David M¨
ockli
ASPECTS OF MULTIGAP AND MULTILAYER
SUPERCONDUCTIVITY
Niter´oi February 2017
M687 Möckli, David.
Aspects of multigap and multilayer superconductivity / David Möckli; orientador: Evandro Vidor Lins de Mello; orientador: Mafred Sigrist. –- Niterói, 2017.
90 p. : il.
Tese (Doutorado) – Universidade Federal Fluminense, Instituto de Física, Niterói, 2017.
Bibliografia: p. 84-90.
1.SUPERCONDUTIVIDADE. 2.MAGNETISMO. 3.TEORIA DE
GINZBURG-LANDAU. 4.SUPER-REDE. I.Mello, Evandro Vidor Lins de, orientador. II.Sigrist, Mafred, co-orientador.
III.Universidade Federal Fluminense. Instituto de Física, Instituição responsável. IV.Eidgenössische Technische Hochschule Zürich. V.Título.
DAVID M ¨OCKLI
PHD DISSERTATION
This DISSERTATION is submitted to the graduate pro-gram in Physics of Universidade Federal Fluminense, in partial fulfilment of the requirements for the degree Doctor in Physics.
Advisor: Prof. Dr. EVANDRO VIDOR LINS DE MELLO (UFF) Coadvisor: Prof. Dr. MANFRED SIGRIST (ETH)
Niter´oi February 2017
Acknowledgements
Writing this dissertation was a privilege. I would have done it for free. This PhD has provided me blissful joy, the satisfaction of learning something new every day, and the opportunity to learn patience. It is a privilege to have the chance the learn how to do a little physics, and enjoy the solitude of research and small mathematical discoveries. I owe this privilege to a few persons who were personally or directly involved with this project to which I must express my gratitude.
I express my gratitude to the author of physics. I thank milady for everything. Happiness for me has a name – the name is Bruna. I thank my family, specially my parents for providing me the heritage of a life of opportunities. My sincere thanks to my advisor Evandro Vidor Lins de Mello for his patience, guidance and dedication. I thank Manfred Sigrist for giving me the opportunity to spend a visiting year at ETH, doing research in my home-country and exceptional hospitality. I thank CNPq for the scholarships provided during the PhD.
Here I repeat my oath from my bachelor graduation:
I promise to execute the duties as a physicist always with respect, dignity, and love towards life. I will not allow knowledge to manipulate or destroy humanity. On the contrary, let knowledge be of service to life and, as an educator, create new worlds towards just societies and free men and women.
Abstract
This dissertation is composed of two parts. The first part was developed at Universidade Federal Fluminense (UFF) under the supervision of Prof. Dr. Evandro Vidor Lins de Mello and addresses the temperature dependence of the superconducting gaps in iron-based superconductors. The second part was developed at the Eidgen¨ossische Technische Hochschule Z¨urich (ETH) during a visiting year under the supervision of Prof. Dr. Mafred Sigrist. This part concerns the effect of vortices on the superconducting properties of a three-layer system. The following paragraph is the abstract of part one. The last paragraph is the abstract of part two.
The temperature dependence of the multiple superconducting gaps in the typical high-Tc iron-based superconductor Ba0.6K0.4Fe2As2 is studied. These multiband
iron-based superconductors display multiple superconducting gaps with different coupling ratios 2Δ0/kBTc, but a single Tc. The guiding question throughout this project is: is it possible to
reproduce the various coupling ratios and single Tc observed in Fe-based superconductors
within a weak-coupling Bogoliubov-deGennes theory? This dissertation proposes two distinct mechanisms through which it is possible to reproduce the temperature dependence of the multigap structure of Ba0.6K0.4Fe2As2. The first proposal shows how intrinsic
charge inhomogeneity might lead to a temperature dependent Cooper pairing potential. The second proposal uses arguments from inhomogeneous superconductivity applied to multiband superconductors to introduce a temperature dependent chemical potential.
In part two, we investigate the pair-density wave phase of multilayer superconductors in the context of a Ginzburg-Landau theory. In multilayer systems, due to local inversion symmetry breaking at the outer layers, a Rashba spin-orbit coupling is induced. This combined with a perpendicular Pauli limiting magnetic field stabilizes a pair-density wave (PDW) phase, which is achieved through a first-order phase transition. The PDW phase is robust against magnetic fields. The central issue discussed in this dissertation is whether orbital limiting (on top of dominant paramagnetic limiting) destroys the PDW phase. We find that orbital limiting does not destroy the the PDW phase and investigate the behavior of a single vortex core through the first-order BCS-PDW phase transition. As a subproject, we also study the paramagnetic effect on the reversible magnetization curves of high-κ type II superconductors by generalizing a circular cell method within a Ginzburg-Landau theory.
Resumo
Esta tese ´e composta por duas partes. A primeira foi desenvolvida na Universidade Federal Fluminense (UFF) sob orienta¸c˜ao do Prof. Dr. Evandro Vidor Lins de Mello. Neste projeto, estudamos como os gaps supercondutores variam com a temperatura em supercondutores a base de ferro. A parte dois desta tese foi desenvolvida na Eidgen¨ossische Technische Hochschule Z¨urich (ETH) durante um est´agio sandu´ıche supervisionado pelo Prof. Dr. Mafred Sigrist. Nesta parte estudamos o efeito de v´ortices sobre as propriedades supercondutoras de um sistema de trˆes camadas. O seguinte par´agrafo corresponde ao resumo da parte um, e o ´ultimo par´agrafo a parte dois.
Aqui estudamos como os m´ultiplos gaps supercondutores variam com a temperatura no supercondutor a base de ferro Ba0.6K0.4Fe2As2. Estes materiais multibanda possuem
m´ultiplos gaps com acoplamentos 2Δ0/kBTc distintos, mas apresentam uma ´unica
tempe-ratura cr´ıtica de transi¸c˜ao Tc. A pergunta guia deste projeto ´e: ´e poss´ıvel reproduzir os
diferentes acoplamentos e ´unico Tc presente em supercondutores a base de ferro, dentro
do contexto de uma teoria de Bogoliubov-deGennes de acoplamento fraco? Nesta tese apresentamos dois mecanismos que permite reproduzir os gaps de Ba0.6K0.4Fe2As2. A
primeira proposta mostra como uma inomogeneidade intr´ınseca destes materiais pode levar a um potencial de pareamento dependente da temperatura. A segunda proposta usa argumentos da supercondutividade inomogˆenea aplicada a supercondutores multibanda para introduzir um potencial qu´ımico dependente da temperatura.
Na parte dois, investigamos a fase PDW (pair-density wave) em supercondutores multicamada no contexto de uma teoria de Ginzburg-Landau. Nestes sistemas, uma intera¸c˜ao spin-´orbita do tipo Rashba ´e induzida devido a quebra local de simetria de invers˜ao nas camadas externas. Isto combinado com um campo magnetico perpendicular externo estabiliza a fase PDW, que ´e obtida atrav´es de uma transi¸c˜ao de fase de primeira ordem. A fase PDW ´e robusta contra a a¸c˜ao de campos magn´eticos. A quest˜ao central discutida nesta tese ´e se a penetra¸c˜ao de v´ortices destr´oi a fase PDW. Como um subprojeto, n´os tamb´em obtivemos as curvas de magnetiza¸c˜ao revers´ıveis para supercondutores do tipo-II com κ grande. Para isto, generalizamos um m´etodo de aproxima¸c˜ao circular dentro do contexto da teoria de Ginzurg-Landau.
