Equivalent electric circuit of a carbon nanotube based molecular conductor

Transcription

1 Equivalnt lctric circuit of a carbon nanotub basd molcular conductor iyung Yam Yan Mo Fan Wang Xiaobo i GuanHua n Dpartmnt of mistry ntr of Tortical and omputational Pysics Univrsity of Hong Kong Hong Kong Xiao Zng Dpartmnt of mistry T Univrsity of Hong Kong Dpartmnt of mistry Hong Kong Univrsity of Scinc and Tcnology Hong Kong Yuki Matsuda Jamil Tair-Kli and William A. Goddard III Matrials and Procss Simulation ntr M 9-7 alifornia Institut of Tcnology Pasadna A 95 Abstract W apply our first-principls mtod to simulat t transint lctrical rspons troug carbon nanotub basd conductors undr tim-dpndnt bias voltags and rport t dynamic conductanc for a spcific systm. W find tat t lctrical rspons of t carbon nanotub dvic can b mappd onto an quivalnt classical lctric circuit. Tis is confirmd by studying t lctric rspons of a simpl modl systm and its quivalnt circuit.

2 As t Moor s aw oadmap for smiconductor industry is followd into t scal of 0 nm it bcoms important to undrstand dynamic rspons of nanomtr scal molcular lctronic dvics.[-] Tis rquirs t us of quantum mcanics to nsur t propr tratmnt of transint and quantum ffcts. For practical us by dsign nginrs it is crucial to cast ts quantum ffcts into t form of classical lctric circuits. An important qustion is wtr suc a mapping is possibl and if so wat ar t forms of t quivalnt circuits? As potntially important componnts of nxt-gnration intgratd circuits carbon nanotubs NTs av bn studid xtnsivly.[-95] Hig frquncy lctrical rspons of micromtr-long individual and bundld NTs av bn masurd[5] and quivalnt lctric circuits av bn proposd.[58-9] In tis work w concntrat on a nanomtr-scal NT-basd lctronic dvic and apply firstprincipls quantum mcanics QM to dtrmin its dynamic lctrical rspons. Our systm is a 55 NT 0.68 nm in diamtr and 0.6 nm in lngt wic is bondd covalntly btwn two aluminum lctrods and sown in Fig. a. Fig. a Prototyp usd for xplicit QM calculations of a carbon nanotub basd conductor: t 55 NT dvic wit aluminum lctrods. b A two-sit systm coupld to t lft and rigt lctrods. / To prdict t transint lctrical rspons of tis molcular dvic w us t rigorous tim-dpndnt dnsity functional tory TDDFT tat w dvlopd rcntly[6] to valuat t tim-dpndnt currnt troug t dvic. W includd xplicitly in t simulation box 8 Al atoms of ac lctrod along wit 60 atoms of t NT.

3 Figurs a and b sow t currnt vrsus tim for two diffrnt typs of bias voltag switcd on at t 0. In Fig. a bias voltag V b is turnd on xponntially. W obsrv tat t currnt racs its stady stat in fs. T tim dpndnt currnt can b fittd by I 0 - -t/ wit.8 fs and I 0.9 na lading to a caractristic tim of.8 fs. T rason for suc a fast switc-on tim is tat t procss involvs only lctrons. Figur a plots t potntial nrgy cang for an lctron along t cntral axis at t 0.0 and fs. T potntial cang is t sum of t applid potntial and t potntial causd by t inducd carg. Our calculation lads to following obsrvations: Aftr turning on t bias voltag at t 0.0 fs t lctrons av not yt rspondd to t applid voltag and t xtrnal fild is ardly scrnd dropping uniformly across ntir Al-NT-Al systm. At t fs t potntial drop occurs mostly on t NT sinc aluminum is mor polarizabl. It taks lss tan fs for t lctrons on t Al lctrods to scrn t applid potntial. In Fig. b w plot inducd carg along Al-NT-Al at t fs. T rd indicats positiv carg wil t blu indicats t ngativ carg. Altrnating positiv and ngativ carg distributions on NT cancl ac otr so tat its nt inducd carg is zro. T xcss carg rsids primarily at two intrfacs and forms an ffctiv capacitor as dpictd scmatically in Fig. c. W also considrd t rspons to a sinusoidal bias voltag turnd on at t 0. Fig. b sows t corrsponding tim-dpndnt currnt. W s a pas dlay in t currnt rspons to bias voltag. Tis implis at tis frquncy t dvic is ovrall inductiv. T Al-NT-Al systm is symmtric. As a consqunc tr is no nt carging of t dvic and t tim dpndnt currnt is consrvd.[7] Tis was confirmd by our numrical simulation. T currnt ntring t systm as t sam magnitud as t currnt laving as sown in Figs. a and b. Trfor t conductanc matrix lmnt G or satisfis G G -G -G G.[78] Taking t Fourir transform t simulatd transint dynamics lads to I and V from wic w obtain t dynamic conductanc G I / V. W find tat bot typs of bias voltags lad to ssntially t sam dynamic conductanc. Tis implis tat t lctrical rspons is in linar rspons rgim and also validats t accuracy of our calculations. Figur c sows t ral and imaginary parts of t rsulting dynamic conductanc.

