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Correlated Topological States in Graphene Nanoribbon
Heterostructures
Joost, Jan-Philip; Jauho, Antti-Pekka; Bonitz, Michael
Published in:Nano Letters
Link to article, DOI:10.1021/acs.nanolett.9b04075
Publication date:2019
Document VersionPeer reviewed version
Link back to DTU Orbit
Citation (APA):Joost, J-P., Jauho, A-P., & Bonitz, M.
(2019). Correlated Topological States in Graphene
NanoribbonHeterostructures. Nano Letters, 19(12), 9045-9050.
https://doi.org/10.1021/acs.nanolett.9b04075
https://doi.org/10.1021/acs.nanolett.9b04075https://orbit.dtu.dk/en/publications/8e5ff993-09ac-4fed-82f4-c219eef74e28https://doi.org/10.1021/acs.nanolett.9b04075
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worksproduced by employees of any Commonwealth realm Crown
government in the courseof their duties.
Communication
Correlated Topological States in Graphene Nanoribbon
HeterostructuresJan-Philip Joost, Antti-Pekka Jauho, and Michael
Bonitz
Nano Lett., Just Accepted Manuscript • DOI:
10.1021/acs.nanolett.9b04075 • Publication Date (Web): 18 Nov
2019
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Correlated Topological States in GrapheneNanoribbon
Heterostructures
Jan-Philip Joost,† Antti-Pekka Jauho,‡ and Michael Bonitz∗,†
†Institut für Theoretische Physik und Astrophysik,
Christian-Albrechts-Universität zu Kiel,D-24098 Kiel, Germany
‡CNG, DTU Physics, Technical University of Denmark, Kongens
Lyngby, DK 2800,Denmark
E-mail: [email protected]: +49 (0)431
880-4122. Fax: +49 (0)431 880-4094
AbstractFinite graphene nanoribbon (GNR) het-erostructures host
intriguing topological in-gapstates (Rizzo, D. J. et al. Nature
2018, 560,204]). These states may be localized either atthe bulk
edges, or at the ends of the struc-ture. Here we show that
correlation effects(not included in previous density
functionalsimulations) play a key role in these systems:they result
in increased magnetic moments atthe ribbon edges accompanied by a
signifi-cant energy renormalization of the topologicalend states –
even in the presence of a metallicsubstrate. Our computed results
are in ex-cellent agreement with the experiments. Fur-thermore, we
discover a striking, novel mech-anism that causes an energy
splitting of thenon-zero-energy topological end states for aweakly
screened system. We predict that sim-ilar effects should be
observable in other GNRheterostructures as well.
Keywordsgraphene nanoribbons, heterostructures, topo-logical
states, electronic correlations, Greenfunction theory
Introduction. Strongly correlated materi-als host exciting
physics such as superconduc-
tivity or the fractional quantum Hall effect.1While in monolayer
graphene electron correla-tions are weak,2 carbon based finite
systemsand heterostructures can exhibit flat bands nearthe Fermi
energy resulting in nontrivial corre-lated phases .3,4 One example
are magic-angletwisted graphene bilayers where localized elec-trons
lead to Mott-like insulator states and un-conventional
superconductivity.5–7 Another ex-citing class of systems that has
been predictedto host strongly localized phases are
graphenenanoribbon (GNR) heterostructures.8–14 Simi-lar to
topological insulators they combine aninsulating bulk with robust
in-gap boundarystates15,16 and are expected to host
Majoranafermions in close proximity to a superconduc-tor.17
Recently, it was confirmed that GNRheterostructures composed of
alternating seg-ments of 7- and 9-armchair GNRs (AGNRs), assketched
in Fig. 1(a), exhibit new topologicalbulk bands and end states that
differ qualita-tively from the band structures of pristine 7-and
9-AGNRs.18Although electronic correlations are expected
to play a crucial role for the localized topolog-ical states in
GNR heterostructures,19 so far,most theoretical work has been
restricted totight-binding (TB) models or density functionaltheory
within the local density approximation(LDA-DFT) which are known to
