Electronic localization in small-angle twisted bilayer graphene
V. Hung Nguyen, D. Paszko, M. Lamparski, B. Van Troeye, V. Meunier, J.-C. Charlier
EElectronic localization in small-angle twistedbilayer graphene
V. Hung Nguyen, ∗ , † D. Paszko, † M. Lamparski, ‡ B. Van Troeye, ‡ V. Meunier, ‡ and J.-C. Charlier ∗ , † † Institute of Condensed Matter and Nanosciences, Universit´e catholique de Louvain(UCLouvain), B-1348 Louvain-la-Neuve, Belgium ‡ Department of Physics, Applied Physics, and Astronomy, Rensselaer PolytechnicInstitute, Troy, New York 12180, United States
E-mail: [email protected]; [email protected]
Abstract
Close to a magical angle, twisted bi-layer graphene (TBLG) systems exhibitisolated flat electronic bands and, ac-cordingly, strong electron localization.TBLGs have hence been ideal platformsto explore superconductivity, correlatedinsulating states, magnetism, and quan-tized anomalous Hall states in reduceddimension. Below a threshold twistangle ( ∼ . ◦ ), the TBLG superlatticeundergoes lattice reconstruction, lead-ing to a periodic moir´e structure whichpresents a marked atomic corrugation.Using a tight-binding framework, this re-search demonstrates that superlattice re-construction affects significantly the elec-tronic structure of small-angle TBLGs.Remarkably, the first magic angle at ∼ . ◦ is found to be the critical case pre-senting globally maximized electron lo-calization and separating reconstructedTBLGs into two classes with clearly dis-tinct electronic properties. While low-energy Dirac fermions are still preservedat large twist angles > . ◦ , small-angle( (cid:46) . ◦ ) TBLG systems present commonfeatures such as large spatial variationand strong electron localization stablyobserved in unfavorable AA stacking re- gions. In very good agreement with exist-ing experiments, these new observationsclarify essentially the reasons of the ab-sence of predicted magic angles below . ◦ previously reported. Ever since its first successful controlled iso-lation, graphene and related materials havebecome the subject of intense research. Forexample, twisted bilayer graphene (TBLG) isobtained when the two monolayers are rotatedrelative to each other by an arbitrary angle θ . TBLGs display many fascinating proper-ties that can be tuned by varying the twist an-gle θ . This possibility has given rise to thenascent field of twistronics . In particular, inthe vicinity of the first magic angle experimen-tally observed at ∼ . ◦ , strong interlayerhybridization leads to the emergence of local-ized flat bands near the Fermi level. Becausethe kinetic energy contribution of these states isnegligible, they are prone to lead to strongly en-hanced many-body effects. This strong elec-tronic correlation in small-angle TBLG has ledto the observation of several novel phenomenasuch as superconductivity, correlated insulatingstates, magnetism, and even quantized anoma-lous Hall states.
In spite of the large body of work devoted tothe electronic properties of TBLGs as a func-tion of twist angle, a comprehensive un-derstanding is still missing, especially concern-1 a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b ng the origin of magic angles. This difficultyis due in large part to the presence of a largenumber of atoms in the supercell of small-angleTBLGs, which precludes a detailed modelingof their properties. For example, a numberof studies using a continuum Dirac Hamilto-nian, predict the existence of a discreteseries of other magic angles < . ◦ , in additionto the first magic angle located around 1 . ◦ .However, these smaller magic angles are notobserved when using more accurate calculationsbased on atomistic tight-binding models param-eterized by density functional theory. Infact, the presence of the first magic angle hasbeen demonstrated in several experiments but there is no experimental evidence pointingto the existence of other magic angles < . ◦ .Small-angle TBLG systems have been foundto be characterized by significant atomic re-construction. In that sense, TBLGs can beconsidered as a network of topological domainwalls ( i.e. , soliton lines) connecting unfavorableAA stacking regions and separating more en-ergetically favorable triangular AB/BA stack-ing domains alternatively arranged.
Thus,these small-angle TBLG lattices exhibit a verydifferent stacking network, compared to large-angle ones where the moir´e superlattice evolvessmoothly. As a consequence, a distinguishedfeature is the observation of low-energy helicaledge states in small-angle TBLGs.
More-over, this specific stacking structure has beenshown to induce a strongly spatial dependenceof the electronic and vibrational properties aswell as related features.
