Table Of Content

Soliton e(cid:27)e ts in dangling-bond wires on Si(001) ∗ † ‡ C.F.Bird, A.J.Fisher, and D.R.Bowler Dept. of Physi s and Astronomy, University College London, Gower Street, London WC1E 6BT, U.K. (Dated: 2nd February2008) 3 Dangling bond wires on Si(001) are prototypi al one dimensional wires, whi h are expe ted to 0 show polaroni and solitoni e(cid:27)e ts. We present ele troni stru ture al ulations, using the tight 0 bindingmodel, of solitons in dangling-bond wires, and demonstratethatthesedefe ts are stable in 2 even-lengthwires,althoughapproximately0.1eVhigherinenergythanaperfe twire. Wealsonote thatin ontrast to onjugated polymersystems,therearetwo typesof soliton andthatthetypeof n solitonhasstrong e(cid:27)e tsontheenergeti s ofthebandgapedges,with formationofintra-gap states a between 0.1eV and 0.2eV from the band edges. These intra-gap states are lo alised on the atoms J omprising the soliton. 0 3 PACSnumbers: 68.65.+k;73.20.-r;71.15.N ;71.38.+k ] i c I. INTRODUCTION In this paper, we examine an alternative pseudo-one- s - dimensional system based on the (cid:16)dangling-bond wire(cid:17). rl There are many motivations to understand the This system is formed on a hydrogen terminated sili on t (001)surfa eviaSTM-indu edsele tivedesorptionofhy- m transport properties of materials in the extreme one- 10 drogen along the edge of a dimer row . The atoms in dimensional limit. Some are te hnologi al: the logi- . at al on lusion to the histori redu tion in size of ele - su h wires have been shown (both theoreti ally and ex- perimentally) to undergo a Peierls/Jahn-Teller type dis- m troni omponents would be a tive devi e elements, and tortion, giving rise to an alternating pattern of atomi passive onne tions between them, that are of atomi - positions with respe t to those of the passivated sur- d s ale. There has been mu h re ent interest in stru tures n that might a t as atomi -s ale wires1 or as atomi - or fa e and produ ing the ele troni stru ture of a one- 10 o dimensional narrow-gapsemi ondu tor . In ontrastto mole ular-s ale swit hes. Other reasons relate to fun- c the ondu ting polymers, this distortion o urs predom- damental physi s: transport in one dimension is very [ inantly perpendi ular to the axis of the `wire'; however, di(cid:27)erent to higher dimensions, be ause the oupling of 1 ele trons to other ex itations, both of the latti e and of as in the ondu tive polymers, the resulting stru ture is 11 v not expe ted to be (cid:16)stati (cid:17). We re ently predi ted , on the ele troni system, is strong. This an lead to insta- 4 2 the basis of tight-binding al ulations, that ele trons or bilitiessu hastheformationofaLuttingerliquid orone 9 3 4 holesintrodu edintosu hwireswillprodu eself-trapped of a Peierls (in(cid:28)nite wire) or Jahn-Teller ((cid:28)nite wire) 5 (cid:16)small polaron(cid:17) defe ts. Despite their lo alisation, we 1 distortion, with orrespondingly strong modi(cid:28) ations to havealsoshownthatthesedefe tsareremarkablymobile 0 transport properties. 12 nearroomtemperatureandabove , raisingtheinterest- 3 One striking example of this type of behaviour is in 0 5 ing possibility of the transport of harge through su h ondu tive polymer systems . These exhibit an alterna- / devi es. t tion of double and single bonds whi h is at least partly a ofthe lassi Peierlstype(drivenbyele tron-latti e ou- It is natural to ask whether there might be solitoni , m pling), although ele tron orrelation e(cid:27)e ts may also be as well as polaroni , defe ts in dangling-bond wires (as d- important6. Carriers introdu ed into these systems are in t-PA); after all, these stru tures are