The Study on the Critical Temperature and Gap-to-T_c Ratio of Yttrium Hydride Superconductors

Abstract

This study investigates the gap-to-$T_c$ ratio (R) of yttrium hydride superconductors within the weak coupling limit. We derived an analytical formula for the gap-to-$T_c$ ratio. The ratio of the gap-to-$T_c$ is dependent on the pressure applied to each superconductor. The maximum ratio, approximately 3.85, is observed in one superconductor, while the lowest ratio, roughly 3.21, is found in another superconductor. Based on the findings of our study, it can be deduced that yttrium hydride superconductors exhibit attributes commonly associated with weak-coupling superconductors. The influence of the Coulomb potential is more pronounced at a critical temperature compared to the ratio of the gap to the critical temperature.

1. Introduction

In early research into high-pressure research on superconductors, it was found that several hydrides, such as hydrogen sulfide $\left(H_{3}S\right)$ [1,2] and lanthanum hydride $\left(La{H}_{10}\right)$ [3,4], have high critical temperatures $\left(T_c\right)$ of 200 K and 250 K, respectively. This was the first evidence that high-critical temperature superconductors under high pressure existed. When searching for superconducting hydrides with high $T_c$, the investigated potentially achievable high-pressure crystal structures of lanthanum hydride (La–H) and yttrium hydride (Y–H) were significant because they can obtain extremely high critical temperatures using a process that seems quite comparable [5]. In their research, Liu et al. [5] employed the electron–phonon-coupling constant, denoted as ${\lambda }_{ph}$, and the effective Coulomb potential, represented by $u^{*}$, with a particular value of $\lambda =0.43$ and $u^{*}=0.1–0.13$ for $La{H}_4$ and $Y{H}_6$ superconductors. It was shown that the La-H and Y-H systems exhibit the presence of a stable hydrogen-rich phase for superconductivity in high-pressure conditions. In their study, Snider et al. [6] achieved the synthesis of a yttrium superhydride on $Y{H}_x, (x\ge 3)$ and observed that the yttrium superhydride has a maximum critical temperature of 262 at 182 GPa for the $Y{H}_9$ superconductor. The coefficient of the isotope effect, denoted as $\alpha$, was determined to be 0.48 by their experimental measurements. In their studies, Kong et al. [7] synthesized the compounds $Y{H}_3, Y{H}_4, Y{H}_6$ and $Y{H}_9$ to investigate their superconducting properties. They reported that the superconducting state in materials $Y{H}_6$ and $Y{H}_9$ occurs at critical temperatures of around 220 K and 243 K, respectively, when subjected to pressures of 183 GPa and 201 GPa. These findings are consistent with the predictions of $Y{H}_9$ with $T_c=253–276$ K at 200 GPa [5] and $Y{H}_6$ with $T_c=251–264$ K at 110 GPa [8]. Nonetheless, these results are inconsistent with the computation that anticipated $Y{H}_{10}$ with $T_c=303$ K at 400 GPa [9] or $T_c=305–326$ K at 250 GPa [5]. The dome-like shape of the critical temperature versus the pressure of $Y{H}_6$ and $Y{H}_9$, and the isotope effect coefficients of 0.39 and 0.5 were found. These values of the isotope effect coefficients of $Y{H}_6$ and $Y{H}_9$ and are in the range of the weak-coupling limit of the BCS theory. The electron–phonon interaction is associated with the pairing mechanism of the superconducting state that can be observed in these hydrides. It was identified and proven that this is the mechanism that causes superconductivity to occur [1,4]. In Ref. [10], it was suggested that the Cooper pairs are associated with the strong electron–phonon constant and that this constant is frequency-dependent. Meanwhile, the isotope effect coefficient of certain hydrides indicates a weak-coupling interaction [7].

