# The Road Map toward Room-Temperature Superconductivity: Manipulating Different Pairing Channels in Systems Composed of Multiple Electronic Components

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## Abstract

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_{C}is possible in a system of multiple electronic components in comparison with a single component system, many different road maps for room temperature superconductivity have been proposed for a variety of multicomponent scenarios. Here we focus on the scenario where the first electronic component is assumed to have a vanishing Fermi velocity corresponding to a case of the intermediate polaronic regime, and the second electronic component is in the weak coupling regime with standard high Fermi velocity using a mean field theory for multiband superconductivity. This roadmap is motivated by compelling experimental evidence for one component in the proximity of a Lifshitz transition in cuprates, diborides, and iron based superconductors. By keeping a constant and small exchange interaction between the two electron fluids, we search for the optimum coupling strength in the electronic polaronic component which gives the largest amplification of the superconducting critical temperature in comparison with the case of a single electronic component.

## 1. Introduction

_{C}of 164 K was held by HgBa

_{2}Ca

_{m}

_{−1}Cu

_{m}O

_{2m+2+δ}with m = 3 and at high pressure [2]. In 2015 Eremets and his collaborators succeeded in observing superconductivity in sulfur hydride with a very high T

_{C}of 203 K at an ultrahigh pressure of 150 GPa [3] which has been confirmed experimentally by several groups [4,5,6,7] and it is the object of high theoretical interest [8,9,10].

_{C}BCS superconductors. For the BCS superconductor the distance between paired electrons can be of the order of several thousands of Angstroms, much larger than the average inter-particle distance whereas, for high-temperature superconductors, this distance is of the order of some unit cells with formation of local molecular-like pairs. From the viewpoint of theory in the strong coupling limit the electron-lattice system enters the polaronic regime with low charge density and low Fermi energy, where phonon frequencies reach an ultralow energy. However, in this regime superconductivity competes with the proximity to a structural lattice instability and localization of charge carriers [11]. While conventional BCS superconductors are stable, high-temperature superconductors have an inherent tendency for structural instabilities, as discussed by Sleight [12].

_{C}superconductivity in a two component system has been the boson-fermion model made of BEC and BCS condensates [13], it was found that the electronic phase of cuprates consists of a first electronic component in the intermediate coupling regime condensing the BCS-BEC crossover which coexists with a second quasi-free electron component in the weak coupling regime [14,15,19,21] which is expected to condense in the BCS limit.

_{F1}to the characteristic pairing energy ω

_{0}, is about E

_{F1}/ω

_{0}= 1.5 and the pairing is in the regime of the BCS-BEC crossover, while the pairing in the second electronic component is in the BCS regime, where the Fermi energy E

_{F2}is a large energy scale, E

_{F2}/ω

_{0}>> 1. The amplification of the critical temperature T

_{C}in comparison with the predictions of the standard single band BCS approximation is optimized for weak coupling strengths, in the λ

_{22}pairing channel and weak exchange interband pairing interaction λ

_{12}, where ${\lambda}_{ii},{\lambda}_{ij}$ correspond to effective coupling constants. In the very strong coupling scenario in these two channels the amplification factor becomes smaller. Therefore, the road map for room temperature superconductivity based on the BPV approach predicts an optimal case where a first electronic system being at particular values of the polaronic regime in the λ

_{11}channel, intermediately between weak and strong coupling. It is interesting to remark that the vertex corrections near the band edge proportional to λ

_{11}≈ sqrt(ω

_{0}/E

_{F1}) are not expected to be large for a coupling constant 0.1 < λ

_{11}< 0.35 and the ratio ω

_{0}/E

_{F1}close to 1.

_{C}ratio [35], the doping-dependent isotope effect [36,37,38,39] measurements, and the interband exchange interaction has been put forward to play a crucial role in enhancing T

_{C}

_{,}as well as in explaining these key experiments [36,37,38,39,40,41]. Therefore, the recipe to realize stable room temperature superconductivity has been suggested for complex systems composed of two different electronic components where instabilities are avoided [27,28,29,36,37,38] through the interactions between them, the polaronic component and the free particles, contributing to a single superconducting phase. The scenario where the unconventional superconductivity emerges from the coexistence of a flat band and a steep band was proposed to occur in the case of superconductivity in CaC

_{6}[42,43] in contrast to elemental calcium.