degenerate (the n lo alised,probablypartlybydisorderbutalsotoasignif- total energy is invariant under ex hanging the (cid:16)up(cid:17) and o i ant degree by self-trapping by the latti e7. There are (cid:16)down(cid:17) atoms). Thisquestionisintimatelyrelatedtothe c two main lasses of harged defe t; one is the polaron, e(cid:27)e tofdi(cid:27)erenttypesofboundary ondition;inthe ase v: whi h is similar in prin iple to ex itations in higher- of t-PA, double bonds are (cid:16)an hored(cid:17) at the ends of the i dimensional systems. Here the arrier is surrounded by hain, so that the ground state of a neutral hain with X a region in whi h the atomi distortions serve to lower odd length ontains a bond-alternation defe t (soliton) r its own energy (in this ase, involving a redu tion in the at the entre. The behaviour in t-PA is simple be ause a double-single bond alternation). The se ond type is the the energy s ales for the binding of double bonds to the soliton; this ano uronlyin `degenerate'systems, su h ends of the hain are mu h larger than any involved in astrans-polya etylene(t-PA),wheretherearetwoequiv- the formation of defe ts along it. alent ground states of an in(cid:28)nitely long hain, in whi h In this paper we expli itly fo us on the behaviour of the double and single bonds are inter hanged. The soli- dangling-bondwiresof(cid:28)nitelength,withaviewtostudy- ton involves a mid-gap state asso iated with a `domain ingtheformationofsolitonsandtheende(cid:27)e ts. We(cid:28)nd wall' between regions of opposite bond alternation. Al- that the situation is more ompli ated than in t-PA, be- though lo alized, both polarons and solitons are highly ausethe energy s ales of the end e(cid:27)e ts and the defe ts mobile, and mu h is known about their e(cid:27)e t on trans- are omparable to one another. This means that there 5,8,9 port in both the oherent and in oherent limits . ismu hmore omplexbehaviourwhendeliberatedefe ts 2 areintrodu ed into the alternating pattern. A ordingly Mesh Size 3−a1t9o6m3.7 8h2a7in 8−a1t9o4m8.6 6h2a5in we pay parti ular attention to the stability of various 1x1x1 −1963.7828 −1948.6625 possible defe ts and their ontributions to the ele troni 1x2x1 −1964.1376 −1949.0197 stru ture of the wire. In light of the large number of 2x1x1 −1964.1377 −1949.0197 possible in>itial on(cid:28)gurations and ne essarily large sys- 2x2x1 −1964.1377 −−− tem size ( 400atoms), we hose to use a semi-empiri al 2x4x1 −1964.1364 −1949.0183 tight-binding approa h13 that was fast enough to allow 4x1x1 −1964.1365 −−− a omprehensive study of possible defe ts and had a - 4x2x1 −1964.1365 −−− 4x4x1 −1964.1365 −−− urate parametrisations available for the system under 4x4x4 onsideration. Table I: Total energies (eV) al ulated for relaxed perfe tly II. TECHNICAL DETAILS ordered 3 and 8 atom long hains at 0 K as a fun tion of k-pointmeshsize. A. Basi on(cid:28)guration & relaxation parameters A sili on slab was onstru ted out of 6 layers, paral- 13 lel to the (001) plane. Ea h layer had a 4 x 12 array binding pa kage . Stru tural relaxations were performed at a system temperature of 0 K until the maxi- of sili on atoms positioned at the appropriate positions for a perfe t (cid:16)diamond(cid:17) latti e, with the (cid:16)bottom(cid:17) layer mum for e per atomrea hed the limit of 0.01eV/Å.The majority of simulations required in the orderof 150iter- terminated with hydrogenatoms. This layerand the asso iatedhydrogenatomswerenotallowedtomoveduring ations to relax. The Hamiltonian was solved using exa t diagonalisation and periodi boundary onditions were relaxation and represented the bulk. The unterminated upper surfa e was