In the present study, the authors are interested in the gap-to-$T_c$ ratio of yttrium hydride superconductors. The experimental results reveal that yttrium hydride superconductors exhibit weak-coupling interactions with electron–phonon interactions. However, there has been limited research on the zero-temperature energy gap of a hydride superconductor with evolving pressure. We first performed the experiment for the conventional s-wave superconductor in the Pb superconductor at 3400 bar and observed that the energy gap of the Pb superconductor dependency on pressure was larger than BCS, and rising pressure led $\frac{\Delta (0, p)}{T_c}$ to approach the BCS value. As the pressure increased, this ratio decreased [11]. The effect of high pressure on the energy gap of indium and thallium superconducting films was also examined. The value of $\frac{\Delta (0, p)}{T_c}$ dropped as the pressure rose [12]. The zero-temperature gap of $H_{3}S$ was estimated from the upper critical field experiment at $\Delta (0)=28.8$ meV, $T_c=190$ K and $R=\frac{2\Delta (0)}{T_c}= 3.55$ at 155 GPa [13]. In the LaH₁₀ superconductor, the theoretical confirmation of an electron–phonon coupling mechanism with a larger gap-to-T_c ratio than the BCS value in the superconductors [14,15] $T_c=214$ K with $R=4.6$ at 300 GPa was observed [14]. In yttrium hydride superconductors, there were reports of $Y{H}_4, Y{H}_6, Y{H}_7$ and $Y{H}_9$ with $R=4.1–4.3, 5.0, 3.7–3.9$ [6,16,17] and $4.7–5.5$ [7,17]. The critical temperature of yttrium hydride superconductors, referred to as $Y{H}_9$, was found to exhibit exceptional properties in experimental studies. Specifically, at a pressure of 182 GPa, the critical temperature reached its maximum, as observed in experiment $T_c=262$ K [6]. Additionally, experiment $T_c=243$ K demonstrated that at a pressure of 250 GPa, yttrium hydride superconductors also exhibit a high critical temperature [7]. However, since this is estimated using predictive models,$Y{H}_{10}$, it suggests that the critical temperature is expected at a pressure of 250 GPa, with its highest at $T_c=305–326$ K [5]. The conventional phonon-mediated superconductor is proposed as a potential candidate for this particular superconductor [6,7].

This study investigates the influence of pressure on the superconducting properties of yttrium hydride materials. We investigated the relationship between the gap-to-$T_c$ ratio and the modified density of states and the energy dispersion relation as a function of pressure in the weak-coupling limit. The experimental findings were compared to the numerical results in order to demonstrate the reliability and precision of our calculations.

2. Materials and Methods

2.1. Materials

Consider the crystal structure of yttrium hydrides and their critical temperature: T_C, in Figure 1a, with I4/mmm-YH₃, had 40 K at 50 GPa [18,19,20,21]. The lattice parameters are $a=b=3.29560$, $c=4.72730$ and $\alpha =\beta =\gamma =90.0000°$ [9]. In Figure 1b, I4/mmm-YH₄ had 85–95 K at 300 GPa. The lattice parameters are $a=b=2.55450$, $c=5.14940$, and $\alpha =\beta =\gamma =90.0000°$ [9]. In Figure 1c, Imm2-YH₇ had 21–43 K at 165 GPa. Figure 1d shows that Im3m-YH₆ had 251–264 K at 200 GPa. The lattice parameters are $a=b=c=3.52690$, and $\alpha =\beta =\gamma =90.0000°$ [9]. Figure 1e shows that P6₃/mmc-YH₉ had 253–276 K at 300 GPa. The lattice parameters are $a=b=5.10410$, $c=3.14060$, with $\alpha =\beta =90.0000°$, $\gamma =120.000°$ [9]. We found that Im3m-YH₆ and P6₃/mmc-YH₉ were attributed to their hydrogen cage structure having a higher T_C than (b) I4/mmm-YH₄ and (d) Imm2-YH₇. In the case of YH10 [9,22], a high TC of about 303 K at 400 GPa was calculated and predicted; the lattice parameters are $a=b=c=4.47880$ and $\alpha =\beta =\gamma =90.0000°$. The calculation used $u^{*}=0.11$ [18], which contrasts with a recent study that used a larger value. However, the peculiar geometry by which yttrium hydrides implement a dense hydrogen lattice compromises H packing and structural stability.

