Casimir Force Control Enabled by 3D Nanostructures

Abstract

The Casimir force dominates interactions between solid objects at sub-micrometer distances and typically limits the smallest distance between micromechanical devices before failure. Here, we experimentally circumvent this limitation by controlling the Casimir force with engineered 3D nanostructures. Using our recently developed method to align and measure the force between two microscale objects on the nanoscale, we characterized the force gradient between spheres and circular pillars, hollow cylinders, and periodic pillar arrays. We demonstrate that the force behavior can be dramatically modified in these geometries, resulting in a suppression of the Casimir force by 10× for a single pillar. We found agreement between theory and experiment, even when the size of the objects was comparable to the surface-to-surface separation (i.e., within a factor of ∼3). We anticipate that our results will impact the design of future micro- and nanoscale actuators, optomechanical devices with increased sensitivities and reduced stiction, and advanced bio-inspired adhesives.

The Casimir force results from the alteration of the zero-point energy of electromagnetic fields in the presence of boundaries. While this force is usually negligible on the macroscale, it becomes increasingly important on the nanoscale and is expected to limit the functionality of next-generation microelectromechanical systems (MEMS). , This force exists independent of any electric charge on the objects and is strongly dependent on the geometry of the surfaces involved. However, today, the most common configuration for measuring the Casimir force is that of a spherical object above a plate, which eliminates the difficulty associated with maintaining parallelism between two flat surfaces at small separations − but limits the ability to study complex geometries. Recent breakthroughs using liquids have enabled more significant modifications to the Casimir force including the characterization of repulsive interactions, , torque between two optically anisotropic materials, and tunable nanolevitation. ,,

To modify the Casimir force without resorting to liquid environments, which is important for a vast range of nanotechnologies, only a few options exist. Experiments have been performed with metallic surfaces composed of different materials, − corrugated surfaces, − and on-chip silicon structures using a MEMS actuator. , Despite these advances and the potential importance of the Casimir force in complex nano-systems, an approach to controlling the force-distance relationship using an arbitrary 3D nanostructure has not been possible due to measurement difficulties associated with maintaining alignment between objects while performing sensitive force experiments. However, complex geometries may provide the opportunity to dramatically change the Casimir force, including the prospect of repulsion without the need for an intervening liquid. ,

Here, we present experiments showing that 3D nanostructures can significantly alter the Casimir force, both increasing and suppressing the interaction. Our measurements are performed between a gold-coated hollow glass sphere and a variety of structures, including circular pillars, hollow cylinders, and periodic arrays at ambient temperature and pressure. Through these different geometries, we demonstrate the ability to engineer the magnitude and force-distance relationship of the Casimir force between identical materials based on their geometry alone. Further, these structures are predominantly isolated, a situation which differs from all previously studied systems, and can thus provide physical insight into the effects of nanostructures separate from the structural periodicity that is typically present. , For example, the proximity force approximation (PFA) has previously been used to extend Casimir–Lifshitz theory for parallel plates to gently curved geometries where the object dimensions are much larger than the separation. , Despite the nanostructures studied here having sharp edges and dimensions comparable to the separations at which measurements are collected, we show a striking agreement between measurement and theory using the PFA. It thus seems reasonable that the PFA can still be used in many situations to guide the design of future nanodevices, which often have edges and small dimensions. Figure shows three distinct nanostructured geometries used to modify the Casimir force (see Supporting Information for fabrication details).

1.

Experimental configuration and geometry of three different nanostructures designed to engineer the Casimir force. Schematic of the experimental configuration used to measure the Casimir force between one gold-coated sphere and (a) a vertical circular pillar, (b) a hollow cylinder, and (c) a periodic pillar array aligned directly below the sphere (out of scale for clarity). d is the separation between the sphere and nanostructure, r _s is the radius of the sphere, r _p is the radius of the pillar, r _h is the radius of the hole located at the center of the cylinder, h _p is the height of the pillar, and h _h is the depth of the hole. SEM images (left) and AFM scans (right) of (d) vertical circular pillars, (e) hollow cylinders, and (f) periodic pillar arrays. Note: the AFM scans were collected using a sharp AFM probe and are presented with a z-range smaller than that for x and y.

