Low Energy Quasiparticle Excitations in Cuprates and Iron-Based Superconductors

University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange

Doctoral Dissertations Graduate School

12-2018

Low Energy Quasiparticle Excitations in Cuprates and Iron-based Superconductors

Ken Nakatsukasa University of Tennessee, [email protected]

Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss

Recommended Citation Nakatsukasa, Ken, "Low Energy Quasiparticle Excitations in Cuprates and Iron-based Superconductors. " PhD diss., University of Tennessee, 2018. https://trace.tennessee.edu/utk_graddiss/5300

This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council:

I am submitting herewith a dissertation written by Ken Nakatsukasa entitled "Low Energy Quasiparticle Excitations in Cuprates and Iron-based Superconductors." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Physics.

Steven Johnston, Major Professor

We have read this dissertation and recommend its acceptance:

Christian Batista, Elbio Dagotto, David Mandrus

Accepted for the Council:

Dixie L. Thompson

Vice Provost and Dean of the Graduate School

(Original signatures are on file with official studentecor r ds.) Low Energy Quasiparticle Excitations in Cuprates and Iron-based Superconductors

A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville

ii Acknowledgments

First, I would like to thank everyone, the school officials, the department staff, committee members, professors who helped me overcome the obstacle I faced in graduate school. A particular thanks to my research advisor Dr. Steven Johnston, along with the research group member, Umesh Kumar, Shaouzhi Li, Phil Dee, who were a graduate student and gave me wonderful physics/science discussions, and also to Dr. Yan Wang, the former postdoctoral researcher in our group who game me grateful insights regarding the spin fluctuation theory and electron-phonon calculations using Density Functional Perturbation Theory. Once again, I thank and give my deep gratitude to all the above mentioned.

iii Abstract

The microscopic properties of the phonon mediated conventional superconductors are well explained by the Bardeen-Cooper-Schrieffer (BCS) theory. However, a comprehensive description of the unconventional superconductivity such as the cuprates and iron-based superconductors is still under considerable debate. One of the theories proposed for explaining the pairing mechanism is the spin fluctuation exchange perhaps playing a leading role in inducing the unconventional superconductivity[1]. These materials are thought to be unconventional because they share certain commonalities that are absent in the conventional superconductors. The materials possess variety of phases upon doping including the superconductivity residing in close proximity to magnetism. They show an anisotropic superconducting order parameter. The transition temperatures tend to be higher than those predicted by the conventional electron-phonon couplings. These observations differ from the conventional superconductors indicating that the electron-phonon interactions may be non-factors or play a minimal role in paring. However, a significant phonon renormalization interpreted from photoemission spectroscopy[2], a pronounced isotope effect in cuprates[3], and the emergence of the replica band due to strong forward scattering phonons[4] raise questions regarding the role of phonons in these magnetic superconductors. McMillan and Rowell have studied the low energy excitations seen from a tunneling experiment to prove the electron-phonon coupling was indeed driving the superconductivity, and thus validated the BCS theory[5]. If the unconventional pairing is explained by the exchange of bosons, then one could use the extended version of the BCS theory such as the Eliashberg theory to describe properties of the superconductivity. In the present analysis, we have investigated the low energy features seen in various unconventional superconductors using a fully self-consistent algorithm. We find that the bosonic excitation picture has

iv correctly explained many experimental features, though there were properties. The present analysis has enlightened the picture that the spin ﬂuctuation exchange plays a pivotal role in mediating the pairing; we may further argue that the phonons can play an important role in renormalizing the electronic structure in cuprates; additionally, the forward scattering phonons can coexist with the spin ﬂuctuation exchange to greatly enhance the transition temperature in a selected layered materials.

v Table of Contents

1 Overview on Unconventional Superconductivity1 1.1 Early History of the Eliashberg Theory...... 2 1.2 Discovery of Magnetic Superconductors...... 4

2 Superconductivity Due to Bosonic Exchange Interactions8 2.1 Eliashberg Theory of Superconductivity...... 10 2.2 Single Band, d-wave Superconductors...... 18 2.2.1 Model...... 18 2.2.2 Spectral Function...... 22 2.2.3 Self-Energy, Transition Temperature, and λ ...... 24 2.2.4 Electron Density of States - Single boson model...... 25 2.2.5 Electron Density of States - Two Boson Model...... 26

3 Spin Fluctuation in High Tc Cuprates 30 3.1 Theory of Spin Fluctuation...... 31 3.2 FLEX Approximations...... 37 3.2.1 FLEX: Normal State...... 38 3.2.2 FLEX: Superconducting State...... 39 3.3 Results...... 45

3.3.1 Tc vs. U ...... 46

3.3.2 Tc Dependence on the Chemical Potential...... 48

3.3.3 Spin Fluctuation Interaction Vs ...... 48 3.3.4 Spectral Function and Density of States...... 52

vi 4 Applications 56 4.1 Forward Scattering Phonons in Iron-based Superconductors...... 57 4.1.1 Model...... 60 4.1.2 Spectral Function...... 63 4.1.3 Quasiparticle Interference...... 66

4.1.4 Sr2VO3FeAs...... 70 4.2 FLEX plus Phonons...... 72

4.2.1 Tc vs. (U, λ) with phonons...... 74 4.2.2 Spectral Function...... 76 4.3 Competition between the Charge Density Wave and Conventional Supercon- ductivity...... 79 4.4 Multi-orbital FLEX Approximations...... 84 4.4.1 Bilayer Hubbard Model...... 90 4.4.2 Pairing with an Incipient Band in a Bilayer System...... 93 4.5 Concluding Remarks...... 95

Bibliography 98

Appendices 124 A Vertex Corrections...... 125

B Fourier Transform between iωn ↔ τ ...... 127 C Pad´eApproximants...... 132 D Anderson Mixing...... 134 E Relevant Formulas...... 135 E.1 Optical Conductivity...... 136 E.2 Speciﬁc Heat...... 137 F Analytic Continuation (Migdal-Eliashberg Theory)...... 137

Vita 140

vii List of Figures

1.1 (a) Neutron scattering data showing the phonon density of states in Pb. (b) α2F (ω) extracted from tunneling data...... 3 1.2 Timeline of newly discovered superconductors...... 6

2.1 (a) Electron density of states with the retardation eﬀect. (b) Conductance ratio for Bi-2212 in the superconducting state...... 9 2.2 (a1) & (a2) Lowest order diagram. (b1) - (b3) Second lowest order diagrams for electron self-energies...... 12 2.3 Electron self-energy within ladder approximations...... 13 2.4 (a) Electron self-energy within Migdal approximations. (b) Self-energy diagrams that are included in Migdal approximations...... 15 2.5 (a) Modeled boson spectrum F (ω) for the simulation. (b) Spin polarized

neutron scattering intensity at q = (π, π) in YBa2Cu3O6.92...... 19 2.6 (a) Calcualted spectral function along the nodal direction in the superconducting state. (b) Calculated spectral function along the anti-nodal direction in the superconducting state. (c) & (d) ARPES intensity taken from optimally doped Bi-2212 compounds in the superconducting state along two diﬀerent directions...... 23 2.7 (a) Calculated real part of the electron self-energies in the superconducting

state. (b) Tc depenedence on λd. (c) Maximum value of the gap as a function

of Tc...... 25 2.8 (a) & (b) Electron density of states using a single boson model in the

superconducting state for three diﬀerent λd values...... 26

viii 2.9 (a) & (b1) - (b3) Electron density of states using a two boson model in the

superconducitng state for three diﬀerent λd values. (c) Modeled function F (ω) used for the two boson model...... 28

3.1 Illustration of dx2−y2 gap symmetry...... 31 3.2 Schematic orbital pictures of Copper-oxygen plane in a typical cuprate.... 33 3.3 Phase diagram for couple cuprate superconductors...... 34 3.4 Illustration of Mott and charge transer insulators...... 36 3.5 Approximation for an electron self-energy that arise from Hubbard U interactions...... 38 3.6 Digramatic represention for the particle-particle vertex within RPA or FLEX approximations...... 40 3.7 Single band FLEX self-energy diagrams in the superconducting state.... 41 3.8 Electron self-energy diagrams for FLEX or RPA approximations...... 41 3.9 Flowchart for the FLEX algorithm...... 44 3.10 (a) Fermi-surface for the current model. (b) Maximum value of the gap as

a function of Tc. (c) Calculated gap function using FLEX showing d-wave

symmetry in the superconducting state. (d) Tc dependence on U and the bandwidth...... 47

3.11 (a) Non-interacting density of states showing the range of µ0 where the system

becomes superconducting. (b) Tc vs. µ0 plot...... 49 3.12 (a) & (b) Imaginary part of the spin fluctuation interaction as a function of energy. (c) Spin fluctuation interaction in the momentum space. (d) Spin fluctuation interaction in the real space...... 50 3.13 (a) Spectral function along the nodal direction in the superconducting state. (b) Imaginary part of the electron self-energy as a function of energy. (c) & (d) Spectral function along the anti-nodal direction...... 53 3.14 (a) - (c) Electron density of states at various U values in the superconducting state...... 54

ix 4.1 (a) ARPES image taken from FeSe-STO showing the replica band. (b) Cystal structure for FeSe-STO...... 58

4.2 Phonon dispersion for bulk SrTiO3 ...... 61 4.3 (a) Spectral function in the superconducting state showing the replica band that originates from strong forward scattering phonons. The correspoding density of states is shown to the right. (b) The corresponding gap function which is highly momentum dependent...... 64 4.4 Spectral function with forward scattering phonons in the system at various

temperatures and q0...... 65 4.5 (a) & (b) Classical wave mechanics illustrating the standing wave. (c) Schematic cartoon picture showing the electrons interference around an impurity. (d) STM image showing the interference pattern taken from copper. 67 4.6 (a1) Spectral function at 10 K. (a2) Corresponding QPI intensity. (b1) & (b2) A(k, ω) at ω = 130, −110 meV. (c1) & (c2) |δρ(q, ω)| at ω = 130, −110 meV. (d1) & (d2) 1-D plot of |δρ(q, ω)| at ω = 130, −110 meV...... 69

4.7 (a) Crystal structure for Sr2VO3FeAs. Red arrows denote the atomic displacements of the phonons that couple to the carriers in FeAs layer. (b) QPI intensity obtained from STM experiments. (c1) Spectral function in the superconducting state showing the renormalization due to strong forward scattering phonons. (c2) The corresponding QPI intensity reproducing the replica band...... 71 4.8 Illustration for the buckling and breathing phonon modes in the copper oxygen plane...... 73 4.9 (a) Feynmann diagrams for the FLEX approximations with phonons. (b) Diagrams that are not included in the current simulation...... 75

4.10 Tc (d-wave) as functions of U and λ for buckling, breathing, and forward scattering phonons...... 76 4.11 (a) ARPES intensity in optimally doped Bi-2212 in the superconducting state. (b) Eﬀective self-energy taken from two diﬀerent Bi-2212 samples consisting of regular and isotope oxygens...... 77

x 4.12 (a) Spenctral function in the superconducting state using FLEX approximations without phonons. (b) Spectral function with phonons. (c) Corresponding eﬀective self-energy with/without phonons. (d) Eﬀective self-

energy with phonons below and above Tc...... 78 4.13 (a) Phase diagram produced from the Migdal-Eliashberg equations that takes renormalized phonon Green’s function into account. (b) Fermi-surface at half- filling in the present model. (c) Auto-correlation function showing a nesting of the fermi-sufrface. (d) & (e) Renormalized Phonon spectral functions at two different temperature...... 83 4.14 Digramatic representation for multi-orbital FLEX...... 87 4.15 (a) Crystal structure for YBCO showing two copper-oxygen planes in the unit cell. (b) Fermi-surface in the current model with/without a finite hopping. (c) Corresponding anomalous self-energy in the superconducting state showing a d-wave character...... 91 4.16 (a) Fermi-surface with a large hopping parameter. (b) The corresponding anomalous energy in the superconducting state showing S± symmetry.... 92 4.17 (a) Spectral function at 20 K without phonons. (b) Spectral function at 20 K with phonons. The mirror band appearing at k = (π, π) slightly above the Fermi-level. (c) Anomalous self-energy as a function of temperature for three different electron-phonon coupling strength...... 95 4.18 Anomalous self-energy obtained from multi-orbital FLEX approximations.. 96 A.1 (a) Self-energy diagram with vertex corrections. (b) & (c) Higher order diagrams that will be calculated. (d) Coupling constants for two different models...... 126 A.2 ∆ vs. T with/without vertex corrections for uniform scattering...... 126 B.1 Three types of self-energies for the uniform sattering phonons...... 128 B.2 Three types of self-energies for the forward scattering phonons...... 129 C.1 A comparison of self-energies with/without Pade approximations...... 134

xi Chapter 1

Overview on Unconventional Superconductivity

The well established theory of superconductivity introduced by Bardeen, Cooper, and Schrieffer in 1957, now known as BCS Theory, had successfully explained the microscopic properties of the so-called conventional superconductors[6,7,8]. The impressiveness of the BCS theory is that it is rather simple in terms of mathematics, but it is highly accurate. While BCS theory was able to describe many important experimental properties for a number of metals such as Al, Sn, In, etc., other materials such as Hg and Pb exhibited deviations from the BCS prediction[9,5]. For example, BCS predicted the ratio of the superconducting gap to the transition temperature 2∆ = (kBTc) to be 3.53 where ∆ is the superconducting gap, kB is the Boltzman constant, and Tc is the sperconducting transition temperature.[10] The measured ratio was in fact 4.3 for Pb and 4.6 for Hg[11, 12]. In addition, electron tunneling data from Pb shows noticeable deviations from BCS predicted behavior[9, 13], which provided strong motivation to extend the original BCS theory[9]. The reason for the observed deviation is due to the strength of the electron-lattice interaction, the electron- phonon coupling, seen in these metals and the retardation effects. One simple way of estimating the strength of the electron-phonon coupling is to take the ratio of the superconducting transition temperature to the Debye temperature. Another way is to evaluate the electron-phonon effective mass enhancement factor as was discussed by McMillan[14]. These criteria indicated the electron-phonon coupling strength for Pb and

1 Hg are relatively larger than other metals such as Al, Sn, and In. Due to this, it is apparent that in order to study materials with a strong electron-phonon coupling, it was necessary to go beyond the BCS theory to address strong coupling superconductors. Here, the coupling strength is considered “strong” if the dimensionless number N(0)V is larger than 1/4, where N(0) is the normal state density of states at Fermi-level and V (see Chapter 10 of Ref. [15]) is the attractive electron-electron interaction between the pair.

1.1 Early History of the Eliashberg Theory

Eliashberg theory of superconductivity is an extended version of the BCS theory that includes retardation effect (renormalization of the electronic structure due to electron- phonon couplings) which is an effect that original BCS theory ignores. Thus, the theory was first developed to include retardation effects to study the strong electron-phonon coupling superconductors[16]. Although, the theory was originally used to study the conventional phonon mediated superconductors, the generalized version of the Eliashberg theory is still used today to study various types of unconventional superconductors such as the cuprates and iron based superconductors. If the superconductivity is driven by some kind of collective excitations such as magnetic excitations for example, then the Eliashberg formalism can be used to study other types of superconductors. McMillan and Rowell have utilized the Eliashberg theory to compare the theory and tunneling results for Pb which showed an excellent agreement, further proving that the electron-phonon coupling was indeed the driving mechanism for the conventional superconductors[9, 13]. Couple remarkable findings from their work were the following. The effective mass enhancement ratio is approximately m∗/m = 1 + λ where λ is given by,1

Z α2(ω)F (ω)dω λ = 2 , (1.1) 2 where α(ω) is the momentum averaged electron-phonon coupling constant, and F (ω) characterizes the phonon density of states. McMillan and Rowell[5] have obtain the electron

1Keep in mind that Eq. 1.1 does not always apply. In other cases, the eﬀective mass is evaluated from averaging the electron self-energy. For a system such as Pb, Eq. 1.1 is approximately true.

2 (a) (b)

Figure 1.1: (a) Neutron scattering data showing the phonon density of states in Pb. (b) Function α2(ω)F (ω) extracted from the tunneling data. Two peaks at 4 meV and 8 meV arise from the transverse and longitudinal phonon modes in FCC structure. Figure courtesy of Ref. [9]. density of states from a tunneling measurement (normal-insulator-superconducting junction). Having obtained the density of states together with the Eliashberg theory, they were able to extract α2(ω)F (ω), and calculated the superconducting transition temperature which showed a good agreement with the experimentally measured Tc. This result is considered a huge success in the superconductivity studies because it eventually led us know that the first principle calculation for superconductivity may be possible. The other finding is that they have compared the results between α2(ω)F (ω) and the phonon density of states from the neutron scattering measurements. The comparison is shown in Fig. 1.1. The impressiveness is that these two results shared similar features particularly the two separate peaks that arise from the transverse and longitudinal phonon modes in the FCC structure of Pb. This agreement is another proof that the phonons play an important role in mediating the cooper pairs. In 1972, Bardeen, Cooper, and Shrieffer

3 equally shared a Nobel prize in physics for successfully describing the microscopic properties of superconductivity.

1.2 Discovery of Magnetic Superconductors

After BCS theory was introduced, Berk and Shrieﬀer continued investigating superconductivity especially its relationship with the magnetism. They concluded that the interaction in ferromagnetism known as the paramagnon spin-ﬂuctuation exchange seemed to suppress Cooper pairing in the phonon mediated conventional superconductors[17]. This likely explains the reason why superconductivity in Fe, Ni, and Cr, which are known to be magnetic is absent. Other materials worth mentioning are Pd, Nb and V. While Pd is known to be a paramagnet, the system slightly fell below the stoner criteria, so that the system can be considered as nearly ferromagnetic. Rieschel and Winter have performed a band structure and frozen phonon calculation to estimate the superconducting transition temperature in

Nb and V, and found that Tc can be as high as ∼ 18 K[18]. Nevertheless, the Tc turned out to be 9 K and 5 K for Nb and V, respectively, which are much lower than the theoretical values. While both materials are paramagnetic, and the paramagnon exchange did not remove superconductivity completely, it seemed it was enough to suppress the transition temperature. At first, the discussion of the phonon mediated superconductivity seemed to have settled. However, the necessity for the quantitative description of superconductivity again emerged as new types of superconductors were later discovered. The first heavy fermion superconductor was discovered in 1978[19]. The intriguing fact was the materials are found to be magnetic which contradicted with the previous assertion that magnetism should suppress Cooper pairing. Previous to these discoveries, the community had looked into the paring due to a spin fluctuation exchange in other systems such as superfluid 3He[20, 21, 22] and organic materials[23]. The proposed scenario was that the antiferromagnetic spin fluctuation

4 exchange could lead to a sign changing d-wave order parameter in the heavy fermion superconductors[24, 25, 26].2 Finally, moving on to the Cuprate era. In 1986, the researchers Bednorz and M¨ullerfrom IBM Laboratory reported a new superconductor from materials that belong to a copper-oxide

family (cuprates) yielding the transition temperature Tc > 30 K at ambient pressure[28].

This transition temperature was the highest Tc observed at this time[28]. Shortly afterwards, several other cuprate superconductors with much higher transition temperature, now referred

to as the “high Tc cuprates” have been discovered[29, 30, 31, 32] including Tc of up to 133 K at ambient pressure as determined from both the magnetic susceptibility and the electrical resistivity measurements[32]. Aside from the pressure induced superconductivity,

the Tc shown in Ref. [32] is now among the highest recorded Tc’s that have ever been reported in scientiﬁc journals[32]. Fig. 1.2 shows the transition temperature for some of the newly discovered superconductors and the year it was discovered. As can be seen, the critical temperature jumps after the cuprate supercondcutors are discovered followed by the discovery of iron-based superconductors. Also note that the heavy-fermion superconductors were discovered prior to the cuprates, though all these superconductors merely produce

Tc < 2.3 K[33]. Most of the remaining discussion in this dissertation will primarily focus on cuprate superconductors. The pairing mechanism in cuprate superconductors are highly controversial. One proposed theory for explaining the cooper pair formation is the antiferromagnetic spin ﬂuctuation exchange that predicts the sign changing d-wave order parameter. In Chapter 2, the formalism for the Eliashberg theory will be introduced, and elucidate how they can be used to study some of the superconducting properties. The results are discussed for a d-wave superconductor in the context of cuprates. In Chapter3, the concept of antiferromagnetic spin ﬂuctuation exchange will be explained using Green’s function and diagrammatic approach as well as how they can be numerically simulated using conserving Flucutaion exchange approximations. In Chapter4, various applications will be presented using the formalsim introduced in Chapter 2 and 3. The necessary backgrounds and basic

2 It was long thought that CeCu2Si2 was a d-wave, but the actual experiment showed it is likely a multiband fully-gapped superconductor.[27]

5 HgBa2Ca2Cu3O8

HgBa2CaCuO10

TiBa2Ca3Cu4O11

YBa2CuO7

BiSr2CaCu2O8

FeSe_STO

SmFeAsO

LaSrCuO CeFeAsOF

V3Si LaBaCuO NbN LaOFFeAs Nb NbAlGe Pb Nb3Ge Nb3Sn Hg UPd2Al3 LaOFeP CeCu2Si2 UBe13

CeCoIn5

Figure 1.2: Transition temperature Tc of some newly discovered superconductors and the year in which they are discovered. The types of superconductors are indicated by the colors shown in the legend (SC stands for superconductors). Tc for FeSe-STO (monolayer FeSe on SrTiO3 substrate) reported various Tc depending on the experiments[4, 34]. All Tc’s are at ambient pressure.

6 concepts will be presented later in each chapters before presenting results. The background may be too brief since the physics of unconventional superconductors are among one of the most outstanding problems in the condensed matter physics. The readers are encouraged to look into each references given therein.

