Unpaired electrons in DFT - VASP

Magnetism 

Martijn MARSMAN 

Institut für Materialphysik and Center for Computational Material Science 

Universität Wien, Sensengasse 8/12, A-1090 Wien, Austria 

b-initio 

ienna 

ackage 

imulation 

M. MARSMAN, VASP WORKSHOP, VIENNA 10-15 FEBRUARY 2003. Page 1

Outline 

(Non)collinear spin-density-functional theory 

On site Coulomb repulsion: L(S)DA+U 

Spin Orbit Interaction 

Spin spiral magnetism 


Spin-density-functional theory 

wavefunction 

¡spinor 

Φ ¢ 

£¥¤ 

Ψ § ¢ 

Ψ ¦ ¢ 

 

£ 

 

 

¡ 

© 

£ 

£ 

density 

2¨2 matrix, 

n 

r 

n αβ 

r ¥¤∑ 

n 

f n 

Ψβn ¢ 

r 

r 

¢ Ψαn £ 


¥¤ 

σy 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n αβ 

r 

n Tr 

r 

δ αβ 

m r 

 

σ αβ 

2 

m r 

¤∑ 

n αβ 

αβ 

r 

 

σ αβ 

n Tr 

r Tr 

n αβ 

r 

α 

¤∑ 

n αα 

r 

Pauli spin matrices, 

σ¤ 

σx 

σ z 

0 1 

0 

i 

1 0 

σ x¤ 

1 0 

σy¤ 

i 0 

σz¤ 

0 

1 


¢ 

 

r 

¢ 

¢ 

 

 

 

 

r 

 

¢ 

 

 

 

 

 

 

The Kohn-Sham density functional becomes 

E 

¤∑ 

α 

∑ 

n 

1 

2 

f n 

and the Kohn-Sham equations 

dr 

Ψα n 

¢ 

1 

2 ∆ 

dr 

¢ Ψαn 

nTr 

 

r 

r 

£drVext 

r 

nTr 

nTr 

r 

Exc 

© 

n 

r 

r 

with the ¨2 2 Hamilton matrix 

∑H αβ 

β 

¢ Ψβn 

£¥¤εnSαα 

¢ Ψαn £ 

H αβ¤ 

1 

2 ∆δ αβ 

Vext 

r 

 

δαβ 

r 

dr 

n Tr 

r 

δ αβ 

V αβ 

xc 

© 

n 

r 

r 


¥¤mz 

 

¥¤ 

© 

 

 

 

 

 

H αα 

V βα 

xc 

V αβ 

xc 

H ββ 

¢ Ψαn £ 

¢ Ψβn £ 

¤εn 

¢ Ψαn £ 

¢ Ψβn £ 

¢ Ψαn £ 

and 

¢ Ψβn £ 

couple over V αβ 

xc 

and V βα 

xc 

V αβ 

xc 

© 

n 

r 

r 

δE xc 

δn βα 

© 

n 

r 

r 

unfortunately, only in case m 

to E xc n r is known 

© 

r 

r n r 

 

diagonal , a reliable approximation 


¦ 

 

¥¤ 

 

¥¤ 

 

¥¤ 

 

 

 

 

 

 

 

¦ 

 

 

 

 

§ 

 

¥¤ 

Uβ j 

¤δi j ni 

nβα 

 

 

 

 

 

 

 

 

 

¦ 

 

¦ 

 

 

 

 

 

 

 

§ 

 

 

 

¥¤ 

 

σzU 

 

 

 

 

¢ 

density matrix can be diagonalized 

∑U iα 

αβ 

r 

r 

r 

r 

where U(r) are spin-1/2 rotation matrices 

V αβ 

xc 

r 

1 

2 

δE xc 

δn 1 

r 

 

δE xc 

δn 2 

r 

 

δ αβ 

1 

2 

δE xc 

δn 1 

r 

 

δE xc 

δn 2 

r 

U 

 

r 

r 

αβ 

or equivalently, using 

n 

r 

1 

2 

n Tr 

r 

¢ 

m 

r 

¢ 

n § 

r 

1 

2 

n Tr 

r 

 

¢ 

m 

r 

¢ 

and ˆm 

r 

¢ 

m 

m 

r 

r 

we can write 

V αβ 

xc 

r 

1 

2 

δE xc 

δn 

r 

 

δE xc 

δn 

r 

 

δ αβ 

1 

2 

δE xc 

δn 

r 

 

δE xc 

δn 

r 

 

ˆm 

r 

 

σ αβ 

where 

E xc¤n Tr 

r 

ε xc 

n 

r 

n § 

r 

dr 


Collinear (spins along z direction): 

H αα 

H ββ 

¢ Ψαn £ 

¢ Ψβn £ 

¤εn 

¢ Ψαn £ 

¢ Ψβn £ 

Noncollinear: 

H αα 

V βα 

xc 

V αβ 

xc 

H ββ 

¢ Ψαn £ 

¢ Ψβn £ 

¤εn 

¢ Ψαn £ 

¢ Ψβn £ 

In the absence of Spin Orbit Interaction (SOI) the spin directions are not linked to the 

crystalline structure, i.e., the system is invariant under a general common rotation of 

all spins 


YMn 2 

Mn sublattice collinear noncollinear 


! 

