Introduction

Atomic force microscopy (AFM) is a powerful tool widely used for imaging surfaces with nanoscale resolution1,2,3. AFM makes use of a nanometric tip for scanning the surface of a given sample while mapping the tip-sample interaction to reproduce its morphology. This enables its application in fields as different as biotechnology4, semiconductors5, photonics6, polymer science7 or 2D materials8. One of the main limitations of the AFM is the lateral resolution which comes from the finite size of the probe tip9,10,11. When the tip and the scanned motifs are comparable in size, the width measurements are dramatically overestimated12, and thus, the shape of the motifs cannot be accurately resolved13,14. This effect is the so-called Tip Convolution (TC), and the estimation of the real dimensions of the surface objects is called tip deconvolution15,16,17. As TC has got a clear dependence on the size and shape of the used probe18, few tip reconstruction techniques have been developed with the aim to allow more accurate tip deconvolution1.

All the approaches for AFM tip reconstruction present pros and cons19. Direct approaches, like for example imaging the probe by electron microscopy20,21, offer an accurate picture of the tip. However, these strategies are time consuming and require undesirable manipulation of the probes22,23. Other methods are based on the use of calibration nano-patterns comparable to the probe tip in size. The resulting images are assumed to contain equal amount of information about the sample and the tip19, to be extracted numerically. However, the resolution is strongly dependent on the nanofabrication limitations24. Finally, we can find a range of Blind Tip Reconstruction (BTR) algorithms which have been developed to estimate the probe geometry without knowing the specific motifs on the sample19. As a disadvantage, BTR methods are based in mathematical morphology25,26,27 and require a wide mathematical background to be understood.

In our previous work13, we proposed a simple algorithm capable of determining convolution effects by means of the numerical simulation of the scan performed by an AFM tip. This simulation was proved to work for diverse nano-objects with different geometries. This way, we could illustrate the influence of the size, shape, height and aspect ratio of different nanometric motifs placed over a flat surface. We also demonstrated that, in most cases, the real width of a surface motif can be estimated from the acquired AFM images without knowing its geometry in detail.

As an extension of our previous work, here we provide a more detailed description of the tip-sample interaction. Based on this description, we can re-define the convolution operation and identify those points at the images containing information about the tip. We demonstrate that in the case of rectangular objects, the information of the tip is acquired because part of the interaction is located at the corners of the object. As a result, the tip geometry is clearly reflected in the resulting AFM profile, with minor distortion. Finally, as experimental proof, tip reconstruction experiments are carried out using cubic nanoparticles (NPs).

Background and experimental approaches

Definition of the AFM lateral resolution in terms of mathematical convolution

As above mentioned, the lateral resolution in AFM images is limited by the area of the tip in contact with the inspected object, which depends on diverse factors such as the size of the probe or the height of the object.

The effect of the TC has been treated from different points of view in literature, using different approximations and algorithms, from simple geometrical considerations of the tip to more sophisticated ones25,28,29,30. As in the case of the BTR, most of the methods for description of TC are based on mathematical morphology or on geometrical considerations, where the tip and surface shapes are approached to analytical functions such as circumferences or parabolas31,32,33,34. Therefore, this limits the number of particular cases, where the TC can be calculated by means of the proper mathematical operation of convolution.

To shed light on this, in Fig. 1 the result of convolution operator and the numerical simulations of AFM profiles are compared in opposite circumstances. In Fig. 1a, the case of an AFM tip scanning a Dirac delta function is considered. As expected, the result consists of an inverted peak function reproducing the AFM tip profile. In this case, the analytical result of the convolution operator perfectly fits with the result of our numerical simulation software (not shown). In terms of AFM imaging, this means that the tip shape can be inspected by studying extremely narrow details. Nevertheless, not many nano-structures fulfil the condition of being extremely narrow, and unfortunately, slight deviations with respect to the delta-like function drive to an important reduction of the lateral resolution. For example, in Fig. 1b the result of the mathematical convolution between the tip and a step-like function (more similar to a real nano-structure), is plotted in dark yellow. The result is compared with the corresponding numerical simulation (Fig. 1c). In this second case, the mathematical convolution does not fit with the numerical simulation. The reason of this mismatch will be further treated later on, first, we need to analyse the numerical simulation in detail to extract additional information from.

Figure 1
figure 1

(a) Mathematical tip-sample convolution between a tip and a delta function: the result fits the numerical simulation of the AFM profile. (b) Mathematical convolution between a tip and a step-like function. (c) Numerical simulation of the profile carried out using software of Ref.13.

