Elemental Plutonium: Electrons at the Edge
Gabriel Kotliar
Physics Department and
Center for Materials Theory
Rutgers University
Colloquium UT July 2003
Outline , Collaborators, References
Los Alamos Science,26, (2000)
S. Savrasov and G. Kotliar Phys. Rev. Lett. 84, 3670-3673, (2000). S.Savrasov G. Kotliar and E. Abrahams, Nature 410, 793 (2001).
X. Dai,S. Savrasov, G. Kotliar,A. Migliori, H. Ledbetter, E. Abrahams Science, Vol300, 954 (2003).
Plutonium PuzzlesSolid State Theory, Old and New (DMFT)ResultsConclusions
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Pu in the periodic table
actinides
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Pu is famous because of its nucleus.
Fission: Pu239 absorbs a neutron and breaks apart into pieces releasing energy and more neutrons.
Pu239 is an alpha emitter, making it into a most toxic substance.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Mott transition in the actinide series (Smith Kmetko phase diagram)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Electronic Physics of Pu
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Small amounts of Ga stabilize the phase (A. Lawson LANL)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Elastic Deformations
In most cubic materials the shear does not depend strongly on crystal orientation,fcc Al, c44/c’=1.2, in Pu C44/C’ ~ 6 largest shear anisotropy of any element.
Uniform compression:p=-B V/V Volume conserving deformations:
F/A=c44 x/L F/A=c’ x/L
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Sommerfeld
Bloch, Landau: Periodic potential, waves form bands , k in Brillouin zone .
The electron in a solid: wave picture 2
2k
k
m
3* 2
B
8p k
3h FV mC T
Landau: Interactions renormalize parameters ,~ const
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Anomalous Resistivity
2 ( )F Fe k k l
h
Maximum metallic resistivity 2
Fe k
h
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Pu Specific Heat
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Electronic specific heat
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Localized model of electron in solids. (Mott)particle picture.Solid=Collection of atoms
•Think in real space , solid collection of atoms•High T : local moments, Low T spin-orbital order
1
T
L, S, J
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Specific heat and susceptibility.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
(Spin) Density Functional Theory.
Focus on the density (spin density ) of the solid. Total energy is obtained by minimizing a functional of the density (spin
density). Exact form of the functional is unknown but good approximations
exist. (LDA, GGA) In practice, one solves a one particle shrodinger equation in a
potential that depends on the density. A band structure is generated (Kohn Sham system).and in many
systems this is a good starting point for perturbative computations of the spectra (GW).
Works exceedingly well for many systems. W. Kohn, Nobel Prize in Chemistry on October 13, 1998 for its
development of the density-functional theory
[ ( ) ]r [ ( ) , ( ) ]r r
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Kohn Sham system
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Delta phase of Plutonium: Problems with LDA
o Many studies and implementations.(Freeman, Koelling 1972)APW methods, ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999).all give an equilibrium an equilibrium volume of the volume of the phasephaseIs 35% lower than Is 35% lower than experiment experiment this is the largest discrepancy ever known in DFT based calculations.
LSDA predicts magnetic long range (Solovyev et.al.) Experimentally Pu is not magnetic.
If one treats the f electrons as part of the core LDA overestimates the volume by 30%
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DFT Studies of Pu DFT in GGA predicts correctly the volume of the
phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is usually taken as an indication that Pu is a weakly correlated system
.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
One Particle Local Spectral Function and Angle Integrated Photoemission
Probability of removing an electron and transfering energy =Ei-Ef,
f() A() M2
Probability of absorbing an electron and transfering energy =Ei-Ef,
(1-f()) A() M2
Theory. Compute one particle greens function and use spectral function.
e
e
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Dynamical Mean Field Theory
Focus on the local spectral function A() of the solid. Write a functional of the local spectral function such that
its stationary point, give the energy of the solid. No explicit expression for the exact functional exists, but
good approximations are available. The spectral function is computed by solving a local
impurity model. Which is a new reference system to think about correlated electrons.
