Understanding Novel Superconductors withAb Initio Calculations

Lilia Boeri

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 A Brief History of Research in Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1 A Short Compendium of Superconductivity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Ab Initio Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Developments in Related Fields: Ab Initio Material Design . . . . . . . . . . . . . . . . . . . . 11

4 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.1 Conventional Superconductors: Search Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Dormant ep Interaction in Graphite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Magnesium Diboride and Other Covalent Superconductors . . . . . . . . . . . . . . . . . . . . 184.4 Intercalated Graphites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.5 High-Tc Conventional Superconductivity in High-Pressure Hydrides . . . . . . . . . . . . 224.6 Unconventional Superconductivity in Fe Pnictides and Chalcogenides . . . . . . . . . . . 25

5 Outlook and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Abstract

This chapter gives an overview of the progress in the field of computationalsuperconductivity. Following the MgB2 discovery (2001), there has been animpressive acceleration in the development of methods based on density func-tional theory to compute the critical temperature and other physical properties ofactual superconductors from first principles. State-of-the-art ab initio methodshave reached predictive accuracy for conventional (phonon-mediated) super-conductors, and substantial progress is being made also for unconventionalsuperconductors. The aim of this chapter is to give an overview of the existing

L. Boeri (�)Dipartimento di Fisica, Sapienza Università di Roma, Roma, Italye-mail: lilia.boeri@uniroma1.it

© Springer International Publishing AG, part of Springer Nature 2018W. Andreoni, S. Yip (eds.), Handbook of Materials Modeling,https://doi.org/10.1007/978-3-319-50257-1_21-1

1

2 L. Boeri

computational methods for superconductivity and present selected examples ofmaterial discoveries that exemplify the main advancements.

1 Introduction

The aim of this chapter is to offer an up-to-date perspective on the field of abinitio superconductivity and of the related development of numerical methods tocompute critical temperatures and other physical properties of superconductors.The material-specific aspect is what distinguishes ab initio (= from first principles)approaches, based on density functional theory (DFT) and its extensions, from othertheoretical approaches to superconductivity, which mainly focus on the generaldescription of the phenomenon. The typical questions addressed by computationalsuperconductivity are as follows: (i) What makes a certain compound a good (orbad) superconductor? (ii) How are its properties modified by external parameters,such as doping, pressure, and strain? (iii) Is it possible to find new materials withimproved superconducting properties compared to existing ones?

The most relevant parameter that defines the performance of a superconductor forlarge-scale applications is its critical temperature (Tc): this means that addressingthe questions above requires the development of methods accurate enough to predictthe Tc of a superconductor and its dependence on external parameters. The progressin this direction, in the last 20 years, has been impressive.

For a large class of superconductors, i.e., conventional, phonon-mediated ones,ab initio methods are now so accurate that the focus of the field is graduallyshifting from the description of existing superconductors to the design of newmaterials. The first successful example was the prediction of high-Tc conventionalsuperconductivity in SH3 (2014) (Drozdov et al. 2015a; Duan et al. 2014).

For unconventional superconductors, which comprise two of the most studiedclasses of materials, the high-Tc cuprates (Bednorz and Mueller 1986) and Fe-based superconductors, (Kamihara et al. 2008) ab initio approaches are still far frombeing predictive, but it is becoming more and more widely accepted that the single-particle electronic structure determines crucial properties of these materials, such asthe symmetry of the superconducting gap and the behavior of magnetic excitations.

The topics and the structure of this chapter have been specifically thoughtto illustrate the parallel progress in ab initio methods and material research forsuperconductors. I have chosen three discoveries that I consider the fundamentalmilestones of this process: (a) the report of superconductivity in MgB2 in 2001,which has disproved the Cohen-Anderson limit for conventional superconduc-tors; (b) the discovery of Fe-based superconductors, which has led to a muchdeeper understanding of the interplay between electronic structure, magnetism, andsuperconductivity in unconventional superconductors; and (c) the discovery of high-temperature superconductivity at megabar pressures in SH3, which has given aspectacular demonstration of the predictive power of ab initio calculations.

Although I will give a general introduction to the theory of superconductivity andbriefly describe the most recent advancements in ab initio methods, methodological

Understanding Novel Superconductors with Ab Initio Calculations 3

developments are not the main topic of this chapter: I refer the interested readerto excellent reviews in literature for a detailed discussion (Giustino 2017; Sanna2017; Sanna et al. 2018). For space reasons, I am also forced to leave out somecurrently very active directions of superconductivity research, such as topologicalsuperconductivity (Sato and Ando 2017) superconductivity in 2D transition metaldichalcogenides (Klemm 2015), and artificial superlattices (Boschker and Mannhart2017), and other more traditional topics, such as cuprates and other oxides (Chuet al. 2015), fullerenes (Gunnarsson 1997), layered halonitrides (Kasahara et al.2015), etc.

On the other hand, I have included at the end a short perspective describingpossible routes to high-Tc superconductivity which exploit novel developments inexperimental and ab initio techniques, since I believe that in the next years, thecombination of the two may lead to the discovery of many new superconductors.

The structure of the chapter is the following: I will start by giving a concisehistorical review of the most important discoveries in Sect. 2. In Sect. 3, I will thenintroduce the basic concepts of superconductivity theory and describe the mostrecent developments in ab initio methods. The main body of the paper is containedin Sect. 4, where, using selected material examples, I will try to give an impressionof the rapid progress of the field in the understanding of both conventional andunconventional superconductors. Finally, in Sect. 5, I will propose possible practicalroutes to high-Tc superconductivity.

2 A Brief History of Research in Superconductivity

Superconductivity was discovered more than 100 years ago when H.K. Onnesobserved that, when cooled below 4 K, mercury exhibits a vanishing resistivity(Onnes 1913). Perfect diamagnetism, which is the second fingerprint of a supercon-ductor, was discovered by Meissner and Ochsenfeld around 20 years later (Meissnerand Ochsenfeld 1933). While it was immediately clear that superconductors couldhave an enormous potential for applications, the low critical temperatures repre-sented an insurmountable obstacle to large-scale applications.

In addition to presenting practical problems, superconductivity proved to be amajor challenge also for theorists: fully microscopic theories of superconductivity –the Bardeen-Cooper-Schrieffer (BCS) and Migdal-Eliashberg (ME) theories – weredeveloped only after almost 50 years after the original discovery (Bardeen et al.1957; Migdal 1958; Eliashberg 1960; Allen and Mitrovic 1982; Scalapino 1969;Carbotte 1990). They describe superconductivity as due to the condensation of pairsof electrons of opposite spin and momentum (Cooper pairs), held together by anattractive glue. In conventional superconductors, the glue is provided by phonons(lattice vibrations), but other excitations such as plasmons, spin fluctuations, etc.can also mediate the superconducting pairing. We will treat here only the case ofthe so-called boson-exchange superconductors, and not other mechanisms, such asresonant valence bond, hole superconductivity, etc..

4 L. Boeri

The understanding of the microscopic mechanism of superconductivity did notlead to any immediate, appreciable progress in the search for new superconductors;this translated into a general skepticism toward theory, which is well exemplifiedby one of Matthias’ rules for superconductivity (stay away from theorists!). Indeed,rather than a predictive theory, ME theory was long considered a sophisticated phe-nomenological framework to describe existing superconductors, while the search fornew materials was (unsuccessfully) guided by semiempirical rules. Even worse, twoleading theorists used ME theory to demonstrate the existence of an intrinsic limitof around 25 K to the Tc of conventional superconductors. Although conceptuallywrong, the Cohen-Anderson limit is still cited today as an argument against high-Tc

superconductivity (Cohen and Anderson 1972).The notion of an upper limit to Tc was first challenged by the discovery of the first

unconventional superconductors, the cuprates, in 1986 (Bednorz and Mueller 1986).In contrast to conventional superconductors, which above Tc behave as ordinarymetals, cuprates exhibit a complex phase diagram, with many coexisting phases andphysical phenomena (charge and spin density waves, metal-insulator transitions,transport anomalies, etc.). In 1987 a cuprate, YBCO, broke the liquid N2 barrier,with a Tc of 92 K (Wu et al. 1987), causing a general excitement in the mediaabout a possible superconducting revolution; the highest Tc ever attained in thisclass is 156 K (Schilling et al. 1993). Despite almost 30 years of research and manydifferent proposals, a quantitative theory of superconductivity in the cuprates is stilllacking; furthermore, their large-scale applicability is also limited due to their highbrittleness and manufacturing costs (Gurevich 2011).

The search for new materials took a different turn at the beginning of this century,when a Tc of 39 K was reported in a simple s–p binary compound, magnesiumdiboride (MgB2) (Nagamatsu et al. 2001). In contrast to the cuprates, MgB2 is aconventional superconductor. In less than 2 years, ab initio calculations provided akey quantitative understanding of very specific aspects of superconductivity in thismaterial, such as two-gap superconductivity, anharmonicity, role of magnetic andnonmagnetic impurities, doping, etc. (An and Pickett 2001; Kong et al. 2001; Kortuset al. 2001; Liu et al. 2001; Choi et al. 2002; Mazin et al. 2002). This stimulateda renewed enthusiasm in the search for new superconductors and, in parallel, thedevelopment of accurate ab initio methods to model them.

Indeed, in the last 17 years, superconductivity has been discovered inB-doped semiconductors (Ekimov et al. 2004; Bustarret et al. 2006), intercalatedgraphites (Weller et al. 2005), unconventional Fe-based superconductors (Kamiharaet al. 2008), and, finally, high-pressure hydrides (Drozdov et al. 2015a,b). TheTcs as a function of the year of their discovery are shown as colored symbols inFig. 1. The most important methodological developments are reported on the topaxis: superconducting density functional theory (Lüders et al. 2005; Marques et al.2005), electron-phonon interaction with Wannier functions (Giustino et al. 2007a;Calandra et al. 2010; Giustino 2017), ab initio anisotropic ME theory (Sannaet al. 2018; Margine and Giustino 2013), (stochastic) self-consistent harmonicapproximation (Errea et al. 2014), and ab initio spin fluctuations (Essenberger et al.2014, 2016).

Understanding Novel Superconductors with Ab Initio Calculations 5

2000 2004 201620122008

year

0

50

100

150

200

250

300

Tc(K)

SCDFT(1)

EP + Wan

(1)

EP + Wan

(2)

Anis. M

E

SCDFT(2)

SSCHA

MgB2

B-diam. CaC6 SrC6 BaC6LaOFeAs

SmOFeAs

BaFe2As2

FeSe@STOPH3

SH3

Liquid N2

0 C

cuprates

Fig. 1 Main developments in the field of superconductivity in the twenty-first century: the Tcsof the most important experimental discoveries as a function of year are shown as coloredsymbols. The top axis reports the most important methodological developments in ab initiosuperconductivity: superconducting density functional theory (SCDFT(1)) (Lüders et al. 2005;Marques et al. 2005), electron-phonon interaction with Wannier functions (EP+WAN(1) andEP+WAN(2)) (Giustino 2017; Giustino et al. 2007a; Calandra et al. 2010), ab initio anisotropicME theory (Anis. ME) (Margine and Giustino 2013; Sanna et al. 2018), (stochastic) self-consistentharmonic approximation (SSCHA) (Errea et al. 2014), ab initio spin fluctuations (SCDFT(2))(Essenberger et al. 2014, 2016)

Thanks to these advancements, ab initio calculations for conventional supercon-ductors have now reached an accuracy which gives them fully predictive power.Methods are being developed also to treat other types of interactions, such asplasmons and spin fluctuations (Essenberger et al. 2014, 2016; Akashi and Arita2013), and parameter ranges where the standard approximations of strong-couplingME theory break down (Grimaldi et al. 1995). Combined with the developmentof efficient methods for ab initio crystal structure prediction and the progress insynthesis and characterization techniques (Woodley and Catlow 2008; Zhang et al.2017), this opens unprecedented possibilities for material discovery.

3 Methods

In this section, I describe the methodological background of computational super-conductivity. The first part introduces the main concepts behind the microscopictheories of superconductivity, i.e., the early Bardeen-Cooper-Schrieffer (BCS)

6 L. Boeri

theory and the strong-coupling Migdal-Eliashberg (ME) theory. Although extremelyaccurate and elegant, ME theory was for a long time employed only as a semi-phenomenological theory, relying on electronic and vibrational spectra extractedfrom experiments. Early attempts of obtaining these quantities from density func-tional theory date back to the early 1980s, but with limited success, due toinadequate computational resources and insufficient accuracy of methods to treatphonons and electron-phonon (ep) interaction (Gaspari and Gyorffy 1972; Changet al. 1985). The required accuracy was only achieved with density functionalperturbation theory (DFPT) (Savrasov and Savrasov 1996; Baroni et al. 2001)and recently substantially increased with Wannier function interpolation methods(Giustino et al. 2007a; Calandra et al. 2010; Marzari et al. 2012).

In a very influential paper, already in 1996, Savrasov and Savrasov demon-strated that linear response calculations combined with Migdal-Eliashberg theorycould reproduce the critical temperatures and other properties of elemental metals(Savrasov and Savrasov 1996). However, since at that time the Tcs of knownconventional superconductors were much smaller than those of the cuprates, thisresult was erroneously perceived as of limited importance.

The report of superconductivity in MgB2 in 2001 gave a strong impulse tothe development of ab initio methods for superconductors, which resulted in twoparallel lines of research: ab initio Migdal-Eliashberg theory and superconductingdensity functional theory, described in Sect. 3.2. In their fully anisotropic versions,including screened Coulomb interactions, they have a comparable accuracy of5–10% on the critical temperature, gap, etc. (Sanna 2017; Sanna et al. 2018;Margine and Giustino 2013). This gives them fully predictive power, and, combinedwith methods for crystal structure prediction, offers the unprecedented possibilityof designing superconductors ab initio, overcoming the practical limitations ofexperiments.

3.1 A Short Compendium of Superconductivity Theory

The first fully microscopic theory of superconductivity was formulated by Bardeen,Cooper, and Schrieffer in 1957 and is known as BCS theory (Bardeen et al. 1957).

BCS theory describes the transition of superconductors from an ordinary metallicstate (normal state) to a new state, characterized by vanishing resistivity and perfectdiamagnetism. In this superconducting state, the electronic spectrum develops agap Δ around the Fermi level, which is maximum at zero temperature and vanishesat the critical temperature Tc. The critical temperature exhibits an isotope effect,i.e., Tc increases (or decreases) upon partial replacement of an element with alighter (heavier) isotope. Isotope effects are also measured for other characteristicproperties of superconductors (gap, specific heat, etc.).