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Remember that AFM images are acquired by monitoring the interaction between the tip and the surface, hence, the image profiles contain the information of both geometries. The shape of the numerical profile in Fig. 1c could be understood either as the broadening of the step-like function (due to the finite size of the tip) or as an inaccurate replica of the AFM probe, where the tip has not been resolved by the step-like shape.

With this in mind, we can re-define the integration limits of the convolution operation to evaluate the step-like function profile at different stages of the scan, see Fig. 2. The first step is correlated with the scan from point O to A. During this step, the contact with the AFM probe is limited to the right side area of the tip, i.e. from T to R, Fig. 2a. The second step begins with the tip reaching and scanning the top of the object, Fig. 2b. Then, the contact is limited to the tip apex (point T) on the region A–B of the sample. The last step is depicted in Fig. 2c, with the tip scanning the region B–C of the sample while making contact in the area comprised between L and T points (left side of the tip).

Figure 2
figure 2

AFM tip operating on a step-like function analysed by steps. Three different interaction areas are defined: (a) from point O to A [R to T in the tip], (b) from A to B [T point in the tip], and (c) from B to C [T to L in the tip]. The analytical result (solid blue lines) recovers the numerical simulation (dashed clear blue lines) when limited to the corresponding interaction areas.

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After evaluating the scan on separated sections, we found that the tip-sample interaction is restricted to certain areas in each step. Importantly, the convolution operator (⨂) recovers the numerical simulation result if the operation is limited to the areas under interaction; i.e. T–R ⨂ O–A, T ⨂ A–B and L–T ⨂ B–C, respectively. The analytical result of these operations (blue solid lines in Fig. 2) is plotted on the profiles previously calculated using our software (light blue dashed lines in Fig. 2).

Maximum expected error and “mirror effect” in rectangular objects

Together with the tip size, the object height is the most influent parameter in the TC error. Considering objects of the same height, the larger TC is expected in the case of rectangular motifs, as demonstrated in Ref.13. The rectangular objects can be evaluated in three steps, as done with the step-like function. Again, the convolution is restricted to half a tip (T–R and L–T) at left and right top corners of the motif during the first and the third steps. In the course of the second step, only the tip apex interacts with the top side of the motif, section A–B. As a result, the second step of the profile is not affected by the tip size in this part of the scan. Indeed, the convolution error is associated to the first and the third steps where a wide area of the tip interacts with the corners of the object. At these points, the rectangular objects act as delta functions revealing the area of the tip where the contact has taken place. This is equivalent to employ the corners for imaging each half-side of the tip.

Thanks to this, one half-side of the tip shape is reflected at each side of the AFM profile, without any distortion. This behaviour gives to rectangular objects the capability of imaging the shape of the tip, as a mirror does. This kind of “mirror effect” is illustrated in Fig. 3 by comparison of three numerical profiles. The numerical simulations have been carried out considering three different tip shapes, scanning a narrow (Fig. 3a) and a wide rectangular object (Fig. 3b). Since both objects have the same height, they produce identical convolution error, i.e. the “tip reflection”. As a result, the tip geometry can be obtained after removing the A–B section of the profile. For more details, a graphical reconstruction of those tips is illustrated Fig. SI1.

Figure 3
figure 3

Numerical simulation of profiles from a narrow (a) and a broad (b) rectangular object (calculations were made using software in Ref.13). Different tip shapes are considered: sharp (black line), round (red line) and flat (blue line). Because of the rectangular geometry, the same profile is obtained after removing section A–B. In addition, the remaining profile fits the corresponding inverted tip-shape giving place to the “mirror effect”. In (c) we illustrate an AFM probe with typical tip parameters such as γ (tip-to-face angle) and rtip, scanning a rectangular object.

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Experimental methods: height to width correlation and averaged “mirror effect”

Based on this finding, here we propose two complementary approaches that can be combined to give rise to a novel tip reconstruction method.

The first approach consists on a statistical study of the experimental width measurements in comparison to the object height (i.e. h-w curve). This will be done by analysing the topography profiles of a large number of cubic NPs. This particular case of rectangular NPs allows settling a one-to-one correspondence between the real object width (w) and the height (h) measurements. Then, the convolution error can be accurately determined as follows13:

$$\frac{1}{2}{w}_{ex\rho }={r}_{tip}+\Delta +\frac{1}{2}w$$
(1)
$$\text{where} \;\;\Delta =\left(h-{r}_{tip}\right)\mathit{tan}\left(\gamma \right)$$
(2)

This way, the relevant tip parameters γ (i.e. the tip side angle, which is also called tip-to-face angle or half cone angle by probe fabricants) and rtip (tip radius) are bound to the real dimensions of the NPs, h and w. Moreover, the experimental width (wexp) is overestimated due to the TC error (rtip + Δ), where Δ represents the convolution effects related with the tip dependent on γ13. Hence, Eqs. (1) and (2) can be employed to estimate the tip parameters from the h-w plot.

