Electronic structure of Strongly
Correlated Materials: A
Dynamical Mean Field
Perspective.
Kristjan Haule,
Physics Department and
Center for Materials Theory
Rutgers University
Collaborators: Ji-Hoon Shim, S.Savrasov, G.Kotliar
ES 07 - Raleigh
Standard theory of solids
Band Theory: electrons as waves: Rigid non-dipersive band
picture: En(k) versus k
Landau Fermi Liquid Theory applicable
Very powerful quantitative tools: LDA,LSDA,GW
Predictions:
•total energies,
•stability of crystal phases
•optical transitions
•……
Strong correlation –
Standard theory fails
• Fermi Liquid Theory does NOT work . Need
new concepts to replace rigid bands picture!
• Breakdown of the wave picture. Need to
incorporate a real space perspective (Mott).
• Non perturbative problem.
Universality of the Mott transition
Crossover: bad insulator to bad metal
Critical point
First order MIT
V2O3
1B HB model
(DMFT):
Ni2-xSex
k organics
Delocalization
Localization
Basic questions
• How to describe the physics of strong correlations close
to the Mott boundary?
• How to computed spectroscopic quantities (single particle
spectra, optical conductivity phonon dispersion…) from
first principles?
• New concepts, new techniques….. DMFT maybe simplest
approach to meet this challenge
DMFT + electronic structure method
Basic idea of DMFT: reduce the quantum many body problem to a problem
of an atom in a conduction band, which obeys DMFT self-consistency condition
(A. Georges et al., RMP 68, 13 (1996)).
DMFT in the language of functionals: DMFT sums up all local diagrams in BK functional
Basic idea of DMFT+electronic structure method (LDA or GW):
For less correlated bands (s,p): use LDA or GW
For correlated bands (f or d): with DMFT add all local diagrams
(G. Kotliar S. Savrasov K.H., V. Oudovenko O. Parcollet and C. Marianetti, RMP 2006).
Effective (DFT-like) single particle
spectrum consists of delta like peaks
DMFT stectral function contains renormalized
quasiparticles and Hubbard bands
LDA+DMFT
(G. Kotliar et.al., RMP 2006).
observable of interest is the "local“ Green's functions (spectral function)
Exact functional of the local Green’s function exists, its form unknown!
Currently Feasible approximations: LDA+DMFT:
LDA functional
ALL local diagrams
Variation gives st. eq.:
Generalized Q. impurity problem!
Exact “QMC” impurity solver,
expansion in terms of hybridization
K.H. Phys. Rev. B 75, 155113 (2007)
P. Werner, Phys. Rev. Lett. 97, 076405 (2006)
General impurity problem
k
Diagrammatic expansion in terms of hybridization D
+Metropolis sampling over the diagrams
•Exact method: samples all diagrams!
•Allows correct treatment of multiplets
Volume of actinides
Trivalent metals with nonbonding f shell
f’s participate in bonding
Partly localized,
partly delocalized
Anomalous Resistivity
Maximum metallic
resistivity:
s=e2 kF/h
Fournier & Troc (1985)
Dramatic increase of specific heat
Heavy-fermion behavior
in an element
NO Magnetic moments!
Pauli-like from melting to lowest T
No curie Weiss up
to 600K
Curium versus Plutonium
nf=6 -> J=0
closed shell
(j-j: 6 e- in 5/2 shell)
(LS: L=3,S=3,J=0)
One hole in the f shell
No magnetic moments,
large mass
Large specific heat,
Many phases, small or large volume
One more electron in the f shell
Magnetic moments! (Curie-Weiss
law at high T,
Orders antiferromagnetically at
low T)
Small effective mass (small
specific heat coefficient)
Large volume
Standard theory of solids:
DFT:
All Cm, Am, Pu are magnetic in LSDA/GGA
LDA: Pu(m~5mB), Am (m~6mB) Cm (m~4mB)
Exp: Pu (m=0),
Am (m=0)
Cm (m~7.6mB)
Non magnetic LDA/GGA predicts volume up to 30% off.
