dielectric constant of metals

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 zDrude model : Lorenz model (Harmonic oscill ator model) without restoration force (that is, free electrons which ar e not bound to a particular nucleus) Linear Dielectric Response of Matter Conduction Current in MetalsConduction Current in Metals 2 2: v : eThe current densit yis de fined CJ N e with units ofsm Substitutin gin the equation o fmotion we obtain dJ N eJEdt mγ⎡⎤=−⎢⎥−⎣⎦ ⎛⎞+=⎜⎟ ⎝⎠r r rrrThe equation of motion of a free electron (not bound to a particular nucleus; ), ( : relaxation time )2 2 140 110rr rr uurr u u re ee eC m dr d r d vmC r e E m m v e Edt dt dt s γ ττγ−= =− − − ⇒ + =− =≈ Lorentz model (Harmonic oscillator model) If C = 0, it is called Drude model Conduction Current in MetalsConduction Current in Metals () () ()() () ()00 0 00 0: exp exp : exp exp exp expAssume that the applied electric field and the conduction current density are given by E E it J J it Substituting into the equation of motion we obtain dJ i t Ji t i Ji t Ji tdtωωωγω ωω γω=− =− ⎡⎤−⎣⎦+− = − − +−rr rr r rr r () () () ()2 0 2 00 2exp exp :e e eNeEi tm Multiplying through by i t NeiJ Em Neor equivalently i J Em ωω ωγωγ⎛⎞=−⎜⎟ ⎝⎠ + ⎛⎞−+ = ⎜⎟ ⎝⎠ ⎛⎞−+ = ⎜⎟ ⎝⎠r rr rrLocal approximation to the current-field relation Conduction Current in MetalsConduction Current in Metals () () ()220: : 1/ ,1 ,eeFor static fields we obtain Ne NeJ E E static conductivitymm For the general case of an oscillating applied field J E E dynamic conductivityi For very low frequencies the dynωω ωσσ γγ σσσ ωγωγ= ⎛⎞== ⇒ = =⎜⎟ ⎝⎠ ⎡⎤== =⎢⎥−⎣⎦ <<rr r rr r . , ,amic conductivity 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 t o the field andthe dynamic conducti (). ,1 , 90 .vityis complex For very high frequencies the dynamic conductivity is purely imaginary and the electron oscillations are out of phasewith the applied fieldωγ>> ° Propagation of EM Waves in MetalsPropagation of EM Waves in Metals () ()2 2 22 2 0 2 2 22 2 0': 11 1/ : 11 1/Maxwell s relations give us the following wave equation for metals EJEct ct But J Ei Substituting in the wave equation we obtain EEEct c i t The wave equation is sεσωγσ εω γ∂∂∇= +∂∂ ⎡⎤=⎢⎥−⎣⎦ ⎡⎤ ∂∂∇= + ⎢⎥∂− ∂⎣⎦rrr rr rrr () ()0 2 22 0 2 00: exp 1 1/atisfied by electric fields of the form EE i k r t where ki cciω σωμ ωωγ εμ⎡⎤=⋅ −⎣⎦ ⎡⎤=+ =⎢⎥−⎣⎦r rr r0 ,0≠= J P Skin DepthSkin Depth () ()2 2 2 0 00 2 0 00 0 0: exp1/ 2 , exp exp cos sin 124 4 4 2 2RI RConsider the case where is small enough that k is given by ki i ici Then k i i i i kk nω σωμ ωπσωμ σωμ ωγσωμ ππ π πσωμ σωμ σωμ σωμ⎡⎤ ⎛⎞=+ ≅ =⎢⎥ ⎜⎟− ⎝⎠ ⎣⎦ ⎡⎤ ⎛⎞ ⎛⎞ ⎛ ⎞ ⎛ ⎞≅== + = +⎜⎟ ⎜⎟ ⎜ ⎟ ⎜ ⎟ ⎢⎥⎝⎠ ⎝⎠ ⎝ ⎠ ⎝ ⎠ ⎣⎦ == =% () ( ) ()2 0 0 0022 ,: exp exp exp expRI IR Rc ckn In the metal for a wave propagating in the z direction zEE k z i k z t E i k z tσμσ ωω ω ε ωωδ⎛⎞== =⎜⎟⎝⎠ − ⎛⎞=− − =− − ⎡⎤ ⎡⎤ ⎜⎟ ⎣⎦ ⎣⎦⎝⎠rr r 2 0 0 2 71 1 72 12: 5.76 10 5.76 10 0.66IcThe skin depth is given byk CsFor copper the static conductivity m mJm ε δδσωμ σω σδμ−−== = −=× Ω=× → =− Plasma FrequencyPlasma Frequency () () ()2 2 0 2 22 2 22 00 2 2 2 0 2 2 22 2 0: 1/ 111/ 1/ 1 : p eNow consider again the general case kici cc cink i ii ii cni The plasma frequency is defined Neccm σωμ ωωγσμ σμγ ωγωω γ ωω γ γσ μωω γ ωγ σ μγγ⎡⎤=+⎢⎥−⎣⎦ ⎧ ⎫⎧ ⎫ ⎪ ⎪⎪ ⎪== + = + ⎨ ⎬⎨ ⎬−−⎡ ⎤⎡ ⎤ ⎪ ⎪⎪ ⎪ ⎣ ⎦⎣ ⎦ ⎩⎭ ⎩⎭ =−+ ⎛⎞== ⎜⎟ ⎝⎠2 0 0 2 2 21e pNe m The refractive index of the medium is given by niμε ω ωω γ= =−+ 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 