = 10 but Z0 = —0.67. Thus paramagnon theory cannot be the right quantitative theory for the bare quasiparticles in liquid 3 He. Even without comparison with experiments the theory has internal inconsis-tencies. For example RPA approximation does not satisfy Pauli principle [64]. Another problem is the absence of Migdal's theorem in the electron-spin fluctuation interaction, first raised by Hertz et al. [85]. Vertex corrections was shown to be of the same order as the bare interaction. This invalidates the Migdals theorem and therefore any Eliashberg type calculation. We will study this point in the context of N A F L theory later. Besides quantitative inconsistency, paramagnon theory gives some qualitatively right results. We will not discuss the applications of paramagnon theory in this thesis. Instead, we proceed to the antiferromagnetic version of the theory which is relevant to high Tc superconduc-tivity. Interested reader should refer to Ref. [63] for more information about paramagnon theory. 3.2 Antiferromagnetic Spin Fluctuations It is more or less well established that high Tc materials are effectively 2-dimensional electron systems close to the antiferromagnetic instability. As we showed in the last chapter, a two-dimensional electron system with a smoothly curved Fermi surface can be a Fermi liquid (at least in weak interaction regime). On the other hand, N M R experiments [14] and inelastic neutron scattering (INS) [86] indicate enhanced staggered susceptibility, even at optimal doping. One therefore might be tempted to generalize paramagnon theory to antiferromagnetic systems to describe high T c superconductors. It is well known that for a nested Fermi surface, the RPA susceptibility (3.105) diverges at the wave vector q = Q = (±7r/2a, ±7r/2a) resulting in spin density wave instability [44]. For a general Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 42 Fermi surface however, microscopic derivation of the spin susceptibility fails to succeed because the incoherent parts of the quasiparticle spectral function play important role in the calculation of the real part of the particle-hole bubble [87]. The low frequency part of the imaginary part of the particle-hole bubble, on the other hand, is given merely by the coherent parts and can be easily calculated. We will discuss this issue when we talk about vertex corrections later. Based on N M R and NIS results, a phenomenological susceptibility analogous to (3.108) was suggested by Millis, Monien and Pines [23] for high Tc systems x(M = ^ (3-110) where XQ ^ Xo is n o w the static staggered susceptibility, £ is the antiferromagnetic correlation length and u>sf specifies low frequency relaxation behavior. As in param-agnon theory, quasiparticles are again assumed to interact with each other via the spin fluctuations. The Hamiltonian that describes this interaction was written as [31] ft = £ + | E A/5 • ( 3 - M ) where g is spin-fermion coupling constant and —2t[cos(pxa) + cos^a)] — At' cos(pxa) cos(pya) — p, (3.112) is the quasiparticle dispersion relation [35, 88], with t = 0.25eV and t' = — 0.451 S(f) is the spin fluctuation density operator with a bosonic propagator given by (3.110). The presence of t' allows "hot spots" [89] on the Fermi surface (Fig. 3.7) which are points that can be connected by Q; this leads to singular behavior when £ in (3.110) di-verges [49, 89]. Quasiparticles near the hot spots are highly scattered and play important role in N A F L calculations. Full analysis of the scattering rate [89, 90] indicates linear T dependence at the hot spots, while T or T 2 dependence at other points, depending on the temperature and the distance from the hot spots. This is used to explain the linear T Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 43 Hot Spot it a / 7 k I n a a k x " a Figure 3.7: Fermi surface in the first Brillouin zone, with the value for t and t' given in the text; and we assume n=0.75. Calculations are presented here for the wave vectors k± and fc^. dependence of the normal state resistivity of cuprates and the crossover between linear T behavior to T 2 behavior [89, 90]. Some other properties of the cuprates such as supercon-ducting Tc [31, 35, 36], Hall conductivity [90], magnetic and scaling properties [91, 92], evolution of Fermi surface [93, 51], Raman scattering [94], etc., have been calculated using N A F L theory. It is not our intention in this thesis to describe these calculations in detail. Instead, we would like to study the self-consistency of the theory. We proceed by briefly discussing how d-wave superconductivity naturally arises from N A F L theory. We then discuss the self-consistency of N A F L and calculations made by this theory. 3.3 d-wave Superconduct iv i ty In N A F L theory, antiferromagnetic spin fluctuations play the same role in the pairing process as phonons do in conventional superconductors. One can therefore apply BCS theory, replacing phonons with magnons. There is however a principle difference here, Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 44 that is the effective interaction potential Ve«=92x(q) (3.113) is always positive (repulsive). Here g = y/3/4<? (the factor comes from the consideration of the Pauli matrices). BCS gap equation for this interaction is A t = V / d ^ - A t , x ( t - f c O t a n h ^ , / 2 T ) (3.114) Since x is positive, for an s-wave gap Ak, both sides of (3.114) appear with different signs and therefore s-wave paring is not possible. However, since x(o) is peaked at q = Q, dx2_y2 pairing is possible because in that case A k + Q = — Ak and this changes the negative sign in (3.114) to positive. Therefore dX2_Y2 pairing is a natural consequence of N A F L theory. Experimental observation of d-wave pairing is considered as one of the triumphs of N A F L theory. In order to get high T c , spin fermion coupling should be strong. At the beginning it was thought that short quasiparticle lifetime due to the strong scattering would reduce Tc dramatically [32]. Weak coupling [31] and also strong coupling (Eliashberg) [35, 36] calculations of Urbana group however, have found a Tc as high as 90K, even including the lifetime effect [36]. Weak coupling calculation in Ref. [31] gives a T c which is well approximated by T c = A Q E ~ 1 / A ( 3 - 1 1 5 ) where T 0.04eV is an energy scale, A = ng2xoN(0) (3.116) is the dimensionless coupling constant, and a and n are dimensionless constants of 0(1). Strong coupling calculation of Tc was also done by Monthoux and Pines (MP) [35, 36] Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 45 using Eliashberg formalism. For t' = 0, a relation similar to (3.115) was again obtained is the tunneling density of state calculated from one particle retarded Green's function at T c [35]. It is interesting that Tc is only sensitive to g and not other parameters. Fitting to the measured value for T c , one can determine g. For Tc = 90K, one gets g « 0.8 [35]. 3.4 Self -Consistency of N A F L Besides the similarities, there is one fundamental difference between N A F L theory and paramagnon theory: charge properties of ferromagnets are still metallic while antiferro-magnets are insulators. The reason behind this difference as pointed out by Anderson [95, 96] is that ferromagnetism is a natural consequence of Hund's rule: exchange inter-action favors spins to be parallel in order to keep electrons as far as possible and make the repulsion energy minimum. Antiferromagnetism on the other hand, is a result of superexchange interaction which is inherently a non-perturbative effect and is a prop-erty of Mott insulators. Going out of the antiferromagnetically ordered state involves a metal-insulator transition which does not have a counterpart in ferromagnetic case (there is a slight difference between strong coupling and weak coupling here. In weak coupling the Neel ordering and metal-insulator transition - which is not of the Mott type - happen at the same time whereas in strong coupling the system is already an insulator above the Neel temperature). It therefore completely makes sense to talk about a nearly ferromagnetic Fermi liquid but one should be very careful to talk about a nearly anti-ferromagnetic Fermi liquid. On the other hand, almost none of the properties of high (3.117) where N(0) = -WlmGR(p,0) (3.118) Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 46 Tc superconductors resembles of a Fermi liquid, even outside the antiferromagnetically ordered (insulating) region of the phase diagram (except for very high dopings). Thus it is hard to believe that the electron system turns into a Fermi liquid state from a highly non-Fermi liquid insulating state by slightly adding doping. Another problem, which is more computational than fundamental, is consideration of the vertex corrections (note that in pseudogap problem the strong frequency dependence of the vertex correction may make a difference between having or not having a pseudogap [64]. I that sense vertex correction is a fundamental problem). In order to calculate Tc in strong coupling limit, Eliashberg formalism [68] is commonly used. A necessary condition for the applicability of Eliashberg formalism is Migdal's theorem [46]; which states that the corrections to the bare spin-fermion vertex are negligible compared to the bare vertex. In case of electron-phonon interaction, the vertex corrections are of order of (LOD/EF) ~ yJm/M <C 1, where m and M are the masses of electron and nucleus respectively. Spin fluctuations however are different from phonons in three aspects. First, phonons are external excitations while spin fluctuations are excitations of the electronic system itself. Secondly, unlike phonons, spin fluctuations are not propagating modes. Thirdly, the ratio J/EF, with J being the exchange coupling responsible for antiferromagnetism, is of order of one, so there is no small parameter like (COD/EF) in spin-fluctuation interaction [47]. Early estimates reported small vertex corrections [97]. Calculations of Ref. [48] and [49] on the other hand, indicated that close to the antiferromagnetic instability point, the first vertex correction contains a logarithmically divergent term. However, sum of a series of divergent diagrams results in a power law behavior for the dressed coupling where 0 is a small positive number (we should mention that in the original publication [49, 50] tftot ~ # ( - ) QJ (3.1.19) Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 47 [49], (3 was reported to be negative but in later publications (3 was claimed to be positive [50]). Thus near the antiferromagnetic instability (f —>• co), gtot become very large. In other words, vertex corrections tend to increase the spin-fermion coupling. On the other hand, a work by Schrieffer [47] showed that right outside the antiferro-magnetically ordered phase, the dressed spin-fermion vertex is given by ^K[tf-3)2 + r 2] 1 / a (3.120) Thus very close to the antiferromagnetic instability (£ —> co), the full vertex vanishes as q —> Q. The vanishing of the dressed vertex is a consequence of Adler principle: in the ordered state, magnons are Goldstone modes and their interaction with other degrees of freedom should vanish at the ordering momentum to preserve the Goldstone modes to all orders in perturbation theory [93]. The above statement means that the vertex corrections should be equal and opposite sign to the bare vertex to make the dressed vertex vanish. Consequently, Migdal's theorem can not exist near the antiferromagnetic instability. However, superconductivity does not happen at very low dopings either. The relevant question therefore is the validity of Migdal's theorem at higher dopings, outside the ordered state where superconducting transition occurs at high temperatures. The first attempt to calculate vertex corrections at optimal doping was made by us [98]. Using the same parameters as used in N A F L theory, we showed that the first correc-tion to the bare vertex is actually the same order as the bare vertex [98]. The sign of the correction is also controversial because it determines whether vertex corrections increase or suppress the effect of the spin fluctuation coupling. Our finding had a sign opposite to the bare vertex. The effect of the correction is therefore to reduce the spin-fermion coupling, in agreement with Schrieffer's result (3.120). In the next section we describe our calculation in detail. We then discuss the effect of inclusion of the quasiparticle renormalization in our calculation. Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 48 To conclude this section we mention one experimental consideration. The susceptibil-ity (3.110) is adopted by N A F L theory only on phenomenological basis and by fitting to experiments. Consequently, N A F L theory is incapable of explaining the incommensurate peaks observed in inelastic neutron scattering (INS) experiments on La 2 _ x Sr x Cu04 [99] and recently on YBa 2 Cu 3 06+ x [13]. The peaks are observed in the magnetic response x"(q,w) at Qi = (1±5, l)u and Qi = (1,1±<J)7T. It should be pointed out that some other theories such as the theory of stripes have natural explanations for the incommensurate peaks [80, 100]. 