Valley notch filter in a graphene strain superlattice: Green's function and machine learning approach

The valley transport properties of a superlattice of out-of-plane Gaussian deformations are calculated using a Green's function and a machine learning approach. Our results show that periodicity significantly improves the valley filter capabilities of a single Gaussian deformation; these manifest themselves in the conductance as a sequence by valley filter plateaus. We establish that the physical effect behind the observed valley notch filter is the coupling between counterpropagating transverse modes; the complex relationship between the design parameters of the superlattice and the valley filter effect make it difficult to estimate in advance the valley filter potentialities of a given superlattice. With this in mind, we show that a deep neural network can be trained to predict valley polarization with a precision similar to the Green's function but with much less computational effort.

Figure 1

(a) Schematic representation of the two-terminal system. The central part of the ribbon has dimensions $L_0\times W_0$ and the zigzag direction is along to the $x$ axis. The terminals are labeled $L$ and $R$. The system is deformed by an array of $N_G$ out-of-plane Gaussian deformations in series. (b) Profile of the pseudomagnetic field for one Gaussian bump in $K$ valley.

Figure 2

Band structure for a zigzag graphene nanoribbon of width $W_0\approx 14.8$ nm around (a) $K^{\prime }$ valley and (b) $K$ valley. (c) Conductance and (d) valley polarization $P_{K^{\prime }}$ and $P_K$ in the presence of one out-of-plane Gaussian deformation for different values of the strain parameter $\alpha$. (d) Valley polarization for the low-energy modes and transmission probability for electrons in (e) $K^{\prime }$ valley and (f) K valley, produced by one Gaussian bump with $\alpha =20\%$.

Figure 3

(a) Conductance for a zigzag graphene nanoribbon with an arrangement of $N_G=15$ Gaussian deformations and $W_0\approx 14.8$ nm. Valley polarization $P_{K^{\prime }}$ and $P_K$ for the same system with (b) $\alpha =7\%$, (c) $\alpha =14\%$, and (d) $\alpha =20\%$.

Figure 4

Left panel: Folded band structure of a zigzag graphene nanoribbon without deformation and supercell lattice constant $d=52a_z$ and $W_0\approx 14.8$ nm, the colors identify the transverse modes of the primitive unit cell presented in Figs. 2 and 2. Right panel: Band structure of an infinite 1D chain of out-of-plane Gaussian deformations with $\alpha =7\%$. (a) Folded band structure, close-up at the energy region corresponding to the second (b) and third valley (c) filter plateau $P_{K^{\prime }}=1$ and $P_K=0$ in Fig. 3.

Figure 5

Average valley polarization $\langle P_{K^{\prime }}\rangle$ for the 1D Gaussian superlattice presented in Fig. 3 $\alpha =14\%$ ($A=1.17\text{nm}$, $b=3.12\text{nm}$, $d=12.79\text{nm}$, and $N_G=15$) with $A$ (height) disorder (red line), $b$ (width) disorder (green line), and $d$ (distance) disorder (blue line). The calculated valley filter capabilities for the disordered systems are $Q_{K^{\prime }}^A=21.7\%$, $Q_{K^{\prime }}^b=22.7\%$, and $Q_{K^{\prime }}^d=20.7\%$.

Figure 6

Feed-forward fully connected deep neural network architecture used for predicting valley polarization. The network consists of an input layer with seven neurons—seven hidden layers with 175 neurons each and one output layer with a single neuron. The neurons have hyperbolic tangent activation with the exception of the output layer neuron that has linear activation.

Figure 7

Variation of the MSE versus number of epochs for training and test data.

Figure 8

Predicted valley polarization $P_K^{\prime }$ as function of $\mathrm{E}({\mathrm{t}}_0$) and number of Gaussians ${\mathrm{N}}_G$ for (a) $\alpha =7\%$ and (b) $\alpha =14\%$. The data set was generated with $N_y=70(W_0=14.8\text{nm})$, $b=3.12\text{nm}$, and $d=12.7\text{nm}$.

Figure 9

(a) $Q_{K^{\prime }}$ as a function of $N_G$ for different values of $\alpha$ calculated by the DNN. (b) Valley polarization calculated by the Green's function (GF) and the DNN for the optimal graphene superlattice parameters $N_G=6$, $\alpha =15\%$, $N_y=70(W_0=14.8\text{nm})$, $b=3.12\text{nm}$, and $d=12.7\text{nm}$.