Anomalously high value of Coulomb pseudopotential for the H₅S₂ superconductor

Abstract

The H₅S₂ and H₂S compounds are the two candidates for the low-temperature phase of compressed sulfur-hydrogen system. We have shown that the value of Coulomb pseudopotential (μ*) for H₅S₂ ([T_C]_exp = 36 K and p = 112 GPa) is anomalously high. The numerical results give the limitation from below to μ* that is equal to 0.402 (μ* = 0.589), if we consider the first order vertex corrections to the electron-phonon interaction). Presented data mean that the properties of superconducting phase in the H₅S₂ compound can be understood within the classical framework of Eliashberg formalism only at the phenomenological level (μ* is the parameter of matching the theory to the experimental data). On the other hand, in the case of H₂S it is not necessary to take high value of Coulomb pseudopotential to reproduce the experimental critical temperature relatively well (μ* = 0.15). In our opinion, H₂S is mainly responsible for the observed superconductivity state in the sulfur-hydrogen system at low temperature.

Introduction

The Eliashberg formalism¹ allows for the analysis of classical phonon-mediated superconducting state at the quantitative level. The input parameters to the Eliashberg equation are: the spectral function, otherwise called the Eliashberg function (α²F(ω)) that is modeling the electron-phonon interaction^2,3, and the Coulomb pseudopotential (μ*), which is responsible for the depairing electron correlations^4,5. Usually, the Eliashberg function is calculated using the DFT method (e.g. in the Quantum Espresso package^6,7). The value of Coulomb pseudopotential is selected in such way that the critical temperature, determined in the framework of Eliashberg formalism, would correspond to T_C from the experiment. Sometimes μ* is tried to be calculated from the first principles, however, this is the very complex issue and it is rarely leading to correct results⁸.

It should be noted that the value of Coulomb pseudopotential should not exceed 0.2. For higher values, μ* cannot be associated only with the depairing electron correlations. In the case at hand, the quantity μ* should be treated only as the parameter of fitting the model to the experimental data, which means that the Eliashberg theory becomes practically the phenomenological approach.

The phonon-mediated superconducting state in the compressed H₂S with high value of critical temperature was discovered by Drozdov, Eremets, and Troyan in 2014⁹ (see also the paper¹⁰). The experimental observation of superconductivity in the compressed dihydrogen sulfide was inspired by the theoretical prediction made by Li et al.¹¹, which is based on the extensive structural study on the H₂S compound at the pressure range of ~10–200 GPa. In the paper¹¹ the following sequence of structural transitions was demonstrated: at the pressure of 8.7 GPa the Pbcm structure is transformed into the P2/c structure. The next transformation occurs for the pressure of 29 GPa (P2/c → Pc). For the pressure of 65 GPa the transformation of Pc structure into the Pmc2₁ structure was observed. It is worth paying attention to the fact that the obtained theoretical results are consistent with the X-ray diffraction (XRD) experimental data^12,13. The last two structural transitions for compressed H₂S were observed for 80 GPa (Pmc2₁ → P-1) and 160 GPa (P-1 → Cmca). Interestingly, the results obtained in the paper¹¹ are in the contradiction with the earlier theoretical predictions, suggesting that H₂S dissociate into elemental sulfur and hydrogen under the high pressure¹⁴. However, it should be noted, that the partial decomposition of H₂S was observed in Raman¹⁵ and XRD studies¹⁶ at the room temperature above 27 GPa. The calculations of the electronic structure made in¹¹ suggest that the H₂S compound is the insulator up to the pressure of 130 GPa. This result correlates well with the value of the metallization pressure of about 96 GPa, observed experimentally¹⁷ (see also related research^{12,13,15,16,18–24}). It is worth noting that in the work¹¹ for the structure of Pbcm (p = 0.3 GPa) the large indirect band gap of ~5.5 eV was determined, which is relatively good comparing with the experimental results (4.8 eV)²⁵. With the pressure increase, the band gap decreases (3.75 eV for 15 GPa, 1.6 eV for 40 GPa, and 0.27 eV for 120 GPa). For the metallization pressure (130 GPa), the value of the electron density of states (ρ(ε_F)) is 0.33 eV⁻¹ per f.u. The sudden increase in ρ(ε_F) to the value of 0.51 eV⁻¹ per f.u. is observed at the structural transition P-1 → Cmca¹¹.

In experiments described in the papers⁹ and¹⁰ are observed two different superconducting states. In particular, superconductivity measured in the low-temperature range (l-T, sample prepared at T < 100 K), possibly relates to the H₂S compound, as it is generally consistent with calculations presented in¹¹ for solid H₂S: both the value of T_C < 82 K and its pressure behavior. In addition, it was demonstrated theoretically²⁶ that the experimental results could be reproduced accurately in the framework of classical Eliashberg equations, whereas the value of Coulomb pseudopotential is low (μ* ~ 0.15).

On the other hand, the result obtained by Ishikawa et al.²⁷ suggest that in the narrow pressure range from 110 GPa to 123 GPa, the H₅S₂ compound, in which asymmetric hydrogen bonds are formed between H₂S and H₃S molecules, is thermodynamically stable and its critical temperature correlates well with experimental results^9,10. However, it should be assumed that the anharmonic effects lower the theoretically determined value of the critical temperature by the minimum of about 20%. This assumption has some theoretical justification^28,29, nevertheless in the paper²⁷, it is not supported by the first-principles calculations.

It is also worth paying attention to the results contained in³⁰, in which it is envisaged that the H₄S₃ is stable within the pressure range of 25–113 GPa. What is important, H₄S₃ coexists with fraction of H₃S and H₂S, at least up to the pressure of 140 GPa. The theoretical results correlate with XRD data, which confirm that above 27 GPa, the dihydrogen sulfide partially decomposes into S + H₃S + H₄S₃. Nevertheless, H₄S₃ is characterized by the very low critical temperature (T_C = 2.2 K for μ* = 0.13), which suggests that kinetically protected H₂S in samples prepared at low temperature is responsible for the observed superconductivity below 160 GPa³⁰.

