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MAGNETISM IN DFT

FROM THEORY TO PRACTICE

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(Copenhagen, Denmark)

MASTANI, Pune, INDIA, July 2014

Cyrille Barreteau

CEA Saclay

2 JUILLET 2014

Saclay

Lyngby

MAGNETISM

2 JUILLET 2014 | PAGE 2

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From the atomic nucleus to the inner core of Earth

time

length

NMR

GMR Read Head

Magnetic rotor

MAGNETISM

| PAGE 3




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Intense activity in which nanomagnetism plays a crucial role

Search for permanent hard magnets without rare earth

Replace Samarium-Cobalt, Neodyum by nanostructured « cheap » TM magnets

Spintronics

micromagnetics

From Fundamentals to applications

DFT is the perfect tool to adress (at least partly) most of these problems

DENSITY FUNCTIONAL THEORY





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Hohenberg & Kohn (1964) + Kohn Sham (1965)

3 3 3

0

1

2

( ) ( ')( ) ( ) '

'ext I I xc

n nE n T n V n d r d rd r E E n

r r

r rr r

22

eff ( ) ( ) ( )2

HKS

V rm

r r

ext Hartree( ) ( ) ( )xcV V V r r r

3

Hartree

( ')( ) '

'

nV d r

r

rr r xc ( )

( )

xcEV

n

r

r

2 2

,

occ occ

( ) ( ) 2 ( )n

r r r

if

ext ( )V r(pseudo)-potental describing the interaction of valence electrons

with the ions (nucleus+core electrons)

Can also include a « true » external potential

2 JUILLET 2014 | PAGE 5 MASTANI, Pune, INDIA, July 2014

3( ) ( ( ))xc xcE n n n d r r r

Local density approximation (LDA)

( ) : exchange correlation energy (per particle) of homogenous gasxc n

xc

( )

( ) (n ( ))xc

n n

dV n

dn

r

r

( ) ( ) ( )xc x cn n n

133 3

( )4

x

nn

Hartree Fock in an homogenous jellium

( ) ( )c n F n Parametrized from QMC


3( ) ( ( ), n(r))xc xcE n n n d r r r

Generalized Gradient approximation (GGA)

hom 3( ) (n( )) F ( ( ), n(r))xc xc xcE n n n d r r r r

hom ( ) : exchange correlation energy (per particle) of homogenous gasxc n

F ( ( ), n(r)) : dimensionless enhancement factorxc n r

Some important sum rules and other relevant conditions should be verified…

But still a large variety of functionals

Most popular

Perdew and Wang (PW91)

…………

Perdew Burke and Enzerhof (PBE)

| PAGE 7 MASTANI, Pune, INDIA, July 2014

DFT implementation diagram

( )n rInitial guess Combination of atomic densities

Effective potential


Solve Poisson equation

but not in Harris Functional scheme

Kohn Sham algorithm

22

eff ( ) ( ) ( )2

Vm

r r r

Solve KS equation

electron density and total energy 2

occ

( ) 2 ( )n

r r ( )totE n r

Converged? Analayse results

Determine the Fermi level

charge mixing

band dcT E E

2 JUILLET 2014


Spin polarization

orbital moment operator

s

s s B B

g 2

g

S

σ

x

0 1

1 0

y

0 i

i 0

z

1 0

0 1

(collinear case)

z σ

spin moment

spin

,

occ

zs zz B

in collinear case

0 or

0

2 2spin

occ occ

( ) ( ) ( )zm n n

r r r

Orbital polarization

L B B

Ll

Average orbital moment: usually small (quenched) in bulk

and strictly null if spin-orbit coupling (SOC) is ignored.

orb

occ

( )

m r L

Spin moment operator

orbital moment


Local Spin density approximation (LSDA)

( , n ) ( , n ) ( , n )xc x cn n n

( , ) ( ,0) ( ,1) ( ,0) ( )x x x x xn n n n f

n n n m

n

4 43 3

13

1 if 11 (1 ) (1 ) 2( )

0 if 02 2 1xf

( , ) ( ,0) ( ,1) ( ,0) ( )c c c c cn n n n f ( ) ( ) (Perdew Zunger)c xf f

eff ext Hartree( ) ( ) ( ( ), ( )) ( )xc xcV V V V n m B r r r r r

Alternative formulation

Spin dependent potential

( (r),m(r))( ) ( )

( )

xcxc

nB r n r

m r

m n n

( ( ),m(r))( ) ( , n ) ( )

