Measurement of non-monotonic Casimir forces between silicon nanostructures

Journal name:
Nature Photonics
Year published:
Published online

Casimir forces are of fundamental interest because they originate from quantum fluctuations of the electromagnetic field1. Apart from controlling this force via the optical properties of materials2, 3, 4, 5, 6, 7, 8, 9, 10, 11, a number of novel geometries have been proposed to generate repulsive and/or non-monotonic Casimir forces between bodies separated by vacuum gaps12, 13, 14. Experimental realization of these geometries, however, is hindered by the difficulties in alignment when the bodies are brought into close proximity. Here, using an on-chip platform with integrated force sensors and actuators15, we circumvent the alignment problem and measure the Casimir force between two surfaces with nanoscale protrusions. We demonstrate that the force depends non-monotonically on the displacement. At some displacements, the Casimir force leads to an effective stiffening of the nanomechanical spring. Our findings pave the way for exploiting the Casimir force in nanomechanical systems using structures of complex and non-conventional shapes.

At a glance


  1. Geometry of interacting surfaces designed to generate non-monotonic Casimir forces.
    Figure 1: Geometry of interacting surfaces designed to generate non-monotonic Casimir forces.

    a, Top-view scanning electron micrograph of arrays of T-shaped protrusions at zero displacement of the bottom movable electrode. The dashed rectangle encloses one unit cell. b, Different regimes of interaction corresponding to various displacements of the movable electrode along y. c, Calculated Casimir force per unit cell along y as a function of displacement (see Methods). For perfectly symmetric structures, the force has no component in the x direction. The red/blue lines represent the Casimir force calculated assuming PFA for regions outlined in red/blue in the right/left inset. The black line represents SCUFF calculations (see Methods). Only numerical calculations are shown (measurements are shown in Fig. 4). Insets: digitization of a typical unit cell. d, Gradient of the force. Scale bars, 1 μm (a) and 2 μm (b).

  2. Detection and actuation scheme.
    Figure 2: Detection and actuation scheme.

    a,b, Scanning electron micrographs of the entire device (a) and part of the comb actuator (b). Scale bars, 100 μm (a) and 10 μm (b). c, Schematic of the detection and actuation scheme. Grey areas are fixed to the substrate via an underlying silicon oxide layer. Blue areas are part of the movable comb. The top part is a colourized scanning electron micrograph of the beam (red) and the movable electrode (blue). There are 31 and 32 units of protrusions on the beam and the movable electrode, respectively. The beam has a width of 1.5 μm (at regions away from the protrusions) and length of 100 μm. It vibrates with a small amplitude and serves as the force gradient detector. The white arrow represents an a.c. current through the beam. Magnetic field B is perpendicular to the substrate.

  3. Calibration by electrostatic force.
    Figure 3: Calibration by electrostatic force.

    a, Mechanical resonance of the fundamental mode of the beam, with ωR/2π = 1,212,849.5 Hz and quality factor 58,600, measured at 4 K and <1 × 10−5 torr (see Methods). b, Measured ΔωR versus applied voltage Ve for different Vcomb. Solid lines are parabolic fits. c, Measured ΔωR versus displacement and Ve. d, Measured electrostatic force gradient at Ve = V0 + 100 mV. Error bars for the force gradient are calculated from the electrical noise in recording the X quadrature of beam vibrations. The error bars in displacement originate from the uncertainty in locating the displacement at which the electrostatic force gradient attains minimum. The red line is a least-squares fit using calculations from finite-element analysis. The pink band shows the range of electrostatic force obtained when the digitized geometry is expanded or shrunk by one pixel of the top-view scanning electron micrographs.

  4. Measured force gradient per unit cell after compensating for residual voltage.
    Figure 4: Measured force gradient per unit cell after compensating for residual voltage.

    Measured data are plotted in black. The red line represents the Casimir force calculated with the boundary elements method. Non-uniformities in the shapes of different T protrusions are taken into account (see Supplementary Section ‘Non-uniformity in the T-protrusions’). The finite conductivity of the silicon is also included (see Methods). The pink band around the red line represents the change in the calculated Casimir force when the digitized geometry is expanded or shrunk by one pixel of the top-view scanning electron micrographs. Error bars for the force gradient are calculated from the electrical noise in recording the X quadrature of vibrations of the beam. Error bars in displacement originate from uncertainty in determining the displacement at which the electrostatic force gradient in Fig. 3d attains its minimum. The ‘Casimir spring’ effect attains its maximum at position II (following the notation in Fig. 1d).


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Author information


  1. Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

    • L. Tang,
    • M. Wang,
    • C. Y. Ng,
    • C. T. Chan &
    • H. B. Chan
  2. Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA

    • M. Nikolic &
    • A. W. Rodriguez


L.T. and M.W. fabricated the devices and conducted the measurements. C.Y.N., M.N., A.W.R. and C.T.C. performed the theoretical calculations. H.B.C. conceived and supervised the experiment. All authors discussed the results and contributed to the writing.

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