Sergey Maksimenko and Gregory Slepyan Institute for Nuclear Problems, Belarus State University,

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1 Sergey Maksimenko and Gregory Slepyan Institute for Nuclear Problems, Belarus State University, Dresden, CoPhen, MPI-PKS, 2004

2 MOTIVATION To stress the role of nanoscale nonhomogeneity of electromagnetic fields in nanostructures To demonstrate the close connection between traditional problems of classical electrodynamics of microwaves and new problems arising in nanostructures To elucidate the peculiarities of electromagnetic problems in nanostructures irreducible to problems in classical ED due to the complex conductivity law

3 S.A.Maksimenko and G.Ya.Slepyan, Electromagnetics of Carbon Nanotubes, in "Introduction to Complex Mediums for Optics and Electromagnetics", SPIE Press Vol. PM 123, 2003.

4 S.A.Maksimenko and G.Ya.Slepyan,, in "Handbook of Nanotechnology: Theory, Modeling and Simulation", Ed. by: A. Lakhtakia, SPIE Press, 2004, pp (in press).

5 NANOSTRUCTURES: quantum wires and quantum dots, fullerenes, nanotubes, sculptured thin films, atomic clusters, nanocrystallites, etc. Spatial nonhomogeneity Confinement of the charge carrier motion Electromagnetic field diffraction complex geometry complex electronics

6 NANOELECTRODYNAMICS

7 Main topics Linear electrodynamical response Nonlinear optics Quantum electrodynamics of CNTs

8 CARBON NANOTUBE Graphene crystalline lattice SWCNT (m,n)

9 Basic concept 3bp z π s 2π s ε ( p, s) = ± γ 1+ 4cos cos + 4cos z 0 2 m m 1/ 2

10 λ >> b, λ >> b = 1.42 A Effective boundary conditions: universal tool for solving of ED problems in CNTs 0 R cn 2 l ( ) 0 4π 1 +, Hφ H (1 / ) R 0 φ = R 0 σ zzez ρ R k i ωτ z ρ = + ρ = = + c H H = 0, E E = 0 z ρ = R 0 z ρ = R+ 0 z, ϕ ρ = R 0 z, ϕ ρ = R+ 0 l ~ too large m for metallic CNTs and not

11 Dynamical conductivity of a single CNT σ zz ( ω, h) = i 2 2 π /3b 2 ( z, ) v z ( z, ) z e F p s p s dp 3πmb ω h v ( p, s) + i / τ ε s 2 π / 3b z z normalized axial conductivity ,1 0, (m,0) CNs 1: Metallic CNs (m=3q) 2: Semiconducting CNs (m 3q) 1E m Phys. Rev. B 60, 17136, 1999

12 Dynamical conductivity of a single CNT: the role of interband transitions normalized axial conductivity CN (9,0) ,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 1 ω/2γ 0 1: Re(σ zz ) 2: Im(σ zz ) Phys. Rev. B 60, 17136, 1999

13 Surface electromagnetic waves in CNT Hertz potential ε = Ae z I I q q ( κρ ) Kq ( κr) ( κr) K ( κρ ) q ihz iqφ e e κ = 2 h k 2 Dispersion equation κ k R ( + i ) ic κ + k 2 Iq ( κ R) Kq ( κ R) = 1 c l πκ cnσ zz ω / τ Slow-wave coefficient k k β = = h h + ih

14 Complex-valued slow-wave coefficient β for a polar-symmetric surface wave β : Re(β) 2: -Re(β)/Im(β) CN (9,0) Dispersionless surface wave nanowaveguide in the IR range E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0,01 kb Phys. Rev. B 60, 17136, 1999 CNT is the optical delay-line: 1/β 100

15 Finite-length effects in CNTs At optical frequencies, the CNT s cross-sectional radius and the length satisfy the following conditions: krcn << 1, klcn ~ 1, k = ω / c Clearly, although the cross-sectional radius is electrically small, the length is electrically large - conditions that are characteristic of wire antennas. Thus, an isolated CNT is a wire nano-antenna at optical frequencies. The key problem for the optical response of isolated CNTs and CNT arrays is the calculation of the scattering pattern of an isolated CNT of finite length.

16 ! analytical Wiener-Hopf technique is applied to a semi-infinite CNT; derivation of the finite-wire scattering amplitude using the edgewave method. Normalized density of the scattered power for the metallic (9, 0) CNT at frequencies of interband transitions " AEU Int. J. Electron. & Commun. 55, 273 (2001).

17 CNT as a travelling wave tube in the IR # $%β &$'' g klcn ϖ ~ lcn β c 2 V. Becker et al, Phys. Rev. A 25, 956 (1982) Estimate: l cn = 30 mkm V~ 10 V I ~ 1 na g ~ 0.3

18 Nonlinearity: motivation Pronounced nonlinearity of CNTs '( -( Electromagnetic effects in nanotubes!"#!$"%%%& )"% *$+,,+& )"!,#!$+,,+&./%,,*!,!$+,,0& Semi-classical theory does not respond to crucial questions.(1 -- "",+%$"%% & 23 4+!*" 0$"%%)& ""0,#*$"%%%& './#" $"%%%& 67./)**+")+,,, './#0,*0),)$+,,"& ,!%$+,,0& 7: ;/ < / =

19 NONLINEAR EFFECTS: general approach Schrödinger equation for electrons in the CNT lattice potential 2 Ψ = Ψ + Ψ i [ ( ) ( )] (, ) t 2m W r e E r p r Bloch wave expansion Standard representation of the density matrix 1 Ψ = C e u ipr / ( p, r) l ( p) l ( r) l, p ρ ( p) = C ( p) C ( p) * ll ' l l '

