Engineering structural modifications of epitaxial perovskite thin films is an effective route to induce new functionalities or enhance existing properties due to the close relation of the electronic ground state to the local bonding environment. As such, there is a necessity to systematically refine and precisely quantify these structural displacements, particularly those of the oxygen octahedra, which is a challenge due to the weak scattering factor of oxygen and the small diffraction volume of thin films. Here, we present an optimized algorithm to refine the octahedral rotation angles using specific unit-cell-doubling half-order diffraction peaks for the *a ^{−}a^{−}c^{+} Pbnm* structure. The oxygen and

*A*-site positions can be obtained by minimizing the squared-error between calculated and experimentally determined peak intensities using the (1/2 1/2 3/2) and (1/2 1/2 5/2) reflections to determine the rotation angle

*α*about in-plane axes and the (1/2 5/2 1), (1/2 3/2 1), and (1/2 3/2 2) reflections to determine the rotation angle

*γ*about the out-of-plane axis, whereas the convoluting

*A*-site displacements associated with the octahedral rotation pattern can be determined using (1 1 1/2) and (1/2 1/2 1/2) reflections to independently determine

*A*-site positions. The validity of the approach is confirmed by applying the refinement procedure to determine the

*A*-site and oxygen displacements in a NdGaO

_{3}single crystal. The ability to refine both the oxygen and

*A*-site displacements relative to the undistorted perovskite structure enables a deeper understanding of how structural modifications alter functionality properties in epitaxial films exhibiting this commonly occurring crystal structure.

## I. INTRODUCTION

Epitaxy can stabilize structures that are beyond the limits of conventional bulk synthesis, thus offering many routes to control the physical properties of the films. The class of *AB*O_{3} perovskites exhibits a wide range of phenomena that arise due to spin, orbital, and electronic degrees of freedom coupling through the lattice environment, which gives rise to novel magnetic phases, ferroic responses, and the competition of diverse ground states resulting in electronic phase transitions, all subject of intense research efforts.^{1} One prominent characteristic of perovskites is that these electronic and ferroic functionalities are intimately coupled to the atomic structure. In particular, many of the functional properties of perovskite oxides can be directly modified through changes to the rotations and distortions of the corner-connected *B*O_{6} octahedra, which determine the *B*-O-*B* bond angles and *B*-O bond lengths. For example, the ordering temperatures of both ferromagnetic and antiferromagnetic perovskites are acutely dependent on octahedral rotations,^{2,3} and a transition between ferromagnetism and antiferromagnetism can be induced through modifications to the *B*-O-*B* bond angle while keeping the nominal *B*-site valence fixed.^{4} Additionally, control of octahedral rotations and distortions provides a means to tune metal-insulator transitions,^{5,6} orbital ordering transitions,^{7,8} and band gaps of perovskites.^{9,10} While rotations and distortions from the cubic perovskite are determined by the size of the ions occupying the *A*-site and *B*-site in bulk perovskites,^{11} epitaxial strain and structural coupling across oxide heterointerfaces can induce non-bulk-like octahedral distortions and rotation patterns in heterostructures and superlattices.^{12} Given the central role of octahedral behavior in the functional properties of perovskites, there are growing efforts to use rotations as a direct means to engineer or even induce new physical properties in material systems that otherwise lack the targeted behavior, which is exemplified by recent work on designing improper ferroelectricity in superlattices made of non-polar constituent layers.^{13–15} Therefore, precisely quantifying the position of the oxygen atoms is critical for understanding behavior in strained films and to experimentally verify predicted properties at heterointerfaces.

The experimental measurement of octahedral rotations in films is challenging due to the weak scattering amplitude of oxygen and the limited sample volume. Currently, approaches include synchrotron diffraction,^{16–23} extended x-ray absorption fine structure,^{24,25} Bragg rod analysis,^{26–28} and electron microscopy/diffraction-based techniques.^{29–32} The latter two techniques have the advantage of providing spatially resolved measurements of rotations, which is particularly useful at interfaces.^{33–37} In contrast, x-ray diffraction provides a fully three-dimensional measurement, is non-destructive, and probes the entire volume of the film providing statistical information over macroscopic length scales. The synchrotron diffraction approach is based on the measurement of half-order Bragg reflections that arise from the doubling of the cubic unit cell due to the octahedral rotations and the associated lowering of the symmetry from the undistorted perovskite structure.^{38} The intensities of these half-order peaks can then be analyzed to refine the atomic positions in the crystal structure responsible for the unit cell doubling.^{16} This approach has been used by multiple groups to determine bond angles and lengths in strained films and superlattices.^{16,18,26,39–44} In the case of perovskites that exhibit the rhombohedral ($R3\xafc$) structure with the *a*^{−}*a*^{−}*c*^{−} rotation pattern, only two fitting parameters for the rotations about the *a*^{−} axes and the *c*^{−} axis are needed to determine the oxygen positions based on relative intensity of specific half-order peaks. For the orthorhombic (*Pbnm*) perovskites with the *a*^{−}*a*^{−}*c*^{+} pattern, however, the *c*^{+} rotation allows for *A*-site displacements that also double the periodicity of the pseudo-cubic unit cell, which significantly increases the complexity to the analysis of half-order diffraction peaks because the origin of the half-order peaks is convoluted by *A*-site displacements as well as octahedral rotations. As the *Pbnm* structure is the most common structural variant found in perovskite oxides^{45} and is the basis for many predictions of new functional behavior in superlattices based on rotations,^{14,15,46–48} developing a robust approach for fitting half-order peaks in films with the *a*^{−}*a*^{−}*c*^{+} rotation pattern is critical for a deeper understanding of oxide heterostructures.

