Quantum size effects in nanostructures

Transcription

1 Qutum s ffts ostutus Ltu ots to th ous Og d Iog Nostutus Kjld Pds Dtmt of Phss d Notholog Albog Uvst Albog August 006

2 . Itoduto O of th most dt ffts of dug th s of mtls to th omt g s th of qutto ffts du to th ofmt of th movmt of ltos. Ths lds to dst g lvls ddg o th s of th stutu s t s ow fom th sml ottl wll ttd toduto qutum mhs. Followg ths l tfl stutus wth ots dfft fom thos of th osodg bul mtls b td. Cotol ov dmsos s wll s omosto of stutus thus ms t ossbl to tlo mtl ots to sf ltos. Both smoduto d mtl ostutus hv b vstgtd ov th s. Whl th ltos of mtl ostutus stll v lmtd smoduto lto d otolto omots bsd o stutus wth qutum s ffts hv b o th mt fo svl s. Th st ots wll thfo tt th s ffts wth smoduto mtls l GAs md. It s v tht th ostutus s f stdg stutus. Rth th mbddd mt mtl tht ssl hs fft o th lto lvls th ostutu. I th th sml ttmts std h t s ot ossbl to t th mt mtls to out. Howv th sml modls stll gv good mssos of th bs ls bhd qutum s ffts vous stutus. Th ostutus ttd h b dvdd to th followg lsss: Two-dmsol objts: Th flms wth thss of th od of fw omts usull dostd o bul mtl. Th ots m b domtd b suf d tf ffts o th m flt th ofmt of ltos th dto dul to th flm qutum wll. I th two dmsos lll to th flm th ltos bhv l bul mtl. O-dmsol objts: Cld-l objts l ws d tubs wth dmts o th osl d lgths tll th momt g. Also mbddd stutus wth tgul oss sto st. Cofmt ffts fo ltos m th tsvs dto whl ltos f to mov o dmso log th stutu. Zo- dmsol objts: Usul ms otls lusts ollods ostls d fulls. Th omosd of svl ts to fw thousd toms. ltos ofd ll th dtos. Cbo otubs: Th b dsbd s gh shts olld to lds. Sgl wlld tubs hv dmts of th od of - m. Th ofmt ffts h b s s th djustmt of lto wvfutos to ft to th umf of th tub. It should b otd tht m ltos of ostutus do ot dd dtl o s ffts o lto ots. Istd th suf to volum to m ld to mw futolts fo st tlt ffts. Itsv sh s dtd towd toduto of w futoltts to mt mtls wthout fftg oth mtl ots b toduto of osd stutus. At th d of ths ots som usful fs wth mo dtld ttmts of ostutus d th otl ots gv.

3 . Qutum wlls ws d dots I ths sto w wll go though squ of stutus wth sg ofmt of ltos fom th ottl bs t th bouds of th stutus.. Two-dmsol objts I ths t of sstms th lto ofd th moto o dto whl th mov s th osodg bul mtl th oth two dtos. Tod ths ffts wdl usd smoduto lss wh th bd stutu of qutum wlls tlod to gv th dsd msso wvlgths. Th lss ml of smoduto qutum wll s th GAs Al G - As multl sstm. H th dth of th wll b djustd though th omosto of th AlGAs l Fg... Gll w dstgush btw qutum wll stutus d sultts. Th two stutus llusttd Fg... If th dst btw th ottl wlls s so lg tht th wv futos of th boud stts hv lttl ovl w tl bout qutum wlls. If o th oth hd th wv futos ovl th stutu s usull lld sultt. Fg.. Qutum wll stutu fomd GAs ls sdwhd btw AlGAs ls. Fg... Qutum wll d su ltt stutus. I th followg w wll osd sgl qutum wll s llusttd o Fg... I od to dsb wv futos d g lvls w stt wth th Shödg quto: h h V φ φ.. m m Fg... A sml ottl wll. Ths s sod-od dfftl quto s oodts. W wll tt th lll d - dtos s ddt d thus st th vbls odg to: φ φ φ.. W wll th hv two ddt dfftl qutos:

4 h m V φ φ. h φ m φ..4 Lt us stt b osdg -ddt t whh luds th s-dd th lto lvls. At fst lt us loo t th soluto fo ft bs tht s Δ C. Th wv futos must th b o t th bods of th wll d outsd th wll: φ L/φ -L/0. Ths boud odtos fulflld b wv futos of th fom π os 5 v L L φ.5 π s 46 odd. L L Fo gowg th wv futos lttvl v d odd futos of Fg..4. Th osodg gs gv b h π fo..6 m L Ths gvs fst msso of th smmts of wv futos d th od of mgtud of th dst g lvls g s sult of th ft thss of th wll. Not tht sto btw g lvls ds s th thss of th wll ss. Th ft wll omto s howv usull f fom usful dtos l sstms. W thus hv to dl wth ft b wlls. 4 I th followg w wll osd boud stts tht s < Δ C. Outsd th wll w t otll dg wv futos whl os d s futos stll wo sd th wll. Th fom of φ wll th b Aos φ.7 B [ α L / ] > l / As B [ α L / ] < L / < L / < l /. v odd -L/ L/ Fg..4. Alttg odd d v wvfutos ftl d wll. 4

5 Th gs gv b h Δ m A h α mb C sd th outsd th wll wll..8 Ids A d B dt tht th fftv msss usull dff th two mtls fomg th wll d ts suoudgs. Zo g s t t th to of th wll. ltos wth gtv g thus ofd to th wll whl thos wth ostv g f to mov though th stutu. To dtm th ostts A d B s wll s th wv vto d dmg t α w todu boud odtos fo th wv futo t th tf ± L/ btw th two mtls: φ otous m dφ d otous.9 Th lst quto stts tht th tl ut s otuous. Fo th v t ths two odtos ld to L B os fom φ otuous.0 A L αb dφ s fom otuous. m m d A B I od to fd solutos fo d α w t th to btw ths two qutos d lso btw th osodg two qutos fo th odd wv futo whh lds to m A m A L α t m B L α ot. m B. Ths two qutos ot b solvd ltll. Th howv qut sml to solv umll. To llustt th solutos ghll lt us wt th qutos b ssg α fom th g quto d ssum m A m B Δ m α C.. h 5

6 osl/ sl/ Istg ths th two tsdt qutos bov lds to L os L s 0 0. mδ wth C 0 s th mmum h vlu of. π/l π/l π/l 4π/L 5π/L Fgu.5 shows lots of ght d lft hd sds of qs... Fg..5. Ghl solutos to qs... Solutos gvg th dst llowd vlus of md wth vtl ls. Wth th vlus of ow th dst g lvls b lultd. s: Wht dtms th umb of dst lvls wll? Hvg solvd th -ddt t of th Shödg quto w wll tu to th lll t. Gll th soluto wll hv th D Bloh fom A B φ U.4 A B wh U s od th ltt of stvl mtl A d B. I th f lto s sml bol bds obtd d th totl g of th ltos s gv b os h..5 m s / 0 Th dst gs wll thus s qudtll wth th lll wv vto... Dst of stts fo f ltos Th dst of stts s v usful qutt wh w wt to loo t otl st. Lt us ssum th ltos ofd to bo of dmsos L ll th dtos Fg..6. Th soluto to th f lto Shödg quto s th fom of l wvs ψ..6 Itodug od boud odtos of th fom Fg..6. 6

