Relativistic effects & Non-collinear magnetism (WIEN2k / WIENncm)

1. 20th WIEN2k Workshop PennStateUniversity – 2013 Relativistic effects & Non-collinear magnetism (WIEN2k / WIENncm) Xavier Rocquefelte Institut des Matériaux Jean-Rouxel (UMR 6502) Université de Nantes, FRANCE

2. 20th WIEN2k Workshop PennStateUniversity – 2013 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) http://www.wien2k.at/reg_user/textbooks/ - Robert Laskowski (Non-collinear magnetic version of WIEN2k package) • Books: • - WIEN2k userguide, ISBN 3-9501031-1-2 • - Electronic Structure: Basic Theory and Practical Methods, Richard M. Martin ISBN 0 521 78285 6 • Relativistic Electronic Structure Theory. Part 1. Fundamentals, Peter Schewerdtfeger, ISBN 0 444 51249 7 • web: • http://www2.slac.stanford.edu/vvc/theory/relativity.html • wienlist digest - http://www.wien2k.at/reg_user/index.html • wikipedia …

3. Few words about Special Theory of Relativity Light Composed of photons (no mass) Speed of light = constant Atomic units: ħ = me = e = 1 c  137 au

4. Few words about Special Theory of Relativity Light Matter Composed of photons (no mass) Composed of atoms (mass) Speed of light = constant v = f(mass) Atomic units: ħ = me = e = 1 Speed of matter mass mass = f(v) c  137 au

5. 1 g = ³ 1 2 æ ö v - ç ÷ 1 c è ø Few words about Special Theory of Relativity Light Matter Composed of photons (no mass) Composed of atoms (mass) Speed of light = constant v = f(mass) Atomic units: ħ = me = e = 1 Speed of matter mass mass = f(v) c  137 au Lorentz Factor (measure of the relativistic effects) Relativistic mass: M = m (m: rest mass) Momentum: p = mv = Mv Total energy: E2 = p2c2 + m2c4 E = mc2 = Mc2

6. e- +Ze = = g g = = H H : : v v ( ( 1 1 s s ) ) 1 1 au au 1 1 . . 00003 00003 e e = = g g = = Au Au : : v v ( ( 1 1 s s ) ) 79 79 au au 1 1 . . 22 22 e e 0 20 40 60 80 100 120 Definition of a relativistic particle (Bohr model) Lorentz factor () Speed of the 1s electron (Bohr model): 10 9 « Non-relativistic » particle:  = 1 8 7 6 5 H(1s) Au(1s) 4 Z Z 3 µ µ v v e e 2 n n 1 Speed (v) c  137 au Details for Au atom: Me(1s-Au) = 1.22me 1s electron of Au atom = relativistic particle

7. +Ze 1) The mass-velocity correction Relativistic increase in the mass of an electron with its velocity (when ve c) Relativistic effects

8. +Ze 1) The mass-velocity correction Relativistic increase in the mass of an electron with its velocity (when ve c) 2) The Darwin term It has no classical relativistic analogue Due to small and irregular motions of an electron about its mean position (Zitterbewegung*) Relativistic effects *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. +Ze 1) The mass-velocity correction Relativistic increase in the mass of an electron with its velocity (when ve c) 2) 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) Relativistic effects

10. 1) The mass-velocity correction Relativistic increase in the mass of an electron with its velocity (when ve c) 2) 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 Relativistic effects +Zeffe

11. One electron radial Schrödinger equation HARTREE ATOMIC UNITS INTERNATIONAL UNITS Atomic units: ħ = me = e = 1 1/(40) = 1 c = 1/  137 au

12. One electron radial Schrödinger equation HARTREE ATOMIC UNITS INTERNATIONAL UNITS In a spherically symmetric potential Atomic units: ħ = me = e = 1 1/(40) = 1 c = 1/  137 au

13. One electron radial Schrödinger equation HARTREE ATOMIC UNITS INTERNATIONAL UNITS In a spherically symmetric potential

14. Dirac Hamiltonian: a brief description Dirac relativistic Hamiltonian provides a quantum mechanical description of electrons, consistent with the theory of special relativity. E2 = p2c2 + m2c4 with

