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The joint modeling of longitudinal and time-to-event outcomes has become a popular tool in follow-up studies. However, fitting Bayesian joint models to large datasets, such as patient registries, can require extended computing times. To speed up sampling, we divided a patient registry dataset into subsamples, analyzed them in parallel, and combined the resulting Markov chain Monte Carlo draws into a consensus distribution. We used a simulation study to investigate how different consensus strategies perform with joint models. In particular, we compared grouping all draws together with using equal- and precision-weighted averages. We considered scenarios reflecting different sample sizes, numbers of data splits, and processor characteristics. Parallelization of the sampling process substantially decreased the time required to run the model. We found that the weighted-average consensus distributions for large sample sizes were nearly identical to the target posterior distribution. The proposed algorithm has been made available in an R package for joint models, JMbayes2. This work was motivated by the clinical interest in investigating the association between ppFEV1, a commonly measured marker of lung function, and the risk of lung transplant or death, using data from the US Cystic Fibrosis Foundation Patient Registry (35,153 individuals with 372,366 years of cumulative follow-up). Splitting the registry into five subsamples resulted in an 85\% decrease in computing time, from 9.22 to 1.39 hours. Splitting the data and finding a consensus distribution by precision-weighted averaging proved to be a computationally efficient and robust approach to handling large datasets under the joint modeling framework.

The superconducting state in Carbonaceous Sulfur Hydride (C-S-H) system is characterized by the record-high critical temperature of $288$~K experimentally observed at $\sim$267 GPa. Herein, we determined the properties of the \mbox{C-S-H} superconducting phase within the scope of both classical Eliashberg equations (CEE) and the Eliashberg equations with vertex corrections (VCEE). We took into account the scenarios pertinent to either the intermediate or the high value of electron-phonon coupling constant ($\lambda\sim 0.75$ or $\lambda\sim 3.3$, respectively). The scenario for the intermediate value, however, cannot be actually realized due to the anomally high value of logarithmic phonon frequency ($\omega_{\rm ln}/k_{B}=7150$~K) it would require. On the other hand, we found it possible to reproduce correctly the value of $T_{C}$ and other thermodynamic quantities in the case of strong coupling. However, the vertex corrections lower the order parameter values within the range from $\sim 50$~K to $\sim275$~K. For the upper critical field $H_{C2}\sim 27$~T, the Ginzburg-Landau parameter $\kappa$ is of the order of $1.7$. This correlates well with the sharp drop of resistance observed by Hirsch and Marsiglio at the critical temperature. The strong-coupling scenario for C-S-H system is also suggested by the high values of $\lambda$ estimated for ${\rm H_{3}S}$ ($\lambda\sim 2.1$, $\kappa\sim 1.5$), ${\rm LaH_{10}}$ ($\lambda\sim 2.8$-$3.9$, $\kappa\sim 1.6$), and ${\rm YH_{6}}$ ($\lambda\sim 1.7$, $\kappa\sim 1.3$) compounds., Comment: 4 pages

In the case of Li-intercalated hexagonal boron nitride bilayer (Li-hBN), the vertex corrections of electron-phonon interaction cannot be omitted. This is evidenced by the very high value of the ratio $\lambda\omega_{D}/\varepsilon_{F}\sim 0.46$, where $\lambda$ is the electron-phonon coupling constant, $\omega_{D}$ is the Debye frequency, and the symbol $\varepsilon_{F}$ represents the Fermi energy. Due to the nonadiabatic effects, the phonon-induced superconducting state in Li-hBN is characterized by the much lower value of critical temperature ($T^{\rm LOVC}_{C}\in\{ 19.1, 15.5, 11.8\}$ K, for $\mu^{\star}\in \{0.1, 0.14, 0.2\}$), than would result from calculations not taking this effect into account: $T^{\rm ME}_{C}\in\{ 31.9, 26.9, 21\}$ K. From the technological point of view, the low value of $T_{C}$ limits the possible applications of Li-hBN superconducting properties. The calculations were carried out under the classic Migdal-Eliashberg formalism (ME) and the Eliashberg theory with the lowest-order vertex corrections (LOVC)., Comment: 8 pages, 5 figures

We analysed the dynamics of the positively charged ions of diatomic molecules (${\rm X_{2}^{+}}$ and ${\rm XY^{+}}$), in which the bond is realised by the single electron. We assumed that the atomic cores separated by the distance $R$ were subjected to the external excitation of the harmonic type with the amplitude $A$ and frequency $\Omega$. We found the ground states of ions using the variational approach within the formalism of second quantization (the Wannier function was reproduced by means of Gaussian orbitals). It occurred that, on the account of the highly non-linear dependence of the total energy on $R$, the chaotic dynamics of cores induced the chaotic evolution of the electronic Hamiltonian parameters (i.e. the energy of the electron orbital $\varepsilon$ and the hopping integral $t$). Changes in cation masses or in the charge arrangement does not affect qualitatively the values of Lyapunov exponents in the $A$-$\Omega$ parameter space.

Thermodynamic parameters of the $\rm LaH_{10}$ superconductor were an object of our interest. $\rm LaH_{10}$ is characterised by the highest experimentally observed value of the critical temperature: $T^{a}_{C}=215$~K ($p_{a}=150$~GPa) and $T^{b}_{C}=260$~K ($p_{b}=190$~GPa). It belongs to the group of superconductors with a~strong electron--phonon coupling ($\lambda_{a}\sim 2.2$ and $\lambda_{b}\sim 2.8$). We calculated thermodynamical 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 ${\rm La_{\delta}X_{1-\delta}H_{10}}$--type (LaXH--type) structure, where ${\rm X}\in\{{\rm Sc},{\rm Y}\}$, would exhibit significantly higher critical temperature than $T_{C}$ obtained for $\rm LaH_{10}$. In the case of LaScH we came to the following assessments: $T^{a}_{C}\in\left<220,267\right>$~K and $T^{b}_{C}\in\left<263,294\right>$~K, while the results for LaYH were: $T^{a}_{C}\in\left<218,247\right>$~K and $T^{b}_{C}\in\left<261,274\right>$~K., Comment: 6 pages, 5 figures

