Inertial imaging with nanomechanical systems

Journal name:
Nature Nanotechnology
Year published:
Published online


Mass sensing with nanoelectromechanical systems has advanced significantly during the last decade. With nanoelectromechanical systems sensors it is now possible to carry out ultrasensitive detection of gaseous analytes, to achieve atomic-scale mass resolution and to perform mass spectrometry on single proteins. Here, we demonstrate that the spatial distribution of mass within an individual analyte can be imaged—in real time and at the molecular scale—when it adsorbs onto a nanomechanical resonator. Each single-molecule adsorption event induces discrete, time-correlated perturbations to all modal frequencies of the device. We show that by continuously monitoring a multiplicity of vibrational modes, the spatial moments of mass distribution can be deduced for individual analytes, one-by-one, as they adsorb. We validate this method for inertial imaging, using both experimental measurements of multimode frequency shifts and numerical simulations, to analyse the inertial mass, position of adsorption and the size and shape of individual analytes. Unlike conventional imaging, the minimum analyte size detectable through nanomechanical inertial imaging is not limited by wavelength-dependent diffraction phenomena. Instead, frequency fluctuation processes determine the ultimate attainable resolution. Advanced nanoelectromechanical devices appear capable of resolving molecular-scale analytes.

At a glance


  1. Superposition of resonator mode shapes.
    Figure 1: Superposition of resonator mode shapes.

    a, Mode shapes of a doubly-clamped beam (from bottom to top) for the first, second, fourth and tenth out-of-plane flexural modes. b, A linear combination of mode shapes intended to yield a superposition g(0)(x) = 1 over an interval Ωl spanning the entire beam [0, 1], using the first mode (black), first two modes (blue), first four modes (red) and first ten modes (green). Significant over- and undershoot is evident. c, Superpositions with a slightly foreshortened interval provide greatly improved convergence. Here, Ωl spans [δ,1 – δ], where δ  =  N/(1  +  N2). For an expansion involving the first ten modes (green), the interval [0.099, 0.901] covers about 80% of the beam. L, cantilever length. N, number of modes used.

  2. Adaptive fitting for enhanced resolution and accuracy.
    Figure 2: Adaptive fitting for enhanced resolution and accuracy.

    Superposition fit to g(0)(x)  =  1 (to calculate mass) using the first four modes of a doubly-clamped beam, showing the effect of measurement zone reduction. a, Initial superposition (black curve) before the initial values for the location and size of the particle are determined. After determination of the location and size of the particle, the measurement zone can be reduced. In this case, the position is determined to be near x/L = 0.35 (dotted vertical line) and a measurement zone smaller than the original is chosen (pink shaded area). This measurement zone (pink shaded area) is centred on the particle position and commensurate with particle size. With this reduction in measurement zone, a new superposition can be calculated (blue curve: solid is within zone, dashed is outside). b, Zoomed-in view of new superposition. Further reduction of the measurement zone by a factor of 10 leads to another superposition (green curve: solid is within zone, dashed is outside). This is again superior to the previous superposition (blue curve). c, Zoomed-in view of final superposition (dotted vertical line is the position of the particle). Error is reduced by six orders of magnitude from a.

  3. Mass and position analysis using published experimental data.
    Figure 3: Mass and position analysis using published experimental data.

    a,b, Mass (a) and position (b) calculations for the experimental data from ref. 7 using two modes of a doubly-clamped beam. The values for mass and position are compared with the previous values from ref. 7 using multimode theory. Position is scaled to the device length. Error bars in inertial imaging theory reflect the total error due to both the fitting residual and frequency fluctuations (2σ, 95% confidence level). c, Analysis of particle mass for different positions using the four-mode measurement of the same particle along a cantilever device16. The particle expected mass is estimated from the scanning electron microscopy (SEM) measurements of ref. 16 (solid red line) with 2% assumed uncertainties in that measurement (dotted red lines). d, Position calculation using the same data, compared with the optically measured position. Red lines in a,b,d represent curves of exact agreement between the two methods. Insets in a and c: Electron micrographs of representative devices used in the two respective studies. Scale bars, 2 μm (a); white, 5 μm and black, 500 nm (c). In all figures, error bars represent the 2σ, 95% confidence level.

  4. Size and shape analysis via frequency-shift measurements of droplet arrays.
    Figure 4: Size and shape analysis via frequency-shift measurements of droplet arrays.

    a, Optical images of liquid droplet arrays deposited on a silicon microcantilever (397 µm long, 29 µm wide, 2 µm thick) using AFM dip-pen lithography. Inverted black–white images serve to highlight the droplets. Inertial imaging using the lowest four vibrational modes resolves these addenda as constituting a composite, thin, spatially distributed analyte that is strongly adherent to the cantilever surface. Measurement details are provided in the Methods and Supplementary Section 11. The numbers of two-droplet rows in the symmetric distributions are specified, and the asymmetric distribution used to assess skewness is indicated. b, Comparison of mean position of the droplet array distribution, measured using both inertial imaging and optical microscopy. c, Comparison of the variance in the droplet array distribution measured using both inertial imaging and optical microscopy. Position and standard deviation are scaled to cantilever length. The single row (denoted ‘1’ in a, does not satisfy the requirement of being a thin and compliant adsorbate and hence is not included in the comparison of variance. Dashed lines are linear regressions of data centred at the origin. Slopes of lines and R2 values are indicated. Vertical and horizontal error bars in b,c represent the 2σ uncertainty levels due to frequency noise and droplet mass variability, respectively.


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Author information

  1. These authors contributed equally to this work

    • M. Selim Hanay,
    • Scott I. Kelber &
    • Cathal D. O'Connell


  1. Kavli Nanoscience Institute and Departments of Physics & Applied Physics and Biological Engineering, California Institute of Technology, Pasadena, California 91125, USA

    • M. Selim Hanay,
    • Scott I. Kelber,
    • John E. Sader &
    • Michael L. Roukes
  2. Department of Mechanical Engineering and National Nanotechnology Research Center (UNAM), Bilkent University, Ankara 06800, Turkey

    • M. Selim Hanay
  3. Bio21 Institute & School of Chemistry, The University of Melbourne, Victoria 3010, Australia

    • Cathal D. O'Connell &
    • Paul Mulvaney
  4. School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia

    • John E. Sader


M.L.R. and J.E.S. supervised the project. J.E.S. provided the principal mathematical idea for mass measurement using mode superposition that was extended to imaging by M.L.R. The resulting theory was further developed by M.S.H., S.I.K., J.E.S., and M.L.R. Droplet measurements were conceived by J.E.S., performed by C.D.O., and supervised by P.M. and J.E.S. The paper was written by M.S.H., S.I.K., C.D.O., J.E.S., and M.L.R. The FE simulations were executed by M.S.H. and S.I.K. All authors analysed the data and contributed to the writing of the paper.

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