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 The equation of motion of a free electron (not bound to a particular nucleus; C = 0), rr r dr2 r m druurruu dv r mCreEmmveE=− −e − ⇒ +γ =− eeedt 2 τ dt dt 1 (τ =≈ : relaxation time 10−14 s) Lorentz model γ (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 − iω t ⎣⎦0 () rrr +−=−−+−γωωωγωJitiJitJitexp() exp () exp () dt 000 ⎛⎞Ne2 r =−⎜⎟Eit0 exp()ω ⎝⎠me Multiplying through byexp()+ iω t : rr⎛⎞Ne2 ()−+iJωγ 00 =⎜⎟ E ⎝⎠me rr⎛⎞Ne2 or equivalently()−+ iωγ J =⎜⎟ E ⎝⎠me ConductionConduction CurrentCurrent inin MetalsMetals For static fields()ω = 0: we obtain rrr⎛⎞Ne22 Ne J==⇒==⎜⎟ Eσσ E static conductivity ⎝⎠mmeeγγ For the general case of an oscillating applied field : rrr⎡⎤σ J==⎢⎥ Eσσωω E = dynamic conductivity ⎣⎦1/− ()iωγ For very low frequencies,1,()ωγ << the dynamicconductivity is purely real and the electrons follow the 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, 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⎡⎤σ But J= ⎢⎥ E ⎣⎦1/− ()iωγ Substituting in the wave equation we obtain : 2 rr 2 r 11∂∂EE⎡⎤σ ∇=E 22 + 2⎢⎥ ct∂−∂εωγ0 c⎣⎦1/() i t The wave equation is satisfied by electric fields of the form : rr r EE=⋅−exp ⎡⎤ ikrtr ω 0 ⎣⎦() where 2 ⎡⎤ 22ω σωμ0 1 ki=+2 ⎢⎥ c = ci⎣⎦1/− ()ωγ εμ00 SkinSkin DepthDepth Consider the case whereω is small enough that k2 is given by : 2 ⎡⎤ 2 ωπσωμ0 ⎛⎞ ki=+2 ⎢⎥ ≅ iσωμ00 =exp⎜⎟ i σωμ ci⎣⎦1/− ()ωγ ⎝⎠ 2 ⎛⎞ππ ⎛⎞⎡⎤ ⎛⎞ ππ ⎛⎞ σ ωμ0 Then, k% ≅==+=+exp⎜⎟ iσωμ00 exp ⎜⎟ i σωμ⎢⎥ cos ⎜⎟ i sin ⎜⎟ σωμ 0() 1 i ⎝⎠24 ⎝⎠⎣⎦ ⎝⎠ 442 ⎝⎠ 2 σωμ0 ⎛⎞c σμc 0 σ kkRI== n R =⎜⎟knRI=== 2 ⎝⎠ωωωε220 In the metal,: for a wave propagating in the z− direction rr r ⎛⎞z EE=−00exp() kzIR exp⎣⎦⎡⎤ ikzt ( −=−ωω ) E exp⎜⎟ exp ⎣⎦⎡⎤ ikzt() R − ⎝⎠δ 122ε c2 The skin depthδδ is given by : == = 0 kI σωμ0 σω Cs2 − For copper the static conductivityσ =×Ω=×5.76 10711−− m 5.76 10 7 →=δμ 0.66 m Jm− PlasmaPlasma FrequencyFrequency Now consider again the general case : 2 ⎡⎤ 2 ω σωμ0 ki=+2 ⎢⎥ ci⎣⎦1/− ()ωγ 2 ⎧ 22⎫⎧ ⎫ 22ci⎪ σμcc00⎪⎪γ σμ ⎪ nk==+2 11 i⎨ ⎬⎨ =+ i ⎬ ωγ⎩⎭⎩⎭⎪ωωγ⎣⎡1/−−()ii⎦⎣⎦⎤⎡⎤⎪⎪i ωωγ 1/() ⎪ γσc2 μ n2 =−1 0 ωωγ2 +i The plasma frequency is defined : 2 2 22⎛⎞Ne 2Ne ωγσμγp ==cc0 ⎜⎟μ0 = ⎝⎠meγ meε0 The refractive index of the medium is given by ω 2 n2 =−1 p ωωγ2 +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 ) / o : electrostatic field by small charge separation δ x δδxx=−opexp( it ω ) : small-amplitude oscillation 222 d()δ x22 Ne Ne meEm2 =−( )sp ⇒ −ωω =− ⇒ ∴ p = dt εεoom PlasmaPlasma FrequencyFrequency 2 22 2 ⎛⎞c icσ μσμγoo c nk==+=−⎜⎟11 ⎝⎠ωωωγ(/)1− i ω 2 + iωγ ω 2 nnin22=+() =−1 p RI ωωγ2 + i ω 2 n2 =−1 p by neglecting γ , valid for high frequency (ωγ >> ). ω 2 For ωω< p , n is complex and radiation is attenuated. For ωω> p , n is real and radiation is not attenuated(transparent). PlasmaPlasma FrequencyFrequency 2πc λc = λp = ω p Born and Wolf, Optics, page 627. PlasmaPlasma FrequencyFrequency Dielectric constant of metal : Drude model 2 εω()=+ εRIin ε = ω 2 =+()1nin2 =− p RI ωωγ2 + i 22 =−+()2nnRI inn RI ⎛⎞⎛⎞ωωγ22 =−1 pp +i ⎜⎟⎜⎟22 3 2 ⎝⎠⎝⎠ω ++γωωγ 1 ⎛⎞⎛⎞ωω22 ω >> γ = εω()=− 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()⎯⎯⎯τ →∞→=− εω () 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 ωp determined by the free electron density and effective mass Ne2 --- ω drude = p mε k 0 +-+ Plasmons confined to surfaces that can interact with light to form propagating “surface plasmon polaritons (SPP)” 2 drude 1 Ne ω 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 k k = zm H = zd H ym yd ε mdε ε m ε d Dispersion relation for surface plasmon polaritons 2 ⎛⎞ω • For any EM wave: 222 k==+εixzixxmxd⎜⎟ 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 1/2 ω ⎛⎞εε x-direction: kkik=+'" = md '" xx x ⎜⎟ε mm=+εεi m c ⎝⎠εεmd+ 1/2 2 2 22⎛⎞ω ω ⎛⎞εi z-direction: kkzii= ε ⎜⎟− x kkikzi=+=±' zi zi c ⎜⎟ ⎝⎠ c ⎝⎠εεmd+ For a bound SP mode: kzi must be imaginary: εm + εd < 0 22 ⎛⎞ω 22 ⎛⎞ωω ⎛⎞ kkikkzi=±εεε 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: ω 2 ε (ω) =1− p m ω 2 • Plot of the dispersion relation: 2 2 ω ε mdε ω (ω −ω )ε k = k = k = p d x c ε + ε x sp 2 2 md c (1+ ε d )ω −ω p • When ε m → −ε d , ω p ⇒ k x → ∞, ω ≡ ωsp = 1+ ε d SurfaceSurface plasmon plasmon dispersiondispersion relation: relation 1/ 2 1/2 ⎛⎞2 ω ⎛ ε mε d ⎞ ω εi k = ⎜ ⎟ kzi = ⎜⎟ x ⎜ ⎟ c εε+ c ⎝ ε 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.