Ferromagnetism and Metal-Insulator Transition in Hubbard Model ...

Ferromagnetism and Metal-Insulator
Transition in Hubbard Model with Alloy
Disorder
Krzysztof Byczuk
Institute of Physics, Augsburg University
Institute of Theoretical Physics, Warsaw University
October 13th, 2005

Main results
• New collective effects induced by correlation and disorder
• Enhancement of Tc in binary alloy ferromagnets
• New Mott–Hubbard metal–insulator transition at n = 1
K. Byczuk, M. Ulmke, D. Vollhardt
Phys. Rev. Lett. 90, 196403 (2003)
K. Byczuk, W. Hofstetter, D. Vollhardt
Phys. Rev. B 69, 04512 (2004)
K. Byczuk, M. Ulmke
Eur. Phys. J. B 45, 449 (2005)

Collaboration
• Walter Hofstetter - Aachen, Germany
• Martin Ulmke - FGAN - FKIE, Wachtberg, Germany
• Dieter Vollhardt - Augsburg University, Germany
Plan of the talk
1. Introduction
• localized vs. itinerant FM
• Mott-Hubbard MIT
• binary alloy disorder
• alloy FM in Nature
2. Earlier result within DMFT on pure FM
3. Our results on binary alloy FM
• enhancement of Tc
• magnetization and Curie–Weiss law
• Mott–Hubbard MIT at n = 1
4. Conclusions

Ferromagnetism of local moments
Exchange (Heisenberg) coupling
H = X
Jij Si · Sj
Saturated magnetization M(T = 0) = µBS
ij
T > Tc
T < Tc
J < 0 → FM appears at Curie temperature Tc

Itinerant Ferromagnetism
Dynamical way to make FM
To reduce interaction energy electrons prefer FM state
due to Pauli principle E FM
int
However, |E FM
kin
| < |EPM|
!
kin
= 0
t

Mott-Hubbard MIT at n = 1
U ≪ |tij|, ∆p = 0 U ≫ |tij|, ∆r = 0
typical intermediate coupling problem Uc ≈ |tij|

Alloy Band Splitting
Binary alloy disorder (alloys A1−xBx, e.g Fe1−xCox)
P(ɛi) = xδ(ɛi + ∆
2 ) + (1 − x)δ(ɛi − ∆
2 )
H = P
i ɛi + t P
ij a†
i aj
alloy band splitting for ∆ W in any dimension
intermediate “coupling” problem !!!
physical quantity: O = R dɛP(ɛ) <
E
∆
Ô(ɛ) >
1−x
DOS
x

Alloy Ferromagnets in Nature I
Piryts: T1−x(T+1)xS2, T=Fe, Co, Ni, Cu, Zn
t 6
2gen g
with n = 0, 1, 2, 3, 4
Fe1−xCoxS2, max Tc(x) @ x ≈ 0.76
Jarrett et al., PRL 1968, Leighton 2004

Alloy Ferromagnets in Nature II
T c (critical temperature: K)
110
109
108
107
106
105
104
103
Theory
Annealed for 14 days at 875 o C
102
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x (amount of Ge)
MT2X2 ternary intermetallic alloys
UCu2Si2−xGex, max Tc(x) @ x ≈ 1.6
Silva Neto et al., PRL 2003
Si and Ge isovalent, only structural disorder

Alloy Ferromagnets in Nature III
Fe weak FM, Co strong FM, bcc alloy
Fe1−xCox, max Tc(x) @ x ≈ 0.5
Pratzer et al., PRL 2003

Alloy Ferromagnets in Nature IV
Alloy ruthenates
SrRu1−xMnxO3, FM Met–AF Ins
Cao et al., cond-mat/0409157

Hubbard model to capture right physics
ɛ0 + U
ɛ0
H = X
ɛ0niσ + X
iσ
t
ijσ
• long history, many contradictions
• exactly solvable in d = 1
U
• exactly solvable in d = ∞ (DMFT)
• how to approximate in 1 < d < ∞?
tij a †
iσ ajσ + U X
t
i
E
U=0
E
E F
ni↑ni↓
DOS: ρ(E) = P R
n dx|ψn(x)|δ(E − ɛn)
t=0
U
2(n/2)
ρ( E)
ρ( E)
2(1−n/2)

