19 th WIEN2k Workshop Waseda University Tokyo Relativistic effects & Non-collinear magnetism. (WIEN2k / WIENncm)

Transcription

1 9 th WIENk Wokshop Wasda Univsity Tokyo Rlativistic ffcts & Non-collina magntism WIENk / WIENncm) Xavi Rocquflt Institut ds Matéiaux Jan-RouxlUMR 65) Univsité d Nants, FRANCE

2 9 th WIENk Wokshop Wasda Univsity Tokyo Talk constuctd using th following documnts: Slids of: Robt Laskowski, Stfaan Cottni, Pt Blaha and Gog Madsn Nots of: - Pavl Novak Calculation of spin-obit coupling) - Robt Laskowski Non-collina magntic vsion of WIENk packag) Books: - WIENk usguid, ISBN Elctonic Stuctu: Basic Thoy and Pactical Mthods, Richad M. Matin ISBN Rlativistic Elctonic Stuctu Thoy. Pat. Fundamntals, Pt Schwdtfg, ISBN wb: winlist digst wikipdia

3 Fw wods about Spcial Thoy of Rlativity Light Composd of photons no mass) Spd of light = constant Atomic units: ħ = m = = c 37 au

4 Fw wods about Spcial Thoy of Rlativity Light Composd of photons no mass) Spd of light = constant Matt Composd of atoms mass) v = fmass) Atomic units: ħ = m = = Spd of matt mass c 37 au mass = fv)

5 Fw wods about Spcial Thoy of Rlativity Light Composd of photons no mass) Spd of light = constant Matt Composd of atoms mass) v = fmass) Atomic units: ħ = m = = Spd of matt mass c 37 au mass = fv) Lontz Facto masu of th lativistic ffcts) γ = v c Rlativistic mass: M = γm m: st mass) Momntum: p = γmv = Mv Total ngy: E = p c m c 4 E = γmc = Mc

6 Dfinition of a lativistic paticl Boh modl) Lontz facto γ) «Non-lativistic» paticl: γ = Hs) Aus) Spd v) Dtails fo Au atom: 79 v s) = c =. 58c 37 c 37 au γ = v c Spd of th s lcton Boh modl): Z v n = -.58) Z H :: v s s) ) = au au v Au : v Au : s ) = 79 au s ) = 79 au v =. γγ= =.3 γ =. γ =. s lcton of Au atom = lativistic paticl M s-au) =.m

7 Rlativistic ffcts Z ) Th mass-vlocity coction Rlativistic incas in th mass of an lcton with its vlocity whn v c)

8 Rlativistic ffcts Z ) Th mass-vlocity coction Rlativistic incas in th mass of an lcton with its vlocity whn v c) ) Th Dawin tm It has no classical lativistic analogu Du to small and igula motions of an lcton about its man position Zittbwgung)

9 Rlativistic ffcts Z ) Th mass-vlocity coction Rlativistic incas in th mass of an lcton with its vlocity whn v c) ) Th Dawin tm It has no classical lativistic analogu Du to small and igula motions of an lcton about its man position Zittbwgung) 3) Th spin-obit coupling It is th intaction of th spin magntic momnt s) of an lcton with th magntic fild inducd by its own obital motion l)

10 Rlativistic ffcts Z ff ) Th mass-vlocity coction Rlativistic incas in th mass of an lcton with its vlocity whn v c) ) Th Dawin tm It has no classical lativistic analogu Du to small and igula motions of an lcton about its man position Zittbwgung) 3) Th spin-obit coupling It is th intaction of th spin magntic momnt s) of an lcton with th magntic fild inducd by its own obital motion l) 4) Indict lativistic ffct Th chang of th lctostatic potntial inducd by lativity is an indict ffct of th co lctons on th valnc lctons

11 On lcton adial Schöding quation HARTREE ATOMIC UNITS INTERNATIONAL UNITS H S Ψ = V Ψ = εψ H S Ψ = h m V Ψ = εψ Atomic units: ħ = m = = /4 πε ) = c = / α 37 au

12 On lcton adial Schöding quation HARTREE ATOMIC UNITS INTERNATIONAL UNITS H S Ψ = V = V Ψ = εψ Z H In a sphically symmtic potntial Ψ = R S Ψ ) Y θ,ϕ) n, l, m n, l l, m = V h m Z = 4πε V Ψ = εψ = ) ) sin θ sin θ θ θ sin θ ) ϕ Atomic units: ħ = m = = /4 πε ) = c = / α 37 au

