Dispersion relation of surface plasmons

“Metal Optics” Prof. Vlad Shalaev, Purdue Univ., ECE Department, http://shay.ecn.purdue.edu/~ece695s/

“Surface plasmons on smooth and rough surfaces and on gratings” Heinz Raether(Univ Hamburg), 1988, Springer-Verlag

“Surface-plasmon-polariton waveguides” Hyongsik Won, Ph.D Thesis, Hanyang Univ, 2005. Metal Optics: An introduction

Photonic functionality based on metals?! What is a plasmon?

Note: SP is a TM wave! Plasmon-Polaritons

Plasma oscillation = density fluctuation of free electrons

+++ Plasmons in the bulk oscillate at ωp determined by the free electron density and effective mass ω Ne2 --- drude = p mε k 0 Plasmons confined to surfaces that can interact with light to +-+ form propagating “surface plasmonω polaritons (SPP)” ω ω ( 2 − 2ε) k = k = ε p d x sp ω 2 2 c (1+ d ) −ω p Confinement effects result in resonant SPP modes in nanoparticles (localized plasmons)ω 2 drude 1 Ne particle = 3 mε 0 Local field intensity depends on wavelength

(small propagation constant, k) (large propagation constant, k) Ekmel Ozbay, Science, vol.311, pp.189-193 (13 Jan. 2006).

Some of the challenges that face plasmonics research in the coming years are

(i) demonstrate optical frequency subwavelength metallic wired circuits with a propagation loss that is comparable to conventional optical waveguides;

(ii) develop highly efficient plasmonic organic and inorganic LEDs with tunable radiation properties;

(iii) achieve active control of plasmonic signals by implementing electro-optic, all-optical, and piezoelectric modulation and gain mechanisms to plasmonic structures;

(iv) demonstrate 2D plasmonic optical components, including lenses and grating couplers, that can couple single mode fiber directly to plasmonic circuits;

(v) develop deep subwavelength plasmonic nanolithography over large surfaces;

(vi) develop highly sensitive plasmonic sensors that can couple to conventional waveguides;

(vii) demonstrate quantum information processing by mesoscopic plasmonics. To derive the Dispersion Relation of Surface Plasmons, let’s start from the Drude Model of dielectric constant of metals. Dielectric constant of metal

z Drude model : Lorenz model (Harmonic oscillator model) without restoration force (that is, free electrons which are not bound to a particular nucleus)

Linear Dielectric Response of Matter Drude model: Metals The equation of motion of a free electron (not bound to a particular nucleus; C = 0 ), GG G dr2 GJJ mdrGGJJ dv m G mCreEmveE=− − − ⇒ + =− dt 2 ττdt dt

τ 1 −14 (=≈ : relaxation time 10 s ) Lorentz model γ JG G (Harmonic oscillator model) The conduction current density J = −Nev. If C = 0, it is called JG Drude model dJ1 JG⎛⎞ Ne2 JJG +=JE⎜⎟ dtτ ⎝⎠ m JJGJJG −itω For a harmonic wave Et ( )= Re{} Eo ( ) e , JG {} the current desity varies at the same rate Jt ( )= Re J ( ) e−it ω o ω 1 JG⎛⎞Ne2 JJG (−+iJωω )() =⎜⎟ ω E () τ ⎝⎠m Local approximation to theω current-field relation JG(/)ne2 m JJG JE()= () (/1 ωω− i ) τω : Static (ω =0) conductivity

those localized in interband (modified Drude model) ∂Dt() ∂ε ε Et() ∇×=H +=JJoB + Maxwell Equations ∂∂tt

∂εεoBEt() ⎡⎤() ∇×HJiEEiE = + =−ωεεoB ω()() ω σω ω ωε + ε()() ω ω =− o⎢⎥ B () − () ∂ti⎣⎦o

ωε σω Dielectric constant of metal : Drude model εωε εεσω() EFF()=+ B i εωo

1 ⎛⎞σ ooσωτ(1+ i ) =+BBii⎜⎟ =+ 22 ε ooωωτ⎝⎠1(1)−+i εωωτ ⎛⎞⎛⎞ε ωτ22 ωτ 22 =−⎜⎟⎜⎟pp +i ⎜⎟⎜⎟B 1++22 33 ⎝⎠⎝⎠ωτ ωτ ωτ 2 ω2 σ o ne where, p == : bulk plasma frequency ( ∼ 10eV for metal) ετ ε m ooe Plasma frequency Al13 : 1S2 2S2 2P6 3S2 3P1

Aluminum

From Microscopic origin of ω-response of matter Ag47 : 1s2 …4s2 4p6 4d10 5s1

Au79 : 1s2 …5s2 5p6 5d10 6s1 Ag47 : 1s2 …4s2 4p6 4d10 5s1

The fully occupied 4d band of Ag can influence the dynamical surface response in two distinct ways: • First, the s-d hybridization modifies the single-particle energies and wave functions. As a result, the nonlocal density-density response function exhibits band structure effects. • Second, the effective time-varying fields are modified due to the mutual polarization of the s and d electron densities.

