Van der Waals-coupled electronic states in incommensurate double-walled carbon nanotubes

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1 Kahu Lu* 1, Chenha Jn* 1, Xapng Hng 1, Jhn Km 1, Alex Zettl 1,2, Enge Wang 3, Feng Wang 1,2 Van der Waals-cupled electrnc states n ncmmensurate duble-walled carbn nantubes S1. Smulated absrptn spectra f SWNT (22,9), (11,11) and (22,9)+(11,11) S2. Thery f ntertube electrnc cuplng S3. Requrement fr strng ntertube electrnc cuplng S4. 11/ EE pattern fr dfferent nantube dameters and transtns S5. Electrn dffractn pattern fr all 28 DWNTs S6. Optcal transtn data Table fr all 28 DWNTs NATURE PHYSICS 1

2 S1. Smulated absrptn spectra f SWNT (22,9), (11,11) and (22,9)+(11,11) The ptcal absrptn spectra f ndvdual sngle-walled nantube were smulated based n Ref If there s n nter-tube cuplng, the absrptn f duble-walled nantubes wll be the sum f the cnsttuent sngle-walled nantube absrptn (Fg. S1). The shape f bserved DWNT (22,9)/(11,11) absrptn spectrum (Fg. 1c) s very smlar t the smulated ne, except that all the transtn peaks are shfted n energy due t tube-tube nteractns. Absrptn (a.u.) SS 3333 MM 1111 SS 4444 (22,9) (11,11) (22,9)+(11,11) SS Energy (ev) Fgure S1. Smulated absrptn spectrum f SWNT (22,9), (11,11) and ther sum. S2. Thery f ntertube electrnc cuplngs T calculate the ntertube electrnc cuplng, we cnsder an electrnc state αµ (, k, σ) n ne cnsttuent SWNT -- wthut lss f generalty, we assume t t be the nner tube -- and treat ts nteractn wth the uter tube as a perturbatn. Frm secnd rder perturbatn thery, the egenenergy f αµ (, k, σ) s: E = E + δ E = E + 0 el 0 α α α α 2 Mαβ, (S1) 0 0 E E β α β where β ( µ ', k ', σ ') labels electrnc states f uter tube and the sum ges ver all the uter tube states. Each state α, β crrespnds t a specfc wavevectr n the 2D graphene 2 NATURE PHYSICS

3 Brllun zne usng the zne-fldng scheme. The matrx element M = Ψ H Ψ αβ α INT β descrbes the cuplng between nner tube state α and uter tube state β. Ψ α, Ψ β are electrnc wavefunctns f unperturbed SWNTs, thus n a π -rbt tght-bndng mdel can be explctly wrtten as: 1 Ψ = ( + )) = Ψ + Ψ. (S2) ( µθ j+ kz j) φ A B e j e α α r R σ r R j τ µ k σ µ k 2N R j Here R = ( θ, z ) j j j s the crdnate f a graphene unt cell n the nantube, whch s summed ver all unt cells (wth N beng the ttal number); τ s the vectr cnnectng sublattce A and B n a graphene unt cell; φα s the phase dfference between tw sublattces. Then we can expand AA AB BA BB M αβ as M + σ ' M + σ M + σσ ' M, each representng the nteractn between ne αβ αβ αβ αβ sublattce f nner tube and ne sublattce f uter tube atms, wth 1 M = e e t( R) AA [( µ ' µθ ) j+ ( k' k) zj] [ µ ' θ+ k' z] αβ 2 NN ' R j R' 1 M = e e e t( R+ τ ') AB φ β [( µ ' µθ ) j+ ( k' k) zj] [ µ ' θ+ k' z] αβ 2 NN ' R j R' 1 M = e e e t( R -τ ) M BA φ [( µ ' µθ ) j+ ( k' k) zj] α [ µ ' θ+ k ' z ] αβ 2 NN ' R j R' BB αβ 1 = e 2 NN ' ( φβ φ α ) [( µ ' µθ ) j+ ( k' k) zj] [ µ ' θ+ k' z] R j e e t( R+ τ '-τ ), R' (S3) n whch R = R' R = ( θ, ) s the crdnate dfference between an nner tube atm at j z R and an uter tube atm at j R ', whse nteractn s descrbed by t( R ). Due t the ncmmensurate nner/uter tube lattce, we can assume that uter wall atms are dstrbutng randmly arund any gven nner-wall atm and apprxmate the sum ver R ' wth an ntegral ver uter tube surface: NATURE PHYSICS 3

