Spin-orbit coupling in Wien2k

Transcription

1 Spin-orbit coupling in Wienk Robert Laskowski Institute of High Performance Computing Singapore

2 Dirac Hamiltonian Quantum mechanical description of electrons, consistent with the theory of special relativity. H D=c α p+β m c +V Pauli matrices: k k= k 1 k= 1 1 = 1, = i, 1 i 3= 1 1 HD and the wave function are 4 dimensional objects

3 Dirac Hamiltonian spin up HD spin down free particle: ( 1 1 = ε mc p^z small components p^ z p^z mc, spin up mc, spin down )( ) ( p^x i p^ y ) ε mc ( p^x +i p^y ) ( p^x i p^y ) ε+mc ( p^x +i p^y ) slow particle limit (p=): large components p^z ε+mc ψ1 ψ = ψ3 ψ4 mc, mc, antiparticles, up, down 3

4 Dirac equation in spherical potential Solution for spherical potential g (r )χ κ κσ Ψ= i f κ (r) χ κ σ ( ) combination of spherical harmonics and spinor (κ+1) dg κ = g κ + Mcf κ dr r df κ 1 κ 1 = (V E ) g κ + fκ dr c r κ= s ( j+1/ ) j =l + s / s=+1, 1 Radial Dirac equation 4

5 Dirac equation in spherical potential Radial Dirac equation (κ+1) dg κ = g κ + Mcf κ dr r df κ 1 κ 1 = (V E ) g κ + fκ dr c r κ dependent term, for a constant l, κ depends on the sign of s substitute f from first eq. into the second eq. [ ] 1 d g κ dg κ l (l +1) dv dg κ 1 κ 1 dv g κ + g +Vg κ κ =Eg κ M dr r dr dr dr 4 M c r dr 4 M c r scalar relativistic approximation spin orbit coupling 5

6 Implementation: core electrons Core states are calculated with spin compensated Dirac equation spin polarized potential spin up and spin down radial functions are calculated separately, the density is averaged according to the occupation number specified in case.inc file 1s1/ p1/ p3/ Relations between quantum numbers j=l+s/ l s=-1 s=+1 =-s(j+½) s=-1 1/ s=+1 occupation s=-1-1 s=+1 s p 1 1/ 3/ 1-4 d 3/ 5/ f 3 5/ 7/ , 1,, 1,, 1,,,4 3, 1, 3, 1, 3,,4 3,,4 3, 3,6 ( ( ( ( ( ( ( ( ( N,KAPPA,OCCUP) N,KAPPA,OCCUP) N,KAPPA,OCCUP) N,KAPPA,OCCUP) N,KAPPA,OCCUP) N,KAPPA,OCCUP) N,KAPPA,OCCUP) N,KAPPA,OCCUP) N,KAPPA,OCCUP) Core levels configuration (case.inc for Ru atom) 6

7 Implementation: valence electrons Valence electrons inside atomic spheres are treated within scalar relativistic approximation (Koelling and Harmon, J. Phys C 1977) if RELA is specified in struct file dp 1 P= McQ dr r (l +1) (V ϵ) dq 1 Q= l Mcr + P dr r c [ radial equations of Koelling and Harmon (spherical potential) ] no κ dependency of the wave function, (l,m,s) are 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 non relativistic 7

8 Effects of RELA contraction of Au s orbitals 1s contracts due to relativistic mass enhancement s 6s contract due to orthogonality to 1s M V /r=z e/r centripetal force ( ) M =m/ 1 v / c v ~ Z: Au Z = 79;M = 1. m 8

9 Effects of RELA orbital expansion of Au d orbitals Higher l quantum number states expand due to better shielding of the core charge from contracted s states (effect is larger for higher states). 9

10 Spin orbit-coupling ℏ H P = V ef l m 1 1 dv MT r = dr Mc r x matrix in spin space, due to Pauli spin operators, wave function is a -component vector (spinor) x= spin up HP Pauli matrices: 1 = 1 spin down y= i i z= 1 1 Spin structure of the Hamiltonian with SOC ℏ V ef m 1 1 l z l x il y = ℏ V ef l x il y l z m 1

