2 The resonators are fabricated from 100 nm thick low-pressure chemical vapor deposited silicon nitride using electron beam lithography and reactive ion etch- ing, and suspended using a dry isotropic release process. The inset in Fig. 1(c) shows two resonators before met- allization. The dimensions are L × w = 200 × 15 µ m $^{2}$. A 5 nm thick chromium adhesion layer is deposited on top, followed by 30 nm of gold. The gold layer is not passivated and is in contact with the water during the experiments. In air, the fundamental resonance mode, measured by the magnetomotive and optical techniques is shown in Fig.1(c). The signal to noise ratio for the optical measurement is 50 times larger, which is due to the approximately 200 times amplification of the resonator displacement by the optical lever. The resonator re- sistance, R ≈ 7 . 5 Ω, is large compared to its motional impedance, Z 0 ≈ 0 . 8 Ω at resonance, and this results in a phase response for the magnetomotive measurement different from the driven harmonic oscillator response obtained by the optical detector. By fitting damped driven harmonic oscillator functions represented by the solid black lines, the natural (undamped) resonance frequency for the fundamental mode is f 1$_{,}$air = 213 kHz, and the quality factor Q 1$_{,}$air = 50. The magnetomo- tive technique can be applied to higher odd modes, though the sensitivity reduces, as will be discussed later. The third resonance mode is also measured with f 3$_{,}$air = 725 kHz. For the n-th mode of vibration, the frequency ratio f n /f 1 = n for a string in tension, and roughly ( 2$_{n}$+1 3 ) 2 for a doubly clamped flexural beam [6]. The measured ratio f 3$_{,}$air /f 3$_{,}$air = 3 . 4, indicates string-like behavior dominates. This occurs when the restoring force from flexure is small compared to that from residual tension. We note that the ratio is 13% larger than for a string in tension, and this is explained by the flexural rigidity which contributes to some increase in effective spring stiffness, whereas the compliance from the large undercuts decreases it slightly. Prior to experiments in water, the flow cell is flushed with ethanol to remove air bubbles. De-ionized water is injected and the optical and magnetomotive measure- ments are repeated on an immersed resonator. Figure 2(a) shows the measured amplitude response using the magnetomotive scheme for the first resonance mode. A harmonic oscillator function is fit through the data with f 1$_{,}$water = 74 kHz, which is identified as a mechanical res- onance by the magnitude and phase response from the optical measurement, shown in the inset. A different Q- factor is found for the magnetomotive and optical de- tector. The magnetomotive quality factor, Q 1$_{,}$water = 1 . 4 is lower than for the optical measurement Q 1$_{,}$water = 2 . 8. This difference is attributed to the complex dielectric constant of water, which presents a frequency-dependent load to the resonator and lowers the Q-factor for the mag- netomotive measurement [7]. The value of the load de- pends on the design of sample and flow cell and on the To compare the shift in resonance frequency and Q- factor upon immersion with theory, we take the viscous and inertial forces into account through a hydrodynamic function, Γ(Re , κ ), which relates the cross section shape of the resonator to a force per unit length acting on the resonator, where its real and imaginary components cor- respond to the inertial and dissipative components of the force [8]. Γ(Re , κ ) is a function of the normalized mode- dependent Reynolds number, Re n = 2 πf n$_{,}$vac ρ/η , and the normalized mode number, κ n = π n w L for our string res- onators. Here ρ is the density and η the dynamic viscosity of water. The hydrodynamic function for a rectangular cross section is given in Ref. [9], and we can calculate the The third flexural mode of the same resonator is plotted in Fig. 2(b). The magnetomotive detected reso- nance frequency equals f 3$_{,}$water = 306 kHz and Q 3$_{,}$water = 2 . 7. The optical measurement reveals a slightly lower resonance frequency, f 3$_{,}$water = 275 kHz. This differ- ence was found independent on sweep time and driving strength, and cannot be explained by electrical loading of the resonator. The optically measured Q-factor equals Q 3$_{,}$water = 6 . 4, again higher than the magnetomotive one. In water the ratio between the resonance frequencies is similar to the ratio in air: f 3$_{,}$water /f 1$_{,}$water = 3 . 8. frequency-dependent dielectric properties of water, and is not further discussed here. FIG. 2: (a) Magnetomotive measurement of the fundamental resonance mode of the string resonator in water. The inset shows the optical measurement. (b) Magnetomotive and op- tical measurement of the third flexural mode.