Physical picture, n = 1
E+U
E
atomic levels
|t|=0 |t|>0
U
UHB
LHB
spin flip on central site
E+U
dynamical processes with spin-flips inject states into correlation gap
giving a quasiparticle resonance
E
at U = Uc resonance disapears
gaped insulator

Route to FM in one-band Hubbard (DMFT)
H = P ta †
iσ ajσ + U P ni↑ni↓
1.4
1.2
1
0.8
N
0.6
0 (ε)
0.4
0.2
a=0
a=0.5
a=0.9
a=0.98
a=1
1.4
N 0 (ε)
0.7
0
−2 −1.8
ε
−1.6
0
−2 −1 0
ε
1 2
Wahle et al. 1998
DOS asymmetry - a
0.06
0.05
0.04
0.03
0.02
a=0.97
0.01
a=0.98
a=0.99
a=1 0
Interaction - U
T
T
0.06
0.05
0.04
0.03
0.02
0.01
F
a=1
a=0.98
a=0.97
0 0.2 0.4 0.6 0.8 1
n
F
U=8
U=6
U=4
0
0 0.2 0.4 0.6 0.8 1
n
P
P
U = 4
a = 0.98

FCC d =∞ FM in one-band Hubbard
M
0.6
0.4
0.2
√ π(1+ √ 2ɛ)
N 0 (ɛ) = exp[−1+√ 2ɛ
2
U=4
n=0.58
0
0.03 0.05 0.07
T
0.09
Ulmke et al. 1998
]
4
χ −1
F
T
N(ω)
0.06
0.04
0.02
0
1.2
0.8
0.4
P
F
0 0.2 0.4 0.6 0.8 1.0
n
U=4
n=0.58
T=0.04
M=0.4
0
−2.0 0.0 2.0
ω−µ
4.0 6.0
U = 2, 4, 5

FM in binary alloy itinerant electrons
Anderson–Hubbard Hamiltonian
H = X
ij,σ
tijĉ †
iσ ĉjσ + X
iσ
ɛiˆniσ + U X
i
ˆni↑ˆni↓
where ɛ is random variable with bimodal PDF
„
P(ɛ) = xδ ɛ + ∆
«
„
+ (1 − x)δ ɛ −
2
∆
«
2
Physical observable averaged arithmetically
Z
〈· · · 〉dis = dɛP(ɛ)(· · · )
d = ∞ FCC DOS stabilizes FM
N 0 (ω) = exp[−1+√ 2ω
2 ]
q
π(1 + √ 2ω)

Dynamical Mean–Field Theory
Local Green function - Hilbert transform of DOS with self–energy
Gσn =
Z
N
dɛ
0 (ɛ)
iωn + µ − Σσn − ɛ
expressed by path integral, which is calculated with
Hubbard–Stratonovich and QMC over auxiliary Ising spins
Gσn = −
*R D [cσ, c ⋆
⋆
σ ] cσncσneAi {cσ,c ⋆ σ ,G−1 σ }
R D [cσ, c ⋆ σ ] eA i {cσ,c ⋆ σ ,G−1
σ }
single impurity action for each ɛi = ±∆/2
Ai{cσ, c ⋆
σ
X
, G−1
σ } = c
n,σ
⋆
σnG−1 σncσn−ɛi X
σ
k–integrated Dyson equation for Weiss function
G −1
σn
= G−1
σn
+ Σσn
Zβ
0
+
dis
dτnσ(τ)− U
2
X
σ
Zβ
0
dτc ∗
∗
σ (τ)cσ(τ)c −σ (τ)c−σ(τ)

T c
T c
T c
T c
Curie temperature
0.04
0.03
0.02
0.01
0
0.05
0.04
0.03
0.02
0.01
0
0.03
0.02
0.01
0
0.08
0.06
0.04
0.02
0
∆=1
∆=4
Mott
n=0.3
U=2
U=6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
concentration, x
n=0.7
U=2
U=6
∆=1
∆=4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
concentration, x
Mott
T c
T c
0.06
0.04
0.02
0
0 1 2 3 4
0.038
0.028
0.018
0.008
(b)
Tc(x) increases !!! at some cases
T c
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
U=2
(a)
U=4
n>x
U=6
n

E
∆
Is there an alloy band splitting at U > 0?
1−x
DOS
x
U = 4, n = 0.3, n = 0.5, T = 0.071, MEM
SPECTRAL DENSITY
LHB
∆=0.0
UHB
∆=0.4
∆=1.0
∆=1.4
∆=1.8
−1 0 1 2 3 4 5
ω−µ
Subtle interplay between ∆ and U increases Tc!