11 Spin-orbit coupling 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) SOC is added in a second variation (lapwso): first diagonalization (lapw1) H 1 ψ1=ε1 ψ1 N second diagonalization (lapwso) ( H 1 + H SO ) ψ=ε ψ second diagonalization ( δ ij ε1 + ψ1 H SO ψ1 ) ψ1 ψ =ε ψ1 ψ i j j i i j sum includes both up/down spin states N is much smaller then the basis size in lapw1!! 11

12 SOC splitting of p states Spin Orbit splitting of l quantum number. orbital moment +e j=3/ p1/ different behavior than non relativistic p state (density is diverging at nucleus), need for extra basis function (p1/ LO) spin e j=1/ +e e Ej=3/ Ej=1/ band edge at?in ZnO 1

13 p1/ orbitals Electronic structure of fcc Th, SOC with 6p1/ local orbital PRB, 64, 1531 (1) p3/states p1/ not included p1/states p1/ included energy vs. basis size DOS with and without p1/ 13

14 Au atomic spectra orbital contraction orbital expansion SOC splitting orbital contractio orbita n l cont ractio n 14

15 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) no time inversion (do not add an inversion for k list) initso_lapw (must be executed) detects new symmetry setting direction of magnetization sym. operation [1] [1] [1] [11] 1 A A A A ma A B B mb B A B - c B B A B 15

16 SOC in Wienk run(sp)_lapw -so script: (increase E-max for more eigenvectors in second diag.) (second diagonalization) (SOC ALWAYS needs complex lapw version) x lapw1 x lapwso x lapw so case.inso file: WFFIL llmax,ipr,kpot emin,emax (output energy window) direction of magnetization (lattice vectors) number of atoms for which RLO is added atom number,e-lo,de (case.in1), repeat NX times number of atoms for which SO is switched off; list of atoms p1/ orbitals, use with caution!! 16

17 Summary relativistic effect are included inside spheres only for core electrons we solve Dirac equation using spherical part of the total potential (dirty trick for spin polarized systems) for valence electrons, scalar relativistic approximation is used as default (RELA switch in case.struct), in order to include SOC for valence electrons lapwso has to be included in SCF cycle (run -so/run_sp -so) limitations: not all programs are compatible with SOC, for instance: no forces with SOC (yet) 17

18 NCM 18

19 Pauli Hamiltonian ℏ H P = V ef B B ef l m x matrix in spin space, due to Pauli spin operators wave function is a component vector (spinor) spin up component HP 1 = 1 V ef =V ext V H V xc Hartee term Pauli matrices: 1= 1 1 = i i 3= 1 1 spin down component B ef =B ext B xc exchange correlation potential exchange correlation field 19

20 Exchange and correlation from DFT LDA exchange correlation energy: E xc n, m = n xc n, m dr 3 definition of Vcx and Bxc: E xc n, m V xc = n local function of n and m E xc n, m B xc = m functional derivatives LDA expression for Vcx and Bxc: xc n, m V xc = xc n, m n n xc n, m B xc =n m m Bxc and m are parallel

21 Non-magnetic case ℏ H P = V ef B B ef l m no magnetization present, Bx, By and Bz=, and spin orbit coupling is not present ℏ V ef m ℏ V ef m =, =, = = solutions are pure spinors degenerate spin solutions 1

22 Non-magnetic calculation spin up and spin down part of the electronic spectrum is the same formally only one spin channel needs to be solved run_lapw script: x x x x x lapw lapw1 lapw lcore mixer calculate potential calculate valence wave functions calculate valence densities calculate core states generete new density m=n n =

23 Collinear magnetism magnetization in z direction, Bx and By= and spin orbit coupling is not present ℏ H P = V ef B B ef l m ℏ V ef B B z m ℏ V ef B B z m = 1, =, = solutions are pure spinors 3

24 Collinear magnetic calculation due to the exchange field the potential is different for up and down spin channels Pauli Hamiltonian is diagonal in spin space, thus up and down spin channels are solved separately ℏ V ef B B z = m ℏ V ef B B z = m runsp_lapw script: x x x x x x x x lapw lapw1 lapw1 lapw lapw lcore lcore mixer up dn up dn up dn m=n n,, 4