List of Figures
1.1 The history of the discovery of new superconducting materials through the years colored by their families. . . 3 1.2 (a) Temperature-doping phase diagram. The undoped parent compound is
an antiferromagnetic spin-density wave. For some doped compounds, this magnetic order coexists with superconductivity. Above the Neel tempera-ture, there is a Nematic order, in analogy to liquid crystals. (b) Considered theoretical crystal unit cells to construct minimal models. . . 6 1.3 Artificial CeCoIn5(n)/YbCoIn5(5) superlattices for n = 5 and n = 3. The
symmetry properties of the superconducting CeCoIn5(n) layers are indicated
by the symbols. Details are explained in the text. . . 8 2.1 Left: The five 3d orbitals of iron. A minimal five-orbital model of Fe-based
superconductors should include all five orbitals per lattice site. Right: Ty-pical four-band structure tight-binding model of Fe-based superconductors plotted along high symmetry lines in the first Brillouin zone (adapted from reference [20]). . . 17 2.2 Three-gap temperature dependence observed in Ba0.6K0.4Fe2As2 [13] of the
α (red), β (blue) and γ (green) bands respectively observed by ARPES. The solid lines are BCS-like fits. . . 18 2.3 Left: Fermi surface of the superconducting bands containing the α-hole
pocket (red), β-hole pocket (blue), and γ-electron pocket (green), as obtained by tight binding-fits to ARPES band dispersions. Right: Correspondent band structure along the Γ→ M line in the first Brillouin zone. . . 19
2.4 Figure extracted from reference [34]. (a) Fermi pockets in the 2D Brillouin zone as observed by ARPES. (b) Grey LDA bands compared to tight-binding ARPES fits (dashed lines). One can see that α and β bands are hole pockets, while the γ forms an electron pocket at the Fermi surface. (c) Fermi surface as obtained from the tight-binding curves. (d) Measured and calculated Fermi velocities. . . 20 2.5 The left panel shows the SC gaps on three non-interacting bands with the
same band parameters of the right panel extracted from [13] and [34]. The big difference is the inter-band hopping. Bands 1 (red), 2 (blue) and 3 (green) have the same hoppings as the α, β and γ pockets respectively listed
in table 2.1 . . . 22 2.6 On the left panel we show the theoretical BdG curves compared to PCAR
[31, 32] for Ba1−xKxFe2As2 with Tc = 23 K. PCAR identifies two SC bands
that may correspond to the nearly coinciding α and γ pocket identified by ARPES. On the right panel we show the BdG curves compared to the three bands from ARPES [28] for the compound with Tc = 32 K. Reference [28]
only estimates gap values for the β pocket. . . 24 2.7 Temperature dependence of (a) the jointly obtained SC gaps Δν(T ) from
multi-band BdG theory (solid lines) in comparison to references [13] (empty points) and [39] (solid points) for Ba0.6K0.4Fe2As2 with Tc = 38 K and
N = 8. We show the three sets of data corresponding to (b) ν = α, (c) ν = β, and (d) ν = γ separately for clarity. The grey curves correspond to the case of a constant potential, which is included for comparison. . . 26 3.1 Plot of Tn(x) with n from 0 to 3. . . 30
3.2 Left: Approximation of δ(x) in a series of Chebyshev polynomials for N = 15 (blue line) and N = 100 (black line). Right: Comparison of the polynomial expansions in terms of φn(x) (blue) and Tn(x) (black). . . 32
3.3 Left: Different Lorentzian curves as η → 0. Right: A comparison between the Dirichlet, Lorentz and Jackson kernels used to improve the expansion of δ(x) (see table 3.2). The Lorentz curve is broader that the Jackson curve. The Jackson kernel is constructed to approximate a Gaussian, which is narrower and higher than a Lorentzian. . . 33 3.4 Evolution of Emax with increasing lattice size n for a typical homogeneous
4.1 Real space superconducting gap profiles with an underlying square lattice (black dots) correspondent to the gap structures (a) s-wave Δ(kx, ky) ∝
const. (b) s-wave Δ(kx, ky)∝ cos kx+ cos ky (c) d-wave Δ(kx, ky)∝ cos kx−
cos ky (d) s-wave Δ(kx, ky) ∝ cos kxcos ky. Lighter colors correspond to
higher gap values. Panels (a) and (d) show the most likely candidates to describe the FeSCs. Red arrows shows a nearest neighbor, while the blue arrow indicates next-nearest neighbors. . . 49 4.2 (a) Behavior of the superconducting gap in the α and β bands with varying
chemical potential. A monotonic loss (or gain) in intra-band electron population causes higher coupling ratios than the usual BCS result of 2Δ/kBT ≈ 3.52. The red and the blue arrows indicate the direction of the
chemical potential when temperature increases. This implies a monotonic reduction of electron population in the α and β bands. See figure c for the 3D version. (b) The gaps in the γ and δ bands as a function of the chemical potential. To conserve the total electron population, the γ and δ bands gain electrons as indicated by the green and blue arrows. (c) The surface of Δα(µ
α, T ) above with the the surface of Δβ(µβ, T ) beneath.
The dashed line shows the constant µ BCS-geodesic Δα(−76, T ). The
solid line shows a geodesic with varying chemical potential, with initial point at Δα(−76, 1) and final point at Δα(−84.5, 37), that reproduces the
temperature dependence of the energy gaps in FeSCs. Geodesics for the other bands are similar. (d) Redistribution of electron population among the superconducting bands with temperature along the four geodesics. In the present case, α and β bands loose electrons, while γ and δ bands gain electrons. The γ and δ curves coincide. (e) Projections of the four geodesics onto the Δν(T ) plane. We show error bars as extracted from ARPES for the
α and β band-gaps. The gap in γ almost coincides with α, and experimental points for the δ-gap were not available. . . 51 6.1 (a) Magnetization curves at T = 0 staring from a κ = 100 superconductor
for different Maki parameters. Bo
c2 is the orbital upper critical field. (b)
Dependence of ψ0 on the H-field for different Maki parameters. ψmax =
�
a(Tc− T )/b. (c) Dependence of the variational parameter ξc (modeling
the vortex core size) on the H-field for different Maki parameters. ξBc2 is
the critical core size for αM = 0. . . 70
7.1 A superconducting three-layer system with its local symmetries. While the system has global inversion symmetry, the outer layers can be viewed as local noncentrosymmetric superconductors, where a Rashba spin-orbit coupling is induced. . . 74
7.2 Typical B-field–temperature phase diagram of a Maki infinity three-layer system obtained within Ginzburg-Landau theory. In this illustrative exam-ple all parameters were set to 1 for simplicity. The density plot shows the intensity of |Δ2| =
�
|ψ2|2+|η2|2. A first order phase transition separates
the low field BCS from the high field PDW region. Here Tc is the
supercon-ducting critical temperature of a single layer, which shows that a three-layer system has its critical temperature increased. . . 78 7.3 Leading order parameter instabilities for an αM = 1.8 case. Varying
temperatures shifts the horizontal axis down or upwards. . . 79 7.4 (a) Plot of|Δi| =
�
|ψi|2+|ηi|2 for the middle and outer layers. |Δ1| = |Δ3|.
