ESTIMATION OF MECHANICAL PARAMETERS OF THIN FILMS USING FINITE ELEMENT ANALYSIS

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1 Update:5/0/209 ESTIMATION OF MECHANICAL PARAMETERS OF THIN FILMS USING FINITE ELEMENT ANALYSIS H. A. Tinoco*,2,3, J. Holze 2, T. Piále 4, Z. Buchta 4, J. Laza 4, A. Chlupová 2, T. Kuml 2, P. Hutař 2 Abstact: This study shows a methodology to estimate mechanical paametes of thin films by means of a bulge test and a numeical appoach. The methodology is based on the combination of finite element analysis with a classical analytical method. Finite element modelling was conducted fo monolaye (Si3N4) membanes of 2x2mm with the aim to appoximate both the load-deflection cuves expeimentally measued and the classical loaddeflection analytical model. Eo functions wee constucted and minimized to delimit a coupled solution space between Young s modulus and Poison s atio. In a taditional bulge test analysis only one of the elastic popeties can be detemined due to that thee is not unique solution in the estimations of these paametes. Howeve, both elastic paametes wee detemined though the poposed numeical pocedue which compaes the defomed sufaces fo a specific set of optimal elastic paametes computed. Results show that the estimated elastic popeties agee with coesponding values detemined by othe methods in the liteatue. Keywods: Bulge test, thin film, finite element analysis, elastic popeties Intoduction Thin films ae used in seveal engineeing fields especially those elated to micoelectonics (ulta lage scale integated cicuits), micoelectomechanical systems (MEMS), nano-devices, coating applications, biomedical devices among othes []. Integated functionality has foced the pocessing of these stuctues to mico and nanomete scales. In this way, thin films equie contolled pocesses and obust instumentation to achieve the geometic conditions equied fo each application. Howeve, the contol of its mechanical behaviou pesents additional challenges if intinsic mechanical popeties of thin films ae unnown since it is not a tivial tas to identify it till this time. Diffeent expeimental techniques have been developed to estimate mechanical popeties of thin films, such as indentation, diffaction-based techniques, Raman spectoscopy, deflection techniques among othes [2-3]. In contast to the techniques mentioned above, a bulge test (deflection technique) can be highlighted and in compaison, with the othe methods it pesents seveal advantages that ae mentioned and discussed in [3]. The bulge test technique consists of applying pessue on a membane fo measuing the displacement field [4-5]. Expeimental load-deflection cuve was analysed using diffeent techniques to detemine the elastic paametes that satisfy the measuements, in which we can point out fitting techniques and finite element analysis [6-8]. These techniques assume that the elastic paametes ae decoupled o these ae mechanically independent since in some cases, Poisson s atio is fixed to obtain a solution fo Young s modulus. CEITEC-Cental Euopean Institute of Technology, Bno Univesity of technology, Technicá 0, Bno, Czech Republic 2 Institute of Physics of Mateials, Czech Academy of Sciences, Žižova 22, 66 62, Bno, Czech Republic 3 Expeimental and Computational Mechanics Laboatoy, Univesidad Autónoma de Manizales. Antigua Estación del Feocail, Edificio Sacatín C.P Manizales, Colombia 4 Institute of Scientific Instuments, Czech Academy of Sciences, Kálovopolsá 47, 66 64, Bno, Czech Republic

2 This pape shows that the coupling conditions of Young s modulus and Poisson s atio can be used fo detemining both elastic paametes combining finite element analysis with the classical identification method. 2 Mateials and Methods 2. Expeimental setup A bulge test appaatus was constucted fo thin films testing as shown in Figue a. The diffeential pessue is applied using an industial gade piston that pesses the ai by a compute-contolled syinge pump. The pessue is measued by a low-pessue tansmitte with maximal eo 0.02% FS which is connected to a data acquisition system. The shape of the membane is captued by an intefeometic system (Twyman-Geen-type intefeomete) in which the light souce is composed by a fibe-coupled HeNe lase with wave-length of 633 nm. The beam is split into a measuing beam that eflects off the measued sample and a efeence beam that eflects off a efeence mio with high suface flatness λ/0. The measuing beam then intefees with the plana efeence beam at the output of the intefeomete foming intefeence finges that ae pojected onto the camea senso using a camea lens (Nion 50mm f/.4 NIKKOR G). Intefeence signals captued on each camea pixel ae then used to detemine the displacement field at the z position which is nomal to the suface of the thin film. Numbe of points fo which the z position was evaluated is dependent on the size, but usually exceeds In a left pat of Figue b, the scheme of the setup is shown. Right pat of Figue b pesents intefeence patten coesponding to the tested membane defomation. The device is also equipped with sensos to measue the ambient tempeatue, pessue and humidity equied fo ai efactive index calculation. Detailed infomation about the expeimental setup can be found in [9]. Figue : a. Expeimental setup of bulge test. b. Scheme of the expeimental setup and image of the membane. 2.2 Classical analytical equation fo estimation of mechanical paametes Let s consides a ectangula thin film 2a 2b pe-stessed by and made of an isotopic elastic mateial that satisfies a linea stess stain elationship. Unde pessue conditions P, the shape of the thin film is defined by the suface defomation. In those conditions, [6] epoted a classical analytical equation that elates the maximum deflection w 0 and pessue P as follows P = C tw + C 2 2( v) Etw, 4 a a ()

