ELECTRIC, OPTICAL & MAGNETIC PROPERTIES OF NANOSTRUCTURES

Transcription

1 ELECTRIC, OPTICAL & MAGNETIC PROPERTIES OF NANOSTRUCTURES THOMAS GARM PEDERSEN LARS DIEKHÖNER AALBORG UNIVERSITY 3

2 Tabl of Cotts Itroductio 4 Tim-dpdt Prturbatio Thory 8 Liar Rspos Thory 8 Prlimiaris 4 A First Exampl: Elctric Polarizability 7 3 Bulk Rspos of Mtals 3 Spi Magtizatio 3 Elctric Currt 5 4 Elctric Currts i Naostructurs 3 4 Matrix Elmts 3 4 Simplificatio: Dcoupld Chals 34 5 Elctro Trasmissio ad Rflctio 39 5 Triagular Barrir 43 6 Elctro Trasmissio i Molculs: Itroductio 48 6 Gral Ladaur Formula 5 7 Elctro Trasmissio i Molculs: Challgs 54 7 Two-dimsioal Lads 59 8 Elctric Proprtis of Smicoductors 6 8 Dopig 66 9 PN ad PIN Juctios 7 9 Aalysis of th PN Juctio 7 9 PIN Juctio Slf-cosistt Pottial ad Trappd Chargs 77 PN Juctio ad Tulig Diods 83 Thr-dimsioal No-dgrat Dvics 84 PN Juctio Diod 86 3 Tulig Currt 89 Mtal-Smicoductor Juctios 95 Schottky Diod 95 Smiclassical Trasport 4 Trasport Cofficits 6 3 Fild Effct Trasistors 3 MOSFET I/V Charactristic 4 3 Modulatio Dopd Fild Effct Trasistors 6 33 MODFET I/V Charactristic 9 34 Trasit Tim 4 Naowir MOSFETs 4 5 Optical Proprtis of Smicoductors 3 5 Two-bad ad Evlop Approximatios 3 6 Optics of Bulk ad Low-dimsioal Smicoductors 37 6 Smicoductor Quatum Wlls 39 6 Smicoductor Quatum Wirs ad Dots 4

3 7 Elctroic ad Optical Proprtis of Graph 45 7 Elctroic ad Optical Proprtis 48 8 Modls of Excitos 55 8 Wair Modl 59 9 Excitos i Bulk ad Two-dimsioal Smicoductors 63 9 Excitos i Quatum Wlls 66 Excitos i Naowirs ad Naotubs 7 s Excitos i Carbo Naotubs 76 Smicoductor Lasrs ad LEDs 83 Gai i Smicoductor Lasrs 86 Solar Clls 9 Ultimat Efficicy 93 Shockly-Quissr Limit 96 3 Photoic Bad Gap Structurs 3 3 O Dimsioal PBG Structurs 4 3 Two Dimsioal PBG Structurs 6 4 Optical Procsss 4 Sigl Dipol 4 Bar ad Drssd Polarizabilitis 5 5 Optical Proprtis of Naosphrs 5 Fidig th Filds 3 5 Scattrig, Absorptio ad Extictio 6 6 Naoparticl Optics i th Elctrostatic Limit 3 6 Cylidrical Naoparticls 3 6 Oblat Sphroids 35 7 Basic Magtism 39 7 Isolatd Momts 4 7 Coupld Momts 45 8 Exchag Itractio for Localizd Momts 5 9 Itirat Exchag Itractio 58 9 Frromagtism of Fr Elctros 58 9 Bad Modl for Frromagtism 6 3 Domais ad Aisotropy 65 3 Aisotropy 65 3 Domai Boudaris Magtostatic Ergy ad th Origi of Domai Formatio Sigl Domai Naoparticls 7 Appdix Naostructurs 73 A Quatum Wlls 73 A Quatum Wirs 76 A3 Quatum Dots 79 Appdix Tight-Bidig Formalism 8 A Tight-Bidig i Priodic Structurs 83 A O-Dimsioal Priodic Structurs 84 A3 Two-Dimsioal Priodic Structurs 87

4 A4 Thr-Dimsioal Priodic Structurs 88 Appdix 3 Dsity Fuctioal Thory 9 A3 Atomic Stats 93 A3 Exampl: Carbo Atom 95 A33 Dsity-Fuctioal Basd Tight-Bidig 96 3

5 Itroductio Proprtis ar sstially about caus ad ffct A matrial lft compltly udisturbd dos t display its proprtis Howvr, if w prob somthig th rspos will rval th charactristic proprtis of th objct Hr, probig should b udrstood i its broadst ss If w look at somthig it is probd by th light rflctd or trasmittd by th objct If w put a objct o a tabl it is probd by cotact forcs If w plac th objct o a hot plat it is probd by th hat A dlss list of xtral prturbatios srv as probs of proprtis that, tak togthr, charactriz th objct At a qualitativ lvl w might labl th objct as shiy, hard, havy tc Mor accuratly, howvr, w should spcify its rflctac, hardss, hat capacity ad so o as prcisly dfid proprtis that ca b masurd i a spcific stup This also allows us to dscrib thortically th phomo i a prcis mar Hc, to this d w must costruct a modl that compasss th objct, th prturbatio ad th rspos Th prst st of lctur ots dals oly with a small part of all th proprtis imagiabl W rstrict ourslvs to th followig prturbatios: lctric pottials, light ad magtic filds Th rsposs to ths stimuli dfi togthr th lctric, optical ad magtic proprtis of a objct Morovr, w xclud cass, whr ay two of ths prturbatios ar prst at th sam tim Fially, w rstrict th aalysis to wak disturbacs that do t prturb sigificatly th objct This might sm as rathr svr rstrictios but, actually, thy ar ot Thy will allow us to udrstad a wid rag of phoma such as th color of a objct Ad withi this rstrictd aalysis w ll still b abl to xplai svral importat faturs of advacd dvics such as diods, lasrs ad magtic hard discs What is charactristic of lctric, optical ad magtic proprtis? To aswr this qustio, w should spcify som appropriat xprimtal stup, i which w wish to masur a giv proprty Focusig o th prturbatios, w might thik of somthig lik th scris show i th Fig I Hr, th bar rprsts a sampl that w wat to charactriz Th lctrical stup i th uppr pal is probably th most familiar typ of stup I this cas, th prturbatio is th voltag applid btw th d poits Th rspos is th currt, which flows i th circuit Hc, prturbatio ad rspos ca b dtrmid xprimtally by mas of a voltmtr ad amp mtr, rspctivly Rally, such a masurmt is tricky bcaus ot oly th sampl plays a rol i dtrmiig th currt Rsistacs i th wirs ad powr supply i gral caot b igord ad so mor advacd mthods (such as four-poit masurmts) may b dd But what is th rol of th sampl? Wll, for a voltag V th currt I is limitd by th lctrical rsistac R= V / I or, if w lik, w could masur th coductac G= I / V= / R Th poit hr is that ths quatitis dpd o both th matrial i th sampl ad shap of th sampl For a larg homogous bar of lgth L ad cross sctio A th 4

6 rsistac is xpctd to vary as R= ρl / A, whr ρ is th spcific rsistac of th matrial I tur, ρ is a charactristic of th matrial ad dpds oly o xtral paramtrs such as tmpratur ad prssur ad so o Figur I Schmatic illustratio of lctric, magtic ad optical prturbatios of sampl Nxt, w tur to th magtic rspos As show i th middl pal of th figur, w could imagi placig a sampl btw th pols of a magt Hc, th prturbatio i this cas is th magtic fild ptratig th sampl What happs isid th sampl? Th aswr to this qustio dpds o th atur of th sampl W cosidr first th cas of so-calld o-magtic matrials Imagi th xtral magtic fild itsity H ptratig th sampl Isid th sampl, th lctros act as tiy bar magts thmslvs A charactristic proprty of o-magtic matrials is that ths tiy bar magts poit i all dirctios with qual probability if th xtral magtic fild is switchd off Howvr, wh th xtral fild H is applid thy will try to alig with this fild Th rsult is that th matrial bcoms magtizd ad w say that a crtai magtizatio M has b iducd Sic th magtizatio is iducd by th magtic fild w xpct a liar dpdc if th 5

7 fild is ot too larg: M= χ M H, whr χ M is th magtic suscptibility Hc, i this cas, χ M is th importat matrial quatity, which w wish to dscrib Masurig χ M is lss simpl tha masurig g th rsistac dscribd abov O possibility is to us a flux mtr to rcord th flux of th total magtic fild B= µ ( H+ M) = µ ( + χ ) H through a wir loop as th xtral fild H is turd o M Th xcptio to this bhavior of o-magtic matrials is foud i magtic matrials Ths ar substacs, i which a fiit magtizatio ca xist v without a xtral fild Th challg i this cas is to udrstad this phomo ad to s how such a matrial will b iflucd by a xtral fild A xtrmly importat aspct of this cas is th chag i th dirctio of th magtizatio that ca b iducd by a xtral fild This forms th basis for magtic storag dvics Th optical rspos is, i fact, quit similar to th magtic o I this cas, a highfrqucy lctric (ad magtic) fild E is icidt o our sampl ad this fild displacs th chargs i th matrial I th optical rgim, th rlvat chargs ar lctros as th ucli rspod primarily to filds of much lowr frqucy Th displacd lctro chargs ar dscribd by th polarizatio P ad i prfct aalogy with th magtic cas w xpct a liar rlatio P εχe, whr χ is ow th lctric suscptibility I tur, χ dtrmis th dilctric costat of th matrial ε= + χ ad also th rfractiv idx = + χ = Hc, ths quatitis ar th rlvat matrial proprtis Masurig th rfractiv idx is i itslf a complicatd task Th imagiary part rvals itslf i absorptio masurmts, whil th ral part is rsposibl for rfractio ad itrfrc ffcts Th tchiqu kow as llipsomtry is dvlopd for th purpos of accuratly masurig ths quatitis So far, th discussio has b quit gral ad applicabl to may circumstacs Th focus of ths ots, howvr, will b o aostructurs Th raso is simply that our ability to dsig ad fabricat structurs o a aomtr scal has xpadd th rag of lctric, optical ad magtic phoma i a truly amazig mar Thr ar svral rasos for this Most importatly, th ruls of physics ar diffrt o th aoscal Prooucd quatizatio ffcts appar ad, hc, scalig bulk rsults such as R= ρl / A ito th aoscal simply dos t work if th cross sctio A bcoms sufficitly small A dramatic cosquc i this particular cas is th apparac of quatizd coductac, i which idividual quatum lvls dtrmi th rsistac Similarly, lctric ad magtic suscptibilitis ar tirly diffrt i low-dimsioal structurs This ops th widow to ovl phoma as wll as brad w applicatios Ev if th matrial proprtis ar almost idtical to th bulk valus, w ffcts ca appar if th siz of th sampl is i th aoscal A importat xampl is foud i aooptics Hr, optical compots of a siz comparabl to th wavlgth ar usd ad this dramatically chags th way light 6

8 propagats, diffracts ad so o Actually, compltly w mtamatrials ca b fabricatd i this fashio Through ths lctur ots w hop to display th diffrcs btw th proprtis of bulk matrials ad aostructurs W do this by dscribig both so that a compariso ca b mad Our approach cosists i th formulatio of a vry gral framwork for th rspos to xtral prturbatios i th bulk as wll as i aostructurs W th tur to all th diffrt lctric, optical ad magtic applicatios of this gral framwork i th subsqut chaptrs Th ots ar itdd for studts i th fial yar of udrgraduat study havig alrady stablishd a solid kowldg i quatum mchaics, statistical mchaics, solid stat physics as wll as optics ad lctromagtism Svral popl mad hlpful commts o th mauscript ad, i particular, w wish to thak Thomas Bastholm Lyg, Kjld Pdrs, Mads Lud Troll ad Jspr Jug 7

9 Tim-dpdt Prturbatio Thory I ordr to dscrib th proprtis of a systm w d to dscrib th rspos of th systm to xtral prturbatios such as a lctromagtic fild Th xtral prturbatio may b tim-dpdt so what is dd is tim-dpdt prturbatio thory I gral, th rspos of th systm ca b vry complicatd I th prst discussio, howvr, w rstrict ourslvs to liar rspos thory, i w oly cosidr chags to th wav fuctio of th systm that ar liar i th prturbatio Th startig poit of th discussio is th uprturbd systm Th uprturbd systm is dscribd by a tim-idpdt Hamiltoia Ĥ W assum that all igstats of Ĥ ar kow, i that w kow th solutios of th statioary Schrödigr quatio Hˆ ϕ = Eϕ, () whr ϕ is th statioary wav fuctio, which dpds o all spac coordiats but ot o tim, ad E is th corrspodig rgy igvalu I a rgy igstat, th full tim-dpdt wav fuctio is giv by ϕ xp( ie t / ħ ) W ow itroduc th prturbatio W assum that th tim-dpdc is harmoic, i charactrizd by a sigl frqucy I fact, th liar rspos to prturbatios with mor complicatd tim-dpdc ca b costructd from a sris of harmoic prturbatios usig Fourir aalysis Th Hamiltoia thrfor chags from Ĥ to ˆ ˆ ω i t ˆ iωt H+ H + H, whr Ĥ cotais th spatial part of th prturbatio, ω is th frqucy ad th cojugatio i th last trm ( ) is i th oprator ss ( Hrmitia cojugatio ) Typical prturbatios ar lctric or magtic filds that itract with lctros via thir charg or spi Liar Rspos Thory Th ky to dscribig th rspos to th prturbatio is tryig to solv th timdpdt Schrödigr quatio ψ { ˆ ˆ iωt ˆ iωt iħ = H + H + H } ψ t This is grally a impossibl task but w r hlpd by th fact that w r oly lookig for th liar chag to th wav fuctio Our kowldg of th ϕ s ca b usd if w writ th ukow ψ i th followig form 8

10 ψ = a iet / ϕ ħ, () whr a is a ukow tim-dpdt cofficit Ay ψ ca b writt i this form sic th ϕ s costitut a complt st W xt isrt this xprssio i th tim-dpdt Schrödigr quatio { ˆ ˆ iωt ˆ iωt iħ aϕ = H+ H + H } aϕ t iet/ ħ iet/ ħ a iet/ / { ˆ ˆ iωt ˆ iωt iet aeϕ i ϕ ħ ħ + ħ = a Hϕ+ H ϕ + H ϕ } t Du to th igstat coditio Eq(), th first trms o th right-had ad lfthad sids cacl W cosqutly fid a t ϕ = a Hˆ + Hˆ ϕ iωt iωt { } iet/ ħ iet/ ħ iħ (3) Nxt, w xploit th orthogoality btw th igstats if = m ϕmϕ = δm=, othrwis whr ϕmϕ is th shorthad Dirac otatio for th itgral = ϕ ϕ ϕϕ d r d r 3 3 m m N ovr all th coordiats of th N lctros If th lctro spi is cosidrd, th itgratio is ovr th spi variabls as wll Hc, w multiply Eq(3) by ϕ ad itgrat to fid m a t iωt iωt { } δ = a ϕ Hˆ ϕ + ϕ Hˆ ϕ a t iet/ ħ iet/ ħ m m m iħ m iωt iωt { ϕ ϕ ϕ ϕ } ˆ ˆ iemt/ ħ a m H m H, iħ = + (4) whr Em= Em E Agai, th Dirac otatio is usd for matrix lmts such as 9

11 = ϕ Hˆ ϕ ϕ Hˆ ϕ d r d r 3 3 m m N To procd, w ow d th importat xpasio of th ukow cofficits a i powrs of th prturbatio For th lctromagtic prturbatios cosidrd i th prst discussio, th prturbig Hamiltoia is proportioal to th fild strgth, i Ĥ E ad Ĥ B for lctric ad magtic prturbatios, rspctivly Hc, ay cofficit a may b cosidrd a fuctio of th fild strgth, g a= a ( E ) i th lctric cas W may cosqutly mak a Taylor xpasio i fild strgth, i () () a= a + a + 'th ordr 'st ordr, whr th suprscript idicats th powr of th prturbatio W ow utiliz a familiar thorm from polyomial sris: p p If, for all x, b x = c x, th b = c p p p p p p Usd i Eq(4) this implis that th powrs of th prturbatio o both sids must b qual, i a t ( p) m iωt iωt { ϕ ϕ ϕ ϕ } ( p ) ˆ ˆ iemt / ħ a m H m H, iħ = + (5) sic th matrix lmts ϕ ˆ m Hϕ ad ϕ Hˆ ϕ o th right-had sid alrady cotai o powr of th prturbatio Th tchiqu ow cosists i startig by sttig p = i this xprssio Subsqutly, w st p= ad so o I fact, for th liar rspos w d ot go byod p= Th so-calld oliar rspos ca b foud by rpatig this xrcis to highr ordrs [] Sttig p=, w fid m () a m t =, i th zro th ordr cofficits ar costat This is quit obviously corrct sic zro th ordr mas that th prturbatio is igord altogthr Havig stablishd this simpl rsult, w ca immdiatly procd to p= for which Eq(5) yilds a t () m iωt iωt { ϕ ϕ ϕ ϕ } () ˆ ˆ iemt / ħ a m H m H iħ = +

12 W itgrat this xprssio to fid iωt iemt/ ħ iωt iemt / ħ { ϕ ϕ ϕ ϕ } a = a Hˆ + Hˆ dt () () m m m iħ / / iωt iemt ħ iωt iemt ħ () ˆ ˆ = a ϕm Hϕ ϕm Hϕ + Em ħω Em+ ħω (6) Th lowr limit of th itgral dos ot cotribut if w assum that th prturbatio was turd off i th ifiit past Physically, th rsult Eq(6) is a idalizatio bcaus w hav igord losss that td to d-xcit th systm, i mak th systm dcay back to th groud stat To icorporat ths ffcts, w may itroduc a crtai dampig ħ Γ i th xprssio abov a ϕ Hˆ ϕ ϕ Hˆ ϕ iωt iemt / ħ iωt iemt/ ħ () () m m m = a + Em ħω iħγ Em+ ħω iħγ (7) W ar ow i a positio to calculat th xpctatio valu of ay tim-idpdt oprator corrspodig to som masurabl quatity W dot this oprator (rspos obsrvabl) by ˆX so that what w wat is ψ ˆX ψ W thrfor us Eq() ad kp oly trms up to liar ordr i th prturbatio { } ˆ () () () () () () ˆ ie mt/ ħ m + m + m ϕ ϕm m, ψ Xψ a a a a a a X To itrprt this rsult, w s what ormalizatio of th total wav fuctio tlls us Usig Eq() w s that ψψ a, = = usig th orthogoality of th w hav ϕ s Thus, if th prturbatio is abst ad a = a () () a = (o prturbatio) As usual, w itrprt stat () a as th probability that th uprturbd systm is i th ϕ I thrmal quilibrium, this mas that a () = f ( E ), whr f is th probability distributio, which dpds o th rgy of th stat oly W us this proprty to postulat th followig:

13 f ( E ) if () () = m a am = othrwis This postulat ca b mad mor rigorous usig th so-calld dsity matrix formulatio of prturbatio thory [] W will simply accpt it as a rasoabl postulat hr W subsqutly apply Eq(7) to show that ψ Xˆ ψ f ( E ) ϕ Xˆ ϕ ˆ iωt ˆ iωt ϕm Hϕ ϕm Hϕ f ( E ) ˆ ϕ Xϕ m + m, Em ħω iħγ Em+ ħω iħγ ˆ iωt ˆ iωt ϕm Hϕ ϕm Hϕ f ( E ) ˆ m ϕ Xϕ m + m, Em ħω+ iħγ Em+ ħω+ iħγ If this rsult is Fourir dcomposd ito frqucy compots it is foud that ˆ ˆ iωt iωt ψ Xψ f ( E ) ϕ Xϕ X( ω) = + + X ( ω) (8) Hr, th tim-idpdt first trm is th prmat cotributio, which xists i th absc of th prturbatio, whil compariso with th prvious rsult shows that X( ω) = m, f m ϕ Hˆ ϕ ϕ Xˆ ϕ m m E m ħω iħγ, (9) whr f f ( E ) f ( E ) This is th tim-dpdt iducd rspos du to th m m prturbatio It is otd that th form Eq(8) surs that th rspos is ralvalud At zro tmpratur, th groud stat is kow with crtaity to b occupid whil all othr stats ar mpty Hc, th oly o-vaishig f m trms ar f m= ad f =, whr ad m ar both largr tha For this raso w fid ϕ ˆ ˆ ˆ ˆ Hϕ ϕ Xϕ ϕ Hϕ ϕ Xϕ X( ω) = + ( T= ) > E ħω iħγ E + ħω+ iħγ ()

14 at zro tmpratur Th xprssios i Eqs(9) ad () ar th fudamtal rsults of liar rspos thory Thy costitut th basis upo which all subsqut rsults ar built Exrcis: High-frqucy limit I this xrcis, w will ivstigat th rspos of a aostructur subjctd to a prturbatio with a vry high frqucy If ħω ħ Γ ad w chag th ordr of crtai trms, w may, first of all, writ Eq() as ϕ Xˆ ϕ ˆ ˆ ˆ ϕ Hϕ ϕ Hϕ ϕ Xϕ X( ω) > ħω E ħω+ E Scodly, w will assum that ħω E for all th importat trasitios i th sum a) Show that usig a gomtric sris E = + ħω± E ħω ( ħω) W also d th compltss proprty of th stats, which mas that ϕ ϕ = If th groud stat = is xcludd from th summatio it follows that ϕ ϕ = ϕ ϕ > b) If oly th first trm of th gomtric sris is rtaid, show that X( ω) ϕ X ˆ, H ˆ ϕ ħω, whr Xˆ, Hˆ is th commutator btw th two oprators Rfrcs [] RW Boyd Noliar Optics (Acadmic Prss, Sa Digo, 99) 3

15 Prlimiaris Bfor w start sriously applyig th rsults of prturbatio thory to ral aostructurs or bulk matrials it is usful to cosidr som gral faturs of prturbatios ad rsposs Th most commo prturbatios ad, also, th os tratd i this work ar: Static or low-frqucy lctric filds Static or low-frqucy magtic filds High-frqucy lctric filds (light) I additio, two or mor of ths prturbatios may b prst simultaously Exampls of prturbatios that ar ot cosidrd hr ar mchaical os such as prssur or mchaical strss Th most importat combiatios of prturbatio ad rspos for solids ar show i th tabl blow Prturbatio Rspos Elctric currt Polarizatio Spi magtizatio Orbital magtizatio Elctric fild Magtic fild Elctric + magtic fild Elctric Hall ffct coductivity Elctric Faraday ffct suscptibility Spi-magtic suscptibility Orbital-magtic suscptibility Th lctric ad magtic prturbatios ar itimatly rlatd to th lctric ad magtic dipol momts Classically, th chags i rgy associatd with a lctric fild E or magtic fild B ar µ E ad µ m B, whr µ ad µ m ar lctric ad magtic dipol momts, rspctivly For a quatum mchaical systm of lctros, th lctric dipol oprator is µ = r, r = r i Hr, th sum is ovr all th lctros Th magtic itractio is a littl mor complicatd bcaus both orbital ad spi agular momtum cotributs Hc, µ = µ + µ with m orb spi i 4

16 µ = µ l, µ = µ s, l = l, s = s, orb B spi B i i i i whr µ B=ħ /m is th Bohr magto Hr, l ad s ar th total agular ad spi agular momtum, rspctivly Th factor i th xprssio for µ spi is actually th g-factor of th lctro ad should b rplacd by g 3 if quatum-lctrodyamic ffcts ar icludd Th rspos obsrvabls, i tur, ar also rlatd to th dipol momts Thus, polarizatio P ( ω) is othig mor that th iducd lctric dipol momt pr volum ad magtizatio M ( ω) is th iducd magtic dipol momt pr volum It follows that th associatd obsrvabls ar ˆ Polarizatio P( ω ) : X= µ / Ω= r / Ω ˆ Orbital magtizatio Morb ( ω ) : X= µ Bl / Ω ˆ Spi magtizatio M ( ω ) : X= µ s / Ω, spi whr Ω is th crystal volum Th oly additios dd to this list ar thos rlatd to lctric currts I this cas, th prturbatio is th actio of th lctrostatic pottial V( r i ) ad th obsrvabl is th currt dsity oprator is i ˆ J = p ˆ / Ωm, whr p ˆ p ˆ is th total momtum oprator As w shall s, = i i howvr, this cas is also dscribd by th lctric cas dscribd abov Th actual calculatio of rspos fuctios always rlis o th computatio of matrix lmts such as ϕ oˆ ϕ, whr ô could b ay of th Hrmtia oprators m coutrd abov May appartly diffrt matrix lmts ar closly rlatd as ca b provd by a clvr trick that gos as follows: W start from th matrix lmt of th commutator btw th Hamiltoia Ĥ ad th oprator ô B ϕ Hˆ, oˆ ϕ ϕ Hˆ oˆ oh ˆ ˆ = ϕ m m = ϕ Hˆ oˆ ϕ ϕ oh ˆ ˆ ϕ m m = ϕ oh ˆ ˆ ϕ ϕ oh ˆ ˆ ϕ, m m whr th last quality follows from th fact that both Ĥ ad ô ar Hrmitia W th utiliz th actio of Ĥ o th igstats 5

17 ϕ Hˆ, oˆ ϕ = E ϕ oˆ ϕ E ϕ oˆ ϕ m m m m = E ϕ oˆ ϕ E ϕ oˆ ϕ = E m m m ϕ oˆ ϕ m m () O th othr had, may commutators ca b calculatd xplicitly W may cosidr just a sigl lctro ad for th z compot of th positio w fid Hˆ, z ħ ϕ=, z ϕ m z ħ z z = ϕ + ϕ+ z ϕ z ϕ m z z z z z ħ = ϕ m z ħ = pˆ zϕ im Thus, i fact Hˆ ˆ, z = ( ħ / im) p z Similarly, for th lctrostatic pottial Hˆ, V ħ ϕ, V = ϕ m ħ = ( ϕ V+ V ϕ+ V ϕ V ϕ) m ħ = ( ϕ V+ V ϕ) m Now, E = V ad if th lctric fild is assumd costat i spac th V = Hc, Hˆ i ˆ, V ϕ ħ = p m ϕ ħ E = E m ϕ As a cosquc, oly momtum matrix lmts ar rally dd i both cass Th rsults ca b summarizd as follows ħ ϕ ˆ m pzϕ = Em ϕm zϕ, im iħ ϕ ˆ me pϕ = Em ϕm Vϕ m () 6

18 A First Exampl: Elctric Polarizability W will ow illustrat som of th cocpts abov by workig our way through a simpl xampl: th lctric polarizability of a sphrical quatum wll with ifiit barrir cofimt This modl ad th first fw igstats ar illustratd i th figur blow I this situatio, th prturbatio is th lctric fild E ad th rspos is th iducd dipol momt d= ψ rψ For simplicity, w assum that oly a sigl lctro is i th systm Also, a isotropic systm is assumd, which mas that a lctric fild alog th z-axis E = z E iducs a dipol momt i th sam dirctio d= d From th gral framwork prstd i th prvious chaptr it th follows that z d( ω) = m, = E f m, m f m ϕ E zϕ ϕ zϕ m m m E m ħω iħγ ϕm zϕ E ħω iħγ Th costat of proportioality btw d ad E is th polarizability αω ( ) giv by ϕ zϕ ħ ħ m ( ) = fm m, Em ω i Γ αω At low tmpratur, w ca similarly to Eq() utiliz th fact that th oly ozro f m trms ar f m= ad f =, whr ad m ar both largr tha ad aftr r-lablig αω ( ) ϕ zϕ ϕ zϕ E ħω iħγ E ħω iħγ = = > ϕ zϕ > E ħ ( ω+γ i ) E This xprssio obviously lads to rsoacs whvr ħ ω= E, i whvr th photo rgy matchs th rgy diffrc btw th groud stat ad a xcitd stat Th dimsiolss quatity m g = ϕ zϕ E ħ 7

19 is kow as th oscillator strgth of th ϕ ϕ trasitio Thus, ħ g αω ( ) = m E i ħ ( ω+γ) Not that w rmovd th rstrictio o i th sum bcaus g = W ow wish to ivstigat th high ad low frqucy limits I th formr cas, w may assum that ħω E for all importat trasitios ad so αω ( ) g mω To valuat th sum ovr oscillator strgths w utiliz Eq() m g = ϕ zϕ E ħ m = ħ = { ϕ ˆ ˆ zϕ ϕ pzϕ ϕ pzϕ ϕ zϕ} iħ { ϕ zϕ ϕ zϕ E E ϕ zϕ ϕ zϕ} Now, applyig th compltss of th stats ϕ ϕ = ad th commutator rlatio [ ] pˆ, z= Nħ / i (for N-lctro oprators), w s that z i g [ ˆ = ϕ pz, z] ϕ = N ħ This gral rsult is kow as th Thomas-Rich-Kuh sum rul I th prst xampl, N = ad g = ad so αω ( ) (high frqucy limit) mω Covrsly, i th low frqucy limit ϕ zϕ ( ) (low frqucy limit) αω E > I ay sphrically symmtric systm (a atom or sphrical aoparticl), th igstats ar of th form ϕ ( r ) = Y ( θφ, ) R ( r) with th agular dpdc i lm lm l 8

20 th form of a sphrical harmoic Th groud stat is ϕ( r ) = Y ( θφ, ) R( r) Morovr, z quals r cosθ i polar coordiats that, i tur, ca b writt as ry ( θφ, ) 4 π /3 Usig Y ( θφ, ) / 4π = w thrfor radily s that 3 lm z R( r) R( r) r drδδ l m ϕ ϕ = 3 Th Krockr dltas follow from th orthogoality of th sphrical harmoics Y Y = δ δ Thus, th groud stat oly coupls to p-typ xcitd stats For th lm l m sphrical quatum wll with ifiit barrirs ad radius a (s Appdix ) βr βr βr { } πr a R( r) = si, R( r) = si cos r a a r β si β a a a Hr, β is th th root of th st sphrical Bssl fuctio, i siβ β cosβ= Numrically, th first fw ar β {44934, 7755, 94,} Th groud stat ad th first two p-typ xcitd stats ar show i Fig Figur Modl of a sphrical quatum wll with ifiit barrirs (lft) ad som of th first igstats displacd vrtically by th rgy (right) From ths stats, th dipol matrix lmts ar asily calculatd 3 R r R r r dr 3 si 3 4πβ a siβ ( ) ( ) = β π β β ( ) 9

21 Also, th rgis ar fid E =ħ π ad /ma g E 6πβ si β 4 = 3 ( β π)( β β) 3 si =ħ β W cosqutly /ma Summd ovr th first 5 stats, w fid g = , which illustrats th Thomas-Rich-Kuh sum rul Similarly, th static polarizability bcoms ma 64πβ si β ma α() = 363 ħ 3 si ħ 5 ( β π)( β β) I trms of th charactristic rgy of th sphrical quatum wll ESQW =ħ /ma, th rsoacs ħ ω = E E ar foud at ħω / ESQW= β π {3, 498, 93,} I th plot blow illustratig th gral rsult, ths rsoacs ar clarly oticabl Figur Th frqucy dpdt polarizability of a sphrical quatum wll Thr rsoacs ar foud i th dpictd rgy rag Exrcis: O-dimsioal quatum wll I a simpl quatum wll strtchig btw z=± L/ with ifiit barrirs, th igstats ar of th form (s Appdix ) cos( zπ / L) odd ϕ( z) =, L, si( zπ / L) v

22 ad th rgis ar =ħ π / E ml a) Vrify that ϕ zϕ 6L =, ϕ zϕ = 3π Th gral rsult is ϕ zϕ 6 L( ) =, ϕ zϕ = π (4 ) + b) Show that, accordigly, th oscillator strgth is g 56 = π (4 ), 3 c) By dirct calculatio, show that g, 96 ad g4, 3 This shows that th sum rul for oscillator strgths is vry arly satisfid with oly a fw trms d) Us th abov rsults to dmostrat that 4 ml α() = πħ (4 ) = Th sum may actually b valuatd xactly with th rsult π (5 π ) = Th t polarizability is thrfor 5 (4 ) 88 = 4 (5 π ) ml α() = 4 πħ

23 3 Bulk Rspos of Mtals I th prst chaptr, w rstrict th atttio to mtals Mtals ar grally dividd ito obl mtals, trasitio mtals ad alkali mtals Tchologically, th first two groups ar th most importat os Th obl mtals iclud coppr (Cu) ad gold (Au), that ar of grat importac for lctroics ad coductors Trasitio mtals cout matrials such as iro (F) ad titaium (Ti) of importac for lctric motors, stl costructios ad high strgth alloys A otabl additio to ths mai classs is alumium (Al), which is a so-calld fr-lctro mtal, ad also of imms importac i may diffrt aras Th bad structur of coppr is show i Fig 3 This pictur is charactristic of obl mtals i that it faturs som rlativly flat d-bads wll blow th Frmi lvl ad a sigl sp-bad crossig th Frmi lvl Figur 3 Bad structur of coppr with th Frmi lvl idicatd by th dashd li A xtral prturbatio cocts a occupid to a mpty lvl Thus, if th typical rgy of th prturbatio is rathr small, oly stats i th viciity of th Frmi lvl ar of importac Exampls of this ar static lctric ad magtic filds of modst fild strgth as wll as low-frquccass, hc, w oly hav to cosidr th sigl sp-bad crossig th Frmi lvl i (ifrard) lctromagtic filds For ths ordr to dscrib th rspos 3 Spi Magtizatio Th simplst xampl of prturbatio thory i bulk mtals is th spi magtizatio I this cas, w cosidr as th prturbatio th couplig btw a magtic fild B ad th magtic momt µ sˆ of a lctro B z Hˆ = µ B sˆ, B z

24 whr B is th magtic fild strgth, ad s ˆz is th z-compot of th lctro spi Th rspos obsrvabl w r lookig for is Xˆ = µ sˆ / Ω, whr Ω is th crystal volum This oprator dscribs th avrag magtic momt pr volum Th rspos itslf is th spi magtizatio for which w us th symbol M ( ω) Bfor actually calculatig th rspos w d to cosidr th followig poit: Th prturbatio ad rspos itroducd i chaptr wr for th tir systm cotaiig may particls O th othr had, th prst xprssios for Ĥ ad ˆX ar for a sigl lctro oly It ca b show, howvr, that if all th lctros ar idpdt th th rspos xprssio B z X( ω) = m, f m ϕ Hˆ ϕ ϕ Xˆ ϕ m m E m ħω iħγ i fact still holds but with a w maig: Now that Ĥ ad ˆX ar sigl-particl oprators, ϕ ad E should b tak as sigl-particl igfuctios ad igvalus Also, svral sigl-particl igstats rathr tha just th mayparticl groud stat ar ow occupid (blow th Frmi lvl) Sic this is th first applicatio of liar rspos thory, w kp thigs simpl ad rstrict ourslvs to th spcial cas of zro frqucy ad zro dampig, i ω=γ= Thus, w wish to calculat th static rspos fm X() ˆ ˆ = ϕm Hϕ ϕ Xϕm E m, m As mtiod abov, th prturbatio rgy is vry small for a static magtic fild ad, hc, w may approximat so that fm f ( E ) f ( E+ Em ) f ( E) = f ( E ), f ( E) = (3) E E E m m X() f ( E ) ϕ Hˆ ϕ ϕ Xˆ = ϕ (3) m, m m Th prst xampl is simplifid furthr by th fact that both Ĥ ad ˆX oly oprat o th spi part of th wav fuctio For a ifiit, priodic solid th igstats ca b labld by thr quatum umbrs: Th bad idx c, th Bloch 3

25 wav vctor k ad th spi σ Hc, th stat idx abov is rplacd by { c kσ } Th igstats thmslvs ar Bloch stats of th form ik r c kσ = ϕ ( r ) σ, ϕ ( r ) = u ( r ), c k c k c k (33) whr u is th lattic-priodic part ad σ cotais th spi part W cosidr c k oly a sigl bad ad so w bgi by valuatig th Ĥ matrix lmt as follows σ ˆ σ µ ϕ ϕ σ σ 3 c k H ˆ c k = BB ( r ) ( r ) d r s c k c k z / σ= µ Bδ δ / = B k, k σσ σ = I xactly th sam mar, ˆ µ σ σ = ϕ ϕ σ σ Ω µ / σ= B = δ k, k δ σσ Ω / σ = B 3 c k X c k ( r ) ( r ) d r sˆ c k c k z I th static rspos Eq(3), th summatios ovr ad m ar ow ovr { kσ } ad { kσ } Each sum ovr wav vctors is rstrictd to th Brilloui zo ad th spi summatios covr th two possibilitis {, } Thr sums ca b carrid out immdiatly ad th static magtizatio is M() = f ( E ) c k σ Hˆ c kσ c kσ Xˆ c k σ kσ, k σ = kσ, k σ µ BB = Ω c k µ B f ( E )µ c k BBδ δ δ δ k, k σσ k, k σσ Ω 4 k f ( E ) c k At sufficitly low tmpraturs, th drivativ of th Frmi fuctio is approximatly a dlta fuctio f ( E ) δ( E E ) ad thrfor c k c k M() µ B D( E ), B F F 4

26 whr ( F ) = Ω δ( F ) k c k D E E E is th dsity of stats at th Frmi lvl This rsult is kow as Pauli paramagtism W fially itroduc th suscptibility via χ = µ M()/ B= µµ D( E ), which is kow as th Pauli paramagtic suscptibility 3 Elctric Currt M B F Th coductivity of lctros i a lctric fild ca b hadld much lik th prvious cas Th lctric fild is tak to b polarizd i th z-dirctio E = z E Th prturbatio o th lctros is giv by V( r ), whr V is th lctric pottial rlatd to th fild via E = V This mas that th currt rspos is giv by χ M ϕ ˆ m Vϕ ϕ Jϕm J( ω) = fm, E ħω iħγ m, m (34) whr th currt dsity oprator is Jˆ = pˆ / Ωm, i qual to charg tims vlocity Th matrix lmts of this oprator is, thrfor, giv by ϕ Jˆ ϕ = ϕ pˆ ϕ m Ω m z m Usig Eq () to valuat matrix lmts of th pottial, th xprssio for th currt bcoms z iħe f ϕ ˆ m m pzϕ mω m, Em Em ħω iħ J( ω) = Γ (35) W ow agai apply Eq(3) du to th smallss of rlvat E m ad, i additio, w glct E m i th frqucy dpdt domiator abov so that ie J( ω) f ( E ) ϕ pˆ ϕ m Ω ( ω +Γ i ) m, m z Nxt, w d th matrix lmts of th momtum oprator Du to th Bloch form of th igstats Eq(33) it follows that 5

27 { ˆ } c k σ pˆ c kσ = δ δ ħ k + u p u (36) z σσ k, k z c k z c k Ths matrix lmts, howvr, ca b rlatd to th rgy usig a fw tricks First, w ot that, as usual, our sigl-particl stats ar igstats of a Hamiltoia ˆ H = pˆ /m+ U so that pˆ ik r U + u ( r ) = E u ( r ) c k c k c k m hˆ ( k ) u ( r ) = E u ( r ), c k c k c k ik r whr pˆ ħ k ħk p ˆ = (37) m m m hˆ ( k ) U ik r This k-dpdt Hamiltoia is obtaid by carryig through th phas factor Hc, multiplyig o both sids by u ad itgratig dmostrats that c k E = u hˆ ( k ) u W ow wish to s how this rgy varis with k z Hc, w c k c k c k tak th drivativ E ˆ c k ˆ h ( k ) ˆ = u h( k) u + u u + u h( k) u c k c k c k c k c k c k k z k z k z k z By mas of th igvalu coditio, this ca b r-xprssd as E hˆ ( k ) k k k c k = E u u + u u c k c k c k c k c k z z z hˆ ( k ) = u u c k c k kz Th first cotributio vaishs sic u u = This rsult is a xampl of th famous Hllma-Fyma thorm It follows usig Eq(37) that c k c k E k c k z { k ˆ z u p c k z u c k } = ħ ħ + (38) m 6

28 This rsult allows us th rwrit th gral matrix lmt Eq(36) as E c k c k σ pˆ z c kσ = δ δ mv ( ), ( ) k, k c k vc k σσ ħ k z (39) This xprssio shows that th momtum matrix lmt is mass tims bad vlocity vc( k ) It should b mphasizd, howvr, that this rsult is oly valid for trasitios withi a sigl bad For trasitios btw bads, so-calld itrbad trasitios, th momtum matrix lmts ar mor complicatd Prcisly as i th magtizatio xampl, th summatios ovr ad m ar rplacd by sums ovr ad { kσ } { kσ} giv by ad upo prformig th sums ovr σσ, ad k th currt is i E c k J( ω) = E f ( E ) c k ħω ( ω+γ i ) k k z Agai, at low ough tmpraturs i E c k J( ω) E δ( E E ) c k F ħω ( ω+γ i ) k kz This rsult may b writt i a brifr form by itroducig th plasma frqucy via ω p ω p E c k ħε Ω k k z δ( E E ) c k F (3) Thus, if th coductivity is dfid via Jω ( ) = σω ( ) E th iεω p σω ( ) = (3) ω +Γ i This is th famous Drud form of th lctric coductivity It rlats th iducd currt dsity to th applid tim-dpdt lctric fild I a similar optical sttig, a pic of mtal might also b subjctd to a tim-dpdt lctric fild du to th light sourc I that cas, w would look for th iducd polarizatio as th rspos Thus, o could xpct that thr is a coctio btw currt ad polarizatio P ad, idd, thr is Th coctio follows from th fact that polarizatio is rlatd to charg dsity ρ ( t) via ρ ( t) = P ( t) Morovr, th 7

29 cotiuity rlatio ρ( t)/ t= J ( t) rlats charg ad currt dsitis Put togthr, J( t) = P ( t)/ t or, for harmoic tim variatio, J( ω) = iωp ( ω) Now, just as Jω ( ) = σω ( ) E w ca writ P( ω) = εχω ( ) E, whr χω ( ) is th suscptibility I tur, χ( ω ) is rlatd to th dilctric costat εω ( ) via εω ( ) = + χω ( ) Puttig it all togthr, it follows that i εω ( ) = + σω ( ) εω I particular, for th itrabad rspos of a mtal dscribd by th Drud xprssio w fid ωp εitra( ω) = ωω ( +Γ i ) As a approximatio th itrbad cotributio may b tak to b roughly idpdt of frqucy ad so a simplifid xprssio for th total dilctric costat is giv by ωp εω ( ) = ε, ωω ( +Γ i ) (3) whr ε is a costat rprstig th high-frqucy itrbad rspos I Fig 3, th optical rspos of bulk silvr is cosidrd [] For this matrial th appropriat paramtrs i th Drud modl ar ħω p = 93 V, ħγ= 3 V ad ε = 5 As th plot dmostrats, th Drud form is quit accurat at low frqucis Figur 3 Compariso of xprimtal [] ad Drud modl dilctric costat of Ag 8

30 Exrcis: Proprtis of fr-lctro mtals For a tru fr-lctro mtal show that a) th dsity of stats is 3 D( EF ) =, whr is th lctro dsity, ad thrfor E F 3µµ χm E F B, b) th squard plasma frqucy is ω p = ε m Rfrcs [] PB Johso ad RW Christy, Phys Rv B6, 437 (97) 9

31 4 Elctric Currts i Naostructurs Th lctric proprtis ad i particular lctric currts ar gratly iflucd by quatum cofimt ffcts i aostructurs ad tirly w faturs aris O of th most strikig faturs is quatizatio of coductac so that udr idal coditios, th coductivity bcoms a itgr multipl of th fudamtal coductivity quatum / h I additio, importat ffcts aris wh th distac travld by charg carrirs bcoms comparabl to or lss tha th ma fr path, i th avrag distac travld btw scattrig vts du to g phoos or impuritis I this ballistic rgim, cohrc of th lctroic wav fuctio is maitaid ad w faturs aris from itrfrc ffcts W bgi our study of coductivity i aostructurs by cosidrig th gral stup illustratd i Fig 4 This figur shows a systm through which a lctric currt is passig from lft to right Th systm, which might b a molcul, a smicoductor slab ad so o, is coctd to idal wirs or lads o both sids Through ths lads, a lctric pottial is applid so that a lctric fild xists isid th systm W suppos that w ar i th ballistic rgim so that a lctro ca b faithfully rprstd as a quatum mchaical wav icidt o th systm This wav is th partly rflctd ad partly trasmittd by th systm Hc, th importat charactristics of th systm li i th rflctio ad trasmissio cofficits Figur 4 Illustratio of th aoscopic coductor cosistig of a systm coctd to lads o both sids As i th prvious chaptr, th lctric currt alog th z-axis is calculatd as th rspos to th applid lctric pottial Hc, th rspos xprssio is giv by Eq(35) iħe f ϕ ˆ m m pzϕ mω m, Em Em ħω iħ J( ω) = Γ 3

32 I th prst chaptr, w ll rstrict th discussio to th DC limit ω= Usig Eq(8) it is clar that th total DC currt JDC is JDC= { J() + J ()} /= R{ J()} Hc, w d to xtract th followig ral part ( ) R i i Em i R ħγ ħγ = = Em iħγ E m+ ħγ Em+ ħγ As w ar i th ballistic rgim, w should tak th limit Γ Usig th idtity ε lim = πδ( x), ε x + ε w hav lim R i = πδ( Em ) Γ Em iħγ so that f J pˆ ( E ) πħe m DC= ϕ m zϕ δ m mω m, Em (4) Th first factor i th sum ca ow b foud by takig th limit as th rgy diffrc E m vaishs fm f ( E ) f ( E+ Em ) f ( E) lim = lim = f ( E ), f ( E) = E E E Em Em m m W thrfor fially gt πħe ˆ DC= ϕm zϕ δ m m Ω m, J f ( E ) p ( E ) (4) W ow d to ivstigat i dtail th rlvat lctroic stats that carry th currt I ay of th two lads, th igstats ar of th form ϕ( r) = φ ν ( r)xp( ± ikz)/ Li, whr L i is th lgth of th i'th lad ad φ ν dscribs th bhavior th stat i th dirctios prpdicular to th lad-dirctio I gral, φν also dpds o z ad has th lattic priodicity i this dirctio Th rgy igvalu corrspodig to this stat is dotd E = E ( k ) ν 3

33 I th prst quasi-o dimsioal problm, it is obviously importat to udrstad how th summatios i Eq(4) should b prformd Primarily, summig ovr mas summig ovr a composit idx = { νk σ}, with σ is th spi idx, ad similarly for m= { µ k m σ } Th spi summatio is radily prformd ad simply producs a factor of two Hc, πħe DC= km z k m mω µν,, k, km ( ) ˆ ν ϕµ ϕν δ( µ ν ) J f E ( k ) p E ( k ) E ( k ) Now, th sum ovr k m ca b covrtd to a itgral usig = L /π dk k I additio, w us th dlta-fuctio proprty G( k ) δ = F ( k ) G( k) ( F( k) ) dk, whr k is th root of F, i F( k ) = This idtity holds udr th assumptio that oly a sigl root lis withi th itgratio itrval Th k drivativ of th rgy is basically th bad vlocity, cf Eq (39) W cosqutly fid Eν ( k ) vν ( k ) ħ k J EL ( ( )) f E k ˆ m ( ) ν DC= ϕ µ k pzϕν k m Ω µν,, k v k ν (43) It is importat to ot that i this xprssio km th coditio E ( k ) = E ( k ) µ m ν is udrstood to b dtrmid by 4 Matrix Elmts W ow s what happs wh th systm rgio is isrtd btw th two lads If a lctro is propagatig to th right i th lft lad it coutrs th systm ad two thigs ca happ: ithr it is rflctd back ito th lft lad OR it is trasmittd ito th right lad I both cass, thr is a chac that if th icidt lctro is i th th mod it will b rflctd/trasmittd ito a diffrt mod, say th l th mod Hc, th total wav fuctio is giv by 3