7 Chapter 2

Superconductivity Due to Bosonic Exchange Interactions

One of the central debate regarding the cuprate superconductors is whether the cooper pairing/renormalizaion is due to the exchange of phonons[35, 36, 37, 38], other bosons such as the spin fluctuation exchange[39, 40, 41, 42], the fluctuation around the quantum critical point[43], resonating valence bond theory[44], or a completely different mechanism[45]. The Eliashberg theory of superconductivity predicts that the electron-boson coupling renormalizes the electronic dispersion at energy ∆ + Ω, where ∆ is the superconducting gap and Ω is the resonance energy of bosons (the energy at which a sharp peak appears in the boson spectrum)[14]. Various spectroscopic measurements in Bi2Sr2CaCu2O8+δ(Bi- 2212) have captured a renormalization of the electronic structure, a possible signature of the bosonic excitations coupling to the electronic states[46, 47, 35, 36]. Scanning Tunneling Microscopy (STM) has shown a modulation in the electron density of states at energy ≈ 70 meV[46, 47]. Angle Resolved Photoemission Spectroscopy (ARPES) has also shown a strong renormalization of the electron dispersion around 70 meV[35, 48], an energy that may be related to ∆ + Ω. Moreover, there seems to be a close connection between the observed renormalization and the superconductivity[49]. For instance, the degree of the renormalization becomes weaker for overdoped cuprates while the superconducting transition temperature also goes down[50]. If the superconductivity in cuprates is driven by coupling of electrons/holes to bosonic excitations, then one could try to describe the properties using the

(a) (b) R

hump hump dip dip

Figure 2.1: (a) Typical electronic density of states N(ω) using the Eliashberg (solid) and BCS (dashed) theory. Figure courtesy of Ref. [9]. Inset shows a typical N(ω) using the BCS model, and the red dashed line is the region of the outer plot. The hump-dip feature is discussed in the text. (b) Conductance ratio R = dI/dV(super) between the super/normal dI/dV(normal) state measured from Bi-2212 at 30 K. ∆ is the superconducting gap. Red arrow is the approximate resonance energy Ω[53] (see the discussion in the text). Figure courtesy of Ref. [46] and Ref. [47] (inset). Inset shows a typical conductance intensity of optimally doped Bi-2212 at 4.2 K. Black arrows point the peak in d2I/dV 2. Compare this to the hump-dip feature in (a). strong coupling theory developed by Eliashberg. This has been done in the past, for example by Carbotte et al.[51] who have used the Eliashberg formalism to study the properties of the superconductivity in the high Tc cuprates. However, the modulation of the electronic density of states at 70 meV seen in Bi-2212 appears to be slightly diﬀerent from the original Eliashberg prediction[52]. The discrepancy between the Eliashberg prediction and the STM data is illustrated in Fig. 2.1. Fig. 2.1(a) shows the modulation of the electronic density of states N(ω) (solid line) and the BCS predicted N(ω) (dashed line). According to the Eliashberg theory, the

ω−∆ modulation due to exchange of bosons is expected at energy ∆ + Ω ( Ω = 1 in Fig. 2.1(a) ). The Eliashberg theory produces a “hump-dip” structure leading to a negative derivative (down-slope) at this energy. Fig. 2.1(b) and the inset shows a typical conductance of an optimally doped Bi-2212 in the superconducting state measured from STM. Since the diﬀerential conductance is proportional to the electronic density of states[54], one can compare the calculated density of states and the conductance directly. One can recognize

9 that the conductance shown in the Fig. 2.1(b) shows a modulation, a “dip-hump” structure, leading to the positive derivative at energy 43 meV (the approximate resonance energy in optimally doped Bi-2212)[53] indicated by the red arrow. This positive derivative (up-slope) at energy ∼ ∆ + Ω is a feature that can be seen in Bi-2212 compounds[46, 47]. This discrepancy between the Eliashberg prediction and the STM data in cuprates will be the main topic in this chapter. We would like to see whether this dip-hump feature seen in STM can be explained using the Eliashberg formalsim. In the next section, the basic Eliashberg theory is introduced. Later, we will show results of the electron density of states using an approach diﬀerent from what Carbotte et al have done in their work[51], and try to interpret the origin of the dip-hump feature.

2.1 Eliashberg Theory of Superconductivity

Ever since the cuprate superconductors were discovered, the physics behind the pairing mechanism in unconventional superconductors have been intensely studied and debated. Various methods have been employed to study the microscopic properties of the superconductivity. In the present case, the Eliashberg theory[16] which is based on a finite temperature, Green’s function formulated by Matsubara[55] will be used. At finite temperature, the Green’s functions are defined over discrete energies along the imaginary axis. Throughout the text, we will use the notation G(k, iωn) for the electron Green’s function and D(q, iΩn) for the (2n+1)π 2nπ boson Green’s function. Here, ωn = β and Ωn = β are the fermion and boson matsubara frequencies, respectively (defined later in the section) with β = 1 being the kB T inverse temperature, k q denote the momentum, n are the integers n ∈ (−∞, ∞). Before we start, we would like to emphasize that the original theory of Eliashberg assumed that the phonons are the main mediators for the cooper pair formation. We will assume that the superconductivity is driven by electrons/holes coupling to some bosons. Detailed examples will be given shortly.

10 First, we begin by considering a simple single band model. Using the second quantization language, the Hamiltonian for the present model is

X † X † 1 X † † H = kckσckσ + Ωqbqbq + √ g(k, q)ck+qσckσ(bq + b−q), (2.1) kσ kσ N kqσ

† † where ckσ (ckσ) and bq (b−q) are the electron and boson creation (annihilaiton) operators respectively with k q σ are the electron, boson momentum, and the electron spin,

respectively. k and Ωq are the electronic and boson band dispersion, respectively, and g(k, q) is the electron-boson coupling constant. The next step is to introduce the single-electron Green’s function deﬁned in the many- body theory[15]. First, let us assume that the system is in the normal state. The detailed derivations of the Green’s function are discussed by Mahan[15]. The electron Green’s

function G(k, iωn) is expanded using the Dyson series,

0 0 0 0 0 0 0 0 1 G(k, iωn) = G + G ΣG + G ΣG ΣG + ... = G + GΣG = −1 , (2.2) G0 − Σ

where

0 0 1 G = G (k, iωn) = (2.3) iωn − k

is the non-interacting (bare) Green’s function, and Σ = Σ(k, iωn) is the electron self-energy.

Here, G(k, iωn) is the interacting (dressed) Green’s function which includes the effect from 0 the second and the third term of the Hamiltonian in Eq. 2.1, while G (k, iωn) is the non- interacting Green’s function that does not include electron-boson couplings. Eq. 2.2 shows that the interacting Green’s function cannot be obtained unless the electron self-energy is known. Normally, the evaluation of the electron self-energy is the first step in this Green’s function approach. One approach to evaluate the self-energy is by summing a subset of Feynman diagrams. The first and the second order correction to the self-energy are shown in Fig. 2.2. Fig. 2.2 (a1) and (a2) are the first order correction to the self-energy and Fig. 2.2 (b1) - (b3) show the second order correction to the self-energy. Fig. 2.2(a2) is the Hartree term which can vanishes, for example, for acoustic phonon modes or other collective excitations, but it

11 (a1) 퐷(퐪, 푖Ω푛) (a2)

퐺(퐤, 푖휔푛) 퐪 = ퟎ

푔(퐤, 퐪)

(b1) (b2) (b3)

Figure 2.2: Black line is the non-interacting Green’s function G0(k, ω). Wavy line is the boson Green’s function D(q, Ωn). The black dots are the electron-boson coupling constant g(k, q). (a) The first order correction to the electron self-energy Σ(k, ω). (b1)-(b3) The second order correction to the electron self-energy. Images created using JaxoDraw[56]. is non-vanishing for an electron-electron interaction and optical phonon modes and others. Because it is not a trivial matter to compute the self-energy exactly, what theorists normally do is to sum particular sets of diagrams to an infinite order. This is the basic idea behind the strong coupling Eliashberg theory of superconductivity. Let us now move on to the superconducting state. If one proceeds and sums specific types of diagrams, called vertex corrections discussed by Shrieffer[57,9], the self-energy diverges at the temperature below some critical temperature. The Feynman diagrams for these self-energies are illustrated in Fig. 2.3. For the conventional phonon mediated superconductors, the type of vertex corrections that diverges are the ladder diagrams shown in Fig. 2.3(b). This divergence is an indication of the phase transition, and in this case, the system settles in the superconducting state. In order to remove this divergence, the electron self-energy is instead expressed using the Nambu formalism, formally known as Nambu- Gor’kov formalism[9, 58]. Within the Nambu formalism, the Green’s function is written in

12 (a) 퐷(퐪, 푖Ω푚)

Σ 퐤, 푖휔푛 = 퐺(퐤, 푖휔푛) 횪 푔(퐤, 퐪)

(b)

횪 = + + + …

Figure 2.3: (a) Feynman diagram for the electron self-energy that contains the vertex function Γ. (b) Ladder diagrams for the vertex function which becomes singular at the critical temperature. terms of a 2×2 matrix. The Green’s function matrix is deﬁned as,

 D † E    T ck↑(τ)ck↑(0) T ck↑(τ)c−k↓(0) Gelectron(k, τ) F (k, τ) Gˆ(k, τ) =   =   , D † † ED † E † T c−k↓(τ)ck↑(0) T c−k↓(τ)c−k↓(0) F (k, τ) Ghole(k, τ) (2.4) where τ is the imaginary time, and T and P are the time ordering and cooper pair operator[9], respectively. The diagonal components Gelectron(k, τ) and Ghole(k, τ) are the electron Green’s function for up-spin electons and down-spin holes, respectively. The oﬀ- diagonal components are the anomalous Green’s function discussed by Abrikosov et al[59]. Note that the anomalous Green’s function is non-zero only in the superconducting state, but zero in the normal state. The usefulness of the Nambu formalism is that the above Green’s function matrix can be written in terms of the Dyson series, and the approach similar to the normal state can be used. The Green’s function in the frequency domain is obtained by the

13 Fourier transform given by,

Z β iωnτ G(k, iωn) = − G(k, τ)e dτ, 0 Z β iωnτ F (k, iωn) = F (k, τ)e dτ, (2.5) 0 Z β † † iωnτ F (k, iωn) = F (k, τ)e dτ, 0 where ωn is the fermion matsubara frequency deﬁned previously. Care needs to be taken when one performs transformation from τ to ωn space since sometimes a function is discontinuous at τ = 0, for example, G(τ = 0+) 6= G(τ = 0−). The Green’s function in the superconducting state is again expressed using the Dyson’s series as, 1 Gˆ(k, iω ) = Gˆ0 + GˆΣˆGˆ0 = , (2.6) n ˆ−1 ˆ G0 − Σ where the non-interacting Green’s function is given by,

ˆ0 ˆ0 1 G = G (k, iωn) = . (2.7) iωnτˆ0 − kτˆ3

ˆ For the self energy matrix Σ(k, iωn), any 2×2 matrix can be written in terms of a linear combination of the pauli matrices and they are given by,

ˆ Σ(k, iωn) = (iωn − iωnZ(k, iωn))ˆτ0 + φ(k, iωn)ˆτ1 + φ2(k, iωn)ˆτ2 + χ(k, iωn)ˆτ3, (2.8)

whereτ ˆ1(i = 1 ∼ 3) are the components of the usual pauli matrices given by,

      0 1 0 −i 1 0 τˆ1 =   , τˆ2 =   , τˆ3 =   (2.9) 1 0 i 0 0 −1

andτ ˆ0 is the identity matrix. The electron self-energy is given by the one-one component of the self-energy matrix,

ˆ Σ11(k, iωn) = iωn − iωnZ(k, iωn) + χ(k, iωn). (2.10)

14 (a) 퐷(퐪, 푖Ω푚)

Σ෠ 퐤, 푖휔푛 = 휏Ƹ3퐺෠(퐤, 푖휔푛)휏Ƹ3

푔(퐤, 퐪) (b)

෠ Σ 퐤, 푖휔푛 = 0 + + + . . . . 휏Ƹ3퐺෠ (퐤, 푖휔푛)휏Ƹ3

Figure 2.4: Self-energy within Migdal approximations. The double line represents the ˆ interacting Green’s function G(k, iωn). (b) Diagrams that are included in the Migdal approximations.

The oﬀ-diagonal elements of the self-energy are the anomalous self-energy given by

±iθ φ(k, iωn)±iφ2(k, iωn) = |φ0(k, iωn)|e . In many applications, θ can be set to zero since they

are simply the phase factor, and do not play a meaningful role, so that only φ(k, iωn) needs to be solved. However, the phase factor indeed becomes important in other applications such as Josephson tunneling[9]. As has been said, it is not easy to sum all the diagrams to an inﬁnite order. Eliashberg instead utilized what is known as the Migdal approximation[60, 61] and proceeded by evaluating the following diagram shown in Fig. 2.4(a). The double line represents the ˆ interacting Green’s function G(k, iωn). Evaluating the diagram this way includes all higher order terms that have the rainbow form shown in Fig. 2.4(b), but ignores other diagrams such as the one shown in Fig. 2.2(b2). This diagram shown in Fig. 2.2(b2) is also known as the ﬁrst order vertex correction. The vertex corrections can be situationally important and sometimes needed to capture some physical phenomena, the features that are not captured by Migdal approximations[62, 63, 64].

15 ˆ Within the Migdal approximation, the matrix form of the self-energy Σ(k, iωn) shown in Fig. 2.4(a) is mathematically expressed as,

1 X Σ(ˆ k, iω ) = − |g(k, k0)|2D(q, iω − iω )τ ˆ Gˆ(k0, iω )ˆτ , (2.11) n Nβ n m 3 m 3 k0m

0 where q = k − k is the momentum transfer, D(q, iΩm) is the boson Green’s function, and Ωn 0 2 is the boson matsubara frequency. Note that D(q, iΩm) and |g(k, k )| are scalar functions. It is often convenient to express the boson Green’s function D(q, iΩm) in terms of the imaginary part of the boson Green’s function F (q, iΩm) using the following relation called a spectral representation, Z ∞ F (q, ω) D(q, iΩm) = dω, (2.12) −∞ iΩm − ω where F (q, ω) = Im[D(q, ω)], and ω is the real energy (not the matsubara frequency). Eq. 2.12 is often preferred because the function F (q, ω) can be estimated from experiments, for example, through tunneling spectroscopy or inelastic scattering data.

Since each self energies are dependent on one another, Z(k, iωn), χ(k, iωn), and φ(k, iωn) have to be solved simultaneously and iteratively until a reasonable convergence is achieved. Once it’s done, one can proceed and evaluate thermodynamic potentials. Afterwards[15, 59, 65], various thermodynamic quantities can be evaluated by taking its derivative. For example, refer to the relevant formula given in Appendix E.2 for evaluating the speciﬁc heat.

The superconducting transition temperature Tc can be calculated by solving the gap function ∆(Tc) = 0 where the gap function is given by,

φ(k, iωn) ∆(k,T ) = . (2.13) Z(k, iωn) n=0

Notice that Eq. 2.13 is not the superconducting gap that experimentalists measure since the equation depends on the matsubara frequency, not the real energy. However, this method is adequate for estimating the transition temperature.

While Eq. 2.11 is adequate for estimating Tc or investigating the thermodynamic properties, it is usually the self-energy with the real frequency dependence that plays a crucial role in studying many other properties. Once the self-energy is obtained, other

16 physical quantities such as the electron density of states, optical conductivity, etc can be obtained. Therefore, it is ideal to have Eq. 2.11 written in terms of the real frequency ω instead of the imaginary matsubara frequency iωn. The process of bringing the self-energy + deﬁned along the imaginary axis to the real axis Σ(k, iωn → ω + i0 ) is called analytic continuation. Normally, the analytic continuation is not easily done. However, for the self- energy given by Eq. 2.11, the analytic continuation has been carried out by hand. The derivation is shown by Marsiglio et al[66] who used the technique described by Baym and Mermin.[67] For more information, please refer to the AppendixF. Here, let us present the ﬁnal expression. Within the Migdal approximation, the self-energy matrix with the real frequency dependence is given by,

ˆ 1 X X 0 2 ˆ 0 Σ(k, ω) = − |g(k, k )| B(q, ω − iωm)ˆτ3G(k , iωm)ˆτ3 Nβ 0 k m=0 (2.14) 1 X Z ∞ + |g(k, k0)|2F (q, ω0)ˆτ Gˆ(k0, ω − ω0)τ ˆ [n (ω0) + n (ω − ω0)]dω0, N 3 3 b f k0 −∞

where Z ∞ 0 0 ω 0 B(q, z) = 2 F (q, ω ) 02 2 dω , (2.15) 0 ω − z

nb and nf are the bose and fermi functions respectively, and F (q, ω) is the imaginary part of the boson Green’s function deﬁned in Eq. 2.12. Note that the boson Green’s function is an odd function F (q, ω) = −F (q, −ω). Obtaining the electron self-energy is usually the starting point in the Green’s function analysis. Once the electron self-energy is determined along the real-axis, one can proceed and evaluate other physical quantities from various correlation functions such as the Kubo formula to study transport properties, charge and magnetic susceptibilities, and others[15]. For evaluating the optical conductivity, refer to Appendix E.1. Finally, the spectroscopic superconducting gap ∆(k,T ) is given by,

Re[φ(k, ω, T )] Re[∆(k,T )] = , (2.16) Re[Z(k, ω, T )] ω=∆

which is a function of momentum. Normally, the gap deﬁned along the fermi-surface is compared with the experiment. In what follows, we use the formalism just discussed to

17 investigate the electronic and superconducting properties for a d-wave superconductor in the context of cuprates.

2.2 Single Band, d-wave Superconductors

In this section, we discuss the application and compare simulated results with the experimental data. Keep in mind that the current simulation is not intended to fully

reproduce the experimental data. So far, estimating accurate Tc for most unconventional superconductors from a ﬁrst principle calculation has not been successful. The following simulation do compare with experimental data to see if the model can capture a particular feature seen from those experiments. First, the electronic/superconducting properties and its relation to the electron-boson coupling g(k, k0) are summarized. Second, the results for the electron density of states are compared with STM data taken from the cuprates, and show how a single boson model is inadequate to reproduce the feature seen in the STM data. We then present the results for the alternative two bosons model and conclude that this model can closely reproduce the STM data.

2.2.1 Model

Our ﬁrst attempt for the self-energy calculation is to assume that the cooper pairing involves electrons/holes coupling to a single type of bosons. The type of bosons that may be involved will be elaborated later in this chapter. Here, we try to model parameters that resembles

the cuprates. What we need before proceeding is the following: the band dispersion k, electron-boson coupling constant g(k, k0), and the imaginary part of the boson propagator F (q, ω). Throughout this chapter, we use a 5-parameter, tight-binding model suggested by

Norman et al[68] for the band dispersion k. The hopping parameters were obtained from ﬁtting the normal-state dispersion of optimally doped Bi-2212 observed by ARPES. The

18 that spectrum wl o hneee fw osdrteaiorp.Temdldbsnspectrum boson modeled The anisotropy. the consider we if even change not will 2.2.5 at peaked Ω particularly but meV, hundred several ω to up scale energy large q 0 by, given is dispersions band the for expression [69]. Ref. of courtesy Figure compound. this for meV at intensity scattering 2.5: Figure hr h opn aaeesadteceia oeta r ( are potential chemical the and parameters hopping the where . sf 0962 ≈ ( = k nlsi eto ctrn xeiet a rvd nomto bu h boson the about information provide can experiments scatering neutron Inelastic sa prxmto,teaiorp ftemmnu eednei atrdotso out factored is dependence momentum the of anisotropy the approximation, an As . ,wihw ee oa h eoac energy resonance the as to refer we which 71], 70, 53, Bi-2212)[69, for meV (43 meV 41 F = ,π π, , ( + q 1 2 − ω , t eni YBa in seen ) 2 1 0 1 F (a) (cos( . t ) 1306 4 ( (cos(2 q → oee oo spectrum boson Modeled ω , k F , x b hw h pnplrzdnurnsatrn nest at intensity scattering neutron spin-polarized the shows 2.5(b) Fig. ). ( a − ω k cos( + ) 0 x sasmd oee,tecnlso htwl epeetdi section in presented be will that conclusion the However, assumed. is ) . 휔 a 0507 )cos( q 2 [ Cu meV ( = , k k 3 y 0 y O a ,π π, . a 0939 )+ )) 6 cos(2 + ) ] . 92 ae rmYBa from taken ) .Tedt eel htsi xiain rsn na on present excitations spin that reveals data The [69]. , t 2 0 cos( . 99 nuiso eV. of units in 0989) k F k y a x ( a )cos( ω )cos( b h pnplrzdneutron spin-polarized The (b) 2.12. Eq. of ) 19 k k x a y 휒′′(휔) a )+ )) 2 Cu + ) 3 t (b) O 2 1 5 cos(2 t 6 3 . 92 (cos(2 h eoac eki t41 at is peak resonance The . t k 1 x t , a k 휔 2 )cos(2 x t , a 3 [ cos(2 + ) t , meV 4 k t , y 5 a µ , + ) ] q ( = ) k ( = y µ, a )) − ,π π, 0 . (2.17) 5908 and ) F ( ω , ) is shown in Fig. 2.5(a). The explicit expression for F (ω) in the current model is given by,

ω/Ω F (ω) ∝ sf + Lor(ω), ω2 + Ω2 sf (2.18) 1 η Lor(ω) = 2 2 . π (ω − Ωsf ) + η

The ﬁrst term represents the imaginary part of the spin susceptibility derived by Millis et al.[51, 72] The second term Lor(ω) is a sharp Lorentzian function which is added to mimic the resonance peak seen in Fig. 2.5(b) at 5 K. The proportionality means the above model R F (ω) fuction is normalized to unity such that 2 ω dω = 1. Ωsf is the resonance energy which is set to be 43 meV estimated from optimally doped Bi-2212[53]. It is a general consensus that the gap structure for the cuprate superconductors are the d-wave (dx2−y2 ) form[73, 74, 75]. Practically, we can enforce any gap symmetry by expanding the coupling constant |g(q)|2 in terms of a linear combination of the Fermi-surface Harmonics

0 2 Y`(k)[76, 77]. In the current simulation, we impose a d-wave solution and replace |g(k, k )| as follow,

 Ys(k) = 1, 0 2 0 0  |g(k, k )| = gsYs(k)Ys(k ) + gdYd(k)Yd(k ) (2.19)  1 Yd(k) = 2 (cos(kxa) − cos(kya)),

0 with the momentum transfer given by q = k − k , and gs gd are the real constants. Note that the above relation only makes sense when gs ≥ gd. With Eq. 2.19 and Eq. 2.18, the momentum dependence of the self-energies can be factored out as follow: Z(k, ω) =

Z(ω)Ys(k), χ(k, ω) = χ(ω)Ys(k), and φ(k, ω) = φ(ω)Yd(k). Each elements of the self-energy matrix can now be expressed as,

¯ gs X iωmZ(ωm)B(ω, ωm) Z(ω) = ω + iδ + 2 2 2 2 Nβ (ωmZ(ωm)) + (k + χ(ωm)) + φ (ωm)Y (k) mk d X Z ∞ F (ω0)Z¯(ω − ω0) − g dω0 (2.20) s Z¯2(ω − ω0) − ( + χ(ω − ω0))2 − φ2(ω − ω0)Y 2(k) k −∞ k d ω − ω0 ω0 × [tanh( β) + coth( β)], 2 2

20 gs X (χ(ωm) + k)B(ω, ωm) χ(ω) = − 2 2 2 2 Nβ (ωmZ(ωm)) + (k + χ(ωm)) + φ (ωm)Y (k) mk d Z ∞ 0 0 X F (ω )(χ(ω − ω ) + k) + g dω0 (2.21) s Z¯2(ω − ω0) − ( + χ(ω − ω0))2 − φ2(ω − ω0)Y 2(k) k −∞ k d ω − ω0 ω0 × [tanh( β) + coth( β)], 2 2

2 gd X φ(ωm)Yd (k)B(ω, ωm) φ(ω) = 2 2 2 2 Nβ (ωmZ(ωm)) + (k + χ(ωm)) + φ (ωm)Y (k) mk d Z ∞ 0 2 X F (ω )φ(ωm)Y (k) − g d dω0 (2.22) d Z¯2(ω − ω0) − ( + χ(ω − ω0))2 − φ2(ω − ω0)Y 2(k) k −∞ k d ω − ω0 ω0 × [tanh( β) + coth( β)] 2 2

where Z ∞ 0 0 0 ω 0 B(ω, ω ) = 2 F (ω ) 02 2 dω , (2.23) 0 ω − (ω − iωm) we have deﬁned Z¯(ω) = ωZ(ω) and δ is a small number which is needed for obtaining a stable solution when evaluating numerically.

In this model, gs and gd are the free parameters. However, it is generally more common

in the superconductivity studies to use the dimensionless coupling constant λ` where ` = s or d[77]. It is evaluated by taking the average of the coupling constant along the fermi-surface deﬁned by[52, 77], 1 0 2 0 P 0 N 2 |g(k, k )| Y`(k)Y`(k )δ(k)δ(k ) k,k0 λ = . (2.24) ` 1 P 2 N Y` (k)δ(k) k

Carbotte et al in their original work have assumed λs = λd[51]. Here, we do not make this assumption. In Section 2.2.5, it will be shown that the value of λd is one important factor for producing the desired result. For the current simulation, Eq. 2.20- 2.22 were self-consistently solved. The modeled function F (ω) shown in Fig. 2.5(a) was integrated up to 400 meV which is an approximate energy range for spin excitations in cuprates[51].