13 On lcton adial Schöding quation Ψ Ψ = = Ψ ε V H S ) l n l n l n R R l l m V d dr d d m,,, = ε h h In a sphically symmtic potntial Ψ Ψ = = Ψ ε V m H S h Z V = Z V 4πε = ) l n l n l n R R l l V d dr d d,,, = ε INTERNATIONAL UNITS ) ) θ,ϕ,,,, m l l n m l n Y = R Ψ ) ) ) = sin sin sin ϕ θ θ θ θ θ HARTREE ATOMIC UNITS

14 Diac Hamiltonian: a bif dsciption Diac lativistic Hamiltonian povids a quantum mchanical dsciption of lctons, consistnt with th thoy of spcial lativity. E = p c m c 4 H Ψ D = ε Ψ with H D = c α p βmc V

15 Diac Hamiltonian: a bif dsciption Diac lativistic Hamiltonian povids a quantum mchanical dsciption of lctons, consistnt with th thoy of spcial lativity. = k k k σ σ α = β k = σ = i i σ = σ 3 ) Pauli spin matics Momntum opato Rst mass Elctostatic potntial V m c p c H D = β α ) unit and zo matics Ψ = D Ψ ε H with E = p c m c 4

16 Diac quation: H D and Ψ a 4-dimnsional Ψ is a fou-componnt singl-paticl wav function that dscibs spin-/ paticls. spin up spin down H D ψ ψ ψ ψ = ε ψ 3 ψ 3 ψ 4 ψ 4 In cas of lctons: Lag componnts Φ) Small componnts χ) Φ ψ = χ facto /m c ) Φ and χ a tim-indpndnt two-componnt spinos dscibing th spatial and spin-/ dgs of fdom Lads to a st of coupld quations fo Φ and χ: c c ) σ p χ = ε V m c )φ ) σ p φ = ε V m c )χ

17 Diac quation: H D and Ψ a 4-dimnsional Fo a f paticl i.. V = ): ) ) ) ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 4 3 = Ψ Ψ Ψ Ψ m c p ip p m c ip p p p ip p m c ip p p m c z y z y z z z y z y x z ε ε ε ε, φ c m, φ m c Paticls: up & down, χ c m χ, m c Antipaticls: up & down Solution in th slow paticl limit p=) Non-lativistic limit dcoupls Ψ fom Ψ and Ψ 3 fom Ψ 4

18 Diac quation: H D and Ψ a 4-dimnsional Fo a f paticl i.. V = ): ) ) ) ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 4 3 = Ψ Ψ Ψ Ψ m c p ip p m c ip p p p ip p m c ip p p m c z y z y z z z y z y x z ε ε ε ε, φ c m, φ m c Paticls: up & down, χ c m χ, m c Antipaticls: up & down Solution in th slow paticl limit p=) Non-lativistic limit dcoupls Ψ fom Ψ and Ψ 3 fom Ψ 4 Fo a sphical potntial V): ) ) Υ Υ = Φ Ψ = κσ nκ κσ nκ χ f i g g nκ and f nκ a Radial functions Y κσ a angula-spin functions s l j = ) = j s κ, = s

19 Diac quation in a sphical potntial Fo a sphical potntial V): Th sulting quations fo th adial functions g and f nκ nκ ) a simplifid if w dfin: ε ' V ε ' = ε m c ) ) Engy: Radially vaying mass: M = m c

20 Diac quation in a sphical potntial Fo a sphical potntial V): Th sulting quations fo th adial functions g and f nκ nκ ) a simplifid if w dfin: ε ' V ε ' = ε m c ) ) Engy: Radially vaying mass: M = m c Thn th coupld quations can b wittn in th fom of th adial q.: h d d dg h l l ) h dv dg h dv κ ) nκ nκ V g nκ M d M 4M d d c 4M c d g nκ = ε ' g nκ Mass-vlocity ffct Dawin tm Spin-obit coupling h m d d dr d n, l l ) h l V m R n, l = ε R n, l On lcton adial Schöding quation in a sphical potntial Not that: κ κ ) = l l )