In atomic limit, Am non-magnetic, but Pu magnetic with spin ~5mB
Many proposals to explain why Pu is non magnetic:
 Mixed level model (O. Eriksson, A.V. Balatsky, and J.M. Wills) (5f)4 conf. +1itt.
 LDA+U, LDA+U+FLEX (Shick, Anisimov, Purovskii) (5f)6 conf.
Cannot account for anomalous transport and thermodynamics


Can LDA+DMFT account for anomalous properties of actinides?
Can it predict which material is magnetic and which is not?
Very strong multiplet splitting
Atomic multiplet splitting crucial -> splits Kondo peak
Atom F2
F4
F6
x
U
8.513 5.502 4.017 0.226
93
Np
9.008 5.838 4.268 0.262
94
Pu
8.859 5.714 4.169 0.276
95
Am
9.313 6.021 4.398 0.315
96
Cm
10.27 6.692 4.906 0.380
Used as input to DMFT calculation - code of R.D. Cowan
Increasing F’s an SOC
N
92
Starting from magnetic solution,
Curium develops antiferromagnetic long range order below Tc
above Tc has large moment (~7.9mB close to LS coupling)
Plutonium dynamically restores symmetry -> becomes paramagnetic
DOS (states/eV)
DOS (states/eV)
4
-Plutonium
3
Total DOS
f DOS
2
1
0
4 -6
-4
Curium
-2
0
Total DOS
f, J=5/2,jz<0
f, J=7/2,jz<0
3
2
2
f, J=5/2,jz>0
f, J=7/2,jz>0
4
6
4
6
1
0
-6
-4
-2
0
ENERGY (eV)
2
J.H. Shim, K.H., G. Kotliar, Nature 446, 513 (2007).
Multiplet structure crucial for correct Tk in Pu (~800K)
and reasonable Tc in Cm (~100K)
Without F2,F4,F6: Curium comes out paramagnetic heavy fermion
Plutonium weakly correlated metal
DOS (states/eV)
DOS (states/eV)
4
-Plutonium
3
Total DOS
f DOS
2
1
0
4 -6
-4
Curium
-2
0
Total DOS
f, J=5/2,jz<0
f, J=7/2,jz<0
3
2
2
f, J=5/2,jz>0
f, J=7/2,jz>0
4
6
4
6
1
0
-6
-4
-2
0
ENERGY (eV)
2
Magnetization of Cm:
Valence histograms
Density matrix projected to the atomic eigenstates of the f-shell
(Probability for atomic configurations)
f electron
fluctuates
between these
atomic states on
the time scale
t~h/Tk
(femtoseconds)
Pu partly f5 partly f6
J=6,g =1
J=5/2, g =0
J=7/2,g =0
J=9/2,g =0
J=0,g =0
J=1,g =0
J=2,g =0
J=3,g =0
J=4,g =0
J=5,g =0
J=4,g =0
0.3
J=5,g =0
Nf =6
Nf =5
Nf =4
J
-Plutonium
J=3,g =1
J=2,g =1
J=1,g =0
J=2,g =0
Probability
0.6
•5 electrons 80%
Nf =6
•6 electrons 20%
J=6,g =0
J=5,g =0
J=4,g =0
J=3,g =0
J=2,g =0
0.3
J=7/2,g =0
Curium
0.6
Probabilities:
Nf =8
Nf =7
J=6,g =0
J=5,g =0
J=4,g =0
J=3,g =0
J=2,g =0
J=1,g =0
J=0,g =0
Probability
0.0
0.9
•4 electrons <1%
0.0
-6
-4
-2
0
ENERGY (eV)
2
4
6
One dominant atomic state – ground state of the atom
J.H. Shim, K. Haule, G. Kotliar, Nature 446, 513 (2007).
Fingerprint of atomic multiplets
- splitting of Kondo peak
Gouder , Havela PRB
2002, 2003
Photoemission and valence in Pu
|ground state > = |a f5(spd)3>+ |b f6 (spd)2>
Af(w)
approximate decomposition
f5<->f6
f5->f4
f6->f7
Probe for Valence and Multiplet structure: EELS&XAS
5f7/2
A plot of the X-ray absorption
as a function of energy
5f5/2
4d5/2->5f7/2 &
4d5/2->5f5/2
hv
4d3/2
4d5/2
Core splitting~50eV
Excitations from 4d core to 5f valence
core
valence
Electron energy loss spectroscopy (EELS) or
X-ray absorption spectroscopy (XAS)
Measures unoccupied valence 5f states
Probes high energy Hubbard bands!
