References: 1. Introduction to Solid State Physics, Kittel 2. Solid State Physics, Ashcroft and Mermin

Transcription

1 Lecture 12 Bn Gp Toy: 1. Seres solutons to the cosne potentl Hmltonn. 2. Dervton of the bngrms the grphcl representton of sngle electron solutons 3. Anlytcl expresson for bngps Questons you shoul be ble to ress fter toy s lecture: 1. Wht s the resultng egenvlues problem n how re ts solutons relte to the egenfuncton of the peroc Hmltonn? 2. How to erve n pproxmte nlytcl expresson for bngps? 3. How s the electronc strbuton relte to the energy of soluton? 4. How oes the bngp scle wth the potentl prmeters? 5. Bngp trens n the peroc tble. 6. How re the bn gps mesure? References: 1. Introucton to Sol Stte Physcs, Kttel 2. Sol Stte Physcs, Ashcroft n Mermn 1

2 Lst tme we hve erve the Centrl Equton: 2 2 C V G C G EC G Ths equton reltes ll the coeffcents C tht re seprte by n nverse lttce vector to ech other formng set of couple lgebrc equtons (whch n essence s recurson formul). Let s conser smple exmple to clrfy the mportnce of centrl equton: Exmple: Cosne Potentl For smplcty lets conser peroc potentl, whch s smple cosne: 2 gx e gx gx gx Vx 2V 0 cos x 2V 0 cos gx V 0 e V 0 e V 0 e We cn plce the two Fourer coeffcents of the potentl on the sme xs we use prevously s both hve nverse length unts: V g =V 0 V g =V 0 V 0 2 g g = 2 We hve use the efnton of the nverse lttce vector G n efne prmtve nverse 2 lttce vector g: G n ng. Then system of the equtons bove cn be wrtten s: 2 2 C VC VC EC g g 2 g 2 C VC VC EC g 2 g g 2 g 2 C VC VC EC g 2g g... Let s rerrnge the equtons bove n couple more just for you to see the pttern: 2

3 ... V 0 C 3g V 0 C 2 g 2 2 2g 2 C 2g V 0 C g EC 2g g 2 C g V 0 C EC g V C 2 C V C EC 2 0 g 0 g V 0 C 2 g 2 Cg V 0 C 2 g EC g V 0 C g 2 2g 2 C2 g V 0 C 3g EC 2 g... C g C C g C 2g 2 g Snce g n V re nown, once s specfe then set (nfnte) of couple lgebrc equtons s completely etermne, thus the problem hs been trnsforme from fferentl opertor to mtrx opertor egenvlues problem. For gven, the egenvectors efne the energy egenfunctons, whle the egenvlues re the corresponng energes. V C C V g V 0 0 2g C g 2 0 V 2 V 0 C E C C 2 g 0 0 V g 2 V C 2g V 2g 2 2 2g 2 2 g C g C g C 2 g The mtrx bove ws truncte fter 5 terms (the effects of ths truncton wll be explore shortly) eep n mn tht the full mtrx s nfnte s ll the coeffcents seprte by n nverse lttce vector re relte. 3

4 Dscusson: 1. The egenfunctons hve the followng form: 2 g x g x x g x 2 g x x...c e C e C e C e C e...,e,n 2g g g 2 g Gx x Gx C G e e C G e G G f x Assocte wth corresponng egenvlue: E n, Whch s just resttement of Bloch s Theorem, where f(x) s peroc functon wth the perocty of the lttce. Ths reflects the fct tht once specfc s efne ll the coeffcents corresponng to tht cn be generte thus efnng soluton lbele by. 2. By conserng the full mtrx form you cn convnce yourself tht choosng two specfc s tht re seprte by ny G woul generte the exct sme mtrx yelng entcl solutons: G x x Ths mens tht the egenvectors n energy egenvlues re peroc n G: E G,n E,n One cn therefore choose s tht re restrcte to the frst BZ: n completely cpture ll the solutons! 3. A herrchy of solutons s generte t ech pont. These solutons re clle bns n lbele by n nex n=1, 2, 3 thus one nees to specfy both n the bn nex n to completely specfy the soluton. 4. The number of nonzero mtrx elements n the equtons s etermne by the number of Fourer components tht re neee to express the potentl functon. 4

