On the use of Artificial Neural Networks in Electrostatic Force Microscopy

We present different applications of Artificial Neural Networks [1] in Electrostatic Force Microscopy [2] [3]. First, a detailed analysis of the electrostatic interaction between an Electrostatic Force Microscope tip and a thin film is presented. By using Artificial Neural Networks, an equivalent semiinfinite sample has been described as an excellent approximation to characterize the whole thin film sample. A useful analytical expression has been also developed. In the case of very small thin film thicknesses (around 1nm), the electric response of the material differs even for very high dielectric constants. This effect can be very important for thin materials where the finite size effect can be described by an ultrahigh thin film dielectric constant.


Background
When electrostatic force microscopy (EFM) [1][2][3][4][5][6] 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 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 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 q i as the weights w i and the potential at the metallic surfaces V j as the expected output values y j (see 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 V 1 and V 2 . 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 (N 1 , N 2 , and N 3 ). 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 R 2 , the thin film thickness h 1 , the thin film dielectric constant E 1 , and the sample dielectric constant E 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, E = 10) as a function of the ANN iterations in the learning process N it . 2D equipotential distribution in the middle of the image corresponds to the results obtained by the standard LSM inset of Figure 1a). The input patterns x ij play the role of the Green functions G i (r ij ,σ i ), where r ij is the distance between the i-charge element and the j-surface point. σ i 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, σ i would represent the length of the line). Following this formalism, the electrostatic potential V j can be expressed as where N C is the number of charged elements q i 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 x ij represents the inputs to the output neuron y j , and w i 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 w i (i.e., q i ) from available x ij data. Previous knowledge can help us to decide the best values for N C (by the selection of the number of neurons) and G i (by the selection of input patterns).
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 q i 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 V 0 , where V 0 is the voltage applied to the tip. In Figure 2, we show the electrostatic potential distribution obtained for   We initialized q i = 0 and fixed the learning rate to 0.1. When N it = 100,000, the equipotential distribution looks identical than that obtained by the LSM. However, the highest error at the tip surface is 0.0076 V 0 (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 q i values obtained by the LSM are not adequate for several physical applications. As we can see in Table 1, the ANN q i absolute values are much smaller than those from LSM. This fact is not important when q i do not have any physical meaning. However, in our case, q i 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 q i 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.

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 E 1 of the thin film with thickness h 1 (see Figure 1b). The thin film will be placed over a semi-infinite dielectric substrate characterized by its dielectric constant E 2 . It is worth noting that in realistic systems where h 1 is very small, E 1 should be considered an effective [6] dielectric constant since several nanoscale effects can modify the response of the thin material Some examples of physical phenomena that could affect E 1 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.  Figure 4a, although the ANN is able to estimate E 1 when F vs. D and h 1 (excluding E 2 ) 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 E 1 values. 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 E 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 E 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 semi-infinite uniform dielectric sample [18]. The fact that different combinations of E 1 , E 2 , and h 1 can correspond to the same effective dielectric constant is in agreement with the results found in Figure 4a since including E 2 and h 1 as input values improves the ANN performance in the E 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.