3 whee C and C () 2 v ae constants that depend on geometic and mateial paametes, diffeent models and numeical estimations have been poposed fo both constants as descibed by [5], [7-8]. epesents the esidual stess, t is the thicness and E Young s modulus. In eal applications, expeimental data obtained fo w 0 and P ae fitted with the aim to detemine, E and Poisson s atio. Howeve, the values of C and C 2 ae assumed accoding to the chosen appoximation techniques as can be seen in [8], [0]. In a pactical sense, these constants should tae fix values but those depend on the techniques developed fo each poblem which in tun affect the identification pocess of popeties. The coupling of the Young s modulus and Poisson s atio have been neglected in the loaddeflection analysis since one of the paametes should be fixed to estimate one of the elastic popeties. To ovecome these challenges, the combination of the analytical model with finite element analysis shows that an estimation of both elastic paametes can be caied out, as it will be descibed in the next sections. 2.3 Numeical appoach fo detemining elastic popeties To detemine Young s modulus and Poisson s atio a sequential numeical pocedue is pesented in Figue 2 by means of a flow diagam. It consists in a set of 0 steps that pemit to obtain both paametes combining finite element analysis with the classical analytical solution (see Equation ). Initially, a finite element model with the equied geometic dimensions should be pefomed including its bounday conditions which ae consideed clamped in the extenal domain of the thin film as illustated both in Figue b and Figue 2. The bulge test modelling by finite element analysis is a well-nown pactice fo many yeas ago (see [6-0]) since the geometies ae vey simplified and in pactice the numeical analysis is not a complex engineeing tas consideing that mechanical paametes ae nown. Howeve, fo the analysis it is impotant to tae into account lage defomations since the thicnesses ae vey thin (nanometic scale). Lage defomations mean that the stiffness changes with the level of input load. Figue 2: Sequential pocedue fo estimating Young s modulus and Poisson s atio.

4 Following the poposed method, fist thee steps deal with the estimation of C and paametes fom the finite element model and expeimental data. C is dependent on the esidual stess as descibed in [3]. Theefoe, if with the finite element model a set of output data m is ceated with input paametes nown ( E j, j, v j j =,2,.. m ), it is possible to detemine C adjusting those output data in Equation. Posteioly, C is used to compute the esidual stess using the expeimental data fo any Young s modulus chosen in Equation. We suggest to choose a value close to mateials with simila mechanical chaacteistics since fo the tue solution is an initial value. In the steps 4, 5 and 6 the main objective is to establish a model fo C () 2 v constant only fom the simulations. Accoding to analytical solutions descibed by [6-0], C ( v) v v 2 = + function is pesented hee, consideing that diffeent values of, have been detemined with numeical and analytical appoximations as summaized by [3]. In ou case, we popose numeical estimations obtaining a set of two paametes E i and vi, i =, 2 that satisfy the load-deflection cuve obtained expeimentally. Then, with all paametes detemined ( Ei,, C ) and expeimental data, C 2( i) is obtained with both Poisson s atio found. So, paametes, ae calculated as follows ( ) ( ) ( ) ( ) () 2(2) 2 2(2) 2 2() = v C v v C v, = C v C v. v v v v (2) Using and values, we can calculate any value of C () 2 v with values of v nown. With all paametes calculated until the step 7, the following eo function can be mapped such as n Pexp( j) P( C2 ( v ), E, wexp( j) ) ep ( )( E, v) = / n P (3) j= exp( j) whee subscipt means a set of paametes E and v detemined fo each load-deflection cuve with n data. The esult of Equation 3 is an eo suface in which the minimum eos should be in the places whee a set of E and v satisfy the expeimental measuements. Accoding to pevious exploatoy data analysis done in ou study, it was found that a linea appoximation can define the set of optimal solutions fo E and v such as descibed in the step 8. Until this step, we found a set of paametes that appoximate the analytical equation and the finite element model, this is due to that between both paametes elastic coupling exists. To compute a unique solution the following index is ceated w ( x, y) w ( x, y) e ( E, v ) = / n, (4) m 3 3 exp( j) FEM c( ) 3 i= wexp( j) ( x, y) Equation 4 was established to compae a 30% of the displacement field between the finite element models and the measued data. The minimum value min ec ( ), =,2..., p indicates that the elastic * * paametes ( E and v ) ae the best appoximations fo the load-deflection cuves obtained expeimentally. 3 Results Fo the application of poposed methodology, expeimental tests wee conducted fo commecial silicon nitide films ( Si3N 4) with 2 2mm of suface and 500nm of thicness. The tests wee caied out in the appaatus descibed in Section 2., detailed infomation about it can be eviewed in [9].

