(spin-orbit) and - WIEN 2k

Magnetism and SOC inWien2kRobert Laskowskirolask@theochem.tuwien.ac.arVienna University of Technology,Institute of Materials Chemistry

SCOPE●magnetism in Wien2k– collinear spins (ferro, ferri, antiferro-magnets)– non-collinear spin (any arrangements),●introduction to WienNCMspin-orbit coupling (SOC) in Wien2kNCM & SOC in Wien2kwien2k workshop 2006 - p2

Pauli HamiltonianH P=− ħ2m ∇2 V ef B ⋅ B ef ⋅l ●●2x2 matrix in spin space, due to Pauli spin operatorswave function is a 2-component vector (spinor)spin upcomponentH P 1 2 = 1 2spin downcomponentNCM & SOC in Wien2kwien2k workshop 2006 - p3

Pauli HamiltonianH P=− ħ2m ∇ 2 V ef B ⋅ B ef ⋅l electrostaticpotentialmagnetic fieldspin-orbit coup.V ef=V extV HV xcB ef=B extB xcHartree termexchange-correlationpotentialexchange-correlationfield●exchange-correlation potential V xcand magneticfield B xcare defined within DFT LDA or GGANCM & SOC in Wien2kwien2k workshop 2006 - p4

Exchange and correlation●from DFT LDA exchange-correlation energy:E xcn , m =∫ n xcn , m dr 3local function of n and m● definition of V cxand B xc:V xc= ∂ E xcn , m ∂ n● LDA expression for V cxand B xc:Bxc= ∂ E xcn , m ∂ mfunctional derivativesB xcand m are parallelV xc= xcn , m n ∂ xcn , m ∂ nBxc=n ∂ xcn , m ∂ mmNCM & SOC in Wien2kwien2k workshop 2006 - p5

Non-collinear case●●H P=− ħ2m ∇ 2 V ef B⋅ B ef ⋅l direction of magnetization vary in spacespin-orbit coupling is present− ħ2m ∇ 2 V ef BB z B B x iB y B B x−iB y − ħ2m ∇ 2 V ef B B z == 1 2 , 1, 2≠0●●solutions are non-pure spinorsnon-collinear magnetic momentsNCM & SOC in Wien2kwien2k workshop 2006 - p6

Collinear caseH P =− ħ2m ∇ 2 V ef B ⋅ B ef ⋅l ● magnetization in Z direction, B xand B y=0●spin-orbit coupling is not present− ħ2m ∇ 2 V ef B B z 00 − ħ2m ∇ 2 V ef BB z= = 10 , = 0 2 , ≠ ●●solutions are pure spinorscollinear magnetic momentsNCM & SOC in Wien2kwien2k workshop 2006 - p7

Non-magnetic caseH P =− ħ2m ∇ 2 V ef B ⋅ B ef ⋅l ● no magnetization present, B x, B yand B z=0●spin-orbit coupling is not present− ħ2m ∇ 2 V ef 00 − ħ2m ∇2 V ef== 0 , = 0 , ●solutions are pure spinors= ●degenerate spin solutionsNCM & SOC in Wien2kwien2k workshop 2006 - p8

Magnetism and Wien2k●Wien2k can only handle collinear or non-magnetic casesrun_lapw script:x lapw0x lapw1x lapw2x lcorex mixernon-magnetic casem=n −n =0NCM & SOC in Wien2kwien2k workshop 2006 - p9

Magnetism and Wien2k●Wien2k can only handle collinear or non-magnetic casesrun_lapw script:x lapw0x lapw1x lapw2x lcorex mixernon-magnetic casem=n −n =0run_lapw script:x lapw0x lapw1 -upx lapw1 -dnx lapw2 -upx lapw2 -dnx lcore -upx lcore -dnx mixermagnetic casem=n −n ≠0NCM & SOC in Wien2kwien2k workshop 2006 - p10

Spin polarized calculations●runsp_lapw script (unconstrained magnetic calc.)– runs lapw1/2 for both spins independently– case.scf contains extra information:●●●grep :MMT case.scf (for total moment)grep :MMI case.scf (for atomic moments)grep :HFF case.scf (for hyperfine fields)NCM & SOC in Wien2kwien2k workshop 2006 - p11

Spin polarized calculations●runsp_lapw script (unconstrained magnetic calc.)– runs lapw1/2 for both spins independently– case.scf contains extra information:●●●●●grep :MMT case.scf (for total moment)grep :MMI case.scf (for atomic moments)grep :HFF case.scf (for hyperfine fields)runfsm_lapw -m value (constrained moment calc.)– for difficult to converge magnetic cases or simply to constrain amoment ( 2 Fermi-energies external magnetic field)runafm_lapw (anti-ferromagnetic calculation)– calculates only spin-up, uses symmetry to generate spin-dnNCM & SOC in Wien2kwien2k workshop 2006 - p12

Spin polarized calculations●●●runsp_lapw script (unconstrained magnetic calc.)runfsm_lapw -m value (constrained moment calc.)runafm_lapw (anti-ferromagnetic calculation)●●spin-orbit coupling can be included in secondvariational stepnever mix polarized and non-polarizedcalculations in one case directory !!!NCM & SOC in Wien2kwien2k workshop 2006 - p13

Non-collinear calculations●in a case of non-collinear spin arrangements WienNCM(Wien2k clone) has to be used– code based on Wien2k (available for Wien2k users)– structure and usage philosophy similar to Wien2k– independent source tree, independent installationNCM & SOC in Wien2kwien2k workshop 2006 - p14