2.2. Methods

In this study, we investigated the influence of pressure on the superconducting characteristics of yttrium hydride materials in the weak-coupling limit, beginning with the BCS gap equation [23,24,25] as follows:

$${\Delta }_k=-{\displaystyle \sum _{k^{\prime }}{V}_{k{k}^{\prime }}}\frac{{\Delta }_{k^{\prime }}}{2\sqrt{{\epsilon }_{k^{\prime }}^2+{\Delta }_{k^{\prime }}^2}}\tanh \left(\frac{\sqrt{{\epsilon }_{k^{\prime }}^2+{\Delta }_{k^{\prime }}^2}}{2T}\right)$$

(1)

Here, the energy of the carrier, ${\epsilon }_k$, is measured relative to the Fermi energy. $V_{k{k}^{\prime }}$ represents the interaction potential. The order parameter ${\Delta }_k$ is dependent on both the temperature (T) and wave vector (k), denoted as ${\Delta }_k\equiv {\Delta }_k(T)$.

In the context of hydride superconductors, it is important to include the Coulomb potential as an additional interaction that should be taken into consideration during calculations. Let $V_{ph}$ represent the attractive–phonon interaction and $U_c$ represent the Coulomb repulsion interaction, either of which possesses a characteristic cutoff energy associated with the Debye phonon $\left({\omega }_D\right)$ and Coulomb $\left({\omega }_c\right)$, respectively. Then, the interaction potential of carrier $V_{k{k}^{\prime }}$ is [26,27,28] $V_{k{k}^{\prime }}=-V_{ph}+U_c$ for $0<\left|{\epsilon }_k\right|<{\omega }_D$ and $V_{k{k}^{\prime }}=U_c$ for ${\omega }_D<\left|{\epsilon }_k\right|<{\omega }_c$. The superconducting order parameter should be expressed in a manner that demonstrates comparable characteristics as ${\Delta }_k={\Delta }_{ph}$ for $0<\left|{\epsilon }_k\right|<{\omega }_D$, and ${\Delta }_k={\Delta }_c$ for ${\omega }_D<\left|{\epsilon }_k\right|<{\omega }_c$. The addition of external high pressure $\left(P\right)$ to a superconductor can result the a deformation of its crystal structure due to energy $\left(PV\right)$, where $\left(V\right)$ indicates the volume of the specific unit cell under consideration. We considered the assumption that all atoms within the unit cell absorb external energy and, thus, reside in an excited state. Consequently, the influence of external pressure $\left(p\right)$ on the unit cell volumes $\left(V\left(p\right)\right)$ was taken into consideration. The newly established equilibrium state of the dispersion relation of the carrier is designated as ${\epsilon }_k(v)$ where $v(p)=\frac{V(p)}{V_0}$, in which $\left(V\left(p\right)\right)$ and $V_0$ present the unit cell volume under pressure $p$ and ambient pressure, respectively. We assume in the relation $p\alpha \frac{1}{v^{\beta }}$ that $\beta$ is a constant related to the Birch–Murnaghan equation of state. The carrier dispersion relation ${\epsilon }_k(v)$ could be analytically represented by an expansion in a power series of ${\epsilon }_k(v)={\epsilon }_k(0)+k\left(\frac{1}{V^{\beta }}-\frac{1}{V_0^{\beta }}\right) {\left[\frac{d{\epsilon }_k(p)}{dp}\right]}_{p=0}+\dots$.

After the higher order is neglected, we obtain the following:

$${\epsilon }_k(v)\approx {\epsilon }_k(V_0)+Q_e(\frac{1}{v^{\beta }}-1)$$

(2)

here $Q_e=\frac{k}{V_0^{\beta }} {\left[\frac{d{\epsilon }_k(p)}{dp}\right]}_{p=0}$ and $Q_e\left(\frac{1}{v^{\beta }}-1\right)$ are assumed to be constants with units of energy, and a higher order is neglected, see Appendix A.