7 ψ L ψ.7 d th osodg odtos th oth dtos lds to dst vlus of odg to L π ψ L ±.8 L wh s tg. Th lto g gows qudtll wth odg to h.9 m wth o dtol dd -s. Th llowd -ots llusttd dmsos o Fg..7. h -ot ots ltos of oost s d ths stts flld fom th lowst g smllst utl ll ltos usd. Th hghst g s lld th Fm g F d th osodg dus F. Ths dus sts oud d uoud lvls. O stt ous π/l -s. Th umb of ltos N sd s wth dus s N 4π V π π L..0 m B stg w gt th dst h of stts th umb of stts g ut: Fg..7. Allowd -ots dmsos D dn d d V m d π h V π m h.. D Dst of stts two dmsos Th D dst of stts must flt th qutto of lvls dsbd th vous sto. Cosd slb of mtl tht hs moso dmsos th - d - dtos whl th thss s smll th omt g. Pod boud odtos th - d - dtos ld to / F Fg..8. F lto dst of stts. 7

8 S π lπ l. Th -ot -s s th L L 4π 4π.. L L S Th umb of stts wth l wth dus s th π S N. 4π π S wh th fto s du to ss. Th umb of stts btw d d s gv b dn ρ d S d..4 d π Smll th umb of stts btw d d s lultd fom dn dn d d d..5 d d d Th umb of stts d g ut s th gv b dn d..6 S d π d Usg th fom of th f lto g to gt d m th lto th D dst of d h stts t th fom dn d m C ρ D..7 S d π d h π It s mott to ot tht th D dst of stts s ddt of th g. Howv ρ dds o th umb of lvls d s thus C sum of th otbutos fom th dst lvls g s sult of th qutto m ρ D H.8 h π Fg..9. g bds s futo of. wh H- s th Hvsd st futo d dots th oduto bd mm of th dst lvls. g D Fg..0. Dst of stts fo D sstm. D D F 8

9 .4 O-dmsol sstms Sstms wh ltos f to mov o dto d ofd th oth two dtos vstgtd tsvl ths s. A lg umb of dfft stutus hv b gow d wd g of tstg ltos hv b suggstd ludg lto omots ssos d lght mttg omots lss. Fg... Shmt llustto of w ld o to of two otts. I fgu b th s sgf molul tos ld o th w od to m t sstv to sf moluls. If th w s v th fw moluls wll b ough to du msubl hg sstvt of th w. Fgu. shows stutu osstg of GAs w wth tgul oss sto suoudd b AlGAs. I th followg ths wll b ttd s sstm wth lto ofmt th - d -dtos d f lto moto th -dto. Th Shödg quto ts th fom Fg... A ow wth tgul oss sto log wth th ottls two dtos. 9

10 0 V m Ψ Ψ.9 wh th ottl th - d -dtos llusttd Fgu.. Assumg f movmt th -dto ddt of th oth two oodts lds to soluto of th fom. L φ Ψ.0 I gl d ot b std fo ft bs d th Shödg quto V m φ φ. s mo dffult to solv th fo th QW s ttd l. If w ssum ft bs th soluto wll b stdg wvs of th fom os os L L L L π π φ. wh φ0 o th bouds d L π. Istg ths th Shödg quto wth V0 gvs dst gs L L m π h.. Alog th -s th ltos f to mov d th totl g s m L L m h h π..4 Th dst of stts s lultd log th sm l s fo th qutum wll m L D πh..5 I ths sml modl D fo. s: Stt wth sml qutum wll wth ft bs todu ostts o lto moto o mo dto. How muh dos th lowst g shft? T 0 m th GAs QW s ml. Th fftv mss th oduto bd s tms th f-lto mss. s: Cosd Ag qutum wll wth ft bs. How m lvls th blow th Fm g f th wll s m th? Us F 5.48 V V J h Js. Fg.. Dst of stts of D sstm. D C C C

11 .5 Zo-dmsol sstms Followg th l of th vous stos w should ow mov dow to 0D sstms wh th ltos ofd th moto ll th dtos. Vous ms fo tht t of sstms suh s qutum dots qutum bos o stls t. Fg..4. Ltt of ldl stutus odud o suf fo st b -bm lthogh. I gl th fftv dmt wll b smll th th hsl dmt du to th st of lold suf stts. Fg..5. No tls mbddd mt mtl fo st smoduto ostls slo dod. Cosd qutum bo wth ft bs d sd lgths L L L. Th wv futo wll hv th fom φ lm φl φm φ.6 wth solutos th fom of stdg wvs φ l s. L.7 Th g s gv b lm g l m.8 wth l h m h π l m L..9 Ths dst g lvls ow dsb th totl g of th sstm d o otuous t fom f lto movmts should b ddd s th D d D ss. I ths ss th sstm s s tfl tom. Th dst of stts thus b wtt u dtl 0D ρ lm δ lm..40 Cosd ub wth sd lgth. Th gs th h π..4 m