34 ϕ νk ikz iklz N φν ( r ) ρν λφ ( r ) + λ z i 'st lad L λ L =, (44) iklz N ( ) z i 'd lad τν λφλ r λ L whr N is a ormalizatio costat ad ρ ad τ ar rflctio ad trasmissio cofficits, rspctivly Th rquirmt that Eq(44) b a igstat of th tir structur mas that k l ad k ar rlatd via E ( k ) = E ( k ) Th thr parts of th wav fuctio (icidt, rflctd ad trasmittd) corrspod to thr currts As usual, thy ca b calculatd as th matrix lmt of th currt oprator Jˆ = pˆ / Ωm tak ovr th appropriat volums, i th st lad z for icidt ad rflctd ad d lad for th trasmittd currt This procdur yilds λ l ν JI = N vν ( k ), JR = N ρν λ vλ ( kl ) ad JT = N τν λ vλ ( kl ), Ω Ω Ω λ λ whr w, agai, d th vlocitis E ( k ) v ( k ) = φ ( r) pˆ φ ( r) = ikz ikz ν ν ν z ν ml ħ k W ca th calculat th rflctac R ν λ as th ratio btw th (absolut) currt rflctd ito th l th mod ad th icidt currt ad, similarly, th trasmittac T ν λ as th ratio btw th currt trasmittd ito th l th mod ad th icidt currt Ths ratios ar R v ( k ) vλ ( kl ), v ( k ) v ( k ) λ l ν λ= ρν λ Tν λ= τν λ ν ν Cosrvatio of currt implis that Rν λ + Tν λ = l I additio, w ca xprss th ormalizatio costat N i trms of ths factors by itgratig th squar of th wav fuctio I th itgral, w should iclud a cotributio from th systm rgio but if this is assumd much smallr tha th lads w may safly igor it ad so l 33

35 / v ν ( k ) vν ( k ) N = + Rν λ + Tν λ l vλ ( kl ) l vλ( kl ) What w rally d is th momtum matrix lmts to b usd i Eq(43) W thrfor us Eq(44) ad a similar xprssio for ϕ to show that ϕ ˆ µ k p ( ) ( ) ( ) m zϕνk = mn mn vν kδµν vλ klρµ λρν λ+ vλ klτµ λτ ν λ λ λ Now, i ordr to valuat Eq(43) w should sum ovr all igstats, ot oly thos giv by Eq(44) I fact, Eq(44) oly provids th right-travllig half th igstats ad thr is aothr lft-travllig half giv by m ϕɶ = ikz iklz N φν ( r ) ρν λφλ ( r ) L l L iklz N τν λφλ ( r ) z i 'st lad l L z i 'd lad Ths w stats ar asily show to b orthogoal to th old os W subsqutly d w momtum matrix lmts ad thy tur out to b ϕɶ ˆ m pzϕɶ = mn mn v ( k ) δ v ( kl ) ρ ρ + v ( kl ) τ τ l l ϕɶ ˆ m pzϕ = mnmn vλ ( kl ) τµ λρν λ+ vλ ( kl ) ρµ λτ ν λ l l ϕ ˆ m pzϕɶ = mnmn vλ ( kl ) τµ λρν λ+ vλ ( kl ) ρµ λτ ν λ l l ν µν λ µ λ ν λ λ µ λ ν λ Ths ar vry gral rsults but, ufortuatly, rathr complicatd i furthr us W cosqutly itroduc a usful simplificatio 4 Simplificatio: Dcoupld Chals I th xprssios abov, w s that lctros icidt i th th mod may scattr ito th l th mod upo rflctio or trasmissio I this coctio, w spak of th mods as chals i which lctros ar trasportd Obviously, ths chals ar coupld ad a lctro that is iitially i a particular chal may scattr ito aothr as a rsult of th couplig W ow mak th simplifyig assumptio that couplig ca b igord so that amog all th rflctacs ad trasmittacs, oly R ν ν ad T ν ν rmai This gratly simplifis th rsults abov ad w hav 34

36 + T = N = { + R + T } = R ν ν ν ν ν ν ν ν / / mvν ( k ) ϕ ˆ m pzϕ = { R + T } δ = mv ( k ) T δ ϕɶ pˆ ϕɶ = mv ( k ) T δ ν ν ν ν µν ν ν ν µν m z ν ν ν µν ϕɶ pˆ ϕ = mv ( k ) τ ρ δ, ϕ pˆ ϕɶ = mv ( k ) τ ρ δ m z ν ν ν ν ν µν m z ν ν ν ν ν µν Aftr summig, this yilds a vry simpl currt dsity xprssio EL J = f ( Eν ( k )) vν ( k ) { Tν ν+ Tν ν Rν ν}, Ω DC ν, k that ca b rducd v furthr usig th cosrvatio rlatio Rν ν+ Tν ν= so that EL J = f ( E ( k )) v ( k ) T Ω DC ν ν ν ν ν, k At this poit, w ca covrt th rmaiig k summatio to a itgral that, i tur, ca b turd ito a rgy itgral EL J f E k v k T dk DC= ( ν ( )) ν ( ) ν ν π Ω ν EL = f ( E) Tν ν de π Ω ħ ν To rlat this rsult to dirctly masurabl quatitis w itroduc th total pottial diffrc (voltag) V= EL btw th two lads ad th total currt qual to currt dsity tims cross sctioal ara of th lads, i I= AJDC Applyig, i additio, th rlatios Ω= AL ad h= πħ w fid th clbratd Ladaur formula for th quatizd currt I th low-tmpratur limit, w th fid V I= f ( E) Tν ν de h (45) ν V I= Tν ν ( EF ) (46) h ν 35

37 It shows that th coductac I G= = Tν ν ( EF ) (47) V h ν is quatizd As a xampl, w may assum ffctiv mass disprsio Eν ( k ) = Eν+ħ k /m, whr E ν= Eν ( k = ) is th quatizatio rgy Th, i th idalizd cas of vaishig rflctio, th trasmittac quals uity whvr th Frmi lvl xcd th bad dg, i T ( E ) = θ( E E ) ad th coductac is simply G= NG, whr G= / h= (964 k Ω) is th fudamtal coductac quatum ad N is th umbr of chals blow th Frmi lvl If th assumptio of compltly ballistic trasport is ot fully applicabl bcaus of scattrig losss, th coductac formula should b covolutd with th broadig fuctio ħγ /( E + ħ Γ )/ π This producs th broadd xprssio ν ν F F ν EF E ν G= ta + h ν π ħγ (48) Figur 4 illustrats th quatizd coductac as a fuctio of Frmi rgy for a fictitious systm i which E ν = ν V, ν=,, 3, ad so o Th idal cas ħ Γ= as wll as a cas of fiit broadig ħ Γ= V is show i th two graphs Figur 4 Illustratio of coductac quatizatio for th idal ad broadd cass Th figur 43 blow shows xprimtal coductac plots for a GaAs/AlGaAs juctio oto which a xtrmly arrow coductig chal is formd by mtallic 36

38 poit cotacts [] Tulig occurs through a ctral mtallic cotact coctd to lads o th lft ad right righ via tulig rgios Th tulig currt is rcordd as a fuctio of a gat pottial applid to th ctral cotact Figur 43 3 Exprimtal coductac tracs for a mtallic poit cotact as a fuctio of gat voltag Tak from [] Exrcis: Mod coutig ad miimum rsistac Ev for a idal coductor havig all trasmittacs Tν ν = thr is a maximal coductac giv by Eq (47) (4 Gmax = # mods h Hr, #mods should b udrstood as th umbr of occupid lctro chals, chal i chals for which Eν EF For a coductor with a squar profil dfid by sid lgths Lx ad Ly ad ifiit surfac surfac barrirs, th mod idx ν is actually twodimsioal ν pq bcaus th rgis dpd o two quatum umbrs (s Appdix ) ℏ π p q + E = m Lx Ly pq W first assum a idal quatum wll, i tak Ly so small that oly a sigl ymod cotributs 37

39 a) Show that by sttig E pq= EF th maximal occupid p-lvl ad, hc, th umbr of occupid mods bcoms x #mods= πħ m L E E ( F ) Nxt, w st Lx= Ly L ad cosidr a squar quatum wir I this cas, sttig E = E mas that ( p + q ) = E m L /( πħ ) Th problm of mod coutig pq F max F thrfor rducs to fidig th umbr of o-gativ itgrs withi a circl of radius E m L/( πħ ) F b) By assumig this radius much largr tha uity, show that #mods EFmL /( π ) = ħ W ow assum that all bads ar parabolic, i that th full rgy disprsio is E = E +ħ k /( m ) Thus, for a sufficitly thick wir w kow that th lctro pq pq dsity is 3/ = ( F / ħ ) /3 E m π c) Show that th miimum rsistac bcoms R π = 3 /3 4ħ L mi Rfrcs [] BJ d Ws t al Phys Rv B43, 43 (99) 38

40 5 Elctro Trasmissio ad Rflctio I th prvious chaptr, th gral formulas for th lctric currt wr dvlopd i trms of lctro trasmittacs Hr, w will cotiu by actually computig th trasmittac for a rag of illustrativ xampls To simplify mattrs, w rstrict ourslvs to a o-dimsioal dscriptio, i w igor what happs i th dirctios paralll to barrirs ad othr pottial stps Throughout, w rly o a ffctiv mass pictur, i which lctros bhav as fr lctros but with a modifid mass m rathr tha th fr lctro mass (xplaid furthr i th xt chaptr) W also assum th pottial rgy to b picwis costat but, as will b show blow, ay tru profil ca b approximatd by a squc of costat pottial stps via th staircasig approach Thus, vry gral cass ca b tratd usig th mthods dvlopd hr Elctros with rgy E movig through a rgio with costat pottial rgy V ar dscribd by th wav fuctio ikx ikx m ψ( x) = A + B, k= ( E V) ħ If w valuat th sam stat at a shiftd positio x+ d w obviously hav ik( x+ d) ik( x+ d) ikx ikx ψ( x+ d) = A + B Hc, if w writ this stat as ψ( x+ d) = C + D, w ca writ ikd A C M, M = = ikd B D (5) Similarly, th lctro may pass from a matrial havig pottial rgy V to a positio with pottial rgy V I this cas, th wav fuctios for th positios ad (locatd immdiatly lft ad right of th itrfac, rspctivly) ar ikx ikx m ψ( x) = A + B, k= ( E V) ħ ikx ikx m ψ ( x) = C + D, k= ( E V) ħ To rlat th cofficits i this situatio, w d to match th wav fuctio ad its drivativ at th juctio A+ B= C+ D k ( A B) = k ( C D) 39

41 By addig ad subtractig ths quatios, it follows that w may writ th rlatios as th matrix xprssio k k + A C k k = B M, M D = k k + k k (5) I this mar, trasmissio through ay pottial rgy profil ca b modld by a sris of matrix multiplicatios As simpl xampls, w cosidr tulig through sigl ad doubl barrirs as illustratd i Fig 5 Figur 5 Sigl ad doubl tulig barrirs For th sigl barrir, trasmissio ivolvs () trig th barrir, () trasvrsig th barrir, ad (3) xitig th barrir If w choos th pottial rgy outsid th barrir as th zro-poit, th wav umbr i this rgio is simply k k m E = = / ħ Isid th barrir th pottial rgy corrspodigly, th wav umbr is k i m V E = β, β= ( )/ ħ ffct of th thr procsss is dscribd by th systm matrix M= M M M i B out Now, i a rflctio-trasmissio xprimt with a icomig wav from th lft, th (u-ormalizd) wav fuctios far to th lft ad right ar, rspctivly, ikx ikx ikx ψ L ( x) = + r, ψ ( x) = t, R is V ad, Th combid 4

42 whr r ad t ar rflctio ad trasmissio cofficits I trms of th systm matrix, w thrfor hav t = M r so that t= / M ad r= M / M Fially, th trasmissio ad rflctio probabilitis ca b computd from T= t = / M ad R= r = M / M, rspctivly As w will s blow, quit complicatd gomtris ca b hadld usig this tchiqu For th particular cas of a sigl rctagular barrir of width d, th gral xprssios abov yild th followig matrics: β β k k i i i + i d + β k k β β M i=, M out, M B βd β β = = k k i + i + i i k k β β Prformig matrix multiplicatios, it follows that th systm matrix for th sigl barrir ( S ) bcoms M ( S) Ad th trasmittac is β β β β ( k )sih d ( + k )sih d coshβd+ i i β β k k = ( β + k )sih βd ( β k )sihβd β i cosh d i β β k k T ( S) 4β k = (53) 4β k cosh βd+ ( β k ) sih βd I th xrcis at th d of th chaptr, a plot of th sigl barrir trasmittac is show For th doubl barrir ( D ) i Fig 5b w d, i additio, th matrix for traslatio btw th barrirs M W ikw = ikw, ad th trasmissio matrix bcoms 4

43 It is asily dmostratd that ( D) M = M M M M M M M ( S) ( S = M M M ) i B out W i B out W M = M M + M M ( D) ( S) ikw ( S) ( S) ikw ( S) Blow, w plot th doubl barrir trasmittac vs rgy usig V = 4 V, d = 6Å ad w = 8Å ad takig m qual to th fr lctro mass Th paks corrspod to rsoat tulig [] ad thy ar locatd, whr tru boud igstats would appar if th barrirs wr ifiitly wid Not that w plot th rsult for E> V also I fact, th oly modificatio for this rgy rag is that β bcoms imagiary Sic th formulas abov rmai valid for imagiary β, th rsults still apply for this rgy rag Figur 5 Tulig trasmittac T for th rsoat tulig structur show i Fig 5b usig V = 4 V, d = 6Å ad w = 8Å Ay full systm matrix built from multiplyig th matrics abov will b of th form M M M= M M Sic T = / M ad = /, w fid for th dtrmiat R M M 4

44 R R dt M= M M = = T T T As lctros do t disappar, w should hav R+ T= ad so ay valid systm matrix must hav uit dtrmiat dt M= 5 Triagular Barrir W will ow attmpt to tackl a much mor complicatd problm: That of a triagular barrir such as th o illustratd i Fig 53 Figur 53 Th triagular tulig barrir (lft) ad th staircasd approximatio (right) Such a barrir aturally ariss i fild missio structurs Hr, lctros ar mittd from a mtal by approachig a positivly biasd lctrod to th mtal surfac For a simpl plaar gomtry, th lctric fild is costat ad, hc, th pottial dcrass liarly with th distac from th surfac (actually th imag charg ffct modifis th pottial ar th surfac [] but this complicatio is igord) Thrfor th lctros hav to ovrcom a triagular barrir, whos hight V quals th mtal work fuctio for lctros at th Frmi lvl This tulig problm ca b attackd i svral ways, ragig from approximat ovr fully umrical to aalytical approachs Th first typ (approximat) is calld th WKB ( S) approximatio aftr Wtzl, Kramrs ad Brilloui Th startig poit is th T xprssio Eq(53) for th sigl barrir, drivd abov If th barrir is sufficitly wid, th argumt β d is larg ad w may approximat T ( S) 6β k 4 β k + ( β k ) ( βd) xp For simplicity, th prfactor ca also b rplacd by uity, as w ar maily ( S) itrstd i th thickss dpdc, i T xp( βd) I this mar, w ar ( S) sur that T as d Now, for this cas it is clarly o approximatio to writ 43

45 d m β d= ( V E) dx ħ Th WKB approximatio cosists i applyig this xprssio to barrirs that ar ot rctagular I gral, th tulig rag through th barrir is dtrmid by th lctro rgy E as illustrats i Fig 54 Hr, th tulig rag is btw x ad x + that ar dfid by th itrsctios V( x ) = ± E Figur 54 Sktch of a arbitrary tulig barrir with classical turig poits x ad x + I this mar, th gral approximat trasmittac bcoms x+ m TWKB= xp ( V( x) E) dx (54) ħ x Th triagular barrir is dfid by th pottial V Fx < x< d V( x) =,, othrwis whr F is th forc proportioal to th lctric fild F= E ad th width d is rlatd to F via d= V / F Also, as log as < E< V w fid x = ad x = ( V + E)/ F I this cas, a lmtary itgral shows that 4 m TWKB= xp V E 3F ħ ( ) 3 (55) For th triagular barrir, howvr, it is possibl to fid a rlativly simpl aalytical solutio Th ky poit is to cosidr th Schrödigr quatio i th barrir rgio < x< d : ħ ( x) ( V Fx) ( x) E ( x) m ψ + ψ = ψ 44

46 Rwritig slightly, w fid mf V E ψ ( x) = ( x x) ψ( x), x = ħ F This rsmbls Airy s diffrtial quatio ψ ( z) = zψ( z) I fact, if w writ z= q( x x), Airy s quatio rads d dx ψ 3 ( q( x x)) q ( x x) ψ ( q( x x)) = This clarly fits if q ( m F / ) /3 = ħ Th two liarly idpdt solutios ar dotd Ai ad Bi ad so th full solutio must b ( ) ( ) ψ ( x) = A Ai q( x x) + B Bi q( x x) Th rflctio ad trasmissio cofficits r ad t alog with A ad B follow from th four boudary coditios ( ) ( ) + r= A Ai qx + B Bi qx ( ) + ( ) ik( r) = q A Ai qx B Bi qx t= A Ai q( x d) + B Bi q( x d) ( ) ( ) ( ) ( ) ikt= q A Ai q( x d) B Bi + q( x d) I this mar, utilizig th Wroskia W[Ai,Bi] = / π, kq t= π ( ika+ qa )( kbɶ + iqbɶ ) ( ikb qb)( ka iqa) + ɶ+ ɶ a= Ai qx, a= Ai q( x d), b= Bi qx, bɶ = Bi q( x d) ( ) ɶ ( ) ( ) ( ) a = Ai qx, a = Ai q( x d), b = Bi qx, bɶ = Bi q( x d) ( ) ɶ ( ) ( ) ( ) (56) Th fully umrical approach to trasmissio through arbitrary barrirs cosists of applyig th matrix formalism prstd abov As th mthod oly applis to pottial profils that ar picwis costat, w d to approximat th barrir usig staircasig As a xampl, a staircasd triagular barrir is show i th right pal of Fig 53 Th ida is that th barrir is choppd ito pics, ach 45

47 havig a crtai width ad costat barrir hight Wh th umbr of stps bcoms larg, th rsult coms clos to th xact o I Fig 55, w hav compard th xact rsult Eq(56) to th WKB approximatio Eq(55) ad to th staircasig approach usig = ad 3 Hr, a barrir of V = 5 V ad a forc of F= V/Å approximats th xact curv ar assumd Notic how wll th = 3 rsult Figur 55 Elctro trasmittac for th triagular barrir of V = 5 V usig () th WKB approximatio, () th staircas mthod, ad (3) th xact solutio Exrcis I Fig 56 blow, w hav plottd th trasmittac of a sigl barrir with V = 4 V ad d = 6Å Similarly to th doubl barrir cas, w ot that th plot xtds to th fly-ovr rag E> V It is also oticd that a uit trasmittac is foud for som vry spcific barrir thicksss a) Show that i th fly-ovr rgy rag kb k k k B + + ikbd k k kb k B M i, M = out, M B ikbd kb k = B k k =, + + k k k k B B whr B= ( )/ ħ k m E V b) Show that th modifid (,) lmt of th systm matrix bcoms 46

48 M k d i ( S) ( + B)si B = cos B kkb k k k d ad, corrspodigly, th trasmittac T ( S) = = ( k + k ) k + k cos si 4 4 B B kbd+ si k Bd + 4k kb 4k kb k d B (57) ( S) Writig th rsult i th scod form clarly dmostrats that T It is also clar that th first form agrs with Eq(53) as follows from th substitutios β ik B, coshβ cos k B ad sihβ i si k B Figur 56Trasmissio of a sigl rctagular barrir with rgis at trasmittac maxima highlightd c) Show that th trasmissio attais its maximum valu of uity at particular valus of th rgy giv by ħ pπ Ep= V +, p=,, 3, m d Rfrcs [] B Ricco ad M Ya Azbl, Phys Rv B9, 97 (984) [] RH Fowlr ad L Nordhim, Proc R Soc A 9, 73 (98) 47

49 6 Elctro Trasmissio i Molculs: Itroductio I this chaptr, w tur to th trasmissio of lctros through molculs W will thrfor ot us ay ffctiv mass approximatio or th rctagular barrirs that wr itroducd i th prvious chaptr Thos cocpts wr appropriat for smicoductor aostructurs but will ot suffic for idividual molculs W will start by cosidrig a simpl xampl that ca b aalyzd aalytically This gomtry is illustratd i Fig 6 ad cosists of a bz molcul attachd to simpl o-dimsioal lads with a sigl atom pr uit cll [] Figur 6 Trasmissio gomtry cosistig of a bz rig ( systm ) attachd to two smi-ifiit moo-atomic chais ( lads ) W aalyz th stats usig th tight-bidig approach dscribd i Appdix Morovr, w will assum that oly a sigl orbital o ach atom cotributs to th trasport This orbital will b dotd if it blogs to th th atom Th quatum stats ca b writt as xpasios i this basis ϕ = c, whr th sum is ovr all sits (atoms) i th structur I our cas, th structur cosists of ifiit lads attachd to a fiit molcul Grally, a larg part of th calculatio is cocrd with hadlig ifiit lads Hr, as a startig poit, w ll cosidr th isolatd molcul For th bz rig chos as our xampl, th molcul has 6 sits For simplicity, assum that oly arst ighbor couplig xists ad dot th hoppig matrix lmt by γ (th mius sig is i ordr to kp γ> ) Also, w ll choos th zro poit of rgy such that th osit matrix lmts Hˆ = Hc, th igvalu problm for th isolatd systm rads γ γ c γ γ c γ γ c 3 HS c Ec, H = S=, c γ γ = c 4 γ γ c 5 γ γ c 6 (6) 48

50 It s a simpl mattr to show that th igvalus ar ± γ ad ± γ with th last pair twic dgrat Th isolatd lads ar v simplr Udr th sam assumptios as for th molcul, w fid ( γc + γc ) = Ec + Th igstats ar simpl propagatig wavs of th form iqa iqa c A B = + ad th associatd rgy is E= γ cos( qa), whr a is th lattic costat of th chai Hc, th rgis form a cotiuous bad with valus i th rag γ E γ W ow aim to coupl systm ad lads togthr I doig so, w will assum that a wav of uit amplitud is icidt from th lft ad th partially rflctd by th systm Dsigatig th rflctio cofficit by r, th full wav i th lft lad is ( i) iqa iqa thus c = + r ( out ) iqa Similarly, i th right lad a trasmittd wav c = t propagats, whr t is th trasmissio cofficit Now, if w aivly trid to st up th igvalu problm for th coupld systm, a ifiit matrix would rsult Fortuatly, th ifluc of th ifiit lads ca b accoutd for via thir lctroic Gr s fuctio I th prst moo-atomic cas, th costructio is iqa iqa particularly simpl First, w ot that c= + r ad c = + r ad, hc, iqa iqa c = + c Scodly, th hoppig itgral couplig th lft lad ad th ( ) systm is dotd τ Th, th Schrödigr quatio for th th sit bcoms γc + τc = Ec Combiig ths rlatios, w ow s that iqa ( E H ) c = H H + τc, H = γ L L L L If th couplig btw right lad ad systm is also τ w fid, similarly, iqa ( E H ) c = τc, H = γ R 7 6 R W ar ow i a positio to writ th matrix quatio for th full coupld problm To kp th otatio as simpl as possibl, w will supprss vctor ad matrix symbols Thus, gl V L c HL H L VL EI HS V R c = VR gr c 7 (6) W strss that i this xprssio, H S ad c ar rally th matrix ad vctor dfid i Eq(6) ad I is th 6 6 uit matrix Also, svral pics of w otatio hav b itroducd: 49

51 V L ( τ ), V ( τ) = = (63) ar couplig matrics coctig th systm with th lft ad right lads, rspctivly Fially, R g = ( E H ), g = ( E H ) (64) L L R R ar th lad Gr s fuctios Now, i ordr to comput th trasmissio cofficit t w d to fid c 7iqa 7= t Th phas factor is irrlvat as w ar rally itrstd oly i th trasmittac T= t = c W could, of cours, simply ivrt th matrix quatio Eq(6) 7 Howvr, to prpar ourslvs for tacklig mor laborat cass w will dmostrat how th calculatio ca b simplifid usig a fw mathmatical maipulatios Primarily, th full Gr s fuctio of th tir dvic (lads plus systm) is just th ivrs of th matrix i Eq(6), i g V G G G L L 3 VL EI HS V R G G G 3 I = V R g R G3 G3 G 33 (65) From Eq(6), it follows that ( c ) 7= G3 HL HL ad, hc, G 3 is th oly importat lmt of th full Gr s fuctio matrix It ca b computd by simpl maipulatios of th first colum of quatios i th matrix xprssio abov Ths ar g G V G = L L V G + ( EI H ) G V G = L S R 3 V G + g G = R R 3 A fw mathmatical opratios dmostrat that G = g V GV g (66) 3 R R L L Hr, G is th dvic Gr s fuctio G= ( EI H Σ Σ ) S L R Σ = V g V, Σ = V g V L L L L R R R R (67) 5

52 Th corrctios Σ L/ R ar th so-calld slf-rgis that icorporat th ffct of couplig th systms to th lads Thus, T= g V GV g ( H H ) W sk to fid R R L L L L th trasmissio T as a fuctio of th lctro rgy E This mas that w should xprss th lad Gr s fuctios i trms of E as wll Howvr, sic iqa H = H = γ ad E= γ cos( qa) it follows that L R / E E L= R= + γ H H i 4 E E gl = gr = i γ 4 / I th prst xampl, th oly o-vaishig lmts of th slf-rgy matrics ar Σ =Σ = τ g Takig τ= γ, it ca actually b dmostratd aalytically that L, R,66 L/ R 4 γ (4 γ E )/(5 γ E ) E < γ T=, othrwis This rsult is plottd i Fig 6 Th maxima of uit trasmittac ar foud at E=± 3γ Th ffct of varyig τ is illustratd i th right-had pal It is clarly s that trasmissio bcoms rstrictd to a arrow rgy rag aroud ± γ, which coicids with th igstats of th isolatd molcul Wh th couplig icrass, so dos th broadig du to th slf rgy util, vtually, th trasmittac is high i th tir rgy rag allowd by th lads Figur 6 Trasmissio spctrum for th dvic illustratd i Fig 6 I th right-had pal, th lad-systm couplig is varid 5

53 6 Gral Ladaur Formula Th xprssio for th trasmittac appars rathr asymmtrical i spit of th clar symmtry btw lft ad right sids Usig som rwritig, howvr, th symmtry ca b rstord First, sic HL= H R w fid that T= ( H H ) g V GV g ( H H ) R R R R L L L L R( R R ) R R L L( L L ) L = g H H g V GV g H H g R R R L L L = ( g g ) V GV ( g g ) Th last quality follows from th idtity g ( H H ) g = g g ad similarly for th lft lad Now, barig i mid that R L VL G VR R R R R R R R V, G ad that ( V GV) = Th, by judicious ordrig of trms, V ar rally matrics it follows L T= ( g g ) V GV ( g g ) V G V R R R L L L L R = Σ Σ ( gr gr ) VRG( L L ) G VR Now, if w could just brig th last factor V to th frot of th xprssio, th fial rsult would b ic ad symmtrical This, though, caot b right sic V R is a matrix ad th fial rsult should b a scalar This ca b rmdid by a at mathmatical trick W ca tak th trac (Tr) of th xprssio, i R { } T= Tr ( g g ) V G( Σ Σ ) G V R R R L L R Takig th trac simply mas summig th diagoal lmts of th matrix i th argumt I our cas, howvr, th argumt is just a scalar ad takig th trac dos t do aythig at all to th xprssio But th trac has a vry importat proprty: it is cyclic, maig that Tr{ ABC} = Tr{ CAB} = Tr{ BCA} as ca asily b vrifid Thus, udr th trac th trms ca b rarragd so that { R R R R L L } { R R G L L G} T= Tr V ( g g ) V G( Σ Σ ) G = Tr ( Σ Σ ) ( Σ Σ ) I much of th litratur [], o itroducs li width fuctios Γ = i( Σ Σ ) that ar sstially th imagiary part of th slf rgy Hc, L/ R L/ R L/ R w fid th fial symmtrical xprssio { } T= Tr Γ GΓ G (68) R L 5

54 A additioal advatag of th trac is that w automatically comput th tir trasmissio, i T= T summd ovr chals This vry hady ad lgat ν ν ν Ladaur xprssio ca b tak as th startig poit for most calculatios of trasport i molcular systms It applis to asymmtric gomtris, whr lft ad right lads diffr It also applis to lads that ar much mor complicatd that moo- ar priodic i th atomic os I fact, it ca v b gralizd to dvics thatt dirctio prpdicular to th trasport Exrcis: Trasmissio i biphyl molculs Th aim of this xrcis is to calculat th lctro trasmittac through a biphyl molcul as illustratd i Fig 63 It is obviously quit similar to th bz cas discussd i th txt a) Bfor spcializig to biphyl, show that Eq(66) actually follows from Eq(65) b) Writ dow xprssios for th matrics H S, V L ad V R c) Writ a computr program to valuat Eq(68) assumig all hoppig lmts to b γ Th rsult should look similar to th spctrum i Fig 63 Rfrcs Figur 63 Trasmissio gomtry ad spctrum for a biphyl molcul [] E Cuasig ad JS Wag, Eur Phys J B69, 55 (9) [] S Datta Elctroic Trasport i Msoscopic Systms (Cambridg Uiv Prss, Cambridg, 997) 53

55 7 Elctro Trasmissio i Molculs: Challgs Basd o th rsults of th prvious chaptr, w ar ow i a positio to tackl mor challgig xampls of lctroic trasmissio through molculs Primarily, w wish to iclud mor complicatd lads with svral atoms pr uit cll Ths lads may coupl to th molcul through svral bods as wll Scodly, gomtris that rpat priodically i th dirctio prpdicular to th trasport will b studid As a startig poit, w will cosidr agai th simpl moo-atomic chai This xampl ca b aalyzd aalytically ad usd to illustrat th rcursiv approach itroducd blow For this simpl chai, th Hamiltoia is γ γ γ H= γ γ γ (7) Not that for otatioal simplicity w agai supprss matrix symbols Th Gr s matrix for this chai is th G= ( EI H) Du to th rpatd structur of H w writ E V G G =, V = ( γ ) V EI H G G I Solvig th quatios rsultig from th first colum w fid EG V ( EI H) VG = Now, G EI H = ( ) ad V GV γ G = ad so EG γ G = with th solutio G E i / E = γ, γ γ 4 i complt agrmt with th prvious chaptr Our aim is to xtd th mthod to mor complicatd o-dimsioal lads Hc, such gral lads could hav svral, possibly diffrt, atoms pr uit cll A xampl is th two-atomic lad i Fig 7 54

56 Figur 7 A two-atomic lad I this cas, th wav fuctio has compots for both th uppr (u) ad lowr (l) rows of atoms ad for th th uit cll ths ca b collctd as a vctor ( u) ( l) (, ) T c = c c I this mar, th chai Schrödigr quatio ca b writt αc + βc + hc = Ec, + whr w itroduc matrics γ γ α= β=, h= γ γ Th raso for kpig sparat symbols α ad β for th coupligs lft ad right, rspctivly, is that i gral ths might b diffrt Th Hamiltoia is ow h β α h β H= α h β α h (7) Similarly to th prvious chaptr, w cosidr th first colum of quatios drivd from th dfiig quatio for th Gr s fuctio ( EI H) G= : E h β G α E h β G α E h β G = α E h G 3 (73) Cosidr th rlatio obtaid from th scod row ( ) αg+ ( E h) G βg= G= gαg+ βg, g= ( E h) Wh combid with th first row it follows that 55

57 ( E h βgα) G βgβg = Th rsultig systm of quatios ca ow b formulatd as E h βgα β gβ G αgα E h αgβ βgα β gβ G αgα E h αgβ βgα βgβ G = 4 α gα E h αgβ βgα G 6 As is appart, th th cll ow coupls to th d, 4 th ad so o If itratd oc mor, couplig will b to clls umbr 4, 8, tc This rcursiv schm [] ca b formulatd as th followig rpatd squc, iitializd by sttig λ= λl= λr= h : g : = ( E λ) λ : = λ+ αgβ+ βgα loop λl : = λl + βgα util λr : = λr+ αgβ covrgc α : = αgα β : = βgβ Aftr covrgc, th lft ad right surfac Gr s fuctios of th lads ar giv by gl= ( E λl ) ad gr= ( E λr ), rspctivly For umrical stability, it s cssary to add a small imagiary part iη ad typically w choos η γ / Figur 7 Compariso of th umrically gratd moo-atomic lad Gr s fuctios at diffrt stags i th itratio procss Th blu curv is th xact rsult 56

58 As a illustratio, th rsult for th simpl moo-atomic chai is show i Fig 7 It is s that 8 itratios produc a rsult vry clos to th xact o apart from th broadig itroducd by th imagiary trm iη For th two-atomic lad, Fig 7, th Gr s fuctio is a matrix G dtrmid by th coditio ( EI h) G αgβg= I This quatio, i fact, has a aalytical solutio It turs out that G= G ad G = G Isolatig, it ca b show that 4γ G 8 Eγ G + (5E + 3 γ ) G E( E / γ + 3) G + E / γ = Dspit apparacs, this quatio has a rlativly simpl solutio ad th sam gos for G Th rsults ar plottd i Fig 73 Th spctral faturs ar foud at ± 3γ ad ± γ Ths faturs corrspod to th igmods of th two-atomic lad It is asily show that th two mods ar ± γ γ cos( ka) Hc, th rgy rags of th mods ar 3γ E γ ad γ E 3γ, rspctivly I th ctral rag γ E γ, th two bads ovrlap, which lads to th additio bump i G Ths xact rsults ca b usd to chck th umrical routi abov i a simpl xampl Figur 73 Compariso of th umrically gratd two-atomic lad Gr s fuctios aftr 5 itratios with th xact rsult 57

59 I ordr to apply th two-atomic lads i a actual trasport calculatio, w imagi that such lads ar attachd to th lft ad right of a vry simpl systm, as show i Fig 74, top pal Figur 74 Two-atomic lads attachd to a systm rprstd by th gr atoms Som of th hoppig itgrals ar idicatd Both a isolatd D dvic (top) ad priodic D dvic (bottom) is show This systm is also two atoms wid ad th oly diffrc compard to th lad atoms is that w lt th dvic atoms b of a diffrt sort This implis two thigs: First th systm Hamiltoia will b tak as H S ε γ =, γ ε whr ε is th o-sit pottial that dscribs th rgy of a lctro o a isolatd systm atom rlativ to th lad atoms Hr, w also assumd that th hoppig itgral withi th systm is th sam as for th lads i γ Scodly, th couplig btw lads ad systms should b cosidrd Agai, thir structur is similar to th couplig matrics of th lads but if w allow for a arbitrary couplig strgth w ca writ thm as V L τ = VR=, τ 58

60 whr τ is th lad-systm couplig Th systm o-sit pottial ε could also b th rsult of a local lctrostatic gat, which would shift th rgy of th systm atoms rlativ to th lads Th trasmittac of this simpl barrir dvic is illustratd i Fig 75 W s som xpctd trds: Wh th couplig btw lads ad systm is rducd, trasmittac drops ad bcoms highly pakd aroud ± γ Similarly, wh th barrir pottial ε is raisd th trasmittac dcrass i a symmtric fashio Figur 75 Trasmittac of th isolatd two-atomic lad dvic I th lft ad right pal, th systm-lad couplig ad systm o-sit pottial is varid, rspctivly 7 Two-dimsioal Lads So far, w hav focusd o o-dimsioal gomtris, whr a small umbr of mods cotribut to th trasport Whil such gomtris ar ralistic rprstatios of crtai xprimtal systms, a mor commo cas is that of much widr lads ad dvics I such cass, it is mor appropriat to viw th systm as a ifiitly wid o Morovr, th systm ca typically b brok dow to a crtai uit that is th rpatd priodically alog th dimsio prpdicular to th trasport dirctio A xampl of such a gomtry is show i Fig 74, bottom pal W s that th systm is sstially idtical to th os w hav b cosidrig so far xcpt for th fact that priodic boudary coditios ar usd to coupl ighborig uits As usual, whvr w coutr priodic boudary coditios, th wav fuctios bcom priodic apart from a Bloch phas factor Hc, if th width of a sigl uit is w, th Bloch factor is xp( ± ikw), whr k is th wav vctor rstrictd to th rag π / w< k< π / w ad th sig dpds o whthr couplig is upwards or dowwards As log as w oly cosidr arst-ighbor couplig, oly th systm ad lad Hamiltoias H ad h ar affctd Th couplig matrics ar ot affctd bcaus S all w couplig itractios ar at last a factor of furthr apart tha arstighbors Thus, addig priodic boudaris simply mas that w should us 59

61 H ikw ikw ε γ( + ) + =, h γ = γ( + ) ε + S ikw ikw I practic, w do th avragig ovr k-vctors by ruig simulatio for a rag of discrt valus ad th simply divid th sum by th umbr of valus N, i k T = T( k) N k k I Fig 76, this avrag has b mad for th structur i Fig 74 (bottom) usig a coars ad fi k-grid I th priodic cas, sic ach atom coupls to 4 arst ighbors, th full rgy rag of th lads is E 4γ Hc, trasmissio is ozro i this rag W also s that discrtizatio rrors ar washd out wh usig th fi grid Figur 76 Trasmittac avragd ovr trasvrs k-poits usig a coars (lft) ad fi (right) grid Exrcis: Trasmissio i graph shts I this xrcis, w cosidr th graph sht show i Fig 77 As show, it ca b costructd as armchair chais joid priodically W thrfor modl th sht as a collctio of coupld chais Figur 77 Graph sht built from coupld armchair chais 6

62 a) Show that th lad Hamiltoia icludig couplig is ikw + ikw + h= γ ikw + ikw + Similarly, th oly o-vaishig tris of th couplig matrics ar α4= β4= γ b) Us th rcursiv schm to costruct th lft ad right surfac Gr s fuctios Tip: sic th k-poits covr th itrval π / w< k< π / w w ca just us th full phas kw as a variabl covrig th rag π< kw< π without worryig about th valu of w c) Comput th k-avragd trasmittac of th sht Tak th systm to b four ctral atoms shiftd by a o-sit pottial ε ad all couplig itgrals to b γ Th rsult should rsmbl th plot blow Figur 78 Trasmittac i graph sht avragd ovr 5 trasvrs k-poits Rfrcs [] T Markuss, RRurali, M Bradbyg, ad A-P Jauho, Phys Rv B74, 4533 (6) 6

63 8 Elctric Proprtis of Smicoductors W ow tur from mtals to smicoductors Smicoductors lik Si ad GaAs ar th activ matrials i lctroic ad optolctroic dvics such as trasistors ad smicoductor lasrs Whil traditioal applicatios of smicoductors us bulk matrials, may rct dvics rly havily o quatum cofid structurs, i particular quatum wlls Also, quatum wirs ad dots ar mrgig i applicatios such as fluorsct aoparticls Fially, two thirds of all possibl carbo aotubs ar smicoductors ad costitut xtrm xampls of odimsioal quatum structurs I th prst chaptr, w will look at dopig i smicoductors of various dimsios i ordr to dscrib th lctric coductivity Figur 8 Bad structur of GaAs with a zoom of th bads i th viciity of th bad gap ad parabolic approximatios to th bads addd I Fig 8, th bad structur of th prototypical smicoductor GaAs is displayd I th udopd cas ad at zro tmpratur, th four lowr bads ar compltly occupid whil th rmaiig ar mpty Cosqutly, thr is o coductio bcaus occupid ad mpty stats ar sparatd by a larg rgy gap (roughly 5 V), which is far mor tha a ormal lctric pottial ca surmout Howvr, two factors ca chag that: dopig ad tmpratur Hc, it is th aim hr to dscrib th ffcts of ths factors o th coductivity Th mchaism of coductio ca b udrstood o th basis of Eq(3): 6

64 iεω p σω ( ) = (8) ω +Γ i I drivig th xprssio for th plasma frqucy i chaptr 3, w took th tmpratur to zro, which is prfctly alright for a mtal I th smicoductig cas, this is o logr justifid ad w should us th mor gral xprssio ω p E c k f E c k εω k k z E c k = ħ ( ) istad As a startig poit, w cosidr th bulk cas Hr, th sum ovr k is radily covrtd ito a itgral ad w gt ω E f ( E ) c k c k 3 p= 3 ħε( π) k z E c k d k W xt apply th chai rul f E f = k k E c k c k c k z z c k to dmostrat that ω c k c k 3 p= 3 ħε( π) kz kz = ħ ε ( π) 3 z E f ( E ) d k E k c k f E 3 ( ) d k c k Th last quality follows by partial itgratio For simplify, w will focus th atttio o th coductio bad Hc, what w dscrib is th chag i coductivity du to addd lctros i this othrwis mpty bad This phomo is kow as -dopig I gral, a additioal cotributio will com from rmovd lctros or hols i th valc bads This cotributio ca b dscribd usig a similar approach ad addd if dd Th bads grally hav a complicatd dpdc o k Howvr, th rlativly fw lctros that ar addd to th coductio bad will occupy stats vry clos to th bottom of th bad as thos ar th lowst i rgy Similarly, th fw lctros rmovd from th valc bad will b tak from th vry top of th bad Hc, as oly a small portio of th bads ar th xtrma mattr, w ca us th parabolic approximatio, which rads for th coductio bad 63

65 E c k ħ k Ec+ m, whr E c dots th rgy at th bottom of th bad ad m is th ffctiv lctro mass dducd from th curvatur of th bad Wh applid i th xprssio abov it follows immdiatly that ω 3 p= f ( E ) 3 c k εm( π) = ε m, d k (8) whr 3 3 ( π) f ( E ) d k c k = is th lctro dsity W otic that this is prcisly as for th fr-lctro cas xcpt that th ffctiv lctro mass has rplacd th fr-lctro mass As usual, w may altrativly xprss th lctro dsity i trms of th dsity of stats, i = f ( E) D( E) de (83) Now, w should also worry about cofimt ffcts First, th sum ovr k should b rplacd by a gral summatio ovr all stats Wh w spak of a D- dimsioal matrial (D =,, or 3), w simply ma that th structur is larg i D out of th 3 possibl spatial dirctios I th rmaiig dirctios, th structur is small ough for quatum ffcts to aris Thus, for th quatizd dirctios th sum ovr k should b rplacd by a sum ovr a idx m lablig th discrt igstats Th total rgy i a particular stat ow bcoms E= ħ k /m+ E, whr k is th magitud of th D-dimsioal wav vctor ad E > is th quatizatio rgy of th lctro Th cosquc of ths chags is that Eq(83) should b rplacd by f ( E) D ( E) de D =, whr D ( E ) is th D-dimsioal dsity of stats Equatio (8) with ω D = / ε m th still applis for th coductivity alog o of th xtdd p D dirctios Th dsity of stats xprssios ar drivd i Appdix ad giv by D m m 64

66 3/ m E E ( ) 3 3 cθ E Ec D= πħ m ( ) θ E Ec Em D= dπħ m DD( E) = / m θ( E Ec Em ) D= Aπ ħ m E Ec E m δ( E Ec Em ) D= Ω m (84) Hr, th width of th D systm (quatum wll) is dotd d, th cross sctioal ara of th D systm (quatum wir) is dotd A ad th volum of th D systm (quatum dot) is Ω Usig th Frmi fuctio w ca ow writ D DD( E) = de xp{( E E )/ kt} + Hr, w follow th usual covtio i solid stat physics ad dot th tmpratur dpdt chmical pottial by E F For th thr cass with at last o xtdd dimsio, w fid F 3/ mkt Li3/( xp{( )/ }) 3 EF Ec kt D= π ħ mkt D= l ( + xp{( EF Ec Em )/ kt} ) D= dπħ m / mkt Li /( xp{( EF Ec Em )/ kt} ) D= A π ħ m (85) I ths xprssios, Li p dots th p th polylogaritm dfid as p x t Li p( x) dt, p>, t Γ( p) x whr Γ ( p) is th gamma fuctio I most situatios, ad, hc, w may xpad th rsults abov usig I this cas, w fid Li p( x) x, l( + x) x, x E F lis svral kt blow E c 65

67 3/ mkt xp{( )/ } 3 EF Ec kt D= π ħ mkt D xp{( E )/ } F Ec Em kt D= dπħ m / mkt xp{( E )/ } F Ec Em kt D= A π ħ m (86) I particular, ths formulas clarly show that providd EF< Ec w hav D at low tmpraturs To add lctros, w d to itroduc dopig 8 Dopig I ordr to add lctros to th coductio bad, w might itroduc som suitabl lctro doatig atoms (door impuritis) to th smicoductor To fuctio as fficit doors, th addd atoms should b charactrizd by a rgy lvl just slightly blow th coductio bad dg Such doors ar calld shallow doors Thus, wh th lctro rsids o th part atom it occupis this shallow rgy lvl This situatio is illustratd i Fig 8 Figur 8 Illustratio of rgy lvls i a smicoductor dopd with door impuritis Th occupatio of th door lvl is a stadard xrcis i statistical mchaics Thr ar four rlvat stats distiguishd by th umbr ad atur of th particls occupyig th stat: () zro lctros, () o spi-dow lctro, (3) o spi-up lctro ad (4) o spi-dow plus o spi-up lctro No mor tha two lctros ca occupy th stat du to th xclusio pricipl Th rgy of a sigl (spi-up or spi-dow) lctro i th stat is dotd E Similarly, th rgy of two lctros occupyig th stat is dotd E dd d Th partitio fuctio is thrfor 66

68 W th hav th ma occupacy Z= + xp{( E E )/ kt} + xp{( E E ) / kt} F d F dd xp{( EF Ed )/ kt} + xp{( EF Edd )/ kt} = + xp{( E E )/ kt} + xp{( E E )/ kt} F d F dd Now, it s importat to raliz that E E ad, i fact, ormally E E Th dd raso is that two lctros localizd o th sam atom will lad to a substatial Coulomb rpulsio rgy Thus, w ca i fact glct this possibility ad so d dd d xp{( EF Ed )/ kt} + xp{( E E ) / kt} = xp{( E E )/ kt} + d F F d (87) Nxt, w wat to st up a balac for th umbr of chargs Equatio (86) givs us th umbr of lctros i th coductio bad ad Eq(87) dscribs th umbr of lctros i th door lvl I additio, w d th umbr of lctros rmovd from th valc bad or, quivaltly, th umbr of hols put ito th valc bad I th simplst cas of a sigl valc bad, th hol dsity p D is giv by a xprssio idtical to Eq(86) xcpt for th followig chags: () th ffctiv lctro mass m is rplacd by th ffctiv hol mass m, () th coductio bad dg c E is rplacd by th valc bad top rvrsd, ad (4) th lctro quatizatio rgis quivalt h E m I this mar, v h E, (3) th sig of all rgis is E m ar rplacd by th hol 3/ mhkt xp{( )/ } 3 Ev EF kt D= π ħ mhkt h pd xp{( E )/ } v+ Em EF kt D= dπħ m / mhkt h xp{( E )/ } v+ Em EF kt D= A π ħ m I aalogy with th lctro cas, it is rquird that th Frmi rgy E F lis wll abov E v for ths xprssios to b corrct W will dot th total dsity of impurity atoms by N I Thus, wh multiplid by Eq(87) this givs th total dsity of lctros rsidig i th door lvl It follows that th grad total lctro dsity is D+ NI Similarly, w should cout th positiv chargs Ths com from two 67

69 sourcs: th hols i th valc bad ad th positiv impurity ios, whos dsity is N Hc, ovrall charg utrality rquirs that I D+ NI= pd+ NI To solv for th Frmi rgy w itroduc th otatio = N xp{( E E )/ kt}, whr D D F c 3/ / mkt mkt mkt 3 m π ħ dπħ m A πħ m N =, N = xp{ E / kt}, N = xp{ E / kt} m Similarly, w writ p = P xp{( E E )/ kt} with D D v F 3/ / mhkt mhkt h mhkt 3= = m = π ħ dπħ m A πħ m h P, P xp{ E / kt}, P xp{ E / kt} m Rathr tha actually solvig for E F, w will itroduc x xp{ EF / kt} ad solv for x Hc, puttig all th pics togthr, w ca rformulat th charg balac as NDx xp{ Ec / kt} = PD xp{ Ev / kt} + N I x xp{ Ed / kt} / x+ ( )( ) N x P xp{( E+ E )/ kt} xp{ E / kt} + x = N x xp{( E+ E )/ kt} D D c v d I c d To simplify v furthr, w itroduc x xp{( E+ E )/ kt}, x xp{( E+ E )/ kt} ad fially x xp{ E / kt} so that cd c d ( )( ) d N x P x x + x = N xx D D cv d I cd Th solutio to this smigly ioct quatio is d cv c v xd x= R+ Q + R + R Q + R 6 NDxd+ 9NIxcd 36PD xcv NDxd+ 6NIxcd+ PD xcv R= xd, Q= 6N 36N D D Oc x is calculatd, th lctro dsity i th coductio bad is foud from = N x xp{ E / kt} To illustrat th rsults, w tak th followig paramtrs D D c appropriat for modratly dopd GaAs: 68