On this basis,an accurate investigation encompassing boththe global and local properties in small-angleTBLG systems is urgently needed to develop animproved understanding of their unusual elec-tronic properties.In this work, a tight-binding approach is em-ployed to compute the electronic properties ofTBLGs around and below the first magic angle( θ MA1 (cid:39) . ◦ ). In contrast to most of the pre-vious reports, this work fully takes the moir´esuperlattice reconstruction into account. Inter-estingly, θ MA1 is found to be a critical case wherethe electron localization is maximum, thus sep-arating TBLG systems into two main classes ( i.e. , large- and small-angles) with clearly dis-tinct electronic features. Particularly and re-markably, for angles below θ MA1 , the local elec-tronic properties in AA stacking regions com-monly display a strong electron localization,similar to that globally observed for θ MA1 , andare found to reach a saturation when furtherreducing θ . Such an electron localization doesnot occur in the AB and BA regions. Instead,the electronic properties of AB/BA stackingzones approach those of Bernal stacking bilayergraphene and helical edge states are observed atlow energies around the soliton lines (SL). Onthe basis of these properties and, including thefact that the contribution of AA zones to theglobal properties of the system is reduced when θ decreases, isolated flat bands as observed at θ MA1 can no longer be obtained for smaller an-gles. In agreement with experiments (particu-larly, those from Refs. ), our numeri-cal analysis demonstrates that the global elec-tron localization is monotonically reduced whenreducing θ below θ MA1 , thus confirming the ab-sence of smaller magic angles.TBLG moir´e superlattices were relaxedto minimize their energy using a second-generation REBO potential for intralayerinteractions and the registry-dependentKolmogorov-Crespi potential to describe theinterlayer ones. This way, all atomic posi-tions and superlattice parameters of the con-sidered TBLG structures were optimized untilall force components are less than 1 meV/atom.The electronic properties of these relaxed lat-tices were then computed using a standard p z tight-binding model. Note that the tight-binding parameters used in the present workhave been adjusted (see Supplementary Infor-mation (SI)) in order to model more accuratelythe first magic angle experimentally observedat ∼ . ◦ . To overcome the numerical chal-lenges related to the very large number ( i.e. ,10 to > ) of atoms in the large supercell ofsmall-angle TBLG, the tight-binding Hamilto-nian was solved using recursive Green’s functiontechniques that enables computing the real-space and momentum dependent local densityof electronic states (see also SI). Additionally,this technique presents the other advantage of2nvestigating both global and local electronicproperties.Our study confirms that atomic reconstruc-tion is negligible for large twist angles but isessential for small ones since it significantlymodifies the stacking structure of TBLGs, com-pared to the ideal isolated graphene lattices.As a consequence, the electronic properties ofsmall-angle TBLG systems are expected to bestrongly affected as well. To confirm thisstatement, electronic band structures and DOSof TBLG featuring three different twist an-gles (3 . ◦ , . ◦ , and 0 . ◦ ) are presented inFig. 1. For twist angles above θ MA1 , the effectof atomic reconstruction is almost negligiblewhereas around and below θ MA1 , the electronicproperties drastically change when the super-lattice relaxation is performed. Around θ MA1 ( i.e. , 1 . ◦ here), the electronic structure of re-constructed TBLG clearly exhibits isolated flatbands and strong electron localization peaks inthe DOS near the Fermi level, in very goodagreement with experiments as well as with pre-vious theoretical works. For angles be-low θ MA1 , while the electronic structure of ideallattices still present some partially flat bands,all low-energy bands of relaxed TBLG systemsare found to be much more dispersive. Notethat in all the reconstructed TBLG systems ex-amined in this work (as more fully presented inFigs. S2a-S2b of SI and for θ down to ∼ . ◦ (not shown)), neither isolated flat bands norhighly localized DOS peaks around the Fermilevel occur again for small angles θ < θ MA1 .We now turn to the local electronic quantitiesof TBLGs when the effect of the moir´e super-lattice relaxation is taken into account. Theprojected densities of states (PDOS) on car-bon sites from different stacking zones and thecorresponding local DOS (LDOS) are shown inFig. 2. For large angles, only a weak spatialdependence is observed. However, this depen-dence is strongly enhanced for angles (cid:46) θ MA1 .More remarkably, in all small-angle cases shownin Fig. 2 (and more fully presented in Figs. S3a-S3b of SI), the low-energy bands exhibit astrong localization in the AA stacking regionsand, accordingly, a high LDOS peak near theFermi level is always observed. Moreover, these