Here we focus on the second mechanism for the following reason: At the parallel wave vectors of interest, the frequency of the Ag surface plasmon lies below the region of interband transitions. Thus, the relevant single-particle transitions all occur within the s-p band close to the Fermi energy where it displays excellent nearly-free-electron character.

A. Liebsch, “Surface Plasmon Dispersion of Ag,” PRL 71, 145-148 (1993). Measured data and model for Ag: ε (ω)

50 Drude model: ε ω 2 ω 2 ε" p ε p '=1− 2 , "= 3 γ 0 ω ω -εd ε Measured data: ε ω Modified Drude model: -50 ε'

ε ε ω2 ε ω 2 ε" ε' '= − p , "= p γ Drude model: ∞ 2 ω3 ε' -100 Bound SP mode : ε’m < -εd ε" Contribution of Modified Drude model: bound electrons ε' -150 Ag: ε ∞ = 3.4 200 400 600 800 1000 1200 1400 1600 1800 WavelengthWavelength (nm) (nm) Ideal case : metal as a free-electron gas

• no decay (γ Æ 0 : infinite relaxation time) • no interband transitions (ε = 1) εωε B σ ()ω EFF()=+ B i εωo εε22 22 2 ⎛⎞⎛⎞⎛⎞ω ppτωττ →∞ ω p =−⎜⎟⎜⎟⎜⎟BEFF +i ⎯⎯⎯→=−1 ⎜⎟⎜⎟⎜⎟22 33εB =1 2 ⎝⎠⎝⎠⎝⎠1++ω τωτωτ ω

2 ω p ε =−1 0 r ω 2 Dispersion relation of surface-plasmon for dielectric-metal boundaries Dispersion relation for EM waves in electron gas (bulk plasmons)

• Dispersion relation:

ω = ω()k Dispersion relation for surface plasmons

εd

εm TM wave

Z > 0

Z < 0

• At the boundary (continuity of the tangential Ex, Hy, and the normal Dz):

EExmxd= HHymyd= ε mzmEE= ε dzd Dispersion relation for surface plasmon polaritons

(−ikzi H yi ,0,ikxi H yi ) (−iωε i Exi ,0,−iωε i Ezi )

kzi H yi = ωε i Exi kzm H ym = ωε m Exm

kzd H yd = ωε d Exd

EExmxd=

HHymyd= kkzm zd kε k = zm H = zd H ym yd ε mdε m ε d Dispersion relation for surface plasmon polaritons

2 ⎛⎞ω • For any EM wave: 222ε k==+ixzixxmxd⎜⎟ k k , where k ≡= k k ⎝⎠c

SP Dispersion Relation ω

ε mdε kx = c ε md+ ε DispersionDispersion relation: relation for surface plasmon polaritons

ω 1/2 ⎛⎞εε x-direction: kkik=+'" = md '" xx x ⎜⎟ε mm=+εεi m c ⎝⎠εεmd+ 1/2 2 ω 2 22ε ⎛⎞ω ⎛⎞εi z-direction: kkzii= ⎜⎟− x kkikzi=+=±' zi zi c ⎜⎟εε ⎝⎠ c ⎝⎠md+ For a bound SP mode: εεε kzi must be imaginary: εm + εd < 0

