Origin of the persistence of soft modes in metallic ferroelectrics

Metallicity and ferroelectriclike polar distortion are mutually noncompatible, and their coexistence in the same system is an intriguing subject of fundamental interest in the field of structure phase transition. However, it is unclear what mechanism may extend the limit of metallicity that allows polar distortion. We investigate the polar instability and soft modes in electron-doped ${\mathrm{PbTiO}}_3$ using linear-response density functional calculations. We find that ferroelectric instability can remarkably sustain up to an electron concentration of $n_e=0.7$ per unit cell, which is beyond the limit that causes the polar catastrophe in ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$. Our study further reveals two discoveries: (i) electron doping can turn nonsoft mode into soft mode, which leads to different microscopic mechanism for ferroelectricity when the system is strongly metallic; (ii) the frequency change $\mathrm{\Delta }\omega /\mathrm{\Delta }{n}_e$ is surprisingly flat at large $n_e$, which is pivotal for the persistence of soft mode and polar distortion at high metallicity. We also provide an interesting physical origin—which is caused by the strong mode-mode interaction—to explain these phenomena, and the finding of this origin may further extend the limit where metallicity and polar distortion coexist.

Figure 1

Phonon eigenvector of the lowest-frequency soft mode (LFSM) at different $n_e$: (a) $n_e=0.0$; (b) $n_e=0.3$; (c) $n_e=0.6$. The phonon eigenvector of the nonsoft $A_{2u}({\mathrm{TO}}_2$) mode at $n_e=0$ is shown in (d). Arrow and its length indicate the vibration direction and amplitude, respectively.

Figure 2

Phonon frequencies at $\mathrm{\Gamma }$ point as a function of $n_e$ in electron-doped ${\mathrm{PbTiO}}_3$. Symbols are the results directly obtained from DFPT calculations; lines that depict the mode evolution between different $n_e$ are obtained by mode projection. Labels of normal modes are given near each curve. The inset shows the mode frequencies for three $n_e$ (i.e., 0, 0.3, 0.6) considered in Table 1.

Figure 3

Projection magnitude $|p_{ml}{|}^2$ for (a) the LFSM at $n_e=0.3$ and (b) the LFSM at $n_e=0.6$. In both cases, the phonon eigenvectors of undoped ${\mathrm{PbTiO}}_3$ are used as the bases (i.e., the horizontal axis) in the mode projection.

Figure 4

Phonon frequencies at $\mathrm{\Gamma }$ point as a function of $n_e$ in electron-doped ${\mathrm{BaTiO}}_3$. Symbols are the results directly obtained from DFPT calculations; lines that depict the mode evolution between different $n_e$ are obtained by mode projection. Labels of normal modes are given near each curve. The inset shows the anticrossings between $A_{2u}({\mathrm{TO}}_1$) and $A_{2u}({\mathrm{TO}}_2$) modes near $n_e=0.2$ and between $E_u({\mathrm{TO}}_1$) and $E_u({\mathrm{TO}}_2$) near $n_e=0.17$.