dn de σ = ρ = ρ i + ρ ph (T) Summary Last Lecture Phys 446 Solid State Physics Lecture 8 (Ch , )

Transcription

1 Phys 446 Solid Stt Physics Lctur 8 Ch , Sury Lst Lctur Fr lctro odl siplst wy to dscrib lctroic proprtis of tls: th vlc lctros of fr tos bco coductio lctros i crystl d ov frly throughout th crystl. Lst ti: Discussd th fr lctro Drud odl pplid to lctroic spcific ht d lctricl coductivity. Tody: Fiish with Fr lctro odl. Thrl coductivity. Motio i gtic fild: cyclotro rsoc d ll ffct Strt w chptr: rgy bds i solids Sury Lst Lctur cotiud Fri rgy - rgy of th highst occupid lctroic lvl t T K; 3D cs: π 3π 3π F F v F Dsity of stts of 3D fr lctro gs: lctricl coductivity: D d d 3 π t cpcity of fr lctro gs t low tprturs B T << F : 3 τ σ ρ ρ i ρ ph T Thrl coductivity: Wid-Frz lw K LσT π B L 3

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3 Motio i gtic fild: cyclotro rsoc Applid gtic fild th Lortz forc: F [v B] Prfct tl, o lctric fild - th qutio of otio is: Lt th gtic fild to b log th z-dirctio. Th dv dt ω v c y dv dt y B ωcv whr ω c - cyclotro frqucy For odrt gtic filds ~ fw G, ω c ~ fw Gz..g. for B. T, f c ω c /π.8 Gz bsorptio Cyclotro rsoc p i bsorptio of lctrogtic wvs t ω c Usd to sur th ffctiv ss i tls d sicoductors

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5 Sury of fr lctro odl Fr lctro odl siplst wy to dscrib lctroic proprtis of tls: th vlc lctros of fr tos bco coductio lctros i crystl d ov frly throughout th crystl. Fri rgy - th rgy of th highst occupid lctroic lvl t T K; π 3π 3π F F v F Dsity of stts of 3D fr lctro gs: D d d 3 π t cpcity of fr lctro gs t low tprturs B T << F : 3 lctricl coductivity: τ σ ρ ρ i ρ ph T Thrl coductivity: Wid-Frz lw K LσT π B L 3 Liittios of fr lctro odl Th fr lctro odl givs good isight ito y proprtis of tls, such s th ht cpcity, thrl coductivity d lctricl coductivity. owvr, it fils to pli ubr of iportt proprtis d pritl fcts, for pl: th diffrc btw tls, sicoductors d isultors It dos ot pli th occurrc of positiv vlus of th ll cofficit. Also th rltio btw coductio lctros i th tl d th ubr of vlc lctros i fr tos is ot lwys corrct. Bivlt d trivlt tls r cosisttly lss coductiv th th oovlt tls Cu, Ag, Au d or ccurt thory, which would b bl to swr ths qustios th bd thory Th probl of lctros i solid y-lctro probl Th full iltoi cotis ot oly th o-lctro pottils dscribig th itrctios of th lctros with toic ucli, but lso pir pottils dscribig th lctro-lctro itrctios Th y-lctro probl is ipossibl to solv ctly siplifid ssuptios dd Th siplst pproch w hv lrdy cosidrd - fr lctro odl Th t stp is idpdt lctro pproitio: ssu tht ll th itrctios r dscribd by ffctiv pottil. O of th ost iportt proprtis of this pottil - its priodicity: Ur Ur T Bloch thor Writ th Schrödigr qutio th pproitio of o-itrctig lctros: ψr wv fuctio for o lctro. Idpdt lctros, which oby o-lctro Schrödigr qutio priodic pottil Ur Ur T - Bloch lctros Bloch thor: th solutio hs th for whr u r u rt - priodic fuctio with th s priod s th lttic Bloch thor itroducs wv vctor, which plys th s fudtl rol i th otio i priodic pottil tht th fr lctro wv vctor plys i th fr-lctro thory. ħ is ow s th crystl otu or qusi-otu

6 Aothr coclusios followig fro th Bloch thor: th wv vctor c lwys b cofid to th first Brilloui zo This is bcus y ' ot i th first Brilloui zo c b writt s ' G Th, if th Bloch for holds for ', it will lso hold for rgy bds Substitut th solutios i th Bloch for Schrodigr qutio, obti: ito th with priodic coditio: u r u rt For y, fid ifiit ubr of solutios with discrt rgis, lbld with th bd id For ch, th st of lctroic lvls spcifid by is clld rgy bd. Th ifortio cotid i ths fuctios for diffrt d is rfrrd to s th bd structur of th solid. ubr of stts i bd Th ubr of stts i bd withi th first Brilloui zo is qul to th ubr of priitiv uit clls i th crystl. Cosidr th o-disiol cs, priodic boudry coditios. Allowd vlus of for uifor sh whos uit spcig is π/l Th ubr of stts isid th first zo, whos lgth is π/, is π//π/l L/, whr is th ubr of uit clls A siilr rgut y b pplid i - d 3-disiol css. Tig ito ccout two spi orittios, coclud tht thr r idpdt stts orbitls i ch rgy bd. rly fr lctro w bidig odl First stp: pty-lttic odl. Wh th pottil is zro th solutios of th Schrödigr qutio r pl wvs: ψ r c i r whr th wv fuctio is orlizd to th volu of uit cll c ow, tur o w pottil. Cosidr it s w priodic prturbtio i iltoi. Fro prturbtio thory hv: ψ U ψ i i i, i, ',, ' U i, whr id i rfrs to i th bd; rfrs to pty-lttic odl. Th first tr is th udisturbd fr-lctro vlu for th rgy. Th scod tr is th vlu of th pottil i th stt i, costt idpdt of c st to zro Th third tr th d ordr corrctio vishs cpt ' G i