( )

xcxc xc

n rV r n n r

n r

( ) exchange correlation magnetic fieldxcB r

( ) external magnetic field can be addedextB r


Spin polarized DFT implementation diagram

12

( ) ( ( ) ( ))n r n r m r

Initial guess

Combination of atomic densities

+ initial moment (symmetry breaking)

Spin dependent potential


eff

22 ( ) ( ) ( )

2V r r r

m

Solve the two KS equation

electron and spin density, total energy

2

( ) ( )occ

n

r r ( ),m( )totE n r r

Converged? Analayse results

1

1

( ) ( ) ( )n n n r r r

m( ) ( ) ( )n n r r r

Determine the common Fermi level

F F FE E E

charge mixing

1

2 JUILLET 2014

ANALYSE RESULTS





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What do we get out of spin-polarized DFT calculation

Total energy → find most stable structure! (not always easy..)

Total (and absolute) spin magnetic moment

tot eq eq,E n m

3( ) n ( )M n d r r r3

abs ( ) n ( )M n d r r r

Pw(scf)

Pw(scf)

Pw(nscf) Spin polarized band structure (for up and down spins)

Pw(scf) +projwf.x +pp;x

Local analysis (many options…)

DOS

2

( , ) ( )n E r r E

2

at

,( )i in E E

2 2at at

, ,

occ

i i im

Spin density 2 2

occ

( ) ( ) ( )m r

r r

Local moment

Atomic moment

PHYSICAL INSIGHT





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DFT is a very powerful tool but there is still a need for simpler phenomenological

models and local analysis to get a deeper physical understanding of the phenomena

Stoner model

Heisenberg/Ising model

Local analysis im


LSDA and the Stoner model

" 2 212

( , ) ( ,0) ( ,0) ( )xc xc xcn n n o

0

eff eff IM2

V V

: magnetic moment per atomM

Rigid shift of up and down eigenvalues

0

2IM

FE

E

0( ) ( )2

n E n E IM

0 01 12 2

(n ( ) ( ))dEFE

N E IM n E IM

0 01 12 2

(n ( ) ( ))dEFE

M E IM n E IM

( )F M M

( )FE M

0( ) ( )2

n E n E IM

3( )M m d r

r"

3( ( ),0)10

( )

xc nI d r

n

r

r

2 JUILLET 2014

"1( ) ( ( ),0) ( )

( )xc xcB n m

n r r r

r


0M

maxM

maxM( )F M

0 ( ) 1FIn E

Non zero solution if '(0) 1F

1M

M ( )F M

1I eV In most elements 0 ( )Fn E plays a crucial role in magnetic

susceptibility and onset of magnetism

Stoner criterion

Criterion can be also derived by analyzing

• magnetic susceptibility 0

0(1 ( ))F

M

H In E

• Total energy 21

2tot i

occ i

E Im

Nowadays Stoner model often used in parametrized Tight-Binding models

0

2ij ij i ijH H IM

2 JUILLET 2014


n( )E

E

W Z

: number of neigborsZ

• Low coordination favors magnetism

n( )FZ E

Spin magnetic moment is generally enhanced on

low coordinated atoms

General trends

2 JUILLET 2014


General trends

• Lattice expansion generally favors magnetism and vice versa…

n( )E

E 0( / ) 1q R RW e

W n( )FR E

( )M d

Fe bcc

Cr bcc

Pd fcc Pd fcc

Rh6

E( )d

2 JUILLET 2014


Multiple magnetic solutions

Stoner criterium does not tell the whole story

( )in outF M M

The iterative scheme converges towards a solution which depends on initial spin moment

How to be sure not to miss the most stable solution?

• Several starting magnetization

(fishing strategy)

• Fix spin moment (FSM) calculation

allow to explore E(V,M) energy surface

?in outM M

stable

unstable stable


Fixed Spin moment

Conventional scheme FSM

iniM MF FE E N N N F FE E N N N M N N

minE

E

M

E

MiniMM

minE

E

M

E

M

M iniM

0FE

one scf loop many calculations

2 ( )FE H M

0FE


Penalization technique

Add a penalization term to the total energy functional to impose a given condition

2 3

0, , (m( ) m ( ))totE n m E n m d r

r r

Minimization → modified KS Hamiltonian

02 ( ( ) ( )) .H H m r m r

• TB

2

0i tot i iE c E c m m

02 ( m ).ij ij i ijH H m

• DFT

at

i i

i

c

SOME IMPORTANT TECHNICAL POINTS





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Before running (or trusting) a DFT calculation you should be aware of

several important technicalities

Pseudopotential (LDA pz /GGA pw91, pbe)

