#### Abstract

The high value of the isotope shift in sulfur hydrides supports a phonon-mediated pairing scenario of superconductivity for these high-temperature superconductors which are consistent with the Bardeen–Cooper–Schrieffer (BCS) framework. Knowing that a large electronic density of states enhances the critical temperature (), generalized Fermi surface topologies are used to increase it. A multicomponent model within the BCS framework is proposed in this work for sulfur hydride superconductors. This model is used to evaluate some properties of the superconductor. Strong and intermediate coupling effects are taken into account with the effective McMillan approximation, and the isotope coefficient is evaluated as a function of the coupling parameter as well as other relevant parameters of the model.

#### 1. Introduction

Research on the possibility of room temperature superconductivity has been invigorated by the discovery of superconducting sulfur hydrides at very high pressure, about 155 gigapascals (GPa) [1, 2]. Given that hydrogen has the smallest atomic mass, it was predicted that metallic hydrogen or hydrogen-rich compounds would be high-*T*_{c} superconductors [3–5]. Light hydrogen atoms provide high-frequency phonon modes and strong electron-phonon coupling due to the lack of an electronic core. It has been shown that is stable below 43 GPa, and at elevated pressures it decomposes into and *S* [6–8]. has been found to be stable up to at least 300 GPa [9]. The superconducting temperature of at pressure of about 150 GPa is as high as 203 K. This superconductor has one of the highest transition temperatures obtained in high-pressure experiments reported up until now [10, 11].

The high value of the isotope exponent in this material shows evidence of a phonon-mediated pairing mechanism, consistent with the BCS framework. The fact that the critical temperature varies with the isotopic mass is the evidence of the interaction between the electrons and lattice vibrations [12–14]. These results are proof that the electron-phonon interaction is an important pairing mechanism in the sulfur hydrides [15–17]. The observation that the superconductivity in these superconductors shows strong sensitivity to the crystal lattice suggests the possibility of unconventional electron-phonon coupling. Phonon properties in sulfur hydride systems are actually responsible for changes in their properties [18].

Understanding the electronic structure at the Fermi level can give some useful clues to unravel the fundamental ingredients responsible for the high transition temperature. However, up to now, the underlying physical process remains unknown. In this context, it seems crucial to study new ideas that use simplified schematic models to isolate the mechanism(s) that generate high transition temperatures. It was known that a large value of the density of states (DOS) at Fermi level increases the critical temperature value. That is, can be enhanced when the Fermi level is at or close to the energy of van Hove’s singularity which provides a peak in the DOS [19–22]. Hirsch and Scalapino signaled the possibility of enhancing the superconducting critical temperature with a two-dimensional structure when the Fermi level was near a singularity [23]. First-principle calculations based on the supercell method have been reported where the authors study the effect of doping on the behavior of its superconductivity [24]. They show that in can be enhanced by hole doping.

A schematic model with generalized Fermi surface topologies, which take into account a two-component system with the same dispersion relation, has been proposed by us to study cuprate superconductors in the weak and intermediate coupling region. The generalized Fermi surface, via band overlapping, is proposed as a way of increasing the density of states at the Fermi energy. This model is based on the idea that the is enhanced when the Fermi level lies at or close to the energy of a singularity in the DOS. When combined with the Cooper pair equation, this schematic model can be taken as a simple device to model the singular behavior in the density of states and to account for the higher values obtained by cuprate superconductors [25–27].

Pairing symmetry is an important element toward understanding the mechanism of high- superconductivity. Calculations with two- and three-dimensional models using the BCS formalism and order parameter with *s*-wave symmetry have been reported in superconducting [28]. It was suggested that the search for better superconductors should be in three-dimensional systems where the thermal fluctuations are less likely to reduce the observed [29].

The Eliashberg equations are strong-coupling extensions to the original BCS theory. Calculations with this theory can be found where they study the high in sulfur hydride as a result of the variability in the DOS within the band [30]. Many theoretical calculations based on Eliashberg theory for can be found in the literature and also in [31], where the authors also provide an optical spectroscopy study for this material and found spectroscopic evidence that the superconducting mechanism in is due to the electron-phonon interaction. It has been shown that is a highly optimized electron-phonon superconductor [31]. As it is known, McMillan numerically solved the Eliashberg nonlinear equations at finite temperature in order to obtain the critical temperature for strong coupled superconductors. He obtained a parametrization which relates the critical temperature to a small number of parameters [32]. This approach, valid for the electron-phonon coupling constant , was later modified by Allen and Dynes to include values in the strong coupling region up to [33–35].

