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Correction of the tip convolution effects in the imaging of nanostructures studied through scanning force microscopy
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Nanotechnology 25 395703 (http://iopscience.iop.org/0957-4484/25/39/395703) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 207.162.240.147 This content was downloaded on 30/06/2017 at 17:28 Please note that terms and conditions apply.
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Nanotechnology Nanotechnology 25 (2014) 395703 (9pp)
doi:10.1088/0957-4484/25/39/395703
Correction of the tip convolution effects in the imaging of nanostructures studied through scanning force microscopy Josep Canet-Ferrer, Eugenio Coronado, Alicia Forment-Aliaga and Elena Pinilla-Cienfuegos Instituto de ciencia molecular (ICMol) de la Universidad de Valencia, c/ Catedrático José Beltrán Martínez num. 2, E46980 Paterna, Spain E-mail: [email protected] Received 26 February 2014, revised 11 June 2014 Accepted for publication 31 July 2014 Published 9 September 2014 Abstract
AFM images are always affected by artifacts arising from tip convolution effects, resulting in a decrease in the lateral resolution of this technique. The magnitude of such effects is described by means of geometrical considerations, thereby providing better understanding of the convolution phenomenon. We demonstrate that for a constant tip radius, the convolution error is increased with the object height, mainly for the narrowest motifs. Certain influence of the object shape is observed between rectangular and elliptical objects with the same height. Such moderate differences are essentially expected among elongated objects; in contrast they are reduced as the object aspect ratio is increased. Finally, we propose an algorithm to study the influence of the size, shape and aspect ratio of different nanometric motifs on a flat substrate. Indeed, with this algorithm, convolution artifacts can be extended to any kind of motif including real surface roughness. From the simulation results we demonstrate that in most cases the real motif’s width can be estimated from AFM images without knowing its shape in detail. S Online supplementary data available from stacks.iop.org/NANO/25/395703/mmedia Keywords: atomic force microscopy, tip convolution effect, scanning artefact, lateral resolution, tip deconvolution, image reconstruction (Some figures may appear in colour only in the online journal) 1. Introduction
artifacts, decreasing their lateral resolution. Such undesired effects result from very different sources such as the intrinsic non-linearity of the scanner, an improper tip-sample feedback, electrical noise or the tip convolution effects [11]. AFM manufacturers have gained much experience in eliminating most of them with the improvement of software, hardware and control electronics, but the tip convolution effects cannot be avoided at all. Convolution effects in AFM arise from the finite dimensions of the probe which determine the extension of the surface area interacting with the tip. The reconstruction of the actual surface from measured images (distorted by tip convolution) has been treated in many articles. The pioneering work in this field was presented by Reiss et al, where the tip size influence on scanning probe images was discussed for the
Since its invention in 1986 [1], atomic force microscopy (AFM) has been extensively used for structural characterization of a wide range of nano-structures including thin films, nano-lithography motifs, polymer composites, semiconductor nano-structures, nanotubes, live cells or biomolecules [2–8]. As a difference with respect to the electron microscopy, AFM offers a non-invasive and accurate measurement of the height of the objects under study, which can be characterized in situ without any sample pre-treatment. This suggests an important advantage for the size characterization of nano-objects and other samples prepared on dielectric surfaces [9, 10]. In spite of high resolution in the vertical direction, AFM images are usually affected by 0957-4484/14/395703+09$33.00
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© 2014 IOP Publishing Ltd Printed in the UK
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first time [12]. Later, Keller established a reconstruction formalism based on the Legendre transforms, where the real sample surface can be estimated from the distorted image by deconvolution of the tip [13]. Until now, the most powerful method was proposed by Villarrubia in the late 90s, based on the concepts of mathematical morphology [14, 15]. This formalism is a branch of set theory dealing with unions and intersections of sets and their translations, which provides a precise language for problems related to convolution effects. In this approach the tip and the sample are represented as a complete set of peaks, and therefore the formalism can be applied over any kind of tip and sample that represents the resolution of the method determined by the number of peaks forming each set. The limits of methods based on mathematical morphology lie in the fact that the reconstruction algorithms are extremely time consuming and detailed information about both the tip and sample is mandatory for obtaining quantitative information from the distorted images. On the other hand, applying mathematical morphology to the tip convolution problem is not a trivial task. The dynamics of the phenomenon occurring during the scan are hindered by the formalism and the discussion of the results obtained demands a considerable mathematical background. For these reasons, the convolution effects are more commonly tackled under the assumption that the surface features have a wellknown symmetrical shape. Such methods are based on geometrical considerations where the tip and the surface shapes are approached to analytical functions (such as a circumference or parabola) [6, 8, 16, 17]. This way the lateral resolution and the real size of surface motifs can be estimated from distorted AFM images. However, some complications can be found due to the simplification of the problem since the real surface reconstruction is hard to do in practice (even considering spherical shapes), and the approaches performed affect the validity range of the methods, which can be only applied to some particular cases [18, 19]. With the same versatility of the mathematical morphology formalism but without demanding deep mathematical knowledge, in this work we present a simple algorithm capable of reproducing convolution effects simulating the scan of any kind of AFM tip working for a large range of nanoobjects with different sizes and shapes. This algorithm presents an important advance with respect to the previous proposals since the tip and the sample surface are defined by means of binary sparse matrices instead of by a linear combination of peaks. In this way, the tip motion can be simulated by substituting the heavy algebra required for the mathematical morphology formalism with soft Boolean operators, thereby saving a considerable amount of time. It is worth mentioning that the accuracy of the method (or its working range) is not limited by mathematical approaches as occurs in methods based on geometrical considerations. Using this algorithm we have obtained the scanning profiles expected for a series of different kinds of objects with noticeable variations in size and shape. The simulation offers a clear picture of the tip convolution phenomenon occurring during the scan of a determined surface; these results are very intuitive for predicting of experimental errors and useful for a qualitative
discussion of the AFM images. This new picture allows distinguishing among the contributions of the different sources responsible for the experimental errors such as object height, shape, aspect ratio and the tip parameters, which have been deeply analyzed and summarized in order to provide a general overview.
