Solution of the Schrödinger equation for quasi-one-dimensional materials using helical waves Shivang Agarwala Amartya S. Banerjeeb

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Solution of the Schrödinger equation for quasi-one-dimensional

materials using helical waves

Shivang Agarwala,, Amartya S. Banerjeeb

aDepartment of Electrical and Computer Engineering, University of California, Los Angeles, CA

90095, U.S.A

bDepartment of Materials Science and Engineering, University of California, Los Angeles, CA 90095,

U.S.A

Abstract

We formulate and implement a spectral method for solving the Schrödinger equation, as

it applies to quasi-one-dimensional materials and structures. This allows for computa-

tion 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 struc-

ture 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 or-

bitals. The basis functions in our method — called helical waves (or twisted waves) —

are eigenfunctions of the Laplacian with symmetry adapted boundary conditions, and

are expressible in terms of plane waves and Bessel functions in helical coordinates.

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 ob-

tain 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 struc-

ture calculations of various quasi-one-dimensional materials through numerous examples

involving nanotubes, nanoribbons and nanowires. We also explore the convergence prop-

erties of our method, and assess its accuracy and computational eﬃciency by comparison

against reference ﬁnite diﬀerence, transfer matrix method and plane-wave results. We

anticipate that our method will ﬁnd applications in computational nanomechanics and

multiscale modeling, for carrying out transport calculations of interest to the ﬁeld of

semiconductor devices, and for the discovery of novel chiral phases of matter that are of

relevance to the burgeoning quantum hardware industry.

Keywords: Helical Waves, Electronic Structure Calculations, Nanomaterials,

Nanostructures, Chiral Materials, Spectral Method.

Preprint submitted to Journal of Computational Physics September 26, 2023

arXiv:2210.12252v2 [physics.comp-ph] 23 Sep 2023

1. Introduction

Low dimensional materials have been intensely investigated in the past few decades

due to their remarkable electronic, optical, transport and mechanical characteristics [1,2].

The properties of these materials often provide sharp contrasts with the bulk phase,

and have led to various technological applications, including e.g., new kinds of sensors,

actuators and energy harvesting devices [3–8]. Quasi-one-dimensional materials — which

include nanotubes, nanoribbons, nanowires, nanocoils, as well as miscellaneous structures

of biological origin [9,10] — are particularly interesting in this regard. This is due to the

unique electronic properties that emerge as a result of the availability of a single extended

spatial dimension in these structures [11–14], the possibility that they are associated

with ferromagnetism, ferroelectricity, and superconductivity [15–18], and the fact that

the behavior of these materials may be readily modulated via imposition of mechanical

deformation modes such as torsion and/or stretching. [19–21]. Quasi-one-dimensional

materials have also been investigated as hardware components for computing platforms

— both conventional [22,23] and quantum [24]. The applications of such materials in

the latter case are connected to anomalous transport (the Chiral Induced Spin Selectivity

eﬀect [25]) and exotic electronic states [26] that can be observed in such systems.

Given the importance of quasi-one-dimensional materials, it is highly desirable to

have available computational methods that can eﬃciently characterize the unique elec-

tronic properties of these systems. However, conventional electronic structure calculation

methods — based e.g. on plane-waves [27,28] — are generally inadequate in handling

them. This is a result of the non-periodic symmetries in the atomic arrangements of such

materials. As a result of these symmetries, the single particle Schrödinger equation asso-

ciated with the electronic structure problem exhibits special invariances [29,30], which

plane-waves, being intrinsically periodic, are unable to handle. For example, ground state

plane-wave calculations of a twisted nanoribbon (see Fig. 1a) will usually involve making

the system artiﬁcially periodic along the direction of the twist axis — thus resulting in

a periodic supercell containing a very large number of atoms, as well as the inclusion

of a substantial amount of vacuum padding in the directions orthogonal to the twist

axis, so as to minimize interactions between periodic images. Together, these conditions

can make such calculations extremely challenging even on high performance computing

platforms, if not altogether impractical. There have been a few attempts to treat quasi-

one-dimensional materials using Linear Combination of Atomic Orbitals (LCAO) based

techniques [31–36]. However, such methods suﬀer from basis incompleteness and super-

position errors [37–39], which can make it diﬃcult to obtain systematically convergent

and improvable results.

