Drude Model for dielectric constant of metals.
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1 Drue Moel for electrc constant of etals. Conucton Current n Metals EM Wave Proagaton n Metals Sn Deth Plasa Frequency Ref : Prof. Robert P. Lucht, Purue Unversty
2 Drue oel Drue oel : Lorenz oel (Haronc oscllator oel) wthout restoraton force (that s, free electrons whch are not boun to a artcular nucleus) Lnear Delectrc Resonse of Matter
3 Conucton Current n Metals The equaton of oton of a free electron (not boun to a artcular nucleus; C 0), r r r r r uur r uur e r v e Cr ee e + eγ v ee t τ t t 1 14 ( τ : relaxaton te 10 s) γ Lorentz oel (Haronc oscllator oel) If C 0, t s calle Drue oel The current ensty s efne : r r J Nev wth unts of C s Substtutng n the equaton of oton we obtan : r J r N e r + γ J E t e
4 Conucton Current n Metals Assue that the ale electrc fel an the conucton current ensty are gven by : r r r r E E ex t J J t ( ) ex( ) 0 0 Local aroxaton to the current-fel relaton Substtutng nto the equaton of oton we obtan : r J0 ex( t) r r r + γ J0ex t J0ex t + γ J0ex t t Ne r E0 ex( t) e ( γ) ( ) ( ) ( ) ( + ) Multlyng through by ex t : r Ne r + J0 E0 e r Ne r or equvalently ( + γ) J E e
5 Conucton Current n Metals For statc fels ( ) 0 we obtan : r Ne r r Ne J E σ E σ statc conuctvty eγ eγ For the general case of an oscllatng ale fel : r σ r r J E σ E σ ynac conuctvty 1 ( / γ) ( γ) For very low frequences, << 1, the ynac conuctvty s urely real an the electrons follow the electrc fel. As the frequency of the ale fel ncreases, the nerta of electrons ntrouces a hase lag n the electron resonse to the fel, anthe ynac conuctvty s colex. ( ) For very hgh frequences, γ >> 1, the ynac conuctvty s urely agnary an the electron oscllatons are 90 out of hasewth the ale fel.
6 Proagaton of EM Waves n 1 E 1 E σ E ( / γ) Metals Maxwell ' s relatons gve us the followng wave equaton for etals : r r r J c t c t P 0, J 0 r σ r But J E 1 ( / γ) Substtutng n the wave equaton we obtan : r r r E E c t c t The wave equaton s satsfe by electrc fels of the for : r r r E E0 ex r ( t) where σμ 0 + c c 1 ( / γ) 1 μ 0 0
7 Sn Deth Conser the case where s sall enough that s gven by σμ 0 π + σμ0 ex σμ0 c 1 ( / γ) 0 Then, % π π π π σ μ ex σμ0 ex σμ0 cos + sn σμ0 ( 1+ ) : σμ0 R I nr σ c μ σ c 0 R n I 0 In the etal, for a wave roagatng n the z recton : r r r z E E0ex( Iz) ex ( Rz t) E0ex ex ( Rz t) δ The sn eth δ s gven by : 1 δ σμ I 0 0c σ C s For coer the statc conuctvty σ Ω δ 0.66μ J
8 Plasa Frequency Now conser agan the general case : σμ + c 1 ( / γ) 0 c σ c μ 0 γ σ c μ 0 n ( / γ) γ 1 ( / γ) n γσc μ + γ 0 1 The lasa frequency s efne : γσ μ Ne γ c 0 c eγ μ 0 Ne e 0 The refractve nex of the eu s gven by n 1 + γ
9 Plasa Frequency If the electrons n a lasa are slace fro a unfor bacgroun of ons, electrc fels wll be bult u n such a recton as to restore the neutralty of the lasa by ullng the electrons bac to ther orgnal ostons. Because of ther nerta, the electrons wll overshoot an oscllate aroun ther equlbru ostons wth a characterstc frequency nown as the lasa frequency. E σ / Ne( δ x) / : electrostatc fel by sall charge searaton δ x s o o o δ x δ x ex( t) : sall-altue oscllaton o ( δ x) Ne Ne ( e) E s t o o
10 Plasa Frequency n c σ c μo σc μoγ 1+ 1 ( 1 / γ) + γ R I n ( n + n ) 1 + γ n 1 by neglectng γ, val for hgh frequency ( >> γ). For <, n s colex an raaton s attenuate. For >, n s real an raaton s not attenuate(transarent).
11 Plasa Frequency λ λ c πc Born an Wolf, Otcs, age 67.
12 Plasa Frequency
13 Delectrc constant of etal : Drue oel ( ) + n R I + ( nr ni) 1 + ( nr ni ) nrni + γ γ γ + γ >> γ 1 τ ( ) / γ
14 Ieal case : etal as a free-electron gas no ecay (nfnte relaxaton te) no nterban transtons ( ) ( ) 1 τ γ 0 r 1 0
15 Plasa waves (lasons) Note: SP s a TM wave!
16 Plaso ns Plasa oscllaton ensty fluctuaton of free electrons Plasons n the bul oscllate at eterne by the free electron ensty an effectve ass rue Ne Plasons confne to surfaces that can nteract wth lght to for roagatng surface lason olartons (SPP) 0 rue 1 Confneent effects result n resonant Ne artcle 3 0 SPP oes n nanoartcles
17 Dserson relaton for EM waves n electron gas (bul lasons) Dserson relaton: ( )
18 Dserson relaton of surface-lason for electrc-etal bounares
19 Dserson relaton for surface lason olartons TM wave Z > 0 Z < 0 At the bounary (contnuty of the tangental E x, H y, an the noral D z ): E x Ex Hy Hy Ez Ez
20 Dserson relaton for surface lason olartons z z x x E E y y H H x y z E H y z y z H H ),0, ( z x E E ),0, ( y x y z H H x y z E H x y z E H
21 Dserson relaton for surface lason olartons For any EM wave: x + z x x x, where c SP Dserson Relaton x c +
22 Dserson relaton: relaton for surface lason olartons x-recton: z-recton: For a boun SP oe: x ' x+ " x c + z x c z ust be agnary: + < 0 1/ + ' " z ' z + z ± c + 1/ z ± x ± x x > c c c x ust be real: < 0 + for z < 0 - for z > 0 So, ' <
23 Plot of the serson relaton 1 ) ( x c + Plot of the electrc constants: Plot of the serson relaton: s + 1,, When x ) (1 ) ( s x c +
24 Surface lason serson relaton: x c + 1/ z c + 1/ + c x c x Raatve oes (' > 0) real x real z Quas-boun oes ( < ' < 0) agnary x real z 1+ z Delectrc: x Metal: ' + " Boun oes (' < ) real x agnary z Re x
6. Dispersion relation of surface plasmons on dielectric-metal boundaries
6. Disersion relation of surface lasons on ielectric-etal bounaries Surface lasons (Gary Wieerrecht, Purue University) Definitions: collective ecitation of the free electrons in a etal Can be ecite by
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