High-energy Landau levels in graphene beyond nearest-neighbor hopping processes Corrections to the eective Dirac Hamiltonian Kevin J. U. Vidarte1and Caio Lewenkopf2 3

2025-05-06 0 0 858.6KB 11 页 10玖币

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High-energy Landau levels in graphene beyond nearest-neighbor hopping processes:

Corrections to the eﬀective Dirac Hamiltonian

Kevin J. U. Vidarte1and Caio Lewenkopf2, 3

1Instituto de F´ısica, Universidade Federal do Rio de Janeiro, 21941-972 Rio de Janeiro - RJ, Brazil

2Instituto de F´ısica, Universidade Federal Fluminense, 24210-346 Niter´oi - RJ, Brazil

3Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany

(Dated: October 10, 2022)

We study the Landau level spectrum of bulk graphene monolayers beyond the Dirac Hamiltonian

with linear dispersion. We consider an eﬀective Wannier-like tight-binding model obtained from

ab initio calculations, that includes long-range electronic hopping integral terms. We employ the

Haydock-Heine-Kelly recursive method to numerically compute the Landau level spectrum of bulk

graphene in the quantum Hall regime and demonstrate that this method is both accurate and

computationally much faster than the standard numerical approaches used for this kind of study.

The Landau level energies are also obtained analytically for an eﬀective Hamiltonian that accounts

for up to third nearest neighbor hopping processes. We ﬁnd an excellent agreement between both

approaches. We also study the eﬀect of disorder on the electronic spectrum. Our analysis helps to

elucidate the discrepancy between theory and experiment for the high-energy Landau levels energies.

I. INTRODUCTION

The pioneering experimental reports [1,2] on the

unique features of the quantum Hall (QH) eﬀect in

graphene systems played a key role to establish the re-

markable massless Dirac electronic properties of this ma-

terial [3–5]. Subsequent investigations on QH eﬀects in

graphene continued to produce fascinating results, such

as Klein tunneling [6,7], fractional quantum Hall eﬀect

[8,9], Hofstadter butterﬂies [10,11], to name a few, at-

tracting a lot of attention to the ﬁeld (see, for instance,

Ref. [12] for a recent review).

In addition to electronic transport properties,

graphene under strong magnetic ﬁelds also shows unique

spectral properties under strong magnetic ﬁelds. The

graphene Landau levels have been theoretically predicted

[4,13] to follow

N= sgn(N)~ωcp|N|(1)

where Nis the Landau level index, ωc=vFp2eB/~is

the cyclotron frequency, Bis the magnetic ﬁeld strength,

and vFstand for the electron Fermi velocity for B= 0.

The LLs spectrum have been experimentally mea-

sured by transmission [14,15] and scanning tunneling

spectroscopy [16,17] for both exfoliated and epithaxial

graphene. These studies have veriﬁed the √BN disper-

sion to a good approximation. A systematic transmis-

sion spectroscopy study [18] focused on the high-energy

LL showed deviations from Npredicted by Eq. (1). The

puzzle is that the disagreement with the experimental

data persists even when one improves the theoretical de-

scription by accounting next-to-nearest neighbor hopping

processes.

In should be stressed that, while the massless Dirac

Hamiltonian has been shown to be very eﬀective in

describing a variety of low energy electronic graphene

properties, the theoretical modeling for higher energies

is not unique. The standard approach uses density

functional theory (DFT) calculations to obtain a tight-

binding model based on Wannier orbitals, that contain

an arbitrary range of hopping terms that ﬁt the nonlinear

features of the dispersion relation [19–24].

Another possibility for the discrepancy between the-

ory and experiment is disorder, which is ubiquitous in

graphene [25]. Numerical investigations have studied the

broadening the Landau subbands [26], but there is no

systematic study of the corresponding disorder-induced

peak shifts. We examine this issue with emphasis on the

large |N|limit.

