Lecture 6. Schrödinger Equation and relationship to electron motion in crystals. Reading: Notes and Brennan Chapter , 7.4, and

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1 Lctur 6 Scrödingr qutin nd rltinsip t lctrn mtin in crystls Rding: Nts nd rnnn Cptr.0-.4, 7.4, nd.5-.6 Grgi Tc

2 Scrödngr qutin S w d w ccunt fr t wvli ntur f smll prticls li lctrns? Scrödngr qutin: n lctricl Prprtis f Mtrils, Slymr nd Wls pint ut tt. Li t 5 Pstults, tr r NO pysicl ssumptins vill t driv t Scrödngr qutin Just li Nwtn s lw f mtin, Fm, nd Mwll s qutins, t Scrödngr qutin ws prpsd t plin svrl srvtins in pysics tt wr prviusly unplind. Ts includ t tmic spctrum f ydrgn, t nrgy lvls f t Plnc scilltr, nn-rditin f lctrnic currnts in tms, nd t sift in nrgy lvls in strng lctric fild. n Scrödingr s furt ppr nds wit: p nd liv tt t v ttmpt will turn ut t usful fr plining t mgntic prprtis f tms nd mlculs, nd ls t lctric currnt in t slid stt. Grgi Tc

3 Scrödngr qutin S w d w ccunt fr t wvli ntur f smll prticls li lctrns? Scrödngr qutin: Lts strt wit t Clssicl Hmiltnin nd sustitut in t prtrs tt crrspnd t t clssicl stt vrils. Kintic nrgy prtr K m P V Ttl i t Ptntil lctrn mvs trug nrgy prtr Grgi Tc

4 Grgi Tc Sinc t lft nd sid vris nly wit psitin, nd t rigt nd sid vris nly wit tim, t nly wy ts tw sids cn qut is if ty r qul t cnstnt ( w will cll tis cnstnt, ttl nrgy,. Tus, w cn r tis qutin int tw qutins: T slv t Scrödingr qutin n must m n ssumptin ut t wv functin. Lts ssum t wv functin s sprt sptil nd tmprl cmpnnts: (,, (,,, ( t w z y t z y φ Plugging tis ( int t Scrödingr qutin nd dividing t sids y ( w rriv t: ( ( ( ( t w t w t i V z y z y m,,,, t w w i V m Cnsidr first t tim vril vrsin (rigt sid tn ltr w will min t sptilly vril prtin. Tis will giv us tim vril slutins nd, ltr, sprt sptilly vril slutin. Scrödngr qutin

5 Grgi Tc Cnsidr t tim vril slutin: t w w i w i t w ( t i t i t w t w ω ( r ( Tis qutin prsss t pridic tim ntur f t wv qutin. ω wr Scrödngr qutin

6 ntrductin t Quntum Mcnics Cnsidr t spc vril slutin: m V T cmind prtr is clld t Hmiltnin V m Hˆ mmntum prtr m ( j V ˆ p m Vˆ Kintic nrgy Ptntil nrgy Ttl nrgy Clssiclly, mmntum, pmv nd intic nrgy is ½ (mv ½ (p /m Grgi Tc

7 ntrductin t Quntum Mcnics Cnsidr spcific slutin fr t fr spc (n lctrsttic ptntil, V0 wv slutin is (lctrn trvling in t dirctin in D nly: wr m V 0 m i ( π λ m i r m Sinc w v t dd ur tim dpndnt prtin (s ( prvius ur ttl slutin is: ( w( t i ( ωt i( ωt Tis is stndrd wv qutin wit n wv trvling in t dirctin nd n wv trvling in t dirctin. Sinc ur prlm sttd tt t lctrn ws nly trvling in t dirctin, 0. Clssiclly, mmntum, pmv nd intic nrgy is ½ (mv ½ (p /m Grgi Tc

8 ntrductin t Quntum Mcnics n intrsting sid: Wt is t vlu f? Sinc is prility, [ d] i ii i d d d Tis rquirs t vnisingly smll (unlss w rstrict ur univrs t finit siz nd is t sm prility fr ll nd t. Mr imprtntly it rings ut quntum pnmn: f w nw t lctrns mmntum, p r, w cn nt nw it s psitin! Tis is rsttmnt f t uncrtinty principl: p ħ/ Wr p is t uncrtinty in mmntum nd is t uncrtinty in psitin Grgi Tc Wv Pct Discussin n rd:

