[PDF] - INTERACTIONS IN MATTER Alexander A. Iskandar Physics of Magnetism PDF Document

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08/01/2018 1

Alexander A. Iskandar Physics of Magnetism and Photonics

FI 3221 ELECTROMAGNETIC INTERACTIONS IN MATTER

Surface

Plasmon

Propagation
f Surface

Plasmon

Localized

Plasmon

SPR

Spectroscopy

SURFACE PLASMON

Alexander A. Iskandar Electromagnetic Interactions in Matter 2

SLIDE 2

08/01/2018 2  Main

▪ S.A. Maier : Section 2.1 – 2.3, 5.1

 Supplementary

▪ J. A. Dionne et.al., Phys. Rev. B72, 075405 (2005)

Alexander A. Iskandar Electromagnetic Interactions in Matter 3

REFERENCES

 Consider the x direction as the effective propagation direction and by symmetry, there is no y-dependence.

Alexander A. Iskandar Electromagnetic Interactions in Matter 4

WAVE EQUATION AT METAL/INSULATOR INTERFACES

z y x metal dielectric

) (

) ( ) , , , (

t x i

e z t z y x

  

 E E

 

) (

2 2 2 2

     E E   k z z  Substituting to the Maxwell’s eq.s, we obtain And similar equation for the H-field.

SLIDE 3

08/01/2018 3  Consider the two special polarization

▪ TE : with only Hx, Hz and Ey are nonzero ▪ TM : with only Ex, Ez and Hy are nonzero

 For the TE mode, we need to solve only for the Ey field component from the wave equation,  And, the Hx, Hz field components can be obtained from the Maxwell’s eq.

Alexander A. Iskandar Electromagnetic Interactions in Matter 5

WAVE EQUATION AT METAL/INSULATOR INTERFACES

 

2 2 2 2

    

y y

E k z E  

y z y x

E H z E i H , 1         Similarly, for the TM mode, we need to solve only for the Hy field component from the wave equation,  And, the Ex, Ez field components can be obtained from the Maxwell’s eq.

Alexander A. Iskandar Electromagnetic Interactions in Matter 6

WAVE EQUATION AT METAL/INSULATOR INTERFACES

 

2 2 2 2

    

y y

H k z H  

y z y x

H E z H i E , 1         

SLIDE 4

08/01/2018 4  Consider the TE mode, solution for region of z > 0 is  And for region z < 0, we obtain

Alexander A. Iskandar Electromagnetic Interactions in Matter 7

TE SURFACE-WAVE SOLUTION

) ( 2 ) ( 2 ) ( 2

) ( , ) ( ] Re[ , ) (

z k x i x i z z k x i d x d z k x i y

d d d

e e A z H e k A z H k e A z E

  

   

   

  

) ( 1 ) ( 1 ) ( 1

) ( , ) ( ] Re[ , ) (

z k x i z z k x i m x m z k x i y

m m m

e A z H e k A z H k e A z E

  

    

  

    Applying continuity condition to the Ey and Hx field components, we arrive at the condition  Since confinement condition requires that Re[km] > 0 and Re[kd] > 0, the above condition can only be satisfied with A1 = 0, which yield a trivial solution.

Alexander A. Iskandar Electromagnetic Interactions in Matter 8

TE SURFACE-WAVE SOLUTION

 

1 2 1

  

d m

k k A and A A

SLIDE 5

08/01/2018 5  On the other hand, for the TM mode, solution for region of z > 0 is  And for region z < 0, we obtain

Alexander A. Iskandar Electromagnetic Interactions in Matter 9

TM SURFACE-WAVE SOLUTION

) ( 2 ) ( 2 ) ( 2

) ( , ) ( ] Re[ , ) (

z k x i d z z k x i d d x d z k x i y

d d d

e A z E e k A z E k e A z H

  

     

  

    

) ( 1 ) ( 1 ) ( 1

) ( , ) ( ] Re[ , ) (

z k x i m z z k x i m m x m z k x i y

m m m

e A z E e k A z E k e A z H

  

    

  

      Applying continuity condition to the Hy and Ex field components, we arrive at the conditions  Hence confinement condition requires that Re[km] > 0 and Re[kd] > 0, and this can be fulfilled by  Further, the wave number in each region is given by

Alexander A. Iskandar Electromagnetic Interactions in Matter 10

TM SURFACE-WAVE SOLUTION

m d m d

k k and A A     

2 1

 

Re  

d m

if  

2 2 2 2 2 2

       

d d m m

k k and k k

SLIDE 6

08/01/2018 6  Combining with the previous condition we have the following dispersion relation for the propagation wavenumber  This surface wave is called Surface Plasmon Wave, that exist only for TM polarization.

