Superconductivity by doping in alkali-metal hydrides without applied pressure: An ab initio study

The electronic, lattice dynamical, electron-phonon coupling, and superconducting properties of alkali-metal hydrides LiH, NaH, and KH, metalized through doping with alkaline-earth metals Be, Mg, and Ca, respectively, are investigated within the framework of density functional perturbation theory. The alloys were modeled by the self-consistent virtual crystal approximation, and the effect of zero-point energy contribution is consistently taken into account. For all three alloys, a steady increase of the electron-phonon coupling constant $\lambda$ is found with progressive alkaline-earth metal doping, reaching values as high as 0.47 for (Li/Be)H, 1.26 for (Na/Mg)H, and 1.69 for (K/Ca)H. The growth of $\lambda$ with doping is the result of two effects: the softening of the phonon spectrum, mainly of the H-optical modes, and the increase of the density of states at the Fermi level. Estimates of the superconducting critical temperature reach values of 2.1 K for ${\mathrm{Li}}_{0.95}{\mathrm{Be}}_{0.05}\mathrm{H}$, 28 K for ${\mathrm{Na}}_{0.8}{\mathrm{Mg}}_{0.2}\mathrm{H}$, and even 49 K for ${\mathrm{K}}_{0.55}{\mathrm{Ca}}_{0.45}\mathrm{H}$, demonstrating that doping is an alternative route to high transition temperatures in this material class without the need to apply high external pressure.

Figure 1

${\mathrm{K}}_{1-x}{\mathrm{Ca}}_x\mathrm{H}$ optimized lattice parameter as a function of Ca content for the static and ZPE schemes. Experimental data was taken from Ref. [41].

Figure 2

Electronic band structure and density of states (DOS), calculated within the ZPE scheme, for (a) ${\mathrm{Li}}_{0.95}{\mathrm{Be}}_{0.05}\mathrm{H}$, (b) ${\mathrm{Na}}_{0.8}{\mathrm{Mg}}_{0.2}\mathrm{H}$, and (c) ${\mathrm{K}}_{0.55}{\mathrm{Ca}}_{0.45}\mathrm{H}$.

Figure 3

Evolution of the total density of states at the Fermi level $N\left(E_F\right)$ as a function of the alkaline-earth metal content ($x$) for ${\mathrm{Na}}_{1-x}{\mathrm{Mg}}_x\mathrm{H}$ [42] and ${\mathrm{K}}_{1-x}{\mathrm{Ca}}_x\mathrm{H}$.

Figure 4

Phonon dispersion, linewidths (vertical lines on top of the phonon branches), and phonon density of states (PDOS) for ${\mathrm{Li}}_{0.95}{\mathrm{Be}}_{0.05}\mathrm{H}$, calculated for the ZPE scheme.

Figure 5

Evolution of the phonon dispersion, linewidths, and related PDOS for ${\mathrm{Na}}_{1-x}{\mathrm{Mg}}_x\mathrm{H}$ at (a) $x=0.05$, (b) $x=0.1$, and (c) $x=0.2$ for the ZPE scheme.

Figure 6

Evolution of the phonon dispersion, related linewidths, and PDOS for ${\mathrm{K}}_{1-x}{\mathrm{Ca}}_x\mathrm{H}$ at (a) $x=0.05$, (b) $x=0.25$, and (c) $x=0.45$ within the ZPE scheme.

Figure 7

Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$ and e-ph coupling constant $\lambda \left(\omega \right)$ for ${\mathrm{Li}}_{0.95}{\mathrm{Be}}_{0.05}\mathrm{H}$, calculated under the ZPE scheme.

Figure 8

Evolution of Eliashberg function and $\lambda \left(\omega \right)$ for ${\mathrm{Na}}_{1-x}{\mathrm{Mg}}_x\mathrm{H}$ at (a) $x=0.05$, (b) $x=0.1$, and (c) $x=0.2$ (ZPE scheme).

Figure 9

Evolution of ${\alpha }^{2}F\left(\omega \right)$ and $\lambda \left(\omega \right)$ for the ${\mathrm{K}}_{1-x}{\mathrm{Ca}}_x\mathrm{H}$ alloy at (a) $x=0.05$, (b) $x=0.25$, and (c) $x=0.45$ within the ZPE scheme.

Figure 10

(a) e-ph coupling parameters $\lambda$ and ${\lambda }_{\omega }$ (where the $x$-dependent phonon spectrum was replaced with the one from $x=0.05$), as a function of Mg-content ($x$) for the ${\mathrm{Na}}_{1-x}{\mathrm{Mg}}_x\mathrm{H}$ alloy (ZPE scheme). (b) Ratio $\lambda /N\left(E_F\right)$ for the quantities shown in (a).

Figure 11

(a) e-ph coupling parameters $\lambda$ and ${\lambda }_{\omega }$ (obtained with a fixed phonon spectrum taken from $x=0.05$ instead of the $x$ dependent one), as a function of Ca-content ($x$) for the ${\mathrm{K}}_{1-x}{\mathrm{Ca}}_x\mathrm{H}$ alloy (ZPE scheme). (b) Ratio $\lambda /N\left(E_F\right)$ for the quantities shown in (a).

Figure 12

Estimates for the superconducting critical temperature $T_c$ as a function of alkaline-earth metal content ($x$) calculated with ${\mu }^{*}=0$ (solid lines) and ${\mu }^{*}=0.1$ (dashed lines), respectively, for (a) ${\mathrm{Na}}_{1-x}{\mathrm{Mg}}_x\mathrm{H}$ and (b) ${\mathrm{K}}_{1-x}{\mathrm{Ca}}_x\mathrm{H}$. Results for both the ZPE (blue) and the static scheme (red) are shown for comparison. $T_c$ smaller than 0.1 K, with ${\mu }^{*}=0$, is obtained for $x\left(\text{Ca}\right)<0.1$, and with ${\mu }^{*}=0.1$ for $x\left(\text{Mg}\right)<0.1$ and $x\left(\text{Ca}\right)<0.15$.