12 Th lowst vlu s. Th osodg gs fo hols gv b h h m h π..4 ρ 0D lm Fg..5. Dst of stts fo 0D sstm. Fg..7. lto d hol stts th qutum dot. Du to th qutto th g g ss omd to bul mtl. Tg both ltos d hols to out th s g g s gv b h π Δ g.4 μ wh s th dud mss. Fo 0-m GAs qutum bos th hg g μ m m h g s 0. V. s: If th qutum bo s v smll th wll ot b ofd lvls th bo. Wht s th odto fo t lst o boud lvl? Δ C h > m π wth m. Ths gvs h π ΔC > m d th h π >. m Δ C Fo Al 0. G 0.7 As th dt of th ofg ottl s of th bo. m. Δ C 0mV osodg to mmum

13 . Otl ots Itbd tstos Som of th most usful thqus fo studs of th ots of ostutus bsd o th otl tstos btw th dst lto lvls. Absoto stoso fo st wll vl th dt bd g d thb lso s g g blu shft du to ofmt ffts. Fg... Dt lft d dt ght bd gs. Ths ould fo st b GAs lft d S ght. Fgu. bov llustts two stutos tht ou smmt dtos smodutos. If th hghst ot th oud vl bd ls dtl ud th lowst ot th mt oduto bd w tl bout dt bd g d th g g osods to th ost of otl bsoto. If o th oth hd mm d mm th vl d oduto bd t dfft ots -s w tl bout dt g. S otl tstos vtl th dgm th bsoto t th dt g wll b w d tstos t th dt g wll domt th bsoto stum. Th tsto t fom tl to fl f stt s gv b π W f M g hω. h wh th mt lmt s gv b M ψ * f H ' ψ d. d g hω s th jot dst of stts whh s foldg of th dst of stts of th tl d fl stts. I od to lult th tsto mt lmt sso fo th tto Hmlto H s dd. I lssl hss th g of tom osdd s dol lt fld ε s ε. As fst omto w th us tht to ss th tto Hmlto o th fom H ' ε. Lt us osd l wv of th fom ε ε 0. d lto wvfuto o th Bloh fom f ψ f u f.4 V

14 wh th futos u T u od th stl ltt dsbd b tslto f f vtos T. Wth ths w gt * f M u f u d V ε 0.5 whh b usg th momtum osvto h h f h s dud to * M u f 0 u d V ε..6 Fgu.. Th tsto fom f toms to toms wth bods sold. Th tsto mt lmt thus ol dds o th s oodt though th ll-od t of th Bloh futo. Ths futos lso ot th dd o th mtl. Th lulto of M thus b do though tgto ov sgl Bllou ll. I od to otu th vluto of th otl tsto t w d to osd th smmt of wv futos solds. Fgu. llustts th tsto fom dst tom lvls to otuous bds of lvls G. Th shm show Fg.. s l th sm fo III-V llos suh s GAs o II-VI llos l CdT. It s foud tht th tsto mt lmt s lg fo s tstos fo st fom th vl bd to th oduto bd of smodutos. N th bd dg Fg.. th 4 bds of smoduto bol wth th fom h.7 m * wh m* s th fftv mss dsbg th uvtu of th dvdul bds s b s fom 4

15 d d h d..8 m * m * h d Fg... Th bd th bd g smoduto. Fg..4. Th dst vlus of th - vto ol s. Fom Fg..4 t b s tht g 4π wh s bfo s th umb of - π π π ots ut volum. Th lto to th dst of stts g ut s s fom g g d g d g..9 d d Istg th sso fo g d usg th bol fom dsbd bov w v t m * g..0 π h W wll osd tstos btw th bds 5

