MODIFIED-OPERATOR METHOD FOR THE CALCULATION OF BAND DIAGRAMS OF CRYSTALLINE MATERIALS ERIC CANCÈS MUHAMMAD HASSAN AND LAURENT VIDAL

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MODIFIED-OPERATOR METHOD FOR THE CALCULATION OF

BAND DIAGRAMS OF CRYSTALLINE MATERIALS

ERIC CANCÈS, MUHAMMAD HASSAN, AND LAURENT VIDAL

Abstract. In solid state physics, electronic properties of crystalline mate-

rials are often inferred from the spectrum of periodic Schrödinger operators.

As a consequence of Bloch’s theorem, the numerical computation of electronic

quantities of interest involves computing derivatives or integrals over the Bril-

louin zone of so-called energy bands, which are piecewise smooth, Lipschitz

continuous periodic functions obtained by solving a parametrized elliptic eigen-

value problem on a Hilbert space of periodic functions. Classical discretization

strategies for resolving these eigenvalue problems produce approximate energy

bands that are either non-periodic or discontinuous, both of which cause dif-

ﬁculty when computing numerical derivatives or employing numerical quad-

rature. In this article, we study an alternative discretization strategy based

on an ad hoc operator modiﬁcation approach. While speciﬁc instances of this

approach have been proposed in the physics literature, we introduce here a

systematic formulation of this operator modiﬁcation approach. We derive a

priori error estimates for the resulting energy bands and we show that these

bands are periodic and can be made arbitrarily smooth (away from band cross-

ings) by adjusting suitable parameters in the operator modiﬁcation approach.

Numerical experiments involving a toy model in 1D, graphene in 2D, and sil-

icon in 3D validate our theoretical results and showcase the eﬃciency of the

operator modiﬁcation approach.

1. Introduction

In solid state physics, macroscopic properties such as the electrical and ther-

mal conductivities, heat capacity, magnetic susceptibility, and optical absorption

of crystalline materials are often explained through the use of an independent elec-

tron model (see, e.g., [16, Part II, Chapter 5], [28, Chapter 12], [24, Chapter 7],

and [15, Chapter 5]). This model consists of treating the crystalline material as

an inﬁnite, perfect crystal and modeling the electrons as independent of each other

(quasiparticle approach) and evolving under the inﬂuence of an eﬀective periodic

potential. The behavior of each electron is thus determined by the spectrum of

an unbounded, self-adjoint, periodic Schrödinger operator acting on L2(R3)(see,

e.g., [34, Chapter XIII]). Although the independent electron assumption might

seem naive, this model has achieved great success in explaining basic phenomena

such as the diﬀerence between conductors, semi-conductors and insulators, as well

as describing the electronic properties of many ubiquitous non-strongly correlated

materials (see, e.g., [16, Part III], [28, Part V], [15, Chapters 10-12], and [24, Chap-

ters 6]). In addition, Kohn-Sham Density Functional Theory (DFT) provides a

method to parameterize this independent-electron model and obtain quantitatively

2020 Mathematics Subject Classiﬁcation. 65N25, 65F15, 65Z05.

Key words and phrases. Periodic Schrödinger Operators, Eigenvalue Problems on the Torus,

Numerical Analysis, Error Analysis.

arXiv:2210.00442v1 [math.NA] 2 Oct 2022

2 ERIC CANCÈS, MUHAMMAD HASSAN, AND LAURENT VIDAL

accurate results for a very large class of materials of practical interest (see, for

instance, [13, 23]).

In the independent electron model, the practical computation of electronic quan-

tities of interest is based on the use of the Bloch-Floquet transform (see, e.g., [34,

Chapter XIII]). The Bloch transform essentially yields an explicit block-diagonal

decomposition of the underlying Schrödinger operator into so-called Bloch ﬁbers,

which are bounded-below, self-adjoint operators acting on a space of periodic square-

integrable functions. Thus, the problem of computing the spectrum of the periodic

Schrödinger operator is reduced to one of calculating the low-lying eigenvalues of

the Bloch ﬁbers. These Bloch ﬁbers are typically indexed by a parameter kthat

belongs to a d-dimensional torus (the Brillouin zone), and therefore each resulting

eigenvalue (often referred to as an energy) can be viewed as a periodic function

on the d-dimensional Brillouin Zone. It is thus common in the solid-state physics

literature to speak of energy bands.

