Magnetism: Spin-orbit coupling magnetic exchange and ... · Magnetism: Spin-orbit coupling magnetic...

Magnetism:

Spin-orbit coupling magnetic exchange and

anisotropy

VASP workshopRennes – 2016

Xavier RocquefelteInstitut des Sciences Chimiques de Rennes

(UMR 6226) Université de Rennes 1, FRANCE

INTRODUCTION

Magnetic properties:

ü  Spin-state (high/low) ü  Long-range/short-range orders ü  Collinear / non-collinear ü  Magnetic anisotropy ü  Magnetic frustration ü  Magnetic exchange

Energy scale (eV) 100 10-3 10-6

Spin-State Magnetic exchange Long-range order

Magnetic anisotropy

INTRODUCTION

Paramagnetic(PM)

Ferrimagneticorder

Ferromagnetic(FM) order

Antiferromagnetic(AFM) order

COLLINEAR MAGNETISM

0 50 100 150 200 250 300

T(K)

χmol(emu/mol)

0

0,1

0,2

0,3

0,4

PM without long range interaction

Magnetic susceptibility of a ferromagnetic (FM) compound

COLLINEAR MAGNETISM

0 50 100 150 200 250 300

T(K)

χmol(emu/mol)

0

0,1

0,2

0,3

0,4

JF

PM without long range interaction

Magnetic susceptibility of a ferromagnetic (FM) compound

COLLINEAR MAGNETISM

Magnetic susceptibility of an antiferromagnetic (AFM) compound

0 50 100 150 200 250 300

T(K)

χmol(emu/mol)

0

0,02

0,03

0,04

PM without long-range interactions

0,01

COLLINEAR MAGNETISM

Magnetic susceptibility of an antiferromagnetic (AFM) compound

0 50 100 150 200 250 300

T(K)

χmol(emu/mol)

0

0,02

0,03

0,04

AF

PM without long-range interactions

0,01 JAF

PM

COLLINEAR MAGNETISM

0 50 100 150 200 250 300

T(K)

χmol(emu/mol)

F

0

0,1

0,2

0,3

0,4

PM

0 50 100 150 200 250 300

T(K)

χmol(emu/mol)0,02

AF0,01

PM

TC TNCurie temperature Néel temperature

Ferromg order↑ when kT ↓

Antiferromagnetic Ferromagnetic

JF JAF

Ferromagnetic exchange: JF < 0 Antiferromagnetic exchange: JAF > 0

NON-COLLINEAR MAGNETISM

AFM with 2 subnetworks having different

magnetization directions

⇒ weak ferromagnetism

Frustrated AFM

Topologic frustration FM-AFM competition

?J1 J1

J2

?J1 : FM J2 : AFM

Illustration of a collinear calculation: NiO

Ni2+: d8 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 µB

Experiment data:


2 x 2 x 2 supercell


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

POSCAR

Ni

O


INCAR: GGA - AFM

KPOINTS:


OSZICAR Total magnetic moment in the cell


Integration of magnetic moment in the PAW sphere (LORBIT = 11 in INCAR file)

OUTCAR

Ni1: 1.34 µB

Ni2: -1.34 µB


KPOINTS:

8 8 8


8 8 8

AND INCAR

… ICHARG = 11 ISMEAR = -5 NEDOS = 1000 EMIN = -10 ; EMAX = 15 …

KPOINTS:

GGA: too small band gap compared to exp. values


OUTCAR

NiO - GGA - AFM

Ni1: 1.24 µB

Ni2: -1.24 µB

Exp.: ±1.7-1.9 µB


INCAR: GGA - FM

KPOINTS:

8 8 8


OUTCAR

NiO - GGA - FM

Ni1: 1.06 µB

Ni2: 1.06 µB


INCAR: GGA+U - AFM

Ueff = U – J = 5 eV


Better k-mesh Higher NEDOS value …

NiO - GGA+U - AFM

Ni1: 1.67 µB

Ni2: -1.67 µB Exp.: ±1.7-1.9 µB


NiO – GGA+U - FM

Ni1: 1.73 µB

Ni2: 1.73 µB

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:

H = H0 + Jij!Si.

i< j∑

!Sj

Jij: spin exchange parameter between the

spin sites i and j

Jij > 0 ⇒ AFM Jij < 0 ⇒ FM

Long-range order


H = H0 + Jij!Si.

i< j∑

!Sj


spin sites i and j


Long-range order

Eα = α H α = E0 +S2 Jiji< j∑ σ iσ j

S: Spin hold by the magnetic center

σi = ±1 (up or down spin)



H = H0 + Jij!Si.

i< j∑

!Sj


spin sites i and j


Long-range order


S: Spin hold by the magnetic center

σi = ±1 (up or down spin)

Example of a spin-half dimer (S = ½) To estimate the J12 value, 2 total energy calculations are needed:

EFM = E0 +14J12 EAFM = E0 +−

14J12

J12 = 2 EFM −EAFM( )σ1 = +1 σ2 = +1 σ1 = +1 σ2 = -1


Ni2+ -> S = 1


2 inequivalent Ni sites in the rhombohedral unit cell (S.G. R-3m)

Estimation of J in NiO

J: magnetic coupling defined by Ni1-O-Ni2 path (angle : 180°)

6J / unit cell

Ni2+ -> S = 1





6J / unit cell

EFM = E0 + 6JEAFM = E0 − 6J-19.54909823 eV -19.30675287 eV

Ni2+ -> S = 1





6J / unit cell

EFM = E0 + 6JEAFM = E0 − 6J

J = (EFM −EAFM ) /12 = 20.2 meV-19.54909823 eV -19.30675287 eV

Exp.: J = 19.01 meV (Hutchings M. T., Samuelsen E. J., Phys. Rev. B 6, 9, 1972, 3447)

Collinear magnetism in VASP

Spin-polarized calculation: ISPIN = 2 Initial magnetic moment: MAGMOM = 2.0 2.0 2*0

INCAR file

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 Ueff values is

meaningless!

VASP can also treat non-collinear magnetic systems!

Noncollinear magnetism in VASP

Replace ISPIN = 2 and MAGMOM = 1.0 by:

INCAR file Illustration with fcc Ni

leads to

or with MAGMOM = 1.0 0.0 0.0

or with MAGMOM = 0.0 1.0 0.0

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]

[1] X. Rocquefelte, P. Blaha, K. Schwarz, S. Kumar, J. van den Brink, Nature Comm. 4, 2511 (2013)

Estimation of the magnetic anisotropy

MAE (μeV)

Magnetization axis

Hard axis

Easy axis

Hard axis

E[uvw] is the energy deduced from spin-orbit calculations with the magnetization along the [uvw]

crystallographic direction

NEED TO SWITCH ON THE SPIN-ORBIT: LSORBIT = .TRUE

Estimation of the Magneto-crystalline Anisotropy Energy (MAE) of CuO

E[uvw] is the energy deduced from spin-orbit calculations with the magnetization along the [uvw]

crystallographic direction

[101]

[-101]

[10-1]

[0-10][-10-1][010]

[1] X. Rocquefelte, P. Blaha, K. Schwarz, S. Kumar, J. van den Brink, Nature Comm. 4, 2511 (2013)


Allows to define the magnetization

easy and hard axes

Here we have considered the following expression:

MAE = E[u v w] – E[easy axis]


LiNbO3-type InFeO3: 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…

If convergence is bad?

Let’s now play with VASP

Magnetism: Spin-orbit coupling magnetic exchange and ... · Magnetism: Spin-orbit coupling magnetic...

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