9 ntrductin t Quntum Mcnics T slutin t tis fr prticl mpl rings ut svrl imprtnt srvtins ut t dul wv-prticl ntur f ur univrs: i( ωt ( w( t Wil prticls ct s wvs, tir crg is crrid s prticl... yu cn nly sy tt tr is prility f finding n lctrn in prticulr rgin f spc, ut if yu find it tr, it will v ll f it s crg tr, nt just frctin. nrgy f mving prticls fllws squr lw rltinsip: Nudc nd Pirrt Fig.3 m p m Clssiclly, mmntum, pmv nd intic nrgy is (mv / (p /m Grgi Tc

10 ntrductin t Quntum Mcnics Wt ffct ds tis - squr lw rltinsip v n lctrn vlcity nd mss? T grup vlcity (rt f nrgy dlivry f wv is: d d v g dp d S t spd f n lctrn in t dirctin dfind y p is fund frm t slp f t - digrm. Similrly, sinc m d m d S t ffctiv mss f n lctrn is rltd t t lcl invrs curvtur f t - digrm Nt: rnnn sctin 8. rigrusly drivs t qutin fr v g nd m Figur ftr Myr nd Lu Fig. Grgi Tc

11 Wt ffct ds n lctrsttic ptntil v n n lctrn? Cnsidr t lctrn mving in n lctrsttic ptntil, V. T wv slutin is (lctrn trvling in t dirctin in D nly: wr m m ( π λ m V ( V i ( V i r 0 m V Sinc w v t dd ur tim dpndnt prtin (s ( prvius ur ttl slutin is: Grgi Tc ( w( t i ( ωt i( ωt Tis is, gin, stndrd wv qutin wit n wv trvling in t dirctin nd n wv trvling in t dirctin. Sinc ur prlm sttd tt t lctrn ws nly trvling in t dirctin, 0. Wn t lctrn mvs trug n lctrsttic ptntil, fr t sm nrgy s in fr spc, t nly ting tt cngs is t wvlngt f t lctrn.

12 Lclizd Prticls Rsult in Quntizd nrgy/mmntum: nfinit Squr Wll First ndd tl: Cnsidr n lctrn trppd in n nrgy wll wit infinit ptntil rrirs. T rflctin cfficint fr infinit ptntil ws s t lctrn cn nt pntrt t rrir. ( 0 m 0 Gnrl Slutin : ( sin 0 Grgi Tc sin ( 0 π m wr r λ undry Cnditins : ( n m (0 n 0 nπ sin V ( cs( 0 m nπ fr n ±, ±, ± 3... n π nd n m ftr Nudc nd Pirrt Figur.4

13 Lclizd Prticls Rsult in Quntizd nrgy/mmntum: nfinit Squr Wll Wt ds it mn? ( n n nπ sin nd n n π m stnding wv rsults frm t rquirmnt tt tr nd t t rrir dgs (i.. C s: (0(0. T wvlngt dtrmins t nrgy. Mny diffrnt pssil stts cn ccupid y t lctrn, c wit diffrnt nrgis nd wvlngts. Grgi Tc ftr Nudc nd Pirrt Figur.4c,d,

14 Lclizd Prticls Rsult in Quntizd nrgy/mmntum: nfinit Squr Wll Wt ds it mn? Slutin fr muc lrgr. Nt: ffst vrticlly fr clrity. ( n n nπ sin nd n n π m Rcll, fr prticl s ~. nstd f ing cntinuus in, is discrt in n!.. t nrgy vlus (nd tus, wvlngts/ f cnfind lctrn r quntizd (t n nly crtin vlus. Nt tt s t dimnsin f t nrgy wll incrss, t spcing twn discrt nrgy lvls (nd discrt vlus rducs. n t infinit crystl, cntinuum sm s ur fr prticl slutin is tind. ftr Nudc nd Pirrt Figur.5 Grgi Tc

