Identification and properties of the non-cubic phases of Mg₂Pb

Alkaline earth lead compounds consist of highly electropositive alkaline earth elements and group IV elements. It’s convenient to describe the electron counting in terms of a nominal ionic Zintl type picture,¹ e.g. $Mg22+$Pb⁴⁻. As potential high performance thermoelectrics, alkaline earth lead compounds have attracted a lot of attention. For example, Ca₂Pb, Sr₂Pb and Ba₂Pb, which form in an orthorhombic structure^2–4 (space group, Pnma), are all narrow band gap semiconductors and may have high thermoelectric figure-of-merit ZT values, especially Ca₂Pb, as well as corresponding tin compounds.⁵ Recent works have highlighted the importance of complex band structures, which arise from spin orbit effects, for thermoelectric performance.^6–8 Mg₂Pb is a semimetal⁹ and therefore uninteresting as a thermoelectric but does show strong spin orbit effects. This is of interest from the point of view of topological materials¹⁰ and also thermoelectrics if a modification with a band gap is found. In this regard, Eldridge et.al. reported a different phase, which however was not characterized.¹¹ Specifically they found a new phase with diffraction lines that are consistent with an orthorhombic structure with a slightly off-stoichiomentric nominal composition. They were neither able to solve this structure due to the small number of lines observed, nor to characterize the properties. A PbCl₂ structure (space group, Pnma) was tentatively assigned based on analogy with other materials. Here we investigate crystal structures and phase stability of the unknown phase with first-principles particle swarm optimization structure search method.^12,13 We identify a P4/nmm structure whose energy is only 2 meV/atom (∼24 K) higher than that of ground state fluorite structure. Dynamic stability analysis indicates it is stable. Obviously, the energy difference between it and the ground state structure is almost indistinguishable and probably it is the unknown structure reported by Eldridge et.al.. Based on the calculated results, one orthorhombic structure is significantly higher in energy than the ground state and has lattice parameters significantly different from those inferred from the diffraction lines reported by Eldridge et.al.. Therefore this is unlikely to be the correct structure. We do find a different orthorhombic structure that does have lattice parameter in accord with the diffraction lines (denoted Pnma^[1], below) while this structure has even higher energy and briefly discuss its properties.

Our search for new bulk Mg₂Pb compounds included cells with up to 24 atoms. We did the search for ambient pressure structures using an unbiased swarm structure search method: CALYPSO methodology.^12,13 In the CALYPSO search for Mg₂Pb compounds, N_p (N_p = 30) random structures firstly are generated with symmetry restriction followed by full structure-relaxation. Then the best 60% structures of the population are generated again which are regarded as the promising area of configuration space. In order to keep the population diversity, 40% structures of the population are generated randomly and must have different symmetries from any of previously generated ones. The lowest energy structure, $Fm3¯m$ phase, is predicted successfully.

The structural relaxations were carried out using DFT with the Perdew-Burke-Ernzerhof generalized gradient approximation (GGA)¹⁴ and the projector-augmented wave method¹⁵ as implemented in the VASP code.¹⁶ We choose 5d¹⁰6s²6p² and 3s² as valence electrons for Pb and Mg, respectively. We used energy cutoff of 600 eV for the plane-wave expansions and a Brillouin zone integration grid spacing of $2π×$0.024 Å⁻¹ for structure relaxation. This produced well converged enthalpies.

As the ³P₀−³P₂ spin-orbit splitting is very large for Pb (1.32 eV),^17,18 spin-orbit coupling (SOC) is included in band structures and density of states (DOS) calculations, which were both performed using GGA method¹⁴ as implemented in the VASP code.¹⁶ Fermi-surfaces were calculated using GGA method¹⁴ as implemented in the WIEN2K code.¹⁹ We also did band structure and total energy calculations to cross-check using the Wien2K code.¹⁹ These all electron calculations are in accord with the pseudo potential VASP calculations supporting the parameter choices.