21 Diac quation in a sphical potntial Fo a sphical potntial V): Th sulting quations fo th adial functions g and f nκ nκ ) a simplifid if w dfin: ε ' V ε ' = ε m c ) ) Engy: Radially vaying mass: M = m c Thn th coupld quations can b wittn in th fom of th adial q.: h d d dg h l l ) h dv dg h dv κ ) nκ nκ V g nκ M d M 4M d d c 4M c d g nκ = ε ' g nκ and κ ) dfnk = V ε ') gnκ d h c Not that: κ f nκ κ ) = l l ) Dawin tm Du to spin-obit coupling, Ψ is not an ignfunction of spin s) and angula obital momnt l). Instad th good quantum numbs a j and κ Spin-obit coupling No No appoximation hav hav bn mad so so fa fa

22 Diac quation in a sphical potntial Scala lativistic appoximation Appoximation that th spin-obit tm is small nglct SOC in adial functions and tat it by ptubation thoy) g ~ ~ No SOC Appoximat adial functions: f n f h d d ~ h dg~ l ) h l nl V g~ nl M d M 4M c dg~ M c d h nκ g nl dv d dg~ d nl κ = ε ' g~ nl and fnl = with th nomalization condition: ~ gnl fnl ) d = nl nl ~

23 Diac quation in a sphical potntial Scala lativistic appoximation Appoximation that th spin-obit tm is small nglct SOC in adial functions and tat it by ptubation thoy) g ~ ~ No SOC Appoximat adial functions: f n f h d d ~ h dg~ l ) h l nl V g~ nl M d M 4M c dg~ M c d h nκ g nl dv d dg~ d nl κ = ε ' g~ nl and fnl = with th nomalization condition: ~ gnl fnl ) d = Th fou-componnt wav function is now wittn as: Φ g Ψ = = χ i f nl nl Φ is a pu spin stat ) Υlm ) Υlm χ is a mixtu of up and down spin stats Inclusion of th spin-obit coupling in scond vaiation on th lag componnt only) H ~ ψ = εψ ~ H ~ SO ψ H SO = h 4M with c dv d nl σl nl ~

24 Rlativistic ffcts in a solid Fo a molcul o a solid: Rlativistic ffcts oiginat dp insid th co. It is thn sufficint to solv th lativistic quations in a sphical atomic gomty insid th atomic sphs of WIENk). Justify an implmntation of th lativistic ffcts only insid th muffin-tin atomic sphs

25 Implmntation in WIENk Atomic sph RMT) Rgion Atomic sph RMT) Rgion Co Co lctons Valnc lctons «Fully» lativistic Scala lativistic no SOC) Spin-compnsatd Diac quation Possibility to add SOC nd vaiational) SOC: Spin obit coupling

26 Implmntation in WIENk Atomic sph RMT) Rgion Atomic sph RMT) Rgion Intstitial Rgion Intstitial Rgion Co Co lctons Valnc lctons Valnc lctons «Fully» lativistic Scala lativistic no SOC) Not lativistic Spin-compnsatd Diac quation Possibility to add SOC nd vaiational) SOC: Spin obit coupling

27 Implmntation in WIENk: co lctons Atomic sph RMT) Rgion Co lctons «Fully» lativistic Spin-compnsatd Diac quation l s=- s= Co stats: fully occupid spin-compnsatd Diac quation includ SOC) Fo spin-polaizd potntial, spin up and spin down a calculatd spaatly, th dnsity is avagd accoding to th occupation numb spcifid in cas.inc fil. j=ls/ κ=-sj/) occupation s=- s= s=- s= s / - p / 3/ - 4 d 3/ 5/ f 3 5/ 7/ cas.inc fo Au atom 7 7..,-,,-,,-,,-,,,,,,-,4,-,4 3,-, 3,-, 3, 3,,, 3,-,4 3,-,4 3, 3,,4,4 3,-3,6 3,-3,6 4,-, 4,-, 4, 4,,, 4,-,4 4,-,4 4, 4,,4,4 4,-3,6 4,-3,6 5,-, 5,-, 4, 4, 3,6 3,6 4,-4,8 4,-4,8