completely ig-nore or underestimate these effects. Here, we
1
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present a systematic analysis of electronic cor-relations in
7-9-AGNRs, based on a Green func-tions method with GW
self-energy20,21 appliedto an effective Hubbard model. We
computethe differential conductance and find excellentagreement
with the experimental measurementsof Ref. 18. Our calculations
reveal that, evenin the presence of a screening Au(111)
surface,local electronic correlations induce a strong en-ergy
renormalization of the band structure. Es-pecially the topological
end states localized atthe heterostructure-vacuum boundary
experi-ence strong quasiparticle corrections which arenot captured
by LDA-DFT. For freestandingsystems, or systems on an insulating
surface,we predict that these states exhibit an energysplitting due
to a magnetic instability at theFermi energy. The local build-up of
electroniccorrelations is further analyzed by consideringthe local
magnetic moment at the ribbon edges.We also examine finite size
effects and the ori-gin of the topological end states by varying
thesystem size and end configuration, respectively.On this basis we
predict that a whole class ofsystems exists that can host end
states withsimilar exciting properties.Model. We consider the
7-9-AGNR het-
erostructure consisting of alternating 7-AGNRand 9-AGNR segments
as depicted in Fig. 1(a).The system was realized experimentally on
ametallic Au(111) surface by Rizzo et al.18 whoobserved topological
in-gap states at the het-erojunction between 7- and 9-AGNR
segments(bulk), and at the termini of the heterostruc-ture (end).
While previous LDA-DFT simula-tions describe reasonably well the
bulk bands,they do not reproduce quantitatively the ex-perimental
energies of the end states (see be-low). To overcome these
limitations, we ap-ply a recently developed Green functions
ap-proach which gives access to spectral and mag-netic properties
of the system, see, e.g. Refs.20,22,23. The electronic system is
describedwith an effective Hubbard model, the Hamil-tonian of which
is expressed in terms of the op-erators ĉ†iα and ĉjα that create
and annihilatean electron with spin projection α at site i and
j, respectively,
Ĥ = −J∑
〈i,j〉,αĉ†iαĉjα + U
∑
i
ĉ†i↑ĉi↑ĉ†i↓ĉi↓ , (1)
where J = 2.7 eV is the hopping amplitude be-tween adjacent
lattice sites,24 and U is the on-site interaction. The edges of the
GNR are as-sumed to be H-passivated. Observables can becomputed
from the Green function G(ω) thatis defined in terms of the
operators ĉ†iα and ĉjα,for details see the Supporting Information
(SI).An approximate solution of Eq. 1 is obtained bysolving the
self-consistent Dyson equation22,23
G(ω) = G0(ω) + G0(ω)Σ(ω)G(ω) , (2)
where the single-particle Green function G con-tains the
spectral and magnetic information ofthe system, and G0 is its
non-interacting limit.Correlation effects are included via the
self-energy Σ. In the present work we report thefirst full GW
-simulations of the system (1, 2)for experimentally realized GNR
heterostruc-tures, as the one shown in Fig. 1(a). The detailsof the
numerical procedure are provided in theSI.
In what follows, we compare the tight-binding(TB), unrestricted
Hartree-Fock (UHF), andfully self-consistent GW approximations for
Σ,in order to quantify electronic correlation ef-fects. The TB
approach, corresponding to set-ting U = 0, is often used to
describe GNRs,due to its simplicity,17,25–27 and here serves asa
point of reference for the uncorrelated sys-tem. For GW the on-site
interaction was cho-sen such that it reproduces the
experimentalbulk band gap of Ref. 18, resulting in U =2.5J , see
SI. This choice of U also takes intoaccount screening effects of
the metallic sub-strate. The description of free-standing
GNRswithin GW requires a larger on-site interac-tion28 which makes
the self-consistent solutionof Eq. (2) more challenging.
