# D zone schemes

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1 Ch. 5. Enegy Bnds in Cysls 5.. -D zone schemes Fee elecons E k m h Fee elecons in cysl sinα P + cosα cosk α cos α cos k cos( k + π n α k + πn mv ob P 0 h cos α cos k n α k + π m h k E Enegy is peiodic funcion of k wih peiodiciy of (π/

2 Disconinuiies of he enegies when cos(k hs mximum o minimum, cos k ± k nπ Peiodic zone scheme - If wo enegy funcions wih equl symmey coss, qunum mechniclly he enegy eigenfuncions e spli, so h hey do no coss. - Elecons in cysl behve like fee elecons excep when k ppoches (nπ/.

3 Ref. Inoducion o Solid Se Physics (Kiel, p. 9

4 Reduced zone scheme secion beween ±π/ Bnd gp Vlence bnd Conducion bnd Exended zone scheme Deviions fom fee elecon pbol e esy o idenify 4

5 Fee elecons in educed zone scheme E π h ( k + m h n 5

6 me h p h π k λ k hs he uni of ecipocl lengh nd is defined in ecipocl spce. Lice el spce, ecipocl spce Ech lice plne in el spce cn be epesened by veco which is noml o his plne nd whose lengh is popoionl o he ecipocl of he inepln disnce. Such vecos fom he poins in ecipocl lices. X-y o elecon diffcion pens is mp of such ecipocl lices 6

7 5.. Billouin Zones The egion in k-spce beween -π/ nd π/ : fis Billouin zone The egion in k-spce beween -π/ nd -π/ o beween π/ nd π/ : second Billouin zone - Individul enegy bnches e peiodic in π/. - By shifing second BZ o lef/igh by π/, one cn obin educed zone scheme. - Do no need ll E-k cuves in ll BZ. - All infomion is ledy conined in he s BZ in educed zone scheme due o π/ peiodiciy. 7

8 In -D. k Veco k x k y Consucion of BZ: s BZ: consuc pependicul bisecos on he shoes ecipocl lice vecos (G nd BZ: second shoed ecipocl lice vecos (G Wigne-Seiz cell Fis fou BZ All he BZ hs he sme e 8

9 - When elecon vels 45 o o he k x -xis: The BZ boundy is π k ci Mximl enegy is E h π π h mx ( m m - Enegy bnds in diffeen diecions in k-spce - When elecon vels pllel o k x (k y -xis: The BZ boundy is π k ci Mximl enegy is E h π π h mx ( m m - When he mximl enegy hs been eched, he elecon wves fom snding wves (elecons e efleced bck ino he BZ. no bndgp, mel 9

10 Elecon wve popging in lice n ngle θ. Bgg lw: consucive inefeence sinθ nλ π π sinθ nλ n k ci n k sinθ A ciicl ngle, he efleced ys will be enhnced considebly. Ciicl k vlues fo eflecion of elecons occu π - When elecon vels θ45 o : k ci - When elecon vels θ0: k π ci This is hppening zone boundy -45 o o he k x -xis: BZ boundy is k ci π h π ( m π h m E mx - 0 o k x (k y -xis: BZ boundy is k ci π h π ( m π h m E mx 0

11 Sudy cysl sucue by diffcion of phoons, neuons, elecons Wh bou visible ligh? Ref. Inoducion o Solid Se Physics (Kiel, p. 8

12 5.. -D BZ 5.4. Wigne-Seiz cells 5.5. Tnslion vecos nd ecipocl lices Cysl Uni cell Pimiive uni cell: smlles possible uni cell, no unique Convenionl uni cell: no pimiive cell, bu esy o ecognize he symmey Wigne-Seiz cell: specil ype of pimiive uni cell (simil o wy o consuc s BZ

13 Lice pimiive uni cell sc: xˆ yˆ zˆ bcc: ( / ( xˆ + yˆ + zˆ ( / ( xˆ yˆ + zˆ ( / ( xˆ + yˆ zˆ z z fcc: ( / ( yˆ + ˆ ( ˆ ( / x + zˆ ( / ( ˆ ˆ x + y z z Any lice poin, R n + n + n n, n, n : inege

14 pimiive uni cell bcc fcc 4

15 Wigne-Seiz cell bcc fcc 5

16 fcc Convenionl uni cell of fcc sucue sc sucue 6

17 n n n R + + n n n : inege Pimiive veco n n n R + + n, n, n : inege Recipocl veco Recipocl veco b h b h h b G + + h, h, h : inege bcc b π bcc ( b h b h h b G + + π In ex, b π b b π 7 volumn cell pimiive uni

