Design and Modelling of MEMS Resonators for an Artificial Basilar Membrane ^†

Abstract

The human cochlea is undeniably one of the most amazing organs in the body. One of its most intriguing features is its unique capability to convert sound waves into electrical nerve impulses. Humans can generally perceive frequencies between 20 Hz and 20 kHz with their auditory systems. Several studies have been conducted on building an artificial basilar membrane for the human cochlea (cochlear biomodel). It is possible to mimic the active behavior of the basilar membrane using micro-electromechanical systems (MEMSs). This paper proposes an array of MEMS bridge beams that are mechanically sensitive to the perceived audible frequency. They were designed to operate within the audible frequency range of bridge beams with 450 µm thickness and varying lengths between 200 µm and 2000 µm. As for the materials for the bridge beam structures, molybdenum (Mo), platinum (Pt), chromium (Cr) and gold (Au) have been considered. For the cochlear biomodel, gold has proven to be the best material, closely mimicking the basilar membrane, based on finite-element (FE) and lumped-element (LE) models.

1. Introduction

Sound can only be heard and manipulated by humans through their auditory system. There are three parts to the human ear: the outer ear, the middle ear, and the inner ear. As sound waves travel from the surrounding area to the middle ear, they are carried by the ear flaps and canals in the outer ear. The anvil, stirrup, and hammer are three miniature ear bones in the middle ear. The eardrum is a thin membrane that the sound waves bump into at this point. The hammer is attached to the eardrum. Therefore, the hammer moves when the eardrum vibrates. The stirrup and anvil are used to transfer these movements. Stirrups are connected to basilar membranes in the inner ear. Consequently, the basilar membrane vibrates due to the movements of the ear bones. In the meantime, the nerve cells detect the movement from the basilar membrane and transmit nerve impulses to the brain [1]. Different biomimetic approaches have also been reported [2,3,4] to detect sound using MEMS technology.

The basilar membrane within the cochlea is one of the essential parts of the hearing process. It may hold the key to understanding the mechanism responsible for the unknown adaptive cochlear mechanism. Researchers have developed artificial basilar membranes, i.e., through cochlear biomodelling, to mimic the characteristics of active cochlea filtering. The basilar membrane has a stiff, narrow base, which is the opening part. As sound waves propagate from the base to the apex, the basilar membrane responds mechanically depending on their frequency, amplitude and time [5]. When high-frequency sounds are received, it responds.

In contrast, the apex is the flexible part of the basilar membrane. There is more flexibility and a larger area in this part. It responds to sound waves with lower frequencies. The sensitivity decreases when the distance between the basilar membrane and the base increases [6]. A microelectromechanical system (MEMS) combines miniaturized mechanical and electro-mechanical elements, such as resonators and microphones [7]. The advantage of MEMS resonators is that they closely mimic the cochlea in terms of measurement and characteristics.

The tonotopic organization factor within the cochlea has been mimicked by artificial basilar membranes [8,9]. Many of them are bulky, heavy, and fluid-surrounded artificial basilar membranes. Based on advances in microfabrication technology, microresonators could be fabricated with a life-size, nonfluidic and unsophisticated surrounding artificial basilar membrane [10,11,12,13].

An array of MEMS bridge beam resonators of various lengths are used in our study to work at audible frequencies of 20 Hz to 20 kHz. Each resonator of the bridge beam series is known to have a thickness of 450 μm and a width of 150 μm, varying in length from 200 μm to 2000 μm. Moreover, four different material structures are investigated for the MEMS bridge beam resonators: platinum (Pt), molybdenum (Mo), chromium (Cr) and gold (Au). The MEMS bridge beam resonators have been designed and analyzed using finite-element (FE) and lumped-element (LE) models. COMSOL Multiphysics is used for FE modelling, and the results are compared with the LE model.

