sdwave multiband Eliashberg theory for the iron pnictides G.A. Ummarino

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s±+dwave multiband Eliashberg theory for the iron

pnictides

G.A. Ummarino

E-mail: giovanni.ummarino@polito.it

Istituto di Ingegneria e Fisica dei Materiali, Dipartimento di Scienza Applicata e

Tecnologia, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy;

Department of Semiconductor Quantum Electronics / N.G.Basov High School of

Physicists / Institute of Engineering Physics for Biomedicine, National Research

Nuclear University MEPhI, Moscow Engineering Physics Institute, Kashira Hwy 31,

Moskva 115409, Russia

D. Torsello

E-mail: daniele.torsello@polito.it

Dipartimento di Scienza Applicata e Tecnologia, Politecnico di Torino, Corso Duca

degli Abruzzi 24, 10129 Torino, Italy; Istituto Nazionale di Fisica Nucleare, Sezione

di Torino, Torino 10125, Italy

Abstract. We calculated the critical temperature in the framework of s±+d-wave

multiband Eliashberg theory. We have solved these equations numerically to see

at what values of the input parameters there is a solution with a non-zero critical

temperature and what is the symmetry of the order parameter of this solution. For

our model we consider the pnictide case with simpliﬁcations that allow us to obtain

the most general possible information. For selected and representative cases in which

the order parameter has s±+dsymmetry, we calculated the superconducting density

of states, the temperature dependence of the gaps, and the superﬂuid density so that

comparison with experimental data can be made. Finally, we show that such a system

has only a twofold in-plane symmetry and undergoes a transition from nodal to fully

gapped with increasing temperature.

arXiv:2210.06231v2 [cond-mat.supr-con] 26 Oct 2022

s±+dwave multiband Eliashberg theory for the iron pnictides 2

1. INTRODUCTION

In the last thirty years, the study of superconducting materials had an astounding

development. The starting point was ordinary low-temperature superconductors with a

electron-phonon mechanism, a conducting band and isotropic order parameter (swave).

Then, high-temperature cuprate superconductors [1] with a non-phonon mechanism

and a non-isotropic order parameter (d-wave)[2, 3] were ﬁrst discovered followed by

multiband phononic materials such as fullerenes [4, 5] and magnesium diboride [6, 7, 8].

Finally, iron-based compounds [9, 10, 11] appeared on the scene: multiband non-

phononic materials with s±wave symmetry of the order parameter. The minimal model

for the latter class of materials consists of only 2 conductivity bands (but even 5 bands

were found to contribute to superconductivity, as in LiF eAs [12]) with an isotropic

order parameter with a phase diﬀerence of πbetween each other [13]. In recent years,

observations of possible mixed sand d−wave behaviour was proposed for non-phononic

multiband superconductors [14, 15, 16, 17], therefore we thought to develop, through

Eliashberg theory, a possible general case considering in each band an order parameter

with two components: one isotropic (swave) and one anisotropic (dwave), realizing

therefore a multiband s+dwave where the two isotropic components (swave) are

out of phase by π, called s±+d. We consider the speciﬁc case of iron-based pnictide

compounds, but our conclusions can be directly generalized to all systems in which the

electron-boson coupling can be described in a similar way. In the following, we will write

the Eliashberg equations for this new situation and we will see if there is a plausible

range of physical input parameters (essentialy the electron-boson intra and inter band

coupling constant λs,d

jk ) where it is actually possible to have one or two order parameters

with the two components (s±and d) at the same time. Then we will try to calculate

physical observables that in the case (s±+d) clearly diﬀer from the pure s±and pure d

cases in order to propose experimental veriﬁcations for this model, and discuss situations

in which they could occur.

2. MODEL: TWO-BAND ELIASHBERG EQUATIONS

We study a superconducting material with two conductivity parabolic bands (the

simplest multiband case): we consider band one to be a hole band and band two an

electron band. The cases with more than two bands can be reduced to eﬀective two bands

systems where the values of the coupling constant loose their physical meaning [18, 19],

therefore this can be considered as a general scheme for multiband superconductors.

