25. Optical properties of materials-Metal

Drude Model

• Conduction Current in Metals • EM Wave Propagation in Metals • Skin Depth • Plasma Frequency Drude model

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

Remind! Linear Dielectric Response of Matter ConductionConduction CurrentCurrent inin MetalsMetals τγ The equation of motion of a free electron (not bound to a particular nucleus; C = 0 ), dr2 m dr dv 1 meEmmveEs=−−Cr e − ⇒ +=γτ − ( = : relaxation time ≈ 10−14 ) eeedt 2 dt dt If Lorentz model C = 0 Drude model (Harmonic oscillator model) (free-electron model)

The current density is defined : ⎡⎤AC JNe=−v = ⎣⎦⎢⎥msm22i

Substituting in the equation of motion we obtain :

dJ⎛⎞ N e2 +=γ JE⎜⎟ dt⎝⎠ me dJ⎛⎞ N e2 +=γ J ⎜⎟E dt⎝⎠ me

Assume that the applied electric field and the conduction current density are given by :

EE=−exp itωω JJ =− exp it 00() () Local approximation to the current-field relation Substituting into the equation of motion we obtain : () dJ⎡⎤exp − i t ⎣⎦0 ω+−=−−+−JitiJitJitexp() exp () exp () dt 000 γωωωγω ⎛⎞Ne2 =−⎜⎟Eit0 exp()ω ⎝⎠me

() Multiplying through byexp+ i t : ωγ ω () ⎛⎞Ne2 2 −+i J = E, or equivalently ⎛⎞Ne 00⎜⎟ ()−+iJωγ = E ⎝⎠me ⎜⎟ ⎝⎠me ⎛⎞Ne2 ()−+iJωγ =⎜⎟ E ⎝⎠me

ω For static fields ()= 0 , 22 ⎛⎞Neγγ Ne J==⎜⎟ Eσσ E , where = : static conductivity ⎝⎠mmee

For the general case of an oscillating applied field : ⎡⎤σ 2 Ne/ me JE= ⎢⎥===ωωE , where : dynamic conductivity 1− ()i / ωγ 1/−−()i i ⎣ ⎦ σσ

For very low frequencies,1()ωγ << , σ the dynamic conductivity is purely real and the electrons followωγ γ the ω electric field.

As the frequency of the applied field increases,the inertia of electrons introduces a phase lag in the electron response to the field, and the dynamicconductivity is complex. ωγ

i π For very high frequencies,1()>> , JiE≈= σσ() e2 E the dynamic conductivity is purely imaginary and the electron oscillations are90° out of phase with the applied field. PropagationPropagation ofof EMEM WavesWaves inin MetalsMetals

Maxwell': s relations give us the following wave equation for metals 11∂∂2 EJ ∇=2 E + 22 2 P = 0, J ≠ 0 ct∂∂ε0 ct

⎡⎤σ But, J= ⎢⎥ E 1/− i ⎣⎦()ωγ

Substituting in the wave equation we obtain : 2 2 11∂∂EE⎡⎤ ∇=E 22 + 2⎢⎥ ct∂−∂0 c⎣⎦1/() i t εωγσ The wave equation is satisfied by electric fields of the form : EE=⋅−exp ⎡⎤ ikrtω 0 ⎣⎦()

ω 2 ⎡⎤ 1 2 σ 0 2 ⇒ ki=+2 ⎢ ωμ⎥ , wherec= c ⎣1− ()i / ⎦ 00 ωγ

ε μ ππ ππ SkinSkin DepthDepth σωμσωμ σωμ atat lowlow frequencyfrequency ωπ Consider the case whereω is small enough that k2 is given by : σωμ 2 ⎡⎤ 2 0 ⎛⎞ ki=+2 ⎢⎥ωγ ≅ i00 =exp⎜⎟ i ci⎣⎦1/− ()σωμ σωμ ⎝⎠ 2

⎛ ⎞ ⎛ ⎞⎡⎤ ⎛⎞ ⎛⎞ σ ωμ0 Then, k≅==+=+ exp⎜ i ⎟00 exp ⎜ i ⎟⎢⎥ cos ⎜⎟ i sin ⎜⎟ 0() 1 i ⎝24 ⎠ ⎝ ⎠⎣⎦ ⎝⎠ 442 ⎝⎠

2 ω σ μ σωμ 00⎛⎞c c kkRI== ⇒ nn RI ==⎜⎟ k RI, = = 222⎝⎠ 0

In the metal,: for aδδ wave propagating in the z− direction ωωωε ⎛⎞z EE==−−=−−00exp() ikzE exp() kzIR exp⎣⎦⎡⎤ ikzt ( E 0 ) exp⎜⎟ exp ⎣⎦⎡⎤ ikzt() Rσ ⎝⎠

12σωμ σω 2ε c2 The skin depth is given by : ==ωω = 0 kI 0 2 δ Cs− For copper the static conductivityσ =×Ω=×5.76 10711−− m 5.76 10 7 →=δμ 0.66 m Jm− RefractiveRefractive IndexIndex ofof aa metalmetal ωγω σμ σμ 2 ⎡⎤σωμ ωωγ2 ωωγ0 Now consider again the general case, k=+2 i ⎢⎥ γσ μ ci⎣⎦1/− () γ ωγ 2 ωωγ⎧⎫⎧⎫22 22ci⎪⎪⎪⎪cc00 nk==+2 11 i⎨⎬⎨⎬ =+ i ⎩⎭⎩⎭⎪⎪⎪⎪⎣⎦⎡⎤1/−−()iii ⎣⎦⎡⎤ 1/() c2 ⇒=−n2 1 0 2 +i

ωγσμγ2 2 22⎛ Ne ⎞ 2 Ne The plasma frequency is defined, p == c 0 ⎜⎟c μ0 = ⎝⎠me meε0 γ The refractive index of the conductive medium is given by ,

2 2 ω p Ne nwhere2 =− 1 , = : Plasma frequency ωωγ2 ε p +ime 0 ω WhatWhat isis thethe plasmaplasma Frequency?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.