Nonetheless, to geta qualitative understanding of the propertiesof
free-standing heterojunctions we employ theUHF approximation which
is known to qualita-tively describe edge magnetism in
free-standing
2
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Bulk Cell End Cell(a)
(b) (c)
−1.0 −0.5 0.0 0.5 1.0(E − EF
)/eV
0.0
0.5
1.0
1.5
2.0
2.5
3.0
dI/
dV/a.u
.
TB
UHFU=1J
GWU=2.5J
experiment
0.74 eV
0.71 eV
0.52 eV
−1.0 −0.5 0.0 0.5 1.0(E − EF
)/eV
0.0
0.5
1.0
1.5
2.0
2.5
dI/
dV/a.u
.
TB
UHFU=1J
GWU=2.5J
experiment
1.32 eV
1.35 eV
1.08 eV
Figure 1: (a) 7-9-AGNR heterostructure containing six unit
cells. The red [blue] cross marks theposition of the dI/dV spectra
shown in (b) [(c)]. The red (blue) dashed rectangle marks the
bulk(end) unit cell referenced in Fig. 3. (b) dI/dV spectrum
measured (red) and simulated (black)at the position in the bulk
region marked by the red cross in (a). (c) dI/dV spectrum
measured(blue) and simulated (black) at the position in the end
region marked by the blue cross in (a). Thecurves in (b) and (c)
are shifted vertically, for better comparison. The experimental
data are takenfrom Ref. 18 and corrected for charge doping
effects.
ZGNRs. For this case the on-site interactionwas chosen as U = 1J
.29 The spatially resolveddI/dV data, recorded in an STM
experiment,are generated by placing 2pz orbitals on top ofthe
atomic sites of the lattice structure, follow-ing the procedures
described in Refs. 30,31.Quasiparticle renormalization. In
Figs. 1(b) and (c) we present differential con-ductance results
for the 7-9-AGNR heterostruc-ture on Au(111), comparing TB, UHF and
GWsimulations to the experiment of Ref. 18.1 Our
1In the experiment the ribbon was slightly dopedwhich shifts the
dI/dV spectrum to higher energies.Our calculations were performed
for half filling. Forcomparison the experimental data was shifted
so thatthe zero-energy peaks of theory and experiment match.
calculations were performed for a system con-taining six unit
cells as shown in Fig. 1(a). Inaccordance with the measurements the
resultsfor the bulk (end) calculation are averaged overthe area
marked by the red (blue) cross. In theexperiment, a band gap of
Eexpg,bulk = 0.74 eV be-tween the bulk bands is observed whereas
thegap between the non-zero energy end peaks isEexpg,end = 1.32 eV.
The TB approximation vastlyunderestimates the gaps, with ETBg,bulk
= 0.52 eVand ETBg,end = 1.08 eV, respectively. In additionto the
two bands in the bulk region an addi-tional unphysical zero-energy
mode appears inthe TB solution.Next, we include mean-field effects
within the
3
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0 1 2 3 4 5x/nm
0
1
y/n
m
UHF
E = EF(a)
0 1 2 3 4 5x/nm
UHF
E = EF ± 0.54 eV
0.0
0.5
1.0
y/n
m
UHF
(b)
0 1 2 3 4 5 6 7 8x/nm
0.0
0.5
1.0
y/n
m
GW
1
2
LD
OS
/a.u
.
0.500
0.501
0.502
0.650
0.675
0.700
loca
lm
om
ent〈m
2 i〉
Figure 2: (a) LDOS of the topological end states at E = EF
(left) and E = EF ± 0.54 eV (right)for UHF as shown in Fig. 1(c).
(b) Local moment 〈m2i 〉, Eq. (3), of the 7-9-AGNR of Fig.
1.(a)calculated within UHF (top) and GW (bottom). Since the ribbon
is symmetric only three of thesix unit cells are shown as indicated
by the three black dots.