70 m = 66 m, m = 5 m (havy hol) E =, E = 5 V, E = E V v c d c N = m h -3 I h D d E E Quatum wll ( = ): = m, = 5 V, = V Quatum wir ( ): 4 m, h D= A= E = 3 V, E = 4 V W oly iclud th lowst quatizd stats for wlls ad wirs as th highr stats will b virtually mpty For this st of paramtrs, th Frmi rgy ad corrspodig lctro dsity vary with tmpratur as illustrats i Fig 83 At low tmpraturs, th Frmi rgy is pid at th midpoit btw th door lvl ad th ffctiv coductio bad dg, i E c for D = 3 ad Ec+ E for D = or Physically, this corrspods to th situatio i which all lctros rsid o th door atoms As th tmpratur is raisd, th Frmi lvl drops blow th door lvl ad lctros ar trasfrrd ito th coductio bad Evtually, most of th door atoms hav giv up thir lctro to th coductio bad ad a platau at -3 D NI= m is rachd for th lctro dsity This tmpratur rag is calld th saturatio rag whras th low tmpratur rag i which th lctro dsity icrass xpotially is kow as th frz out rag If mor lctros should b addd to th coductio bad, thy must b tak from th valc bad, which is oly fasibl at rathr high tmpraturs about 8 K as s i th graph Figur 83 Frmi rgy ad lctro dsity vs tmpratur for dopd GaAs i, ad 3 dimsioal structurs It is otd, that at rlativly low tmpraturs th lctro dsity i th quatizd structurs is lss tha for th bulk This is maily du to th icrasd sparatio btw th door lvl ad th ffctiv coductio bad dg du to th cofimt rgy Howvr, at high tmpraturs (abov th saturatio rag) th trd is rvrsd Th raso is that th 3D Frmi lvl icrass stply abov th saturatio rag ad much lss so for th D ad D cass Th origi of this 69

71 diffrc is th way th prfactors i th lctro dsity Eq(86) vary with D/ tmpratur, i th T bhavior of th factors i frot of th xpotials Exrcis: Proprtis of D dimsioal smicoductors Cosidr a matrial with a simpl parabolic coductio bad a) Show that th dsity of stats xprssios i Eq(84) ar corrct Now, assum that w r dalig with a u-dopd smicoductor, i tak N I= b) Providd oly th lowst quatizd lctro ad hol lvls ar icludd, show that h D m h EF= ( Ec+ E+ Ev+ E) + kt l 4 m 7

72 9 PN ad PIN Juctios I this chaptr, w tur to som basic but importat lctroic applicatios of smicoductors ad ivstigat th pculiar faturs that appar i th aoscal rgim Th basic lmt of (bipolar juctio) trasistors is th PN juctio, which itslf acts as a currt rctifir ad, hc, is a importat dvic i its ow right W thrfor discuss this dvic i dtail blow As th am says, a p juctio diod cosists of a juctio btw a p-dopd ad a -dopd smicoductor I th itrfac rgio, a lctric fild ariss du to ubalacd chargs Th fild xists i a arrow rgio calld th dpltio layr, which forms a barrir for th charg carrirs Hr, w aalyz th juctio i ordr to comput th lctrostatic pottial corrspodig to th fild Subsqutly, w look at a rlatd structur, i th pi juctio, i which a itrisic layr is isrtd btw th dopd rgios Figur 9 Illustratio of th p juctio Th bottom part shows th spac charg dsity W rstrict th aalysis to thr dimsioal structurs so that carrir coctratios, lctric filds tc vary oly alog x Our startig poit is th p juctio illustratd i Fig 9 I th last chaptr, w discussd -dopd matrials, i which a larg coctratio of door impuritis supplid th xtra lctros Similarly, p-dopd matrials cotai a larg coctratio of accptor impuritis supplyig additioal hols to th coductio bad To distiguish btw th two typs of impuritis, w dot thir coctratios by N DI ad N AI, rspctivly Th basic physics of th p juctio is rathr simpl: O th -sid thr is a surplus of mobil lctros ad o th p-sid thr is a surplus of mobil hols A crtai umbr of lctros will thrfor diffus from th - ito th p-sid ad vic vrsa for th hols W ow mak th followig simplifyig assumptio: Th lctros missig from th -sid ar tak from a layr of width W, which is compltly dpltd of fr lctros Similarly, th hols ar tak from a dpltd layr of width W p o th p-sid W saw i th prvious chaptr that for tmpraturs i th dpltio rag, th lctro 7

73 coctratio i th -dopd matrial is approximatly qual to th door coctratio N I aalogy, w ll hav p N for a bulk p-dopd matrial DI Hc, aftr formatio of th dpltio layr at th itrfac btw th two sids, th t charg dsity will corrspod to th illustratio i th lowr part of Fig 9 Ovrall charg utrality of cours dmads that AI NDIW= N AIWp (9) Locally, chargs ar obviously ot balacd ad so thr xists a spac charg dsity This charg dsity will produc a lctric fild E( x ) dirctd from th positiv charg towards th gativ, i from right to lft This fild acts to prvt furthr lctros from diffusig ito th p-sid ad hols from diffusig ito th - sid Hc, o ca imagi th formatio of th juctio as follows: Start from two sparatd pics of - ad p-dopd smicoductors Upo brigig th two togthr, lctros will diffus to th lft ad hols to th right This cotius util a sufficit lctric fild has b stablishd ad furthr diffusio is prvtd Th dpltio layr thrfor forms a barrir for th chargs ad our aim is to dscrib th widths W ad W p 9 Aalysis of th PN Juctio I th aalysis, w first solv th Poisso quatio to fid th fild W tak x= at th itrfac ad so th boudary coditios for th fild ar that E( W ) = E( W ) = With th charg dsity i Fig 9 w th fid p de = NDI dx εε N AI < x< W W < x< NDI( x W ) < x< W E( x) = εε N ( + ) < < AI x Wp Wp x Hr, ε is th rlativ dilctric costat Sttig th two xprssios for th fild qual at x= is asily s to lad to th charg cosrvatio coditio Eq(9) Nxt, th lctric pottial V is rlatd to E via E = dv / dx Outsid th dpltio layrs, th pottial must b costat ad o th two sids th costat valus ar dotd V ad V p, rspctivly Aftr itgratio, w cosqutly hav p N V x W < x< W V( x) = N V + x+ W W < x< DI ( ), εε AI p ( p ), p εε 7

74 Agai, w rquir cotiuity ad st th two xprssios for th pottial qual at x= i N N V W = V + W DI AI p p εε εε N N V V V = W + W DI AI D p p εε εε, (9) whr V D is th so-calld diffusio pottial or built-i pottial Schmatically, th pottial varis across th juctio as illustratd by th solid li i Fig 9 Figur 9 Th lctric pottial V across th juctio Tratig V D as a kow quatity, Eqs(9) ad (9) provid two quatios for th two ukows W ad W p that w ca solv to giv W / / εεn AIV D εεn DIV D =, Wp= NDI( NDI+ N AI ) N AI( NDI+ N AI ) I additio, thir sum W= W+ Wp is W εε ( N + N ) DI AI = VD NDIN AI / (93) To b of ay us, howvr, w still d th valu of V D To this d, w first stablish th law of mass actio, which is th smicoductor quivalt of th chmical 73

75 quilibrium coditio of statistical mchaics It is asily drivd usig th dsity xprssios from last chaptr = N3 xp{( EF Ec )/ kt} ad p= P3 xp{( Ev EF )/ kt} By formig th product it is s that th ukow Frmi rgy cacls ad w gt p= N P xp{( E E )/ kt} = N P xp{ E / kt}, 3 3 v c 3 3 g i whr Eg= Ec Ev is th rgy gap ad i is th dsity of both lctros ad hols i th itrisic (u-dopd) cas Th law of mass actio thrfor stats that o mattr what happs to th Frmi rgy, th product of ad p rmais a costat -3 For GaAs aroud room tmpratur, m [] Now, to th far lft th i spac chargs hav producd a pottial V p ad to th far right th pottial is V Th pottial rgy of a lctro is chagd by a amout V by placig it i a pottial V O th othr had, without V th pottial rgy must b E sic this is th total rgy if k= Ec Hc, addig V simply corrspods to rplacig V for lctros Similarly, for th valc bad c E c by V E v is to b rplacd by Ev = N xp{( E E+ V )/ kt} It follows that th lctro coctratio must vary as 3 F c W ow form th ratio btw lctro coctratios o th - ad p-sids: ( -sid) N xp{( E E+ V )/ kt} ( p-sid) N xp{( E E+ V )/ kt} 3 F c = = 3 F c p xp{ V / kt} D W saw alrady that ( -sid) N if w ar far from th juctio Additioally, th law of mass actio tll us that DI ( p-sid) p( p-sid) i = Thus, usig p( p-sid) N w s that AI N N kt N N xp{ VD / kt} = V = l DI AI DI AI D i i Usd i Eq(93), this dmostrats that th total dpltio layr width W is W εεkt( NDI+ N AI ) N l DIN AI = NDIN AI i / Th dopig dpdc of th dpltio width is show i Fig 93 udr th assumptio of qual door ad accptor dopig NDI= NAI It is s that W may bcom as small as m or v lss if th dopig coctratio is sufficitly high 74

76 Figur 93 Th total dpltio width W as a fuctio of dopig coctratios for GaAs assumig idtical door ad accptor coctratios A word of cautio is rquird hr, howvr: At such high impurity coctratios, th assumptios usd for th calculatio of carrir dsitis i th prvious chaptr ar likly to brak dow I particular, for g havily -dopd matrials th Frmi rgy will li abov th bad dg E ad, similarly, E is likly to b blow E o c th p-sid Hc, th actual dpltio width W at high dopig dsity is xpctd to dviat from th plot i Fig 93 W will ot try to costruct a mor laborat dscriptio hr, howvr, bcaus w xpct th simpl thory to b at last qualitativly corrct F v 9 PIN Juctio A pi juctio is similar to a p juctio but diffrs by th isrtio of a itrisic (u-dopd) ctral rgio, as illustratd i Fig 94 Th thickss of th itrisic layr is d, ad th costat lctric fild i this rag is dotd E I Th aalysis of th lctrostatic pottial follows th prvious sctio closly, ad w oly list th fial rsult NDI V ( x d W ), d< x< d+ W εε N AI V( x ) = Vp+ Wp EIx, < x< d εε N AI Vp+ ( x+ Wp ), Wp< x< εε W 75

77 Figur 94 Schmatic illustratio of a pi juctio gomtry By costructio, th pottial is cotiuous across th boudary at x= To sur cotiuity at x= d w rquir that N N V W = V + W E d DI AI p p I εε εε I additio, th slop must b cotiuous ad w fid N εε W N = E, W = E (94) DI AI I p I εε Ths rsults just dmostrat that Eq(9) is still valid, as xpctd W ca combi th boudary coditios to form a sigl quatio for W p N N N V W W W d AI AI AI D= p+ p+ p εεn DI εε εε with th solutio W p / NDId εεndiv D NDId = + N N + N ( N + N ) N + N DI AI AI DI AI DI AI This rsult is s to agr with that of th p juctio if d = Also, from Eq(94) it is a simpl mattr to comput th built-i fild A particular cas ariss wh th itrisic rgio is wid, i wh d W, Wp I this cas, th quatio for W p rducs to VD N AIWpd /( εε ) ad combid with Eq(94) it follows that 76

78 E / I V D d Hc, th fild is simply th diffusio pottial dividd by th thickss Th thickss dpdc of th fild i a silico juctio is compard to th /d approximatio i Fig 95 Figur 95 Magitud of th built-i lctric fild of a pi juctio i th dpltio approximatio 93 Slf-cosistt Pottial ad Trappd Chargs Th prvious aalysis rlid havily o th dpltio approximatio I this sctio, w wish to study th xact, umrical solutio to th Poisso quatio without ivokig this approximatio Morovr, our umrical schm allows us to iclud mor advacd ffcts such as trappd chargs To kp thigs o a rasoably simpl lvl, w rstrict th aalysis to compltly symmtrical pi juctios That is, th two halvs of th juctio ar complt imags of o aothr This mas that N AI= NDI ad w us idtical widths of th dopd rgios W choos th pottial at th midpoit as our zro poit ad so V = V = V / Accordig to th p D discussio abov, th lctro ad hol dsitis i th prsc of a lctric pottial ar giv by { } { } = N xp ( E E+ V )/ kt, p= P xp ( E E V )/ kt, 3 F c 3 v F rspctivly Our assumptio of complt lctro-hol symmtry implis that w ca writ ϕ ϕ V = i, p= i, ϕ kt 77

79 Not that th commo pr-factor must cssarily b mass actio p= i i to comply with th law of W ow tur to th slf-cosistt solutio of th Poisso quatio Th ivstigatd gomtry is show blow Figur 96 Fully symmtric pi juctio gomtry W focus o th rgio x for which w ca writ th Poisso quatio dϕ ϕ ϕ = { i i NDIθ( x d /)} dx εε kt Th boudary coditios hr ar ϕ () = ad ϕ( L) = ϕ l( N / ) Th lattr is DI i quivalt to V = V / Bfor attmptig a solutio, w rwrit as D dϕ N = = i DI { sih ϕ θ( x d /)}, dx εεkt i Also, it is covit to masur x i uits of th scrig lgth ( εε ) / / i l= kt so that dϕ = sih ϕ θ( x d /) (95) dx This diffrtial quatio is a xampl of a broad class of oliar problms that ca b formulatd grally as dϕ = F( ϕ( x), x), a x b, F( ϕ( a), a) = F, ( ( ), ) a Fϕ b b = Fb dx Th last two rlatios ar mat to idicat that boudary coditios ar spcifid i th form of fuctio valus at th d poits Th gral stratgy for solvig such quatios is through discrtizatio Hc, w divid th x-rag ito N+ poits via 78

80 b a xi= a+ i, i=,, N, = N Similarly, w dfi ϕ ϕ( x ) ad F F( ϕ, x ) A simpl mthod ivolvs discrtizig th scod drivativ as follows i i i i i dϕ ϕ( x+ + ) ϕ( x ) ϕ( x) dx I this way, w ca writ ϕ + ϕ ϕ= F i N i+ i i i,, This st of oliar quatios ca b rformulatd as a matrix quatio by itroducig ϕ F ϕ ϕ F ϕ =, F, H = =, ϕ F ϕ N N N FN ϕn so that H ϕ= F ϕ= H F It s importat to rmmbr that th F s dpd o th ϕ s so that mthod must b itratd util covrgc For istac, if w writ two succssiv itratios as ϕ old ad ϕ w w hav ϕ w = H F( ϕold ) This simpl approach, howvr, oft fails to covrg A dramatically improvd vrsio is th matrix Nwto mthod that procds as follows First, w writ th discrtizd st of quatios as H ϕ = F( ϕ ) Scodly, w xpad aroud ϕ old, i Solvig, w th fid H ϕ + δϕ F( ϕ ) + G( ϕ ) δϕ, δϕ= ϕ ϕ ( ) old old old w old 79

81 ϕ = ϕ + ϕ ϕ ϕ ( H G( )) ( F( ) H ) w old old old old I th prst cas, th matrix G is a simpl diagoal matrix with lmts G F ii= ϕ ϕ = ϕi For th pi juctio, illustratd as th black curv i Fig 97 G ii = coshϕ Th covrgd rsult for Si (l = 9 µ m ) is i Figur 97 Normalizd lctrostatic pottial of a symmtric Si pi juctio with varyig trap coctratios takig d = 34 m ad L = m W fially wish to discuss th rol of traps i such a juctio Traps captur lctros ad hols i localizd stats ad prvt thm from cotributig to th currt Morovr, trappd chargs td to scr th built-i lctric fild, which may b dtrimtal for charg sparatio i a pi juctio solar cll I th prvious chaptr, w ivstigatd lctro doors ad drivd thir occupacy udr th assumptio that two lctros i a sigl sit is highly improbabl du to th rpulsiv rgy I a compltly lctro-hol symmtric modl this dos t work, howvr Th trap stat has thr possibl charg valus:, +, ad Th utral trap corrspods to o lctro i th stat ad ± corrspod to a additioal hol or lctro To maitai symmtry, it is cssary to rquir that th rgy of th stat is prcisly twic th rgy of th stat Thus, th lctro occupacy of a sigl trap bcoms 8

82 T xp{( EF ET )/ kt} + xp{( EF ET )/ kt} = + xp{( E E )/ kt} + xp{( E E )/ kt} F T F T If th trap coctratio is ( ) N = N T T T N T it follows that th trappd charg coctratio is xp{( EF ET )/ kt} + xp{( E E )/ kt} + xp{( E E )/ kt} ET E F = NT tah kt F T F T Agai, i th prsc of a pottial V, w must rplac ET ET V Morovr, th lctro-hol symmtry rquirs E T= V Fially, th Frmi lvl (which is obviously idpdt of x) must b fixd at E F= V as wll This implis that w ca writ th modifid Poisso quatio as dϕ ϕ NT = sihϕ+ tah ( /), T θ x d T= dx i Naturally, i th Nwto mthod, w ow iclud th additioal trm ad tak G = cosh ϕ+ /[ cosh ( ϕ /)] Som rsults for various trap dsitis ar { } ii i T i show i Fig 97 Notic how trappd chargs td to scr th pottial ad thrby rduc th lctric fild Exrcis: Slf-cosistt p juctio modl Th p juctio modl usd abov is oly approximatly corrct Th mai problm lis i our assumptio of a simpl rctagular charg distributio as show i Fig 9 To rmdy this, o should calculat th actual charg distributio rathr tha postulatig it I this xrcis, w cosidr a p juctio udr th simplifyig assumptio of symmtry btw N- ad P-sids ad btw lctros ad hols Hc, th startig poit is Eq(95) but with d = dϕ NDI = sih ϕ, =, x> dx i This is a complicatd, scod-ordr oliar diffrtial quatio but ca b rformulatd as a somwhat simplr first-ordr quatio usig a mathmatical trick First, w multiply by dϕ / dx : dϕ dϕ dϕ = { sihϕ } dx dx dx 8

83 a) Show that this is quivalt to d dϕ d = { coshϕ ϕ} dx dx dx b) Itgrat this rsult ad us th boudary coditios to show that dϕ = { coshϕ ϕ cosh ϕ } / + ϕ, ϕ= l( ) dx This rsult, which is still xact, provids a aalytical xprssio for th ormalizd lctric fild E ( x) = dϕ / dx orm at th juctio { ϕ ϕ} / ( ) E orm() = cosh + l( ) Th actual ad ormalizd filds ar rlatd via = ( εε) / E( x) kt / E ( x) i orm c) Show that th dpltio approximatio prdicts a ormalizd fild of E () = l( ) orm Rfrcs [] M Balkaski ad RF Wallis Smicoductor Physics ad Applicatios (Oxford Uivrsity Prss, Nw York, ) 8

84 PN Juctio ad Tulig Diods I th prvious chaptr, w aalyzd th p ad pi juctios without bias Hc, w could comput lctrostatic pottials ad carrir coctratios basd o thrmal quilibrium statistical mchaics I th prst chaptr, w wish to study th bhavior udr a bias voltag producig a currt flowig through th juctio Th purpos is to comput this currt udr gral assumptios ad ultimatly udrstad th rctifyig bhavior of p juctio diods I ordr to fully dscrib th dvic, howvr, w wat to graliz th rsults of chaptr 4 Throughout this book, w hav rlid o liar rspos thory ad so th calculatd currt cssarily varis liarly with th applid voltag This mas that w rstrict ourslvs to th ohmic rgim, i which currt ad voltag ar proportioal To fully dscrib th rctifyig bhaviour of a diod, w clarly d to go byod th liar rspos ad ivstigat th o-liar rlatio btw currt I ad applid voltag V A Our approach will rly o plausibl argumts i ordr to graliz th rsults of chaptr 4 rathr tha rigorous aalysis W bgi by rcallig Eq(45) for th ohmic rgim VA I= f ( E) T ( E) de πħ, () whr th assumptio of dcoupld chals has b mad Hr, th summatio is ovr th trasvrs igstats labld by th idx Each igstat corrspods to a coductio chal with trasmittac T ad trasvrs rgy E Th total rgy of a lctro i this stat is thrfor E= E + E, whr E is th kitic rgy associatd with th motio paralll to th wir axis To procd, w ow cosidr th spcific gomtry illustratd i Fig Hr, th systm agai is positiod btw two lads: lft (L) ad right (R) Th right lad is groudd ad th lft lad is biasd at a pottial V A Figur Biasd juctio with th right-had sid groudd Th Frmi fuctios with Frmi lvls ϕ for th two sids ar show blow L / R 83

85 I th prsc of a pottial V, th lctro rgis ar shiftd by a amout V Hc, i th lft ad right lad, th lctro rgis ar rally EL= E VA ad ER= E, rspctivly This mas that th statistical distributios for th two halvs ca b writt fl( E) =, f R( E) = xp{( E V E )/ kt} + xp{( E E )/ kt} + A F F Naturally, f R( E ) is simply th usual, ubiasd Frmi fuctio f ( E ) Now, if V A is sufficitly small it is obvious that fl( E) fl( E) fl( E) fr( E) VA = VA V E V = A VA= A O this groud, w postulat [] that th currt flowig i Fig udr a fiit bias voltag should b calculatd by th followig gralizatio of Eq(): I= { fl( E) fr( E) } T ( E) de π ħ () Hr, w chagd itgratio variabl from E to E sic thir diffrc E is just a costat This xprssio is a vry gral rsult applyig qually wll to low dimsioal structurs for which th summatio is ovr a rstrictd st of discrt trasvrs stats as to bulk dvics for which th summatio bcoms a itgral ovr a cotiuum of trasvrs stats W ow focus o th lattr situatio Thr-dimsioal No-dgrat Dvics If th trasvrs ara of th juctio A is sufficitly larg, th summatio ovr i Eq(3) bcoms a itgral ovr a cotiuum of trasvrs igstats Ths stats ar lablld by th trasvrs wav vctor k ( k, k ), which rplacs th = subscript Thus, th sum is actually ovr ( k, k ) ad writig de dk =ħ k m w fid / / y z y z E =ħ k /m to gt A Am () = () = () dk () ydkz= de 4π πħ ky, kz By writig E =ħ k /m w hav implicitly chos th smicoductor coductio bad dg E as th zro-poit for th rgy Summig ovr all trasvrs stats is c cosqutly rplacd by a itgral ovr th trasvrs rgis E Also, th - 84

86 dpdc of T ( E) ow should b thought of as a dpdc o th cotiuous paramtr E ad, thrby, T bcoms a fuctio of th both rgis E ad E ad so Eq() bcoms Am I= 3 { fl( E) fr( E) } T( E, E ) de de π ħ Hr, of cours E= E + E Not that by itgratig oly ovr E w rstrict th calculatio to stats with rgy abov th smicoductor coductio bad dg I th prst sctio, w wat to focus o o-dgrat situatios similar to th tratmt i chaptr 8 No-dgracy mas that th ma occupacy of stats i th coductio bad is much lss tha uity This is th cas whvr th Frmi rgy is far blow all rgis E i th bad I this cas, th distributio fuctios ca b approximatd as i chaptr 8 so that f ( E) xp{( E + V E)/ kt}, f ( E) xp{( E E)/ kt} (3) L F A R F This approximatio is applicabl to both th mtal i th lft lad ad to th smicoductor sic all rgis cosidrd li abov E ad, hc, far abov E Cosqutly, ad so ( V A / kt ) f ( E) f ( E) xp{( E E)/ kt} L R F Am / / / 3( V A kt ) E F kt I= E kt T( E, E ) de de π ħ As w hav idicatd, th trasmittac may, i gral, dpd o both compots of th kitic rgy I systms with prfct traslatioal ivariac alog th itrfacs, howvr, phas-matchig rquirs cosrvatio of th prpdicular compot of th wav vctor k ad accordigly of E =ħ k /m Sparatio of prpdicular ad paralll variabls th shows that oly th lattr ar ivolvd i th tullig procss Cosqutly, T is idpdt of E ad itgratig ovr this dgr of frdom lads to c F AmkT V / / / A kt E E kt F kt I= 3 ( ) T E de π ħ ( ) (4) This xprssio is a valuabl tool for th calculatio of currts i various typs of VA / kt juctios It is also clar that w xpct a rathr uivrsal bhaviour I for 85

87 diffrt dvics As w shall s xt, a fully classical aalysis of th p juctio diod agrs prcisly with this xpctatio PN Juctio Diod I diod applicatios, a xtral bias voltag V A is applid across th diod For dfiitss, w tak th -sid to b groudd so that this pottial rmais fixd at a valu V Hc, by a positiv bias w udrstad a positiv voltag applid to th p-sid raisig th pottial hr from V p to Vp+ VA I th biasd cas, th pottial follows th dashd li i Fig Figur Th lctric pottial V across th juctio Th solid ad dashd lis illustrat th cass without ad with ad xtral bias V assumig th -sid groudd Th full aalysis of th biasd p juctio is a complicatd umrical problm, v for th prst o-dimsioal cas It cosists i solvig th coupld drift-diffusio quatios Th first of ths is th Poisso quatio for th pottial A d V dx = ρ( x), εε whr ρ is th charg dsity Scodly, w d th xprssios for lctro ad hol currts d dp J( x) = σe+ D, Jh( x) = σhe Dh dx dx Hr, E is th lctric fild whras σ i ad D i ar coductivitis ad diffusivitis of lctros (i = ) ad hols (i = h) Udr th ffctiv mass approximatio, th coductivitis ca b assumd proportioal to carrir dsitis as show i Eq(8) Hc, w writ σ = µ ad σ h= µ hp, whr µ i is th mobility Th diffusivitis ar coctd to th mobilitis via th Eisti rlatio D= µ kt / [] To complt th systm of quatios, w d to look at th way currts chag with positio i i 86

88 Th total currt J flowig through th juctio is a sum J= J+ Jh of lctro ad hol currts ad, morovr, idpdt of positio x Hc, J Jh + = x x Howvr, th idividual currts do vary across th juctio This follows from th cotiuity quatio, which for lctros rads as whr J = + G, t x G is th t gratio rat of lctros Similarly for hols p Jh = + Gh, t x whr G h is th t gratio rat of hols I stady stat, subtractig th two quatios shows that G= Gh G This maks ss, as lctros ad hols ar always producd i pairs Svral diffrt mchaisms cotribut to carrir gratio ad rcombiatio Th most importat os ar (i) bad to bad, (ii) trap mdiatd ad (iii) Augr procsss W will focus o th importat cas of trap mdiatd carrir gratio ad rcombiatio I this cas, a aalysis of th rlvat trasitio rats (s th xrcis) lads to th rsult i p G= τ ( + ) + τ ( p+ p ) h Hr, τ ad τ h ar lctro ad hol liftims ad ad p ar factors proportioal to th coductio ad valc stat dsitis subjct to th coditio p = This is th famous Shockly-Rad-Hall xprssio for th carrir i gratio rat To simplify mattrs v furthr, w may assum that thigs ar roughly symmtrical i that τ τh= τ ad p= i I that cas, i p G= τ + p+ i (5) Wh combid udr stady-stat coditios, th drift-diffusio quatios thrfor rad 87

89 d V dx { ( x) p( x) N ( x) N ( x) } Ths must b supplmtd by th dsity xprssios = DI + εε d J ( x) = µ ( x) E( x) + ktµ, J ( x) = µ p( x) E( x) ktµ dx dj djh = G( x), = G( x) dx dx h h h AI dp dx (6) ( x) = N xp{[ E E+ V( x)]/ kt} ad p( x) = P xp{[ E V( x) E ]/ kt} 3 F c 3 v F Solvig th full st is difficult ad ca oly b do usig umrical tools Howvr, a simpl approximatio ca b costructd by th followig rasoig: Far to th lft of th dpltio rgio i a wid dvic thr is practically o lctric fild Hc, th currt is purly diffusiv By combiig th last two lis of quatios, it th follows that d d kt µ G( x) D ( ) G x dx = dx = Morovr, far to th lft w may tak p NAI ad so usig th gratio rat Eq(5) ad th fact that N, lads to AI i d D dx N = i AI τn AI At th xtrm lft, ar th mtal cotact, thrmal quilibrium is rstord bcaus all xcss carrirs ar forcd to rcombi via mtal ad itrfac stats Hc, th lctro dsity is / N Th simpl scod ordr quatio abov ca ow b solvd udr th boudary coditios W N V kt ( p ) = i / AI xp{ A / } i AI that lad to i V ( )/ A / kt x+ Wp L ( x) = { + ( ) }, N AI = N ad ( ) i / whr L= τd is th lctro diffusio lgth A similar xprssio is foud for th hol dsity to th right of th dpltio zo Takig th gradits, w thrfor fid th currts AI 88

90 Di V / ( )/ A kt x+ Wp L J( x) = ( ), x< W L N h i VA / kt ( x W )/ Lh h( ) = ( ), > LhNDI Th total currt is J= J ( x) + J ( x) Th problm, howvr, with ths xprssios is h AI D J x x W that thy caot simply b addd bcaus thy ar valid i diffrt rgios Of cours, w ca always writ J= J ( W ) + J ( W ) + J, J = J ( W ) J ( W ) h p scr scr p Hr, J scr is th spac charg rgio currt that accouts for th chag i lctro currt ovr th dpltio rgio W ca rwrit this quatity usig th gratio rat W dj( x) i p Jscr= dx= dx dx τ + p+ Wp W Wp i p At both dgs of th dpltio rgio w hav p= i xp{ VA / kt} Hc, as a simpl stimat w will tak p xp{ V / kt} throughout th zo so that i As a cosquc, this cotributio varis as xp{ VA / kt } at larg bias i cotrast to th diffusiv part varyig as xp{ V / kt } Tak togthr th total currt ca b approximatd as J W Wi = τ VA / kt i scr VA / kt τ + A A ( η ) I( V ) = I xp{ V / kt}, (7) A A VA / kt ( ) whr η is a idality factor ad of th matrials I is a costat rlatd to th lctric proprtis 3 Tulig Currt I a low or modratly dopd p juctio diod, th dpltio rgio is sufficitly thick that carrirs caot tul dirctly btw valc ad coductio bads I cotrast, i highly dopd structurs th barrir thickss is gratly rducd Cosqutly, wh th barrir bcoms sufficitly thi (about m) tulig of carrirs is xpctd to occur This phomo is th basis of th tulig diod, also kow as th Esaki diod aftr its ivtor Bcaus carrirs simply tul through a 89

91 thi barrir, th tul diod is a xtrmly fast dvic capabl of opratio at svral hudrd GHz Hc, it is a importat applicatio of aoscal faturs i lctroics It is otd that th built-i voltag V D is rducd to VD VA i th prsc of th bias I tur, rplacig V D by VD VA i Eq(93) shows that V A tds to rduc th dpltio width, i / ( N N ) W V εε + V V V V DI AI ( A ) = ( D A ), A D NDIN AI (8) This xprssio braks dow if VA> VD ad is, thrfor, rstrictd to th low bias rag VA VD Th dpltio rgio forms a quatum mchaical barrir for lctros to tul dirctly btw valc ad coductio bads W will aalyz this problm usig th WKB tulig formula drivd i chaptr 5 x m T( E) = T xp ( U( x) E) dx, (9) ħ x i which U( x ) is th varyig pottial rgy ad x, ar th dpoits of th barrir giv by th solutios of U( x, ) E= I th prst cotxt, th tulig barrir ca b udrstood from Fig 3 Figur 3 Tulig gomtry showig th pottial rgy of lctros i th two bads Th gr ad blu aras ar occupid by lctros ad th rd triagl is th tulig barrir This figur illustrats th x-dpdc of th pottial rgy across th juctio For a lctro i th coductio bad, th pottial rgy is Uc= Ec V ad similarly for th valc bad Uv= Ev V Ths rgis ar displayd as th solid lis i th figur usig th simplificatio that th lctric pottial varis roughly liarly across th barrir rathr tha havig th actual curvd shap show i Fig 9

92 Th two bads ar sparatd by th bad gap E g Also, th gr ad blu aras illustrat occupid lctro stats I gral, th Frmi lvls o th two sids of th juctio will ot b qual sic quilibrium is disturbd wh a currt is flowig W igor this complicatio hr, howvr, ad tak a sigl valu E F as th Frmi rgy throughout th structur Cosidr a lctro with rgy qual to th Frmi rgy movig from right to lft i th -sid It obviously blogs to th coductio bad At a crtai poit isid th dpltio layr, th lctro coutrs a rgio that is classically forbidd, that is, th pottial rgy U xcds th total rgy E of th lctro This is c idicatd as th rd ara i th figur Th lctro caot mov furthr to th lft withi th coductio bad ad ormally would b rflctd back Thr is, howvr, a possibility that th lctro tuls through th rd barrir ito th valc bad, whr it is fr to kp movig to th lft This is th origi of th tulig currt As show i th figur, th full hight of th triagular barrir is th bad gap E ad th width is dotd B This width B is ot kow but by g comparig cogrut triagls i th figur it is s that With a x-coordiat ruig from right to lft, th barrir hight is giv by U ( x) E = E x / B Thus, th trasmittac i Eq(4) is asily obtaid as c F g B W W = B= Eg E ( V V ) ( V V ) g D A D A F B m E g 4 me g T( EF ) = T xp x dx T xp B = B 3 ħ ħ If fially th xprssio for th barrir width B is isrtd w fid 3 4 me g W T( EF ) = T xp 3 ħ ( VD VA ) Wh w combi with Eq(93) for th dpltio width w ca rwrit this xprssio i a simpl form highlightig th bias dpdc: 3 V 64 εεmeg( NDI+ N AI ) B T( EF ) = T xp, VB= 3 VD V A 9ħ NDIN AI () Havig dtrmid th barrir trasmittac allows us to calculat th tulig currt I T by mas of th low-bias Ladaur formula Eq(46): 9

93 IT ( VA ) = VAT( EF ) h pr occupid chal I fact, w do t fulfill th coditios applid i drivig th Ladaur formula i th prst cas Th problm is that lctros ar icidt i o chal (coductio bad) ad tulig ito a compltly diffrt o (valc bad) Howvr, w xpct a similar bhavior ad w might thrfor assum I ( V ) V T( E ) Thus, w ca formulat th tulig currt as T A A F V B IT ( VA ) = GTVA xp, VD V A whr G T is a ffctiv coductac rlatd to T To gt a sizabl tulig currt w d a small V B From Eq() is it clar that this rquirs a small ffctiv mass m ad a larg dopig coctratio Thus, havily dopd GaAs is a good 6-3 cadidat ad i Fig 5 blow, w hav tak NDI= N AI= m This choic yilds VB 3 V ad VD 6 V Th figur illustrats th tulig currt as wll as th ormal currt giv by Eq(7) ad th total currt Figur 5 Tul, ormal ad total currt vs applid voltag for havily dopd GaAs Exprimtally, th tulig ffct was first obsrvd i Grmaium p juctios by L Esaki Evtually, Esaki was rwardd with th Nobl Priz for this work Th plot blow is tak from his origial publicatio from 958 9

94 Figur 6 Exprimtal tulig currt as a fuctio of voltag for a arrow G p juctio Tak from [] Exrcis: Trap-assistd carrir gratio ad rcombiatio I this xrcis, w aalys lctro ad hol gratio ad rcombiatio via a trap stat locatd i th bad gap Th situatio is illustratd i Fig 7 Figur 7 Schmatic of th four procsss by which carrirs hop btw bads ad trap stats Four procsss, dotdd R through R 4 i th figur, cotribut to th charg balac Thy ar giv by R = C ( f ), R = C f t t R = C p ( f ), R = C pf 3 p t 4 p t 93

95 I ths xprssios, f t dots th Frmi fuctios valuatd at th trap lvl, i th probability that a trap is occupid Also, C ad C p dot charactristic trap captur/missio cofficits that ar proportioal to trap dsity Fially, ad p dot th dsity of availabl lctro stats i th coductio bad ad availabl hol stats i th valc bad, rspctivly Obviously, G= R R ad G = R R h 3 4 a) Show that by quatig ths rats, w ar lad to th balac C f C ( f ) = C p ( f ) C pf t t p t p t f t C+ Cpp = C ( + ) + C ( p+ p ) p b) Show that applyig th rsults immdiatly provids th carrir gratio rats as CCp( p p) G= Gh= G C ( + ) + C ( p+ p ) p Now, this xprssio should hold also i th ubiasd casd for which G = G = ad p= i Hc, it follows that τ / C ad τ / C so that p h p = Also, w will itroduc carrir liftims i h i p G= τ ( + ) + τ ( p+ p ) h This famous xprssio for th carrir gratio rat was first drivd by Shockly, Rad, ad Hall Rfrcs [] M Balkaski ad RF Wallis Smicoductor Physics ad Applicatios (Oxford Uivrsity Prss, Nw York, ) [] L Esaki, Phys Rv 9, 63 (958) 94

96 Mtal-Smicoductor Juctios Juctios btw mtallic lads ad smicoductors ar of grat practical importac As w shall s, udr appropriat coditios currts flowig across th juctio ar rctifid i that currt flow is oly possibl if lctros mov from th smicoductor ito th mtal This is th basis for Schottky diods ad th physical barrir prvtig currts flowig i th opposit dirctio is kow as a Schottky barrir W will cosidr th spcific gomtry illustratd i Fig Hr, th systm agai is positiod btw two lads: lft (L) ad right (R) Th right lad is groudd ad th lft lad is biasd at a pottial V A Figur Biasd mtal-smicoductor juctio with th smicoductor sid groudd Schottky Diod As i th prvious chaptr, w bas th aalysis o th gral currt-voltag rlatio for thr-dimsioal dgrat dvics, cf Eq(4) rpatd hr: AmkT V / / / A kt E E kt F kt I= 3 ( ) T E de π ħ ( ) () Wh a mtal is brought ito cotact with a smicoductor, charg will b rdistributd ar th itrfac If th smicoductor is -typ, lctros will flow ito th mtal util a balac is rachd Elctros mov ito th mtal bcaus lowrgy stats ar availabl thr Th flow soo stops, howvr, bcaus a surplus of gativ charg is build up o th mtallic sid of th juctio This gativ spac charg producs a lctric fild that coutracts th lctro flow Statd i a diffrt way, a pottial rgy barrir is formd ad th tullig trasmittac T of this barrir is prcisly th o dd to calculat th I / V charactristic of th Schottky diod usig Eq() W ow sk to dscrib th barrir i th u-biasd cas, i assumig vaishig applid voltag 95

97 Figur Schmatic of th mtal-smicoductor juctio formd by cotactig mtal ad smicoductor pics Aftr cotact, charg rdistributio lads to bad bdig To comput shap ad hight of th barrir, w cosidr Fig, which illustrats th situatio bfor ad aftr a mtal is brought ito cotact with a -typ smicoductor A importat cosquc of th cotact is that th Frmi lvls of th two matrials quilibrat This is a rquirmt if thrmal quilibrium without charg flow is to stablish Th pottial curvs rally display th lctro rgis i th prsc of th lctrostatic pottial i th juctio Hc, th coductio bad rgy Uc= Ec V cotais th costat bad dg E c ad th lctrostatic bad bdig V of th lctros Similarly, th vacuum lvl cosists of th ovrall vacuum lvl ad th bad bdig U vac= Evac V As idicatd i th figur, o bad bdig is assumd i th mtallic rgio Th raso for this is th much highr dsity of scrig chargs i th mtal that, as w shall s blow, lads to a gligibl variatio of th pottial isid th mtal Isid th mtal, th vacuum lvl is sparatd from th Frmi lvl by th bulk mtal work fuctio Φ Similarly, wll ito th smicoductor sid, all bads ar flat ad th distac from th Frmi lvl to th vacuum lvl Φ S is a sum of th lctro affiity EA= Evac Ec ad th coductio bad off-st Ec EF Ths obsrvatios allow us to calculat th barrir hight if w mak th followig assumptios: () th vacuum lvl is compltly flat isid th mtal ad () th vacuum lvl i cotiuous across th juctio I rality, assumptio () is of cours always fulfilld Howvr, chargs may accumulat i a xtrmly thi layr at th itrfac du to dfct ad itrfac stats Such a localizd charg layr will cotribut to th lctrostatic pottial V ad ffctivly caus all bads to appar discotiuous at th juctio Udr th two assumptios, though, Fig immdiatly shows that th barrir for a lctro i th smicoductor coductio bad to tul ito th mtal is =Φ Φ M S M 96

98 Th spatial width of th barrir is of importac as wll To fid it, th spatial dpdc of th lctrostatic pottial V must b calculatd As i chaptr 5, th basic rlatio to b usd is th Poisso quatio d V dx ρ =, εε whr ρ is th spatially varyig charg dsity ad ε is th rlativ dilctric costat of th matrial As i th aalysis of th p juctio i chaptr 5, w will assum total dpltio of lctros i a layr of width W isid th -typ smicoductor As w hav s, this amout to sayig that th pottial is a costat V for x> W ad follows a quadratic bhaviour N V x V x W DI ( ) = ( ), εε S for < x< W Hr, ε S is th rlativ dilctric costat of th smicoductor ad N DI is th dsity of door impuritis O th mtallic sid of th juctio, a stimat of th charg dsity is foud usig th Thomas-Frmi modl Hr, th mtal is assumd to b a simpl fr-lctro mtal with a dsity of stats giv by 3/ 3 Eq(44), i D( E) = η E with η= m / πħ It follows that th lctro coctratio is = ηe udr ormal circumstacs Howvr, if a 3/ 3 F lctrostatic pottial V is applid whil th Frmi lvl is kpt fixd, all rgis ar lowrd by V, which amouts to raisig th Frmi lvl by a amout + V just as i Eq(3) Now, i a utral bulk matrial th dsity of positiv charg must b to compsat th lctros Hc, th t charg i prsc of th pottial must b { } ρ= η( E + V) ηe ηve = V / E 3/ 3/ / 3 3 F 3 F F F, () whr w hav Taylor-xpadd to first ordr udr th assumptio that Wh usd i th Poisso quatio, it follows that V E F = 3 = 3, (3) d V V k, TFV ktf dx εmεef εmεef whr ε is th rlativ dilctric costat of th mtal ad M k TF is th Thomas-Frmi wav umbr W choos as a boudary coditio V ( ) =, i a vaishig pottial i th bulk of th mtal ad so th solutio to this quatio is a simpl xpotial 97

99 V( x) = V xp( k x), TF whr V is a (yt udtrmid) costat To dtrmi V ad th rmaiig costat W w ow compar th chargs isid th mtal to that i th smicoductor Itgratig Eq() for th mtal w fid a total charg pr ara of 3 ktf x M= ρ = / F = εmε TF Q dx E V dx k V As i chaptr 9, w fid for th smicoductor QS= NDIW Puttig QM+ QS= provids o quatio for th ukows A scod is obtaid from th rquirmt of cotiuity of th pottial at x=, which is quivalt to th coditio V = V N W /εε Tak togthr, w fid DI S W ε εε W + W V = S S εmktf NDI ε S εs εε S = + V εmk + TF εmk TF N DI / (th mius-solutio is discardd as w r obviously lookig for a positiv quatity) To valuat this rsult w d to compar two lgth scals: k / ad ( V / N ) Usig th dfiitio i Eq(3) w s that k E TF / TF ( F / ) DI Sic for ay ral / mtal-smicoductor juctio NDI, w must hav ( V / NDI ) k TF ad so W εε S V N DI / By th sam tok, V = V N W /εε follows by isrtio Thus, as DI S promisd abov, bad bdig isid th mtal is gligibl A dirct cosquc is that V = ad so W εε N S DI / (4) Havig stablishd hight ad width of th tullig barrir w may ow comput th tullig currt usig Eq() If th barrir is approximatd by a triagular o of hight ad width W ad th WKB-approximatio is applid, w ca writ 98

100 4W m xp ( E) 3/ E T( E ) < 3 ħ E> Hc, th total currt ca b split ito two parts: A gui tullig cotributio from stats with E< ad a so-calld thrmioic currt producd by stats with rgy E> Th lattr is asily valuatd ad w fid AmkT V / / / A kt E E kt F kt Ithrmioic= 3 ( ) de π ħ Am( kt) VA / kt ( EF )/ kt = 3 ( ) πħ Am ( kt) I I = πħ VA / kt ( EF )/ kt thrmioic( ), thrmioic 3 (5) Th costat R m kt πħ is kow as Richardso s costat ad for fixd 3 ( ) /( ) tmpratur it oly dpds o th ffctiv mass of th smicoductor matrial Th tullig currt is foud from a similar itgral AmkT V / / / 4 ( ) xp ( ) 3/ A kt E E kt F kt W m Itul= E de 3 π ħ 3 ħ To valuat this xprssio, w first itroduc a w variabl z= ( E )/ kt ad a w dimsiolss costat magitud of β= ħ Cosidrig th typical 3/ 4 m W( kt) /3 W (- m) it is clar that ormally β I trms of ths quatitis, th tullig currt ca b writt as / kt Am 3 / ( kt) A / ( EF )/ kt 3 ( V kt ) z β z tul= πħ I dz Th itgral caot b valuatd aalytically but thaks to th larg valu of β, th uppr limit may b xtdd to ifiity with littl loss of accuracy Th rsultig itgral ca b xprssd xactly i trms of hyprgomtric fuctios To simplify th fial rsult, howvr, a Taylor xpasio i β ca b applid whvr β, i for wid barrirs I this mar, 3 / z βz Γ( 3) 5 Γ( 3) 7 dz + / /3 3β 3β 7β 9β 54β 99

101 Th fial tullig currt th bcoms I tul VA kt ( ) I = I, tul / tul Am ( )/ ( 3) ( kt) E 5 ( 3) 7 F kt Γ Γ = 3 + / /3 + πħ 3β 3β 7β 9β 54β (6) It follows that th ratio btw tullig ad thrmioic currt is simply I Γ( 3) 5 Γ( 3) 7 tul = + / /3 I 3β 3β 7β 9β 54β thrmioic W tak th xampl of GaAs ( m= 66m ) to valuat th currt ratio as a fuctio of barrir width For room tmpratur ad = V, th rsult is show i Fig 3 As xpctd, th tullig currt domiats for a sufficitly arrow barrir ( W < 5 m ) Fially, for compltss, w ot that a drift-diffusio VA / kt currt xists as wll It also follows th form Id d Id d( ) prov this = but w will ot Figur 3 Ratio btw tulig ad thrmioic currts i a GaAs Schottky diod at room tmpratur Accordig to our rsults, th ratio btw tullig ad thrmioic currts is idpdt of th applid voltag I rality, th barrir itslf is iflucd by th voltag ad so th currt ratio may dpd o voltag as wll

102 Exrcis: Nao-scal Schottky diod This xrcis is built o Rf [] W cosidr th gomtry i Fig 4 cosistig of a circular mtallic pad o top of a -typ smicoductor Th radius a of th pad is i th - m rag Th problm of th xrcis is to calculat th lctrostatic pottial for this gomtry i th cas of vaishig bias voltag Th boudary coditios for th pottial ar () V= o th cotact ara btw mtal ad smicoductor, i for { z= r a} ad () V= V outsid th bordr Γ i th figur Figur 4 A ao-scal Schottky diod formd as a circular mtallic pad o top of a -typ smicoductor Th hatchd ara i th right pal illustrats th xtt of th dpltd rgio with boudary curv Γ Th Poisso quatio for this problm is N DI V= εε S abov curv Γ blow curv Γ To simplify th mathmatics, w switch to so-calld oblat sphroidal coordiats ( s, t ) rlatd to ( r, z ) via th trasformatios r= a ( + s )( t ), z= ast s<, t Th Laplacia i ths coordiats rads as { ( + s ) + ( t ) } = a ( s + t ) s s t t a) Cosidr first th cas of vaishig dopig for which V= throughout Show that th solutio is of th form V= f ( s) g( t) ad that f ( s) = A+ Bta ( s) ad g( t) = + C tah ( t) Hit: d ta ( s)/ ds= ( + s ) ad d tah ( t)/ dt= ( t )

