Elemental Plutonium: Electrons at the Edge

Elemental Plutonium: Electrons at the Edge

Gabriel KotliarPhysics Department and

Center for Materials TheoryRutgers University

SFU September 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

http://www.physics.rutgers.edu/~kotliar/papers/prl84_3670.pdf



http://www.physics.rutgers.edu/~kotliar/papers/nature.pdf

THE STATE UNIVERSITY OF NEW JERSEY

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Pu in the periodic table

actinides


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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.


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Mott transition in the actinide series (Smith Kmetko phase diagram)


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Electronic Physics of Pu


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Small amounts of Ga stabilize the phase (A. Lawson LANL)


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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


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Sommerfeld

Bloch, Landau: Periodic potential, waves form bands , k in Brillouin zone .

The electron in a solid: wave picture 2

2kkm

3* 2

B

8p k3h FV mC T

Landau: Interactions renormalize parameters ,~ const


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Anomalous Resistivity

2 ( )F Fe k k lh

Maximum metallic resistivity 2Fe k

h


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Pu Specific Heat


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Electronic specific heat


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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

1T

L, S, J


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Specific heat and susceptibility.


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(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


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Kohn Sham system


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Delta phase of Plutonium: Problems with LDAo 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%


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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

.


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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


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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 2001-2002 )


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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( )( )[ ]

[ ]n

kn k

n

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=-

effh0 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


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Canonical Phase Diagram of the Localization Delocalization Transition.


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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.


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One electron spectra near the Mott transition.


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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


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Pu: DMFT total energy vs Volume (Savrasov Kotliar and Abrahams 2001)


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Double well structure and Pu Qualitative explanation

of negative thermal expansion

Sensitivity to impurities which easily raise the energy of the -like minimum.


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Generalized phase diagram

T

U/WStructure, bands,

orbitals


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Minimum in melting curve and divergence of the compressibility at the Mott endpoint

( )dT Vdp S

Vsol

Vliq


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Cerium


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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 ionizationand affinity energy of the atom.


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Lda vs Exp Spectra


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Pu Spectra DMFT(Savrasov) EXP (Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000)


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Alpha and delta Pu


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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.


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Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003


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Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev.

E = Ei - EfQ =ki - kf


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Expt. Wong et. al.


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Wong et. al.


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Expts’ Wong et. al.


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Epsilon Plutonium.


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Phonon frequency (Thz ) vs q in epsilon Pu.


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Phonons epsilon


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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.


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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, ……..


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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, ….


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What do we want from materials theory?

New concepts , qualitative ideas Understanding, explanation of existent

experiments, and predictions of new ones. Quantitative capabilities with predictivepower.

Notoriously difficult to achieve in strongly correlated materials.


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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.


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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.




http://www.physics.rutgers.edu/~kotliar/papers/nature.pdf


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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


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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)


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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 theElectronic entropy: TC 600 K.

ResultsforNiO:PhononsResultsforNiO:PhononsSolidcircles–theory,opencircles–exp.(Roy et.al, 1976)

DMFT Savrasov and GK PRL 2003


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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.


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LDA+DMFT functional2 *log[ / 2 ( ) ( )]

( ) ( ) ( ) ( )

1 ( ) ( ')( ) ( ) ' [ ]2 | ' |

[ ]

R R

n

n KS

KS n ni

LDAext xc

DCR

Tr i V r r

V r r dr Tr i G i

r rV r r dr drdr Er r

G

a b ba

w

w c c

r w w

r rr r

- +Ñ - - S -- S +

+ + +-F - F

åòò òå

Sum of local 2PI graphs with local U matrix and local G

1[ ] ( 1)2DC G Un nF = - ( )0( ) i

ababi

n T G i ew

w += å

KS ab [ ( ) G V ( ) ]LDA DMFT a br r


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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

1T

•Superexchange

Summary

LDA

LDA+U

DMFT

Spectra Method E vs V


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For future reference.


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Shear anisotropy. C’=(C11-C12)/2 4.78

C44= 33.59 19.70

C44/C’ ~ 6 Largest shear anisotropy in any element!


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Electronic specific heat


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DMFT BOX


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Anomalous Resistivity2 ( )F Fe k k l

h Maximum metallic resistivity 200 mohm cm


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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).

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Elemental Plutonium: Electrons at the Edge