Magnetism in the Density Functional Theory Mebarek Alouaniqmmrc.net/winter-school-2009/Alouani.pdf · Magnetism in the Density Functional Theory IPCMS, 23 rue du Loess, UMR 7504,

QMMRC Winter School, 9-13 February, 2009, Ewha Woman’s University, Seoul, S. Korea 1/107

Magnetism in the Density Functional Theory

IPCMS, 23 rue du Loess, UMR 7504, Strasbourg, France

Mebarek Alouani


Course content

I. Magnetic moment and magnetic field

II. No magnetism in classical mechanics

III. Where does magnetism comes from?

IV. Crystal field, superexchange, double exchange

V. Free electron model: Spontaneous magnetization

VI. The local spin density approximation of the DFT

VI. Beyond the DFT: LDA+U

VII. Spin-orbit effects: Magnetic anisotropy, XMCD

VIII. Bibliography


Course content









VIII. Bibliography


Magnetic momentsIn classical electrodynamics, the vector potential, in the magnetic dipole approximation, is given by:

0

2

ˆ( )

4

r

r

µπ

×=

µA r

where the magnetic moment µ is defined as:

1( ' )

2I d= ×∫ r l�µµµµ

It is shown that the magnetic moment isproportional to the angular momentum

γ= Lµµµµ (γ the gyromagnetic factor)

The projection along the quantification axis z gives

z Bg mµ µ= − ( / 2 Bohr magneton)B eeh mµ =


Magnetization and Field

The magnetization M is the magnetic moment per unit volume

In free space B = µ0 H because there is no net magnetization.

Inside a magnet, the relation is more complicated B = µ0 (H +M).

For linear materials, the magnetic susceptibility χ is defined as M= χ H.In this case there is still a linear relationship between B and H (B =µ0 (1 + χ )H=µ0 µr H).

relative permeability

Example: For a cylinder Bi = Ba+ µ0 M. so that H =Ha =Ba/ µ0In this case it is the free current (coil) that controls the magnetization and hence controls the magnetic field inside thecylinder.

In other more complex geometry, one has to subtract out the demagnetizing field to find H.


Course content









VIII. Bibliography


The magnetization is zero in classical mechanicsThis is because the application of a magnetic field amounts in changing the kinetic energy to : ( )2

2K

eW

m

−=

p A

3 3 3 3

1 1 1 1exp( ( ,..., , ,..., )) ... ...N N N N

Z E p p r r d p d p d r d rβ−∫∼Since the integral goes all over the phase space, the effect of a magnetic field is just to shift the momentum zero. The partition function is therefore independent of the magnetic field (Bohr-van Leeuwen theorem)

, ,

ln0B

T V T V

F ZM Nk T

B B

∂ ∂ = − = = ∂ ∂

This can be easily seen in a finite magnetic sample where the orbital current thatscatter at the surface cancels out the orbital volume current.


Course content









VIII. Bibliography


Two particle system

A single particle wave function: A Two-particle wave function:

The time evolution is determined by the non relativistic Schrödinger equation (SE):

1 2( , , )r r tΨ( , )r tΨ

i Ht

∂Ψ = Ψ

∂�

The Hamiltonian is given by

2 2

1 2 1 2( , , )2 2

H V r r tm m

= − ∆ − ∆ +� �

23 3

1 2( , , ) ' 1r r t d rd rΨ =∫∫For time-independent potential the wave function is given by

1 2 1 2( , , ) ( , )Et

i

r r t r r eψ−

Ψ = �

where E is the total energy of the system given by

1 2 1 2( , ) ( , )H r r E r rψ ψ=


BOSONS AND FERMIONS

Symmetrization of the wave functionDistinguishable particles: particle 1 in ψ1 and particle 2 in ψ2. The state of the 2 particles is:

1 2 1 1 2 2( , ) ( ) ( )r r r rψ ψ ψ=

Indistinguishable particles: In QM all particles are identical. We construct a wave functionthat is noncommittal as to which particle is which state:

[ ]1 2 1 1 2 2 2 1 1 2( , ) ( ) ( ) ( ) ( )r r C r r r rψ ψ ψ ψ ψ± = ±

QM accommodates 2 kinds of identical particles, for bosons we use + sign, for fermions the minus sign.

Spin statistics axiom: particles with integer spin are bosons, and particles with half-integer spin are fermions (W. Pauli, PR 1940).

Note: This spin statistics axiom can be proved in relativistic QM. Pauli exclusion principle (PEP) is a consequence of this axiom. PEP: Two identical fermions can not occupy the same state.


The exchange operator P

The exchange operator P interchanges the two particles:1 2 2 1( , ) ( , )P r r r rψ ψ=

It is then obvious that P2 =1 and P and H commute (compatible observables).

We can find a complete set of functions that are simultaneous eigenstatesof H and P. The eigenvalues of P are +1 and -1.

1 2 2 1( , ) ( , )r r r rψ ψ= ± + for bosons and – for fermions.

If the system starts in a given symmetrized state, it remains in such state during its evolution.

The wave function of a QM system is required to be symmetric or antisymmetric.


Exchange forceThe expectation value of the square of the separation value of two bosons is reduced compared to that of distinguishable particles, and that of two identicalfermions is increased. It is like there is an exchange force between identical particles.Proof (one dimension case)

Distinguishable particles: 1 2 1 1 2 2( , ) ( ) ( )D

x x x xψ ψ ψ=

22 2

1 1 1 2 1 2( , )Dx x x x dx dxψ= ∫∫ 2 22

1 1 1 1 2 2 2( ) ( )x x dx x dxψ ψ= ∫ ∫ 2

1x=

( )2 2 2

1 2 1 2 1 22x x x x x x− = + −and

2 2

2 2x x=

1 2 1 2x x x x=Similarly, and

Identical particles: [ ]1 2 1 1 2 2 2 1 1 2

1( , ) ( ) ( ) ( ) ( )

2r r r r r rψ ψ ψ ψ ψ± = ±

( )2 2 2

1 2 1 21 22x x x x x x− = + −so that

2 2 2

1 1 2

1,

2x x x = +

2 2 2

2 2 1

1,

2x x x = + 1 2 1 2 1,2

x x x x x= ±and

where *

1 21,2( ) ( )x x x x dxψ ψ= ∫

( )2 2 2

1 2 1 2 1,21 22x x x x x x x

±− = + − ∓ ( )2

1 2 1,2D

x x x= − ∓


Exchange force (continued)

++ e-e-

+ +e- e-

Symmetric configurationproduces attractive exchange force

Anitisymmetric configuration produces at repulsive exchange force

The exchange force it is a purely geometrical consequence of the symmetrizationrequirement. It is not a real force because no energy is spent in moving the particles.

