Fabrication and geometry of honeycomb plate metamaterials

Figure 1a shows a cantilevered (single-clamped) rectangular plate based on a hexagonal honeycomb metamaterial design. The plates were fabricated out of ALD alumina films with uniform thickness, which varied between 25 and 100 nm in different fabrication runs, using optical lithography and a combination of wet and dry etching techniques (see the Supplementary Note 1, Supplementary Figs 1 and 2, and (ref. 31) for detail of the fabrication process). Alumina material was chosen as the cell wall material for its high stiffness, chemical and high-temperature resistance, as well as the highly conformal and pinhole-free nature of its ALD films.

Figure 1: Geometry of honeycomb plate metamaterials. (a) An scanning electron microscopic (SEM) image of a 1-mm-long cantilevered plate made using a hexagonal honeycomb metamaterial geometry. The 35-nm-thick ALD alumina film is coloured for clarity. Scale bar, 100 μm. (b) An SEM image of the detail of the cell structure. Scale bar, 10 μm. (c) A schematic representation showing an individual cell with its characteristic dimensions. (d) An SEM image of a 50-nm-thick cantilever without the honeycomb pattern, illustrating the warping typical of suspended flat ultrathin films. Scale bar, 200 μm. (e) An SEM image of a 1-mm-long 25-nm-thick plate bent by more than 90° using a micromanipulator without any fracture. After the tip of the micromanipulator was removed, the plate recovered its original shape (see Supplementary Movie 1, Supplementary Note 2 and Supplementary Figs 3 and 4). Scale bar, 200 μm. (f) Schematic illustrating the requirement that any plane perpendicular to the plate will intersect vertical walls. The insets show zoomed-in detail of how the plane intersects individual cells. Full size image

As shown in Fig. 1b,c, the plate is able to maintain its shape because of its three-dimensional honeycomb microstructure, which increases the flexural stiffness of the plate by one to three orders of magnitude depending on the used cell parameters (see Supplementary Note 3 for detail). When bending stresses are applied, rigidity is provided by the vertical walls and the horizontal segments between the ‘hexagonal cups’ and the ‘cup’ bottoms similar to the way that C-beams or corrugated sheets resist deformation. This keeps the sheet flat even when the film has significant internal stresses due to fabrication-induced stress gradients that generally cause unpatterned films to curl up with a small radius of curvature (Fig. 1d).

Robustness of plate metamaterials

In contrast to macroscale honeycomb sandwich structures, our plates can sustain large bending deformations because these top and bottom surfaces are not continuous, allowing the structures to fold with a very small radius of curvature without fracture, as shown in Fig. 1e. Despite the brittleness of the cell wall ceramic material (aluminium oxide), these ultrathin structures are robust and typically recover their original shapes even after undergoing extreme bending deformations multiple times (Supplementary Movie 1). As with previously reported bulk metamaterials5, structures made out of thinner films were typically more robust, that is, able to withstand sharper bends. This is simply because of the fact that the maximum strain along the fold lines formed in these plates at large bending deformations is proportional to the thickness of the solid cell wall film. In this aspect, plate metamaterials are very different from plates with effectively uniform cross-sections, such as corrugated sheets or sandwich plates, for which the classical plate theory holds and the maximum strains are therefore proportional to the curvature multiplied by the height of the plate rather than the thickness of the face sheets. If a sandwich plate undergoes a sharp bend (that is, with radius of curvature similar to the plate height), the induced strains are of the order of one, which would lead to failure of most structural materials. This remains true for sandwich plates made of films with macro-, micro- or nanoscale thickness. In contrast, the maximum strains in our plates are determined by the ratio of film thickness to plate height, which is in the order of 10−3 for the thinnest ALD plates we fabricated, explaining their observed robustness.

The design of the particular features and dimensions of the plates was guided by the results of COMSOL finite element simulations (see Supplementary Note 3). Based on the results of the simulations, we have focused our experiments on plates with hexagonal honeycomb geometry because of their approximately isotropic bending stiffness for small deformations6,32. Other periodic geometries are possible, but to achieve high flatness and bending stiffness, it is crucial that any plane perpendicular to the plate must intersect vertical walls (Fig. 1f). Otherwise, the plane will bend very easily along a line that contains only horizontal elements. Among regular polygon tiling patterns, only hexagonal (honeycomb) patterns can satisfy this requirement. Other, lower-symmetry tiling patterns, such as basket-weave or rhombille, also satisfy this requirement, but generally have a lower bending stiffness (see Supplementary Note 3, Supplementary Fig. 7, and Supplementary Table 1).

