Microscopic origin and experimental validation of the decoupling between lattice tetragonality and ionic displacements (local polarization) in ultrathin epitaxial PbTiO3-based ferroelectric films

In bulk PbTiO3-based ferroelectrics, spontaneous polarization is assumed to scale tightly with ionic off-centre displacements and hence with the unit-cell tetragonality ($c/a$). Unit-cell-resolved transmission electron microscopy of an ~8 nm epitaxial PbZr0.2Ti0.8O3 film on a SrRuO3 electrode found the opposite in the film interior: tetragonality $c/a \approx 1.08$, larger than bulk PbTiO3, coexisting with Zr/Ti and (inferred) oxygen displacements smaller than the bulk PbTiO3 values. The microscopic mechanism of this decoupling, its validation by direct oxygen-position measurement, the effect of composition-dependent effective ionic charges on the quantitative polarization, the role of electrical boundary conditions in the interfacial suppression, and the generality of the effect across growth conditions and thickness remain unresolved. This problem asks for a validated, mechanism-level, quantitative account of the tetragonality–polarization decoupling and of interfacial polarization suppression in ultrathin PbTiO3-based ferroelectric films, establishing which of growth kinetics, defect structure, interfacial chemistry, partial ionic sharing with electrode or substrate, and electrostatic compensation (stripe-domain or flux-closure) physics dominates, and whether the decoupling is a general feature of such films.

Background

A ferroelectric is a crystal with a switchable spontaneous polarization $P_{S}$. In the perovskite oxide PbTiO3 (PTO) — a crystal in which Ti sits inside an octahedral cage of oxygen ions with Pb at the cell corners — and in its solid solution $\mathrm{PbZr}_{x}\mathrm{Ti}_{1-x}\mathrm{O}_{3}$ (PZT), $P_{S}$ arises from small off-centre displacements of the Pb, Ti/Zr and oxygen ions away from their centrosymmetric cubic positions; the tetragonal phase distorts the pseudo-cubic cell so that the out-of-plane lattice parameter $c$ exceeds the in-plane parameter $a$, and the ratio $c/a$ ('tetragonality') is widely used as a structural proxy for the polarization magnitude. A classical empirical relation connects displacements, tetragonality and $P_{S}$ in displacive ferroelectrics (Abrahams, Kurtz and Jamieson, 1968), and quantitative $P_{S}$ values can be computed from ionic displacements through the charge-sum formula $P_{S} = (1/v)\sum_{i} \delta_{i} Z_{i}$, where $v$ is the unit-cell volume, $\delta_{i}$ the displacement of ion $i$ and $Z_{i}$ its Born effective charge — the proportionality between the polarization change and the displacement of an ion, obtainable from first-principles calculations. In bulk PTO the oxygen displacement $\delta_{O}$ and the Ti displacement $\delta_{\mathrm{Ti}}$ are directly measured quantities (e.g. Glazer and Mabud, 1978), and first-principles effective charges for PbTiO3 and related perovskite alloys are available (Bellaiche and Vanderbilt, 1999).

When such ferroelectrics are grown as epitaxial films only a few nanometres thick, the polarization state becomes a competition between ferroelectric distortion and electrostatics: the unscreened bound surface charge creates a depolarizing field that suppresses out-of-plane polarization. Theory predicted a critical thickness below which ferroelectricity disappears (Junquera and Ghosez, 2003), while synchrotron x-ray scattering on PbTiO3/SrTiO3 showed stable ferroelectricity and 180-degree stripe domains that periodically compensate the surface charge (Streiffer et al., 2002; Fong et al., 2004). The thickness and boundary-condition dependence of tetragonality itself was mapped by x-ray diffraction (Lichtensteiger, Triscone, Junquera and Ghosez, 2005), and the standard continuum description of interfacial suppression is the Kretschmer–Binder model, in which the order-parameter profile near a boundary is characterized by an extrapolation length $\lambda$ and a bulk correlation length $\xi$; interfacial layers with strongly reduced polarization are commonly called 'dead layers'.

