Computational Materials Science Magnetism and LSDA Peter Mohn Center for Computational Materials...

ce Magnetism and LSDAMagnetism and LSDA

Peter Mohn

Center for Computational Materials Science

Vienna University of Technology

Vienna, Austria

Outline:Outline:

• Trivia

• Fe and ist alloys• Magnetism and crystal structure• noncollinearity• Where it works and where not…

ceItinerant electron magnetismItinerant electron magnetism

Experimental facts:

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

exchange interaction

Stoner susceptibility Stoner criterion

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

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.)

ceIron and its alloysIron and its alloys

Fe: weak ferromagnet (almost)

Co: strong ferromagnet

ceIron and its alloysIron and its alloys

Itinerant or localized?

ceFe-Ni Invar alloysFe-Ni Invar alloys

„classical“ Fe-Ni Invar

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.

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

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

average number of valence electrons

Slater-Pauling plot

ceMagnetism and crystal structureMagnetism and crystal structure

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

Covalent magnetism, FeCo:

ceNon-collinearityNon-collinearity

ASA of muffin-tin geometry, potential spherically symmetric

’’ ’’’

The effective local potential

is diagonal with

respect to the spin

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

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:

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

cebcc-Fe spinspiralbcc-Fe spinspiral

q=2 / [0,0,0.5]

-1,0 -0,5 0,0 0,5 1,0-50

100150200250300350

bcc Fe

[ ] spin spiral q-vector [0,0, ] , ,

ceBand structure and non-collinearityBand structure and non-collinearity

/a /aq/2

antiferromagnetic orderantiferromagnetic order

/a /2a /2aq/2

ceThe groundstate of fcc Fe

M. Uhl et al. JMMM 103 314 (1992)

--FeFe

Band structure of non-magnetic -Fe

q=[0,0,0.6]

just shifted

fully selfcon-sistent result with magnetic moment 1.8B

/a /aq/2

spin down

spin up

Mixing of spin-up and spin-down states

ceNon collinear states in bcc Mn

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

approximating

allows to write the dispersion as

0,0 0,2 0,4 0,6 0,8 1,00,000

0,025bcc Fe

]spin spiral q-vector [0,0, ]

q=2 / [0,0,0.5]

ceOrdering temperature for MF Ordering temperature for MF HeisenbergHeisenberg

For a fcc and bcc lattice:

-1,0 -0,5 0,0 0,5 1,0-50

100150200250300350

bcc Fe

[ ] spin spiral q-vector [0,0, ] , ,

Magnon density of states for bcc Fe

-1,0 -0,5 0,0 0,5 1,0

bcc Fe

[ ] spin spiral q-vector [0,0, ]

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

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…

ceCorrelation effects in FeAlCorrelation effects in FeAl

eg eg*

eg*narow bands:

Correlation effects ?

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)

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

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

ceTHE END ...THE END ...

I gratefully acknowledge support by the Austrian Science Foundation FWF within the

Wissenschaftskolleg

“Computational Materials Science”

Computational Materials Science Magnetism and LSDA Peter Mohn Center for Computational Materials...

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