The superconductivity with the record high value of critical temperature (T_C = 203 K and p = 155 GPa), obtained for the sample prepared at high temperatures (h-T), relates to the decomposition of starting material above 43 GPa³¹ (see also^28,30,32,33): 3H₂S → 2H₃S + S. We notice that the experimental values of critical temperature and its pressure dependencies are close to the values of T_C predicted theoretically by Duan et al.^31,34 (T_C ∈ 〈191–204〉 K at 200 GPa), or later by Errea et al.²⁹ for the cubic Im-3m structure with H₃S stoichiometry. The physical mechanism underlying the superconductivity of H₃S is similar to that in the MgB₂ compound: metallization of covalent bonds. The main difference from the magnesium dibore is that the hydrogen mass is 11 times smaller than the mass of boron³². Considering the electron-phonon interaction, it was noted that in the case of H₃S, the vertex corrections in the local approximation have the least meaning³⁵. However, in the static limit with finite q, the vertex corrections change the critical temperature by −34 K³⁶. The result above correlates well with the data for SiH₄³⁷. For the H₃S compound, the thermodynamic parameters are also affected by the anharmonic effects. It was shown that for 200 GPa and 250 GPa, the anharmonic effects lower substantially the value of electron-phonon coupling constant^28,36. It is worth noting that the pressure which increases above 250 GPa does not increase the critical temperature value in H₃S³⁸. However, we have recently shown that the value of T_C increases (242 K), if we use the sulfur isotope ³⁶S³⁹. Therefore, it can be reasonably supposed that it is possible to obtain the superconducting condensate close to the room temperature (see also related research⁴⁰).

In the presented paper, we analyze precisely the thermodynamic properties of superconducting state in the H₅S₂ system²⁷. According to what we have mentioned before this compound is thermodynamically stable in the narrow pressure range from 110 GPa to 123 GPa. As the part of the analysis, we will prove that for the pressure at 112 GPa, the superconducting state is characterized by the anomalously high value of μ* (also after taking into account the vertex corrections to the electron-phonon interaction). The above result is not consistent with the result obtained by Ishikawa et al.²⁷, where ${\mu }^{⁎}\in \langle 0.13,0.17\rangle$. The paper contains also the parameter’s analysis of superconducting state, that is induced in the H₄S₃ compound³⁰. The results obtained by us were compared with the results for the H₂S compound, which the thermodynamic properties of superconducting state naturally explain the properties of low-temperature superconducting phase in the compressed hydrogen sulfide^11,26.

Formalism

Let us take into account the Eliashberg equations on the imaginary axis ($i=\sqrt{-1}$):

$$\begin{array}{lll}{\varphi }_n& =& \pi k_{B}T\sum _{m=-M}^M\frac{{\lambda }_{n,m}-{\mu }_m^{⁎}}{\sqrt{{\omega }_m^2{Z}_m^2+{\varphi }_m^2}}{\varphi }_m-A\frac{{\pi }^3{(k_{B}T)}^2}{4{\epsilon }_F}\\ & & \times \sum _{m=-M}^M\sum _{m\prime =-M}^M\frac{{\lambda }_{n,m}{\lambda }_{n,m\prime }}{\sqrt{({\omega }_m^2{Z}_m^2+{\varphi }_m^2)({\omega }_{m\prime }^2{Z}_{m\prime }^2+{\varphi }_{m\prime }^2)({\omega }_{-n+m+m\prime }^2{Z}_{-n+m+m\prime }^2+{\varphi }_{-n+m+m\prime }^2)}}\\ & & \times \left[{\varphi }_m{\varphi }_{m\prime }{\varphi }_{-n+m+m\prime }+2{\varphi }_m{\omega }_{m\prime }{Z}_{m\prime }{\omega }_{-n+m+m\prime }{Z}_{-n+m+m\prime }-{\omega }_m{Z}_m{\omega }_{m\prime }{Z}_{m\prime }{\varphi }_{-n+m+m\prime }\right],\end{array}$$ 1

and

$$\begin{array}{lll}{Z}_n& =& 1+\frac{\pi k_{B}T}{{\omega }_n}\sum _{m=-M}^M\frac{{\lambda }_{n,m}}{\sqrt{{\omega }_m^2{Z}_m^2+{\varphi }_m^2}}{\omega }_m{Z}_m-A\frac{{\pi }^3{(k_{B}T)}^2}{4{\epsilon }_F{\omega }_n}\\ & & \times \sum _{m=-M}^M\sum _{m\prime =-M}^M\frac{{\lambda }_{n,m}{\lambda }_{n,m\prime }}{\sqrt{({\omega }_m^2{Z}_m^2+{\varphi }_m^2)({\omega }_{m\prime }^2{Z}_{m\prime }^2+{\varphi }_{m\prime }^2)({\omega }_{-n+m+m\prime }^2{Z}_{-n+m+m\prime }^2+{\varphi }_{-n+m+m\prime }^2)}}\\ & & \times \left[{\omega }_m{Z}_m{\omega }_{m\prime }{Z}_{m\prime }{\omega }_{-n+m+m\prime }{Z}_{-n+m+m\prime }+2{\omega }_m{Z}_m{\varphi }_{m\prime }{\varphi }_{-n+m+m\prime }-{\varphi }_m{\varphi }_{m\prime }{\omega }_{-n+m+m\prime }{Z}_{-n+m+m\prime }\right]\mathrm{.}\end{array}$$ 2

In the case when A = 1, the Eliashberg set was generalized to include the lowest-order vertex correction - scheme VCEE (Vertex Corrected Eliashberg Equations)⁴¹. On the other hand (A = 0), we get the model without the vertex corrections: the so-called CEE scheme (Classical Eliashberg Equations)¹. Note that in the considered equations, the momentum dependence of electron-phonon matrix elements has been neglected, which is equivalent to using the local approximation.

