Young’s moduli for different values of porosity for porous silicon layers were investigated by X-ray diffraction, acoustic techniques, nanoindentation and Brillouin scattering. The obtained values are summarized and discussed in [8] . A good approximation for the Young’s modulus in a p⁺-type PS material containing three-dimensional arrangement of open pores is given by

E p = C E b ρ r 2 = C E b ( 1 − p ) 2 , (1)

where E_p and E_b stand for the Young’s moduli of porous and bulk silicon, respectively, C is a constant including all the geometric scaling factors, ρ r is the relative density and p the porosity. This approximation is consistent with the literature [9] [10] and was confirmed in [11] in the particular case of nano-porous silicon containing pores with cubic symmetry. Obtained from a discrete model including near-neighbors (NN) and next-to-near-neighbors (NNN) interactions the model proposed in [11] fits particularly well previous mentioned experimental results. Moreover, the model is able to predict the more complex anisotropic behavior in situations where, due to directional anodization, the resulting macroscopic PS is strongly anisotropic. Back to the cubic case, it has been shown in [11] that the shear modulus and the Poisson ratio obey

G p = G b ( 1 − p ) 4 ,   ν p = ν b ( 1 − p ) . (2)

Let us consider a simple cantilever clamped at one end without any seismic mass. Denoting by L the cantilever length, by w and t the width and the thickness of its cross section respectively, by ρ b the mass density and E_b the Young’s modulus of the bulk silicon, the first eigenfrequency is given by

f b = α 2 t 4 π L 2 E b 3 ρ b , (3)

where α ≃ 1.875 is the first positive solution of the (characteristic) equation cos α = − 1 / cosh α . For a PS cantilever with the same geometry, as the Young’s modulus follows the rule given in (1) but ρ p = ρ b ( 1 − p ) we obtain

f p = f b 1 − p . (4)

It follows that using porous silicon, one can decrease the eigenfrequencies of the cantilever and this remark was the starting point of our study. For technological reasons (fabrication limitations) the cantilevers we studied were merely two-layer materials: Si/PS at various thicknesses. For that case the derivation of eigenfrequencies and comparison with numerical values provided by finite element method computations under more general assumptions (finite strains in the fully three-dimensional framework) will be presented in the next section.

In a fully functional energy-harvesting device, a bottom electrode, a piezoelectric material and a top electrode will cover the upper side of the cantilever. In order to investigate the mechanical behavior of the harvesting device we shall focus in the following only on a Si/PS cantilever and a generic geometry of the structure is illustrated in Figure 1.

In a bilayer beam, the position of the neutral axis, denoted z 0 and measured from the bottom face of the cantilever, is given by

z 0 = E b t b ( t p + t b / 2 ) + E p t p 2 / 2 E b t b + E p t p = E p t p 2 + E b t b ( t b + 2 t p ) 2 ( E b t b + E p t p ) , (5)

where t_p and t_b are the thicknesses of porous (bottom) layer and bulk (upper) layer, respectively. Using I b = w t b 3 / 12 and I p = w t p 3 / 12 for the quadratic moment of inertia with respect to their centroid axis for the bulk and porous layers respectively, the flexural stiffness of the bilayer, further denoted EI can be computed as

E I = E b ( I b + w t b ( t p + t b / 2 − z 0 ) 2 ) + E p ( I p + w t p ( z 0 − t p / 2 ) 2 ) = w 12 ( E b t b 2 + E p t p 2 ) 2 + 4 E b E p t b t p ( t b 2 + t b t p + t p 2 ) E b t b + E p t p . (6)

As expected, the last relation, symmetric with respect to interchange of subscripts p and b, reduces to E I = E w ( t b + t p ) 3 / 12 in the case of equal elastic moduli.

For completeness, we shall present in this subsection analytical results for the deflection of a generic bilayer cantilever in two cases (a) a first one when when the cantilever is loaded by a point load at its end and the mass of the cantilever is neglected (b) a second case that generalizes the first one by including the effect of its own weight and that of the seismic mass.

