T3 Fooui-loome alias mopous Dispesio Fomula Spetosopi ellipsomety (S) is a teique based o te measuemet of te elatie pase ae of efleted ad polaized lit i ode to aateize ti film optial futios ad ote popeties. Te measued data ae used to desibe a model wee ea laye efes to a ie mateial. Te model uses matematial elatios alled dispesio fomulae tat elp to ealuate te tiess ad optial popeties of te mateial by adusti speifi fit paametes. Tis appliatio ote deals wit te oiial Fooui & loome dispesio fomula also ow as «mopous» (ame used i te DeltaPsi softwae). We eommed te use of te updated «ew mopous» fomulatio, please efe to te oespodi teial ote. Fooui-loome alias «mopous» teoetial model Te Fooui-loome fomulatio (986, 988) is appliable to amopous semiodutos ad dieletis ad is based o te quatum-meaial teoy of absoptio. It taes ito aout te optial bad ap i te ite-bad eio. It is supposed tat te alee ad odutio bads ae paaboli ad ae sepaated by a fobidde bad wose widt is. Peas tat a be see i te optial spetum oespod to tasitios of eletos betwee two states. Fo a amopous mateial, a sile pea is peset i te optial spetum wi efes to te tasitio betwee te bodi state i te alee bad ad te ati-bodi state i te odutio bad. Fo ystallie semiodutos dieletis ad metals, seeal peas a be obseed, idiati te tasitios oui betwee te itial-poit i te alee ad odutio bads. I te itebad eio, metals, semiodutos ad dieletis ae simila stutue i tei optial futios. Tus, te same pysial poesses ae ioled fo tese tee types of mateials. Optial popeties deped o ite-bad tasitios of eletos tat ae elated to poto absoptio. Te equatio of α(ω) below deies fom te quatum expessio of te absoptio. quatio of α(ω) ΔI ω α( ω) lim x Δ I x Δ I Φ( ω) ω wee: - (ω) is te total umbe of ways a poto, of eey ω is emoed fom te iidet itesity i a uit olume ad a laye of ifiitesimal tiess Δx; (ω) () aies wit te type (amopous, dieleti, metalli) of te mateial osideed. - Φ(ω) is te tasitio pobability ate at wi a poto is absobed fom te iidet itesity i te fequey ae [ω ; ω dω]. -I is te iidet poto itesity. Wit Φ(ω) defied as: Te equatio of Φ(ω) is dedued fom fist-ode timedepedat petubatio teoy fo diet ad fiite eleto tasitios. Φ(ω) depeds o te dipole positio matix elemet squaed betwee iitial fial states ad also o te lifetime of te exited state tou te followi equatio. Fo a amopous mateial, te expessio of Φ(ω) is ie by: π Φ( ω) e ω I 3 TIODIG ODIG * x ( * ω) η() ODUTIO D VL D Fi. Obital states of te eeies of te eeies ad * (Ref. ) (.)
T3 wee: - e is te eleto ae - is te lit speed i auum - is te Pla ostat -ω is te poto eey - is te eey of te iitial state - * is te eey of te fial exited state - x is te dipole matix elemet squaed betwee te bodi ad ati-bodi states. I te ase of ystallie semiodutos, dieletis ad metals, Φ(ω) is te sum oe te umbe of types of tasitios. wee: - te sum oe efes to te sum oe te umbe of obseed peas assoiated wit te tasitios. - Wit (ω) defied as: (ω) is popotioal to te umbe of possible tasitios fo a ie poto of eey ω to be emoed. (ω) depeds o te podut of te umbe of oupied eleto states i te alee bad times te umbe of uoupied eleto states i te odutio bad. Te desities of states i te alee ad odutio bads ae assumed to ae a paaboli sape. Teefoe, (ω) is witte as wee: π Φ( ω) e ω I 3 {[ ( ) ( )] ω} it top - η st te alee bad. it it x it K (.) it x it is te dipole matix elemet squaed betwee te i t itial-poit state i te alee bad ad te i t itial-poit state i te odutio bad. ( ω) d η ( ) η ( ω)( 3.) / ( 3.) is te desity of states i bottom - η st 3.3 states i te odutio bad. - te bad ap eey is Fo amopous ad ystallie mateials te fial expessio of a (ω) is Fo metals, m(ω) as te followi expessio: quatio of (ω) Kowi te equatios of (ω), Φ(ω) ad α(ω) it is te possible to detemie (ω): y eleti all te seod ode poesses ad eplai all a (ω) ad Φ a (ω) by tei espetie expessios te oespodi extitio oeffiiet is deied tou tese equatios: Te fial equatio fo te extitio oeffiiet of amopous mateials is witte as: wit / a bottom is te desity of ( 3. ). Te fial expessio fo te extitio oeffiiet of ystallie mateials is witte tis way: top ( ω) ( ω) st ( ω ) ( 3.5) m ( ω) st ( ω) ( 3.6) ω I ( ω) α( ω) Φ( ω) ( ω) (.) ω ( ) st x ( ) ( ) ( ) ; fo ; fo > (.3) ; fo > ; fo (.) (.)
