Stresses in deposited thin films can have serious effects on the properties, performance, and long term stability of the device. Developments in the fabrication of freestanding micromechanical structures using surface micromachining techniques require knowledge of and control over the internal stresses.
Residual stress can be divided into thermal stress and intrinsic stress according to its source. The formula of the thermal stress is shown in Eq.2-1.
σth = E∆α∆T (2-1)
where E is the elastic coefficient, ∆α is the difference in coefficients of thermal expansion coefficient between the film and the substrate and ∆T is the difference between the deposition temperature and the room temperature. If the temperature of the fabrication is higher, the main stress is produced by thermal stress.
The intrinsic stress, σi, reflects the internal structure of a material and is less clearly understood than the thermal stress, which it often dominates. Several phenomena may contribute to σi, making its analysis very complex. Intrinsic stress
depends on thickness, deposition rate (locking in defects), deposition temperature, ambient pressure, method of film preparation, type of substrate used (lattice mismatch), incorporation of impurities during growth, etc.
The types of stress in the films are separated into two kinds, one is tensile stress, and the other is compressive stress. In bimorph beams, the upper layer that has a tensile stress and the lower layer that has zero stress make the cantilever beam to curve upwards, as shown in Figure 2-1(a). If the upper layer has a compressive stress and the lower layer has zero stress, they make the cantilever beam bow downwards, as shown in Figure 2-1(b). Excessive compressive or tensile strain fields result in splintering, cracking and adhesion problems of the film to the substrate.
Figure 2-1: Two types of stress. (a) tensile Stress (b) compressive Stress
Stress is a serious problem in surface micromachining. But here the drawback is transferred to be an advantage to apply a force to pop up the structure. In this thesis, the residual stress beams are used to supply the force to lift up the plate of the Fresnel lens. The theory and calculation of the lift up height of the stress-induced beam will be discussed.
2-1.1 Analytical solution
The cantilever becomes curved from the bending moment caused by the variation in stress between the upper and lower films. The beam bends upward because the lower film has a compressive residual stress while the upper film has a tensile residual stress. The curvature of the cantilever and the maximum beam
deflection are determined by the physical dimensions and material properties. The dimension parameters of a cantilever beam are labeled in Figure 2-2 [15].
Figure 2-2: Dimensions of a cantilever beam
When two films of different residual stresses share an interface at equilibrium, the induced forces P1 and P2, and the moments M1 and M2, must be balanced [16]: moment-curvature relations for each material, an equivalent beam strength (EI) equiv
can be defined [17], [18], film and the upper film, respectively. The relationship between the stress-induced internal force (P) and the radius of curvature (ρ) can be determined:
ρ The radius of curvature (ρ) is constant along the beam, since the internal force (P) and the beam geometry do not vary.
An additional condition is that of zero slip at the interface. The strain in each
film is composed of three components: one due to the residual stresses σ1 and σ2, one due to the axial force P, and one due to the curvature ρ of the beam. Setting the sum of the three components in one material equal to that of the other at the interface is that:
The radius of curvature of the pre-biased flexure is obtained by solving Eqs. (2-3), (2-5) and (2-6),
With the radius of curvature known, the end deflection of the beam can be calculated from trigonometry. The deflection perpendicular to the unreleased position for a given beam length, L, with radius of curvature ρ is given by
The bending height of the bimorph residual stress beam is calculated by utilizing the above theory and simulated by Coventorware. The material and geometric parameter of a Si3N4/Poly-Si bimorph are E1 = 161Gpa (polysilicon), E2 = 270Gpa (silicon nitride) [19], h1 = 2µm, h2 = 0.5µm, σ1 = 0Mpa and σ2 = 100Mpa. The material and geometric parameter of a Gold/Poly-Si bimorph are E1 = 161Gpa (polysilicon), E2 = 78Gpa (gold) [19], h1 = 2µm, h2 = 0.5µm, σ1 = 0Mpa and σ2 = 270Mpa. The residual stresses are obtained from Ref [20] and measurement results.
The deflection of a Si3N4/Poly-Si and a gold/Poly-Si residual stress beams, calculated from Eqs. (2-4) and (2-7), and of a Si3N4/Poly-Si residual stress beam, simulated by CoventorWare, is shown in Figure 2-3.
Figure 2-3: Tip Deflection v.s. Beam Length
Figure 2-4 is the 3D plot of the CoventorWare simulation result. The length and width of the beam are 750µm and 50µm, respectively. The beam is a Si3N4/Poly-Si bimorph. The thickness of Si3N4 and Poly layers are 0.5µm and 2µm, and the stress of Si3N4 and Poly layers are 100Mpa and zero, respectively. The simulation result of the end deflection to substrate is 23µm.
From above calculation, the tip displacement of a 750µm long beam is 30µm.
There is an error value between simulation result and calculation result. The error value is 23%. And because the Young’s modulus and residual stress can change with fabrication process, the calculation and simulation results are not the accurate values of the experiment result.
Figure 2-4: The 3D plot of the CoventorWare simulation result 2-1.3 Residual Stress Beam Design
The residual stress beams are the assemblers and the fixers in this system.
Several residual stress beams are designed. There are ten different beam dimensions, two kinds of shapes in the tip and three different widths of necks in funnel shape, as shown in Figure 2-5.
The beam dimensions are L W = 500µm 200µm, 500µm 400µm, 500µm×600µm, 700µm×200µm, 700µm×400µm, 700µm×600µm, 1000µm×200µm, 1000µm×400µm, 1000µm×600µm and 1000µm×1000µm. The 1000µm×1000µm is the limit of the residual stress beam dimension. The V and funnel shapes are two kinds of the beam tips to fix the device in after popping up. The widths of the neck in the funnel shape tips are 3µm, 5µm and 7µm. The total width of the neck is the 2µm thickness of the poly2 plate plus 1µm, 3µm or 5µm tolerance for the movement during
× × ×
assembly.
The residual stress beam is composed of two Poly layers and one residual stress layer. The Poly1 links with Poly2, and Poly1 area is under the Poly2 plate to pop up the plate. The residual stress layer is on the Poly2 layer.
W
Figure 2-5: Residual stress beam. (a) with V shape (b) with funnel shape