The paper presents our verification of induction of the superconducting state on a square lattice by the linear electron-phonon interaction for values of the unbalance parameter ($\gamma=\lambda_{D}/\lambda_{ND}$) less than $\gamma_{C}=0.42$. Symbols $\lambda_{D}$ and $\lambda_{ND}$ denote the values of the coupling constant in the diagonal and the non-diagonal channel of the self-energy. Calculations were carried out using the Eliashberg equations, in which the order parameter ($\Delta_{\bf k}\left(i\omega_{n}\right)$) and the wave function renormalising factor ($Z_{\bf k}\left(i\omega_{n}\right)$) depend explicitly on the Matsubara frequency ($\omega_{n}$) and the wave vector (${\bf k}$). The value of $\gamma_{C}$ in the static boundary ($\Delta_{\bf k}\left(i\omega_{n}\right)\rightarrow \Delta_{\bf k}\left(i\omega_{n=1}\right)$), equal to ($0.93$), is significantly greater than the obtained limit value of $0.42$. Values of the thermodynamic functions of the superconducting state determined for our assumptions are significantly different from the values calculated in accordance with the BCS theory. The results were obtained for the electron-phonon interaction function explicitly dependent on the momentum transfer between electron states., Comment: 12 pages, 9 figures

We study the stability of the hydrogen molecule interacting with the environment according to the balanced gain and loss energy scheme. We determined the properties of the molecule taking into account all electronic interactions, where the parameters of the Hamiltonian have been computed by using the variational method. The interaction of the hydrogen molecule with the environment was modeled parametrically ($\gamma$) with the help of the non-hermitian operator. We have shown that the hydrogen molecule is dynamically unstable. The dissociation time ($T_{D}$) decreases, if the $\gamma$ parameter increases (for $\gamma\rightarrow 0$, we get $T_{D}\rightarrow +\infty$). At the dynamic instability of the hydrogen molecule overlaps its static instability as the coupling constant $\gamma$ increases. We observed the decrease in the dissociation energy and the existence of the metastable state of the molecule ($\gamma_{MS}=0.659374$~Ry). The hydrogen molecule is statically unstable for $\gamma >\gamma_{D}=1.024638$~Ry. One can also observed the $\mathcal{PT}$ symmetry breaking effect for the electronic Hamiltonian ($\gamma_{\mathcal {PT}}=0.520873$~Ry). However, it does not affect the properties of the hydrogen molecule, such as: the electronic Hamiltonian parameters, the phonon and rotational energy, and the values of the electron-phonon coupling constants., Comment: 16 pages, 8 figures

In the present study, we investigate the thermodynamic properties of the Ba$_{x}$K$_{1-x}$BiO$_{3}$ (BKBO) superconductor in the under- ($x=0.5$) and over-doped ($x=0.7$) regime, within the framework of the Migdal-Eliashberg formalism. The analysis is conducted to verify that the electron-phonon pairing mechanism is responsible for the induction of the superconducting phase in the mentioned compound. In particular, we show that BKBO is characterized by the relatively high critical value of the Coulomb pseudopotential, which changes with doping level and does not follow the Morel-Anderson model. In what follows, the corresponding superconducting band gap size and related dimensionless ratio are estimated to increase with the doping, in agreement with the experimental predictions. Moreover the effective mass of electrons is found to take on high values in the entire doping and temperature region. Finally, the characteristic dimensionless ratios for the superconducting band gap, the critical magnetic field and the specific heat for the superconducting state are predicted to exceed the limits set within the Bardeen-Cooper-Schrieffer theory, suggesting pivotal role of the strong-coupling and retardation effects in the analyzed compound. Presented results supplement our previous investigations and account for the strong-coupling phonon-mediated character of the superconducting phase in BKBO at any doping level., Comment: 6 pages, 4 figures

We have determined the entropy, the total energy, and the specific heat of the systems consisting of $M\geq 3$ hydrogen molecules. The calculations were conducted in the framework of the nonextensive Tsallis statistics. The relation between $M$ and the entropic index $q$ is given by $q = 1 + 1/M$, which results from the fact that the temperature of the nanosystems fluctuates around the temperature of the reservoir (Wilk and W{\l}odarczyk, Phys. Rev. Lett. {\bf 84}, 2770 (2000)). The electron energy states of the hydrogen molecule have been determined with the help of the Hubbard Hamiltonian, which models all two-body interactions. The Hubbard Hamiltonian integrals have been calculated by using the variational method, whereas the Wannier function has been associated with $1s$ Slater-type orbitals. We have included the contributions to the energy of the hydrogen molecule coming from the oscillatory (either harmonic or anharmonic), rotational and translational degrees of freedom. In addition, we have investigated the impact of the external force ($F$) or the magnetic field ($h$) on the thermodynamic parameters of the systems. In each case the noticeable deviation from the results of the classical statistical physics can be observed for the systems consisting of $M

We theoretically investigate the possibility of achieving a superconducting state in transition-metal dichalcogenide bilayers through intercalation, a process previously and widely used to achieve metallization and superconducting states in novel superconductors. For the Ca-intercalated bilayers MoS$_2$ and WS$_2$, we find that the superconducting state is characterized by an electron-phonon coupling constant larger than $1.0$ and a superconducting critical temperature of $13.3$ and $9.3$ K, respectively. These results are superior to other predicted or experimentally observed two-dimensional conventional superconductors and suggest that the investigated materials may be good candidates for nanoscale superconductors. More interestingly, we proved that the obtained thermodynamic properties go beyond the predictions of the mean-field Bardeen--Cooper--Schrieffer approximation and that the calculations conducted within the framework of the strong-coupling Eliashberg theory should be treated as those that yield quantitative results., Comment: 7 pages, 7 figures