BCS theory reconciles all the above experimental observations in a consistentframework, based on three key concepts:

Understanding Novel Superconductors with Ab Initio Calculations 7

(i) A Fermi sea of electrons in the presence of an attractive interaction is unstabletoward the formation of a pair of electrons with opposite spin and momentum(Cooper pair), which effectively behaves as a boson. In a superconductor,below Tc, a small but macroscopic fraction of electrons, of order Δ/EF �10−3, forms Cooper pairs – this is sometimes referred to as the condensatefraction of a superconductor.

(ii) A variational many-body wavefunction for the electrons is constructed froma superposition of ordinary single-particle states and Cooper pairs (BCSwavefunction). The existence of a condensate fraction leads to the appearanceof a gap in the electronic spectrum εk, which satisfies the self-consistentequation:

Δk = −1

2

∑

k,k′

Vk,k′Δk′√

ε2k + Δ2

k

· tanh

⎛

⎝

√ε2

k + Δ2k

2T

⎞

⎠ . (1)

iii) Using a simple model for the electron-electron interaction Vk,k′ , the so-calledBCS potential, which is attractive only if the two electrons with wavevectork, k′ both lie in a small energy shell ωD around the Fermi energy EF , Eq. 1 canbe solved analytically, and the gap and Tc are given by:

Δ(T = 0) � 2ωD exp

(− 1

N(EF )V

), kbTc=1.13ωD exp

(− 1

N(EF )V

),

(2)

The original idea of Bardeen, Cooper, and Schrieffer is that the attractiveinteraction V between electrons is mediated by lattice vibrations (phonons). In thiscase, ωD is a representative phonon energy scale, such as the Debye frequency;N(EF ) is the electronic density of states at the Fermi level.

One of the first successes of BCS theory has been the explanation of the isotopeeffect on Tc, αTc = − d ln(Tc)

d ln(M)= 0.5, where M is the ionic mass, as well as the

prediction of several magic ratios, satisfied in most elemental superconductors: themost famous is probably the ratio 2Δ(0)

kBTc= 3.53.

However, BCS theory is valid only at weak-coupling (λ = N(EF )V < 0.2 −0.3) and instantaneous interactions; these assumptions are not verified in manyconventional superconductors, where the actual values of Tc, isotope effect, magicratios, etc. are spectacularly different from the BCS predictions (Marsiglio andCarbotte 1986).

A quantitative description of the strong-coupling, retarded regime is given bythe many-body ME theory, based on a set of self-consistent coupled diagrammaticequations for the electronic and bosonic propagators (Allen and Mitrovic 1982;Scalapino 1969; Carbotte 1990; Margine and Giustino 2013). The bosons thatmediate the superconducting pairing can be phonons or other excitations ofthe crystal, such as plasmons or spin fluctuations (Berk and Schrieffer). Below

8 L. Boeri

the critical temperature, electrons are described by a normal and an anomalouselectronic propagator, the latter accounting for Cooper pairs. ME equations are thenobtained from Dyson’s equations for the normal and anomalous propagators; in theirmost commonly used, T -dependent form, they can be written as:

Z(k, iωn) = 1 + πT

ωn

∑

k′n′

δ(εk′)

N(EF )

ωn′√

ω2n′ + Δ2(k, iωn′)

λ(k, k′, n − n′) (3)

Z(k, iωn)Δ(k, iωn) = πT∑

k′n′

δ(εk′)

N(EF )

Δ(k′, iωn′)√ω2

n′ + Δ2(k, iωn′)2×

× [λ(k, k′, n − n′) − μ(k − k′))

](4)

where Z(k, iωn) and Z(k, iωn)Δ(k, iωn) are the self-energy of the normal andanomalous electronic propagators, respectively; iωn = i(2nπT + 1) are Matsubarafrequencies, and k, k′ are the electronic momenta; and the δ function restricts thesum over k’ only to electronic states at the Fermi level. The electrons interactthrough a retarded attractive interaction λ(k, k′, n − n′) and an instantaneousCoulomb repulsion μ(k − k′). The interaction λ(k, k′, n − n′) is usually expressedin terms of an electron-boson spectral function α2F(k, k′, ω) as :

λ(k, k′, n − n′) =∫ ∞

0dω

2ω

(ωn − ωn′)2 + ω2α2F(k, k′, ω) (5)

Equations (3) and (4) can be solved numerically to obtain the gap and otherthermodynamic quantities. It is very common, and in most cases sufficientlyaccurate, to approximate the more general form with an isotropic version, replacingthe sums on the electronic momenta (k, k′) with Fermi surface averages. If oneis only interested in the Tc, there are excellent approximation formulas; a verypopular choice for phonon-mediated superconductors is the McMillan-Allen-Dynesexpression (Allen and Dynes 1975):

Tc = ωlog

1.2kB

exp

[− 1.04(1 + λ)

λ − μ∗(1 + 0.62λ)

], (6)

where λ = 2∫

dωα2F(ω)

ωand ωlog = exp

[2λ

∫dωω

α2F(ω) ln(ω)]

are the ep

coupling constant and logarithmic averaged phonon frequency, respectively; μ∗is the so-called Morel-Anderson pseudopotential, obtained by screening the fullCoulomb potential up to a characteristic cutoff energy (Morel and Anderson 1962).

Understanding Novel Superconductors with Ab Initio Calculations 9

3.2 Ab Initio Methods

The two methods described in this section, ab initio anisotropic Migdal-Eliashbergtheory (DFT-ME in the following) (Margine and Giustino 2013; Sanna et al. 2018)and superconducting density functional theory (SCDFT) (Oliveira et al. 1988;Lüders et al. 2005; Marques et al. 2005), represent a fundamental step forward in thestudy of actual superconductors, because they permit to obtain a full characterizationof the normal and superconducting state of a system from the sole knowledge of thechemical composition and crystal structure. Although there are fundamental andpractical differences between the two, both methods rely crucially on the ability ofDFT of providing accurate electronic and bosonic spectra for most materials at anaffordable computational cost (Jones and Gunnarsson 1989; Baroni et al. 2001).

The basic assumptions are the following:

1. Electronic quasiparticles appearing in Eqs. (3) and (4) are replaced by the Kohn-Sham quasiparticles.

2. Bosonic excitation energies and electron-boson spectral functions are obtainedfrom density functional perturbation theory (DFPT) (Baroni et al. 2001).

For phonons, the spectral function is:

α2Fph(k, k′, ω) = N(EF )∑

ν

gkk′,νδ(ω − ωk−k′,ν), (7)

where gk,k′ =⟨k′|δV ν,k′−k

scf |k⟩

is the ep matrix element for the mode ν and δVν,qscf

is the variation of the Kohn-Sham self-consistent potential due to an infinitesimaldisplacement along the eigenvector of the phonon mode ν with wavevectorq = k′ − k.

For spin fluctuations, the spectral function is proportional to the imaginarypart of the longitudinal interacting spin susceptibility χzz(q = k−k′, ω)(Vignaleand Singwi 1985), which, using linear response within the time-dependentdensity functional theory (TDDFT) framework (Runge and Gross 1984), can bewritten as:

χzz(q, ω) = χKS(q, ω)

1 − fxc(q, ω)χks(q, ω), (8)

where χKS(q, ω) is the Kohn-Sham susceptibility and fxc(q, ω) is the exchangeand correlation kernel (Essenberger et al. 2012).

3. The Coulomb potential μ(k−k′) which, in most empirical approaches, is treatedas an adjustable parameter within the Morel-Anderson approximation (Moreland Anderson 1962), is treated fully ab initio by screening the bare Coulombpotential within RPA. Substantial deviations from typical values of μ∗ = 0.1 −−0.15 are found in strongly anisotropic systems such as MgB2 and layeredsuperconductors; in some cases, such as alkali metals at high pressure, the effect

10 L. Boeri

of Coulomb interactions is even stronger, giving rise to plasmonic effects (Akashiand Arita 2013).

Once the spectra of the quasiparticles and the interactions between them areknown from first principles, DFT-ME or SCDFT can be applied to describe thesuperconducting state. DFT-ME theory amounts to solving the fully anisotropic MEequations, for electronic and bosonic spectra computed in DFT; the current imple-mentations solve the equations in Matsubara frequencies and use Padé approximantsto continue them to real space. The obvious advantage of this method is that allquantities have an immediate physical interpretation through many-body theory.

SCDFT is a fundamentally different approach, which generalizes the originalHohenberg-Kohn idea of one-to-one correspondence between ground-state densityand potential (Hohenberg and Kohn 1964), introducing two additional densities (andpotentials) for the ionic system Γ (Ri )(Vext (Ri )) and the superconducting electronsχ(r, r′)(Δ(r, r′)) and finding the values that minimize a suitable energy functional.This permits to derive a gap equation which is analogous to the BCS one butinstead of an empirical potential contains a kernel with all the relevant physicalinformation on the system. I refer the reader to the original references for a fullderivation (Lüders et al. 2005; Marques et al. 2005; Oliveira et al. 1988) and toSanna (2017) for an excellent pedagogical introduction.

SCDFT equations are more easily solvable on a computer than fully anisotropicME equations because they do not require expensive sums over Matsubara fre-quencies; however, the interpretation of many physical quantities, including thefrequency dependence of the gap, is not equally transparent and straightforward.Another intrinsic limitation is that, like in all DFT-like methods, the quality of theresults depends strongly on the quality of the functional.

The latest-developed functionals yield results with an accuracy comparable tothat of the best DFT-ME calculations, which for most conventional superconductorsis between 5 and 10% of the critical temperature. The most severe source ofinaccuracy in DFT calculations for superconductors is usually an underconvergedintegration in reciprocal space in Eqs. (3) and (4), an issue that has considerablyimproved thanks to the use of Wannier interpolation techniques.

Achieving quantitative accuracy for conventional pairing also encouraged toaddress ab initio effects, which are often disregarded even in model approaches,such as anharmonicity, vertex corrections, and zero-point effects (Pietronero et al.1995; Boeri et al. 2005; Lazzeri et al. 2003).

These turned out to be relevant for a wide variety of compounds, particularly forthe newly discovered superconducting hydrides, where the energy scales of phononsand electrons are comparable (Errea et al. 2014, 2016; Sano et al. 2016). I refer thereader to the relevant references for an in-depth discussion.

For unconventional superconductors, on the other hand, the most severe source ofinaccuracy is intrinsic and is the possible divergence in the spin-fluctuation propaga-tor, which completely destroys the predictive power of DFT approaches for currentlyavailable functionals. Moreover, most unconventional superconductors suffer fromthe lack of accuracy of DFT in strongly correlated systems (Yin et al. 2013).

Understanding Novel Superconductors with Ab Initio Calculations 11

3.3 Developments in Related Fields: Ab Initio Material Design

The term ab initio material design indicates the combination of methods for abinitio crystal structure prediction and thermodynamics to predict the behavior ofmaterials at arbitrary conditions of pressure and temperature, knowing only theinitial chemical composition of the system. The development of these methodsrepresents a substantial step forward in computational condensed matter research,as it overcomes one of its biggest limitations, exemplified by the Maddox Paradox(1988): “One of the continuing scandals in the physical sciences is that it remainsin general impossible to predict the structure of even the simplest crystalline solidsfrom a knowledge of their chemical composition.”

The basic working principle of ab initio crystal structure prediction is quitesimple. Predicting the crystal structure of a material for a given regime of chemicalcomposition and pressure amounts to finding the global minimum of a complicatedlandscape, generated by the ab initio total energies (or enthalpies) of all possiblestructures. The number of possible configurations for a typical problem is so largethat a purely enumerative approach is unfeasible; in the last years, several methodshave been devised to make the problem computationally manageable, such as abinitio random structure search, minima hopping, metadynamics, genetic algorithms,particle swarm algorithms, etc. (Woodley and Catlow 2008).

Once the most favorable crystal structure for a given composition and pressureis known, the ab initio CALPHAD (CALculation of PHAse Diagrams) approachpermits to predict accurate phase diagrams (Lukas et al. 2007), as illustrated inFig. 2. The binary Li-S phase diagram in panel (a) shows the stability ranges ofdifferent Li-S compositions and has been constructed repeating several convex hullcalculations at different pressures.

The convex hull construction is shown in panel (b). The points represent thelowest-energy structure predicted by an evolutionary search for a given composition;for a binary phase with composition AxBy , the formation enthalpy ΔH is definedas: ΔH = H(AxBy) − [xH(A) + yH(B)] . If this quantity is negative, the phaseAxBy is stable with respect to elemental decomposition; if it is positive, the phaseis highly (or weakly) metastable, i.e., if formed, it will decompose into its elementalconstituents in finite time. However, the decomposition into the two elements isoften not the relevant one, as a compound could decompose into other phases,preserving the correct stoichiometry. To estimate all possible decompositions forthe binary system, one constructs the most convex curve connecting the formationenthalpy of all known phases for a given stoichiometry. Points on this convexhull represent stable compositions into which phases which lie above the convexhull will decompose into given a sufficient interval of time. The figure also showsthat, as pressure increases, the diversity of the phase diagram increases, i.e., off-stoichiometry compositions become possible (forbidden chemistry). The convexhull construction can be easily extended to multinary systems (Gibbs diagrams) andfinite temperatures including entropic effects.

12 L. Boeri

S8S

a

b

R3m (b-Polonium)

R3m

SII, S3-polymer

Im3m (bcc)

Fm3m

Pm3m

cI16

Im3m

Fm3m

I4/mmm

I4/mmm

ImmmCmmm

Pnma

P63/mmc

P4132Cmca-24fcc

bcc fcc+R3m

LiS3

LiS2

Li2S

Li3S

Li4S

Li5S

Li

0 100 200 300

0 0.1 0.2 0.3

form

atio

n en

thal

py (

ev/a

t.)

0.4 0.5

xS/(xLi + xS)

0.6

0 GPa50 GPa100 GPa200 GPa300 GPa400 GPa500 GPa600 GPa700 GPa

0.7 0.8 0.9 1–2.5

–2

–1.5

–1

–0.5

0

LiS3LiS2

Li5S

Li4S

Li3S

Li2S

external pressure (GPa)

400 500 600 700

Fig. 2 Basic steps of ab initio materials prediction: the complete phase diagram of a binary alloyas a function of pressure (a) can be constructed combining several convex hull constructions atdifferent pressures (b) (see text). (Readapted from Kokail et al., Phys. Rev. B 94, 060502(R) (2016)– Copyright (2016) by the American Physical Society)

4 Materials

The aim of this section is to illustrate how, within a little bit more than a decade,an increased understanding of material-specific aspects of superconductivity gained

Understanding Novel Superconductors with Ab Initio Calculations 13

from ab initio calculations has permitted to replace empirical rules to search for newsuperconductors with quantitative strategies.

