· classical ED due to the complex conductivity law. ... S.A.Maksimenko and G.Ya.Slepyan,...

Sergey Maksimenko and Gregory Slepyan

Institute for Nuclear Problems, Belarus State University,

[email protected]

Dresden, CoPhen, MPI-PKS, 2004

Electromagnetic effects in nanotubes

MOTIVATION To stress the role of nanoscale nonhomogeneity of

electromagnetic fields in nanostructures

• To demonstrate the close connection betweentraditional 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


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.


S.A.Maksimenko and G.Ya.Slepyan, , in "Hand-book of Nanotechnology: Theory, Modeling and Simulation", Ed. by: A. Lakhtakia, SPIE Press, 2004, pp. 145-206 (in press).


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 geometrycomplex electronics


NANOELECTRODYNAMICS

Linear electrodynamical response

Nonlinear optics

Quantum electrodynamics of CNTs


Main topics


Graphene crystalline lattice SWCNT (m,n)

CARBON NANOTUBE


Basic concept

20

1/ 23

( , ) 1 4cos cos 4cos2

zz

bp s sp s

m mπ πγε = ± + +


Effective boundary conditions:universal tool for solving of ED problems in CNTs

( )20

2 2 2 0 0

0 0 , 0 , 0

41 ,

(1 / )

| | 0, | | 0

zz z RR R

z R z R z R z R

lH H E

k i z c

H H E E

φ φ ρρ ρ

ρ ρ ϕ ρ ϕ ρ

π σωτ == + = −

= − = + = − = +

∂+ − = + ∂

− = − =

l0 ~ 10-5 for metallic CNTs and not too large m

042.1

,

Ab

Rb cn

=

>>>> λλ


Dynamical conductivity of a single CNT

0 20 40 60 80 100 120 1401E-3

0,01

0,1

1

10

100

(m,0) CNsn

orm

aliz

ed a

xial

co

nduc

tivit

y

2

1

1: Metallic CNs (m=3q)2: Semiconducting CNs (m≠3q)

m

( )2 /32 2

2 / 3

( , ) v ( , ),

v ( , ) /3

bz z z z

zzs z zb

e F p s p s dph i

h p s imb

π

π

σ ωω τπ ε−

∂= −∂ − +

Phys. Rev. B60, 17136,1999


Dynamical conductivity of a single CNT:the role of interband transitions

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5-10

-5

0

5

10

15

20

CN (9,0)

2

1

1: Re(σzz

)2: Im(σ

zz)

no

rmal

ized

axi

al c

ond

uct

ivit

y

ωωωω////2222γγγγ0000

Phys. Rev. B60, 17136, 1999


Surface electromagnetic waves in CNT

( ) ( )( )

2 2 22

021 .4 /

q qcn zz

ic kI R K R c l

k R i

κ κκ κπκ σ ω τ

+ = − +

22 kh −=κ( ) ( )( ) ( )

φκρκκκρ

εiqeihze

KRI

RKIA

qq

qqz

= eHertz potential

Dispersion equation

k k

h h ihβ = =

′ ′′+Slow-wave coefficient


Complex-valued slow-wave coefficient ββββ for a polar-symmetric surface wave

1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0,01

10-2

100

102

104 1: Re(ββββ)2: -Re(ββββ)/Im(ββββ)

CN (9,0)

2

1

ββββ

kb

Dispersionlesssurface wave nanowaveguidein the IR range

Phys. Rev. B60, 17136, 1999

CNT is the optical delay-line: 1/β∼100


Finite-length effects in CNTs

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.

ckklkR cncn /,1~,1 ω=<<

At optical frequencies, the CNT’s cross-sectional radius and the length satisfy the following conditions:


!• analytical Wiener-Hopf technique is applied to a semi-infinite

CNT;• derivation of the finite-wire scattering amplitude using the edge-

wave 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).


CNT as a travelling wave tube in the IR

#$%β β β β &$''

V. Becker et al, Phys. Rev. A 25, 956 (1982)

Estimate:lcn= 30 mkmV~ 10 VI ~ 1 nA

0.3g ~

2

~

cn

cn lc

klg

ϖβ

Pronounced nonlinearity of CNTs !"#!$"%%%&'( )"% *$+,,+& )"!,#!$+,,+&-( . / %,,* !,!$+,,0&

Semi-classical theory does not respond to crucial questions .(1 -- "",+%$"%% &23 4+!*" 0$"%%)&23 -53""0,#*$"%%%&' . /#" $"%%%&67 . / )**+")+,,,' . /#0,*0),)$+,,"&3689 -64+,!%$+,,0&


Nonlinearity: motivation7: ; /

< / =


NONLINEAR EFFECTS: general approach

Schrödinger equation for electrons in the CNT lattice potential

Bloch wave expansion

Standard representation of the density matrix

2

[ ( ) ( )] ( , )2

i W et m

∂ Ψ = − ∆Ψ + − ⋅ Ψ∂

r E r p r

/

,

1( , ) ( ) ( )i

l ll

C e uΨ = pr

p

p r p r

*' '( ) ( ) ( )ll l lC Cρ =p p p

Restrictions and approximations


• infinitely long rectilinear single-wall CN exposed to E(r,t) = ezE(x,t)

• tight-binding approximation for ππππ-electrons

• quantum-mechanical dispersion law

• CN radius is small compared with the driving field wavelength, Rcn << λλλλ1

• To avoid CNT damage by the driving field, its amplitude is accepted to be much less than the interatomic field strength:

|E| << m2e5/ 4= 5x109 V/cm

E

k


( )