4 Fig. a and b Transint currnt rd lins and squars and applid bias voltag grn lins for Al-NT-Al systm. a Bias voltag is turnd on xponntially Vb V0 - -t/a wit V 0 0. mv and a tim constant a fs. Blu lin in a is a fit to transint currnt. b Bias voltag is sinusoidal wit a priod of 5 fs. d lin is for currnt from rigt lctrod and squars ar currnt from lft lctrod. c Dynamic conductanc calculatd from t xponntial bias voltag turnd on at t0 solid squars and t sinusoidal bias voltag solid triangl turnd on at t0. T rd lins ar t fittd rsults. T uppr curvs ar t ral part of t conductanc wil t lowr ons ar t imaginary part.

5 Fig.. a Elctrostatic potntial nrgy distribution along t cntral axis at t 0.0 and fs. b arg distribution along Al-NT-Al at t fs. c Scmatic diagram sowing t inducd carg accumulation at two intrfacs wic forms an ffctiv capacitor. Now t qustion is ow to modl t Al-NT-Al dvic. Our abov rsults and analysis sow tat our molcular dvic as bot inductiv and capacitiv componnts. As t currnt ntrs into t dvic rgion from t lft lctrod a part of it I c cargs t lft intrfac s Fig. a. T rmaining currnt I gos straigt troug t dvic and is joind by I c at t rigt intrfac. Trfor our dvic can b modld by t

6 classical circuit dpictd in Fig. b. At zro frquncy t stady currnt gos only troug t - branc. is simply t stady stat rsistanc [90] wic w calculat to b 7.9 k. A similar circuit was proposd to fit t numrical dynamic conductanc of a modl quantum wir [] wit an additional inductor to account for a rsonanc at a ig frquncy. Büttikr and coworkrs [0] studid a msoscopic capacitor mad of two plats wit ac coupld to an lctron rsrvoir via a narrow lad. Ty discovrd tat t carg rlaxation rsistanc is univrsal indpndnt of transmission dtails lading to wit and givn by c EF E D n n D n n wr D E n F is t dnsity of stats DOS at Frmi nrgy for nt spin-spcific carging cannl of plat and n was confirmd by a rcnt xprimnt. [] F is ovr all carging cannls for. Tis In our Al-NT-Al systm two intrfacs corrspond to two plats of t capacitor s Fig. c and coupl to t lctrods via Al lads. Our NT as two dgnrat orbitals for transmission. Bot spin-up and spin-down lctrons contribut to t transmission. Trfor tr ar four carging cannls for ac intrfac and according to Eq. t carg rlaxation rsistanc for t Al-NT-Al systm is. W tun t valus of and to fit t calculatd dynamic conductanc wil fixing and 7.9 k. T rsulting valus of and ar 6.6 ph and 0.07 af rspctivly. d lins in Fig. c ar t ral and imaginary valus of t dynamic conductanc of t quivalnt lctric circuit wic agr wll wit our calculatd dynamic conductanc up to 50 THz. W can stimat t capacitanc dirctly from t xcss carg Q at t intrfacs. For bias voltag V b 0. mv w find tat at t stady stat Q is ~0.0. Trfor w stimat t capacitanc as Q/V b 0.05 af wic is of t sam magnitud as t calculatd 0.07 af. Sinc tr is uncrtainty to dfin t xcss carg at t intrfacs t valu 0.05 af is mant to b an stimation only. According to Eqs. 5 7 blow t inductanc is ~.

