Machine learning based prediction of the electronic structure of quasi-one-dimensional materials under strain

We present a machine learning based model that can predict the electronic structure of quasi-one-dimensional materials while they are subjected to deformation modes such as torsion and extension/compression. The technique described here applies to important classes of materials systems such as nanotubes, nanoribbons, nanowires, miscellaneous chiral structures, and nanoassemblies, for all of which, tuning the interplay of mechanical deformations and electronic fields, i.e., strain engineering, is an active area of investigation in the literature. Our model incorporates global structural symmetries and atomic relaxation effects, benefits from the use of helical coordinates to specify the electronic fields, and makes use of a specialized data generation process that solves the symmetry-adapted equations of Kohn-Sham density functional theory in these coordinates. Using armchair single-wall carbon nanotubes as a prototypical example, we demonstrate the use of the model to predict the fields associated with the ground-state electron density and the nuclear pseudocharges, when three parameters (namely, the radius of the nanotube, its axial stretch, and the twist per unit length) are specified as inputs. Other electronic properties of interest, including the ground-state electronic free energy, can be evaluated from these predicted fields with low-overhead postprocessing, typically to chemical accuracy. Additionally, we show how the nuclear coordinates can be reliably determined from the predicted pseudocharge field using a clustering-based technique. Remarkably, only about 120 data points are found to be enough to predict the three-dimensional electronic fields accurately, which we ascribe to the constraints imposed by symmetry in the problem setup, the use of low-discrepancy sequences for sampling, and efficient representation of the intrinsic low-dimensional features of the electronic fields. We comment on the interpretability of our machine learning model and anticipate that our framework will find utility in the automated discovery of low-dimensional materials, as well as the multiscale modeling of such systems.

Figure 1

Rollup construction of an undeformed armchair carbon nanotube, starting from a graphene sheet. The four atoms shown in the shaded region are used for the data generation process using Helical DFT. The parameter $a$ represents the planar interatomic distance of 1.407 Å .

Figure 2

Schematic of the present machine learning (ML) model and the data generation process via DFT simulations. The firm arrows show the steps for data generation and training, and the dashed arrows show the steps for prediction via the ML model.

Figure 3

Atomic pseudocharge as a function of distance (in bohrs) from the atom for the Troullier-Martins pseudopotential for carbon used in this work. The dashed red line indicates the truncation level employed before the DBSCAN procedure is used.

Figure 4

Cluster formation from nuclear pseudocharge field to determine nuclei position. A slice of the pseudocharge field at the average radial coordinate of the atoms in the fundamental domain is shown. Red clusters show the positive charge around the nucleus and the black dots are nuclei. The pseudocharge field on the fundamental domain is expanded to a supercell to avoid domain edge effects, a truncation is implemented to discard secondary peaks in the atomic pseudocharges, the DBSCAN procedure is then applied on the supercell, and, finally, the nuclear coordinates within the fundamental domain are identified.

Figure 5

Cumulative percentage of variance vs principal components for $\rho$ (left) and $b$ (right). The red dashed line shows $99.99\%$ variance.

Figure 6

Parity plots for (a) test data of $\rho$ ($R=0.9949$), (b) test data of $b$ ($R=0.9983$).

Figure 7

Comparison between ML predicted and DFT simulation obtained electronic fields for a test data point with all unknown input parameters ($R_{\mathrm{avg}}=49.51\text{bohrs}$, $\alpha =0.00125$, $\tau =4.5552\text{bohrs}$). A slice of the electronic fields at the average radial coordinate of the atoms in the fundamental domain is shown. The error is computed as $\frac{|{\rho }^{\text{DFT}}-{\rho }^{\text{ML}}|}{|\max \left({\rho }^{\text{DFT}}\right)-\min \left({\rho }^{\text{DFT}}\right)|}$, similarly for $b$. Here, max($·$) and min($·$) denote maximum and minimum over the fundamental domain.

Figure 8

Comparison of symmetry-adapted band diagrams produced by the Helical DFT method and the machine learning model (with postprocessing) for the unknown test data point with $R_{\mathrm{avg}}=49.51\text{bohrs},\alpha =0.00125$, and $\tau =4.5552\text{bohrs}$. The agreement appears excellent and the postprocessed ML model is also able to precisely predict the location of the band gap (at $\eta =\frac{1}{3},\nu =2$) as well as its value (0.128 eV from Helical DFT) to about $6\%$ accuracy in this case. Note that ${\lambda }_{\text{F}}$ denotes the system's Fermi level.

Figure 9

First two principal components for $\rho$ (top) and $b$ (bottom). A slice of the PCA modes at the average radial coordinate of the atoms in the fundamental domain is shown.

Figure 10

Mean of NRMSE for test data when the machine learning model is trained using Sobol sets. (Left) Error bars for $\rho$, (right) error bars for $b$.

Figure 11

(a) Learning curve for ${\mathcal{N}}_1$, (b) learning curve for ${\mathcal{N}}_2$.

Figure 12

(Top) Test error ($\times {10}^{-5}$) for various architectures of ${\mathcal{N}}_1$ (trained for $\rho$). (Bottom) Test error ($\times {10}^{-4}$) for various architectures of ${\mathcal{N}}_2$ (trained for $b$.)

Figure 13

Error in predicted atomic coordinates (in bohrs), i.e., the distance between true and predicted nucleus positions, using a neural network and the DBSCAN-based clustering approach.