# A Schematic Two Overlapping-Band Model for Superconducting Sulfur Hydrides: The Isotope Mass Exponent

Authors: R. M. Méndez-Moreno
Journal: Advances in Condensed Matter Physics
 (2019)

---

## 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 (Tc), 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 H3S 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.


---

## Body

## 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-Tc 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 H2S is stable below 43 GPa, and at elevated pressures it decomposes into H3S and S [6–8]. H3S has been found to be stable up to at least 300 GPa [9]. The superconducting temperature of H3S 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,Tc 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 H3S on the behavior of its superconductivity [24]. They show that Tc in H3S 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 theTc 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 Tc values obtained by cuprate superconductors [25–27].Pairing symmetry is an important element toward understanding the mechanism of high-Tc superconductivity. Calculations with two- and three-dimensional models using the BCS formalism and order parameter with s-wave symmetry have been reported in superconducting H3S [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 Tc [29].The Eliashberg equations are strong-coupling extensions to the original BCS theory. Calculations with this theory can be found where they study the highTc in sulfur hydride as a result of the variability in the DOS within the band [30]. Many theoretical calculations based on Eliashberg theory for H3S 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 H3S is due to the electron-phonon interaction. It has been shown that H3S 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 λ<1.5, was later modified by Allen and Dynes to include values in the strong coupling region up to λ≃2 [33–35].H
3
S is a multiband metal [36]. It was found that the optical properties of H3S are different from those in conventional superconductors which are in the dirty limit [37]. H3S is in the clean, phonon-dominated regime because of the very large energy scale associated to its phonon [37]. The electronic structure of H3S 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 Tc 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, Eph. 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, H3S, 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 equation(1)Δk′=∑kVk,k′ΔktanhEk/2kBT2Ek,with Vk,k′ the pairing interaction, kB the Boltzmann constant, and Ek2=εk2+Δk2, where ϵk=ℏ2k2/2m are the self-consistent, single-particle energies.Then, for the electron-phonon interaction, we have consideredVk,k′=V0, where V0 is a constant if ϵk≤Eph and ϵk′≤Eph and 0 elsewhere. Eph 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, EF. The superconducting order parameter Δk=ΔT if ϵk≤Eph 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:(2)nk=ΘγkF−k+ΘγkF−kΘk−βkF,with kF the Fermi momentum and 0<β<γ<1. The distribution in momentum induces another distribution in energy, Eβ<Eγ, where Eβ=β2EF and Eγ=γ2EF [47]. In order to keep the average number of electron states constant, the parameters in the system are related by the equation 2γ2−β2=1; 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 1−γ2EF≤Eph. This can be rewritten as 1−γ2=2νδ, where 2ν=Eph/EF and 0<δ<1. The minimum γ2 value consistent with our model is γph2=1−Eph/EF. While EF−Eγ≤Eph, 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(3)1=V0DE2∫Eγ−EphEγ+EphtanhΞk2kBTdεkΞk+V0DE2∫EβEFtanhΞk2kBTdεkΞk.In this equation,Ξk=ϵk−EF2+ΔT2, V0 is the effective attractive interaction of the BCS model, and DE 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 Eγ in the first band is restricted to states in the interval Eγ−Eph≤Ek≤Eγ+Eph. In the second band, in order to conserve particle number, the integration is restricted to the interval Eβ≤Ek≤EF, if Eγ+Eph>EF, with Eβ=2γ2−1EF, in terms of the parameter γ in our model.The critical temperature is introduced via equation (3) at T=Tc, where the gap becomes ΔTc=0. At this temperature, equation (3) is reduced to(4)1=V0DE2∫Eγ−EphEγ+Ephtanhϵk−EF2kBTcdϵkϵk−EF+V0DE2∫EβEFtanhϵk−EF2kBTcdϵkϵk−EF,which is to be numerically evaluated. Taking into account that V0DE=λ−μ∗/1+λ, where μ∗ is the Coulomb pseudopotential, the last equation relates Tc to the electron-phonon coupling parameter λ and to the anomalous occupancy γ2 parameter. This relationship determines the γ2 values which reproduce the critical temperature of H3S in the intermediate and strong coupling region.AtT=0K, equation (3) will be evaluated and λ values, consistent with the model, and which reproduce the zero-temperature superconducting gap (Δ0 values), will be obtained:(5)1=V0DE2sinh−1Eph+γ2−1EFΔ0+V0DE2sinh−1Eph+1−γ2EFΔ0+V0DE2sinh−121−γ2EFΔ0.