15. Nanoparticle Optics in the Electrostatic Limit

Transcription

1 5 aopartcle Optcs the lectrostatc Lmt I the prevous chapter, we were restrcted to aospheres because a full soluto of Maxwell s equato was eeded Such a aalytc calculato s oly possble for very smple geometres However, f partcles become suffcetly small so that the spatal varato of the electromagetc feld ca be smplfed, more complcated shapes ca be appled Hece, ths chapter, we adopt the electrostatc lmt that s equvalet to gorg the spatal varato of the cdet electrc feld sde the partcle Fgure 5 Geometry of metal aopartcle havg a delectrc costat ε embedded a homogeeous medum wth a delectrc costat ε We cosder a aopartcle such as the oe depcted Fg 5 ad apply the electrostatc method preseted Ref [] I the electrostatc pcture, a electrc feld oly duces charges o the surface of the partcle Hece, the total electrostatc potetal Φ( r ) s the sum of a cdet part Φ( r ) ad the cotrbuto geerated by the surface charges If the surface charge desty s σ( r ), the potetal s gve by σ( r ) Φ ( r) =Φ ( r) + 4 πε ds, r r where the tegral s over the etre surface of the partcle We ca ow use ths to compute the electrc feld va the relato ( r) = Φ( r) I partcular, we wsh to compute the electrc feld o the surface I ths stuato, case has to be take because the ormal compoet s dscotuous The ormal compoet s gve by ( r) = e Φ( r), where e s the outward ut ormal vector at posto r The out dscotuty meas that the ormal compoet just outsde the partcle s related to the oe just sde va the relato out σ( r) σ( r ) ( r) = ( r) + ε ε

2 Usg these relatos, we fd that out / σ( r ) ( ) = + σ( ) (, ) r e ± 4πε r g r r ds, ε where + ad go wth the felds outsde ad sde the partcle, respectvely Also, s the (costat) cdet feld ad g deotes the so-called Gree s fucto r r gr (, r ) = e = e 3 r r r r () However, we also kow that the ormal compoets are related va the delectrc out costats, e ε ( r) = ε ( r) Ths meas that wth a bt of rearragemet, q (3) ca be reformulated as λ σ( r) = ελe + σ( ) (, ) π r g r r ds, () where λ= ( ε ε )/( ε+ ε ) Ths formulato s very coveet because t allows us to compute the dstrbuto of surface charges from a sgle equato 5 Cyldrcal aopartcles The framework above apples to aopartcles of completely geeral shape It eve works for collectos of aopartcles f the surface s take as the sum of surfaces I practce, however, q (5) s dffcult to solve the geeral case Fortuately, may mportat cases are much smpler I partcular, may relevat aopartcle geometres have cyldrcal symmetry, e they have a rotatoal symmetry axs as llustrated Fg 5 I ths case, the geeral problem ca be reduced sgfcatly Fgure 5 Cyldrcally symmetrc aopartcle

3 Usg the geometry of Fg 5, we ca wthout loss of geeralty choose to keep the cdet feld the (x,z)-plae The x- ad z-axes are the horzotal (h) ad vertcal (v) drectos, respectvely, ad so we decompose = + h v e x e z Due to the superposto prcple we ca, fact, treat the horzotal ad vertcal cases separately We the beeft sgfcatly from the smple agular depedece of these cases I a cyldrcal geometry, we may express the geometrcal vectors usg polar agles a smple maer, Hece, r = r(s θcos ϕ,s θs ϕ,cos θ) e = ( x cos ϕ, xs ϕ, z), (3) r = r (s θ cos ϕ,s θ s ϕ,cos θ ) where r, x, z are all fucto of θ ad r s a fucto of θ It follows that e s x cosϕ ad z the horzotal ad vertcal cases, respectvely It s the readly show that the surface charge follows exactly the same depedece o the agle ϕ Due to the symmetry, the surface area elemet ds must be depedet of ϕ ad we may wrte ds = S( θ ) dθ dϕ We wll retur to the θ depedece later We also eed the followg relatos to reduce the Gree s fucto: e ( ) ( cosθ cos θ ) ( s θ s θ cos( r r = z r r + x r r ϕ ϕ )) (4) r r = r + r rr cosθcosθ rr s θs θ cos( ϕ ϕ ) The smplest case s that of vertcal polarzato, for whch e s depedet of ϕ We wsh to prove that the surface charge s a fucto of θ oly ad so we wrte σ( r ) = σ( θ) The charge balace q(5) therefore reduces to where π ( v) Vertcal: σθ ( ) = ελ d ( θ) λ σθ ( ) ( θθ, ) ( θ z + G S ) θ, (5) G ( v) ( θθ, ) = π e ( r r ) d ϕ 3 π r r Performg the tegral usg the geometrcal relatos we the fd ( θθ, ) = ( cosθ cos θ) s θ, s θ, π + { z x,( ) x,( )} ( v) G r r r F x y r F xy, (6) 3