Our analysis uses the Haydock-Heine-Kelly (HHK) re-

cursive method [27–29], also called the Haydock method,

an order N[30] real-space computational approach de-

veloped to study local spectral functions. It transforms

an arbitrary sparse Hamiltonian matrix in a tridiagonal

form and evaluates the diagonal Green’s function by a

continued fraction expansion, avoiding the need of solv-

ing the full eigenvalue problem. The HHK method has

been successfully used to compute the LDOS of diﬀer-

ent compounds [31,32] and more recently in the study

of carbon nanotubes [33,34], and disordered graphene

systems [35–37].

In addition to its eﬃciency, another attractive feature

of the HHK method is that, since it relies on the concept

of nearsightedness [38], it does not use periodic boundary

conditions [39]. Hence, it can be applied to study local

spectral properties of disordered systems, quasicrystals

[40,41], and systems with very large primitive unit cells,

such as twisted graphene layers and 2D systems under

realistic magnetic ﬁelds B. Surprising the HHK method

has only been employed once in a case where B6= 0,

namely, in the investigation of Hoftstadter butterﬂy en-

ergy gaps in square lattices [42]. To the best of our knowl-

edge, so far no one has realized that the method is also

fast and very accurate (as we show) for the study of dis-

crete QH spectra.

Regarding our results, we numerically show that the

large-|N|LL spectrum analytical solution of the contin-

arXiv:2210.03636v1 [cond-mat.mes-hall] 7 Oct 2022

uum (long wavelength) eﬀective graphene Hamiltonian

is very accurate up to |N| ≤ 25 and B= 25 T. We in-

clude DFT-ﬁtted third nearest hopping matrix elements

into the graphene tight-binding model Hamiltonian and

show that agreement between theory and experiment is

signiﬁcantly improved.

This paper is organized as follows. In Sec. II, we

present the model Hamiltonian we use to describe the

electronic properties in graphene, namely, a tight-binding

Hamiltonian that accounts for up to third nearest neigh-

bor hopping processes. Next, we discuss disorder eﬀects

and review the SCBA predictions for the shifts in the LL

subband peak energies and widths. Finally, we brieﬂy

present the main steps of the implementation of the HHK

method, discuss its computational cost and benchmark

its accuracy. In Sec. III we present our results. We ex-

pand previous analytical results for Nand show that, de-

spite the approximations involved (discussed in App. A),

the agreement with the numerical values is remarkable.

We further show that the inclusion of third nearest neigh-

bor matrix elements helps to improve the agreement be-

tween theory and experiment and that disorder plays a

minor role. We summarize our conclusions in Sec. IV.

II. THEORY AND METHODS

A. Model Hamiltonian

The tight-binding Hamiltonian that describes the elec-

tronic structure of graphene monolayers reads [3,20]

H=X

i,j tij c†

icj+ H.c,(2)

where tij stands for the hopping matrix element between

the Wannier electronic orbitals centered at the carbon

sites iand j. Most studies consider only ﬁrst nearest

neighbor hopping processes, a simple model that is able

to describe the low energy properties of bulk graphene

[3,25]. Tight-binding parameterizations based on den-

sity functional theory (DFT) [19,21,22,24] show the

necessity to including hopping terms beyond ﬁrst nearest

neighbors for a more accurate modeling of the electronic

dispersion, particularly when addressing higher energies.

Here, we consider ﬁrst t(1), second t(2), and third t(3)

nearest neighbor hopping terms. Within this approxima-

tion it is convenient to write the graphene Hamiltonian

in a sublattice matrix representation that in reciprocal

space reads [4,43]

Hk≡t(2)|γk|2t(1)γ∗

k+t(3)γ0

t(1)γk+t(3)γ0∗

kt(2)|γk|2,(3)

with

γk≡1 + eik·a2+eik·(a2−a1)(4)

and

γ0

k≡1 + ei2k·a2+ei2k·(a2−a1),(5)

where a1=√3a0ˆ

exand a2=√3a0/2ˆ

ex+√3ˆ

eyare

the primitive vectors honeycomb lattice and a0= 1.4

is the carbon-carbon bond length [3]. The ﬁrst and

third nearest neighbor hopping terms, that connect dif-

ferent sublattices [3,4], correspond to the oﬀ-diagonal

matrix elements, while the second-nearest hoppings are

related to the diagonal ones. This apparent correspon-

dence between even-odd nearest neighbors and diagonal

oﬀ-diagonal matrix elements breaks down for fourth near-

est neighbors and beyond [23]. In Eq. (3) we neglect a

constant diagonal term that shifts the energy spectrum

by −3t(2).

