h Summary Chapter 7.

Transcription

1 Summry Chptr 7. In Chptr 7 w dscussd byond th fr lctron modl of chptr 6. In prtculrly w focusd on th nflunc of th prodc potntl of th on cors on th nrgy lvl dgrm of th outr lctrons of th toms. It wll hlp us bttr undrstnd why th lctrc conductvty vrs btwn dffrnt mtrls: not tht good nsultor cn hv rsstvty s hgh s 10 Ohm.cm whl good conductor hs rsstvty of Ohm.cm. In prvous courss you mght hv bn ntroducd to th nrgy bnd dgrms of nsultors, mtls, nd smconductors (s Fg. 1 of chptr 7). In ths chptr w wll lrn mor bout th rsons bhnd th bnd-dgrm. Th nrgy of fr lctron cn b found by solvng for th Schrodngr quton ssumng th potntl nrgy s vrywhr zro,..: h h U E E m m [1] In quntum w hv lrnd tht th solutons r compl pln wvs,.. r A [] Th fr lctrons n pc of coppr of sz L r no longr fr s thy r contnd n th bloc of coppr, so thr wv functon s pctd to b zro outsd th mtrl. Snc good wv-functon s contnuous, w hv to pply th boundry condtons t =0 nd =L. Smlr to th phonon cs dscussd n chptrs 4 nd 5, w cn pply zro boundry condtons for =0 nd =L, or bttr w cn pply prodc boundry condton nd ssum (0)-(L). For th prodc boundry condton, th gnfunctons r stll pln wvs s n th compltly fr lctron cs, but now not ll wvvctors r llowd, only th followng -vlus r llowd: 4, y, z 0,,,... [3] L L Also th nrgy vlus r gvn by: E h y z [4] m Not tht th =, nd thus s rltd to th momntum of th fr lctron ( p h ). Th rlton btwn th momntum nd nrgy for fr lctrons s gvn by prbol (s lso dshd curv n Fg. 4). Strctly spng th conducton lctrons n prodc crystl structur r not fr but prnc th lctrc potntl of th postv on cors. So th potntl t th bottom of th wll of lngth L, whr L s th lngth of th bloc, s not zro but vrs prodclly. W pct th potntl nrgy of th lctron to b low nr th postv on cors nd hgh n btwn th ons. A stch s md n Fg. 3 on

2 pg 166. S lso th fgur 1 blow whch shows th tomc potntl n sold (nr th top), th potntl nrgy of th fr lctron gs (FEG) of chptr 6 (mddl), nd th potntl nrgy of th nrly fr lctron modl (bottom). Fg. 1: From top to bottom: tomc potntl n sold (top); potntl nrgy functon of conducton lctron n th fr lctron modl (mddl); potntl nrgy functon of conducton lctron n th nrly fr lctron modl (bottom) Notc tht th potntl nrgy of n lctron n hydrogn tom s dscrbd by Coulomb s lw for V; n SI unts: U 1 qv ` [5] 4 r o Snc conducton lctrons r solutons cn b rprsntd by wvs (soluton of th Schrodngr quton) thy cn dffrct from ths prodc potntl just l photons dd n chptr. So w pct tht whn th wvlngth of th conducton lctrons s smlr to th prodcty of th lttc tht dffrcton wll t plc nd lctron wll b dffrctd of th lttc structur (rd -vctor of lctron wll chng). I pct strong constructv ntrfrnc of th wvs scttrd of two conscutv toms f th pth lngth dffrnc of th scttrd wvs s ctly n, whr n s n ntgr nd s th wvlngth of th pln wv. Ths condton s mt whn th wvlngth of th conducton lctron s ctly qul to whr s th lttc dstnc. S lso th fgur stchd blow. So whn th wv-vctor of th conducton lctron s qul to ()=, I pct strong ntrcton btwn th lctron s wv-functon nd th crystl lttc. So for vlus t th frst Brlloun zon boundry pln wv trvllng towrds th lft wll b Brgg rflctd towrds th rght. So I pct tht th soluton of th Schrodngr quton wll consst of lnr combnton of both pln wv solutons nr th zon boundrs.. Or: [6] [7]

