Octahedral rotations in strained LaAlO₃/SrTiO₃ (001) heterostructures

The perovskite family of oxides shows a wide range of functionalities enabled by strong electron-spin-orbital-lattice interactions. This is reflected in the different structural instabilities found in these systems, e.g., oxygen octahedral tilting, octahedral distortions, and cation displacements,¹ all of which are associated with phase transformations when cooling from the typically undistorted, high temperature cubic phase. While substitutional doping has long been used to tailor the properties of these systems, researchers are now taking advantage of thin film synthesis techniques to examine the effects of epitaxial strain and thickness on perovskite structure and properties.^2,3 Such previous work has, for example, led to an improved understanding of ferroelectricity and enabled fine control over ferroelectric properties with strain engineering.⁴ However, recent studies have shown that progress in the field of oxide electronics will also require better understanding the interactions between strain, crystal symmetry, and octahedral behavior, such that this knowledge can be utilized to construct new materials exhibiting a myriad of different electronic and ferroic properties.^5–10 

Here, we report an investigation into the effects of biaxial strain and film thickness on octahedral rotations in LaAlO₃/SrTiO₃ (LAO/STO) (001) heterostructures. This system has stimulated a large number of experimental and theoretical studies (see, e.g., Refs. 11 and 12, and references therein) but few have considered octahedral rotations and their effect on material properties.^13,14 We find that LAO strained in biaxial tension favors the Imma phase, with an elevated transition temperature with respect to bulk LAO for thicker films. The rotations are suppressed near the STO interface, with a transition region approximately 4 unit cells thick, which may have implications on polarization^15,16 and the magnitude of the dipole field in films below the critical thickness for 2D electron gas formation.¹⁷ 

In bulk form, LAO cools from a high temperature cubic

At room temperature, bulk LAO has a pseudocubic lattice parameter of a_pc = 3.791 Å with α = 5.7°.²¹ When lattice-matched to STO (a = 3.905 Å), LAO is in biaxial tension with a strain of ∼3.0%. The STO substrate maintains the space group

We employed the pulsed laser deposition technique to grow LAO films on TiO₂-terminated, low miscut (<0.1°) STO (001) substrates, with a substrate temperature of 700 °C and an oxygen pressure of 1.3 × 10⁻⁶ mbar. A KrF excimer laser (λ = 248 nm) with a fluence of ∼1 J/cm² was used to ablate a single crystal LAO target. Reflection high energy electron diffraction was used to monitor growth, allowing sub-nanometer control over thickness. The electrical characteristics of LAO/STO heterostructures grown under identical conditions have been reported elsewhere.^22,23

Synchrotron X-ray measurements of the LAO/STO samples were conducted at the Advanced Photon Source (APS) at Sector 12ID with an X-ray energy of 24 keV. Specular crystal truncation rod (CTR) 00L and symmetry-inequivalent 10L, 11L, and 20L CTRs were recorded with the maximum value for the vertical reciprocal space coordinate L_max = 2.5 (∼4 Å⁻¹ in Q_z) and the sampling density of 100–200 points per reciprocal lattice unit. The samples were placed in a quartz-walled chamber, permitting measurements in a wide range of environmental conditions. Here, the sample temperature was controlled between 25 °C and 800 °C with a fixed oxygen partial pressure of 200 mbar. Several film thicknesses were studied, and detailed measurements of the film structure were conducted for heterostructures with 9- and 24-unit-cell-thick LAO films.

Room temperature scans along the specular 00L rod, where L is parallel to the [001] direction, for the 9- and 24-unit-cell films are shown in Figs. 2(a) and 2(b). The high uniformity and smooth interfaces of the samples are evidenced by the pronounced thickness fringes. Both films were coherently strained to the STO substrate as determined by scans (not shown) along the in-plane directions.

In addition to the 00L rod, several non-specular CTRs were measured at room temperature, and COherent Bragg Rod Analysis (COBRA) was used to reconstruct the 3D electron density distributions of both samples. COBRA is a phase-retrieval algorithm that has been proven useful in the study of buried interfaces in epitaxial heterostructures.^24–28 The resulting electron density profiles and extracted integrated electron numbers for both the AO and BO₂ layers (where A = Sr or La and B = Ti or Al) along the out-of-plane direction, z, are shown in Figs. 2(c)–2(f) for the 9- and 24-unit-cell-thick LAO films. The out-of-plane c lattice parameter across the LAO/STO interface, taken as the distance between consecutive A-site cations, can be extracted from the density profiles, with the methodology used for determining the error bars described in the supplementary material.²⁹ As seen in Figs. 2(g) and 2(h), c shows a gradual transition from the bulk value of STO to a relatively constant value in the LAO layer. Due to Poisson contraction, the c lattice parameter of coherently strained LAO should decrease to a value of 3.73 Å. While c for the 24-unit-cell sample matches that this prediction, the average lattice spacing of the 9-unit-cell sample is slightly larger with c ∼ 3.77 Å. Cancellieri et al.³⁰ have shown that the larger lattice spacing is due to electrostriction from an intrinsic dipole field within the LAO layer. When the field is fully compensated, the electrostriction-induced lattice expansion is suppressed. A small residual field thus appears to remain within the 9-unit-cell-thick film while it is screened for the thicker sample.