4 1 e t( θ, z) = rdθdze t( θ, z) (S4) [ µ ' θ + k' z] [ µ ' θ+ k' z] R ' S 0 utertube whch s smply the Furer cmpnent f par nteractn ptental t( θ, z) and s ndependent n R ( S j 0 s the graphene unt cell area). S3. Requrement fr strng ntertube electrnc cuplng Here we use expnental decay frm r / λ r ( r θ) z / λ θ = γ γ + + t descrbe the t(, z) e e par nteractn, wth the nteratm dstance r, the ntertube spacng r, the nteractn strength γ ev and the characterstc length λ = nm (Ref. 4, 5). AA M αβ can then be calculated as: 2 λ r 2 µ ' 2 ( k ' ) AA πλ r + r/ λ 2 rr Mαβ = δ q+g, q ' γe e S0. (S5) Where µ ˆ µ ' r ˆ µ ' q = θ + k z ˆ, q' = θ + k' z ˆ = η ˆ θ + k' z ˆ. Smlar calculatn can be dne fr AB M r r r r αβ, BA M αβ and BB M αβ t gve M αβ : M = C M AA αβ αβ αβ ϕ αβ α α β β ϕ C = (1 + σe α )(1 + σ ' e β ), ϕ = φ q τ, ϕ = φ q' τ'. (S6) The requrement fr strng cuplng s gven by equatn (S5). The Drac delta functn strctly, requres q' = q + G, crrespndng t the sld dts n Fg. 3b. In addtn, the term e 2 λ r 2 µ ' ( k ' + ) 2 rr ndcates that the cuplng decays very fast wth large q'. As a result, nly the three q' (red dts n Fg. 3b) can cuple strngly wth the nner wall state at q. Equatn (S1) can then be wrtten as: 4 NATURE PHYSICS

5 δ E el α = 3 E M αβ 2 E 0 0 β = 1 α β. (S7) In whch we nly need cnsder the cuplng wth three states (nly σ = σ ' case s ncluded fr small energy dfference). The transtn energy shft fr a gven ptcal transtn E at wavevectr ( µ, k) can be calculated as: δδee eeee eeee = δδee μμμμ+ eeee δδee μμμμ 3 MM μμμμ+,μμββ μμββ MM μμμμ,μμββ μμββ 2 = EE μμμμ+ EE μμββ μμ EE ββ + μμμμ EE μμββ μμ ββ ββ=1 ββ=1 3 MM μμμμ+,μμββ μμββ MM μμμμ,μμββ μμββ = AA ββ 2 EE μμμμ+ EE μμββ μμ ΔΔEE ββ + ββ ββ=1 ββ=1 (S8) where A β 2 = M µk+,µ k M µk,µ k 2 s the matrx element cmbnng cntrbutn 0 frm bth valence and cnductn band states; ΔE β = E µk+ 0 ntrduced the apprxmatn E µk+ 0 E µβ kβ + symmetry fr states clse t the K and K pnt. 0 (E µk 0 E µ k +. In the thrd step, we have 0 E µβ kβ ) due t the electrn-hle S4. 11/ EE Pattern fr dfferent nantube dameters and transtns Equatn S8 shws that the energy shft el δ E s determned by bth matrx element M β and energy dfference E β. M β has a smple frm and s nly senstve t ntertube spacng (Fg. 3c), therefre the rch behavr f energy shft rgns largely frm the E β term. Here we nvestgate n detal the pattern f 1/ E β fr ptcal transtns f semcnductng nner-wall tubes. The results can be drectly extended t metallc and uter-wall tubes. NATURE PHYSICS 5