22 ⎛⎞ω 22 ⎛⎞ωω ⎛⎞ kkikkzi=± i⎜⎟ − x =± x − i ⎜⎟ ⇒ x > i ⎜⎟ ⎝⎠ccc ⎝⎠ ⎝⎠

+ for z < 0 - for z > 0 k’x must be real: εm < 0

' So, ε md< −ε ω 1/2 ω ⎛⎞εε '" md '" kkikxx=+ x =ε ⎜⎟ ε mm=+εεi m ε c ⎝⎠εεmd+ ε ε ε ε 1 ε ε 1 ω 2 4 2 2 ⎡ ⎤ 2 ⎡ + + ()" ⎤ ' d ⎢ e e m d ⎥ kx = ⎢ ε ⎥ c ' 2 " 2 ⎢ 2 ⎥ ⎣⎢(ε m + d ) + ()m ⎦⎥ ε ⎣ ⎦ ε 1 1 ⎡ ε ε ⎤ 2 ε ⎡ ⎤ 2 ⎢ ε " 2 ⎥ ε ε " d ()m d kx = ⎢ ⎥ ⎢ ε ⎥ ε ' 2 " 2 ()2 2 2 2 c ⎢( + ) + ()⎥ ⎢ ⎧ 2 4 ε " ⎫⎥ ' " ε ' ε⎣ m d m ⎦ 2⎨ e + e + mε d ⎬ where, e = ( m ) + ( m ) + dε m ε ⎣⎢ ⎩ ⎭⎦⎥ , ε ' ' ' " m < 0, ωm > d , and m >> ε m in most of metals, ε ε ε 1/ 2 ⎛ ε' ⎞ k ' ≈ ⎜ m d ⎟ x ωc ⎜ ' + ⎟ ⎝ε mε d ⎠ 3/ 2 ⎛ε ' ⎞ ε " k " ≈ ⎜ m εd ⎟ m x c ⎜ ' + ⎟ ' 2 ⎝ d d ⎠ 2()ε m Plot of the dispersion relation

• Plot of the dielectric constants: ε ω ω 2 ( ) =1− p m ω 2

• Plot of the dispersion relation: ω ω ω ω ε ε ( 2 − 2ε) k = md ε p d x kx = ksp = ω c ε + ε 2 2 md c (1+ d ) −ω p

• When ε m → −ε d , ω p ⇒ k x → ∞, ω ≡ ωsp = 1+ ε d SurfaceSurface plasmon plasmon dispersiondispersion relation: relation ω ε 1/ 2 ω 1/2 ε ⎛⎞2 ⎛ m d ⎞ εi k = ⎜ ⎟ kzi = ⎜⎟ x ⎜ ε ⎟ c εε+ c ⎝ m + ε d ⎠ ⎝⎠md

2222ck ω ωω=+p ckx x Radiative modes real k ε x d (ε'm > 0) real kz

ωp Quasi-bound modes imaginary kx (−ε < ε' < 0) d m real kz ω p

1+ ε d Dielectric: Bound modes real k εd x z imaginary kz x (ε'm < −εd) ' Metal: εm = εm + " εm

Re kx X-ray wavelengths at optical frequencies Very small SP wavelength

λvac=360 nm

SiO2 Ag Dispersion relation for bulk and surface plasmons ω ε 1/ 2 ⎛ ε ⎞ ⎜ m d ⎟ kx = ⎜ ε ⎟ c ⎝ m + ε d ⎠

ε ⎛⎞⎛⎞ωτ ωτ22 22 =−1 pp +i m ⎜⎟⎜⎟22 33 ⎝⎠⎝⎠1++ω τωτωτ

ε ω

ω ε Cut-off frequency of SP ω ε ω

2 ε2 ω p 2 2 2 p p D When m =1− 2 = − d , ω − p = − d ⇒ ω = ⇒ ωsp = 1+ d 1+ ε d Measured data and model for Ag: ε (ω)

50 Drude model: 2 2 ε ω p ε ω p ε" '= 1− , "= γ 2 2 0 ω ω -εd

Measured data: Modified Drude model: -50 ε'

ε ε ε 2 2 ε" ε' ε ω p ε ω p '= ∞ − , "= γ Drude model: ω 2 ω 2 ε' -100 Bound SP mode : ε’m < -εd ε" Contribution of Modified Drude model: bound electrons ε' -150 Ag: ε ∞ = 3.4 200 400 600 800 1000 1200 1400 1600 1800 WavelengthWavelength (nm) (nm) Propagation length

The length after which the intensity decreases to 1/e :

ω '"3/2 −1 ""⎛⎞εε ε12 1 Lkix==() 2 , where k x⎜⎟''2 c ⎝⎠12εε+ ε2( 1 ) Propagation length : Ag/air, Ag/glass

εε ε ε

⎛⎞⎛⎞ωτ ωτ22 22 =+'"ii = −pp + mm m⎜⎟⎜⎟ B 22 33 ⎝⎠⎝⎠1++ω τωτωτ Penetration depth

'  1 At large kx (εε12→− ), zi ≈ . Strong concentration near the surface in both media. kx Ezx= ±+iE(:, air i metal:-) i

' E At low k (ε >> 1), z ' x 1 =−i ε1 in air : Larger Ez component Ex

Ez 1 = i in metal : Smaller E component ' z Ex ε1 Gooood waveguide!

