The inverse-perovskites have recently attracted attention as candidate materials with quasi-relativistic three-dimensional Dirac electrons . Such systems are currently intensely studied for their potential unconventional electronic properties such as giant linear magneto-resistance, high mobility of charge carriers, new physics in the ultra quantum limit and topologically protected surface states . The inverse-perovskites of the series M3TtO (M = Ca, Sr, Ba, Eu; Tt = tetrel element: Si, Ge,Sn, Pb) crystallize in the anti-perovskite type of structure, with the reverse occupancy of cations and anions to the perovskite structure. Although the series of M3TtO compounds are structurally Zintl phases and expected to be semiconducting, the completely filled p-orbitals of the Tt4– anions, and the empty d-orbitals of the M2+ cations are very close in energy. Indeed the valence and conductance bands are about to touch or slightly overlap and the family of M3TtO representatives therefore are supposed to be narrow gap semiconductors or semimetals. Detailed band structure calculations  showed that indeed band inversion takes place at the Γ-point of the Brillouin zone. The p-and d-orbital derived bands subsequently hybridize opening up a gap of up to 300 meV except along the Γ-X line in the Brillouin zone, which is to first order protected by symmetry properties of the M2+ cation d-orbitals and tetrel p-orbitals at the Fermi energy. This leads to the emergence of six symmetry related Dirac electrons. A more detailed analysis unveils the importance of spin-orbit coupling in combination with higher orbitals in inducing a small mass-gap of a few meV. The magnitude of this Dirac mass term depends on the respective compound . The physical properties of such materials with an almost zero or negative energy gap are known to be extremely sensitive to even minor lattice distortions. Therefore to understand the novel electronic states in these M3TtO compounds, a detailed knowledge of crystal structures over a wide temperature range is highly desirable .
Single crystal X-ray diffraction experiments were performed for all members of the series of inverse-perovskites, M3TtO (M = Ca, Sr, Ba, Eu; Tt = tetrel element: Si, Ge, Sn, Pb) in the temperature range of 500–50 K. For Tt = Sn, Pb, they crystallize as inverted cubic perovskites (space group Pm3m), and their lattice constants are in the range of a = 4.8–5.5 Å. Figure 1, top-right, shows the structure with the M atoms located at the 3d site (½, 0, 0), Tt atoms at 1b (½, ½, ½), and O atoms at 1a (0, 0, 0). The oxygen atoms are at the centers of undistorted M6 octahedra, which are condensed to a 3D arrangement by sharing common corners. The O—M—O bonding angles are fixed by crystal symmetry to 180°, and the Tt atom is located at the center of the unit cell, surrounded by twelve M atoms in the shape of a cuboctahedron; these are the basic structural elements of the perovskite type of structure. However, all of them show distinct anisotropies of the thermal ellipsoids of the M atoms at room temperature, where the ratio q = U22/U11 is always bigger than one (Fig. 1). This behavior vanishes upon cooling for M = Ca, Sr, Eu, and the structures can be regarded as ‘ideal’ cubic perovskites at 50 K. The anisotropies of the displacement ellipsoids are much more enhanced in the case of the Ba compounds (Fig. 1, bottom-right) and the value of q ≈ 20 for Ba3SnO at 50 K indicates a pronounced deviation from the "ideal" perovskite. Indeed, the Ba-containing compounds undergo a structural phase transition at app. 150 K from cubic to orthorhombic (Ibmm). As a consequence, the OBa6 octahedra are tilted, and the O—Ba—O angles show distinct deviations from 180°, with the average angles at 50 K being 173.5° and 174.1° for Ba3SnO and Ba3PbO, respectively.
For the bigger Ba2+ cations, the structural changes are in agreement with smaller tolerance factors t as defined by Goldschmidt (Fig. 2). A similar structural behavior is observed for Ca3TtO. Smaller Tt4–anions (Si, Ge) also give rise to reduced tolerance factors. Both compounds, Ca3SiO and Ca3GeO, with cubic structures at 500 K, have an orthorhombic symmetry (Ibmm) at room temperature. Ca3SiO is the only representative within the M3TtO family where three polymorphs can be found within the temperature range 500–50 K: Pm3m — Ibmm — Pbnm. These structures show tiny differences in the tilting of the OCa6 octahedra, expressed by the O—Ca—O bonding angles of 180° (500 K), ≈174° (295 K), and ≈170° (100 K).
Compounds containing larger M (Sr, Eu, Ba) in combination with smaller Tt (Si, Ge) atoms have tolerance factors significantly smaller than one. These representatives show pronounced deviation from the ‘ideal’ perovskite structure (Fig. 2). The average O—M—O bonding angles, which are a measure of the tilting of the OM6 octahedra, are all about 160°, even at room temperature. They crystallize in the anti-GdFeO3 type of structure (Pbnm) as observed for Ca3SiO at low temperatures, confirming the overall trend of the phase diagram. No phase transitions occur between 500 and 50 K (Fig. 2) in this subgroup of compounds.
Overall our X-ray diffraction studies show that the undistorted, "ideal" cubic inverse-perovskites (Pm3m) are realized with a tolerance factor t ≥ 0.97 (e.g Sr3SnO and Sr3PbO). For those members with a smaller tolerance factor temperature dependent distortion variants are found: Ibmm for 0.93 ≤ t ≤ 0.97 and Pbnm for t ≤ 0.93.
The physical properties of these materials are expected to be exceptional among 3D Dirac electron systems [1,2]. First of all the Dirac electrons are (i) nearly isotropic, (ii) by charge neutrality forced to be at the Fermi energy with (iii) no other band / charge reservoir existing. In order to characterize the physical properties of the inverse-perovskite series we initially concentrated on Sr3PbO. It is a typical member of the family expected to have six Dirac electrons at the Fermi energy and a Dirac mass term of less than 10meV (inset Fig. 3). As-synthesized samples show metallic transport with estimated carrier densities corresponding to a doping of less than 10-5 electrons per unit cell. This is confirmed by our magneto-transport (Fig. 3(a)) and quantum oscillation measurements (Fig. 3(b)). The former first of all shows extremely large magneto-resistance which is furthermore crossing over to the high-field limit at magnetic fields as low as a few tesla. Crucially we observed quantum oscillations in magneto-torque measurements which unveil the existence of small Fermi surface pockets corresponding to a carrier density as small as 3·1017cm-3 with effective masses as small as 0.016 me. Such properties can only be consistent with a lightly doped Dirac electron band structure with the Dirac point being within 50 meV of the Fermi energy.
The existence of near-isotropic isolated Dirac electrons makes Sr3PbO together with the other members of this family an ideal playground for the study of the intrinsic properties of such an exceptional band dispersion. As the chemical composition can have crucial effects on the lattice structure it is expected to be a relevant tuning parameter for both the effective velocities and Dirac-mass of the electronic states in these materials.