wee, ad ae ie by: Te fial expessio fo te extitio oeffiiet of metals is witte as: wee, ad ae ie by: ystallie semiodutos ad metals may ae seeal peas peset i te stutue of te extitio oeffiiet wi eeals te pesee of seeal osillatos. Te umbe of osillatos detemies te ame of te dispesio fomula as sow below: quatio of (ω) Te Kames-Koi (K-K) elatio is used to detemie te efatie idex tou a Hilbet tasfom of (): wee P is te auy piipal alue otaii te esiduals of te itead at poles loated o te lowe alf of te omplex plae ad alo te eal axis. Te deiatio of K-K yields te fial expessio fo te efatie idex: I te ase of a amopous mateial: wee I te ase of a ystallie semiodutos ad metals: wee: Fomula umbe of osillatos umbe of paametes mopous 5 Double mopous 8 Tiple mopous 3 uate mopous.5 it it it it it it x st.6 fo ; fo ; >.7 it it it it it it x st 5. d P π 5. ε 5.3 5. ε 5.5 T3
T3 Te paametes of te equatio Paamete desibi te efatie idex Te tem ε() is a additioal paamete oespodi to te i-fequey dieleti ostat. It is at least supeio to oe ad equal to te alue of te dieleti futio we ω. t least paametes desibe te extitio oeffiiet.,, ae positie o zeo paametes efei to te eletoi stutue of te mateial. Fo ystallie semiodutos, dieletis ad metals te subsipt (,, 3) fo te paametes,, efes to te umbe of osillatos. (i ev) is elated to te dipole matix squaed ad ies te stet (amplitude) of te extitio oeffiiet pea. Due to oelatio betwee ad, te ie is te ie te absoptio pea ad te smalle its widt is. Geeally, < <. / (i ev) is appoximately te eey at wi te extitio oeffiiet is maximum (pea of absoptio). s te alue of ieases te absoptio pea is sifted towads te UV eio. Geeally, 3< <3. (i ev ) is elated to te boadei tem of te absoptio pea. It depeds o te eey diffeee betwee diffeet states ad o te lifetime of tasitio. If tee is o lifetime otibutio i te expessio of te /. Te lae is te lae te absoptio pea but te smalle its amplitude. Geeally, 3< <5. (i ev) is te eey bad ap. It is equal to te miimum of eey equied fo a tasitio fom te alee bad to te odutio bad. It is te eey fom wi te absoptio stats to be o-zeo: ( ). < fo all ases. Limitatio of te model Te Fooui-loome dispesio as some dawbas due to te eeatio of upysial esults. Tus aodi to Jelliso & Modie (Ref. ): Te Fooui-loome model auay is ate bad below te bad ap. fte te fit, seeal eey aps i te Fooui- loome equatios mit be eatie. Te limit of () st as is iosistet wit wat is pedited by expeimets, tat is (). Fooui-loome does ot espet time eesal symmety beause () (-). Te ad paametes ae ofte oelated. Paamete set up ote tat: - Te aps below sow te diffeet otibutios (i ed dased lies) to te imaiay pat of te tiple mopous dieleti futio (ed bold lie). - Te si befoe a ie paamete meas tat eite te amplitude o te boadei of te pea is lied to tat paamete. mopous futio Optial popeties of a-si ie by te mopous futio Tiple mopous futio Optial popeties of lq3 ie by Tiple mopous futio
T3 ppliatios to mateials Te amopous model wos patiulaly well fo amopous mateials exibiti a absoptio i te isible ad/o FUV ae (absobi dieletis, semiodutos, polymes). Mateials followi te mopous model Te asteis efes to paametes tat ae eatie ad tus do ot * ae ay pysial meai but epeset ood stati alues to pefom te fit o te mateial. Mateials ε? S. R. (ev) l.7.66 7.67 8..89.5-3 ligap 5.5.9 6.7.8.6.75-3 a-.75.9 7.3 8.63.3.5-3.5 a-gas.766.7 5.87 9.65.8.6-3 a-si 3.53.865 6.73 3.37.96.6-5 a-si:h 3.7.9 6.67.7.7.6-5 a-sige.37.63 8.85 9.5 -.89*.8-3 Si 3.569.5 3.9 8.69.99.6-6 TaOx 3.3.9.95 3.53 3.8. - 5 TiOx.7.7 8.3 7.799 3.9.6-5 WO x 3.96.6 7.756 5.75.895.5 -.5 ZSe 5.5. 6.7.8.6.6-3 Mateials followi te Tiple mopous model Mateials Gas 8.68.8 6.8.37 GaP 3.967. 7.96 5.957 Ge 5.8.3 6.789.69 IGaP.97.6 3.395 3. -Si.778.5 7.36 3.65 Mateials 3 3 3 Gas.6 9.79..8 GaP.9.9 6.699.5 Ge.3 8.7 9.6. IGaP. 8.33 6.93.98 -Si.53 8.63 8.87.7 Mateials S. R. (ev) Gas.5.969.6-3 GaP.79.58.6-3 Ge 9.997 5.5.5-3 IGaP 3.8 39.6.6-3 -Si.3 6.859.6-3 Refeees.. D. Pali, Hadboo of Optial ostats of Solids II, ap. 7... R. Fooui, I. loome, Pys. Re. 3, 78-76 (986). 3.. R. Fooui, I. loome, Pys. Re. 38, 865 (988).. G. Jelliso, F.. Modie, Pys. Re., 69(3), 37-37 (996). Mateials ε? lq.36.39.5 9.38..6 Mateials 3 3 3 S. R. (ev) lq 6.79 9.93.3.8 3..75 - Mateials followi te uate mopous model Mateials ε? Gas.3.53. 5.87 GaP.8.93. 7.7 Ge.879.78..56 IGaP.996.96. 6.68 -Si 5.66.65. 6.86 Tis doumet is ot otatually bidi ude ay iumstaes - Pited i Fae - /6