21 2.2.2 Spectral Function

First, let us brieﬂy review the concept of the spectral function. The spectral function is simply given by the imaginary part of the electron Green’s fucntion,

 1 Σ2(k,ω) − 2 2 (Normal state), A(k, ω) = π (ω−k−Σ1(k,ω)) +Σ(k,ω) (2.25)  1 ωZ(k,ω)+k+χ(k,ω) − π Im[ (ωZ(k,ω))2−(k+χ(k,ω))2−φ2(k,ω) ] (Superconducting state),

where Σ1(ω) and Σ2(ω) are the real and imaginary part of the electron self-energy respectively. It is fairly easy to understand what the spectral function can show when plotted. Basically, the spectral function shows the band structure that includes many-body eﬀects. The normal state A(k, ω) has the form of Lorentzian function. The intensity of

A(k, ω) is peaked when ω = k + Σ1(k, ω) with the width characterized by Σ2(k, ω). If the electron-boson coupling is ﬁnite, k is renormalized. The renormalized dispersion is given by 0 k = k +Σ1(k, ω) where the real part of the self-energy shifts the non-interacting band by an amount Σ1(k, ω). So, when A(k, ω) is plotted with respect to both energy and momentum, it shows the renormalized band dispersion while the intensity is controlled by Σ2(k, ω). Since the spectral function shows a probability of ﬁnding an electron having k and ω,Σ2(k, ω) is often regarded as the lifetime of the electron. The calculated spectral function A(k, ω) at 10 K (superconducting) is shown in Fig. 2.6

(a) and (b). The dimensionless coupling constants defined in Eq.2.24 are λd = λs = 1.5. Fig. 2.6 (a) and (b) show the spectral function plotted along the nodal (diagonal) and anti-nodal (off-diagonal) directions respectively. The inset shows the normal state fermi-surface of the band dispersion k. The gap opens along the anti-nodal direction which is expected for a ˆ d-wave superconductor. The corresponding electron self-energy Σ11(ω) is also shown to the right of each panel. The energy indicated by the red-dashed line in which we call “kink- energy” is defined as ωkink = ∆ + Ωres = 58.6 meV where Ωres = 43 meV is the resonance energy[53], and ∆ is the maximum value of the gap on the fermi-surface. As discussed in

Fig. 2.1, ωkink is the energy we expect to see the modulation in the electron density of states. Fig. 2.6 (c) and (d) show the intensity of the Angle Resolved Photoemission Spectroscopy (ARPES) along two diﬀerent directions[48, 35]. Often times, literatures show a direct

22 (c) (d)

Figure 2.6: Calculated spectral function A(k, ω) along the nodal (a) and anti-nodal (b) direction indicated by the red line in the inset. The corresponding self-energy Σ11(ω) is also shown to the right. The red-dashed line is the calculated kink energy ωkink. The yellow-dashed line is the non-interacting band k. (c) ARPES intensity of optimally doped, superconducting Bi-2212 at 20 K along T-X (nodal) direction. Figure courtesy of Ref. [48]. (d) ARPES intensity of optimally doped Bi-2212 at 15 K along the anti-nodal direction indicated in the right panel. The solid red line is at 70 meV which is an approximate kink energy. Figure courtesy of Ref. [35]

23 comparison between the ARPES intensity and the calculated spectral function upon assuming sudden approximations[78]. The kink is apparent in both images, and becomes robust in the anti-nodal direction at around 70 meV[35]. This low energy kink has attracted numerous investigations[79, 46, 47, 48, 80, 81, 82] because of the possibility that the renormalization may be linked to the cooper pairing as was the case for the conventional superconductors[5]. A number of literatures claim the origin of the kink is due to coupling to antiferromagnetic spin ﬂuctuations[49, 83, 84, 85, 86, 87, 88] while others attributed to electron-phonon couplings[35,3, 89, 90, 91]. The calculated spectral function A(k, ω) shown in Fig. 2.6 (a) and (b) clearly illustrate

a robust renormalization near the kink energy ωkink. One can see the diﬀerence between

the region above and below the kink energy: high intensity and small value of Σ2(k, ω) at

ω > ωkink, and low intensity and large Σ2(k, ω) at ω < ωkink. Another feature is the degree of renormalization along diﬀerent momentum directions. Weaker renormalization seen along the nodal direction whereas stronger renormalization along anti-nodal direction. Since the self-energy is momentum independent in the present model, the diﬀerence is coming from

∂k the band structure eﬀect in which the eﬀective velocity ∂k is lower along the anti-nodal direction.

2.2.3 Self-Energy, Transition Temperature, and λ

Let us discuss the dimensionless coupling contants λ` introduced in Eq. 2.24 qualitatively and quantitatively. It is important to explore the physical meaning of each coupling constants before we move on because this will help us understand the result when we examine the

electron density of states in the next two sections. Here, we examine how λ` would play a role in the electron self-energy and superconducting transition temperature Tc. In the present analysis, λs is ﬁxed at 1.5 and we choose λs ≥ λd.

Fig. 2.7(a) shows the electron self-energy and the dependence on λd values. One can see some shifts in the peak energy (50 ∼ 60 meV) by a few meV, yet the overall shape and the magnitude are relatively unchanged. Tc vs. λd is given in Fig. 2.7(b). Tc is estimated from Eq. 2.13 and extrapolating ∆(T ) ≈ 0 by ﬁtting with the BCS gap formula

24 (a) (b) (c)

Figure 2.7: (a) Calculated electron self-energy Σ1(ω) for three different values of λd. λs is fixed at 1.5. Inset shows the imaginary part Σ2(ω). (b) Tc as a function of λd. (c) 2∆ as a function of kBTc. The dashed line is the linear fit to the data.

q ∆(T ) = C tanh(C 1 − T ). One can see that λ has a strong inﬂuence on the transition 1 2 Tc d

temperature. It shows that λd and Tc are linearly related at least down to λd = 0.8, and its superconductivity vanishes when λd is approximately 0.56.

Based on the above results, an interpretation of the coupling constants is as follow. λs for the most part dictates the overall magnitude of the electron self-energy Σ(ω), whereas λd characterizes the magnitude of the anomalous self-energy φ(ω), which is closely associated with the binding energy of the cooper pair. A decrease in λd means weaker binding energy, and the result is a drop in the transition temperature. Fig. 2.7 (c) shows the superconducting gap as a function of kBTc. Once again, it shows nearly a linear relation yielding the ratio

2∆0/(kBTc) ≈ 5.21, but deviates away at large Tc value. This nonlinearity is a usual characteristic for the strong coupling superconductors. The ratio is no longer constant at a large Tc value.

2.2.4 Electron Density of States - Single boson model

This and the next section focus on the electron density of states. This section deals with a single boson model followed by a two boson model in the next section. As was mentioned (refer to Fig. 2.1), the original Eliashberg theory predicts that the electron-phonon coupling modulates the electron density of states creating the hump-dip structure leading to “down- slope” at the kink energy. According to STM data, the modulation in cuprates appears as dip-hump structure producing “up-slope” near the kink energy. Let us keep this in mind as

25 (a) (b)

Figure 2.8: (a) (b) Calculated electron density of states N(ω) for three diﬀerent values of λd at 10 K. The dashed line is the resonance energy Ωres = 43 meV for the current model. we move on to discuss the results for the calculated electron density of states. The electron 1 P ˆ density of states N(ω) is calculated from N(ω) = − π k Im[G11(k, ω)], and they are shown in Fig. 2.8 for three diﬀerent values of λd at 10 K in which all cases are superconducting.

λs is set to 1.5. The density of states in Fig. 2.8(a) shows a modulation at ∼ 50 meV. Fig. 2.1(b) shows the same density of states zoomed in near the kink. While it does show the dip-hump structure which is similar to STM data, the important remark is as follow. First, the hump is absent because the kink is too close to the gap where there is the coherent peak in the density of states. More importantly, the slope appears to be “down-slope” at the kink energy indicated by the dashed line. This clearly contradicts with the STM data which shows “up-slope” feature. We take this contradiction as an implication that the current model is inadequate to explain the observed dip-hump feature seen in STM data. In the next section, we will present a two boson model to try to produce the observed “up-slope” feature.

2.2.5 Electron Density of States - Two Boson Model

The previous model assumed that the superconductivity is driven by a single boson mode. However, ARPES has shown a signature of to multi-modes couplings in Bi-2212 compounds1[92, 82]. In the present model, we propose that there are two kinds of bosonic

1The current model may not directly reﬂect the references shown here, but the idea is that multiple modes can exist in real systems.

26 modes: the ﬁrst type of mode denoted by ‘a’ and the second type of mode denoted by ‘b’. For the current single band model, the boson spectrum becomes the sum of two contributions

F (ω) = Fa(ω) + Fb(ω). Here, each terms of Eq. 2.18 are assigned to each type of bosons as follow: ω/Ωa Fa(ω) ∝ 2 2 , (2.26) ω + Ωa for the ﬁrst type of bosons, and

1 η Fb(ω) ∝ Lor(ω) = 2 2 , (2.27) π (ω − Ωa) + η for the second type of bosons where Lor(ω) is a Lorentzian function to represent the sharp resonance mode. These modeled functions are separately normalized to unity

R Fa(ω)dω R Fb(ω)dω 1 = 2 ω = 2 ω . They are shown in Fig. 2.9(c). Once again, the function

Fa(ω) is integrated up to 400 meV. Since this excitation exists up to such higher energy, much higher than the resonance energy ∼ 43 meV, we refer this ﬁrst type of bosons as “high energy excitations”, and call the second type of bosons the “resonance mode”. Since there are two types of bosons, we need to separately deﬁne the coupling constant of Eq. 2.19 as,

n 0 2 n n 0 |g (k, k )| = gs + gd Yd(k)Yd(k ), (2.28) where n = a or b. Therefore, the current two boson model has four dimensionless coupling

a b a b constants λs , λs, λs , λd that need to be assigned. The total coupling constants are simply a b a b given by λs = λs + λs and λd = λd + λd. An analysis similar to the previous one boson model is performed by assigning coupling constants to each λ values. Speciﬁcally, we choose

a a b b λs = λd = 2.0, λs = 1.0, and compare results for λd ≤ 1.0. The calculated electron density of states N(ω) are shown in Fig. 2.9(a) for three diﬀerent

b λd values. Fig. 2.9 (b1)-(b3) show a closer view of the renormalization around the kink b energy. For λd = 1.0, clearly the result produces the down-slope at kink energy which is b b analogous to the previous one boson model. If the coupling constants are both equal λs = λd, the situation is same as the one boson model. The result is there to compare with the other

27 Figure 2.9: (a) and (b1)-(b3) Calculated electron density of states N(ω) for three diﬀerent b λd values. The dashed line is the calculated kink energy ωkink = ∆ + Ωa. (c) Two modeled function Fa(ω) and Fb(ω) used to obtain the electron density of states.

28 b b two cases. For λd = 0.5 and λd = 0.0, the dip feature has slightly shifted towards the fermi- b b level leading to “up-slope” at kink energy. This is particularly evident for λd = 0.0, but λd = 0.5 also slightly produces this up-slope. This up-slope feature is particularly similar to what is observed in STM data shown in Fig. 2.1(b). Let us qualitatively summarize the result we have just shown. The coupling constants are

a b varied λd ≥ λd. The meaning of λs and λd are previously discussed; except this time, they {a,b} are referring to the speciﬁc modes. λd is closely related to the binding energy of cooper a b pairs originating from either mode a or b. If λd >> λd such as in the present simulation, majority of the pairing contribution comes from the mode a, the high energy excitations. On

b the other hand, λs = 1.0 is still ﬁnite. This resonance mode still renormalizes the electronic structure strongly at the resonance energy Ωa + ∆. In summary, the observed feature seen in STM data could not be explained by the single boson mode. At least two boson modes are needed to reproduce the quantitative feature seen in STM data with the condition that

a b λd >> λd is satisﬁed. In the upcoming chapter, we will look into the origin of this spin resonance mode more rigorously within conserving approximations.

29 Chapter 3

Spin Fluctuation in High Tc Cuprates

In the previous chapter, we studied the single band, d-wave superconductors assuming a cuprate like system. This is done by evaluating the self-energy diagram using Migdal approximations, but additionally, a d-wave symmetry was imposed in the coupling constant |g(k, k0)|2. Furthermore, the momentum dependence of the boson spectrum was factored out such that F (q, ω) → F (ω) was assumed. We assumed that these collective excitations came from magnetism because the spin-polarized inelastic neutron-scattering data showed a spectrum of magnetic excitations shown in Fig. 2.5(b). The neutron data reveals a sharp peak intensity at a momentum transfer q = (π, π), but only present in the superconducting state. This signals that these excitations are closely related to the superconductivity. One has to also remember that the compelling feature appearing in ARPES measurements as a kink at energy around ∼ 70 meV in Bi-2212 sample[48], particularly robust in the superconducting state, but nearly absent in the normal state. Furthermore, as we saw in Fig. 2.1, STM reveals a modulation in the electron density of states that signals a bosonic-like glue near the kink energy ∆ + Ω except that it has the shape of peak-dip-hump as opposed to the BCS’s peak-hump-dip modulations[46, 47]. For these reasons, we justiﬁed ourselves that the superconductivity in cuprates was driven by some collective excitations, although, instead of phonons, it is replaced with some sets of bosons given by F (q, ω) without having a deep understanding of the actual physics behind. As was mentioned, we split the modeled boson spectrum into two parts (see Eq. 2.18): the spin excitations from the spin susceptibility derived by Millis et al[72] and the sharply peaked Lorentzian function to represent the

30 1 Δ 퐤 = cos 푘 − cos(푘 ) 2 푥 푦

1 Figure 3.1: Illustration of d-wave dx2−y2 ∼ 2 (cos(kx) − cos(ky)) gap structure. The white line represents the normal state Fermi-surface taken from Ref. [68]. resonance mode[53, 69,3]. We will have more discussions of this origin of the resonance peak later in this chapter. The origin of the resonance mode is confusing because of the fact that both the magnetic resonance and the buckling mode phonon coincidentally exist fairly close to each other in energy with 43 meV for the magnetic mode[53] and ∼ 40 meV for the buckling mode in Bi-2212 sample[35]. In the previous chapter, we have modeled the function F (q, ω) without having a clear understanding of how this resonance mode emerged in the ﬁrst place. In the remainder of this chapter, we will try to analyze this low energy excitation from the perturbative spin ﬂuctuation theory point of view.

3.1 Theory of Spin Fluctuation

Ever since the ﬁrst discovery of cuprates, many theories were introduced to address the pairing mechanism in these superconductors, and the theory of spin ﬂuctuation is just one

31 of them. One of the reasons why this theory receives attention is because it predicts the sign changing d-wave pairing symmetry, which is consistent with the experiment. The justiﬁcation for the sign changing gap can be explained as follow. Let us consider the usual BCS gap equation, 0 X Veff (k − k )∆k0 ∆k = − , (3.1) 2Ek0 k0

p 2 2 where ∆k is the gap and Ek is the excitation energy[15] given by Ek = k0 + ∆k0 . Veff (k) is the effective pairing interaction. For the phonon mediated superconductors, Veff is negative in certain frequency range signifying the attractive interaction between the pair. For the spin fluctuation mediated superconductors, the effective interaction is closely related to the imaginary part of the spin susceptibility, and is positive in the momentum space. Because of the minus sign in front of the summation, one hypothesis for the equality in Eq. 3.1 to make sense is the sign changing gap function. The strong nesting at q = (π, π) according to the neutron scattering[53, 93, 71, 70] and the large Fermi-surface found in Cuprates may

∆0 infer that one plausible gap symmetry is dx2−y2 form given by ∆k = 2 (cos(kx) − cos(ky)) which is shown in Fig. 3.1. One can see that the gap changes sign between k = (0, π) and (π, 0) with a node in between (along the diagonal). Today, it is a general consensus that cuprate superconductors are d-wave superconductors[75, 74, 73, 94, 95]. In this section, we will look into the theory of spin ﬂuctuation mediated pairing, particularly the antiferromagnetic spin ﬂuctuation which invoked for explaining the pairing mechanism in cuprate superconductors (See Ref. [96] and references therein). P. W. Anderson has suggested that the essential physics required to describe the superconductivity in the cuprates is a single band Hubbard model[97] given by,

X † X X H = tij(ciσcjσ + h.c.) + U ni↑ni↓ − µ niσ (3.2) ijσ i iσ

† † where ciσ (ciσ) are the creation (annihilation) operators at a Copper site i, tij is the hopping † of an electrons/hole between sites i and j, niσ = ciσciσ is the occupation number operator, U is the on-site coulomb repulsion, the Hubbard U, and µ is the chemical potential.

32 Figure 3.2: (Left) Copper-oxygen plane consists of Copper (black circle) dx2−y2 orbital and Oxygen (gray circle) px and py orbitals comprise of bonding, anti-bonding, no bonding combination. (Right) Relevant eﬀective single orbital having a dx2−y2 symmetry centering at the Copper site with extended lobes. Figure courtesy of Ref. [98].

The justification for the single band model is given from the first principle calculation[98] and illustrated in Fig. 3.2. The relevant physics is mainly contained in the copper-oxygen plane where the superconductivity takes place.1 The interlayer hopping is often neglected, so the system is assumed in 2-dimensional space. There are actually evidence of hopping between the copper-oxygen plane which are seen as a bilayer splitting in spectroscopic data[99, 100, 101, 102]. The discussion of the effects of bilayer splitting is in section 4.4.1.

The plane consists of the copper dx2−y2 orbitals and the oxygen px and py orbitals with a combinations of bonding, anti-bonding, and non-bonding shown in Fig. 3.2(left). The dominant orbitals near the Fermi-level are the anti-bonding configuration in which copper d’s and oxygen p’s form an effective single orbital having dx2−y2 symmetry centering at the copper site with extended lobes along Cu-O-Cu directions shown in Fig. 3.2(right). In real cuprate superconductors, it is known from experiments that the system is rich in phase diagrams as shown in Fig. 3.3. For a discussions of pseudogap phase, please refer to Ref. [104, 105, 106, 107, 108, 109]. Also not shown explicitly in the figure but evidently exist are the stripes, or spin/charge density wave [86, 110, 111, 112, 113, 114, 115]. This

1Fig. 4.15 shows the crystal structure for a YBCO cuprate which contains two copper oxygen planes in the unit cell.

33 Figure 3.3: Schematic phase diagram for couple diﬀerent cuprate superconductors. Figure courtesy of Ref. [96, 103].

Hubbard model can capture many observed properties in cuprates such as the spin/charge anti-ferromagnetic stripes[116, 117], as well as the superconducting phase including the pseudogap[118, 119, 120, 121, 122, 123]. While the present single band model may be inadequate to reproduce other physical phenomena such as the anomaly in the phonon dispersion[124, 125], bilayer splitting[99, 100, 101, 102], etc[126, 127], it will be shown

that the Hubbard model can induce superconductivity that predicts a sign changing dx2−y2 superconducting order parameter if certain conditions are met. The second term in the Hamiltonian of Eq. 3.2 is the on-site coulomb repulsion, the Hubbard U. It is this Hubbard U that is responsible for driving the superconductivity. Simple picture can be illustrated as follow. Suppose the system is metallic. When an electron hops onto another atomic site that is already occupied by another electron, then, the carriers should scatter away due to the strong repulsion. The scattering can be inelastic, so the scattering is intervened by collective excitations. As revealed by the neutron scattering experiments[53], these scattering is often peaked at a large momentum transfer q = (π, π) which means the carriers have a tendency to scatter in a particular direction (as opposed

34 to the uniform scattering assumed in BCS theory). These inelastic scatterings are mostly dominant at temperature below or slightly above Tc, but absent at higher temperatures[53, 93, 128, 71]. The highly peaked scattering at a large momentum transfer tells that these collective excitations are a short range. The interaction between the carriers is repulsive in momentum space but becomes attractive in real space between the nearest neighboring copper site (this will be illustrated in section 3.3.3) which leads to form a bound pair at lower temperatures. Qualitatively the above describes a simple picture of the spin fluctuation interactions. In the next section, we will describe the theory of spin fluctuation quantitatively using a diagrammatic Green’s function formalism together with the Feynman diagrams. Before we move on, let us briefly examine Eq. 3.2 on a case-by-case basis. Assuming a single band with the chemical potential somewhere in the band, the conventional band theory predicts a metal. For U >> t (where t is the hopping parameter), a gap opens around the Fermi-level and the system becomes an insulator.2 The band splits into two bands called lower and upper Hubbard band. Bare in mind that the real cuprates are a charge-transfer insulator due to the existence of the Oxygen p orbitals lying closer to the fermi level than the lower Hubbard band. This is schematically illustrated in Fig. 3.4. With a moderate Hubbard U (compared to the hopping t) at an appropriate carrier concentration, the system is metallic and give rise to an itinerant magnetism. Itinerant carriers give rise to damped collective modes. Carriers can move around while its spin can flip, so the system is in a spin fluctuation state. At a particular doping range, the system can be superconducting at low temperature. The above description may explain at least in a qualitative level why the superconductivity has to reside in close proximity to the magnetism. In a real system, the antiferromagnetic and superconducting phases are not competing with each other, but often found to coexist[129, 110, 86] indicating the antiferreromagnetic spin fluctuation may be an important ingredient for the superconductivity.

2For half-ﬁlling, the system is in an antiferromagnetic mott-insuling state.

35 (a)

Lower Hubbard-band Upper Mott insulator Oxygen Hubbard-band p

휇0

(b)

Charge transfer Lower Hubbard-band Upper insulator Oxygen Hubbard-band p

휇0

Figure 3.4: Schematic density of states for two diﬀerent types of insulators. µ0 denotes the fermi-level. (a) Mott Hubbard insulator. (b) Charge transfer insulator.

36 3.2 FLEX Approximations

In Chapter2, we studied the superconducting properties of high Tc cuprates using the generalized Migdal-Eliashberg theory together with the modeled boson spectrum. Based on the neutron scattering data, we assumed that the superconductivity was induced from magnetic excitations, and therefore the boson spectrum was simply replaced with the phenomenological boson function F (ω) that mimics the dynamical structure factor of the neutron experiments[69]. We did not bother to analyze the origin of the boson spectrum seen in the neutron data. There are several downsides to this approach. First, it factors out the momentum dependence in the self-energies Z and χ. As was shown in Chapter2 (also refer to Fig. 2.5.(b)), neutron scattering data shows an intensity peaked at q = (π, π)[69, 53, 70, 71] indicating that inclusion of the momentum dependence is essential for an in-depth analysis. Second, we had to impose the d-wave symmetry manually in the electron-boson coupling constant. In this chapter, we will show couple important results. First, it will be shown that the above Hubbard model can produce the d-wave pairing symmetry. Second, readers can remember that McMillan and Rowell have compared α2F (ω) with the neutron data, and saw a remarkable similarities between them (refer to Fig. 1.1). In section 3.3.3, we will show that the calculated effective interaction reveals features that are similar to the neutron data supporting that the spin fluctuation exchange plays a pivotal role in mediating the paring. While there are various methods for studying the Hubbard model, we will stick to the Green’s function method which is similar to what was done in Chapter2. One way to implement the spin fluctuation exchange is to consider the electron self-energy of the diagram shown in Fig. 3.5. Veff (q, iΩm) is the effective interaction that accounts for the coulomb repulsion, the Hubbard U. To a first approximation, one can estimate Veff (q, iΩm) using Random Phase approximations (RPA)[25, 40]. For a spin singlet cooper pair, it is given by,

3 V (q, Ω = 0) ≈ U 2χ(q), eff n 2 χ0(q) χ(q) = , (3.3) 1 − Uχ0(q) 1 X f(k+q) − f(k) χ0(q) = , N k − k+q, k

37 푉푒푓푓(퐪, 푖훺푚)

Σ 퐤, 푖휔푛 =

퐺(퐤, 푖휔푛)

Figure 3.5: Electron self-energy diagram with an eﬀective interaction Veff (q, iΩm) that arises from the Hubbard U repulsion.

where k is the non-interacting electron dispersion, and f(k) is the fermi-function. One can see that for a sufficiently large U, the denominator of χ(q) can diverge, indicating an instability. The stoner criteria requires that Uχ0(q) < 1 to avoid this instability. For a cuprate like band structure with the effective interaction Veff (q), one will find the form dx2−y2 is one solution to the gap equation.