RELAXEDIDEAL θ ≈ 3.15° θ ≈ 1.08° θ ≈ 0.50° -0.4-0.2-0.0-0.2-0.4-0.2-0.1-0.0-0.1-0.2-0.1-0.05-0.0-0.05-0.1 -0.1-0.05-0.0-0.05-0.1-0.2-0.1-0.0-0.1-0.2-0.4-0.2-0.0-0.2-0.4-0.2-0.2-0.2
DOS (a.u.) E n er gy ( e V ) E n er gy ( e V ) E n er gy ( e V ) K Γ M K’
Figure 1: Electronic band structures and corre-sponding DOS of TBLG systems with differenttwist angles: above ( θ (cid:39) . ◦ ), in the vicin-ity of ( θ (cid:39) . ◦ ) and below ( θ (cid:39) . ◦ ) thefirst magic angle, with (red curves) and with-out (blue curves) atomic structure relaxation ofthe moir´e superlattice.localized bands are mostly isolated from high-energy bands by a few bands with relativelylow LDOS in the energy range of [ ± . , ± . DOS intensity (a.u.)0 high θ ≈ 3.15° θ ≈ 1.08°
AAAB/BASL E n er gy ( e V ) E n er gy ( e V ) E n er gy ( e V ) θ ≈ 0.50° LDOS (a.u.) -0.4-0.2-0.0-0.2-0.4-0.2-0.1-0.0-0.1-0.2-0.2-0.1-0.0-0.1-0.2 -0.4-0.2-0.0-0.2-0.4-0.2-0.1-0.0-0.1-0.2-0.2-0.1-0.0-0.1-0.2-0.0-0.0-0.0
K M K’ΓK M K’ΓK M K’Γ
AA AB/BA SLAA AB/BA SLAA AB/BA SLAA AB/BA SL
Figure 2: Projected DOS computed in AA zones (first column), AB/BA zones (second column)and at the center of soliton lines (third column) and the corresponding local DOS in TBLG fordifferent twist angles as investigated in Fig.1.angle TBLG systems present the following com-mon features: (i) a strong spatial variation and(ii) a strong electron localization in AA stack-ing regions, similar to the associated flat bandsobserved for θ MA1 .To further clarify the spatial dependence ofthe electronic properties of these small-angleTBLGs, the LDOS at the most strongly lo-calized peaks in AA stacking regions ( i.e. , vanHove singularities at large angles and zero en-ergy at small angles), are represented in realspace in Fig. 3(a). The inset is a typicalreal-space image depicting low-energy LDOSin these TBLG systems where a high peakis present in the AA regions and where the AB/BA zones and SLs are marked. Here,LDOS along a straight line from the center ofAA region to the center of the SL is investigatedfor different θ . First, these results confirm thatthe spatial dependence and the electron local-ization in AA zones are weak for large twist an-gles but are strongly enhanced when reducing θ .Most interestingly, the electron localization inAA stacking regions is found to saturate when θ passes below θ MA1 . Whereas other stackingzones with small low-energy LDOS are contin-uously enlarged when reducing θ in the regime (cid:46) θ MA1 , the electron localization peak in theAA stacking zones is almost unchanged. As aconsequence, the contribution of AA stacking4egions to the global properties of the TBLGsystem is monotonically reduced, thus explain-ing the disappearance of the localized peak inthe total DOS at small twist angles as presentedin Fig. 2 even though a high localized peak inLDOS is still present in AA stacking region. Inother words, although not observed directly inthe global electronic properties, the strong elec-tron localization, similar to that associated tothe presence of flat bands at θ MA1 , is preservedlocally in the AA stacking zones of TBLG sys-tems below θ MA1 .Remarkably, the variation (at large angles)and stability (at small angles) of the electronlocalization in AA zones presented in Fig. 3(a)are found to be essentially due to the depen-dence of stacking structure on the twist angle.This can be understood by analysing the localstructural deformations ( i.e. , the spatial vari-ation of C-C bond length) for different θ (seeFig. 3(b)). These local strain fields are a directconsequence of the stacking structure of recon-structed TBLG superlattices as further demon-strated in the insets of Fig. 3(b) and discussedin details in Fig. S5 of SI. This analysis showsthe strong correlation between the electron lo-calization in the AA stacking regions and theformation of those regions when varying thetwist angle. In particular, in the large-angle( > θ MA1 ) regime, the AA stacking regions (seethe images on top of Fig. 3(b)) and the elec-tron localization ( i.e. , height and width of local-ization peak in Fig. 3(a)) are concurrently en-larged when reducing θ . Both the size of the AAstacking regions and the electron localizationpeaks in these regions however reach a satura-tion point when approaching θ MA1 . Indeed, theyare almost unchanged (as illustrated in Fig. 3(a)and the images in the bottom of Fig. 3(b)) whenfurther reducing θ in the small-angle ( (cid:46) θ MA1 )regime. We conclude that the atomic recon-struction is the essential factor governing theobserved properties related to the electron lo-calization in AA stacking regions as well asother common properties of small-angle TBLGmoir´e superlattice presented above (as furtherdiscussed in Fig. S4 of SI).We will now discuss the consistency betweenthe present theoretical results and the available (a) = 3.48° 2.0° 0.5° 1.1° ≈ C-C bond length (Å)
