Band structures calculations

Transcription

1 lultos reprol spe of -vetors, Brllou zoe seulr equto, vrtol method Ttle pge bd struture, perod potetl desty of sttes, Ferm eergy lmost free eletros method tght bdg method, MO-LCO, Bloh futo

2 Lterture T.. lbrght, J. K. Burdett, M.-. Whgbo, Wley 3 Orbtl Itertos I Chemstry. J. K. Burdett, Progress Sold Stte Chemstry From Bods to Bds d Moleules to Solds.. Cdell, M.-. Whgbo, Chem. Rev Coeptul spets of struture-property orreltos d eletro stbltes, wth ppltos to low-dmesol trsto-metl odes. R. offm, gew. Chem. It. d. gl ow hemstry d physs meet the sold stte.. L. Messler, P. J. Fsher, D.. Trr: Iorg hemstry 5th ed., hp.5 Moleulr Orbtls. S. Cotteer 3 Desty Futol Theory d the Fmly of LPW-methods: step-by-step troduto.

3 3 Reprol spe -vetors spe spe of -vetors - reprol spe R r z y 3 3 b b b b b b g l h w v u r r r b b b r r b b b dret ltte: r rystl ltte reprol spe: reprol ltte

4 Brllou zoes holds: - = - - Oe vlue of for eh wth oe bd - s perodl futo of, t s suffet to be dsplyed wth the tervl -/ ; / the frst Brllou zoe the frst Brllou zoe Wger-Setz ell the reprol ltte Wger-Setz ell s lwys prmtve d t lwys hs the sme symmetry s the ltte prmtve rystlogrph ell my hve lower symmetry th the ltte Costruto: the ples orml to b, b, b 3 through the pots ±b, ±b, ±b 3 /b Y -/ / -X X -/b -Y 4

5 Brllou zoes 5

6 Brllou zoes s R M X smple ub b b dret spe orrespods to f reprol spe f f dret spe orrespods to b reprol spe 6

7 Brllou zoes Brllou zoes of hgher order: the sme volume s the. Brllou zoe. the sme symmetry s the. Brllou zoe. trslto by reprol ltte vetor shfts them to the. Brllou zoe.. Brllou zoe. Brllou zoe 3. Brllou zoe 7

8 Brllou zoes Trl / Trgol /6 Mool /m /4 / Orthorhomb mmm /8 egol 6/m / Tetrgol 4/m /8 6/mmm /4 4/mmm /6 Cub m3m /4 /48 3 m3

9 Brllou zoes = X= Y= Z= S= T= U= R=

10 Shrödger equto Shrödger equto ydroge tom: spherl oordtes: m r etá. eergy e 4 o r y z r poteálí potetl eergy. R, l r Yl, r r,, l, m m Crystl feld m: eletro mss o : permtvty of vuum : wve futos e: eletro hrge : eergy ħ: Pl s ostt R: rdl futo Y: gulr futo r, l, m, l, m L Yl, m l l Yl, m T L zyl, m mlyl, m : prpl qutum umber l: orbtl qutum umber determe the orbtl gulr mometum l =... - m l : mget qutum umber proeto of the gulr mometum to z-s m l = -l l

11 Nerly free eletros Tght bdg model Ĥ Ĥ N : et wve futo : ppromte wve futo epressed the bss = for N : e.g. tom orbtls, ple wves,... Nerly free eletros : Ket eergy predomtes over potetl eergy Bss = ple wves metll bod, eletro gs ep[ ] Tght bdg: Potetl eergy predomtes over et eergy Bss = tom orbtls Covlet d o bods

12 Ple wve Ple wve: ostt frequey spreds le fte prllel ples orml to vetor of moto. ep[ ]

13 3 Seulr equto System of equtos hs o-trvl soluto, oly f the mtr determt = : S S S S S S S det S S S S S S S Multply equto 3 from left subsequetly by futos,,...,, d rete system of equtos: Covert to mtr form, for the ostt t holds = : Covert ll o oe sde d o to mtr: Mtr ege vetors: Symmetry s vetor: Ԧv = Ԧv ge-futos: Φ = Φ Φ = Φ, Φ = σ = φ By substtuto of to 3 3 φ + φ + + φ = φ + φ + + φ Uows:,

14 Retrevl of ege-vlues d ege-vetors Ԧ = Ԧ, = B = P P: B = Ԧ B = P Ԧ Ԧ B = P Ԧ Job s method, ves mtres P B = Φ = Φ, Φ = σ = φ By substtuto of to 3 3 φ + φ + + φ = φ + φ + + φ Uows:, Ԧ B =,,,, Ԧ B =,,,, Ԧ = Ԧ Ԧ Ԧ, = I Ԧ, = I:utry mtr B B B = R,I Comple mtr: R - I I R 4

15 rtol method seulr equto System of equtos for =,,..., N [ [ [ det ] S S ] ] [ [ [ S ] ] S ] [ [ S S S N S [ S ] S = S [ S S ] ] ] System of equtos hs soluto, oly f determt S = : S S Clulto of the determt seulr equto of the N.order. The soluto s N ege-vlues eergy. For eh we get N oeffets ege-vetors by solvg the system of equtos. : eergy of the futo = If the potetl depeds o the wve futos,.e. o the serhed oeffets, the seulr equto must be solved tertvely, by the so-lled SCF method selfosstet feld. S d d : ehge tegrl =: o-ste eergy of dvdul bse sttes. S : overlp tegrl. S = =, S. 5

