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The use of artificial neural networks in electrostatic force microscopy


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.


When electrostatic force microscopy (EFM) [16] is working at the nanoscale, several interacting parameters have a strong influence in the signal [7]. Since the electrostatic force is a long-range 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 semi-infinite 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.


Artificial neural network formalism for the calculation of electric fields

To briefly illustrate how ANNs can be related to the problem of estimating unknown parameters in EFM setups, let us focus in the scheme shown in Figure 1a. Here, we have a set of metallic objects that are connected to a battery that provides a constant electric potential. The calculation of electric magnitudes such as the electrostatic potential or the force between these elements is, in general, very difficult, and only a few specific geometries can be analytically calculated [13]. To solve electrostatic problems with arbitrary geometries, an algorithm called the generalized image charge method [14, 15] (GICM) has been developed. The GICM replaces the surface charge density by a set of charges inside the metallic objects. The value, position, and number of charges are obtained after a standard least-squares minimization (LSM) routine for the electrostatic potential at the metallic surfaces. An alternative to the LSM is to use the ANN formalism by considering the value of the charges qi as the weights wi and the potential at the metallic surfaces Vj as the expected output values yj (see inset of Figure 1a). The input patterns xij play the role of the Green functions Gi(rijσi), where rij is the distance between the i-charge element and the j-surface point. σi represents the geometrical parameters that may be used to adequately calculate the electrostatic potential generated by the i-charge element (for example, if the element is a charged line, σi would represent the length of the line). Following this formalism, the electrostatic potential Vj can be expressed as

V j = i = 1 N C q i G i ( r ij , σ i ) = i = 1 N C w i x ij ,

where NC is the number of charged elements qi inside the tip. The most right-hand-side term in Equation 1 represents the electrostatic potential in the notation of a single-output ANN, where xij represents the inputs to the output neuron yj, and wi are the connection weights from the inputs (i = 1,…,) to this neuron (see Figure 1a). A neural network learns by example. The task of the learning algorithm of the network (i.e., the delta rule, backpropagation, etc. [12, 16]) is to determine wi (i.e., qi) from available xij data. Previous knowledge can help us to decide the best values for NC (by the selection of the number of neurons) and Gi (by the selection of input patterns).

Figure 1
figure 1

Two metallic objects with different applied voltages and electrostatic force microscope tip and sample. (a) Schematic representation of a system of two metallic objects with different applied voltages V1 and V2. Its equivalent ANN is also shown as an inset. (b) Scheme of an electrostatic force microscope tip and sample. The tip surface has been divided in three regions with a finite number of points (N1, N2, and N3). The tip and sample are characterized by eight parameters: the tip sample distance D, the tip apex radius R, the tip length L, the cone half-angle θ, the radius R2, the thin film thickness h1, the thin film dielectric constant ϵ1, and the sample dielectric constant ϵ2.

To compare both minimization techniques (LSM and ANN), we have simulated the EFM shown in Figure 1b. To illustrate the advantages of using the ANN minimization routine, we have calculated the qi coefficients for the tip-sample distance described in Figure 1b with both the ANN and standard LSM routines. We have used the winGICM software v1.1 [17] which also uses ANNs to estimate the number of punctual charges and the number of segments (4 and 12, respectively). The LSM lowest error (located at x = 0.9876 R z = 0.8896 R) at the tip surface was 0.0019 V0, where V0 is the voltage applied to the tip. In Figure 2, we show the electrostatic potential distribution obtained for different numbers of iterations Nit in the training process. We initialized qi = 0 and fixed the learning rate to 0.1. When Nit = 100,000, the equipotential distribution looks identical than that obtained by the LSM. However, the highest error at the tip surface is 0.0076 V0 (four times larger than that from LSM). At this point, it seems that LSM is a better minimization technique since it gives a lower error and does not use any iterative process. However, the qi values obtained by the LSM are not adequate for several physical applications. As we can see in Table1, the ANN qi absolute values are much smaller than those from LSM. This fact is not important when qi do not have any physical meaning. However, in our case, qi correspond to the charge inside the tip and must be used to calculate electric magnitudes like the capacitance or the electrostatic force F (used in the following section). By using ANN qi values, these magnitudes can be calculated with improved accuracy since the low values of the charges strongly reduce the numerical noise. In conclusion, the ANN minimization allows the user to choose the balance between numerical accuracy and physical meaning of the simulations to adapt them to the necessity of the problem.