(b) Plot of the singlet layer dependent order parameters as a function of the B-field. (c) Plot of the triplet layer dependent order parameters as a function of the B-field. (d) Plot of the core size ξc as a function of the
List of Tables
1.1 Comparison between the cuprates and iron-based superconductors. . . 5 2.1 First and second nearest neighbour intra tν and inter-band hoppings tµν in
meV used in the present calculations. Intra-band hoppings were extracted from ARPES and inter-band hoppings closely agree with band dispersion [34]. The parameters of this table are used for all three compound discussed in this paper. . . 23 2.2 Phenomenological values of the electronic phase transition temperature TPS
and potential |V0| in equation (2.23) used to model Ba1−xKxFe2As2 with
different Tc’s. . . 23
3.1 The first few Chebyshev polynomials of the first Tn(x) and second kind
Un(x). Polynomials with even n are even functions, whereas polynomials
with odd n are odd functions. . . 29 3.2 Summary of different kernels studied throughout the literature to improve
the quality of an order N Chebyshev series. The Dirichlet and the Fej´er kernels are mainly of academic interest. The Lorentz and Jackson kernels are positive, and useful for most applications. Other kernels are the Lanczos and Wang, but are not optimal for most applications. . . 34 4.1 Effective hopping parameters t1 (nearest neighbor) and t2 (next-nearest
neighbor) consistent with APRES band dispersion fit [29]. U0 is the on-site
s-wave pairing potential used in the BdG calculations of Δν
i. . . 46
4.2 Correspondence between different gap structures in momentum space and lattice space. The vector x (y) connects two neighboring horizontal (vertical) lattice sites. The vector d = x + y and g = y− x. . . 50
5.1 Table of unconventional superfluid order-parameters. Here we substituted eiϕΔ by Δ/k
F. . . 61
5.2 Even-parity basis gap functions ˆΔ(Γ, m; k) = iσyψ(Γ, m; k) for tetragonal
Contents
1 Introduction 2
1.1 Historical background . . . 2
1.2 Part I: iron-based superconductors . . . 4
1.3 Part II: artificial multilayer superconductors . . . 6
1.4 Overview . . . 8
I
Temperature dependence of the superconducting gaps in
iron-based superconductors
11
2 The multiband Bogoliubov deGennes model 12 2.1 Introductory remarks . . . 122.2 The model . . . 12
2.3 Multi-band BdG exact diagonalization . . . 14
2.4 Self-consistent physical quantities . . . 16
2.5 Effective three-band structure in Fe-based superconductors . . . 17
2.5.1 Introduction . . . 17
2.5.2 Multi-band superconducting theory . . . 19
2.5.3 Temperature dependence of the three-gap structure . . . 23
2.6 Concluding remarks . . . 25
3 The kernel polynomial method 27 3.1 Introductory remarks . . . 27
3.2 Basic concepts . . . 28
3.3 The moments . . . 30
3.3.2 Gibbs oscillations and Kernels . . . 31
3.4 Computational implementation details . . . 33
3.4.1 Rescaling of the energy . . . 33
3.4.2 Recursive calculation of moments . . . 34
3.4.3 Fast Fourier transforms . . . 36
3.4.4 Chebyshev-Gauss integration . . . 37
3.4.5 The stochastic evaluation of traces . . . 37
3.5 Concluding remarks . . . 40
4 The Chebyshev Bogoliubov deGennes method 41 4.1 Introductory remarks . . . 41
4.2 Green’s functions . . . 41
4.3 Expansion of Green’s functions . . . 43
4.4 The superconducting four-gap temperature evolution . . . 44
4.4.1 Introduction . . . 44
4.4.2 The model . . . 45
4.4.3 The multiband CBdG method . . . 47
4.4.4 Temperature dependence of the four-gap structure . . . 50
4.5 Concluding remarks . . . 52
II
Vortices in the pair-density wave phase of multilayer
su-perconductors
53
5 Rudiments of unconventional superconductivity 55 5.1 Introductory remarks . . . 555.2 The instability of the Fermi surface . . . 55
5.3 BCS theory of anisotropic superconductivity . . . 57
5.3.1 Diagonalization of the Hamiltonian . . . 58
5.3.2 The gap matrix: singlet and triplet superconductivity . . . 59
5.4 Ginzburg-Landau theory . . . 62
5.5 Concluding remarks . . . 63
6 Paramagnetic effect on the magnetization curves of type-II supercon-ductors 64 6.1 Introductory remarks . . . 64
6.2 The paramagnetic effect within GL theory . . . 65
6.3 The Maki parameter . . . 66
6.4 Circular cell method for the vortex lattice . . . 67
7 The pair-density wave phase in multilayer superconductors 72
7.1 Introductory remarks . . . 72
7.2 Non-centrosymmetric superconductors . . . 72
7.3 Formulation . . . 73
7.4 No orbital effect: the Maki infinity case . . . 77
7.4.1 Leading superconducting instabilities . . . 77
7.5 A vortex in a Maki infinity trilayer system . . . 79
7.6 Concluding remarks . . . 80
CHAPTER
1
Introduction
1.1
Historical background
Superconducting phenomena has not ceased to amaze researchers since its discovery in 1911 by Heike Kamerlingh Onnes. More than hundred years have passed, and what was once the best understood many-body problem in physics, is now a thriving research field. Topics such as high temperature, unconventional, and non-centrosymmetric superconductivity, and many other classes of superconductors gain continuous attention of the community, because of its wealth of new physics.
We survey the highlights of the history of superconductivity by reviewing the Nobel prizes granted to research related to superconductivity and superfluidity. In 1913, Heike Kamerlingh-Onnes was awarded for his investigations on the properties of matter at low temperatures, which led to the production of liquid helium. This allowed for the serendipitous discovery of superconductivity. At this time, the London brothers developed an electrodynamic theory of superconductivity. However, a microscopic theory of super-conductivity had to wait until the development of quantum mechanics a few decades later. It took almost half a century after the discovery of superconductivity for the emergence of a satisfying quantum theory. In 1972, John Bardeen, Leon Neil Cooper, and John Robert Schrieffer (the BCS trio) received the Nobel prize for their jointly developed theory of superconductivity, nowadays called the BCS-theory. In 1973, Leo Esaki and Ivar Giaever were awarded for their experimental discoveries regarding tunneling phenomena in semiconductors and superconductors. In the very same year Brian David Josephson also was awarded for his theoretical predictions of the properties of a supercurrent through a tunnel barrier, in particular those phenomena which are generally known as the Josephson effect. Superconductors dominated Nobel prizes in the early seventies. More than ten
Figure 1.1: The history of the discovery of new superconducting materials through the years colored by their families.
years later Johannes Georg Bednorz and Karl Alexander M¨uller were awarded for their important break-through in the discovery of superconductivity in ceramic materials – the era of high-Tc superconductivity was born. In 1996, David Morris Lee, Douglas
D. Osheroff and Robert Coleman Richardson received the prize for their discovery of superfluidity in helium-3. The development of a generalized BCS theory was necessary to treat the physics of helium-3. In 2003, Vitaly Lazarevich Ginzburg, Anthony James Leggett and Alexei Alexeyevich Abrikosov won the Nobel prize for pioneering contributions to the theory of superconductors and superfluids. In particular, the development of the phenomenological Ginzburg-Landau theory of superconductivity alongside BCS theory was of huge importance. Ginzburg-Landau theory provides an elegant bridge between the microscopic quantum theory and material engineering. In 2016, David J. Thouless, F. Duncan M. Haldane and John M. Kosterlitz shared the Nobel prize for theoretical discoveries of topological phase transitions and topological phases of matter, including topological superconductors. No other field of physics has received that many Nobel prizes.
The history of superconductivity surpasses now a century. Still, many mysteries remain: the biggest probably being the underlying pairing mechanism in high-Tc superconductors.
There are many families of superconducting materials discovered thorough the decades; see figure 1.1. The first superconductors to be discovered were metals of very simple chemical structure – the conventional superconductors. Conventional superconductors are the ones that are well described by BCS theory and have a typical coupling ratio 2Δ0/kBTc = 3.52. These materials are shown in green in figure 1.1. Later, copper based
superconductors (the cuprates) were discovered (in blue). They receive enormous attention because of their high critical temperatures as compared to conventional superconductors. However, note that their chemical structure is much more complicated and these materials
display several coexisting electronic orders, which makes them interesting and hard to understand. More recently, a new family of high-Tc superconductors were discovered: the
iron-based superconductors (in orange). This caused some degree of perplexity among the community, because it was long believed that magnetic materials – such as iron – destroy superconductivity. At last, we would like to call attention to the light green family, which usually have very low Tc. These are unconventional superconductors with rich magnetic
features. Among them are heavy-fermion materials, where spin-orbit coupling plays a crucial role. Superconductivity is fairly well understood for the conventional family. The same cannot be affirmed for the remaining families. Looking at the common features of all families is what boosts the hope for the development of a microscopic theory of superconductivity in the next few decades. Most now agree that the underlining pairing mechanism is purely electronic [1].
In part I of this dissertation we focus on iron-based superconductors. In part II superconductors we concentrate on perhaps a completely new family, which are the artificially engineered superconductors. The following sections overview part I and part II of this dissertation.
1.2
Part I: iron-based superconductors
Iron based superconductors trail right behind the cuprates in their superconducting critical temperature. Despite their different chemical composition, they share many features with the cuprates. Understanding the commonalities and differences might help unveiling the mechanism behind high-Tc superconductivity [2]. As it is with the cuprates, the materials
are obtained upon electron or hole doping (charge injection via chemical substitution) starting from an antiferromagnetically ordered parent compound. This may suggest that magnetism, instead of destroying superconductivity, might be the secret ingredient of high-Tc materials. Much is hotly debated about this issue. Because electron-electron
correlations are suspected to be not as strong as in the cuprates, hopes are that these materials are simpler to study. For instance, mean-field density functional theories (DFT’s) are unsuccessful when applied to cuprates, whereas good predictions are achieved for iron-based materials [3]. In table 1.1 we compare a few aspects between the cooper and iron-based materials. Note that both families of materials, even though having a 3D unit cell, have their magnetic and superconducting properties mainly determined by 2D planes.
