Quantum electron transport in toroidal carbon nanotubes with metallic leads Authors: Mark Jack and Mario Encinosa Dept. of Physics, Florida A&M University, 205 Jones Hall, Tallahassee, FL 32307. Email: [email protected], [email protected]. Gordon Research Conference ‘Quantum Control of Light and Matter’ Aug. 12-17, 2007 Salve Regina University Newport, RI

Abstract The authors have extended earlier studies for metallic toroidal nanostructures to calculating the density-of-states and transmission function for toroidal carbon nanotubes with attached metallic or carbon nanotube leads. The Green's function of the nanodevice region is calculated in tight-binding approximation. Initial investigations for a smaller model (i.e. a graphene sheet of 150 layers of a 12-carbonatom super cell wrapped to a nanotorus) have been expanded to simulations of larger, more realistic systems with different geometric attachments of the leads on a parallel computer cluster. In earlier work, the existence and relevance of solenoidal surface currents on a metallic nanotorus surface in a microwave field and their effects on the total magnetic moment have been calculated. Substantial enhancements of the magnetic moment’s solenoidal mode versus the dipole mode had been achieved in simulations. Discussion of these effects is important for spintronics applications of nanotube structures.

Introduction Carbon nanot ubes [ 1] and graphene [ 2] are t he unique mat erials t hat carry t he promise of defining t he key t echnological advancement in nanoelect ronics in t he 21st cent ury. Their unique st ruct ural, mechanical, opt ical and elect ronic propert ies wit h f ascinat ing new mult i-quasipart icle coherent quant um phenomena visible at room t emperat ure also make t hem t he mat erials of choice f or st udies of coherent quant um t ransport and quant um cont rol of elect rical charge and spin [ 3-6] . Toroidal carbon nanot ube st ruct ures, also known as carbon nanot ori, could play a part icularly int erest ing role in nanoelect ronics, quant um comput ing realizat ions or as biosensors [ 7-10] . S ome of t he applicat ions t hat come t o mind, in which t he unique elect ronic quant um t ransport f eat ures of t hese nanoscale t oroidal geomet ries are ut ilized, are: • C arbon nanot ori as t hree-dimensional molecular rings wit h persist ent current ef f ect s and azimut hal and dipole-t ype elect ronic excit at ions in ext ernal microwave fields; • C arbon nanot ori as molecular A haronov-B ohm oscillat or wit h magnet ic flux t hreaded t hrough t he t orus; • Carbon nanot ori wit h def ect s in biosensor applicat ions af t er covalent ly at t aching a biopolymer. We want t o calculat e densit y-of -st at es D(E) and t ransmission f unct ion T(E) f or elect ron t ransport t hrough t he t orus bet ween t wo at t ached met allic leads under a small volt age bias [ 11-15] . The t orus modelled here consist s of 1800 carbon at om sit es in a graphene lat t ice. The t orus measures a = 4A in azimut hal widt h and has a cent ral diamet er of D = 116A.

Non-equilibrium Green’s function method (NEGF) and recursive algorithm (RGF) For a small bias volt age bet ween t he met allic cont act s, densit y-of -st at es D (E), t ransmission f unct ion T(E) and conduct ivit y dI dV can be calculat ed f or t he modified CNT in t ight -binding approximat ion [ 11-15] . A ll quant it ies can be derived f rom t he (ret arded/ advanced) Green’s f unct ion G d (E) of t he device region which is comput ed in a reverse Green’s f unct ion algorit hm [ 12, 14, 15 ] .

! ! ! $% E ( k ) I ! H ( k ) ! " L ! " R ± i# &' Gd( a,r ) ( k ) = I (E q. 1)

The basic equat ion f or G d (E) is illust rat ed in E q. 1 where t he ef f ect of t he lef!t / right met allic lead can be f olded int o t he Hamilt onian H ( k ) f or t he device region in f orm of t he self -energy correct ions ! L, R [ 11] . In t ight -binding, only t he overlap int egrals of t he nearest neighbor p z- orbit als are considered in t he Hamilt onian f or conduct ion and are described in t erms of a general hopping paramet er t. Wit h t he recursively det ermined Green’s f unct ion, t he current ISD c an be calculat ed wit h t he t ransmission f unct ion T(E) f rom E q. 2:

2e dE T (E) "# fL ( E ) ! fR ( E ) $% , ! & T ( E ) = Trace "# ' L G r ' RG a $% , I SD =

(E q. 2)

wit h fL , R(E) as Fermi dist ribut ions at t he t wo cont act s and ΓL ,R as nanot ube-t o-lead couplings [ 11] .

