Many-body Electronic Structure
of Actinides: A Dynamical Mean
Field Perspective.
Kristjan Haule,
Physics Department and
Center for Materials Theory
Rutgers University
Collaborators: G.Kotliar, Ji-Hoon Shim, S. Savrasov
Half Moon Bay
Overview
• DMFT in actinides and their compounds (Spectral
density functional approach).
Examples:
– Plutonium, Americium, Curium.
– Compounds: PuO2, PuAm
Observables:
– Valence, Photoemission, and Optics, X-ray absorption
• Extensions of DMFT to clusters.
Examples:
– Coherence in the Hubbard and t-J model
New general impurity solver (continuous time QMC)
developed (can treat clusters and multiplets)
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
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).
DMFT + electronic structure method
Basic idea of DMFT: reduce the quantum many body problem to a one site
or a cluster of sites problem, in a medium of non interacting electrons obeying a
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
Effective (DFT-like) single particle
Spectrum consists of delta like peaks
Spectral density usually contains renormalized
quasiparticles and Hubbard bands
How good is single site DMFT for f systems?
Elements:
Compounds:
f5
L=5,S=5/2 J=5/2
PuO2
PuAm
f7
L=0,S=7/2 J=7/2
f6
L=3,S=3 J=0
Overview of actinides
Many phases
Two phases of Ce, a and g
with 15% volume difference
25% increase in volume between a and d phase
Overview of actinides?
Trivalent metals with nonbonding f shell
f’s participate in bonding
Partly localized,
partly delocalized
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
Plutonium puzzle?
Ga doping stabilizes d-Pu
at low T, lattice contraction
Am doping -> lattice expansion
Expecting unscreened moments!
Does not happen!
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
Density functional based electronic structure calculations:
All Cm, Am, Pu are magnetic in LDA/GGA
LDA: Pu(m~5mB), Am (m~6mB) Cm (m~4mB)
Exp: Pu (m=0),
Am (m=0)
Cm (m~7.9mB)
Non magnetic LDA/GGA predicts volume up to 30% off.
Treating f’s as core overestimates volume of d-Pu,
reasonable volume for Cm and Am
Can LDA+DMFT predict which material
is magnetic and which is not?
Very strong multiplet splitting
Atomic multiplet splitting crucial
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
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
d-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
2
4
6
1
0
-6
-4
-2
0
ENERGY (eV)
cond-mat/0611760
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
d-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
Valence histograms
Density matrix projected to the atomic eigenstates of the f-shell
(Probability for atomic configurations)
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
d-Plutonium
J=3,g =1
J=2,g =1
J=1,g =0
J=2,g =0
Probability
0.6
Nf =6
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
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
0.0
-6
-4
-2
0
ENERGY (eV)
2
4
6
f electron
fluctuates
between these
atomic states
on the time
scale t~h/Tk
(femtoseconds)
One dominant atomic state – ground state of the atom
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
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]
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
B=B0 - 4/15<l.s>/(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).
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
Pu-Am mixture, 50%Pu,50%Am
Lattice expands -> Kondo collapse is expected
Could Pu be close to f6 like Am?
Inert shell can not account for large cv anomaly
Large resistivity!
Absence of preadge structure in XAS
Our calculations suggest
charge transfer
Pu d 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!
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. Haule, 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 a Ce or d 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. Haule, 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
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
k
General impurity problem
Contains all: “Non-crossing” and all crossing diagrams!
Multiplets correctly treated
Phys. Rev. B 75, 155113 (2007)
Hubbard model self-energy on imaginary axis, 2x2
Low frequency
very different
Far from Mott transition
coherent
Close to Mott transition
Very incoherent
Optimal doping in
the t-J model
(d~0.16)
has largest low energy
self-energy
Very incoherent
at optimal doping
Optimal doping in the
Hubbard model (d~0.1)
has largest low energy
self-energy
Very incoherent
at optimal doping
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