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 λ<1.5.The isotope coefficient in the harmonic approximation is given by(6)α=−dlnTcdlnM.Hence, we obtain(7)α=0.51+δ2Y−1+δ+2Y−1/Dδ1−2Y−1−D,with δ=1−γ2EF/Eph, where(8)D=δ2+22Y−1δ+1,and Y is given by(9)Y=exp1.041+λλ−μ∗1+0.62λ.Equations (7)–(9) allow one to study the effect of intermediate and strong coupling up to λ=1.5 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 0.4<λ<1.5 in the intermediate and strong coupling regions, compatible with the McMillan approximation, is taken [32]. The λ range could be extended up to λ=2 with the Allen–Dynes approximation [33, 34].Different values ofμ∗, as reported in the literature [6, 12, 39], are used in our calculations. Overlapping parameter values γ2, consistent with the model, are obtained for the material. The relationship between the characteristic parameters will be obtained for H3S 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 ofH3S obtained with our model. Two extreme values of the Coulomb pseudopotential μ∗=0.1 and μ∗=0.16 are taken from references [6, 12, 39]. The high-energy stretching mode Eph=200meV is assumed to hold. The Migdal approximation, Eph/EF≪1, might not be satisfied when the multiband electronic structure of H3S is taken into account. Then, the ratio Eph/EF in the range 0.5≥Eph/EF≥2/9, with Eph/EF=2/9 as the threshold to the multiband region [17, 51], will be selected in our calculations.In Figure1, with the help of equation (4), the behavior of the coupling parameter λ, as a function of the band overlapping γ2, is shown for H3S at 150 Gpa. At this pressure, the critical temperature Tc is about 203 K. We select Eph=200meV. The values μ∗=0.1 and μ∗=0.16 are taken in the calculations, and the full line represents μ∗=0.1, while the dashed one is for μ∗=0.16. The Fermi energy satisfies Eph/EF=0.222. The range of λ values obtained with our model is in the intermediate and strong coupling region λ≤1.5, 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.Figure 1
Values of the coupling constant,λ, in terms of the parameter γ2 obtained in this work. The Tc curves are given for the critical temperature of 203 K. The Coulomb pseudopotential values μ∗=0.1 (solid line) and μ∗=0.16 (dashed line) are taken. The ratio Eph/EF=0.222.In Figure2, the coupling parameter λ as a function of the band overlapping γ2 for H3S, at the critical temperature Tc=203K, is shown for two values of Eph. In the top curve Eph=100meV and Eph/EF=0.248, and in the bottom one Eph=200meV and Eph/EF=0.285. The Coulomb pseudopotential μ∗=0.1 is satisfied in both cases. The λ values obtained are within the intermediate coupling region.Figure 2
Values of the coupling constant,λ, in terms of the parameter γ2 obtained in this work at Tc=203K. The Coulomb pseudopotential is μ∗=0.1 for both curves; Eph=200meV for the lower curve and Eph=100meV for the upper one. Eph/EF=0.285 and 0.248, respectively.In Figure3, the curves for the zero-temperature gap, Δ0, consistent with those in the literature, are shown. The curves for Δ0=60,40,and30meV are shown from top to bottom, the Coulomb pseudopotential was taken to be μ∗=0.1, and Eph/EF=0.4. Comparison with other multicomponent models we observe that the upper curve in Figure 3 is compatible with the value of Δ0=65meV with a multiband model in reference [36]. The middle curve in our figure is similar to Δ0=43.8meV, 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 γ2 of our model. For a larger value of the gap, larger coupling constant values are required, but still consistent with the McMillan approximation.Figure 3
The curves for three values of the zero-temperature gap,Δ0, are shown; μ∗=0.1 and Eph/EF=0.40. The shown curves are for Δ0=60,40, and 30 meV, from top to bottom.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 withμ∗=0.1, they got a superconducting energy gap of Δ0=62meV [52]. While in reference [18], they got a value of Δ0=43meV with the phonon-mediated Eliashberg theory. On the other hand, in reference [53], a smaller value of Δ0=28meV was obtained.In Figure4, the isotope mass exponent, αλ,γ2, for the value of the ratio Eph/EF=0.40, is evaluated via the relation δ=1−γ2EF/Eph in equation (7). The curves for δ=0.03,0.06,and0.1 are shown from top to bottom, where μ∗=0.1 in all the curves. With λ>0.7, we get 0.15<α<0.37, and these values are smaller but close to the BCS result, α=0.5. 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 H3S. 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 λ=2 [33, 34].Figure 4
The isotope mass exponent,α, is shown as a function of the coupling constant, λ, using δ values obtained with our model, where δ=1−γ2EF/Eph. The curves for δ=0.03,0.06, and 0.1 are plotted from top to bottom.


## 4. Conclusions

Based on the electronic structure forH3S [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-Tc mechanism (clean limit). The energy scale of the band overlapping is of the order of the phonon energy, 1−γ2EF≤Eph. A nonstandard electron-phonon coupling is considered, and the minimum value of the band overlapping parameter, γ2, is consistent with this value. This energy is then the overall scale that determines the highest Tc 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, Δ0, are in the intermediate coupling region. The H3S 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 H3S superconductor and can be a guide for a more detailed study of these materials with, for example, the Eliashberg theory.



---

*Source: 6795250-2019-10-10.xml*