elect rons will overshoot and oscillate around their equilibrium positions with a characteristic frequency known as the plasma frequency . / ( ) / : electrostatic field by small char ge separation exp( ) : small-amplitude oscillation ()( ) 22 2 22 2so o o op sp p ooE Ne x x xx i t d x Ne Neme E mm dt σεδ ε δδδ ωδωωεε== =− =− ⇒ − = − ⇒ ∴ =Plasma Frequency Plasma Frequency (/ ) ()2 22 2 2 2 22 2111 1oo p RIic c cnki i nn i n i σμσ μ γ ωω ω γωωγ ω ωω γ⎛⎞== + = −⎜⎟− + ⎝⎠ =+ = − + by neglecting , valid for high frequency ( ). For , is complex and radiation is attenuated. For , is real and radiation is not attenuated(transparent).2 2 21p p pn n n ωγωγ ω ωωωω=− > > < >Plasma Frequency Plasma Frequency Plasma FrequencyPlasma Frequency Born and Wolf, Optics, page 627.pp cc ωπ λ λ2== Plasma FrequencyPlasma Frequency 2 2 2 2 22 22 22 3 2() () 1 () 2 1RI p RI RI R I ppin ni ni nn i n n i εω ε εω ωω γ ωω γ ωγω ω γ=+= =+ = −+ =−+ ⎛⎞ ⎛ ⎞ =− +⎜⎟ ⎜ ⎟⎜⎟ ⎜ ⎟++⎝⎠ ⎝ ⎠ Dielectric constant of metal : Drude modelτ γ ω1=>>22 23() 1/ppi ωω εωωωγ⎛⎞ ⎛⎞ =− +⎜⎟ ⎜⎟⎜⎟ ⎜⎟⎝⎠ ⎝⎠ Ideal case : metal as a free-electron gas • no decay (infinite relaxation time) • no interband transitions 2 0 2() () 1p τγω εω εωω→∞ →⎛⎞ ⎯⎯⎯→= −⎜⎟⎜⎟⎝⎠ 2 21p rω εω=− 0 Note: SP is a TM wave!Plasma waves (plasmons) Plasmo ns Plasmons in the bulk oscillate at ωpdetermined by the free electron density and effective mass Plasmons confined to surfaces that can interact with light to form pro pagating “surface plasmon polaritons (SPP)” Confinement effects result in resonant SPP modes in nanoparticles +++ --- +-+kPlasma oscillation = density fluctuation of free electrons 02ε ωmNe drude p= 02 31ε ωmNe drude particle= Dispersion relation for EM waves in electron gas (bulk plasmons) ()k ωω= • Dispersion relation: Dispersion relation of surface-plasmon for dielectric -metal boundaries Dispersion relation for s urface plasmon polaritons TM wave εmεd xmx dEE=ymy dHH=mz m dz dEE εε=• At the boundary (continuity of the tangential Ex, Hy,and the normal Dz):Z > 0 Z < 0 Dispersion relation for s urface plasmon polaritons zm zd mdkkεε= xmx dEE= ymy dHH= xm m ym zm E Hkωε= yd dzd ym mzmHkHkε ε=) ,0, (zii xii Ei Eiωε ωε− −) ,0, (yi xi yi zi Hik Hik− xii yi zi E Hkωε= xdd yd zd E Hkωε= Dispersion relation for surface plasmon polaritons • For any EM wave:2 22 2 ix z i x x m x d k k k , where k k kcω ε⎛⎞== + ≡ =⎜⎟⎝⎠ md x mdkcεε ωεε=+SP Dispersion Relation 1/2 '"md xx x mdkki kcεε ωεε⎛⎞=+ = ⎜⎟+⎝⎠x-direction: For a bound SP mode: kzimust be imaginary: εm+ εd < 0 k’xmust be real: εm < 0 So, 1/22 'i zi zi zi mdkki kcε ωεε⎛⎞=+= ± ⎜⎟+⎝⎠z-direction:2 22 zii xkkcω ε⎛⎞=−⎜⎟⎝⎠Dispersion relation:Dispersion relation for surface plasmon polaritons '" mm m i εεε=+ ' mdεε<−22 22 zi i x x i x ikk i kkcc cωωω εε ε⎛⎞ ⎛⎞ ⎛⎞=± − =± − ⇒ >⎜⎟ ⎜⎟ ⎜⎟⎝⎠ ⎝⎠ ⎝⎠ + for z < 0 - for z > 0 Plot of the dispersion relation 22 1)(ωω ω εp m−= md x mdkcεε ωεε=+• Plot of the dielectric constants: • Plot of the dispersion relation: dp spd m ωωε ω ε ε +=≡∞→⇒−→ • 1 , k , When x 2 22 2 ) 1() ( p dd p sp xck kω ω εε ω ω ω −+−== Surface plasmon dispersion relation: 2/1 ⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ += d mdm xckε εε ε ω ωωp dpε ω +1 Rekxreal kx real kz imaginary kx real kz real kx imaginary kzdxckε Bound modesRadiative modes Quasi-bound modes Dielectric: εd Metal: εm= εm'+ εm"xz(ε'm> 0) (−εd < ε'm< 0) (ε'm< −εd)22 2 2 p xck ωω=+Surface plasmon dispersion relation 1/22 i zi mdkcε ωεε⎛⎞=⎜⎟+⎝⎠