Ecient Chebyshev polynomial approach to quantum conductance calculations Application to twisted bilayer graphene Santiago Gim enez de Castro1 2 3Aires Ferreira1and D. A. Bahamon2 3y

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Eﬃcient Chebyshev polynomial approach to quantum conductance calculations:

Application to twisted bilayer graphene

Santiago Gim´enez de Castro,1, 2, 3 Aires Ferreira,1, ∗and D. A. Bahamon2, 3, †

1School of Physics, Engineering and Technology and York Centre for Quantum Technologies,

University of York, York YO10 5DD, United Kingdom

2School of Engineering, Mackenzie Presbyterian University, S˜ao Paulo - 01302-907, Brazil

3MackGraphe – Graphene and Nanomaterials Research Institute,

Mackenzie Presbyterian University, S˜ao Paulo -01302-907, Brazil

In recent years, Chebyshev polynomial expansions of tight-binding Green’s functions have been

successfully applied to the study of a wide range of spectral and transport properties of materials.

However, the application of the Chebyshev approach to the study of quantum transport proper-

ties of noninteracting mesoscopic systems with leads has been hampered by the lack of a suitable

Chebyshev expansion of Landaeur’s formula or one of its equivalent formulations in terms of Green’s

functions in Keldysh’s perturbation theory. Here, we tackle this issue by means of a hybrid approach

that combines the eﬃciency of Chebyshev expansions with the convenience of complex absorbing

potentials to calculate the conductance of two-terminal devices in a computationally expedient and

accurate fashion. The versatility of the approach is demonstrated for mesoscopic twisted bilayer

graphene (TBG) devices with up to 2.3×106atomic sites. Our results highlight the importance of

moir´e eﬀects, interlayer scattering events and twist-angle disorder in determining the conductance

curves in devices with a small twist angle near the TBG magic angle θm≈1.1◦.

I. INTRODUCTION

The quantum scattering model due to Landauer [1] has

become a central tool in mesoscopic physics because it al-

lows for a clear interpretation of phase-coherent electron

transport in terms of a transmission problem [2]. Within

this framework, the conductance of a mesoscopic system

coupled to ideal leads reads G= (e2/h)Pn,m Tnm, where

Tnm is the transmission probability to scatter elastically

across the system from channel non the source lead to

channel mon the drain lead. Thus, for an ideal con-

ductor, the Landauer formula predicts that changes in

low-temperature conductance occurs in discrete steps of

e2/h (per spin) each time a new transport channel be-

comes accessible at the Fermi level. This unique ﬁnger-

print of noninteracting one-dimensional (1D) conductors

was ﬁrst observed in semiconductor ballistic point con-

tacts more than thirty years ago [3,4], and subsequently

in a variety of systems, including nanowires [5–7], carbon

nanotubes [8–11] and graphene devices [12,13].

Meanwhile, the development of eﬃcient tight-binding

frameworks for numerical quantum transport simulation

has been receiving considerable interest because they can

be used to handle realistic geometries as well as to elu-

cidate the role of imperfections and disorder [14–19].

Among these, tight-binding Green’s function (TBGF)

methods have become a standard class of tools owing to

their ﬂexibility and computational eﬃciency [20–23]. In

addition to providing a convenient framework to calculate

the conductance in multi-terminal devices, the TBGF

approach allows determination of current distributions,

∗aires.ferreira@york.ac.uk

†dario.bahamon@mackenzie.br

local density of states and other quantities of interest,

and can be extended to incorporate the eﬀect of inter-

actions [24,25]. Notwithstanding its proven merits, the

standard implementations of the TBGF method suﬀer

from cubic algorithmic complexity, which severely lim-

its the system sizes attainable. The popular recursive

Green’s function (RGF) technique [26–28] partly miti-

gates this issue by partitioning the computational do-

main into small unit transverse sections whose Green’s

functions are recursively generated, but still requires the

inversion of matrices whose size scale with the width of

the transverse section. This technical challenge has not

precluded the study of ballistic transport through a vari-

ety of nanostructures (including quantum dots [29], inter-

faces between bulk crystals [30], and disordered topolog-

ical insulators [31]), but presents a signiﬁcant hurdle for

performing large-scale simulations beyond the ballistic

regime as well as for tackling complex devices composed

of many diﬀerent materials or with sub-units displaying

large unit cells.

