AIP ADVANCES 6, 125108 (2016)

Identification and properties of the non-cubic phasesof Mg2Pb

Yuwei Li,1,2 Guang Bian,2 and David J. Singh2,a1College of Materials Science and Engineering and Key Laboratory of Automobile Materialsof MOE, Jilin University, Changchun 130012, China2Department of Physics and Astronomy, University of Missouri, Columbia 65211-7010,Missouri, USA

(Received 15 November 2016; accepted 12 December 2016; published online20 December 2016)

Mg2Pb occurs in the cubic fluorite structure and is a semimetal with a band struc-ture strongly affected by spin-orbit interaction on the Pb p states. Its properties aretherefore of interest in the context of topological materials. In addition a differentphase of Mg2Pb was experimentally reported, but its crystal structure and proper-ties remain unknown. Here we determine the structure of this phase using ab initioevolutionary 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 struc-ture supporting the experimental observation. We find this tetragonal phase to be acompenstated anisotropic metal with strong spin orbit effects. Many other metastablestructures have also been identified, especially one orthorhombic structure, spacegroup Pnma, of which energy is 17 meV per atom higher than that of ground statestructure and which perhaps could be the phase that was reported based on sim-ilarity of lattice parameters. © 2016 Author(s). All article content, except whereotherwise 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]

Alkaline earth lead compounds consist of highly electropositive alkaline earth elements andgroup IV elements. It’s convenient to describe the electron counting in terms of a nominal ionicZintl type picture,1 e.g. Mg2+

2 Pb4�. As potential high performance thermoelectrics, alkaline earthlead compounds have attracted a lot of attention. For example, Ca2Pb, Sr2Pb and Ba2Pb, which formin an orthorhombic structure2–4 (space group, Pnma), are all narrow band gap semiconductors andmay have high thermoelectric figure-of-merit ZT values, especially Ca2Pb, as well as correspondingtin compounds.5 Recent works have highlighted the importance of complex band structures, whicharise from spin orbit effects, for thermoelectric performance.6–8 Mg2Pb is a semimetal9 and thereforeuninteresting as a thermoelectric but does show strong spin orbit effects. This is of interest from thepoint of view of topological materials10 and also thermoelectrics if a modification with a band gap isfound. 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 orthorhombicstructure with a slightly off-stoichiomentric nominal composition. They were neither able to solvethis structure due to the small number of lines observed, nor to characterize the properties. A PbCl2structure (space group, Pnma) was tentatively assigned based on analogy with other materials. Herewe investigate crystal structures and phase stability of the unknown phase with first-principles particleswarm optimization structure search method.12,13 We identify a P4/nmm structure whose energy isonly 2 meV/atom (∼24 K) higher than that of ground state fluorite structure. Dynamic stability analysisindicates it is stable. Obviously, the energy difference between it and the ground state structure isalmost indistinguishable and probably it is the unknown structure reported by Eldridge et.al.. Basedon the calculated results, one orthorhombic structure is significantly higher in energy than the ground

aElectronic mail: singhdj@missouri.edu

2158-3226/2016/6(12)/125108/6 6, 125108-1 © Author(s) 2016

125108-2 Li, Bian, and Singh AIP Advances 6, 125108 (2016)

FIG. 1. The low-energy metastable structures P4/nmm (a), C2/m (b), Pnma[2] (c), P21/m (d), Cmmm (e) and Pnma[1] (f)identified by structure searches. The grey spheres represent Pb atoms and the blue spheres represent Mg atoms.

state and has lattice parameters significantly different from those inferred from the diffraction linesreported by Eldridge et.al.. Therefore this is unlikely to be the correct structure. We do find a differentorthorhombic structure that does have lattice parameter in accord with the diffraction lines (denotedPnma[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 thesearch for ambient pressure structures using an unbiased swarm structure search method: CALYPSOmethodology.12,13 In the CALYPSO search for Mg2Pb compounds, Np (Np = 30) random structuresfirstly are generated with symmetry restriction followed by full structure-relaxation. Then the best60% structures of the population are generated again which are regarded as the promising area ofconfiguration space. In order to keep the population diversity, 40% structures of the population aregenerated randomly and must have different symmetries from any of previously generated ones. Thelowest energy structure, Fm3m phase, is predicted successfully.

The structural relaxations were carried out using DFT with the Perdew-Burke-Ernzerhof gener-alized gradient approximation (GGA)14 and the projector-augmented wave method15 as implementedin 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 gridspacing 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 performedusing GGA method14 as implemented in the VASP code.16 Fermi-surfaces were calculated usingGGA method14 as implemented in the WIEN2K code.19 We also did band structure and total energycalculations to cross-check using the Wien2K code.19 These all electron calculations are in accordwith the pseudo potential VASP calculations supporting the parameter choices.

We performed electrical transport property calculations on a regular Γ-centred k-points grid of2π×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 PHONOPYcode.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

125108-3 Li, Bian, and Singh AIP Advances 6, 125108 (2016)

TABLE I. Structural data and density of states (DOS) at Fermi level of the five lowest-energy metastable phases and anotherPnma[1] phase identified from structure searches. The unit of DOS is States/(eV*atom). The enthalpy difference from groundstate is abbreviated as dH and the unit of it is meV/atom.

