Condensed Matter Physics - Introduction

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1 Poblem of ondutivity: Dude model m ee m (L) E is the elxtion time. t e t (L) Stedy stte, times muh longe thn : e m (L3) Cuent theefoe is j ne ne m ne m E (L4) (L5) is the eletil ondutivity 3, Mihel Mde

2 Poblem of ondutivity: Dude model m ne n 3 55 m 3 3 s m (L6) Exeise:. Estimte typil vlue of.. How does this ompe with te t whih lssil theml eletons stte off nulei? 3. How does this ompe with te t whih eletons t Femi veloity stte off nulei? 4. Wht hppens if one stts ove nd tes the elxtion time popotionl to the eleton veloity? 3, Mihel Mde

3 Bloh s solution 3 Peiodi funtion u exp i u Single ptile in peiodi potentil U: U U (L7) Solve 3, Mihel Mde

4 Bloh s solution 4 P m U (L8) WONG: (L9) Cn see this is wong fom se U e i (L) 3, Mihel Mde

5 Tnsltion opetos 5 Let T tnslte wve funtion by T e ip h (L) Theoem: if one hs olletion of Hemitin opetos tht ommute with one nothe, they n be digonlized simultneously Suppose hs uniue eigenveto with eigenvlue. so is eigenveto of ; by uniueness, must be some onstnt times In se of degenete eigenvlues, one opeto my tegoize futhe sttes of othe; pity. Use theoem: T e ip h C. (L) (L3) C (L4) 3, Mihel Mde

6 Tnsltion opetos 6 e i C (L5) eithe C ei o (L6) : Bloh wve veto h: Cystl momentum 3, Mihel Mde

7 estte s o 3, Mihel Mde Bloh s Theoem 7 T n n n n ei n (L7) (L7b) n ei n (L8) u n e i n (L9) u u nd n ei u n (L)

8 Enegy Bnds 8 LMTO Plne wve 4 Enegy (ev) (A) -8 - L X W L (B) L X W L u h m i u U u u (L) 3, Mihel Mde

9 Allowed vlues of 9 If ystl is peiodi with (mosopi) dimensions M M M3 3 then euiing exp i to be peiodi onstins to 3 l m l M l b l m lm l (L) b b 3 b l l ll (L3) Peiodi boundy onditions ple ondition on how smll n be. Demnding tht C exp i be uniue ples onditions on how big n be. Numbe of points in ystl euls numbe of uniue Bloh wve vetos. 3, Mihel Mde

10 Billouin Zone L Q X W Z U S X 3, Mihel Mde

11 Density of Sttes b b b 3 M M M 3 (L4) 3 b b M M M 3 (L5) M M M (L6) 3 (L7) F d F (L8) Define D s befoe: D n d n (L9) 3, Mihel Mde

12 3, Mihel Mde Dynmil impotne of enegy bnds n h n (L3) Wve pet: W t d e i i t h e i (L3) e i i t h d e i t h e i (L3) e i i t h t h (L33)

13 d d 3, Mihel Mde Vn Hove Singulities 3 D d (L34) d (L36) (L35) D mx (L37) D d Ld d (L38) D ln o (L39)

14 Vn Hove Singulities 4 D fo o else (L4) fo else d 3, Mihel Mde

15 Vn Hove Singulities 5 Phonon density of sttes Model Expeiment Feueny (THz) 3, Mihel Mde

16 Uniueness of Bloh vetos 6 e i ei (L4) it follows tht n n (L4) n eiei n (L43) edued zone sheme Extended zone sheme 3, Mihel Mde

17 Uniueness of Bloh vetos 7 epeted Zone edued Zone Extended Zone 3 3, Mihel Mde

18 Expliit onstution of Bloh funtions 8 e i ei (L44) d e i U unit ell d e i U e i (L45) e i U (L46) wheeis the volume of the unit ell, nd U unit ell d e i U (L47) is volume of unit ell e i N (L48) 3, Mihel Mde

19 3, Mihel Mde Expliit onstution of Bloh funtions 9 d e i U U (L49) U e i U e i U (L5) Peiodi boundy onditions imply e i (L5) d e i (L5) U e i (L53) e i e i U (L54)

20 3, Mihel Mde Expliit onstution of Bloh funtions d e i U e i U (L55) (L56) U (L57) e i (L58) U (L59)

21 Stutue of eutions 3 M 3 M M MM 3 M 3 M M MM 3 M 3 M 3, Mihel Mde

22 onig Penney model U x (L6) U U (L6) U (L6) Q (L63) Then E. (L6) beomes U Q (L64) Note fom its definition (63) tht Q Q (L65) 3, Mihel Mde

23 onig Penney model 3 U Q (L66) U Q (L67) Q U Q (L68) U S (L69) 3, Mihel Mde

24 m m m m h m h m 3, Mihel Mde onig Penney model 4. U h U h S -5 5 U U h h

25 h m h m 3, Mihel Mde 3 onig Penney model 5 8 Extended Zone 3 8 epeted Zone 8 edued Zone h m 3 3

26 Billouin zones nd ottionl symmety 6 L Q U S X W Z X In units of 4 3 3, 4, nd U, X 4 4., L, W, 3, Mihel Mde

27 Billouin zones nd ottionl symmety 7 P D F G N H In units of, H, N nd P. 3, Mihel Mde

28 Billouin zones nd ottionl symmety 8 b 3 A L S S H U T P M T b b In units of 4, A L. 3, 4 3, nd, M, long the thee pimitive vetos, 3 3, H 3 b, b, nd b 3 ; 3, nd 3, Mihel Mde

中华人民共和国时期$1952年$

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