Esoo==σ /εδεNe ( x ) / δ o : electrostatic field by small charge separation x

xx=−exp( it ) : small-amplitude oscillation δδop ω

d222() x Ne Ne meE=−( ) ⇒ −m 22 =− ⇒ = 2 δ sp p dt o m o

Ne2 p = ωω m o ε

ω ε ε CriticalCritical wavelengthwavelength (or,(or, plasmaplasma wavelength)wavelength)

λ 2πc c = λp = ω p

Born and Wolf, Optics, page 627. RefractiveRefractive IndexIndex ofof aa metalmetal

ω 2 n2 =− 1 p ω 2 +iωγ

For a high frequency (ω >> γ ), ω

2 n2 ≈−1 p by neglecting γ . ω 2

ω < ω p : n is complex and radiation is attenuated. Æ EM waves with lower frequencies are reflected/absorbed at metal surfaces. ωω > p : n is real and radiation is not attenuated(transparent). Æ EM waves with higher frequencies can propagate through metals. DispersionDispersion ofof RefractiveRefractive IndexIndex forfor coppercopper Dielectric constant of metal given by Drude model

2 εω() ε ε=+RIin = 2 =+()1nin2 =− p RI 2 + i =−+()2nn22 inn RI RIω 22 ⎛⎞⎛⎞ppωωγ =−1 +i ⎜⎟⎜⎟22 3 2 ⎝⎠⎝⎠ω ++γωωγ ωωγ

εω ω 1 ⎛⎞⎛⎞ωω22 >> γ = ()=− 1 pp +i ⎜⎟⎜⎟23 τ ⎝⎠⎝⎠ω ωγ/ Ideal case : metal as a free-electron gas

• no decay (infinite relaxation time) • no interband transitions ⎛⎞ω 2 εω() εω⎯⎯⎯τ →∞→=− ()⎜⎟ 1 p γ→0 ⎜⎟2 ⎝⎠ω

2 ω p ε =−1 0 r ω 2 AnAn applicationapplication ofof DrudeDrude model model :: SurfaceSurface plasmonsplasmons Plasma wave (oscillation) = density fluctuation of charged particles Plasmon = plasma wave with well defined oscillation frequency (energy) Plasmon in metals = collective oscillation of free electrons with well defined energy Surface plasmons = Plasmons on metal surfaces Plasma waves (plasmons)

Plasma oscillation = density fluctuation of free electrons

+++

Plasmons propagating through bulk media with a resonance at ωp ω Ne2 --- Æ Bulk Plasmons drude = p mε k 0 Plasmons confined but propagating on surfaces +-+

Æ Surface Plasmons (SP)

Plasmons confined at nanoparticles with a resonanceω at 2 drude 1 Ne Æ Localized Surface Plasmons particle = 3 mε 0 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 plasmons

(−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 plasmons

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

kx SP Dispersion Relation ω

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

ω 1/2 ⎛⎞εεmd '" x-direction: kkikxx=+'" 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 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 dispersion dispersion 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

λ ~λ Λ <<λ x x Re kx = 2π / λ Surface plasmon 응용

Barnes et al., Nature (2003) Metal waveguides

Surface plasmonic waveguides

Several전자신호 1 cm long, 15 nm thin and 8 micron wide gold stripes guiding LRSPPs전자신호 3-6 mm long control electrodes low driving powers (approx. 100 mW) and high extinction ratios (approx. 30 dB) response times (approx. 0.5 ms) 금속선 광신호 total (fiber-to-fiber)광신호 insertion loss of approx. 8 dB when using single-mode fibers Asymmetric mode : field enhancement at a metallic tip

E r Er Ez

Ez

M. I. Stockman, “Nanofocusing of Optical Energy in Tapered Plasmonic Waveguides,” Phys. Rev. Lett. 93, 137404 (2004)] Nano-scale light guiding MetalMetalMetal NanoparticleNanoparticleNanoparticleWaveguidesWaveguides Waveguides

Maier et al., Adv. Mater. 13, 1501 (2001) NanoNanoNanoPlasmonicsPlasmonics Plasmonics NanoNanoNano-Photonics--PhotonicsPhotonics BasedBasedBased ononon PlasmonicsPlasmonicsPlasmonics

By Prof. M. Brongersma

Nanoscale integrated circuits with the operating speed of photonics can be possible!

A World of NanoPlasmonics

Long-range(~ cm) waveguides On-chip light source Short-range(~ nm) ~ cm waveguides

Photonic integrated circuit

Nano- photonics Nano-electronics Harry Atwater, California Institute of Technology

Could such an Architecture be Realized with Metal rather than Dielectric Waveguide Technology?