a r Fermi Surface With periodic boundary conditions: 2D k space: Area per k point: 3D k space: Area per k point:

Transcription

1 Chm 53, UC, Bly Fmi Sufac E F K F K m E x ) ( a a 0.4 Chm 53, UC, Bly With piodic bouday coditios: z z y y x x L i L i L i z z z y y y x x x L L L D spac: Aa p poit: L x L y 3D spac: Aa p poit: V L L L z y x 3 8 A gio of spac of volum will cotai: allowd valus ) 8 ( V V

2 Chm 53, UC, Bly Fo divalt lmts: f lcto modl a 0.56a Chm 53, UC, Bly

3 Chm 53, UC, Bly Fo aly f lcto:. Itactio of lcto with piodic pottial ops gap at zo bouday. Almost always Fmi sufac will itsct zo Boudais ppdiculaly. 3. Th total volum closd by th Fmi sufac dpds oly o total lcto coctatio, ot o itactio Chm 53, UC, Bly Alali Mtal Na, Cs: sphical Fmi sufac a 0.4a Al. Eath mtal: B, Mg:: aly sphical Fmi sufac D cas a 0.56a 3

4 Chm 53, UC, Bly Chm 53, UC, Bly 4

5 Chm 53, UC, Bly Chm 53, UC, Bly 5

6 Chm 53, UC, Bly Chm 53, UC, Bly 6

7 Chm 53, UC, Bly Chm 53, UC, Bly 7

8 Chm 53, UC, Bly Billoui Zo of Diamod ad Zicbld Stuctu (FCC Lattic) Sig Covtio Zo Edg o sufac : Lati alphabts Itio of Zo: G alphabts Ct of Zo o oigi: Chm 53, UC, Bly Bad Stuctu of 3D F Elcto i FCC i ducd zo schm Notatio: <=>[00] dictio X<=>BZ dg alog [00] dictio <=>[] dictio E()=( /m) ( x + y + z ) L<=>BZ dg alog [] dictio 8

9 Chm 53, UC, Bly Compaiso btw F Elcto ad Ral Elcto Bad Stuctu of Si Chm 53, UC, Bly v d E m* 9

10 Chm 53, UC, Bly h Chm 53, UC, Bly 0

11 Chm 53, UC, Bly E E E E Paabolic c v m * m * h Wav pact mad of wavfuctios a a paticula wavvcto Motio of cai i fild: Goup vlocity: tasmissio vlocity of a wav pact v g d d de d Acclatio: a d dt g d E ddt d E d d dt Chm 53, UC, Bly E E E E Paabolic c v m * m * h Wav pact mad of wavfuctios a a paticula wavvcto Motio of cai i fild: Goup vlocity: tasmissio vlocity of a wav pact v g d d de d Acclatio: Effctiv mass p m* v dv d dt m* dt a d g dt m * d E ddt d E d d E d d dt

12 Chm 53, UC, Bly Chm 53, UC, Bly m * d E d Positiv m*: th bad has upwad cuvatu d E 0 d If th gy i a bad dpd oly waly o, th m* vy lag d E d m* / m Wh vy small. Havy cai

13 3 Chm 53, UC, Bly ) ( ) ( z y x m m E * d E d m Chm 53, UC, Bly * d E d m E

14 Chm 53, UC, Bly Havy hol Light hol Chm 53, UC, Bly Goup Matial Elcto m Hol m h IV III-V II-VI Si (300K) G GaAs ISb ZO 0.9. ZS

15 Chm 53, UC, Bly Excitos Th aihilatio of a photo i xcitig a lcto fom th valc bad to th coductio bad i a smicoducto ca b witt as a quatio: +h. Sic th is a Coulomb attactio btw th lcto ad hol, th photo gy quid is lowd tha th bad gap by this attactio To coctly calculat th absoptio cofficit w hav to itoduc a two-paticl stat cosistig of a lcto attactd to a hol ow as a xcito Chm 53, UC, Bly Excitos pst th lmtay xcitatio of a smicoducto. I th goud stat th smicoducto has oly filld o mpty bads. Th simplst xcitatio is to xcit o lcto fom a filld bad to a mpty bad ad so catig a lcto ad a hol Excito is utal ov all but cais a lctic dipol momt ad thfo ca b xcitd by ith a photo o a lcto 5

16 6 Chm 53, UC, Bly 4 h m E m h v v m 0 4 a m h =Boh adius =a0=0.59 A o m v E ) ( ) ( = V Chm 53, UC, Bly Excitos 0 4 a m h * * 4 ) ( h m m E E U Bidig gy: m v E E Excito Boh Radius: h x m m m h ) ( 4 * * 4 h m E H ) ( ) ( z y x m m E Rducd mass