I will start with a general discussion that determine the Tc of conventionalsuperconductors (Sect. 4.1), introducing the concepts of dormant ep interactionsand lattice instabilities and showing how these can be used to interpret both theold empirical knowledge (Matthias’ rules and Cohen-Anderson limit) and the latestexperimental discoveries (Savrasov and Savrasov 1996; Boeri et al. 2002, 2007,2008; Profeta et al. 2012).

I will then use a toy numerical model (simple graphite), to see how these conceptsare realized in an actual physical system, and a very simple approximation to doping(rigid-band) to simulate the effect of physical doping and detect the sources of ep

interaction in graphite-like materials. Both models are useful for a first explorationbut inadequate to make accurate predictions for actual superconductors, wherethe doping is usually obtained via chemical substitution, which causes a majorrearrangement of phononic and electronic states and hence sizable changes in thevalues of the ep interaction and Tc.

For simple graphite, nature provides two simple realizations of two of its sourcesof dormant ep interactions: MgB2, a prototype of covalent superconductors, andgraphite intercalated compounds, where superconductivity is correlated with thefilling of interlayer states. In Sects.4.3 and 4.4, I will discuss these two examplesin detail and also indicate the main theoretical predictions and experimentaldiscoveries which were inspired by them.

In particular, the line of research on covalent superconductors culminated inthe discovery of high-Tc conventional superconductivity at extreme pressures inSH3, discussed in Sect. 4.5, which also represents a fundamental step forward in thedirection of the search of new superconductors using first-principles methods.

After describing the incredible evolution of the state of research in con-ventional superconductors, in Sect. 4.6, I will present a representative exampleof unconventional superconductors, iron pnictides, and chalcogenides (Fe-basedsuperconductors), discovered in 2008, which shares many similarities with the high-Tc cuprates. This example will allow me to give an idea of the many challenges thattheory faces in the description of unconventional superconductors, already in thenormal state, and currently represent a fundamental obstacle to the derivation ofnumerical methods to compute Tcs.

4.1 Conventional Superconductors: Search Strategies

Matthias’ rules The so-called Matthias’ rules are a set of empirical rules thatsummarize the understanding of superconductors in the 1970s. The rules wereallegedly formulated, and revised several times, by Bernd Matthias, one of theleading material scientists in superconductivity: they are usually cited in thisform:

(continued)

14 L. Boeri

1. High symmetry is good; cubic symmetry is the best.2. High density of electronic states is good.3. Stay away from oxygen.4. Stay away from magnetism.5. Stay away from insulators.6. Stay away from theorists.

Matthias’ rules were inspired by A15 superconductors (cubic transition metalalloys, which can be easily doped, exhibit sharp peaks in the electronic DOS, andare prone to lattice and magnetic instabilities) and clearly disproved by subsequentexperimental discoveries: cuprates and pnictides (first, third, fourth, and fifthrule) (Bednorz and Mueller 1986; Kamihara et al. 2008) but also conventional super-conductors such as MgB2 (Nagamatsu et al. 2001) (first and second rule) (Drozdovet al. 2015a). However, their impact on superconductivity research has been soimportant that they are still sometimes cited as arguments against conventionalsuperconductivity or the possibility of theoretically predicting new superconductors,together with another old quasi-empirical rule, the Cohen-Anderson limit.

In order to derive more general, nonempirical strategies to search for newsuperconductors, I will begin with a simple analytical model. Instead of taking intoaccount complexity of the electronic and vibrational properties of real superconduc-tors, for the moment I consider an ideal case, in which a single phonon branch withfrequency ω and a single electronic band with density of states at the Fermi levelN(EF ) is coupled through an average matrix element I . In this case, the Tc is welldescribed by the McMillan-Allen-Dynes formula (Eq. 6) with all constant factorsset to one, ωln = ω, and the coupling constant is given by the Hopfield expressionλ = (N(EF )I 2)/(Mω2):

Tc = ω

kB

exp

[− (1 + λ)

λ − μ∗(1 + λ)

], (9)

These formulas indicate that there are three main strategies to optimize thecritical temperature of a conventional superconductor: (i) maximize the value ofthe electronic DOS at the Fermi level N(EF ) (first two Matthias’ rules); (ii)select compounds which contain light elements (and stiff bonds), to maximize thecharacteristic lattice energy scales (ω); and (iii) increase the ep matrix elements (I ).

While the first two strategies were already understood in the early 1970s, itbecame apparent only with the MgB2 discovery that it is possible to find compoundswhere the intrinsic ep matrix elements I are much larger than in transition metalsand their alloys, where the maximum Tc does not exceed 25 K. In a seminal paper,An and Pickett pointed out that the (relatively) high Tc of MgB2 occurs because ofcovalent bonds driven metallic (An and Pickett 2001).

The reason why covalent metals have larger intrinsic ep coupling than ordinarymetals can be intuitively understood looking at Fig. 3, which shows isosurfaces of

Understanding Novel Superconductors with Ab Initio Calculations 15

Fig. 3 Electronic localization function in conventional superconductors: bcc Nb (Tc = 9 K); MgB2(Tc = 39 K); SH3 (Tc = 203 K)

the electronic localization function (ELF) for three different superconductors: Nb(Tc = 9 K), MgB2 (Tc = 39 K), and SH3 (Tc = 203 K). The ELF indicates regionswhere electrons are concentrated. It is clear that in MgB2 and SH3, the electronslocalize along the bonds, while in Nb they are delocalized over the whole volume.When atoms undergo phonon vibrations, electrons localized along a bond will feel amuch stronger perturbation than those spread out over the whole crystal. However,the arguments above are oversimplified, as the existence of strong directional bondsis a necessary prerequisite for large ep coupling, but it is not sufficient: due to thesmall energy scales involved in the superconducting pairing, it is also essential thatthe electronic states which contribute to this bond lie at the Fermi level; otherwise,they remain dormant and do not contribute to the superconducting pairing.Moreprecisely, this means that shifting the position of the Fermi level (EF ) selectsdifferent matrix elements g in Eq. (7) when performing averages over the Fermisurfaces in in Eqs. (3) and (4), and the ep interaction λ, and hence Tc, is appreciableonly for some positions of EF .

A possible strategy to search for new conventional superconductors thus amountsto identifying, first, possible dormant ep interactions within a given materialclass and, second, physical mechanisms to activate them, such as doping, pressure,and alloying. Ab initio approaches permit to explore both steps of this process, atdifferent levels of approximation. In the following, I will illustrate the basic workingprinciple, using an example (simple graphite) and two practical realizations (MgB2and intercalated graphites); I will also refer to the same principles when discussingpossible perspective of future research in Sect. 5.

Before moving on with the discussion, I need to introduce a second conceptwhich is crucial in the search for new conventional superconductors: latticeinstabilities. This argument is important because it is at the heart of the Cohen-Anderson limit (Cohen and Anderson 1972).

According to Eq. (9), Tc can apparently be increased indefinitely, increasingλ, which contradicts what is observed in practice, since the critical temperaturesof actual superconductors are limited. However, in my discussion, I have so fardisregarded the feedback effect between phonon frequencies and ep interaction,which is one of the main limiting factors to high Tc in actual materials. Indeed,

16 L. Boeri

the same ep interaction that pairs electrons leading to superconductivity also causesa decrease (softening) of the phonon frequencies. This means that the frequency ω

appearing in McMillan’s formula for Tc (Eq. 9) should be more correctly rewrittenas: ω2 = (Ω0)

2(1 − 2λ0), where Ω0 is the bare frequency of the lattice, in theabsence of ep interaction, and λ0 is the corresponding coupling constant. It is noweasy to see that the Tc for a fixed Ω0 has a maximum for a given value of λ0 andthen decreases approaching a lattice instability (ω → 0); this means that within agiven material class, Tc can only be increased up to a threshold value determinedby λ0, before incurring in a lattice instability. Using typical parameters for theA15 compounds, which were the best known superconductors when the Cohen-Anderson limit was formulated, gives a maximum value of Tc of ∼ 25 K. However,covalent superconductors such as MgB2, doped diamond, and SH3 have much largercharacteristic phonon scales (Ω0, λ0) and can sustain much larger Tc.

4.2 Dormant ep Interaction in Graphite

To put general arguments on more physical grounds, we now consider a toy modelbased on an actual physical system, simple graphite, which is realized stackingseveral layers of graphene on top of each other (the stacking is thus AAA, incontrast to the ABA and ABC stacking of Bernal and rhombohedral graphite); forthis experiment, we keep the interlayer distance equal to that of Bernal graphite.

The unit cell contains two inequivalent carbon atoms, which form a hexagonallattice. This means that the four orbitals of carbon will split into three sp2 hybridsand a pz orbital, forming six σ and two π bands, of bonding and antibonding char-acter. The blue arrow points to a fifth band, which has no carbon character: this isthe so-called interlayer (IL) band, which is essentially a free-electron state confinedbetween the carbon layers. The relative Wannier function, shown in the inset ofthe figure, is indeed centered in the middle of the interstitial region between thegraphitic layers and has spherical symmetry (Boeri et al. 2007; Csaniy et al. 2005).

The phonon spectrum of pure graphite (not shown) is even simpler than itselectronic structure: in-plane bond-stretching optical phonons form a rather narrowband between 150 and 200 meV, while optical out-of-plane and acoustical modesare more dispersive and extend up to 100 meV.

For the doping of pure graphite, corresponding to four electrons/carbon, theFermi level cuts the band structure near the crossing point between π and π∗ bands;the ep coupling is extremely low and gives rise to a negligible Tc. However, differentdormant ep interactions can be activated if, with a gedankenexperiment, the Fermilevel is shifted to higher or lower energies. This rigid-band shift is the simplestapproximation to physical doping within an ab initio calculation.

The right panel of Fig. 4 shows a rigid-band calculation of the Tc of simplegraphite, in an energy range of ±8 eV around the Fermi level, corresponding toa doping of ±1.5 electrons/carbon; in this calculation, the phonons and ep matrixelements are computed only once, for the physical doping of four electrons/carbon,

Understanding Novel Superconductors with Ab Initio Calculations 17

8

0

-8

M

E (

eV)

K

Tc(K)

A 0 50 100

–1.5 e-–1.0 e-–0.5e-

+1.5 e-

+1.0 e-

+0.5e-

Γ Γ

Fig. 4 Rigid-band study of superconductivity in simple graphite; dashed lines indicate the positionof the Fermi level for a semi-integer doping of holes or electrons; the blue arrow marks the positionof the interlayer band, whose Wannier function is shown in the inset

but the averages and sums on the Fermi surface in Eqs. (3) and (4) are recomputedfor different positions of the Fermi level (EF ).

The figure clearly shows that Tc is still negligible for small variations of energyaround the original Fermi level but has a marked increase as soon as holes orelectrons are doped into the σ (bonding or antibonding) or IL bands, reaching amaximum of ∼100 K, when the Fermi level reaches a large van Hove singularity at∼8 eV, corresponding to the bottom of the σ ∗ band. Apart from causing substantialdeviations in Tcs, the coupling of phonons to σ , IL, and π electrons is also quali-tatively different. σ and π electrons couple mostly to high-energy optical phonons,which modulate the interatomic distances in the hexagonal layers, and hence thehopping between neighboring carbon sites; IL electrons, instead, respond to out-of-plane phonons, which modulate their overlap with the π wavefunctions that stickout-of-plane. This effect causes a substantial variation in the shape of the Eliashbergfunctions (not shown) and ωlog . The right scale on Fig. 4 indicates the position of theFermi level for semi-integer values of hole(−) or electron(+) doping; in both cases, afinite Tc is obtained for a minimum doping of half an electron/carbon if needed, butobtaining a high Tc requires a much higher doping (∼1 e−/C atom). Note that thehighest doping levels that can be obtained with field effect using liquid electrolytesare much lower, i.e., of the order of a few tenths of e−/carbon, so that the only viablealternative to realize superconductivity in doped graphite is via chemical doping.

The rigid-band approximation is instructive to identify dormant ep interactions,but the calculated Tcs are often severely overestimated compared to other exper-iments or more sophisticated approximations for doping. In fact, the rigid-bandapproach neglects important effects due to the self-consistent rearrangement ofelectrons produced by doping, such as band shifts, renormalization of the phonon

18 L. Boeri

frequencies, and screening of the ep matrix elements (Subedi and Boeri 2011;Casula et al. 2012).

For simple graphite, nature provided two extremely ingenious realizations of ourpredictions, discussed in the next two sections.

4.3 Magnesium Diboride and Other Covalent Superconductors

With a Tc of 39K, magnesium diboride (MgB2) currently holds the record for con-ventional superconductivity at ambient pressure (Nagamatsu et al. 2001; Bhaumiket al. 2017). Its crystal structure is layered: boron forms graphite-like hexagonalplanes; magnesium is placed in between, at the center of the hexagons. Mg iscompletely ionized (Mg2+) and thus MgB2 is not only isostructural but alsoisoelectronic to graphite. However, due to the attractive potential of the Mg ions,the center of mass of the π bands is shifted up with respect to that of the σ bandscompared to simple graphite, so that MgB2 behaves effectively as a compensated(semi)-metal. The σ holes and π electrons form two cylinders around the centerand a 3D tubular network around the corners of the Brillouin zone, respectively, asshown in Fig. 5a.

The two groups of electronic states experience a rather different coupling tophonons: over 70% of the total ep coupling, in fact, comes from σ holes and bond-stretching phonons; the rest is distributed over the remaining phonon modes andelectronic states (Kong et al. 2001). The simplest approximation to account for thestrong anisotropy of the ep coupling over the Fermi surface is to replace the ep

coupling constant λ in Eq. (3) and (4), with 2 × 2 matrices of the form (Mazin et al.2002):

λ =(

λσσ λσπ

λπσ λππ

)=

(1.02 0.300.15 0.45

)(10)

When the interband coupling is finite but appreciably smaller than the intrabandone, |λσπ + λπσ | < |λσσ + λππ |, the theory predicts that experiments shouldobserve two distinct gaps, closing at the same Tc (Suhl et al. 1959). Two-gap superconductivity was indeed observed by different experimental techniques:specific heat, tunneling, and angle-resolved photoemission spectroscopy (ARPES)for the first time in MgB2 (Mazin et al. 2002).

Indeed, in most superconductors, multiband and anisotropic effects are extremelydifficult to detect because they are suppressed by the interband scattering induced bysample impurities; in MgB2 the real-space orthogonality of the σ and π electronicwavefunctions prevents interband scattering, and two-gap superconductivity can bedetected also by techniques with a limited resolution (Mazin et al. 2002).

