Quantum electron transport in toroidal carbon nanotubes with metallic leads M. Jack and M. Encinosa Department of Physics, Florida A&M University, Tallahassee, FL 32307, USA. Electronic address: [email protected]. 2007 NSTI Nanotechnology Conference, Santa Clara, CA, May 20-24.

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Motivation • Carbon nanotorus as 3-dim mesoscopic ring: Persistent currents and azimuthal and dipole-type electronic excitations possible. • Carbon nanotorus as molecular Aharonov-Bohm oscillator: Modulation of current with magnetic flux. • Carbon nanotorus as possible biosensor: Change in conductivity after covalent attachment of biopolymer (e.g. protein/DNA) at defect site. 2

Toroidal Geometry and Graphite Lattice

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

Device Geometry Sketch:

Right lead

α w R

Left lead

L. Liu et al. 2002 2a

– Nanotorus with attached metallic leads. – System: (3,3) armchair torus; 1800 C atoms = 150 layers of 12 C atoms per azimuthal ring. ! ! – Torus: R = 116 A! ; a = 2 A! . – Semi-infinite leads: w = 100 A; h = 4 A . – 4 symmetric atomic contact sites at each lead.

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Theory: Non-equilibrium Green’s Function Method (NEGF) Hamiltonian H for electron transport in tight-binding approximation:

(

)

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

i> j

Tight-binding scheme: Example: Single layer graphene A. Siber 2003, A. Castro Neto 2006.

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Theory: NEGF ( a,r ) G Retarded and advanced Green’s functions d

for the device region:

[E ! H ! "

! " R ± i# ] Gd

( a,r )

L

=I

! L, R : left/right self-energy corrections due to lead-torus coupling.

Coupling matrix ! k ( E ) between left/right lead and torus:

! k ( E ) = 2" i %& # k ( E ) $ #†k ( E ) '( = 2" Vk† Im %& gkr ( E ) '( Vk ; k = L, R. gkr ( E ) : Green’s functions of semi-infinite metallic leads.

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Theory: NEGF Local density-of-states D(E):

D ( E ) = Gd ( ! L + ! R ) Gd† Transmission function T(E):

T ( E ) = Trace "# ! L Gd ! RGd†

%$(

)

Electron density ! ( f0 E ! µ1,2 : Fermi distributions):

!=

)

† † % dE G " G # f E $ µ + G " G ( ) 1 d R d # f0 ( E $ µ 2 ) ' ( * & d L d 0

$)

Source-drain current I : &

2e I= dE T (E) "# f0 ( E ! µ1 ) ! f0 ( E ! µ2 ) $% ' ! !&

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Recursive Green’s Function Algorithm (RGF) H !Gd = [ E " H " # L " # R ± i$ ] Gd = I Effective Hamiltonian H ! on l.h.s (150x150 matrix):

# A1 % "V % % 0 H! = % % " % 0 % $U

"V † A2 "V ! O 0

0 … 0 "V † ! O ! ! ! ! ! …

! ! ! ! 0 "V

& ( ( ( U = !V ( ; (A ,U,V :12x12) i 0 ( "V † ( ( An ' U† 0 "

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RGF Algorithm Recursively determine matrix Gd in several steps (N=1800): • Forward propagation: recursive calculation of g!i +1,i +1 from g!i,i ( i < N ) . • Forward propagation vertically and horizontally from g! N , N to g!i, N and g! N ,i respectively. • Include U,U † and propagate back vertically or horizontally to g! N! , N " GN , N . • Propagate back to calculate G j, j and Gi < j +1, j +1 ,G j +1,i < j +1 from G j +1, j +1 ( j < N ) . --> Fast Algorithm! 9

RGF Algorithm • Equations for ! k ( E ) , " k ( E ) only include surface term !" gkr #$11 which can be determined recursively:

I !" g #$ = E % H + i& % V !" gkr #$11 V † r k 11

• Include constant magnetic field B0 :