In this paper, we revisit the linear-response transport

framework and formulate a Chebyshev polynomial-based

spectral technique that will allow us to bypass altogether

expensive matrix inversions in the numerical evaluation

of the conductance of mesoscopic systems. The approach,

which makes use of a complex absorbing potential (CAP)

to alleviate the computational resources needs [32,33], is

applied to two-terminal twisted bilayer graphene (TBG)

devices containing in excess of a million orbitals. Our re-

sults show that the spatial modulation of the interlayer

couplings that is responsible for the dramatic modiﬁca-

tion of the band structure of TBG [34–38] translates into

important features in the conductance curves, including

the appearance of symmetric peaks located at energies of

the van Hove singularities which merge into a single peak

(centered at zero energy) as the twist angle approaches

arXiv:2210.11227v3 [cond-mat.mes-hall] 2 Feb 2023

a magic angle. This article is structured as follows: Sec-

tion II lays out the spectral approach to calculating the

two-probe conductance. We also discuss the CAP strat-

egy employed to handle the leads eﬃciently and show

how it can be implemented by means of a simple mod-

iﬁcation of the Chebyshev recursion relations. Section

III presents our results for the ballistic transport regime

of TBG nanoribbons with a 3 ×104nm2cross-section

area, which is, to our knowledge, the largest such sys-

tem simulated with a real-space TBGF method so far.

Section III B investigates the impact of twist-angle dis-

order in the quantum transport properties. Our results

are summarized in Sec. IV.

II. MODEL AND METHODS

We consider a two-terminal device setup composed of

a central region of length LSconnected to leads of length

LC(Fig. 1). The corresponding atomistic tight-binding

Hamiltonian may be written as

H=

b

HLb

VL0

V†

HCb

V†



,(1)

where b

HR(L)and b

HCare the Hamiltonians of the right

(left) lead and central region, respectively, and b

VR(L)de-

scribes the coupling of right (left) contacts to the central

region.

The linear-response conductance is obtained via the

Kubo-Greenwood formula

G(E) = 4e2

hL2

Tr h~ˆvxIm b

G(E)~ˆvxIm b

G(E)i,(2)

where Eis the Fermi energy, ˆvx= (i/~)[ b

HC,ˆx] is the ve-

locity operator in the xdirection and b

G= (E−b

H+i0+)−1

is the TBGF of the full system. Note that ˆvxhas support

only on sites within the central region, so that Eq. (2) cor-

rectly describes the total electric current (I) ﬂowing in

response to constant voltages applied at its boundaries.

The linear-response formulation is preferred here over the

more commonly employed non-equilibrium Keldysh tech-

nique [39,40] since it is amenable to a spectral represen-

tation in terms of Chebyshev polynomials similar to the

bulk longitudinal conductivity [41,42] as shown below.

We note that the equivalence between Landauer-type and

Kubo approaches to linear-response transport is well es-

tablished, and we refer the interested reader to Refs. [43–

45] for additional details. To make use of the spectral

machinery, we start by expanding the TBGF in terms of

Chebyshev polynomials of the ﬁrst kind [46]. To this end,

we apply the linear transformation ˆ

h= ( b

H−E+1)/E−,

with E±= (E>±E<)/2, 1is the identity operator de-

ﬁned on the Hilbert space of the lattice and E>(<)indi-

cates the largest (smallest) eigenvalue of b

H. Note that

FIG. 1. Two-terminal TBG device considered in this work.

