Neural network-based recognition of multiple nanobubbles in graphene
Subin Kim Nojoon Myoung Seunghyun Jun Ara Go



1]Subin Kimcor1
2,3]Nojoon Myoungcor1
2,3]Seunghyun Jun


[1]
	organization=Department of Physics, Chonnam National University,
	city=Gwangju,
	postcode=61186,
	country=Republic of Korea

[2]
	organization=Department of Physics Education, Chosun University,
	city=Gwangju,
	postcode=61452,
	country=Republic of Korea
[3]
	organization=Institute of Well-Aging Medicare & Chosun University LAMP Center, Chosun University,
	city=Gwangju,
	postcode=61452,
	country=Republic of Korea

[cor1]These authors contributed equally to this work.
[cor2]Corresponding author: arago@jnu.ac.kr



We present a machine learning method for swiftly identifying nanobubbles in graphene, crucial for understanding electronic transport in graphene-based devices. Nanobubbles cause local strain, impacting graphene's transport properties. Traditional techniques like optical imaging are slow and limited for characterizing multiple nanobubbles. Our approach uses neural networks to analyze graphene's density of states, enabling rapid detection and characterization of nanobubbles from electronic transport data. This method swiftly enumerates nanobubbles and surpasses conventional imaging methods in efficiency and speed. It enhances quality assessment and optimization of graphene nanodevices, marking a significant advance in condensed matter physics and materials science. Our technique offers an efficient solution for probing the interplay between nanoscale features and electronic properties in two-dimensional materials.



 
Machine Learning, Neural Networks, Graphene, Nanobubble, Strain Effects

Neural network-based recognition of multiple nanobubbles in graphene
    Ara Gocor2
    April 29, 2024
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§ INTRODUCTION



Structural imperfections, including atomic defects and ripples <cit.>, represent diverse factors contributing to disorder in graphene. The formation of graphene nanobubbles, characterized by the elastic deformation of crystal structures, is an intrinsic phenomenon observed during sample transfer onto substrates <cit.>. Extensive research has explored the impact of nanobubbles on the reduction of carrier mobility <cit.>. As a result, researchers strive to avoid regions prone to nanobubble formation when fabricating graphene devices, aiming to mitigate potential, unidentified influences. Recognizing nanobubbles in a graphene sample is thus pivotal for fabricating high-quality devices <cit.>. Nanobubble location being a local feature necessitates the use of scanning probe techniques that focus on processing local information.

Substantial research has focused on the impact of elastic strain induced by nanobubbles, which create a characteristic, non-uniform profile of pseudomagnetic fields (PMFs) <cit.>. These PMFs play a critical role in the transport of Dirac fermions through graphene, facilitating the emergence of phenomena such as zero-field Landau levels <cit.>, valley-selective transport behaviors <cit.>, and strain-induced quantum interference <cit.>. Moreover, nanobubbles can host localized states that exhibit conductance resonances, indicating that the presence of nanobubbles significantly influences the detailed structure of the density of states (DOS) <cit.>. Although the DOS provides comprehensive information about the system, directly extracting specific physical details about nanobubbles from the DOS is challenging. This difficulty arises from the simultaneous presence of both local and global features within the aggregated data.



This study introduces a novel strategy for detecting and identifying nanobubbles in monolayer graphene by leveraging machine learning (ML) techniques to navigate the complexities inherent in processing data from electrical measurements. Building on prior research in nanobubble identification <cit.> and the advancement of a graphene nanobubble sensor using quantum interferometry <cit.>, our approach emphasizes the identification of multiple nanobubbles. It aims to develop an accurate predictive model that directly correlates the geometric characteristics of nanobubbles with the DOS in graphene samples  <cit.>. This direct correlation enables the precise assignment of measurements to individual nanobubbles within a sample, even in scenarios involving multiple nanobubbles. Given that nanobubbles can significantly alter the electrical properties of graphene, particularly in cases of degenerate defect states, the exploration of an ML approach to recognize multiple nanobubbles is deemed essential. Our investigation into the effects of nanobubble multiplicity on the ML recognition algorithm reveals an effective technique for accurately identifying individual nanobubbles. The proposed ML-based method for nanobubble recognition is poised to enhance the pure-electrical evaluation of graphene and other 2D materials, potentially streamlining the mass production of these materials for device fabrication.