3.5 Ca lcu la t ion of Ver tex Cor rect ion Let us first assume that the quasiparticle residue is Z « 1. We will discuss the effect of the quasiparticle renormalization later. The first vertex correction is shown in Fig. 3.8 and can be written as *A(3) = £ / n ( l - 2 ) C ( l - 3 ) G ( 3 - 2 ) (3.121) where numbers are representing the coordinates and II is defined by n = <s{si>=%J2(2Pj-M)** <s(s32> 1 3=1 1 j=l = 9-G{ [2 <S\Si>-J2< S{Si > j = - A o x ( l - 2) (3.122) where A 0 = (g/2)al is the bare vertex. In the last step we assumed three equivalent modes of spin fluctuation, reflecting the rotational symmetry of the system. In momentum representation the first vertex correction can be calculated for general external momenta. Here we concentrate on the interaction between a Fermi surface electron and a spin —* fluctuation with q — Q. We have 5^ = ~^V^f-0^^x(Q' + q^)G(k+ (3.123) Q' Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 49 Figure 3.8: First correction SA^ to the bare vertex, for an incoming fermion with momentum k and energy 0 (relative to the Fermi energy), interacting with a fluctuation of wave-vector Q and zero energy. where 8A^ = 5A(k, e = 0; q = Q, Q, = 0) and Q" = Q + Q'; the sum over Q' takes care of Umklapp processes. In the reduced Brillouin zone Q" — (0,0). Note that the overall sign of the graph in Fig. 3.8 is negative, i.e., Eq. (3.123) is negative. —t We concentrate on SA^, with k being a point on the Fermi surface, since it gives —* an indication of the size of vertex corrections. This is because in general \5A(k, e; q, Q)\ exceeds \SAA (in fact it diverges along a surface in k-space, for given values of q, Q and e); thus one cannot argue, even after integrating over one or more of its arguments, that \SA(k,e;q,Q)\ will lead to corrections smaller than one would get from just using 5A^ (our argument here parallels Migdal's [46]). More generally one finds that if Q <C qv F , A , where A is the spin gap, then 5A^ is a good approximation to 5A(k,e;q,Q,). Writing (3.123) as <% = 9 2 v y [ d 2 i ^ x"(Q' + q,u) Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 50 x ( fk+Q'+q _j_ 1 /k+Q'+q _ fk+Q"+q _ 1 fk+Q'+q\ \ek+Q'+q~U €k+Q'+q + UJ e f e+Q"+9 _ A ; ek+Q'+q+UJ = - 9 2 7 ^ T - ( — ) 2 l k (3-124) 127r3a;s/ u where g2 = | g 2 and is dimensionless; we define g to correspond directly with the coupling constant g used in Monthoux and Pines [35, 36]. At zero temperature one has /27T r2ir i ^ ^ dqx / dg„- —[Sgn(ei)Gi(|, ei) - Sgn(e 2)Gi(f, e2) - G 2 ( | , ex) + G 2 ( f , e2)] -2TT y-27r t l — e2 (3.125) where qx = qxa, etc., e\ and e2 are e1 = ^ ± £ , e2 = = f*±Z (3.126) A* A* A* and G i and G 2 are defined by ^ = • G » ( M = !?TP (3'127) with X = [1 + ( f ) ' (g + Z% can be investigated both analytically and numerically. Here we calculate it for two different points in k-space, both on the Fermi surface (see Fig. 3.7); k\ makes a 45° angle with kx, and kh is & "hot spot" wave vector. In order to obtain a value for 5A^, we need values for the spin fluctuation energy usf, the correlation length £, the susceptibility X Q , the coupling constant g, and the chemical potential // (which is determined by the electron filling factor n). There are different values reported in the references [23, 31, 32]. We have evaluated SA^ for k = ki, kh, in two ways, viz. (a) by assuming various published values for the different parameters, and (b) by making the simple assumption that one is very close to an A F M instability, and then, in the spirit of RPA, imposing the condition \(g/2)x^(Q, 0)| « 1 where x ^ ( Q i u ) is the electron-hole bubble. Numerical calculation gives |x^(Q>0)l = 2 .6(eV) - 1 and thereby Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 51 0(eV) Losf (meV) % / A o 5AL/A0 M P I 1.36 7.7 78.6 -1.81 105.6 -2.43 MPII 0.64 14 49.6 -0.46 73.4 -0.68 "RPA" 0.67 7.7 78.6 -0.44 105.6 -0.59 14 49.6 -0.50 73.4 -0.74 Table 3.1: Calculated values of the vertex correction 5A^ for two different wave-vectors ki and kh on the Fermi surface (columns 4 and 6 in the table). Ref. [35] and [36] give different values for g, and different values for LJSJ. From the values for these two models one calculates 1^ in equation (3.125), and thence 6A^/A0. The third model is the naive "RPA" model described in the text, for which g is determined; we have calculated 1^ and SA^/A0 for two values of ujsf given in MPI and MPII respectively. g = 0.67 eV, assuming n=0.75 and the band structure in (3.112). Since u)sf/\u\ <C 1, this value of g should be a very good guess, within a naive R P A scheme. The results are summarized in Table 3.1. We use two different values for tosf ; these are the two different values quoted by Monthoux and Pines et al. [35, 36]. We also use two different values for the coupling constant g, quoted from Ref. [35] and [36]. We use values of £ = 2.5a, XQ — 80 states/eV (from [35, 36]) and n — 0.75 , appropriate to Y B a 2 C u 3 0 7 (again quoted from [35, 36]). This value of n corresponds, with the band structure in (3.111), to a value of \p\ ~ 1.46t = 0.365 eV. We see that even the values for the vertex correction calculated from the simple R P A model (b) are not small; as in the standard discussion of Migdal's theorem, the importance of vertex corrections appears in the ratio \8A%\/A0. If one takes values of g from the literature [35, 36], this ratio is quite unreasonably large (as large as 2.43 for the hot spots in the model used by Ref. [35]). Thus vertex corrections are clearly very important. The values we quote for |<5A^|/A0 are considerably larger than previous estimates [23, 97, 101]. The reason for this difference with previous work can be tracked back to the factor X^, which is impossible to guess from purely dimensional arguments. Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 52 In fact if we drop the factor from 6Ak, we get an order of magnitude estimate for ^A^ given by ^ - ^ i ? ^ 1 ( 3 - 1 2 8 ) which is broadly in agreement with previous estimates (see e.g. Millis [97]) ; XQ> WSJ and g must be redefined to conform with the parametrizations in this paper). In fact however Z^ is surprisingly large, and also shows a significant variation around the Fermi surface, with a maximum at the hot spots, and a minimum at intermediate wave-vector like k\. We should emphasize here that analytic calculations of 8A^ have to be approximated rather carefully in order to give reasonable agreement with the nu-merical results in Table 3.1. Approximations such as those of Hertz et al. [85] (see also [101]), which try to separate off a rapidly-varying (in (f-space) contribution from x"(<f, cu), give quantitatively incorrect results (including a completely unphysical ln[(k — kh)a] di-vergence as one approaches the hot spot). One might suppose that Tk is large simply because of the band structure (i.e., because of van Hove singularities, or the hot spots). If this were true one could argue that the quasiparticle weight ought to be renormalized down near these singular points in the Brillouin zone, and that this would considerably reduce the vertex correction. In fact however we find this is not the case; this can be checked analytically by suppressing the regions immediately around the hot spots in the integral for Z^, or by simply redoing the numerical calculation for a slightly different band structure. We find that suppressing the hot spots entirely, reduces the vertex correction by a factor which is everywhere less than 2 (and which differs very little from unity when k is far from a hot spot). Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 53 3.5.1 The Effect of Quasipart ic le Renormal i zat ion Z Our calculation of the vertex correction was criticized by Chubukov, Monthoux and Morr (CMM) [50] on the basis of inappropriate quasiparticle renormalization factor Z. As we mentioned earlier the value we used 1 which is the value used in N A F L calculations of Tc [31, 35, 36]. C M M ' s criticism was based on the fact that self consistent consideration of spin fluctuation damping gives spin fluctuation frequency cosf in terms of other parameters. More precisely, one can write [93] u>sf = c2w/2^2^y, where csw is the spin wave velocity, £ is the correlation rate and 7 is the spin damping rate. In fact two points of views can be taken here. One can assume the spin environment as independent degrees of freedom. In that case u>s/ will be an independent parameter. On the other hand if one considers a one band model then assuming that the damping of the spin excitations is dominantly due to the spin-fermion interaction (and not other sources such as spin-spin exchange, impurity, etc.), the damping rate 7 can be obtained from the imaginary part of the particle-hole bubble at transfer momentum Q. The result of the calculation given in [50] is where v is the Fermi velocity at the hot spots and <j>o is the angle between normals to the Fermi surface at the hot spots. C M M claim that if one uses the parameters reported in [36] and at the same time requires Z = 1, one finds ojsf « l.OGmeV which is one order of magnitude smaller than u>sf = limeV used in Ref. [36] and our calculations. C M M then conclude that this is a sign of inconsistency in our calculations. Using the same parameters and solving (3.129) for Z , with uisf = l4meV one finds Z = 0.28. Substituting this value of Z in the calculations, the vertex correction will be reduced by a factor of Z2 = 0.08. This way the magnitude of the correction will be one order of magnitude smaller than what we obtained, and therefore the vertex corrections will be — I sin(j)o\-4 g (3.129) Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 54 small. We do not dispute the fact that all the calculations should be self-consistent. What we argue actually is that the self-consistency should be considered everywhere throughout the calculation. We also agree that the Hamiltonian (3.111) is valid for renormalized quasiparticles and therefore it is necessary to include the renormalization factor Z in all calculations. A l l the parameters we used in our calculation (including Z) were based on the reported values by Monthoux and Pines [35, 36]. We emphasize again that the coupling constant g in Ref. [35, 36] is obtained by fitting the calculated T c to the measured value (~ 90K). If Z, as claimed in Ref. [50] is much smaller than one, then this value should be also considered in Tc calculations and would affect the magnitude of g which gives a reasonable Tc. In a strong coupling calculation, this means that all the fermionic lines have to be replaced by renormalized Green's functions. This multiplies the coupling constant g by a factor Z which exactly compensates the reduction of the vertex correction discussed in [50]. This is more clear by looking at (3.118). One realizes that a renormalized GR is going to reduce iV(0) by a factor of Z. This factor multiplies g in the exponent in (3.117). Keeping all the other parameters fixed the coupling constant should be increased to preserve the value of Tc. The new coupling constant is now given by fc = § ( 3 - 1 3 ° ) where g is the (old) coupling obtained using Z = 1 (i.e. the values reported in Ref. [35] and [36]). The same statement holds for weak coupling calculation (3.115) keeping in mind that A oc g2N(0)2. Substituting Z and gz into the calculation of the vertex correction we get Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 55 The factor 1/Z in (3.130) therefore exactly cancels the effect of Z2 in vertex correction calculations and our calculated value remains unchanged. Looking back at (3.129) but now remembering that gZ is actually what one obtains from fitting to Tc, we find tosf = 1.06raeV even after the inclusion of the renormalization factor Z. The mismatch of this value with the reported value of Ref. [36] reflects the inconsistency of the theory rather than a mistake in our calculations. 3.5.2 Sign of the Vertex Corrections The sign of the vertex correction is important because if it is positive, the vertex correc-tions tend to increase the effect of the spin fluctuations on the electron system resulting in a higher T c . On the other hand if it is negative, it decreases the spin-fermion coupling and reduces the Tc. We saw that at the optimal doping, our calculation indicates a nega-tive sign for the first vertex correction. On the other hand, Ref. [93, 50] report a positive sign at the optimal doping but negative sign near the antiferromagnetic instability. In this section we discuss what a change in the sign of the vertex correction between strong and weak doping would entail physically - essentially it implies a phase transition of some sort for which there is no experimental evidence that we know of. We give our argument based on a fact with which everyone agrees: near the antiferromagnetic instability the sign of the vertex correction has to be negative to suppress the effect of spin-fermion coupling [47]. If the sign of the vertex corrections at the optimal doping is negative, as we found, then the vertex corrections tend to decrease the dressed coupling. Decreasing doping, increases the effect of spin fluctuations and therefore increases the magnitude of the vertex corrections. This can be checked by Eq. (3.124). Decreasing doping would increase XQ (oc £ 2 ) and thereby the vertex correction. Consequently, the dressed coupling would be decreased even further. This trend continues until the total vertex vanishes completely Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 56 at the edge of the ordered phase. Thus the flow of the renorrnalized coupling when changing the doping is continuous. On the other hand if the vertex corrections have positive sign at the optimal doping, decreasing the doping would increase the dressed vertex. Continuation of this trend towards zero doping would give a very large dressed coupling which is unphysical. Interestingly it was speculated by Chubukov et al [51, 93] that perhaps a weak phase transition (a Lifshitz transition ) might exist in the intermediate doping regime. The transition would change the big Fermi surface, compatible with Luttinger's theorem, to small hole pockets centered at (ir/2a, ir/2a) and symmetry related points. If this were really true, and moreover followed from the Hubbard model, it might save the consistency of the N A F L approach, in that it would make it possible for a sign change of the vertex corrections, at some critical g = gc. To our knowledge, no experimental justification exists yet [52] and the evolution of the Fermi surface is not widely believed. Without this evolution, the positive sign of the vertex correction at optimal doping is not physical. At the end we point out that the suppression of the spin-fermion coupling due to the vertex corrections, also gives a natural way to explain why Tc reduces at low dopings in spite of the increase in the spin susceptibility [93] (although other reasons can also be thought of, e.g. reduction of charged carriers, etc.). 3.6 Concluding Remarks We re-emphasize here that these results do not necessarily invalidate the internal consis-tency of the Fermi liquid starting point, in N A F L theory. However they do show that the theory cannot be trusted quantitatively, at least in the usual R P A form. As is well known the R P A is not a "conserving approximation", and for spin fluctuation theories this makes it unreliable (cf. ref. [63], especially section 3). It is useful to compare the Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 57 case of nearly ferromagnetic 3 He liquid, where vertex corrections are also quite large, and where use of the paramagnon model yields values for m*/m which are off by a large factor [102]. Thus if we use melting curve Landau parameters, Z0 ~ 0.75 and F\ ~ 15, we infer a value for the Stoner factor S ~ 24 which yields m*/m = | In 5 ~ 15 , in the paramagnon model. This is roughly 2.5 times the correct value of ~ 6 (note that the first vertex correction is 5A/1 ~ In S ~ 3 in this model), and no amount of self-consistent summing of diagrams can cure this numerical problem. Similar problems can clearly occur in the present N A F L model. We believe this is the main reason for the difficulty one encounters in the M P models, in determining a value for g that (a) gives the correct superconducting T c , and (b) is consistent with the observed spin susceptibility. To check the structure at higher order, we have also estimated the contributions from the graphs containing 2 spin fluctuation lines (there are actually 7 distinct graphs at this level), and found that some of them are also large for the values of g used above. Thus, just as for the case of nearly ferromagnetic 3He, we see no reason to believe, for the values of the parameters given in the table, that performing infinite graphical sums will lead to results which are numerically more reliable, even if they do converge to some smaller renormalized vertex-there will always be other diagrams with large values, which will in general give uncontrolled contributions. It is interesting to compare our results with some other investigations. As we men-tioned briefly, in the weak-coupling limit, Chubukov [49, 93] has calculated the leading vertex corrections to g, concentrating on the gapless case; in the case where there is a gap, he finds that the renormalized coupling is large (for a large Fermi surface consid-ered in our investigation). On the other hand Schrieffer [47] has argued that a correct formulation of the theory, even in the weak-coupling limit, must take account of the short-range local antiferromagnetic order even in the normal state-if this done, he finds Chapter 3. Nearly Antiferromagnetic