Short communication Enhancing accuracy to Stoney equation J.M. Pureza a, *, M.M. Lacerda b , A.L. De Oliveira b , J.F. Fragalli b , R.A.S. Zanon b a Department of Mechanical Engineering, Santa Catarina State University, UDESC, Joinville 89223-100, Brazil b Department of Physics, Santa Catarina State University, UDESC, Joinville, Brazil 1. Introduction Stress and strain of materials permeate several areas of science related to technological advance, in special for layer-structured samples in micro- and nanotechnology [1–3], such as graphene [4] and multilayer structures [5–7]. These subjects also play an important role in applications in micro- and nanoelectromecha- nical systems (MEMS and NEMS, respectively) [8], anti-corrosive barriers [9], protective surface coatings [10], opto-electronic devices [11], nano-porous metals [12–14] and bulk metallic glasses [15]. Differences between thermal expansion coefficients may generate huge intrinsic stress in the film, being the analysis essential to prevent cracks and failures on the film–substrate interface [2,9,16]. The first and most important description of the problem was proposed by Stoney [17] who considered an one-dimensional sample and obtained a linear relationship between its bending and the film stress (s). Stoney equation can be straightforwardly modified for two-dimensional systems with small deformation by including the substrate Poisson ratio (n s ), as seen in Eq. (1) [18]: s ¼ E s t 2 s 6ð1  n s Þt f K (1) where E s is the substrate Young modulus, t s and t f are the substrate and film thickness and K is the film curvature. This equation has become the standard expression for the analysis of this problem with reasonable agreement with experimental results. In fact, regarding the accuracy of the procedure (t s > 10t f ), the Stoney equation appears to hold for large thickness ratios, which is attributed to self-compensating errors in its derivation [1]. Stoney equation does not take into account some relevant aspects like the non-uniformity of the stresses, the tri-dimension- ality of the real samples, the boundary conditions of the samples and multilayer films [5–7]. Such aspects give rise to semi-empirical modifications, with little experimental agreement [2,19,20]. Moreover, a series of theoretical works based on Timoshenko and Woinowski-Krieger formulations [21,22] generated more elaborated models (see, e.g. [1] and references therein [23]), as well as finite-element simulations that present local dependence to stress distribution [5,7]. For instance, Finot et al. [7] using a finite element analysis identified three distinct regimes for the evolution of curvature, which are dependent on the quantity A ¼ st f l 2 s t 3 s , where l s is the characteristic size of the sample. The regimes are distinguished (I) for lower values of A, the deformation has a spherical shape and Stoney equation is satisfied; (II) as the parameter A increases, deformation maintains the spherical shape but Stoney equation looses validity and (III) for even larger values of A, Stoney equation is no longer valid and the sample undergoes two abrupt changes, initially to an ellipsoidal shape and finally to a cylindrical shape. Applied Surface Science 255 (2009) 6426–6428 ARTICLE INFO Article history: Received 19 September 2008 Received in revised form 28 January 2009 Accepted 30 January 2009 Available online 11 February 2009 PACS: 62.20.Dc 81.40.Jj 68.60.Bs Keywords: Stress Strain Stoney equation Deformation energy ABSTRACT This paper proposes a three-dimensional system for modelling stress in thin films deposited on plane substrates much thicker than the film itself. The approach is the minimization of the deformation energy of the sample (substrate + film), considering the deformed substrate as a small spherical surface with large curvature radius. Results of the model show the validity limits of the well-established Stoney equation that satisfy the upper theoretical limit of Poisson ratio (n) for isotropic materials. Our main result is that the stress values obtained in general literature using Stoney equation are underestimated when considering a typical Poisson ratio for substrates in the range of 0.25  n s  0.4. ß 2009 Elsevier B.V. All rights reserved. * Corresponding author. Tel.: +55 47 4009 7821; fax: +55 47 4009 7940. E-mail address: pureza@joinville.udesc.br (J.M. Pureza). Contents lists available at ScienceDirect Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc 0169-4332/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2009.01.097