Figure 5b shows in- and out-of-plane tunneling spectra of MgB2, which permitto unambiguously identify a σ (large) and π (small) gap (Gonnelli et al. 2002). Theexperimental spectra compare extremely well with the theoretical prediction of two

Understanding Novel Superconductors with Ab Initio Calculations 19

1.7ba1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.9

0.8M

M

MK

H

L

MK

A

L

Γ

Γ

Γ

Γ

Γ

Γ

Γ

–20 –10 0

c axisNor

mal

ized

con

duct

ance

c axis

ab plane

ab plane

Voltage (mV)10 20

Fig. 5 Two-gap superconductivity in MgB2: anisotropic distribution of the gap on the Fermisurface, predicted by DFT (a) (Figure reprinted with permission from Nature publishing group,H.J. Choi et al., The origin of the anomalous superconducting properties of MgB2, Nature, 418, 758(2002); Copyright (2002); evidence of two-gap behavior from tunneling experiments (b) (Figurereprinted with permission from R.S. Gonnelli et al., Phys. Rev. Lett. 89, 247004 (2002); Copyright(2002) by the American Physical Society)

different gaps on the σ and π sheets of the Fermi surface; the image in the left panelof the figure shows an anisotropic DFT-ME calculation of the gap; in the figure, thecolor of the Fermi surface is proportional to the size of the gap on that sheet (Choiet al. 2002).

Besides providing the first clear case of two-gap superconductivity, MgB2 isthe first example of superconductivity from doping covalent bonds. The mostspectacular realization of this possibility came in 2004, with the report of a Tc of 4 Kin heavily boron-doped diamond, raised to 11 K in thin films (Ekimov et al. 2004;Takano et al. 2004).

Diamond is a wideband insulator (Δ ∼ 5.5 eV); when boron is doped at smallconcentrations, an acceptor band forms within the gap. At the much higher dopinglevels (∼1−10%) realized in superconducting samples, the impurity band overlapsso strongly with the valence band of diamond, that the net effect of B doping is tocreate holes at the top of the valence band (Yokoya et al. 2005). This band is formedby the bonding combination of the four carbon sp3 hybrids. Boron-doped diamondcan thus be seen as a 3D analogue of MgB2, where, similarly to MgB2, a smallfraction of σ holes created by B doping exhibits an extremely strong coupling tobond-stretching phonons.

20 L. Boeri

1.50a

b

1.25

1.00

0.75λ

0.50

0.25

0

0

20

40

60

80

Tc

(K)

100

0 1 2 3 4 5Hole doping (%)

6 7

Graphene

Diamond

Diamond

CaC6

CaC6

MgB2

MgB2

Graphane

Graphane

8 9 10

0 1 2 3 4 5Hole doping (%)

6 7 8 9 10

Fig. 6 Electron-phonon coupling (top) and Tc (bottom) as a function of doping in differentcarbon-based superconductors. (Figure reprinted with permission from G. Savini et al., Phys. Rev.Lett. 105, 037002 (2010), Copyright by the American Physical Society 2010)

Since the ep coupling is concentrated in a single type of phonon modes andelectronic states, the simplified formulas introduced in Sect. 4.1 give a reasonableapproximation to Tc for both MgB2 and diamond (Boeri et al. 2004; Lee and Pickett2004; Blase et al. 2004; Giustino et al. 2007b; Hoesch et al. 2007). Furthermore,they permit to understand why, even though the C-C bonds in diamond are actuallystiffer than the B-B bonds in MgB2, the measured Tcs are much lower. In fact, nearthe bottom (or top) of a band, i.e., at low dopings, the DOS increases as

√E in 3D,

while it is virtually constant in 2D. This implies that, as holes or electrons are dopedinto the system, N(EF ), and hence λ, increases much more slowly in 3D systemsthan in 2D ones. Thus, for physical ranges of dopings, the maximum Tc in dopedsemiconductors is typically very low (Boeri et al. 2004; Bustarret et al. 2006).

A very ingenious strategy that attains high-Tc conventional superconductivityin a doped sp3 system avoiding negative dimensionality effects was proposed afew years later by Savini et al. (2010), i.e., realizing superconductivity in p-dopedgraphane (fully hydrogenated graphene). Graphane can be considered a 2D versionof diamond, in which the bonding is still sp3; hence, the matrix elements are aslarge as those of diamond, but the DOS is 2D; hence, λ is sizable already at low

Understanding Novel Superconductors with Ab Initio Calculations 21

dopings. For this compound, the authors of Savini et al. (2010) estimate a Tc of∼90 K already for 1% doping. Figure 6, from the original reference, compares thebehavior of Tc with doping in the difference class of carbon-based superconductorsdiscussed so far.

Although not as spectacular as graphane, in general, many carbon-based com-pounds doped with boron are good candidates for superconductivity with relativelyhigh temperatures. For example, recent experiments indicate that Q-carbon, anamorphous form of carbon, in between diamond and graphite, can achieve Tcs ashigh as 56 K upon boron doping (Bhaumik et al. 2017).

A complementary idea is that of doping boron-rich phases with carbon; boron,being an electron-deficient material, forms a variety of structures with two- andthree-center bonds. One of the most common motifs is the icosahedron (B12), found,for example, in the α and β phases of elemental boron, as well as in superconductingdodecaborides (Matthias et al. 1968). Boron icosahedra doped with carbon arepredicted to superconduct with Tcs as high as 37 K (Calandra et al. 2004).

4.4 Intercalated Graphites

While MgB2 can be considered a natural realization of hole-doped graphite σ

bonds, superconducting graphite intercalation compounds (GICs) are the practicalrealization of superconductivity by doping into the interlayer states. Although theTcs are substantially lower than in covalent metals, this type of superconductivityis quite interesting, because it can be more easily manipulated by external means(doping, pressure) and also realized in “artificial” systems, such as mono- andbilayers of graphene decorated with alkali or noble metals.

Intercalated graphites had been extensively studied in the 1970s, because oftheir high mobilities, but the only known superconducting member of the family,KC8, had an extremely small Tc (<1K). Only in 2005, superconductivity withrelatively high Tcs was reported in several AC6 compounds: CaC6 (Tc = 11.5 K),YbC6 (Tc = 6.5 K), and later SrC6 (Tc = 1.6 K) (Weller et al. 2005; Kim et al. 2007).At the same time, it was observed that for all newly discovered members of thefamily, the occurrence of superconductivity clearly correlates with the filling of theinterlayer band, which is empty in non-superconductors (Csaniy et al. 2005).

A strong correlation is also found between the Tc and the distance between twosubsequent graphitic layers in different AC6 compounds – Fig. 7a. This can be easilyexplained within the conventional ep scenario (Calandra and Mauri 2005; Kim et al.2006, 2007; Boeri et al. 2007), since, as demonstrated in Sect. 4.1, when localizeddue to doping and confinement effects, interlayer electrons can couple to π electronsthrough out-of-plane phonons of the carbon layers (in GICs, additional coupling isalso provided by intercalant modes).

Like in MgB2, the distribution of the ep interaction, and hence of the supercon-ducting gap, is very anisotropic on the Fermi surface, being much larger for theinterlayer electrons – central sphere in Fig. 7b – than for π ones, which form theouter tubular manifold.

22 L. Boeri

Fig. 7 Physical properties of superconducting graphite intercalation compounds: (a) dependenceof Tc on the interlayer separation in different GICs (Figure reprinted with permission from Kim etal., Phys. Rev. Lett. 99, 027001 (2007), Copyright by the American Physical Society (2007)); (b)anisotropic superconducting gap predicted from SCDFT calculations in CaC6 (Figure reprintedwith permission from A. Sanna et al., Phys. Rev. B 75, 020511(R) (2007), copyright by theAmerican Physical Society (2007)); (c) superconducting gap measured by ARPES in Li-decoratedgraphene (Figure reprinted with permission from B.M. Ludbrook et al., PNAS 112, 11795 (2015))

The idea of superconductivity due to the localization of interlayer states wasexploited in Profeta et al. (2012), proposing to achieve superconductivity in Li-decorated graphene. A Tc of 5.9 K, close to the theoretical prediction of 8.6 K, wasreported in 2015 by Ludbrook et al. (2015). Figure 7c shows the superconductinggap, measured by ARPES. Note that also in this case there is a visible variation ofthe gap between π and IL sheets of the Fermi surface.

4.5 High-Tc Conventional Superconductivity in High-PressureHydrides

The most spectacular realization of high-Tc conventional superconductivity fromdoped covalent bonds can be found in SH3, which, so far, holds the Tc the recordamong all (conventional and unconventional) superconductors.

Indeed, superconductivity at high pressures is a rather ubiquitous phenomenon,because high pressures tend to increase the hopping between neighboring sitesand hence metallicity. Almost all the elements of the periodic table can be madesuperconducting; the typical Tcs are rather low, reaching a maximum of ∼ 20 K

Understanding Novel Superconductors with Ab Initio Calculations 23

in Li, Ca, Sc, Y, V, B, P, and S (Hamlin 2015), well reproduced by ab initiocalculations (Profeta et al. 2006; Yao et al. 2009; Monni et al. 2017; Flores-Livaset al. 2017).

Hydrogen and its compounds represent a notable exception, having provided, in2014, the first example of high-Tc conventional superconductivity. This discoverywas the coronation of a 50-year-long search, inspired by two insightful papers byNeil Ashcroft, predicting high-Tc superconductivity in metallic hydrogen (1968)and covalent hydrides (2004) (Ashcroft 1968, 2004). Both predictions rely onthe general arguments for high-Tc conventional superconductivity introduced inSect. 4.1, i.e., large phonon frequencies due to the light hydrogen mass (ii) combinedwith large ep matrix elements due to the lack of screening from core electrons (iii)can yield remarkable superconducting transition temperatures, even if the density ofstates at the Fermi level is moderate (i).

In 1968, the first of Ashcroft’s predictions, superconductivity in metallic hydro-gen, seemed merely an academic speculation. The pressure required to metallizehydrogen, which is a large-gap insulator at ambient pressure (Δ ∼ 10 eV), wasclearly beyond reach for the experimental techniques of the time. However, 50 yearslater, high-pressure research has advanced to such a point that at least three groupshave reported evidences of hydrogen metallization, at pressures ranging from 360 to500 GPa (3.6 to 5 Mbar) (Dalladay-Simpson and Howie 2016; Eremets et al. 2017;Dias and Silvera 2017).

These reports are still controversial, since the pressure ranges are close to thelimit of current high-pressure techniques (Eremets and Drozdov 2017; Goncharovand Struzhkin 2017). The insulator-to-metal transition has two possible origins:band overlap, in the molecular CmCa phase, or a structural transition to an atomicβ − Sn phase. The two phases are almost impossible to discern experimentallybecause hydrogen is a poor X-ray scatterer, and theoretical studies also predictan unusual spread of values (300–500) GPa for the transition boundary betweenthe two phases, due to the different approximations used to account for quantumlattice effects (McMahon et al. 2012). However, according to DFT calculations, boththe CmCa and the β − Sn structures should become superconducting at or aboveambient temperature, suggesting that the first of Ashcroft’s predictions may soon berealized (Cudazzo et al. 2008; McMahon and Ceperley 2011; Borinaga et al. 2016).

The second prediction, superconductivity in covalent hydrides, was experimen-tally verified at the end of 2014. The underlying idea is that in covalent hydrides,metallization of the hydrogen sublattice should occur at lower pressure than in purehydrogen, because the other atoms exert an additional chemical pressure.

Indeed, in 2008, superconductivity was measured in compressed silane (SiH4) at120 GPa, but the measured Tc (17 K) was disappointingly low compared to theoret-ical prediction of 100 K (Eremets et al. 2008; Li et al. 2010). However, this findingproved that the experimental knowledge to make covalent hydrides was available,and it was only a matter of time before high-Tc conventional superconductivitywould actually be observed. Ab initio calculations revealed two essential missingpieces of the puzzle: (a) megabar pressures can stabilize superhydrides, i.e., phaseswith much larger hydrogen content than the hydrides stable at ambient pressure, and

24 L. Boeri

(b) some of these superhydrides are metals with unusually strong bonds, which canlead to high-Tc superconductivity (Zurek et al. 2009).

The first high-Tc superconductor discovered experimentally is a sulfur super-hydride (SH3), which is stabilized under pressure, by compressing gaseous sulfurdihydride (SH2) in a hydrogen-rich atmosphere. The compound metallizes at∼100 GPa, where it exhibits a Tc of 40 K. The Tc increases reaching a maximum ofover 203 K at 200 GPa; isotope effect measurements confirmed that superconductiv-ity is of conventional origin. I refer the reader to the original experimental (Drozdovet al. 2015a; Goncharov et al. 2016; Einaga et al. 2016; Capitani et al. 2017) andtheoretical (Duan et al. 2014; Heil and Boeri 2015; José et al. 2016; Errea et al.2015, 2016; Bernstein et al. 2015; Quan and Pickett 2016; Akashi et al. 2016; Sanoet al. 2016) references for a more detailed discussion of specific aspects of the SH3discovery, such as the nature and thermodynamics of the SH2 to SH3 transition andanharmonic and nonadiabatic effects (Gor’kov and Kresin 2018).

In the history of superconducting materials, SH3 stands out for one main reason:it is the first example of a high-Tc superconductor whose chemical composition,crystal structure, and superconducting transition temperature were predicted fromfirst principles before the actual experimental discovery. In fact, a few months beforethe experimental report, Duan et al. (2014) predicted that SH2 and H2 would reactunder pressure and give rise to a highly symmetric bcc structure, later confirmed byX-ray experiments (Einaga et al. 2016), which could reach a Tc as high as 200 K at200 GPa.

From the point of view of electronic structure, SH3 is a paradigmatic example ofhigh-Tc conventional superconductivity. Indeed, in this case, all three conditionsreported in Sect. 4.1 are verified: SH3 is a hydride, and hence its characteristicvibrational frequencies are high (i); the Fermi level falls in the vicinity of a vanHove singularity of the DOS of the bcc lattice (ii); and the ep matrix elements arehigh due to the unusual H-S covalent bonds stabilized by pressure (iii).

Replicating a similar combination is not simple, even in other high-pressurehydrides, where one can hope that high pressure may help stabilize unusualbonding environments and hydrogen-rich stoichiometries. Despite an intense the-oretical exploration of the high-pressure superconducting phase diagrams of binaryhydrides (Zurek et al. 2009), only a few candidates match or surpass the Tc of SH3:Ca, Y, and La (Wang et al. 2012; Liu et al. 2017).

In fact, the formation of covalent directional bonds between hydrogen and otheratoms appears to be very sensitive to their electronegativity difference (Bernsteinet al. 2015; Fu et al. 2016), and in binary hydrides, the possibilities to optimize thisparameter are obviously limited. A possible strategy to overcome this limitation is toexplore ternary hydrides, where the electronegativity, atomic size, etc. can be tunedcontinuously combining different elements. However, since ternary Gibb’s diagramsare computationally much more expensive than binary convex hulls, fully ab initiostudies of ternaries are rare. Two recent studies explore two different strategiestoward high Tc in ternary hydrides: doping low-pressure molecular phases ofcovalent binary hydrides, like water (Flores-Livas et al. 2017), and off-stoichiometry

Understanding Novel Superconductors with Ab Initio Calculations 25

phases of alkali-metal alanates and borates, which permit to independently tune thedegree of metallicity and covalency (Kokail et al. 2017).