Our investigation starts from the consideration that, at the moment, the most studied

multiband superconductors are the iron-pnictides where the mediation for the Cooper

pairs is provided by antiferromagnetic spin ﬂuctuactions, so this is the speciﬁc system

we consider. Their lattice is described by the tetragonal symmetry at high temperature

and by the orthorhombic symmetry in cryogenic conditions, which we study. In this

case the isotropic part is repulsive (in the iron pnictides the contribution of phonons

s±+dwave multiband Eliashberg theory for the iron pnictides 3

is very small [20] so we will put, in ﬁrst approximation, this contribution equal to

zero). The electron-boson spectral functions (one for each band) have two components:

one isotropic (swave) and one anisotropic (dwave), yielding an overall anisotropic

s+delectron-boson interaction that is allowed only in the orthorhombic state. This

happens because both the s- and d-wave states in the orthorhombic case belong to the

same irreducible representation (A1g) [21]. Solving the Eliashberg equations yields the

superconducting gaps, and the more general solution has a two sand two dcomponents

(one for each band for each symmetry) where the two scomponents are opposite in

sign (s±wave), so the general solution for the order parameter is an s±+dwave. Both

the sand dwave states are also compatible with a tetragonal symmetry of the system,

whereas the mixed order can exist only in the orthorhombic phase. It is important to

note that the k-dependent Eliashberg equations are nonlinear, for this reason the gaps

can deviate from the symmetry of the interaction. Therefore, despite the symmetry of

the interaction being ﬁxed, that of the superconducting state is not. However the result

must still be compatible with the orthorhombic state. We calculated the experimental

critical temperatures and the superconducting gaps by solving the s±+d-wave two-band

Eliashberg equations [22, 23, 24, 25, 26, 27]. In this case, four coupled equations for the

gaps ∆s,d

k(iωn) and four for the renormalization functions Zs,d

k(iωn) have to be solved

self consistently (ωndenotes the Matsubara frequencies and k= 1,2 the band index).

The s±+d-wave two-band Eliashberg equations (assuming that the Migdal theorem

works [28]) in the imaginary axis representation, and in the compact shape (where the

sand dcomponents are not separated) read:

ωnZk(ωn, φ) = ωn+πT X

j,m Z2π

dφ0

2πΛkj (ωn, ωm, φ, φ0)NZ

j(ωm, φ0) (1)

Zk(ωn, φ)∆k(ωn, φ) = πT X

j,m Z2π

dφ0

2π[Λkj (ωn, ωm, φ, φ0)−µ∗

kj (φ, φ0)] ×

×Θ(ωc− |ωm|)N∆

kj (ωm, φ0) (2)

where Θ(ωc−ωm) is the Heaviside function, ωcis a cut-oﬀ energy and

Λkj (ωn, ωm, φ, φ0)=2Z+∞

ΩdΩα2Fkj (Ω, φ, φ0)/[(ωn−ωm)2+ Ω2] (3)

j(ωm, φ) = ωm

pω2

m+ ∆j(ωm, φ)2(4)

N∆

j(ωm, φ) = ∆j(ωm, φ)

pω2

m+ ∆j(ωm, φ)2(5)

We assume [22, 23, 24, 25, 26] that the electron-boson spectral functions

α2(Ω)Fkj (Ω, φ, φ0) and the Coulomb pseudopotential µ∗

kj (φ, φ0) at the lowest order to

contain separated sand d-wave contributions,

α2Fjk (Ω, φ, φ0) = λs

jk α2Fs

jk (Ω) + λd

jk α2Fd(Ω)cos(2φ)cos(2φ0) (6)

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摘要：

s+dwavemultibandEliashbergtheoryfortheironpnictidesG.A.UmmarinoE-mail:giovanni.ummarino@polito.itIstitutodiIngegneriaeFisicadeiMateriali,DipartimentodiScienzaApplicataeTecnologia,PolitecnicodiTorino,CorsoDucadegliAbruzzi24,10129Torino,Italy;DepartmentofSemiconductorQuantumElectronics/N.G.BasovHighS...

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sdwave multiband Eliashberg theory for the iron pnictides G.A. Ummarino

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