TOPICS IN CUPRATE SUPERCONDUCTORS LIH YIR SHIEH

Transcription

1 c Copyright by Lih Yir Shieh, 1998

2 TOPICS IN CUPRATE SUPERCONDUCTORS BY LIH YIR SHIEH B. S., National Tsinghua University, 1981 M. S., National Tsinghua University, 1983 DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 1998 Urbana, Illinois

3 TOPICS IN CUPRATE SUPERCONDUCTORS Lih-Yir Shieh, Ph.D. Department of Physics University of Illinois at Urbana-Champaign, 1998 Anthony J. Leggett, Advisor In this thesis, we focus on two specific problem in the high temperature cuprate superconductors. One is the tunneling behavior along the c-axis in the normal state. We use two-layer system coupling with fluctuation to study this problem. The other on the crossover behavior from p-wave Copper pairing to Boson condensation in 2D case as the attractive interaction increases. First we discuss the tunneling behavior along the c-axis in the case of static impurities on the layers. We found out the symmetric impurities play no role in the damping of the coherent tunneling. However the anti-symmetric impurities gradually destroy the interference of the consecutive tunneling wave function. The tunneling behavior changes from coherent tunneling to incoherently tunneling smoothly as the density of impurities increases. Later we discuss the three possible sources of dynamic fluctuation in the high temperature cuprate superconductors: the phonons, the plasmons and the spin fluctuation. The explicit form the spectral function of those three fluctuations are obtained. And we reduced them into two-layer system for the purpose of studying c-axis tunneling in the future. We integrate out the q variable to give the spectral function of ω variable to investigate the low temperature properties. We also use the single parameter, the fluctuation at the same point and at the same time, to compare the strengths of those three fluctuations. We also study the p-wave pairing in 2D case. The crossover from Cooper pairing to Boson condensation is a weak transition at the chemical potential µ = 0. If the gap function is angledependent, the energy gap of quasiparticle change from angle dependence to angel independence as the chemical potential µ changes from above zero to below zero. iii

4 T o my parents. iv

5 Acknowledgments First of all I have to thank for my advisor, Professor A. J. Leggett, for being patient for so many year to lead me to do the research of the topics of superconductivity. Without his constant encouragement, I can not reach this point to present my thesis. I thank Mohit Randeria and Jimin Duan for the cooperation together that I have the first paper published. I also want to thank my colleague Ping Ao. The cooperation with him and Jimin Duan help me to understand physics much deeper. He also help me to glimpse what our great master Tony was thinking about. I thank all the colleague in Tony s group that we can discuss physics and encourage each other, especially Rachel Wortis, Ioan Kosztin and Michael Turlakov. Special thank to my officemate Thao Tran for constantly to be there when I need him. In my private life, I thank all the brothers and sisters in the Illini Chinese Christian Fellowship to help me recover from a personal crisis and gradually become a normal person. Especially I thank Juifen Huang to lead me to Christ and help me grow. I have to thank those brother Jen-Yuen Ho, Eddie Tsai, Norman Chen, Kuo-Chen Fu and Song-Wu Lu and sister Hei-Fen Wang, Shin-Shin Lou, Li-Ru Wu, Gek-Woo Tan, Lily Lau and Kai-Li Wu to be there, to listen my complaint and to give me advises. Also I thank Mike McQueen and his family to accept me to be their housemate. There I enjoy a family life. I definitely have to thank my father and my mother for their unconditional love. No matter how worst I am they accept me completely. My elder sister Dai-Yi and Younger brother Shin-Yi support me all the time. I give this honor to my God. For He gives me the strength to continue and stop me from giving up. Also He help me to grow and to be a normal person even under the pressure. I can only say Thank You. This work presented here was partially supported by National Science Foundation under Grant v

6 DMR Additional support was provided by a teaching assistantship from the Department of Physics at the University of Illinois at Urbana-Champaign. vi

7 Table of Contents List of Tables ix List of Figures x Chapter 1 Introduction D P-wave pairing C-axis resistivity The crystal structure Estimation The two-layer model Chapter 2 Static Impurity Case Introduction Hamiltonian Probability of finding the extra particle Underdamped Case The Hamiltonian Probability The equations of motion for Green s functions Approximation for two-particle Green s function The calculation of In(1; 1, k : ω) The overdamped case One-particle Green s function Probability The calculation of two-particle Green s function The final result The impurities between the layers Hamiltonian One-particle Green s functions Probability The final result Chapter 3 Sources of Fluctuation Introduction Electron-phonon interaction Deformation potential Reducing into two-layer system The plasma modes in multilayer system vii

8 3.3.1 The response function Equations of motion Reducing to two-layer system The spectral function of voltage fluctuation Spin fluctuation Two-layer model Numerical results Electron-phonon interaction Plasma modes in 2D case The spin fluctuation Comparison Chapter 4 P-Wave Pairing in 2D Case Introduction Model and calculation Interaction in two-dimensional case The pairing ansatz The P-wave pairing Conclusion: P-wave Chapter 5 Conclusion Appendix A The Overdamped Case A.1 Some remarks A.2 One-particle Green s function A.3 Alternative method: spectral Function A.4 Two-particle Green s function Appendix B Interlayer case Appendix C Plasma modes C.1 In multilayer system C.2 Fourier transformation w.r.t. z C.3 The response function C.4 Reducing to two-layered system Appendix D P-Wave Pairing in 2D Case References Author s Biography viii

9 List of Tables 3.1 The comparison of the strengths of three dynamic fluctuations ix

10 List of Figures 1.1 The tetragonal(left) and orthorhombic (right) forms of YBa 2 Cu 3 O 7 [5] The phase diagram of YBa 2 Cu 3 O 6+δ stoichiometry; AF = antiferromagnet, SC = superconducting[6] The resistivity of YBa 2 Cu 3 O 7 δ [7] The spectrum function of phonon fluctuation as a function of variable ω with with q = 0.1π/a The spectrum function of potential fluctuation as a function of variable ω The spectrum function of spin fluctuation as a function of variable ω x

11 Chapter 1 Introduction Since the discovery of La 2 x Sr x CuO 4 [1], the first of high temperature cuprate superconductors, scientists all over the world have tried to find new cuprate superconductors to push the transition temperature of superconductors T c to be higher. Up to now several classes of cuprate superconductors have been discovered[2]. All possess quite a few similar properties. Also scientists have made systematic efforts, experimentally and theoretically, to understand the origin of the interaction that cause the superconducting transition and numerous other important physical properties[3]. Here we present our studies about cuprate superconductors. In this thesis we study two properties: First is the 2D p-wave pairing, and second the study of nature of the c-axis resistivity. motivation of 2D p-wave pairing. In the following section I will give brief introduction of the In the next few sections we give a lengthy discussion on the motivation for the study of c-axis resistivity, the naive experimental and theoretical implications, and the model we use to study the c-axis resistivity. It is still on going research and we will present the results we have up to now in this thesis D P-wave pairing Our early work focused on the 2D pairing of superconductor. We have done the work on s-wave pairing and p-wave pairing from Cooper pair, where the attractive interaction is weak, to Boson condensation, where the attractive interaction is extremely strong[4]. My own work mainly focus on p-wave pairing. The motivation of this work is based on: (1) all the cuprate superconductors are layered structure and several experiments indicate a 2D properties; (2) the coherent length of the Cooper pair is small, around a few lattice constant. Then there are few electrons within 1

12 the distance of the radius of Cooper pair. So it is worthwhile to investigate how the Cooper pair changes as the attractive interaction increases in strength. The detail of the calculation, results and the discussion can be found in Chap. 4 and the paper attached in the appendix D. 1.2 C-axis resistivity We want to study the nature of c-axis resistivity. For simplicity we focus on YBa 2 Cu 3 O 7 δ. In the first subsection, we review the crystal structure of YBa 2 Cu 3 O 7 δ and some of its general properties. In the latter section we review some of the experimental data concerning the c-axis transport properties and make a simple estimation of a few physical quantities, which would give us the hint of the underlying physics of c-axis transport properties. In the last subsection we present a simple model that we can use to study c-axis transport properties. The outline of our research in this topic is also listed in the subsection The crystal structure The crystal structure of YBa 2 Cu 3 O 7 is shown in fig.(1.2.1). As shown in the fig.(1.2.1), there are two CuO 2 planes, separated about by 3.1 Å, and one CuO chain in a unit cell. The lattice constants along the a-axis are 3.8 Å, along b-axis 3.8 Å, and along c-axis Å. As one can see the lattice constant along the c-axis is much longer than the other two lattice constants. The oxygen deficiency changes the properties of YBa 2 Cu 3 O 7 δ dramatically[6]. For δ = 0.1 YBCO is a superconductor with T c = 92 K, the highest for this material. As the oxygen vacancies increase to the case δ , the superconducting temperature T c drops to 60 K at this region. Finally the superconducting state of YBa 2 Cu 3 O 7 δ vanishes around δ 0.6. With increase of the oxygen deficiency, i.e. δ 0.6, YBa 2 Cu 3 O 7 δ becomes anti-ferromagnetic state and it is a insulator. The transition temperature of the anti-ferromagnetism T N increases as the δ increases from 0.6. The phase diagram for YBa 2 Cu 3 O 7 δ stoichiometry is shown in Fig.(1.2.1). Because of this unique feature that anti-ferromagnetic state is right next to the superconducting state, many physicists think that the fluctuation of antiferromagnetic spin wave might be the source of the interaction that leads to superconducting state of YBa 2 Cu 3 O 7 δ. Many experiments support this argument. 2

13 Figure 1.1: The tetragonal(left) and orthorhombic (right) forms of YBa 2 Cu 3 O 7 [5] Estimation One of the striking features about high temperature cuprate superconductors is the resistivity along the c-axis. Earlier data[7], shown in fig.(1.2.2), shows that resistivity along the c-axis is a semiconductor like behavior, i.e. at the low temperature the resistivity increases as the temperature decreases. More recently the best data[8] however shows that the resistivity along the c-axis is linear in temperature. Nevertheless all data shows a large anisotropy between the resistivity along the c-axis and the one along the ab-plane. In YBa 2 Cu 3 O 7 δ case, the ratio of the anisotropy of the resistivity is around 100. First let us make a rough estimation of some physical quantities. We assume the Free electron gas model with the anisotropic effective mass for simplicity 1. We use Drude model[10] to do the estimation. 1 From band structure calculation and confirmed by photoemission experiment[9], the electron in YBa 2 Cu 3 O 7 is hole-like with around half filled of the reciprocal Brillouin zone. 3

14 Figure 1.2: The phase diagram of YBa 2 Cu 3 O 6+δ stoichiometry; AF = antiferromagnet, SC = superconducting[6] ρ = m ne 2 τ (1.1) We assume the electron density n is around 0.5 electron/unit cell and the electron mass along ab-plane m to be 1m e. From the available data, ρ ab is around µωcm. Then we obtain that h/τ ab is roughly 10 mev. Fitting into the result of tight-binding approximation by using the theoretical band structure calculation the coupling along ab plane t a must be approximately 0.35 ev, and the coupling along c-axis t c must be approximately 0.03 ev[11]. One might say along the ab-plane the Bloch-wave description is still valid. However along the c-axis the band width is at the same order of the relaxation rate. The Bloch-wave picture might not valid alone c-axis. Let us calculate another physical quantity, the mean free path along the c-axis l c. We know the relaxation time along the c-axis can be written as τ c = l c /v F c and the effective mass m c = p F c /v F c. 4

15 Figure 1.3: The resistivity of YBa 2 Cu 3 O 7 δ [7] So the eqn.(1.1), for the case along the c-axis, can be written as ρ c = p F c ne 2 l c (1.2) Now from the best experimental data the resistivity along c-axis is around Ωcm and Fermi momentum along c-axis p F c is always less than the reciprocal lattice constant along c-axis π/c. Then the mean free path along the c-axis l c is less than 1.2 Å, which is much less than the lattice constant along the c-axis c = Å. So this strongly implies that the Bloch-wave description fails along the c-axis. That gives us the motivation to study the tunneling behavior along the c-axis. 5

16 1.2.3 The two-layer model As indicated by the result of band structure calculation and estimation of relaxation time along the ab-plane and c-axis, the electrons on the ab-plane might be well described by the Bloch wave picture. However it fails along the c-axis. So we assume the electrons on the ab-plane is a 2D Fermi gas on the CuO 2 plane with a small tunneling matrix along the c-axis. The electrons can freely move along the ab-plane and tunnels between the layers. Since the tunneling matrix element is small, the electrons stay on 2D layer most of the time. We are inspired by idea of the tunneling behavior of two-well system, described by the spin- Boson Hamiltonian[13], that the out-of-phase on-site fluctuation might destroy the coherent tunneling behavior between the two wells. As one particle, initially in one of the wells, tunnels between the the wells, the consecutively tunneling waves should be added together to construct the final form of wavefunction. And the probability of the particle is the absolute square value of this final wavefunction. But as the fluctuation is introduced, each consecutively tunneling wave can be detuned by the fluctuation. The outcome is the gradual destruction of the interference between the consecutive tunneling waves. If the interference is completely destroyed, the probability of the particle is the sum of each absolute square value of the consecutive tunneling waves. This process is the so called phase detuning. The spectrum density of the fluctuation and the temperature are crucial in determining whether the tunneling of the particle in this two-well system is coherent or incoherent. We expand the idea of two-well system into two-layer system with the electron on the layer that can be describe as 2D free electron gas. The dynamic fluctuation on the layers couples with the electrons and we want to study how the dynamic fluctuation affect the tunneling behavior between the layers. The Hamiltonian of this two-layer system can be written as H = k,q;i=1,2 (a + 1,k a 2,k + a + 2,k a 1,k) (1.3) ɛ k a + i,k a i,k + ( 1 2 h ) k,i=1,2 k f i(q){a + i,k+q a i,k}(b i,q + b + i, q ) + hω(b + i,q b i,q ) q;i=1,2 a (+) i,k is the annihilation (creation) operator of the electrons and b(+) i,q the annihilation (creation) 6

17 operators of the bosonic fluctuation. The index i refers to the index of layers. In the next chapter first we study the simplest case, static impurity case. We learn how the tunneling behavior of the electron changes as the impurity density increases and what kind of impurities play an important role. This information provides us the insight about tunneling behavior in the dynamic fluctuation case. In the following chapter we will discuss three possible sources of dynamic fluctuations in multilayer system. Eventually we reduce the spectrum function of dynamic fluctuation in multilayer system into the one in two-layer system. Note that in all these calculation we deal with one electron in the system. We are still working on many-electron system and will present the result in the future. 7

18 Chapter 2 Static Impurity Case 2.1 Introduction Let us first study the tunneling behavior along the c-axis under the effects of static impurities on the layers. This case is the simplest case. It provide some information about the effects of dynamic fluctuations Hamiltonian Taking suggestion from experiments for cuprate superconductors, we study the layered structure with electrons on the layers. However, for simplicity, we study two-layer system instead of multilayer system since we are interested in the question of whether the tunneling behavior between the layers is coherent or incoherent. We assume that the electrons on the layers can be described as a free Fermi gas, i.e. there is no interaction between electrons and the effect of band structure is neglected here. Along the c-axis, the electrons can be described as tunneling with a tunneling matrix element, which is constant for any coordinate variable in the ab-plane. The impurities are randomly distributed on the layers and they act as the scattering centers for the electrons along the ab-plane. We focus on how the impurities on the layers affect the tunneling behavior along the c-axis. This is the simplest case and the result might shed some light on the dynamic fluctuation case. According to the description above, we can construct the Hamiltonian in the following way, H = H k + H T + H imp 8

