Theory in Materials Science

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Theory in Materials Science

• 9th lecture
• Magnetism of the Hubbard model

Associate Professor Koichi Kusakabe
Graduate School of Engineering Science, Osaka
University
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What is the electron correlation effect?
Physical phenomena appearing in many-electron
systems are called the Electron correlation
effect, when the phenomena do not appear within
the Hartree-Fock approximation (a mean-field
approximation).
Examples of correlation effects

• Difference between the CI energy and the
  Hartree-Fock energy, electronic states by CI,
  difference in molecular structures from those
  given by HF.
• Phase transitions of electron gas Paramagnetic
  (gt Perfect diamagnetic) gt ferromagnetic gt
  Wigner crystal
• Motts metal-insulator transition
• Magnetism in the Mott insulator
• Superconductivity around the Mott insulator
• Quantum phase transitions Critical phenomena
  around the quantum critical point, including
  magnetism and superconductivity

Some of the phase transitions are analyzed by
considering model systems in place of the
original Coulomb system, whose phase space is
restricted.
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A model for the superexchange

• Renormalization process (three steps)
• The LCAO model of electrons with overlap
  integrals
• The tight-binding model written in the Wannier
  orbitals
• The Heisenberg spin Hamiltonian

Here, depending on the modeling, i.e.
construction of Wannier functions, the
perturbative descript ion of the superexchange
processes changes its appearance.
ts
ts
ts
Degenerate Hubbard model
d-p model
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The extended Hubbard model (EHM) magnetic
interactions in EHM
t
U Intra-orbital repulsion U Inter-orbital
repulsion V Inter-orbital repulsion J
Inter-orbital exchange
V
fi
U
U-J

• Direct exchange
• Kinetic exchange
• (AF channel)
• Kinetic exchange
• (F channel)
• Double exchange

i
i
-J
J
t
j
i
UJ
t
U-J
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Superexchange
Half-filled d-shell
U
d5
d5
d5
d6
d5
d5
t
t
overlap
U
Formation of Wannier orbital
overlap
Existence of the transfer
Antiferromagnetic exchange interaction
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Superexchange
d3-d3 (less than half-filled)
d3
d3
d3
d4
d3
d3
t
t
J
overlap
J
No overlap
Existence of the transfer
Ferromagnetic direct exchange
Antiferromagnetic exchange interaction
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Superexchange
d8-d8 (more than half-filled)
d8
d8
d8
d9
d8
d8
t
t
J
overlap
J
Overlap
Existence of the transfer
Anti-Ferromagnetic exchange
Antiferromagnetic exchange interaction
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Superexchange
d3-d8 (less than half-filled more than
half-filled)
d3
d8
d8
d4
d3
d8
t
t
J
overlap
J
Overlap
Existence of the transfer
Anti-Ferromagnetic exchange
Ferromagnetic exchange interaction
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Double exchange
d3- d4-shell
d3
d4
d4
d4
d4
d3
t
t
overlap
overlap
Localized electrons
Itinerant electrons
Ferromagnetic exchange interaction
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Anti-ferromagnetic magnetic structures
Some magnetic structures of La1-xSrxMnO3
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Stoner Theory v.s. Slater-Kanamori Theory

• The mean-field approach gives the Stoner theory
  of ferromagnetism in the Hubbard model.
• The exchange splitting is derived from the
  Hubbard interaction.

D exchange splitting
Magnetic
The Slater-Kanamori Theory

• If we consider the correlation-effect for
  paramagnetic states, the Stoner criterion should
  be qualitatively modified.

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The Mean-Field Method I.
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The Mean-Field Method II.
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The Mean-Field Method III.
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The Mean-Field Method IV.
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The Slater Theory I.
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The Slater Theory II.
E
Energy levels of triplets
Energy levels of singlets
U/N
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Effect of other electrons

• Existence of the Fermi sea restricts the states
• (scattered states k1-q and k2q ) to the
  unoccupied states.
• Effect of the other electrons is treated by
• the effective potential energy.
• This potential energy is included in e(k,
  k).
• The effective value of U in the paramagnetic
• state can be defiend as,

k3
k4

k1
k2
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The Kanamori Theory I.