103 b) Th cotact ara btw mtal ad smicoductor { z= r a} corrspods to { s= t } Also, z is quivalt to s Us th boudary coditios to show that A= C= ad B= V Hc, th full solutio for th u-dopd cas is V= V ta ( s) π Th fact that th u-dopd solutio is idpdt of t idicats that th gral solutio will dpd oly wakly o t Hc, i th Poisso quatio w will igor V / t ad so th full problm for th dpltio rgio (rgio abov curv Γ ) is approximatly π V NDIa ( + s ) = ( s + t ) v, v= s s εε S c) Whil th homogous solutio is still of th form that th particular o is { } V A t B t s = ( ) + ( )ta ( ), show V= ( 3 t )l( + s ) s v /6 Th full solutio is th sum of ths Us th boudary coditio for th cotact ara to show that A( t ) = d) O th boudary curv Γ th boudary coditio is V= V I additio, V / s= o th curv Γ W dot by s ( ) t th valus of s foud by tracig th curv Γ as t varis btw - ad Show that th coditio V / s= o Γ implis B( t) = s ( s + 3 t ) v /3 Th coditio V= V o Γ ow mas that 6 v V = s ( s + 3 t )ta ( s ) + ( 3 t )l( + s ) s ) Takig V = 4 V ad v= 5 V/ µ m a, which ar ralistic valus, solv this quatio umrically for t Us th coordiat trasformatio to calculat corrspodig ( r, z ) poits I this mar, th curv Γ is computd Tak a = µ m, 3µ m ad µ m ad plot th Γ curvs that, if succssful, should look lik th os i Fig 5

104 Figur 5 Calculatd boudary curvs Γ for thr diffrt radii of th mtallic pad Rfrcs [] S Datta Elctroic Trasport i Msoscopic Systms (Cambridg Uiv Prss, Cambridg, 997) [] C Doolato, J Appl Phys 95, 84 (4) 3

105 Smiclassical Trasport I this chaptr, w try to stablish th lik btw quatum ad classical approachs to trasport As w will dmostrat, th two agr for mtallic structurs providd momtum ad vlocity ar tratd smiclassically, i dtrmid from th slop of th tru rgy bads I our xpositio, w will focus o a sigl-bad mtal Howvr, for multi-bad structurs, th fial rsult should simply b summd ovr bads W start by itroducig th lctro distributio fuctio g that govrs th umbr of lctros at a giv positio, momtum ad tim: dn( r, k, t) = g( r, k, t) 3 3 d rd k 4π 3 Hr, th ormalizatio is clarly such that th total umbr of lctro is N tot = g( r, k, t) 3 3 d rd k I thrmal quilibrium at a giv positio charactrizd by a local Frmi lvl E ( r ) ad tmpratur T( r ), th distributio fuctio is just th Frmi-Dirac distributio 4π g( r, k, t) f ( r, k) = E( k) EF ( r ) xp + kt( r ) W rstrict ourslvs to th rlaxatio tim approximatio, i which th rat of chag of th distributio fuctio is dtrmid by a charactristic rlaxatio tim towards quilibrium 3 d g( r, k, t) f ( r, k) g ( r, k, t ) = () dt τ If w momtarily igor all but th xplicit tim dpdc, w hav d g( t) f g ( t ) =, dt τ F 4

106 with th simpl solutio ( ) [ ( ) ] g t f g t f t / τ = + as ca asily b vrifid Hc, th distributio rlaxs xpotially toward thrmal quilibrium Mor grally, th rat of chag is d g( r+ dr, k+ dk, t+ dt) g( r, k, t) g ( r, k, t ) = dt dt g g dr+ g dk+ dt k t () dt dk g = g v+ g + k dt t Hr, w Taylor xpadd i th ifiitsimals i th scod li ad applid v= dr / dt i th third This diffrtial quatio for th lctro distributio is kow as Boltzma s trasport quatio W ow rtur to Eq() ad rwrit as d g( r, k, t) = f ( r, k) τ g( r, k, t) (3) dt Udr th assumptio that w ar clos to quilibrium, w ca writ dow a ordrby-ordr xpasio for g I th ordr, w clarly hav g ( r, k, t) = f ( r, k ) To costruct th st ordr rsult, w ot that by isrtig th th ordr trm i Eq() it follows that d dk g( r, k, t) f v+ f, k dt dt (4) as f / t= W th fid dk g( r, k, t) = f ( r, k) τ f v τ f k dt Th last trm ca b valuatd by appalig to smiclassical argumts First, p = ħ k is th momtum, which i a lctric fild E is govrd by Nwto s law dp dt = E Hc, w fid 5

107 τ g( r, k, t) = f ( r, k) τ f v+ f E (5) k ħ W will apply this xprssio as a approximatio for th gral distributio fuctio Sic f is kow xplicitly, all drivativs ar asily calculatd Focusig o th x-dirctio w hav Similarly, for th k-drivativ f E E F f = kt x x kt E EF f E E F T f = x E T x E Smiclassically, togthr, w th fid f E f = k k E x x E= p /m with p = ħ k so that E/ k = ħp / m= ħ v Puttig it all x x x f E E F g( r, k, t) = f+ τ + E F+ T E v E T Th first two drivig trms may b groupd ito a ffctiv lctric fild E = E + E / ad so ff F Trasport Cofficits f E E F g( r, k, t) = f+ τ ff+ T E v E T (6) W ow wat to apply th distributio fuctio to valuat crtai trasport cofficits To simplify mattrs, w will assum trasport i th x-dirctio oly so that Eff= E ff x ad v = vx ad so x f E E F T g( r, k, t) = f+ τ E ff+ vx E T (7) x Grally, both charg ad rgy flows Thus, w ca itroduc th (familiar) lctric currt as wll as a w rgy (or hat) currt dsity giv by 6

108 3 d k J= vxg( r, k, t) 4 3 π 3 d k JQ= [ E EF] vxg( r, k, t) 3 4π Th raso that E F is subtractd i th last xprssio is that th hat currt is rally th flow of fr rgy [] Approximatig g by Eq(7) w first fid a cotributio from g= f This cotributio clarly vaishs as it rprsts th currt flow i th uprturbd stat For th rmaiig cotributios w ca grally writ Th cofficits that J = L E L Q ff J = L L T x T x Eff L ij ar th co-calld trasport cofficits ad by isrtio it follows f 3 f 3 L= τv, [ ], 3 x d k L τv 3 x E EF d k 4π = E 4π T E f 3 L= TL, L= τv[ ] 3 x E EF d k 4π T E (8) Throughout, th vlocity should b valuatd from vx= ħ E/ kx Amog ths cofficits, L is rcogizd as th usual itrabad coductivity Also, th ratio Q= L / L is oft calld th thrmopowr or Sbck cofficit ad Π= L / L is th Pltir cofficit Not that Π= TQ W will ow valuat th trasport cofficits for a fr lctro mtal Such a matrial is charactrizd by a uiform lctro dsity giv by 3 3 k = f ( E) d k F 3 4π =, (9) 3π whr k = me / ħ is th Frmi wav umbr Th dsity of stats is simply F F / 3/ ( ) 3 /( ) D E = E E F Fially, for this isotropic matrial w mak th substitutio v v ( v + v + v )/3 As E= m( v + v + v )/ it follows that v = E/3m x x y z x y z Thus, covrtig th itgratios from k to E mas that 7

109 E f f 3/ = τ = τ 3/ 3m E me F E L D( E) de E de If w ow () assum τ idpdt of rgy, ad () tak th low tmpratur limit f / E δ( E E F ) w s that L = τ / m i agrmt with th quatum rsult For L w fid by aalogy f 3/ L= τ 3/ [ E EF] E de me T E F This itgral is trickir bcaus th rgy factor E EF cacls th aïv low tmpratur cotributio Howvr, usig th so-calld Sommrfld xpasio it ca b show that [] For τ idpdt of rgy this yilds E F f h E π k T h E h( E) de de E E 6 E ( ) ( ) E= EF L τ π k T π τk T / 3 3/ EF = mef T 6 mef, which mas Q π k T E F approximatio udr ralistic coditios /( ) As th figur shows, this is a xcllt Figur Compariso of th umrically valuatd thrmopowr to th lowtmpratur approximatio for a fr lctro gas 8

110 Exrcis: Trasport i graph Graph is a two-dimsioal matrial ad so it is mor appropriat to talk of sht trasport cofficits giv by (with th otatio f ( E) = f / E ) L= τv [ ] x f ( E) d k, L τv ( ), x E EF f E d k π = π T L= TL, L= τv[ ] ( ) x E EF f E d k π T () Morovr, th rgy spctrum is spcially simpl i th so-calld Dirac approximatio, whr th actual bad structur is rplacd by two Dirac cos with 6 th disprsio E=± εε, =ħ vfk, whr vf m/s is th Frmi spd (s chaptr 7 for dtails) To accout for th two cos, all cofficits should simply b multiplid by two Fially, th ± solutios for th rgy should b summd ovr W tak th graph sht to li i th (x,y) pla ad for such a isotropic matrial w may mak th rplacmt v v ( v + v )/ a) Show that v = v F / x x y Utilizig this rsult ad th fact that v d k= v πkdk= πεε d / ħ F F mas that L= τ { f ( ε) + f ( ε) } εdε πħ To xprss th rsults w itroduc ϕf EF / kt Also, to valuat th rquird itgrals you will d whr Li is polylogarithm b) Assum τ idpdt of rgy to show that Quit similarly, w fid [ ϕ ] f ± ( εεε ) d = kt l + xp( ± ) ( ± ε EF ) f ( ± εεε ) d = ( kt) { ϕf l[ + xp( ± ϕf )] ± Li[ xp( ± ϕf )]}, L τkt = l cosh( F /) π ħ [ ϕ ] F 9

111 τ L= {( ε EF ) f ( ε) ( ε+ EF ) f ( ε) } εε d πħ T c) Show that τk L = E + kt kt πħ { l[ cosh( ϕ /)] Li [ xp( ϕ )] Li [ xp( ϕ )]} F F F F For positiv Frmi rgy ad modratly low tmpraturs ϕf Thus, w may xpad usig ϕf ϕf π l[ cosh( ϕf /)], Li[ xp( ϕf )], Li[ xp( ϕ F )] 6 d) Us th xpasio to dmostrat that at low tmpratur τe πτ k T π k T L L Q F,, πħ 3ħ 3EF Ths rsults could also hav b obtaid dirctly usig th Sommrfld xpasio Rfrcs [] NW Ashcroft ad ND Mrmi Solid Stat Physics (Saudrs Collg, Philadlphia, 976)

112 3 Fild Effct Trasistors Fild ffct trasistors ar arguably th most importat smicoductor dvics vr fabricatd Thy form th basis of logical circuits ad largly pavd th way for th computr rvolutio This chaptr is aimd at providig a physical udrstadig of fild ffct trasistors ad, i particular, th dpdc of currt o gat ad drai voltags Both thr-dimsioal (MOSFETs) ad two-dimsioal (MODFETs) trasistors will b studid Th MOSFET gomtry is illustratd i Fig 3 Figur 3 Schmatic of th MOSFET Lft: Th ctral part is th chal coctig sourc ad drai Th chal is sparatd from th mtallic gat by a thi oxid Tak from [] Right: Simplifid modl Th sourc lctrod ad substrat ar groudd but gat ad drai ar biasd W d to calculat th lctrostatic pottial isid th smicoductor To this d, w cosidr th pottial diagram i Fig 3 Figur 3 Variatio of th lctrostatic pottial across mtal, oxid ad smicoductor Th spatial variatio of th lctrostatic pottial ψ isid th smicoductor is govrd by th Poisso quatio dψ ρ =, dx εε

113 whr ρ is th spatially varyig charg dsity ad ε is th rlativ dilctric costat of th matrial Th boudary coditios ar ψ() = ψs (th surfac pottial) ad ψ( ) = W cosidr a p-typ smicoductor so that th charg dsity is giv as ρ= { p+ N A }, (3) whr N A is th dsity of accptor impuritis Th hol dsity follows th pottial accordig to p= p xp( ψ / kt ) I th absc of a sourc-drai voltag, th lctro dsity would simply vary as = xp( ψ / kt ) By rquirig charg utrality at ifiity i this situatio, it follows from th boudary coditio ψ ( ) = that p+ N A= Howvr, with th sourc groudd ad th drai at a pottial V D, a drai currt flows ad w o logr hav thrmal quilibrium As th am says, th drai cotact drais lctros from th smicoductor As a cosquc, th lctro dsity at th sourc S is still giv by S= xp( ψ / kt ) At th drai, howvr, th ffctiv Frmi lvl (or quasi-frmi lvl) is lowrd by V D ad so th lctro dsity is rducd to = xp( ( ψ V )/ kt ) Btw sourc ad drai, th dsity varis accordig to D D = xp( ( ψ V )/ kt ), whr th coutr pottial V varis btw at th V D sourc ad at th drai (hols ar ot affctd du to th + rgios that block hol trasport cf Fig 3) Hc, writig N = p th charg dsity is ψ ψ ρ ( ) = xp V p xp (3) kt kt At this poit it is highly covit to itroduc ormalizd pottials ϕ= ψ / kt, ϕ = ψ / kt, v= V / kt ad v = V / kt Hc, th Poisso quatio yilds S S G G dϕ = dx εε kt This diffrtial quatio is ot solvabl but a first itgral ca b obtaid by th stadard procdur Thus, / (33) Hr ad throughout, it is udrstood that th positiv sig of th squar root i g is to b usd if ψ S> ad th gativ if ψ S< Th boudary coditio rlatig th pottials i th oxid ad smicoductor is A v { ϕ p ϕ } dϕ = g( ϕ, v), g( ϕ, v) ϕ + p / + ϕ dx εε kt ϕ v v ϕ { } /

114 ε OX dϕ dx dϕ = ε dx x= x= + Th slop of th ormalizd pottial isid th oxid is simply ( ϕ v )/ d Hc, ε / ( ) OXε dϕ dx = C, whr is th capacitac pr ara of OXϕS vg C x= OX= εoxε / d th oxid layr Th right-had sid is giv by Eq(33) ad i combiatio w gt v = ϕ + γg( ϕ, v), (34) G S S whr γ= ( εε ) / / kt / C OX For giv paramtrs, Eq(34) must b solvd umrically i ordr to dtrmi th surfac pottial W ow wat to dtrmi th lctro fractio of th total charg At ay giv y poit, th lctro charg pr ara is giv by Q S G ϕ v Q = dx= dx By substitutio, this rsult ca b rwritt as / ϕs ϕ v ϕ v dx εεkt ϕ Q= d = dϕ dϕ g( ϕ, v) ϕs (35) I Fig 33, th solutio of Eq(34) is plottd togthr with th lctro charg Eq(35) assumig room tmpratur (kt = 6 V) ad usig silico paramtrs: ε = 9, p = m, = m ad γ= Figur 33 Surfac pottial ad lctro charg vs coutr pottial for a Si MOSFET 3

115 3 MOSFET I/V Charactristic At ay giv poit, th y compot of th currt dsity a diffusio part dψ d Jy= µ + µ kt dy dy dv = µ, dy J y cosists of a drift ad usig = xp( ( ψ V )/ kt ) Hc, for a dvic of thickss Z i th z-dirctio w fid that th drai currt follows as dv µ ZkT dv ID= Z Jydx= µ ZQ = Q dy dy Our problm hr is that w do t kow how v dpds o y Followig Pao ad Sah [], o possibl way aroud this obstacl is to itgrat ovr th lgth of th chal L L vd µ ZkT dv µ ZkT D= = I Q dy Q dv L dy L / vd ϕs ϕ v µ ZkT εε kt = dϕdv L g( ϕ, v) Not, that this xprssio is xact bcaus is idpdt of y ad so avragig ovr y dos t chag th rsult Thus, w ca fially writ vd ϕs ϕ v 3 / µ Z εε ( kt) D= MOSFET ϕ, MOSFET g( ϕ, v) L I I d dv I (36) Hc, to comput th currt i Eq(36) w first d to solv Eq(34) for a giv v to obtai ϕ as a fuctio of v This mthod provids a vry accurat xprssio G S for th currt but du to th two itgrals that hav to b do umrically, this mthod is also rathr cumbrsom A altrativ approximat xprssio ca b foud by obsrvig that th total charg Q is th sum of th lctro charg ad a dpltio charg Th lattr is approximatly Q εε p ψ Thus, Q ( ) / d I D d S Q 4

116 VD V ( ) / D µ Z ID QdV εεp ψs dv + L Now, Q is giv as Q= COX( ψs VG ) W still do t kow th dpdc of ψ S o V, howvr Cosqutly, as a simplificatio w ll assum that th surfac pottial varis i dirct proportio to th coutr-pottial, i dψ / dv This implis that w chag itgratio variabl from V to, providd itgratio limits ar also chagd Hc, as V varis btw ad, th surfac pottial varis btw ad It follows that th currt itgral bcoms ψ S ψ SL ψ ψ µ SL SL Z / I ψ ψ ( εε ) ψ ψ D C OX + ( S VG ) d S p s d S L ψs ψs µ / ( εε ) ZC p OX = ( ψ ψ ) V ( ψ ψ ) + ψ ψ L 3C OX V D ψ S S ( ) 3/ 3/ SL S G SL S SL S I trms of I MOSFET dfid i Eq(36), th currt fially bcoms I / I p v 3/ 3/ ( ϕ ϕ ) ( ) ϕ ϕ γ ( ϕ ϕ ) MOSFET D G SL S SL S SL S γ 3 (37) It is clar that to valuat this xprssio oly th surfac pottials at th ds of th chal ar dd A compariso of full ad approx xprssios is show i Fig 34 Figur 34 Drai currt vrsus drai voltag for diffrt valus of th gat pottial Solid ad dashd curvs rprst full ad approximat calculatios, rspctivly 6 8 Paramtrs ar γ= ad p / = 5

117 It is clar from Fig 34 that a thrshold coditio for th gat voltag xists I th figur, th thrshold appartly is about 6 V Th rquirmt for ivrsio is that ϕ ϕf p /, whr ϕ = l( p / ) = l( N / ) is th bulk Frmi lvl F A i Followig Eq(34), th ormalizd gat voltag dd to achiv ivrsio is F v v ϕ + γg( ϕ,) ϕ + ϕ γ ϕ G GT F F F F I fact, a slightly diffrt thrshold of is closr to th umrical valu For th paramtrs of Fig 33 w fid v 5 or 64 V Abov thrshold, th saturatio valu of th drai currt 8 matchd by th parabolic approximatio I ( ) D, Sat IMOSFET vg vgt γ 3 Modulatio Dopd Fild Effct Trasistors is quit closly Modulatio dopd fild ffct trasistors (MODFETs) also go by th ams HEMT (High Elctro Mobility Trasistor) ad HFET (Htrostructur FET) Svral diffrcs btw MOSFETs ad MODFETs ar worth otig: Th oxid is rplacd by a -dopd AlxGa-xAs layr Th smicoductor is udopd GaAs 3 Bad bdig is sufficitly strog that lctros doatd by th AlxGa-xAs layr accumulat i a vry thi layr o th GaAs sid of th juctio Th gomtry of th structur is illustratd i Fig 35 v = ϕ + ϕf γ ϕ GT F F GT I D, Sat V GT Figur 35 Schmatic structur of a MODFET Th right-had diagram illustrats th profil of th lctrostatic pottial Th door dsity i th AlxGa-xAs layr is dotd ad w will assum that all doors ar ioizd ad th lctros trasfrrd to th GaAs sid For simplicity, w will assum that all lctros occupy a sigl stat with quatizatio rgy Also, w will igor th variatio of th pottial across th quatum wll ad N D E 6

118 simply tak it to b ara is ψ S, th surfac pottial I this cas, th lctro dsity pr mkt ψs+ EF E = l xp + πħ kt mkt ϕ E E S+ ϕ F = { + } πħ kt F l, ϕf A cosquc of th AlxGa-xAs dopig is that th layr bcoms a imprfct capacitor I a ormal capacitor, th pottial drops liarly across th layr but i th prst cas th Poisso quatio for th layr (usig th ormalizd pottial) bcoms dϕ N = D dx εε kt Hr, w hav igord th small diffrc i dilctric costat btw th two smicoductors ad dotd th commo valu by ε Th gral solutio is obviously a parabola ϕ ( x) = a+ bx+ cx With th boudary coditios ϕ ( d) = vg ad ϕ() = ϕs it follows that x+ d ND ϕ( x) = vg+ ( ϕs vg ) x( x+ d) d εε kt As th dilctric costat is assumd idpdt of positio, it follows that th drivativ of th pottial is cotiuous across th boudary Hc, just isid th GaAs sid, th pottial drivativ ϕ S ϕ ( + ) is giv by ϕ v ϕ N d ϕ = S S G D D, ϕd d d εεkt (38) If a drai voltag is applid, charg utrality is brok I this cas, th coutr voltag V varis btw ad V D at sourc ad drai, rspctivly Accordigly, th lctro charg is ow mkt S F = l + πħ, (39) This charg is th total charg pr ara i th GaAs layr W may cosqutly apply Gauss law to a box closig th GaAs layr ad writ ϕ = Q / ktεε Combiig th rsults abov w fially fid Q v { ϕ + ϕ } S 7

119 ϕ v ϕ m = S F l + S G D d d πħεε ϕ v+ ϕ { } Itroducig th auxiliary pottial ϕ md π ħ εε th balac quatio ca b writt This is th quatio w d to solv to comput (3) as a fuctio of v First, howvr, th locatio of th Frmi lvl must b dtrmid This is a complicatd problm rlatd to piig by th AlxGa-xAs doors For simplicity, thrfor, th Frmi rgy ca b dtrmid by rquirig charg balac whvr th gat ad drai biass ar abst I this cas, lctro chargs must b balacd by door chargs i a AlxGa-xAs layr of thickss d, ad so w hav dnd=, which mas that whr ϕ S, (3) is th surfac pottial i th absc of biass I trms of th auxiliary pottials dfid abov, this mas that O th othr S F had, Eq(3) with v v shows that S D l ϕ + ϕ ϕ = ϕ + ϕ Tak togthr, it follows that ϕ = ϕ S G D ad so th balac quatio fially bcoms (3) A xampl of th surfac pottial profil is show i Fig 36 Hr, th followig paramtrs rprstativ of GaAs MODFETs hav b applid: m= 67 m 3-3 N D= m ε = 9 Al Ga As layr thickss: d= 3 m x -x { ϕ v + ϕ } S F ϕ v ϕ = ϕ l + G S dn mkt S F = l + πħ { ϕ } + ϕ Ths valus corrspod to ϕ = 43 ad ϕ = 87 D S F D / l{ ϕ + ϕ ϕ = + ϕ } = = { } D S G D ϕ S ϕs { v + ϕd( ϕd / ϕ )} ϕ v ϕ = ϕ l + D 8

120 Figur 36 Surfac pottial ad lctro charg vs coutr pottial for a GaAs MODFET 33 MODFET I/V Charactristic Similarly to th MOSFET, th drai currt is giv as avoid computig dv / dy w agai itgrat ovr y ad fid I = µ ZQ kt / dv / dy To D L ZkT dv ZkT ID= µ Q dy Qdv L = µ dy L v D Now, usig ϕ = S Q / ktεε as wll as Eq(38) w ca writ v D εεµ Z( kt) D= MODFET { G+ ϕd ϕs}, MODFET Ld I I v dv I (33) I Fig 37, th I/V charactristic for th GaAs MODFET is illustratd 9

121 Figur 37 Drai currt vrsus drai voltag for diffrt valus of th gat pottial i a GaAs MODFET at room tmpratur Agai, a prooucd saturatio of th currt is foud as th drai voltag icrass As for th MOSFET, a aalytical stimat of th saturatio currt ca b giv First, w ot from Fig 36 that to a good approximatio th surfac pottial profil is picwis liar Igorig th factor isid th curly brackts of Eq(3) w thrfor fid ϕ S ϕ ( v ϕ ) + v F G, v< vg+ + ϕ v, v v + ϕ F G G F ϕ If this approximatio is usd ad vd vg+ ϕf is assumd, w fid for th saturatio currt ID, Sat IMODFET ( vg+ ϕf ) ϕ /( + ϕ ) as ca asily b vrifid Hc, just as for th MOSFET, th saturatio currt varis quadratically with gat voltag Comparig Figs 34 ad 37 it would appar that th MOSFET currt is much largr tha th MODFET currt Howvr, it is asily show that I I MODFET MOSFET / N D = ϕ D Usig th sam paramtrs as abov, this ratio quals ar, i fact, roughly qual 5 ad so th two currts

122 34 Trasit Tim For both MOSFETs ad MODFETs, th trasit tim τ btw sourc ad drai ca b computd as th itgral of a lctro travlig L V D τ= dy dy dv u = u dv Hr, u is th lctro vlocity u= µ dv / dy Usig I = µ ZQ dv / dy, it th follows that D ID V D µ Z τ= QdV W wish to rformulat i trms of ormalizd pottials ad dimsiolss quatitis It turs out that for both MOSFETs ad MODFETs w ca writ whr v D vd qdv qdv L τ= µ kt (34) ϕs ϕ v dϕ, MOSFET q g v = ( ϕ, ) vg+ ϕd ϕs, MODFET To illustrat th rsults w agai cosidr a Si MOSFET ( µ = 45 m /Vs ) ad a GaAs MODFET ( µ = 85 m /Vs ) W tak i both cass th dvic lgth to b L = µ m ad othrwis us paramtrs as abov Th rsults ar show i Figs 38 ad 39 blow It should b otd that th trasit tim provids th ultimat physical limit for th cut-off frqucy of th dvics giv by f cut off = πτ Hc, for th two dvics w fid approximat cut-offs of 6 GHz ad 6 GHz, rspctivly I a ral dvic, th valu will b somwhat lowr du to capacitiv ffcts

123 Figur 38 Trasit tim vs drai voltag for diffrt valus of th gat pottial i a Si MOSFET Figur 39 Trasit tim vs drai voltag for diffrt valus of th gat pottial i a GaAs MODFET Exrcis: Variatio of th coutr pottial A usolvd problm is th y-dpdc of th coutr pottial Hr, w will show how this may b computd a) Show that for both MOSFETs ad MODFETs

124 dv ID= IMOSFET / MODFET Lq, dy whr ϕs ϕ v dϕ, MOSFET q g v = ( ϕ, ) vg+ ϕd ϕs, MODFET b) Show by rarragmt ad itgratio that y = L v vd q dv q dv c) Itgrat umrically to fid V( y) If succssful, you should fid two highly similar curvs as illustratd i Fig 3 Rfrcs Figur 3 Coutr voltag vs y-positio for both MOSFET ad MODFET [] SM Sz Physics of Smicoductor Dvics (Wily, Nw York 98) [] HC Pao ad CT Sah, Solid Stat Elctro 9, 97 (966) 3

125 4 Naowir MOSFETs Th gomtry of a cylidrical aowir MOSFET is show i Fig 4 Sourc ad drai cotact ar attachd at th ds ad alog th aowir a oxid isolats th smicoductor cor from th surroudig mtallic gat Figur 4 Schmatic of th aowir MOSFET Th smicoductor cor is surroudd by a thi oxid ad a cylidrical mtallic gat Our aalysis will b basd o crtai assumptios about th structur ad th mod of opratio: () Th smicoductor cor is udopd ad () th gat is positivly biasd so that (practically) oly lctros d to b cosidrd ad (3) th odgrat limit is applicabl I this cas, th coupld Poisso-Boltzma quatios rad d ψ d ψ ( ) xp ψ V + = dr r dr εε kt Th boudary coditios for th pottial ( r) ar ψ () = ad ψ() = ψ, whr th ctr valu ψ will b dtrmid latr As usual w itroduc ormalizd pottials ϕ= ψ / kt ad v= V / kt so that w obtai th rducd Poisso- Boltzma quatio ψ dϕ dϕ ϕ v ϕ v + = N, N= dr r dr εεkt εεkt Boudary coditios: ϕ () =, ϕ() = ϕ (4) It ca b show by ispctio that a aalytical solutio to this quatio is giv by ϕ v ( 8 ) ϕ( r) = ϕ l N r (4) W also d th pottial isid th oxid W dot radius of th smicoductor by R ad th oxid thickss by d Thus, w rquir ϕ( R) = ϕs ad ϕ ( R+ d) = vg 4

126 Th pottial isid th (utral) oxid is giv by th homogous solutio to th Poisso quatio ϕ ( r) = al r+ b Subtractig th boudary coditios for th two facs of th oxid th mas that a= ( vg ϕs )/l( + d / R) I additio, th slop of th oxid pottial is dϕ / dr= a/ r ad w ca thrfor rlat slops o both sids of th oxid-smicoductor itrfac via ε OX dϕ dr dϕ = ε dr r= R+ r= R, whr ε ad ε ar dilctric costats of th oxid ad smicoductor, OX rspctivly This, i tur, mas that εox ( vg ϕs ) dϕ = ε Rl( + d / R) dr Th pottial drivativ isid th smicoductor is foud from Eq(4) ad quals r= R dϕ 4N ϕ v = v dr 8 ϕ N r r (43) O th othr had, it also follows from Eq(4) that ϕ v ( ) ϕs= ϕ l N R 8 Itroducig th oxid capacitac C / l( / ) OX= εoxε R + d R ad puttig vrythig togthr w fially obtai a quatio for th ctr pottial εε v = ϕ N R + G 4N ϕ v ϕ v l( 8 ) ϕ v COX 8 N R R (44) This is th quatio to solv to fid th ctr pottial profil alog th aowir as a fuctio of th coutr voltag v that varis btw ad W d, i additio, th variatio of th lctro charg alog th wir Rathr tha itgratig th lctro dsity ovr th cross sctio, w us Gauss thorm for a surfac boudig a small slab of th cylidrical cor: dϕ = Q Q dr εε kt εε kt r= R Q v D 5

127 Hr, Q is th charg pr ara ad Q Q Alog with Eq(43) this mas that ϕ v N R ϕ v 8Q ϕ v 8 N R Q+ Q Q= Q N R =, whr Q = 4 εε kt /( R) It follows that ϕ 8Q = v+ l (45) NR ( Q+ Q ) This mas that th pottial balac Eq(44) ca b rformulatd as a charg balac v G 8 Q( Q+ Q ) Q = v+ l + NR Q ktc OX (46) From a umrical stad poit, it is advatagous to solv Eq(46) ad subsqutly us Eq(45) to comput th pottial profil For latr purposs, w ot that diffrtiatig this rlatio lads to dv Q+ Q dq Q Q Q ktc = ( + ) OX (47) A xampl of a actual xampl is show i Fig 4 Hr, th followig Si/SiO 6-3 aowir paramtrs hav b usd: ε =9, = m, R = 5 m, d = 5 m ad 3 ε OX= 39 ladig to C OX = 757 F/m Figur 4 Th charg ad pottial profil of a aowir MOSFET 6

128 W ow procd to calculat th drai currt Similarly to th MOSFET ad MODFET cass w writ V D D R R kt ID= π µ QdV π µ Qdv L = L It turs out that itgratios ar asir if w us charg rathr that coutr pottial as itgratio variabl [] Hc, w utiliz Eq(47) ad writ v D π Rµ kt dv ID= Q dq L dq Q QS π µ = L + Q Q Q + ktc QS R kt Q+ Q Q dq D OX Hr, Q ad Q ar (positiv) chargs at sourc ad drai, rspctivly Aftr S simpl itgratios w fially fid D ( S QD) π Rµ kt QS+ Q Q ID= ( QS QD ) Q l + L QD Q + ktc (48) This rsult is illustratd i Fig 43 blow usig µ = 45 m /Vs ad L = µ m OX Figur 43Drai I/V charactristic of a micro aowir MOSFET 7

129 W d this chaptr by calculatig th trasit tim τ I a mar compltly aalogous to th MOSFET ad MODFET cass it turs out that τ vd vd L = Qdv Qdv µ kt Th itgral i th umrator is asily calculatd usig th tchiqu xplaid abov As a rsult, w fid 3 3 ( S QD) QS+ Q Q Q S QD Q ( QS QD ) + Q l QD Q + L + 3kTC τ= µ kt Q ( Q ) S Q S Q + D ( QS QD ) Q l + QD Q + ktc OX OX (49) For a µm dvic, th rsult is as show blow Figur 44Drai ad gat dpdc of th aowir MOSFET trasit tim Exrcis: IAs aowir trasistors Th Physics Group at Lud Uivrsity producs aowir trasistors mad from IAs ( ε = 45), s Rf [] Th oxid i thir structurs is actually silico itrid ( = 63) with a thickss d = 5 m Also, radius ad ffctiv lgth of th ε OX 8

130 aowirs ar approximatly 4 m ad µm, rspctivly W will tak th 3-3 mobility to b µ m /Vs ad th ubiasd lctro dsity to b = m Also, masurmts ar mad o 4 trasistors i paralll a) To quatify th dvic ssitivity o somtims masurs th saturatio currt (th currt at high drai voltag) as a fuctio of gat voltag W xpct a quadratic dpdc I V, so that plottig I vrsus V should produc D, sat G D, sat a straight li Us th thory of this chaptr to mak th plot blow by fixig th drai voltag at V D= 4 V ad calculatig for 4 aowirs i paralll Compar to th xprimtal plot G Figur 45Calculatd (lft) ad xprimtal (right, tak from []) ssitivity plot for a 4-aowir array Rfrcs [] B Iñíguz t al, IEEE Tras Elctro Dvics 5, 868 (5) [] T Bryllrt t al, IEEE Elctro Dvics Ltt 7, 33 (6) 9

131 5 Optical Proprtis of Smicoductors Th optical proprtis of smicoductors ar applid i a imprssiv list of dvics: lasrs, light-mittig diods, CCD camras, solar clls to am a fw Rctly, th list has b xtdd to optical smicoductor dvics that activly utiliz quatum cofimt: quatum wll lasrs, fluorsct quatum dots, solar clls ad photocatalysts basd o smicoductig TiO aoparticls ad so o Also, th traditioal iorgaic smicoductors such as Si, GaAs ad GaP hav s comptitio from orgaic or carbo basd os, most otably cojugatd polymrs ad carbo aotubs Ths ovl matrials ar o-dimsioal smicoductors with proprtis that dviat sigificatly from bulk iorgaic smicoductors Dvics such as displays ad light-mittig diods basd o ths matrials ar mrgig ow ad may wll play a importat rol i futur applicatios du to rducd cost ad possibl molcular dsig I this chaptr, w ivstigat th optical rspos of bulk ad low-dimsioal smicoductors i ordr to display thir diffrcs By ow, th fudamtal approach should b familiar: W d th prturbatio ad th rspos obsrvabl to calculat th iducd rspos Th prturbatio is th wll-kow itractio btw th optical lctric fild E = z E oscillatig at a frqucy ω ad th lctric dipol momt r giv by Ĥ= E r= E z Also, th rspos obsrvabl is th dipol momt dsity z/ Ω, whr, as always, Ω is th volum W oly cosidr th z compot of th dipol momt, i w rstrict th discussio to co-liar cass whr th iducd dipol momt is paralll to th icidt fild Th masurabl rspos is th polarizatio P( ω) that w cosqutly fid from ϕ E zϕ ϕ zϕ P( ω) = Ω fm E ħω iħ Γ m, ϕm zϕ = E Ω fm E ħω iħ Γ m, m m m Th ratio btw th polarizatio ad th lctric fild is th lctric suscptibility χω ( ) multiplid by ε ad so m ϕm zϕ χω ( ) = fm (5) ε Ω E ħ ω i ħ Γ m, I th prst chaptr, w oly wish to dscrib itrisic smicoductors, i which th mpty stats ar sparatd from th occupid os by a larg rgy gap Th m E g 3

132 Frmi lvl i this cas lis clos to th middl of th gap ad providd may safly tak th tmpratur to zro I Eq(5) abov, th sums ovr m ad ar ovr both occupid ad mpty stats W dot collctivly th occupid stats by v (for valc) ad th mpty os by c (for coductio) Hc, th doubl summatio may schmatically b split accordig to Eg kt w = m, m v v m c v m v c m c c Now, out of thos four cass, th occupatio factor f = f ( E ) f ( E ) m m zro i th first ad fourth os Thus, th rspos ca b writt is practically ϕm zϕ ϕm zϕ χω ( ) = fm + f m ε Ω m c v Em ħω iħγ m v c Em ħω iħγ Nxt, w itrchag m ad i th last sum ad group th two trms If it is usd that f f ad E = E w ca rarrag as m= m m m ϕm zϕ ϕ zϕ m χω ( ) = fm f m ε Ω m c v Em ħω iħγ v m c Em ħω iħγ Em ϕm zϕ = fm ε Ω E ħ ( ω +Γ i ) m c v m (5) Fially, i th low tmpratur limit f as log as v ad m c ad so m Em ϕm zϕ χω ( ) = ε Ω E ( ω +Γ i ) ħ m c v m Th oprator z is difficult to hadl i xtdd systms ad w thrfor prfr to rformulat th suscptibility xprssio To this d, w apply th commutator trick drivd i chaptr, Eq(): ϕ zϕ = ħ ϕ pˆ ϕ m m z imem This, vtually, provids th dsird rformulatio of th suscptibility as 3

133 ħ ϕ ˆ m pzϕ χω ( ) = εmω m c v E m Em ( ω+γ i ) ħ (53) This rsult is compltly gral ad ca b applid i accurat umrical calculatios if rgis ad momtum matrix lmts ar calculatd from a st of igstats obtaid from g ab iitio or mpirical quatum mthods Howvr, w wish to gai som basic isight ito th optical proprtis of smicoductors i various dimsios For this purpos w ll d som approximatios to mak th calculatio tractabl 5 Two-bad ad Evlop Approximatios Th first simplificatio w ca mak is that oly two bads ar cosidrd: o occupid valc bad (v) ad o mpty coductio bad (c) Hc, i th bulk, w ca writ gral occupid ad mpty igstats as u ( r ) ik r ik r, u m ( r ) ϕ = σ ϕ = σ vk ck Ω Ω, whr, agai, u ( r ) ad u( r ) ar th lattic-priodic parts ad σ is th spi part vk ck Now, v i low-dimsioal gomtris w may writ th igstats i th form ϕ = u r F r σ ϕ = u r F r σ ( D) ( D) ( ) ( ), m ( ) ( ) vk ν k ck µ k ( D) F ν F µ ( Th fuctios ad D ) ik r that rplac ar ow labld by composit idics k k { ν k } ad { µ k } This otatio is to b udrstood as follows: k is ow th D dimsioal wav vctor that cotais th quatum umbrs for th D xtdd dimsios ad ν =,,3, ad µ =,,3, labl th quatizd igstats for th rducd dimsios As a xampl, th mpty stats i a quatum wll ar labld by k = ( k, k ) for th i-pla motios ad ν =,,3, is th quatum umbr for x y th quatizd stats prpdicular to th quatum wll pla Th vlop ( D) F ν k ( approximatio cosists i takig for ad D ) th wav fuctios dtrmid xclusivly by th quatum cofimt, i igorig th lattic-priodic part of th pottial rgy Hc, w ar assumig that th total wav fuctio ca b factord ito a rapidly varyig lattic-priodic part ad a slowly varyig vlop fuctio This maks ss if th cofiig pottial is slowly varyig compard to th lattic costat Th vlop fuctio is a mix of ruig wavs for th xtdd dirctios ad stadig wavs for th cofid os As th simplst possibl cas, w may tak th cofimt to b a rctagular pottial wll with zro pottial rgy isid th wll ad ifiit outsid (s Appdix ) For F µ k 3

134 smicoductors i,, ad 3 dimsios this corrspods to a cubic box, a rod with squar cross sctio, a slab, ad ifiit spac, rspctivly Hc, if th width of th pottial wll is d i all cass, th occupid vlop fuctios i D dimsios ar 3/ νπ x x νyπy νπ z z si( d )si( d )si( d ) D= νπ x x νπ y y ikzz si( )si( ) ( D) d d D= ( r ) = νk νπ x i( kyy+ kzz) Ω si( ) D d = ik r D= 3 F Not that w hav chos to ormaliz th vlop fuctios withi th total volum Ω i ach cas W will xt assum that th lattic-priodic parts do ot dpd too strogly o wav vctor i th rlvat part of th Brilloui zo so that w may tak u ( r ) u ( r ) ad u( r ) u ( r ) This mas that i th gral cas vk v ck c ħ ( D) ( D) 3 ϕ ˆ m pzϕ = uc ( r ) F ( r ) uv( r ) F ( r ) d r k k i µ ν z ħ ( D) ( D) 3 ħ ( D) ( D) 3 = uc ( r ) uv( r ) F ( r ) F ( r ) d r+ F ( r ) F ( r ) uc ( r ) uv( r ) d r µ k ν k µ k νk i z i z Th itgrad hr is comprisd of a rapidly varyig lattic-priodic part ad a slowly varyig vlop part To s how such itgrals ar valuatd w cosidr th product of a rapid lattic-priodic fuctio R( x ) ad a slow fuctio S( x) i o dimsio If w ar to itgrat this product ovr N uit clls ach of a siz a w ca split th itgral as follows Na N la = l= ( l ) a R( x) S( x) dx R( x) S( x) dx Nxt, as th slow fuctio barly varis across a sigl uit cll w rplac it by th valu foud at th midpoit x= ( l ) a i ach uit cll: N la N la l= ( l ) a l= ( l ) a R( x) S( x) dx S(( l ) a) R( x) dx a = R( x) dx S(( l ) a), N l= whr th priodicity of R( x) is usd i th last li Fially, ruig th argumt backwards, th sum ovr l ca b approximatd by a itgral 33

135 N N la l a = l= ( l ) a S(( l ) a) S( x) dx = a Na S( x) dx Puttig thigs togthr, w s that Na a Na R( x) S( x) dx R( x) dx S( x) dx a Th ssc of this rsult is that a itgral of a product btw a rapid latticpriodic part ad a slow vlop part is approximatly qual to th avrag of th rapid part ovr th uit cll tims th itgral of th slow part ovr th tir volum of itgratio If w apply this to th momtum matrix lmt w fid 3 ( D) ( D) 3 ϕ ˆ ˆ m pzϕ uc ( r ) uv( r ) d r F ( r ) pzf ( r ) d r Ω µ k νk UC UC + u r pˆ u r d r F r F r d r Ω UC UC 3 ( D) ( D) 3 ( ) ( ) c z v ( ) ( ), µ k νk whr Ω UC is th volum of th uit cll ad th first itgral i ach trm is ovr Ω Th orthogoality btw u ( r ) ad u ( r ) mas that th first trm vaishs UC ad so v c ( D) ˆ m pz pcvs µν, ϕ ϕ whr p u r pˆ u r d r S F r F r d r 3 ( D) ( D) ( D) 3 cv= c( ) z v( ), µν = ( ) ( ), µ k νk ΩUC UC ar th itrbad momtum matrix lmt ad vlop ovrlap, rspctivly This otatio allows us to rformulat Eq(54) as ( ) D ħ p S cv µν ε mω µν,, k Eµν ( k) Eµν ( k) ħ ( ω+γ i ) 4 χω ( ) = (54) 34

136 Hr, th xtra factor of is from summatio ovr spi Th xcitatio rgy is th diffrc btw th rgy of th mpty stat E ν k ad th rgy of th occupid stat participatig i th trasitio Ths quatitis both iclud a quatizatio cotributio i additio to th usual k -dpdt kitic part Thy ar giv by E µ k E µν ( k) ħ k E E E E E E h = c+, k µ + = k v+ µ ν ν m mh ħ k, h whr th quatizatio rgis of lctros ad hols ar dotd ad, rspctivly I tur, thir diffrc is E µ E ν ħ k ħ k E k E E E E E E h µν ( ) = = k k c+ µ + µ ν v ν+ m mh ħ k Egµν+ m h h Hr, E = E+ E E E is th ffctiv bad gap ad m = m m /( m + m ) is gµν c µ v ν th rducd mass of a lctro-hol pair I th xt chaptr, w succssivly trat 3,, ad dimsioal smicoductors ad for ths cass valuat Eq(55) for th optical suscptibility Exrcis: Evlop fuctios i parabolic cofimt Suppos that th quatizig pottial of a quatum wll is parabolic Th curvatur of th cofimt may b diffrt for lctros ad hols just as thy will hav diffrt ffctiv masss This mas that th vlop fuctios will b diffrt as wll From basic quatum mchaics w kow that th first two igfuctios i a parabolic pottial ar /4 3 β 4β ϕ( z) = xp{ βz }, ϕ( z) = zxp{ βz }, π π whr β= mω / ħ dpds o mass ad curvatur Elctros ad hols will cosqutly b dscribd by ths fuctios but diffrt xprssios for β should b usd: β = mω / ħ for lctros ad β = mω / ħ for hols h h h a) Calculat th partial vlop ovrlaps giv by /4 h h h 35

137 = () S ϕ ( z) ϕ ( z) dz, µν µ ν i trms of β ad β Hlp: h ax π ax π dx=, x dx a = a a b) What happs if β = β? h 36

138 6 Optics of Bulk ad Low-dimsioal Smicoductors W ow wat to apply th gral thory of th optical proprtis of smicoductors dvlopd i th prvious chaptr to som itrstig cass Th gral thory applis qually wll to bulk ad low-dimsioal matrials ad by ivstigatig thir optical rsposs w ca highlight thir diffrcs As will bcom appart, th mathmatical tchiqus for dalig with th diffrt cass ar highly similar Physically, th diffrc btw bulk ad low-dimsioal structurs origiats i quatizatio ffcts Th quatizd motio of carrirs iflucs both slctio ruls, strgth of trasitios ad positio of rsoacs Most obviously, quatizatio tds to blu-shift th absorptio dg, i th photo rgy thrshold for absorptio, du to th addd quatizatio rgy Hc, th ffctiv bad gap is icrasd i a quatizd gomtry As w will s blow, howvr, by goig from 3D to D matrials th shap of th spctra chags dramatically as wll For a bulk smicoductor, th absorptio strgth is a smooth fuctio of frqucy abov th bad gap Wh quatizatio icrass, a strog absorptio fatur dvlops dirctly abov th ffctiv bad gap Aothr importat fatur is th apparac of multipl rsoacs i th spctra du to trasitios btw may subbads W ow systmatically study th optical rspos of 3,, ad -dimsioal smicoductors usig th two-bad modl ad th vlop approximatio dscribd i chaptr 5 Th gral startig poit is Eq(55), which w rpat hr: ( ) D ħ p S cv µν ε mω µν,, k Eµν ( k) Eµν ( k) ħ ( ω+γ i ) 4 χω ( ) = (6) Th summatios hr ar ovr a D-dimsioal k-vctor k ad th idics of th occupid ( ν ) ad mpty ( µ ) quatizd stats Th summatio ovr k ca i ach cas b prformd aalytically lavig us with a summatio of th quatizatio idics For a bulk or 3-dimsioal smicoductor thr ar o quatizatio ffcts ad, hc, th summatio i Eq(6) is, i fact, oly ovr k As always, w covrt th k summatio ito a itgral ad so 3 ħ pcv d k 3 m ( ) E ( k) µν Eµν ( k) ħ ( ω+γ i ) 4 χω ( ) = ε π Th xcitatio rgy for trasitios btw th two parabolic bads is 37

139 ħ k ħ k Eµν ( k) = Eg+ + m m ħ k = Eg+ m h, h whr m = m m /( m + m ) h h h valuat th itgral by itroducig is th rducd mass of a lctro-hol pair W x=ħ k /m h, which mas that 3 m x m x m d k= 4πk dk= πkd( k ) = π h d h π h xdx = ħ ħ ħ I this mar, th suscptibility itgral bcoms 3/ 3/ 4ħ pcv m h xdx εm ( π) ħ ( E ) ( ) ( ) g+ x Eg+ x ħ ω+γ i χω ( ) = Th x-itgral ca ow b valuatd To this d, w itroduc th complx frqucy w ω+γ i ad th ormalizd 3D suscptibility fuctio X ( w) 3 as X E xdx 3/ πeg = { Eg Eg ħw Eg+ ħw} ħ w 3/ g 3( w) ( E ) ( ) g+ x Eg+ x ħ w (6) This lads to th rsult 3/ ħ pcv m h 3/ m E g ħ χω ( ) = X3( w) πε Th limitig valu of th suscptibility fuctio is Fig 6 illustrats this rsult usig th valus X () = π /8 393 Th plot i that ar rprstativ of GaAs A fw thigs ar worth poitig out about this figur First, th absorptio is dtrmid by th imagiary part of th rfractiv idx ( ω) Sic = χ, w hav =χ / so that th absorptio is dirctly proportioal r i i i i r to th imagiary part of th suscptibility From th figur, it is obvious that th absorptio thrshold is locatd at th rgy gap, i if a frqucy sca is mad, strog absorptio will st i at ħω= E g 3 E = 5 V ad ħγ= 5 V g i 38

