2.1. Optimal Design of an Electromagnetically Actuated Cantilever: Direct Problem
The direct (or analysis) problem reads as follows given the shape g of the cantilever end, current I, and magnetic induction B, find:
the stiffness k of the cantilever;
the resonance frequency f of the cantilever;
the force Fz acting on the end region and its displacement Δz;
the electric resistance R (power-loss related) of the Lorentz loop.
The stiffness
k of the cantilever can be calculated [
17] as follows:
where
E is the Young’s modulus,
b is the cantilever width,
w the arm width,
t is the thickness equal to 1.5 µm,
L1 is the cantilever length and
L2 is the tip length (see
Figure 1).
The resonance frequency
f can be evaluated with the following approximate formula [
18]:
where
ρ is the mass density equal to 2330 kgm
−3.
The force
Fz and its displacement Δ
z can be calculated as follows:
Equation (3), which is derived from the Lorentz’s equation, is under the assumption that the cantilever, i.e., the plane in which the current flows, is perpendicular to the magnetic induction field.
Finally, the electric resistance
R can be calculated as the series of three electric resistances of the three path components (two arms, with the same resistance value
R1 and the tip, with resistance
R2):
where σ is the electric conductivity of the boron-doped silicon (without metal layer) equal to 6.67 × 10
4 Sm
−1.
2.2. Optimal Design of an Electromagnetically Actuated Cantilever: Inverse Problem
If the shape of the cantilever end is defined by means of a n-dimensional vector g = (g1, …, gk, …, gn) of geometric variables (e.g., for a polygonally-shaped end region, the coordinates of the relevant vertices), the inverse (or design) problem reads: given current I and magnetic induction B, find the shape g = (g1, …, gk, …, gn) of the cantilever end region such that:
the stiffness k(g) of the cantilever is minimized;
the resonance frequency f(g) is maximized;
the displacement Δz(g) of the end region is maximized;
the electric resistance R(g) of the Lorentz loop is minimized.
A multi-objective optimization problem characterized by four objective functions [k(g), f(g), Δz(g), R(g)] is originated. When more than one objective function is considered in the optimization, more solutions, belonging to the so-called Pareto front, are obtained. In particular, a solution is called Pareto optimal if there does not exist another solution that dominates it i.e., a solution that cannot be improved in any of the objectives without degrading at least one of the other objectives.
Considering
n-objective functions, a solution
g1 is said to dominate another solution
g2, if:
A solution
g1 is called Pareto indifferent with respect to a solution
g2 if:
In our inverse problem it turns out to be n = 4 and f1 = k(g), f2 = f(g), f3 = Δz (g) and f4 = R(g). Our goal is, starting from a prototype geometry g0, to find a new geometry improving g0 against the four objectives, according to Equations (6)–(9).
When many objective functions (say more than two) are considered, it is very common to find solutions of the optimization problem which are indifferent in the Pareto sense to the starting point. However, these solutions are nevertheless interesting because they improve at least one objective function.
It can be noted that Equations (1)–(5) define an analytical model for the direct problem, however, Equations (6)–(9) prevent from an analytical solution of the inverse problem and therefore the numerical method is in order.
The shape of the cantilever is defined by four design variables, as shown in
Figure 1:
w, arm width.
L1, cantilever length.
L2, tip length.
b, cantilever width.
The variation range for each design variable is shown in
Table 1. The chosen values for the boundaries are based on the experience.
In order to guarantee a geometrical congruency, the following constraint (units in µm) is set:
A series of optimizations are subsequently run, considering one (Opt1), two (Opt2) or three (Opt3) objective functions at a time:
Opt1—each objective function i.e., k, f, Δz and R, is individually optimized in four different single-objective optimizations (Opt1k, Opt1f, Opt1z, Opt1R). In each case, the activated objective function is the leading one, while the remaining three objective functions are updated, depending on the current value of the design vector g;
Opt2—the following optimizations are run: k and R are optimized (Opt2kR), f and z are optimized (Opt2fz), f and R are optimized (Opt2fR), z and R are optimized (Opt2zR). Each pair of objective function is optimized in the Pareto sense, the remaining two functions are updated, depending on the current value of the design vector g;
Opt3—k, z and R are optimized (Opt3) in the Pareto sense, while the frequency f(g) is simply updated.
In order to solve these optimization problems, an evolutionary algorithm of lowest order is applied [
19]. This algorithm is able to solve single-objective problems (in this case it is called “ESTRA method” [
20,
21]) and multi-objective problems (“MOESTRA method” [
22]). The search in the design space begins in a region of radius
d0 (standard deviation) centered at the initial point
m0 (mean value);
m0 is externally provided, while
d0 is internally calculated on the basis of the bounds boxing the variation of the design variables.
