Effect of surface stress induced curvature on the eigenfrequencies of microcantilever plates

Ultrasensitive physical, chemical and biological sensors have emerged in the last decade based on the measurement of the eigenfrequencies of micro- and nanosized cantilever plates. Surface stress is omnipresent in these devices due to a variety of factors such as the fabrication process, temperature variations and analyte adsorption. How surface stress influences on the eigenfrequencies of cantilever plates has remained as an unsolved question in physics that has raised a long debate since first experiments in 1975. Recent theoretical models have shed light on the role of the net surface stress. Still, there exists a discrepancy between theory and some experimental reports, affecting to the capability for quantification of these sensors. In this Letter, we present a theoretical framework that demonstrates that the cantilever bending due to differential surface stress between opposite faces of the cantilever, a neglected effect in classical beam theory, plays a relevant role in the stiffness and eigenfrequencies of cantilevers. We develop a new theoretical framework that provides analytical equations that accurately describe the effect of surface stress on the first three vibration modes of cantilevers. Our findings provide the final piece of the puzzle for solving this long-standing problem in physics.