Handout on. Crystal Symmetries and Energy Bands

Transcription

1 dou o Csl s d g Bds I hs lu ou wll l: Th loshp bw ss d g bds h bs of sp-ob ouplg Th loshp bw ss d g bds h ps of sp-ob ouplg C 7 pg 9 Fh Coll Uvs d g Bds gll hs oh Th sl pol ss ddo o h l slo s: Fo pl h D pol of squ o l s show hs h followg ss: ) ud oos b 9 8 d 7 dgs b) ud flos w -s d -s ) ud flos w h wo dgols Ŝ b h opo ( pso) fo o of hs s opos h: ' ' Ŝ = oo b 9 o C 7 pg 9 Fh Coll Uvs

2 Csl Po-Goup Po-Goup Th po goup s opo of l ll hos opos h lv h l uhgd d ls o po of h l s uovd ud h opo Po goup s opos lud: ) oos (w o s of oo) ) flos (oss ls o pls) ) Ivsos (w o po) Ŝ b h opo fo po-goup s opo suh h: ' Ŝ = oo b 9 o Th opo Ŝ s u: u T C 7 pg 9 Fh Coll Uvs Csl Po-Goup d g Bds Ŝ b h opo fo po-goup s opo suh h: T ' u uppos o hs solvd h hodg quo d obd h g d wvfuo of Bloh Now pl b vwh h hodg quo: pl s v C 7 pg 9 Fh Coll Uvs

3 Csl Po-Goup d g Bds Th bov quo ss h h fuo s lso Bloh s wh h s g s (w hv foud w gfuo!) Th quso s f w ll hv foud w gfuo o o d f so wh s h wvvo of hs w gfuo W ow h Bloh fuos hv h pop h: o w hs o : s lso l vo s Bloh fuo wh wvvo d g C 7 pg 9 Fh Coll Uvs Csl Po-Goup d g Bds o w fll hv fo h s opo Ŝ : W lso ow h h gg of s Thfo: O quvll: Ipo ssos: ) If Ŝ s s of h pol suh h l-sp w hv: h h g bds lso o h s of h pol suh h -sp: ) Dgs h g bds hfo s fo sl po-goup ss! C 7 pg 9 Fh Coll Uvs 3

4 C 7 pg 9 Fh Coll Uvs T vsl d g Bds uppos w hv solvd h dpd hodg d obd h Bloh s wh g : Af pluggg h soluo h -dpd quo w g: If w h opl oug of h bov quo w g: W hv foud oh Bloh fuo wh h s g s Quso: Wh s h phsl sgf of h s? C 7 pg 9 Fh Coll Uvs s s f w fd soluo ud -vsl ( wh s pld b ): Th bov dos o loo l hodg quo so w opl oug : Ths s h s h -vsd s ospodg o h s Th fuo s h -vsd Bloh s ospodg o uppos w hv solvd h dpd hodg d obd h Bloh s wh g : T vsl d g Bds

5 5 C 7 pg 9 Fh Coll Uvs T vsl d g Bds W hv foud oh Bloh fuo wh h s g s Th quso s f w ll hv foud w gfuo o o d f so wh s h wvvo of hs w gfuo W ow h Bloh fuos hv h pop h: o w hs o : s Bloh fuo wh wvvo d g d Ipo sso: T vsl s pls h v f h sl ls spl vso s (g GAs IP ) C 7 pg 9 Fh Coll Uvs p-ob Io olds A lo ovg l fld ss ffv g fld gv b: P B ff Th ddol fo of s og fo Thos psso Th lo hs g o ld o s sp gul ou b: g B B g B Th o bw h lo sp d h ffv g fld dds w o h lo: P P B B B ff B ff so

6 6 C 7 pg 9 Fh Coll Uvs p-ob Io d Bloh Fuos I h bs of sp-ob o w hd: lo ss wh sp-up d sp-dow w dg o I h ps of sp-ob ouplg h lo bos: so so o P h lo s ow sp-dpd pu sp-up o pu sp-dow ss o log h gss of h lo Th gss b w os gll s supposo of up d dow sp ss o: =Quu ub fo h wo sp dgs of fdo C 7 pg 9 Fh Coll Uvs p-ob Io d Bloh Fuos Fo h wvvo h FBZ d fo h bd d o wll ob wo soluos of h bov quo W lbl h wh ddol subsp h wo dff vlus:

7 7 C 7 pg 9 Fh Coll Uvs p-ob Io d Bloh Fuos I h ps of sp-ob ouplg sp s o good quu ub! gss o b lbld b h sp quu ub o wh do w do?! d d P d d so N h os wh h pol s ppol dl o w: I gl h Bloh wvfuo s supposo of o obls: d d d N A h o ( po) h Bloh wvfuo s: d d N Th ol spl dpd h o os fo h o obls C 7 pg 9 Fh Coll Uvs p-ob Io d Bloh Fuos Th wo Bloh soluos fo bd h o b hos o ospod o wo dff gvlus of d h s vlu of : o Ĵ o h o h o obls b hos o b gss of h ol gul ou od o dgol h sp-ob lo If h Bloh fuos h o b hos o b gvlus of d h so popool o: wll b dgold! Ĵ Ĵ Ĵ If h sl hs vsl s h h o o lws hoos:

8 p-ob Io d Bloh Fuos pl l Bds of GAs d IP ( Cod): Bd: 3 3 Bd: O Bd: O O O O O O lghl w fo o o us p pubo ho wh h sp-ob ouplg ludd h pubg lo: P p Thus s sf o ssu h h Bloh gss hos s bov wll sobl good gss of Ĵ v slghl w fo h o C 7 pg 9 Fh Coll Uvs p-ob Io d s I h ps of sp-ob o w hv h hodg quo: Tslo : oo : Ŝ b opo blogg o h oo subgoup of h sl po-goup suh h: T u (Th s of vso s wll b d spl) C 7 pg 9 Fh Coll Uvs 8

9 9 C 7 pg 9 Fh Coll Uvs uppos w hv foud h soluo o h hodg quo: p-ob Io d oo W pl b vwh h hodg quo: Ad h soluo s: C 7 pg 9 Fh Coll Uvs Th bov quo dos o loo l h hodg quo! W df u sp oo opo h ops h lb sp of sps d os sp ss h ss of h opo Cosd sp vo pog h do: b b b b b b b b Th sp oo opos hv h pop: Ŝ p-ob Io d oo

10 C 7 pg 9 Fh Coll Uvs fo: Iodu sp oo opo ospodg o h oo gd b h : Ŝ Th bov quo shows h h w s: ssfs h hodg quo d hs h s g s h s: p-ob Io d Po-Goup C 7 pg 9 Fh Coll Uvs ' : Th w s s Bloh s wh wvvo u: If s opo fo po-goup s opo h h wo ss gv b: Ŝ ' hv h s g: p-ob Io d Po-Goup Ths pss od ( sp) vso of h ogl Bloh s v h sp s od ppopl b h sp oo opo

11 C 7 pg 9 Fh Coll Uvs p-ob Io d Ivso uppos w hv foud h soluo o h hodg quo: W pl b vwh h hodg quo: Ad h soluo s: uppos h sl pol hs vso s: C 7 pg 9 Fh Coll Uvs p-ob Io d Ivso Th bov quo shows h h w s: ssfs h hodg quo d hs h s g s h s: : h w s s Bloh s wh wvvo Also s h ol gul ou opos v ud spl vso:

12 C 7 pg 9 Fh Coll Uvs Thfo w w: p-ob Io d Ivso o h w gs foud hs h s gul ou s h ogl s! u: If h sl pol hs vso s h h wo ss gv b: hv h s g: C 7 pg 9 Fh Coll Uvs p-ob Io d T vsl I h ps of sp-ob o w hv h -dpd hodg quo: s s f w fd soluo ud -vsl ( wh s pld b ): Th bov dos o loo l hodg quo so w opl oug : Ad sll dos o loo l h ogl hodg quo! oluo s:

13 3 C 7 pg 9 Fh Coll Uvs p-ob Io d T vsl Gv gvlu quo: O lws pfo u sfoo wh T d ob: Av v u Bu Tv Tv TAT Tv u TAT B o sfoo wh h u wh h quo: Th bov quo ow loos l h -dpd hodg quo C 7 pg 9 Fh Coll Uvs p-ob Io d T vsl Cospodg o h Bloh s: wh g: h -vsd Bloh s s: d h -vsd s hs h s g s h ogl s Q: Wh h quu ubs of h -vsd s? u:

14 C 7 pg 9 Fh Coll Uvs Cosd h Bloh fuo: uppos h Bloh fuo ospods o h sp pog h do of h u vo h loo : Wh f w w h s wh h oppos sp h s loo? Th sw s: Poof: p-ob Io d T vsl C 7 pg 9 Fh Coll Uvs I h ps of sp-ob o w hv h hodg quo: W opl oug : uppos w hv solvd d foud h soluo: I dos o loo l h ogl hodg quo! No h: p-ob Io d T vsl

15 5 C 7 pg 9 Fh Coll Uvs p-ob Io d T vsl Gv gvlu quo: O lws pfo u sfoo wh T d ob: Av v u Bu Tv Tv TAT Tv u TAT B o sfoo wh h u wh h quo: W hv foud w soluo: wh h s g s h ogl soluo: No: h w soluo s h -vsd soluo foud pvousl! C 7 pg 9 Fh Coll Uvs p-ob Io d T vsl Ud l slo w g fo h w soluo: o h w soluo s Bloh s wh wvvo :? No h h w soluo foud lso b w s: Bu s show l hs s h h bov s hs sp oppos o h s: Bu wh bou:?

16 6 C 7 pg 9 Fh Coll Uvs? p-ob Io d T vsl fo: C 7 pg 9 Fh Coll Uvs Thfo h w soluo s h Bloh s : Ad w hv lso foud h s g s h s s h of h s : p-ob Io d T vsl Thfo h w soluo s lso gfuo of wh gvlu : Ĵ

17 Csl Ivso d T vsl T vsl s pls: Ivso s pls: I sls whh hv vso d vsl ss h bov wo pl: Th s sp (o h ol gul ou) dg! Csls whh do o hv vso s sp dg of h bds s o gud C 7 pg 9 Fh Coll Uvs Csl Ivso d T vsl Coo (d uh ggd) shs of h oduo bds of G d GAs show blow: G GAs C 7 pg 9 Fh Coll Uvs 7