Modeling of Substrate-Induced Anisotropy in Through-Plane Thermal Behavior of Polymeric Thin Films

Transcription

1 Modeling of Sbstrate-ndced Anisotropy in Throgh-Plane Thermal Behavior of Polymeric Thin Films JONC B. L,'** MARK C. ALLN,' THOMAS C. HODC,' SU ANN BDSTRUP,' and PAUL A. KOHL' 'School of lectrical and Compter ngineering, 'School of Chemical ngineering, Georgia nstitte of Technology, Atlanta, Georgia SYNOPSS Polymeric thin films are widely sed in microelectronic applications for a variety of prposes. These films may possess completely isotropic material properties and yet still exhibit anisotropic effects de to the constraining inflence of the sbstrate copling into the film behavior via the film Poisson ratio. A theoretical model of this effect on the throgh-plane thermal properties of isotropic thin films for single layer (thin film rigidly clamped) and bilayer (thin film on sbstrate, e.g., silicon wafer) has been developed based on the assmption that the material follows Hooke's law in all directions. Finite element analyses sing ANSYS 5.OA have also been performed to confirm theoretical reslts both for singlelayer and bilayer models. n the case of Poisson ratio of 0.5, the effective coefficient of thermal expansion (CT) in the thickness direction can be as high as three times that of the nconstrained film John Wiley & Sons, nc. Keywords: anisotropy polymeric thin film coefficient of thermal expansion (CT) Poisson's ratio throgh-plane direction finite element modeling NTRODUCTON Varios kinds of polymeric thin films are sed in microelectronic applications. xamples inclde spin-coated polymeric thin films as interlayer dielectrics in mltichip modle-deposited (MCM-D) applications. Blk material properties of polymers are relatively easy to measre and are sally well known. n contrast, thin film material properties are relatively hard to measre. Several investigations have been performed to characterize the thin film material properties becase in sit material properties are vital for practical device design. The throgh-plane thermal properties of thin films are of great interest and have been investigated experimentally.'p2 These investigations have been performed on polyimide films which have been deposited and sbseqently removed totally' or partially2 * To whom correspondence shold be addressed at Pettit Microelectronics Research Center, Georgia nstitte of Technology, 791 Atlantic Drive NW, Atlanta, GA Jornal of Polymer Science: Part B: Polymer Physics, Vol. 34, (1996) John Wiley & Sons, nc. CCC OSa7-6266/96/09159l-O6 from a sbstrate, redcing or eliminating any sbstrate constraining effect. These investigations have yielded higher coefficient of thermal expansion (CT) in the throgh-plane direction than that of the in-plane direction. One possible reason for these deviations is in-plane orientation of moleclar chains in spin-coated polymeric thin films. Since most polymeric thin films sed in MCM-D strctres are deposited on sbstrates, the sbstrate constraining effect shold also be considered in practical device design. For highly anisotropic polymeric thin films, however, sch as BPDA//PPD (biphenyl dianhydride//para-phenylenediamine) polyimide which showed anisotropy of 49.3' (the ratio of the throgh-plane CT to the in-plane CT), the moleclar orientation effect cold be a dominating effect in the thermal behavior of the films. n contrast, for relatively less anisotropic polymeric thin films, sch as PMDA//ODA (pyromellitic dianhydride//oxydianiline), BTDA//ODA/ MPD (benzophenone-tetracarboxylic dianhydridel /oxydianiline/meta-phenylenediamine), and Kapton which showed anisotropy of 3.8,' 1.2,' and 2.4,2 respectively, the sbstrate constraining effect cold 1591

2 1592 L T AL. / / : normal strain where,, ey,, and ax, ay, a, are normal strains and stresses in x, y, and z direction, respectively, is the Yong's modls, and v is the Poisson ratio. qation (1) can be rearranged for normal stresses a, or, and a, yielding: Figre 1. Deposited thin film on a sbstrate (bilayer). contribte to the anisotropy in the thermal behavior of the film. The prpose of this article is to qantify throgh analytical models as well as finite element models the effect of this sbstrate-indced anisotropy alone based solely on the assmption that the thin film follows Hooke's law and possesses isotropic, blk material properties. For isotropic films, this effect will dominate the effective anisotropy of the film/sbstrate system; any inherent material anisotropy will additionally affect the system anisotropy. t is shown that the effect of Poisson's ratio on the constrained thin polymeric films can explain how mch the sbstrate-indced effects contribte to the anisotropy of thermal expansion behavior of films. Since the anisotropy described here is de to the constraining inflence of the sbstrate copling into the film behavior via the film Poisson ratio, it shold be noted that this anisotropy is separate and distinct from anisotropy de to crystalline morphology in crystallizable thermoplastic^.^ f a crystallizable thin film is deposited on a sbstrate reslting in sbseqent anisotropy in thermal behavior, this anisotropy cold reslt from two cases: the anisotropy de to the presence of a rigid sbstrate alone (the sbject of this article) and the anisotropy cased by phase changes in the p01ymer.~ Theory f a material follows Hooke's law, the three-dimensional normal strains can be expressed by eq. (1) X a, = [( - (1 + v)(l - 2v) a, = [( - (1 + v)(l - 2v) qations (1) and (2) are called the generalized Hooke's law.4 The throgh-plane normal strain can be expressed by (Fig. 1):, = : - l;', (3) where : is the throgh-plane strain indced by throgh-plane stress, l;' is the throgh-plane strain indced by in-plane stress, and the mins sign indicates that if the in-plane stress is tensile, the film shrinks in the throgh-plane direction. The strain, is the strain that is measred when the film thickness changes de to thermal expansion. The throgh-plane strain indced by in-plane stress l;' can be calclated by setting the throgh-plane stress az eqal to zero (since the top of the film is a free srface) in eq. (2): O = (1 + v)(l - 2v) [(l - v)l;' + (c, + &,). (4) f there is no applied stress in any direction other than thermally indced stresses, and if the material is completely isotropic, then:, = ey = : = aat, (5) where a is the CT, and the AT is the temperatre change. qation (4) can be rearranged by sbstittion of eq. (5) as follows: -v -2v aat. (6) l;'=- (ex + y) = - (1- v) (1- v) Ths, the throgh-plane normal strain,, represented by: can be

