Lattice planes. Lattice planes are usually specified by giving their Miller indices in parentheses: (h,k,l)

Transcription

1 Ltte ples Se the epol ltte of smple u ltte s g smple u ltte d the Mlle des e the oodtes of eto oml to the ples, the use s ey smple lttes wth u symmety. Ltte ples e usully spefed y gg the Mlle des petheses: h,,l

2 Cheml Appoh I ode to udestd the og of some popetes of semodutos t ws eessy to wt the det of qutum mehs. We osde two toms eh hg oe eleto. As the toms e ought togethe the eletos oud eh of the ule wll eg to feel the potetl of the othe ule. hs potetl s petuto whh lfts the degeey moe d moe s dste etwee toms s deesed. If we osde ystl wth toms, whe the toms e ey lose the degeey s lfted ut seel eegy leels of the sme toms e mxed.

3 Eleto Peod Potetl: : Bd heoy he os pefet ystl e ged egul y, the we e led to osde the polem of eleto peod potetl U: U U e the Bs ltte etos λ el de Bogle welegth Og of U: U V o V el me dpedet eleto ppoxmto eleto-eleto teto s ot luded me feld e - o lulte V el me V o. s omplted, we just ssume tht U hs the sme peody of

4 Shödge Equto he Shödge equto fo sgle eleto s: h m U ψ Eψ fee eleto s spel se wth U0 Idpedet eletos, eh of whh oeys oe-eleto Shödge equto wth peod potetl, e ow s Bloh eletos. he sttoy sttes of Bloh eletos he the followg ey mpott popety s geel osequee of peodty of the potetl U: the egesttes of oe-eleto mlto wth peod potetl e hose ψ exp u u Bloh s heoem u whee fo ll the Bs ltte

5 Bloh s s heoem ote the mpltos: Bloh s heoem ψ exp u ψ exp u exp exp u exp ψ he pots d thus he the sme physl popetes, the futos dffeg oly y phse fto dpedet of. Cosequetly ψ exp ψ exp u exp exp u the t esults tht u u he Bloh heoem s oously tue fo empty ltte U 0 ψ exp

6 Bloh Bloh s heoem s heoem Poof of Bloh s theoem Fo eh Bs eto we defe tslto opeto f f. Se the s peod: ommutes wth 0. d ommute the ppltos of two tsltos does ot deped fom the ode. fo ll Bs ltte etos d fom set of ommutg opetos.

7 Bloh Bloh s heoem s heoem heefoe fom fudmetl theoem of qutum mehs tht the egesttes of e hose to e smulteous egesttes of. he e hose to stsfy smulteous egesttes of ll the Popetes of the egelues of the tslto opetos: Let us wte wth, thee pmte etos fo the Bs ltte. We lwys wte the the fom hs s equlet to: E eh fo the fo sutle hoe of exp x x π

8 Bloh s s heoem poded tht x x πδ j exp, j x Summzg, we he show tht we hoose the egesttes of so tht fo eey Bs lte eto, exp hs s the Bloh s heoem

9 Bo-o Km Boudy Codto Whh e the possle lues of? By mposg ppopte oudy odtos o the we futos we demostte the must e el d e t odto esttg the llowed lues of. Bo-o Km oudy odtos smulte fte sold wth fte smple otg ut ells.,, B.. Usg the Bloh s heoem: B K exp exp,, If x the oe eq. eomes expπ x m wth heefoe x o m tegl m

10 Bo Bo-o Km Boudy Codto o Km Boudy Codto he olume Δ of -spe pe llowed lue of olume of pmte ell epol ltte he oe equto ssets tht the ume of llowed we etos pmte ell of the epol ltte s equl to the ume of stes the ystl. Volume of pmte ell the epol ltte s π / whee V/ heefoe Δ Δ V π π Δ

11 Bloh we Bloh we s popetes s popetes Bloh s wes e lelled y qutum umes: we eto eflets the tsltol e of the mlto d-dex eegy leels t ge e dsete _ geelzes p htest of the tslto symmety the se of peod potetl lwys e ofed the th B.Z. ths euse y ot t the th B. Z. e wtte s G f G elogs to the epol ltte. the dex ppes the B.. euse fo fo ge thee e my solutos to the Shödge Eq. emt egelue polem wth fxed olume -> desete set of solutos le ptle---ox polem exp exp exp G wth the oudy odto exp u u u E u U m u u E h

12 Bloh we s s popetes s otuous pmete the Shödge equto ut llowed lues e ge y Bo Km oudy odto. he futos E e lled ds. hey e peod the epol spe: E G E > they he mmm d mxm

13 Bloh we s s popetes he th B. Z. s D spe. Bds e ofte plot log les of ptul symmety, fom 0 to spef oudes o log oudes. Spef detos d pots te spef mes te fom goup theoy. All futos wth the podty of the ltte ludg u, U he Foue tsfom whh ots oly the G s: f f f f exp f exp f G G exp G Bloh s wes e sttoy sttes tme-dpedet of the Shödge equto wth peod potetl. heefoe, the uet s ot degded y the fxed ltte os fxed os e ot soue of stteg, otst to ele theoes of metls Dude. hose theoes, howee, wo well poded stteg s tepeted s due ot to os, ut to deto fom pefet peodty > me fee pth >>.

14 Fem Eegy Fee eletos: E_ /m whee E F s detemed y equg tht the totl ume of oe-eleto leels wth eeges less th E F to e equl to the totl ume of eletos Smly Bloh eletos: E, wth ofed to sgle pmte ell of the epol ltte. Isultos/semodutos A et ume of ds e ompletely flled, ll othes e empty. Flled d empty ds e septed y the d gp. Metls Some ds e ptlly flled. E F les wth oe o moe ds. All suh tht E E F osttute the Fem sufe. he Fem sufe fo fee eletos s sphee.

15 Fem Eegy Why the Fem eegy s so mpott? j E h d e 4π E h Fo flled ds j 0 se the tegl oe pmte ell of the gdet of peod futo m*>0 must sh. Metls heefoe, flled ds e et. Coduto s well s my othe eleto popetes of solds s due oly to eletos ptlly flled ds. I metls o eegy s equed fo oduto whle fo semoduto mmum eegy s equed Semodutos