Mott insulator

Material Science
I. d Electron systems
1. Electronic structure of transition-metal ions
(April 9)
2. Crystal structure and band structure (April16)
3. Mott insulators (April 23)
4. Metal-insulator transition (April 30, May 14)
5. High-temperature superconductivity (May 21)
6. Spin-related phenomena (July 23)
Electronic phase diagram of strongly correlated
electron system at T = 0 K, n = 1
: Electron chemical potential
= Fermi level
1 electrons/atom
T = 0 K
s orbital
1

3. Mott insulators
3.1 Hubbard model
3.2 Hartree-Fock approximation
3.3 Orbital ordering
3.4 Mott-Hubbard vs chargetransfer
type
3.5 Cluster model
3. Mott insulators
3.1 Hubbard model
Hubbard model for single kind of atomic orbitals
(Single-band Hubbard model)
Kinetic energy
Potential energy
= ,
-t
c a + , c a : annihilation, creation operators
t: transfer integral
n a : number operator
U: atomic Coulomb integra
+U
a b a
e.g., d x2-y2 band of CuO 2 plane in high-Tc cuprates
2

3. Mott insulators
3.1 Hubbard model
Super-exchange interaction
-t
X
a b a X
-t
b
Effective spin Hamiltonian = Heisenberg model
Antiferromagnetic
coupling
>0: exchange interaction constant
Various lattices
3. Mott insulators
3.1 Hubbard model
Antiferromagnetic
spin ordering
Frustrated spins
Cu in CuO 2 plane of
high-Tc cuprates
B-site ions of perovskite
lattice
Triangular
lattices
Kagome lattices
B-site ions of spinel
and pyrochlore lattices
3

3. Mott insulators
3.1 Hubbard model
Various lattices
B-site ions of perovskite
lattice
Cu in CuO 2 plane of
high-Tc cuprates
B-site ions of spinel
and pyrochlore lattices
3. Mott insulators
3.1 Hubbard model
U/t 1
(U/2Zt = U/W >> 1)
Metal
Mott insulator
4

3. Mott insulators
3.1 Hubbard model
U/t 1
(U/2Zt = U/W >> 1)
W
Metal
Density of
states
U
W
W
Upper
Hubbard
band
E(d 2 )+E(d 0 )-2E(d 1 ) = U
in general E(d Mott n+1 insulator )+E(d n-1 )-2E(d n ) = U
U
Lower
Hubbard
band
Density of
states
3. Mott insulators
3.1 Hubbard model
3.2 Hartree-Fock approximation
3.3 Orbital ordering
3.4 Mott-Hubbard vs chargetransfer
type
3.5 Cluster model
5

2. Crystal structure and band structure
2.5 Magnetically ordered states
Non-magnetic or Pauliparamagnetic
v MF (r)
state
d x2-y2 band on square lattice
Antiferromagnetic state
v MF (r)
=
r
r
2
=
Fermi surf
AFM
Brillouin
zone
k = dx2-y2 - 2t(cos k x a + cos k y a)
2. Crystal structure and band structure
2.5 Magnetically ordered states
Wave function: Bloch orbital
Mean-field Hamiltonian:
(One-electron Hamiltonian)
Orbital
part
Nuclei +
core electrons Valence
electrons
Spin
part
Schroedinger equation
Eigenfunction:
Eigenvalue:
=n: Band index
= ,
- ~ - J H ( - )
6

1. Electronic structure of transition-metal ions
1.3 Coulomb-exchange interaction
Many-electron system
(e.g., 3 electron system)
Expectation
value:U ’ -J ’
U ”
’
U ’”
”
Coulomb integral
Exchange integral
~ 4-8 eV
Pauli principle ~ 0.5-1 eV
3. Mott insulators
3.2 Hartree-Fock approximation
Wave function: Bloch orbital
Hartree-Fock
Hamiltonian:
(One-electron Hamiltonian)
Hartree-Fock equation
Eigenfunction:
Potential due to
Orbital
part
Nuclei +
core electrons
Spin
part
Valence
electrons
Exchange
potential
(Non-local)
Eigenvalue:
=n: Band index
= k’’ (U kk’’ - J kk’’ )
=0,1
7

3. Mott insulators
3.2 Hartree-Fock approximation
W
W
Upper
Hubbard
band
d0 +U
U
0
Lower d
Hubbard
Density of band
states
Mott insulator
U/t >> 1
Upper
Hubbard
band
W
d0 +U
Density ofU
W
states
0
Lower d
Hubbard
band
3. Mott insulators
3.1 Hubbard model
3.2 Hartree-Fock approximation
3.3 Orbital ordering
3.4 Mott-Hubbard vs chargetransfer
type
3.5 Cluster model
8