The superconducting state in vanadium characterizes with the critical temperature ($T_{c}$) equal to $5.3$~K. The Coulomb pseudopotential, calculated with the help of the Eliashberg equations, possesses anomalously high value $\mu^{\star}\left(3\Omega_{\rm max}\right)=0.259$ or $\mu^{\star}\left(10\Omega_{\rm max}\right)=0.368$ ($\Omega_{\rm max}$ denotes the maximum phonon frequency). Despite the relatively large electron-phonon coupling constant ($\lambda=0.91$), the quantities such as: the order parameter ($\Delta$), the specific heat ($C$), and the thermodynamic critical field ($H_{c}$) determine the values of the dimensionless ratios not deviating much from the predictions of the BCS theory: $R_{\Delta}=2\Delta\left(0\right)/ k_{B}T_{c}=3.68$, $R_{C}=\Delta C\left(T_{c}\right)/ C^{N}\left(T_{c}\right)=1.69$, and $R_{H}=T_{c}C^{N}\left(T_{c}\right)/ H^{2}_{c}\left(0\right)=0.171$. This result is associated with the reduction of the strong-coupling and the retardation effects by the high value of the Coulomb pseudopotential. It has been shown that the results of the Eliashberg formalism can be relatively precisely reproduced with the help of the semi-analytical formulas, if the value of $\mu^{\star}$ is determined on the basis of the $T_{c}$-Allen-Dynes expression ($\mu^{\star}_{AD}=0.198$). The attention should be paid to the fact that in the numerical and in the semi-analytical approach the comparable values of the thermodynamic parameters for the same $\mu^{\star}$ have been obtained only in the vicinity of the point $\mu^{\star}=0.1$., Comment: 6 pages, 5 figures

The unconventional superconductivity in the noncentrosymmetric LaNiC$_2$, and its evolution with pressure, is analyzed basing on the {\it ab initio} computations and the full Eliashberg formalism. First principles calculations of the electronic structure, phonons and the electron-phonon coupling are reported in the pressure range 0-15 GPa. The thermodynamic properties of the superconducting state were determined numerically solving the Eliashberg equations. We found that already at $p=0$ GPa, the superconducting parameters deviate from the BCS-type, and a large value of the Coulomb pseudopotential $\mu^{\star}=0.22$ is required to get the critical temperature $T_c = 2.8$~K consistent with experiment. If such $\mu^{\star}$ is used, the Eliashberg formalism reproduces also the experimentally observed values of the superconducting order parameter, the electronic specific heat jump at the critical temperature, and the change of the London penetration depth with temperature. This shows, that deviation of the above-mentioned parameters from the BCS predictions do not prejudge on the triplet or multiple gap nature of the superconductivity in this compound. Under the external pressure, calculations predict continuous increase of the electron-phonon coupling constant in the whole pressure range 0-15~GPa, consistent with the experimentally observed increase in $T_c$ for the pressure range 0-4~GPa, but inconsistent with the drop of $T_c$ above 4~GPa and the disappearance of the superconductivity above 7~GPa, reported experimentally. The disappearance of superconductivity may be accounted for by increasing the $\mu^{\star}$ to 0.36 at 7~GPa, which supports the hypothesis of the formation of a new high-pressure electronic phase, which competes with the superconductivity., Comment: 15 pages, 13 figures, to be published in Phys. Rev. B

The high-temperature superconducting state in sulfur trihydride ($T_{C}=203$~K) has been investigated in the context of the non-adiabatic and anharmonic effects. The Migdal-Eliashberg equations and the extended Eliashberg equations, which include the lowest-order vertex corrections, have been solved numerically in the self-consistent way. For $R3m$ crystal structure, the lowest-order vertex corrections decrease the value of the Coulomb pseudopotential from $0.123$ to $0.108$. The anharmonic effects work antagonistically in relation to the vertex corrections shifting the value of $\mu^{\star}$ to $0.156$. The studies conducted for the structure $Im\overline{3}m$, where the Eliashberg function includes both the non-adiabatic and anharmonic effects, prove the even higher value of $\mu^{\star}=0.185$. Independently of the assumed method of the analysis, the nearly identical no mean-field dependence of the order parameter on the temperature was obtained: $2\Delta(0)/k_{B}T_{C}\sim 4.7$ - due to the significant strong-coupling and retardation effects: $\lambda\sim 2$ and $k_{B}T_{C}\slash \omega_{\rm \ln}\sim 0.15$-$0.19$. It means that the classical equations of Migdal-Eliashberg can be treated as a correct effective model for the superconducting state in $\rm H_{3}S$. This paper has shown that the McMillan or Allen-Dynes formulas substantially lower the value of the critical temperature in relation to the result obtained with the Eliashberg equations.

The paper determines the thermodynamic parameters of the superconducting state in the metallic atomic hydrogen under the pressure at $1$ TPa, $1.5$ TPa, and $2.5$ TPa. The calculations were conducted in the framework of the Eliashberg formalism. It has been shown that the critical temperature is very high (in the range from $301.2$ K to $437.3$ K), as well as high are the values of the electron effective mass (from $3.43$ $m_{e}$ to $6.88$ $m_{e}$), where $m_{e}$ denotes the electron band mass. The ratio of the low-temperature energy gap to the critical temperature explicitly violates the predictions of the BCS theory: $2\Delta\left(0\right)/k_{B}T_{C}\in\left<4.84,5.85\right>$. Additionally, the free energy difference between the superconducting and normal state, the thermodynamic critical field, and the specific heat of the superconducting state have been determined. Due to the significant strong-coupling and retardation effects those quantities cannot be correctly described in the framework of the BCS theory., Comment: 6 figures

The thermodynamic properties of the superconducting state in the amorphous ${\rm Sn_{1-x}Cu_{x}}$ thin films have been characterized. The concentration of copper changes in the range from $0.08$ to $0.41$. The calculations have been conducted in the framework of the strong-coupling formalism, wherein the Eliashberg functions determined in the tunnel experiment (Phys. Rev. B {\bf 51}, 685 (1995)) have been used. The value of the Coulomb pseudopotential equal to $0.1$ has been adopted. It has been found that the critical temperature ($T_{C}$) decreases from $7$ K to $3.9$ K. The ratio of the energy gap to the critical temperature: $R_{\Delta}=2\Delta\left(0\right)/k_{B}T_{C}$, differs significantly from the BCS value: $R_{\Delta}\in\left<4.4,3.95\right>$. Similarly behaves the ratio of the specific heat jump to the specific heat of the normal state: $R_{C}=\Delta C\left(T_{C}\right)/C^{N}\left(T_{C}\right)$, and the parameter: $R_{H}=T_{C}C^{N}\left(T_{C}\right)/H^{2}_{C}\left(0\right)$, where $H_{C}\left(0\right)$ is the thermodynamic critical field. In particular, $R_{C}\in\left<2.2,1.75\right>$ and $R_{H}\in\left<0.141,0.154\right>$.

