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
UC Davis
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 expansion
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?
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)
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).
Pu dioxide
d-Pu nf~5.2
PuO2 nf~4.3
3spd electrons + 1f electron of Pu
is taken by 2 oxygens-> f^4
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 for 20% -> Kondo collapse is expected
Could Pu be close to f6 like Am?
Inert shell can not account for large cv anomaly
Large resistivity!
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!
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
What can we learn from “small”
Cluster-DMFT?
Phase diagram for the t-J model
t’=0
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!
No problem with multiplets
k
General impurity problem
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
Insights into superconducting state
(BCS/non-BCS)?
BCS: upon pairing potential energy of
electrons decreases, kinetic energy
increases
(cooper pairs propagate slower)
Condensation energy is the difference
non-BCS: kinetic energy decreases upon
pairing
(holes propagate easier in superconductor)
J. E. Hirsch, Science, 295, 5563 (2001)
Optical weight, plasma frequency
Weight bigger in SC,
K decreases (non-BCS)
~1eV
Bi2212
Weight smaller in SC,
K increases (BCS-like)
F. Carbone et.al, cond-mat/0605209
Hubbard versus t-J model
Kinetic energy in Hubbard model:
•Moving of holes
•Excitations between Hubbard bands
Hubbard model
U
Drude
t2/U
Experiments
Excitations into upper
Hubbard band
Kinetic energy in t-J model
•Only moving of holes
Drude
J
intraband
interband
transitions
t-J model
no-U
~1eV
Kinetic energy change
Kinetic energy increases
cluster-DMFT, cond-mat/0601478
Kinetic energy decreases
Kinetic energy increases
cond-mat/0503073
Phys Rev. B 72, 092504 (2005)
Exchange energy decreases and gives
largest contribution to condensation energy
Kinetic energy upon condensation
underdoped
overdoped
J
J
electrons gain energy due to exchange energy electrons gain energy due to exchange energy
holes gain kinetic energy (move faster)
hole loose kinetic energy (move slower)
J
same as RVB (see P.W. Anderson Physica C, 341, 9 (2000),
or slave boson mean field (P. Lee, Physica C, 317, 194 (1999)
J
BCS like
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
– Evolution from kinetic energy saving to BCS kinetic
energy cost mechanism
41meV resonance
•Resonance at 0.16t~48meV
•Most pronounced at optimal doping
•Second peak shifts with doping (at
0.38~120meV opt.d.) and changes
below Tc – contribution to
condensation energy
local susceptibility
YBa2Cu3O6.6 (Tc=62.7K)
Pengcheng et.al.,
Science 284, (1999)
Optics mass and plasma frequency
Extended Drude model
•In sigle site DMFT plasma frequency
vanishes as 1/Z (Drude shrinks as Kondo
peak shrinks) at small doping
•Plasma frequency vanishes because the
active (coherent) part of the Fermi
surface shrinks
•In cluster-DMFT optics mass constant at
low doping doping ~ 1/Jeff
line: cluster DMFT (cond-mat 0601478),
symbols: Bi2212, F. Carbone et.al, cond-mat/0605209
Optical conductivity
optimally doped
overdoped
cond-mat/0601478
D van der Marel, Nature 425, 271-274 (2003)
Partial DOS
4f
5d
6s
Z=0.33
Pseudoparticle insight
N=4,S=0,K=0
N=4,S=1,K=(p,p)
N=3,S=1/2,K=(p,0)
N=2,S=0,K=0
A(w)
S’’(w)
PH symmetry,
Large t
The simplest model of high Tc’s
t-J, PW Anderson
Hubbard-Stratonovich ->(to keep some out-of-cluster quantum fluctuations)
BK Functional, Exact
cluster in k space
cluster in real space
Cerium
Ce overview
 isostructural phase transition ends in a critical
point at (T=600K, P=2GPa)
 g (fcc) phase
[ magnetic moment
(Curie-Wiess law),
large volume,
stable high-T, low-p]
 a (fcc) phase
[ loss of magnetic
moment (Pauli-para),
smaller volume,
stable low-T, high-p]
with large
volume collapse
Dv/v  15
volumes exp.
28Å3
a
34.4Å3
g
LDA
24.7Å3
•Transition is 1.order
•ends with CP
LDA+U
35.2Å3
LDA and LDA+U
ferromagnetic
volumes exp.
28Å3
a
34.4Å3
g
LDA
24.7Å3
LDA+U
35.2Å3
f DOS
total DOS
LDA+DMFT alpha DOS
TK(exp)=1000-2000K
LDA+DMFT gamma DOS
TK(exp)=60-80K
Photoemission&experiment
•A. Mc Mahan K Held and R. Scalettar (2002)
•K. Haule V. Udovenko and GK. (2003)
Fenomenological approach
describes well the transition
Kondo volume colapse (J.W. Allen, R.M. Martin, 1982)
Optical conductivity
+
*
+ K. Haule, et.al.,
Phys. Rev. Lett. 94, 036401 (2005)
* J.W. van der Eb, A.B. Ku’zmenko, and D. van der Marel,
Phys. Rev. Lett. 86, 3407 (2001)
Monotonically increasing J an SOC
Americium
Americium
f6 -> L=3, S=3, J=0
Mott Transition?
"soft" phase
f localized
"hard" phase
f bonding
Density functional based electronic structure calculations:
 Non magnetic LDA/GGA predicts volume 50% off.
A.Lindbaum, S. Heathman, K. Litfin, and Y. Méresse,
J.-C. Griveau, J. Rebizant, G. H. Lander, and G.Kotliar
Rev. Lett.
097002
Phys.
Rev. B 63, 214101

Magnetic
GGA(2001)
corrects most of error inPhys.
volume
but94,gives
m(2005)
~6mB
(Soderlind et.al., PRB 2000).
 Experimentally, Am has non magnetic f6 ground state with
J=0 (7F0)
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 very different from g Ce
not in local moment regime since J=0 (no entropy)
•"Hard" phase similar to a Ce,
Kondo physics due to hybridization, however,
nf still far from Kondo regime
Different from Sm!
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)
Trends in Actinides
alpa->delta volume collapse transition
F0=4,F2=6.1
F0=4.5,F2=7.15
Curie-Weiss
Same transition in Am under pressure
F0=4.5,F2=8.11
Curium has large magnetic moment and orders antif.
Tc
LS versus jj coupling in
Actinides
•Occupations non-integer except Cm
•Close to intermediate coupling
•Delocalization in U & Pu-> towards LS
•Am under pressure goes towards LS
•Curium is localized, but close to LS!
m=7.9mB not m=4.2mB
K.T.Moore, et.al.,PRB in press, 2006
G. Van der Laan, et.al, PRL 93,27401 (2004)
J.G. Tobin, et.al, PRB 72, 85109 (2005)
d
a
Localization versus itinerancy in the
periodic table?