Denoting by F the load due to the seismic mass and assuming that: 1) this load acts at x = L + L S M / 2 and 2) the cantilever is rigid for x > L , the classical Euler-Bernoulli theory gives the static vertical deflection of the massless cantilever as

d = F 12 E I ( x 2 [ 3 L S M + 2 ( 3 L − x ) ]                                             0 ≤ x ≤ L L [ 6 ( L + L S M ) x − L ( 2 L + 3 L S M ) ]         L ≤ x ≤ L + L S M (7)

If the mass of the bilayer cantilever cannot be neglected (this is the case of a thick beam) assuming an uniform load q = − g ( ρ b t b + ρ p t p ) w the vertical deflection obtained is given by

d = 1 24 E I { [ q x 2 − 4 ( q L + F ) x + 6 ( q L 2 + F ( 2 L + L S M ) ) ] x 2                                   0 ≤ x ≤ L L [ ( 4 q L 2 + 12 F ( L + L S M ) ) x − L ( q L 2 + 2 F ( 2 L + 3 L S M ) ) ]       L ≤ x ≤ L + L S M (8)

In both cases, for L < x < L + L S M , the seismic mass is assumed to be rigid so that the deflection for x > L is completely determined by the values of d ( L ) and d ′ ( L ) obtained for x ≤ L .

In order to estimate the effect of the cantilever bending on a thin piezo-layer one can compute the axial strain at the upper surface. The longitudinal deformation along the beam, at a distance h from the neutral axis, is obtained as ε x x = − κ h , where the (linearized) curvature of the cantilever is given by κ = ∂ 2 d / ∂ x 2 . At the upper free-surface of the cantilever, we obtain

ϵ x x = − ( t b + t p − z 0 ) ∂ 2 d ∂ x 2 . (9)

The fundamental frequency of the massless cantilever beam with a concentrated mass at x = L + L S M / 2 (see notations in Figure 1) is given by [12]

f 0 = 1 2π ( 3 E I M S M L 3 ( 1 + η / 2 ) 3 ) 1 / 2 , (10)

where η represents the non-dimensional number: η = L S M / L . The result overestimates the first eigenfrequency of the cantilever since in contrast to our previous assumptions, in (10) one does not assume that the part of the beam located between L and L + L S M / 2 is rigid.

An alternative approach consists in using the solution for the cantilever static deflection in order to provide a better estimate (more exactly, an overestimate of it [13] ) of the first eigenfrequency by using the Rayleigh principle. A similar approach was successfully used to calculate the eigenfrequency of a silicon accelerometer [14] . Using (7), neglecting the rigidity of the cantilever for x > L and assuming that the seismic mass is much larger than the cantilever mass, the first eigenfrequency of the system including the seismic mass located at x = L + L S M / 2 is given by

f 0 = 1 2π ( E I M S M L 3 12 ( 4 + 6 η + 3 η 2 ) ( 16 + 48 η + 63 η 2 + 42 η 3 + 12 η 4 ) ) 1 / 2 . (11)

As an example, if t b = 18   μ m , t p = 2   μ m (i.e., only 10% of the total cantilever thickness is a PS layer) at 50% porosity, if the bilayer dimensions are L = 2   mm , w = 500   μ m and seismic mass dimensions are L S M = w S M = 1   mm and t S M = 450   μ m , Equation (11) leads to f 0 = 454   Hz for the bilayer cantilever. Same dimensions for a bulk silicon cantilever ( t b = 20   μ m , t p = 0   μ m ) provide f 0 = 507   Hz , so that porosification of 10% of the cantilever thickness decreases the first eigenfrequency of about 10%. We conclude that even a relatively thin PS layer could lead to a measurable change in the first eigenfrequency of the structure.

Before fabrication and discussion of the experimental results, one must ensure of the reliability of the given analytical models in the case of some common geometries found in MEMS devices, under actual simplifying assumptions (one dimensional model for bending under Euler-Bernoulli hypothesis, neglecting the rigidity of the cantilever for x > L and accounting only for the seismic mass) can be confirmed in a more general framework (finite strains, three-dimensional elasticity). For this reasons the next subsection compares the analytical results for the deflection, surface strain and first eigenfrequency for different bilayer cantilevers to finite element simulations of fully three-dimensional structures in finite strains.

Finite element simulations were performed using Ansys 14 and the three-dimensional model of the cantilever was built using 3D, 20-nodes, structural solid (SOLID186) elements. These elements have three degrees of freedom per node (translations in the nodal x, y and z directions) and they support stress stiffening, large deflection and large strain capabilities. The structure was meshed with quadrilateral-shaped elements and clamped at one end. For modal analysis, the default Block Lanczos mode-extraction method was used. Model’s mechanical parameters are summarized in Table 1, assuming C = 1 in (1).