Because electrons are fermions, in molecules it is the antisymmetric configuration that should happen. This is true but one has to take into account the spin of the electrons.

1 2, 1 2 1 2 1 2( , ) ( , ) ( , )s s r r r r s sψ ψ χ=

The spatial part can be symmetric if the spin part is antisymmetric.

Note: In molecules, the bonding state is a singlet (antisymmetic spin state) and the anti-bonding a triplet state (antisymmetric spatial wave function, but symmetric spin function).


Molecules: The hydrogen molecule

It is difficult to get the correct GS of H2 because of the e-e interaction. Instead, a simple molecular orbital approach captures most of the physics.

Let ψ denote the state of an electron in the molecule, it can be written as a linear combination of the state of the isolated atoms:

1 2| |1 | 2c cψ ⟩ = ⟩ + ⟩ and assume that,|

i ji j δ⟨ ⟩ =

The SE for the molecule is : | |H Eψ ψ⟩ = ⟩

for simplification.

0 1 12 2 1

21 1 0 2 2

E c H c Ec

H c E c Ec

+ =

+ =

To solve it we project it into the atomic basis and obtain:

with E0 = H11 = H22different from the atomic s level.

The fact that the Hamiltonian is hermitian and the s orbital is real implies that H12=H21=β.The solutions are: Eb = E0 + β et Ea = E0 - β and the eigenvectors are:

( ) ( )1 1| |1 | 2 | |1 | 2

2 2b a

andψ ψ⟩ = ⟩+ ⟩ ⟩ = ⟩− ⟩

β < 0 the molecular state |ψb> has the lowest energy Eb.The spin state is therefore antisymmetric and correspond to a singlet state.


Representation of bonding and antibonding states

a. Bonding ψbρb

b. Antibonding ψa

wave function

ρa

Charge density

2

1 2( ) 2 | ( ) | ( ) ( ) ( )b b bond

r r r r rρ ψ ρ ρ ρ= = + + where 1 2( ) 2 ( ) ( )bond r r rρ ψ ψ=

The chemical bond in molecules is a direct consequence of QM interferencebetween the wave functions of the constituent atoms. There is no analogue inclassical physics.


Importance of the off-diagonal Hamiltonian matrix elements H12 and H21

We start from the time-dependent SE for the state Ψ:

( , ) ( , )ih r t H r tt

∂Ψ = Ψ

∂

10 1 2

21 0 2

dcih E c c

dt

dcih c E c

dt

β

β

= + = +i t

i ic Ae

ω−=If we set and insert them into these coupled equations, we obtain:

0 0andb a

E E

h h

β βω ω

+ −= =

1

2

( )

( )

b a

b a

i t i th h

i t i th h

c t ae be

c t ae be

ω ω

ω ω

− −

− −

= +

= −

and the coefficients c1 et c2 given by

where a and b are arbitrary constants to be determined by the initial conditions.

If we know that at t=0s, the molecule is in the state |1> so that c1(0)=1 and c2(0)=0, then a=b=1/2. The coefficients c1 and c2 are then given by:

0 0

1 2( ) cos and ( ) sin

E Ei t i t

h hc t e t c t e th h

β β− − = =

The amplitudes oscillate harmonically with time.


Probabilities |c1(t)|2 et |c2(t)|2

0

1

2 22 2

1 2( ) cos et ( ) sinc t t c t th h

β β = =

4

h

β 2

h

β

3

4

h

βThe probability starts at 1 and then decreases to 0 in time h/4β, and returns toone in h/2β. At the same time the probability that the molecule is in state |2>

is exactly out of phase.

The frequency of the molecule passing from state |1> to state |2> and back again to state |1> is 2β/h. An electron in this molecular orbital is vibrating between the two atoms.

The electron tunnel from atom 1 to atom 2 back and forth despitethe energy barrier of about 13.6 eV corresponding to the ionization energy.

The probability per unit time that the electron tunnels or “hops” from one atom tothe other is 2β/h. For this reason the off-diagonal elements of the Hamiltonian

are called hopping integrals.


Heteronuclear diatomic moleculeThe problem is similar to that of the H2 molecule except that the H11 is now different from the H22. Let us call them E1 and E2, respectively.

1 1 2 1

1 2 2 2

E c c Ec

c E c Ec

β

β

+ =

+ =

The SE of the molecule’s gives:

The secular determinant leads to a bonding state (BS) and antibonding state (ABS)eigenvalues:

2 21 2

2 21 2

( )

2

( )

2

b

a

E EE

E EE

β

β

+ = − ∆ +

+ = + ∆ +

where 1 2

2

E E−∆ =

Notice that the difference in the on-site energies E1 et E2 is to increase thesplitting between the bonding and antibonding states.

In the homomuclear case the charge density associated with each atom in both the BS and ABS are equal.

For the heteronuclear molecule, the fact that (for example E1 > E2) results in it being energetically favorable in the BS for some charge density to be transferred from atom 1 to atom 2 and is in reverse direction for the ABS.


Indeed if we insert the eigenvalues of BS or the ABS in the secular equation we find:

2

1

2 2 22

1,

1 2 2 1

c

c ε ε ε=

+ ± +respectively. Here є=∆/β.

ABS

BS

1

0 єAs є increases all the charge density is transferred to atom 2 for the BS and allcharge is transferred to atom 1 for the ABS.

Conclusion, when ∆ is not zero the bond becomes partially ionic because some charge transfer takes place in the molecule. These simple ideas hold also for alloys.

Heteronuclear diatomic molecule


[ ]

[ ]

1 2 1 1 2 2 2 1 1 2

1 2 1 1 2 2 2 1 1 2

1( , ) ( ) ( ) ( ) ( )

2

1( , ) ( ) ( ) ( ) ( )

2

S S

T T

r r r r r r

r r r r r r

ψ ψ ψ ψ ψ χ

ψ ψ ψ ψ ψ χ

= +

= −

The wave functions of the singlet and triplet states are:

* * 3 3

1 1 2 2 1 2 2 1 1 2ˆ2 ( ) ( ) ( ) ( )

S TE E r r H r r d rd rψ ψ ψ ψ− = ∫

1 2

1 ˆ ˆˆ ( 3 ) ( )4

S T S TH E E E E S S= + − − ⋅1 2

1

4ˆ ˆ3

4

S S

⋅ = −

triplet state

singlet state

Defining J =ES-ET, the spin dependent term of the effective Hamiltonian is:

1 2ˆ ˆH JS S= − ⋅ J is the exchange parameter

Heisenberg Hamiltonian

The effective spin Hamiltonian is therefore given by:

this is because

The generalization to N spins leads to the Heisenberg Hamiltonianˆ ˆˆ

ij i j

i j

H J S S>

= − ⋅∑Notice if J>0 Es>ET and the triplet state is favored

if J<0 Es<ET and the singlet state is favored

(ferromagnetic coupling)

(antiferromagnetic coupling)



Hund’s rules

Hund’s first rule: The state with the highest total spin has the lowest energy, andfor the largest multiplicity of S, it has the lowest energy for the largest L value.