2. Lumped-Element Model

An analysis of the dynamic behavior of a bridge beam structure using lumped-element models may be represented as a vibrating system with a single degree of freedom. The resonating structure represents a lumped mass, spring and damper within the model. Using Equation (1), a series of bridge beams can be designed that resonate within a certain frequency range, where the fundamental mode vibration γ is equal to 4.73; the cross-sectional area is Ab = w_bt_b, where t_b and w_b are the bridge beam thickness and width, respectively; E is Young’s modulus of the material being used to construct the bridge beam structure; $\mathit{I}=\frac{{\mathit{w}}_{\mathit{b}}{\mathit{t}}_{\mathit{b}}^{\mathbf{3}}}{\mathbf{12}}$ is the moment of inertia; ρ indicates the material density; and l_b is the bridge beam length. Equation (1) can be simplified to Equation (2), through which the resonant frequency f₀ can be observed to have an inverse proportional and direct proportional relationship with ${\mathit{l}}_{\mathit{b}}^{\mathbf{2}}$ and $\sqrt{\frac{\mathit{E}}{\mathit{\rho }}}$, respectively. In our work, we have used t_b = 450 μm and w_b = 150 μm with l_b = 200–2000 μm.

$$\mathit{f}\mathit{o}=\frac{{\mathit{\gamma }}^{\mathbf{2}}}{\mathbf{2}\mathit{\pi }}\sqrt{\frac{\mathit{E}\mathit{I}}{\mathit{\rho }{\mathit{A}}_{\mathit{b}}{\mathit{l}}_{\mathit{b}}^{\mathbf{2}}}}$$

(1)

$$\mathit{f}\mathit{o}=\mathbf{1.028}\frac{{\mathit{t}}^{\mathit{b}}}{{\mathit{l}}_{\mathit{b}}^{\mathbf{2}}}\sqrt{\frac{\mathit{E}}{\mathit{\rho }}}$$

(2)

3. Finite-Element Model

The novel array of bridge beam resonators shown in Figure 1 resembles the basilar membrane in the human cochlea in terms of its characteristics. Bridge beams with a length of 200 m indicate the opening area of the membrane (base), which will be highly responsive to high-frequency sound waves. The longest bridge beam, which has a length of 2000 m, indicates where the membrane ends (apex), which is responsive to the lowest frequency of the audible sound wave, and moves upwards [14]. COMSOL Multiphysics 4.3 was used to construct the finite-element models, and the resonators’ desired frequency responses were verified and designed.

The material structure for the MEMS bridge beams in this study includes platinum (Pt), molybdenum (Mo), chromium (Cr) and gold (Au). Each material has different mechanical/material properties [15], and these must be considered. MEMS bridge beams might be able to operate at desired audible frequencies with these proposed materials, given their small E/ρ ratios.

Table 1. Geometrical dimensions of MEMS bridge beams.

Beam	Size (µm)
Length	200–2000
Width	150
Thickness	450

summarizes the geometrical dimensions of the designed MEMS bridge beams, while

Table 2. MEMS bridge beams’ mechanical properties.

Material	Mass Density (g cm−3)	Young’s Modulus (GPa)
Platinum	21.45	168
Molybdenum	10.10	315
Chromium	7.20	140
Gold	19.3	79

shows the mass density and Young’s modulus of the materials considered. Finite- and lumped-element models have been developed based on these data.

4. Results and Discussion

The MEMS bridge beam resonance frequencies for all four materials are shown in Figure 2, with bridge length as a function of the resonance frequency. The designed MEMS bridge beams resonate close to the audible frequency range, as shown via the simulation. Based on their design, the MEMS bridge beam resonators mimic the apex-to-base characteristics of basilar membranes.

For a bridge length of lb = 200–2000 μm, the simulated resonance frequencies for platinum (Pt), molybdenum (Mo), chromium (Cr), and gold (Au) are 32,399.11–350.42 Hz, 64,623.43–698.34 Hz, 51,067.66–550.90 Hz and 23,434.89–251.90 Hz, respectively. It has been observed that the gold MEMS bridge beams offer the best performance due to their proximity to audible frequencies.

Here, a comparison is made between the simulation results from the FE modelling and those from the lumped-element modelling. The materials were analyzed based on their dimensions and mechanical properties.

