of Physics, Florida A&M University, 205 Jones Hall, Tallahassee, FL 32307. Email: [email protected], [email protected]

SC’06 International Conference on High Performance Computing, November 11-17, 2006, Tampa, FL.

Abstract The time-dependent Schrödinger equation inclusive of curvature effects is developed for a spinless electron constrained to motion on a toroidal surface and subjected to circularly and linearly polarized waves in the microwave regime. Typical nanotori major radii studied here are 50 nm with circularly symmetric cross sections. A seven-state basis set is employed with the goal of determining the character of the surface currents as the system is driven at a resonance frequency that selects for a solenoidal mode. Trajectory methods are used as a means of visualizing the character of the induced surface currents. Optical transitions into solenoidal modes of excitation can be observed. The magnetic moment which is induced by surface currents on the torus shows an expected dipole component parallel to the toroidal central symmetry axis together with additional components in radial and/or azimuthal direction. Size and relative magnitude of the different components can be steered by adjusting magnitude, polarization, and phase information of the electromagnetic field of a single linear or circular polarized microwave or of the field created by the interference of two equally polarized waves.

Introduction

Methodology

The quant um cont rol of nanoscale object s and ot her st ruct ures t hrough elect romagnet ic int eract ions is of f arranging f undament al and pract ical int erest . Considerable ef f ort has been direct ed t owards quant um rings, or more generally, flat annular st ruct ures which because of t heir t opology give rise t o A haronov-B ohm and persist ent current ef f ect s. Toroidal nanost ruct ures present int riguing possibilit ies f or use as nanoscale device element s. In addit ion t o A haronov-B ohm and persist ent azimut hal current ef f ect s, t oroidal st ruct ures allow f or mot ions around t he minor radius of t he t orus subject t o boundary condit ions which are dist inct f rom t hose f or flat t wo-dimensional rings. For hollow t ori, elect rons are t hought t o be localized near t he surf ace of t he object . The rest rict ion t o mot ion near a surf ace has int erest ing manif est at ions. Recent work on a t orus of major radius R and circular cross sect ion of radius a, has shown t hat a surf ace dependent geomet ric pot ent ial VC is import ant even f or elect rons t hat can wander subst ant ial dist ances away f rom t he surf ace and should be employed as an ef f ect ive pot ent ial when considering t wo dimensional problems on curved surf aces. S urf ace curvat ure ef f ect s (S C) are more pronounced f or ellipt ical t ori which can behave in dif f erent ways t han a t orus wit h circular cross sect ion.

When rest rict ing a part icle init ially in t he neighborhood of a surf ace t o t he surf ace proper, t he inclusion of an ef f ect ive geomet ric pot ent ial t erm VC is needed t o capt ure t he f ull t hreedimensional s pect ra and wave f unct ions wit h good fidelit y. In t his work, VC will not be t he only geomet ric pot ent ial; t he normal component A N of t he vect or pot ent ial also couples t o S C. To evaluat e E q. 2, point s in t he neighborhood of a t oroidal surf ace of major radius R and minor radius a are paramet erized in cylindrical coordinat es wit h:

In t his work, rat her t han assuming a flat annular geomet ry, a t oroidal st ruct ure in a microwave field is considered. In cont rast t o flat st ruct ures, it is possible t o excit e modes which circulat e around t he minor radius of T2 (called here in what f ollows solenoidal modes) in addit ion t o t he azimut hal modes around t he major radius (ref erred t o here as dipole modes). The goal of t his work is t o underst and t he t imedependent charact erist ics of a spinless elect ron on T2 upon being subject ed t o various f orms of elect romagnet ic signals, and t o t hen employ t raject ory met hods as an aid t o visualize t he behavior of surf ace current s. The approach adopt ed here will be t o writ e (1) wit h H 0 as Hamilt onian f or a spinless elect ron const rained t o mot ion on T2 inclusive of surf ace curvat ure ef f ect s and V(t) being t he t ime-dependent elect romagnet ic int eract ion. The S chrödinger equat ion f or a spinless charged part icle in t he presence of a vect or pot ent ial A(t) is (2)

wit h J(r) as elect ron current calculat ed f rom t he wave f unct ion in E q. 4. It is suf ficient t o f ocus on t he cases of linearly polarized wave(s) shown in Fig. 2. The t hree component s in direct ion of e z e φ , and e ρ are shown in Fig. 1 and 2 wit h t he dipole t erm (prop. t o e z) being st rongly suppressed while one of t he solenoidal t erms is st rongly enhanced.