Due to the intrinsic difficulty of reaching megabar pressures and the limited infor-mation that can be extracted from X-ray spectra, the experimental information onhigh-pressure hydrides is much scarcer than theoretical predictions. Nevertheless,there has been a substantial progress in the last 5 years: metallic superhydridespredicted by theory have been reported in hydrides of alkali metals (Li,Na), (Pépinet al. 2015; Struzhkin et al. 2016) transition metals (Fe), (Pépin et al. 2017) groupIV elements, etc. For some of these systems, first-principles calculations predictsubstantial superconducting temperatures (Zurek et al. 2009), while other cases arecontroversial (Majumdar et al. 2017; Kvashnin et al. 2018; Heil et al. 2018).

Resistivity and susceptibility measurements required to detect superconductivityunder pressure are even more challenging, and therefore the available informationon superconductivity in high-pressure hydrides is still very scarce: besides SH3, onlyone other hydride, PH3, has been shown to superconduct at high pressures, albeitwith a lower Tc (∼100 K) (Drozdov et al. 2015b). At variance with SH3, PH3 ishighly metastable, and samples rapidly degrade over time, consistently with ab initiopredictions of metastability (Fu et al. 2016; Flores-Livas et al. 2016; Shamp et al.2016). SeH3, which should exhibit Tcs comparable to SH3, has been successfullysynthesized at the end of last year (Pace et al. 2017), but superconductivity has notbeen measured yet.

The same arguments that motivated the search for high-Tc conventional super-conductivity in high-pressure hydrides can be applied to other compounds thatcontain light elements, such as Li, B, C, etc. Some of these elements form covalentstructures with strong directional bonds already at ambient pressure, and highpressures could be used to optimize doping, stoichiometry, etc. However, element-specific factors can unpredictably affect properties relevant in high-pressure super-conductivity. For example, elemental phases of alkali metals at high pressure exhibita characteristic interstitial charge localization, due to avoided core overlap. Thesame effect, which is extremely detrimental for conventional superconductivity(charge localized in the interstitial regions has an intrinsically low coupling tophonons), also occurs in many of their compounds. This was shown, for example,for the Li-S system, whose behavior at high pressure is remarkably different fromH-S (Kokail et al. 2016).

4.6 Unconventional Superconductivity in Fe Pnictides andChalcogenides

After illustrating the remarkable progress in the research on conventional super-conductors, I have chosen to discuss the single largest class of unconventionalsuperconductors, formed by Fe pnictides and chalcogenides (Fe-based superconduc-tors, FeSC). The Tcs of these compounds, discovered in 2008, go up to 56 K in thebulk, and allegedly up to 100 K in monolayers of FeSe grown on SrTiO3 (Kamiharaet al. 2008; Ge et al. 2014; Paglione and Greene 2010; Johnston 2010; Huang and

26 L. Boeri

Hoffman 2017). Their rich phase diagram, the quasi-2D crystal structure and theunconventional behavior of the superconducting gaps, is strongly reminiscent of thehigh-Tc cuprates. Thus, I will use FeSC as a representative example to illustrate thechallenges faced by ab initio methods in unconventional superconductors, both inthe normal and in the superconducting state (Basov and Chubukov 2011).

Figure 8 shows the main features, common to most compounds:

• (a)–(b) A common structural motif, consisting of square planes of iron atoms,and X4 tetrahedra; X is either a pnictogen (As,P) or a chalchogen (Se,Te).

• (c) A quasi-two-dimensional Fermi surface, comprising two hole and twoelectron sheets, strongly nested with one other.

• (d) A phase diagram, in which superconductivity sets in after suppressing a spindensity wave (SDW) ordered state, by means of doping or pressure. The mostcommon SDW pattern is a stripe one, in which the Fe spins are aligned ferro-magnetically along one of the edges of the Fe squares and antiferromagneticallyalong the other. The SDW transition is usually preceded by a structural-nematictransition (Fernandes et al. 2014).

FeSC can be grouped in different families, depending on the nature of the FeX layersand of the intercalating blocks; the most common are 11 Fe chalcogenides (FeS,FeSe, FeTe), 111 alkali-metal pnictides (LiFeAs, LiFeP, NaFeAs, NaFeP, etc.),122 pnictides (Ba/KFe2As2, BaFe2P2, CaFeAs2, EuFe2As2, etc.), 1111 pnictides(LaOFeAs, LaOFeP, etc.), and 122 chalcogenides (KFe2Se2, etc.). In most of thesecases, superconductivity appears around a Fe d6 configuration, but it survives up tohigh hole or electron dopings in 122 K pnictides and chalcogenides.

In most FeSC, the superconducting gap exhibits a feature which is distinctiveof unconventional superconductors, i.e., changes sign over the Fermi surface.According to Eq. 1, this is only possible if, unlike ep interaction, the pairinginteraction Vk,k′ is repulsive over some regions of reciprocal space.

In contrast to the cuprates, where the Fermi surface topology favors d-wavesuperconductivity, the gap of most FeSC exhibits a characteristic s± symmetry,with opposite signs on the hole and electron sheets of the Fermi surface. However,substantial variations in the symmetry and magnitude of the superconducting gap areobserved among and within different FeSC families, which can be related to changesin the general shape and orbital distribution of Fermi surface sheets (Hirschfeld et al.2011).

The general topology of the Fermi surface is well described by DFT calculations;however, quasiparticle bands measured by ARPES exhibit renormalizations andshifts, which hint to strong local correlation effects beyond DFT, which require theuse of specialized methods, such as DFT+DMFT (Kotliar et al. 2006).

Figure 9a, from Yin et al. (2011), shows the Fermi surfaces calculated withinDFT (upper panel) and DFT+DMFT (lower panel) for representative FeSC:LaFePO, BaFe2As2, LiFeAs, and KFe2As2. The first three are d6 pnictides, withthe typical hole-electron topology shown in Fig. 8, eventually deformed due tointerlayer hopping (Andersen and Boeri 2011). In KFe2As2, where the electron

Understanding Novel Superconductors with Ab Initio Calculations 27

Fig. 8 Common features of Fe-based superconductors (FeSC): FeX layers, seen from the top (a)and side (b); Fe and X atoms are red and green, respectively. (c) Two-dimensional Fermi surfaceof LaOFeAs, from Andersen and Boeri (2011), with the hole and electron pockets centered at X,Y , and M , respectively. Here a third hole pocket at Γ is also present. (d) Phase diagram, showingthe transition from SDW to superconducting (SC) state; x is an external tuning parameter (doping,pressure)

count is d5.5, the hole pocket is expanded, and the electron pocket has lost itstypical circular shape.

The figure shows that, in most cases, the inclusion of electronic correlationmodifies the Fermi surface quantitatively, but not qualitatively, with respect to theDFT prediction. The renormalization of the quasiparticle band dispersion dependson their orbital character, being stronger for Fe t2 (xz, yz, xy) than for e (x2 − y2,3z2 − 1) orbitals, and on the nature of the X atom; correlation effects are in factweakest for X=P and increase moving to As,S,Se and T e (de’ Medici et al. 2014;Yin et al. 2011). Orbital-selective mass renormalizations and poor Fermi liquidbehavior are characteristic of a special correlation regime, known as Hund’s metalregime, which has been extensively studied by several authors (Werner et al. 2008;Haule and Kotliar 2009; de’ Medici et al. 2011; Yin et al. 2011); excellent reviewscan be found in Georges et al. (2013) and de’ Medici (2017).

A very important consequence of the interplay between metallicity, strongcorrelations, and multiband character in FeSC is their anomalous magnetic behavior.Early DFT studies pointed out that magnetism cannot be consistently describedwithin neither the itinerant (Slater) nor the localized (Heisenberg) scenario (Mazinet al. 2008b; Mazin and Johannes 2008). A regime which is intermediate between

28 L. Boeri

the two cannot be treated in DFT, which is a mean-field theory, but requires methodsable to capture the dynamics of the magnetic moments at different frequencyscales (Yin et al. 2011; Aichhorn et al. 2011; Hansmann et al. 2010; Toschi et al.2012; Skornyakov et al. 2011; Schickling et al. 2012).

Panel (c) shows a DFT+DMFT calculation of the magnetically ordered state for avariety of FeSC (Yin et al. 2011). Circles (and stars) indicate the theoretical (exper-imental) value of the ordered SDW magnetic moment, which shows large variationsamong different compounds. Chalcogenides exhibit large ordered moments, while111 and 1111 pnictides and phosphides exhibit smaller values. (Gray) Squaresindicate the value of the local magnetic moment, which oscillates over a muchsmaller range of values around 2.5 μB . In the SDW phase, the two moments coexist;when the itinerant moment is suppressed with doping or pressure, only the localmoment survives, meaning that in the nonmagnetic regions of the phase diagram,FeSC is in a paramagnetic state.

The superconducting state that emerges from this complicated normal stateis clearly unconventional. According to linear response calculations, ep couplingplays a marginal role in the superconducting pairing, since the values of thecalculated coupling constant λ in all FeSC are extremely small, even includingmagnetoelastic effects1 (Boeri et al. 2008, 2010; Subedi et al. 2008). The strongestcandidate as superconducting mediator are spin fluctuations, as suggested by theunconventional symmetry of the superconducting gap and the proximity of SDWand superconductivity in the phase diagram (Chubukov et al. 2008; Kuroki et al.2008; Mazin et al. 2008a). As discussed in Sect. 3.2, there is currently no first-principles theory of spin fluctuations with an accuracy comparable to that for ep

interaction. Most studies of superconductivity in FeSC have therefore adopted ahybrid approach, in which the electronic structure in the vicinity of the Fermi levelis downfolded or projected to an effective analytical tight-binding model (Graseret al. 2009; Kuroki et al. 2009; Miyake et al. 2010; Andersen and Boeri 2011), andelectron-electron interactions leading to spin and charge fluctuations are included ata second stage with many-body techniques. The most common are the random phaseapproximation (RPA), fluctuation exchange (FLEX), and functional renormalizationgroup (FRG) (Chubukov et al. 2008; Thomale et al. 2009; Hirschfeld et al. 2011;Platt et al. 2012). Although not quantitatively predictive, these weak-couplingapproaches have provided a detailed understanding of the superconducting gap sym-metry, competition of superconductivity with other ordered phases, and occurrenceof nematic order and traced their common origin to multi-orbital physics (Yi et al.2017). Figure 9c, from Kuroki et al. (2009), shows that the variations in Tc across

1(This result is generally accepted, although DFT+DMFT studies evidenced a strong renormaliza-tion of some phonon modes, due to strong electronic correlations (Mandal et al. 2014) ep couplinghas also been suggested to play a primary role in the enhancement of the superconducting Tc inFeSe monolayers grown on SrTiO3 (Huang and Hoffman 2017), although in this case, the modesinvolved in the pairing belong to the substrate.)

Understanding Novel Superconductors with Ab Initio Calculations 29

1.0a

bc

d

b

0.5 0

00.

51.

0X

XM

LaF

ePO

Γ

1.0

0.5

2.5

SD

W D

MF

TD

SD

W D

MF

T

DS

DW

EX

PE

XP

PM

fluc

tuat

ing

mom

ent

SD

W E

XP

2.0

1.5

1.0

0.5

Magnetic moment (mB)

0

0

00.

51.

0X

xy

XM

LaF

ePO

Γ

1.0

0.5 0

00.

51.

0X

xz

XM

BaF

e 2As 2

LiF

eAs

LiF

eAs

Γ

1.0

0.5 0

00.

51.

0X

xz

XM

Γ

1.0

0.5 0

00.

51.

0

pnic

toge

n he

ight

hP

n

Ky

Kx

0.45

0.45

0

FeTe CsF

e2Se

2

CsFeA

sF SrFeA

sF SrFe2AsF

2CaF

e2As2

LaFeP

O

KFe2Se

2 FeSe LiF

eAs NaF

eAs KFe

2As2 BaFe2As2 La

FeAsO

0

2.55.

0 ⎥Δ(k)⎢ (meV)

Δ − Δ +

–0.5

–0.5

X

Equ

al x

z/yz

XM

Γ

1.0

0.5 0

00.

51.

0X

XM

BaF

e 2As 2

ΓΓ

1.0

0.5 0

00.

51.

0X

XM

1.0

lattice constants a,c

0.5 0

00.

51.

0XP

0.2

2noda

l 2242

6P0.

37La

0.9

0node

less

Nd

1.20

high

er T

cN

d-p

0.67

XM

KF

e 2As 2

KF

e 2As 2

Fig

.9

Nor

mal

and

supe

rcon

duct

ing-

stat

epr

oper

ties

ofFe

SC,

pred

icte

dby

DFT

+m

any-

body

met

hods

.(a

)R

enor

mal

izat

ion

ofth

eFe

rmi

surf

aces

and

(b)

vari

atio

nof

the

orde

red

and

loca

lm

agne

ticm

omen

tac

ross

diff

eren

tfa

mili

esof

FeSC

,pr

edic

ted

byin

DFT

+D

MFT

(Fig

ures

repr

inte

dw

ithpe

rmis

sion

from

Nat

ure

publ

ishi

nggr

oup,

Y.

Yin

etal

.,N

at.

Phys

.7,

294

(201

1),

Cop

yrig

htN

atur

e(2

011)

);(c

)su

perc

ondu

ctin

gtr

ends

in11

11pn

ictid

espr

edic

ted

byD

FT+

RPA

(Fig

ures

repr

inte

dw

ithpe

rmis

sion

from

K.K

urok

ieta

l.,Ph

ys.R

ev.B

(200

9),C

opyr

ight

Am

eric

anPh

ysic

alSo

ciet

y(2

009)

);(d

)su

perc

ondu

ctin

gga

pof

FeSe

pred

icte

dby

SCD

FT(F

igur

ere

prin

ted

with

perm

issi

onfr

omF.

Ess

enbe

rger

etal

.,Ph

ys.R

ev.B

79,2

2451

1(2

009)

,Cop

yrig

htA

mer

ican

Phys

ical

Soci

ety

(200

9)).

30 L. Boeri

0

–20

–40

–60

–80

–100

–120

0

80 G

Pa

α-L

iB

β-Li

B

a) α

-LiB

b) β

-LiB

MS

1-Li

BM

S2-

LiB

100

GP

a12

0 G

Pa

–20

–40

–60

–80

–100

–120

–5

HSi

Pb

cb

de

a

5010

015

0

Pre

ssur

e (G

Pa)

Pre

ssur

e (G

Pa)

Pre

ssur

eEnthalpy (eV/H2O)

Relative enthalpy ΔH (meV/atom)

200

250

300

–4.5

–3.5 –4

02

46

810

1214

HH

H

HO

O

Cov

alen

tbo

nd

phas

e V

III

phas

e V

III

phas

e X

phas

e X

Hyd

roge

nbo

nd

Transition region (exp.)

