ELECTRON TRANSPORT THROUGH ONE AND FOUR-CHANNEL DNA MODELS

Transcription

1 ELECTRON TRANSPORT THROUGH ONE AN FOUR-CHANNEL NA MOELS A THESIS SUBMITTE TO THE GRAUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE EGREE MASTER OF SCIENCE BY SUN-HEE LEE R YONG JOE BALL STATE UNIVERSITY MUNCIE, INIANA JULY, 21

2 To my Faher

3 ABSTRACT NA molecules possess hgh densy genec nformaon n lvng bengs, as well as self-assembly and self-recognon properes ha make hem excellen canddaes for many scenfc areas, from medcne o nanoechnology The process of elecron ranspor hrough NA s mporan because NA repar occurs sponaneously va he process ha resores msmaches and lesons, and furhermore, NA-based molecular elecroncs n nano-boelecroncs can be possble hrough he process In hs hess, we sudy heorecally he ranspor properes hrough a one-dmensonal one-channel NA model, a quas-one-dmensonal one-channel NA model, and a wo-dmensonal four-channel NA model by usng he Tgh-Bndng Hamlonan mehod We show graphcal oupus of he ransmsson, overall conour plos of ransmsson, localzaon lenghs, he Lyapunov exponen, and curren-volage characerscs as a funcon of ncomng elecron energy and magnec flux whch are obaned usng Mahemaca run on he CSH Beowulf Cluser Our resuls show ha he semconducor behavor can be observed n he I-V characerscs The curren hrough a quas-one-dmensonal one-channel NA model sars o flow afer he breakdown volage and remans consan afer hreshold volage The varaons of he emperaure make he flucuaons of he sysem As he emperaure ncreases, he sharp ransmsson resonances are smeared ou and he localzaon lenghs are also decreased ue o a magnec feld penerang a he cener of he wodmensonal NA model, he Aharonov-Bohm (AB) oscllaons can be observed

4 ACKNOWLEGEMENTS I would lke o hank my advsor, r Yong Joe, for hs help, advce, suppor, and work wh me durng he research process I really apprecae hs paen nsrucon and enhusasm n hnkng of problems and soluons, whch nspred me n he followng sudy and research I would also lke o hank boh r Yong Joe and r Erc Hedn for her asssance hroughou my research and her suggesons for preparng presenaons a conferences I also wan o express my apprecaon o my hess commee members, r Thomas Roberson, r Yong Joe, r Erc Hedn, and r Muhammad Maqbool for her concern and for supporng me o connue o do my research Thanks also o Ball Sae Unversy and he eparmen of Physcs and Asronomy I wll always remember and apprecae he prayers and encouragemen from my famly and frends n Munce, For Wayne, New Jersey, and Korea

5 CONTENTS Page ABSTRACT ACKNOWLEGEMENTS LIST OF FIGURES v CHAPTERS 1 Inroducon and Overvew 1 2 Sequence dependen elecron ranspor hrough a one-dmensonal, one-channel NA model 8 21 Inroducon 8 22 One-dmensonal, one-channel TB model 1 23 Sequence dependen elecron ranspor Poly sequence Perodc sequence Fbonacc sequence Random sequence Poson varables 19 3 Backbone-nduced effecs on charge ranspor hrough a quas one-dmensonal NA molecule Inroducon The quas one-dmensonal, one channel gh-bndng model One-sep renormalzed onse energy Sequence dependen charge ranspor Varaon of he band-gap Temperaure effecs on he ransmsson Two-sep renormalzed onse energy Varaon of he hoppng srengh 33

6 342 Varaon of backbone onse energy Varaon of couplng beween leads-na 43 4 Temperaure and magnec felds effecs on he elecron ranspor hrough wodmensonal and four-channel NA model Inroducon Two-dmensonal, four-channel model The effecs of he varaon of he parameers on he ransmsson and I-V characerscs Couplng beween he NA bases and he backbone Hydrogen bonds Inro-srand couplngs along he backbone Temperaure effecs Magnec flux effecs 6 5 Summary and Conclusons 64 6 References 67 7 Appendx A 71 8 Appendx B 74

7 LIST OF FIGURES Fgure 11 Schemac of NA srucure The srucure of NA consss of wo polymer chans whch s called a duplex There are four knds of bases, guanne (G), adenne (A), cyosne (C), and hymne (T) They form pars wh each oher, G wh C, and A wh T, by hydrogen bonds The NA helx s abou 2 nm n dameer wh a vercal dsance of abou 34 nm beween layers of he base-pars and abou 1 base-pars for each complee urn of he helx 2 Fgure 12 (a) ouble-sranded poly (G)-poly (C) NA molecules whch are placed beween wo meal nanoelecrodes, wh semconducor behavor shown on I-V curves (b) 15 base-par NA n a dry sae s deposed and rapped beween gold elecrodes and observed o have elecronc conducon 4 Fgure 13 (a) Expermenal layou of a sngle NA conducance measuremen by Xu e al [4] (b) Schemac drawngs The upper (lower) fgure s for a sngle (double)-sranded NA molecule [5] The upper (lower) fgure represens an arbrarly shaped ssna (dsna) molecule, whch s sreched and aached o he nanoelecrodes whch are separaed by a gap of 27±2 nm (8 bp NA molecule, ~27 nm) 5 Fgure 14 Curren-volage characerscs a low emperaure (18 K: crcles and 36 K: crosses) Theorecal resuls are ploed as sold lnes Boh expermenal and heorecal resuls mach well The upper nse shows he ransmsson as a funcon of energy for a 3-poly G-C NA molecule The normalzed dfferenal conducance s shown n he lower nse [17] 6 Fgure 21 Au/ds-NA/Au srucure A-A' (no A-T base-pars), B-B (one A-T), C-C (hree A-T), and - (fve A-T a he cener of he double-srand NA) [13] 9 Fgure 22 The curren-volage characerscs are measured hrough mmoblzaon of A-A, B-B, C-C, and - NA sequences The conducance decreases as he number of A-T base-pars ncreases [13] 9 Fgure 23 A 1- TB model for elecron ranspor along he long axs of he NA; he fragmens of 2 poly NA bases are aached o leads Lnes denoe couplngs ( L,, and ) and he crcles gve he NA base ses (e ) and lead ses (e ) 11 Fgure 24 The ransmsson coeffcens of poly 2 bases Base G, A, C, and T have onzaon poenals, 775, 824, 887, and 914 ev, respecvely (e = 775, = 1, L(R) = 5, and = 4 ev) 14

8 Fgure 2 5 Energy-dependen ransmsson coeffcen for (a) perodc poly G-C, (b) perodc poly A- T, (c) perodc G-C-A-T, and (d) perodc A-T-G-C wh N=2 bases 15 Fgure 26 Transmsson coeffcen for Fbonacc poly G-C wh 2 bases, (a) sarng from G base, and (b) from C base 16 Fgure 27 Energy dependen ransmsson coeffcen for random G-C sequences; (a) GGCGCCGGGGCCCCCCGGCG, (b) CGCCCCGGCGCCGCGCCGGC, and random G-C-A-T sequences; (c) GGAGCTCGGGAATTTAGCTT, (d) TATTCGTCTGAATGGATAAC The un of E s ev 17 Fgure 28 Transmsson coeffcen (lef frames), Lyapunov coeffcen (mddle frames), and localzaon lengh (rgh frames) for (a) he poly 2 G bases, (b) perodc GC sequence, (c) Fbonacc GC sequence, and (d) random GC sequence 19 Fgure 29 Transmsson coeffcen for (a) CCGGGGGGGGGGGGGGGGGG, (b) GGGGGGGGGGGGGGGGGGCC, (c) GGCCGGGGGGGGGGGGGGGG, and (d) GGGGGGGGGGGGGGGGCCGG 2 Fgure 31 Schemac llusraon of he fshbone model The lef and rgh ends of he NA are conneced o he elecrode, lnes denoe hoppng ampludes, crcles denoe base-pars, and penagons are backbone srucure 23 Fgure 32 Transmsson coeffcen as a funcon of a renormalzed energy-dependen onse poenal deermned from Eq(3 3) There s a gap beween wo mn-bands and fve peaks arse n he frs mn-band n overall However, he shapes of he ransmsson, he dsance of he band-gap, he wdh of he each mn-band are changed dependng on he sequence of he base pars; (a) G/C-G/C-G/C-G/C-G/C, (b) A/T-A/T-A/T-A/T-A/T, (c) G/C-A/T-G/C-A/T-G/C, and (d) A/T- G/C-A/T-G/C-A/T 27 Fgure 33 Curren as a funcon of he source-dran appled volage for dfferen sequence order, where Ferm level energy s 54 ev and he emperaure s 3 K; Poly(G/C) sequence (blue, doed lne), Poly(A/T) sequence (black, do-dashed lne), alernae (G/C-A/T) sequence (green, dashed lne), and alernae (A/T-G/C) sequence (red, sold lne) The nse shows expermenal resul for 14-nm-long NA molecule [3 base-pars, double-sranded poly(dg)-poly(dc)] The upper nse s a schemac dagram of he es devce o measure conducon a room emperaure [7, 37] 29

9 Fgure 34 (a) The ransmsson T(E) for a one-chan TB model wh poly G/C sequence (blue, doed lne) and G/G msmached sequence, G/C-G/G-G/C-G/C-G/C (orange, sold lne) (b) The correspondng I-V characerscs wh Ferm Energy ( E = 54 ev) 3 F Fgure 35 (a) Transmsson coeffcen as a funcon of he elecron energy wh varaons of hoppng srengh (b) The ransmsson gap funcons due o he backbone effec Blue dos correspond o he dfferen values of gap (or separaon beween wo mn-bands) due o he changng of hoppng srengh beween neares-neghborng base pars, (c) The wdh of he frs mnband n he ransmsson vs 32 Fgure 36 Temperaure dependence of he ransmsson as a funcon of elecron energy wh fxed σ α = 85, ε = 7 75, = 1, L ( R ) = 5, = 5, and α = 7, for dfferen emperaure T =, 25, 1, and 3 K 33 Fgure 37 Conour plos of ransmsson as a funcon of elecron energy and hoppng srengh G and (a) when boh and C G are changed smulaneously (symmerc sysem), wo mnbands wh a gap n he ransmsson are merged o a sngle mn-band whch s localzed n a C small wndow of energy (b) In a symmery-breakng sysem wh a modulaon of he hoppng srengh C for fxed G = 35, a sngle-merged mn-band n he ransmsson compleely collapses when C 58 or C d Fgure 38 (a) Conour plo of ransmsson as a funcon of elecron energy and symmerc backbone onse energes ( σ G andσ C ) Fve dsncve resonan peaks wh a full ransmsson n each mn-band are shown n he nse (b) Curren (I) as a funcon of he source-dran appled volage Vsd wh k B T º 26 mev and E f = 62 ev for dfferen values of σ G = σ C = 7 (black, doed lne), 75 (green, dashed lne), 8 (yellow, sold lne), 85 (red, do-dashed lne), and 9 (blue, hck-dashed lne), where a ypcal feaure of semconducor s shown 37 Fgure 39 Conour plos of he ransmsson of he NA wh asymmerc backbone onse energy When he symmery of NA backbone onse energy s broken, wo ransmsson mn-bands overlap n he lower energy and an exra sharp resonance peak appears n he range of 745< E <76 38 Fgure 31 The resonance characerscs of he ransmsson as a funcon of elecron energy are ploed for varous backbone onse energy values The mergng and collapse of he wo mnbands as well as he appearance of an exra resonance peak are shown n (a)-(d) The enlarged plo of he exra resonance peak, depced n (), shows o be a sngle mn-band wh fve welldefned resonance peaks The sequence of overlappng of he wo BW resonances no a sngle

10 BW resonance, formng an under-uny resonance whch evenually collapses as he backbone asymmery ncreases, s shown n (e)-(h) 41 Fgure 311 The curren-volage characerscs calculaed from ransmsson T(E) of Fg 3 1(a)-(d) a 3 K where he Ferm energy s 62 ev The nonlnear I-V curves exhb a curren gap a low appled bas, whch s an ndcaon of semconducor behavor The nse shows he dfferenal conducance di/dv versus appled volage Vsd correspondng o I-V curves for σ C = 724 (black, dashed lne) and σ = 75 (blue, sold lne) 43 C Fgure 312 Conour plos of he ransmsson vs elecron energy and (a) symmerc couplng srengh L = R and (c) asymmerc couplng srengh L wh a fxed R = 5 A specfc feaure of ransmsson n he weak couplng regme: (b) well-arranged resonan peaks n each mn-band for L = R = 2 and (d) under-uny resonance n he ransmsson due o nerference effecs for L = 2 and R = 5 45 Fgure 41 The chess model for elecronc ranspor hrough 4C NA model: Crcles denoe onses; = 775 (lead onse), a = 85 (backbone onse), = 775 ( = 1-5, G base), = 887 ( = 6-1, C base) have he dfferen onse energes, Lnes; = 1 (ner-lead couplng), L1,2, R1,2 = 3 (hoppng amplude beween lead onse and he end bases), and,+1,t,+1 = 2 (nra-chan of he neares neghborng bases, h ( = 1-5, hydrogen bonds) denoe hoppng ampludes 48 Fgure 42 (a) Transmsson as a funcon of elecron energy for α = 1, 1, 5, and 8 (from boom o op), and (b) conour plo of he ransmsson wh fxed = 1, L(R) = 3, ( T), + 1= 2, B a = h = 1, ε = 775, and σ α = 85, for dfferen hoppng amplude α from 1 o 9 The backbone effec causes he exra mn-bands on he ransmsson 52 Fgure 43 (a) Conour plo of ransmsson as a funcon of elecron energy and hydrogen bonds ( h ) The lef resonan peaks are geng merged and reach o un ransmsson, on he oher hand, fve resonan peaks on he rgh mn-band are more pronounced and shfed oward hgher energes, as h s ncreased from 1 o 9 (b) The resonan characerscs of he ransmsson as a funcon of elecron energy are ploed for varous ner-couplng beween base-pars: h = 1, 3, 6, and 9 (from boom o op) 54 Fgure 44 (a) Conour plo of ransmsson as a funcon of elecron energy and nra-couplng Ba beween backbone onse As B a s ncreased, he peaks on he exra mn-bands are exended ou and become overlapped wh man mn-bands (b)the ransmsson as a funcon of small elec-