3 Not tht th soluton of th Schrodngr quton nr th Brlloun zon boundry s no longr trvlng wv but stndng wv. Fg. : Intrfrnc of two trvlng wvs rflctd of nghborng toms. Th probblty dnsty functon of both wv-functons dscrbd by quton [6] nd [7] cn b clcultd from * (whr th str ndcts th compl conjugtd),.. 4cos cos 1 1 * * [8] And for th wv-functon of quton [7]: 4sn cos 1 1 * * [9] Both solutons r stchd n fgur 3 on pg 166 of Kttl. Th probblty dnsty of th (+) soluton s mmum nr th postv on cors, whl th probblty dnsty of th (-) soluton s mmum n btwn th two on cors (dshd functon). Th fgur lso shows th probblty dnsty functon of trvlng pln wv soluton (constnt ln). Th (+) soluton hs th lowst nrgy whl th (-) soluton hs th hghst nrgy of th thr wv-functon. Notc tht th pln wv s not sttonry stt s t wll b mmdtly Brgg rflctd f t trs to propgt through th prodc crystl. Anothr ntrstng ths bout th wv-functon of Fg. 3 s tht thy loo th sm from ny lttc pont (ms sns!). Th pctton vlu of th nrgy of both stts cn b clcultd usng th nrgy oprtor nd th wv-functon,.. sndwch th nrgy oprtor btwn * nd nd thn ntgrt ovr ll spc (0-L). I pct th ntc nrgy of both solutons to b comprbl s thy both hv th sm wvlngth, nd th wvlngth tlls m normlly somthng bout th momntum of th prtcl, whch for fr prtcl s rltd to th ntc nrgy (KE=p m). I pct th potntl nrgy of both wv-functons howvr to b qut dffrnt. Th potntl nrgy oprtor s gvn by prsson [5]. So:

4 U * 1 1 d d [10] 4 o 4 o So th pctton vlu of th potntl nrgy s nd of wghtd ntgrl. Snc s lrg for th rs of low potntl nrgy I pct tht th +wv-functon to hv low potntl nrgy. On th contrry nspctng Fg. 3 blow I s tht s lrg for th rs whr th potntl nrgy s hgh, so (-) wll hv lrgr potntl nrgy. Fg. 3: Potntl nrgy (dshd blu curv) functon of conducton lctron (nrly fr lctron modl) nd probblty dnsty functon of lctrons wth wv-vctor t th frst zon boundry. Th orng dots rprsnt th postv on cors. Summrzng: nr th Brlloun zon boundrs I pct tht th - dsprson of th nrly fr lctron dffrs from th - dsprson of fr lctron. Both dsprson curvs r stchd n fgur on pg 164 of Kttl. Pont A corrsponds to (+) nd pont B corrsponds to (-). In btwn both nrgs, no sttonry stts sts. Hv lso loo t th - dsprson rlton stchd n fgur 6 on pg 170 nd n th fgur blow. Not tht th -s of fgur 6 s not but. You cn s tht bcus of th ntrcton of th lctron wth th postv nucl nr th zon boundrs bndgps r ntroducd n th - dsprson (s lso fgur blow). Fg. 4: Enrgy dsprson for fr lctron modl (FEG: dshd curv) nd nrly fr lctron modl (NFE: blu sold ln). Not tht th quntty long th -s s th wv-vctor nd th quntty long th y-s s th totl nrgy. Th dshd lns t =p nd =-p rprsnt th boundrs of th frst Brlloun zon.