→ q→ k′ 2

→ k′ 1→ k 1 → k 2

Fig

.1

0Se

lect

edlit

erat

ure

exam

ples

ofpo

ssib

lest

rate

gies

toro

om-t

empe

ratu

resu

perc

ondu

ctiv

ityat

ambi

entp

ress

ures

:(a)

first

-pri

ncip

les

pred

ictio

nof

αan

dβ

-LiB

(Fig

ure

repr

inte

dw

ithpe

rmis

sion

from

A.K

olm

ogor

ovan

dS.

Cur

taro

lo,P

hys.

Rev

.B74

,224

507

(200

6),C

opyr

ight

byth

eA

mer

ican

Phys

ical

Soci

ety

(200

6))

(Des

ign

byA

nalo

gy);

(b)

supe

rcon

duct

ivity

indo

ped

ices

(Flo

res-

Liv

aset

al.2

017)

(Che

mic

alD

opin

gof

mol

ecul

arcr

ysta

ls);

(c)

char

ge-d

ensi

typr

ofile

ofsu

perc

ondu

ctin

gPb

@Si

(111

)su

rfac

e(L

insc

hied

etal

.201

5a)

(Ato

mic

-sca

lede

sign

and

Dim

ensi

onal

ity)

;(d)

high

-pre

ssur

esu

perc

ondu

ctin

gph

ase

diag

ram

ofph

osph

orus

(Fig

ure

repr

inte

dw

ithpe

rmis

sion

from

Flor

esL

ivas

etal

.,Ph

ys.

Rev

.M

at.

102

4802

(201

7),

Cop

yrig

htby

the

Am

eric

anPh

ysic

alSo

ciet

y(2

017)

);(Q

uenc

hing

ofH

igh-

pres

sure

met

asta

ble

phas

es);

(e)

Tc

vsfie

ldef

fect

and

chem

ical

dopi

ngin

mon

olay

erM

oS2

(W.

Shi

etal

.Sc

i.R

ep.

5,12

534

(201

5))

(Dop

ing

byfie

ldef

fect

)

Understanding Novel Superconductors with Ab Initio Calculations 31

different 1111 pnictides can be reproduced within RPA and derive from changes inorbital composition of the Fermi surface.

A first, very elegant, fully first-principles study of superconductivity in FeSCwas carried out by the Gross group, who applied their recently derived ab initiotheory for spin fluctuations to FeSe – see Sect. 3.2 for details. Figure 9d showsthe superconducting gap calculated in SCDFT, which exhibits an s± symmetry,consistently with previous hybrid studies. It is important to remark, however, thatthe gap in Fig. 9d was calculated treating the contributions of all pairing channels(phonons, spin fluctuations, charge fluctuations) on equal footing and withoutany intermediate mapping on an effective many-body model. However, even thiscalculation cannot be considered fully ab initio as, in order to obtain a physicallymeaningful value for the superconducting gap and Tc, the authors had to introducean artificial scaling factor in the spin-fluctuation propagator, which would otherwisediverge (Essenberger et al. 2014; Ortenzi et al. 2012).

The divergence is associated to the specific choice of TDDFT exchange func-tional ( adiabatic local density approximation, ALDA) made by the authors which,being the time-dependent equivalent to the standard local density approximation(LDA), overestimates the tendency to magnetism in FeSC (Mazin et al. 2008b).Proposals to cure this critical divergence in a nonempirical fashion are under-way (Sharma et al. 2018) and could lead to a very important step forward in theunderstanding of unconventional superconductors. However, it is also important tostress that, even if divergence issues are solved, Tc computed within DFT may stillbe inaccurate, because it neglects renormalization effects of quasiparticle energiesand interaction vertices due to strong local electronic correlations, which, in someFeSC, may be sizable (Yin et al. 2013; de’ Medici et al. 2017).

5 Outlook and Perspectives

The results described in this chapter show that in the last 20 years, there hasbeen a substantial advancement in the understanding of superconductors, drivenby the progress of ab initio electronic structure methods. The SH3 discovery hasdemonstrated that room-temperature superconductivity can be attained, at leastat extreme pressures (Drozdov et al. 2015a), and that ab initio methods can bevery effectively employed to predict new superconductors. The next challenge inthe field is clearly to devise practical strategies to replicate the same result atambient conditions, exploiting novel synthesis and doping techniques. Obviously,the development of ab initio methods that can treat these regimes is a crucial step inthis direction.

The aim of this final section is to give a short overview of promising strategies tohigh-Tc superconductivity, exemplified in Fig. 10a–e.

(a) Design by Analogy: This is one of the most common routes to search fornew superconductors, i.e., achieve high-Tc designing materials with a similargeometry, chemistry, electronic structure, etc. as the best existing superconduc-

32 L. Boeri

tors. This route has lead to several predictions of high-Tc superconductivityin layered borides and carbides after the discovery of MgB2, as well as hole-doped LiBC, and, more recently, doped graphane (Satta et al. 2001; Rosneret al. 2002; Savini et al. 2010). The main drawback of many of these earlyworks is that many of these hypothetical compounds are thermodynamicallyunstable. Nowadays, studies of the thermodynamic stability of compounds arewell established but were very rare at the time of the MgB2 discovery. One of thefirst studies to take this aspect into account is the prediction of superconductivityin LiB, (Kolmogorov and Curtarolo 2006) shown in Fig. 10a.

(b) Chemical Doping of Molecular Crystals: Similar to covalent solids, molecularones exhibit stiff bonds, and hence their electronic states can couple strongly tolattice vibrations, which is a prerequisite for high-Tc conventional superconduc-tivity. However, similar to covalent solids, molecular ones are usually insulatingat ambient pressures. Making them superconducting requires either very highpressures or chemical doping, which is often hard to control experimentally andto model ab initio.

One of the most complete studies of the effect of doping on supercon-ductivity in molecular crystals is the study of superconductivity in doped ice,by Flores-Livas et al. (2017) Here, analyzing the effect of different dopantsusing supercells, the authors found that nitrogen acts as an effective dopantat the oxygen site, leading to a Tc of 60 K. The study nicely evidences thecrucial difference between actual doping, modeled with supercells, and moreapproximate approaches, as rigid-band or jellium doping.

(c) Atomic-Scale Design and Dimensionality: Another recent trend in condensedmatter research opened by the development of novel techniques, such asmolecular beam epitaxy (MBE), or exfoliation (Novoselov et al. 2004), is thedesign of materials at the atomic scale, through heterostructuring, controlleddeposition, strain engineering, etc. This permits to tune superconductivityin existing materials; induce it in semiconductors and semimetals throughdoping, like Li-decorated graphene and phosphorene; or create completelyartificial superconductors, depositing superconducting elements on semicon-ductor surfaces, as in Fig. 10c, which shows a SCDFT calculation for Pb onthe 111 surface (Zhang et al. 2010; Linschied et al. 2015a). For this type ofproblems, real-space approaches may be more appropriate than reciprocal-spaceones (Linscheid et al. 2015b).

(d) Quenching of High-Pressure Metastable Phases down to ambient pressure isanother very attractive route to stabilize unusual bonding environments, enabledby recent developments in high-pressure techniques. In fact, controlled heatingand cooling cycles permit to selectively stabilize different metastable phases,realizing pressure hysteresis cycles. Figure 10d, from Flores-Livas et al. (2017),shows the crystal structures of the different phases that form the complicatedsuperconducting phase diagram of elemental phosphorus. In this element, ahigh-Tc branch, clearly distinct from the low-Tc ground-state one, can beaccessed by laser heating and is associated with metastable black phosphorus;

Understanding Novel Superconductors with Ab Initio Calculations 33

similar branchings between high-Tc and low-Tc phases have been predicted athigher pressures.

(e) Doping via Field Effect is an attractive route to tune the properties of materialscontinuously, without impurities and distortions associated to chemical doping.Modern techniques based on liquid electrolytes permit to achieve doping levelsas high as 1015carriers/cm3, corresponding to tenths of electrons, aroundthree orders of magnitude larger than standard solid-state techniques. The firstsuccessful applications have been to cuprates, ZrNiCl, and 2D materials –Fig. 10e. Rigorous modeling of field-effect devices has been derived in Sohieret al. (2015), Sohier et al. (2017).

Although not an experimental technique, methods for machine learning, whichcan be used to prescreen ab initio proposals, will most likely play a bigger andbigger role in the design and discovery of new materials (Curtarolo et al. 2013).

Note that all of the specific examples discussed above are based on theassumption of conventional (phonon-mediated) pairing. The same techniques havebeen, and can be, applied also to unconventional superconductors, which remain thelikeliest candidates for high-Tc superconductivity. However, without a quantitativetheory of the superconducting pairing, it is at the moment impossible to formulatereliable predictions of Tc and other superconducting properties. A progress in thisdirection is therefore an essential prerequisite to any meaningful computationalsearch.

Acknowledgements There are many people who, over the years, helped me to shape my view onsuperconductivity. Many of these encounters turned into friendships, and I am very grateful for that.A special thank goes to my mentors in Rome (Luciano Pietronero, Giovanni Bachelet) and Stuttgart(Jens Kortus and Ole Krogh Andersen), who introduced me to the field of superconductivity andelectronic structure, as well as to all my collaborators and students, with whom I had the pleasureto work and argue on many topics. Thanks to José Flores-Livas, Christoph Heil, Renato Gonnelli,Bernhard Keimer, Jun Sung Kim, Rheinhard Kremer, Igor Mazin, Paolo Postorino, Gianni Profeta,and Antonio Sanna for the many discussions and projects we shared over the years.

I would never have completed this chapter without the help of my current office neighbor,Paolo Dore, who inquired about the status of the project almost every day, and of Luca de’ Medici,Christoph Heil, Antonio Sanna, and Alessandro Toschi, who gave me suggestion on parts of themanuscript at different stages. Finally, I would like to dedicate this work to the memory of twovery special people, Sandro Massidda and Ove Jepsen, whom I will always remember for theirkindness, culture, and enthusiasm for physics. I miss them both.

References

Aichhorn M, Pourovskii L, Georges A (2011) Importance of electronic correlations for structuraland magnetic properties of the iron pnictide superconductor lafeaso. Phys Rev B 84:054529

Akashi R, Arita R (2013) Development of density-functional theory for a plasmon-assistedsuperconducting state: application to lithium under high pressures. Phys Rev Lett 111:057006

Akashi R, Sano W, Arita R, Tsuneyuki S (2016) Possible “magnéli” phases and self-alloying in thesuperconducting sulfur hydride. Phys Rev Lett 117:075503

34 L. Boeri

Allen P, Dynes R (1975) Transition temperature of strong-coupled superconductors reanalyzed.Phys Rev B 12:905

Allen P, Mitrovic B (1982) Theory of superconducting tc. In: Solid State Physics, vol 37.Academic, New York, pp 1–92

An JM, Pickett WE (2001) Superconductivity of MgB2: covalent bonds driven metallic. Phys RevLett 86(19):4366–4369

Andersen O, Boeri L (2011) On the multi-orbital band structure and itinerant magnetism of iron-based superconductors. Annalen der Physik 523(1–2):8–50

Ashcroft N (1968) Metallic hydrogen: a high-temperature superconductor? Phys Rev Lett21:1748–1749

Ashcroft N (2004) Hydrogen dominant metallic alloys: high temperature superconductors? PhysRev Lett 92:187002

Bardeen J, Cooper LN, Schrieffer JR (1957) Theory of superconductivity. Phys Rev 108:1175Baroni S, de Gironcoli S, Corso AD, Giannozzi P (2001) Phonons and related crystal properties

from density-functional perturbation theory. Rev Mod Phys 73:515Basov DN, Chubukov AV (2011) Manifesto for a higher Tc. Nat Phys 7:272 EP PerspectiveBednorz J, Mueller K (1986) Possible high Tc superconductivity in the Ba-La-Cu-O system. Zeit

Phys B 64:189Berk NF, Schrieffer JR (1966) Effect of ferromagnetic spin correlations on superconductivity. Phys

Rev Lett 17:433–435Bernstein N, Hellberg CS, Johannes MD, Mazin II, Mehl MJ (2015) What superconducts in sulfur

hydrides under pressure and why. Phys Rev B 91:060511Bhaumik A, Sachan R, Narayan J (2017) High-temperature superconductivity in boron-doped q-

carbon. ACS Nano 11(6):5351–5357. PMID:28448115Blase X, Adessi C, Connétable D (2004) Role of the dopant in the superconductivity of diamond.

Phys Rev Lett 93:237004Boeri L, Bachelet G, Cappelluti E, Pietronero L (2002) Small fermi energy and phonon anhar-

monicity in mgb2 and related compounds. Phys Rev B 65:214501Boeri L, Kortus J, Andersen OK (2004) Three-dimensional mgb2-type superconductivity in hole-

doped diamond. Phys Rev Lett 93:237002Boeri L, Cappelluti E, Pietronero L (2005) Small fermi energy, zero-point fluctuations, and

nonadiabaticity in Mgb2. Phys Rev B 71:012501Boeri L, Bachelet GB, Giantomassi M, Andersen OK (2007) Electron-phonon interaction in

graphite intercalation compounds. Phys Rev B 76:064510Boeri L, Dolgov OV, Golubov AA (2008) Is lafeaso1−x fx an electron-phonon superconductor?