11 ron energy range (7 o 8) are ploed for varous couplng B a = 3, 4, and 5 for fxed = 1, L(R) = 3, ( T), + 1= 2, α = 3, h = 5, ε = 775, and σ α = Fgure 45 The resonance characerscs of he ransmsson as a funcon of elecron energy are ploed for varous B = 3, 4, and 5 (boom o op) Curves are shfed vercally by one un a for clary The swng of a Fano resonance n he ransmsson appears Ba = 3 ev: peak dp, Ba = 4 ev: dp peak, and Ba = 5 ev: peak dp 56 Fgure four-channel TB model for elecron ranspor along he long axs of he NA n he presence of hermal flucuaon Thermal effec nduces ws-angle flucuaon on he hoppng negrals and srucural dsorder 58 Fgure 47 Conour plo of he ransmsson vs elecron energy and emperaure and ransmsson coeffcen for K and 3 K We apply emperaure-dependen hoppng negrals n order o see hermal flucuaon effecs n our sysem 59 Fgure 48 Localzaon lenghs as a funcon of energy are ploed for dfferen emperaure, K and 3 K 59 Fgure 4 9 Schemac for AB-rng nerference wh he ds poly(g)-poly(c) NA 6 Fgure 41 Transmsson conour plos vs elecron energy and magnec flux ( Φ / Φ ) wh fxed = 1, L(R) = 3, ( T), + 1= 2, α =, h =, B a =, ε = 775, and σ α = 85 whou consderng backbone effec and hydrogen bonds for dfferen number of base-pars, from one, wo, fve base-pars (op o boom) Fve ransmsson coeffcen vs elecron energy for fxed ncomng energy (E = 8) shown n below he conour plo As a number of base-pars s ncreased, he amplude of oscllaons s decreased However, here are sll oscllaons n he ransmsson shown n he nse for fve base-pars 62 Fgure 411 Flux dependence of he ransmsson for one base-par (op) and fve base-pars (boom), showng perodc AB oscllaons The perodcy s he same as he number of loops hrough he NA 63

12 Chaper 1: Inroducon and Overvew In he nanoechnology era, people wan o make elecronc componens smaller and faser and o pu more nformaon n a very hgh densy Bu, n fac s hard o make hngs smaller whou error, and s me-consumng and dffcul Thus, organc maerals whch have self-organzng or self-assembly characerscs are consdered as good canddaes for nano-elecronc devces NA s he one of he more suable soluons for nano-fabrcaon echnology Generally, as we know, NA (deoxyrbonuclec acd) s he molecule whch sores genec nformaon n cells The srucure of NA consss of wo polymer chans of nucleode uns, whch are called bases There are four knds of bases, guanne (G), adenne (A), cyosne (C), and hymne (T) Along each backbone, he sugar lnks ogeher wh he phosphaes and bases The NA helx s abou 2 nm n dameer wh a vercal dsance of abou 34 nm beween layers of he base-pars and abou 34 nm for each complee urn of he helx, as shown n Fg 11 The wo chans ws around each oher hrough base parngs by hydrogen bonds The parng occurs only beween G and C or beween A and T; e here are only wo knds of base-pars, (G/C) and (A/T) One srand of NA bnds o anoher complemenary srand wh a hgh probably, gvng he propery of self-assembly Self-assembly s why NA could be useful n nanoechnology for elecrc crcus, for nsance, as a NA chp In addon, we can also synhesze NA o conan whaever

13 2 sequence we wan o have NA can provde pahways for charge ransfer processes because of he formaon of p-sacked base-pars n s double-helx srucure The process of charge ranspor, lke oxdave hole ransfer and reducve elecron ransfer hrough NA, s an mporan process for deecng msmaches and for NA repar [1] Based on hs naural charge ransfer process n NA, some applcaons have been developed n bochemsry and nanoechnology 34 nm 34 nm 2 nm Fg 11 Schemac of NA srucure The srucure of NA consss of wo polymer chans whch s called a duplex There are four knds of bases, guanne (G), adenne (A), cyosne (C), and hymne (T) They form pars wh each oher, G wh C, and A wh T, by hydrogen bonds The NA helx s abou 2 nm n dameer wh a vercal dsance of abou 34 nm beween layers of he base-pars and abou 1 base-pars for each complee urn of he helx (wwwbranncacom)

14 3 Movaed by hese poenal properes, many recen expermenal and heorecal sudes of charge ranspor n NA have been carred ou The queson of wheher NA s nrnscally conducng s an unsolved problem Because he expermenal oucomes are amazngly dfferen, NA mgh serve as nsulaor [2-5], semconducor [6-7], conducor [8-11], or even superconducor [12] There are numerous varables whch affec expermens n nanoscale dmensons In parcular, Porah e al drecly measured he elecrcal ranspor hrough 1 nm long (3-basepars), double sranded poly (G)-poly (C) NA molecules (Fg 12(a)) [7] The curren-volage curves ha hey measured beween wo meal nanoelecrodes show an asymmerc sharp rse of he curren a a hreshold volage Therefore, NA molecules can ranspor hgh currens a low emperaure Mahaparo e al observed elecronc conducon hrough 15 base NA olgonucleode pars whch were n a dry sae (Fg 12 (b)) [13] The sngle ds-na s mmoblzed n a defne nanogap whch s he separaon beween he elecrodes A conducance of ~1-9 S was esmaed for 15-base-par G-C rch ds-na Xu s group drecly measured he conducance of sngle NA molecules n aqueous soluon [14] Fgure 13 (a) s he expermenal layou by Xu e al n whch he curren s measured beween wo gold elecrodes hrough a NA duplex They observed ha he measured conducance of a sngle NA molecule depends on he base-pars sequence and he NA lengh Roy s group measured elecrcal ranspor n sngle and double-sranded NA molecules [15] They performed an expermen wh wo arbrary samples whch represen a sngle-sranded NA molecule (upper), and a double-sranded one (lower) whch s conneced o a par of funconalzed sngle-walled carbon nanoubes (black recangular) n Fg 13(a) Abou 25-4 pa of curren was measured for he ds-na, and ~1

15 4 pa or less for he ss-na whou base-pars Hence, he conducance of NA s seen o be dverse, dependng on expermenal condons and dfferen samples (a) (b) Fg 12 (a) ouble-sranded poly (G)-poly (C) NA molecules whch are placed beween wo meal nanoelecrodes, wh semconducor behavor shown on I-V curves (b) 15 base-par NA n a dry sae s deposed and rapped beween gold elecrodes and observed o have elecronc conducon

16 5 (a) (b) Fg 13 (a) Expermenal layou of a sngle NA conducance measuremen by Xu e al [4] (b) Schemac drawngs The upper (lower) fgure s for a sngle (double)-sranded NA molecule [5] The upper (lower) fgure represens an arbrarly shaped ss-na (ds-na) molecule, whch s sreched and aached o he nanoelecrodes whch are separaed by a gap of 27±2 nm (8 bp NA molecule, ~27 nm) Theorecal effecs on NA ranspor also have been suded by many groups For example, Roche e al have repored energy and emperaure-dependen ransmsson coeffcens for wo dfferen cases, whch are poly (dg)-poly (dc) and an aperodc λ -phase NA sequence [16] By usng gh-bndng (TB) calculaons, wo bands were found, separaed by a specfc energy gap, wh he number of resonan energy values dependen

17 6 on he lengh of he NA molecules Furhermore, hey observed ha emperaure effecs nduce hermal flucuaons of consecuve bases and reduce coheren ransmsson Cunber e al also observed semconducng behavor n NA ranspor by sudyng low emperaure curren-volage (I-V) curves hrough 3-poly (G)-poly (C) NA olgomers [17] Charges can propagae along he p-orbal sack va he neares-neghbor bases or he backbone They found ha he backbone couplng deermned he openng of a bandgap n he ransmsson In addon, he calculaon of he curren shows good agreemen wh he expermenal resuls, shown as closed crcles and crosses on he I-V curves n Fg 14 Fg 14 Curren-volage characerscs a low emperaure (18 K: crcles and 36 K: crosses) Theorecal resuls are ploed as sold lnes Boh expermenal and heorecal resuls mach well The upper nse shows he ransmsson as a funcon of energy for a 3-poly G-C NA molecule The normalzed dfferenal conducance s shown n he lower nse [17] Inspred by hese neresng resuls, we model one-dmensonal, quas-onedmensonal, and wo-dmensonal NA srucures n order o beer undersand elecron ranspor hrough NA for fuure applcaons n nano-echnology The ranspor properes of he elecronc ransmsson coeffcen are compued as a funcon of elecron ncomng energy Furhermore, we plo curren-volage (I-V curves) characerscs and

18 7 conour plos of he ransmsson In Chaper 2, we descrbe one-dmensonal NA model wh 2 bases We nvesgae he ransmsson coeffcen and localzaon lengh for dfferen sequences of NA bases In Chaper 3, we deal wh he quas-one-dmensonal NA model, whch conans an energy-dependen onse energy, ncludng he srucure of he base-pars and backbone effecs We show mpury and emperaure effecs on he ransmsson In addon, energy-dependen hoppng negrals are ncluded n he TB quas-one-dmensonal model Chaper 4 conans a wo-dmensonal, four-channel NA model, where we consdered all possble pahways ha an elecron can ranspor, and show he resuls of he effecs of he varaon of several parameers, ncludng an exernal magnec flux Fnally, n Chaper 5 we conclude and summarze our projecs

19 8 Chaper 2: Sequence dependen elecron ranspor hrough a one-dmensonal one-channel NA model 2 1 Inroducon NA consss of four nucleodes, known by her abbrevaons A, T, G, and C Wh jus four leers, hey are used o deermne he sequence of NA, whch s called he genec code The order of he sequence of NA provdes he genec nformaon for he lvng cell The dea of sudyng sequence specfc elecronc conducon came from an observaon made n several references Transpor expermens wh 15-base-par double-sranded NA molecules beween wo gold elecrodes by Mahaparo e al are shown n Fg 21 [13] Elecron ranspor hrough he p-orbal occurs by molecule-omolecule hoppng The onzaon poenals, whch are onse energes, correspond o quasbound saes, whch manfes as resonances n he ransmsson as a funcon of energy The sronger he hydrogen bondng beween base-pars, he more charge ranspor occurs The hydrogen bonds beween A-T base-pars are relavely weaker han hose beween G- C base-pars Mahaparo e al sysemacally changed he cenral fve base pars from G-C o A-T base-pars As a resul, he conducance s decreased exponenally by subsung A-T base-pars a he cener of he NA molecule Measured I-V characerscs are presened wh four knds of NA sequences n Fg 22 The conducance (G NA ) of he A- A confguraon, whch s homogeneous G-C base-pars, s measured around 1-9 S Ths conducvy decreases wh he addon of more A-T base-pars

20 9 Fg 21 Au/ds-NA/Au srucure A-A' (no A-T base-pars), B-B (one A-T), C-C (hree A-T), and - (fve A-T a he cener of he double-srand NA) [13] Fg 22 The curren-volage characerscs are measured hrough mmoblzaon of A-A, B-B, C-C, and - NA sequences The conducance decreases as he number of A-T base-pars ncreases [13] ong e al also nvesgaed he effecs of he sequence on ransmsson wh a double helx NA model (N = 3 base-pars) [18] Usng he ransfer marx mehod,

21 1 hey obaned ransmsson coeffcens and curren-volage curves for four dfferen sequences of NA models, such as a homogeneous poly(g)-poly(c) sequence, a perodc poly(g)-poly(c) sequence, a Fbonacc poly(g)-poly(c), and a quas-perodc Rudn- Shapro sequence As he poenal barrers change (e, he order of he sequence becomes more randomzed), coheren charge unnelng s decreased whch resuls n a lower conducvy In hs chaper, we consder a one-dmensonal (1-) one-channel NA of 2 bases beween elecrodes usng TB models Here, we do no consder a double-helx (base-pars and backbone) on our one srand NA model We nvesgae he ransmsson coeffcen, Lyapunov coeffcen, and localzaon lengh n order o sudy he nfluence of NA sequences Furhermore, we observe he varaon of he resonan peaks on he ransmsson by changng he poson of base C on a homogeneous poly(g) NA srand one-channel TB model Our model consss of 2 bases whch are aached o wo elecrodes In Fg 23, he crcles represen NA bases and he dscree ses of he elecrodes, and he lnes beween ses ndcae hoppng ampludes e (e ) s he onse poenal energy of he NA (lead), ( ) s he hoppng probably beween neares-neghbor bases (lead ses), and R ( L ) s he hoppng amplude beween he rgh (lef) lead and he end NA bases We assume ha he elecron ranspor occurs along he long axs of he NA molecule because of π -orbal overlap beween consecuve bases Hence, elecrons pass from he lef lead and hrough 2 NA bases and o he rgh lead

