Ab initio study of Li-Mg-B superconductors

LiB, a predicted layered compound analogous to the ${\mathrm{MgB}}_2$ superconductor, has been recently synthesized via cold compression and quenched to ambient pressure, yet its superconducting properties have not been measured. According to prior isotropic superconductivity calculations, the critical temperature ($T_{\mathrm{c}}$) was expected to be only 10–15 K. Using the anisotropic Migdal-Eliashberg formalism, we show that the $T_{\mathrm{c}}$ may actually exceed 32 K. Our analysis of the contribution from different electronic states helps explain the detrimental effect of pressure and doping on the compound's superconducting properties. In the search for related superconductors, we screened Li-Mg-B binary and ternary layered materials and found metastable phases with $T_{\mathrm{c}}$ close to or even 10–20% above the record 39 K value in ${\mathrm{MgB}}_2$. Our reexamination of the Li-B binary phase stability reveals a possible route to synthesize the LiB superconductor at lower pressures readily achievable in multianvil cells.

Figure 1

Location of the known phases (in black) and some of the hypothetical phases (in red) on the Li-Mg-B ternary phase diagram. The open crossed symbols correspond to metastable phases at (0 K, 0 GPa). The gray dashed lines are the ternary convex hull facet boundaries projected onto the composition plane. The ${\mathrm{LiB}}_{y\approx 0.9}$ compound with incommensurate one-dimensional (1D) B chains and the LiB compound with randomly stacked two-dimensional (2D) B layers are shown with short-period hP15-${\mathrm{Li}}_8{\mathrm{B}}_7$ and hP8-LiB unit cells, respectively.

Figure 2

Formation enthalpies of ${\mathrm{Li}}_x{\mathrm{B}}_{1-x}$ compounds near $x=0.5$ illustrating high-pressure LiB synthesis routes [(a)–(c)] inferred from a previous experiment [24] and [(d)–(f)] proposed in this study. The use of a delithiated ${\mathrm{LiB}}_y$ starting material (${\mathrm{Li}}_{16}{\mathrm{B}}_{15}\approx {\mathrm{LiB}}_{0.94}$ instead of ${\mathrm{Li}}_9{\mathrm{B}}_8\approx {\mathrm{LiB}}_{0.89}$) is expected to lower the LiB formation pressure and lead to a higher fraction of LiB in the resulting multiphase sample (0.52LiB $+ 0.48{\mathrm{L}}_8{\mathrm{B}}_7$ instead of 0.35LiB $+ 0.65{\mathrm{Li}}_6{\mathrm{B}}_5$). The dashed curves are parabolic fits to the DFT formation energies of the ${\mathrm{LiB}}_{y\approx 0.9}$ compounds with linear boron chains. The straight lines are tangents to the parabolas defining the range of ${\mathrm{LiB}}_y$ stability. ${\mu }_{{\mathrm{LiB}}_3}^{\mathrm{Li}}, {\mu }_{\mathrm{bcc}-\mathrm{Li}}^{\mathrm{Li}}$, and ${\mu }_{\mathrm{LiH}}^{\mathrm{Li}}$ are chemical potentials of Li in ${\mathrm{LiB}}_3$, bcc-Li, and LiH, respectively.

Figure 3

Structure and stability of layered Li-Mg-B phases obtained by combining the LiB and ${\mathrm{MgB}}_2$ blocks. To illustrate the relation between the morphological and thermodynamic properties, the compounds are displayed along the four composition lines LiB-MgB-${\mathrm{MgB}}_2$-LiB and ${\mathrm{Li}}_2{\mathrm{MgB}}_4\text{−}{\mathrm{Mg}}_3{\mathrm{B}}_4$ highlighted in Fig. 1, with the intermediate phases marked as fractions of the end-point phases. [(a)–(d)] The distance to the ternary convex hull ($\mathrm{\Delta }{G}_{\mathrm{hull}}$) at different ($T,P$) conditions. [(e)–(h)] The relative Gibbs free energies ($\mathrm{\Delta }G$) of composite structures with respect to the end-point phases (e.g., $\mathrm{\Delta }{G}_{{\mathrm{LiMgB}}_2}=G_{{\mathrm{LiMgB}}_2}-0.5G_{\mathrm{LiB}}-0.5G_{\mathrm{MgB}}$). All end-point and select intermediate compounds are depicted above the panels, with the corresponding compositions marked by the dashed gray lines.

Figure 4

Summary of LiB properties at 0 GPa. (a) Calculated band structure and DOS [states/(eV f.u.)]. The size of the symbols is proportional to the contribution of each orbital character. (b) Calculated phonon dispersion, phonon DOS, and Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$. The phonon branches are broadened by the $e$-ph coupling strength ${\lambda }_{\mathbf{q}\nu }$, showcasing the modes and high-symmetry directions that have the strongest $e$-ph coupling. (c) The distribution of superconducting gap at 4 K on the cross-section Fermi surface plane through $z=0$. (d) Calculated anisotropic superconducting gaps ${\mathrm{\Delta }}_{\mathbf{k}}$ as a function of temperature (red dashed curves are a guide to the eye).

Figure 5

(a) Calculated total and projected DOS at the $E_{\mathrm{F}}$, (b) the total $\lambda$ and partition of the $\lambda$ according to the labeled frequency range, and (c) the anisotropic and isotropic $T_{\mathrm{c}}$ of LiB as a function of pressure.

Figure 6

Summary of MgB properties at 0 GPa. (a) Calculated band structure and DOS [states/(eV f.u.)]. The size of the symbols is proportional to the contribution of each orbital character. (b) Calculated phonon dispersion, phonon DOS, and Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$. The phonon branches are broadened by the $e$-ph coupling strength ${\lambda }_{\mathbf{q}\nu }$, showcasing the modes and directions that have the strongest $e$-ph coupling. (c) The distribution of superconducting gap at 4 K on the cross-section Fermi surface plane through $z=0$. (d) Calculated anisotropic superconducting gaps ${\mathrm{\Delta }}_{\mathbf{k}}$ as a function of temperature (red dashed curves are a guide to the eye).

Figure 7

Summary of ${\mathrm{LiMgB}}_2$ properties. (a) Calculated band structure and DOS [states/(eV f.u.)]. The size of the symbols is proportional to the contribution of each orbital character. (b) Calculated phonon dispersion, phonon DOS, and Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$. The phonon branches are broadened by the $e$-ph coupling strength ${\lambda }_{\mathbf{q}\nu }$, showcasing the modes and directions that have the strongest $e$-ph coupling. (c) The distribution of superconducting gap at 4 K on the cross-section Fermi surface plane through $z=0$. (d) Calculated anisotropic superconducting gaps ${\mathrm{\Delta }}_{\mathbf{k}}$ as a function of temperature (red dashed curves are a guide to the eye).

Figure 8

Comparison of $T_{\mathrm{c}}$ estimated for observed (LiB and ${\mathrm{MgB}}_2$) and metastable (${\mathrm{LiMgB}}_2$ and MgB) layered borides at different levels of theory and different ${\mu }^{*}$ values.