7 Avrag lin widt </> at t Frmi nrgy is ~0.8 V. Tus 8.8 ph. Tis is clos to t calculatd 6.6 ph. Fig.. a urrnt flow in t paralll circuit. b T quivalnt lctric circuit. 7.9 k 6.6 ph 6.5 k and 0.07 af. To confirm furtr t quivalnt lctric circuit for our NT basd molcular conductor w dsignd a simpl modl: a two-sit systm in contact wit t lft and rigt lctrods s Fig. b. T two sits ar dgnrat in nrgy 0. For t capacitor limit wr d 0<<< and 0 µ µ w driv analytically t frquncy dpndnt conductanc as follows 8 ˆ Î ˆ Tr O i E G i E G E f E f de G a q r q * * -. / : ; Expanding t dynamic conductanc of t lctric circuit in Fig. b in lads to a dynamic conductanc:

8 O i G omparing Eqs. and w find tat. 8 8 d 5 For t ballistic transport limit wr ± d and 0 < << and 0 µ µ wic is similar to our systm t dynamic conductanc is 0 O i G * 6 omparing Eq. 6 wit Eq. w find tat. 5 5 c * * 7 T abov calculation sows tat nar bot capacitiv and inductiv limits t lctrical rspons of two-sit systm can b modld by t classical circuit in Fig. b and its carging rlaxation rsistanc is wic agrs wit Eq.. If ac sit is two-fold dgnrat t carg rlaxation rsistanc is rducd to alf tis valu i.. / wic is xactly t sam as our Al-NT-Al systm. An - circuit was proposd for long NT wit as t kintic inductanc.[589] In t prsnc of a substrat xtra paralll capacitors ar introducd btwn t tub and substrat.[89] Wn a NT sits on top of an lctrod via van dr Waals attraction an ffctiv capacitor is introducd btwn t NT and lctrod in

9 addition to a paralll contact rsistor.[5] T kintic inductanc and quantum capacitanc of a long NT ar intrinsic proprtis of t tub bing dtrmind by t DOS at t Frmi nrgy or t Frmi vlocity v f. In our cas t NT is muc sortr and is wldd to t lctrods covalntly. T lctrical rsponss of t intrfacs and tub cannot b sparatd. Our inductanc is dtrmind by its slf-nrgy du to t coupling to t lctrods or t dwll tim of t conducting lctron insid t dvic.[] Our capacitanc is mostly dictatd by t local DOS at intrfacs. As t lngt of NT incrass t capacitanc du to t intrfacs dcrass and t kintic inductanc of t tub dominats. As a rsult our quivalnt circuit is rducd to t - branc only wic is consistnt wit t quivalnt lctric circuit proposd for t long NTs. [589] Wang and coworkrs[] introducd t dwll tim to unify t inductanc xprssion for sort and long tubs as ~ d. For a long D systm of lngt l ~ l / v f lading to an xprssion for t kintic inductanc pr lngt /l ~ v f. d Our nanoscal dvic as vry small valus of and lading to t sort switcing tim. Suc a fast switcing spd for lctronic dvics basd on nanomtrsiz NTs indicats tat ts dvics will not limit switcing spds in t forsabl futur. T quivalnt lctric circuit of t paralll - and - circuit in Fig. b is not limitd to t NT-basd conductor studid in tis work. It also applis to otr twotrminal molcular nanoscopic and msoscopic lctronic dvics. is givn by andaur-buttikr formula for stady stat currnt[90] and c is t univrsal carg rlaxation rsistanc dpnding only on t numbr of carging cannls and spin polarization.[0] is dtrmind by t dwll tim of t lctrons insid t dvic as.[] is t lctro-cmical capacitanc wic is dtrmind by t gomtry ~ d and t DOS at intrfacs.[0] Ts rsults sould b usful in dsigning t nanoscal lctronics systms rquird ovr t nxt dcad. d Acknowldgmnts: T autors tank Hong Guo Jian Wang and YiJing Yan for stimulating discussions. T altc tam was supportd partly by Intl omponnts sarc Portland O and by NSF F T HKU tam was supportd by t Hong Kong sarc Grant ouncil HKU 70/06P N_HKU 76/05 HKUST /0. gc@vrst.ku.k xzng@yangtz.ku.k wag@wag.caltc.du frnc:

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