Relativistic effects & magnetism. in WIEN2k

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1 4 th WIENk Workshop Vienna 7 Relativistic effects & magnetism in WIENk Xavier Rocquefelte Institut des Sciences Chimiques de Rennes (UMR 66) Université de Rennes, FRANCE

2 4 th WIENk Workshop Vienna 7 Talk constructed using the following documents: Slides of: Robert Laskowski, Stefaan Cottenier, Peter Blaha and Georg Madsen Notes of: - Pavel Novak (Calculation of spin-orbit coupling) - Robert Laskowski (Non-collinear magnetic version of WIENk package) Books: - WIENk userguide, ISBN Electronic Structure: Basic Theory and Practical Methods, Richard M. Martin ISBN Relativistic Electronic Structure Theory. Part. Fundamentals, Peter Schewerdtfeger, ISBN web: wienlist digest wikipedia

3 Few words about Special Theory of Relativity Light Composed of photons (no mass) Speed of light = constant Atomic units: ħ = m e = e = c 37 au

4 Few words about Special Theory of Relativity Light Composed of photons (no mass) Speed of light = constant Matter Composed of atoms (mass) v = f(mass) Atomic units: ħ = m e = e = Speed of matter mass c 37 au mass = f(v)

5 Few words about Special Theory of Relativity Light Composed of photons (no mass) Speed of light = constant Matter Composed of atoms (mass) v = f(mass) Atomic units: ħ = m e = e = Speed of matter mass c 37 au mass = f(v) Lorentz Factor (measure of the relativistic effects) γ = - v c Relativistic mass: M = γm (m: rest mass) Momentum: p = γmv = Mv Total energy: E = p c m c 4 E = γmc = Mc

6 Definition of a relativistic particle (Bohr model) Lorentz factor (γ) «Non-relativistic» particle: γ = H(s) Au(s) Speed (v) Details for Au atom: 79 v e ( s) = c =. 58c 37 c 37 au γ = v e c Speed of the s electron (Bohr model): v Z e n = e - (.58) Ze : ( ) H : v e ( s s ) = au au v e Au : v Au : e ( s ) = 79 au ( s ) = 79 au v e =. γ γ = = γ =. γ =. s electron of Au atom = relativistic particle M e (s-au) =.m e

7 Relativistic effects Ze ) The mass-velocity correction Relativistic increase in the mass of an electron with its velocity (when v e c)

8 Relativistic effects Ze ) The mass-velocity correction Relativistic increase in the mass of an electron with its velocity (when v e c) ) The Darwin term It has no classical relativistic analogue Due to small and irregular motions of an electron about its mean position (Zitterbewegung*) *Analysis of Erwin Schrödinger of the wave packet solutions of the Dirac equation for relativistic electrons in free space:the interference between positive and negative energy states produces what appears to be a fluctuation (at the speed of light) of the position of an electron around the median.

9 Relativistic effects Ze ) The mass-velocity correction Relativistic increase in the mass of an electron with its velocity (when v e c) ) The Darwin term It has no classical relativistic analogue Due to small and irregular motions of an electron about its mean position (Zitterbewegung) 3) The spin-orbit coupling It is the interaction of the spin magnetic moment (s) of an electron with the magnetic field induced by its own orbital motion (l)

10 Relativistic effects Z eff e ) The mass-velocity correction Relativistic increase in the mass of an electron with its velocity (when v e c) ) The Darwin term It has no classical relativistic analogue Due to small and irregular motions of an electron about its mean position (Zitterbewegung) 3) The spin-orbit coupling It is the interaction of the spin magnetic moment (s) of an electron with the magnetic field induced by its own orbital motion (l) 4) Indirect relativistic effect The change of the electrostatic potential induced by relativity is an indirect effect of the core electrons on the valence electrons

11 One electron radial Schrödinger equation HARTREE ATOMIC UNITS INTERNATIONAL UNITS H S Ψ = V Ψ = εψ H S Ψ = m e V Ψ = εψ Atomic units: ħ = m e = e = /(4 πε ) = c = / α 37 au

12 Ψ Ψ = = Ψ ε V H S Ψ Ψ = = Ψ ε V m H e S r Z V = r Ze V 4πε = INTERNATIONAL UNITS In a spherically symmetric potential Atomic units: ħ = m e = e = /(4 πε ) = c = / α 37 au ( ) ( ) θ,ϕ,,,, m l l n m l n Y r = R Ψ ( ) ( ) ( ) = sin sin sin ϕ θ θ θ θ θ r r r r r r HARTREE ATOMIC UNITS One electron radial Schrödinger equation