K-points sampling: denser mesh for metals

note that in QE the #k points is doubled in spin-polarized system

Smearing (smearing, degausss) Marzari Vanderbilt recommended

Energy cut-off (ecutwfc, ecutrho≥8ecutwfc for US pseudo)

Initial magnetization (starting_magnetization)

Number of computed eigenvalues (nbnd) often needs to be increased

Convergence threshold (conv_thr)


E(V)

hcp NM

bcc FM

fcc NM

bcc FM

pseudopotential

As a rule of thumb: aLDA <aexp<aGGA

LDA bad for 3d but OK for 5d→ problematic for alloys (FePt, CoPt)

| PAGE 22


Initial magnetization Cr bcc AF

Define the system as simple cubic with 2 types of atoms

per unit cell and opposite starting magnetization

ibrav=1

ntyp=2

nat=2 starting_magnetization(1)= 0.5

starting_magnetization(2)=-0.5

0

0

1 220 (1 )

20L a q

a

Cr SDW 0 cos( . )m m q R0

20.953 0.95 in unit of q

a

2 JUILLET 2014

NON COLLINEAR MAGNETISM





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Why should we care about non collinearity?

Spin excitation

Non-collinearity does exist!

The future is in skyrmions

Mn


2 2

22

2 2

2 2

2 2

cos( ) sin( )( , )

sin( ) cos( )

yz

i i

ii

i i

e eU e e

e e

( , , ) global axisx y z

Spin ½ rotation matrix

sin cos

sin sin

cos

m

m

( ", ", ") local axisx y z

" "

"

Spin gymnastics

2 JUILLET 2014 "i ij jRσ σ

, , also transforms like a space vectorx y z σ

†

" ( , ) ( , )i iU U σ σ

0

" 0

1

m

m


2x2 matrices algebra

†0

0

aU U

a

bM

b

2 2

2 2

2 2

2 2

cos( ) sin( )

sin( ) cos( )

i i

i i

e eU

e e

2 ( )MM

.a M I b σ 12

( )a Tr M 12

( . )Trb M σ

2 eigenvalues a b

If is hermitian then and are real numbersaM b

b b n

sin cos

sin sin

cos

n

diagonalization

2 JUILLET 2014


From electron density/magnetization vector to the Density matrix

' '( ) ( ) ( )f

r r r

,

( ) ( ) ( )n f

r r r† '

'

, , '

( ) ( ) ( ) ( ) ( )f f

m r Ψ r σΨ r r σ r

1 1

( ) ( ) I . ( )2 2

z x y

x y z

n m m imn

m im n m

ρ r r σm r

Ψ

( )n Tr ρ ( )Trm ρσ

,n m

( )

ρ r

Density matrix

n,m →

→ n,m

2 JUILLET 2014


The non-collinear Stoner Hamiltonian

"

Stoner "

1 0

0 12 2z

I IH m m

" †

Stoner

cos sin

2 sin cos

i

Stoner i

eIH UH U m

e

Local spin axis

Global spin axis

Stoner

0 1 0 1 0sin cos sin sin cos

1 0 0 0 12

iIH m

i

Stoner .2

IH m σ

A lot of fuss for a straightforward result!

2 JUILLET 2014


Kohn Sham Hamiltonian

eff

2' 2 ' ( )

2KSH V r

m

'

' 3 '

eff ext '

( )

( ')( ) ( ) ' ( )

'xc

xc

E

nV V d r V r

ρ r

rr r ρ

r r

In LDA (n) (n)xc f

In GGA

Additional complication since

(n) (n, )xc f n

andρ ρ cannot be diagonalized in the same basis

off diagonal terms of are neglected

' 3

eff ext '

( )

( )

( )( ')( ) ( ) ' .

' ( )xc

xc

xc

E

r

EnV V d r

n r

ρ r

m

ρ rrr r σ B

r r

2 JUILLET 2014

( ( ), ( ), ( )) at each position in spacem r r r r

rotate "back" with matrix to get ( ( ))xc U ρ r

(n ) 0diagonalization of diagonalization of ( )

0 (n )

xc

xc

xc

ρ ρ

PHYSICAL INSIGHT





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Classical Heisenberg Hamiltonian

,

1

2ij i j

i j

E J e e

ie

je

ijJijJ Can be obtained by various approaches from DFT

(not yet available in QE)

0ijJ

Fairly happy

0ijJ Favors AF order

Frustration

Favors FM order


Some examples

Cr/Cu(111) Fe/Cr(001) interface Fe/Cr(110) interface

Fe bibyramidal cluster

col ncolE Ecol ncolE E

Cr

Fe

2 JUILLET 2014

bccFe fccNi

SPIN-ORBIT COUPLING

| PAGE 31




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Why should we care about SOC?