The individual symbols in equations (1 and 2) have the following meaning: φ_n = φ(iω_n) is the order parameter function and Z_n = Z(iω_n) represents the wave function renormalization factor. The quantity Z_n describes the renormalization of the thermodynamic parameters of superconducting state by the electron-phonon interaction². This is the typical strong-coupling effect, because Z_n=1 in the CEE scheme is expressed by the formula: $Z_{n=1}\sim 1+\lambda$, where λ denotes the electron-phonon coupling constant: $\lambda =2{\int }_0^{{\omega }_D}d\omega {\alpha }^{2}F\left(\omega \right)/\omega$, and ω_D is the Debye frequency. The order parameter is defined as the ratio: ${\mathrm{\Delta }}_n={\phi }_n/Z_n$. Symbol ω_n is the Matsubara frequency: ω_n = πk_BT(2n + 1), while $n$ is the integer. Let us emphasize that the dependence of order parameter on the Matsubara frequency means that the Eliashberg formalism explicitly takes into account the retarding nature of electron-phonon interaction. In the present paper, it was assumed M = 1100, which allowed to achieve convergent results in the range from T₀ = 5 K to T_C (see also⁴²).

The function ${\mu }_n^{⁎}$ modeling the depairing correlations has the following form: ${\mu }_n^{⁎}={\mu }^{⁎}\theta ({\omega }_c-|{\omega }_n|)$. The Heaviside function is given by θ(x) and ω_c represents the cut-off frequency. By default, it is assumed that $w_c\in ⟨3w_D,10w_D⟩.$

The electron-phonon pairing kernel is expressed by the formula:

$${\lambda }_{n,m}=2{\int }_0^{{\omega }_D}d\omega \frac{\omega }{{\omega }^2+4{\pi }^2{\left(k_{B}T\right)}^2{\left(n-m\right)}^2}{\alpha }^{2}F(\omega ).$$ 3

The higher order corrections are not included in equation (3). The full formalism demands taking into consideration the additional terms related to the phonon-phonon interactions and the non-linear coupling between the electrons and the phonons. This has been discussed by Kresin et al. in the paper⁴³.

The Eliashberg spectral function is defined as:

$${\alpha }^{2}F(\omega )=\frac{1}{2\pi \rho ({\epsilon }_F)}\sum _{\mathbf{q}\nu }\delta (\omega -{\omega }_{\mathbf{q}\nu })\frac{{\gamma }_{\mathbf{q}\nu }}{{\omega }_{\mathbf{q}\nu }},$$ 4

with:

$$\begin{array}{lll}{\gamma }_{\mathbf{q}\nu }& =& 2\pi {\omega }_{\mathbf{q}\nu }\sum _{ij}\int \frac{d^{3}k}{{\mathrm{\Omega }}_{BZ}}|g_{\mathbf{q}\nu }(\mathbf{k},i,j{)|}^2\\ & & \times \delta ({\epsilon }_{\mathbf{q},i}-{\epsilon }_F)\delta ({\epsilon }_{\mathbf{k}+\mathbf{q},j}-{\epsilon }_F),\end{array}$$ 5

where ω_qν determines the values of phonon energies and γ_qν represents the phonon linewidth. The electron-phonon coefficients are given by g_qν(k, i, j) and ε_k,i is the electron band energy (ε_F denotes the Fermi energy).

For the purpose of this paper, the spectral functions obtained by: Li et al.¹¹ (H₂S), Ishikawa et al.²⁷ (H₅S₂), and Li et al.³⁰ (H₄S₃), have been taken into account. The Eliashberg functions were calculated in the framework of Quantum Espresso code^6,7.

Results

The pseudopotential parameter

Figure 1(a) presents the experimental dependence of critical temperature on the pressure for the sulfur-hydrogen systems¹⁰ (see also³³). Additionally, we included the theoretical results obtained for the H₅S₂²⁷, H₂S¹¹, H₄S₃³⁰, and H₃S³⁴.

Figure 1.

(a) The experimental values of critical temperature as a function of pressure for the selected hydrogen-containing compounds of sulfur. The following designation was concluded in the legend: exp. - experimental data taken from the paper¹⁰. The gray line represents the dependence of T_C(p) in the pressure range from 110 GPa to 123 GPa, for which Ishikawa et al.²⁷ predicted the stability of H₅S₂ compound. The black line with the circles indicates the theoretical results, obtained for H₂S¹¹. The violet and magenta dash lines are the predictions at the phenomenological level based on the classical theory of Eliashberg. In addition, we have included theoretical results for H₅S₂²⁷ (green squares), H₄S₃³⁰ (orange triangle) and H₃S (blue stars)³⁴. (b) The dependence of order parameter (Δ_n=1) on the Coulomb pseudopotential for H₅S₂ - selected values of cut-off frequency (T = T_C). The spheres represent the results obtained with the aid of Eliashberg equations with the vertex corrections (VCEE), the squares correspond to the results obtained in the framework of Migdal-Eliashberg formalism (CEE).

It is worth to notice that H₅S₂ is thermodynamically stable in the fairly narrow pressure range (110–123 GPa). Especially at pressure 110 GPa there is the transformation: H₄S₃ + 7H₃S → 5H₅S₂. On the other hand, breakup H₅S₂ for p = 123 GPa takes place according to the scheme: 3H₅S₂ → 5H₃S + S.