22 11 Fgure 23 A 1- TB model for elecron ranspor along he long axs of he NA; he fragmens of 2 poly NA bases are aached o leads Lnes denoe couplngs ( L,, and ) and he crcles gve he NA base ses (e ) and lead ses (e ) There are several mehods o calculae he ransmsson coeffcen of a 1- srucure The TB mehod s he mos useful for descrbng he perodc poenal n whch he wave funcons are overlapped beween he laces In a perodc lace, he neracon beween neares neghbors s he same over he sysem Usng he TB approxmaon, we can wre he Schrödnger equaon: where he marx elemens n, mψ m + enψ n = Eψ n, (21) n, m are couplngs beween ses m and n wh he sngle-se poenal of se n, he sum runs over he neares neghbors of n, E s he elecron energy, and e n s he se energy The oal Hamlonan of he sysem can be wren as H To = H + H + H (22) Lead NA Lead NA I s classfed as hree pars ha descrbe he Hamlonan for he NA molecule, for he leads hemselves, and for he hoppng ampludes beween he end NA bases and rgh (lef) lead: H NA = e d d ( d d H H Leads = e l l ( l l+ 1 + c h c) h ) NA = LlLd1 RlRd N h c (23) Leads +

23 where d ( d ) and -h of he leads Eq (21) l ( l ) are he creaon (annhlaon) operaors a he -h base se and Accordng o he Bloch heorem, we propose a perodc soluon n k-space from 12 ψ n nθ = Ae ( = ka) θ, (24) where a s a lace consan, whch s he dsance beween neares-neghbors and k s he wave vecor ha s conneced wh he energy by he dsperson relaon for he Bloch saes, E = 2 cos ka + e For a 1- sysem, he general ncomng and ougong wave funcon n he leads may be wren as ψ n nθ nθ = e + re ( ) n, ψ n nθ = e ( 1) n, (25) where r s he reflecon amplude and s he ransmsson amplude [19, 2] Applyng he wave funcons no he TB Schrödnger equaon, we form a 22 by 22 marx (Eq 26) n order o oban he ransmsson amplude as a funcon of he ncomng elecron energy, E, wh oher varables as parameers Therefore, we can oban he desred ransmsson coeffcen by akng square of he ransmsson amplude, T = (E) 2

24 13 (26) 2 3 Sequence dependen elecron ranspor Poly sequence We nvesgae he ransmsson coeffcen as a funcon of ncomng energy wh four dfferen NA sequences, whch are homogeneous poly bases, perodc, Fbonacc, and random sequences usng he 1- sngle-sranded NA molecule [16, 21] No only he ransmsson coeffcen s ploed, bu also he Lyapunov coeffcen and localzaon lengh are generaed as a funcon of ncden energy based on he Anderson localzaon heory [22] Furhermore, we nvesgae he ransmsson coeffcen accordng o he dfferen locaons of wo C bases n a srand wh 18 G bases NA bases G, A, C, and T are gven by he onzaon poenals, aken as 775, 824, 887, and 914 ev, respecvely Fgure 24 shows he ransmsson coeffcen of four knds of NA bases (N = 2 bases) Curves are shfed vercally by one un for clary Noce ha he paern of ransmsson probably s shfed o hgher energy as he base onse energy s ncreased We can also see ha he number of peaks s he same ψ ψ ψ ψ r = L e θ R R L e e e e e e e e e e e e θ θ

25 as he number of NA bases (2 bases) and he energy of he mddle of resonance peaks n each mn-band s approxmaely he same as he onzaon poenal of each base 14 Base T Base C Base A Base G Fg 24 The ransmsson coeffcens of poly 2 bases Base G, A, C, and T have onzaon poenals, 775, 824, 887, and 914 ev, respecvely (e = 775, = 1, L(R) = 5, and = 4 ev) Perodc sequence We apply four perodc sequences o he 1- NA model: poly G-C (G-C-G-C- G-C- ), poly A-T (A-T-A-T-A-T-A-T- ), perodc G-C-A-T (G-C-A-T-G-C-A-T- ), and perodc A-T-G-C (A-T-G-C-A-T-G-C- ) Fgure 25 shows he ransmssons of hese four perodc sequences wh specfc parameers (e = 775, = 1, L(R) = 5, and = 4 ev) whch are obaned from ab no calculaon [23] The number of mnbands s he same as he number of dfferen bases, and each only reaches a maxmum value of less han one If we choose specfc values of he hoppng ampludes, such as L(R) = = 9 ev, may reach o one We found ha he number of ransmsson peaks s N-2 [24] Comparng G-C- wh G-C-A-T- sequences (Fg 25 (a) and Fg 25 (c)), he wdh of he mn-bands for he laer are narrower, because as he energy levels a whch

26 he elecrons can hop (resonan unnelng) are no he same all along he srand, elecron propagaon s hndered and he ransmsson s decreased, even n a perodc sequence 15 (a) (b) (c) (d) Fg 25 Energy-dependen ransmsson coeffcen for (a) perodc poly G-C, (b) perodc poly A-T, (c) perodc G-C-A-T, and (d) perodc A-T-G-C wh N = 2 bases Fbonacc sequence The Fbonacc poly G-C sequence s formed by sarng from a G or C base and followng he nflaon rule G GC (C CG) and endng wh C G (G C) Each subsequen number s he sum of he prevous wo Fgure 26 presens he ransmsson coeffcen for wo dfferen confguraons of Fbonacc sequences, whch are G-GC- GCG-GCGGC-GCGGCGCG-G (N = 2) and C-CG-CGC-CGCCG-CGCCGCGC-C The hegh and wdh of he Fbonacc ransmsson mn-bands are smaller han for he perodc G-C sequences, shown n Fg 25(a) The ransmsson shows quas-perodc feaures whch are neher compleely perodc nor random sequences

27 16 (a) G-GC-GCG-GCGGC- GCGGCGCG-G (b) C-CG-CGC-CGCCG- CGCCGCGC-C Fg 26 Transmsson coeffcen for Fbonacc poly G-C wh 2 bases, (a) sarng from G base, and (b) from C base Random sequence Expermenally, he conducvy s measured wh arfcal or genomc NA sequences, such as human chromosome 22 and bacerophage l-na, whch s exraced from E col [11] These sequences are close o random sequences For example, he l 1 - chan s he frs 6-base-par of he phase sequence; GGGCGGCGACCTCGCGGGTTTTCGCTATTTAT- GAAAATTTTCCGGTTTAAGGCGTTTCCG and he l 2 -chan s he nex 6-base-par; TTCTTCTTCGTCATAACTTAATGTTTTTATTTAAAATACCCTCTGAAAA- GAAAGGAAACG [16] Alhough hese sequences appear random, hey are par of he specfc codng n NA whch conans he nsrucon se for buldng proens In Fg 27, we randomly chose 2 bases, each wh 5% G and C bases (he upper (a) and (b)) and ploed T(E) The boom frames ((c) and (d)) are he random G-C-A-T sequences When he sysem s dsordered by randomly shfng he energes of he varous rappng ses, he wave funcon exponenally decays Therefore, he ransmsson mn-bands

28 17 become more localzed [25] We noe ha n Fg 27 he dsorder effecs, or Anderson localzaon, on he ransmsson coeffcen of random sequences occur Anderson localzaon s he quanum nerference effec beween many scaered waveles In oher words, an elecron hops from one se o anoher hrough quanum unnelng If each lace se has he same poenal wh all wells he same, he elecron would be compleely moble Whereas, f he lace ses change randomly, he elecron can become rapped, or localzed, and he ransmsson s decreased Ths means ha he wave funcon of he elecron exponenally decays n space due o he ncreased backscaerng [26] (a) (b) (c) (d) Fg 27 Energy dependen ransmsson coeffcen for random G-C sequences; (a) GGCGCCGGGGCCCCCCGGCG, (b) CGCCCCGGCGCCGCGCCGGC, and random G-C-A-T sequences; (c) GGAGCTCGGGAATTTAGCTT, (d) TATTCGTCTGAATGGATAAC The un of E s ev Fgure 28 shows he energy-dependen ransmsson coeffcen as well as he Lyapunov coeffcen and he Localzaon lengh The hree fgures n each row n Fg 28 ndcae ransmsson coeffcen, Lyapunov coeffcen, and Localzaon lengh for (a) he homogeneous poly (G) bases sequence, (b) he perodc G-C sequence, (c) he Fbo-

29 18 nacc G-C sequence, and (d) he random G-C sequence, respecvely The Lyapunov coeffcen and he localzaon lengh are calculaed usng he ransmsson coeffcens n order o compare he ransmsson properes dependng on he dfferen sequences The mddle column depcs he Lyapunov coeffcen, γ = ln[ T( E)]/(2N), and shows specfc paerns accordng o he dfferen sequences [27, 28] Especally, he selfsmlary feaure (he seres of ellpc bumps) s shown on he Fbonacc sequence compared o he random sequence In addon, he Lyapunov coeffcen s nversely relaed = N 1 o he localzaon lengh, ξ ( E) [ γ ] [23, 26-36] Le us ake a look a he hrd column n Fg 28(a, b, c, and d) If he sysem becomes randomzed, he localzaon lengh becomes smaller because he ransmsson has poor behavor accordng o he relaonshp, T exp[ L / ξ ], where L s oal lengh of he sysem In our case, s fxed a 2 bases N

30 19 (a) (b) (c) (d) Fg 28 Transmsson coeffcen (lef frames), Lyapunov coeffcen (mddle frames), and localzaon lengh (rgh frames) for (a) he poly 2 G bases, (b) perodc GC sequence, (c) Fbonacc GC sequence, and (d) random GC sequence 2 4 Poson varables We nvesgae he ransmsson coeffcen by changng he poson of C bases In Fg 29, by changng he poson of wo C bases n a ceran NA sequence (18 G bases and 2 C bases), we oban some neresng resuls Wheher wo C bases are on he fron or he las poson, he ransmsson of hese wo sequences of NA s he same

31 2 (Fg 29 (a) and (b)) When wo C bases are on he hrd and forh posons (Fg 29 (c)) or on he 17 h and 18 h posons ou of 2 bases oal (Fg 29(d)), he curves of he ransmsson coeffcen are he same Therefore, we can conclude ha he ransmssons are he same f wo NA sequences have mrror symmery paerns The ransmsson fgures are dfferen beween (a, b) and (c, d) because of dfferen sequences Fgure 29 (c) and (d) are more randomzed han (a) and (b) so ha he resonan peaks are lower han n he upper frames (a) and (b) (a) (b) (c) (d) Fg 29 Transmsson coeffcen for (a) CCGGGGGGGGGGGGGGGGGG, (b) GGGGGGGGGGGGGGGGGGCC, (c) GGCCGGGGGGGGGGGGGGGG, and (d) GGGGGGGGGGGGGGGGCCGG

32 21 Chaper 3: Backbone-nduced effecs on charge ranspor hrough a quas-one-dmensonal NA molecule 3 1 Inroducon We have suded charge ransfer hrough sngle-sranded NA n Chaper 2, bu real NA has double-srands (ds) whch nclude base-pars formed by complemenary bases (A wh T or C wh G) and sugar-phosphae backbones o hold n he double-helx srucure Many heorecal models have been formulaed o explan he expermenal resuls, such as 1- TB model and ds-tb model The elecrc conducance of NA s domnaed by he srucure and he base sequence Furher, overlappng p-orbals locaed on he sacked base-pars are consdered by many heorecal calculaons as he way n whch charge ranspor can occur However, Cunber e al have repored charge ranspor hrough a shor poly(g)-poly(c) double-sranded NA molecule (N = 3 base-pars long) by consderng he backbone effec [17] They showed ha he backbone couplng, he hybrdzaon of he overlappng p-orbals n he base-par o he backbone, conrols he openng of a gap n he ransmsson They also found semconducng behavor n he I-V characerscs whch shows good agreemen wh he expermenal resuls by Porah e al [7] Maca e al have suggesed a major role of he backbone-nduced effecs n he charge ransfer wh poly(g)-poly(c) and poly(a)-poly(t) chans They nroduced a wo-sep renormalzaon process n order o descrbe he realsc double-sranded NA

