Interplay between polarization, strain, and defect pairs in Fe-doped ${\mathrm{SrMnO}}_{3-\delta }$

Defect chemistry, strain, and structural, magnetic, and electronic degrees of freedom constitute a rich space for the design of functional properties in transition-metal oxides. Here, we show that it is possible to engineer polarity and ferroelectricity in nonpolar perovskite oxides via polar defect pairs formed by anion vacancies coupled to substitutional cations. We use a self-consistent site-dependent approach employing with a correction that accounts for local structural and chemical changes upon defect creation and which is crucial to reconcile predictions with the available experimental data. Our results for Fe-doped oxygen-deficient $\mathrm{Sr}\mathrm{Mn}{\mathrm{O}}_3$ show that substitutional Fe and oxygen vacancies can promote polarity due to an off-center displacement of the defect charge resulting in a net electric dipole moment, which polarizes the lattice in the defect neighborhood. The formation of these defects and the resulting polarization can be tuned by epitaxial strain, resulting in enhanced polarization also for strain values lower than the ones necessary to induce a polar phase transition in undoped $\mathrm{Sr}\mathrm{Mn}{\mathrm{O}}_3$. For high enough defect concentrations, these defect dipoles couple in a parallel fashion, thus enabling defect- and strain-based engineering of ferroelectricity in $\mathrm{Sr}\mathrm{Mn}{\mathrm{O}}_3$.

Figure 1

($2\times 2\times 2$) Pnma supercell of stoichiometric $\mathrm{Sr}\mathrm{Mn}{\mathrm{O}}_3$.

Figure 2

Schematic representation of the possible relative arrangements of ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}$ defect pairs in the 40-atom SMO cell in the case of an (a) out-of-plane (OP) or (b) in-plane (IP) oxygen vacancy. Purple, orange and brown, and green refer to configurations in which ${\mathrm{Fe}}_{\mathrm{Mn}}$ is in next-neighbor (NN), next-nearest-neighbor (NNN), and next-next-nearest-neighbor (NNNN) position relative to the ${\mathrm{V}}_{\mathrm{O}}$. As opposed to (a), the two Mn sites at the same distance from the ${\mathrm{V}}_{\mathrm{O}}$ in (b) are not symmetry equivalent as indicated by the slightly different shade for each color.

Figure 3

Formation energy $E_{\text{f}}$ computed for ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}$ configurations in unstrained (a) AFM and (b) FM SMO as a function of the ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}$ distance. Circles and squares refer to configurations with ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ and ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$, respectively. See color code in Fig. 2.

Figure 4

Strain-dependent formation energy $E_{\text{f}}$ of (a) ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$ and (b) ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ defect pairs in AFM SMO. See color code in Fig. 2.

Figure 5

Total energy differences $\left[\mathrm{\Delta }E\right(\text{FM-AFM}\left)\right]$ per formula unit between the defective cells with FM and AFM order. AFM is more stable for positive differences, and FM is more stable for negative differences. (a) $\mathrm{\Delta }E\left(\text{FM-AFM}\right)$ reported with respect to the ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}$ distance in each defective configuration in the unstrained SMO structure. (b) Changes in $\mathrm{\Delta }E\left(\text{FM-AFM}\right)$ as a function of strain for all the considered configurations. Circles and squares refer to data obtained for ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ and ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$, respectively. See color code in Fig. 2.

Figure 6

(a) Angle between the total dipole ${\vec{D}}_{\text{tot}}$ and the computed polarization $\vec{P}$ with respect to the angle between the ${\mathrm{Fe}}_{\mathrm{Mn}}^{\prime }\text{−}{\mathrm{V}}_{\mathrm{O}}^{••}$ defect dipole $\vec{D}$ and the ${\mathrm{Mn}}_{\mathrm{Mn}}^{\prime }\text{−}{\mathrm{V}}_{\mathrm{O}}^{••}$ dipole $\overrightarrow{D^{\prime }}$ for the different defect-pair configurations. Circles and squares refer to data obtained for ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ and ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$, respectively. See color code in Fig. 2. (b)–(d) Schematic representation of the $\vec{D}$, $\overrightarrow{D^{\prime }}$, and $\vec{P}$ vectors for the cases in which the angle between $\vec{D}$ and $\overrightarrow{D^{\prime }}$ is about ${60}^{\circ }$ (b), ${120}^{\circ }$ (c), or ${180}^{\circ }$ (d).