5 Th lrgr th vrton of th potntl nrgy of th postv on cors, th lrgr th bndgp. In clss U U cos, th w clcultd tht for snusodl potntl nrgy wth mpltud U,. bndgp s qul to E g U. To rlly undrstnd th lctronc bndstructur of sold w hv to solv th Schrodngr quton for crtn prodc potntl nrgy functon. In gnrl ths rqurs dvncd computr progrms to do: Dr. Scolfro uss Dnsty Functonl Thory (DFT) for clculton of th bnd-structur of solds. DFT s bsd on th Bloch thorm: Th gnfunctons of th wv quton for prodc potntl r th product of pln wv.r tms prodc functon u (r) tht hs th sm prod s th crystl lttc. Or n othr words: f th potntl nrgy s prodc functon wth prodcty T, th soluton of th Schrodngr quton cn b wrttn s th product of compl pln wv nd functon u (r) whch hs th sm prodcty s th potntl nrgy functon,..: [11] r r u r Not tht th compl pln wv prt of quton [11] s th wv-functon of fr prtcl. So w s tht wvfuncton s modultd compl pln wv. Not tht u (r) n gnrl hs much smllr prod thn th pln wv (s lso th fgur blow). Fg. 5: Bloch functon for 3s lctron n sodum: notc tht th wv-functon s product of pln wv wth long prod nd functon u (r) whch hs th sm prodcty s th lttc. Th blc dots rprsnt th poston of th toms n th crystl lttc. W dscussd two proofs of Bloch s thorm: 1. Assum tht th soluton of th Schrodngr quton s gvn by: r [1] r C Whr (r) mtchs th prodc boundry condtons for bloc of mtrl wth lngth L, so (0)=(L) nd L=N, whr s th lttc constnt, N s th numbr of toms, nd L s th totl lngth of th bloc of mtrl. Not tht n lctron cn b consdrd to consst of sum of pln wvs ch wth thr own phs nd mpltud. Almost ny s llowd but strctly spng s dscrtzd to 0, +-L, +- 4L, tc. Equton [1] cn b consdrd to b Fourr srs. W cn rwrt th sum by splttng th summton n two prts,..

6 r [13] G r C 1 BZ G G Whr th frst summton s only ovr th frst Brlloun zon (1BZ), nd th nd summton s ovr -vctors n hghr Brlloun zons: not tht for ch n th frst Brlloun zon, thr s pont n th nd Brlloun zon tht dffrs from by G. Lt us focus on just on of th pln wvs of quton [13]: r r Gr C [14] G r C G G G G Not tht th summton on th lft s Fourr srs nd s thus prodc wth th lttc (compr quton [14] wth quton [9] of chptr ). So rplcng th summton on th lft by u (r) gvs us Bloch s thorm.. Bloch s thorm mpls tht r T u r T r T r r T r T u r r T T [15] Or n othr words th wv-functon t tom poston r+t (whr T s crystl lttc vctor) cn b clcultd from th wv-functon t r by multplcton wth phs fctor.t. So f I cn proof 15 I bsclly proof Bloch s thorm. Consdr N dntcl lttc ponts sprtd by. Th symmtry of th prodc boundry condton mpls tht w cn fnd soluton to th wv qutons tht loos l: C [16] bc of prodc boundry condton: N C N [17] As th wv functon s sngl vlud: C N n n N 1 C [18] If w ssum tht =n(n) thn w cn rwrt [18] s: [19] Whch s th sm quton [15]. W dscussd n clss th Krong Pnny modl dscrbd on pgs of Kttl. Th modl ssums squr potntl nrgy dstrbuton cusd by th postv on cors. It thn solvs th Schrodngr quton for th two typs of rs. Th mth s smlr to th prtcl n th squr bo

