Thermoelastic stresses in multilayered beams

Thin Solid Films 515 (2007) 8402 – 8406 www.elsevier.com/locate/tsf

Thermoelastic stresses in multilayered beams Neng-Hui Zhang ⁎ Department of Mechanics, College of Sciences, Shanghai University, Shanghai 200444, China Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, China Received 13 September 2006; received in revised form 21 March 2007; accepted 3 May 2007 Available online 22 May 2007

Abstract The functionality and reliability of multilayered systems are strongly influenced by thermoelastic stresses. Recently Hsueh formulated a closedform solution [Hsueh, Thermal stresses in elastic multilayer systems, Thin Solid Films 418 (2002) 182] by decomposing the total strain into a uniform strain component and a bending strain component in order to overcome the complexity of the traditional analytical models. The present study develops an alternative analytical model in terms of the curvature radius of the neutral axis for zero normal strain and the normal strain at the interface between the substrate and the films. Numerical results are calculated for thermoelastic stresses in five-layer (AlGa)As laser diodes. Also two approximate models for the case of multilayered films on a thick substrate are obtained. © 2007 Elsevier B.V. All rights reserved. Keywords: Gallium arsenide; Multilayers; Thermoelasticity; Stress; Modeling

1. Introduction Multilayered structures have been used widely in microelectronic, optical, and structural components and protective coatings [1]. The functionality and reliability of multilayered systems are strongly influenced by thermoelastic stresses. Thermoelastic stresses are inevitably generated because of: (i) growth stresses [2,3]; and (ii) the thermal mismatch between the films and the substrate when the system is subjected to temperature change during its fabrications and applications. Up to now, many efforts have been devoted to the analysis of thermoelastic stresses in multilayered systems [4–13]. Since Stoney [4] formulated a simple relationship between the thin film stress and the curvature of the substrate when the film is ultrathin, the equation has been used widely in many applications. But his assumption on a neutral axis for zero bending moment does not exist in a bilayer system bent by internal stresses. A general solution for the bending of bilayers due to thermal stresses was first derived by Timoshenko [5].

⁎ Department of Mechanics, College of Sciences, Shanghai University, Shanghai 200444, China. Tel.: +86 021 66134790; fax: +86 021 66134021. E-mail address: [email protected]. 0040-6090/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2007.05.003

However in his calculation both layers were assumed to have the same curvature and there were three unknowns to be solved for the bilayer cases. And the numbers of both the unknowns and continuity conditions at interfaces increase with the number of layers [4–9]. As a result, obtaining a closed-form solution is a formidable task, and the analysis is either left to the computer [6,7] or simplified by assuming a constant elastic modulus throughout the system [7–9]. By decomposing the total strain into a uniform strain component and a bending strain component, an exact closed-form solution for thermoelastic stress in multilayer systems was derived by Hsueh et al. [10–13]. And there are only three unknowns to be solved and three boundary conditions to be satisfied for any multilayer systems. The paper is devoted to derive an alternative two-variable analytical model for distributions of thermoelastic stresses in multilayered beams. The two unknowns are the curvature radius of the neutral axis for zero normal strain and the normal strain at the interface between the substrate and the films. First an exact closed-form solution for thermoelastic stress distributions in multilayer beams is derived. Then the solution is reduced to those for multilayer systems with a thick substrate and thin films. Finally numerical results are given out for five-layer (AlGa)As laser diodes and the accuracy of its approximate solutions is discussed.

N.-H. Zhang / Thin Solid Films 515 (2007) 8402–8406

2. Analyses An elastic multilayer beam with length l and width b is schematically shown in Fig. 1, where n layers of film with individual thickness, hi (i = 1, 2,… , n), are bonded sequentially to a substrate with thickness, ts. The first layer is in direct contact with the substrate. The coordinate axis x is located at the interface between the substrate and layer 1 of the film. The thermoelastic properties of the substrate and films are Es, αs and Ei, αi, where E is Young's modulus, α the thermal expansion coefficient. Assume that the beam bends downwards due to the temperature fluctuation. If Δ is the distance from the neutral axis for zero normal strain to the coordinate axis x, and ρ is the curvature radius of the deformed neutral axis, then the normal strain at the location y is

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For zero axial force and zero bending moment, the balance equations of the multilayer beam are given as Z n Z X rs dAs þ ri dAi ¼ 0; ð6aÞ As