allowed to re onstru t to form a pair applied to the simulation. Spin-polarisation was not in- luded in the al ulations. The sili on parametrisation of dimer rows. This (001) surfa e was then terminated 16 with hydrogen atoms to produ e a fully terminated slab used was that of Bowler et al , whi h was spe i(cid:28) ally (cid:28)tted for the Si(001) surfa e and hydrogen on that sur- that served as a basis for all the simulations. This unit fa e. It is worth noting that this parameterisation did ell ontained 432 atoms. not (cid:28)t to the ondu tion band, so that absolute values Ea h slab wasseparated from its neighbours by a va - shouldnotbetrusted,thoughgeneri behaviour(su has uumgapof15Åon eperiodi boundary onditionswere shifts in levels) are likely to be orre t. applied. For the maximum length onsidered of eight danglingbonds, ea hwire wasisolated by four hydrogen Convergen e of the total energy was tested with re- terminated atoms from its virtual neighbours along the spe t to the k-point mesh. An initial sear h was per- wire dire tion, while shorter wires had proportionately formed with a 3 atomwire to minimise al ulation time, greater numbers of hydrogen terminated atoms at ea h and sele ted results repli ated with an 8 atom wire. Ta- end. All wires were isolated by a ompletely terminated ble I shows the results of the onvergen e al ulations. dimer row from adja ent wires on e periodi boundaries Results were adequately onverged (<0.01eV) using a were onsidered as this has been found to provide ade- 17 14 Monkhorst-Pa k 2x1x1 grid, and all simulations pre- quate isolation . sented below used su h a mesh. Wires were reated by removing the required number of terminating hydrogen atoms from the upper surfa e, along one side of a dimer row. The referen e point for distortions was the average position of hydrogen terminated sili on atoms in the passivatedsurfa e, labelled as the baseline or (cid:16)level(cid:17) position. Displa ements normal to the surfa e in the dire tion of the bulk were (cid:16)down(cid:17) B. Con(cid:28)gurations onsidered and awayfrom the bulk were(cid:16)up(cid:17). Before relaxation,up atomsweredispla edbyanadditional0.3Å omparedto terminatedatomsofthenormalspa inganddownatoms Dangling bond wires 4, 6 and 8 atoms long were on- by 0.4Å less than terminated atoms. sidered. Some simulations of odd length wires were per- 15 Previous work had examined the e(cid:27)e t of slightly formed,butthesewerenotpursuedforreasonsdis ussed perturbing the starting positions to determine whether belowinSe tionIIIA. Ea hwirewas onsideredinitially the (cid:28)nal results were stable with respe t to small dis- with perfe t stru ture, i.e. the initial starting positions tortions and found this was the ase. A ordingly only were alternately up then down (or vi e versa). Defe ts one magnitude of starting displa ement was onsidered werethenintrodu ed onsistingoftwoadja entupatoms in this study. or two adja ent down atoms at all possible positions in All relaxations and ele troni stru ture al ulations the wire with the remaining atoms alternating appropri- wereperformedusingtheOxfordOrder-N(OXON)tight- ately. 