13 Ψ Ψ = = Ψ ε V H S ( ) l n l n e l n e R R r l l m V dr dr r dr d r m,,, = ε In a spherically symmetric potential Ψ Ψ = = Ψ ε V m H e S r Z V = r Ze V 4πε = ( ) l n l n l n R R r l l V dr dr r dr d r,,, = ε INTERNATIONAL UNITS ( ) ( ) θ,ϕ,,,, m l l n m l n Y r = R Ψ ( ) ( ) ( ) = sin sin sin ϕ θ θ θ θ θ r r r r r r HARTREE ATOMIC UNITS One electron radial Schrödinger equation

14 Dirac Hamiltonian: a brief description Dirac relativistic Hamiltonian provides a quantum mechanical description of electrons, consistent with the theory of special relativity. E = p c m c 4 HDΨ = εψ with H D = c α p βmec V

15 Dirac relativistic Hamiltonian provides a quantum mechanical description of electrons, consistent with the theory of special relativity. = k k k σ σ α = β k = σ = i i σ = σ 3 ( ) Pauli spin matrices Momentum operator Rest mass Electrostatic potential V m c p c H e D = β α ( ) unit and zero matrices Ψ = DΨ ε H with E = p c m c 4 Dirac Hamiltonian: a brief description

16 Dirac equation: H D and Ψ are 4-dimensional Ψ is a four-component single-particle wave function that describes spin-/ particles. spin up spin down H D ψ ψ ψ ψ = ε ψ 3 ψ 3 ψ 4 ψ 4 In case of electrons: Large components (Φ) Small components (χ) Φ ψ = χ factor /(m e c ) Φ and χ are time-independent two-component spinors describing the spatial and spin-/ degrees of freedom Leads to a set of coupled equations for Φ and χ: c c ( ) ( σ p χ = ε V m c )φ ( ) ( σ p φ = ε V m c )χ e e

17 Dirac equation: H D and Ψ are 4-dimensional For a free particle (i.e. 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 e z y z e y z z z y z e y x z e ε ε ε ε, φ c m e, φ m e c Particles: up & down, χ c m e χ, m e c Antiparticles: up & down Solution in the slow particle limit (p=) Non-relativistic limit decouples Ψ from Ψ and Ψ 3 from Ψ 4

18 For a free particle (i.e. 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 e z y z e y z z z y z e y x z e ε ε ε ε, φ c m e, φ m e c Particles: up & down, χ c m e χ, m e c Antiparticles: up & down Solution in the slow particle limit (p=) Non-relativistic limit decouples Ψ from Ψ and Ψ 3 from Ψ 4 For a spherical potential V(r): ( ) ( ) Υ - Υ = Φ Ψ = κσ nκ κσ nκ χ r f i r g g nκ and f nκ are Radial functions Y κσ are angular-spin functions s l j = ( ) = j s κ, = s Dirac equation: H D and Ψ are 4-dimensional

19 Dirac equation in a spherical potential For a spherical potential V(r): The resulting equations for the radial functions (g nκ and f nκ ) are simplified if we define: Energy: ε ' = ε m e c Radially varying mass: M e( r) = m e ε ' V c ( r)

20 Dirac equation in a spherical potential For a spherical potential V(r): The resulting equations for the radial functions (g nκ and f nκ ) are simplified if we define: Energy: ε ' = ε m e c Radially varying mass: M e( r) = m e ε ' V c Then the coupled equations can be written in the form of the radial eq.: d dr dg l ( l ) dv dg dv ( κ ) nκ nκ r V g nκ M e r dr M e r 4M dr dr e c 4M e c dr r g ( r) nκ = ε' g nκ Mass-velocity effect Darwin term Spin-orbit coupling Note that: κ ( l ) d drn, l l One electron radial r V Rn, l = ε R n, l me r dr dr me r ( κ ) = l( l ) Schrödinger equation in a spherical potential

21 Dirac equation in a spherical potential The resulting equations for the radial functions (g nκ and f nκ ) are simplified if we define: Energy: d dr dg ε ' = ε m e c Radially varying mass: M e( r) l = m e ε ' V c Then the coupled equations can be written in the form of the radial eq.: ( l ) dv dg dv ( κ ) nκ nκ r V g nκ M e r dr M e r 4M dr dr e c 4M e c and For a spherical potential V(r): ( κ ) df nk = ( V ε ') gn κ dr c r f nκ Darwin term Due to spin-orbit coupling, Ψ is not an eigenfunction of spin (s) and angular orbital moment (l). Instead the good quantum numbers are j and κ Note that: κ ( κ ) = l( l ) dr r Spin-orbit coupling g ( r) nκ = ε' g nκ No approximation has been made so far

22 Dirac equation in a spherical potential Scalar relativistic approximation Approximation that the spin-orbit term is small neglect SOC in radial functions (and treat it by perturbation theory) g ~ ~ No SOC Approximate radial functions: f n f d dr ~ dg~ ( l ) l nκ g nl nl nl r V g~ nl ' g~ = ε nl M e r dr M e r 4M e c dg~ nl fnl = ( ~ ) ~ and with the normalization condition: = M ec dr gnl fnl r dr dv dr dg~ dr κ nl