is a multiband metal [36]. It was found that the optical properties of are different from those in conventional superconductors which are in the dirty limit [37]. is in the clean, phonon-dominated regime because of the very large energy scale associated to its phonon [37]. The electronic structure of has several Fermi surface sheets in the pressure range where high-temperature superconductivity appears [38]. However, it was found that the contribution to the total DOS, at the Fermi level, mostly comes from two Fermi surface sheets [39]. Based on the electronic band structure obtained from this material and on the results mentioned in this paragraph, we propose to evaluate the properties of sulfur hydrides with the topology of generalized Fermi surfaces modeled with band overlapping. A multiband structure within the BCS framework is proposed. This model, which can be taken as a minimal singularity in the density of states, together with the BCS framework, predicts higher values given that the energy band overlapping increases the DOS near the Fermi level [40–42]. The experimental results support the fact that the pairing mechanism is via phonon exchange. For physical consistency, an important requirement introduced in our model is that the band overlapping parameter is not larger than the phonon energy, . The model with generalized Fermi surface topologies is now extended to intermediate and strong coupling with the use of the McMillan effective approach to the Eliashberg equations. In this work, we use a momentum-independent pairing interaction, supposing that the superconducting order parameter has s-wave symmetry. We will evaluate some properties of the sulfur hydride superconductor, , as the isotope mass exponent, *α*, in terms of the coupling constant and the parameters of the model. The multiband scenario of superconductivity had previously been used to describe the isotope coefficient in other superconductor materials such as cuprate superconductors [43], hydrides [36, 44], and organic materials [45].

#### 2. Model and Calculation

Next, we describe the schematic overlapping-band model which has been successfully used by us to describe properties, in particular the isotope mass exponent, of cuprate superconductors and iron-based oxypnictide superconductors [40, 42].

We begin with the gap equationwith the pairing interaction, the Boltzmann constant, and , where are the self-consistent, single-particle energies.

Then, for the electron-phonon interaction, we have considered , where is a constant if and and 0 elsewhere. is the electron-phonon coupling energy. As usual, the attractive BCS interaction is nonzero only for unoccupied orbitals in the neighborhood of the Fermi level, . The superconducting order parameter if and 0 elsewhere.

With these considerations, we propose a two-component model with overlapping bands. The generalized Fermi sea proposed here consists of concentric spherical shells separating occupied orbitals. We propose a Fermi surface with two bands as a prototype of band overlapping [46]. As a particular distribution in momentum space, the following form has been considered:with the Fermi momentum and . The distribution in momentum induces another distribution in energy, , where and [47]. In order to keep the average number of electron states constant, the parameters in the system are related by the equation ; then, only one of the relevant parameters is independent. We consider a high-frequency electron-optical-phonon coupling and require the band overlapping to be of order (or smaller) than the cutoff energy. That means . This can be rewritten as , where and . The minimum value consistent with our model is . While , the energy difference between the normal and the anomalously occupied states must be provided by the material itself.

In the last framework, the summation in equation (1) is changed to an integration which is done over the (*symmetric*) generalized Fermi surface defined above. One gets

In this equation, , is the effective attractive interaction of the BCS model, and is the electronic density of states, which will be taken as a constant in the integration range.

The two integrals correspond to the bands proposed by equation (2). The integration over the surface at in the first band is restricted to states in the interval . In the second band, in order to conserve particle number, the integration is restricted to the interval , if , with , in terms of the parameter *γ* in our model.

The critical temperature is introduced via equation (3) at , where the gap becomes . At this temperature, equation (3) is reduced towhich is to be numerically evaluated. Taking into account that , where is the Coulomb pseudopotential, the last equation relates to the electron-phonon coupling parameter *λ* and to the anomalous occupancy parameter. This relationship determines the values which reproduce the critical temperature of in the intermediate and strong coupling region.

At , equation (3) will be evaluated and *λ* values, consistent with the model, and which reproduce the zero-temperature superconducting gap ( values), will be obtained:

Because theoretical results in the literature are quite different among themselves [48–50], the isotope exponent in the harmonic approximation, *α*, is also evaluated. The two-band model in this work, along with the Cooper pair equation, is used in order to obtain *α*. Strong and intermediate coupling effects are taken into account with the effective McMillan approximation which is valid for values of the coupling constant .

The isotope coefficient in the harmonic approximation is given by

Hence, we obtainwith , whereand *Y* is given by

Equations (7)–(9) allow one to study the effect of intermediate and strong coupling up to in our energy band overlapping model.

The model presented in this section can be useful to describe sulfur hydride superconductors. As was discussed in references [25, 47], tighter-bound Cooper pairs arise by generalizing the assumed spherical Fermi sea without invoking neither stronger electron-phonon coupling nor unconventional interaction mechanisms. Then, the range for the coupling parameter in the intermediate and strong coupling regions, compatible with the McMillan approximation, is taken [32]. The *λ* range could be extended up to with the Allen–Dynes approximation [33, 34].