§ METHODS



 §.§ Generating Raw Nanobubble Data

We approximated the nanobubble using a Gaussian distribution located at r⃗_0 as

    h( r⃗ ) = h_0e^-(r⃗-r⃗ _0)^2/2σ^2,

where is the maximum height of the vertical deformation and is the standard deviation corresponding to the width of the nanobubble.
This nanobubble introduces a PMF, which can be described by the same mechanism as electron confinement in real magnetic fields <cit.>. The C_3v symmetric PMF is described by

    B⃗_ps( r⃗ ) = νħβe a_0h_0^2σ^6 r^3 e^-(r-r_0)^2/2σ^2sin 3θẑ,

where β=3.37, a_0=0.146 nm is the nearest carbon-carbon bonding length, and ν=±1 for different valleys of graphene, respectively.
We set its maximum strength B̃_max at r-r_0=√(3/2)σ as the representative quantity of nanobubbles, which is formulated as 

    B̃_max(σ, h_0) ≡h_0^2σ^3.


In the presence of nanobubbles, DOS spectra exhibit peaks at specific energies, resulting from PMF-induced localized states within nanobubbles. As increases, a greater number of localized states emerge, and their corresponding DOS peaks tend to shift toward lower energy. This correlation between DOS and is a primary focus of this work, analyzed through a supervised machine learning approach. As a theoretical study, we obtained the DOS spectra through tight-binding calculations of a graphene sheet with nanobubbles, employing kwant codes <cit.>. While the number of bubbles could be further extended, our study focuses on N_max = 3 case, indicating that the maximum number of bubbles considered is 3. This consideration is justified as the standard preprocessing involving heating and cooling of the graphene leads to the coalescence of adjacent nanobubbles, thereby limiting the total number of bubbles within a single sheet.



 §.§ Preprocessing Nanobubble Data

The raw DOS spectra consist of 20,001 data points spanning an energy range from -0.1 to 0.1 eV; subsequently, we performed interpolation on the DOS to standardize the energy grid, resulting in a range from -0.099 to 0.099 eV with 10,001 data points  <cit.>. Each DOS spectrum, D(E)_int, is characterized by two important parameters, σ and h, corresponding to the geometrical factors of nanobubbles, as previously mentioned. We considered σ ranging from 20 to 30 a and h from 1 to 25a, with a step of a, where a = √(3)a_0 = 0.246 nm is the lattice constant of graphene. Consequently, the total dataset comprises 275 samples (= 11 × 25), marking a significant expansion compared to previous research <cit.>, which utilized 75 samples.

In this work, we generate a multibubble dataset by combining data from the single-bubble dataset. The multibubble DOS is equivalent to the average of the corresponding single-bubble DOS data, within numerical accuracy, provided that the bubbles are sufficiently distant from each other. If the bubbles are spatially close to each other, they merge into a single bubble for stabilization. Therefore, for our analysis, we can simply aggregate single-bubble DOS data to generate a multibubble dataset.

For N_bubble = 1, 2, we include all 275 single-bubble samples and all possible combinations between them, totaling 275 × 274 / 2 = 37,675. For N_bubble = 3, the number of available combinations exceeds our computational capacity, so we randomly select 37,675 samples from the three-bubble combinations. Additionally, we include a dataset where N_bubble = 0, as a reference to determine whether a given DOS contains nanobubbles or not.

We introduce a unified labeling scheme for varying numbers of nanobubbles, as illustrated in Table <ref>. In practice, the accidental occurrence of two nanobubbles of identical size on a single graphene sheet is extremely rare. Therefore, we treat a set of nanobubbles with the same index as a single nanobubble.

We divided the dataset into training and testing sets with a train-test ratio of 7:3 for each N_bubble case to mitigate the risk of an imbalanced dataset, which could adversely affect the model's performance. To prevent overfitting, we employed data augmentation techniques, such as min-max scaling and the addition of appropriate white noise. This ensured that each N_bubble occupies the same partition in the training and test set, resulting in a total sample ratio of approximately 7:3 (527,100 samples in the training set and 226,400 samples in the test set). The DOS spectra, D̃(E), serving as features for the supervised machine learning, are defined as follows  <cit.>:

    D̃(E) = D(E) - min(D(E))/max(D(E)) - min(D(E)) + ηξ(E)

where η denotes the amplitude of noise, and ξ(E) represents a random variate sampled from a normal distribution 𝒩(0,1). By adjusting η, we introduce controlled measurement noise into the dataset.




 §.§ Deep learning model parameters



Figure <ref> illustrates the architecture of our deep learning model. As outlined in Table <ref>, we utilize a discrete label and 2N_bubble continuous labels to represent the number of nanobubbles and the geometrical shapes of the nanobubbles, respectively.
The model serves dual functions: it operates as a classifier using the discrete label and as a regressor for the continuous labels.