E
∆
Why is Curie temperature enhanced?
1−x
DOS
x
n < 2x → neff = n
x
Tc = xT p
c
T c
T
p
c
(n)
Tc(x) increases !!! at some cases
n > 2x → neff = n−2x
1−x
Tc = (1 − x)T p
c
T c
T
p
(
c n eff )
(n)
Tc
x
n n eff
U1
U
0.0 0.5 1.0
T c (n)
n
eff
1−x
Tc p (neff )
U 1
Tc p (n)
2
U 2
0.0 0.5 1.0
n
T
0.06
0.04
0.02
n
n
P
0
0 0.2 0.4 0.6 0.8 1.0
n
F
Good for large U
Good for small U

T c
T c
T c
T c
Magnetization and Curie-Weiss law
0.04
0.03
0.02
0.01
0
0.05
0.04
0.03
0.02
0.01
0
0.03
0.02
0.01
0
0.08
0.06
0.04
0.02
0
∆=1
∆=4
Mott
n=0.3
U=2
U=6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
concentration, x
n=0.7
U=2
U=6
∆=1
∆=4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
concentration, x
Mott
M(T) , χ −1 (T)
0.6
0.5
0.4
0.3
0.2
0.1
0
x=0.6, U=6, n=0.3
x=0.1, U=2, n=0.7
If ∆ ≫ W and n < 2x → Ms = n but n > 2x → Ms = n − 2x
∆=4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
kT
M(T)
Ms
= tanh[TcM(T)
TMs ]
χ(T) = C
T −Tc
, where C ≈ Ms
C1 C = 0.623 close to
2 3
5

T c
T c
T c
T c
Mott–Hubbard metal–insulator transition
0.04
0.03
0.02
0.01
0
0.05
0.04
0.03
0.02
0.01
0
0.03
0.02
0.01
0
0.08
0.06
0.04
0.02
0
∆=1
∆=4
Mott
n=0.3
U=2
U=6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
concentration, x
n=0.7
U=2
U=6
∆=1
∆=4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
concentration, x
Mott
A( ω)
µ
LHB
µ
2N L
ω
xNL
µ µ
U
METAL
U
µ
LAB UAB
2xNL
2(1−x)NL
µ
xNL
∆
INSULATOR
If n = x (or 1 + x) Mott-Hubbard MIT occures for ∆ > √ x and U > 6 √ x (or √ 1 − x and 6 √ 1 − x)
SPECTRAL DENSITY
UHB
∆ c
U = 6, x = 0.5, n = 0.5, T = 0.071, MEM
0.8
0.6
0.4
0.2
TEMPERATURE T
0.08
0.06
0.04
0.02
0
PM
FM
0 1 2
DISORDER ∆
∆
U c
0
−2 −1 0 1 2
ω−µ
3 4 5 6 7
PI

Correlated insulators
• alloy Mott insulator
• alloy charge transfer insulator
ε+
∆
ε+
U
U< ∆ U> ∆
UHB
ε + U
ε + ∆
UHB
UAB UAB
ε LHB ε LHB
alloy Mott
insulator
alloy charge transfer
insulator

T c
T c
Quantum critical points
0.04
0.03
0.02
0.01
0
0.05
0.04
0.03
0.02
0.01
0
∆=1
∆=4
Mott
n=0.3
U=2
U=6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
concentration, x
T c
T c
0.03
0.02
0.01
0
0.08
0.06
0.04
0.02
0
n=0.7
U=2
U=6
∆=1
∆=4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
concentration, x
At T = 0 quantum phase transitions: FM met → PM ins or FM met → PM ins (Mott).
Correlation (band-width) controlled, Filling controlled, Alloy concentration controlled Mott MITs.
Is non-Fermi liquid in d < ∞ ? Role of correlations in space.
Mott

Summary
• New collective effects induced by correlation and disorder
• Possibilities of Tc increase in binary alloy ferromagnet
• New Mott–Hubbard metal–insulator transition at n = 1
• Alloy Mott insulator vs. Alloy charge transfer insulator
• Alloy concentration controlled Mott MIT
Outlook
• Tc(x) - QPT ? 2nd vs 1st order PT ?
• Multi-band Hubbard model, role of Hund and exchange coupling,
which from our findings are generic for many orbitals ?
• Material specific models ?? LDA+DMFT+disorder ???