Ab initio linear response and frozen phonons for the relaxor $\mathrm{Pb}{\mathrm{Mg}}_{1∕3}{\mathrm{Nb}}_{2∕3}{\mathrm{O}}_3$

We report first principles density functional studies using plane wave basis sets and pseudopotentials and all-electron linear augmented plane wave (LAPW) of the relative stability of various ferroelectric and antiferroelectric supercells of PMN $\left(\mathrm{Pb}{\mathrm{Mg}}_{1∕3}{\mathrm{Nb}}_{2∕3}{\mathrm{O}}_3\right)$ for 1:2 chemical ordering along [111] and [001]. We used linear response with density functional perturbation theory as implemented in the code ABINIT to compute the Born effective charges, electronic dielectric tensors, long wavelength phonon frequencies and longitudinal optic-transverse optic (LO-TO) splittings. The polar response is different for supercells ordered along [111] and [001]. Several polar phonon modes show significant coupling with the macroscopic electric field giving giant LO-TO splittings. For [111] ordering, a polar transverse optic mode with $E$ symmetry is found to be unstable in the ferroelectric $P3m1$ structure and the ground state is found to be monoclinic. Multiple phonon instabilities of polar modes and their mode couplings provide the pathway for polarization rotation. The Born effective charges in PMN are highly anisotropic and this anisotropy contributes to the observed huge electromechanical coupling in PMN solid solutions.

Figure 1

The polyhedral representation of the supercells for [111] FE and [001] AFE structures. The coordination polyhedra around the Mg (light) and Nb (dark) are shown. The Pb atoms are shown as spheres (distinct shade and sizes represent different Wyckoff positions).

Figure 2

Frozen phonon energy surfaces for various ordered supercells. (a) The changes in total energy involving structural changes from $P\overline{3}m1$ AFE to $P3m1$ FE obtained using all electron LAPW and pseudopotential ABINIT. In (b) and (c) the double wells corresponding to the lattice instabilities from $P3m1$ to $Cm\left(E\right)$ and from $Cm\left(E\right)$ to $Cm\left(R\right)$ are shown.

Figure 3

Representation quadric of the Born effective charge tensor as a “charge ellipsoid” generated from the software xtaldraw (Ref. 50). (a) and (c) denote the [111] FE structures, and (b) and (d) are [111] AFE. (a) and (b) give the view with the hexagonal $c$-axis as vertical, and viewed down the $a$-axis, while (c) and (d) show the view in the horizontal $ab$-plane. The lightest shade corresponds to Pb atoms, and the Mg atoms are located midway between the Pb atoms.

Figure 4

(a) The computed zone center phonon frequencies of various ordered supercells and their comparisons with available Raman [star (Ref. 30), triangle (Ref. 8)] and infrared (Ref. 30) (hexagonal TO, triangle LO) data. (b) The polar phonon frequencies and mode assignments in [111] FE structure. The $A_1$ and $E$ modes are both Raman and infrared active. LO1 gives the LO frequencies using the IR oscillator strength (Refs. 43, 44) approach, and LO2 and LO3 give the LO frequencies obtained from linear response of electric field perturbations. LO3 involves symmetrized charge tensors for the oxygens.

Figure 5

The long wavelength frequency distribution (total) and the displacement weighted spectra giving the dynamical contributions from various atoms in the $Cm\left(R\right)$ (a)–(e), $P3m1$ (f)–(j), $P\overline{3}m1$ (k)–(o), and $P4∕mmm$ (p)–(t) structures. These provide a graphical representation of the long wavelength phonon eigenvectors and show that despite the complex structure of PMN, the vibrational contributions from various cations are quite well separated.

Figure 6

(a) The computed infrared oscillator strengths (atomic units: $1\mathrm{a.u.}=253.2638413{\mathrm{m}}^3∕{\mathrm{s}}^2$) resolved along (perpendicular to) the tetragonal axes corresponding to phonon modes with $A_{2u}\left(E_u\right)$ symmetry in the $P4∕mmm$ structure. (b) and (c) The IR oscillator strengths along (perpendicular to) hexagonal axes as dashed (full) lines, and they represent the $A_1\left(E\right)$ and $A_{2u}\left(E_u\right)$ types polar phonon modes in the $P3m1$ and $P\overline{3}m1$ structures, respectively. (d) The oscillator strengths along (perpendicular to) the monoclinic axis in the $Cm\left(R\right)$ structure corresponding to the $A^{″}\left(A^{\prime }\right)$ modes. The associated TO phonon frequencies $\left({\mathrm{cm}}^{-1}\right)$ are indicated on top.