5 Bn gps: wht hppens ner the ege of the BZ? The ege of the BZ s efne by: g 2 Revew the mster equton for Vx 2V cos gx: E C VC VC 0 g g An conser only the frst terms n the sum n focus on C n C +g : 2 2 VC 2 E 2 g C VC g 0 E V C C g 0 VC 2 2 g E C g VC 2 g 0 V g E 2 2 g E V 2 For g we cn wrte: g C V E 2 C g g E V 2 C Anlogously for g 2 : g 2 2 C g 2 2 V g E 2 g The only wy ths homogeneous system of equton cn hve nonzero soluton; the etermnnt of the mtrx hs to be equl to zero. 2 2 g E 2 V V 2 g E Then the energy egenvlues re: E 2 2 g V 2 5

6 Thus the energy hs two roots, whch re seprte by E E E 2V. Keep n mn tht ths result s vl for the prtculr form of the potentl tht we use: 2V cos gx V e gx e gx V x So we re ble to prect the exstence of bn gp, whch s pproxmtely proportonl to the mgntue of the peroc potentl. E 2 2 g 2 V 2 E 1 2 g 2 V 2 2V The form of the egenfunctons ner the bn ege s: One soluton gves the egenfunctons ner the bottom of the gp the other gves t ner the top of the gp. The fferences n energes cn be explne by loong t the chrge strbuton reltve to the on postons. x x x u x e e 2cos x x x u e x e 2sn The two stnng wves concentrte ther electrons n fferent regons the probblty ensty for prtcle s: u * 2 x x u x cos Ths functon ples electrons (wth negtve chrge) on the postve ons centere t x=0,, 2 where the potentl energy s lowest. u * x For the other stnng wve: u x sn 2 x Ths functon concentrtes the electrons precsely n the mle between the ons,.e. on x=/2, 3/2 6

7 Fgure remove ue to copyrght restrctons. Fg. 3: Kttel, Chrles. Introucton to Sol Stte Physcs. 8th e. Wley, 2004, p Wht nformton cn we get from the bn grms? Let s focus on the bn shpe ner the bn eges. If we shft the xs by g/2 so tht the bn ege s t =0 (see the pcture bellow), then we cn pproxmte the frst two bns wth prbols ner the bn ege: Then the smplest form of the bn grm hs the followng shpe: 1. Allowe n forben bns. Ientfcton of the gp energy. 2. Slope of the bns group velocty. 3. Curvture of bns effectve mss. 7

8 1. Mgntue of the bn gps Crystl Gp E g, ev 0 K 300 K Crystl Gp E g, ev 0 K 300 K Dmon 5.4 HgTe 0.30 S PbS Ge PbSe αsn PbTe InSb CS InAs CSe InP CTe GP ZnO GAs ZnS GSb SnTe AlSb AgCl 3.2 SC(hex) 3.0 Agl 2.8 Te 0.33 Cu 2 O ZnSb TO Imge by MIT OpenCourseWre. Observtons: 1. Lrger toms smller bngps. Why? Lrger toms hve more electron shells n the onc chrge s screene from the free flyng electron. Consequently electrons wll be experencng lower effectve potentl n lttces of lrge toms. From our smple moel we hve seen tht n the smplest cse of cosne potentl the bngp s proportonl to the mpltue of the potentl. Consequently: V E g 2V. Whle n rel crystls the reltonshp between the potentl n the bngp s more complex, the sme logc pples: lrger toms smller potentl smller bngp. 2. Energy gps spn ev. 3. The wer the gp the hever the electron. 8

9 Optcl mesurement of the bngp. Bngps re relte to the optcl trnsprency n electrcl conuctvty of the mterl. Electronc bngps re generlly mesure by optcl bsorpton spectroscopy: Fgure remove ue to copyrght restrctons. Fg. 45: Kttel, Chrles. Introucton to Sol Stte Physcs. 8th e. Wley, 2004, p (from Introucton to Sol Stte Physcs by Kttel) Whle n clss we hve scusse very smple cosne potentl, n rel mterls potentls loo sgnfcntly more complcte leng to more complcte forms of bn grm. For some mterls t turns out tht the vlence bn mxmum oes not conce wth the conucton bn mnmum n. Ths mens tht n orer for n optcl trnston to hppen (.e. for electron to jump from vlence to conucton bn wth help of photon), the electron nees to lso obtn extr crystl momentum. Where woul the electron get extr crystl momentum? Lttce vbrtons, phonons hve very low energy comprng to photons but lrge crystl momentum. However ths mens tht for bsorpton to hppen the photon, the electron n the phonon nee to ll meet t the sme plce n tme, whch s low probblty occurrence n hence the bsorpton s poor for the nrect bngp mterls between E g n E vert (vertcl gp the energy fference between the conucton n vlence bns t = 0). 9

10 The expermentl setup: hc E ph h 10

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