Itinerant Ferromagnetism: mechanisms and models

1. Itinerant Ferromagnetism: mechanisms and models J. E. Gubernatis,1 C. D. Batista,1 and J. Bonča2 1Los Alamos National Laboratory 2University of Ljubljana

2. Magnetism

3. Outline • Basic models • Traditional mechanism • Interference (Nagaoka) mechanisms • Mixed valent mechanism: strong ferromagnetism • Relevance to experiment • Summary

4. Approach • Analytic theory • Generate effective Hamiltonians • Usually, 2nd order degenerate perturbation theory • Identify physics by • Inspection • Exact solutions • Numerical simulations • Guide and interprets simulations of the original Hamiltonian • Numerical Simulation • Compute ground-state properties • Constrained-Path Monte Carlo method • Extend analytic theory

5. Standard Models f electrons • Kondo Lattice Model • Periodic Anderson Model

6. d V f Periodic Anderson Model

7. Hubbard Segmented Band weak U/t >> 1 |EF-f| ~ 0 1< n <2 Mixed valent U/t >>1 |EF-f|<<V 0 <nd <2 <nf >=1 strong Anderson U/t >>1 |EF-f| >> V 0< nd <1 nf=1 Heisenberg RKKY Hubbard Infinite U Kondo J/t >>1 0< nc <2 J/t<<1 0< nc <1 Standard Models

8. Standard Models • Small J/t KLM and large U/t PAM connected by a canonical transformation, when |EF-f| >>V, nf=1 • Emergent symmetry: [HPAM,nif]=0. • Result: JV2/ |EF-f| • Mixed valence regime: |EF-f| 0.

9. Traditional Mechanism • Competing energy scales • TRKKY J² N(EF) • TKondo EF exp(-1/JN(EF)) • Approximate methods support this. • Kondo compensation explains • moment reduction • heavy masses. • T=0 critical point at J  EF. • Mixed valent materials are paramagnetic. O(1)

10. Traditional Mechanism “The fact that Kondo-like quenching of local moments appears to occur for fractional valence systems is consistent with the above ideas on the empirical ground that only when the f-level is degenerate with the d-band is the effective Schrieffer-Wolff exchange interaction likely to be strong enough to satisfy the above criterion for a non-magnetic ground state of JN(0)=O(1).” Doniach, Physica 91B, 231 (1977).

11. Ce(M1-xXx)3 B2 CeRh3(N1-yYy)2 Quantum Monte Carlo Bonca et al Cornelius and Schilling, PRB 49, 3955 (1994)

12. Single Impurity Compensation … … Compensation cloud

13. Exhaustion in strong mixed valent limit

14. Nagaoka-like mechanism for Ferromagnetism

15. Nagaoka Mechanism • Relevant for holes away from half-filing in a strongly correlated band (U/t >> 1). • Holes can lower their kinetic energy by moving through an aligned background. • Hole can cycle back to original configuration. • Ground state wavefunction results from the constructive interference of many hole-configurations.

16. Nagaoka-like Mechanism in Weak Mixed Valent Regime • Adding tf << td embeds a Hubbard model in the PAM. • When U/tf>>1, the physics of the Nagaoka mechanism applies. • In polarized regime, conduction band is a charge reservoir for localized band • Increasing pressure, converts f’s to d’s, • Decreasing pressure, converts d’s to f’s. Holes U= Hubbard Model Becca and Sorella, PRL 86, 3396 (2001)

17. Periodic Anderson Model • Mixed valent regime • U/t>>1, |EF-f|~0 • Observations at U=0 • Two subspaces in each band. • Predominantly d or f character. • Size of cross-over region  V²/W. • Very small. mixed valent regime Localized moment regime

18. Mixed Valent Mechanism • Take U = 0, EF f and in lower band. • Electrons pair. • Set U 0. • Electrons in mixed valent state spread to unoccupied f states and align. • Anti-symmetric spatial part of wavefunction prevents double occupancy. • Kinetic energy cost is proportional to .

19. Mixed Valent Mechanism • A nonmagnetic state has an energy cost  to occupy upper band states needed to localize and avoid the cost of U. • Ferromagnetic state is stable if   . • TCurie   • By the uncertainty principle, a state built from these lower band f states has a restricted extension. • Not all k’sare used.

20. Numerical Consistency: 2D

21. Some Other Numerical Results Local Moment Compensation • In the single impurity model, a singlet ground state implies • In the lattice model, it implies • In the lattice, the second term is more significant than the first. • Mainly the f electrons, not the d’s, compensate the f electrons.

22. Experimental Relevance • Ternary Ce Borides (4f). • CeRh3B2: highest TCurie (115oK) of any Ce compound with nonmagnetic elements. • Small magnetic moment. • Unusual magnetization and TCurie as a function of (chemical) pressure. • Uranium chalcogenides (5f) • UxXy , X= S, Se, or Te. • Some properties similar to Ce(Rh1-x Rux)3B2

23. Challenges • Expansion case • Doping removes magnetic moments • Increases overlap • Tc decrease while M increases • Compression case • M does not increases monotonically • Specific heat peaks where M peaks

24. (LaxCe1-x)Rh3B2 • Increase of M. • If CeRh3B2 is in a 4f-4d mixed valent state and EF f, TCurie  . • With La doping, • f electron subspace increases so M increases. • Localized f moment regime reached via occupation of f states in upper band.

25. Ce(Rh1-xRux)3B2 • Reduction of M. • If CeRh3B2 is in a 4f-4d mixed valent state and EF f, TCurie  . • With Ru doping, EF < f ,  increases, and eventually  ~ . • Peak in Cp • Thermal excitations will promote previously paired electrons into highly degenerate aligned states.

26. Summary • We established by analytic and numerical studies several mechanisms for itinerant ferromagnetism. • A novel mechanism operates in the PAM in the mixed valence regime. • It depends of a segmentation of non-degenerate bands and is not the RKKY interaction. • The segmentation of the bands is also relevant to the non-magnetic behavior. • Non-magnetic state is not a coherent state of Kondo compensated singlets.

27. Summary • In polarized regime, we learned • Increasing pressure, converts f’s to d’s, • Decreasing pressure, converts d’s to f’s. • The implied figure of merit is |EF - f|. • If large, local moments and RKKY. • If small, less localized moments and mixed valent behavior.

28. Summary • In the unpolarized regime, the d-electrons are mainly compensating themselves; f-electrons, themselves. • In the polarized regime, more than one mechanism produces ferromagnetism. • The weakest is the RKKY, when the moments are spatially localized. • The strongest is the segmented band, when the moments are partially localized.