5 Applying the pocedues descibed until the step 6 (see Section 2.3), the following constants wee detemined; = GPa, C = 3.373, =.9690 and = Figue 3a shows the eo function ep ( E, v ) established in Equation 2 which in tun was computed with the paametes anteioly expessed in the domains E (200, 280) GPa and v (0.2,0.4). It is obseved that thee is a egion in which the elastic values minimize the function ep ( E, v ). As we poposed, these values listed in Table ae appoximated by a linea elationship. The values indicate that all pais satisfy load-deflection cuve with good accuacy since these ae mechanically coupled. E [GPa] v Table : Elastic paametes that minimize ep ( E, v ). Figue 3: a. e ( E, v ) eo function. b. RMS cubic eo scheme of the expeimental setup. c. Load-deflection p cuve compaisons. As descibed in step 9, values listed in Table ae used to pefom finite element simulations to find an optimal solution inside the chosen values. These values epesent a specific case of the expeiments. Then, an optimal solution is found in the minimum of e ( E, v ), fo ou test, it was detemined as c

6 E = GPa and v = with five expeiments analysed. The esults ae shown in Figue 3b. In Figue 3c, it can be obseved that the numeical solutions agee with the expeimental data, computations wee done to veify that calculated solution adjusts the measued data. Additionally, fo the maximum state of pessue (9.57 KPa), the absolute eo between finite element solution and expeimental displacement field is evidenced. It is seen that esults agee vey well with the measuements since displacement eos ae less than % in the majoity of the bulge domain. Conclusions A numeical appoach fo identifying the elastic popeties of thin films was descibed and applied, it based on bulge test analysis, classical analytical methods and finite element analysis. The main benefit lies in the detemination of both popeties Young s modulus and Poisson s atio since in a taditional bulge test analysis only one of these can be detemined. The poposed numeical pocedue showed that compaing the defomed sufaces optimal elastic paametes can be found. Results showed that the estimated elastic popeties agee with coesponding values epoted by othe methods in the liteatue. Acnowledgement This eseach has been suppoted by the Ministy of Education, Youth and Spots of the Czech Republic unde the poject m-ipminfa (CZ.02..0/0.0/0.0/6_03/000823) and the equipment and the base of eseach infastuctue IPMinfa wee used duing the eseach activities. Refeences [] Poelma, R. H., Sadeghian, H., Noijen, S. P. M., Zaal, J. J. M., & Zhang, G. Q. (20). A numeical expeimental appoach fo chaacteizing the elastic popeties of thin films: application of nanocantileves. Jounal of Micomechanics and Micoengineeing, 2(6), [2] Pantano, M. F., Espinosa, H. D., & Pagnotta, L. (202). Mechanical chaacteization of mateials at small length scales. Jounal of Mechanical Science and technology, 26(2), [3] Mitchell, J. S., Zoman, C. A., Kiche, T., Roy, S., & Mehegany, M. (2003). Examination of bulge test fo detemining esidual stess, Young s modulus, and Poisson s atio of 3C-SiC thin films. Jounal of Aeospace Engineeing, 6(2), [4] Paul, O., & Gaspa J. (2007) Thin-film chaacteization using the bulge test, in Reliability of MEMS, O. Tabata and T. Tsuchiya, Eds. Weinheim, Gemany: Wiley-VCH ch. 3, pp [5] Xiang, Y., Chen, X., & Vlassa, J. J. (2005). Plane-stain bulge test fo thin films. Jounal of mateials eseach, 20(9), [6] Tabata, O., Kawahata, K., Sugiyama, S., & Igaashi, I. (989). Mechanical popety measuements of thin films using load-deflection of composite ectangula membanes. Sensos and actuatos, 20(-2), [7] Vlassa, J. J., & Nix, W. D. (992). A new bulge test technique fo the detemination of Young's modulus and Poisson's atio of thin films. Jounal of Mateials Reseach, 7(2), [8] Maie-Schneide, D., Maibach, J., & Obemeie, E. (995). A new analytical solution fo the loaddeflection of squae membanes. Jounal of micoelectomechanical systems, 4(4), [9] Holze, J.; Piále, T.; Buchta, Z.; Laza, J.; Tinoco, H. A.; Chlupová, A; Kuml, T. (207, May). Development of the bulge test equipment fo measuing mechanical popeties of thin films, In: Poceedings of 26th Intenational Confeence on Metallugy and Mateials. Bno, Czech Republic. pp. -6. Bno, Czech Republic. [0] Pan, J. Y., Lin, P., Maseeh, F., & Sentuia, S. D. (990, June). Veification of FEM analysis of load-deflection methods fo measuing mechanical popeties of thin films. In: Solid-State Senso and Actuato Woshop, th Technical Digest., IEEE (pp ).