19 = k ɛ k (a + 1k a 1k + a + 2k a 2k) + ( 1 2 h ) k (a + 1k a 2k + a + 2k a 1k) + k,q ρ 1 (q)v(q)a + 1k+q a 1k + k,q ρ 2 (q )v(q )a + 2k +q a 2k (2.1) The first term, H k, is the kinetic energy term. ɛ k is the kinetic energy of a free electron with momentum k. One might replace ɛ k with more complicated band structure to include the interaction between electrons and ions. But this is not the focus here. The indices 1, 2 refer to the layer number. T here are two layers in this model and k is the momentum along the ab-plane. The second term, H T describes the tunneling between the layers. Here we assume that the overlap of wavefunction along the c-axis is very small and the transport property of the electrons along the c-axis is better described by tunneling. Essentially it is the tight-binding approximation. Another assumption is that the tunneling matrix element is translational invariance along ab-plane. So the momentum along the ab-plane is conserved after the tunneling 1, which is expressed with the same k in H T. The third term, H imp, describes the scattering of electrons due to the existence to static impurities on the layers where ρ i (q) is the impurity density on the i-th layer and v(q) is the impurity potential. For simplicity, we assume the impurity potential to be point-like with a strength v 0 in the coordinate space. We will study how the impurities on the layer affect the tunneling behavior along the c-axis. Let us first give a simple description here. If there are no impurities on the layers, one can immediately obtain the Bloch-wave state as the eigenstates for the multilayer system, and in this special twolayer system they are even and odd states. In other words, it forms an energy band structure and the eigenfunction is a Bloch wave function. If the layers have been doped by high density of impurities, then the appropriate state to describe the state on the layer is localized eigenstate. And the tunneling behavior should be a decaying tunneling, described by Fermi golden rule. It is the character of incoherent tunneling. Here we use this Hamiltonian to study how this tunneling behavior changes quantitatively as we change the density of impurities. 1 This assumption is totally different from the case of impurities between the layers, where there is no momentum conservation for the tunneling mechanism introduced by the impurities between the layers. The latter case is discussed in Sec

20 2.1.2 Probability of finding the extra particle Before we go any further, we need to define an appropriate quantity which can reveal the nature of coherent and incoherent tunneling. As shown in the latter calculation, we have to choose appropriate zeroth-order Hamiltonian and designate the rest as first-order Hamiltonian in different regions. That leads to different representations and approximations in different regions. So we need to construct the same appropriate quantity that we may use in different regions. Taking the suggestion from the paper by Leggett et al.[13], we can formulate the problem in the following way. Assume there are electrons on each layer filling the energy level up to Fermi energy ɛ F. Now add one extra electron to layer 1 with state index n at time t and measure it later at layer 1 with state index m at time t, where m, n are indices of representation we choose in different region. The probability amplitude for this process is θ(t t ) G ã 1,m (t)ã 1,n (t ) G where ã ( ) i,m = eiht/ h â ( ) i,m e iht/ h in the Heisenberg representation. If we take one electron out from layer 1 with state index m at time t and put it back to layer 1 with state index n at time t, the probability amplitude is θ(t t) G ã 1,n (t )ã 1,m (t) G Actually, these two processed can be described by one Green s function defined as G(1, n, t : 1, m, t ) i [ã 1,m (t)ã 1,n (t )] (2.2) So the probability 2 of putting (taking) an electron to (from) layer 1 with state index n (m) to time t (t) and taking (putting) it out (back) from (into) layer 1 with state index m (n) to time t (t ) can be obtained. Actually what we want to measure is the probability of finding this extra electron (or hole) in layer 1 no matter what state it is in. So we have to sum over all the final states in layer 1. 2 One may use density matrix method to obtain the same result 10

21 G(1, m, t : 1, n, t ) 2 = m G(1, m, t : 1, n, t )G (1, m, t; 1, n, t ) m (2.3) where G is the hermitian conjugate of the one-particle Green s function G. However the probability defined in eqn.(2.3) is sample-dependent, since the exact form of initial state 1, n depends on the choice of the representation for exact configuration of the impurity distribution which is different from sample to sample. We need to redefine an appropriate probability which is independent of the choice of the representation of the system. We assume that we put an electron with energy between ɛ and ɛ + ɛ, where ɛ is very small, on layer 1. Those states within the energy range have equal opportunity at the initial states. Then the initial density of matrix should be defined as 3 ρ(t = 0) = n,ɛ ɛ n ɛ+ ɛ â 1n G G â 1n / n,ɛ ɛ n ɛ+ ɛ 1 The in indicates that we sum over only those states in layer 1. In the limit ɛ 0, the equation above becomes ρ(t = 0) = 1 â 1n G G â 1n δ(ɛ ɛ n ) (2.4) N 0 n where N 0 is the density of states in a two-dimensional case which is constant. Now we can use this density matrix to define the probability in eqn.(2.3). The probability defined in this way is independent of the configuration of impurities and therefore is sample-independent. In the latter stage, we will take ensemble average over impurities mention in Sec.(2.2.3) to do the further calculation. We now may rewrite the equation for the probability in the following way, P (1, 1; ɛ, t t ) = 1 G(1, m, t : 1, n, t )G (1, m, t; 1, n, t )δ(ɛ ɛ n ) (2.5) N 0 mn 3 Instead of the step function described here, one may use the following definition dɛ f(ɛ ) / dɛ f(ɛ ) where f(ɛ ) is a smooth distribution function with maximum center at energy ɛ and a narrow half-height width ɛ. It serves the same purpose. 11

22 This probability is independent of the choice of representation and we may use it in a one-particle system as well as a many-particle system. Later, all the Green s functions are taken Fourier transformation into frequency variable, so we will take the Fourier transformation for the probability P (1, 1; ɛ, t t ), P (1; 1; ɛ, ω) = mn dω 2π G(1, m; 1, n; ω )G (1, n; 1, m; ω ω)δ(ɛ ɛ ) (2.6) where we have used the following equation in eqn.(2.6) F{G (1, m, t; 1, n, t )} = G (1, n; 1, m; ω) (2.7) where F{ } represents the Fourier transformation. We have summed over all the transverse degrees of freedom in the final state since we are only interested in the probability of finding the electron on layer 1 at time t but not interested in the information on the transverse degrees of freedom. 2.2 Underdamped Case In this section we will discuss the tunneling behavior along the c-axis when the impurity density is not high enough to destroy coherent tunneling. We need to rewrite the Hamiltonian in order to choose the appropriate approximation in this region. In the next subsection we state briefly about the probability we need to calculate. The equations of one-particle Green s function, the assumption of ensemble average, and the approximation for two-particle Green s function are developed. In the last subsection we present the final result The Hamiltonian The Hamiltonian that describes the dynamics of a two-layer system with impurities on each layer can be found in eqn.(2.1). In the underdamped case we assume that the tunneling term is much large than the effect of the impurity term. So we take the kinetic energy term and the tunneling term as the zeroth-order Hamiltonian, and treat the impurity term as a perturbation. Then the 12

23 Hamiltonian in this case can be written as H = k (ɛ k 1 2 h )â e,kâe,k + k (ɛ k h )â o,kâo,k + ρ s (q)[â e,k+qâe,k + â o,k+qâo,k] k,q + ρ a (q)[â e,k+qâo,k + â o,k+qâe,k] (2.8) k,q where â ( ) (e,o),k = 1 2 (â ( ) 1,k ± â( ) 2,k ) ρ (s,a) = 1 2 (ρ 1 ± ρ 2 ) One immediately can see that without the anti-symmetrically distributed impurities, represented as ρ a, there is no channel to tunnel between the even and odd states. Then there exist no mechanism to destroy the coherent tunneling. As the anti-symmetrically distributed impurities are introduced, the interference between the even and odd states gradually destroy the coherent tunneling behavior. But here we first study the underdamped case, i.e. the impurities do not completely destroy the coherent tunneling behavior. Later we discuss the overdamped case where the impurities are strong enough to destroy the coherent tunneling behavior. Then we should take impurity term into the zeroth-ordered Hamiltonian instead of the tunneling term Probability The probability of finding an extra electron is shown in eqn.(2.5) and its form in Fourier transformation in eqn.(2.6). We just need to mention a few things: the ground state in this one-particle system is a vacuum, i.e. G = vacuum. The state index m and n in this underdamped case is the transverse momentum k and k. So the probability in the form of Fourier transformation in this underdamped case is 13

24 P (1; 1; ɛ, ω) = kk dω 2π G(1k ; 1k; ω )G (1k ; 1, k; ω ω) (2.9) So we need to develop the equations of motion for these one-particle Green s function and make assumption about ensemble average in order to calculate this probability of finding the electron at later time t The equations of motion for Green s functions We define the Green s function in the usual way: G(i, k, t; j, k, t ) = i T[ã i,k (t)ã j,k (t )] (2.10) where (i, j) is referred to (e, o) in this case. So there are four kinds of Green s functions according to the index of (e, o). With the equation of motion for the field operator, one can write down the equation of motion of one-particle Green s function as i h t G((e, o), k, t; j, k, t ) = (ɛ k 1 2 h )G((e, o), k, t; j, k, t ) + δ(t t )δ k,k δ (e,o),j + q + q ρ s (q)v 0 G((e, o), k q, t; j, k, t ) ρ a (q)v 0 G((o, e), k q, t; j, k, t ) (2.11) Note we put G(e) and G(o) together in this equation. For the case of no impurities on the layers, the equation of motion of the free-particle Green s functions become i h t G(0) ((e, o), k, t t ) = (ɛ k 1 2 h )G(0) ((e, o), k, t t ) + δ(t t ) (2.12) So the integral equation of the one-particle Green s function can be written as G((e, o), k, t; j, k, t ) = G (0) ((e, o), k, t t )δ k,k δ (e,o),j 14

25 + + dt q dt q G (0) ((e, o), k, t t )ρ s (q)v 0 G((e, o), k q, t ; j, k, t ) G (0) ((e, o), k, t t )ρ a (q)v 0 G((o, e), k q, t ; j, k, t (2.13) ) Let us take the Fourier transformation of this integral equation, and the equation becomes G((e, o), k; j, k ; ω) = G (0) ((e, o), k; ω)δ k,k δ (e,o),j + q + q G (0) ((e, o), k; ω)ρ s (q)v 0 G((e, o), k q; j, k ; ω) G (0) ((o, e), k; ω)ρ a (q)v 0 G((o, e), k q; j, k ; ω) (2.14) Then we may use eqns. (2.14) to expand one-particle Green s function G in series of small parameter ρ i (q)v 0. One may obtain the equation of motion for the complex conjugate of the one-particle Green s function G by applying the complex conjugate to the eqns. (2.14) which represent Green s functions backward in time. The detail is not shown here. Ensemble average According to the Hamiltonian defined in (2.8), the density of the symmetrically and the antisymmetrically distributed impurities are defined as ρ e,o (q) = 1 2 (ρ 1(q) ± ρ 2 (q)) (2.15) We assume 4 that the impurity potential is weak enough so that we may neglect the high order of ensemble average over impurity density. The first order of ensemble average over impurity is just a constant potential that we may set to zero[14], i.e. ρ i (q 1 ) = 0 ρ i (q 1 )ρ j (q 2 ) = n i δ i,j δ q1,q 2 (2.16) 4 We have assumed that the impurity potential is a point-like potential without any structure in coordinate space. 15

26 ρ i (q 1 )ρ j (q 2 )ρ k (q 3 ) = 0 = 0 (2.17) where i, j, k, are either (s, a), and n s, n a are the average number of density of symmetrically and anti-symmetrically distributed impurities and the symbol represents ensemble average. Now we can calculate the self-energy of the one-particle Green s function. We take the rainbow approximation in the calculation of the self-energy term 5. The detail of the calculation to the one-particle Green s function can be found in [14]. Here we just quote some of their results. The one-particle Green s functions become G (0) ((e, o), k, ω) = And the one-particle Green s functions for impurity case are 1 ω (ɛ k 1 2 h ) + iη (2.18) G((e, o), k, ω) = where the damping constant Γ is defined as 1 ω ( ɛ k 1 2 h ) + iγ/2 (2.19) Γ s = 2πN(ɛ F )n s v 2 0 Γ a = 2πN(ɛ F )n a v 2 0 Γ = Γ s + Γ a (2.20) ɛ k is the renormalized energy of the electron with the momentum k. N(ɛ F ) is the density of states at the Fermi surface Approximation for two-particle Green s function The quantity that we want is the probability of finding the electron on layer 1 at a later time t which shows the tunneling behavior of the electron between the layers, as we stated in the beginning of 5 One may use Migdel s theorem to prove that there is no crossing-line terms in this static impurity case with the assumption of eqn.(2.17). 16

27 the Sec.(2.2.2). P (1, 1 : ɛ, t t ) = kk G(1, k, t; 1, k, t )G (1, k, t ; 1, k, t) (2.21) However in the underdamped case it is more appropriate to use { (e, o), k } representation to do the perturbation calculation. There are sixteen terms for the probability in this representation. In the Sec.(2.2.3) we have assumed that only the second order term remains. Then the total number of the scattering events due to each kind of the impurities is a even number for the Green s functions before ensemble average. In this case the number of terms for the probability in this representation is reduced to eight and they are shown in the following equation. P (1, 1 : ɛ, t t ) = 1 δ(ɛ ɛ k ){ G(e, k, t; e, k, t )G (e, k, t ; e, k, t) 4 kk 1 + G(o, k, t; e, k, t )G (o, k, t ; e, k, t) + G(o, k, t; o, k, t )G (o, k, t ; o, k, t) + G(e, k, t; o, k, t )G (e, k, t ; o, k, t) + G(e, k, t; e, k, t )G (o, k, t ; o, k, t) + G(o, k, t; o, k, t )G (e, k, t ; e, k, t) + G(o, k, t; e, k, t )G (e, k, t ; o, k, t) + G(e, k, t; o, k, t )G (o, k, t ; e, k, t) } (2.22) Now we have to do the calculation of the eight terms in the eqn.(2.22). Before we do any calculation, let us first observe a few things about this equation. The first two terms in the eqn.(2.22) can be interpreted in this way: at the beginning we put an electron (or a hole) in the e, k state and later we find the probability of finding this electron (or hole) in the even and odd states, namely all the possible final states. Then the probability is 1 (or 1 for a hole) within the assumption that we take only the terms of second order in the ensemble average. 6 In the similar argument one can obtain the sum of third and fourth terms in eqn.(2.22) is equal to 1 for electron (or 1 for hole). 6 This point of view may be much clear in the formulation of a density matrix since these two terms are the sum of the diagonal terms of a 2 2 matrix in the representation of (e, o), k with the initial density matrix ρ i = e, k e, k. Then the probability is 1. See ref. [15] 17

28 As a result of this we only have to calculate the last four terms which are the interference terms within the approximation of keeping second order in the ensemble average 7. Let us express those four interference terms in the frequency variable since it is much easier to calculate. In(1, 1; ω) = 1 4 kk dω 2π { G(e, k; e, k : ω )G (o, k ; o, k; ω ω) + G(o, k; o, k ; ω )G (e, k ; e, k; ω ω) + G(o, k; e, k ; ω )G (e, k ; o, k; ω ω) + G(e, k; o, k ; ω )G (o, k ; e, k; ω ω) } (2.23) We have use the property of the eqn.(2.7) in this equation. Then the probability P can be expressed as P (1, 1 : ɛ, t t ) = 1 2 {θ(t t ) θ(t dω t)} + 2π e iω(t t ) In(1, 1 : ω) (2.24) The first term in the brace is for the case of an electron and the second term in the brace for the case of a hole. The last term in the eqn.(2.24) is the interference term, In(1; 1, k : ω), that we need to calculate The calculation of In(1; 1, k : ω) Let us do the calculation of the first term in the eqn.(2.23), i.e. I(1) = k δ k,k k dω 2π G(e, k; e, k ; ω )G (o, k ; o, k; ω ω) First let us use the eqn.(2.14) to make the perturbation expansion of Green s functions G and G before any ensemble average. Note that the perturbation expansion of G can be obtained by taking complex conjugate of the perturbation expansion of G. Note that the impurity density equation for G is ρ i (q) which is equal to ρ i( q) since the impurity density is a real function in coordinate variable. 7 Again it is much clearer in the 2 2 matrix in the representation of (e, o), k. Those four terms are the off-diagonal terms with the initial states ρ i = e, k o, k and ρ i = o, k e, k. 18