Another possible consequence of exerting pressure on a hydride superconductor is the possible fluctuation of the density of states, resulting in the formation of a narrow peak in the density of states. This peak is seen to occur somewhat below the Fermi level [17,28,29]. In our suggested model, we also made the assumption that the fluctuation in carriers under pressure leads to a single narrow fluctuation constant in the density of states. The parameter $N(0)\chi$ represents the magnitude of the variation in height, and the parameter $\chi$ is dimensionless. To facilitate the analysis, the position of distortion in the unpressured condition is denoted as ${\epsilon }_0$, thereby establishing the relationship $\rho (\epsilon )=\delta (\epsilon -{\epsilon }_0)$. As a result, we achieved the following:

$$N(\epsilon )=N(0)(1+\chi \delta (\epsilon -{\epsilon }_0)).$$

(3)

In our framework, we suggest that for the impact of high pressure on the physical properties of hydride superconductors, we make the assumption that the modification of the dispersion relation, Equation (2), and the presence of a delta peak in the density of states, Equation (3), should be incorporated into our calculations. Substituting Equations (2) and (3) into the integral form of Equation (1) for a temperature equal to the critical temperature $\left(T=T_c\right)$, we obtain the matrix from the gap equation as follows:

$$\left(\begin{array}{c}{\Delta }_{ph}\\ {\Delta }_c\end{array}\right)=\left(\begin{array}{cc}\begin{array}{c}-(-{\lambda }_{ph}+u_c)I_{11}^c\\ -u_c{I}_{22}^c\end{array}& \begin{array}{c}-u_c{I}_{12}^c\\ -u_c{I}_{21}^c\end{array}\end{array}\right)\left(\begin{array}{c}{\Delta }_{ph}\\ {\Delta }_c\end{array}\right)$$

(4)

here ${\lambda }_{ph}=N(0)V_{ph}$, $u_c=N(0)U_c$, $I_{11}^c=I_{22}^c={\displaystyle \underset{0}{\overset{{\omega }_D}{\int }}d\epsilon }\frac{\left(1+\chi \delta (\epsilon -{\epsilon }_0)\right)}{\epsilon +Q_e\left(\frac{1}{v^{\beta }}-1\right)}\tanh \left(\frac{\epsilon +Q_e(\frac{1}{v^{\beta }}-1)}{2T_c}\right)$ and $I_{12}^c=I_{21}^c={\displaystyle \underset{{\omega }_D}{\overset{{\omega }_c}{\int }}d\epsilon }\frac{\left(1+\chi \delta (\epsilon -{\epsilon }_0)\right)}{\epsilon +Q_e\left(\frac{1}{v^{\beta }}-1\right)}\tanh \left(\frac{\epsilon +Q_e\left(\frac{1}{v^{\beta }}-1\right)}{2T_c}\right)$.

For ${\omega }_D>>Q_e\left(\frac{1}{v^{\beta }}-1\right)$, we obtain the critical temperature equation as follows:

$$T_c=1.13\left({\omega }_D+Q_e\left(\frac{1}{v^{\beta }}-1\right)\right)\exp \left\{-\frac{1}{{\lambda }_{ph}-u^{*}}+\frac{\chi }{{\epsilon }_0+Q_e\left(\frac{1}{v^{\beta }}-1\right)}\tanh \left(\frac{{\epsilon }_0+Q_e(\frac{1}{v^{\beta }}-1)}{2T_c}\right)\right\}$$

(5)

here $u^{*}=\frac{u_c}{1+u_c{I}_{12}^c}$.