Space group Lattice WyckoffdH parameters (Å) DOS positions Atoms x y z

P4/nmm a = 4.5396 2a Mg1 0.0000 0.0000 0.00002 c = 7.5278 0.3811 2c Mg2 0.0000 0.5000 0.6790

2c Pb 0.5000 0.0000 0.7195

a = 10.2034 4i Mg1 0.1711 0.5000 0.3244C2/m b = 4.6616 0.3919 2c Mg2 0.0000 0.0000 0.50007 c = 6.5547 2a Mg3 0.0000 0.0000 0.0000

β = 104.4036◦ 4i Pb 0.6546 0.0000 0.8069

Pnma[2] a = 6.5265 Mg1 0.1387 0.7500 0.925510 b = 4.6955 0.4831 4c Mg2 0.6026 0.7500 0.2420

c =9.7956 Pb 0.3489 0.2500 0.0988

Mg1 0.7402 0.7500 0.3198Mg2 0.9262 0.2500 0.7662Mg3 0.0910 0.7500 0.8990

a = 10.2736 Mg4 0.2405 0.2500 0.3494P21/m b = 4.6863 0.4364 2e Mg5 0.4112 0.7500 0.114011 c = 9.7765 Mg6 0.4141 0.7500 0.4471

β = 73.1623◦ Pb1 0.2344 0.2500 0.0259Pb2 0.5635 0.2500 0.2355Pb3 0.0955 0.7500 0.5718

a = 14.2749 4h Mg1 0.3355 0.0000 0.5000Cmmm b = 4.7120 0.4598 2d Mg2 0.0000 0.0000 0.500013 c = 4.4852 2b Mg3 0.0000 0.5000 0.0000

4g Pb 0.1583 0.0000 0.0000

Pnma[1] a = 7.7577 Mg1 0.3710 0.2500 0.552817 b = 4.5080 0.3120 4c Mg2 0.0446 0.7500 0.6676

c = 8.7791 Pb 0.7553 0.2500 0.6359

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 structuresis the Mg-Pb polyhedron in which Pb is coordinated by Mg atoms and these form the frameworksof the structures. The electropositive Mg atoms donate electrons and stabilize lattices via Madelung

FIG. 2. Calculated phonon dispersion curves of the P4/nmm structure.

125108-4 Li, Bian, and Singh AIP Advances 6, 125108 (2016)

FIG. 3. Calculated band structures of the P4/nmm structure. The Fermi level is denoted by a dashed line.

FIG. 4. The Fermi surface for the P4/nmm structure.

potential. Therefore in general these metastable structures belong to the category of Zintl phasematerials.1 The averaged Mg-Pb bond lengths of these structures are 3.14 Å for P4/nmm, 3.25 Å forC2/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 longerthan that of ground states (3.01 Å). Considering the usual overestimation of lattice constants by theDFT-GGA method, the actual bond lengths should be smaller than the real ones.

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

125108-5 Li, Bian, and Singh AIP Advances 6, 125108 (2016)

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/atomhigher than that of ground state structure, as seen in Table I. Absence of any imaginary phononmode under dynamic stability simulation, as seen in Fig. 2, indicates its stability. So it is likely to besynthesizable due to the relatively low energy and dynamic stabilities. The Pb atoms are nine-foldcoordinated by Mg, which is higher than the coordination in the Pnma[1], or Fm3m structure. Therelatively long average Pb-Mg bond length of it, 3.14 Å, suggests weak bonding and very soft phononsas seen in the Fig. 2.

The electronic structure studies show that it is a metallic phase, as seen in Fig. 3. The Fermisurface 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 beexperimentally synthesized. The energies of other metastable structures are higher substantially asseen in Table I. As mentioned, we obtained one orthorhombic structure (Pnma[1]) with lattice parame-ters well matched to that of unknown experimental phase. Based on lattice parameters, the metastablePnma[1] phase may be the unknown phase. In the experiment of Eldridge for off-stoichiomentricMg2Pb, 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 Fm3m lattice parameters betweencalculated in this work and experiment.22 The metastable Pnma[1] phase has higher enthalpy, 17 meVper atom than that of ground state, which amounts to ∼ 210 K. We simulated the dynamic stabilityof this structure. Fig. 6 shows its phonon dispersion curves. The absence of any imaginary phononmode in the whole Brillouin zone shows its lattice stability. It is a PbCl2-type structure.23 The averagePb-Mg bond length is 3.08 Å, which is a little longer than that of ground state, suggestive of weakbonding 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.

125108-6 Li, Bian, and Singh AIP Advances 6, 125108 (2016)

FIG. 8. The DOS of five lowest-energy metastable structures and Pnma[1] structure. The result of the ground-state Fm-3mstructure 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 semimetallicground 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 inthe Fig. 8. Those are all metals in contrast to the semimetallic ground state. This is different fromother alkaline earth lead compounds,5 such as Ca2Pb, Sr2Pb and Ba2Pb, which are semiconductorand potential high performance thermoelectrics.

In conclusion, using first principles structural prediction and electronic structure calculationmethods, we explored the Mg2Pb phases. First principles total energy calculations indicate theenthalpy of one metallic tetragonal (P4/nmm) compound is only 2 meV per atom higher than that ofground 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 wellmatched to that of report diffraction lines of Eldridge et.al. They are both metallic phases.

ACKNOWLEDGMENTS

This work was supported by the Department of Energy, Office of Basic Energy Sciences throughthe MAGICS center, award DE-SC0014607.

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