Translation symmetry, Space groups, Bloch functions, Fermi energy

Transcription

1 9/7/4 Trnsltion symmetry, Spe groups, Bloh funtions, Fermi energy Roerto Orlndo Diprtimento di Chimi Università di Torino Vi ietro Giuri 5, 6 Torino, Itly roerto.orlndo@unito.it Outline The rystllogrphi model of rystl: trnsltion invrine nd lttie point symmetry nd rystllogrphi point groups Brvis ltties nd spe groups Our model of perfet rystl The k-monkhorst net Lol sis sets nd Bloh funtions The lultion of the Crystlline Oritls with lol sis set

2 9/7/4 Crystl strutures O Si Qurtz rystls Qurtz struture How do we generte rystl from struturl motif? The infinite rystl of qurtz n e generted from two toms, Si O, y the pplition of symmetry opertions The lttie , : sis vetors Every lttie vetor is liner omintion of the sis vetors A lttie is trnsltion invrint: trnsltions long lttie vetors re symmetry opertions

3 9/7/4 Brvis lttie Unit ell: volume in spe tht fills spe entirely when trnslted y ll lttie vetors. 3 The ovious hoie: prllelepiped defined y,, 3, three sis vetors with the est,, 3 re s orthogonl s possile the ell is s symmetri s possile (4 types) g A unit ell ontining one lttie point is lled primitive ell. oint symmetry opertions When point symmetry opertion is pplied to lttie, the trnsformed unit ell is indistiguishle from the originl unit ell. Rottion round n xis: n = 36 / xis order 3 During the tion of point symmetry opertion t lest one point stys put 3

4 9/7/4 Rottions Reflexion m Inversion Rottion with inversion,, 3, 4, 6 ml eix to the xis Comining rottion xes Comining rottion xes nd inversion 3 rystllogrphi point goups Trilini Monolini Orthorhomi Trigonl Tetrgonl Hexgonl Cui m /m 4/m 6/m m 3 mm 3m 4mm 6mm m 4m 6m 43m mmm 4/mmm 6/mmm m3m 4

5 9/7/4 The 7 Crystl Systems Trilini:, Monolini:, m, /m Orthorhomi:, mm, mmm C Tetrgonl: 4, 4, 4, 4m, 4mm, 4/mmm, 4/m C I F I Cui: 3, 43, 43m, m3, m3m Trigonl: 3, 3, 3, 3m Hexgonl: 6, 6, 6mm, 6, 6/m, 6/mmm, 6m R F I rimitive nd entered ltties rimitive lttie m 8 8 Centered ltties: Fe-entered F, A, B, C Body-entered t I t I (,,),, m 8 8 (x, y, z) x, y, z ½ 5

6 9/7/4 The 4 Brvis Ltties Trilini:, Monolini:, m, /m Orthorhomi:, mm, mmm C Tetrgonl: 4, 4, 4, 4m, 4mm, 4/mmm, 4/m C I F I Cui: 3, 43, 43m, m3, m3m Trigonl: 3, 3, 3, 3m Hexgonl: 6, 6, 6mm, 6, 6/m, 6/mmm, 6m R F I Srew xes 6

7 9/7/4, n d Glide plnes ROJ. 4 4 SIMBOLS d 4 4 I I ROJ. d } n () types 4 4 } n d () types n}types prllel to () The 3 Spe Groups All possile omintions of symmetry opertions omptile with the 4 Brvis ltties originte the 3 spe groups Clssifition nd detils out spe groups n e found in the Interntionl Tles for Crystllogrphy vol. A ed. Theo Hhn, Kluwer Ademi ulishers Hermnn-Muguin symols digrms of the spe groups symmetry opertions generl nd speil positions sugroups supergroups refletion onditions 7

8 9/7/4 Defining the rystl struture: α-qurtz Crystl system: trigonl Spe group: 3 (o. 54) Unit ell prmeters (γ= ): = 4.96 Å Formul unit: SiO = Å o. of formul units per ell: 3 SiO Frtionl oordintes of the toms in the symmetri unit: x Si.4697 y z x O.435 y.669 z.9 (x, y, z tom oordintes in terms of the sis vetors - in Å) Hermnn-Muguin symol 8

9 9/7/4 Generting the unit ell of α-qurtz Symmetry opertions: + 3 / 3-3 / 3 / 3 / 3 Si frtionl oordintes O frtionl oordintes (generl position) long the two-fold xes (speil position) 9

10 9/7/4 The Wigner-Seitz ell Wigner-Seitz ell: the portion of spe whih is loser to one lttie point thn to nyone else. There re 4 different types of Wigner-Seitz ells. A Wigner-Seitz ell is primitive y onstrution. The reiprol lttie A reiprol lttie with lttie sis vetors,, 3 orresponds to every rel (or diret) lttie with lttie sis vetors,, 3. Bsis vetors oey the following orthogonlity rules: = π = π 3 3 =π = 3 = 3 = 3 = = 3 = or, equivlently, = π/v Λ 3 = π/v 3 Λ 3 = π/v Λ V = Λ 3 V* = (π) 3 /V

11 9/7/4 The first Brillouin zone trnsformtion to the reiprol spe Unit ells for feentered-ui rystls First Brillouin zone for f lttie Brillouin zone: Wigner-Seitz ell in the reiprol spe. Speil points in the Brillouin zone hve een lssified nd lelled with letters, used in the speifition of pths. Wigner-Seitz ell for ody-entered-ui rystls The model of perfet rystl A rel rystl: mrosopi finite rry of very lrge numer n of toms/ions with surfes the frtion of toms t the surfe is proportionl to n -/3 (very smll) if the surfe is neutrl, the perturtion due to the oundry is limited to few surfe lyers rel rystl mostly exhiits ulk fetures nd properties A mro-lttie of unit ells is good model of suh rystl s is very lrge nd surfe effets re negligile in the ulk, the mro-lttie n e repeted periodilly under the tion of trnsltion vetors,, 3 3 without ffeting its properties thus, our model of perfet rystl oinides with the rystllogrphi model of n infinite rry of ells ontining the sme group of toms