Energy bands provide both qualitative and quantitative information about the

electronic properties of the crystalline material being studied (see, e.g., the ref-

erences quoted above). Insulators and conductors for instance, are characterized

by the presence or absence, respectively, of an energy band gap. Other electronic

quantities of interest can be expressed in terms of integrals (over the Brillouin zone)

or derivatives involving the energy bands (see, e.g., [9]). In order to estimate im-

portant quantities such as the integrated density of states or the integrated density

of energy (see Section 2 for precise deﬁnitions of these quantities), it is therefore

necessary to

•sample the energy bands at diﬀerent k-points which corresponds to solving

approximately the k-ﬁber eigenvalue problems posed on a periodic domain;

•use suitable numerical quadrature to approximate integrals involving these

energy bands.

Concerning the ﬁrst step, the famous Monkhorst-Pack numerical scheme [30]

is widely used to select the speciﬁc k-points at which the eigenvalue problem is

to be solved. For the second step, a number of numerical quadrature methods

for integration in the Brillouin zone have been proposed including the well-known

linear tetrahedron method (see, e.g., [27]) and the improvement due to Blöchl et

al. [7], and smearing methods (see, e.g., [18, 29, 31,33]).

From a mathematical and computational point of view, two natural questions

now arise. First, which discretization method should be employed in the actual

numerical resolution of the k-ﬁber eigenvalue problems, and second, what can be

said about the convergence rate of the various numerical quadrature methods that

are in use? For technical reasons, these questions become particularly relevant

for metallic systems (see, e.g., [14] for an analysis involving insulators and semi-

conductors), and in this case, the latter question has recently been addressed by

the ﬁrst author and coworkers in [9]. The analysis carried out in [9] revealed that

the periodicity (with respect to the Brillouin zone) and regularity properties of

the energy bands play a crucial role in the quadrature convergence rates, which of

course is consistent with the experience of classical integration schemes in numerical

analysis. Given that diﬀerent eigenvalue discretization methods can conceivably

produce (and in fact do produce, as we show in Section 3) energy bands that

possess diﬀerent regularity properties or may be altogether aperiodic, the choice of

MODIFIED-OPERATOR METHOD FOR BAND DIAGRAMS 3

discretization scheme becomes vitally important. This article is concerned precisely

with the study of approximation strategies for energy bands in the Brillouin zone.

The remainder of this article is organized as follows: We begin in Section 2 by

introducing our notation and stating precisely the problem setting and governing

equations. We then present in Section 3 two classical Galerkin discretization strate-

gies for approximating the k-ﬁber eigenvalue problems, and we show the problems

associated with the energy bands produced by these classical approaches. Next,

in Section 4, we present an alternative discretization scheme, systematizing ideas

ﬁrst introduced in the physics literature (see Remark 4.2 below), which is based on

modifying in a controlled manner the underlying k-ﬁber operator. In Section 5, we

present our two main results on the analysis of this alternative approach: we derive

a priori error estimates with respect to a discretization cutoﬀ for the modiﬁed en-

ergy bands, and we show that these bands are periodic and can be made arbitrarily

smooth (away from band crossings) by adjusting suitable parameters in the oper-

ator modiﬁcation approach. Numerical experiments in Section 6 involving a 1D

toy model, and two real materials (graphene and face-centered cubic silicon), vali-

date our theoretical results and showcase the eﬃciency of the operator modiﬁcation

approach. Finally, in Section 7, we present the proofs of our main results.

2. Problem Formulation and Setting

Perfect crystals are structures composed of a periodic arrangement of atoms.