Enhanced bending stiffness of plate metamaterials

Traditional prismatic corrugated sheets with a periodic cross-section (Fig. 2a) offer a greatly enhanced bending stiffness in one direction but almost no change in the bending stiffness in the perpendicular direction22,23. The enhancement factor, EF, which can be used to describe the increase in bending stiffness relative to a completely planar sheet, is determined by the moment of inertia of its constant cross section. Typically, it scales quadratically with the ratio of height of the corrugation, h, to the thickness of the sheet, t, and depends only weakly on the other geometric parameters (for example, period) of the corrugation pattern: .

Figure 2: Simulated bending properties of traditional corrugated sheets and plate metamaterials. (a) Illustration of the anisotropic bending stiffness of periodic corrugated sheets with a constant cross-section. (b) Isotropic enhancement factor of honeycomb corrugated sheets versus the ratio of cell diameter to rib width. (c) COMSOL finite element simulation illustrating the concentration of elastic energy in the areas where the ribs intersect. The colour represents the elastic energy density, with red areas corresponding to the highest elastic energy density and blue to the lowest. Full size image

In contrast, the hexagonal corrugation used in this work offers an enhanced bending stiffness in all directions, and its isotropic enhancement factor depends on all parameters of the unit cell. We have studied this dependence through extensive finite element simulations of small-deformation response in COMSOL (see Supplementary Note 3 and Supplementary Figs 5 and 6). Briefly, the enhancement factor of ultrathin plates initially increases quickly with height, just like in simple corrugated sheets, but then saturates for heights much greater than the film thickness. This maximum saturated value of the enhancement factor depends on the ratio of two in-plane parameters of the cell geometry: , where D is the inner diameter of the hexagons and w is the width of the ribs (Fig. 2b).

An elastic energy analysis can be used to explain these results for the maximum enhancement factor, as well as the overall dependence of the bending stiffness on the height of our honeycomb plates (see Supplementary Note 3 for detail). Briefly, for sufficiently large heights, the hexagons with their attached vertical ribs become very rigid, and practically do not bend. As a result, when one of our honeycomb plates is bent, the majority of the elastic energy is concentrated in the areas near the hexagon vertices, where the strengthening ribs intersect. Figure 2c shows one representative case, where ∼95% of the total elastic energy is concentrated in these triangular regions according to COMSOL simulations. Thus, the total area undergoing bending in a honeycomb plate is only a small fraction of the total area, and the bending stiffness is enhanced proportionally to the ratio of these areas.

More specifically, consider a unit cell of area A of a honeycomb patterned plate. Let the area occupied by the small triangular regions (near each vertex) per unit cell be A 2 and therefore A 2 =A−A 1 is the area of the remaining part of the unit cell. The corresponding mean curvature bending moduli of these regions are K 1 and K 2 , respectively. If a uniform isotropic moment per unit length m is acting on the unit cell, then the elastic energy stored in the patterned plate per unit cell can be written as . Equating , where K eff is the effective (average) bending modulus for the plate metamaterial, we obtain . Now, and . In the limit of large plate heights, K 1 →∞, and the effective bending modulus is given by , which gives for . The latter agrees with the enhancement factor obtained by numerical simulations. More detailed analysis can be found in Supplementary Note 4 and Supplementary Fig. 8.

Measurements of bending stiffness

To validate the finite element model used for metamaterial designs and determine the Young’s modulus of the cell wall ALD alumina material, we have measured the small-deformation bending stiffness of honeycomb plates using an atomic force microscope (AFM). In particular, the spring constants of the cantilevered plates were inferred from the force-displacement measurements obtained using an AFM.

Somewhat unexpectedly, the measured stiffness of the honeycomb cantilevers depends very strongly on whether continuous vertical walls are present on the sides of the cantilevers (compare Fig. 3a,b). Such walls can be formed at cantilever edges either intentionally by designing them into the lithography mask or unintentionally due to fabrication imperfections. When sidewalls are absent, the cantilever’s spring constant is directly proportional to the bending stiffness of the metamaterial, K eff , introduced in the previous section, and the cantilever width W cant . As a result, the spring constant scales cubically with the film thickness, and is almost independent of the height of the structure (Fig. 3c), as discussed in the previous section. In contrast, when sidewalls are present, the force-displacement response of the cantilevers is dominated by the stiffness of the two C-beams formed by these walls at each side of the cantilever. As a result, the corresponding spring constants scales linearly with thickness and quadratically with plate height, , as expected from the stiffness scaling of conventional C-beams and corrugated sheets (Fig. 3d). Figure 3e compares these experimental results to the spring constants from finite element simulations for plates with a cell height of 10 μm, a cell diameter of 50 μm, and ALD alumina thickness between ∼25 and ∼100 nm. The finite element simulations agree with experimental results best if we assume a Young’s modulus of 130 GPa, which is within the range of elastic moduli reported for ALD alumina films in the literature33.

