Effect of the Constrained Fixed End on the Structure Stability of Ti/Au Single Layered Cantilever Single Layered Cantilever

In practical applications of the Ti/Au multi-layered design in MEMS devices, constraint conditions at fixed end of the micro-cantilever structure could vary. Top surface at the fixed end can be either free or exposing to air, and the bottom surface can be constrained to a structure. Also, both the top and bottom surface can be sandwiched between two other structures. Constraint conditions at the fixed end will affect deformation behavior and structure stability of the micro-cantilever, which would influence practical implantation of the Ti/Au multi-layered structure in MEMS devices. Therefore, in this section, effects of constraint conditions at the fixed end, such as constraining both the top and the bottom surfaces or the bottom surface only, on structure stability of micro-cantilevers comprised of the Ti/Au single layered structure will be reported. The evaluations are carried out by measuring the tip deflection (∆htip) using results obtained from a 3D optical microscope (OM) and finite element method (FEM) simulations.

Fig. 1.23 shows SEM images of the Ti/Au single layered micro-cantilevers fabricated by a series of lithography and electroplating processes. More details of the fabrication processes are provided in a previous study [6]. A schematic diagram of the Ti/Au micro-cantilever is shown in Fig. 1.24. Two types of the micro-cantilever are studied.

Type one is with constraint on bottom surface of the fixed end only. Type two is with constraints on both top and bottom surfaces of the fixed end. Micro-cantilevers with different dimensions are prepared. Width (w) of the micro-cantilevers is fixed at 15 μm.

Thickness of the Au layer (tAu) is 15 μm. Thickness of the Ti layer (tTi) is 0.1 μm. Various length (l) of the micro-cantilevers are used, which are 50, 100 and 200 μm.

Fig. 1.23. SEM image of the (a) Ti/Au micro-cantilevers array with the length varied from 50 µm to 200 µm, width of 15 µm and Au thickness of 15 µm, the (b) type one and the (c) type two Ti/Au micro-cantilevers with a length of 50 µm, width of 15 µm and Au thickness of 15 µm.

Fig. 1.24. Schematic diagrams of the (a) type one and the (b) type two Ti/Au micro-cantilevers.

Structure stability of the Ti/Au micro-cantilevers are evaluated by observing height profile of the micro-cantilevers using an OM equipped with a 3D measurement function. Height of top surface of the micro-cantilever (hx) at a position x away from the fixed end is determined along long-side of the micro-cantilever in a step size (d) of 0.1 μm by the OM.

The structure stability is quantified by calculating the height difference along the micro-cantilever body (Δhx), which is the difference between the hx and the height at the fixed end (h0). The equation is shown in the following:

0

x x

h h h

   . (1.6)

For the type two micro-cantilevers, because of the constraint on top surface of the fixed end, height of the top surface of the micro-cantilever body at a location very close to the fixed end is used as the h0. The deflection at the tip (Δhtip) and average deformation of the entire micro-cantilever (Δhave,l) are also calculated. The equation for Δhave,l is shown in the following:

 

1 ave,

ld x x l

h d

h l

  

 



Inertial Sensors

where l is the total length of the micro-cantilever. d is the step size of the measurement, which is 0.1 μm in this case. Fig. 1.25 shows height difference profile of the two types of the Ti/Au micro-cantilevers with the l of 50, 100, and 200 μm observed by the OM. The red horizontal line represents the height at the fixed end, which is used as the baseline to determine deformation of the micro-cantilevers. In Fig. 1.25 (a), height of the type one (only bottom constrained) micro-cantilever at different location along the long-side deviates significantly from the baseline. It gradually bends downward from the fixed end to the tip. Downward deflections of 1.55 to 2.26 µm are observed at the tip as the l increases from 50 to 200 µm as shown in Table 1.3. The Δhave,l increases from 0.883 to 1.560 µm as the l increases from 50 to 200 µm. The results indicate structure stability of the Ti/Au micro-cantilever is lowered with an increase in the l. This behavior occurs because of the increase in the beam’s own weight as the l increases, which follows the Euler-Bernoulli beam theory well. Nerveless, the deformation is still insignificant when compared with length of the micro-cantilever, which demonstrates the Ti/Au multi-layered design’s positive contribution to the structure stability.

Table 1.3. Structure stability determined from the OM results.

Fig. 1.25. Height difference, Δh, profile of the (a) type one and the (b) type two micro-cantilever with a length of 50, 100, 200 µm measured by the OM; the red line represent Δh = 0

along the entire l.

