Superconductivity on ScH₃ and YH₃ hydrides: Effects of applied pressure in combination with electron- and hole-doping on the electron-phonon coupling properties

We performed structural optimizations of the fcc (cubic NaCl (B1)) structure ($Fm\bar{3}m$ space group) with a primitive cell of four atoms (one metal and three hydrogen) for the two Sc_1-xM_xH₃ (M=Ca, Ti) and Y_1-xM_xH₃ (M=Sr, Zr) solid solutions at different values of metal M content $x$. For both systems, the equations of state were constructed from the minimum pressure, at which each system becomes stable at the cubic structure, to 100 GPa as maximum pressure value.

For Sc_1-xM_xH₃, the equation of state was determined for concentrations up to $x=0.5(0.4)$ of electron(hole) doping, while for Y_1-xM_xH₃, the range was for concentrations up to $x=0.05(0.4)$ of electron(hole) doping. Electron- and hole-doping thresholds and the minimum pressures where the systems are stable were determined through dynamical instabilities, observed as imaginary frequencies in the phonon dispersion for larger $x$ contents and smaller pressure values. Phonon instabilities in metal hydrides induced by alloying have been reported previously in literature [51, 52, 30], where such dynamical behavior have been related to an increase of the heat of formation, meaning that the solid solutions become less stable.

In Fig. 2, we show the evolution of the equation of state of each metal content $x$. For both systems, Sc_1-xM_xH₃ and Y_1-xM_xH₃, increasing the electron content leads to a monotonous reduction of the volume, at a given pressure, as well as an increment in the minimum stable pressure value. For hole doping, the volume tendency is opposite, while the minimum stable pressure values are not following any specific trend. This behavior indicates a strengthening of the chemical bonding as the electron-content is increased, given by the increment of Zr- and Ti-content, suggesting a hardening of phonon frequencies. Similarly, as the hole-content grows, by the increase of Sr- and Ca-content, the chemical bonding gets weaker, implying a softening of the phonon frequencies. A complete set of structural parameters, that is, the equilibrium volume ($V_{0}$), bulk modulus ($B_{0}$), and its pressure derivative ($B^{{}^{\prime}}_{0}$) are given in Tab. 1 (Appendix A), for both systems, of all M content ($x$) values studied. It is worth noting that our YH₃ results are in excellent agreement with the experimental data reported by Machida et al.[53].

In Fig.3 we show the calculated cohesive energy ($E_{coh}$) of the two systems, within their respective electron and hole stability-range at their corresponding minimum pressure threshold. This quantity is used to characterize the stability of alloys and solid solutions, and is given by the following:

where $E_{sys}^{tot}$ is the total energy of the N_1-xM_xH₃ solid solution at content $x$, while $E_{N}^{a}$, $E_{M}^{a}$, and $E_{H}^{a}$ are the calculated total energies of the isolated atoms N = Y, Sc; M = Sr, Zr, Ca, Ti; and hydrogen, respectively. In general, for the two solid solutions, the hole-doped systems are less stable than their corresponding pristine systems ($x=0$) (the larger the $E_{coh}$ absolute value, the more stable the system is), nevertheless, they are still in the stability range (negative $E_{coh}$). For the case of electron-doped regime, while Y_1-xZr_xH₃ follows the same observed tendency than the hole-doped systems, we found that Sc_1-xTi_xH₃ is more stable than the pristine one, indicating the possibility to synthesize experimentally such solid solutions.

With the optimized lattice parameters and the corresponding equation of state for each system, at different content for their electron- and hole-doping regions, we proceeded to calculate their electronic and lattice dynamical properties for different applied pressure values at each $x$. Furthermore, we are presenting results obtained by the ZPE scheme. While the ZPE effects on the electronic properties are hardly visible, comparing with the static scheme, on the lattice dynamical ones, there is a noticeable softening as a general effect. This tendency comes mainly from the unit cell expansion as the ZPE contribution to the energy is taken into account.

The evolution of the electronic band structure and the density of states at the Fermi level, $N\left(0\right)$, is analyzed in order to evaluate the effects of electron- and hole-content and pressure on the electronic properties of the solid-solutions.

