Lecture 21 : Graphene Bandstructure

Transcription

1 Fundmnls of Nnolcronics Prof. Suprio D C 45 Purdu Univrsi Lcur : Grpn Bndsrucur Rf. Cpr 6. Nwor for Compuionl Nnocnolog

2 Rviw of Rciprocl Lic :5 In ls clss w lrnd ow o consruc rciprocl lic. For D w v: Rl-Spc: -Spc: - π/ BZ π/ R In gnrl for priodic srucurs w cn wri sis vcors suc n poin in lic cn wrin s linr cominion of m wi condiion cofficins mus ingrs. R m n p K π / Similrl n poin in rciprocl lic cn wrin s: K MA NA PA How r vcors A rld o vcors? T dfining condiion is: A j i πδ ij T significnc of rciprocl lic vcors A is poins in spc wic r pr from c or n ingr mulipl of Ai s, giv is sm wvfuncion soluion. δ ij ij for i δ for i j j

3 Grpn 6:5 Grpn is md up of cron oms ondd in gonl D pln. Grpi is D srucur is md up of wl coupld Grpn ss. Tis is of priculr impornc cus cron nnous r md up of Grpn s is rolld up li clindr. Cron nnous mslvs r of inrs cus popl liv cn m ll inds of Nno dvics wi m.

4 Rciprocl Lic in D 8: Smiconducors of inrs o us v w is clld dimond srucur. T dimond srucur is composd of o inrpnring FCC lics following w: Imgin wo FCC lics suc c om of c lic is on op of corrsponding om of or lic. You sould onl sing FCC lic s of now. Tn fi on lic nd mov or on in dircion of od digonl of fid on ¼ of od digonl. Now ou v ourslf dimond lic. If wo FCC lics r md up of wo diffrn ps of oms, srucur is n clld Zinclnd lic. To visuliz rciprocl lic focus onl on on FCC lic in dimond srucur. FCC in Rl Spc BCC in Rciprocl Spc Brillouin Zon in Rciprocl Lic

5 - Digrms for D Rciprocl Lics :6 Sinc rciprocl spc is now dimnsionl, o drw - digrm w v coos priculr dircions nd drw - digrm long os dircions: Som usful informion: T op of vlnc nd usull occurs Gmm poin (). T oom of conducion nd owvr dos no lws li. For mpl considr Silicon: L Γ X If o conducion nd minimum nd vlnc nd mimum li sm vlu of, mril is clld dircion ndgp smiconducor. Or wis mril is indirc li Si.

6 Prolic Approimion 7:8 Usull, i is ncssr o driv n prssion for (,, z ) ou conducion poins of ul solid Silicon Prolic Conducion Bnd Approimion For silicon, us prolic pproimion c m* ( m* z ) Approimion wr m* is ffciv mss. For nnous w cn driv similr prolic prssion vi Tlor sris pnsion pproims sunds nr conducion vlls

7 - Rlion for Grpn :5 L s g c o Grpn. Firs idnif sic uni cll T lic srucur onl rps in pirs of! Bsic Uni Cll Rmmr gnrl rsul of principl of ndsrucur: { φ } [ ( )]{ } φ [ ( )] i [ ] ( d m d n H ) nm m To wri () considr on uni cll n is nrs nigors. Figur sows r will 5 rms in summion for ().

8 Grpn - Digrm :45 Rmmr gnrl rsul of principl of ndsrucur: { φ } ( ) To wri () considr on uni cll n is nrs nigors. Figur sows r will 5 rms in summion for (). [ ]{ } φ [ ( )] i [ ] ( d m d n H ) m nm Wriing summion rms nd dding m up w g: Wr ε ( ) * T ignvlus of is mri r givn : ε ε ( ) i i ( ) ε ± ( ) Conducion Poin { filld ss Conducion Poin

9 Mgniud of () 9:5 N w li o loc conducion poins in dimnsionl spc: To find conducion poins w nd o s (). So w nd o find () : Uni Cll ( ) ( ) ( ) i i ( ) i i ( ) i cos ( ) * 4 cos cos 4 cos ) ( so, 4 cos cos 4 cos

10 Conducion Vlls 8:5 Now l ( ) 4 cos cos 4 cos L nd invsig () s funcion of. ( cos ) for π o g () L pi nd invsig () s funcion of. ( cos ) for π o g () π Conducion Vll Conducion Vll (, π/) (π/,π/ ) (π/,-π/)

11 Two Full Vlls 4:45 T si Brillouin vlls rll onl giv indpndn vlls,.g. in c group of r in picur wo of vlls r w form or rciprocl lic uni vcor; nc rprsn sm s. On cn in c cornr in s Brillouin zon conrius / rd./ 6 (lf figur). Alrnivl w cn rnsl wo of cornrs in c group o g full vlls on rig. Disprsion rlion long. cos for ( ) ε Trnsling wo of cornrs in c group of ε ± Conducion Vll ( ) ε Conducion Vll ε-