Letue 8 Optal oesses 7. henomenologal Theoy 7. Optal popetes of Cystals 7.. Maxwell s quatons 7.. Delet Funtons 7..3 Kames Kong Relatons 7.3 Optal oesses n semondutos - Det and Indet Tanstons Refeenes: Buns, Chapte 3 Kttel, Chapte, pp. 7-77, 3 Zman, Chapte 8. 4 Ashoft & Memn, Chapte 7 5 Made, Chapte - Letue 8 Lght and letomagnet Spetum Compae the length of the pobe and extatons: Vsble lght: letomagnet adaton wth wavelength 4-75 nm λ >> dmensons of the unt ell λ << dmensons of sample dmensons Inteaton of hν s aveaged ove many unt ells Ultavolet : 4 nm Infaed: 75 6 nm Lght quanta - photon, haatezed by photon waveveto λ, λ Letue 8
8.henomenologal Theoy Inomng lght eletons osllate letons esponse depends upon nteatons wthn a sold Smple models: - effetve mass - delet funton Letue 8 3 Delet olazaton Assume that the sold has no net maosop hage It s omposed of postvely and negatvely haged enttes When elet feld appled, these hages move, ntodung a dpole moment, µ Cumulatve effet of the moton an be summazed by polazaton, µ pe unt volume µ V u. ell o V whee ρ s the hage densty, n p - numbe of n q u u the dsplaement of the hage due to, dsete hages Letue 8 4
Delet suseptblty In geneal the dsplaement of hage an be wtten n sees n : µ α + β + γ 3 +... If elet feld ae small that the hghe tems an be gnoed, the delet suseptblty χ s defned as: When vayng elet feld s appled, a uent densty, J, s defned as J t χ Fom Gauss theoem the aveage elet feld s elated to the polazaton: 4 Ths esult leads to the defnton of a new feld, D, alled the elet dsplaement D + 4 and D Letue 8 5 Delet onstant Sne, D delet onstant, D + 4χ Fnally, the ondutvty, σ, onnets J to Ohm s Law: J σ ffets of ystal symmety: - χ,, σ ae tensos n geneal they onnet two vetos; - when the pnple axes of χ,, σ onde, the poblem of popagaton of lght s smplfed Letue 8 6 3
4 Letue 8 7 Osllatng Feld When the felds,, vay n tme, a phase shft between and ae onsdeed χ χ χ + when ~, polazaton ompletely adjusts to the appled feld χ χ when, thee wll be a phase shft χ χ + χ Smlaly, we defne a omplex delet onstant : D 4 + + A omplex ondutvty, σ, an be defned: J σ σ σ + Reall: t J J χ χ χ Letue 8 8 7.. Maxwell s quatons Classal Long ange nteatons between vaous patles nto ontnuous funtons t 4 t B 4 + j B B en Dstngush hages outsde the system n ext and nsde n nt Fo hages nsde: substtutng plane wave equatons nto Maxwell s equatons, we get, t t e e D Reall: b a b a b a
Optal onstants It s ommon to use omplex ndex of efaton, n, defned as: n n + n In fequeny egons whee and n s eal, / s eal; the phase veloty s n The attenuaton oeffent fo the ntensty loss of a popagatng wave s di I z I e αz α o o I dz n / o n / α σ α 4 Note: the wave s attenuated as long as n > thee s a loss of powe that aompanes the attenuaton, o n >, n ; so σ, attenuaton wthout powe loss e.g., n metals. When the wave s attenuated by efleton Letue 8 9 Refletvty at an ntefae Useful expemental method to detemne the fequeny dependene of the optal onstants s to measue the efletvty of a sample as a funton of fequeny Reall: - Fesnel s law onnets values of the ampltude of the efleted and efated waves at bounday between two meda z n na n + n - Refletvty, R, fo nomal ndene depends on the squae of the felds and R By measung the fequeny dependene of the attenuaton oeffent, n an be obtaned next n an be obtaned fom the fequeny dependene of R a n + n n + + n Letue 8 5