Such structures can therefore be described very conveniently through the use of a

suitable lattice: assuming a d-dimensional lattice with d∈N∗={1,2,3, . . .}, we

denote by {ai}d

i=1 a collection of dlinearly independent primitive vectors in Rd,

and we denote by {bi}d

i=1 ⊂Rdthe corresponding reciprocal vectors, i.e., vectors

in Rdthat satisfy ai·bj= 2πδij ∀i, j ∈ {1, . . . , d}. The primitive lattice L⊂Rd

and reciprocal lattice L∗⊂Rdare then deﬁned as

L:= {Za1+. . . +Zad}and L∗:= {Zb1+. . . +Zbd}.

We denote by Ω⊂Rdand Ω∗⊂Rdthe ﬁrst Wigner-Seitz unit cell of the

primitive and reciprocal lattice respectively. Recall that the ﬁrst Wigner-Seitz unit

cell of a lattice in Rdis the locus of points in Rdthat are closer to the origin of

the lattice than to any other lattice point. The ﬁrst Wigner-Seitz cell Ω∗of the

reciprocal lattice is called the (ﬁrst) Brillouin zone.

Finally for clarity of the subsequent exposition, let us introduce the so-called

translation operator and the related notion of lattice periodicity: Given any y∈Rd

and denoting D(Rd) := C∞

c(Rd)the space of complex-valued smooth compactly-

supported functions on Rd, we deﬁne the translation operator τy:D(Rd)→D(Rd)

as the mapping with the property that

∀Φ∈D(Rd): τyΦ(x) := Φ(x−y)for a.e. x∈Rd.

It follows that for any y∈Rd, the translation operator extends by duality as a

mapping τy:D0(Rd)→D0(Rd).

Given now some Φ∈D0(Rd), we will say that Φis L∗-periodic (resp. L-periodic)

if τGΦ=Φfor all G∈L∗(resp. τRΦ=Φfor all R∈L).

4 ERIC CANCÈS, MUHAMMAD HASSAN, AND LAURENT VIDAL

2.1. Function spaces and norms.

We deﬁne the function space L2

per(Ω) as the set of (equivalence classes of) func-

tions given by

per(Ω) := f∈L2

loc(Rd)such that fis L-periodic,

equipped with the inner-product

∀f, g ∈L2

per(Ω) : (f, g)L2

per (Ω) := ˆΩ

f(x)g(x)dx,

where L2

loc(Rd)denotes the space of complex-valued, locally square-integrable func-

tions on Rd, and f(·)indicates the complex conjugate of f(·). The spaces Lp

per(Ω),

p∈[1,2) ∪(2,∞]are deﬁned analogously.

We denote by B, the orthonormal Fourier basis of L2

per(Ω), i.e.,

B:= eG(x) := 1

|Ω|1

eıG·x:G∈L∗.

Given any f∈L2

per(Ω), we will frequently express fin the form

f=X

G∈L∗b

fGeG,where b

fG:= ˆΩ

f(x)eG(x)dx,and we have

G∈L∗b

fG2=ˆΩ|f(x)|2dx<∞.

Periodic Sobolev spaces of positive orders are constructed analogously. Indeed,

we deﬁne for each s > 0the set

per(Ω) := (f∈L2

per(Ω) : X

G∈L∗1 + |G|2sb

fG2<∞),

equipped with the inner-product

∀f, g ∈Hs

per(Ω) : (f, g)Hs

per (Ω) := X

G∈L∗1 + |G|2sb

fGbgG.

Naturally, we have H0

per(Ω) := L2

per(Ω), and we deﬁne periodic Sobolev spaces

of negative orders through duality, i.e., for each s > 0we deﬁne H−s

per(Ω) :=

Hs

per(Ω)0, and we equip H−s

per(Ω) with the canonical dual norm.

Finally, given a Banach space X, we will write L(X)to denote the Banach space

of bounded linear operators from Xto X, equipped with the usual operator norm.

2.2. Governing operators and quantities of interest.

In this section, we assume that the electronic properties of the crystal that we

study are encoded in an eﬀective one-body Schrödinger operator

(2.1) H := −1

2∆ + Vacting on L2(Rd)with domain H2(Rd),

where V∈L∞

per(Ω) is an L-periodic eﬀective potential. Many electronic properties

of the crystal we study can be computed from the spectral decomposition of this

one-body Hamiltonian operator H, and we are therefore interested in its analysis

and computation. The classical approach to this problem relies on the use of the

Bloch-Floquet transform (see, e.g., [34, Chapter XIII]), which we will now brieﬂy

present. The following exposition is based on the article [10].