Ref: A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod
Phys 68,1 (1996) . Generalizations to realistic electronic structure. (G. Kotliar and S. Savrasov in )
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Mean-Field : Classical vs Quantum
Classical case Quantum case
Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992)
†
0 0 0
( )[ ( ')] ( ')o o o oc c U n nb b b
s st m t t tt ¯
¶+ - D - +
¶òò ò
( )wD
†( )( ) ( )
MFL o n o n HG c i c iw w D=- á ñ
1( )
1( )
( )[ ][ ]
nk
n kn
G ii
G i
ww e
w
=D - -
D
å
,ij i j i
i j i
J S S h S- -å å
MF eff oH h S=-
effh
0 0 ( )MF effH hm S=á ñ
eff ij jj
h J m h= +å
† †
, ,
( )( )ij ij i j j i i ii j i
t c c c c U n n
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Canonical Phase Diagram of the Localization Delocalization Transition.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT has bridged the gap between band theory and atomic physics.
Delocalized picture, it should resemble the density of states, (perhaps with some additional shifts and satellites).
Localized picture. Two peaks at the ionization
and affinity energy of the atom.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
One electron spectra near the Mott transition.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
What is the dominant atomic configuration? Local moment?
Snapshots of the f electron Dominant configuration:(5f)5
Naïve view Lz=-3,-2,-1,0,1 ML=-5 B
S=5/2 Ms=5 B Mtot=0
More refined estimates ML=-3.9 Mtot=1.1 This bit is quenches by the f and spd electrons
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Pu: DMFT total energy vs Volume (Savrasov Kotliar and Abrahams 2001)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Double well structure and Pu Qualitative explanation
of negative thermal expansion
Sensitivity to impurities which easily raise the energy of the -like minimum.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Generalized phase diagram
T
U/WStructure, bands,
orbitals
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Minimum in melting curve and divergence of the compressibility at the Mott endpoint
( )dT V
dp S
Vsol
Vliq
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Cerium
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Photoemission Technique
Density of states for removing (adding ) a particle to the sample.
Delocalized picture, it should resemble the density of states, (perhaps with some satellites).
Localized picture. Two peaks at the ionization
and affinity energy of the atom.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Lda vs Exp Spectra
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Pu Spectra DMFT(Savrasov) EXP (Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Alpha and delta Pu
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Phonon Spectra
Electrons are the glue that hold the atoms together. Vibration spectra (phonons) probe the electronic structure.
Phonon spectra reveals instablities, via soft modes.
Phonon spectrum of Pu had not been measured until recently.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev.
E = Ei - EfQ =ki - kf
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Expt. Wong et. al.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Expts’ Wong et. al.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Conclusions Pu is a unique ELEMENT, but by no means unique
material. It is one among many strongly correlated electron system, materials for which neither the standard model of solids, either for itinerant or localized electrons works well.
The Mott transition across the actinide series [ B. Johansson Phil Mag. 30,469 (1974)] concept has finally been worked out!
They require, new concepts, new computational methods, new algorithms, DMFT provides all of the above, and is being used in many other problems.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Conclusions Constant interplay between theory and
experiment has lead to new advances. General anomalies of correlated electrons and
anomalous system specific studies, need for a flexible approach. (DMFT).
New understanding of Pu. Methodology applicable to a large number of other problems, involving correlated electrions, thermoelectrics, batteries, optical devices, memories, high temperature superconductors, ……..
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Conclusions DMFT produces non magnetic state, around a
fluctuating (5f)^5 configuraton with correct volume the qualitative features of the photoemission spectra, and a double minima structure in the E vs V curve.
Correlated view of the alpha and delta phases of Pu. Interplay of correlations and electron phonon interactions (delta-epsilon).
Calculations can be refined in many ways, electronic structure calculations for correlated electrons research program, MINDLAB, ….
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
What do we want from materials theory?
New concepts , qualitative ideas Understanding, explanation of existent
experiments, and predictions of new ones. Quantitative capabilities with predictive
power.
Notoriously difficult to achieve in strongly correlated materials.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Some new insights into the funny properties of Pu
Physical anomalies, are the result of the unique position of Pu in the periodic table, where the f electrons are near a localization delocalization transition. We learned how to think about this unusual situation using spectral functions.
Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha). Negative thermal expansion, multitude of phases.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Quantitative calculations Photoemission spectra,equilibrium volume, and
vibration spectra of delta. Good agreement with experiments given the approximations made.Many systematic improvements are needed.
