We consider a beam with a multilayer structure of n layers with Young's modulus Ei, Poisson's ratio νi, thermal expansion coefficient (TEC) αi, intrinsic residual stress σi, and homogeneous thickness hi for layer i, where i=1,2,…,n. Due to the residual stresses presented in the layers, the multilayer structure may experience a bending with radius R to balance the residual stresses. The origin of the y axis is at the bottom of the first layer (Fig. ). The system is supposed to be at a temperature ΔT above the reference temperature Tr. Then, in the isotropic linear elastic approximation, the elastic stress σxi at layer i can be expressed as 1σxi(y)=Ei(ϵ0-αiΔT)+Eiκy+σi(i=1,2,…,n), where κ=1R is the curvature of the bent structure, and ϵ0 is the net strain. For a circular wafer with cylindrical symmetry, the biaxial modulus Ei′=Ei1-νi is used for Ei. According to the virtual work principle, the system satisfies the equilibrium conditions of the total force and bending moment (). Solving the equilibrium equations, we obtain the following two parameters: 2κ=P2I1-P1I2I1I3-I22,3ϵ0=P1I3-P2I2I1I3-I22, where 4I1=∑i=1nEihi,5I2=12∑i=1nEi(yi2-yi-12),6I3=13∑i=1nEi(yi3-yi-13),7P1=∑i=1n(EiαiΔT-σi)hi,8P2=12∑i=1n(EiαiΔT-σi)(yi2-yi-12).

For a two-layer system, n=2, one obtains the following at the reference temperature, ΔT=0: 9κ=-6(E1σ2-E2σ1)(h1h22+h12h2)(E1h12+E2h22)2+4E1E2h1h2(h12+h1h2+h22). Furthermore, for a single thin film deposited on a substrate at the reference temperature, if the top layer (i=2) is much thinner than the substrate (i=1), i.e., h2≪h1, and the substrate is free of residual stress, σ1=0, the above formula is reduced to the Stoney formula. 10κ=-6σ2h2E1h12 A situation of interest is a thin multilayer on a thick substrate. If n thin layers are deposited on a residual-stress-free substrate with Young's modulus Es and thickness hs, such that hs≫∑i=1nhi, then 11κ=-12Mt+6FthsEshs3. The average elastic stress in layer i is approximately 12σxi=6Mt-4FthsEshs2Ei+σi-Ei(αi-αs)ΔT(i=1,2,…,n), where Ft and Mt are the total linear force density and the total linear moment density of the thin-layer system: 13Ft=∑i=1n(σi-Ei(αi-αs)ΔT)hi,14Mt=∑i=1n(σi-Ei(αi-αs)ΔT)hiyic, with yic being the center position of thin layer i. This can be used in the sensor design to optimize the chip deformation and elastic stress in the function layer. It can also be used to design optical thin-film interference filters. By adjusting the thickness and residual stress of the individual films, a flat filter can be ensured. This is because the alternative tensile and compressive residual stresses can be compensated for to achieve a small curvature.

To verify the theory, we simulate a double-layer cantilever composed of a 10 µm silicon layer and a 1 µm nitride layer with COMSOL. The cantilever is 200 µm in length. Assuming a residual stress of 1×108 Pa in the nitride layer, we obtain the elastic distribution at position x=90 µm, as shown in Fig. . According to the Saint-Venant principle, the effects of the boundary constraints can be neglected in a region sufficiently far from the boundary (). This elastic stress distribution can be compared with the results calculated using our theory (Fig. ), demonstrating good agreement between the calculated and simulated results.

If the residual stresses are kept constant after deposition, they can be extracted with wafer bow measurements. We consider a MEMS structure involving n layers with l unknown residual stresses that need to be extracted. Then, l≤n, because the residual stress in some layers may be known or assumed to be stress free. One designs an experiment with m wafers with different layer structures from the same process flow. These different film-stacking structures can be achieved by inserting or taking out wafers from the same process flow. This is the successive deposition approach, where the deflection of each deposited wafer is measured. The same can be achieved after the entire process by etching the deposited wafer layer by layer. This is the successive-etching approach, where the deflection of each etched wafer is measured. This yields a data set of m measurements. The flat wafer bends due to the residual stress of the deposited thin films. The bending profile can be approximated by a spherical cap with curvature κ. The wafer center shifts a deflection g (wafer bow) relative to the wafer rim. The deflections of the wafers are measured using 3D laser line scanning based on the principle of optical triangulation (). For a wafer of diameter d, one can obtain the deflection from the bending curvature (κ) as 15g=-d28κ if the bending radius is much bigger than the wafer size. One can thus get m equations for the l unknown residual stresses. The residual stress of a given layer is assumed to be the same when it appears in different combinations of layer stacks because it is formed in the same process and under the same conditions. With an optimization method, one can extract the l residual stresses if m is bigger than or equal to l. A Python program was written for data analysis. The fsolve and minimize functions of the scipy module optimize are used for stress extraction in the cases of m=l and m>l, respectively.

