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Com
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ce Magnetism and LSDAMagnetism and LSDA
Peter Mohn
Center for Computational Materials Science
Vienna University of Technology
Vienna, Austria
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Outline:Outline:
• Trivia
• Fe and ist alloys• Magnetism and crystal structure• noncollinearity• Where it works and where not…
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ceItinerant electron magnetismItinerant electron magnetism
Experimental facts:
Com
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ceStoner theory of itinerant electron Stoner theory of itinerant electron magnetismmagnetism
1. The carriers of magnetism are the unsaturated spins in the d-band.
2. Effects of exchange are treated within a molecular field term.
3. One must conform to Fermi statistics.
Stoner, 1936
Com
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ceStoner theory of itinerant electron Stoner theory of itinerant electron magnetismmagnetism
exchange interaction
Stoner susceptibility Stoner criterion
Com
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ceStoner theory of itinerant electron Stoner theory of itinerant electron magnetismmagnetism
Exchange splitting ∆E and Stoner factor Is for closed packed cobaltfor various models of the local density approximation for exchange and correlation. Despite of the large scattering found for ∆E and Is the calculated magnetic moments are all between 1.55 and 1.7µB (exp: 1.62µB).
X after Wakoh et al.LA local correlations (Oles and Stollhoff)HL Hedin-Lundquist
vBH von Barth-Hedin
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ceStoner theory of itinerant electron Stoner theory of itinerant electron magnetismmagnetism
The Stoner exchange parameter describes intraatomic exchange.
For the transition metals Is is of comparable order of magnitude ~ 70mRy (1 eV).
Fulfilling the Stoner criterion does not tell us anything about the long range magnetic Structure (ferro, antiferro, etc.)
Com
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ceIron and its alloysIron and its alloys
Fe: weak ferromagnet (almost)
Co: strong ferromagnet
Com
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ceIron and its alloysIron and its alloys
Com
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ceIron and its alloysIron and its alloys
Itinerant or localized?
Com
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ceFe-Ni Invar alloysFe-Ni Invar alloys
„classical“ Fe-Ni Invar
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ceMagnetostriction and Invar Magnetostriction and Invar behaviourbehaviour
What is magnetostriction?
Magnetostriction s0 is the diffe-Rence in volume between the Volume in the magnetic ground state and the volume in a hypothetical non-magnetic state.
Above the Curie temperature theMagnetic contribution m vanishes.
Tc
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ceInvarInvar
Fe74Pt26:
so(exp)=1.7% so(calc)=1.9%
Maximum for s0 at 8.4 e/a
„Disordered Local Moment“ DLM calculations for Fe-Co, Fe-Pd, Fe-Pt
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ceMagnetostriction und Invar behaviourMagnetostriction und Invar behaviour
8 9 100.00.51.01.52.02.5
NiCoFeMn
Fe-Cr, Fe-Ni Fe-Co, Ni-Co Fe-V, Ni-Cu Ni-Zn, Co-Cr Co-Mn, Ni-Mn Ni-Cr, Ni-V pure metals
averagema
gneticmome
ntperatom
[ B]
average number of valence electrons
Slater-Pauling plot
Com
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ceMagnetism and crystal structureMagnetism and crystal structure
V. Heine: „metals are systems with unsaturated covalent bonds“
Com
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ceMagnetism and crystal structureMagnetism and crystal structure
Covalent magnetism, FeCo:
Com
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ceMagnetism and crystal structureMagnetism and crystal structure
Com
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ceNon-collinearityNon-collinearity
ASA of muffin-tin geometry, potential spherically symmetric
’
’’ ’’’
j
The effective local potential
is diagonal with
respect to the spin
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are the spin ½ rotation matrices
The single particle WF is now a two component spinorfunction, which produces a charge density matrix whichis also not spin diagonal
Com
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ceSpin Spiral StatesSpin Spiral States
Given that the angle changes proportional to a lattice vector Rj
allows to separate in a lattice periodic part
and a lattice independent part:
Com
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ceGeneralized Bloch theoremGeneralized Bloch theorem
