Casimir Force Control Enabled by 3D Nanostructures

Casimir Force Control Enabled by 3D Nanostructures

Calum Shelden, Benjamin Spreng, Joseph L. Garrett, Tahmid S. Rahman, Jongbum Kim and Jeremy N. Munday

Nano Letters 2025. Published open access by the American Chemical Society.

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 times 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 about 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.

Introduction

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 1 shows three distinct nanostructured geometries used to modify the Casimir force.

Figure 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-sub-s is the radius of the sphere, r-sub-p is the radius of the pillar, r-sub-h is the radius of the hole located at the center of the cylinder, h-sub-p is the height of the pillar, and h-sub-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.

Geometries and measurement method

As depicted in Figure 1, 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 equals zero position lies along the tops of each nanostructure. The radius of the sphere is 33.1 plus or minus 0.2 micrometres, determined from a scanning electron micrograph, and the nominal spring constant of the cantilever is 0.3 newtons per metre. The height of the nanostructures is 120 nm to minimize the interaction between the sphere and supporting plate. The geometry depicted in Figure 1a consists of a sphere and an upright pillar. Four different pillars are fabricated on the same plate with radii of 300, 600, 1200, and 3000 nm, separated by several millimetres so that they can be studied independently. The geometry shown in Figure 1b consists of a sphere and a hollow pillar, which has a circular hole in the center. The radius of the hole 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 between adjacent pillars of 600, 900, and 1200 nm. It should be noted that the schematics shown in Figure 1a, 1b and 1c, 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 between the two surfaces is accurately controlled by a piezoelectric transducer in discrete steps. For each sphere-and-nanostructure configuration, we collect data at about 400 individual separations, from 3 micrometres down to 30 nm, for each approach and retraction, fifteen cycles in total. We preserve horizontal alignment between the sphere and nanostructure by performing topographical scans every five measurements and minimize the electrostatic contribution to the total force signal at each separation by a technique similar to amplitude-modulated Kelvin probe force microscopy. In this step, an AC bias is applied between the sphere and nanostructure at a frequency chosen to electrostatically drive the cantilever at its resonance angular frequency. A feedback loop then applies a slowly varying DC bias in order to minimize the oscillation at that frequency, thereby minimizing the electrostatic force. The value of the DC bias at which the electrostatic force is minimized is maintained during the Casimir force measurement step. After determining it, the DC bias is slowly varied about that value while the plate supporting the nanostructure oscillates along the z-axis at a drive angular frequency, 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. Modification of the capacitance due to an expected water layer of 1.5 nm on each surface due to ambient humidity is also included.

Pillars and hollow cylinders

Figure 2a and 2b show how the Casimir force can be engineered between a sphere and a pillar, or a pillar with a hole, by changing the radius of the pillar or of the hole. As expected, when the interacting areas of the involved bodies are reduced — for example 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 of two parallel plates over local distances described by a local-distance function across the interacting surface area — the first named result of the paper. The Casimir force gradient is then obtained by taking the derivative of the force with respect to the separation. We calculate the plate-to-plate Casimir pressure using the Lifshitz formalism with published optical data extrapolated to zero frequency using the Drude model with a plasma frequency of 8.84 eV and a damping frequency of 42 meV. Because the Casimir pressure for real materials at finite temperature does not obey a simple power law, the integral over the surface needs to be calculated numerically. The local-distance function between the surface of the sphere and the single vertical pillar is given by the second named result; the local-distance function for the pillar-with-hole geometry is similar but contains an additional term for the hole depth multiplied by a Heaviside step function of the hole radius minus the radial coordinate. 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-to-plate pressure by a factor of one plus ten times the sum of the squared roughness amplitudes divided by the squared separation, where both measured root-mean-square roughness amplitudes are 3 nm. The calculated Casimir force for each geometry, shown as dashed lines in Figure 2a and 2b, agrees well with experimental results for all data above the noise level, which are generally present at separations below 200 nm.