29 The next step is ensemble average over impurities. As mentioned in Sec.(2.2.3) we assume that only the second order terms in ensemble average remain and other order terms vanish, as shown in eqn.(2.22). we do not take account of the vertex correction term for simplicity. This excludes lots of Feynman diagrams. There is another requirement for this two-particle Green s function, namely the conservation of probability[17]. In order to fulfill this requirement the approximation of the two-particle Green s function should be chosen appropriately according to the approximation of the one-particle Green s function. Since we have chosen the rainbow approximation for the one-particle Green s function, the appropriate approximation for two-particle Green s function should be the random-phase approximation. Then after those steps the first term in eqn.(2.23), represented as I(1), becomes I(1) dω 2π { 1 ω ( ɛ /2) + iγ/2sgn(ω ɛ F ) 1 (ω ω) ( ɛ + /2) iγ/2sgn((ω ω) ɛ F ) + G(e, k ; ω ) G (o, k ; ω ω) [ k n 2 ev 0 G(e, k + k ; ω ) G (o, k + k ; ω ω) ] + G(e, k ; ω ) G (o, k ; ω ω) [ k n 2 ev 0 G(e, k + k ; ω ) G (o, k + k ; ω ω) ] [ k n 2 ev 0 G(e, k + k + k ; ω ) G (o, k + k + k ; ω ω) ] + } (2.25) Let us define the following quantities and do the calculation in order to simplify the eqn.(2.25) shown above. F (s) eo (ω) k (n s v 2 0) G(e, k + k ; ω ) G (o, k + k ; ω ω) = iγ s ω + + iγ 19

30 F (s) oe (ω) k (n s v 2 0) G(o, k + k ; ω ) G (e, k + k ; ω ω) = iγ s ω + iγ F (a) eo (ω) k (n a v 2 0) G(e, k + k ; ω ) G (o, k + k ; ω ω) = iγ a ω + + iγ F (a) oe (ω) k (n a v 2 0) G(o, k + k ; ω ) G (e, k + k ; ω ω) = iγ a ω + iγ (2.26) where Γ (s,a) is defined in eqn.(4.4). One may notice that F (s,a) ij is independent of the frequency variable ω. One may integrate out the functions containing the variable ω, which are G(ω ) G (ω ω), and obtains dω 2π G(e, k ; ω ) G (o, k ; ω ω) = i ω + + iγ From eqn.(2.25) one may notice that there are two kinds of ensemble average over impurities. One is for the symmetric impurities which do not change both the Green s functions G G after the scattering. So we may sum over all the possible scatterings due to symmetric impurities. The other is for the anti-symmetric impurities which, however, change even (or odd) state into odd (or even) state. So it needs even number of scattering of anti-symmetric impurities in order to return to even (or odd) state for this term I(1). We need to deal with these two cases differently. Under careful classification of those two different impurity case eqn.(2.25) can be written as I(1) dω 2π G(e, k ; ω ) G (o, k ; ω i (s) ω) {1 + F eo ω + h + iγ + (F (s) eo ) 2 + } {1 + F oe (a) [1 + F oe (s) + (F oe (s) ) 2 + ]F eo (a) [1 + F eo (s) + (F eo (s) ) 2 + ] + } After simple calculation one can obtain I(1) i ω + + iγ 1 1 F (s) eo 1 1 F oe (a) 1 F oe (s) F (a) eo 1 F (s) eo 20

31 Since the explicit form off (s,a) ij s can be found in eqn.(2.26), one can carry out the simple calculation and the result is ω + iγ a I(1) i ω 2 2 (2.27) + 2iωΓ a In the same way, one can obtain the rest of the three terms in eqn.(2.23), dω 2π G(o, k; e, k ; ω )G (o, k ; e, k; ω iγ a ω) i ω iωΓ a k k k dω 2π G(o, k; o, k ; ω )G (e, k ; e, k; ω ω + + iγ a ω) i ω iωΓ a dω 2π G(e, k; o, k ; ω )G (e, k ; o, k; ω iγ a ω) i ω 2 2 (2.28) + 2iωΓ a The summation of all the four terms gives the following result In(1, 1; ω) i ω + 2iΓ a 2 ω 2 2 (2.29) + 2iωΓ a One immediately note that the symmetric impurities play no role in destroying the phase coherence of the tunneling wavefunction. This result is obtained by taking care of the approximation we make for one-particle and two-particle Green s functions in order to preserve the conservation of probability. Final result The final result of the calculation of P (1, 1 : ɛ, ω) is i P (1, 1; ω) 2ω + iη + i ω + 2iΓ a 2 ω 2 2 (2.30) + 2iωΓ a What we want is the probability of the electron in layer 1 as time evolves, i.e. k P (1, 1 : ɛ, t t ). With eqns.(2.24) and (2.29) one can express the probability in the following way for the underdamped case, P (1, 1 : ɛ, t t ) = 1 2 θ(t t ) + i 2 dω ) 2π eiω(t t 21 ω + 2iΓ a (ω + iγ a ) 2 ( 2 Γ 2 a)

32 Since in the beginning of this calculation we have assumed that the impurities are not strong enough to destroy the description of k, i.e. the underdamped case where > Γ a, then the probability can be written as P (1, 1 : ɛ, t t ) = 1 2 θ(t t ){1 + cos[ (t t ) θ]e Γ a(t t ) } (2.31) where = 2 Γ 2 a > 0, tan θ = Γ a, cos θ = As one can see that in the final result, the behavior of the probability is like an underdamped oscillation with a frequency and a damping constant Γ a. So the transport behavior along the c-axis in this underdamped case is a coherent one. 2.3 The overdamped case As we see in the underdamped case, we have to use perturbation in order to develop the method to do the calculation. Because the effect of scattering due to impurities is much stronger than the tunneling behavior in the overdamped case, the impurity term is included in the zeroth order of the Hamiltonian and the tunneling term is treated as a perturbation. The Hamiltonian in this case can be written as H = n ɛ n â 1,nâ1,n + n ɛ n â 2,n â 2,n +( 1 2 h ) n,n { φ n χ n â 1,nâ2,n + χ n φ n â 2,n â 1,n } = H 0 + H T (2.32) where 1, n and 2, n are the eigenstate of H 0 with eigenvalue ɛ n and ɛ n and the transverse eigenfunction φ n on layer 1 and χ n on layer 2 respectively. For the specific distribution of impurities, the eigenstates 1, n and 2, n are uniquely determined[18]. The term φ n χ n (or χ n φ n ) is the overlap of two transverse eigenfunctions on layer 1 and 2 respectively due to the tunneling process. This is the so-called exact-eigenstate representation. 22

33 Naturally one can see that if the impurities are distributed in the exactly the same way on both layers, the transverse eigenfunctions are the same for both layers and the overlap of the two transverse wavefunctions is a delta-function which means that there is only one channel for each of eigenstate to tunnel. Then it is a coherent tunneling in this case. As the distributions of the impurities on each layer are different from each other, the eigenfunctions in the transverse direction, represented as φ n on layer 1 and χ n on layer 2 respectively, are different from each other. The overlap of two eigenfunction, i.e. φ n χ n, is no longer a delta-function but a function over a range of energy scale. Then there are many channels for each state to tunnel according to the quantity of φ n χ n. It is this effect of opening many channels to tunnel that gradually destroys the coherent tunneling. In overdamped case we assume that this effect of impurities is strong enough to destroy the coherent tunneling and we should take the impurity term in the zeroth order Hamiltonian and treat the tunneling term as perturbation, as shown in eqn.(2.32) One-particle Green s function We define the one-particle Green s function in the usual way but using { 1, n, 2, n } as the representation. G(1, n; 1, n; ɛ 1 ) = 1, n ɛ 1, n (2.33) H + iη However this quantity is not appropriate since it depends on the specific set of eigenstates which is determined by specific configuration of impurity distribution. In order to avoid this dependence of specific configuration of impurities, we choose the following quantity to investigate, G(ɛ, ɛ ) = 1 G(1, n; 1, n; ɛ )δ(ɛ ɛ n ) (2.34) N 0 n where N 0 is the density of states which is constant in a 2D case. By expressing delta-function in terms of Green s functions, one find out that G(ɛ, ɛ ) = 1 G(1, n; 1, n; ɛ ) i N 0 n 2π [G(0) R (1, n; ɛ) G(0) (1, n; ɛ)] A where 23

34 G (0) R.A (1, n; ɛ) = 1, n 1 1, n ɛ H 0 ± iη The Hamiltonian H 0 is defined in eqn.(2.32). This newly defined Green s function can also be written as G(ɛ, ɛ ) = = i 1 { 1, n [ 2πN 0 ɛ H 0 + iη 1 ɛ H 0 iη ] 1 1, n } (2.35) ɛ H + iη n i 1 Tr 1 {[ 2πN 0 ɛ H 0 + iη 1 ɛ H 0 iη ] 1 ɛ H + iη } (2.36) where Tr 1 { } = n 1, n 1, n = k 1, k 1, k which is independent of the choice of the representation in the transverse direction. So the Green s function G(ɛ, ɛ ) is independent of the specific configuration of the impurities. The contribution from the first term is zero since both Green s function are retarded ones. So we only have to calculate the second term. The detail calculation can be found in appendix A. Here we just quote the conditions and the results. The approximation we use here is the same one we use in the underdamped case, i.e. rainbow approximation for one-particle Green s function and random-phase approximation for two-particle Green s function. And the final result is G(ɛ, ɛ ) = ɛ ɛ 1 (2.37) ( 1 2 h )2 ɛ ɛ+2iγ a The term ( 1 2 h )2 ɛ ɛ+2iγ a can be viewed as the first order self-energy, as shown in appendix A.2. The normalized spectral function between the two transverse eigenfunctions of layer 1 and layer 2 can be written as F (ɛ, ɛ ) = 1 π 2Γ a (ɛ ɛ ) 2 + (2Γ a ) 2 (2.38) Once again one can see that the symmetrically distributed impurities play no role in this function of two overlapping wavefunctions. 24

35 2.3.2 Probability We may rewrite the equation for the probability in the following way, P (1, 1; ɛ, t) = n P (1, 1n; t)δ(ɛ ɛ n ) = i [G R 0 (1n; ɛ) G A 0 (1n; ɛ)] G R (1m; 1n; t)g A (1n; 1m; t) (2.39) 2πN 0 nm where G R,A 0 are in usual definition which is 1 0 (1n; ɛ) = 1n ɛ H 0 ± iη 1n G R,A and in eqn.(2.39) we have use the following property, δ(ɛ ɛ n ) = i 2πN 0 [G R 0 (1n; ɛ) G A 0 (1n; ɛ)] H 0 is the zeroth order of the Hamiltonian defined in eqn.(2.32). In the following calculation, it is more convenient to use frequency variable. Here we take Fourier transformation of the probability and the result is P (1, 1; ɛ, ω) = i [G R 0 (1n; ɛ) G A 0 (1n; ɛ)] 2πN 0 nm dɛ 2π GR (1m; 1n; ω + ɛ )G A (1n; 1m; ɛ ) (2.40) Before we proceed to do any calculation, let us first explore some properties of P (1, 1; ɛ, ω). The eqn.(2.40) can be rewritten as P (1, 1; ɛ, ω) = i dɛ 1 1m 2πN 0 2π nm ω + ɛ H + iη 1n 1 [ 1n ɛ H 01 + iη 1n 1n 1 ɛ H 01 iη 1n ] 1n 1 ɛ H iη 1m 25

36 where H 01 is the zeroth Hamiltonian of layer one. Since 1n 1 ɛ H 01 ±iη 1l = 1n 1 ɛ H 01 +iη 1n δ n,l, we may change one of 1n into 1l and sum over 1l. Then we can find that the result is P (1, 1; ɛ, ω) = i dɛ 2πN 0 2π Tr 1 1{ ω + ɛ H + iη 1 [ ɛ H 01 + iη 1 ɛ H 01 iη ] 1 ɛ H iη } = P (1, 1; R : ɛ, ω) P (1, 1; A : ɛ, ω) (2.41) where Tr 1 = m 1m 1m. One can see that this formulation is independent of the representation in the transverse direction. In the following calculation we will use 1k, 2k as the representation to solve this problem. Note that up to this stage we have not used ensemble average over impurities The calculation of two-particle Green s function In the eqn.(2.41) we use H 0 as the zeroth order Hamiltonian and treat the tunneling Hamiltonian H T as the perturbation. After a simple calculation one can obtain the expansion of the Green s function in terms of tunneling Hamiltonian as shown below 1 (ω+)ɛ ( ) H ± iη = 1 (ω+)ɛ ( ) H 0 ± iη 1 {H T (ω+)ɛ i=0 ( ) H 0 ± iη }i (2.42) Then the probability can be written as P (1, 1; ɛ, ω) = i dɛ 2πN 0 2π Tr 1 1 (ω+)ɛ ( ) H 0 ± iη { 1 {H T (ω+)ɛ i=0 ( ) H 0 ± iη }i 1 [ ɛ H 01 + iη 1 ɛ H 01 iη ] 1 1 (ω+)ɛ ( ) {H T H 0 ± iη (ω+)ɛ ( ) H 0 ± iη }j } (2.43) j=0 Since in the Sec.(2.3.1) we have shown that this quantity is independent of the choice of representation in the transverse direction, we will choose { 1k, 2k as the representation in doing 26

37 the calculation. Then we will take ensemble average of impurities. The whole procedure is very complicated and so we will describe the calculation in the Appendix.(A.4) The final result After a lengthy calculation shown in appendix, one can obtain the final result of the probability as P (1, 1; ɛ, ω) = i 2ω + i ω + 2iΓ a 2 ω(ω + 2iΓ a ) ( ) 2 (2.44) This result is exactly the same form we obtained in the underdamped case however in the different limit, i.e. the damping constant Γ a is larger than the tunneling matrix element. The first term in the equation above should be replaced by i 2ω+iη where η 0+ due to the requirement of causality. It is better to view this result in t-variable, instead of ω-variable. After the Fourier transformation, one obtains where Γ 2 a P (1, 1; ɛ, t) = 1 2 θ(t){1 + e Γ at [cosh Γ at + Γ a Γ sinh Γ at]} (2.45) a = Γ 2 a 2. The first term is the equilibrium value after long period of time. As one can see in the limit of Γ a the probability becomes P (1, 1; ɛ, t) = 1 2 θ(t) θ(t)e 1 2 ( Γa )2 t (2.46) which is the result obtained from second order perturbation theory for tunneling case. 2.4 The impurities between the layers As shown in the experiment, all the high temperature cuprate materials are doped with some alloy. So they are not in perfect crystal structure. Now most believe that the copper-oxide planes are metallic. The c-axis transport is very weak and here we use tunneling to describe it. Some of alloys go to layers and some goes between the layers. The alloy between the layers stimulates us to consider the tunneling behavior along the c-axis due to the impurities between the layers. Here we study this case. 27

