Quantitative analysis of nonadiabatic effects in dense H₃S and PH₃ superconductors

Abstract

The comparison study of high pressure superconducting state of recently synthesized H₃S and PH₃ compounds are conducted within the framework of the strong-coupling theory. By generalization of the standard Eliashberg equations to include the lowest-order vertex correction, we have investigated the influence of the nonadiabatic effects on the Coulomb pseudopotential, electron effective mass, energy gap function and on the 2Δ(0)/T_C ratio. We found that, for a fixed value of critical temperature (178 K for H₃S and 81 K for PH₃), the nonadiabatic corrections reduce the Coulomb pseudopotential for H₃S from 0.204 to 0.185 and for PH₃ from 0.088 to 0.083, however, the electron effective mass and ratio 2Δ(0)/T_C remain unaffected. Independently of the assumed method of analysis, the thermodynamic parameters of superconducting H₃S and PH₃ strongly deviate from the prediction of BCS theory due to the strong-coupling and retardation effects.

Introduction

The first-principles theoretical studies of the metallization and high-temperature superconductivity of dense hydrogen sulfide were reported for the first time by Li et al.¹. This directly initiated the experimental work of Drozdov et al. who found that H₂S compressed in a diamond anvil cell exhibits the superconductivity ranging from 30 to 150 K measured in the low-temperature runs^2,3, which is consistent with calculations mentioned¹. Furthermore, the results achieved in samples prepared at high-temperature showed that the record critical temperature of 164 K for cooper-oxide system HgBa₂Ca₂Cu₃O_8+δ under quasihydrostatic pressure⁴ has been trumped. Based on a sharp drop of the resistivity to zero and an expulsion of the magnetic field, Drozdov et al. observed a transition from metal to superconducting state at 203 K in H₂S sample compressed up to 155 GPa^2,5. Subsequent theoretical^6,7,8 and experimental^9,10 studies suggested that at high pressure, the phase diagram favors decomposition of H₂S into H₃S and elemental sulfur. This result means that the superconducting state observed at 203 K comes from a decomposition product H₃S. More recently, referring to the theoretical crystal structure searches performed by Duan et al.¹¹, the stability of high-pressure cubic structure of H₃S was confirmed by Li et al. in first-principles DFT structure searches joined with high-pressure X-ray diffraction experiments⁹, and then by Einaga et al. in synchrotron X-ray diffraction measurements combined with the electrical resistance measurements¹⁰. In contrast to cuprates where the nature of superconductivity is still not fully understood^12,13,14, the presence of a strong isotope effect in H₃S clearly suggests the electron–phonon origin of the superconducting state^2,15,16,17. In addition, Jarlborg and Bianconi predict that the Fermi surface of H₃S consists of multiple sheets similar to those in MgB₂¹⁸. However, the existence of multi-gap superconductivity in H₃S has not been so far confirmed.

The above theoretical and experimental discovery have stimulated significant interest in finding new hydrogen-containing superconductors^19,20,21. Very recently, Drozdov et al. reported superconductivity in compressed PH₃ with a T_C above 100 K²². The pressure dependence of the experimental critical temperature for H₃S and PH₃ compounds together with the relevant crystal structures is presented in Fig. 1. For H₃S the second-order structural phase transition from trigonal R3m to cubic is experimentally observed at pressure close to 150 GPa^10,23. However, according to the static lattice calculations the phase transition from R3m to occurs at ~180 GPa¹¹. A recent theoretical study found that quantum nuclear motion lowers the transition pressure to 103 GPa²⁴. In the case of PH₃, a theoretical search of crystal structure reveals two phases with lowest energy: orthorhombic P2₁2₁2₁ and monoclinic C2/m. The DFT studies realized by Liu et al. indicate that both structures are dynamically stable and superconducting but C2/m phase are in a better agreement with an experimental results²⁵. Unfortunately, the lack of structural informations on the superconducting phases from suitable measurements do not allow at this moment for the unambiguous verification of these assumptions.

Figure 1

The experimental data of critical temperature as a function of pressure for H₃S^2,10 and PH₃²².

In addition the stable structures of H₃S¹¹ are presented.

Motivated by a recent significant experimental and theoretical progress in chemistry and physics of hydrogen-dense materials, we have carried out calculations to explore in detail the thermodynamic properties of superconducting hydrogen sulfide H₃S and hydrogen phosphide PH₃ at extremely high pressure (p = 200 GPa). The very large values of electron-phonon coupling interaction observed in these systems, caused that our investigations were performed within the framework of the Migdal-Eliashberg (ME) theory of superconductivity²⁶, which goes beyond the BCS model²⁷ by taking into account the retardation and strong-coupling effects.