The energy dispersion reads

kλ=t(2)|γk|2+λ|t(1)γk+t(3)γ0∗

k|,(6)

where λlabels the valence (λ=−1) and conduction

bands (λ= +1). Note that the second-neighbors hop-

ping contributions do not depend of λand, thus, break

the electron-hole symmetry.

The presence of an external magnetic ﬁeld Bcan be

accounted by the Peierls substitution [44,45], that is, by

the transformation

tij −→ tij exp "ie

}ZRj

dr·A(r)#,(7)

where Rnis the lattice vector associated with the site

n,eis the electron charge. Here, we choose the vector

potential A(r) = (0, Bx, 0) that gives a magnetic ﬁeld

B=∇ × A=Bˆ

ezperpendicular to the graphene layer.

B. Numerical method

We compute the Landau level spectra using the

Haydock-Heine-Kelly (HHK) recursion technique [27–

29]. The latter has been developed to calculate the local

properties of electronic systems represented on a basis of

localized (orthogonal) states |ii, like the one used in the

tight-binding Hamiltonian of Eq. (2). The HHK method

provides a very eﬃcient O(N) recursive procedure to

transform a given Hamiltonian matrix into a tridiagonal

one, that is much more amenable for numerical calcula-

tion. As mentioned introduction, the HHK method has

been successfully used to compute the LDOS of contin-

uous spectra of several systems [31–37]. Here, we use

it to compute the LDOS of a Hamiltonian with discrete

eigenvalues.

Let us quickly review the main ingredients of the

HHK method before discussing its use in computing the

graphene Landau level spectrum.

The recursion method starts by targeting a given state

|0}=|ji. The method generates a hierarchy of states

|n}based on the three-term recursion relations [27–29]

H|n}=an|n}+bn|n−1}+bn+1|n+ 1},(8)

with the recursive coeﬃcients

an={n|H|n},(9)

bn+1 =k(H−an)|n} − bn|n−1}k,(10)

and the orthogonal basis element

|n+ 1}=1

bn+1

[(H−an)|n} − bn|n−1}],(11)

where b0= 0 and | − 1}= 0. Thus, by construction,

the Hamiltonian matrix in the orthogonal basis {|n}}

is tridiagonal. In turn, the basis functions |n}can be

expressed in terms of Wannier-like states |ji,

|n}=

i=1

Ani|ii.(12)

Here, we assume that the electronic wave functions are a

superposition of Pstates centered at the atomic sites i

and the atomic orbitals are orthogonal to each other, in

line with the tight binding model of Eq. (2).

The diagonal Green’s function for the seed state |0}is

given by continued fraction [27–29]

G00() = {0|1

−H|0}

−a0−b2

−a1−b2

−a2−b2

...

(13)

expressed in terms of the matrix elements of the tridiag-

onal Hamiltonian in the basis |n}. The LDOS at any site

jcan be written as

LDOS(j, ) = −1

πIm Gr

jj () (14)

≡ −1

πlim

η→0+[Im Gjj (+iη)] .

In practice, a ﬁnite ηserves as a convenient regularization

parameter. For continuous spectra, setting η≈2D/M,

where Dis the bandwidth, guarantees a nice smooth ap-

proximation to LDOS(j, ) [27].

For pristine systems, due to translational symmetry,

the LDOS(j, ) at any jis proportional to the (total)

density of states ρ() [46]. In our calculations we ﬁx jat

the center of the honeycomb lattice of size P.

