Rhombohedral perovskites like La_0.7Sr_0.3MnO₃ (space group R-3c) are characterized by equal antiphase tilts of the BO₆ octahedra about the three Cartesian axes, α = β = γ (see Figure 1A), resulting in a net rotation ω about the pseudocubic [111] direction α = β = γ = ω/3^1/3 (Haun et al., 1989). According to Glazer, 1972, this octahedral tilt pattern is noted as a⁻a⁻a⁻, where the components of the triad describe the octahedral tilts about the unit cell axes of the undistorted cubic aristotype (Pm-3m, a⁰a⁰a⁰). As a result of this tilting, the unit cell interaxial angles, α_c, are no longer 90° (Megaw and Darlington, 1975), inducing a misfit shear strain χ = tan(α_c − 90°) relative to the underlying (001) plane of the cubic substrate. Note that since the substrate does not impose constrains on the inclination of the out-of-plane pseudocubic axis, the initial −3m rhombohedral point symmetry is reduced to a 2/m monoclinic one (Daumont et al., 2010). Therefore, the total strain energy of the rhombohedral film, E_T, splits into two contributing elastic energies, a shear and a normal one, E_T = E(χ) + E(δ). In the absence of further perturbations at the interface, and assuming that the lattice distorts elastically (i.e., that strain couples with the B–O bonds, with no octahedral rotations), dislocation models (Matthews, 1975) predict a relaxation of E(δ) at critical thickness of ~10 nm (Abellán et al., 2009), in strong contrast with the observation of fully strained in-plane lattice parameters up to 475 nm (Sandiumenge et al., 2013). Conversely, E(χ) is expected to be relieved from the initial growth stages by twinning on the (100) and (010) planes perpendicular to the interface (Farag et al., 2005), again in conflict with experimental observations pointing the occurrence of a shear transition at a critical thickness of about 2.5 nm (Sandiumenge et al., 2013; Santiso et al., 2013). Such discrepancies constitute in fact a dramatic manifestation of the ability of octahedral frameworks to propagate interfacial perturbations over length scales exceeding a few 100 nm at a lower energy cost than that associated with the formation of the misfit dislocation network needed to plastically relax an equivalent elastic strain energy.

Misfit strains have important consequences on the electronic structure of the films. As shown in Figure 1C, the octahedral coordination of the Mn-sites splits the 3d orbitals into a degenerate t_2g triplet and a degenerate e_g doublet with xy/yz/xz and x² − y²/3z² − r² symmetries, respectively. Misfit strains induce further lowering of the symmetry that also unfolds the e_g doublet. This is commonly achieved by half-filling of the e_g orbitals, in which case the strong electron-phonon coupling induces a Jahn–Teller distortion of the coordination environment that breaks the degeneracy of the x² − y² and 3z² − r² orbitals, thus modifying their electron occupancy and eventually leading to orbital reconstructions. In the La_1-xSr_xMnO₃ solid solution, the electron occupancy of the e_g doublet depends on the strength of the Jahn–Teller distortion, which in turn is determined by the hole doping level, x (Tokura and Nagaosa, 2000). In epitaxial thin films, however, similarly to the Jahn–Teller distortion, misfit strains have the ability to break the x² − y²/3z² − r² orbital degeneracy (Fang et al., 2000) as recently confirmed by X-ray linear dichroism investigations (Huijben et al., 2008; Tebano et al., 2008; Pesquera et al., 2012). A thorough investigation taking into account the attenuation of the signal with the depth in the sample, suggests that the observed X-ray linear dichroism is rather a consequence of the superposition of interfacial and free surface effects (Pesquera et al., 2012). As shown in Figure 1C, those results suggest that a tensile strain would favor the occupancy of the in-plane x² − y² orbitals, while a compressive strain would induce the occupancy of the out-of-plane 3z² − r² ones, as expected from electrostatic arguments. On the other hand, the Mn³⁺ cations located at the free surface appear to exhibit a preferred 3z² − r² occupancy regardless of the sign of the interfacial biaxial strain. According to theoretical work (Calderon et al., 1999), this scenario is promoted by the absence of the apical oxygen in the coordination environment of surface cations and the resulting reduction of the repulsive electron–electron interaction between the manganese 3d (3z² − r²) and oxygen 2p (z) orbitals.

Despite the surprising adaptability of the octahedral framework, there are situations where plastic relaxation is energetically favored. As an example, La_0.7Sr_0.3MnO₃ films grown on LaAlO₃ substrates under a compressive strain of −2.3% are relaxed by misfit dislocations at a critical thickness of about 3 nm. This indicates that there is a limited range of rigid octahedral tilts within which the film can deform to fit the substrate. In this case, the value of the out-of-plane lattice parameter prior to the onset of plastic relaxation, 3.98(1) Å, is consistent with the value derived from the Poisson’s ratio ν = 0.33, 3.975 Å, indicating that the film behaves elastically. Note that on a rigid octahedron basis, see Figure 1A, a shrinking of the in-plane lattice parameters can be either accomplished by increasing the octahedral tilt γ about the c-axis, or by increasing the octahedral tilts simultaneously about the in-plane a and b axes. While the first operation would leave the length of the c-axis unaltered, the latter would even shrink it resulting in a strong contraction of the unit cell volume. Figure 1D shows the strain pattern associated with the misfit dislocation structure of a 7.1 nm thick film projected on the interface plane, using annular dark field scanning transmission electron microscopy at low beam convergence angles (Fitting et al., 2006). The density of the pattern can be manipulated through the thickness of the film. Thus, misfit dislocations in such compounds provide a means for the self-organization and spatial modulation of strain patterns in a strongly susceptible functional matrix, expanding the possibilities of interfacial engineering beyond homogeneous manipulations of the octahedral pattern. Moreover, dislocations in complex oxides are known to exhibit distinct electrical properties compared to the host material (Szot et al., 2006). Although the control of their density and distribution has been extensively studied in other functional oxides in bulk form (Sandiumenge et al., 2000), the spontaneous formation of regular patterns at interfaces in heteroepitaxial systems offers unique opportunities for miniaturization.