15. Dirac Hamiltonian: a brief description Dirac relativistic Hamiltonian provides a quantum mechanical description of electrons, consistent with the theory of special relativity. Momentum operator E2 = p2c2 + m2c4 Rest mass with Electrostatic potential (22) unit and zero matrices (22) Pauli spin matrices

16. Dirac equation: HD and  are 4-dimensional  is a four-component single-particle wave function that describes spin-1/2 particles. In case of electrons: spin up Large components () F æ ö factor  1/(mec2) ç ÷ y = ç ÷ c Small components () è ø spin down  and  are time-independent two-component spinors describing the spatial and spin-1/2 degrees of freedom Leads to a set of coupled equations for  and :

17. Dirac equation: HD and  are 4-dimensional For a free particle (i.e. V = 0): Solution in the slow particle limit (p=0) Antiparticles: up & down Particles: up & down Non-relativistic limit decouples 1 from 2 and 3 from 4

18. ( ) U r æ ö æ ö g F nk ks ç ÷ ç ÷ Y = = ç ÷ ç ÷ ( ) c - U r i f è ø è ø nk ks gn and fn are Radial functions Y are angular-spin functions Dirac equation: HD and  are 4-dimensional For a free particle (i.e. V = 0): Solution in the slow particle limit (p=0) Antiparticles: up & down Particles: up & down Non-relativistic limit decouples 1 from 2 and 3 from 4 For a spherical potential V(r):

19. Dirac equation in a spherical potential For a spherical potential V(r): The resulting equations for the radial functions (gn and fn) are simplified if we define: Radially varying mass: Energy:

20. Dirac equation in a spherical potential For a spherical potential V(r): The resulting equations for the radial functions (gn and fn) are simplified if we define: Radially varying mass: Energy: Then the coupled equations can be written in the form of the radial eq.: Mass-velocity effect Darwin term Spin-orbit coupling One electron radial Schrödinger equation in a spherical potential Note that:

21. Dirac equation in a spherical potential For a spherical potential V(r): The resulting equations for the radial functions (gn and fn) are simplified if we define: Radially varying mass: Energy: Then the coupled equations can be written in the form of the radial eq.: Darwin term Spin-orbit coupling and Due to spin-orbit coupling,  is not an eigenfunction of spin (s) and angular orbital moment (l). No approximation have been made so far Instead the good quantum numbers are j and  Note that:

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) No SOC  Approximate radial functions: and with the normalization condition:

23. ~ F ~ c 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) No SOC  Approximate radial functions: and with the normalization condition: The four-component wave function is now written as: Inclusion of the spin-orbit coupling in “second variation” (on the large component only) ~ ( ) ~ U r æ ö æ ö g ~ F nl lm ç ÷ ç ÷ Y = = ~ ~ ç ÷ ç ÷ ( ) c - U r i f è ø è ø nl lm with is a pure spin state is a mixture of up and down spin states

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 WIEN2k). Justify an implementation of the relativistic effects only inside the muffin-tin atomic spheres

25. Implementation in WIEN2k Atomic sphere (RMT) Region Core electrons Valence electrons « Fully » relativistic Scalar relativistic (no SOC) Spin-compensated Dirac equation Possibility to add SOC (2nd variational) SOC: Spin orbit coupling

26. Interstitial Region Valence electrons Not relativistic Implementation in WIEN2k Atomic sphere (RMT) Region Core electrons Valence electrons « Fully » relativistic Scalar relativistic (no SOC) Spin-compensated Dirac equation Possibility to add SOC (2nd variational) SOC: Spin orbit coupling

