Ferroelectricity of stress-free and strained pure $\mathrm{SrTi}{\mathrm{O}}_3$ revealed by ab initio calculations with hybrid and density functionals

The properties of stress-free and biaxially strained stoichiometric $\mathrm{SrTi}{\mathrm{O}}_3$ in the absence and presence of antiferrodistortive (AFD) distortion were calculated ab initio. To obtain reliable results, multiple exchange correlation (XC) functionals, including the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional, were used. HSE was the primary XC functional, while another functional provided a good lower bound of ferroelectricity (FE). The reliability of the calculations was further reinforced by the calculations of the strain and AFD dependence. In contrast with previous works, we show that the ferroelectric phase ($C_{4\mathrm{v}}^1$, $C_{4\mathrm{v}}^{10}$) is more stable in the absence of quantum and thermal fluctuations than the paraelectric phase ($O_{\mathrm{h}}^1$, $D_{4\mathrm{h}}^{18}$), even for the stress-free case, and clarify the properties of these FE phases. The energy gain of stress-free FE, in comparison with the thermal and quantum fluctuation energy, indicates that a paraelectric phase emerges at room temperature by thermal fluctuations, but is near 0 K marginally close to the FE phase, which aligns with the experimental incipient FE. This implies that the paraelectricity of stress-free $\mathrm{SrTi}{\mathrm{O}}_3$ in experiments contains incoherent atomic-scale FE regions. These results are consistent with the FE microregions (FMR), signatures of polar disorders, and the emergence of FE in $\mathrm{SrTi}{\mathrm{O}}_3$ due to impurities and defects. The value of spontaneous polarization $P_{\mathrm{S}}$ could reach $10\mu \mathrm{C}/\mathrm{c}{\mathrm{m}}^2$ in the absence of fluctuations, even for the stress-free case. In view of the earlier theory of the carrier layer at polar discontinuities, the present results may explain the conduction at the interfaces of the $\mathrm{LaAl}{\mathrm{O}}_3/\mathrm{SrTi}{\mathrm{O}}_3$. In addition, an “enhancement of FE due to symmetry constriction” is proposed as an additional mechanism to the strain-enhanced FE in the epitaxial effect. For large compressive strain, e.g., 2%, the Perdew-Burke-Ernzerhof density functional for solids (PBEsol) yielded properties with $P_{\mathrm{S}}>20\mu \mathrm{C}/\mathrm{c}{\mathrm{m}}^2$, agreeing with HSE, and therefore is usable as a practical substitute of HSE for $\mathrm{SrTi}{\mathrm{O}}_3$.

Figure 1

(a) Top and (b) side schematic view of 105 K PE phase $\mathrm{SrTi}{\mathrm{O}}_3{D}_{4\mathrm{h}}^{18}$ ($I4/mcm$) unit cell, showing the AFD distortion of octahedral oxygens around the $c$ axis with rotation angle φ. Orange square frames show a “pseudocubic unit cell,” and the letters $a_{\mathrm{pc}}$ and $c_{\mathrm{pc}}$ indicates the $a$ and $c$ axis of the pseudocubic unit cell, respectively. The $z$ axis is parallel to the $c$ axes. Large and small spheres and diamonds represent Sr, O, and Ti atoms, respectively, where Ti atoms are indicated with “Ti.” Two inequivalent positions of O atoms are indicated by “O1” and “O2.”

Figure 2

(a) Comparison of theoretical lattice constant of RT PE phase $\mathrm{SrTi}{\mathrm{O}}_3$ (cubic: $O_{\mathrm{h}}^1$) with experimental average lattice constant of pseudocubic unit cell $a^{2/3}{c}^{1/3}$. Filled red circles, filled green diamonds, dark half-filled green diamonds, light blue squares, purple triangles, and inverted violet triangles represent the results of PBEsol, HSE, LDA, TPSS+$U$, and PBE, respectively. This convention for symbols is used throughout. The lines represent experimental values [23, 24, 25]. (b) Enlarged plot. In (a) and (b), the experimental average lattice constants of the 105 K PE phase ($D_{4\mathrm{h}}^{18}$) are compared with the theoretical ones of the $O_{\mathrm{h}}^1$ PE phase. This is because the $D_{4\mathrm{h}}^{18}$ PE phase is very close to the $O_{\mathrm{h}}^1$ PE phase having the ratio of the experimental $c$- and $a$-lattice constant <1.0016 [23, 25], and the effect of the unit-cell volume is known to dominate in the results of ab initio calculations. In Appendix pp1, the experimental lattice constants of the $D_{4\mathrm{h}}^{18}$ PE phase are compared with those theoretical ones.