Green regions denote the leads. The largest systems consid-

ered here have W= 100 nm and L=LS+ 2LC= 300 nm,

corresponding to a total of 2.2 million orbitals.

this procedure maps the eigenvalues of the Hamiltonian

onto the canonical interval of the Chebyshev polynomi-

als i.e., I= [−1,1]. Likewise, the Fermi energy variable

is transformed according to ε≡(E−E+)/E−. To esti-

mate the end points, E±, we use a power method [47],

and a ‘safety factor’ is included to ensure that no spec-

tral weight falls outside I. This is achieved by means of

a simple uniform re-scaling, E±→(1 + αSF)E±, with

αSF >0 (in this work we use αSF = 0.1).

In terms of the rescaled quantities introduced above,

the imaginary part of the full TBGF admits the following

Chebyshev decomposition [41]

Im b

G(ε) = 2

π√1−ε2

∞

m=0

Tm(ε)

(δm,0+ 1) b

Tm(ˆ

h),(3)

where Tm(ε) are Chebyshev polynomials of the ﬁrst kind,

and the operators b

Tm(ˆ

h) (m∈Z+) satisfy the Chebyshev

recurrence relations: T0(ˆ

h) = 1,T1(ˆ

h) = ˆ

h, and

Tm+1(ˆ

h)=2ˆ

Tm(ˆ

h)−b

Tm−1(ˆ

h).(4)

By virtue of these relations, Eq. (3) and hence Eq. (2)

can be computed by means of an eﬃcient iterative scheme

based on computations of Chebyshev moments (see Sec.

II B for details). Once the Chebyshev expansion [Eq. (3)]

has been evaluated to the desired precision, the TBGF

of the original system is obtained by a simple rescaling

Im b

G(E) = E−1

−Im b

G(ε).

A. CAP and modiﬁed Chebyshev polynomials

Next, we discuss the handling of the ﬁnite-size contacts

in our implementation. As customary, the leads should

be suﬃciently large to behave as proper reservoirs of elec-

trons. In practice, this is a demanding computational

task, especially in nanostructures with large unit cells,

such as the case of the TBG system of interest to this

work. In order to reduce the computational overhead,

we make use of a CAP approach [32,33]. The CAP is a

phenomenological damping term,

Σ≡ −ib

Γ = diag {− b

iΓL,0,−ib

ΓR}(5)

included in the Green’s function that generates absorp-

tion of propagating waves across the contacts, thus min-

imizing reﬂections and emulating the behavior of a semi-

inﬁnite contact. The explicit form of the CAP self-energy

for the setup in Fig. 1can be obtained by means of the

Wentzel–Kramers–Brillouin semiclassical approximation

as detailed in Ref. [33], and is discussed in Sec. II C.

Here, it is important to recognize that the presence of a

self-energy term in the TBGF invalidates the Chebyshev

expansion (3) (this is the very reason why a standard

self-energy formulation describing semi-inﬁnite leads is

avoided in our spectral approach), but can be conve-

niently handled by means of modiﬁed Chebyshev poly-

nomials,b

Qn(ˆ

h, ˆγ) (n∈Z+), which are functions of the

rescaled Hamiltonian, ˆ

h, and the damping operator, ˆγ.

This technique, originally devised for scattering calcula-

tions in molecular systems [48,49], allows to reconstruct

the CAP Green’s function, b

GCAP ≡(E−b

H−b

Σ)−1, via

the modiﬁed recurrence relations: b

Q0(ˆ

h) = 1,b

Q1(ˆ

h) =

e−ˆγˆ

hand

Qm+1(ˆ

h) = 2e−ˆγˆ

Qm(ˆ

h)−e−2ˆγb

Qm−1(ˆ

h) (6)

The formal relation between ˆγ—the main feature of the

new recursion rule that fully encapsulates the eﬀects of

the CAP—and the original damping operator, b

Γ, is de-

rived in the Appendix A for clarity.