2. Simulation details In this work, 2D simulation of the AFM tip motion is performed for different kinds of objects in order to study tip convolution effects on AFM images. The code is briefly discussed here, while the computational aspects of the proposed algorithm are described in detail in the supporting information. The tip and the surface motifs are drawn and saved as binary matrices. For that purpose, a heather file is generated to simulate the AFM probe using the tip radius and the tip-to-face angle as input parameters while the sample consists of a flat surface with a geometrical object on top. The algorithm allows the simulation of more complicated profiles if the surface motif and/or the AFM tip are transformed from a grey scale image or even from real SEM micrographs. The resolution of the simulation will be determined by the number of pixels (nxn for the tip matrix and nx2n for the surface matrix, where n is an odd number) and the image scale (in nm/pixel), which will be the same for both matrices. See supporting information for more details. The scan area is represented as another matrix whose dimension doubles that of the first one. At the beginning, the scan area matrix is defined as a 2nx2n null matrix, then, the elements of the tip matrix are introduced at the top-left side of the scan area matrix while the ones corresponding to the surface matrix are inserted at the bottom side. The surface scan is simulated by the ordered translation of the elements corresponding to the tip matrix from the top-left to the bottom-right side. Different from a real AFM, where a proportional-integral gain control is used to correct the feedback distance while the tip is moved forward, in our simulation algorithm the vertical and horizontal motions are done separately. The scan is started in the vertical direction and during this motion the tip-sample distance is evaluated at every pixel, until it reaches the tip-sample contact. Once the contact point is found, the tip is displaced a single pixel forward along the horizontal direction. After this step, the tip-sample distance is re-evaluated: the tip is either withdrawn if it crashed after the forward step, or, if required, it is approached for contacting the surface again. This way, the most common scanning artifacts related to feedback correction are prevented while maintaining the signatures arising from the tip convolution effects. The translation of the tip elements at the 2nx2n matrix is registered to monitor the tip-sample distance. With this information, the tip motion can be saved (like a movie) and after that, repeated to observe the scan details. Scan profiles can be extracted from the registered data as done by the conventional AFM acquisition software. The tip-sample distance at every point (pixel) of the surface is depicted in these 2
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3D simulation can be also performed by the use of ordered arrays of matrices, although it implies an extra waste of computation capabilities. However, this could be interesting in future works because it enables the simulation of full AFM images. In any case, quantitative information from AFM measurements is usually analyzed by means of the AFM image profiles. For this reason, in the next section the tip convolution effects are quantified from the difference between the measured width (wexp) and the real width (w1, which is a simulation input) obtained from profiles in the 2D simulations (Conv = wexp − w). The influence of the surface motif shape is analyzed by comparison between simulations performed on 2D rectangular and elliptical objects. With these results we can discuss the contribution of the different parameters involved in the tip-convolution phenomenon: the importance of the object height is pointed out by comparison of simulation profiles among objects with different sizes, while the results of scanning different shapes is observed by comparison between rectangular and elliptical motifs. Finally, the effects attributable to the tip parameters are analyzed by repeating those simulations for a wide range of rtip and γ.
Figure 1. Representative tip and sample parameters.
profiles which represent the experimental width and height expected in AFM images. In figure 1 the AFM tip is depicted to show some geometrical aspects of this kind of simulation. As in previous works, the probe is represented as a round apex cone whose main parameters consist of the tip radius (rtip) and the tip-toface angle (γ). As will be shown below, such parameters are the most influential on the tip convolution effects. From scanning electron micrographs rtip = 10 nm and g = 19.4° can be estimated for the case of a standard AFM probe (Point Probe Plus from Nanosensors™). Other parameters, for instance the tilt with respect to the sample, are not considered even they could be simulated without calculation costs by introducing a small modification in the surface matrix. Our discussion requires defining the contact height (or effective height, heff) and the effective width (weff) which differ from the real height and width; see figure 1. Notice that the tip position is referred to its apex, because this is the part of the tip interacting with the sample during the scan of a flat substrate. In the presence of an object, the tip must be displaced in the vertical direction to surpass the surface motifs. We use contact point (C) and substrate return (C’) to refer to the points where the tip first makes contact with the object and the substrate (once scanned the object in the forward direction), respectively. The origin of convolution effects in AFM is explained because at C, the tip apex is withdrawn before it arrives at the object, due to the fact that the tip contacts the object away from the apex. Since the tip position is referred to its apex, this motion leads to an overestimation of the object width in the scan profile. The heff represents the distance from C to the flat surface which can be related with the area of the tip interacting with the sample. In an analogue way, the object area interacting with the tip is determined by weff, i.e. the distance between the C and C’. In addition defining L and T as the leftmost and topmost points of the object will be useful for further discussions.
3. Results and discussion It is well known that together with the tip parameters, the convolution effects on AFM topography images are mainly dependent on h and on the object shape. In this work, the ratio between h and w is defined as the aspect ratio, A = h/w. The introduction of this parameter will be useful to distinguish between the height and shape contribution to the convolution error. Our discussion starts by providing fundamental insights of the convolution effects as studied by means of geometrical considerations. Depending on the object height and shape, four possible cases are proposed, as shown in figure 2: case (a), corresponding to C at the linear side region of the tip (h > rtip) and case (b), below the tip round end (h < rtip), both for rectangular shaped objects. For (c) and (d), analogue situations will be presented for round shaped objects. This way, from figure 2(a) we see that ½wexp = rtip + Δ + ½weff
(1)
being
(
)
Δ = h − rtip tan (γ ),
(2)
the basis of the blue shadowed triangle. In this expression, Δ represents the convolution effects that depend on γ, and also to the rtip that must be included to obtain the whole experimental error. From figure 2(b) we determine that: ½wexp = Δ + ½w eff
1
(3)
Note that, properly speaking, in AFM the width is referred to the projection of the object size in the scan plane. For symmetric objects such as rectangles and ellipses the width coincides with the base or with the diameter (parallel to the horizontal plane) of the object, respectively.
3
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Figure 2. Scheme of the tip convolution effects described by means of trigonometric considerations for rectangular and round objects at the linear side region (a) and (c), or below the tip round end (b) and (d).
with
3.1. Height influence on the convolution error
Δ = rtip Cos
{ArcSin ⎡⎣ (r
tip
− h rtip ⎤⎦
)
}
The dependence of the convolution error for a rectangular motif with different h and a given rtip is analyzed with equations (1–4) in five possible cases. In the first case, h ≫ 2rtip, we are in the linear side region of the tip, so we use equation (2) and we find that Δ will be mainly dependent on h with an important influence of γ, and therefore convolution errors on the order of 2 h tan(γ) are expected from equation (1). In the second case, h = 2rtip, we still are in the linear side region of the tip so using equation (2), we find Δ ∼ rtip with a convolution error around 4rtip. In the third case, h = rtip, we also find that Δ = rtip without any influence of γ, and a convolution error of 2rtip. Hence, as the height of the object under study is reduced, the influence of the rtip increases. For the fourth case, h = ½ rtip, using equation (4), we can predict a convolution error reduction from 2rtip to 1.7rtip. Finally, for very low objects, h = 0.1rtip, usually below
(4)
representing the horizontal side of the shadowed triangles. In this case, γ has no influence on the resulting profiles since the objects are lower than rtip. Therefore, the whole experimental error depends on the rtip and the convolution is attributed only to the size of the tip radius. If the round objects in figures 2(c) and (d) are treated in a similar way, analogue expressions are found but now with Δ = (heff-rtip)tan(γ) and Δ = rtip Cos(ArcSin((rtip-heff)/rtip)) for objects above and below the tip round end, respectively. Equations reveal that, independently of object shape, when C is located at the linear side region of the tip, the convolution error is mainly affected by γ and rtip, but when C is located below the tip round end, the convolution error depends only on the rtip, for a given h. 4
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the nanometer height, the convolution error is around 0.9rtip. These results point out the importance of the tip sharpness for imaging nanometric objects or surfaces with low roughness.