28 Implmntation in WIENk: co lctons Atomic sph RMT) Rgion Co lctons «Fully» lativistic Spin-compnsatd Diac quation l s=- s= Co stats: fully occupid spin-compnsatd Diac quation includ SOC) Fo spin-polaizd potntial, spin up and spin down a calculatd spaatly, th dnsity is avagd accoding to th occupation numb spcifid in cas.inc fil. j=ls/ κ=-sj/) occupation s=- s= s=- s= s / - p / 3/ - 4 d 3/ 5/ f 3 5/ 7/ s / s / p / p 3/ 3s / 3p / 3p 3/ 3d 3/ 3d 5/ 4s / 4p / 4p 3/ 4d 3/ 4d 5/ 5s / 4f 5/ 4f 7/ cas.inc fo Au atom 7 7..,-,,-,,-,,-,,,,,,-,4,-,4 3,-, 3,-, 3, 3,,, 3,-,4 3,-,4 3, 3,,4,4 3,-3,6 3,-3,6 4,-, 4,-, 4, 4,,, 4,-,4 4,-,4 4, 4,,4,4 4,-3,6 4,-3,6 5,-, 5,-, 4, 4, 3,6 3,6 4,-4,8 4,-4,8

29 Implmntation in WIENk: valnc lctons Valnc lctons INSIDE atomic sphs a tatd within scala lativistic appoximation [] if RELA is spcifid in cas.stuct fil by dfault). Titl Titl F F LATTICE,NONEQUIV.ATOMS: LATTICE,NONEQUIV.ATOMS: 5 5 Fm-3m Fm-3m MODE MODE OF OF CALC=RELA CALC=RELA unit=boh unit=boh ATOM : X=. Y=. Z=. ATOM : X=. Y=. Z=. MULT= ISPLIT= MULT= ISPLIT= Au NPT= 78 R=.5 RMT=.6 Z: 79. Au NPT= 78 R=.5 RMT=.6 Z: 79. LOCAL ROT MATRIX:... LOCAL ROT MATRIX: NUMBER OF SYMMETRY OPERATIONS 48 NUMBER OF SYMMETRY OPERATIONS no κ dpndncy of th wav function, n,l,s) a still good quantum numbs all lativistic ffcts a includd xcpt SOC [] Kolling and Hamon, J. Phys. C 977) Atomic sph RMT) Rgion small componnt nts nomalization and calculation of chag insid sphs augmntation with lag componnt only SOC can b includd in «scond vaiation» Valnc lctons Scala lativistic no SOC) Valnc lctons in in intstitial gion a a tatd classically

30 Implmntation in WIENk: valnc lctons SOC is addd in a scond vaiation lapwso): - Fist diagonalization lapw): - Scond diagonalization lapwso): H Ψ ε Ψ = ε = Ψ H H SO ) Ψ Th scond quation is xpandd in th basis of fist ignvctos Ψ ) N i ) j j i i j δ ε Ψ Ψ Ψ Ψ = ε Ψ Ψ ij H SO sum includ both up/down spin stats N is much small than th basis siz in lapw Atomic sph RMT) Rgion Valnc lctons Scala lativistic no SOC) Possibility to add SOC nd vaiational)

31 Implmntation in WIENk: valnc lctons SOC is addd in a scond vaiation lapwso): - Fist diagonalization lapw): - Scond diagonalization lapwso): H Ψ ε Ψ = ε = Ψ H H SO ) Ψ Th scond quation is xpandd in th basis of fist ignvctos Ψ ) N i ) j j i i j δ ε Ψ Ψ Ψ Ψ = ε Ψ Ψ ij H SO sum includ both up/down spin stats N is much small than th basis siz in lapw Atomic sph RMT) Rgion Valnc lctons Scala lativistic no SOC) Possibility to add SOC nd vaiational) SOC is activ only insid atomic sphs, only sphical potntial V MT ) is takn into account, in th polaizd cas spin up and down pats a avagd. Eignstats a not pu spin stats, SOC mixs up and down spin stats Off-diagonal tm of th spin-dnsity matix is ignod. It mans that in ach SCF cycl th magntization is pojctd on th chosn diction fom cas.inso) V MT : Muffin-tin potntial sphically symmtic)