33 22 molecule n erms of an effecve 1- TB model They descrbed sgnfcan changes wh subsequen srong mpac n he ransmsson and I-V characerscs, such as he volage hreshold and he urn-on curren capably [37] Movaed by hs dea, we nroduce a quas-1- TB model for charge ranspor In par 1, we use a sngle sep renormalzed onse energy and show he varaon of he ransmsson, dependng on he sequence and emperaure effec In he second par, we apply a renormalzed onse energy and hoppng negral and show he effecs of nhomogeneous backbone onse energes, asymmerc energy-dependen hoppng amplude beween NA base-pars and he backbone, and he asymmerc conac couplng beween he leads and NA base-pars, llusrang he dverse crcumsances whch may affec expermen resuls We show he overall conour plo of he ransmsson, he currenvolage characerscs, and he dfferenal conducance 3 2 The quas-1- one-channel TB model We consder a sngle-channel model for charge carrer propagaon hrough he NA duplex, shown schemacally n Fg 31 The elecron ranspor n he NA molecule, conneced beween wo sem-nfne elecrodes, arses hrough he cenral conducon channel whch consss of base-pars and s conneced o he upper and he lower backbone ses Ths model s called fshbone model Usng a quas-1- TB model, a sngle and effecve Hamlonan for charge ranspor hrough he ds-na beween wo meallc leads can be wren as [16, 17, 25, 37], H To = H + H + H (31) Lead NA Lead NA

34 23 Fg 31 Schemac llusraon of he fshbone model The lef and rgh ends of he NA are conneced o he elecrode, lnes denoe hoppng ampludes, crcles denoe base-pars, and penagons are backbone srucure Here, he Hamlonan for a shor NA molecule s descrbed by H NA = ε α, α = G, C, A, T ( b α d d α d + h c), ( d d h c) + σ α, α = G, C, A, T b α b α (32) where d ( d ) and b ( b α α ) are he creaon (annhlaon) operaors a he -h base-pars and he -h upper and lower backbone se, ε s he average of NA base-pars, and s he hoppng negral beween neares-neghbor base-pars The nfluence of he backbone s consdered n he hrd and fourh erms n Eq (32), where σ α (α = A, T, G, and C) s he backbone onse energy (85 ev) and α s he hoppng negral from each base (G, C, A, or T) o he upper and lower backbone se In order o map he orgnal doublechan no he equvalen sngle-channel, we nroduce an effecvely renormalzed and energy-dependen onse poenal ε (E) : 2 2 α α ε ( E ) = ε + +, (33) E σ E σ α α

35 where ε s he average of wo complemenary bases whch are pared each oher, such asε = ( ε + ε ) G / 2 Our NA molecule s lnked o wo sem-nfne meallc leads by C he unnelng Hamlonan H NA = LlLd1 RlRd N h c, (34) Leads + L ( R where ) s he couplng srengh beween he lef (rgh) lead and he end NA basepars, and l ( l ) s he creaon (annhlaon) operaor a he -h se of he leads The leads hemselves are modeled by anoher TB Hamlonan as H ε l l l l h ), (35) Lead = ( c where he lead onse energy s ε = 775 ev and he hoppng amplude s aken as = 1 ev By dscrezng he sysem spaally wh lace consan a and denong he wave funcon on se n by ψ, he Schrödnger equaon n he TB approxmaon can be wren as n n, mψ m + ε nψ n = Eψ n, (36) 24 where he marx elemens n m, are hoppng negrals beween ses m and n wh he sngle-se poenal of se n, he sum runs over he neares neghbors of n, E s he elecron energy, and ε n s he se energy Hence, he general ncomng and ougong wave funcons n he leads from he soluon of Eq (36) may be wren as ψ n nθ nθ = e + re, n, ψ n nθ = e, n 1, (37)

36 wh θ = ka Here, k s he wave vecor ha s conneced wh he energy by he dsperson relaon for he Bloch saes E = 2 cos ka + ε, and and r are he ransmsson 25 and reflecon ampludes, respecvely The Schrödnger equaon for ampludes n he leads and NA molecules can be obaned as e 5 (1 + r) θ ψ + e R L ψ + re 2θ ψ + e θ + ψ =, + ( E ε ) e 2 ψ + ( E ε ) ψ ( E ε ) ψ =, 5 1 =, ψ + ( E ε ) ψ =, θ θ ψ + ψ + ( E ε ) ψ =, ψ + ψ + ( E ε ) ψ =, R L =, (38) where he energy-dependen onse poenal ε s defned by Eq (33) Rearrangng Eq (38) n a marx form and nverng hs marx, we oban he ransmsson amplude (E) of he sysem and he ransmsson coeffcen by akng he square of he amplude, T = (E) 2 In he followng, we demonsrae he openng of a ransmsson band gap, he varaon of he ransmsson wh base-par sequence, I-V characerscs, and he emperaure effecs For he second model, we use a wo-sep renormalzaon process for a shor poly(g)-poly(c) NA molecule The frs sep of he renormalzaon s he energydependen ransfer negral ( τ α ) from a G or C base o he backbone se: τ ε ( E σ ) α α α = α + (39) α The second sep s he energy-dependen onse poenal ε (E) whch ncorporaes he exsence of he backbone and renormalzed hoppng negrals:

37 τ G τ C ε ( E) = ε GC + +, (31) E σ E σ G C where ε ( ) / GC = GC + ε G + ε C 2 wh he onse energy of G or C bases, ε C = 887 ev and ε G = 775 ev, gven by he onzaon poenals of he respecve bases, and he hoppng negral GC = 4 ev, whch descrbes he hydrogen bonds connecng G-C base-pars We oban he ransmsson amplude (E) of he sysem: where 4 2θ L R (1 e ) E) =, (311) θ 2 θ P( E) e Q( E) ( + ) e R( E) ( 2 L R L R P( E) = ( E ε ){ ( E ε ) Q( E) = ( E ε ){( E ε ) R( E) = {( E ε ) ( E ε ) + 4( E ε ) }, } 3 4 }, (312) Eq (311) allows us o fnd he conducance hrough he NA molecules by he Landauer-Buker approach [38, 39]: 2 G = (2e / h) T, where T = (E) 2 We show he resul of he second model for he backbone conrbuon and conac effecs wh a fxed hoppng probably beween neares-neghborng poly(g)-poly(c) base-pars ( = 4 ev) More specfcally, o smulae he complcaed expermenal suaons, we modulae parameers of he sysem such as he onse energes of he backbone ( σ ), he hybrdzed hoppng amplude ( α ) beween a G (or C) base and he backbone, and he conac couplng srengh ( L and R ) boh symmercally and asymmercally In our numercal calculaons, we use he re-scaled parameers, ), σ σ ), ), and he elecron energy G ( C G ( C α L ( R E, all of whch are normalzed wh respec o he hoppng negral of he leads = 1 ev

38 One-sep renormalzed onse energy Sequence dependen charge ranspor The elecrcal ranspor properes of NA have been wdely suded wh dfferen sequences of NA usng drec measuremens [1], ab-no calculaons [4], and he ransfer marx mehod [18, 37], where he srong sequence dependence of resuls s ndcaed Based on hs dea, we consder four sequences of a NA sysem: a poly(g)- poly(c) sequence, a poly(a)-poly(t) sequence, an alernae G/C-A/T sequence, and an A/T-G/C sequence In Fg 32, four all ransmsson coeffcens have wo mn-bands and a band gap However, he wdh of each mn-band, he wdh of he band-gap, and he shape of he resonan peaks are dfferen In Eq (33), we fxed he followng parameers; σ α = 85, ε = 7 75, = 1, = 1, L ( R ) = 5, α = 7; bu ε s dfferen for a G/C base-par, ε + ε ) / 2 = 831, or for an A/T base-par, ε + ε ) / 2 = 869 As he ( G C ( A T onse energy changes from G/C base-pars ( ε = 831) o A/T base-pars ( ε = 869), wo mn-bands are shfed oward hgher energy (a) (b) (c) (d) Fg 32 Transmsson coeffcen as a funcon of a renormalzed energy-dependen onse poenal deermned from Eq (33) There s a gap beween wo he mn-bands and fve peaks oal arse n he frs mn-band However, he shapes of he ransmsson, he dsance of he band-gap, and he wdh of each mn-band vary dependng on he sequence of he base pars; (a) G/C-G/C- G/C-G/C-G/C, (b) A/T-A/T-A/T-A/T-A/T, (c) G/C-A/T-G/C-A/T-G/C, and (d) A/T-G/C-A/T- G/C-A/T

39 The curren s defned as he rae of charge ranspor, and s of drec neres snce corresponds o an expermenally observable quany The ransmsson funcon, whch has dfferen resonan shapes due o he varable sequences of he base-pars, s drecly relaed o he curren [6] (see Eq (313)) The I-V characerscs are obaned from he Landauer-Buker (scaerng) formula a room emperaure and expressed as 2e I = det ( E)[ f L ( E) f R ( E)] (313) h β ( E µ L ( R ) Here, f (E) s he Ferm-rac dsrbuon gven by f ( E) = 1/{ e + 1}, L( R) ) 28 where β = 1/ k B T and µ L(R) sands for he elecrochemcal poenal of he lef (rgh) leads whose values depend on he appled bas volage We choose µ = E + ( 1 η) ev L f sd and R = E f ηevsd, where sd µ + V s he source-dran appled volage, E f s he equlbrum Ferm energy, and η s a parameer descrbng he possble asymmery of conac o leads, chosen here as E f = 54 ev and η = 1/2, respecvely [38, 41] The shape of he I-V curves are changed very sensvely dependng on he relave values of he Ferm energy, because he Ferm energy wndow manly conrbues o he conducon Therefore, we adjus he Ferm energy as 54 ev n order o compare our resul wh he expermenal one Fgure 33 shows he curren-volage curves of a poly(g)-poly(c) sequence (blue doed lne), accordng o Eq (313) The exsence of he gap n he I-V curves ndcaes a semconducor-lke behavor Several expermenal measuremens, drecly probng he elecrc curren as a funcon of he poenal appled across synhec NA molecules, have evdenced he presence of conducon n he curren-volage feaures a room emperaure The upper nse n Fg 33 s a schemac dagram of he es devce o

40 29 measure curren hrough a 14-nm-long NA molecule whch s rapped beween wo elecrodes [7] The blue curve s he resul whch s measured afer rappng a NA molecule as shown n he upper nse Our numercal I-V calculaons n Fg 33 f well wh he expermenal resul Fg 33 Curren as a funcon of he source-dran appled volage for dfferen sequences, where he Ferm level energy s 54 ev and he emperaure s 3 K; Poly(G/C) sequence (blue, doed lne), Poly(A/T) sequence (black, do-dashed lne), alernae (G/C-A/T) sequence (green, dashed lne), and alernae (A/T-G/C) sequence (red, sold lne) The nse shows he expermenal resul for a 14-nm-long NA molecule [3 base-pars, double-sranded poly(dg)-poly(dc)] The upper nse s a schemac dagram of he es devce o measure conducon a room emperaure [7, 37] Snce he nfluence of a msmached (or mpury) NA sequence s mporan n bology or medcal research, we nvesgae he ransmsson and I-V characerscs for he poly(g)-poly(c) base-pars wh one msmached base-par (G wh G) Transmsson vs elecron ncomng energy s presened n Fg 34(a) wh (orange, sold lne) and whou (blue, doed lne) an mpury base-par The exsence of he mpury n a shor NA sequence leads o a dsoron of he ransmsson bands and a reduced peak hegh n he ransmsson We also calculae he I-V curve from Eq (313) n he presence of a

41 3 msmached base-par I s clearly seen from he I-V curve n Fg 34(b) ha he curren gap wh an mpury (orange, sold lne) s reduced n comparson wh normal poly(g)- poly(c) base-pars (blue, doed lne) I s neresng o noe ha when we change he locaon of he msmached base-par, here s no sgnfcan change n he I-V characerscs (a) (b) Fg 34 (a) The ransmsson T(E) for a one-chan TB model wh a poly G/C sequence (blue, doed lne) and a G/G msmached sequence, G/C-G/G-G/C-G/C-G/C (orange, sold lne) (b) The correspondng I-V characerscs wh Ferm Energy ( E = 54 ev) F Varaon of he band-gap The nucleobase n NA can nerac wh he sugar-phosphae backbone by means of he hoppng negral beween he base and he backbone Therefore, he onse energy of he base-pars wh a fne backbone couplng s renormalzed as expressed n Eq (33) Snce he backbone couplng conrols an openng of he gap n he ransmsson, he gap wdh beween he wo mn-bands can be wren as 2 = 2 + 2α 2, (314) gap 2

42 where s he hoppng amplude beween neghborng base-pars and α s he couplng beween he base-pars and he sugar-phosphae backbone [17, 37] In Fg 35(a), he ransmsson for poly(g)-poly(c) versus elecron energy s ploed wh fxed σ α = 85, ε = 775, = 1, ) L ( R 31 = 5, and α = 7 for dfferen hoppng srenghs = 1, 3, 6, and 9 (boom o op) beween neares-neghborng NA base-pars Fgure 35(b) presens he gap, whch s he dsance beween wo mn-bands n he ransmsson, as a funcon of he value of (from 1 o 9) The blue dos are measuremens of he dsance beween he wo mn-bands and he lne plo (red, sold lne) s obaned from he Eq (314), where α s fxed a 7 As you can see, he daa f he equaon well The wdh of he frs mn-band as a funcon of he hoppng srengh ( ) s shown n Fg 35(c) The wdh of he mn-band s broadened as he couplng ncreases When = 9, for example, he wdh of he frs mn-band s exended by around 1 mes n comparson wh he case of = 1