38 2.4.1 Hamiltonian Here again we first have to define the Hamiltonian in a proper way. Let us simplify the problem by assuming that the impurities between the layer only provide the random tunneling channel between the two neighboring layers. Furthermore we assume that the tunneling provided by the interlayer impurities is a point-to-point tunneling without complicated structure. Let us again use the two-layer system to investigate the tunneling property in this case. The Hamiltonian in this case can be written as H = i=1,2;k ɛ k (â i,kâi,k) + k ( 1 2 h 0)(â 1,kâ2,k + â 2,kâ1,k) + k,q ( 1 2 h 1)ρ(q)(â 1,k+qâ2,k + â 2,k+qâ1,k) (2.47) where ρ(q) is the Fourier transformation of the impurity density ρ(x) = i δ(x x i). In this Hamiltonian we still assume that there exist a coherent tunneling matrix element 0 which is momentum conserved in tunneling process. The impurities between the layers just provide the extra channel to tunneling with the tunneling matrix element 1. But in this tunneling channel the momentum is not conserved. We make the problem extremely simple by assuming that 1 is real 8 and static (i.e. no frequency dependence). Without the random tunneling channels, the tunneling along the c-axis is a coherent one. Now we want to study the effect on the tunneling behavior by bringing these random tunneling channels into the system. As we have done in underdamped case, we can rewrite the Hamiltonian in the representation of even and odd states, that is H = k (ɛ k 1 2 h )[â e,kâe,k] + k (ɛ k h )[â o,kâo,k] + k,q ( 1 2 h 1)ρ(q)[â e,k+qâe,k â o,k+qâo,k] (2.48) One immediately sees that those random tunneling matrix element does not change the even state 8 If the impurities are right in the middle between the layers, one can use symmetry to show that 1 can be real. 28

39 to odd state and vice versa. This is a special property due to the fact that 1 is real number in this case. We will use this Hamiltonian to do further calculation One-particle Green s functions One can derive the equation of motion for one-particle Green s function. Without going into detail, we here present the Dyson s equation of one-particle Green s function as where G((e, o), k, k ; ω) = G (0) ((e, o), k; ω) ± 1 2 h 1 ρ(q)g((e, o), k q, k ; ω) (2.49) q G (0) ((e, o), k; ω) = 1 1 ω (ɛ k 2 h ) + iη where η 0 +. This is the result without ensemble average. Now we will use the assumption of the ensemble average described in the underdamped case here. Then after ensemble average, one-particle Green s functions become G((e, o), k; ω) = 1 ω (ɛ k 1 2 h ) + iγ /2 (2.50) where Γ = 2π( 2 h 1 1) 2 ρ imp N 0, ρ imp the density of impurities per unit area, and N 0 the density of states. This one-particle Green s function is similar to the one in underdamped case except the detail form of decaying constant Γ. One clear result is that the introducing of impurities as the random tunneling channel brings the decaying behavior of electrons. However we are interested in transport property along the c-axis, we will study the probability in the next subsection Probability Now we want to study how the probability of finding the electron on layer 1 at later time t is affected by the introducing of impurities between the layer as random tunneling channels. One can expand the probability in the representation of even and odd states. However from eqn.(2.48), one knows that there is no crossing term between the even and odd states. So the expansion of the 29

40 probability is much simpler and it is P (1, 1 : ɛ, t t ) = 1 δ(ɛ ɛ k ){ G(e, k, t; e, k, t )G (e, k, t ; e, k, t) 4 kk 1 + G(o, k, t; o, k, t )G (o, k, t ; o, k, t) + G(e, k, t; e, k, t )G (o, k, t ; o, k, t) + G(o, k, t; o, k, t )G (e, k, t ; e, k, t) (2.51) As seen in underdamped case, the first term is the probability of finding the electron in all even state when the initial state is an even states. The probability of this term is 1 since there is no crossing term between the even and odd state in the Hamiltonian shown in eqn.(2.48). In the same way the second term is also equal to 1 too. So we just have to calculate the last two terms The final result We leave the detail of the calculation in appendix B. Here we point out some of the important assumptions we use in the calculation. We have used the rainbow approximation for one-particle Green s function and the corresponding random-phase approximation for two-particle Green s function in order to insure that the probability is conserved. The result of the last two term are P (3) + P (4) = i 4 { 1 ω + 2iΓ + 1 ω + + 2iΓ } (2.52) One notice that this result is independent of the energy of the electron that we put in the system. As we take the Fourier transformation the probability becomes P (1, 1 : ɛ, t t ) = 1 2 θ(t t ){1 + cos (t t )e 2Γ (t t ) } (2.53) One can see that the initial energy of the electron ɛ is unrelated with the probability. From the eqn.(2.53) one can see that the introducing of the impurities between the layers as the random tunneling channels brings the exponentially decaying term. However it does not destroy the the coherent tunneling between the nearest neighboring layers. A study on the multilayer system is needed to see whether it is just a special property of two-layer system. 30

41 Chapter 3 Sources of Fluctuation 3.1 Introduction In the last chapter we discussed the tunneling behavior along the c-axis in the static impurity case. Eventually we need to study the dynamic fluctuation cases. But let us first examine the possible sources of dynamic fluctuation. In this chapter we are going to examine three possible sources of dynamic fluctuation to our knowledge. One is caused by the existence of phonons through the electron-phonon interaction. This interaction causes the effective electron-electron interaction through the exchange of the phonons. It is the essential interaction of the traditional superconductors. We will discuss this interaction in Sec.(3.2) The second one is the direct Coulomb interaction between the electrons. One knows that in three dimensional case it costs a huge amount of energy to excite a plasma mode due to the strong Coulomb interaction. But here we may treat superconducting cuprate materials as a multilayer system in the zeroth order due to the fact of difficulty of transporting along the c-axis. Then the nature of the plasma modes is totally different from the three dimensional case. The detail of study will be presented in Sec.(3.3). The last one is the most discussed spin-spin interaction. Today most physicists believe that it is the spin-spin interaction, existing in the antiferromagnetic state in appropriate doping, which induces those cuprate materials to become superconductors. Here we take the final form of the spin-spin interaction to study the fluctuation difference between two nearest neighboring layers. The results is presented in Sec.(3.4) In the last section of this chapter we will put reasonable quantities for each parameters shown 31

42 in the expressions of the imaginary part of the response functions of those three sources of dynamic fluctuation. Then we can compare the strength of these three fluctuation sources. 3.2 Electron-phonon interaction Traditional superconductors are caused by the electron-phonon interaction which produces the attractive effective electron-electron retarded interaction[19]. We consider this interaction as one of the possible source to induce superconductivity. However here we want to study the effect of this interaction on the transport properties along c axis in order to compare it with the effects of other interactions. Here we use the simplest assumption. Again we assume the the electron in two-dimensional layer forms a Fermi disk and there is no tunneling between the layer 1. The phonon we consider here are isotropic 2 and acoustic. In the next subsection we use the simple method of deformation potential to derive the the interaction between electrons and phonons. Then we write the result in form of the effective interaction Hamiltonian, which serves as the Hamiltonian of fluctuation in the spin-boson Hamiltonian[13]. In the last subsection, we use available experimental data to estimate the coupling constant Deformation potential In order to find the coupling interaction between this electron gas and the acoustic phonons, we use deformation potential to formulate the coupling interaction[20]. In order to use the deformation potential formulation there are several restrictions. One is that it has to be in the long-wavelength limit such that wavevector of the phonon is smaller than the mean free path of electrons. Then the system can be treated as a homogeneous system within one wavelength. Secondly we assume that the motion of the phonon is very slow comparing with the motion of electrons so that we may treat the phonon as a static one and the electrons reach the local equilibrium according to the motion of phonon. As one can see in this method this is the interaction between the electrons and the acoustic phonons. 1 we ignore the small tunneling matrix in the beginning. 2 The anisotropic case, which is much realistic in superconducting cuprate material, can be modified into isotropic by rescaling the wave vector by a ratio of sound velocity. 32

43 Before discussion, we need to point out one feature. Since there is no tunneling in the c-axis for electrons, the electrons won t form a three-dimensional Fermi sea but a two-dimensional one. As a phonon passes the region and changes the lattice constant along the c-axis, it has no effect on the electrons. So the dilation A( r) is strictly related with the ratio of change of area along the ab plane. The electrons on the two dimensional layers are a two-dimensional free Fermi gas and form a symmetric Fermi circle in two dimension. The energy spectrum is free electron kinetic energy and we ignore the effect of band structure for simplicity. As a slow acoustic phonon goes through the lattice, the area of the unit lattice changes by an ratio A( r). Then the density of the electrons also changes from n 0 to n 0 (1 A( r)), since the number of the electrons in each unit lattice does not change 3. As the number of electrons change, the electron in the region immediately reach the new equilibrium situation and form the new Fermi surface. The new Fermi vector, related with the density of electrons, changes by an amount δk F 1 2 A( r)k F (3.1) for this two dimensional case. And the energy of an electron near at the Fermi surface change by the amount δɛ A( r)ɛ F (3.2) This the amount of energy change of an electron on the Fermi surface when a phonon passing through the region. {The effective interaction Hamiltonian Let us first write down the effective Hamiltonian to represent the electron-phonon interaction, H int = d rv ( r)ψ ( r)ψ ( r) (3.3) where V ( r) contains the information about phonon and Ψ ( ) ( r) the annihilation (creation) operator of electrons. Here we did not include the kinetic energy of the electrons which does not change during the deformation of potential caused by phonons. The change of the energy comes from this 3 Here we have to assume that the diffusion of electrons is too slow compared with the acoustic phonon. 33

44 term which connects with the parameter of phonons. Now we want to use the information that we obtain from the deformation method to derive the appropriate coefficient. Let us assume the dilation A( r) is a universal one, i.e. all the lattices are changed by same ratio of area, A. Then one can take the Fourier transformation to momentum space. From the derivation of deformation potential, one knows that the energy of an electron on the Fermi surface changes by an amount δɛ in eqn.(3.2). By comparing the result obtained from the deformation method and the effective Hamiltonian in the universal case one can immediately show that V = Aɛ F. Applying this to a general case, it should be V ( r) = A( r)ɛ F (3.4) We have the right coefficient for the effective Hamiltonian. Now we need to change dilation A( r) into phonon operators. First let us define variable ε j ( r) as the displacement in j direction for a lattice at r position. Then one can define the strain W ij as W ij = ε j / x i Strain W ij can be understood as how much changes of lattice constant in j direction for a unit length in i direction. Then one can see that relation between dilation A( r) and strain function W ij as A( r) = W ij ( r)δ ij ij The over the summation means that i, j are only along the ab plane, since the change in c-direction does not cause the change of energy of electrons on the Fermi surface. After quantization, the displacement ε j ( r) can be expressed as ε j ( r) = q h (ê) j ( 2ω q ρ )1/2 (b a + b ei q r q) V where ê is the unit vector of the polarization of the phonon, ρ the mass density of ions, ω q the energy spectrum of phonons, b ( ) q the annihilation (creation) operator for the phonons. 34

45 Putting all the information together, one can find out that the interaction Hamiltonian can be written as H int = i D q â k+qâk(b a + b q) (3.5) k,q h where D q = ( 2ω q ρv )1/2 (ê q )ɛ F where q is the component vector of q along the ab plane. So this Hamiltonian describes the interaction between the electrons and acoustic phonons Reducing into two-layer system What we want is the spectrum density of fluctuation in two layer system. So we will do the same thing described in Sec in order to reduce the result into a two-layer system. First let us add the Hamiltonian of phonon H pho = q (n ) hω qb qb q (3.6) into the interaction Hamiltonian we have just obtained. ω q is the energy spectrum of the acoustic phonon. Then we replace k â k+qâk+q by a variable F (q), which can be view as the external force. Then one can find the response function of the phonon. The imaginary part of the response function is Imχ(q, ω) = π h D q 2 δ(ω ω q ) (3.7) However what we want is the fluctuation difference between two nearest neighboring layers. First let us assume the phonon energy spectrum is a linear one, i.e. ω q = cq, c the sound velocity. One can utilize the eqn.(3.16) in Sec and takes the imaginary part on both side. After a few step of calculation one can find the correct spectrum function is Imχ (q, ω) = q4 ɛ2 F ω 2 ρa 1 cos( (ω/c) 2 q 2 d) θ((ω/c) 2 q 2 (ω/c) 2 q 2 ) (3.8) 35

46 where A is the area of ab plane of the sample, a normalization factor. One should notice that for each selecting value of ω and q, there exists two unique q z, which are ± ω 2 q 2 c2. Then one can see that when q z = 0 this term is zero, since there is no fluctuation difference between two neighboring layers. Another distinguished feature is when q = 0. Then there is no contribution, since there is no deformation potential formed in this case. From the above argument, one knows that the low frequency region is also the long wavelength region since the frequency sets up the maximum wavevector as (q ) max = ω/c. So we can integrate out the q variable and find out the power law of ω in the spectrum function of this fluctuation. After appropriate approximation, one can find Imχ (ω) ω 5 (3.9) In the model of the spin-boson Hamiltonian one knows that this is in th weakly damped region. So it is coherent tunneling for two-layered system with the electron-phonon interaction. 3.3 The plasma modes in multilayer system Here we are going to consider the case of fluctuation of plasma modes. In the usual threedimensional metal it need a definite energy, which the minimum energy is E min = hω p where ω p is the plasma frequency, to a excite plasma excitation. Those excitations are rare since minimum required energy ranges from 0.1 to several ev which is extremely high comparing with room temperature. However all the new found cuprate superconductors are layered structure, which is quite different from the usual 3-dimensional metals. The ratio of resistivity along c-axis and the abplane of cuprate superconductors is extremely high, and in some cases the temperature dependence along the c-axis is different from those along ab-plane. This indicates that transport properties might be different along c-axis and ab-plane. And then the plasma modes in such layered-structure system might be quite different from the plasma modes in usual 3-dimensional metals. Now we will explore the physical properties of plasma modes in this kind of layered structure. For simplicity, we assume a multi-layer system with single conduction plane in each lattice. As indicated in the experiment, we assume that electron moves freely on the layer and form a Fermi gas. However we assume there is no electron tunneling between layers for the zeroth order approximation 36

47 since the transport along the c-axis is more difficult than along the ab-plane in cuprate superconductors. The kind of plasma modes in this multilayer system has been studied by Fetter[21]. Here we are interested in the fluctuation of the voltage difference between two nearest neighboring layer in the low frequency and long wavelength limit. In the next few subsections, first we will define the related response function in this multilayer system,which can be understood straight; the exact response function is related to this one. Later we will use the classical dynamics of fluid to describe this system. The calculation has been done by Fetter[21], and we will give brief derivation and the final form of response function. In the following subsection, we will define the response function of voltage difference between two nearest neighboring layers and relate it to the one we have derived. In order to be used in two-layered system, we will reduce the response function into a two dimensional case. In the last subsection, the spectral function of the fluctuation of voltage difference is given The response function Let us first define the appropriate physical quantity that we need to calculate. It is the voltage fluctuation on each layer in which we are interested. Then we may put an external test charge to measure the response of the induced potential. The Hamiltonian that describes this behavior can be written as 4 H ext = d 3 xen ext ( x, t)φ ind ( x, t) Here we are seeking the response function of the induced voltage due to the external charge density 5, i.e. χ V V ( k, z j, z i, ω) = δeφ in( k, z j, ω) δn ex ( k, z i, ω) (3.10) where k is the wave vector along the two dimensions. Note that we have used voltage V, which is eφ in, instead of the potential φ in in this equation. One can use quantum mechanics to derive the 4 We did not write this equation in the usual way because it is more convenient in deriving the response function of the voltage due to external charge density in this specific form 5 I choose z = z j, z = z i in the response function χ V V ( k, z, z, ω) because we are interested in the potential difference between two adjacent layer (then i = j, j ± 1). 37

