Measurement of the Casimir Force between a Gold Sphere and a Silicon Surface with Nanoscale Trench Arrays

We report measurements of the Casimir force between a gold sphere and a silicon surface with an array of nanoscale, rectangular corrugations using a micromechanical torsional oscillator. At distances between 150 and 500 nm, the measured force shows significant deviations from the pairwise additive formulism, demonstrating the strong dependence of the Casimir force on the shape of the interacting bodies. The observed deviation, however, is smaller than the calculated values for perfectly conducting surfaces, possibly due to the interplay between finite conductivity and geometry effects.

Figure 1

(a) Cross section of rectangular trenches in silicon, with periodicity of 400 nm and depth of $0.98\mu \mathrm{m}$ (sample $B$). (b) Top view of the structure. (c) Schematic of the experimental setup (not to scale) including the micromechanical torsional oscillator, gold spheres, and silicon trench array. (d) Measurement scheme with electrical connections. Excitation voltages $V_{ac1}$ and $V_{ac2}$ are applied to the bottom electrodes.

Figure 2

Gradient of the electrostatic force $V=V_0+300\mathrm{mV}$) on the flat silicon surface (solid circles) and corrugated silicon structure (hollow squares). The solid line is a fit using Eq. (2) for the flat surface. The dashed line is a fit using the force gradient from finite element analysis for the corrugated structure. Inset: The space between the corrugated structure and a flat surface is divided into triangular mesh to solve the Poisson equation in 2D ($z=200\mathrm{nm}$).

Figure 3

Measured Casimir force gradient between the same gold sphere and (a) a flat silicon surface, $F_{c,\mathrm{flat}}^{\prime }$, (b) sample $A$, $F_{c,A}^{\prime }$ ($\lambda =1\mu \mathrm{m}$), and (c) sample $B$, $F_{c,B}^{\prime }$ ($\lambda =400\mathrm{nm}$). In (a), the line represents the theoretical Casimir force gradient including finite conductivity and surface roughness corrections. In (b) and (c), the lines represent the force gradients expected from PAA ($p{F}_{c,\mathrm{flat}}^{\prime }$). (d) Ratio $\rho$ of the measured Casimir force gradient to the force gradient expected from PAA, for samples $A$ ($\lambda /a=1.87$, hollow circles) and $B$ $\lambda /a=0.82$, solid squares), respectively. Theoretical values [22] for perfectly conducting surfaces are plotted as the solid ($\lambda /a=2$) and dashed lines ($\lambda /a=1$).