Ai

i¼1

Z rs ydAs þ As

n Z X

ri ydAi ¼ 0;

ð6bÞ

Ai

i¼1

where As and Ai are, respectively, the cross section areas of the substrate and the films. Substituting Eqs. (5a) and (5b) into Eqs. (6a) and (6b) yields n X e0 ¼ ½ðEs as hs þ Ei ai hi ÞbDT  M T A=C=ðbBÞ; ð7aÞ i¼1

1=q ¼ M =C;

ð7bÞ

T

e ¼ ½ðq þ D þ yÞdh  qdh=ðqdhÞ;

ð1Þ

in which in which dθ is the relative rotation angle of the two adjacent cross sections. Similarly, the normal strain at the coordinate axis x is e0 ¼ ½ðq þ DÞdh  qdh=ðqdhÞ;

ð2Þ

n X

A ¼ Es Ss þ C ¼ ðEs Is þ

i¼1 n X i¼1

M ¼ ðEs as Ss þ T

Eq. (1) minus Eq. (2) yields

Ei Si ; B ¼ Es hs þ

n X

Ei Ii Þ  A2 =ðbBÞ;

n X

Ei ai Si ÞDT  ðEs as hs þ

i¼1

e ¼ e0 þ y=q:

ð3Þ

The thermoelastic constitutive relation between stress σ and strain ε is given as r ¼ Ee  EaDT ;

ð4Þ

where ΔT is the temperature variation, and is unrelated to y. Combining Eqs. (3) and (4), one can get rs ¼ Es y=q þ Es e0  Es as DT ;

ð5aÞ

ri ¼ Ei y=q þ Ei e0  Ei ai DT ; ðfor i ¼ 1 to nÞ;

ð5bÞ

E i hi ;

i¼1

n X

Ei ai hi ÞDTA=B;

i¼1

where Ss and Si are, respectively, the static moment of the substrate and the films with regard to the coordinate axis x; Is and Ii, are, respectively, the second axial moment of the substrate and the films on the coordinate axis x. By using the following formula i1 X ð8aÞ Ss ¼ bh2s =2; Si ¼ bhi ð hj þ hi =2Þ; h0 ¼ 0; j¼0 i1 X Is ¼ bh3s =3; Ii ¼ bhi ð hj þ hi =2Þ2 þ bh3i =12;

ð8bÞ

j¼0

in which σs and σi are the normal stresses in the substrate and the films, respectively.

Eqs. (6a) and (6b) can be transformed into e0 ¼ fðEs as hs þ þ

n X i¼1 i1 X

n X

E i hi ð

i¼1

Ei ai hi ÞDT  ½Es h2s =2 hj þ hi =2Þ=qg=B;

1=q ¼ fDT ½

n X

Ei ai hi ð

i¼1

 e0 ½

n X

i1 X hj þ hi =2Þ  Es as h2s =2 j¼0

E i hi ð

i¼1

 Es h2s =2g=f

i1 X

j¼0 n X i¼1

þ Es h3s =3g; Fig. 1. Schematics showing a multilayer beam and the coordinate system.

ð9aÞ

j¼0

hj þ hi =2Þ

i1 X Ei hi ½ð hj þ hi =2Þ2 þ h2i =12 j¼0

ð9bÞ

When the thicknesses of the film layers are much less than that of the substrate, the solutions can be reduced. Ignoring

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N.-H. Zhang / Thin Solid Films 515 (2007) 8402–8406

Fig. 2. The cross-section of a typical (AlGa)As laser diode.

terms with orders of hi higher than one in Eqs. (9a) and (9b), we can get a first-order approximation e0 ¼ as DT þ

n X

Ei hi ðai  as ÞDT =ðEs hs Þ  hs =ð2qÞ;

ð10aÞ

Fig. 3. Comparison between the exact solution and first-order approximation for the stress distributions in the substrate.

i¼1

1=q ¼ 6

n X

Ei hi ðai 

as ÞDT =ðEs h2s Þ:

ð10bÞ

i¼1

Based on Eq. (10b), the resultant curvature 1 / ρ, may be decomposed into components contributed by thermal mismatch between the substrate and the individual film layer, such that 1=q ¼

n X

1=qi ;

ð11Þ

i¼1

in which the component curvature, 1 / ρi, is denoted by 1=qi ¼ 6Ei hi ðai  as ÞDT =ðEs h2s Þ:

n X

2Ei hi =ð3qÞ;

