Predicting the transport properties of silicene nanoribbons using a neural network

T. Župančić, I. Stresec, and M. Poljak* Computational Nanoelectronics Group

Micro and Nano Electronics Laboratory, Faculty of Electrical Engineering and Computing University of Zagreb, Zagreb, HR 10000, Croatia

*Corresponding author: mirko.poljak@fer.hr

Abstract—Atomistic quantum transport simulations are used to generate the electronic and transport properties of 10,000 realistic silicene nanoribbons (SiNRs) with edge defects. This ensemble of 20 nm-long and 2.1 nm-wide SiNRs is divided into the training and inference set for the artificial neural network (ANN) employed for the prediction of edge-defect-limited carrier mobility from the known values of bandgap and nanoribbon conductance. We find that an optimized ANN with 3 hidden layers can predict SiNR mobility values and variability histograms with acceptable accuracy, thus providing a useful supplement to atomistic quantum transport simulations that take several hours or days for large device ensemble sizes.

Keywords—quantum transport; silicene nanoribbon; mobility; bandgap; neural network; TensorFlow; Keras

I. INTRODUCTION The traditional approach of semiconductor research

was in large part based on laboratory experiments, leading to an expensive development process, primarily because of the high cost of clean room maintenance. With the development of computer-aided design (CAD) software, the research and development process today includes material and device simulation, which give researchers a possibility to gain deeper physical insights, and to design and optimize materials and devices without the need for costly fabrication, especially at the nanoscale [1], [2]. This paradigm has greatly decreased research costs, as well as reduced the total development time at different parts of the semiconductor electronics industry.

Over the last decade, quantum transport formalism for nanodevices has gained a tremendous interest due to its ability to capture atomistic resolution of the material used in the device, and due to its ability to incorporate the effects of real-world contacts. Among these, the non-equilibrium Green's function (NEGF) formalism, in its matrix formulation, has become the de-facto standard due to its relative simplicity and versatility [3], [4]. Atomistic NEGF presents a bottom-up approach in the device research community, and it is able to tackle different problems from classical FETs [5], nanotube and nanoribbon devices [6], to molecular sensors. However, computational burden increases significantly when realistically-sized nanostructures (containing several thousands of atoms) are simulated within the NEGF approach [7]. This problem becomes even more pronounced when statistical simulations of a large number

of devices are needed for proper assessment of device variability.

Recently, artificial neural networks (ANNs) have been proven useful in device analysis in terms of statistical variability of e.g. Si junctionless nanowire transistors [8]. The ANNs are collections of nodes called neurons that are interconnected via synapses and designed to mimic a (very small part of a) biological brain. The unique design of ANNs has proven to be successful in solving various problems, being especially effective in pattern recognition, classification and image processing [9], [10]. Recent advancements in ANN design suggest a great potential of their large-scale application, both in consumer software business [11]–[13], but also directly in scientific research in various fields [14]–[16].

In this paper, we present a machine learning approach for predicting a chosen transport parameter from the known several electronic and transport parameters of realistic silicene nanoribbons (SiNRs) with edge defects. The ANN is trained on the data obtained from 10,000 atomistic quantum transport simulations of randomly-defective SiNRs. The analysis provided herein gives a detailed assessment of ANN prediction performance (i.e. comparing predicted vs. true values). We show that a relatively simple ANN with 3 hidden layers can predict averaged values and variability properties of nanoscale SiNRs with a reasonable accuracy.

II. MODELING AND SIMULATION APPROACH

A. Quantum Transport For the SiNRs illustrated in Fig. 1, we employ an

atomistic tight-binding (TB) Hamiltonian that accounts for the nearest-neighbor interactions

( )† † †

,

,i i i i j j ii i j

H c c t c c c cε= + +∑ ∑ (1)

where εi is the on-site energy, and t is the hopping parameter between neighboring Si atoms. Using t = 1.03 eV for internal Si-Si bonds and a larger t' = 1.15 eV for edge bonds reproduces the electronic structure of hydrogen-terminated SiNRs obtained by ab initio simulations [17]. Edge defects are implemented by randomly removing single atoms from the top and bottom edges of the nanoribbon shown in Fig. 1. Edge atoms are removed in a given percentage (Pdef) which equals 10% in this work. The total Hamiltonian is then adjusted by

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setting the interaction parameters of defects (i.e. missing atoms) to zero.

