Anomalously high value of Coulomb pseudopotential for the H5S2 superconductor

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 ([TC]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.

The Eliashberg formalism 1 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 (α 2 F(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 results 8 .
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 H 2 S with high value of critical temperature was discovered by Drozdov, Eremets, and Troyan in 2014 9 (see also the paper 10 ). 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 H 2 S compound at the pressure range of ~10-200 GPa. In the paper 11 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 Pmc2 1 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 H 2 S were observed for 80 GPa (Pmc2 1 → P-1) and 160 GPa (P-1 → Cmca). Interestingly, the results obtained in the paper 11 are in the contradiction with the earlier theoretical predictions, suggesting that H 2 S dissociate into elemental sulfur and hydrogen under the high pressure 14 . However, it should be noted, that the partial decomposition of H 2 S was observed in Raman 15 and XRD studies 16 at the room temperature above 27 GPa. The calculations of the electronic structure made in 11 suggest that the H 2 S 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 experimentally 17 (see also related resea rch 12,13,15,16,[18][19][20][21][22][23][24]. It is worth noting that in the work 11 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 → Cmca 11 .
In experiments described in the papers 9 and 10 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 H 2 S compound, as it is generally consistent with calculations presented in 11 for solid H 2 S: both the value of T C < 82 K and its pressure behavior. In addition, it was demonstrated theoretically 26 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 H 5 S 2 compound, in which asymmetric hydrogen bonds are formed between H 2 S and H 3 S 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 paper 27 , it is not supported by the first-principles calculations.
It is also worth paying attention to the results contained in 30 , in which it is envisaged that the H 4 S 3 is stable within the pressure range of 25-113 GPa. What is important, H 4 S 3 coexists with fraction of H 3 S and H 2 S, 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 + H 3 S + H 4 S 3 . Nevertheless, H 4 S 3 is characterized by the very low critical temperature (T C = 2.2 K for μ* = 0.13), which suggests that kinetically protected H 2 S in samples prepared at low temperature is responsible for the observed superconductivity below 160 GPa 30 .
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 GPa 31 (see also 28,30,32,33 ): 3H 2 S → 2H 3 S + 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 H 3 S stoichiometry. The physical mechanism underlying the superconductivity of H 3 S is similar to that in the MgB 2 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 boron 32 . Considering the electron-phonon interaction, it was noted that in the case of H 3 S, the vertex corrections in the local approximation have the least meaning 35 . However, in the static limit with finite q, the vertex corrections change the critical temperature by −34 K 36 . The result above correlates well with the data for SiH 4 37 . For the H 3 S 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 H 3 S 38 . However, we have recently shown that the value of T C increases (242 K), if we use the sulfur isotope 36 S 39 . Therefore, it can be reasonably supposed that it is possible to obtain the superconducting condensate close to the room temperature (see also related research 40 ).
In the presented paper, we analyze precisely the thermodynamic properties of superconducting state in the H 5 S 2 system 27 . 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 μ ∈ 〈 .
. 〉 ⁎ 0 13, 0 17 . The paper contains also the parameter's analysis of superconducting state, that is induced in the H 4 S 3 compound 30 . The results obtained by us were compared with the results for the H 2 S 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 1 ): In the case when A = 1, the Eliashberg set was generalized to include the lowest-order vertex correctionscheme 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 interaction 2 . This is the typical strong-coupling effect, because Z n=1 in the CEE scheme is expressed by the formula: , where λ denotes the electron-phonon coupling constant: and ω D is the Debye frequency. The order parameter is defined as the ratio: φ Δ = Z / n n n . Symbol ω n is the Matsubara frequency: ω n = πk B T(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 T 0 = 5 K to T C (see also 42 ).
The function μ ⁎ n modeling the depairing correlations has the following form: The Heaviside function is given by θ(x) and ω c represents the cut-off frequency. By default, it is assumed that The electron-phonon pairing kernel is expressed by the formula: 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 paper 43 .
The Eliashberg spectral function is defined as: 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 (H 2 S), Ishikawa et al. 27 (H 5 S 2 ), and Li et al. 30 (H 4 S 3 ), 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 systems 10 (see also 33 ). Additionally, we included the theoretical results obtained for the H 5 S 2 27 , H 2 S 11 , H 4 S 3 30 , and H 3 S 34 . It is worth to notice that H 5 S 2 is thermodynamically stable in the fairly narrow pressure range (110-123 GPa). Especially at pressure 110 GPa there is the transformation: H 4 S 3 + 7H 3 S → 5H 5 S 2 . On the other hand, breakup H 5 S 2 for p = 123 GPa takes place according to the scheme: 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 H 5 S 2 compound with value of T C = 36 K, we have calculated the pseudopotential parameter. To this end, we used the equation:

44
. We obtained the very high value of μ* in both considered approaches: μ ω = . ⁎
, whereby we have chosen the following cut-off frequency: 27 . Additionally for H 5 S 2 , 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: 45 . We encounter the similar situation for lithium: μ ω = .
= . ⁎ , while ω D = 82.7 meV 46 . Interestingly, for H 3 S compound under the pressure at 150 GPa, the value of Coulomb pseudopotential is low: 0.123 47 . However, after increasing the pressure by 50 GPa, this value is clearly increasing: μ ∼ . ⁎ 0 2 35 . In the case of another compound (PH 3 ), for which we also know the experimental critical temperature (T C = 81 K), the much lower value of Coulomb pseudopotential was obtained: 35 . It is also worth mentioning the paper 44 , where the properties of superconducting state were studied in the SiH 4 compound for μ ∈ . .  49 , while it is argued that the experimental data does not refer to SiH 4 but to the PtH compound (the hydrogenated electrodes of measuring system).
Considering the facts presented above, it is easy to see that the H 5 S 2 compound, even in the group of high-pressure superconductors, has the unusually high value of Coulomb pseudopotential. This is the feature of H 5 S 2 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: c VCEE (2) and μ ω = .
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 paper 27 , where μ ∈ . . ⁎ 0 13, 0 17 . This result is related to the fact that in the publication 27 the critical temperature was calculated using the simplified Allen-Dynes formula (f 1 = f 2 = 1) 50 , which significantly understates μ*, for values greater than 0.1 51 . 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: Parameters Λ 1 and Λ 2 were calculated using the formulas: The second moment is given by the expression: The quantity ω ln stands for the phonon logarithmic frequency: K. From the physical point of view, the values of Coulomb pseudopotential calculated for H 5 S 2 , 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 paper 27 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 Hamiltonian 52 the non-linear terms of electron-phonon interaction 43 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 H 2 S compound (see black line in Fig. 1). In fact, the predicted thermodynamically stable pressure range of H 5 S 2 is really narrow (110-123 GPa), and the inevitable kinetic barrier may prevent the decomposition of H 2 S into H 5 S 2 in such narrow pressure range. It should also be mentioned that in the following two experiments 30 and 33 , the XRD measurements on compressed H 2 S do not observe the formation of H 5 S 2 . On the other hand the XRD results in the paper 30 indeed observed the residual H 2 S coexist with dissociation products H 3 S and H 4 S 3 at least up to 140 GPa. It should also be emphasized that the anomalously high μ* obtained for H 5 S 2 are extremely incompatible with the estimation of μ* value (0.1-0.13) contained in Ashcroft's fundamental paper 55 concerning the superconducting state in the hydrogen-rich compounds.
The value of Coulomb pseudopotential observed in H 5 S 2 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 form 5 :   (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: μ α ε ω = = .
⁎ [ ] 1 /ln( / ) 0 295 F max l n . 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 formula 56 :  Taking into account all the facts given above, it is unlikely for the low-temperature phase of compressed sulfur-hydrogen to be tied with H 5 S 2 . The dependence of critical temperature on the pressure could be explained according to the H 2 S compound, whereas the low value of Coulomb pseudopotential is assumed (μ = . At the phenomenological level, one can even extend the T C (p) curve calculated in the paper 11 . 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 in 57 ). As the result for p = 112 GPa, we received ∈ 〈 .
⁎ 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 H 4 S 3 . The data obtained for the pressure of 140 GPa gives the following characteristics of the electron-phonon interaction 30 : 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 formula 58 (f 1 = f 2 = 1). Assuming that μ = .
⁎ 0 13, we get T C = 2.2 K. This result means that the superconducting state inducing in H 4 S 3 cannot be equated with the low-temperature superconducting phase of compressed sulfur-hydrogen system. We probably have the analogous situation here as for the H 5 S 2 compound, where kinetically protected H 2 S in samples prepared at low temperature is responsible for the observed superconductivity below 160 GPa 30 .
In the following chapters, for illustrative purposes, we calculated the thermodynamic parameters for the H 5 S 2 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 H 2 S (p = 130 GPa) and H 4 S 3 (p = 140 GPa) compounds.
The order parameter and the electron effective mass. Figure 2 presents for H 5 S 2 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 H 5 S 2 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.
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: 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: = . e T e C . 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: ). The insertion presents the ratio ⁎ m m / e 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: where the traditional markings were introduced: Δ n=1 (0) = Δ n=1 (T 0 ) and Z n=1 (0) = Z n=1 (T 0 ). 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 = 3 60 . 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 B T C /ω ln , which for the H 5 S 2 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: 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 solutions 61 . 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 B T 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 . For the H 2 S and H 4 S 3 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 difference between the superconducting and normal state has been calculated in agreement with the formula 64 : 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 H 5 S 2 . 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 meV 2 .
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: C We presented the results obtained for the H 5 S 2 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 H 5 S 2 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 equation 65 : 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:

B B
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.
The values of ΔC have been calculated using the formula: In addition, the specific heat of normal state is determined based on: where the Sommerfeld constant equals: . 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 H 5 S 2 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 H 5 S 2 compound, it was obtained R C = 1.69. Hence, the value of R C for H 5 S 2 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 H 5 S 2 perfectly fits the general trend anticipated by the classic Eliashberg formalism.
In the case of H 2 S compound the ratio R C = 1.63 is close to The parameter R C for H 4 S 3 is equal to the value predicted by the BCS theory.

Summary
We have calculated the thermodynamic parameters of superconducting state for the H 5 S 2 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: μ ω = .
, while ω ω = 3 c D (1) . Note that increase in the cut-off frequency only increases value of μ ⁎ (e.g. for VCEE case μ = . ). The calculated values of μ ⁎ 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 H 5 S 2 compound. In our opinion, experimentally was observed the superconducting state in the H 2 S compound, which is kinetically protected in the samples prepared at the low temperature 11,30 . It should be emphasized that in the case of H 2 S reproducing the experimental dependence of critical temperature on the pressure does not require anomalously high value of Coulomb pseudopotential 26 .
In our paper, we also analyzed the thermodynamic properties of superconducting state in the H 4 S 3 compound. It is characterized by the very low value of critical temperature ( ∼ T [ ] 2 C max 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 H 5 S 2 compound, where kinetically protected H 2 S 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 H 5 S 2 , H 2 S, and H 4 S 3 compounds. In the case of H 5 S 2 , 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 H 5 S 2 compound is not the state of BCS type. The superconducting state for H 2 S has the thermodynamic parameters with values are close to the ones determined for H 5 S 2 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 H 4 S 3 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.