Lctur contnts Bloch thorm -vctor Brillouin zon Almost fr-lctron modl Bnds ffctiv mss Hols
Trnsltionl symmtry: Bloch thorm On-lctron Schrödingr qution ch stt cn ccommo up to lctrons: If Vr is priodic function: p m V V r R V r R m m m33 Th solution is : ir u whr u r is priodic function: u u r R From: Linrity of th Schrödingr qution Fourir thorm Importnt : u might b not singl vlnc lctron function but is clos to linr combintion of vlnc lctron wvfunctions Qusi-wvvctor is nlogous to wvvctor for fr lctrons V=const r R
Bloch thorm: consquncs 3 Introducd -vctor quntum numbr for priodic potntil to numrt stts Momntum is not consrvd not quntum numbr, howvr qusi-momntum is consrvd -vctor cn b considrd to li in th first Brillouin zon Solution with priodic boundry conditions givs ign-functions u n, for givn which forms orthogonl bsis compr with Fourir pnsion n vlus numrt bnds lctron occupying lvl with wvvctor in th bnd n hs vlocity compr to group vlocity p m ir u m i v u V r u r R n m u n,, n, V u n n u
Rciprocl spc D 4 Wvfunction of n lctron in crystl : ir u D fr lctrons bnd structur is: m -b u u u i ' r ir ibr ir ' ' ', D rciprocl lttic vctor : b m priodic function nd bnd First Brillouin zon:
Dimond or zinc-blnd structurs 4G + 4As=8 toms in cubic unit cll + = toms in primitiv unit cll 5 Primitiv unit cll in rcipricl spc st Brillouin zon b m b mb m3b3 Brillouin zon FCC: 3 b, b, b3... 3 3 Primitiv unit cll FCC:,,,,,, 4
6 Two wlls: Illustrtion of Bloch Thorm V V L r u r ir m im m How? m m i i m m i m m quivlnt r nd priodic u!,, =, =
Tim-indpndnt Schrödingr qution: Solution - pln wv Fr lctrons r Though fr lctron wv functions do not dpnd on th structur of solid, thy cn b writtn in th form of Bloch functions For ny propgtion vctor w cn find ' G in th first Brillouin zon ˆ H V m i ' r with nrgy V 7 V r V const ' m Thn wv function Bloch function nd nrgy: priodic function ir igr r V G m For ths wv functions w cn plot th bnd digrm, which bcom priodic with /
Nrly fr lctrons: bndgp 8 Introduc w priodic potntil Lt s simplify th problm: D potntil with just on Fourir componnt: V V cos lctrons r wvs : Brgg rflction occurs t n, p,... In quntum mchnics dgnrt stts cn split whn prturbtion is pplid: Wv functions corrsponding to split stts will b linr combintions of : V m or in Fourir sris V r g V V r VG G igr m
Nrly fr lctrons: Bndgp 9 By first-ordr prturbtion thory: V cos Clculting th intgrl, find bndgp: Fr lctrons pln wvs don t intrct with th lttic much until wvvctor bcoms comprbl with /, thn thy r Brgg rflctd nd w hv intrfrnc btwn pln wv nd its oppositly dirctd countrprt. Ths suprpositions r stnding wvs with th sm intic nrgy, but totl nrgy is diffrnt L V g cos cos sin d V L
Nrly fr lctrons: D bnds Th first thr Brillouin zons of simpl squr lttic - curvs for thr diffrnt dirctions for prbolic bnd Irrlvnt to dimnsionlity, th following proprtis r vlid: Within th first zon li ll points of llowd rducd wv vctor On-zon nd mny zon dscriptions r ltrntivs All th zons hs th sm volum Th zon boundris r th points of nrgy discontinuity From Cusc 963
Nrly fr lctrons: 3D bnds First Brillouin zons for vrious 3D structurs From Cusc 963
First Brillouin zon for fcc structur Nrly fr lctrons: 3D bnds Fr lctron bnds of fcc structur lctron bnds in fcc Al comprd to fr lctron bnds dshd lins From Humml,
Bnd structur for svrl fcc smiconductors 3 With dimond structur With zinc-blnd structur From Burns, 985
Bnd-structurs of Si nd G 4
Most ssntil bnds in dimond/zb smiconductors 5 GAs From www.ioff.ru
Fr lctrons nd crystl lctrons 6 Wv function: Kintic nrgy: Fr lctrons Vlocity or group vlocity: Dynmics F forc: Forc qution: V m i v * dr m F dv F m ir m Wv function: lctrons in solid Disprsion nr bnd trmum isotropic nd prbolic: ir u m* Group vlocity: v Vlocity t bnd trmum: Dynmics in bnd: dv v m* Fv dp d dv * m F ; d Forc qution: d m* d Fv F if m* isotropic nd prbolic F d
Hols 7 It is convnint to trt top of th upprmost vlnc bnd s hol stts Wvvctor of hol = totl wvvctor of th vlnc bnd =zro minus wvvctor of rmovd lctron: nrgy of hol. nrgy of th systm incrss s missing lctron wvvctor incrss: Mss of hol. Positiv! lctron ffctiv mss is ngtiv! m * * h m h h h h h v m v * m h * h Hol nrgy: Missing lctron nrgy: Group vlocity of hol is th sm s of th missing lctron v h h h v Chrg of hol. Positiv! d d h h h
mpl: lctron-hol pirs in smiconductors 8 Hol h + Si tom g c v lctron - HP gnrtion : Minimum nrgy rquird to br covlnt bonding is g.
Chrg crrirs in crystl 9 F m q F hol m q Si tom lctron Chrg crrirs in crystl r not compltly fr. Nd to us ffctiv mss NOT RST MASS!!! V - +