Identification and properties of the non-cubic phases of Mg2Pb

Mg2Pb occurs in the cubic fluorite structure and is a semimetal with a band structure strongly affected by spin-orbit interaction on the Pb p states. Its properties are therefore of interest in the context of topological materials. In addition a different phase of Mg2Pb was experimentally reported, but its crystal structure and properties remain unknown. Here we determine the structure of this phase using ab initio evolutionary methods and report its properties. The energy of one tetragonal phase, space group P4/nmm, is 2 meV per atom higher than that of the ground state structure supporting the experimental observation. We find this tetragonal phase to be a compenstated anisotropic metal with strong spin orbit effects. Many other metastable structures have also been identified, especially one orthorhombic structure, space group Pnma, of which energy is 17 meV per atom higher than that of ground state structure and which perhaps could be the phase that was reported based on similarity of lattice parameters. © 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4972957]

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 Mg 2 Pb 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. 12,13In the CALYPSO search for Mg 2 Pb 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, Fm 3m 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 method 15 as implemented in the VASP code. 16We choose 5d 10 6s 2 6p 2 and 3s 2 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×0.024Å 1 for structure relaxation.This produced well converged enthalpies.
As the 3 P 0 3 P 2 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 method 14 as implemented in the VASP code. 16Fermi-surfaces were calculated using GGA method 14 as implemented in the WIEN2K code. 19We also did band structure and total energy calculations to cross-check using the Wien2K code. 19These 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×0.016Å 1 using the Boltztrap code. 20The phonon dispersions of predicted P4/nmm and Pnma [1] phases are calculated by the supercell finite difference method as implemented in the PHONOPY code. 21e 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 TABLE I. Structural data and density of states (DOS) at Fermi level of the five lowest-energy metastable phases and another Pnma [1] phase identified from structure searches.The unit of DOS is States/(eV*atom).The enthalpy difference from ground state is abbreviated as dH and the unit of it is meV/atom.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 potential.Therefore in general these metastable structures belong to the category of Zintl phase materials. 1The 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 P2 1 /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.FIG. 6. Calculated phonon dispersion curves of the Pnma [1] structure.
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 Fm 3m 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 Mg 2 Pb, 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 Fm 3m lattice parameters between calculated in this work and experiment. 22The 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 PbCl 2 -type structure. 23The 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.FIG. 7. Calculated band structures of the Pnma [1] structure.The Fermi level is denoted by a dashed line.FIG. 8.The DOS of five lowest-energy metastable structures and Pnma [1] structure.The result of the ground-state Fm-3m structure is shown for comparison (in black line).Fermi level is at the energy zero.
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 Ca 2 Pb, Sr 2 Pb and Ba 2 Pb, which are semiconductor and potential high performance thermoelectrics.
In conclusion, using first principles structural prediction and electronic structure calculation methods, we explored the Mg 2 Pb 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.

FIG. 5 .
FIG.5.The electric transport properties (σ/τ) of P4/nmm structure as a function of the carriers concentration along XX, YY and ZZ direcitons respectively.Where σ is conductivity and τ is the inverse scattering rate.