The tilting of the oxygen octahedra effectively doubles the lattice parameter (as shown in Fig. 1) so Bragg peaks from the LAO appear at both integer and half-order positions when referenced to STO reciprocal lattice units (r.l.u.). Scans along the in-plane H-direction are shown in Fig. S1 of the supplementary material,²⁹ displaying

As discussed by Glazer,¹ the intensities of the half-order peaks are directly related to the magnitude of octahedral rotations, and the absence or existence of peaks at

For quantitative modeling of the half-order intensities and the determination of α = β and γ, we measured 14 peaks from the 9-unit-cell-thick film and 15 peaks from the 24-unit-cell-thick film. These intensities were compared with those from a model assuming scattering only from oxygen atoms. Following Ref. 33, the structure factor of these reflections is given by

where f_O(Q) is the Q-dependent form factor for oxygen, B_O is the Debye-Waller factor, and r_n is the position of each oxygen in the 2 × 2 × 2 unit cell. If one considers this unit cell (Fig. 1), it is apparent that there are four inequivalent ways of orientating the cell on the STO substrate with the O₆ octahedra rotated about the any of the four ⟨111⟩_pc body diagonals. The total intensity at each half-order position then corresponds to the incoherent sum of scattered intensities from the four domains, i.e.,

where S is a scale factor and ν_j is the volume fraction of the jth domain. Similar to the procedure of May et al.,³³ we do not directly compare the measured intensities to the modeled intensities but rather the intensities relative to one of the peaks (here, the

Comparisons between the experimental and modeled intensities are shown in Figure 3. Fits to the a⁻a⁻c⁻ symmetry give γ = 1.3 ± 0.3° and γ = 0.0 ± 0.3° for the 9- and 24-unit-cell data, respectively, indicating that the system tends toward the a⁻a⁻c⁰ tilt system. Reducing the number of fitting parameters by constraining γ to zero gave better fits, resulting in α = β = 5.5 ± 0.7° for the 9-unit-cell film and α = β = 5.1 ± 0.3° for the 24-unit-cell film. Furthermore, there were no indications of diffuse scatter adjacent to the half-order reflections (or integer reflections), as shown in the H-scans of Fig. S1 of the supplementary material,²⁹ unlike the case for epitaxial manganite thin films.^34–36 This suggests that the LAO films do not relax into a monoclinic structure, which would lead to monoclinic domain formation.³⁷ Consequently, LAO forms the Imma phase when strained to STO, as predicted in Ref. 14.

As was shown in previous reports,^24,38,39 the structure factor of a CTR from an epitaxial heterostructure corresponds to the Fourier transform of the “folded” electron density distribution, ρ:

where R_{‖i, j} and r_‖ refer to the in-plane position of the (i, j) unit cell and the in-plane position of an atom within that cell, respectively. This electron density can be depicted by laterally translating all of the atoms in the film's unit cells into that of the substrate-defined unit cell using the substrate's in-plane unit cell vectors; in this way, detailed information regarding the octahedral rotations is encoded into the CTRs. As shown in Fig. 4(a), oxygens displaced by octahedral rotations would appear as doublets in the folded structure. The atomic positions determined from COBRA can therefore be used to quantitatively determine the degree of rotation as functions of x, y, and z, similar to the method for determining spatial distribution of ferroelectric displacements in structures with 180° stripe domains.²⁵ 

A slice through the COBRA-derived electron density distribution along the (200) plane is shown in Fig. 4(b) for the 9-unit-cell sample. As seen, the equatorial oxygen ions (i.e., the O_II ions in the BO₂ plane) appear broader along z than any other ion, but only within the LAO film. The effect is elucidated in Fig. 4(c), where the full width at half maximum (FWHM) of different ions are plotted as a function of z. Thus, while the sampled in-plane scattering range was too small to allow the resolution of individual oxygen ions within a doublet, information regarding the octahedral rotations is contained in the full width of the O_II electron density peaks. Since the same peaks within the STO substrate are known to stem from single O_II ions, they can be used as a reference to convert the broadened O_II peaks into out-of-plane displacements, and thus into the rotational angle α. The resulting z-dependent angles are shown in Fig. 4(c) with the right-hand axis. As seen, α increases gradually from zero at the interface and nears the bulk value 5.7° once reaching ∼4 unit cells, with an average value of 4.1 ± 1.9° that is in good agreement with the value determined from the fitting techniques described above. The possibility of non-zero values of γ at the interface was also investigated, but due to the poor in-plane resolution, values smaller than ∼1° were difficult to resolve.