As depicted in Figure , the Casimir force is measured between a sphere and each nanostructure by oscillating the plate supporting the nanostructure along the z-axis, where the z = 0 position lies along the tops of each nanostructure. The radius of the sphere is 33.1 ± 0.2 μm (determined from a scanning electron micrograph), and the nominal spring constant k of the cantilever is 0.3 N/m. The height of the nanostructures (h _p) is 120 nm to minimize the interaction between the sphere and supporting plate (see S3 in Supporting Information for further details of the sample geometries). The geometry depicted in Figure a consists of a sphere and an upright pillar. Four different pillars are fabricated on the same plate with radii of r _p = 300, 600, 1200, and 3000 nm, separated by several mm so that they can be studied independently. The geometry shown in Figure b consists of a sphere and a hollow pillar, which has a circular hole in the center. The radius of the hole (r _h) is varied from 600 to 1500 nm with a constant pillar radius of 3000 nm. The depth of the hole varies slightly from 55 to 65 nm due to charging effects during e-beam lithography. Lastly, we prepare a periodic array of pillars with a constant radius of 300 nm, a height of 100 nm, and a gap (g) between adjacent pillars of 600, 900, and 1200 nm. It should be noted that the schematics shown in Figure a,b,c, which represent the object geometries, and similar ones shown in later figures are not to scale.

To determine the Casimir force between the sphere and the various nanostructures, we measure the spatial derivative of the Casimir force in an ambient environment utilizing a force modulation measurement technique. ,, This process allows us to determine the surface separation and spring constant, while also eliminating hydrodynamic and electrostatic forces from the Casimir force data. The separation (d) between the two surfaces is accurately controlled by a piezoelectric transducer in discrete steps. For each sphere–nanostructure configuration, we collect data at ∼400 individual separations (3 μm to 30 nm) for each approach and retraction (15 total cycles). We preserve horizontal alignment between the sphere and nanostructure by performing topographical scans every five measurements (see S4 in Supporting Information) and minimize the electrostatic contribution to the total force signal at each separation by a technique similar to amplitude-modulated Kelvin probe force microscopy (AM-KPFM). In this step, an AC bias (V _ac) is applied between the sphere and nanostructure at a frequency of $\frac{{\omega }_{\mathrm{A}}}{2\pi }=77\mathrm{Hz}$ to electrostatically drive the cantilever at an angular frequency of ω_A. A feedback loop then applies a slowly varying DC bias (V _dc) in order to minimize the oscillation at ω_A, thereby minimizing the electrostatic force. The value of V _dc at which the electrostatic force is minimized (V _dc = −V ₀) is maintained during the Casimir force measurement step. After determining V ₀, V _dc is slowly varied about V ₀ while the plate supporting the nanostructure oscillates along the z-axis at an angular frequency ω_pz, resulting in an oscillation of the AFM probe. The signal corresponding to probe oscillation amplitude is proportional to the second order separation-derivative of the capacitance, and is used to determine the absolute separation and AFM probe sensitivity by fitting the measured data to the expected value obtained between a sphere and the nanostructure using the PFA (see Section 2 of the Supporting Information). Modification of the capacitance due to an expected water layer of 1.5 nm on each surface due to ambient humidity is also included. ,

Figure a,b shows how the Casimir force can be engineered between a sphere and a pillar (or pillar with a hole) by changing the radius of the pillar (or hole). As expected, when the interacting areas of the involved bodies are reduced (e.g., by decreasing the pillar radius or increasing the hole radius), the Casimir force is decreased; however, the separation dependence of the force is notably different in these two cases. To further understand these results, we compare the experimental data to numerical calculations using the PFA. Within the PFA, the Casimir force is calculated by integrating the Casimir pressure P _{Plate–Plate} of two parallel plates over local distances described by a local-distance function H(x) across the interacting surface area S:

$$F^{\mathrm{PFA}}={\int }_S{\mathrm{d}}^{2}x{P}_{\mathrm{Plate}-\mathrm{Plate}}(H(x))$$ 1

The Casimir force gradient is then obtained by taking the derivative of the force with respect to the separation, i.e. δF ≡ ∂F/∂z. We calculate the Casimir pressure P _{Plate–Plate} using the Lifshitz formalism with optical data from ref . extrapolated to zero-frequency using the Drude model with plasma frequency ω_p = 8.84 eV and damping frequency ω_τ = 42 meV. Because the Casimir pressure for real materials at finite temperature does not obey a simple power law, the integral over the surface in eq needs to be calculated numerically. The local-distance function between the surface of the sphere and the single vertical pillar is

$$H(x)=d+r_{\mathrm{s}}-\sqrt{{r_{\mathrm{s}}}^2-{|x|}^2}+h_{\mathrm{p}}\mathrm{\Theta }(|x|-r_{\mathrm{p}})$$