7.. Mehanal Osllato as Delet Funtons Ths mpotant elatonshp ae shown by a smple model ntended to mm sold wth one optal absopton band Absobng goup may be a haged eleton vbatng on a spng The foe onstant of the spng gves se d d m + mγ + m e dt dt to a vbatonal fequeny Inteaton wth the suoundng damps the osllaton wth a haatest deay tme τ Fo a dlute olleton of N suh osllatng eletons n the gas phase o n soluton the absopton oeffent s gven by: Ne τ α om o + τ Wdth of the lne ~/τ Letue 8 Delet Funtons Consde N osllatos pe unt volume n a sold; the eletostat alulaton leads to the expesson fo the delet funton Regon of stongest absopton oesponds p + to pea n at < o p 3 τ Ne p m eal < magnay gh efletane at hgh w Tansmsson at long wavelength sueeded by an absopton band, oespondng to band gap Letue 8 6
7 Letue 8 3 Mehanal Osllatos as Delet Funton Chaatest shapes of the eal and magnay pats of the delet funton desbed plasma fequeny equaton: 4 4 γ γ γ + + + mv N e n n mv N e n n Letue 8 4 7..3 Kames Kong Relatons Assumng an osllato ente n a sold model, and an be alulated The Kame-Kong elatonshp elates the eal and the magnay pats of the delet onstant ' ' ' ' ' ' ' d d Cauhy Cauhy Cauhy - the Cauhy value : the sngulaty n the ntegal at omtted fom ntegaton
Band Stutue and Temnology Semondutos and nsulatos have qualtatvely smla band stutues, wth the quanttatve dstnton that the band gap g > 3. ev n nsulatos. onduton band valene band onduton band edge v valene band edge negy band dagam n - spae Flat-band dagam n eal spae Letue 8 5 7.3 Optal Absopton by Semondutos xamne the followng alulated 3-D band stutues fo semondutos S and GaAs. S GaAs Indet band gap gap Det band gap gap Letue 8 6 8
Optal Absopton and Consevaton Laws Absopton of a photon by a semonduto an pomote an eleton fom the valene to the onduton band, but both enegy and momentum must be onseved: + hν f ν photon fequeny v v v v + ν photon waveveto f ν Fo semondutos g ev so the photon waveveto an be estmated: ν p m h h 3. m / s.5 Js 9 / ev.6 J / ev 6 5 8 34 But ths s uttely tny ompaed to a typal BZ dmenson: So essentally we have: v v f A det-gap vetal tanston Letue 8 7 Optal Absopton: xpemental Results xpemental absopton oeffents α ae measued to be: α d α g hν g h q hν α A ν h / d g h α T A T ν g Whle fo an ndet gap mateal wth a det tanston at a slghtly hghe enegy: α g hν hq Letue 8 8 9
Det vs. Indet Gap Semondutos But fo ndet gap semondutos t s lea that: v v f So fo an ndet gap tanston momentum an only be onseved by absopton o emsson of a phonon latte vbaton + hν ± hq f q phonon fequeny v v v + q v q phonon waveveto f To estmate a typal phonon enegy, we now: q 4 s Letue 8 9 Aonyms of uent expemental methods fo band stutue studes AS LS XAFS IS STMS VsUV abs US XS XAS Auge eleton spetosopy leton enegy loss spetosopy xtended x-ay absopton fne stutue Invese hotoemsson Spetosopy Sannng Tunnelng Mosopy Spetosopy Vs./ UV absopton Ultavolet photoeleton spetosopy X-ay photoeleton spetosopy X-ay absopton hoton eleton n, eleton out, nne shells leton n/out; Conduton eletons hoton n; flled bands leton n, photon out; mpty Levels letons n, onduton and valene bands hoton n; band gap, defets hoton n, eleton out, flled bands hoton n, eleton out, flled bands hoton n, empty levels Kttel, p. 38 - modfed Letue 8