those mentioned above for a standard AFM tip (rtip = 10 nm and γ = 19.4°). We set the scale of the simulation to 0.2 nm/ pixel (for objects with sizes below 50 nm); in this way, the round-off effect on the discretization of the standard tip radius is minimized. Working at this scale, we can expect a dispersion error for the simulated profiles of about 0.5 nm (see Supplementary Information). Figures 3(a) and (b) show the contact points for the particular cases of square and circumference shapes respectively. From the analysis of the following scan steps we can reproduce the profile expected for these objects in AFM topography images [blue and red dashed lines in figures 3(a) and (b)]. By comparison among profiles of different size objects we can observe the convolution dependence on the height which is attributed to the fact that C is found at larger distances from the tip apex as raising the object height. This increase is reflected in the profiles and compared with the object diameter to estimate the experimental error in the AFM images. The results are summarized in figures 3(c) and (d) where the experimental diameter and the convolution error (Conv = wexp − w) are represented. For example, when scanning a feature of h = 10 nm (w = 10 nm), the measured width is expected to be close to wexp ≈ 30 nm due to convolution, Conv ≈ 20 nm. This value is in agreement with the geometrical model which predicts Conv = 2ritp. In the simulation it is clearly shown that in this situation the tip contacts the square at the tip round end, or very close to this point in the case of the circumference. The rest of cases can be distinguished between objects larger or shorter than the tip radius. For larger objects (h > rtip) the contact point is found above the tip round end, in the linear side region of the tip, while this separation is reduced in the case of shorter objects (h < rtip), since in that case the contact point occurs into the round side region, providing insight into the real nature of the convolution effect.
3.2. Shape influence on the convolution error
After describing the height dependence, we can relate the shape influence on the convolution with the deviation of heff and weff with respect to the real values (h and w). By definition, the variation of heff (and weff) with respect to h (and w) is determined by the contact point location. For the symmetric objects studied here, C should be found at some point between the leftmost (L) and the topmost (T) points of the object (see figure 1). Actually, the rectangle is a special case where heff = h and weff = w. Since at the rectangle L coincides with T at the left side corner, the contact point is necessarily going to occur there (L = T = C) independently of the tip and the object aspect ratio, as will be described below. Similarly, a small shape influence will be expected in non-rectangular narrow objects (with A > 1), since in this type of structure, L is necessarily close to T, reducing the possible differences between weff and heff with respect to the real values. For the same reasons, at the lower and wider objects (with A ∼ 0.25) L and T are separated by larger distances, and as result, C (and hence the convolution error) can take a wide range of values depending on the object shape. From the above discussion we conclude that for objects with a high A, the convolution error depends on the shape in a minor way while for low A, the convolution error is highly dependent on the shape of the object under study. 3.3. Comparison between the geometrical considerations and the simulations
The most important point in the above discussion is that the convolution error of a real object can be estimated using the above expressions if the aspect ratio is known. The validity of this approach will depend on how the real objects approach to the elliptical or rectangular shapes. This means an accurate estimation for rectangular and elliptical shapes, while in the case of objects with exotic geometries we expect a better estimation from the algorithm described in section 2. This is because the algorithm allows for the representation of the surface motifs as they are, without being approached to any mathematical expression. To demonstrate the goodness of our simulations we have applied our algorithm on the geometrical objects, where equations (1–4) are expected to work more properly (see more in supporting information). In addition we found that the simulation results yield complementary information about the nature of the convolution phenomenon since the tip motion and the convolution profile are completely reproduced. In figure 3 the relationship between the height and the convolution effects is illustrated. In this case, the data in this picture are extracted from a simulation series of squares and circumferences with different height. The object aspect ratio (A = 1) is maintained by simultaneously increasing both the object height and width. The tip parameters employed are
3.4. Importance of the aspect ratio
We can find major differences comparing objects with different aspect ratios, as done in the simulations presented in figure 4. In this figure, simulations for rectangular and elliptical structures with different A = 0.25-4 are compared. As in figure 3, the data are extracted from simulation series using the standard parameters for the tip matrix. Some representative cases extracted from figures 4(a) and (b) are shown in table S2 in the supporting information. The convolution error is not completely independent from the aspect ratio in the case of elliptical shapes, but even so, similar values of the convolution error are observed between rectangular and elliptical shapes of the same height (see supporting information). As expected the major difference in the convolution error value between rectangular and round objects (attributable to the aspect ratio) is observed for the highest and most elongated objects (e.g. h = 20 nm and A = 0.25). It is worth noting that for low A values such differences are maintained even below the tip round end (h ∼ 5 nm). This fact could be deduced from equations (3) and (4), but is better illustrated by comparison of the corresponding profiles arising from the simulations, 5
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Figure 3. Topography profiles obtained from simulations of a square (a) and a circumference (b) motif considering rtip of 10 nm and γ of 19.4°. (c) and (d) plots of the experimental width and convolution error [Conv = wexp − w (nm)] with respect to the height for both kinds of objects. Inset in (c) illustrates the small convolution dependence on the object shape for a square and a circumference of the same height, due to the proximity of C.
treating the rectangular or elliptical case similar to a first approach.
which indicates that this feature is strongly correlated with the non-linear behavior of the wexp vs h curves in figure 4. This can be considered as a proof of the additional information extracted from the simulations. While equations (1–4) represent the evolution of the convolution error as the object height rises, the simulations show the complete profile and how it is followed by the tip during the scan. With that result we conclude that the shape dependence is generally minor or negligible with two clear exceptions determined by the aspect ratio: i) the very high objects where the contact is found at the linear side region; and ii) the elongated objects (with A < 1) with C below the tip round end, where the contact point is quite sensitive to the shape variations. For example, a recent publication based on a geometrical model obtains similar results by treating DNA chains in both ways, as a rectangular and as a circular motif [8]. This is consistent with our results since the aspect ratio of such chains is around A = 1 and belongs to the low working range (h < ½ rtip), but the study presented here suggests that for more elongated structures, as could be quantum nanostructures or local oxidation nano-lithography motifs in this working range [23–27], the object shape must be considered,
3.5. Tip influence on the convolution error
The effects of working with different rtip are shown in figure 5(a) (maintaining γ = 19.4°). The tip radius takes values from 10 nm (coinciding with the standard value) to 50 nm, which is the radius estimated for the metal coated tips usually employed for scanning near-field optical, electrostatic force or magnetic force microscopy [20–22]. In this figure the curvature of the depicted data is directly related to shape of the tip. The round side region of the tip leads to curved lines in the plot below h = rtip while the linear side region of the tip produces straight lines with identical slopes. This is because for higher objects the convolution is not related to rtip but to γ (which is constant in this simulation). In figure 5(b) the series are repeated for different γ and we observe the opposite trend. Since rtip is not changed during that simulation (rtip = 10 nm) the data are overlapped below 10 nm height. The differences appear above this value, situated at the linear side region of the tip, because higher slopes appear in the case of larger 6
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Figure 4. Results from rectangular (a) and elliptical (b) simulations representing the expected motif widths with respect to the height for
different aspect ratios. In the simulations the selected rtip is fixed to 10 nm while the γ is set to 19.4°. They are accompanied by the convolution error (c) and (d), respectively. In the inset of (c) it is illustrated that convolution effects are independent of the aspect ratio in the case of rectangular motifs. In the inset of (d) it is shown that aspect ratio dependence is common in round objects.