UHF approximation. This does not lead toa considerable energy
renormalization for thebulk and end states, but to a splitting of
allthree topological states which are localized atthe end of the
heterostructure, cf. Fig. 1(c).The behaviour of the two non-zero
energy endpeaks is particularly surprising. While the split-ting of
the zero-energy edge peaks in ZGNRsis well known32 and is
attributed to magneticinstabilities at the Fermi level,33 the
splitting ofstates at non-zero energy cannot be understoodin this
picture. In the experimental data wherethe system is on top of a
screening Au(111)surface this effect is not observed.
However,metallic substrates are known to suppress thesplitting of
zero-energy states in finite lengthpristine AGNRs34 compared to
insulating sub-strates.35 The UHF result indicates that a
split-ting of all three end peaks, as seen in Fig. 1(c),will emerge
in measurements for an insulatedheterojunction.Finally, including
quasiparticle correctionswithin the GW approximation results in a
con-siderable correlation-induced renormalizationof both the bulk
and end states. The observedgaps of EGWg,bulk = 0.71 eV and
EGWg,end = 1.35 eVare in excellent agreement with the experi-
mental findings. Additionally, GW correctlyreduces the
unphysical zero-energy contribu-tion in the bulk and prevents the
splitting ofthe end states through self-consistent screening2.Local
correlations. To understand the
mechanisms causing the above mentionedrenormalization and
splitting of the topolog-ical states, we next consider local
correlationsand magnetic polarizations. In Fig. 2(b) wecompare the
local moment at site i
〈m̂2i〉
=〈(n̂↑,i − n̂↓,i)2
〉= ρi − 2Di , (3)
from UHF- and GW -simulations for the samesystem as in Fig. 1.
The local moment quanti-fies the local interaction energy and is a
measureof the local magnetic polarization of the sys-tem. It is
directly related to the local density ρiand double occupation Di at
site i [cf. Eq. (3)]which are strongly affected by electronic
corre-lations. Consequently, the local moment is, in
2One should note that the extreme broadening of theupper bulk
band seen in the experiment is not capturedby our simulations. The
origin of this broadening isunclear at present, however it is
probably caused by theexperimental setup, i.e. the substrate or the
tip.
4
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general, higher for GW than for UHF. In thelatter case the local
moment is peaked exactlyat the sites where the zero-energy end
state islocalized, which can be confirmed by comparingto the LDOS
in Fig. 2(a). Considering the split-ting of the zero-energy end
peak in Fig. 1(c) thisis in agreement with previous mean-field
cal-culations for ZGNRs33,36 where the magneticinstability of the
zero-energy edge state givesrise to an antiferromagnetic ordering
at oppos-ing zigzag edges. However, such an instabilitydoes not
occur for the non-zero end states forwhich the local distribution
only partially co-incides with the local moment, cf. Fig
2(a).Instead, the splitting of these states observedin Fig. 1(c)
originates from their hybridizationwith the zero-energy zigzag
state which is fur-ther investigated in the discussion of Fig.
4.Strikingly, for GW the local moment is in-creased on all edges of
the heterostructure,which coincides with the regions where
thetopological states are localized, cf. Fig. 4(b).Consequently,
the renormalization of the topo-logical bulk and end peaks can be
attributed tostrongly localized correlations at the edges ofthe
heterostructure. The topological states thatextend across the
boundary of the heterostruc-ture result in increased magnetic
polarizationeven at the armchair edges of the ribbon.
