Handout 7. Properties of Bloch States and Electron Statistics in Energy Bands

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1 Hdout 7 Popts of Bloch Stts d Elcto Sttstcs Eg Bds I ths lctu ou wll l: Popts of Bloch fuctos Podc boud codtos fo Bloch fuctos Dst of stts -spc Elcto occupto sttstcs g bds ECE 407 Spg 009 Fh R Coll Uvst Bloch Fuctos - Su Elcto gs d solutos wtt s ( d Th solutos stsf th Bloch s tho: R R s stctd to th fst BZ): d c b wtt s supposto of pl wvs s show blow fo D: A lttc vcto d cpocl lttc vcto c b wtt s: R b b b olu of th dct lttc ptv cll d th cpocl lttc fst BZ : E c b b b ECE 407 Spg 009 Fh R Coll Uvst

2 ECE 407 Spg 009 Fh R Coll Uvst Bloch Fucto Poduct Fo Epsso A Bloch fucto cospodg to th wvvcto d g bd c lws b wtt s supposto ov pl wvs th fo: c Th bov psso c b -wtt s follows: u c c Wh th fucto s lttc podc: u u c c R u R stsfs Bloch s tho R R u ot tht: ECE 407 Spg 009 Fh R Coll Uvst Allowd Wvvctos fo F-Elctos (Sofld Modl) W usd podc boud codtos: Th boud codtos dctt tht th llowd vlus of d such tht: p = 0 ± ± = 0 ± ± p = 0 ± ± Th gd pots p ut volu of -spc

3 Bloch Fuctos Podc Boud Codtos b b A vcto Dct lttc th fst BZ c b wtt s: b b b Rcpocl lttc fo D lttc wh d g fo -/ to +/: ECE 407 Spg 009 Fh R Coll Uvst Bloch Fuctos Podc Boud Codtos Fo th D cstl : A Cosd D cstl d up of ptv clls th dcto ptv clls th dcto d ptv clls th dcto olu of th t cstl s: Assug podc boud codtos ll th dctos w ust hv: ECE 407 Spg 009 Fh R Coll Uvst

4 4 ECE 407 Spg 009 Fh R Coll Uvst Bloch Fuctos Podc Boud Codtos Th podc boud codto th dcto pls: : tht cll tg s b Sc: c hv dfft tgl vlus btw / d + / Rcpocl lttc fo D lttc Dct lttc b b b b b ECE 407 Spg 009 Fh R Coll Uvst Bloch Fuctos Podc Boud Codtos Sll th podc boud codtos th dctos of d pl: & & & c hv dfft tgl vlus c hv dfft tgl vlus c hv dfft tgl vlus Rcpocl lttc fo D lttc Dct lttc b b b b b Sc -vcto th FBZ s gv s: th dfft llowd -vlus th FBZ Th s dfft llowd -vlus th FBZ s th ub of ptv clls th cstl &

5 Rcpocl lttc fo D lttc b Dst of Stts -Spc Qusto: Sc b s llowd to hv ol dsct vlus how llowd -vlus th p ut volu of th -spc? D Cs: olu of th fst BZ s: b b b b b b I ths volu th llowd -vlus Th ub of llowd -vlus p ut volu -spc : wh s th volu of th cstl ECE 407 Spg 009 Fh R Coll Uvst Dst of Stts -Spc D Cs: gth of th cstl: gth of th fst BZ s: b I th fst BZ th llowd -vlus Th ub of llowd -vlus p ut lgth -spc : D Cs: A of th cstl: A A of th fst BZ s: b b I th fst BZ th llowd -vlus Th ub of llowd -vlus p ut -spc : A ECE 407 Spg 009 Fh R Coll Uvst b b b 5

6 D Cs: Rcpocl lttc s: Stts -Spc d ub of Ptv Clls Eg ˆ b gth of th cstl: gth of th fst BZ s: b I th fst BZ th llowd -vlus Th ub of llowd -vlus p ut lgth -spc : Th llowd -vlus -spc Th llowd -vlus p g bd Th s llowd -vlus p g bd s th ub of ptv clls th t cstl ECE 407 Spg 009 Fh R Coll Uvst D Cs: Stts -Spc d ub of Ptv Clls Eg Rcpocl lttc s: b b I th fst BZ th llowd -vlus (0 ) FBZ ( 0) Th llowd -vlus p g bd Th s llowd -vlus p g bd s th ub of ptv clls th t cstl ECE 407 Spg 009 Fh R Coll Uvst 6