3.2.1 FLEX: Normal State

The RPA assumes a system in a weak coupling limit, and therefore, a more rigorous method is necessary for a system beyond a weak interaction. A method that goes beyond RPA is the ﬂuctuation exchange (FLEX) approximations[130, 131, 132, 133, 134, 135, 136, 137, 138, 139]. When investigating the superconducting properties, one can either approach from the normal state (T > Tc) or the superconducting state (T < Tc). If one proceeds from the normal state, it is commonly done by solving the Bethe-Salpeter eigenvalue equation in what is referred to as the particle-particle channel. Again, assuming a spin singlet cooper pair, this eigenvalue equation is given by,

1 X − Γ(k, k0)G (k0)G (−k0)ψ(k0) = λ ψ(k), (3.4) Nβ ↑ ↓ k k0 where Γ(k, k0) is the particle-particle vertex function, ψ(k) is the eigen function, and λ is the eigenvalue. In RPA, G(k0) is treated in the non-interacting limit. In FLEX approximations,

38 the Green’s function is fully dressed (renormalized), and is obtained self-consistently. In RPA or FLEX, assuming the spin singlet pair, the particle-particle vertex function that enters into Eq. 3.4 consists of the longitudinal and transverse spin ﬂuctuation terms which are shown in Fig. 3.6. Mathematically, they are expressed as[96],

2 0 U U χ0(q, Ωn) Γ(q = k − k, Ωn) = 2 2 + , 1 − U χ (q, Ωn) 1 − Uχ0(q, Ωn) 0 (3.5) 1 X χ (q, Ω ) = − G(k + q, ω + Ω )G(k, ω ), 0 n Nβ m n m k,m where the first term comes from the bubble diagram (longitudinal spin fluctuation), and the second term comes from the ladder diagram (transverse spin fluctuations). For RPA, non-interating Green’s function are used. For FLEX approximations, the Green’s function is the dressed Green’s function, and should be obtained self-consistently. How Eq. 3.4 works is as follow. One can choose a particular gap symmetry of their choice, then proceeds to obtain the eigenvalue. The superconducting transition temperature is determined when the largest eigenvalue λ becomes one. Likewise, one can try various gap symmetries to see which gap symmetry is most favored in the system. One final remark regarding the function Γ(q) in Eq. 3.5 is that it can also be written algebraically as, 3 χ (q) U 2χ (q) Γ(q) = U 2 0 + 0 + U. (3.6) 2 1 − Uχ0(q) 1 + Uχ0(q) The first term refers to the spin fluctuation and the second term refers to the charge

ﬂuctuation. For cuprates at T & Tc, the spin ﬂuctuation term dominates over the charge

ﬂuctuation owing to the denominator being close to zero (Uχ0(q) ≈ 1), meaning the system lies in close proximity to the stoner instability. The superconductivity is induced by the spin ﬂuctuation term. This discussion can be found from Ref. [96, 140].

3.2.2 FLEX: Superconducting State

In general, Eq. 3.4 is more commonly used especially within RPA. However, because the method is limited in a sense that the system is studied in the normal state and the method is unable to study the superconducting properties in details. Therefore, it is highly

39 + + …

Γ(퐤, 퐤′) =

… + + + …

Figure 3.6: Particle-particle vertex function Γ(k, k0) for a spin singlet pair entering in Eq. 3.4. Dashed line is the Hubbard U interaction. Blue arrow represents the spin. In RPA or FLEX approximations, the Γ(k, k0) consists of the ladder diagram (transverse spin fluctuations), and the bubble diagram (longitudinal spin fluctuation). Note that in FLEX approximations, the Green’s function are dressed (interacting). recommended that the self-energies are obtained in the superconducting state, so that one can actually compare real physical quantities with experimental data since oftentimes data are available in both the normal and superconducting state. It is those self-energies in the superconducting state that can give further insights regarding the superconducting properties. Details of the numerical implementations of FLEX approximations in the superconducting state are documented in Ref. [141]. We do not need to go over the whole theory again since much of the formalism introduced in Chapter2 can still be used, and the self-energies are evaluated in a similar fashion; that is, we again evaluate the rainbow diagram, but instead of the phenomenological bosons, we use the effective interaction Veff . In FLEX, the effective interactions have different algebraic expressions for each of the self-energies. Here, they are denoted by Vn(q, Ωn) for the normal self-energy and Va(q, Ωn) for the anomalous self-energy. The Feynman diagrams for each of the self-energies are shown in Fig. 3.7. These effective interactions include scattering processes shown in Fig. 3.6, though this Γ(q) only enters into the two particle scattering diagram. For the self-energy, one has to consider a diagram with

40 푉푛(퐪, 푖Ω푚) 푉푎(퐪, 푖Ω푚)

Σ 퐤, 푖휔푛 = 휙 퐤, 푖휔푛 =

퐺 퐤, 푖휔푛 F 퐤, 푖휔푛

Figure 3.7: Single band FLEX diagrams in the superconducting state. Σ(k, ωn) and Φ(k, ωn) are the normal and anomalous self-energies respectively. G(k, ωn) and F (k, ωn) are the normal and anomalous dressed Green’s function. Vn,a(q, Ωn) is the eﬀective interactions for the normal (n) and anomalous (a) self-energies. The double line in the eﬀective interaction informs that they are fully dressed. The inward arrows seen in F (k, ωn) symbolizes particle destructions at both points in time[15] (see Eq. 2.4).

Σ 퐤, 푖휔푛 = + + …

… + + + …

Figure 3.8: Normal state self-energy that are included in the approximations. The dashed line is the Hubbard U interaction. As noted, the above diagrams assume a spin singlet pair.

41 a closed loop. How these eﬀective interactions enter in the self-energy diagram is illustrated in Fig. 3.8. All higher order diagrams consist of the ladder and bubble diagrams.

Just as we saw in Chapter2, the anomalous self-energy is introduced to remove the divergence in the self-energy, and the Green’s function is expanded in terms of the Dyson’s series. The basic procedure for evaluating the self-energy is similar to Eq. 3.3. First, we

s c determine the irreducible spin χ0 and charge χ0 susceptibilities in the superconducting state. They are given by,

s 1 X χ (q, Ωn) = − G(k + q, ωm + Ωn)G(k, ωm) + F (k + q, ωm + Ωn)F (k, ωm), 0 Nβ k,m (3.7) s 1 X χ (q, Ωn) = − G(k + q, ωm + Ωn)G(k, ωm) − F (k + q, ωm + Ωn)F (k, ωm). 0 Nβ q,m

The spin and charge ﬂuctuation interaction are given by,

s 3 2 χ0(q, Ωn) 1 2 s Vs(q, Ωn) = U s − U χ0(q, Ωn), 2 1 − Uχ0(q, Ωn) 2 c (3.8) 1 2 χ0(q, Ωn) 1 2 c Vc(q, Ωn) = U c − U χ0(q, Ωn). 2 1 + Uχ0(q, Ωn) 2

Notice that the upper-left diagram in Fig. 3.8 belongs to both the ladder and bubble diagram. This is why the second term in the above equations is subtracted to avoid double counting this diagram. The eﬀective interactions in Fig. 3.7 are given by,

Vn(q, Ωn) = Vs(q, Ωn) + Vc(q, Ωn), (3.9) Va(q, Ωn) = Vs(q, Ωn) − Vc(q, Ωn), then, the normal and anomalous self-energies are given by,

1 X Σ(k, ω ) = V (q, Ω )G(k − q, ω − Ω ), n Nβ n m n m q,m (3.10) 1 X φ(k, ωn) = Va(q, Ωm)F (k − q, ωn − Ωm), Nβ q,m which is just the diagram shown in Fig. 3.7. The components of the self-energies can be ¯ obtained from Z(k, ωn) = ωn−Im[Σ(k, ωn)] and χ(k, ωn) = Re[Σ(k, ωn)] as shown in Chapter

42 2. Finally, the Hartree terms can be separately added. They are given by,

1 X U Σ = UG(k, ω ) + , HF Nβ n 2 k,n (3.11) 1 X φHF = UF (k, ωn). Nβ k,n

A reader can notice that this FLEX approximations also neglect the vertex corrections just like the Migdal approximations. Due to the absence of the vertex corrections, we must ﬁx the particle number n by updating the chemical potential µ after each iteration. Since

ΣHF is just the number, when we ﬁx the particle number, ΣHF can be absorbed into the chemical potential; namely, each iteration will simply produce the new chemical potential

0 µ + ΣHF → µ . Basically, if n is ﬁxed at each iteration, ΣHF can be neglected for a single- band case. However, for the multi-orbital case, Hartree term is needed. Also note that for the d-wave (dx2−y2 ) gap symmetry, φHF will also vanish. Finally, we use the following expression for obtaining the particle number,

2 X X 2 X n = [G(k, ω ) − G (k, ω )] + n ( ). (3.12) Nβ n 0 n N F k k |ωn|<ωcut k

The above expression includes the spin, so n = 1 corresponds to half-ﬁlling. Eq. 3.7 - Eq. 3.12 form a set of equations that need to be solved self-consistently until a good convergence is achieved. One can see that Eq. 3.7 and Eq. 3.10 are the convolution sum in both the momentum and frequency space. Applying the Fast Fourier Transform (FFT) to carry out the convolution sum is highly suggested since it will drastically reduce the calculation time. Applying the FFT in momentum space is trivial since the system is assumed to be periodic in momentum/real space, though the Fourier transform in the frequency/tau space requires care when one is performing the summation. For details, refer to AppendixB on how the Fourier transformation is performed. The ﬂow chart for the iterative loop is shown in Fig. 3.9. Note that the Hartree terms are calculated in the early steps, then added since the interaction is considered instantaneous.

43 Guess Σ푖푛 푘 , 휙푖푛(푘)

Obtain 퐺푖푛 푘 , F푖푛(푘)

Obtain Σ퐻퐹, 휙퐻퐹 Apply FFT to get 퐺 휏 , F (휏)

0 0 Apply FFT back to 휒푠,푐(푞) Obtain 휒푠,푐(휏)

Obtain 푉푛(푞), 푉푎(푞) Apply FFT to get 푉푛(휏), 푉푎(휏) Obtain 퐺푖푛 푘 , F푖푛(푘) Apply FFT back to Σ표푢푡(푘), ϕ표푢푡(푘), with mixing Obtain Σ표푢푡(휏), 휙표푢푡(휏) and add Σ퐻퐹, 휙퐻퐹

Obtain Δ = Σ푖푛 − Σ표푢푡, Δ = 휙푖푛 − 휙표푢푡

Δ < 푇표푙푒푟푎푛푐푒 Exit False True

Figure 3.9: Flowchart for the FLEX algorithm. The notations are denoted by k = (k, ωn), q = (q, Ωn) and τ = (r, τ). Before initializing, a user should choose the temperature, Hubbard U, and the band dispersion of their interests.

44 Lastly, it is quite common that the above FLEX calculations encounter issues with the convergence, especially at larger Hubbard U values. At the end of each iteration, one has to update the self-energy and Green’s function for the next iterations using some mixing scheme. If the mixing is not properly handled, the code converges very slowly, or it will not converge. In our case, the current code implemented the Green’s function mixing as well as Anderson mixing[142, 143]. For details, please refer to AppendixD. 3

3.3 Results

First, we try the single-band Hubbard model. In the present case, the non-interacting band dispersion k is taken from Norman et al[68] as was done in Chapter2. We choose this band as the default except only in section 3.3.1, where we vary the bandwidth to compare Tc. Note that the band structure taken from ARPES already includes the eﬀect of Hubbard U,

3 This footnote explains further numerical details regarding the Fourier transform algorithm.

1. Fourier transform G(k, ωn) = Gij,n → G(rx, ry, τ`) = Gxy,` is given by,

N −1 i( m )π` Nk−1 Nm−1 (β−τ )ξ e Nm ix+jy n` e ` k X X −i( N + N )2π Gxy,` = e k m Gij,n − δx,0δy,0 (3.13) N 2 eβξk + 1 k ij=0 n=0

where ξk = k − µ, Nm is the number of matsubara frequency, and Nk is the number of k points

along one axis. For ` = Nm, we use Gxy,Nm = −Gxy,0 − δi,0δj,0. To obtain Fxy,`, we use the same expression, but without the second term. (This is because F (ωn) decays faster than 1/ωn).

2. For the Fourier transform, χ(r, τ`) = χxy,` → χ(q, Ωn) = χij,n, ﬁrst we determine

χ(τ) = − G(τ)G(−τ) ± F (τ)F (τ) = G(τ)G(β − τ) ± F (τ)F (τ) (3.14) χxy,` = Gxy,`Gxy,Nm−` ± Fxy,`Fxy,`

Now, we may use Eq. 60 to obtain χij,n. (Here, we need to obtain χij,n for Ωn > 0 and Ωn < 0 separately by exploiting the periodic property.) Notice that χ(Ωn) decays faster than 1/Ωn, so χ(0+) = χ(0−).

3. Fourier transform V (q, Ωn) = Vij,n → V (r, τ`) = Vxy,` is given by,

iπ` Nk−1 Nm−1 e X X −i( xi+yj + n` )2π V = e Nk Nm V . (3.15) xy,` N 2 ij,n k ij=0 `=0

For ` = Nm, we set Vxy,Nm = Vxy,0. N A 4. Lastly, we obtain self-energies from Σxy,` = Vxy,`Gxy,`, and φxy,` = Vxy,`Fxy,`. Then, using Eq. 60 yields expressions for Σij,n and φij,n. The normal self-energy is discontinuous at τ = 0 given by + − Σxy(0 ) − Σxy(0 ) = −Vij,0δi,0δj,0. For the anomalous self-energy, it is continuous at τ = 0. Also, note we need to obtain self-energies for ωn > 0 and ωn < 0 as was done in step 2.

45 though this tight binding model is still suﬃcient and serves our purpose. Other studies can tune the bandwidth to approximately model the non-interacting band dispersion. Here, we are testing the band structure with a cuprate-like fermi-surface. It will be shown that the topology of the band structure and the Fermi-surface is critical for inducing a certain gap symmetry in the spin ﬂuctuation mediated superconductors.

3.3.1 Tc vs. U

Here, we would like to show the dependence on Hubbard U in regards to the superconducting

transition temperature. Tc was obtained in the same manner by extrapolating zero of the gap function with the fit (an example of the fit is shown in Fig. 3.10(b)). In Fig. 3.10(c), the calculated gap function ∆(k, π/β) for U = 0.6 eV at T = 10 K (Tc = 30 K) is plotted, showing the dx2−y2 symmetry with a node along the diagonal (nodal) direction. The s-wave symmetry was also tested, but the current code did not converge after 500 iterations whereas the d-wave case converged within less than 40 iterations for most cases. For sufficiently large U value, for example U = 1.0 eV, the code will find the d-wave solution after enough iterations even if the code starts with an s-wave gap symmetry. The current simulation shows the system favors d-wave pairing rather than s-wave. In Fig. 3.10(d), Tc (dx2−y2 ) as a function of U is plotted for two equivalent band dispersions but with a different bandwidth W . One of the factors for the Tc enhancement is that the system has a large bandwidth can bare large U value before reaching the instability. Strong Hubbard U can enhance the superconducting transition temperature as long as the system falls below the stoner instability. This is in contrast to the BCS theory in which the bandwidth plays a minimal role with the assumption that the phonon energy is much smaller than the Fermi-energy. Also note that the Tc reaches maximum value for a particular U value then begins to decline as U is increased until the system hits the stoner instability implying the suppression of the pairing near the magnetic transition.

46 (a) (b)

Figure 3.10: All results presented above are dx2−y2 gap symmetry. (a) Fermi-surface for the current simulation. (b) Gap function ∆(T, k = (π, 0), π/β) for U = 0.6 eV. Function p C1tanh[C2 1 − T/Tc] is used to fit the data. From the fit, Tc is found to be 30.6 K. (c) Calculated Gap function ∆(k, π/β) producing dx2−y2 symmetry. Compare this to the d-wave in Fig. 3.1(b). Relevant parameters are U = 0.6 eV at T = 10 K. (d) Comparison of Tc vs. U for two different bandwidth W . W1 = 1.3 eV. W2 = 1.5 ×W1. Here, for the band 2, all the hopping parameters including the initial chemical potential were set to be 1.5 times that of the band 1. For all U values, dx2−y2 symmetry is realized.

47 3.3.2 Tc Dependence on the Chemical Potential

In BCS theory, the transition temperature depends on the electron density of states at the fermi-level. It predicts that the larger density of states yields higher Tc. More carriers at the Fermi-level means extra cooper pairing. In cuprates and iron-based superconductors, it is known that the superconducting transition temperature has strong dependence on the doping concentration. As was shown in Fig. 3.3, these unconventional superconductors possess multiple phases with respect to the doping. Only in a certain range of doping can the system settle in the superconducting state. Here, we simulate the same model as was done in the previous section, but vary the fermi-level to see what range of chemical potential the system becomes superconducting. Fig. 3.11(a) shows the electron density of states N(ω) for the non- interacting band dispersion. The peak in the density of states is the Van-Hove singularity (VHS). The dashed-red line shows the range of chemical potential where the system realizes d-wave superconductivity. As can be seen, the system only finds superconductivity only within the limited range of the chemical potential, and no superconductivity is found outside this range. Again, this is in contrast to the phonon mediated superconductors in which the superconductivity only emerges away from VHS.4 On the other hand, the current model predicts the superconductivity emerges only near VHS. Fig. 3.11(b) shows Tc as a function 0 of the chemical potential µ where µ is defines as k = k(µ=0) − µ. The superconductivity is mostly found in the range above VHS, and its Tc reaches the maximum value around the µ = − 0.075 eV (VHS is at µ = − 0.13 eV). This implies the certain degree of nesting vector is necessary for the superconductivity, though the maximum Tc is not achieved at VHS, but slightly higher. The origin of these nesting can arise from either the spin fluctuation exchange or the VHS effect. One should be aware that these two effects are different physics.

3.3.3 Spin Fluctuation Interaction Vs

In Chapter1, we have shown that the electron-phonon coupling strength is related to the function α2F (ω) by, Z α2(ω)F (ω)dω λ = 2 . (3.16) ω

4This will be discussed in section 4.3

48 Superconductivity (a) (b)

Figure 3.11: U = 0.6 eV. (a) Electron density of states N(ω) for the non-interacting band k. Dashed-red line shows the range of non-interacting chemical potential where the system becomes superconducting. (b) Tc as a function of µ in the region indicated in (a). µ is 0 deﬁned as k = k(µ=0) − µ. The Van-Hove singularity is at µ = −0.13 eV.

Fig. 1.1 also revealed similarities between the neutron scattering data and the extracted function α2(ω)F (ω) in Pb. For the spin ﬂuctuation exchange, one can also deﬁne the coupling strength according to the following relation,

Z Im[V (q, ω)]dω λ = 2 s . (3.17) q ω

Based on the previous studies, it may be a worthwhile to compare Im[Vs(q, ω)] with the neutron scattering data. As was shown in Fig. 2.5, the spin polarized neutron scattering shows an intensity peaked particularly at q = (π, π) at resonance energy[69]. Here, we can check the spin ﬂuctuation interaction Vs(q, ω) and see if they follow the same characteristic.

The spin ﬂuctuation interaction Vs(q, Ωn) in Eq. 3.8 is given in terms of the imaginary

matsubara frequency Ωn. Since the analytic continuation is not given algebraically, the real

frequency dependence Vs(q, ω) is obtained using Pad´eapproximants[144], which allows to go to the real axis Ωn → ω + iδ. Fig. 3.12(a) shows Vs(q, ω) at three diﬀerent momentum at q = (0, 0), q = (π, 0), and q = (π, π). Here, the Hubbard U and the temperature

49 (a) (b)

Figure 3.12: Imaginary part of the spin ﬂuctuation interaction Im[Vs(q, ω)]. U = 0.8 eV and Tc = 37.8 K. (a) Comparison at three diﬀerent momentum q = (π, π), (π, 0), (0, π), and T = 0.26Tc. (b) A comparison between the normal and superconducting state at q = (π, π) (c) Vs(q, Ωn = 0) in momentum space. It is positive (repulsive) for all momenta. (d) Vs(r) in real space. The sign of the interaction oscillates from one site to the nearest site. The interaction is a short-rage and decays rapidly away from r = (0, 0).

50 were set to be 0.8 eV at T = 0.26Tc respectively. Among the three momentum points, the intensity is particularly peaked at q = (π, π). Here, the peak appears at ω = 12 meV which is substantially smaller than 2∆max = 32 meV and agrees with the previous studies[145]. The peak can be understood from the singularity in the formula. Deﬁning

s 0 00 χ0(q, ω) = χ (q, ω) + iχ (q, ω), the formula for Im[Vs(q, ω)] is given by, 3 χ00(q, ω) 1 Im[V (q, ω)] = U 2 − U 2χs(q, ω). (3.18) s 2 [1 − Uχ0(q, ω)]2 + [Uχ00(q, ω)]2 2 0

The singular behavior comes from the ﬁrst term. The temperature dependence of the peak is shown in Fig. 3.12(b) in which the robustness of the peak suddenly drops in the normal state.

If we compare Im[Vs(q, ω)] with the neutron data from Fig. 2.5(b), it does show similar characters such as the peak at q = (π, π) and the temperature dependence, especially the shift in the resonance peak to lower energy in the normal state is noteworthy. Nevertheless, there are features that cannot be answered with the current model such as the doping dependence (See Ref. [146]), or the resonance peak appearing at such low energy (∼ 12 meV in the simulation) as opposed to ∼ 41 meV according to the neutron data. As previously mentioned, the d-wave gap symmetry arises from the repulsive interaction (V (q) > 0) of the spin ﬂuctuation exchange in the momentum space. Fig. 2.5(c) shows

Vs(q, Ωn = 0) with the intensity peaked at q = (π, π), but also being positive for all

momenta which shows its repulsive nature. However, the Fourier transform of Vs(q, Ωn = 0)

into the real space Vs(r) shows oscillatory interactions consisting of both repulsive and attractive interactions. This is illustrated in Fig. 3.12(d) which clearly shows strong repulsive interaction at r = (0, 0) (on-site), but becomes negative at nearest neighboring sites, and oscillates from one site to the nearest site with its intensity dies oﬀ rapidly as the relative distance between the pair increases. The attractive interaction is most robust when the distance between the pair is at the nearest neighboring site. The pairs can become bound owing to this attractive interaction if temperature is suﬃciently low.