Figure 3

Strain $u$ dependence of RT phases: (a) $P_{\mathrm{S}}//c$ vs $u$, (b) $c$-lattice constant vs $u$, (c) $c/a$ ratio vs $u$, and (d) $V/V_0$ vs $u$, where $V/V_0$ is the unit-cell volume normalized by that of the stress-free cubic ($O_{\mathrm{h}}^1$) phase. In (a), the vertical dash-dotted lines represent experimental critical strain at 2 K below which FE ($P_{\mathrm{S}}//c$) appears by a uniaxial stress along ${\left[1,1,0\right]}_{\mathrm{pc}}$ [28]. Filled red circles, filled green diamonds, half-filled dark green diamonds, and light blue squares represent the results of PBEsol, HSE, HSEsol, and LDA, respectively. These LDA results in (a) agree with the previous LDA ones [39, 40, 41] shown by open black asterisks [40] and black plus signs [39]. In (b) and (c), filled purple asterisks represent experimental values [23, 50]. The red circle and green diamond encircled by blue open circles in (a) represent $P_{\mathrm{S}}$ of the lattice for the minimum free energy under the stress-free condition by PBEsol and HSE, respectively, and are at $u<0$ because of the spontaneous strain.

Figure 4

Strain $u$ dependence (105 K phases) of (a) $P_{\mathrm{S}}//c$. (b) Difference of the free energy per pseudocubic cell from that of stress-free PE phase ($D_{4\mathrm{h}}^{18}$) $\mathrm{\Delta }{F}_{\mathrm{pc}}$. (c) AFD rotation angle φ and (d) $c$-lattice constant. Filled green diamonds and filled red circles represent HSE and PBEsol results, respectively. Small filled green diamond at $u=0$ in (b)–(d) represents stress-free PE phase ($D_{4\mathrm{h}}^{18}$) calculated with HSE. Vertical dash-dotted lines in (a) represent experimental critical strain at 2 K below which FE ($P_{\mathrm{S}}//c$) appears by a uniaxial stress along ${\left[1,1,0\right]}_{\mathrm{pc}}$ [28]. The green diamond encircled by a blue open circle in (a) represents $P_{\mathrm{S}}$ for the minimum free energy by HSE under the stress-free condition and is at $u<0$ because of the spontaneous strain.