B. CAP-Chebyshev conductance algorithm

To evaluate Eq. (2), the TBGF of the device is approx-

imated by means of the modiﬁed Chebyshev polynomials

introduced above. To reduce the cost associated with the

trace operation in Eq. (2), we make use of a stochastic

trace evaluation technique [50]. It consists of replacing

the exact trace by the average expectation value over an

ensemble of random vectors |zrias follows

G(ε) = 4e2

hL2

R−1

r=0 hzr|~ˆvxIm b

G(E)~ˆvxIm b

G(E)|zri,

(7)

where |zri=PN

i=1 χr|iiis a vector with random ampli-

tude on each lattice site, Nis the total number of orbitals

and χrare random (real) variables satisfying white-noise

statistics (i.e., χr= 0 and χrχ0

r=δr,r0, where the bar

denotes the average over the random vector ensemble and

δr,r0is the Kronecker delta symbol). The relative error in

this approach scales favourably as 1/√NR [41], insofar

as the operator being traced remains sparse [42]. As a

rule of thumb, we set the number of random vectors such

that R×Nis on the order of 108, which will aﬀord us

high accuracy in the evaluation of the conductance.

Next, we expand the TBGFs in Eq. (7) in terms of the

modiﬁed Chebyshev polynomials [Eq. (17)] to obtain the

M-order spectral approximation to G(E). Following Ref.

[42], it is convenient to express the conductance as follows

GCAP

M(ε) = 4e2

hL2

R−1

r=0 hφ(r)

+(ε)|φ(r)

−(ε)i,(8)

with the single-shot vectors

|φ(r)

+(ε)i=

M−1

m=0

fm(ε)b

Qm(ˆ

h)ˆvx|zri,(9)

|φ(r)

−(ε)i=

M−1

m=0

fm(ε)ˆvxb

Qm(ˆ

h)|zri,(10)

where

fm(ε) = km

(2 −δm,0)Tm(ε)

√1−ε2(11)

and kmare Jackson kernel coeﬃcients [41] introduced to

suppress Gibbs oscillations generated by the truncation

of the formal inﬁnite series in Eq. (3). In this work, we

will use Mup to 20000 which corresponds to a smearing

of the delta functions (i.e. energy resolution) of δE =

πE−/M ≈1 meV at the band center.

The single shot vectors are constructed on the ﬂy

via a sequence of standard matrix-vector multiplications.

First, by deﬁning the vector |zm

ri ≡ b

Qm(ˆ

h)|zri, Eq. (6)

can be used to yield the sequence

|zm

ri= 2e−ˆγˆ

h|zm−1

ri − e−2ˆγ|zm−2

ri(12)

which is initiated with |z0

ri≡|zriand |z1

ri=e−ˆγˆ

h|zri.

This process is iterated to obtain the M-th order approx-

imation deﬁned as |φ(r)

−(ε)i=PM−1

m=0 fm(ε)ˆvx|zm

ri. A

similar procedure, but with starting vectors |z0

ri= ˆvx|zri

and |z1

ri=e−ˆγˆ

hˆvx|zri, yields the remaining single shot

vector |φ(r)

+(ε)i=PM−1

m=0 fm(ε)|zm

ri.

The numerical determination of the single-shot vec-

tors, |φ(r)

±(ε)i, is the most demanding part of the CAP-

Chebyshev algorithm. However, the complexity of this

approach grows only linearly (see Fig. 2(b)) with the

system size because the relevant matrices in the Cheby-

shev iteration, ˆ

hand γ, are sparse. All together, the

number of operations required by the algorithm scales as

N×E×R×M, where Eis the number of energy points

being considered.

The single-shot algorithm adapted here to the Lan-

dauer problem provides a particularly eﬃcient scheme for

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EcientChebyshevpolynomialapproachtoquantumconductancecalculations:ApplicationtotwistedbilayergrapheneSantiagoGimenezdeCastro,1,2,3AiresFerreira,1,andD.A.Bahamon2,3,y1SchoolofPhysics,EngineeringandTechnologyandYorkCentreforQuantumTechnologies,UniversityofYork,YorkYO105DD,UnitedKingdom2SchoolofEngi...

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Ecient Chebyshev polynomial approach to quantum conductance calculations Application to twisted bilayer graphene Santiago Gim enez de Castro1 2 3Aires Ferreira1and D. A. Bahamon2 3y

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