The role of the depolarization field

For many years, it has been known that reducing the thickness of ferro-electric materials leads to a degradation of their ferroferro-electric properties.

This is explained as being due to bound charges at the surfaces of a ferro-electric slab creating an ferro-electric field, called thedepolarization field, which points in the opposite direction to the spontaneous polarization and hence destabilizes it. The bound charges can be screened by free charges at the surfaces of the ferroelectric, thereby reducing the depolarization field.

Such free charges can be carriers in the metallic electrodes, ions from the atmosphere or simply mobile charges from within the semiconduct-ing ferroelectric itself. Their screensemiconduct-ing ability (namely how close they can come to the surfaces modulated by the dielectric constant of the inter-face layer) is described by an effective screening lengthλ_eff. In a simple model considering the interfaces as parallel plate capacitors, and under short circuit boundary conditions, the depolarization fieldE_d is given by:

2.3 Depolarization field as the origin of 180^◦ferroelectric domains

E_d∼ −^{P λeff}

0t

4, for a film of thicknesstand polarizationP.

Note that throughout this entire thesis, the depolarization field we are referring to, is the field that would be generated by a uniform polarization in a capacitor with a finite screening length, and that serves as the driving force for domain formation; it is not the stray field associated with the actual polydomain ground state.

λeff reflects the screening properties of the electrode, which strongly depend on the quality of the interface between the film and the electrode as well as on interface chemistry [21]. Ideal metallic electrodes would cor-respond toλeff = 0, resulting in a perfect cancellation of the depolariza-tion field. In reality, however, even for structurally perfect metal-insulator interfaces, the screening charges spread over a finite length λI, and the finite dielectric constant of the interface layer_I contributes to the effec-tive screening length viaλ_eff =λ_I/_I. More details about this model can be found in Refs. [22–25]. Ab initio calculations by Stengel et al. [23]

highlight the importance of interface chemistry in affecting the screening.

For instance, although SrRuO₃ is considered a good metal with a good ionic polarizability, its electronic capabilities have been proved not to be the best, resulting into a so-called dead-layer at the interface. Junquera et al. [26] have shown that BaTiO3 in between SrRuO3 electrodes loses ferroelectricity below 6 unit cells due to the presence of dipoles at the in-terfaces generating a depolarization field. Such results imply that even though incredible improvements have been made in the growth quality of oxide interfaces, their screening properties are still intrinsically limited.

The calculations show that for screening often the best electrodes to use are single elements such as platinum or gold. Saiet al.[27] have reported an enhancement of the polarization in thin films of PbTiO3 and BaTiO3

with platinum electrodes compared to oxide metallic electrodes. Stengel et al.[21] have theoretically demonstrated the importance of the chemi-cal bonds at the metal-oxide interface. The ferroelectricity is found to be enhanced whenever those bounds are polarizable. Experimentally, in ad-dition to the interface chemistry, strain, defects and/or grain boundaries also contribute to imperfect screening that leads to depolarization fields.

Because of a strong depolarization field, the system might lose its

po-4A potential drop is induced at each interface. Because of short-circuit boundary condi-tions, this results in a voltage drop across the sample equal to the sum of the voltage drops at each interface (but of opposite sign). This voltage drop depends therefore only on the two interfaces and is independent of the film thickness. It thus results in a stronger electric field for thinner samples (∆V =E·t).

Figure 2.6: The different responses of a ferroelectric to the presence of a depolarization field. The screening charges from atmospheric adsor-bates, metallic electrodes or defects in the film itself can screen the bound charges, reducing the depolarization field. In the absence of screening, the ferroelectric can either lose ferroelectricity (P = 0) or form domains.

Reproduced from Ref. [25].

larization. Nevertheless, even in the absence of good screening, ferro-electrics can find other ways of preserving their polarization as illustrated in Fig. 2.6 [25]. For example, the system could “break” into domains of opposite polarization forming so-called 180^◦ domains known as Kittel or Landau-Lifshitz domains. Such domains were observed in ultathin films of PbTiO₃ using synchrotron X-ray diffraction by S.K. Streiffer et al. in 2002 [28].

2.3 Depolarization field as the origin of 180^◦ferroelectric domains

When a sample forms domains, although the polarization within each domain may remain large, its macroscopic properties are significantly al-tered. It is therefore important to understand and control the formation of these domains. For example, the effects of screening on domain forma-tion were studied in samples with different electrodes [28–30], dielectric spacers [31, 32] and ionic adsorbates [33, 34]. Photochemical switch-ing of stripe domains was reported by Takahashi et al. in Ref. [35], whereas Ref. [36] investigated the possibility of polarization switching without domain formation in ultrathin films near the ferroelectric criti-cal temperature TC. More complex PbTiO3-based heterostructures, such as ferroelectric-dielectric (PbTiO3-SrTiO3) superlattices [8, 37–39], have been particularly useful for investigating the response of nanoscale stripe domains to applied fields and their effect on the macroscopic electrical properties [40, 41]. The formation of 180^◦ domains typically leads to the reduction or even suppression of the piezoelectric response and of the macroscopic polarization [29]. In addition, many theoretical studies have addressed the formation - as well as the microscopic and macro-scopic properties - of nanoscale stripe domains, revealing complex, inho-mogeneous polarization profiles [42–45] and unusual switching dynam-ics [46–48].