Itinerant Ferromagnetism:

mechanisms and models

J. E. Gubernatis,1 C. D. Batista,1

and J. Bonča2

1Los Alamos National Laboratory2University of Ljubljana

Magnetism

Outline

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

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

Standard Models f electrons

Kondo Lattice Model

Periodic Anderson Model

† †

, ,

†' ' '

', ,

( )

0 2

1; : Pauli matrices; J >0

2

iKLM i j j i ii j i

c

i is s s is s ss s

H t c c c c J S

n

c c

† †,

,

† †

2

0 2, 0 2, 0 4

f fPAM d i j j i i i

i j i

fi i i i f i

i i

d f d f

UH t d d d d n n

V d f f d n

n n n n n

Periodic Anderson Model

† †,

,

† †

2f f

d r r r r r rr r r

fr r r r f r

r r

UH t d d d d n n

V d f f d n

V

d

f

Standard Models

Anderson

Kondo

HubbardInfinite U

J/t >>10< nc <2

U/t >>1|EF-f|<<V0 <nd <2<nf >=1

J/t<<10< nc <1

U/t >>1|EF-f| >> V0< nd <1nf=1

Heisenberg RKKY

HubbardSegmented Band

U/t >> 1|EF-f| ~ 01< 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,ni

f]=0.

Result: JV2/ |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

Cornelius and Schilling, PRB 49, 3955 (1994)

Quantum Monte CarloBonca et al

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.

U= Hubbard ModelBecca and Sorella, PRL 86, 3396 (2001)

Holes

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.

Localized moment regime

mixedvalentregime

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.

2z z zf d f

i

S j S i S j

2z z z z zf d f f f

i i j

S j S i S j S i S j

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 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. 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.