Using the same process for the zero-temperature (T = 0) consideration, and the similar matrix with ${\omega }_D+Q_e\left(\frac{1}{v^{\beta }}-1\right)>>{\Delta }_{ph}(0)$, we obtain the zero-temperature gap as follows:

$${\Delta }_{ph}(0)=2\left({\omega }_D+Q_e\left(\frac{1}{v^{\beta }}-1\right)\right)\exp \left\{-\frac{1}{{\lambda }_{ph}-u^{*}}+\frac{\chi }{\sqrt{{\left({\epsilon }_0+Q_e\left(\frac{1}{v^{\beta }}-1\right)\right)}^2+{\Delta }_{ph}^2(0)}}-{\sinh }^{-1}\left(\frac{Q_e\left(\frac{1}{v^{\beta }}-1\right)}{{\Delta }_{ph}(0)}\right)\right\}.$$

(6)

Here, the gap-to-T_c can be found, $R=\frac{2{\Delta }_{ph}(0)}{T_c}$, therefore,

$$R=3.53 \left[1+RE+\frac{\chi }{\left({\epsilon }_0+Q_e\left(\frac{1}{v^{\beta }}-1\right)\right)}\left[1-\tanh \left(\frac{Q_e\left(\frac{1}{v^{\beta }}-1\right)+{\epsilon }_0}{2T_c}\right)\right]\right]$$

(7)

where $RE={\displaystyle \underset{0}{\overset{+Q_e(\frac{1}{v^{\beta }}-1)}{\int }}dx}\frac{1}{x}\tanh \left(\frac{x}{2T_c}\right)$.

At this point, the variables depend on the volume ratio as $T_c\equiv T_c(v)$, ${\Delta }_0\equiv {\Delta }_0(v)$ and $R\equiv R(v)$. In order to establish a connection with the experimental data concerning hydride superconductors, it is preferable to present the curve depicting the relationship between these parameters and the external pressure. The Birch–Murnaghan equation of state [30,31,32,33] is employed to analyze the impact of external pressure on the expansion of a new state in terms of reduced volume $\left(v=\frac{V}{V_0}\right)$. However, an experiment employing the Birch–Murnaghan equation of state for the yttrium hydride superconductor has not been located. Nonetheless, a comparable form of the equation is referenced in Ref. [7]. The data points were extracted from their respective curves and subsequently fitted using a polynomial equation. Following that, we could determine the relationship between external pressure and reduced volume $\left(P=P(v)\right)$.

3. Results and Discussion

In this paper, we performed calculations to support the abovementioned results using the gap-to-$T_c$ ratio. Several phases of Y-H superconductors were laid out under pressure, including the $Y{H}_3$, $Y{H}_4$, $Y{H}_6$, $Y{H}_7$, and $Y{H}_9$ superconductors [7,16,22]. Among these, the $Y{H}_9$ superconductor was identified to have the greatest critical temperature. The aim of the present research is to investigate the critical temperature and the gap-to-$T_c$ ratio in the Y-H superconductor. The equations for the critical temperature and the gap-to-$T_c$ ratio may be determined by evaluating Equations (5) and (7), respectively. However, the Birch–Murnaghan equation was not obtained. Therefore, we initially computed the $P=P(v)$ relation of each YH superconductor using the data from Reference [7]. The data points were detached from their respective curves and subsequently fitted with a second-order polynomial equation in order to determine the connection between external pressure and decreased volume $\left(P=P(v)\right)$. The Debye cutoff temperature was reported as ${\omega }_D=1082$ K for $Y{H}_4$, ${\omega }_D=1333$ K for $Y{H}_6$, and ${\omega }_D=684–1333$ K for $Y{H}_9$ [16]. In our analysis, we utilized the parameter (${\omega }_D=$ 684 K, 1333 K) to represent the whole range of potential values for the Debye cutoff temperature in the Y-H superconductor. The coupling constant ($\lambda$) was utilized in the weak-coupling limit with the effective Coulomb potential ($u^{*}$) employed in the range $u^{*}=0.1–0.13$ [5].