23 Dirac equation in a spherical potential Scalar relativistic approximation Approximation that the spin-orbit term is small neglect SOC in radial functions (and treat it by perturbation theory) g ~ ~ No SOC Approximate radial functions: f n f d dr ~ dg~ ( l ) l nκ g nl nl nl r V g~ nl ' g~ = ε nl M e r dr M e r 4M e c dg~ nl fnl = ( ~ ) ~ and with the normalization condition: = M ec dr gnl fnl r dr dv dr dg~ dr The four-component wave function is now written as: Φ Ψ = = χ - i g f nl nl Φ is a pure spin state ( r ) Υ lm ( ) r Υ is a mixture of up and down spin states χ lm Inclusion of the spin-orbit coupling in second variation (on the large component only) H ~ ψ = εψ ~ H ~ SO ψ H SO κ with " dv σl = 4M c r dr e nl

24 Relativistic effects in a solid For a molecule or a solid: Relativistic effects originate deep inside the core. It is then sufficient to solve the relativistic equations in a spherical atomic geometry (inside the atomic spheres of WIENk). Justify an implementation of the relativistic effects only inside the muffin-tin atomic spheres

25 Implementation in WIENk Atomic sphere (RMT) Region Core electrons Valence electrons «Fully» relativistic Scalar relativistic (no SOC) Spin-compensated Dirac equation Possibility to add SOC ( nd variational) SOC: Spin orbit coupling

26 Implementation in WIENk Atomic sphere (RMT) Region Interstitial Region Core electrons Valence electrons Valence electrons «Fully» relativistic Scalar relativistic (no SOC) Not relativistic Spin-compensated Dirac equation Possibility to add SOC ( nd variational) SOC: Spin orbit coupling

27 Implementation in WIENk: core electrons Atomic sphere (RMT) Region Core Core electrons «Fully» relativistic Spin-compensated Dirac equation l s=- j=ls/ s= Core states: fully occupied spin-compensated Dirac equation (include SOC) For spin-polarized potential, spin up and spin down are calculated separately, the density is averaged according to the occupation number specified in case.inc file. κ=-s(j/) occupation s=- s= s=- s= s / - p / 3/ - 4 d 3/ 5/ f 3 5/ 7/ case.inc for Au atom 7.,-, ( n,κ,occup),-, ( n,κ,occup),, ( n,κ,occup),-,4 ( n,κ,occup) 3,-, ( n,κ,occup) 3,, ( n,κ,occup) 3,-,4 ( n,κ,occup) 3,,4 ( n,κ,occup) 3,-3,6 ( n,κ,occup) 4,-, ( n,κ,occup) 4,, ( n,κ,occup) 4,-,4 ( n,κ,occup) 4,,4 ( n,κ,occup) 4,-3,6 ( n,κ,occup) 5,-, ( n,κ,occup) 4, 3,6 ( n,κ,occup) 4,-4,8 ( n,κ,occup)

28 Implementation in WIENk: core electrons Atomic sphere (RMT) Region Core Core electrons «Fully» relativistic Spin-compensated Dirac equation l s=- j=ls/ s= Core states: fully occupied spin-compensated Dirac equation (include SOC) For spin-polarized potential, spin up and spin down are calculated separately, the density is averaged according to the occupation number specified in case.inc file. κ=-s(j/) 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/ case.inc for Au atom 7.,-, ( n,κ,occup),-, ( n,κ,occup),, ( n,κ,occup),-,4 ( n,κ,occup) 3,-, ( n,κ,occup) 3,, ( n,κ,occup) 3,-,4 ( n,κ,occup) 3,,4 ( n,κ,occup) 3,-3,6 ( n,κ,occup) 4,-, ( n,κ,occup) 4,, ( n,κ,occup) 4,-,4 ( n,κ,occup) 4,,4 ( n,κ,occup) 4,-3,6 ( n,κ,occup) 5,-, ( n,κ,occup) 4, 3,6 ( n,κ,occup) 4,-4,8 ( n,κ,occup)

29 Implementation in WIENk: valence electrons Valence electrons INSIDE atomic spheres are treated within scalar relativistic approximation [] if RELA is specified in case.struct file (by default). Title F LATTICE,NONEQUIV.ATOMS: 5 Fm-3m MODE OF CALC=RELA unit=bohr ATOM : X=. Y=. Z=. MULT= ISPLIT= Au NPT= 78 R=.5 RMT=.6 Z: 79. LOCAL ROT MATRIX: NUMBER OF SYMMETRY OPERATIONS Atomic sphere (RMT) Region Valence electrons Scalar relativistic (no SOC) no κ dependency of the wave function, (n,l,s) are still good quantum numbers all relativistic effects are included except SOC small component enters normalization and calculation of charge inside spheres augmentation with large component only SOC can be included in «second variation» [] Koelling and Harmon, J. Phys. C (977) Valence electrons in interstitial region are treated classically