16 h g m. h h mh whh gvs dff g btw th two bds of th fom h h h h ω g g.. m mh μ μ m mh To gt sso fo th jot dst of stts w thus just hv to l th fftv mss b th dud mss q..0 bov hω < hω g g : : g hω 0 μ g π h hω g.. S th bsoto s ootol to th tsto t w hv α hω M g hω..4 As M s umb th stl dd of th bsoto s gv b th jot dst of stts. Sml gumts l to low-dmsol stutus. Blow th foms of bsoto st fom dfft sstms sthd. Bul Qutum wll D α D Fg..5. Dst of stts dfft sstms. α hω KD hω g α h hω KDH ω g g Qutum W α hω hω KD hω g 4 g Qutum dot α hω h hω K0Dδ ω g α g hω 4 g hω 4 6

17 4. Hbdto of bo toms Cbo toms th bs buldg blos of og soft mtls. At th sm tm th fom dmod th hdst ow mtl wh th gd th dmod stl stutu. Th ots of bo stutus stogl otd to th m dfft ofgutos of th lto stts of bo toms usull fd to s th hbdto. Cbo toms hv 6 ltos oug s s d obtls. Th fou s d vl ltos fom ovlt bods fo st th dmod stutu wh h tom hs 4 st ghbous. Howv th g dff btw th lowst s lto d th hghst lto s qut smll whh lds to s mg of obtls d fomto of hbdtos of s lto wth o ltos: s s s. Lt us fst osd s hbdto wh s stt s ombd wth stt od to dsb bodg btw two toms lg o th -s. As llusttd Fg. 4. th s two ossbl ombtos dsbd b s C s C 4. sb C s C4. Othoomlto qus s sb 0 4. s s sb sb whh lds to s s 4. sb s. Fg. 4.. Hbdto of s d stts. Th sm osdtos holds fo th - d -dtos. Mg of obtls wth h oth gvs s to otto of th sultg obtl. A wvfuto C C C 4.4 wh C C C osods to wvfuto dsbg bod th dto C C C. A sml ml of molul otg bo toms showg s hbdto s tl C H wth tl bods btw bo toms oft std b HC CH. I ths l molul th s obtl of o C tom foms ovlt bod wth th sb obtl of th oth C tom. Ths s lld σ bod d s t s usull th s wth ovlt bods t s th stog. Two muh w so lld π bods fomd b th mg d wvfutos. Poltl s wll-ow smodutg olm tht wth dfft stutus. All of thm ot howv s hbdd bods. I s hbdto s obtl s ombd wth two obtls fo st d. Ths foms th stog σ bods of th olm h. Th lst bod foms w π bods th dto dul to th l of th molul. Th ombto of two bods th hbdto lds to th g-g stutu of th h. 7

18 8 Th σ bods ot th dtos Th s stts fomd fom s d tom lvls wth th omlto odto s s. 4.6 Usg ths th bods th dtos gv bov b std b [ ] [ ]. ½ ½ b s s s s s s 4.7 Fg. 4.. Poltl. Dstbuto of th s-t othoomlto qus tht. Th followg odtos ld us to th offts. 0 ½ 0 0 ½ 0 s s s s b 4.8 Ths gvs th th l ombto of th tom obtls tht b dsbd s tgul bods wth th lgst mltud log th dtos to th st ghbous. As mls of s hbdto w wll osd Als C H d tul th Mth molul CH 4. I ths s th s stt s md wth stts. Th dtos of th bods gv b d -- d th ombto of obtls b wtt u dtl: [ ] [ ] [ ] [ ]. ½ ½ ½ ½ d b s s s s s s s s 4.9 Fg. 4.. Posto of C toms CH 4.

