Breakdown of topological protection due to non-magnetic edge disorder in two-dimensional materials in the Quantum Spin Hall phase Leandro R. F. Lima1and Caio Lewenkopf2 3

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Breakdown of topological protection due to non-magnetic edge disorder in

two-dimensional materials in the Quantum Spin Hall phase

Leandro R. F. Lima1and Caio Lewenkopf2, 3

1Departamento de F´ısica, Instituto de Ciˆencias Exatas,

Universidade Federal Rural do Rio de Janeiro, 23897-000 Serop´edica - 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 11, 2022)

We study the suppression of the conductance quantization in quantum spin Hall systems by a

combined eﬀect of electronic interactions and edge disorder, that is ubiquitous in exfoliated and

CVD grown 2D materials. We show that the interplay between the electronic localized states due

to edge defects and electron-electron interactions gives rise to local magnetic moments, that break

time-reversal symmetry and the topological protection of the edge states in 2D topological systems.

Our results suggest that edge disorder leads to small deviations of a perfect quantized conductance

in short samples and to a strong conductance suppression in long ones. Our analysis is based on

on the Kane-Mele model, an unrestricted Hubbard mean ﬁeld Hamiltonian and on a self-consistent

recursive Green’s functions technique to calculate the transport quantities.

Introduction.– The study of topological phenomena has

grown enormously over the last years in condensed mat-

ter and material sciences, with signiﬁcant impact in both

fundamental and applied research [1–4]. Of particular

interest are two-dimensional (2D) topological insulators

(TIs) characterized by robust edge states with a helical

spin texture. These systems, also called quantum spin

Hall (QSH) systems [1,2], are promising platforms for

transistor and spintronic applications [4]. Theory pre-

dicts that QSH phases require, among other properties,

a strong spin-orbit (SO) interaction. The later can be in-

trinsic, like in inverted band semiconductor heterostruc-

tures [2] and in a variety of 2D materials [5], or extrinsic,

generated by adatom doping [6] or proximity eﬀects [7].

Accordingly, experiments reported QSH realizations in

semiconductor quantum wells [8–12], 2D crystals [13–16],

and graphene with adsorbed clusters [17].

Time-reversal symmetry and momentum-spin locking

make the edge states robust against disorder, prevent-

ing backscattering and causing conductance quantiza-

tion, G0= 2e2/h. There are successful observations of lo-

calized edge states [9,18,19] and spin polarization [20] in

2DTIs. However, the unexpected general experimentally

determined ﬁnite deviations from G0in small systems and

the conductance suppression in larger samples remain as

a long standing and important puzzle [16,21,22].

The proposed backscattering mechanisms in 2DTIs can

be divided into two main categories: interedge hybridiza-

tion and intraedge spin-ﬂip scattering processes. Since

the edge states typically have a penetration depth ξthat

is much smaller than the experimental sample widths W,

interedge hybridization is usually discarded. However,

recent studies speculate that interface roughness in semi-

conductor heterostructures leads to chiral disorder that

can create percolating paths enabling interedge scatter-

ing [21]. In turn, since the magnetic impurities are rare

in molecular beam epitaxy grown semiconductors as well

as in exfoliated 2D materials, the simplest mechanism

for spin-ﬂip scattering to explain the lack of topologi-

cal protection is also ruled out. This motivated several

studies to explore a variety of ingenious mechanisms that

eﬀectively break time-reversal symmetry, namely, noise

[23], edge reconstruction [24], Rashba SO interaction [25],

phonons [26], nuclear spins [27,28], charge puddles [29],

scattering processes due to adatoms [30], to name a few.

Some of those give a temperature dependence at odds

with the experimental ﬁndings [11,16] and, more im-

portantly, most are only suited for semiconductor het-

erostructures [8–11]. To the best of our knowledge, this

is the ﬁrst study to propose a breakdown of topological

protection speciﬁc to 2D crystals.

The combination of localization and electron-electron

(e-e) interactions can also give rise to local magnetic mo-

ments. This feature is quite general and has been ex-

tensively studied in 2D materials, in particular the prop-

erties of vacancy induced localized states [31–33] and of

systems with zigzag terminated edges [34,35]. Recently,

Novelli and coauthors [33] have shown that vacancies-

induced magnetic moments destroy the topological pro-

tection. However, this eﬀect occurs only within narrow

energy resonances. Hence, despite being very insightful,

this mechanism fails to explain the weak dependence of

the conductance on gate potential observed in 2DTIs ex-

periments [8–12,16].

In this Letter we put forward a new non-magnetic dis-

order mechanism to explain the breakdown of the topo-

logical protection in exfoliated and chemical vapor depo-

sition (CVD) grown 2D materials. We show that edge

disorder [36], which is ubiquitous in exfoliated and CVD

grown 2D materials, can lead to localization. We ﬁnd

that short sequences of zigzag edge terminations com-

bined with e-einteractions drive the formation of local

magnetic moments that cause backscattering and destroy

the conductance quantization in 2DTIs. We argue that

the conductance suppression is small in short samples

and can be large in longer ones, in line with experiments.

Model – We describe the system electronic properties

within the topological gap using the Kane-Mele model

arXiv:2210.03846v1 [cond-mat.mes-hall] 7 Oct 2022

Hamiltonian with a Hubbard term [33,37]

H=H0+HSO +HU.(1)

Here H0is the tight-binding Hamiltonian

H0=−tX

hi,ji,α

c†

iαcjα + H.c.(2)

where c†

iα(ciα) creates (annihilates) an electron of spin

αat the honeycomb lattice site iand hi, jilimits the

hopping integrals to nearest neighbor sites.