Another representation of SP dispersion relation

ε =ε ' + " = − 2 m m ε m ν k p = ω p / c Excitation of surface plasmons Excitationω of surface plasmon by prism ε ε ε ε 1/ 2 ω ω ⎛ ' ⎞ ' m d 2222 k = ⎜ ⎟ > k = ε ωω=+p ck x ⎜ ' ⎟ d c ⎝ m + d ⎠ c c cε ω = k o ω = k x sinθ sp d p ε ω p c ω ω = k o ω p sp = p 1 + ε d ε

ω excitation ω p sp = 1 + ε p

εd

' k x

εp ' ⎛⎞εθω kkx ==⎜⎟pspspsin ⎝⎠c Generalization : Surface Electric Polaritons and Surface Magnetic Polaritons : Energy quanta of surface localized oscillation of electric or magnetic dipoles in coherent manner

Surface Electric Polariton (SEP) Surface Magnetic Polariton (SMP) K K E H

+q -q +q -q N S N S

Coupling to TM polarized EM wave Coupling to TE polarized EM wave

Common Features - Non-radiative modes → scale down of control elements - Smaller group velocity than light coupling to SP - Enhancement of field and surface photon DOS Generalization :

Surface Electric Polaritonsβ and Surface Magnetic Polaritons

Dispersion Relation & Decay Constantsγ εεμ εμ

εε''(''12 ε μ 2 εμ 1− ' 1 ') 2 ⎛⎞ε ""μ SEP =+22 kO0 ⎜⎟, ε ''21− εεμ⎝⎠ '' εμ For (εε"/ ' , μμ "/ ')<< (1,1) , '( ' '− ' ') ⎛⎞"" γγ { i 11 2 2 SEP,1 SEP ,2 SEP,0 i =+=22 kO⎜⎟,, ε ''21−εεμεε⎝⎠ '' 1 ' 2 '

J. Yoon, et al., Opt Exp 2005. In Summary

Permittivity of a metal

22 ωωpp⎛⎞γ εωm ()=− 1 22 +i 22⎜⎟ ωγ++ ωγ ⎝⎠ω 22 ≈−1/ωωp

Dispersion relations

ω 1/2 ⎛⎞εεdm kSPP = ⎜⎟ c ⎝⎠εεdm+ Type-A : low k

Type-A

- Low frequency region (IR)

- Weak field-confinement H. Won, APL 88, 011110 (2006).

- Most of energy is guided in clad

- Low propagation loss

► clad sensitive applications

► SPP waveguides applications Type-B : middle k

Type-B

- Visible-light frequency region Nano-hole - Coupling of localized field and propagation field

- Moderated field enhancement

► Sensors, display applications

► Extraordinary transmission of light Type-C : high k Ag (20nm) p-GaN (20nm, 120nm)

QW Type-C Λ n-GaN

- UV frequency region Light emission

- Strong field confinement QW - Very-low group velocity

► Nano-focusing, Nano-lithography

► SP-enhanced LEDs τω ε SE Rate : 11 2 Rfi==pE ⋅ρ()ω () 2= 0 Photon DOS Electric field strength (Density of States) of half photon (vacuum fluctuation) Importance of understanding the dispersion relation : Broadband slow and subwavelength light in air Mode conversion: negative group velocity

dω < 0 dk SiO dω ω 2 = 0 p dk 1+ ε SiO2

Si3N4 ω p

1+ ε Si3N 4 Re kx

SiO2

Si3N4 Al Appendix

Plasma frequency

If the electrons in a plasma are displaced from a uniform background of ions, electric fields will be built up in such a direction as to restore the neutrality of the plasma by pulling the electrons back to their original positions. Because of their inertia, the electrons will overshoot and oscillate around their equilibrium positions with a characteristic frequency known as the plasma frequency.

Eso= (surface charge density) / ε

= Nex (δ ) /εδo : electrostatic field by small charge separation x δδ ω

xx=−opexp( it ) : small-amplitude oscillation 222δ dx() 22 Ne Ne meEm2 =−( )sp ⇒ − =− ⇒ ∴ p = dt ε oomε

ωω