3 Figure 1: A DB wire of length eight, with a DD soliton in themiddle(markedwith darker atoms). Theimage shows only the topfour layers of sili on, and limitedterminating hydrogens(one on theleft and two ontheright). III. RESULTS entral atom, in qualitative agreement with experiment and DFT modelling. We also modelled the length four A. End E(cid:27)e ts wire with LCN, and found little hange(cid:22)thealternating up/down pattern was still observed. However, the LCN simulationswereextremelyhardto onverge,anddidnot We have previously tested the use of tight binding for 11,12 show parti ularly improved results. We on lude that, in(cid:28)nite dangling bond wires , obtaining good agree- for odd length wires, there is spurious harge transfer ment with experiment and ab-initio results. However, that makes modelling of these systems potentially ina - when we ame to examine (cid:28)nite wireswith an odd num- urate in tight binding, and we will not onsider them ber of sites, we found an anomaly: one site on the wire 20 further . However, for even length wires there is no be ame(cid:16)level(cid:17) (i.e. atthesameheightasifitwerehydro- problem and we will ontinue to model these using tight genated) and the restof the wire behaved asa wire with binding with on(cid:28)den e. an even number of sites, in ontrast to ab-initio results. Thelo al hargeonthelevelatomwaszero,againin on- trasttotherestofthewire(wheredownatomsshoweda B. Solitons in even length wires hargede(cid:28) itandtheupatomsshoweda hargeex ess). In this se tion, weexplorethe reasonsfor this behaviour and investigate whether tight binding is a viable te h- Simulations were performed for hain lengths of 4, 6 nique for su h systems. and8dangling-bondsystems. Initial on(cid:28)gurationswere The shortest, and simplest, (cid:28)nite wires of odd and hosen so as to in lude all possible permutations of sin- even length for whi h there is experimental data are glepositionorderingdefe ts,forbothupanddowntypes. thosewiththree and foursites respe tively10 (thelength All relaxations su essfully onverged to the for e limit. two wire might be too short, as it e(cid:27)e tively onsists of The results are shown in detail in Table II, but in sum- only end atoms). We have already performed extensive mary both UU and DD soliton defe ts ould be formed investigations of this system using spin-polarised den- in the relaxed wire. Compared to a perfe tly ordered sity fun tional theory, and found good agreement with wire, those ontaining solitons are approximately 0.1eV experiment18. In parti ular, we found that the length less stable. No soliton defe ts formed on the end atoms three wire hasarathersmall displa ementpattern (with and there was strong periodi ity in possible lo ations of the two end atoms 0.11Å higher than the entral atom) the defe ts, as seen in the table. in loseagreementwithexperiment(wheredispla ements As well as showing the (cid:28)nal, relaxed on(cid:28)guration of of 0.15Å are measured), and that the length four wire thewires(withupatomsnotated(cid:16)u(cid:17),downatoms(cid:16)d(cid:17)and an form alternating up/down or down/up patterns, or atomsasso iatedwithasolitoninbold apitals)andvar- an up/down/down/up pattern. ious energies,Table II shows the initial on(cid:28)gurationfor Ourtightbindingsimulationsofevenlengthwireswere ea hsystem. Itisimportanttonotethatseveraldi(cid:27)erent in good agreement both with the experimental results initial on(cid:28)gurations led to the same (cid:28)nal on(cid:28)guration and the DFT simulations. In orderto better understand (for reasons dis ussed below). the pe uliar behaviour for the odd length wires, we used The e(cid:27)e ts of the ends of the wire are rather small 19 lo al harge neutrality (LCN) (where the on-site en- (only a few meV) but seem to exert a small attra tive ergies of individual atoms are adjusted until their net for e on the solitons. More interesting is the apparent hargeiszero)toremoveany hargetransfere(cid:27)e tsinthe instability of solitons of both kinds exa tly at the ends tight-bindingsimulations(itisworthnotingthatitisstill of the wire (and of ertain on(cid:28)gurations with the soli- perfe tly possible