19 I gl fo s hbdto w hv: ltos σ obtls ltos π obtls Thus w hv: s : σ π s : σ π. 4. s : 4σ 0π s: Dv th ssos fo th offts d fo th s hbdto. 9

20 5. Tght-bdg lultos Cosd stll mtl wth od ltt. Th wvfutos Ψ Bloh futos fulfllg th odto ψ T ψ 5. whh lso b ssd s ψ ψ. H T s tsltol oto d s ltt vto. 5. W wll t tom obtls s th sttg ot d ostut futo tht stfs th Bloh odto: N R Φ j φ R N R j 5. wh R : osto of toms th φ j : tom wvfuto j stt : umb of tom obtls ut ll N : umb of 4 ut lls ~ 0. Usull s smll umb. Th fom q. 5. stsfs th Bloh odto: Φ j N N R N N R Φ. j R φ R R φ R j 5.4 Th lst st q. 5.4 follows b lg tht subttg ltt vto fom th tom osto vto just hgs th od of th summto. Lt us osd od boud odtos of th fom Φ j M Φ j M N. 5.5 Usg ths boud odto togth wth th Bloh odto q. 4 gvs us dst vlus of th wvvto. M π 0 M. 5.6 M Lt us ow wt th wvfuto th sold mtl s suosto of th futos q. 5.: ψ j jj' Φ j'. 5.7 j' S 5. stsfs Blohs thom so dos th sum 5.7. Th g gvlus gv b 0

21 j H j j ψ ψ ψ j ψ j wh H s th Hmlto oto. B stg th fom 5.7 w gt j j' jj' j j' 5.8 H jj' jj' j j' 5.9 S wh H S jj' jj' Φ Φ j j H Φ Φ j' j' th tsto mt lmts d ovl mt lmts stvl. Fo gv mts H jj d ρ jj d gv w should dtm j suh tht th g s mmd. Th odto s mmbg tht j s oml j j' j j' H jj' j' s S jj' If w multl b S H j j' j j' j j' jj' jj' jj' j S jj' j' B dfg C W gt th followg lgb st of qutos t t HC SC o t t [ H S] C 0. H jj' jj' ' ' 0 S jj j. 5.0 j' S ' jj jj' d us th sso fo 5.9 w gt Th odto fo solutos s t t Dt[ H S] Ths s gvlu oblm wth gvls fo h. Th odu ldg to lultos of g lvls s th followg:. Dtm ut ll d bss ltt vtos

22 . Rol ltt b. Chos smmt dtos lult H j Sj t t H S C od to fd. 4. Solv [ ] 0 It should b otd tht ths s ot slf osstt odu. Th sult dds omltl o th tl ho of tom wvfutos. j As ml w lult th lto bds of Poltl. Th ut ll s show o Fg. 5.. Th molul b ttd s o-dmsol sstm wth two C toms A d B th ut ll. Thus 00 d b 0 0 d th Bllou o s π gv b π < < π. Fg. 5.. Ut lls oltl. Th sum ov tom wvfutos us ov th two dfft toms A d B: R Φ j αφ Rα α A B. 5.5 N Rα Th dgol mt lmt s gv b H AA N N N ε N R R' R R' R R' ± ' R' RR' A R' φ R' H φ R φ R H φ R ± A φ R' H φ R hgh od tms. A φ R' H A A A A R R φ R dv 5.6 Th sm holds fo th oth dgol lmt: H BB ε hgh od tms. I th off-dgol lmts ol th st ghbous t ostos R ' ± osdd: H AB φa R H φa R φa R H φa R N R' 5.7 t os wh th gtv qutt t s gv b t φ R H φ R 5.8 d A A ± H H H BA AB AB.

23 If w ssum omld tom wvfutos w gt S S AA BB S AB SBA s os Wth s φ A R φa R ±. 5.0 Th dtmt quto gvg us th gs 5.9 H H AA BA S S AA BA H H AB BB S S AB BB Istg th ssos fo th mt lmts lds to ε t os t os ε Th solutos fo th g ± t os ε ± s os ε 5. Fgu 5. shows shmtll th sh of th two bds sultg fom q. 5.. h bd hv two stts fo h - vlu. h ll hs π-ltos A d B. Thus both ltos th low bd d th g of lto s low th fo th ltos f toms. Ths dmostts tht th sstm gs g b fomg th bods th molul. Th low d u bds oft lld bodg d t-bodg stts. Fg. 5.. g bds of oltl. s: Dv th sso q. 5.