4d3/2->5f5/2
Core splitting~50eV
Energy loss [eV]
Branching ration B=A5/2/(A5/2+A3/2)
B=B0 - 4/15<l.s>/(14-nf)
2/3<l.s>=-5/2(B-B0) (14-nf)
LDA+DMFT
One measured quantity B, two unknowns
Close to atom (IC regime)
Itinerancy tends to decrease <l.s>
[a] G. Van der Laan et al., PRL 93, 97401 (2004).
[b] G. Kalkowski et al., PRB 35, 2667 (1987)
[c] K.T. Moore et al., PRB 73, 33109 (2006).
[d] K.T. Moore et al., PRL in press
Specific heat
Could Pu be close to f6 like Am?
(Shick, Anisimov, Purovskii) (5f)6 conf
Purovskii et.al. cond-mat/0702342:
f6 configuration gives smaller g
in  Pu than a Pu
Americium
f6 -> L=3, S=3, J=0
Mott Transition?
"soft" phase
f localized
"hard" phase
f bonding
A.Lindbaum, S. Heathman, K. Litfin, and Y. Méresse,
Phys. Rev. B 63, 214101 (2001)
J.-C. Griveau, J. Rebizant, G. H. Lander, and G.Kotliar
Phys. Rev. Lett. 94, 097002 (2005)
Am within LDA+DMFT
Large multiple effects: F(0)=4.5 eV
S. Y. Savrasov, K.H., and G. Kotliar
Phys. Rev. Lett. 96, 036404 (2006)
F(2)=8.0 eV
F(4)=5.4 eV
F(6)=4.0 eV
Am within LDA+DMFT
from J=0 to J=7/2
Comparisson with experiment
V=V0 Am I
V=0.76V0 Am III
V=0.63V0 Am IV
nf=6.2
nf=6
•“Soft” phase not in local moment regime
since J=0 (no entropy)
•"Hard" phase similar to  Pu,
Kondo physics due to hybridization, however,
nf still far from Kondo regime
Exp: J. R. Naegele, L. Manes, J. C. Spirlet, and W. Müller
Phys. Rev. Lett. 52, 1834-1837 (1984)
Theory: S. Y. Savrasov, K.H., and G. Kotliar
Phys. Rev. Lett. 96, 036404 (2006)
What is captured by single
site DMFT?
•Captures volume collapse transition (first order Mott-like transition)
•Predicts well photoemission spectra, optics spectra,
total energy at the Mott boundary
•Antiferromagnetic ordering of magnetic moments,
magnetism at finite temperature
•Branching ratios in XAS experiments, Dynamic valence fluctuations,
Specific heat
•Gap in charge transfer insulators like PuO2
Beyond single site DMFT
What is missing in DMFT?
•Momentum dependence of the self-energy m*/m=1/Z
•Various orders: d-waveSC,…
•Variation of Z, m*,t on the Fermi surface
•Non trivial insulator (frustrated magnets)
•Non-local interactions (spin-spin, long range Columb,correlated hopping..)
Present in DMFT:
•Quantum time fluctuations
Present in cluster DMFT:
•Quantum time fluctuations
•Spatially short range quantum fluctuations
Plaquette DMFT for the Hubbard model
as relevant for cuprates
Large onsite
component
anomalous
SE-SC
Small next-nearest neighbor component
(except in the underdoped regime)
Complicated Fermi surface
evolution with temperature
underoped phase
“fermi arcs”
“arcs” decrease
with T
Superconducting phasebanana like Fermi surface
Conclusion
• Pu and Am (under pressure) are unique
strongly correlated elements. Unique mixed
valence.
• They require, new concepts, new
computational methods, new algorithms,
DMFT!
