SUPPLEMENTARY INFORMATION

Transcription

1 SUPPLEMENTARY INFORMATION doi: /PHYS180 Supplmtary Iformatio to Cohrt lctro-uclar couplig i oligothioph molcular wirs Jascha Rpp 1,, Ptr Liljroth 1,3, Grhard Myr 1 1 IBM Rsarch Zurich, 8803 Rüschliko, Switzrlad Istitut of Exprimtal ad Applid Physics, Uivrsity of Rgsburg, 9300 Rgsburg, Grmay 3 Dby Istitut for Naomatrials Scic, Uivrsity of Utrcht, PO Box 80000, 3508TA Utrcht, th Nthrlads Sampl prparatio Th molculs wr studid o ultrathi isulatig films (NaCl, RbI, ad X) o coppr sigl crystals by ultra-high vacuum low-tmpratur STM opratd at T = 5 K. Cu sigl crystals wr clad by svral sputtrig-aalig cycls. Alkali halid films wr grow by thrmal vaporatio of NaCl o Cu(111) or RbI o Cu(331) such that dfct-fr, (100)-trmiatd islads of up to thr atomic layrs wr formd. X films wr formd by adsorptio o Cu(111) hld at low-tmpratur ad this rsultd i moo- ad bilayrs of X. Idividual oligothioph molculs (s blow) wr adsorbd at a sampl tmpratur of T = 5 K, with th sampl locatd i th STM. Th covrag was chos to b far blow 1 moolayr such that th molculs ar wll sparatd. Bias voltags rfr to th sampl voltag with rspct to th tip. Th oligothioph molculs ar a mixtur of bromid trmiatd chais of diffrt chai lgths ad wr bought from Sigma-Aldrich. Th rlativ quatitis of th diffrt lgths i th origial matrial ar ukow. Th molculs wr sublimd i UHV. Thus, th oligothioph molculs o th sampl could b ithr th shortr os still trmiatd by bromid, or fractios of logr chais, which would th ot b bromid-trmiatd. W hav compard our rsults to pur alpha- atur physics 1

2 supplmtary iformatio doi: /phys180 sxithioph molculs to sur that irrspctiv of th trmiatio of th chais th oligothioph structur itslf is still itact. Idd w fid that th alpha-sxithioph rsults fit vry wll ito th othr data. di/dv spctroscopy Th molcular rsoacs ar at rlativly larg voltags ad hc, th tulig barrir dcrass strogly for highr lyig rsoacs. To compsat for that ffct, w liarly icrasd th tip-sampl distac with icrasig voltag. This liar tip displacmt was adjustd such that w obsrv similar pak itsitis for all lctroic stats (at positios, at which thir spatial dpdc has a maximum). This rsults ito a tip displacmt of roughly 1 Å pr 1 V voltag icras. A o-prfct compsatio of th abov discussd dcras of th tulig barrir may rsult i slight iaccuracis i th dtrmiatio of th pak positios ad satllit itsitis with rspct to th lastic paks. Th pak positio shifts ar small compard to th ucrtaity associatd with th fittig procdur of th paks. Sic all paks hav similar shap, such a pak shift would affct all paks th sam way ad rsult oly ito a gligibl offst of th voltag scal. Likwis, th ffct o th dtrmiatio of th lctro-vibro (-) couplig strgth is small compard to th scattr of ths valus. DFT calculatios Th orbital imags corrspodig to th LUMO of sxithioph show i Fig. 1 wr calculatd by dsity fuctioal thory (DFT) with B3LYP fuctioal ad DZVP basis as implmtd i NWChm 5.0 (s rfs. S1, S). Th orbitals wr visualizd usig Molkl 5. (Swiss Natioal Suprcomputig Ctr, Mao, Switzrlad). Th orbital structur at low dsity was somwhat ssitiv to th fuctioal ad basis st usd, whil th rgis ad orbitals at highr dsity wr ot. Th ovrall atur physics