43 32 Fg 35 (a) Transmsson coeffcen as a funcon of he elecron energy wh varaons of hoppng srengh (b) The ransmsson gap funcons due o he backbone effec Blue dos correspond o he dfferen values of gap (or separaon beween wo mn-bands) due o he changng of hoppng srengh beween neares-neghborng base pars, (c) The wdh of he frs mn-band n he ransmsson vs Temperaure effecs on he ransmsson The varaon of he emperaure nduces hermal vbraons and wsng of NA molecules Thus, elecron ranspor can be changed sgnfcanly due o he srucural dsorder Here, we nvesgae fne emperaure effecs on he ransmsson hrough a shor poly(g)-poly(c) NA molecule In prncple, nelasc elecron-phonon scaerng can occur because of he hermal vbraons n he sysem However, we gnore nelasc scaerng effecs because hey play a mnor role on he conducance [4] Neglecng nelasc scaerng effecs, he ransmsson ( T emp ) a emperaure T can be calculaed by hermally averagng he zero-emperaure resuls wh he approprae Ferm-facor, and can be wren as

44 33 ~ df T Temp ( E) = T ( E, emp = ) ( ) de, (315) de where ( df de ) = β 2 β 4 cosh ( ( E E ~ )) 2 wh β = k B 1T and 1 f ( E) = ~ ( E E ) / kt e + 1 In Fg 36, we show he oal emperaure-dependen ransmsson coeffcen as a funcon of elecron energy for he varous emperaures, T =, 25, 1, 3 K I s clearly seen ha as he emperaure s ncreased from zero o room emperaure, he sharp ransmsson oscllaons are smeared ou and he resonan peak heghs are suppressed (T(E)<1) Fg 36 Temperaure dependence of he ransmsson as a funcon of elecron energy wh fxed σ α = 85, ε = 7 75, = 1, L ( R ) = 5, = 5, and α = 7, for dfferen emperaure T =, 25, 1, and 3 K 3 4 Two-sep renormalzed onse energy Varaon of he hoppng srengh Frs, we examne he ransmsson characerscs along he long axs of he fve poly(g)-poly(c) NA base-pars whch have a wo-sep renormalzed onse energy

45 (energy-dependen ransfer negral from a G (or C) base o he backbone, and energydependen nucleode onse poenal) We vary he hoppng negral beween he NA base-pars and he upper and lower backbone boh symmercally and asymmercally Fgure 37(a) shows a conour plo of he ransmsson as a funcon of boh elecron energy (E) and hoppng srengh ( G and C ) for fxed σ G = σ C = 8 5 and = 4 L = R 34 When = 3, s seen ha here are wo mn-bands wh a gap n he ransmsson, G C whch s a ypcal semconducng feaure As boh G and C are decreased, however, an overlappng of he wo mn-bands occurs and a sngle merged mn-band appears The sngle mn-band whou a gap n he ransmsson becomes localzed n a small wndow of energy a hgher elecron energy, E º 84 We also show a conour plo of ransmsson for fxed G = 35 n Fg 37(b) by modulang he hoppng srengh C beween base C and he lower backbone In hs symmery-breakng NA srucure, wo ransmsson mn-bands wh a gap progressvely approach each oher and evenually merge no a sngle mn-band as he dfference of he hoppng ampludes - G C becomes larger I s neresng o noe, however, ha he ransmsson n hs asymmerc sysem dsappears compleely when C 58 or C d 27 Therefore, he backbone couplng o NA base-pars conrols he openng of a gap and he mergng and a collapsng of a mn-band n he ransmsson

46 35 (a) (b) Fg 37 Conour plos of ransmsson as a funcon of elecron energy and hoppng srengh G and C (a) When boh G and C are changed smulaneously (symmerc sysem), wo mnbands wh a gap n he ransmsson merge no a sngle mn-band whch s localzed n a small wndow of energy (b) In a symmery-breakng sysem wh a modulaon of he hoppng srengh for fxed C G = 35, a sngle-merged mn-band n he ransmsson compleely collapses when C 58 or C d Varaon of backbone onse energy We also modulae he onse energes of he backbone ( σ ) boh symmercally and asymmercally We frs examne he ransmsson of NA molecules wh a symmerc varaon of he backbone onse energes σ G and σ C In Fg 38(a), we show a conour plo of he ransmsson as a funcon of elecron energy (E) and backbone onse energes ( σ G and σ C ) for fxed G = C = 15 and = 4 When 6d σ G (= σ C ) d 78, wo ransmsson mn-bands wh a gap arse n he lower elecron energy wndow L = R α

47 As he backbone onse energy ncreases, he ransmsson bands are shfed owards hgher energes and he wo mn-bands n he ransmsson merge no a sngle mn-band, snce he onse poenal energy of he NA s affeced by he backbone onse energes The nse of Fg 38(a) shows ha each mn-band has fve dsncve resonan peaks, where each peak reaches full ransmsson Access o ransmsson properes can be provded by measurng I-V curves In order o compare our resuls wh expermenally measured daa [7], we evaluae he I-V characerscs of he sysem wh he ransmsson funcon T(E) usng Eq (313) wh E f = 62 ev and η = 1/2 In Fg 38(b), he curren (I) as a funcon of he appled volage ( V sd ) a room emperaure s shown for dfferen values of σ G = σ C = 7 (black, doed lne), 75 (green, dashed lne), 8 (yellow, sold lne), 85 (red, do-dashed lne), and 9 (blue, hck-dashed lne) The I-V curves show neglgble curren up o a hreshold volage followed by a sharp rse of he curren, whch s a ypcal feaure of a semconducor We noe ha he curren gap n he I-V curve wdens on ncreasng he backbone onse energy 36

48 37 Fg 38 (a) Conour plo of he ransmsson as a funcon of elecron energy and symmerc backbone onse energes ( σ G andσ C ) Fve dsncve resonan peaks wh full ransmsson n each mn-band are shown n he nse (b) Curren (I) as a funcon of he source-dran appled volage Vsd wh k B T º 26 mev and E f = 62 ev for dfferen values of σ G = σ C = 7 (black, doed lne), 75 (green, dashed lne), 8 (yellow, sold lne), 85 (red, do-dashed lne), and 9 (blue, hck-dashed lne), where ypcal semconducng characerscs are shown Nex, we examne he asymmercal effecs of he backbone onse energy on he ransmsson A conour plo of he ransmsson wh varaon of he lower backbone onse energy σ C, for fxed σ G = 75, G = C = 15, and = 4 s shown n Fg 39 When σ C s equal o σ G = 75, we see wo mn-bands wh a gap n he ransmsson as L = R shown before As he dfference G C σ σ beween he wo values of he backbone onse energes s ncreased, however, an overlappng of he wo mn-bands occurs and he

49 sngle-merged mn-band evenually dsappears Mos mporanly, s clearly seen ha an addonal sharp resonance peak appears a hgher energy (E º 75) as soon as he symmery of he NA backbone onse energy s broken Ths exra resonance peak, whch n urn s a mn-band wh fve dsncve resonance peaks [see Fg 31()], remans even when he sngle-merged mn-band dsappears a σ C 764 and σ C d 725 Noce here ha we have generaed each conour plo of he ransmsson wh a dfferen energy scale and combned hese ogeher because hs exra resonance peak s so narrow and sharp 38 Fg 39 Conour plos of he ransmsson of he NA wh asymmerc backbone onse energy When he symmery of NA backbone onse energy s broken, wo ransmsson mn-bands overlap n he lower energy and an addonal sharp resonance peak appears n he range of 745 < E < 76 In order o undersand hese resonance phenomena of NA ranspor more clearly, we plo he ransmsson as a funcon of elecron energy n Fg 31 for a varaon of he backbone onse energy For a fxed σ G = 75, he resonance characerscs of he ransmsson are shown n Fg 31 as (a) σ C = 75, (b) σ C = 74, (c) σ C = 73, and (d) σ C =

50 724 When σ G = σ C = 75, he ransmsson T of he srucure exhbs weakly spl groups of ransmsson resonances n each mn-band due o he ner-base-par unnelng As he sysem moves away from he symmery pon abou he ranspor drecon (namely, when σ G σ C = 74), he wo mn-bands are shfed wh a reduced gap o lower energy, and a pronounced resonance peak appears a E º 747 n Fg 31(b) From he enlarged plo of hs resonance peak, depced n Fg 31(), s seen o be a sngle mn-band wh fve well-defned resonance peaks Ths exra mn-band shfs gradually o lower energy, and s wdh ncreases slghly as σ C decreases Ths exra mn-band remans as long as σ σ, unlke he wo prmary mn-bands whch evenually dsappear G C The wo prmary mn-bands wh a gap become overlapped and her gap dsappears a σ C = 73 [Fg 31(c)] Ths combned mn-band compleely dsappears a σ C = 724 [Fg 31(d)] In order o make sure ha hs sngle-merged mn-band ndeed dsappears (and s no jus shfed o a lower energy wndow), we change he backbone onse energy over a small scale n Fg 31(e)-(h) When σ C = 7255, he merged mnband acqures he form of wo pronounced Bre-Wgner (BW) resonances a E º 734 and E º 754 When σ C reaches he crcal value σ C = 39 σ cr = 7253, oal overlappng of he BW resonances resuls n a sngle BW resonance Ths can be qualavely nerpreed by sayng ha he varaon of σ C effecvely makes he energy levels n he NA base-pars degenerae, and nduces a srong neracon beween hem When σ < σ, he amplude of he BW resonance n he ransmsson s less han uny, as seen n Fg C cr

51 31(g) The appearance of an under-uny resonance (less han full ransmsson, called a quas-resonance wh nonzero reflecon), s also observed n asymmercal quanumdo sysems [42, 43] I s consdered here ha hs under-uny resonance may occur when he effecve couplng beween he do and he nearby lef and rgh wells becomes weaker Hence, he varaon of he lower backbone onse energy for a fxed σ G deermnes he degree of asymmery of he NA molecule and herefore, he modulaon of σ G σ C has he equvalen effec of conrollng he couplng beween he leads and NA base-pars I s also found n Fg 31(h) ha he peak value of he ransmsson coeffcen decreases and evenually becomes zero wh decreasngσ, whch s equvalen o he absence of ransmng saes C 4

52 41 Fg 31 The resonance characerscs of he ransmsson as a funcon of elecron energy are ploed for varous backbone onse energy values The mergng and collapse of he wo mnbands as well as he appearance of an exra resonance peak are shown n (a)-(d) The enlarged plo of he exra resonance peak, depced n (), shows o be a sngle mn-band wh fve welldefned resonance peaks The sequence of overlappng of he wo BW resonances no a sngle BW resonance, formng an under-uny resonance whch evenually collapses as he backbone asymmery ncreases, s shown n (e)-(h) Usng Eq (313) and he ransmsson T(E) of Fg 31(a)-(d), we nvesgae he characerscs of I-V curves a room emperaure ( k B T º 26 mev) for an asymmerc NA srucure Fgure 311 demonsraes nonlnear I-V curves, whch exhb a varable curren gap a low appled bas, wh a varaon of σ C = 73 (red, dashed-do lne), σ C = 74 (green, doed lne), σ C = 75 (blue, sold lne), and σ C = 724 (black, dashed lne)

53 for fxed σ G = 75 I s clearly seen ha he volage hreshold s modulaed as he backbone onse energy changes In oher words, he curren gap ges wder as σ C ncreases Ths can be qualavely explaned as follows: as he backbone onse energy, σ α, ncreases, he man NA onse energy ε (E) also ncreases When ε (E) becomes larger, he mn-bands n he ransmsson are shfed o hgher elecron energy and herefore, he onse of curren arses a a hgher source-dran volagev sd 42 Ths requres a hgher volage hreshold o observe a curren and a wder curren gap n he I-V curves When σ C = 724 (black, dashed lne), on he oher hand, he volage hreshold ncreases o V sd (V) and he curren remans consan afer a volage of V sd º14 vols º18 V, because he man conrbuon o he ransmsson from he merged mn-band has dsappeared, as shown n Fg 31(d) In all cases, he sysem behaves as a semconducor wh a curren gap ha vares wh he modulaon of σ σ (or he effecve couplngs) In addon, we calculae G C he dfferenal I-V curve (di/dv) as a funcon of V sd for σ C = 724 (black, dashed lne) and σ C = 75 (blue, sold lne) n he nse of Fg 311 The dfferenal conducance for σ C = 724 exhbs a double-peak srucure wh an amplude of 3 na/v and a peak wdh of ~ 2 V

54 43 Fg 311 The curren-volage characerscs calculaed from he ransmsson T(E) of Fg 31(a)-(d) a 3 K where he Ferm energy s 62 ev The nonlnear I-V curves exhb a curren gap a low appled bas, whch s an ndcaon of semconducor behavor The nse shows he dfferenal conducance di/dv versus appled volage V sd, correspondng o I-V curves for σ C = 724 (black, dashed lne) and σ C = 75 (blue, sold lne) Varaon of couplng beween leads and NA The role of conacs deserves parcular aenon because he precse deals of NA-lead conacs n mos expermens are no unformly known or repored In many expermenal measuremens, s dffcul o prove ha he NA molecule s n drec conac wh he elecrodes because he conac wh he meal elecrodes s acheved by layng down he molecules drecly ono he elecrodes Such an unceran expermenal suaon wh regards o NA-elecrode conacs makes dffcul o se he bass for a meanngful heorecal approach o sudy nrnsc NA elecrcal ranspor properes In order o specfcally address NA-lead conac effecs on charge ranspor, we examne he ransmsson effecs of NA-lead couplng by varyng he couplng srenghs beween he molecule and he leads, boh symmercally and asymmercally In Fg 312(a), we presen a conour plo of he ransmsson as a funcon of he elecron energy and he n-