18 i b j πδ ij R G π ( h n + h n + h n Konecke-Del δ ij if i j 0 if i j b π 8

19 ecipocl lice pimiive uni cell ecipocl lice pimiive uni vecos fo sc lice sc wih cubic side of π/ ecipocl lice fo bcc lice fcc wih cubic side of 4π/ ecipocl lice fo fcc lice symmey bcc wih cubic side of 4π/ 9

20 Now, one cn consuc Wigne-Seiz pimiive cell fo ecipocl lices. This cell is efeed o he fis BZ. diec lices ecipocl lices fcc bcc 0

21 5.6. Fee elecon bnd Cysl: Peiodic lices Bloch wvefuncion in he fom of plne wve muliplied by peiodic funcion (peiodiciy of lice consn is he soluion of Schődinge equion in peiodic poenil. Bloch heoem ψ k ( exp( ik uk ( v u ( u ( + R R : diec lice k k ψ k ( + R exp( ik ( + R uk ( + R exp( ik R ψ ( k This ells h he elecon wvefuncion in ny pimiive uni cell of he diec lice diffes fom h in ny ohe cell by fco exp( ik R

22 Fo wveveco h is equl o ecipocl lice veco, ψ ( + R exp( ig R ψ ( G When n elecon hs wveveco ψ k ( + R exp( i( G + k ' R ψ k ( exp( ik ' R ψ ( k G k G R G π ( hn + hn + hn Elecon wvefuncions e peiodic in R. k G + k ' The wvefuncions ψ k obey Bloch heoem s if hey hd wveveco. k ' Thus, we cn define wveveco uniquely by educing i wih he ppopie ecipocl lice veco o he fis Billouin zone.

23 All infomion is conined in he s BZ. The enegy E(k fo k ouside he s BZ is idenicl o enegy E(k wihin s BZ. k ' k + G Fee elecon bnds in -D h m E k ( k + G ' Fee elecon bnd in bcc cysl ( s BZ Γ [0,0,0] H [,0,0] N [,,0] P [,,] bcc s BZ

24 Γ-H diecion k x vies beween 0 nd π/ ( s BZ boundy E h π ( x xˆ + G m x vies beween 0 nd G 0 E π ( x xˆ C x m π h h C ( m G π hb + hb + hb (h,h,h (0,-,0 G ( xˆ + zˆ E h π π [ x xˆ ( xˆ + zˆ] m C [( x + ] C [ x x + ] x 0 E C X E C C [( x xˆ zˆ] Fee elecon bnd in bcc cysl ( s BZ G h b + hb + hb π (h,h,h (,-, G ( x ˆ + y ˆ z ˆ..? 4

25 Fee elecon bnd in fcc cysl ( s BZ Γ [0,0,0] X [,0,0] K [,,0] L [,,] fcc s BZ 5

26 5.7. Bnd sucues fo some mels nd semiconducos Aluminum enegy bnd sucue (fcc - Simil o fee elecon bnds (Γ-X, Γ-K - Suggesing elecons in Al behve fee-elecon like. -bndgp beween X4 nd X, bu individul bnds ovelp in diffeen k-diecion, so h hee is no bndgp: mel! - he lowe pbol shped bnds e ssocied wih Al s elecons (behves fee elecon like Al: s s p 6 s p 6

27 Coppe enegy bnd sucue (fcc -lowe fl unning bnds: due o d-bnds -Cu: s s p 6 s p d 0 4s - bnd ss Γ s s-elecon like, becomes d-elecon like ppoching o X. - no bndgp: mel 7

28 Silicon enegy bnd sucue (fcc - bndgp: semiconduco popeies - indiec bndgp semiconduco 8

29 GAs enegy bnd sucue (fcc - bndgp: semiconduco - III-V compound semiconduco - diec bndgp semiconduco 9

30 5.8. Cuves nd plnes of equl enegy - in -D, hee is only one +k vlue fo given enegy - in -D, hee e moe hn one k vlues fo given enegy s BZ fo -D lice (fo smlle enegy equl enegy: cicles Deviion fom cicul shpes BZ boundies 0

31 Cu s BZ (fcc nd equl enegy sufce

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