AA stacking (b)
Figure 3: (a) Local DOS (with the inset illus-trating a real-space image of zero-energy localDOS at θ ≈ . ◦ ) extracted along the line froma AA zone (star) to the center of soliton line(circle) passing though AB/BA zone (square)as indicated in the inset for different twist an-gles. Local DOS is normalized to its maximumvalue obtained in AA zone. Energy at van Hovesingularity point of the DOS is considered forangles > . ◦ while zero energy is computed forangles (cid:46) . ◦ . (b) Spatial variation of the C-Cbond length computed for one layer of TBLGwith large (top) and small (bottom) twist an-gles. Insets on top illustrate the strong correla-tion between weakly strained (light blue) zonesand AA stacking ones.experimental data from the literature. First,the first magic angle obtained here and accord-ingly the maximum electron localization at ∼ θ MA1 are in very good agreement with the exper-imental measurements presented in Refs.,
Itis important to note that the size evolution(at large angles > θ
MA1 ) and the stability (atsmall angles (cid:46) θ MA1 ) of the AA stacking re-gions as well as of low-energy localization peakcorrespond to STM/STS results when varyingthe twist angle. More specifically, the width( ∼ −
10 nm) of the localization peak at smallangles shown in Fig. 3(a) matches the experi-mental data presented in Ref. Finally, as em-phasized above, one of the most important re-sults is that even though it is not observed inthe global electronic properties, the strong elec-tron localization is preserved in the AA stack-ing regions of small-angle TBLG systems. Thisresult is in agrement with the recent Ramanstudy performed on small-angle TBLGs. Inparticular, this study shows that the full-widthat half maximum of the G band (Γ G ) measuredby micro-Raman techniques is maximal around θ MA1 and monotonically reduced when reducing θ . These spectroscopic results thus confirm thatindeed, the electron localization (the direct ori-gin of large Γ G ) is globally maximal at θ MA1 , butalso demonstrate that such electron localizationdecreases monotonically for smaller values of θ .Moreover, Γ G measured locally in the AA stack-ing regions of small-angle TBLGs ( i.e., by thenano-Raman technique) remains large and sim-ilar to the Γ G value obtained at θ MA1 .To summarize, the electronic properties ofsmall-angle TBLGs with fully reconstructedmoir´e superlattices have been investigated us-ing an atomistic tight-binding approach. Thefirst magic angle obtained at θ MA1 (cid:39) . ◦ is found to be a critical case globally rep-resenting the maximum electron localizationand, remarkably, separating TBLGs into large-and small-twist angle classes which exhibitdistinct electronic properties. In contrast tothe large-angle cases where the model of low-energy Dirac fermions is conserved, small-angleTBLGs present strong spatial variation andstrong electron localization (both in energy and real-space) in the AA stacking regions. In thesmall-angle regime, when θ is reduced below θ MA1 , the contribution of AA stacking regionsto the global properties of TBLG gradually re-duces. Consequently, isolated flat bands as wellas strong electron localization can no longer beobserved in the global electronic properties ofsmall-angle TBLGs, implying that magic anglesbelow 1 . ◦ previously predicted by other theo-retical models do not practically exist. Theseresults are in very good agreement with exist-ing experiments (both Raman and STM/STSspectroscopies) and thus provide a more com-prehensive and accurate understanding of theelectronic properties of small-angle TBLG sys-tems. Acknowledgments - V.-H.N. and J.-C.C. acknowledge financial support from theF´ed´eration Wallonie-Bruxelles through theARC on 3D nano-architecturing of 2D crystals(N ◦ ◦ ◦ R.8010.19), and from the Bel-gium FNRS through the research project (N ◦ T.0051.18). V.M. acknowledges support fromNY State Empire State Development’s Divisionof Science, Technology and Innovation (NYS-TAR). Computational resources have been pro-vided by the CISM supercomputing facilities ofUCLouvain and the C ´ECI consortium fundedby F.R.S.-FNRS of Belgium (N ◦ References (1) Novoselov, K. S.; Geim, A. K.; Mo-rozov, S. V.; Jiang, D.; Zhang, Y.;Dubonos, S. V.; Grigorieva, I. V.;Firsov, A. A. Electric Field Effect inAtomically Thin Carbon Films.