Here, we describe an experimental approach to quantify the positions of oxygen and *A*-site atoms within *Pbnm*-type perovskite films based on x-ray diffraction measurements and analysis of half-order diffraction peaks. We show how the various atomic displacements individually contribute to the intensity of particular half-order peaks and use simulations and a nonlinear fitting routine to disentangle the relative contribution from the oxygen octahedral rotation and *A*-site displacements. We then identify which combinations of diffraction peaks lead to the most robust fitting results, as verified by numerical simulations, and experimentally using bulk NdGaO_{3}, which will serve as a guide for future experimental studies aimed at quantifying octahedral rotations and *A*-site displacements in *Pbnm*-type perovskite thin films, heterostructures, and superlattices.

## II. MODEL DESCRIPTION

The undistorted perovskite structure shown in Fig. 1(a) can be described using a single cubic unit cell with the *A*-site cation at the corners and the *B*-site cation at the center coordinated by six oxygen located at the face centers forming an oxygen octahedron. Deviations from this base structure can be accommodated by rotating the oxygen octahedra about the three cubic axes [100], [010], and [001] by the angles *α*, *β*, and *γ*, respectively, as shown in Fig. 1(a). To maintain corner connectivity, adjacent octahedra must rotate as well, which reduces the symmetry. Of interest here are the perovskites falling into the *Pbnm* space group which have an orthorhombic unit cell that is shown in Fig. 1(b), in which the orthorhombic main axes [100] and [010] coincide with the [110] and [$11\xaf0$] cubic axes, and the [001] is parallel to the [001] cubic axis. An alternative and commonly used description of the orthorhombic crystal is to use a larger non-primitive unit cell that is shown in Fig. 1(c), the pseudocubic unit cell, which is constructed out of eight unit cells of the undistorted cubic perovskite. This pseudocubic unit cell provides a very intuitive structural model that spans many space groups and is particularly useful for epitaxial strained thin films. We will use this pseudocubic unit cell throughout this work.

There are 23 possible rotation patterns by which these unit cell doubling rotations can maintain corner connectivity.^{49} They are determined by both the magnitude of the rotation and if the rotation of the adjacent octahedron is in-phase or out-of-phase with respect to the adjacent octahedron along a given pseudocubic direction.^{49–51} In the standard Glazer notation, the +, −, or 0 superscripts are used to denote in-phase, out-of-phase, and absence of rotation along each of the three pseudocubic axes. For example, *a*^{−}*b*^{+}*c*^{0} indicates an out-of-phase rotation along the *a*-axis, in-phase rotation along the *b*-axis, and no rotation along the *c*-axis. Alternatively, the *a*^{+}*a*^{+}*a*^{+} pattern indicates in-phase rotations of equal magnitude along all three axes.

In general, refining a crystal structure requires parameterizing the atomic positions within the unit cell, and varying these parameters to match the experimentally measured diffraction intensities. The symmetry of the perovskite structure significantly simplifies this general refinement procedure and the atomic positions can be reduced to a few specific modes described by a small set of parameters. The structure model requires analytic functions describing the unique positions within the pseudocubic unit cell of the 24 oxygen anions, 8 *B*-site cations, and 8 *A*-site cations. Figures 2(a)–2(e) show the individual displacements of the oxygen resolved layer-by-layer along the *c*^{+} [001] axis. As shown in Fig. 2(b) for *z* = 0, the change in position of the 4 oxygen atoms is only due to the *α* and *β* rotations, which only alter the *x* and *y* coordinates. The change in *x* is proportional to *c*/(2*a*)tan(*β*), whereas the change in *y* is proportional to *c*/(2*b*)tan(*α*). Two things need to be noted: (1) the *a*^{−}*a*^{−}*c*^{+} rotation pattern requires *α* = *β*, but, the angles and lattice parameters are kept unique to conveniently account for possible rotational domains, as will be explained in detailed later and (2) the prefactors of *c*/(2*a*) and *c*/(2*b*) are required so that the rotation angle of the unit cell used in the structure-factor calculation matches that of the real-space cell. This rationale applies to the remaining oxygen positions for the unit cell shown in Figs. 2(b)–2(e) as well and can be similarly generalized for any of the 23 tilt systems by changing the sign of the distortions for selected oxygen positions accordingly.

In the case of epitaxial films considered here, different orientations of the *Pbnm* crystal are possible relative to the underlying substrate. Specifically, the in-phase rotation axis can lay in the plane of the film (along the pseudocubic *a-* or *b-*axis) or oriented perpendicular to the film plane along the pseudocubic *c-*axis. Defining the *c*-axis as the out-of-plane pseudocubic direction for a film grown on a (001)-oriented pseudocubic perovskite substrate, the latter case is described by the *a*^{−}*a*^{−}*c*^{+} pattern, while the two former cases correspond to *a*^{−}*a*^{+}*c*^{−} and the equivalent *a*^{+}*a*^{−}*c*^{−} rotation pattern. Density functional calculations have found that epitaxial films with the *a*^{−}*a*^{+}*c*^{−} or *a*^{+}*a*^{−}*c*^{−} rotation pattern belong to the *P*2_{1}/*m* space group, while films with the *a*^{−}*a*^{−}*c*^{+} pattern retain the *Pbnm* space group.^{52–54} Strained films with these rotation patterns retain these space groups even when the in-plane lattice parameters are equal (*a* = *b*).^{52,54} The orientation of the in-phase axis does not alter the relationship of α, β, and γ to the crystallographic axes; γ always describes the magnitude of the rotation along the *c*-axis independent of whether the pattern is *a*^{−}*a*^{−}*c*^{+} or *a*^{−}*a*^{+}*c*^{−}. However, the direction of the *A*-site displacements is related to the orientation of the in-phase axis, and the displacement of the *A*-site cations occurs within the plane perpendicular to the in-phase axis.