140 Figur 6 Complx suscptibility fuctio of a 3D smicoductor with a bad gap of 5 V A scod poit is that th suscptibility at low photo rgy (wll blow th rgy gap) is purly ral-valud ad corrspods to th usual rfractiv idx of th matrial via = + χ so that = + χ r 6 Smicoductor Quatum Wlls r r r A quatum wll is a smicoductor slab havig a smallr bad gap tha th surroudigs ad thrfor big abl to cofi carrirs to th slab Th cofimt producs a sris of subbads offst by th quatizatio rgy Hc, for th µν th subbad th xcitatio rgy is r ħ k Eµν ( k) = Eg µν+, m whr k is th dimsioal wav vctor i th pla ad is th ffctiv rgy gap icludig th quatizatio rgy Hc, if ifiit squar-wll cofimt with a width d is assumd h E g µν E gµν µ ħπ νħπ = Eg+ + m d m d h I aalogy with th 3D cas, w ow covrt th D ad subsqutly writ th suscptibility as k - summatio ito a itgral ħ pcv () d k Sµν m ( ) d, E ( k) µν µν Eµν ( k) ħ ( ω+γ i ) 4 χω ( ) =, ε π 39

141 usig Ω= Ad, whr A is th ara of th slab Agai, w us x=ħ k m so that mh d k= πkdk= πd( k ) = π dx ħ / h This allows us to writ th suscptibility as th itgral 4ħ pcv πm () h dx Sµν m ( ) d ħ µν, ( E ) ( ) ( ) gµν + x Egµν+ x ħ ω+γ i χω ( ) = ε π Hc, w aturally itroduc th D suscptibility fuctio dpdt part X gµν ( w) ( Egµν + x) ( Egµν+ x) ħ w E dx E gµν Egµν = l, ħ w Eg µν ħ w X ( w) as th frqucy (63) with th limitig bhavior X () = / It follows that th D suscptibility is () pcv m S h µν πεm d µν, Egµν χω ( ) = X( w) Figur 6 Complx suscptibility fuctio of a quatum wll with a ffctiv bad gap of 6 V 4

142 This w suscptibility fuctio is dpictd i Fig 6 Hr w hav agai tak data for GaAs ad assumd a quatizatio rgy of V so that th ffctiv gap is E µν = 6 V By compariso to Fig 6 it is s that th bad gap fatur is g sigificatly stpr i th quatum wll cas 6 Smicoductor Quatum Wirs ad Dots Quatum wirs ar o-dimsioal i th ss that thir xtsio alog o axis is much gratr tha th othr two, which w assum to hav aoscal dimsios Th calculatio of th suscptibility is highly similar to th quatum wll cas, xcpt that ow th k-vctor is o-dimsioal ad th sum ovr µν covrs th twodimsioally quatizd statd W bgi by otig that / / mhx mh dk d( k ) ħ = = d dx k = m x ħ ħ x h This, i tur, yilds a suscptibility giv by (rmmbrig a factor of from ±k ) / ħ pcv m () h dx S µν m d ħ µν, x( E ) ( ) ( ) gµν + x Egµν+ x ħ ω+γ i 8 χω ( ) = ε π () / ħ cv h Sµν 5/ m d ħ µν, Egµν 4 p m = πε X ( w) I this cas, th corrspodig suscptibility fuctio is X E dx 5/ πe g µν = + + ħ w Egµν Egµν ħw Egµν+ ħw 5/ gµν ( w) x( E ) ( ) gµν + x Egµν+ x ħ w (64) I this cas, th DC limit is X() = 3 π /8 78 Th rsult is illustratd blow assumig E µν = 8 V Th trd from th quatum wll cas is cotiud ad ow g th absorptio pak at th bad dg is furthr sharpd I fact, if th broadig ħγ gos to zro th imagiary part of th suscptibility fuctio will divrg This is radily s from Eq(64) Wh Γ=, th imagiary part ca oly com from th scod trm i th curly brackt This trm will b purly imagiary if ħω>e g ad cosqutly 4

143 ( ħ Egµν) 5/ πe θ ω gµν lim Im{ X( w)} = Γ ħω ħω E gµν Figur 63 Complx suscptibility fuctio of a quatum wir with a ffctiv bad gap of 8 V It follows that as th absorptio dg is approachd from abov, th imagiary part will divrg as a ivrs squar root of th frqucy Quatum dots ar smicoductor aoparticls with o xtdd dimsios Hc, th spctrum of igvalus is purly discrt If a box shap with sid lgth d is assumd for th particl, th xcitatio rgis ar E gµν µ ħπ νħπ = Eg+ + m d m d h Hc, thr is o k-itgratio is this cas ad w simply fid () ħ p S cv µν 3 3 m d µν, Egµν 4 χω ( ) = X( w), ε whr X E gµν ( w) = Egµν ħ w (65) is th -dimsioal suscptibility fuctio with th limit X () = as illustratd i Fig 64 blow Hr, E µν = V is assumd i ordr to follow th trd of g 4

144 icrasd quatizatio rgy as w go through wlls, wirs ad ow dots Th rspos of th D systm is a isolatd rsoac at th ffctiv gap It is otd that th imagiary part is compltly symmtric i cotrast to th highr-dimsioal structurs Figur 64 Suscptibility fuctio of a quatum dot with a ffctiv bad gap of V I th plots abov, w hav displayd th cotributio from a sigl rsoac I gral, th spctrum will cosist of a sris of rsoacs du to trasitios btw multipl subbads Th wight of ach trasitio is giv by th vlop ovrlap ( D) S µν i Eq(6) If th vlop fuctios of coductio ad valc bad stats ( D) wr igfuctios of th sam Hamiltoia th ovrlap would simply b S µν =δ µν bcaus of orthogoality Howvr, th Hamiltoias of th two bads ar grally diffrt du to diffrt ffctiv masss ad diffrtly shapd pottial wlls A ( D) spcial cas is th ifiit rctagular wll, for which =δ still holds Th S µν µν raso is that, i this cas, th igstats ar simpl stadig wavs that do ot dpd o ffctiv masss aywhr Hc, th wav fuctios rmai orthogoal As a xampl of svral subbads cotributig to th total rspos w cosidr a quatum wir with thr allowd trasitios ν = µ =, ν = µ = ad ν = 3 µ = 3 W tak E = 8 V, E = 5 V ad E 33= 35 V ad kp g ħγ=5 V Th rsultig spctrum for th imagiary part of th suscptibility fuctio X ( w) is illustratd i Fig 65 Sic th sparatio btw th rsoacs is far gratr tha th broadig of th idividual paks, th thr rsoacs ar wll rsolvd If th spacig btw th paks bcoms comparabl to ħγ th paks will ovrlap ad b hard to distiguish g g 43

145 Figur 65 Absorptio spctrum of a quatum wir with thr allowd subbad trasitios Exrcis: Limits of th suscptibility fuctios a) Show that th suscptibility fuctios hav th DC limits statd i th txt, that is, show that X () = π /8 3 X () = / X () = 3 π /8 X () = b) Fid limitig xprssio for th imagiary parts of th suscptibility fuctios as th broadig ħγ gos to zro 44

146 7 Elctroic ad Optical Proprtis of Graph Graph is a ovl wodr matrial that has stimulatd a trmdous amout of xprimtal ad thortical work For thortical aoscic, it is a grat matrial bcaus of its xtrm simplicity but, othlss, itriguig proprtis A lot of this fasciatio drivs from th fact that carrirs i graph ar th itrisic Frmi lvl bhav similarly to masslss rlativistic particls, albit movig with a ffctiv spd of light of roughly c/3 W will start this chaptr by a brif rviw of th lctroic structur of this matrial ad subsqutly cosidr th spcific proprtis drivd from th lctroic igstats Mor dtails ar giv i Appdix Graph is a truly twodimsioal matrial cosistig of a hoycomb lattic of carbo atoms A pic of th lattic is show i th lft pal of Fig 7 Figur 7 Hoycomb lattic icl lattic vctors (lft) ad Brilloui zo icl rciprocal lattic vctors (right) Th lattic costat of graph is lattic vctors ar a=46 Å Corrspodigly, th lmtary a a 3 a 3 =, a = Clarly, thr ar two atoms pr uit cll Also, th rciprocal lattic vctors ar asily foud from th rquirmt g a = πδ to b g i j ij π π =, g 3a = 3 3a 3 Th Brilloui zo spad by ths vctors is show i th right pal of Fig 7 Hr, th so-calld irrducibl Brilloui zo is also highlightd as th rd triagl alog with high symmtry poits 45

147 Th simplicity of graph drivs from th fact that ach carbo atom coms with 4 valc orbitals: s, px, py,ad pz Amog ths, pz has odd parity with rspct th rflctios i th (x,y) pla whras th rmaiig os ar v Thus, - stats compltly dcoupl from th rst As a cosquc, xtdd Bloch stats formd by couplig p - orbitals (calld π z - stats) also dcoupl from thos formd by th rmaiig orbitals ( σ - stats) Th π - stats ar loosly boud ad rsposibl for all lctroic ad optical proprtis i th usual low-rgy rag Covrsly, th σ - stats ar largly rsposibl for holdig th matrial togthr but ar vry difficult to xcit ad, thrfor, irrlvat for most rspos proprtis Th simplst, ralistic lctroic modl is costructd by assumig that ach - orbital is coupld to its arst ighbors oly As a basis for xtdd Bloch igstats of th full lattic, w tak Bloch sums formd by summig - orbitals blogig to th two atoms A ad B i th uit cll (so-calld sublattics), sparatly I a uit cll at positio R, th positio of th atoms A ad B will b dotd R A ad R B, rspctivly To simplify th otatio, w will dsigat th pz - orbitals A B blogig to th two sublattics by π pz( r RA ) ad π p ( ), R R z r RB rspctivly Formig th two Bloch sums α ad β for th A ad B sublattics, w fid p z p z p z A B α = π, β = π R N N R ik R A ik R B R R, whr N is formally th umbr of uit clls that is vtually tak to ifiity Th Hamiltoia i this basis is giv by α Hˆ α α Hˆ β H= β Hˆ α β Hˆ β Assumig oly arst ighbor couplig, th diagoal lmts both qual th osit rgy ˆ A A π Hπ that w will st to zro by choosig it as our rgy zro R R poit Th off-diagoal lmts ar o-zro, howvr Th arst ighbor couplig, somtims dotd th hoppig itgral or tul couplig, is A ˆ B γ π Hπ (th mius sig is itroducd for umrical covic as th R R matrix lmt itslf is actually gativ) Hc, it is straightforward to show that α Hβ = γh k h k = + k a ( y ) ˆ ikxa/ 3 ikxa/ 3 ( ), ( ) cos /, (7) 46

148 ad β Hˆ α = α Hˆ β At this poit, it is th clar that th igvalus ar E =± ± γ h( k), i E ± k ya kya 3kxa =± γ + 4 cos 4 cos cos + (7) Th bad structur is illustratd i Fig 7 Importatly, th bads touch at th K poit K= π / a( / 3,/3) T I fact, ar th K poit, th bads form a co This is radily s if w xpad th igvalus aroud K, i by writig k= K+ q ad i subsqutly xpad i I this mar, w fid that h( k) 3 a/ q + iq π i / Th costat phas ca b absorbd by rdfiig π i / α α ad β β π This mas that th Hamiltoia bcoms q ( ) /6 x y qx iq y 3aγ H= ħvf, vf= qx+ iqy ħ Hc, th igvalus ar of th form E =±ħ ± vfq Th quatity vf is th Frmi 6 vlocity Th hoppig itgral is foud to b aroud γ 3 V implyig that vf m/s Thus, carrirs rally rsmbl masslss rlativistic os (sic th rgy disprsio is liar i th momtum ħ q ) movig with a vlocity v c /3 This liarizd approximatio is usually calld th Dirac modl of graph Each carbo atom supplis prcisly o lctro to th π - bads (th othr 3 ar boud i chmical bods with th ighbor atoms) Thus, for itrisic graph, th lowr bad is filld whil th uppr is mpty at low tmpraturs This mas that th Frmi lvl coicids with th so-calld Dirac poit, at which th bads cross Not / that th igvctors ar simply v = ± iϕ, T ± with ϕ th polar agl of q ( ) F Figur 7 Full tight-bidig bad structur compard to th Dirac modl (lft) I th liarizd modl, a Dirac co sits i ach corr of th Brilloui zo (right) 47

149 7 Elctroic ad Optical Proprtis I th followig, w will focus o th Dirac modl bcaus th simplicity allows us to comput may quatitis aalytically At various poits, w ca th compar to mor complicatd rsults obtaid for th full tight-bidig thory First, w will cosidr th dsity of stats I ordr to prform th computatio, w ot that th cotributio from a sigl Dirac co should b multiplid by a factor of 4: first thr ar two cos pr uit cll (vally dgracy) ad scodly thr is spi dgracy Hc, dsity of stats pr ara for a ara A ad rgy E > is 4 E D( E) = δ( E ħvfq) = δ( E vfq) d q δ( E vfq) qdq A π ħ = = π ħ πħ v q F (73) A similar rsult is foud for E < ad grally w ca writ D( E) = E / πħ vf I a mor laborat tratmt usig th full tight-bidig modl o fids [, ] 3 γ 3Ω (3 Ω )( +Ω) D( E) = R K, (74) πħ v F 6Ω whr K is th complt lliptic itgral of th first kid ad w hav itroducd ormalizd rgis E / γ Usig th limit lim R K( z) = π / z shows that Ω= [ ] th two xprssios agr i th low-rgy rag Th rsults ar compard i th figur blow z Figur 73 Compariso of th tight-bidig ad Dirac modl rsults for th dsity of stats 48

150 Bcaus graph has a vaishig bad gap but, at th sam tim, a vaishig dsity of stats at th Dirac poit, it ca b classifid as a matrial i btw smicoductors ad mtals, i a so-calld smimtal Th obtaid dsity of stats abls us to comput th xcss lctro dsity As mtiod abov, itrisic graph is charactrizd by a Frmi lvl locatd at th rgy zro poit W thrfor fid th xcss carrirs, that is th umbr of additioal lctros pr ara, by th itgral { } = D( E) f ( E) f ( E) de Hr, f ( E) = (xp{( E E )/ kt} + ) is th Frmi-Dirac distributio ad F f( E) = (xp{ E/ kt} + ) is th rfrc for th itrisic cas, E F= Isrtig ad rformulatig, it turs out that kt x = sih{ EF / kt} dx π v ħ cosh x+ cosh{ E / kt} F kt = { Li( xp{ EF / kt} ) Li( xp{ EF / kt} )} π v ħ F F (75) Hr, Li is th scod polylogarithm At low ad high dopig lvls, w fid kt l EF / kt EF / kt, π v ħ ± ( EF / kt) EF / kt F Nxt, w go o to calculat th coductivity of graph First, howvr, a word about th currt rspos of D systms I th usual 3D cas, w writ th rspos to a lctric fild E as j= σ E, whr σ is th 3D coductivity ad j is th 3D 3D currt dsity dfid as currt pr ara For a vry thi D sht, w may itgrat ovr th thickss assumig that th fild dos ot vary across th sht Hc, w dfi a sht coductivity σ= σ dz 3D that rlats th sht currt dsity (currt pr lgth) to th lctric fild I this mar, w d a sht plasma frqucy ω p 4 E + E = f E f E + ħεa q q x q x Agai, th factor of 4 is du to spi- ad vally-dgracy Also, w summd ovr th two bads Covrtig to a itgral, it is foud that 3D ( ) ( ) + 49

151 Th optical rspos grally cosists of a itrabad itrbad cotributio valuatd, cf Eq(3) cotributio ad a Giv th plasma frqucy, th formr is asily iεω p σitra( ω) =, ω +Γ i whr Γ is a phomological broadig paramtr Th itrbad trm is sigificatly mor complicatd I th limit of vaishig broadig, th gral rsult aftr summatio ovr spi for a two-dimsioal matrial is σitr ( ω) = { f ( E ) ( )} ( ) v f Ec Pvc δ Ec Ev ω d k (76) πmω ħ Hr, th sum is ovr occupid valc bads (v) ad mpty coductio bads (c) Also, P vc is th momtum matrix lmt W valuat this matrix lmt usig th commutator pˆ = im / ħ [ Hˆ, x] ad th compltss of th - stats, i x ω p σ itr = { f ( E) + f ( E) } EdE πħε kt = l( cosh{ E / }) F kt πħε v, c σ itra p z = π π + R π π A A B B R R R R R Thus, im ˆ im α pˆ [, ] ˆ ˆ xβ = α H x β = α Hπ π xβ α xπ π Hβ R R R R ħ ħ A A A A { } R im { ˆ B B B B + α Hπ π xβ α xπ π Hˆ β } R R R R ħ R W cotiu assumig that our orbitals ar sufficitly localizd that oly o-sit trms mattr Hc, π A xβ = A ik R ad α xπ = X A / N tc W thrfor s that R ik RA A ˆ ˆ B ik RB { π β α π R R } im α pˆ xβ = XA H + H XB ħ N R R A 5

152 ik RA ik RA A importat poit, ow, is that X = i / k ad ik RB X = i / k B ik RB x Thus, m ik RA A ˆ ˆ ˆ B ik R B α pxβ = π Hβ α Hπ + R R ħ N R k x k x m = α Hˆ β ħ k x As a cosquc, th itrbad momtum oprator ca b rplacd by ( m / ħ ) H / k x For th Dirac modl, this implis that i px mv = F, py= mv F i Th matrix lmts, thrfor, ar xcdigly simpl ˆ ( ) i mv ϕ F iϕ + p, si, ˆ x = = imvf ϕ + py = imvf cosϕ Wh this rsult is usd i th itrbad xprssio w s that (rmmbrig to iclud vally dgracy) A x vf σitr ( ω) = { f ( E ) f ( E+ )} δ( E+ E ω) qdq ω ħ = { f ( E) f ( E) } δ( E ω) EdE ω ħ ħ = { f ( ħω /) f ( ħω /)} 4ħ sih{ ħω / kt} = 4ħ cosh{ ħω / kt} + cosh{ E / kt} F (77) If th full tight-bidig modl is cosidrd, a aalytical calculatio shows that i th low-tmpratur, itrisic limit T, E [, ] F Ω+Ω (6 Ω )( +Ω) (6 Ω )( +Ω) σitr ( ω) = R K E, (78) 3/ 4πħΩ 4 8Ω 8Ω whr Ω=ħω / γ ad K ad E ar lliptic itgrals It ca b show that takig th zro frqucy limit of th abov xprssio lads to a miimum graph 5

153 coductivity of σ /4ħ A compariso of th two modls for T, EF is show i Fig 74 Figur 74 Compariso of th tight-bidig ad Dirac modl rsults for th itrbad coductivity I actual masurmts, oly th sum of itra- ad itrbad trms is masurd As a xampl, this sum (ral part) is illustratd i Fig 75 for som rasoabl valus of th rlvat paramtrs (compar to xprimts i Fig 76) Figur 75 Full coductivity (ral part) for a graph sampl dopd to E F = mv 5

154 I a trasmissio xprimt, th trasmittac of light with diffrt wavlgths is masurd Hc, a masurd spctrum ca b rlatd to our calculatios i th followig mar First, for a graph slab of thickss d, th dilctric costat is ω ε= + iσ /( dεω ) Scodly, th trasmittac is T= xp{ id}, whr i is th imagiary part of th rfractiv idx = ε Assumig wak absorptio, w ca approximat + iσ /( dεω ) ad, hc, i σr /( dεω ), whr σr is th ral part of th coductivity Approximatig furthr, w fid that ω ω T= xp{ d} d σ /( ε c) (79) i c i r Thus, for graph i th low-rgy itrbad rag, whr c, w fid T /(4 ε cħ) = πα, whr α /(4 πε cħ) /37 is th fi-structur costat It follows that th absorptio loss for moolayr graph is roughly 3% I Fig 76, this is s to b cofirmd xprimtally Not also th absorptio pak aroud 5 m that corrspods to th tight-bidig rsoac i Fig 74 c σ /4ħ Figur 76 Exprimtal optical spctra i th UV/VIS rag [VG Kravts t al, Phys Rv B8, 5543 ()] (top) ad ifrard [ZQ Li t al, Nat Phys 4, 53 (8)] (bottom) 53

155 Exrcis: Gappd graph May idas hav b suggstd to provid graph with a bad gap, so that a tru smicoductor would rsult Notabl os iclud itractios with substrats, addig lctric filds to multilayr structurs ad atidot lattics [3] A commo phomological modl of such gappd graph is obtaid with th Hamiltoia α vf( qx iqy ) H ħ = vf( qx+ iqy ) α ħ a) Show that for this modl th igvalus ar E =± ( ħ vfq) + ± α so that a bad gap of α rsults Th corrspodig igvctors ar v iϕ iϕ ( E α)/ E ( E α)/ E =, v = ( E α)/ E + + ( E α)/ E + b) Usig th w igvctors, dmostrat that E +α + pˆ ˆ x + + py = m vf E c) Show that th itrbad coductivity is giv by th itgral E + α σitr ( ω) = { f ( E) f ( E) } δ( E ω) EdE ω ħ ħ E α ħω + 4α = { f ( ħω /) f ( ħω /)} θ( ħω α) 4ħ ħω sih{ ħω / kt} ħω + 4α = ( ) θ ħ ω α 4ħ cosh{ ħω / kt} + cosh{ E / kt} ħω F (7) Rfrcs [] T G Pdrs, Phys Rv B67, 36 (3) [] T G Pdrs, A-P Jauho, ad K Pdrs, Phys Rv B 79, 346 (9) [3] T G Pdrs, C Flidt, J Pdrs, A-P Jauho, NA Morts ad K Pdrs, Phys Rv Ltt, 3684 (8) 54

156 8 Modls of Excitos At this poit, o might wodr about th accuracy of th optical rspos calculatd i th prvious chaptrs So far, w hav b tratig th lctros as idpdt particls ad th qustio is to what xtt this is sufficit For bulk smicoductors, th sigl-particl calculatios prdict a absorptio dg that is sstially a squar root, cf Fig 6 As a classic xampl of th failur of this prdictio, Fig 8 shows a compariso btw xprimtal spctra [] ad thortical sigl-particl spctra computd from Eq(53) for th wurtzit smicoductor ZO Figur 8 Masurd spctra (lft pal) ad calculatd sigl-particl spctra (right pal) for ZO Th ordiary ad xtraordiary spctra corrspod to light polarizd prpdicular ad paralll to th crystal c-axis, rspctivly It is obvious that sigl-particl thory fails misrably i this cas I most matrials, howvr, th discrpacy is lss prooucd but still oticabl Th aim of this chaptr is to dscrib a mthod for th iclusio of ffcts byod th sigl-particl rspos It ivolvs a much mor accurat calculatio of may-body xcitd stats usually rfrrd to as xcitos I subsqut chaptrs, th ffcts of xcitos i lowdimsioal smicoductors will b ivstigatd W will dmostrat that xcitos ar v mor importat for thos cass Applyig th sigl-particl approximatio mas, i ffct, approximatig alllctro wav fuctios by Slatr dtrmiats To dmostrat this fact, w tur to th mor gral xprssio for th optical suscptibility 55

157 ˆ ħ Pz xc χω ( ) = εmω xc E xc Exc ( ω+γ i ) ħ (8) This xprssio diffrs from th sigl-particl rsult Eq(53) i that th sum is ovr all xcitd stats xc with xcitatio rgy E, i rgy masurd rlativ to th groud stat Th prfactor of (rathr tha 4) is usd bcaus th summatio also covrs spi Also, th oprator is th may-body momtum oprator, which for a systm with N lctros is giv as th sum of sigl-lctro oprators whr z, stat is a Slatr dtrmiat pˆ = ħ i d / dz Pˆz Pˆ N z= = pˆ xc z,, oprats o th th lctro coordiat oly Th groud = ( v ) ( v ),( v ) ( v ), N, N with all sigl-lctro valc stats occupid by spi-up ad dow lctros Th total spi of th groud stat is zro ad sic optical xcitatios do t flip spis w look for xcitd stats with vaishig spi Ths ar so-calld siglt stats To costruct thm, w first xami two typs of sigly-xcitd stats ( v ) ( c ) =,( c ),( v ), i j j i ( v ) ( c ) =,( v ),( c ), i j i j i which a sigl occupid spi-up or dow orbital is rplacd by uoccupid (coductio) stats with similar spi Nithr of ths stats hav dfiit total spi Howvr, th combiatio { ) ) ) )} v c ( v ( c + ( v ( c / i j i j i j is a siglt with total spi S = W ow us th ruls for matrix lmts btw Slatr dtrmiats [] to calculat for th momtum Pˆ v c = v pˆ c z i j i z j Morovr, th rgy diffrc btw th siglt ad th groud stat is simply E E Hc, Eq(8) rducs xactly to Eq(53) i this cas c j vi W ow wish to b somwhat mor accurat To this d, w writ th xcitd stats as liar combiatios of th siglts abov, i 56

158 xc = Ψ v c ij ij i j, whr Ψ ij ar ukow xpasio factors Th problm is how to fid matrix lmts of th total Hamiltoia for ay two siglts H = v c Hˆ v c ij, kl i j k l Th total Hamiltoia is giv by N N ˆ ˆ H= h+ V( r rm ), V( r ) =, 4πε r whr hˆ is th sigl-lctro Hamiltoia As a start w look at th rgy of th groud stat Not that th additioal factors of appar bcaus th spi-summatio has alrady b prformd Nxt, w look at th diagoal lmts for th stat ( v ) ( c ) Compard to th groud stat, v should b rplacd by c It follows that W ow itroduc th quasi-particl rgis N = < m N Hˆ = v hˆ v + v v V v v v v V v v Not that thr is o rstrictio o th summatios, i all valc stats ar summd I trms of ths quatitis w hav { } m m m m =, m i ( v ) ( c ) Hˆ ( v ) ( c ) = Hˆ + c hˆ c v hˆ v i j i j j j i i { vc j V vc j vc j V c jv} vic j V vic j { vvi V vvi vvi V viv} + + i N { } Eɶ c hˆ c + v c V v c v c V c v c j j j j j j j = N Eɶ v hˆ v + v v V v v v v V v v { } vi i i i i i i = j i j v c Hˆ v c = Hˆ + Eɶ Eɶ v c V v c + v c V c v ( ) ( ) ( ) ( ) j i i j i j c v i j i j i j j i, 57

159 whr th last two trms srv to corrct th urstrictd summatios i th quasiparticl rgis Th xact sam xprssio is obtaid if th spi-dow Slatr dtrmiat is cosidrd Th cross-trm yilds ( v ( c Hˆ ( v ( c v c V c v i ) j ) i ) j ) = i j j i Combiig, w fid th full diagoal matrix lmt for th siglt xcitatio v c Hˆ v c = Hˆ + Eɶ Eɶ v c V v c + v c V c v i j i j c j vi i j i j i j j i It ca b show that couplig btw sigly xcitd stats ad th groud stat is idtically zro, i that Hˆ v c = [3] Th o-zro off-diagoal trms follow i much th sam styl as th diagoal os ( ( ˆ ( ( i j v ) c ) H v ) c ) = c v V v c c v V c v i j k l j k i l j k l i v c Hˆ v c = c v V v c ( ) ( ) ( ) ( ) i j k l j k i l As Hˆ fially fid is th groud stat rgy, which w us as a zro-poit of rgy, w H = Eɶ Eɶ δδ c v V c v + c v V v c ij, kl c j vi ik jl j k l i j k i l (8) Th matrix problm th rads as kl H Ψ = E Ψ ij, kl kl xc ij from which xcito wav fuctios ad rgis ar computd I tur, th xcito momtum matrix lmts bcom, Pˆ xc = Ψ v pˆ c z ij i z j ij (83) W ow spcializ to priodic solids for which orbitals ar labld by a bad idx (v or c) ad a wav vctor k I a optical procss, th oly rlvat xcitatios ar thos that prsrv k (glctig th small momtum lost/gaid by th photo) Thus, th siglts ar of th typ vck vk ck I tur, th sought matrix lmts ar 58

160 H E E δ δ δ vck V vck, v c k ck vk vv cc kk C v c k vck Vx v = ɶ ɶ + c k, (84) whr Coulomb ad xchag matrix lmts ar dfid as 3 3 vck VC v c k = ψ( r ) ψ ( r ) V( r r ) ψ ( r ) ψ ( r ) d rd r ck c k vk v k vck V v c k = ψ rψ r V r r ψ rψ r d rd r 3 3 ( ) ( ) ( ) x ( ) ( ) ck c k vk v k I a mor rigorous drivatio [4], it turs out that th Coulomb itractio should b scrd by surroudig chargs, so that itroducig th dilctric costat ε w fid 3 3 vck VC v c k = ψ( r ) ψ ( r ) V( r r ) ψ ( r ) ψ ( r ) d rd r ck c k vk v k ε This full matrix quatio (usig th scrd Coulomb itractio) is kow as th Bth-Salptr quatio 8 Wair Modl Th framwork abov is trribly complicatd ad xtrmly difficult to hadl umrically Fortuatly, a much simplifid vrsio ca b applid i may cass providd th Coulomb itractio is ot too strog To driv this Wair modl w first ot that th igstats of a priodic solid ca b writt as ψ vk ( ) ( ) ik r = u r r, Ω vk whr u vk is th lattic-priodic part ormalizd so that u ( r ) = vk Ω UC UC with th itgral tak ovr th uit cll volum Ω UC W first tur to th Coulomb matrix lmt Th product ψ ( r ) ψ ( r ) V( r r ) has a rapidly varyig priodic vk v k part ad slow part I aalogy with chaptr 5, w will approximat th itgral 59

161 ψ ψ 3 3 i( k k ) r 3 ( r ) ( r ) V( r r ) d r u ( r ) u ( r ) d r V( r r ) d r vk v k vk v k ΩΩUC ( ), = ( ) ( ) Ω i( k k ) r 3 3 I V r r d r I u r u r d r v k, vk v k, vk v k vk Ω UC Makig a similar approximatio for th r-itgratio, w th fid εω i( k k ) r 3 = I I V( r ) d r v k, vk c k, ck ε Ω i( k k ) ( r r ) 3 3 vck VC v c k I I = V( r r ) d rd r v k, vk c k, ck I a compltly aalogous mar, th xchag itgral bcoms vck V v c k I I V 3 x ( r = ) d r Ω ck, vk c k, v k Now, at k= k w hav I =δ Hc, if th k-dpdc is ot too svr w αk, βk αβ may assum that I hold approximatly to a rasoabl dgr v wh αk, βk =δαβ k k I this cas, w fid th much simplr approximatios vck V v c k V r d r ε Ω vck V v c k i( k k ) r 3 C ( ) δ δ vv cc x Hc, th bads dcoupl ad w ca focus o a sigl pair v ad c Th Hamiltoia matrix lmts bcom i( k k ) r 3 H = E E δ V( r ) d r, k, k ck vk kk εω ɶ ɶ whr w hav skippd th bad idics o th matrix lmts Also, th ukow xpasio cofficits ca b r-labld accordig to Ψ Ψ It follows that th xcito igvalu problm is ow i( k k ) r 3 E E ( ) ck vk Ψ V r d rψ = E k k xcψ ɶ ɶ k εω k ij k Hr, th k summatio ca b turd ito a itgral, i 6

162 ( π) ε i( k k ) r 3 3 Eɶ Eɶ Ψ V( r ) d rψ d k= E ck vk k 3 k xcψ k Th fial approximatio of th Wair modl cosists i applyig th ffctiv mass disprsio for both bads so that k k k Eɶ ħ Eg+, Eɶ ħ Eɶ Eɶ ħ E ck vk ck vk g+ m m m h h I this way, th k-spac igproblm ca b trasformd ito physical spac by mas of a simpl ivrs Fourir trasform: 3 3 ( ), ( ) 3 3 ( ) Ω ik r d k r ik r k xc k xc ( ) Ω Ψ =Ψ π π k Ψ d k = Ψ r Th Ω factors ar isrtd to sur that Ψ ( r ) is ormalizd Morovr, th Coulomb trm abov is simply th covolutio btw th wav fuctio ad th Coulomb pottial ad, hc, w fially fid ħ E g Ψ xc( r ) V( r ) Ψ xc( r ) = ExcΨ xc( r ) (85) m ε h It is appart that this so-calld Wair quatio is mathmatically similar to th Schrödigr quatio for th hydrog atom Th diffrcs ar that m h rplacs th rducd lctro-uclus mass ad that ε scrs th Coulomb trm Th physical itrprtatio is that th positiv hol ad gativ lctro itract via a attractiv Coulomb pottial W ot that oly th rlativ motio of th lctrohol pair is prst i th problm, so that th stats hav a vaishig ctr-of-mass momtum This is a cosquc of our rtaiig oly vk ck xcitatios i th xpasio, i glctig photo momtum Hc, th ctr-of-mass momtum must vaish both bfor ad aftr th photo is mittd/absorbd To vtually calculat th optical proprtis, w d th momtum matrix lmt Eq(83), which ow rads as ˆ Ω 3 P ˆ ˆ z xc = Ψ vk pz ck = vk p 3 z ck Ψ d k k k ( π) k As a simplificatio, w may tak th sigl-lctro momtum matrix lmt idpdt of k so that vk pˆ z ck p vc, which mas that xc 6

163 ˆ Ω 3 Pz xc p () 3 vc Ψ d k= Ω pvcψxc, k ( π) whr Ψ xc () is th xcito wav fuctio i physical spac valuatd at th origi This lads to a simpl xprssio for th optical rspos ħ p Ψ () vc xc εm xc E xc Exc ħ ( ω+γ i ) 4 χω ( ) = (86) I this approximatio, oly xcito stats that ar fiit at th origi ( s-typ ) cotribut to th rspos I th followig, w valuat th imagiary part i th limit of vaishig broadig π p Im χω ( ) = Ψxc() δ( Exc ħω) ε ω vc m xc (87) for bulk ad low-dimsioal cass Exrcis: Natural xcito uits Th Wair quatio Eq(85) is formulatd i SI uits ad it is advatagous to switch to mor atural uits a) Show that usig a = 4 πεεħ / m as th uit of lgth ad Ry =ħ m a as th rgy uit, th Wair quatio rducs to Eg Ψ xc( r ) Ψ xc( r ) = ExcΨ xc( r ) r b) Show that a = 59Å εm / m ad Ry = 36V m / ε m (m is th fr lctro mass) ad valuat both for GaAs: ad for ZO: m = 8 m, m = 59 m (havy hol), ε= 67 (icludig ffctiv phoo cotributio) Rfrcs B B ( ) h h h / h B [] GE Jlliso ad LA Boatr, Phys Rv B58, 3586 (998) [] JP Dahl Itroductio to th Quatum World of Atoms ad Molculs (World Scitific, Sigapor, ) [3] L Salm Th Molcular Orbital Thory of Cojugatd Systms (WA Bjami, Nw York, 966) [4] M Rohlfig ad SG Loui, Phys Rv B6, 497 () h m = 66 m, m = 5 m (havy hol), ε= 9 h 6

164 9 Excitos i Bulk ad Two-dimsioal Smicoductors Th Wair modl drivd i th prvious chaptr provids a simpl framwork for th iclusio of xcitos i th optical proprtis of smicoductors I this chaptr, w will valuat th optical rspos for 3D ad D smicoductors As will bcom appart, xcitoic ffcts i low-dimsioal smicoductors ar hugly hacd Th raso is that xcitoic ffcts origiat from th attractiv itractio btw lctros ad hols Th strogr th attractio, th mor prooucd th xcitoic corrctios to th rspos But a additioal cofimt will also td to localiz lctros ad hols i th sam rgio of spac ad, hc, icras th ovrlap of thir wav fuctios This may icras th Coulomb attractio sigificatly W bgi by ivstigatig th 3D or bulk cas Th startig poit is th Wair quatio writt i atural xcito uits (usig a ad Ry as th uits of lgth ad rgy, rspctivly) Eg Ψ xc( r ) Ψ xc( r ) = ExcΨ xc( r ) r Th xprssio for th suscptibility Eq(86) shows that oly s-typ igstats ar rlvat Ths ar othig but th usual igstats of th hydrog atom For th boud stats with rgy blow w hav i trms of th pricipl quatum umbr =,, 3,, ( ) E g B Ψ ( r ) = L ( r / ), E = E π r / 5/ g m Hr, L is a associatd Lagurr polyomial As w us Ry as th uit of rgy it follows that ths stats form a sris of rsoacs locatd spctrally btw ad Eg Ry Hc, boud xcitos lad to discrt absorptio paks blow th bad gap Th cotiuum stats with ar hardr, i part bcaus thy caot b ormalizd i th usual ss To circumvt this problm, w clos th xcito i a larg sphr of radius R [] W lt th wav fuctio vaish o th surfac of th sphr ad vtually lt th radius go to ifiity W writ α = E E g so that th radial Schrödigr quatio bcoms A solutio is E> E g d r Ψ( r ) Ψ ( r ) = αψ( r ) r dr r E g 63

165 iαr Ψ ( r) = N F( + i / α, ; iαr), α whr F is calld a coflut hyprgomtric fuctio (s g [] for dtails) W d to study th bhavior as r approachs R to fulfill th boudary coditios Sic R is vtually tak vry larg, w ca us th asymptotic xprssio for F W ca also apply th asymptotic xprssio to dtrmi th ormalizatio costat N bcaus w maily itgrat ovr a rgio with r larg Th asymptotic limit is ( iαr) ( iαr) F( + i / α, ; iαr) + Γ( i / α) Γ ( + i / α) π/ α i/ α π( i / α) iαr + i/ α sih( π / α) cos ( αr ) παr Hc, to ormaliz w itgrat withi th sphr sih( π / α) = 4 π Ψ α( r) r dr= N R π/ α α It follows that th wav fuctio at th origi is giv by R, E= E boud stats E< E 3 g π Ψ () = π/ α α = g+ α >, E E cotiuum E Eg R sih( π / α) g Th mai rmaiig problm lis i summig th cotiuum solutios whil takig R For this purpos w itroduc th wightd joit dsity of stats S( ω) = Ψ() δ( E ħω), cot whr th sum is ovr all cotiuum stats From th asymptotic xprssio for th wav fuctio abov, it is clar that th allowd valus of α fulfill αr= π( p+ /), whr p is a itgr Hc, th distac btw allowd valus of α is δα= π /R W ca thrfor writ th sum as π/ α α S( ω) = δ( Eg+ α ħωδα ) π sih( π / α) α Takig R mas δα ad so, covrtig to a itgral, w fid th simpl rsult 64

166 π / α α ( ω) = ( g+ ħ ) S δ E α ω dα π sih( π / α) π/ ħω Eg θ( ħω Eg ) =, 4π sih( π / ħω E ) g (9) whr θ is th uit stp fuctio Rvrtig to SI uits mas ritroducig ad Ry 3 Sic S has uits of lgth rgy w d to divid by a factor a Ry =ħ /( m ) / Ry 3 3 3/ B h writ th fial rsult as with usig th rlatio a Ry ad w ca th 3/ π/ pvc Ry mh 4 θ( ) Im χω ( ) = δ( / ) + + 3, εmω ħ sih( π / ) = ( ħω E )/ Ry g This formula is calld th 3D Elliott formula Basd o this rsult, th ral part ca b obtaid usig th Kramrs-Kroig rlatios ad broadig ca b itroducd through covolutio with a Lortzia broadig fuctio This lads to th xprssio 3/ ħ pvc m h χω ( ) = X3( w), 3/ πε m E ħ whr X ( w) 3 is ow th xcitoic suscptibility fuctio giv by [3] / 3/ πry E g 8 Ry ( w) Ry Ry Ry X3( w) ħ = + g 3 + g g ( w) ħ E E ( w) Eg w Eg w E ħ ħ + ħ g g( z) l( z) + ψ( z) +, ψ( z) : digamma fuctio, z Ry whr w= ω+γ i ad E= Eg is th rgy of th th boud xcito I Fig 9, th ral ad imagiary parts ar plottd for a 3D matrial with = 5 V ad varyig valus of th ffctiv Rydbrg ad broadig g B =ħ / To s that th Elliott formula has th corrct limitig bhavior if Coulomb ffcts bcom gligibl (if g ε bcoms vry larg) w should tak Ry ad accordigly W ot that m h E g a B 65

167 / π lim = sih( π / ) π Also, th cotributio from boud xcitos vaishs Hc, w fid th approximat xprssio p m Im χω ( ) = ω Egθ( ω Eg ) πε ω ħ ħ vc h m ħ 3/ This xprssio is idtical to th imagiary part of th idpdt-lctro rsult Eq(6) if vaishig broadig is assumd Figur 9 Th xcitoic suscptibility fuctio for a bulk smicoductor Th plots illustrat th ffct of varyig th strgth of th Coulomb itractio (lft pals) ad broadig (right pals) 9 Excitos i Quatum Wlls W will cosidr th cas of a lctro-hol pair cofid to a thi quatum wll It is assumd that light is polarizd alog th quatum wll pla so that th motio prpdicular to th pla (tak as th z dirctio) is ot xcitd As i th 3D cas, w thrfor rquir a vaishig ctr-of-mass momtum i th pla Hc, for th i-pla motio oly th rlativ part is rtaid ad th lctro-hol pair is charactrizd by a Hamiltoia (i polar coordiats) 66

168 ˆ ˆ ˆ d d d Hh= h( z ) + hh( zh ) + Eg dr r dr r dθ r z z + ( h ) (9) Hr, ˆ ( ) / h ( ) z = m d dz+ V z is th Hamiltoia for th z motio of th lctro cofid by th quatum wll pottial V ad hˆ ( z ) is th aalogous trm for th hol W will ow spcializ to th idalizd cas of a xcito cofid to a xtrmly thi quatum wll Th strog cofimt mas that z z ad so w will us th simplifyig approximatio (93) r + ( z z ) r This mas that th Hamiltoia bcoms a sum Hˆ ˆ ˆ ˆ h= h( z ) + hh( zh ) + H( r), whr H ˆ ( r ) dscribs th i-pla rlativ motio As a cosquc, th xcito wav fuctio is a product Ψ ( r, r ) = ϕ ( z ) ϕ ( z ) Ψ( r), whr is a igstat of h xc h h h ad similarly for th hol W assum that th z-motio of both lctros ad hols ar froz i th lowst igstats This amouts to rplacig th tru bad gap h by th ffctiv o Eɶ h g= Eg+ E+ E, whr E + E is th sum of lctro ad hol quatizatio rgis (i prvious chaptr w usd th otatio E for E ɶ ) Th ipla rlativ motio th lads to a purly two-dimsioal Wair quatio giv i polar coordiats by d d d Eɶ g Ψ( r ) Ψ ( r ) = EΨ( r ) dr r dr r dθ r This problm is mathmatically idtical to a hydrog atom i two dimsios Th s-typ boud stats (with o agular dpdc) ar quit similar to th 3D cas ad ca b writt h h h ϕ ˆ g h g E g Ψ ( r ) = L ( r /( + )), E = Eɶ + r /( + ) 3/ g π( + ) ( ) Hr, th pricipl quatum umbr is agai a itgr, but ow th allowd valus iclud zro, i =,,, Th most importat fatur of this rsult compard to th 3D cas is that th bidig rgy of th lowst xcito is ow 4Ry whras xcitos i bulk wr boud by o mor that Ry This is a dirct maifstatio of th icrasd lctro-hol ovrlap i low-dimsioal gomtris Th cotiuum stats ar also highly similar to th bulk cas ad foud to b 67

169 with a ormalizatio coditio that is ow iαr Ψ ( r) = N F( + i / α,; iαr), α R cosh( π / α) = π Ψ α( r) rdr= N R π/ α α This rsult corrspods to ormalizatio withi a circl of radius R Evtually, it follows that th wav fuctio at th origi is giv by, E= Eɶ boud stats E< Eɶ 3 g π( + ) ( + ) Ψ () = π / α α = ɶ g+ α, E E cotiuum E> Eɶ R cosh( π / α) g g Summig ovr th cotiuum stats lads to a rsult idtical to Eq(9) xcpt that sih is rplacd by cosh Fially, w arriv at a D Elliott formula dscribig th xcitoic absorptio i a ultrathi quatum wll of width d: / pvc m π h 4 θ( ) Im χω ( ) = ( /( δ ) ) + εm ω d ħ ( + ) cosh( π / ) Hr, = ( ω E ɶ ħ )/ Ry ad th rlatio a Ry =ħ /m has b utilizd Also, it is clar that i th limit of idpdt lctros w fid pvc mh Im χω ( ) = θ( ħω E ) g, ε m ω d ɶ ħ i agrmt with chaptr 6 Agai, broadig ca b itroducd ad th ral part ca b addd so that th full D xcitoic suscptibility bcoms with th suscptibility fuctio [4] g pvc mh χω ( ) = X ( w ), πε m Eɶ d B g h 68

170 Eɶ g 8 Ry ( w) Ry Ry Ry X( w) ħ = + f 3 + f f ( w) ħ ( ) E E ( w) Eg w Eg w E + ħ ɶ ħ ɶ + ħ ɶ g f ( z) l( z) + ψ( z+ ), ψ( z) : digamma fuctio, whr w= ω+γ i ad E = Eɶ Ry /( + ) is th rgy of th th boud xcito g I Fig 9, w hav plottd this rsult It is clarly s that xcitos lad to a complt rorgaizatio of th spctra A hug xcitoic rsoac locatd 4Ry blow th bad gap mrgs but also th cotiuum part of th spctrum is svrly modifid Figur 9 Th xcitoic suscptibility fuctio for a smicoductor quatum wll I th plots, th ffctiv bad gap is tak as 6 V Exrcis: Variatioal tratmt of quatum wll xcitos I this xrcis, w will rtur to th quatum wll Hamiltoia Eq(9) ad look for mor accurat solutios Primarily, w will ot apply th simpl approximatio Eq(93) Howvr, w will still look for solutios of th form Ψ ( r, r ) = ϕ ( z ) ϕ ( z ) Ψ( r) xc h h h Hc, th improvmt lis i fidig a bttr stimat for purpos w will us th variatioal mthod with ϕ a igstat of hˆ ad similarly for th hol Ψ( r ) ad for this 69

171 a) Show usig Eq(9) that th xpctatio valu for th rgy is d d d E= Eɶ g+ Ψ( r ) + V( r) Ψ( r ) dr r dr r dθ, with a ffctiv pottial V( r) = ϕ( z ) ϕh( zh ) dzdz r + ( z z ) h h, As a simpl xampl, w cosidr a quatum wll of width d with ifiit barrirs πz This mas that th igstats for th z-motio ar ϕ ( z) = si( ) b) Usig u= ( z z )/ d ad v= ( z + z )/ d show that Evaluatig th v itgral, w fid h h u u+ v ( ) u v 8 si si ( ) V( r) = dvdu d ( r / d) + u u d d h( u) V( r) = d ( r / d) + u du 3 with h( u) = ( u)[+ cos( πu)] + π si( πu) As a particular variatioal asatz, w will try th form Ψ ( r ) = αr π α c) Show that th asatz is ormalizd ad that th kitic rgy is α Us ths rsults to dmostrat that αr 8 r whr W ( u) = dr d ( r / d) + u α E= Eɶ g+ α + h( u) W( u) du, I ca b show that W ( u) = 4 πuα d{ π+ Y ( uαd) H ( uαd) }, whr Y ad H ar Bssl ad Struv fuctios, rspctivly If d is sufficitly small, this rathr 7

172 complicatd xprssio may b approximatd by th ordr xpasio W( u) 4α+ 8uα d d) Show that 8 E Eɶ g+ α 4α+ α d, 3 π α ad that by miimizig with rspct to that th optimal rgy E is giv by 4 E Eɶ g 8 + d 3 π I th plot blow, th xcito bidig rgy is plottd vrsus th width of th quatum wll, both i atural xcito uits Figur 93 Variatioal calculatio of th bidig rgy of xcitos i a smicoductor quatum wll Rfrcs [] H Haug ad SW Koch Quatum Thory of th Optical ad Elctroic Proprtis of Smicoductors (World Scitific, Sigapor, 993) [] IS Gradsty ad IM Ryzhik Tabl of Itgrals, Sris ad Products (acadmic Prss, Sa Digo, 994) [3] C Taguy, Phys Rv Ltt 75, 49 (995) [4] C Taguy, Solid Stat Commu 98, 65 (996) 7