Science , , 666–669.(2) Ferrari, A. C. et al. Science and technol-ogy roadmap for graphene, related two-dimensional crystals, and hybrid systems. Nanoscale , , 4598–4810.(3) Andrei, E. Y.; MacDonald, A. H.6raphene bilayers with a twist. Nat.Mater. , , 1265–1275.(4) Wang, J.; Mu, X.; Wang, L.; Sun, M.Properties and applications of new super-lattice: twisted bilayer graphene. Mater.Today Phys. , , 100099.(5) Carr, S.; Fang, S.; Kaxiras, E. Electronic-structure methods for twisted moir´e lay-ers. Nat. Rev. Mater. , , 748–763.(6) MacDonald, A. H. Bilayer Graphene’sWicked, Twisted Road. Physics , ,12.(7) Bistritzer, R.; MacDonald, A. H. Moir´ebands in twisted double-layer graphene. PNAS , , 12233–12237.(8) Cao, Y.; Fatemi, V.; Fang, S.; Watan-abe, K.; Taniguchi, T.; Kaxiras, E.;Jarillo-Herrero, P. Unconventional super-conductivity in magic-angle graphene su-perlattices. Nature , , 43–50.(9) Yankowitz, M.; Chen, S.; Polshyn, H.;Zhang, Y.; Watanabe, K.; Taniguchi, T.;Graf, D.; Young, A. F.; Dean, C. R. Tun-ing superconductivity in twisted bilayergraphene. Science , , 1059–1064.(10) Utama, M. I. B. et al. Visualization ofthe flat electronic band in twisted bilayergraphene near the magic angle twist. Nat.Phys. , , 184–188.(11) Choi, Y. W.; Choi, H. J. Strong electron-phonon coupling, electron-hole asymme-try, and nonadiabaticity in magic-angletwisted bilayer graphene. Phys. Rev. B , , 241412.(12) Kerelsky, A.; McGilly, L. J.;Kennes, D. M.; Xian, L.; Yankowitz, M.;Chen, S.; Watanabe, K.; Taniguchi, T.;Hone, J.; Dean, C.; Rubio, A.; Pa-supathy, A. N. Maximized electroninteractions at the magic angle in twistedbilayer graphene. Nature , ,95–100. (13) Cao, Y.; Fatemi, V.; Demir, A.; Fang, S.;Tomarken, S. L.; Luo, J. Y.; Sanchez-Yamagishi, J. D.; Watanabe, K.;Taniguchi, T.; Kaxiras, E.; Ashoori, R. C.;Jarillo-Herrero, P. Correlated insulatorbehaviour at half-filling in magic-anglegraphene superlattices. Nature , , 80–84.(14) Lu, X.; Stepanov, P.; Yang, W.; Xie, M.;Aamir, M. A.; Das, I.; Urgell, C.; Watan-abe, K.; Taniguchi, T.; Zhang, G.; Bach-told, A.; MacDonald, A. H.; Efetov, D. K.Superconductors, orbital magnets andcorrelated states in magic-angle bilayergraphene. Nature , , 653–657.(15) Jiang, Y.; Lai, X.; Watanabe, K.;Taniguchi, T.; Haule, K.; Mao, J.; An-drei, E. Y. Charge order and broken ro-tational symmetry in magic-angle twistedbilayer graphene. Nature , , 91–95.(16) Xie, Y.; Lian, B.; J˜ack, B.; Liu, X.;Chiu, C.-L.; Watanabe, K.; Taniguchi, T.;Bernevig, B. A.; Yazdani, A. Spectro-scopic signatures of many-body corre-lations in magic-angle twisted bilayergraphene. Nature , , 101–105.(17) Choi, Y.; Kemmer, J.; Peng, Y.; Thom-son, A.; Arora, H.; Polski, R.; Zhang, Y.;Ren, H.; Alicea, J.; Refael, G.; von Op-pen, F.; Watanabe, K.; Taniguchi, T.;Nadj-Perge, S. Electronic correlations intwisted bilayer graphene near the magicangle. Nat. Phys. , , 1174–1180.(18) Balents, L.; Dean, C. R.; Efetov, D. K.;Young, A. F. Superconductivity andstrong correlations in moir´e flat bands. Nat. Phys. , , 725–733.(19) Serlin, M.; Tschirhart, C. L.; Polshyn, H.;Zhang, Y.; Zhu, J.; Watanabe, K.;Taniguchi, T.; Balents, L.; Young, A. F.Intrinsic quantized anomalous Hall effectin a moir´e heterostructure. Science , , 900–903.720) Ma, C.; Wang, Q.; Mills, S.; Chen, X.;Deng, B.; Yuan, S.; Li, C.; Watanabe, K.;Taniguchi, T.; Du, X.; Zhang, F.; Xia, F.Moir´e Band Topology in Twisted BilayerGraphene. Nano Lett. , , 6076–6083.(21) Lopes dos Santos, J. M. B.; Peres, N.M. R.; Castro Neto, A. H. Graphene Bi-layer with a Twist: Electronic Structure. Phys. Rev. Lett. , , 256802.(22) Trambly de Laissardi`ere, G.; Mayou, D.;Magaud, L. Localization of Dirac electronsin rotated graphene bilayers. Nano Lett. , , 804–808.(23) Moon, P.; Koshino, M. Energy spectrumand quantum Hall effect in twisted bilayergraphene. Phys. Rev. B , , 195458.(24) Nguyen, V. H.; Dollfus, P. Strain-inducedmodulation of Dirac cones and van Hovesingularities in a twisted graphene bilayer.