The work describes the properties of the high-pressure superconducting state in phosphor: $p\in\{20, 30, 40, 70\}$ GPa. The calculations were performed in the framework of the Eliashberg formalism, which is the natural generalization of the BCS theory. The exceptional attention was paid to the accurate presentation of the used analysis scheme. With respect to the superconducting state in phosphor it was shown that: (i) the observed not-high values of the critical temperature ($\left[T_{C}\right]_{p=30{\rm GPa}}^{\rm max}=8.45$ K) result not only from the low values of the electron - phonon coupling constant, but also from the very strong depairing Coulomb interactions, (ii) the inconsiderable strong - coupling and retardation effects force the dimensionless ratios $R_{\Delta}$, $R_{C}$, and $R_{H}$ - related to the critical temperature, the order parameter, the specific heat and the thermodynamic critical field - to take the values close to the BCS predictions., Comment: 6 pages, 6 figures

We investigate the dependence of the energy gap ($G$) on the temperature ($T$) for the electron-doped high-temperature superconductors. The following compounds, in the form of the thin films, have been taken into consideration: $\rm La_{2-x}Ce_xCuO_{4}$ (LCCO), $\rm Pr_{2-x}Ce_xCuO_{4}$ (PCCO), and $\rm Nd_{2-x}Ce_xCuO_4$ (NCCO). It was found that $G\left(T\right)$ deviates from the BCS prediction more, if a concentration of cerium assumes the lower values. For the lowest concentration (in the case of LCCO and NCCO), the function $G\left(T\right)$ is not quite like the BCS curve, which is connected with the existence of the residual Nernst region. Next, it has been pointed out that the NCCO superconductor becomes structurally unstable for the maximum concentration of cerium, which is leading to the anomalous dependence of the energy gap on the temperature and the induction of the wide Nernst region.

The experimental value of ${\rm H_{3}S}$ isotope coefficient decreases from $2.37$ to $0.31$ in the pressure range from $130$ GPa to $200$ GPa. We have shown that the value of $0.31$ is correctly reproduced in the framework of the classical Eliashberg approach. On the other hand, the anomalously large value of the isotope coefficient ($2.37$) may be associated with the strong renormalization of the normal state by the electron density of states.

Hydrogen-rich compounds are extensively explored as candidates for a high-temperature superconductors. Currently, the measured critical temperature of $203$ K in hydrogen sulfide (H$_3$S) is among the highest over all-known superconductors. In present paper, using the strong-coupling Eliashberg theory of superconductivity, we compared in detail the thermodynamic properties of two samples containing different hydrogen isotopes H$_3$S and D$_3$S at $150$ GPa. Our research indicates that it is possible to reproduce the measured values of critical temperature $203$ K and $147$ K for H$_3$S and D$_3$S by using a Coulomb pseudopotential of $0.123$ and $0.131$, respectively. However, we also discuss a scenario in which the isotope effect is independent of pressure and the Coulomb pseudopotential for D$_3$S is smaller than for H$_3$S. For both scenarios, the energy gap, specific heat, thermodynamic critical field and related dimensionless ratios are calculated and compared with other conventional superconductors. We shown that the existence of the strong-coupling and retardation effects in the systems analysed result in significant differences between values obtained within the framework of the Eliashberg formalism and the prediction of the Bardeen-Cooper-Schrieffer theory., Comment: 7 pages, 9 figures

The properties of the superconducting and the anomalous normal state have been described by using the Eliashberg method. The pairing mechanism has been reproduced with help of the Hamiltonian, which models the electron-phonon and electron-electron-phonon interaction (EEPh). The set of the Eliashberg equations, which determines the order parameter function ($\varphi$), the wave function renormalization factor ($Z$), and the energy shift function ($\chi$) has been derived. It has been proven that for the sufficiently large values of EEPh potential, the doping dependence of order parameter ($\varphi/Z$) has the analogous course to that observed experimentally in cuprates. The energy gap in the electron density of states is induced by $Z$ and $\chi$ - the contribution from $\varphi$ is negligible. The electron density of states possesses the characteristic asymmetric form and the pseudogap is observed above the critical temperature., Comment: 12 pages, 8 figures

In the present paper we report comprehensive analysis of the thermodynamic properties of novel hydrogen dominant potassium hydride superconductor (KH$_{6}$). Our computations are conducted within the Eliashberg theory which yields quantitative estimations of the most important thermodynamic properties of superconducting phase. In particular, we observe, that together with the increasing pressure all the thermodynamic properties decrease, e.g. $T_{C} \in \left< 72.91 , 55.50 \right>$ K for $p \in \left< 166 , 300 \right>$ GPa. It is predicted that such decreasing behavior corresponds to the decreasing hydrogen lattice molecularization with increasing pressure value. Futhermore, by calculating the dimensionless thermodynamic ratios, familiar in the Bardeen-Cooper-Schrieffer theory (BCS), it is proved that KH$_{6}$ material is a strong-coupling superconductor and cannot be quantitatively described within the BCS theory.

