Semiconductor Quantum Structures (Summer, 2015)

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1 Smiconducor Quanum Srucurs (Summr by I. Kamiya T Aim of is Cours Undrsanding lcronic propris of quanum srucurs mainly using Quanum Mcanics & Saisical Pysics Plans # Topics Conns Covrag Wa ar quanum srucurs? Quanum srucurs and ir prparaion principl of saisical pysics HO Wavs and Paricls Dualy of lig and lcrons Cap. 3 T Scrödingr quaion Wavs and diffrnial quaions dnsiy and currn Cap. 4 Infini and fini ponial wlls Cap. 5 Triangular wlls coupld quanum wlls Cap. 6 Tim variaion numrical soluions Cap. 7 Tunnlling Tunnl barrirs compl barrirs doubl barrirs Cap. 3 8 WKB approimaion unnlling dvics Cap. 3 9 Tunnling dvics (con. Landaur forumula Cap. 3 Priodic ponials singl-lcron unnlling Cap. 3 T armonic oscillaor Hrmi polynominals gnraing funcions opraors Cap. 4 Vibraing laics lcrons undr magnic fild Cap. 4 3 Saionary prurbaion ory Prurbaion sris Sark ffc Cap. 6 4 Variaional mod Cap. 6 5 Moion in cnrally symmric ponials Two dimnsional armonic oscillaor Cap. 8 6 Final Eam

2 Rfrncs ( David K. Frry Quanum Mcanics an inroducion for dvic pysiciss and lcrical nginrs nd d. Taylor & Francis ISBN ( 原康夫著 量子力学 ( 岩波書店 994 年 ISBN ( 岡崎誠著 物質の量子力学 ( 岩波書店 995 年 ISBN (3 Jaspri Sing Elcronic and Opolcronic Propris of Smiconducor Srucurs Cambridg 3 ISBN ( Carls Kil: Inroducion o Solid Sa Pysics 8 d. Wily 5; ISBN X (3 Jon Singlon: Band Tory and Elcronic Propris of Solids Oford ; ISBN (5 Nil W. Ascrof/N. David Mrmin: Solid Sa Pysics W. B. Saundrs 976; ISBN (6 Walr A. Harrison Elcronic Srucur and Propris of Solids Dovr 989; ISBN (7 J. C. Pillips Bonds and Bands in Smiconducors Acadmic Prss 973; ISBN J. C. Pillips and G. Lucovsky Bonds and Bands in Smiconducors nd d. Momnum Prss 9; ISBN 小笠原毅一訳 半導体結合論 ( 吉岡書店 976 年 ( 絶版 (8 浜口智尋 半導体物理 ( 朝倉書店 年 ISBN (9 黒沢達美 物性論 ( 改訂版 ( 裳華房 年 ISBN

3 Capr. Wavs and paricls. Inroducion Classical mcanics : Principia Mamaica Issac Nwon 686 Quanum mcanics : La 9C - Black-body radiaion Planck 89s i Low-frquncy viw : radiaion incrasd as a powr of frquncy ii Hig-frquncy viw : radiaion dcrasd rapidly wi frquncy Unificaion I( f df ~ df f 3 f p k BT (. I: innsiy of radiaion f : frquncy T: mpraur [ J s] : Planck s consan.38 [ J K ] E f (. k B : Planck s consan Einsin s Tory of poolcric ffc : Gnrally accpd as proof a radiaion is du o quanum paricls of nrgy in (. Bor s quanum modl of aom : Elcrons is in discr slls wi wll dfind nrgy lvls Hisnbrg & Scrödingr (ca.97 : Formal quanum ory d Brogli : d Brogli wav (mar wavs λ (.3 p m v λ : d Brogli wavlng of paricls Quanum Mcanics Mamaical dscripion of pysical sysms wi non-commuing opraors AB BA Complmnariy : dualiy of wavs and paricls Corrspondnc principl : Doubl sli : Classical Howvr quanum ffcs sould no b ignord MOS ransisor affcd by quanum ffcs Tunnl diod : fundamnal propry 3

4 . Lig as paricls poolcric ffc Vlociy of mid lcrons dpnds only on wavlng of incidn lig and no on radiaion (innsiy (T nrgy of mid lcrons varis invrsly on wavlng of incidn lig Numbr of mid lcrons dpnds on innsiy of radiaion and no on wavlng. Einsin : Tramn of lig in rms of is corpuscular naur poons E ν ω (. wr (Dirac consan. 4

