Strain-phonon coupling in (111)-oriented perovskite oxides

Strain-phonon coupling, in terms of the shift in phonon frequencies under biaxial strain, is studied by density functional theory calculations for 20 perovskite oxides strained in their (111) and (001) planes. While the strain-phonon coupling under (001) strain follows the established, intuitive trends, the response to (111) strain is more complex. Here we show that strain-phonon coupling under (111) strain can be rationalized in terms of the Goldschmidt tolerance factor and the formal cation oxidation states. The established trends for coupling between (111) strain and in-phase and out-of-phase octahedral rotational modes as well as polar modes provide guidelines for rational design of (111)-oriented perovskite thin films.

Figure 1

Strain-phonon coupling for (001) strain. The sketches show the different phonon modes in $2\times 2\times 2$, supercells, A cations and oxygen atoms omitted for clarity. The modes discussed in this paper are out-of-phase rotations denoted $\mathrm{Ro}{\mathrm{t}}^{-}$, in-phase rotations denoted $\mathrm{Ro}{\mathrm{t}}^{+}$, and polar distortions of the B cation denoted $P$. Subscript $\parallel$ and $\perp$ denotes in-plane and out-of-plane modes, respectively. Arrow up (down) denotes that the preceding mode type is softer (harder) for that type of strain.

Figure 2

(a)–(c) Three important phonon modes for perovskite oxides, here depicted in $2\times 2\times 2$ supercells which were used for the phonon calculations. (a) Visualization of the out-of-phase rotations at the $R$ point of the Brillouin zone, denoted $\mathrm{Ro}{\mathrm{t}}^{-}$. (b) Visualization of the in-phase rotations at the $M$ point of the Brillouin zone, denoted $\mathrm{Ro}{\mathrm{t}}^{+}$. (c) Visualization of the polar distortions of the B cation at $\mathrm{\Gamma }$ point of the Brillouin zone, denoted $P$. In (a)–(c) the oxygen atoms are omitted for clarity. (d) and (e) Calculation cells used for (001) and (111) strain, respectively, along with the relations between the lattice vectors and the stacking sequence.

Figure 3

The frequencies of the three different phonon modes considered as a function of in-plane strain in the (a) (001) plane and (b) (111) plane for $\mathrm{SrTi}{\mathrm{O}}_3$ (STO). Under strain the phonon frequencies are split into two degenerate perpendicular in-plane modes denoted with a subscript $\parallel$, and one out-of-plane mode denoted with a subscript $\perp$. The three different modes considered are out-of-phase rotations denoted by $\mathrm{Ro}{\mathrm{t}}^{-}$, in-phase rotations denoted by $\mathrm{Ro}{\mathrm{t}}^{+}$, and polar displacement of the B cation denoted by $P$. For (001) strain the in-plane directions are $\left[100\right]$ and $\left[010\right]$, while the out-of-plane direction is $\left[001\right]$. For (111) strain the in-plane directions are $\left[1\overline{1}0\right]$ and $\left[11\overline{2}\right]$ while the out-of-plane direction is $\left[111\right]$. The dashed lines are guides to the eye.

Figure 4

The frequencies of the three different phonon modes considered as a function of in-plane strain in the (a) (001) plane and (b) (111) plane for $\mathrm{NaTa}{\mathrm{O}}_3$ (NTO). Under strain the phonon frequencies are split into two degenerate perpendicular in-plane modes denoted with a subscript $\parallel$, and one out-of-plane mode denoted with a subscript $\perp$. The three different modes considered are out-of-phase rotations denoted by $\mathrm{Ro}{\mathrm{t}}^{-}$, in-phase rotations denoted by $\mathrm{Ro}{\mathrm{t}}^{+}$, and polar displacement of the B cation denoted by $P$. For (001) strain the in-plane directions are $\left[100\right]$ and $\left[010\right]$, while the out-of-plane direction is $\left[001\right]$. For (111) strain the in-plane directions are $\left[1\overline{1}0\right]$ and $\left[11\overline{2}\right]$, while the out-of-plane direction is $\left[111\right]$. The dashed lines are guides to the eye.

Figure 5

Strain-phonon coupling for (111) strain. The schematic shows the different phonon modes in $\sqrt{2}\times \sqrt{2}\times \sqrt{3}$ supercells, A cations and oxygen omitted for clarity. Arrow up (down) denotes that the preceding mode type is softer (harder) for that type of strain. The strain-phonon coupling for (111) strain is different for high ($t\sim 1$) and low $(t\sim 0.9$) tolerance factors, as discussed in the text. The exact crossover depends on the relative charge of the A and B cations. For low tolerance factors, the response is similar to what is known for (001) strain, except for in-plane rotations, which are not split for (111) strain, while for high tolerance factors, different responses are seen.

Figure 6

The degree of splitting $\mathrm{\Delta }{f}_{\mathrm{rot}-}$ between in-plane and out-of-plane out-of-phase rotations as a function for $\pm 1\%$ (a) (001) strain and (b) (111) strain as a function of tolerance factor $t.$ A positive $\mathrm{\Delta }{f}_{\mathrm{rot}-}$, yellow area, indicates (001)-like splitting where out-of-plane rotations are softer for compressive (comp) strain and out-of-plane rotations are softer for tensile (tens) strain. In the green area an opposite splitting is seen, where in-plane rotations are softer for comp strain, and out-of-plane rotations are softer for tens strain. See Table 1 for abbreviations.

Figure 7

Derivative of the in-plane out-of-phase rotational modes with respect to (a) (001) strain and (b) (111) strain. A positive $df/d\epsilon$, yellow area, means that the respective mode is softened by compressive (comp) strain and hardened by tensile (tens) strain, while a negative $df/d\epsilon$, green area, means that the respective mode is softened tens strain and hardened for comp strain and softened for tens strain. See Table 1 for abbreviations. The dashed lines are guides to the eye, based on linear fits to the data.

Figure 8

Derivative of the out-of-plane out-of-phase rotational modes with respect to (a) (001) strain and (b) (111) strain. A positive $df/d\epsilon$, yellow area, means that the respective mode is softened by compressive (comp) strain and hardened by tensile (tens) strain, while a negative $df/d\epsilon$, green area, means that the respective mode is softened tens strain and hardened for comp strain and softened for tens strain. See Table 1 for abbreviations. No trend lines are added, as no given trend is seen.

Figure 9

Derivative of the in-phase rotational modes with respect to (111) strain as a function of tolerance factor $t$. Note that since the in-phase rotations are not split under (111) strain, this plot shows both in-plane and out-of-plane modes, as they are degenerate. A positive $df/d\epsilon$, yellow area, means that the respective mode is softened by compressive (comp) strain and hardened by tensile (tens) strain, while a negative $df/d\epsilon$, green area, means that the respective mode is softened by tens strain and hardened for comp strain and softened for tens strain. See Table 1 for abbreviations. The dashed lines are guides to the eye, based on linear fits to the data.

Figure 10

Derivative of the out-of-plane polar modes as a function of tolerance factor $t$ with respect to (a) (001) strain and (b) (111) strain as a function of tolerance factor $t$. A positive $df/d\epsilon$, yellow area, means that the respective mode is softened by compressive (comp) strain and hardened by tensile (tens) strain, while a negative $df/d\epsilon$, green area, means that the respective mode is softened by tens strain and hardened for comp strain and softened for tens strain. See Table 1 for abbreviations. The dashed lines are guides to the eye, based on linear fits to the data.