48 general form of this response function[22]. However we won t use quantum mechanics to do the detailed calculation. Instead we will use classical mechanics, as described in Sec.(3.3.2), to do the derivation Equations of motion There are three equations that govern the classical charged fluid in order to describe the dynamics of the electron gas in this two-dimensional multilayer system. We assume the variation of the electron density is small and then we can make the first order approximation on those three equations. The first equation is continuity equation which describes the number conservation of electrons, i.e. t n j( r, t) + n 0 v j ( r, t) = 0 (3.11) where n j ( r, t) is the two-dimensional electron density on jth layer derivation from the mean electron density n 0 and v j ( r, t) the electron velocity density on jthe layer. Note that r is the vector on twodimensional layer. In this equation we have neglect the small contribution from the variation of the electron density in the second term which give second order correction in comparison with the those two terms shown in the eqn.(3.11). The next equation is the classical hydrodynamic equation, t v j( r, t) + v j( r, t) = s2 τ n 0 n j( r, t) e m E t( r, z j, t) (3.12) where τ is the relaxation time of the electron motion due to inelastic scattering, and it may be taken from the experimental data, s the adiabatic sound velocity in two-dimensional system, z j the coordinate in the z direction for jth layer. The Coulomb force is expressed by the term E t, the electric field in the tangential direction, i.e. along the two-dimensional surface 6. This equation is essentially the Newton s second law for the fluid. The second term on the LHS of the equation is the flux term of the velocity density of electrons, which will be ignored in the following calculation because it is a second order term. We have ignored the magnetic interaction in this approximation because it is of order (v/c), again a higher order term. This approximate equation can be viewed 6 Since we have assumed that the electrons are not moving along the z-direction, only the tangential component of the electric field remains. 38

49 as following physically: The change of the velocity density of electron is due to several sources of force. One is the Coulomb force acting on it because it is a charged fluid. The other is the pressure difference, arisen from the variation of electron density, acting on the density due to the fact it is a fluid. However there are impurities or fluctuations that scatter electrons and eventually slow down its motion, which is expressed in terms of decaying term. The last equation concerns the Coulomb interaction, 2 φ( r, z, t) = 4πρ tot = 4π[ρ ext ( r, z, t) e j n j( r, t)δ(z z j )] (3.13) where φ is the electric potential field. In this equation we have included the induced electron density n j ( r, t). It is clear that it is the inclusion of this induced electron density that guarantee the existence of plasma modes. We can see that from the eqn.(3.12), one of the forces acting on the electron velocity density is the Coulomb force. It causes the change of velocity of electron density. If the change is not spatially uniform, we know that there is a change of electron density according to the eqn.(3.11). This induced variation of electron density in turn raises the Coulomb interaction, acting as a restoring force, which acts on the electron velocity density. The whole process gives rise to the plasma modes. Without including the term of the induced electron density in eqn.(3.13), it will give rise to the diffusion mode in the low frequency limit as we usually find in the neutral fluid. Appendix C gives a brief and much easier derivation of the response function that we want. The final result is χ V V ( k, z j, z i, ω) = dq 2π eiq(z j z i ) 4π 2πne 2 k sinh kd k 2 + q 2 m cosh kd cos qd [ω(ω + i/τ) s 2 k 2 2πne2 k sinh kd m cosh kd cos qd ] 1 (3.14) Now we shall reduced this result into two-layer system. This will be discussed in next subsection. 39

50 3.3.3 Reducing to two-layer system As one may notice that what we want is the response function of the voltage difference between two adjacent layers at the same position on the two dimensional plane (since potential along z direction is translationally invariant along the x-y plane, the tunneling matrix element is not a function of x and y coordinate variable but a constant), i.e. χ V V = θ(t t ) [ V, V ] = θ(t t ) [V (r, z i, t) V (r, z i+1, t)), (V (r, z i, t ) V (r, z i+1, t )] (3.15) After expansion and taking Fourier transformation in the two dimension and the z direction, and using the relation V (r, z i+1, t)/v (r, z i, t) = e iqd we can obtain the following equation χ,v V ( k, q, ω) = 2(1 cos qd)χ V V ( k, q, ω) (3.16) As one may see, the response function is a function of variables k, the tangential momentum along two dimensional layer, q the momentum in the z direction, and the frequency of the fluctuation ω. However we need to integrate out the variable q since it is irrelevant in our two-layer model. For simplicity. let us first define the following quantity ω(k, q) as ω(k, q) 2 = s 2 k 2 + 4πn0 e 2 m kd 2 sinh kd cosh kd cos qd (3.17) which is the plasma frequency of the infinite-layer system if there is no damping term. Now we perform the integration over the variable q. After the tedious and tricky derivation, shown in Appendix C.3, one may obtain the appropriated response function which is χ V V = 4π k (1 cosh kd) 4π k sinh kd [ [ω(ω + iτ) s 2 k 2 ](cosh kd 1) (2πn 0 e 2 ] 1/2 k/m) sinh kd [ω(ω + iτ) s 2 k 2 ](cosh kd + 1) (2πn 0 e 2 (3.18) k/m) sinh kd 40

51 3.3.4 The spectral function of voltage fluctuation Using the fluctuation-dissipation theorem, one knows that the imaginary part of the response function fluctuation that is induced by the external potential is exactly the spectrum density of the instant fluctuation with an appropriate constant. After tedious and careful calculation, one can obtain the spectrum function as F co (k, ω) = 2π k cosh kd 1 4 B 2 + C 2 { A 2 + B 2 A C 2 + B 2 + C A 2 + B 2 + A C 2 + B 2 C} (3.19) where A = ω 2 s 2 k 2 2πne2 k m B = ω 2 s 2 k 2 2πne2 k m sinh kd cosh kd 1 sinh kd cosh kd + 1 C = ω/τ (3.20) 3.4 Spin fluctuation Since the discovery of cuprate superconductors, one of the important features of this class of superconductors is its antiferromagnetic properties in materials La 2 CuO 4 and Y Ba 2 Cu 3 O 6. And as the doping of Sr in La 2 x Sr x CuO 4 and oxygen in Y Ba 2 Cu 3 O 6+x are increased the materials change from antiferromagnetism to superconductor, then to metal. Some physicists take it as a clue that spin fluctuation might be the mediated interaction between the electrons instead of traditional phonon interaction. From the NMR experimental data, one of the very striking features is non-korringa temperature dependence of 63 Cu spin-lattice relaxation rate. On the other hand the temperature dependence of 17 O and 89 Y spin-lattice relaxation rate in Y Ba 2 Cu 3 O 7 is Korringa-like. In order to explain this experimental data, Millis, Monien, and Pines[23] proposed a phenomenological spin-spin correlation function along the plane as following χ MMP ( q, ω) = χ Q 1 + ξ 2 (q Q) 2 iω/ω SF (3.21) where χ Q is the static spin susceptibility at the wavevector Q = (π/a, π/a), ξ is the antiferromagnetic correlation length, and ω SF is the decaying time for the relaxation of an antiferro- 41

52 magtism. Essentially it is a relaxation mode of antiferromagtism which is peaked at the wavevector Q = (π/a, π/a). However there is no spin correlation between lattices along the c direction. This model reasonably explains the conflict temperature dependence of spin-lattice relaxation rate at different nuclei sites. The values of χ Q, ξ, and ω SF are taken from fits to NMR experiment. Now the best fit parameters for Y Ba 2 Cu 3 O 7 δ, determined from experiments, is χ Q 80 states/ev, ω SF 14 mev, and ξ 2.3a at T c where a is the lattice constant along the ab plane. Using this spin-spin correlation function, Pines group then developed the electron system with spin fluctuation as the mediated interaction[24]. The Hamiltonian can be written as H = k,σ ɛ k ψ kσ ψ kσ + ḡ 2 q,k,αβ ψ k+qα ψ kβ σ αβ S( q) (3.22) where ɛ p is the dispersion relation of electrons. They called this electron system as nearly antiferromagnetic Fermi liquid (NAFL). In this equation, they have assumed that the coupling constant ḡ is a structureless constant, independent of momentum q and energy transfer ω. And the effective interaction between electrons on the plan is V spin = ḡ 2 χ MMP ( q, ω) σ 1 σ 2 (3.23) Using this Hamiltonian and the dispersion relation of electron, derived from tight binding approximation including next nearest neighboring binding, Monthoux and Pines[24, 25] obtained the d x 2 y 2 pairing superconductivity for cuprate superconductors. This d x 2 y2 pairing superconductivity has been supported by many experiments[26] while it is not finally concluded yet. Furthermore, they use the calculation of T c from the weak-coupling theory and the calculation of resistivity in the ab plane to fix the coupling constant ḡ consistently. Later they use strongcoupling theory to find a better fit ḡ. However T c, the transition of temperature of superconductor, also depends on the detail of the structure of the Fermi surface in their model. The best fit for Y Ba 2 Cu 3 O 7 δ at T c = 90K is a hole concentration n h = 0.25 and a coupling constant ḡ = 0.64eV. 42

53 3.4.1 Two-layer model Since there is no correlation between the lattices along the c-axis, the spin fluctuation on each lattice is independent of each other. So the response function of fluctuation difference is simply χ = 2V spin (3.24) And the imaginary part, by fluctuation-dissipation theorem, gives us the spectrum density of spin fluctuation which is Imχ (q, ω) = 2ḡ 2 ω/ω SF χ Q (1 + (q Q) 2 ξ 2 ) 2 + (ω/ω SF ) 2 (3.25) which is what we want in the two-layer system. 3.5 Numerical results Before we do any numerical calculations of those dynamic fluctuations, let us first review what we want and what have done, and put them together in a consistent form that we may compare. We want to study, in the future, the effect of the dynamic fluctuations on the c-axis resistivity in the high temperature cuprate superconductors. So we start from a multilayer system with single layer in each lattice for simplicity. After we obtain the explicit form of fluctuations, we reduce it into a two-layer system and find the energy difference of the fluctuation between these two layers since we are only interested in the c-axis properties due to these fluctuations. In these criteria, we may write down the effective Hamiltonian 7 that we want to study in the future, H = k,i=1,2 + k,q ɛ k a + i,k a i,k + ( 1 2 h ){a+ 1,k a 2,k + a + 2,k a 1,k} 1 2 f(q){a+ 1,k+q a 1,k a + 2,k+q a 2,k}(b q + b + q ) + q hω(b + q b q ) (3.26) Here we only discuss bosonic fluctuation. This Hamiltonian is similar to the spin-boson Hamiltonian except that the Hamiltonian here has the 2D transverse degrees of freedom. One can obtain the 7 This effective Hamiltonian is specifically for the purpose stated above. For different purpose the effective Hamiltonian might be different. 43

54 spectrum function of the fluctuation difference, seen by the electrons, of this Hamiltonian, J(q, ω) = π h f 2 (q)δ(ω ω q ) (3.27) What we have done in this chapter is to find out the explicit form of the spectrum function in each fluctuation case. Instead of using the spectrum function J(q, ω), I use the fluctuation correlation function F (q, ω), which is equal to h π J(q, ω), in all three fluctuation cases in order to compare with each other. The mean square value of fluctuation may derived from this function. Now we may discuss the orders of magnitude and the shapes of the those three fluctuations that we have discussed in previous three sections. We will discuss each interaction separately and then put them together in one table to compare their strength of the fluctuation Electron-phonon interaction In this section we are going to put in the reasonable physical quantities in order to obtain the numerical result. Note that we just want the right order of magnitude of all the physical quantities here in order to estimate the strength of this fluctuation to be used for comparison later. We focus on material YBa 2 Cu 3 O 7 δ. And the lattice constants of this material along the a- and b-axis are cm, and cm along the c-axis [2]. So the volume of the unit cell is cm 3, and the density n 0 is lattices/cm 3. With the known atomic mass of each element of YBa 2 Cu 3 O 7 δ, one can calculate the mass density, ρ, of this material to be 6.42 gm/cm 3. We assume the electron on the two dimensional layer is a free electron gas with 0.3 to 0.5 electrons per unit cell. Then the Fermi energy is around 0.48 to 0.80 ev. Since we use deformation potential to derive this electron-phonon interaction, the phonon we discuss here is acoustic longitudinal sound wave. So we assume the energy spectrum of this acoustic sound wave to be a linear one, i.e. ω = v L q. From experiment [27] we find out the longitudinal sound velocity v L to be cm/sec. The upper bound of wavevector q, decided by the lattice constant on the ab plane, is cm 1 in one direction. So the maximum ω, which is 3q v L in three dimension, is sec 1, which is around 400 K in temperature or 35 mev in energy scale. This result is quite reasonable. From the method of deformation potential and numerous assumptions one can obtain the the 44

55 Figure 3.1: The spectrum function of phonon fluctuation as a function of variable ω with with q = 0.1π/a. spectrum function of dynamic fluctuation difference of the phonons between two nearest neighboring layers, which is 8 F ph (q, ω) = h π q 4 ɛ2 F ω 2 ρ 1 cos( (ω/v L ) 2 q 2 d) θ((ω/v L ) 2 q 2 (ω/vl ) 2 q 2 ) (3.28) Here we have used the fluctuation-dissipation theorem to replace the imaginary part to the response function to obtain the fluctuation spectrum function. We take the parameter d as the lattice constant along a-axis in order to prevent some unphysical result due to the assumption of isotropic sound wave. Let us examine a figure, as shown as Fig.(3.5.1), of this spectrum function as a function of variable q with ω = sec 1, which corresponds to wavevector q = π/a. 8 I have eliminate the factor of area A in this equation and treat q as a continuous variable, not a discrete variable used in Sec.(3.2). I use this one in order to have consistent expressions in all three sources of dynamic fluctuation. 45

56 One can see that at long wavelength limit the spectrum function F ph is proportional to q 4 since in this region the contribution from the out-of-phase factor, (1 cos( (ω/v L ) 2 q 2 d)), is almost one. As q approaches ω/v L, this out-of-factor approaches zero faster than the divergence of factor in denominator. This feature holds for all ω v L π/a case. For the region ω higher than this v L π/a the situation is complicated by the edge of the reciprocal lattice and therefore there exists a upper cutoff for variable q. The detail will not be discussed here. For ω v L π/a case, one can integrate out the variable q, and obtain the frequency dependence of fluctuation difference at the same position. And the result is F ph (ω) = 2 h { 105π 2 ɛ2 F a2 ρvl 7 ω ( ( ω 2 ( ) )} ω 4 a) + O a 36 vl v L (3.29) Note that ω/v L is equal to the total wavevector q. If we put those physical quantities into eqn.(3.29) the prefactor is erg 2 sec 6 or ev 2 sec 6. However the maximum of ω is around 35 mev only for this acoustic phonon. One notices that this fluctuation difference of phonon at the same position is proportional to ω 5 at the low frequency limit. If we naively take the spin-boson model, used to study the tunneling behavior between two points with the coupling with environment, one can see that this is in the underdamped region and it should be coherent tunneling between those two point. However because we are discussing a two-layer system, instead of two-point system, the result should be served as an implication, not as a correct result. It is more convenient to compare the strength of the fluctuation source on one simple number. So we integrate out the variable ω within a reasonable range, which the upper limit is 3q v L, and the variable q within the reciprocal lattice and obtain F ph = (ev ) 2 (3.30) This quantity is the fluctuation of phonon at the same position at the same time. The square of this value, ev, gives the mean-square value of the fluctuation of phonon, which is roughly 85 K in temperature. Note that here we just give roughly right order of magnitude. 46

57 3.5.2 Plasma modes in 2D case Let us first check a few physical quantities. n is the surface density of electrons and then the density of electron in 3D is n/d, where d is the lattice constant along the c-axis and it is cm for YBa 2 Cu 3 O 7 δ. From Sec.(3.5.1), one knows that the density of electron in 3D is cm 3 for 0.3 electrons per unit cell. The adiabatic speed of sound[21], s, for electron gas is cm/s for 0.3 electrons per unit cell at temperature T = 0. Taking the best experimental result on resistivity along the c-axis on material YBa 2 Cu 3 O 7 δ [8] and other relative quantities from naive estimation one can estimate the relaxation time τ by using Drude model. And the result is τ to be sec. or h/τ to be 7.02 mev which is around the transition temperature T c. These are the physical quantities we need here. From the Sec.(3.3), one know that the spectrum function of the fluctuation difference due to Coulomb interaction between the electrons in the multilayer system is F coul (k, ω) = h 4πe 2 4 A (cosh kd 1) 2 + C 2 ( ) π k B 2 + C 2 sin θ1 θ 2 2 where A = ω 2 s 2 k 2 2πne2 k m and sin θ 1 = B = ω 2 s 2 k 2 2πne2 k m C = ω/τ C A 2 + C 2, sin θ 2 = sinh kd cosh kd 1 sinh kd cosh kd + 1 C B 2 + C 2 (3.31) I have rewritten the equation in order to show the physical meaning of each term in a better way. Note that F coul is the potential fluctuation, instead of the voltage fluctuation mentioned in Sec.(3.3). I have shown this spectrum function of the potential fluctuation with momentum k = 0.5/a in fig.(3.5.2). This figure can be understood in the following way: If we take τ to be zero, the maximum is at the position of ω 2 = s 2 k 2 + 2πne2 k sinh kd m cosh kd+1, i.e. the q z = π/d case. And there is a minimum at the position of ω 1 = s 2 k 2 + 2πne2 k sinh kd m cosh kd 1, i.e. the q z = 0 case. The region in between is the contribution from each q z term of the plasma modes in the multilayer system. One can easily see that the main contribution comes from the region near q z = π/d, as indicated in 47