Fermi Liquid Theory 58 that a weak-coupling calculation shows very strong suppression of the vertex when one. is close to the antiferromagnetic transition. This theory seems rather interesting-note that a related calculation by Vilk and Tremblay [64, 103]finds that the existence of such a short-range antiferromagnetic order in 2-d will cause a breakdown of the Fermi-liquid starting point itself when the correlation length becomes larger than the single particle de Broglie wavelength! Thus the question of what is the correct theory itself is rather confused, even in the weak-coupling regime. It is certainly not clear how any of these arguments will work in the regime discussed in this paper, when g is not small enough to control the magnitude of the vertex corrections. It is of course crucial that these higher-order corrections also be included in any version of this theory that tries to reconcile different experiments - as emphasized by Pines [32], the justification of the theory stands or falls on its ability to do this quantitatively. It is possible that such a program might succeed if one can show that the actual parameters g, cosf, and XQ a r e s u c n t n a t . |<5A^|/AQ is considerably less than one (i.e., if one is genuinely in the weak-coupling regime). This would also be true of versions of the theory in which cosf depends on g, whilst the spin gap becomes an independent parameter [49]; or of the theory of Schrieffer cited above [47]. On the other hand if \5Ak\/A0 > 0(1), we see no hope that such a scheme could succeed quantitatively (in, e.g., the calculation of T c ), since the vertex corrections become large. Part II Vortices in d-wave Superconductors 59 Chapter 4 Introduction Although at a microscopic level, theories of high Tc superconductors, especially for nor-mal state properties, are not well established, superconducting properties of cuprates are better understood. Almost everybody now agrees that the symmetry of the order parameter is d I2_y2-wave, and the existence of nodes in the superconducting gap is well established [33, 34]. A highly anisotropic dx2_j/2-wave order parameter, with four nodes in the superconducting gap is a unique property of cuprates ans also some organic super-conductors, with no analog in conventional superconductors. The effect of this anomalous symmetry on different observable quantities has been a subject of investigation in recent years. An early theoretical investigation of the weak-field response of a dx2_y2 superconduc-tor by Yip and Sauls [104] predicted a direction dependent non-linear Meissner effect, associated with the quasiclassical shift of the excitation spectrum due to the superflow created by the screening currents. Maeda et al. [105] reported experimental evidence for such an effect in Bi 2 Sr 2 CaCu20j / , but subsequent experiments [106] failed to confirm their findings and the situation remains controversial. A similar effect was also studied independently by Volovik [107]. In the mixed state, he predicted a contribution to the residual density of states (DOS) proportional to the inter-vortex distance ~ \fH. Such a contribution was identified in the specific heat measurements on YBa 2 Cu 3 07_5 (YBCO) by Moler et al. [108] but later this interpretation was disputed by Ramirez [109] who found evidence for a similar effect in a conventional superconductor VaSi and by others[110]. 60 Chapter 4. Introduction 61 Kosztin and Leggett [111] predicted that the nonlocal response at very low temperatures will lead to a T 2 dependence of the penetration depth in clean samples in contrast to the linear T-dependence obtained from the local theory for d-wave materials. Another important characteristic of high Tc compounds is their short coherence length £ 0; responsible for strong fluctuation effects near Tc [112]. The large ratio K = A0/£o 3> 1 (A 0 is the magnetic penetration depth) also makes them extremely type-II superconduc-tors. Existence of a mixed state, characterized by a regular array of magnetic flux lines penetrating the material, is perhaps one of the most striking properties of type-II su-perconductors. The original pioneering work of Abrikosov [113], based on the solution of Ginsburg-Landau (GL) equations [114] near the upper critical field Hc2, predicted a triangular flux lattice. This prediction was subsequently verified by low field magnetic decoration experiments on a variety of conventional superconductors. In stronger fields neutron scattering experiments revealed significant deviations from perfect triangular lattices [115] which where attributed to anisotropics in the electronic band structure and other effects. These were modeled by G L theories containing additional higher order derivative terms reflecting the material anisotropics [115]. Based on the experience with conventional superconductors one would expect even richer behavior of flux lattices in the copper-oxide superconductors. In high-Tc cuprates much of the experimental and theoretical effort has been focused on the sizable region of the phase diagram just below TC(H) in which the vortex lattice properties are completely dominated by thermal fluctuations [116]. While understanding the physics of this fluctu-ation dominated regime poses a very difficult statistical mechanics problem, investigation of the equilibrium vortex lattice structures at low temperatures may provide clues about the microscopic mechanism in these materials. Neutron scattering [117] and S T M [118] experiments on YBa2Cu 3 0 7_,j compound, revealed vortex lattices with centered rectangular symmetry and different orientations Chapter 4. Introduction 62 with respect to the ionic lattice. These have been modeled by phenomenological G L theo-ries appropriate for anisotropic superconductors, containing additional quartic derivative terms [119] or a mixed gradient coupling to an order parameter with different symmetry [120, 121, 122, 123]. These works found structures in qualitative agreement with exper-iment, but their inherent shortcoming is a large number of unknown phenomenological parameters and subsequent lack of predictive power. Also, the G L theory is only solvable for a vortex lattice near Hc2, which is experimentally inaccessible in cuprates away from T c . The observed behavior of the vortex lattice may also be understood by incorporating penetration depth anisotropy and twin-boundary pinning without involving any effects associated with mixing symmetries or gap anisotropy [124]. Muon-spin-rotation (/xSR) experiments [125, 126, 127, 128], on the other hand, show an unusual magnetic field dependence in their line-shapes for the magnetic field distribu-tion. This has been attributed to a field dependent penetration depth - which is expected in the Meissner state because of quasiparticle accumulation at gap nodes [104]. It was also modeled using an approach based on the Bogoliubov- de Gennes (BdG) equations in a square lattice tight-binding model [129]. At intermediate fields Hc\ < H <C Hc2, properties of the flux lattice are determined primarily by the superfluid response of the condensate, i.e. by the relation between the supercurrent j and the superfluid velocity vs. In conventional isotropic strongly type-II superconductors this relation is to a good approximation that of simple proportionality, j = -epsvs, (4.132) where ps is a superfluid density. More generally, however, this relation can be both nonlocal and nonlinear. The concept of nonlocal response dates back to the ideas of Pippard [130] and is related to the fact that the current response must be averaged over the finite size of the Cooper pair given by the coherence length £0- In strongly type-II Chapter 4. Introduction 63 materials the magnetic field varies on a length scale given by the London penetration depth Ao, which is much larger than £ 0 and therefore nonlocality is typically unimportant unless there exist strong anisotropies in the electronic band structure [131]. Nonlinear corrections arise from the change of quasiparticle population due to superflow which, to leading order, modifies the excitation spectrum by a quasiclassical shift [130] Sk = Ek + vf • vs, (4.133) where Ek = \Je\ + A 2 , is the BCS energy. Again, in clean, fully gapped conventional superconductors this effect is typically negligible except when the current approaches the pair breaking value. In the mixed state this happens only in the close vicinity of the vortex cores which occupy a small fraction of the total sample volume at fields well below HC2. The situation changes dramatically when the order parameter has nodes, such as in dx2_y2 superconductors. Nonlocal corrections to (4.132) become important for the response of electrons with momenta on the Fermi surface close to the gap nodes even for strongly type-II materials. This can be understood by realizing that the coherence length, being inversely proportional to the gap [130], becomes very large close to the node and formally diverges at the nodal point. Thus, quite generally, there exists a locus of points on the Fermi surface where £ 3> A 0 and the response becomes highly nonlocal. This effect was first discussed by Kosztin and Leggett [111] for the Meissner state and by us [132] in the mixed state. Similarly, the nonlinear corrections become important in a d-wave superconductor. Eq. (4.133) indicates that finite areas of gapless excitations appear near the node for arbitrarily small va. One of the simplest theories to study the magnetic behavior of the type-II supercon-ductors is the London model [133]. In London theory the Free energy is.written only in terms of magnetic field and is therefore very easy for calculation. Unlike in G L theory, the order parameter does not appear explicitly in London free energy. The effect of the Chapter 4. Introduction 64 symmetry of the order parameter or other Fermi surface anisotropies, is therefore not contained in the London model. Simple generalization of the model however can incor-porate these effects. The generalized model contains anisotropic higher order and higher derivative terms reflecting the symmetries of the order parameter and the anisotropic Fermi surface. In the next chapter, we first briefly introduce Ginsburg-Landau (GL) theory, and London theory emphasizing the solutions for a vortex lattice. Starting from a G L model, we then derive the leading fourfold anisotropic corrections to the London equation mak-ing the usual assumption that the free energy was an analytic functional of the order parameter and field. The number of new parameters in this model is far smaller than in the G L approach (a reasonable model contains only one new parameter which controls the strength of the symmetry breaking term) and numerical simulations are consider-ably easier. This provides a useful tool to study vortex lattice structure, pinning by twin boundaries and the magnetic field distribution measured in fj,SR experiments. The model is suitable to study the intermediate field region Hci <C H <C Hc2 which is experimentally most relevant but traditionally difficult to handle within the G L theory. Furthermore, this approach can be extended to T = 0 where G L theory breaks down and the supercur-rent becomes singular. With increasing magnetic field this model predicts a transition from triangular to centered rectangular and eventually a square vortex lattice. While no direct experimental evidence exists in cuprates at present to confirm such a prediction, a similar transition has been recently observed in a boro-carbide material E r N i 2 B 2 C [134] and has been described by a similar London model [131]. In the last chapter of this thesis, we again derive a generalized London free energy, but this time completely from a weak coupling microscopic model, including the nonlinear and nonlocal effects mentioned above. We show that the dominant effect that determines the vortex lattice geometry and the effective penetration depth as defined in //SR experiments Chapter 4. Introduction 65 is the nonlocal corrections, while the nonlinear corrections play a secondary role at low T. At high temperatures we obtain a nonlocal correction similar to the one suggested in chapter 5. At low temperatures however, we find a novel singular behavior directly related to the nodal structure of the gap which completely changes the form of the London equation. This singular behavior has profound implications for the structure of the vortex lattice which, as a function of decreasing temperature, undergoes a series of sharp structural transitions and attains a universal limit at T = 0. Our theory is now completely parameter free. The London free energy at low T is non-analytic and its long wavelength part is fully determined by the nodal structure of the gap function. Such behavior is caused by the low-lying quasiparticle excitations within the nodes and thus could never occur in conventional superconductors with anisotropic band structures. Finally, in the last section of chapter 6, we talk about the experimental justifica-tion of our low temperature nonlinear nonlocal theory. We especially emphasize //SR experiments. Recent //SR data at high fields have justified our prediction for the Field dependence of the penetration depth in vortex state. Chapter 5 Phenomenological Model 5.1 Ginsburg-Landau Theory In 1950 Ginsburg and Landau [114] proposed a phenomenological theory to describe the behavior of superconductors near their transition temperature. At a mean field level, the phase transition of superconductors is second order (gauge field fluctuations however are believed to change the order of the transition to a weak first order [135]). A free energy density that can describe such a transition was proposed to be ^ = 2 ^ l ( - f l l v - 7 ^ * l > + o | * l ' + i l * | 4 + 8 7 ( 5 ' 1 3 4 ) where e* (= 2e) and m* (= 2m) are the charge and effective mass of a Cooper pair, H and A are magnetic field and vector potential respectively. Superfluid density ns is related to the order parameter by |\I , | 2 = ns/n, where n is the total density of electrons. Ginsburg-Landau (GL) equations can be obtained by minimizing / with respect to ^ and A aV + m\2y + — \(-ihV - -A)2V = 0 (5.135) 2m c j = - V x B = — ( f - $ V $ * ) - — ^ M (5.136) J 4TT 2im*V ; m*c • where j is the current density. In the absence of magnetic field (i.e. H = A = 0) and for uniform the minimum of / occurs at | * | = 0 when a > 0, but at | ^ | = yJ-a/0 when a < 0. A continuous transition therefore happens as a goes from positive to negative values. Taking aoc (T - T c ), one can study the critical behavior of the superconductor 66 Chapter 5. Phenomenological Model 67 and calculate all the critical exponents near Tc. G L theory has been studied extensively in the literature in both Meissner and vortex states of superconductors. It is not our attempt in this thesis to discuss the solutions of G L equations for different problems. Instead, we shall focus on deriving a London theory from G L theory, especially in the vortex state. Interested readers should refer to Ref. [136, 130] or any other standard text books in the subject, for more detailed information. 