From LaH10 to room–temperature superconductors

Thermodynamic parameters of the LaH10 superconductor were an object of our interest. LaH10 is characterised by the highest experimentally observed value of the critical temperature: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{T}}}_{{\boldsymbol{C}}}^{{\boldsymbol{a}}}={\bf{215}}$$\end{document}TCa=215 K (pa = 150 GPa) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{T}}}_{{\boldsymbol{C}}}^{b}={\bf{260}}$$\end{document}TCb=260 K (pb = 190 GPa). It belongs to the group of superconductors with a strong electron-phonon coupling (λa ~ 2.2 and λb ~ 2.8). We calculated the thermodynamic parameters of this superconductor and found that the values of the order parameter, the thermodynamic critical field, and the specific heat differ significantly from the values predicted by the conventional BCS theory. Due to the specific structure of the Eliashberg function for the hydrogenated compounds, the qualitative analysis suggests that the superconductors of the LaδX1−δH10-type (LaXH-type) structure, where X ∈ {Sc, Y}, would exhibit significantly higher critical temperature than TC obtained for LaH10. In the case of LaScH we came to the following assessments: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{T}}}_{{\boldsymbol{C}}}^{{\boldsymbol{a}}}\in \left\langle {\bf{220}},{\bf{267}}\right\rangle $$\end{document}TCa∈220,267 K and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{T}}}_{{\boldsymbol{C}}}^{{\boldsymbol{b}}}\in \left\langle {\bf{263}},{\bf{294}}\right\rangle $$\end{document}TCb∈263,294 K, while the results for LaYH were: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{T}}}_{{\boldsymbol{C}}}^{{\boldsymbol{a}}}\in \left\langle {\bf{218}},{\bf{247}}\right\rangle $$\end{document}TCa∈218,247 K and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{T}}}_{{\boldsymbol{C}}}^{{\boldsymbol{b}}}\in \left\langle {\bf{261}},{\bf{274}}\right\rangle $$\end{document}TCb∈261,274 K.

