Thermomechanical stabilization of electron small polarons in $\mathrm{SrTi}{\mathrm{O}}_3$ assessed by the quasiharmonic approximation

We predict a predominance diagram for electron defects in the temperature-hydrostatic stress space for $\mathrm{SrTi}{\mathrm{O}}_3$ by combining density functional theory and the quasiharmonic approximation. We discovered two regimes where small polarons dominate: under tensile stress at lower temperature due to a larger relaxation volume of the defect $\mathrm{\Omega }$, and under compressive stress at higher temperature due to a smaller $\mathrm{\Omega }$ and larger formation entropy. This provides a means to modulate the electronic conductivity via controlling the underlying charge carrier. Furthermore, the results challenge the common association between larger $\mathrm{\Omega }$ and charge localization by demonstrating that at high temperature the free electron can induce larger $\mathrm{\Omega }$ compared to the small polaron. This finding is attributed to the ability of the free electron to generate greater vibrational entropy upon finite isothermal expansion.

Figure 1

Visualization of a spin density isosurface, shown in yellow, of (a) a free electron taken at $0.0018e/{\AA }^3$ and (b) a small polaron taken at $0.0030e/{\AA }^3$. For illustration purposes only, the free-electron spin density was generated using a low accuracy single $k$ point. Accurate calculations used for the rest of the work do not yield a net spin for the free electron inside the PAW sphere. Blue (large), gray (medium), and magenta (small) balls represent Sr, Ti, and O, respectively. Visualization rendered with the software vesta [40].

Figure 2

(a) Predominance map of electronic defects as a function of temperature and pressure in cubic $\mathrm{SrTi}{\mathrm{O}}_3$ calculated quasiharmonically based on self-trapping Gibbs free energy. (b) $G_{\mathrm{self}\text{−}\mathrm{trap}}$ in meV as a function of $T$ and $P$ corresponding to the map in (a). Predominance maps based on other thermodynamic potentials are shown in (c) internal energy, (d) Helmholtz free energy, and (e) enthalpy. In (a), (c), (d), and (e) green and red indicate small polaron and free-electron predominance zones, respectively. The black dashed line in all panels represents the experimental boundary between the cubic and tetragonal phases of $\mathrm{SrTi}{\mathrm{O}}_3$ from Ref. [41]. Cubic $\mathrm{SrTi}{\mathrm{O}}_3$ is stable above the boundary.

Figure 3

Pressure dependence of relaxation volume $\mathrm{\Omega }$ isotherms for (a) free electron and (b) small polaron. (c) The ratio of the relaxation volumes of both defects as a function of $T$ and $P$.

Figure 4

Phonon density of states in arbitrary units (a.u.) for $\mathrm{SrTi}{\mathrm{O}}_3$ containing free electron and small polaron at $T=800\mathrm{K}$ and (a) $P=3\mathrm{GPa}$ and (b) $P=0\mathrm{GPa}$. These plots were obtained by spline interpolation. The black line represents 800 K in energy units of Boltzmann constant $\left(800k_{\mathrm{B}}\right)$. Visualizations of the lengthened (red) and shortened (blue) Ti-O bonds around a small polaron are shown in (c) and (d), respectively. Sr ions were removed for clarity and the rest of the color code is the same as in Fig. 1.