15 Wt ut n lctrsttic ptntil stp? Cnsidr t lctrn incidnt n n lctrsttic ptntil rrir, V. T wv slutin (D nly: V(<00 V(>0V V Rgin 0 Rgin W v lrdy slvd ts in rgins nd. T ttl slutin is: ( w ( w ( t ( t i i ( ωt i( ωt ( ωt i( ωt wr π λ m nd π λ m ( V Grgi Tc

16 Wt ut n lctrsttic ptntil stp? V cnt d... ( w ( t i ( ωt i( ωt Rgin 0 Rgin ( w ( t i ( ωt i( ωt wr π λ m nd π λ m ( V Wn t wv is incidnt n t rrir, sm f it is rflctd, sm f it is trnsmittd. Hwvr, sinc tr is nting t t rflct t wv c, 0. Grgi Tc ncidnt Wv Rflctd Wv Trnsmittd Wv ( ω t i( ωt i( ωt i nrgy is cnsrvd crss t undry s, ncidnt Wv Rflctd Wv Trnsmittd Wv m ϖ incidnt rflctd ϖ trnsmittd V m ϖ

17 Wt ut n lctrsttic ptntil stp? V cnt d... 0 Rgin Rgin W cn pply ur prility currnt dnsity cncpt, ρ mi ρ ( j 0 ( r ( r j ( r ( r ( r ( r t t dfin trnsmissin, T, nd rflctin cfficints, R, suc tt t trnsmissin nd rflctin prility is J J trnsmittd rflctd T T nd R R J J J ncidnt mi J ncidnt mi Similrly, J Trnsmittd mi ( i d d ( i nd J ( i d d mi incidnt R flctd i( ωt i( ωt i( ωt i( ωt ( i ( i mi incidnt Grgi Tc

18 Wt ut n lctrsttic ptntil stp? V cnt d... Rgin 0 Rgin Tus, T T J J trnsmittd incidnt nd R R J J rflctd incidnt ( ( nd ( ( T T R R ( ( Tis trnsmissin prility is drmticlly diffrnt frm tt qutd in MNY quntum mcnics tts NCLUDNG rnnn s. T v frm (sd n currnt flw is vlid fr ll css, ut t ttm frm is vlid fr nly V. fr w cnsidr t cs f <V, it is wrt cnsidring wr t rrr in ts tt (including urs cms frm. Grgi Tc

19 Wt ut n lctrsttic ptntil stp? Cnsidr diffrnt pprc... wr ( w ( w π λ ( t ( t m nd i i ( ωt i( ωt ( ωt i( ωt Rgin π λ m 0 ( V V Rgin Wn t wv is incidnt n t rrir, sm f it is rflctd, sm f it is trnsmittd. Hwvr, sinc tr is nting t t rflct t wv c, 0. f w us t simplr undry cnditin, ψ is wv, t ψ nd it s first drivtiv must cntinuus crss t undry t 0 fr ll tim, t. Tus, ( 0 nd ( 0 ( 0 ( 0 i nd ( i Grgi Tc

20 cnt d... Wt ut n lctrsttic ptntil stp? i i nd i ( i ( i ( i i i i i Rgin 0 V Rgin W cn dfin rflctin cfficint s t mplitud f t rflctd wv rltiv t t incidnt wv, Sm s prvius cs. R nd R R Grgi Tc

21 Wt ut n lctrsttic ptntil stp? cnt d... V 0 Rgin Rgin nd liwis, w cn dfin trnsmissin cfficint s t mplitud f t trnsmittd wv rltiv t t incidnt wv, T / T i i i ( ( ( ( i nd nd T T i i i π λ m Grgi Tc nd π λ m ( V Ntic tt tis pprc is missing t fctr ( /. G RROR! Fr tis rrnus pprc, TT RR.