7 problm tht ws dscussd n quntum. Solutons for th low potntl nrgy rgons r compl pln wvs trvlng n both drctons. Solutons for th hgh potntl nrgy rgons r ponntl dcyng. Th constnts n th solutons cn b found by nforcng th boundry condtons t th ntrfcs btwn both rgons,.. (1) good wv functon s contnuous nd () good wvfuncton n n r wth fnt nrgy s dffrntbl. Togthr wth Bloch s thorm ths lds to four qutons nd four unnown. W r loong for th non-trvl solutons, so w st th dtrmnnt qul to zro. Pls chc quton (1) on pg 169 nd m sur tht you now th orgn of K nd lowr cs, nd how thy rlt to th physcl proprts of th systm. Also chc th chptr 7 hndout s tht contnd svrl prtty slds tht mght b usful whn studyng th Krong-Pnny mthod. Th Krong Pnny modl cn b usd to clcult th bnd-structur of prodc crystl. Th fgur blow shows two rprsnttons of th bnd structur clcultd wth th Krong-Pnny modl: th tndd zon schm (blc) nd th rducd zon schm (rd). Th tndd zon schm s sngl vlud functon tht strtchs from mnus nfnty to nfnty. Along th -s s th wv-vctor (ssoctd wth th momntum but not qut th momntum) nd long th y-s s th nrgy of th lctron. Th tndd zon schm shows th nd E combntons of sttonry stts. Th fgur shows tht thr r nrgy vlus for whch no sttonry stts sts. Th fgur lso shows th rducd zon schm. For th rducd zon schm on plots th - dsprson only for th frst Brlloun zon. Prts of th tndd zon schm tht fll outsd th frst Brlloun zon r shftd ovr on or mor rcprocl lttc vctors so thr dsprson curv s dpctd n th frst Brlloun zon. Not tht for th rducd zon schm th dsprson s no longr sngl vlud functon. W us th rducd zon schm to smplfy llowd trnstons. An lctron tht jumps vrtclly n th rducd zon schm dgrm hs th sm crystl momntum n th ntl nd fnl stt. Pls s lso th dscusson blow tht summrzs wv-vctor consrvton ruls dscussd n th frst 7 chptrs of th boo. Fg. 6: Etndd zon schm for dsprson rlton of fr lctron (blc dshd) nd nrly fr lctron (blc sold); Rducd zon schm for nrly fr lctron modl s stchd n sold rd.

8 Erlr n th boo w hv notcd tht n prodc lttc th momntum consrvton ruls cn b wrttn s wv vctor consrvton ruls nvolvng th rcprocl lttc vctors: 1. Th dffrcton condton of lctrons, nutrons, or -ry photons of prodc crystl could b wrttn s (chptr, pg 31): ' G [0] Whr s th wv-vctor of th scttrd prtcls, th wv-vctor of th ncdnt prtcls nd G rcprocl lttc vctor. Multplyng both sds of th quton by h br wll rsult n: h ' h hg [1] Not tht th lft sd s th chng off momntum of th ncdnt prtcls upon dffrcton, so th rght prt should b th rcol of th totl crystl. Not lso tht f th dffrcton condton s not mt, thr wll not b dffrctd bm nd thr s no chng of momntum btwn nd s ncdnt wv s not dffrctd.. Also lctrons n th mtrl hv to oby quton [0] upon scttrng. As th lft prt of quton [1] s th crystl momntum chng of th scttrd lctron, th rght prt of quton [] cn b consdrd to b th momntum chng of th rst of th mtrl,.. ll othr lctrons nd th on-lttc. Equton [0] s oftn rfrrd to s th wv-vctor slcton rul (s lso Kttl pg 100). Not tht whn phonon s nvolvd n th scttrng nd s bsorbd tht thn th wv-vctor slcton rul wll bcom: ' G K [] Whr s th wv-vctor of th scttrd lctron, s th wv-vctor of th ncdnt lctron, G s rcprocl lttc vctor, nd K s th wv-vctor of th bsorbd phonon. Rorgnzng quton [] nd multplyng by h br gvs: h ' h hg hk h G K [3] Th lft prt of quton [3] s th crystl momntum chng of th lctron. Th rght prt s th crystl momntum chng of ll othr lctrons nd th postv on cors. Th phonon cn b consdrd to chng th prodcty of th lttc nd thus th ffctv rcprocl lttc vctor: so w cn consdr quton [3] s dffrcton condton smlr to quton [0]. Kttl mphszs tht phonon dos not crry physcl lnr momntum, whch s thus dffrnt from photons, lctrons, nd nutrons. Th fctor hk s rfrrd to s th crystl momntum on th phonon. 3. Whn two phonons colld nd r convrtd to sngl phonon th crtd phonon hs wv-vctor qul to th wv-vctor of th two ntl phonons (pg 13 of Kttl),..