As mentioned previously, the in-phase rotation of the *a*^{−}*a*^{−}*c*^{+} rotation pattern allows the *A*-site cation to displace in a unit-cell-doubling manner. Alteration of the *A*-site position allows for optimization of the local *A*-O bond environment, by changing the *A*-O distances and thus covalency, which lowers the energy of the structure.^{51} The *A*-site positions can be captured using two fitting parameters, *d*_{1} and *d*_{2}, as illustrated in Fig. 2(f). The *d*_{1} parameter describes the out-of-phase displacement of *A*-site cations along a ⟨110⟩ direction, i.e., the opposite directions in adjacent layers along the + rotation axis. The *d*_{2} parameter slightly shifts the *A*-site cations either towards the [100] or [010] direction to align them along the empty rhombus-like tunnels created by the *B*O_{6} octahedra, consistent with the bulk *Pbnm* structure. Table I lists how the atomic positions are calculated from *α*, *β*, *γ*, *d*_{1}, and *d*_{2} assuming the *a*^{−}*a*^{−}*c*^{+} rotation pattern; atomic positions within the *a*^{−}*a*^{+}*c*^{−} pattern can be obtained from Table I by switching the *y* and *z* coordinates, and then exchanging angles β and γ and lattice parameters *b* and *c*. The same process is followed for *a*^{+}*a*^{−}*c*^{−}, except instead the *x* and *z* coordinates are switched, and the α and γ angles are exchanged, as are *a* and *c* (as shown in Table S1 of the supplementary material).

