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*T _{c}* uperconductors can be described by a one band d wave Eliashberg theory where the mechanism of superconducting coupling is mediated by antiferromagnetic spin fluctuations and whose characteristic energy Ω

Eliashberg's theory [1] was born as a generalization of the BCS theory to explain some anomalies in the experimental data concerning lead. Subsequently, it was seen that the theory can be successfully applied to explain the experimental data of practically almost all superconducting materials [2,3], first of all low *T _{c}* phononic superconductors [4], then magnesium diboride [5,6], graphite intercalated compound CaC6 [7], iron-based superconductors [8-12]. This theory can be applied to describe particular systems such as proximized systems [13] and field effect junctions [14-16]. For what concerns the high

The one-band d-wave Eliashberg equations [23,30-35] are two coupled equations: one for the gap ∆(i*ω _{n},φ*) and one for the renormalization functions Z(i

${\omega}_{n}Z\mathrm{(}{\omega}_{n}\mathrm{,}\varphi \mathrm{)=}{\omega}_{n}+\pi T{\displaystyle \sum _{m}}{\displaystyle {\int}_{0}^{2\pi}}\frac{d{\varphi}^{\prime}}{2\pi}\Lambda \mathrm{(}{\omega}_{n}\mathrm{,}{\omega}_{m}\mathrm{,}\varphi \mathrm{,}{\varphi}^{\prime}\mathrm{)}{N}_{Z}\mathrm{(}{\omega}_{m}\mathrm{,}{\varphi}^{\prime}\mathrm{)}\text{(1)}$

$Z\mathrm{(}{\omega}_{n}\mathrm{,}\varphi \mathrm{)}\Delta \mathrm{(}{\omega}_{n}\mathrm{,}\varphi \mathrm{)=}\pi T{\displaystyle \sum _{m}}{\displaystyle {\int}_{0}^{2\pi}}\frac{d{\varphi}^{\prime}}{2\pi}\mathrm{[}\Lambda \mathrm{(}{\omega}_{n}\mathrm{,}{\omega}_{m}\mathrm{,}\varphi \mathrm{,}{\varphi}^{\prime}\mathrm{)}-{\mu}^{\mathrm{*}}\mathrm{(}\varphi \mathrm{,}{\varphi}^{\prime}\mathrm{)]}\times $

$\times \Theta \mathrm{(}{\omega}_{c}-\mathrm{|}{\omega}_{m}\mathrm{|)}{N}_{\Delta}\mathrm{(}{\omega}_{m}\mathrm{,}{\varphi}^{\prime}\mathrm{)}\text{(2)}$

where
$\Theta \mathrm{(}{\omega}_{c}-{\omega}_{m}\mathrm{)}$
is the Heaviside function, *ω _{c}* is cut-off energy and

$\Lambda \mathrm{(}{\omega}_{n}\mathrm{,}{\omega}_{m}\mathrm{,}\varphi \mathrm{,}{\varphi}^{\prime}\mathrm{)=2}{\displaystyle {\int}_{0}^{+\infty}}\Omega d\Omega {\alpha}^{2}F\mathrm{(}\Omega \mathrm{,}\varphi \mathrm{,}{\varphi}^{\prime}\mathrm{)/[(}{\omega}_{n}-{\omega}_{m}{\mathrm{)}}^{2}+{\Omega}^{2}\mathrm{]}\text{(3)}$

${N}_{Z}\mathrm{(}{\omega}_{m}\mathrm{,}\varphi \mathrm{)=}\frac{{\omega}_{m}}{\sqrt{{\omega}_{m}^{2}+\Delta {\mathrm{(}{\omega}_{m}\mathrm{,}\varphi \mathrm{)}}^{2}}}\text{(4)}$

