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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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is published by the American Chemical Society. 1155 Sixteenth Street N.W.,Washington, DC 20036Published by American Chemical Society. Copyright © American Chemical Society.However, no copyright claim is made to original U.S. Government works, or 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: bonitz@theo-physik.uni-kiel.dePhone: +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

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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

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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

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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.

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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.

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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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