The electron-doped silicene under the influence of the biaxial tensile strain is predicted to be the phonon-mediated superconductor. By using the Eliashberg formalism, we investigate the thermodynamic properties of the superconducting silicene in the case when the tension is $5\%$ and the electron doping equals $3.5\times10^{14}~{\rm cm^{-2}}$. Under such conditions, silicene monolayer is expected to exhibit the highest superconducting transition temperature ($T_C$). In particular, based on the electron-phonon spectral function and assuming wide range of the Coulomb pseudopotential values ($\mu^{\star}\in\left\langle0.1,0.3\right\rangle$) it is stated that the superconducting transition temperature decreases from $18.7$ K to $11.6$ K. Similar behavior is observed in the case of the zeroth temperature superconducting energy gap at the Fermi level: $2\Delta(0)\in\left\langle6.68, 3.88\right\rangle$ meV. Other thermodynamic parameters differ from the predictions of the Bardeen-Cooper-Schrieffer theory. In particular, the ratio of the energy gap to the critical temperature changes in the range from $4.14$ to $3.87$. The ratio of the specific heat jump to the specific heat in the normal state takes the values from $2.19$ to $2.05$, and the ratio of the critical temperature and specific heat in the normal state to the thermodynamic critical field increases from $0.143$ to $0.155$. It is also determined that the maximum value of the electron effective mass equals $2.11$ of the electron band mass.

The paper analyzes the influence of the hole density, the out-of-plane or in-plane disorder, and the isotopic oxygen mass on the zero temperature energy gap ($2\Delta\left(0\right)$) for $\rm{Y}_{1-x}\rm{Ca}_{x}\rm{Ba}_2\left(\rm{Cu}_{1-y}\rm{Zn}_{y}\right)_{3}\rm{O}_{7-\delta}$ (YCBCZO) and $\rm{La}_{1.96-x}\rm{Sr}_{x}\rm{Ho}_{0.04}\rm{CuO}_{4}$ (LSHCO) superconductors. It has been found that the energy gap is visibly correlated with the value of the pseudogap temperature ($T^{\star}$). On the other hand, no correlation between $2\Delta\left(0\right)$ and the critical temperature ($T_{C}$) has been found. The above results mean that the value of the dimensionless ratio $2\Delta\left(0\right)/k_{B}T_{C}$ can vary very strongly together with the chemical composition, while the parameter $2\Delta\left(0\right)/k_{B}T^{\star}$ does not change significantly. In the paper, the analytical formula which binds the zero temperature energy gap and the pseudogap temperature has been also presented., Comment: 7 pages, 4 figures, 3 tables

It has been recently observed that the conventional electron-phonon mediated superconducting phase in graphene can be easily induced by doping its surface with the lihitum adatoms. Due to the emerging interest in this field of research we present our theoretical discussion of the thermodynamic properties of this novel structures within the strong-coupling limit. We show, that together with the increase of the adatom doping, from one (LiC$_{6}$) to two (Li$_{2}$C$_{6}$) lithium atoms per unit cell, the critical temperature ($T_C$) changes from 8.55 K to 21.83 K. Such trend is also observed for other thermodynamic properties which moreover differ from the predictions of the Bardeen-Cooper-Schrieffer theory. In particular, the zero-temperature energy gap to the critical temperature ratio: R$_{\Delta}^{{\rm LiC}_6}$=3.72 and R$_{\Delta}^{{\rm Li}_{2}{\rm C}_6}$=4.21; the ratio of the specific heat for superconducting and the normal state: R$_{\rm C}^{{\rm LiC}_6}$=1.47 and R$_{\rm C}^{{\rm Li}_{2}{\rm C}_6}$=1.79; and the parameter connected with the zero-temperature thermodynamic critical field: R$_{\rm H}^{{\rm LiC}_6}$=0.167 and R$_{\rm H}^{{\rm Li}_{2}{\rm C}_6}$=0.144. Finally the electron effective mass at $T_c$ is calculated to be: $(1.62m_e)_{{\rm LiC}_6}$ and $(2.48m_e)_{{\rm Li}_{2}{\rm C}_6}$., Comment: 5 pages, 4 figures

The ab initio calculations suggest that the superconducting state in CaH6 under the pressure at 150 GPa has the highest critical temperature among the examined hydrogen-rich compounds. For this reason, the relevant thermodynamic parameters of the superconducting state in CaH6 have been determined; a wide range of the Coulomb pseudopotential has been assumed: $\mu^{\star}\in\left<0.1,0.3\right>$. It has been found that: (i) The critical temperature (Tc) changes in the range from 243 K to 180 K. (ii) The values of the ratio of the energy gap to the critical temperature ($R_{\Delta}\equiv 2\Delta\left(0\right)/k_{B}T_{C}$) can be found in the range from 5.42 to 5.02. (iii) The ratio of the specific heat jump ($\Delta C\left(T_{C}\right)$) to the value of the specific heat in the normal state ($C^{N}\left(T_{C}\right)$), which has been represented by the symbol Rc, takes the values from 3.30 to 3.18. (iv) The ratio $R_{H}\equiv T_{C}C^{N}\left(T_{C}\right)/H^{2}_{C}\left(0\right)$, where $H_{C}\left(0\right)$ denotes the critical thermodynamic field, changes from 0.122 to 0.125. The study has brought out the expressions that correctly predict the values of the thermodynamic parameters for the superconducting state in CaH6 and for the compounds: SiH4(H2)2, Si2H6, B2H6, SiH4, GeH4, and PtH. Next, in the whole family of the hydrogen-rich compounds, the possible ranges of the values have been determined for Tc, $R_{\Delta}$, Rc, and $R_{H}$. It has been found that the maximum value of the critical temperature can be equal to 764 K, which very well correlates with Tc for metallic hydrogen (p=2 TPa). Other parameters ($R_{\Delta}$, Rc, and $R_{H}$) should not deviate from the predictions of the BCS theory more than the analogous parameters for CaH6., Comment: 12 pages, 15 figures

The thermodynamic parameters of the superconducting state in PtH at the pressure 76 GPa have been examined. The calculations have been carried out in the framework of the Eliashberg formalism. It has been found that the critical temperature changes in the range from 30.6 K to 16.8 K, depending on the assumed value of the Coulomb pseudopotential (0.1, 0.2 and 0.3). The other thermodynamic quantities differ significantly from the predictions of the classical BCS theory. In particular: (i) The parameter which corresponds to the superconducting band gap reaches the values from 4.25 to 3.98. (ii) The ratio of the specific heat jump to the specific heat in the normal state takes the values from 2.07 to 1.97. (iii) Finally, the parameter for the low-temperature thermodynamic critical field is in the range from 0.145 to 0.156.

