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
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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).
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