by SHIPRA CHOUDHARY

14/MAP/007 M.Sc. (Applied Physics)

Under the guidance of

Dr. Manmohan Singh Shishodia Gautam Buddha University, Greater Noida

Otical propertIes of dimer of plasmonIc nanosphere

Why Nanomaterials? Advantages of Dimer over Single Nanosphere Introduction

Multipole Spectral Expansion method (MSE) MSE method for single nanoshere MSE method for dimer of nanaosphere

Dimer Matrix ElementsTranslated Eigenstates

Future plan

outlines

Why nanomaterials?• Material that has unique or novel properties, due to the

nanoscale ( nano metre- scale) structuring.

• The properties of the nanomaterials can be different from bulk material:

Larger surface area Quantum effect begins to dominate

Solar cells Nanoantenna’s (Metal Nanoparticles)

Nanoantenna’s (Silicon Nanoparticles)

Advantages of dimer over single nanoparticles

Dimer provides a stronger electric field in than gap region than a single metallic nanoparticle does in its proximity.

Dimer plays the role of a nanolens to focus the incident wave into a small hotspot re- gion around the gap.

Dimer plays the role of an antenna.

Lesser the gap, greater is the electric field enhancement factor.

dm m

b

LR RR

**[ref: Jiunn-Woei Liaw, Jeng-Hong chen, chi-San, and Mao-Kuen kuo, Opt.Express 16, 13532-13540 (2009).]

3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4

0

1000

2000

3000

4000

5000

6000

7000

Fiel

d en

hanc

emen

t ()

Frequency (eV)

a/r = 0.6 a/r = 0.7 a/r = 0.8 a/r = 0.9 a/r = 1.0

Introduced by Fuchs and further developed by Bergman, Milton and stockman.

Analytical approach for calculating potential at any point.

Separates the geometrical and dielectric properties and can be extended to arbitrary combination of nanoparticles.

Extendible to dimers and multimers.

Dimer nanostructures may induce a relatively intense local EMF within the dimer gap region and in the proximity of MNS.

**[ref: D. J. Bergman, Phys. Rep. 43, 377 (1978).]

MULTIPOLE SPECTRAL EXPANSION METHOD

The overall potential expression in this approach

External potential

MSE METHOD FOR Single nanosphere

**[fig ref:Manmohan S. Shihodia, Boris D. Fainberg, and Abraham Nitzen, “Theory of energy transfer interactions near sphere and nanoshell based plasmonic nanostructures”, SPIE 0277-786X (2011).]

)()(φ|)()(s)(s

s)(φ)φ( extext rrrrrr mlmlml l

l

surface) on theR(r θ cosREθcosrEΦ 00ext

R

P

zO

r

E0 z

ε(ω) hε

θ

h

h

ε2ε(ω)εε(ω)

s)(ωss

The dielectric part

The total potential outside the sphere

)1for(3/112s lll

hεε(ω)11)(ωs

,

1

12

(1/2)12

m,m, r

R

R

φ),(θY)r(ψ

l

l

l

ll

l θcosr

Rπ3

21)r(ψ 2

2/3

,01

The potential eigenfunctions

The induced potential

Using Green’s first identity in the overlap integral3/2RE

34πI 0m, l

θcosrRE

ε2ε(ω)εε(ω)

)r(Φ 2

3

0h

hinduced

θcosrRE

ε2ε(ω)εε(ω)cosθrE)r(Φ 2

3

0h

h0out

Nanosphere dimer

)r(ψ|ψ))(()(

)r()r( '0'

,,

,',

'b,la,lablal

RLala

RLblb

blext BB

sss

The potential for two sphere geometry

Eigenvalue equation)(ψ)(ψs rr

Eigenvalues and eigenvectors of gamma

'',,,,,'',,,,,

'',,,,,'',,,,,

mlRightmlRightmlLeftmlRight

mlRightmlLeftmlLeftmlLeft

m d

'rr

LR

LORR

RO

b

L R

P

Z

zE ˆ0

)b-r(ψψrθd1'2

'm'l',L

3'',;,

*

l,mV

mlml ll

Using Green’s identity

dimer matrix elementS

)()(

12''

L

'' ,

*,

Rr

2'

'

,,,br

rr

dRll

mlml

Lmlml

)]()([)(

)()]([dV

*,

2,

3*,

,Rr

2,

*,

V

3''''

L

'' rbrrdrr

brdRbrr mlmlLml

mlLmlmlL

),(1)( ,12, mll

lL

ml YrlR

r

),(1)( *

,12

*, ml

l

lL

ml YrlR

r

),(1)( *,

1

12

*,

ml

l

lL

ml YrlRr

r

Eigenstates of left sphere

dimer matrix elementS Eigenstates of right sphere

)(1)( '''