The experimental discovery of the high-temperature superconducting state in the compressed hydrogen and sulfur systems H 2 S (T C = 150 K for p = 150 GPa) and H 3 S (T C = 203 K for p = 150 GPa) [1,2] accounts for carrying out investigations, which can potentially lead to the discovery of a material showing the superconducting properties at room temperature.
For the first time, the possibility of existence of the superconducting state in hydrogenated compounds was pointed out by Ashcroft in 2004 [3].It was stated in his second fundamental work concerning the hightemperature superconductivity, following his first work written in 1968, in which he propounded the existence of the high-temperature superconducting state in metallic hydrogen [4].
The superconducting state in hydrogenated compounds is induced by the conventional electron-phonon interaction.This fact made possible the theoretical description of the superconducting phase in H 2 S and H 3 S even prior to carrying out the suitable experiments [5,6].The detailed discussion with respect to the thermodynamic properties of the superconducting state occurring in H 2 S and H 3 S one can find in references [7,8,9,10,11,12,13,14,15,16,17].
In 2018, there were held the groundbreaking experiments, which confirmed the existence of the superconducting state of extremely high values of the critical temperature in the LaH 10 compound: T a C = 215 K for p a = 150 GPa and T b C = 260 K for p b ∈ (180 − 200) GPa (and then T c C ∼ 250 K for p c ∼ 170 GPa [18]).It was proved on the theoretical basis [19] that the results achieved by Drozdov et al. [20] can be related to the induction of the superconducting phase in the R3m structure (T C = 206 − 223 K).The experimental results re-ported by Somayazulu et al. [21] should be related to the superconducting state induced in the F m3m structure, where the critical temperature can potentially reach even the value of 280 K.
From the materials science perspective, the achieved results imply that all possible actions should be taken in order to examine the hydrogen-containing materials with respect to the existence of the high-temperature superconducting state in room temperature.Attention should be paid to the importance of the discovery of the high-temperature superconducting state in LaH 10 , because La can form stable hydrogenated compounds with other metals.Such materials can exhibit so large hydrogen concentration, that they are presently taken into account as basic components of the hydrogen cells intended for vehicle drives [22].
The purpose of this work is, firstly, to present the performed analysis of the thermodynamic properties of the superconducting state in the LaH 10 compound.We took advantage of the phenomenological version of the Eliashberg equations, for which we fitted the value of the electron-phonon coupling constant on the basis of the experimentally found T C value.
Our next step consisted in examining the hydrogenated compounds of the La δ X 1−δ H 10 -type (LaXH-type) on the basis of the achieved results in order to find a system with even higher value of the critical temperature.Taking into account the structure of the Eliashberg function for hydrogenated compounds, with its distinctly separated parts coming from the heavy elements and from hydrogen, we assumed X to be Sc or Y, what would, in our opinion, fill the gap in the Eliashberg function occurring within the range from about 40 meV to 100 meV.A significant increase in the value of critical temperature should take place as a consequence.
The thermodynamic parameters of the LaH 10 superconductor were calculated by means of Eliashberg equations on the imaginary axis [23]: and The symbols ∆ n = ∆ (iω n ) and Z n = Z (iω n ) denote the order parameter and the wave function renormalization factor, respectively.The quantity ω n represents the Matsubara frequency: , where k B is the Boltzmann constant.The pairing kernel is defined by: , where λ denotes the electron-phonon coupling constant.We determined the value of λ on the basis of experimental data [20,21] and the condition: [∆ n=1 ] T =TC = 0.The fitting between the theory and the experimental results is presented in Fig. 1.We obtained λ a = 2.187 for p a = 150 GPa and λ b = 2.818 for p b = 190 GPa.The symbol Ω C represents the characteristic phonon frequency, its value being assumed as Ω C = 100 meV.The repulsion between electrons is modeled by the function: , where µ ⋆ is the Coulomb pseudopotential (µ ⋆ = 0.1).The quantity ω C denotes the cut-off frequency (ω C = 1 eV).The Eliashberg equations were solved for the Matsubara frequency equal to 1000.We used numerical methods presented in the previous paper [24].In the considered case, we obtained stable equation solutions for T ≥ T 0 = 15 K. Fig. 2 illustrates the full dependence of the order parameter on temperature.Physical values of the order parameter were calculated from the equation: ∆ (T ) = Re [∆ (ω = ∆ (T ))], while the function of the order parameter on the real axis (∆ (ω)) was determined using the solutions of the Eliashberg equations on the imaginary axis and the analytical continuation method described in the reference [25].It can be easily seen that the order parameter curves determined within the Eliashberg formalism differ significantly from the curves resulting from the BCS theory [26,27].These differences arise from the very high value of the electron-phonon coupling constant of the superconductor, what is mirrored by the high value of the dimensionless Let us recall that within the BCS theory we come to the result: [R ∆ ] BCS = 3.53, however the BCS theory approximates well the experimental results for λ < 0.5.
We plotted the temperature dependence of the effective electron mass (m ⋆ e ) to the band electron mass (m e ) ratio in the insets in Fig. 2. The value of the m ⋆ e /m e ratio is given with good approximation by the value of 1 + λ [28].
Fig. 3 presents the results achieved for the difference in free energy between the superconducting and the normal state (∆F ), the thermodynamic critical field (H C ), and the specific heat in both the superconducting (C S ) and the normal (C N ) states.The values of the considered quantities were calculated on the basis of formulae given in reference [28].Deviations from the results of the BCS theory can be traced in the easiest way by determining the values of dimensionless ratios: The subsequent last part of the paper discusses the question of induction of the superconducting state in a group of compounds of the La δ X 1−δ H 10 -type (or LaXH-type for short).Firstly, we are going to give some criteria, which can potentially make easier the search for a material showing the required high-temperature superconducting properties.To do this, let us take into account the formula for the critical temperature valid for the BCS theory: where Ω max denotes the Debye frequency and V stands for the pairing potential value.It can be easily noticed that the critical temperature is the higher, the greater are the values of the electron density of states at the Fermi surface, the pairing potential, and the maximum phonon frequency.Therefore it should be supposed, even at such an early stage of analysis, that special attention is to be paid to these hydrogenated compounds, for which the respective non-hydrogenated compounds (La δ X 1−δ ) or hydrides XH exhibit the high density of electron states at the Fermi surface.Considerations given to the pairing potential at the phenomenological level do not get us very far because this quantity is calculated in a rather complicated way, usually by means of the DFT (Density Functional Theory) method.Nevertheless, a sensible qualitative analysis can be made with respect to the influence of the atomic mass of the X element on a value of the critical temperature (since the mass of the X element determines Ω max ).In this regard, let us refer to the theoretical results obtained within the Eliashberg formalism for H 2 S and H 3 S superconductors [6], [5].They prove that contributions to the Eliashberg function (α 2 F (Ω)) coming from sulphur and from hydrogen are separated due to a huge difference between atomic masses of these two elements.To be precise, the electron-phonon interaction derived from sulphur is crucial in the frequency range from 0 meV to Ω S max equal to about 70 meV, while the contribution derived from hydrogen (Ω H max = 220 meV) is significant above ∼ 100 meV.It is noteworthy that we come upon a similar situation in the case of the LaH 10 compound [30].Therefore the following factorization of the Eliashberg function for the LaXH compound can be assumed: where λ La , λ X , and λ H are the contributions to the electron-phonon coupling constant derived from both metals (La, X) and hydrogen, respectively.Similarly, the , symbols Ω La max , Ω X max , and Ω H max represent the respective maximum phonon frequencies.The value of the critical temperature can be assessed from the generalised formula of the BCS theory [7]: while the symbols appearing in Eq. ( 4) are defined in Tab.I. Let us calculate explicitly the relevant quantities: and .
We are going to consider the case Ω La max ∼ 40 meV < Ω X max < 100 meV.It means that we are interested in such an X element, the contribution of which to the Eliashberg function fills the gap between contributions coming from lanthanum and from hydrogen.It can be assumed that 0 < λ X < 1, while keeping in mind that λ La = 0.68 [31].Additionally, the previous calculations discussed in the work allow to write that λ La +λ H is equal to λ a = 2.187 for p a = 150 GPa or to λ b = 2.818 for p b = 190 GPa.The quantity µ ⋆ occurring in the Eq. ( 4) serves now as the fitting parameter.One should remember that the formula for the critical temperature given by the Eq. ( 4) was derived with the use of significant simplyfing assumptions (the value of the cut-off frequency is neglected, as well as the retardation effects modelled by the Matsubara frequency).Therefore the value of the Coulomb pseudopotential determined from the full Eliashberg equations usually differs from the value of µ ⋆ calculated analytically.The experimental data for the LaH 10 superconductor can be reproduced using Eq. ( 4) and assuming that µ ⋆ a = 0.170 and µ ⋆ b = 0.276.The achieved results are presented in Fig. 4. It is evident that taking into consideration the additional X element, which enriches the LaH 10 composition, leads to a large increase in the critical temperature value.The estimated upper limit of the T a C value is equal to 288 K for p a = 150 GPa, while for p b = 190 GPa we obtain T b C = 315 K. Therefore the superconducting state can potentially exists in room temperature for both cases.Now, let us take into account elements with the identical electron configuration at the valence shell as lanthanum, but lighter than lanthanum: scandium and yttrium, both being selected as X.Attention should be paid to the fact that the electron configuration of X, identical as in lanthanum, should minimize such changes in properties of the obtained compound which could result from changes in both the electron dispersion relation and the matrix elements of the electron-phonon interaction.Applying the formula: Ω X max /Ω La max ∼ M La /M X we get Ω Sc max ∼ 70 meV and Ω Y max ∼ 50 meV (M La and M X denote atomic mass of lanthanum and the element X, i.e.Sc or Y, respectively).Apparently, the significant increase in the critical temperature value should be observed in both cases.The effect of growth in the value of the critical temperature results from filling the gap in the Eliashberg function between the contributions coming from La and H, as was already stated above.
To summarize, the experimental results obtained for the LaH 10 compound get us much closer to the purpose of obtaining the superconducting state at room temperature.The huge difference between atomic masses of lanthanum and hydrogen results in the characteristic structure of the Eliashberg function modeling the electronphonon interaction in the considered compound, with distinctly separated parts proceeded either from lanthanum or from hydrogen.The proper selection of the additional element (X) in the LaXH compound is expected to fill the 'empty' range of the Eliashberg function between the parts coming from La and H.In our opinion, good candidates are scandium and yttrium.These elements have the electron configuration at the valence shell exactly the same as lanthanum, and yet they are considerably lighter.Our numerical calculations suggest the possible growth in the critical temperature of the LaScH compound equal to about 52 K (150 GPa) or to about 79 K (190 GPa) as compared to the T C value for the LaH 10 compound.As far as the LaYH compound is concerned, the pertinent increase in T C value can reach about 32 K for 150 GPa or about 59 K for 190 GPa.Our results can be a starting point for the advanced DFT calculations or perhaps provide inspiration for carrying out the appropriate experimental measurements.
One needs to recognise that the exact quantitative analysis of the problem discussed in our work would require to be carried out using the Eliashberg equations including the anharmonicity of the phonon system and the non-linear terms of the electron-phonon-phonon interaction, especially for such high values of the critical temperature as are observed for LaH 10 .Presently we work upon the derivation of suitable equations.