Hund’s second rule: If a subshell (n,l) is no more than half filled, then the lowest energy level has the total angular momentum J=|L-S|; if it is more than half filled, then J=|L+S| has the lowest energy.

The ground state of an atom is then denoted: 2S+1LJ

S is the total spin momentum, J is the total angular momentum and L is the orbitalangular momentum written S, P, D, F, G, H, I, K, L, M, N, etc.Examples:

He 1s2 1S0O 1s2 2s2 2p4 3P2Si (Ne)3s2 3p2 3P0Fe (Ar)4s23d6 5D4Co (Ar)4s23d7 4F9/2Ni (Ar)4s23d8 3F4Cu (Ar)4s13d10 1S0

maximize ML


Magnetic ground states of 4f and 3d ions using Hund’s rules

9.59.594I15/215/263/24f11Er3+

9.779.727F66334f8Tb3+

7.987.948S7/27/207/24f7Gd3+

2.512.542F5/25/23½4f1Ce3+

Expt.µeff/µBtermJLSshellion

( 1)eff B J

g J Jµ µ= +

gJ is the Landé factor

0

3

eff

B

n

k T

µ µχ = (Brillouin)

1.831.733.552D5/25/221/23d9Cu2+

5.825.925.926S5/25/205/23d5Fe3+

5.364.906.705D44223d6Fe2+

1.701.731.552D3/23/221/23d1Ti3+

Exptµ’eff/µBµeff/µBtermJLSshellion

2 ( 1)eff B

S Sµ µ′ = +

3d orbital quenching

because the crystal field splitting is muchlarger than the SOC.


Course content









VIII. Bibliography


Crystal field

spherical negative field

The d orbitals form two types of irreducible representations (eg and t2g) in cubic Oh symmetry.


Octahedral crystal fieldIn octahedral field the degenerecy of the d-orbitals is lifted.

The d-orbitals interact differently with the point charges located at +x, -x, +y,-y, +z and -z

The eg orbitals will lie along these axes are affected more with the ionic electrostatic interaction and move higher in energy.

∆0

∆0 depends on:1. Nature of the ligands2. The charge on the metal ion3. The nature of the orbital (3d, 4d, 5d)

If ∆0 < 3 eV: Weak crystal field and if ∆0 > 3 eV Strong crystal field


Tetrahedral crystal field

0

4

9t∆ = ∆

There are only 4 ligands in the tetrahedral complex and therefore the electrostatic potential is reduced by roughly 2/3 from its octahedron complexvalue. The tetrahedron crystal field is therefore reduced by (2/3)2:

This make all octahedron complexes high spin due to the reduced crystalfield splitting compared to the pairing energy (U).


Pairing energy (U) versus eg-t2g splitting

The colors of metallic ions is a consequence of the eg and t2g energy splitting

U>∆U<∆

1st Hund’s rule


z elongation2 long and 4 shortbonds

z compression 2 short and 4 long bonds

Jahn-Teller distorsionJahn-Teller theorem, 1937:

xxxx2222----yyyy2222

Any non-linear molecular system in a degenerate electronic state will be unstable and will undergodistortion to form a system of lower energy thereby removing the degeneracy.

The opposite happen for (dZ2)0 (dx2-y2)1 configuration

octahedron compressed along the z direction.

For (dZ2)1 (dx2-y2)0: ligands along z, -z will be repelled more and bonds elongated

The octahedron will be elongated along the z direction.


Examples of Jahn-Teller distortions


Magnetic orderCrNi CrMn

orthogonality overlap

AntiFerro

↑ ↑↓ ↓

= ↑↓ ↑↓ ↑↓ ↑↓

Superexchange mechanism

{Ferro = ↑ ↑↓ ↑

The superexchange reduces the kinetic energy of electrons and stabilizesthe antiferromagnetism

M. Verdaguer, Paris VI


Magnetic order in NiO

The AF2 structure is energetically the most stable because of the superexchange mediated by the oxygen atoms.


Double exchnage

In some mixed valence oxydes (example: Mn3+ and Mn4+ in La1-xSrxMnO3)

eg

t2g

eg

t2g

eg

t2g

eg

t2g

yes

No

Mn3+ Mn4+

The double exchange mechanism produces ferromagnetic coupling in La1-xSrxMnO3


Course content









XIII. Bibliography


∆E/2

Free-electron model: magnetism of metals (Stoner Creterion)

Paramagnetic DOS g(є) splitinto two subbands for the2 spin directions.

Spontaneous spin split DOS and appearance ofmagnetism.

Question: Can a material save energy by becoming ferromagnetic?The total kinetic energy cost due to spin split is: 21

( )( )2

K FW g E E∆ = ∆

This energy cost is compensated by an energy reduction of the magnetic momentwith the molecular field:

2 2 2 2

0 0 0

0

1 1 1( ') ' ( ) ( ( ) )

2 2 2

M

MF B FW M dM M n n U g E Eµ λ µ λ µ µ λ + −∆ = − = − = − − = − ∆∫

where 2

0 BU µ µ λ=21

( )( ) (1 ( )) 02

K MF F FW W W g E E Ug E∆ = ∆ + ∆ = ∆ − < ( ) 1F

Ug E >


Stoner Criterion (continued)

( ) 1FUg E <

( ) 1FUg E >

( ) 1F

Ug E =

E U∆ >

E U∆ <

Paramagnetic

Ferromagnetic

In the ferromagnetic ground state, the DOS at the Fermi level are split by the exchange splitting ∆E.If we add the effect of an applied magnetic field then we can determine the magnetic susceptibility

22

2

1( )( ) (1 ( )) (1 ( ))

2 2 ( )K MF F F F

B F

MW W W g E E Ug E MB Ug E MB

g Eµ∆ = ∆ + ∆ = ∆ − − = − −

The minimization of ∆W with respect to M leads to:

0

1 ( )

P

F

M M

H B Ug E

µ χχ = ≈ =

−where

2

0( )

P B Fg Eχ µ µ=(Stoner enhancement)


Course content









VIII. Bibliography


Nobel lectures by W. Kohn and J. A. Pople

Nobel prize in Chemistry 1998


PROPERTIES of materials

Goal: Describe properties of matter from ab initio methods.

magnetic

optical

electronic

structural

vibrational

Quantum transport


Main approximations:Born-Oppenhaimer

Decouple the movement of the electrons and the nuclei.

Hartree-Fock Approximation

Treatment of the electron ─ electron interactions.