_{2}, where in a superlattice of atomic boron layers intercalated by magnesium layers a wide band of boron π electrons coexists with boron σ electrons with a small Fermi energy, since the chemical potential is near the σ band edge in the energy scale of its energy fluctuations due to zero point motion [44,45,46,47,48,49]. A very large amplification of the critical temperature appears as a function of the shift of the chemical potential below the σ band edge. It has been explained by Bussmann-Holder et al. [50] using the multiband approach of Suhl et al. [51], by Ummarino et al. [52] using the multi-band Eliashberg theory, and by the Innocenti et al. [53] using the BPV theory.

_{C}materials should be formed in a nanoscale superlattice of atomic layers and/or second that multiple Fermi surface spots should coexist and give different superconducting condensates for each electronic component. Finally, the fine tuning of the chemical potential should be achieved in different ways by pressure, by chemical doping, or by misfit strain of charge injection (or by a combination of these) in order to cross an electronic topological transition (ETT) called a Lifshitz transition [54,55], which is a type of quantum criticality [56]. In fact, the tuning of the chemical potential near a band edge, called a Lifshitz transition for the appearance of a new Fermi surface spot, leads to resonant phenomena between different pairing channels. In particular the shape resonances between superconducting gaps are Fano resonances or Feshbach resonances between the superconducting first condensate in the BCS-BEC crossover regime and second condensates in the BCS regime.

_{2}and cuprates.

_{C}ratio as in the experiments [35,53].

## 2. Results and Discussion

_{C}is given by the condition ${\Delta}_{1},{\Delta}_{2}$ to yield the linearized gap equations:

_{C}.

_{C}caused by the interband interaction is calculated, as well as the temperature dependence of the related gaps and the isotope effects on T

_{C}which helps in understanding the experimental results and leads to conclusions concerning the mechanism realized in sulfur hydride.

_{C}is calculated as a function of ${\lambda}_{11}$ for the three cases given above (Figure 1a). An amazing and generic dependence of T

_{C}on ${\lambda}_{11}$ is clearly seen since T

_{C}remains negligibly small up to values of ${\lambda}_{11}=0.2$ with the exception of the increased interband coupling ${\lambda}_{12}=0.1$ which also induces for small values of ${\lambda}_{11}$ considerable increases in T

_{C}. This does not hold for ${\lambda}_{22}$ which—when doubled—does not affect much the T

_{C}versus ${\lambda}_{11}$ dependence suggesting that its role is almost negligible. For all values ${\lambda}_{11}>0.2$ a rapid increase in T

_{C}takes place to reach values of more than 300 K for ${\lambda}_{11}=0.6$.

_{C}enhancements relates to the intraband polaron coupling. In order to evidence the increases in T

_{C}in better detail, the values shown in Figure 1a have been normalized to the single band T

_{C}values when only the polaronic band is considered (Figure 1b). While, for small ${\lambda}_{11}$, T

_{C}is very small and, correspondingly, the normalized value is large, a strong depression of T

_{C}with increasing ${\lambda}_{11}$ takes place highlighting its crucial role. All three cases follow an almost unique dependence for ${\lambda}_{11}\ge 0.25$.

_{C}and ${\lambda}_{11}$ are shown for the same parameter combinations in Figure 2a,b. Apparently, the larger polaron related superconducting gap increases in a unique manner as a function of T

_{C}, whereas the second gap strongly depends on the interband interaction which induces strong increases in it with increasing ${\lambda}_{12},$ emphasizing its importance for the itinerant band. The BCS ratio 2Δ

_{1,2}/kT

_{C}is shown in Figure 3a–c. As expected, the leading gap to T

_{C}ratio is enhanced as compared to the BCS value as long as ${\lambda}_{11}>0.25$ whereas, for values ${\lambda}_{11}<0.25$, the leading gap is reversed. This feature is generic and independent of the parameter choice. Their average remains, however, always smaller as compared to the BCS value, rather analogous to Al-doped MgB

_{2}[50,53,63].

_{C}, respectively, to almost saturate for ${\lambda}_{11}>0.4$. This can be more clearly seen by plotting the gap ratio ${\Delta}_{2}/{\Delta}_{1}$ versus ${\lambda}_{11}$ as is done in Figure 5. In spite of the fact that ${\Delta}_{2}$ remains small even at large values of T

_{C}, it is essential for enhancing T

_{C}since the interband coupling is a basic ingredient in the model.

_{2}as compared to Δ

_{1}, whereas an increased interband interaction removes it.

