Magnetism: Spin-orbit coupling magnetic exchange and anisotropy

VASP workshop Rennes 2016 Magnetism: Spin-orbit coupling magnetic exchange and anisotropy Xavier Rocquefelte Institut des Sciences Chimiques de Rennes (UMR 6226) Université de Rennes 1, FRANCE

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 10 0 10-3 10-6 Energy scale (ev)

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

COLLINEAR MAGNETISM Magnetic susceptibility of a ferromagnetic (FM) compound 0,4 χ mol (emu/mol) 0,3 0,2 0,1 PM without long range interaction 0 0 50 100 150 200 250 300 T(K)

COLLINEAR MAGNETISM Magnetic susceptibility of a ferromagnetic (FM) compound 0,4 χ mol (emu/mol) 0,3 J F 0,2 0,1 PM without long range interaction 0 0 50 100 150 200 250 300 T(K)

COLLINEAR MAGNETISM Magnetic susceptibility of an antiferromagnetic (AFM) compound 0,04 χ mol (emu/mol) 0,03 PM without longrange interactions 0,02 0,01 0 0 50 100 150 200 250 300 T(K)

COLLINEAR MAGNETISM Magnetic susceptibility of an antiferromagnetic (AFM) compound 0,04 χ mol (emu/mol) 0,03 PM without longrange interactions 0,02 J AF 0,01 0 AF PM T(K) 0 50 100 150 200 250 300

COLLINEAR MAGNETISM Ferromagnetic Antiferromagnetic 0,4 χ mol (emu/mol) 0,02 χ mol (emu/mol) 0,3 Ferromg order when kt 0,2 0,1 0 F PM T(K) AF PM 0 50 100 150 200 250 300 0 50 100 150 200 250 300 T C Curie temperature T N Néel temperature 0,01 T(K) J F J AF Ferromagnetic exchange: J F < 0 Antiferromagnetic exchange: J AF > 0

NON-COLLINEAR MAGNETISM AFM with 2 subnetworks having different magnetization directions Frustrated AFM weak ferromagnetism Topologic frustration FM-AFM competition? J 1 J 1 J 2? J 1 : FM J 2 : AFM

Illustration of a collinear calculation: NiO Experiment data: Ni 2+ : d 8 electronic configuration Octahedral environment Rock-salt structure Space group: Fm-3m (#225) Optical gap: 4-4.3 ev Magnetic properties: AFM order µ(ni) = 1.7-1.9 µ B

Illustration of a collinear calculation: NiO 2 x 2 x 2 supercell

Illustration of a collinear calculation: NiO POSCAR Ni O Exercises: GGA calculations for AFM and FM orders GGA+U calculations for AFM and FM orders Comparison: Density of states Total energy Estimation of magnetic exchange

Illustration of a collinear calculation: NiO INCAR: GGA - AFM KPOINTS:

Illustration of a collinear calculation: NiO OSZICAR Total magnetic moment in the cell

Illustration of a collinear calculation: NiO OUTCAR Integration of magnetic moment in the PAW sphere (LORBIT = 11 in INCAR file) Ni1: 1.34 µ B Ni2: -1.34 µ B

Illustration of a collinear calculation: NiO

Illustration of a collinear calculation: NiO KPOINTS: 8 8 8

Illustration of a collinear calculation: NiO KPOINTS: 8 8 8 INCAR AND ICHARG = 11 ISMEAR = -5 NEDOS = 1000 EMIN = -10 ; EMAX = 15 GGA: too small band gap compared to exp. values

Illustration of a collinear calculation: NiO NiO - GGA - AFM Ni1: 1.24 µ B Ni2: -1.24 µ B Exp.: ±1.7-1.9 µ B OUTCAR

Illustration of a collinear calculation: NiO INCAR: GGA - FM KPOINTS: 8 8 8

Illustration of a collinear calculation: NiO NiO - GGA - FM Ni1: 1.06 µ B Ni2: 1.06 µ B OUTCAR

Illustration of a collinear calculation: NiO INCAR: GGA+U - AFM U eff = U J = 5 ev

Illustration of a collinear calculation: NiO NiO - GGA+U - AFM Better k-mesh Higher NEDOS value Ni1: 1.67 µ B Ni2: -1.67 µ B Exp.: ±1.7-1.9 µ B

Illustration of a collinear calculation: NiO NiO GGA+U - FM Ni1: 1.73 µ B Ni2: 1.73 µ B Oxygen magnetic moment Estimation of magnetic exchange?