Figure 7

Magnitude of (a) the total polarization vector, as well as its (b) in-plane ($|P_{ac}|$) and (c) out-of-plane ($|P_b|$) components for the different defect configurations in the AFM phase. Circles and squares refer to data obtained for ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ and ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$, respectively. See color code in Fig. 2.

Figure 8

Total off-centering along the $b$ axis for Mn atoms lying within a sphere of radius $r$ and centered on the center of mass of the defect pair. Results are for different cell sizes in both unstrained (a) AFM and (b) FM SMO.

Figure 9

$4\times 2\times 2$ supercells with two aligned [(a) and (c)] and antialigned [(b) and (d)] ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}$ defect pairs. Illustrations are for (a) and (b) NNN and (c) and (d) NN ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$. Gold and purple arrows indicate the direction of the ${\mathrm{Fe}}_{\mathrm{Mn}}^{\prime }\text{−}{\mathrm{V}}_{\mathrm{O}}^{••}$ and ${\mathrm{Mn}}_{\mathrm{Mn}}^{\prime }\text{−}{\mathrm{V}}_{\mathrm{O}}^{••}$ defect dipoles, respectively.

Figure 10

Strain dependence of the $a$ [(a) and (b)], $b$ [(c) and (d)], and $c$ [(e) and (f)] components of the polarization for the different defect-pair configurations in AFM SMO. Plots are for (a), (c), and (e) ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ and (b), (d), and (f) ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$ defect pairs. Circles and squares refer to data obtained for ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ and ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$, respectively. See color code in Fig. 2.

Figure 11

Strain dependence of the average Mn off-centering along the $a$ axis [(a) and (b)], $b$ axis [(c) and (d)], and $c$ axis [(e) and (f)] for the different defect-pair configurations in the AFM phase of SMO. Plots are for (a), (c), and (e) ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ and (b), (d), and (f) ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$ defects. Circles and squares refer to data obtained for ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ and ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$, respectively. See color code in Fig. 2. The white triangles correspond, instead, to the Mn off-centerings in the stoichiometric SMO cells.

Figure 12

$U_{\text{SC-SD}}$ values reported as a function of the distance of each Hubbard site from the ${\mathrm{V}}_{\mathrm{O}}$ and the ${\mathrm{Fe}}_{\mathrm{Mn}}$ defects, computed for all Mn atoms in SMO with a NN (top), NNN (middle), and NNNN (bottom) ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ defect pair in the (a) AFM and (b) FM phases. The oxidation state is explicitly indicated for (partially) reduced Mn sites. The insets show the structure of the corresponding configuration, where the red, violet, gold, and dashed spheres represent the O, Mn, and Fe atoms and the ${\mathrm{V}}_{\mathrm{O}}$, respectively. The values of $U_{\text{SC}}$ for the stoichiometric phases are indicated by horizontal dashed lines for comparison.

Figure 13

(a) and (b) Defect-pair configurations as in the main text, numbered according to the location of the Fe substitution site.

Figure 14

$U_{\text{SC-SD}}$ values reported as a function of the distance of each Hubbard site from ${\mathrm{Fe}}_{\mathrm{Mn}}$ and ${\mathrm{V}}_{\mathrm{O}}$ defects computed for the Mn atoms in AFM SMO structures in each of the 12 ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}$ configurations shown in Fig. 13. The oxidation state is indicated for the reduced Mn sites.

Figure 15

Oxidation state reported as a function of the distance of each site from ${\mathrm{Fe}}_{\mathrm{Mn}}$ and ${\mathrm{V}}_{\mathrm{O}}$ defects computed for the Mn or Fe atoms in AFM SMO structures in each of the 12 ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}$ configurations shown in Fig. 13.

Figure 16

$U_{\text{SC-SD}}$ values reported as a function of the distance of each Hubbard site from ${\mathrm{Fe}}_{\mathrm{Mn}}$ and ${\mathrm{V}}_{\mathrm{O}}$ defects computed for the Mn atoms in FM SMO structures in each of the 12 ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}$ configurations shown in Fig. 13. The oxidation state is indicated for the partially reduced Mn sites.