values of γ. The range of angles selected for these simulations (γ = 5°−50°) was chosen in accordance with typical values observed from SEM micrographs, around 19.4° in the front face of the tip (normal plane with respect to the cantilever), or close to 32° if the scan is done with the tip aligned in the perpendicular direction; see figures 5(c) and (d). In addition, these values could vary from 12° to more than 40° if tilts between the tip and the sample were considered. There are also special probes where γ could be smaller than 10°. Hence, we can conclude that all the representative angles are included in the simulation. The tip radius effects are observed even for the smallest objects losing importance with respect to γ effects as the height of the motifs increases and C moves into the linear side region of the tip. However, γ has no influence on the convolution error when h < rtip.
sample. In 2D this is a friendly method that illustrates the problem of the convolution in a clear way. In addition, the code employed is not time consuming since complex operations are not required, unlike, for instance, in methods based on the Legendre transform. AFM topography profiles have been simulated in order to be compared with the expressions describing convolution effects for rectangular and elliptical geometries. The simulations are performed using a sequential algorithm to reproduce the tip motion while preventing the scanning artifacts usually found in real images. From comparison among simulation series, the tip convolution effects are studied for different surface motifs and tips. Focusing on the influence of the motif properties such as size, shape and aspect ratio we demonstrate that: i) the convolution error depends mainly on γ for objects with h ≫ rtip, since in that cases the tip contacts the object in the linear side region, in contrast as h decreases the influence of the rtip increases and the influence of γ becomes negligible; ii) the shape influence is always more important in the case of larger objects, while for objects with sizes comparable with rtip the convolution effects are clearly around 2 rtip, and in general have minor shape dependence. Therefore,
4. Conclusions. In this work, we have presented a new algorithm for resolving the convolution effects in AFM imaging. It is a versatile method because it can be applied to any kind of tip and 7
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nanostructured devices for spintronics or opto-electronics, fabricated by means of lithography techniques onto epitaxial growth wafers, with almost rectangular profiles. Currently, improvements are being made to the code for its application to new algorithms programmed for tip and sample reconstruction, simulating tip erosion, and for obtaining the deconvolution of distorted images. Acknowledgments Financial support from the EU (ERC Advanced Grant SPINMOL), the Spanish MINECO (MAT2011-22785, cofinanced by FEDER) and the Generalidad Valenciana (Prometeo and ISIC-Nano programmes) are gratefully acknowledged. The corresponding author J C-F thanks the Generalitat Valenciana for his grant funded by the Vali+d program (APOSTD/2013/076). A F-A thanks the Spanish MINECO for her R y C grant. The authors are also grateful to Dr A Gaita for helpful discussions. Supporting information Computational aspects of the algorithm used are presented in detail accompanied by an example. The resolution of the simulations is also discussed and a code optimization route is pointed out. We have added a comparison table between geometrical expressions and simulation results. A simulation of an AFM profile from a SEM image is shown as a proof of the algorithm’s capabilities. References [1] Binning G, Quate C F and Gerber C 1986 Atomic force mircroscope Phys. Rev. Lett. 56 930 [2] Alen B, Fuster D, Munoz-Matutano G, Martinez-Pastor J, Gonzalez Y, Canet-Ferrer J and Gonzalez L 2008 Exciton gas compression and metallic condensation in a single semiconductor quantum wire Phys. Rev. Lett. 101 067405 [3] Martin-Sanchez J et al 2009 Single photon emission from sitecontrolled InAs quantum dots grown on GaAs(001) patterned substrates ACS Nano 3 1513 [4] Abargues R, Gradess R, Canet-Ferrer J, Abderrafi K and Valdes J L 2009 Scalable heterogeneous synthesis of metallic nanoparticles and aggregates with polyvinyl alcohol New J. Chem. 33 913 [5] Munoz-Matutano G et al 2011 Charge control in laterally coupled double quantum dots Phys. Rev. B 84 041308 [6] Wang W, Niu D X, Jiang C R and Yang X J 2013 The conductive properties of single DNA molecules studied by torsion tunneling atomic force microscopy Nanotechnology 25 025707 [7] Cadena J M, Misiego R, Smith K C, Avila A, Pipes B, Reifenberger R and Raman A 2013 Sub-surface imaging of carbon nanotube-polymer composites using dynamic AFM methods Nanotechnology 24 135706 [8] Winzer A T, Kraft C, Bhushan S, Stepanenko V and Tessmer I 2012 Correcting for AFM tip induced topography convolutions in protein–DNAsamples Ultramicroscopy 121 8 [9] Tawil N, Sacher E, Mandeville R and Meunier M 2013 Strategies for the immobilization of bacteriophages on gold
Figure 5. (a) Experimental width with respect to the height for square motifs obtained using tips of different radius. The tip-to-face angle is set to 19.4°. (b) Experimental width with respect to the height for square motifs for different tip-to-face angles (γ). The tip radius (rtip) is fixed to 10 nm. (c) and (d) SEM micrographs showing the tip-toface angle from the front and lateral views of an standard AFM tip (Point Probe Plus from NanosensorsTM). SEM images courtesy of NanosensorsTM.
we can conclude that in such a working range, the experimental error can be estimated without knowing the object shape in detail, since for objects as different as squares and circumferences comparable convolution errors are obtained; iii) the convolution error for rectangular objects is aspect ratio independent at a constant h; however, that is not the case for rounded motifs which dramatically increases for the most elongated structures (A ≪ 1). All these results point out the importance of the convolution error for the study of objects with sizes comparable to the tip radius (molecules, proteins, DNA, NPs, etc), as well as for larger objects that are found in samples such as 8
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of protamine-complexed DNA fibers Ultramicroscopy 42 1095 Canet-Ferrer J, Martin-Carron L, Valdes J L and Martinez-Pastor J 2006 Scanning probe microscopy applied to the study of domains and domain walls in a ferroelectric KNbO3 crystal BSECV 45 218 Ibrahim I, Ruemmeli M H, Ranjan N, Buechner B and Cuniberti G 2013 Spatial recognition of defects and tube type in carbon nanotube field effect transistors using electrostatic force microscopy Nanotechnology 24 235708 Sun L X and Chen Q W 2009 Core–shell cylindrical magnetic domains in nickel wires prepared under magnetic fields J. Phys. Chem. C 113 2710 Mott D, Cotts B, Lim I-I S, Luo J, Park H-Y, Njoki P N, Schadt M J and Zhong C-J 2008 Size determination of nanoparticles based on tapping-mode atomic force microscopy measurements J. Scann. Probe Microsc. 3 1 Iqbal M Z, Iqbal M W, Lee J H, Kim Y S, Chun S H and Eom J 2013 Spin valve effect of NiFe/graphene/NiFe junctions Nano Research 6 373–80 Canet-Ferrer J et al 2012 Purcell effect in photonic crystal microcavities embedding InAs/InP quantum wires Opt. Expr. 20 7901 Seravalli L, Trevisi G, Frigeri P, Rivas D, Munoz-Matutano G, Suarez I, Canet-Ferrer J, Alen B and Martínez-Pastor J 2011 Single quantum dot emission at telecom wavelengths from metamorphic InAs/InGaAs nanostructures grown on GaAs substrates Appl. Phys. Lett. 98 173112 Garcia R, Martinez R V and Martinez J 2006 Nano-chemistry and scanning probe nanolithographies Chem. Soc. Rev. 35 29
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Correction of the tip convolution effects in the imaging of nanostructures studied through scanning force microscopy
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Nanotechnology 25 395703 (http://iopscience.iop.org/0957-4484/25/39/395703) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 207.162.240.147 This content was downloaded on 30/06/2017 at 17:28 Please note that terms and conditions apply.