On the other hand, the type two (top and bottom constrained) micro-cantilevers deform mostly upward from the baseline, but a downward deflection is observed at the tip as shown in Fig. 1.25(b). The upward deformation is suggested to be simply reflecting surface roughness on top surface of the micro-cantilever as shown in Figs. 1.22(b) and (c). In general, all of the type two micro-cantilevers do not deform much from the fixed end to the tip. The ∆htip’s are all negative indicating a downward deflection and vary from -0.32 to -1.06 µm as the l varies from 50 to 200 µm as shown in Table 1.3. The Δhave,l

varies from 0.315 to 0.447 µm as the l varies from 50 to 200 µm. From the result of the type two micro-cantilever, the l becomes less influential on the structure stability, which is different from the result of the type one micro-cantilever. In addition, both the Δhtip and the Δhave,l are all lower for the type two micro-cantilever when compared with the type one cantilever.

FEM simulations are carried out using a simulation software (COMSOL Multiphysics) to analyze deformation behaviors of the micro-cantilevers. The micro-cantilever analyzed is composed of two parts, one part is the fixed end and the other is the beam body. For the type one micro-cantilevers, the fixed end is modeled as having the same width, which is 15 μm, as width of the beam body. On the other hand, for the type two micro-cantilevers, width of the fixed end is set to be 30 μm. For the type one, only bottom surface of the fixed end is restrained. For the type two, both of upper and lower surface of the fixed end are restrained. The equations of linear elastic material are selected in the category of solid mechanics. Constants of linear elastic materials such as Young's modulus, thermal expansion coefficient, Poisson's ratio and density are applied in the simulation. The constants are provided by the database embedded in the COMSOL Multiphysics. Young’s modulus of Ti and Au are 115.7 and 70 GPa, respectively. Thermal expansion coefficient of Ti and Au are 8.6×10^-6 and 14.2×10^-6 K^-1, respectively. The effect of an increase in the temperature from 20 to 310 °C on deformation behaviors of the micro-cantilevers is simulated. Based on the conditions mentioned above, the FEM simulations are performed.

The Δhtip and the Δhave,l are also calculated to quantify the structure stability.

Fig. 1.26 shows FEM simulation results of the type one micro-cantilevers with the tAu of 15 µm, the w of 15 µm and the l varied from 50 µm to 200 µm. Length of the fixed end part is set to be 34 µm long, hence total lengths of the 50, 100, and 200 µm specimens shown in Fig. 1.26 are 84, 134, and 234 µm long, respectively. For the three type one micro-cantilevers, upward deformations at ca. x = 20 µm are observed. As mentioned before, the fixed end is from x = 0 to 34 µm. Hence, the upward deformation mainly occurs at the fixed end part, and the micro-cantilever body mostly shows a downward deformation. Degree of the downward deflection increases as the l increases. FEM simulation results of the type two micro-cantilevers are shown in Fig. 1.27. Similar to FEM simulations of type one cantilevers, length of the fixed end part is set to be 34 µm long. Upward deformation are observed on top surfaces of the micro-cantilever at location near the fixed end, but downward deflection is observed when approaching the tip.

Height difference profiles of the type one and type two micro-cantilevers are shown in Fig. 1.28. Information of the ∆htip and the Δhave,l are shown in Table 1.4. Generally, the results are similar to the results obtained from the OM observations. For the type one,

Inertial Sensors

downward ∆htip’s are obtained, and degree of the deflection and the Δhave,l increase as the l increases. For the type two, the deformations are all smaller when compared to the type one. As indicated by the FEM result of the type one micro-cantilever shown in Fig. 1.26, the deformation takes place at the fixed end leads to significant negative effects on the overall structure stability. Hence, when the fixed end is sandwiched between two structures that is constrained on both top and bottom surfaces of the fixed end, the overall structure stability is improved.

Fig. 1.26. FEM simulation results of the type one micro-cantilever with the Au thickness of 15 µm, the width of 15 µm and the length of (a) 50 µm, (b) 100 µm, and (c) 200 µm.

Fig. 1.27. FEM simulation results of the type two micro-cantilever with the Au thickness of 15 µm, the width of 15 µm and the length of (a) 50 µm, (b) 100 µm, and (c) 200 µm.

Fig. 1.28. Height difference, Δh, profile of the (a) type one and the (b) type two micro-cantilever with a length of 50, 100, 200 µm evaluated by the FEM; the red line represent Δh = 0

along the entire l.

Table 1.4. Structure stability determined from the FEM results.

Structure stability of the Ti/Au single layered micro-cantilevers with two types of constraint at the fixed end is evaluated from OM observation and FEM simulation. The