to obtain bu kled dimers on the lean tons in the middle of the wire). For instan e, the third Si(001) surfa e with a LCN ondition(cid:22)we will onsider and fourth lines of table II show that the initial on(cid:28)g- the e(cid:27)e ts on an even length wire below). The length urations DDud and UUdu relaxed ba k to perfe t wires, three wire be ame almost (cid:29)at under this ondition(cid:22) while the eighth line shows that duDD relaxed ba k to though the two end atoms were 0.06Å higher than the dUUd. This pattern an be easily understood in terms 4 Final E¯di(cid:27) VB¯O(cid:27)set CB¯O(cid:27)set Initial Length Defe t V−B7E.6d6ge C−B6E.3d5ge Ban1d.3g1ap udud ¯ ¯ ¯ udud 4 Perfe t −7.66 −6.46 1.20 dudu ¯ ¯ ¯ dudu dd −7.44 −6.33 1.11 udud ¯ ¯ ¯ ddud uu −7.66 −6.36 1.30 udud 0.111 −0.003 −0.109 uduu 6 Perfe t −7.66 −6.46 1.20 uddu 0.111 −0.003 −0.110 uddu dd −7.43 −6.35 1.08 uddu 0.102 0.224 0.012 uudu uu −7.66 −6.36 1.30 duud 0.101 0.224 0.012 duud 8 Perfe t −7.66 −6.46 1.20 duud ¯ ¯ ¯ dudd dd −7.43 −6.35 1.08 dududu ¯ ¯ ¯ dududu uu ududud ¯ ¯ ¯ ududud ududud ¯ ¯ ¯ ddudud ududud 0.114 −0.003 −0.106 ududuu Table III: Values of the bandgap in even-length wires on- uddudu 0.114 −0.002 −0.106 uddudu taining various defe ts. VB stands for Valen e Band, CB for uddudu 0.114 −0.002 −0.106 uududu Condu tion Band, all energies in eV. ududdu 0.114 −0.002 −0.106 uduudu ududdu 0.104 0.226 0.007 ududdu -6.0 duudud 0.104 0.226 0.006 duddud --66..21 aaaaaaaaaaaaaaaaaaaaaaaaa))))))))))))))))))))))))) pppppppppppppppppppppppppeeeeeeeeeeeeeeeeeeeeeeeeerrrrrrrrrrrrrrrrrrrrrrrrrfffffffffffffffffffffffffeeeeeeeeeeeeeeeeeeeeeeeeecccccccccccccccccccccccccttttttttttttttttttttttttt bbbbbbbbbbbbbbbbbbbbbbbbb))))))))))))))))))))))))) UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU sssssssssssssssssssssssssooooooooooooooooooooooooollllllllllllllllllllllllliiiiiiiiiiiiiiiiiiiiiiiiitttttttttttttttttttttttttooooooooooooooooooooooooonnnnnnnnnnnnnnnnnnnnnnnnn ccccccccccccccccccccccccc))))))))))))))))))))))))) DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD sssssssssssssssssssssssssooooooooooooooooooooooooollllllllllllllllllllllllliiiiiiiiiiiiiiiiiiiiiiiiitttttttttttttttttttttttttooooooooooooooooooooooooonnnnnnnnnnnnnnnnnnnnnnnnn duudud 0.104 0.226 0.007 duudud -6.3 duduud 0.104 0.226 0.007 dududd --66..54 duduud ¯ ¯ ¯ duduud -6.6 dudududu ¯ ¯ ¯ dudududu -6.7 -6.8 udududud ¯ ¯ ¯ udududud -6.9 dudududu ¯ ¯ ¯ dudududd -7.0 -7.1 dudududu 0.115 0.000 −0.102 uudududu -7.2 uddududu 0.115 0.000 −0.103 uddududu -7.3 -7.4 uddududu 0.117 −0.001 −0.105 uduududu -7.5 ududdudu 0.117 0.000 −0.105 ududdudu -7.6 ududdudu 0.115 0.000 −0.102 ududuudu -7.7 -7.8 udududdu 0.115 0.001 −0.102 udududdu -7.9 udududdu 0.104 0.228 0.003 udududuu -8.0 -8.1 duududud 0.105 0.227 0.004 ddududud -8.2 duududud 0.106 0.230 0.009 duududud duduudud 0.106 0.230 0.009 duddudud duduudud 0.104 0.227 0.003 duduudud Figure 2: Energies (eV) of states adja ent to the band gap dududuud 0.104 0.228 0.003 dududdud fora)perfe tlyordered8atomwire,b)8atomwirewithUU dududuud dududuud defe t, ) 8 atomwire ontaining DD defe t. TableII:Finalstru ture,energy hange,valen eand ondu - tionbando(cid:27)setsandstartingstru tureforsoliton- ontaining the soliton was of the DD type, the bottom of the on- even-length wires. Edi(cid:27) represents di(cid:27)eren e in total en- du tionbandwasloweredby0.1eV,whilstthetopofthe ergy relative to a perfe t wire. All energy di(cid:27)eren es are in valen eband wasraisedby 0.22eVby the presen eof an ele tron-volts. Solitondefe tsareshowninbold apitals. VB UU defe t. Both these e(cid:27)e ts were due to the introdu - stands for valen e band, CB for ondu