# Design of a Piezoelectrically Actuated Ultrananocrystalline Diamond (UNCD) Microantilever Biosensor
## 1 Introduction
This implies that microcantilevers fabricated with UNCD are expected to exhibit superior sensitivity and mass resolution, especially when operating in dynamic mode. For a cantilever operating in static mode, the sensing method relies on the flexural single bending of the beam (static mode) [4]. When a cantilever oscillates at its resonance frequency, induced by electrical or physical excitation, usually at its first resonance mode or eigenfrequency (dynamic mode) [5], the change in mass produced by the absorption of particles or molecules on the cantilever's surface induces a change in the natural frequency [13]. Static mode is described by the Stoney Equation (1) [14],  
\[\Delta z=\frac{3(1-\
u)L^{2}}{Et^{2}}\Delta\sigma \tag{1}\]  
where \(\Delta z\) is the bending amplitude of the microcantilever, \(\Delta\sigma\) is the surface stress change, \(\
u\) is the [PERSON]'s ratio, \(L\) is the length of the microcantilever, \(E\) is the Young's modulus of the base material used to make the microcantilever, and \(t\) is thickness of the microcantilever. Thus, in operation mode, the sensing is based on the differential stress produced by the addition of particles or molecules on the surface of the cantilever and the consequent amplitude of the bending. On the other hand, in dynamic mode, the sensing is related to the resonance frequency shift produced by the addition of particles to the surface of the vibrating microcantilever [9]. The resonance frequency, in terms of geometry and mechanical properties of the microcantilever, is given by Equation (2),  
\[f_{n,0}=\frac{(\beta_{n})^{2}}{2\pi}\frac{t}{L^{2}}\sqrt{\frac{E}{12\rho}} \tag{2}\]  
where \(f_{n,0}\) is the natural resonance frequency of the microcantilever, with no damping for the \(n\) mode, \(\beta\) is a constant that appears during modal analysis, \(\rho\) is the density of the base material, and \(E\), \(L\), and \(t\) represent the same parameters as in Equation (1). For the first eigenfrequency, \(\beta_{1}\) is 1.875, which originates from the solution of the Euler-Bernoulli beam theory, as shown in Equation (2) [3; 15]. Furthermore, the mass sensitivity per unit mass of load on a rectangular microcantilever is given by Equation (3) [16],  
\[\frac{df_{1,0}}{dm}\simeq\frac{f_{n,0}}{2m_{c}} \tag{3}\]where \(m_{c}\) is the total mass of the microcantilever. Equation (3) can be rewritten in terms of geometrical characteristics and material properties, replacing \(f_{n,0}\) in Equation (3) with the parameters shown in Equation (2), giving Equation (4). \[\frac{df_{1,0}}{dm}=0.279\frac{1}{L^{3}W\rho}\sqrt{\frac{E}{12\rho}} \tag{4}\]  
Equation (4) enables the calculation of the shift in the microcantilever vibration frequency in the first resonance mode \(f_{1,0}\) produced by the addition of mass on the microcantilever. The frequency shift increases with the squared root of \(E\) and decreases with an increased density \(\rho\) of the beam's material and an increased width \(W\) and length \(L\) of the cantilever. Another important aspect to consider is the mass resolution, which is given by Equation (5) [14; 16],  
\[\delta\bigg{(}\frac{df}{dm}\bigg{)}=\sqrt{\frac{f_{n,0}k_{B}TB}{\pi kQA^{2}}} \tag{5}\]  
where \(\delta\bigg{(}\frac{df}{dm}\bigg{)}\) is defined as the minimum mass detectable by the minimum frequency shift, \(k_{B}\) is the Boltzmann constant, \(T\) is the absolute temperature, \(B\) is the detection bandwidth, \(k\) is the spring constant of the microcantilever, \(Q\) is the quality factor, and \(A\) is the amplitude of the oscillation. Equation (5) is a general equation for micro-resonators for mass sensing. This equation shows that as the amplitude (\(A\)) of the oscillations decreases, the mass resolution of the resonator increases. On the other hand, the increase in resonant frequency (\(f_{n,0}\)) increases the mass resolution. As seen in Equations (4) and (5) the mechanical properties of the material play an important role in the mass sensitivity of microcantilevers.