3 doi: /phys180 supplmtary iformatio shap of th wavfuctio was uchagd; howvr, with som basis st / fuctioal combiatios, th outr lobs did ot fully mrg ad hid th ir lob of opposit wavfuctio sig. This simply rflcts th fact that th calculatd rgy (ad hc DFT optimizatio) is ot vry ssitiv to th orbital shap far outsid th molcul. Th disprsio rlatio was stimatd basd o th calculatd rgy lvls of sxithioph, dcathioph, ttradcathioph ad octadcathioph with B3LYP fuctioal. I this cas, th gomtry optimizatio was carrid out with 3-1G basis followd by a sigl poit rgy calculatio with DZVP basis. Plottig th rgis of th uoccupid orbitals as a fuctio of k = π / l, s Fig. S1, w fid a disprsio of th form E( k) = E + A + B k A, with A 0. V ad B 5.8 VÅ 0 as fittig paramtrs. Modllig W us th typical lctro-vibratio Hamiltoia H i our calculatios with = H + H + Η H = ε c c, H H = h a a λ λ λ = γ λ, m a + a c, ( λ λ ) cm, λ,, m λ, whr, m ad λ dot th diffrt lctroic ad vibratioal lvls of th systm, rspctivly, ε rfrs to th sigl lctro rgis, ad c ( a λ ) ad c ( a λ ) ar th lctro (vibratio) cratio ad aihilatio oprators. Bfor goig ito th dtails of our modl, w would lik to mphasis that th qualitativ rsult of a avoidd crossig is rlativly idpdt from th dtails of th calculatio. Th aim of our calculatio is oly to cofirm that th xprimtal rsults ar idd idicativ for th atur physics 3

4 supplmtary iformatio doi: /phys180 brakdow of th Bor-Opphimr-approximatio. Thrfor w kpt our calculatios simpl rathr tha tryig to accout for all dtails. I agrmt with th xprimtally obsrvd vibroic paks, crtai aromatic C-C-strtch-drivd vibratios ar xpctd to coupl strogst to th lctros o th dlocalizd molcular orbitals. A vibratio couplig two diffrt lctroic stats, m has to hav suitabl symmtry. It is importat to ot, that similar to th lctroic stats, th vibratioal stats of th subuits will coupl to form bads of vibratioal mods. Out of ths mods thr will b always o that matchs th symmtry rquirmts to coupl two spcific lctroic stats. Howvr, th rgis of ths mods will all b similar bcaus thy ar all drivd from th sam vibratioal stat of th subuits. Thrfor w assum that oly o vibratio is rlvat for vry pair of lctroic stats, m. W assumd a crtai fixd couplig costat γ, frqucy, ad rgy h for thos vibratios. Fially, w oly cosidr th couplig btw th LUMO ad othr stats. Udr ths assumptios, H simplifis to H N = γ a + a) ( c c LUMO + = whr N dots th umbr of lctroic lvls. ( c c ), Morovr, w cosidr th spcial cas with xactly o xcss lctro i th systm, thus o ad LUMO oly o of th lctroic lvls is occupid. With that w ca dscrib th stats, by two quatum umbrs: rfrs to th lctroic stat ( = 1 for th LUMO) ad rfrs to th umbrs of vibratioal quata xcitd i th molcul. W cosidr up to t vibratioal quata, that is, w trucat th Hamiltoia at = 1 0. Th lgth l of th molcular wirs is dtrmid by th umbr of thioph uits N, ach havig a lgth of Å. Th umbr of LUMO-drivd stats quals th umbr of uits ad th rgis of atur physics