and the local-distance function for the pillar with hole geometry is similar but contains an additional term h _hΘ­(r _h – |x|) due to the hole, where Θ­(x) is the Heaviside step function. Utilizing polar coordinates for those two cylindrically symmetric geometries greatly increases the numerical performance. We note that surface roughness leads to an increase in the Casimir interaction, which we take into account by multiplying the plate–plate pressure in eq by a factor of 1 + 10­(δ₁ ² + δ₂ ²)/d ², where δ₁ = δ₂ = 3 nm are the measured rms roughness amplitudes. The calculated Casimir force for each geometry (dashed lines in Figure a,b) agrees well with experimental results for all data above the noise level, which are generally present at separations d < 200 nm (see S6 in Supporting Information for Casimir gradient with error bars and ref for more information about noise, errors, and sensitivities).

2.

Casimir force gradient between a sphere and two pillar-like geometries. (a, b) Spatial derivative of the force measured between a sphere and a pillar and between a sphere and a hollow cylinder, respectively. All the individual measurements (light dots) are shown. The force gradients and separations of individual measurements are binned and averaged (solid lines). The bin sizes are 2 nm below 80 nm separation and 4 nm above. The calculated force gradients using Lifshitz’s theory with the PFA (dashed lines) are also shown. The roughness of each sample is incorporated using a perturbative roughness correction. (c, d) Ratio of force gradient between the sphere–plate configuration and the sphere–pillar and sphere–hole configurations, respectively. The solid lines correspond to the ratio of the binned measurement data. The maximum and minimum of the standard deviation of the ratio are depicted by the shaded areas. The calculated force ratio is determined using Lifshitz’s theory with the PFA (dashed lines). Pressure gradient map for three (e) pillar and (f) hole radii at several separations showing which surfaces dominate the interaction within the parameter space. The percentage contribution of the plate surface to the total force gradient is noted in each map.

The effect of the nanoscale geometry on the distance dependence of the force gradient can be seen in Figure c,d (the reference Casimir gradient between the sphere and the plate is provided in Figure S7). For large-diameter pillars, the Casimir force in the sphere–pillar configuration is similar to that of the sphere–plate system, as expected. However, as the radius of the pillar is decreased from 3 μm, we find that the force gradient decreases more rapidly with increasing separation than the sphere–plate interaction. For a fixed pillar radius, we also find that at larger separations, we recover the sphere–plate results due to the increased interaction with the substrate. For the pillar with hole configuration, we find that the force decreases less rapidly than for the sphere–plate case, which is opposite of the close-range behavior found from the sphere–pillar configuration. This behavior is because of the multiple interacting surfaces (bottom of hole, top of pillar, and substrate), which give important contributions to the total force at a variety of surface separations. By considering the different potential regimes of operation for these geometries, one can control the force gradient to either increase or reduce the effect of the Casimir interaction.

The force gradient is dominated by the pillar structure or plate surface depending on the size of the structure and separation, as shown in the spatial distribution of the force gradient for the pillar (hole) structure depicted in Figure e (Figure f) for a few pillar (hole) radii. Said data were calculated by numerically evaluating Casimir–Lifshitz theory approximated for each structure using the PFA in the form of finite-element simulations (see Section 2 of the Supporting Information for the description of a similar calculation of the electrostatic force). Generally, the pillar dominates the interaction at short separations and for large pillar radii. As the separation increases from 10 to 300 nm between the sphere and a pillar with r _p = 300 nm, the plate’s contribution to the force gradient increases from 0.5% to 94%; however, the contribution from the plate surface surrounding a pillar with r _p = 3 μm only increases by ∼11% over the same separation range. For a pillar with hole where r _p = 3 μm, a small hole (r _h = 300 nm) results in a pressure gradient distribution similar to that of the pillar with no hole; however, as r _h increases, so does the contribution from both the plate and hole. Similarly, the contribution from the plate increases with separation, where the scaling between plate contribution and separation becomes more dramatic for larger hole radii. We also note that despite the fabrication leading to a ∼10 nm variation in the hole depth, there remains good agreement between measurement and theory in Figure b, which assumes no variation in hole depth (h _h = 60 nm). This apparent insensitivity of the total force gradient to variations in the hole depth is a result of the hole making a rather small contribution to the total force gradient at short separations and for the fabricated hole sizes, as seen in Figure f.

