# Lecture 11 Waves in Periodic Potentials Today: Questions you should be able to address after today s lecture:

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1 Lctur 11 Wvs in Priodic Potntils Tody: 1. Invrs lttic dfinition in 1D.. rphicl rprsnttion of priodic nd -priodic functions using th -xis nd invrs lttic vctors. 3. Sris solutions to th priodic potntil Hmiltonin gnrl form. Qustions you should b bl to ddrss ftr tody s lctur: 1. Wht is th dfinition of n invrs lttic?. Wht is th form of Fourir sris in trms of th invrs lttic vctor. 3. How is priodic function rprsntd grphiclly on th -xis? 4. How is n rbitrry function xprssd s linr combintion of pln wvs nd how is this dscription diffrnt from Fourir sris? 5. Know how to xpnd complx function in complx Fourir sris. 6. Know how to gnrt th sris of qutions tht lin th cofficints sprtd by invrs lttic vctors. 7. Wht is th rsulting ignvlus problm nd how r its solutions rltd to th ignfunction of th priodic Hmiltonin? 1

2 Dirct nd Rciprocl lttic in 1D nd 3D A 1D Brvis lttic is dfind by th collction of vctors tht hv th form: R n is clld primitiv vctor nd th st of {R} form th Brvis lttic. Clrly ny point on th x-xis cn b writtn s: x' x n, x 0, Suppos w found st of vctors tht stisfy th following rltion: in 1 m (m...1,0,1,...): This st of s lso compris Brvis lttic. Th rciprocl vctors form lttic with rciprocl lngth dimnsions (lrg sprtions btwn lttic points in dirct lttic lds to smll sprtion in rciprocl lttic nd vic vrs.) Any point in th rciprocl spc (not ncssrily lttic point) cn b xprssd s: m, whr (1D): This rng is clld th first Brilllouin Zon (BZ), it is th Wignr-Sitz cll of th rciprocl lttic. A similr pproch is usd to dfin D nd 3D lttic: R n n 1 1 n 3 3 Suppos w found st of vctors tht stisfy th following rltion: ir 1 Ths s r clld th rciprocl lttic vctors, nd cn b found in th following wy: m 1 b 1 m b m b3 3 Hr s r th st of primitiv vctors for th dirct lttic nd th b s r th primitiv vctors for th rciprocl lttic givn by: 3 b , b, b Exmpl simpl cubic lttic: 1 xˆ, ŷ, 3 ẑ b 1 xˆ, b ŷ, b 3 ẑ i R 1 Proprtis of Crystl momntum. Clrly is numbr (or vctor) rltd to th ignvlus of th discrt trnsltionl oprtor nd hs units of invrs lngth. W mntiond bov tht ny vctor of invrs lngth units cn b writtn s sum of rciprocl lttic vctor nd vctor rstrictd to th first BZ.

3 m, whr th first Brillouin zon dfind by: oing bc to th ignfunctions of th discrt trnsltionl oprtor. Suppos w pt our in th rng spcifid bov (BZ) nd ddd to it rciprocl ix lttic vctor : u ix x f x This ignfunction still corrsponds to n ignvlu of i sinc by construction i 1 nd thus dding to my modify th functionl form of th priodic trm but will not obviously chng th fct tht it is priodic in s ix is itslf priodic function in (thrfor it cn b tn to b prt of f x ). In othr words in trms of lbling th stts it is sufficint to considr only s tht r rstrictd to th first BZ s only thy corrspond to distinct ignvlus of th discrt trnsltionl oprtor. 3

4 Numricl solution of th priodic potntil: Th ignvlu problm W r considring priodic potntil: V x n V x Sinc th function is priodic with priod it cn b xpndd in Fourir sris whr s r th st of rciprocl lttic vctors, i.. priodic function cn b xpndd in trms of sin cosin functions such tht: V 0 n n Vx A cos x B sin x n n n1 n n A n Vxcos x dx 0 n B n Vxsin x dx 0 n Th rciprocl lttic s th collction of points dfind by th rltion: ir 1 n gn Thn w cn rwrit th Fourir sris s: A 0 cos V x V cos x V g cos V Vxcosx dx 0 sinx sin V V x dx 0 sin sinx Th cofficints of th sris cn b rprsntd grphiclly on th xis. V g V g V g V 3g g 0 g g 3g Sinc our functions r complx vlud w will us th complx Fourir sris: VxV ix 4

5 Now th Hmiltonin for th priodic potntil hs th following form: d ix V x Eu x m dx Th ignfunctions cn b writtn s sris. Sinc thr is no rson to ssum tht th ix ignfunctions r priodic in lttic priod w xpnd thm in : x C C C 1 1 Any prticulr solution φ(x) is compltly dfind by spcifying th st of cofficints C. All s r llowd so th sprtion btwn two trms cn b infinitsiml. Substituting φ(x) into th priodic potntil Hmiltonin: d ix ix ix V C E C m dx i x m ix ix C V C E C Sinc th sum ovr is infinit nd is just running indx w cn chng our vribl in th scond sum from : C V C EC ix ix ix m ix C m V C EC ix EC V C 0 m In ordr for this sum to b qul to zro vry componnt hs to b zro: EC V C 0 C V C EC m m ix Unli th sitution w strtd with whr th sprtion btwn two trms in th xpnsion could b infinitsiml this qution rlts ll th cofficints C tht r 5

6 sprtd by n invrs lttic vctor to ch othr forming st of coupld lgbric qutions (which in ssnc is rcursion formul). Now ll you nd to do is to solv this infinit systm of qutions i.. find th nrgy ignvlus E nd th ignvctors C composd of componnts C, which dfin our tril wvfunctions: x C ix 6

7 MIT OpnCoursWr Elctronic, Opticl nd Mgntic Proprtis of Mtrils Spring 013 For informtion bout citing ths mtrils or our Trms of Us, visit:

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