Similar information could not be retrieved from the 24-unit-cell-thick film. The COBRA procedure considers only coherently scattered intensities, where the total scattering amplitude is the summation of amplitudes from each domain. As described in the supplementary material,²⁹ the sum over the amplitudes of the scattered oxygen ions cancel when the domain fractions are equal, leaving zero coherent intensity at each half-order position. Under this particular condition, octahedral rotations cannot be observed in the folded structure. However, due to the significant incoherent fraction of the incoming beam and the relatively large beam spot on the sample (∼1 mm²), Eq. (2) can still be used to determine the average rotations. The COBRA results from the 9-unit-cell film suggest that either the octahedral domain fractions are not exactly equal in this sample or other structural deviations between the different domains prevent phase cancellation, a point that will be discussed further below.

In order to better understand the nature of Imma phase in these heterostructures, we measured the evolution of the

Upon cooling, the 24-unit-cell-thick sample exhibits a phase transition from the high symmetry (strained and unrotated) tetragonal phase, P4/mmm, to the tilted octahedra phase, Imma, at ∼695 °C, near the growth temperature. This elevated transition temperature, about 150 °C above the bulk value, results from the biaxial strain state of the film, which favors γ → 0 and enlarges the Imma phase field. In contrast, the 9-unit-cell-thick film has a transition temperature of approximately 540 °C, although it shares the same strain state as the thicker film. A dependence of the tilt transition temperature on film thickness was also observed by He et al.⁴⁰ for STO films, who attributed the phenomenon to the different unit cell volume in ultrathin films. From the COBRA results, the difference in unit cell volume between the 9- and 24-unit-cell samples is 0.8%.

When the LAO film is grown on STO at 700 °C, the results in Fig. 5 imply that the initial deposit remains in the P4/mmm phase until reaching a thickness of ∼24-unit-cells, at which point α begins to increase. The higher mobilities at elevated temperature favor the formation of an equilibrium octahedral domain fraction, with larger in-plane correlation lengths. However, if a thinner film is grown and cooled through the Imma transition temperature, such as for the 9-unit-cell film, domains of equal volume fraction and large correlation lengths may not be able to form because of decreased boundary mobility.

Another key difference between the 9- and 24-unit-cell-thick films is the larger c lattice constant in the former, implying the presence of a polar field.³⁰ Since octahedral rotations originate from a desire to shorten the A–O bond length while maintaining the octahedral environment of the B-cation, the rotation magnitude and transition temperature are strongly strain dependent.^40–42 Electrostriction can thus lead to a change in strain state (i.e., unit cell volume) and lower the transition temperature in ultrathin films. While intermixing of Sr and Ti into LAO would produce a similar effect, the COBRA results indicate that the intermixing lies primarily on the STO side of the interface, in agreement with previous studies.²³ 

The appearance of tilted octahedra in the 3-unit-cell-thick LAO film is not surprising since biaxial tension stabilizes in-plane rotations for this particular tilt system. Indeed, rotations were previously observed in LaNiO₃ layers only 1-unit-cell thick in LaNiO₃/SrMnO₃ superlattices (where bulk LaNiO₃ is a⁻a⁻a⁻ and SrMnO₃ exhibits no rotations),⁷ but the rotation magnitude was substantially reduced, returning to bulk-like values only after reaching a distance of ∼4 unit cells from the interface, much like the results in Fig. 4(c). (The thickness of this transition region also contributes to lowering of T_t in the 9-unit-cell film.) Using scanning transmission electron microscopy (STEM), similar interfacial coupling lengths were found by Borisevich et al.,⁴³ who directly observe a transition region at the BiFeO₃/La_0.7Sr_0.3MnO₃ interface, and by Jia et al.,¹³ who studied LAO/STO superlattices and discovered LAO-induced rotations in the STO, unlike the findings presented here.

In general, the interfacial coupling length depends on both the difference in rotational patterns between the two parent systems and the difference in the cationic species.⁹ While the relevant density functional calculation has not yet been performed for LAO/STO, He et al.⁴⁴ calculated a coupling length of ∼4 unit cells for a chemically sharp La_0.75Sr_0.25MnO₃/STO interface, which exhibits the same rotational mismatch as for the current study. This consistency between the experimental and theoretical results indicates that 4 unit cells may be a typical coupling length for many a⁻a⁻a⁻/a⁰a⁰a⁰ interfaces.