30 Implementation in WIENk: valence electrons SOC is added in a second variation (lapwso): - First diagonalization (lapw): - Second diagonalization (lapwso): H Ψ ε Ψ = ε = Ψ H H SO ( ) Ψ The second equation is expanded in the basis of first eigenvectors (Ψ ) N i ( ) j j i i j δ ε Ψ Ψ Ψ Ψ = ε Ψ Ψ ij H SO sum include both up/down spin states N is much smaller than the basis size in lapw Atomic sphere (RMT) Region Valence electrons Scalar relativistic (no SOC) Possibility to add SOC ( nd variational)

31 Implementation in WIENk: valence electrons SOC is added in a second variation (lapwso): - First diagonalization (lapw): - Second diagonalization (lapwso): H Ψ ε Ψ = ε = Ψ H H SO ( ) Ψ The second equation is expanded in the basis of first eigenvectors (Ψ ) N i ( ) j j i i j δ ε Ψ Ψ Ψ Ψ = ε Ψ Ψ ij H SO sum include both up/down spin states N is much smaller than the basis size in lapw Atomic sphere (RMT) Region Valence electrons Scalar relativistic (no SOC) Possibility to add SOC ( nd variational) SOC is active only inside atomic spheres, only spherical potential (V MT ) is taken into account, in the polarized case spin up and down parts are averaged. Eigenstates are not pure spin states, SOC mixes up and down spin states Off-diagonal term of the spin-density matrix is ignored. It means that in each SCF cycle the magnetization is projected on the chosen direction (from case.inso) V MT : Muffin-tin potential (spherically symmetric)

32 Controlling spin-orbit coupling in WIENk Do a regular scalar-relativistic scf calculation save_lapw initso_lapw case.inso: WFFIL 4 llmax,ipr,kpot -..5 emin,emax (output energy window)... direction of magnetization (lattice vectors) NX number of atoms for which RLO is added NX atom number,e-lo,de (case.in), repeat NX times number of atoms for which SO is switch off; atoms case.in(c): ( ).3.5 CONT.3. CONT K-VECTORS FROM UNIT: emin/emax/nband symmetso (for spin-polarized calculations only) run(sp)_lapw -so -so switch specifies that scf cycles will include SOC

33 Controlling spin-orbit coupling in WIENk The wweb interface is helping you Non-spin polarized case

34 Controlling spin-orbit coupling in WIENk The wweb interface is helping you Spin polarized case

35 Relativistic effects in the solid: Illustration LDA overbinding (7%) No difference NREL/SREL hcp-be Z = 4 Bulk modulus: - NREL: 3.4 GPa - SREL: 3.5 GPa - Exp.: 3 GPa

36 Relativistic effects in the solid: Illustration LDA overbinding (7%) No difference NREL/SREL hcp-be Z = 4 Bulk modulus: - NREL: 3.4 GPa - SREL: 3.5 GPa - Exp.: 3 GPa hcp-os Z = exp SREL NREL LDA overbinding (%) Clear difference NREL/SREL Bulk modulus: - NREL: 344 GPa - SREL: 447 GPa - Exp.: 46 GPa

37 Relativistic effects in the solid: Illustration hcp-be Z = Scalar-relativistic (SREL): hcp-os SREL exp - LDA overbinding (%) - Bulk modulus: 447 GPa Z = 76 NREL spin-orbit coupling (SRELSO): - LDA overbinding (%) - Bulk modulus: 436 GPa SRELSO Exp. Bulk modulus: 46 GPa

38 Relativistic effects Z eff e ) The mass-velocity correction Relativistic increase in the mass of an electron with its velocity (when v e c) ) The Darwin term It has no classical relativistic analogue Due to small and irregular motions of an electron about its mean position (Zitterbewegung) 3) The spin-orbit coupling It is the interaction of the spin magnetic moment (s) of an electron with the magnetic field induced by its own orbital motion (l) 4) Indirect relativistic effect The change of the electrostatic potential induced by relativity is an indirect effect of the core electrons on the valence electrons

39 () Relativistic orbital contraction r ρ (e/ bohr ) 5 Non relativistic (l=) Radius of the s orbit (Bohr model): 4 e - Ze 3. Au s r ( bohr ) n a ( s) = AND a = =bohr Z m cα r r( s) = =.3 bohr 79 e Atomic units: ħ = m e = e = c = / α 37 au

40 () Relativistic orbital contraction r ρ (e/ bohr ) 5 4 Non relativistic (l=) Relativistic ( κ = - ) Radius of the s orbit (Bohr model): e - Ze 3. r ( s ) = % Orbital contraction r ( bohr ) n Z a γ = 79 Au s =.. bohr a [ RELA] n a ( s) = AND a = =bohr Z mcα r = = M cα e r( s) = =.3 bohr 79 In Au atom, the relativistic mass (M) of the s electron is % larger than the rest mass (m) M = γ m e =. m e a γ