3 ANSOTROPY N POLYMRC THN FLMS 1593 Table. Parameters Used in Finite lement Modeling (FM) Parameters Used in FM Literatre Vale Free CT Thickness Free CT (ppm/"c) Poisson Ratio (rm) (ppm/"c) Thin film Sbstrate BCB abot Si typical 2.37 Ref. 5: BCB with antioxidant; ref. 6: BCB withot antioxidant., = &: - = ()a~~. (7) Ths, the effective throgh-plane CT of the thin film on a sbstrate, sch as a Si wafer, can be represented as a fnction of the free CT of the thin film, the CT of the sbstrate, and the Poisson ratio of the thin film by: Therefore, the effective throgh-plane CT (alz, in of the deposited thin film on a rigid sbstrate (i.e., a sbstrate which does not warp ot of the plane and has zero CT) can be represented as a fnction of the free CT and Poisson ratio of the film by: where afree is the CT of free standing film, and eqal to a in eq. (7). n the case of a bilayer strctre, and neglecting the effect of sbstrate crvatre and assming that the sbstrate is isotropic, the thin film strctre can also expand or shrink in the in-plane direction by the amont which the sbstrate expands or contracts in the in-plane direction de to the sbstrate CT. The in-plane expansion of the sbstrate also affects the stress in the thin film. Ths, eq. (5) can be modified for a bilayer by: where 1 denotes the thin film and 2 denotes the sbstrate. This eqation holds as long as the free CT of the thin film (alfree) is greater than or eqal to the CT of sbstrate (a2). qation (6) can be modified for a bilayer by: (alat - a2at). (10) (1 - v) qation (11) gives the measred CT, alzin as a fnction of the free film and sbstrate parameters. t shold be noted that the measred CT can be as high as three times the free standing film vale for the case of an incompressible film ( = 0.5) and low CT sbstrate (a2 = 0). FNT LMNT MODLNG Finite element modeling (FM) has been performed sing three dimensional ( 3-D ) tetrahedron and block elements in ANSYS 5.OA for single and bilayer strctres in order to confirm the analytical derivation. The test case modeled sing FM was spincoated benzocyclobtene (BCB) as the thin film and a Si wafer as the sbstrate. Parameters sed in FM are listed in Table. Those parameters are based on previos literatre vale^.^-^ The free CT of the thin film (BCB in this FM) is abot 60 ppm/"c based on the literatre vales for BCB with antioxidant5 and withot antioxidant.6 The model vales for the free CT of the sbstrate range from the free CT of Si (2.3 ppm/"c7) to the free CT of the thin film (60 ppm/ "C ). Since the maximm Poisson ratio is 0.5, the Poisson ratio chosen for the thin film is ranging from 0 to Since this FM is a general model, it can be applicable to varios materials with different material properties. The bottom of the thin film in the single layer and the bottom of the sbstrate in the bilayer were clamped as bondary conditions so that the bottom cannot