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Nanotechnology Nanotechnology 25 (2014) 395703 (9pp)
doi:10.1088/0957-4484/25/39/395703
Correction of the tip convolution effects in the imaging of nanostructures studied through scanning force microscopy Josep Canet-Ferrer, Eugenio Coronado, Alicia Forment-Aliaga and Elena Pinilla-Cienfuegos Instituto de ciencia molecular (ICMol) de la Universidad de Valencia, c/ Catedrático José Beltrán Martínez num. 2, E46980 Paterna, Spain E-mail: [email protected] Received 26 February 2014, revised 11 June 2014 Accepted for publication 31 July 2014 Published 9 September 2014 Abstract
AFM images are always affected by artifacts arising from tip convolution effects, resulting in a decrease in the lateral resolution of this technique. The magnitude of such effects is described by means of geometrical considerations, thereby providing better understanding of the convolution phenomenon. We demonstrate that for a constant tip radius, the convolution error is increased with the object height, mainly for the narrowest motifs. Certain influence of the object shape is observed between rectangular and elliptical objects with the same height. Such moderate differences are essentially expected among elongated objects; in contrast they are reduced as the object aspect ratio is increased. Finally, we propose an algorithm to study the influence of the size, shape and aspect ratio of different nanometric motifs on a flat substrate. Indeed, with this algorithm, convolution artifacts can be extended to any kind of motif including real surface roughness. From the simulation results we demonstrate that in most cases the real motif’s width can be estimated from AFM images without knowing its shape in detail. S Online supplementary data available from stacks.iop.org/NANO/25/395703/mmedia Keywords: atomic force microscopy, tip convolution effect, scanning artefact, lateral resolution, tip deconvolution, image reconstruction (Some figures may appear in colour only in the online journal) 1. Introduction
artifacts, decreasing their lateral resolution. Such undesired effects result from very different sources such as the intrinsic non-linearity of the scanner, an improper tip-sample feedback, electrical noise or the tip convolution effects [11]. AFM manufacturers have gained much experience in eliminating most of them with the improvement of software, hardware and control electronics, but the tip convolution effects cannot be avoided at all. Convolution effects in AFM arise from the finite dimensions of the probe which determine the extension of the surface area interacting with the tip. The reconstruction of the actual surface from measured images (distorted by tip convolution) has been treated in many articles. The pioneering work in this field was presented by Reiss et al, where the tip size influence on scanning probe images was discussed for the
Since its invention in 1986 [1], atomic force microscopy (AFM) has been extensively used for structural characterization of a wide range of nano-structures including thin films, nano-lithography motifs, polymer composites, semiconductor nano-structures, nanotubes, live cells or biomolecules [2–8]. As a difference with respect to the electron microscopy, AFM offers a non-invasive and accurate measurement of the height of the objects under study, which can be characterized in situ without any sample pre-treatment. This suggests an important advantage for the size characterization of nano-objects and other samples prepared on dielectric surfaces [9, 10]. In spite of high resolution in the vertical direction, AFM images are usually affected by 0957-4484/14/395703+09$33.00
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first time [12]. Later, Keller established a reconstruction formalism based on the Legendre transforms, where the real sample surface can be estimated from the distorted image by deconvolution of the tip [13]. Until now, the most powerful method was proposed by Villarrubia in the late 90s, based on the concepts of mathematical morphology [14, 15]. This formalism is a branch of set theory dealing with unions and intersections of sets and their translations, which provides a precise language for problems related to convolution effects. In this approach the tip and the sample are represented as a complete set of peaks, and therefore the formalism can be applied over any kind of tip and sample that represents the resolution of the method determined by the number of peaks forming each set. The limits of methods based on mathematical morphology lie in the fact that the reconstruction algorithms are extremely time consuming and detailed information about both the tip and sample is mandatory for obtaining quantitative information from the distorted images. On the other hand, applying mathematical morphology to the tip convolution problem is not a trivial task. The dynamics of the phenomenon occurring during the scan are hindered by the formalism and the discussion of the results obtained demands a considerable mathematical background. For these reasons, the convolution effects are more commonly tackled under the assumption that the surface features have a wellknown symmetrical shape. Such methods are based on geometrical considerations where the tip and the surface shapes are approached to analytical functions (such as a circumference or parabola) [6, 8, 16, 17]. This way the lateral resolution and the real size of surface motifs can be estimated from distorted AFM images. However, some complications can be found due to the simplification of the problem since the real surface reconstruction is hard to do in practice (even considering spherical shapes), and the approaches performed affect the validity range of the methods, which can be only applied to some particular cases [18, 19]. With the same versatility of the mathematical morphology formalism but without demanding deep mathematical knowledge, in this work we present a simple algorithm capable of reproducing convolution effects simulating the scan of any kind of AFM tip working for a large range of nanoobjects with different sizes and shapes. This algorithm presents an important advance with respect to the previous proposals since the tip and the sample surface are defined by means of binary sparse matrices instead of by a linear combination of peaks. In this way, the tip motion can be simulated by substituting the heavy algebra required for the mathematical morphology formalism with soft Boolean operators, thereby saving a considerable amount of time. It is worth mentioning that the accuracy of the method (or its working range) is not limited by mathematical approaches as occurs in methods based on geometrical considerations. Using this algorithm we have obtained the scanning profiles expected for a series of different kinds of objects with noticeable variations in size and shape. The simulation offers a clear picture of the tip convolution phenomenon occurring during the scan of a determined surface; these results are very intuitive for predicting of experimental errors and useful for a qualitative
discussion of the AFM images. This new picture allows distinguishing among the contributions of the different sources responsible for the experimental errors such as object height, shape, aspect ratio and the tip parameters, which have been deeply analyzed and summarized in order to provide a general overview.