173 Excitos i Naowirs ad Naotubs W hav s that cofimt i quatum wlls lads to hacd xcitoic ffcts i th optical rspos of smicoductors Th bidig rgy of th strogst boud xcitos icras by a factor of 4 i th idal D cas Cosqutly, o xpcts this trd to cotiu to D-structurs with v strogr bidig of xcitos As w will s, this is prcisly what happs v to th poit, whr xcitos compltly domiat th rspos Followig th prvious chaptr, w will limit ourslvs to variatioal calculatios of xcitos i D-structurs Ths structurs ar assumd to b ifiit alog th logaxis dirctio ad strogly cofiig i th two trasvrs dimsios Also, th systm is xcitd alog th log-axis so that ctr-of-mass momtum for this dirctio is to rmai zro throughout W tak th z-axis as th log-axis ad so th cofid lctro-hol pair is dscribd by th Hamiltoia ˆ ˆ ˆ d Hh= h( x, y ) + hh( xh, yh ) + Eg dz x x y y z ( h ) + ( h ) + () Hr, ˆ (, ) ( / / h ) (, ) x y = m d dx+ d dy + V x y is th Hamiltoia for th trasvrs motio of th lctro cofid by th pottial ad h is th aalogous trm for th hol I this xprssio, th ffctiv lctro ad hol masss should b tak i uits of th rducd lctro-hol pair mass, which is tak as th uit of mass I a purly variatioal tratmt, w attmpt to dscrib th xcito stat by th variatioal asatz Ψ ( r, r ) = ϕ ( x, y ) ϕ ( x, y ) Ψ( z) Th corrspodig xpctatio valu for th rgy is th V ˆh xc h h h h d E= Eɶ g+ Ψ( z) + V( z) Ψ( z) dz, () with Eɶ = E + E + E g g h ad a ffctiv pottial V( z) = ϕ ( x, y ) ϕ ( x, y ) h h h ( x x ) + ( y y ) + z h h dx dy dx dy h h (3) At this poit, it is istructiv to cosidr som spcific xampls of trasvrs cofimt: 7

174 A Rctagular aowir with trasvrs dimsios d d (Rf []): πx ϕ( x, y) = si si πy d d d B Hxagoal aowir with diamtr d (Rf []): 8π ϕ( x, y) 3 (6+ 5 π ) /4 / πx π( x 3 y) ( x 3 cos π cos cos + y) d d d d C Circular aotub with radius r (Rf [3]): ϕ(, ) ( πr x y = δ x + y y r) W ot that i th aotub modl, th wav fuctio is compltly localizd to th cylidr wall ad rathr tha givig th wav fuctio itslf, w giv th ormalizd squar I cass A ad B, th ffctiv pottial ca oly b computd umrically Howvr, for th aotub modl a aalytic rsult ca b foud To this d, w itroduc polar trasvrs coordiats x= ρ cosθ ad y= ρ siθ Du to th complt localizatio o th cylidr wall, w always hav ρ= = r I this mar, x x + y y = r ( h ) ( h ) 4 si (( θ θ )/) Hc, i polar coordiats, h V( z) = ( πr) = π π π π 4r si ( θ /) + z 4 4r = K π z z δρ ( r) δρ ( r) 4r si (( θ θ )/) + z h dθ h ρρ d ρ dθ dρ dθ h h h (4) I th last li, K is a so-calld complt lliptic itgral of th st kid dfid by K( x) π / x si dθ θ From this dfiitio, it is s that K() = π / ad th pottial approachs th bar D Coulomb pottial V( z) = / z i th limit r From Eq(3) it is clar that this must always b th limit of a D ffctiv pottial whvr th cofiig pottial bcoms sufficitly arrow that x xh ad y y h bcaus of th cofimt 73

175 Figur Effctiv Coulomb pottial for thr diffrt D cofimts compard to th bar pottial Th actual ffctiv pottials for th thr modls listd abov ar illustratd i Fig Wh compard to th bar Coulomb pottial, it is clar that th bhaviour as z is must lss sigular Hc, for modls A ad B, th sigularity is compltly rmovd ad for C, th sigularity is ow logarithmic istad of / z Howvr, i all cass th bar pottial / z is foud as a limit wh th diamtr of th aowir or tub bcoms vry small It might th b thought that a viabl ad simpl modl for D xcitos would rsult from usig th pur D pottial V( z) = / z i Eq() similarly to th D quatum wll cas To look at th proprtis of this simpl modl w d to start with a rgularizd pottial, howvr For this purpos, w will tak V( z) = /( z + c) as our pottial, whr c is a positiv costat, which should vtually b tak to zro This Loudo modl was origially aalyzd by R Loudo [4] Th form is mathmatically simplr tha ay of th altrativs abov As our variatioal asatz w will, as usual, try th xpotial ( z) α α z Ψ = Diffrtiatig twic lads to dψ ( z)/ dz = αψ( z) αψ() δ( z) Thus, th xpctatio valu Eq() is αz E= Eɶ g+ α 4α dz z+ c Evaluatig this itgral lads to yt aothr complicatd fuctio: th xpotial itgral Ei : αc E= Eɶ + α + 4 α Ei( αc) g 74

176 Howvr, as c should go to zro w ca us a xpasio basd o partial itgratio valid for small c E= Eɶ + α 4α g = Eɶ αz αz g+ α 4α l( z c) + 8α l( z +c) dz = Eɶ αz + α + 4α l( c) 8α l( z+ c) dz g Eɶ αz + α + 4α l( c) 8α l( z) dz g = Eɶ + α + 4 αγ ( + l( αc)), g αz dz z+ c (5) whr th itgral z β l( z) dz = ( γ + l( β))/ β with γ = 577 as Eulr s costat has b applid Diffrtiatig, o fids = α+ 4( γ+ l( αc)) + 4 With a fw maipulatios this coditio ca b rformulatd as α α xp = 4c + γ Th formal solutio to th quatio wxp( w) = x is somtims calld th product logarithm (pl), i w= for α is pl( x) Thus, th solutio l l + γ + γ, 4c 4c α = pl l + 4c γ 4 whr th scod xprssio is th xpasio for low c Accordigly, th rgy is E= Eɶ g 8pl 4pl + γ + γ 4c 4c As th plot of th product logarithm to th right shows, it is a mootoically icrasig fuctio thatt divrgs logarithmically as th argumt icrass Th xtrmly importat coclusio is this: as c gos to zro, w fid α ad E Hc, th wav fuctio bcoms compltly localizd to th poit z= This is ot a artfact of th variatioal approach bcaus th variatioal stimat for 75

177 th rgy is always highr tha th tru valu W thrfor coclud that th D Coulomb modl is pathological i that th groud stat collapss ad th groud stat rgy divrgs I Fig, w illustrat th bhaviour of th rgy as c bcoms smallr Figur Bidig rgy ad variatioal paramtr (ist) i th Loudo modl as a fuctio of th pottial cut-off s Excitos i Carbo Naotubs Ev though th pur D Coulomb modl is clarly uphysical, it is still corrct that th tru pottial for all ralistic modls approachs this strag situatio as th cofimt bcoms strogr Hc, v if actual xcito bidig rgis obviously do ot divrg, thy ca still grow xtrmly larg compard to bulk valus As a xampl of this, w ow cosidr th tru aotub pottial giv by Eq(4) With th sam xpotial asatz as abov, w fid th rgy xpctatio valu 3α r E= Eɶ g+ α + παj( αr) Y( αr) F 3,;,, ; 4α r π whr ad ar Bssl fuctios of first ad scod kid, rspctivly, ad J Y 3 is a gralizd hyprgomtric fuctio [5] Th similarity with th Loudo modl abov bcoms appart if w agai us partial itgratio to approximat to lowst ordr i r Usig th dfiitio of th lliptic itgral, w hav π / αz 8α E= Eɶ g+ α dzdθ π z + 4r si θ, F 76

178 Th idfiit itgral of / z + x is l( z+ z + x ) ad it follows that π / 8α αz αz E= Eɶ g+ α l( z z 4r si θ ) α l( z z 4r si θ ) dz dθ π π / 8α αz = Eɶ g+ α l(r si θ) α l( z z 4r si θ ) dz dθ π π / 6α αz Eɶ g+ α + 4α l( r) l( z) dzdθ π = Eɶ + α + 4 αγ ( + l( αr)) g Comparig to Eq(5), w s that r taks th plac of c W thrfor xpct to fid prcisly th sam bhavior as abov wh r gos to zro It should b otd that this similarity is obtaid v though th aotub pottial is actually (logarithmically) divrgt at th origi whras th Loudo modl pottial is fiit Th xcito bidig rgy for th aotub modl is illustratd i Fig 3 Figur 3 Bidig rgy ad variatioal paramtr (ist) of xcitos i carbo aotubs as a fuctio of aotub radius, all i atural xcito uits From Fig 3 it is appart that th xcito bidig rgy may bcom vry larg if th aotub radius is sufficitly small Th qustio is th: what ar th actual valus of a ad Ry? W rcall that a = 59Å εm / m ad Ry = B 36V mh / mε Hc, to aswr this qustio w d th rducd ffctiv lctro-hol pair mass ad a valu of th dilctric costat Th lattr is rlativly straight-forward sic most xprimts ar prformd i liquid suspsios ad a B h 77

179 rasoabl valu dscribig th scrig i this cas is ε= 35 [3] To comput mh w d to cosidr th bad structur For aotubs xcitd alog th log-axis th allowd trasitios ar btw bads symmtrically positiod abov ad blow th Frmi lvl I a simpl arst-ighbor tight-bidig modl, th trasitio rgy is giv by Eq(7) k ya kya 3kxa Ecv ( k) = γ + 4 cos + 4 cos cos, (6) whr γ 3 V is th hoppig itgral ad a = 46 Å is th lattic costat Th aotub is charactrizd by th chiral idics (, m) ad i trms of ths, th compots of th k-vctor ar Hr, k is th cotiuous log-axis compot of th k-vctor ad q is th quatizd short-axis compot giv by q= p / r, whr p is a itgr ad r is th radius Also, L= + m + m is th radius i uits of a/π, i π r= al A importat poit about th rgy disprsio Eq(6) is that ( ) Ecv K = with K= ( 3,) π /3a Thus, th bad gap is foud at th allowd k-poit closst to K To simplify th aalysis, w xpad th disprsio i th viciity of K ad fid ( ) 3 E cv k aγ k K W itroduc ˆl ad ŝ as uit vctors for th log-axis ad short-axis, rspctivly If w xprss K ad k i trms of th projctios alog ths dirctios w fid k= klˆ+ qsˆ ad similarly K= Klˆ+ Qsˆ, whr it ca b show that Q= ( + m)/3r Hc, sttig q = Q, lads to th coditio b fulfilld if ( + m)/3 th miimum diffrc bcoms trasitio rgy, i th bad gap 3( + m) m kx= k q L L m 3( + m) ky= k + q L L p= ( + m)/3 This coditio ca oly is a itgr i which cas th aotub is a mtal If ot, q Q = /3r I this cas, th miimum mi E ɶ g, th bcoms Eɶ g= 3 aγ q Q mi = aγ 3r To fid th ffctiv mass, w cosidr a approximatly parabolic disprsio 78

180 so that, to lowst ordr, th squar bcoms O th othr had, th squar of th rgy disprsio for th aotub ar th miimum is Hc, a compariso dmostrats that Cosqutly th ffctiv mass is proportioal to th bad gap Pluggig i th umbrs it turs out that E ɶ = 4 VÅ / r ad m / m= 9Å / r A xtrmly importat poit is th g that th ffctiv Bohr radius Ecv ( k) Eɶ ħ g+ ( k K) m E ( k) 3 aγ ( k K) q Q bcoms a liar fuctio of r giv by a B= 97r As a cosquc, th radius r masurd i uits of a B is always roughly! At this valu, th xcito bidig rgy as computd abov ad illustratd i Fig 3 is aroud -744 Ry A mor accurat calculatio [3] fids a bidig rgy of approximatly -8 Ry It is otd that ths valus ar substatially highr tha th maximum valu -4 Ry foud for D structurs I aalogy with th ffctiv Bohr radius, th ffctiv Rydbrg also dpds o r ad isrtig valus w fid Ry = VÅ / r It thrfor follows that th ratio btw xcito bidig rgy ad bad gap is a ar costat of aroud -4% This is obviously a hug valu, which will compltly rarrag th optical rspos As a xampl, w cosidr th (7,6) aotub with a radius of r = 44 Å For this structur, th xcito bidig rgy is th -39 V Ufortuatly, thr is o simpl way to sum all th cotributios to th optical rspos aalytically Istad, a umrical calculatio of boud ad uboud xcitos ca b mad usig a fiit basis st [6] Summig th diffrt trms lads to th spctrum show i Fig 4, whr th idpdt-particl rsult is icludd for compariso Th vry larg rdshift of th rsoac corrspods to th valu of th xcito bidig rgy Also, it is oticd that th pak is ow much mor symmtric tha th ivrs squar-root of th idpdt-particl spctrum h E ( k ) Eɶ + ħ ( k K) Eɶ / m cv g g h { + mi} cv h = Eɶ + 3 aγ ( k K) g m = ħ E ɶ /3aγ h a = 59Å εm / m B h g 79

181 Figur 4 Normalizd absorptio spctra of a (7,6) carbo aotub Th curvs show th spctra icludig ad glctig xcitoic ffcts, rspctivly Exrcis: p aotub xcitos I this xrcis, w will attmpt to comput th rgy of th p xcito i a aotub ad for this purpos th asatz Ψ ( z) = β z β is always orthogoal to th groud stat 3 z a) Show that Ψ( z) is ormalizd ad that th kitic rgy is β will b usd Not that it Th difficult part lis i dtrmiig a approximat xprssio for th Coulomb rgy valid for small but fiit r It is giv by 8 U= π π / z Ψ ( z) + 4r si θ dzdθ As a start, w will cosidr th dfiit itgral W= z f ( z) + x dz Th first fw idfiit itgrals of th squar-root ar dotd S (, ) z x so that 8

182 S ( z, x) z + x S ( z, x) S ( z, x) dz= l z+ z + x ( ) ( ) ( ) S ( z, x) S ( z, x) dz= zl z+ z + x z + x 3 S z x S z x dz z x z z x z z x 4 4 3(, ) (, ) = ( )l b) Udr th assumptio that f ( ) =, show by rpatd us of partial itgratio that W= f ()l x f ( z) S ( z, x) dz = f ()l x f () x+ f ( z) S ( z, x) dz ()l () () l ( ) 3(, ) = f x f x+ f x x f z S z x dz 4 At this stag, o approximatios hav b mad Howvr, to actually calculat th itgral, w will ow us th small-x xpasio S3( z, x) ( z x )l( z) (3 z + x ) 4 4 I th prst xampl, f is th fuctio f ( z) =Ψ ( z) = β z 3 βz ad so f z = z z f z = z+ z f z = z+ z 3 βz 3 βz 4 βz ( ) 4 β ( β ), ( ) 4 β ( 4β β ), ( ) 8 β (3 6β β ) With this form ad th approximat S ( z, x) 3 w fid 3 β β x f ( z) S3( z, x) dz { + γ+ lβ} I tur, W bcoms 3 β β x W + { + γ+ lβx} ad w th hav π / 8 β 3 U= + r si + + l( r si ) d π { γ βr} = r β 8β l( ) { } β θ γ β θ θ 8

183 3 Th total rgy is thrfor E = Eɶ + β β 8β r { + γ+ l( βr) } c) Show that th miimum rgy is approximatly This rsult is plottd blow p g E Eɶ 8r { + γ+ l r} p g Rfrcs Figur 5 Variatioal p xcito bidig rgy [] TG Pdrs, PM Johas ad H C Pdrs, Phys Rv B6, 54 () [] TG Pdrs, Phys Stat Sol (c), 46 (5) [3] TG Pdrs, Phys Rv B67, 734 (3) [4] R Loudo, Am J Phys 7, 649 (959) [5] IS Gradsty ad IM Ryzhik Tabl of Itgrals, Sris ad Products (Acadmic Prss, Sa Digo, 994) [6] TG Pdrs, Carbo 4, 7 (4) 8

184 Smicoductor Lasrs ad LEDs Lasrs ad light-mittig diods (LEDs) ar arguably amog th most sigificat optolctroic applicatios of smicoductors Ths dvics ar ctral i optical commuicatio, LED displays, lasr pritrs ad CD/DVD drivs I fact, lasrs ad LEDs ar ot fudamtally diffrt ad a lasr ru at a currt blow lasig thrshold will oprat as a LED I both dvics, spotaous missio is prst This light is diffus ad covrs a rlativly broad frqucy rag I a lasr, howvr, populatio ivrsio producs gai isid th light mittig matrial If th optical gai xcds th optical losss at som frqucy, light will b amplifid as it propagats alog th matrial As light of this particular frqucy is amplifid, th amout of stimulatd missio at th sam frqucy is gratly hacd sic th stimulatd missio rat is proportioal to th light itsity This, i tur, stimulats missio v furthr ad, vtually, arly all availabl rgy is chald ito light mittd with o spcific frqucy, dirctio ad polarizatio mod Thus, a prrquisit for lasig is gai ad i a LED th ijctd currt just is t ough to produc amplificatio ad, cosqutly, oly spotaous missio is obtaid I th prst chaptr, w will first ivstigat spotaous missio ad study th spctrum of th missio Scodly, w will cosidr th coditio for gai i bulk smicoductors ad quatum wlls Th structur of lasrs ad LEDs is similar to p juctio diods discussd i chaptrs 9 ad Hc, w cosidr a structur comprisd of two halvs: to th right th -dopd half havig a xcss of lctros ad to th lft th p-dopd half cotaiig xcss hols A sstial poit i this rspct is that lasrs ad LEDs ar ot i quilibrium This mas that th populatio of lctros i diffrt stats dos ot follow th usual statistical mchaical Frmi distributio Th raso is that w hav placd th dvic i a circuit ad kp ijctig lctros from th right ad xtractig lctros (ijctig hols) from th lft d of th juctio This costat supply of carrirs drivs th dvic out of quilibrium Cosidr ow th -dopd half, i which additioal lctros ar ijctd If all th lctros hr thrmaliz, i stablish a local quilibrium amog thmslvs, this amouts to pushig up th Frmi lvl locally Similarly, addig hols to th p-sid amout to a lowrig of th Frmi lvl thr Hc, udr th assumptio of sparat quasi-quilibria o both sids th lctro coctratio i th -sid ad th hol coctratio p i th p- sid ar charactrizd by spcific lctrochmical pottials dotd ad µ, rspctivly Clarly, i tru quilibrium w would hav µ = µ = E h F µ h Th actual o-quilibrium situatio is illustratd i Fig This figur shows th bad dgs U ad U i th juctio ad th positio of th lctrochmical pottials i cass c v with low (a) ad high (b) ijctio Pal (a) corrspods to th LED rgim, for which ijctio is too small to achiv populatio ivrsio Pal (b) is th lasr situatio, whr th lctrochmical pottials ar pushd ito th bads ad a substatial dsity of xcss lctros ad hols is prst 83

185 Figur Biasd p juctio i o-quilibrium Th rd arrows idicat lctro-hol rcombiatio across th gap I (a), th ijctd currt is small ad w ar i th LED rgim I (b), w ar i th lasr rgim for which th lctrochmical pottials may b pushd ito th bads A spcial situatio ariss i th itrfac whr th two halvs mt Hr, xcss lctros i th coductio bad hav a chac to rlax dow ito mpty stats i th valc bad This rlaxatio procss is oft rfrrd to as lctro-hol rcombiatio If th rgy lost by th lctro is giv off i th form of radiatio, th dvic is a light-mittig diod Th miimum rgy loss is th bad gap rgy E ad w thrfor xpct missio with a spctrum ctrd slightly abov E If g th probability of radiativ missio is assumd idpdt of th iitial ad fial rgy of th lctro, a simpl calculatio of th spctrum ca b do To this d, cosidr a lctro jumpig from th coductio bad ito th valc bad whil mittig a photo with rgy qual to that lost by th lctro If th fial rgy of th lctro i th valc bad is E ad th mittd photo rgy is ħω, it follows that th iitial lctro rgy of th trasitio must b E+ħω Th umbr of mittd photos will b proportioal to th umbr of rcombiig lctro-hol pairs Thus, w ca valuat th itsity distributio as th itgral Uv I( ω) ħω p( E) ( E+ ħω) de, Uc ħω whr th fial lctro rgy E is rstrictd to th valc bad E U v Not that this simplifid modl compltly igors slctio ruls i that all trasitios ar qually probabl Th lowr limit i th itgral is drivd from th rquirmt that g 84

186 th iitial rgy lis i th coductio bad, i Uc E+ħ ω I this itgral, p( E) should b udrstood as th umbr of hols i th itrval [ E, E+ de] ad similarly for th lctros With a lctrochmical pottial, th Frmi fuctio for th lctros is giv by µ f( E) = xp{( E µ )/ kt} + () Th lctro dsity pr uit rgy is giv by this fuctio, valuatd at a rgy E+ħω, tims th dsity of stats i th coductio bad Hc, i a bulk smicoductor th rsult is ( E+ ħω) = 3/ m E+ ħω U c 3 πħ xp{( E+ ħω µ )/ kt} + takig th lctro dsity of stats from Eq(84) I th LED rgim, th positio of µ surs that ( E+ ħω µ )/ kt ad so 3/ m ( E + ħω) E ω U xp{ ( )/ } 3 c E ω µ kt π + ħ + ħ ħ Th probability that a lvl is occupid by a hol must qual o mius th probability that th lvl is occupid by a lctro Hc, with a lctrochmical pottial th hol Frmi fuctio is µ h fh( E) = = xp{( E µ )/ kt} + xp{( µ E)/ kt} + h h () Usig argumts similar to thos usd for lctros, th hol dsity pr rgy itrval is 3/ mh p( E) U xp{( )/ } 3 v E E µ h kt πħ Takig th product of lctro ad hol dsitis w fid th followig rsult for th mittd spctrum Uv ħω { } v kt Uc ħω ħω π { } ( E) g I( ω) ħω xp U E E+ ħω U de = ħω xp ħω θ( ħω Eg ) kt 8 c (3) 85

187 Th stp-fuctio hr idicats that th rgy of th mittd light must b at last th bad gap rgy, as xpctd Th spctral shap basd o this rsult is dpictd i Fig for a GaAs basd LED opratig at room tmpratur Gai i Smicoductor Lasrs Figur LED spctrum at room tmpratur If th ijctd currt is sufficitly high it may lad to gai isid th smicoductor I this situatio, light is amplifid as it travls alog rathr tha big partially absorbd as it ormally would To udrstad th coditios for this spcial situatio, w d to dscrib th propagatio of light i a matrial If th complx rfractiv idx is + i th complx amplitud of a wav propagatig i th positiv z dirctio volvs as whr λ is th wavlgth Th itsity of th wav is proportioal to E( z) giv by r i { π ( ) } E( z) = Axp i r+ ii z λ 4 π { } E( z) = A xp iz λ It is s that th bhavior of th itsity is dtrmid by th imagiary part of th rfractiv idx : If > th fild is dampd ad if < th fild is amplifid i i As χ / it follows that th sig of is giv by th sig of χ th imagiary i= i r i r part of th suscptibility ( is always positiv i smicoductor lasr matrials), i i 86

188 Hc, th coditio for gai, which is a prrquisit for lasr actio, is that χ< To ivstigat this coditio w rtur to th suscptibility xprssio I th prst discussio, w wish to iclud a fiit tmpratur ad so th appropriat xprssio is i i which f νµ ( k ) ( ) D ħ p S cv µν fνµ k ε mω µν,, k Eµν ( k) Eµν ( k) ħ ( ω+γ i ) 4 χω ( ) = ( ) is ow th gralizd occupatio factor giv by, (4) This fact that f ( ) f ( k) = f ( E ) f ( E ) is th lctro Frmi factor is usd to obtai this rsult It is s that for th usual cas of matrials i thrmal quilibrium at low tmpraturs with ithr mobil lctros or hols w ll hav f ( k ) i agrmt with th rsults of th last two chaptrs W wish to simplify th discussio by takig th broadig to zro, i Γ= Furthrmor, w will focus o th imagiary part of th suscptibility xclusivly Th startig poit is thrfor to xtract th imagiary part of Eq(4) i th followig mar: Usd i Eq(4) w fid that h E ν k νµ h ν k µ k µν νµ = lim Im = lim Im + Γ Γ Eµν ( k) ( ω i ) E ( k) µν Eµν ( k) ( ω i ) Eµν ( k) ( ω i ) ħ +Γ ħ +Γ + ħ +Γ lim ħγ ħγ = Γ Eµν ( k) ( Eµν ( k) ω) + Γ ( Eµν ( k) + ω) + Γ ħ ħ ħ ħ π = δ( Eµν ( k) ħω) E ( k) π p ( ) cv D fνµ k Sµν Eµν k mω µν,, k χi ( ω) = ( ) δ ( ) ω ε ω ( ħ ) (5) It is this gral but simpl rsult w ow wish to valuat for smicoductors of various dimsios just as w did for th ordiary suscptibility i th quilibrium situatio 87

189 W bgi by lookig at bulk smicoductors, for which siz quatizatio is abst ad stats ar labld by k alo Th suscptibility abov th bcoms π p cv 3 χi( ω) = f ( k) ( E ( k) ) d k 3 νµ δ µν ω ε mω ( π) ħ To valuat th itgral, w rly o th tchiqu usd i th prvious chaptr ad itroduc x=ħ k m Now, i trms of x w hav / h ħ k mh ħ k m h E = Ec+ = Ec+ x, E = Ev = Ev x, E ( k) Eg x µ k νk µν = + m m m m h h Thus, upo chagig th itgratio variabl from k to x w fid 3/ pcv mh i( ) h v c g πεmω ħ 3/ pcv m h πεmω ħ mh mh { ( m ) ( )} ( ) h m ħ χ ω = f E x f E+ x δ E + x ω xdx mh mh { fh( Ev m ħω E ) ( )} ( ) h g f Ec m ħω E g ħω Egθħω Eg = ( ) + ( ) Th two lctrochmical pottials ar ot compltly idpdt bcaus w xpct partial utrality If w assum that th lctro ad hol dsitis ar qual, th th rsults of chaptr 8 for a 3-dimsioal smicoductor show that, as far as w ar i th low-ijctio rgim, 3 m µ + µ h= Ec+ Ev+ kt l m h imagiary part of th suscptibility is always o-gativ as show i th figur This corrspods to th usual matrial absorptio, for which th spctrum follows a squar root ħω E g bhavior as drivd i th prvious chaptr Wh µ is pushd abov µ h W us this rlatio blow v if w ar ot strictly i th low-ijctio rgim to dtrmi oc µ has b spcifid For GaAs at room tmpratur, th righthad sid of this rlatio is approximatly 58 V if E v= Th spctra for four cass covrig low, modrat ad high ijctio ar show i Fig 3 I th lowijctio limit, is wll blow th coductio bad dg E I this cas, th µ c E c a rgio of gativ absorptio or, quivaltly, positiv gai dvlops This rag bgis prcisly at th bad gap ad th dpoit ca b foud by sttig th gai xprssio to zro, which corrspods to th coditio 88

190 Hc, th spctral gai rag is mh mh ( m ħω ) ( m ħω ) f E ( E ) = f E+ ( E ) h v h g c g ħω = µ µ E ω µ µ This clarly shows that at last o of th lctrochmical pottials must b pushd ito th bad i ordr to achiv gai Th coditio for gai f > f is prcisly that of populatio ivrsio: Th populatio of lctros i th coductio bad must xcd that of th valc bad g< ħ < h h h Figur 3 Spctra of th imagiary part of th suscptibility for diffrt lctro lctrochmical pottials Positiv ad gativ valus corrspod to absorptio ad gai, rspctivly Nxt, w tur to a smicoductor quatum wll for compariso Th calculatio procds prcisly as for th suscptibility i th prvious chaptr Covrtig th sum ovr th two-dimsioal k to a itgral, Eq(5) yilds π p () cv χi ( ω) = S f ( k) ( E ( k) ) d k µν νµ δ µν ω ε m d( π) ω ħ µν, Writig agai Eµν ( k ) = Egµν+ x w hav π pcv πmh m d( ) ħ χi( ω) = ε π ω µν, () µν h mh mh { h( v ν m ) ( )} ( ) h c µ m δ gµν ħω f E + E x f E+ E + x E + x dx S 89

191 Th fial rsult is thrfor pcv mh εmħω d µν, ( Eg ) χi( ω) = S ħ () µν θ ω µν h mh mh { fh( Ev Eν m ħω E ) ( )} h gµν f Ec Eµ m ħω E gµν + ( ) + + ( ) I Fig 4, w plot th gai spctra for th quatum wll similar to thos of th bulk matrial W hav cosidrd oly th lowst subbad trasitio ν = µ = i th plot ad tak th bad dg as E g= 8 V To rlat th lctrochmical pottials w ow us µ µ m h h + h= Ec+ E+ Ev+ E+ kt l m Th most strikig diffrc btw th D ad 3D cass is th abrupt jump at th bad dg i th D cas This diffrc simply rflcts th diffrc i dsity of stats: For 3D ad D cass, th dsity of stats varis as a squar-root ad a stp fuctio of th rgy masurd from th bad dg, rspctivly Hc, th stp profil i Fig 4 is asily xplaid If a quatum wir wr cosidrd, a v mor abrupt ivrs squar-root bhavior would hav rsultd Figur 4 Absorptio/gai spctra for a quatum wll assumig diffrt valus of th lctro lctrochmical pottial Thus, th lasr gai dirctly rflcts th dsity of stats I ordr to rduc th ijctio thrshold for lasig i a giv smicoductor structur it is obviously crucial that gai is maximizd As a cosquc, lasig i low-dimsioal structurs is achivd at lowr ijctio lvls ad mor fficit lasrs ca b producd from ths structurs At prst, xtrmly fficit quatum wll lasrs ar producd ad srious fforts ar put ito dvlopig lasrs i quatum wir ad 9

192 v quatum dot structurs i ordr to xploit thir pottially larg gai Morovr, th low-dimsioal structurs may b usd as wavguids for th optical fild that will focus th itsity i th gai rgio This will lowr th lasig thrshold v furthr Exrcis: LED pak wavlgth It ca b show, that for a D-dimsioal matrial, th LED spctrum is giv by th followig gralizatio of Eq(3): ħω { }( g) Th diffrt spctra ar illustratd i Fig 5 To simplify th aalysis, w itroduc a ormalizd photo rgy z=ħω / kt ad th ormalizd bad gap g= E / kt Not that g I this mar, th spctrum ca b writt g a) Show that for a thr-dimsioal LED th spctrum paks at a ormalizd photo rgy z giv by D I( ω) ħω xp ħω E θ( ħω Eg ) kt { }( ) D I( z) z xp z z g θ( z g) 3+ g+ 9+ g + g z= g+ b) Rpat th qustio for a two-dimsioal LED ad show that hr + g+ 4+ g z= g+ Figur 5 LED missio spctra for 3, ad dimsioal cass 9

193 Solar Clls Th solar cll combis svral aspcts of both optical ad lctric proprtis of smicoductors Th optical aspcts ivolv, obviously, absorptio of solar radiatio But mor subtl issus such as radiatio producd by lctro-hol rcombiatio ar importat as wll Th lctric proprtis ar qually sigificat as th bulk of th cll is a p juctio ad th trasport of carrirs is clarly of grat importac for th cll I this chaptr, w ivstigat th idal solar cll This mas that all imprfctios of ral solar clls that could, i pricipl, b limiatd ar assumd limiatd Nvrthlss, thr ar strict limits to th fficicy of th solar cll Th startig poit is th flux of photos from th su Th su is approximatly a black body radiator with a surfac tmpratur of T 58 K corrspodig to ktsu giv by, whr E is th photo rgy ad h= πħ is Plack s costat Th radiatio sprads out as it travls from th su, ad o arth (outsid th atmosphr) w rciv a flux () ( E) giv by 6 whr R ad R is th radius of th su ( 696 m) ad th radius of arth s orbit 9 aroud th su ( 5 m) W thrfor fid that th umbr of photos pr ara ad tim rachig us is whr 5 V Th photo flux (photos mittd pr ara tim rgy) is th su ζ( x) E E 3 () () Rsu 4 π( ktsu ) ζ(3) NE = E ( E) de= 3, R h c is th Rima zta fuctio As ach photo carris a rgy E th itsity of th solar radiatio bcoms πe su( E) = 3 E/ kt su h c R R πe E E, R R h c () su su E ( ) = su( ) = 3 E/ ktsu E E E su () () E E R π ( kt ) I = ( E) EdE= R 5 4 su su 3 E 5h c Pluggig i umbrs, w fid 9

194 N = 64 photos/m s, I = 38 Watt/m () () E E I rality, th su is oly approximatly a black body radiator Morovr, th ifluc of th atmosphr o arth mas that a strogly modifid spctrum is rcivd Th flux corrspodig to th actual spctrum is dotd ( E E ) ad looks as show i Fig Th spctrum rcivd o arth wh th su is at a agl of 48 dgrs rlativ to th ormal is calld th AM5 spctrum [] Numrical itgratio rvals that for this spctrum N E 43 photos/m s, I Watt/m E () Figur Exprimtal solar spctra abov ad blow th atmosphr Th black li is th black body Plack distributio Ultimat Efficicy A practical solar cll cosists of smicoductor matrial charactrizd by a bad gap I ordiary clls, oly a sigl layr is usd but i mor advacd tadm solar clls, svral diffrt layrs ar combid with th highst bad gap matrial o top, as illustratd i Fig Th layrs ar dopd to form p juctios that produc built-i lctric filds, which srv to dissociat th photo-xcitd lctro hol pairs If a solar cll is hld at a tmpratur of K, o rgy is lost to radiativ rcombiatio A calculatio of th fficicy i this situatio lads to th so-calld ultimat fficicy Idally, if vry photo is absorbd, this might lad to a fficicy of % Howvr, immdiatly aftr absorptio, a lctro is a hot carrir with a rgy highr tha th coductio bad dg A crtai tim is 93

195 dd bfor th lctro is collctd i a cotact ad durig this tim, th lctro is practically crtai to los this xcss rgy to hat (phoos) bcaus this procss taks plac o a xtrmly short tim scal As a cosquc, th usabl lctro rgy is rducd to that of th bad dg A aalogous procss applis to th hols ad, i tur, th usabl rgy of ach lctro-hol pair quals th bad gap Hc, th bad gap should b larg to maximiz this rgy Howvr, with a larg gap oly a tiy fractio of th solar spctrum will b absorbd E g Figur Schmatic illustratio of a ordiary sigl-layr solar cll (lft) ad a two-layr tadm solar cll (right) Followig this li of thought, th ultimat fficicy for a sigl-layr dvic is η ( K ) E ( E) de ( E) EdE () Eg = g E E Not that th lowr limit of th itgral i th umrator is illustratd i Fig 3 E g Th rsult is Figur 3 Ultimat fficicy of a idal solar cll opratd at zro tmpratur simulatd usig ral (rd) ad black body (black) solar spctra 94

196 Usig th Plack distributio, a maximum of 44% is foud at a bad gap aroud V, which is vry clos to th valu i silico To icras fficicy, multi-layr clls ca b costructd For xampl, a two-layr tadm dvic (Fig ) cosists of two diffrt matrials with th top layr havig th highst bad gap ad th bottom layr havig a lowr valu bcoms ( K ) I this structur, th ultimat fficicy As show i Fig 4, a maximum fficicy of aroud 6% is obtaid with = E gh η 76 V ad = 64 V With v mor layrs, th fficicy icrass furthr ad vtually rachs % for ifiitly may layrs = E gl E ( E) de+ E ( E) de Egh Egh gh E gl E E Egl ( E) EdE E gh E gl Figur 4 Ultimat fficicy of a idal tadm solar cll opratd at zro tmpratur Th maximum fficicy of 64 % is foud for low ad high bad gaps of 76 ad 64 V, rspctivly It ca b otd that bad gaps of 76 ad 64 V match quit closly th valus of th dirct bad gaps i G ad GaAs, rspctivly 95

197 Shockly-Quissr Limit A solar cll at a tmpratur caot attai th ultimat fficicy for a simpl raso: th icrasd umbr of lctros ad hols producd by absorptio will icras th umbr of rcombiatio vts Th smallst possibl rat is obtaid wh oly radiativ rcombiatio occurs Hc, rgy is lost as thrmal radiatio mittd by th cll This cas, which was first aalyzd by Shockly ad Quissr [], lads to th maximum fficicy for a cll at a tmpratur T I th litratur, it is somtims rfrrd to as th radiativ or dtaild balac limit Th usabl powr i th ultimat fficicy calculatio is drivd i trms of optical proprtis It has, howvr, a qually simpl physical itrprtatio i lctrical trms Hc, i th idal limit ach absorbd photo producs o lctro-hol pair This mas that th cll producs a lctric currt dsity, which is just th absorbd photo flux tims th lmtary charg This maximal currt, which is calld th short circuit currt Tcll ( K ) J sc, is thrfor giv by cll ( K ) J ( E) de It follows from Eq() that for a zro bad gap matrial with prfct absorptio a currt dsity of N E= 69 ma/cm ca b producd O th othr had, ach lctro-hol pair carris a rgy E or, quivaltly, a voltag of E / This ( K ) voltag would b th op circuit voltag of a idal cll at K, i V = E / ( Thus, th powr producd by th idal solar cll is simply th product K ) ( K ) J V This product obviously quals th umrator i Eq() as it should W ow tur to calculatig th rgy lost by radiativ rcombiatio isid th solar cll To aalyz this ffct, w first cosidr th situatio, i which th cll is ot illumiatd I this cas, th product of lctro ad hol dsitis is simply, i th squar of th itrisic dsity, cf chaptr 9 Th rat of radiativ rcombiatio vts is proportioal to th product of lctro ad hol dsitis so that R rad R rad = B i This rgy is mittd as thrmal radiatio charactrizd by a Plack distributio with a tmpratur I this xprssio, β( E) sc T cll = g Eg followig th xprssio 4 E cll( ) = β( ) π 3 E/ kt cll E E h c is th missivity for photos of rgy E Morovr, th factor of 4 (istad of ) accouts for th fact that th plaar cll has two surfacs (frot ad back) I our modl of a solar cll with prfct absorptio abov th bad E oc g sc i g oc 96

198 gap E g ad prfct trasparcy blow, th missivity is simply β( E) = θ( E E g ) By balacig rcombiatio vts ad thrmal missio w thrfor fid that R rad 4πE = 3 h c Eg E/ ktcll de 4πkTcllEg Eg / ktcll J, J 3 h c (3) Hr, th fact that has b usd Now, wh th cll is illumiatd, a bias V dvlops across th cll I th cas, th carrir product icrass from V / ktcll i Accordigly, th total radiativ rcombiatio currt bcoms ( Th total currt is th K ) ( K ) J= J J It is covit to writ J = J J so that It is clar that J() = Jsc is th actual short circuit currt at Tcll Similarly, w ca fid th op circuit voltag from th zro-crossig of th I/V charactristic: Usig simpl maipulatios it follows that J sc J( Voc ) = Voc= ktcll l + J 3 h c Voc= Eg+ ktcll l ( ) E E de 4πkTcllE g E g Th quatity isid th curly brackts is much lss tha uity ad so th corrctio is gativ, i V E Th tmpratur dpdc of th op circuit voltag is show i Fig 5 It is appart, that it dcrass roughly liarly with tmpratur ad for = V th op circuit voltag has droppd to 93 V at room E g tmpratur oc E g kt g cll sc Jrad rad = J V / kt cll V / kt { cll } J( V ) = J J sc sc sc i to 97

199 Figur 5 Op circuit voltag as a fuctio of tmpratur ad bad gap Th I/V charactristic ca b writt i th simpl form { V / kt V / kt } oc cll cll J( V ) = J (4) This I/V charactristic is illustratd i Fig 6 Th powr pr ara that is xtractd from a cll opratig at a crtai voltag V is giv by P= J( V) V Th optimum workig poit (max powr poit) is foud by diffrtiatig dp/ dv= ad th voltag at this poit is th max poit voltag V mp Figur 6 Illustratio of th I/V charactristic Th maximum powr is xtractd wh th cll voltag is at th max poit V mp 98

200 Usig Eq(4) w fid V V J{ } J = + = kt cll kt cll V / mp / cll mp / cll mp / oc kt V kt mp V kt mp V kt cll cll Voc / ktcll Th solutio is giv i trms of th product logarithm pl (s chaptr ): V kt { pl Voc ( / ktcll+ ) } cll mp= This rsult for th powr dsity is usually writt i trms of th fill factor FF J( Vmp ) V Pmp= J( Vmp ) Vmp= FF Jsc Voc, FF J V sc oc mp (5) I tur, th actual fficicy bcoms J η= η FF J V ( K ) sc oc ( K ) sc E g (6) I Fig 7, th tmpratur dpdc of th fficicy is illustratd for thr charactristic valus of th bad gap Figur 7 Solar cll fficicy as a fuctio of tmpratur ad bad gap It is s that th fficicy drops roughly liarly with tmpratur i th rlvat rag Also, it is clar that at high ough tmpraturs, a highr bad gap bcoms favorabl For xampl, th fficicis corrspodig to bad gaps of ad 5 V 99

201 cross aroud a tmpratur of 4 K Fixig th tmpratur at room tmpratur T = 3 K, th fficicy as a fuctio of bad gap is illustratd i Fig 8 A cll maximum of 33% is foud at a bad gap of 35 V Figur 8 Room tmpratur fficicy as a fuctio of bad gap calculatd i th Shockly-Quissr limit Exrcis: Tadm solar cll i th Shockly-Quissr limit I this xrcis, w will study a tadm solar cll at fiit tmpratur Th total voltag across th dvic is V ad th voltag drops across th low ad high gap rgios ar V ad, rspctivly, such that V + V= V h Vl h l a) Show that th radiativ rcombiatio currts gratd i th highr gap frot ( cll h ) ( ad th lowr gap back cll l ) ar J rad J rad 4πkT ( ) ( ) / ( ) clle h h Vh ktcll h gh Jrad= J, J 3 h c 4πkT ( l) ( l) V / ( ) cllegl Egl / kt l ktcll l cll Jrad= J, J 3 h c Egh / ktcll To comput th total currts, it is importat to raliz that ach layr illumiats th othr layr Hc, half of th rcombiatio radiatio of th frot cll is mittd i th dirctio of th back cll ad vic vrsa Hc, th total currt i th high gap layr is

202 πe h E( ) 3 E/ ktcll h c Egh Egh J = E de+ de J ( h) Vl / ktcll ( h) Vh ( ) / ktcll E E de+ J J Egh Vl / ktcll ( h) Vh / ktcll b) Show that th currt gratd i lowr gap back cll is Egh πe l E( ) 3 E/ ktcll h c Egl Egh J= E de+ de J E gh ( h) Vh / ktcll ( l) Vl ( ) / ktcll E E de+ J J Egl Vh / ktcll ( l) Vl / ktcll Th powr gratd by th tadm cll is v V kt l, h l, h / cll P= Vh Jh+ Vl Jl W itroduc c) Show by diffrtiatig that th coditios for obtaiig maximum powr ar Jh vl ( h) v J h l vh ( h) vl Jh+ vh + J =, Jl+ vl + J = v v Ths simultaous quatios rduc to h ( h) vl vh ( h) vh E( E) de+ J { + vl } J { + vh} = Egh E gh ( h) vh vl ( l) vl E( E) de+ J { + vh } J { + vl} = Egl Thir solutio ca oly b obtaid umrically Howvr, from ths solutios th fficicy is immdiatly calculatd Th rsult for room tmpratur coditios is illustratd i Fig 9 Th corrspodig aalysis for mor tha two layrs ca b foud i Rf [3] l

203 Figur 9 Room tmpratur fficicy of a tadm solar cll Rfrcs [] [] W Shockly ad HJ Quissr, J Appl Phys 3, 5 (96) [3] A D Vos, J Phys D3, 839 (98)

204 3 Photoic Bad Gap Structurs Th prvious chaptrs hav dalt with th microscopic backgroud for th suscptibility of matrials Th suscptibilityχ dtrmis th rfractiv idx = +χ ad thrby th optical proprtis of th matrial Sic thr is this lik btw th lctroic structur ad th optical proprtis w ca to som xtt dsig th optics by mas of th atomic costituts of th matrial I th discussio so far, howvr, w hav focusd o homogous matrials If w wat v gratr flxibility i th dsig w should go to ihomogous matrials That is, w should pic togthr rgios of diffrt matrials i ordr to produc brad w composit matrials with proprtis that ar tirly diffrt from thos of th costituts Th simplst such xampl is a stack of altratig layrs of two optically diffrt matrials As w shall s, th proprtis of such a stack ar radically diffrt from a homogous matrial To hav a prooucd ifluc o th optical proprtis, th altratio of th matrials should happ o a scal comparabl to th optical wavlgth I a layrd structur, for istac, th thickss of th diffrt layrs should b lss tha a wavlgth ad grally th typical lgth scal is aroud a fw hudrd aomtrs A spcially importat applicatio of such artificial optical matrials is as rflctors Typically, mtals ar usd as rflctors (basically mirrors) but mtals ar ot prfct rflctors, i particular ot for small wavlgths A diffrt problm is that mtals ar highly absorbig matrials that caot b usd with high powr optical filds i g lasrs bcaus thy ar dstroyd by th hatig Fially, mtallic rflctors caot asily b dsigd to srv mor gral purposs For istac, you might wat a structur that rflcts 95% at a particular wavlgth but oly 5% at a arby wavlgth Such a clvr mirror ca oly b mad usig artificial matrials To s how, w should thik about bad structurs of ordiary matrials such as GaAs show i Fig 8 Th bad structur is charactrizd by altratig rgios of allowd rgy bads ad forbidd rgy gaps If a lctro has a rgy lyig isid o of th gaps, it caot propagat isid th matrial This mas that if such a lctro is icidt o a pic of th matrial, it will b rflctd back Th ida bhid photoic bad gap (PBG) structurs is to xploit this ida for optical filds Hc, th priodic pottial that is rsposibl for th bad structur of lctros is mimickd by a priodically varyig rfractiv idx By a propr dsig of th uit cll, optical filds of particular wavlgth, propagatio dirctio ad polarizatio will b prvtd from propagatig i th structur Thy ar, thrfor, cssarily rflctd if th structur is composd from o-absorbig costituts I th prst chaptr, w study th gral pricipls bhid th dsig ad proprtis of PBG structurs For simplicity, th discussio is rstrictd to o ad two dimsioal PBG structurs For ths cass, th rfractiv idx varis alog oly o ad two dirctios, rspctivly 3

205 Figur 3 A simpl o dimsioal PBG structur with altratig layrs of diffrt thickss ad rfractiv idx 3 O Dimsioal PBG Structurs A priodic arragmt of dilctric slabs ca clarly b vry complicatd Howvr, th PBG ffct appars v i th simpl doubl-layr arragmt i Fig 3 Hr, th priod cosists of two slabs: o of thickss a ad rfractiv idx a ad aothr of thickss b ad rfractiv idx Th total priod is Λ= a+ b Th startig poit for th aalysis is th wav quatio for th lctric fild W tak th z-axis as th dirctio, i which th rfractiv idx varis Furthrmor, w mak th simplifyig assumptio that th fild propagats alog z, i prpdicular to th slabs Hc, th polarizatio vctor of th fild is paralll to th slabs ad, thrfor, th fild is tirly tagtial I this cas, th wav quatio for th amplitud of th fild rads as E( z) + ( z) ke( z) =, z whr k= π / λ= ω / c is th fr-spac wav umbr ad ( z) is th z-dpdt rfractiv idx Th ky to solvig this quatio is th Bloch thorm, wll kow from lctro wav fuctios i priodic solids Th thorm says that for a priodically varyig pottial th wav fuctio ϕ( z) satisfis th coditio ϕ( z+λ ) = ϕ( z) ik Λ Λ I prfct aalogy, th lctric fild satisfis E( z+λ ) = E( z) ik, whr k is a wav umbr lablig a particular solutio ad rstrictd to th Brilloui zo π / Λ k< π / Λ Now, isid slabs of typ a ad b havig rfractiv idics ad, rspctivly, th complt solutio is a b Axp( ikaz) + Bxp( ikaz), < z< a E( z) = C xp( ikbz) + Dxp( ikbz), b < z <, whr ka, b= a, bk W d to dtrmi th cofficits A, B, C ad D ad for this purpos w rquir cotiuity of th fild ad its drivativ at z = ad z= a Wh cosidrig th lattr poit z= a, w should kow th fild to th right of this b 4