32 Contolling spin-obit coupling in WIENk Do a gula scala-lativistic scf calculation sav_lapw initso_lapw cas.inso: WFFIL WFFIL 4 4 llmax,ip,kpot llmax,ip,kpot min,max min,max output output ngy ngy window) window) diction diction of of magntization magntization lattic lattic vctos) vctos) NX NX numb numb of of atoms atoms fo fo which which RLO RLO is is addd addd NX NX atom atom numb,-lo,d numb,-lo,d cas.in), cas.in), pat pat NX NX tims tims numb numb of of atoms atoms fo fo which which SO SO is is switch switch off; off; atoms atoms cas.inc): ) ) CONT CONT CONT CONT K-VECTORS K-VECTORS FROM FROM UNIT:4 UNIT: min/max/nband min/max/nband symmtso fo spin-polaizd calculations only) unsp)_lapw -so -so switch spcifis that scf cycls will includ SOC

33 Contolling spin-obit coupling in WIENk Th wwb intfac is hlping you Non-spin polaizd cas

34 Contolling spin-obit coupling in WIENk Th wwb intfac is hlping you Spin polaizd cas

35 Rlativistic ffcts in th solid: Illustation LDA ovbinding 7%) No diffnc NREL/SREL hcp-b Z = 4 Bulk modulus: - NREL: 3.4 GPa - SREL: 3.5 GPa - Exp.: 3 GPa

36 Rlativistic ffcts in th solid: Illustation LDA ovbinding 7%) No diffnc NREL/SREL hcp-b Z = 4 Bulk modulus: - NREL: 3.4 GPa - SREL: 3.5 GPa - Exp.: 3 GPa LDA ovbinding %) hcp-os Z = 76 Cla diffnc NREL/SREL Bulk modulus: - NREL: 344 GPa - SREL: 447 GPa - Exp.: 46 GPa

37 Rlativistic ffcts in th solid: Illustation hcp-b Z = 4 Scala-lativistic SREL): hcp-os Z = 76 - LDA ovbinding %) - Bulk modulus: 447 GPa spin-obit coupling SRELSO): - LDA ovbinding %) - Bulk modulus: 436 GPa Exp. Bulk modulus: 46 GPa

38 ) Rlativistic obital contaction ρ /boh) 5 Non lativistic l=) Radius of th s obit Boh modl): 4 - Z 3. Au s boh) n a h s) = AND a = =boh Z m cα s) = =.3 boh 79 Atomic units: ħ = m = = c = / α 37 au

39 ) Rlativistic obital contaction ρ /boh) 5 4 Non lativistic l=) Rlativistic κ=-) Radius of th s obit Boh modl): - Z 3. % Obital contaction boh) n a s) = = Z γ 79 Au s. =. boh a [ RELA] n a h s) = AND a = =boh Z mcα h = = M cα s) = =.3 boh 79 In Au atom, th lativistic mass M) of th s lcton is % lag than th st mass m) M = γ m =. m a γ

40 ) Rlativistic obital contaction ρ /boh) Au 6s Non lativistic l=) Rlativistic κ=-) Obital contaction. 4 6 boh) γ = Z 79 v 6s) = = = 3.7 =. 96c n 6 v c =.96) =.46 Dict lativistic ffct mass nhancmnt) contaction of.46% only Howv, th lativistic contaction of th 6s obital is lag >%) ns obitals with n > ) contact du to othogonality to s

41 ) Obital Contaction: Effct on th ngy ρ /boh) ρ /boh) Rlativistic coction %) E E ) RELA E NRELA NRELA Non lativistic l=) Rlativistic κ=-) Obital contaction Non lativistic l=) Rlativistic κ=-) Obital contaction s s. Au s boh). 4 6 boh) 3s 4s 5s 6s. Au 6s

42 «Taagona» Intlud Tokyo

43 «Taagona» Intlud Lt s tavl in spac and tim...