Figure 5

Analyses of free energy (a) Δ$F$ vs $u$ for the RT-phase $\mathrm{SrTi}{\mathrm{O}}_3$ and $\mathrm{BaTi}{\mathrm{O}}_3$, where Δ$F$ is the change of free energy from the stress-free $O_{\mathrm{h}}^1$ PE phase. The orange solid line shows the fitting with $\mathrm{\Delta }F=k_{\mathrm{v}}{u}^2/2$ to the $\mathrm{SrTi}{\mathrm{O}}_3$ data. (b) $\mathrm{\Delta }{F}_{\mathrm{f}-\mathrm{p}}^{\mathrm{f}}$ vs $u$ of the RT and the 105 K phase $\mathrm{SrTi}{\mathrm{O}}_3$, where $\mathrm{\Delta }{F}_{\mathrm{f}-\mathrm{p}}^{\mathrm{f}}$ is free-energy difference between the FE and PE phases that have the same $a$ and $c$ lattice constants. The meanings of the symbols for $\mathrm{SrTi}{\mathrm{O}}_3$ are the same as Fig. 3. For the RT phases, the internal coordinates of the PE phase ($D_{4\mathrm{h}}^1$) at $u<0$ are the same as those of the $O_{\mathrm{h}}^1$ PE phase. For the 105 K phases, the internal coordinates of the PE phase ($D_{4\mathrm{h}}^{18}$) at $u<0$ are the same as those of the stress-free $D_{4\mathrm{h}}^{18}$ phase. In an enlarged view in the inset of (b), the stress-free RT phase at the free-energy minimum ($C_{4\mathrm{v}}^1$) calculated with PBEsol and HSE are encircled by a pink and a green open circle, respectively. The stress-free 105 K phase at the free-energy minimum ($C_{4\mathrm{v}}^{10}$) calculated with HSE is encircled by dark green open circles. (c) The difference of free energy per pseudocubic cell from that of the stress-free PE phase ($D_{4\mathrm{h}}^{18}$) $\mathrm{\Delta }{F}_{\mathrm{pc}}$ vs Ti displacement of the Slater-mode distortion from the PE phase position calculated with HSE for 105 K phase $\mathrm{SrTi}{\mathrm{O}}_3$. In these frozen phonon calculations, four Ti's, eight O2's, and four O1's in Fig. 1 are displaced along the $z$ axis with other atoms being fixed at the $D_{4\mathrm{h}}^{18}$ positions. The maximum $|\mathrm{\Delta }{F}_{\mathrm{pc}}|$ is smaller than $|\mathrm{\Delta }{F}_{\mathrm{f}-\mathrm{p}}^{\mathrm{f}}|$ of the stress-free condition (∼0.74 meV) in Fig. 5 because the lattice constants and φ are fixed at the values of the $D_{4\mathrm{h}}^{18}$ PE phase (i.e., not relaxed). The ratio of displacements of Ti, O2, and O1 are 1 : −3.71 : −3.95. Solid lines are fitting to the data.

Figure 6

(a) $\mathrm{\Delta }{F}_{\mathrm{f}-\mathrm{p}}^{\mathrm{f}}$ vs $P_{\mathrm{S}}//c$ for $\mathrm{SrTi}{\mathrm{O}}_3$, where $\mathrm{\Delta }{F}_{\mathrm{f}-\mathrm{p}}^{\mathrm{f}}$ is the free-energy difference between the FE and PE phases that have mutually the same $a$- and $c$-lattice constants. The internal coordinates of the PE phase are the same as those explained in Fig. 5. (b) $\mathrm{\Delta }{F}_{\mathrm{f}-\mathrm{p}}^{\mathrm{f}}$ vs $P_{\mathrm{S}}$ for $\mathrm{BaTi}{\mathrm{O}}_3$, with $\mathrm{\Delta }{F}_{\mathrm{f}-\mathrm{p}}$ vs $P_{\mathrm{S}}$ calculated with six different XC functionals (orange filled squares), where $\mathrm{\Delta }{F}_{\mathrm{f}-\mathrm{p}}$ is the free-energy difference between the stress-free FE phase and stress-free PE phase [49]. The blue solid lines show $\mathrm{\Delta }{F}_{\mathrm{f}-\mathrm{p}}$ by the GL theory for $\mathrm{BaTi}{\mathrm{O}}_3$ [49]. In (a) and (b), red filled circles, green filled diamonds, purple filled triangles, and dark-green half-filled diamonds represent PBEsol, HSE, TPSS+$U$, and HSEsol for $\mathrm{\Delta }{F}_{\mathrm{f}-\mathrm{p}}^{\mathrm{f}}$ of $\mathrm{SrTi}{\mathrm{O}}_3$ or RT phase $\mathrm{BaTi}{\mathrm{O}}_3$. In (a), small dark-green filled diamonds represent $\mathrm{\Delta }{F}_{\mathrm{f}-\mathrm{p}}^{\mathrm{f}}$ calculated with HSE for 105 K phases. The black dashed lines represent fitting to the data. These results show that $\mathrm{\Delta }{F}_{\mathrm{f}-\mathrm{p}}^{\mathrm{f}}$ is a measure of the energy gain of the FE phase with $P_{\mathrm{S}}//c$.

