High-speed multiple-mode mass-sensing resolves dynamic nanoscale mass distributions

Simultaneously measuring multiple eigenmode frequencies of nanomechanical resonators can determine the position and mass of surface-adsorbed proteins, and could ultimately reveal the mass tomography of nanoscale analytes. However, existing measurement techniques are slow (<1 Hz bandwidth), limiting throughput and preventing use with resonators generating fast transient signals. Here we develop a general platform for independently and simultaneously oscillating multiple modes of mechanical resonators, enabling frequency measurements that can precisely track fast transient signals within a user-defined bandwidth that exceeds 500 Hz. We use this enhanced bandwidth to resolve signals from multiple nanoparticles flowing simultaneously through a suspended nanochannel resonator and show that four resonant modes are sufficient for determining their individual position and mass with an accuracy near 150 nm and 40 attograms throughout their 150-ms transit. We envision that our method can be readily extended to other systems to increase bandwidth, number of modes, or number of resonators.

Amplitude response of the closed loop system as a function of frequency normalized to desired bandwidth, β when loop parameters given in Supplementary Table 1 are evaluated in (S13). The calculated responses are identical to that of a Butterworth low-pass filter of given order.

Supplementary Note 1 Phase-domain transfer function
As noted in the main text, for deriving the transfer function of the phase of a resonator, we analyze its time-domain response to an abrupt change in the excitation frequency. Prior to the frequency change, we assume that the resonator is oscillating in steady state with amplitude A(ω 0 ) and phase-delay θ(ω 0 ) as given in references [2,3], where F 0 is the amplitude of the driving force, m is the effective mass of the resonator, ω 0 is the resonant frequency and Q is the quality factor of the resonator. When the excitation instantaneously changes its frequency, the resonator response is described by the sum of the zero-input response and the zero-state response as follows: where ω 1 is the new excitation frequency and τ = 2Q/w 0 . If we write (S3) in the form of gives the phase of the resonator as a function of time. Therefore, we will use the following identity We will entirely ignore the amplitude term c, other than noting that it is a time-dependent function.
We'll use the following definitions to coerce (S3) into the form of (S4): Therefore we are left with: Multiplying the arctan argument by e t/τ /e t/τ and dividing by A(ω 0 )/A(ω 0 ), we get: where the approximation is valid for very small frequency deviations.
If the the frequency step ω 0 − ω 1 is very small, then (ω 0 − ω 1 )t is a very slow term compared to e t/τ , and we can approximate the time-varying α as the constantα. Then the equation reduces to: . Noting thatα is typically small justifies dropping all but the linear term, yielding Noting that θ(ω) is also a tan function, we approximate that with its 1 st -order Taylor series around the resonant frequency, θ(ω) = θ(ω 0 ) − τ (ω − ω 0 ): To verify that our approximations were reasonable, we compared our model predictions to time- For calculating the phase domain transfer function of a resonator, we used the approximate expression in (S8), where the response of the resonator phase, φ(t), to a phase step of −τ (ω 1 − ω 0 ) at t = 0 is given as: Note that θ(ω 0 ) is the initial condition of the phase at t = 0. After normalizing the time-varying response in (S9) with the phase-step amplitude, we took its derivative to calculate the impulse response in time domain. Finally, the phase domain transfer function of a resonator to changes in its driving frequency is calculated by taking the Laplace-transform of the impulse response as:

Supplementary Note 2 Closed loop system function
We first model the resonator-PLL systems in phase domain using a commonly practiced 2 nd -order Type-2 PLL [4] and the resonator response in (S10). We implement a model such that we can access the resonant frequency of the resonator as an input as well, by making the following modification: which can be represented as shown in the blue box of Figure 1d. This model can then be further simplified by noting that the resonator has a positive feed-forward path that cancels with the PLL's negative feedback path. Furthermore, the PLL integrator cancels with the resonator differentiator.
This allowed us to derive a simple, computationally tractable model (Supplementary Figure 2a) for the entire system. We derived the Laplace-domain transfer function of this closed loop system as: We first analyzed (S12) computationally by assuming different resonators for a given set of loop coefficients (solid lines in Figure 1e). We found that the resonator itself can substantially affect the system dynamics, which underlines the requirement of setting the loop coefficients for each resonator separately and carefully.