## 3. Conclusions

_{C}to values smaller than 30 K due to the inherent inverse relation between phonon energy and the electron-phonon coupling constant. In addition, isotope effects are not pressure dependent as long as the structure remains unchanged. even though the value of α can be smaller than the BCS value. The scenario outlined in this work shows that a two-band/two-gap theory can easily lead to values of T

_{C}> 200 K using rather moderate inter- and intraband coupling constants. By extending the SMW model to a combination of a flat band, i.e., dispersion-less polaronic type, and a steep itinerant one, moderate coupling is realized in combination with weak coupling. Such an approach resembles the BEC-BCS crossover discussed broadly in the literature for ultracold fermionic atoms and for the high-T

_{C}superconductors [73,74], however, with the distinction that not a single crossover is considered, but the coexistence of different pairing regimes in the two different bands. The strong coupling flat band can be related to polaron and bipolaron physics, whereas the itinerant electrons delocalize the localized ones through pair-exchange interband interactions. The isotope effect arising from this model can—when combined with experimental data—distinguish what scenario is realized in the respective compound. For sulfur hydride apparently a flat polaronic band combined with a steep itinerant band accounts for the observed experimental data. We have suggested that not only sulfur hydride, but also cuprates, pnictides, and MgB

_{2}are multiband systems where the coexistence of bosonic-like composite pairs (small Cooper pairs) in terms of bipolarons and fermions coupled through interband interactions are the important ingredients to increase T

_{C}to the observed large values. The special case of the combination of strong coupling polaronic and weak coupling itinerant case has been investigated in deeper detail by concentrating on the role of the intraband coupling related to the bosonic band. In contrast to the well-known SMW model, new physics emerge from this approach, namely, a considerable enhancement of T

_{C}caused by the intraband coupling only. Additinoally, the gap to T

_{C}ratios are substantially modified as compared to the standard SMW model. The isotope effect stemming from this combination decreases abruptly when the intraband couplings become pronouncedly different, indicating a crossover from BCS to coexistence of BCS and BCS-BEC crossover physics [75,76]. From the above results, certain directions in the search for new superconductors can be deduced, namely that layered compounds with distinctly different properties of the constituting layers are well suited to meet the above criteria. Further heterostructures at the atomic limit, so called superstripes materials like cuprates, diborides, and iron based superconductors, are ideal candidates to observe high-temperature superconductivity.

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**) T

_{C}as a function of the intraband coupling λ

_{11}which is related to the strong coupling flat polaronic band. The black squares refer to the parameters ${\lambda}_{22}=0.1,{\lambda}_{12}=0.03,$ red circles: ${\lambda}_{22}=0.2,{\lambda}_{12}=0.03$, and blue triangles: ${\lambda}_{22}=0.2,{\lambda}_{12}=0.1$. (

**b**) Double logarithmic plot of the normalized T

_{C}as a function of the interband coupling.

**Figure 3.**2Δ

_{1,2}/kT

_{C}as a function of T

_{C}for the three parameters sets discussed in the text. (

**a**) ${\lambda}_{22}=0.2,{\lambda}_{12}=0.1$. (

**b**) ${\lambda}_{22}=0.2,{\lambda}_{12}=0.03$. (

**c**) ${\lambda}_{22}=0.1,{\lambda}_{12}=0.03$.

**Figure 4.**The isotope exponent α

_{1,2,3}(1, 2, 3 refer to the three cases of coupling constant combinations discussed above) as a function of ${\lambda}_{11}$ panel (

**a**) and of T

_{C}panel (

**b**).

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Bussmann-Holder, A.; Köhler, J.; Simon, A.; Whangbo, M.-H.; Bianconi, A.; Perali, A.
The Road Map toward Room-Temperature Superconductivity: Manipulating Different Pairing Channels in Systems Composed of Multiple Electronic Components. *Condens. Matter* **2017**, *2*, 24.
https://doi.org/10.3390/condmat2030024

**AMA Style**

Bussmann-Holder A, Köhler J, Simon A, Whangbo M-H, Bianconi A, Perali A.
The Road Map toward Room-Temperature Superconductivity: Manipulating Different Pairing Channels in Systems Composed of Multiple Electronic Components. *Condensed Matter*. 2017; 2(3):24.
https://doi.org/10.3390/condmat2030024

**Chicago/Turabian Style**

Bussmann-Holder, Annette, Jürgen Köhler, Arndt Simon, Myung-Hwan Whangbo, Antonio Bianconi, and Andrea Perali.
2017. "The Road Map toward Room-Temperature Superconductivity: Manipulating Different Pairing Channels in Systems Composed of Multiple Electronic Components" *Condensed Matter* 2, no. 3: 24.
https://doi.org/10.3390/condmat2030024