Phys Rev Lett 101:026403Boeri L, Calandra M, Mazin II, Dolgov OV, Mauri F (2010) Effects of magnetism and doping on

the electron-phonon coupling in bafe2as2. Phys Rev B 82:020506Borinaga M, Errea I, Calandra M, Mauri F, Bergara A (2016) Anharmonic effects in atomic

hydrogen: superconductivity and lattice dynamical stability. Phys Rev B 93:174308Boschker H, Mannhart J (2017) Quantum-matter heterostructures. Ann Rev Condens Matter Phys

8(1):145–164Bustarret E, Marcenat C, Achatz P, Kacmarcik J, Lévy F, Huxley A, Ortéga L, Bourgeois E, Blase

X, Débarre D, Boulmer J (2006) Superconductivity in doped cubic silicon. Nature 444:465Calandra M, Mauri F (2005) Theoretical explanation of superconductivity in c6Ca. Phys Rev Lett

95:237002Calandra M, Vast N, Mauri F (2004) Superconductivity from doping boron icosahedra. Phys Rev

B 69:224505Calandra M, Profeta G, Mauri F (2010) Adiabatic and nonadiabatic phonon dispersion in a wannier

function approach. Phys Rev B 82:165111Capitani F, Langerome B, Brubach JB, Roy P, Drozdov A, Eremets MI, Nicol EJ, Carbotte JP,

Timusk T (2017) Spectroscopic evidence of a new energy scale for superconductivity in h3s.Nat Phys 13:859 EP Article

Carbotte JP (1990) Properties of boson-exchange superconductors. Rev Mod Phys 62:1027–1157

Understanding Novel Superconductors with Ab Initio Calculations 35

Casula M, Calandra M, Mauri F (2012) Local and nonlocal electron-phonon couplings in k3 piceneand the effect of metallic screening. Phys Rev B 86:075445

Chang KJ, Dacorogna MM, Cohen ML, Mignot JM, Chouteau G, Martinez G (1985) Supercon-ductivity in high-pressure metallic phases of si. Phys Rev Lett 54:2375–2378

Choi JH, David R, Hong S, Cohen ML, Louie SG (2002) The origin of the anomaloussuperconducting properties of MgB2. Nature 418:758

Christoph Heil LB, Bachelet GB (2018) No superconductivity in iron polyhydrides at highpressures. arXiv preprint, arXiv:1804.03572

Chu C, Deng L, Lv B (2015) Hole-doped cuprate high temperature superconductors. Phys CSupercond Appl 514:290–313. Superconducting materials: conventional, unconventional andundetermined

Chubukov AV, Efremov DV, Eremin I (2008) Magnetism, superconductivity, and pairing symmetryin iron-based superconductors. Phys Rev B 78:134512

Cohen M, Anderson P (1972) Superconductivity in d− and f − band metals. Amer Inst of Phys,New York

Csaniy G, Littlewood PB, Nevidomskyy AH, Pickard CJ, Simons BD (2005) The role of theinterlayer state in the electronic structure of superconducting graphite intercalated compounds.Nat Phys 1:42

Cudazzo P, Profeta G, Sanna A, Floris A, Continenza A, Massidda S, Gross EKU (2008) Ab initiodescription of high-temperature superconductivity in dense molecular hydrogen. Phys Rev Lett100:257001

Curtarolo S, Hart GLW, Buongiorno Nardelli M, Mingo N, Sanvito S, Levy O (2013) The high-throughput highway to computational materials design. Nat Mater 12:191–201

Dalladay-Simpson EG, Howie RT (2016) Evidence for a new phase of dense hydrogen above 325gigapascals. Nature 529:63–67

de’ Medici L (2017) Hund’s metals, explained. arXiv/cond-mat/1707.03282de’ Medici L, Mravlje J, Georges A (2011) Janus-faced influence of Hund’s rule coupling in

strongly correlated materials. Phys Rev Lett 107:256401de’ Medici L, Giovannetti G, Capone M (2014) Selective Mott physics as a key to iron

superconductors. Phys Rev Lett 112:177001de’ Medici L (2017) Hund’s induced fermi-liquid instabilities and enhanced quasiparticle interac-

tions. Phys Rev Lett 118:167003Dias RP, Silvera IF (2017) Observation of the Wigner-Huntington transition to metallic hydrogen.

Science 355(6326):715–718Drozdov AP, Eremets MI, Troyan IA, Ksenofontov V, Shylin SI (2015a) Conventional supercon-

ductivity at 203 kelvin at high pressures in the sulfur hydride system. Nature 525:73–76Drozdov A, Eremets MI, Troyan IA (2015b) Superconductivity above 100 K in PH3 at high

pressures. arXiv/cond-mat/1508.06224Duan D, Liu Y, Tian F, Li D, Huang X, Zhao Z, Yu H, Liu B, Tian W, Cui T (2014) Pressure-

induced metallization of dense (h2s)2h2 with high-tc superconductivity. Sci Rep 4:6968Einaga M, Sakata M, Ishikawa T, Shimizu K, Eremets MI, Drozdov AP, Troyan IA, Hirao N,

Ohishi Y (2016) Crystal structure of the superconducting phase of sulfur hydride. Nat Phys12:835 EP

Ekimov EA, Sidorov VA, Bauer ED, Mel’nik N, Curro NJ, Thompson J, Stishov S (2004)Superconductivity in diamond. Nature 428:542

Eliashberg GM (1960) Interactions between electrons and lattice vibrations in a superconductorSov Phys JETP 11:696

Eremets MI, Trojan IA, Medvedev SA, Tse JS, Yao Y (2008) Superconductivity in hydrogendominant materials: silane. Science 319(5869):1506–1509

Eremets M, Troyan I, Drozdov A (2017) Low temperature phase diagram of hydrogen at pressuresup to 380 GPa. A possible metallic phase at 360 GPa and 200 K. arXiv/cond-mat 1601.04479

Eremets M, Drozdov AP (2017) Comments on the claimed observation of the wigner-huntingtontransition to metallic hydrogen. arxiv-condmat/1702.05125

36 L. Boeri

Errea I, Calandra M, Mauri F (2014) Anharmonic free energies and phonon dispersions fromthe stochastic self-consistent harmonic approximation: application to platinum and palladiumhydrides. Phys Rev B 89:064302

Errea I, Calandra M, Pickard CJ, Nelson J, Needs RJ, Li Y, Liu H, Zhang Y, Ma Y, Mauri F (2015)High-pressure hydrogen sulfide from first principles: a strongly anharmonic phonon-mediatedsuperconductor. Phys Rev Lett 114:157004

Errea I, Calandra M, Pickard CJ, Nelson JR, Needs RJ, Li Y, Liu H, Zhang Y, Ma Y, MauriF (2016) Quantum hydrogen-bond symmetrization in the superconducting hydrogen sulfidesystem. Nature 532:81–84

Essenberger F, Buczek P, Ernst A, Sandratskii L, Gross EKU (2012) Paramagnons in FeSe closeto a magnetic quantum phase transition: ab initio study. Phys Rev B 86:060412

Essenberger F, Sanna A, Linscheid A, Tandetzky F, Profeta G, Cudazzo P, Gross EKU (2014)Superconducting pairing mediated by spin fluctuations from first principles. Phys Rev B90:214504

Essenberger F, Sanna A, Buczek P, Ernst A, Sandratskii L, Gross EKU (2016) Ab initio theory ofiron-based superconductors. Phys Rev B 94:014503

Fernandes RM, Chubukov AV, Schmalian J (2014) What drives nematic order in iron-basedsuperconductors? Nat Phys 10:97 EP Review Article

Flores-Livas JA, Sanna A, Graužinyte M, Davydov A, Goedecker S, Marques MAL (2017)Emergence of superconductivity in doped h2o ice at high pressure. Sci Rep, 7:6825 https://doi.org/10.1038/s41598-017-07145-4 arxiv-condmat/1610.04110

Flores-Livas JA, Amsler M, Heil C, Sanna A, Boeri L, Profeta G, Wolverton C, Goedecker S,Gross EKU (2016) Superconductivity in metastable phases of phosphorus-hydride compoundsunder high pressure. Phys Rev B 93:020508

Flores-Livas JA, Sanna A, Drozdov AP, Boeri L, Profeta G, Eremets M, Goedecker S (2017)Interplay between structure and superconductivity: metastable phases of phosphorus underpressure. Phys Rev Mater 1:024802

Fu Y, Du X, Zhang L, Peng F, Zhang M, Pickard CJ, Needs RJ, Singh DJ, Zheng W, Ma Y (2016)High-pressure phase stability and superconductivity of pnictogen hydrides and chemical trendsfor compressed hydrides. Chem Mater 28(6):1746–1755

Gaspari GD, Gyorffy BL (1972) Electron-phonon interactions, d resonances, and superconductiv-ity in transition metals. Phys Rev Lett 28:801–805

Ge JF, Liu ZL, Liu C, Gao CL, Qian D, Xue QK, Liu Y, Jia JF (2014) Superconductivity above100 k in single-layer FeSe films on doped srtio3. Nat Mater 14:285 EP

Georges A, de’ Medici L, Mravlje J (2013) Strong correlations from Hund’s coupling. Ann RevCondens Matter Phys 4(1):137–178

Giustino F (2017) Electron-phonon interactions from first principles. Rev Mod Phys 89:015003Giustino F, ML Cohen, Louie SG (2007a) Electron-phonon interaction using wannier functions.

Phys Rev B 76:165108Giustino F, Yates JR, Souza I, Cohen ML, Louie SG (2007b) Electron-phonon interaction

via electronic and lattice Wannier functions: Superconductivity in boron-doped diamondreexamined. Phys Rev Lett 98:047005

Goncharov AF, Struzhkin VV (2017) Comment on observation of the wigner-huntington transitionto metallic hydrogen. arXiv preprint arXiv:1702.04246

Goncharov AF, Lobanov SS, Kruglov I, Zhao XM, Chen XJ, Oganov AR, Konôpková Z,Prakapenka VB (2016) Hydrogen sulfide at high pressure: change in stoichiometry. Phys Rev B93:174105

Gonnelli RS, Daghero D, Ummarino GA, Stepanov VA, Jun J, Kazakov SM, Karpinski J (2002)Direct evidence for two-band superconductivity in mgb2 single crystals from directional point-contact spectroscopy in magnetic fields. Phys Rev Lett 89:247004

Gor’kov LP, Kresin VZ (2018) Colloquium: high pressure and road to room temperaturesuperconductivity. Rev Mod Phys 90:011001

Graser S, Maier TA, Hirschfeld PJ, Scalapino DJ (2009) Near-degeneracy of several pairingchannels in multiorbital models for the fe pnictides. New J Phys 11(2):025016

Understanding Novel Superconductors with Ab Initio Calculations 37

Grimaldi C, Pietronero L, Straessler S (1995) Nonadiabatic superconductivity: electron-phononinteraction beyond Migdal’s theorem. Phys Rev Lett 75:1158

Gunnarsson O (1997) Superconductivity in fullerides. Rev Mod Phys 69:575–606Gurevich A (2011) To use or not to use cool superconductors? Nature Mat 10:255Hamlin J (2015) Superconductivity in the metallic elements at high pressures. Phys C Supercond

Appl 514:59–76. Superconducting materials: conventional, unconventional and undeterminedHansmann P, Arita R, Toschi A, Sakai S, Sangiovanni G, Held K (2010) Dichotomy between

large local and small ordered magnetic moments in iron-based superconductors. Phys Rev Lett104:197002

Haule K, Kotliar G (2009) Coherence-incoherence crossover in the normal state of iron oxypnic-tides and importance of Hund’s rule coupling. New J Phys 11(2):025021

Heil C, Boeri L (2015) Influence of bonding on superconductivity in high-pressure hydrides. PhysRev B 92:060508

Hirschfeld PJ, Korshunov MM, Mazin II (2011) Gap symmetry and structure of fe-basedsuperconductors. Rep Prog Phys 74(12):124508

Hoesch M, Fukuda T, Mizuki J, Takenouchi T, Kawarada H, Sutter JP, Tsutsui S, Baron AQR,Nagao M, Takano Y (2007) Phonon softening in superconducting diamond. Phys Rev B75:140508

Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev 136:B864Huang D, Hoffman JE (2017) Monolayer FeSe on srtio3. Ann Rev Condens Matter Phys 8(1):311–

336Johnston DC (2010) The puzzle of high temperature superconductivity in layered iron pnictides

and chalcogenides. Adv Phys 59(6):803–1061Jones RO, Gunnarsson O (1989) The density functional formalism, its applications and prospects.

Rev Mod Phys 61:689–746José FL, Sanna A, Gross EKU (2016) High temperature superconductivity in sulfur and selenium

hydrides at high pressure. Eur Phys J B 89(3):63Kamihara Y, Watanabe T, Hirano M, Hosono H (2008) Iron-based layered superconductor

LaOFeAs. J Am Chem Soc 130(11):3296–3297Kasahara Y, Kuroki K, Yamanaka S, Taguchi Y (2015) Unconventional superconductivity in

electron-doped layered metal nitride halides mnx (m=ti, zr, hf; x=cl, br, i). Phys C SupercondAppl 514:354–367. Superconducting materials: conventional, unconventional and undeter-mined

Kim JS, Boeri L, Kremer RK, Razavi FS (2006) Effect of pressure on superconducting ca-intercalated graphite Cac6. Phys Rev B 74:214513

Kim JS, Boeri L, O’Brien JR, Razavi FS, Kremer RK (2007) Superconductivity in heavy alkaline-earth intercalated graphites. Phys Rev Lett 99:027001

Klemm RA (2015) Pristine and intercalated transition metal dichalcogenide superconductors. PhysC Supercond Appl 514:86–94. Superconducting materials: conventional, unconventional andundetermined

Kolmogorov AN, Curtarolo S (2006) Prediction of different crystal structure phases in metalborides: a lithium monoboride analog to Mgb2. Phys Rev B 73:180501

Kuroki K, Onari S, Arita R, Usui H, Tanaka Y, Kontani H, Aoki H (2008) Unconventional pairingoriginating from the disconnected fermi surfaces of superconducting lafeaso1−x fx . Phys RevLett 101:087004

Kuroki K, Usui H, Onari S, Arita R, Aoki H (2009) Pnictogen height as a possible switch betweenhigh-Tc nodeless and low-Tc nodal pairings in the iron-based superconductors. Phys Rev B79:224511

Kokail C, Heil C, Boeri L (2016) Search for high-Tc conventional superconductivity at megabarpressures in the lithium-sulfur system. Phys Rev B 94:060502

Kokail C, von der Linden W, Boeri L (2017) Prediction of high-Tc conventional superconductivityin the ternary lithium borohydride system. Phys Rev Mater 1:074803

Kong Y, Dolgov OV, Jepsen O, Andersen OK (2001) Electron-phonon interaction in the normaland superconducting states of MgB2. Phys Rev B 64(2):020501

38 L. Boeri

Kortus J, Mazin II, Belashchenko KD, VP Antropov, Boyer LL (2001) Superconductivity ofmetallic boron in mgb2. Phys Rev Lett 86:4656–4659

Kotliar G, Savrasov SY, Haule K, Oudovenko VS, Parcollet O, Marianetti CA (2006) Electronicstructure calculations with dynamical mean-field theory. Rev Mod Phys 78:865–951

Kvashnin AG, Kruglov IA, Semenok DV, Oganov AR (2018) Iron superhydrides feh5 and feh6:stability, electronic properties, and superconductivity. J Phys Chem C 122(8):4731–4736

Lazzeri M, Calandra M, Mauri F (2003) Anharmonic phonon frequency shift in mgb2. Phys RevB 68:220509

Lee KW, Pickett WE (2004) Superconductivity in boron-doped diamond. Phys Rev Lett 93:237003Li Y, Gao G, Xie Y, Ma Y, Cui T, Zou G (2010) Superconductivity at ˜100 k in dense sih4(h2)2

predicted by first principles. Proc Nat Acad Sci 107(36):15708–15711Linschied A, Sanna A, Gross EKU (2015a) Ab-initio calculation of a pb single layer on a si

substrate: two-dimensionality and superconductivity. airXiv cond-mat/1503.00977Linscheid A, Sanna A, Floris A, Gross EKU (2015b) First-principles calculation of the real-space

order parameter and condensation energy density in phonon-mediated superconductors. PhysRev Lett 115:097002

Liu AY, Mazin II, Kortus J (2001) Beyond Eliashberg superconductivity in mgb2: anharmonicity,two-phonon scattering, and multiple gaps. Phys Rev Lett 87:087005

Liu H, Naumov II, Hoffmann R, Ashcroft NW, Hemley RJ (2017) Potential high-tc super-conducting lanthanum and yttrium hydrides at high pressure. Proc Natl Acad Sci USA114(27):6990–6995

Ludbrook BM, Levy G, Nigge P, Zonno M, Schneider M, Dvorak DJ, Veenstra CN, ZhdanovichS, Wong D, Dosanjh P, Straßer C, Stöhr A, Forti S, Ast CR, Starke U, Damascelli A(2015) Evidence for superconductivity in li-decorated monolayer graphene. Proc Nat Acad Sci112(38):11795–11799

Lüders M, Marques MAL, Lathiotakis NN, Floris A, Profeta G, Fast L, Continenza A, MassiddaS, Gross EKU (2005) Ab initio. Phys Rev B 72:024545

Lukas H, Fries SG, Sundman B (2007) Computational thermodynamics: the Calphad method.Cambridge University Press, Cambridge

Majumdar A, Tse JS, Wu M, Yao Y (2017) Superconductivity in feh5. Phys Rev B 96:201107Mandal S, Cohen RE, Haule K (2014) Strong pressure-dependent electron-phonon coupling in

FeSe. Phys Rev B 89:220502Margine ER, Giustino F (2013) Anisotropic migdal-Eliashberg theory using wannier functions.