2D Mater. , , 035005.(25) Leconte, N.; Javvaji, S.; An, J.; Jung, J.Relaxation Effects in Twisted BilayerGraphene: a Multi-Scale Approach. arXiv:1910.12805 ,(26) Carr, S.; Fang, S.; Zhu, Z.; Kaxiras, E. Ex-act continuum model for low-energy elec-tronic states of twisted bilayer graphene. Phys. Rev. Research , , 013001.(27) Tarnopolsky, G.; Kruchkov, A. J.; Vish-wanath, A. Origin of Magic Angles inTwisted Bilayer Graphene. Phys. Rev.Lett. , , 106405.(28) Ren, Y.; Gao, Q.; MacDonald, A. H.;Niu, Q. WKB Estimate of BilayerGraphene’s Magic Twist Angles. Phys.Rev. Lett. , , 016404.(29) Walet, N. R.; Guinea, F. The emergenceof one-dimensional channels in marginal-angle twisted bilayer graphene.
2D Mater. , , 015023. (30) Cantele, G.; Alf`e, D.; Conte, F.;Cataudella, V.; Ninno, D.; Lucignano, P.Structural relaxation and low-energyproperties of twisted bilayer graphene. Phys. Rev. Research , , 043127.(31) Lamparski, M.; Troeye, B. V.; Meunier, V.Soliton signature in the phonon spectrumof twisted bilayer graphene.
2D Mater. , , 025050.(32) Gargiulo, F.; Yazyev, O. V. Structuraland electronic transformation in low-angletwisted bilayer graphene.
2D Mater. , , 015019.(33) Nam, N. N. T.; Koshino, M. Lattice re-laxation and energy band modulation intwisted bilayer graphene. Phys. Rev. B , , 075311.(34) Yoo, H. et al. Atomic and electronic recon-struction at the van der Waals interfacein twisted bilayer graphene. Nat. Mater. , , 448–453.(35) Efimkin, D. K.; MacDonald, A. H. He-lical network model for twisted bilayergraphene. Phys. Rev. B , , 035404.(36) San-Jose, P.; Prada, E. Helical networks intwisted bilayer graphene under interlayerbias. Phys. Rev. B , , 121408.(37) Huang, S.; Kim, K.; Efimkin, D. K.;Lovorn, T.; Taniguchi, T.; Watanabe, K.;MacDonald, A. H.; Tutuc, E.; LeRoy, B. J.Topologically Protected Helical Statesin Minimally Twisted Bilayer Graphene. Phys. Rev. Lett. , , 037702.(38) Rickhaus, P.; Wallbank, J.; Slizovskiy, S.;Pisoni, R.; Overweg, H.; Lee, Y.; Eich, M.;Liu, M.-H.; Watanabe, K.; Taniguchi, T.;Ihn, T.; Ensslin, K. Transport Througha Network of Topological Channels inTwisted Bilayer Graphene. Nano Lett. , , 6725–6730.(39) Xu, S. G.; Berdyugin, A. I.; Kumar-avadivel, P.; Guinea, F.; Krishna Ku-mar, R.; Bandurin, D. A.; Moro-zov, S. V.; Kuang, W.; Tsim, B.;8iu, S.; Edgar, J. H.; Grigorieva, I. V.;Fal’ko, V. I.; Kim, M.; Geim, A. K. Giantoscillations in a triangular network of one-dimensional states in marginally twistedgraphene. Nat. Commun. , , 4008.(40) De Beule, C.; Dominguez, F.; Recher, P.Aharonov-Bohm Oscillations in MinimallyTwisted Bilayer Graphene. Phys. Rev.Lett. , , 096402.(41) Sunku, S. S.; Ni, G. X.; Jiang, B. Y.;Yoo, H.; Sternbach, A.; McLeod, A. S.;Stauber, T.; Xiong, L.; Taniguchi, T.;Watanabe, K.; Kim, P.; Fogler, M. M.;Basov, D. N. Photonic crystals for nano-light in moir´e graphene superlattices. Sci-ence , , 1153–1156.(42) Gadelha, A. C. et al. Localization of lat-tice dynamics in low-angle twisted bilayergraphene. Nature , , 405–409.(43) Brihuega, I.; Mallet, P.; Gonz´alez-Herrero, H.; Trambly de Laissardi`ere, G.;Ugeda, M. M.; Magaud, L.; G´omez-Rodr´ıguez, J. M.; Yndur´ain, F.;Veuillen, J.-Y. Unraveling the Intrin-sic and Robust Nature of van HoveSingularities in Twisted Bilayer Grapheneby Scanning Tunneling Microscopy andTheoretical Analysis. Phys. Rev. Lett. , , 196802.(44) Huder, L.; Artaud, A.; Le Quang, T.;de Laissardi`ere, G. T.; Jansen, A. G. M.;Lapertot, G.; Chapelier, C.; Renard, V. T.Electronic Spectrum of Twisted GrapheneLayers under Heterostrain. Phys. Rev.Lett. , , 156405.