Alexander A. Iskandar Electromagnetic Interactions in Matter 11

SURFACE PLASMON WAVE

d m d m

k         Plasmons:

▪ collective oscillations of the “free electron gas” density, often at

ptical frequencies.

 Surface Plasmons:

▪ plasmons confined to surface (interface) and interact with light resulting in polaritons. ▪ propagating electron density waves occurring at the interface between metal and dielectric.

 Surface Plasmon Resonance:

▪ light () in resonance with surface plasmon oscillation

Alexander A. Iskandar Electromagnetic Interactions in Matter 13

SURFACE PLASMON WAVE

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08/01/2018 7

Alexander A. Iskandar Electromagnetic Interactions in Matter 14

SURFACE PLASMON DISPERSION RELATION

 p

d p sp

     1 Re  (or kx)

real  real kz (km) imaginary  real kz real  imaginary kz

d x

ck 

Bound modes Radiative modes Quasi-bound modes

Dielectric: d Metal: m = ′m + i″m ″m << 1

x z

′m > 0) d < ′m < 0) (′m < d)

2 / 1

         

d m d m

c      

2 2

1   

p m

 

2 2 2

   

m m

k k  Radiative solution (kx and km real) lies the left of the light-line. This happens for  > p.  Bound solution (kx real and km imaginary) lies to the right of the light-line.  Between the regime of the bound and radiative modes, a frequency gap region with purely imaginary  prohibiting propagation exists.

Alexander A. Iskandar Electromagnetic Interactions in Matter 15

SURFACE PLASMON DISPERSION RELATION

2 2 2

   

m m

k k

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08/01/2018 8

Alexander A. Iskandar Electromagnetic Interactions in Matter 16

0.05 0.1 0.15 0.2 0.25 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 kx (1/nm) Energi (eV) Kurva Dispersi - Model Gas Elektron Bebas Re[kx] Im[kx] light line

1 15

10 85 . 8



  s

P



2

1     

P SP

5.82 eV / 213 nm 3.28 eV / 378 nm

0.01 0.02 0.03 0.04 0.05 0.06 0.07 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 kx' (1/nm) Energi (eV) Kurva Dispersi - Komponen Re[kx] dengan Data Empiris kx' light line

Modus Radiatif (RPP) Modus Terikat (SPP)

3.73 eV / 332 nm 3.44 eV / 360 nm

Modus Quasi-bound (QB)

EXAMPLE OF Ag-SiO2 INTERFACE

 For real metals, ″m is not negligible, hence  (= kx) will be a complex quantity.  The propagation of surface plasmon wave will be

attenuated. Define

propagation length L as the distance when the intensity has become 1/e

f the initial intensity.

Alexander A. Iskandar Electromagnetic Interactions in Matter 17

PROPAGATION LENGTH AND SKIN DEPTH

] Im[ 2 1   L

200 400 600 800 1000 1200 1400 1600 1800 2000 10

9

10

8

10

7

10

6

10

5

10

4

10

3

10

2

Lambda Vakum (nm) Panjang Propagasi (m) Kurva Propagasi untuk Tiga Konstanta Dielektrik yang Berbeda SiO2 Udara Si

SPP RPP

SLIDE 9

08/01/2018 9  Confinement of the surface plasmon wave near the interface can be characterize by its skin depth, defined as the length d when the intensity has become 1/e of the initial intensity.

Alexander A. Iskandar Electromagnetic Interactions in Matter 18

PROPAGATION LENGTH AND SKIN DEPTH

 

i

k Im 1  d

200 400 600 800 1000 1200 1400 1600 1800 2000 500 1000 1500 2000 2500 Panjang Gelombang Vakum (nm) Skin Depth (nm) Skin Depth SiO2 Vs. Lambda 200 400 600 800 1000 1200 1400 1600 1800 2000

250
200
150
100
50

Panjang Gelombang Vakum (nm) Skin Depth (nm) Skin Depth Ag

SiO2 Ag

360 nm (3.44 eV) 360 nm (3.44 eV)

Alexander A. Iskandar Electromagnetic Interactions in Matter 19

Ex DISTRIBUTION AND ENERGY DENSITY

50 100 150 200

200
100

100 200

1
0.5

0.5 1 x(nm) Distribusi Medan Ex, untuk Lambda = 476 nm z(nm) Ex(N/C) SiO2 Ag 20 40 60 80 100 120 140 160 180 200

200
150
100
50

50 100 150 200 x(nm) Distribusi Medan Ex, untuk Lambda = 476 nm z(nm) SiO2 Ag

200
150
100
50

50 100 150 200 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 z(nm) U Distribusi Rapat Energi pada x = 0 SiO2 Ag