5 Mal Workfuncion E W Frmi nrgy E F Wn incidn poon nrgy saisfis T kinic nrgy of lcrons mid is E > E W poolcrons ar mid. E E E k W Accping lig as paricls momnum! In classical mcanics momnum is associad wi mass Mass of lig In fild ory for masslss wav in scalar v p (.3 c λ And nc p k (.4 λ conncing momnum of paricl and wavlng of wav. ( k or mor gnrally k is calld wav vcor d Brogli rfrrd o s E ν ω (. v p (.3 c λ as Einsin rlaions ( Elcrons as wavs Traing lcromagnic wavs as flow of paricls (poons Traing paricls as wavs Innsiy of wav Numbr of paricls Composi wav funcion of wo non-inracing wavs (paricls ( ( + ( (.5 Probabiliy wavs Using wav vcors is can b rwrin as i( kω i( k ω ( A + B (.6 Considr a saic cas wiou im dpndnc A saic magnic fild is radd roug Au loop in Fig..3. Classically no lcric fild is inducd. 5

6 L s assum a an lcric fild is inducd along along ring (CCW Two lcron wavs nr rig dg a φ T lcron a movs roug uppr sid (CCW is acclrad T lcron a movs roug lowr sid (CW is dclrad Tn k k E d (.7 Trfor pas diffrnc inducd a oupu is φ + d Edl Edl d Edl Φ d E n da B n da Φ (.8 wr Φ : quanum uni of flu ( 磁束量子. Saic magnic fild inducing pas diffrnc : Aaronov-Bom ffc (959. Wn magnic fild is varid oscillaion in conducanc is obsrvd..4 Posiion and momnum Can paricls b dscribd solly by wavs? No. Wavs (wavfuncions: compl disribud Carg & posiion of paricls : ral localizd 6

7 Probabiliy of finding an lcron (paricl a a posiion a im ( (.9 Normalizaion of wavfuncion ( d (. Limiing spac o a bo (for 3D L ( d lim (. L L Linariy of wavfunion (allows suprposiion ( c ψ ( + c ( (. ψ Trfor wr c ( d c ψ ( + c ψ ( ψ ( d + c ψ ( d d ( ψ ( d ψ ( ψ ( d ψ (.4 (.3 In is cas wavfuncions ar orogonal..4. Epcaion of posiion Epcaion valu or avrag valu can b obaind as Sor-and noaion ( d ( ( d (.5 ( ( ( d ( (.6 (.7 Eampl: Gaussian wav funcion i ( ω A p (.8 Normalizaion of is funcion (by drmining A ( p( iω d A d A { p( } A p( d A d (.9 7

8 4 A Trfor pcaion valu of posiion is ( d p( p( d Hnc avrag posiion is a. Now pcaion valu of is givn by d p d ( p( d Posiion rprsnaion of wav funcion d (. (. Posiion opraor ( ( (. wr is ignvalu. i.. In Fig..5 is pak pcaioin valu? Probably clos bu no ncssarily. Wid of wav funcion ( ( (.3 : uncrainy in posiion Sinc wav funcion is i ( ω A p (.8 uncrainy is (.4 To obain a wav pack a dscribs posiion of paricl acly i.. Dirac s dla funcion 8

9 Igonoring im (aloug i can b ndd o im dpndn ( δ ( (.5 pak.4. Momnum Wav funcion in Fig..5 : variaion in spac (no uniform Localizd paricl : muc smallr variaion in spac Dscribing wav funcion in rms of spaial frquncis as an invrs ransform φ( k ik dk φ ( k : Fourir ransform of k : spaial frquncy ; (.6 or momnum wav funcion p k (.4 λ Calculaing avrag momnum p ( φ kφ φ kφ dk (.7 As an ampl of wav funcion w us i ( ω A p (.8 in wic cas momnum wav funcion is φ ( k ( 3 4 k ik d + p 3 4 k ( ik d 4 ik d (.8 using formula Sinc is is basically idnical o (.8 k a k To obain pcaion valu of momnum from posiion rprsnaion wav: Saring from (.7 and subsiuing for wav funcion d a p ( φ kφ φ kφ dk (.7 9

10 p i ik φ k ( d (.8 ik' ( φ kφ dk ( ' d' k i dk dk d' ik' ( ' d' k d ik d ik' ik ( ' d' d ( ' δ ( ' ( i d ( ' ( (.9 Hr Fourir ransform of dla funcion ik ' δ ' dk (.3 as bn usd. Tus momnum opraor can b prssd in posiion rprsnaion p i (.3 ik d.4.3 Non-commuing opraors As w saw abov momnum opraor in posiion rprsnaion as bn don by a diffrnial opraor. Trfor opraors for posiion and momnum will no commu. [ p] p p (.3 Commuaor brack g. ( [ p] + i i (.33 Wn wo variabls or opraors do no commu y canno b masurd simulanously. For diffrnial opraor o produc simpl ignvalu wav funcion nds o b ip p Tis funcion is no ingrabl canno yild ignvalus for bo posiion and momnum. Fourir ransform pair of posiion and momnum wav funcions: If posiion is known ( δ ( (.5 pak

