Università Cattolica del Sacro Cuore A.A From weak periodic potentials to band structure

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1 nestà Cattola del Sao Cuoe A.A. 5-6 Fo wea peod potentals to band stutue Fsa dello Stato soldo - I Modulo Bande elettone

2 nestà Cattola del Sao Cuoe A.A. 5-6 One an gan substantal nsgt nto te stutue posed on te eleton enegy leels by a peod potental f tat potental s ey wea. Wen te peod potental s zeo te soluton to te Soednge uaton ae plane waes unpetubed states. A easonable statng plae fo te teatent of wea peod potental s teefoe te expanson of te exat soluton n plane waes. We now pode a seond poof of te Blo Teoe tat wll allow us to loo fo solutons of te Soednge euaton wt peod n tes of plane waes. Ts foulaton of te Blo teoe wll be useful to fnd a soluton wen an be teated as a petubaton. Ψ fee eletons wt and - d ell sne s eal * Fsa dello Stato soldo - I Modulo Bande elettone

3 Lets us substtute te fst two euatons n te Södnge euaton Te ognal poble s splt up n N ndpendent pobles one fo ea. ouples wt all wt -. Ts s te seond Poof of te Blo Teoe... ultplng by Ψ st B Z d p ε ε ε ψ ψ Bande elettone Fsa dello Stato soldo - I Modulo nestà Cattola del Sao Cuoe A.A. 5-6

4 Te ognal poble s splt up n N ndpendent pobles one fo ea. ouples wt all wt -. Fo a gen n te st BZ tee ae nfnte solutons to te syste of nfnte euatons we ae obtaned. Tese solutons ae labelled by n te band ndex. If we wte ts as ten ts s te Blo fo wt te peod funton u gen by: Ψ exp exp Ψ u Case s any lnea obnaton of fnte set of fo a If fee spae solutons fo any If Ψ Ψ Bande elettone Fsa dello Stato soldo - I Modulo nestà Cattola del Sao Cuoe A.A. 5-6

5 nestà Cattola del Sao Cuoe A.A. 5-6 Wat appens wen we onsde as a wea petubaton to te fee eleton beao? We an teat te Soednge euaton soluton wt statonay petubaton teoy. Two ases an be onsdeed.e. te non-degeneate and te degeneate ases see. Asoft and Men Capt.9. We now dsuss te degeneate ase. Wen te alue of s su tat tee ae epoal latte etos K..K wt all ε wtn ode of ea ote but fa apat fo te ote L ε K K eneges on te sale of ε... K K L ε >>... K K K te appoxate soluton to leadng oetons n an be sown to be: ε ε... K K j K j K K j Fsa dello Stato soldo - I Modulo Bande elettone

6 Te splest and ost potant exaple s wen two fee eleton leels ae wtn ode of ea ote but fa opaed wt fo all ote leels. At ½ we ae: -. We eep only te tes wt and - : We sole te seula euaton wt - / 4 ± Bande elettone Fsa dello Stato soldo - I Modulo nestà Cattola del Sao Cuoe A.A. 5-6

7 Bande elettone Fsa dello Stato soldo - I Modulo nestà Cattola del Sao Cuoe A.A. 5-6 Te enegy oeton due to te wea petubaton ae easly alulated at te Bagg plane. At te ½ pont n -spae we ae: -. and teefoe One te egenalues ae been extated we an alulate te oespondng egenetos toug te. We ae: and ± os sn sn os sgn Wen Wen < > ± ψ ψ ψ ψ sn -> p-le waes os -> s-le waes

8 nestà Cattola del Sao Cuoe A.A. 5-6 It s easy to efy tat wen s on te Bagg plane fo te euaton one obtans: ± 4.e. wen s on a Bagg plane te gadent of s paallel to te plane. / / -/ - / Sne te gadent s pependula to te sufaes on w a funton s onstant te ontant-enegy sufaes at te Bagg plane ae pependula to te plane. Fsa dello Stato soldo - I Modulo Bande elettone

9 nestà Cattola del Sao Cuoe A.A. 5-6 Nealy Fee leton Model Te pesene of a peod potental leads to te exstene of a fobdden enegy egon o band gap nea / / Fsa dello Stato soldo - I Modulo Bande elettone

10 nestà Cattola del Sao Cuoe A.A. 5-6 negy bands Te two uantu nubes n and ae ute dffeent eanngs. Te band ndex s an ntege. Te allowed alues of wae eto ae fxed by te bounday ondtons n te sold. Fsa dello Stato soldo - I Modulo Bande elettone

11 nestà Cattola del Sao Cuoe A.A. 5-6 D negy Band Fee eleton paabol enegy dspeson ue Dstoton of te fee paabola due to a wea peod potental xtended-zone see Redued-zone see Repeated-zone see Fsa dello Stato soldo - I Modulo Bande elettone

12 nestà Cattola del Sao Cuoe A.A. 5-6 negy bands n 3 Densons Fee eleton enegy leels fo an f Baas latte Fsa dello Stato soldo - I Modulo Bande elettone