Competition between Polar and Nonpolar Lattice Distortions in Oxide Quantum Wells: New Critical Thickness at Polar Interfaces

Two basic lattice distortions permeate the structural phase diagram of oxide perovskites: antiferrodistortive (AFD) rotations and tilts of the oxygen octahedral network and polar ferroelectric modes. With some notable exceptions, these two order parameters rarely coexist in a bulk crystal, and understanding their competition is a lively area of active research. Here we demonstrate, by using the ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ system as a test case, that quantum confinement can be a viable tool to shift the balance between AFD and polar modes and selectively stabilize one of the two phases. By combining scanning transmission electron microscopy (STEM) and first-principles-based models, we find a crossover between a bulklike ${\mathrm{LaAlO}}_3$ structure where AFD rotations prevail, to a strongly polar state with no AFD tilts at a thickness of approximately three unit cells; therefore, in addition to the celebrated electronic reconstruction, our work unveils a second critical thickness, related not to the electronic properties but to the structural ones. We discuss the implications of these findings, both for the specifics of the ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ system and for the general quest towards nanoscale control of material properties.

Figure 1

(a) Schematic structures of the polar displacements and octahedral tilts in ${\mathrm{LaAlO}}_3$ film of thicknesses below and above the critical one for 2DEG formation in ${\mathrm{SrTiO}}_3$ in ${\mathrm{LaAlO}}_3$/${\mathrm{SrTiO}}_3$ heterostructures: $<4\mathrm{u}.\mathrm{c}.$ (left panel) and $>4\mathrm{u}.\mathrm{c}.$ (right panel), respectively. (b),(c) Atomic-scale mapping of the interface. Upper panel, (b),(c) $Z$ contrast (right) and ABF (left) simultaneously acquired STEM images of 3- and 7-u.c.-thick ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ samples, respectively, viewed along the [110] zone axis. The $[110{]}_{\text{pc}}$ projection allows visualizing the O sublattice buckling. Lower panel, zoomed out regions of the ABF images of the 3- and 7-u.c.-thick ${\mathrm{LaAlO}}_3$ layers along with schematics of the ${\mathrm{LaAlO}}_3$ structure showing the revealed distortions. Scale bar, 1 nm.

Figure 2

(a),(b) Schematics of the distortions revealed by the ABF images shown in Fig. 1. The perovskite $AB{\mathrm{O}}_3$ unit cells viewed along the [110] zone axis show the distance $\delta$ from the center of mass of the unit cell (marked with a cross) and that of the $B$ and O1 and O2 sites for $<4\mathrm{u}.\mathrm{c}.$ (a) and $>4\mathrm{u}.\mathrm{c}.$ (b) thick ${\mathrm{LaAlO}}_3$ layers. The schemes show the displacements (not in scale) of $B$ and O atoms from the center of the unit cell. From left to right, (c) and (d) show ABF and inverted ABF images along with the averaged values of the distance $\delta$ of the $B$ (black squares) and O (red circles) sites from the center of mass of each unit cell (right panel) as a function of position of the 3- and 7-u.c.-thick samples, respectively. Scale bar, 1 nm.

Figure 3

(a) [110] view of the ${\mathrm{LaAlO}}_3$ structure showing octahedral tilts. (b) $B\text{-O-}B$ tilt angle along the $z$ axis, as a function of thickness. The tilting of the 3-, 5-, and 7-u.c.-thick samples are shown in red, blue, and black, respectively. The error bars show the standard deviation with respect to the average for each atomic layer. (c) Energy per unit cell associated to the tilts as a function of the dielectric displacement $D_z$. The solid blue circles show the first-principles results for a uniform tilt pattern, calculated as the energy difference between a distorted and undistorted geometry. The solid and dashed blue curves are fits, respectively including or excluding the additional ${\varphi }^2{D}_z^4$ term. The empty symbols are model results for confined tilt patterns: Results for 2-u.c. (black circles), 3-u.c. (red squares), and 5-u.c. (green diamonds) films are shown. The electric displacement $D_z$ is in units of $e/2a_0^2$. (d) Phonon dispersion curves of the reference undistorted (but elastically strained) cell along the RM segment in the Brillouin zone. The dashed red curve shows a quadratic fit of the lowest unstable branch. (e) Distortion amplitude as a function of film thickness [same color and symbol code as in (b)], as predicted by the model Eq. (1). The 2-u.c. film with asymmetric boundary conditions (solid line) is shown as an equivalent 5-u.c. segment with symmetrical blocking boundary conditions (dashed line). The solid symbols correspond to the amplitudes extracted from an explicit first-principles calculation (see Supplemental Material [22]). (f) AFD distortions (black symbols) in a thin (3-u.c.-thick) ${\mathrm{LaAlO}}_3$ layer deposited on a ${\mathrm{SrTiO}}_3$ substrate, with the internal field neutralized via external surface charges (see Supplemental Material [22]). In the “pristine” system (red symbols), the tilts as they are suppressed by the strong internal field.