(45) Brenner, D. W.; Shenderova, O. A.; Har-rison, J. A.; Stuart, S. J.; Ni, B.; Sin-nott, S. B. A second-generation reactiveempirical bond order (REBO) potentialenergy expression for hydrocarbons. J.Phys.: Condens. Matter , , 783–802.(46) Kolmogorov, A. N.; Crespi, V. H.Registry-dependent interlayer potential for graphitic systems. Phys. Rev. B , , 235415.(47) Trambly de Laissardi`ere, G.; Mayou, D.;Magaud, L. Numerical studies of confinedstates in rotated bilayers of graphene. Phys. Rev. B , , 125413.(48) Thorgilsson, G.; Viktorsson, G.; Er-lingsson, S. Recursive Green’s functionmethod for multi-terminal nanostructures. J. Comput. Phys. , , 256–266.9 upplementary Information for ”Electronic localization in small-angle twisted bilayer graphene” V. Hung Nguyen , D. Paszko , B. Van Troeye , M. Lamparski , V. Meunier , and J.-C. Charlier Institute of Condensed Matter and Nanosciences,Université catholique de Louvain (UCLouvain), B-1348 Louvain-la-Neuve, Belgium Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, United States
1. Electronic model and calculations
To compute the electronic properties of twisted bilayer graphene (TBLG), the structural relaxation obtained bycalculations as fully described in [1] was taken into account and the p z tight-binding (TB) Hamiltonian H tb = ∑ n,m t nm a n† a m + h .c (S.1)was then employed. In particular, the hopping energies t nm between C-atoms are given by the standard Slater-Kosterformula t nm = cos ϕ nm V pp σ ( r nm )+( − cos ϕ nm ) V pp π ( r nm ) (S.2)where the direction cosine of ⃗ r nm =⃗ r m −⃗ r n along Oz axis cos ϕ nm = z nm / r nm . Similarly as in Ref. [2], the distance-dependent Slater-Koster parameters are determined as V pp π ( r nm )= V pp π exp [ q π ( − r nm a ) ] F c ( r nm ) , (S.3a) V pp σ ( r nm )= V pp σ exp [ q σ ( − r nm d ) ] F c ( r nm ) (S.3b)with a smooth cutoff function F c ( r nm )= + exp ( r nm − r c λ c ) . To model more accurately the first magic angle (with the most flat bands around the Fermi level) experimentallyobserved in the vicinity of 1.1° [3] (see Fig.S1 below) when the atomic reconstruction is practically taken intoaccount, our calculated TB parameters were obtained by slightly adjusting those reported in [2]. In particular, V pp π =− eV , V pp σ = meV ,q π a = q σ d = Å − , a = Å , d = Å , r c = Å , λ c = Å .
Basically, the electronic bandstructures and corresponding electronic quantities can be computed by diagonalizing theHamiltonian (S.1). However, diagonalization calculations in the case of small twist angles (<< the first magic one)are very numerically challenging due to a huge number of atoms in their supercell. In particular, there are more than10 atoms in the supercell at ~ 1.1° and this number rapidly increases when reducing θ, i.e., to more than 140 000 toms for θ ~ E , r , k ).Actually, this function represents its finite and high values only when the electron energy satisfies E ≡ ω n ( k ) (ω n ( k ) isthe nth eigenvalue of the Hamiltonian (S.1) at the momentum k ). The presented method thus allows to model theelectronic structure using the real-space/momentum maps of LDOS( E , r , k ) (see the example in Fig.S1). The mainadvantages of this Green’s function method, compared to direct diagonalization, include: (i) minimizing the size ofcalculated matrices by using recursive techniques [4] and (ii) efficient for calculating both global and local electronicproperties. The former enables calculations of extremely small twist angles, i.e., large Moiré superlattices. Fig. S1 : Bandstructures of reconstructed TBLG computed by exact diagonalization(for the top-left panel) and Green’s function method (for the rest).