Iron-based superconductors come in various chemical flavors. For details on the various families we refer the reader to review papers [4, 5]. In this dissertation, we mainly focus on optimally potassium hole doped Ba0.6K0.4Fe2As2, which is frequently referred to as a
prototypical pnictide regarding its superconducting properties. In figure 1.2 we show a typical illustrative phase diagram and a unit cell of Ba0.6K0.4Fe2As2.
Characteristic Cuprates Iron pnictides High-Tc ? Yes, up to 155K Yes, up to 55K
2D layers? Yes Yes
Building
blocks 3d transition metal ions: Cu 3d transition metal ions: Fe 2p-ligands Very near the Cu ions Offset with respect the
Fe plane Dominant
orbitals dx2−y2 All five Fe 3d orbitals Minimal
model
Low energy effective
one-band model Multi orbital/band model Undoped characteristic Antiferromagnetic Mott insulator Antiferromagnetic metal Superconducting
dome? Yes Yes
Strong
correlations? Yes ?
Success of
DFT Very restrictive Good Normal state
pseudogap? Explicit Maybe Pairing symmetry Universally singlet d-wave Possible singlet mixtures Pairing mechanisms
proximity to magnetism and quantum criticality, anisotropy of the Fermi surface, CDW, SDW,
intrinsic electronic phase separation, spin fluctuations.
Figure 1.2: (a) Temperature-doping phase diagram. The undoped parent compound is an antiferromagnetic spin-density wave. For some doped compounds, this magnetic order coexists with superconductivity. Above the Neel temperature, there is a Nematic order, in analogy to liquid crystals. (b) Considered theoretical crystal unit cells to construct minimal models.
occur in the Fe square planes, iron-based materials have complicated multiband structures, where each band receives weights from different 3d orbitals. Consequently, since five bands cross the Fermi level, the materials are multiband superconductors – a major difference with respect to the cuprates. Also, in cuprates, dominant dx2−y2-gap symmetry is fairly
well established [6]. In the iron based materials the gap symmetry is still elusive. It may be doping dependent, and might have contributions from more than one even parity channels. At the present moment, the most likely superconducting gap symmetry is the s± state [7]. However, here we will assume that the gap symmetry is an isotropic s-wave for all bands.
1.3
Part II: artificial multilayer superconductors
All high-temperature superconductors to date are multi-layered materials. Thus, under-standing 2D superconductivity has always remained an important issue. On the other hand, heavy-fermion superconductors have an essentially 3D electronic structure. Besides this, one knows that low-dimensional strongly interacting systems display unusual and exotic electronic properties. Therefore, artificially lowering the dimensionality of heavy-fermion systems could lead to a better understanding of strong correlations in 2D.
A two-dimensional heavy fermion system was recently achieved by epitaxially growing an artificiall superlattice [8]. In such a superlattice, thin layers of the heavy fermion compound CeIn3 are inter-spaced between the conventional metal LaIn3. The controlled reduction
of the dimensionality of the CeIn3 causes the suppression of magnetic order, and the
striking deviations from the standard Fermi-liquid low temperature electronic properties. Another interesting example is the behaviour of the upper critical field Hc2in CeCoIn5/
YbCoIn5 artificial superlattices [9]. The epitaxially grown superlattices are formed by
alternating n layers of the heavy fermion superconductor CeCoIn5 that displays a strong
Pauli effect, and a 5 unit cell thick normal metal YbCoIn5. By the Pauli effect (or
paramagnetic effect), we mean polarization of the electrons’ spins due to the presence of a magnetic field. In this dissertation, we use the terms Pauli effect and paramagnetic effect interchangeably. This effect is negligible in type-I superconductors. Such artificial superlattices display a smooth angular dependence of the upper critical field Hc2(θ) (θ = 0
is the parallel field with respect to the layers). However, when n is reduced to 3, a cusp appears in Hc2(θ), which is interpreted as a dominance of the orbital effect with respect to
the paramagnetic effect. By orbital effect, we mean the effects caused by the magnetic Lorentz force, such as the penetration of magnetic vortices in the sample. The suppression of the paramagnetic effect is attributed to local inversion symmetry breaking at the outer layers of CeCoIn5. In figure 1.3 we illustrate the cases of a n = 5 and n = 3 superlattice.
The red superconducting subsystem displays the features of local inversion symmetry breaking, which is indicated by the circles and crosses. In the n = 5 case, the only CeCoIn5
layer that fully retains inversion symmetry (and also mirror) is the middle layer, which is indicated by the open circles. The outer layers with a cross do not have inversion symmetry. The layers with a filled circle also break inversion symmetry, but in a lower degree. As is seen in the n = 3 case, the effects of local inversion symmetry breaking are more striking. At the outer layer, a Rashba spin-orbit coupling is induced (indicated by αR) [10]. This
splits the bands in the outer layers giving them a helical spin structure. The big novelty and advantage in these superlattices, is that the number of superconducting layers (unit cells) n is tunable. In other words, this allows one to experimentally tune the spin-orbit Rashba interaction and the nature of the pair breaking (Pauli or orbital) [11].
Superlattices and heterostructures The terms ”superlattice” and ”heterostructure” are often used interchangeably in the literature. In this dissertation we use superlattice for a periodic structure of layers of the same material. Each layer is a unit cell thick. On the other hand, we use heterostructure to refer to a periodic structure of layers of two (or more) materials. In the present case, the subsystem CeCoIn5(n) is a superlattice
of n layers, while the whole system CeCoIn5(n)/YbCoIn5(5) is a heterostructure.
In this dissertation’s second project, we study the n = 3 case with a perpendicular magnetic field in more detail. In particular, theory predicts a pair-density wave phase in the singlet order parameter at high magnetic fields [12]. Here we formulate this phase in the context of the phenomenological Ginzburg-Landau formalism, and investigate the role of orbital limiting on the pair-density wave phase.
Figure 1.3: Artificial CeCoIn5(n)/YbCoIn5(5) superlattices for n = 5 and n = 3. The
symmetry properties of the superconducting CeCoIn5(n) layers are indicated by the
symbols. Details are explained in the text.
1.4
Overview
In this dissertation’s first project, we investigate the temperature dependence of the superconducting gaps in Ba0.6K0.4Fe2As2. For single band conventional BCS
supercon-ductors, the gap’s temperature dependence is characterized by a coupling ratio around 2Δ(T = 0)/kBTc = 3.52. In Ba0.6K0.4Fe2As2, all gaps vanish at a single Tc, but the
coupling ratios are band dependent and range from weak coupling values of below 3.52 to strong coupling values of almost 8. We investigate if it is possible to reproduce the various coupling rations with a weak-coupling BdG theory.
In chapter 2 we introduce a real space Bogoliubov-deGennes (BdG) theory, where the bands communicate via an inter-site hybridization. With this, we then model the Ba0.6K0.4Fe2As2 system to address the temperature dependence of the superconducting
gaps. We show that the single particle inter-band scattering is sufficient to merge all gaps at a single Tc. Nevertheless, it is not possible to reproduce the experimental coupling
ratios using a constant attractive pairing potential as is used in BCS theory. Because of this, motivated on arguments from inhomogeneous superconductivity, we introduce a temperature dependent attractive potential derived from a Landau theory, which then allows us to reproduce the experimental coupling ratios. To do this we used exact diagonalization of the BdG Hamiltonian, which imposes severe restrictions on the system’s size and quantity of bands one can treat numerically. This is then circumvented using a powerful computational technique introduced in the next chapter.
In chapter 3 we explain the most important elements of the kernel polynomial method, which is a computational algorithm that allows to calculate spectral quantities of big matrices. Unlike exact diagonalization, which has a computation time that scales with the cube of the matrix size, the KPM method scales linearly. This allowed us to treat a bigger system with more bands. The basic idea is to expand the physical quantities of interest in
terms of orthogonal polynomials. Even tough it is an approximate method, convergence is stable and can be carried out to arbitrary precision.