The second term describes the spin-orbit interaction

due to adsorbed adatoms [6]

HSO = +iλ X

hhi,jii,αβ

νp

ij c†

iασz

αβciβ (3)

where σ= (σx, σy, σz) stand for 2 ×2 Pauli matrices in

the spin space, hhi, jii restricts the sum to second neigh-

bor sites, and λis the hopping integral energy. We as-

sume that the adatoms are adsorbed at the so-called hol-

low positions (centers of the hexagons) of the honeycomb

lattice [6], that we denote by p. Accordingly, νp

ij =±1

distinguishes clockwise (νp

ij = 1) and counterclockwise

(νp

ij =−1) hopping directions with respect to pif the

latter corresponds to an adsorbed adatom position, oth-

erwise νp

ij = 0. The topological gap ∆Tis proportional to

the adatom concentration, namely, ∆T= 6√3λnad [38].

In the limit of nad = 1 all p’s are ﬁlled and one recovers

the original Kane-Mele model [37].

Finally, we account for the e-einteraction using an un-

restricted Hartree-Fock approximation [39] to the Hub-

bard Hamiltonian HU, namely,

HHF

U=U

i,αβ

c†

iα(ni1αβ −mi·σαβ )ciβ −U

(n2

i−|mi|2),

(4)

where Urepresents the on-site (local) e-erepulsive inter-

action, 1is the 2 ×2 identity matrix, while

ni=X

αDc†

iαciαE(5)

is the mean electron occupation of the i-th site and

mi=X

αβ

Dc†

iασαβ ciβE(6)

is related to the local electronic mean spin polarization,

accordingly we refer to mias local magnetic moments.

We consider a system of width Wand length L

with armchair edges along the transport direction, see

Fig. 1(a). Left (L) and right (R) contacts connect the

system with source and drain reservoirs. For simplicity,

we model the contacts by semi-inﬁnite ribbons, with the

same width as the central region and doped at EF≈t

to maximize the number of available propagating modes,

mimicking metallic contacts.

Methods – We study the electronic transport using

the nonequilibrium Green’s-functions formalism (NEGF)

[40–42]. We use the spin-resolved linear conductance

Gαβ as Gαβ(µ)=(e2/h)R∞

−∞ (−∂f0/∂E)Tαβ (E) where

f0(E) = [1 + e(E−µ)/kBT]−1is the Fermi-Dirac distribu-

tion, µis the equilibrium chemical potential and Tαβ is

the transmission coeﬃcient given by [40]

Tαβ(E) = Tr ΓR(E)Gr

αβ(E)ΓL(E)Ga

βα(E).(7)

Here Gr(E)=[E−H−Σr

R(E)−Σr

L(E)]−1and

Ga(E)=[Gr(E)]†are, respectively, the retarded and

advanced Green’s functions in the site representation

and Σr

R(L)is the embedding self-energy that depends

on the retarded contact Green’s functions and the cou-

pling between the contacts R(L) and the central region

[41,42]. The Rand L-terminal linewidths are given by

ΓR(L)(E) = i[Σr

R(L)−(Σr

R(L))†]. Since in our model

T↓↑ =T↑↓ = 0, the total transmission is T=T↑↑ +T↓↓.

We also analyze the nonequilibrium local spin resolved

conductance injected by the R(L) terminal e

GR(L)

iα,jβ , that

is given by e

GR(L)

iα,jβ (µ) = (e2/h)R∞

−∞(−∂f0/∂E)e

TR(L)

iα,jβ (E)

where [43]

TR(L)

iα,jβ (E) = 2 Im hGrΓR(L)Gajβ,iα Hiα,jβ i(8)

is the local transmission of electrons ﬂowing from site j

with spin βto the site iwith spin α. The spin resolved

local current is obtained using the local version of the

Landauer-B¨utiker equation e

iα,jβ VR+e

iα,jβ VL, where

VRand VLare the terminal voltages [43].

We calculate Grusing the recursive Green’s-function

(RGF) technique [44–46], and ΓR,L(E) by decimation

[45,47]. The system Green’s function depends self-

consistently on niand mithat we obtain using optimized

methods (see Supplemental Materials [48] for details).

For technical reasons [48,49] we perform our calcula-

tions at ﬁnite temperature keeping kBT∆T, in line

with all situations of interest. Hence, in what follows we

neglect thermal smearing eﬀects and take µ=EF.

Vacancy-induced magnetic moments.−The occurrence

of vacancies in honeycomb lattices gives rise to quasi-

localized states [50–52], For a suﬃciently strong e-ein-

teraction, due to the Stoner instability, these localized

states lead to the formation of local magnetic moments

[53–55]. For the Hubbard mean ﬁeld approximation, it

has been shown that any ﬁnite Ucauses magnetization

[56]. Vacancy-induced magnetism has been recently pro-

posed as a mechanism to explain the breakdown of the

conductance quantization in TIs [33]. Here, we brieﬂy re-

view this setting and argue that this is hardly a suitable

mechanism to explain the lack of conductance quantiza-

tion observed in experiments.

We consider a system with a vacancy near the bottom

edge, see Fig. 1(a). We take W= 27˚

A, that is suﬃ-

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Breakdownoftopologicalprotectionduetonon-magneticedgedisorderintwo-dimensionalmaterialsintheQuantumSpinHallphaseLeandroR.F.Lima1andCaioLewenkopf2,31DepartamentodeFsica,InstitutodeCi^enciasExatas,UniversidadeFederalRuraldoRiodeJaneiro,23897-000Seropedica-RJ,Brazil2InstitutodeFsica,UniversidadeFe...

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Breakdown of topological protection due to non-magnetic edge disorder in two-dimensional materials in the Quantum Spin Hall phase Leandro R. F. Lima1and Caio Lewenkopf2 3

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