51 3.3.4 Spectral Function and Density of States

We now present results for the spectral function. As was mentioned in Chapter2, the feature that we are most interested is the low energy kink like the one shown in Fig. 2.6 (c) and (d). Just as McMillan and Rowell[5] have done to prove that phonons were the mediators for the supercondtivity in Pb, studying these low energy features, possibly the bosonic glue, might provide clues for the pairing mechanism in some unconventional superconductors. By obtaining the self-energy in the superconducting state, one can study various superconducting properties. The analytic continuation within FLEX is not possible algebraically, so once again, we rely on the Padéapproximants. The spectral functions at three different momentum cuts are shown in Fig. 3.13 (a), (b) and (c). In (a), one can see the known waterfall dispersion along the nodal direction. The dispersion is weakly renormalized and no pronounced kink can be seen in this direction. In (c) and (d), the renormalization becomes more robust as we plot the dispersion in the anti-nodal direction. In Fig. 3.13(b), the imaginary part of the electron self-energy is plotted at three high symmetry points. One can easily confirm that the self-energy develops a pronounced peak at k = (π, 0) at energy

∼ 2∆max. The spin ﬂuctuation exchange create a highly momentum dependent electron self-energy. The dispersion is weakly renormalized along the nodal directions, but becomes stronger in the anti-nodal directions owing to the peak in self-energy at this momentum. The current simulations are not intended to fully reproduce the experiments (for various reasons such as the absence of the vertex corrections), but the readers can compare similar features that can be seen from other ARPES data such as Ref. [147, 148, 149]. Next, let us investigate the electron density of states. As was shown in Fig. 2.1, STM results from cuprate superconductors show a modulation in the density of states with a dip- hump structure, the “up-slope”, at the kink energy ωkink = Ω + ∆[46, 47]. The simulated density of states N(ω) using FLEX for three diﬀerent U values are shown in Fig. 3.14. One can see the dip-hump features appearing in all three cases. The dashed line denotes the calculated kink energy ωkink = Ω + ∆ where Ω is the peak in Vs(ω) at q = (π, π) and ∆ is the maximum value of the gap along the fermi-surface. At least in the present model, the “down-slope” appears at the kink energy in all three cases which contradict with the

52 (a) (b)

Figure 3.13: (a) Spectral function along the nodal direction. U = 0.6 eV and T = 0.33Tc with Tc = 30.6 K. (b) −Im[Σ(k, ω)] at three diﬀerent momentum points indicated in the legend. (c) (d) Same spectral function along the anti-nodal cuts indicated at the bottom. The kink strengthens as it approaches (π, 0) where the peak in ImΣ(k, ω) appears.

53 (a) (b)

(c)

Figure 3.14: (a) - (c) Electron density of states at three diﬀerent values of U. Red dash line is the kink energy deﬁned as ∆ + Ω. All showing the down-slope feature at these energy. (see discussions in the texts)

54 STM results which show the “up-slope”. One can remember from Chapter2 that the two boson model was needed to produce the up-slope; this FLEX model alone did not reproduce the up-slope feature. At this point, it is still inconclusive and require further investigations whether the up-slope feature is the result of the multi-bosons model, inclusion of the vertex corrections, or due to other reasons.

55 Chapter 4

Applications

In Chapter2 and3, the basic formalism of the Migdal-Eliashberg theory and FLEX were given. In this chapter, we will continue investigating other applications using the same formalism introduced in the previous chapters. We will look into a momentum dependent electron-phonon coupling, spin ﬂuctuation plus phonon mediated superconductivity, the competition between the charge density wave (CDW) order and superconductivity, and more. Before we move on, let us introduce once again the Migdal-Eliashberg theory but with the full momentum dependence reinserted into the equation. As we saw in Chapter2 that each self-energies expressions given in Eq. 2.20- 2.22 factored out the momentum dependence. For a system with a simple momentum dependence in the electron-boson coupling constant, this approximation is valid just as was done in the BCS theory or traditional Eliashberg theory[5]. However, as we saw in Chapter3, the spin ﬂuctuation exchange is highly momentum dependent. Furthermore, there exists various phonon modes that are associated with momentum dependent coupling. In such a case, Eq. 2.14 needs to be used instead of the momentum averaged scheme. The momentum independent electron- phonon coupling, normally called the Holstein model, is simply one general case that may arise in a real system while many other systems are highly anisotropic and momentum dependent.

56 4.1 Forward Scattering Phonons in Iron-based Super- conductors

A Couple of diﬀerent iron-based superconductors have shown unique electronic structures that caught attention by the scientiﬁc community. Those are the monolayer FeSe grown on

SrTiO3 substrate (FeSe-STO)[4, 150, 151] and a heterostructured bulk Sr2VO3FeAs[152]. An

interesting fact is that bulk FeSe merely exhibits Tc of ∼ 8 K at ambient pressure; however,

when it is grown on SrTiO3 substrate, it yields Tc = 55 ∼ 77 K (up to 107 K according to transport measurements[34]), the highest among all iron based superconductors. This discovery motivated researchers to try out other interface systems using combinations of

various layered materials. It turns out the mechanism for the drastic enhancement in Tc remains highly controversial. Some possible explanations are the charge transfer effect[153, 154, 155], an induced electric field due to charge transfer[156], or strain effects[157]. Another possibility is due to interactions between the carriers in the FeSe layer and the forward scattering phonons in the STO substrate[4].

Both FeSe-STO and Sr2VO3FeAs materials show a band structure known as a replica band, a same exact replica of the electron band but shifted by the phonon energy, which has been revealed by the band structure image from ARPES for FeSe-STO[4] or Quasiparticle interference (QPI) image from STM for bulk Sr2VO3FeAs[152]. Fig. 4.1(a) shows the ARPES images taken from FeSe-STO at various temperatures. One can visualize the presence of the replica band at all temperatures. The simulation that will be shown shortly reveals that the replica band can exist in the normal state. The origin of the replica band is attributed to the strong electron-phonon coupling with its scattering intensity sharply peaked at a momentum transfer q = 0. FeSe-STO is particularly interesting since the Fermi-surface revealed by ARPES shows only small electron pockets at M points in the Brillouin zone while the hole pocket at Γ point is 60−80 meV below the Fermi-level[158]. This contradicts with the standard picture of spin ﬂuctuation mediated pairing which requires strong nesting with a large momentum transfer near the Fermi-level. Coupling with an incipient band is another candidate that explains the reasoning for the pairing in FeSe-STO[159, 160].

57 (a)

(b)

풉 휹풉

Figure 4.1: (a) ARPES image taken from FeSe-STO at various temperature around M point of the Brillouin zone. Figure courtesy of Ref. [4]. Solid blue line is the electron main band. Blue dashed line is the replica band. (b) Crystal structure for FeSe-STO. h is the distance between FeSe layer and TiO2 layer. δh is the displacement associated with the forward scattering phonon mode. Image generated using XCrySDen[161].

58 The origin of the forward scattering phonons generally depends on the material and crystral structure. Forward scattering phonons tend to emerge when the system is strongly quasi-two dimension in a layered system. In such materials, the system can yield

highly anisotropic dielectric response with perpendicular (⊥) and parallel components (k) significantly differ from each other. When a monolayer FeSe is grown on STO substrate, charges from the substrate are transfered to the monolayer. This charge transfer shifts the chemical potential of the FeSe leaving the hole pocket below the Fermi-level while electron pockets still remain. Additionally, the charge transfer effect sets up an electric field which creates dipole moments at each atomic sites. The relevant phonons are those that couples

to the vibration of oxygen dipoles at the interface (TiO2 layer) since other dipoles down

below are eﬀectively screened. The Coulomb potential between the vibrating dipoles qeff δh

in TiO2 layer and carriers in FeSe plane is given by, 1/2 Z ∞ 0 0 0 0 ndqeff h0k δh(x , y )dx dy Φ(x, y) = 3/2 3 , (4.1) −∞ k 2 ⊥ h2 + (x − x0)2 + (y − y0)2 ⊥ 0 0 0 where x y are the position in FeSe plane, x y are the postion in TiO2 plane, nd is the number of the oscillating dipoles per unit FeSe surface, h is the distance between FeSe and

TiO2 layer. Applying the Fourier transform to the above potential and inserting it into the ˆ † ˆ ˆ† Hamiltonian H = Σq eΦ(q)ˆρ(q) = Σk,q g(q)ˆck+qcˆk(bq +b−q), we obtain the electron-phonon coupling constant given by[162,4, 163],

s 2πqeff nd g(q) = ~ e−|q|/q0 , (4.2) ⊥ MNΩph

where M is the mass of the oxygen, and Ωph is the phonon frequency. Fig. 4.1(b) shows the crystal structure for this FeSe-STO system. The important quantity here is the q0 = ⊥/(kh) which decides the degree of the forward scattering. Since the carriers are conﬁned in the quasi-two dimensional FeSe plane, one may naively expect k to be much larger than ⊥ because k contains contributions from both FeSe plane and the STO substrate, whereas ⊥ is contributed only from STO substrate. Therefore, q0 = ⊥/(kh) is expected to be small leading to a sharp intensity peak in the electron phonon coupling constant around q = (0, 0); thus, the scattering tends to be in the forward direction.

59 4.1.1 Model

To investigate the forward scattering phonons, we use the formulas similar to Eq. 2.14, but assume dispersionless phonons for simplicity. Here, we consider a single band for the FeSe electron given by k = −2t[cos(kxa) + cos(kya)] − µ with t = 75 meV and µ = − 235 meV which yields an electron pocket and Fermi-velocity close to the observed dispersion seen from ARPES measurements[150, 151]. The expression for the self-energy along the imaginary axis is given by,

1 X Σ(ˆ k, iω ) = − |g(q)|2D(iω − iω )τ ˆ Gˆ(k + q, iω )ˆτ , (4.3) n Nβ n m 3 m 3 qm

2Ωph where D(q, iω`) = − 2 2 is the bare phonon propagator. The analytic continuation for Ωph+ω` the above self-energy is done algebraically. The self-energy along the real axis is given by,

∞ Z ˆ ˆ 1 X X 2 0 τˆ3G(k + q, iωm)ˆτ3 0 Σ(k, ω) = − |g(q)| F (ω ) 0 dω Nβ ω − iωm − ω q m=0−∞ (4.4) 1 X Z ∞ + |g(q)|2F (ω0)ˆτ Gˆ(k0, ω − ω0)τ ˆ [n (ω0) + n (ω − ω0)]dω0, N 3 3 b f k0 −∞

2 2|q| where we replace the electron-phonon coupling constant with |g(q)| = g0 exp(− ) adopted q0 from Eq. 4.1. Again, let us deﬁne the electron-phonon coupling strength λ in a same fashion as Eq. 2.24 evaluated for ` = s channel since we will show that the gap function in the present case is going to be an s-wave superconductor. λ in this case is given by,

2 X 2 λ = |g(q)| δ(k)δ(k+q), (4.5) N(0)Ωph k,q where N(0) is the electron density of states at the fermi-level. Keep in mind that the eﬀective

∗ ∂Σ(k) mass enhancement factor m /m = 1+λm is given by λm = −( ∂ω )ω=0 k where the bracket means the fermi-surface averaging. Generally, λ ≈ λm for the uniform scattering, but λ 6= λm for the strong forward scattering.

We have run the ﬁrst principle calculations for the bulk SrTiO3 using Quantum Espresso[164] to estimate the phonon energy for the model calculations. Fig. 4.2 shows

60 Figure 4.2: Phonon dispersion for the bulk SrTiO3. The calculation was carried out using Quantum Espresso[164] with PBE functional and GBRV ultrasoft pseudopotential[165]. The cubic structure with the lattice constant a = 3.905 A˚ was used for the simulation. The negative phonon frequency indicates an instability of the structure. The real SrTiO3 undergoes an antiferroelectric distortion (tetragonal) at ∼ 105 K but remains paraelectric, then stabilize to quantum paraelectric below ∼ 4 K[166].

61 the phonon dispersion for bulk SrTiO3. The highest longitudinal optical phonon mode is slightly dispersive and found at 90 ∼ 110 meV which is well separated from the lower phonon

branches. For the present model, the phonon energy is set to be Ωph = 90 meV. Before we move on, it is worth the time to examine whether or not the Migdal approximations are valid in the strong forward scattering limit. One criteria that is used to

Ωph validate Migdal approximations is give by the ratio where EF is the Fermi-energy. If EF

we assume Ωph = 90 meV and EF = 40 meV for this FeSe-STO system, then the ratio is obviously not small. It does seem that Migdal approximations are violated. However, for

the perfect forward scattering case, the vertex corrections are bounded by λm. To illustrate (1) 0 0 this, we ﬁrst consider the lowest order vertex correction Γ (k, k , ωn, ωn). Here, the perfect forward scattering means the electron-phonon coupling constant is replaced with the delta

2 2 function given by |g(q)| = g0Nδ(q). The vertex correction becomes,

g2 X Γ(1)(k, k0, ω , ω0 ) = − 0 D(0)(ω )G(0)(k, ω0 − ω )G(0)(k, ω − ω ), n n β m n m n m m 2 (4.6) g0 X 2Ωph 1 = 2 . β Ω + ω [i(ω 0 − ω ) − ][i(ω − ω ) − ] m ph n m k n m k

We only need to consider electrons near the Fermi level, so we let k = 0 with k = kF . The summation over the matsubara frequencies can be evaluated yielding,

2 2 3 0 (1) 0 h Ωphβ Ωph(Ωph − ωnωn ) Ωph β i 0 Γ (kF , ωn, ωn) = λm coth 2 2 2 2 − δn,n 2 2 , (4.7) 2 (Ωph + ωn)(Ωph + ωn0 ) Ωph + ωn 2

2 g0 where we redeﬁned λm = Ω2 to be the coupling strength. For the inelastic scattering ωn 6= (1) ωn0 ,Γ falls within the range,

Ω β 0 < Γ(1)(k , ω , ω0 ) ≤ λ coth ph . (4.8) F n n m 2

Ωphβ For temperature T << Ωph, coth 2 ≈ 1, so the vertex correction is less than λm which is estimated to be 0.15 ∼ 0.2 according to ARPES measurements[150,4].

For the elastic scattering ωn = ωn0 , we need to proceed carefully since the ﬁrst order vertex correction diverges at low temperatures. It is found that these contributions from

62 elastic scatterings diminishes when higher order corrections are included. Here, we expand the vertex corrections within the ladder approximations. We have the following expression for the vertex corrections given by,

g2 X Γ(k , ω ) = Γ(0)(k , ω ) − 0 D0(ω )[G0(k , ω − ω )]2Γ(k , ω − ω ), F n F n β m F n m F n m m 2 (4.9) g X 2Ωph 1 = 1 − 0 Γ(k , ω − ω ). β (Ω2 + ω2 ) (ω − ω )2 F n m m ph m n m

Evaluating the Matsubara frequency summation, we have

2 (0) Ωphβ Ωph (0) Γ (kF , ωn) = 1 − λm 2 2 Γ (kF , 0), 2 Ωph + ωn 2 (4.10) λm Ωph = 1 − 2 2 2 . + λm Ω + ω Ωphβ ph n

Notice that the final expression contains no diverging behavior, and the elastic vertex function is bounded within 0 ≤ Γ ≤ 1. One can also see the final expression approaches zero at low temperature, and therefore, the contribution from the elastic scattering does not play a role significantly in the limit of perfect forward scatterings. For the contribution from

the inelastic scattering, the vertex correction is smaller than λm which is estimated to be 0.15∼ 0.2[150]. To summarize, the Migdal approximations are still valid in a strong forward

scattering case provided that λm is small.

4.1.2 Spectral Function

The spectral function and the density of states are shown in Fig. 4.3(a) at T = 10 K

(Tc = 54 K) for λ = 0.6 and q0 = 0.3. The replica band emerges at energy ∆Ωreplica = 2 Ωph + 2λmΩph + O(λm) below the main band, but with a reduced spectral weight. Here, ∗ λm is the general deﬁnition of the eﬀective mass enhancement factor m /m = 1 + λm. The ratio of the spectral weight between the main and replica band gives the direct measure

2 of the coupling strength given by, A(k, ωmain)/A(k, ωreplica) = λm + O(λm), where k = 0 is taken to be the bottom of the band. The observed ratio from ARPES is approximately

63 (a) Replica (b)

Kink

풌푭

Main band 훥훺푟푒푝푙푖푐푎

Replica

Figure 4.3: (a) Spectral function A(k, ω) along ky = 0 at 10 K in the superconducting state. Replica band appears below ∆Ωreplica relative to the main band. To the right, the corresponding electron density of states is plotted. The kink at ω ∼ 0.1 eV is also visible as a sudden drop in the spectral weight. (b) Gap function ∆(k, ωn = π/β) is shown. It is an isotropic s-wave, but the intensity is peaked at k = kF .

0.15 ∼ 0.2[150], while the current simulation yields ∼ 0.18. The replica band also emerges for energy ω > 0 in the unoccupied states along with the kink seen at ω + ∆ ∼ 0.099 eV. In Fig. 4.3(b), the gap function as a function of momentum is shown. The red dashed line represents the fermi-surface. The gap is an isotropic s-wave structure along the Fermi- surface; however, the real FeSe-STO system shows an anisotropic superconducting gap[167]. The current model produces an anisotropic gap structure in momentum when the Fermi- level is close to the van-hove singularity, though such scenario would be unlikely because the Fermi-surface topology near the van-hove significantly differs from the observed small circular, electron pocket. The current model neglects the hybridization from orbitals which might be needed to reproduce the observed momentum dependence. Another remark one can make is that the gap structure is sharply peaked at k = kF as opposed to the uniform gap expected for normal phonons with uniform scattering. This shows that the gap affects the electronic structure only at the Fermi-momenum where the gap opens in the superconducting state.

64 푞0 = 0.3 푞0 = 0.3 푞0 = 0.3 푇 = 0.55푇푐 푇 = 0.93푇푐 푇 = 1.3푇푐

푞0 = 1.0 푞0 = 2.0 푞0 = 3.0 푇 = 0.55푇푐 푇 = 0.58푇푐 푇 = 0.59푇푐

Figure 4.4: Spectral function with forward scattering phonons in the system at various temperatures and q0. The replica band remains above Tc. For higher values of q0, the replica band starts to disperse.

Next, the spectral function for various temperatures and q0 are compared and presented in Fig. 4.4. The replica band clearly remains above Tc as well as the kink in the unoccupied region. Looking at the occupied region (ω < 0), the replica band starts to disperse with larger q0 value which is expected. In the limit q0 → ∞, the electron-phonon coupling constant becomes uniform scattering |g(q)|2 = constant. For the unoccupied region (ω > 0),

the replica bands are visible for q0 = 0.3 at all temperature, but for q0 > 1.0, they disperse away and the main and replica bands are no longer distinguishable. These are the typical features one should expect from the forward scattering phonons. The idea of the present simulation is that observing the renormalization features and spectral weights of each bands can give an estimate of the electron-coupling strength and the value

of q0. The simulation can easily plot the electron dispersion in the unoccupied region in which ARPES generally does not have access to. However, using QPI imaging, one can also investigate the electronic structure in the unoccupied region. This will be discussed in the next section.

65 4.1.3 Quasiparticle Interference

The concept of QPI is similar to the standing wave in classical wave mechanics despite QPI being completely quantum mechanical phenomena. This is illustrated in Fig. 4.5(a). Some wave (e.g. rope or air) is traveling towards the rigid wall on the left. In wave mechanics, we deﬁne a wave number given by k = 2π/λ where λ is the wavelength. Suppose the wave

having kin is completely reﬂected from the wall. At the right frequency, it forms a standing

wave with a new wave number knew = 2kin. In a real system, all materials contain some degree of impurities due to imperfections such as atomic vacancies. Although it is circumstantial, a similar phenomena can happen in a real metallic system owing to the particle-wave duality of electrons. Electrons can hit the impurity and scatter away. Fig. 4.5(c) describes the case for the point-like impurity. If no other scatterings are dominant (exceptions are for example cuprates[168, 169, 170, 171]), the most probable scattering is given by the backward scattering with the momentum transfer given by q = 2k. The incoming and outgoing electrons interferes to form an oscillation pattern similar to the standing wave. This interference pattern will appear as an electron density oscillation and can be directly observed by STM imaging. The example is shown in Fig. 4.5(d). Because these oscillations have periodicity, the Fourier transform of the STM image will show a peak at a momentum associated with twice the electron momentum 2k. One can vary the bias voltage to get the electron momentum at diﬀerent energy. Finally, the energy vs. momentum can be plotted to see the electron band dispersion. The advantage of this technique over ARPES is that one can probe the electron dispersion in the unoccupied states[172, 173, 174]. To formulate QPI, we calculate the change in the local density of states due to a single impurity using T-matrix approach[175, 176, 177, 178]. We assume the concentration of the impurity to be dilute. No combined interferences from multiple impurities are considered. The Hamiltonian due to the impurity is given by,

X † ˆ Himp = ΦkVkk0 Φk0 , (4.11) k,k0

66 (a) 2휋 푘 = 휆 푘표푢푡 푘푖푛

휆 (b) 휆 = 푖푛 → 푘 = 2푘 2 푖푛

Incoming electron Outgoing electron

Standing wave

Figure 4.5: (a) Classical wave mechanics illustrating the standing wave. Initially, a wave with a wavelength λ travels from the right to left. The incoming wave with a wave number kin = 2π/λ hits the left wall and reﬂects back. (b) At the right frequency, the standing wave is formed with the wave number k = 2kin. (c) Schematic picture describing the interference. Incoming electron scatters from the impurity in the backward direction. Outgoing electron interferes with the incoming electron. (d) STM image showing the interference pattern taken from Cu(111) at -5mV and 150 K. Figure courtesy of Ref. [179].

67 ˆ R ˆ i(k−k0)·r where Vkk0 = V (r)e dr is the Fourier transform of the impurity potential, and Φk = † [ck↓, c−k↑] is the Numbu spinor[58]. The impurity-renormalized 2×2 Green’s function matrix is,

ˆ(imp) 0 ˆ ˆ ˆ ˆ 0 ˆ ˆ ˆ 00 ˆ ˆ 0 G (k, k ) = G(k) + G(k)Vkk0 G(k ) + G(k)Vkk00 G(k )Vk00k0 G(k ) + ... (4.12) ˆ ˆ ˆ ˆ 0 = G(k) + G(k)Tkk0 G(k ),

ˆ where Tkk0 is the T-matrix given by,

ˆ ˆ 1 X ˆ ˆ 00 ˆ Tkk0 = Vkk0 + Vkk00 G(k )Vk00k0 + ... N 00 k (4.13) 1 X 00 = Vˆ 0 + Vˆ 00 Gˆ(k )Tˆ 00 0 . kk N kk k k k00

Note that the ω dependence has been omitted for simplicity. The above T-matrix can be solved self-consistently. However, if the impurity potential is a delta function (a point-like ˆ ˆ impurity), then the potential becomes independent of momentum Vkk0 → V , and the T- matrix can be solved explicitly. Inserting ω dependence, we obtain,

X Tˆ(ω) = [Vˆ −1 − Gˆ(k, ω)]−1. (4.14) k

For a non-magnetic impurity,τ ˆ3 component contributes, so that the potential is given by ˆ V = V τˆ3. Finally, the change in the local density of states is given by,

i X δρ(q, ω) = [δGˆ (k, q,ω) + δGˆ (k, q, − ω) − δGˆ∗ (k, −q,ω) − δGˆ∗ (k, −q, − ω)], 2πN 11 22 11 22 k (4.15) where we have deﬁned δGˆ(k, q,ω) = Gˆ(imp)(k, q, ω) − Gˆ(k, ω), and the momentum transfer q = k − k0 as usual. First, let us begin with the same FeSe-STO model introduced in the previous section 4.1.1, and compare the spectral function A(k, ω) and QPI intensity. The comparisons are presented in Fig. 4.6. (a1) shows the spectral function in the superconducting state. (a2) shows the corresponding QPI intensity |δρ(q, ω)|. Notice that band dispersion in QPI image appears at q = 2k as expected, which indicates the dominant scattering is the backward

68 Figure 4.6: (a1) Spectral function A(k, ω) plotted along the diagonal direction k = (q, q). (a2) Corresponding QPI intensity |δρ(q, ω)|. White dashed lines are the energy where the intensity slice are plotted in (b) & (c). Arrows denote diﬀerent types of scatterings (see texts for further details). (d1) & (d2) Intensity cuts along the diagonal direction taken from |δρ(q, ω)|. scattering. (b1) & (b2) show A(k, ω) at ω = 130 meV (unoccupied states) and −110 meV (occupied states) respectively. For the unoccupied states shown in (b1), one can clearly distinguish both the main band (outer) and replica band (inner). Due to the existence of the replica band, there can be three types of scattering: scattering between the main band denoted by “m-m” (white arrow), scattering between the main and replica band “m-r” (yellow arrow), and scattering between the replica band “r-r” (red arrow). The corresponding |δρ(q, ω)| are plotted in (c1). It shows a peak originating from the intra “m-m” scattering (white arrow). It also shows a weaker intensity peaked at a momentum (yellow arrow) arising from the inter “m- r” scattering. The “m-r” scattering process is clearly seen in the 1-D plot shown in (d1) which is plotted along the diagonal direction q = (q, q). Another peak from the intra “r-r” scattering is also expected at a momentum indicated by the red arrow, though it is too weak to spot because of the reduced spectral weight of the replica band compared to the main band. Due to this, it is expected that these features arising from the “r-r” scattering, or even “m-r” scattering are diﬃcult to resolve in an actual experiment. On the other hand,

69 the story is diﬀerent for the occupied states. (b2) shows A(k, ω) at ω = − 110 meV where only the replica band emerges. The corresponding |δρ(q, ω)| in (c2) clearly shows the replica band with its radius given by 2k (red arrow) with additional scattering concentrated near q = 0 (we will explain later). If the phonon energy is large enough that the replica band is well separated from the main band, then the QPI should be able to probe the replica band assuming no other scatterings are dominant1. In summary, a signature of the forward scattering phonons can be probed using QPI imaging, just as ARPES is capable of capturing a replica band; ideally, the replica band needs to be isolated from other types of scattering in order to clearly resolve its spectral weight. Ref. [151] is one of earlier results for simulating QPI and forward scattering phonons. These predictions are later conﬁrmed by Ref. [152] which will be discussed next.