206 poit This fild is foud dirctly from th fild spcifid abov togthr with th Λ coditio E( z+λ ) = E( z) ik Thus, th four boudary coditios ar A+ B= C+ D { } Axp( ik a) + Bxp( ik a) = C xp( ik b) + Dxp( ik b) xp( ikλ) a a b b k ( A B) = k ( C D) a { } { } k Axp( ik a) Bxp( ik a) = k C xp( ik b) Dxp( ik b) xp( ikλ) a a a b b b Th simplst way of dalig with this st of quatios is to rwrit thm i th form M { A, B, C, D} =, whr M is a 4 by 4 matrix cotaiig th cofficits of A, B, C ad D i th quatios abov For o-trivial solutios w rquir dt M=, which radily lads to th coditio b ka+ kb cos kλ= cos kaa cos kbb si kaasi kbb k k a b a+ b = cos( aka)cos( bkb) si( aka)si( bkb) a b (3) What dos this quatio tll us? Wll, th lft-had sid clarly oly ivolvs th wav umbr k Th right-had sid, o th othr had, dos ot ivolv k but rathr k =ω / c Hc, w may viw this quatio as a rlatio btw th wav umbr k of a particular fild mod ad th frqucy ω of this mod A st of solutios to th quatio is asily obtaid: W simply fid k from kλ= cos (right-had sid)! This provids us with k as a fuctio of frqucy If w th ivrt this st of data w gt frqucy as a fuctio wav umbr, just as a rgular bad structur A xampl of such a calculatio is show i Fig 3 Figur 3 Photoic bad structur for a o dimsioal PBG structur with paramtrs as listd i th lgd 5

207 Actually, if w simply valuat k from Eq(3) w fid that it is somtims complx Thos solutios clarly caot b usd bcaus thy do t corrspod to propagatig wavs but, rathr, xpotially dampd wavs Hc, what is plottd i Fig 3 is solly th purly ral-valud solutios What is crucially importat about this rsult is th apparac of gaps i th bad structur: For som valus of th frqucy thr ar o propagatig solutios! Th physical cosquc of this fact is clar: If light of this frqucy is icidt o th structur it is crtai to b rflctd sic it is ot allowd to propagat ito th matrial From th plot, w s that th first gap is coutrd if ωλ/c lis i th rag from ωλ/ c 4 to ωλ/ c I fact, thos limits ca b foud aalytically sic th paramtrs chos oby th coditio b / a= Hc, usig som trigoomtric qualitis [] it ca b show that th b a bad gap covrs th rag Λ b ωλ Λ a ta < < ta a + c a + a a b a b a ωλ 4< < 5, c whr th spcific paramtrs of th xampl ar isrtd i th last li This is a rathr substatial gap amoutig to roughly 38% of th frqucy at th midpoit of th gap It is s from th limits of th itrval that such a larg gap rquirs a larg dilctric cotrast, i th diffrc btw ad must b larg If th two idics ar arly qual th siz of th gap is b / a= b a if th coditio still holds Hc, if w wat a prfct mirror ovr a larg rag of frqucis w should choos layrs with vry dissimilar rfractiv idics It is otabl, howvr, that a bad gap ca always b obtaid v if th idics ar almost idtical This is ot th cas i highr dimsios, for which a miimum idx diffrc is rquird i ordr to produc a full bad gap Similarly, thr ar strict rquirmts for th gomtris capabl of producig a full bad gap i ad 3 dimsioal structurs I D, slabs of all thicksss ca produc gaps, albit ot cssarily larg os 3 Two Dimsioal PBG Structurs A pottial problm with th o dimsioal PBG structur is that th bad gap is highly dirctioal If light is icidt at a agl rathr tha prpdicular to th stack of slabs th siz of th gap shriks ad vtually disappars at sufficitly larg agls of icidc A solutio to th problm is to mak th priodic arragmt two dimsioal as illustratd i Fig 33 Hc, w imagi colums of o matrial mbddd i aothr matrial i a priodic fashio a b Λ /( 3 a) b a a 6

208 Figur 33 Two dimsioal PBG structur with a squar gomtry Th aalysis of this situatio is much mor ivolvd ad w bgi by cosidrig th Maxwll quatios i E= i ZkH, H= ke, (3) Z whr Z = µ / ε is th impdac of vacuum W ca isolat H i ths quatios if w divid th lattr by, tak th curl ad us th formr to substitut: i H = k E= kh (33) Z This wav quatio is simplr tha th corrspodig o for th lctric fild For simplicity w cosidr a cas for which th lctric fild vctor E is prpdicular to H th colums, i lis i th xy pla of Fig 33 Accordigly, will b dirctd alog th z axis ad, thrfor, prpdicular to th pla i Fig 33 This cofiguratio of th fild is somtims kow as TE polarizatio ad othr tims as p-polarizatio A straightforward way of solvig is by Fourir trasformig H as ik r wll as η / I our particular cas, H is of th form H= zh( x, y), whr k= ( k, k ) is a two dimsioal Bloch wav vctor Th amplitud H( x, y) is a x y lattic-priodic fuctio that ca b Fourir trasformd accordig to = ig r H G G H ( x, y ), 7

209 whr { G} is a complt st of rciprocal lattic vctors I th simpl squar lattic, G= ( p, q) π / a with p ad q itgrs Similarly, w writ th ivrs squar of th rfractiv idx as = G G ig r η( x, y) η Th cofficits i this xpasio follow from th Fourir trasform of η ( x, y) as (34) = η( x, y)xp{ i( G )} xx+ Gyy dxdy (35) ( a) To s th us of ths xpasios w go through thir us i th wav quatio Eq(33) stp by stp First, w apply th curl opratio to H Scod, w multiply this rsult by η ( x, y) rlablig from G to G to avoid cofusio: Third, w apply th curl to this xprssio usig th xpasio Eq(34) but Bcaus all vctors G, G ad k ar prpdicular to it is asily dmostratd that Ad so η G Fourth, w isrt this xprssio as wll as th xpasio of Eq(33) a a a a + = + i ( ) ( G k ) r H i G k H G i( G + G+ k ) r η( x, y) H= iη ( G+ k) H G, G { η H} η { z} G, G G i( G + G+ k ) r ( x, y) = ( G+ G+ k) ( G+ k) H G ( G + G+ k) ( G+ k) = ( G + G+ k) ( G+ k) { η H} { z} i( G + G+ k ) r ( x, y) = η ( G + G+ k) ( G+ k) H z z G z G z z G G G, G H G i th wav quatio 8

210 i( G + G+ k ) r i( G + k ) r η ( G + G+ k) ( G+ k) H = k H z G G z G G, G G Fifth, w s by compariso that trm-wis G = G+ G G = G G ad thrfor quatig idtical trms yilds G ( ) ( ) η G + k G+ kh= k G G G H G or, quivaltly, Rlablig yt aothr tim provids th fial rsult ( ) ( ) η G+ k G+ kh = k G G G H G G (36) I this form, it is appart that k is a igvalu of a matrix havig G ad G as idics ad lmts giv by η ( G + k ) ( G + k ) Thus, th rlatio btw wav G G vctor ad frqucy is ow formulatd as a igvalu problm that ca b solvd usig stadard umrical routis if a fiit st of rciprocal lattic vctors is slctd Typically, a fw hudrd is sufficit for covrgc i such twodimsioal problms Grally, Eq(35) is difficult to valuat but i th prst simpl xampl illustratd i Fig 33 th Fourir itgral ca b carrid out aalytically ad th rsult is η G = si G + xasi Gya si G xb si Gyb a G xgy a b a Th bad structur calculatd from this xprssio is illustratd i Fig 34 To produc th plot all G vctors of th form G= ( p, q) π / a with 7 p, q 7 hav b icludd givig a total of 5 vctors Th matrix problm is thrfor of dimsio 5 tims 5 Th paramtrs chos ar = 35 ad = so that, actually, th structur cosists of a array of air hols i a high-idx matrial such as GaAs Th ratio btw hol siz ad lattic costat is tak to b b/ a= 8 Agai, th apparac of a bad gap is otd, just as for th D cas I th prst cas, th magitud of th gap corrspods to roughly 3% of th midpoit frqucy Du to th D atur of th structur, howvr, th gap will xist for light propagatig i all dirctios withi th xy pla If light is icidt at a agl to this pla, th gap may disappar ad othrs may appar Also, our rsults ar valid oly for light polarizd with th lctric fild lyig i th xy pla For light of th prpdicular polarizatio, o gap is obsrvd a b 9

211 Figur 34 Bad structur of a D PBG structur with a squar array of hols i a high-idx matrial Light is TE polarizd ad th otatio o th horizotal axis rprsts high symmtry poits of th D Brilloui zo Exrcis: TM polarizd solutios By combiig th Maxwll quatios i Eq(3) so that wav quatio for th lctric fild = E k E E is isolatd w gt th Bcaus E= ( E ) E= E usig E =, this rducs to η = E k E I this xrcis, w look at TM polarizd solutios, for which E ca b writt i th gral form E= E( x, y) ik r ig r a) Show by applyig Eq(34) ad th xpasio E( x, y) E that th followig igvalu problm is obtaid: z ( + ) = η G k E k G G G E G G = b) Show usig F= G+ k E that this quatio ca b rcast i th Hrmitia form G G G G Rfrcs η G+ k G+ k F = k G G G F G G [] F Szmulowicz, Am J Phys 7, 39 (4) [] MR Spigl Mathmatical Hadbook (McGraw-Hill, Nw York, 99)

212 4 Optical Procsss Th optical proprtis of aostructurs (as wll as atoms ad molculs) maifst thmslvs via optical procsss Most promit amog ths is absorptio, which is associatd with th imagiary part of th suscptibility as discussd i th prvious chaptrs But othr procsss such as scattrig ad fluorscc ar of importac Th procsss ar ot idpdt Th rgy mittd i as scattrig ad fluorscc procsss must origiat from rgy trasfrrd from light to mattr, ultimatly absorptio i a broad ss I this chaptr, w will study th balac btw ths procsss ad study thir coctio Quit grally, th xchag of rgy btw light ad mattr is govrd by th balac of lctromagtic powr dsity (th Poytig vctor S= E H) rachig mattr, o th o had, ad rgy stord i th filds (lctromagtic rgy dsity u) plus absorbd powr, o th othr Th rgy balac is xprssd as u S= + j E t Th lctromagtic rgy dsity u is giv by = ( u E D + B H ) W will mak th simplifyig assumptio that all procsss ar lastic Hc, if w rstrict ourslvs to moochromatic icidt filds with a frqucy ω all filds vary with this frqucy ad w fid that u cotais trms varyig at twic th frqucy ω as wll as tmporally costat trms Takig th tim drivativ, th costat trms vaish I additio, w will avrag th rgy balac ovr o priod of th fild This kills off th ω trms as wll It should b otd that th lastic assumptio mas that fluorscc is igord Also, covrsio of lctromagtic rgy ito hat is glctd W writ th tim-avragd quatitis usig poitd brackts such as < S > ad fid < S>=< j E> Nxt, w itgrat this rlatio ovr a fiit volum V ad us Gauss thorm to trasform ito a itgral ovr th boudig surfac S < > = < > V 3 3 S d r j Ed r < S> ds= < j E> d r S V V 3,

213 whr is th outward poitig ormal Th lft-had sid has a simpl itrprtatio as th t lctromagtic itsity radiatd ito th volum Also, th j E trm is th absorbd optical powr isid th volum Both lctric ad magtic filds cotai a icidt (subscript ) ad a scattrd (subscript scat ) part ad so th Poytig vctor has thr cotributios Th itgral ovr < S> vaishs bcaus of cosrvatio of lctromagtic powr i th absc of mattr Th itgral of < S > is th powr radiatd by scattrig Hc, th rgy balac yilds whr th thr trms ar, rspctivly, th xtictio, scattrd, ad absorbd powr giv by 3 P = < S > ds, P = < S > ds, P = < j E> d r 4 Sigl Dipol S= S+ Sscat+ Sxt S = E H, S = E H, S = E H + E H scat scat scat xt scat scat Pxt= Pscat+ Pabs xt xt scat scat abs S S V W ow spcializ to a sigl poit dipol, that is, a sigl aostructur, atom or molcul that is sufficitly small compard to th optical wavlgth that it ca b rgardd as a poit sourc Th dipol momt varis with tim as p( t) = ( ) ad takig th dipol positio as th origi, th associatd p( ω) ω i t + c c dsity is p ( t) δ( r ) Th accompayig currt dsity is, p( t) iωt j( r, t) = δ( r ) = ( iωp( ω) + c c ) δ( r ) (4) t scat With a lctric fild E ( r, t) = ( r, ) + c c ( E ω ω i t ) it th radily follows that ( ω ω E E ) 3 Pabs= 4 i p( ) ( r, ω) + iωp ( ω) ( r, ω) δ( r ) d r V = ω Im ( ω) (, ) { p E ω}

214 I our cas, th dipol is iducd by th lctric fild ad w thrfor writ p( ω) = α( ω) E (, ω), whr α is th polarizability (tsor) Th subscript sigifis that it is th bar polarizability, which rlats th dipol momt to th total drivig fild Writig th fild vctor as E ( r, ω) = E( r, ω) w subsqutly hav Pabs= ω Im { α( ω) } E(, ω) (4) This rsult dmostrats that th absorbd powr is proportioal to th imagiary part of th polarizability Our xt stp is to comput th filds radiatd by th dipol, i th scattrd filds Th simplst stratgy is to obtai th vctor pottial A (choosig Lortz gaug) ad th fid th magtic fild via B = A ad fially th lctric fild from th magtic fild [] W writ (, ) (, ω) ω i t A r t = A r + c c For a dipol sourc scat ( scat ) mbddd i a homogous mdium with rfractiv idx ω ( ) (assumd ral i without absorptio), A ( r, ω) is govrd by th wav quatio scat ω iω Ascat( r, ω) + A (, ) (, ) scat r ω = j r ω, c ε c whr th currt amplitud is giv by solutio is th j( r, ω) = iωp( ωδ ) ( r ) cf Eq(4) Th A ik ikr scat( r, ω) = p( ω), k, k ( ω) k 4πεc r c ω (43) I tur, th magtic fild bcoms B scat i t ( r, t) = ( r, ) + c c ( B ω ω scat ) with ik B ikr scat( r, ω) = Ascat( r, ω) = p( ω) 4πεc r ik ikr = r 4πεc r r kk ikr r 4πεcr p( ω) p( ω), 3

215 ikr whr w hav applid th far-fild assumptio that is much fastr varyig tha r, which is clarly th cas whvr kr Similarly, th lctric fild i th farfild limit bcoms ikr ikr k k scat( r, ω ) ic /( ω ) scat( r, ω = ) r [ r p( )] [ U rr ] p( ) 4πε r ω E B = 4πε r ω Th last form is obtaid by cosidrig th dirctio of [ pω ( )] From ths filds w ow comput th tim-avragd Poytig vctor of th dipol radiatio εc εc < Sscat>= R { Escat ( r, ω) Bscat ( r, ω) } = R { Escat ( r, ω) Bscat ( r, ω) } r Th last quality follows from th dirctio of th cross product btw r pω ( ) ad r [ r pω ( )] Itroducig θ as th agl btw r ad pω ( ) as illustratd i Fig 4 w raliz that p( ω) = [ p( ω)] = p( ω)siθ ad so r r r ck si θ < Sscat>= p 3πε 4 ( ω) r This rsult ca b itgratd ovr a larg sphr to provid th scattrd powr r r r ck si θ ck Pscat= p ds= p 3πε S 4 4 ( ω) ( ω) r πε 4 ck = α( ω) E(, ω) πε (44) si θ usig th fact that ds= 8 π /3 S r Figur 4 Vctor diagram for th dirctios of scattrd vctor pottial ( ), p magtic fild ( ) ad lctric fild ( ) r [ p ] r r p 4

216 4 Bar ad Drssd Polarizabilitis Will a particl b iflucd by th scattrd fild that it mits? This may soud as a strag qustio but, i fact, it s importat for a full udrstadig of optical procsss Th aswr is ys ad th basic raso is simpl I th procss of mittig light, momtum is lost ad so a forc acts o th particl This ffct, howvr, ca b icorporatd ito th rspos of th particl Hc, istad of a bar particl itractig with both icidt ad scattrd filds w fid a quivalt pictur of a drssd particl itractig with th icidt fild oly To st up this quivalt pictur, a carful aalysis of xtral ad local filds is dd Primarily, w d to cosidr th filds without usig th far-fild approximatio To this d, w us th idtity ikr iq r = r π q k 3 d q ad writ th vctor pottial as iq r ik 3 Ascat( r, ω) = d q p( ω) 3 8πε c q k Takig various curls, th lctric fild is th radily foud as iq r 3 scat(r, ω) = q q p( ω) d q 3 8π ε q k iq r 3 q U q q = d q p 3 8π ε q k E ( ω) This fild, which is radiatd by th dipol, also acts o th dipol! I fact, th total local fild is th sum of th icidt fild ad this radiatd fild valuatd at r= Th imagiary part is particularly importat bcaus it acts as a dampig forc big 9 dgrs out of phas with th dipol Th ral part (which, icidtally, caot b corrctly hadld i a classical schm but rquirs a fully quatumlctrodyamical thory) lads to a frqucy shift of th rsoac, th so-calld Lamb shift Th imagiary part of th fild valuatd at r= is calld th Radiatio- Ractio fild E ( ω ) It is obtaid from th imagiary part of th pol, i RR E i 3 RR( ω) = δ( q k ) q U qq d q p( ω) 8π ε ikk = 6πε p ( ω) (45) 5

217 Th ffcts of radiatio ractio ar clar if w ot that th total fild drivig th dipol bcoms E (, ω) = E(, ω) + E RR ( ω) Hc, th iducd dipol momt is ( ω) α ( ω) (, ω) ( ω) { } p = E + E RR Applyig th xprssio for th radiatio-ractio fild, this rsult ca b rarragd as ikk p( ω) U α( ω) = α( ω) E(, ω) 6πε αω ( ) E (, ω) Hr, α is th so-calld drssd polarizability rlatig th dipol momt to th icidt fild alo Hc, th ffcts of radiatio ractio hav b absorbd ito th drssd polarizability Th bar polarizability was calculatd prviously i chaptr α ( ω) = D D E ħω Hr, E is th may body rgy igvalu of th th stat masurd rlativ to th groud stat Also, D = r is th may-lctro trasitio dipol momt W ow mak th simplifyig assumptio of isotropy i th polarizability as appropriat for atoms or sphrical aostructurs Also, th approximatios applid abov ar oly xpctd to hold wh w ar clos to a rsoac ω / ħ ω Hc, E E D α( ω) U ħω ω ad, thrby, with k ω / c 3 ik D D αω ( ) = U 6πε ħω ω ħω ω Usig simpl maipulatios, this rsult ca b rwritt as 6

218 αω ( ) = ħ D ω ω i Γ U, (46) whr k D Γ = 3πεħ 3 (47) This quatity, which is actually th spotaous dcay rat of th th xcitd stat [], th rsults i a fiit li width qual to for th rsoac Without, th rsoac would divrg prcisly at ω ω vry trasitio ad so αω = Γ ( ) = ħ ω ω i Γ This modificatio should b do for D U Γ W hav dfid th drssd polarizability so that α ( ω) E (, ω) = αω ( ) E (, ω) mas that th corrct vrsios of Eq(4) ad (44) ca b writt as This P = ω Im αω ( ) E (, ω) abs { } 4 ck Pscat= αω ( ) E (, ω) πε W wish to itroduc cross sctios for th two procsss ad, to this d, d th itsity of th icidt fild I = S = ε c E Dividig th powr xprssios ic abov by this itsity provids absorptio ad scattrig cross sctios σ σ Now, if rgy is ot accumulatd i th particl ad it is ot dissipatd by othr mas, th absorbd powr must qual th scattrd powr Thus, quatig th abov cross sctios yilds { αω } abs= Im ( ) ε scat k k = αω ( ) 6πε 4 3 k Im { αω ( ) } αω ( ) (48) 6πε 7

219 This coditio is oly xpctd hold xactly at a rsoac ω= ω so that Eq(46) yilds αω ( ) = i D /( ħγ ) U Pluggig this ito Eq(48) shows that th rquirmt is prcisly obyd if is giv by th xprssio Eq(47) Hc, our xprssio for th spotaous dcay rat is cosistt with all absorbd powr vtually big r-mittd as scattrig Exrcis: Nar-fild rlatios Γ Th calculatio of th scattrd fild abov rlid o th far-fild approximatio kr If all trms ar rtaid, a somwhat tdious computatio shows that E ikr k i 3i 3 ( r, ω) U = + + p( ω) 4πεr kr k r kr k r scat r r a) Show by xpasio aroud x = that ix ix i i x x x x x 3 3i x x x x x b) Us this rsult to show that th radiatio-ractio fild is prcisly th imagiary part of th scattrd fild i th limit r c) Show that th ar-fild, i th lctric fild vry clos to th dipol, is approximatly giv by E ( r, ω) { 3 U} p( ω) 3 4πε r ar fild r r Th sam xprssio is obtaid for th lctrostatic fild producd by a dipol I a static calculatio, w put E= Φ, whr Φ is th lctrostatic pottial d) Show that th accompayig ar-fild pottial is Φ (, ) ( ) ar r ω pω fild 4πε r r W ow cosidr a small aosphr of radius a ad rfractiv idx mbddd i a mdium with rfractiv idx subjctd to a costat icidt fild E = E Hc, th icidt pottial must b Φ = E I polar coordiats, z z 8

220 Φ = Er cosθ ad w thrfor writ th full solutio as Φ ( r, θ) = f ( r)cosθ Laplac s quatio for th pottial r ( rf )/ r = f cosqutly simplifis to d) Show that f ( r) = r ad f ( r) = / r ar solutios to Laplac s quatio ) Of th abov, oly th first typ is allowd isid th sphr whil both forms ar applicabl outsid Apply th boudary coditios f ( a ) = f ( a ) ad f ( a ) = f ( a ) + Φ=Φ +Φscat with to dmostrat that th full solutio outsid th sphr is 3 a ( r, θ) E cosθ r + Comparig to th gral xprssio for th ar-fild pottial, this dmostrats 3 that th polarizability is αω ( ) 4πεa + Rfrcs Φ scat Φ = ( rθ, ) + [] JR Ritz, FJ Milford, ad RW Christy Foudatios of Elctromagtic Thory (Addiso-Wsly, Massachustts, 979) [] R Loudo Th Quatum Thory of Light (Oxford Uivrsity Prss, Oxford, 99) 9

221 5 Optical Proprtis of Naosphrs Thr ar two limits, i which th optical proprtis of a objct ar rlativly simply dscribd: Eithr th objct should b much smallr tha th optical wavlgth or th objct should b much largr I th formr cas, th lctric fild practically dos t vary across th objct ad th fild ca b approximatd by a costat i spac I th lattr cas, w tr th macroscopic rgim ad so-calld gomtric (or ray) optics ca b applid This is th rgim of ordiary lss ad similar optical compots Th scitific fild of aooptics dals with th complicatd rgim i btw ths xtrms I aooptics, objcts ar typically comparabl to th wavlgth ad so th optical filds vary isid ad aroud th objct i a vry complicatd fashio This lads to quit ovl phoma such as th xtraordiarily larg trasmissio through sub-wavlgth hols i mtallic films [] To tr th fild of aooptics, w will rstrict ourslvs to a simpl but importat xampl: A aosphr illumiatd by a pla wav This cas has a sufficitly simpl gomtry that it is possibl to calculat th optical fild xactly At th sam tim, howvr, it is a cas of grat practical rlvac For istac, it dscribs th optical proprtis of colloidal mtal particls Also, atural phoma such as th raibow ad light scattrd by fog is xplaid by this xampl I th prst chaptr, w formulat a thory for such phoma Basically, light itractig with a aosphr ca b ithr scattrd or absorbd by th particl If a dtctor is placd bhid a sampl cotaiig such sphrs it will rcord th amout of light lost i both procsss Th combid ffcts (scattrig ad absorptio) ar kow as xtictio ad to compar with masurmts w should discuss how scattrig, absorptio ad xtictio dpd o particl siz, optical wavlgth ad rfractiv idics of sphrs ad th surroudig mdium Our stratgy is as follows: Giv a icidt fild, w d to fid th scattrd fild as wll as th fild isid th sphr W do this by dcomposig th icidt fild i a sum of sphrical wavs Th, w writ th ukow filds as similar xpasios but cotaiig ukow xpasio cofficits Fially, th usual boudary coditios ar ivokd i ordr to dtrmi th xpasio cofficits Oc th filds ar dtrmid, th scattrd ad absorbd powr ca b calculatd This approach was stablishd by G Mi i 98 ad th scattrig of light from particls comparabl to th wavlgth i siz is kow as Mi scattrig W start by itroducig gral solutios to th Hlmholtz quatio whr k = k ( + k ) g( r ) =, is th wav umbr of a fild i a homogous mdium with a rfractiv idx Latr, w will us ths solutios to build th diffrt filds i th

222 problm For a vry small particl, th rlvat solutio is th simpl outgoig sphrical wav g( r) = xp( ikr)/ r I a mor gral calculatio, though, w should cosidr all possibl solutios W obviously should tak advatag of th simpl sphrical gomtry ad cosqutly fid solutios with a simpl sphrical symmtry Th appropriat sphrical coordiats ar { r, θφ, } ad w sk solutios that ar sparabl i ths variabls By dirct calculatio it ca b show that ths solutios ar () m () m gm( r ) = cos mφp (cos θ) j( kr), gom( r ) = si mφp (cos θ) j( kr) () m () m g ( r ) = cos mφp (cos θ) h ( kr), g ( r ) = si mφp (cos θ) h ( kr), m om (5) whr P m (cos θ) is a associatd Lgdr polyomial, j ( kr) is a sphrical Bssl fuctio ad h ( kr ) is a sphrical Hakl fuctio Th suprscripts ar usd to distiguish btw ths two kids of radial bhavior Also, subscripts ad o ar usd for v ad odd fuctios of φ, rspctivly Th radial fuctios ar dfid by π π j( x) = J+ /( x), h( x) = ( J+ /( x) + iy+ /( x) ), x x whr J ( +/ x ) ad Y ( ) +/ x ar ordiary Bssl fuctios of first ad scod kid, rspctivly At larg valus of x ths fuctios bhav as + π ( ) ix j( x) cos x, h( x), x larg + x i x (5) O th othr had, wh x is small j ( x ) whil lim h( x) = i This bhavior mas that oly th j( kr) - typ solutio ca b usd isid th sphr For th scattrd fild, oly th h( kr) - typ solutio ca b usd bcaus it approachs a outgoig sphrical wav at larg r Th xt stp is to costruct solutios for th actual vctorial filds i a sphrical gomtry I rgios of spac with a costat, isotropic rfractiv idx, th vctorial E ad H filds satisfy th wav quatios as wll as th trasvrsality coditios x ( + k ) E ( r ) =, ( + k ) H( r ) = E ( r ) =, H( r ) =

223 Also, w should mak sur that E ad H ar rlatd by th Maxwll quatios ik E ( r ) = ikzh( r ), H( r ) = E ( r ), Z (53) whr Z= µ / ε is th impdac of th mdium whos rfractiv idx is W may ow apply th st of g s to costruct solutios to th fild quatios i sphrical coordiats To gt from th scalar g s to th vctorial filds w itroduc th oprator ˆ A = r θ φ (54) siθ φ θ Th xprssio of this oprator i sphrical coordiats is dmostratd usig th rlatio = r + θ + φ r r θ r siθ φ I ths formulas, r, θ ad φ ar th usual uit vctors of sphrical gomtry Th ˆ ˆ oprator commuts with th Laplacia, i A = A Hc, it follows dirctly that if ( + k ) g( r ) = th ˆ M Ag( r ) fulfils th quatio ( + k ) M= Howvr, if for istac E cotais M fuctios th Eq(53) shows that H will cotai N fuctios giv by N M k Ths fuctios also satisfy th wav quatio ad ca b writt i trms of th g s accordig to ˆ N Bg ( r ), whr th w oprator is giv by ˆ ˆ = B A θ r+ φ r θ + r si θ θ φ k kr r kr si r kr siθ θ θ si θ φ Usig th Laplacia i sphrical coordiats = r+ siθ + r r r siθ θ θ r si θ φ

224 ad th fact that ( + k ) g( r ) =, th B oprator ca b writt as ˆ B= θ r+ φ r+ r r+ rk kr θ θ φ (55) r kr si r k r It might b thought that, i tur, this would itroduc tirly w fuctios N / k ad so o without d Howvr, usig th fact that th divrgc of a curl vaishs 5 Fidig th Filds N= M k k = k = M k = M, { ( M) M} As xplaid abov, th aalysis procds by (i) xprssig th icidt fild i trms of sphrical wavs, i th M ad N fuctios itroducd abov, (ii) xprssig th ukow filds i a similar form, ad (iii) dtrmi th ukows by mas of th boudary coditios of th problm W choos th gomtry (agls) as idicatd i Fig 5 blow Figur 5 Illustratio of th sphrical coordiats ad thir rlatio to th lctric fild Th icomig fild is rprstd by a stadard pla wav propagatig alog z ad polarizd alog x as show i Fig 5 ad, cosqutly, giv by 3

225 E = E xp( i kz) = E xp( i kr cos θ) ic x x Th phas xp( i kr cos θ) ca b rwritt usig th idtity (usig th positiv sig P covtio for, i P ( x) = x dp ( x)/ dx ) xp( i kr cos θ) = i (+ ) P (cos θ) j( kr) ikr siθ = I additio, w d to r-xprss th polarizatio vctor x usig th sphrical uit vctors Th trasformatio btw Cartsia ad sphrical coordiats is giv by = siθ cosφ + cosθ cosφ siφ ad cosqutly x r θ φ E E = siθ cosφ+ cosθ cosφ si φ i (+ ) P (cos θ) j ( kr) { θ φ} ic r ikr siθ = (56) Th crucial poit is ow to xprss E ic i trms of M ad N fuctios, i as a sum of ˆ ˆ trms giv by Ag( r ) ad Bg( r ) Lookig at th xprssio abov, it is obvious that w do t d all th g s dfid i Eq(5) to costruct th fild Th simplifyig poit is that amog all th associatd Lgdr polyomials oly P appars i Eq(56) As a cosquc, w should put m= i Eq(5) ad focus o th corrspodig M ad N fuctios giv by ad () ˆ () siφ P (cos θ) M Ag( r ) = P (cos θ) j( kr) θ cos φ j( kr) siθ θ () ˆ () cosφ P (cos θ) Mo Ago( r ) = P (cos θ) j( kr) θ si φ j( kr) φ, siθ θ φ () () N M( r ) k θ θ j( kr) [ rj( kr)] = φ θ + + P (cos ) P (cos ) cos P (cos ) ( ) cosφ θ siφ r φ kr kr r θ siθ () () No Mo( r ) k θ θ j( kr) [ rj( kr)] = φ θ + + P (cos ) P (cos ) si P (cos ) ( ) siφ θ+ cos φ r φ kr kr r θ siθ 4

226 I additio, typ- M ad N fuctios (with suprscript ) ar giv by idtical xprssios xcpt that j ( kr ) is rplacd by h( kr) Now, a dirct compariso btw ths rsults ad Eq(56) dmostrats that W ow writ th rmaiig filds i aalogous ways As argud abov, oly typ- solutios ar applicabl for th scattrd fild E scat whil th fild isid th sphr E sph must b costructd from typ- solutios i ordr to rmai fiit at th origi Thus, ad E (+ ) () () ic( r ) = E i Mo in = ( + ) { } + ( ) () () scat( r ) = E i amo ibn = ( + ) E { } + ( ) () () sph( r ) = E i cmo idn = ( + ) E { } Th ukow cofficits a, b, c ad d must ow b dtrmid by matchig boudary coditios for th radial ad tagtial compots of th filds: ad ( Eic+ Escat) r= Esph r ( Eic+ Escat) θ= Esph θ whr ad ar rfractiv idics outsid ad isid th sphr, rspctivly I this mar, for a sphr with radius a th boudary coditios rquir that c j ( k a) = j ( k a) + a h ( k a) c [ k a j ( k a)] = [ k a j ( k a)] + a [ k a h ( k a)] d k j ( k a) = k j ( k a) + b k h ( k a) d = + [ ka j( ka)] [ ka j( ka)] b [ ka h( ka)] k a k a k a ( xf ( x)) Hr, th otatio ki= iω / c ad [ xf ( x)] is usd Solvig th four x coupld quatios yilds, 5

227 b a j( ka)[ ka j( ka)] j( ka)[ ka j( ka)] = j ( k a)[ k a h ( k a)] h ( k a)[ k a j ( k a)] j ( k a)[ k a j ( k a)] j ( k a)[ k a j ( k a)] = h ( k a)[ k a j ( k a)] j ( k a)[ k a h ( k a)] (57) 5 Scattrig, Absorptio ad Extictio To calculat th scattrd itsity, w d th scattrd magtic fild as dtrmid by Eq(53) + = E = ( ) () () scat( r ) Escat ( r ) i an o ibm ikz iz = ( + ) H { } Th corrspodig itsity is th radial compot of th Poytig vctor I = R { E ( r ) H ( r ) } scat scat scat r Th powr scattrd ito a solid agl dω at a distac r from th scattrr is giv by I r d Ω Also, th itsity of th icomig fild is I = E /Z Th scattrd scat powr pr solid agl dividd by th itsity of th icomig bam is th so-calld diffrtial scattrig cross sctio, i th cross sctio pr solid agl, ad it ca b obtaid via th rlatio ic σscat ( θφ, ) IscatZr = Ω E Th cross sctio should b idpdt of distac from th scattrr ad for that raso w may look at th filds far away from th sphr whr th asymptotic xpasios Eq(5) apply Hc, w fid whr ikr E + ( ) Escat r a + b a + b ikr = ( + ) ikr E + ( ) Hscat r a + b + a + b Zikr = ( + ) { θ π τ φ φ τ π φ} ( ) ( )cos ( )si { φ π τ φ θ τ π φ} ( ) ( )cos ( )si, π P (cos θ) P (cos θ) =, τ= siθ θ 6

228 W subsqutly fid σscat( θφ, ) = γp( θ)cos φ+ γs( θ)si φ, (58) Ω whr γ p ad γ s ar th cross sctios for two spcial cass: scattrig obsrvd i th pla cotaiig th x-axis ( γ p ) ad scattrig obsrvd i th pla prpdicular to ( γ s ) Ths cass corrspod to φ= ad φ= π/, rspctivly Thir xprssios ar x (+ ) (+ ) γp( θ) = ( aπ + τ ), γ ( θ) = ( τ + π ) b s a b k ( + ) k ( + ) = = (59) Figur 5 blow illustrats γ ( θ) for two cass Hr, a wavlgth of 5 m is p µ assumd ad th rfractiv idics ar tak as thos of latx sphrs i watr This is a commo tst systm for th agular dpdc of optical scattrig Figur 5 Diffrtial scattrig cross sctio for sphrs of two diffrt radii ad wavlgth 5 µ m Th rfractiv idics corrspod to latx sphrs i watr I additio, w wish to calculat th total (itgratd ovr solid agl) scattrig cross sctio To this d, w d th rsult W thrfor fid π ( + ) ππ + ττ siθdθ= + ( ) m m m δ 7

229 I a similar mar, th total xtictio cross sctio is giv by [] (5) π = (+ )R( a + b ) (5) k Th absorptio cross sctio is giv by th diffrc btw ths rsults If a sris xpasio with rspct to a is mad th first o-vaishig trm coms from : b Th ral part is th σ σ { } π = (+ ) a + b k scat = xt = i 3 + ( k a) ad th xtictio (actually absorptio) cross sctio bcoms b Im( ) R( b ) ( k a) + π Im( ) σxt k a + (5) (53) This is th clbratd Rayligh cross sctio, which is valid for small particls W s that it mrgs as a spcial cas of th gral thory Also, sic π = ad τ = cosθ w fid i this limit σscat( θφ, ) 9 b + cos cos si Ω 4k which shows that scattrig from vry small particls prdomiatly is i th pla prpdicular to th polarizatio of th icidt light Th plots blow show umrical spctra (xtictio cross sctio vs photo rgy) calculatd for diamtrs of 4 m, 8 m ad 6 m assumig = 35 ad usig a fr lctro modl for, cf chaptr 3 3 { θ φ φ} 9 b = { ( r x ) }, 4k 3 3 8

230 ω ( ω ) p = ε (54) ωω ( +Γ i ) with ħω p = 93 V, ħγ= 5 V ad ε = 6 Ths paramtrs ar appropriat for silvr aosphrs Lookig at Eq(5) it is clar that i th Rayligh limit, a rsoac appars wh + is small Th approximat miimum valu is foud at th frqucy, for which th ral part vaishs I th fr lctro modl, th happs wh + ε ω ω p / = so that ω= ω p / + ε For silvr th rsoac occurs at ħω 3 V corrspodig to a wavlgth of 45 m This valu is oly corrct for vry small sphrs, howvr From Fig 53 it is s that v for 4 m sphrs th rsoac is slightly shiftd (by roughly 5 m) compard to th Rayligh limit Figur 53 Total xtictio cross sctio for silvr aosphrs of diffrt radii Exrcis: Itsity hacmt At th surfac of a mtallic sphr, th optical itsity ca b sigificatly largr tha th icidt itsity a) Show from th dfiitio of th diffrtial scattrig cross sctio that at th surfac I I scat ic = a σ ( θφ, ) Ω scat It follows that th agular avrag of this ratio is 9

231 Iscat σscat( θφ, ) σscat = Ω= 4π d I a Ω 4πa ic b) Show that i th Rayligh limit, whr Eq(5) applis Iscat 3 b = = I k a 3 ic + ( k a) 4 Th maximum hacmt is foud at th rsoac frqucy at which + iω Γ / ω 3 p ω= ω / + p ε c) Show that providd 3 ωγ / ω p I I scat ic 6 ω p ( ka) ( + ε ) Γ d) Evaluat this ratio for silvr sphrs of radius, ad 3 m i a mdium whos rfractiv idx is 35 Rfrcs [] TW Ebbs t al Natur 39, 667 (998) [] JD Jackso Classical Elctrodyamics (Wily, Nw York, 999) 3

232 6 Naoparticl Optics i th Elctrostatic Limit I th prvious chaptr, w wr rstrictd to aosphrs bcaus a full solutio of Maxwll s quatio was dd Such a aalytic calculatio is oly possibl for vry simpl gomtris Howvr, if th particls bcom sufficitly small so that th spatial variatio of th lctromagtic fild ca b simplifid, mor complicatd shaps ca b studid Hc, i this chaptr, w adopt th lctrostatic limit that is quivalt to igorig th spatial variatio of th icidt lctric fild isid th particl Figur 6 Gomtry of mtal aoparticl havig a dilctric costat homogous mdium with a dilctric costat ε ε mbddd i a W cosidr a aoparticl such as th o dpictd i Fig 6 ad apply th lctrostatic mthod prstd i Rf [] I th lctrostatic pictur, a lctric fild oly iducs polarizatio chargs o th surfac of th particl Hc, th total lctrostatic pottial Φ( r ) is th sum of a icidt part Φ ( r ) ad th cotributio gratd by th surfac chargs If th surfac charg dsity is is giv by σ( r ) Φ ( r ) =Φ ( r ) +, 4 πε ds r r, th pottial whr th itgral is ovr th tir surfac of th particl W ca ow us this to comput th lctric fild via th rlatio E ( r ) = Φ( r ) I particular, w wish to comput th lctric fild o th surfac I this situatio, car has to b tak bcaus th ormal compot is discotiuous Th ormal compot is giv by E ( r) = Φ( r), whr is th outward uit ormal vctor at positio r Th discotiuity mas that th ormal compot just outsid th particl rlatd to th o just isid Evia th rlatio i out σ( r ) i σ( r ) E ( r ) = E ( r ) + ε ε σ( r ) out E is 3

233 Usig ths rlatios, w fid that E out / i σ( r ) ( r ) = + σ( ) (, ) E ±, (6) 4πε r g r r ds ε whr + ad go with th filds outsid ad isid th particl, rspctivly Also, E is th (costat) icidt fild ad g dots th so-calld Gr s fuctio r r g( r, r ) = = 3 r r r r Howvr, w also kow that th ormal compots ar rlatd via th dilctric out i costats, i ε E ( r ) = εe ( r ) This mas that with a bit of rarragmt, Eq (6) ca b rformulatd as λ σ( r ) = ελ + σ( ) (, ) E, (6) π r g r r ds whr λ= ( ε ε )/( ε+ ε ) This formulatio is vry covit bcaus it allows us to comput th distributio of surfac chargs from a sigl quatio 6 Cylidrical Naoparticls Th framwork abov applis to aoparticls of compltly gral shap It v works for collctios of aoparticls if th surfac is tak as th sum of surfacs I practic, howvr, Eq (6) is difficult to solv i th gral cas Fortuatly, may importat cass ar much simplr I particular, may rlvat aoparticl gomtris hav cylidrical symmtry, i thy hav a rotatioal symmtry axis as illustratd i Fig 6 I this cas, th gral problm ca b rducd sigificatly Figur 6 Cylidrically symmtric aoparticl 3

234 Usig th gomtry of Fig 6, w ca without loss of grality choos to kp th icidt fild E i th (x,z)-pla Th x- ad z-axs ar th horizotal (h) ad ( h) ( v) vrtical (v) dirctios, rspctivly, ad so w dcompos E = E + E Du to th suprpositio pricipl w ca, i fact, trat th horizotal ad vrtical cass sparatly W th bfit sigificatly from th simpl agular dpdc of ths cass I a cylidrical gomtry, w may xprss th gomtrical vctors usig polar agls i a simpl mar Hc, r= r(siθ cos ϕ, siθ si ϕ, cos θ) = ( x cos ϕ, x si ϕ, z ), r = r (siθ cos ϕ, siθ si ϕ, cos θ ) whr r, x, ar all fuctios of θ z ad r is a fuctio of θ It follows that E is E x cosϕ ad Ez i th horizotal ad vrtical cass, rspctivly It is th radily show that th surfac charg follows xactly th sam dpdc o th agl ϕ Du to th symmtry, th surfac ara lmt ds must b idpdt of ϕ ad w may writ ds = S( θ ) dθ dϕ W will rtur to th θ dpdc latr W also d th followig rlatios to rduc th Gr s fuctio: x ( ) ( cosθ cos θ ) ( siθ siθ cos( ϕ ϕ r r = z r r + x r r )) r r = r + r rr cosθ cosθ rr siθ siθ cos( ϕ ϕ ) Th simplst cas is that of vrtical polarizatio, for which E is idpdt of ϕ W wish to prov that th surfac charg is a fuctio of θ oly ad so w writ σ( r) = σθ ( ) Th charg balac Eq(6) thrfor rducs to z whr ( v) Vrtical: σθ ( ) = ελ E ( θ) + λ σθ ( ) ( θθ, ) ( θ ) θ z G S d, (63) G ( v) π ( r r ) ( θθ, ) = dϕ 3 π r r Prformig th itgral usig th gomtrical rlatios w th fid π { z x,( ) x,( )} ( v) G ( θθ, ) = ( r cosθ r cos θ ) + r si θ F x, y r si θ F x, y π, (64) 33

235 with x= r + r rr cosθ cos θ, y= rr siθ siθ ad itroducig th fuctios F ( x, y) π cos m, ( )/ ( x+ y cos ϕ ) + m ϕ dϕ Th first fw of ths fuctios ar whr K ad E ar complt lliptic itgrals Highr trms ca b gratd usig F = ( F xf )/ y Th fact that ths fuctios dpd oly o θ ad θ complts th proof For th horizotal cas, w procd i almost complt aalogy but ow th surfac charg is foud to follow th cosϕ bhaviour of Hc, w writ E σ( r) = σθ ( )cosϕ i this cas ad usig lmtary mathmatical maipulatios i( ϕ ϕ ) iϕ (rwritig as R ad doig th itgral bfor takig th ral part) fid that whr F, ( x, y) K x y F, x y E x y, 4 y = x + y + 4 y (, ) = ( x y ) x + y + 4 x+ y y 4x y F, ( x, y) = E K, y x y y x y x y x y 4 y F, ( x, y) = E K y( y x) x y x y + y x y x y F m, m, m, 4y 8x y 8x y ( x, y) = E K y ( y x) x y x y y x y x y cosϕ { } π ( h) Horizotal: σθ ( ) = ελ E ( θ) + λ σθ ( ) ( θθ, ) ( θ ) θ x G S d, (65) G ( h) π ( )cos( ϕ ϕ r r ) ( θθ, ) = dϕ 3 π r r 34

236 ca b xprssd as = π + 6 Oblat Sphroids { z x,( ) x,( )} ( h ) G ( θθ, ) ( r cosθ r cos θ ) r si θ F x, y r si θ F x, y (66) As a rlativly simpl but still tchologically importat xampl w will cosidr oblat sphroids: pacak shapd particls obtaid by flattig sphrs alog o dirctio Such a particl is illustratd i Fig 63 Figur 63 Cross sctio of a oblat sphroid Th gomtry of th sphroid is tak such that th thickss of th pacak is uity ad th radius is d Hc, all distacs ar actually masurd i uits of half th particl hight A particular poit o th surfac obys th llips paramtrizatio x r d = z + = r cos θ+ si θ r= d d d cos θ+ si θ Also, th surfac ormal is calculatd from th rquirmt that ( dr / dθ) = Hc, diffrtiatig ad surig ormalizatio it is foud that x siθ d cosθ =, 4 z= 4 d cos θ+ si θ d cos θ+ si θ I gral, th surfac aral fuctio S( θ) is to b calculatd as 3 S( θ) = r si θ / r 4 / ad i th sphroid cas S( θ) = d si θ(si θ+ d cos θ) /(si θ+ d cos θ) To solv quatios lik Eq(63) ad Eq(65) umrically w d to discrtiz th agl θ O th itrval θ [, π[ w thrfor slct N discrt valus with θ i 35

237 sparatios i= θi+ θi (for i = N w tak N= π θi ) Hc, th quatios ar rformulatd as + N i E i j G i j S j j j= σθ ( ) ελ ( θ ) λ σθ ( ) ( θ, θ ) ( θ ), (67) whr th appropriat Gr s fuctio ad ormal vctor compot should b chos for th two polarizatios Equatios of this sort ar asily covrtd ito a tractabl form by itroducig vctors σθ ( ) ( θ ) σθ ( ) ( θ ) σ =, ε = E σθ ( ) ( θ ) N N as wll as a matrix ( θ, θ ) ( θ ) ( θ, θ ) ( θ ) ( θ, θ ) ( θ ) G S G S G N S N N ( θ, θ ) ( θ ) ( θ, θ ) ( θ ) ( θ, θ ) ( θ ) G S G S G N S N N G= ( θ, θ ) ( θ ) ( θ, θ ) ( θ ) ( θ, θ ) ( θ ) G S G S G S I trms of ths quatitis, th discrtizd quatio rads as λ σ, σ λ U G = = U G (68) Thus, th ukow surfac chargs i vctor σ ar foud by ivrtig a matrix ad multiplyig oto a kow vctor Furthrmor, it is ralizd that crtai igmods of th surfac charg ca b foud whvr th dtrmiat λ U G vaishs This is bcaus this coditio corrspods to a situatio, i which a surfac charg xists v with a vaishigly small icidt fild This is clarly a mathmatical abstractio but th sigificac is that, i actual calculatios, rsoacs i absorptio or scattrig cross sctios may appar ar ths igmods From th form of th matrix it is also vidt that igmods ar foud whvr λ is a igvalu of G I practic, th matrix G is slightly problmatic bcaus th diagoal lmts divrg! By clvr usag of th gral proprtis of th Gr s fuctio, howvr, appropriat valus of th diagoal lmts ca b foud (s th xrcis) If th dilctric costat of th mtal aoparticl is assumd to b of th losslss Drud form εω ( ) = ε ω / ω th rlatio btw igvalu ad rsoac frqucy is giv by N N N N N N { } { } p 36