5.2 London Theory in Vortex State Although we will use G L equations to derive the London equation, London theory was proposed long before GL, by F. and H. London in 1935 [133]. The basic assumption in their theory is a local proportionality relation between supercurrent j and vector potential —* A -> n«e 2 r j = —-—A. 5.137 mc This equation is usually taken as the definition of superfiuid density ns. Eq. (5.137) is not a gauge invariant equation. However, to satisfy current conservation, one has to fix the gauge to the London gauge V - i 4 = 0. (5.138) This way (5.137) and all the results following from it are gauge independent and therefore physical. Eq. (5.137) plays a very important role in our theory for the vortex lattice. Generalization of this equation will be used in this chapter and also in the next chapter to develop a generalized London theory to describe the vortex lattice properties of d-wave superconductors. Taking the curl of the both sides of (5.137) and using the Maxwell's equation V x B = (47r/c)j we get A 2 V x V x B + fi = 0 (5.139) Chapter 5. Phenomenological Model 68 which is the famous London equation. A is called the London penetration depth and is given by A=\S (5-140) A free energy density that gives rise to (5.139) via minimization with respect to the. —* magnetic field B, is called the London free energy density / L = ^ ( B 2 + A 2 | V x B | 2 ) (5.141) We now obtain London equations (5.139) from G L equations (5.135) and (5.136). 5.2.1 Derivation of the London Equation from GL Theory Let us begin our derivation of the London equation by writing the order parameter as v[/ = ipel<t>, with ip and (5.142) 27T where A = ^m*c2/A.-ne*2ip2 is the London penetration depth and $o = (hc/e*) is the flux quantum. In type II superconductors tp varies over a length scale £ much smaller than the penetration depth A. £ is usually called the coherence length and is of the order of the extent of a cooper pair. In the Meissner state therefore, ip (and consequently A) stays constant in the bulk of the superconductor away from the boundaries. The London equation (5.139) can therefore be derived easily by taking the curl of Eq. (5.142) and keeping in mind that V x = 0. In the vortex state on the other hand, ip is not a constant and actually vanishes at the center of each vortex. Taking this into account gives rise to the effect of the vortex core which is the subject of the next subsection. Furthermore, in the derivation of (5.139), we used V x V(p = 0 which is not true in the vortex state. The reason is that around a Chapter 5. Phenomenological Model 69 vortex, <f> has a topological winding number equal to 1. In other words, 4> increases by 2n in every turn around a vortex core. Integrating (V x V</>) • dS over a. small region around the vortex center we get J(V x V4>) • dS = j> • rff = (j)f - 4>i = 2im (5.143) where n is the topological winding number; which is one in our case. This equation is satisfied for every infinitesimal region around the center of the vortex r 0 , only if V x = 2^8^ (f — fo). The London equation in the presence of a vortex will therefore become (assuming constant tp again) A 2 V x V x B + B = < ± V ( 2 ) ( r - f 0 ) (5.144) To find a solution for a single vortex, it is easy to take the magnetic field in ^-direction and write B = B(r)z. Eq. (5.144) then becomes r dr dr which is actually a Bessel's differential equation. The solution to this equation is B W = ( 5 : 1 4 6 ) where KQ is the zero-order modified Bessel function. Important to notice here is that K0 diverges logarithmically as r —> 0; which is apparently non-physical. The reason behind this non-physical behavior lies in the fact that we did not incorporate the effect of the vortex core properly. In other words, the order parameter ip always vanishes at the center of the vortex. Therefore, it is not a good approximation to consider it to be homogeneous everywhere in the lattice. We shall take into account this effect in more detail in the next subsection. Chapter 5. Phenomenological Model 70 5.2.2 The Effect of the Vortex Core Let us start from eq. (5.136) again by writing it as j(r) = e*ns{r)vs{r) (5.147) where ns = ip2 is the superfluid density and vs is the superfluid velocity defined by < 5 - 1 4 8 ) At the center of a vortex, ns vanishes. We can therefore write ns(r)=n0n(r) (5.149) where rj (r) is a function with the property (5.150) rj(r) = 0 at r = 0 r)(r) —• 1 for r / f » 1 and £ is the coherence length of the superconductor. The exact form of rj should come from a microscopic theory or a complete solution of G L equation. Here, we do not derive rj. Instead, we find the source function for a few assumed forms for tj. Before that, let us take the curl of both sides of (5.147), keeping in mind that V x (r)vs) = —ritfWlf) - + Vn x vs = ^ - B + Vnxvs (5.151) m* $o m*$o and also from (5.147) \e*n0n J y47re*n0n The generalized London equation therefore becomes 1 V = " (-7-L- 1 V x B (5.152) A 2 V x V x B + B = p(r) (5.153) Chapter 5. Phenomenological Model 71 with a source term p(f) defined by p(f) = (1 -77)73 + >?~ x V x B (5.154) Equations (5.153) and (5.154) can be viewed as a set of self-consistent equations to be solved for B and p. However, p(f) can be simplified significantly in extremely type II superconductors which have large K = A/£. First notice that p is non-zero only in a region of size £ around the center of the vortex. In this region, both B and r\ scale with £. Therefore the first term in the left hand side of (5.153) and the second term on the right hand side of (5.154) are both 0(\2/£2)B and therefore are the dominant terms in (5.153). Ignoring the subdominant terms, (5.153) can be written as V X T 3 = ^ X T I V x - = 0 (5.155) V \vj where u = V x B. If we assume rotational symmetry and let B = B(r)z and therefore u = u(r)9 where 9 is the azimuthal angle, then (5.155) leads to ~ ( r ^ ) = 0 => u = CV- (5.156) r dr \ rj J r C is a constant that can be found, using the fact that / p-dS = $ 0 5 to be C = ($ 0/27rA 2). Substituting all into (5.154), the source function now can be written as p(r) = p{r)z, with P(r) = P-P- (5-157) r v ; 2?rr dr v ' Now let's calculate p for different choices of rj. Example 1: Clem's Ansatz Let's assume that r?(r) = - ^ with R2 = r2 + ev (5.158) Chapter 5. Phenomenological Model 72 where is some parameter of the order of the coherence length £. Substituting into (5.157), we find the source term to be P(r) = ^f4 (5.159) In order to calculate the magnetic field, it is easier to find the source term in terms of the magnetic field, using (5.154). Ignoring the first term we get * > ~ ^ . Substituting back into (5.153), with a little algebra, we get Comparing (5.161) with (5.145), the solution would be * < r > = 2 f W l ) ( 5 ' 1 6 2 ) This result was first obtained by Clem [137], directly by solving G L equation. It was later shown by Brandt [138] that the magnetic field obtained this way in a vortex lattice is consistent with the exact solution of the G L equations. In Clem's approach, is a variational parameter to be determined by minimizing the free energy. It turns out that in extremely type II superconductors f„ = \/2£- As is clear from (5.162), B(r) is not divergent as r 0. In fact, B(0) = ($Q/27r\2)KQ(t/\). Another way to find the magnetic field is by Fourier transforming (5.153) to get d2k F(k)ei%f where X2k2B(k) + B(k) = F(k) or B{r) = J ^ y f ^ (5-163) F(k) = J d2re-&fp{r) (5.164) Chapter 5. Phenomenological Model 73 is the Fourier transform of the source function which sets an ultraviolet cutoff in momen-tum integrations. In the present case, it is not difficult to show that F{k) = QotvkKifak) (5.165) where K\ is the first order modified Bessel function. Substituting (5.165) into (5.163), we get back to (5.162). Example 2: Gaussian Source Function Now let us assume that 7/(r) = 1 - e~r2M2 (5.166) The resulting source function will then be p ( r ) = = *o c - r W (5.167) The corresponding cutoff function will also have a Gaussian form F(k) = $ o e - W 2 (5.168) This form of momentum cutoff was first proposed by Brandt [139]. This is actually the form of cutoff we are going to use in our calculations throughout the rest of this thesis. We will talk about the sensitivity of our solutions to this particular choice of cutoff function, when appropriate later. 5.2.3 Solution for a Vortex Lattice In a vortex lattice, taking the magnetic field in z-direction (B = Bz), equation (5.153) becomes -X2V2B + B = y£p{r~fi) (5.169) Chapter 5. Phenomenological Model 74 where u are lattice vectors indicating the position of vortices. Taking the Fourier trans-form of both sides of (5.169), we get F(k) k (5.170) 1 + \2k2 where k are now reciprocal lattice wave vectors, B is the average magnetic field given by B = §o/Q, with f2 being the area of a unit cell. Unlike G L , the London approach does not automatically determine the form of the lattice for us. Instead, one has to calculate the free energy for different configurations of the lattice and find the one that minimizes the free energy. Although this sounds a formidable job, symmetries of the free energy help us in our initial guess. In other words, the lattice that satisfies the symmetries of the free energy the most, is the most likely lattice for the vortices to exist on. Figure 5.9: Magnetic field distribution in a triangular lattice, using the ordinary London equation with Gaussian cut off. Using (5.141), the free energy per unit cell is given by fL=f d2r[B2 + (VB)2} = £ ( 1 + X2k2)B2 = B2J2-J u c n . k Substituting the Gaussian cutoff function (5.168), we find F(kf + \2k2 B(f) 0ik-rp-ek2/2 + X2k2 (5.171) (5.172) Chapter 5. Phenomenological Model 75 These sums can be performed numerically quite easily. The lattice configuration that minimizes the free energy (5.173) is a triangular lattice. The contour plot of the magnetic field in this lattice is shown in Fig 5.9. A n important thing to notice is that the free energy (5.173) is isotropic and therefore does not determine the orientation of the lattice. In other words, there is an infinite degeneracy in the orientation of the lattice. This degeneracy will be removed as soon as symmetry breaking terms are added to the London equation. We will discuss this issue in more detail in the next two sections. 5.3 M i x i n g of d-wave w i t h s-wave So far we have established a London theory derived from G L theory, but we didn't mention anything about the symmetry of the order parameter. Since an s-wave order parameter is isotropic, whatever we discussed in the previous sections is expected to hold for an s-wave superconductor. A d-wave order parameter on the other hand, is not isotropic. Instead, it has fourfold anisotropy, and consequently, it is natural to expect this anisotropy to influence the structure of the vortex lattice. Surprisingly, the G L equations, as written in (5.135) and (5.136), have exactly the same form for a d-wave order parameter as s-wave. A l l the statements in the previous sections should therefore hold for a d-wave superconductor as well. A way out of this dilemma, is to notice the fact that the G L free energy written in (5.134), is an approximation and contains only the lowest order contributions. In other words, we have ignored all higher order derivative terms. We have also overlooked the possibility of existence of a second order parameter that in general can couple to the main (d-wave) order parameter. It is conceivable that including these effects will introduce some fourfold anisotropic terms to the G L equations, reflecting the Chapter 5. Phenomenological Model 76 d-wave nature of the superconductor. In this section, we shall first consider the second possibility, namely including an s-wave order parameter coupled to the original d-wave order parameter. We shall see that the effect of this inclusion is to introduce higher order derivative terms in the effective G L free energy written only in terms of the d-wave order parameter. We shall then show how this additional high-derivative term will change the London free energy. This way we actually cover both possibilities mentioned above. The material discussed in this section has been published in Ref. [140]. A G L free energy density to describe mixing of d-wave and s-wave order parameters was first proposed by Joynt [141] as / = as|s|2 + ad|d|2 + 7 s|ns|2 + 7 d |rid| 2 + /4 + iI2/87r +jv[(nys)*(Jlyd) - (Uxs)*(Uxd) + cc.]. (5.174) where II = — zV — e*A/hc and f$ contains the quartic terms, d and s represent d-wave and s-wave order parameters respectively. In the bulk of the superconductor and in the absence of magnetic field, we want only a d-wave order parameter to exist. Therefore below Tc, we expect to have a<j < 0, but as > 0. In finite field (H > Hci) on the other hand, a small s-component with a highly anisotropic spatial distribution is nucleated in the vicinity of a vortex [120, 121]. This free energy has been extensively studied in the literature [120, 121, 122, 123, 142]. It is well known that it gives rise to non-triangular equilibrium lattice structures [120, 121]. Our objective here, is not to discuss the solution to the G L equations resulting from this free energy, but to focus on deriving the corrections to the London equation that result from this free energy. Our strategy will be to simplify the free energy (5.174) by integrating out the s-component in favor of higher order derivative terms in d. In this process, some short length-scale information on the order parameter is lost but the magnetic field distribution is described accurately. Using its Euler-Lagrange equation, s can be expressed to the leading order in (1 — Chapter 5. Phenomenological Model 77 T/Tc) as S = ( 7 , M ) ( n 2 - n 2 ) d . (5.175) Substituting this into / gives the leading derivative terms in d of the form: f = 7d[\Ud\2-(^hdaMK-^2y)d\2} + ... (5.176) Various additional corrections to the free energy are obtained from integrating out s more accurately, taking into account the 7 s|ITs| 2 term and quartic terms. However these all —* involve higher powers of n or other terms that will not concern us. The coefficient of the second term has dimensions of (length)2. For future convenience, we will write it in the form e£ 2 / 3 , where e = 3(adj2 / agjf) is a dimensionless parameter which controls the strength of the s-d coupling and £ = \J^d/Wd\ is the G L coherence length. We henceforth assume e < l . As we will discuss in the next section, neutron scattering and S T M experiments probably support this assumption. As mentioned before, a term of the form |(n2—n2)d|2 could arise without invoking s-d mixing. It can result from a systematic derivation of higher order terms in the G L free energy starting with a BCS-like model and taking into account the square symmetry of the Fermi surface [143, 119, 144, 145]. The free energy (5.176) is not bounded below, exhibiting runaway behavior for rapidly varying d-fields. This is in fact cured by keeping additional higher derivative terms that also arise from integrating out s. In fact, the approximation of Eq. (5.176) will be sufficient for our purposes, yielding a local minimum which we expect would become a global minimum upon including the additional terms. In the case of extremely type II superconductors - which we are considering here -the penetration depth A » £. We may therefore assume that \d[f)\ « d0, the zero field equilibrium value, almost everywhere in the vortex lattice, except within a distance of 0(£) of the cores. This gives the London free energy, h = (1/8TT)(73)2 + 7<^K - K 2 /3)[(^ - ^ 2 ) 2 + (dyvy - dxvx)2}}, (5.177) Chapter 5. Phenomenological Model 78 written in terms of the superfluid velocity (we set h — m* = 1), v = V9 - [e*/c)A, (5.178) where 6 is the phase of d. The corresponding London equation, obtained by varying fL with respect to A, is: c {2G*\ 2 — V x S = J 1dl{v--^2[{yvy-xvx){vl-vl)-{ydy- (5.179) For many purposes it is very convenient to express v in terms of B and its derivatives, and then substitute this expression for v back into / L , giving an explicit expression for —* fi as a functional of B only. For e = 0 this gives v<°) = V x B/BQ, (5.180) where B0 = $0/27TAQ is a characteristic field of order Hc\, and / » = ( l M B 2 + A g ( V x B f ] . (5.181) which is actually the ordinary London free energy (i.e. Eq. (5.141)). Here the penetration depth, for e = 0 is AQ2 = 87r7d(e*d0//ic)2. It is presumably not possible to solve Eq. (5.179) in closed form for v as a function of B for e ^ 0. However, this can be done readily in a perturbative expansion in e. The first order correction is: tfW = (2eem{(H0) - ^ ) H ° ] ) 2 - (^ 0 )) 2] - (Vfy - ^x){dyvf - dxv®)} (5.182) with {^ °^ given by Eq. (5.180). The London free energy density, up to 0(e) is then: h = fl + ^ m d y B ) ' + ((dxBr-(dyB)y/Bl}. (5.183) Note that we could have arrived at a similar conclusion by simply writing down all terms allowed by symmetry in fi, expanding in number of derivatives and powers of B. Square Chapter 5. Phenomenological Model 79 anisotropy is first possible in the fourth derivative terms. In principle, we should also include all isotropic terms to order B4 and V 4 . However, assuming that these have small coefficients, they will not be important. As we shall see in the next chapter, similar results can also be obtained from nonlocal effects due to the divergence of the coherence length along the node directions [132] and also from considering generation of quasi-particles near gap nodes [104, 146], in a range of temperature and field where the supercurrent can be Taylor expanded in the superfluid velocity. More generally, the quadratic and quartic terms in (5.183) have independent coefficients. The corresponding London equation is obtained by varying fL with respect to B(r). For B along the z direction, one obtains [1 - A 2 V 2 + U\ie(dxdy)2]B - eQ[B] = 0, (5.184) where Q[B] = 2A 2 )£ 2£? 