41 () Relativistic orbital contraction r ρ (e/ bohr ) Au 6s Non relativistic (l=) Relativistic ( κ = - ) Orbital contraction. 4 6 r ( bohr ) γ = Z 79 v e ( 6s) = = = 3.7 =. 96c n 6 v e c = (.96) =.46 Direct relativistic effect (mass enhancement) contraction of.46% only However, the relativistic contraction of the 6s orbital is large (>%) ns orbitals (with n > ) contract due to orthogonality to s

42 () Orbital Contraction: Effect on the energy r ρ (e/ bohr ) r ρ (e/ bohr ) Relativistic correction (%) ( E E ) RELA E NRELA NRELA Non relativistic (l=) Relativistic ( κ = - ) Orbital contraction Non relativistic (l=) Relativistic ( κ = - ) Orbital contraction s s. Au s r ( bohr ) r ( bohr ) 3s 4s 5s 6s. Au 6s

43 () Spin-Orbit splitting of p states r ρ (e/ bohr ).7 Non relativistic (l=) Au 5p r ( bohr ).5

44 () Spin-Orbit splitting of p states Spin-orbit splitting of l-quantum number r ρ (e/ bohr ).7.6 Non relativistic (l=) Relativistic ( κ = - ) E j=3/ (κ=-).5 l=.4.3 Au 5p. j=/=3/ r ( bohr ).5 orbital moment e spin -e p 3/ (κ=-): nearly same behavior than non-relativistic p-state

45 () Spin-Orbit splitting of p states Spin-orbit splitting of l-quantum number r ρ (e/ bohr ).7 Non relativistic (l=) E.6 Relativistic ( κ =) Au 5p l= j=-/=/ j=/ (κ=) r ( bohr ).5 j=-/=/ orbital moment e -e p / (κ=): markedly different behavior than non-relativistic p-state g κ= is non-zero at nucleus spin

46 () Spin-Orbit splitting of p states Spin-orbit splitting of l-quantum number r ρ (e/ bohr ).7.6 Non relativistic (l=) Relativistic ( κ = - ) Relativistic ( κ =) E j=3/ (κ=-) Au 5p l= j=-/=/ j=/ (κ=). j=/=3/ j=-/=/ r ( bohr ).5 orbital moment e spin -e orbital moment e -e E j=3/ E j=/ spin p / (κ=): markedly different behavior than non-relativistic p-state g κ= is non-zero at nucleus

47 () Spin-Orbit splitting of p states Relativistic correction (%) ( E E ) RELA E NRELA NRELA r ρ (e/bohr) Non relativistic (l=) Relativistic (κ=-) Relativistic (κ=) Au 5p κ= κ=- 3p / 3p 3/ 4p / 4p 3/ 5p / 5p 3/ p / p 3/ r (bohr) Scalar-relativistic p-orbital is similar to p 3/ wave function, but Ψ does not contain p / radial basis function

48 (3) Orbital expansion: Au(d) states Higher l-quantum number states expand due to better shielding of nucleus charge from contracted s-states Non-relativistic (NREL) -e Ze -e -e

49 (3) Orbital expansion: Au(d) states Higher l-quantum number states expand due to better shielding of nucleus charge from contracted s-states Non-relativistic (NREL) -e Z eff = Z- σ(nrel) Ze -e -e Z eff e -e

50 (3) Orbital expansion: Au(d) states Higher l-quantum number states expand due to better shielding of nucleus charge from contracted s-states Non-relativistic (NREL) Relativistic (REL) -e -e Ze -e Ze -e -e -e Z eff = Z- σ(nrel) Z eff > Z eff Z eff = Z- σ(rel) Z eff e Z eff e -e -e

51 (3) Orbital expansion: Au(d) states Higher l-quantum number states expand due to better shielding of nucleus charge from contracted s-states Non-relativistic (NREL) Relativistic (REL) -e -e Ze -e Ze -e -e -e Z eff = Z- σ(nrel) Z eff > Z eff Z eff = Z- σ(rel) Z eff e Z eff e -e Indirect relativistic effect -e

52 (3) Orbital expansion: Au(d) states Relativistic correction (%) ( 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= r ρ (e/ bohr ) 4 3 Non relativistic (l=) Relativistic ( κ =) Relativistic ( κ = - 3) r ρ (e/ bohr ).4.3 Non relativistic (l=) Relativistic ( κ =) Relativistic ( κ = - 3) Au 3d....3 r ( bohr ).4.. Au 5d 3 r ( bohr ) Orbital expansion 4

53 Relativistic effects on the Au energy levels Relativistic correction (%) ( 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