7 Th third tr c b rwritt s U G - Fourir trsfor of th crystl pottil U Filly w obti for th rgy: owvr, th prturbtio thory cot b pplid wh th pottil cot b cosidrd s sll prturbtio This hpps wh th gitud of th pottil bcos coprbl with th rgy sprtio btw th bds, i.. I this cs w hv to solv th Schrödigr qutio plicitly Thr r spcil poits for which th rgy lvls bco dgrt d th rltioship holds for y o-zro vlu of th pottil: This coductio iplis tht ust li o Brgg pl bisctig th li oiig th origi of spc d th rciprocl lttic poit G w priodic pottil hs its or ffct o thos fr lctro lvls whos wv vctors r clos to os t which th Brgg rflctio c occur. I ordr to fid th rgy lvls d th wv fuctios r ths poits w d to ivo th dgrt prturbtio thory. Th rsult: Illustrt this bhvior usig o-disiol lttic Th splittig of th bds t ch Brgg pl i th tdd-zo sch rsults i th splittig of th bds both t th boudris d t th ctr of th first Brilloui zo. This rsults is prticulrly sipl for poit lyig o th Brgg pl: Obti Th gitud of th bd gp is qul to twic th Fourir copot of th crystl pottil.

8 Itrdit Sury Th Bloch thor: th wv fuctio for lctro i priodic pottil c b writt i th for: whr u r u rt - priodic fuctio with th priod of th lttic Origi of th rgy gps Th rgy spctru of lctros cosists of st of cotiuous rgy bds, sprtd by rgios with o llowd stts - gps Fuctio stisfis th sytry proprtis of crystl, i prticulr, th trsltiol ivric: G This llows cosidrig th first Brilloui zo oly. Also, ivrsio sytry: - rly fr lctro odl w crystl pottil. lctro bhvs sstilly s fr prticl, cpt th wv vctors clos to th boudris of th zo. I ths rgios, rgy gps ppr: g U G W focusd o th rgy vlus got d r th zo dgs. wy fro th zo dgs ow, lt's s how th wv fuctios r odifid by w crystl pottil. Fro th prturbtio thory, hv for th first bd wy fro th zo dg: UG ψ ψ ψ r, gi, w lv oly th rst-bd d tr, s w did for th rgy lvls bcus of th lrg doitor for highr bds. Fuctios ψ i r thos of fr lctros: ψ ~ If is ot clos to th zo dg, th cofficit of ψ is vry sll So, i ψ ψ - bhvs li fr-lctro / L owvr, r th dg, th doitor bcos vry sll ust us th dgrt prturbtio thory th fuctios ψ d ψ r trtd qully For siplicity w cosidr o-disiol lttic, for which th zo dgs r ½G π/ Brgg rflctio occurs Th rsult: ψ / L i π i π [ ψ π ± ψ π ] ± ± / t th zo dg, th scttrig is so strog tht th rflctd wv hs th s plitud s th icidt wv th lctro is rprstd thr by stdig wv, uli fr prticl Th distributio of th chrg dsity is proportiol to ψ, so tht π ψ ~ cos - highr rgy ψ ~ si π - lowr rgy L

9 Assuptios: Tight bidig odl toic pottil is strog, lctros r tightly boud to th ios th probl for isoltd tos is solvd: ow wv fuctios d rgis of toic orbitls w ovrlppig of toic orbitls Strt with D cs i Bloch fuctio i th for: ψ, / whr positio of th th to, lttic costt; ψ - toic orbitl ctrd roud th th to lrg r, but dcys rpidly vy fro it. Sll ovrlp ists oly btw th ighborig tos

10 priodic fuctio, / / i i i ψ Th fuctio chos stisfis th Bloch thor: r, ψ, i - ~ - - bhvs li toic orbitl Th rgy of th lctro dscribd by ψ is ψ ψ Obti ' ', ' i Sutio ovr, ' covrs ll th tos i th lttic. For ch ', th su ovr givs th s rsult gt qul trs / / i put ' / / i got sprt th tr fro th othrs: sigifict oly for tulig ffcts icluds i isoltd to lctro rgy of ' ± i Writ th iltoi d d with crystl pottil s su of toic pottils: ' v v v pottil du to th to i th origi; ' du to ll th othrs ' is sll coprd to v r th origi. Rtur to : ' i Th first tr: d ' β β ' toic lvl rgy of v d d sig is chos so tht β is positiv, sic ' is gtiv v ' i β β is sll: r lrg oly r th origi, whr ' is sll Lt's cosidr th itrctio tr hs two trs i th su: ± ' ovrlp sll bcus of gligibl - hv For v d d ' i β Rtur to d ' ovrlp itgrl th toic fuctios r sytric gt th s rsult for - ow hv i i cos β β β Lt th si 4

