Itinerant%20Ferromagnetism:%20mechanisms%20and%20models

1
Itinerant Ferromagnetism mechanisms and models

• J. E. Gubernatis,1 C. D. Batista,1
• and J. Bonca2
• 1Los Alamos National Laboratory
• 2University of Ljubljana

2
Magnetism
3
A Quote about Magnetism
Readers who wish to be spared the rather
compromised exposition forced on us by the
difficulty and incompleteness of the subject
might merely want to note the following two
points 1. By far the most important source
of magnetic interaction is the ordinary
electrostatic electron- electron interaction.
2. To explain magnetic ordering in solids, in
the great majority of cases it necessary to go
beyond the independent particle approximation,
upon which band theory is based Ashcroft and
Mermin, Chapter 32.
4
Outline

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

5
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

6
2nd Order Perturbation Theory
7
Standard Models f electrons

• Kondo Lattice Model
• Periodic Anderson Model

8
Periodic Anderson Model
9
Standard Models
10
Standard Models

• Small J/t KLM and large U/t PAM connected by a
  canonical transformation, when EF-?f gtgtV, nf1
• Emergent symmetry HPAM,nif0.
• Result J?V2/ EF-?f
• Mixed valence regime EF-?f ?0.

11
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.
• T0 critical point at J ? EF.
• Mixed valent materials are paramagnetic.

O(1)
12
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).
13
Traditional Mechanism
14
Ce(M1-xXx)3 B2 CeRh3(N1-yYy)2
Quantum Monte Carlo Bonca et al
Cornelius and Schilling, PRB 49, 3955 (1994)
15
Single Impurity Compensation


Compensation cloud
16
Exhaustion in strong mixed valent limit
17
Nagaoka-like mechanism for Ferromagnetism
18
Nagaoka Mechanism

• Relevant for holes away from half-filing in a
  strongly correlated band (U/t gtgt 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.

19
Nagaoka-like Mechanism in Weak Mixed Valent Regime

• Adding tf ltlt td embeds a Hubbard model in the
  PAM.
• When U/tfgtgt1, the physics of the Nagaoka
  mechanism applies.
• In polarized regime, conduction band is a charge
  reservoir for localized band
• Increasing pressure, converts fs to ds,
• Decreasing pressure, converts ds to fs.

Holes
U? Hubbard Model Becca and Sorella, PRL 86, 3396
(2001)
20
Periodic Anderson Model

• Mixed valent regime
• U/tgtgt1, EF-?f0
• Observations at U0
• 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
21
Two-Dimensional Bands
M
G
X
22
Some Observations V??td ?f

• Two subspaces in each band.
• Predominantly d or f character.
• Size of cross-over region ? V/td.
• Very small.
• First approximation assume d and f subspaces are
  invariant under H.

23
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 ?.

24
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 ks are used.

25
Numerical Consistency 1D
26
Numerical Consistency 2D
27
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 ds, compensate
  the f electrons.

28
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

29
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

30
Ce(M1-xXx)3 B2 CeRh3(N1-yYy)2
Quantum Monte Carlo for the Periodic Anderson
Model Bonca et al
Cornelius and Schilling, PRB 49, 3955 (1994)
31
(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.

32
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 lt ?f , ? increases, and
  eventually ? ?.
• Peak in Cp
• Thermal excitations will promote previously
  paired electrons into highly degenerate aligned
  states.

33
Relevance to UTe

• Ferromagnetic with Tc104K.
• Tc increases and then decreases with pressure.
• Elastic and transport properties suggest mixed
  valency.
• Properties not described well by LSDA and LDAU.
• Sharp but dispersive peak near EF
• New Observation Binding energy proportional to M.

Durakiewicz et al., preprint
34
Some Final Remarks

• Appearance of anti-ferromagnetic state
• Adding non-zero tf might also generate an anti-
  ferromagnetic state.
• Anti-ferromagnetism otherwise occurs only when
  there is nesting.
• Stability of ferro-magnetic state
• Adding non-zero tf can stabilize (or destroy)
  the ferromagnetic state.

35
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.

36
Summary

• In polarized regime, we learned
• Increasing pressure, converts fs to ds,
• Decreasing pressure, converts ds to fs.
• The implied figure of merit is EF - ?f.
• If large, local moments and RKKY.
• If small, less localized moments and mixed valent
  behavior.

37
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.

38
Summary

• Mixed valency and itinerant ferromagnetism can
  co-exist.
• Such co-existence observed in UTe.
• Nagaoka-like constructive inference mechanisms
  for ferromagnetism are consistent with physics of
  the standard models of f-electron materials.
• Inherently non-local in character,
• Non-exchange in origin.
• Simple signature for itinerant ferromagnetism in
  mixed valent materials was found.
• Magnetization proportional to binding energy.

39
References

• Constrained-Path Method
• S. Zhang, J. Carlson, and J. E. Gubernatis, Phys.
  Rev. Lett. 74, 3652 (1997) Phys. Rev B 55, 7464
  (1997).
• J. Carlson, J. E. Gubernatis, G. Ortiz, S. Zhang,
  Phys. Rev. B 59, 12788 (1999).
• Itinerant Ferromagnetism
• C. D. Batista, J. Bonca, and J. E. Gubernatis,
  Phys. Rev. B 63, 184428 (2001).
• C. D. Batista, J. Bonca, and J. E. Gubernatis,
  Phys. Rev. Lett. 88, 187203 (2002).
• C. D. Batista, J. Bonca, and J. E. Gubernatis,
  Phys. Rev. B 68, 06443 (2003).
• C. D. Batista, J. Bonca, and J. E. Gubernatis,
  Phys. Rev. B., to appear (cond-mat/0308002).

40
Numerical MethodConstrained Path Monte Carlo

• General many-body wave-function
• Successively sample the Slater determinants from
  the distribution
• Result for i-th sample

41
Numerical MethodConstrained Path Monte Carlo

• Projection of ground state
• Steady-state iteration
• Constraint on path to remove sign problem,
  eliminate any random walker that violates