In Fig. 4 we show the band structure for Sc_1-xM_xH₃ and Y_1-xM_xH₃ of the pristine system and of their corresponding threshold electron- (Ti, Zr) and hole- (Ca, Sr) doping content, each of them at the minimum pressure (where the system is dynamically stable) and the maximum pressure considered ($100$ GPa).

It can be seen that the main pressure effect on the band structure, independent of the electron- or hole-content, is a slight increase of the bandwidth. That is, the low-energy bands are push to even lower energy, and the high-energy ones to higher values, while keeping almost unaffected the bands around the Fermi level ($E_{F}$). The band structure of the pristine ScH₃ shows a threefold degenerate state on the L-point (giving place to one hole-like and two electron-like bands) and a twofold degenerate state on the $\Gamma$-point (electron-like bands), both located at $E_{F}$. In comparison, the pristine YH₃ compound shows a twofold state on the L-point, also at the Fermi level, while the twofold state on the $\Gamma$-point is located just above $E_{F}$. As the electron content is increased for ScH₃ (by Ti), the degeneracy at the L-point breaks, giving place to a new twofold state that continue to move far away from $E_{F}$, rising as an electron-like band and a hole-like band. Instead, at $\Gamma$-point the twofold state is intact, moving as a whole to lower energy values, indicating that electronic topological transitions (ETT) take place. In the case of electron-doping for YH₃ (by Zr), the degenerate states at L- and $\Gamma$-points are maintained, while they undergo a minor shift to lower energies. Regarding the hole-content increase on both systems, SH₃ by Ca and YH₃ by Sr, its essential effect is the shift of the band structure to higher energies, without noticeable changes on the degenerate states discussed before.

Analyzing the evolution of $N\left(0\right)$, as a function of M content ($x$) for the minimum and maximum pressure values of both systems, Sc_1-xM_xH₃ and Y_1-xM_xH₃ (see Fig. 5), it can be observed that, in general, $N(0)$ reduces at $x=0.1$ on the hole-doping regime, and in a more drastic way for the ScH₃ case. As the hole-content increases, $N(0)$ starts to rise slowly, getting close to its value at $x=0$ for the YH₃ system. For the electron doping regime, while YH₃ undergoes a minor increase, mainly by the threshold limit on the Zr-content, $N(0)$ of ScH₃ grows rapidly, doubling its pristine value at $x(\mbox{Ti})=0.3$. As $x$ increases even more, the $N(0)$ grow ratio slows considerably, tending to saturate. As the applied pressure raises, $N(0)$ shows an small reduction, which is due to the expand of the bandwidth discussed previously.

The phonon dispersion is presented on Fig. 6, including their respective phonon linewidth $\gamma_{\vec{q}\nu}$ and the phonon density of states (PHDOS), for Sc_1-xM_xH₃ and Y_1-xM_xH₃ at the pristine ($x=0$) and the threshold electron- (Ti,Zr) and hole- (Ca,Sr) contents. In general, for both systems, the optical branches soften slightly as the hole-content increases, while they are shifted to higher frequencies as the electron-content rises. In particular for the high-frequency optical branches, while they shift almost on a rigid way above the frequencies of the pristine systems for the electron-doping case, on the hole-doping solid-solutions they show, in addition to the softening, a small renormalization in the $T_{2g}$ and $T^{1}_{1u}$ optical branches at $\Gamma$. While the mid-frequency region shows subtle renormalizations at $\Gamma$- and L-point for electron-content regime, they are stronger for the same high-symmetry points, in addition to the X-point, for hole-content systems. Regarding the acoustic branches, they remain almost unchanged for electron-doping, whereas for hole-doping they lightly harden. Interestingly, the phonon linewidths $\gamma_{\vec{q}\nu}$ (vertical lines along the phonon branches), mainly localized around $\Gamma$ at the $T_{2g}$ and $T^{1}_{1u}$ optical phonon branches for $x=0$, remain almost unchanged for the hole-doping regime in both alloys, while for the electron-doping regime it slightly reduces in Sc_1-xM_xH₃, and increases in Y_1-xM_xH₃. In general, for both solid solid-solutions, independent of doping-scheme, the observed effect of applied pressure is a rigid hardening of the phonon frequencies and a lifting of phonon anomalies, that leans to instabilities at K and X-point, for the acoustic branches.