55 comng (ougong) couplng srengh L ( R ) for a fxed hoppng negral beween NA base-pars In he weak couplng regme, elecron unnelng beween he leads and he NA decreases and he ransmsson shows sharp and narrow un resonances n each mn-band due o he localzaon of saes, depced n Fg 312(b) for L = R = 2 For he srong couplng regme, on he oher hand, overlappng of he wave funcons ncreases due o mxng of energy saes beween he molecule and he elecrodes, and wellseparaed resonan peaks n each mn-band become overlapped n he ransmsson as L (= R ) ncreases Fnally, we consder an asymmerc NA srucure wh varaon of he ncomng couplng srengh L for fxed ougong couplng, R = 5 The resuls are presened n a conour plo of he ransmsson as a funcon of energy and L n Fg 312(c) In he srong conac couplng regme, he general rend of he ransmsson, whch s an ncreased overlappng of he wave funcons due o he mxng of energy saes, s he same as n he symmerc NA sysem When L becomes smaller ( L d 2), however, a dsnc and under-uny resonance n each band appears n Fg 312(d) Ths under-uny ransmsson, whch s a drec consequence of asymmerc conac effecs, can be nerpreed as resulng from nerference beween he NA molecular bands and he elecronc srucure of he leads a he NA-lead nerface 44

56 45 Fg 312 Conour plos of he ransmsson vs elecron energy and (a) symmerc couplng srengh L = R and (c) asymmerc couplng srengh L wh a fxed R = 5 Specfc feaures of he ransmsson n he weak couplng regme: (b) well-separaed resonan peaks n each mnband for L = R = 2 and (d) under-uny resonance n he ransmsson due o nerference effecs for L = 2 and R = 5

57 46 Chaper 4: Temperaure and magnec feld effecs on elecron ranspor hrough a wo-dmensonal fourchannel NA model 4 1 Inroducon NA srucure was dscovered by Wason and Crck n 1953 and he ner-base hybrdzaon of p-orbals perpendcular o he planes of he sacked base-pars n double-sranded (ds) NA was found by Eley and Spvey n 1962 Boh posve charges (holes) and elecrons can propagae hrough p-sacks of NA bases Thus, he dea of usng NA as a componen of fuure molecular elecronc devces has been repored and s sll beng explored n nanoechnology and nanoelecroncs [44-47] Varous expermenal resuls [3, 7, 9] have been observed and have movaed a number of heorecal sudes of he elecronc properes of NA, for nsance, one-dmensonal (1-) and wodemensonal (2-) gh-bndng (TB) models [17, 48, 49], and densy-funconal mehods [5, 51, 52, 53] Mos prevous research has used a 1- wo-channel TB model whch gnored he possbly of elecron ranspor along he backbone Klosa e al [25] performed wo TB models of NA, whch are a one-channel fshbone model and a wochannel ladder model, and obaned he elecronc properes n erms of localzaon lenghs There s a sugar-phosphae backbone of NA, bu elecron ranspor s no allowed along he backbone n her model They showed ha as backbone dsorder ncreased, he localzaon lenghs ncreased and hus, larger currens flow They also sug-

58 47 gesed ha he more approprae model for her sudy was he ladder model, whch s a more realsc model han he fshbone model Iguch nvesgaed he semconducvy of NA by usng a ladder sysem whch has wo man chans wh hoppng beween nearesneghbor ses, and nerchans beween he ds-na [54] Iguch also suggesed he backbone chans and hydrogen bonds lead o elecronc properes of NA In hs chaper, we presen charge ransmsson properes of a 2- four-channel NA model, whch s more represenave of he acual NA molecule We parcularly consder he nhomogeneous hoppng srengh beween base and backbone ses, he hydrogen bonds beween basepars, and he nra-couplng along he backbone n our new model Y e al demonsraed he elecronc properes of NA molecules wh wsed harpnlke shapes n he presence of a magnec feld [55] They showed ha he curren was oscllaed by he appled feld Roche repored elecronc conducon hrough poly(dg)-poly(dc) and λ -phase sequences as a funcon of emperaure-dependen basebase couplngs [16] A hgher number of ransmng saes have been shown n he ransmsson specrum a low emperaure Movaed by hese neresng resuls, we employ TB model and calculae he overall conour plo of he ransmsson, emperaure effecs, and magnec feld effecs n order o undersand he possble condons for usng NA as an elecronc componen We show ha he behavor of he ransmsson coeffcen vares dependng on he changng parameers Furhermore, suppresson and oscllaons can be seen n he ransmsson due o he Aharonov-Bohm (AB) magnec flux effec and due o he flucuaons n he wsng angle from hermal effecs

59 four-channel TB model We consder four possble conducon channels for charge carrer propagaon by ncorporang ner-backbone couplngs [9-11] and hydrogen bonds, shown schemacally n Fg 41 whch s called he chess model Elecron ranspor hrough he NA molecule, conneced beween wo sem-nfne elecrodes, arses hrough four dfferen channels whch conss of p-orbal overlappng hrough he neares neghborng-bases from wo man conducon chans and along he upper and lower backbones Each nucleode s represened by a se whch has an energy gven by he onzaon poenals of respecve bases, aken as G = 775, C = 887, A = 824, and T = 914 ev These are nerconneced and lnked o he backbone, leads, and neares-neghbor nucleodes by p-p sackng neracon, and hydrogen bonds Every lne beween ses denoes a hoppng amplude Fg 41 The chess model for elecronc ranspor hrough 4-channel NA model: Crcles denoe ses; = 775 (lead ses), a = 85 (backbone ses), = 775 ( = 1-5, G base), = 887 ( = 6-1, C base) whch have dfferen onse energes, Lnes; = 1 (ner-lead couplng), L1,2, R1,2 = 3 (hoppng amplude beween he lead onse and he end bases),,+1,t,+1 = 2 (nra-chan of he neares neghborng bases), and h ( = 1-5, hydrogen bonds) denoe varous hoppng ampludes

60 In Fg 41, we show a shor poly(g)-poly(c) chess model The ndvdual upper purple crcles (lower pnk crcles) represen NA G (C) bases, he green hexagons are sugar-phosphae backbone ses, and he yellow crcles are he lead ses These are nerconneced by he couplngs,, + 1 ( T, + 1 ), beween bases along he long axs and lnked o he backbone ( σ α ) by he hoppng ampludes, α, and couplngs, L1(2) ( R1(2) ) o he leads from he end bases The backbone and leads are also ner-lnked by he hoppng ampludes, wren as B a and Each base-par ( ε - ε + 5 ) s coupled by he hydrogen bonds, h Usng a 2- four-channel TB model, he Hamlonan for he chess model can be 49 H To = H + H + H (41) Lead NA Lead NA Here, he Hamlonan for a shor poly(g)-poly(c) NA molecule s descrbed by H NA ( =, j c ( ε c c c, ε d d, + 1 d j d σ α σ α + B a a a a a B b b b ) a b + 1 ( c α + h c), a + d b + h c d + h c) α (42) where c ( c ), d ( d ), a ( a ), and b ( b ) are he creaon (annhlaon) operaors a he -h G/C base and he -h upper and lower backbone, and ε ( j) s he onse poenal energy of NA G(C) base Hamlonan The NA molecule s coupled o wo sem-nfne meallc leads by he unnelng H NA = L1l c1 L 2l d 1 R1l1 c N R 2l1 d N h c, (43) Leads +

61 5 where ) L1(2 ( ) R1(2 ) are he hoppng srenghs beween he lef (rgh) lead and he end NA bases, and l ( l ) s he creaon (annhlaon) operaor a he -h se of he leads The leads hemselves are modeled by anoher TB Hamlonan as ) ( 1 c h l l l l H Leads + = + ε, (44) where he lead onse energy s ε = 775 ev and he hoppng amplude s aken as =1 ev, aken from Ref [23] In our numercal calculaons, we use re-scaled parameers and he elecron energy E, all of whch are normalzed wh respec o he hoppng negral of he leads = 1 ev We apply he wave funcons no he TB Schrödnger equaons for 22 ses (5 G bases, 5 C bases, 1 sugar-phosphae backbone ses, and 2 lead dscree ses) and combne hem no he followng he 22µ22 marx form Therefore, by solvng he marx equaon for he lnearzed TB Hamlonan we oban he ransmsson amplude as a funcon of he ncomng elecron energy, E The desred ransmsson coeffcen s obaned by akng he square of he ransmsson amplude, 2 (E) T = B E B E T E h h E e a a L L L L α α α α α α θ σ σ ε ε

62 4 3 The effecs of varous parameers on he NA ransmsson and I-V characerscs 4 31 Couplng beween he NA bases and he backbone Frs, we nvesgae he ransmsson characerscs along he long axs of he NA molecule wh fve poly(g)-poly(c) base-pars by changng he hoppng amplude ( α ) beween NA bases and he upper and lower backbone, symmercally Fgure 42(a) shows he ransmsson coeffcen as a funcon of he ncomng energy for dfferen hoppng srenghs, α = 1, 1, 5, and 8 (boom o op) For clary, each curve s vercally offse by one un In addon, we show a conour plo of he ransmsson as a funcon of boh elecron energy (E) and hoppng srengh ( α ) wh fxed = 1, L(R) = 3, ( T), + 1= 2, ε = 775, and σ α = 85 for dfferen hoppng srengh from 1 o 9 n Fg 42(b) Elecron ranspor hrough he sugar-phosphae backbone and hydrogen bonds are negleced n order o focus on he effecs of varyng α ( B a = h = 1) Even n he absence of he couplng beween he backbone and he bases ( α = 1), wo mn-bands arse wh a gap n he ransmsson These mn-bands are due o he dfferen se energes of he G and C bases n he upper and lower srands, where each provde a pahway for elecron ranspor and lead o conducng behavor When α 1, exra peaks appear a E = 85, whch s he same as he backbone onse energy As he couplng ges sronger, he gap beween he wo exra mn-bands ncreases and he wdh of exra mn-bands broadens When he ncomng energy maches wh he onse energes of ds-na bases, n oher words, when he ncomng wave funcon s close o he 51

63 value of he energy level of he bases, resonan unnelng occurs wh full un ransmsson conssng of 5-peak, sem-overlapped band srucure Once he backbone onse energy s coupled no he sysem hrough α (e, he role of he brdge o he backbone), he ncomng wave funcon overlaps a he energy level of he backbone and anoher mn-band s developed n he ransmsson probably [53, 56] 52 (a) (b) Fg 42 (a) Transmsson as a funcon of elecron energy for α = 1, 1, 5, and 8 (from boom o op), and (b) conour plo of he ransmsson wh fxed = 1, L(R) = 3, ( T), + 1= 2, B a = h = 1, ε = 775, and σ α = 85 for dfferen hoppng ampludes α from 1 o 9 The backbone effec causes exra mn-bands n he ransmsson

64 Hydrogen bonds The complemenary bases (A wh T or G wh C) are held ogeher by hydrogen bonds and form a unque NA double-helx srucure Iguch found ha he ransmsson band-gap was reduced by he hoppng amplude beween adjacen chans, whch means ha elecrons ravel hrough he ransverse axs of wo man chans of he NA [54, 57, 58] Jauregu e al suded he ransverse elecron ranspor hrough ds-na nucleodes (A/T nucleode molecule and G/C nucleode molecule) [59] They showed asymmercal curren (pa) n he I-V characerscs for drec and reverse bases Movaed by hese resuls, we examne how he hydrogen bonds affec elecron ranspor hrough NA by modulang he couplng ( h ) n he absence of backbone effecs In addon, hydrogen bonds cause small loops nsde double-chans of NA, for example sx small loops are made by hydrogen bonds n fve poly(g)-poly(c) base-pars Fgure 43(a) shows a conour plo of he ransmsson as a funcon of boh elecron energy (E) and hydrogen bonds ( h, = 1~5) wh fxed = 1, L(R) = 3, ( T), + 1= 2, B a = α = 1, ε = 775, and σ α = 85 for dfferen h from 1 o 9 In Fg 4 3(b), we show plos of he ransmsson versus E for varous hydrogen bonds h = 1, 3, 6, and 9 (from boom o op) For clary, each curve s vercally offse by one un I s clearly seen ha as hydrogen bonds are ncreased from almos zero (1), he fve peaks n he lef mnband merge and shf oward lower energes, whle he oher peaks n he rgh mn-band become more pronounced For weak ner-couplng beween base-pars, meanng a hgh barrer, he elecrons separaely unnel hrough he G bases n he upper and C bases n he lower srand By ncreasngh, he solaed quas-bound saes n each base-par grad-