ð12Þ

ð13aÞ

i¼1

ri ¼ Es h2s =ð6qi hi Þ þ 2Ei hs =ð3qÞ:

e ¼ c þ ðy  tb Þ=qb ;

ð13bÞ

ð15Þ

where c is the uniform strain component, tb indicates the location of the bending axis (which is defined as the line in the cross section of the system where the bending strain component is zero), ρb is the curvature radius of the bending axis. For small deformation problems, if ρ ≈ρb, then from Eqs. (3) and (15), we have e0 ¼ c  tb =q:

Eqs. (11) and (12) have also been deduced by Townsend et al. [14]. For the first-order approximation, the stress distribution can be derived from Eqs. (5a–6a), (10a) and (10b), such that rs ¼ Es ðy þ 2hs =3Þ=q 

In Hsueh's analysis [10–13], the strain field is assumed as follows:

ð16Þ

Compared with the strain filed (15) proposed by Hsueh [11], three unknowns in the strain filed (3) are reduced to two unknowns in the present model. Unlike previous works, it is unnecessary to find the location of the neutral axis in another twovariable model or the bending axis in the three-variable model [10–13,15]. As Hsueh [15] pointed out that the essence of twovariable model is to define a reference plane with a reference strain and curvature. While in the present model (3), the reference strain ε0 at the film/substrate interface and the reference curvature 1 / ρ of the neutral axis are taken as unknowns. So there are two

In Eq. (13b), the approximation relation y ≈ 0 has been used when y ∈ [0,∑in= 1 hi]. The formulas (13a) and (13b) are identical to the Hsueh's first-order approximation [11]. The above first-order approximate for the film layers can further be reduced to the zero-order approximate, such that ri ¼ Es h2s =ð6qi hi Þ:

ð14Þ

Obviously the stress distribution (13a) in the substrate is related to the curvature of the coordinate axis, 1 / ρ, the stress distribution (14) in each film layer is related to its curvature component 1 / ρi. Also from Eq. (13a), the neural axis for zero normal stress is located at 2/3 of the substrate thickness underneath the film/substrate interface, which has also been obtained by Hsueh [11] and Townsend et al. [14].

Fig. 4. Comparison between the exact solution, first-order approximation and zero-order approximation for the stress distributions in the films.

N.-H. Zhang / Thin Solid Films 515 (2007) 8402–8406

Fig. 5. The first-order approximate thermal stress in the AlxGa1−xAs active layer as a function of the Al content, x, in (AlGa)As laser diodes.

reference planes in the present model (3). In addition, the two unknowns in Freund's model [16] and the present model are solved simultaneously, while the three unknowns in Hsueh and Evans' model [10–13,15] can be solved sequentially. 3. Results It is important to determine film stresses in microelectronics and microelectromechanical systems for various purposes such as structural integrity, electrical functionality, and structural dynamics characteristics. The bending beam method based on the measured curvature, layer thickness, and elastic properties of the substrate is traditionally the most widely acknowledged method for thermal stress characterization due to its simplicity and nondestructive nature. For the lacking of direct stress measurements for multilayers, the examples used by Olsen, Ettenberg [9] and Hsueh [11] are hence chosen for comparison in the present study. A typical cross-section of the laser diode is shown in Fig. 2, which consists of an 80 μm GaAs substrate, a 0.2 μm active GaAs layer sandwiched between two 1 μm Al0.25Ga0.75As confining layers and a 1 μm GaAs Cap [9]. The related material properties were reported: EGaAs = 100 GPa [9], EAlAs = 83.5 GPa [17], αGaAs = 6.86 × 10− 6/°C [8], and αAlAs = 5.2 × 10− 6/°C [18]. Both Young's modulus and coefficient of thermal expansion of AlxGa1−xAs were found to obey Vegard's rule, such that [17] EAlx Ga1x As ¼ xEAlAs þ ð1  xÞEGaAs ; :

ð17aÞ

aAlx Ga1x As ¼ xaAlAs þ ð1  xÞaGaAs :

ð17bÞ

In computation, the temperature variation ΔT = − 785 °C in Olsen and Ettenberg's analysis [9] is taken here. The assumption of constant Young's modulus throughout the films in Olsen and Ettenberg's analysis [9] is abandoned and different elastic constant for each layer is considered. In order to validate the accuracy of the exact solution, the first-order and the zeroorder approximations, comparisons of thermal stress distributions in the substrate and films are shown in Figs. 3 and 4, respectively.