• Cluster extensions of DMFT can describe
many features of cuprates including
superconductivity and gapping of fermi
surface (pseudogap)
Many strongly correlated compounds await
the explanation:
CeCoIn5, CeRhIn5, CeIrIn5
Photoemission of CeIrIn5
Photoemission of CeIrIn5
LDA+DMFT DOS
Comparison
to experiment
Optics of CeIrIn5
LDA+DMFT
Experiment:
K.S. Burch et.al.,
cond-mat/0604146
Optimal doping: Coherence
scale seems to vanish
underdoped
scattering
at Tc
optimally
Tc
overdoped
New continuous time QMC, expansion in terms of hybridization
Diagrammatic expansion in terms of hybridization D
+Metropolis sampling over the diagrams
Contains all: “Non-crossing” and all crossing diagrams!
Multiplets correctly treated
k
General impurity problem
Conclusions
• LDA+DMFT can describe interplay of lattice and electronic
structure near Mott transition. Gives physical connection
between spectra, lattice structure, optics,....
– Allows to study the Mott transition in open and closed
shell cases.
– In actinides and their compounds, single site LDA+DMFT
gives the zero-th order picture
• 2D models of high-Tc require cluster of sites. Some aspects
of optimally doped regime can be described with cluster
DMFT on plaquette:
– Large scattering rate in normal state close to optimal doping
Basic questions
• How does the electron go from being localized to
itinerant.
• How do the physical properties evolve.
• How to bridge between the microscopic
information (atomic positions) and experimental
measurements.
• New concepts, new techniques….. DMFT
simplest approach to meet this challenge
Coherence incoherence crossover in the
1B HB model (DMFT)
Phase diagram of the HM with partial frustration at half-filling
M. Rozenberg et.al., Phys. Rev. Lett. 75, 105 (1995).
Singlet-type Mott state (no entropy) goes mixed valence under pressure
-> Tc enhanced (Capone et.al, Science 296, 2364 (2002))
Overview
• DMFT in actinides and their compounds
(Spectral density functional approach).
Examples:
– Plutonium, Americium, Curium.
– Compounds: PuAm
Observables:
– Valence, Photoemission, and Optics, X-ray
absorption
Why is Plutonium so special?
Heavy-fermion behavior
in an element
Typical heavy fermions
(large mass->small Tk
Curie Weis at T>Tk)
No curie Weiss up to 600K
Why is Plutonium so special?
Heavy-fermion behavior
in an element
Overview of actinides
Many phases
Two phases of Ce, a and g
with 15% volume difference
25% increase in volume between a and  phase
f-sumrule for core-valence conductivity
Similar to optical conductivity:
Current:
Expressed in core valence orbitals:
The f-sumrule:
can be expressed as
Branching ration B=A5/2/(A5/2+A3/2)
B=B0 - 4/15<l.s>/(14-nf)
B0~3/5
Branching ratio depends on:
•average SO coupling in the f-shell <l.s>
•average number of holes in the f-shell nf
B.T. Tole and G. van de Laan, PRA 38, 1943 (1988)
4d5/2->5f7/2
4d3/2->5f5/2
Core splitting~50eV
Energy loss [eV]
A5/2 area under the 5/2 peak
Optical conductivity
2p->5f
5f->5f
Pu: similar to heavy fermions (Kondo type conductivity)
Scale is large MIR peak at 0.5eV
PuO2: typical semiconductor with 2eV gap, charge transfer
Spectral density functional theory
(G. Kotliar et.al., RMP 2006).
observable of interest is the "local“ Green's functions (spectral function)
Currently feasible approximations: LDA+DMFT:
Variation gives st. eq.:
Generalized Q. impurity problem!
Pu-Am mixture, 50%Pu,50%Am
Lattice expands -> Kondo collapse is expected
Our calculations suggest
charge transfer
Pu  phase stabilized by shift to
mixed valence nf~5.2->nf~5.4
f6: Shorikov, et al., PRB 72, 024458 (2005);
Shick et al, Europhys. Lett. 69, 588 (2005).
Pourovskii et al., Europhys. Lett. 74, 479 (2006).
Hybridization decreases, but nf
increases,
Tk does not change
significantly!