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Asymmetric resonance frequency analysis of in-plane electrothermal silicon cantilevers for nanoparticle sensors

The asymmetric resonance frequency analysis of silicon cantilevers for a low-cost wearable airborne nanoparticle detector (Cantor) is described in this paper. The cantilevers, which are operated in the fundamental in-plane resonance mode, are used as a mass-sensitive microbalance. They are manufactured out of bulk silicon, containing a full piezoresistive Wheatstone bridge and an integrated thermal heater for reading the measurement output signal and stimulating the in-plane excitation, respectively. To optimize the sensor performance, cantilevers with different cantilever geometries are designed, fabricated and characterized. Besides the resonance frequency, the quality factor (Q) of the resonance curve has a high influence concerning the sensor sensitivity. Because of an asymmetric resonance behaviour, a novel fitting function and method to extract the Q is created, different from that of the simple harmonic oscillator (SHO). For testing the sensor in a long-term frequency analysis, a phase- locked loop (PLL) circuit is employed, yielding a frequency stability of up to 0.753 Hz at an Allan variance of 3.77 × 10-6. This proposed asymmetric resonance frequency analysis method is expected to be further used in the process development of the next-generation Cantor.


Introduction
Since the last few years, more and more types of nanoparticles (NPs) have been used in different consumer goods, resulting in an increasing awareness towards releasing these NPs in air. Because of their small size, possessed hazard, and presence in both indoor and outdoor environments, they can become a huge danger or risk when they are exposed to human body via either direct deposition on the skin or inhalation through respiratory systems [1]. Therefore, there have been many studies concerning the characteristics and health effects of different types of airborne NPs in the environmental communities as well as the strategies to manage their exposure [2].
To ensure a safety of the workers who directly handle and interact with NPs during the manufacturing of the NP-based devices or products, it is necessary to equip them with a low-cost wearable directreading tool for continuously monitoring the airborne NP mass concentration levels [3]. Therefore, owing to their detection simplicity and cost-effective batch fabrication, various types of resonant micro/nanoelectromechanical systems (M/NEMS) were developed as NP mass-sensitive sensors by several world-leading research groups to meet the demands (e.g., microfluidic film bulk acoustic resonator (FBAR) [4], electrothermal silicon cantilever [5], filter-fiber nanoresonator [6], and vertical silicon nanowire resonator [7]). The NP measurement principle is based on the monitoring of the resonance frequency shifts of the M/NEMS caused by the NPs additionally deposited on their surfaces. In our second generation of the pocket-sized cantilever-based airborne NP detector (Cantor-2), the inplane electrothermal piezoresistive silicon cantilever is connected to a phase-locked loop (PLL) circuit to realize a real-time tracking system of the resonance frequency [8,9]. However, the phenomenon of the asymmetric resonance frequency signals produced by the employed cantilevers has not been analyzed in details. Thus, in this work, several designs of the electrothermal cantilever resonators are investigated in terms of their resonance characteristics, which will then be used as a basis for further improvement of the Cantor. Moreover, their frequency stability is also measured to define the Allan variance of the device in a long-term test.

Sensor preparations
The electrothermal silicon cantilever resonators with different free-end geometries (Figures 1(a)-(d)) were simulated using the finite element modelling (FEM) tool of COMSOL Multiphysics 4.3b to determine their figure of merit (i.e., FoM = mc/(f0×Ac×Q) [10]) as listed in Table 1. In this case, mc, f0, and Ac denote the mass of the cantilever, the resonance frequency, and the area of the collection surface, respectively. It should be noted that all these cantilevers have a total length of 1000 µm and a beam spring width of 170 µm. By exciting the heating resistor on the cantilever with a DC bias of 5 V, the temperature (T) increases from 293.15 K to 306.1 K (i.e., ΔT = 12.95 K) resulting in a cantilever bending ( Figure 2(a)). The calculated maximum stress obtained from FEM is 3.63 × 10 7 N/m 2 [9].    Subsequently, these in-plane cantilevers were fabricated using bulk silicon wafers and bulk micromachining techniques, leading to lower cost of materials and fabrication processes, respectively (Figures 2(b) and (c)). The detailed fabrication steps have been described previously, which mainly employ photolithography, dopant diffusions using borofilm 100 and phosphorosilica (Emulsitone), thermal oxidation, metal evaporation, and inductively coupled plasma (ICP) cryogenic dry etching processes [9].