 

¢ 

 

r 

¢ 

On site Coulomb repulsion 

L(S)DA fails to describe systems with localized (strongly correlated) d and f 

electrons ¡wrong one-electron energies 

Strong intra-atomic interaction is introduced in a (screened) Hartree-Fock like 

manner ¡replacing L(S)DA on site 

E HF¤ 

1 

2 ∑ 

γ 

" 

Uγ1 γ3 γ2γ4 

Uγ1γ3 γ4 γ2 

ˆn γ1 γ 2 ˆn γ3 γ 4 

determined by the PAW on site occupancies 

ˆn γ1 γ 2¤ 

Ψs2 m2 

¢ 

£m1 

and the (unscreened) on site electron-electron interaction 

U γ1 γ 3 γ 2 γ 4¤ 

m1 m3 

r 

1 

¢ ¢ 

m2m4 £ 

¢ Ψs1 £ 

δ s1 s 2 

δ s3 s 4 

( 

¢ m £ 

are the spherical harmonics) 


¤ 

Uγ1 γ3 γ2 γ4 given by Slater’s integrals F 0, F2, F4, and F 6 (f-electrons) 

Calculation of Slater’s integrals from atomic wave functions leads to a large 

overestimation because in solids the Coulomb interaction is screened (especially 

F 0 ). 

In practice treated as fitting parameters, i.e., adjusted to reach agreement with 

experiment: equilibrium volume, magnetic moment, band gap, structure. 

Normally specified in terms of effective on site Coulomb- and exchange 

parameters, U and J. 

For 3d-electrons: U 

F 0 , J¤ 

1 

14 

F2 

F4 

, and F4 0#65 

F 2¤ 

U and J sometimes extracted from constrained-LSDA calculations. 


EHF 

 

 

Edc 

 

Total energy and double counting 

E tot 

n 

ˆn 

Total energy 

n 

¥¤EDFT 

ˆn 

ˆn 

Double counting 

LSDA+U 

E dc 

ˆn 

¥¤ 

U 

2 ˆn tot 

ˆn tot 

1 

J 

2 ∑ σ ˆn σ tot 

ˆn σ tot 

1 

LDA+U 

E dc 

ˆn 

¥¤ 

U 

2 ˆn tot 

ˆn tot 

1 

J 

4 ˆn tot 

ˆn tot 

2 

Hartree-Fock Hamiltonian can be simply added to the AE part of the PAW 

Hamiltonian 


Orbital dependent potential that enforces Hund’ s first and second rule 

– maximal spin multiplicity 

– highest possible azimuthal quantum number L z 

(when SOI included) 


U 

 

 

' 

$ 

U 

 

 

& 

Dudarev’s approach to LSDA+U 

E LSDA U¤E LSDA 

J 

2 

∑ 

σ 

∑ 

m 1 

n σ m 1 

% m1 

∑ 

m 1 

% m2 ˆnσm 1 

%%m2 ˆnσm 2 m1 

n−n 2 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

0 

0 0.2 0.4 0.6 0.8 1 

n 

Penalty function that forces idempotency of 

the onsite occupancy matrix, 

ˆn σ¤ˆn σ ˆn σ 

real matrices are only idempotent, if their 

eigenvalues are either 1 or 0 

(fully occupied or unoccupied) 

E LSDA U¤E LSDA 

ε i 

( 

J 

2 

∑ 

σ %%m1 m2 ˆnσm 1 

%%m2 ˆnσm 2 m1 


) 

An example: NiO, a Mott-Hubbard insulator 

Rocksalt structure 

AFM ordering of Ni (111) planes 

Ni 3d electrons in octahedral crystal field 

t 2g (3d xy , 3d xz , 3d yz ) 

e g (3d x 2 

y 2 , 3d z 2) 


LSDA 

Dudarev U=8 J=0.95 

n(E) (states/eV atom) 

4 

2 

0 

−2 

−4 

t 2g 

e g 

n(E) (states/eV atom) 

4 

2 

0 

−2 

−4 

t 2g 

e g 

−4 −2 0 2 4 6 8 10 

−4 −2 0 2 4 6 8 10 

E(eV) 

E(eV) 

¢¢mNi 

= 1.15 µ B 

¢¢mNi 

= 1.71 µ B 

E gap = 0.44 eV E gap = 3.38 eV 

Experiment 

= 1.64 - 1.70 µ B E gap = 4.0 - 4.3 eV 

¢¢mNi 


¤ 

2me c 

% 

¢ 

¢ 

 

 