Density Functional Theory

Treatment of the electron ─ electron interactions.

Pseudopotentials or all electron potentials

Treatment of the (nuclei + core) ─ valence.

Basis set

To expand the eigenstates of the hamiltonian.

Numerical evaluation of matrix elements

Efficient and self-consistent computations of H and S.

Supercells

Makes the calculation of materials possible (Bloch theorem)


Adiabatic or Born-Oppenheimer approximation decouple the electronic and nuclear degrees of freedom

(1)

Solve electronic equations assuming fixed positions for nuclei

(2)

Move the nuclei as classical particles in the potential generated by the e-

Electrons in their ground state for any instantaneous ionic configuration.

⇒ Nuclei much slower than the electrons

Born-Oppenheimer Approximation

Mp >> Me

ve >> vp


If the nuclear positions are fixed, the wave function can be decoupled

Fixed potential “external” to e-

Electrons

NucleiClassical displacement

Born-Oppenheimer Approximation



III.1 IntroductionThe HF method is essentially used by chemists because it can be improvedby adding the so called correlation energy. It results in very accurate totalenergy for small molecules.

The HF method uses wave functions called orbitals, obtained from the effective one electron Hamiltonian of atoms or Gaussian orbitals.

The HF method approximates the interacting electrons in terms of an effective single-particle problem, i.e., each electron moves in a mean-field potential created by the other electrons.

III.2 Atomic unitsWe use h=me=e2=1 so that the unit of distance is the Bohr radius, the massin unit of me, the electric charge in unit of e, the spin in unit of h, and the energy in Hartree unit.

2 4

21H 2Ry 27.2eVee m e

a h= = = �

210

21a.u.= 0.529 10 m 0.529

e

h

m e

−= × = Åand

The SE:2 2

0

1

2 4

eE

m rψ ψ

πε − ∆ − =

�becomes

1E

rψ ψ −∆ − =



III.3 Slater determinantThe wave function is a Slater determinant

( )1 11

1 2

1

( ) ( )1

, ,...,!

( ) ( )N N

v N

N

v N

x x

x x xN

x x

ν

ν

φ φ

ψ

φ φ

=

…

� � �

The exact Hamiltonian of the system is given by

( ) ,

1

1ˆ ˆ ˆ2

N N

i i i j

i i j

H k u v= ≠

= + +∑ ∑where ˆ ,i ik = −∆

1

ˆ (r )r R

N

i i

j i j

Zu

=

= −−

∑ and ,

1ˆ

r ri j

i j

v =−

We approximate this two particle Hamiltonian by a single particle one:

( )( ) ( )

1 1

ˆ ˆˆ ˆN N

i i

HF HF i HF

i i

H h k v= =

= = +∑ ∑ so that ˆHFH Eψ ψ= and ˆ

HF i i ih φ ε φ=


The Ritz variational theorem will alow us to vary the expectation value of the exact Hamiltonian in a single Slater determinant

( )*1

ˆ| | | 1 0( )

N

Hx

β β ββα

δψ ψ ε φ φ

δφ =

⟨ ⟩ − ⟨ ⟩ − =

∑Le Lagrange multipliers єβ insure that the are orthogonal.βφ

( )1 ,

1ˆˆ ˆ ˆ ˆ| | | | | | | |2

N N

HFE H k u v vα α α α α β α β α β β αα α β

ψ ψ φ φ φ φ φ φ φ φ φ φ=

= ⟨ ⟩ = ⟨ + ⟩ + ⟨ ⟩ − ⟨ ⟩∑ ∑


The variational theorem yields:

( )( ) ( ) ( , ) ( ) ( , ) ( , ) ( ) ( )u x dy y v x y x dy x y v x y y xα α α αρ φ ρ φ ε φ−∆ + + − =∫ ∫where the density and the density matrix are defined respectively by

*

1

( ) ( ) ( )N

y y yβ ββ

ρ φ φ=

=∑ *

1

( , ) ( ) ( )N

x y y xβ ββ

ρ φ φ=

=∑and


Hartree-Fock ApproximationWe define the HF potential as:

( ) ( ) ( ) ( ) ( ) ( , ) ( ) ( , ) ( , ) ( )HF

v x x u x x dy y v x y x dy x y v x y yα α α αφ φ ρ φ ρ φ= + −∫ ∫We solve the SE in the HF approximation (HFA):

( )( ) ( ) ( )HF

v x x xα α αφ ε φ−∆ + =The HF Hamiltonian is hermitian and therefore the Lagrange multipliers єα(as eigenvalues of this Hamiltonian) are real numbers, and the eigenvectorsare orthogonal.

,|α β α βφ φ δ⟨ ⟩ =

We select the lowest orbitals as solutions and restart again until selfconsistency is achieved:

Because the orbitals are not known, we start from the hydrogenoid orbitalsas starting point, and we calculate the HF potential from the density and thedensity matrix. We then solve the SE in the HFA.

αφ

( )2 23 ( ) ( 1)

1

( ) ( )N

n nd x x xα αα

φ φ ε−

=

− ≤∑∫where є is the precision of the calculation achieved at the nth iteration.



When the selfconsistency is achieved one calculates the total energy.

Since the eigenvalues are given by

( )ˆ ˆ ˆ ˆ| | | | | |N

k u v vα α α α α α β α β α β β αβ

ε φ φ φ φ φ φ φ φ φ φ= ⟨ + ⟩ + ⟨ ⟩ − ⟨ ⟩∑

( )1 ,

1ˆ ˆ| | | |

2

N N

HFE v vα α β α β α β β α

α α β

ε φ φ φ φ φ φ φ φ=

= − ⟨ ⟩ − ⟨ ⟩∑ ∑

We can rewrite the total energy as:

1 1

1 1 ˆ ˆ| |2 2

N N

k uα α α α αα α

ε φ φ= =

= + ⟨ + ⟩∑ ∑Example: HF equation for an atom of Z electrons:

3 3

'

(r') (r ,r' ')( ') (r, ) (r', ') ' (r, )

r r ' r r 's

Z s sd r s s d r s

rα α α α

ρ ρφ φ ε φ−∆ − + − =

− −∑∫ ∫



Koopman’s Theorem

The ionization energies Iλ of a weakly bounded state k is given by

( 1 ) ( )HF HFI E N E Nλ λ λε= − − = −

where EHF(N-1λ) is the HF total energy of the system without the λth electron.

