The use of artificial neural networks in electrostatic force microscopy
 Elena CastellanoHernández^{1},
 Francisco B Rodríguez^{1},
 Eduardo Serrano^{1},
 Pablo Varona^{1} and
 Gomez Monivas Sacha^{1}Email author
DOI: 10.1186/1556276X7250
© CastellanoHernandez et al.; licensee Springer 2012
Received: 6 January 2012
Accepted: 15 May 2012
Published: 15 May 2012
Abstract
The use of electrostatic force microscopy (EFM) to characterize and manipulate surfaces at the nanoscale usually faces the problem of dealing with systems where several parameters are not known. Artificial neural networks (ANNs) have demonstrated to be a very useful tool to tackle this type of problems. Here, we show that the use of ANNs allows us to quantitatively estimate magnitudes such as the dielectric constant of thin films. To improve thin film dielectric constant estimations in EFM, we first increase the accuracy of numerical simulations by replacing the standard minimization technique by a method based on ANN learning algorithms. Second, we use the improved numerical results to build a complete training set for a new ANN. The results obtained by the ANN suggest that accurate values for the thin film dielectric constant can only be estimated if the thin film thickness and sample dielectric constant are known.
PACS: 07.79.Lh; 07.05.Mh; 61.46.Fg.
Keywords
Electrostatic force microscopy Thin films Artificial neural networksBackground
When electrostatic force microscopy (EFM) [1–6] is working at the nanoscale, several interacting parameters have a strong influence in the signal [7]. Since the electrostatic force is a longrange interaction, macroscopic parameters such as the shape of the tip or the sample thickness can strongly modify the electrostatic interaction [8, 9]. However, in many experimental situations, it is not possible to obtain accurate values for all of these parameters, and it is very difficult to achieve quantitative experimental results [10]. Previous results [11] have shown that artificial neural networks (ANNs) [12] are a useful tool to characterize dielectric samples in highly undetermined EFM systems. Using known force vs. distance curves as inputs for their training, ANNs have been able to estimate the dielectric constant of a semiinfinite sample in a system where the tip radius and shape were not known.
In this paper, we demonstrate that ANNs can be used to improve the accuracy of numerical simulations in EFM and to quantitatively estimate the thin film dielectric constant from vertical force curves. First, we compare standard minimization and ANN techniques, demonstrating that ANN techniques provide a better control of the final result of the simulation. The improved numerical results are also used to create a complete training set of an ANN that estimates the dielectric constant of a thin film placed over a dielectric sample.
As it has been shown before [11], ANNs are able to estimate physical magnitudes in highly undetermined systems. In this article, we train an ANN with a complete thin film sample to study the necessity of knowing the geometry of the sample in the estimations of the thin film dielectric constant. Although the influence of the thin film thickness is much larger than that of the substrate dielectric constant, we demonstrate that accurate values of the thin film dielectric constant can only be obtained when both magnitudes are known.
Methods
Artificial neural network formalism for the calculation of electric fields
Coefficients obtained by the ANN and LSM algorithms for an EFM system
Coefficients  ANN  LSM 

q _{i}  
q1  −0.26  417.40 
q2  1.75  −2.31 
q3  −1.54  2.43 
q4  0.98  −0.82 
L _{i}  
L1  −0.032  −12,582.71 
L2  −0.15  2,194.47 
L3  −0.30  −157.42 
L4  0.90  5.51 
L5  0.41  0.015 
L6  0.13  0.48 
L7  1.32  −0.39 
L8  −7.31  26.26 
L9  10.16  −445.25 
L10  11.51  3,126.11 
L11  7.75  −9,298.52 
L12  4.42  9,311.48 
Results and discussion
Thin film dielectric constant estimation
The ANN can be used with realistic experimental curves without any previous treatment, which is one of the advantages of using this technique [11]. In this case, experimental curves with a high error could make the ANN give wrong ϵ_{1} estimations. This problem can be easily solved by training the ANN with a mixture of experimental and numerical F vs. D curves. This strategy would make the ANN more robust against experimental noise (by the use of experimental curves) and still effective on the ϵ_{1} estimations (by the use of a whole set of numerical curves).
Recently, a simple analytical expression has been developed that demonstrates that a sample composed by a thin film over a dielectric substrate gives the same response as that of a semiinfinite uniform dielectric sample [18]. The fact that different combinations of ϵ_{1}ϵ_{2}, and h_{1} can correspond to the same effective dielectric constant is in agreement with the results found in Figure 4a since including ϵ_{2} and h_{1} as input values improves the ANN performance in the ϵ_{1} estimations.
Conclusions
We have demonstrated that ANNs can strongly improve the efficiency of the characterization of samples by electrostatic force microscopy. First, we have demonstrated that the generalized image charge method can be modified to use a neural network minimization algorithm. Using this technique, we have increased the accuracy of the electrostatic force and capacitance calculations. By using electrostatic force simulations, we have been able to train an ANN to estimate the dielectric constant of thin films. The analysis of the results of the ANN suggests that the thin film dielectric constant can only be obtained when the thin film thickness and the dielectric nature of the sample are known. Note that the methods explained in this paper can be easily applied to experimental data by providing this kind of input to the ANN. If enough data are available, experimental curves can be used for the ANN training alone or together with theoretical curves.
Abbreviations
 ANNs:

Artificial neural networks
 EFM:

Electrostatic force microscopy
 F vs. D:

Force vs. tipsample distance
 GICM:

Generalized image charge method
 LSM:

Leastsquares minimization.
Declarations
Acknowledgments
This work was supported by TIN201019607 and BFU200908473. GMS acknowledges support from the Spanish Ramón y Cajal Program.
Authors’ Affiliations
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