27. j=l+s/2 =-s(j+1/2) occupation l s=-1 s=+1 s=-1 s=+1 s=-1 s=+1 s 0 1/2 -1 2 p 1 1/2 3/2 1 -2 2 4 d 2 3/2 5/2 2 -3 4 6 f 3 5/2 7/2 3 -4 6 8 Implementation in WIEN2k: core electrons case.inc for Au atom Core states: fully occupied  spin-compensated Dirac equation (include SOC) 17 0.00 0 1,-1,2 ( n,,occup) 2,-1,2 ( n,,occup) 2, 1,2 ( n,,occup) 2,-2,4 ( n,,occup) 3,-1,2 ( n,,occup) 3, 1,2 ( n,,occup) 3,-2,4 ( n,,occup) 3, 2,4 ( n,,occup) 3,-3,6 ( n,,occup) 4,-1,2 ( n,,occup) 4, 1,2 ( n,,occup) 4,-2,4 ( n,,occup) 4, 2,4 ( n,,occup) 4,-3,6 ( n,,occup) 5,-1,2 ( n,,occup) 4, 3,6 ( n,,occup) 4,-4,8 ( n,,occup) 0 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.

28. j=l+s/2 =-s(j+1/2) occupation l s=-1 s=+1 s=-1 s=+1 s=-1 s=+1 s 0 1/2 -1 2 p 1 1/2 3/2 1 -2 2 4 d 2 3/2 5/2 2 -3 4 6 f 3 5/2 7/2 3 -4 6 8 Implementation in WIEN2k: core electrons case.inc for Au atom Core states: fully occupied  spin-compensated Dirac equation (include SOC) 1s1/2 2s1/2 2p1/2 2p3/2 3s1/2 3p1/2 3p3/2 3d3/2 3d5/2 4s1/2 4p1/2 4p3/2 4d3/2 4d5/2 5s1/2 4f5/2 4f7/2 17 0.00 0 1,-1,2 ( n,,occup) 2,-1,2 ( n,,occup) 2, 1,2 ( n,,occup) 2,-2,4 ( n,,occup) 3,-1,2 ( n,,occup) 3, 1,2 ( n,,occup) 3,-2,4 ( n,,occup) 3, 2,4 ( n,,occup) 3,-3,6 ( n,,occup) 4,-1,2 ( n,,occup) 4, 1,2 ( n,,occup) 4,-2,4 ( n,,occup) 4, 2,4 ( n,,occup) 4,-3,6 ( n,,occup) 5,-1,2 ( n,,occup) 4, 3,6 ( n,,occup) 4,-4,8 ( n,,occup) 0 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.

29. Atomic sphere (RMT) Region Atomic sphere (RMT) Region Atomic sphere (RMT) Region Atomic sphere (RMT) Region Valence Valence Valence Valence electrons electrons electrons electrons Scalar Scalar relativistic relativistic (no SOC) (no SOC) Implementation in WIEN2k: valence electrons Valence electrons INSIDE atomic spheres are treated within scalar relativistic approximation [1] if RELA is specified in case.struct file (by default). Title F LATTICE,NONEQUIV.ATOMS: 1 225 Fm-3m MODE OF CALC=RELA unit=bohr 7.670000 7.670000 7.670000 90.000000 90.000000 90.000000 ATOM 1: X=0.00000000 Y=0.00000000 Z=0.00000000 MULT= 1 ISPLIT= 2 Au1 NPT= 781 R0=0.00000500 RMT= 2.6000 Z: 79.0 LOCAL ROT MATRIX: 1.0000000 0.0000000 0.0000000 0.0000000 1.0000000 0.0000000 0.0000000 0.0000000 1.0000000 48 NUMBER OF SYMMETRY OPERATIONS  no  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 » Valence electrons in interstitial region are treated classically [1] Koelling and Harmon, J. Phys. C (1977)

30. Implementation in WIEN2k: valence electrons SOC is added in a second variation (lapwso): - First diagonalization (lapw1): - Second diagonalization (lapwso): The second equation is expanded in the basis of first eigenvectors (1) sum include both up/down spin states  N is much smaller than the basis size in lapw1

31. Implementation in WIEN2k: valence electrons SOC is added in a second variation (lapwso): - First diagonalization (lapw1): - Second diagonalization (lapwso): The second equation is expanded in the basis of first eigenvectors (1) sum include both up/down spin states  N is much smaller than the basis size in lapw1  SOC is active only inside atomic spheres, only spherical potential(VMT) 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) VMT: Muffin-tin potential (spherically symmetric)