24 6. Cbo ostutus A bo otub b s s gh sht olld u to ld. Bfo w od to CNT s w thus hv to m dsto of two-dmsol bo sht. 6. Two-dmsol ght Fgu 6. shows both th dt d th ol ltt d ut vtos of gh sht osstg of ght toms ld t th os of th hgos. Fg. 6.. Dt d ol ltt wth ut vtos fo ghm. Th ltt ostt th l s.4 Å d th dst btw ght ls s.5 Å. It s thus ossbl to tt ght s two-dmsol sstm wh th w ttos mog ls gltd. Th bs vtos dsbg th dt d ol ltt gv b s Fg. 6. π π b b 6.. Fg. 6.. Th ltt ostt th ol ltt s 4 π d th ltt s ottd 90 ltv to th dt ltt. Th D Bllou o b tsltd to gv th whol ol ltt. Fom th stutu w s tht h C tom hs st ghbous wth whh t s gog to m σ-bods s ofguto. W th lft wth sgl π-bod s -obtl th dto dul to th l of th ght sht. It s ths π-ltos tht sosbl fo th lto d otl ots of ght. Lt us th fous o th π-ltos. W wll osd two toms A d B d ol loo t st ghbou tto. I ths s 4

25 5 H AA H BB s fo olthl. Followg th l of lultos fo olthl ghbous t th ostos R R R wll otbut to H AB : [ ]. os t t f t t H R R R AB 6. B usg tht f s S S S S BA AB BB AA w gt th mts. sf sf S tf tf H ε ε 6. Sttg th dtmt of [ ] S H t t 0 gvs th g w s w t D ± ± ε 6.4 wth 4os os 4os f w. 6.5 Fg. 6.. Th bodg d t-bodg π-bds of gh. Fom f 4.

26 As fst omto o osd vshg ovl s0 to v t g ltv to ε of th fom 4os D ± t os 4os 6.6 Th lultd gs fom ths modl show Fg. 6.. Wth two ltos h -ot th low bd bodg s full d th u π* tbodg o s mt. Th ght sht s thus o g smoduto. Th wdth of th bds s foud to b ± t t 0. W wll ot dsb th σ-bds h. Th wll b toms ll h otbutg to obtls totl 6 6 mt oblm. Ths dds 6 bds to th two π-bds. 6

27 6. Cbo otubs W wll ow osd bo otubs s ght shts olld u to lds wth lmtd umb of bo toms log th umf. Tl dmts fo sgl wlld otubs th g fom 0.7 to m whl th lgth b tms lg. I ddto to th dmt th ots of th CNTs dd o th otto of th C-C bods ltv to th s of th tub. Cosd th tgl wth o t quvlt ots OABB th gh sht Fg Th vto T s log th s of th tub whl h otg ots O d A s log th umf of th tub. Th gl θ dsbs th dto of oss sto of th tub ltv to hgh smmt dto th sht. Tubs wth θ 0 lld gg tubs whl thos wth θ 0 lld mh both ss fg to th stutu of oss sto of th tub. Tubs wth θ th g o 0 < θ < 0 lld hl. A CNT s dsbd b two tg umbs m Fg. 6.4 Vtos dsbg th ostuto of CNT fom gv b h m. It s s gh sht. dtl tht th CNT fomd b th vtos Fg. 6.4 s std b. Tubs of th t of th mh t d 0 s of th g-g t. I tms of m th dmt of th tub s h d m m π π wh Å. 49 Å. Also th gl θ b ssd b m: h m os θ. 6.8 h m m Futhmo wth T o th fom T t 6.9 t wh t d t tgs th othogolt wth h lds to 7