The Schrödinger's equation with open boundary conditions is solved directly by employing the NEGF formalism [4], [18]. Retarded Green's function of the device is calculated with

( ) 1 2( ) 0 ( ) ( ) ,R R RG E E i I H E E+ = + − − Σ − Σ

where H is the device Hamiltonian defined in (1), and matrices designate the retarded contact selfaccount for the coupling of the device to the two contacts (1 and 2, or source and drain). The retarded and advanced Green's functions of the device are then used to find the transmission function, extract the bandgap (the ON- and OFF-state conductance (GONratio (R = GON/GOFF) [19], and edgemobility (µED), according to the following approach. Total resistance of a SiNR can be expressed as where RB is the length-independent ballistic resistance, and RD is the length-dependent diffusive resistance. Sithe latter can be found from RD = L/(qµ[20]–[22], we can extract µED using

( )1

EDD TOT B

Lqn R R

µ =−

for a given line charge density (n1D).

Figure 2 reports the transmission functN = 10,000 defective 2.1 nm-wide and 20and also contains the transmission of an ideal (defectSiNR and an averaged transmission. We note the decrease of transmission, leading to the possibility of extraction of RD and µED, and the increase of the transport energy gap in

Fig. 3. Model of the artificial neural network used for the depicted by Wi and bi, respectively, fi is the activation function

Fig. 1. Top and side view of a monolayer silicene The side view shows that silicene is not flat but periodically buckled. setting the interaction parameters of defects (i.e. missing

The Schrödinger's equation with open boundary conditions is solved directly by employing the NEGF

. Retarded Green's function of the

1

1 2( ) 0 ( ) ( ) ,R R RG E E i I H E E−

= + − − Σ − Σ (2)

is the device Hamiltonian defined in (1), and Σ matrices designate the retarded contact self-energies that account for the coupling of the device to the two contacts

and drain). The retarded and advanced Green's functions of the device are then used to find the transmission function, extract the bandgap (EG), calculate

ON, GOFF) and their , and edge-defect limited

, according to the following approach. Total SiNR can be expressed as RTOT = RB + RD,

independent ballistic resistance, dependent diffusive resistance. Since

qµEDn1D) = RTOT − RB

, (3)

2 reports the transmission functions for and 20 nm-long SiNRs,

and also contains the transmission of an ideal (defect-less) SiNR and an averaged transmission. We note the decrease of transmission, leading to the possibility of extraction of

d the increase of the transport energy gap in

defective SiNRs. The data reported in Fig.training set for the ANN described in the following Subsection.

B. Neural Network In order to obtain meaningful data about the electronic

and transport properties of realistic SiNRs with edge defects we need to simulate hundreds or thousands of nanoribbons to study their average properties and variability of their parameters caused by rDue to numerical complexity of atomistic NEGF simulations, these simulations exhibit wallorder of ~hours or ~days, even for 202.1 nm-wide SiNRs that contain only 720 atomsTherefore, this work aims to exploreANNs for the prediction of certain SiNRs properties such as µED, thus alleviating the need for timequantum transport simulations.

The feed forward regression Python using Keras, a highTensorFlow, and Pandas and NumPy libraries manipulation. The neural network, illustrated in Fig.fully connected and has three inputs: R = GON/GOFF; three hidden layers neurons, respectively; and one output: electron mobility (µED). Hence, the 3predict µED values from the known triplets ofR = GON/GOFF. The activation function used in

Fig. 2. Transmission for 10,000 defective SiNRs with 10% of edge defects. For comparison, averaged transmission and the transmission of an ideal SiNR are also inserted. W = 2.1

f the artificial neural network used for the prediction of edge-defect-limited electron mobility. The weight and the bias functions are

activation function (ReLU), while the output of each layer is annotated by

monolayer silicene armchair nanoribbon.The side view shows that silicene is not flat but periodically buckled.

defective SiNRs. The data reported in Fig. 2 serve as the NN described in the following

In order to obtain meaningful data about the electronic and transport properties of realistic SiNRs with edge defects we need to simulate hundreds or thousands of nanoribbons to study their average properties and variability of their parameters caused by random defects. Due to numerical complexity of atomistic NEGF simulations, these simulations exhibit wall-times on the

ys, even for 20 nm-long and wide SiNRs that contain only 720 atoms [23].

aims to explore the applicability of NNs for the prediction of certain SiNRs properties such

, thus alleviating the need for time-consuming

regression NN is implemented in a high-level NN API built on

and Pandas and NumPy libraries for data network, illustrated in Fig. 3, is

fully connected and has three inputs: EG, GON and ; three hidden layers of 32, 64, and 32

and one output: edge-defect-limited Hence, the 3-layer NN is used to

values from the known triplets of EG, GON and The activation function used in all hidden

Transmission for 10,000 defective SiNRs with 10% of edge

defects. For comparison, averaged transmission and the transmission of 2.1 nm, L = 20 nm.