Small quantity (at least in 3d) with huge physical consequences

• At the origin of magneto-crystalline anisotropy

• At the origin Anisotropic Magneto-Resistance

…..and therefore of the stability of magnets!

current

B

hcpCo

,E

2 JUILLET 2014

R


Relativistic effects

2( , ). ( , )

r ti c mc V r t

t

α p β

0

0

i

i

i

α

0

0

Iβ

I

Dirac equation

1

2

3

4

( , )

( , )( , )

( , )

( , )

r t

r tr t

r t

r t

( , )r t

( , )r t

Large component

Small component

Small v/c limit ( , ) ( , )v

r t r tc

Scalar relativistic

22 ( ) ( 2 ).

2

BH V r Bm

L S

4

3 28

p

m c

122 2

1 1. ( ) . ( ) .

2

dVr r

m c r dr L S l s l σ

22

2 28V

m c

Schrödinger + Zeeman

Mass-velocity

Spin-Orbit Coupling

Darwin

Contraction and stabilization

of s and p shells + expansion

and destabilization of d and f

Splitting of orbitals with

angular momentum.

2 JUILLET 2014


The good basis

•Without SOC L et S commute with H

•With SOC soL et S no longer commute with H H

J L S

Basis diagonalizing 2 2

z zL ,L ,S ,S l,m

2

zJ , J

22

z

L l,m l(l 1) l,m

L l,m m l,m

22

z

3S

4

S

we consider

22

j J

z j j

J j,m j(j 1) j,m

J j,m j m j,m

jj, m

j

2 j 1

l s j l s

m j, j 1, , j

2

22 2

j j j

1L.S j,m J L S j,m j(j 1) l(l 1) s(s 1) j,m

2 2

Basis diagonalizing

2 JUILLET 2014

•From one basis to the other = Clebsh Gordan


12

12

l (2l+2)j 2(2 l 1)

l (2l)

2 2 2

1 1j j j2 2

l.s j,m j,m j(j 1) l(l 1) ( 1) j,m2 2

j l s

2 JUILLET 2014

1 (p orbitals)l

3/22

pEp

1/2 pEp 2 (d orbitals)l

3/2

3

2dEd

5/2 dEd

p( 6)

( 4)

( 2)

d( 10)

( 6)

( 4)

1s

2


Relativistic pseudo-potentials

•Without SOC

•With SOC

pseudo loc , ,

,

( )l l

l

NL

I I I I I

l l l m l l m

I l m

V

V V r E Y Y

pseudo loc1

2

( ) j

NL

j l

V V r V

But including other (scalar) relativistic effects

For technical details please contact Andrea dal Corso….

Fe.pbe-nd-rrkjus.UPF

Pseudopotential type: ULTRASOFT

Method: Rappe Rabe Kaxiras Joannopoulos

Functional type: Perdew-Burke-Ernzerhof (PBE) exch-corr

Nonlinear core correction

scalar relativistic

Fe.rel-pbe-spn-rrkjus_psl.0.2.1.UPF

Pseudopotential type: ULTRASOFT

Method: Rappe Rabe Kaxiras Joannopoulos

Functional type: Perdew-Burke-Ernzerhof (PBE) exch-corr

Semi-core state in valence

Nonlinear core correction

full relativistic 2 JUILLET 2014

.SO

l l lm lm

lm

V V Y Y Id L S

SOME IMPORTANT TECHNICAL POINTS





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Relativistic Pseudopotential (LDA pz /GGA pw91, pbe)

Very dense K-points mesh

Check very carefully the influence of degauss

Use penalization when necessary (constrained_magnetization)

Very strict convergence threshold (conv_thr)

'

.

