Cantilever-free probes as a distributed optical lever

As a foundation for cantilever-free AFM, we postulated that vertically moving a rigid probe on a compliant backing layer will result in a visible deflection of the backing layer (Fig. 1b). Combining contact mechanics and an analytical estimate for specular reflection off a tilted surface, we developed a model of this effect, which we term a distributed optical lever (Supplementary Fig. 1 and Supplementary Information). Two key relationships emerge from this analysis. First, assuming that the force acting on the probe leads primarily to a deflection of the backing layer, contact mechanics indicates that the cantilever-free probe spring constant k_cf is only dependent upon the probe base radius R and backing layer effective elastic modulus E and is found to be

It should be noted that while this relationship is expected to accurately relate the deformation of the backing layer to the probe–sample force, this does not preclude the probe or surface from deforming during an imaging experiment, an occurrence that is common to all forms of AFM. Such deformation effectively softens the probe–sample spring constant (i.e. the proportionality between the vertical motion of the z-piezo and the probe–sample force). For the system considered here, simulation predicts that this softening will be less than 10% provided the substrate stiffness is greater than 1 GPa (Supplementary Fig. 2). The fact that the probe is expected to behave as a linear spring is empowering as this is a foundational property of cantilever-based probes. In order to understand the optical consequences of deforming the probe, we combined an estimate of surface deflection from contact mechanics and a ray optics model of light reflected off the backing layer to quantify the change in reflected intensity I as a fraction of the maximum intensity I_max, which was found to only depend on the probe motion δ₀ and angular aperture β of the optics. Specifically, at the perimeter of the probe, we compute (See Supporting Information Section 1 for derivation)

Interestingly, this model predicts that the change in reflected light intensity will be linear in probe motion and can provide ~1 nm precision in determining the vertical position of the probe. However, for operation without complications from the sample in contact with the probe array, the compliant backing layer must be rendered reflective. To realize rigid probes on compliant films, we used two-photon polymerization direct laser writing (2PP-DLW) to write rigid conical probes⁶ directly on a backing layer composed of a polydimethylsiloxane (PDMS) film on a sapphire wafer. These structures were subsequently rendered reflective using an aluminum coating (Fig. 1c—see ‘Methods’ for details). In contrast with prior methods for fabricating rigid probes^25,26,27, 2PP-DLW enables arbitrary probe geometries with optically pristine interfaces. Inspection using scanning electron microscopy (SEM) confirmed that the final structures consisted of planar arrays of polymeric probes (Fig. 1d, e). Mechanical characterization of these probes revealed that they behaved as linear springs with ~10 N/m spring constants, which is in the range of that used in cantilever-based AFM (Supplementary Fig. 3).

In order to evaluate whether cantilever-free AFM can function in a massively parallel format, we constructed an array of 1088 probes and mounted it in a scanning probe instrument. Bright-field optical images taken through the sapphire wafer depicted an array of dark spots, each corresponding to a single probe (Fig. 2a). Subsequently, the probe array was brought in proximity with a flat silicon wafer and leveled with respect to the surface using force feedback, as is common in cantilever-free SPL²⁸. To calibrate the distributed optical lever associated with each probe, the probe array extension Z was increased until the force feedback registered probe array–sample contact. Notably, the size and intensity of the dark spot corresponding to each probe changed drastically upon contact (Fig. 2b). To more quantitatively analyze the change in optical contrast with increasing Z, image processing was used to identify the center of each probe and average the pixel brightness in a 15 µm diameter circle centered on each probe. This average pixel brightness, when normalized using an image of the probe array out of contact, was defined to be the intensity I (Fig. 2c). As predicted by our model, I did not change with respect to Z prior to probe–sample contact and decreased linearly with increasing Z upon contact (Fig. 2d), enabling a direct translation between the optical signal and surface height. Specifically, we fit this decreasing region to I = α(h − Z) with slope α and h denoting the point at which the probe makes contact with the sample—a direct measure of sample height (Fig. 2e). Critically, once probe–sample contact is made for a calibrated probe, one measurement of I is sufficient to compute h.