The deep learning model is developed using TensorFlow packages and consists of three primary components: a 1D Convolutional Neural Network (CNN), 1D Global Average Pooling (GAP), and a Multi-Layer Perceptron (MLP)<cit.>. Linear and Rectified Linear Unit (ReLU) activation functions are employed within our model. The CNN layers are organized into blocks as follows: [1D CNN layer - Batch Normalization (BN) <cit.> - ReLU], with filter sizes of [8 – 16 – 32 – 64], kernel sizes of [15 – 45 – 75 – 105], and all strides set to 10. This arrangement of CNN layers facilitates the extraction of the feature tensor from the input data. Subsequently, a 1D GAP layer is used to flatten the feature tensor, preparing it for the MLP component, which is composed of blocks structured as [MLP layer - BN - Linear]. The unit sizes in the MLP layers are determined to be [32 – 16 – 8 – (2+1)], where denotes the maximum number of nanobubbles, set at 3 in this instance.

To facilitate effective training, we employ the Adaptive Moment Estimation (Adam) optimizer <cit.> along with a Mean Squared Error (MSE) loss function. For validation, 20% of the training set is randomly selected and shuffled at the beginning of each epoch. Utilizing callback functions from TensorFlow, we monitor the validation loss value for each epoch and optimize our model by aiming to achieve the minimum validation loss.









§ RESULTS AND DISCUSSION


Figure <ref> showcases the performance results for both regression and classification. The regression analysis evaluates the precision of predicted values for , , and of individual nanobubbles, compared to their true values, denoted by superscripts (i), (ii), and (iii) respectively.

In Figure <ref> panels (a-c), the regression outcomes for , , and are presented. Overall, the mean values and standard deviations closely align with the diagonal lines, illustrating the model's accurate prediction of nanobubble parameters. Additionally, the precision of prediction performance is further assessed by calculating the mean absolute errors (MAE) for the nanobubble parameters , , and for individual bubbles,

    MAE = 1N∑_i=1^N |x_i - x̂_i|,

where N denotes the total number of datasets, x_i represents the true value, and x̂_i is the predicted value. As demonstrated in Figure <ref>(a-c), the prediction errors for the nanobubble parameters when N_bubble=1 are found to be minimal, attributed to the uniformity of labels (i.e., [1, A, A, A]). Conversely, for N_bubble>1, the labels within the dataset are not uniform but distinct, leading to larger MAEs.


In addition to and , we also analyze the performance of our deep learning model based on the parameter . In the classification of the DOS spectra, we find that serves as the representative parameter, rather than  and . Given that the electronic states of strained graphene are determined by characteristic PMF profiles, it is logical to predict that graphene nanobubbles with similar values will produce analogous DOS spectra, despite variations in and values.

Figure <ref>(c) illustrates the prediction accuracy of such a representative physical quantity, , which we calculated using predicted and values. The results show a good agreement between the predicted and true values, surpassing the prediction accuracy for and . Additionally, it is observed that the prediction accuracy slightly declines for larger values. Nanobubbles corresponding to these larger cases are generally characterized by a high aspect ratio of and , that is, smaller and larger . From a theoretical perspective, such high nanobubbles are likely to result in significant errors, as nanobubbles with small cannot be accurately modeled by elastic theory under the assumption of a continuous graphene sheet. Therefore, we focus on moderate or small cases, where the accuracy of our machine learning recognition remains notably high.



The classification performance of is evaluated using a confusion matrix. As displayed in Figure <ref>(d), the confusion matrix from the classification results clearly demonstrates that our model effectively recognizes the number of nanobubbles present in a graphene sample by analyzing given DOS spectra. For this confusion matrix, we employ the conventional evaluation metrics as follows  <cit.>:

    A = Σ_iM_iiΣ_iΣ_jM_ij, 
    P_i = M_iiΣ_jM_ji, 
    R_i = M_iiΣ_jM_ij,

where A represents accuracy, P_i denotes precision, and R_i signifies recall for each = i (0 ≤ i ≤ 3). Precision measures the proportion of retrieved values that are relevant, while recall quantifies the proportion of relevant values that are successfully retrieved. Table <ref> showcases the computed evaluation metrics for our model. Our model demonstrates superior performance with smaller N_bubble, yet remains effective even with larger numbers of bubbles. Furthermore, we can systematically improve the performance by providing additional training samples for a larger number of bubbles.