6 The whte cntur lne n Fg. 3d,e crrespnds t the largest 1/ E β (.e. strngest electrnc cuplngs), and t shws an nterestng dependence n the nner and uter tube chral angles fr a gven ptcal transtn and nner-wall tube dameter. Here we study hw ths strng-cuplng cntur lne evlves wth dfferent ptcal transtns (Fg. S2a) and wth varyng nantube dameters (Fg. S2b). In Fg. S2a we examne ts dependence n dfferent ptcal transtns fr a fxed nantube dameter (d =1nm). The strng-cuplng cntur pattern lne shrnks wth decreasng transtn ndex (r equvalently lwer transtn energy). It suggests that the prbablty t have strng nter-tube cuplng becmes lwer fr lw energy transtns. Fgure S2b shws the dameter dependence f the strng-cuplng cntur lne fr the S22 transtn wth dd = 0.5, 1.0, 2.0 nnnn. The strng-cuplng cntur lne pattern shrnks wth ncreasng dameter, largely due t the lwer transtn energy f S 22 fr a larger dameter tube. Besdes, we ntce that fr the large dameter lmt, ths cntur lne appraches t the requrement f θ twst =0. But fr small dameter nantubes, the cntur pattern lne rtates away frm a lne f cnstant twst angle. It shws that the electrnc cuplng n DWNT s nt nly determned by the nter-tube twst angle, but als the chral angle f each tube. Ths feature cmes frm the unque stretchng characterstc f the 1D blayer system. 6 NATURE PHYSICS

7 Fgure S2. (a) Dependence f the strng-cuplng cntur lne n transtn energy wth dd = 1.0 nnnn. (b) Dependence f the strng-cuplng cntur lne n nner-tube dameter dd fr S 22 transtn. The ntertube spacng s fxed t 0.35 nm. NATURE PHYSICS 7

8 References: 1 Lu, K. et al. An atlas f carbn nantube ptcal transtns. Nature Nantechnlgy 7, (2012). 2 Ch, S., Deslppe, J., Capaz, R. B. & Lue, S. G. An Explct Frmula fr Optcal Oscllatr Strength f Exctns n Semcnductng Sngle-Walled Carbn Nantubes: Famly Behavr. Nan Letters 13, (2013). 3 Lu, K. et al. Systematc Determnatn f Abslute Absrptn Crss-sectn f Indvdual Carbn Nantubes. arxv, (2013). 4 Sat, R., Dresselhaus, G. & Dresselhaus, M. S. Electrnc-Structure f Duble-Layer Graphene Tubules. Jurnal f Appled Physcs 73, (1993). 5 Rche, S., Trzn, F., Rub, A. & Mayu, D. Cnductn mechansms and magnettransprt n multwalled carbn nantubes. Physcal Revew B 64, (2001). 8 NATURE PHYSICS

9 Supplementary Electrn Dffractn Pattern SUPPLEMENTARY INFORMATION (27,5)/(18,5) (22,8)/(16,4) (25,2)/(12,8) NATURE PHYSICS (20,12)/(15,7) (22,9)/(11,11) (17,13)/(15,3) (25,8)/(15,9) (19,14)/(18,3) (18,11)/(12,7) 9

10 10 (16,14)/(17,1) (21,3)/(13,2) (23,18)/(16,15) (25,8)/(15,9) (27,6)/(15,10) (21,13)/(12,12) (13,12)/(9,6) (27,17)/(25,7) (21,18)/(19,8) NATURE PHYSICS

11 (16,15)/(13,8) (15,11)/(12,3) (17,16)/(14,9) (14,10)/(7,7) (21,6)/(11,7) (16,12)/(12,5) (23,3)/(13,5) (24,10)/(16,9) (16,11)/(12,4) NATURE PHYSICS 11

12 (s=-1) Supp trans s the r zg (20,14)/(18,5) n NATURE PHYSICS

13 lementary Table S1. Chral ndces and ptcal transtn data fr DWNT (n,m )/(n,m ). E DW and E SW are the p tn energes f nantubes n DWNTs and as slated SWNTs, respectvely. δe Exp =E DW -E SW, δe tt = δe el 55, where theretcal electrnc cuplng nduced ptcal transtn change; and 55 mev cmes frm the delectrc screenng effects ndcates the nner and uter tubes have the same (ppste) handedness (left-handed r rght-handed). If ne wall s arm zag the DWNT has n handedness, and s=0. m n m ndex E DW (ev) E (ev) δe Exp (mev) s δe el (mev) δe tt (me S S S S M M S S M M S S M M S S NATURE PHYSICS 13

14 S M M M S S S S M M S S S S S M M M S M M S S S S M MM SS NATURE PHYSICS

15 SS SS MM MM SS SS SS SS SS SS SS S MM SS SS SS SS SS SS SS SS SS MM MM SS SS SS SS NATURE PHYSICS 15

16 SS SS SS SS SS SS MM MM SS SS SS SS SS SS SS SS MM SS SS SS MM SS SS SS MM SS SS NATURE PHYSICS