• For the many-electron systems, Kanamori showed
  that U should be rewritten by Ueff in the Stoner
  criterion.
• Ueff comes from effects of multiple scatterings
  and can be written as,
• If we note that. when U?8,
• we see that
• does not hold for usual band structures with
  DF1/W.

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The Kanamori Theory II.

• However, the Slater-Kanamori criterion may be
  satisfied, if the density of states has a form
  like (a) below.
• Such DOS can be found in Ni as shown in (b).

(b)
(a)
DOS
E
Spectral function of Ni (d band at X). Springer
et al., PRB 80 (1998) 2389.
Schematic picture of DOS in Ni.
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The Hubbard models with various lattice structures
Kanamori theory at low density Kanamori (1963)
Nagaoka theory for 1 hole from h.f. Nagaoka (1964)
Ferrimagnetism Lieb(1989) Flatband
ferromagnetism Mielke(1991) Tasaki(1992). Ferroma
gnetism in Multi-band models Tanaka Idogaki
Nagaoka ferromagnetism at finite density Kohno
(1997)
Double exchange Nagaoka Kubo, Tsunetsugu,
YanagisawaShimoi
Flatband ferromagnetism to Nagaoka F Penc. et al.
SakamotoKubo WatanabeMiyashita, DaulNoak
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Phase diagram of the Hubbard model inthe
strong-correlation limit

• Liebs theorem ? S0 (or NA-NB/2) for bipartite
  lattices.
• Nagaokas theorem ? SSmaxNe/2 for U8

t/U
S0 (or NA-NB/2)
??
The half filling
0
1
2
3
Nh
SSmaxNe/2
number of holes in a half-filled system
Nh
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Stability of ferromagnetic phase in the Hubbard
model
Hanish, Uhrig Muller-Hartmann, PRB 56 (1997)
13960.
2D t-t-U model
HM on FCC
(a) Identical DOS for the fcc and hcp lattices
(t-1) and DOS for the fcc lattice with
additional next-nearest neighbor hopping
(tt/2), (b) phase diagram (ngt1)Nagaoka
instability lines on the fcc lattice for Gw and
NN and on the t-t-fcc lattice with tt/2 for Gw.
t-t-U model on the square lattice for
t?t/2 t1-t0,0.1,0.2,0.3,1/3 (a)
DOS rt-t(e) for t,tgt0, (b) Nagaoka
instability lines for a Gutzwiller single spin
flip.
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Ferromagnetic phase of the t-J model
Putikka-Luchini-Ogata PRL 69 (1992) 2288.
The result of the high-temperature expansion
method concludes that the fully polarized
ferromagnetic phase is limited to the Nagaoka
state.
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The Nagaoka theorem

• Consider a Hamiltonian for the Hubbard model with
  U8.
• Here, N, Ne and S denote the total number
  of sites, the total number of electrons and the
  total spin. PG is the Gutzwiller projection.
• Theorem 1
• If tij?0 and if NeN-1, one of the lowest energy
  eigen states is a fully spin-polarized state with
  SSmaxNe/2.
• Theorem 2
• Furthermore, if a graph defined by tij satisfy
  the connectivity condition, the lowest state
  given by the theorem 1 is the unique ground state
  up to trivial (2S1)-fold degeneracy.
• If one finds a proper unitary transformation,
  these theorems are applicable for models with
  tij?0.
• The connectivity condition guarantees that every
  possible spin configuration within Szconstant is
  generated by motion of a hole.

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The s-d exchange model

• Exchange interaction between the impurity spin S
  and a spin s of a conduction electron

Effective magnetic field by S (assumed to be a
classical spin)
Magnetic impurities in a metal A magnetic moment
on the impurity induce magnetic polarization in
the conduction electrons in the metal, which
becomes an effective magnetic field on other
atoms.
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Derivation of integral I(r)
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Derivation of integral I(r)
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Derivation of integral I(r)
CVI
Fig. 1 Closed path C for integration.
R
CII
CIV
CVI
CI
CV
1
-1
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Derivation of integral I(r)
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Derivation of integral I(r)
Derivation of integral I(r)
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Derivation of integral I(r)