28 T h t ½t m ½t t m 0 0 Ths s stsfd b R d R. 6.0 m m t t 6. d wh d R s th lgst ommo dvso fo m d m. Tg th ml o Fg. 6.4 wth m w hv m5 d m7 w hv d R d thus t 5 d t -7. Th ut ll th otub s OABB. I gh th of th ut ll s gv b. Th umb of gh ut lls N o CNT ut ll s thus h T m m N. 6. d R Wth tom gh ut ll w hv N obtls h CNT ut ll. Fgu 6.5 shows CNT wth th ut lls sd b h d T h otg N bo toms. Th osto of th toms s gv b th osto vto R. I th ol ltt ottd 90 ltv to th dt ltt th ltt vto K s log th umf ot ll vto but th mgtuds of K m ss d K s log th s of th tub. As t s th s bul stls vtos dt d ol s must stsf th odto R K j πδ j 6. wh δ f j d o othws. Usg ths w gt th qutos h h K K j π 0 T K T K 0 π 6.4 whh wth th foms h m d T t t lds to tb tb m b b K N 6.5 K N Fg CNT ut ll. Fgu 6.6 shows ml of ol ltt vtos fo 4 CNT wth N8 h 4 d T 4 5.W hv N dst vlus of th K vto: μk μ 0 N 6.6 8

29 Ths gvs us dst vlus of th wvvto th dto log th umf of th CNT. Alog th s of th tub th lgth of K s π T. If th lgth of th tub s L t th dst btw K -vlus s π. Fo log tubs ths lds to otuous st of K - vlus. Th g bds of th CNT s obtd Fg Rol ltt vtos of 4 CNT. dtl fom th bds lultd fo th D gh sht. Comd to th qutum wlls ttd l wh dst vlus of th wvvto osodd to stdg wvs th ottl wll th qutto CNT s osods to wvs tht ft to th umf of th tub. Istd of th otuous g suf of gh show Fg. 6. w gt st of uvs o ths suf osodg to th dst vlus of K. Lt us t th omto q. 6.6 s th two g sufs of th gh sht. Th dst g uvs of th CNT th gv b K K μ g D μ K K 6.7 π π μ 0 N < <. T T Fg Smmt ots th ol Bllou o. Ths dtms th N dst g uvs. At th K-ot Fg. 6.7 th π d th π bds dgt th touh. If th dst ls sss though th K-ot th CNT wll b mtl. Othws t wll b smoduto. Th odto fo ths s tht YK tg. Fom Fg. 6.8 t s s tht Γ K b b. S Fg K L t Γ K h π j b b m π j 6.8 m j m wh j s tg w gt th odto tht CNT s wth qul to tg mtls whl th st smodutos. Ths ms tht ll mh tubs mtls s wll s th g-g tubs About / d of th CNT s mtll s show Fg

30 Fg Rstto of mtll d smodutg CNT s. Fom f 4. Cosd th hgh-smmt mh stutu wth th dmt d m m. Fttg wv to th umf of th tub qus tht q d q π q o q π q. Istg ths q. 6.6 gvs π q q ± t 4os os 4os Lt b log th ΓK - dto. At th o boud. 6.9 π whh lds us to t q ±. W thus hv bd g t th o boud. Fgus 6.9 d 6.0 show th g bds fo two dfft CNT s. Th 4 tub Fg. 6.9 s smodutg s o bds ossg. Th tub show Fg. 6.0 o th oth hd hs bds ossg th Fm lvl d thus mtll. s: Dv th ssos fo th ol ltt vtos q

31 Fg g lvls of th 4 smodutg CNT. Fom f 4.

32 Fg g lvls of th mtll CNT. Fom f 4.

33 Rfs:. P. K. Bsu Tho of otl osss smodutos Ofod S Publtos P. Hso Qutum wlls ws d dots Wl R. Sto G&M. Dsslhus Phsl ots of bo otubs Iml Collg Pss J. Jug J. Bo T. Holmgd N. A. Kotb Sgl-wlld bo otubs 8 th smst ojt Thl Phss 005.