16 6 rtol method : et wve futo : ppromte wve futo epressed the bss = for N : e.g. tom orbtls, ple wves,... Ĥ N,,,, N N N N N N N N S S d d d d d d d d S : ehge tegrl =: o-ste eergy of dvdul bse sttes. S : overlp tegrl. S = =, S. Ĥ

17 Nerly free eletros ep[ ] 4 ep[ ] ep[ : Potetl s rel : ] Ltte vetors. For D =,,,... t potetl: huge ttrto fore er the ore. If we re terested the potetl whh the eletros espelly the vlee move, we eglet the vty of the uleus. L ep[ Futo: ] L l For D l =,,,..., L/ Potetl s repeted fter perod, futo s repeted fter perod L. The.Brllou zoe s / the reprol spe, the futo s lulted /L. L L L L 7

18 Nerly free eletros Wve futo d potetl put to Shrödger equto ep[ ] m m e e e m e, e I order the sum to be =, eh term [] must be =. ep[ ] m Mster equto: system of L equtos, formulto of seulr equto for the ple wve bss. rous solutos wth. Brllou zoe. -/ / - / l/l / 8

19 9 Nerly free eletros mster equto system of L equtos, vrous solutos wth. Brllou zoe - / /. m s B os o o o m B B, : e m m mv p, for >

20 -vetors, bd wdth, bd gp - qutum umber wve vetor p mv h umber of llowed vlues = umber of elemetry ells rystl free eletros: mv p m m e e e -/ -/ / / -/ / bd wdth: determed by the overlp of tertg orbtls s for MO

21 Desty of sttes DOS, g - umber of llowed eergy levels per eergy tervl holds: gd = umber of levels the tervl ; +d dmeso: geerlly: g g BZ S ds, e s s -/. umerlly: g / e, DOS

22 Desty of sttes -D 3-D e e X M Ne X M R Ne s M R X M X

23 Ferm level Ferm level the hghest ouped level t T= K T>: Ferm-Dr sttst holds: ouped sttes DOSf f ep / T F B Ferm ple the set of -spe, for whh holds = F ChemPot.ee 3

24 4 MO-LCO = Moleulr orbtls ler ombto of tom orbtls : moleulr orbtl, : tom orbtl N B S S R R R R S R R R R R R B B B B B B B B B B B B BB B, S S S S BB B B B B Cell otg detl orbtls N S ] [

25 5 MO-LCO = Moleulr orbtls ler ombto of tom orbtls det S S S, β S = : oulomb eergy eergy of O < = t : ehge eergy degree of bodg eergy S - : overlp tegrl,, S S B B B B R R S R R R R, S : :, det

26 Bd struture Bloh orbtls... BO N r, r ep N BO : Bloh orbtl, : tom orbtl = = = N BO = r + r-e + r-e r-e N ep os s = e = =/ os/ =,,-,,... s/ =,,,-,... - =/ X e = - =,-,... + / X

27 Symmetry of orbtls / e s s /. / e p p / -/. / 7

28 Symmetry of orbtls e os s p y X s -/. / e os d y p X. -/ / 8

29 Formto of bds orbtls p y 9

30 Formto of bds orbtls p y Reltve otrbuto of dvdul orbtls to eh eergy levels s lso dsplyed 3

31 Formto of bds orbtls p Reltve otrbuto of dvdul orbtls to eh eergy levels s lso dsplyed 3

32 Bd wdth Bd wdth W z W p > W s p orbtls reh loser to eh other, bgger overlp W z > W,W y -bodg > - bodg vlee > ore Delolzto of orbtls: W5d > W4d > W3d Smll dfferee of orbtls eerges WCo-O > WT-O 3

33 33 Tght bdg method CO-LCBO Bloh orbtls: - bss BO N ep, R R r r Crystl orbtls: CO, =?,, r r l l S l l S l l l l S l l l l t prmeters: mtr elemets: d l d l l l

34 Tght bdg method CO-LCBO Oly the terto wth the erest eghbours re te to out: oly the ehge tegrl wth the erest eghbour ~, t~,s<< e e os y e e e e os os y os z y e z e z os y,,,,,

35 Ler rystl, toms bss = =, t = t = t - - X X - - MO ~ = X X 35

36 Ler rystl, toms bss = =, t = t = t - - X X - - MO ~ = X X 36

37 Ler rystl, toms bss geerl formuls t p -p t - - p e e p p p e e p e p e e p e p e 37

38 Ler rystl, toms bss geerl formuls t t BO CO = +?,, = 38

39 Ler rystl, toms bss = =, t < t < <, t = t = t < w = t e g = t t w = t e g = 39

40 CuO - ple bodg b g p = t pp = -. 3 d = -.9 t pd = -.5 p [e] tot d -y p p y = X M DOS [e - ] M X [e] 4

41 CuO - ple X X M 4

42 CuO - ple M X M 4

43 Formto of bd gp o sultors 8 6 Cl - 3s NCl Cl - 3p N-3s DOS [e - ] 4 N - 3s Cl-3p ovlet sultors [e] C - dmod C-p C-s DOS [e - ]..5 C - s C - p [e] 43

44 Some bs reltos O f f Of Of ; Of, B B B, L y z Of, p L L, L L ler opertor ommutg opertors Fl pv F m p mv Ft K K K K K K K d d S p T T b; K K K T d K ; b b K, t. K K, t. K T b K K s ermt opertor K : Comple ougte ermt mtr utry mtr ortogol mtr S = : ormlsed futo S = : orthogol futo S = : orthoorml futo 44