Illumination with  = 476 nm (2.61 eV) – SPP mode

SLIDE 10

08/01/2018 10

Alexander A. Iskandar Electromagnetic Interactions in Matter 20

Ex DISTRIBUTION AND ENERGY DENSITY

Illumination with  = 370 nm (3.35 eV) – SPP mode

50 100 150 200

200
100

100 200

1
0.5

0.5 1 x(nm) Distribusi Medan Ex, untuk Lambda = 370 nm z(nm) Ex(N/C) Ag SiO2 20 40 60 80 100 120 140 160 180 200

200
150
100
50

50 100 150 200 x(nm) Distribusi Medan Ex, untuk Lambda = 370 nm z(nm) Ag SiO2

200
150
100
50

50 100 150 200 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 z(nm) U Distribusi Rapat Energi pada x = 0 Ag SiO2

Alexander A. Iskandar Electromagnetic Interactions in Matter 21

Ex DISTRIBUTION AND ENERGY DENSITY

Illumination with  = 315 nm (3.93 eV) – RPP mode

50 100 150 200

200
100

100 200

1
0.5

0.5 1 x(nm) Distribusi Medan Ex, untuk Lambda = 315 nm z(nm) Ex(N/C) Ag SiO2 20 40 60 80 100 120 140 160 180 200

200
150
100
50

50 100 150 200 x(nm) Distribusi Medan Ex, untuk Lambda = 315 nm z(nm) Ag SiO2

200
150
100
50

50 100 150 200 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 z(nm) U Distribusi Rapat Energi pada x = 0 Ag SiO2

SLIDE 11

08/01/2018 11  Similar mode analysis can be made for multilayer system.  Field solution for region z > a and z < –a is similar as before, while field in the region between –a < z < a is With the electric field components can be obtained from the Maxwell’s equations.

Alexander A. Iskandar Electromagnetic Interactions in Matter 22

MULTILAYER SYSTEM

] Re[ , ) (

) ( 2 ) ( 1

  

  m z k x i z k x i y

k e B e B z H

m m

 

Alexander A. Iskandar Electromagnetic Interactions in Matter 23

MULTILAYER SYSTEM

 Solving the continuity conditions yields two dispersion relation for km (symmetric and antisymmetric mode).  For symmetric configuration (I = III = d), it is found These dispersion relations have to be solved numerically. 2 tanh :         

  m m d d m

k d i k k L    2 coth :         

  m m d d m

k d i k k L   

Anti-symmetric Symmetric

SLIDE 12

08/01/2018 12

Alexander A. Iskandar Electromagnetic Interactions in Matter 24

Anti-symmetric mode Illumination with  = 400 nm Ag film thickness 50 nm Propagation length L = 1388.31 nm

500 1000 1500

400
200

200 400

1
0.5

0.5 1 x(nm) Profil Medan Listrik Tangensial Untuk Tebal Film = 50 nm, Lambda = 400 nm z(nm) Ex(N/C) SiO2 SiO2 Ag 500 1000 1500

400
300
200
100

100 200 300 400 x(nm) Profil Medan Listrik Tangensial Untuk Tebal Film = 50 nm, Lambda = 400 nm z(nm) Ag SiO2 SiO2

400
300
200
100

100 200 300 400

1
0.8
0.6
0.4
0.2

0.2 0.4 0.6 0.8 1 z (nm) Ex(z) Cross-Section Profil Medan E pada x=0, Tebal Film = 50, Lambda = 400 nm

Ex DISTRIBUTION AND FIELD CROSS-SECTION

Alexander A. Iskandar Electromagnetic Interactions in Matter 25

Ex DISTRIBUTION AND FIELD CROSS-SECTION

Symmetric mode Illumination with  = 400 nm Ag film thickness 50 nm Propagation length L = 434.81 nm

500 1000 1500

400
200

200 400

1
0.5

0.5 1 x(nm) Profil Medan Listrik Tangensial Untuk Tebal Film = 50 nm, Lambda = 400 nm z(nm) Ex(N/C) SiO2 SiO2 Ag 500 1000 1500

400
300
200
100

100 200 300 400 x(nm) Profil Medan Listrik Tangensial Untuk Tebal Film = 50 nm, Lambda = 400 nm z(nm) Ag SiO2 SiO2

400
300
200
100

100 200 300 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z (nm) Ex(z) Cross-Section Profil Medan E pada x=0, Tebal Film = 50, Lambda = 400 nm

SLIDE 13

08/01/2018 13  Consider a small metallic sphere (a << ), so that we can apply a quasi-static approximation, in a region of constant external electric field E0.  Solve the Laplace equation for the electric potential,

Alexander A. Iskandar Electromagnetic Interactions in Matter 26

LOCALIZED SURFACE PLASMON

2

  

d m()

 The potential is found to be  The dipole moment can be written in terms of atomic polarizability as

Alexander A. Iskandar Electromagnetic Interactions in Matter 27

LOCALIZED SURFACE PLASMON

3 3 2 3

2 4 , 4 cos cos 2 cos cos 2 3 E p r p

d m d m d d d m d m

ut

d m d in

a r r E r a E r E r E                                    

d m d m d

a             2 ) ( ) ( 4 ) ( ,

3

    E p

SLIDE 14

08/01/2018 14  Note that the atomic polarizability experiences a resonant enhancement under the condition that |m + 2d | is a minimum, which for the case of small or slowly-varying Im[m] around the resonance simplifies to  This condition is called Fröhlich condition.