Work is at the early stages, only a few quantities in one phase have been considered.
Other phases? Metastability ? Effects of impurities? What else, do electrons at the edge of a localization localization do ? [ See epsilon Pu spectra ]
Collaborators, Acknowledgements References
Los Alamos Science,26, (2000)
S. Savrasov and G. Kotliar Phys. Rev. Lett. 84, 3670-3673, (2000). S.Savrasov G. Kotliar and E. Abrahams, Nature 410, 793 (2001).
X. Dai,S. Savrasov, G. Kotliar,A. Migliori, H. Ledbetter, E. Abrahams Science, Vol300, 954 (2003).
Collaborators: S. Savrasov ( Rutgers-NJIT)
X. Dai ( Rutgers), E. Abrahams (Rutgers), A. Migliori (LANL),H Ledbeter(LANL).
Acknowledgements: G Lander (ITU) J Thompson(LANL)
Funding: NSF, DOE, LANL.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Acknowledgements: Development of DMFT
Collaborators: V. Anisimov, R. Chitra, V. Dobrosavlevic, X. Dai, D. Fisher, A. Georges, H. Kajueter, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, I. Yang, X.Y. Zhang
Support: NSF DMR 0096462
Support: Instrumentation. NSF DMR-0116068
Work on Fe and Ni: ONR4-2650
Work on Pu: DOE DE-FG02-99ER45761 and LANL subcontract No. 03737-001-02
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Wong et. al.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
The delta –epsilon transition
The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase.
What drives this phase transition?
Having a functional, that computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Phonon frequency (Thz ) vs q in epsilon Pu.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Epsilon Plutonium.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Phonons epsilon
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Phonon entropy drives the epsilon delta phase transition
Epsilon is slightly more metallic than delta, but it has a much larger phonon entropy than delta.
At the phase transition the volume shrinks but the phonon entropy increases.
Estimates of the phase transition neglecting the
Electronic entropy: TC 600 K.
ResultsforNiO:PhononsResultsforNiO:Phonons
Solidcircles–theory,opencircles–exp.(Roy et.al, 1976)
DMFT Savrasov and GK PRL 2003
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Two models of a solid. Itinerant and localized. Mott transition between the two. Spectral function differentiates between the two
phases. Insert the phase diagram that I like.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDA+DMFT functional2 *log[ / 2 ( ) ( )]
( ) ( ) ( ) ( )
1 ( ) ( ')( ) ( ) ' [ ]
2 | ' |
[ ]
R R
n
n KS
KS n n
i
LDAext xc
DC
R
Tr i V r r
V r r dr Tr i G i
r rV r r dr drdr E
r r
G
a b ba
w
w c c
r w w
r rr r
- +Ñ - - S -
- S +
+ + +-
F - F
åò
ò òå
Sum of local 2PI graphs with local U matrix and local G
1[ ] ( 1)
2DC G Un nF = - ( )0( ) iab
abi
n T G i ew
w+
= å
KS ab [ ( ) G V ( ) ]LDA DMFT a br r
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
The electron in a solid: particle picture.
NiO, MnO, …Array of atoms is insulating if a>>aB. Mott: correlations localize the electron
e_ e_ e_ e_
•Think in real space , solid collection of atoms•High T : local moments, Low T spin-orbital order
1
T
•Superexchange
Summary
LDA
LDA+U
DMFT
Spectra Method E vs V
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
For future reference.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Shear anisotropy.
C’=(C11-C12)/2 4.78
C44= 33.59 19.70
C44/C’ ~ 6 Largest shear anisotropy in any element!
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Electronic specific heat
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT BOX
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Anomalous Resistivity2 ( )F Fe k k l
h Maximum metallic resistivity 200 mohm cm
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Magnetic moment
L=5, S=5/2, J=5/2, Mtot=Ms=B gJ =.7 B
Crystal fields
GGA+U estimate (Savrasov and Kotliar 2000) ML=-3.9 Mtot=1.1
This bit is quenched by Kondo effect of spd electrons [ DMFT treatment]
Experimental consequence: neutrons large magnetic field induced form factor (G. Lander).
Top Related