Atom . | x
. | y
. | z
. |
---|---|---|---|

O1 | 0.25 + c/(4a) tan(β) | 0.25 − c/(4b) tan(α) | 0 |

O2 | 0.25 − c/(4a) tan(β) | 0.75 + c/(4b) tan(α) | 0 |

O3 | 0.75 + c/(4a) tan(β) | 0.75 − c/(4b) tan(α) | 0 |

O4 | 0.75 − c/(4a) tan(β) | 0.25 + c/(4b) tan(α) | 0 |

O5 | 0 | 0.25 + a/(4b) tan(γ) | 0.25 − a/(4c) tan(β) |

O6 | 0.25 − b/(4a) tan(γ) | 0 | 0.25 + b/(4c) tan(α) |

O7 | 0.5 | 0.25 − a/(4b) tan(γ) | 0.25 + a/(4c) tan(β) |

O8 | 0.25 + b/(4a) tan(γ) | 0.5 | 0.25 − b/(4c) tan(α) |

O9 | 0 | 0.75 − a/(4b) tan(γ) | 0.25 + a/(4c) tan(β) |

O10 | 0.5 | 0.75 + a/(4b) tan(γ) | 0.25 − a/(4c) tan(β) |

O11 | 0.75 − b/(4a) tan(γ) | 0.5 | 0.25 + b/(4c) tan(α) |

O12 | 0.75 + b/(4a) tan(γ) | 0 | 0.25 − b/(4c) tan(α) |

O13 | 0.25 − c/(4a) tan(β) | 0.25 + c/(4b) tan(α) | 0.5 |

O14 | 0.25 + c/(4a) tan(β) | 0.75 − c/(4b) tan(α) | 0.5 |

O15 | 0.75 − c/(4a) tan(β) | 0.75 + c/(4b) tan(α) | 0.5 |

O16 | 0.75 + c/(4a) tan(β) | 0.25 − c/(4b) tan(α) | 0.5 |

O17 | 0 | 0.25 + a/(4b) tan(γ) | 0.75 + a/(4c) tan(β) |

O18 | 0.25 − b/(4a) tan(γ) | 0 | 0.75 − b/(4c) tan(α) |

O19 | 0.5 | 0.25 − a/(4b) tan(γ) | 0.75 − a/(4c) tan(β) |

O20 | 0.25 + b/(4a) tan(γ) | 0.5 | 0.75 + b/(4c) tan(α) |

O21 | 0 | 0.75 − a/(4b) tan(γ) | 0.75 − a/(4c) tan(β) |

O22 | 0.5 | 0.75 + a/(4b) tan(γ) | 0.75 + a/(4c) tan(β) |

O23 | 0.75 − b/(4a) tan(γ) | 0.5 | 0.75 − b/(4c) tan(α) |

O24 | 0.75 + b/(4a) tan(γ) | 0 | 0.75 + b/(4c) tan(α) |

A1 | d_{1} | d_{1} + d_{2} | 0 |

A2 | 0.5 + d_{1} + d_{2} | d_{1} | 0 |

A3 | 0.5 + d_{1} | 0.5 + d_{1} + d_{2} | 0 |

A4 | d_{1} + d_{2} | 0.5 + d_{1} | 0 |

A5 | −d_{1} | −d_{1}− d_{2} | 0.5 |

A6 | 0.5 − d_{1} − d_{2} | −d_{1} | 0.5 |

A7 | 0.5 − d_{1} | 0.5 − d_{1} − d_{2} | 0.5 |

A8 | −d_{1} − d_{2} | 0.5 − d_{1} | 0.5 |

Atom . | x
. | y
. | z
. |
---|---|---|---|

O1 | 0.25 + c/(4a) tan(β) | 0.25 − c/(4b) tan(α) | 0 |

O2 | 0.25 − c/(4a) tan(β) | 0.75 + c/(4b) tan(α) | 0 |

O3 | 0.75 + c/(4a) tan(β) | 0.75 − c/(4b) tan(α) | 0 |

O4 | 0.75 − c/(4a) tan(β) | 0.25 + c/(4b) tan(α) | 0 |

O5 | 0 | 0.25 + a/(4b) tan(γ) | 0.25 − a/(4c) tan(β) |

O6 | 0.25 − b/(4a) tan(γ) | 0 | 0.25 + b/(4c) tan(α) |

O7 | 0.5 | 0.25 − a/(4b) tan(γ) | 0.25 + a/(4c) tan(β) |

O8 | 0.25 + b/(4a) tan(γ) | 0.5 | 0.25 − b/(4c) tan(α) |

O9 | 0 | 0.75 − a/(4b) tan(γ) | 0.25 + a/(4c) tan(β) |

O10 | 0.5 | 0.75 + a/(4b) tan(γ) | 0.25 − a/(4c) tan(β) |

O11 | 0.75 − b/(4a) tan(γ) | 0.5 | 0.25 + b/(4c) tan(α) |

O12 | 0.75 + b/(4a) tan(γ) | 0 | 0.25 − b/(4c) tan(α) |

O13 | 0.25 − c/(4a) tan(β) | 0.25 + c/(4b) tan(α) | 0.5 |

O14 | 0.25 + c/(4a) tan(β) | 0.75 − c/(4b) tan(α) | 0.5 |

O15 | 0.75 − c/(4a) tan(β) | 0.75 + c/(4b) tan(α) | 0.5 |

O16 | 0.75 + c/(4a) tan(β) | 0.25 − c/(4b) tan(α) | 0.5 |

O17 | 0 | 0.25 + a/(4b) tan(γ) | 0.75 + a/(4c) tan(β) |

O18 | 0.25 − b/(4a) tan(γ) | 0 | 0.75 − b/(4c) tan(α) |

O19 | 0.5 | 0.25 − a/(4b) tan(γ) | 0.75 − a/(4c) tan(β) |

O20 | 0.25 + b/(4a) tan(γ) | 0.5 | 0.75 + b/(4c) tan(α) |

O21 | 0 | 0.75 − a/(4b) tan(γ) | 0.75 − a/(4c) tan(β) |

O22 | 0.5 | 0.75 + a/(4b) tan(γ) | 0.75 + a/(4c) tan(β) |

O23 | 0.75 − b/(4a) tan(γ) | 0.5 | 0.75 − b/(4c) tan(α) |

O24 | 0.75 + b/(4a) tan(γ) | 0 | 0.75 + b/(4c) tan(α) |

A1 | d_{1} | d_{1} + d_{2} | 0 |

A2 | 0.5 + d_{1} + d_{2} | d_{1} | 0 |

A3 | 0.5 + d_{1} | 0.5 + d_{1} + d_{2} | 0 |

A4 | d_{1} + d_{2} | 0.5 + d_{1} | 0 |

A5 | −d_{1} | −d_{1}− d_{2} | 0.5 |

A6 | 0.5 − d_{1} − d_{2} | −d_{1} | 0.5 |

A7 | 0.5 − d_{1} | 0.5 − d_{1} − d_{2} | 0.5 |

A8 | −d_{1} − d_{2} | 0.5 − d_{1} | 0.5 |

Within the *a*^{−}*a*^{−}*c*^{+} structure exist 8 rotational domains and 4 equivalent *A*-site displacements, which need to be considered when the film is epitaxially fixed to the substrate. The 8 rotational domains arise from the possibility of *α*, *β*, and *γ* being positive or negative, which are energetically degenerate when nucleating a film on a cubic substrate. We refer to them as rotational domains with a combination of + and – signs, listed in order of *α*, *β*, and *γ*. In the absence of *A*-site displacements, each of these domains forms a pair of equal intensity. For example, the domain in which *α*, *β*, and *γ* are all positive (+++ domain) yields the same intensity as that in which *α*, *β*, and *γ* are all negative (− domain); hence, only 4 of these rotational domains give unique diffraction intensities.^{16} Furthermore, within each rotational domain, we find four equivalent *A*-site configurations satisfying the condition of displacing the cations along the long diagonal of the rhombus, which are labeled as 1R, 2R, 1L, and 2L, as depicted in Fig. 2(f). Figure 2(f) illustrates all four possible displacements viewed along the *c*-axis for *γ* > 0. Noting the position of the cations relative to the oxygen atoms, one can see that for *γ* > 0, only 1R and 2R are energetically favorable, because the *A*-site cations in 2L and 1L are pushed close to the oxygen atoms; for *γ* < 0 this is reversed. The eight *A*-site positions for 1R are listed in Table I, and the coordinates for other patterns are listed in the supplementary material. Combining the rotational domains with the *A*-site displacement patterns yields a total of 32 possible domains within a single *Pbnm*-type rotation pattern. Of these 32 possible domains, only 8 are found to be unique, +++1R, +++2R, −++1R, −++2R, +−+1R, +−+2R, ++−1L, and ++−2L. The space group associated with each of the 8 domains was determined under a variety of displacement parameters within the *a*^{−}*a*^{−}*c*^{+} structure. Using the FINDSYM program,^{55} we find that 4 of the 8 domains yield *Pbnm* structures: +++_1R, +−+_2R, ++−_1L, and −++_2R, while the other 4 domains result in *P*2_{1}/*m* structures: +++_2R, +−+_1R, ++−_2L, and −++_1R. Therefore, when refining films exhibiting the *a*^{−}*a*^{−}*c*^{+} pattern, only the 4 domains that result in the *Pbnm* structure need to be considered. Similarly for films with the *a*^{−}*a*^{+}*c*^{−} pattern, 4 domains yield *Pbnm* structures (+++_2R, +−+_2L, ++−_1R, and −++_1R) and 4 domains produce *P*2_{1}/*m* structures (+++_1R, +−+_1L, ++−_2R, and −++_2R). More details related to rotational domains can be found in the supplementary material. As discussed later, the relative volume fraction, *D _{j}*, of these various domains can be measured and accounted for in the refinement procedure.

