# Inverse eigenvalue sensing with corner coupled square plate MEMS resonators array

## Abstract

Monitoring the collective behaviour of coupled micro/nano resonators array provides a distinct opportunity for high resolution multi-function sensing. We report an inverse eigenvalue analysis based sensing approach for large array of coupled micro/nano resonators. A new characterization algorithm is proposed to precisely extract the system matrix of those multiplexed sensors with reduced algorithmic complexity and below 1% relative error. The method has been verified experimentally using five corner coupled square plate MEMS resonators with a natural frequency close to 0.85 MHz. The method is also capable of characterizing the fabrication process and important sensor parameters such as the spring constant and coupling ratio.

## Keywords

MEMS sensor readout Coupled resonators array Inverse eigenvalue analysis Process variability## 1 Introduction

Micro/nano resonator has been widely implemented as inertial sensors for mass, force and acceleration sensing applications [1, 2, 3, 4]. Coupled resonators have been suggested to improve the sensitivity and functionality of those sensors. There are several advantages of mechanically or electrically coupling them together. First, the mode localisation effect opens up new sensing methods, such as multi-mode sensing and the use of eigenvector (amplitude), to push down the limit by several orders of magnitude [5, 6, 7, 8]. Second, it is possible to drive and readout from only one single resonator to find all eigenvalues (resonance frequencies) of a large array thanks to the collective behaviour [9]. This reduces the actuation complexity and turns the system into a single-input-single-output (SISO) multiplexed sensors array. In addition, coupling facilitates the characterization of large resonators array utilizing an inverse eigenvalue analysis (IEA) method. By perturbing the terminal element of a coupled array, two unique sets of eigenvalues are recorded to extract the system matrix using inverse problem theory. The system matrix is associated with important device properties such as Young’s modulus [10], mass sensitivity [11] and process variability [12]. However, existing IEA techniques require solving complicated eigenvectors or orthogonal polynomials, and hence become complicated when the array size is large.

Herein, we present a simplified IEA technique to circumvent this issue, thereby providing a new solution to system matrix determination. An array of five nearest-neighbour coupled square plate resonators were fabricated via the multi-user MEMS processes (MUMPs) to validate the ideas. Using the new IEA method, important sensor information including sensitivity and coupling ratio can be provided, which are of substantial interest for resonator designers.

The article is organised as follows: Section 2 provides the mathematical foundation of the simplified IEA technique to extract eigenvectors and system matrix, and analyses its complexity and accuracy. Section 3 describes the MEMS square plate resonator used in this research, including the fabrication process, electromechanical properties, experimental setup and result of a prototype sensor. Section 4 studies the collective behaviour of coupled systems using an example of five coupled resonators. Section 5 characterizes the sensor using perturbation analysis and the proposed IEA technique. System matrix and other parameters such as spring constant and coupling ratio are therefore determined. Section 6 gives the conclusion.

## 2 Inverse eigenvalue analysis

### 2.1 Coupled systems

*n*nearest-neighbour coupled resonators can be modelled by the following equation

*i*th mode, \({\mathbf {S}}\) has eigenvalues \(\lambda _i = \omega _i^2\) and eigenvectors \(\mathbf {x}_{\mathbf {i}}\) that are related to resonance frequencies and amplitudes. Ideally, there are

*n*eigenvalues already available from the

*n*resonance peaks, which can be observed from any element of the array due to collective behaviour. A mass or stiffness perturbation upon one resonator would lead to another set of

*n*unique eigenvalues \(\lambda _i^*\).

### 2.2 New characterization algorithm

*n*becomes large. The algorithm can be linearised by ignoring high order terms while preserving adequate precision. When a perturbation happens on the

*p*th element, the new system matrix becomes

*p*,

*p*)th entry \(\varDelta S_{p,p}\), which can be calculated using the trace property of the system matrix before and after perturbation

### 2.3 Complexity and accuracy

*O*(

*n*), which facilitates signal processing requirement and paves the way for rapid characterization of massive sensors array. With the amplitudes of the

*p*th resonator known for all modes, the (

*p*,

*p*)th element of the system matrix is determined by

## 3 Square plate resonator

### 3.1 Device architecture and fabrication process

Square plate resonator is a simple structure that utilises the transverse mode vibration of a centre-stemmed square membrane, and has already been applied in the fields of radio frequency and communication [16]. The design benefits from high quality factor *Q*, low impedance and convenient eigenvalue readout channel, which can be potentially used as sensors or transducers.