4.1.4 Sr2VO3FeAs

Let us now look at another iron based superconductor, the heterostructured bulk Sr2VO3FeAs (Sr-FeAs). FeSe-STO was a mono-layer FeSe on a paralectric STO substrates. Sr-FeAs is

a bulk material that consists of the FeAs layer sandwiched by the insulating Sr2VO3 layer. The crystal structure is shown in Fig. 4.7(a). The red arrows indicate the displacement pattern for an oxygen phonon mode that exists at ∼ 14 meV according to a Density Functional Theory (DFT) calculation[152]. The DFT calculation also shows the electron- phonon coupling constant is strongly peaked at the Γ point suggesting that this is likely a forward scattering phonon. Fig. 4.7(b) shows the QPI image taken from STM experiments

at T = 4.6 K (Tc ≈ 33 K). The experimental image reveals a parabolic band α-α made up of Fe orbitals according to the DFT calculations. α-α denotes the intra-scattering between the electron pocket α at Γ point. The system do consist of other electron pockets at the M point, though this α-α band is less likely originated from interband-scatterings because of the small pocket size and its eﬀective mass compared with M band, so there is no need to consider mixed scattering types. What to notice is the possible replica of the α-α band approximately below ∼ 14 meV denoted by α00-α00 which is the signature of the forward scattering phonons shown in (b).

1In a multi-band system, one has to consider all scattering types into considerations.

70 (b) (a)

(c1) (c2) Kink

Main band Main band

Replica Replica

Figure 4.7: (a) Crystal structure for Sr2VO3FeAs. Red arrows denote the atomic displacements for the phonons that couple to the carriers in FeAs layer. (b) QPI intensity obtained from STM experiments. Figure courtesy of Ref. [152]. α-α denotes the intra- scattering between the electron pocket α at Γ point, and its replica band denoted by α00-α00. α0-α0 is also the replica band, though too close to the the main band, so it appears as one band. (c1) Spectral function using the model discussed in the texts at T = 10 K (Tc = 20 K). Robust kink appearing at ωkink = ∆+Ωph = 14.2 meV. Replica band appears at roughly below the phonon energy. (c2) Corresponding QPI intensity using the T-matrix approach. Red circle shown is additional q = 0 scattering that is a general feature in QPI arising from the large joint density of states where the gap opens.

71 For the QPI simulation, we use a simple tight binding model given by k = −2t[cos(kx)+ cos(ky)] − µ with the band parameter (t, µ) = (0.38, −1.485) eV which resembles the band dispersion found in this material[152] near the fermi-level at Γ point where we found the replica band. The phonon energy is set to be Ωph=10 meV energy, q0 = 0.1, and λm = 0.69 which yields Tc = 20 K. The results for the spectral function and QPI intensity |δρ(q, ω)| at T = 10 K are shown in Fig. 4.7(c). Both the spectral function and QPI intensity show clearly the main and replica band well separated from each other in the occupied region. The intensity at q = 0 in the QPI image (denoted by the red circles) is a normal feature that appears due to opening of the gap around the fermi-level where scattering is concentrated at large joint density of states ∝ 1 (E = ±p2 + ∆2). Strong kink is also visible at energy |∇~ Ek| k k

ωkink = ∆+Ωph in the unoccupied region. The missing replica band in the unoccupied region is due to the replica band being too close to the main band for this highly dispersive band. If we imagine that the kink is solely coming from the electron-phonon coupling, then, the robust kink appearing in the unoccupied states indicates substantially strong interactions between the electrons and phonons. It explains why the present model produces superconductivity despite the small phonon energy being only 10 meV as opposed to ∼ 100 meV in FeSe-STO

system. The current simulation yields Tc ≈ 20 K while the experimental value is Tc ≈ 30 K indicating that substantial contribution is coming from this forward scattering phonons while other types of pairings are also contributing for the superconductivity.

4.2 FLEX plus Phonons

The role of phonons in high Tc cuprates has been controversial[35,3, 89, 90, 91, 180, 181]. A small effect in YBa2Cu3O7 due to oxygen isotope suggested the phonons play relatively a minor role[182]. On the other hand, a robust effect away from optimal doping[183], pronounced isotope effects in Bi-2212[3], substantial renormalization in the electronic structure which are attributed to phonons signify that phonons may have larger influence in the high Tc cuprates[3, 184]. It has been shown that the magnetism can greatly enhance the electron-phonon coupling strength[185]. This is important because strong electron-phonon interactions can compete with other ordered phases such as magnetism[186]. Whether

72 CuO2 Plane

Buckling mode Breathing mode

Oxygen Copper

Figure 4.8: CuO2 plane in a typical Copper-Oxide superconductors showing atomic displacements for the buckling and breathing phonon modes (not to be confused with the magnetic moments). phonons conflict or collaborate with the spin fluctuation exchange, it should be emphasized that these two interactions can renormalize with each other through their impact on the self- energy which can also have non-negligible effects. It was pointed out that the enhancement or suppression of Tc actually depends on the type of phonons that are involved in mediating the pairing[187]. For example, there exists the out-of-plane buckling mode around 40 meV and the in-plane breathing mode around 70 meV[35, 188, 189,2]. It is shown within conserving approximations that the buckling mode can enhance dx2−y2 pairing[190, 191] while the breathing mode leads to a suppression of Tc[192]. The buckling and breathing mode phonons are schematically illustrated in Fig. 4.8. For the buckling mode, the electron-phonon coupling constant is approximately written as,

2 2 2 |g(q)| ∝ [cos (qx/2) + cos (qy/2)]. (4.16)

For the breathing mode, they are given by,

2 2 2 |g(q)| ∝ [sin (qx/2) + sin (qy/2)]. (4.17)

73 To understand why these two phonon modes either enhance or suppress the pairing, we once

again evaluate the coupling strength analogous to Eq. 2.24 for the d-wave (dx2−y2 ) channel. Setting ` = d, it is given by,

1 P 2 N 2 |g(q)| Yd(k)Yd(k + q)δ(k)δ(k+q) k,q λ = . (4.18) d 1 P 2 N Yd (k)δ(k) k

1 where Yd(k) = 2 (cos(kx) − cos(ky)). λd can be interpreted as a pairing strength in the d- wave channel. Let us just plug in numbers and compare the values using the cuprate band

2 from Eq. 2.2.1. For the uniform scattering (|g(q)| = constant), one can easily see λd = 2 0. Thus, it does not contribute to d-wave pairing. For the breathing mode with Ωph = 50

meV and λ = 1.0 (see Eq. 4.5), λd = −0.015 < 0 which is negative, so the breathing mode

seem to suppress the d-wave pairing. For the buckling mode with the same parameter, λd

yields 0.015 which is positive. Finally, for the forward scattering phonon with q0 = 0.3, λd = 0.028. These simple analysis predict the buckling mode and forward scattering phonons may help the d-wave pairing to some degree. It can be understood that the buckling mode is an

2 analog of the weak (large q0) forward scattering phonons. When q0 ≈ 2.3, |g(q)| becomes

a lot more like the buckling mode. For stronger forward scattering (small q0), the pairing enhancement is larger for the d-wave symmetry.

4.2.1 Tc vs. (U, λ) with phonons

To formulate phonons with the spin fluctuation exchange, the following Feynman diagram is proposed. We evaluate self-energies using FLEX approximations from Eq. 3.10 for the spin fluctuations and Migdal-Eliashberg theory from Eq. 4.3 for the electron-phonon coupling fully self-consistently capturing full momentum dependence in the gap function. The diagrammatic representation for the present model is shown in Fig. 4.9. The self- energies in (a) are self-consistently evaluated as was done before. The self-energies that consider cross coupling (spin fluctuations and phonons) such as the one shown in (b) are not

2We have run the simulation and found that the uniform coupling also suppresses d-wave pairing due to mass enhancements, but to a lesser degree than the breathing mode.

74 (a) 퐷 퐪, 푖Ω푚 푉푛 퐪, 푖Ω푚 (b)

Σ 퐤, 푖휔푛 = +

퐺 퐤, 푖휔푛 퐺 퐤, 푖휔푛

퐷 퐪, 푖Ω푚 푉푎 퐪, 푖Ω푚

휙 퐤, 푖휔푛 = +

F 퐤, 푖휔푛 F 퐤, 푖휔푛

Figure 4.9: (a) Diagrams that are considered in the present FLEX plus phonon models. Again, the vertex corrections are neglected. (b) Diagram representing the coulomb-phonon cross diagrams such as this one are not considered. included. For the band structure, let us use the same tight binding model taken from Eq. 2.2.1 as before to model cuprates.

Tc dependence on Hubbard U and electron-phonon coupling strength λ (see Eq. 4.5) is

shown in Fig. 4.10. Here, ∆Tc means Tc(U, λ) − Tc(U, λ = 0), and U/W is the ratio between

U and the bandwidth W = 1.3 eV. The intensity map shows Tc for the d-wave pairing while the black shaded region is where an s-wave gap is realized. The red dashed region is where

Tc was too small and the favored gap symmetry is inconclusive. The system favors d-wave pairing for the most part, but particularly discernible for the forward scattering. Even modest Hubbard U (U/W < 0.1) already favors d-wave pairing. One can see from the colorbar that

∆Tc is positive for the buckling and forward scattering phonons which implies that these two phonon modes enhance the d-wave pairing. Nevertheless, the Tc enhancement is rather concentrated for small U values (U/W . 0.6), and this enhancement is rather minimal for large U. For the breathing mode, the results indicate ∆Tc is negative, so the pairing is suppressed. Here, Tc suppression is most pronounced for small U. The conclusions that buckling/breathing mode enhances/suppresses Tc are consistent with the previous studies, but these enhancement/suppression seem to heavily depend on the type of phonons and the values of U and λ. We showed that phonons with such forward scattering peak can co-exist with the spin-ﬂuctuation exchange whose pairing requires nesting with a large momentum

75 Ω푝ℎ = 35푚푒푉 Ω푝ℎ = 70푚푒푉 Ω푝ℎ = 70푚푒푉

S-wave

Figure 4.10: Tc dependence on the electron-phonon coupling strength λ (see Eq. 4.5) and U/W ratio. (U=Hubbard U and W = Bandwidth = 1.3 eV ) ∆Tc = Tc(U, λ) − Tc(U, λ = 0). The band structure from Eq. 2.2.1 was used. Black shaded region is approximately where s-wave symmetry is realized. Red dashed region is where Tc is small and inconclusive. The dx2−y2 symmetry is realized for the rest of the intensity. For the Forward scattering, q0 = 0.5 was chosen. transfer. Eventually, it is important to consider all contributions including charge transfer eﬀects and strain distortions which can shape the electronic structure aﬀecting the nesting of Fermi-surface.

4.2.2 Spectral Function

One common methodology for interpreting ARPES data is to extract electron self-energies. The method for estimating the self-energy is illustrated in Fig. 4.11(a). The difference between the renormalized and bare dispersion gives the real part of the self-energy approximately. In (b), the self-energy is extracted from two different Bi-2212 samples using O16 for one of the sample, and O18 for the other sample. Results show a shift in the peak near the nodal kink at ∼ 70 meV suggesting that these renormalizations are attributed to phonons. In the present case, we added a buckling mode phonon at 35 meV with λ = 0.3 and a breathing mode phonon at 70 meV with λ = 0.2 in addition to Hubbard U = 0.6 eV. The results for the spectral fuction are shown in Fig. 4.12 (a) and (b). We have also extracted the effective self-energy using the same method used in ARPES by taking the

76 h hf ntepa ugssrnraiainatiue opoos iuecuts of courtesy Figure phonons. to attributed renormalization directions). suggests nodal [3]. the peak oxygens to isotope Ref. the close diﬀerent in (all two cuts shift from momentum extracted The various Self-energies along line (b) superconductors blue ∆ Bi-2212 [193]. dashed in diﬀerence the Ref. The and of dispersion, courtesy renormalized dispersion. Figure the bare represents the line Red is K. 20 at direction 4.11: Figure

[eV] (a) a yia RE nest o pial oe i21 ln h nodal the along Bi-2212 doped optimally for intensity ARPES Typical (a) = Δ 퐸 E = stknt etera ato h self-energy. the of part real the be to taken is Re 77 [ Σ ( 휔 ) ] (b) (a) (b)

푇 = 0.33푇푐 푇 = 0.42푇푐

Figure 4.12: (a) Spectral function without phonons for U = 0.6 eV. (b) Spectral function with phonons. Yellow arrows point the kink features that emerge due to phonons. (c) Eﬀective self-energy extracted from (a) and (b). Black arrows point peaks that originated from phonons. (d) Eﬀective self-energy with phonons in the superconducting and normal state denoted in the legend.

78 difference between the renormalized and the bare dispersion. The renormalized dispersion was estimated from the peak in the Momentum distribution curve (MDC), and the bare dispersion was modeled to be linear. In (a) and (b), the spectral functions with/without phonons in the superconducting state are shown. Clearly, (b) shows double kink features emerging from the two phonon modes at approximately ∆ + Ωph indicated by the yellow arrows. In (c), the effective self energy extracted from (a) and (b) are plotted. One can see the two pronounced peaks appearing from the two phonon modes while no discernible peaks are seen without phonons. This seems to be true for other values of U. Based on the above results, a conclusion can be drawn that the low energy discernible peak in the self-energy can appear only if phonons are added in the system. Finally, in (d), we have compared the self energy with phonons in the superconducting/normal state. It can be seen that the two peaks appearing in the superconducting state are greatly smeared out in the normal state in contrast to the replica band being present in the normal state (see Fig. 4.4). This is also consistent with the ARPES results that this low energy peak becomes appreciably weaker in the normal state. (See for example Ref. [194]) While adding phonons showed the low energy peaks in the self-energy as well as their temperature dependence, it still did not explain the doping dependence. Unfortunately, the present model imposes a difficulty in simulating the doping dependence simply because varying the chemical potential is not equivalent to changing the doping concentrations in the full Hubbard model. Investigating the doping dependence has to be done by other means.

4.3 Competition between the Charge Density Wave and Conventional Superconductivity

The self-consistent formalism we have developed also allows us to examine competition between the superconductivity and charge density wave (CDW). Here, we will not consider Hubbard U, but include electron-phonon interaction within a Holstein model which neglects momentum dependence of the coupling constant g(q) → g. The approach is similar to what was done in the previous sections except the phonon propagator now becomes a dressed

79 (renormalized) phonon Green’s function which needs to be updated at each iterations. In many applications, the phonon Green’s function is constructed from a model or estimated from experiments, and treated as non-interacting. Here, the phonon Green’s function is necessary to be dressed in order to see the CDW transition. At CDW transition, it’s the bubble diagrams that diverges while the ladder diagrams diverges at the superconducting transition. The system will settle whichever diverges faster. As before, we only consider a single band in a two dimensional system, and a dispersionless optical phonon mode. The method is reviewed by Ref. [195]. In this method, we work in the normal state, so there are no anomalous Green’s functions included. The set of equations that need to be solved are as follows. The phonon self-energy (also called a polarization) is given by,

2 2λ0MΩ X Π(q, Ω ) = G(k,ω )G(k + q, ω + Ω ), (4.19) n Nβ m m n k,m

2g2 where M is the ion mass, Ω is the phonon energy, and λ0 = Ω . The phonon self-energy comprises of a single irreducible bubble diagram. The dressed phonon Green’s function is given by, 1 D(q, Ωm) = 2 2 , (4.20) −M(Ω + Ωn) − Π(q, Ωn) and the electron self-energy is,

2 λ0MΩ X Σ(k, ω ) = − D(q, Ω )G(k − q, ω − Ω ). (4.21) n Nβ m n m k0,m

The electron Green’s function is same as before which is given by,

1 G(k, ωn) = . (4.22) iωn − k − Σ(k, ωn)

The Hartree term is given by,

2 2λ0MΩ X Σ = λ MΩ2D(0, 0) + G(k0, ω ). (4.23) HF 0 Nβ m k0,m

80 Again, the particle number is ﬁxed as before. Therefore, the Hartree term can be neglected in the current scheme since it can be absorbed into the new chemical potential that is updated during iterations. The above Eq. 4.19 - Eq. 4.22 form a set of equations that can be solved self-consistently. Once the self-energy is obtained, one can move on to evaluate the particle-particle vertex equation given by,

2 λ0MΩ X Λ(k, ω ) = 1 − D(q,Ω )f(k − q, ω − Ω )Λ(k − q, ω − Ω ). n Nβ m n m n m q,m (4.24) 0 0 0 0 0 0 f(k , ωm) = G(k , ωm)G(−k , −ωm),

When the temperature approaches the superconducting transition temperature, Λ(k, ωn) becomes much greater than one, and Eq. 4.24 becomes the eigenvalue problem, analogous to the Bethe-Salpeter equation in the particle-particle channel. Eq. 4.24 is the second self- consistent equation that needs to solved. Notice that Eq. 4.19, Eq. 4.21, and Eq. 4.24 are of the convolution form. One can implement the Fourier transform algorithm to speed

3 up the calculation. Finally, one can calculate CDW susceptibility χcdw(q,T ) and pairing

susceptibility χsc(T ) given by,

χ0(q) χcdw(q,T ) = , (4.25) 1 − λ0χ0(q)

Π(q,0) where χ0(q) = − 2 , and λ0MΩ

1 X χ (T ) = G(k, ω )G(−k, −ω )Λ(k, ω ). (4.26) sc Nβ m m m k,m

χcdw consists of the bubble diagrams, and χsc contains the ladder diagrams. Going from the high to low temperature, a susceptibility begins to diverge. The divergence in the susceptibility signals the phase transition. If both diverges, the phase is determined by the susceptibility that diverges faster (divergence at higher temperature). To estimate Tc, we −γ have tried to ﬁt χcdw(T ) with a model function ∝ (T − Tc) for the CDW transition where the constants including γ is obtained from minimizing the least square. Note that we have

3 Note that f(k − q)Λ(k − q) decays faster than 1/ωn, so there is no discontinuity at τ = 0.

81 relaxed the parameter γ rather than fixed value γ = 1.75 known from 2-D Ising universality class[196] since we are dealing with the finite momentum grid rather than infinite lattice

−1 points. For the superconducting transition, Tc is extrapolated by tracing zeros of χsc (T ) from the cubic spline data fit[197]. For the simulation, we considered a simple tight binding model in a two dimensional system given by k = −2t(cos(kx) + cos(ky)) − µ, and set (λ0, Ω) = (1.5, 1.0) in units of t. The analytic continuation was carried out using Padéapproximants. The phase diagram as a function of the particle number n is shown in Fig. 4.13(a). The particle number is obtained from Eq. 3.12. Here, n = 1 is half filling. The phase diagram for n > 1 is exactly the mirror image owing to the particle-hole symmetry N(−ω) = N(ω) in the electron density of states. From low filling to approximately 0.875, the system favors s-wave

superconductivity. The superconducting Tc monotonically increases up to n ≈ 0.8 because the electron density of states at Fermi-level N(0) increases with larger ﬁlling. From n ≈ 0.8 to 0.875, a partial gap gradually starts to open up at lower temperature suppressing

N(0) which lowers superconducting Tc. This opening of the gap signals the system is close to the CDW transition. Then, near the half filling does CDW take over. At half filling, the density of states is a van-hove singularity, and the topology of the fermi-surface is a square shape with its vertices at k = (0, π) shown in Fig. 4.13(b). A strong nesting occurs near the van-hove singularity where the dominant scatterings are the momentum transfer between the high joint density of states denoted in red circle. The nesting vector in such a case would therefore be the scattering from one vertex to the other vertex which is the momentum transfer given by q = (π, π) shown in red arrow. The consequence of the strong nesting is manifested in the phonon dispersion. Fig. 4.13 (d) and (e) show the phonon dispersion given by the imaginary part of the dressed phonon Green’s function at two different temperatures. One can see the sharp softening of the phonon dispersion, an anomaly, appearing at q = (π, π). It is particularly robust at lower temperature near the CDW transition. In the present case, such anomaly appears from a strong Fermi-surface nesting. This is illustrated in (c) which shows the auto-correlation function at fermi-level. Here, the auto-correlation function is defined as the convolution of

the two spectral function ΣkA(k)A(k + q) where A(k) = Im[G(k, ω = 0)] (Note here that

82 (b) 퐺0(퐤, 휔 = 0)

(d) 푇 = 2.37 푇푐 (a)

CDW SC

Figure 4.13: (a) The phase diagram as a function of the particle number n. Here, n = 1 is half filled. (b) Non-interacting Green’s function at Fermi-level. Red circle denotes where the joint density of states is largest. Red arrow shows the momentum transfers (π, π) which is supposed to be dominant at CDW phase. (c) Auto-correlation function defined by ΣkA(k)A(k + q) with A(k) = Im[G(k, ω = 0)] at T = 1.4Tc. The peak at q = (π, π) signify the fermi-surface nesting. (d) and (e) Phonon dispersion given by the imaginary part of dressed phonon Green’s function at two different temperature. Anormaly appears at q = (π, π) due to fermi-surface nesting.

83 G(k, ω) is a dressed Green’s function). One can see that the auto-correlation function is peaked at q = (π, π) indicating that this is the dominant nesting vector. In one dimension,

Kohn anomaly[198] appears at q = 2kF in the phonon dispersion because there are only two k points present at the fermi-level, and the only nesting that can occur is q = 2kF . In two dimension, the nesting is given by q = (π, π) where the softening of the phonon dispersion appears.