The measuring points were used to determine the curve T_C(p). We used the approximating function, matching the polynomial with the least squares method, respectively. On this basis, the estimated value of critical temperature for the pressure at 112 GPa is equal to 36 K. In the case of H₅S₂ compound with value of T_C = 36 K, we have calculated the pseudopotential parameter. To this end, we used the equation: ${\left[{\mathrm{\Delta }}_{n=1}\left({\mu }^{⁎}\right)\right]}_{T=T_C}=0$⁴⁴. We obtained the very high value of μ* in both considered approaches: ${\left[{\mu }^{⁎}({\omega }_c^{(1)})\right]}_{VCEE}=0.589$ and ${\left[{\mu }^{⁎}({\omega }_c^{(1)})\right]}_{CEE}=0.402$, whereby we have chosen the following cut-off frequency: ${\omega }_c^{\left(1\right)}=3{\omega }_D$, where ${\omega }_D=237.2$ meV²⁷. Additionally for H₅S₂, we have ε_F = 22.85 eV.

Note that the high value of μ* is relatively often observed in the case of high-pressure superconducting state. For example, for phosphorus: ${\left[{\mu }^{⁎}\left(5{\omega }_D\right)\right]}_{CEE}^{p\mathrm{=20}\mathrm{G}Pa}=0.37$, where ${\omega }_D=59.4$ meV⁴⁵. We encounter the similar situation for lithium: ${\left[{\mu }^{⁎}\left(3{\omega }_D\right)\right]}_{CEE}^{p\mathrm{=29.7}\mathrm{G}Pa}=0.36$, while ω_D = 82.7 meV⁴⁶. Interestingly, for H₃S compound under the pressure at 150 GPa, the value of Coulomb pseudopotential is low: 0.123⁴⁷. However, after increasing the pressure by 50 GPa, this value is clearly increasing: ${\mu }^{⁎}\sim 0.2$³⁵. In the case of another compound (PH₃), for which we also know the experimental critical temperature (T_C = 81 K), the much lower value of Coulomb pseudopotential was obtained: ${\left[{\mu }^{⁎}\left(10{\omega }_D\right)\right]}_{CEE}^{p\mathrm{=200}\mathrm{G}Pa}=0.088$³⁵. It is also worth mentioning the paper⁴⁴, where the properties of superconducting state were studied in the SiH₄ compound for ${\mu }^{⁎}\in \langle 0.1,0.3\rangle$. The result was the decreasing critical temperature range from 51.7 K to 20.6 K, what in relation to the results contained in the experimental work⁴⁸ suggests ${\mu }^{⁎}\sim 0.3$. However, it should be remembered that the results presented in the publication⁴⁸ are very much undermined⁴⁹, while it is argued that the experimental data does not refer to SiH₄ but to the PtH compound (the hydrogenated electrodes of measuring system).

Considering the facts presented above, it is easy to see that the H₅S₂ compound, even in the group of high-pressure superconductors, has the unusually high value of Coulomb pseudopotential. This is the feature of H₅S₂ system, because values of μ* cannot be reduced by selecting another acceptable cut-off frequency. On the contrary, increasing ω_c leads to the absurdly large increase in the value of Coulomb pseudopotential: ${\left[{\mu }^{⁎}({\omega }_c^{(2)})\right]}_{VCEE}=1.241$ and ${\left[{\mu }^{⁎}({\omega }_c^{(2)})\right]}_{CEE}=0.671$, where ${\omega }_c^{(2)}=10{\omega }_D$. The dependence courses of Δ_n=1 on ${\mu }^{⁎}$, characterizing the situation discussed by us, are collected in Fig. 1(b).

It should be emphasized that value of μ* calculated by us for the CEE case (0.402) significantly exceeds the value of Coulomb pseudopotential estimated in the paper²⁷, where ${\mu }^{⁎}\in \langle 0.13,0.17\rangle$. This result is related to the fact that in the publication²⁷ the critical temperature was calculated using the simplified Allen-Dynes formula (f₁ = f₂ = 1)⁵⁰, which significantly understates μ*, for values greater than 0.1⁵¹. This is due to the approximations used in the derivation of Allen-Dynes formula. Among other things, the analytical approach does not take into account the effects of retardation and the impact on the result of the cut-off frequency. Below is the explicit form of Allen-Dynes formula:

$$k_B{T}_C=f_1{f}_2\frac{{\omega }_{\mathrm{l}\mathrm{n}}}{1.2}\exp \left[\frac{-1.04\left(1+\lambda \right)}{\lambda -{\mu }^{⁎}\left(1+0.62\lambda \right)}\right],$$ 6

where:

$$f_1={\left[1+{\left(\frac{\lambda }{{\mathrm{\Lambda }}_1}\right)}^{\frac{3}{2}}\right]}^{\frac{1}{3}},$$ 7

$${\mathit{f}}_2=\left[1+\frac{\left(\frac{\sqrt{{\mathit{\omega }}_2}}{{\mathit{\omega }}_{\ln }}-1\right){\mathit{\lambda }}^2}{{\mathit{\lambda }}^2+{\mathrm{\Lambda }}_2^2}\right].$$ 8

Parameters Λ₁ and Λ₂ were calculated using the formulas:

$${\mathrm{\Lambda }}_1=2.46\left(1+3.8{\mu }^{⁎}\right),$$ 9

$${\mathrm{\Lambda }}_2=1.82\left(1+6.3{\mu }^{⁎}\right)\frac{\sqrt{{\omega }_2}}{{\omega }_{\mathrm{l}\mathrm{n}}}\mathrm{.}$$ 10

The second moment is given by the expression:

$${\omega }_2=\frac{2}{\lambda }{\int }_0^{{\omega }_D}d\omega {\alpha }^{2}F\left(\omega \right)\omega \mathrm{.}$$ 11

The quantity ω_ln stands for the phonon logarithmic frequency:

$${\omega }_{\ln }=\exp \left[\frac{2}{\lambda }{\int }_0^{{\omega }_D}d\omega \frac{{\alpha }^{2}F(\omega )}{\omega }\ln (\omega )\right]\mathrm{.}$$ 12

For the H₅S₂ compound, we have obtained: ω₂ = 112.88 meV and ω_ln = 77.37 meV. Assuming f₁ = f₂ = 1, we reproduce the result contained in²⁷:  ${\left[{\mathrm{T}}_{\mathrm{C}}\right]}^{{\mu }^{⁎}=0.17}=58.3$ K and ${\left[{\mathrm{T}}_{\mathrm{C}}\right]}^{{\mu }^{⁎}=0.13}=70.1$ K. However, the full Allen-Dynes formula gives: ${\left[{\mathrm{T}}_{\mathrm{C}}\right]}^{{\mu }^{⁎}=0.17}=62.5$ K and ${\left[{\mathrm{T}}_{\mathrm{C}}\right]}^{{\mu }^{⁎}=0.13}=76.1$ K.