Figure 7

AFD rotation angle φ dependence (105 K phases) of (a) $a$- and $c$-lattice constant, (b) Δ$F$ (change of free energy from the PE phase at $\phi =0$), (c) difference of free energy between PE and FE phases with $P_{\mathrm{S}}//c$ at the same φ ($\mathrm{\Delta }{F}_{\mathrm{f}-\mathrm{p}}$), and (d) $P_{\mathrm{S}}$. Large red and small pink filled circles represent the PE and FE phases ($P_{\mathrm{S}}//c$) calculated with PBEsol, respectively. Filled green diamonds in (a) represent the PE phase calculated with HSE. Large light-blue and small blue filled squares in (b) and (d) represent the PE and FE phases ($P_{\mathrm{S}}//c$) calculated with LDA, respectively. Asterisks represent experiments [23, 25]. The red dashed and the green dash-dotted vertical lines show φ at the minimum Δ$F$ by PBEsol and the experimental φ, respectively. The lattice constants at the free-energy minimum by PBEsol are encircled by red open circles in (a) and are much more different from experimental ones than those given by HSE. The lattice constants by LDA are outside of the range of the plot of (a).

Figure 8

Correlation of $P_{\mathrm{S}}$ with atomic displacements (a) $\mathrm{\Delta }{z}_{\mathrm{Ti}}$, (b) $\mathrm{\Delta }{z}_{\mathrm{Ti}-\mathrm{O}2}$, and (c) $\mathrm{\Delta }{z}_{\mathrm{Ti}-\mathrm{O}1}$ in stress-free RT and l05 K phase $\mathrm{SrTi}{\mathrm{O}}_3$. Sr atom is at (0, 0, 0). The displacements of Ti, O1, and O2 ($\mathrm{\Delta }{z}_{\mathrm{Ti}}$, $\mathrm{\Delta }{z}_{\mathrm{O}1}$, and $\mathrm{\Delta }{z}_{\mathrm{O}2}$) are measured from the PE phase ($O_{\mathrm{h}}^1$, $D_{4\mathrm{h}}^{18}$) position of each ion along the $c$ axes ($z$ axis) in fractions of the $c$-lattice constant of a cubic or a pseudocubic unit cell, i.e., $c=1$. O1 and O2 are defined in Fig. 1. $\mathrm{\Delta }{z}_{\mathrm{Ti}-\mathrm{O}2}$ and $\mathrm{\Delta }{z}_{\mathrm{Ti}-\mathrm{O}1}$ are the differences between $\mathrm{\Delta }{z}_{\mathrm{Ti}}$ and $\mathrm{\Delta }{z}_{\mathrm{O}2}$ and between $\mathrm{\Delta }{z}_{\mathrm{Ti}}$ and $\mathrm{\Delta }{z}_{\mathrm{O}1}$, respectively. Filled light-blue squares, filled red circles, filled green diamonds, half-filled dark-green diamonds, small filled purple circles, and small filled dark-green diamonds represent the results of LDA, PBEsol, HSE, and HSEsol for the RT phase, and PBEsol and HSE for the 105 K phase, respectively. The open purple circle represents the PBEsol results for the 105 K phase with the AFD angle fixed at $2.{26}^{\circ }$. The straight lines are a guide to the eye.

Figure 9

Distribution of valence charge density ρ of PE phase $\mathrm{SrTi}{\mathrm{O}}_3$ calculated ab initio with (a)–(c) PBEsol, (d)–(f) HSE, and (g)–(i) LDA. The first and second rows [(a),(d),(g),(b),(e), and (h)] show the RT PE phase ($O_{\mathrm{h}}^1$), and the third row [(c),(f), and (i)] shows the 105 K PE phase ($D_{4\mathrm{h}}^{18}$). The first and third rows show ρ in the [100] plane, and the second row shows ρ in the [110] plane. The right bar shows the color code, where ρ increases from blue to red. The red color represents $\rho >0.5e/{\AA }^3$ in the [100] planes [(a), (d),(g), (c),(f), and (i)] and $\rho >0.1e/{\AA }^3$ in the [110] planes [(b),(e), and (h)]. O1 and O2 are oxygen in the TiO2 and SrO layers, respectively [Fig. 1]. The open dashed white squares show the area where the difference between the XC functional is clearly visible.

Figure 10

Schematic explanation of strain-induced $P_{\mathrm{S}}$ in the strained $\mathrm{SrTi}{\mathrm{O}}_3$ surface due to the atomically thin $\mathrm{LaAl}{\mathrm{O}}_3$ layer. Horizontal black and vertical red arrows show the compressive strain and $P_{\mathrm{S}}$, respectively.