238 ω= + ω p ε ε λ λ+ As a xampl, w ow tak Ag sphroids ( ε = 6, ħω = 93 V) mbddd i Si ( ε = = / ) For a particular gomtry, th sphroid is charactrizd by its llipticity d, which rags btw for a sphr ad for a pla I Fig 64, rsults for th rsoac wavlgths i this cas ar dpictd It is otd that th fudamtal horizotal ad vrtical mods shift towards logr ad shortr wavlgths, rspctivly, as th particl is flattd p Figur 64 Rsoac wavlgths of Ag sphroids mbddd i Si Solid ad dashd curvs illustrat vrtical ad horizotal rsoacs vrsus aoparticl llipticity Th prst tchiqu ca b xtdd i svral dirctios Primarily, basd o solutio of th ihomogous quatio Eq(68), th absorptio cross sctio is calculatd via ω σ ( ω) Im p abs =, (69) cε ε E whr p is th iducd dipol momt that is asily calculatd usig th formulas 37

239 4 r ( θ) Vrtical: p= πεz σθ ( ) cosθ siθdθ r I additio, aoparticls positiod o a surfac or mbddd i thi layrs ca b hadld by propr modificatios to th Gr s fuctios [] Exrcis: Proprtis of th Gr s fuctio π 4 r ( θ) p= πεx σθ d r Horizotal: ( ) si π a) Usig Gauss thorm, show that (rmmbrig that th sigularity of g lis o th boudary ad thrfor oly cotributs half) θ θ g( r, r ) ds= π b) Basd o this rsult, show that th rducd Gr s fuctio for vrtical polarizatio satisfis th coditio π ( v) G ( θθ, ) S( θ) dθ= This rsult ca b usd to hadl th sigularity of th Gr s fuctio I a discrtizd vrsio, it rads as It follows that th diagoal lmt must b ( θ, θ ) S( θ ) = ( ) ( ) ( θ, θ ) = ( θ, θ ) ( θ ) ( θ ) N v v G j j G i j S i i S j j i=, i j For th horizotal cas, ufortuatly, o such simpl rsult applis Hc, spcial umrical hadlig of th diagoal trms is dd Rfrcs N i= G ( v) i j i i [] ID Mayrgoyz, DR Frdki, ad Z Zhag, Phys Rv B7, 554 (5) [] J Jug, TG Pdrs, T Sødrgaard, K Pdrs, A Nyladstd Lars, ad B Bch Nils, Phys Rv 8, 543 () 38

240 7 Basic Magtism Magtism is a fudamtal proprty of matrials W will i this chaptr rviw th most importat aspcts of magtism ad dscrib para- ad frro-magtism i som dtail Lt us first brifly dfi a fw trms B: Magtic flux dsity H: Magtic fild strgth M: Magtisatio, (dipol momt dsity) I fr spac w hav th followig rlatio: B=µ H, whr µ is th vacuum prmability I a matrial w hav to tak ito accout th magtizatio, thus B= H+ M µ ( ) (7) W ca ow dfi th magtic suscptibility χ, which is th rspos of th matrial to a applid magtic fild, via th followig rlatio M = χh (7) Or mor corrctly χ= M H Strictly spakig χ is a tsor ad th magtizatio is, for som matrials, ot poitig i th sam dirctio as th applid fild W thus hav rlativ prmability B= µ ( H+ M) = µ ( H+ χh) = µ ( + χ) H, which also dfis th fild ca b classifid ito th followig Paramagtic matrials: χ~+ -5 Diamagtic matrials: χ ~- -5 Frromagtic matrials: χ > µ = + χ How a matrial rspods to a applid magtic r Th valus ar ordrs of magitud ad ca vary strogly Furthrmor thr is a prooucd tmpratur variatio for para- ad frromagtic matrials which will b discussd i dtail latr Not that ths valus ar oly valid for static magtic filds sic tim-varyig applid filds ca caus ddy currts, which produc a magtic fild poitig i th opposit dirctio of th applid fild Th matrial thus bhavs strogly diamagtic This ffct is dpdt o th lctrical coductivity Aothr class of matrials is suprcoductors, which act as prfct diamagts, i χ= It is obsrvd that th flux lis of a applid magtic fild 39

241 ar ot allowd to ptrat th suprcoductig matrial This is also calld th Missr ffct W will ow stablish a microscopic dscriptio of magtism Th sstial paramtr is th magtic dipol momt µ Classically, for a currt loop, this is giv as µ= IA, (73) whr I is th currt i th loop ad A th ara Th dirctio of th vctor is giv by th right had rul W ca xprss th currt as I= dq / dt= q / T, whr T is th volutio tim of a sigl charg (a lctro, i q = -) Look at th orbit of a lctro i th currt loop giv blow Fig 7 A classical currt loop with a orbitig lctro Th ifiitsimal ara of th triagl is giv by da= r dr ad w ca th writ da r p L dr L = r = = =, (74) dt dt m m m whr p is th momtum ad L th agular momtum of th lctro Furthrmor w ca writ da dt= A T for a lliptic orbit ad thus A L =, (75) T m which is qual to Kplr s scod law for platary motio! Isrtig Eq (75) ad I= / T ito Eq (73), w ca ow xprss th magtic dipol momt i trms of th agular momtum µ= L (76) m Th origi of magtism is thus th lctros: orbitig ad spiig W ca graliz th abov xprssio ad trasfr it ito a quatum mchaical 4

242 dscriptio by usig th agular momtum oprator istad of th classical agular momtum Lˆ ψ= l( l+ ) ħψ ad Lˆ zψ= mlħψ whr l is th quatum umbr dscribig th agular momtum, ml th quatum umbr dscribig th projctio of L oto th z-axis ad ψ th igfuctio of th oprators Th classical agular momtum vctor L i Eq (76) is th to b rplacd by th igvalu of th corrspodig quatum mchaical oprator For a gral orbital agular momtum (spi, orbital or a combiatio) w ca xprss th magtic momt as gµ BJ µ=, (77) ħ whr µ B= ħ is th Bohr magto ad J = L + S th total, L th orbital ad S m th spi agular momtum Th rlvat quatum umbrs ar j, m,, ad, Th so-calld Ladé g-factor is giv by j l ml s ms g= + j( j+ ) l( l+ ) + s( s+ ) (78) j( j+ ) Not that g= for a pur agular momtum ad g= for a pur spi Th z-compot of th magtic momt th bcoms gµ gµ µ = J = mħ= gµ m ħ ħ B B z z j B j (79) Ad spcifically for a pur spi or orbital momt w fid spi orbital µ = µ m =± µ ad µ = µ m z B s B z B l 7 Isolatd Momts Lt us ow look at isolatd magtic momts placd i a magtic fild It is assumd that thr is o itractio btw momts, but oly itractio btw ach momt ad th applid fild For simplicity w oly look at spi momts Th itractio rgy dpds o th mutual dirctio of th fild ad momt as 4

243 E = µ B (7) Assumig that th fild is i th z-dirctio w ca xprss th rgy as E = µ gm B=± µ B, B s B (7) whr w hav usd g= ad th projctio of our spi oprator o th z-axis which has th two igvalus m =± ½ Our two-lvl-systm dscribd by a spi s up ad a spi dow stat ow xprics a splittig i rgy dpdig o th oritatio of th dipol momt with rspct to th applid magtic fild This is also kow as th Zma ffct For a spi-systm thr ar oly two oritatios possibl but for a gral agular momtum thr ca b may (agai oly thos allowd by th quatizatio coditios of J : m= j, + j,, j+, j ) Th lowst rgy is for th magtic dipol momt aligd paralll to th fild (i th spi atiparalll to th fild) That all momts do ot alig i that dirctio is du to thrmal fluctuatios which will td to rdistribut th momts Th avrag momt ca b foud usig simpl statistical mchaics z J Sˆz µ z = i µ i xp( Ei kbt), Z (7) whr k B is Boltzma s costat, T th tmpratur ad Z= xp E k T ( ) p i B th partitio fuctio For i= (spi up) w hav µ µ, E = µ B ad for i= (spi dow) w hav µ i= µ B, E= i= µ B B W fid that i= B i B µ = µ tah( x), z B (73) whr x = µ B k T B B Fig 7 Magtizatio as a fuctio of ivrs tmpratur 4

244 W s that for x, i for low tmpraturs all spis ar i th sam dirctio whras for high tmpraturs th thrmal rgy is too high for th spi aligmt to b of ay favour W ca look at th fr rgy of th systm: F= E TS, whr S is th tropy, which is a masur of th disordr of th systm Th radom distributio of spis thus icrass S ad th trm TS bcoms importat at icrasig T ad will rduc th fr rgy Th magtizatio of a matrial cosistig of atoms (or mor prcisly: magtic momts) which ca b dscribd by two-lvl spi stats ca ow b foud M= µ = µ tah( x), z z B (74) whr is th atom dsity Th suscptibility ca b calculatd from ad is giv as to T) x χ=µ B H cosh ( x) χ= M H plottd blow as a fuctio of /x (proportioal Fig 73 Magtic suscptibility as a fuctio of tmpratur Th avrag rgy dsity ca b foud lik th avrag dipol momt ad is giv by E= µ B tah( x) B (75) 43

245 Fig 74 Avrag rgy as a fuctio of tmpratur (arb uits) W obsrv that th avrag rgy du to itractio with th applid fild approachs as th tmpratur icrass W ca also calculat th hat capacity of this systm (s th xrcis) I th limit of high tmpraturs (or low filds) w xpad C χ= T tah( x) x ad fid (76) µ µ This is th Curi law ad th costat C= k B is plottd th simpl dpdc o tmpratur B is calld th Curi costat Blow Fig 75 Ivrs magtic suscptibility as a fuctio of tmpratur For a gral agular momtum J w fid 44

246 µ = µ gjb ( y), (77) j B j whr j( j+ ) j+ y Bj( y) = coth y coth j j j j (78) gµ B j is th Brilloui fuctio, ad y= B k T B For multi lctro atoms it is cssary to us th total, orbital ad spi agular momtum as dtrmid by Hud s thr ruls It should b otd though that i solids th orbital momtum is oft quchd du to crystal fild ffcts ad it ca thus b sufficit oly to cosidr th total spi momt 7 Coupld Momts W ow itroduc a couplig btw dipol momts ad s how this lads to prmat magtizatio dscribig g frromagts Th first couplig to cosidr is th dirct dipol-dipol couplig O dipol momt (with strgth µ ) crats a magtic fild ad a ighbourig dipol momt ( µ ) will xpric a chag i rgy giv by Eq (7) Th magtic fild from a dipol is o th ordr of µ µ µ µ µ µ µ B=, ad th rgy is thus E=± =± B = ±5 µv wh usig a 4πr 4πr 4πr typical xt-arst ighbour distac of r ~ Å ad a dipol momt of µ B This quals a thrmal rgy of 5 K ad is thus gligibl at all rlvat tmpraturs A couplig causd by dirct dipol-dipol itractio ca thus ot xplai why th dipol momts i som matrials td to alig Th origi has to b foud i th socalld xchag couplig, which is a purly quatum mchaical ffct W will dscrib this i dtail i th two followig chaptrs ad will hr mrly writ th ffct as follows Th xchag rgy btw two ighbourig particls (i ad j) is giv by E = JS S x i j, (79) whr J is th so-calld xchag itgral Followig th sig of th dot product btw th spis, w fid that J< favours atiparalll spi aligmt, thus dscribig a atifrromagt ad if J> paralll spi aligmt is favourd thus For historical rasos th lttr J is usd for th xchag itgral This is ot to b cofusd with th agular momtum J 45

247 ladig to a frromagtic cofiguratio Th magtic ordr is thus causd by th xchag couplig For ow w simplify it v furthr ad dscrib th couplig btw ighbours i a ma-fild modl as follows Th total magtic fild s by o spi is giv as H = H+ H tot ir (7) H = λm, (7) whr H is th applid fild ad Hir is th fild producd by th ighbourig spi Now, this is ot just qual to th fild du to th dipol (i ~ M ) which was s to b far from sufficit, but istad it is dscribd hr i a phomological way through th couplig paramtr λ W ot that th rquird ir fild is vry high ( B ~ 3 T), but should ot b s as a ral fild but th fild which would b cssary to achiv th sam ffct as th quatum mchaical xchag couplig Th modl dscribd hr is calld th ma-fild modl ad is du to Wiss Lt us first dscrib our frromagt at tmpraturs abov th trasitio tmpratur (th so-calld Curi tmpratur blow which frromagtic ordr occurs) Th magtizatio is giv by Eq (7) ow usig th magtic fild of Eq (7) whr ir χ p ir p, (7) is th paramagtic suscptibility w foud arlir Not that w ar lookig at th stat abov TC, whr th matrial is i th paramagtic stat Still th itractio is tak ito accout via th λ M trm Isolatig M from Eq (7) ad usig th dfiitio for th suscptibility χ= MH w fid th Curi-Wiss law T C M = χ H + λm ( ) χp C χ= = λχ T T p C (73) Hr w usd χ = C T p (valid at high tmpraturs ), whr C is th Curi costat ad T λc, whr T is th Curi tmpratur This ca b plottd lik i Fig 75 C= C whr th graph is simply shiftd to th right by By masurig th suscptibility as a fuctio of tmpratur o ca thus mak th distictio btw a paramagt ad a frromagt (or a atifrromagt, whr a shift i th othr dirctio with rspct to T= is xpctd) T C 46

248 Lt us ow look at T< < T C, ad fid a xprssio for th magtizatio of th frromagtic stat W kow that for a smbl of dipol momts th magtizatio is giv by M= µ = µ gjb ( α), j B j (74) whr H µ µ Bgj, with k T α= tot B H tot giv by Eq (7) Sic th ir fild is much largr tha th applid fild w simplify th quatios by sttig th applid fild H= ad d up with two quatios, which both hav to b satisfid ad M= M s B j ( α) M = c Tα, (75) (76) whr c is a costat W ca isolat α from Eq (76), isrt it i Eq (75) ad solv for M umrically Altrativly th combiatio ca b solvd graphically (s Fig 76) ad has solutios wh th Brilloui fuctio itrscts th straight li giv by Eq (76) (xcpt for th trivial crossig i ) Thr will oly b solutios for sufficitly low tmpraturs, i th slop of th straight li has to b smallr tha th slop of th Brilloui fuctio at α= This also dfis th trasitio tmpratur btw th ordrd frromagtic stat ad th disordrd stat ad is giv by T C = gµ B( j+ ) λms 3k B (77) Fig 76 Graphical solutio of Eq (75) ad Eq (76) at th itrsctio Thr ar oly solutios for tmpraturs blow th trasitio tmpratur TC 47

249 Th magtizatio is thus tmpratur dpdt, saturatig at T= K ad dcrasig gradually to wh T T is approachd Hr a sharp phas trasitio = C occurs (s Fig 77) W ot that thr is o abrupt phas trasitio wh a applid fild is prst Not that scals dirctly with th couplig paramtr λ T C Th xtsio of th ma-filmatrials is i pricipl straightforward modl to atifrromagtic ad frrimagtic Fig 77 Magtizatio of a frromagt as a fuctio of tmpratur Exrcis 7 Hat capacity i a two-lvl systm Look at a systm with oly two rgy lvls, whr th diffrc btw th rgy lvls is µb B Th rgy splittig thus dpds o th applid magtic fild B ad quals th systm w lookd at arlir i this chaptr Th populatio of th two lvls dpds o th tmpratur ad th systm thrfor xhibits a hat capacity Show that th hat capacity C of th two-lvl systm is giv by µ BB kbt 4 µ B C B = k B T + µ BB kbt ( ) 48

250 ad plot th hat capacity as a fuctio of tmpratur T Discuss th bhavior, spcially i th limits of low ad high tmpraturs Assum a costat magtic fild B Exrcis 7 Magtism of a molcul Look at a systm which has a siglt stat (S=) ad a triplt stat (S=) sparatd by a rgy This could b a molcul Assum that th xcitd stat is th triplt ) Fid a xprssio for th avrag magtic momt wh th systm is placd i a magtic fild B, which poits i th z-dirctio ) Show that th paramagtic suscptibility χ is giv by a simpl Curi law (~ /T) i th limit of high tmpraturs ( k T µ, B ) B B 49

251 8 Exchag Itractio for Localizd Momts Th xchag itractio is a quatum ffct ad is th fudamtal proprty dtrmiig th magtic bhaviour of a solid W usd th phomological molcular fild i th prvious chaptr but will hr look at a modl that ca xplai th xchag couplig btw two lctros Th modl is basd o th approach of Hitlr ad Lodo Lt us look at a Hydrog molcul: H I th figur blow th rlvat distacs btw th two ucli ad th two lctros of th molcul ar dfid Fig 8 dfis th gomtry ad th distacs of a H molcul: a ad b ar th ucli ad ad th lctros Th Schrödigr quatio for th molcul taks th form ħ ħ + E m m 4πε + Ψ = Ψ rab r ra rb ra r b m m m, (8) whr Ψ is th igfuctio ad E th igvalu of th molcul Th idx m m ad rfrs to th rspctiv lctro ad th idx a ad b rfr to th two ucli For th isolatd H-atom w ca writ ħ = m 4 ħ = m 4 ϕa Eϕa πεr a ϕb Eϕb πεr b (8) 5

252 ϕ whr is th atomic wav fuctio ad E th rgy of th fr atom W will ow fid Ψ ad E Upo formig a molcul th sphrical symmtry xistig i m m th H-atom is go ad this complicats th problm W will hr us th approach of LCAO: Liar Combiatio of Atomic Orbitals whr w writ th molcular orbital as a liar combiatio of atomic orbitals, which ar kow It is importat also to cosidr th spi part of th wav fuctio W us th followig otatio: α is th spi-up ( m = ½ ) fuctio ad β th spi-dow ( m = ½ ) fuctio, whr m s s rprsts th spi quatum umbr Th total orbital ca b writt as a product of a spatial part ad a spi part, sic th spatial ad spi coordiats ar idpdt For a systm of two idtical frmios th total wav fuctio has to b atisymmtric udr xchag of particls, i th sig chags A lgat costructio is th Slatr dtrmiats giv blow s Ψ m ϕα a ( ) ϕα b () ψ= MS= ϕα( ) ϕα( ) a b (83) ψ ψ 3 ϕβ a ( ) ϕα b ( ) = MS= ϕβ( ) ϕα( ) a b ϕα a ( ) ϕβ b ( ) = MS= ϕα( ) ϕβ( ) a b (84) (85) whr g ϕα() a ϕβ a ( ) ϕβ b ( ) = MS=, (86) ϕβ( ) ϕβ( ) mas lctro at atom a with spi-up Not that thy ar all atisymmtric Th total spi is idicatd for ach fuctio Th approximatio to th molcular wav fuctio is th writt as a liar combiatio of th abov dtrmiats W ow hav to solv th Schrödigr quatio ψ 4 a b 4 Ψ= cψ i i (87) i= ( ) E + V Ψ= EΨ ( ) E E Ψ= VΨ (88) 5

253 whr V= rprsts th itractio part ad is th πε + r r r r E 4 ab b a rgy of th two atoms at ifiit sparatio Lt us dfi th followig orthoormal spi fuctios σ = α( ) α( ) M = σ = β( ) α( ) M = σ = α( ) β( ) M = 3 σ = β( ) β( ) M = 4 S S S S (89) This allows us to writ th 's ψ i as ψ= σ ϕa( ) ϕb( ) ϕa( ) ϕ( ) b ψ= σϕ a( ) ϕb( ) σϕ 3 a( ) ϕb() (8) ψ = σϕ( ) ϕ( ) σϕ( ) ϕ( ) 3 3 a b a b ψ4= σ 4ϕa( ) ϕb( ) ϕa( ) ϕb( ) Multiplyig Eq (88) with gt σϕ( ) ϕ( ) k a b from th right had sid ad itgratig w E E Ψ σϕ ϕ dστ d = VΨσϕ ϕ dστ d, (8) whr th itgratio ovr spi is idicatd by σ ad itgratio ovr th spatial part by τ W ow isrt Eq (87) ad gt th followig xprssio whr k=,,3,4 ad ( ) ( ) ( ) ( ) ( ) k a b k a b τ σ τ σ 4 i= ( ) c i Hik E E S + ik =, (8) Sik= σψϕ k i a( ) ϕb( ) dσdτ τ σ (83) H = σψvϕ( ) ϕ( ) dσdτ ik k i a b τ σ W ow hav to solv th st of quatios (i =,,3,4 ad k =,,3,4) 5

254 whr H+ US H+ US H3+ US3 H4+ US4 c H US H US H3 US3 H 4 US 4 c , H3 US3 H3 US3 H33 US33 H 43 US = 43 c H + US H + US H + US H + US c U= E E This st of quatios has solutios whvr th dtrmiat vaishs May of th matrix lmts ar du to orthoormality of th spi fuctios ad th dtrmiat w hav to solv is giv by H + US H + US H + US 3 3 H + US H + US H + US = (84) This ca b solvd i thr parts, startig with th uppr lft block H + US= S = σσ ϕa( ) ϕb( ) ϕa( ) ϕ( ) b a( ) b( ) d d τ σ ϕ ϕ σ τ = σdσ( ϕa( ) ϕb( ) dτ ϕa( ) ϕb( ) ϕb() ϕa() dτ) σ τ τ Th first two itgrals ar du to orthoormality ad th last itgral is th ovrlap A = τ ( ) ( ) ( ) ( ) ϕ ϕ ϕ ϕ dτ a b b a Thus S = A Lt us ow fid H H ( ( ) ( ) ( ) ( ) ( ) ( ) ) = σdσ ϕa ϕb V dτ ϕa ϕb ϕb ϕa Vdτ, σ τ τ whr w dfi th Coulomb part as C ad th xchag itgral as Thus allowig us to writ H = C J J = τ = τ ϕ ( ) ϕ ( ) V dτ a b ( ) ( ) ( ) () ϕ ϕ ϕ ϕ V dτ a b b a 53

255 W ca ow solv th first block of th dtrmiat ad fid C J E= E+, MS=, S= (85) A Likwis for th last block (i=4, k=4) w fid C J E4= E+, MS=, S= (86) A Rtaiig th ctral part of th dtrmiat whr w us H + US H + US 3 3 H + US H + US =, S = S = σ dσ ϕ ( ) ϕ ( ) dτ= 33 3 σ H = H = σ dσ ϕ ( ) ϕ ( ) V dτ= C 33 3 a b σ τ S3= σ σϕ 3 a( ) ϕb( ) σϕ a( ) ϕb( ) ϕa( ) ϕb( ) dσdτ= A S 3 H3= σ σϕ 3 a( ) ϕb( ) σϕ a( ) ϕb( ) ϕa( ) ϕb( ) Vdσdτ= J H 3 σ τ = A = J σ τ τ a Sic th parts cotaiig diffrt spi fuctios vaish W th gt b ( ) ( ) ( ) + ( ) C+ E E J E E A J E E A C E E =, which lads to th solutios C J E= E+, MS=, S= A (87) C+ J E3= E+, MS=, S= (88) + A W thus hav four solutios; a triplt with total spi S= (i th two spis ar paralll) dscribig th frromagtic stat ad a siglt with total spi S=, 54

256 dscribig th atifrromagtic stat Not that E rprsts th rgy of th two sparatd atoms For a frromagtic groud stat th rgy of th triplt has to b lowr tha th rgy of th siglt brigig us to th coditio ( CA J) E( S= ) E( S= ) = < (89) A If w glct th ovrlap, w ca dirctly s that th coditio for a frromagtic couplig is J> (8) W thus d a positiv xchag itgral Th magitud of th xchag rgy, i th rgy diffrc btw th siglt ad triplt is ca J ad is proportioal to th molcular fild of th Wiss modl discussd i chaptr 7 Th modl ca b xtdd to solids with som cautio sic it oly dscribs localizd lctros, i orbitals which ar still atomic lik It works for g th rar arth lmt ad oxids W will i th xt chaptr s that th bad lctros also play a importat rol i dscribig frromagtism of mtals lik Co, F ad Ni Exrcis 8 Exchag ad frromagtism i -dimsioal chais Th gral Hisbrg pictur allows us to writ th xchag rgy as E = JS S as discussd i chaptr 7 Lt us ow look at a simpl -dimsioal x i j Isig modl, whr th spis ar oly allowd to poit up or dow Oly icludig arst ighbor itractios w ca writ th Hamiltoia for a chai of N spis as N H = J S S x i i+ i= (8) a) Assum that th spis ar S= for all i ad show that th rgy of th frromagtic groud stat (for J>) is E= J( N ) i b) Show that th rgy cost of flippig o spi isid th chai is giv by E= J 55

257 Th fr rgy of th systm is giv by F= E TS ad th chag i tropy upo flippig o spi is giv as S= kb l( N) sic thr ar N possibl ways to plac th flippd spi If th chag i fr rgy is gativ th systm prfrs to lav its frromagtic stat This happs for N (sic S ) at fiit tmpraturs c) Show that th critrio for th -dim chai to stay i its frromagtic cofiguratio is giv by J N< xp k T Obviously, this dpds o th strgth of th xchag couplig J ad th tmpratur T d) Assum J=5 mv ad fid th critical lgth for T = 3, 3 ad 3 K B Not, that th sam argumts holds if th xchag couplig J< ad th chai cosists of atifrromagtically aligd spis Exrcis 8 Spi clustr Look at a clustr of spis (s=/) i a cofiguratio lik th followig Th ctral atom has z ighbors, of which x (ms=/) hav spi up ad y hav spi dow (ms = -/), z=x+y a) Show that th total rgy du to th couplig btw th ctral spi ad it s ighbors for th two cass whr th ctral atom has spi up ad spi dow is giv by E =±zjη, η=(x-y)/(x+y) is th rlativ magtizatio Us E=E+(C±J)/(±A), whr A is th ovrlap, C is th coulomb part ad J th xchag couplig Also, ± rfrs to th spi oritatio btw th ctral atom ad its ighbors Nglct th ovrlap b) Show that by usig simpl statistical avragig, η ca b writt as 56

258 η=tah(zjη/kbt) Hit: look at a fixd surroudig ad lt th ctral atom vary btw th two possibl spi dirctios W ca writ th Curi tmpratur as T = gµ ( J+ ) λm/3 k = µ λm/k η=tah(η T C /T), ot that η=m/ms, whr Ms is th saturatio magtizatio, ad c) Show that w ca fid th followig rlatio btw th Curi tmpratur ad th xchag itgral k T / J= z, ad giv a stimat of J, wh usig a typical = 5 K B C C B s B B s B T C Rfrcs [8] P Moh, Magtism i th Solid Stat (Sprigr, 6) 57

259 9 Itirat Exchag Itractio To dscrib frromagtism of mtals it is importat also to cosidr th valc lctros which ar mor or lss fr to mov aroud i th crystal W alrady saw i chaptr that th coductio lctros cotribut to th paramagtic suscptibility ad hr w will look at th ffct bad lctros has o th frromagtic stat of som mtals This is also calld itirat frromagtism 9 Frromagtism of Fr Elctros Fr lctros i a solid ca b dscribd i th most simpl way as pla wavs with a wav fuctio of th form Look ow at two lctros (i ad j) which ar assumd to b i th sam spi stat, thus rprstig a frromagtic cofiguratio Th total wav fuctio ca b writt as a product of a spi part ad a spatial part Sic lctros ar frmios th wav fuctio has to b atisymmtric udr xchag of particls ad sic th spi part is symmtric for th giv cofiguratio th spatial part has to b atisymmtric This ca b rwritt as xp( ikr) ikiri Ψ ij= V ik r ik r ik r j j i j j i ik i( ki k j)( ri ri) iri+ ik jr j Ψ ij= V (9) Th probability of simultaously fidig lctro i i th volum lmt lctro j i th volum lmt dr * is th giv by Ψ Ψ dr dr j ij ij i j dr i ad Ψ ij dri drj= cos (( ki k j)( ri rj) ) dri drj (9) V Not that th probability of fidig two lctros with th sam spi at th sam plac ( r = r ) vaishs for all k ad lctros of sam spi thus prfr to stay apart i j This o th othr had rducs th Coulomb rpulsio ad thus lowrs th rgy This is du to th coditios outlid abov, amly that lctros with th sam spi d to hav a total wav fuctio big atisymmtric udr xchag Th xchag itractio thus lads to a lowr rgy for th frromagtic cofiguratio Blow w will quatify this 58

260 Look ow at a giv spi-up lctro Th k-avragd probability of fidig aothr spi-up lctro i th volum lmt dr ca b foud by avragig ovr th Frmi sphr ad ca b writt as whr r = r r is th rlativ coordiat btw th two spi-up lctros ad is th dsity of spi-up lctros, which is half of th total coctratio of all lctro, i / W ca also spak of a ffctiv lctro charg dsity actig o th giv spi-up lctro which w dot xchag itractio i j = W ow hav to avrag ovr th Frmi sphr by ρ P( r) dr= dr cos( ( ki kj) r), (93) ρ ρx du to its origi i th r = cos( k k r) (94) ( ) ( ) x i j i ( ki k j) r i( ki k j) r r = + dk dk, V FS ( ) x i j whr th cosi has b rplacd by th xpotials to as itgratio ovr ki 3 ad kj V = πk is th volum of th Frmi sphr ad kf is th Frmi wav vctor W fid FS 4 3 Th total ffctiv dsity F ρ x ( r) = 9 ρ ff ( si( kfr) kfr cos( kfr) ) 6 ( k r) s b th spi-up lctro is th dsity abov, i origiatig from othr spi-up lctros, togthr with th dsity of spi-dow lctros Ths ar ot xchag-hidrd to com clos ad w ca thus simply add th trm / to i ordr to fid th total dsity ρ x ρ ff 9( si( kfr) kfr cos( kfr ) = 6 ( kfr) ( r) (95) For larg distac this approachs th avrag dsity, but clos to th spi-up lctro th dsity dcrass sigificatly as s i Fig 9 F 59

261 Fig 9 Th lctro dsity aroud a giv spi-up lctro Th xchag hol is idicatd [9] Th ara with th rducd charg dsity is calld th xchag-hol ad dscribs th dpltd rgio of lctros with th sam spi thus ladig to a lowr rgy of th total systm du to rducd lctrostatic itractios A typical Frmi wav - vctor is k F= Å ad th xtsio of th xchag hol is th r Å A positiv xchag couplig with a frromagtic cofiguratio du to th fr lctros is th rsult of this modl calculatio With this w ar abl to dscrib frromagtism of mtals I th prvious chaptr w lookd at xchag btw two isolatd ighbourig atoms but this is isufficit to dscrib frromagtism of mtals which to a larg xtd is govrd by fr or rathr bad-lctros I th followig w will dscrib a bad modl for frromagtism 9 Bad Modl for Frromagtism Th modl is du to Stor Assum that w ca writ th ad Wohlfart ad starts with th followig Asatz rgy bads for spi-up ad spi-dow lctro as IN E ( k) = E( k) N IN E ( k) = E( k) N, (96) whr N ( N ) is th umbr of spi-up (spi-dow) lctros, N th total umbr of atoms ad I is th Stor paramtr which dscribs th rgy corrctio du to th xchag itractio dscribd abov E( k ) is th o-lctro bad disprsio Th lctro rgis ar thus split ito two bads, o cotaiig spi-up lctros ad th othr cotaiig th spi-dow lctros I Fig 9 w display th calculatd total dsity of stats for Ni, whr th lft had sid shows th spi-dow dsity ad th right had sid shows th spi-up dsity 6

262 Th two curvs ar similar but shiftd by a amout (which is rlatd to I) brigig part of th spi-up abov th Frmi lvl, thus big uoccupid This givs a ubalac i th total spi dsity ad lads to a magtizatio of th sampl Th ffct dpds o th strgth of th xchag couplig ad o th dsity of stats ar th Frmi lvl Fig 9 Th total dsity of stats for Ni Th lft part shows th spi-dow dsity ad th right part th spi-up dsity [9] Th spi-split bads ca also b writt as IR E ( k) = Eɶ ( k), IR E ( k) = Eɶ ( k) + (97) N N whr R is a xprssio proportioal to th magtizatio ad N I( N N ) Eɶ ( k) = E( k) Th splittig is assumd to b idpdt of k, which is N of cours a approximatio Th rror w itroduc is withi a factor of two W s that th splittig dpds o R which w ca fid usig Frmi statistics 6

263 R= ( f( k) f( k) ) = N k N k xp( ( E ( k) EF) kbt) + xp( ( E ( k) EF) kbt) + R= N k xp( ( Eɶ ( k) EF IR ) kbt) + xp( ( Eɶ ( k) EF+ IR ) kbt) + Th critrio for frromagtism is that R> also wh o xtral fild is applid If w assum that IR Eɶ ( k) EF w ca xpad R Sttig α=ir ad x= Eɶ k E w fid ( ) F f( x α) f( x+ α) = f '( x) α f '''( x) α 3, 6 whr f( x± α) f( x) + f '( x)( ± α) + f ''( x)( ± α) + f '''( x)( ± α) 3 6 ad w gt 3 f k f k R= IR IR 3 N k Eɶ k N 4 k Eɶ k ( ) ( ) ( ) ( ) ( ) 3 (98) or 3 f( k I ) f k R + = 3 N k Eɶ ( k) 4N k Eɶ k ( ) ( ) ( IR ) 3 Sic f ''' > th right had sid is gativ ad sic f ' < th coditio for frromagtism bcoms I f( k) + < (99) N k Eɶ k ( ) Lt us assum that th tmpratur T = K Hr w fid f ' bcoms a dlta fuctio ad 6

264 f k N k Eɶ k N k ( ) ( ) ( ) f k dk= E k E Eɶ k ( ) V δ ( ɶ( ) ( ) N( π) 3 3 ( ( F) ) V ) dk= D( EF) whr D( E F ) is th total lctroic dsity of stats at th Frmi lvl Th factor ½ coms from that w oly sum ovr o spi dirctio Isrtig ito Eq (99) ad lttig V D( EF) = D( EF F) big th dsity of stats pr atom pr spi dirctio w N fid th Stor critrio for frromagtism N, ID ( E F) > (9) Importat is that th dsity of stats at th Frmi lvl is high as s so oft for may othr physical ffcts i solids (g coductivity, lctroicc hat capacity or th Pauli paramagtic suscptibility (chaptr )) ad mor spcifically th strgth of th xchag couplig giv by I I th figur blow typical valus for I, D ad ID ar plottd W s that th wll kow mtallic frromagts F, Co ad Ni satisfy th Stor critrio Fig 93 Th Stor paramtr I, th dsity of stats ad th product DI is plottd for various lmts [9] 63

265 Rfrc [9] H Ibach ad H Lüth, Solid Stat Physics, d (Sprigr, 995) Exrcis Go through th dtails from Eq(9) to Eq (95) ad discuss Fig 9 64

266 3 Domais ad Aisotropy W will i this chaptr ivstigat thr importat magtic rgy trms, amly xchag, aisotropy ad magtostatic rgy ad discuss how thy ifluc th domai structur obsrvd for frromagtic matrials Th xchag rgy forcs th momts to poit i th sam dirctio thus makig a sampl which oly cotais o domai This o th othr had lads to larg stray filds which icrass th so-calld magtostatic rgy Th lattr ca b rducd wh th sampl is brok up ito smallr domais poitig i diffrt dirctios cacllig out th stray filds Th balac btw xchag ad magtostatic rgy (also calld th dipol trm) thus dtrmis th domai structur A third rgy trm, th so-calld aisotropy, plays a importat rol i th dtaild shap ad siz of th domai boudaris ad also i th dirctio of th magtizatio of ach domai with rspct to th crystal axs 3 Aisotropy Lt us start with th aisotropy which dtrmis th asy-axis of magtizatio, i th dirctio i th crystal i which it is asist to magtiz th matrial Fig 3 shows th magtizatio as a fuctio of applid magtic fild for iro, ickl ad cobalt It is clar that thr ar crtai dirctios which ar asir to magtiz tha othrs Sic th aisotropy rgy (also calld th magtocrystalli rgy) usually is small compard to th xchag rgy th absolut magitud of th magtizatio is idpdt of th dirctio Fig 3 Magtizatio of F, Ni ad Co show for diffrt crystallographicc dirctios as a fuctio of applid magtic fild [3] Th origi of th aisotropy lis i th spi-orbit couplig Du to this th charg distributio is o logr sphrical but sphroidal (crud approximatio - spcially for atoms i a crystal) Th asymmtry is tid to th dirctio of th spi so that a rotatio of th spi dirctios rlativ to th crystal axis chags th ovrlap btw ach atomic orbital ad thus th xchag ad th lctrostatic itractio This th lads to a chag i th total rgy I a crystal th orbital agular momtum is 65

267 quchd du to crystal fild itractios which lock th orbitals ito spcific dirctios ad th magtizatio of a solid is thrfor ormally dtrmid by th spi momt Not that th spi-orbit couplig is mostly small compard to crystal- highly fild ffcts ad th idicatdd chag of th orbital i Fig 3 is thrfor xaggratd Fig 3 Asymmtry of th ovrlap of lctro distributios of ighborig atoms provid a mchaism for th magtocrystalli aisotropy rgy [3] How asy it is to magtiz a crystal thus dpds o which dirctio w choos ad th crystal is thrfor aisotropic Th aisotropy rgy thus dscribs th diffrc i rgy dpdig upo dirctio of magtizatio Lt us first look at a uiaxial systm lik hcp Co which has o asy axis alog th c-axis Assum th magtizatio M poits i a dirctio θ with rspct to th c-axis W ca ow projct M oto th asy ad hard axis Th part M cos( θ) poits i th asy axis dirctio whras th part M si( θ) poits i th hard axis dirctio ladig to a icras i th total rgy of th systm big proportioal tomsi( θ) W ca xprss th aisotropy rgy i trms of a Taylor xpasio i si(θ) 3 4 E= c si( θ ) + c si ( θ) + c3 si ( θ) + c4 si ( θ) + (3) whr ci ar costats Du to th symmtry of th crystal thr is o diffrc i rgy with rspct to rflctio i th clos packd plas of th crystal, i E( θ) = E( θ+ π) ad all trms with odd ordr must vaish ad w gt th followig for th aisotropy rgy 4 E= K si ( θ) + K si ( θ) (3) Th costats K i ar tmpratur dpdt, dcrasig with icrasig tmpratur such that ar th Curi tmpratur thr is o prfrrd dirctio for domai magtizatio For Co at room tmpratur K = 4 x 5 J/m 3 ad K = x 5 J/m 3 For a cubic crystal lik F with th asy axis alog th cub dgs th aisotropy rgy is mor complicatd W fid that 66

268 ( ) E= = K m m + m m + m m + K m m m x y y z z x x y z, (33) whr m= m, m, m = MM givs th dirctios w xpad i Agai oly v ( x y z) powrs ar allowd for symmtry rasos Th lowst ordr trm m + m + m x y z idtical to uity ad dos ot dscrib aisotropy ffcts At room tmpratur w hav for F K = 4 x 4 J/m 3 ad K = 5 x 4 J/m 3 ad for Ni K = -57 x 3 J/m 3 (ot th sig ad compar with th asy dirctios for bcc F ad fcc Ni show i Fig 3) Aisotropy rgis ar usually i th rag - 7 J/m 3 corrspodig to a rgy pr atom of V This is idd vry small compard to th xchag rgy is 3 Domai Boudaris W will latr discus th rgtic advatags upo havig mor tha o domai i a frromagtic sampl Hr w will look i dtail at th boudary btw two domais Blow is show a 9 ad a 8 domai boudary Fig 33 Lft: 9 ad a 8 domai boudaris Right: A Bloch domai wall spis ovr a 8 domai boudary [3] Lt us look at a 8 domai boudary ovr which th spis hav to r-oritat Btw ach spi pair th xchag rgy is Ex= JS S= JS cos( ϕ), whr ϕ is th agl btw two ighborig spis Th xchag itractio will kp th spis aligd ad th rgy diffrc du to turig o spi ϕ is ( ) ( ) showig th rotatio of ( ) ( ) ( ) Ex ϕ = Eϕ Eϕ= JS ϕ JS = JSϕ, whr w hav xpadd th cosi Th total rotatio π is takig plac ovr N atoms, thus ϕ= π N 67

269 ad thrby E = JSπ N for ach spi pair Thr is li of atoms pr a, whr x a is th lattic costat ad th rgy dsity (pr ara) is thus σ = x JSπ Na (34) Hr w hav tak ito accout that thr ar N spi pairs i total W s that σ for N ad th domai wall will thus td to b vry larg by x distributig th cost of misaligd spis ovr as may pairs as possibl This o th othr had lads to may spis poitig i dirctios whr th aisotropy rgy is ' high (Fig 3), thus icrasig th aisotropy rgy giv by E = K si ( ϕ) i th simplst form I total w gt θ= π N NK Ea= K si ( θ) K si ( θ) dθ=, whr θ N Not that this is a volum dsity ad th aral dsity du to aisotropy is thus foud by multiplicatio with a NKa σ a= (35) I total w hav th domai wall rgy dsity JSπ NKa σdw= σx+ σa= + (36) Na Th first trm dcrass with N ad th scod trm icrass with N W ow just d to miimiz σ with rspct to N i ordr to fid th domai wall width δ W fid dw 3 N= Sπ J Ka ad δ= Na i= π a δ= Sπ J Ka (37) Isrtig approximat valus: S=½, J=mV, a=87å, K= 4x 4 J/m 3 (F) w fid = m, thus th wall xtds ovr ca 8 atoms δ 33 Magtostatic Ergy ad th Origi of Domai Formatio Th raso for domai formatio at all has to b foud i a third rgy trm, amly th magtostatic rgy of th particl W kow that B= µ ( H+ M) ad B = which lads us to 68

270 Look at a homogously magtizd sampl At th boudary M divrgs ad M< at th uppr ad M> at th lowr boudary rspctivly This rsults i H > at th uppr ad H < at th lowr boudary, i as if magtic chargs (or pols) xistd at th surfac: positiv at th uppr ad gativ at th lowr Th rsultig fild btw thm is calld th dmagtizig fild Hd Not that H poits i th opposit dirctio as M ad thus works agaist a applid d fild tryig to magtiz th sampl For a logatd sampl this ffct is mor prooucd i th short dirctio, sic i th log dirctio th pols ar far apart ladig to a low H H d H= M Fig 34 Magtizatio (M), dmagtizig fild (H d) ad th magtic chargs or pols at th boudary [3] W ca also look at th stray filds from a frromagt This dpds strogly o th umbr ad cofiguratio of domais as s i th figur blow W s that th magtic chargs cacl out wh th cofiguratio is favorabl (d ad ) thus rducig th magtostatic rgy Fig 35 Stray filds aroud diffrt domai cofiguratios i a frromagt [3] I gral w ca xprss dpdt tsor T H d i trms of th magtizatio ad a matrials shap H = T M d Th magtostatic rgy, which is giv blow, ca ow b valuatd 69

271 E µ ms= d V H MdV (38) Not that this looks lik a dipol-dipol trm itgratd up (sic M is a dsity) Th factor ½ is to avoid doubl coutig of dipol pairs For particls with high symmtry, g a llipsoid, i th ovrall shap ad M poitig alog o of th pricipal axs w ca writ th tsor as Tx T = T y T For a sphr w hav Tx= Ty= Tz= ad = M H 3 d - For a vry logatd llipsoid 3 (i a rod which is log i th z-dirctio) w hav, Tx= Ty= Tz= ad Hd will work agaist M oly i th x ad y dirctio, but ot i th z dirctio It is thus x, y asir to magtiz th sampl alog z tha alog x or y sic Hd = M ad z H This ffct is also calld shap aisotropy Th diffrc i asy/hard axis d = du to shap is usd g i data storag whr w wat magtic particls with two wll dfid oritatios A logatd llipsoid is th prfct choic sic th magtizatio prfrs to b i o of th two log axis dirctios, ad ot i th orthogoal short axis dirctios For a flat sampl (oblat) w hav T= T =, T= ladig to a asy axis of magtizatio i th pla This agrs x y z with obsrvatios for thi films which ca b dscribd by a aisotropy trm µ Ms cos ( θ), whr M is th saturatio magtizatio ad θ s is th agl btw th surfac film ad th surfac ormal It should b otd that surfac aisotropy trms ca altr this for vry thi films Thr is thus a dlicat balac btw xchag, magtocrystalli aisotropy ad magtostatic rgy (or shap aisotropy) which dtrmis th domai structur of a frromagtic particl z 34 Sigl Domai Naoparticls Fially w will look at a sphrical particl ad s wh a sigl domai is favorabl This has may applicatios i data storag usig th magtic stat of a particl (up/dow) for storig bits of iformatio Hr it is importat to hav wll dfid domai structurs which bst is achivd wh oly a sigl domai xist 7

272 Look at a small frromagtic particl with volum V ad radius r havig ithr or domais Fig 36 A sigl domai ad a two-domai frromagtic ao particl For th sigl domai particl th rgy is giv by th magtostatic rgy from Eq (38), whr th dmagtizig fild for a sphr is giv by H M Isrtig ad itgratig ovr th volum of a sphr w fid For th -domai particl th magtostatic rgy is roughly halvd thus rducig th total rgy O th othr had w hav to crat a domai wall which costs rgy σ dw A, whr th ara is A=πr Th total rgy is th Isrtig σ from Eq (36) (with th valu of N miimizig σ ) w fid that th dw critrio for a sigl domai particl is E = µ πm r 9 3 E = µ πm r + σ πr 3 9 dw dw d = 3 E < E 9πS JK a r< µ M r< 3m, (39) whr w hav isrtd th paramtrs from arlir = x -3 J/m ad assumd a magtizatio of µ M= T This is actually slightly smallr tha th domai wall boudary width of m ad might b cofusig, but th abov should mor b s as a rough stimat Th trasitio btw sigl-domai ad two-domai structur is gradual ad thr will thus b a itrmdiat stat whr th magtizatio xhibits a vortx lik structur Rfrcs [3] C Kittl, Itroductio to solid stat physics, 8d (Wily, 5) [3] S Bludll, Magtism i codsd mattr (Oxford Uivrsity Prss, ) σ dw 7

273 Exrcis: Look at a small frromagtic particl with ithr or domais a) At which particl radius is th -domai cas rgtically favorabl? (th µ magtostatic rgy: Ems=, whr th dmagtizig fild for a Hd MdV sphr is giv by H M Ems is ca rducd to half for th -domai cas d = 3 Assum a domai wall rgy dsity of ca x -3 J/m ad a magtizatio µ M= T ) Now a -domai particl is placd i a magtic fild B i th z dirctio Th magtizatio M of th particl poits i a dirctio with a agl θ to th z-axis Th particl has volum V b) What is th rgy of th particl, wh takig ito accout th dipol itractio with th fild ad th aisotropy which w assum ca b dscribd simply by K si ( θ)? (whr K is th aisotropy costat, ot that th uits ar J/m 3, i a rgy dsity) c) Fid th first two miima ad th first maxima ad plot th rgy for V=5MBV i th rag θ [ ; π+ ] K d) Assum that th magtizatio of th particl will jump btw th stats dscribd by th two miima i rgy Writ dow a xprssio for th rat of that procss Idicat th cssary coditios for th matrials paramtrs (M, K,) if th particl should b usd for data storag applicatios 7

274 Appdix Naostructurs Much of ths lctur ots prsupposs som dgr of familiarity with th basic proprtis of aostructurs It is assumd that lctroic stats of quatum wlls, wirs ad dots i th ifiit stp barrir ad ffctiv mass approximatios ar kow For compltss, howvr, ad to rfrsh th radr s mmory, w prst i this appdix a brif ovrviw of som commo aostructurs, show i Fig A Th ffctiv mass approximatio is assumd throughout ad oly simpl gomtris ar cosidrd W cosidr th quatum stats i various dimsios ad cofimt pottials as wll as thir accompayig dsity of stats W focus o lctros rathr tha hols sic stats for th lattr follows asily from th formr Figur A A slctio of aostructurs Top from lft to right: quatum wll, wir ad dot with rctagular cofimt Bottom from lft to right: quatum wir ad dot with circular cofimt A Quatum Wlls Quatum wlls ar D structurs, for which th boud stats ar charactrizd by stadig wavs i th cofid dirctios ad ruig i th othr two Takig th cofimt to b alog z w ca writ th Hamiltoia i th ffctiv mass approximatio as ˆ ħ H= + V( z) m 73