5 doi: /phys180 supplmtary iformatio thos stats wr calculatd by usig a hyprbolic disprsio of th form E( k) = E 0 + A + B k A with k = π / l as foud i our DFT calculatios. W usd A 0. V as dducd from ths calculatios, but slightly adjustd B 5.5 VÅ that yildd bttr agrmt to th xprimtal rsults. Furthr, w usd h = 00 mv for th vibratio rgy quatum. Th - couplig trm γ ca b dducd from th closst distac i rgy of two stats at th avoidd crossig as E = γ. Our xprimtal valus yild γ 60 mv. Motivatd by th xprimtal rsults for short molculs, w assum that th -v couplig strgth for th couplig btw LUMO ad LUMO+1 ( γ ) is twic as larg as btw LUMO ad th othr lctroic stats ( γ γ / with > ). For short oligothiophs, w fid that th vibroic satllits of th highr lvls (LUMO+1 ) hav a lowr amplitud (by at last a factor of ) compard to th LUMO. = Cohrt couplig For th udrstadig of th physics bhid th avoidd crossig of th stats 1,1 ad,0 ad th cohrt couplig, it is mor istructiv to simplify th problm furthr by just cosidrig ths two stats ad glctig all othrs (w icludd all LUMO-drivd stats i th calculatios show i Figs. 3a ad 3b). W ar itrstd i th cas, for which th rgy diffrc of th lctroic stats quals th vibratioal rgy quatum h. Thus 1,1 ad,0 hav th sam rgy ε. Th two-lvl Hamiltoia th simplifis to ε γ H =. γ ε atur physics 5

6 supplmtary iformatio doi: /phys180 Th rsultig stats ar sparatd i rgy by E = γ ad hav th form (, 1, 0 ) 1 ±. Thy ar cohrtly coupld stats btw th lctroic ad vibratioal dgrs of frdom. Th smiclassical aalogu would b th lctromchaical oscillator. I this aalogy, th lctro oscillats btw th lft ad th right of th molcul at th frqucy of th vibratio. Th vibratio has to hav a od at th ctr of th molcul, such that th prsc of th vibratio lowrs th rgy of th lctro. Th vibratio ad th movmt of th lctro hav th sam frqucy bcaus ε ε 1 = h. I th pur quatum mchaical dscriptio, th wav fuctios drivd abov ar statioary stats ad hav o tim dpdc. To udrstad this pictur, w cosidr two w stats that ar suprpositios of th LUMO ad th LUMO+1. W dfi: r :=1/ ( 1 ), l :=1/ ( 1 + ). Th subscript dots that w oly rfr to th lctroic stats. If w calculat th spatial distributio of th lctro wav fuctio for ths stats usig th LUMO ad LUMO+1 stats from th particli-a-box modl, w fid that th lctro is locatd o th right of th molcul for r ad o th lft of th molcul for l, s Fig. S. With ths w stats, 1 ad ca b udrstood as th bodig ad atibodig combiatios of th stats r ad l, rspctivly. Similarly, w dfi for th vibratios (cf. Fig. S): r :=1/ ( 0 + ), 1 l := 1/ ( 0 ). 1 Hr, th subscript rfrs to vibratioal stats. Th qustio of th spatial distributio of th vibratio is slightly mor complx. Th vibratio is always locatd ovr th tir molcul, but for r ad l, th displacmt of th ios is i opposit dirctios. Morovr, o th lft ad right sid of th 6 atur physics

7 doi: /phys180 supplmtary iformatio molcul, th displacmt of th ios (of th sam typ) is also opposit, sic th couplig vibratio has a odal pla at th ctr of th molcul. Ths two stats ca b also udrstood as follows: r is a displacmt of th ios i such a way that th rgy of th lctro is lowrd, if th lctro is o th right sid of th molcul, but icrasd if it is o th lft sid. Thus, Importat for us ar th stats 1,1 ad,0, which ar giv by ( + r, r r, l + l, r l, )/ 1, 1 = l, ( r, r r, l + l, r + l, )/, 0 = l. ( 1, 1, 0 ) = ( + r, r l, l ), ( 1, 1, 0 ) = ( r, l + l, r ) +. This shows th quatum mchaical sid of th smi-classical aalogu: Th cohrt couplig of th stat (, 1, 0 ) 1 is such that if w obsrv th lctro o o sid of th molcul, th ios ar displacd such that th rgy of th systm is lowrd. For th stat (, 1, 0 ) displacd such that th rgy is icrasd. 1 +, thy ar Elctro distributio of mixd stats What is th lctro distributio of ths stats i ral spac? For this w hav to calculat Ψ Ψ as a fuctio of th lctroic coordiat. Ψ = 1 / ( 1, 1 ±, 0 ) ( 1, 1 ±, 0 ) Ψ r ) ( ( 1, 1 1, 1 ±, 0 1, 1 ± 1, 1, 0 +, 0, ) = 1 / 0. atur physics 7