58 Figure 3.2: The spectrum function of potential fluctuation as a function of variable ω. eqn.(3.16), where the potential difference is largest. θ 1, as related to ω 1, can be viewed as the phase shift of function A 2 + C 2, and so as θ 2 to the function B 2 + C 2. (θ 1 θ 2 )/2 shows that only those states with the range between ω 2 and ω 1 are allowed and the contribution of others are zero. As for the case of τ 0, the edge of the open window between ω 2 and ω 1 is modified smoothly. One also notices that as momentum k increases, ω 2 is gradually closer to ω 1 and so the window gets narrower. We have integrate out the k-variable to give F coul (ω), the fluctuation at the same position in coordinate space. However the result can not be expressed in a simple analytic form. F coul (ω) rises quickly as ω increases. The maximum range of ω is around 5 ev, which is much larger than the range of phonon spectrum. Since we are more interested in the frequency in the region of room temperature, we show here the approximate fitting function for ω smaller than 35 mev. 48

59 F coul (ω) = ω{ ω ω 4 + } ev 2 sec (3.32) ω is in units of rad/sec. Finally we integrate the ω-variable and the result is F coul = 1.5 (ev ) 2 (3.33) If we take the square root of this result, it will give us the mean square value of the fluctuation. And the result is 1.2 ev. This value is in the range of Coulomb energy between two electrons on two nearest neighboring layer respectively The spin fluctuation As we find out in Sec.(3.4), the spectrum function of the spin fluctuation difference between two layers can be written as F spin (k, ω) = 2 h π ω/ω SF ḡ2 χ Q (1 + (k Q) 2 ξ 2 ) 2 + (ω/ω SF ) 2 (3.34) where χ Q is approximately 75 states/ev, ω SF, the relation rate of the spin mode, is approximately 14 mev, and ξ, the coherent length of the spin modes, is approximately 2.3a at T c where a is the lattice constant along the ab plane. Once those numerical numbers are put into eqn.(3.34), we can see the spectrum of this spin fluctuation, as shown in fig.(3.5.3). This spectrum function is completely different from the ones shown in previous two section. The two previous spectra have the feature that there is no contribution from the low ω region until some beginning ω 2. Here the spectrum starts from ω = 0 and there is no cutoff frequency. It is a smooth spectrum function. We first integrate out the momentum variable k of the spectrum function over the first reciprocal lattice. The outcome is quite different from the previous two fluctuation. It rises quickly to maximum at ω about 0.1 ev, and then it decreases slowly. The range of the correlation function may 49

60 Figure 3.3: The spectrum function of spin fluctuation as a function of variable ω. be up to 10 ev. However in doing the integration of the frequency variable ω we take a reasonable cutoff to be 8 ev. In the frequency of the range around 35 mev, close to room temperature, we fit the result into the following function, F spin (ω) = ω{ ω ω ω 6 } ev 2 sec (3.35) Finally we integrate out the frequency variable ω and take the cutoff to be 8 ev. Then we obtain the following result F spin = (ev ) 2 (3.36) If we take root mean square value, we obtain that F spin is equal to 0.31 ev. This value is smaller 50

61 Table 3.1: The comparison of the strengths of three dynamic fluctuations phonon plasmon spin F (ev 2 ) F (ev) than the contribution from the plasma modes in multilayer system Comparison After all these calculations, we can compare the strength of those three fluctuation sources. We present it in the table(3.1). where F is the correlation function of the fluctuation and F (ev) is the mean square value of fluctuation at the same point and at the same time. Obviously plasmons is the strong one and spin fluctuation is one fourth of the strength of plasmons. The phonon fluctuation is at least two orders of magnitude smaller than the other two fluctuations. 51

62 Chapter 4 P-Wave Pairing in 2D Case 4.1 Introduction Since the discovery of high temperature cuprate superconductors, one of the striking features is the existence of a copper-oxide plane in all cuprate superconductors. Further experimental data and theoretical calculation on band structure strongly suggest the copper-oxide plane is mainly responsible for transport properties along the ab plane. On the other hand the properties along c-axis showed quite distinctive behavior with the ones along ab plane[3]. It leads quite a number of physicists, including us, to think that the cuprate superconductors are strongly two-dimensional materials with weak coupling along the c-axis. From the early data on upper critical magnetic field, the rough estimation of the coherent length ξ 0, which is the radius of Cooper pair at temperature T = 0, is only a few lattice constants. This result is different from the traditional superconductors, which have a coherent length ξ 0 of 10 3 to 10 4 times of lattice constants. Combining with the rough estimation of Fermi wavevector k F, one can obtain the dimensionless quantity k F ξ 0, which gives the ratio between the coherent length and the distance between electrons. This may be used indirectly as the parameter to show how many electron are within the radius of Cooper pair. For La 1.85 Sr 0.15 CuO 4, the parameter k F ξ 0 is roughly and for YBa 2 Cu3O 7 the parameter is 5-10, while for the traditional BCS superconductors k F ξ 0 is around [19]. This might suggest that the pairing in cuprate superconductors is away from BCS region where there are thousands of electrons within a radius of Cooper pair, but also not in the region of Bose condensation where the pair of Fermions forms a composite Boson, i.e. no other electrons within this composite Boson. The character of two dimensional properties and the intermedium value of k F ξ 0 in cuprate 52

63 superconductors gives us the motivation to investigate how the pairing of electron gas goes from Cooper pairing to Bose condensation as the attractive interaction between the electrons increases at T = 0 in the two dimensional case. In the paper we published[4] we studied the s-wave pairing and the p-wave pairing from weak interaction to strong interaction at T = 0 in the two dimensional case[4]. My main contribution is on p-wave pairing. In the following section I will discuss the main assumptions of the model and the important techniques in the calculation of the p-wave pairing. Some of the main results will be discussed in the last section. Detail can be found in the Appendix D. 4.2 Model and calculation In this section I will divided the discussion into several subsections to make the whole scheme much clearer. First we discuss the interaction in two dimensional case. Then we state the assumption of the model and then derive the two main equations: the gap equation and the number equation. The following subsection is about the main techniques for the calculations in the gap equation and the number equation to resolve the ultraviolet divergences in each equation. Of course, all the detail can be found in Appendix D, and I shall list the main ideas structurally here Interaction in two-dimensional case Here we discuss a model of 2D Fermi gas with an interaction between the electrons, which we assume to be a static potential V (r). Here we limit ourself to the dilute gas case, i.e. the distance between the electron, characterized by k 1, which is much larger than the range of the F potential V (r). Then we can do the analytical calculation in this condition in the low energy limit. The results are independent of the detailed shape of the potential. This two-body static potential V (r) may have a repulsive hard core at short distance and a long-range attractive interaction. The Fourier transformation of this potential may be ill defined due to the existence of hard core at short distance. So we use the two-body T-matrix to replace the two-body potential V (r). The expression can be found in eqn. (2.1) in Appendix D. Later we expand the matrix elements in terms of angular momentum eigenfunctions. Different channels decouple with each other and then we can deal with s-wave and p-wave channels respec- 53

64 tively. Furthermore we take the T-matrix in the low energy limit and express it in terms of the phase shift in two-body problem. For the p-wave case the expression is shown in eqn. (2.10) and (2.11) in Appendix D. Of course the parameters about the problem of two particle in the vacuum with the same interaction in 2D case will enter this expression through the low-energy phase shift in 2D The pairing ansatz We follow BCS method to construct the many-body pairing ground-state wave function. And then use the variation method to obtain the usual gap equation. However there are two things we need to modify. One is due to the fact that the chemical potential µ is no longer equal to Fermi energy ɛ F [28] as the attractive interaction between electrons becomes stronger. So we need the number equation, besides the gap equation, in order to obtain the chemical potential self-consistently. The number equation is N = k [ 1 ɛ k µ E k ] (4.1) where N is the number of electrons in an unit area and the excitation energy E k of the superconducting ground state is E k [ (ɛ k µ) 2 + k 2] 1/2 (4.2) The other one is about the ill defined potential V kk in the gap equation. We need to replace this potential with the well-defined two-body T-matrix. First we use the psudopotential technique[29] to integrate out the contribution from the high momentum region. Then we obtain an effective low-energy kernel, replacing V kk in the gap equation, k = < k Γ kk k 2E k (4.3) where the superscript < represents the summation over k < Λ, the momentum cutoff. However Γ kk is still in terms of V kk. Now we can use the two-body T-matrix to replace V kk. After some calculation one can obtain the kernel Γ in terms of two-body T-matrix as 54

65 Γ (l) kk = T (l) dq kk q<λ 2π T (l) kq (E)(G 0(E)) qq Γ (l) qk (E) (4.4) where we have written this equation in terms of angular component. So after variation of the assuming many-body pairing ground-state wavefunction, we obtain the number equation and the gap equation in terms of the low-energy kernel Γ kk, which connects to the two-body T-matrix. Now we can do the calculation to find the gap function k and the chemical potential µ self-consistently The P-wave pairing Since the electrons are Fermions, the total wavefunction should be antisymmetric. If the pair wavefunction is in p state (l = 1), then the spin state of the pair of electrons must be triplet. Here we have done two cases: one with the gap function to be the angular dependence k e iθ, the other one with k sin θ. In the calculation, we first take the low-energy limit in all quantities and then proceed the calculation. One can integrate the gap equation and the number equation, and solve them self-consistently. However both equations have ultraviolet divergence, where the contribution comes from the region of the short distance in the coordinate space. In the case of gap equation, one can choose the free parameter, introduced by the replacement of the two-body T-matrix, to be an appropriate quantity to eliminate the ultraviolet divergence. The result is presented in eqn. (6.8) in Appendix D. The case for number equation is a little complicated. The logarithmic divergence is very similar to the divergence of the normalization of the two-body Schrodinger wavefunction in the momentum space. So we subtract both sides of the number equation with the chosen factors times the normalization of the wavefunction to eliminate the unwanted divergence. The detail of the mathematics can be found in Appendices B and C in Appendix D. The result is shown in eqn. (6.16) in Appendix D. Both gap equation and number equation are independent of the cutoff after taking care of ultraviolet divergence. 55

66 4.3 Conclusion: P-wave We have studied two cases in p-wave pairing: one with gap function e iθ, and the other with gap function cos θ. θ is the angular variable. Once the gap equation and the number equation are known, one can solve these two equations in terms of the parameters that characterize the two-body interaction between the electrons. First we obtain the analytic result in the BCS limit and the composite Boson limit. The results are consistent with the ones obtained in the traditional way in both limits respectively. In the region between those two limits we can only solve it numerically and obtain the numerical expression of gap function and the chemical potential in terms of the strength of the attractive potential between the electrons. The results and detailed discussion can be found in Sec. VI (C) in Sec.VIII and IX in Appendix D. Once we know the gap function and the chemical potential, we can obtain the spectrum of excitation energy of the superconductors by using the equ. (4.2). For the case of gap function e iθ, the magnitude of the energy gap, i.e. minimum energy to excite a quasiparticle, may change according to the change of the chemical potential but nevertheless has angular independence. However in the case of gap function cos θ, in the BCS limit, where the chemical potential is nearly equal to Fermi energy, there is a angular dependence of the energy gap. Actually there exists a node line to this specific case. While in the composite Boson limit, this angular dependence disappears. The reason is due to the fact that in this region the chemical potential is negative. There exists a transition region from the angle-dependent region to the no-angle-dependent region. The Figures of this transition and the other discussion is in Sec. VII in Appendix D. 56

67 Chapter 5 Conclusion In this chapter I will summarize all the work I have done in this thesis and point out the future work that should be pursued to extend the understanding of the subject that we have studied here. In order to understand several properties of high temperature cuprate superconductors, we use simplified models to study the properties we want to study. We use two-layered system to study the c-axis transport properties. First we study the static impurity case to study how the density of impurities change the tunneling behavior from coherent to incoherent one. We find out there are three energy scales in this problem: the kinetic energy ɛ, the tunneling matrix element and the relaxation rate Γ a which characterized the density of antisymmetric impurities and the strength of impurity potential. We discuss only the case that ɛ >> hγ a where the electron gas on the layers can be treated as Fermi gas. As we find out the transport property along the c-axis depends on the ratio of Γ a / : if this ratio smaller than 1, it is a coherent tunneling along the c-axis; otherwise it is a incoherent tunneling. The crossover is a smooth one according to the calculation. we also show that the symmetric impurities play no role in the tunneling behavior along c-axis. The result can be phenomenologically written as d 2 P dt 2 + 2iΓ dp a dt + 2 P = 0 (5.1) where P represents the probability of finding the extra particle in layer 1. However the calculations are done here on the two limits: Γ a / < 1 case and Γ a / > 1 case. A consistent method is needed to study the tunneling behavior around the crossover region. Furthermore, we have only studied the one-electron system. A extending study on the manyelectron system will be pursued in the future. From the beginning we have assume the potential 57

68 wells is fixed. So the result will give us the DC conductivity. In order to obtain result on the AC conductivity, one might apply the external periodic voltage on the wells in order to study the tunneling behavior between the wells. It is totally different from the DC case and has been little studied. For the case of impurities between the layers we assume that those impurities provide extra tunneling centers besides existing tunneling matrix element. According to the calculation on the two-layer system, it does not change the coherent tunneling behavior but provide extra decaying behavior. The only situation that the tunneling behavior is an incoherent one is when the tunneling matrix element is zero, i.e. there is no tunneling between the layers unless there exists impurities between the layers playing the role as the random tunneling centers. However these results might be a special case due to two-layer system. We need to extend the study into multilayer system to see whether the results above is valid or not. We have also studied three sources of dynamic fluctuation: the phonons, the plasmon, and spin fluctuation. Those are studied in the multilayer system and reduced the results into two-layer system. We have derived the spectrum function of the fluctuation difference of those three sources. We use a single parameter to compare the strength of those three fluctuations. From the result, we find out that the plasmons in multilayer system is the strongest one. And the following one is the spin fluctuation, roughly in the same order. The weakest one, two orders of magnitude smaller, is the phonon. In the future, we will study the effect of those dynamic fluctuations on the tunneling behavior along the c-axis. It is much more complicated than the case of static impurities since there is energy transfer during the scattering for the case of dynamic fluctuation. And we have done the study of p-wave pairing in 2D case. We use scattering length in 2D to characterize the strength of potential between two electrons. We have studied how paring function change from weak interaction to strong attractive interaction. Also we have studied how the energy gap of quasiparticles change for anisotropic gap function as the chemical potential changes. 58