0- 2[(d 2 - d2y)B + dxBdx - dyBdy][(dxB)2 - (dyB)2} (5.185) is the non-linear term arising from the last term in Eq. (5.183). Our numerical calculation (next section) shows that, the effect of the non-linear term is negligible (Contrary to naive expectation, it doesn't become more important with increasing applied field because the field becomes nearly constant in the vortex lattice when the applied field is large.) Thus to an excellent approximation one may neglect Q[B] in the London equation (5.184). To get a feeling for the effect of the extra nonlocal (four derivative) term, consider a semi-infinite superconductor in its Meissner state with an interface parallel to the yz plane. The magnetic field in this case depends only on x; all the y-derivatives and thereby the extra term vanish. The penetration depth will then remain unchanged and equal to A 0 . On the other hand, if the interface is 45° rotated with respect to x and y axis (a and b directions), the magnetic field will be a function Chapter 5. Phenomenological Model 80 of (x + y). Substituting B = B0e (x+v)/^x into (5.184), we get (neglecting the nonlinear part) (5.186) A = A 0 i ir^1'2 2+U A§ j Here we keep only the larger solution because it is the one that determines the decay rate of the field and therefore can be identified as the penetration depth. The penetration depth is now longer along the crystal axes as is clear from (5.186). To determine vortex lattice structure, we insert source terms p(r—rj) at the vortex core positions, fj, on the right hand side of Eq. (5.184). The source term we use is the Gaussian source term introduced in eq. (5.167). Taking Fourier transform, the magnetic field may be written explicitly as: eik-re-k2£2/2 B^ = B^TTWT^WW- (5'187) Here the sum is over all wave-vectors in the reciprocal lattice and B is the average field. The lattice constant is determined by the condition that BQ, = $ 0 ) where Q is the area of the unit cell. The lattice symmetry is then determined by minimizing the Gibbs free energy QL = FL~ HB/A-K, where •v e 1 + XU,2 k ^ = 5 2 ? l + A ^ + 4 e A ^ ( W ( 5 J 8 8 ) 5.4 N u m e r i c a l Results and Exper imenta l Real izat ions Ignoring the nonlinear term, one can perform the sums in (5.187) and (5.188) numerically to find the magnetic field and lattice structure. The correction due to the nonlinear term can then be found iteratively. To do so, we start by writing (5.184) in Fourier transformed form (with Gaussian source) as h " l + A ^ + 4 e A ^ ( W , {0-1™> Chapter 5. Phenomenological Model 81 where Q[Bk] is the Fourier transform of Q[B] and the superscript n represents the stage of iteration. The initial B% is found by neglecting the nonlinear term. At each following step we calculate Q[Bk] using the known value for B^, substitute it into (5.189) to find the new Bk. We continue this iteration until we get the desired accuracy. Having B, we then can calculate the Gibbs free energy QL = FL — HB/4n using (5.183). We calculate the Free energy this way for different configurations of the vortex lattice. We find that a centered rectangular flux lattice, with principal axes aligned with the ionic crystal lattice minimizes the free energy. An angle (3 between unit vectors (Fig. 5.10) characterizes the dependence of the lattice on e and the magnetic field. A n example of such a centered rectangular lattice is shown in Fig. 5.10. Figure 5.10: Distribution of magnetic field in a vortex lattice for e = 0.3 and H = 6.8T, leading to an angle of (3 ~ 74°. We use A 0 = 1400A and K = A 0 /£ = 68. In agreement with earlier results within G L [121] and Eilenberger [147] formalisms, individual vortices are elongated along the crystalline axes. Figure 5.11(a) shows the dependence of Gibbs free energy on f3 for various values of e at fixed applied field H = 400i?o — 6.8T. For e = 0 minimum occurs for /?MIN = 60°, corresponding to a hexagonal lattice. As e increases, /?MIN continuously increases and for sufficiently large e, the flux Chapter 5. Phenomenological Model 82 lattice becomes tetragonal with /?MIN = 90°. For / ? M I N 90°, there are always two solutions, related by a 90° rotation, in which the long axis of the centered rectangle is aligned with either the x or y axis. The degeneracy is much larger for e = 0, when the flux lattice may have an arbitrary orientation relative to the ionic crystal lattice. 8.5 8.0 7.5 7.0 6.5 6.0 S ' 1 1 1 1 1 1 1 1 6=0.00 , 0.15 40 50 60 70 80 90 P[deg] 0.0 3.0 6.0 9.0 12.0 H[T] Figure 5.11: a) Gibbs free energy as a function of P for the same parameters as Fig. 5.10 and various values of e. Arrows indicate positions / ? M I N of the minima and G0 = —H2/8TT. b) Equilibrium angle PMIN a s a function of H for several values of e. The dependence of /?MIN on the applied field for various values of e is displayed in Fig. 5.11(b). Clearly the anisotropic term becomes more important at larger fields. Our perturbative elimination of v in favor of B breaks down when e and H are sufficiently large that Pum differs significantly from 60°. Furthermore, we might expect higher order corrections to (5.177) to be important in this regime. By fitting Fig. 5.11(b) to experimental data on tetragonal materials such as Tl 2 Ba 2 Cu0 6 +d (once such data become available) one can directly assess the magnitude of e, the only unknown parameter in the model. Chapter 5. Phenomenological Model 83 The analysis presented here, can be easily extended to take into account effective mass (i.e. penetration depth) anisotropy. In a simple one-component G L model, the derivative term is generalized to: / = £ 7 * | I M 2 . . (5.190) i=x,y,z We restrict our attention to fields along the z-axis. The anisotropy then can be removed by a rescaling of the rr-coordinate and a corresponding rescaling of the magnetic field. The coherence length and penetration depth anisotropies are considered the same: Cy/Cx = Xx/Xy. Making a simplifying assumption that the higher derivative and mixed derivative terms in T are also simply modified by a rescaling by a common factor, it then follows that the flux lattice shape is obtained by stretching along the rr-axis by the factor Xx/Xy. We now obtain two possible vortex lattices, both of centered rectangular symmetry, aligned with the ionic lattice, with different angles, /3. (Relaxing our simplifying assumption may split the degeneracy between these two lattices.) On the other hand, when e = 0, we may rotate the hexagonal lattice by an arbitrary angle before stretching. This gives an infinite set of oblique lattices with arbitrary orientation. To compare theory with Y B C O we should take into account twin boundaries; which are the boundaries between different orientations of crystal lattice. They may also tend to align the vortex lattice by pinning vortices to the twin boundaries, at ±45° to the z-axis. This effect competes with alignment to the ionic lattice which we have been discussing. Only in the special case of a square vortex lattice does a line of vortices occur at ±45°. If this is not the case, and if pinning by twin boundaries is significant, then we should expect that the vortex lattice will align with the ionic lattice far from twin boundaries but will be deformed in the vicinity of a twin boundary in an effort to align itself with the twin boundary. On the other hand, for e = 0, the vortex lattice would remain aligned with the twin boundaries everywhere except within vortex lattice domain Chapter 5. Phenomenological Model 84 boundaries which necessarily exist roughly midway between the twin boundaries. Neutron scattering experiments on Y B C O [117] suggest that the vortex lattice is well-aligned with the twin boundaries and is close to being centered rectangular (the ratio of lattice constants is about 1.04) with (3 « 73°, with weak dependence on H. This corresponds to a rotation away from alignment with the ionic lattice by 9°. Four dif-ferent orientational domains, related by reflection in the (1,1,0) axis and 90° rotation were reported. These results can be rather well fitted [124] by the basic London model (e = 0) with mass anisotropy. For \ x / \ y = 1.5, a value roughly consistent with infrared experiments, this lattice has about the right shape. Taking into account the two crystal-lographic domains (related by interchanging A^ and Xy) there are altogether four vortex lattice domains, as seen experimentally. The experimental fact that the vortex lattice appears to be well aligned with the twin boundaries suggests that the tendency to align with the ionic lattice is small. No evidence for a bending of the vortex lattice (by 9°) into alignment with the ionic lattice far from the twin boundaries has so far been found. S T M imaging of the Y B C O vortex lattice [118] also suggests that the (highly disor-dered) lattice has approximately centered rectangular symmetry with / ? « 77°. However, no evidence for the 9° tilt into alignment with the twin boundaries was reported. Con-sidering the observed anisotropy of the vortex cores it has been concluded that the mass anisotropy alone cannot account for the measured 77° angle of the vortex lattice. It has been suggested that a mechanism related to the internal symmetry of the order param-eter (such as the one discussed in the present thesis) needs to be invoked in order to reconcile these observations. Low field Bitter decoration data on Y B C O [148] show vortex lattice geometry with a very small distortion from hexagonal, consistent with a much smaller anisotropy Xx/Xy = 1.11 — 1.15. One may be tempted to attribute this apparent field dependence of f3 to the effects discussed above in connection with Fig. 5.11(b). A n alternative explanation Chapter 5. Phenomenological Model 85 is a poor quality of samples used in the Bitter decoration experiments that may have resulted in partial washing out of the a-b plane anisotropy [149]. 6.795 6.800 6.805 6.810 6.815 B m Figure 5.12: Magnetic field distribution function; Solid line: using the same parameters as in Fig. 5.10. Dashed line: the same parameters except for e = 0. //SR experiments measure the field distribution n(B) = (I/O) / S[B — B(f)]dxdy. Our calculation of this quantity is shown in Fig. 5.12. For comparison we show our results with (solid line) and without (dashed line) the extra nonlocal term. For pi ^ 60°, B(f) has two inequivalent saddle points leading to two peaks in n(B). n(B) is unaffected by effective mass anisotropy, as can be shown by the rescaling transformation, mentioned above. Existing pSR experiments show only a single peak [150] but this might be due to the large broadening as a result of finite muon lifetime. The weak field dependence of (3, the alignment with twin boundaries in the neutron scattering experiments, and the single peak in n(B) suggest that e is small in Y B C O and that the normal London model, together with twin boundary pinning, provides a good fit to the data. S T M and Bitter decoration data on the other hand, seem to favor Chapter 5. Phenomenological Model 86 finite e and weak pining to twin boundaries. Further experimental work, preferably on untwinned Y B C O or other tetragonal superconductors, will probably be necessary to clarify the importance of square lattice anisotropy in high-Tc superconductors. Most of the experiments mentioned above are done at temperatures far below Tc, i.e. beyond the validity range of G L theory. Thus our derivation of the generalized London model from G L theory is not valid for most experimental situations. However, as we emphasized, the corrections we introduce to the London free energy are the most general analytic corrections possible by symmetry to the order of our concern. It is therefore reasonable that even at lower temperatures all the lowest order analytic corrections come with these forms. We will see in the next chapter that at very low temperatures, the lowest order corrections are actually non-analytic. This is because of the existence of the gap nodes; a unique feature of d-wave superconductors. Even those effects turn into the analytic forms discussed here at higher temperatures. Chapter 6 Microscopic Model 6.1 Gorkov Theory and Quasi-Classical Approximation Ginsburg Landau theory, discussed in the last chapter, is a theory that is valid only near the transition temperature Tc. Most of the experiments however, are done at tempera-tures far below Tc. Also, one of the most essential characteristics of d-wave supercon-ductors, i.e. the existence of nodes in the superconducting gap, does not show a direct effect on the G L equations. Thus a more sophisticated microscopic approach is necessary to describe the vortex lattice properties of high Tc superconductors. The microscopic theory we present here is based on the theory developed by Gorkov [151] in 1959. In his paper, Gorkov showed how G L theory results from pairing theory at temperatures close to T c . Here, we don't give the detailed derivation of Gorkov's equations from the first principles. Our goal instead, is to derive relations between supercurrent j and superfluid velocity vs (or vector potential A) starting from these equations. This relation is then used to obtain a generalized London equation. For a detailed derivation of the Gorkov equations, interested reader should refer to ref. [136, 152]. Let us start from an interaction Hamiltonian of the form Hint = j d3rdW^)M?)4(^MrlV(?