Setting m = m0 and d = d0, the generation of the design vector x = m + u d then proceeds, resorting to a normal sample . It is verified that x fulfils bounds and constraints (i.e., that x is feasible), otherwise a new design vector is generated until it falls inside the feasible region.
The associated objective function f(x) is then evaluated and the test if f(x) dominates f(m) (Equations (6) and (7)) is performed; if the test is successful, m is replaced by x (the so-called selection process), otherwise m is retained.
The next step is concerned with the size of the search region that will be used for the successive iterations. The underlying rationale is that when a point better than the current one is found, the radius of the search region is increased around the new point to search for further improvements; if no improvement is found, the radius of the search region is gradually decreased up to convergence (annealing process).
In this respect, the evolutionary algorithm substantially differs from a deterministic one e.g., Nelder and Mead algorithm [
23], in which the search region would be narrowed around the better point in order to converge towards the corresponding, nearest minimum. The drawback is that this minimum might be a local one. On the contrary, the evolutionary algorithm, if successful in finding a better point, covers a larger region of search in order to see if there would be another good candidate in the neighborhood, and then does the opposite when this is not deemed possible. This way, there is a non-zero probability of finding the region where the global optimum of the objective function is located. To assess the optimization results, a set of prototypes has been fabricated based on the technology described in the subsequent Section.
2.3. Fabrication Process
The fabrication process of the microcantilevers used for a radiation pressure sensing was based on a double side micromachining concept [
24]. However, in contrast to the typical technology based on bulk silicon substrates, in this case the silicon on insulator (SOI) wafers with 1 and 1.5 micrometers thick buried oxide and cantilever layer, respectively, were used as the input material. Despite the fact that the use of SOI substrates is more expensive, this solution has many advantages compared to the use of the bulk wafers. Two advantages of using the SOI substrates are particularly important.
The first advantage is a significant simplification of the microcantilever production process. The second advantage is a guarantee that all cantilevers defined on one wafer are characterized by uniform thickness regardless of its shape and size (the thickness depends only on the SOI wafer cantilever layer properties).
Therefore, using the SOI substrate, the production technology consists of only four technological steps: high p doping of the whole cantilever layer, definition of a gold contacts and mirrors, definition of the shape of the cantilever and finally the releasing of the cantilevers.
Figure 2 presents the scanning electron microscopy (SEM) image of the cantilever after three steps, i.e., after plasma etching processes. Upon magnification (
Figure 2b,c) the four layers can be observed: gold (which serves as a mirror), a 1.5 μm silicon layer, a buried silicon dioxide and a handle silicon wafer. The example final cantilevers matrix after released operation is presented in
Figure 3.
The presented construction was optimized for the electromagnetic actuation, first proposed by Buguin group [
25]. The use of SOI wafer makes it easy to obtain a homogeneous doping of the cantilever defined in the SOI cantilever layer, which significantly reduced the thermal actuation effect.
The homogeneous high-p-doped cantilever layer was obtained by using boron doped layer deposited by Low Pressure Chemical Vapor Deposition (LPCVD) method [
26]. In this method there are two technological stages: boron source layer deposition using B
2H
6 5% in N
2 at 270 °C (pre-diffusion) and high temperature annealing (diffusion of the boron dopant from previously deposited layer into silicon). The boron dopant profile can be optimized by controlling the duration time of pre-diffusion and duration time and temperature of the diffusion step.
By changing the pre-diffusion time we can control the surface boron concentration. In our experiments, the duration of the pre-diffusion was 55 min. Relatively long times allow one to obtain a high boron concentration of 1 × 1020 cm−3.
On the other hand, changing the temperature and duration time of diffusion allows one to control the final dopant profile in the device layer. In our case the temperature and duration time of the diffusion were 1100 °C and 25 min, respectively. Further increases in value of these parameters would cause an undesirable effect of reducing the surface boron concentration. In this case the dopant profile would not be homogeneous and would increase the thermal actuation effect. On the other hand, decreasing the diffusion temperature causes incomplete oxidation of the layer used as a source of boron, increasing its etch resistance, and makes its removal difficult.
The results of simulation of the obtained dopant profile for the listed above process parameters are presented in
Figure 4. The simulation confirmed that the boron dopant profile is uniform along the cantilever layer thickness; thus, the thermal expansion coefficient should be also constant [
27].