This paper is organized as follows. In Section II, we introduce the theoretical model used to determine the quantities characterizing the superconducting state. Moreover, we present details of the first-principles calculations carried out to study the phonon properties and electron-phonon interactions. Then, in Section III, we report and compare the thermodynamic properties of superconducting H₃S and PH₃ systems at 200 GPa. We discuss the validity of the conventional Migdal-Eliashberg theory by introduce the lowest-order vertex correction and we examine its effect on Coulomb pseudopotential, energy gap, 2Δ(0)/T_C ratio and electron effective mass. Finally, we give a summary of this study in Section IV.

Theoretical model and computational methods

Besides the experimental works, most of the theoretical studies concluded that compressed H₃S and PH₃ are phonon-mediated strong-coupling superconductors^{8,11,25,28,29}. Thus, the superconducting state of these compounds can be accurately described by the Migdal-Eliashberg theory^30,31. The Eliashberg equations for the superconducting order parameter function φ_n ≡ φ(iω_n) and the electron mass renormalization function Z_n ≡ Z(iω_n) written in the imaginary-axis formulation are given by^26,32:

and

where the pairing kernel for the electron-phonon interaction is given by:

Moreover, β = 1/k_BT and k_B = 0.0862 meV/K states the Boltzmann constant. Symbols μ* and θ denote the screened Coulomb repulsion and the Heaviside function with cut-off frequency ω_c equal to ten times the maximum phonon frequency: ω_c = 10ω_D.

The application of the above Eliashberg equations to describe the electron-phonon superconductivity is justified for systems in which the value of the phonon energy scale (Debye frequency, ω_D) to the electron energy scale (Fermi energy, ε_F) ratio is negligible. Otherwise the Eliashberg equations should be generalized by taking into account the lowest-order vertex correction^{33,34,35,36,37}:

and

where, the modified electron-phonon pairing kernel takes the following form:

The Eliashberg spectral function, one of the main input element to the Eliashberg equations, is defined as:

with

where N(0), γ_qν and g_qν(k, i, j) denote the density of states at the Fermi energy, the phonon linewidth and the electron-phonon coefficients, respectively. The α²F(ω) functions for H₃S and PH₃ were calculated in this paper using density functional perturbation theory and the plane-wave pseudopotential method, as implemented in the Quantum-Espresso package³⁸. We assume that H₃S has cubic crystal structure with lattice parameter a = 2.984 Ź¹, whereas PH₃ crystallize in monoclinic C2/m structure with lattice parameter a = 5.152 Å, b = 2.961 Å, c = 2.960 Å, α = γ = 90° and β = 90.23° ³⁹. The Vanderbilt-type ultrasoft pseudopotentials for S, P and H atoms were employed with a kinetic energy cutoff equal to 80 Ry. The phonon calculations were performed for 32 × 32 × 32 Monkhorst-Pack k-mesh with gaussian smearing of 0.03 Ry. The electron-phonon coupling matrices were computed using 8 × 8 × 8 q-grid for H₃S and 4 × 4 × 4 q-grid for PH₃. The calculated phonon band structures together with the Eliashberg spectral functions and electron-phonon integrals for both investigated hydrides are presented in Fig. 2. The absence of imaginary frequencies in the full phonon spectra indicates that both systems are dynamically stable.

Figure 2

Phonon dispersion and the Eliashberg spectral function α²F(ω) with electron-phonon integral λ(ω) for H₃S and PH₃ superconductors at 200 GPa.

The electronic band structures of H₃S and PH₃ at 200 GPa were also explored. The investigated compounds are good metals with a large density of states (DOS) at the Fermi level. To clarify the difference of the electronic structures between H₃S and C2/m PH₃, the DOS calculated for these two crystal structures are shown in Fig. 3. A strong peak around the Fermi level (the Van Hove singularities) is favorable for a strong electron-phonon coupling and thus really high superconducting temperature in the case of H₃S⁴⁰.

Figure 3

Density of states (DOS) for H₃S () and PH₃ (C2/m) at 200 GPa.

The dotted line at zero indicates the Fermi level.

Our ab-initio studies showed that ratio λω_D/ε_F is equal to 0.020 for H₃S and 0.014 for PH₃. These values are rather small in comparison to fullerene compounds or high-T_c cuprates, however, are not zero. Due to the above, we decided to conduct our calculations simultaneously using the conventional Eliashberg equations and equations with the lowest-order vertex correction, which allows us to examine the influence of nonadiabatic effects on the thermodynamic properties in studied compounds. The numerical analysis was performed using a self-consistent iteration methods⁴¹, which were implemented successfully in our previous papers^31,42,43,44. The convergence and precision of our results are controlled by assuming the sufficiently high number (M = 1100) of Matsubara frequencies ω_n ≡ (π/β)(2n − 1), where n = 0, ±1, ±2, …, ±M.