In the paper, the thermodynamic properties of the superconducting state in the $\rm{B_{2}H_{6}}$ compound have been characterized. The pressure of 360 GPa has been taken into account. The calculations have been carried out in the framework of the Eliashberg formalism for the wide range of the Coulomb pseudopotential: $\mu^{\star}\in<0.1,0.3>$. It has been found that the critical temperature ($T_{C}$) varies in the range from 147 K to 87 K, depending on the assumed value of the Coulomb pseudopotential. The ratio of the energy gap to the critical temperature ($R_{\Delta}\equiv 2\Delta(0)/k_{B}T_{C}$) significantly exceeds the value predicted by the BCS theory: $R_{\Delta}\in<4.24,3.98>$. In the similar manner behaves the ratio of the specific heat jump to the heat of the normal state ($R_{C}\equiv\Delta C(T_{C})/C^{N}(T_{C})$), namely: $R_{C}\in<2.33,2.17>$. The parameter $R_{H}\equiv T_{C}C^{N}(T_{C})/H^{2}_{C}(0)$, where $H_{C}(0)$ is the thermodynamic critical field, ranges from 0.144 to 0.168.

In the paper, the thermodynamic parameters of the high-pressure superconducting state in the SiH$_4$ compound have been determined ($p=250$ GPa). By using the Eliashberg equations in the mixed representation, the critical temperature, the energy gap, and the electron effective mass have been calculated. It has been stated that the critical temperature ($T_{C}$) decreases from 51.65 K to 20.62 K, if the Coulomb pseudopotential increases ($\mu^{\star}\in 0.1,0.3$). The dimensionless ratio $2\Delta\(0\)/k_{B}T_{C}$ decreases from 4.10 to 3.84, where the symbol $\Delta\(0\)$ denotes the value of the order parameter close to the zero temperature. The ratio of the electron effective mass to the band electron mass is high, and it reaches maximum equal to 1.95 for the critical temperature., Comment: 5 pages, 5 figures

The structure of the gap parameter ($\Delta_{k}$) for the hole-doped cuprates has been studied. The obtained results indicate that the antinodal part of $\Delta_{k}$ is very weakly temperature dependent and above the critical temperature ($T_{C}$), it extends into the anomalous normal state to the pseudogap temperature. On the other hand, the values of $\Delta_{k}$, which are close to the nodal part, are strongly temperature dependent. The model has been tested for the ${\rm YBa_{2}Cu_{3}O_{7-\delta}}$ superconductor. It has been shown that the theoretical results agree with the experimental data., Comment: 9 pages, 5 figures

In the paper, the thermodynamic properties of the CaLi$_2$ superconductor have been studied. It has been assumed the high value of the pressure: $p=100$ GPa. The thermodynamic critical field ($H_{C}$) and the specific heats for the superconducting and normal state ($C^{S}$ and $C^{N}$) have been obtained. The calculations have been made within the framework of the Eliashberg approach for the wide range of the Coulomb pseudopotential: $\mu^{\star}\in<0.1,0.3>$. It has been shown that the low-temperature critical field and the specific heat jump at $T_{C}$ strongly decrease if $\mu^{\star}$ increases. The value of the dimensionless ratio $R_{C}\equiv(C^{S}-C^{N})/C^{N}$ at the critical temperature differs significantly from the value predicted by the BCS model. In particular, the parameter $R_{C}$ decreases from 1.83 to 1.73 with the increase of the Coulomb pseudopotential value. In the paper, the interpolation formulas for the functions $T_{C}(\mu^{\star})$ and $R_{C}(\mu^{\star})$ have been also given., Comment: 5 figures

A superconducting state, that induces in ${\rm GeH_{4}}$ under the pressure at 20 GPa, is being characterized by a high critical temperature $T_{C}\in<54.54,31.23>$ K, for a wide range of the Coulomb pseudopotential $\mu^{\star}\in<0.1,0.3>$. Other thermodynamic parameters for the considered range of the Coulomb pseudopotential significantly differ from the predictions of a classical BCS theory. In particular: (i) the ratio of the energy gap to the critical temperature ($R_{\Delta}\equiv 2\Delta(0)/k_{B}T_{C}$) changes from 4.07 to 3.88; (ii) the value of the parameter $R_{C}\equiv\Delta C(T_{C})/C^{N}(T_{C})$ decreases from 1.85 to 1.69, where $\Delta C$ denotes the specific heat's jump, and $C^{N}$ is the specific heat of the normal state; (iii) the ratio $R_{H}\equiv T_{C}C^{N}(T_{C})/H^{2}_{C}(0)$, where $H_{C}(0)$ represents the low - temperature thermodynamic critical field, increases from 0.147 to 0.155., Comment: 6 pages, 5 figures

The thermodynamic parameters of the superconducting state, that induces in ${\rm Si_{2}H_{6}}$ under the pressure at 275 GPa, have been calculated. In the framework of the Eliashberg formalism, it has been shown that the critical temperature can attain extremely high values: $T_{C}(\mu^{\star})\in<173.36, 99.99>$ K, where the Coulomb pseudopotential $(\mu^{\star})$ belongs to the range from 0.1 to 0.3. The ratio of the energy gap to the critical temperature ($R_{\Delta}$) significantly exceeds the value predicted by the BCS theory: $R_{\Delta}(\mu^{\star})\in<4.40, 4.04>$. Additionally, it has been stated that in the whole range of the superconducting state's existence, the electron effective mass ($m^{\star}_{e}$) is large; $[m^{\star}_{e}]^{\rm max}_{T=T_{C}}=2.397m_{e}$, where the symbol $m_{e}$ denotes the electron band mass., Comment: 7 pages, 5 figures