( ),

ˆ ˆ ˆ ˆ( 1 ) | | | | | |N N

HFE N k u v vλ α α α α α β α β α β β αα λ α β

λ

φ φ φ φ φ φ φ φ φ φ≠

≠

− = ⟨ + ⟩ + ⟨ ⟩ − ⟨ ⟩∑ ∑

In fact:

( )

( )1

1

1ˆ ˆ ˆ ˆ( ) | | | | | |2

1ˆ ˆ| | | |

2

N

HF

N

E N k u v v

v v

λ λ λ λ α λ α λ α λ λ αα

λ β λ β λ β β λβ

φ φ φ φ φ φ φ φ φ φ

φ φ φ φ φ φ φ φ

=

=

= − ⟨ + ⟩ − ⟨ ⟩ − ⟨ ⟩

− ⟨ ⟩ − ⟨ ⟩

∑

∑

( )1

ˆ ˆ ˆ ˆ( ) | | | | | |N

HFE N k u v vλ λ λ λ α λ α λ α λ λ α

α

φ φ φ φ φ φ φ φ φ φ=

= − ⟨ + ⟩ + ⟨ ⟩ − ⟨ ⟩

∑ ( )HFE N λε= −



Restricted HFA

The non-relativistic many-body Hamiltonian commutes with the square and the z-component of both the total angular momentum and total spin of theof the N-electrons.

The HF Hamiltonian however does not commute with all these operators andtherefore the HF solutions are not symmetric (not rotationally invariants).

In fact the unrestricted variation of the HF assumption led to orbitalsgiven by

(r ) (r ) ( ) (r ) ( )v v vs s s s sφ φ χ φ χ+ + − −= + with1

( )2

ssχ±

±=

To construct HF orbitals wave functions with better symmetry is to restrictthe form of single particle orbitals:

( ) ( )

(r ) (r) ( ) for 1,...,

(r ) (r) ( ) for 1,...,

v v

N v N v

s s v N

s s v N

φ φ χ

φ φ χ+ +

+ + +

+ + − − −

= =

= =with N=N++N-



Using this restriction, we have now:

1| , {1,..., }

2ˆ |

1| , { 1,..., }

2

z

N

s

N N

α

α

α

φ αφ

φ α

+

+

+ ⟩ ∈⟩ =

− ⟩ ∈ +

and therefore ( )ˆ | |2

z

N NS ψ ψ+ −−

⟩ = ⟩

In order to distinguish this restriction from other further restrictions, the HFmethod using this restriction is called the unrestricted HF method.

3 3

1

3

_

1ˆ ˆ( ) | | | | (r) (r) ' (r) (r,r') (r')2

1' (r,r') (r,r') (r',r) (r,r') (r,r') (r',r)

2

N N

HF

N

E N k k d r u d r d r v

d r d r v v

α α α α α αα α

φ φ φ φ ρ ρ ρ

ρ ρ ρ ρ

+

+

3+ + − −

= =

3+ + −

= ⟨ ⟩ + ⟨ ⟩ + +

− +

∑ ∑ ∫ ∫ ∫

∫ ∫

*(r,r') (r') (r)α αα

ρ φ φ± ± ±=∑where and (r) (r) (r)ρ ρ ρ+ −= +

Notice that only the interaction between particles with the same spins contributes to the exchange energy.


Hartree-Fock ApproximationTo find the HF equations, we need this time to vary the total energy withrespect to αφ + αφ −or

:3 3(r) ' (r') (r,r') (r) ' (r,r') (r,r') (r') (r)u d r v d r vα α α αρ φ ρ φ ε φ+ + + +

−∆+ + − = ∫ ∫3 3(r) ' (r') (r,r') (r) ' (r,r') (r,r') (r') (r)u d r v d r vα α α αρ φ ρ φ ε φ− − − −

−∆+ + − = ∫ ∫These two HF SE are coupled by the total density.

It is interesting to notice that the exchange energy can be rewritten as anelectrostatic potential

( )3 3

r,' (r,r') (r,r') (r')= ' (r') (r,r') (r)d r v d r vα α αρ φ ρ φ± ± ± ±− −∫ ∫where

r,

(r') (r, r')(r')

(r)

αα

α

φ ρρ

φ± ±

±±

=3

r,' (r') 1d r αρ ± =∫and it is easily checked that

Each electron is therefore surrounded by an exchange hole of charge +e resulting of the repulsion between electrons with the same spin.



If we further restrict the HF orbitals by assuming that angular momentumis a good quantum number

( )

,

( ) ,

ˆ(r ) ( ) (r) ( ) for 1,...,

ˆ(r ) ( ) (r) ( ) for 1,...,

v v

v v

v v l m

N v l mN v

s r Y s v N

s r Y s v N

φ φ χ

φ φ χ+ +

+ + +

+ − −+ −

= =

= =

This time we obtain the so called restricted HR (RHF) method.

The solutions have better symmetry properties:

ˆ | |z v

l mα αφ φ⟩ = ⟩ and therefore ˆ | |z v

v

L mψ ψ

⟩ = ⟩ ∑

To use the HF method one has first to select a basis-set. The mostly usedare: (1) atomic orbitals, (2) plane waves, Slater orbitals exp(-αr), and theGaussina orbitals exp(-αr2).



The most difficult part the calculation of the 4-centered integrals:

3 *

A B(r-R ) (r-R )r R

nv

n n

Zd r βφ φ

−∆ + −

∑∫

Difficulties with the HF methodWhen the system consists of many atoms, one has to compute 1, 2 and 4-centers integrals:

However, the values of the integrals depend verymuch on the spacial overlap of the wave functions.

One then expects that the 4-centered integrals willmuch lower than the one centered integrals. Even the2-centered integrals can be small if the nuclei arefar from each other.

3 3 * *

A B C D

1' (r'-R ) (r-R ) (r-R ) (r'-R )

r-r'v

d r d r β γ δφ φ φ φ∫ ∫

To learn more about calculating see: R.Poirer, et al. Handbook of Gaussian Basis sets,Elvevier, Amsterdam, 1985.


DFT: primary tool for calculation of electronic structure in condensed matter

A special role can be assigned to the density of particles in the ground-state of a quantum many-body system

Many electron wave function

Satisfies the many-electron Schrödinger equation

Contains all information about the system

3N degrees of freedom for N electrons

One electron density

Integrates out this information

One equation for the density is simpler than the full many-body Schrödinger equation

All properties of the system can be considered as unique functionals of the ground state density



First theorem of Hohenberg-Kohn:

Theorem I: For any system of interacting particles in an external potential, this potential is determined uniquely, except for a constant, by the ground-particle density .




Proof by reductio absurdum:i. Let’s suppose that we have an exact ground state density n0(r).

ii. Let’s assume that there is only one wave function for this ground state (nondegenerate state)

iii. Let’s suppose that we have 2 external potentials Vext(r) and V’ext(r) forthis ground state density.

These 2 potentials produce 2 different wave functions ψ and ψ’, and2 total energies E0=<ψ|H’|ψ> and E’0=<ψ’|H|ψ’> respectively.

