Magnetism
Martijn MARSMAN
Institut für Materialphysik and Center for Computational Material Science
Universität Wien, Sensengasse 8/12, A-1090 Wien, Austria
b-initio
ackage
ienna
M. M ARSMAN ,
VASP WORKSHOP, VIENNA
10-15 F EBRUARY 2003.
imulation
Page 1
Outline
(Non)collinear spin-density-functional theory
On site Coulomb repulsion: L(S)DA+U
Spin Orbit Interaction
Spin spiral magnetism
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Spin-density-functional theory
wavefunction
spinor
Ψ
Φ
Ψ
2 2 matrix, n r
density
Ψβn r r Ψαn
∑ fn
nαβ r
n
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α
r
∑ nαα
Tr nαβ r
nTr r
σ x σy σz
Pauli spin matrices, σ
2
αβ
i
0
0
M. M ARSMAN ,
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1
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1
0
i
1
σy
0
1
0
σx
r σαβ
∑ nαβ
mr
m r σαβ
nTr r δαβ
nαβ r
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The Kohn-Sham density functional becomes
Exc n r
dr
nTr r nTr r
dr
r r
1
2
drVext r nTr r
α n
1
∆ Ψαn
2
∑ ∑ fn Ψαn
E
and the Kohn-Sham equations
β
εn Sαα Ψαn
∑ H αβ Ψβn
with the 2 2 Hamilton matrix
r
αβ
Vxc n r
nTr r
δαβ
r r
dr
Vext r δαβ
1
∆δαβ
2
H αβ
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αβ
εn
β
Ψn
αβ
β
β
Ψn
H ββ
Ψαn
βα
Vxc
Ψαn
Vxc
H αα
βα
Ψαn and Ψn couple over Vxc and Vxc
δExc n r
δnβα r
r
αβ
Vxc n r
n r diagonal , a reliable approximation
mz r
unfortunately, only in case m r
to Exc n r is known
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10-15 F EBRUARY 2003.
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density matrix can be diagonalized
∑ Uiα
δ i j ni r
αβ
r nβα r Uβ j r
where U(r) are spin-1/2 rotation matrices
U
r σ zU r
αβ
δExc
δn2 r
1 δExc
2 δn1 r
δExc
δ
δn2 r αβ
r
1 δExc
2 δn1 r
αβ
Vxc
or equivalently, using
mr
mr
and m̂ r
mr
1
nTr r
2
n r
mr
1
nTr r
2
n r
we can write
δExc
m̂ r σαβ
δn r
1 δExc
2 δn r
δExc
δ
δn r αβ
1 δExc
2 δn r
r
αβ
Vxc
where
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nTr r εxc n r n r dr
Exc
10-15 F EBRUARY 2003.
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Collinear (spins along z direction):
H αα
β
Ψn
εn
H ββ
Ψαn
Ψαn
β
Ψn
Noncollinear:
αβ
εn
β
Ψn
β
Ψn
H ββ
Ψαn
βα
Vxc
Ψαn
Vxc
H αα
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
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Page 8
YMn2
Mn sublattice
collinear
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noncollinear
Page 9
On site Coulomb repulsion
L(S)DA fails to describe systems with localized (strongly correlated) d and f
wrong one-electron energies
electrons
Strong intra-atomic interaction is introduced in a (screened) Hartree-Fock like
replacing L(S)DA on site
manner
Uγ1 γ3 γ4 γ2 n̂γ1 γ2 n̂γ3 γ4
EHF
1
Uγ1 γ3 γ2 γ4
2∑
γ
"
!
determined by the PAW on site occupancies
Ψ s 2 m2 m1 Ψ s 1
n̂γ1 γ2
and the (unscreened) on site electron-electron interaction
1
r
r
m2 m4 δ s 1 s 2 δ s 3 s 4
m1 m3
Uγ1 γ3 γ2 γ4
( m are the spherical harmonics)
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10-15 F EBRUARY 2003.
Page 10
Uγ1 γ3 γ2 γ4 given by Slater’s integrals F 0 , F 2 , F 4 , 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.
F4
F2
0 65
#
F 4 , and
F2
1
14
F 0, J
For 3d-electrons: U
U and J sometimes extracted from constrained-LSDA calculations.
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VASP WORKSHOP, VIENNA
10-15 F EBRUARY 2003.
Page 11
Total energy and double counting
Edc n̂
Etot n n̂
Total energy
EDFT n EHF n̂
Double counting
2
n̂tot
J
4 n̂tot
1
∑σ n̂σtot n̂σtot
1
J
2
n̂tot
U
2 n̂tot
Edc n̂
LDA+U
1
n̂tot
U
2 n̂tot
Edc n̂
LSDA+U
Hartree-Fock Hamiltonian can be simply added to the AE part of the PAW
Hamiltonian
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VASP WORKSHOP, VIENNA
10-15 F EBRUARY 2003.