In chapter 4 we then apply the kernel polynomial method to Ba0.6K0.4Fe2As2. However,
this chapter is not simply a reproduction of the same physics of chapter 2 using a new computational tool. Now, unlike chapter 2, we use a constant attractive potential. We observed that if one allows the electrons to redistribute among the bands with evolving temperature, then this might cause an increase in the band dependent coupling ratios. Postulating that this mechanism might occur, we show that using a constant attractive potential and a small redistribution of charge carriers between the bands, it is possible to reproduce the experimental coupling ratios measured in Ba0.6K0.4Fe2As2. Part I of this
dissertation proposes to distinct ways to reproduce the experimental coupling ratios of the Ba0.6K0.4Fe2As2 system within a weak coupling BdG theory.
In part II of this dissertation we investigate the exotic pair-density wave that is theoretically predicted to occur in Pauli limited multilayer superconductors. In real experiments, both Pauli and orbital limiting effects are expected to play important roles. The relative importance of the Pauli effect with respect to the orbital effect is measured by the Maki parameter. The guiding question is: is there a critical Maki parameter value for the survival of the pair-density wave phase? This is an ongoing project, and the question is not yet fully answered.
In chapter 5 we introduce a generalized BCS theory to treat unconventional super-conductors, that is, superconductors that have a non-trivial momentum dependence of the gap. We show that even if the underlying pairing potential is unknown, the system’s symmetries impose restrictions on the symmetries that the gap function is allowed to assume. A menu of possible gap structure arises from an underlying crystal group. In particular, if the system has an inversion center, then the superconducting wavefunction must be either a singlet or a triplet state, but not a superposition. If an inversion center is lacking, then group theory allows for a mixing between even singlet and odd triplet order parameters. At the end of the chapter we show how one can treat unconventional superconductivity in the context of Ginzburg-Landau theory.
Before one studies the effect of paramagnetic limiting on multilayer superconductors, it is a good idea to understand the influence of the Pauli effect on high κ type-II supercon-ductors, where vortices play the crucial role. For this reason, in chapter 6, we study the effect of paramagnetic limiting on the reversible magnetization curves of type-II super-conductors. Some powerful computational techniques exist that allow for the calculation of magnetization curves. We choose a method that still allows for some analytic insight. Therefore, we generalize a circular cell method applied to the superconducting vortex lattice to include the effect of paramagnetic limiting. We introduce the Maki parameter within Ginzburg-Landau theory and calculate the magnetization curves for various Maki parameters.
In chapter 7 we study the pair-density wave phase of a trilayer system. This has already been investigated using BdG theory. The pair-density wave phase depends on the concomitant action of a perpendicular external magnetic field, Rashba spin-orbit coupling and inter-layer hopping. We show that this phase can also be studied within a Ginzburg-Landau theory. The advantage of doing so is that vortex physics can be more easily included than in microscopic theories. We study the behavior of a single vortex threading the three layers in a Maki infinity system (no orbital effect). This is a useful toy model to later construct a full underlying vortex lattice within a circular cell method.
Part I
Temperature dependence of the
superconducting gaps in iron-based
CHAPTER
2
The multiband Bogoliubov deGennes model
2.1
Introductory remarks
In this chapter we develop a model to obtain multi-band Bogoliubov deGennes (mbBdG) equations, which is a generalization of the usual Bogoliubov deGennes (BdG) approach to multi-band systems. The motivation to do so is the three-bandgap structure observed in Ba0.6K0.4Fe2As2 [13] – a Fe-based superconductor frequently preferred to study universal
properties of Fe-based superconductors. This compound appears to have a three-gap structure, each gap belonging to a different band. Various strategies were available to formulate a mbBdG approach. A common approach used is to consider Cooper pair mixing between overlapping bands at the Fermi energy. A simpler single particle mixing approach was used by Troper et al [14, 15], and provided a suitable starting point to construct the mbBdG approach for the application to multi-bandgap structures in Fe-based superconductors.
We start presenting a generalized multi-band BdG Hamiltonian. Then we develop the exact diagonalization approach to the multi-band model. In the last section of this chapter, we apply this approach to explain the three-gap structure of Ba1−xKxFe2As2 compounds.
2.2
The model
We seek a model capable of treating real space inhomogeneities that also addresses the multi-gap structure of Fe-based superconductors. Also, the Hamiltonian should provide some insight into the inter-band interactions. Therefore, we propose a Hamiltonian of the
form
H = Hintra+Hinter, (2.1)
where the intra-band dynamics is modelled by Hintra = � �i�=j�,ν,σ tν(i, j) c†νσ(i)cνσ(j)− � i,ν,σ µν(i)nνσ(i) +� i,ν
Uν(i)nν↓(i)nν↑(i) +
�
ij,ν
Vν(i, j)nν↓(i)nν↑(j). (2.2)
The Greek letter ν is a band index. The tν(i, j) are intra-band hoppings between 2D
lattice sites i and j. The on-site potential µν(i) regulates the local density of electrons. We
also include the repulsive Coulomb term Uν(i) and an inter-site interaction Vν(i, j) that
– when attractive – gives rise to superconductivity. The creation operator c†νσ(i) creates
an electron at site i with spin projection σ (↑ or ↓) in band ν. The operator nνσ(i) is the
intra-band number operator.
The inter-band Hamiltonian in equation (2.2) includes the inter-band mixing of single particles (not Cooper pairs) at points where different bands overlap [14–16]. A more detailed justification of this term is given in the application section of this chapter. It is given by Hinter = � �i�=j� � µ�=ν � σ tµν(i, j) c†µσ(i)cνσ(j). (2.3)
A model such as (2.1) is useful when the multi-orbital structure of the material of interest is too complicated to account for. Instead, one concentrates on specific band properties, such as the superconducting gap, and models the complex multi-orbital contribution to a single band via the interaction (2.3).
In order to apply the BdG machinery to our model, we give equation (2.1) a mean-field treatment. After mean-field decoupling with no spin-flip, one can shorten the notation for the purpose of neat calculations. Then
HMF = � �i�=j�,µν,σ Tµν(i, j) c†µσ(i)cνσ(j) + � ij,ν � Δν(i, j)c†ν↑(j)c†ν↓(i) + h.c. � , (2.4) with Tµν(i, j) = tµν(i, j)− �µν(i)δµν, where
� µν(i) = � µν(i)− Uν(i) nν(i) 2 � . (2.5)
The term Uν(i)nν(i)/2 rescales the chemical potential due to the Coulomb repulsion and
is called a Hartree-shift [17]. Also, Δν(i, j) = Vν(i, j)�cν↓(i)cν↑(j)� and the band density
n(i) = �νnν(i). In summary, the Tµν(i, j) include: intra-band hoppings, inter-band
hoppings (non-local band-mixing between orbitals), the chemical potential, and the Hartree-shift due to Coulomb repulsion after mean-field decoupling. The hoppings (nearest neighbours, next-nearest neighbours, etc.) are constant, but the chemical potential µν(i)
will vary from site to site if nν(i) is inhomogeneous.
Finally, equation (2.4) is bilinear in the operators, and therefore is diagonalizable by a unitary transformation. The method we choose to do so is the self-consistent BdG formalism. According to it, we introduce the unitary Bogoliubov transformation that writes the site operators in terms of new quasi-particle operators γnσ, which are labelled
by the quantum number n associated with the quasi-particle excitation energy En.