With the electronic and the lattice dynamics information, the electron-phonon spectral functions $\alpha^{2}F\left(\omega\right)$ were calculated for the entire range of hole- and electron-content stable regimes in a broad pressure range. The $\alpha^{2}F\left(\omega\right)$ for the threshold electron- and hole-content, as well as the pristine cases, at the minimum pressure where the systems are dynamically stable, can be seen in Fig. 7. As the electron(hole)-content increases on both systems, the high-frequency optical region of the Eliashberg function shifts to higher(lower) frequencies, while the mid-range frequency region shows an opposite behavior, and the acoustical one practically does not shift. For both systems, the weight of the spectral function is incremented as the doping goes from the electron- to the hole-content thresholds, passing through the pristine one ($x=0$). Regarding the pressure effects, in general, for all different $x$ content cases (electron- and hole-doping), at both solid-solutions, the spectra shift to a higher frequency region as the pressure arises, going from the minimum dynamically stable value up to 100 GPa.

As the Eliashberg spectral function determines the electron-phonon coupling parameter $\lambda$ (see Eq. 3), the $\alpha^{2}F(\omega)$ observed shift due to both, doping and applied pressure, has an impact on $\lambda$ as well. The evolution of the electron-phonon coupling constant as a function of frequency, $\lambda(\omega)$, is shown in Fig.7. For the pristine ($x=0$) and the electron-doped Sc_1-xTi_xH₃ solid solution, it can be observed that the main contribution to $\lambda$ comes from the acoustic region. However, for the hole-doped Sc_1-xCa_xH₃, the main contribution comes from the mid-range frequency optical phonons. The behavior of $\lambda(\omega)$ for the Y_1-xM_xH₃ solid solution is slightly different. In this case, both regions, the acoustic and the mid-range frequency optical ones, contribute almost at the same rate to $\lambda(\omega)$. It is worth noting that, for both solid-solutions, the high-frequency optical phonons have marginal contribution to $\lambda$.

In order to analyze the behavior of $\lambda$ as a function of pressure and electron- or hole-content for the solid-solutions, we present it as a color-map plot in Fig. 8. In general, it can be observed that for most of the scanned pressure values, on both doping schemes (electron and hole) $\lambda$ hardly goes beyond $1$, regardless of the solid-solution. It is worth to mention that higher $\lambda$ values are obtained for pressure close to dynamical instabilities at each electron- or hole-threshold content. Interestingly, only for very specific combination of pressure and hole-content it is possible to reach higher $\lambda$ values, like $1.8$ for Sc_0.7Ca_0.3H₃ at $10.8$ GPa, and $2.0$ for Y_0.7Sr_0.3H₃ at $5.8$ GPa. In particular for the latter solid-solution, the region of pressure and hole-content that could provide $\lambda$ values close to $2$ is spread between $x=0.2$ and $0.3$, and around $1$ to $6$ GPa on applied pressure.

In a similar fashion as $\lambda$, the evolution of the Allen-Dynes characteristic phonon frequency, $\omega_{ln}$, as a function of pressure and electron- and hole-content is shown in Fig. 9. It can be observed a hardening as the pressure increases, specially noticeable for the hole-doping regime and more subtle for the electron-doping one, while the lower $\omega_{ln}$ values are located at the pressure and hole-content regions where both solid-solutions present their maximum $\lambda$ values.