Next, we measure the Casimir force gradient between the sphere and several pillar arrays, as depicted in Figure a. The periodic pillar array results in strong modification of the Casimir force as a function of the separation. At the smallest separations, the interaction is still dominated by the single pillar and the sphere. However, as the separation is increased, adjacent pillars also contribute, leading to a force gradient that is larger than expected for an isolated pillar (Figure , inset). This trend continues until separations of ∼80–100 nm, where the enhancement of the force gradient compared to a single pillar decreases. At the largest separations, the pillars appear as a slight perturbation on the substrate, and the force gradient converges to a sphere and a rough plate configuration.

3.

Casimir force gradient between a sphere and various periodic pillar arrays as a function of their separation. (a) Spatial derivative of the force measured between the sphere and the periodic pillar array. The force gradients and separations of individual measurements (light dots) are binned and averaged (solid lines). The bin sizes are 2 nm below 80 nm separation and 4 nm above. The force gradients are calculated using Lifshitz’s theory (dashed lines). The inset shows the ratio of the force gradient between the sphere–pillar array and the sphere–single-pillar pair. (b) Pressure gradient map for three pillar array gap sizes at several separations showing which surfaces dominate the interaction within the parameter space. The percentage contribution of the plate surface to the total force gradient is noted in each map.

As the pillar array gets denser, the force gradient increases as a result of the interactions of adjacent pillars with the sphere. For a pillar radius to gap ratio of 0.5 (i.e., r _p/g = 0.5), we find that the force gradient is nearly a factor of 3 greater than that of a single, isolated pillar for a surface separation of ∼85 nm. The force gradient between the sphere and the pillar array, presented in Figure b, is calculated by finite-element simulations, identical to the calculations in Figure e,f and is in general agreement with the experimental data (additional details and error bars are provided in the Supporting Information). At the shortest separations, surface roughness causes a slight increase in the force, but the overall agreement indicates that the PFA accurately describes variations in the Casimir force between these more complex structures on planar surfaces, as long as the separation is sufficiently small compared to the radius of the sphere and any of the lateral dimensions of the structure. The spatial distributions of the force gradient shown in Figure b are consistent with the observations of the experimental data. As expected, we note that (i) at short separations, a single pillar dominates the interaction between array and sphere, (ii) as the separation increases, nearby pillars cause an increase in force gradient relative to that of a single pillar (leading to a decrease in percentage contribution from the plate), and (iii) the force gradient (and percentage contribution from the pillars) scales with the density of the pillars.

While the experimental results appear to be well described by the PFA theory outlined above, alternative versions of the PFA exist but deviate from the experimental measurements. Reference outlines two alternatives to eq . The first is a variant of the PFA where we keep the interaction from the periodic surface exact and only average the Casimir force over local distances across the sphere surface. Denoting this variant as Derjaguin approximation (DA), the resulting Casimir force gradient can then be expressed as

$$\delta F^{\mathrm{DA}}(d)=2\pi r_{\mathrm{s}}{P}_{\mathrm{Plate}-\mathrm{Array}}(d)$$ 2

with the Casimir pressure between the surface of a plate and a pillar array P _{Plate–Array}(d). We numerically calculate P _{Plate–Array}(d) using the Fourier modal method. , The other alternative approach is based on eq and approximates the Plate–Array pressure further by using the PFA. We denote this variant as piece-wise Derjaguin approximation (PWDA), and it results in

$$\delta F^{\mathrm{PWDA}}(d)=2\pi r_{\mathrm{s}}{P}_{\mathrm{Plate}-\mathrm{Array}}^{\mathrm{PFA}}(d)=2\pi r_{\mathrm{s}}[f{P}_{\mathrm{Plate}-\mathrm{Plate}}(d)+(1-f)\times P_{\mathrm{Plate}-\mathrm{Plate}}(d+h_{\mathrm{p}})]$$ 3

with the filling factor f = πr _s ²/D ², where D = 2r _p + g is the period length of the pillar array.

Figure shows a comparison between the experiment and the three approaches to the PFA. Interestingly, PFA and PWDA agree well with the experimental data, while DA deviates significantly. For comparison, the theoretical result in the sphere–plate geometry is shown as the gray line. Notice that only PFA is sensitive to the lateral in-plane positioning of the sphere above the array pillars. We thus show two different curves for the PFA. The upper dotted curve in each panel corresponds to the sphere centered above a pillar as it is the case in the experiment. As expected, it shows better agreement with the experimental data. For the lower curves, the sphere is centered in the middle between two neighboring pillars. The difference between the two PFA curves becomes more noticeable for shorter separations and for increasing gap size between the pillars, which can be explained by the fact that the effective interacting area, which scales as ∼dr _s, becomes comparable to the area of a unit cell of the pillar array (g + 2r _p)². The two areas are equal for a separation of 44, 68, and 98 nm for the gap sizes between the pillars in Figure a,b,c, respectively, below which the splitting of the two PFA curves becomes more visible for the geometries with the larger two gap sizes.

