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Simultaneous determination of the residual stress, elastic modulus, density and thickness of ultrathin film utilizing vibrating doubly clamped micro-/nanobeams
9.C. M. Stafford, C. Harrison, K. L. Beers, A. Karim, E. J. Amis, M. R. VanLandingham, H.-C. Kim, W. Volksen, R. D. Miller, and E. E. Simonyi, Nature Mat. 3, 545 (2004).
12.P. Blake, E. W. Hill, A. H. C. Neto, K. S. Novoselov, D. Jiang, R. Yang, T. J. Booth, and A. K. Geim, Appl. Phys. Lett. 91, 063124 (2007).
36. Micro-/nanosized beams with dominated flexural rigidity fulfill the following inequality IcEcr(υ, η) > > (FT + Fres) L2 (see Refs. 33–35).
37. If |b| > 1.5, then the higher order polynomial dependency of dimensionless frequency on b is needed, i.e , where N is an order of the polynomial,33 and, consequently, to determine Fres requires the numerical computations.38
39.S. Neumark, Solution of Cubic and Quartic Equations (Pergamon Press, Oxford, UK, 1965).
43. In general, F1 and F2 for beam with and without deposited film can be different from each other.
44.P. Blake, E. W. Hill, A. H. C. Neto, K. S. Novoselov, D. Jiang, R. Yang, T. J. Booth, and A. K. Geim, Appl. Phys. Lett. 91, 063124 (2007).
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Measurement of ultrathin film thickness and its basic properties can be highly challenging and time consuming due to necessity of using several very sophisticated devices. Here, we report an easy accessible resonant based method capable to simultaneously determinate the residual stress, elastic modulus, density and thickness of ultrathin film coated on doubly clamped micro-/nanobeam. We show that a general dependency of the resonant frequencies on the axial load is also valid for in-plane vibrations, and the one depends only on the considered vibrational mode. As a result, we found that the filmelastic modulus, density and thickness can be evaluated from two measured in-plane and out-plane fundamental resonant frequencies of micro-/nanobeam with and without film under different prestress forces. Whereas, the residual stress can be determined from two out-plane (in-plane) measured consecutive resonant frequencies of beam with film under different prestress forces without necessity of knowing film and substrate properties and dimensions. Moreover, we also reveal that the common uncertainties in force (and thickness) determination have a negligible (and minor) impact on the determined filmproperties. The application potential of the present method is illustrated on the beam made of silicon and SiO2 with deposited 20 nm thick AlN and 40 nm thick Au thin films, respectively.
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