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+<text xmlns="http://www.tei-c.org/ns/1.0"><p>thermodynamic parameters of the LaH 10 superconductor were an object of our interest. LaH 10 is characterised by the highest experimentally observed value of the critical temperature: = T 215 C a K (p a = <measure type="value">150 GPa</measure>) and = T 260 C b K (p b = <measure type="value">190 GPa</measure>). It belongs to the group of superconductors with a strong electron-phonon coupling (λ a ~ <measure type="list">2.2</measure> and λ b ~ 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 La δ X 1-δ H 10 -type (<quantifiedObject id="56844c1a-aaf3-4f85-a615-690d6d358864">LaXH-type</quantifiedObject>) structure, where X ∈ {Sc, Y}, would exhibit significantly higher critical temperature than T C obtained for LaH 10 . in the case of LaScH we came to the following assessments: <measure type="value" ptr="#56844c1a-aaf3-4f85-a615-690d6d358864">∈ T 220 267</measure> , C a K and ∈ T 263 294 , C b K, while the results for LaYH were: ∈ T 218 247 , C a K and ∈ T 261 274 , C b K.</p><p>The experimental discovery of the high-temperature superconducting state in the compressed hydrogen and sulfur systems H 2 S (T C = <measure type="value">150 K</measure> for p = <measure type="value" ptr="#53e9fc2f-f956-412d-822b-e28d286a5926">150 GPa</measure>) and H 3 S (T C = <measure type="value">203 K</measure> for p = <measure type="value">150 GPa</measure>) 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 the existence of the superconducting state in hydrogenated compounds was pointed out by <quantifiedObject id="a23eb78e-14ce-41e4-a358-b1b306381090">Ashcroft</quantifiedObject> in <measure type="value" ptr="#a23eb78e-14ce-41e4-a358-b1b306381090">2004 3</measure> . It was stated in his second fundamental work concerning the high-temperature superconductivity, following <quantifiedObject id="a5f251e8-c6a0-4db0-a633-48cc64478226">his first work written</quantifiedObject> in <measure type="value" ptr="#a5f251e8-c6a0-4db0-a633-48cc64478226">1968</measure>, 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]--> .</p><p>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: K for p c ~ <measure type="value">170 GPa</measure> 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 R m 3 structure (T C = <measure type="interval">206-223 K</measure>). The experimental results reported by Somayazulu et al. 21 should be related to the superconducting state induced in the Fm m 3 structure, where the critical temperature can potentially reach even the value of <measure type="value" ptr="#ad0366bf-482b-4c3f-8060-899748bdcac4">280 K</measure>. 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 at 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 .</p><p>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 an 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 <quantifiedObject id="86f1ef0d-1ce4-40a9-a9c8-66fbf249f1c1">the range from</quantifiedObject> about <measure type="interval" ptr="#86f1ef0d-1ce4-40a9-a9c8-66fbf249f1c1">40 meV to 100 meV</measure>. A significant increase in the value of critical temperature should take place as a consequence.</p><p>. The fitting between the theory and the experimental results is presented in Fig. 1. We obtained λ a = <measure type="value">2.187</measure> for p a = <measure type="value">150 GPa</measure> and λ b = <measure type="value">2.818</measure> for p b = <measure type="value">190 GPa</measure>. The symbol Ω C represents the characteristic phonon frequency, its value being assumed as Ω C = <measure type="value">100 meV</measure>.</p><p>, where  μ is the Coulomb pseudopotential (  μ = <measure type="value">0.1</measure>). The quantity Ω C denotes the cut-off frequency (Ω C = <measure type="value">1 eV</measure>). The Eliashberg equations were solved for <quantifiedObject id="23990b8f-df6a-49af-871c-568ce87e5407">the Matsubara frequency equal</quantifiedObject> to <measure type="value" ptr="#23990b8f-df6a-49af-871c-568ce87e5407">1000</measure>. We used numerical methods presented in the previous paper 24 . In the considered case, we obtained stable equation solutions for T ≥ T 0 = <measure type="value">15 K</measure>.</p><p>BCS , however the BCS theory approximates well the experimental results for <quantifiedObject id="bf2395b3-61d2-490b-aa23-7dd63ca3135a">λ</quantifiedObject> &lt; <measure type="interval" ptr="#bf2395b3-61d2-490b-aa23-7dd63ca3135a">0.5</measure>.</p><p>where ρ(0) denotes the value of electronic density of states at Fermi surface; Z n S and Z n N are the wave function normalization factors for the superconducting and the normal state, respectively. Note that ΔF is equal to <measure type="value">zero</measure> exactly for T = T C . This fact results from the overt dependence of free energy on solutions of Eliashberg equations (Δ n and Z n ) that have been adjusted to the experimental value of critical temperature by appropriate selection of electron-phonon coupling constant (see Fig. 1). Thermodynamic critical field should be calculated from the formula:</p><p>) coming from sulphur and from hydrogen are separated due to a huge difference between atomic masses of these <measure type="value" ptr="#6cebcf45-47a3-4a09-aadb-3b143788e2d8">two</measure> <quantifiedObject id="6cebcf45-47a3-4a09-aadb-3b143788e2d8">elements</quantifiedObject>. To be precise, the electron-phonon interaction derived from sulphur is crucial in <quantifiedObject id="295f994b-689a-4801-b788-69304105569a">the frequency range from</quantifiedObject> <measure type="interval" ptr="#295f994b-689a-4801-b788-69304105569a">0 meV</measure> to <quantifiedObject id="e28b4eda-6af1-4073-ab29-f54f338af700">Ω max S equal</quantifiedObject> to about <measure type="value" ptr="#e28b4eda-6af1-4073-ab29-f54f338af700">70 meV</measure>, while the contribution derived from hydrogen (Ω = <measure type="value">220</measure> max H meV) is significant above ~<measure type="value">100 meV</measure>. 