Figure 2
figure 2

Potential distribution around an EFM tip. 3D representation of the potential distribution around an EFM tip (L = 10 R, θ = 20°) over a dielectric sample (D = 0.1 R, ϵ = 10) as a function of the ANN iterations in the learning process Nit. 2D equipotential distribution in the middle of the image corresponds to the results obtained by the standard LSM.

Table 1 Coefficients obtained by the ANN and LSM algorithms for an EFM system

Results and discussion

Thin film dielectric constant estimation

In this section, we are going to use the GICM force vs. tip-sample distance (F vs. D) curves for an EFM tip over a thin film to train an ANN that will be able to estimate the dielectric constant ϵ1 of the thin film with thickness h1 (see Figure 1b). The thin film will be placed over a semi-infinite dielectric substrate characterized by its dielectric constant ϵ2. It is worth noting that in realistic systems where h1 is very small, ϵ1 should be considered an effective [6] dielectric constant since several nanoscale effects can modify the response of the thin material and change the ϵ1 value. Some examples of physical phenomena that could affect ϵ1 are the roughness of the thin film surface, the presence of a water layer over the thin film [5], or the finite amount of free charge due to the small size of the film. The ANN architecture is shown in Figure 3. We used a multilayer perceptron with sigmoid activation functions. The input layer is composed of 20 neurons for the F vs. D curves that are calculated for D = {2.5, 5,…,50} nm. Additional neurons are added in the cases where ϵ2 and h1 are included as input values. We used two hidden layers composed of 10 neurons with no bias applied. The output layer contains a single neuron which provides the estimate values for ϵ1. We have considered three different inputs: the F vs. D curves, h1, and ϵ2. GICM F vs. D curves included in the training were calculated for ϵ1 = {5, 15, 25,…,105}, ϵ2 = {5, 15, 25,…,105}, and h1 = {1, 2,…,10}. The ANN has been tested with 100 F vs. D curves (not used during the training) with randomly selected ϵ1ϵ2 (between 5 and 105), and h1 (between 1 and 10) values. As we can see in Figure 4a, although the ANN is able to estimate ϵ1 when F vs. D and h1 (excluding ϵ2) are used as input parameters, it gives the best results when all the input parameters are included. In Figure 4b, we show the error obtained by the ANN when all the inputs are included. The error is always smaller than 9% for all the ϵ1 values.

Figure 3
figure 3

Scheme of the ANN used to estimate ϵ 1 . Inputs are (a) F vs. D curves; (b) F vs. D curves and h1 thickness; and (c) F vs. D curves, h1 thickness, and ϵ2.

Figure 4
figure 4

Test error vs. number of iterations and ANN estimation. (a) Test error vs. number of iterations in the ANN learning process for different ANN inputs. (b) ANN estimation of the thin film dielectric constant for the test set for the case where F vs. D, ϵ2, and h1 are included as inputs.

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 ϵ1 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 ϵ1 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 semi-infinite uniform dielectric sample [18]. The fact that different combinations of ϵ1ϵ2, and h1 can correspond to the same effective dielectric constant is in agreement with the results found in Figure 4a since including ϵ2 and h1 as input values improves the ANN performance in the ϵ1 estimations.


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.



Artificial neural networks


Electrostatic force microscopy

F vs. D:

Force vs. tip-sample distance


Generalized image charge method


Least-squares minimization.


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This work was supported by TIN2010-19607 and BFU2009-08473. GMS acknowledges support from the Spanish Ramón y Cajal Program.

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Correspondence to Gomez Monivas Sacha.

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Authors' contributions

ECH carried out the numerical simulations. FBR and ES participated in the design of the artificial neural networks and mathematical formalism. PV participated in the design of the artificial neural networks and helped draft the manuscript. GMS conceived the study, participated in its design and coordination, and drafted the manuscript. All authors read and approved the final manuscript.

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Castellano-Hernández, E., Rodríguez, F.B., Serrano, E. et al. The use of artificial neural networks in electrostatic force microscopy. Nanoscale Res Lett 7, 250 (2012).

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  • Electrostatic force microscopy
  • Thin films
  • Artificial neural networks