The variational theorem insures that

0 | | ' | | 'E H Hψ ψ ψ ψ= ⟨ ⟩ < ⟨ ⟩ ' | ' ' | 'H H Hψ ψ= ⟨ − + ⟩ ( )3 ' '

0 0(r) (r) (r)ext ext

d rn V V E= − +∫Similarly:

'

0 ' | ' | ' | ' |E H Hψ ψ ψ ψ= ⟨ ⟩ < ⟨ ⟩ | ' |H H Hψ ψ= ⟨ − + ⟩ ( )3 '

0 0(r) (r) (r)ext extd rn V V E= − − +∫By adding these 2 equations we obtain:

' '

0 0 0 0E E E E+ < +

The outcome is absurd, so there cannot be two different external potentialsthat produce the same density for their ground state.


S Second theorem of Hohenberg-Kohn: econdtheorem of Hohenberg-Kohn

Theorem II: A universal functional for the energy E[n] in terms ofthe density n can be defined, valid for any external potential Vext. For any particular Vext, the exact ground state of the system is the global minimum value of this functional, and the density n that minimizes the functional is the exact ground state density n0.



Density Functional TheoryThis second theorem can be rewritten as:

For a trial electron density n’(r) ≥0 and ∫d3r n’(r)=N, E0 ≤E[n’].

3[ ] [ ] (r) (r)HK extE n F n d rn V= + ∫The ground-state total energy calculated from the trial density n’(r) cannotbe lower that the exact energy E0.

This theorem is easily proved since:to the density n’ correspond a wave function ψ’ and an external potential V’ext,and to n correspond the external potential Vext then:

3

0' | | ' [ '] '(r) (r) [ '] [ ]HK extH F n d rn V E n E nψ ψ⟨ ⟩ = + = ≥∫This is because there is one to one correspondence between the wave function and the electronic density.


The Kohn-Sham ansatz replaces the many-body problem with an independent-particle problem

All the properties of the system are completely determined givenonly the ground state density n

But no prescription to solve the difficult interacting many-body hamiltonian

Ground state density of the many-body interacting system

Density of an auxiliary non-interacting independent particle system=

Kohn-Sham ansatz

(never proven in general)



One electron or independent particle model

We assume that each electron moves independently in a potential created by the nuclei and the rest of the electrons.

Actual calculations performed on the auxiliary independent-particle system




The Kohn-Sham (KS) equationsLike the derivation of the HF equations, we can use the variational theoremto find out the KS equations. This time we take a variation with respect tothe electron density:

{ }3[ ] (r) 0E n d rn Nδ ε − − = ∫

If we write *(r) (r) (r)n ψ ψ= and knowing that[ ]

[ ]E n

E n nn

δδ δ

δ=

We then obtain from the variational theorem the KS SE:[ (r)]

( (r)) ( ) ( )(r)

HKext

F nV r r

n

δψ εψ

δ+ =

If the exact form of the FHK is known then the problem is solved, sincethe SE will produce a selfconsistent ψ that determines the density,and hence the ground state total energy.The trick is to use the independent-particle kinetic energy which is given explicitly as a functional of the orbitals:


Then one has to rewrite the functional as

The rest:

Exchange-correlation


The exchange-correlation energy has to obey certain rules:

The variational theorem gives then:

1. Like in the HFA, EXC can be written in terms of the exchange-correlationhole nxc:

2. Again, like in the HFA, the exchange-correlation hole has to fulfill the sum rule:

This last sum rule help constrain the search for EXC.


Density functional theory is the most widely used method today for electronic structure calculations because of the

approach proposed by Kohn and Sham




The success of the LDA is certainly due to the fact that it satisfy the nxc sum rule, and being constructed from a homogeneous electron gas, it depends only on the spherical average of this hole density.

Local density approximation (LDA):

The general form of EXC: 3[ ] ( ) [ , ]XC XCE n n r n r d rε= ∫єXC at r depends on the density shape n in the whole space.

3[ ] ( ) ( ( ))XC XCE n n r n r d rε= ∫єXC at r depends only on the density at r (exact for a homogeneous electron gas).It is a function of the density and not a functional!

Generalized Gradient Approximation (GGA)

3[ ] ( ) ( ( ), ( ),...)XC XC

E n n r n r n r d rε= ∇∫єXC at r depends on the density and its gradient at r.



Local spin density approximation

In the collinear case, we assume that the direction of the magnetization does notdepend on the position. We can always use the z-direction for convenience.

Here the functional is of the density n(r) and the magnetization m(r).LSDA was first introduced byVon Barth and Hedin, J. Phys. C 5, 1629 (1972).

The wave function for each spin direction (+ or -) is given by:

The spin-up (+) and down (-) electron densities are given by:

The total charge density and the magnetization are given by:


The rest:

Exchange-correlation

Density Functional TheoryThe exchange-correlation per spin is determined by functional derivative:

The variational theorem gives the one particle equation for each spin direction:

The weights wi are determined so that the total density of states integrated up to the Fermi energy yields the number of valence electrons in the unit cell:


The Kohn-Sham equations must be solved self-consistently The potential (input) depends on the density (output)

No

Output quantities

Energy, forces, stresses …

Yes


I. Input: Structure, Atomic species

II. Guess for input

III. Compute the potential

IV. Solve the Ks equations

V. Compute the output density

VI. If selfconsistent: output the tolal energy,forces, etc…


Used approximations, basis-sets, potentials, etc

, PAWPAWPAWPAW


(Full-Potential Linear Augmented Plane-Wave method- LSDA, GGA, or LDA+U- Different treatment of the muffin-tin spheres and the interstitial region

Methods of calculation

Partial waves in the muffin-tin spheres

Plane waves in the interstitial: ( ).

,

G i k G r

n kG

C e+∑

, ,

,ˆ( ) ( ) ( ) ( ) ( )n k n k m

lm l l m l l

lm

A G U r B G U r Y r + ∑

�


Accuracy of the XC functionals in the structural and electronic properties

-50%-50%Egap

-5%+15%Ec

-20%,

+10%+10, +40%B

+1%-1% , -3% a

GGALDA

LDA: simplest approximation but accurate enough (structural properties, …).

GGA: usually tends to overcompensate LDA results, but not always better.



In some cases, GGA is a must: DFT ground state of iron

LSDA– NM – fcc

GGA– FM – bcc – Correct

lattice constant

Experiment– FM– bcc

GGA

GGA

LSDA

LSDA

Results obtained with Wien2k.