The second approach consists on the reconstruction of the tip shape, as done with the numerical profiles in Fig. 3. This task results a bit more demanding when working with experimental profiles since they can be affected by surface tilt, roughness or shape irregularities. In the next section, we will show how the influence of these undesired effects can be identified and partially prevented by averaging a great number of profiles.

Results and discussion

Cubic NPs as a reference

For this study we chose molecular-based nanoparticles of the bimetallic cyanide-bridged coordination networks known as Prussian blue analogues35,36,37. Specifically, we have used K0.22Ni[Cr(CN)6]0.74 nanoparticles (KNiCr-NPs) that present the face centred cubic structure (Fm3m) and grow as almost perfect cubic crystals an average size of 25 nm (23.9 ± 5.5 nm) and an average aspect ratio (AR) of 1.1. For further details about synthetic approach and physical characterization see our previous publications38,39. These NPs can be easily prepared in a large amount within a short time, by a straightforward solution method in water. They are obtained with a narrow size distribution and moreover, their size can be tuned at will by changing the alkaline cation40. Furthermore, thanks to their negative charge, they are directly stabilized in water as bare particles, without any organic capping. This last point is relevant for the correct performance of profile studies carried out for the development of this work because no additional adhesion forces between tip and organic capping are expected.

KNiCr-NPs were randomly deposited on a silicon substrate previously functionalized with a protonated amminopropyl-triethoxysilane (APTES) self-assembled monolayer. KNiCr-NPs solution was drop casted on top of the modified substrates which were copiously rinsed with mili-Q water after 1 min to remove not attached NPs. Electrostatic driving forces between the cationic layer and the anionic nanoparticles drive the successful deposition of the particles38.

h-w plot approach

Figure 4a shows an AFM image of a representative area of the sample used for the tip characterization. From this kind of images, we can extract the height to width dispersion statistics, (Fig. 4c). The dispersion suggests certain size anisotropy that we attribute to deviation of the NP dimensions with respect to the cubic geometry and to the acquisition error. In the case the NPs here studied (AR = 1.1) a deviation of about 10% in the measurement of the experimental width might be expected as we are considering AR = 1. This deviation is perfectly acceptable for NPs about 25 nm in height, given the discretization of the AFM image is also around 10%, (i.e. 1 × 1 µm image with 512 × 512 pixel resolution).

Figure 4
figure 4

(a) AFM image of a representative area of the NPs sample under study. Few profiles can be measured in each image if focusing on isolated NPs with regular shape and avoiding aggregates. (b) Profiles of different height exhibiting a progressive experimental width reduction. (c) height-to-width plot generated with data from 46 NPs in three different images. The red line corresponds to the best fitting to Eq. (1).

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The images are acquired in tapping mode with a scan rate of about 1 Hz using the AFM Nanoscope IV from Veeco, now a Bruker’s Company. The tip-sample distance is monitored by keeping constant the amplitude of the tip oscillation with a typical set point of about 1 V (20% under the free oscillation amplitude) with a P-I feedback rate of 0.3–0.2. According to the manufacturer, the tip (PPP-NCH from Nanosensors™) presents a rtip = 5–10 nm and γ = 19° (32°) for the fast (slow) axis. The data analysis is carried out using just isolated NPs and avoiding elongated profiles. Up to certain point we have included in our study NPs with slightly asymmetric profiles or moderate tilting, see some examples in Fig. 4b. These low asymmetries can be explained by a low tilt of the sample or by slight deviations from the rectangular geometry, even in some cases we could also identify certain contribution of scanning artefacts like parachuting effect. Consequently, more than 40 profiles are analysed in order to reduce the impact of those defects on the determination of the tip parameters.

The best fitting of Eq. (1) to the experimental data occurs for γ  = 19 ± 3° and the rtip = 15 ± 6 nm. We can conclude that the tip-to-face angle is successfully determined while the tip radius is clearly overestimated by means of this approach. The explanation is intrinsic to the h-w curve which allows to determine γ directly from the slope. In contrast, rtip must be obtained by extrapolation, and hence, it can be dramatically affected by small changes in the slope of the curve. In this situation, we should look for a better estimation of the tip radius by exploiting the “mirror effect”.

“Mirror effect” approach

In this section the “mirror effect” is studied on the same profiles employed on the h-w curve. In Fig. 5a we show three examples: the top profile (in dark cyan) corresponds to a 20 nm height NP. At the upper side of this profile we clearly observe a plateau (marked in the figure with small black arrows), which would be related with the A–B section. This plateau is also observed in the numerical simulation of square objects (see red line in Fig. 2b). The light blue profile (at the bottom), corresponds to a 15 nm height NP, and interestingly, it looks rather similar to the top one. In contrast, the A–B section cannot be easily distinguished in the slightly rounded profile at the centre, which corresponds to a 20 nm height NP (in blue). However, it still presents symmetry enough to be processed considering the maximum of the profile as the centre of the NP.