2. Simulation details In this work, 2D simulation of the AFM tip motion is performed for different kinds of objects in order to study tip convolution effects on AFM images. The code is briefly discussed here, while the computational aspects of the proposed algorithm are described in detail in the supporting information. The tip and the surface motifs are drawn and saved as binary matrices. For that purpose, a heather file is generated to simulate the AFM probe using the tip radius and the tip-to-face angle as input parameters while the sample consists of a flat surface with a geometrical object on top. The algorithm allows the simulation of more complicated profiles if the surface motif and/or the AFM tip are transformed from a grey scale image or even from real SEM micrographs. The resolution of the simulation will be determined by the number of pixels (nxn for the tip matrix and nx2n for the surface matrix, where n is an odd number) and the image scale (in nm/pixel), which will be the same for both matrices. See supporting information for more details. The scan area is represented as another matrix whose dimension doubles that of the first one. At the beginning, the scan area matrix is defined as a 2nx2n null matrix, then, the elements of the tip matrix are introduced at the top-left side of the scan area matrix while the ones corresponding to the surface matrix are inserted at the bottom side. The surface scan is simulated by the ordered translation of the elements corresponding to the tip matrix from the top-left to the bottom-right side. Different from a real AFM, where a proportional-integral gain control is used to correct the feedback distance while the tip is moved forward, in our simulation algorithm the vertical and horizontal motions are done separately. The scan is started in the vertical direction and during this motion the tip-sample distance is evaluated at every pixel, until it reaches the tip-sample contact. Once the contact point is found, the tip is displaced a single pixel forward along the horizontal direction. After this step, the tip-sample distance is re-evaluated: the tip is either withdrawn if it crashed after the forward step, or, if required, it is approached for contacting the surface again. This way, the most common scanning artifacts related to feedback correction are prevented while maintaining the signatures arising from the tip convolution effects. The translation of the tip elements at the 2nx2n matrix is registered to monitor the tip-sample distance. With this information, the tip motion can be saved (like a movie) and after that, repeated to observe the scan details. Scan profiles can be extracted from the registered data as done by the conventional AFM acquisition software. The tip-sample distance at every point (pixel) of the surface is depicted in these 2
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3D simulation can be also performed by the use of ordered arrays of matrices, although it implies an extra waste of computation capabilities. However, this could be interesting in future works because it enables the simulation of full AFM images. In any case, quantitative information from AFM measurements is usually analyzed by means of the AFM image profiles. For this reason, in the next section the tip convolution effects are quantified from the difference between the measured width (wexp) and the real width (w1, which is a simulation input) obtained from profiles in the 2D simulations (Conv = wexp − w). The influence of the surface motif shape is analyzed by comparison between simulations performed on 2D rectangular and elliptical objects. With these results we can discuss the contribution of the different parameters involved in the tip-convolution phenomenon: the importance of the object height is pointed out by comparison of simulation profiles among objects with different sizes, while the results of scanning different shapes is observed by comparison between rectangular and elliptical motifs. Finally, the effects attributable to the tip parameters are analyzed by repeating those simulations for a wide range of rtip and γ.
Figure 1. Representative tip and sample parameters.
profiles which represent the experimental width and height expected in AFM images. In figure 1 the AFM tip is depicted to show some geometrical aspects of this kind of simulation. As in previous works, the probe is represented as a round apex cone whose main parameters consist of the tip radius (rtip) and the tip-toface angle (γ). As will be shown below, such parameters are the most influential on the tip convolution effects. From scanning electron micrographs rtip = 10 nm and g = 19.4° can be estimated for the case of a standard AFM probe (Point Probe Plus from Nanosensors™). Other parameters, for instance the tilt with respect to the sample, are not considered even they could be simulated without calculation costs by introducing a small modification in the surface matrix. Our discussion requires defining the contact height (or effective height, heff) and the effective width (weff) which differ from the real height and width; see figure 1. Notice that the tip position is referred to its apex, because this is the part of the tip interacting with the sample during the scan of a flat substrate. In the presence of an object, the tip must be displaced in the vertical direction to surpass the surface motifs. We use contact point (C) and substrate return (C’) to refer to the points where the tip first makes contact with the object and the substrate (once scanned the object in the forward direction), respectively. The origin of convolution effects in AFM is explained because at C, the tip apex is withdrawn before it arrives at the object, due to the fact that the tip contacts the object away from the apex. Since the tip position is referred to its apex, this motion leads to an overestimation of the object width in the scan profile. The heff represents the distance from C to the flat surface which can be related with the area of the tip interacting with the sample. In an analogue way, the object area interacting with the tip is determined by weff, i.e. the distance between the C and C’. In addition defining L and T as the leftmost and topmost points of the object will be useful for further discussions.
3. Results and discussion It is well known that together with the tip parameters, the convolution effects on AFM topography images are mainly dependent on h and on the object shape. In this work, the ratio between h and w is defined as the aspect ratio, A = h/w. The introduction of this parameter will be useful to distinguish between the height and shape contribution to the convolution error. Our discussion starts by providing fundamental insights of the convolution effects as studied by means of geometrical considerations. Depending on the object height and shape, four possible cases are proposed, as shown in figure 2: case (a), corresponding to C at the linear side region of the tip (h > rtip) and case (b), below the tip round end (h < rtip), both for rectangular shaped objects. For (c) and (d), analogue situations will be presented for round shaped objects. This way, from figure 2(a) we see that ½wexp = rtip + Δ + ½weff
(1)
being
(
)
Δ = h − rtip tan (γ ),
(2)
the basis of the blue shadowed triangle. In this expression, Δ represents the convolution effects that depend on γ, and also to the rtip that must be included to obtain the whole experimental error. From figure 2(b) we determine that: ½wexp = Δ + ½w eff
1
(3)
Note that, properly speaking, in AFM the width is referred to the projection of the object size in the scan plane. For symmetric objects such as rectangles and ellipses the width coincides with the base or with the diameter (parallel to the horizontal plane) of the object, respectively.
3
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Figure 2. Scheme of the tip convolution effects described by means of trigonometric considerations for rectangular and round objects at the linear side region (a) and (c), or below the tip round end (b) and (d).