Figure 17

Oxidation state reported as a function of the distance of each site from ${\mathrm{Fe}}_{\mathrm{Mn}}$ and ${\mathrm{V}}_{\mathrm{O}}$ defects computed for the Mn or Fe atoms in FM SMO structures in each of the 12 ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}$ configurations shown in Fig. 13.

Figure 18

$U_{\text{SC-SD}}$ values on the ${\mathrm{Fe}}_{\mathrm{Mn}}$ atoms computed for all the considered ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}$ configurations in (a) AFM and (b) FM SMO. Circles and squares refer to data obtained for ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ and ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$, respectively. See Fig. 2 of the main text for the color code. The horizontal dashed line indicates the $U_{\text{SC}}$ value computed for Fe ions in the (a) G-AFM and (b) FM phases of ${\mathrm{LaFeO}}_3$ as a reference.

Figure 19

Strain-dependent formation energy $E_{\text{f}}$ of (a) ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$ and (b) ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ defect pairs in FM SMO. See color code in Fig. 2 of the main text.

Figure 20

Total energy differences $\left[\mathrm{\Delta }E\right(\text{FM-AFM}\left)\right]$ per formula unit between the defective cells with FM and AFM order computed with the $\mathrm{DFT}+U_{\text{SC}}$ approach. AFM is more stable for positive differences, and FM is more stable for negative differences. $\mathrm{\Delta }E\left(\text{FM-AFM}\right)$ is reported with respect to the ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}$ distance in each defective configuration in the unstrained SMO structure. Circles and squares refer to data obtained for ${\mathrm{VO}}_{\mathrm{O}}^{\mathrm{OP}}$ and ${\mathrm{VO}}_{\mathrm{O}}^{\mathrm{IP}}$, respectively. See color code in Fig. 2 of the main text.

Figure 21

Angle between (a) the ${\mathrm{Fe}}_{\mathrm{Mn}}^{\prime }\text{−}{\mathrm{V}}_{\mathrm{O}}^{••}$ defect dipole $\vec{D}$ or (b) the total defect dipole ${\vec{D}}_{\mathrm{tot}}$ and the polarization $\vec{P}$ as a function of the distance between ${\mathrm{Fe}}_{\mathrm{Mn}}$ and ${\mathrm{V}}_{\mathrm{O}}$ for the different configurations in the AFM phase. Circles and squares refer to data obtained for ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ and ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$, respectively. See Fig. 2 in the main text for the color code.

Figure 22

Modulus of (a) the total polarization vector and (b) the in-plane ($|{\vec{P}}_{ac}|$) and (c) the out-of-plane ($|{\vec{P}}_b|$) components for the different defect configurations in the FM phase. Circles and squares refer to data obtained for ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ and ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$, respectively. See Fig. 2 in the main text for the color code.

Figure 23

Evolution of the $a$, $b$, and $c$ components of the polarization for the NN ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ defective supercell as a function of the cell size in (a) the AFM and (b) the FM phase.

Figure 24

Epitaxial-strain-driven changes of the (a) and (b) $a$, (c) and (d) $b$, and (e) and (f) $c$ components of the polarization for the different defect-pair configurations in the FM phase of SMO. Plots are for (a), (c), and (e) ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ and (b), (d), and (f) ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$ defects. Circles and squares refer to data obtained for ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ and ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$, respectively. See Fig. 2 in the main text for the color code.

Figure 25

Strain dependence of the average Mn off-centering along the (a) and (b) $a$ axis, (c) and (d) $b$ axis, and (e) and (f) $c$ axis for the different defect-pair configurations in the FM phase of SMO. Plots are for (a), (c), and (e) ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ and (b), (d), and (f) ${\mathrm{Fe}}_{\mathrm{Mn}}\text{−}{\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$ defects. Circles and squares refer to data obtained for ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{OP}}$ and ${\mathrm{V}}_{\mathrm{O}}^{\mathrm{IP}}$, respectively. See Fig. 2 in the main text for the color code.