8 supplmtary iformatio doi: /phys180 Sic th STM rspos is ot dpdt o th uclar displacmts, w hav to itgrat ovr th spac of th vibratioal coordiats. Th vibratioal stats ar orthogoal, thrfor th mixd trms cacl out as a rsult of this itgratio. This yilds ( ) Ψ Ψ( r ) = 1 /. This rsult shows that th spatial distributio of th lctro has th apparac of a icohrt suprpositio of th two lctroic stats 1 ad, or LUMO ad LUMO+1, rspctivly. This is i good agrmt with th spatial distributio of th stats show i Fig. 3 of th mai txt. I cotrast, for short molcular wirs for which th Bor-Opphimr approximatio holds, th stats factoriz ito a lctroic ad a vibratioal cotributio as Ψ = Ψ Ψ. I this rgim wll-dfid lctroic lvls (.g. 1,0 ) ar followd by satllit paks (.g. 1,1 ). Thus, th lctroic part of th satllit wav fuctio is th sam as for th corrspodig zro-vibro pak. This bhaviour ca b obsrvd i corrspodig costat-hight di/dv maps for short molcular wirs ad is distictly diffrt from th o for log molcular wirs. Both cass ar displayd i Fig. S3. Rfrcs S1 Bylaska, E. J. t al. NWChm, A Computatioal Chmistry Packag for Paralll Computrs, Vrsio 5.0 (Pacific Northwst Natioal Laboratory, Richlad, Washigto, USA, 007). S Kdall, R. A. t al. High prformac computatioal chmistry: a ovrviw of NWChm a distributd paralll applicatio. Computr Phys. Comm. 18, (000). 8 atur physics

9 doi: /phys180 supplmtary iformatio E (V) E = 0.3 V -.5 E = 0. V k = /l (Å ) Fig. S1 Disprsio rlatio drivd from th calculatd rgis of th lowst uoccupid orbitals i oligothiophs of 10, 1, ad 18 thioph uits lgth. Th rgis rfr to th rgis of Koh- Sham stats ad ar giv with rspct to th vacuum lvl. Th disprsio is parabolic clos to k = 0, but liar othrwis. Th data poits corrspodig to th chai cotaiig 1 uits ar show i rd. For this lgth th rgy diffrc btw LUMO ad LUMO+1 is alrady cosidrably smallr as compard to othrs of two subsqut orbitals. atur physics 9

10 supplmtary iformatio doi: /phys180 particl-i-a-box harmoic oscillator > 1> 1> 0> 1>+ > 1>- > 0>- 1> 0>+ 1> Fig. S Illustratio of th spatial dpdc of suprpositios (purpl) of th lowst (blu) ad scod-lowst (rd) stats i a particl-i-a-box-modl (lft) ad quatum mchaical oscillator (right). Th thick black lis illustrat th pottial. O of th suprpositios (dashd li) is primarily cofid to th lft, th othr (dottd li) to th right sid. 10 atur physics

11 doi: /phys180 supplmtary iformatio di/dv (arb. uits) di/dv (arb. uits) short wir log wir Voltag (V) Voltag (V) Fig. S3 Exprimtal di/dv spctra showig th diffrc i th vibroic structur for short ad log oligomrs masurd o a RbI bilayr o Cu(331) ad a X moolayr o Cu(111), rspctivly. Not that th voltag offst is a rsult of th diffrt work fuctios. For both di/dv spctra th corrspodig costat-hight spatially rsolvd di/dv maps at voltags corrspodig to th first four paks i th spctrum ar show. Th molculs ar approximatly 30 ad 65 Å log. Whras for th short molcul th vibroic satllits show th sam spatial distributio as thir zro-vibro coutrparts, for th log wir, th four maps show diffrt distributios that ar suprpositios of svral diffrt lctroic stats. atur physics 11