-f) (6.191) The Gorkov equations for this Hamiltonian are OT L C 87 Chapter 6. Microscopic Model 88 + J A(r,f').F t(r''T,fV)dV = 5^(f-f")6(r - r') (6.192) - j A*(fy)g(f'TyT')d3r" = o (6.193) where Q and F are regular and anomalous parts of the Nambu-Gorkov Green function respectively, S(rr,fV)<W = - < T T ^ ( F r ) ^ , ( f V ) > (6.194) ^ ( f r . f V ) = - < TT^{(rr)V>I(fV) > (6.195) Here, and also throughout the rest of this chapter, we choose h = m = 1. The order parameter A is related to the interaction potential V by A ( f - f ' ) = V ( f - f ) < ^ ( r ^ f ) > . . (6.196) Equations (6.192) and (6.193) have to be solved for Q and T self-consistently with A . The self-consistency equation follows from (6.195) and (6.196) to be A*(f-fO = V(f-7)^(rTj+yO) (6.197) Solving these equations self-consistently, in a general case at the presence of Magnetic field is tedious if not impossible. However, substantial simplification can be achieved using local approximation or gradient expansion. The method we use here (with some minor differences), is a generalization of the gradient expansion method discussed in Ref. [152]. The generalization is necessary to study d-wave superconductivity. 6.1.1 Local Approximation Gorkov equations (6.192) and (6.193) take much simpler forms in Fourier space. If the system is translationally invariant in time, Q and T are only functions of (r — r ') . We Chapter 6. Microscopic Model 89 can therefore take their Fourier transform with respect to (r — r') and define Qu{f,f) = J G(fT,T*0)e-iuj»TdT ?l{r,7) = j T^rr^tye-^dT (6.198) where ton = 7rT(2n — 1) are fermionic Matsubara frequencies. We also change the co-ordinates to R = r' and p = r — r' and take the Fourier transform with respect to p to get GU{R, k) = f G„{R + p, R)e-i%"dzp (6.199) J*(R\ k) = j Tl(R + p, R)e-^d3p (6.200) A{R, k)= f A{R + | , R - he'^d^p (6.201) (6.192) therefore becomes {iun + hik--A(R + i^)]2 + p}g„(R,k)+ f dzp f d3rne-^i\{R + py)J*{<r*',R) = 1 2 c dk J J (6.202) The integral on the left hand side of (6.202) can be written as J dzPl j d3p2e-^+^A(R + p1 + p2,R + p2)Fl(R + p2, R) = lim / d 3 P l / d'p^'^e-^AdR + p2 + f ) + £ (R + ih + f ) - f )^l(R + P2, R) k'->kJ J 2 2 2 2 = Jim A{R + i ^ + l-^r, k')Fl(R,k) (6.203) k'-+k dk *dk' If A(R) and A(R, k) are slowly varying functions of R, then we can neglect all -J* and -4; which are at most of the order of £. We can also follow the same procedure for (6.193). The resulting Gorkov equations in this limit will be { ^ n _ I ^ _ ^ i ( ^ ) ] 2 + ^ ( ^ ^ + A ( ^ ^ ) j r t ( ^ f c ) = ! (6.204) 2 c {- iw n - i [ fc + ^ ( £ ) ] 2 + ^^ = 0 (6.205) Chapter 6. Microscopic Model 90 These are now algebraic equations instead of integral equations. This approximation is usually referred to as quasi-classical approximation. The reason is that the equations —* —* involve position R and momentum k at the same time, similar to classical equations. The self-consistency equation also takes the form A*(R, k) = \Y,i d3k'V(k - k')Fl(R, k') (6.206) P n J where V(k) is the Fourier transform of V(r). The solutions to (6.204) and (6.205) are GU(R, k) = --iojn-\[k + ZA(R)}2 + y {iujn - \[k - \A(R)Y + »}{-iun - \[k + ~cMR)]2 + »} + &2(k) A * (A?) {iujn - \[k - \A{R)f + a){-iun -\[k + \A{R)Y + u} + A*(k) In the absence of magnetic field these solutions will be simplified to k * ® = J r l i (6-207) k with H = \ k 2 - i i and E% = y)^ + £s?(k) (6.208) The self-consistency equation (6.206) also will be A%R,k)= fd3k'V(k-k')A*(k')W-Y^ (6-209) Summing over Matsubara frequencies, the self-consistency equation becomes A(k) = -Jd*k'V(k-k')^tanh(^) (6.210) which is the well known BCS gap equation. Chapter 6. Microscopic Model 91 In the presence of magnetic field, simplifications can be achieved if we introduce new Green's functions Q and T by G(fr, r V ) = ^(fr, ^ r ' ) e - W - x ^ » and also a new pairing potential (6.211) (6.212) where x ( 0 i s the phase of the operator ip(f) at some fixed gauge. The Gorkov equations (6.204) and (6.205) for Q and T become {iujn-^[k + vs(R)]2 + p}Gu{R,k) + ^ = 1 (6.213) { - iw n -^ [ fc - t7 , ( J? ) ] 2 + ^^ = 0 (6.214) where vs is the superfluid velocity defined by (6.215) This is the same definition as in (5.148) if we notice that x = 4>/2- Shifting the chemical potential p by v2/2, we have 6 -» vs(r) = V x(r) - -A(f) c -(k ± Vs)2 - fj, = ± • u s (6.216) Gorkov equations will then be simplified to —* — iuon - €£ - k • vs A A* and their solutions will be V G Gu{R,k) = iu)n + e^ - k- vs —* icon - k • vs(R) + 6% (6.217) [iujn-k-vs(R)]2-El(R) -A*{R,k) [iun-k-vs(R)]2-El{R) (6.218) Chapter 6. Microscopic Model 92 Notice that the only difference between these and (6.207) is a Doppler shift (by k • vs) in the Matsubara frequencies. We again sum over the Matsubara frequencies in the gap equation to get A(R, k) = - /d*k'V{k - k')^l^{f[k' • Vs(R) - E-k,(R)] - f[k' • vs(R) + E-k,(R)}} J ^Ek\R) (6.219) where /^) = ^ T T ( 6 - 2 2 0 ) is the Fermi distribution function and is the inverse temperature 1/T. Note that as vs —> 0, this reduces to (6.210). 6.1.2 Calculation of Current in a Superconductor The current density, in terms of the Green functions, can be written as j(r) = % lim £ ( V , - VR-) < j>ti?)M?) > £ < tiWMV > 2 r -+f"^ C c 1 IP . P _* _ = ^ l i m £ ( V R - , - V ^ ( f D > f O + ) - - A ( f ) E ^ ( f D , H ) + ) = ie lim (VP - Vf)G{rO, f"0+) 4- 2ei7,(r)0(fO, rt)+) (6.221) f—>f The current in this form is manifestly gauge invariant. Keeping in mind that the super-fluid density is ns(f) = E<r < ^(0^(0 > = z 2£(r t ) , r t ) + ) , the current can be written as J(r) = ensvs(f)+jqp(?) (6.222) where jqp(r) = ie lim (VR-> - V r - )£( fD, r*0+) (6.223) Chapter 6. Microscopic Model 93 jgp turns out to be the current due to quasiparticle excitations in the superconductor. Fourier transforming, using the same techniques discussed in the last subsection, we get lv(R^ = ief^{VA-2ik)-^gMk) (6.224) Again under quasi-classical approximation, one can neglect and write UR) = j ^ j ^ g ^ k ) T^HO- + £)f[Ei + k • MR)] - (1 - f)m - k • v-s(R]$.22h) In the last step, we have performed a Matsubara sum. Combining the two terms the current can be written as jqp(R) = -2c j -0pkf[Ei ~ k • vs(R)} (6.226) or jqp(R) - -4eNF (vF | o ° ° ^ / ( V ^ + IAI 2 - vF • vs^j } (6.227) where NF is the density of states at the Fermi surface and (. . .) F S represents averaging over the Fermi surface. Note that A , in general, can depend on the point on the Fermi surface. Equation (6.227) has a natural interpretation. The argument of the Fermi distribution function / is the Doppler shifted quasiparticle energy at the particular point on the Fermi surface. Thus (6.227) is actually a Fermi surface average of the current due to the quasiparticles excited via thermal excitation plus the Doppler shift. This can be viewed as a normal fluid in the opposite direction of the superfluid that tends to reduce the supercurrent density. At zero temperature, the Fermi function is a step function which is zero unless its argument is negative. Therefore (6.227) is zero unless vF • vs > | A | (6.228) Chapter 6. Microscopic Model 94 Eq. (6.228) actually defines a critical value vc = | A| m i n/uF for the superfluid velocity vs, beyond which the superconducting gap vanishes at some regions of the Fermi surface. Increasing vs beyond vc increases the number of quasiparticle excitations and eventually destroys the superconductivity. Therefore, there is an interval of vs over which we can have superconductivity with a gap that vanishes in some region of the Fermi surface. This is usually known as gapless superconductivity. For an s-wave superconductor, (6.228) gives a finite critical value for the superfluid velocity. For a d-wave superconductor (or any superconductor with nodes in the superconducting gap) on the other hand, the gap vanishes at the nodes even at zero vs. In other words, quasiparticles exist at node lines even for infinitesimal vs and zero temperature. This phenomenon is highly anisotropic because the nodes exist only at four symmetric points on the Fermi surface. The effect of these nodal quasiparticles in the Meissner state was studied by Yip and Sauls [104] and others [153]. Here, we focus on the effect in the vortex state; which has been studied in [132, 146]. We again derive a generalized London equation and solve it numerically for a vortex lattice to calculate physical quantities. 6.2 Generalized London Equation As we mentioned earlier, the fourfold anisotropic nature of the gap nodes is expected to influence the magnetic properties of a d-wave superconductor. This influence will be via two distinct effects: a nonlinear effect due to quasiparticle generation because of Doppler shift in the quasi-particle spectrum, and nonlocal effect due to divergence of the coherence length along the nodal directions. In this section we study these effects in two separate subsections. In what follows we assume that these two effects can be studied independently and their effects are additive. In other words, we assume that any inter-connection between these two effects produces higher order corrections to our generalized Chapter 6. Microscopic Model 95 London equation. In general the excitation of quasiparticles at the gap nodes can affect the coherence length and therefore the nonlocality of the system. Similarly nonlocalities can change the form of Eq. (6.227). This effect which is neglected in our calculations has been studied extensively in the context of superfluid 3 He [154]. 6.2.1 Nonlinear Corrections Let us first neglect any nonlocal effect and focus on the quasiparticles generated at the gap nodes. As we mentioned at the end of the last section, excitation of quasiparticles at the gap nodes produces a current density flowing in the direction opposite to the superfluid velocity - sometimes called back-flow. The total current is given by (6.222). We again define the superfluid velocity by vs = l(V<P-—A). (6.229) with 4> being the phase of the order parameter. In this definition, vs is in the direction of the superfluid, i.e. in the opposite direction of motion of the Cooper pairs (which is used in [104, 146] as the definition for the direction of vs). The contribution of the quasiparticles generated at the nodes to the total current is given by (6.227) which we rewrite it as jqp = -AeNF jps dsvF(s) d£f(y/e + A(s ) 2 - vF(s) • vs) (6.230) where s parameterizes a point on the Fermi surface. A(s) is the superconducting gap which in general can have s-wave, d-wave, or other symmetries. At zero temperature (6.230) leads to jqp = -4eNF j dsvF(s)0(vF • vs - \A\)y/(vF • vs)2 - A 2 (6.231) At higher temperatures however, this gives the first term in the Sommerfeld expansion; which is a good approximation as long as T < T* = TC(H/H0), where H0 is of order of the Chapter 6. Microscopic Model 96 Figure 6.13: Circular Fermi surface with a dx2_y2 gap. Quasiparticles will be excited at nodes marked by VFI and vp2 opposite to vs. thermodynamic critical field, Hc [104]. The presence of the ^-function in (6.231), results in excitations only at the nodes which are in the direction of va. Fig. 6.13 illustrates a circular Fermi surface with a d-wave gap. The quasiparticles are excited at the nodes marked by vpi and vp2 in the direction of vs. For a small enough vs, the excitations stay very close to the gap nodes. Therefore, one can linearize the gap function near the nodes writing A(9) ~ ^/A09, with A 0 the maximum gap and 7 defined by 7 = ^ [ ^ A ( 0 ) ] n o d e (6.232) The most commonly used form for a d-wave gap is A(9) = A0cos(2#). In that case, (6.232) leads to 7 = 2. The component of jqp along the x'-direction which is diagonal to the x and y (a and b) directions (as illustrated in Fig. 6.13) is then ,9C d9 I jqpx' = -AeNFvF —^/(vFvscosa)2-\jA09\2 = — ens— cosa cosa = — ens (0.266) v0 v0 a is the angle between vs and x'-axis, ns = NFvp is the superfluid density, v0 = 'jAo/vp is some characteristic velocity and 9C is a cut off imposed by the ^-function in (6.231). Chapter 6. Microscopic Model 97 Similarly the y'-component is UvV = - e n s V s y ' M (6.234) and the total current thereby becomes j = ens[vs - [vsxi\vsxl\x' + vsy/\vsyl\y')/v0]. (6.235) The nonanalytic nature of the effect is evident from this equation. It is now possible to write a free energy in such a way that (6.235) can be obtained by minimization with respect to A. Keeping in mind that j = (c/47r)V X B = (c/47r)V x V x A and d/dva = —-JdjdA, the corresponding free energy density can be written as / = ns L 2 " ' - 3 ^ ( l ' ' » ' | 3 + K ' ' ' | 3 ) + ^ (6.236) In general, it is possible to solve (6.235) for vs in terms of j = (c/47r)V x B, substitute it into (6.236) and write down a London free energy only in terms of B and its derivatives. However, instead of solving (6.235) exactly, we find vs perturbatively assuming that the nonlinear part is much smaller than the linear part. This way we get a polynomial correction to the London equation which is convenient for numerical purposes. To first order in perturbation theory we have c Vs = . V x B + — ((V x B)X,\V x B\x,x' + (V x B)y,\V x B\y/y')) (6.237) Substituting this into (6.236) and keeping the lowest order terms, the London free energy density becomes h = ^ [ B 2 + Ag(V x Bf + ( ^ ) ^ ( I ( V x B)x,\* + | (V x B)y,\3)} (6.238) where A 0 = ^/c2/47re2n s is the zeroth order penetration depth, £ 0 = vp/^Ao is the coherence length, B0 = (j)o/2^Xl is a characteristic field of the order of Hcl and = W e Chapter 6. Microscopic Model 98 is the flux quantum. For magnetic fields in the z-direction, (6.238) becomes h = 871*' + A ° ( V 5 ) 2 + ( 3 ^ ) ^ ( I ^ ' j B | 3 + l ^ ' j B | 3 ) l = f l + / n l ( 6 - 2 3 9 ) with / ° and /„/ representing the ordinary London free energy density and the leading nonlinear correction to the free energy density respectively. The corresponding London equation is given by -X2V2B + B- (—)^L(d2x,B\dx,B\ + d2y,B\dy,B\) = 0 (6.240) 7 B0 A similar London equation is also derived by Zutic and Vails who investigated the effect in the Meissner state [153]. In order to find the magnetic field distribution in a vortex lattice, one has to insert a source term Y>jp{f — f3) on the right hand side of (6.240) with fj being the position of the vortices in the lattice. The function p{f) takes into account the vanishing of the order parameter at the center of the vortex cores. Numerically, it is more convenient to work in Fourier space rather than real space. Fourier transforming (6.240) with a proper source term on the right hand side yields B-k + X2k2BR - Gn,{k, BK) = BF{k) (6.241) where k is a reciprocal lattice wave vector, Gn\ is the Fourier transform of the nonlinear term and B is the average magnetic field. The cut off function F(k) comes from the Fourier transformation of the source term and removes the divergences by cutting off the momentum sums. We are going to use Gaussian cutoff (5.168) in our numerical calculation in the next section. 