69 Appendix A The Overdamped Case In this appendix we will describe the procedure to calculate (1) the one-particle Green s function and (2) the probability shown in eqn.(2.41). First we need to discuss some remarks about the Green s functions after ensemble average which will be useful in both calculations. A.1 Some remarks First we expand the Green s functions in terms of tunneling Hamiltonian, shown in eqn.(2.43). Then we use representation of { 1k, 2k } inserting in every Hamiltonians and Greens functions in this equation. One may find out that the tunneling Hamiltonian H T is in diagonal form in this representation but not the zeroth order Hamiltonian H 0, defined in eqn.(2.32). Second, we expand the zeroth order Hamiltonian in terms of kinetic energy term and impurity term. In this stage we can take ensemble average of impurities on this one-particle Green s function. We make a few assumptions to simplify the calculation. First we assume the vertex correction term for the external impurity can be neglected in our claculation. Then there is no line-line crossing term in one-particle Green s function. Then we can obtain the ensemble average of the one-particle Green s function defined in zeroth order Hamiltonian H 0, 0 (ik, jk 1 ; ɛ) = ik ɛ H 0 ± iη jk G (R,A) = δ i,j δ(k k 1 ) ɛ ɛ k ± iγ/2 (A.1) where (i, j) can be (1, 2). Second we assume the vertex correction for the external tunneling 59

70 correction can be neglected in the zeroth order approximation. Then there is no line crossing between two one-particle Green s functions connecting by a tunneling process. It implies that we can treat the zeroth-order one-particle Green s function and the tunneling term independently. It greatly simplifies the calculation. Then the one-particle Green s function in the full Hamiltonian after complete ensemble average of impurities can be derived, G (R,A) (1k, 1k ; ɛ) = δ(k k ) G (R,A) 0 (1k; ɛ) [ ( 1 2 h ) G(R,A) 0 (1k; ɛ) ( 1 2 h ) G(R,A) 0 (2k; ɛ) i=0 = 1 [ ] 2 δ(k 1 k ) ɛ ɛ k + h /2 ± iγ/2 + 1 ɛ ɛ k h /2 ± iγ/2 = G (R,A) (2k, 2k ; ɛ) (A.2) G (R,A) (1k, 2k ; ɛ) = δ(k k ) G (R,A) 0 (1k; ɛ) ( 1 2 ) h ) G(R,A) 0 (1k; ɛ) [ ( 1 2 h ) G(R,A) 0 (1k; ɛ) ( 1 2 h ) G(R,A) 0 (2k; ɛ) n=0 = 1 2 δ(k 1 k )[ ɛ ɛ k + h /2 ± iγ/2 1 ɛ ɛ k h /2 ± iγ/2 ] = G (R,A) (2k, 1k ; ɛ) (A.3) ] n ] n However, in deriving two-particle Green s functions, we need to take ensemble average over those lines within the same one-particle Green s function, but leave those lines that connect to different one-particle Green s function and do the ensemble average later. Then, in this case, the one-particle Green s function can be written as G (R,A) (1k, 1k ; ɛ) = G (R,A) (1k; ɛ) δ k,k + G (R,A) (1k; ɛ) q ρ 1 (q)v 0 G (R,A) (1k + q, 1k ; ɛ)δ k+q,k + G (R,A) (1k, 2k; ɛ) q ρ 2 (q)v 0 G (R,A) (2k + q, 1k ; ɛ)δ k+q,k G (R,A) (1k, 2k ; ɛ) = G (R,A) (1k, 2k; ɛ) δ k,k + G (R,A) (1k; ɛ) q ρ 1 (q)v 0 G (R,A) (1k + q, 2k ; ɛ)δ k+q,k 60

71 + G (R,A) (1k, 2k; ɛ) q ρ 2 (q)v 0 G (R,A) (2k + q, 2k ; ɛ)δ k+q,k (A.4) One can use iteration to obtain the higher order terms. We will use these results intensively in the following calculation. A.2 One-particle Green s function From the Sec one know that the appropriate one-particle Green s function, after some reduction, is G(ɛ, ɛ ) = = i 1 1 Tr 1 { 2πN 0 ɛ H 0 iη ɛ H + iη } i G A 0 (1k, 1k ; ɛ)g R (1k, 1k; ɛ ) 2πN 0 k,k (A.5) The expansion of G R (1k, 1k; ɛ ) can be found in eqn.(a.4). Using the same argument, one can obtain the similar equation for G A 0 (1k, 1k ; ɛ), i.e. G (R,A) 0 (1k, 1k ; ɛ) = G (R,A) 0 (1k; ɛ) δ k,k + G (R,A) 0 (1k; ɛ) q ρ 1 (q)v 0 G (R,A) 0 (1k + q, 1k ; ɛ)δ k+q,k (A.6) where the expression of G (R,A) 0 (1k; ɛ) can be found in eqn.(a.1). Now we have both Green s functions expanding in terms of impurities. We may take ensemble average of the impurity line that connecting these two Green s function. Again we use the randomphase approximation in this calculation. Let us define those quantities which are needed in calculation of G (R,A) 0 (1k, 1k ; ɛ) and in later calculation of two-particle Green s function. I(ii; ɛ, ɛ ) k G A 0 (1k, 1k; ɛ) G R (ik, ik; ɛ ) 61

72 = i2πn 0 1 { 2 ɛ ɛ + /2 + iγ + 1 ɛ ɛ /2 + iγ } = I(22) I(12) k G A 0 (1k, 1k; ɛ) G R (1k, 2k; ɛ ) = i2πn { ɛ ɛ + /2 + iγ 1 ɛ ɛ /2 + iγ } = I(21) (A.7) If we catalog the terms according to the number of tunneling, there are two parts in G(ɛ, ɛ ) in eqn.(a.5). One is the even number of tunneling which is composed by I(11) and can be written as R e = I(11) + I(11)(n s + n a )v 2 0R e = I(11) 1 (n s + n a )v 2 0 I(11) However G(ɛ, ɛ ) can also be composed of I(12) and I(21), odd number in tunneling, together and can be written R o = = 1 (n s n a )v0 2 1 (n s + n a )v0 2I(11)I(12) 1 (n s n a )v0 2I(11) (n s + n a )v0 2 (n s n a )v0 2 {I(21) 1 (n s + n a )v0 2I(22)I(12) 1 (n s n a )v0 2I(11)}l l=0 1 I(21) 1 (n s + n a )v0 2I(11) I(12) (n s n a )v 2 1 (n s +n a )v0 2I(11) 0 1 (n s n a )v0 2I(11) (n s+n a)v 2 0 I(21) 1 (n s +n a )v0 2I(11) I(21) 1 (n s+n a)v0 2I(22)I(12) (n s n a)v0 2 1 (n s n a)v 2 0 I(11) So the complete form of G(ɛ, ɛ ) is composed by R e and R o. The explicit form of I(ij) can be found in eqn.(a.7) and Γ s in eqn.(2.20). The reduction of G(ɛ, ɛ ) is extremely tedious. But the form of the final result is extremely simple, shown as 62

73 G(ɛ, ɛ i ) = ɛ ɛ ( 1 2 h )2 ɛ ɛ+2iγ a (A.8) The term, ( 1 2 h )2 ɛ ɛ+2iγ a can be viewed as the self-energy term of the one-particle Green s function. A.3 Alternative method: spectral Function There is another method to derive the one-particle Green s function G(ɛ, ɛ ) which might give use a much clearer meaning of the self-energy term. Let us start from the eqn.(2.33). Since the Hamiltonian in the exact-eigenstate representation, shown in eqn.(2.32), is composed by the zeroth-ordered Hamiltonian H 0 and the tunneling Hamiltonian H T, one can obtain Dyson s equation of one-particle Green s function and it is G(1n, 1n; ɛ ) = G (0) (1n; ɛ ) + G (0) (1n; ɛ )( 1 2 h ) φ n χ n G(2n, 1n; ɛ ) where G (0) (1n; ɛ ) = G (0) (2n ; ɛ ) = 1 ɛ ɛ n + iη 1 ɛ ɛ n + iη One can write down the perturbation expansion of G(1n, 1n; ɛ ). We can also write the Dyson s equation in term of self-energy term, i.e. G(1n, 1n; ɛ ) = G (0) (1n; ɛ ) + G (0) (1n; ɛ )Σ(1n; ɛ )G(1n, 1n; ɛ ) (A.9) Here we introduce the self-energy Σ(1n; ɛ ) for state 1, n. The lowest order of this self-energy term is Σ (1) (1n; ɛ ) = n ( 1 2 h )2 φ n χ n G (0) (2n ; ɛ ) χ n φ n 63

74 = ( 1 2 h )2 f n (ɛ ) (A.10) In this way we have defined a new function f n (ɛ ). The imaginary of this function is Imf n (ɛ ) = π n φ n χ n 2 δ(ɛ ɛ n ) (A.11) This is the spectral function, except a constant 1/π, of transverse wavefunction of state 1, n mapping into the transverse wavefunction of the complete eigenstate set { 2, n }. However this spectral function is n-dependent, i.e. it is sample-dependent. The appropriate spectral function, after ensemble average, is F (ɛ, ɛ ) = 1 N 0 n 1 π Imf n(ɛ )δ(ɛ ɛ n ) av. (A.12) where N 0 is the average density of state. This is the similar function defined in paper by Abrahams and et al.[30]. The self-energy term can be defined in the same fashion in order to be sampleindependent, i.e. Σ (1) (ɛ, ɛ ) = 1 Σ (1) (1n; ɛ )δ(ɛ ɛ n ) N 0 n (A.13) This expression is independent of the choice of representation. Then we use the similar method shown in the calculation of one-particle Green s function here. Then the self-energy term, after ensemble average, becomes Σ (1) (ɛ, ɛ ) = ( 1 i 2 h )2 { G R 0 (1k, 1k ; ɛ)g R 0 (2k, 2k; ɛ ) 2πN 0 kk G A 0 (1k, 1k ; ɛ)g R 0 (2k, 2k; ɛ ) } (A.14) The first term is zero since both terms are retarded Green s functions. We can use the similar method of calculating I(ij) in the previous section to calculate the second term. We will not repeat the calculation here even though the result is a little different. The approximations we use are rainbow approximation for one-particle Green s function and random-phase approximation for two-particle Green s function. The final result of the first order self-energy term is 64

75 and the appropriate spectral function is Σ (1) (ɛ, ɛ ) = ( 1 2 h )2 (ɛ ɛ ) + 2iΓ a (A.15) F (ɛ, ɛ ) = 2Γ a (ɛ ɛ ) 2 + (2Γ a ) 2 (A.16) So one can see that the spectral function is a Lorentz-shaped function with a width 2Γ a. One may go beyond first order self-energy term. But special care should be taken and it is not the focus here. A.4 Two-particle Green s function Now we turn to the calculation of probability which is essential a two particle Green s function. From eqn.(2.41), one knows that the probability can be composed by two parts, P (1, 1; R : ɛ, ω) and P (1, 1; A : ɛ, ω). We calculate P (1, 1; A : ɛ, ω) first. From eqn.(2.41), P (1, 1; A : ɛ, ω), after ensemble average, can be written as P (1, 1; A : ɛ, ω) = i dɛ G R (1k, 1k ; ω + ɛ ) 2πN 0 2π kk k G A 0 (1k, 1k ; ɛ)g A (1k, 1k; ɛ ) (A.17) In this equation, we have used { 1, k, 2, k } as the representation since this expression is independent of the choice of representation. Essentially, it is a three-particle Green s function. We follow the same scheme we use in the calculation of one-particle Green s function. First let us expand G R and G A in terms of order of tunneling matrix element. Later, we expand each term in terms of order of impurities. Then we take partial ensemble average described in Sec.A.1 1, except those lines that connecting different one-particle Green s function. Again we can regroup the tunneling terms together since there is no impurity line crossing the tunneling term. Then we obtain those expression of G R and G A shown in eqn.(a.4). It is the same thing for G0 A, with the result shown in eqn.(a.6). Now let us connect those lines between those three one-particle Green s functions. 1 Of course, we use rainbow approximation for these two one-particle Green s function because of Migdel s theorem. 65

76 There are too many diagrams. However one might notice that the diagrams containing lines between G A and G A 0 do not contribute since both Green s functions are advanced ones. Only those diagrams that contain lines between G R and (G A, G A 0 ) remain. It greatly simplifies the diagrams. Now we further assume that we choose random-phase approximation for this three-particle Green s function. If one rearrange the diagrams, one may obtain the following equation, P (1, 1; A : ɛ, ω) = dɛ 2π a,b,c=1,2 +R(a)S(a, b; 1, c)t (b, c) (A.18) where those connecting diagrams are S(a, b; 1, c) = k G R (ak, bk; ɛ + ω) G A 0 (1k, 1k; ɛ) G A (1k, ck; ɛ) (A.19) and the diagrams R, in front of S, can be written as R(a) = δ 1,a kqq [ ρ a (q)ρ 1(q ) v 2 0] G R (1k, ak + q; ɛ + ω)g A 0 (1k, 1k + q ; ɛ) (A.20) where i is either (1, 2), the index of layers, and we may use the property described in Sec.2.2.3, i.e. ρ 1 (q)ρ 2 (q ) = (n s n a )δ(q q ). The last set of the diagrams T, after the diagram S, are T (b, c) = δ 1,b δ 1,c + kqq [ ρ b (q)ρ c(q ) v 2 0] G R (bk + q, 1k; ɛ + ω)g A (ck + q, 1k; ɛ) (A.21) So we have to calculate each term of R, S and T. First let us calculate R(i). In order to calculate these terms, we need the function I(ij) defined in eqn.(a.7). The method to construct R(1) is the same as the one in construct one-particle Green s function in this appendix. R(1) is composed by two parts. One consists of even number of tunneling events. The other is odd number of tunneling. We won t repeat the calculation but just show the final result, R(1) = 1 1 I(11)(n s + n a )v

77 + 1 1 I(11)(n s +n a )v0 2 I(12)(n s n a)v I(22)(n s n a )v I(21)(ns+na)v2 0 1 I(11)(n s+n a)v 2 0 I(21)(n s+n a)v I(11)(n s +n a )v 2 0 I(12)(n s n a)v I(22)(n s n a)v 2 0 (A.22) After putting I(ij) shown in eqn.(a.7), one can obtain the complete form of R(1) as R(1) = (ɛ ɛ + ω + 2iΓ a )(ɛ ɛ + ω + iγ) ( /2) 2 (ɛ ɛ + ω + 2iΓ a )(ɛ ɛ + ω) ( /2) 2 (A.23) In the similar way, one can obtain the quantity R(2), which is odd number of tunneling in total, in the following way R(2) = = 1 I(12)(n s n a )v 2 1 I(11)(n s+n a)v I(22)(n s n a)v0 2 1 I(21)(n s+n a )v I(11)(n s +n a )v 2 0 I(12)(n s n a )v I(22)(n s n a )v 2 0 i(γ s Γ a )( /2) (ɛ ɛ + ω + 2iΓ a )(ɛ ɛ + ω) ( /2) 2 (A.24) We now turn to calculate four terms of T. Similar in the case of R, we need to define the following sixteen terms in order to simplify the calculation. J(ii, jj) k G R (ik, ik; ɛ + ω) G A (jk, jk; ɛ) = i2πn 0 4 J(12, jj) k = i2πn 0 4 = J(21, jj) 2 { ω + iγ + 1 ω + iγ + 1 ω + + iγ } G R (1k, 2k; ɛ + ω) G A (jk, jk; ɛ) 1 { ω + iγ + 1 ω + + iγ } J(ii, 12) k G R (ik, ik; ɛ + ω) G A (1k, 2k; ɛ) = i2πn 0 4 = J(ii, 21) 1 { ω + iγ 1 ω + + iγ } J(12, 12) k G R (1k, 2k; ɛ + ω) G A (1k, 2k; ɛ) = i2πn { ω + iγ 1 ω + iγ 1 ω + + iγ } 67