The thermodynamic parameters of the superconducting state, that gets induced in the metallic molecular hydrogen under the influence of the pressure at 414 GPa ($T_{C}=84$ K), have been determined. The calculations have been conducted in the framework of the three-band Eliashberg equations. The order parameters ($\Delta^{\alpha}$) and the wave function renormalization factors ($Z^{\alpha}$) have been derived; the symbol $\alpha$ denotes the band index: $\alpha\in{a,b,c}$. It has been stated that the dimensionless ratios $2\Delta^{\alpha}(0)/k_{B}T_{C}$ are equal to: 5.55, 3.96 and 3.53, respectively. Next, the total normalized function of the density of states, the free energy, the thermodynamic critical field and the specific heat have been determined. The obtained results differ significantly from the results achieved in the framework of the one-band Eliashberg model for the comparable value of the pressure ($p=428$ GPa). In the last step, the values of the wave function renormalization factors have been estimated for $T_{C}$. It allowed us to calculate the maximum value of the electron effective mass in a given band [($[m^{\star}_{e}]_{\alpha}$). The following results have been achieved: $[m^{\star}_{e}]_{a}=2.99 [m_{e}]_{a}$, $[m^{\star}_{e}]_{b}=2.10 [m_{e}]_{b}$ and $[m^{\star}_{e}]_{c}=1.94 [m_{e}]_{c}$, where $[m_{e}]_{\alpha}$ stands for the electron band mass.

The critical temperature ($T_{C}$) and the energy gap ($2\Delta(T)$) for the superconductor SiH$_4$(H$_2$)$_2$ at 250 GPa have been calculated. The wide range of the Coulomb pseudopotential's values has been considered: $\mu^{\star}\in<0.1,0.3>$. It has been stated that $T_{C}$ decreases together with the increase of $\mu^{\star}$ from 129.83 K to 81.40 K. The low-temperature energy gap ($T\sim 0$ K) decreases together with the increase of the Coulomb pseudopotential from 50.96 meV to 30.12 meV. The high values of $2\Delta(0)$ mean that the dimensionless ratio $R_{\Delta}\equiv 2\Delta(0)/k_{B}T_{C}$ significantly exceeds the value predicted by the classical BCS theory. In the considered case: $R_{\Delta}\in<4.55,4.29>$. Due to the unusual dependence of the critical temperature and the energy gap on $\mu^{\star}$, the analytical expressions for $T_{C}(\mu^{\star})$ and $\Delta(\mu^{\star})$ have been given., Comment: 7 pages, 5 figures

The model for the cuprates based on the modified electron-phonon pairing mechanism has been tested. For this purpose, the superconductors with high value of the critical temperature have been taken into consideration. In particular: ${\rm YBa_{2}Cu_{3}O_{7-y}}$, ${\rm HgBa_{2}CuO_{4+y}}$, ${\rm HgBa_{2}Cu_{1-x}Zn_{x}O_{4+y}}$, and ${\rm HgBa_{2}Ca_{2}Cu_{3}O_{8+y}}$. It has been shown that the dependence of the ratio $R_{1}\equiv 2\Delta_{tot}^{(0)}/k_{B}T_{C}$ on the doping ($p$) can be properly predicted in the framework of the presented theory; the symbol $\Delta_{tot}^{(0)}$ denotes the energy gap amplitude at the temperature of zero Kelvin, and $T_{C}$ is the critical temperature. The numerical results have been supplemented by the formula which describes the function $R_{1}(p)$., Comment: 4 figures

The unknown quantum electronic conductance across nanojunctions made of silicon-doped carbon wires between carbon leads is investigated. This is done by an appropriate generalization of the phase field matching theory for the multi-scattering processes of the electronic excitations at the nanojunction, and the use of the tight-binding method. Our calculations of the electronic band structures for carbon, silicon and diatomic silicon carbide, are matched with the available corresponding density functional theory results to optimize the required tight-binding parameters. The silicon and carbon atoms are treated on the same footing by characterizing each with their corresponding orbitals. Several types of nanojunctions are analyzed to sample their behavior under different atomic configurations. We calculate for each nanojunction the individual contributions to the quantum conductance for the propagating $\sigma$, $\pi$, and $\sigma^{*}$ electrons incident from the carbon leads. The calculated results show a number of remarkable features, which include the influence of the ordered periodic configurations of silicon-carbon pairs, and the suppression of the quantum conductance due to minimum substitutional disorder and to artificially organized symmetry on these nanojunctions. Our results also demonstrate that the phase field matching theory is an efficient tool to treat the quantum conductance of complex molecular nanojunctions., Comment: 13 pages, 7 figures, 1 table

The properties of the d-wave superconducting state in the two-dimensional system have been studied. It has been assumed, that the pairing mechanism is based on the electron-phonon and the electron-electron-phonon interactions. The obtained results have shown the energy gap amplitude ($\Delta_{tot}$) crossover, from the BCS to non-BCS behavior, as the value of the electron-electron-phonon potential increases. The model has been tested for the ${\rm La_{2-x}Sr_{x}CuO_{4}}$ and ${\rm Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta}}$ high-$T_{C}$ superconductors. It has been shown, that the dependence of the $2\Delta^{(0)}_{tot}/k_{B}T_{C}$ ratio on the hole density is in agreement with the experimental data.

The thermodynamic parameters of the superconducting state in calcium under the pressure at 161 GPa have been calculated within the framework of the Eliashberg approach. It has been shown that the value of the Coulomb pseudopotential is high (0.24) and the critical temperature (25 K) should be determined from the modified Allen-Dynes formula. In addition, it has been found that the basic dimensionless ratios of the thermodynamic parameters significantly diverge from the BCS predictions, and take the following values: (i) The zero temperature energy gap to the critical temperature(R1) is equal to 4.01. (ii) The ratio R2 equals 2.17. (iii) The quantity R3=0.158. Finally, it has been proven that the electron effective mass is large and takes the maximum of 2.32*me at TC., Comment: 8 pages, 9 figures; Phys. Status Solidi B (2012)

In the paper, we have discussed the problem of the existence of the metallic hydrogen in the superconducting state inside the cold giant planet. We have shown that the {\it cold} planet represents the final state of the thermal evolution of free-floating giant planet., Comment: 4 pages, 2 figures