2.3
Multi-band BdG exact diagonalization
In order to diagonalize the mean-field Hamiltonian (2.4), we introduce the unitary Bogoliubov-transformation for multi-band systems,
cνσ(i) = � n � unν(i)γnσ− sgn(σ)v∗nν(i)γn,−σ† � , (2.6)
where unν(i) and vnν(i) are probability amplitudes associated with the destruction and
creation of a quasi-particle fermion γnσ, respectively. The term sgn(σ) is the sign function
and accounts for the sign of the spin. The spin σ as usual can assume two possibilities: up↑ with a plus sign, and down ↓ with a minus sign. Note that the quasi-particles γnσ
do carry a spin index. We demand (this is possible for bilinear Hamiltonians) that the transformation (2.6) diagonalizes HMF into
HMFdiag =
�
nσ
Enγnσ† γnσ, (2.7)
where the eigenvalues of HMF
diag are measured from the ground state. A crucial aspect of
equation (2.7) is that it does not carry the band index ν. Quasi-particles γnσ are not
confined specific bands, but are overall entities of the system. In order to determine the amplitudes unν(i) and vnν(i) that diagonalize HMF, we write the Heisenberg equation of
motion for a conveniently chosen operator for both diagonalizedHMF
diag and non diagonalized
HMF Hamiltonians and compare the commutators. This determines the self-consistent
condition (characteristic of mean-field theories) that fixes unν(i) and vnν(i). Therefore we
compare � c1↓(i),HdiagMF � =�c1↓(i),HMF � , (2.8)
and extract the self-consistent condition for unν(i) and vnν(i). This ensures proper
time-savers: �γnσ,HMFdiag � = Enγnσ and � γ† nσ,HMFdiag �
= −Enγnσ† , where the anti-commutation
rule�γnσ, γmσ† �
�
= δnmδσσ� was used, which guarantees that quasi-particles are fermions.
Then, � c1↓(i),HdiagMF � =� n En � un1(i)γn↓− vn1∗ (i)γn†↑ � . (2.9)
Now we do the same withHMF using �c
µσ(i), c†νσ�(j) � = δijδσσ�δµν. This gives � c1↓(i),HMF � =� j,ν T1ν(i, j)cν↓(j)− � j Δ1(i, j)c†1↑(j). (2.10)
In order to compare this with (2.9) we must Bogoliubov transform it yielding � c1↓(i),HMF� = � j,ν T1ν(i, j) � n � unν(j)γn↓+ vnν∗ (j)γn↑† � −� j Δ1(i, j) � n � u∗n1(j)γn†↑− vn1(j)γn↓ � . (2.11)
Now we compare the coefficients of γn↓ and γn†↑ and we arrive at the multiband BdG
equations, which when solved determine the unν(i) and vnν(i) that diagonalize HMF:
Enun1(i) = � j,ν T1ν(i, j)unν(j) + � j Δ1(i, j)vn1(j); (2.12) Envn1(i) = � j Δ∗1(i, j)un1(j)− � j,ν T1ν∗(i, j)vnν(j). (2.13)
A similar set of equations are obtained for �c2↓(i),HMF
�
. The BdG equations (2.12) and (2.13) are usually put into matrix form as
En � unµ(i) vnµ(i) � =� j ��
ν(tµν(i, j)− ˜µν(i)δµν) Δµ(i, j)δµν
Δ∗ µ(i, j)δµν −�ν � t∗ µν(i, j) + ˜µν(i)δµν � � � unν(j) vnν(j) � . (2.14) Each eigenvalue En will have its correspondent eigenvector composed of unν(i) and
vnν(i), which determine the system’s properties. This is shown in the next section. The
present form of the multi-bands BdG equation, despite its similarity, should absolutely not to be confused with the multi-orbital BdG equations found in the literature [18].
2.4
Self-consistent physical quantities
Now we turn to the physical quantities that we will extract from this formalism: the inter-site gap function Δν(i, j), and the electron site density n(i), which is limited to the
interval [0, 2], since no more than two electrons with opposite spins can occupy a single lattice site. Since we deal with mean-fields, the gap function
Δν(i, j) = Vν(i, j)�cν↓(i)cν↑(j)�, (2.15)
is obtained by self-consistent computational convergence. To connect Δν(i, j) to the unν
and vnν’s, we Bogoliubov transform (2.15) assuming that thermal contractions such as
�γn↓γm↑� do not occur. Also, we take the chemical potential of quasi-particles to be µγ = 0,
that is � γnσ† γmσ � = δnm exp En kBT + 1 ≡ fnδnm, (2.16)
where kB is the Boltzmann constant, and fn is the Fermi distribution function. With this
one obtains
Δν(i, j) =
�
n
[unν(i)vnν∗ (j) (fn− 1) + unν(j)vnν∗ (i)fn] . (2.17)
This equation can be further simplified by using relations between the amplitudes unν and
vnν’s, which are fixed by the Fermion anti-commutation relations. For instance,
{cµσ(i), cνσ�(j)} = δijδσσ�δµν ⇒ unν(i)vnν∗ (j) = unν(j)vnν∗ (i). (2.18)
Using this on equation (2.17) we get Δν(i, j) =−Vν(i, j) � n unν(i)v∗nν(j) tanh � En 2kBT � . (2.19) The band density nν(i) =
�
σ�nνσ(i)� is determined from
nν(i) = 2 � n � |unν(i)|2fn+|vnν(i)|2(1− fn) � , (2.20)
where the factor of 2 arises due to spin degeneracy. This degeneracy would be lifted in the presence of a magnetic field. In the next section we apply this method to address the temperature dependence of the three superconducting gaps experimentally observed in Ba1−xKxFe2As2.
Figure 2.1: Left: The five 3d orbitals of iron. A minimal five-orbital model of Fe-based superconductors should include all five orbitals per lattice site. Right: Typical four-band structure tight-binding model of Fe-based superconductors plotted along high symmetry lines in the first Brillouin zone (adapted from reference [20]).
2.5
Effective three-band structure in Fe-based
super-conductors
The contents of this section is largely the same in reference [19].
2.5.1
Introduction
Here we present self-consistent calculations of the multi-gap temperature dependence measured in some Fe-based superconductors. These materials are known to have structural disorder in real space and a multi-gap structure due to the 3d Fe-orbitals contributing to a complex Fermi surface topology with hole and electron pockets. Different experiments identify three s-wave like superconducting gaps with a single critical temperature (Tc)
(Figure 2.2). We now investigate the temperature dependence of these gaps by the multi-band Bogoliubov-deGennes theory developed in the previous sections in this chapter. We model the intra-band gaps at different pockets in the presence of effective hybridizations between some bands and an attractive temperature dependent intra-band interaction. We show that this approach reproduces the three observed gaps and single Tc in different
compounds of Ba1−xKxFe2As2, providing some insights on the inter-band interactions.
First principle band theory calculations, such as local density approximation [21, 22] on Ba1−xKxFe2As2, predicted five bands of the Fe 3d orbitals across the Fermi surface forming
three hole-like Fermi surfaces centred at the zone center and two electron-like Fermi surfaces centred at the zone corner. In figure 2.3 we show a typical superconducting band dispersion and Fermi surface. This Fe-based superconductor (FeSC) revealed to be a highly
Figure 2.2: Three-gap temperature dependence observed in Ba0.6K0.4Fe2As2 [13] of the α
(red), β (blue) and γ (green) bands respectively observed by ARPES. The solid lines are BCS-like fits.
complex high critical temperature (Tc) superconducting (SC) material with multi-band
and multi-gap structure [13]. Furthermore, it also possess an unusual anisotropy in the ab-plane resistivity [23–25] just above Tc, possibly related to an electronic phase separation
transition [26] or a nematic phase [27].
What are Fermi Pockets? Pockets are enclosed hole or electron-like Fermi surfaces. If the states inside the pockets are empty – the band is above the Fermi surface – then it is said to be a hole pocket. If the enclosed states are occupied – the enclosed band is below the Fermi surface – then it is said to be an electron pocket. Electron pockets grow with electron doping, while hole pockets decrease simultaneously.
Experiments such as angle-resolved photoemission spectroscopy (ARPES) [13, 28, 29], point-contact Andreev reflection spectroscopy (PCAR) [30–32] and muon-spin rotation (µSR) [33] identify two or more nodeless s-wave like SC gaps. Here we want to address the classical ARPES experiment that identified nearly two coinciding gap structures in the α and γ, and another one in the large β pocket of Ba1−xKxFe2As2 with Tc = 38 K [13, 34].
Unlike than PCAR, ARPES is able to associate the gaps with specific pockets or bands in the Brillouin zone.
