Young’s modulus of multi-layer microcantilevers

A theoretical model for calculating the Young’s modulus of multi-layer microcantilevers with a coating is proposed, and validated by a three-dimensional (3D) finite element (FE) model using ANSYS parametric design language (APDL) and atomic force microscopy (AFM) characterization. Compared with typical theoretical models (Rayleigh-Ritz model, Euler-Bernoulli (E-B) beam model and spring mass model), the proposed theoretical model can obtain Young’s modulus of multi-layer microcantilevers more precisely. Also, the influences of coating’s geometric dimensions on Young’s modulus and resonant frequency of microcantilevers are discussed. The thickness of coating has a great influence on Young’s modulus and resonant frequency of multi-layer microcantilevers, and the coating should be considered to calculate Young’s modulus more precisely, especially when fairly thicker coating is employed.

A theoretical model for calculating the Young's modulus of multi-layer microcantilevers with a coating is proposed, and validated by a three-dimensional (3D) finite element (FE) model using ANSYS parametric design language (APDL) and atomic force microscopy (AFM) characterization. Compared with typical theoretical models (Rayleigh-Ritz model, Euler-Bernoulli (E-B) beam model and spring mass model), the proposed theoretical model can obtain Young's modulus of multilayer microcantilevers more precisely. Also, the influences of coating's geometric dimensions on Young's modulus and resonant frequency of microcantilevers are discussed. The thickness of coating has a great influence on Young's modulus and resonant frequency of multi-layer microcantilevers, and the coating should be considered to calculate Young's modulus more precisely, especially when fairly thicker coating is employed. © 2017 Author(s). All  Microcantilevers have been widely applied in microelectromechanical system (MEMS) and nanoelectromechanical system (NEMS), such as micro-force sensors, biosensors and microactuators, etc. [1][2][3][4][5] Most microcantilever-based transducers are multi-layer structures including piezoelectric layers, sensitive layers, microelectrodes and reflectors, 6,7 in which the microcantilevers can be utilized for detecting specific mass, chemical compounds and DNA sequences with special coatings. 8,9 Thin coatings on surface of microcantilevers are essential in any applications, such as biological detection and chemical analysis, improving measurement accuracy by increasing the reflectivity. 8 Although the coating would have influences on calculation of the Young's modulus of multilayer microcantilevers, sometimes it is difficult to accurately calculate the Young's modulus and resonant frequency of multi-layer microcantilevers with thin coating, as the dimension of coating decreases to micro/nano scale. For this calculation, the influences of the coating should be investigated.
High resonant frequency is required in many applications of microcantilevers, especially for ultrahigh sensitivity or resolution applications. [10][11][12][13][14][15][16][17] As the critical issues of mechanical performance of microcantilevers, Young's modulus [18][19][20][21][22][23] and resonant frequency [24][25][26][27][28][29][30][31] have been studied by various characterizations. In this research, atomic force microscopy (AFM) measurement is conducted to characterize resonant frequency of multi-layer microcantilevers, and a theoretical model is proposed and discussed including the impact of coating on both Young's modulus and resonant frequency. In addition, a three-dimensional (3D) finite element (FE) simulation is performed by ANSYS parametric design language (APDL) to verify the accuracy of the proposed theoretical model. In comparisons of the proposed theoretical models and three typical theoretical models (Rayleigh-Ritz model, 32 a Authors to whom correspondence should be addressed. Electronic mail: hel@whut.edu.cn; jldeng@whut.edu.cn Euler-Bernoulli (E-B) beam model, [33][34][35] and spring mass model 32,36 ), the results proved that our model can accurately calculate the Young's modulus and resonant frequency of multi-layer microcantilevers. Furthermore, the impact of the width, length and thickness of the coating on Young's modulus is discussed since the geometry would have great influence on properties of microcantilevers. [37][38][39] The microcantilever is schematically shown in Fig. 1 (a), three theoretical models without considering the impact of coating are given in equations. (1)- (3), and one theoretical model considering the impact of coating is shown in Eq. (4). These models are introduced to calculate the Young's modulus of microcantilevers.