Material 1: Platinum (Pt)

Due to the small E/ρ ratio, platinum is one of the top materials for fabricating MEMS bridge beams because of its unique properties. The finite-element model of platinum MEMS bridge beams with resonance frequencies between 32,399.11 and 350.42 Hz is shown in

Table 3. Comparison of the simulated and calculated resonance frequency values of MEMS bridge beams built from platinum (Pt), molybdenum (Mo), chromium (Cr) and gold (Au) and the error percentage of each.

Length (µm)	Platinum (Pt)			Molybdenum (Mo)			Chromium (Cr)			Gold (Au)
	FE	LE	Error (%)	FE	LE	Error (%)	FE	LE	Error (%)	FE	LE	Error (%)
200	32,399.11	32,365.80	0.10	64,623.43	64,586.24	0.05	51,067.66	50,996.79	0.13	23,434.89	23,398	0.15
400	8140.32	8091.45	0.60	16,188.32	16,139.58	0.30	12,793.34	12,743.68	0.38	5887.33	5846.99	0.68
600	3634.28	3596.20	1.04	7208.90	7176.24	0.45	5699.98	5666.20	0.59	2623.87	2599.78	0.91
800	2092.87	2022.86	3.34	4098.87	4036.57	1.15	3218.87	3184.90	1.05	1501.45	1462.35	2.60
1000	1333.99	1294.63	2.95	2612.12	2583.44	1.09	2089.98	2039.87	2.39	998.34	935.92	6.25
1200	945.78	899.05	4.94	1819.81	1794.06	1.41	1479.47	1416.56	4.25	699.56	649.94	7.09
1400	700.46	660.46	5.71	1384.65	1317.97	4.81	1092.56	1040.66	4.75	500.76	477.47	4.65
1600	533.87	505.70	5.27	1078.46	1009.14	6.42	845.76	796.81	5.78	393.43	365.58	7.07
1800	419.89	399.36	4.88	837.67	796.92	4.86	679.78	629.68	7.37	310.44	288.91	6.93
2000	350.42	323.51	7.67	698.34	645.58	7.50	550.90	509.74	7.47	251.90	233.87	7.15

. A comparison of FE and LE models for platinum MEMS bridge beam resonance frequencies is shown in Figure 3a. This figure shows the difference between the FE and LE models for the resonance frequency of platinum MEMS bridge beams.

Material 2: Molybdenum (Mo)

As shown in Table 3 and Figure 3b, the resonance frequency of the MEMS bridge beams made of molybdenum ranges from 64,623.43 to 698.34 Hz (finite-element model). The percentage errors between the FE and LE models are also acceptable as the highest percentage error is 7.50%.

Material 3: Chromium (Cr)

As shown in Table 3 and Figure 3c, the resonance frequency of the MEMS bridge beams made of copper ranges from 51,067.66 to 550.90 Hz (finite-element model). The percentage errors between the FE and LE models are also acceptable as the highest percentage error is 7.47%. Having a smaller E/ρ ratio, chromium is better than molybdenum as it operates closer to the audible frequency range [16,17].

Material 4: Gold (Au)

In Table 3, the lumped-element model of the MEMS bridge beams made of gold show that the resonance frequency ranges from 23,434.89 to 251.90 Hz. The highest error is 7.15% at lb = 2000 μm, and the lowest is 0.15% at lb = 200 μm. Figure 3d shows the comparison of both the simulated (FE model) and calculated (LE model) values for the resonance frequencies.

5. Conclusions

In this work, MEMS bridge beam resonators have been designed to mimic the cochlear basilar membrane to operate in the audible frequency range. Important factors have to be taken into account when designing the MEMS bridge beams of the future, and these include the geometry of the beam and the material used in the beam structure. Based on FE and LE models, an array of MEMS bridge beams with dimensions of 450 µm thickness, 150 µm width, and 200 µm to 2000 µm length have been designed using platinum (Pt), molybdenum (Mo), chromium (Cr) and gold (Au) as the materials. According to the functions of the base and apex in the basilar membrane, the resonant frequencies have been shown to decrease with increasing bridge lengths. Gold provides a resonance frequency that is closest to the desired audible range, making it the ideal material for an artificial basilar membrane. A MEMS bridge beam resonator can be accurately designed with both FE and LE models with very small percentage differences.