17 Chm 53, UC, Bly Rx E 3.6 v m m Chm 53, UC, Bly Smicoducto Egy gap (V) at 73 K Effctiv mass m*/m Dilctic costat Elctos Hols G Si ISb IAs IP GaSb GaAs

18 Chm 53, UC, Bly Excito 3.6v m E m Smicoducto Eg... / m Rx o Ex * * ( m / m ; m / m ) h h mv m Si. 0.33; G ; GaAs (0.066, 0.5) CdS.74 (0.3, 0.45) 5 5. Bi small >50 ZO 3.4 (0.7,?) 59 3 GaN 3.4 (0.9, 0.60) 5 x Chm 53, UC, Bly Excito bidig gy h h h Rlatd to DOS othwis dissociats 8

19 Chm 53, UC, Bly h Chm 53, UC, Bly 9

20 Chm 53, UC, Bly h Chm 53, UC, Bly 0

21 Chm 53, UC, Bly Chm 53, UC, Bly h h h

22 Chm 53, UC, Bly Implicatio i sola cll Chm 53, UC, Bly Thi film PV Si PV

23 Chm 53, UC, Bly Chm 53, UC, Bly 3

24 Chm 53, UC, Bly Chm 53, UC, Bly 4

25 Chm 53, UC, Bly Chm 53, UC, Bly h 5

26 Chm 53, UC, Bly Fmi s Gold Rul: Tim-dpdt ptubatio thoy: tat xcitatios which dpd o tim Optical tasitio: viw th solid with uptubd Hamiltoia H0 as big ptubd by th tim-dpdt EM fild H (t) gatd by th icidt photo flux. H H 0 H '( t) Tasitio Rat ml m H ' l ( E E l m ) is th photo gy +: missio -: absoptio Atis, Molcula Quatum Mchaics, Oxfod Chm 53, UC, Bly 6

27 Chm 53, UC, Bly Fmi s Gold Rul: Assum th stat m ad l a th valc ad coductio bad stats, th m H ' l v H ' c H ' vc Dfi: Joit dsity of stats Chm 53, UC, Bly Fmi s Gold Rul: S is th sufac of all possibl dict optical tasitios with h = Ec - Ev S Lt's itoduc a gy sufac S i -spac such that Ec Ev = d d =dsd vc( ) ( ( ) ( ) ) 3 8 d Ec Ev 7

28 Chm 53, UC, Bly Fmi s Gold Rul: At citical poit wh Lag JDOS cotibutio. E Ec m * E Ev m * h Chm 53, UC, Bly Fmi s Gold Rul: 8

29 Chm 53, UC, Bly h h h Chm 53, UC, Bly Fmi s Gold Rul: Spotaous missio at ( ) M if () Dsity of stats M: tasitio matix lmts M if V dv i f Opato fo th physical itactio that coupls th iitial ad fial stats 9

30 Chm 53, UC, Bly Slctio Rul: Elctic Dipol (E) Tasitio Light itactio with dipol momt (p=x): H E p Ex i( x E( x, t) E0 I gal, th wavlgth of th typ of lctomagtic adiatio which iducs, o is mittd duig, tasitios btw difft atomic gy lvls is much lag tha th typical siz of a atom. x x x 0 Elctic dipol appoximatio Tasitio dipol momt M M M H P m ix (0) E V 3 x d 3 y d 3 z d m x dv Fo x, y, z polaizd light Light as hamoic EM pla wav t ) ( ) M Slctio Rul: Elctic Dipol (E) Tasitio 3 M x d 3 Dipol Momt M y d 3 M z d Chm 53, UC, Bly Matix lmt (dipol momt) is o-zo allowd lctic dipol tasitio Paity of wavfuctio: sig chag ud ivsio about th oigi v paity: f(-x)=f(x) odd paity: f(-x)=-f(x) Iitial/fial wavfuctios must hav difft paitis fo allowd lctic dipol tasitio! 30

31 Chm 53, UC, Bly Elctoic Tasitios i H atoms Hydog atom: lowst stat S, optical tasitio btw S & S? Both stats a symmtic, agula momtum l=0 No lctoic tasitio btw S ad S! Chm 53, UC, Bly Elctoic Tasitios i H atoms Hydog atom: lowst stat S, optical tasitio btw S & P? P is asymmtic, agula momtum l= Elctoic tasitio btw S ad P is allowd! 3

32 Chm 53, UC, Bly Slctio Ruls Symmtic fuctio (gad): Asymmtic fuctio (ugad): ( x) ( x) ( x) ( x) Th opato of th lctic fild: -(x)=-x Tasitio btw two gad fuctios: (0) Tasitio btw gad ad ugad fuctios: H H P P (0) gugdv uugdv udv 0 gdv 0 Fobidd Allowd Slctio ul fo lctoic tasitio: l Chm 53, UC, Bly Elctoic Tasitios: Paticl i a box 3