'

'' ,1'

12

, brmll

lR

mlY

brlRbr

),(),(!)!12(!)!12(

!]!1)(2[000

)12](1)(2)[12(4)1()(1),,1'

'

''

''''''

,)1( '''

''

''' bbmlllm

brmllYY

br

ll

mmllll

llYbr

),(),(!)!12(!)!12(

!]!1)(2[000

)12](1)(2)[12(4)1()( '''

''

'

' ,,1'

'

''

''''''

'

12

, bbmll

lmlR

mlYY

br

ll

mmllll

lllR

br

),(),(!)!12(!)!12(

!]!1)(2[000

)12](1)(2)[12(4)1(),(12

'''

''

'

'' ,,1'

'

''

''''''

'

12*,

1

12

2'

'

,,, bbmll

lmlR

Rrml

l

lL

LmlmlYY

br

ll

mmllll

lllR

YrlR

ldRll

L

),(!)!12(!)!12(

!]!1)(2[000

)12](1)(2)[12(412

)1( ''

'

''

'' ,'

'

''

''''''

)2/1()2/1(

'

'

,,, bbmmll

lR

lLlm

mlmlY

ll

mmmmllllllll

llbR

bR

lll

dimer matrix elementS

!!)!(

]!1)(2[)!2()!2()1(

000 '

'

'

''''

llll

llllllll ll

)!()!()!()!(]!1)(2[)!()!()!2()!2()1(

)( ''''

'''''

''

''''

mlmlmlmlllmmllmmllll

mmmmllll mmll

bmmimmllbbmmll

ePmmllmmllll

)(

''

'''

,

''

''' )(cos)!()!(

4]1)(2[),(Y

Using properties of Wigner 3j symbols

Relation b/w Spherical harmonics & Legendre functions

bmmimmll

lR

lLlm

mlml

ePllll

ll

ll

llll

mlmlmlml

mmllbR

bRll

lllll

)('

'

'

'

'

'

''''

'')2/1()2/1('

'

''

,,,

''

'

'

'

''

!!)(

!)!12()!2(

!)!12()!2(

]!1)(2[!]!1)(2[

)!()!()!()!(

)!()12)(12(12

]1)(2[)1(

bmmimmll

lR

lLlm

mlmleP

mlmlmlmlmmll

llll

bR

bR )'('

'''''

''

'

')2/1(')2/1(

)!()!()!()!()!(

)12)(12()1(

'

''

TRANSLATED EIGENStates

Outside sphere eigenstates

1,

12

b-r

),()b-r(

l

mllR

lm

YlR

1,

12 ),()b-r(

l

mllR

lm RY

lR

b,r en wh1b,r when1

cosbrbr

b,r n whe0b,r when

m d

'rr

LR

LORR

RO

b

L R

P

Z

zE ˆ0

1

012

41)1()b-r(

l

R

R

ll br

Rll

R

1

12

10,

12

01

412)1(

)0,()b-r(

l

lRl

ll

lR

lbrl

Rlbr

YlR

Inside sphere eigenstatesl

lR

mllmlmR

lR

Yb-r

),()b-r(

12

,,

l

lR

llR br

lRl

12,

14

12)1()r(

l

RR

llR R

brll

R

12

41)1()r(,

Future plan To Calculate the external potential and overlap integral for a pair of

metallic nanoparticles (dimer) to obtain the overall potential in the gap region.

To study the effect of Electric Field Enhancement, polarizability and plexcitonic interactions in the vicinity and the gap region of a pair of metallic nanoparticles (dimer).

To explore different plasmonic materials other than metals(Au or Ag).

To extend Multipole Spectral Expansion Method to treat Nanoshells.

references Manmohan S. Shishodia, Boris D. Fainberg, and Abraham Nitzen, “Theory of energy

transfer interactions near sphere and nanoshell based plasmonic nanostructures”, SPIE 0277-786X (2011).

J.D. Jackson, “Classical Electrodynamics”, John Wiley & Sons, (1998). D.J. Bergman, “Dielectric constant of a two-component granular composite: A

practical scheme for calculating the pole spectrum”, phys. Rev. B, 19, 2359 (1979). M. Danos, and L.C. Maximon, “Multipole matrix elements of the translation

operator”, J. Mathematical Phys. 6, pp. 766-778 (1965). http://functions.wolfram.com/Polynomials/SphericalHarmonicY/20/01/02 http://mathworld.wolfram.com/Polynomials/Wigner3j-Symbol.html Jiunn-Woei Liaw, Jeng-Hong chen, chi-San, and Mao-Kuen kuo, “Purecell effect of

nanoshell dimer on single molecule’s fluoresecnce”, Opt.Express 16, 13532-13540 (2009).

THANKYOU