From the physical point of view, the values of Coulomb pseudopotential calculated for H₅S₂, in the framework of full Eliashberg formalism, are so high that μ* cannot be associated only with the depairing Coulomb correlations. In principle, it should be treated only as the parameter to fit the model to the experimental data. Of course, there can be very much reasons that cause anomalously high value of μ*. The authors of paper²⁷ pay special attention to the role of anharmonic effects. It should be emphasized that even if it were this way, the anharmonic nature of phonons should be taken into account not only in the Eliashberg function, but also in the structure of Eliashberg equations itself (this fact is usually omitted in the analysis scheme). In our opinion, the probable cause may also be non-inclusion in the Fröhlich Hamiltonian⁵² the non-linear terms of electron-phonon interaction⁴³ or possibly anomalous electron-phonon interactions related to the dependence of full electron Hamiltonian parameters of system on the distance between the atoms of crystal lattice^53,54.

However, in our opinion it is more likely that the experimental results obtained for the low temperature superconducting state of compressed sulfur-hydrogen system are largely related to the condensate being induced in the H₂S compound (see black line in Fig. 1). In fact, the predicted thermodynamically stable pressure range of H₅S₂ is really narrow (110–123 GPa), and the inevitable kinetic barrier may prevent the decomposition of H₂S into H₅S₂ in such narrow pressure range. It should also be mentioned that in the following two experiments³⁰ and³³, the XRD measurements on compressed H₂S do not observe the formation of H₅S₂. On the other hand the XRD results in the paper³⁰ indeed observed the residual H₂S coexist with dissociation products H₃S and H₄S₃ at least up to 140 GPa. It should also be emphasized that the anomalously high μ* obtained for H₅S₂ are extremely incompatible with the estimation of μ* value (0.1–0.13) contained in Ashcroft’s fundamental paper⁵⁵ concerning the superconducting state in the hydrogen-rich compounds.

The value of Coulomb pseudopotential observed in H₅S₂ compound is so high that it is worth to confront it with other microscopic parameters obtained from ab initio calculations: (ε_F and ω_ln)²⁷. The most advanced formula on the Coulomb pseudopotential takes the following form⁵:

$${\mu }^{⁎}=\frac{\mu +a{\mu }^2}{1+\mu \ln \left(\frac{{\epsilon }_F}{{\omega }_{\ln }}\right)+a{\mu }^2\ln \left(\frac{\alpha {\epsilon }_F}{{\omega }_{\ln }}\right)},$$ 13

where a = 1.38 and α ≃ 0.1. Symbol μ denotes product of the electron density of states at the Fermi level ρ(ε_F) and the Coulomb potential U > 0. Note that the expression (13) is the non-trivial generalization of Morel-Anderson formula⁴:

$${\mu }^{⁎}=\frac{\mu }{1+\mu \ln \left(\frac{{\epsilon }_F}{{\omega }_{\ln }}\right)}\mathrm{.}$$ 14

In particular, with the help of equation (13), it can be proven that the retardation effects associated with the electron-phonon interaction reduce the original value of μ to the value of μ*, but to the much lesser extent than expected by Morel and Anderson.

Using the formula (13), it is easy to show that the anomalously high value of μ* cannot result only from the strong electron depairing correlations. Namely in the limit μ → +∞, we obtain:  ${\left[{\mu }^{⁎}\right]}_{\max }=\mathrm{1/}\ln \left(\alpha {\epsilon }_F/{\omega }_{\ln }\right)=0.295$. It is worth noting that the Morel-Anderson model is completely unsuitable for the analysis, because for μ* = 0.402 we get the negative (non-physical) value of μ = −0.48.

The value of Coulomb pseudopotential can also be tried to be estimated using the phenomenological Bennemann-Garland formula⁵⁶:

$${\mu }_{\mathrm{BG}}^{⁎}\sim 0.26\rho \left({\epsilon }_F\right)\mathrm{/[1}+\rho \left({\epsilon }_F\right)\mathrm{].}$$ 15

For the H₅S₂ compound subjected to the action of pressure at 112 GPa, we obtain ${\mu }_{\mathrm{BG}}^{⁎}=0.23$, which also proves the breakdown of the classical interpretation of μ*.

Taking into account all the facts given above, it is unlikely for the low-temperature phase of compressed sulfur-hydrogen to be tied with H₅S₂. The dependence of critical temperature on the pressure could be explained according to the H₂S compound, whereas the low value of Coulomb pseudopotential is assumed (${\mu }^{⁎}=0.15$)^11,26. In particular, let’s focus on the lowest pressure considered in¹¹ (130 GPa). The calculations within the paper, are performed according to the CEE scheme and they give the critical temperature value of $\sim 30.6$ K²⁶, which correlates well with the experimental value of T_C = 36 K for p = 112 GPa. It is worth noting that the electron-phonon interaction in H₂S for the pressure of 130 GPa, is characterized by the following set of parameters: λ = 0.785, ω_ln = 81.921 meV, and $\sqrt{{\omega }_2}=112.507$ meV. Physically, this means the intermediate value of coupling between the electrons and phonons. H₅S₂ is the system characterized by the strong electron-phonon coupling (λ = 1.186). At the phenomenological level, one can even extend the T_C(p) curve calculated in the paper¹¹. For this purpose, with the help of the approximation function, we determined the relationship λ(p), which we used in the appropriately modified Eliashberg equations (the case of one-band equations presented in⁵⁷). As the result for p = 112 GPa, we received $T_C\in \langle 30.7,22.3\rangle$ K assuming that ${\mu }^{⁎}\in \langle 0.1,0.15\rangle$. The full form of T_C(p) curves, for ${\mu }^{⁎}=0.1$ and ${\mu }^{⁎}=0.15$ in the pressure range from 112 GPa to 130 GPa, is presented in Fig. 1(a).