44 «Taagona» Intlud Th aim of th tavl is to find som «daily lif» analogus of quantum physics and lativity concpts! Fanc Cosica Spain Wlcom to Taagona Mditanan Sa

45 «Taogona» Intlud

46 «Taagona» Intlud In-t-ludDfinition). an intvning pisod, piod, spac, tc.. a shot damatic pic, sp. of a lighto facical chaact, fomly intoducd btwn th pats o acts of miacland moalityplays o givn as pat of oth nttainmnts. 3. on of th aly English facs o comdis, as thos wittn by John Hywood, which gw out of such pics. 4. any intmdiat pfomanco nttainmnt, as btwn th acts of a play. 5. an instumntal passago a pic of music nddbtwn th pats of a song, chuch svic, dama, tc. Human pyamid at Taagona Spain) Santa Tcla fstival

47 «Taagona» Intlud n = 6 n = 5 n = 4 n = 3 n = n = In-t-ludDfinition). an intvning pisod, piod, spac, tc.. a shot damatic pic, sp. of a lighto facical chaact, fomly intoducd btwn th pats o acts of miacland moalityplays o givn as pat of oth nttainmnts. 3. on of th aly English facs o comdis, as thos wittn by John Hywood, which gw out of such pics. 4. any intmdiat pfomanco nttainmnt, as btwn th acts of a play. 5. an instumntal passago a pic of music nddbtwn th pats of a song, chuch svic, dama, tc. Human pyamid at Taagona Spain) Santa Tcla fstival

48 «Taagona» Intlud n = 6 Indict Impact n = 5 n = 4 n = 3 n = n = Dict Impact Human pyamid at Taagona Spain) Santa Tcla fstival «Rlativistic» ptubation!

49 «Taagona» Intlud n = 6 Indict Impact n = 5 n = 4 n = 3 n = n = Dict Impact Human pyamid at Taagona Spain) Santa Tcla fstival «Rlativistic» ptubation!

50 ) Spin-Obit splitting of p stats ρ /boh).7 Non lativistic l=) Au 5p boh).5

51 ) Spin-Obit splitting of p stats Spin-obit splitting of l-quantum numb ρ /boh) Non lativistic l=) Rlativistic κ=-) l= E j=/=3/ j=3/ κ=-).4.3 Au 5p. j=/=3/ boh).5 obital momnt spin - p 3/ κ=-): naly sam bhavio than non-lativistic p-stat

52 ) Spin-Obit splitting of p stats Spin-obit splitting of l-quantum numb ρ /boh).7 Non lativistic l=) E.6 Rlativistic κ=) Au 5p l= j=-/=/ j=/ κ=) boh).5 j=-/=/ obital momnt - p / κ=): makdly diffnt bhavio than non-lativistic p-stat g κ= is non-zo at nuclus spin

53 ) Spin-Obit splitting of p stats Spin-obit splitting of l-quantum numb ρ /boh) Non lativistic l=) Rlativistic κ=-) Rlativistic κ=) l= Au 5p E j=/=3/ j=-/=/ j=3/ κ=-) j=/ κ=). j=/=3/ j=-/=/ boh).5 obital momnt spin - obital momnt - E j=3/ E j=/ spin p / κ=): makdly diffnt bhavio than non-lativistic p-stat g κ= is non-zo at nuclus

54 ) Spin-Obit splitting of p stats Rlativistic coction %) E E ) RELA E NRELA NRELA ρ /boh) Non lativistic l=) Rlativistic κ=-) Rlativistic κ=) Au 5p κ= κ=- 3p / 3p 3/ 4p / 4p 3/ 5p / 5p 3/ p / p 3/ boh) Scala-lativistic p-obital is simila to p 3/ wav function, but Ψ dos not contain p / adial basis function

55 3) Obital xpansion: Aud) stats High l-quantum numb stats xpand du to btt shilding of nuclus chag fom contactd s-stats Non-lativistic NREL) - Z - -

56 3) Obital xpansion: Aud) stats High l-quantum numb stats xpand du to btt shilding of nuclus chag fom contactd s-stats Non-lativistic NREL) - Z Z ff = Z- σnrel) - - Z ff -

57 3) Obital xpansion: Aud) stats High l-quantum numb stats xpand du to btt shilding of nuclus chag fom contactd s-stats Non-lativistic NREL) Rlativistic REL) - - Z - Z Z ff = Z- σnrel) Z ff > Z ff Z ff = Z- σrel) Z ff Z ff - -

58 3) Obital xpansion: Aud) stats High l-quantum numb stats xpand du to btt shilding of nuclus chag fom contactd s-stats Non-lativistic NREL) Rlativistic REL) - - Z - Z Z ff = Z- σnrel) Z ff > Z ff Z ff = Z- σrel) Z ff Z ff - Indict lativistic ffct -