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 = Ĥ0 + J ij Si. S! j i<j Long-range order J ij > 0 AFM J ij < 0 FM E α = α H α = E 0 + S 2 J ij σ i σ j i<j S: Spin hold by the magnetic center σ i = ±1 (up or down spin) Example of a spin-half dimer (S = ½) To estimate the J 12 value, 2 total energy calculations are needed: σ 1 = +1 σ 2 = +1 σ 1 = +1 σ 2 = -1 J 12 = 2( E FM E AFM ) E FM = E 0 + 1 4 J 12 E AFM = E 0 + 1 4 J 12

Estimation of J in NiO Ni 2+ -> S = 1 E α = α H α = E 0 + S 2 J ij σ i σ j i<j 2 inequivalent Ni sites in the rhombohedral unit cell (S.G. R-3m) J: magnetic coupling defined by Ni 1 -O-Ni 2 path (angle : 180 ) 6J / unit cell

Estimation of J in NiO Ni2+ -> S = 1 2 inequivalent Ni sites in the rhombohedral unit cell (S.G. R-3m) Eα = α H α = E 0 + S2 J ijσ iσ j J: magnetic coupling defined by Ni1-O-Ni2 path (angle : 180 ) i< j 6J / unit cell E AFM = E 0 6J E FM = E 0 + 6J -19.54909823 ev -19.30675287 ev

Collinear magnetism in VASP INCAR file Spin-polarized calculation: ISPIN = 2 Initial magnetic moment: MAGMOM = 2.0 2.0 2*0 Warning: Too small initial magnetic moments will/may lead to a non-magnetic solution Badly initialized calculations take longer to converge (local minima) Convergency of k-mesh, ENCUT and choice of POTCAR Comparing the total energies from calculations with different U eff values is meaningless! VASP can also treat non-collinear magnetic systems!

Noncollinear magnetism in VASP INCAR file Illustration with fcc Ni Replace ISPIN = 2 and MAGMOM = 1.0 by: leads to or with MAGMOM = 1.0 0.0 0.0 or with MAGMOM = 0.0 1.0 0.0

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 MAE (μev) Hard axis Easy axis Hard axis Magnetization axis NEED TO SWITCH ON THE SPIN-ORBIT: LSORBIT =.TRUE [1] X. Rocquefelte, P. Blaha, K. Schwarz, S. Kumar, J. van den Brink, Nature Comm. 4, 2511 (2013)

Estimation of the magnetic anisotropy Estimation of the Magneto-crystalline Anisotropy Energy (MAE) of CuO Allows to define the magnetization easy and hard axes [10-1] Here we have considered the following expression: MAE = E[u v w] E[easy axis] [010] [-10-1] [101] [0-10] E[uvw] is the energy deduced from spin-orbit calculations with the magnetization along the [uvw] crystallographic direction [-101] [1] X. Rocquefelte, P. Blaha, K. Schwarz, S. Kumar, J. van den Brink, Nature Comm. 4, 2511 (2013)

Estimation of the magnetic anisotropy LiNbO 3 -type InFeO 3 : Room-Temperature Polar Magnet without Second-Order Jahn Teller Active Ions Fujita, T. Kawamoto, I. Yamada, O. Hernandez, N. Hayashi, H. Aakamatsu, W. Lafargue-Dit-Hauret, X. Rocquefelte, M. Fukuzumi, P. Manuel, A. J. Studer, C. Knee, K. Tanaka Chemistry of Materials accepted (2016).

AND MORE VASP allows to constrain the magnetic moment using the following lines in INCAR: u Switch on constraints on magnetic moments u Integration radius to determine local moments u Weight in penalty function u Target direction A penalty function is added to the system which drives the integrated local moments into the desired direction Warning: The penalty function contributes to the total energy.

AND MORE

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