2. Electronic bandstructures and DOS with different twist angles
In Figs. S2a and S2b, we present the electronic bandstructures and corresponding DOS obtained in TBLG systemswith different twist angles ranging from 3.15° down to 0.30°. Similar to those in Fig.1 of the main text, bothcalculations without and with the structural relaxation are presented to illustrate the effects of the latticereconstruction at small twist angles. ig. S2a : Bandstructures and corresponding DOS with different twist angles. The blue andred curves present the results obtained without and with the structural relaxation, respectively. ig. S2b : Bandstructures and corresponding DOS with different twist angles. The black andblue curves present the results obtained without and with the structural relaxation, respectively.
3. Projected DOS and local DOS with different twist angles
In Figs. S3a and S3b, projected DOS (PDOS) on carbon sites in AA, AB/BA stacking zones and soliton lines andcorresponding local DOS (LDOS) are computed and presented, similar to those in Fig.2 of the main text, for differenttwist angles ranging from 3.15° down to 0.40°. ig. S3a : Projected DOS computed in AA zones (first column), AB/BA zones (second column) and at the center of soliton lines (third column) and the corresponding local DOS in TBLG for different twist angles. ig. S3b : Projected DOS computed in AA zones (first column), AB/BA zones (second column) and at the center of soliton lines (third column) and the corresponding local DOS in TBLG for different twist angles. . Electronic properties of TBLG systems with a tiny twist angle In Fig. S4, local DOS at low energies and at a tiny angle (particularly, ~ 0.40° here) is computed and presentedwithout and with a gate bias. These results confirm and illustrate that in addition to the strong electron localization inAA stacking zones, small-angle TBLG represents the following characteristics- At tiny angles, when the atomic reconstruction is extremely strong and the superlattice periodicity is concurrentlylarge, TBLG is really considered as a network of domain walls (i.e., soliton lines) connecting AA stacking domainsand separating alternative triangular AB/BA stacking ones.- On this basis, the characteristics of Bernal stacking bilayer graphene can be recovered in AB/BA stacking domainsand helical edge states are concurrently observed around the soliton lines. When a gate bias is applied, an energy gapopens in AB/BA stacking zones and accordingly the helical edge states around the soliton lines are more pronounced.These results, together with the strong electron localization in AA stacking zones, emphasize the highly distinctelectronic properties of TBLG systems, compared to the large angle ones.
Fig. S4 : Local DOS at low energy of TBLG lattice with θ ≈ 0.40° iscomputed and presented without (left) and with (right) a gate bias. . Correlation between local strain and stacking structure of reconstructedTBLG systems When the atomic reconstruction plays important roles at small twist angles, the AA and AB/BA stacking zones ofTBLG becomes well constructed whereas local strains occur around the soliton lines. Notice that different from theAA stacking one as discussed in the main text, AB/BA staking zones are m onotonically enlarged when reducing the twistangle. Importantly, the C-C bond length strongly varies around the soliton lines, due to the effects of local strains, whereas it isalmost homogeneous in AA and AB/BA stacking regions. In particular, among three bond vectors of a carbon atom around thesoliton lines, at least, one bond vector is enlarged and another one is reduced, and the mixture of colors (enlarged and reducedC-C distances) generates Fig.S5. These properties thus induce a strong correlation between the profile of local strains andstacking structure of reconstructed TBLG systems, as illustrated in Fig.S5 below and in Fig.3b of the main text.
Fig. S5 : Spatial variation of C-C bond length and an illustration of the strong correlation between the local strain profile and stacking structure of reconstructed TBLG lattices (θ ≈ 0.50° is presented here). eferences [1] M. Lamparski, B. Van Troeye, and V. Meunier, Soliton signature in the phonon spectrum of twisted bilayer graphene ,
2D Materials , 025050 (2020).[2] G. Trambly de Laissardière, D. Mayou, and L. Magaud, Numerical studies of confined states in rotated bilayers ofgraphene , Phys. Rev. B , 125413 (2012).[3] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, P. Jarillo-Herrero, Unconventionalsuperconductivity in magic-angle graphene superlattices , Nature , 43-50 (2018).[4] G. Thorgilsson, G.Viktorsson, and S. I.Erlingsson,
Recursive Green s function method for multi-terminal ʼ nanostructures , J. Comput. Phys. , 256 (2014)., 256 (2014).