It is also worth mentioning the superconducting state, which can potentially be induced in H₄S₃. The data obtained for the pressure of 140 GPa gives the following characteristics of the electron-phonon interaction³⁰: λ = 0.42 and ω_ln = 71.869 meV (typical limit of weak coupling). Due to the low value of λ, the critical temperature can be calculated using the McMillan formula⁵⁸ (f₁ = f₂ = 1). Assuming that ${\mu }^{⁎}=0.13$, we get T_C = 2.2 K. This result means that the superconducting state inducing in H₄S₃ cannot be equated with the low-temperature superconducting phase of compressed sulfur-hydrogen system. We probably have the analogous situation here as for the H₅S₂ compound, where kinetically protected H₂S in samples prepared at low temperature is responsible for the observed superconductivity below 160 GPa³⁰.

In the following chapters, for illustrative purposes, we calculated the thermodynamic parameters for the H₅S₂ compound. Note, that even for anomalously high value of μ*, the classical Eliashberg equations allow to set thermodynamic functions of superconducting state correctly, but at the phenomenological level^2,59. The presented discussion is complemented by the results obtained in the CEE scheme for H₂S (p = 130 GPa) and H₄S₃ (p = 140 GPa) compounds.

The order parameter and the electron effective mass

Figure 2 presents for H₅S₂ the plot of the dependence of order parameter for the first Matsubara frequency on the temperature. It is clearly visible that, in the low temperature range, the values of order parameter calculated with regard to the vertex corrections are much higher than the values of Δ_n=1 designated within the framework of classic Eliashberg scheme. This is the very interesting result, because the ratio v_s = λω_D/ε_F for H₅S₂ compound is not high (v_s = 0.012)²⁷. This result, with the cursory analysis, could suggest the slight influence of vertex corrections on the thermodynamic parameters. It is not, however, because it should be commemorated that v_s defines only the static criterion of the impact of vertex corrections. This means that the differences observed between the results obtained in the VCEE and CEE schemes are associated with dynamic effects, i.e. the explicit dependence of order parameter on the Matsubara frequency.

Figure 2.

The influence of temperature on the maximum value of order parameter in the VCEE and CEE scheme for H₅S₂ (${\omega }_c={\omega }_c^{\left(1\right)}$). The insertion presents the ratio $m_e^{⁎}/m_e$ as a function of temperature. The open symbols represent the numerical results, the gray lines correspond to the results obtained with the help of analytical formulas ((16) or (17)). The red spheres were obtained in the BCS scheme assuming that: 2Δ_n=1(0)/k_BT_C = 3.53^62,63.

On the insertion in Fig. 2, we presented the influence of temperature on the value of the ratio of electron effective mass ($m_e^{⁎}$) to the electron band mass (m_e). In the Eliashberg formalism this ratio can be estimated using the following formula: $m_e^{⁎}/m_e=Z_{n\mathrm{=1}}\left(T\right)$². It can be seen that, in the entire temperature range analyzed by us, the effective mass of electron is high however slightly dependent on the temperature. In particular, in the VCEE scheme we get: ${\left[m_e^{⁎}\right]}_{T_0}=2.10m_e$ and ${\left[m_e^{⁎}\right]}_{T_C}=2.14m_e$. On the other hand, the classic Eliashberg approach gives: ${\left[m_e^{⁎}\right]}_{T_0}=2.16m_e$ and ${\left[m_e^{⁎}\right]}_{T_C}=2.19m_e$. When comparing the above results, we conclude that the vertex corrections have the very little effect on the value of the effective mass of electron, contrary to the situation with the order parameter.

The functions plotted in Fig. 2 can be characterized analytically by means of formulas:

$${\mathrm{\Delta }}_{n=1}\left(T\right)={\mathrm{\Delta }}_{n=1}\left(0\right)\sqrt{1-{\left(T/T_C\right)}^{\mathrm{\Gamma }}}$$ 16

and (only for the CEE scheme):

$$m_e^{⁎}/m_e=\left[Z_{n=1}\left(T_C\right)-Z_{n=1}\left(0\right)\right]{\left(T/T_C\right)}^{\mathrm{\Gamma }}+Z_{n=1}\left(0\right),$$ 17

where the traditional markings were introduced: Δ_n=1(0) = Δ_n=1(T₀) and Z_n=1(0) = Z_n=1(T₀).

For the order parameter, we obtained the following estimation of temperature exponent: [Γ]_VCEE = 1.55 and [Γ]_CEE = 3.15. Physically, this result means that in the VCEE scheme the temperature dependence of order parameter differs very much from the course anticipated by the mean-field BCS theory, where Γ_BCS = 3⁶⁰. On the other hand, not very large deviations from the predictions of BCS theory at the level of classical Eliashberg equations can be explained by referring to the impact of retardation and strong-coupling effects on the superconducting state. In the simplest way, they are characterized by the ratio r = k_BT_C/ω_ln, which for the H₅S₂ compound equals 0.0401. On the other hand, in the BCS limit we obtain: r = 0. Of course, in the case of the VCEE scheme, the deviations from the BCS predictions should be interpreted as the cumulative effect of vertex corrections, strong-coupling, and retardation effects influencing the superconducting state.

