1/f noise considerations for the design and process optimization of piezoresistive cantilevers

Piezoresistive cantilevers are limited by two major noise sources: Johnson noise, which is independent of frequency, and conductance fluctuation noise, which has a 1/f spectrum. The 1/f fluctuations of piezoresistive cantilevers are shown to vary inversely with the total number of carriers in the piezoresistor, as formulated by Hooge in 1969. Therefore, while 1/f noise is reduced for large heavily doped cantilevers, sensitivity considerations favor thin lightly doped cantilevers. Balancing these conflicting constraints produces optima for many design and processing parameters. For a cantilever with specified spring constant and bandwidth requirements, optima are identified for the beam thickness and length, and it is shown that the legs should be between 1/3 and 2/3 of the total length with a doping depth that is 1/3 of the beam thickness. Additionally, an optimal doping concentration is identified as a function of the cantilever volume and the measurement bandwidth. Annealing reduces 1/f noise, but causes a loss in sensitivity due to dopant diffusion, and an optimal anneal is computed with a typical diffusion length 10/sup -8/ cm. The analysis, methods, and some of the conclusions of this paper are also applicable to other types of piezoresistive sensors.