6.2.2 Nonlocal Corrections In our derivation of the corrections to the London equation in the previous subsection, we derived a local relation between current density j and superfluid velocity vs (or vector Chapter 6. Microscopic Model 99 —* _ potential ^4). This was a result of the local approximation we used in our derivation. In fact, this relation is always nonlocal over the length scale of the coherence length £ 0; which is of the order of the finite spatial extent of the Cooper pair [130]. Magnetic field in a superconductor varies in the length scale given by London penetration depth A 0 and therefore nonlocal corrections to physical quantities, such as the effective penetration depth, will be of order K ~ 2 , where K = A0/£o is the G L ratio. For strongly type II materials (K 1) such corrections are negligible. Since cuprate superconductors fall well within this class (K is of 50 for most) local electrodynamics is usually used. However, a closer examination suggests that this might not be justified in all situations, if, as it is widely believed, these materials exhibit nodes in the gap. In such a case in place of the usual coherence length £ 0 = VF/ITA0 one is forced to define an angle dependent quantity, £o(p) = VF/TTAP, which diverges along the nodes. Clearly, in the vicinity of nodes the condition A0/£o(p) 3> 1 is no longer satisfied and, in fact, the extreme nonlocal limit is achieved. Nonlocal corrections therefore cannot be dismissed in unconventional superconductors. The effect of these corrections on the temperature dependence of penetration depth at low T was studies by Kosztin and Leggett [111]. Here, we discuss about the effect on the vortex lattice properties. What we follow here is based on a work published in Ref. [132]. From the above arguments, it is clear that corrections due to the nonlocal effects at the gap nodes will be highly anisotropic and will in general break the rotational symmetry of the flow field around the vortex, contributing an anisotropic component to the inter-vortex interaction in the mixed state. Let us now find these nonlocal corrections to the London model neglecting all the nonlinear effects discussed in the previous subsection. In the next subsection, we will combine these two effects into a single London equation to be used in numerical evalua-—* —* tions. Nonlocal relation between j and A is conveniently written in the Fourier space as Chapter 6. Microscopic Model 100 [130] jk = -{c/4n)Q(k)Ak. (6.242) Here Q(k) is the electromagnetic response tensor which can be computed, within the weak coupling theory, by generalizing the standard linear response treatment of Gorkov equations [155] to an anisotropic gap. We find Qv® = *g.H , ^F,i>F> V (6.243) where 7^ = vp • k/2, A 0 = \Jc2/4.-Ke2vFNF is the London penetration depth as before, u>n are the Matsubara frequencies and the angular brackets again mean the Fermi surface averaging. Derivation of (6.243) involves a calculation of the perturbative correction to the Green's functions to linear order in A using Gorkov's equations. The result is then used to calculate the supercurrent j [155]. Eq. (6.243) is valid for arbitrary Fermi surface and gap function. For isotropic gap one recovers an expression derived by Kogan et al. from the Eilenberger theory [156]. One may simplify solving the London equation again by writing it in terms of magnetic field only. Eliminating j from (6.242) using the Ampere's law j — (c/47r)V x B, one obtains Bk-k x [Q-\k){kx Bk)] = 0. (6.244) To study vortex lattice using this formalism, it is necessary to insert a cutoff function F(k) on the right-hand side of the Eq. (6.244). In our numerical calculations in the next section we again use Gaussian cutoff function (5.168). For many purposes it is also convenient to write down the corresponding London free energy, such that SJ^L/SBk = 0 gives the above London equation: J~L = Y $ t + (kx Bk)Q-\k)(k x Bk)]/8ir. . (6.245) Chapter 6. Microscopic Model 101 At long wavelengths Q(k) can be evaluated by expanding expression (6.243) in powers of 7jg. The zeroth order term clearly recovers the ordinary London free energy. Here, A = A(T) is the temperature dependent penetration depth which, at low temperatures, has the well known T-linear behavior [157, 33]; for a dx2_y2 superconductor with A p = A d ( p 2 — p2) and a cylindrical Fermi surface (from now on we shall focus on this simple case) A - 2 « AQ 2 (1 — 21n2T/Ad). The leading nonlocal term is quadratic in k: ( 2 ) _ 4 T T T / AjvFlvFj \ Q L 3 A 2 B ^ o \ ( ^ + A j ) » / 2 7 * / - [ b - M 7 ) The expression Qij = 8ijX~2 + Cj\f is easily inverted to leading order in k to get Q ^ 1 « X2[Sij - X2Q{f]. Substituting this into (6.245) and specializing to fields along the z-direction we have ? L = £ + X2k2 + A 2 e 2 ( C l f c 4 + c2k2xk2y)}/8n. (6.248) k Here £ = vF/7tAd and A^ is assumed to be a temperature dependent solution to the appropriate gap equation. Dimensionless coefficients C\ and c2 are given by *=5^£sf"R^- ( 6 ' 2 4 9 ) where wx = v2Fxv2Fy, w2 = (t)^ - t) 2^) 2 - 4 ? } ^ ^ and the Fermi surface has been explicitly parameterized by the angle 9 between p and x-axis; vF = (cos 9, sin 9) and A p = Ad cos 29. Coefficients ci and c2 depend on temperature through a dimensionless parameter t = T / A d . From (6.249) one can deduce their leading behavior in the two limiting cases: for t <C l we find C i = y f i ' C 2 = _ 4 c i ' ( 6 - 2 5 0 ) Chapter 6. Microscopic Model 102 and for t ^ > 1 (i.e., near Tc) A 2 1 c 2 = 8ci (6.251) where a = C ( 5 ) ( l - 2~5)/87r2 = 0.01272. In the above A also depends on t, but this will be unimportant for the following qualitative discussion. The free energy (6.248) formally coincides with (5.188), deduced previously in the last chapter with phenomenological considerations. The present model however, allows us to evaluate the coefficient of the symmetry breaking term c 2 (or e in (5.188)). At high temperatures, c 2 is found to be positive; in agreement with our previous result. As we showed in the last chapter, such a term leads to a centered rectangular vortex lattice structure with principal axes oriented along x or y axes of the ionic lattice (see Fig. 6.14a). The magnitude of distortion from a perfect triangular lattice is controlled by the magnitude of c 2 and grows with increasing magnetic field. Eq. (6.251) shows that at fixed field this distortion will initially grow with decreasing temperature. At low temperatures (6.250) predicts c 2 < 0. This will lead to the same centered rectangular lattice but rotated by 45° (Fig. 6.14b). Numerical evaluation of (6.249) shows that c 2 passes through zero at t* ~ 0.19. At this temperature the free energy (6.248) is isotropic and the lattice will be triangular at all fields. The change of sign of c 2 reflects the competition between the two terms of different symmetry in w 2 . At t <C 1 only the region around the node is important where (vFx — vFy)2 vanishes, while at £ 3> 1 the average involves the entire Fermi surface to which both terms in iy 2 contribute. This change of sign is a unique consequence of nodes in the gap function and would not occur in conventional superconductors. Another consequence of nodes is the fact that, as can be seen from (6.250), both c\ and c 2 diverge as 1/t for t —> 0. This divergence signals that the response tensor Q(k) is a non-analytic function of k at T = 0, and the expansion in powers of 7 | breaks down. Chapter 6. Microscopic Model 103 Figure 6.14: Two high symmetry orientations of centered rectangular unit cell. Formally this is caused by the fact that at T = 0, at the nodal point, the expression (6.243) for Qij(k) contains a term proportional to 1/7|- At T = 0 the frequency sum in (6.243) becomes an integral which can be evaluated exactly with the result: ^ /r\ 1 / - ~ 2arcsinh , y\ ,„ Q v { k ) = - ^ V F i V F j - ^ . y (6.252) where y = For small k the dominant contribution to the angular average comes from the close vicinity of nodes and can be evaluated by linearizing Ap around the nodes. One finds that the leading nonlocal contribution is linear in k rather than quadratic. For Qij = S^XQ2 4- Q\f, we have Qx1} = Q^J = -/j.(kM€o) and — Qylx = -At(^m£o)sgn(A;a:fcy), where kM = max(|fcx|, \ky\) and km = min(|A;x|, \ky\). Prefac-tor n = 7 r 2 /8 \ /2 = 0.8723 is exact in the sense that all corrections to are 0(k2). The resulting free energy at T = 0 is ^ = E + X2k2 + iiX2SkM{k2M - k2J]/87r. (6.253) k The nonlocal term is clearly non-analytic in k. Its functional form is universal in the sense that it is independent of the Fermi surface structure (as long as it has the same symmetry as the order parameter) and the prefactor \i only depends on the angular slope of the gap function and Fermi velocity at the node. Chapter 6. Microscopic Model 104 The present calculation can be easily generalized to treat the effects of Fermi surface anisotropy. As mentioned above tetragonal anisotropy will not modify the T —> 0 univer-sal behavior but may lead to quantitative changes at higher temperatures. Orthorhombic anisotropy, on the other hand, will modify even the T —>• 0 limit. We expect that it will , to leading order, merely rescale the coordinate axes, leading to the same structures as described above stretched by the axes, leading to the same structures as described above stretched by the appropriate factor [140]. It may further remove the degeneracy between two equivalent lattices related by 90° rotation. Also neglected in our calculation is the effect of electronic disorder, which will remove the non-analyticity of Q(k) at longest wavelengths, just as small finite temperature would. Since the lattice structure is most sensitive to Q(k) at finite k ~ l/l (I is the vortex spacing), we expect our predictions to be robust with respect to weak disorder. 6.2.3 Combining Nonlinear and Nonlocal Corrections Having established theories for nonlinear and nonlocal effects independently, we now would like to combine them in one London equation to be used in numerical calculations. Nonlocal effect can be added to the generalized London equation (6.241), simply by replacing the second term with its nonlocal counterpart Bj; + Cij{k)kikjB-k-Ga,(k,B~k) = BF(k). (6.254) —* Sums over i and j are implicit here. Cij(k) is related to the electromagnetic response tensor Q(k) by . Cij{k) = Qij(k)/detQ{k). (6.255) The electromagnetic response tensor Q(k) is given by (6.243) for T ^ 0, and by (6.252) a t T = 0. Chapter 6. Microscopic Model 105 In the local case, we have Cij(k) = A 2 , ^ which leads back to (6.241). Taking Gn\ to the right hand side of (6.254), can be obtained by flF(3+ft,(£,Bri . (6.266)-1 ~\~ )C*ij (&) k%kj We will use this equation, in the next section, to find Bk iteratively for a specific lattice geometry. Having B^, the free energy can be easily calculated using •T7 = .Fni + £ [ 1 + dj^kik^Bl (6.257) where Tn\ is the free energy due to the nonlinear term in Eq. (6.239) = 0 ^ Jd\ (|cV/3|3 + \dy,B?) (6.258) 6.3 N u m e r i c a l Calculat ions Unlike Ginsburg-Landau, the London free energy cannot completely determine the vortex lattice by a simple minimization. Instead, one has to impose a set of source terms located at the position of the vortices on an assumed lattice. The functional form of the source terms does not come from the London theory and requires more fundamental treatments; as we discussed in Sec. (5.2.2). Information about the functional form of the source term is included in (6.256) via F(k), and information about the positions of the vortices is included in the reciprocal lattice vectors k. The free energy must then be minimized with respect to the positions of the vortices in order to find the equilibrium lattice configuration. In general, a 2D lattice can be determined by four parameters. However, it turns out that a centered rectangular lattice is energetically more favorable than an oblique lattice. On the other hand, the vortex lattice spacing is fixed by the average magnetic field B (B & H away from Hci). Thus we are left with two variational parameters, i.e. the lattice orientation with respect to a and b directions and the apex Chapter 6. Microscopic Model 106 angle /5 - the angle between the two basic vectors of the lattice. We therefore find the vortex lattice geometry by minimizing T in with respect to the apex angle /3 for different orientations of the lattice. In our numerical calculation, we first neglect the nonlinear effects and consider only the nonlocal corrections. At the end, we will add the effect of the nonlinear corrections to the results. We shall see that the nonlinear corrections are subdominant to the nonlocal ones. Therefore the results we obtain, ignoring the nonlinear effects, are actually very close to the reality. In our calculations, we first find Bk using (6.256) but ignoring Gn\ (i.e. ignoring the nonlinear corrections). Substituting (6.256) into (6.257), and also ignoring we can calculate the free energy. At T = 0, we can alternatively use (6.253) to calculate the free energy. Numerical evaluation, shows that the free energy 70.0 65.0 P 60.0 Figure 6.15: Equilibrium angle /? as a function of reduced temperature t = T/Aj for various fields. Open symbols mark lattice with orientation along x or y direction while solid symbols mark the lattice rotated by 45°. We use A 0 = 1400Aand K = 68. Inset: (3 as a function of field at fixed T. (6.253) gives rise to a centered rectangular vortex lattice, aligned with a; or y axes, but Chapter 6. Microscopic Model 107 now with the apex angle pi < 60°; unlike (5.188) which gives a pi > 60°. In order to. map out the complete equilibrium H-T phase diagram we have carried out a numerical computation of the vortex lattice structure using the full expression for the response tensor Q(k), as given by Eqs. (6.243) and (6.252). We find that the free energy has two local minima for centered rectangular lattices aligned with two high symmetry directions (see Fig. 6.14), as expected from the tetragonal symmetry of the problem. Which of the two becomes the global minimum depends on temperature and field. The results are summarized in Fig. 6.15. For high temperatures, the exact result agrees well with the one obtained from the long wavelength free energy (6.248). The deformation of the lattice from perfect triangular, grows with decreasing temperature, reaches maximum, and then falls. Maximum distortion occurs around t ~ 0.3, attaining pi ~ 70° at 10T. Extrapolating this field dependence (see inset to Fig. 6.15), the lattice should become square lattice around H « 30T, but this field is outside the domain of validity of the London model. At lower temperatures the distortion decreases but instead of going all the way back to triangular at t*, the lattice undergoes a first order phase transition to another centered rectangular lattice rotated by 45° and with j3 < 60°. Further decrease of temperature causes the angle to grow again. We note that the precise temperature at which it crosses 60° depends on field, but for all fields it is close to t* = 0.19, as predicted by the long wavelength approximation. At yet lower temperature we predict another first order transition to a centered rectangular lattice along x (or y) with pi < 60°. It has to be noted that the free energy difference between the two minima is very small in the region where the 45° rotated lattice wins. It is thus likely that real system, in which vortex pinning to various defects occurs, will prefer to remain in the metastable state at these intermediate temperatures. Instead of two consecutive first order transitions the experiment would detect only a smooth crossover from a lattice with pi > 60° to the one with pi < 60°. Alternatively domains with various orientations may