Device and measurement scheme

Figure 1a shows a simplified schematic of the structure and a scanning electron micrograph of sample A, respectively. The element for sensing the force is a doubly clamped silicon beam that is 100 μm long and~1.42 μm wide, depicted in the top parts of Fig. 1a. The silicon is p-doped with a high carrier concentration of 7.0 × 1018 cm−3. Figure 1c zooms in on the micro-beam. A silicon electrode of width 2.80 μm is positioned close to the beam, as shown in the lower part of Fig. 1c. The beam and the electrode have the same thickness (2.65 μm) and distance to the substrate (2 μm) (see Methods, Supplementary Fig. S1 and Supplementary Methods for fabrication details). Electrostatic and/or Casimir forces are exerted by the electrode on the beam depending on the voltage V e between them. The gap between the beam and the electrode is created by deep reactive ion etching (DRIE) while the beam and the electrode are protected with an etch mask. With the etch mask defined by electron-beam lithography, a high degree of parallelism is ensured between the beam and the electrode, without any need for manual alignment before force measurement. By viewing the beam and the electrode from the top with a scanning electron microscope, their separation is found to remain constant to within ~15 nm along the entire length of the beam, yielding an upper bound of 150 μrad for the angle between the lithographic patterns of the beam and the electrode.

Figure 1: The setup of the experiment and device. (a) A simplified schematic (not to scale) of the beam (red), movable electrode and comb actuator supported by four springs (blue), with electrical connections. The current amplifier provides a virtual ground to the right end of the beam. Suspended and anchored parts of the comb actuator are shown in blue and dark grey, respectively. The separation d between the beam and the movable electrode was controllably reduced so that the Casimir force can be detected. (b–e) Scanning electron micrographs of the entire micromechanical structure (b) and close-ups of the doubly clamped beam (c), the comb actuator (d) and the serpentine spring (e). The close-ups in c–e zoom into the top, middle and bottom white-dashed boxes in b, respectively. Scale bars, 50 μm (b) and 10 μm (c–e). Full size image

The electrode is attached to a comb actuator so that it can be controllably moved along the y direction, reducing the separation d between the electrode and the beam from an initial distance of d 0 =1.92 μm (measured with a scanning electron microscope) down to ~260 nm while maintaining parallelism (see Methods, Supplementary Figs S2, S3 and S4, and Supplementary Methods). Figure 1d shows a close-up of part of the comb actuator. The comb actuator consists of a set of movable comb fingers supported by four serpentine springs (Fig. 1e), one at each corner of the structure (Fig. 1a). A second set of comb fingers (the solid structures in Fig. 1d with no etch holes) is fixed to the substrate on one end. When a voltage V comb is applied to the fixed comb relative to the movable comb (in the experiment, a negative V comb is used), an electrostatic force parallel to the substrate is generated. The movable combs are displaced towards the beam until the restoring force from the four springs balances the electrostatic force. As a result, the separation d is reduced as |V comb | increases (see Supplementary Methods and Supplementary Movies 1 and 2). It should be noted that the electrostatic force between the fixed and movable combs merely serves to set d. As explained below, the potential difference between the beam and the movable electrode V e can be controlled separately, independent of V comb .

The suspended beam acts as a resonant force sensor. As shown in Fig. 1a, a small alternating voltage (V ac =5.7 μV) is applied to one end of the beam, producing an alternating current. In the presence of a 5 T magnetic field perpendicular to the substrate, the beam is subjected to a periodic Lorentz force. The frequency of the alternating voltage ω D is chosen such that the beam vibrates along the y direction in its fundamental mode. Vibrations of the beam in the magnetic field generate an induced electromotive force that is detected with a current amplifier. In Fig. 2a, the resonant frequency of the beam ω R is measured to be 7.26185 × 106 rad s−1. All measurements were performed at 4 K and <10−5 Torr. When the Casimir force and/or electrostatic forces are exerted on the beam, the resonant frequency ω R decreases due to the spring softening effect, by an amount Δω R that is proportional to the force gradient F′(d):

Figure 2: Calibration of the device using electrostatic force gradient. (a) Oscillation amplitude of the beam and its X quadrature that is in phase with the periodic driving force. Fitting to the driven underdamped oscillator model (lines) gives a damping coefficient of 30 rad s−1. (b) Measured frequency shift Δω R as a function of electrode voltage V e , at d=1.403 μm, 1.065 μm, 865 nm, 643 nm, 450 nm and 349 nm, from top to bottom. (c) Measured dependence of the residual voltage as a function of d. (d) Measured electrostatic force gradient on the beam (circles) at V e =V 0 +100 mV. The line represents a fit to the values calculated using finite element analysis. Inset: cross-sectional schematic of the beam, electrode and substrate with direct current (dc) electrical connections. Error bars represent ±1 s.e. Full size image

where K is a positive proportionality constant and F′(d)<0.