## III. ANALYSIS OF HALF-ORDER PEAK INTENSITIES

The refinement procedure is based on minimizing the square error of the calculated and measured intensity of a specific set of diffraction peaks with respect to parameters *α*, *β*, *γ*, *d*_{1}, and *d*_{2}. This is implemented by calculating the structure factor (*F _{hkl}*) given by

where *f _{A}* is the atomic form factor of element

*A*;

*H*,

*K*, and

*L*are the Miller indices for a given Bragg peak, and

*u*,

*v*, and

*w*denote the atomic positions calculated from

*α*(

*=β*),

*γ*,

*d*

_{1}, and

*d*

_{2}, as shown in Table I. The atomic form factor for O

^{2−}is reported in Ref. 56, and the form factors for various metal cations can be found in Ref. 57. It is important to note that the Miller indices in Eq. (1) refer to a pseudocubic unit cell that has been doubled in each direction with respect to the undistorted cubic perovskite crystal to account for the symmetry lowering structural distortions. Therefore, half-order reflections are calculated from odd integer Miller indices. For example,

*F*for the (1/2 1/2 3/2) peak is calculated using

_{HKL}*H*= 1,

*K*= 1, and

*L*= 3. Throughout the rest of the text, the

*H*,

*K*, and

*L*values are discussed with respect to the cubic perovskite structure and it is only in Eq. (1) that the Miller indices are doubled from these values. The peak intensity of a given reflection was obtained from

where $FHKL*$ is the complex conjugate of the structure factor, *D _{j}* is the volume fraction of the four possible domains, and

*L*is the Lorentz polarization correction. For experiments in which the incident angle is not fixed at a constant value for all measured peaks, the beam footprint correction (=1/sin(

_{p}*η*)) is necessary, where

*η*is the incident angle of the x-ray beam on the sample. In the case of a four-circle diffraction measurement, $sin\u2009(\eta )=sin\u2009(\omega +\theta )\u2009sin\u2009(\chi )$, where

*ω*accounts for slight adjustments of the

*θ*motor with respect to half of the 2

*θ*position and

*χ*rotates the sample about the axis parallel to the straight-through beam.

^{58}

In Eq. (2), it is assumed that the rotational orientation is uniform throughout the film, for example, the entire film exhibits the *a*^{−}*a*^{−}*c*^{+} pattern. Note that *D _{j}* only accounts for rotational domains, which all occur within the same rotation pattern, i.e.,

*a*

^{−}

*a*

^{−}

*c*

^{+}or

*a*

^{−}

*a*

^{+}

*c*

^{−}. However, in heterostructures with the in-phase rotation axis lying in the plane of the film, a spatially inhomogeneous mixture of

*a*

^{−}

*a*

^{+}

*c*

^{−}/

*a*

^{+}

*a*

^{−}

*c*

^{−}structures may be presented, as has been experimentally observed in numerous systems.

^{18,59–62}In this scenario, one region of the film exhibits

*a*

^{−}

*a*

^{+}

*c*

^{−}while another region is oriented as

*a*

^{+}

*a*

^{−}

*c*

^{−}; the direction of the in-phase axis is rotated by 90° but the symmetry of the two regions is the same. A film may even exhibit a mixture of all three possible orientations of the in-phase axis with regions of

*a*

^{+}

*a*

^{−}

*c*

^{−},

*a*

^{−}

*a*

^{+}

*c*

^{−}, and

*a*

^{−}

*a*

^{−}

*c*

^{+}all present in the same film.

^{62–64}In this case, Eq. (2) must be augmented by summing over the volume fractions of the regions with different in-phase axis orientations leading to $I=V1Ia\u2212a\u2212c++V2Ia\u2212a+c\u2212+V3Ia+a\u2212c\u2212$, where

*V*indicates relative volume fractions of the three possible orientations and the

_{i}*I*terms are individually calculated from Eq. (2). The values of

_{aac}*V*can be directly extracted from the intensities of systematic sets of half-order diffraction peaks.