Besides misfit strains, polar discontinuities are a common driving force for interfacial charge and orbital reconstructions. They appear as a result of the fact that individual atomic layers AO and BO₂ in ABO₃ perovskites can be electrically charged, as is the case of La_0.7Sr_0.3MnO₃ (see Figure 1B). The net charge per area unit is +2/3e and −2/3e in La_0.7Sr_0.3MnO₃, while it is null in SrTiO₃ (e is the electron charge). Polar discontinuities lead to a diverging electrostatic potential which in the absence of any reconstruction results in the so-called polar catastrophe (Nakagawa et al., 2006). Since La_0.7Sr_0.3MnO₃ exhibits metallic behavior, mobile charges can screen the diverging potential (Dagotto, 2009), resulting in a compensating charge transfer of 1/3e. This would lead to a shift of the Mn valency at the interface over a screening length of a few nanometers toward 3+, as confirmed by electron energy loss (Riedl et al., 2009), X-ray absorption (Lee et al., 2010), and x-ray photoemission (Sandiumenge et al., 2013) spectroscopies. According to the bulk La_1−xSr_xMnO₃ phase diagram (Dabrowski et al., 1999), the transfer of charge toward the interface results in the formation of a non-ferromagnetic and insulating wetting layer called ‘‘dead-layer.” As films thicken beyond ~2.5 nm, the number of e_g electrons decreases and become delocalized with strong consequences on the deformation behavior of the films; the electronic energy gained by removing the e_g orbital degeneracy becomes progressively exceeded by the elastic energy opposing a similar expansion of the equatorial Mn–O distances needed to keep the in-plane lattice parameters fully strained (see Figure 1C). This unbalance evolves until a fully elastically strained rhombohedral phase condenses at a thickness of 10–20 nm, similar to the critical thickness for plastic relaxation. Strikingly, above this thickness the elastic energy is released by a composite octahedral tilting mechanism, which allows to maintain fully strained in-plane lattice parameters up to at least 475 nm (Sandiumenge et al., 2013). This complex behavior clearly reflects the competing nature of the different degrees of freedom involved in the relaxation mechanism of such materials.

Thanks to the extraordinary progress achieved in the atomic control of interfaces in oxide heterostructures, the formation of the dead-layer can be avoided by precise modifications of the interfacial architecture (Boschker et al., 2012), or even polar discontinuities can be exploited to intentionally induce hole doping avoiding disorder effects typically associated with chemical doping (Dagotto, 2009).

A more subtle dissimilarity is that between the octahedral tilt systems of the film and the substrate. The concept of interfacial octahedral coupling resides in the idea that the substrate tends to transmit its octahedral pattern to the film. This concept has motivated an intense research on its use for the engineering of novel tilt patterns with specific functionalities (Rondinelli and Spaldin, 2011; Rondinelli et al., 2012; Moon et al., 2014). Excitingly, recent advances in electron microscopy have provided direct access to oxygen positions (Jia et al., 2009; Borisevich et al., 2010a), and though direct experimental evidences are still very scarce, indeed they support the idea of octahedral coupling across the interfaces.

As an example, Figure 1E shows the variation of octahedral tilt angles in a BiFeO₃ film grown on a 5 nm La_0.7Sr_0.3MnO₃ buffer layer fully elastically strained on top of a SrTiO₃ substrate, as determined by bright field scanning transmission electron microscopy imaging (Borisevich et al., 2010a). As can be seen in the right panel, the in-plane octahedral tilts of the La_0.7Sr_0.3MnO₃ buffer are suppressed by the influence of the cubic SrTiO₃ substrate, and this pattern is transmitted to the first unit cells of the bismuth ferrite film. Tilt angles in the BiFeO₃ layer almost recover their bulk values at ~10 u.c. ahead of the interface. Similarly, the propagation of the octahedral tilt pattern of a rhombohedral LaAlO₃ substrate into an epitaxial cubic SrTiO₃ film has been directly determined using the negative spherical aberration imaging technique (Jia et al., 2009). These experiments provide direct evidence for the coupling of the antiferrodistortive order parameter (octahedral tilting) at interfaces, previously postulated for the SrTiO₃/LaTiO₃ (Hamann et al., 2006; Okamoto et al., 2006) and SrTiO₃/LaAlO₃ systems (Willmott et al., 2007) systems. Notably, a similar octahedral tilt suppression has also been determined inside the twin walls of BiFeO₃ (Borisevich et al., 2010b), thus pointing an increased orbital overlap and thus bigger bandwidth as responsible for their enhanced conductivity (Catalan and Scott, 2009).