UvA-VU Master Course: Advanced Solid State Physics
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1 UvA-U Maste ouse: Advaced Sold State Pyscs otets 005: Dffacto fo peodc stuctues wee 6, Ad lectoc bad stuctue of solds wee 7, Ad Moto of electos ad taspot peoea wee 8, Ad Supecoductvty wee 9&10, RW Magets wee 11&1,JB Ae de sse Re Wjgaade Jüge Buscow
2 Lteatue, softwae ad oewo Te couse s based o te boo: H. Ibac ad H. Lüt: Sold State Pyscs 3 d edto Spge-elag, Bel, 003 ISBN X See also: N.W. Ascoft ad N.D. Me: Sold State Pyscs Saudes ollege Publ. ISBN opute sulatos fo a essetal pat of te couse: R.H. Slsbee ad J. Däge: Sulatos fo Sold State Pyscs abdge Uvesty Pess, abdge 1997 ISBN Softwae feewae: Hoewo execses wll be dstbuted tougout te couse opletg te couse gves 6 TS ~ 6 x 8 ous
3 Pctues ae tae fo te Sold State ouse by Ma Jael cat Uvesty, fo te boo of Ibac ad Lüt, fo te boo of Ascoft ad Me ad fo seveal souces o te web. ouse : lectoc bad stuctue of solds u u e u
4 ouse : lectoc bad stuctue of solds Rede: oupled quatu wells Rede: Fee electos solds ffect of te lattce potetal eeal syety Bloc states Te ealy-fee electo appoxato Te tgt bdg appoxato xaples: Fe sufaces / Bad stuctue Pctues ae tae fo te Sold State ouse by Ma Jael cat Uvesty, fo te boo of Ibac ad Lüt, fo te boo of Ascoft ad Me ad fo seveal souces o te web.
5 Rede: oupled QWs; 1- ad -ato electo states d 4 H [ ] Scödge eq. dx x 0.5
6 ffect of lattce spacg o electo states a0.7 a1 a d dx x 4 a 0.5 x 4 a 0.5
7 N states fo N atos states 4 states
8 Rede: Fee electos solds Aold J.W. Soefeld Fee electo gas Paul excluso pcple Quatu Fe-Dac dstbuto N electos volue L 3 Haltoa wt potetal 0 Peodc bouday codtos Popagatg electo waves :,, :,,,, : 1 : : z y x z z y y x x geeeges L L L wave vecto opoets z y x L z L y L x codtos Peodc bouday e soluto eeal equato Scödge π π π fxed bouday codtos
9 Fe spee -space Fe-Dac dstbuto: µ s cecal potetal F F B F F F F F v velocty Fe T tepeatue Fe N wave vecto Fe eegy Fe : / : 3 : : 1/3 π 1 ] / exp[ 1 4 / T v v f B FD µ π fxed bouday codtos peodc bouday codtos
10 ffect of te lattce potetal X Axs Y Axs Fe gas costat potetal 0 Fe gas peodc potetal
11 Solve Scödge eq. wt potetal wt potetal as Foue sees ad geeal soluto Scödge eq. ecpocal space couples oly wt -, -, -, etc. fo eac -vecto ut cell N pobles dex egevalues by : eeal syety Bloc states H e e 0 dect lattce vecto ecpocal lattce vecto
12 soluto fo eac Foue sees ove ecpocal lattce pots soluto s odulated plae wave Bloc teoe Bloc waves Peodc Peodcty owledge 1st Bllou zoe s suffcet e e e u u e u e u H
13 xaple Bloc wave peodc lattce potetal Bloc wave u e eal pat Bloc fucto u plae wave e eal pat
14 Te ealy-fee electo appoxato: epty lattce case 0; syety eques: 1 D case; π/a fo vasg potetal epty lattce educto to 1st BZ , , D cubc case; evolvg paabola s coplex beavou 1st BZ 000
15 Degeeacy: At ZB π/a electo state volves at least two values: 0, π/a plae waves fo exact soluto ay solve: lagest cotbutos fo 1 fo wc: - Bagg codto ase of wea 1 ; 0 ; 0 1 >> ote / /,, x x x x e e e e
