Ferroelectric Instability Under Screened Coulomb Interactions

We explore the effect of charge carrier doping on ferroelectricity using density functional calculations and phenomenological modeling. By considering a prototypical ferroelectric material, ${\mathrm{BaTiO}}_3$, we demonstrate that ferroelectric displacements are sustained up to the critical concentration of 0.11 electron per unit cell volume. This result is consistent with experimental observations and reveals that the ferroelectric phase and conductivity can coexist. Our investigations show that the ferroelectric instability requires only a short-range portion of the Coulomb force with an interaction range of the order of the lattice constant. These results provide a new insight into the origin of ferroelectricity in displacive ferroelectrics and open opportunities for using doped ferroelectrics in novel electronic devices.

Figure 1

DOS of ${\mathrm{BaTiO}}_3$ for different electron doping concentrations $n$ given in units of $e/\mathrm{u}.\mathrm{c}$. The shaded plot is the DOS of undoped ${\mathrm{BaTiO}}_3$. The vertical dashed line denotes the Fermi energy. The inset shows the Thomas-Fermi screening length $\lambda$ as a function of $n$.

Figure 2

$M$-O ($M=\mathrm{Ti}$, Ba) relative displacements in ${\mathrm{BaTiO}}_3$ (a) and the ratio of out-of-plane lattice constant $c$ and in-plane lattice constant $a$ (b) as a function of electron doping concentration $n$. The dashed line indicates the critical value $n_c$.

Figure 3

Occupation numbers for Ti-$3d$ and O-$2p$ orbitals as a function of electron concentration $n$. ${\mathrm{O}}_1$ (${\mathrm{O}}_2$) correspond to O atoms lying in (off) the ${\mathrm{TiO}}_2$ plane.

Figure 4

$M$-O ($M=\mathrm{Ti}$, Ba) relative displacements and phonon frequency of the soft mode at the $\Gamma$ point in cubic ${\mathrm{BaTiO}}_3$ as a function of electron concentration. The negative sign of the frequency indicates an imaginary value of the frequency.

Figure 5

$M$-O ($M=\mathrm{Ti}$, Ba) relative displacements (in relative units) in cubic ${\mathrm{BaTiO}}_3$ as predicted by the phenomenological model (solid lines) and DFT calculation (open symbols). The latter are the same as those in Fig. 4 but plotted versus $\lambda /{\lambda }_c$ according to the Thomas-Fermi relationship between $\lambda$ and $n$ displayed in the inset of Fig. 1. The inset shows the total energy versus relative polarization for different values of $\lambda$, as follows from the phenomenological model.