The complete Ref [1] is also included in the discussion of this erratum. The central conclusions of our work (e.g. enhanced size-dependent piezoelectricity in nanostructures due to flexoelectricity) remain the same. In fact, corrected theoretical results in the case of BaTiO 3 in piezoelectric phase compare better with atomistics than the original publication [1]. We provide here the corrected equations and for completeness, the revised figures as well.

An error was found in the constitutive equations Eqs. ͑19͒ and ͑20͒ in our paper. Here we correct them. The central conclusions of our work ͑e.g., enhanced size-dependent piezoelectricity in nanostructures due to flexoelectricity͒ remain the same. In fact, corrected theoretical results in the case of BaTiO 3 in piezoelectric phase compare better with atomistics than in our original publication. We provide here the corrected equations and for completeness, the revised figures as well.The correct constitutive equations are 11 = YS 11 + dP 3 − fP 3,3 ͑1͒ E 3 = aP 3 + dS 11 + fЈS 11,3 ͑2͒By means of Poisson's equation and in the absence of free charges and applied voltage ͑open circuit condition͒ the electric displacement is D 3 = 0 E 3 + P 3 = 0 ͑3͒Equations ͑2͒ and ͑3͒ lead to:Hence, the correct effective beam bending rigidity ͑Eq. ͑38͒ in our paper͒ becomesThe beam bending rigidity has the elastic, the piezoelectric, and the size dependent flexoelectric contributions.The effective electromechanical coupling factor k ef f can be defined from energy consideration as the square root of the ratio of the convertible energy ͑electric energy͒ to the total input energy ͑mechanical energy͒ ͑see, e.g., Ref. 1 and 2͒.By means of Eqs. ͑2͒ and ͑4͒, the effective electromechanical coupling factor k ef f reduces to:Hence, the normalized effective piezoelectric constant ͑with bulk piezoelectric constant͒ isWe used the flexoelectric constants values estimated by one of us 3 from ab initio calculations on BaTiO 3 ͑BT͒ as f BT = 5.46 nC m . The piezoelectric constant of BT is taken from Ref. 4 d BT = −4.4 C m 2 . We note that the flexoelectric constants values from ab initio calculations are three orders of magnitude lower than the experimental estimates reported by Cross et al. 5 In addition, the existence of such a large discrepancy between the ab initio calculations 3 and the experimental values 5 was also confirmed by the work of another independent group from Cambridge. 6 The possible reasons behind this discrepancy are discussed in details in Ref. 3.The piezoelectric-flexoelectric interaction term incorrectly found in our paper vanishes in the revised solution. Hence, our nonpiezoelectric results in our paper remain valid and the size-dependent behavior due to pure flexoelectricity is seen at the nanoscale ͑in good agreement with atomistic simulations see Fig. 1͒. However, in the piezoelectric case ͑see Fig. 2͒, the size dependency is also due to the existence of flexoelectricity and is appreciable down to few nanometers instead of micrometers ͑as was found in our paper͒.In Fig. 2, the effective piezoelectric response is between 3 to 4 times the bulk values at sizes around 2 nm. Our theoretical results show that the effective piezoelectric response is doubled at 2 nm and increase up to four times the piezoelectric constant PHYSICAL REVIEW B 79, 119904͑E͒ ͑2009͒

Thin films of high-permittivity dielectrics are considered ideal candidates for realizing high charge density nanoscale capacitors for use in next generation energy storage and nanoelectronics applications. The experimentally observed capacitance of such film nanocapacitors is, however, an order of magnitude lower than expected. This dramatic drop in capacitance is attributed to the so-called "dead layer" -a low-permittivity layer at the metal-dielectric interface in series with the high-permittivity dielectric. Recent evidence suggests that this effect is intrinsic in the sense that its emergence is evident even in "perfectly" fabricated structures. The exact nature of the intrinsic dead-layer and the reasons for its origin still remain somewhat unclear. Based on insights gained from recently published ab initio work on SrRuO 3 /SrTiO 3 /SrRuO 3 and our first principle simulations on Au/MgO/Au and Pt/MgO/Pt nanocapacitors, we construct an analytical model that isolates the contributions of various physical mechanisms to the intrinsic dead layer. In particular we argue that strain-gradients automatically arise in very thin films even in complete absence of external strain inducers and, due to flexoelectric coupling, are dominant contributors to the dead layer effect. Our theoretical results compare well with existing, as well as our own, ab initio calculations and suggest that inclusion of flexoelectricity is essential for qualitative reconciliation of atomistic results. Our results also hint at some novel remedies for mitigating the dead layer effect.