54 Atomic spectra of gold Orbital contraction Orbital expansion SO splitting SO splitting

55 Ag Au: the differences (DOS & optical prop.) Ag Au

56 Relativistic semicore states: p / orbitals Electronic structure of fcc Th, SOC with 6p / local orbital Energy vs. basis size DOS with and without p / 6p / 6p 3/ p / not included 6p / p / included 6p 3/ J.Kuneš, P.Novak, R.Schmid, P.Blaha, K.Schwarz, Phys.Rev.B. 64, 53 ()

57 SOC in magnetic systems SOC couples magnetic moment to the lattice direction of the exchange field matters (input in case.inso) Symmetry operations acts in real and spin space number of symmetry operations may be reduced (reflections act differently on spins than on positions) time inversion is not symmetry operation (do not add an inversion for k list) initso_lapw (must be executed) detects new symmetry setting Direction of magnetization [] [] [] [] m x m y z A A A A A B B - B A B - B B A B

58 Relativity in WIENk: Summary WIENk offers several levels of treating relativity: non-relativistic: select NREL in case.struct (not recommended) standard: fully-relativistic core, scalar-relativistic valence mass-velocity and Darwin s-shift, no spin-orbit interaction fully -relativistic: adding SO in second variation (using previous eigenstates as basis) adding p / LOs to increase accuracy (caution) x lapw (increase E-max for more eigenvalues, to have x lapwso basis for lapwso) x lapw so -c SO ALWAYS needs complex lapw version Non-magnetic systems: SO does NOT reduce symmetry. initso_lapw just generates case.inso and case.inc. Magnetic systems: symmetso detects proper symmetry and rewrites case.struct/in*/clm*

59 4 th WIENk Workshop Vienna 7 Magnetic coupling & Magnetic anisotropy Xavier Rocquefelte Institut des Sciences Chimiques de Rennes (UMR 66) Université de Rennes, FRANCE

60 INTRODUCTION Magnetic properties: ü ü ü ü ü ü Spin-state (high/low) Long-range/short-range orders Collinear / non-collinear Magnetic anisotropy Magnetic frustration Magnetic exchange Spin-State Magnetic exchange Long-range order Magnetic anisotropy -3-6 Energy scale (ev)

61 INTRODUCTION Paramagnetic (PM) Ferromagnetic (FM) order Ferrimagnetic order Antiferromagnetic (AFM) order

62 COLLINEAR MAGNETISM Magnetic susceptibility of a ferromagnetic (FM) compound,4 χ mol (emu/mol),3,, PM without long range interaction T(K)

63 COLLINEAR MAGNETISM Magnetic susceptibility of a ferromagnetic (FM) compound,4 χ mol (emu/mol),3,, PM without long range interaction T(K)

64 COLLINEAR MAGNETISM Magnetic susceptibility of a ferromagnetic (FM) compound,4 χ mol (emu/mol),3,, PM without long range interaction T(K)

65 COLLINEAR MAGNETISM Magnetic susceptibility of a ferromagnetic (FM) compound,4 χ mol (emu/mol),3,, PM without long range interaction T(K)

66 COLLINEAR MAGNETISM Magnetic susceptibility of a ferromagnetic (FM) compound,4 χ mol (emu/mol),3 J F,, PM without long range interaction T(K)

67 COLLINEAR MAGNETISM Magnetic susceptibility of an antiferromagnetic (AFM) compound,4 χ mol (emu/mol),3 PM without longrange interactions,, T(K)

68 COLLINEAR MAGNETISM Magnetic susceptibility of an antiferromagnetic (AFM) compound,4 χ mol (emu/mol),3 PM without longrange interactions,, T(K)

69 COLLINEAR MAGNETISM Magnetic susceptibility of an antiferromagnetic (AFM) compound,4 χ mol (emu/mol),3 PM without longrange interactions, J AF, AF PM T(K)

70 COLLINEAR MAGNETISM Ferromagnetic Antiferromagnetic,4 χ mol (emu/mol), χ mol (emu/mol),3 Ferromg order when kt,, F PM T(K) AF PM T C Curie temperature T N Néel temperature, T(K) J F J AF Ferromagnetic exchange: J F < Antiferromagnetic exchange: J AF >

71 NON-COLLINEAR MAGNETISM AFM with subnetworks having different magnetization directions Frustrated AFM weak ferromagnetism Topologic frustration FM-AFM competition? J J J? J : FM J : AFM

72 NON-COLLINEAR MAGNETISM AFM with subnetworks having different magnetization directions Frustrated AFM weak ferromagnetism Topologic frustration FM-AFM competition? J J J? J : FM J : AFM

73 Estimation of magnetic coupling parameters Estimation of J can be done by mapping energy differences onto the general Heisenberg Spin Hamiltonian: J ij : spin exchange parameter between the spin sites i and j Ĥ = Ĥ J ij Si. S j i<j Long-range order J ij > AFM J ij < FM