Thissurprising finding is in contrast to the predic-tion of the UHF
simulation in this work andother mean-field theory
results29,33,37–40 wheretypically considerable magnetic
polarization isonly observed at the zigzag edges of GNRs.Finite
size effects. The GNR heterostruc-
ture discussed in Figs. 1 and 2 contains six unitcells, exactly
as in the experiments. In the fol-lowing we explore the effect of
the system sizeon the topological states. In Fig. 3 the
localdensity of states (LDOS) of heterostructurescontaining one to
eight unit cells is shown forthe bulk and end cells comparing TB
and GWresults. For the eight unit cell system, addition-ally, the
LDA-DFT result of Ref. 18 is plottedto allow for a direct
comparison to our results.The observed effects differ for the two
regionsof the system. The spectral weight of the bulkbands
increases with system size. In fact, theTB results, that show the
total DOS, indicate
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75(E − EF
)/eV
0
2
4
6
8
10
12
LD
OS/a.u
.
8 cells
6 cells
3 cells
2 cells
1 cell
GW (U=2.5J):
DFT+LDA:
End Cell
End Cell
Bulk Cell
Bulk Cell
TB
GW (U=2.5J):
DFT+LDA:
End Cell
End Cell
Bulk Cell
Bulk Cell
TB
Figure 3: LDOS for 7-9-AGNR heterostruc-tures consisting of one
to eight unit cells. Thecase of six unit cells is the one shown
inFig. 1(a). Red (blue) lines: bulk (end) cell re-sults as shown in
Fig. 1(a). Solid (dashed): GW(LDA-DFT) calculations. The DFT
results aretaken from Ref. 18. Solid grey lines: total DOSfrom TB
calculations. The lines for differentsystems are shifted vertically
for better com-parison.
that the number of peaks in the bulk bandscorresponds to the
number of bulk unit cells.This is in agreement with the idea that
the bulkbands form by hybridization of heterojunctionstates of
adjacent bulk cells. For GW the en-ergies of the bulk states are
renormalized andbroadened by electronic correlations, as shownin
Fig. 1.
In the end cell the three topological statesare stable for
systems of three or more unitcells. For these large systems the
states onboth ends of the ribbon are separated by bulkcells.
However, for smaller systems the statesof opposing termini overlap
and result in anadditional splitting of the end states. In
ad-dition to this topological effect, the energiesof the end states
are also renormalized due toelectronic correlations, in general
resulting inhigher energies of the states. However, inter-estingly,
for the intermediate system of three
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unit cells the splitting of the zero-energy stateis reduced for
the correlated GW -result as com-pared to TB. This is the result of
a competitionof both aforementioned effects. Within GWthe spatial
extension of the zero-energy state isstrongly reduced, cf. Fig 1(b)
and SI, resultingin a significantly smaller overlap and
finite-sizesplitting of the states on both ends of the sys-tem.
Comparing the results of GW and LDA-DFTfor the eight unit cell
system reveals that LDA-DFT captures reasonably well the
renormaliza-tion of the bulk band energies whereas it com-pletely
fails to describe the shift of the endstates. This indicates that a
correct charac-terization of these topological end states is
par-ticularly challenging and requires an accuratedescription of
the underlying electronic correla-tions.Topological end states.
Since the topolog-
ical states at the end of the heterostructure arefound to be
particularly sensitive to electron-electron interactions it is
important to examinein detail the origin of these states. The
emer-gence of topological end states at the terminiof GNR
heterostructures can be explained bythe Z2 invariant and the
bulk-boundary cor-respondence of topological insulator
theory.10However, to get a better understanding of theirproperties,
in the following, the specific mech-anism that leads to the
existence of these mul-tiple end states will be analyzed. For
this,in Fig. 4 the total DOS (a) and the dI/dVmaps (b) of the
heterostructure containing sixunit cells (green) are compared to
the samesystem with an additional ten zigzag lines oneach side of
the ribbon (orange) for GW . Thetwo systems are depicted in the
topmost panelof Fig. 4(b). Comparing the DOS it standsout that the
high-energy end peaks, that arepresent in the six unit cell system,
are stronglysuppressed in the longer system. Instead a
newhigh-energy peak emerges around ±1 eV. Addi-tionally, the
spectral weight of the bulk bandsis increased for the longer
system. All these ob-servations can be understood from the
dI/dVmaps of the states in Fig. 4(b). For the longersystem the
zigzag edge at the end of the system
−1.0 −0.5 0.0 0.5 1.0(E − EF
)/eV
0
1
2
DO
S/a.u
.