22 Wt ut n lctrsttic ptntil stp? cnt d... V Finl dtils: Rgin 0 Rgin ( ( T T R R nd ( ( Sinc nd T T T T 4 T T R R R R π λ m nd π λ m ( V Grgi Tc

23 cnt d... Wt ut n lctrsttic ptntil stp? Cnsidr css: Cs : >V t nd r rl nd tus, t prticl trvls s wv f diffrnt wvlngt in t tw rgins. Hwvr, RR is finit. Tus, vn tugt t lctrn s n nrgy,, grtr tn V it will v finit prility f ing rflctd y t ptntil rrir. f >>V, tis prility f rflctin rducs t ~0 ( R R λ λ T T 4 V π λ m nd π λ m( V Rgin 0 Rgin Grgi Tc

24 Wt ut n lctrsttic ptntil stp? cnt d... Cs : <V is rl ut is imginry. Wn n imginry is plcd insid ur pnntil, (i, dcying functin f t frm, (- rsults in rgin. Trnsmittd Wv cn cmpl i ( ωt ( i t ω Hwvr, TT is nw finit ut vnscnt. vnscnt wvs crry n currnt (s mwr. S vn tug t lctrn s n nrgy,, lss tn V it will v finit prility f ing fund witin t ptntil rrir. T prility f finding t lctrn dp insid t ptntil rrir is ~0 du t t rpid dcy f ψ. R R i i z z T T 0 V Grgi Tc Du t J Trnsmittd 0 π λ m nd π λ m ( V Rgin 0 Rgin

25 Wt ut n lctrsttic ptntil stp? Cnsidr t fllwing ptntil prfil wit n lctrn f nrgy <V. λ λ λ >V <V V Rgin 0 Rgin Rgin T lctrn s finit prility t tunnl trug t rrir nd will d s if t rrir is tin nug. Onc trug, it will cntinu trvling n it s wy. Grgi Tc

26 Wt ut n lctrsttic ptntil stp? Cnsidr t fllwing ptntil prfil wit n lctrn f nrgy <V. λ λ λ ncidnt Wv Rflctd Wv frm - >V ( ωt i( ωt i <V V Wv Rgin Rflctd Wv frm ( ωt i( ωt i C D Rgin 0 Rgin Rgin Trnsmittd Wv ( ωt i F nrgy is cnsrvd crss t undry s, m m V m π λ m nd π λ m( V Grgi Tc

27 s ws fund fr fr t currnts, J mi J mi Similrly, Slving fr Grgi Tc ( i i( ωt i( ωt i( ωt i( ωt ( i ( i ( i nd J ( i CC nd J ( i DD nd J ( i FF ( CC ( DD nd CC ( DD CC ( DD J mi mi Currnt cntinuity t t undris ± implis, J J RR J TT FF d d J Wt ut n lctrsttic ptntil stp? FF ut, nd d d J mi J J mi frm t first qutin nd sustituting int t scnd, r rrrnging nd dividing trug y T T F Rgin i( ωt ( t i ω R R FF V Rgin ( t i ω D 0 Rgin i( ω t i( ωt C F mi

28 Wt ut n lctrsttic ptntil stp? ( t i ω ( t i ω V i( ω t i( ωt C ( t i ω D F 0 Rgin Rgin Rgin T T? R R? Hmwr! Hint: Us cmintin f t cntinuity f t wv functin nd cntinuity f t prility dnsity currnt (r quivlntly in tis cs, t cntinuity f t drivtiv. Grgi Tc

29 Multipl/Pridic Ptntil rrirs: Krnig-Pnny Mdl Rsnnt rflctnc/trnsmissin crts stnding wvs in t crystl. Only crtin wvlngts (nrgis cn pss trug t D crystl. y nlgy, multipl lyr pticl cting s similr rflctin/trnsmissin crctristics. T rsult is t sm, nly crtin wvlngts (nrgis r trnsmittd trug t pticl stc. n sinc, w v n pticl ndgp. Grgi Tc

30 Nw cnsidr n pridic ptntil in D Krnig-Pnny Mdl: lc Functins plind Sinc c unit cll is indistinguisl frm t nt, t prility f finding n lctrn in n unit cll is idnticl t tt f finding it in n djcnt unit cll. T lc trm stts tt sinc t ptntil rpts vry lngts, t mgnitud f t wvfunctin (ut nt ncssrily t ps must ls rpt vry lngts. Tis is tru cus t prility f finding n lctrn t givn pint in t crystl must t sm s fund in t sm lctin in ny tr unit cll. Grgi Tc