Figure 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 are shown as light dots. The force gradients and separations of individual measurements are binned and averaged as solid lines. The bin sizes are 2 nm below 80 nm separation and 4 nm above. The calculated force gradients using Lifshitz theory with the PFA are shown as dashed lines. The roughness of each sample is incorporated using a perturbative roughness correction. (c, d) Ratio of force gradient between the sphere-and-plate configuration and the sphere-and-pillar and sphere-and-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 theory with the PFA. 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 2c and 2d. For large-diameter pillars, the Casimir force in the sphere-and-pillar configuration is similar to that of the sphere-and-plate system, as expected. However, as the radius of the pillar is decreased from 3 micrometres, we find that the force gradient decreases more rapidly with increasing separation than the sphere-and-plate interaction. For a fixed pillar radius, we also find that at larger separations, we recover the sphere-and-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-and-plate case, which is opposite of the close-range behavior found from the sphere-and-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 structure in Figure 2e and for the hole structure in Figure 2f, for a few 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. 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 of radius 300 nm, the plate’s contribution to the force gradient increases from 0.5 percent to 94 percent; however, the contribution from the plate surface surrounding a pillar of radius 3 micrometres only increases by about 11 percent over the same separation range. For a pillar with hole of radius 3 micrometres, a small hole of radius 300 nm results in a pressure gradient distribution similar to that of the pillar with no hole; however, as the hole radius 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 roughly 10 nm variation in the hole depth, there remains good agreement between measurement and theory in Figure 2b, which assumes no variation in hole depth. 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.

Periodic pillar arrays

Next, we measure the Casimir force gradient between the sphere and several pillar arrays, as depicted in Figure 3a. 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, as shown in the inset. This trend continues until separations of about 80 to 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.

Figure 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 are binned and averaged. The bin sizes are 2 nm below 80 nm separation and 4 nm above. The force gradients are calculated using Lifshitz theory. The inset shows the ratio of the force gradient between the sphere-and-pillar-array and the sphere-and-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 one half, 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 about 85 nm. The force gradient between the sphere and the pillar array, presented in Figure 3b, is calculated by finite-element simulations identical to the calculations in Figure 2e and 2f, and is in general agreement with the experimental data. 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 3b are consistent with the observations of the experimental data. As expected, we note that at short separations a single pillar dominates the interaction between array and sphere; 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 the force gradient, and the percentage contribution from the pillars, scales with the density of the pillars.

Comparing the approximations

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. Two alternatives have been outlined in the literature. 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 the Derjaguin approximation (DA), the resulting Casimir force gradient is expressed as the third named result, using the Casimir pressure between the surface of a plate and a pillar array, which we calculate numerically using the Fourier modal method. The other alternative approach approximates the plate-to-array pressure further by using the PFA. We denote this variant as the piece-wise Derjaguin approximation (PWDA), giving the fourth named result, with a filling factor equal to pi times the squared sphere radius divided by the squared period length, where the period length is twice the pillar radius plus the gap.

Figure 4 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-and-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 the separation times the sphere radius, becomes comparable to the area of a unit cell of the pillar array. The two areas are equal for separations of 44, 68, and 98 nm for the gap sizes between the pillars in Figure 4a, 4b and 4c respectively, below which the splitting of the two PFA curves becomes more visible for the geometries with the larger two gap sizes.

Figure 4. Comparison of various theoretical approaches with the experimental results 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 the three approaches, respectively. The gray line indicates the corresponding theoretical result in the plane-and-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 dash-dotted lines visibly converge as separation increases for all structures.

The insets in Figure 4 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. 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 about 3 micrometres, 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 to 4 micrometres, DA matches the exact calculations best, while it fails to match experimental results at separations shorter than 300 nm and exact calculations at large separations of 4 to 100 micrometres. The reason for this behavior at short separations 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 of 4 to 10 micrometres, which is largely a consequence of a difference in limiting behaviors. PWDA underestimates the experimental results in the small separation limit near 40 nm, whereas it overestimates the exact calculations in the large separation limit near 100 micrometres. The force gradient curve approximated by PWDA thus intersects the exactly calculated force gradient curve at a separation of about 4 micrometres, resulting in a visible dip in the error curve for PWDA. At the largest separations of 10 to 100 micrometres, 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. Contrarily, DA and PWDA assume that the effective interaction area is proportional to the separation times the sphere radius, which exceeds the sphere cross section at larger separations, thereby increasingly overestimating the Casimir interaction.

Conclusion

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, figures and reference list are at the source.)