Calibration

Similar to conventional experiments on Casimir forces, we also need to perform a calibration procedure by applying a voltage V e between the beam and the movable electrode to generate an electrostatic force F e between them. F e is proportional to (V e −V 0 )2, where V 0 is the residual voltage. Figure 2b shows parabolic fits to Δω R versus V e . Each parabola corresponds to Δω R recorded at a fixed d that is set by V comb . There are two contributions to Δω R : the electrostatic part that depends quadratically on V e −V 0 and a vertical offset that is independent of V e −V 0 . The latter becomes more negative as d decreases. As described later, we will compare this vertical offset to the Casimir force gradient and remnant force gradients due to patch potentials. The electrostatic part will be used for force calibration.

The residual voltage V 0 is measured by identifying V e at which the maximum of the parabolic dependence of Δω R occurs. Figure 2c shows that V 0 is measured to be about −25 mV at small d. Over the full range of distances, V 0 changes by about 15 mV, comparable to previous experiments in the lens-plate23 and sphere-plate24 geometries. Even though both the beam and the electrode are made of single-crystal silicon on the same wafer, the residual voltage V 0 is non-zero and shows distance dependence. Based on current experimental data, we cannot convincingly identify the origin of these effects. We suspect that the non-zero residual voltage is possibly due to solder contacts in the electrical leads at different temperatures. The distance dependence likely originates from adsorbed impurities and/or the etching profile (see Supplementary Methods) exposing patches of different crystal orientations at the sidewalls with non-uniform potentials.

In conventional Casimir force experiments, the extension of the piezoelectric element is either pre-calibrated or directly measured. At the same time, the initial distance between the two interacting surfaces is an unknown that needs to be determined by the application of electrostatic forces. In our experiment, distance calibration is performed using a slightly different procedure. Here d is given by:

where d 0 is the initial separation, V comb is the voltage of the fixed combs relative to the movable combs and is a proportionality constant to be determined by fitting. k // is the spring constant of the serpentine springs along , the spatial derivative of the capacitance between the fixed and the movable combs, remains almost constant as the movable combs are displaced. One main difference from previous experiments is that d 0 is not a fitting parameter. Instead, d 0 is accurately measured to be 1.92±0.015 μm using a scanning electron microscope. The dependence of F e on the distance d is calculated using finite element analysis with our device geometry. By fitting to the calculated electrostatic force gradient F′ e (d) given by equations (1) and (2) for six sets of data with V e ranging from V 0 +100 to V 0 +150 mV, we obtain α=10.55±0.04 nm V−2 and K=5.38±0.10 × 104 rad m (s N)−1. Figure 2d plots the electrostatic force gradient on the beam as a function of d at V e =V 0 +100 mV, where d is controlled by increasing |V comb | from 0 to 11.375 V according to equation (2). When V e is applied, the effective distance for the electrostatic force changes because of carrier depletion. However, because of the high carrier concentration (7 × 1018 cm−3), this change is small (<1 nm) and negligible compared with the uncertainty in d.

Casimir force measurement and calculations

Next, we set V e =V 0 (d) for each distance d and measure the force gradient F′ c between the beam and the electrode as a function of d (Fig. 3a). The red line in Fig. 3a represents the theoretical values of the Casimir force calculated for silicon structures of such geometry, with no fitting parameters. The theoretical calculation involves a boundary-element method (BEM) discretization of the beam and substrate surfaces, combined with a recent fluctuating-surface-current formulation of the Casimir force between dielectric bodies that writes the full Casimir-energy path integral as a simple expression in the classical BEM interaction matrix14,25. It includes the contributions of the finite conductivity of silicon and the imperfect etching profiles on the sidewalls of the beam and the electrode (~88° from the substrate surface, see Supplementary Methods). Despite the imperfect agreement between measurement and theory, it is clear that the Casimir force becomes the dominant interaction between the beam and the movable electrode at small d. Unlike the sphere-plate configuration, the roughness on the sidewalls cannot be directly measured. From the top-view micrograph of the beam and electrode, we determine the root-mean-square (rms) roughness of the edges to be 12 nm, mainly due to non-uniformity of the electron-beam lithography. The roughness correction is estimated to be about 3% of the Casimir force at the closest distance26. We further note that the calculation assumes the lithographic patterns of the beam and the electrode to be parallel. Using an upper bound of 150 μrad for the angle between them, the calculated Casimir force increases by up to 1.1% at the closest distance26.

Figure 3: Measured force gradient F′ c between the beam and the movable electrode as a function of separation d after compensating for the residual voltage. (a) The red line represents the calculated Casimir force gradients between an electrode and a beam made of silicon. The purple line includes possible contributions from patch potentials. Inset: the ratios of the calculated Casimir force between the beam and the electrode to the forces given by the PFA are plotted as the red (with substrate) and blue lines (without substrate). (b) Deviations of the measured force gradient from the purple line in a. Error bars represent ±1 s.e. Full size image