Computational Materials Science Magnetism and LSDA Peter Mohn Center for Computational Materials Science Vienna University of Technology Vienna, Austria

Computational Materials Science Outline: Trivia Fe and ist alloys Magnetism and crystal structure noncollinearity Where it works and where not…

Computational Materials Science Itinerant electron magnetism Experimental facts:

Computational Materials Science 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 within a molecular field term. 3. One must conform to Fermi statistics. Stoner, 1936

Computational Materials Science Stoner theory of itinerant electron magnetism exchange interaction Stoner susceptibility Stoner criterion

Computational Materials Science Stoner theory of itinerant electron magnetism Exchange splitting ∆E and Stoner factor I s for closed packed cobalt for 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

Computational Materials Science Stoner theory of itinerant electron magnetism The Stoner exchange parameter describes intraatomic exchange. For the transition metals I s 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.)

Computational Materials Science Iron and its alloys Fe: weak ferromagnet (almost) Co: strong ferromagnet

Computational Materials Science Iron and its alloys

Computational Materials Science Iron and its alloys Itinerant or localized?

Computational Materials Science Fe-Ni Invar alloys „classical“ Fe-Ni Invar

Computational Materials Science Magnetostriction and Invar behaviour 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 the Magnetic contribution  m vanishes. TcTc

Computational Materials Science Fe-Pt Invar Fe 74 Pt 26 :  so (exp) =1.7%  so (calc) =1.9% DLM calculations

Computational Materials Science Invar  x Fe 74 Pt 26 :  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

Computational Materials Science Magnetostriction und Invar behaviour Slater-Pauling plot

Computational Materials Science Magnetism and crystal structure V. Heine: „metals are systems with unsaturated covalent bonds“

Computational Materials Science Magnetism and crystal structure Covalent magnetism, FeCo:

Computational Materials Science Magnetism and crystal structure

Computational Materials Science Non-collinearity ASA of muffin-tin geometry, potential spherically symmetric The effective local potential is diagonal with respect to the spin

Computational Materials Science are the spin ½ rotation matrices The single particle WF is now a two component spinor function, which produces a charge density matrix which is also not spin diagonal

Computational Materials Science Spin Spiral States Given that the angle  changes proportional to a lattice vector R j allows to separatein a lattice periodic part and a lattice independent part:

Computational Materials Science Generalized Bloch theorem The Hamilonian for a spin-spiral now reads The helix-operatorsform a cyclic abelian group 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

Computational Materials Science bcc-Fe spinspiral

Computational Materials Science Collective excitations within the Heisenberg Model Solution: with:

Computational Materials Science expanding the cosine: for a cubic lattice with a 0

Computational Materials Science Classical representation of a spin wave Dispersionrelation of a Heisenberg ferromagnet D[meV C 2 ] Fe 280 Co 510 Ni 455

Computational Materials Science D. McKenzie Paul et al. PRB (1988) Landau damping

Computational Materials Science Approximations for the Heisenberg model Ising model: XY model:

Computational Materials Science Mean field solution for the Heisenberg model Replacing the spin operator by ist mean value plus the deviations form it (fluctuations) yields

Computational Materials Science Exchange coupled spin fluctuations

Computational Materials Science Susceptibility and Curie temperature

Computational Materials Science Band structure and non-collinearity

Computational Materials Science antiferromagnetic order

Computational Materials Science The groundstate of fcc Fe M. Uhl et al. JMMM (1992)

Computational Materials Science  - Fe Band structure of non- magnetic  -Fe q=[0,0,0.6] just shifted fully selfcon- sistent result with magnetic moment 1.8  B

Computational Materials Science spin down spin up Mixing of spin-up and spin-down states

Computational Materials Science Non collinear states in bcc Mn

Computational Materials Science q=[0,0,0.35] q=[0,0,0.70] q=[0,0,0.875] P. M. Solid State Commun (1997) Non collinear states in bcc Mn

Computational Materials Science approximating allows to write the dispersion as

Computational Materials Science Ordering temperature for MF Heisenberg For a fcc and bcc lattice:

Computational Materials Science Magnon density of states for bcc Fe

Computational Materials Science The Curietemperatur of Fe and Ni Fe: local moments dominate Distributions almost equal! T c =1065K (exp. 1040K) Ni: longitudinal fluctuations dominate for T>Tc. Distributions are different! T c =615K (exp. 630K) A.Ruban, S. Khmelevskyi, P. Mohn, B. Johansson, PRB, 2006

Computational Materials Science The 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…

Computational Materials Science Correlation effects in FeAl egeg eg*eg* t 2g egeg eg*eg* narow bands: Correlation effects ?

Computational Materials Science Correlation effects in FeAl non magnetic for U>4.5 eV Stoner criterion I Fe N(  F )>1 no longer fulfilled. Phys. Rev. Letters, (2001)

Computational Materials Science Some 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.) ZrZn Ni 3 Al Sc 3 In m (LDA,  B /f.u.) m (expt.,  B /f.u.) FeAl Ni 3 Ga Sr 3 Ru 2 O Na 0.5 CoO  (LDA, emu/mol)  (expt., emu/mol) Pd

Computational Materials Science Quantum 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 tendency is a signature of quantum critical fluctuations – neglected fluctuations suppress magnetism

Computational Materials Science THE END... I gratefully acknowledge support by the Austrian Science Foundation FWF within the Wissenschaftskolleg “Computational Materials Science”