This formalism can be generalized to analyze inhomogeneous residual stresses in a multilayer system. In this case, small cantilever test structures should be implemented on the wafer. In , cantilevers were fabricated, and the local residual stress in a single overlay was analyzed using the Stoney formula. Our formalism can be applied to a multilayered cantilever by replacing the wafer radius with the cantilever length. Instead of using the biaxial modulus for the wafer bow, the original Young's modulus should be used for the cantilever deflection.

The precision of the fitted residual stress depends not only on the fitting residuals themselves, but also on the measurement errors of deflection and thickness, as well as the parameter errors of Young's modulus and the thermal expansion coefficient of the films. Here, we neglect the effect of possible uncertainties in Poisson's ratios. These errors can propagate to the uncertainty in the extracted residual stresses. For measurement j, the theoretical deflections are 16gj=f(σ1,…,σl,h1,..,hn,E1,…,En,α1,…,αn)(j=1,2,…,m). The total standard deviation of deflection δt(gj) is therefore 17δt2(gj)=res2(gj)+δ2(gj)+∑i=1n∂gj∂hiδ(hi)2+∑i=1n∂gj∂Eiδ(Ei)2+∑i=1n∂gj∂αiδ(αi)2(j=1,2,…,m), where res(gj) is the fitting residual, which directly determines the fitting error; δ(gj) is the measurement error of the deflection of the jth measurement; δ(hi) is the thickness error of the ith layer; and δ(Ei) and δ(αi) are the parameter errors of Young's modulus and the thermal expansion coefficient of the ith layer, respectively. They determine the propagated error of the fitted residual stresses. One can obtain the uncertainties in the l fitted unknown residual stresses from sensitivity analysis (). The m×l Jacobian matrix J, also known as the sensitivity matrix, can be calculated numerically with the following elements: 18Jjk=∂gj∂σk(j=1,2,…,m)(k=1,2,…,l). The l×l Fisher information matrix of the fitted parameters is approximately () 19F=JTGJ, where 20G=diag1δt2(g1),1δt2(g2),…,1δt2(gm). The inversion of Fisher's information matrix is the covariance matrix C of fitting parameters C=F-1. The standard deviations, the correlation matrix, and the confidence intervals of the fitted residual stresses can be calculated from the covariance matrix (). The relative error of the kth fitted residual stress is 21Ckkσk(k=1,2,…,l). With nonzero errors, δ(gj), δ(hi), δ(Ei), and δ(αi), one can see the error propagation.

We designed an experiment (experiment I) to study the residual stresses of four thin films – oxidation layer (FOX), MoSi, CVD oxide, and stress-reduced LPCVD nitride – to investigate the generation and evolution of residual stresses in different layers involved in a MEMS production process. Since the MoSi layer will be annealed, we designed a five-step experiment using five starting wafers of (100)-Si with a thickness of 390 µm. After the thermal oxidation process, wafer 1 was removed from the flow. A MoSi layer was then deposited on the remaining wafers, after which wafer 2 was removed from the flow. The remaining three wafers were annealed, and wafer 3 was removed from the flow. LPCVD oxide deposition was performed on the remaining two wafers, after which wafer 4 was removed from the flow. Finally, a stress-reduced LPCVD nitride deposition was performed on wafer 5. The process steps and the wafers used are shown in Table . The wafer bows of the five wafers with different layer stacking were measured using 3D laser line scanning. As an example, Fig.  shows the wafer scans for wafer 2 (with the MoSi layer as deposited) and wafer 3 (after high-temperature annealing of the deposited MoSi layer). The wafer bending profiles aligned well with the arc of a circle, except for the rim of the wafer. This agrees with the Saint-Venant principle. This demonstrates that the curvature can be easily calculated from wafer deflection using Eq. (). The results indicate that the stress in the MoSi layer changed drastically after annealing. The mechanical properties of the layers involved in this paper are shown in Table . For stress extraction, the wafer bows shown in Table , which have been corrected with the initial wafer bow, are used. Using the error analysis theory (Sect. ), we calculated the relative fitting errors with ideal experimental parameters and the total relative errors, which took into account the propagated error from the 1 % relative error in wafer deflection, and the same relative error in thickness, Young's modulus, and the thermal expansion coefficient of all involved layers. The layer thicknesses, the extracted residual stresses, the relative fitting errors, and the total relative errors of fitted residual stresses are shown in Table . The results show that the residual stress of oxide depends on the preparation method and preceding layers. The exact kinetics are still unknown and must be studied using more systematic technological parameters. Furthermore, the correlation matrix of the fitted parameters is shown in Table , which also includes the propagated errors. A large anticorrelation coefficient is observed between the residual stresses of annealed MoSi and oxide. Although the total relative error of the fitted parameters is small, this correlation distorts the confidence ellipse from an ideal circle and will increase the size of the profile confidence interval ().