with
3.1. Height influence on the convolution error
Δ = rtip Cos
{ArcSin ⎡⎣ (r
tip
− h rtip ⎤⎦
)
}
The dependence of the convolution error for a rectangular motif with different h and a given rtip is analyzed with equations (1–4) in five possible cases. In the first case, h ≫ 2rtip, we are in the linear side region of the tip, so we use equation (2) and we find that Δ will be mainly dependent on h with an important influence of γ, and therefore convolution errors on the order of 2 h tan(γ) are expected from equation (1). In the second case, h = 2rtip, we still are in the linear side region of the tip so using equation (2), we find Δ ∼ rtip with a convolution error around 4rtip. In the third case, h = rtip, we also find that Δ = rtip without any influence of γ, and a convolution error of 2rtip. Hence, as the height of the object under study is reduced, the influence of the rtip increases. For the fourth case, h = ½ rtip, using equation (4), we can predict a convolution error reduction from 2rtip to 1.7rtip. Finally, for very low objects, h = 0.1rtip, usually below
(4)
representing the horizontal side of the shadowed triangles. In this case, γ has no influence on the resulting profiles since the objects are lower than rtip. Therefore, the whole experimental error depends on the rtip and the convolution is attributed only to the size of the tip radius. If the round objects in figures 2(c) and (d) are treated in a similar way, analogue expressions are found but now with Δ = (heff-rtip)tan(γ) and Δ = rtip Cos(ArcSin((rtip-heff)/rtip)) for objects above and below the tip round end, respectively. Equations reveal that, independently of object shape, when C is located at the linear side region of the tip, the convolution error is mainly affected by γ and rtip, but when C is located below the tip round end, the convolution error depends only on the rtip, for a given h. 4
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the nanometer height, the convolution error is around 0.9rtip. These results point out the importance of the tip sharpness for imaging nanometric objects or surfaces with low roughness.
those mentioned above for a standard AFM tip (rtip = 10 nm and γ = 19.4°). We set the scale of the simulation to 0.2 nm/ pixel (for objects with sizes below 50 nm); in this way, the round-off effect on the discretization of the standard tip radius is minimized. Working at this scale, we can expect a dispersion error for the simulated profiles of about 0.5 nm (see Supplementary Information). Figures 3(a) and (b) show the contact points for the particular cases of square and circumference shapes respectively. From the analysis of the following scan steps we can reproduce the profile expected for these objects in AFM topography images [blue and red dashed lines in figures 3(a) and (b)]. By comparison among profiles of different size objects we can observe the convolution dependence on the height which is attributed to the fact that C is found at larger distances from the tip apex as raising the object height. This increase is reflected in the profiles and compared with the object diameter to estimate the experimental error in the AFM images. The results are summarized in figures 3(c) and (d) where the experimental diameter and the convolution error (Conv = wexp − w) are represented. For example, when scanning a feature of h = 10 nm (w = 10 nm), the measured width is expected to be close to wexp ≈ 30 nm due to convolution, Conv ≈ 20 nm. This value is in agreement with the geometrical model which predicts Conv = 2ritp. In the simulation it is clearly shown that in this situation the tip contacts the square at the tip round end, or very close to this point in the case of the circumference. The rest of cases can be distinguished between objects larger or shorter than the tip radius. For larger objects (h > rtip) the contact point is found above the tip round end, in the linear side region of the tip, while this separation is reduced in the case of shorter objects (h < rtip), since in that case the contact point occurs into the round side region, providing insight into the real nature of the convolution effect.
3.2. Shape influence on the convolution error
After describing the height dependence, we can relate the shape influence on the convolution with the deviation of heff and weff with respect to the real values (h and w). By definition, the variation of heff (and weff) with respect to h (and w) is determined by the contact point location. For the symmetric objects studied here, C should be found at some point between the leftmost (L) and the topmost (T) points of the object (see figure 1). Actually, the rectangle is a special case where heff = h and weff = w. Since at the rectangle L coincides with T at the left side corner, the contact point is necessarily going to occur there (L = T = C) independently of the tip and the object aspect ratio, as will be described below. Similarly, a small shape influence will be expected in non-rectangular narrow objects (with A > 1), since in this type of structure, L is necessarily close to T, reducing the possible differences between weff and heff with respect to the real values. For the same reasons, at the lower and wider objects (with A ∼ 0.25) L and T are separated by larger distances, and as result, C (and hence the convolution error) can take a wide range of values depending on the object shape. From the above discussion we conclude that for objects with a high A, the convolution error depends on the shape in a minor way while for low A, the convolution error is highly dependent on the shape of the object under study. 3.3. Comparison between the geometrical considerations and the simulations
The most important point in the above discussion is that the convolution error of a real object can be estimated using the above expressions if the aspect ratio is known. The validity of this approach will depend on how the real objects approach to the elliptical or rectangular shapes. This means an accurate estimation for rectangular and elliptical shapes, while in the case of objects with exotic geometries we expect a better estimation from the algorithm described in section 2. This is because the algorithm allows for the representation of the surface motifs as they are, without being approached to any mathematical expression. To demonstrate the goodness of our simulations we have applied our algorithm on the geometrical objects, where equations (1–4) are expected to work more properly (see more in supporting information). In addition we found that the simulation results yield complementary information about the nature of the convolution phenomenon since the tip motion and the convolution profile are completely reproduced. In figure 3 the relationship between the height and the convolution effects is illustrated. In this case, the data in this picture are extracted from a simulation series of squares and circumferences with different height. The object aspect ratio (A = 1) is maintained by simultaneously increasing both the object height and width. The tip parameters employed are
3.4. Importance of the aspect ratio
We can find major differences comparing objects with different aspect ratios, as done in the simulations presented in figure 4. In this figure, simulations for rectangular and elliptical structures with different A = 0.25-4 are compared. As in figure 3, the data are extracted from simulation series using the standard parameters for the tip matrix. Some representative cases extracted from figures 4(a) and (b) are shown in table S2 in the supporting information. The convolution error is not completely independent from the aspect ratio in the case of elliptical shapes, but even so, similar values of the convolution error are observed between rectangular and elliptical shapes of the same height (see supporting information). As expected the major difference in the convolution error value between rectangular and round objects (attributable to the aspect ratio) is observed for the highest and most elongated objects (e.g. h = 20 nm and A = 0.25). It is worth noting that for low A values such differences are maintained even below the tip round end (h ∼ 5 nm). This fact could be deduced from equations (3) and (4), but is better illustrated by comparison of the corresponding profiles arising from the simulations, 5
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Figure 3. Topography profiles obtained from simulations of a square (a) and a circumference (b) motif considering rtip of 10 nm and γ of 19.4°. (c) and (d) plots of the experimental width and convolution error [Conv = wexp − w (nm)] with respect to the height for both kinds of objects. Inset in (c) illustrates the small convolution dependence on the object shape for a square and a circumference of the same height, due to the proximity of C.
treating the rectangular or elliptical case similar to a first approach.
