br Acknowledgements The authors thank Brian Murphy

Acknowledgements The authors thank Brian Murphy, Barry Ives, Jared Lowe, Suzy Hong, and Kirsten Seagers for technical support and discussions. The authors thank the Bill & Melinda Gates Foundation, Grant #OPP1086152, for their generous support.
Introduction Capacitive micromachined ultrasonic transducers (CMUTs) have been making significant impact in many fields such as medical diagnostic ultrasound, structural health detecting and real-time monitoring of machinery operation [1,2]. CMUT-based sensors are ideal for these purposes due to their high sensitivity, small size, low mass, long lifetime and low power requirement. A CMUT with a rectangular neutrophil elastase inhibitor is shown in Fig. 1. It consists of a dielectric spacer-supported clamped rectangular diaphragm and a fixed back plate, which are separated by a thin gap. When an external pressure is exerted onto the sensor, the top membrane will deform, leading to a dynamic change in the capacitance between the deformed membrane and the fixed back plate [3]. Highly accurate analytical deflection shape functions that describe the deflection of deformed CMUT membranes will not only provide the insight into the CMUT design methodology but also ascertain the effect of specific geometry parameters [4–11]. For these reasons, deflection shape functions of square and circular membranes have been widely studied by many authors. Plate theory is often applied to capture the functional form of the deformation curve. The transverse deflection ω(x, y) of any point (x, y) on a uniformly loaded diaphragm can be obtained by energy minimization method [12]. However, this method is computationally intensive as solutions of numerous simultaneous nonlinear equations are required. Some simple analytical deflection models, the accuracy of which depends mainly on the deformed membrane shape, have been used to predict the deformed curves of square membranes. The deflection model of a rigidly clamped square membrane under a uniform external pressure was first presented as a cosine-like function as [12]where a is half the side length of the diaphragm and ω0 is the center deflection. This function describes the general membrane deflection shape, but not accurately. In order to achieved desirable accuracy, authors in [13] expanded the function with two more terms to yield However, authors in [14] pointed out that function (2) fails to catch the deflection profiles of thick membranes, although it agree well with those of thin and large ones. And a new deflection model was introduced in [14] to cover different cases by squaring the cosine terms and adding a new term. Authors in [15] further improved the accuracy by developing a new deflection model for square membranes following a two-step process. In the process, the center deflection ω0 is first obtained by solving a load–deflection model, then the deformed diaphragm shape ω′(x, y), which is independent from the center deflection, is calculated and multiplied by ω0 to obtain the complete deflection profile. The load–deflection model of a square diaphragm under a uniform pressure can be expressed as [15]where σ0 is residual stress, h is membrane thickness, ν is Poisson’s ratio. The Poisson’s ratio dependent function (ν) is given as . D is the flexural rigidity of the membrane expressed as , where E is the Young’s modulus of the diaphragm material. The constants , and are determined by comparing the deflection profiles in (3) with finite element analysis (FEA) results [16]. Following the two-step process, the analytical deflection model for CMUT with square membrane was presented as [15] The polynomial basis function was substituted for cosine basis function and higher accuracy was obtained. Function (4) was subsequently used to calculate the capacitance values of CMUTs by formula aswhere d0 is the gap thickness, and ɛ0 is the permittivity of free space, given as ɛ0=8.85×10−12F/m. It has been shown that FEA provides highly accurate deflection profiles for CMUT membranes and other MEMS-based transducers [17–21]. But it does not give an insight into the influences of the device geometries on CMUTs as analytical models do. The capacitance calculated by the analytical deflection models (2) and (4) has a good agreement with the FEA results for square diaphragms [15]. However, deflection shape function of rectangular membranes has never been studied, probably due to their complexities in deflection shape. In fact, rectangular membranes are worth studying because they have shown the potential in improving the fill factor and the performance of CMUTs compared with square ones [22,23].