Ch. 8: Electron Levels In Periodic Potential

Transcription

1 C. 8: Elto Lvls I Pod Pottal Wat a t oss of a pod pottal o t fo of t wav ftos sptv of t xat fo of t pottal ad sptv of wt t atal s a odto o a slato? s t fftv olto pottal. Blo s To: T gstats of t o-lto altoa wt a pod pottal a Bavas latt a os to av t fo of a pla wav ts a fto wt t podty of t Bavas latt. O valtly st Poof Of Blo s To Df taslato opato T : altoa s pod Bt also f f T T T T T T T T T S ad all T s ot wt o aot gstat of a os to gstat of all T s. T a

2 st Poof Of Blo s To T Lt x a π a a a ptv vto a oplx x x x π π π x x x To s t ostats o w d to tod oday odtos. Bo vo aa Boday Codto a PBC t a a x π x x x x / tg vol p allowd stat x / Δ V ll π Δ sa as f lto gas!

3 d Poof of Blo s To Pla Wav Expaso Wog wt t Bo-vo aa pod oday odto w xpad ot t olto wav fto ad t fftv pottal pla wavs s a all t wav vtos allowd y t PBC t s a oly poal latt vtos as t pottal s pod. d V ll ll Plggg to Sodg s. to loo fo g stats * S s fo a oplt otogoal st offt fo a spf pla wav o LS st al offt fo t sa o t S. - Pla Wav Expaso ot. T C offt fo pla wav wt a spf s opld to t offts fo all wav vtos latd to y w s a poal latt vto ad s o-zo. ot tat a always os sftd so tat. Pla wavs ot dffg y a poal latt vto a latd. O f a pla wav s to opl t a oly opl to ot pla wavs wt wav vtos dffg y fo ts ow. Ga all pla wavs wt wav vtos latd to ad fo a sst lt spa tat s dpdt of ot ssts. ow a pla wav a oly opl to wavs wt ts ow sst. To dstgs ssts w a dtfy t y Fo a spf ts ato as o-tval soltos oly fo a fw s o fo o. Wat dos t a pysally? If vass fo lag s say >G? If vass fo a ss of s.g. G w s a odd tg? op o w vs w s ow ss. o d s g s ss s w d y y t o wav vto a sst tat logs to t fst Bllo zo.

4 4 Pla Wav Expaso ot. T s a ft of pla wavs a sst as t of poal latt pots s ft. owv t of ssts s ft of allowd of t lls t Bo vo aa ox. Wt a sst t aov atx a dagoalzd to yld a ft of dst gvals... Ea gstat s oposd of a sst of pla wavs fo o sst wt t pop C offts w satsfs t Sodg ato Wat a w say aot t C s f t pod pottal s wa? Egstats wt sa t dfft a ad to otogoal to a ot. Ptato toy tlls s tat fo a tal ag.. gog ssts t st allowd gvals tat a tal agd fo a ot. Ts stats a vwd as fog ads ad s t t ad dx. Pla Wav Expaso ot. Loog at t Sodg ato ad stg tat Ψ a gstat wt gval w s tat oly a vy sall of ssts old av as o of t gvals. A spf Ψ a xpadd pla wavs fo ts fw dpdt ssts wt a spf allowd gval. a s a dpdt ad atay a δ fw ts a W av as ay os as t a ts t s to df gstats. Ctaly ay gfto a os to av oly ottos fo o.

5 5 d Poof of Blo s To ay g fto of a pod latt all ts pod wt latt Vloty of Blo Eltos... * d Bt w also ow tat... v d * ow tat

6 6 Aot Blo s To. Allowd wav vto follow sa PBC as f l t t l t / ltos t s o log ot/ a.. a always ofd to t fst Bllo zo.. ad dx 4 patd zo pstato 4. patd zo pstato 5. vloty of lto 6. Flld ads pty ads patally flld ads ad F sfa. Dsty of Lvls 8 Q d V Q Q O s oft tstd sg atts fo opd dvdal lto lvls. I t lt of lag ystals t sato a l d t l ov Bllo zo 8 π plad y a tgal. If t tstd atty oly dpds o tog gy t t tgal a ad ot Q g d V Q 4 d g g δ π Cosd -gy sfas poal spa. Fo sall og gy dff w gt ds d g δ 4 d g S δ π d δ 4 ds g S π