Non-collinear calculations●●case of non-collinear spin arrangements WienNCM(Wien2k clone) has to be used– code based on Wien2k (available for Wien2k users)– structure and usage philosophy similar to Wien2k– independent source tree, independent installationWienNCM properties:– real and spin symmetry (simplifies SCF, less k-points)– constrained or unconstrained calculations (optimizesmagnetic moments)– SOC in first variational step, LDA+U– spin spiralsNCM & SOC in Wien2kwien2k workshop 2006 - p15

WienNCM - implementation● basis set – mixed spinors (Yamagami, PRB (2000); Kurtz PRB (2001)interstitials:spheres: G =ei Gk ⋅r APW =∑G ∑lm A lmG u l B lmG ˙u l Y lm APW =G G Alm uG l Blm ˙uG l Clm u2, l Y lm = 10 , 0 1●real and spin space parts of symmetry op. are bounded– symmetry treatment like SOC always on– tool for setting up magnetic configurationm– concept of magnetic and non-magnetic atomsNCM & SOC in Wien2kwien2k workshop 2006 - p16

WienNCM implementation● sphere Hamiltonian: H =− ħ2m ∇2 V H so H orb H cAMA and full NCcalculationSOC in firstdiagonalizationV FULL=V V V V =V V 0AMAlH so= ⋅l =zl x−i l yl xi l y−l z0 V diagonal orbital fieldconstraining fieldH orb=∑m m 'H c = B ⋅B c =∣m 〉 V mm '〈 m '∣ 00 ∣m 〉 V mm '〈m '∣0 B B cx−iB cy B B cxiB cy 0NCM & SOC in Wien2kwien2k workshop 2006 - p17

WienNCM – spin spirals●transverse spin wave=R⋅q●●generalized Bloch theorem– generalized translations– group of T nis AbelianRT n={−q⋅R n∣∣R n }T n k r =U −q⋅R k r R = k r efficient way for calculation of spinwaves, only one unit cell is necessaryfor even incommensurate wave k r =e i k⋅r i q⋅re−i q⋅r2e2 u r u r NCM & SOC in Wien2kwien2k workshop 2006 - p18

WienNCM – case.inncm file●case.inncm – magnetic structure fileFULL0.000000 0.000000 0.00000045.00000 54.73561 0135.00000 125.26439 0-135.00000 54.73561 0-45.00000 125.26439 045.00000 54.73561 045.00000 54.73561 0315.00000 125.26439 0315.00000 125.26439 0135.00000 125.26439 0135.00000 125.26439 0225.00000 54.73561 0225.00000 54.73561 00.50000mixing forconstraining fieldU, magneticatomsO, non-magneticatomsq spiral vectorpolar angles of mmoptimization switchNCM & SOC in Wien2kwien2k workshop 2006 - p19

SOC in Wien2k

SOC in Wien2k●Non-relativistic limit of Dirac equation[ p22m V −p48m 3 c 2−ħ2 dV4m 2 c 2 dr∂∂ r 1 12m 2 c 2 rdVdr l s ] Φ=εΦSchrödingerEquationmass enhancement +Darwin termspin-orbit coupling●SOC mixes up and down states, j=l+s is good quantum numberj=l+s/2κ=-s(j+½)occupationls=-1s=+1s=-1s=+1s=-1s=+1s01/2-12p11/23/21-224d23/25/22-346f35/27/23-468NCM & SOC in Wien2kwien2k workshop 2006 - p21

Relativistic orbital contraction●Au s orbitals (no SOC)• 1s contracts due to relativistic mass enhancement• 2s - 6s contract due to orthogonality to 1sM =m/1− v/c 2v proportional Z: Gold: Z = 79;M = 1.2 mNCM & SOC in Wien2kwien2k workshop 2006 - p22

SOC splitting of p states• Spin Orbit splitting of l-quantum number.• p 1/2(κ=1): markedly different behavior than non-relativistic p-state• u κ=1: non-zero at nucleusNCM & SOC in Wien2kwien2k workshop 2006 - p23

Relativistic orbital expansion• Higher l-quantum number states expand due to bettershielding of core charge from contracted s-states.NCM & SOC in Wien2kwien2k workshop 2006 - p24

orbital contractionAu atomic spectraorbital contractionorbital expansionorbital contractionSOC splitingNCM & SOC in Wien2kwien2k workshop 2006 - p25

SOC in Wien2k●WIEN2k 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 asbasis)●●●adding p-1/2 LOs to increase accuracy (caution!!!)Non-magnetic systems:– SO does NOT reduce symmetry. initso_lapw just generates case.insoand case.in2c.Magnetic systems:– symmetso detects proper symmetry and rewrites case.struct/in*/clm*NCM & SOC in Wien2kwien2k workshop 2006 - p27

SOC in Wien2k●●run(sp)_lapw -so script:x lapw1(increase E-max for more eigenvectors in second diag.)x lapwso(second diagonalization)x lapw2 –so –c (SO ALWAYS needs complex lapw2 version)case.inso file:WFFIL4 1 0 llmax,ipr,kpot-10.0000 1.50000 emin,emax (output energy window)0. 0. 1. direction of magnetization (lattice vectors)1 number of atoms for which RLO is added2 -0.97 0.005 atom number,e-lo,de (case.in1), repeat NX times0 0 0 0 0 number of atoms for which SO is switched off; atomsNCM & SOC in Wien2kwien2k workshop 2006 - p28

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