Courtesy of Karl H. Schwartz


P. Bagno et al., PRB 40, 1997 (1987)

GGA


Non-collinear magnetism, spin-spiral: Exchange parametersand TC

The magnetization is position dependent vector. In cellular methods, it is defined as:

The LAPW (for example) wave function is defined as:

Global frame

Global frame

Local spin frame


First do LDA then GGA

photo taken by Claudia Ambrosch


Noncollinear ground state of FeMn

relaxation of spin directions


Spin spiral ground stateUsing the generalized Bloch theorem, make it possible to study spin spiral in the Bravaislattice unit cell.

Using a second neighbor spin interactions in a Heisenberg model, if the angle between the two first neighbors is φ, the total energy is given by

( )2

1 22 cos cos2E NS J Jϕ ϕ= − +Its minimization gives:

( )1 24 cos sin 0J J ϕ ϕ+ =Solutions:

φ=0 ferromagnetic order; φ=π Antiferromagnetic order

1

2

cos4

J

Jϕ = − Spin-spiral order

Ph. Kurtz et al., PRB 69, 024415 (2005).


Generalized Bloch theorem to spin-spiral ground state

combines a lattice translation and a spin rotation.


3d transition metals

FM AFM

bcc Fe

fcc Fe

hcp Co

fcc Cofcc Ni

J. Kübler, Theory of Itinerant Electron magnetism, Oxford Science Publications, 2000


Spin-spiral ground state

S. Di Napoli et al, PRB 70, 174418 (2004)

La

SSDW, with θ=58O, and q=2π(0, 0, 0.7)/c.


Spin-spiral ground state

The spin-spiral magnetism can also be used to determine the Curie temperature.

In the meanfield approximation (overestimation of TC by up to 50%):

0 0( 1)n

n

J S S J= + ∑where

In the RPA (TC much better):

Padja et al, PRB 64, 174402 (01).Rosengaard and Johansson, PRB 55, 14975 (97).Kurz et al, J. Phys. Cond. Matter14, 6353 (02).Turek et al, ibid, 15, 2771 (03)


Easy axis

Without spin-orbit coupling

With spin-orbit coupling The generalized Bloch theoremdoes not hold.


Course content









XIII. Bibliography


Self-interaction

Does not cancel for the sameorbital like in the HFA.Example : position of the 1s level of hydrogen using

different approximation to EXC:

J.P. Perdew and A. Zunger, PRB 23, 5048 (1981) proposed the first SIC.

SVWN: Slater, Vosko, Wilk, Nusair, BLYP: Becke, Lee, Yang, Parr,BPW91: Becke, Perdew, Wang,B3LYP: Like BLYP, but with 3 parameters,HF: Hartree Fock


Electronic structure of Nio within the LSDA

Band gap Magnetic moment


The LDA+U approach (see the talk of B. Amadon)

LDA+U in a simple form (Anisimov et al, 91) takes into account the Hubbard interaction in localised orbitals:

The correlated energy levels and LDA+U potential are given by:

The effect of the Hubbard U is therefore to open a band gap between the upper and low Hubbard bands.


Is electronic correlation important for half-metals such as CrO2?

( ) ( ) ( ) ( )LSDA U LDA dc UE E E Eσ σ σ σρ ρ ρ ρ+ = − +

{ }1 3 1 32 4 2 4 4 2

1 3 2 4 1 3 1 3 2 4, , ,

1( )

2

m m m mm m m m m m

U m m m m m m m m m mE V V Vσ σ

σ σρ ρ ρ ρ−= − −

[ ]1 1( 1) ( 1) ( 1)

2 2dcE U N N J N N N N

↑ ↑ ↓ ↓ = − − − + −

The variationnel principe produces a Schrödinger like equation:

, '

, ' *, ,

,

( )( ) ( ) ( ) 0

( )

m m

lda m mn k n k

n k

n rV r r V r

r

σσ σ σ σ

σ

δε

δ −∆ + − Ψ + = Ψ

In the LDA+U general form, one takes into account the U and J and the non-sphericity of the interaction among the correlated orbitals:

What is the ground state of NiO within this method?

LDA+U (continued)


Ground state of NiO within the LDA+U method

U Gap M

To improve further the electronicstructure of NiO, we need to go

beyond the LDA+U, like the DMFT(see A. Georges, et al., RMP 68, 13, 1996)

O. Bengone et al., PRB 62, 16392 (00)


Density of states of CrO2, GGA (red) and LSDA+U (black)


The SOC reduces the spin polarization of CrO2First order perturbation theory

LSDA

Expt.

GGA.

LSDA+U (U=2eV)

LSDA+U (U=3eV)

Spin Polarization near EF

99.91%0.000170.408U= 3eV +

SOC

93.86%0.01600.505LSDA+SOC

100%6.8 10-100.375Without

SOC

Spin polar.DnUp

The SOC reduces the polarizationwhile the Hubbard interaction increases it.


Course content









VIII. Bibliography


What is the spin-orbit coupling?You can of course start from the Dirac equation and make a Foldy-Wouthuysentransformation, but if you are like me, you will still want a simple picture.A simple picture

-+

Point of view of p+

-

Point of view of e

The electron sees the proton orbitingaround it, and experience a magneticfield BThis field exerts a torque on the spinning electron to align its magnetic moment µealong the field.The Hamiltonian is H= -µe.B.This field is given by Biot-Savart law: B=µ0I/2r

The current is I=e/T (T orbit period)

The angular momentum of the electron (in the rest frame of the proton) L=rmv = 2πmr2/T point in the direction of B.:

2 3

0

1

4

e

mc rπε=B L

.

The spin moment of the electon is :e

m= − Seeeeμμμμ

So that the spin-orbit coupling is given by2

2 2 3

0

1

4

eH

m c rπε= ⋅S L

The missing factor ½ is due to Thomas precession.


Magnetic-anisotropy energy

Usually we use the force theorem because the small spin-orbit coupling can becomputed using perturbation theory.

The force theorem assumes that the effect of changing the spin-orientation does not affect the potential. The potential remains frozen during the change of the orientation of the magnetic moment.

The magnetic anisotropy energy which is the difference of the total energy between the hard and easy axis is then given by the difference of the sum of the eigenvalues for the hard and easy axis:

, , , ,

( ) ( ) ( ) ( )a kn kn kn kn

k n k n

E E hard axis E easy axis f hard f easyσ σ σ σ

σ σ

ε ε= − = −∑ ∑Bruno’s model (P. Bruno, PRB 39, 865 1989)

C. Kittel, 1968


2.7-0.5Fcc Ni

1.80.5fcc Co

-65-29hcp Co

-1.4-0.5bcc Fe

Expt.Ean (LSDA)Atom

Trygg et al. , PRL, 75,1995

S. Abdelouahed and M.A (to be published)

MAE of Fe, Co and Ni

MAE Gd (FLAPW and Bruno model)


MAE from GGA, GGA-core, and GGA+U

The energy position of the f statesmonitor the strength of the MAE.