*t*of the square plate (\(t = 1.5\,\upmu {\hbox {m}}\)) and the gap distance

*g*between the top plate and bottom electrode (\(g = 0.75\,\upmu {\hbox {m}}\)) were defined by the process. As Poly 0 layer was not used in our design, the device was anchored to the Nitride substrate by filling an anchor hole with Poly 1. The hole was formed by patterning silicon dioxide with a mask layer and reactive ion etching (RIE).

### 3.2 Electromechanical properties

*E*and density \(\rho\) is given by [18]

*L*is designed to be 100 \(\upmu {\hbox {m}}\) to provide around 1.2 MHz resonance. The mass and stiffness of the resonator are calculated by

### 3.3 Single resonator

Figure 6 shows the conductance and phase response of a prototype sensor by applying \(V_{DC}=10\) V and \(v_{in} = 50\) mV under a low pressure level \(P < 100\) mPa. Fabricated resonators demonstrate a natural frequency between 0.8 and 0.85 MHz, which is approximately 30% less than the nominal frequency 1.2 MHz due to fabrication tolerance. This was likely to be caused by the extra side length when depositing Poly 2 layer onto the uneven silicon dioxide formed during the patterning process, thereby increasing the effective length by approximately 15%.

*Q*and ambient pressure

*P*is plotted in Fig. 7(a). The quality factor varies between 10 and \(1.6 \times 10^4\) when the pressure drops from 3.6 kPa to 27 mPa. Figure 7(b) shows the square root of the electrostatic stiffness as a linear function of \(V_{DC}\). The resonator collapses at a pull-in voltage about 25 V, which causes short circuit between the top plate and bottom electrodes, therefore burning the device. To prevent such irreversible process from happening and ensure good linearity, the recommended DC operating range is within 10 V, which allows a maximum 2.1% perturbation. The lower-than-expected pull-in voltage is attributed to the same over-sizing effect of the side length.

## 4 Coupled resonators array

Weakly coupled MEMS resonators demonstrate collective resonance behaviour at a unique set of eigen-modes that are different from a single resonator, and have been extensively explored as sensors with enhanced sensitivity, reduced connections and bandwidth, as well as the capability of multi-sensing. Coupling is first seen in micro cantilevers array, where the natural overhang structure introduces parasitic crosstalk between neighbouring elements.

### 4.1 Eigen-mode analysis

### 4.2 Array configuration

Dimensions for the proposed corner coupled square plate MEMS resonators array

Parameter | Value | Unit |
---|---|---|

Plate side length, | 100 | \(\upmu {\hbox {m}}\) |

Plate thickness, | 1.5 | \(\upmu {\hbox {m}}\) |

Coupling beam length, \(L_C\) | 20 | \(\upmu {\hbox {m}}\) |

Coupling beam width, \(W_C\) | 10 | \(\upmu {\hbox {m}}\) |

Gap distance, | 0.75 | \(\upmu {\hbox {m}}\) |

## 5 Sensor characterization

### 5.1 Perturbation analysis

### 5.2 System matrix determination

### 5.3 Spring constant and coupling ratio

Sensor parameters such as mass, stiffness and coupling ratio can be derived from the extracted system matrix. The mass normalised spring constants of resonator 1–5 are \((2.55, 2.31, 2.51, 2.69, 2.70) \times 10^{13}\), which correspond to natural frequencies 0.804, 0.765, 0.797, 0.826 and 0.827 MHz. Hence the actual frequency variation of the PolyMUMPs fabrication is determined to be 3.2%.

## 6 Conclusion

We introduce a new inverse eigenvalue technique to investigate coupled micro/nano systems. With reduced algorithmic complexity and less than 1% system matrix error, the method has been experimentally verified using an example of five coupled square plate MEMS resonators with eigenfrequencies around 0.85 MHz. By obtaining two sets of eigenvalues before and after terminal perturbation, the IEA has been performed to derive the system matrix and hence important sensor and process information. The method offers improved accuracy and simplified readout, which can be used to actuate and characterize large array of coupled resonators, including ultrasound transducers and multi-function inertial sensors. As a proof of concept, only five resonators are examined in this paper. Theoretically, the maximum number of resonators in a coupled array is limited by the mode liaison effect, i.e. \(n < \kappa Q \approx 100\) [20]. Hence we have fabricated larger arrays of twenty to fifty coupled resonators, which will be tested and used as sensors in our future work.

## Notes

### Acknowledgements

We thank Ching-Mei Chen for designing the PCB. We acknowledge the help from Madhav Kumar in taking the SEM images. Authors would like to acknowledge the financial support by the European Commission, through FP7 Project iARTIST, Grant Agreement Nos. 611362, FP7-PEOPLE.

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