7 D Cs: Sttstcs of Elctos Eg Bds Suppos I wt to fd th totl ub of lctos th -th bd how should I fd t? Th pobblt tht th qutu stt of wvvcto s th -th g bd s occupd b lcto s gv b th F-Dc dstbuto: f E E K T Th th totl ub of lctos th -th bd ust qul th followg su ov ll th llowd vlus - spc th fst BZ: f ll FBZ sp Th ub of llowd -vlus p ut lgth - spc s / thfo: d f f ll FBZ f Eg E E E ECE 407 Spg 009 Fh R Coll Uvst d to fd th totl ub of lctos th -th bd f D Cs: Sttstcs of Elctos Eg Bds ll FBZ Th ub of llowd -vlus p ut -spc s: A Thfo: d f A f ll FBZ FBZ D Cs: Th ub of llowd -vlus p ut volu -spc s: Thfo: d f f ll FBZ FBZ ECE 407 Spg 009 Fh R Coll Uvst 7

8 Bd Fllg t T0K fo D lttc Suppos th ub of ptv clls = Eg Qusto: suppos w hv lctos p ptv cll How wll th bds fll up t T0K? Wh wll b th F lvl? lctos p ptv cll totl ub of lctos E f fo 4 lctos p ptv cll ub of -vlus p bd = ub of qutu stts p bd = sp Fst bd wll b copltl flld All hgh bds wll b pt Qusto: Suppos w hv lctos p ptv cll How wll th bds fll up t T0K? lctos p ptv cll totl ub of lctos Fst bd wll b copltl flld Scod bd wll b hlf flld All hgh bds wll b pt E f fo lctos p ptv cll E f fo lctos p ptv cll ECE 407 Spg 009 Fh R Coll Uvst Bd Fllg t T0K fo D lttc Suppos th ub of ptv clls = Qusto: suppos w hv lctos p ptv cll How wll th bds fll up t T0K? Wh wll b th F lvl? lctos p ptv cll totl ub of lctos ub of -vlus p bd = ub of qutu stts p bd = sp Fst bd wll b copltl flld All hgh bds wll b pt Ipott lsso: I g bd (whth D D o D) th totl ub of qutu stts vlbl s twc th ub of ptv clls th dct lttc How th bds gt flld dpds o th ub of lctos p ptv cll ECE 407 Spg 009 Fh R Coll Uvst 8

9 F Sufcs (D) d Cotous (D) Solds Eg Eg Fst g bd of D lttc E F F FBZ F ccl fo f lcto gs D Wht hpps solds wh th g bds o copl? ECE 407 Spg 009 Fh R Coll Uvst F Sufcs (D) d Cotous (D) Solds c b ˆ ˆ c cˆ Eg FBZ Fst g bd of D lttc b ˆ c F cotous fo dfft lcto dsts cospodg to th g bd show o th lft FBZ FBZ ECE 407 Spg 009 Fh R Coll Uvst 9

10 F Sufcs (D) d Cotous (D) Solds F sufc of spl cubc dct lttc show sd th fst BZ F sufc of FCC lttc show sd th fst BZ (th fgu shows th F sufc of Copp) ECE 407 Spg 009 Fh R Coll Uvst Bd Fllg t T0K fo Slco Slco: Atoc ub: 4 Elcto Cofguto: s s p 6 s p ub of lctos th outost shll: 4 Th lctos th outost shll c ov fo to to to th lttc th ot cofd to dvdul to Th gs dscbd b th g bds Th lctos th shlls cofd to dvdul tos Slco lttc s FCC Th Slco tos p ptv cll ( bss tos) ECE 407 Spg 009 Fh R Coll Uvst Th 4 lctos cotbutd b ch Slco to d so th 8 lctos p ptv cll tht vlbl to fll th g bds 0

11 Bd Fllg t T0K fo Slco Th 8 lctos p ut cll vlbl to fll th g bds Rcll tht ch g bd th ub of stts vlbl s twc th ub of ptv clls th cstl I Slco th lowst 4 g bds wll gt copltl flld t T0K d ll th hgh g bds wll b pt E f FBZ (fo FCC lttc) Slco Eg Bds ECE 407 Spg 009 Fh R Coll Uvst Eg Bds Slco Th hghst flld g bd s clld th vlc bd I slco th vlc bd s doubl dgt t ost pots th fst BZ Th lowst pt g bd s clld th coducto bd I g th vlc bd u d th coducto bd u d ot hpp t th s pot -spc (s s th cs Slco) Th lowst g of th coducto bd s clld E c d th hghst g of th vlc bd s clld E v Coducto bd lc bd E f FBZ (fo FCC lttc) Slco Eg Bds ECE 407 Spg 009 Fh R Coll Uvst

12 Eg Eg FBZ ECE 407 Spg 009 Fh R Coll Uvst

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