¢ 

Spin Orbit Interaction 

Relativistic effects, in principle stemming from 4-component Dirac equation 

Pseudopotential generation: Radial wave functions are solutions of the scalar 

relativistic radial equation, which includes Mass-velocity and Darwin terms 

Kohn-Sham equations: Spin Orbit Interaction is added to the AE part of the PAW 

Hamiltonian (variational treatment of the SOI) 

H αβ 

SOI 

¯h 2 

2 ∑ 

i j 

φi 

1 

r 

dV spher 

dr 

¢ φ j £ 

˜p i 

£ 

σ L αβ 

i j 

˜p j 


Consequences: 

Mixing of up- and down spinor components, noncollinear magnetism 

Spin directions couple to the crystalline structure, magneto-crystalline anisotropy 

Orbital magnetic moments 


¤ 

 

An example: CoO 

Rocksalt structure 

AFM ordering of Co (111) planes 

Experiment: 

¢ mCo 

112 

Orbital moment in t 2g manifold of 

Co 2 [3d 7 ] ion not completely quenched 

by crystal field 

LDA+U (U=8 eV, J=0.95 eV) + SOI 

,enforce Hund’ s rules 

Calculations yield correct easy axis, 

and c/a ratio 

¢ mS 

m L 

µB ¢¢¤2.8 

-1 (magnetostriction) 

¢ ¤1.4 µB 

¢+*3.8 µB along 


¦ 

§ 

 

 

 

 

 

¤ 

¤ 

. 

. 

) 

 

 

 

 

1 

1 

 

 

 

 

¦ 

§ 

 

 

qR 

qR 

 

 

/ 

/ 

Spin spirals 

q 

m rR 

Generalized Bloch condition 

m x 

m x 

r 

r 

cos 

sin 

qR 

qR 

m z 

r 

my 

my 

r sin 

r cos 

Ψ 

Ψ 

k 

k 

r 

r 

e 

iq0R 

2 

0 

0 e iq 0R 

2 

Ψ 

Ψ 

k 

k 

rR 

rR 


¦ 

 

¦ 

¢ 

¢ 

¡ 

¡ 

¡ 

 

G 

§ 

 

¢ 

 

¢ 

 

§ 

) 

 

 

keeping to the usual definition of the Bloch functions 

Ψ 

k 

r 

¥¤∑ 

G 

C 

kG ei 

k G 0r 2 

3 

and 

Ψ 

k 

r 

G 

¤∑ 

C 

kG ei 

k G 0r 2 

3 

the Hamiltonian changes only minimally 

H αα 

V βα 

xc 

V αβ 

xc 

H ββ 

H αα Vxc αβ e 

Vxc βα e iq 0r Hββ 

iq0r 

where in H αα and H ββ the kinetic energy of a plane wave component changes to 

¢ k 

G 

2 

¢ k 

G 

2 

q 

2 

in H αα 

¢ k 

G 

2 

¢ k 

2 

q 

2 

in H ββ 


¤ 

¤E 

E 

q θ 

 

 

Primitive cell suffices, no need for supercell that contains a complete spiral period 

Adiabatic spin dynamics: Magnon spectra 

for instance for elementary ferromagnetic metals (bcc Fe, fcc Ni) 

ω 

q 

¥¤lim 

θ40 

4 

M 

∆E 

sin 2 θ 

M 

θ 

q 

∆E 

q θ 

 

q 

θ 

0 

θ 

see for instance: 

“Theory of Itinerant Electron Magnetism”, J. Kübler, Clarendon Press, Oxford (2000). 


0 

 

 

 

 

 

Spin spirals in fcc Fe 

−8.13 

−8.15 

10.45 Å 3 

10.63 Å 3 

10.72 Å 3 

11.44 Å 3 

Γ (FM) = 

2π 

q 1 = 

2π 

a 0¨ 

X (AFM) = 

2π 

q 2 = 

2π 

a 0¨ 

0 a0¨ 

 

0 

0 a0¨ 

 

0#15 

0 

0 

0 

0 

0#6 

1#0 

0 

1#0 

E [eV] 

−8.17 

−8.19 

−8.21 

Γ q 1 X q 2 W 

q exp¤ 

2π 

0#1 a0¨ 

 

#0 1 


Magnetization at q 1 


Some references 

Noncollinear magnetism in the PAW formalism 

– D. Hobbs, G. Kresse and J. Hafner, Phys. Rev. B. 62, 11 556 (2000). 

L(S)DA+U 

– I. V. Solovyev, P. H. Dederichs and V. I. Anisimov, Phys. Rev. B. 50, 16 861 

(1994). 

– A. B. Shick, A. I. Liechtenstein and W. E. Pickett, Phys. Rev. B. 60, 10 763 

(1999). 

– S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys and A. P. 

Sutton, Phys. Rev. B. 57, 1505 (1998). 

Spin spirals 

– L. M. Sandratskii, J. Phys. Condens. Matter 3, 8565 (1993); J. Phys. Condens. 

Matter 3, 8587 (1993)

Unpaired electrons in DFT - VASP

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