The numerical calculations of $T_c$ and $R$ of the high-pressure superconductor of the $Y{H}_9$ superconductor are shown in Figure 2. The experimental data for $T_c$ were compared to our numerical calculation, as shown in Figure 2a. Within the same parameters of calculation, Figure 2b presents the prediction for the gap-to-$T_c$ ratio $\left(R\right)$. The parameters were ${\omega }_D=$ 684 K, 1333 K, $u^{*}=0.1, 0.13$, $\chi =550$, $\lambda =0.44$, $Q_e=100$, ${\epsilon }_0=50$, $\beta =5/3$. The $P=P(v)$, read out from Ref. [7], was found to be $P(v)=4884.4v^2-8900v+4189.4$ for $0.66<v<0.82$. In our investigation of ${\omega }_D=$ 684 K, 1333 K and $u^{*}=0.1, 0.13$, it was observed that the $Y{H}_9$ superconductor exhibits the critical temperature $T_c\approx 150–240$ K, while the gap-to-$T_c$ ratio $\left(R\approx 3.76–3.85\right)$ was determined. As the values of ${\omega }_D$ decreased and $u^{*}$ increased, there was a corresponding decrease in the value of $R$. As the pressure decreased, the critical temperature and the gap-to-$T_c$ ratio $\left(R\right)$ increased. The Coulomb potential had a greater impact on the critical temperature compared to its influence on the gap-to-$T_c$ ratio.

Figure 3 presents the $T_c$ and $R$ versus pressure of the $Y{H}_6$ superconductor. The parameters were ${\omega }_D=$ 684 K, 1333 K, $u^{*}=0.1, 0.13$ $\chi =550$, $\lambda =0.44$, $Q_e=100$, ${\epsilon }_0=50$, and $\beta =5/3$. The $P=P(v)$, read out from Ref. [7], was found to be $P(v)=6801.7v^2-10697v+4367.4$ for $0.58<v<0.80$. In Figure 3a, our numerical calculation of the critical temperature agrees with the experimental data for ${\omega }_D=1333$ K and $u^{*}=0.1$. It was observed that the $Y{H}_6$ superconductor exhibits a critical temperature of $T_c\approx 135–235$ K, while the gap-to-$T_c$ ratio ($R\approx 3.57–3.83$) was determined. As the values of ${\omega }_D$ decrease and $u^{*}$ increase, there is a corresponding decrease in the value of $R$. As the pressure decreases, the critical temperature and the gap-to-$T_c$ ratio $\left(R\right)$ rises. The impact of pressure on the gap-to-$T_c$ ratio may be observed via an elevation in the Coulomb potential, accompanied by a corresponding reduction in ${\omega }_D$. In particular, the decrease in the gap-to-$T_c$ ratio is clearer than other gap-to-$T_c$ ratios. The influence of the Coulomb potential on the critical temperature is stronger in comparison to its effect on the gap-to-$T_c$ ratio.

The numerical calculations for the quantities of $T_c$ and $R$ in the high-pressure $Y{H}_4$ superconductor are presented in Figure 4. The parameters were ${\omega }_D=$ 684 K, 1333 K, $u^{*}=0.1, 0.13$ $\chi =300$, $\lambda =0.34$, $Q_e=-100$, ${\epsilon }_0=50$, $\beta =5/3$. The $P=P(v)$, read out from Ref. [7], was found to be $P(v)=-2238.7v^2+2486.4v-175.01$ for $0.78<v<0.98$. Based on the restrictions caused by our available data, we aim to demonstrate that curve $Y{H}_4$ serves as an appropriate model for curve $Y{H}_3$. The $Y{H}_3$ exhibits a comparable $P=P(v)$ to that of the $Y{H}_4$ [7], but there is a lack of data on the critical temperature versus pressure relationship in $Y{H}_3$. $Y{H}_4$ is depicted as a representation of the $Y{H}_3$ superconductor. The $T_c$ experimental data of the $Y{H}_4$ superconductor ($T_c\approx$70–88 K) fit well with our parameters as ${\omega }_D=1333$ K $u^{*}=0.13$ and ${\omega }_D=684$ K, $u^{*}= 0.1$. In our investigation of ${\omega }_D=$ 684 K, 1333 K and $u^{*}=0.1, 0.13$, it was observed that the $Y{H}_4$ superconductor exhibits a critical temperature when $T_c\approx 64–104$ K, while the gap-to-$T_c$ ratio ($R\approx 3.79–3.82$) was determined. When the pressure increases, the critical temperature also increases, but the gap-to-$T_c$ value decreases. This behavior is contradictory to $Y{H}_6$ and $Y{H}_9$.