In view of these limitations of conventional methods, a series of recent contributions

has explored the use of real space techniques to study quasi-one-dimensional materials

and their natural deformation modes [20,30,40,41]. Speciﬁcally, this line of work incor-

porates the helical interaction potentials present in such systems using helical Bloch waves

and employs higher order ﬁnite diﬀerences to discretize the single particle Schrödinger

equation in helical coordinates. While this technique shows systematic convergence, and

has enabled the exploration of various fascinating electromechanical properties, it also has

a number of signiﬁcant drawbacks. First, due to the curvilinearity of helical coordinates,

Email address: asbanerjee@ucla.edu (Amartya S. Banerjee)

the discretized Hamiltonian appearing in these calculations is necessarily non-Hermitian

[42,43]. This complicates the process of numerical diagonalization and makes many of

the standard iterative eigensolvers [44] unusable. Second, the discretized equations have

a coordinate singularity along the system axis which restricts the use of the methods to

tubular structures and prevents important nanomaterials such as nanowires and nanorib-

bons from being studied. The presence of the singularity also tends to ill condition the

discretized Hamiltonian, which further restricts the applicability of the method to sys-

tems in which the atoms lie far enough away from the system axis (e.g. larger diameter

nanotubes). Finally, while the ﬁnite diﬀerence approach does allow for the simulation of

materials with twist (intrinsic or applied), the sparsity pattern of the discretized Hamil-

tonian worsens upon inclusion of twist, making simulations of such systems signiﬁcantly

more burdensome.

In this work we formulate and implement a novel computational technique that reme-

dies all of the above issues and allows one to carry out systematic numerical solutions of

the Schrödinger equation, as it applies to quasi-one-dimensional materials and structures.

The technique presented here can be thought of as an analog of the classical plane-wave

method, and is similar in spirit to the spectral scheme for clusters presented in [45].

Like the classical plane-wave method, a single parameter (the kinetic energy cutoﬀ) dic-

tates the overall quality of solution of our numerical scheme. We present a derivation

of the basis functions of our method — called helical waves (or twisted waves) — as

eigenfunctions of the Laplacian under suitable boundary conditions. We describe how

helical waves may be used to discretize the symmetry adapted Schrödinger equation for

quasi-one-dimensional materials, and how matrix-free iterative techniques can be used

for diagonalization. A key feature of our technique is the handling of convolution sums

through the use of fast basis transforms, and we describe in detail how these transforms

are formulated and implemented. We also discuss various other computational aspects,

including the choice of eigensolvers and preconditioners, and the handling of oscilla-

tory radial integrals that appear in our method. We have implemented these techniques

into a MATLAB [46] package called HelicES (Helical Electronic Structure), which we

use for carrying out demonstrative electronic structure calculations of various quasi-one-

dimensional materials. We also present results related to the convergence, computational

eﬃciency and accuracy properties of our method, while using ﬁnite diﬀerence, transfer

matrix and plane-wave methods for reference data.

We remark that our technique has connections with methods presented in earlier work

concerning electronic structure calculations in cylindrical geometries [47–51], but is more

general in that the use of helical waves automatically allows both chiral (i.e., twisted) and

achiral (i.e., untwisted) structures to be naturally handled. Additionally, some of these

earlier studies have employed the strategy of setting up of the discretized Hamiltonian

explicitly and then using direct diagonalization techniques, which scales in a signiﬁcantly

worse way (both in memory and computational time) compared to the transform based

matrix-free strategies adopted by us. We also note in passing that the basis functions

presented here appear to be scalar versions of twisted wave ﬁelds explored recently in the

x-ray crystallography [52,53] and elastodynamics [54,55] literature.

The rest of this paper is organized as follows. In Section 2, we specify the class

of systems of interest to this work, formalize the relevant computational problem, and

describe our discretization strategy. Numerical techniques and algorithms are presented

in Section 3, following which we present results in Section 4. We conclude in Section 5and

also discuss the future outlook of the work. Miscellaneous derivations and computational

details are presented in the Appendices.

2. Formulation

In what follows, eX,eY,eZwill denote the standard orthonormal basis of R3. Position

vectors will be typically denoted using boldface lower case letters (e.g., p)and rotation

matrices using boldface uppercase (e.g., Q).The atomic unit system of me= 1,ℏ=

1,1

4πϵ0= 1 will be used throughout the paper, unless otherwise mentioned. Cartesian

and cylindrical coordinates will be typically denoted as (x, y, z)and (r, ϑ, z)respectively.