_{i}^{62}

There are assumptions inherent to this approach. First, we assume that the *B*-site cations remain at the center of the octahedra. Second, all sites are completely occupied—the model would need to be modified accordingly to address oxygen vacancy ordering in the film by changing Eq. (1) to account for the partial occupancy of particular oxygen sites. A large concentration of disordered vacancies would be expected to create local octahedral distortions by displacing ions from the uniform positions assumed here, thus reducing the expected accuracy of this approach. Third, it is assumed that the film is strained to a pseudocubic substrate yielding *α* = *β* within the film. This leads to equal in-plane bond angles along the [100] and [010] directions. For films where this condition does not hold and partial or complete strain relaxation has occurred, *α* and *β* must be treated as independent fitting parameters. In addition, it is assumed that *A*-site displacements in strained films occur along the same crystallographic directions, as they do in the bulk. For example, if the + rotation axis is orientated along the [001] direction, i.e., normal to the film plane, then *A*-site displacements are along the ⟨110⟩ directions with some deviation captured by the *d*_{2} parameter. This assumption is supported by high resolution scanning transmission electron microscopy results obtained from *Pbnm*-type layers, in which the *A*-site displacements can be observed within the plane perpendicular to the in-phase rotation axis.^{23,33,35} Finally, in simplying adding intensities from the different rotational domain populations, this approach does not take into account any coherent scattering effects of adjacent domains or domain boundaries.

Using Eqs. (1) and (2), the relative contributions of *A*-site displacements and octahedral rotations for various half-order peaks can be calculated (herein, we use *D _{j}* = 1/4 for all

*j*assuming equal domain populations). An overview of the peak intensities arising from these displacements can be seen in reciprocal space maps (RSMs) shown in Fig. 3. A structure with the

*a*

^{−}

*a*

^{−}

*c*

^{+}pattern was assumed and cuts through the

*L*= 1 and 3/2 reciprocal lattice planes were calculated. Both the oxygen and

*A*-site displacements result in numerous families of peaks, including those consisting of three half-order indices [ex. (1/2 1/2 3/2)], two half-order indices [ex. (1/2 3/2 1)], and one half-order index [ex. (1 1 1/2)]. It is instructive to discuss some of the general features. For the

*L*= 1 RSM cut, all peaks that appear for rotations only also appear for

*A*-site displacements only, whereas the peaks with

*H*=

*K*=

*n*/2 [ex. (1/2 1/2 1)] are extinct. For the

*L*= 3/2 cut, the

*H*=

*K*=

*L*[ex. (3/2 3/2 3/2)] is extinct for the octahedral rotations only and has finite intensity for

*A*-site displacements.

Given the large number of half-order peaks that arise from the distortions within *Pbnm*-type perovskites, it is critical to analyze the contribution of individual displacements (α, γ, *d*_{1}, and *d*_{2}) to the peaks in order to identify key peaks allowing to efficiently measure and analyze peak intensities for accurate structural refinement. The reflections to be analyzed must include those that are the most sensitive to *α*, *γ*, *d*_{1}, and *d*_{2}. For an *a*^{−}*a*^{−}*c*^{+} pattern, the *a*^{−} rotations result in peaks consisting of three half-order intensities (where two of the three are equal), for example (1/2 1/2 3/2) or (1/2 3/2 3/2), while the *c*^{+} rotations result in peaks consisting of half-order (but unequal) indices on *H* and *K* with *L* equal to an integer, for example (1/2 3/2 1) or (1/2 5/2 2). When there are no in-phase rotations along one of the in-plane directions, the octahedral rotations do not result in any half-order peaks where only *H* or *K* is an integer and the other two indices are half-order [ex. (1/2 1 3/2)]. The above conditions are all consistent with those first reported by Glazer in 1975.^{38} Unexpectedly, we also find that octahedral rotations result in weak (*n n n*/2) peaks, such as (1 1 1/2), when *α*, *β*, and *γ* are all non-zero. Unlike the peaks with at least two half-order indices described above, the (*n n n*/2) reflections only arise from the cooperative motion of all three rotations, and, as such, are completely absent when any of the individual *α*, *β*, or *γ* rotations are equal to 0°. The *d*_{1} parameter acts to displace the *A*-sites along opposite ⟨110⟩-directions along the + rotation axis. In the *a*^{−}*a*^{−}*c*^{+} pattern, this leads to a unit cell doubling along the [001] direction, which produces intense (*n n n*/2) reflections seen in the second column of Fig. 3. The *d*_{2} displacement, shown in Fig. 2(f), is the main contribution for the *H* = *K* = *L* half-order reflections, such as (1/2 1/2 1/2).

Figure 4 illustrates how the intensity of important Bragg reflections for the *a*^{−}*a*^{−}*c*^{+} structure is affected by the different parameters *α*, *γ*, *d*_{1}, and *d*_{2}. Here, peak intensity is plotted as a function of a single parameter while keeping the other three fixed (herein, we take the *A*-site cation to be La^{3+}, and the *B*-site cation to be V^{3+}). The (0 0 4) peak [Fig. 4(a)] is invariant to *γ*, *d*_{1}, and *d*_{2} and exhibits a small decrease in intensity with increasing *α*. This can be understood as the *γ*, *d*_{1}, and *d*_{2} displacements do not shift atomic positions along the [001] direction, and, thus, do not alter scattering from (0 0 *L*) reflections, whereas the *α* and *β* rotations displacing oxygen along [001]. The (1/2 1/2 3/2) and (1/2 1/2 5/2) peaks [Figs. 4(b) and 4(c)] exhibit no contribution from *γ* or *d*_{1}, but predominantly depend on *α*. The *d*_{1} parameter most strongly affects the (1 1 1/2) intensity [Fig. 4(d)], while *d*_{2} is the only parameter that contributes to the (1/2 1/2 1/2) reflection [Fig. 4(e)]. Finally, (*n*/2 *n*/2 *n*) peaks [Figs. 4(f)–4(h)] are only weakly dependent on *d*_{2} and change with *γ*, albeit in a non-monotonic manner.