. The weight and the bias functions are (ReLU), while the output of each layer is annotated by ai.

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layers is the rectified linear unit function (ReLU) and back propagation is used for training. For the optimization of the network we used the ADADELTA method [24] with a starting learning rate of 0.1 and a decay factor (ρ) of 0.95. All other parameters of the network were left to their default values [25]. This NN-based approach has two goals: (1) for a given SiNR size, demonstrate the possibility of mobility prediction from the large existing set of SiNR data for EG, GON and R = GON/GOFF, and (2) demonstrate the same possibility for another SiNR size using the same NN found in the previous step without additional training or any parameter adjustment.

III. RESULTS AND DISCUSSIONS

A. NN training and mobility prediction for W = 2.1 nm The data used in training of the network consists of

9524 sets of EG, GON, R, and µED values from a Matlab simulation of 2.1 nm-wide and 20 nm-long SiNRs with 10% edge defects. The original 10,000 sets simulated in Matlab is filtered to remove physically unrealistic mobility values. Statistical parameters for this set are summarized in Table I, reporting the mean and median value, and standard deviation for each SiNR parameter. The data is then split as follows: 85% for learning (of which 85% was used for training and 15% for training validation) and 15% for testing.

Optimization with the ADADELTA method demonstrated greater training success compared to Adam, RMSProp and SGD optimizers. These results can be attributed to the adaptive learning rate approach which seems to be the best fit for our data set, presumably due to its comparatively unstable parameters. In general, stated optimizers can serve as good alternatives for training, with small performance differences which in our case suggest the possibility of them outperforming ADADELTA on similar data sets.

The NN architecture used in this work can be considered relatively simple. With only three hidden layers, the network takes significantly less time to train compared to more complex networks. Furthermore, a smaller size of the network did not show significant performance decrease compared to larger ones, rendering more complex NNs as non-cost-effective.

Mean absolute error (MAE) and mean square error (MSE) were used as training metrics, with MSE as the loss function. As mentioned earlier, three inputs (EG, GON, R) are used in order to predict the mobility (µED). Fig. 4 shows the MAE of training and validation sets through 400 training steps (epochs). MAE decreases quickly in the first few epochs, later stabilizing just above 20 cm2/Vs. As expected, the validation MAE is slightly higher than that of the training, indicating a successful learning process.

The results of training can be seen in Fig. 5, which shows the distribution of prediction errors (i.e. differences between predicted and true mobility values). The prediction error exhibits a normal (Gaussian) distribution with a mean value close to the ideal 0 cm2/Vs (actual value is 2.1 cm2/Vs) and a standard deviation of 27.6 cm2/Vs, which is considerably lower than the standard deviation of the original data for µED (i.e. 40.2 cm2/Vs) reported in Table I.

Furthermore, the NN's tendency to predict values closer to the mean can be seen in Fig. 6, which show the distribution of true (Fig. 6a) and predicted (Fig. 6b) edge-defect-limited mobility values. True values seem to exhibit a Poisson distribution, while predicted values follow a more Gaussian-like distribution. Mean values in both cases are almost the same, with the predicted mean being less than 5% higher than the true mean. Lower number of predicted values in the high-mobility region and the region near 0 cm2/Vs results in standard deviation of predicted values being 24% lower than the standard deviation of true values.

The success of the NN-based predictions is demonstrated in Fig. 7 that reports the predicted-true value scatter plot for 2.1 nm-wide SiNRs. The plot contains a

TABLE I. Statistical parameters for W = 2.1 nm (N = 9524)

Stat. values

SiNR parameters

EG GON R µED

std 0.072 0.172 249654 40.2

median 0.336 1.912 35031 31.8

mean 0.339 1.924 117498 45.2

Fig. 4. Mean absolute error of training and training validation versus training epoch number.

Fig. 5. Distribution of prediction errors for 2.1 nm-wide SiNR.