Figure 3: Measured bending response of honeycomb cantilevers with and without sidewalls. Scanning electron micrographs of a 1-mm-long cantilevers (a) without and (b) with vertical sidewalls, which dominate the spring constant of cantilever when present. Scale bar, 50 μm in both micrographs. The insets show the vertical sidewall detail. (c) Force-displacement curves measured for 1-mm-long ALD honeycomb cantilevered plates (without vertical sidewalls) for different thicknesses and heights. Note that cantilevers with a thickness of ∼100 nm and heights of 7 and 10 μm have essentially the same spring constant. (d) Same for cantilevers with vertical sidewalls, which increase the overall stiffness of the cantilever by up to two orders of magnitude. Note that the cantilevers with a thickness of ∼50 nm and heights of 10 and 14 μm have dramatically different spring constants, in accordance with the scaling .(e) A comparison between experimental results and simulations for honeycomb structures with 10 μm height and different ALD film thicknesses (with and without sidewalls). The curves show the actual predictions of the finite-element simulations (solid lines) along with ±20% confidence interval (dashed lines). The symbols show experimental data. The experimental errors bars denote s.e.m. in c,d and s.d. in e. For the top curve (with sidewalls) in e, the vertical error bars are smaller than the symbols. Full size image

To illustrate the repeatability and stability of the elastic behaviour of the ALD honeycomb plates, we also performed cyclic loading tests: a plate with a honeycomb depth of 10 μm and an ALD layer thickness of 72 nm was bent and unloaded 400 times consecutively without exhibiting any significant changes in its spring constant (see Supplementary Note 5 and Supplementary Fig. 9 for detail).

Properties of centimetre-scale plates

The robustness and flatness of the honeycomb periodic metamaterial allowed us to fabricate honeycomb plates on the macroscopic scale with lateral dimensions on the centimetre scale (Fig. 4). While the ALD layer thickness is only ∼50 nm, the devices do not show any visible warping or deformation. The plates fully recover their shape after bending (Fig. 4a,b). They are also extremely lightweight (Fig. 4c) and float in air, descending very slowly (see Supplementary Movie 2).

Figure 4: Photographs of large-area honeycomb plates. Before (a) and after (b) pictures of a bent 0.5 × 1.0 cm honeycomb plate. (c) Same plate placed on flower petals does not cause any bending of the petals. Scale bar, 3 mm. The inset zooms in on a section of the plate showing the honeycomb microstructure. Full size image

For the largest honeycomb plates we made (Fig. 4), the cell parameters shown in Fig. 1c must be simultaneously optimized for high bending stiffness, robustness to defects, and ease of fabrication. In particular, we fixed the minimum feature size at 10 μm to maximize the yield of optical lithography and etching fabrication steps, and that determined the rib width w=10 μm in all fabricated samples. While the bending stiffness increases with increasing cell diameter (see Supplementary Note 3), we limited cell diameters to D=50 μm in fabricated samples to maximize the robustness of the films. Plates using cells with larger diameters are more prone to damage when probed with a micromanipulator tip or when individual fabrication defects occur because damaging a single cell affects a significant portion of the total structure. In contrast, in our fabricated plates, the fabrication defects or external damage due to, for example, a puncture by an AFM tip, was localized to within a few cells because the cracks typically terminated at the vertical walls (see Supplementary Note 6 and Supplementary Figs 10–12 for detail).

The cell height, which can be viewed as the effective thickness of the plate metamaterial, was varied between 1 and 14 μm, corresponding to a range of etching depths easily accessible with standard semiconductor reactive ion etching (RIE) tools. The primary effect of increasing the cell height is an increase in the bending stiffness and flatness (see Supplementary Note 3). The plates with the largest heights of 10−14 μm were remarkably flat, with the typical vertical displacement of only ±4 μm over the 0.8 × 1.2 mm field of view of the profilometer (Fig. 5a). The measured radii of curvature were in the order of 10 cm (Fig. 5b,c).