In the case of a film with the *a*^{−}*a*^{+}*c*^{−} rotation pattern or mixed *a*^{−}*a*^{+}*c*^{−}/*a*^{+}*a*^{−}*c*^{−} domains expected to form when grown on a substrate with square in-plane lattice, similar physical arguments can be made to determine the displacement contributions to various peaks. The in-phase *α* rotations would produce (*n*_{1}/2 *n*_{2} *n*_{3}/2) peaks, such as (1/2 1 3/2) and (1 1/2 3/2) where *n*_{1} ≠ *n*_{3}. The out-of-phase *α* and *γ* rotations would yield (*n*_{1}/2 *n*_{2}/2 *n*_{3}/2) peaks, except for *n*_{1}* = n*_{2}* = n*_{3} and the *n* integers are odd. The *A*-site displacements would be along ⟨101⟩ directions and therefore the (*n n*/2 *n*) and (*n*/2 *n n*) peaks would be most sensitive to the *d*_{1} parameter. The *H* = *K* = *L* half-order peaks would continue to arise from the *d*_{2} parameter. The (0 0 *L*) peaks would now depend on α, *d*_{1}, and *d*_{2} as the *A*-site displacements contain motion along the [001] direction.

Based on the different intensity contributions to the half-order peaks, we propose and validate a strategy for refining atomic structure based on a judicious selection of Bragg peaks

An integer (0 0

*L*) peak should be measured. We recommend*L*= 4 as the increased separation in reciprocal lattice vector,*q*, between the film and substrate peak, compared to*L*= 2, is useful in minimizing intensity contributions from the substrate. The intensity of the half-order peaks should be normalized to this (0 0 4) peak, as it has minimal intensity dependence on the oxygen and*A*-site displacements.The next step is to identify the orientation of the in-phase axis. This is achieved by measuring (1 1/2 3/2), (1/2 1 3/2), and (1/2 3/2 1), or any other series of constant-

*q*peaks with one integer and two unequal half-order indices. The presence of the integer index along*H*,*K*, or*L*indicates the orientation of the in-phase axis. For example, if the (1/2 3/2 1) is present but the (1/2 1 3/2) and (1 1/2 3/2) are absent, then the film has an*a*^{−}*a*^{−}*c*^{+}rotation pattern.Having identified the in-phase rotation axis, a series of peaks with positive and negative indices on

*H*and*K*should next be collected to probe the volume fraction of rotational domains,*D*. The (1/2 1/2 3/2), (−1/2 1/2 3/2), (1/2 −1/2 3/2), and (−1/2 −1/2 3/2) peaks should be measured; equal intensities of these peaks are indicative of equal rotational domain populations._{j}Continuing to assume an

*a*^{−}*a*^{−}*c*^{+}pattern, two additional (*n*/2*n*/2*n*) peaks—(1/2 5/2 1) and (1/2 3/2 2)—should be measured to allow for*γ*to be accurately refined.The (1 1 1/2) and (1/2 1/2 1/2) peaks should be measured allowing for

*d*_{1}and*d*_{2}parameters to be extracted. If the rotational domain populations are not equal, then these peaks should also be measured with different combinations of positive and negative*H*and*K*values.To obtain

*α*, the (1/2 1/2 3/2) and (1/2 1/2 5/2) peaks should be measured.

Assuming an *a*^{−}*a*^{−}*c*^{+} pattern and an equal volume *D _{j}* of the 4 rotational domains, it is demonstrated how Eqs. (1) and (2) can be used to determine the oxygen and

*A*-site positions from the following 8 peaks: (0 0 4), (1/2 1/2 3/2), (1/2 1/2 5/2), (1/2 1/2 1/2), (1 1 1/2), (1/2 3/2 1), (1/2 3/2 2), and (1/2 5/2 1). An effective algorithm has been found by minimizing the square error with respect to each parameter individually. The total squared error was calculated using

where *I _{C,j}* is the calculated intensity of the

*j*th reflection using Eq. (2),

*I*is the measured intensity of the reflection, and

_{M,j}*j*is summed over the set of peaks proposed to refine the parameter,

*P*=

*α*,

*γ*,

*d*

_{1}, or

*d*

_{2}. The prefactor of 1/

*I*is used to equally weight the error from peaks of varying intensity, for example, the (1/2 1/2 3/2) versus the (1/2 1/2 5/2) reflection that typically differ by nearly one order of magnitude. First, by fixing

_{M,j}*α*=

*γ*=

*d*

_{1}= 0, the

*d*

_{2}value is determined by minimizing

*E*

_{d}_{2}using the (1/2 1/2 1/2) peak. Second, fixing

*d*

_{2}at this value and using γ =

*d*

_{1}= 0, the

*α*value is refined by minimizing

*E*for the (1/2 1/2 3/2) and (1/2 1/2 5/2) peaks. Similarly, the (1 1 1/2) peak intensity is used to obtain

_{α}*d*

_{1}, followed by the (1/2 3/2 1), (1/2 5/2 1), and (1/2 3/2 2) peaks to obtain

*γ*. This sequence (

*d*

_{2}→

*α*→

*d*

_{1}→

*γ*→

*d*

_{2}…) is repeated until values for all the parameters converged, which is typically achieved after 4–5 iterations.