We performed electrical transport property calculations on a regular $Γ$-centred k-points grid of $2π×$0.016 Å⁻¹ using the Boltztrap code.²⁰ The phonon dispersions of predicted P4/nmm and Pnma^[1] phases are calculated by the supercell finite difference method as implemented in the PHONOPY code.²¹ 

We begin with the identification of unknown experimental structure based on structure prediction. In addition to the ground state structure, many low-energy metastable structures are searched, for example five lowest-energy metastable structures and another Pnma^[1] structure as seen in the Fig. 1, and their explicit structural information is listed in Table I. The basic motif forming this structures is the Mg-Pb polyhedron in which Pb is coordinated by Mg atoms and these form the frameworks of the structures. The electropositive Mg atoms donate electrons and stabilize lattices via Madelung potential. Therefore in general these metastable structures belong to the category of Zintl phase materials.¹ The averaged Mg-Pb bond lengths of these structures are 3.14 Å for P4/nmm, 3.25 Å for C2/m, 3.12 Å for Pnma^[2], 3.23 Å for P2₁/m, 3.26 Å for Cmm and 3.08 Å for Pnma^[1] and all longer than that of ground states (3.01 Å). Considering the usual overestimation of lattice constants by the DFT-GGA method, the actual bond lengths should be smaller than the real ones.

An interesting metastable structure is the P4/nmm phase of which energy is only 2 meV/atom higher than that of ground state structure, as seen in Table I. Absence of any imaginary phonon mode under dynamic stability simulation, as seen in Fig. 2, indicates its stability. So it is likely to be synthesizable due to the relatively low energy and dynamic stabilities. The Pb atoms are nine-fold coordinated by Mg, which is higher than the coordination in the Pnma^[1], or $Fm3¯m$ structure. The relatively long average Pb-Mg bond length of it, 3.14 Å, suggests weak bonding and very soft phonons as seen in the Fig. 2.

The electronic structure studies show that it is a metallic phase, as seen in Fig. 3. The Fermi surface is shown in Fig. 4. It shows significant anisotropy, which is reflected in the transport functions, as seen in Fig. 5.

Based on the energies, P4/nmm is the metastable non-ground state structure most likely to be experimentally synthesized. The energies of other metastable structures are higher substantially as seen in Table I. As mentioned, we obtained one orthorhombic structure (Pnma^[1]) with lattice parameters well matched to that of unknown experimental phase. Based on lattice parameters, the metastable Pnma^[1] phase may be the unknown phase. In the experiment of Eldridge for off-stoichiomentric Mg₂Pb, the lattice parameter errors between Pnma^[1] and unknown phase are only 1.3% for a, 1.0% for b and 0.5% for c. This is very similar to the error (1.4%) of $Fm3¯m$ lattice parameters between calculated in this work and experiment.²² The metastable Pnma^[1] phase has higher enthalpy, 17 meV per atom than that of ground state, which amounts to ∼ 210 K. We simulated the dynamic stability of this structure. Fig. 6 shows its phonon dispersion curves. The absence of any imaginary phonon mode in the whole Brillouin zone shows its lattice stability. It is a PbCl₂-type structure.²³ The average Pb-Mg bond length is 3.08 Å, which is a little longer than that of ground state, suggestive of weak bonding and hence soft phonons as seen in Fig. 6.

As seen in Fig. 7, it is metallic phase and has larger DOS at the Fermi level than semimetallic ground state (see below). So it should have more excellent electrical transport properties.

We show the DOS of five lowest-energy metastable structures and Pnma^[1] structure as seen in the Fig. 8. Those are all metals in contrast to the semimetallic ground state. This is different from other alkaline earth lead compounds,⁵ such as Ca₂Pb, Sr₂Pb and Ba₂Pb, which are semiconductor and potential high performance thermoelectrics.

In conclusion, using first principles structural prediction and electronic structure calculation methods, we explored the Mg₂Pb phases. First principles total energy calculations indicate the enthalpy of one metallic tetragonal (P4/nmm) compound is only 2 meV per atom higher than that of ground state structure and it may be synthesized. There is also an orthorhombic structure, Pnma^[1], with energy 17 meV per atom higher than that of ground state structure and lattice parameters well matched to that of report diffraction lines of Eldridge et.al. They are both metallic phases.