70 research outputs found
Ab initio framework for systems with helical symmetry: theory, numerical implementation and applications to torsional deformations in nanostructures
We formulate and implement Helical DFT -- a self-consistent first principles
simulation method for nanostructures with helical symmetries. Such materials
are well represented in all of nanotechnology, chemistry and biology, and are
expected to be associated with unprecedented material properties. We rigorously
demonstrate the existence and completeness of special solutions to the single
electron problem for helical nanostructures, called helical Bloch waves. We
describe how the Kohn-Sham Density Functional Theory equations for a helical
nanostructure can be reduced to a fundamental domain with the aid of these
solutions. A key component in our mathematical treatment is the definition and
use of a helical Bloch-Floquet transform to perform a block-diagonalization of
the Hamiltonian in the sense of direct integrals. We develop a symmetry-adapted
finite-difference strategy in helical coordinates to discretize the governing
equations, and obtain a working realization of the proposed approach. We verify
the accuracy and convergence properties of our numerical implementation through
examples. Finally, we employ Helical DFT to study the properties of zigzag and
chiral single wall black phosphorus (i.e., phosphorene) nanotubes. We use our
simulations to evaluate the torsional stiffness of a zigzag nanotube ab initio.
Additionally, we observe an insulator-to-metal-like transition in the
electronic properties of this nanotube as it is subjected to twisting. We also
find that a similar transition can be effected in chiral phosphorene nanotubes
by means of axial strains. Notably, self-consistent ab initio simulations of
this nature are unprecedented and well outside the scope of any other
systematic first principles method in existence. We end with a discussion on
various future avenues and applications
A spectral scheme for Kohn-Sham density functional theory of clusters
Starting from the observation that one of the most successful methods for
solving the Kohn-Sham equations for periodic systems -- the plane-wave method
-- is a spectral method based on eigenfunction expansion, we formulate a
spectral method designed towards solving the Kohn-Sham equations for clusters.
This allows for efficient calculation of the electronic structure of clusters
(and molecules) with high accuracy and systematic convergence properties
without the need for any artificial periodicity. The basis functions in this
method form a complete orthonormal set and are expressible in terms of
spherical harmonics and spherical Bessel functions. Computation of the occupied
eigenstates of the discretized Kohn-Sham Hamiltonian is carried out using a
combination of preconditioned block eigensolvers and Chebyshev polynomial
filter accelerated subspace iterations. Several algorithmic and computational
aspects of the method, including computation of the electrostatics terms and
parallelization are discussed. We have implemented these methods and algorithms
into an efficient and reliable package called ClusterES (Cluster Electronic
Structure). A variety of benchmark calculations employing local and non-local
pseudopotentials are carried out using our package and the results are compared
to the literature. Convergence properties of the basis set are discussed
through numerical examples. Computations involving large systems that contain
thousands of electrons are demonstrated to highlight the efficacy of our
methodology. The use of our method to study clusters with arbitrary point group
symmetries is briefly discussed.Comment: Manuscript submitted (with revisions) to Journal of Computational
Physic
Periodic Pulay method for robust and efficient convergence acceleration of self-consistent field iterations
Pulay's Direct Inversion in the Iterative Subspace (DIIS) method is one of
the most widely used mixing schemes for accelerating the self-consistent
solution of electronic structure problems. In this work, we propose a simple
generalization of DIIS in which Pulay extrapolation is performed at periodic
intervals rather than on every self-consistent field iteration, and linear
mixing is performed on all other iterations. We demonstrate through numerical
tests on a wide variety of materials systems in the framework of density
functional theory that the proposed generalization of Pulay's method
significantly improves its robustness and efficiency.Comment: Version 2 (with minor edits from version 1
Solution of the Schrodinger equation for quasi-one-dimensional materials using helical waves
We formulate and implement a spectral method for solving the Schrodinger
equation, as it applies to quasi-one-dimensional materials and structures. This
allows for computation of the electronic structure of important technological
materials such as nanotubes (of arbitrary chirality), nanowires, nanoribbons,
chiral nanoassemblies, nanosprings and nanocoils, in an accurate, efficient and
systematic manner. Our work is motivated by the observation that one of the
most successful methods for carrying out electronic structure calculations of
bulk/crystalline systems -- the plane-wave method -- is a spectral method based
on eigenfunction expansion. Our scheme avoids computationally onerous