But most of the anisotropy is dueto the anisotropy of 5d states asunderstood from Bruno’s model.

The MAE is similar to the of Co HCP

MAE of GdN and GdFe2

In both materials, the easy axis is alongone of the symmetric axis (100), (010) or (001)

The hard axis is along the (111) direction.

The MAE is different from that of FCC Ni.


Advantage of XMCD- Atomic species selectivity All atoms can be measured

- Orbital selectivity Electric-dipole selection rule

First row transition metals:

(1) 2p 3d : Magnetic orbitals and magnetic coupling

(2) 1s 4p : Delocalized orbitals and magnetic coupling

Rare earths:

(1) 3d 4f : Magnetic orbitals

(2) 2p 5d : delocalized orbitals and magnetic coupling


The intensity of absorbed x-ray radiation, at a distance d from the surface, isgiven by Beer’s law :

0( ) dI d I e µ−=

XAS and XMCD

(((( ))))Im nc

ωωωωµµµµ = −= −= −= −

where the cross section µ is given by:

2 2( ) 0ij i j ij jn n s s Eδ ε− − =

For a monochromatic radiation 0

ni ( r .s ) t

cE E eωωωω −−−− ====

of refraction n by solving the Fresnel’s equation

we can determine the index


Fresnel’s equation:

1

1

0

0

0 0 '

ε ε

ε ε ε

ε

= −

1

1

0

0 ' 0

0

ε ε

ε ε

ε ε

= −

1

1

' 0 0

0

0

ε

ε ε ε

ε ε

= −

0

sin

cos

t

t

Ks

Kθ

θ

= = −

(a) Polar geometry

(b) Longitudinal geometry :

(c) Transversal geometry :

4 2 2 2 2 2 2 2 2 2 2

1 1( 'cos sin ) ( sin 'cos ' sin ) ' ' 0n nε θ ε θ ε θ εε θ εε ε θ ε ε ε ε+ + + + + + + =

4 2 2 2 2 2 2 2 2 2 2

1 1( 'sin cos ) ( cos 'sin ' cos ) ' ' 0n nε θ ε θ ε θ εε θ εε ε θ ε ε ε ε+ + + + + + + =

2 2 2 2

1( ')( ) 0n nε ε ε ε− − − =


1

0

l

m λ

σ

∆ = ±

∆ =

∆ =

Selection rules

XMCD

4( , ) Im ( , ) Re ( , )q q q

c cλ λ λ

ω πµ ω ε ω σ ω= =

{ },

,

,

2

2

, ,

( ,0) Im 1 ( )j n k

n k

j jn k

m

m k n m

cf

mV

λ σ

σλ σ

σ

φπµ ω ε

ω ε ε ω+

∇ Ψ= −

− −∑ ∑

�

Polar geometry for cubic crystals


Allowed electronic transitions towards the d states

at the L2 edge


( ) ( )

( )

( )

3 2

3 2

0( ) ( ) ( ) , 10

( ) 2 ( ) 73

( ) ( )2

C

C

C

E

h d

E

z

E

z

L L

L

z

h

h

L

N E nE E dE n

NE E dE S

Ln

T

NE E E

n

d

µ µ µ

µ µ

µ µ

+ −= + + = −

∆ − ∆ = +

∆ + ∆ =

∫

∫

∫

XMCD sum rules for L23 edges

XMCD sum rules for K edge( )

, 62

C

Kp

E

z h

h

E NL ndE n

E n

µ∆= = −∫

Assumptions for deriving the sum rules:1. Neglect the energy dependence of the conduction wave functions

2. Neglect the exchange-splitting for the core levels

3. Neglect the p to s transitions

4. Neglect the difference between d3/2 and d5/2 wave functions

(Thole et al, PRL 1992)


For non cubic materials like CrO2, in the longitudinal geometry, extra dichroism is present due to the birefringence of the material.

When this extra dichroism is included, the calculation reproduces nicely the experimental data

XMCD spectra at the L23 edges of Cr in CrO2.

(expt): Goering et al, Phys. Rev. Lett. 88, 207203 (2002).


The XMCD sum rules are not fulfilled when the dichroicspectrum contains the birefringence contribution.

Total GGATotal LSDA+UMinority spin GGA

Majority spin GGA

Majority spin LSDA+U

Minority spin LSDA+U


Calculated and experimental XMCD oxygen K edge Spectra

(expt): Goering et al, Phys. Rev. Lett. 88, 207203 (2002).


[ ]1ˆ 3 ( . )2

z r r zT u uσ σ= −

Calculated Tz for the magnetization along the c direction

, ',ˆ ˆTRz z zn k n k

T T T = = Ψ Ψ

Dipolar magnetic moment Tz of CrO2

LSDA+U


-0.037

-0.064

-0.053

-0.066

1.89

2.04

GGA

U=3eVKomelj et al.

-0.003± 0.001(0.4)-0.06± 0.02(0.43)2.0 (1.2)Expt.

-0.001

-0.002

-0.045

-0.114

-0.03

-0.05

-0.053

-0.012

1.82

1.94

GGA

U=3eV

Theory

-0.001-0.03

-0.05

1.82

1.94

1.452

1.86

GGA

U=3eV

XMCD Sum

rules

Lz

O

Mz

O

Lz

Cr

Tz

Cr

Mz

Cr

Mzeff

Cr

Using only the XMCD part of the dichroism one gets, through the use of the sum rules, the correct spin and orbital magneticmoments.

Spin and orbital magnetic moments from the XMCD sum rules and direct calculations


Sr2FeMoO6 (SFMO)SFMO crystallizes in a bct, a=b= 5.55Å and c=7.90 Å

DOS

XAS and XMCD at Fe and Mo L23 edges

Spin and orbital moments of Fe and Mo

-

Kanchana et al., PRB 75, 22040(R), 2007


Course content









VIII. Bibliography


BIBLIOGRAPHY�The Theory of Magnetism Made Simple

D. C. Mattis, World Scientific

�Principles of the Theory of SolidsJ. M. Ziman, Wiley

�Solid State PhysicsC. Kittel, Wiley

�Magnetism in Condensed Matter,S. Blundell, Oxford Univ. Press

�Solid State Physics,N. W. Aschroft and N. D. Mermin

�Density functional Theory,,R.M. Dreizler and E.K.U. Gross, Spinger-Verlag, 1990

�The Fundamentals of Density functional Theory,,H. Eschrig, http://www.ifwdresden.de/institutes/itf/members/helmut

�Theory of Itinerant Electron magnetism,J. Kuebler, Oxford science publications

Documents

Magnetism in the Density Functional Theory Mebarek Alouaniqmmrc.net/winter-school-2009/Alouani.pdf · Magnetism in the Density Functional Theory IPCMS, 23 rue du Loess, UMR 7504,