Alexander A. Iskandar Electromagnetic Interactions in Matter 28

LOCALIZED SURFACE PLASMON

 

d m

   2 ) ( Re  

Absolute value and phase of the polarizability α of a sub- wavelength metal nanoparticle with respect to the frequency of the driving field (expressed in eV units). Here, ε(ω) is taken as a Drude fit to the dielectric function

f experimental data of silver.

 From the potential, it is easy to find the field

Alexander A. Iskandar Electromagnetic Interactions in Matter 29

LOCALIZED SURFACE PLASMON

3

4 ) ( 3 2 3 r

d

ut

d m d in

     p p n n E E E E      

a = 8nm εd = 1 I = 1 W/cm2 ħω = 1.76 eV

SLIDE 15

08/01/2018 15  From the potential, it is easy to find the field

Alexander A. Iskandar Electromagnetic Interactions in Matter 30

LOCALIZED SURFACE PLASMON

3

4 ) ( 3 2 3 r

d

ut

d m d in

     p p n n E E E E      

a = 8nm εd = 1 I = 1 W/cm2 ħω = 2.43 eV

 From the potential, it is easy to find the field

Alexander A. Iskandar Electromagnetic Interactions in Matter 31

LOCALIZED SURFACE PLASMON

3

4 ) ( 3 2 3 r

d

ut

d m d in

     p p n n E E E E      

a = 8nm εd = 1 I = 1 W/cm2 ħω = 2.88 eV

SLIDE 16

08/01/2018 16

Alexander A. Iskandar Electromagnetic Interactions in Matter 32

SIZE DEPENDENT SURFACE PLASMON RESONANCE

 Plasmons of gold nanoparticles in glass reflect green, transmit red.  The red color is due to tiny gold particles embedded in the glass, which have an absorption peak at around 520 nm.

Alexander A. Iskandar Electromagnetic Interactions in Matter 33

NANOTECHNOLOGY IN ROMAN TIMES: THE LYCURGUS CUP AND STAINED-GLASS

SLIDE 17

08/01/2018 17

Alexander A. Iskandar Electromagnetic Interactions in Matter 34

SURFACE PLASMON RESONANCE SPECTROSCOPY

 prism

reflectance 1 angle

c  prism

Alexander A. Iskandar Electromagnetic Interactions in Matter 35

SURFACE PLASMON RESONANCE SPECTROSCOPY

 prism evanescent field evanescent wave: - nearfield standing wave,

extends about 1/2 ,
decays exponentially with the distance

reflectance 1 angle

c

SLIDE 18

08/01/2018 18

Alexander A. Iskandar Electromagnetic Interactions in Matter 36

SURFACE PLASMON RESONANCE SPECTROSCOPY

 detector 50 nm Au

reflectance 1 angle

c o (Kretschmann configuration)

Alexander A. Iskandar Electromagnetic Interactions in Matter 37

SURFACE PLASMON RESONANCE SPECTROSCOPY

 detector Au analyte

reflectance 1 angle

c o 1 To measure:

thickness changes,
density fluctuation,
molecular adsorption

SLIDE 19

08/01/2018 19

Alexander A. Iskandar Electromagnetic Interactions in Matter 38

SURFACE PLASMON RESONANCE SPECTROSCOPY

Alexander A. Iskandar Electromagnetic Interactions in Matter 39

SURFACE PLASMON RESONANCE SPECTROSCOPY

Reflected intensity / %

SLIDE 20

08/01/2018 20

 And for TM wave propagation, show that the electric field components can be obtained from

Alexander A. Iskandar Electromagnetic Interactions in Matter 40

HOMEWORK

y z y x

E H z E i H , 1       

y z y x

H E z H i E , 1         

z y x metal dielectric

 For TE wave propagation at the boundary between the media with given geometry, show that the magnetic field components can be

btained from

 Derive the dispersion relation for the propagation wavenumber of a surface plasmon Explain the meaning of this dispersion relation.

Alexander A. Iskandar Electromagnetic Interactions in Matter 41

HOMEWORK

d m d m