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) [16] 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
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 leastsquares 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 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 icharge element and the jsurface point. σ_{i} represents the geometrical parameters that may be used to adequately calculate the electrostatic potential generated by the icharge 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 righthandside term in Equation 1 represents the electrostatic potential in the notation of a singleoutput 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).
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 halfangle θ, the radius R_{2}, the thin film thickness h_{1}, the thin film dielectric constant ϵ_{1}, and the sample dielectric constant ϵ_{2}.
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 tipsample 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 Rz = 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 different numbers of iterations N_{it} in the training process. 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 Table1, 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.
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 N_{it}. 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. tipsample 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 ϵ_{1} of the thin film with thickness h_{1} (see Figure 1b). The thin film will be placed over a semiinfinite dielectric substrate characterized by its dielectric constant ϵ_{2}. It is worth noting that in realistic systems where h_{1} is very small, ϵ_{1} should be considered an effective [6] dielectric constant since several nanoscale effects can modify the response of the thin material and change the ϵ_{1} value. Some examples of physical phenomena that could affect ϵ_{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. 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 ϵ_{2} and h_{1} 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 ϵ_{1}. We have considered three different inputs: the F vs. D curves, h_{1}, and ϵ_{2}. GICM F vs. D curves included in the training were calculated for ϵ_{1} = {5, 15, 25,…,105}, ϵ_{2} = {5, 15, 25,…,105}, and h_{1} = {1, 2,…,10}. The ANN has been tested with 100 F vs. D curves (not used during the training) with randomly selected ϵ_{1}ϵ_{2} (between 5 and 105), and h_{1} (between 1 and 10) values. As we can see in Figure 4a, although the ANN is able to estimate ϵ_{1} when F vs. D and h_{1} (excluding ϵ_{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 ϵ_{1} values.
Figure 3 . Scheme of the ANN used to estimateϵ_{1}. Inputs are (a) F vs. D curves; (b) F vs. D curves and h_{1} thickness; and (c) F vs. D curves, h_{1} thickness, and ϵ_{2}.
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, ϵ_{2}, and h_{1} 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 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.
Competing interests
The authors declare that they have no competing interests.
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.
Acknowledgments
This work was supported by TIN201019607 and BFU200908473. GMS acknowledges support from the Spanish Ramón y Cajal Program.
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