Size effects in nanoscale ferroelectrics

Abstract

Ferroelectrics are among the most advanced candidates of fast non-volatile memory materials. How do the properties of the commonly used perovskites such as PbTiO₃, Pb(Zr_xTi_1−x)O₃ (PZT) and BaTiO₃ change with size? Is there a fundamental limit showing up below which ferroelectricity irrevocably ceases? While the operating voltage as the predominant driving force for commercial applications shifted the thickness down to a few unit cells, ferroelectrics are now on the verge of true nanoscale integration of laterally confined structures. Top-down, bottom-up approaches and their combination provide samples far below 100 nm and indicate that the interaction of electrode and ferroelectric becomes increasingly relevant in terms of strain, screening of the depolarization field and fatigue resistance. As the qualitative understanding of nanoscale ferroelectricity advances the ferroelectric limit appears to be below 10 nm thus paving the road for further miniaturization.

Introduction

Ferroelectrics make use of two thermodynamically equivalent groundstates of the spontaneous polarization that can be reversed by and external electric field. This type of memory is therefore non-volatile. As the structures are shrinking in thickness, the operation voltage has dropped below one volt, and the switching speed is only limited by the RC time constants of the circuit [1]. So we practically have a fast and energy-efficient non-volatile but charge-based memory. How small can this device become and still provide a detectable amount of charge? The commonly used perovskites are wide-bandgap semiconductors so dc-conductivity can be omitted for bulk samples where the detected switching current is solely displacive. The polarization as expressed by a bound surface charge density (C/m²) is thickness independent but rapidly shrinks with area (see Fig. 1). For typical materials we are left with only 6000 electrons for a capacitor of 100 nm × 100 nm. A device of 10 nm × 10 nm will therefore have to operate with 60 electrons.

Irrespective of all challenges to operate at these current levels, a fundamental question arose from the analogy to ferromagnetism. How small can such a structure be made and still be ferroelectric [2]? The situation is fundamentally different from superparamagnetism as the coupling force has a very long interaction length as compared to magnetism. The unit cell dipoles are electrostatically coupled and strain also propagates several unit cells across the material. This is why the phenomenological mean-field Landau–Ginzburg–Devonshire theory was successfully employed to describe many features of ferroelectricity and its phase transitions on a macroscopic scale. But as the structures shrink down to the length-scales of ferroelectric coupling we should wonder to what extent the assumptions of mean-field theories still hold true. And indeed, in order to account for some size-driven effect, the polynomial description of the free-energy had to be expanded by two more terms: a surface energy term and a polarization gradient term both of which contain parameters that are not experimentally accessible [3], [4], [5], [6]. Ab initio calculations are on the way together with a comprehensive survey on thin films provided by Dawber et al. [7]. The surface plays a major role as its volume fraction increases with decreasing overall volume: the polarization gradient is a direct consequence of a modified surface. There is plenty of experimental evidence that the surface behaves different from the bulk [8]. Talking about the surface it has to become clear that ferroelectricity in contrast to mere pyroelectricity does not only involve a structural prerequisite (polar axis) but the need for reversibility which goes far beyond the structure. So the mere existence of a non-centrosymmetric phase with a polar axis is necessary, but not sufficient to pinpoint ferroelectricity. A pyroelectric in that sense is a ferroelectric where the breakdown-voltage is smaller than the coercive field. Once we want to show functionality, we have to apply electrodes which inherently modify all relevant electromechanical boundary conditions: strain is applied to the system, the depolarization field that promotes the formation of ferroelectric domains is partially screened in the electrodes and we have to consider a system of electrode–ferroelectric–electrode. In all cases where we cannot be certain about the quality of our fabrication it is required to also consider two interfaces between electrode and ferroelectric. These interfaces have turned out to be responsible for many effects in ferroelectric devices (imprint, parasitic capacitances, dielectric losses, etc.).

What should a comprehensive theory of nanoscale ferroelectricity take into account? Clearly, there will be a structural change as the surface gains volume fraction. But what else may be important? Some recent experiments point into two more directions that have not been considered so far. For one, the chemical composition of a three-component system like all perovskites (ABO₃) is a lot more difficult to control than in most one-component nanomagnets. Growth itself already is an issue [9] but the exact chemical composition is almost unknown and difficult to access as the volumes of ferroelectric nanostructures are small and often averaging is not feasible. The second issue directly originates from the analogy to the superparamagnetic limit: The soft-mode phonon (TO) (in analogy to the Larmor-frequency) tries to perform just the movement that is required to switch the structure. With an increasing number of ferroelectric unit cells it becomes exponentially unlikely that for a given temperature the structure will ever spontaneously switch. As we talk of maybe a thousand unit cells left (4 nm)³ this exponential term may no longer be lasting for ages but come to the timescale of our experiment since the soft-mode phonon will try at about 10¹² s⁻¹ to flip the polarization state. For ferromagnets this effect is known and clearly puts the thermodynamic limit to the miniaturization of magnetic memories.

All the aforementioned three contributions: structure, composition and time will also depend on the shape of our structures as a dipole–dipole interaction is highly anisotropic and as strain can be used to intensify the tetragonality and therefore the polarization.