78 = J(12, 21) = J(21, 12) = J(21, 21) (A.25) where (i, j) can be either (1, 2). Only J(ii, jj) are even number of tunneling in both retarded and advanced Green s functions. Others are odd number of tunneling in either or both Green s functions. One can use single J(ii, jj) to form an iteration equation without changing the layer indices of the Green s function. So we can define T 0 (ii, jj) = (n s ± n a )v 2 0J(ii, jj) + (n s ± n a )v 2 0J(ii, jj)t 0 (ii, jj) (A.26) where + sign for i = j case and sign for i j case. The other 12 terms of J can change the layer indices of the Green s functions. We can write the equations for the 4 terms of T that we are interested in 2 T (11, 11) = (1 + T 0 (11, 11)) +(1 + T 0 (11, 11))(n s + n a )v 2 0J(12, 11)T (21, 11) +(1 + T 0 (11, 11))(n s + n a )v 2 0J(11, 12)T (11, 21) +(1 + T 0 (11, 11))(n s + n a )v 2 0J(12, 12)T (21, 21) T (21, 11) = (1 + T 0 (22, 11))(n s n a )v 2 0J(21, 11)T (11, 11) +(1 + T 0 (22, 11))(n s n a )v 2 0J(22, 12)T (21, 21) +(1 + T 0 (22, 11))(n s n a )v 2 0J(21, 12)T (11, 21) T (11, 21) = (1 + T 0 (11, 22))(n s n a )v 2 0J(12, 22)T (21, 21) +(1 + T 0 (11, 22))(n s n a )v 2 0J(11, 21)T (11, 11) +(1 + T 0 (11, 22))(n s n a )v 2 0J(12, 21)T (21, 11) T (21, 21) = (1 + T 0 (22, 22))(n s + n a )v 2 0J(21, 22)T (11, 21) +(1 + T 0 (22, 22))(n s + n a )v 2 0J(22, 21)T (21, 11) +(1 + T 0 (22, 22))(n s + n a )v 2 0J(21, 21)T (11, 11) (A.27) 2 One might use iteration method, like the one we use in drive R(i), to derive T. But it is more complicated and we may miss a few terms easily. So we choose this algebra method. 68

79 There are four equations for four unknown variables T with other quantities shown in eqns.(a.25) and (A.26). So one can carry out the calculation of these algebra. And the results are T (11, 11) = 1 + i 2 (Γ 1 s + Γ a ){ ω + iη + ω + 2iΓ a ω(ω + 2iΓ a ) 2 } T (21, 11) = i 2 (Γ s Γ a ) ω(ω + 2iΓ A ) 2 T (11, 21) = i 2 (Γ s Γ a ) ω(ω + 2iΓ A ) 2 T (21, 21) = i 2 (Γ 1 s + Γ a ){ ω + iη + ω + 2iΓ a ω(ω + 2iΓ a ) 2 } (A.28) where η 0 +. Now let us calculate the center part S(ab; 1c). We just have to use integration contour in the complex plan to calculate S(ab; 1c) and the result are S(11, 11) = i2πn 0 1 { 4 ɛ + ω ɛ + /2 + iγ [ 1 ω + iγ + 1 ω + + iγ ] 1 + ɛ + ω ɛ /2 + iγ [ 1 ω + iγ + 1 ω + iγ ]} = S(22, 11) S(12, 11) = i2πn 0 1 { 4 ɛ + ω ɛ + /2 + iγ [ 1 ω + iγ + 1 ω + + iγ ] 1 ɛ + ω ɛ /2 + iγ [ 1 ω + iγ + 1 ω + iγ ]} = S(21, 11) S(11, 12) = i2πn 0 1 { 4 ɛ + ω ɛ + /2 + iγ [ 1 ω + iγ 1 ω + + iγ ] 1 + ɛ + ω ɛ /2 + iγ [ 1 ω + iγ 1 ω + iγ ]} = S(22, 12) S(12, 12) = i2πn 0 1 { 4 ɛ + ω ɛ + /2 + iγ [ 1 ω + iγ 1 ω + + iγ ] 1 ɛ + ω ɛ /2 + iγ [ 1 ω + iγ 1 ω + iγ ]} = S(21, 12) (A.29) Since we have all the elements, as R in eqn.(a.23) and (A.24), S in eqn.(a.29) and T In 69

80 eqn.(a.28), we may use eqn.(a.18) to obtain P (1, 1; A : ɛ, ω), and the result, after integrating out the variable ɛ, is P (1, 1; A : ɛ, ω) = i 4 { } 2 ω + ω + 2iΓ a ω(ω + 2iΓ a ) ( ) 2 + ω + + 2iΓ a ω(ω + 2iΓ a ) ( ) 2 (A.30) Now we have done the calculation P (1, 1; A : ɛ, ω), part of the requirement in eqn.(2.41). The other part, P (1, 1; R : ɛ, ω), in this equation can be obtained from eqn.(a.30), by taking the complex conjugate of P (1, 1; A : ɛ, ω) in the form of definition shown in eqn..(2.41) and interchanging the variables between (ɛ + ω) and ɛ, later integrating out the variable ɛ, then one may find that P (1, 1; R : ɛ, ω) = P (1, 1; A : ɛ, ω) (A.31) And the final result for P (1, 1; ɛ, ω) is P (1, 1; ɛ, ω) = i 2ω + i ω + 2iΓ a 2 ω(ω + 2iΓ a ) ( ) 2 (A.32) 70

81 Appendix B Interlayer case In this section we will calculate the last two terms in eqn.(2.51). Let us first calculate the third term, i.e. P (3) = 1 4 dω 2π δ(ɛ ɛ k ) G(e, k; e, k ; ω )G (o, k ; o, k; ω ω) kk 1 (B.1) We first expand the Green s function according to the Dyson s eqn. (2.49). Then we need to take ensemble average. As we have done in underdamped case, we use rainbow approximation for one-particle Green s function where the result is shown in eqn. (2.50), and the random-phase approximation for two-particle Green s function. Those two approximations are consistent with the requirement to insure that the probability is conserved. Before we present the result of the calculation for P (3), let us first calculate the following two quantities which simplify the expression of P (3). F eo (s) (ω) (n imp ( 1 2 h 1) 2 ) G(e, k + q; ω ) G (o, k + q; ω ω) q = iγ ω + + iγ F (s) oe (ω) q (n s v 2 0) G(o, k + q; ω ) G (e, k + q; ω ω) = iγ ω + iγ (B.2) Then the final result of P (3), after the ensemble average, is 71

82 P (3) = 1 4 dω 2π δ(ɛ ɛ k ) G(e, k; ω ) G (o, k; ω 1 ω) (B.3) 1 F kk 1 eo One notices that F eo is independent of variable ω. So we can integrate out the variable ω by using the contour integral in the complex-ω plane. And the final result is P (3) = i 4 1 ω + + 2iΓ (B.4) In the similar way, one can obtain P (4) as P (4) = i 4 1 ω + 2iΓ (B.5) So we have obtained the last two terms in eqn.(2.51) 72

83 Appendix C Plasma modes C.1 In multilayer system Here we give a brief derivation of the induced potential in terms of an external charge density, which we need in the calculation of response function. There are three classical equations in describing the motion of the electrons. Let us take Fourier transformation of those equations. The first one is the equation of continuity, ωn j( k, ω) k v j ( k, ω) = 0 (C.1) where k is the wavevector along the two-dimensional space and the index j refers to the j-th layer. The second one is the classical hydrodynamic equation, (ω + i/τ) v j ( k, ω) = s2 n 0 kn j( k, ω) e m kφ( k, z j, ω) (C.2) Note that φ is a function of three-dimensional coordinate and we need to specify the z variable in this equation. We may use the first two equation to eliminate the velocity variable v j and obtain the following equation [ω(ω + i/τ) s 2 k 2 ]n j( k, ω) = ne m k2 φ( k, z j, ω) (C.3) In this equation, the RHS is the term related with the force. If the particles in the discussion carry no charge, then this equation is the result of the neutral fluid. One can find out the energy spectrum 73

84 of this fluid is [ω(ω + i/τ) s 2 k 2 ], which is the sound wave with a damping term. However what we are discussing here is electrons. So we need to write down the equation of Coulomb interaction which includes the force caused by the induced charge density, ( d2 dz 2 k2 )φ( k, z, ω) = 4π(ρ ex ( k, z, ω) + ρ in ( k, z, ω)) (C.4) where the induced charge density is the variation of the surface charge density on each layer, and can be expressed in the following way, ρ in ( k, z, ω) = e l n l( k, ω)δ(z z l ) (C.5) For further calculation, we need to take Fourier transformation w.r.t. coordinate variable z. But special care is needed. C.2 Fourier transformation w.r.t. z The definition of the Fourier transformation is as usual, ρ ex ( k, z, ω) = dq 2π eiqz ρ ex ( k, q, ω) (C.6) where ρ ex ( k, q, ω) = dze iqz ρ ex ( k, z, ω), < q < Here we insist that the variable q from to since this is an infinite-layer system and Coulomb force is a long range interaction[31]. Then the Gauss law of the Coulomb interaction, after the Fourier transformation, turns to be φ tot ( k, q, ω) = 4π k 2 + q 2 [ρ ex( k, q, ω) + ρ in ( k, q, ω)] where < q < (C.7) However, we need to pay special attention to the induced charge density ρ in ( k, q, ω). After a few steps of derivation, one can obtain 74

85 ρ in ( k, q, ω) = e l n l( k, ω)e iqz j (C.8) where the position of j-th layer is z j = j d, j is an integer and d the space distance between two nearest neighboring layers. One immediately notices that this induced charge density ρ in ( k, q, ω) is a periodic function since ρ in ( k, q + Q, ω) = ρ in ( k, q, ω) where Q = 2πp/d and p is an integer. Now we want to take Fourier transformation of eqn.(c.3). Since eqn.(c.8), we may multiply eqn.(c.3) with ( e)e iqz j and sum over j. Then we obtain the following equation, [ω(ω + i/τ) s 2 k 2 ]ρ in ( k, q, ω) = ne2 m k2 j e iqz j φ( k, z j, ω) (C.9) So one just has to take the Fourier transformation of φ( k, z j, ω). However in the infinite layer system we need to use the following equation in order to obtain the correct answer. e i(q q) z j = 2π d j m= δ[(q q) 2mπ/d], z j = j d (C.10) And sine ρ in is a periodic function, one may obtain the following equation, ρ in ( k, q, ω) = 4πne2 k 2 md Q ρ ex ( k, q + Q, ω) k 2 + (q + Q) 2 [ω(ω + i/τ) s 2 k 2 2πne2 k m sinh kd cosh kd cos qd ] 1 (C.11) where Q = 2πm/d and the sum is over all the integer. In deriving this equation, we have used the following equation, Q 1 k 2 + (q + Q) 2 = d sinh kd 2k cosh kd cos qd (C.12) Now we will start from this eqn.(c.11) to obtain the result we want. 75

86 C.3 The response function What we want to know is the induced potential φ in ( k, z j, ω) due to the existence of the external charge density ρ ex ( k, z j, ω). So we just have to take the Gauss law for the induced potential φ in to replace the induced charge density ρ in in eqn.(c.11). Then taking the Fourier transformation for induced potential φ in and the external charge density ρ ex we will obtain φ in ( k, z j, ω) = Q dq 2π eiqz j 1 k 2 + (q + Q) 2 4π 4πne 2 k 2 k 2 + q 2 md [ω(ω + i/τ) s 2 k 2 2πne2 k m dz e i(q+q)z ρ ex ( k, z, ω) sinh kd cosh kd cos qd ] 1 (C.13) In this expression, one notices the summing over Q term. This is because the induced potential term is a periodic function and the Coulomb interaction is a long range interaction. So all the force induced by the ρ ex ( k, q + Q, ω) will act on the induced potential φ in ( k, q, ω). From the Sec one know that response function we are interested in is in the form of eqn.(3.3.1). Then using the equation above, one can obtain the explicit form of the response function 1 as χ V V ( k, z j, z i, ω) = dqe iq(z j z i ) 1 4πne 2 k k 2 + q 2 m [ω(ω + i/τ) s 2 k 2 2πne2 k m sinh kd cosh kd cos qd sinh kd cosh kd cos qd ] 1 (C.14) C.4 Reducing to two-layered system Form the eqn.(3.15) one knows the relationship between the response function of potential difference and the the response function of potential. In a two-layer system we don t need variable q, which characterizes the wavevector along the z-direction. In order to include all the possible fluctuation we need to integrate out the variable q, i.e. 1 The general form of χ V V ( k, z, z, ω), which is more complicated, can also be obtained. 76

87 χ V V ( k, ω) = dq2(1 cos qd) 4πne2 k m [ω(ω + i/τ) ω(k, q) 2 ] 1 1 k 2 + q 2 sinh kd cosh kd cos qd (C.15) The definition of ω(k, q) can be found in eqn.(3.17), which is the energy spectrum of the plasma modes for multilayer system without the damping term. Now the rest of the calculation is just a mathematical technique. First of all, Let us rearrange the eqn. (C.15) as following χ V V ( k, ω) = (8πne2 k/m) sinh kd ω(ω + i/τ) s 2 k 2 where C = cosh kd (2πne2 k/m) sinh kd ω(ω + i/τ) s 2 k 2 1 (1 C) dq k 2 {1 + q2 cos qd C } (C.16) This integral can be solved by using the complex variable q. We take the upper semicircle contour in the complex-q plane. One can find the the integral along the semicircle arc approaches zero as the radius of the semicircle approaches infinity. So the integral of eqn.(c.16) is equal to the sum of the residues in the upper plane of complex-q plane. There are two kinds of residues. We explore them one by one. 1. q = ik case: After a few steps of calculation one can obtain the contribution from this residue as Re(1) = 4π k (cosh kd 1) (C.17) So this term is independent of frequency. 2. cos qd = C case: This is the case that the poles exist at cos qd = cosh kd (2πne2 k/m) sinh kd ω(ω + i/τ) s 2 k 2 (C.18) Let us set cos(k r + ik i )d = C and require that π/d k r π/d and k i > 0. The constant C is 77

88 shown in eqn.(c.16). Since the imaginary part of C is always positive, that further restricts the region of k r to be π/d k r 0. In principle one can find out the unique solution for k r and k i. Then the poles of this case are those 2 q = (k r + ik i ) + Q (C.19) where Q = 2nπ/d and n is an integer. After careful calculation one can obtain the contribution from these residue as Re(2) = i 4π k [ ] sinh kd 1 cosh kd + (2πne2 k/m) sinh kd sin(k r + ik i )d ω(ω + i/τ) s 2 k 2 (C.20) In deriving this result we have used the eqn.(c.16). Now we need to identify the value of sin(k r + ik i )d. In the limit of ω, one knows from eqn.(c.18) that k r = 0 and k i = k. And so sin(k r + ik i )d = i sinh kd in the limit of ω. This result will help us to identify the sign when we derive unknown sin x from known cos x. One can use the usual rectangular triangle theorem to derive the sin(k r + ik i )d, and the result is sin(k r + ik i )d = ±i{sinh 2 kd 2 cosh kd (2πne2 k/m) sinh kd ω(ω + i/τ) s 2 k 2 +[ (2πne2 k/m) sinh kd ω(ω + i/τ) s 2 k 2 ]2 } 1/2 (C.21) From the requirement of the sigh in the limit of ω we find out the positive sign is the correct result. So one can put this result in eqn.(c.20) to obtain the contribution from the poles in the second case. Combining the contribution from two type of poles and making some arrangement one would obtain χ V V = 4π k (cosh kd 1) 2 There is another set of poles below the real axis, but we are interested in the poles in the upper complex-q space. 78

89 4π k sinh kd (cosh kd 1) (2πn0 e 2 k/m) sinh kd ω(ω+i/τ) s 2 k 2 (cosh kd + 1) (2πn0 e 2 k/m) sinh kd ω(ω+i/τ) s 2 k 2 1/2 (C.22) This is the final result we want. 79

90 Appendix D P-Wave Pairing in 2D Case In this appendix we give the detail ca lculation in Chap.4. This is a reprint of the article published in Physical Review B41, 327 (1990), a work done in collaboration with Mohit Randeria and Ji-Min Duan. 80

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