The very accurate values of order parameter can be calculated by referring to the equation: Δ(T) = Re[Δ(ω = Δ(T))], where the symbol Δ(ω) denotes the order parameter determined on the real axis. The form of function Δ(ω) should be determined using the method of the analytical extension of imaginary axis solutions⁶¹. The sample results have been posted in Fig. 3, where it can be immediately noticed that the function of order parameter takes the complex values. However, for low values of frequency, only the real part of order parameter is non-zero. Physically, this means no damping effects, or equally, the endless life of Cooper pairs. Having the open form of order parameter on the real axis, we have calculated the value of ratio: R_Δ = 2Δ(0)/k_BT_C. The obtained result was compared with the data for other superconductors with the phonon pairing mechanism (see Fig. 4). It is easy to see that the result of CEE very well fits in the general trend anticipated by the classic Eliashberg formalism (R_Δ = 3.77). On the other hand, the impact of vertex corrections on the value of parameter R_Δ is strong (R_Δ = 4.85). Let us recall that the mean-field BCS theory predicts: R_Δ = 3.53^62,63.

Figure 3.

The real and imaginary part of H₅S₂ order parameter on the real axis for the selected values of temperature. The sample results obtained in the CEE scheme.

Figure 4.

The value of ratio 2Δ(0)/k_BT_C in the dependence on the parameter r = k_BT_C/ω_ln. The blue line represents the general trend reproduced by the formula: 2Δ(0)/k_BT_C = 3.53[1 + 12.5(r)²ln(1/2r)] (CEE scheme²).

Particularly interesting is the comparison between the calculated values of R_Δ for H₅S₂ with results obtained for H₂S and H₄S₃. In the first step, it is noticeable that the value of parameter r for H₂S and H₄S₃ is 0.0362 and 0.0014, respectively. It is evident that ${\left[r\right]}_{{\mathrm{H}}_2\mathrm{S}}$ is close to the value of r obtained for H₅S₂, while in the case of H₄S₃ the retardation and strong-coupling effects do not play the major role. In the CEE scheme, the dimensionless ratio R_Δ for H₂S and H₄S₃ assumes the values: ${\left[R_{\mathrm{\Delta }}\right]}_{{\mathrm{H}}_2\mathrm{S}}=3.67$ and ${\left[R_{\mathrm{\Delta }}\right]}_{{\mathrm{H}}_4{\mathrm{S}}_3}=3.53$. For the H₂S and H₄S₃ compounds, the vertex corrections do not change R_Δ. They do not influence the function Δ(T), which is why they are not taken into account in the further part of paper.

The free energy, thermodynamic critical field, entropy, and the specific heat jump

The thermodynamics of superconducting state is fully determined by the dependence of order parameter on the temperature. Based on the results posted in Fig. 2, it can be seen that in the temperature area $T_C-T\ll T_C$, the values of Δ(T) obtained for H₅S₂ in the framework of VCEE and CEE schemes are physically indistinguishable. This result means that the thermodynamics of superconducting state near the critical temperature can be analyzed with the help of classical Eliashberg approach without vertex corrections.

The free energy difference between the superconducting and normal state has been calculated in agreement with the formula⁶⁴:

$$\begin{array}{l}\frac{\mathrm{\Delta }F}{\rho \mathrm{(0)}}=-2\pi k_{B}T\sum _{m\mathrm{=1}}^M[\sqrt{{\omega }_m^2+{\left({\mathrm{\Delta }}_m\right)}^2}-|{\omega }_m|]\\ \times [Z_m^S-Z_m^N\frac{|{\omega }_m|}{\sqrt{{\omega }_m^2+{\left({\mathrm{\Delta }}_m\right)}^2}}],\end{array}$$ 18

whereas $Z_m^N$ and $Z_m^S$ denote respectively the wave function renormalization factor for the normal state ($N$) and for the superconducting state (S).

In Fig. 5 (lower panel), we have plotted the form of function ΔF(T)/ρ(0) for H₅S₂. It can be seen that the free energy difference takes negative values in the whole temperature range up to T_C. This demonstrates thermodynamic stability of superconducting phase in the compound under investigation. For the lowest temperature taken into account, it was obtained: [ΔF/ρ(0)]_{T=30 K} = −2.94 meV².

Figure 5.

(lower panel) The free energy difference between the superconducting and normal state as a function of temperature, and (upper panel) the thermodynamic critical field. The charts were obtained as the part of classical Eliashberg formalism without the vertex corrections (for the temperature area in which the results obtained with the help of VECC and CEE models are identical (T > 30 K)).

On the basis of the temperature dependence of free energy difference, it is relatively easy to determine the values of thermodynamic critical field (H_C), the difference in entropy (ΔS) between the superconducting state and the normal state, and the specific heat difference (ΔC).

The thermodynamic critical field has been calculated on the basis of formula:

$$\frac{H_C}{\sqrt{\rho \left(0\right)}}=\sqrt{-8\pi \frac{\Delta F}{\rho \left(0\right)}}\mathrm{.}$$ 19

We presented the results obtained for the H₅S₂ compound in Fig. 5 (upper panel). We see that as the temperature rises, the critical field decreases so that in T = T_C its value equaled zero. Assuming for the H₅S₂ compound the following designation H(0) = H(T = 30 K) = 8.95 meV, it is easy to see that the critical field decreases in the proportion to the square of temperature, which is well illustrated by the parabola plotted on the basis of equation⁶⁵:

$$H_C\left(T\right)=H\left(0\right)\left[1-{\left(T/T_C\right)}^2\right]\mathrm{.}$$ 20

The dependence H_C(T) determined using this formula is represented by the red spheres in Fig. 5.

The difference in the entropy between the superconducting and normal state has been estimated based on the expression:

$$\frac{\mathrm{\Delta }S}{k_B\rho \left(0\right)}=-\frac{d\left[\mathrm{\Delta }F/\rho \left(0\right)\right]}{d\left(k_{B}T\right)}\mathrm{.}$$ 21

We presented the obtained results in Fig. 6(a). Physically increasing are the value of entropy up to T_C proves the higher ordering of superconducting state in relation to the normal state.