59 3) Obital xpansion: Aud) stats Rlativistic coction %) E E ) RELA E NRELA NRELA 4f 5/ 4f 7/ κ=3 κ=-4 5d 3/ 5d 5/ 3d 3/ 3d 5/ 4d 3/ 4d 5/ κ= κ= ρ /boh) 4 3 Non lativistic l=) Rlativistic κ=) Rlativistic κ=-3) ρ /boh).4.3 Non lativistic l=) Rlativistic κ=) Rlativistic κ=-3) Au 3d....3 boh).4.. Au 5d 3 boh) Obital xpansion 4

60 Rlativistic ffcts on th Au ngy lvls Rlativistic coction %) E E ) RELA E NRELA NRELA s s p / p 3/ 3d 3/ 3d 5/ 4d 3/ 4d 5/ 4p / 4p 3/ 5p / 5p 3/ 4f 5/ 4f 7/ 5d 3/ 5d 5/ 3p / 3p 3/ 3s 4s 5s 6s

61 Atomic spcta of gold SO splitting SO splitting Obital contaction Obital xpansion

62 Ag Au: th diffncs DOS & optical pop.) Ag Au

63 Rlativistic smico stats: p / obitals Elctonic stuctu of fcc Th, SOC with 6p / local obital Engy vs. basis siz DOS with and without p / 6p / 6p 3/ p / not includd 6p / p / includd 6p 3/ J.Kunš, P.Novak, R.Schmid, P.Blaha, K.Schwaz, Phys.Rv.B. 64, 53 )

64 SOC in magntic systms SOC coupls magntic momnt to th lattic diction of th xchang fild matts input in cas.inso) Symmty opations acts in al and spin spac numb of symmty opations may b ducd flctions act diffntly on spins than on positions) tim invsion is not symmty opation do not add an invsion fo k-list) initso_lapw must b xcutd) dtcts nw symmty stting Diction of magntization [] [] [] [] m x m y z A A A A A B B - B A B - B B A B

65 Rlativity in WIENk: Summay WIENk offs sval lvls of tating lativity: non-lativistic: slct NREL in cas.stuct not commndd) standad: fully-lativistic co, scala-lativistic valnc mass-vlocity and Dawin s-shift, no spin-obit intaction fully -lativistic: adding SO in scond vaiation using pvious ignstats as basis) adding p / LOs to incas accuacy caution!!!) x lapw x lapwso x lapw so -c incas E-max fo mo ignvalus, to hav basis fo lapwso) SO ALWAYS nds complx lapw vsion Non-magntic systms: SO dos NOT duc symmty. initso_lapw just gnats cas.inso and cas.inc. Magntic systms: symmtso ddcts pop symmty and wits cas.stuct/in*/clm*

66 8 th WIENk Wokshop PnnStat Univsity USA Rlativistic ffcts & Non-collina magntism WIENk / WIENncm) Xavi Rocquflt Institut ds Matéiaux Jan-RouxlUMR 65) Univsité d Nants, FRANCE

67 Pauli Hamiltonian fo magntic systms )... = l B V m H ff B ff P h σ ζ σ µ x matix in spin spac, du to Pauli spin opatos = σ = i i σ = σ 3 ) Pauli spin matics

68 Pauli Hamiltonian fo magntic systms )... = l B V m H ff B ff P h σ ζ σ µ x matix in spin spac, du to Pauli spin opatos Wav function is a -componnt vcto spino) It cosponds to th lag componnts of th diac wav function small componnts a nglctd) Ψ Ψ = Ψ Ψ ε H P spin up spin down = σ = i i σ = σ 3 ) Pauli spin matics

69 Pauli Hamiltonian fo magntic systms x matix in spin spac, du to Pauli spin opatos H P h = Vff µ Bσ Bff ζ l m Effctiv lctostatic potntial V ff Effctiv magntic fild = Vxt VH Vxc B ff = Bxt Bxc σ )... Exchang-colation potntial Exchang-colation fild

70 Pauli Hamiltonian fo magntic systms x matix in spin spac, du to Pauli spin opatos H P h = Vff µ Bσ Bff ζ l m σ )... Effctiv lctostatic potntial V ff Effctiv magntic fild = Vxt VH Vxc B ff = Bxt Bxc Exchang-colation potntial Exchang-colation fild ζ = Spin-obit coupling h M c dv d Many-body ffcts which a dfind within DFT LDA o GGA