4 1594 L T AL. be displaced in any direction. Both 3-D tetrahedron elements for cylindrical-shaped single and bilayer strctres, as well as 3-D block elements for sqareshaped single and bilayer strctres, have been modeled. For the cylindrical shaped model, several different radii have been tested ranging from 100 to 5,000 pm, reslting in less than 1% discrepancy of effective throgh-plane CT when the radis of the model was larger than 1000 pm. Several different anglar sections of the model also have been tested ranging from 7.5" to 360" (i.e., the fll circlar sbstrate) reslting in less than 1% discrepancy when the angle of the model was larger than 15'. Figre 2 shows the generated meshed element diagrams for 3-D tetrahedron elements for the cylindrical shaped bilayer FM. The angle of the model is 15", the radis is 3000 pm, the thickness of the sbstrate is 500 pm, and the thickness of the thin film is 30 pm. The FM reslts and the theoretical models are well matched to each other and also the FM reslts for both 3-D tetrahedron and block elements are matched within 1%. The effective throgh-plane CT of the thin film for both single layer and bilayer theoretical models and FM as a fnction of the Poisson ratio of the thin film are shown in Figre 3. As seen in Figre 3, the effective throgh-plane CT of the film is the same as that of the nconstrained film in the case of Poisson ratio of 0, dobles in the case of Poisson ratio of 0.33, and can be three times higher in the case of Poisson ratio of 0.5. n this figre, the thin film modeled is BCB (CT of 60 ppm/ "C) and the sbstrate modeled is Si (2.3 ppm/ "C ). Since the sbstrate modeled has relatively low vale of CT (2.3 ppm/"c), the effective throgh-plane CT vales for single and bilayer models are similar. n the case of a sbstrate having relatively larger vales of CT, the effective throgh-plane CT vales for the single-layer and ANSYS 5.0 A SP : 28: 17 PLOT NO. 1 LMNTS TYP NUM xv =1 w =-2 zv =2 DST=O XF =0.05 YF = ZF = PRCS HDDN Figre 2. A meshed diagram for bilayer FM sing 3-D tetrahedron elements in ANSYS 5.OA. Fifteen degrees of a cylindrical shape bilayer strctre is modeled.

5 F 140 g 120 m = 100 M s 80 e 5 60 P x FMl + Theory2 o FM Poisson ratio of thin film Figre 3. ffective throgh-plane CT of the film as a fnction of the Poisson ratio of the film; 1 denotes the single layer and '2 denotes the bilayer modeling; in this case the thin film is BCB (CT of 60 ppm/"c) and the sbstrate is Si (CT of 2.3 ppm/"c). bilayer models wold no longer be similar, as predicted by eq. ( 11 ). The dependence of the effective throgh-plane CT of the film as a fnction of the CT of the sbstrate is shown in Figre 4. The thin film can expand or shrink in the in-plane direction by the amont which the sbstrate expands or contracts in the in-plane direction, reslting in the change of the effective throgh-plane CT of the R & 140 W P & c.c 40 Y * Theory0 FM0.33 * Theory0.33 A FM Thoery ' CT of the sbstrate [ppm/cl Figre 4. ffective throgh-plane CT of the film as a fnction of the CT of sbstrate; nmbers indicate the Poisson ratio of the film. ANSOTROPY N POLYMRC THN FLMS a R ; 100 ; 80 P s 60 e 5 40., > U 4 2o Thickness of Film [ml Figre 5. ffective throgh-plane CT of the film as a fnction of the thickness of the film; in the case of thin film Poisson ratio of 0.33, ths the effective throgh-plane CT is abot doble (120 ppm/"c) that of the free film. film. When the CT of the sbstrate is relatively low, the effective throgh-plane CT of the film can be determined by the Poisson's ratio of the thin film. As the sbstrate CT approaches the free film CT, the effective throgh-plane film CT approaches the free standing film CT (60 ppm/oc). Figre 5 shows the effective throgh-plane CT of the film which is independent of the thickness of the film (to within 1-2%, the precision of the finite element simlator) when the thickness of the film is very small compared to the length and width of the film (less than 1 : 500). As the thickness of the film increases, the effective throgh-plane CT gets smaller and finally, if the thickness approaches the magnitde of the length and width of the film (i.e., no longer the thin film case), the effective throgh-plane film CT approaches the free standing film CT. CONCLUSONS A theoretical model and a finite element model for a completely isotropic polymeric thin film which is deposited on a relatively rigid sbstrate has been developed and shows sbstrate-indced effective anisotropy in the thermal expansion behavior of the polymeric thin film, even when the intrinsic film properties are isotropic. ven thogh the film sed in this stdy is isotropic, it is clear that when anisotropic polymeric thin films are deposited on a sbstrate, we can expect the effective anisotropy will

6 1596 L T AL. be affected by intrinsic film anisotropy as well as this sbstrate-indced effect. The athors wish to acknowledge Dr. Agaram Abhiraman at Georgia Tech for valable technical discssions. This work was spported in part by ARPA, throgh Office of Naval Research nder Grant N J4008. RFRNCS AND NOTS 1. M. T. Pottiger, J. C. Cobrn, and J. R. dman, J. Polym. Sci. B, 32,825 ( 1994). 2. H. M. Tong, H. K. D. Hsen, K. L. Saenger, and G. W. S, Rev. Sci. nstrm., 62(2), 422 (1991). 3. N. A. Mehl and L. Rebenfeld, J. Polym. Sci. B, 31(12), 1687 (1993). 4. J. M. Gere and S. P. Timoshenko, Mechanics of Materials, PWS-KNT, Boston, P.. Garro et al., Trans. CHMT, 16( 1), 46 ( 1993). 6. D. Brdeax et al., J. lec. Mter., 19(12), 1357 (1990). 7. J. Bryzek, K. Peterson, and W. McClley, Spectrm, 31(5), 20 (1994). Received Jne 8, 1995 Revised Janary 26, 1996 Accepted Febrary 1, 1996