4.4 Multi-orbital FLEX Approximations

So far, we have only dealt with the system in a single band case which is not necessary practical since in many other cases, the system consists of multi-orbitals or multi-bands whichever basis you choose to set up. The single-band model refers to a very speciﬁc case. The Migdal-Eliashberg equations introduced in Eq. 2.11 can easily be generalized to a multi-band case. They are given by,

1 X X 0 2 0 Σˆ (k, iω ) = − |g 0 (k, k )| D 0 (q, iω − iω )τ ˆ Gˆ 0 (k , iω )ˆτ , (4.27) l n Nβ ll ll n m 3 l m 3 k0m l0 where l is the band indices. However, the multi-orbital Hubbard model is more cumbersome. The Hamiltonian with the Hubbard interactions in an orbital basis is given by,

H = H0 + Hint,

X l1l2 † H0 = t − µδl l δij c c , ij 1 2 il1σ jl2σ (4.28) ij,l1l2σ

1 X X σ1σ2σ3σ4 † † Hint = U c c c c , 4 l1l2l3l4 il1σ1 il2σ2 il3σ3 il4σ4 i l1l2l3l4 σ1σ2σ3σ4 where the Hubbard U now has orbital/spin indices l and σ respectively, t is the hopping term with ij being the lattice site indices, and the rest obeys the usual notations. Here, the spin-orbit coupling will be ignored. The spin dependent part is split into the spin and charge terms as,

σ1σ2σ3σ4 1 s 1 c U = − U σσ σ · σσ σ + U δσ σ δσ σ , (4.29) l1l2l3l4 2 l1l2l3l4 1 4 2 3 2 l1l2l3l4 1 4 2 3

84 where σ is the pauli matrices. Each speciﬁc interactions are assigned as,

 U if l = l = l = l  1 2 3 4   0 U if l1 = l3 6= l2 = l4 s  Ul1l2l3l4 = (4.30) J if l = l 6= l = l  1 4 2 3   0 J if l1 = l2 6= l3 = l4 and,  U if l = l = l = l  1 2 3 4   0 −U + 2J if l1 = l3 6= l2 = l4 c  Ul1l2l3l4 = (4.31) 2U 0 − J if l = l 6= l = l  1 4 2 3   0 J if l1 = l2 6= l3 = l4,

Using the identity σσ1σ4 · σσ2σ3 = 2δσ1σ3 δσ2σ4 − δσ1σ4 δσ2σ3 , the Hamiltonian can be rewritten as,

1 X X s † † 1 s c † † H = U c c 0 c 0 c + (U + U )c c 0 c 0 c , int l1l2l3l4 il1σ il2σ il4σ il3σ l1l2l3l4 l1l2l3l4 il1σ il2σ il3σ il4σ 4 0 2 i l1l2l3l4 σσ

1 X X 0 † † 0 † † = Un n 0 + U n n 0 0 − Jc c 0 c 0 0 c 0 + J c c 0 c 0 0 c 0 , 2 ilσ ilσ ilσ il σ ilσ ilσ il σ il σ ilσ ilσ il σ il σ i l6=l0 σσ0 (4.32) where U, U 0 are the intra/inter-orbital Coulomb interactions, J is the Hund’s coupling, and U 0 is the pair-hopping exchange. Examples of multi-orbital FLEX method can be found in Ref. [199, 200, 201, 202, 141] For a system with M orbitals, the Green’s function will be M × M matrix. If we adopt the

85 Nambu formalism, then the dimension of the matrix will be 2M × 2M given by,4

 0  Gˆσ(k) Fˆσσ (k) G¯(k) = , (4.34)  0  Fˆσσ (k)∗ −Gˆσ(k)∗

where the hat signiﬁes M × M matrix, σ is the electron spin, and k = (k, ωn). As before, we expand the above Green’s function matrix in terms of the Dyson’s equation as,

¯ ¯ ¯ ¯ ¯ G = G0 + G0ΣG, (4.35)

¯ ¯ where G0 is the non-interacting Green’s function, and Σ is the Self-energy matrix. They are written as,   Gˆ (k) 0 ¯ 0 G0 =   , (4.36) ˆ ∗ 0 −G0(k)

  Σ(ˆ k) φˆ(k) Σ¯ =   . (4.37) φˆ(k)∗ −Σ(ˆ k)∗

After comparing the left and right hand side of Eq. 4.35, a pair of equations can be extracted. They are,

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ∗ G = G0 + G0ΣG + G0φF , (4.38) ˆ ˆ ˆ ˆ ˆ ˆ ˆ∗ F = G0ΣF − G0φG ,

4The process of getting to Eq. 4.34 takes some symmetry operations. Starting with the formal deﬁnition,

† ! σ σσ0 − < T c (τ)c (0) > − < T c (τ)c 0 (0) > G (k, τ) (k, τ) kl1σ kl2σ kl1σ −kl2σ l1l2 Fl1l2 G¯ = = 0 . (4.33) † † † ˜σ σ ˜σ − < T c 0 (τ)c (0) > − < T c (τ)c (0) > (k, τ) G (k, τ) −kl1σ kl2σ −kl1σ −kl2σ Fl1l2 l1l2

0 We can see with the fermionic property that G˜σ (k, τ) = −Gσ (k, −τ) = Gσ (k, τ)∗ and σσ (k, τ) = l1l2 l2l1 l2l1 Fl1l2 0 0 − σ σ(−k, τ) = σ σ(k, τ)∗ obeys (since the Hamiltonian is hermitian). Next, we consider the Fl2l1 Fl2l1 0 spin rotational symmetry around y-axis by 180 degree which yields, Gσ (k, τ) = Gσ (k, τ), and l1l2 l1l2 0 0 0 σσ (k, τ) = − σ σ(k, τ) for the singlet pairing. With the time reversal symmetry, we have G˜σ (−k, τ) = Fl1l2 Fl1l2 l1l2 0 0 0 −Gσ (−k, −τ)∗ and ˜σ σ(k, τ) = σσ (−k, −τ)∗, or in the frequency domain, G˜σ (−k, iω ) = l2l1 Fl1l2 Fl1l2 l1l2 n 0 0 −Gσ (−k, iω )∗ and ˜σ σ(k, iω ) = σσ (−k, iω )∗. Lastly, we assume the inversion symmetry, so that l1l2 n Fl1l2 n Fl1l2 n 0 Gσ (k, iω ) and σσ (k, iω ) are even in k. With these, we recover Eq. 4.34. We started with four Green’s l1l2 n Fl1l2 n function blocks, but now the number is reduced to two. Another word, we only need to evaluate two Green’s function blocks.

86 which are shown in Fig. 4.14 diagrammatically. Solving for Gˆ(k) yields,

ˆ ˆ−1 ˆ ˆ ˆ∗ −1 ˆ ∗ −1 ∗−1 G = G0 − Σ − φ(−G0 + Σ ) φ , −1 (4.39) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ∗ −1 ∗ = iωnI − (ˆk − µI) − Σ − φ[iωnI + (ˆk − µI) + Σ ] φ , and solving for φˆ yields,

ˆ ˆ ˆ ˆ∗−1 ˆ ∗ F = − Gφ(G0 − Σ ), −1 (4.40) ˆ ˆ ˆ ∗ ˆ−1 ˆ ˆ ˆ ∗ = [iωnI + (ˆk − µI) + Σ ]φ [iωnI − (ˆk − µI) − Σ] − φ .

So, the original 2M × 2M matrix now became two M × M matrices for the electron and anomalous Green’s function.

= + Σ + 휙

= Σ − 휙

Figure 4.14: Diagrammatic representation of Eq. 4.38.

The irreducible spin and charge susceptibilities for the multi-orbital case follow the same logic as the one-band case, but now contains the orbitals dependence. They are deﬁned as,

0,s 1 X ∗ χ (q) = − [Gl l (k + q)Gl l (k) + Fl l (k + q)F (k)], l1l2,l3l4 Nβ 1 3 4 2 1 4 l3l2 k (4.41) 0,c 1 X ∗ χ (q) = − [Gl l (k + q)Gl l (k) − Fl l (k + q)F (k)], l1l2,l3l4 Nβ 1 3 4 2 1 4 l3l2 k where q = (q, Ωn), and the superscripts s and c stand for the spin and charge respectively. These susceptibilities are rank 4 tensor, but normally reshaped into a matrix with M 2 × M 2

87 dimension denoted asχ ˆ0,s andχ ˆ0,c. The spin and charge FLEX susceptibilities are,

χˆs(q) = [Iˆ− χˆ0,s(q)Uˆ s]−1χˆ0,s(q), (4.42) χˆc(q) = [Iˆ+χ ˆ0,c(q)Uˆ c]−1χˆ0,c(q), where Uˆ s = U s and Uˆ c = U c are the spin and charge interactions deﬁned in Eq. l1l2l3l4 l1l2l3l4 4.29. The spin and charge eﬀective interactions are,

3 Vˆ s(q) = Uˆ s[ˆχs(q) − χˆ0,s(q)]Uˆ s, 2 (4.43) 1 Vˆ c(q) = Uˆ c[ˆχc(q) − χˆ0,c(q)]Uˆ c. 2

The eﬀective interactions for the normal and anomalous self-energies denoted by “n” and “a” are,

ˆ n ˆ s ˆ c ˆ n V (q) = V (q) + V (q) + V(2)(q), (4.44) ˆ a ˆ s ˆ c ˆ a V (q) = V (q) − V (q) + V(2)(q),

ˆ a,n where V(2) (q) are the second order corrections given by,

ˆ n 3 ˆ s 0,s 0,c ˆ s 1 ˆ c 0,s 0,c ˆ c V(2)(q) = − U [ˆχ (q) +χ ˆ (q)]U − U [ˆχ (q) +χ ˆ (q)]U , 8 8 (4.45) 3 1 Vˆ a (q) = − Uˆ s[ˆχ0,s(q) − χˆ0,c(q)]Uˆ s − Uˆ c[ˆχ0,s(q) − χˆ0,c(q)]Uˆ c, (2) 8 8

As before, the second order corrections are added separately to avoid double counting the ladder and bubble diagrams. Finally, the normal and anomalous self-energies are given by,

1 X n Σ (k) = V 0 0 (q)G 0 0 (k − q), lm Nβ ll ,mm l m q l0m0 (4.46) 1 X a φlm(k) = V 0 0 (q)Fl0m0 (k − q). Nβ ll ,m m q l0m0

88 Lastly, the Hartree terms are added separately. They are given by,

h i lm X 1 ˆ s ˆ 1 X 0 ΣHF = (3U − Uc)ll0,mm0 δl0m0 + Gl0m0 (k ) , 0 0 2 Nβ 0 l m k (4.47) lm X 1 ˆ s ˆ 1 X 0 φ = (U + Uc)ll0,m0m Fl0m0 (k ). HF 2 Nβ l0m0 k0

Just like it was for the single-band case, the chemical potential is adjusted by ﬁxing the particle number; thus, we instead use the modiﬁed Hartree term for the normal self-energy given by, 1 X Σlm = Σlm − δ Σnn . (4.48) mod−HF HF lm M HF n Eq. 4.39- 4.48 comprises of a set of equations that can be solved self-consistently. We will also need formulas for the particle number and the chemical potential. The particle number is determined from,

Nc−1 2 X X h ˆ 1 i n = Gll(k, ωm, µ) − + 2MnF (ξ), (4.49) Nβ iωm − (ξ − µ) k,l m=−Nc

1 M where we choose ξ = β ln n − 1 [141]. The equation for the chemical potential is given by,

N −1 2 X Xc h 1 i Gll(k, ωm, µ) − = 0. (4.50) Nβ iωm − ξ k,l m=−Nc

These are the basic tools for the FLEX formalism. There are subtleties, for example when working between τ ωn space, but again, implementing the fast Fourier transform greatly improves the speed, so it is worth the eﬀort. See footnotes for further details (continued on the next page).5

5 The basic procedure for the Fourier transform is mostly equivalent to the singe band case. Below are the some lists to keep in mind. 1. Fourier transform Gl1l2 → Gl1l2 can be performed using Eq. 3.13. For ` = N , we have Gl1l2 = ij,n xy,` m xy,Nm −Gl1l2 − δ δ and l1l2 = − l1l2 . xy,0 x0 y0 Fxy,Nm Fxy,0 2. Irreducible susceptibilities are symmetric matrices and obtained from

χl1l2,l3l4 (τ) = −Gl1l3 (τ)Gl4l2 (−τ) ± F l1l4 (τ)F l3l2 (τ). (4.51)

89 4.4.1 Bilayer Hubbard Model

One of the simplest cases for the multi-band model is the bilayer Hubbard model. The physics of the cuprate superconductors is thought to be contained in quasi-two-dimension where the superconductivity takes place in the copper oxygen (CuO) plane. However, there are evidence that the hopping between the planes do exist [99, 203, 204] introducing a third dimension in the quasi-two-dimensional system. Fig. 4.15(a) shows the crystal structure for Yttrium barium copper oxide (YBCO) superconductors. As has been said, the copper oxygen plane becomes superconducting, and contains most of the essential physics, but when a small hopping parameter is introduced, the band can split into two bands called bonding and anti-bonding band resulting in a Fermi-surface shown in Fig. 4.15(b). In the following simulation, we run a simple bilayer Hubbard model which considers a ﬁnite Hubbard U, but set U 0 = J = J 0 = 0. The band dispersion is a 2 × 2 matrix given by,

  k −th ˆk =   , (4.54) −th k

where k is the in-plane band-dispersion, and th is the simple hopping between the plane. Several examples of theoretical studies for such bilayer system are given in Ref. [205, 206] using FLEX approximations and Ref. [96] (and references therein) using other techniques. Fig. 4.15(b) compares the Fermi-surface with and without the hopping parameter using the

We also have the properties that Gl1l2 (−τ) = −Gl1l2 (β − τ) and F l1l2 (τ) = F l2l1 (−τ) = l2l1 l1l2,l3l4 + l1l2,l3l4 − l1l3 −F (β − τ). The discontinuities are given by, χxy (0 ) − χxy (0 ) = −Gxy (0)δx0δy0δl4l2 + l4l2 l1l2,l3l4 Gxy (0)δx0δy0δl1l3 . Then, use Eq. 60 to obtain χ (k).

3. V l1l2,l3l4 (τ) is obtained from Eq. 3.15. Then, we may proceed to ﬁnd self-energies from

l1l2 X l1l3,l2l4 l3l4 Σ (τ) = VN (τ)G (τ), l l 3 4 (4.52) l1l2 X l1l3,l4l2 l3l4 φ (τ) = VA (τ)F (τ). l3l4 The discontinuity in the normal self-energy is,

l1l2 + l1l2 − X l1l3,l2l3 Σx y(0 ) − Σx y(0 ) = VN (r = 0, τ = 0). (4.53) l3

Finally, one can perform the Fourier transform Eq. 60 to obtain Σl1l2 (k) and φl1l2 (k).

90 풕 풕 = ퟎ 풉 = ퟎ. ퟏퟓ (a) (b) 풉 푾

풕 = Hopping CuO plane 풉

Figure 4.15: (a) Crystal structure for Yttrium barium copper oxide (YBCO) showing two CuO planes in the unit cell. Hopping between the CuO plane characterized by th. (b) Fermi-surface for the non-interacting band dispersion k [68] with/without the hopping. (c) Diagonal component of the anomalous self-energy at T/Tc = 0.33 for (U/W, th/W, n) = (0.5, 0.02, 1.7). White lines show the fermi-surface k = 0.

91 (a) (b)

Figure 4.16: Fermi-surface for th/W = 0.38 using the in-plane band dispersion k = −2t(cos(kx) + sin(ky)) − 4t2 cos(kx) cos(ky) − µ with (t, t2, µ) = (0.2, −0.02, 0.05) eV. (b) The diagonal components of the anomalous self-energy at T/Tc = 0.17 showing S± gap symmetry. The relevant parameters are n = 2.0 and U/W = 0.5. in-plane band-dispersion taken from Ref. [68]. It can be seen that the introduction of a small hopping parameter induces a splitting of the band. Here, the ratio th/W is 0.02 where

W is the bandwidth of k. Fig. 4.15(c) shows the diagonal components of the anomalous self-energy φ11 obtained for (U/W, th/W, n) = (0.5, 0.02, 1.7) in the superconducting state

T/Tc = 0.33. Here, n = 2.0 is the half ﬁlling. The intensity is maximum/minimum at k = (±π, 0) and (0, ±π) with its node along the diagonal showing d-wave gap structure.

However, slight diﬀerences can be seen when one compares with dx2−y2 shown in Fig. 3.1 indicating the topology of the fermi-surface played a role in shaping the gap function. An interesting transition can be seen when the hopping parameter is large. The Fermi- surface for th/W = 0.38 is shown in Fig. 4.16(a). Increasing the hopping parameter further splits the band such that the system shown in (a) now possesses an electron pocket at Γ point and a hole pocket at M point (the corner of the Brillouin zone) which reminds oneself the multi-band iron-based superconductors. We run the FLEX code again to obtain the anomalous self-energy. Here, we choose (U/W, th/W, n) = (0.5, 0.38, 2.0), The diagonal

92 component of the anomalous self-energy is shown in Fig. 4.16(b). The self-energy has a node and changes sign between the two pockets which is required for the repulsive pairing

interaction. The intensity is isotropic along the band which signifies this is a simple s± gap symmetry. These simple analysis conveys that the topology of the Fermi-surface plays an important role in shaping the gap structure. Finally, the results for a general two orbital model with finite J, U 0,J 0 are briefly presented in the next section along with the concluding remark of the dissertation.

4.4.2 Pairing with an Incipient Band in a Bilayer System

As was mentioned, another leading candidate for explaining the pairing mechanism in FeSe- STO is the coupling of electrons with an incipient band. In FeSe-STO system, the ARPES reveals the Fermi-surface consisting of couple electron pockets at M point while the maximum of the hole band at Γ point is 60-80 meV below the Fermi-level[158]. This has questioned the validity of the spin fluctuation pairing which is thought that the nesting of a large momentum transfer at the Fermi-level is required to induce superconductivity. However, this is not necessary true as we shall demonstrate that the spin fluctuation exchange can still induce superconductivity as long as the other band is close to the Fermi-level. The model we consider here is again the bilayer system with band parameters and U values chosen appropriately to induce pairing with an incipient hole band similar to the band structure observed in the FeSe-STO system. Keep in mind that we are not trying to fully simulate the real system, but we do model the band structure to resemble such kind of system. The purpose of the current simulation is to use the present FLEX code to show that the pairing with an incipient band can induce superconductivity; furthermore, we will demonstrate that by adding forward scattering phonons, the superconducting transition temperature can be enhanced just as presented in section 4.2 for the purpose of emphasizing that phonons with a small momentum transfer can work with the spin fluctuation pairing. Since we are only considering phonons with the strong forward scattering limit (q = 0), we will only consider the intraband components for the phonon self-energy. Similar to what was done in section 4.2, the self-energies from the spin fluctuation exchange and phonons were added and self-consistently iterated until the convergence is achieved. The self-energy

93 matrix is given by,   Σ (k) 0 ˆ ˆ ph Σ(k) = Σsf (k) +   , (4.55) 0 Σph(k) for the normal self-energy, and,

  φ (k) 0 ˆ ˆ ph φ(k) = φsf (k) +   , (4.56) 0 φph(k)

ˆ ˆ for the anomalous self-energy. Σsf and φsf are the self-energies from the spin ﬂuctuation exchange evaluated from Eq. 4.46.Σph and φph are given by,

1 X Σ (k) = − |g(q)|2D(q)G (k − q), ph Nβ 11 q,m (4.57) 1 X 2 φph(k) = |g(q)| D(q)F11(k − q), Nβ q,m

2Ωph 2 where D(q, iω`) = − 2 2 is the bare phonon propagator. |g(q)| is the electron-phonon Ωph+ω` coupling constant in the strong forward scattering limit and given by,

2 −2q/q0 |g(q)| = g0e , (4.58)

where q0 = 0.3 is chosen for the present model. The in-plane band structure is modeled to

be k = −2t(cos(kx) + cos(ky)) − µ. The parameters are (t, th,U) = (0.1, −0.238, 0.72) eV, and the ﬁlling is set to be n = 2.1.

1 The spectral function A(k, ω) = − π Im[G11(k, ω)] at T = 20 K without/with phonons is shown in Fig. 4.17 (a) and (b) respectively. One can conﬁrm the electron band with the superconducting gap opening around Fermi-level while the maximum of the hole band is ∼ 60 meV below the Fermi-level. One can also spot some feature appearing in the unoccupied region (ω > 0) at k = (π, π) where the mirror of the hole band has emerged. This is the feature we may expect to see if the spin ﬂuctuation exchange is involved in the pairing, though the spectral weight from the mirror band is expected to be much weaker than the original band, so that this feature may not be guaranteed to be observed using QPI imaging.

94 (a) No phonon (b) With phonon (c) Mirror band Mirror band

Figure 4.17: (a) Spectral function at 20 K without phonons. (b) Spectral function at 20 K with phonons. The mirror band appearing at k = (π, π) slightly above the Fermi- level. (c) Maximum value of the diagonal element of anomalous self-energy φ11(kmax, π/β) as a function of temperature for three diﬀerent electron-phonon coupling strength g0. Tc is enhanced with increasing g0.

The results for Tc at three diﬀerent values of g0 is shown in Fig. XX(a). One can see the

enhancement in Tc with increasing the electron-phonon coupling strength g0. The message from the simulation is that the forward scattering phonons can eﬀectively work with various gap symmetries such as the single-band d-wave or S± wave and are capable of enhancing the transition temperature.

4.5 Concluding Remarks

We will not go into details of the multi-orbital FLEX code because the general two orbital model has already been reviewed by Ref. [141, 201, 202]. Fig. 4.18 shows the anomalous

0 0 self-energy Σiiφii(k, π/β) for (U, U , J, J ) = (1.5, 1.0, 1.0, 1.0) eV at T = 30 K. Each elements

of orbitals ˆk were taken from Ref. [201, 202]. The sign changing gap at two pockets shows the typical s± pairing known from the spin fluctuation theory. These multi-orbital Hubbard model introduces additional degrees of freedom with orbital fluctuations, so there is a lot more to explore. The downside of the current model is that it only consists of two orbitals whereas the minumum model to represent iron-based superconductors generally requires three orbitals, but five orbitals are highly suggested[207]. Therefore, studying such systems in diagrammatic approach is often done in the normal state using the simple RPA. 5-orbital

95 1 0.2 0.8 0 -0.2 [eV]

/a] 0.6

[2 y

k 0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 k [2:/a] x

0 0 Figure 4.18: Anomalous self-energy φii(k, π/β) using (U, U , J, J ) = (1.5, 1.0, 1.0, 1.0) eV at T = 30 K, and orbitals ˆk adopted from Ref. [141]. Fermi-surface is denoted by the solid white line. Filling set to n = 1.88. self-consistent calculations are difficult at this stage. Other suggested future studies can be: adding phonons to the multi-orbital FLEX code, calculating optical conductivities, and the specific heat using the formulas provided in AppendixE. In Chapter2, the basic formalism of the Migdal-Eliashberg theory is reviewed. We deduce that the low energy modulation in the electron density of states seen in STM data originated from some collective excitations. Motivated by the neutron data, we simulated a single-band d-wave superconductor using the boson spectrum similar to what the neutron experiments have shown, then the results were compared with STM data. A conclusion drawn from this analysis is that the present simulation requires at least two types of bosons to explain the observed STM data with the condition that one boson should have a pairing strength much smaller than the second boson. In Chapter3, we studied the superconductivity arising from the single band Hubbard model within diagrammatic FLEX approximations. The corresponding electron density of states did not reproduce the STM data perhaps due to lack of vertex corrections or because additional bosons are needed, or other physics which we are not aware. The STM data revealed in high Tc cuprates is still worth further investigations. Finally, in Chapter4, we studied various applications using the technique introduced in the previous chapters. While the scientific community has made a tremendous progress towards understanding the physics of high Tc superconductivity, yet predictive models especially

96 from the ﬁrst principle perspective is still lacking in this ﬁeld. The physics of high Tc superconductivity is still considered one of the most outstanding problem in condensed matter physics.