In previous applications [31,32,42] it has been ob-

served that for a suﬃciently large M, the recursive coeﬃ-

cients converge towards their asymptotic values, namely,

aM→a∞and bM→b∞. The asymptotic value of

a∞is associated with the center of the energy band 0,

namely, 0=a∞/2 [27,28]. In graphene systems, a∞/2

corresponds to the Dirac point energy. For the nearest-

neighbor tight-binding Hamiltonian, the center band en-

ergy is 0= 0 and all ancoeﬃcients are zero. In this

case, by inspecting Eq. (13), one immediately ﬁnds that

G00() = −G00(−). This implies that the LDOS(j, )

has electron-hole symmetry for any jand ρ(0) = 0,

a condition that deﬁnes the so-called Dirac points [3].

When t(2) 6= 0, the coeﬃcient anare no longer zero,

the Dirac points are energy shifted, and the electron-

hole symmetry broken. These simple properties are in

line with well established literature results [3], as they

should.

Let us now apply the HHK method to graphene sys-

tems. We begin showing results of the LDOS for bulk

graphene in the absence of external magnetic ﬁelds.

Throughout this paper we use the tight-binding param-

eters given in Table I.

The optimal number of iterations Mdepends on P, the

size of the system chosen to represent the bulk, as well

as on the desired accuracy. A detailed analysis about the

dependence of the HHK method accuracy on Mfor con-

tinuum spectra can be found in Ref. [28]. For graphene

in the absence of magnetic ﬁeld, we ﬁnd that M≈√P

guarantees a good accuracy. At the end of this section

we study the accuracy for the case where B6= 0.

TABLE I. Tight-binding hopping parameter sets in eV.

Parameterization t(1) (eV) t(2) (eV) t(3) (eV)

A [3] -2.7

B [19] -3.0 0.3

C [22] -3.0933 0.19915 -0.16214

Figure 1shows the LDOS of a graphene monolayer ob-

tained from the HHK method for P= 2.5×107carbon

atoms and M= 5000 iterations. We have contrasted the

the latter with ρ() computed by a direct numerical eval-

uation of ρ() = 1

ABZ Pkδ(−kλ), where kλis given

by Eq. (6) and the wavevectors kare sampled over the

Brillouin zone of area ABZ. By accounting for the nor-

malization factor that relates LDOS(j, ) with ρ(), we

observe a nice agreement within the numerical precision

imposed by the regularization parameter η. We ﬁnd that

diﬀerences are only appreciable, as expected, at the band

edges as well as at the van Hove singularities.

Let us now consider the case of a pristine graphene

sheet under an external perpendicular magnetic ﬁeld B.

The electronic spectrum becomes discrete and strongly

degenerate forming a sequence of Landau levels. Fig-

ure 2(a) shows the graphene LDOS for the tight-binding

parameterizations A, B, and C computed with the HHK

method for B= 25 T, P= 2.25 ×106,M= 1500, and

η= 0.1 meV. Due to the ﬁnite η, the LL are broadened

and become Lorentzian distributions, whose energy peaks

(obtained by ﬁtting) are associated with the LL energies

N. Here, we chose η |N−N−1|, guaranteeing that

the |N| ≤ 30 lowest LL peaks are nicely resolved.

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High-energyLandaulevelsingraphenebeyondnearest-neighborhoppingprocesses:CorrectionstotheeectiveDiracHamiltonianKevinJ.U.Vidarte1andCaioLewenkopf2,31InstitutodeFsica,UniversidadeFederaldoRiodeJaneiro,21941-972RiodeJaneiro-RJ,Brazil2InstitutodeFsica,UniversidadeFederalFluminense,24210-346Niteroi...

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High-energy Landau levels in graphene beyond nearest-neighbor hopping processes Corrections to the eective Dirac Hamiltonian Kevin J. U. Vidarte1and Caio Lewenkopf2 3

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