32. Controlling spin-orbit coupling in WIEN2k  Do a regular scalar-relativistic “scf” calculation  save_lapw  initso_lapw  case.inso: WFFIL 4 1 0 llmax,ipr,kpot -10.0000 1.50000 emin,emax (output energy window) 0. 0. 1. direction of magnetization (lattice vectors) NX number of atoms for which RLO is added NX1 -4.97 0.0005 atom number,e-lo,de (case.in1), repeat NX times 0 0 0 0 0 number of atoms for which SO is switch off; atoms  case.in1(c): (…) 2 0.30 0.005 CONT 1 0 0.30 0.000 CONT 1 K-VECTORS FROM UNIT:4 -9.0 4.5 65 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 WIEN2k The w2web interface is helping you Non-spin polarized case

34. Controlling spin-orbit coupling in WIEN2k The w2web interface is helping you Spin polarized case

35. Relativistic effects in the solid: Illustration LDA overbinding (7%) No difference NREL/SREL hcp-Be • Bulk modulus: • NREL: 131.4 GPa • SREL: 131.5 GPa • Exp.: 130 GPa Z = 4

36. Relativistic effects in the solid: Illustration LDA overbinding (7%) No difference NREL/SREL hcp-Be • Bulk modulus: • NREL: 131.4 GPa • SREL: 131.5 GPa • Exp.: 130 GPa Z = 4 LDA overbinding (2%) Clear difference NREL/SREL hcp-Os • Bulk modulus: • NREL: 344 GPa • SREL: 447 GPa • Exp.: 462 GPa Z = 76

37. Relativistic effects in the solid: Illustration hcp-Be Z = 4  Scalar-relativistic (SREL): • LDA overbinding (2%) • Bulk modulus: 447 GPa hcp-Os + spin-orbit coupling (SREL+SO): Z = 76 • LDA overbinding (1%) • Bulk modulus: 436 GPa  Exp. Bulk modulus: 462 GPa

38. e- +Ze (1) Relativistic orbital contraction Radius of the 1s orbit (Bohr model): r r r r r r (e/ (e/ (e/ bohr bohr bohr ) ) ) 2 2 2 50 50 50 Non Non relativistic relativistic (l=0) (l=0) 40 40 40 30 30 30 Au 1s 20 20 20 AND 10 10 10 0 0 0 0.00 0.00 0.00 0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.04 0.05 0.05 0.05 0.06 0.06 0.06 r r r ( ( ( bohr bohr bohr ) ) ) Atomic units: ħ = me = e = 1 c = 1/  137 au

39. k k Relativistic Relativistic ( ( = = - - 1) 1) e- +Ze 2 a n 1 1 = = = 0 r ( 1 s ) 0 . 010 bohr g Z 79 1 . 22 (1) Relativistic orbital contraction Radius of the 1s orbit (Bohr model): r r r r r r (e/ (e/ (e/ bohr bohr bohr ) ) ) 2 2 2 50 50 50 Non Non Non relativistic relativistic relativistic (l=0) (l=0) (l=0) k k Relativistic Relativistic ( ( = = - - 1) 1) 40 40 40 30 30 30 Au 1s 20 20 20 AND 10 10 10 20% Orbital contraction 0 0 0 0.00 0.00 0.00 0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.04 0.05 0.05 0.05 0.06 0.06 0.06 r r r ( ( ( bohr bohr bohr ) ) ) In Au atom, the relativistic mass (M) of the 1s electron is 22% larger than the rest mass (m)

40. k Relativistic ( = - 1) (1) Relativistic orbital contraction r r r r (e/ (e/ bohr bohr ) ) 2 2 0.5 0.5 Non Non relativistic relativistic (l=0) (l=0) k k Relativistic Relativistic ( ( = = - - 1) 1) 0.4 0.4 0.3 0.3 Orbital contraction 0.2 0.2 0.1 0.1 Au 6s 0.0 0.0 0 0 2 2 4 4 6 6 r r ( ( bohr bohr ) ) Direct relativistic effect (mass enhancement)  contraction of 0.46% only However, the relativistic contraction of the 6s orbital is large (>20%) ns orbitals (with n > 1) contract due to orthogonality to 1s