Recent work suggests that flexoelectricity causes significant enhancement of electromechanical coupling of nonuniformly strained piezoelectric and nonpiezoelectric nanostructures below a material-dependent length scale. In the present work, employing an atomistically informed dynamical continuum model that accounts for flexoelectricity, we argue that in a narrow range of geometric dimensions, piezoelectric nanostructures can dramatically enhance energy harvesting. Specifically, in the case of lead zirconate titanate ͑PZT͒ material employed in the form of cantilever beams, our results indicate that the total harvested power peak value can increase by 100% around 21 nm beam thickness ͑under short circuit conditions͒ and nearly a 200% increase may be achieved for specifically tailored cross-section shapes. The key ͑hereto undiscovered͒ insight is that the striking enhancement in energy harvesting is predicted to rapidly diminish ͑compared to bulk͒ both below and above a certain nanoscale structural length thus providing a rather stringent condition for the experimentalists.

In a recent erratum, 1 we fixed an error found in the constitutive Eqs. ͑19͒ and ͑20͒ in Ref. 2. As a consequence, we proceed to give the revised expressions for Eqs. ͑9͒ and ͑10͒ in Ref. 3. The fundamental conclusions of our work ͑e.g., dramatic enhancement in energy harvesting for a narrow range of dimensions in piezoelectric nanostructures͒ remain valid. The revised beam "renormalized" bending rigidity ͓Eq. ͑9͒ in Ref. 3͔ and the effective electromechanical coupling coefficient k ef f ͓Eq. ͑10͒ in Ref. 3 redefined from energy considerations͔ as established in Ref. 1 are, respectivelyThe piezoelectric-flexoelectric interaction term incorrectly found in Ref. 3 vanishes in the revised solution. The sized dependency does occur only because of flexoelectricity. It is worth to mention that there exists a discrepancy between the flexoelectric constants values determined from ab initio calculations and those obtained from experimental data. For example, in the case of BaTiO 3 ͑BT͒, the flexoelectric constants estimated from ab initio calculations 4 are three orders of magnitude lower than the experimental estimates reported by Ma and Cross. 5 In addition, the existence of such a large discrepancy between the ab initio calculations 4 and the experimental values 5 was also confirmed by the work of another independent group from Cambridge. 6 The piezoelectric constant of lead zirconate titante ͑PZT͒ is taken from Ref. 7 as d = −5.4 C m 2 . Here, we use the flexoelectric constants values estimated from both ab initio calculations on lead titanate in the order of f =1 nC m as reported by Ref. 6 and from experimental estimates by Ma and Cross 8 as f = 0.5 C m . We also report the results for the total harvested power in both cases ͑with flexoelectric constants estimated from ab initio calculations 6 ͑see Fig. 1͒ and from experiments 8 ͑see Fig. 2͒.Results ͑Figs. 1 and 2͒ indicate the same maximum enhancement ͑200% for short circuit and 30% for open circuit͒ for the harvested power due to flexoelectricity for both scenarios ͑with flexoelectric constants estimated from ab initio calculations 6 and from experiments 8 ͒. However, the enhancement is seen at different sizes in both cases. In the case of the ab initio calculations, 6 the enhancement start at around tens of nanometers and the harvested power peak occurs at around 2 nm. In the 0 5 10 15 20 0 0.5 1 1.5 2 Thickness h (nm) Normalized Power Harvested power for PZT beam (short and open circuits) short circuit with flexoelectricity short circuit without flexoelectricity open circuit with flexoelectricity open circuit without flexoelectricity FIG. 1. ͑Color online͒ Harvested power as function of beam thickness for short and open circuit resonances with flexoelectric constants estimated from ab initio calculations ͑Ref. 6͒. Solid lines correspond to the harvested power for classical piezoelectric beam. The dashed and dotted lines show a size dependency of the harvested power which nearly doubles for the short circuit ͑green or light gray͒ and is enhanced by 30% for the op...