12 9/7/4 The mro-lttie A A T ii T i i A Commuttive group of trnsltion opertors T i i Born-Von Kármn oundry onditions Our model of rystl onsists of n infinite rry of mro-ltties. Every mro-lttie ontins unit ells Every funtion or opertor defined for the rystl then dmits the following oundry onditions: trnsltion symmetry f (r + i i ) = f (r) for i =,, 3 new metri is defined A i = i i for i =,, 3

13 9/7/4 From rel to reiprol spe: metris A Rel spe lttie: spnned y, Rel spe mro-lttie: spnned y A, A Rel spe A Reiprol spe lttie: spnned y, Reiprol spe B B Reiprol spe miro-lttie: spnned y B, B k-monkhorst net A reiprol lttie n e ssoited with the mro-lttie with metri A: the reiprol miro-lttie with metri B = - this miro-lttie is muh denser thn the originl reiprol lttie with metri the unit ell of the originl reiprol lttie with metri n e prtitioned into miro-ells, eh loted y vetor (wve vetor) k m : m m m k m When referred to the st Brillouin zone, the omplete set of the k m vetors form the k-monkhorst net m, m, m integers Integers i re lled the shrinking ftors 3

14 9/7/4 Chrter Tle of the trnsltion group k m k m trnsltion opertions lsses one-dimensionl irreduile l l exp( ik l ) exp( ik l ) m m exp( ikm l ) exp( ikml ) representtions ssoited with k m vetors of Brillouin Zone Due to the orthogonlity etween rows nd olumns exp ik k l δ m m mm & exp ik m ( l l ) m Atomi oritl sis set Choie of set of lol funtions L.C.A.O method (Liner Comintion of Atomi Oritls) the lol sis set M tomi oritls ( r - l ) M tomi oritls () r per ell representtion of opertor f f ( l, l ) f l l r -l f r r l dr * (, ) ( ) ( ) ( - ) V 4

15 9/7/4 Trnsltion invrine of mtrix elements If in f l l r l f r r l dr * (, ) ( ) ( ) ( ) V the origin is trnslted y l *, ( ) ( ) f l l r f r l r l l dr V ut, for trnsltion invrine nd if f(r) is periodi with the sme periodiity of the diret lttie, r l () r r l r l l r l f ( r) f ( r l ) so tht mtrix elements re lso trnsltion invrint l, l, l l, l f f f Representtion of opertors in the AO sis Consequenes of trnsltion invrine: f ( l, l ) f (, l l ) l Squre Mtrix: loks of size M l 5

16 9/7/4 Bloh funtions The lol sis M tomi oritls ( r - l ) In this sis set the mtrix element f μν depends on two new indies: k m nd k m whih re vetors of the reiprol spe: Bloh funtion sis sis for the irreduile representtion of the trnsltions group ( k, r) ( k, r) exp( ik l ) ( r l ) m m m ikml kml * f ( k m, k m) e f ( ) d r l r r l r, V f ( k, k ) e e f l, l Bloh theorem ikml ikml m m for trnsltion invrine nd fter multiplying nd dividing y i m m i m f ( k, k ) e k k l e k l l f, l l m m e ikml ut, s the sum over extends to ll diret lttie vetors, it is equivlent to summing over ll vetors l = l l ikm km l ikml f ( k m, k m) e e f, l nd relling the hrter orthogonlity reltion for rows, we get f i m m m f m m k l k k ( k, k ) e ( l ) 6

17 9/7/4 Consequenes of Bloh theorem the AO sis ( r - l ) Bloh funtion sis ( k, r) m loks of size M ssoited with k m Clultion of the Crystlline Oritls with the Self-onsistent-field pproximtion Representtion of one-eletron opertor F in the sis of the AOs : Clultion of (M ) elements of F: F () l Representtion in the sis of Bloh funtions: At every point k m of k-monkhorst net: F (k ) exp( ik l ) F ( l ) m m At every k m, the eigenvlue eqution of size M is solved: M rystlline oritls F( k ) C( k ) S( k )C( k )E( k ) m m m m m ( k, r ) ssoited with ( k ) M m ( k, r) C ( k ) ( k, r) m m m m 7

18 Energy (hrtree) 9/7/4 Clultion of the Crystlline Oritls with the Self-onsistent-field pproximtion Aufu priniple in populting eletroni nds Fermi Energy F Clultion of density mtrix elements in the AO sis from oupied CO in reiprol spe in diret spe M totl * ( k m) C ( k m) C ( k m) ) oupied totl totl (l ) exp( ik m l ) ( k m) m redy for the lultion of F in the AO sis during the next itertive step Bnd struture representtion: α-qurtz ondution nd Fermi energy gps p O } 3p Si s Si p Si s O s O vlene nds s Si top vlene nd k pth 8

19 9/7/4 Fermi energy n sttes M ( F ( km)) / m with θ(x)= if x < nd θ(x)= if x n sttes M ( F ( )) d BZ BZ k k definition of new funtion : Beryllium Fermi energy n sttes M ( ) ( ( )) d BZ BZ k k itertive evlution of F (n sttes ()-n sttes ) Definition of the density mtrix M totl ( l) (, ) ( F ( )) d k l k k BZ BZ * (, ) C ( ) C ( )exp i k l k k k l 9