20 Restrictions and approximations infinitely long rectilinear single-wall CN exposed to E(r,t) = e z E(x,t) tight-binding approximation for π-electrons quantum-mechanical dispersion law E CN radius is small compared with the driving field wavelength, R cn << λ 1 k To avoid CNT damage by the driving field, its amplitude is accepted to be much less than the interatomic field strength: E << m 2 e 5 / 4 = 5x10 9 V/cm

21 Electromagnetic effects in nanotubes ( ) ρcc t ρcv t ρ i + ee = ee R R cc * z z( cvρcv cvρvc ), pz ρcv i + eez = eez[ Rcv ( ρvv ρcc) ( Rcc Rvv ) ρcv] iωcv ρcv, p z i u u R u u d * * l l 3 ll = l l 2 pz p Ω z r Indices l and l take the values v and c which correspond to valence and conduction bands Explicit expressions for R ll are available

22 Axial current in CNT j = j + j (1) (2) z z z, > ; > ; 4e ε 8e j = d, j R Im( ) d p ρ p = ε ρ p (1) c 2 (2) 2 z 2 cc z 2 c cv cv (2 π) z (2 π) A r m c h a i r ( 5,5 ) Z i g z a g ( 9,0 ) j z (ω) / j ω /ω ω / ω 0 Induced current spectrum of CNTs illuminated by a Ti:Sapphire laser pulse

23 Third-order harmonics j z ( Nω) ~ E, N p p 0 Figure: Amplitude of the third harmonic current as a function of the driving field strength ( 5

24 TH: Theory and experiment I (arb.u.) x W/cm x W/cm 2 A (abr.u) j z Armchair 5 nm Armchair 20 nm Zigzag 5 nm Zigzag 20 nm A Experiment: Max-Born Institute, Berlin, Germany Chalmers University of Technology, Sweden I (arb.u.) a b c TH yield (arb.u.) MWCNTs 20 nm, aligned MWCNTs 20 nm MWCNTs 5 nm d Wavelength (nm) Broad background and TH-signal; (A) theory, (B) experiment B J(W/cm 2 ) TH generation efficiency; (A) theory, (B) experiment B Appl. Phys. Lett. 81, 4064, 2002

25 Plasma resonance axial conductivity ,0 0,5 1,0 1,5 2,0 ω/2γ *"+"#,"-,"!."/"0'$1112 3$4425 ω = 2γ π-plasmon p 0 67 / : /

26 Manifestations 11 ω p = 2γ 0 Exact resonance HH spectra for different carrier frequencies. w=w p (a), w=w p /3(b), w= 1.27 w p (c); J=5x W/cm 2. Interband transitions Total current Pulse evolution in (9,0) zigzag CNT at the plasma resonance ) ) "

27 Purcell effect: Realizations: Example: Microcavity. enhancement of the spontaneous decay rate of an atom located near media interface and/or optical nonhomogeneity microcavities, optical fibers, photonic crystals atom in a microcavity ξ = Γ Γ 0 = 6πQ 3 k V Q Atom 4 Γ 0 = k µ A 4 Q ~ Pioneering experiments P.Goy et al. Phys. Rev. Lett. 50, 1903, 1983 G.Gabrielse, et al. Phys. Rev. Lett. 55, 67, 1985

28 Decay rate ratio At different distances outside the (9,0) zigzag CNT Phys. Rev. Lett. 89, , 2002 perfectly conducting cylinder Γ/Γ Contribution of the radiative channel Γ β v I ( v R ) K ( v r ) dh ξ ( ωa) = = 1+ Im Γ Rcn A A p A cn p A A 3 2 2π k C A p= 1 Rcn β AvAI p ( varcn ) K p ( varcn )

29 NEXT STEPS Antenna (finite-length) effects monomolecular travelling wave tube (ampl and genert) x-ray transportation and control CNT-based composites Finite length of a single CNT Interaction of electronic subsystems Evolution in CNT ensembles of femtosecond pulses To incorporate relaxation in the theory To develop homogenization procedure Instabilities in CNTs Interaction with quantum states of light

30 Acknowledgment: Collaboration in ED of nanostructures PENN S TATE A.Lakhtakia Department of Engineering Science and Mechanics Heat-and Mass Transfer Institute NAS Belarus I.Hertel A.Hoffmann, D.Bimberg : INSTITUT FUR FESTKORPERPHYSIK : Usikov Institute O.Yevtushenko For Radiophysics And Electronics Ukraine, Kharkov N.Ledentsov I.Krestnikov

31 THANKS The research is partially supported through INTAS projects and and BMBF (Germany) projects WEI and BEL NATO Science for Peace Program SfP Collaborative Linkage Grant PST.CLG Belarus Foundation for Fundamental Research projects F and F02 R-047 SPIE, E-MRS

32 You Are Welcome NanoModeling (AM221) SPIE's 49th Annual Meeting, 2-6 August 2004 Denver, USA Chairs: Akhlesh Lakhtakia, The Pennsylvania State Univ.; Sergey A. Maksimenko, Belarus State Univ. Invited speakers: M. C. Demirel (Biodetection and biomolecules), PennState T. G. Mackay (Unusual metamaterials), Univ. of Edinburgh T. S. Rahman (Atomistic modeling of thin films), Kansas Univ. V. Shchukin (Semiconductor diode lasers in photonic bandgap crystals), Nanosemiconductor GmbH and Ioffe Institute V. B. Shenoy (Nanomechanics), Indian Institute of Science, Bangalore