Earlier it was predicted by Suhl et al [35] that for the case in which two independent bands develop SC gaps with different values, two Tc’s exist. Nonetheless, a small amount
of interband scattering causes the two to merge at a single Tc. This result suggests a clear
interaction between the bands forming the hole and electron pockets in the Ba1−xKxFe2As2
system. An additional difficulty to treat this system is the multi-orbital character of the bands or pockets at the Fermi surface [3] instead of the single structure like the s-d superconductors [35].
Figure 2.3: Left: Fermi surface of the superconducting bands containing the α-hole pocket (red), β-hole pocket (blue), and γ-electron pocket (green), as obtained by tight binding-fits to ARPES band dispersions. Right: Correspondent band structure along the Γ→ M line in the first Brillouin zone.
The main point tackled here concerns the k-dependent hybridization among the multi-bands forming the pockets where SC gaps were measured by ARPES [13, 34]. Since the bands forming the pockets can be model by two [36] or more orbitals [3], we describe the inter-band scattering by the hybridization of these orbitals. This generates a k-dependent interaction arising from a nonlocal character of the mixing [15]. In the real calculations we consider a special case of a local constant multi-orbital hybridization between the α-β and β-γ pockets, representing an average over the Brillouin zone.
2.5.2
Multi-band superconducting theory
As mentioned, LDA calculations on the Ba1−xKxFe2As2 family yielded five multi-orbital
bands across the Fermi surface [21, 22]. On the other hand, ARPES experiments measured three s-wave like SC gaps at different pockets with quite different low temperature intensity and the same Tc [13, 34]. This behavior is likely to occur due to interband interactions
[35]. Thus, we rewrite the mean-field Hamiltonian (intra and inter terms) (2.4) as Hintra = � �i�=j� � ν=α,β,γ � σ tν(i, j)c†νσ(i)cνσ(j)− � i,ν=α,β,γ,σ ˜
µν(i)c†νσ(i)cνσ(i)
+ � ij,ν=α,β,γ � Δν(i, j)c†ν↑(j)c†ν↓(i) + h.c. � , (2.21)
Figure 2.4: Figure extracted from reference [34]. (a) Fermi pockets in the 2D Brillouin zone as observed by ARPES. (b) Grey LDA bands compared to tight-binding ARPES fits (dashed lines). One can see that α and β bands are hole pockets, while the γ forms an electron pocket at the Fermi surface. (c) Fermi surface as obtained from the tight-binding curves. (d) Measured and calculated Fermi velocities.
which contains the information of the three separate hole and electron pockets α, β and γ [13, 34]. The tν(i, j) are intra-band hoppings between lattice sites i and j up to second
nearest neighbors as derived by the ARPES band dispersion [34], based on their results and on the theory of magnetic excitation on a four orbital model [20]. We introduced a short-hand notation for the shifted chemical potential ˜µν(i) = µν(i)− Uν(i)nν(i)/2, which
includes: the local chemical potential µν(i), the on-site Coulomb repulsion Uν(i), that due
to the mean-field treatment just enters as a rescaling factor, and the band charge density nν(i). The spin index σ assumes either ↑ or ↓, and the creation and annihilation operators
obey the Fermi anti-commutation relation {cµσ(i), c†νσ�(j)} = δijδµνδσσ�.
The second term in equation (2.4) includes the inter-band contribution along the same lines of the theory developed by Kishore et al. [14, 16], and Caixeiro et al. [15], where the hybridization between overlapping bands near the Fermi surface is approximated by a constant (representing an average over the Brillouin zone) nearest neighbour hopping
Hinter= � i�=j � µ�=ν � σ tµν(i, j) c†µσ(i)cνσ(j). (2.22)
The band density nν(i) is self-consistently regulated by adjusting µν(i) until nν(i) converges
to some fixed value. Due to its local expression this approach can be applied to cases where there is some degree of disorder in the electronic density. The local SC band gap is Δν(i, j) = −V (T )�cν↓(i)cν↑(j)�, where �· · · � represents a thermal average. V (T ) is a
temperature dependent potential derived from a two-phase separated system. Using a typical Ginzburg-Landau free energy expansion it is possible to show that free energy barrier between the two phases is proportional to (TPS− T )2, where TPS is the phase
separation critical temperature. This approach has been applied to cuprates and in this case TPS is associated with the pseudogap temperature T∗ [37].
Phase separation
In superconducting Ginzburg-Landau (GL) theory with the superconducting wave-function ψ as the order parameter, the system’s free-energy is minimized at two specific symmetric values of ψ, separated by a free-energy wall of height proportional to (Tc−T )2.
A theory of electronic phase separation can be constructed is an analogous manner, by using the deviation of the charge carrier concentration as the order parameter in the GL expansion. With TPS being the onset temperature of phase transition, the domain
wall height separating the two regions with minimized free-energy (one with low and the other with high carrier concentration) is proportional to (TPS− T )2.
As mentioned, there is experimental evidence that the Ba1−xKxFe2As2 system may
Figure 2.5: The left panel shows the SC gaps on three non-interacting bands with the same band parameters of the right panel extracted from [13] and [34]. The big difference is the inter-band hopping. Bands 1 (red), 2 (blue) and 3 (green) have the same hoppings as the α, β and γ pockets respectively listed in table 2.1
observations, we take V (T ) =−|V0| � 1− T TPS �2 . (2.23)
The Hamiltonian defined by (2.4) is diagonalized by the unitary Bogoliubov transfor-mation cνσ(i) = � n � unν(i)γnσ− sgn(σ)v∗nν(i)γn,†−σ � , (2.24)
which leads to the multi-band Bogoliubov-deGennes (BdG) equations where tνν ≡ tν in
equation (2.14). The full BdG matrix has dimension 6N2× 6N2 in the three-band case,
where N × N is the lattice’s dimension. The positive quasi-particle excitations {En} are
used to self-consistently calculate the temperature dependent local s-wave gaps Δν(i) for
each band using equation (2.19), and the local band density nν(i) = �σ
� c†
νσ(i)cνσ(i)
� according to equation (2.20). Equation (2.19) allows for the simultaneous determination of Δα(T ), Δβ(T ), and Δγ(T ).
A typical solution of equation (2.19) with a single constant attractive potential for all three bands with independent band dynamics (no inter-band hoppings) is shown on the left panel of figure 2.5. The intra-band hoppings are those derived from Ding et al [34] that reproduces the three α, β and γ pockets and are listed in table 2.1. Notice that lower intra-band hoppings or lower kinetic energies yield bigger gaps and larger Tc’s. However,
as predicted by Suhl et al. [35] for two bands, non-zero inter-band hoppings modify the three-gap structure drastically as shown on the right panel of figure 2.5. The crucial
Hoppings (meV) tα tβ tγ tαβ tαγ tβγ
First 160 13 380 165 0 140 Second -52 42 80 0 0 0
Table 2.1: First and second nearest neighbour intra tν and inter-band hoppings tµν in meV
used in the present calculations. Intra-band hoppings were extracted from ARPES and inter-band hoppings closely agree with band dispersion [34]. The parameters of this table are used for all three compound discussed in this paper.
Compound Tc (K) TPS (K) |V0| (meV)
Ba0.6K0.4Fe2As2 38 165 400
Ba1−xKxFe2As2 32 130 360
Ba1−xKxFe2As2 23 80 300
Table 2.2: Phenomenological values of the electronic phase transition temperature TPS
and potential |V0| in equation (2.23) used to model Ba1−xKxFe2As2 with different Tc’s.
difference in the second case is a single value for Tc for all three bands, in accordance with
experimental observations [13, 28–32].
2.5.3
Temperature dependence of the three-gap structure
Now we apply multi-band BdG theory to model three compounds of Ba1−xKxFe2As2 with
Tc = 23, 32 and 38 K to reproduce the SC gaps measured by different techniques. PCAR
techniques [31, 32] find two SC bands, whereas ARPES [13] detects SC gaps in three bands or pockets in the Brillouin zone. It is possible that the bigger gap observed by PCAR may correspond to the α and γ bands from ARPES, since they have about the same intensity.