In practical experimental situations, we expect the DOS signals to contain random noise from measurements. Therefore, the model's performance in the presence of noise is a key component for applying the model to real-world samples.
We examine the effects of noise on our model's prediction and classification performance. We introduce white noise by adding a uniform random distribution to the DOS spectra, with an amplitude η. Figure <ref> demonstrates that both the regression and classification performance deteriorate as η increases. The analysis shows that the lower the noise level η, the better the model performs, as expected. While solely based on this noise level dependency, it might seem that training a model with higher noise strength is not beneficial, cross-noise level testing reveals different conclusions.




After training a model with the η=0.05, we apply the model to the DOS dataset with different noise levels. To visualize the recognition performance for different noise levels, we choose specific samples h_0=13 and σ=20,25,30, but the overall tendency is independent of the choice of the samples. The regression and classification evaluations are carried out with the same dataset by varying the noise level 0≤η≤ 0.1. As shown in Figure <ref>(a), it is evident that our model exhibits good performance when the noise amplitudes for the evaluation are equal to or less than the trained model's η. Interestingly, the prediction performance does not deteriorate much even if the noise level is slightly higher than the trained η=0.05.

Similarly, as shown in Figure <ref>(b), the classification performance of our model exhibits high accuracy when the noise levels of the evaluation datasets do not exceed that of the trained dataset. A notable aspect of the classification results is the optimization of model performance at noise levels in the evaluation dataset that match the trained dataset's noise level. This observation suggests that our deep learning model is adept at effectively detecting nanobubbles in when operating within an appropriate noise amplitude, η, consistent with the model's training conditions. Consequently, it is crucial to choose an appropriate noise amplitude during training, considering an optimal range of η to ensure high accuracy.





§ RESULTS OF = 5 MODEL























For our next step, we aim to generalize our recognition model to handle larger cases beyond 3. To achieve this, we apply the same methodology to a dataset with = 5, using the same algorithm as described in the previous section. With =5, the number of possible combinations from the dataset exceeds the predefined maximal combination _275C_2. Consequently, we augment the dataset to ensure that the maximum number of combinations is effectively doubled for N_bubble=4,5, compensating for the relatively small portion of all possible combinations included in the training set.





With the augmented dataset, we assess the regression performance — finding an accuracy of 0.71. While the accuracy of our model for =5 may not be significantly higher compared to the =3 scenarios, this moderate level of accuracy could still be meaningful for generalization purposes. The classification performance for =5 with the augmented dataset is illustrated in Figure <ref>. Our model performs effectively for a larger number of cases, yet exhibits lower performance for scenarios with fewer cases. The decrease in classification performance for fewer cases is attributed to the limited variety within the dataset for = 1 and = 2, even after dataset augmentation. Providing sufficiently large datasets for training could further enhance our model’s ability to effectively recognize . In essence, our deep learning approach remains valid when is less than or equal to the of the training dataset.





§ SUMMARY

We have explored the recognition of multiple nanobubbles in graphene using DOS spectra through neural network techniques, aiming to expand upon the previously demonstrated single-bubble recognition <cit.>. To facilitate the recognition of multiple nanobubbles, we devised a labeling rule based on the premise that multiple nanobubbles with identical indices can be treated as a single bubble, given their low likelihood of simultaneous occurrence in practical scenarios.

Our study demonstrates commendable performance in both regression and classification for a model with = 3. Notably, the trained model shows high proficiency in distinguishing whether a given DOS spectrum represents graphene with or without nanobubbles. Moreover, the model maintains effective recognition capabilities for nanobubbles when the DOS spectra are subjected to noise levels with an amplitude that is equal to or less than the model's trained noise level.

We also endeavored to extend our model to accommodate larger values to generalize its application. Although the model's performance diminishes slightly in comparison to the =3 scenario, achieving a moderate level of accuracy, we identified that the diminished performance for larger values could be mitigated by increasing the dataset size, thus enabling significant enhancements in nanobubble recognition accuracy. Consequently, by incorporating additional training datasets into the ML model, our approach can be adapted to manage larger values effectively.



§ ACKNOWLEDGMENTS

SK and AG are supported by the National Research Foundation under grant numbers NRF-2021R1C1C1010429 and NRF-2023M3K5A1094813. NM and SJ acknowledge the support of the National Research Foundation through the Korean governments NRF-2022R1F1A1065365 (MIST) and RS-2023-00285353 (MOE).



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