Title: Fundamental Limitations of Chemical and Biological Sensors
Author: Alma L. Chung
Last saved by: James B. Lewis

Fundamental Limit of Chemical and Biological Resonant Micro/Nano-Sensors
Peter H. Handel
Physics Department, Univ. of Missouri, St. Louis, MO63121
E-mail: handel@umsl.edu

Piezoelectric sensors used for the detection of chemical agents and for electronic nose instruments are based on BAW and SAW quartz resonators.  They are also useful in the detection of biological agents, and can be specific, particularly if the sensitive surface is activated with the right antigen. The MEMS resonators are shaped as a silicon bar of nanometric dimensions incased (fixed) at both ends, subject to bending strain.  They are best driven by an AC current flowing in a very thin straight wire attached along the bar, in the presence of a strong constant magnetic field, perpendicular to it. 
The BAW resonators are used for instance in the "quartz crystal microbalance" (QCM).  This is usually a small polymer-coated resonating quartz disk, with smaller diameter metal electrodes on each side and with quality factor Q.  The resonance frequency is usually in the 50 MHz range.  Absorption of gas molecules with mass m on the surface of the polymer coating gets detected by a reduction y=m/mof the resonance frequency of the quartz disk, subject to frequency fluctuations.  The quantum 1/f limit of detection is given by
^2S_m/m(f)=S_y(f), with S_y(f) = 'V/fQ^4, for V≤	and	S(f) = '^2/fVQ^4, for V≥

where  is the phonon coherence volume, first introduced by T. Parker et. al. as a noise coherence volume.  To optimize the device we must avoid closeness of V with  which corresponds to the maximum error and minimal sensitivity situation. Differential measurements that include a reference resonator without polymer coating can effectively eliminate temperature fluctuations, power supply instability, etc.  Adsorbed masses below the pg=10^-12g range can be detected.  With <>=10^8/s being the average circular frequency of a thermal phonon interacting with phonons in the main resonator mode, with n=kT/<> being the average number of phonons in that typical thermal phonon mode, and with T=300K,  we obtain approximately the quantum 1/f noise coefficient
	' =(N/V)<>/12ng^2mc^2 = 10^22(1/137)(10^-2710^8)^2/12kT10^-27.9^.10^20 ≈1.
For V<, his is in good agreement with the known data for quartz resonators of very high Q as experimental results obtained by Ferre-Pikal et al., T. Parker et al., J.R. Vig et al., as well as other research groups indicate. Here m is the reduced mass of the elementary oscillating dipoles, N their number in the quartz resonator volume, and g a polarization constant of the order of the unity.  The formulas given above are derived from the equation 
	S_(f) = 4)^2/3fe^2c^2.	
of the conventional quantum 1/f effect applied to the energy dissipation  of a quartz crystal.  Here /V is the change in the quartz polarization rate caused by the elementary dissipative process of removing a phonon from the main resonator mode.
Similarly, SAW resonators are used at about ten times higher frequencies in SAW sensors that operate in the V> regime with ’=120.  The discussion is similar, but applied to the surface and to surface waves.