The ×sign will be reserved for denoting dimensions of matrices (e.g. using M×Nto

denote the dimensions of a matrix with Mrows and Ncolumns), while ∗will be used to

explicitly denote multiplication by or in between scalars, vectors and matrices.

2.1. Description of Physical System and Computational Problem

We consider a quasi-one-dimensional nanostructure of inﬁnite extent aligned along eZ

(see Fig. 1). We assume the structure to be of limited extent along eXand eY. Let the

atoms of the structure have coordinates:

S={p1,p2,p3,· · · :pi∈R3}.(1)

Quasi-one-dimensional structures in their undeformed states, or while being subjected

to natural deformation modes such as extension, compression or torsion, can often be

described using helical (i.e., screw transformation) and cyclic symmetries [10,20,29,30].

Accordingly, we may identify a ﬁnite subset of atoms of the structure with coordinates:

P={r1,r2,r3,...,rM:ri∈R3},(2)

and a corresponding set of symmetry operations:

G=nΥζ,µ =R(2πζα+µΘ)|ζτeZ) : ζ∈Z, µ = 0,1,...,N−1o,(3)

such that:

S=[

ζ∈Z

µ=0,1,...,N−1

[

i=1

R(2πζα+µΘ)ri+ζτeZ.(4)

Here, the Υζ,µ are symmetry operations of the structure — speciﬁcally, each Υζ,µ is an

isometry whose action on an arbitrary point x∈R3(denoted as Υζ,µ ◦x) is to rotate it by

the angle 2πζα +µΘabout eZ, while simultaneously translating it by µτ about the same

axis. The natural number Nis related to cyclic symmetries in the nanostructure about

the axis eZ, with Θ = 2π/Ndenoting the cyclic symmetry angle. The quantity τis the

pitch of the screw transformation part of Υζ,µ, the parameter αtakes values 0≤α < 1,

and β= 2πα/τ captures the rate of twist (imposed or intrinsic) in the structure. The

case α= 0 usually represents achiral or untwisted structures (see Fig. 1) .

The electronic properties of a quasi-one-dimensional material under study can be

investigated by calculating the spectrum of the single particle Schrödinger operator:

H=−1

2∆ + V(x),(5)

𝝉

𝟐𝝅𝜶Twist of

radians over

length 𝝉

(a) A twisted nanoribbon

𝝉

(b) An armchair nanotube

Figure 1: Examples of the type of nanostructures that can be investigated using the computational

framework presented in this work. Helical and cyclic symmetry parameters associated with the geometries

of the structures are shown.

associated with the system. Determination of the spectrum in an eﬃcient manner, espe-

cially for realistic quasi-one-dimensional nanomaterials serves as the primary computa-

tional problem of interest in this work. Here, V(x)represents the “eﬀective potential” as

perceived by the electrons. The potential can be computed through self-consistent means

(for example, as part of Density Functional Theory calculations [20,30]), or through the

use of empirical pseudopotentials [56,57], as done here. Due to the presence of global

structural symmetries, the potential is expected to obey:

V(x) = V(Υζ,µ ◦x),∀Υζ,µ ∈ G .(6)

As a consequence of the quasi-one-dimensional nature of the system, and the above

symmetry conditions, the eigenstates of the Hamiltonian can be characterized in terms

of Helical Bloch waves [29,30]. Speciﬁcally, solutions of the Schrödinger equation:

−1

2∆ + V(x)ψ=λ ψ , (7)

can be labeled using band indices j∈N, and symmetry adapted quantum numbers

η∈−1

2,1

2,ν∈ {0,1,2,...,N−1}. Moreover, these solutions obey the following

condition for any symmetry operation Υζ,µ ∈ G:

ψj(Υζ,µ ◦x;η, ν) = e−2πiζη+µν

Nψj(x;η, ν).(8)

The above relation can be used to reduce the computational problem of determining the

eigenstates of the Schrödinger operator over all of space, to a fundamental domain or

symmetry-adapted unit cell.

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SolutionoftheSchrödingerequationforquasi-one-dimensionalmaterialsusinghelicalwavesShivangAgarwala,,AmartyaS.BanerjeebaDepartmentofElectricalandComputerEngineering,UniversityofCalifornia,LosAngeles,CA90095,U.S.AbDepartmentofMaterialsScienceandEngineering,UniversityofCalifornia,LosAngeles,CA90095,U.S....

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Solution of the Schrödinger equation for quasi-one-dimensional materials using helical waves Shivang Agarwala Amartya S. Banerjeeb

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