Erratum: Compositional and phase dependence of elastic modulus of crystalline and amorphous Hf_1-xZr_xO₂ thin films (Appl. Phys. Lett. (2021) 118 (102901) DOI: 10.1063/5.0044702)

The following Erratum has been prepared by the authors: In the original published article,1 calculations of elastic moduli did not appropriately account for the uniaxial strain state imposed by the constraining Al transducer, TaN bottom electrodes, and silicon substrates probed with picosecond acoustic measurements. The original calculation stated that, according to Eq. (1) in the manuscript, Young’s modulus could be calculated directly using only longitudinal sound velocity and film density. However, this equation is only valid if (Table Presentec) the material probed is in a uniaxial stress state, in which the solid is free to move in the plane orthogonal to the wave propagation direction. Given that the vibrating atoms are constrained by a surrounding material, it is more accurately approximated as a uniaxial strain state, and so Eq. (1) was incorrectly utilized in the manuscript. Instead, the elastic moduli may be calculated as (Formula presented) where C11 is an axial normal component of the elastic stiffness tensor. Given that the material is assumed to be isotropic, only two elastic constants are required to specify all stress-strain relationships. Using Poisson’s ratio of 0.292–4 already assumed for crystallized HZO in the original manuscript, Young’s modulus, E, can be estimated as shown in Eq. (2). The ultimate effect of this correction is a reduction in each reported elastic moduli by 24%. This reduction does not change the conclusions or implications of the article. An updated Table I is provided in this Erratum. Table S3 in the supplementary material has been updated to reflect these changes as well. An updated Fig. 3 from the manuscript is also shown in this Erratum, which accounts for the changes in moduli calculation. Furthermore, as shown in Table I, the tensile biaxial stressmagnitudes resultant from processing, calculated from sin2ðwÞ analyses of area detector diffraction patterns, are also lower than reported from the manuscript due to the corrected moduli calculation. An updated Fig. 4 from the manuscript is provided in this Erratum, which reflects the stress calculations with the correctly calculated elastic moduli. In tandem, supplementary Fig. S13 has been updated in the supplementary material file to reflect the corrected modulus calculation. The reductions in elastic moduli by 24% resulted in commensurate reductions in calculated biaxial stresses. All biaxial stresses remain tensile, and the relative orders of elastic moduli and tensile biaxial stress magnitude in relation to HZO composition and phase remain unchanged. coauthored by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The authors acknowledge Professor Sean R. Agnew from the University of Virginia for his feedback and guidance in making this correction..