Drude Model for dielectric constant of metals.

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

Ref : Prof. Robert P. Lucht, Purdue University 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)

Linear Dielectric Response of Matter ConductionConduction CurrentCurrent inin MetalsMetals τ The equation of motion of a free electron (not bound to a particular nucleus; C = 0 ), 2 rr r drr me drτ uurruu dv r mCreEmmveEeee2 =− − − ⇒ +γ =− dt dtγ dt 1 (=≈ : relaxation time 10−14 s ) Lorentz model (Harmonic oscillator model)

If C = 0, it is called Drude model

The current density is defined : r ⎡⎤C J=− N evr with units of ⎣⎦⎢⎥sm− 2

Substituting in the equation of motion we obtain : r dJrr⎛⎞ N e2 +=γ JE⎜⎟ dt⎝⎠ me ConductionConduction CurrentCurrent inin MetalsMetals

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

rr rr Local approximation E=− Eexp() itωω J =− J exp it () 00to the current-field relation

Substituting into the equation of motion we obtain : () r dJ⎡⎤exp − i t ⎣⎦0 ω rrr +−=−−+−JitiJitJitexp() exp () exp () dt 000 γωωωγω ⎛⎞Ne2 r =−⎜⎟Eit0 exp()ω ⎝⎠me

() Multiplyingωγ through byexp+ i t : () ω rr⎛⎞Ne2 −+iJ00 =⎜⎟ E ⎝⎠me rr⎛⎞Ne2 or equivalently()−+ i J =⎜⎟ E ωγ⎝⎠me ConductionConduction CurrentCurrent inin MetalsMetals For static fields()ω = 0: we obtain

rrr⎛⎞Ne22 Ne J==⇒==⎜⎟ Eγγ Eσσ static conductivity ⎝⎠mmee

For the general case of an oscillating applied field : rrr⎡⎤σ J==⎢⎥ Eωω E = dynamic conductivity ⎣⎦1/− ()i ωγ σσ

For very low frequencies,1,()<< the dynamicconductivity is purely real and the electrons follow the electric field. ωγ As the frequency of the applied field increases, the inertia of electrons introduces a phase lag in the electron response to the field, andthe dynamic conductivity is complex.

For very high frequencies,1,()ωγ >> the dynamicconductivity is purely imaginary and the electron oscillations are90° out of phasewith the applied field . PropagationPropagation ofof EMEM WavesWaves inin

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

rr⎡⎤ But J= ⎢⎥σ E ⎣⎦1/− ()i ωγ Substituting in the wave equation we obtain :

2 rr 2 r 11∂∂EE⎡⎤ ∇=E 22 + 2⎢⎥ ct∂−∂0 c⎣⎦1/() i t εωγσ The wave equation is satisfied by electric fields of the form : rr r EE=⋅−exp ⎡⎤ ikrtr ω 0 ⎣⎦() where ω 2 22⎡⎤σωμ 1 ki=+⎢⎥0 c = ci2 1/− () ⎣⎦ωγ εμ 00 Skin Depth ππSkin Depth ππ ω ω σωμσωμ σωμ ωπ Consider the caseσωμ 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⎜⎟ i00 exp ⎜⎟ i⎢⎥ cos ⎜⎟ i sin ⎜⎟ 0() 1 i ⎝⎠24 ⎝⎠⎣⎦ ⎝⎠ 442 ⎝⎠

2 0 ⎛⎞cωωωεσμc 0 kkRI== n R =⎜⎟knRI=== 2 22 ⎝⎠ σ 0

In the metal,:δδ for a wave propagatingωω in the z− direction rr r ⎛⎞z EE=−00exp() kzIR exp⎡⎤⎣⎦ ikzt ( −=− E ) exp⎜⎟ exp ⎣⎦⎡⎤ ikzt() R − ⎝⎠δ

12σωμ σω 2ε c2 The skin depth is given by : == = 0 kI 0 Cs2 − For copper the static conductivityσ =×Ω=×5.76 10711−− m 5.76 10 7 →=δμ 0.66 m Jm− PlasmaPlasma FrequencyFrequency

Nowω consider again the general case : 2 2 ⎡⎤σωμ ki=+⎢⎥0 ci2 1/− () ⎣⎦ωγ

2 ⎧ 22⎫⎧ ⎫ 22ci⎪ cc00⎪⎪ ⎪ nk==+2 ωγ11 i⎨ ⎬⎨ =+ i ⎬ ⎩⎭⎩⎭⎪ ⎣⎡1/−−()iiσμ σμ⎦⎣⎦⎤⎡⎤⎪⎪i 1/() ⎪ 2 ωωγ ωωγ c γσ μ n2 =−1 0 2 +i ωωγ γ

The plasma frequency is defined :

2 2 22ωγσμγ⎛⎞Ne 2Ne p ==cc0 ⎜⎟μ0 = ⎝⎠me me 0 ε The refractive indexω of the mediumγ is given by 2ωωγ n2 =−1 p 2 +i PlasmaPlasma FrequencyFrequency

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 /εδε= Ne ( x ) / δ o : electrostatic field by small charge separation x

xx=−exp( it ) : small-amplitude oscillation δδop ω 222 d() x22 Ne Ne meEm2 δ=−( )sp ⇒ − =− ⇒ ∴ p = dt oom

ωω

εε PlasmaPlasma FrequencyFrequency ωωωγ 2 22 2 ⎛⎞c icσ μσμγoo c nk==+=−⎜⎟112 ⎝⎠ (/)1− i ω ω + iωγ ω 2 22ωωγp nnin=+()RI =−1 2 ωω + i

ωω2 2 ω p n =−1 2 by neglecting γ , valid for high frequency (ωγ >> ).

For < p , n is complex and radiation is attenuated.

For > p , n is real and radiation is not attenuated(transparent). PlasmaPlasma FrequencyFrequency

λ 2πc c = λp = ω p

Born and Wolf, Optics, page 627. PlasmaPlasma FrequencyFrequency Dielectric constant of metal : 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 Plasma waves (plasmons)

Note: SP is a TM wave! Plasmo ns Plasma oscillation = density fluctuation of free electrons

+++ Plasmons in the bulk oscillate at ω determined by ω p the free electron density and effective mass Ne2 --- drude = p mε k 0

+-+ Plasmons confined to surfaces that can interact with light to form propagating “surface plasmon polaritons (SPP)”

ω 2 drude 1 Ne particle = Confinement effects result in resonant3 mε 0 SPP modes in nanoparticles Dispersion relation for EM waves in electron gas (bulk plasmons)

• Dispersion relation:

ω = ω()k Dispersion relation of surface-plasmon for dielectric-metal boundaries Dispersion relation for surface plasmon polaritons

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 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