Such a transition region can exhibit significant distortions in the octahedra in order to maintain octahedral connectivity, including Jahn-Teller elongation and polar displacements.⁴⁴ In the present case, the appearance of atomic buckling within the BO₂ planes can be extracted from the COBRA-derived 3D electron density profiles. As shown in Fig. 4(d), the 9-unit-cell sample displays negligible buckling within the STO substrate but gradually more buckling starting from the interface, reaching a value of ∼0.13 Å at 3 unit cells. This evolution of atomic buckling as a function of z inside LAO is similar to those observed in previous surface x-ray diffraction and STEM studies^45–47 and again indicates the presence of a residual field within the 9-unit-cell-thick film, in agreement with the electrostrictive behavior shown in Fig. 2(g). Thus, while the polar displacements within the first 4 unit cells of LAO may be affected by the rotational transition layer, the octahedra are nevertheless distorted by the intrinsic field, which is expected to be uniform throughout the film thickness. We note that while this film is larger than the 4-unit-cell critical thickness for 2D electron gas formation⁴⁸ and expected to be fully screened, the degree of charge compensation can depend on several factors, including the magnitude of the ionic displacements¹⁷ and redox processes occurring at the surface^49,50 and interface.⁵¹ Since there may be several different contributors to interfacial conduction in this system, the effects of our structural observations on electronic properties will require additional studies isolating the conduction mechanism. Furthermore, while there have been consistent findings of a polar distortion within the LAO, contrasting observations have been reported for buckling in the STO unit cells below the interface,^28,45–47 again reflecting some of the complexities of this system and indicating possible sample dependent behavior, as discussed in the supplementary material.²⁹ 

Since the larger B–O–B bond angles in the initial layers of LAO are expected to lead to wider bands and a narrower band gap, the present findings contribute to the on-going discussion concerning the critical thickness necessary for 2D electron gas formation.¹² In addition, while electronic structure studies were not performed here, coupling between the observed structural distortions and order parameters in the transition region, e.g., roto-flexoelectric coupling,¹⁶ may lead to considerable differences in the local density of states as compared with those calculated when rotations and displacements are ignored. Local polarization can also stem from the octahedral domain boundaries,⁵² and the subtle amounts of buckling and tilts at the immediate STO interface can be crucial to the resulting interfacial transport and magnetic behaviors.⁵³ Regardless, further computational studies incorporating the rotational and displacement patterns observed here will be necessary to determine the impact of the transition region on the electronic properties of the interface.

Finally, we recall that substitutional doping has long been used to tailor properties in oxide systems. For instance the substitution of Pr for La in LaAlO₃ is known to lead to Imma phase stabilization.⁵⁴ Based on our results, we find that strain is able to effectively mimic this doping effect. However, since a change in oxide composition is unnecessary, strain provides an additional and independent parameter for phase modification in perovskites.

Using synchrotron x-ray scattering, we have conducted detailed investigations into the atomic-scale structure of LAO/STO (001) epitaxial systems. The coherently-strained LAO films are in the Imma phase at room temperature, in agreement with Ref. 14, but a 4-unit-cell transitional layer with reduced rotation magnitudes exists at the interface. The transition temperature from

The observation of half-order reflections in a 3-unit-cell-thick film indicates that octahedral rotations about the in-plane ⟨100⟩ directions are stable in ultrathin LAO films, but the in-plane correlation length is relatively short as compared with thick films due to the lower transition temperature.

While these structural results add to the growing literature concerning the complexity of the LAO/STO interface, several additional studies should be conducted as points of comparison. For instance, a detailed study of (110) interface that lacks a polar discontinuity may prove illuminating.⁵⁵ In addition, the surface x-ray diffraction technique used here averages over the in-plane directions such that the out-of-plane transitional layer can be easily observed but in-plane inhomogeneities cannot. Due to the relatively short correlation length, an octahedral domain structure exists in these films, with greater domain boundary densities at lower thickness; others have seen that such local disorder can lead to interfacial magnetism.⁵⁶ Future studies employing nanobeam x-ray probes may be useful in mapping the mesoscale in-plane structure. As others have recently discovered, such inhomogeneities may provide the key to understanding emergent behavior in complex oxide systems.^57,58

The authors thank J. M. Rondinelli and P. Zapol for valuable discussions. This work was supported by the U.S. Department of Energy, Basic Energy Sciences, Materials Sciences and Engineering Division. Use of the Advanced Photon Source was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.