31 Nw cnsidr n pridic ptntil in D Krnig-Pnny Mdl: lc Functins plind T civ tis prprty, t MGNTUD f t wvfunctin (ut nt ncssrily t wvfunctin must v t sm pridicity s t lttic. Tus, w cs wvfunctin tt is mdultd y t pridicity f t lttic. Sinc V(r w cs pln wvs mdultd y pridic functin, (r Tus, (r u (r Tus, Grgi Tc ir u n n i V(r (r wr u (r ( r ir [ u (r ] n i n (r i u n (r T wvfunctin in n unit cll is mrly ps siftd vrsin f t wvfunctin in n djcnt unit cll. (r r mrly ps siftd vrsin f (r -i i (r (r (r (r (r (r Tus, t prility f finding n lctrn in n unit cll is idnticl t tt f finding it in n djcnt unit cll s pctd.

32 n mprtnt sid: ffct f lc Functins ssuming lrg numr f unit clls in mtril, N, t undry cnditin fr t systm is N trnsltins must rsult in t wvfunctin ing trnsltd t rturn t itslf? (T prility t t mtril dgs must symmtric nd qul. ( N Tus, in i s ting t Nt rt, ( ( ( ( inπ N ( nπ N S t llwd stts f π n N in inπ ( r : Tus, if N (t numr f unit clls vill is vry lrg, li in smicnductr, t spcing twn t llwd -vlus (n-n tc... r lmst cntinuus, justifying t trtmnt f t -stts s cntinuum. Grgi Tc N ( N

33 Nw cnsidr n pridic ptntil in D Krnig-Pnny Mdl Cnsidr wt ptntils n lctrn wuld s s it mvs trug t lttic (limitd t D fr nw. T lctrsttic ptntil, V( is pridic suc tt V(LV(. W MUST v stnding wvs in t crystl tt v prid qul t multipl f t prid f t crystl s lctrsttic ptntil. (Similr t multilyr ntirflctin cting in ptics t is imprtnt t nt tt sinc, t wvfunctin rpts c unit cll, w nly v t cnsidr wt ppns in n unit cll t dscri t ntir crystl. Tus, w cn rstrict urslvs t vlus f suc tt π/ t π/ (implying r (π/λ Grgi Tc ftr Nudc nd Pirrt Fig 3.

34 ssumptins f Krnig-Pnny Mdl: Simplifying t ptntil t tt swn r: D nly Nw cnsidr pridic ptntil in D Krnig-Pnny Mdl ssum lctrn is simpl pln wv f t frm, i...mdultd y functin wit t sm pridicity s t pridic crystllin ptntil, U( T crystllin ptntil is pridic, U(U(L Tus t wv functin is simpl plin wv mdultd y functin wit t sm pridicity s t pridic crystllin ptntil: ( u n i ( T qustin w ds t prsnc f pridic ptntil ffct t fr lctrn cn tus cnvrtd t t qustin wt ffct n t lctrn ds t cnging f n lctrn s fr stt (pln wv t lc stt? Grgi Tc Nudc nd Pirrt Fig 3.

35 Grgi Tc Krnig-Pnny Mdl wr 0 m V m α α ( ( U 0 U 0 < < > fr m U fr U m i V m β β β 0 Fr 0<<: Fr -<<0: cs( sin( ( α α cs( sin( ( D C β β

36 Krnig-Pnny Mdl Fr 0<<: ( sin( α cs( α Fr -<<0: ( C sin( β D cs( β pplying t fllwing undry cnditins: ( 0 d ( d 0 ( 0 d ( d 0 C fr cntinuus wv functin t t undry ( d ( d i ( i ( ( d ( d C fr pridic wv functin t t undry Grgi Tc

37 Grgi Tc Krnig-Pnny Mdl [ ] [ ] sin( cs( sin( cs( cs( sin( cs( sin( ( ( D C D C C D i i β β β β α α α α β β α α β α pplying t undry cnditins, w gt: liminting t vrils C nd D using t v qutins, w gt: [ ] [ ] [ ] 0 sin( sin( cs( cs( 0 cs( cs( sin( sin( ( ( ( ( i i i i β β α α β α α α β α β β α α nd r nly nn-zr (nn-trivil slutin wn t dtrmint f t v st is qul t zr. Tis qutin st frms mtri f t frm: 0 0 z y w