74 Estimation of magnetic coupling parameters Estimation of J can be done by mapping energy differences onto the general Heisenberg Spin Hamiltonian: J ij : spin exchange parameter between the spin sites i and j Ĥ = Ĥ J ij Si. S j i<j Long-range order J ij > AFM J ij < FM E α = α H α = E S J ij σ i σ j i<j S: Spin hold by the magnetic center σ i = ± (up or down spin)

75 Estimation of magnetic coupling parameters Estimation of J can be done by mapping energy differences onto the general Heisenberg Spin Hamiltonian: J ij : spin exchange parameter between the spin sites i and j Ĥ = Ĥ J ij Si. S j i<j Long-range order J ij > AFM J ij < FM E α = α H α = E S J ij σ i σ j i<j S: Spin hold by the magnetic center σ i = ± (up or down spin) Example of a spin-half dimer (S = ½) To estimate the J value, total energy calculations are needed: σ = σ = σ = σ = - J = ( E FM E AFM ) E FM = E 4 J E AFM = E 4 J

76 Estimation of magnetic coupling parameters Illustration with NiO: NaCl structure, A-type AFM along [] Ni -> S = E α = α H α = E S J ij σ i σ j i<j

77 Estimation of magnetic coupling parameters Illustration with NiO: NaCl structure, A-type AFM along [] Ni -> S = E α = α H α = E S J ij σ i σ j i<j inequivalent Ni sites in the rhombohedral unit cell (S.G. R-3m) J: magnetic coupling defined by Ni -O-Ni path (angle : 8 ) J / unit cell J / f.u.

78 Estimation of magnetic coupling parameters Illustration with NiO: NaCl structure, A-type AFM along [] Ni -> S = Eα = α H α = E S J ijσ iσ j i< j inequivalent Ni sites in the rhombohedral unit cell (S.G. R-3m) J: magnetic coupling defined by Ni-O-Ni path (angle : 8 ) J / unit cell J / f.u. E FM = E J J = (E FM E AFM ) / 4 E AFM = E J

79 Magnetic structure of NiO GGAU calculations FM Ni M.M. (exp.) =.64.9 μ B AFM PRB 6, 639 ()

80 Magnetic structure of NiO GGAU calculations E g (exp.) = ev AFM FM PRB 6, 639 ()

81 Magnetic structure of NiO GGAU calculations FM AFM (more stable) Exp. Ground State: AFM

82 Magnetic structure of NiO GGAU calculations J (exp.) = 9 mev PRB 6, 3447 (97)

83 Estimation of the magnetic anisotropy Do a regular scalar-relativistic scf calculation save_lapw initso_lapw case.inso: WFFIL 4 llmax,ipr,kpot -..5 emin,emax (output energy window)... direction of magnetization (lattice vectors) NX number of atoms for which RLO is added NX atom number,e-lo,de (case.in), repeat NX times number of atoms for which SO is switch off; atoms case.in(c): ( ).3.5 CONT.3. CONT K-VECTORS FROM UNIT: emin/emax/nband symmetso (for spin-polarized calculations only) run(sp)_lapw -so -so switch specifies that scf cycles will include SOC

84 Estimation of the magnetic anisotropy Estimation of the Magneto-crystalline Anisotropy Energy (MAE) of CuO Allows to define the magnetization easy and hard axes Here we have considered the following expression: MAE (μev) Hard axis Hard axis MAE = E[u v w] E[easy axis] E[uvw] is the energy deduced from spin-orbit calculations with the magnetization along the [uvw] crystallographic direction Easy axis Magnetization axis [] X. Rocquefelte, P. Blaha, K. Schwarz, S. Kumar, J. van den Brink, Nature Comm. 4, 5 (3)

85 Estimation of the magnetic anisotropy Estimation of the Magneto-crystalline Anisotropy Energy (MAE) of CuO Allows to define the magnetization easy and hard axes [-] Here we have considered the following expression: MAE = E[u v w] E[easy axis] [] [--] [] [-] E[uvw] is the energy deduced from spin-orbit calculations with the magnetization along the [uvw] crystallographic direction [-] [] X. Rocquefelte, P. Blaha, K. Schwarz, S. Kumar, J. van den Brink, Nature Comm. 4, 5 (3)

86 4 th WIENk Workshop Vienna 7 Relativistic effects & Non-collinear magnetism (WIENk / WIENncm) Xavier Rocquefelte Institut des Sciences Chimiques de Rennes (UMR 66) Université de Rennes, FRANCE

87 Pauli Hamiltonian for magnetic systems ( )... = l B V m H eff B eff e P " σ ζ σ µ x matrix in spin space, due to Pauli spin operators = σ = i i σ = σ 3 ( ) Pauli spin matrices

88 Pauli Hamiltonian for magnetic systems ( )... = l B V m H eff B eff e P " σ ζ σ µ x matrix in spin space, due to Pauli spin operators Wave function is a -component vector (spinor) It corresponds to the large components of the dirac wave function (small components are neglected) Ψ Ψ = Ψ Ψ ε H P spin up spin down = σ = i i σ = σ 3 ( ) Pauli spin matrices