tionband. tion of a intra-gap state, as illustrated in Figure 2. We re allthatthesystemwasnotparameterisedforthe ondu tion band and a ordingly absolute energies relating of harge balan e: an up atom is asso iated with an ex- to it should be treated with aution. ess of harge, while a down atom is asso iated with a A soliton is generallyasso iatedwith a mid-gapstate, de(cid:28) itof harge. The on(cid:28)gurationswhi hwereunstable whi h is seen here in the hange of the valen e and on- (whi h in ludes all systems with a single soliton at the du tionbands. Thesee(cid:27)e ts anbeunderstoodin terms end of the wire) had di(cid:27)erent numbers of up and down ofthe hybridisationofthe sili onmakin3gupthe surfa e: atoms, whi h would lead to harge imbalan e, so they inideal,bulk positionstheatomsaresp hybridised,but hanged to a stable on(cid:28)guration with equal numbers of re onstru tioninto dimersat thesurfa epullstheatoms up and down atoms, either removingthe soliton entirely awayfromthis state. When Jahn-Tellerdistorted,either or hanging it. during bu kling of dimers or forming a (cid:28)nite length DB3 Looking at the ele troni stru ture of relaxed wires wire as here, the atoms displa ed up move loser to sp ontainingsolitons,the presen eofasolitonredu edthe hybridisation and gain harge (tending to a lone 2pair) bandgap. The general pattern is that a DD type defe t while the atoms displa ed down move loser to sp hy- lowered the bottom of the ondu tion band, whilst an bridisationwithalonep-orbitalandlose harge(tending UU type defe t raised the top of the valen e band. The to an empty dangling bond). Thus the up atoms are as- valuesfordi(cid:27)erentlengthwiresareshownintableIII. If so iatedwith(cid:28)lled statesat thetopofthevalen eband, 5 0.35 ba)) CVBB udduudDUDUudduud IV. DISCUSSION AND CONCLUSIONS 0.3 Although the dangling-bond system is too rea tive to 0.25 bea omponentofnanos ale ir uitry,itdoesserveboth as a potential model for the study of one-dimensional 0.2 ondu tors and as a stepping stone to realisable useful systems, for example as a template for the deposition of 0.15 metal atoms in linear features. As su h it is important that systems an be modelled a urately and qui kly. 0.1 In this paper, we have demonstrated that tight-binding simulations are suitable for even-length wires and used 0.05 su h te hniques to explore the stability of ordering defe ts in the Jahn-Teller alternating displa ement seen in 0 5 10 15 20 25 30 35 40 perfe t dangling bond wires. We note that modelling Atom position (Angstroms) of odd-length wires is ompli ated by the development of (cid:16)level(cid:17) atoms in the relaxed system. This appears Figure3: E(cid:27)e ts of solitonformationontheele troni stru - to be related to di(cid:30) ulties in adequately modelling un- ture of (cid:28)nite DB wires. We plot ontributions to the harge paired ele trons using the tight-binding formalism; we density for spe i(cid:28) bands from atoms most strongly a(cid:27)e ted have modelled these systems using spin-polarised DFT, as a fun tion of distan e along the wire; a) highest o upied 18 ele troni statefora entralUUsoliton,b)lowestuno upied reported elsewhere . ele troni state for a entral DD soliton. In the even length system, we demonstrate that (cid:16)soliton(cid:17) type defe ts, where alternation between∼up0.1and down displa ements is interrupted, are only eV while the down atoms are asso iated with empty states lessstable than the perfe t wire, but areasso iatedwith atthebottomofthe ondu tionband. Theupanddown stronge(cid:27)e tsontheedgesonthevalen eand ondu tion displa ements a(cid:27)e t the underlying substrate (in parti - bands,asshowninFigure2, asa onsequen eof hanges ularthese ondandthirdlayeratoms)insu hawaythat in hybridisation state. The presen e of an (cid:16)UU(cid:17) defe t alternating up and down displa ements (either along or leads to the formation of an isolated state 0.22eV above a ross a dimer row) are energeti ally favourable. When the original valen e band. A (cid:16)DD(cid:17) defe t is asso iated this is interfered with3, as in2a soliton, the extent of the with a state 0.1eV below the original ondu tion band, relaxationtowardssp orsp +pisredu ed,andtheband althoughthe ondu tionbandenergiesshouldbetreated edges are a(cid:27)e ted. This is in marked ontrast to onju- with aution due to the nature of the parametrisation. gated polymers, as will be dis ussed below. Wealsodemonstrate thatthesestates ontainlarge on- The solitons are broadly lo alised as expe ted, whi h tributions from the atoms asso iated with the defe t i.e is shown in Figure 3. This is similar to the polaron seen lo alisation. 11,12 in the same system and indi ates that the soliton is similarly weakly oupled to the bulk. The overall behaviour and form of the soliton in the DB wire is very A knowledgments di(cid:27)erent to the onjugated polymer system. In parti u- lar,thesolitonisasso iatedwithapairofatomsandhas two forms, while in onjugated polymers it is asso iated We thank the UK Engineering and Physi al S ien es with the bond lengths near a single atom, and has only Resear hCoun ilforfunding (CFB) andtheRoyalSo i- one form. ety for a University Resear h Fellowship (DRB). ∗ † .birdu l.a .uk (1937). 5 andrew.(cid:28)sheru l.a .uk; Also at London Centre for Nan- A. J. Heeger, S. Kivelson, J. R. S hrie(cid:27)er, and W. P. Su, ‡ ote hnology, Gordon St, London WC1E 6BT Rev. Mod. Phys.60, 781 (1988). 6 david.bowleru l.a .uk; Also at London Centre for Nan- G. König and G. Stollho(cid:27), Phys. Rev. Lett. 65, 1239 ote hnology, Gordon St, London WC1E 6BT (1990). 1 7 J. Lyding, T.-C. Shen, J. Huba ek, J. Tu ker, and L. Landau, Phys.Z. Sowjetunion 3, 664 (1933). 8 G. Abeln, Appl.Phys.Lett. 64, 2010 (1994). H. Ness and A. Fisher, Phys.Rev. Lett. 83, 452 (1999). 2 9 F. Haldane, J. Phys.C 14, 2585 (1981). H. Ness and A. Fisher, Europhys.Lett. 57 (2002). 3 10 R. E. Peierls, Quantum Theory of Solids.(OxfordUniver- T.Hitosugi, S. Heike, T.Onogi, T.Hashizume,S. Watan- sityPress,2001), p.108 (cid:27).,OxfordClassi s Series, reprint abe, Z. Q. Li, K. Ohno, Y. Kawazoe, T. Hasegawa, and of 1936 original. K. Kitazawa, Phys.Rev. Lett. 82, 4034 (1999). 4 11 H. A. Jahn and E. Teller, Pro . Roy. So . A 161, 220 D. R. Bowler and A. J. Fisher, Phys. Rev. B 63, 035310 6 17 (2001). H. J. Monkhorst and J. D. Pa k, Phys. Rev. B 13, 5188 12 M. Todorovi , A. Fisher, and D. Bowler, J. Phys.: Con- (1976). 18 dens. Matter 14, L749 (2002). C.F.BirdandD.R.Bowler,Surf.S i.submitted(2003), 13 C. M. Goringe, D. R. Bowler, and E. Hernandez, Rep. ond-mat/0301592. 19 Prog. Phys. 60, 1447 (1997). A. P. Sutton, M. W. Finnis, D. G. Pettifor, and Y. Ohta, 14 S.B.Healy,C.Filippi,P.Kratzer,E.Penev,andM.S hef- J. Phys. C: Solid State Physi s 21, 35 (1988). 20 (cid:29)er, Phys.Rev. Lett. 87, 16105 (2001). If two length three wires are modelled on adja ent dimer 15 C. F. Bird, MS i Proje t Report, CMMP, Physi s rows, one shows an up/down/up pattern and the other Dept., University College London (2001), URL down/up/down, showing that, while tight binding allows http://www. mmp.u l.a .uk/~ fb/DB_wire_Si.pdf. harge transfer too easily, it an model the basi physi s 16 D. R. Bowler, M. Fearn, C. M. Goringe, A. P. Horsfeld, of the system orre tly. and D. G. Pettifor, J. Phys.: Condens. Matter 10, 3719 (1998).