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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:

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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

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ceStoner theory of itinerant electron Stoner theory of itinerant electron magnetismmagnetism

exchange interaction

Stoner susceptibility Stoner criterion

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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.)

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ceIron and its alloysIron and its alloys

Fe: weak ferromagnet (almost)

Co: strong ferromagnet

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ceIron and its alloysIron and its alloys

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ceIron and its alloysIron and its alloys

Itinerant or localized?

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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

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ceMagnetism and crystal structureMagnetism and crystal structure

V. Heine: „metals are systems with unsaturated covalent bonds“

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ceMagnetism and crystal structureMagnetism and crystal structure

Covalent magnetism, FeCo:

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ceMagnetism and crystal structureMagnetism and crystal structure

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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

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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:

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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, ] , ,

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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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ce

ener

gy

/a /aq/2

EEF

F

E

spin down

spin up

Mixing of spin-up and spin-down states

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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

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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]

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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…

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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”