Microcantilever can be considered as a spring with a mass at its end (mass of the spring could be ignored), 32,36 which is referred as the spring-mass model. Young's modulus of the spring model can be calculated by following Eq. (1): 36 where E 1 is the calculated Young's modulus, and M is the effective mass of the microcantilever. L, W and t are the length, width and thickness of the microcantilever, respectively. f is the resonant frequency of the microcantilever. According to Work-Energy Principle, the change in kinetic energy of an object is equal to the net work done on the object. Therefore, the Rayleigh-Ritz model 31 can be applied to calculate the Young's modulus, as expressed in Eq. (2): where E 2 is the Young's modulus of the microcantilever calculated using the Rayleigh-Ritz model, and m is the mass of the microcantilever. For a single degree of freedom model, the microcantilever could be considered as ideally clamped E-B cantilever beams. [33][34][35] The mass loading of a cantilever is simplified as a point load at the free end. Therefore, the microcantilever is considered to execute simple harmonic motion in a vacuum. Young's modulus of a rectangle cantilever is calculated by Eq. (3): where E 3 is the calculated Young's modulus of the microcantilever based on E-B beam model. n g is a geometry constant of 0.24, 35 and ρ is the mass density of the microcantilever.  As shown in Fig. 1 (a), the thickness, density and Young's modulus of the coating are different from those of the base in rectangular multi-layer microcantilever, while the width and length are the same. Young's modulus of this cantilever considering the coating is obtained from Eq. (4): where z is the position along z direction as described in Fig. 1 (a), z 0 is the position of neutral axis defined by Eq. (5): One end of the microcantilever is defined as a fixed support, which indicates that the contact faces are prevented from moving or deforming, and the microcantilever can vibrate freely only along the z-direction under its working state. Vibration of the microcantilever at working state can be expressed in Eq. (6): where ω is the angular frequency, and t i (i = 1, 2) is the thickness of the base and coating respectively. t is the total thickness of base and coating. ρ i (i = 1, 2) is the mass density of base and coating respectively. ψ (t, y) is the function that describes resonant mode changing with time. E i (i = 1, 2) is the Young's modulus of base and coating respectively. M is the moment of force, which can be derived from Eq. (7): And ψ (t, y) in Eq. (5) can be expressed as: where ψ (t, y) is the function describing bending of the microcantilever along length direction. ω n is the n th degree angular frequency harmonic. θ is the phase. By combining Eqs. (6) and (7), we can obtain following Eq. (9): Model parameter δ n is given by following Eq. (10): Therefore, the n th degree angular frequency harmonic of the multi-layer microcantilever is calculated by following Eq. (11): where δ n L are 1.875, 4.694, 7.854, 10.995, 14.137 with n value of 1, 2, 3, 4 and 5 respectively, therefore the first degree angular frequency harmonic of the multi-layer microcantilever is given by following Eq. (12): By combining Eqs. (2) and (12), the Young's modulus of multi-layer microcantilever can be obtained by Eq. (4).
To verify the accuracy of theoretical models described above, the length, width and thickness of coating and base were characterized by scanning electron microscope (SEM), then a 3D FE simulation of multi-layer microcantilever was performed by APDL. As shown in Fig. 1 (a), one end region is fully constrained. In this simulation, the parameters of the base (Si) and the coatings (Al and Au) are shown in Table I, and five samples of Tap300AI-G (Budget Sensors) with Al coating were investigated. The base and coating were meshed using all-hexahedral element Solid186 and Solid185 respectively, the size of element for meshing was program-controlled. Conta174 element and targe179 element were employed to simulate the contact behavior between the base and coating. The resonant frequency obtained from ANSYS can be used to calculate Young's modulus by Eq. (3).
The AFM sweep-frequency technique was utilized to characterize the resonance frequency ( Fig. 1  (b), which matches well with the results from FEM and theoretical models (Fig. 1 (c)). Residual stress (thermal stress and internal stress) resulting from fabrication process and the tip of microcantilever is not considered in these models, which has influence on the resonant frequency and Young's modulus. Therefore, the error will be introduced, demonstrating that the theoretical models are capable of calculating the resonant frequency and Young's modulus of multi-layer microcantilevers.
Comparisons between proposed model and typical models are shown in Figs Fig. 2 illustrates that only the theoretical models considered the impact of coating presented in this work could obtain accurate results compared with commonly used theoretical models (Rayleigh-Ritz model, E-B beam model and spring mass model). This suggests that coating should be taken into account for this calculation.
As shown in Fig. 2 (b) and 2(d), for Tap300GD and Multi75AI, the Young's modulus obtained from the theoretical model (∼145 GPa) is almost the same as the Young's modulus of the base (Si, 145 GPa) with or without fairly thin coating, and in good agreement with the results in which the impact of coating is not considered. This result verifies the accuracy of theoretical models in view of coating, and demonstrates that the model presented in this study is also capable of calculating the Young's modulus of microcantilevers without coating or with fairly thin coating. Furthermore, the Young's modulus of these multi-layer microcantilevers calculated according to the presented model considering the coating (80 and 70 GPa) are approaching those of the coating (78 and 69 GPa) when the thickness of coating is extremely high, which also validates the accuracy of the presented model. For Tap300GD and Multi75AI, lowest Young's modulus and resonant frequency of multi-layer microcantilevers could be obtained when the thickness of coating was around 4000 nm and 3000 nm respectively, which is almost the same as the thickness of the base of Tap300GD and Multi75AI, as  The resonant frequency increases due to weaker influence of interface when the thickness of coating is larger than that of base, and the resonant frequency increases with the thickness, as shown in theoretical models. The thickness of coating has a large influence on Young's modulus and resonant frequency of multi-layer microcantilevers, indicating that the impact of coating should be taken into account, therefore the models presented in this work will obtain accurate calculation results.
The length of coating has little influence on Young's modulus whether the coating is Al or Au, as shown in Fig. 3. However, it has large influence on the resonant frequency, and the results of our model is almost the same as those of others. Because for Tap300GD and Multi75AI, the thicknesses of coating are fairly thin (around 30 nm and 70 nm respectively).
As shown in Fig. 4, the width has almost no influence on the Young's modulus and resonant frequency of microcantilevers.
The results calculated from the theoretical models are in good agreement with those of FE simulation and AFM, indicating that the theoretical models considering the impact of coating are capable of calculating the Young's modulus of multi-layer microcantilevers. By comparing the results of presented theoretical models introducing the impact of coating and three typical theoretical models (Rayleigh-Ritz model, E-B beam model and spring mass model), it can be concluded that the presented models are more precise than other models.