The calculated electron-phonon coupling properties were used to obtain estimates for the superconducting critical temperature, $T_{c}$, as a function of applied pressure and content $x$ for both solid solutions. The isotropic Migdal-Eliashberg gap equations where solved numerically with two different values of the Coulomb pseudopotential ($\mu^{*}$): $\mu^{*}=0$ (which provides an upper limit for $T_{c}$) and $0.15$, in order to get an idea of how strong $T_{c}$ could be affected by the $\mu^{*}$ variation. In general, we get the maximum $T_{c}$ at the minimum pressure values were we found dynamically stable solid solutions for each content ($x$) of both regimes, electron- and hole-doping, as can be observed in Fig. 10. Comparing doping regimes, the hole-doping one reports the highest critical temperature, with values of $T_{c}=92.7(67.9)$ K and $T_{c}=84.5(60.2)$ K at $x=0.3$ for Sc_1-xCa_xH₃ and Y_1-xSr_xH₃ respectively, with $\mu^{*}=0(0.15)$. These maximum $T_{c}$ values corresponds to the highest $\lambda$, the lowest $\omega_{ln}$, and a comparatively low $N(0)$ (related to its corresponding electron-doping values). Such behavior shows that the tuning of the lattice dynamics is the path to enhance superconductivity in both systems. It is worth to mention that Y_0.8Sr_0.2H₃, the only system that is dynamically stable at ambient pressure ($0$ GPa), presents $T_{c}=65.4(44.3)$ K for $\mu^{*}=0(0.15)$.

Regarding the pristine systems, we are fully aware that have not been experimental observation of superconductivity for fcc-phase YH₃ and SH₃ compounds. In particular, Kong et al.[14] reported no superconductivity for temperatures above $5$ K for pure-fcc metallic YH₃, at pressures values going from $40$ GPa up to $180$ GPa. Then, in Fig. 11 we show our $T_{c}$ results for ScH₃ and YH₃, as a function of applied pressure, and from there it can be seen that for pressure values above $40$ GPa, $T_{c}$ goes below $5$ K, and gets even smaller as pressure arises, in good agreement with the experimental reports. Additionally, comparing with previous results calculated by Kim et al.[28], it can be observed that the downward tendency is reproduced on both systems.

Finally, to get more insight about the contribution of the different phonon-frequency intervals to the superconductivity state, we have calculated the Rainer and Culleto differential isotope effect coefficient, $\beta\left(\omega\right)$ (Eq. 5), and the partial isotope effect $\alpha(\omega)$ (Eq. 8), which give information on how strong can $T_{c}$ be modified due to an infinitesimal ion-mass change in a specific phonon interval.

In Fig. 12 we present $\beta(\omega)$ and $\alpha(\omega)$ for both solid-solutions at the pressure and hole-content that presents the highest $T_{c}$ ($\mu^{*}=0.15$): Sc_0.7Ca_0.3H₃ at $10.8$ GPa and Y_0.7Sr_0.3H₃ at $5.8$ GPa. We found $\alpha(\omega)$ to be $0.21(0.15)$ at the acoustic interval, and $0.27(0.33)$ for the mid-frequency region (shadow interval) for Sc_0.7Ca_0.3H₃(Y_0.7Sr_0.3H₃), values that contribute to the $41(30)\%$ and $53(65)\%$, respectively, of the total isotope coefficient $\alpha=0.49$, showing the importance of the mid-frequency region to the superconducting state.

In order to get an idea of how these different phonon regions contribute to $T_{c}$, we calculated the critical-temperature change, $\Delta T_{c}$, when a phonon region is suppressed by means of the Bergmann and Rainer formalism[46, 47]:

where $\Delta\alpha^{2}F\left(\omega\right)=\alpha^{2}F^{{}^{\prime}}\left(\omega\right)-\alpha^{2}F\left(\omega\right)$. Here, $\alpha^{2}F\left(\omega\right)$ is the total Eliashberg function and $\alpha^{2}F^{{}^{\prime}}\left(\omega\right)$ is the one with a specific phonon region suppressed. Then, applying it to the same cases as before (the ones with the highest obtained $T_{c}$ for $\mu^{*}=0.15$) we found $\Delta T_{c}$ to be $-18.7(-11.1)$ K for the acoustic region and $-52.3(-56.6)$ K for the mid-frequency region for Sc_0.7Ca_0.3H₃(Y_0.7Sr_0.3H₃), representing the later a reduction in $T_{c}$ of $77(94)\%$ when the mid-frequency region is suppressed in $\alpha^{2}F\left(\omega\right)$.