The tight-binding method

Transcription

1 Th tight-idig thod Wa ottial aoach: tat lcto a a ga of aly f coductio lcto. ow aout iulato? ow aout d-lcto? d Tight-idig thod: gad a olid a a collctio of wa itactig utal ato. Ovla of atoic wav fuctio i ough to chag th ictu of iolatd ato ut ot o uch a to d th atoic dcitio coltly ilvat. ful fo d-lcto d i taitio tal ato. Sv a a coltay ictu to th aly f lcto dcid y Bloch lvl.

2

3 Gal foulatio I th liit of iolatd ato Th yt of a igl ato at I th xt ca wh th lattic acig i uch lag tha th ag of atoic lvl th igtat with igvalu of th cytal ailtoia ca aoxiatly witt a th wav fuctio fo all i th Bavai lattic. Th coctio to th xt ca Th cytal ailtoia at ottial co fo oth io

4 cotai all coctio to th atoic ottial quid to oduc th full iodic ottial of th cytal.

5 Th Bloch coditio Th Bloch coditio i Pood Pood ig ig wav fuctio wav fuctio ' ' ' ' ' ' ' ' '

6 Th idicatio of Bloch tho W ay a olutio to th full cytal Schödig quatio that tai th gal fo of ot cay to a xact atoic tatioay-tat wav fuctio ut o to dtid. If i xcdigly all i xctd d to quit clo to th atoic wav fuctio o to wav fuctio with which i dgat. xad i t of a lativ all u of Lia coiatio of atoic oital LCAO localizd atoic wav fuctio

7 Th cytal Sch Th cytal Schödig quatio dig quatio at d d at d d d d ] [ d d d

8 d d d A zo od aoxiatio w ay glct all th t i ig A zo od aoxiatio w ay glct all th t i ight had id. ht had id. ul Scodly w hall lvat atoic lvl.g. Scodly w hall lvat atoic lvl.g. - - d d-lvl o thi hyidizatio lvl o thi hyidizatio th olv coodig cula ol. th olv coodig cula ol.

9 Alicatio: Alicatio: -ad ad d d d K oly K oly -lvl ad do all oth t lvl ad do all oth t Lt d d d 2 1

10 Silificatio of Silificatio of Syti: Syti: wav - Bavai lattic of Ivio yty K oly th t fo at igho od K oly th t fo at igho od Igo th t i Igo th t i co.. 1 Q: ow to div ffctiv a? Q: ow to div ffctiv a?

11 Gal a Bad lot alog ytic dictio i th fit Billoui zo Th followig ovla itgal dti adwidth d ij i i Syty will ud to duc colicity i LCAO ca i-ddt: i-oit coulig Slat ad Kot Phy.v Itactio tw lcto: Mott taitio

12 Scod quatizatio Scod quatizatio ˆ ˆ ˆ ˆ ˆ ˆ d d at io c ˆ ˆ No No-itactig igl itactig igl-aticl ailtoia aticl ailtoia xad i t of atoic oital xad i t of atoic oital wavfuctio wavfuctio ˆ ' ' ' ˆ ˆ ˆ ˆ at at c c d d

13 ' ' ' ˆ ' ˆ ' d t c c t at To th ladig od To th ladig od ' ' ' ' ' d t ' ˆ ' ˆ ˆ ˆ ' c c c c aiig iu: aiig iu: Atoic wav fuctio at difft it a Atoic wav fuctio at difft it a ot othogoal to ach oth. ot othogoal to ach oth.

14 Wai Wai fuctio fuctio K Piodicity of Bloch wav fuctio i Piodicity of Bloch wav fuctio i -ac ac f d v d v f d v f f i i i i f Wai Wai fuctio fuctio

15 Fatu: Wai wav fuctio at difft it o ad a a othogoal. Wai wav fuctio v a altativ t of local ai. Alicatio of Wai fuctio: To thoy wh ad how th iclaical thoy of taot y Bloch lcto a dow. Iuity lvl. Magtic hoa. Taitio tal oxid. Itactig lcto ic ot itactio a local itactio.

16 Ovla itgal i LCAO Q1: ow to otai ij fo atoic olcula oital? Q2: ow to ilify th calculatio of? Ky iu: ovla of oital wav fuctio: odig ad atiodig ij -oital: σ-od

17 -oital: σ-od -oital: π-od

18 gy lvl

19 - hyidizatio hyidizatio z z z x y x y y x x z z y x z V V V V

20 Suay Tight-adig thod: lia coiatio of atoic oital LCAO a fw of aat to dci th whol gy ad. ow ovla itgal giv i to cula quatio which dti gy lvl. ow to choo liitd u of atoic oital. ow to ilify th calculatio of ovla itgal. Syty chical od ad ilifid ovla itgal. Wai fuctio