71 Exchang and colation Fom DFT xchang colation ngy: E xc ) ) ) hom [ ) ] 3 ρ, m = ρ ε ρ, m d xc Local function of th lctonic dnsity ρ) and th magntic momnt m) Dfinition of V xc and B xc functional divativs): V xc = E xc ρ, m) E ρ, m) ρ B xc = xc m V xc LDA xpssion fo V xc and B xc : = ε hom xc ρ, m) ρ m) hom ε xc, ρ ρ B xc ε = ρ hom xc ρ, m) m ˆ m B xc is paalll to th magntization dnsity vcto m) ^

72 Non-collina magntism Diction of magntization vay in spac, thus spin-obit tm is psnt )... = l B V m H ff B ff P h σ ζ σ µ ) ) εψ ψ µ µ µ µ = z B ff y x B y x B z B ff B V m ib B ib B B V m h h Non-collina magntic momnts = ψ ψ ψ Ψ and Ψ a non-zo Solutions a non-pu spinos

73 Collina magntism Magntization in z-diction / spin-obit is not psnt )... = l B V m H ff B ff P h σ ζ σ µ εψ ψ µ µ = z B ff z B ff B V m B V m h h Collina magntic momnts Solutions a pu spinos = ψ ψ = ψ ψ ε ε Non-dgnat ngis

74 Non-magntic calculation No magntization psnt, B x = B y = B z = and no spin-obit coupling )... = l B V m H ff B ff P h σ ζ σ µ ψ = εψ ff ff V m V m h h = ψ ψ = ψ ψ ε = ε Solutions a pu spinos Dgnat spin solutions

75 Magntism and WIENk Wink can only handl collina o non-magntic cass DOS non-magntic cas m = n n = magntic cas m = n n DOS E F un_lapw scipt: un_lapw scipt: E F x lapw x lapw x lapw x lco x mix x lapw x lapw up x lapw -dn x lapw up x lapw -dn x lco up x lco -dn x mix

76 Magntism and WIENk Spin-polaizd calculations unsp_lapw scipt unconstaind magntic calc.) unfsm_lapw -m valu constaind momnt calc.) unafm_lapw constaind anti-fomagntic calculation) spin-obit coupling can b includd in scond vaiational stp nv mix polaizd and non-polaizd calculations in on cas dictoy!!!

77 Non-collina magntism In cas of non-collina spin aangmnts WIENncm WIENk clon) has to b usd: cod basd on Wink availabl fo Wink uss) stuctu and usag philosophy simila to Wink indpndnt souc t, indpndnt installation WIENncm poptis: al and spin symmty simplifis SCF, lss k-points) constaind o unconstaind calculations optimizs magntic momnts) SOC in fist vaiational stp, LDAU Spin spials

78 Non-collina magntism Fo non-collina magntic systms, both spin channls hav to b considd simultanously Rlation btwn spin dnsity matix and magntization DOS unncm_lapw scipt: xncm lapw xncm lapw xncm lapw xncm lco xncm mix m z = n n m x = ½n n ) m y = i½n - n ) E F

79 WinNCM: Spin spials Tansvs spin wav α α = R q m n = m R cos ), sin ) sinθ, cosθ ) n n q R q R spin-spial is dfind by a vcto q givn in cipocal spac and an angl θ btwn magntic momnt and otation axis. Rotation axis is abitay no SOC) fixd as z-axis in WIENNCM Tanslational symmty is lost! But WIENncm is using th gnalizd Bloch thom. Th calculation of spin wavs only quis on unit cll fo vn incommnsuat modulation q vcto.

80 WinNCM: Usag. Gnat th atomic and magntic stuctus Cat atomic stuctu Cat magntic stuctu S utility pogams: ncmsymmty, polaangls,. Run initncm initialization scipt) 3. Run th NCM calculation: xncm WIENncm vsion of x scipt) unncm WIENncm vsion of un scipt) Mo infomation on th manual Robt Laskowski) olask@thochm.tuwin.ac.at

81 9 th WIENk Wokshop Wasda Univsity Tokyo Xavi Rocquflt Institut ds Matéiaux Jan-RouxlUMR 65) Univsité d Nants, FRANCE