Ferroelectric instability and topological crystalline insulating nature in PbPo

We have investigated the lattice instability and the topological property of the electronic structure in PbPo in comparison with other IV-VI semiconductors, SnTe and PbTe. In the conventional exchange-correlation schemes of the density functional theory, the fcc structure of PbPo tends to be unstable in the presence of spin-orbit coupling (SOC) under [111] distortion so as to have a ferroelectric instability. This feature is revealed in the calculated phonon dispersion of PbPo by the phonon softening instability at $\mathbf{k}=\mathrm{\Gamma }$. But, in the modified Becke-Johnson (mBJ) potential scheme, we have shown that the tendency for SOC-driven ferroelectric instability is suppressed, and the fcc structure becomes stabilized. We have demonstrated that PbPo is a semiconductor having a band inversion at $\mathbf{k}=L$, which leads to a topological crystalline insulator (TCI) phase of PbPo, and the TCI and the ferroelectric states coexist for moderate [111] distortions.

Figure 1

(a) NaCl-type crystal structure of PbPo. Black and red lines represent conventional and primitive unit cells, respectively. (b) Bulk and surface Brillouin zones (BZs) for a primitive unit cell of PbPo. A band inversion at $L$ of bulk BZ leads to double Dirac cones at $\overline{X}$. (c) Massless double Dirac cones near $\overline{X}$. The hybridization between double Dirac cones is not allowed along the $\overline{\mathrm{\Gamma }}-\overline{X}$ direction due to crystal mirror symmetry, and so two Dirac points emerge at ${\overline{\mathrm{\Lambda }}}_1$ and ${\overline{\mathrm{\Lambda }}}_2$. (d) Massive double Dirac cones near $\overline{X}$ in the presence of [111] distortion. Since [111] distortion breaks (110) crystal mirror symmetry, the double Dirac cones become massive.

Figure 2

(a) Phonon dispersions of PbPo in the PBEsol without and with the SOC schemes. The SOC makes the $\mathrm{\Gamma }$-point optical phonon modes have imaginary phonon frequencies. The corresponding normal modes are shown in the inset of (b), which corresponds to [111] distortion. (b) Total energy vs [111] distortion. The PBEsol+SOC scheme yields the instability of the NaCl-type structure under [111] distortion, with minimum energy at finite distortion (gray star), and a metal to insulator transition for distortions of larger than 0.16 Å. The PBEsol (without the SOC) and mBJ+SOC schemes, however, yield a stable NaCl-type structure against [111] distortion.

Figure 3

(a) Band structures in the PBEsol and mBJ schemes with the SOC. (b), (c), and (d) are band structures of PbPo along $K-L-\mathrm{\Gamma }$ in PBEsol, PBEsol+SOC, and mBJ+SOC, respectively. Red and blue dots represent Po $p$ and Pb $p$ characters, respectively. Green dots in (d) represent the $G_0{W}_0$ quasiparticle energies.

Figure 4

Bulk and surface band structures of PbPo with (001) termination. (a) Band structure in the mBJ without SOC. Since no band/parity inversion occurs without SOC, no surface states emerge. (b) Band structure in the mBJ+SOC scheme. Since band/parity inversion occurs at $L$, the topologically protected surface states appear near $E_F$. The surface states in (b) are visualized in Fig. 1 in the three-dimensional (3D) version.