65 54 ually mx ogeher The dsnc resonan peaks n he lef mn-band overlap and merge, whle he peaks n he rgh mn-band become sharper, ndcang ncreased localzaon of he hgher energy saes (a) (b) Fg 43 (a) Conour plo of he ransmsson as a funcon of elecron energy and he hydrogen bondng ( h ) beween he bases The lef resonan peaks merge and reach o un ransmsson, whereas, he fve resonan peaks n he rgh mn-band are more pronounced and shfed oward hgher energes as h s ncreased from 1 o 9 (b) The resonan characerscs of he ransmsson as a funcon of elecron energy are ploed for varous ner-couplngs beween basepars: h = 1, 3, 6, and 9 (from boom o op) Inra-srand couplngs along he backbone Nex, we consder all he couplngs beween neares-neghborng onses wh varaon of he nra-srand couplng along he backbone B Elecron ranspor hrough a one or wo-sranded NA molecule usng he TB model has been suded by many groups whou consderng he sugar-phosphae channel [7, 9, 25, 6] We show a conour plo of he ransmsson as a funcon of elecron energy (E) and couplng a B a from 1 o 9 for fxed = 1, L(R) = 3, ( T), + 1= 2, α = 3, h = 5, ε = 775, and σ α = 85 n Fg 44(a) As B a s ncreased, he resonan peaks of he exra mn-bands

66 spread ou and collapse no he man wo mn-bands I s clearly seen ha he nrasrand couplngs along he backbone sysem affec he four mn-bands on he ransmsson very sensvely We also plo he ransmsson whn a small elecron energy wndow (E = 7 o E = 8) for a varaon of 55 B a = 3, 4, and 5 (boom o op) The behavor of he ransmsson zero and pole, called a Fano resonance, s shown n Fg 44(b) [61] The Fano resonance s a manfesaon of nerference beween he localzed quasbound saes of he quanum do n one arm and he connuum saes n he oher arm, characerzed by a zero-pole par n he complex-energy plane [61-63] As nra-couplng along he backbone s ncreased, he sharp zero-pole par peak s shfed oward lower energy I s also found n Fg 45 ha he paern of he zero-pole par s changed from peak o dp a B a = 3, o peak o dp a B a = 5) n he energy wndow E = 75 o 79 as he couplng B a s ncreased Ths s called he swng of Fano resonances B a = 4, o dp o peak a

67 56 (b) (a) B a = 5 B a = 4 B a = 3 Fg 44 (a) Conour plo of he ransmsson as a funcon of elecron energy and nra-couplng Ba beween backbone ses As B a s ncreased, he peaks on he exra mn-bands exend ou and become overlapped wh he man mn-bands (b)the ransmsson whn a small elecron energy range (7 o 8), ploed for varous couplngs, B a = 3, 4, and 5 for fxed = 1, L(R) = 3, ( T ), + 1 = 2, α = 3, h = 5, ε = 775, and σ α = 85 Fg 45 The resonance characerscs of he ransmsson as a funcon of elecron energy are ploed for varous B a = 3, 4, and 5 (boom o op) Curves are shfed vercally by one un for clary The swng of a Fano resonance n he ransmsson appears Ba = 3 ev: dp peak, Ba = 4 ev: peak dp, and Ba = 5 ev: dp peak

68 Temperaure effecs Charge ranspor n NA s a complex phenomenon because he envronmen plays a sgnfcan role n deermnng he conducvy of NA Temperaure s one of he mporan facors n expermens wh bo-maerals, snce varaon of he emperaure nduces srucural dsorder and flucuaons of he sysem Fgure 46 shows a schemac for a 2-, four-channel NA model wh appled hermal flucuaons Here, we nvesgae he ranspor behavor for elecrons hrough a shor poly(g)-poly(c) NA molecule by applyng emperaure-dependen hoppng srenghs n order o observe he effecs of emperaure n our sysem Hence, we apply he varaon of he emperaure no he hoppng negrals n erms of ws-angle flucuaons These are based on a Gaussan dsrbuon wh average ws angle < θ I, I + 1 >=, where θ, + 1 s a relave ws angle devaed from s equlbrum value beween and +1 wh he help of he equparon 2 2 heorem, < θ, + 1 >= k B T / IΩ, where IΩ / k B = 25K, T s emperaure, I s he reduced momen of nera for relave roaon of he wo adjacen bases, and Ω s he oscllaor frequency of he mode [11, 16, 3, 64-66] Usng hese formulas, he emperauredependen hoppng negrals can be obaned as ( T ), + 1 ( T ), + 1 cosθ, + 1, 2 θ k cos θ BT α α, + 1, and h h cos θ, + 1 Inserng cos θ, χ 2, 2 2IΩ we can now form he fnal hoppng negrals as ( T ) ( T ) + {1 ( k T / 2IΩ ) χ } 2, + 1, 1 B 2 α α {1 ( k BT / 2IΩ ) χ } 2 h h{1 ( k T / 2IΩ ) χ }, B

69 58 where χ s a facor gvng he random flucuaon such as χ = 5, -3,, 3, -5 Fg four-channel TB model for elecron ranspor along he long axs of he NA n he presence of hermal flucuaons Thermal effecs nduce srucural dsorder and ws-angle flucuaons on he hoppng negrals Fgure 47 shows a conour plo of he ransmsson as a funcon of boh elecron energy (E) and emperaure (Temp) and wo plos of he ransmsson coeffcen as a funcon of elecron energy a K and 3 K wh fxed = 1, L(R) = 3, ( T), + 1= 2, α = 3, h = 5, B a = 1, ε = 775, σ α = 85, and a random facor χ = 5, -3,, 3, -5 I s clearly seen ha as emperaure s ncreased from K o 3 K, he resonan peaks on he four mn-bands, whch nally had un ransmsson, become suppressed and smear ou below uny due o he random varaon of he hoppng negrals We also show he localzaon lengh as a funcon of elecron energy for wo dfferen emperaures, K and 3 K, n Fg 48 Localzaon lengh s nversely relaed o he Lyapunov coeffcen and calculaed usng ransmsson coeffcens o compare he ransmsson properes The localzaon lengh can be wren asξ ( E) = N [ γ ] 1, where γ = ln[ T( E)]/(2N) and N s he number of base-pars [23, 26-36] An asympoc be- N

70 59 havor of he localzaon lengh ndcaes ha he hermal flucuaons can desroy coheren charge ranspor and reduce he mean ransmsson coeffcen In oher word, hgh emperaure leads o he dsorder of he sysem and a reducon of he localzaon lengh and consequenly a reducon of he elecron conducance accordng o he relaonshp, T exp[ L / ξ ], where L s oal lengh of he sysem 3 K K Fg 47 Conour plo of he ransmsson vs elecron energy and emperaure, and wo plos of he ransmsson coeffcen for K and 3 K We apply emperaure-dependen hoppng negrals n order o see hermal flucuaon effecs n our sysem K 3 K Fg 48 Localzaon lenghs as a funcon of energy are ploed for dfferen emperaure, K and 3 K

71 6 4 5 Magnec flux effecs Fnally, we nvesgae he elecronc properes of a shor poly(g)-poly(c) NA molecule n he presence of an exernal magnec feld Our model becomes a sngle loop or a rng due o he ds-na srucure, whch s aached o he sem-nfne elecrodes and lnked by couplngs beween adjacen bases and ner-base hydrogen bonds n he absence of he backbone effec, as shown n Fg 49 We color he backbone ses and couplngs lgh gray n order o ndcae lack of consderaon of he backbone effec Fg 49 Schemac for AB-rng nerference wh he ds poly(g)-poly(c) NA Aharonov and Bohm heorzed ha a beam of elecrons approachng a long solenod would spl n wo, and recombne ou of phase by a facor proporonal o he enclosed flux A magnec feld of flux densy penerang a he cener of he 2- NA srucure nduces AB phase dfference beween he elecron wave funcons of he upper and lower NA srands and produces AB oscllaons n he ransmsson T(E) The hoppng negrals are mulpled by a phase facor n order o observe he quanum nerference hrough he double-helx NA, defned as ± φ ( T ), + 1 ( T ), + 1e, h h e ± φ, and ± φ α α e,

72 where φ = πφ /( NΦ ) denoes he oal phase shf Φ measures he oal flux hrough 2 our sysem n uns of he flux quanum, Φ (h/e), and he plus or mnus sgns n he exponenal phase facor are appled when he elecron moves n he couner-clockwse or clockwse drecon, respecvely N s he number of ses; for example, a 2-5 base-par NA molecule whch s aached o he lef and rgh leads has 12 ses (N = 12; 5 G bases, 5 C bases, and 2 lead onses) Fgure 41 shows conour plos of he ransmsson as a funcon of magnec flux ( Φ / Φ ) and elecron energy (op) and he ransmsson vs elecron energy (boom) wh fxed = 1, L(R) = 3, ( T), + 1 = 2, α =, h =, 61 B a =, ε = 775, and σ α = 85 for dfferen numbers of base-pars: 1 base-par, 2 base-pars, and 5 base-pars (op o boom), n he absence of backbone effec and hydrogen bonds As he number of base-pars ncreases, he amplude of he AB oscllaons n he ransmsson probably decreases The AB nerference s small for he case of fve base-pars due o he ransmsson beng resrced more o only one or he oher srand, gvng less neracon beween he wo srands As a resul, only small oscllaons n he ransmsson can be seen n he nse of Fg 41 for fve base-pars [64, 65]

73 62 Fg 41 Transmsson conour plos vs elecron energy and magnec flux ( Φ / Φ ) wh fxed = 1, L(R) = 3, ( T), + 1= 2, α =, h =, B a =, ε = 775, and σ α = 85 whou consderng backbone effec and hydrogen bonds for dfferen number of base-pars, from one, wo, fve base-pars (op o boom) Transmsson coeffcen plos vs elecron energy for fxed ncomng energy (E = 8) are shown below he conour plos As he number of base-pars ncreases, he amplude of he AB oscllaons decreases However, here are sll small oscllaons n he ransmsson, as shown n he nse for fve base-pars Nex, we nvesgae he phase shf wh hydrogen bonds hrough shor poly(g)- poly(c) NA molecules As he number of base-pars changes, he number of loops va-

74 res For nsance, a sngle base-par has 2 loops, and 5 base-pars has 6 loops due o he hydrogen bonds ( h, = 1,,5) connecng bases across he long axs of NA We show he ransmsson T(E) vs magnec flux ( Φ / Φ ) wh fxed = 1, L(R) = 3, ( T), + 1= 2, h = 5, α =, B a =, ε = 775, 63 σ α = 85 and ncomng energy (E = 75) for dfferen numbers of base-pars, 1 base-par (op) and 5 base-par (boom) I s neresng o noe from Fg 411 ha he perodcy of he AB oscllaons s drecly proporonal o he number of loops n he NA molecules One base-par (2 loops, green) has a perodcy of 2 Φ and fve base-pars (6 loops, green) has a perodcy of 6 Φ n he ransmsson vs flux Fg 411 Flux dependence of he ransmsson for one base-par (op) and fve base-pars (boom), showng perodc AB oscllaons The perodcy n erms of Φ s he same as he number of loops hrough he NA

75 64 Chaper 5: Summary and Conclusons In hs hess, we have numercally suded elecron ranspor properes hrough a 1- one-channel NA model, a quas-1- one-channel NA model, and a 2- fourchannel NA model We nvesgaed he ransmsson, conour plos of ransmsson, localzaon lengh, and I-V characerscs as a funcon of ncomng elecron energy and magnec feld by solvng he TB Schrödnger equaon In Chaper 2, we sared wh a sngle srand, 2 base NA model We analyzed he elecron ransmsson wh varyng sequences, such as he homogeneous poly G bases sequence, he perodc G-C sequence, he Fbonacc G-C sequence, and he random G-C sequence The Lyapunov coeffcen and localzaon lengh, whch can be deermned from he ransmsson characerscs, were calculaed as well n order o undersand Anderson localzaon or dsorder effecs We have shown ha as he sysem becomes more randomzed by arbrarly choosng he 2 bases, he ransmsson resonances become narrower and he localzaon lengh s decreased Nex, we examned he ransmsson properes of ds NA by usng a one-channel TB model n Chaper 3 We ulzed he renormalzed onse poenal energy n order o ncorporae he effecs of he backbone n our calculaons The backbone-nduced effecs produce a gap n he ransmsson, whch ndcaes a semconducor-lke behavor of he NA Our numercal calculaons of he I-V characerscs are n agreemen wh he ex-

76 65 permenal resul [7] We have also nvesgaed he characerscs of elecron ranspor hrough boh symmerc and asymmerc NA molecules havng energy-dependen renormalzed onse energes and hoppng negrals from NA base-pars o he backbone We have shown ha as he asymmerc backbone onse energes are ncreased, overlappng of he wo ransmsson mn-bands occurs, and he merged sngle mn-band evenually dsappears and an exra resonance band remans In he I-V curve, he volage hreshold ncreases and he curren remans consan a a crcal asymmerc backbone onse energy When he asymmerc conac couplng s decreased, elecron unnelng s also decreased and a dsnc and under-uny resonance s shown n each band Fnally, n Chaper 4 we nvesgaed charge ranspor hrough a 2- four-channel poly(g)-poly(c) NA model by varyng parameers; he couplng beween NA bases and he backbone, hydrogen bonds beween bases, and ner-backbone couplng Exra ransmsson mn-bands appeared when he backbone couplngs exs The ransmsson resonan peaks merge and shf o lower energy as he hydrogen bonds are ncreased, snce he solaed quas-bound sae n each base-par mxes ogeher By consderng he ner-backbone couplngs we obaned he swng of Fano resonances n a specfc small energy wndow We appled a varaon of he emperaure and AB phase shfs no he hoppng negrals n order o consder hermal flucuaons and magnec feld effecs on he ransmsson As emperaure ncreases, he ransmsson oscllaons are suppressed and he localzaon lenghs are decreased A magnec flux penerang a he cener of he 2- NA nduces AB oscllaons whch are drecly proporonal o he number of loops, and whch have an oscllaon amplude nversely proporonal o he number of base-pars