25 anti-ferromagnetic systems Cr has AFM bcc structure Cr1 Cr Cr1 spin-up Cr spin-up EF spin-down spin-down there is a symmetry relation between spin up and spin down densities (one atoms spin up is another atoms spin down) therefore it is enough to do the spin up calculation, the spin down results can be generated afterwards use runafm_lapw script in such cases (stabilizes and speeds up the SCF convergence) 5

26 fix spin moment calculations constrain value of magnetic moments and dependence of energy on magnetic moment A.R.Williams, V.L.Moruzzi, J.Kübler, K.Schwarz, Bull.Am.Phys.Soc. 9, 78 (1984) K.Schwarz, P.Mohn J.Phys.F 14, L19 (1984) P.H.Dederichs, S.Blügel, R.Zoller, H.Akai, Phys. Rev, Lett. 53,51 (1984) M H H under certain conditions magnetization can be multivalued function of H M interchanging the dependent and independent variable makes this function is single valued (unique) 6

27 Fixed spin moment (FSM) method Conventional scheme constrained (FSM) method EF E F M N N Zv N N Zv N N output M N N one SCF Simple case: bcc Fe E F E F input output many calculations difficult case: Fe3Ni poor convergence good convergence 7

28 Non-collinear magnetism direction of magnetization vary in space and/or spin orbit coupling is present ℏ H P = V ef B B ef l m ℏ V ef B B z m B B x ib y B B x ib y ℏ V ef B B z m = 1, 1, = solutions are non pure spinors 8

29 Non-collinear magnetism Wienk can only handle collinear or non magnetic cases in NCM case both spin channels have to be considered simultaneously runncm_lapw script: xncm xncm xncm xncm xncm lapw lapw1 lapw lcore mixer nk n= nk nk nk nk m z=n n 1 m x = n n 1 m x =i n n relation between spin density matrix and magnetization NCM case 9

30 Non-collinear calculations in the case of non collinear arrangement of spin moment WienNCM (Wienk clone) has to be used code is based on Wienk (available for Wienk users) structure and usage 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 is applied in the first variational step, LDA+U spin spirals are available 3

31 WienNCM - implementation basis set mixed spinors (Yamagami, PRB (); Kurtz PRB (1) = 1, 1 i G k r =e interstities: G spheres: APW G = A APW G lm G = A lm G lm l l u B u B G lm G lm l u l Y lm u C G lm u, l Y lm real and spin space parts of symmetry op. are not independent m symmetry treatment like for SOC tool for setting up magnetic configuration concept of magnetic and non magnetic atoms 31

32 WienNCM implementation Hamiltonian inside spheres: ℏ = orb H c H V H so H m AMA and full NC calculation V V V FULL= V V SOC in first diagonalization l z l x i l y so = H l = l x i l y l z diagonal orbital field constraining field orb= H mm' V V AMA = V m V mm ' m ' m V mm ' m ' c = B H Bc = B Bcx ib cy B B cx ibcy 3

33 NCM (SOC) Hamiltonian Wienk WienNCM size of the Hamiltonian/overlap matrix is doubled comparing to Wienk computational cost increases!!! 33

34 WienNCM spin spirals transverse spin wave q = R R n=m cos q R n,sin q R n sin,cos m 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), hard coded as Z Translational symmetry is lost!!! 34

35 WienNCM spin spirals generalized Bloch theorem generalized translations are symmetry operation of the H n T n = { q R Rn } T n H r T n =U q R n H r R n U q R n group of Tn is Abelian k r =e r i k e e i q r u r i q r u r 1 d representations, Bloch Theorem =e i k r T n k r =U q R r R r k k efficient way for calculation of spin waves, only one unit cell is necessary for even incommensurate wave 35

36 Usage generate atomic and magnetic structure 1) create atomic structure ) create magnetic structure need to specify only directions of magnetic atoms use utility programs: ncmsymmetry, polarangles,... run initncm (initialization script) xncm ( WienNCM version of x script) runncm (WienNCM version of run script) find more in manual runncm_lapw script: xncm xncm xncm xncm xncm lapw lapw1 lapw lcore mixer 36

37 WienNCM case.inncm file case.inncm magnetic structure file FULL q spiral vector polar angles of mm optimization switch U, magnetic atoms O, non magnetic atoms mixing for constraining field 37