Resonance frequency analysis
To analyze the resonance behaviors of the fabricated cantilevers (Figures 3(a)-(d)), a frequency sweep was carried out using the measurement setup described in [10]. The main relevant parameters of the measured resonance curves are the f0, the Q and the relative amplitude of the peak, which can be directly extracted from a fitting procedure. Because of the asymmetry of the resonance curve, a fitting function in relation to the Fano resonance is developed replacing the commonly used simple harmonic oscillator (SHO) fitting function [11]. The Fano resonance is based on a system of two coupled vibrations where one is forced to its oscillation directly and their superposition will result in an asymmetric resonance behavior ( Figure 4) [12].   The Fano resonance frequency response can be described implicitly by [12]: ( ) where q and f are the asymmetry factor and the frequency, respectively. The frequency sweep response σ around the f0 is then given by: with line width and gain parameters g and H, respectively, and an offset comprising a constant (σ0) and linearly varying term (t × f). The proposed Q is calculated by combining two established methods (i.e., Qq=0 = f0/2g[(2) 0.5 -1] 0.5 and Qq=1 = (2f0 2 +2g 2 )/(4g×f0) [13]) with their newly introduced correction factors of (1-|q| 0.5 ) and |q| 0.5 as: (3) A simulation of an electrical RLC circuit using MATLAB is used as a reference to validate Equation (3) replicating the different resonance curves. In this case, an excitation is generated to bring the circuit to resonance followed by a cut off at steady state. The yielded decaying response amplitude can then be fitted over the time. The damping of the oscillation defines the quality factor of the system [13,14]. Figures 5(a)-(c) depict the resonance curves with different asymmetry levels calculated using Equation (2). For the validation, the frequency responses of the fitted RLC circuit are added. The relative deviation between the calculated and simulated f0 and Q are listed in Table 2. Small differences between these two curves ( Figure 5(a)) causes the deviation of f0 and Q at q = 1. We find that inaccuracies of the Equation (3) at q = 0.5 and q = 0.25 are still comparable with that of q = 1 (i.e., ΔQ ˂ 0.7), proving its reliability.    Table 3 shows good agreement of the f0 obtained from FEM and the fitting of the measurements with four different cantilever sensors using Equations (2) and (3). However, the Q values of the designed cantilevers with strivings (Cts and Crs) are much lower than expected, which can be due to the simplification of the 3-D model geometries used in the FEM. Moreover, in the long-term stability tests using a homebuilt PLL circuit [15], the fabricated cantilevers without strivings (Cs and Ct) have demonstrated a frequency stability of up to 0.753 Hz at an Allan variance of 3.77 × 10 -6 ( Figure 6). Although the proposed sensors have shown promising results in terms of the simulation and measurement (i.e., high Q > 2000), some geometry modifications as well as fabrication parameter adjustments are still required to enhance the performance of the devices. Besides that, the described asymmetric resonance frequency fitting method (i.e., based on Fano resonance method) is expected to be employed supporting the optimization of the next-generation Cantors to precisely estimate their resonance characteristics (i.e., f0 and Q).

Conclusions
Electrothermal silicon piezoresistive cantilever sensors with different free-end geometries have been analyzed in terms of their asymmetric resonance behaviors. A novel fitting model based on the Fano resonance approach has been developed to precisely extract the values of the resonance frequency and quality factor (Q) of the devices, replacing the simple harmonic oscillator (SHO) fitting model. Moreover, despite the promising results from long-term stability tests, further experiments of the cantilever resonators in airborne nanoparticle (NP) exposure are still needed to justify the resonance characteristics of the devices in the conditions of before and after being polluted with NPs.