L S Is not diagonal in spins → impossible to split up and down spins

= drastic increase of the computational cost

0

0

collinear Non-collinear

,E

A FEW THINGS ABOUT SOC

2 JUILLET 2014

| PAGE 37




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2 JUILLET 2014

( )r Is short range (atomic-like)

4 ( ) Zr Increases drastically with atomic number

2

' '

0

( ) ( ) (r) rat at

ll l lR r R r dr

0.05 for 3d eV d

0.5 for 5d eV d

• The band structure depends on the magnetization axis

• SOC is at the origin of the magnetocrystalline anisotropy

,E

in TB (or LCAO) on site term .I II

L S

• SOC is at the origin of the anisotropic magneto_resistance

2 310 10 / atom in bulk Fe, Co, NiMCA meV

/ atom in bulk FePt L10MCA meV

R

Much larger in nanostructures

1% in bulkAMR

larger in atomic wires

• SOC is at the origin of the orbital moment

MAGNETOCRSYTALLINE ANISOTROPY





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How to calculate MCA?

•Brute force method (self-consistent) 1 2

tot totMCAE E E n n

where and are obtained from SCF calculation including SOCE E1 2n n

•Force Theorem method

In principle « exact » but very time consuming and hard to converge

One should use penalization techniques to obtain En for any direction

1 2

1 2

band band ( ) ( )F FE E

MCAE E E En E dE En E dE 1 2n n

band band and are band energies of NSCF calculations including SOCE E1 2n n

Initial density is obtained from a SCF spin-collinear calculation and spin-moment

further rotated to appropriate spin direction

Very stable numerically but cannot be applied to systems with too large SOC.


MCA: the local picture

•Force Theorem in grand canonical ensemble

1 2 0

1 2 0( ) ( ) ( ) ( )F F FE E E

FE En E dE En E dE E E n E dE

0

0( ) ( )FE

gc

i F iE E E n E dE

0

0( ) ( )FE

gc FE E E n E dE

•Local decomposition

Projection onto atomic orbitals

Real space picture 0

0( ) ( ) ( , )FE

gc

FE r E E n E r dE

The local decommposition in « canonical » picture leads to spurious oscillations

due to long range Friedel-like charge oscillations.. But the total MCA value is

kept due to charge conservation


MCA: the local picture

MCA of a slab orginates from surface atoms of the outermost layers

fccCo

2 JUILLET 2014

Fe pyramid

i

i

MCA MCA

PHYSICAL INSIGHT

| PAGE 41




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Extended Heisenberg Hamiltonian

,

1( ) .

2ij i j i i ij i j

i j i ij

E J F e e e D e e

2( ) ( . )i iF K e n e Uniaxial anisotropy

Dzyaloshinskii-Moriya interaction

MCA ( ) t

i i i iiF Ke e e

DM

(only exists in the absence of inversion symmetry)

DM=0 DM=0 DM≠0

only between first neighboursD

DM favors non-collinear configurations

At the origin of skyrmion structures

SHAPE ANISOTROPY





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•Classical dipole-dipole interaction

01 2 1 2

3 2

3( ) . . .

4dipE r m m m r m r

r r

Colllinear magnetism

01 23

( ) 1 3cos4

dipE m mr

For <54.74° FM For >54.74° AF

In thin films: in-plane magnetization is always favored

shape VolumeE

MCA SurfaceE For ultra-thin films MCA is generally dominant

while the shape anisotropy always ends up by

dominating for thicker films

1m

2m

r

•Breit interaction

(2 body relativistic term)





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IMPORTANT THINGS I DID NOT TALK ABOUT

•Orbital moment

orb

occ

( )

m r L orb

occ

( )i i

m r L

For modern theory of orbital magnetization see David Vanderbilt

•DFT+U and orbital polarization

When using the rotationally invariant scheme of Liechtenstein (lda_plus_u_kind=1) the

Racah B parameter plays a crucial role in orbital magnetization

•Spin-spirals and generalized Bloch theorem

•Mapping DFT on model Hamiltonian Jii

•Spin dynamics etc….

•Spin-polarized transport





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ACKNOWLEDGEMENTS

Alexander SMOGUNOV

Sylvain LATIL

Yannick DAPPE

Dongzhe LI (PhD)

TEAM

Chu Chu FU

Romain SOULAIROL (PhD)

Pascal THIBAUDEAU

David BEAUJOUAN (PhD)

COLLEAGUES COLLEAGUES (Julich)

Stefan BLUGEL

Gustav BIHLMAYER

Timo SCHENA (Master)





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BIBLIOGRAPHY

Electronic Structure

Basic Theory and Practical Methods

Richard Martin

Cambridge University Press

Theory of itinerant electron magnetism

Jürgen Kübler

Oxford Science Publications

Nouvelles méthodes pour le calcul ab-initio des propriétés

statiques et dynamiques des matériaux magnétiques.

Ralph Gebauer PhD thesis 1999

(in english)

2 JUILLET 2014

| PAGE 46


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