4 wth x= r + r rr cosθcos θ, y= rr sθsθ ad troducg the fuctos F π m cos ϕdϕ ( x, y), m, ( )/ ( x+ ycos ϕ ) + The frst few of these fuctos are F,( x, y) K x y F,( x, y) x y, 4 y = x + y + 4 y = ( x y ) x + y + 4 x+ y y 4x y F,( x, y) = K y x y y x y x y, x y 4 y F,( x, y) = K yy ( x) x y + x + y y x y x + y + + F 4y 8x y 8x y ( x, y) = K y ( y x) x y x y y x y x y where K ad are ellptc tegrals Hgher terms ca be geerated usg F = ( F xf )/y The fact that these fuctos deped oly o θ ad θ m, m, m, completes the proof For the horzotal case, we proceed almost complete aalogy but ow the surface charge s foud to follow the cosϕ behavour of e Hece, we wrte σ( r ) = σ( θ)cosϕ ths case ad usg elemetary mathematcal mapulatos (rewrtg cosϕ as Re{ ( ϕ ϕ ) ϕ e e ad dog the tegral before takg the real part) fd that } where π ( h) Horzotal: σθ ( ) = ελ d ( θ) λ σθ ( ) ( θθ, ) ( θ x + G S ) θ, (7) G ( h) ( θθ, ) = π e ( )cos( ϕ ϕ r r ) d ϕ 3 π r r ca be expressed as 4

5 ( θθ, ) ( cosθ cos θ) s θ, s θ, π { z x,( ) x,( )} ( h ) G = r r + r F x y r F x y (8) 5 Oblate Spherods As a relatvely smple but stll techologcally mportat example we wll cosder oblate spherods: pacake shaped partcles obtaed by flatteg spheres alog oe drecto Such a partcle s llustrated Fg 53 Fgure 53 Cross secto of a oblate spherod The geometry of the spherod s take such that the thckess of the pacake s uty ad the radus s d Hece, all dstaces are actually measured uts of half the partcle heght A partcular pot o the surface obeys the ellpse parameterzato x r = z + = r cos + s r= d d d θ θ cos d θ+ s θ Also, the surface ormal s calculated from the requremet that dfferetatg ad esurg ormalzato t s foud that e ( dr/ dθ) = Hece, x s θ d cosθ =, 4 z = 4 d cos θ+ s θ d cos θ+ s θ 3 I geeral, the surface areal fucto S( θ) s to be calculated as S( θ) = r s θ/ e r ad 4 / the spherod case we have S( θ) = d s θ(s θ+ d cos θ) /( s θ+d cos θ) To solve equatos lke q(55) ad q(57) umercally we eed to dscretze the agle θ O the terval θ [, π[ we therefore select dscrete values wth separatos Δ = (for = we take θ + θ Δ = π θ ) Hece, the equatos are reformulated as θ σθ ( ) ελ ( θ) + λ σθ ( ) G( θ, θ) S( θ) Δ, (9) j j j= j j 5