Against this background, Jia, Nagarajan, He, Houben, Zhao, Ramesh, Urban and Waser (Nature Materials 6, 64–69, 2007; DOI 10.1038/nmat1808) mapped, at the unit-cell scale, the lattice parameters and the cation-column displacements in an ~8 nm pulsed-laser-deposited epitaxial PbZr0.2Ti0.8O3 film on a SrRuO3 (SRO) metallic electrode on SrTiO3 (STO). Their protocol combined 400 kV high-resolution transmission electron microscopy (HRTEM), least-squares Gaussian fitting of cation contrast minima averaged along the interface, and an iterative quantum-mechanical image-simulation/model-refinement loop correcting aberration, defocus and tilt artefacts. Because oxygen columns were not directly imaged under those conditions, oxygen positions were imposed through the linear relation $\delta_{O} = 2.96\,\delta_{\mathrm{Ti}}$ derived from bulk PbTiO3 data (Abrahams et al. 1968; Bellaiche and Vanderbilt 1999; Glazer and Mabud 1978), and $P_{S}(x)$ was computed with effective charges taken from PbTiO3. Their findings were: the out-of-plane lattice parameter peaks at $c = 0.422$ nm ($c/a \approx 1.08$, above the bulk PbTiO3 value of about 1.05) roughly four unit cells from the SRO interface; the Zr/Ti displacements (and the inferred oxygen displacements) peak in the interior but are significantly smaller than the corresponding bulk PbTiO3 values, so the polarization–tetragonality coupling fails; $P_{S}$ decreases toward both interfaces, with visibly stronger suppression at the free surface than at the SRO electrode; the thickness-averaged polarization (about 40 $\mu$C cm$^{-2}$, i.e. a switchable polarization of about 80 $\mu$C cm$^{-2}$) agrees with electrical measurements on comparable films; and a Kretschmer–Binder fit gives $\lambda \approx 1.9$ nm and $\xi \approx 3.35$ nm at the PZT/SRO interface. Direct imaging of oxygen atomic columns in perovskites was already known to be feasible with negative spherical-aberration imaging (an aberration-corrected transmission-electron-microscopy mode in which a negative lens spherical-aberration coefficient makes light oxygen columns appear as strong, directly measurable contrast; Jia, Lentzen and Urban, 2003), and later with annular bright-field scanning transmission electron microscopy (STEM), a mode in which even light oxygen columns are imaged directly; however, neither method had been applied to this heterostructure. The study itself flagged the issues that remain open and that constitute this problem: the unknown mechanism of the decoupling; the unmeasured oxygen positions; the use of PbTiO3 effective charges for a PbZr0.2Ti0.8O3 film; the unquantified electrical boundary conditions (leakage, finite electrode screening length, dynamic compensation); and the absence of systematic variation of growth parameters, thickness, composition and electrode configuration.

Problem Statement

For epitaxial ultrathin PbTiO3-based ferroelectric films (the archetype being ~8 nm PbZr0.2Ti0.8O3 on a SrRuO3 bottom electrode on SrTiO3), determine the microscopic origin of the experimentally observed decoupling between lattice tetragonality and ionic off-centre displacements, and validate it quantitatively. Specifically, establish which of the candidate mechanisms — growth kinetics, defect structure, interfacial chemistry, partial ionic sharing with the electrode or substrate (sharing of the polar ionic displacements by ions of the electrode or substrate, which screens the ferroelectric bound charge ionically and incompletely rather than as a rigid electronic conductor), or electrostatic compensation physics such as stripe domains or flux-closure patterns (arrangements in which the polarization alternates or rotates to close on itself so that the bound surface charge is compensated) — dominates the reduction of ionic displacements relative to the measured tetragonality in the film interior, and how the interfaces (ferroelectric/electrode and free surface) and the electrical boundary conditions (leakage, finite electrode screening length, and dynamic compensation processes) control the depth profile and the observed stronger suppression of the local spontaneous polarization near the free surface compared with the electrode interface. The validation requires: (a) direct imaging of the oxygen atomic columns in the relevant heterostructure (by any method that resolves the light oxygen columns), rather than estimating oxygen positions from the linear relation $\delta_{O} = 2.96\,\delta_{\mathrm{Ti}}$ derived from bulk PbTiO3 data, in order to test the inferred oxygen-displacement profile and the quantitative local spontaneous polarization $P_{S}(x)$; (b) quantification of the impact of composition-dependent (and strain-dependent) Born effective ionic charges of PbZr0.2Ti0.8O3 versus the bulk PbTiO3 values on the absolute $P_{S}$ magnitudes; and (c) systematic variation of growth parameters, film thickness, composition and electrode/boundary configuration, together with chemically resolved interface characterization, to establish whether the decoupling of tetragonality and polarization is a general feature of PbTiO3-based ultrathin films or specific to particular growth, defect and electrode conditions. The single overarching objective is a validated, mechanism-level, quantitative account of how local ferroelectric polarization, ionic displacements and tetragonality are related, and how interfaces and electrical boundary conditions suppress them, in ultrathin PbTiO3-based films — the deliverable being that account itself, to which each of the component measurements and calculations above is necessary but not separately sufficient, and for which a direct oxygen-position measurement that instead finds the ionic displacements scaling with the measured tetragonality resolves the decoupling sub-question by falsifying the reported decoupling.

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