Page 12
Orbital dependent potential that enforces Hund’s first and second rule
– maximal spin multiplicity
– highest possible azimuthal quantum number Lz
(when SOI included)
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Dudarev’s approach to LSDA+U
U
J
%
σ
m1 m2
%
m1
n̂σm1 m2 n̂σm2 m1
Penalty function that forces idempotency of
the onsite occupancy matrix,
n̂σ n̂σ n̂σ
0.25
0.2
n−n2
&
$
0.3
%
2
∑
%
∑ nσm1 m1
∑
ELSDA
U
ELSDA
0.15
real matrices are only idempotent, if their
eigenvalues are either 1 or 0
(fully occupied or unoccupied)
0.1
0.05
0
0
0.2
0.4
0.6
0.8
1
n
U
J
(
'
ELSDA εi
U
ELSDA
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%
M. M ARSMAN ,
%
%
σ m 1 m2
n̂σm1 m2 n̂σm2 m1
%
2
∑
10-15 F EBRUARY 2003.
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An example: NiO, a Mott-Hubbard insulator
Rocksalt structure
AFM ordering of Ni (111) planes
Ni 3d electrons in octahedral crystal field
t2g
(3dxy , 3dxz , 3dyz )
eg
(3dx2
)
y2 ,
3dz2 )
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VASP WORKSHOP, VIENNA
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Page 15
Dudarev U=8 J=0.95
t2g
eg
4
n(E) (states/eV atom)
n(E) (states/eV atom)
LSDA
2
0
−2
−4
−4
−2
0
2
4
E(eV)
=
0.44 eV
−2
−4
10
−4
−2
0
2
4
E(eV)
6
mNi
=
1.71 µB
Egap
=
3.38 eV
8
10
Experiment
= 1.64 - 1.70 µB
Egap = 4.0 - 4.3 eV
mNi
0
Egap
8
2
1.15 µB
=
mNi
6
t2g
eg
4
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VASP WORKSHOP, VIENNA
10-15 F EBRUARY 2003.
Page 16
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)
p̃i σαβ Li j p̃ j
%
i j
1 dVspher
φj
r dr
∑ φi
2
h̄2
2me c
αβ
HSOI
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Page 17
Consequences:
Mixing of up- and down spinor components, noncollinear magnetism
Spin directions couple to the crystalline structure, magneto-crystalline anisotropy
Orbital magnetic moments
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Page 18
An example: CoO
Rocksalt structure
AFM ordering of Co (111) planes
3.8 µB along 112
+*
Experiment: mCo
Orbital moment in t2g manifold of
Co2 [3d7 ] 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,
mS
2.8 µB mL
1.4 µB
and c/a ratio 1 (magnetostriction)
-
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VASP WORKSHOP, VIENNA
10-15 F EBRUARY 2003.
Page 19
Spin spirals
q
mx r cos q R
/ /
my r cos q R
mx r sin q R
. .
R
mr
my r sin q R
mz r
Generalized Bloch condition
iq R 2
0
)
0
1
R
Ψk r
R
Ψk r
e
Ψk r
iq R 2
1
e
0
0
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Ψk r
Page 20
keeping to the usual definition of the Bloch functions
∑ CkG ei k
2
G
G r
0 3
Ψk r
and
G r
0 3
2
∑ CkG ei k
Ψk r
G
the Hamiltonian changes only minimally
βα
H ββ
Vxc
Vxc e
αβ
iq r
Vxc e
0
H αα
H ββ
iq r
0
βα
αβ
Vxc
)
H αα
where in H αα and H ββ the kinetic energy of a plane wave component changes to
G
k
G2
k
q 22
in H αα
q 22
in H ββ
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G
k
G2
k
10-15 F EBRUARY 2003.
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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)
lim
θ 0
4
θ
E 0θ
E qθ
∆E q θ
q
ωq
M
4 ∆E q θ
M sin2 θ
see for instance:
“Theory of Itinerant Electron Magnetism”, J. K übler, Clarendon Press, Oxford (2000).
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10-15 F EBRUARY 2003.
Page 22
−8.13
Spin spirals in fcc Fe
E [eV]
−8.17
#
#
−8.19
#
0 15 0 1 0
#
=
0 0 10
q2
=
X (AFM)
0 0 06
=
−8.15
000
q1
2π
a0
2π
a0
2π
a0
2π
a0
=
Γ (FM)
−8.21
Γ
q1
X
q2
W
#
01 0 10
#
2π
a0
qexp
10.45 Å33
10.63 Å3
10.72 Å3
11.44 Å
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Magnetization at q1
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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)
M. M ARSMAN ,
VASP WORKSHOP, VIENNA
10-15 F EBRUARY 2003.
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