approximations involving periodic supercells often employed in conventional
plane-wave calculations of quasi-one-dimensional materials, and also overcomes
several limitations of other discretization strategies, e.g., those based on
finite differences and atomic orbitals. We describe the setup of fast
transforms to carry out discretization of the governing equations using our
basis set, and the use of matrix-free iterative diagonalization to obtain the
electronic eigenstates. Miscellaneous computational details, including the
choice of eigensolvers, use of a preconditioning scheme, evaluation of
oscillatory radial integrals and the imposition of a kinetic energy cutoff are
discussed. We have implemented these strategies into a computational package
called HelicES (Helical Electronic Structure). We demonstrate the utility of
our method in carrying out systematic electronic structure calculations of
various quasi-one-dimensional materials through numerous examples involving
nanotubes, nanoribbons and nanowires. We also explore the convergence, accuracy
and efficiency of our method. We anticipate that our method will find numerous
applications in computational nanomechanics and materials science
Density functional theory method for twisted geometries with application to torsional deformations in group-IV nanotubes
We present a real-space formulation and implementation of Kohn-Sham Density
Functional Theory suited to twisted geometries, and apply it to the study of
torsional deformations of X (X = C, Si, Ge, Sn) nanotubes. Our formulation is
based on higher order finite difference discretization in helical coordinates,
uses ab intio pseudopotentials, and naturally incorporates rotational (cyclic)
and screw operation (i.e., helical) symmetries. We discuss several aspects of
the computational method, including the form of the governing equations,
details of the numerical implementation, as well as its convergence, accuracy
and efficiency properties.
The technique presented here is particularly well suited to the first
principles simulation of quasi-one-dimensional structures and their
deformations, and many systems of interest can be investigated using small
simulation cells containing just a few atoms. We apply the method to
systematically study the properties of single-wall zigzag and armchair group-IV
nanotubes, as they undergo twisting. For the range of deformations considered,
the mechanical behavior of the tubes is found to be largely consistent with
isotropic linear elasticity, with the torsional stiffness varying as the cube
of the nanotube radius. Furthermore, for a given tube radius, this quantity is
seen to be highest for carbon nanotubes and the lowest for those of tin, while
nanotubes of silicon and germanium have intermediate values close to each
other. We also describe different aspects of the variation in electronic
properties of the nanotubes as they are twisted. In particular, we find that
akin to the well known behavior of armchair carbon nanotubes, armchair
nanotubes of silicon, germanium and tin also exhibit bandgaps that vary
periodically with imposed rate of twist, and that the periodicity of the
variation scales in an inverse quadratic manner with the tube radius
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Symmetry-adapted real-space density functional theory for cylindrical geometries: Application to large group-IV nanotubes
We present a symmetry-adapted real-space formulation of Kohn-Sham density functional theory for cylindrical geometries and apply it to the study of large X (X=C, Si, Ge, Sn) nanotubes. Specifically, starting from the Kohn-Sham equations posed on all of space, we reduce the problem to the fundamental domain by incorporating cyclic and periodic symmetries present in the angular and axial directions of the cylinder, respectively. We develop a high-order finite-difference parallel implementation of this formulation, and verify its accuracy against established plane-wave and real-space codes. Using this implementation, we study the band structure and bending properties of X nanotubes and Xene sheets, respectively. Specifically, we first show that zigzag and armchair X nanotubes with radii in the range 1 to
5
nm
are semiconducting, other than the armchair and zigzag type III carbon variants, for which we find a vanishingly small bandgap, indicative of metallic behavior. In particular, we find an inverse linear dependence of the bandgap with respect to the radius for all nanotubes, other than the armchair and zigzag type III carbon variants, for which we find an inverse quadratic dependence. Next, we exploit the connection between cyclic symmetry and uniform bending deformations to calculate the bending moduli of Xene sheets in both zigzag and armchair directions, while considering radii of curvature up to
5
nm
. We find Kirchhoff-Love type bending behavior for all sheets, with graphene and stanene possessing the largest and smallest moduli, respectively. In addition, other than graphene, the sheets demonstrate significant anisotropy, with larger bending moduli along the armchair direction. Finally, we demonstrate that the proposed approach has very good parallel scaling and is highly efficient, enabling ab initio simulations of unprecedented size for systems with a high degree of cyclic symmetry. In particular, we show that even micron-sized nanotubes can be simulated with modest computational effort. Overall, the current work opens an avenue for the efficient ab initio study of 1D nanostructures with large radii as well as 1D/2D nanostructures under uniform bending