! ! 1 A ( r ) = B0 !eˆ" ; B = B0 eˆz ; 2 % ie ! ! ! ! Vij # Vij exp ' A rij ri $ rj &"

( )(

)

( *) . 10

Results: Local density-of-states D(E) - no B-field (B0 = 0)

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Results: Local density-of-states D(E) - different B-fields B0

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Results: Transmission function T(E) - no B-field (B0 = 0)

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Results: Transmission function T(E) - different B-fields B0

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Results: T(E) and D(E) for different electronic hopping parameters thop (torus-metallic lead coupling)

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Results: T(E) and D(E) for different electronic hopping parameters thop (torus-metallic lead coupling)

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Results: Coherence in electronic transport - Interference plateaus in T(E) for different relative lead positions

‘Back-of-the-envelope estimate’: Interference of left/right electronic pathways on torus from lead to lead for wavelengths ! ! o ( 2" R ) ; R: major radius; α: angle between leads. $ !2 k 2 E = 0.01eV = ! " " 12.2nm & # " 2 # ! ( ) = 360 * = 30.2 2me % u & R = 11.6nm ! u = 2# R = 72.9nm '

 Destructive interference for !s = ( 2n + 1) " 2 ; n = 0,1, 2, …  Minima (and maxima) of T(E) as a function of α for E ! 10…100meV . 17

Results: Coherence in electronic transport - Interference plateaus in T(E) for different relative lead positions

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Results: Coherence in electronic transport - Interference plateaus in T(E) for different relative lead positions

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Results: Coherence in electronic transport - Interference plateaus in T(E) for different relative lead positions

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Summary 1.

Fast and precise recursive algorithm to calculate electronic transport with non-equilibrium Green’s function method (tight-binding).

2.

Strong enhancement of T(E) and D(E) locally in (static) magnetic field B.

3.

Coherent interference phenomena in T(E) strongly dependent on relative orientation α of metallic leads for energies near E = 0 (device Fermi level).

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Outlook 1.

Study bigger systems (> 10000 atoms) and go beyond tight-binding approximation (e.g. next-tonearest neighbor contributions).

2.

Better (= closer) lead attachment to increase number of atomic contacts.

3.

Alternative lead attachment geometries e.g. put torus flat onto (thin) leads.

4.

Self-consistent treatment with corrections e.g. e-phonon coupling, Coulomb blockade etc. 22

References 1. 2. 3. 4. 5. 6. 7. 8.

S. Iijima, Helical microtubules of graphitic carbon. Nature (London) 354, 56 (1991). L. Liu et al., Colossal Paramagnetic Moments in Metallic Carbon Nanotori. Phys. Rev. Lett. 88, 217206 (2002). S. Datta, Electronic Transport in Mesoscopic Systems. Cambridge Univ. Press (1995). M.P. Anantram and T.R. Govindan, Conductance of carbon nanotubes with disorder: A numerical study. Phys. Rev. B 58 (8), 4882-4887 (1998). M. Encinosa, Application of a modified recursive Green’s function method to toroidal carbon nanotube electronic properties. 2000 NASA-ASEE San Jose State University Summer Faculty Fellowship Final Report. M. Encinosa and M. Jack, Excitation of surface dipole and solenoidal modes on toroidal structures. E-print archive: physics/0604214. M. Encinosa and M. Jack, Elliptical tori in a constant magnetic field. Phys. Scr. 73, 439-442 (2006). E-print archive: quant-ph/0509172. M. Encinosa and M. Jack, Dipole and solenoidal magnetic moments of electronic surface currents on toroidal nanostructures. Journal of ComputerAided Materials Design (Springer), May 2006. Proceedings of the conference ‘Synergy between Experiment and Computation in Nanoscale Science’ at Harvard University’s Center for Nanoscale Systems, Cambridge, MA, May 31 – June 03, 2006. 23

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Results: Transmission function T(E) as a function of B0

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