Spin, charge, orbital, and lattice degrees of
freedom of strongly correlated electron system
p, d, f orbitals
3. Mott insulators
3.1 Hubbard model
Super-exchange interaction
-t
X
a b a X
-t
b
Effective spin Hamiltonian = Heisenberg model
Antiferromagnetic
coupling
>0: exchange interaction constant
9

orbital1
orbital2
orbital1
orbital2
Super-exchange interaction for degenerate orbitals
-t
X
a
a
-t
-t
-t
b
b
3. Mott insulators
3.3 Orbital ordering
Intermediate-state
energy E =
• 2 nd -order perturbation energy = -2t 2 /E
• U > U’ > U’-J
U
a
U‘ U‘ - J
a
X
-t
b
b
forbidden
Ferro-spin
Antiferro-orbital
1. Electronic structure of transition-metal ions
1.2 Crystal-field splitting
Crystal fields due to anions (e.g., oxygen ions)
Octahedral
coordination
Tetrahedral
coordination
: One-electron energy
( ): Degeneracy (inc. spin)
10Dq ~ 0.5-2 eV
10

3. Mott insulators
3.3 Orbital ordering
W
W
yz
yz
xy , zx
Ferro-spin, antiferro-orbital ordering in YTiO 3 : Ti 3+ (d 1 )
Upper
Hubbard band
Lower
d0 +U’-J H
U’- J H
d
0
Hubbard band
Density of
states
distorted perovskite structure
Y
Ti
O
W
W
Upper Hubbard
band
d0 +U’-J H
zx
xy , yz
U’- J H
zx
0
d
Lower Hubbard
band
Density of
states
yz
zx
3. Mott insulators
3.2 Hartree-Fock approximation
Wave function: Bloch orbital
Hartree-Fock
Hamiltonian:
(One-electron Hamiltonian)
Hartree-Fock equation
Potential due to
Orbital
part
Nuclei +
core electrons
Spin
part
Valence
electrons
Exchange
potential
(Non-local)
Eigenfunction:
Eigenvalue:
Atomic orbitals
=n: Band index
= k’’ (U kk’’ - J kk’’ )
=0,1
11

3. Mott insulators
3.1 Hubbard model
3.2 Hartree-Fock approximation
3.3 Orbital ordering
3.4 Mott-Hubbard vs charge-transfer
type
3.5 Cluster model
3. Mott insulators
3.4 Mott-Hubbard vs charge-transfer type
Hubbard model
p-d model
Transition-metal ion (d orbitals)
Non-metal anion (p orbitals)
12

3. Mott insulators
3.4 Mott-Hubbard vs charge-transfer type
Hubbard model
p-d model
=
3. Mott insulators
3.4 Mott-Hubbard vs charge-transfer type
Mott-Hubbard-type
insulator
Charge-transfer-type
insulator
3d
W
3d
3d
W
U
O 2p
U
O 2p
Neglected
in Hubbard model
3d
U <
Gap ~ U - W
U >
Gap ~ - W
W: Band width
U : Atomic Coulomb energy (Coulomb integral)
: Charge-transfer enrgy
13

2. Crystal structure and band structure
2.1 What determines crystal structure?
Electronegatigvity ~ [Ionization energy I + Electron affinityA]/2
1.0 1.5
0.8 1.0
Energy
Vacuum level
A
I
1.5 1.6 1.5 1.8
1.2 1.8
2.2
3.5 4.0
2.5
S
2.1
large
small
Number of orthogonal R nl (r)
small
large
3. Mott insulators
3.4 Mott-Hubbard vs charge-transfer type
small
Charge-transfer energy
Valence: 2- 1- …. + 2+ 3+ 4+ 5+
1.0 1.5
large
small
3.5 4.0
large
Transition-metal ions
1.5 1.6 1.5 1.8
2.5
S
0.8 1.0
2.1
1.2
1.8
2.2
large
large
small
Non-TM ions
small
small large
14

3. Mott insulators
3.4 Mott-Hubbard vs charge-transfer type
Mott-Hubbard-type
insulator
Charge-transfer-type
insulator
3d
W
3d
3d
W
U
O 2p
U
O 2p
Neglected
in Hubbard model
3d
U <
Gap ~ U - W
U >
Gap ~ - W
W: Band width
U : Atomic Coulomb energy (Coulomb integral)
: Charge-transfer enrgy
2. Crystal structure and band structure
2.1 What determines crystal structure?
Ionic radius
Valence: 2- 1- …. + 2+ 3+ 4+ 5+
large
Units: A
0.74 0.35
1.40 1.33
+ 2+
2- -
1.84 2-
0.67 0.79 0.83 0.69
S
1.38 1.00
+ 2+
3+ 2+ 2+
2+
0.90 0.60 2.24
3+ 6+
2+
2-
0.86
small
small
large
Number of orthogonal R nl (r)
large
small
15