11 T Fourir ransform as uni ampliud vrywr. T rlaionsip bwn wo uncrainis in posiion and momnum p. g. Gaussian wav funcion p 4 (.34 ( 4 T wav pack is cnrd a pak and pcd valu is d ( p d (.35 T uncrainy in posiion is d p d (.36 ( ( T appropria momnum wav funcion is φ ( k ik p d ( 4 ( 3 4 ( ( ik d 4 4 k k 3 4 p 4 k ik p d ( i k 3 4 (.37 4 From is rsul i is sn a momnum wavfuncion is also cnrd abou zro momnum. d Tn uncrainy in momnum can b found as 4 k ( p k d (.38 Trfor uncrainy in momnum is p. Trfor uncrainy rlaionsip bcoms p

12 T uncrainy principl: dscribs conncion bwn wo uncrainis (in drminaion of pcaion valus for wo non-commuing opraors For wo opraors A and B wic do no commu A B [ A B] (.39 p [ p] p p i + i + i i i Classical masurmn problm Considr simpl im-varying ponnial T frquncy conn is ( ωτ τ F ( ω (.4 + To rproduc simpl ponnial wi our lcronics w rquir a bandwid ω wic is a las of ordr τ. Tis is o say ω > > τ E ( Rurning o mporal baviour Momnum wav funcion : NOT ncssarily cnrd a zro momnum If momnum wav funcion is cnrd a k k nir posiion rprsnaion wav funcion movs a an avrag vlociy of k m. v Tn dos pak of posiion funcion pak mov a is vlociy? Dos is affc uncrainy in posiion? Fourir Invrs Transform φ( k Taking im dpndn componn ino accoun ik i( kω ( φ( k dk (.6 dk (.4 If frquncy ω is a funcion of k

13 ω ω ( k ω( k + ( k k + L (.43 k k k Hnc posiion wav funcion is composd of a group of closly rlad wavs propagaing in sam dircion (assuming ( k ( φ for < is dfind as wav pack. k. L s considr influnc of disprsion on propagaion of wav funcion. Inroducing a variabl u k k and (.43(.4 bcoms wr i( k ω i ( u ω u ( φ( u + k i ( k ω i u( ω φ u + k ω ω and igr ordr rms in (.43 ar nglcd. k Tn using a rm ik ( ω ik ( ω i( kω ik ( ω i u ( ω ( φ( u + k i i i du ( ω ω k i ( u+ k ( ω ( ω ω k i ( u+ k ( ω ( ω ω k ( ω φ u + k φ u + k φ k dk (.4 i( kω Rf. du du du (.44 du (.45 Trm i( ω k ω squar of magniud. sows pas sif in posiion wav funcion. (Bu as no influnc on T nir wav funcion movs wi a vlociy givn by ω or group vlociy v g ω ω (.46 k k k Hr k canno b akn arbirarily sinc pak in momnum disribuion mus rla o avrag moion of wav pack in posiion spac. Tr mus b a valu for k a acually saisfis v g k m ω k (.47 3

14 Ingraing (.47 by k k p E ω (.48 m m T group vlociy of wav pack: Avrag momnum of momnum wav funcion Rlas vlociy/momnum o nrgy of paricl Now considr im variaion. For a cnrd wav pack (.37 φ (.4 urns ino ( k 4 k (.37 4 u + iuiω ik du (.49 From (.48 and (.47 k u k u k ω ( u + k + uk + + uvg + (.5 m m m m m m m By convring (.4 iu vg du (.4 ( φ( u Insring (.5 ino (.49 saic ffciv momnum wav funcion 4 v g ik ( u p u + i φ (.5 m from wic p and p can b drivd. For driving posiion rprsnaion wav funcion by inroducing ino (.49 + i (.5a m v (.5b g 4

15 5 [ ] p p du iu u v ik v ik g g (.53 Tis as ac idnical form as original wav funcion in posiion rprsnaion (.34 cp a i is unnormalizd du o im variaion. So l s normaliz. Considring in mind a is a compl numbr S m + Wi is normanlizaion pcaion valu for is givn by v g Similarly sandard dviaion for is givn by + m S Sinc p 3 4 m m p + + Sowing a uncrainy incrass as a funcion of im or wav pack gs widr as i propagas. (Sif of cnroid + broadning....5 Summary Summary Summary Summary