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Fig. 3 shows that the top of the substrate is subjected to tension and its bottom is subjected to compression, and the first-order approximation curve is very close to the exact one. The superiority of the first-order approximation to the zeroorder approximation can be clearly seen in Fig. 4. Specifically, thermal stress in Al0.25Ga0.75As confining layers is always − 30.1594 MPa based on the zero-order approximation, or − 27.2679 MPa based on the first-order approximation, and variates linearly from − 27.5891 MPa to − 27.5796 MPa. The same approximate prediction of stress distributions in two confining layers is caused by the reason that the two confining layers have the same thermal and physical parameters and the same geometrical parameters. Degradation of (AlGa)As laser diodes is a major concern in applications. Investigations have revealed that the degradation was caused mainly by crystalline layer, and addition of Al or P to the GaAs active layer can significantly increase the diode's operating life [19,20]. Fig. 5 shows that the stress in the doped active layer decreases almost linearly with the increasing of Al contents. The stress is tensile for x b 2.4%, and is zero when x = 2.4%, and becomes compressive for x N 2.4%. This reveals that the stress distribution can be changed by the modulation of the Al contents in the active layer to prolong the diode's operating life. 4. Conclusions An exact closed-form solution (7a) and (7b) for elastic deformation of multilayered beams due to thermal stresses is derived in terms of the curvature radius of the neutral axis for zero normal strain and the normal strain at the interface between the substrate and the films. Compared with Hsueh's model, there are only two unknowns in the present analysis. Based on the present model (7a) and (7b), the first-order approximation of stress distribution [Eqs. (13a) and (13b)], and the zero-order approximation of stress distribution in the films (i.e. Stoney's equation) [Eq. (14)] can also be obtained. The present formulas are used to predict thermal stresses in five-layer (AlGa)As laser diodes, which have been frequently quoted. Numerical results reveal that the first-order approximation is superior to the zero-order approximation for the prediction of thermal stress in thin films. Acknowledgments The authors thank Dr. C.H. Hsueh for providing his related PDF papers. This research was sponsored by the Outstanding Youth Program of Shanghai Municipal Commission of Education under grant No. 04YQB088 and the Shanghai Leading Academic Discipline Project under grant No. Y0103. References [1] S.M. Hu, J. Appl. Phys. 70 (1991) R53. [2] R.O.E. Vijgen, J.H. Dautzenberg, Thin Solid Films 270 (1995) 264. [3] M.D. Tran, J. Poublan, J.H. Dautzerberg, Thin Solid Films 308–309 (1997) 310. [4] G.G. Stoney, Proc. R. Soc. Lond. 82 (1909) 172. [5] S. Timoshenko, J. Opt. Soc. 11 (1925) 233. [6] H.C. Liu, S.P. Murarka, J. Appl. Phys. 72 (1992) 3458.

8406 [7] [8] [9] [10] [11] [12] [13]

N.-H. Zhang / Thin Solid Films 515 (2007) 8402–8406 V. Teixeria, Thin Solid Films 392 (2001) 276. R.H. Saul, J. Appl. Phys. 40 (1969) 3273. G.H. Olsen, M. Ettenberg, J. Appl. Phys. 48 (1977) 2543. C.H. Hsueh, A.G. Evans, J. Am. Ceram. Soc. 68 (1985) 241. C.H. Hsueh, Thin Solid Films 418 (2002) 182. C.H. Hsueh, J. Appl. Phys. 91 (2002) 9652. C.H. Hsueh, S. Lee, T.J. Chuang, J. Appl. Mech. 70 (2003) 51.

[14] [15] [16] [17] [18] [19] [20]

P. Towsend, D. Barnett, T. Brunner, J. Appl. Phys. 62 (1987) 5487. C.H. Hsueh, J. Cryst. Growth 258 (2003) 302. L.B. Freund, J. Mech. Phys. Solids 44 (1996) 723. S. Adachi, J. Appl. Phys. 58 (1985) R1. M. Ettenberg, R.J. Paff, J. Appl. Phys. 41 (1970) 3926. Y. Nannichi, I. Hayashi, J. Cryst. Growth 27 (1974) 126. M. Ettenberg, H. Kressel, H.F. Lockwood, Appl. Phys. Lett. 25 (1974) 82.