3. Mott insulators
3.4 Mott-Hubbard vs charge-transfer type
large
Atomic Coulomb energy U
Valence: 2- 1- …. + 2+ 3+ 4+ 5+
Units: A
0.74 0.35
1.38 1.00
small large
0.67 0.83 0.69
2+
0.90 0.60
Transition-metal ions
1.40
1.84
2.24
1.33
0.86
small
small
large
3. Mott insulators
3.4 Mott-Hubbard vs charge-transfer type
Mott-Hubbard-type
insulator
Charge-transfer-type
insulator
3d
W
3d
3d
W
U
O 2p
U
O 2p
Neglected
in Hubbard model
3d
U <
Gap ~ U - W
U >
Gap ~ - W
W: Band width
U : Atomic Coulomb energy (Coulomb integral)
: Charge-transfer enrgy
16

3. Mott insulators
3.4 Mott-Hubbard vs charge-transfer type
CT
Mott-Hubbard vs charge-transfer
Valence: 2- 1- …. + 2+ 3+ 4+ 5+
MH
CT
MH
Transition-metal ions
2+
MH
MT
CT
Non-TM ions
CT
MH
CT
3. Mott insulators
3.4 Mott-Hubbard vs charge-transfer type
Zaanen-Sawatzky-Allen phase diagram
p-band metal
Schematic
Charge-transfer
insulator
Mott-Hubbard
insulator
d-band metal
Negative
charge-transfer
energy insulator
p-band metal
Real materials
Charge-transfer
insulator
Mott-Hubbard
insulator
d-band metal
17

3. Mott insulators
3.1 Hubbard model
3.2 Hartree-Fock approximation
3.3 Orbital ordering
3.4 Mott-Hubbard vs chargetransfer
type
3.5 Cluster model
1. Electronic structure of transition-metal ions
1.4 Multiplet splitting
Many-electron system (4 electron system)
3J H -10Dq
3J H
10Dq
n = 4
E: n-electron energy
: one-electon energy
18

3. Mott insulators
3.4 Cluster model
Spinel-type
structure
Cluster model
Perovskite-type
structure
Atomic Coulomb energy: U = E(d n+1 )+E(d n-1 )-2E(d n )
Charge-transfer enrgy: = E(d n+1 L)-E(d n )
Transfer integral: T pd = L: Ligand (p) hole
3. Mott insulators
3.4 Cluster model
Atomic d orbitals
Molecular orbitals consisting of
ligand p orbitals
19

3. Mott insulators
3.4 Cluster model
Optical absorption spectrum of Ni 2+ (d 8 ) ion in NiO
E – N
Charge-transfer
Optical absorption
d n+1 L
Ni
O
Optical absorption
Optical
absorption
p-d hybridization Reduction of U, U’, J H
Finite 10Dq
d n
Cluster model CI theory
Ligand-field theory
M. Imada, A. Fujimori and Y. Tokura, Rev. Mod. Phys. 1998
A. Fujimori and F. Minami, PRB 1983
3. Mott insulators
3.4 Cluster model
CI description of Ni 2+ (d 8 ) ion in NiO
Antibonding states
v
Lu
u
Lv
E – N
6 t pd
d n+1 L
d n
v
u
Hole
p-d hubridization n=8
u = d 3z2-r2 Bonding states Ground state
v = d x2-y2
20

3. Mott insulators
3.4 Mott-Hubbard vs charge-transfer type
Mott-Hubbard-type
insulator
Charge-transfer-type
insulator
3d
W
3d
3d
W
U
O 2p
U
O 2p
Neglected
in Hubbard model
3d
U <
Gap ~ U - W
U >
Gap ~ - W
W: Band width
U : Atomic Coulomb energy (Coulomb integral)
: Charge-transfer enrgy
3. Mott insulators
3.4 Cluster model
A hole doped in the CuO 2 plane: Zhang-Rice singlet
Antibonding states
Hole
v
E – N
6 t pd
d n
U
d n+1 L
v
Lv
S=0
v
Lv
~0.5-1 eV !
p-d hubridization
n=8
Bonding states Ground state
v = d x2-y2
21

3. Mott insulators
3.4 Cluster model
A hole doped in the CuO 2 plane: Zhang-Rice singlet
O
Hole
Cu
3. Mott insulators
3.4 Cluster model
A hole doped in the CuO 2 plane: Zhang-Rice singlet
O
Hole
Cu
22