7 7 va ov Sglats va ov sglats D: g s ft. owv slop dvgs dvgs. C. 9: Eltos I Wa Pod Pottal Assg wa pod pottal so tals a asoal statg pot fo gstats of odto ltos s pla wavs. Wy do w v xpt t pod pottal xpd y so odto ot-sll ltos to wa so tals? Ot sll ltos a fodd fo og los to t o os. Otsd t os pottal s fftvly sd y ol ltos. Blo s to Wa Pottal: s ~ << typal

8 8 Ext Cas: F Eltos I t as of ltatly wa pottal f lto gas... Fo f ltos o aly f ltos gy s latd to t pla wav vto a ovos way. Iovt to s ad dx. Covt to s xtd zo s.. fll wav vto. dgat as a a s a dpdt ad atay aly F Eltos: o-dgat o Fst o-dgat ass y w w a all fo >> Cotat o gstat dvd fo - pla wav Expt gs ad wav ftos to oly slgtly odfd fo FEG as. fato of / sall ta - sg ov all o To fd w wt t topost. gval dvd fo - ot - o

9 9 o-dgat FE To od of O... opd fo last pag sall O Sft gy s to od of. plld y ot gy lvls. Ptato Toy Evaltly w old av sd ptato toy ad wold av gott t asw... > < > < o o... > > > w /

10 aly F Eltos: -Fold a-dgay a dgay as: >>... I o-dgat as t s fo t pla wav opot to t t pla wav opot to t stat dvd fo - s I as of a-dgay s a ot ssaly sall. Eg stats a ot ssaly doatd y o pla wav opot v fo vy wa pottal. Dgat FE fo pla wav xpaso d t t dstgs two gops O...

11 Dgat FE O... O To fst od oly d to osd t aly dgat stats To fst od oly d to osd t aly dgat stats. Egval volvs wt wav vto. W dgay os fo ta - t os fo a sall vol aod tat pot t -spa. Splttg a Sgl Bagg Pla Oly two vtos ad lads to a-dgay wt of t - lvl a a sgl Bagg pla. aly ot - o o-fold dgat. Coos s tat. fo >> < * Wy a dgat FE stats always o so Bagg plas?

12 Sgl Bagg Pla Fo o-tval soltos to fod * ± ± Fo lyg t Bagg pla Addtoal Cots: Sgl Bagg Pla 4 ± 4 Gadt ls t Bagg pla. Costat gy sfa pp. to pla. statoay stat stadg wav ± sg Ass to al pottal as vso syty g ad low ad

13 at of Stats ± sg > s os Io os at a/. p-l / os ~ / / s ~ / Dto of popagato? < os s Io os at a. s-l p Egy Bads I D I o dso o dgay g ta -fold s possl. Extdd Zo S dd Zo S patd Zo S

14 D Exapls FEG gy ads. ot dgay. Egy gaps at Bagg plas. T a as ay -stats a Bllo zo as t a lls t Bo vo aa ox. Two ltos sp p ad dow to a -stat. At lto p t ll t lowst ad sold xatly alf-flld o so xt ass aot alf-flld. E Bllo Zos Ad F Sfas E ot Daw t f lto F sfa Dfo t slgtly a Bagg pla Taslat t poto wt -t Bllo zo tog all poal latt vtos. Ts ostts a patd zo F sfa. 4

15 Sap of -t Bllo Zos F Elto F Sfa val 4 5

16 6 FCC Val ad Latts wt Bass Idtal Atos d φ ll d d v φ Gotal Stt Fato φ * S v φ spa d d v φ d S Stt Fato d spa φ φ o splttg o Bagg plas wt vasg stt fatos.g. asal plas of p tal. Covt to s Jos zos : zogap plas god.

17 Ot Cottos To Dgay oval ξ L S Sp-ot oplg Sp-ot oplg s potat fo avy lts ad fo dgay ot ovd y wa pod pottal. owo Capt 8 - Capt 8: Pols Capt 9: Pols Capt : Pol D /6/ 7