Magnetism (FM, AFM, FSM) Karlheinz Schwarz Institute of Materials Chemistry TU Wien Localized vs. itinerant systems In localized systems (e.g. some rare earth) the magnetism is mainly governed by the atom (Hunds rule) In itinerant (delocalized) systems (many transition metals) magnetism comes from partial occupation of states, which differ between spin-up and spin-down.

Boarderline cases (some f-electron systems) details of the structure (e.g. lattice spacing) determine whether or not some electrons are localized or itinerant. Ferro-, antiferro-, or ferri-magnetic Ferromagnetic (FM) (e.g. bcc Fe) M>0 Antiferromagnetic (AFM) (e.g. Cr) M=0

Ferrimagnetic cases the moments at different atoms are antiparallel but of different magnitude M>0 Non-collinear magnetism (NCM) the magnetic moments are not ligned up parallel. Itinerant electron magnetism Experimental facts: Curie temperature Stoner theory of itinerant electron magnetism 1. The carriers of magnetism are the unsaturated spins

in the d-band. 2. Effects of exchange are treated with a molecular field term. 3. One must conform to Fermi statistics. Stoner, 1936 Stoner model for itinerant electrons In a non magnetic (NM) case N = N (spin-up and spin-down) ferromagnetic (FM) case N > N (majority and minority spin) Exchange splitting spin-down

spin-up the moments at all sites are parallel (collinear) the (spin) magnetic moment m m = N - N its orientation with respect to the crystal exchange axes is only defined by interaction spin orbit coupling. there can also be an orbital

moment it is often suppressed in 3d transition metals Stoner criterion Stoner model for itinerant electrons The existence of ferromagnetism (FM) is governed by the (a) Fe 1 IFe

Stoner criterion I . N(EF) > 1 N(EF) DOS at EF I (b) Ni 1 INi (of NM case) Stoner parameter ~ independent of structure Ferromagnetism appears when the gain in exchange energy is larger than the loss in kinetic

energy P.James, O.Eriksson, B.Johansson, I.A.Abrikosov, Phys.Rev.B 58, ... (1998) bcc Fe Non magnetic case spin-up EF spindown ferromagnetic case spin-up EF

spindown Exchange splitting EF at high DOS DFT ground state of iron LSDA GGA LSD A

GGA GGA FM bcc Correct lattice constant Experiment LSDA

NM fcc in contrast to experiment FM bcc Iron and its alloys Fe: weak ferromagnet (almost) Co: strong ferromagnet Magnetism and crystal structure V. Heine: metals are systems with unsaturated covalent bonds

Covalent magnetism Fe-Co alloys e.g. Fe-Co alloys Wigner delay times Spin projected DOS of Fe-Co alloys % Co The alloy is represented by ordered structures

Fe3Co and FeCo3 (Heusler) FeCo Zintl or CsCl Fe, Co bcc Iron and its alloys Itinerant or localized? Magnetization density in FeCo Magnetization density difference between Majoity spin Minority spin

m(r)= (r)- (r) Localized around Fe and Co slightly negative between the atoms Itinerant electrons K.Schwarz, P.Mohn, P.Blaha, J.Kbler, Electronic and magnetic structure of bcc Fe-Co alloys

from band theory, J.Phys.F:Met.Phys. 14, 2659 CsCl structure Bonding by Wigner delay time single scatterer (Friedel) V(r)=0 solution: Rl joined in value and slope defines phase shift : Friedel sum Wigner delay time Bessel Neumann

Phase shifts, Wigner delay times of Fe, Co, Ni resonance states Wigner delay time Phase shifts Covalent magnetism in FeCo Wigner delay time For spin up

Fe and Co equivalent partial DOS similar typical bcc DOS For spin down Fe higher than Co antibonding Co Fe Co bonding No charge transfer between Fe and Co

Magnetism and crystal structure Covalent magnetism, FeCo: Antiferromagnetic (AFM) Cr Cr has AFM bcc structure Cr1 Cr1 spin-up Cr2 Cr2 spin-up EF

spin-down There is a symmetry it is enough to do the spin-up calculation spin-down can be copied Cr1 = Cr2 Cr2 = Cr1 spin-down Zeolite, sodalite Al-silicate corner shared

SiO4 tetrahedra AlO4 tetrahedra cage Al / Si ratio 1 alternating ordered (cubic) 3 e- per cage Si Al SES Sodium electro sodalite

Si-Al zeolite (sodalite) Formed by corner-shared SiO4 and AlO4 tetrahedra Charge compensated by doping with 4 Na+ one e- (color center) antiferromagnetic (AFM) order of e-

Energy (relative stability) color center e- SES AFM order between color centers (e-) Spin density - G.K.H. Madsen, Bo B. Iversen, P. Blaha, K. Schwarz, Phys. Rev. B 64, 195102 (2001)

INVAR alloys (invariant) e.g. Fe-Ni Such systems essentially show no thermal expansion around room temperature INVAR (invariant) of Fe-Ni alloys Ch.E.Guillaume (1897)

The thermal expansion of the Eifel tower Measured with a rigid Fe-Ni INVAR wire The length of the tower correlates with the temperature Fe65Ni35 alloy has vanishing thermal expansion around room temperature Magnetostriction and Invar behaviour What is magnetostriction? Magnetostriction s0 is the difference in volume between the volume in the magnetic ground state and the volume in a hypothetical non-magnetic state. Above the Curie temperature the magnetic contribution m vanishes. Tc