1
2 3 4
5
(a)
(b)
1
2
3
4
5
Figure 4: (a) Total DOS calculated with GWand U = 2.5J for the
two systems indicatedin the top panel of (b). The green
(orange)line corresponds to the left (right) system. Theleft system
contains exactly six unit cells whilethe right system is extended
additionally by tenzigzag lines on both sides. (b) dI/dV maps
forthe same two systems. The maps labelled 1-5correspond to the
labelled peaks in (a). Only asmall section of the ribbon is shown
indicatedby the three black dots in the top panel.
is separated from the heterojunction of the firstunit cell by a
long 7-AGNR section. Due to thisclear separation the three end
states (1, 3, 5)resemble states observed in pristine 7-AGNR,see SI.
Furthermore, the heterojunction statesof all six unit cells
hybridize to form the bulkbands (2, 4). In contrast, for the
shorter systemthe adjacent states localized at the zigzag edgeand
at the heterojunctions of the first unit cellcan overlap leading to
three hybridized statesin the termini of the system. As a
consequencethe bulk bands are localized only in the fourinner bulk
unit cells and do not occupy the endcells resulting in a reduced
spectral weight inthe DOS.
Splitting of peaks induced by the spatial over-lap of the
associated states typically appears insmall finite systems, cf.
Fig. 3 for one unit cell.
6
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Strikingly, due to the close proximity of finitesubstructures,
i.e. heterojunctions and ribbonedges, the same effect emerges at
the terminiof large heterostructures containing hundredsto
thousands of atoms. Consequently, this ob-servation is not
restricted to the present 7-9-AGNR heterostructures. Instead, we
predictthat similar strongly correlated states exist alsoin an
entire class of systems which meet the cri-teria for hosting
hybridized end states, i.e. pos-sess topological states close to
localized edgestates. The existence of such states is deter-mined
by the topology of the system.10,41,42Summary and discussion. We
analyzed
the influence of electronic correlations on thetopological
states of 7-9-AGNR heterostruc-tures on Au(111). While the general
topolog-ical structure of the system, previously pre-dicted on the
TB level,10 remains stable, ad-ditional new effects connected to
the topolog-ical states emerge. Our GW simulations re-veal that
strong local electronic correlations arepresent in both the edges
of the bulk and theend region of the heterostructure resulting
inincreased magnetic moments in the zigzag andarmchair edges.
Strikingly, the spatially con-fined topological states of the
termini are moreseverely affected by these correlations than
theextended topological bulk bands. For the lat-ter we found, by
comparison to our GW re-sults, that LDA-DFT is able to reproduce
theexperimental dI/dV measurements since quasi-particle corrections
are weak, in this case. Incontrast, the topological end states,
emergingdue to the hybridization of zigzag-edge andheterojunction
states, are strongly renormal-ized and screened due to electronic
correlationswhich gives rise to a large discrepancy betweenLDA-DFT
and experimental energies. For free-standing heterostructures we
find a new mecha-nism that leads to the splitting of non-zero
en-ergy states due to hybridization with a zero-energy state. These
findings are not restrictedto the specific system considered here
but in-stead are expected to be present in similar
GNRheterostructures that exhibit strongly localizedtopological
states.
Acknowledgement We acknowledge helpful
discussions with Niclas Schlünzen. APJ is sup-ported by the
Danish National Research Foun-dation, Project DNRF103.
Supporting Information Avail-ableThe following files are
available free of charge.
• suppinfo.pdf: Details on the Green func-tion approach, the
fitting of the Hub-bard interaction U , damping of the zero-energy
state, and a comparison to 7-AGNR states.
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