89 Pauli Hamiltonian for magnetic systems x matrix in spin space, due to Pauli spin operators H P " = Veff µ Bσ Beff ζ l m Effective electrostatic potential V eff e Effective magnetic field = Vext VH Vxc B eff = Bext Bxc ( σ )... Exchange-correlation potential Exchange-correlation field

90 Pauli Hamiltonian for magnetic systems x matrix in spin space, due to Pauli spin operators H P " = Veff µ Bσ Beff ζ l m e ( σ )... Effective electrostatic potential V eff Effective magnetic field = Vext VH Vxc B eff = Bext Bxc Exchange-correlation potential Exchange-correlation field ζ = Spin-orbit coupling M e c r dv dr Many-body effects which are defined within DFT LDA or GGA

91 Exchange and correlation From DFT exchange correlation energy: E xc ( ( ) ) ( ) hom [ ( ) ] 3 ρ r, m = ρ r ε ρ r, m dr xc Local function of the electronic density (ρ) and the magnetic moment (m) Definition of V xc and B xc (functional derivatives): V xc = E xc ( ρ, m) E ( ρ, m) ρ B = xc xc m V xc LDA expression for V xc and B xc : = ε hom xc ( ρ, m) ( ρ m) hom ε xc, ρ ρ hom ε xc ρ Bxc = ρ m (, m) m ˆ B xc is parallel to the magnetization density vector (m) ^

92 Non-collinear magnetism Direction of magnetization vary in space, thus spin-orbit term is present ( )... = l B V m H eff B eff e P " σ ζ σ µ ( ) ( ) εψ ψ µ µ µ µ = z B eff e y x B y x B z B eff e B V m ib B ib B B V m Non-collinear magnetic moments = ψ ψ ψ Ψ and Ψ are non-zero Solutions are non-pure spinors

93 Collinear magnetism Magnetization in z-direction / spin-orbit is not present ( )... = l B V m H eff B eff e P " σ ζ σ µ εψ ψ µ µ = z B eff e z B eff e B V m B V m Collinear magnetic moments Solutions are pure spinors = ψ ψ = ψ ψ ε ε Non-degenerate energies

94 Non-magnetic calculation No magnetization present, B x = B y = B z = and no spin-orbit coupling ( )... = l B V m H eff B eff e P " σ ζ σ µ ψ = εψ eff e eff e V m V m = ψ ψ = ψ ψ ε = ε Solutions are pure spinors Degenerate spin solutions

95 Magnetism and WIENk Wienk can only handle collinear or non-magnetic cases DOS non-magnetic case m = n n = magnetic case m = n n DOS E F run_lapw script: run_lapw script: E F x lapw x lapw x lapw x lcore x mixer x lapw x lapw up x lapw -dn x lapw up x lapw -dn x lcore up x lcore -dn x mixer

96 Magnetism and WIENk Spin-polarized calculations runsp_lapw script (unconstrained magnetic calc.) runfsm_lapw -m value (constrained moment calc.) runafm_lapw (constrained anti-ferromagnetic calculation) spin-orbit coupling can be included in second variational step never mix polarized and non-polarized calculations in one case directory

97 Non-collinear magnetism In case of non-collinear spin arrangements WIENncm (WIENk clone) has to be used: code based on Wienk (available for Wienk users) structure and usage philosophy similar to Wienk independent source tree, independent installation WIENncm properties: real and spin symmetry (simplifies SCF, less k-points) constrained or unconstrained calculations (optimizes magnetic moments) SOC in first variational step, LDAU Spin spirals

98 Non-collinear magnetism For non-collinear magnetic systems, both spin channels have to be considered simultaneously Relation between spin density matrix and magnetization DOS runncm_lapw script: xncm lapw xncm lapw xncm lapw xncm lcore xncm mixer m z = n n m x = ½(n n ) m y = i½(n - n ) E F

99 WienNCM: Spin spirals Transverse spin wave α = α R q m n = m R ( cos( ), sin( ) sinθ, cosθ ) n n q R q R spin-spiral is defined by a vector q given in reciprocal space and an angle θ between magnetic moment and rotation axis. Rotation axis is arbitrary (no SOC) fixed as z-axis in WIENNCM Translational symmetry is lost But WIENncm is using the generalized Bloch theorem. The calculation of spin waves only requires one unit cell for even incommensurate modulation q vector.

100 WienNCM: Usage. Generate the atomic and magnetic structures Create atomic structure Create magnetic structure See utility programs: ncmsymmetry, polarangles,. Run initncm (initialization script) 3. Run the NCM calculation: xncm (WIENncm version of x script) runncm (WIENncm version of run script) More information on the manual (Robert Laskowski) rolask@theochem.tuwien.ac.at