Two-level Chebyshev filter based complementary subspace method: pushing the envelope of large-scale electronic structure calculations
We describe a novel iterative strategy for Kohn-Sham density functional
theory calculations aimed at large systems (> 1000 electrons), applicable to
metals and insulators alike. In lieu of explicit diagonalization of the
Kohn-Sham Hamiltonian on every self-consistent field (SCF) iteration, we employ
a two-level Chebyshev polynomial filter based complementary subspace strategy
to: 1) compute a set of vectors that span the occupied subspace of the
Hamiltonian; 2) reduce subspace diagonalization to just partially occupied
states; and 3) obtain those states in an efficient, scalable manner via an
inner Chebyshev-filter iteration. By reducing the necessary computation to just
partially occupied states, and obtaining these through an inner Chebyshev
iteration, our approach reduces the cost of large metallic calculations
significantly, while eliminating subspace diagonalization for insulating
systems altogether. We describe the implementation of the method within the
framework of the Discontinuous Galerkin (DG) electronic structure method and
show that this results in a computational scheme that can effectively tackle
bulk and nano systems containing tens of thousands of electrons, with chemical
accuracy, within a few minutes or less of wall clock time per SCF iteration on
large-scale computing platforms. We anticipate that our method will be
instrumental in pushing the envelope of large-scale ab initio molecular
dynamics. As a demonstration of this, we simulate a bulk silicon system
containing 8,000 atoms at finite temperature, and obtain an average SCF step
wall time of 51 seconds on 34,560 processors; thus allowing us to carry out 1.0
ps of ab initio molecular dynamics in approximately 28 hours (of wall time).Comment: Resubmitted version (version 2
Bicrystallography-informed Frenkel-Kontorova model for interlayer dislocations in strained 2D heterostructures
In recent years, van der Waals (vdW) heterostructures and homostructures,
which consist of stacks of two-dimensional (2D) materials, have risen to
prominence due to their association with exotic quantum phenomena. Atomistic
scale relaxation effects play an extremely important role in the electronic
scale quantum physics of these systems. We investigate such structural
relaxation effects in this work using atomistic and mesoscale models, within
the context of twisted bilayer graphene -- a well-known heterostructure system
that features moire patterns arising from the lattices of the two graphene
layers. For small twist angles, atomic relaxation effects in this system are
associated with the natural emergence of interface dislocations or strain
solitons, which result from the cyclic nature of the generalized stacking fault
energy (GSFE), that measures the interface energy based on the relative
movement of the two layers. In this work, we first demonstrate using atomistic
simulations that atomic reconstruction in bilayer graphene under a large twist
also results from interface dislocations, although the Burgers vectors of such
dislocations are considerably smaller than those observed in small-twist
systems. To reveal the translational invariance of the heterointerface
responsible for the formation of such dislocations, we derive the translational
symmetry of the GSFE of a 2D heterostructure using the notions of coincident
site lattices (CSLs) and displacement shift complete lattices (DSCLs). The
workhorse for this exercise is a recently developed Smith normal form
bicrystallography framework. Next, we construct a bicrystallography-informed
and frame-invariant Frenkel-Kontorova model, which can predict the formation of
strain solitons in arbitrary 2D heterostructures, and apply it to study a
heterostrained, large-twist bilayer graphene system
Electronic Structure Prediction of Multi-million Atom Systems Through Uncertainty Quantification Enabled Transfer Learning
The ground state electron density - obtainable using Kohn-Sham Density
Functional Theory (KS-DFT) simulations - contains a wealth of material
information, making its prediction via machine learning (ML) models attractive.
However, the computational expense of KS-DFT scales cubically with system size
which tends to stymie training data generation, making it difficult to develop
quantifiably accurate ML models that are applicable across many scales and
system configurations. Here, we address these fundamental challenges using
Bayesian neural networks and employ transfer learning to leverage the
multi-scale nature of the training data. Our ML models employ descriptors
involving simple scalar products, comprehensively sample system configurations
through thermalization, and quantify uncertainty in electron density
predictions. We show that our models incur significantly lower data generation
costs while allowing confident - and when verifiable, accurate - predictions
for a wide variety of bulk systems well beyond training, including systems with
defects, different alloy compositions, and at unprecedented, multi-million-atom
scales
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