4.

Comparison of various theoretical approaches with the experimental results from Figure for the gap sizes between adjacent pillars of (a) 600 nm, (b) 900 nm, and (c) 1200 nm. The dots represent the experimental data, and the solid line their mean value for bin sizes of 2 nm below 80 nm separation and bin sizes of 4 nm above. The dotted, dashed, and dash-dotted lines correspond to the approximations of the Casimir force gradient based on eqs , , and , respectively. The gray line indicates the corresponding theoretical result in the plane–sphere geometry. The lower-left inset of each plot shows an AFM scan of the structure using a sharp AFM probe. The upper-right inset in each plot shows the relative error made by the approximations compared with numerically exact results at larger separations. Please note that the dashed and dashed-dotted lines visibly converge as separation increases for all structures.

The insets in Figure show the relative deviations of the approximations PFA, DA, and PWDA with the numerically exact results based on the scattering formalism. The gray solid line shows the corresponding relative deviation for a sphere and a plate with the exact results. As discussed in detail in the Supporting Information, the exact calculations for the sphere and pillar array become so numerically demanding that we are only able to find converged results for separations larger than ∼3 μm, clearly above the experimental regime.

Surprisingly, there is not a single PFA theory that works best for all separations. The one that best matches the experimental results (or the exact calculations), depends upon the range of surface separations under consideration. For separations of about 3–4 μm, DA matches the exact calculations best, while it fails to match experimental results at shorter separations (<300 nm) and exact calculations at large separations (4–100 μm). The reason for this behavior at short separations (<300 nm) is that DA keeps the interaction from the periodic surface exact and yields a more precise result when the separation between the periodic array and the sphere is large enough to be agnostic to lateral displacements. At shorter separations, the force depends upon the alignment between the pillars and the closest point on the sphere, which is better described by PFA. PWDA offers the lowest error at separations 4–10 μm, which is largely a consequence of a difference in limiting behaviors. PWDA underestimates the experimental results in the small separation limit (∼40 nm), whereas it overestimates the exact calculations in the large separation limit (∼100 μm). The force gradient curve approximated by PWDA thus intersects the exactly calculated force gradient curve at a separation of ∼4 μm, resulting in a visible dip in the error curve for PWDA. At the largest separations (10–100 μm), PFA again produces the lowest error when compared to the exact calculation, as it considers the interacting area to be bounded by the sphere’s cross section, R ². Contrarily, DA and PWDA assume that the effective interaction area is proportional to dR, which exceeds the sphere cross section at larger separations, thereby increasingly overestimating the Casimir interaction.

In conclusion, we have experimentally demonstrated an approach to engineer the Casimir interaction through nanostructured metal surfaces. By considering various shapes including cylindrical pillars, holes, and periodic pillar arrays, we have experimentally demonstrated the ability to tailor the strength and power law of the Casimir effect through geometry. Interestingly, comparison of measurements on isolated structures (pillar and pillar with hole) with simple PFA calculations shows good agreement. This agreement provides evidence supporting the notion that it is the confinement of modes between nanostructures, not the nanostructure itself, that causes strong deviations between experiment and PFA, consistent with previous observations in systems involving high aspect ratio gratings. , While a simple PFA calculation is well-matched to the experimental data, validating its use with these geometries, we find that alternative versions of the PFA more closely match exact calculations at larger separations. Surprisingly, there is not a single version of the PFA calculation that works best for all separations. Controlling the strength of the Casimir force through geometry in experimentally feasible configurations opens the door to future nanoscale technologies that incorporate quantum electromagnetic fluctuation forces in customizable actuators and optomechanical systems.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.nanolett.5c01101.

• Nanostructured sample preparation process; determination of electrostatic force using the proximity force approximation; numerically exact Casimir force calculations between sphere and array of pillars; design of circular pillar and hollow cylinder systems; horizontal alignment using the atomic force microscope; Casimir gradient with error bars; Casimir gradient between the sphere and the plate (PDF)

The authors declare no competing financial interest.