FIG. 1 :
FIG. 1: The dependence of the maximum value of the order parameter on the electron-phonon coupling constant.We consider two cases: T a C = 215 K (pa = 150 GPa) and T b C = 260 K (p b = 190 GPa).

FIG. 2 :
FIG. 2: The dependence of the order parameter on temperature.The insets present the influence of temperature on the value of effective electron mass to the band electron mass ratio.Blue or red disks represent numerical results.Black curves were obtained from the analytical formulae: ∆ (T ) = ∆ (T0) 1 − (T /TC ) Γ and m ⋆ e /me = [Z (TC ) − Z (T0)] (T /TC) Γ +Z (T0), where Z (TC ) = 1 + λ, Γa = 3.5 and Γ b = 3.4.The predictions of the BCS theory we marked with grey circles.

FIG. 3 :
FIG.3:(Bottom panels:) The difference in free energy between the superconducting and the normal state versus temperature.The symbol ρ (0) represents the value of the electron density of states at the Fermi surface.(Top panels:) Thermodynamic critical field.(Insets:) The specific heat in the superconducting and the normal states.

Fig. 5
presents the expected range of the critical temperature values for the LaScH and the LaYH compounds.We took into account two pressure values: p a = 150 GPa and p b = 190 GPa.For LaScH we got: T a C ∈ 220, 267 K and T b C ∈ 263, 294 K, while the results for LaYH are as follows: T a C ∈ 218, 247 K and T b C ∈ 261, 274 K.