Nanomechanical mass measurement using nonlinear response of a graphene membrane

We propose a scheme to measure the mass of a single particle using the nonlinear response of a 2D nanoresonator with degenerate eigenmodes. Using numerical and analytical calculations, we show that by driving a square graphene nanoresonator into the nonlinear regime, simultaneous determination of the mass and position of an added particle is possible. Moreover, this scheme only requires measurements in a narrow frequency band near the fundamental resonance. Copyright c © EPLA, 2010 Introduction. – Nanoelectromechanical (NEM) resonators hold promise as ultrasensitive mass detectors [1,2]. NEM mass sensors (NEM-MS) rely on a resonant frequency shift ∆ω due to an added mass ∆M . However, as opposed to detecting a single adsorbed particle, to actually measure its mass ∆M from ∆ω, the position of the particle must be known. Proposed position determination schemes [3–6] rely on detectors to measure the frequency shifts of several vibration modes. While this poses no problems in principle, it causes practical difficulties for NEM-MS operating in the GHz regime. We propose a detection scheme that only requires measurements in a single narrow band centered at the fundamental mode resonance frequency of a square 2D resonator. Our method uses the nonlinear response of the resonator by exploiting the interaction between vibration modes to make information about higher modes available at the fundamental frequency. We illustrate by showing, analytically and numerically, how the nonlinear response of micrometer-size graphene resonators [7,8] can be used for single-particle mass measurements with zeptogram precision at room temperature. Several other technology tracks are being considered for NEM-MS devices. One is downscaling of Si-MEMS [9–13] where the present state-of-the-art give a minimum detectable mass of ∼ 10 zg [9]. Another track relies on carbon nanotubes (CNTs) [14] and has already reached sub-zg levels [15–18]. However, after the discovery of graphene [19], novel 2D NEMS devices have been (a)E-mail: [email protected] explored [20–23], including mass detectors with zg sensitivity [7]. Apart from increasing the adsorbtion crosssection, 2D NEMS can also have degenerate flexural modes. As we show, this degeneracy makes possible to distinguish single-particle from multi-particle adsorption. Graphene also represents the ultimate material for 2D NEMS through its combination of large strength and low mass. System. – We consider a square graphene sheet with mass M and side length L0 suspended in the XY -plane above an actuation gate (see fig. 1). The sheet is simply clamped at all edges. The gate geometry, which has a symmetry line parallel to the Y -axis, is chosen such that the fundamental and higher-order modes can be excited. The transverse deflection w(X, t) of the membrane is given by [8]