develop. Chapter 6. Microscopic Model 108 60.0 58.0 h P 54.0 [ 56.0 h 52.0 V 50.0 10.0 0.0 2.0 4.0 6.0 8.0 B[T] Figure 6.16: Apex angle (3 as a function of magnetic field B at T = 0. Circles represent the result of the calculation using only the nonlocal corrections. Squares correspond to. the calculations considering both nonlinear and nonlocal corrections. At this point, we can add the nonlinear corrections to our numerical calculations. The general strategy is to use (6.256) to calculate iteratively. At each step Gn\ has to be calculated numerically. The non-analytic form of G n i , prevents us from using usual convolution integrals in our Fourier transformation. Instead, we first calculate Gn\ on a lattice in a unit cell of the vortex lattice in position space. Then we use Fast Fourier Transformation (FFT) to calculate the Fourier transformed Gn\ at the reciprocal lattice wave vectors k. In our numerical calculation, we use A 0 = 1400A, K = Ao/£o = 68 as before and also 7 = 2. At T = 0, the stable orientation of the lattice is again aligned with a and b axes. Fig. 6.16 shows the results of our numerical calculations for (3 as a function of magnetic field. The upper curve (squares) is the result of combined calculation, considering both nonlocal and nonlinear corrections, i.e. using (6.256) and (6.257). The lower curve (circles) on the other hand, corresponds to taking into account only the nonlocal corrections (this curve is the same as the t = 0 curve in the inset of Fig. 6.15). As is clear from Fig. 6.16, the Chapter 6. Microscopic Model 109 1.5 1.4 1.0 0.0 2.0 4.0 6.0 8.0 10.0 B[T] Figure 6.17: The effective penetration depth as a function of the magnetic field. Circles represent the result of the calculations considering only the nonlocal effects whereas squares are the result of combined calculation considering both nonlocal and nonlinear effects. difference between the two cases is small and about one or two degrees. Therefore, the phase diagram given in Fig. 6.15 retains its validity qualitatively, even after adding the nonlinear correction to the generalized London free energy. We also calculate the effective penetration depth Aeff for different magnetic fields in almost the same way as it is calculated from / J S R data [125]. In these experiments, the / / S R precession signal obtained from experiment is fit to a signal obtained by Fourier transforming a theoretical magnetic field distribution function n(B) defined by with £1 denoting the area of a unit cell. The magnetic field B(r) in (6.259) is calculated on a hexagonal vortex lattice with the same average magnetic field as the experimental field and using the ordinary London model with some cutoff function. The A which provides the best fit to data is considered as Aeff. Here, we calculate Aeff in a different way (but similar in spirit), using the fact that in the ordinary London model, for a general vortex (6.259) Chapter 6. Microscopic Model 110 lattice and for a large enough field, (B- B)2 = AB2 = B2 (1 + X2k2)2 r (0A - 4 (6.260) were T(£) is a function that depends on the structure of the vortex lattice. Associating all the field dependence of AB2 in our calculation with the field dependence of an effective penetration depth Aeff, we can define Aeff by where ABQ is the mean squared value of the magnetic field (Bo(r) — B) evaluated using with the penetration depth A 0 . Fig. 6.17 shows the result of our numerical calculation for Aeff using (6.261). The lower curve corresponds to the calculations including only the nonlocal correction. The upper curve corresponds to the result of the calculations using both nonlinear and nonlocal terms. The effect of the nonlinear term to the field dependence of Aeff is almost nothing but an overall shift. Fig. 6.18 exhibits magnetic field distribution n(B) at the average magnetic field B — 5.9T. The solid line in Fig. 6.18 represents the magnetic field distribution calculated from the nonlinear-nonlocal London equation. The double peak feature is a sign of existing two different saddle points in a unit cell of the lattice which is a result of having a pi ^ 60°. This line-shape is then compared with another line-shape (dashed line) obtained from an ordinary London calculation but with a larger value of A 0 . Similar line shapes with some additional broadening can also be produced from p$R data. The broadening is due to lattice disorder, interaction of muons with nuclear dipolar fields and finite lifetime of muons. The resolution of the magnetic field as a result of this broadening is 8B ~ 10 _ 3 T. The difference between the solid line and dashed line in Fig. 6.18 as well as the double peak feature of the solid line is therefore not observable by (6.261) the ordinary London model on a hexagonal lattice with the same average field B and Chapter 6. Microscopic Model 111 0.08 E9- 0.04 0.06 0.00 0.02 -0.001 0.000 0.001 AB/B Figure 6.18: Solid line: Magnetic field distribution obtained from the nonlinear-nonlocal London equation with B — 5.9T and A 0 = 1400 A. Dot-dashed line: field distribution obtained from an ordinary London equation on a hexagonal lattice with the same B and A 0 . Dashed line: Result of the same ordinary London calculation but with A 0 = 1850 //SR experiments because of these broadening effects. Thus as far as these line-shapes are concerned, it is difficult to distinguish a nonlinear-nonlocal effect from a simple shift in the magnetic penetration depth in the ordinary London model. Fig. 6.19 compares the effect of including both nonlinear and nonlocal corrections to the London equation with the effect of including only the nonlocal term. Comparing the two line-shapes, it is apparent that the effect of the nonlinear term is small compared to the nonlocal term as was emphasized before. 6.4 Conclusion Ordinary London theory is not adequate to describe all the different properties of a vortex lattice in high T c superconductors; especially the properties resulting from the presence of the superconducting gap nodes or other anisotropies on the Fermi surface. However, A. Chapter 6. Microscopic Model 112 0.08 £ . 0.04 0.02 0.06 0.00. -0.001 0.000 0.001 0.002 AB/B Figure 6.19: Solid line: Magnetic field distribution obtained from the nonlinear-nonlocal London equation. Dashed line: Magnetic field distribution obtained from a London equation including only the additional nonlocal term using the same parameters. as we established throughout this thesis, a generalized London model with appropriate higher order corrections which take into account these anisotropic effects, can still provide a fairly simple way to calculate different properties of a vortex lattice. The corrections we found to the London equation were classified into nonlocal and nonlinear ones. In both analytic and non-analytic (low T) cases, nonlocal corrections play the dominant role in determining the vortex lattice properties. Equilibrium vortex lattice geometry exhibits novel field and temperature dependence owing to the fourfold anisotropic effects expressed by the corrections to the London equation (Fig. 5.11, Fig. 6.15 and Fig. 6.16). Numerical calculation of the lattice geometry is rather insensitive to the details of the vortex cores. The reason is that the details of the magnetic field inside the vortex cores mainly affect the magnetic self-energies of the vortices. In magnetic fields far below HC2 on the other hand, the vortex lattice geometry is mostly determined by the magnetic interaction energy between vortices which is not sensitive to the precise shape of the core. Therefore, our replacement of the vortex Chapter 6. Microscopic Model 113 core by a simple Gaussian source term should be adequate for the vortex lattice structure calculations. The effective penetration depth A es also exhibits field dependence at low temperatures as illustrated in Fig. 6.17. Some important points need to be emphasized here: (i) The field dependence of Aeff is not linear as is evident from Fig. 6.17. Variation of Aeff with magnetic field is faster at lower fields. At low magnetic fields, the relative variation of the effective penetration depth in our calculation is about 7% for an increase of IT in the magnetic field which is close to 7.3% variation obtained from //SR data for an optimally doped and 9.5% variation for a detwinned underdoped Y B C O single crystal using the same cutoff function as (5.168) for fitting calculations [125, 127]. Most recent //SR results [158] extrapolated to T = 0, has provided a measurement of Aeff as a function of magnetic field up to 7.5T. A n excellent agreement between this result and our nonlinear-nonlocal theory is presented in Fig. 6.20. (ii) More importantly, this field dependence of Aefr has a predominantly nonlocal origin rather than a nonlinear one; contrary to what is generally believed. The contribution of the nonlinear term to the total (minimized) free energy is almost one order of magnitude smaller than the nonlocal term. What is more important however, is the field dependence and (3 dependence of these terms not their orders of magnitude at fixed J3 and B. As we mentioned earlier, we consider /5 as a variational parameter to be fixed by minimizing the London free energy. As can be inferred from Fig. 6.16 and Fig. 6.17, the field dependence and /? dependence of the nonlinear term in the free energy is also smaller than the nonlocal term. It is worth noting that in the Meissner state, a linear nonlocal term can never produce field dependence in the penetration depth (as it is usually defined in that state) and therefore a nonlinear term is necessary for such an effect [104]. In the vortex state on the other hand, a nonlocal term can result in a field dependent effective penetration depth, if we define it the way it is defined in //SR experiments. We should Chapter 6. Microscopic Model 114 Figure 6.20: Comparison between our nonlinear-nonlocal theory and a recent pSR result at T — 0. Squares represent our calculation of the effective penetration depth using nonlinear-nonlocal model (cf. Fig. 6.17). Circles correspond to the recent //SR data [158]. Our A 0 (= 1400 A) is a fitting parameter which might be different from the measured value. emphasize here that in this way of definition, does not directly give the superfluid density as it does in the Meissner state.. To understand this better, let's neglect the nonlinear term and assume a linear but nonlocal London equation. The total magnetic field, in this case, will be the superposition of the fields around individual vortices. The magnetic field around an isolated vortex is given by d2k F(k)eiR-f B « = *»/(2rfr , N O - " (6.262) (27r)2l + / v i ( f c ) ^ where $o is the flux quantum and F(k) is the cutoff function resulting from the source term. £ij(k) is defined in (6.255). The field decays with some constant (field independent) decay rate A; which is the Meissner state penetration depth (see Fig. 6.21). Increasing the magnetic field does not affect the profile of the magnetic field and therefore A. Rather it squeezed the vortices together. For small values of k, £ij(k) « A2<%. Since small k Chapter 6. Microscopic Model 115 2 I . , . , , : , 0.00 0.05 0.10 0.15 0.20 Figure 6.21: Magnetic field as a function of the distance from the center of the vortex (a-direction) for an isolated vortex. The solid line corresponds to the nonlocal London equation whereas the dashed line represents an ordinary London calculation with the same value of A 0 . BQ = B(r — 0) for the ordinary London vortex. corresponds to large r, one expects an isotropic field, similar to the local London case, far away from the vortex core. For large values of k on the other hand, Cij(k) has strong ^-dependence with four-fold anisotropy. Thus the closer to the vortex core, the more deviation form an isotropic ordinary London single vortex is expected, as is clearly shown in Fig. 6.21. At low magnetic fields, the vortices are far apart and their magnetic fields overlap in regions far away from their cores. The properties of the vortex lattice should then be similar to the ordinary london hexagonal lattice. As the magnetic field is increased, the vortices come closer to each other. Although the profile of the magnetic field around each vortex remains unchanged, the overlap regions will be closer to the vortex cores and will be more affected by the nonlocal term. Therefore, it is conceivable that at large magnetic fields, the vortex lattice properties such as the magnetic field distribution will be affected by the nonlocal term in the London equation. The magnetic field near the vortex core is always reduced by the nonlocal term as is clear in Fig. 6.21. Chapter 6. Microscopic Model 116 This is because £ij(k) — A 2 ,^- is a positive definite tensor for all k and therefore the denominator of the integrand in (6.262) is always larger than the corresponding ordinary London one. As a result, the magnitude of AB2 is smaller for the nonlocal case and therefore Aeff defined by (6.261) tends to be larger. This explains why A eff/A 0 is always greater than one in Fig. 6.17. Fig. 6.18 exhibits the resemblance between a change in the magnetic field distribution due to the nonlocal term and due to a shift in the ordinary London penetration depth. The slight difference between the solid and dashed lines in Fig. 6.18 would be unobservable in //SR experiments as a result of the broadening effects. Since no field dependence due to the nonlocal term is expected in the Meissner state, Aeff as defined here and also in //SR experiments is expected to be conceptually different from what is usually defined as penetration depth in the Meissner state, although they are closely related. (iii) The magnitude and field dependence of A eg is not so sensitive to the apex angle 0. In other words, a few degrees change in the variational parameter /3 does not modify the magnetic field distribution as much as a variation in the average magnetic field does. (iv) Calculation of Aeff is rather sensitive to the form of the vortex source term. The importance of the source term in the calculation of AB2 has already been emphasized by Yaouanc et a/.[159]. In Ref.[125], Aeff is obtained by fitting to the //SR data using both a Gaussian cutoff ((5.168)) and also the cutoff function proposed by Hao et a/.[160, 159]. The difference between the two cases is significant and about 30% for the magnitude of Aeff and even more (for detwinned sample) for the relative variation with respect to the magnetic field. This can explain the importance of the source term in calculations of the effective penetration depth. //SR experiments [125, 128] on NbSe 2, which is believed to be a conventional super-conductor, also show a field dependence in the effective penetration depth; although it Chapter 6. Microscopic Model 117 is much weaker than what is observed in high Tc compounds. Since there is no node in the superconducting gap of these materials, the theory presented in this thesis cannot explain this field dependence. However, since the size of the vortex core in these ma-terials is large and comparable to the vortex lattice spacing for the magnetic fields of experimental interest, it is conceivable that a significant effect can come from the cores as is pointed out in Ref. [159]. 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Vertices and vortices in high Tc superconductors
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