MODIFIED-OPERATOR METHOD FOR BAND DIAGRAMS 5

We begin by introducing for each G∈L∗, the unitary multiplication operator

TG:L2

per(Ω) →L2

per(Ω) deﬁned as

∀v∈L2

per(Ω) : TGv(x) = e−ıG·xv(x)for a.e. x∈Rd.

Next, we introduce the Hilbert space of L∗-quasi-periodic, L2

per(Ω)-valued func-

tions on Rdas the vector space

qp(Rd;L2

per(Ω)) := nRd3k7→uk∈L2

per(Ω) : ˆΩ∗kukk2

per (Ω)dk<∞and

uk+G=TGuk∀G∈L∗and a.e. k∈Rdo,

equipped with the inner product

∀u, v ∈L2

qp(Rd;L2

per(Ω)) : (u, v)L2

qp(Rd;L2

per (Ω)) = Ω∗

(uk, vk)L2

per (Ω) dk,

where we have denoted ﬄΩ∗:= 1

|Ω∗|´Ω∗and we have used the subscript ‘qp’ to

highlight quasi-periodicity.

The Bloch-Floquet transform is now the unitary mapping from L2(Rd)to

qp(Rd;L2

per(Ω)) with the property that any u∈D(Rd)is mapped to the element

of L2

qp(Rd;L2

per(Ω)) deﬁned as

Rd3k7→ uk:= X

R∈L

u•+Re−ık·(•+R)∈L2

per(Ω).

Any bounded linear operator A: L2(Rd)→L2(Rd)that is L-periodic, i.e., one

that commutes with the translation operator τRfor all R∈L, is decomposed by

the Bloch transform in the following sense: there exists a function k7→ Akin

L∞

qp Rd;LL2

per(Ω)such that for any u∈L2(Rd), all G∈L∗and a.e. k∈Rdit

holds that

(Au)k= Akuk,and Ak+G=TGAkT∗

G.(2.2)

where the operators (Ak)k∈Rd∈LL2

per(Ω)are called the Bloch ﬁbers of A.

The Bloch decomposition (2.2) can also be extended to unbounded,L-periodic

self-adjoint operators such as the one-body electronic Hamiltonian deﬁned through

Equation (2.1). In this case, the ﬁbers Hk,k∈Rdof the electronic Hamiltonian

Hare unbounded operators on L2

per(Ω) given by

(2.3) Hk:= 1

2(−ı∇+k)2+V, with domain H2

per(Ω).

A detailed proof of this technical result can be found in [34, Chapter XIII].

Thanks to the Bloch-Floquet decomposition (2.2), the spectral properties of the

Hamiltonian Hcan be deduced using properties of the ﬁbers Hk,k∈Rd. Indeed,

it is a classical result (see, e.g., [34, Chapter XIII]) that

•each Hkis a self-adjoint operator on L2

per(Ω) with domain H2

per(Ω) and form

domain H1

per(Ω). Additionally, each Hkis bounded below and has compact

resolvent so that each Hkhas a purely discrete spectrum with eigenvalues

accumulating at +∞and eigenfunctions that form an orthonormal basis

for L2

per(Ω);

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MODIFIED-OPERATORMETHODFORTHECALCULATIONOFBANDDIAGRAMSOFCRYSTALLINEMATERIALSERICCANCÈS,MUHAMMADHASSAN,ANDLAURENTVIDALAbstract.Insolidstatephysics,electronicpropertiesofcrystallinemate-rialsareofteninferredfromthespectrumofperiodicSchrödingeroperators.AsaconsequenceofBloch'stheorem,thenumericalcomput...

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MODIFIED-OPERATOR METHOD FOR THE CALCULATION OF BAND DIAGRAMS OF CRYSTALLINE MATERIALS ERIC CANCÈS MUHAMMAD HASSAN AND LAURENT VIDAL

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