Figure 5
figure 5

(a) Experimental profiles of three representative NPs. The arrows point the A–B section. (b) Reconstructed tip shape after removing the corresponding A–B section. The offset of the resulting profiles can be fixed either to the substrate surface or to the tip apex. (c) Averaging of the 46 profiles extracted from three different images. The best fitting to Eq. (1) is plot in red to quantify the results.

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The results of removing the A–B section are shown in Fig. 5b. We have used a simple sequential code of logic and arithmetic operations to identify and remove a segment of w = h ± 2 nm (i.e. ± 1 pixel) width from each profile. This way, we also prevent a subjective assessment due to manual treatment on the data. Finally, the resulting profiles are inverted and biased with different offsets (the NP height) for a better comparison. The details of the procedure are offered in the Supporting Information. The processed data are shown as a diluted scatter plot in Fig. 5c. The broadening can be attributed to the rounding during the tip reconstruction data process. The red curve corresponds to the best fitting to the tip geometry which offers a good estimation of the relevant parameters despite the broadening: (i) rtip = 7 ± 1 nm on both sides, close to the 5 nm estimated by the manufacturer; and (ii) γ  = 20 ± 8° also in good agreement but with an important relative error.

After comparing the result from different samples by using several tips, we can conclude that even the scatters in Fig. 5c clearly reproduce the tip shape and size, it must be taken as an upper bound since logical operations tend to produce important deviations in case of tilted profiles or defects (see Fig. SI2 for more details). A much better estimation can be obtained doing a manual treatment to minimize contribution of the artefacts and reduce the rounding error.

Final remarks

To conclude, the results obtained in 3.2 y and 3.3 are summed up for comparison. As expected, the values obtained through both methods result complementary, see Table 1. An overestimated value of rtip = 15 ± 6 nm is obtained from the h-w curve while the mirror effect points to rtip = 7 ± 1 nm, almost within the experimental error. On the other hand, we find good agreement in the determination of the tip-to-face angle, however, in the case of γ the error is clearly higher for the mirror effect results. As mentioned above, the reason of the higher error in γ would be related with the operation range, as we are taking profiles of objects of about 20 nm. We believe that this error could be reduced by imaging taller objects (e.g. 60 nm height) which will provide more information about the tip side.

Table 1 Values of rtip and γ (tip-to-face angle) obtained from both methods: w–h curve and mirror effect.
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Notice that our experiments have afforded the convolution in the most problematic range. In the case of NPs around 20 nm the height is too high for extracting information from the tip radius from h-w curves, but not enough for the proper estimation of γ by means of the “mirror effect”. The use of metrology patterns, instead of NPs, will definitely reduce the experimental data dispersion for both methods. However, we believe that using real samples results more interesting for the reader. Moreover, all the employed NPs can be easily synthetized in different sizes by a quick and inexpensive solution method. This represents the possibility of preparing multiple reference samples for the evaluation of different kind of tips by an approach that, in contrast to sophisticated lithography methods, is available to a broader number of research groups that commonly use AFM techniques in different fields. However, it is also worth mentioning that the above discussed sources of error in our results, could be reduced to the single pixel level by using lithographic patterns.

Conclusions

In this work, a deep discussion on the tip convolution effect is employed to illustrate the difficulties of analytical methods for estimation of convolution errors. For the sake of simplicity our discussion is focused on rectangular geometries which exhibit a prominent interaction with the tip at the corners of the object, leading to a high convolution effect. During this interaction the corners behave as delta-like functions, each one operating over a half-side of the tip, as we demonstrated by reproducing numerically an inverted image of the tip from simulated AFM profiles.

The experimental realization of our numerical predictions has been carried out using a dispersion of cubic molecular-based NPs as a reference sample for the tip reconstruction of a conventional AFM probe. We have evaluated the relation between the height and the width observed at the NP profiles, which ratio has got analytical solution for rectangular shapes. This way, we can obtain a good estimation of the tip-to-face angle but a poor estimation of the tip radius. In a second approach, we have directly identified those points of the experimental profile containing the information of the tip (“mirror effect”). To ensure an objective assessment of tip reconstruction, in this step the experimental profiles have been treated by means of a sequential algorithm. In this way, we have found the proper estimation of the tip radius. Therefore, both approaches presented in this work are complementary and used for double checking the tip reconstruction.