Figure 6.

(a) The dependence of difference in the entropy on the temperature, and (b) the specific heat in the superconducting and normal state (the Eliashberg formalism without the vertex corrections).

The values of ΔC have been calculated using the formula:

$$\frac{\mathrm{\Delta }C}{k_B\rho \left(0\right)}=-\frac{1}{\beta }\frac{d^2\left[\mathrm{\Delta }F/\rho \left(0\right)\right]}{d{\left(k_{B}T\right)}^2}\mathrm{.}$$ 22

In addition, the specific heat of normal state is determined based on:

$$\frac{C^N}{k_B\rho \mathrm{(0)}}=\gamma k_{B}T,$$ 23

where the Sommerfeld constant equals: $\gamma =\frac{2}{3}{\pi }^2\mathrm{(1}+\lambda )$. The influence of temperature on the specific heat in the superconducting state and the normal state has been presented in Fig. 6(b). The characteristic specific heat jump visible in the critical temperature is noteworthy. For the H₅S₂ compound its value equals 75.34 meV.

The thermodynamic parameters determined allow to calculate the dimensionless ratio: R_C = ΔC(T_C)/C^N(T_C). As part of classic BCS theory, the quantity R_C takes the universal value equal to 1.43^62,63. In the case of H₅S₂ compound, it was obtained R_C = 1.69. Hence, the value of R_C for H₅S₂ deviates from the predictions of BCS theory. Additionally, Fig. 7 presents the dependence of dimensionless parameter R_C on r. The chart shows that the value of R_C obtained for H₅S₂ perfectly fits the general trend anticipated by the classic Eliashberg formalism.

Figure 7.

The value of ratio R_C in the dependence on the parameter r. The blue line represents the general trend obtained using the formula: R_C = 1.43[1 + 53(r)²ln(1/3r)] (CEE scheme²).

In the case of H₂S compound the ratio R_C = 1.63 is close to ${\left[R_C\right]}_{{\mathrm{H}}_{\mathrm{5}}{\mathrm{S}}_{\mathrm{2}}}$. The parameter R_C for H₄S₃ is equal to the value predicted by the BCS theory.

Summary

We have calculated the thermodynamic parameters of superconducting state for the H₅S₂ compound under the pressure at 112 GPa. We have solved the Eliashberg equations on the imaginary axis, both including the first-order vertex corrections to the electron-phonon interaction, as well as without vertex corrections. For both cases, we have obtained anomalously high values of Coulomb pseudopotential: ${\left[{\mu }^{⁎}\left({\omega }_c^{\left(1\right)}\right)\right]}_{VCEE}=0.589$ and ${\left[{\mu }^{⁎}({\omega }_c^{(1)})\right]}_{CEE}=0.402$, while ${\omega }_c^{\left(1\right)}=3{\omega }_D$. Note that increase in the cut-off frequency only increases value of ${\mu }^{⁎}$ (e.g. for VCEE case ${\mu }^{⁎}=1.241$, where ${\omega }_c^{\left(2\right)}=10{\omega }_D$). The calculated values of ${\mu }^{⁎}$ proves that this parameter cannot be associated only with the depairing Coulomb correlations - in principle, it should be treated as the effective parameter of fitting the model to experimental data. The above results mean that it is very unlikely that the low-temperature superconducting state of compressed sulfur-hydrogen system is induced in the H₅S₂ compound. In our opinion, experimentally was observed the superconducting state in the H₂S compound, which is kinetically protected in the samples prepared at the low temperature^11,30. It should be emphasized that in the case of H₂S reproducing the experimental dependence of critical temperature on the pressure does not require anomalously high value of Coulomb pseudopotential²⁶.

In our paper, we also analyzed the thermodynamic properties of superconducting state in the H₄S₃ compound. It is characterized by the very low value of critical temperature (${\left[T_C\right]}_{\max }\sim 2$ K)³⁰, and it cannot be associated with the low temperature superconducting state of compressed dihydrogen sulfide. Probably, we have the analogous situation here like for the H₅S₂ compound, where kinetically protected H₂S in the samples prepared at the low temperature is responsible for the observed superconductivity.

As part of the analysis, we calculated the thermodynamic parameters of superconducting state for the H₅S₂, H₂S, and H₄S₃ compounds. In the case of H₅S₂, we have shown that the first order vertex corrections to the electron-phonon interaction play the important role, i.e. they significantly change the thermodynamics of superconducting state in the low temperature range, while nearby T_C they are practically irrelevant. For this reason, the dimensionless parameter R_Δ in the VCEE scheme is equal to 4.85, and in the CEE scheme it equals 3.77. On the other hand, for both cases it was obtained R_C = 1.693. The above values prove that the superconducting state in the H₅S₂ compound is not the state of BCS type. The superconducting state for H₂S has the thermodynamic parameters with values are close to the ones determined for H₅S₂ in the CEE scheme. In particular, we got: R_Δ = 3.67 and R_C = 1.626. On the other hand, the superconducting state of H₄S₃ compound is the BCS type.

Regarding the results presented by Ishikawa et al.²⁷, in respect to the Eliashberg function (harmonic limit), it should be assumed that these are the results proving the impossibility of correct description of the low-temperature superconducting phase of compressed sulfur-hydrogen system.

Author Contributions

Małgorzata Kostrzewa wrote the code for numerical calculations (VCEE scheme), carried out the calculations (VCEE scheme), and participated in writing the manuscript. R. Szczęśniak wrote the code for numerical calculations (CEE scheme) and participated in writing the manuscript. Joanna K. Kalaga collected data and drafted the final version of the manuscript. Izabela A. Wrona carried out the calculations (CEE scheme). All authors reviewed the manuscript.

The authors declare no competing interests.