Results Density-of-states D(E) and transmission function T(E):

Figure 1: Sketch of device setup with toroidal carbon nanotube and metallic leads. Figure 2: a. Density-of-states D(E) for different magnetic fields B 0 . b. Transmission function T(E) vs. B 0 .

Quantum electron transport in tight-binding approximation The lef t and right semi-infinit e met allic leads t ouch t he t orus surf ace in f our at omic cont act sit es at each lead. The at omic posit ions of t hese cont act s can be det ermined easily wit h t he t oroidal vect or r , mapping t he surf ace of t he t orus ,

Coherence in electronic transmission - Plateaus in T(E) for E = o(0.01 … 0.1 eV) for different lead angles α:

! r (! , " ) = ( R + a cos! ) eˆ# + a sin ! eˆz ,

and wit h t he relat ive posit ions of t he t wo leads. The opening angle α bet ween t he t wo leads can be varied f rom 180 o ( back-to-back configuration) down t o 45o (315 o ). The Hamilt onian H f or elect ron t ransport t hrough t he t orus is prepared in t ight -binding approximat ion f ollowing [ 11, 12] :

(

)

H = ! Ei ci†ci + ! tij ci†c j + h.c. . i

i> j

Carbon nanotorus as Aharonov-Bohm oscillator:

Figure 3: Coherence in electronic transport: a. T(E) at E = 0.01eV as a function of magnetic field B 0 for different angles α between metallic leads. b. T(E) at E = 0.02eV.

Figure 4: B ack-to-back leads: a) Source-drain current ISD as function of magnetic field B 0 . $90o angle btw. leads: b) ISD vs. B 0 . Chemical potential µ1 ,2 at left/right lead: +/- V SD/2.

Conclusions Coherent int erf erence could be observed in elect ronic t ransmission T(E) as a f unct ion of t he opening angle α bet ween met allic leads and in t he source-drain current ISD as a f unct ion of magnet ic flux Φ. To be considered In f ut ure are an ext ension t o larger syst ems wit h more t han 10000 at oms, a calculat ion beyond t ight -binding approximat ion t o include nearest and next -t o-nearest neighbor cont ribut ions and a more realist ic device set up wit h t he t orus lying flat on t he t wo planar leads in order t o significant ly increase t he number of at omic cont act sit es incl. a self -consist ent t reat ment of e-e Int eract ion, e-phonon c oupling and Coulomb blockade ef f ect s. Acknowledgments M. E . would like t o t hank M. P . A nant ram and M. Meyyapan f or init ially suggest ing t his project and f or helpf ul support at t he NA S A A mes Research Cent er. B ot h aut hors would also like t o t hank L. Johnson and N. Christ opher at Florida A & M Universit y's Laser Remot e S ensing Laborat ory (LRS L) Comput er Clust er. References 1. S . Iijima, Nat ure (London) 354, 56 (1991). 2. K . Novoselov et al. , S cience 306, 666 (2004). 3. B . Trauzet t el et al. , Nat ure P hysics 3, 192 (2007). 4. A . Rycerz et al. , Nat ure P hysics 3, 172 (2007). 5. Y. -W. S on et al. , Nat ure 444, 347 (2006). Corrigendum, Nat ure 446, 347 ( 2007). 6. P . Recher et al. , E -print archive: cond-mat . meshall/ 0706. 2103v. 1. 7. M. E ncinosa and M. Jack, E -print archive: physics/ 0604214. P hys. S cr. 73, 439-442 (2006). Dipole and solenoidal magnet ic moment s of elect ronic surf ace current s on t oroidal nanost ruct ures. Journal of Comput er-A ided Mat erials Design (S pringer), May 2006. 8. L. Liu et al. , P hys. Rev. Let t . 88, 217206 (2002). 9. K . S asaki and Y. K awazoe, E -print archive: condmat . mes-hall/ 0307339v1. 10. B . R. Goldsmit h et al. , S cience 315, 77-81 (2007). 11. S . Dat t a, E lect ronic Transport in Mesoscopic S yst ems. Cambridge Univ. P ress (1995). 12. M. P . A nant ram and T. R. Govindan, P hys. Rev. B 58(8), 4882-4887 (1998). 13. Y. Xue, S . Dat t a, and M. Rat ner, Chem. P hys. 281 (2002) 151-170. 14. M. E ncinosa, S urf ace charge ef f ect s f rom t he coupling of a carbon nanot ube t o met allic leads. 1999 NA S A A S E E S t anf ord S ummer Fellowship Final Report . 15. M. E ncinosa, A pplicat ion of a modified recursive Green’s f unct ion met hod t o t oroidal carbon nanot ube elect ronic propert ies. 2000 NA S A -A S E E S an Jose S t at e Univ. S ummer Facult y Fellowship Final Report .