6 where the approprate Gree s fucto ad ormal vector compoet should be chose for the two polarzatos quatos of ths sort are easly coverted to a tractable form by troducg vectors as well as a matrx σθ ( ) ( θ) σθ ( ) ( θ) σ =, e = ε σθ ( ) ( θ ) ( θ, θ) ( θ) ( θ, θ) ( θ) ( θ, θ ) ( θ ) Δ G S Δ G S Δ G S ( θ, θ) ( θ) ( θ, θ) ( θ) ( θ, θ ) ( θ ) Δ G S Δ G S Δ G S G = ( θ, θ ) ( θ ) Δ ( θ, θ ) ( θ ) Δ ( θ, θ ) ( θ ) Δ G S G S G S I terms of these quattes, the dscretzed equato reads as { λ } σ, σ { λ } U G = e = U G e () Thus, the ukow surface charges vector σ are foud by vertg a matrx ad multplyg oto a kow vector Furthermore, t s realzed that certa egemodes of the surface charge ca be foud wheever the determat λ U G vashes Ths s because ths codto correspods to a stuato, whch a surface charge exsts eve wth a vashgly small cdet feld Ths s clearly a mathematcal abstracto but the sgfcace s that actual calculatos, resoaces absorpto or scatterg cross sectos may appear ear these egemodes From the form of the matrx t s also evdet that egemodes are foud wheever λ s a egevalue of G I practce, the matrx G s slghtly problematc because the dagoal elemets dverge! By clever usage of the geeral propertes of the Gree s fucto, however, approprate values of the dagoal elemets ca be foud (see the exercse) If the delectrc costat of the metal aopartcle s assumed to be of the lossless Drude form εω ( ) = ε ωp / ω the relato betwee egevalue ad resoace frequecy s gve by ω = ε + ω p ε λ λ + 6

7 As a example, we ow take Ag spherods ( ε = 6, ω = 93 ev) embedded S ( ε = ) For a partcular geometry, the spherod s characterzed by e= / p ts ellptcty d, whch rages betwee for a sphere ad for a plae I Fg 54, results for the resoace wavelegths ths case are depcted Fgure 54 Resoace wavelegths of Ag spherods embedded S Sold ad dashed curves llustrate vertcal ad horzotal resoaces versus aopartcle ellptcty The preset techque ca be exteded several drectos Prmarly, based o soluto of the homogeeous equato q(5), the absorpto cross secto s calculated va σ abs ω ( ω) Im p =, () cε ε where p s the duced dpole momet that s easly calculated usg the formulas π 4 r ( θ) Vertcal: p= πzˆ σ( θ) cosθs θdθ e r () π 4 r ( θ) Horzotal: p= πxˆ σ( θ) s θdθ e r I addto, the aopartcles postoed o a surface of embedded th layers ca be hadled by proper modfcatos to the Gree s fuctos [] 7

8 xercse: Propertes of the Gree s fucto a) Usg Gauss theorem, show that gr (, r ) ds = 4π b) Based o ths result, show that the reduced Gree s fucto for vertcal polarzato satsfes the codto π θθ θ θ= ( v) G (, ) S( ) d Ths result ca be used to hadle the sgularty of the Gree s fucto I a dscetzed verso, t reads as ( ) v G ( θ, θj) S ( θ) Δ= = It follows that the dagoal elemet must be G ( θ, θ ) ( θ, θ ) ( θ ) j S Δ Δ ( ) ( ) = ( θ ) v v j j G S j j =, j For the horzotal case, ufortuately, o such smple result apples Hece, specal umercal hadlg of the dagoal terms s eeded Refereces [] ID Mayergoyz, DR Fredk, ad Z Zhag, Phys Rev B7, 554 (5) [] J Jug, TG Pederse, T Sødergaard, K Pederse, A yladsted Larse ad B Bech else lectrostatc plasmo resoaces of metal aospheres layered geometres, Accepted Phys Rev B 8