Supplementary Note 3 Tailoring the desired system response
In this work, for oscillating a resonator at multiple of its resonant modes simultaneously, we used a dedicated PLL in closed loop with each mode. In order to track the oscillation frequency of a mode precisely, we need a flat amplitude and linear phase response in the pass-band for minimum distortion of the frequency modulation signal and high enough rejection in the stop-band for minimum cross-talk between different modes and maximum noise rejection. Hence, we shall design the PLL such that we achieve a closed-loop transfer function identical to a Butterworth low-pass filter, since it has maximally flat gain and linear phase response. Since an n th order Butterworth filter includes n poles and no zeros in its transfer function, we start with the generalized system given in Supplementary Figure 2b utilizing an additional loop filter with n − 1 poles in its forward path compared to the simplified model in Supplementary Figure 2a. We derived the closed loop transfer function of the system in Supplementary Figure 2b as: where τ k are the poles introduced by the additional filter section in the forward path. Since there are one zero and n + 1 poles in (S13), we can cancel the zero with one of the poles and position S10 the rest of the poles carefully to achieve an n th order Butterworth-type response (or similarly a Chebyshev-type or Elliptic response). We equate (S13) to a Butterworth filter transfer function with a 3-dB bandwidth, β and solve for the unknown parameters, k p , k i and τ k .
where B n is the normalized Butterworth polynomial of n th order, which is given in [5]: π + 1 if n is odd. (S15) The solution of (S14) up to third order is given in Supplementary Table 1

Supplementary Note 4 Testing the system implementation
For testing the operation of the PLL system, we first developed a z-domain model of the PLL implementation on the FPGA board. The full-scale nonlinear system implementation is given in given in Supplementary Figure 2c. For z-domain analysis we approximated the multiplier in the PLL with a subtraction operation as signals are now in the phase domain, because the multiplier and low-pass filter effectively yield the phase difference between the input and the internal oscillator.
We also omit the automatic gain control, as it operates only on the amplitude of the incoming signal, and does not affect the phase. We derived the z-domain response of the CIC filter [6] as: where R and N are the rate factor and the order of the CIC, respectively. The second term is the transfer function of a zero-order hold [4] for modeling the effect of the rate change in the CIC filter.
Similarly, the loop filter and the NCO are represented by H LF and H N CO , respectively as follows: where T S is the sampling period of the fastest rate in the loop, which is 10 ns in this case. Note that the CIC filter and the loop filter operates at a slower rate than the NCO. Furthermore, we modeled the resonator response (S11) in z-domain as follows: where α is T s /(τ + T s ).
To measure the response of our PLL, we add a 10 degree

Supplementary Note 5 Frequency content of the particle signals
To determine the necessary bandwidth for each resonator mode, we calculate the frequency modulation signal given in [1] for a particle moving at uniform speed towards the tip of the cantilever and then returning back to the base. We then calculate the cumulative energy density for varying frequencies and determine the frequency at which 99.99% of signal energy will be retained (Supplementary Figure 5). For a 100 ms peak, these frequencies are roughly 70, 150, 185 and 210 Hz, respectively. Conservatively, we set the 3 dB bandwidth of our resonator-PLL system to a little over twice these values, yielding bandwidths of 150, 335, 435 and 505 Hz for the transfer functions shown in Figure 2b.

Supplementary Note 6 Precision of the position estimation
We calculated the root-mean-square (RMS) error of the position estimation of a 30 fg (∼ 150 nm gold) particle as a function of the position along the resonator length by using the fitting algorithm S12 described in the Methods section. We added experimentally measured noise waveforms to the simulated signals of identical particles passing through the resonator in 150 ms. Then, we estimated the position of each particle throughout their 150-ms transits. We calculated the RMS error of positions estimated by the fitting algorithm when the signals from first two, three and four modes as a function of particle position (Supplementary Figure 6a). The results show that the number of modes improves the precision of the position estimation.
It is known that the mass sensitivity of a mechanical resonator improves with miniaturization.
In order to determine the effect of miniaturization on position precision for less dense, biologically relevant nanoparticles, we calculated the precision that could be achieved by a hypothetical SNR device utilizing a 20 µm long, 0.5 µm thick and 5 µm wide cantilever with an integrated fluidic channel (250 nm by 1 µm in cross-section). We calculate the fundamental resonant frequency of such a cantilever when it is filled with water as 1.8 MHz. Currently, our oscillator system can operate with 5 ppb frequency stability around 1.8 MHz [7]. We assumed with improved transduction techniques and high-frequency (HF) and very-high-frequency (VHF) control electronics, the same stability level could be achieved for the higher order modes as well. Under these assumptions we calculated the position precision for a human immunodeficiency virus, HIV (80-attogram buoyant mass and ∼100 nm size) using the same algorithm for estimating the particle position that was used for Fig. 5 of the main text. To demonstrate the effect of the increased number of modes, we expoited the first two, four and eight modes of the hypothetical SNR for the calculations. The highest frequency of operation for these three cases are 11, 61, 281 MHz, respectively. The resulting position precision is plotted in Supplementary Figure 6b.