Results and Discussion

To study the thermodynamic properties of phonon-mediated superconductors on the quantitatively level in the first step we determine the critical value of the Coulomb pseudopotential . For this purpose, in the Eliashberg equations we replace T by the experimental value of critical temperature: T_C = 178 K for H₃S² and T_C = 81 K for PH₃²². Then, starting from zero, we increase the value of μ* until we reach the equality , where the order parameter is defined as: Δ_m = φ_m/Z_m⁴⁵. The obtained results are presented in Fig. 4. In particular, on the basis of the full dependence of Δ_m=1(μ*) we can conclude that if we take into account the conventional Eliashberg equations, the Coulomb pseudopotential takes a relatively high critical value for H₃S () and low value for PH₃ () at 200 GPa. Moreover, at this point we can found that, for a fixed value of critical temperature, the lowest-order vertex correction changes by −9.3% for H₃S and −5.7% for PH₃.

Figure 4

The order parameter on the imaginary axis as a function of m and μ* for (A) H₃S and (B) PH₃ at 200 GPa - the results was obtained using conventional Eliashberg (E) equations. (C) The full dependence of the first value of the order parameter as a function of Coulomb pseudopotential.

In the next step, by using the analytical continuation of the imaginary-axis solutions to the real frequency axis⁴⁶, we calculate the temperature dependence of superconducting energy gap Δ(T) = Re[Δ(ω = Δ(T), T)]⁴⁵. The obtained results are presented in Fig. 5. Knowledge of the energy gap value at zero temperature allowed us to calculate the dimensionless ratio R_Δ ≡ 2Δ(0)/T_C for which the BCS theory predicts universal value [R_Δ]_BCS = 3.53. In the case of studied hydrides, R_Δ exceed the value of 4, in particular by using the conventional Eliashberg equations we received and . This behavior is connected with the strong-coupling and retardation effects, which in the framework of the Eliashberg formalism can be characterized by the ratio T_C/ω_ln, where ω_ln is the logarithmic phonon frequency, correspondingly 131 meV for H₃S and 79 meV for PH₃. Thus, the considered ratio equals 0.12 and 0.09, respectively, while in the weak-coupling BCS limit we have: T_C/ω_ln → 0. Moreover, we find that although the vertex corrections seriously reduce for H₃S and PH₃, the ratio 2Δ(0)/T_C remains practically unaffected. Similar situation is observed in the case of electron effective mass at T_C calculated from , where m_e denotes the electron band mass. In our case for H₃S and for PH₃, regardless of the equations applied. This means that the thermodynamic properties of phonon-mediated superconductors can be successfully obtained in the framework of the conventional Migdal-Eliashberg formalism with the proviso that has to be accurately determined.

Figure 5

The temperature dependence of the superconducting energy gap.

The numerical results can be reproduced using the analytical formula , where α = 3.36.

In the last step, to investigate the specific heat and thermodynamic critical field behavior, the condensation energy was numerically calculated:

where and denote the mass renormalization functions for the normal and superconducting states, respectively. The specific heat difference between superconducting and normal state was then obtained from the second derivative of E_cond(T):

and thermodynamic critical field is defined as:

The temperature dependence of ΔC, with characteristic specific heat jump at T_C marked by vertical line, is presented in Fig. 6. The inset shows the thermodynamic critical field for investigated hydrides. These results allow us to determine the other two fundamental dimensionless ratios: and R_C ≡ ΔC(T_C)/C^N(T_C), where the specific heat for the normal state is defined as C^N = γT, and γ denotes the Sommerfeld constant: . It is noteworthy that, in the framework of the BCS theory, these ratios adopt universal values of 0.168 and 1.43, respectively⁴⁵. We emphasize that these dimensionless ratios, similar to that of energy gap, take non-BCS values, in particular: R_H = 0.136, R_C = 2.47 for H₃S and R_H = 0.150, R_C = 1.99 for PH₃. We can see that with increasing values of T_C/ω_ln ratio the thermodynamic properties take increasingly non-BCS behavior.

Figure 6

The specific heat difference between the superconducting and the normal state as a function of temperature.

Inset presents the temperature dependence of the thermodynamic critical field.

Conclusions

In this work, using the first-principles calculations and Eliashberg theory with and without vertex corrections, we systematically study the nonadiabatic effects on the superconductivity of compressed H₃S and PH₃ compounds. We find that for a fixed experimental value of critical temperature the lowest-order vertex correction reduces the Coulomb pseudopotential (from 0.204 to 0.185 in the case of H₃S and from 0.088 to 0.083 for PH₃), however, the energy gap, electron effective mass and ratio 2Δ(0)/T_C remain unaffected. It means that the superconducting behavior can be properly determined even in the framework of the classical Migdal-Eliashberg formalism, as long as the value of the Coulomb pseudopotential is correctly determined. Moreover, we calculated the specific heat and thermodynamic critical field and we proved that strong-coupling and retardation effects caused that thermodynamic properties of H₃S and PH₃ at high pressures cannot be correctly estimated in the framework of the BCS theory.