16 ase of wea Stadg waves: supeposto of cog ad Bagg eflected wave ~ e ~ e x / x / e e x / x / ~ cosπ x ~ sπ x a a ρ ρ * * ~ cos ~ s π x π x a a eegy < - gap pobablty desty gap desty of states
17 Lagest devato fo fee electo odel at ZB we - Bagg codto eta lagest tes wt ad - wt At ZB gap eeal teatet [ ] ± ± 0 0
18 egy dspeso 1 D lattce exteded zoe scee Bagg codto at BZ bacscatteg destoys electoc states N depedet states eac eegy bad
19 xaple - xecse osde a squae lattce two desos wt a bacgoud potetal x x, y 0 cos cos e a - x,y dstubs te syste - gaps ae foed at te tesecto of fee electo bads π πy a Set up a x secula eq. to calculate te badgap at te pot π/a, π/a of te Bllou zoe solve Scödge eq.: H H 0 x, y ε
20 - cystal electo state s lea supeposto of atoc obtals - potetal A of fee ato at - stat fo atoc poble wt valece egestates ϕ ad eegy upetubed Scödge eq. fo ato clude petubato due to atoc potetal of all ote atos Te tgt-bdg appoxato A H ϕ ϕ A A v v H H A v
21 Te tgt-bdg appoxato atoc X Axs Y Axs Y Axs
22 H H tas: solve Haltoa solve fo φ wt wt φ s Bloc state: φ φ wte φ as lea cobato of atoc obtals ϕ e a e a ; ϕ ϕ φ d v e N ] [ 1 *, ϕ ϕ N d d e * *, ϕ ϕ ϕ ϕ φ φ eglect teacto: clude oly eaest egbou teacto ϕ - 1 ϕ -
23 Sple cubc case: - sotopc ybdzato - ϕ specal syety, s-wave wt A B * ϕ * ϕ A B v ϕ d v ϕ d te case of sple cubc lattce e oly eaest egbou of cubc tgt-bdg lattce A Bcos a cos a cos a x y ovelap tegal B ± a,0,0 ; 0, ± a,0 ; 0,0, ± a z - A, B >0; bad cete -A - wdt of te bad 1B - fo sall ea Γ-pot: wt x y z A 6B Ba
24 Sple cubc case: A Bcos a cos a cos a x y z tgt-bdg bads 1B A 1 atoc levels 1, A 1B 1 - eegy wdt lage as ovelap ybdzato ceases - N electos pe bad
25 xaples: Fe sufaces / Bad stuctue D squae lattce exteded zoe scee fee electo gas wt lage Fe spee Bllou zoes educed zoe scee ole le electo le FS 3d zoe peodc zoe scee
26 D squae lattce effect of lattce potetal
![Moovalet etals Alal etals: L 1s s 1, Na [Ne]3s 1,K [A]4s 1, Rb [K]5s 1, s [Xe]6s 1 bcc lattce: desty /a 3 F3 /3π F 0.](/docs-images/93/111820746/images/27-0.jpg "6 π/a ΓΝ π π 1 1 0 0.")
27 Moovalet etals Alal etals: L 1s s 1, Na [Ne]3s 1,K [A]4s 1, Rb [K]5s 1, s [Xe]6s 1 bcc lattce: desty /a 3 F3 /3π F 0.6 π/a ΓΝ π π a a 3 3 4π F F ube of states 3 3 8π 6π N desty of electos 6π 3 F 3π 3 F Fe spee cotaed fst BZ F
![4s 1 Au [Xe]4f 14 3d 10 4s](/docs-images/93/111820746/images/28-1.jpg "1 Badstuctue of coppe fcc")
28 Moovalet etals Noble etals: u [A]3d 10 4s 1 Ag [K]4d 10 4s 1 Au [Xe]4f 14 3d 10 4s 1 Badstuctue of coppe fcc lattce: 6 bads s bad fee electo-le FS d bad fee electo bads
29 A tetavalet fcc fee electo etal bcc fcc Fe spee 1st d 3d exteo sufaces Bllou zoes ole-le electo-le Fe sufaces fo fcc lattce educed zoe scee
30 valece 1-4 valet fee electo sufaces 1 ole le FS electo le 3 4 Fe sufaces fo fcc lattces educed zoe scee
31 xaple badstuctue: 5d etal tugste bcc lattce DOS F bcc BZ 3d bad 4t bad
32 aote ce Fe suface webste: ttp://
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