Topographical image reconstruction

The connection between optical contrast and probe deformation indicates that it is possible to use cantilever-free AFM to generate topographical images of a surface. In particular, if the probe array is brought into contact with a surface and an optical picture is taken, this can be considered a “frame” that contains information about the height of the sample beneath each probe. Since the sample can be raster scanned in the x–y plane with respect to the probe array, the process of bringing the probe array into contact and taking an optical image can be repeated such that subsequent frames are collected while the probe array visits a grid of points. In this way, the topographical information about the entire sample area beneath the probe array can be retrieved with a pixel density that is determined by the number of frames. It is important to highlight that the probes never move laterally while in contact with the sample. Specifically, the probes move vertically into contact with the sample, are held fixed for a contact time, and then are withdrawn until the instrument’s force sensor registered that they were out of contact prior to the lateral move that brings them into their next lateral position. Because of this, abrasion that is common in contact mode imaging is not present.

To accurately reconstruct the series of optical images into a topographical image, we developed a reconstruction algorithm. First, the locations of the probes in the optical image were determined using a Hough transform (an image-processing technique that identifies circular features) and used to compute an I matrix based on averaging the pixel brightness around each probe (Fig. 3a). Next, I was converted into an h matrix using the probe array calibration, which was unchanging as the probe array did not move relative to the optics. This process was repeated for each frame until each probe had visited a 15 × 15 µm² field-of-view (Fig. 3b). Since the probes in the array were hexagonally packed with a 15 µm probe-to-probe spacing, the square fields-of-view captured by each probe overlapped with those of four neighboring probes, which facilitates stitching them into a continuous image (Fig. 3c).

Cantilever-free AFM introduces a few classes of imaging artifacts that must be corrected to accurately image a surface (Fig. 3d). For instance, the probes in the array may vary in height. However, the field-of-view overlap between neighboring probes provides an avenue for addressing this potential variability in probe height. Specifically, for a location (x, y) on the sample that is visited by two probes (e.g. probes 1 and 2), we may specify the deviation in sample height Δh_1,2 = h₁(x, y) − h₂(x, y) = H₁ − H₂ where H_i is the height of probe i. For a probe array with k total probes, there are ~2k distinct overlapping regions, so this presents a linear system of equations that can be solved using a least-squares method. Thus, we compute a deviation vector Δh and a connectivity matrix K for defining Δh = KH that can be solved as H = (K^TK)⁻¹K^TΔh. For this initial imaging experiment, the standard deviation of H was found to be 1.4 μm. In addition to variations in probe height, Z can vary frame-to-frame due to the repeatability of stage motion. As a measure of this, the mean of h for each frame was computed and found to have a 32 nm standard deviation. This offset was removed by shifting the mean of each frame to be zero.

As a final step in the image reconstruction process, we noted that the probes were not mechanically isolated from one another due to their close proximity. Thus, deforming one probe physically moves neighboring probes, resulting in proximal probes registering an artificial change in h. From an image-processing perspective, this artifact is a sharpening process as it enhances the contrast between neighboring probes and is visible as a border between regions with different heights (Fig. 3e). This artifact was removed by estimating the point spread function (PSF) and using deconvolution to recover the original image. Once complete, the final reconstructed image is produced (Fig. 3f). Importantly, this empirical measure of crosstalk in which probes move vertically 35% the distance as their neighbors is in good agreement with the 29% value predicted by finite element simulations (Supplementary Fig. 4). These simulations confirm a critical feature of crosstalk, specifically that probes move a consistent fraction of the distance of their neighbors. This linearity is key as it allows imaging artifacts from crosstalk to be removed using linear image processing, thus enabling the production of an image free from crosstalk artifacts. It is worth noting that in our prior work using polymeric probes prepared using DLW, we found that continuous imaging in contact mode for 8 h did not produce a degradation in image quality⁶. Further, the intermittent contact imaging described here is known to produce less probe wear due to the lack of lateral motion during contact²⁹. Thus, while tip wear remains a consideration that requires further study, it is not likely to be a major limiting feature. While we did not observe the breakage of any probes during imaging, we note that the ability of neighboring probes to overlap their imaging areas provides this approach a potential method for dealing with broken probes by increasing the imaging area such that nominally each region is imaged by multiple probes.