38 Grgi Tc Krnig-Pnny Mdl ( ( cs cs( cs( sin( sin( β α β α αβ β α Ting t dtrmint nd simplifying w gt: Plugging in t dfinitins fr α nd β w gt: ( U fr ( cs cs cs sin sin > U mu U mu U mu U mu U U U ( U fr 0 ( cs cs cs sin sin < < U mu U mu U mu U mu U U U T rigt nd sid is cnstrind t rng f /- nd is functin f nly. T limits f t rigt nd sid (/- ccurs t 0 nd /- π/( wr is t prid f t crystl ptntil. T lft nd sid is NOT cnstrind t /- nd is functin f nrgy nly.

39 Krnig-Pnny Mdl T rigt nd sid is cnstrind t rng f /- nd is functin f nly. T limits f t rigt nd sid (/- ccurs t 0 t /- π/(. T lft nd sid is NOT cnstrind t /- nd is functin f nrgy nly. Lft r Rigt nd sid f Krnig-Pnny Slutin Witin ts friddn nrgy rngs, n slutin cn ist (i.. lctrns cn nt prpgt. Vrius nds r llwd nrgy ist wr t nrgy is functin f t cic f (s slutin qutin /U fr t spcific cs wr, Grgi Tc mu mu π

40 Krnig-Pnny Mdl Rpltting t prvius rsult in ntr frm rcgnizing t lwr limit is srd y nd π/( wil t uppr limit is fr 0. Tr r t mst -vlus fr c llwd nrgy, T slp, d/dk is zr t t -zn undris t 0, π/( nd π/( Tus w s tt t vlcity f t lctrns pprcs zr t t zn undris. Tis mns tt t lctrn trjctry/mmntum r cnfind t sty witin t llwl -zns. Nt: -vlu slutins diffring y π/( r indistinguisl Grgi Tc Figurs ftr Nudc nd Pirrt

41 Rpltting t prvius rsult in ntr frm... Krnig-Pnny Mdl Fr Spc - digrm 3π/( π/( π/( 0 π/( π/( 3π/( T prsnc f t pridic ptntil rs t fr spc slutin up int nds f llwd/disllwd nrgis. T undris f ts nds ccurs t ±π/( 3 rd rillin Zn nd rillin Zn st rillin Zn nd rillin Zn 3 rd rillin Zn Nt: -vlu slutins diffring y π/( r indistinguisl. ls du t nimtins printd vrsin ds nt rflct sm infrmtin. Grgi Tc

42 Nw cnsidr n pridic ptntil in D Krnig-Pnny Mdl Grgi Tc ftr Nudc nd Pirrt Fig 4.

43 Wt is t mprtnc f - Spc undris t (/-π/? Crystl Structurs, rilluin Zns nd rgg Rflctin Grgi Tc

44 Cncpt f Rciprcl Spc (Rltd t t Furir Trnsfrm Rl Spc Rciprcl Spc π/ Grgi Tc

45 mprtnc f -Spc undris t (/-π/ Crystl Structurs, rilluin Zns nd rgg Rflctin T crystl rciprcl lttic cnsists f pridic rry f invrs-tms ( mr plntin t cm ltr. G G G ½G rgg Rflctin Cnditin fr ttl intrnl rflctin Surfc md frm ll suc plns is t st rilluin Zn. T rilluin zn cnsists f just ts vctrs fr wic rgg Rflctin ccurs. Sinc G π/ tn ½G π/ Grgi Tc

46 mprtnc f -Spc undris t (/-π/ Crystl Structurs, rilluin Zns nd rgg Rflctin ( Tus, ( Tus, π fr π fr ( ( iπ π cs π fr iπ cs r r sin π iπ π r sin iπ π lctrns r rprsntd y stnding wvs f wvfunctins lclizd nr t tm crs (i.. vlnc lctrns nd s fr wy frm t crs s pssil (i.. fr lctrns. Givn t diffrnt ptntil nrgis in t f ts rgins, t nrgis f t lctrns wy frm t tm crs is igr. Tis is n plntin fr t rigin f t nrgy ndgp. Grgi Tc Rl Spc Crystl