11 si 4 Origil rgy lvl hs brodd ito rgy bd. Th botto of th bd is - loctd t Th bd width 4 proportiol to th ovrlp itgrl For sll, / << r th zo ctr - qudrtic disprsio, s s for fr lctro whr - ffctiv ss Grlly, d d For π/, you will fid tht Th rsults obtid c b tdd to 3D cs. For sipl cubic lttic, gt si si si 4 z y locity of th Bloch lctro: v d d Grlly, isotropic: z y i, i i,, - ivrs ffctiv ss tsor ffctiv ss is dtrid by th curvtur of disprsio v 3D: Mtls, Isultors, sitls, sicoductors isultor si tl tl Dsity of stts ubr of lctroic stts pr uit rgy rg, d: Dd D dsity of stts. I th ffctiv ss pproitio, d d d stts π π π ubr of stts d stts i th shll, d: Tig ito ccout spi, ultiply by. Gt 3 D π

12 Sury Tight bidig odl strog crystl pottil, w ovrlp. Th bd width icrss d lctros bco or obil sllr ffctiv ss s th ovrlp btw toic wv fuctios icrss Cocpt of ffctiv ss: i priodic pottil lctro ovs s i fr spc, but with diffrt ss: i,, y, z Mtls: prtilly filld bds; isultors t K th vlc bd is full, coductc bd is pty. Sicoductors d sitls. locity of th Bloch lctro: v ris costt i prfctly priodic lttic 3 Dsity of stts. Sipl cs: D π i i locity of th Bloch lctro: Sury v I th prsc of lctric fild th lctro ovs i -spc ccordig to th rltio: This is quivlt to th wto s scod lw if w ssu tht th lctro otu is qul to ħ Dyicl ffctiv ss: d d is ivrsly proportiol to th curvtur of th disprsio. I grl cs th ffctiv ss is tsor: i,, y, z i i p c ħ is clld th crystl otu or qusi-otu. Th ctul otu is giv by ψ i ψ C show tht p v, whr is th fr lctro ss, v is giv by th bov prssio Physicl origi of th ffctiv ss Sic p v - tru otu, o c writ: Th totl forc is th su of th trl d lttic forcs. But So, w c writ dv dt F t Ft F F t L dv F dt tot F t F Th diffrc btw d lis i th prsc of th lttic forc F L L Currt dsity Fr lctro odl: v ; - th ubr of vlc lctros pr uit volu, d v - th vlocity of lctros. Grliz this prssio to th cs of Bloch lctros. I this cs th vlocity dpds of th wv vctor d to su up ovr vctors for which thr r occupid stts vilbl: Covit to rplc th sutio by th itgrtio. Th volu of -spc pr llowd vlu is 8π 3 / w c writ th su ovr s obti for th currt dsity:

13 ols Alrdy ow tht copltly filld bds do ot cotribut to th currt Thrfor, c writ: w hd for th currt dsity: c qully wll writ this i th for: th currt producd by lctros occupyig spcifid st of lvls i bd is prcisly th s s th currt tht would b producd if th spcifid lvls wr uoccupid d ll othr lvls i th bd wr occupid but with prticls of chrg - hols. Covit to cosidr trsport of th hols for th bds which r lost occupid, so tht oly fw lctros r issig. C itroduc th ffctiv ss for th hols. It hs gtiv sig. lctricl coductivity τ Fr lctro odl: F σ Lt's obti th corrspodig prssio withi th bd thory. S id: o lctric fild - th Fri sphr is ctrd t th origi. Th totl currt of th syst is zro. Applid fild th whol Fri sphr is displcd Th vrg displct is δ τ Th currt dsity: -v g ε F v F, ucop. δ F, ε F vf, v τ F g ε F F, σ vfτ F g ε F 3 F Cyclotro rsoc Lortz forc: F v B qutio of otio: d v B dt Chg i i ti itrvl δt: δ v Bδt W hd R ll ffct logously, if oly hols r prst, hv R h h Priod of cyclic otio: T Grlizd cyclotro frqucy for Bloch lctro: Wh ffctiv ss pproitio is pplicbl, v ħ/ δt B δ v πb δ ωc v ω c B if both lctros d hols r prst R σ Rhσ R σ σ h h You will show this i your t howor

14 Sury Physicl origi of ffctiv ss: crystl fild Cocpt of th hol: cosidr trsport of th hols for th bds, which r rly occupid. lctricl coductivity: Grlizd cyclotro frqucy: ll cofficit for tls with both lctros d hols: g v F F F ε τ σ 3 v B c δ π ω h h h R R R σ σ σ σ