*( ),

[ ( ) ( ) ] ,

cc ccz z cv cv cv vc

z

cv cvz z cv vv cc cc vv cv cv cv

z

ieE eE R R

t p

ieE eE R R R i

t p

ρ ρ ρ ρ

ρ ρ ρ ρ ρ ω ρ

∂ ∂+ = − −∂ ∂

∂ ∂+ = − − − − −∂ ∂

** 3

2l l

ll l lz z

u uiR u u d

p p′

′ ′Ω

∂ ∂= − ∂ ∂ r

Indices l and l’ take the values v and c which correspond to valence and conduction bands

Explicit expressions for Rll’ are available


Axial current in CNT(1) (2) ,z z zj j j= +

(1) 2 (2) 22 2

4 8, Im( )

(2 ) (2 )c

z cc z c cv cvz

e ej d j R d

pρ ρ

π πε ε∂= =

∂ p p

0 2 0 4 01 0 - 2

1 00

1 02

1 0 4

A r m c h a i r ( 5 ,5 )

ω /ω0

| jz( ω

) | /

j 0

0 2 0 4 01 0 - 2

1 0 0

1 02

1 04

Z i g z a g ( 9 ,0 )

ω / ω0

>; > ;

Induced current spectrum of CNTs illuminated by a Ti:Sapphire laser pulse


Third-order harmonics

0( ) ~ ,zpj EN

N p

ω≠

Figure:Amplitude of the third harmonic current as a function of the driving field strength

(

5


TH: Theory and experiment

Broad background and TH-signal; (A) theory, (B) experiment

350 400 450 500 550 600 650 700 750

10-1

100

d

c

b

a

B

I (a

rb.u

.)

I (a

rb.u

.)

Wavelength (nm)

10-4

10-3

10-2

10-1

100

101

102

0.54 x 1011 W/cm2

A

2.17 x 1011 W/cm2

10-11

10-10

10-9

10-8

2.43

2.04

A

Armchair 5 nm Armchair 20 nm Zigzag 5 nm Zigzag 20 nm

2.15

2.58

j z2 (ab

r.u)

1010

1011

10-3

10-2

10-1

100

J(W/cm2)

B

MWCNTs 20 nm, aligned MWCNTs 20 nm MWCNTs 5 nm

3.1

2.3

2.3

TH

yie

ld (

arb.

u.)

TH generation efficiency; (A) theory, (B) experiment

Experiment:Max-Born Institute, Berlin, GermanyChalmers University of Technology, Sweden

Appl. Phys. Lett.81, 4064, 2002


Plasma resonance

0,0 0,5 1,0 1,5 2,0-10

-5

0

5

10

15

20

2

1

axia

l condu

ctiv

ity

ωωωω////2222γγγγ0000

π-plasmon 02pω γ=

*"+"#,"-,"!. "/"0'$11123$4425

67 / : /


Manifestations

Pulse evolution in (9,0) zigzag CNT at the plasma resonance

Interband transitions

Total current

11

Exact resonance) ) "

02pω γ=

HH spectra for different carrier frequencies. w=wp (a), w=wp /3(b), w= 1.27 wp (c); J=5x 1011

W/cm2.


.Purcell effect: enhancement of the spontaneous decay rate of an atom

located near media interface and/or optical nonhomogeneity

Realizations: microcavities, optical fibers, photonic crystals

Example: atom in a microcavity

3 20

4| |

3 Ak µΓ =

Atom

MicrocavityQ

Vk

Q ≅=ΓΓ=

30

6πξ

Pioneering experimentsP.Goy et al. Phys. Rev. Lett. 50, 1903, 1983G.Gabrielse, et al. Phys. Rev. Lett. 55, 67, 198584 1010~ −Q


Decay rate ratio Phys. Rev. Lett.89, 115504, 2002

perfectly conductingcylinder

Γ/Γ0 ∼105

At different distances outside the (9,0) zigzag CNT

Contribution of the radiativechannel

4 2 2

3 20

( ) ( )3( ) 1 Im

2 1 ( ) ( )A A p A cn p A Acn

A CpA cn A A p A cn p A cn

v I v R K v r dhR

k R v I v R K v R

βξ ω

π β

∞

=−∞

Γ= = +Γ +


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


Acknowledgment: Collaboration in ED of nanostructures

PENN S TATE

Department of Engineering Science and Mechanics

INSTITUT FUR FESTKORPERPHYSIK

: :

Usikov InstituteFor Radiophysics And Electronics Ukraine, Kharkov

A.Lakhtakia

I.Hertel

A.Hoffmann,D.Bimberg

O.Yevtushenko

N.LedentsovI.Krestnikov

Heat-and Mass Transfer Institute NAS Belarus


THANKSThe research is partially supported through

INTASprojects 96-0467 and 97-2018 and 03-50-4409

BMBF (Germany)projects WEI-001-98 and BEL-001-01

NATO Science for Peace Program SfP-972614

Collaborative Linkage Grant PST.CLG.980375

Belarus Foundation for Fundamental Researchprojects F02-176 and F02 R-047

SPIE, E-MRS


NanoModeling(AM221) www.spie.org

SPIE's 49th Annual Meeting, 2-6 August 2004 Denver, USAChairs: Akhlesh Lakhtakia, The Pennsylvania State Univ.;

Sergey A. Maksimenko, Belarus State Univ.

Invited speakers:M. C. Demirel (Biodetection and biomolecules), PennStateT. 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

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