For each compound we solve a three-band 6N2 × 6N2 BdG matrix in a square lattice
with periodic boundary conditions. Following the LDA calculations [21, 22] and the derived bands by ARPES [34] we can obtain an estimation of the inter-band hoppings shown in the first three columns of table 2.1. For instance, there is negligible overlap between and no intersection between α and γ and so we take tαγ = 0 and larger inter-band hoppings to
reflect band intersections between the α and β and β and γ pockets [34, 38], which are shown in the last three columns of table 2.1.
To model the experimental results we perform calculations with the hopping parameters listed in table 2.1, used for all three compounds. The charge distribution in each pocket (nα, nβ, nγ) = (0.192, 0.084, 0.12) are proportional to specific Fermi areas [38], and Fermi
velocities [13, 33]. Even if the system has a disordered electronic density, the average gap is the same of that of a homogeneous one with the same average density. Furthermore, ARPES and PCAR measure only the average gap.
To obtain the SC gaps of different Ba1−xKxFe2As2 compounds with distinct Tc’s we
used the band parameters discussed before and estimate the two-body attractive potential in equation (2.23) as listed in table 2.2. The Δν(T ) results for the compound with Tc = 23
Figure 2.6: On the left panel we show the theoretical BdG curves compared to PCAR [31, 32] for Ba1−xKxFe2As2 with Tc = 23 K. PCAR identifies two SC bands that may
correspond to the nearly coinciding α and γ pocket identified by ARPES. On the right panel we show the BdG curves compared to the three bands from ARPES [28] for the compound with Tc = 32 K. Reference [28] only estimates gap values for the β pocket.
K is depicted on the left panel of figure 2.6. The coloured solid lines are the theoretical BdG curves obtained from (2.19). We observe that PCAR [31, 32] (black triangles) scales more closely with the BdG α/γ bands. In the case of the compound with Tc = 32 K
we show also experimental points from ARPES [28] in comparison to the theoretical BdG curves on the right panel of figure 2.6. Experimental points for the β bands were unavailable and estimated under 4 meV [28], consistent with the blue β BdG curve.
The results for Ba0.6K0.4Fe2As2 with Tc = 38 K are shown in figure 2.7, with all bands
(left panel) and with each band shown separately for clarity. Two different sets of ARPES points are displayed denoted by ARPES1 referring to [13] and ARPES2 according to [39]. It is interesting to note that the measured band’s correspondent SC gaps are reproduced by a temperature-dependent attractive potential within a three-band BdG theory. This is indicated by the grey curves in figure 2.7, which correspond to the case where a constant potential that gives the same Δν(T = 0) was used.
Despite the real and k-space complexity of the system, we obtain a reasonable agreement with the experimental values, using a single attractive potential (equation (2.23)) for the three bands for each compound. The misfit of the experimental points with respect to the theoretical BdG curves may be due to differences in the tight binding hoppings and, more importantly, in the finite size used in the calculations with respect to the real system.
Our matrix is only 384× 384 and it is solved by exact diagonalization. New calculations using a more powerful method [40, 41] that allows investigation of much larger arrays are introduced in the next chapter. However, we believe that the essential properties of these compounds are captured by the present calculations.
2.6
Concluding remarks
We have discussed the evidence of electronic disorder complexity in real and k-space of the Ba1−xKxFe2As2 compounds. A possible electronic phase separation or a nematic
order together with the multi-orbital band structure make this system quite unusual. As expected the description to its superconducting multigap properties is not a simple task, unlike MgB2 with two bands and two conventional BCS-like gaps [42].
The superconducting multigap structure revealed by ARPES [13, 34] with three different gaps on distinct locations of the Brillouin zone, but with a single Tc, in light of the two
band theory of Suhl et al [35], is an indication of inter-band interaction or hybridization. This was confirmed by self-consistent calculations with the BdG theory with inter-band hoppings. These hoppings describe phenomenologically the interaction between bands, mediated by the hybridized iron orbitals.
The anisotropy in real space was also taken into consideration. In analogy with the cuprate superconductors, which displayed many indications of charge disorder [43], we also use the observed charge disorder or nematic order as the origin of a two-body attractive pair potential with a temperature dependence derived from a GL theory.
We conclude that the multi-band scattering via multi-orbital hybridizations and in plane charge disorder are well known important properties of high complexity of these materials that are essential to quantify the pairing amplitudes and the overall superconducting properties.
In this chapter we showed that a phenomenological introduction of a temperature dependent pairing potential, allows for the reproduction of the coupling ratios in iron-based superconductors. In chapter 4 we propose another alternative – a small redistribution of electronic band population – as a possible mechanism for steeper drops in Δν(T ).
Figure 2.7: Temperature dependence of (a) the jointly obtained SC gaps Δν(T ) from
multi-band BdG theory (solid lines) in comparison to references [13] (empty points) and [39] (solid points) for Ba0.6K0.4Fe2As2 with Tc = 38 K and N = 8. We show the three
sets of data corresponding to (b) ν = α, (c) ν = β, and (d) ν = γ separately for clarity. The grey curves correspond to the case of a constant potential, which is included for comparison.
CHAPTER
3
The kernel polynomial method
3.1
Introductory remarks
In this chapter we introduce an efficient numerical approach capable of retiring the Bogoliubov deGennes method of the previous chapter. Here we present a modern and stable algorithm, very useful for the calculation of spectral quantities and correlation functions; a key aspect in condensed matter physics. The kernel polynomial method (KPM), as we shall see, is characterized by a resource consumption that scales linearly with the problem’s dimension. Therefore, this method has gained popularity in the fields of disordered systems, strongly correlated electrons, electron-phonon interaction, and quantum spin systems [41].
Most physical properties depend on the distribution of the eigenenergies of the Hamil-tonian. If the matrix representation of the Hamiltonian has dimension D, then finding its eigenenergy distribution is traditionally achieved via matrix diagonalization, whose computation time scales with D3. This imposes severe restrictions on the systems sizes
of interested, and one may loose interesting physics that emerge in bigger systems. The triumphant aspect of the KPM algorithm is that it escapes from diagonalization by redirect-ing the most time-consumredirect-ing process to matrix-vector multiplications – and computation time scales with D.
This chapter is structured as follows. In the first section the basics of orthogonal Chebyshev polynomials is introduced. In the second section we develop on the central aspect of the KPM method – the moments of the expansion series. In section three we present computational details that are indispensable to guarantee the methods efficiency, such as fast Fourier transforms and the stochastic evaluation of traces.
3.2
Basic concepts
We begin discussing the basic features of the KPM method. The contents of this chapter are largely found in reference [41]. The basic idea is to expand a function of interest f (x) in orthogonal polynomials. We therefore write a scalar product of two continuous functions f (x) and g(x) defined on the interval [a, b] as
�f|g� = � b
a
dx w(x)f (x)g(x), (3.1) where w(x) is a weight function defined on the same interval. With respect to each choice of w(x) there exists a complete set of orthogonal polynomials{pn(x)} that fulfill
�pn|pm� = δnm�pn|pn�. (3.2)
This allows for the expansion of a generic function f (x) (shortly to be the Green’s function G(ω)) in terms of the orthogonal polynomials pn(x):
f (x) = ∞ � n=0 µnpn(x), with µn = �f|pn� �pn|pn� . (3.3)
The determination of f (x) translates into calculating the set of expansion coefficients{µn},
which in the KPM context are called moments, as inspired by the statistical distribution of moments. All sorts of orthogonal polynomials can be used. We chose the Chebyshev polynomials because of their good convergence properties, recursive definition, relation to Fourier transforms, and their mainstream application in the literature [40, 41, 44–48].
There are two sets of Chebyshev polynomials: the ones of the first kind Tn(x), and
of the ones of the second kind Un(x). They are defined on the interval [a, b] = [−1, 1],
which is imposed by their weight functions. The polynomials of the first kind Tn(x) are
orthogonal with respect to the weight function w1(x) =
1
π√1− x2, (3.4)
and the polynomials of the second kind Un(x) are orthogonal with respect to
w2(x) = π
√
1− x2. (3.5)
Evidently, both weight are related to each other by their inverse. Using the change of variable x = cos ϕ one can verify that
Tn(x) = cos nϕ and Un(x) =
sin [(n + 1)ϕ]