9 K [4] 1 K K3 W sw tht nhrmonc ffcts cn cus ntrcton btwn two phonons. But lso crystl dfcts cn mdt ntrcton btwn phonons. Kttl dd not proof ths quton, t ws just prsntd. It sys tht crystl momntum s consrvd wth two phonons ntrct. Not tht phonon mods byond th frst Brlloun zon boundry r dntcl to mods n th frst Brlloun zon. So f K 1 +K s n th nd Brlloun zon, w cn subtrct rcprocl lttc vctor to fnd th dntcl mod n th frst Brlloun zon (s lso dscusson on pg 93 nd 94 of Kttl). So: K1 K K3 G [5] W rfrrd to ths s n umlpp procss. 4. In chptr 6 umlpp procsss wr lso ntroducd for lctron phonon ntrcton to bttr undrstnd lctrcl conductvty. S Fgur 13 on pg 15. Assumng tht n lctron wth wv-vctor bsorbs phonon wth wv-vctor q. Th wv-vctor of th scttrd lctron should b qul to: ' q [6] If s stutd n th frst Brlloun zon w sp of norml procss. If s stutd n -spc byond th zon boundrs w cn consdr t to b scttrd from phonon wth wv-vctor q+g: ' q G [7] W rfrrd to such procss s n umlpp procss snc drcton of s nd of oppost to (s lso Fg. 13 on pg 15). So summrzng, w found tht for th ntrcton of lctrons, phonons, nd photons n prodc crystl, wv-vctor consrvton ruls sts: s for mpl qutons [5], [7], nd [10] provdd bov. Not tht for fr lctron th wv-vctor s proportonl to th momntum of th lctron, so wv-vctor consrvton ruls r smlr to momntum consrvton ruls. So whn prtcls ntrct n prodc crystl: 1. Consrvton of nrgy s stll vld, so th totl nrgy of th prtcls bfor th collson should b qul to th totl nrgy of th prtcls ftr th collson.. Consrvton of momntum s rplcd by wv-vctor consrvton lws nvolvng th rcprocl lttc vctor. Th fgur blow shows trnston tht occurs whn conducton lctron n crystl bsorbs photon. Both consrvton ruls lstd bov pply. Although photons crry momntum, th momntum of photon s so smll tht t cn normlly b gnord. Consdrng th consrvton ruls ctd bov, ftr bsorpton I pct th ctd lctron to hv n nrgy qul to th sum of ts

10 orgnl nrgy nd th nrgy of th photon. I lso pct tht momntum consrvton s obyd nd tht th ctd lctron hs wv-vctor tht s qul to th wv-vctor of th unctd lctron plus or mnus rcprocl lttc vctor (s quton [5] bov). So only f n unoccupd stt sts wth tht prtculr nrgy nd tht prtculr wv-vctor, th lctron cn bsorb th photon. Th fgur blow shows th trnstons for th tndd zon schm (blu rrows) nd th trnstons for th rducd zon schm (rd). Plottng th dsprson rltons n th rducd zon schm wll smplfy th ndcton of trnstons tht do not nvolv momntum chng. For mpl th bsorpton of photon by n lctron wll not sgnfcntly chng th -vctor of th lctron s th photon hs such smll momntum comprd to th lctron. So ths trnstons r ndctd by vrtcl lns n th rducd zon schm. Not tht n most lctroncs ttboos th -dpndnc of th nrgy s omttd nd th bnddgrm shows only th vlbl sttonry nrgy stt but now s functon of th poston n th dvc (s th plots of Fg. 1 of chptr 7). Fg. 7: Allowd trnstons for bsorpton of photon n tndd zon schm (blu rrows) nd rducd zon schm (rd rrows).