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

Magnetism

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

Approach Analytic theory Numerical Simulation 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

Standard Models f electrons Kondo Lattice Model Periodic Anderson Model

Periodic Anderson Model V d f

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

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.

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)

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).

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

Single Impurity Compensation … … Compensation cloud

Exhaustion in strong mixed valent limit

Nagaoka-like mechanism for Ferromagnetism

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.

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)

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

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 .

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’s are used.

Numerical Consistency: 2D

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.

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

Challenges Expansion case Compression 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

(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.

Ce(Rh1-xRux)3B2 Reduction of M. Peak in Cp 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.

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.

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.

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.