For experiment II, we start from a 100 mm diameter n-type (100) silicon wafer, which was oxidized with 60 nm silicon dioxide and deposited with 1 µm silicon nitride with LPCVD. The silicon nitride layer was deposited with a high DCS(dichlorosilane):NH3 flow ratio to effectively reduce the residual stress (). To extract the residual stress in the thin layers, the layer thickness and wafer bow are measured at different stages of multilayer etching. The first bow measurement, D1, was made for the initial processed wafer without any etching. The second bow measurement, D2, was made after the backside nitride layer was removed. The third bow measurement, D3, was made after the backside oxide layer was removed. The fourth bow measurement, D4, was made after the topside nitride layer was removed. Finally, the fifth bow measurement, D5, was made after the topside oxide layer was removed. The experimental results, the extracted residual stresses, the relative fitting errors, and the total relative errors in the fitted residual stresses (as in Sect. ) are shown in Table . The results show that the relative fitting error is very small. The fitted residual stress of the nitride film, 1.3×108 Pa, agrees well with the value of 1.35×108 Pa presented in . This also confirms the effectiveness of the high DCS:NH3 ratio in reducing the residual stress of LPCVD nitride films. The residual stresses of normal LPCVD nitride films is 1.3×109 Pa (Sect. ), while the residual stress of nitride prepared in experiment I (Sect. ) is only reduced to 4.9×108 Pa.

For experiment III, two 100 mm diameter n-type (100) silicon wafers, both 498 µm thick, are deposited with 300 nm of silicon dioxide and 100.7 nm of silicon nitride, respectively. The deposited film on the backside is etched away. The Tencor FLX2900 system is used to perform wafer curvature measurements with temperatures scanned from 50 to 350 °C. For stress extraction, it is assumed that the intrinsic residual stresses and thermal expansion coefficient (TEC) of the films remain constant during wafer heating measurements. Figure  shows the measured and simulated wafer curvature as a function of temperature. The fitted residual stress of silicon dioxide film is -1.3×108 Pa, and that of silicon nitride film is 1.3×109 Pa. The relative fitting errors are 8.4×10-5 and 4.3×10-4 for silicon dioxide and silicon nitride films, respectively. The corresponding total relative errors are 1.3×10-3 and 2.3×10-3, respectively, if the propagated errors are taken into account like in Sect. . The thermal stresses are induced by the TEC differences between the film and substrate. Because the experiments contain errors in both the curvature measurement and the temperature reading, the extracted residual stresses also contain propagated uncertainty. Using these parameters, one can calculate the average elastic stress in the thin layer, as illustrated in Fig. .

It has been demonstrated that this formalism can address both intrinsic and extrinsic residual stresses in a layered structure. This is a useful extension of established approaches. For example, the widely used Stoney formula () can be applied to a single layer deposited on substrate. Thermal stresses can be analyzed in a layered structure with any number of layers (), but intrinsic stresses induced by manufacturing are not incorporated. The equivalent reference temperature method () for dealing with intrinsic residual stresses can also model multilayer systems; however, no closed-form solution can be obtained. Therefore, it is difficult for them to be used in the reverse problem of general residual stress extraction. Furthermore, a systematic methodology for parameter extraction is presented using either successive deposition or etching methods, and a parameter estimation method is proposed. Additionally, this closed-form formalism can be incorporated into established device design flows and online stress-monitoring software as a software gadget to improve device performance and process yield.