Structural and electronic properties of the alkali metal incommensurate phases

Under pressure, the alkali elements sodium, potassium, and rubidium adopt nonperiodic structures based on two incommensurate interpenetrating lattices. While all elements form the same “host” lattice, their “guest” lattices are all distinct. The physical mechanism that stabilizes these phases is not known, and detailed calculations are challenging due to the incommensurability of the lattices. Using a series of commensurate approximant structures, we tackle this issue using density functional theory calculations. In Na and K, the calculations prove accurate enough to reproduce not only the stability of the host-guest phases, but also the complicated pressure dependence of the host-guest ratio and the two guest-lattice transitions. We find Rb-IV to be metastable at all pressures, and suggest it is a high-temperature phase. The electronic structure of these materials is unique: they exhibit two distinct, coexisting types of electride behavior, with both fully localized pseudoanions and electrons localized in 1D wells in the host lattice, leading to low conductivity. While all phases feature pseudogaps in the electronic density of states, the perturbative free-electron picture applies to Na, but not to K and Rb, due to significant $d$-orbital population in the latter.

Figure 1

Incommensurate structures of the alkali metals, host/guest atoms are indicated by yellow/black circles. (Top) The tetragonal host lattice unit cell common to all elements, seen along the $c$ axis. (Bottom) The four different guest unit cells seen in various elements; the space group and atomic positions are indicated, the $(a,b)$ axes are rotated relative to the upper panel, as indicated therein.

Figure 2

The K-IIIa structure as seen in experiment at 27.7 GPa, along the [001] (left) and the $\left[1\overline{1}0\right]$ directions (right) [9]. Host/guest atoms are denoted by yellow/black spheres. Side view shows multiples of host (x3) and guest unit cells (x5) along the $c$ axis; the incommensurate nature of the two sub-lattices is evident.

Figure 3

Sodium formation enthalpies per atom with respect to the fcc structure, from PBE calculations. For all $tI19$ structures, the enthalpy at the optimal host-guest ratio at every pressure is shown. Inset shows pressure region 140–300 GPa, relative to the Rb-IV-type guest structure of $tI19$.

Figure 4

Upper panel: PBE formation enthalpies of Na-V approximants (using K-IIIa guest lattice), for three pressures, and relative to the value of the 5g/3h structure $\left(c_H/c_G=1.67\right)$. Solid lines are parabolic fits. Lower panel: optimal values for $c_H/c_G$ in Na-V as function of pressure, obtained from parabolic fits and compared to experimental data [30], from both PBE and PBEsol optimizations. Dashed lines are guides to the eye.

Figure 5

PBE enthalpies of formation of various potassium structures relative to fcc as a function of pressure. The inset shows the pressure regime relevant for $tI19$ formation, relative to the Rb-IV-type guest structure of $tI19$. For all $tI19$ phases, the enthalpy at the optimal host-guest ratio is shown.

Figure 6

Optimal $c_H/c_G$ ratio for K-IIIa as a function of pressure, using both PBE and PBEsol. Calculated results (green diamond symbols) are from parabolic fits to approximants' data, and experimental data (red circles) are from Ref. [9]. Dashed lines are guides to the eye.

Figure 7

K-III host lattice ratio $c_H/a_H$ as a function of pressure, as obtained from experiment (red circles, Ref. [9]) and two relevant different approximants (orange: 8g/5h, $c_H/c_G=1.60$; blue: 5g/3h, $c_H/c_G=1.67)$.

Figure 8

PBE enthalpies of formation of various rubidium structures relative to fcc as a function of pressure. Inset shows pressure region where Rb-IV is least unstable, and relative to the Rb-IV structure. For all $tI19$ phases, the enthalpy at the optimal host-guest ratio is shown.

Figure 9

Optimal $c_H/c_G$ ratios for Rb-IV, from both PBE and PBEsol optimisations. Calculated results (green diamonds) are from parabolic fits to approximants' data, experimental data (red circles and error bars) is from Ref. [6]. Dashed lines are guides to the eye.

Figure 10

ELF plots for the 5g/3h approximant of Na-V at 160 GPa, with the K-IIIa guest lattice and host/guest atoms shown as yellow/black spheres. (a) The ELF=0.85 isosurface in orange, seen along (001); red lines mark the positions of cross sections along $\left(\overline{1}10\right)$ and (100) shown in (b,c). These planes contain the “host” ELF maxima (b) and the “channel” ELF maxima (c). ELF values are shown from 0 (blue) to 1 (red).

Figure 11

Projection of the electronic wave functions onto localized $s,p$, and $d$ orbitals for three different structures of Na, K, and Rb. The exact magnitude of the projection depends on the precise definition of the localized orbitals, hence the method counts slightly fewer than six semicore $p$ electrons. Within uncertainty, the data implies that projected densities are independent of crystal structure.

Figure 12

From top to bottom: electronic DOS for Na (8g/5h, 160 GPa), K (8g/5h, 30 GPa), and Rb (5g/3h, 20 GPa) as grey shaded areas, with their Fermi energies (black solid lines). Also shown are the DOS of bcc/fcc phases at the same valence electron densities (blue/orange lines) with their Fermi energies (black dashed lines), and the diffraction peaks of the incommensurate structures (solid red bars) as a function of wave vector $k$ expressed as $E\left(k\right)={\hslash }^2{k}^2/2m_e$. All energies are normalized to coincide with the valence band onset of the incommensurate phases.