The Hamilonian for a spin-spiral now reads
The helix-operators form a cyclic abeliangroup and commute with the hamiltonian and are isomorphous with the lattice-translation operator
C. Herring, in: Magnetism IV (G. Rado, H. Suhl eds.) Acad.Press 1966
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cebcc-Fe spinspiralbcc-Fe spinspiral
q=2 / [0,0,0.5]
=qRj
-1,0 -0,5 0,0 0,5 1,0-50
050
100150200250300350
bcc Fe
to
tal e
nerg
y [m
eV/a
tom
]
[ ] spin spiral q-vector [0,0, ] , ,
Com
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ceBand structure and non-collinearityBand structure and non-collinearity
ener
gy
/a /aq/2
EEF
F
E
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antiferromagnetic orderantiferromagnetic order
ener
gy
/a /2a /2aq/2
E EF
FE
a 2a
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ceThe groundstate of fcc Fe
M. Uhl et al. JMMM 103 314 (1992)
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--FeFe
Band structure of non-magnetic -Fe
q=[0,0,0.6]
just shifted
fully selfcon-sistent result with magnetic moment 1.8B
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ener
gy
/a /aq/2
EEF
F
E
spin down
spin up
Mixing of spin-up and spin-down states
Com
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ceNon collinear states in bcc Mn
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ce q=[0,0,0.35]
q=[0,0,0.70]
q=[0,0,0.875]
P. M. Solid State Commun. 102 729 (1997)
Non collinear states in bcc Mn
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approximating
allows to write the dispersion as
0,0 0,2 0,4 0,6 0,8 1,00,000
0,005
0,010
0,015
0,020
0,025bcc Fe
tota
l ene
rgy
[Ry/
atom
]spin spiral q-vector [0,0, ]
q=2 / [0,0,0.5]
=qRj
Com
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ceOrdering temperature for MF Ordering temperature for MF HeisenbergHeisenberg
For a fcc and bcc lattice:
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-1,0 -0,5 0,0 0,5 1,0-50
050
100150200250300350
bcc Fe
to
tal e
nerg
y [m
eV/a
tom
]
[ ] spin spiral q-vector [0,0, ] , ,
Magnon density of states for bcc Fe
-1,0 -0,5 0,0 0,5 1,0
1,0
1,2
1,4
1,6
1,8
2,0
2,2
bcc Fe
[ ] spin spiral q-vector [0,0, ]
mag
netic
mom
ent [
B/Ato
m]
Com
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ceThe Curietemperatur of Fe and NiThe Curietemperatur of Fe and Ni
Fe: local moments dominateDistributions almost equal!
Tc=1065K (exp. 1040K)
Ni: longitudinal fluctuationsdominate for T>Tc.Distributions are different!
Tc=615K (exp. 630K)
A.Ruban, S. Khmelevskyi, P. Mohn, B. Johansson, PRB, 2006
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ceThe limitations of LSDAThe limitations of LSDA
FeAl forms an intermetallic compound and crystallizes in the CsCl structure. The phase is highly ordered ~98%.
Experiment: FeAl is a paramagnet
Calculation: DFT calculations yield a ferromagnetic ground state with a rather stable moment of 0.8!
FeAl a seemingly simple alloy…
Com
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ceCorrelation effects in FeAlCorrelation effects in FeAl
eg eg*
t2g
eg
t2g
eg*narow bands:
Correlation effects ?
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ceCorrelation effects in FeAlCorrelation effects in FeAl
non magnetic for U>4.5 eVStoner criterion IFe N(F)>1 no longerfulfilled.
Phys. Rev. Letters, 87 196401 (2001)
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ceSome Metals Where the LSDA Overestimates
Ferromagnetism
Class 1: Ferromagnets where the LDA overestimates the magnetization.
Class 2: Paramagnets where the LDA predicts ferromagnetism
Class 3: Paramagnets where the LDA overestimates the susceptibility.
m (LDA, B/f.u.) m (expt., B/f.u.)
ZrZn2 0.72 0.17Ni3Al 0.71 0.23Sc3In 1.05 0.20
m (LDA, B/f.u.) m (expt., B/f.u.)
FeAl 0.80 0.0Ni3Ga 0.79 0.0Sr3Ru2O7 0.9 0.0Na0.5CoO2 0.50 0.0
(LDA, 10-4 emu/mol) (expt., 10-4 emu/mol)
Pd 11.6 6.8
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ceQuantum Critical Points and the LDA
Density Functional Theory: LDA & GGA are widely used for first
principles calculations but have problems:
•Mott-Hubbard: Well known poor treatment of on-site Coulomb correlations.
•Based on uniform electron gas. Give mean field treatment of
magnetism: Fluctuations missing.
LDA overestimate of ferromagnetic LDA overestimate of ferromagnetic
tendency is a signature of tendency is a signature of
quantum critical fluctuations – quantum critical fluctuations –
neglected fluctuations suppress neglected fluctuations suppress magnetismmagnetism
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ceTHE END ...THE END ...
I gratefully acknowledge support by the Austrian Science Foundation FWF within the
Wissenschaftskolleg
“Computational Materials Science”