Orbitally-Resolved Mechanical Properties of Solids from Maximally Localized Wannier Functions

Abstract

We present a technique for partitioning the total energy from a semi-local density functional theory calculation into contributions from individual electronic states in a localized Wannier basis. We use our technique to reveal the key role played by the $s$ and $p$ orbitals of the apical oxygen atoms in a curious elastic anomaly exhibited by ferroelectric PbTiO$_3$ under applied stress, which has so far gone unexplained. Our technique enables new insights into the chemical origins of the mechanical properties of materials, or any property given by an energy derivative.

Figures (4)

• Figure 1: a) Ferroelectric PbTiO3 in the tetragonal $P4mm$ space group with equatorial (O_eq) and apical (O_ap) oxygens highlighted. There is one short and one long Ti-O_ap bond along the tetragonal $c$-axis. b) Variation in the $c$-axis lattice parameter with applied stress along the tetragonal axis ($\sigma_{zz}$) shows significant non-linearity at small stresses. The strain induced by $\sigma_{zz}$ is plotted on the top $x$-axis. c) Electronic ($E_{\rm elec}$, closed circles) and ion-ion ($E_{\rm ext}^{\rm (I-I)}$, open circles) energy differences between equilibrium PbTiO3 and PbTiO3 under applied stress. d) Internal stresses (cf. Eq. \ref{['eq:internal']}) associated with $E_{\rm elec}$ and $E_{\rm ext}^{\rm (I-I)}$ for PbTiO3 in response to an applied stress. Vertical lines at $c = 4.242$ Å denote the computed equilibrium $c$-axis lattice parameter for PbTiO3. In c) and d), the $c$-axis lattice parameters resulting from the applied stress are plotted on the $x$-axis, rather than the applied stress itself.
• Figure 2: (Top) Variations in the orbitally resolved internal stresses (summed over atoms in the primitive cell) felt by the MLWFs associated with Pb, Ti, O_eq, and O_ap in PbTiO3 with applied stress. To show how the structure is changing with applied stress, the $c$-axis lattice parameters resulting from the applied stress are plotted on the $x$-axis, rather than the applied stress itself. Positions A, B, and C denote the $c$-axis lattice parameters for which the plotted O_ap-$2s$ MLWF is shown in the top inset. The circled areas on each image are intended to draw the eye to the Ti atom, in particular the loss of hybridization with the O_ap-$2s$ states as the applied stress increases and the $c$-axis lengthens. (Bottom) Ionically resolved internal stress associated with $E_{\rm ext}^{\rm (I-I)}$. Vertical lines in each plot correspond to the computed equilibrium $c$-axis lattice parameter for PbTiO3. The lines are guides for the eyes.
• Figure 3: (Left) Variation in the $c$-axis lattice parameter of various perovskites in the $P4mm$ space group with applied stress from our first-principles calculations. (Right) The strain value of the elastic anomaly compared with its predicted value using MLWF stress alone. The elastic anomaly is defined from structural data as the inflection point of the stress-strain curve, corresponding to the peak compliance, and is predicted from the O_ap MLWF stress as the (local) minimum. The line is a linear fit to our data and is shown as a guide to the eye.
• Figure 4: (Left) Change in the total energy of cubic paraelectric BaTiO$_3$ ($Pm\bar{3}m$) as a function of the amplitude of the unstable $\Gamma_4^-$ phonon. (Right) Change in energy partitioned into contributions from MLWFs centered on the atoms. Note the difference in energy units between the left and right panels. The lines are guides for the eyes.