To further test the robustness of this algorithm, we simulated intensities (*I _{S,j}*) for each of these 8 peaks using

where *I _{C,j}* is the intensity calculated from Eq. (2) using input values of

*α*,

*γ*,

*d*

_{1}, and

*d*

_{2};

*R*is a random number from −1 to 1; and

_{rand}*R*is the maximum amount of deviation introduced to the calculated intensity. The addition of a random error to the calculated intensities is to mimic experimental error. The results of this fitting routine, carried out for 4200 generated data sets, are shown in Fig. 5 with each point representing a separate data set and fit; nine different combinations of

_{j}*α*and

*γ*were used in the simulations with fixed

*d*

_{1}= 0.015 and

*d*

_{2}= 0.01 and with

*R*= 0.03, 0.10, 0.15, and 0.25. It can be seen that the obtained

_{j}*α*and

*γ*parameters cluster around the correct values. The standard deviations (

*Δα*,

*Δγ*) are plotted for

*R*= 0.10 in Fig. 5(b), and it can be seen that they are on order of 5% of the

_{j}*α*and

*γ*values. The standard deviation increases with increasing

*α*and

*γ*but

*Δα/α*and

*Δγ/γ*decrease with increasing

*α*and

*γ*. We find that the standard deviations for the refined

*α*values are less than those of the

*γ*values indicating that the determination of

*α*is more robust against random error than

*γ*, when refining the structure based on the peaks we have identified.

To confirm the validity of our approach, we have measured and analyzed the half-order diffraction peaks identified above to refine the structure of a NdGaO_{3} (001) single crystal. The measurements were performed at the Advanced Photon Source at Sector 33-BM. We obtain Ga-O-Ga bond angles of 153.4° and 149.0°, which compare well with the previously reported bulk values of 153.9° and 154.0°.^{65} The magnitude of the Ga displacements away from the ideal perovskite corner positions is 0.33 Å in our refinement, compared to 0.23 Å in the bulk. If the structure is refined without accounting for *A*-site displacements, by setting *d*_{1} = *d*_{2} = 0, then the refinement yields much larger values of α and γ as the total intensity of the half-order peaks must be accounted for solely through oxygen displacements. Our refinement without *A*-site displacements results in Ga-O-Ga bond angles of 125.2° and 133.2°, illustrating the importance of including both *A*-site and oxygen displacements in analysis of half-order peaks from *Pbnm*-type films. In these refinements, we assumed the presence of only two rotational domain populations; further improvements to the refinement may be possible by including additional volume fractions of the two other domains. Additional details of the NdGaO_{3} refinements can be found in the supplementary material.

The approach outlined here is applicable not only for refining the structure of a single sample but can also be used in experimental planning for hypothesized scenarios prior to carrying out the diffraction measurements. Furthermore, data sets can be kept small and can even be analyzed during ongoing experiments, thus allowing for the determination of structural modifications as a function of continuous parameters. For example, with the recent improvement in pump-probe experiments, there is great interest in exploring structural modification on ultrafast time scales, to discover hidden or metastable phases.^{66} By utilizing advanced ultrafast linear- and synchrotron-beamlines with high-brightness sources and temporal resolution on the sub-picosecond time scale, the proposed method will be key in correlating structural dynamics to underlying changes in the electronic ground state on these timescales. Further, there are a host of temperature-dependent phase transitions known in *Pbnm*-type perovskites coupled to electronic ground states, for example, the metal-insulator transition in rare earth nickelates, which has been shown to be susceptible to strain or other types of structural modification.^{67,68} Therefore, there is a need to reliably track the structure around the critical temperature of phase transitions using a general model that allows for spatially inhomogeneous films. We believe that this optimized approach for refining the structure of *Pbnm* perovskite thin films can play a key role in developing a deeper understanding of coupled structural-electronic phase transitions in oxide films, heterostructures, and superlattices.

## IV. CONCLUSIONS

An approach was outlined to refine the structure of strained perovskites by analyzing the contributions of oxygen octahedral rotations and *A*-site displacements to a variety of half-order diffraction peaks particular to the *a*^{−}*a*^{−}*c*^{+} structure. A specific set of peaks arising from individual displacements of the oxygen and *A*-site ions has been identified allowing for the quantification of these displacements through the analysis of selected diffraction peaks. The accuracy of this approach has been verified with numerical simulations. We anticipate that the approach presented here can be widely adopted by the oxide thin film community to better understand the local structural origins of strain-induced electronic and magnetic phenomena and to effectively pursue experimental validation of first principles-based predictions of new ferroic phases in *Pbnm*-type superlattices.

## V. SUPPLEMENTARY MATERIAL

See supplementary material for additional information on atomic positions and calculated intensities.

## ACKNOWLEDGMENTS

M.B. and R.E.-H. were supported by the Department of Energy (Grant No. DE-SC0012375). A.K.C., C.R.S., and S.J.M. were supported by the National Science Foundation (DMR-1151649). We acknowledge Hirofumi Akamatsu, James Rondinelli, Phil Ryan, and Jong-Woo Kim for valuable discussions and Jenia Karapetrov for assistance with synchrotron diffraction measurements. This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357.