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123 Appendices

124 A Vertex Corrections

So far, we have not included the vertex corrections in any of the calculations. The diagrammatic approach in these calculations becomes expensive if the vertex corrections are taken into account, so oftentimes, they are neglected. However, Migdal approximations are still widely used because the calculation allows to go down to a low temperature while using a dense momentum/frequency-grid often needed to capture low energy features such the kink in spectral functions. At the same time, it is important to consider whether the system is in a strong coupling regime or not. There are literatures that try to address this questions[208, 209, 210, 211]. For most iron-based superconductors, the system is considered in a intermediate coupling regime[209], while a strong coupling theory is needed for high Tc cuprates[211]. In order to include vertex corrections, one has to evaluate the diagram shown in Fig. A.1(a). For an

Ω uniform electron-phonon coupling constant, λm is a measure of the vertex correction. EF Fig. A.2 shows the superconducting gap as a function of temperature with and without the vertex corrections for an uniform scattering case. Tc is suppressed if the vertex corrections are included. For the phonon mediated superconductors, there are situations such as the dilute n-type SrTiO3[212], or FeSe-STO in which the ratio Ω/EF is not small. In these cases, one has to examine carefully if the vertex corrections are necessary or not. For the forward scattering phonons in the perfect forward scattering limit |g(q)|2 ∼ δ(q), it has been shown that vertex corrections are bounded by the coupling strength λm (Refer to section 4.1.1). To demonstrate how the vertex corrections are compared with other diagrams, below we have computed the first and second order corrections to the self-energies in the non- interacting limit (no self-consistent calculations) for the uniform and forward scattering phonons. Here, the first order correction is the single rainbow diagram denoted by “sr”. The second order corrections are the double rainbow “dr” and the leading order vertex diagram “vx” which are shown in Fig. A.1 (b) and (c). We compare two types of coupling: uniform scattering and forward scattering with q0 = 0.3. The dimension coupling constant λ defined in Eq. 2.24 with ` = s is equal to 1.0. The electron phonon coupling constant for the present model is shown in Fig. A.1(d). For the band structure, we used k = −2t cos(k) − µ with

125 (a) (d)

Σ 퐤, 푖휔푛 = +

(b) (c) Σ푑푟 퐤, 푖휔푛 = Σ푣푥 퐤, 푖휔푛 =

Figure A.1: (a) Electron self-energy with the vertex corrections. (b) Double rainbow “dr” diagram. (b) Leading order vertex “vx” diagram. (d) Electron phonon coupling constant used for the present model.

0.02 fit Vertex 0.015 fit

Migdal

) [eV] ) -

/ 0.01

( " 0.005

0 0 20 40 60 80 Temperature [K]

Figure A.2: ∆ vs. T with/without vertex corrections for uniform scattering. k taken from FeSe-STO model in section 4.1.1. λ = 0.6 Ωph = 80 meV. 16 × 16 momentum grid.

126 (t, µ) = (−0.075, −0.085) [eV] which is analogous to the earlier band structure calcualation for FeSe-STO except in one dimension in this case.

The results for the uniform scattering is shown in Fig. B.1 at T = 10 K. First, one can compare the overall magnitude between the first and second order corrections. For the first order correction, the real part of Σsr has maximum or minimum of ∼ 30 meV while the imaginary part is around ∼ 15 meV, and Im[Σsr] is still finite up to ∼ n = 100 matsubara frequency. For the second order corrections, all of the results show magnitude around 5 meV and saturate to zero no more than n ∼ 50. The results for the forward scattering is shown in Fig. B.2. For the first order correction, the overall magnitude is around ∼ 15 meV, and Im[Σsr] is again non-zero at n = 100. For the second order correction, all of the intensities are rather peaked at k = kF and zero elsewhere, among which Re[Σdr] yields the largest magnitude ∼ 15 meV extending to n ∼ 50 while vertex diagrams show finite magnitude only near n = 0. Based on the above results, one can see that it is still inconclusive whether the vertex corrections are indeed negligible. As has been discussed in section 4.1.1, the forward scattering phonons are shown to be negligible only if the higher order corrections are included. Considering only the first order vertex correction is not necessary a good indicator whether they can be neglected or not.

B Fourier Transform between iωn ↔ τ

This section covers how to implement the Fourier transform between iωn ↔ τ space. For the Fourier transform between the momentum k and real r space, they are rather trivial since the functions we are dealing with are normally assumed a periodic boundary condition, so the Fast Fourier Transform can be easily implemented[213, 214, 215, 216, 217].

For the Fourier transform τ → iωn, we need to evaluate the following integral expression,

Z β iωn f(iωn) = f(τ)e dτ. (59) 0

Numerically, we are only provided with discrete data sets f(τ`) with τ` = `β/Nm where

0 ≤ ` ≤ Nm where Nm is the number of matsubara frequency. For a better accuracy,

127 Uniform scattering phonons

Figure B.1: Electron self-energy calculated from the uniform scattering phonons. The single rainbow “sr”, double rainbow “dr”, and vertex “vx”. See texts for the discussions.

128 Forward scattering phonons

Figure B.2: Electron self-energy calculated from the forward scattering phonons with q0 = 0.3. Colorbars have units in eV. See texts for the discussions.

129 evaluating the integral using Riemann sum is deﬁnitely inadequate. Here, the technique normally used is known as Filon’s method[218]. The basic idea behind this approach is to

approximate the function for arbitrary τ near τ`. Namely, we need to obtain f(τ) in an interval τ` < τ < τ`+1 using some interpolation scheme. The linear interpolation scheme is known as the trapezoidal order (second order). Other intepolation schemes are the Lagrange polynomial or cubic spline. Once the intepolation method is finalized, one can integrate by hand respect to τ. The final fourmulas for the trapezoidal[141, 219] and cubic order using the Lagrange polynomial interpolation are given in Ref. [219]. In our case, we have used the Lagrange polynomial up to the quadratic order. The final expression is given by,

Nm−1 X 0 0 0 0 0 0 f(iωn) = (a1+a2+a3) f`−(b3+a3)δf+(c1−a2+b3)f1+(c2−a3)f2+c3f3+fNm−1+(b2−c1)fNm , `=0 (60) where

0 iωnτ` f` = e f(τ`), δf = f(τ = 0+) − f(τ = 0−), δ a = [I (2δ) − 3I (2δ) + 2I (2δ)], 1 4 2 1 0 δe−iωnδ a = − [I (δ) − 2I (δ)], 2 2 2 1 δe2iωnδ a = [I (δ) − I (δ)], 3 4 2 1 δeiωnδ b = [I (δ) − I (δ)], 1 4 2 0 δ b = − [I (δ) − I (δ)], 2 2 2 0 δe−iωnδ b = [I (δ) + I (δ)], 3 4 2 1 δ c = [I (δ) − 3I (δ) + 2I (δ)], 1 4 2 1 0 δe−iωnδ c = − [I (δ) − 2I (δ)], 2 2 2 1 δe−iωnδ c = [I (δ) + I (δ)], 3 4 2 1

130 and

esiωnδ − 1 I0(sδ) = , iωnδ sesiωnδ esiωnδ − 1 I1(sδ) = − 2 , iωnδ (iωnδ) s2esiωnδ 2sesiωnδ 2(esiωnδ − 1) I2(sδ) = − 2 + 3 , iωnδ (iωnδ) (iωnδ)

iωnτ` and τ`+1 − τ` = δ. The summation Σ` e f` can be evaluated using the same Fast Fourier transform algorithm, while other terms are simply just numbers. Note that the above expression contains f(`=Nm) which has to be obtained by some other means. For this, refer to the footnote 3.2.2.

For the Fourier transform iωn → τ, we need to evaluate the sum over the matsubara frequency. Namely, the expression is given by,

Nm/2−1 1 X f(τ ) = e−iωnτ` f(iω ). (61) ` β n n=−Nm/2

Ideally, the number of matsubara points Nm needs to be as large as possible, but numerically, we are dealing with finite numbers of matsubara frequency. Thus, we split the summation into two parts: one for the normal summation up to a finite cutoff, and two for the high frequency summation. For n suffciently large, we may approximate the function as,

f∞ f(iωn>>1) = , (62) iωn − ξ where f∞ is a number in the limit when n is large, though keep in mind that f∞ can have momentum dependence. The summation can be rewritten as,

N /2−1 N /2−1 1 m m f X −iωnτ` X ∞ f(τ`) = e f(iωn) − . (63) β iωn − ξ n=−Nm/2 n=−Nm/2

131 The second sum can be analytically handled. The above expression becomes,

Nm/2−1 (β−τ`ξ) 1 X f∞e f(τ ) = e−iωnτ` f(iω ) − . (64) ` β n eβξ n=−Nm/2

Finally, shifting the frequency from n = [−Nm/2,Nm/2 − 1] to [0,Nm − 1] yields the ﬁnal expression, N −s m Nm−1 i( N )π` (β−τ`)ξ e m X − i2π` e f∞ f(τ ) = e Nm f(iω ) − , (65) ` β n eβξ + 1 n=0 where s = 1 for fermions and s = 0 is for bosons. For example, for the electron Green’s function, ξ can be estimated as follow. Assuming n is suﬃciently large, then self-energies are approximately Z → 1 and (χ, φ) → 0. The Green’s function becomes,

iZ(k, iωn)ωn + k − µ + χ(k, iωn) 1 G(k, iωn) = 2 2 2 → , (66) (iZ(k, iωn)ωn) − (k − µ + χ(k, iωn)) − φ (k, iωn) iωn − (k − µ) so ξ is taken to be k − µ, and f∞ = 1 in this case.

C Pad´eApproximants

There are occasions when a function is known only along the imaginary axis, but no way to obtain at arbitrary complex numbers. We have seen that the Green’s function method discussed in the main text begins by obtaining self-energies along the imaginary axis in terms of the matsubara frequency. While many thermodynamic properties can be studied just from the knowledge of self-energies along the imaginary axis, what is physically practical is the self-energies deﬁned along the real axis. There are many measurable quantities that are a function of energy. Here, the process of brining the function from the imaginary axis to real axis is the analytic continuation. If the analytic continuation is not available algebraically, then one has to make approximations to obtain the real energy dependence. Typically, there are two methods for the analytic continuation. If a function is obtained with statistical errors such as the case in a Monte Carlo method, then the analytic continuation is done

132 using Maximum Entropy Method. For other cases, Pade approximation is another common method[144], while other methods utilize Laplace integral transform[139].

Suppose we know the function along the imaginary axis given by f(zn) where zn=1,2,...N is the imaginary number, and we want to obtain the function f(z) where z is an arbitrary complex number. The method of Pade begins by evaluating the matrix pi,j where it is obtainable from

p1,j = f(zj), j = 1,...,N (67) pi−1,i−1 − pi−1,j pi,j = . i, j = 2,...,N (zj − zi−1)pi−1,j

Then, the following recursion formula,

An+1 = An + (z − zn)pn+1,n+1An−1, (68) Bn+1 = Bn + (z − zn)pn+1,n+1Bn−1,

with A0 = 0,A1 = p1,1,B0 = B1 = 1 is initially used to obtain AN and BN . The desired function is obtained from f(z) = AN . BN For Migdal Eliashberg theory, the analytic continuation is done algebraically and given by Eq. 2.14. One can compare the self-energy between Eq. 2.14 and Pade approximations.

Here, we took band parameters from the FeSe-STO simulation, and set λ = 0.8, Ωph = 30 meV, at T = 10 K (supercnducting). We have tried applying Pade approximations to the electron Green’s function, but no satisfactory results were produced. It is highly suggested that one should apply Pade approximations to the electron self-energies instead, then construct Green’s function from it. However, for the phonon Green’s function, the Pade approximations worked fairly well. Fig. C.1 shows a comparison between the two methods: Eq. 2.14 and Pade approximations. Overall features and the magnitude of peaks were mostly captured, though few minor peaks were not perfectly reproduced such as those found at low energy in ωΣ2.

133 No Pade Pade Approximation No Pade Pade Approximation No Pade Pade Approximation 0.02 0.04

0.05

)

) )

! 0 0.02

! (k,

0 (k,

(k, 1

' ?

@ -0.02 0 !

-0.05 -0.04 -0.02 k=(0,0) -1 0 1-1 0 1 -1 0 1-1 0 1 -0.2 0 0.2 -0.2 0 0.2 k=(:,0) k=(:,:) No Pade Pade Approximation No Pade Pade Approximation No Pade Pade Approximation 0.06 0.04 0.04

) 0.02 0.02

) )

! 0.04

! (k,

(k, 0 0

(k, 2

' ? 0.02 @ ! -0.02 -0.02

0 -0.04 -0.04 -1 0 1-1 0 1 -1 0 1-1 0 1 -0.2 0 0.2 -0.2 0 0.2 ! [eV] ! [eV] ! [eV] ! [eV] ! [eV] ! [eV] Figure C.1: A comparison of electron self-energies between Pad´eapproximants and Eq. 2.14. Blue, red, and yellow lines are for k = (0, 0), (π, 0), and (π, π) respectively.

D Anderson Mixing

A self-consistent calculation starts by ﬁrst initializing input values. In our case, it is the

(1) (1) (1) Green’s function |Gin >= | · · · G (k, ωn) ··· F (k, ωn) ··· > which is written in a vector format. The superscript 1 means it’s the ﬁrst iteration. The size of the vector is determined (1) by the number of k points, number of ωn all included in a single vector form. This |Gin > (1) is used to output another |Gout >. Now, for the next iteration, one can simply proceed by (1) (2) (`) (`) letting |Gout >= |Gin >. One can keep doing this `th times until |Gout > −|Gin >≈ 0. Then, the Green’s function has converged, and the solution has been found. However, this method often leads to instability, and one can never reach the converging solution. The next step is to implement some mixing scheme to prevent from the divergence. The most simplest mixing scheme is given by,

(`) (`) (`) where we have deﬁned the residual vector |F >= |Gout > −|Gin >. a ∈ [0, 1] is the mixing parameter chosen by the user. In many instances, this simple mixing is enough to reach the convergence, although there are situations when a is forced to be smaller than 0.1 or even much smaller. Then, the convergence proceeds very slow. Therefore, more sophisticated

134 method is necessary to speed up the convergence. Here, we will discuss Anderson’s mixing scheme[220, 221, 222]. Other common methods are the Broyden mixing which is widely used in electronic calculations for Density Functional Theory. The reason for the faster convergence in Anderson’s mixing is because it considers vectors from the previous iterations. The vectors are mixed with up to Mth previous iterations using the mixing parameter γ in the following way, M ¯(`) (`) X ` (`−j) (`) |Gin >= |Gin > + γj (|Gin > −|Gin >). (70) j Likewise, the residual vectors are deﬁned in the same manner,

M ¯(`) (`) X ` (`−j) (`) |F >= |F > + γj (|F > −|F >). (71) j

` ¯(`) ¯(`) The values γj are found by minimizing the norm of the residual vector < F |F > which is what we need. This will lead to a system of linear equations given by,

M X ¯(`) ¯(`−i) ¯(`) ¯(`−j) ` ¯(`) ¯(`−i) ¯(`) < F − F |F − F > γj =< F − F |F >, (72) j

` where i = 1, 2,...M which has to be solved to obtain the coeﬃcients γj . Finally, the vector for the next iteration can be found from,

(`+1) ¯(`) ` ¯(`) |Gin >= |Gin > +a |F >, (73) where a` is a user input mixing parameter which can be varied for `th iterations. Anderson[222] has pointed out that it is not highly suggested to choose large M values which can inhibit from ﬁnding the accurate minimum, so M = 3 ∼ 5 are normally chosen.

E Relevant Formulas

In the main section, we have introduced a method for calculating electron self-energies in the superconducting state. While the electron spectral function and the density of states can

135 be obtained rather easily, calculation of other observables needs extra work. Here, we will introduce couple formulas, but will not discuss derivations or physics since the main theme is about the self-energy. Once the self-energies are obtained, one can use the formulas shown below. The formulas that will be introduced are the optical conductivity and speciﬁc heat.

E.1 Optical Conductivity

The optical conductivity measurement is another powerful method for probing quasiparticle excitations[223, 224]. The optical conductivity for d-wave supercondcutors is evaluated using Kubo formula. In this formalsim, the induced electric current density J is assumed to be

linear to the external electric ﬁeld E by Ji(q, ω) = Σijσij(q, ω)Ej(q, ω), where i j ∈ x, y, z are the cartesian cordinates, and q and ω are the momentum and energy of the applied ﬁeld. For optics, we consider a long wavelength limit q → 0. The optical conductivity σ is given by,

2 2 2e i X ∂k ∂k ˆ ˆ ∂ k σij(iωn) = Tr[G(k, iωm + iωn)G(k, iωm)] − G11(k, iωm) , (74) N βω ∂ki ∂kj ∂ki∂kj ~ k,m where the hat denotes 2 × 2 matrix obeying Nambu formalism. Note the above expression neglects the vertex corrections. The analytic continuation can be carried out algebraically to obtain the real frequency dependence. The real and imaginary parts are given by,

2 2 Z ∞ 2e π X ∂k ˆ ˆ 0 0 0 0 Re[σxx(ω)] = 2 Tr[A(k, ω)A(k, ω + ω)][n(ω ) + n(ω + ω)]dω , (75) Nω ∂kx ~ k −∞

and,

" ∞ ∞ 2 2 Z Z ˆ 0 ˆ 00 0 00 2e X 1 ∂k Tr[A(k, ω )A(k, ω )][n(ω ) − n(ω )] 0 00 Im[σxx(ω)] = 2 0 00 dω dω ~ Nω π ∂kx ω + ω − ω k −∞ −∞ ∞ (76) 2 Z # X ∂k 0 0 0 − 2 A11(k, ω )n(ω )dω , ∂kx k −∞

136 where Aˆ = Im[Gˆ], and n(ω) is the Fermi-dirac function. The real and imaginary parts are related by the Kramer-Kronig relation which is one way to check the results. The above expression considers a system with a ﬁnite band width. An example for the inﬁnite band model is given in Ref. [51].

E.2 Speciﬁc Heat

Assuming that the self-energies are already calculated using Eliashberg equations, for example using Eq. 4.27, then one can use the formulas given below. Calculation of the specific heat begins with first obtaining the thermodynamic potential. The derivation in earlier steps are given in Ref. [59] followed by intermediate steps shown in Ref. [225]. The final expression is,

" ¯s 2 2 2 ! ¯s ¯s 2 2 1 X (Zi (k)) + ξi,k + φi (k) −Z (k)ωn + (Z (k)) + φ (k) ∆Ω(T ) = − ln − i i i Nβ (Z¯n(k))2 + ξ2 (Z¯s(k))2 + ξ2 + φ2(k) k,i j i,k i i,k i # (77) −Z¯n(k)ω + (Z¯n(k))2 + i n i , ¯n 2 2 (Zi (k)) + ξi,k where ∆Ω(T ) is the diﬀerence between the superconducting and normal state, i is the band indices, the superscript s n denote the superconducting and normal state respectively, ξi,k =

i,k − µi, and k = (k, ωn) as before. Once the thermodynamic potential is obtained, the ∂2(∆Ω(T )) speciﬁc heat can be obtained from the second derivative ∆C(T ) = T ∂T 2 .

F Analytic Continuation (Migdal-Eliashberg Theory)

Normally the analytic continuation to bring the imaginary axis to the real axis is done from approximations, but for certain cases such as the Migdal approximations with non-interacting bosons, the analytic continuation can be done algebraically. In this section, we will go over the derivation of Eq. 2.14. Let us start with Eq. 2.11 which is,

1 X Σ(ˆ k, ω ) = − |g(q)|2D(q, iω − iω )ˆτ Gˆ(k + q, ω )ˆτ . (78) n Nβ n m 3 m 3 k,m

137 We ﬁrst introduce the spectral representations for Green’s functions. They are,

1 Z ∞ Aˆ(k, z) Gˆ(k, ω ) = − dz, n π iω − z −∞ n (79) Z ∞ ˆ F (q, z) D(q, Ωn) = dz, −∞ iΩn − z

ˆ ˆ ˆ ˆ where A(k, ωn) = Im[G(k, ωn)] and F (q, Ωn) = Im[D(q, Ωn)] as deﬁned before. Inserting the above two representations into Eq. 78 yields,

∞ ∞ Z Z ˆ 0 ˆ 1 X 2 τˆ3A(k + q, z)ˆτ3 F (q, z ) 0 Σ(k, ωn) = |g(q)| 0 dzdz . (80) πNβ iωm − z iωn − iωm − z k,m −∞ −∞

The summation over m can be carried out. See Ref. [15] section 3.5. The result is,

∞ Z ˆ 0 0 ˆ 1 X τˆ3A(k + q, z)ˆτ3F (q, z ) zβ z β Σ(k, ωn) = 0 tanh + coth . (81) 2πN iωn − z − z 2 2 q −∞

We may now perform the analytic continuation ωn → ω + iδ. Now, utilizing the identity 1 ∗ ∗ Im[f]h = 2 (Im[f(h + h )]) − iRe[f(h − h )]), we obtain,

∞ ! 1 X Z Σ(ˆ k, ω ) = − |g(q)|2F (q, z0)dz0 × ... n 4πN q −∞ ! " ∞ # Z τˆ Gˆ(k + q, z)ˆτ τˆ Gˆ(k + q, z)ˆτ zβ z0β ... = Im 3 3 + 3 3 tanh + coth dz ω + iδ − z − z0 ω − iδ − z − z0 2 2 −∞ " ∞ # Z τˆ Gˆ(k + q, z)ˆτ τˆ Gˆ(k + q, z)ˆτ zβ z0β − iRe 3 3 − 3 3 tanh + coth dz . ω + iδ − z − z0 ω − iδ − z − z0 2 2 −∞ (82)

138 The above integral respect to z has simple poles. One can perform the integration over the ˆ upper half plane since G is analytic for ωm > 0. Carrying out the integration yields,

∞ ! 1 X Z Σ(ˆ k, ω ) = − |g(q)|2F (q, z0)dz0 × ... n 2N q −∞ ! " (ω − z0)β z0β ... = Im − iτˆ Gˆ(k + q, ω − z0 + iδ)ˆτ tanh + coth 3 3 2 2 (83) ∞ ˆ # 2i X τˆ3G(k + q, ωm)ˆτ3 + 0 β ω − iωm − z ωm>0 " # (ω − z0)β z0β − iRe − iτˆ Gˆ(k + q, ω − z0 + iδ)ˆτ tanh + coth . 3 3 2 2

∗ Utilizing the relation G (iωn) = G(−iωn) ﬁnally yields,

∞ Z ∞ ˆ ˆ 1 X 2 0 X τˆ3G(k + q, iωm)ˆτ3 0 Σ(k, ωn) = − |g(q)| F (q, z ) 0 dz βN ω − iωm − z q −∞ ωm=−∞ ∞ 1 Z (ω − z0)β z0β + |g(q)|2F (q, z0)ˆτ Gˆ(k + q, ω − z + iδ)ˆτ tanh + coth , 2N 3 3 2 2 −∞ (84) which is the desired result.

139 Vita

Ken Nakatsukasa was born and raised in Japan. Later, he has made a transition in his life to study abroad and attended the University of California, Los Angeles where he earned his Bachelor’s degree in Physics. Then, he moved on to pursue Ph.D degree in Physics at University of Tennessee, Knoxville. After his second year in graduate school, he decided to join Dr.Johnston’s research group who is specialized in theory and computations for studying strongly correlated materials and superconductivity. In July 2018, he has successfully defended his thesis on his ﬁrst attempt.

140