The comparison between analytical models and FEM results for different PS porosities is shown for three cantilever geometries that differ only by the beam thickness: t b = t p = 10   μ m , 25 μm or 50 μm. Other common dimensions are: L = 2   mm and w = 500   μ m for the cantilever and L S M = w S M = 1   mm and t S M = 450   μ m for the seismic mass. The evolution of the first eigenfrequency as a function of the porosity is plotted on Figure 2. Both models, (11) and (10), slightly overestimate the first eigenfrequency, but the Rayleigh’s model gives values which are closest to those obtained by FEM. Eigenfrequency values calculated by (11) are between 1.24% (for the thinnest beam) up to 2% (the thickest beam) agreement with those obtained by FEM. These relative errors are, at least, about 1.5 times more important, when compared with the Timoshenko’s model (the difference in both models is obvious only for the thickest cantilever in Figure 2).

For the thinnest cantilever which consists of 10 Si on top of 10 PS, the simulated and calculated static deflections are plotted in Figure 3. The difference between the results obtained for cantilevers without mass (7) and cantilevers with mass (8) is undistinguishable (less than 2%). Thus, the massless analytical model is in agreement within 2% with FEM simulations.

The simulated and calculated (9) longitudinal deformations ϵ L on the surface of the cantilever are shown in Figure 4. For L ≤ x ≤ L + L S M , the deformation is null: the hypothesis of a rigid seismic mass is confirmed by FEM.

A series of devices were fabricated from a p-type (boron doped), 4 inches, double-side polished, (100)-oriented silicon wafer with a resistivity of 0.1 - 10 Ω∙m. The process steps are summarized in Figure 5. On the frontside, a 300 nm aluminum layer was deposited by evaporation then patterned to define the geometry of the cantilevers (S1813 photoresist and Al etchant). This was followed by a standard annealing for 30 minutes at 450˚C in nitrogen ambient in order to form an Ohmic contact [15] . Cavities with a mesa were achieved on the backside of the wafer by Deep Reactive Ion Etching (DRIE) using a thick (20 μm) AZ40XT photoresist mask. The nano-porous (pore width smaller than 2 nm)

silicon layer was formed on the backside of the obtained membranes by electrochemical etching (anodization) of silicon in a mixture of hydrofluoridric acid (HF) and ethanol (9 mL HF 48% + 1 mL ethanol) for 20 minutes at a constant current density of 43 mA/cm². The aluminum pattern on the frontside of the wafer acted as a positive electrode for the anodization process of silicon. A platinum wire, in contact with the HF solution, was used as a negative electrode. Each membrane was processed individually. A picture of the silicon wafer during the anodisation process is shown in Figure 6. Several tentatives were made in order to define the right current density such that to obtain the PS formation

regime and avoid the electropolishing regime. Finally, the Al patterns were used as a mask for a second Si DRIE step in order to release the cantilevers from the substrate. The bulk density of PS (density of the material including pores and interparticle voids [16] ) was estimated to be about 50%.

After dicing, each sample was mounted on a vibrating pot and analyzed by Laser Doppler Vibrometry (Polytec MSA-500 micro-system analyzer). The measurement system and a close view of a sample are shown in Figure 7. The measured fundamental frequencies and quality factors for different Si cantilevers and one composite Si/PS cantilever are reported in Table 2. We obtained an average of the first eigenfrequency for the six silicon cantilevers: ( 5447 ± 120 )   Hz , and the quality factor Q was measured to ( 2870 ± 400 ) . For the Si/PS bilayer cantilever (sample #1) the first measured eigenfrequency was f 0 ≈ 5198   Hz with a quality factor Q = 1529 . The thickness of all cantilevers was measured at ( 106 ± 3 )   μ m . By observing the side of the beam using a microscope for the Si/PS cantilever the

PS thickness was measured as ( 6.0 ± 0.5 )   μ m . The cantilever geometry was L = 1.3   mm and w = 400   μ m , respectively while the seismic mass was estimated at ( 1.7 ± 0.1 )   mg . With that geometry, (11) leads to f 0 = 5818   Hz for a Si cantilever and f 0 = 5462   Hz for a Si/PS cantilever, which is about 5% agreement with measurements. The difference may be explained by geometrical uncertainties, particularly on the seismic mass.

We can conclude that, in agreement with the theoretical estimate, the first eigenfrequency of the Si/PS cantilever decreases by about 250 Hz compared to that of Si cantilevers (same cantilever thickness) while the quality factor was divided by almost a factor two. Measurements were also made on samples #2 and #7 after aluminum removal: the change in f₀ was limited to less than 0.2% whereas Q increased by about 15%. This shows that any non-uniformity of the aluminum thickness cannot explain the observed difference between the Si/PS sample and other Si samples, which is clearly the result of the stiffness decrease in the Si/PS sample with respect to bulk Si.