${N}_{\Delta}\mathrm{(}{\omega}_{m}\mathrm{,}\varphi \mathrm{)=}\frac{\Delta \mathrm{(}{\omega}_{m}\mathrm{,}\varphi \mathrm{)}}{\sqrt{{\omega}_{m}^{2}+\Delta {\mathrm{(}{\omega}_{m}\mathrm{,}\varphi \mathrm{)}}^{2}}}\text{(5)}$

We assume [2,23,30-35] that the electron boson spectral function
${\alpha}^{2}\mathrm{(}\Omega \mathrm{)}F\mathrm{(}\Omega \mathrm{,}\varphi \mathrm{,}{\varphi}^{\prime}\mathrm{)}$
and the Coulomb pseudopotential
${\mu}^{\mathrm{*}}\mathrm{(}\varphi \mathrm{,}{\varphi}^{\prime}\mathrm{)}$
at the lowest order contain separated *s* and *d* -wave contributions,

${\alpha}^{2}F\mathrm{(}\Omega \mathrm{,}\varphi \mathrm{,}{\varphi}^{\prime}\mathrm{)=}{\lambda}_{s}{\alpha}^{2}{F}_{s}\mathrm{(}\Omega \mathrm{)}+{\lambda}_{d}{\alpha}^{2}{F}_{d}\mathrm{(}\Omega \mathrm{)}\sqrt{2}cos\mathrm{(2}\varphi \mathrm{)}\sqrt{2}cos\mathrm{(2}{\varphi}^{\prime}\mathrm{)}\text{(6)}$

${\mu}^{\mathrm{*}}\mathrm{(}\varphi \mathrm{,}{\varphi}^{\prime}\mathrm{)=}{\mu}_{s}^{\mathrm{*}}+{\mu}_{d}^{\mathrm{*}}\sqrt{2}cos\mathrm{(2}\varphi \mathrm{)}\sqrt{2}cos\mathrm{(2}{\varphi}^{\prime}\mathrm{)}\text{(7)}$

as well as the self-energy functions:

$Z\mathrm{(}{\omega}_{n}\mathrm{,}\varphi \mathrm{)=}{Z}_{s}\mathrm{(}{\omega}_{n}\mathrm{)}+{Z}_{d}\mathrm{(}{\omega}_{n}\mathrm{)}cos\mathrm{(2}\varphi \mathrm{)}\text{(8)}$

$\Delta \mathrm{(}{\omega}_{n}\mathrm{,}\varphi \mathrm{)=}{\Delta}_{s}\mathrm{(}{\omega}_{n}\mathrm{)}+{\Delta}_{d}\mathrm{(}{\omega}_{n}\mathrm{)}cos\mathrm{(2}\varphi \mathrm{)}\text{(9)}$

We put the factor
$\sqrt{2}$
inside the definition ∆_{d}(*ω _{n}*) because, experimentally, the peak of the density of the state is, usually, identified ∆

We fix three different critical temperatures (70 K, 90 K and 110 K) and for any particular critical temperature, we choose different values *λ _{s}* and determine which value

${\lambda}_{d}\mathrm{=0.616}{\lambda}_{s}+0.732\text{(10)}$

These results are general and do not depend on the particular shape of the electron-boson spectral function. If we change the shape of the electron-boson spectral function and we choose, for example,
${\alpha}^{2}{F}_{s\mathrm{,}d}\mathrm{(}\Omega \mathrm{)=0.5}{\Omega}_{0}\delta \mathrm{(}\Omega -{\Omega}_{0}\mathrm{)}$
we find that the linear link between *λ _{d}* and

${k}_{B}{T}_{c}\mathrm{=}{\Omega}_{0}exp\mathrm{(}-\frac{1+{\lambda}_{s}}{2{\lambda}_{d}}\mathrm{)}\text{(11)}$

The problem is that the MacMillan equation works just in a weak coupling regime. Now we solve, for each couple of *λ _{d}* and

In this article, it has been shown that one band *d*-wave Eliashbeg's theory presents universal aspects as the linear link between *λ _{d}* and

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