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y = x line, meaning the prediction quality can be evaluated by the proximity of blue dots to the diagonal line. In a perfect scenario, where every prediction is equal to the real value, all dots would be placed on the line, however in our case results close to the perfect scenario are not possible due to defect-induced stochastic nature of the data. As expected, the NN predicts values on which it was predominantly trained most accurately, while for values above 100 cm2/Vs, which were fewer in number in the training dataset, the NN prediction exhibits greater errors. This results in a tighter scatter plot in the lower-mobility region, and a looser scatter plot in the higher-mobility region, as noticeable in Fig. 7.

The training takes around 65.5 s on an Intel i7-8750H CPU and the prediction itself only 36 ms, as calculated using Python’s time module.

B. NN mobility prediction for W = 3.2 nm To further test the capabilities of the NN in predicting

mobility values, 200 sets of data were simulated in Matlab for 3.2 nm-wide and 20 nm-long defective SiNRs with other parameters unchanged. Statistical parameters of EG, GON, R = GON/GOFF, and µED for 3.2 nm-wide SiNRs are listed in Table II. For these wider nanoribbons, µED values

are predicted using the previously trained NN (trained on the data belonging to 2.1 nm-wide SiNRs).

As reported in Table II, the wider defective SiNRs exhibit a lower average bandgap and a higher average µED value (average of 89.2 cm2/Vs, and a standard deviation of 82.8 cm2/Vs). With a larger MAE of 55 cm2/Vs, the distribution of which can be seen in Fig. 8, the NN shows promising results and predicts µED values surprisingly well despite high defect-induced variability in the original set, and despite being trained on data obtained for a different SiNR width. Just like with the 2.1 nm-wide SiNRs, the prediction error graph exhibits a normal (Gaussian) distribution. The prediction error mean is close to ideal 0 cm2/Vs, being only slightly higher at 3.4 cm2/Vs. In this case, the standard deviation of the prediction error is 95.9 cm2/Vs and is slightly higher than the standard deviation of the true µED values (see Table II).

The promising prediction abilities of the neural network are further affirmed by the predicted mean of 92.6 cm2/Vs, which is less than 5% different from the real value of 89.2 cm2/Vs. Figure 9 shows the scatter graph of the predicted and true µED values for 3.2 nm-wide SiNRs, which acts as expected: namely, mobility values near the mean of the original data are being predicted most successfully, and those further away less so. Nevertheless, the predicted mean and variability of µED in the case of W = 3.2 nm demonstrate the suitability of our optimized 3-hidden-layer NN for predicting the transport physics in defective SiNRs irrespective of their size.

The prediction of the sets took about 4 ms, which corresponds to a smaller data set, roughly 8 times smaller than the previous one (for 2.1 nm-wide SiNRs).

Fig. 6. Distribution of (a) true µED values, and (b) predicted µED values for 2.1 nm-wide SiNRs.

Fig. 7. Scatter graph of predicted and true µED values for 2.1 nm-wide SiNRs.

TABLE II. Statistical parameters for W = 3.2 nm (N = 200)

Stat. values

SiNR parameters

EG GON R µED

std 0.052 0.195 8167 82.8

median 0.194 3.626 3216 67.2

mean 0.204 3.624 5672 89.2

Fig. 8. Distribution of prediction errors for the 3.2 nm-wide SiNRs.

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IV. CONLUSIONS We have demonstrated the potential of neural network

usage for predicting the transport properties of defective SiNRs. The NN with relatively simple architecture (three inputs, three hidden layers with 32, 64, and 32 neurons, respectively, and one output) has been constructed and has predicted average µED values and its variability within 5% difference from the true values, with average errors significantly smaller than the standard deviation of the data set. We have also shown that the 3-hidden-layer NN, trained on smaller nanoribbons, can also be used for the prediction of device parameters for larger SiNRs. This finding is quite useful since NN training and validation take less than a minute, whereas atomistic quantum transport simulations of a large set of SiNRs take ~hours or even ~days, depending on nanoribbon dimensions and ensemble size. Therefore, this work shows that NNs can be a useful supplement to time-consuming NEGF simulations of nanoscale 2D material-based devices.

ACKNOWLEDGMENTS This work is sponsored by the Croatian Science

Foundation (HRZZ) under the project CONAN2D (UIP-2019-04-3493).

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Fig. 9. Scatter graph of predicted and true µED values in the case of3.2 nm-wide SiNRs.

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