Different values of , as reported in the literature [6, 12, 39], are used in our calculations. Overlapping parameter values , consistent with the model, are obtained for the material. The relationship between the characteristic parameters will be obtained for and used to evaluate the isotope mass exponent. The pairing in the superconducting state is taken as due, principally, to high-frequency optical modes.

#### 3. Results and Discussion

In this section, we report the results for the properties of obtained with our model. Two extreme values of the Coulomb pseudopotential and are taken from references [6, 12, 39]. The high-energy stretching mode is assumed to hold. The Migdal approximation, , might not be satisfied when the multiband electronic structure of is taken into account. Then, the ratio in the range , with as the threshold to the multiband region [17, 51], will be selected in our calculations.

In Figure 1, with the help of equation (4), the behavior of the coupling parameter *λ*, as a function of the band overlapping , is shown for at 150 Gpa. At this pressure, the critical temperature is about 203 K. We select . The values and are taken in the calculations, and the full line represents , while the dashed one is for . The Fermi energy satisfies . The range of *λ* values obtained with our model is in the intermediate and strong coupling region where the McMillan approximation is valid. For a greater value of the Coulomb pseudopotential, a stronger electron-phonon coupling is required, as seen in this figure.

In Figure 2, the coupling parameter *λ* as a function of the band overlapping for , at the critical temperature , is shown for two values of . In the top curve and , and in the bottom one and . The Coulomb pseudopotential is satisfied in both cases. The *λ* values obtained are within the intermediate coupling region.

In Figure 3, the curves for the zero-temperature gap, , consistent with those in the literature, are shown. The curves for are shown from top to bottom, the Coulomb pseudopotential was taken to be , and . Comparison with other multicomponent models we observe that the upper curve in Figure 3 is compatible with the value of with a multiband model in reference [36]. The middle curve in our figure is similar to , obtained with a two Fermi surface sheet model at 200 GPa by Flores-Livas et al. [39]. These curves allow us to observe the behavior of the coupling constant in terms of the parameter of our model. For a larger value of the gap, larger coupling constant values are required, but still consistent with the McMillan approximation.

Similar results are obtained by using different models. For example, with an analysis based on density-functional theory of electronic and vibrational properties of hydrogen and a Coulomb pseudopotential with , they got a superconducting energy gap of [52]. While in reference [18], they got a value of with the phonon-mediated Eliashberg theory. On the other hand, in reference [53], a smaller value of was obtained.

In Figure 4, the isotope mass exponent, , for the value of the ratio , is evaluated via the relation in equation (7). The curves for are shown from top to bottom, where in all the curves. With we get , and these values are smaller but close to the BCS result, . At a given *λ* value, the mass exponent is larger for smaller values of *δ*. Our results for the isotope mass exponent agree with those obtained using other models in the literature, but are lower than the experimental value [50, 54]. The *α* values obtained in this work are in the range of those in references [6, 18, 54, 55], evaluated with different models for . As seen in the graphic, for greater *λ* values, the *α* results do not change notoriously. The *λ* range could be extended with the Allen–Dynes approximation up to [33, 34].

#### 4. Conclusions

Based on the electronic structure for [36, 39] and in order to increase the density of states at the Fermi level for high-temperature superconductors, we presented a two-component model with generalized Fermi surface topologies obtained via band overlapping. We considered two overlapping bands at the Fermi level, where the order parameter is taken to have s-wave symmetry. The band overlapping allowed for improvement on the results obtained with an s-wave mean-field approximation in a scheme in which the electron-phonon interaction is the relevant one for the high- mechanism (clean limit). The energy scale of the band overlapping is of the order of the phonon energy, . A nonstandard electron-phonon coupling is considered, and the minimum value of the band overlapping parameter, , is consistent with this value. This energy is then the overall scale that determines the highest and gives consistency to the model because it requires an energy scale accessible to the lattice. We used this model to describe sulfur hydride superconductors, within the BCS framework, and the model was extended to include intermediate coupling with the McMillan approximation. As mentioned before, tighter-bound Cooper pairs arise by generalizing the assumed spherical Fermi sea without invoking neither stronger electron-phonon coupling nor unconventional interaction mechanisms. In fact, the *λ* values obtained in this work for the critical temperature and the zero-temperature gap, , are in the intermediate coupling region. The isotope exponent values we calculated agree with those reported in the literature using other models, but are lower than the experimental value. We were not able to get a larger value of the isotope mass exponent. That suggests the necessity to introduce modifications in order to account for additional elements. However, this schematic model has the capability to generically describe some properties of the superconductor and can be a guide for a more detailed study of these materials with, for example, the Eliashberg theory.

#### Data Availability

The data used to support the findings of this work are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

It is a pleasure to thank S. Orozco, M. Moreno, and E. Moreno for their valued comments.