77 66 Our fndngs provde possble characerscs and applcaons for usng NA as a componen n molecular elecroncs For fuure sudy, we wll apply a magnec feld hrough a 2- four-channel NA model wh backbone effec Ths mgh be more complcaed because many loops can be formed by he backbone couplngs We wll also nvesgae curren-volage characerscs whch are he measurable quany for conducvy of NA n he presence of magnec feld Furhermore, dfferen sequences of NA and longer NA base-par chans can be consdered n our fuure work

78 67 REFERENCES [1] H-A Waqenknech, Charge ransfer n NA: From Mechansm o Applcaon, Nanoboechnology (Wley, 25), and references heren [2] E Braun, Y Echen, U Svan, and G Ben-Yoseph, Naure (London) 391, 775 (1998) [3] PJ de Pablo, F Moreno-Herrero, J Colchero, J Gomez Herrero, P Herrero, A M Baro, P Ordejon, J M Soler, and E Aracho, Phys Rev Le 85, 4992 (2) [4] J Sorm, J van Noor, S de Vres, and C ekker, Appl Phys Le 79, 3881 (21) [5] Y Zhang, RH Ausn, J Kraef, EC Cox, and NP Ong, Phys Rev Le 89, (22) [6] KH Yoo, H Ha, JO Lee, JW Park, J Km, JJ Km, HY Lee, T Kawa, and HY Cho, Phys Rev Le 87, (21) [7] Porah, A Bezryadn, S de Vres, and C ecker, Naure (London) 43, 635 (2) [8] L Ca, H Tabaa, and T Kawa, Appl Phys Le 77, 315 (2) [9] HV Fnk, and C Schonerberg, Naure (London) 398, 47 (1999) [1] A Rakn, P Ach, C Papadopoulos, Y Kobzar, AS Vedeneev, JS Lee, and JM Xu, Phys Rev Le 86, 367 (21) [11] P Tran, B Alav, and G Gruner, Phys Rev Le 85, 1564 (2) [12] Y Kaxumov, M Kocak, S Gueron, B Reule, and V T Volkov, Scence 291, 28 (21) [13] AK Mahaparo, KJ Jeong, GU Lee, and B Janes, Nanoechnology 18, (27) [14] B Xu, P Zhang, X L, and N Tao, Nano Leers 4, 115 (24)

79 68 [15] S Roy, H Vedala, A Roy, Km, M oud, K Mahee, H Shn, N Shmamoo, V Prasad, and W Cho, Nano Leers 8, 26 (28) [16] S Roche, Phys Rev Le 91, 1811 (23) [17] G Cunber, L Craco, Porah, and C ekker, Phys Rev B 65, (22) [18] R ong, X Yan, and B Yang, Commun Theor Phys (Bejng, Chna), 5, 532 (28) [19] YS Joe, ER Hedn, and AM Saann, Phys Rev B 76, (27) [2] ER Hedn, YS Joe, and AM Saann, J Phys:Condens Maer 21, 1533 (29) [21] S Roche, Bcou, E Maca, and E Kas, Phys Rev Le 91, (23) [22] PW Anderson, Phlos Mag B 52, 55 (1985) [23] E Marca, F Trozon, and S Roche, Phys Rev B 71, (25) [24] AM Guo and H Xu, Physca B 391, 292 (27) [25] Klosa, RA Romer, and MS Turner, Bophys J 89, 2187 (25) [26] A Lagendjk, B van Tggelen, and S Wersma, Physcs Today 62, 24 (29) [27] M Zwolak and M Venra, Nano Leers 5, 421 (25) [28] G Xong and XR Wang, Phys Les A 344, 64 (25) [29] H Yamada, Inernaonal Journal of Modern Phys B 18, 1697 (24) [3] ZG Yu and X Song, Phys Rev Le 86, 618 (21) [31] P Lee, Revews of Modern Phys 57, 287 (1985) [32] CM Soukouls and EN Economou, Waves Random Meda 9, 255 (1999) [33] M Sorzer, P Gross, C Aegerer, and G Mare, Phys Rev Le 96, 6394 (26) [34] S Roche and E Maca, Modern Phys Les B 18, 847 (24) [35] H Yamada, Phys Les A 332, 65 (24) [36] H Yamada, M Goda, and Y Azawa, J Phys:Condens Maer 3, 143 (1991)

80 69 [37] E Maca and S Roche, Nanoechnology 17, 32 (26) [38] R Landauer, Phlos Mag 21, 863 (197) [39] M Buker, Phys Rev B 35, 4123 (1987) [4] M Hjor and S Safsrom, Phys Rev Le 87, (21) [41] K Ferry and SM Goodnck, Transpor n Nanosrucure (Cambrdge unversy press, New York, 1997) [42] YS Joe, S Ikeler, RM Cosby, AM Saann, and CS Km, J Appl Phys 88, 274 (2) [43] YS Joe, JS Km, and AM Saann, J Phys : Appl Phys 39, 1766 (26) [44] H Tabaa, L-T Ca, J-H Gu, S Tanaka, Y Osuka, Y Sacho, M Tanguch, and T Kawa, Synh Meals (23) [45] H-A Wagenknech, Cheme n unserer Ze 36, 318 (22) [46] A Csak, G Maubach, Born, J Recher, and W Frzsche, Sngle Mol 3, 275 (22) [47] C ekker and M Raner, Phys World, Augus 29 (21) [48] P Carpena, P Bernaola-Galvan, PCh Ivanov, and HE Sanley, Naure (London) 418, 955 (22) [49] W Zhang and SE Ulloa, Phys Rev B 69, (24) [5] H Wang, JP Lews, and OF Sankey, Phys Rev Le 93, 1641 (24) [51] Ch Adess, S Walch, and MP Ananram, Phys Rev B 67, 8145(R) (23) [52] E Aracho, M Machado, Sanchez-Poral, P Ordejon, and JM Soler, Molecular Phys 11, 1587 (23) [53] JX Zhong, Nanoech, 2, 15 (23) [54] K Iguch, J Phys Soc Jpn, 7, 593 (21) [55] J Y and H Orland, J Appl Phys 99, (26) [56] YS Joe, SH Lee, and ER Hedn, Phys Le A 374, 2367 (21)

81 7 [57] K Iguch, In J Mod Phys B 11, 245 (1997) [58] K Iguch, J Phys Soc Jpn, 7, 593 (21) [59] LA Jauregu, K Salazar-Salnas, and JM Semnaro, J Phys Chem B 113, 623 (29) [6] E az, A Sedrakyan, Sedrakyan, and F omnguez-adame, Phys Rev B 75, 1421 (27) [61] U Fano, Phys Rev 124, 1866 (1961) [62] W Porod, Z Shao, and CS Len, Phys Rev B 48, 8495 (1991) [63] ML Ladon de Guevara, F Claro, and PA Orellana, Phys Rev B 67, (23) [64] JF Feng, XS Wu, SJ Xong, and SS Jang, Sold Sae Communcaons 139, 452 (26) [65] W Ren, J Wang, Z Ma, and H Guo, Phys Rev B 72, (25) [66] S Roche and E Maca, Modern Phys Le B 18, 847 (24)

82 71 APPENIX A: Sample Mahemaca 7 Program for quas 1 NA model and 2- four-channel NA model Clear["Global`*"] n2:=7; m1=array[m,{n2,n2}]; o[m[,j]:=,{,1,n2},{j,1,n2}]; Array[vd,n2]; Array[en,n2-2]; M[1,1]:=; M[2,2]:=-vL; M[1,3]:=vL; M[n2,n2]:=-vR; en[1]:=e1; en[2]:=e2; o[m[,]:=vd[],{,3,n2-1}]; M[n2-1,1]:=-vR*Exp[I*θ]; M[n2,1]:=v; M[2,3]:=e1-e;M[3,4]:=e2-e;M[4,5]:=e3-e;M[5,6]:=e4-e;M[6,7]:=e5-e; (*o[m[+1,+3]:=vd[+2],{,1,n2-3}];*) M[3,3]:=-va;M[4,4]:=-vb;M[5,5]:=-vc;M[6,6]:=-vd; M[2,4]:=-va;M[3,5]:=-vb;M[4,6]:=-vc;M[5,7]:=-vd; M[1,2]:=-v*Exp[-I*θ]; MarxForm[m1] c1:=array[cons,n2]; o[cons[]:=,{,1,n2}]; cons[1]:=v*exp[i*θ]; cons[2]:=vl; MarxForm[c1] Mc:=Inverse[m1]; rp:=mcc1; Par[rp,1]; =%; Par[rp,2]; r=%; T2[en_]:=Module[{},e:=en;θ:=ArcCos[(-e+ε)/(v 2`)];1:=;Reurn[1]]; R1[en_]:=Module[{},e:=en;θ:=ArcCos[(-e+ε)/(v 2`)];r1:=r;Reurn[r1]]; v:=1; vl:=5; vr:=5; ε:=775; b:=7; εb:=85;

83 72 εa:=824; εt:=914; εg:=775; εc:=887; ε1:=(εa+εt)/2; ε2:=(εg+εc)/2;ε3:=(εg+εg)/2; va:=1;vd:=1;vb:=1;vc:=1; e1:=ε2-b^2/(εb-e)-b^2/(εb-e);e2:=ε2-b^2/(εb-e)-b^2/(εb-e);e3:=ε2- b^2/(εb-e)-b^2/(εb-e);e4:=ε2-b^2/(εb-e)-b^2/(εb-e);e5:=ε2-b^2/(εbe)-b^2/(εb-e); Plo[Abs[T2[e]]^2,{e,575,975},Axes False,Frame True,FrameSyle Thck, PloRange {{575,975},{-1,1}},PloSyle {AbsolueThckness[3]}, FrameLabel {"E","T(E)"},BaseSyle {"Helveca",22}] Clear["Global`*"] v:=1; vl1:=3;vl2:=3;(* couplng beween NA and lef elecrode vl *) vr1:=3;vr2:=3;(* couplng beween NA and rgh elecrode vr *) 12:=2;T12:=2; h1:=1;(*hydrogen bonds*) va:=1;(*couplng beween NA base and backbone*) Ba:=1;(*ner-backbone couplng*) ε:=775;(* elecrode onse energy *) (*εa:=824; εt:=914; εg:=775; εc:=887;*) e1:=775;e6:=887; ea:=85; n2:=7; m1:=array[m,{n2,n2}]; o[m[,j]:=,{,1,n2},{j,1,n2}]; o[m[,+1]:=e-e1,{,2,n2-4,4}]; o[m[,+1]:=e-e6,{,3,n2-3,4}]; o[m[,+1]:=e-ea,{,4,n2-2,4}]; o[m[,+1]:=e-ea,{,5,n2-1,4}]; o[m[,-3]:=12,{,6,n2-4,4}]; o[m[,+5]:=12,{,2,n2-8,4}]; o[m[,-3]:=t12,{,7,n2-3,4}]; o[m[,+5]:=t12,{,3,n2-7,4}]; o[m[,-3]:=ba,{,8,n2-2,4}]; o[m[,-3]:=ba,{,9,n2-1,4}]; o[m[,+5]:=ba,{,4,n2-6,4}]; o[m[,+5]:=ba,{,5,n2-5,4}]; o[m[,-1]:=va,{,4,n2-2,4}]; o[m[,-1]:=va,{,5,n2-1,4}]; o[m[,+3]:=va,{,2,n2-4,4}]; o[m[,+3]:=va,{,3,n2-3,4}]; o[m[,]:=h1,{,3,n2-3,4}]; o[m[,+2]:=h1,{,2,n2-4,4}]; M[1,1]:=; M[1,2]:=-v*Exp[-I*θ]; M[1,3]:=vL1; M[1,4]:=vL2; M[2,2]:=vL1; M[3,2]:=vL2; M[n2,1]:=-v; M[n2,n2-3]:=vR1; M[n2,n2-2]:=vR2; M[n2-4,1]:=vR1*Exp[I*θ];

84 73 M[n2-3,1]:=vR2*Exp[I*θ]; MarxForm[m1]; c1:=array[cons,n2]; o[cons[]:=,{,1,n2}]; cons[1]:=v*exp[i*θ]; cons[2]:=-vl1; cons[3]:=-vl2; MarxForm[c1]; JBF:=LnearSolve[m1,c1,Mehod "vsonfreerowreducon"]; :=Par[JBF,1]; T[en_]:=Module[{},e:=en;θ:=ArcCos[(-e+ε)/(v*2)];Reurn[]]; Plo[Abs[T[e]]^2,{e,575,975},Axes False,Frame True,FrameLabel {"E","T (E)"},BaseSyle {"Helveca",2},PloSyle {AbsolueThckness[23]}, PloRange {{575,975},{-1,11}}]

85 APPENIX B:

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91