Massively parallel imaging

In order to evaluate the imaging process and image reconstruction algorithm, we ran test scans on a number of regions of an AFM calibration sample. In particular, we began by imaging a fiducial arrow feature that had a 110 nm depth, as determined by AFM (Fig. 4a). A line scan of this sample was taken at 100 nm lateral spacing, which revealed a set of discrete vertical jumps with an average step height of 126 nm, within 15% of the AFM measurement (Fig. 4b). It is worth emphasizing that this line scan represented an aspect ratio of over 10,000 where the vertical precision is estimated as 9 nm (from the root-mean-square error in flat regions) over the >0.4 mm horizontal span. This approximated precision is in agreement with an ~6 nm estimate of the precision estimated using AFM to deform a single probe (Supplementary Fig. 5). Finally, the imaging and reconstruction process was repeated on an intricate region of the calibration sample (Fig. 4c), showing the capability of this approach to generate images of multiscale surfaces. Analysis of this image by exploring the step heights measured by four probes that measured comparable regions revealed an ~6% variation in step height measured for a single probe and an ~3% variation in the average step height measured by all four probes (Supplementary Fig. 6). These metrics begin to address the question of repeatability in cantilever-free AFM, but their modest size in this proof-of-concept study shows the potential for this technique to provide reliable measurements of nanoscale structures.

While this work provides evidence that cantilever-free AFM can provide high-resolution topographical imaging, it is important to consider the opportunities and fundamental limitations of this approach. For example, one potential benefit is cantilever-free AFM could provide more information from each frame than simply topography. For instance, initial experiments using an AFM have found that torques acting on the probe can lead to asymmetric deformation profiles that enable the measurement of lateral forces. Critically, this could allow one to measure the gradient of sample topography (Supplementary Fig. 7), in contrast to cantilever-based AFMs that are at most only sensitive to lateral forces in one direction. Further, the size of the probe array discussed here is not a fundamental limit. In particular, the maximum probe array size is limited by the optical field-of-view, which can be nearly 0.5 cm for large format sensors with 5× optical magnification. Thus, we anticipate that arrays with 100× larger area should be readily attainable. The large working area and increased throughput that are achieved by CF-AFM through parallelization come at the cost of vertical range. While the largest contrast in height that can be measured will depend upon the details of sample topography, the probe height represents an absolute maximum vertical range that can be accommodated without contact between the backing layer and the sample. This limit indicates that vertical range can be increased at the cost of lower throughput by using larger probes that are spaced further apart. Samples that deviate from planarity, such as those with substantial bowing, would be difficult to measure using CF-AFM, as all probes are subjected to the same vertical range. Taking inspiration from AFM studies that image samples multiple times with different probe tilts for accurate sidewall measurement³⁰, CF-AFM measurements could be repeated at multiple tilt angles to capture regions of a bowed surface.

While the experiments here were performed on stiff samples, namely silicon wafers (substrate stiffness E_s > 100 GPa), it is important to evaluate how this technique would perform on samples with different E_s. To explore this possible generalization, we performed a series of finite element simulations to compute the deformation of the backing layer-probe–sample system (Supplementary Fig. 2). In particular, high-resolution topographic imaging is possible when the sample and probe deformations are very small compared with the backing layer deformation as this will preserve a small tip–sample contact area. If, on the other hand, one wishes to deform the sample in order to study its nanomechanical properties, a more substantial sample deformation is desired. Following these guidelines, the probes discussed herein are ideal for high-resolution topographical imaging when E_s ≳ 1 GPa and could be useful for nanomechanical studies in the range 10 MPa < E_s < 1 GPa. Similar to how the stiffness range of cantilever-based probes can be chosen by selecting a probe with the correct spring constant, these ranges can be shifted to higher or lower values of E_s by increasing or decreasing the backing layer stiffness, respectively.

A driving consideration for parallelization is throughput; thus, imaging speed is the main concern. Here, each frame was acquired in 1 s, which indicates that the imaging bandwidth of this system (defined as the number of points in a frame divided by the time to acquire one frame) was over 1 kHz. To provide a frame of reference for this number, the imaging bandwidth of a cantilever-based system is bounded by its mechanical bandwidth, which is approximated by the cantilever resonance frequency divided by the cantilever quality factor. For standard cantilever-based probes, this can range from ~200 to 1 kHz and is a fundamental limitation of the cantilever, although it is worth highlighting achievements in the field of high-speed cantilever-based AFM that utilize high operating frequencies and custom-designed stages for fast motion⁷. In contrast, the bandwidth of cantilever-free systems can be increased through parallelization and shortening frame acquisition time. As previously discussed, scaling to arrays that are 100× larger can be achieved with no change to the optical system. Due to the simplicity of the optical measurement, we predict that the measurement scheme may be reduced to 100 ms or less without specialized optics. With these improvements, MHz bandwidth is attainable. We note that total imaging time will also be defined by the number of frames in an image and processing time for the software. For instance, the data in Fig. 4c took 256 s to collect and 54 s to process using the image-processing algorithm described in Fig. 3.