(5) wit h unit vect or e n everywhere normal t o T2 and q being t he normal coordinat e measuring t he dist ance f rom T2. The vect or pot ent ial A( t) in E q. 3 corresponds t o a circularly polarized wave. The linearly polarized case can be obt ained t rivially. The geomet ric pot ent ! e ial Vc is f ound by reducing E q. 2 f urt her by t he well known procedure of assuming a suit able confining pot ent ial Vn (q) in t he normal direct ion and demanding conservat ion of t he norm in t he limit q → 0. There is a second geomet ric pot ent ial t hat arises f rom t he normal part of t he vect or pot ent ial in t his limit . E q. 3 may be solved by rout ine met hods. Here a basis set expansion in GramS chmidt (GS ) f unct ions ort hogonal over t he int egrat ion measure F=1+αcosθ used in previous work was chosen. B ecause it is t he low-lying st at es t hat are of int erest here, a 30-st at e expansion f or each θ- parit y proves suf ficient . z

Results On resonance f or t ransit ions f rom t he ground st at e t o t he first negat ive θ-parit y st at e, i. e. ω 1 7 = ω 1 - ω 7 , t he coef ficient d 7 ν (t) f or t he negat ive θ-parit y was calculat ed f or numerically t he opt imal mirror posit ion z0 = 0.0027λ (see E q. 4) wit h an increase of up t o 3 t o 4 orders of magnit ude compared t o values at posit ions around z0 = a ( λ: wavelengt h). This observat ion indicat es t hat measurable solenoidal current s can be generat ed by a suit ably prepared microwave field.

(3)

E q. 2 is t hen solved wit h st andard met hodology, i. e. t he d n ν (t) in t he expansion (4) are solved f or wit h t he eigenst at es ψν (θ) of H 0 . A n addit ional st at ic magnet ic field B parallel t o t he cent ral symmet ry axis can easily be included and will be discussed lat er. The Hamilt onian on T2 is developed by beginning wit h a t hreedimensional f ormulat ion and proceeding t o rest rict t he part icle (account ing f or curvat ure) t o T2. The magnet ic moment µ(t) which is creat ed by t he surf ace current s induced by t he microwave field is calculat ed and it s component s parallel t o t he cent ral symmet ry axis, in azimut hal and in radial direct ion are present ed. In order t o see whet her t he int erf erence of t wo linearly or circularly polarized waves at t he quant um ring locat ion allows f or opt ical t ransit ions int o a solenoidal mode we have considered an incoming microwave t o be reflect ed at a complet ely reflect ing flat surf ace parallel t o t he horizont al symmet ry plane of t he t orus. The mirror is posit ioned at variable dist ances z0 f rom t he cent ral symmet ry plane in order t o st udy ef f ect s of dif f erent phase dif f erences φ 0 bet ween t he t wo waves on t he relat ive amplit ude of t he solenoidal mode.

(5)

Fig. 2: Including a static external B -field: Magnetic moment µ(t) of T2 surface currents for a) single linearly polarized wave and b) interference of incoming and reflected linearly polarized wave. From top to bottom: Components in direction of ez, eφ and eρ .

S ubst ant ial enhancement of s olenoidal modes µ φ and µ ρ c an be observed of up t o 10% of t he dipole mode µ z w hen an addit ional st at ic B -field along t he z-axis is swit ched on.

Conclusion and Outlook The existence and relevance of solenoidal surface currents on T2 in a microwave field was demonstrated and their effects on the total magnetic moment discussed. Substantial enhancement of the magnetic moment’s solenoidal mode versus the dipole mode, which is in turn suppressed, can be achieved through on-resonance microwave interference and proper choice of the phase difference between the two waves. In a new project the authors are extending their studies for the metallic nanotorus to calculating density-of-states and transmission coefficients through toroidal c arbon nanotubes with attached metallic or carbon nanotube leads. In this case a Green’s function approach in a tight-binding approximation is used for the investigation of a smaller model (i.e. a graphene sheet of 150 layers of a 12-carbon-atom super cell wrapped to a nanotorus ) . This shall be expanded to s imulations of larger, more realistic systems with different lead attachments on a parallel 128-node MAC computer cluster. Effects on quantum transport properties from e.g. electron-electron Coulomb and exchange interaction, electron-phonon coupling, defect scattering etc. are of particular interest for nanoscale device applications.

Acknowledgements This work has been funded in parts by NIH Grant HD41829 (M. Jack). The authors would like to thank B. Etemadi for helpful discussions, L. Johnson for generously providing resources on his LRSL MAC cluster and N. Christopher for technical support. Fig. 1: Magnetic moment µ(t) of T2 surface currents for a) single linearly polarized wave and b) interference of incoming and reflected linearly polarized wave. No static external B field. From top to bottom: Components in direction of ez, eφ and eρ .

The ef f ect s of s olenoidal s urf ace current s on t he calculat ed magnetic moment µ(t) are even more pronounced where µ(t) is derived in E q. 5 as surf ace int egral over T2,

References •

M. Encinosa and M. Jack, ‘Elliptical tori in a constant magnetic field’, Phys. S cr i. 73 (2006) 439 - 442. • M. E ncinosa and M. Jack, ‘Excitation of surface dipole and solenoidal modes on toroidal structures’, submitted to P hotonics and Nanostructures (Elsevier) in April 2006, http://www.ar Xive.or g/physics/0604214. • M. Encinosa and M. Jack, ‘Dipole and solenoidal magnetic moments of electronic surface currents on toroidal nanostructures’, Journal of Computer-Aided Materials Design (Springer), May 2006. (In Press)