The thermodynamic parameters of the superconducting state in Calcium under the pressure at 200 GPa have been determined. The numerical analysis by using the Eliashberg equations in the mixed representation has been conducted. It has been stated, that the critical temperature ($T_{C}$) decreases from 36.15 K to 20.79 K dependently on the assumed value of the Coulomb pseudopotential ($\mu^{*}\in<0.1,0.3>$). Next, the order parameter near the temperature of zero Kelvin ($\Delta(0)$) has been obtained. It has been proven, that the dimensionless ratio $2\Delta(0)/k_{B}T_{C}$ decreases from 4.25 to 3.90 together with the growth of $\mu^{*}$. Finally, the ratio of the electron effective mass to the electron bare mass ($m^{*}_{e}/m_{e}$) has been calculated. It has been shown, that $m^{*}_{e}/m_{e}$ takes the high value in the whole range of the superconducting phase's existence, and its maximum is equal to 2.23 for T=T_{C}., Comment: 5 pages, 5 figures

The thermodynamic parameters of the superconducting state in Calcium under the pressure at 200 GPa were calculated. The Coulomb pseudopotential values ($\mu^{\star}$) from 0.1 to 0.3 were taken into consideration. It has been shown, that the specific heat's jump at the critical temperature and the thermodynamic critical field near zero Kelvin strongly decrease with $\mu^{\star}$. The dimensionless ratios $r_{1}\equiv \Delta C(T_{C})/C^{N}(T_{C})$ and $r_{2}\equiv T_{C}C^{N}(T_{C})/H^{2}_{C}(0)$ significantly differ from the predictions based on the BCS model. In particular, $r_{1}$ decreases from 2.64 to 1.97 with the Coulomb pseudopotential; whereas $r_{2}$ increases from 0.140 to 0.157. The numerical results have been supplemented by the analytical approach., Comment: 7 pages, 6 figures

The thermodynamic properties of the superconducting state induced in metallic molecular hydrogen under the influence of pressure 347 GPa were determined. In particular, it has been shown that the critical temperature ($T_{C}$) changes in the range from 120 K to 90 K for $\mu^{*}\in<0.08,0.15>$, where $\mu^{*}$ is the value of the Coulomb pseudopotential. Next, the energy gap near the temperature of zero Kelvin ($2\Delta(0)$) was calculated. It has been stated, that the dimensionless ratio $2\Delta(0)/k_{B}T_{C}$ slightly decreases with the increase of $\mu^{*}$ from 3.98 to 3.84. In the last step, the ratio of effective electron mass ($m^{*}_{e}$) to the bare electron mass ($m_{e}$)) was determined. It has been proved that $m^{*}_{e}/m_{e}$ takes its highest value equal to 1.96 for $T=T_{C}$.

The simple microscopic model for the high-T_{C} superconductors based on the electron-phonon (EPH) and electron-electron-phonon (EEPH) interactions has been presented. On the {\it fold} mean-field level, it has been shown, that the obtained model supplements the predictions based on the BCS van Hove scenario. In particular: (i) For strong EEPH coupling and T

In the framework of the Eliashberg formalism the free energy difference between the superconducting and normal state for the molecular metallic hydrogen was calculated. The pressure values $p_{1}=347$ GPa and $p_{2}=428$ GPa were taken into consideration. It has been shown, that together with the increase of the pressure, grows the value of the specific heat jump at the critical temperature and the value of the thermodynamic critical field near zero Kelvin: $[\Delta C(T_{C})]_{p2}/[\Delta C(T_{C})]_{p1}\simeq 2.33$ and $[H_{C}(0)]_{p2}/[H_{C}(0)]_{p1}\simeq 1.74$. Next, it has been stated, that the ratio $\Delta C(T_{C})/C^{N}(T_{C})$ also increases from 1.91 to 2.39; whereas $T_{C}C^{N}(T_{C})/H^{2}_{C}(0)$ decreases from 0.152 to 0.140. The last results prove that the considered parameters significantly diverge from the prediction based on the BCS model.

The detailed study of the selected thermodynamic properties of the superconducting phase in the molecular hydrogen under the pressure at 428 GPa has been presented. For the increasing value of the Coulomb pseudopotential, $\mu^{*}\in<0.08,0.15>$, the following results have been obtained: (i) the critical temperature decreases from 179 K to 141 K, (ii) the ratio $R_{1}\equiv\Delta(0)/k_{B}T_{C}$ differs noticeably from the BCS value: $R_{1}\in<4.71,3.60>$; (iii) the electron effective mass is large and grows slightly together with the temperature ($[m^{*}_{e}/m_{e}]_{{\rm max}} = 2.2$ for $T = T_{C}$).

The basic thermodynamic parameters of the superconducting state in $\rm{YNi_{2}B_{2}C}$ were calculated in the framework of the one-band Eliashberg model. The effective Eliashberg function, determined on the basis of the transport function (R.S. Gonnelli, {\it et al.}, Physica C \textbf{341}, 1957 (2000)), was used during calculations. It was shown that the dimensionless ratios are equal to: $R_{1}\equiv 2\Delta(0)/k_{B}T_{C}=3.87$, $R_{2}\equiv\Delta C(T_{C})/C^{N}(T_{C})=1.79$ and $R_{3}\equiv T_{C}C^{N}(T_{C})/H_{C}^{2}(0)=0.159$. The value $R_{1}$ fairly agrees with the experimental data whereas $R_{2}$ and $R_{3}$ agree very well.

The free energy difference between the superconducting and normal state for Calcium under the pressure at 120 GPa has been determined. The numerical calculations have been made in the framework of the imaginary axis Eliashberg approach. On the basis of the obtained results the specific heat in the superconducting C^{S}(T) and normal C^{N}(T) state, as well as, the thermodynamic critical field H_{C}(T) have been obtained. It has been shown that the characteristic values of the considered thermodynamic quantities do not obey the BCS universal laws. In particular, {\Delta}C(T_{C})/C^N(T_{C})=2.48 and T_{C}C^{N}(T_{C})/H_{C}^{2}(0)=0.154., Comment: 3 pages, 4 figures