which indicates that this feature is strongly correlated with the non-linear behavior of the wexp vs h curves in figure 4. This can be considered as a proof of the additional information extracted from the simulations. While equations (1–4) represent the evolution of the convolution error as the object height rises, the simulations show the complete profile and how it is followed by the tip during the scan. With that result we conclude that the shape dependence is generally minor or negligible with two clear exceptions determined by the aspect ratio: i) the very high objects where the contact is found at the linear side region; and ii) the elongated objects (with A < 1) with C below the tip round end, where the contact point is quite sensitive to the shape variations. For example, a recent publication based on a geometrical model obtains similar results by treating DNA chains in both ways, as a rectangular and as a circular motif [8]. This is consistent with our results since the aspect ratio of such chains is around A = 1 and belongs to the low working range (h < ½ rtip), but the study presented here suggests that for more elongated structures, as could be quantum nanostructures or local oxidation nano-lithography motifs in this working range [23–27], the object shape must be considered,
3.5. Tip influence on the convolution error
The effects of working with different rtip are shown in figure 5(a) (maintaining γ = 19.4°). The tip radius takes values from 10 nm (coinciding with the standard value) to 50 nm, which is the radius estimated for the metal coated tips usually employed for scanning near-field optical, electrostatic force or magnetic force microscopy [20–22]. In this figure the curvature of the depicted data is directly related to shape of the tip. The round side region of the tip leads to curved lines in the plot below h = rtip while the linear side region of the tip produces straight lines with identical slopes. This is because for higher objects the convolution is not related to rtip but to γ (which is constant in this simulation). In figure 5(b) the series are repeated for different γ and we observe the opposite trend. Since rtip is not changed during that simulation (rtip = 10 nm) the data are overlapped below 10 nm height. The differences appear above this value, situated at the linear side region of the tip, because higher slopes appear in the case of larger 6
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Figure 4. Results from rectangular (a) and elliptical (b) simulations representing the expected motif widths with respect to the height for
different aspect ratios. In the simulations the selected rtip is fixed to 10 nm while the γ is set to 19.4°. They are accompanied by the convolution error (c) and (d), respectively. In the inset of (c) it is illustrated that convolution effects are independent of the aspect ratio in the case of rectangular motifs. In the inset of (d) it is shown that aspect ratio dependence is common in round objects.
values of γ. The range of angles selected for these simulations (γ = 5°−50°) was chosen in accordance with typical values observed from SEM micrographs, around 19.4° in the front face of the tip (normal plane with respect to the cantilever), or close to 32° if the scan is done with the tip aligned in the perpendicular direction; see figures 5(c) and (d). In addition, these values could vary from 12° to more than 40° if tilts between the tip and the sample were considered. There are also special probes where γ could be smaller than 10°. Hence, we can conclude that all the representative angles are included in the simulation. The tip radius effects are observed even for the smallest objects losing importance with respect to γ effects as the height of the motifs increases and C moves into the linear side region of the tip. However, γ has no influence on the convolution error when h < rtip.
sample. In 2D this is a friendly method that illustrates the problem of the convolution in a clear way. In addition, the code employed is not time consuming since complex operations are not required, unlike, for instance, in methods based on the Legendre transform. AFM topography profiles have been simulated in order to be compared with the expressions describing convolution effects for rectangular and elliptical geometries. The simulations are performed using a sequential algorithm to reproduce the tip motion while preventing the scanning artifacts usually found in real images. From comparison among simulation series, the tip convolution effects are studied for different surface motifs and tips. Focusing on the influence of the motif properties such as size, shape and aspect ratio we demonstrate that: i) the convolution error depends mainly on γ for objects with h ≫ rtip, since in that cases the tip contacts the object in the linear side region, in contrast as h decreases the influence of the rtip increases and the influence of γ becomes negligible; ii) the shape influence is always more important in the case of larger objects, while for objects with sizes comparable with rtip the convolution effects are clearly around 2 rtip, and in general have minor shape dependence. Therefore,
4. Conclusions. In this work, we have presented a new algorithm for resolving the convolution effects in AFM imaging. It is a versatile method because it can be applied to any kind of tip and 7
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nanostructured devices for spintronics or opto-electronics, fabricated by means of lithography techniques onto epitaxial growth wafers, with almost rectangular profiles. Currently, improvements are being made to the code for its application to new algorithms programmed for tip and sample reconstruction, simulating tip erosion, and for obtaining the deconvolution of distorted images. Acknowledgments Financial support from the EU (ERC Advanced Grant SPINMOL), the Spanish MINECO (MAT2011-22785, cofinanced by FEDER) and the Generalidad Valenciana (Prometeo and ISIC-Nano programmes) are gratefully acknowledged. The corresponding author J C-F thanks the Generalitat Valenciana for his grant funded by the Vali+d program (APOSTD/2013/076). A F-A thanks the Spanish MINECO for her R y C grant. The authors are also grateful to Dr A Gaita for helpful discussions. Supporting information Computational aspects of the algorithm used are presented in detail accompanied by an example. The resolution of the simulations is also discussed and a code optimization route is pointed out. We have added a comparison table between geometrical expressions and simulation results. A simulation of an AFM profile from a SEM image is shown as a proof of the algorithm’s capabilities. References [1] Binning G, Quate C F and Gerber C 1986 Atomic force mircroscope Phys. Rev. Lett. 56 930 [2] Alen B, Fuster D, Munoz-Matutano G, Martinez-Pastor J, Gonzalez Y, Canet-Ferrer J and Gonzalez L 2008 Exciton gas compression and metallic condensation in a single semiconductor quantum wire Phys. Rev. Lett. 101 067405 [3] Martin-Sanchez J et al 2009 Single photon emission from sitecontrolled InAs quantum dots grown on GaAs(001) patterned substrates ACS Nano 3 1513 [4] Abargues R, Gradess R, Canet-Ferrer J, Abderrafi K and Valdes J L 2009 Scalable heterogeneous synthesis of metallic nanoparticles and aggregates with polyvinyl alcohol New J. Chem. 33 913 [5] Munoz-Matutano G et al 2011 Charge control in laterally coupled double quantum dots Phys. Rev. B 84 041308 [6] Wang W, Niu D X, Jiang C R and Yang X J 2013 The conductive properties of single DNA molecules studied by torsion tunneling atomic force microscopy Nanotechnology 25 025707 [7] Cadena J M, Misiego R, Smith K C, Avila A, Pipes B, Reifenberger R and Raman A 2013 Sub-surface imaging of carbon nanotube-polymer composites using dynamic AFM methods Nanotechnology 24 135706 [8] Winzer A T, Kraft C, Bhushan S, Stepanenko V and Tessmer I 2012 Correcting for AFM tip induced topography convolutions in protein–DNAsamples Ultramicroscopy 121 8 [9] Tawil N, Sacher E, Mandeville R and Meunier M 2013 Strategies for the immobilization of bacteriophages on gold
Figure 5. (a) Experimental width with respect to the height for square motifs obtained using tips of different radius. The tip-to-face angle is set to 19.4°. (b) Experimental width with respect to the height for square motifs for different tip-to-face angles (γ). The tip radius (rtip) is fixed to 10 nm. (c) and (d) SEM micrographs showing the tip-toface angle from the front and lateral views of an standard AFM tip (Point Probe Plus from NanosensorsTM). SEM images courtesy of NanosensorsTM.
we can conclude that in such a working range, the experimental error can be estimated without knowing the object shape in detail, since for objects as different as squares and circumferences comparable convolution errors are obtained; iii) the convolution error for rectangular objects is aspect ratio independent at a constant h; however, that is not the case for rounded motifs which dramatically increases for the most elongated structures (A ≪ 1). All these results point out the importance of the convolution error for the study of objects with sizes comparable to the tip radius (molecules, proteins, DNA, NPs, etc), as well as for larger objects that are found in samples such as 8
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