Phys Rev B 87:024505Marsiglio F, Carbotte JP (1986) Strong-coupling corrections to Bardeen-Cooper-Schrieffer ratios.

Phys Rev B 33:6141–6146Marques MAL, Lüders M, Lathiotakis NN, Profeta G, Floris A, Fast L, Continenza A, Gross EKU,

Massidda S (2005) Ab initio. Phys Rev B 72:024546Marzari N, Mostofi AA, Yates JR, Souza I, Vanderbilt D (2012) Maximally localized Wannier

functions: theory and applications. Rev Mod Phys 84:1419–1475Matthias BT, Geballe TH, Andres K, Corenzwit E, Hull GW, Maita JP (1968) Superconductivity

and antiferromagnetism in boron-rich lattices. Science 159(3814):530–530Mazin II, Johannes MD (2008) A key role for unusual spin dynamics in ferropnictides. Nat Phys

5:141 EP ArticleMazin II, Andersen OK, Jepsen O, Dolgov OV, Kortus J, Golubov AA, Kuz’menko AB, van der

Marel D (2002) Superconductivity in mgb2: clean or dirty? Phys Rev Lett 89:107002Mazin II, Singh DJ, Johannes MD, Du MH (2008a) Unconventional superconductivity with a sign

reversal in the order parameter of lafeaso1−x fx . Phys Rev Lett 101:057003Mazin II, Johannes MD, Boeri L, Koepernik K, Singh DJ (2008b) Problems with reconciling

density functional theory calculations with experiment in ferropnictides. Phys Rev B 78:085104Meissner W, Ochsenfeld R (1933) Ein neuer effekt bei eintritt dernsupraleitfŁaehigkeit. Naturwis-

senschaften 21:787McMahon JM, Ceperley DM (2011) High-temperature superconductivity in atomic metallic

hydrogen. Phys Rev B 84:144515

Understanding Novel Superconductors with Ab Initio Calculations 39

McMahon JM, Morales MA, Pierleoni C, Ceperley DM (2012) The properties of hydrogen andhelium under extreme conditions. Rev Mod Phys 84:1607–1653

Migdal A (1958) Migdal’s theorem. Sov Phys JETP 34:996Miyake T, Nakamura K, Arita R, Imada M (2010) Comparison of ab initio low-energy models for

LaFePo, lafeaso, bafe2as2, LiFeAs, FeSe, and FeTe: electron correlation and covalency. J PhysSoc Japan 79(4):044705

Monni M, Bernardini F, Sanna A, Profeta G, Massidda S (2017) Origin of the critical temperaturediscontinuity in superconducting sulfur under high pressure. Phys Rev B 95:064516

Morel P, Anderson PW (1962) Calculation of the superconducting state parameters with retardedelectron-phonon interaction. Phys Rev 125:1263–1271

Nagamatsu J, Nakagawa N, Muranaka T, Zenitani Y, Akimitsu J (2001) Superconductivity at 39 Kin magnesium diboride. Nature (London) 410:63

Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, Grigorieva IV, Firsov AA(2004) Electric field effect in atomically thin carbon films. Science 306(5696):666–669

Oliveira LN, Gross EKU, Kohn W (1988) Density-functional theory for superconductors. PhysRev Lett 60:2430–2433

Onnes HK (1913) The resistance of pure mercury at helium temperatures. Commun Phys Lab UnivLeiden 133:37

Ortenzi L, Mazin II, Blaha P, Boeri L (2012) Accounting for spin fluctuations beyond local spindensity approximation in the density functional theory. Phys Rev B 86:064437

Pace EJ, Binns J, Alvarez MP, Dalladay-Simpson P, Gregoryanz E, Howie RT (2017) Synthesisand stability of hydrogen selenide compounds at high pressure. J Chem Phys 147(18):184303

Paglione J, Greene RL (2010) High-temperature superconductivity in iron-based materials. NatPhys 6:645 EP Review article

Pépin C, Loubeyre P, Occelli F, Dumas P (2015) Synthesis of lithium polyhydrides above 130 GPAat 300 K. Proc Nat Acad Sci 112(25):7673–7676

Pépin CM, Geneste G, Dewaele A, Mezouar M, Loubeyre P (2017) Synthesis of feh5: a layeredstructure with atomic hydrogen slabs. Science 357(6349):382–385

Pietronero L, Straessler S, Grimaldi C (1995) Nonadiabatic superconductivity I. Vertex correctionsfor the electron-phonon interactions. Phys Rev B 52:10516

Platt C, Thomale R, Honerkamp C, Zhang SC, Hanke W (2012) Mechanism for a pairing statewith time-reversal symmetry breaking in iron-based superconductors. Phys Rev B 85:180502

Profeta G, Franchini C, Lathiotakis NN, Floris A, Sanna A, Marques MAL, Lüders M, MassiddaS, Gross EKU, Continenza A (2006) Superconductivity in lithium, potassium, and aluminumunder extreme pressure: a first-principles study. Phys Rev Lett 96:047003

Profeta G, Calandra M, Mauri F (2012) Phonon-mediated superconductivity in graphene by lithiumdeposition. Nature Phys 8:131–134

Quan Y, Pickett WE (2016) Van hove singularities and spectral smearing in high-temperaturesuperconducting h3S. Phys Rev B 93:104526

Rosner H, Kitaigorodsky A, Pickett WE (2002) Prediction of high Tc superconductivity in hole-doped libc. Phys Rev Lett 88:127001

Runge E, Gross EKU (1984) Density-functional theory for time-dependent systems. Phys Rev Lett52:997–1000

Sanna A (2017) Introduction to superconducting density functional theory. https://www.cond-mat.de/events/correl17/manuscripts/sanna.pdf

Sanna A, Profeta G, Floris A, Marini A, Gross EKU, Massidda S (2007) Anisotropic gap ofsuperconducting cac6: a first-principles density functional calculation. Phys Rev B 75:020511

Sanna A, Flores-Livas JA, Davydov A, Profeta G, Dewhurst K, Sharma S, Gross EKU (2018) Abinitio Eliashberg theory: making genuine predictions of superconducting features. J Phys SocJapan 87(4):041012

Sano W, Koretsune T, Tadano T, Akashi R, Arita R (2016) Effect of van hove singularities onhigh-Tc superconductivity in h3S. Phys Rev B 93:094525

Sato M, Ando Y (2017) Topological superconductors: a review. Rep Prog Phys 80(7):076501

40 L. Boeri

Satta G, Profeta G, Bernardini F, Continenza A, Massidda S (2001) Electronic and structuralproperties of superconducting mgb2, casi2, and related compounds. Phys Rev B 64:104507

Savini G, Ferrari AC, Giustino F (2010) First-principles prediction of doped graphane as a high-temperature electron-phonon superconductor. Phys Rev Lett 105:037002

Savrasov SY, Savrasov DY (1996) Electron-phonon interactions and related physical properties ofmetals from linear-response theory. Phys Rev B 54:16487

Scalapino D (1969) Superconductivity. Dekker, New York, pp 1–92Schilling A, Cantoni M, Guo JD, Ott HR (1993) Superconductivity above 130 K in the hg-ba-ca-

cu-o system. Nature 363:56 EPSchickling T, Gebhard F, Bünemann J, Boeri L, Andersen OK, Weber W (2012) Gutzwiller theory

of band magnetism in laofeas. Phys Rev Lett 108:036406Shamp A, Terpstra T, Bi T, Falls Z, Avery P, Zurek E (2016) Decomposition products of

phosphine under pressure: Ph2 stable and superconducting? J Am Chem Soc 138(6):1884–1892.PMID:26777416

Sharma S, Gross EKU, Sanna A, Dewhurst JK (2018) Source-free exchange-correlation magneticfields in density functional theory. J Chem Theory Comput 14(3):1247–1253. PMID:29420031

Skornyakov SL, Katanin AA, Anisimov VI (2011) Linear-temperature dependence of staticmagnetic susceptibility in lafeaso from dynamical mean-field theory. Phys Rev Lett 106:047007

Sohier T, Calandra M, Mauri F (2015) Density-functional calculation of static screening intwo-dimensional materials: the long-wavelength dielectric function of graphene. Phys Rev B91:165428

Sohier T, Calandra M, Mauri F (2017) Density functional perturbation theory for gated two-dimensional heterostructures: theoretical developments and application to flexural phonons ingraphene. Phys Rev B 96:075448

Struzhkin VV, Kim DY, Stavrou E, Muramatsu T, Mao Hk, Pickard CJ, Needs RJ, PrakapenkaVB, Goncharov AF (2016) Synthesis of sodium polyhydrides at high pressures. Nat Commun7:12267 EP Article

Subedi A, Boeri L (2011) Vibrational spectrum and electron-phonon coupling of doped solidpicene from first principles. Phys Rev B 84:020508

Subedi A, Zhang L, Singh DJ, Du MH (2008) Density functional study of FeS, FeSe, and FeTe:electronic structure, magnetism, phonons, and superconductivity. Phys Rev B 78:134514

Suhl H, Matthias BT, Walker LR (1959) Bardeen-Cooper-Schrieffer theory of superconductivityin the case of overlapping bands. Phys Rev Lett 3:552–554

Takano Y, Nagao M, Sakaguchi I, Tachiki M, Hatano T, Kobayashi K, Umezawa H, KawaradaH (2004) Superconductivity in diamond thin films well above liquid helium temperature. ApplPhys Lett 85(14):2851–2853

Thomale R, Platt C, Hu J, Honerkamp C, Bernevig BA (2009) Functional renormalization-groupstudy of the doping dependence of pairing symmetry in the iron pnictide superconductors. PhysRev B 80:180505

Toschi A, Arita R, Hansmann P, Sangiovanni G, Held K (2012) Quantum dynamical screening ofthe local magnetic moment in fe-based superconductors. Phys Rev B 86:064411

Vignale G, Singwi KS (1985) Effective two-body interaction in coulomb fermi liquids. Phys RevB 32:2156–2166

Wang H, Tse JS, Tanaka K, Iitaka T, Ma Y (2012) Superconductive sodalite-like clathrate calciumhydride at high pressures. Proc Natl Acad Sci USA 109(17):6463–6466

Weller TE, Ellerby M, Saxena SS, Smith RP, Skipper NT (2005) Superconductivity in theintercalated graphite compounds C6Yb and C6Ca. Nature Phys 1:39

Werner P, Gull E, Troyer M, Millis AJ (2008) Spin freezing transition and non-fermi-liquid self-energy in a three-orbital model. Phys Rev Lett 101:166405

Woodley SM, Catlow R (2008) Crystal structure prediction from first principles. Nature Mat7:937–946

Wu MK, Ashburn JR, Torng CJ, Hor PH, Meng RL, Gao L, Huang ZJ, Wang YQ, Chu CW(1987) Superconductivity at 93 K in a new mixed-phase y-ba-cu-o compound system at ambientpressure. Phys Rev Lett 58:908–910

Understanding Novel Superconductors with Ab Initio Calculations 41

Yao Y, Klug DD, Sun J, Martonák R (2009) Structural prediction and phase transformationmechanisms in calcium at high pressure. Phys Rev Lett 103:055503

Ye JT, Zhang YJ, Akashi R, Bahramy MS, Arita R, Iwasa Y (2012) Superconducting dome in agate-tuned band insulator. Science 338(6111):1193–1196

Yi M, Zhang Y, Shen ZX, Lu D (2017) Role of the orbital degree of freedom in iron-basedsuperconductors. npj Quantum Mater 2(1):57

Yin ZP, Haule K, Kotliar G (2011) Kinetic frustration and the nature of the magnetic andparamagnetic states in iron pnictides and iron chalcogenides. Nat Mater 10:932

Yin ZP, Kutepov A, Kotliar G (2013) Correlation-enhanced electron-phonon coupling: applicationsof gw and screened hybrid functional to bismuthates, chloronitrides, and other high-Tc

superconductors. Phys Rev X 3:021011Yokoya T, Nakamura T, Matsushita T, Muro T, Takano Y, Nagao M, Takenouchi T, Kawarada

H, Oguchi T (2005) Origin of the metallic properties of heavily boron-doped superconductingdiamond. Nature 438:647 EP

Zhang T, Cheng P, Li WJ, Sun YJ, Wang G, Zhu XG, He K, Wang L, Ma X, Chen X, Wang Y, LiuY, Lin HQ, Jia JF, Xue QK (2010) Superconductivity in one-atomic-layer metal films grown onsi(111). Nat Phys 6:104 EP

Zhang L, Wang Y, Lv J, Ma Y (2017) Materials discovery at high pressures. Nat Rev Mat 2:17005EP. Review article

Zurek E, Hoffmann R, Ashcroft N, Oganov AR, Lyakhov AO (2009) A little bit of lithium does alot for hydrogen. Proc Natl Acad Sci USA 106(42):17640–17643