41. Non Non relativistic relativistic (l=0) (l=0) k k Relativistic Relativistic ( ( = = - - 1) 1) (1) Orbital Contraction: Effect on the energy r r r r (e/ bohr ) r r (e/ (e/ bohr bohr ) ) 2 2 2 50 0.5 Relativistic correction (%) 40 0.4 30 0.3 Orbital contraction Orbital contraction 20 0.2 20 10 0.1 Au 1s Au 6s 0 0.0 10 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0 2 4 6 r ( bohr ) r r ( ( bohr ) ) 1s 2s 3s 4s 5s 6s 0 -10 -20 -30 -40

42. (2) Spin-Orbit splitting of p states r r r r (e/ (e/ bohr bohr ) ) 2 2 Non Non relativistic relativistic (l=1) (l=1) 0.7 0.6 0.5 0.4 Au 5p 0.3 0.2 0.1 0.0 0.0 0.5 1.0 1.5 2.0 2.5 r r ( ( bohr bohr ) )

43. (2) Spin-Orbit splitting of p states  Spin-orbit splitting of l-quantum number E r r r r (e/ (e/ bohr bohr ) ) 2 2 Non Non relativistic relativistic (l=1) (l=1) 0.7 j=1+1/2=3/2 k k Relativistic Relativistic ( ( = = - - 2) 2) j=3/2 (=-2) 0.6 l=1 0.5 0.4 Au 5p 0.3 j=1+1/2=3/2 0.2 orbital moment 0.1 spin 0.0 0.0 0.5 1.0 1.5 2.0 2.5 +e -e r r ( ( bohr bohr ) )  p3/2 (=-2): nearly same behavior than non-relativistic p-state

44. (2) Spin-Orbit splitting of p states  Spin-orbit splitting of l-quantum number E r r r r (e/ (e/ bohr bohr ) ) 2 2 Non Non relativistic relativistic (l=1) (l=1) 0.7 0.6 k k Relativistic Relativistic ( ( =1) =1) l=1 0.5 j=1/2 (=1) j=1-1/2=1/2 0.4 Au 5p 0.3 j=1-1/2=1/2 0.2 orbital moment 0.1 0.0 0.0 0.5 1.0 1.5 2.0 2.5 +e -e r r ( ( bohr bohr ) ) spin  p1/2 (=1): markedly different behavior than non-relativistic p-state g=1 is non-zero at nucleus

45. (2) Spin-Orbit splitting of p states  Spin-orbit splitting of l-quantum number E r r r r (e/ (e/ bohr bohr ) ) 2 2 Non Non relativistic relativistic (l=1) (l=1) 0.7 j=1+1/2=3/2 k k Relativistic Relativistic ( ( = = - - 2) 2) j=3/2 (=-2) 0.6 k k Relativistic Relativistic ( ( =1) =1) l=1 0.5 j=1/2 (=1) j=1-1/2=1/2 0.4 Au 5p 0.3 j=1+1/2=3/2 j=1-1/2=1/2 0.2 orbital moment orbital moment 0.1 spin 0.0 0.0 0.5 1.0 1.5 2.0 2.5 +e +e -e -e r r ( ( bohr bohr ) ) spin Ej=3/2 Ej=1/2  p1/2 (=1): markedly different behavior than non-relativistic p-state g=1 is non-zero at nucleus

46. (2) Spin-Orbit splitting of p states Relativistic correction (%) 20 10 2p1/2 2p3/2 3p1/2 3p3/2 4p1/2 4p3/2 5p1/2 5p3/2 =1 =-2 0 -10 -20 -30 Scalar-relativistic p-orbital is similar to p3/2 wave function, but  does not contain p1/2 radial basis function -40

47. (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

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 Zeff1 = Z-(NREL) +Zeff1e -e

49. -e (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 +Ze -e -e -e -e Zeff1 = Z-(NREL) Zeff2 = Z-(REL) Zeff1 > Zeff2 +Zeff1e +Zeff2e -e

50. Indirect relativistic effect (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 +Ze -e -e -e -e Zeff1 = Z-(NREL) Zeff2 = Z-(REL) Zeff1 > Zeff2 +Zeff1e +Zeff2e -e -e