The numerical values of $T_c$ and $R$ for the $Y{H}_7$ superconductor are shown in Figure 5. The parameters were ${\omega }_D=$ 684 K, 1333 K, $u^{*}=0.1, 0.13$ $\chi =300$, $\lambda =0.30$ $Q_e=100$, ${\epsilon }_0=50$, $\beta =5/3$. We lacked $P$ versus $v$ data for the $Y{H}_7$ superconductor, so we employed a linear fit on the $Y{H}_6$ data while assuming that the components of the $Y{H}_7$ superconductor and $Y{H}_6$ superconductor possess the same phase characteristics as those found in the $Y{H}_4$ superconductor and $Y{H}_3$ superconductor [7]. Then, the $P=P(v)$ read out from Ref. [7] was found to be $P(v)=-1149.9v+1160.9$ for $0.68<v<0.86$. The $T_c$ experimental data for the $Y{H}_7$ superconductor were found in a low-temperature region at about $T_c\approx 29–32$ K. Our computation should be appropriately correlated with variables ${\omega }_D=684$ K and $u^{*}>0.13$. However, we did not include the specific number in order to maintain consistency with other Y-H superconductors. It was observed that the $Y{H}_7$ superconductor exhibits a critical temperature when $T_c\approx 33–67$ K, while the gap-to-$T_c$ ratio ($R\approx 3.21–3.68$) was determined. The $Y{H}_7$ phase shows the lowest critical temperature, but the gap-to-$T_c$ ratio is closer to the BCS value than the other.

In our model, the parameter of YH₇ under high pressure had a lower density of state than that of YH₆ and YH₉. Therefore, a lower critical temperature was found in YH₇, which agrees with the calculation and experimental result in Ref. [16]. The higher critical temperature of YH₆ and YH₉ was due to the large contribution of H to the electronic density of states, resulting in a larger density of states [9,22]. The high T_c of Im3m-YH₆ and P6₃/mmc-YH₉ was attributed to their hydrogen cage structure and especially to the large contribution of the H-derived electronic density of states at the Fermi level. This was not successful for the present YH₁₀ to be synthesized [7,34], but YH₁₀ had symmetrical hydrogen cages and a high density of states the same as YH₉. That could be a candidate for the high T_c [22].

4. Conclusions

This research investigated the properties of $Y{H}_9$, $Y{H}_6$, $Y{H}_4$, and $Y{H}_7$ superconductors, specifically focusing on the analysis of $T_c$ and $R$. The obtained results provide insights into the underlying crystal structural similarities seen in $Y{H}_3$. The data of $T_c$ were utilized to determine the pressure parameters, which, in turn, allowed for the estimation of $R$. The gap-to-$T_c$ ratios are 3.76–3.85, 3.57–3.83, 3.79–3.82, and 3.21–3.68 for the $Y{H}_9$, $Y{H}_6$, $Y{H}_4$, and $Y{H}_7$ superconductors, respectively. The phase exhibiting a lower critical temperature $\left(Y{H}_7\right)$ indicates a gap-to-T_c ratio that is closer to the BCS value compared to the other phases. Based on our findings, it may be inferred that the Y-H superconductor exhibits characteristics typically associated with weak-coupling superconductors. Meanwhile, the dominant effect of the Coulomb potential is more pronounced at the critical temperature compared to the gap-to-T_c ratio. Finally, we evaluated the McMillan methods [34,35] that used numerical methods to determine strong electron–phonon coupling and high T_c but which were neglected from the BCS-based mechanism [23,24,25,36,37,38,39] and the anti-adiabatic scenario [40] for all types of superconductors. In our model, we used $\lambda <0.5$, $u^{*}=0.1–0.13$ and ${\omega }_D=684–1333$ K, which ensured weak coupling and were preserved with BCS via the rising ${\omega }_D$.