Fe-Ni Invar alloys classical Fe-Ni Invar Fe65Ni35 alloy has vanishing therma expansion around room temperature Early explanations of INVAR R.J.Weiss Proc.Roy.Phys.Soc (London) 32, 281 (1963) FCC fcc Fe small moment small volume

50% Fe high spin m=2.8 B FM a = 3.64 low spin m=0.5 B AF a = 3.57 75% Fe 1 AF kT

100% Fe high moment FM large volume 2 60 70 80 volume (Bohr)3 A.R.Williams, V.L.Moruzzi, G.D.Gelatt Jr., J.Kbler, K.Schwarz, Aspects of transition metal magnetism, J.Appl.Phys. 53, 2019 (1982) Energy surfaces of Fe-Ni alloys

This fcc structure from non magnetic Fe (fcc) to ferromagnetic Ni Fe-Ni alloy % Fe 100% as the composition changes

At the INVAR composition There is a flat energy surface 75% as function of volume and moment 50% 0% Finite temperature Energy surface at T=0 (DFT)

as a function of volume and moment using fixed spin moment (FSM) calculations Finite temperature Spin and volume fluctuations Ginzburg-Landau model T

439 K TC 300 K 200 K 100 K 0 K FSM calculations fixed spin moment (FSM)

e.g. Fe-Ni alloy allows to explore energy surface E(V,M) as function of volume V magnetic moment M Fixed spin moment (FSM) method There are systems (e.g. like fcc Fe or fcc Co), for which the magnetization shows a hysteresis, when a magnetic field is applied (at a volume V). The volume of the unit cell defines the Wigner-Seitz radius rWS 3 4rWS

V 3 The hysteresis causes numerical difficulties, since there are several solutions (in the present case 3 for a certain field H). In order to solve this problem the FSM method was invented Hysteresis V

3 4RWS 3 Fixed spin moment (FSM) method Conventional scheme method constrained (FSM) E F E F E F E F Z v N N

Z v N N output M N N Simple case: bcc Fe one SCF M N N output input many calculations difficult case:

Fe3Ni poor convergence good convergence FSM Physical situation: One applies a field H and obtains M but this functions can be multivalued Computational trick (unphysical): One interchanges the dependent and independent variable this function is single valued (unique) i.e. one chooses M and calculates H afterwards

FSM key references A.R.Williams, V.L.Moruzzi, J.Kbler, K.Schwarz, Bull.Am.Phys.Soc. 29, 278 (1984) K.Schwarz, P.Mohn J.Phys.F 14, L129 (1984) P.H.Dederichs, S.Blgel, R.Zoller, H.Akai, Phys. Rev, Lett. 53,2512 (1984) Unusual magnetic systems GMR (Giant Magneto Resistance)

half-metallic systems e.g. CrO2 important for spintronics Once upon a time, Once upon a time, in the early 1980s What happens if I bring two ferromagnets close I mean really close together? Peter Grnberg N S

S N ? Giant magnetoresistance (GMR) Ferromagnet Metal Ferromagnet Electrical resistance: RP <(>) RAP The electrical resistance depends on

the relative magnetic alignment of the ferromagnetic layers GMR RAP RP RP 19% for trilayers @RT 80% for multilayers @ RT GMR is much larger than the anisotropic magnetoresistance (AMR) 1988: simultaneously, but independent Does the electrical resistance depend on the magnetization alignment? Albert Fert Peter Grnberg

http://www.kva.se/ Scientific background CrO2 half-metallic ferromagnet CrO2 (rutile structure) spin-up metallic spin-down gap important for CrO2 DOS

The DOS features of CrO2 are qualitatively like K.Schwarz, CrO2 predicted as a half-metallic ferromagnet, J.Phys.F:Met.Phys. 16, L211 (1986) 4 TiO2 (for spin-down) spin RuO2 (for spin-up) 5

6 7 8 Ti V Cr Mn Fe Zr Nb

Mo Tc Ru gap metallic all three compound crystallize in the rutile structure Half-metallic ferromagnet CrO2 (rutile structure) spin-up

metallic spin-down gap CrO2 spin-down (TiO2) spin-up (RuO2) Magnetic structure of uranium dioxide UO2 R.Laskowski G.K.H.Madsen P.Blaha K.Schwarz topics

non-collinear magnetism spin-orbit coupling LDA+U (correlation of U-5f electrons) Structure relaxations electric field gradient (EFG) U O R.Laskowski, G.K.H.Madsen, P.Blaha, K.Schwarz: Magnetic structure and electric-field gradients of uranium dioxide: An ab

initio study Phys.Rev.B 69, 140408-1-4 (2004) Atomic configuration of uranium (Z=92) [Rn] U [Xe] 4f14 7s2 core nre l n 7s 6d 5f Ej (Ryd) j

(relativ.) -s -0.29 -0.17 6p 6s 5d -1.46 5p 18.05 5s 4f 5d10 6s2 6p6

-7.48 27.58 semi-core 5f3 6d1 valence +s -0.25 -0.25 -0.11 -2.10 -3.40 -6.89 delocalized -14.06

-22.57 -26.77 core-like non-collinear magnetism in UO2 collinear 1k- non-collinear 2k- or 3k-structure UO2 2k structure, LDA+SO+U Magnetisation direction perpenticular at the two U sites (arrows) Magnetisation density (color)

U O Magnetism with WIEN2k Spin polarized calculations Run spin-polarized, FSM or AFM calculations Various magnetism cases Thank you for your attention

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