Handout 11. Energy Bands in Graphene: Tight Binding and the Nearly Free Electron Approach
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1 Hdout rg ds Grh: Tght dg d th Nrl Fr ltro roh I ths ltur ou wll lr: rg Th tght bdg thod (otd ) Th -bds grh FZ C 407 Srg 009 Frh R Corll Uvrst Grh d Crbo Notubs: ss Grh s two dsol sgl to lr o rbo tos rrgd Hoob ltt (whh s ot rvs ltt) Th udrlg rvs ltt s show b th loto o th bl dots d s hgol ltt Thr r two rbo tos r rtv ll, d (show blu d rd olors, rstvl) Grh b rolld to tubs tht r lld rbo otubs (CNTs) ˆ ˆ =.46 ˆ ˆ CNT CNT C 407 Srg 009 Frh R Corll Uvrst
2 S hbrdzto rbo: Grh: S Hbrdzto r s r r r 6 r s r r r 6 r s r r ll rbo tos r ll s hbrdzd (o s orbtl togthr wth th d th orbtls grt thr s orbtls) ll s orbtls or -bods wth th s orbtls o th gborg rbo tos Th bodg orbtl ssotd wth h -bod s oud b two ltros (s-u d s-dow) Thr s o ltro r rbo to lt th z orbtl C 407 Srg 009 Frh R Corll Uvrst Grh: z Orbtls -bodg: h rbo to otrbuts o z -orbtl tht rtts bodg h rtv ll otrbuts two z -orbtls tht rtt bodg Th z orbtl st out o th l o th h d or -bods wth gborg z orbtls Th -bodg rsults rg bds (-bds) tht w wll stud v tght bdg C 407 Srg 009 Frh R Corll Uvrst
3 ss vtors: d d d ˆ d Grh: So Usul tors Nrst ghbor vtors: ˆ ˆ ˆ ˆ ˆ ˆ =.46 Ths wll b usul or wrtg th l soluto ot or C 407 Srg 009 Frh R Corll Uvrst Grh: Tght dg Soluto h bss to otrbuts o z -orbtl tht rtts bodg h rtv ll otrbuts two z -orbtls tht rtt bodg r r O th wrt th trl tght-bdg soluto or wvvtor s: =.46. d. d r r R d r R d. R N C 407 Srg 009 Frh R Corll Uvrst
4 Plug th soluto to th Shrodgr quto: Hˆ r r d th, o b o, ultl b ro th lt b th br s orrsodg to vr orbtl o rtv ll to grt s qutos s th ubr o orbtls r rtv ll St : Multl th quto wth r d d: th rg tr lts or orbtls tht r rst ghbors, d ssu tht th orbtls o drt tos r orthogol... Grh: Tght dg Soluto =.46 Not tht th l rsult b wrtt trs o th rst ghbor vtors C 407 Srg 009 Frh R Corll Uvrst Grh: Tght dg Soluto =.46 St : Multl th quto wth r d d: th rg tr lts or orbtls tht r rst ghbors, d ssu tht th orbtls o drt tos r orthogol... Not tht th l rsult b wrtt trs o th rst ghbor vtors C 407 Srg 009 Frh R Corll Uvrst 4
5 Grh: Tght dg Soluto =.46 Wrt th qutos obtd tr or: Whr th uto s:... Solutos r: d th orrsodg gvtors r: C 407 Srg 009 Frh R Corll Uvrst K Grh: -rg ds rg K K M K FZ M K K FZ dgs o t th M-ots btw th rst d th sod bds No bdgs o t th K-ots d th K -ots C 407 Srg 009 Frh R Corll Uvrst 5
6 Grh: -rg ds S grh hs two ltros r rtv ll otrbutg to -bodg, th lowr -bd wll b oltl lld t T 0K Th loto o Fr lvl r T 0K s show b th dshd urv I grtg th lots I hos rg zro suh tht: d or grh: 0.0 C 407 Srg 009 Frh R Corll Uvrst Grh: Corso o NF d T Sl orlzd to : d ost b o To or th rl-r-ltro roh (NF) to tght-bdg (T) I ssud th DC ottl NF to b: o d grh:.0 C 407 Srg 009 Frh R Corll Uvrst 6
7 7 C 407 Srg 009 Frh R Corll Uvrst Wh th Zro dg Grh? =.46 Th swr ro tght bdg: Th two tos rtv ll r dtl. I th wr drt th thr would b o-zro bdg: g t th K(K )-ots Th swr ro th rl-r-ltro roh: s ou sw our howor, th rstl ottl ld vrso str w.r.t. th -s (.. ) th thr would b o-zro bdg. O ours, th two tos th rtv ll wr drt th th rstl would l vrso str! So both th rohs lg th zro bdg r osstt.,, C 407 Srg 009 Frh R Corll Uvrst Psudos Grh... Solutos r: d th orrsodg gvtors r: =.46 Cor wth th s o / s rtls wth ss th - l:
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