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 max La , Ω max X , and Ω max H represent the respective maximum phonon frequencies. The value of the critical temperature can be assessed from the generalised formula of the BCS theory 7 :</p><p>We are going to consider the case Ω &lt; Ω &lt; ~<measure type="value">40 meV</measure> <measure type="value">100 meV</measure> max La max X</p><p>. 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 hydrogen. It can be assumed that 0 &lt; λ X &lt; <measure type="interval">1</measure>, while keeping in mind that λ La = <measure type="value">0.68 31</measure> . Additionally, the previous calculations discussed in the work allow to write that λ La + λ H is equal to λ a = <measure type="value">2.187</measure> for p a = <measure type="value">150 GPa</measure> or to λ b = <measure type="value">2.818</measure> for p b = <measure type="value">190 GPa</measure>. The quantity  μ occurring in the Eq. ( 8) serves now as the fitting parameter. One should remember that the formula for the critical temperature given by the Eq. ( 8) was derived with the use of significant simplifying assumptions (the value of the cut-off frequency is neglected, as well as the retardation effects modeled 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. ( 8) and assuming that K. Therefore the superconducting state can potentially exist at 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:</p><p>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 electron-phonon 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 <quantifiedObject id="7f894382-375f-4745-b036-7599af4592b8">the LaScH compound equal</quantifiedObject> to about <measure type="value" ptr="#7f894382-375f-4745-b036-7599af4592b8">52 K</measure> (<measure type="value" ptr="#70168bdc-22ba-4105-ad71-522a8ef6bdf9">150 GPa</measure>) or to about <measure type="value" ptr="#37ab9379-9e39-402c-a60e-85906d9a2efe">79 K</measure> (<measure type="value" ptr="#db4168a7-5dd7-4332-b996-5afe06b49808">190 GPa</measure>) 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 <measure type="value">32 K</measure> for <measure type="value">150 GPa</measure> or about <measure type="value">59 K</measure> for <measure type="value">190 GPa</measure>.</p><p>• In the paper, we assume the relatively simple form of Eliashberg function, which is the linear combination of each of the contributions from La, H and X. Does this mean that any contribution related to Sc or Y will be positive? Of course, this doesn't have to be the case. For example, the properly selected concentration of Sc or Y atoms can lead to a decrease in the electron-phonon coupling constant. On the other hand, one should remember the results obtained for <quantifiedObject id="49c731f3-e355-4685-a248-2fcba20d3c8a">YH 10 compound</quantifiedObject> <measure type="interval" ptr="#49c731f3-e355-4685-a248-2fcba20d3c8a">32,33</measure> . Based on the DFT method, it was found that the critical temperature for ∈ p (<measure type="value">250, 300</measure>) GPa can exceed the room temperature ( ) roughly correspond to the physical values of this parameters? In particular, are these values too low, which would lead to the significant overestimation of the critical temperature in our paper. In this case, it is worth referring to the recently obtained DFT results for LaH 10 . In the publication 19 , the authors showed that qualitative compliance with experimental data can be obtained assuming  μ = <measure type="value">0</measure>. K for the crystal structure Fm<measure type="value">3m</measure>). In the first case, the experimental critical temperature was underestimated by <measure type="value">18 K</measure> (too high value of  μ ), in the second case, T [ ] C exp was revalued by <measure type="value">11 K</measure> (too low value of  μ ). Comparing the results obtained in the paper 19 with ours, it is clearly seen that μ a  and μ b  are fairly well-obtained. In the most interesting case for LaH 10 corresponds to <quantifiedObject id="7b26cb29-4d51-4062-98a2-487e45e424f0">the pressure of</quantifiedObject> <measure type="value" ptr="#7b26cb29-4d51-4062-98a2-487e45e424f0">190 GPa</measure>, taking into account the possible reduction of μ b  suggested in <measure type="value">19</measure> , the increase in the critical temperature value for the LaScH and LaYH compounds can be expected. It is worth noting that our results also correlate well with the data obtained in the <quantifiedObject id="84b0652d-f513-4e78-a813-b4f0c2e19a25">paper 30 ,</quantifiedObject> where  μ = <measure type="value" ptr="#84b0652d-f513-4e78-a813-b4f0c2e19a25">0.22</measure> was assumed, which allowed to reproduce the experimental critical temperature for LaH 10 (p = <measure type="value" ptr="#ec711713-6a56-4dff-baff-c93e01df6185">190 GPa</measure>).</p><p>Scientific RepoRtS |(2020) 10:1592 | https://doi.org/<measure type="interval">10.1038/s41598</measure>-<measure type="interval">020</measure>-58065-9</p></text>