Elemental Plutonium: Electrons at the Edge The Mott transition across the actinide series.

1
Elemental Plutonium Electrons at the EdgeThe
Mott transition across the actinide series.

• Gabriel Kotliar
• Physics Department and
• Center for Materials Theory
• Rutgers University

Santa Fe November 2003

2
Outline , Collaborators, References

• Physical properties of plutonium.
• Dynamical Mean Field Theory (DMFT)
• DMFT study of elemental plutonium.
• Conclusions

Los Alamos Science,26, (2000). S. Savrasov and
G. Kotliar PRL 84 3670 (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).

3
Pu in the periodic table
actinides
4
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.
5

Mott transition in the actinide series
(Smith-Kmetko phase diagram)
6
Phases of Pu (A. Lawson LANL)
7
Small amounts of Ga stabilize the d phase (A.
Lawson LANL)
8
Elastic Deformations
Uniform compressionDp-B DV/V
Volume conserving deformations
F/Ac44 Dx/L
F/Ac Dx/L
In most cubic materials the shear does not depend
strongly on crystal orientation,fcc Al,
c44/c1.2, in Pu C44/C 7 largest shear
anisotropy of any element.
9
The electron in a solid wave picture
Sommerfeld

Bloch, Landau Periodic potential, waves form
bands , k in Brillouin zone . Density functional
theory
Landau Interactions renormalize parameters .
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Anomalous Resistivity
Maximum metallic resistivity
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Pu Specific Heat
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Electronic specific heat(J Lashley et.al. LANL)
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Localized model of electron in solids. (Peierls
Mott)particle picture.SolidCollection of atoms
L, S, J

• Think in real space , solid collection of atoms
• High T local moments, Low T spin-orbital order

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Specific heat and susceptibility.
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Density Functional Theory and Kohn Sham
Reference System.

• Total energy is minimizes 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).

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Delta phase of Plutonium Problems with LDA

• 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 volume of the d phase Is 35
  lower than experiment this is the largest
  discrepancy ever known in DFT based calculations.
• LSDA predicts magnetic long range (Solovyev
  et.al.) Experimentally d 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 a
  phase of Pu, when full potential LMTO (Soderlind
  Eriksson and Wills) is used. This is usually
  taken as an indication that a Pu is a weakly
  correlated system
• Alternative models1) For the delta phase a model
  with 4 5f electrons localized and 1 electron as
  itinerant was proposed by Wills et. al, in the
  spirit of SIC corrected LDA. This model produces
  correct volume of delta. 2) Strong random
  potential. (B. Cooper).
• .

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Dynamical Mean Field Theory

• Basic idea reduce the quantum many body problem
  to a one site or a cluster of sites, in a medium
  of non interacting electrons obeying a self
  consistency condition.A. Georges and GK 1992
• Basic idea instead of using functionals of the
  density, use more sensitive functionals of the
  one electron spectral function. density of
  states for adding or removing particles in a
  solid, measured in photoemission GK R. Chitra
  and S. Savrasov 2000,2002

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DMFT
Reference System
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
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One Particle Local Spectral Function and Angle
Integrated Photoemission
e

• Probability of removing an electron and
  transfering energy wEi-Ef,
• f(w) A(w) M2
• Probability of absorbing an electron and
  transfering energy wEi-Ef,
• (1-f(w)) A(w) M2
• Theory. Compute one particle greens function and
  use spectral function.

n
n
e
21

• Simple interface with electronic structure. Treat
  the spd electrons within LDA (static self energy
  approximated by xc potential). Treat the f
  electrons with DMFT. LDADMFT.
• Extensions. Treat the electric field and the
  electronic fields using DMFT. E-DMFT

22
DMFT functional formulation.

• Focus on the local spectral function A(w) 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, by
  making systematic truncations in the range of the
  BK functional.
• 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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Canonical Phase Diagram of the Localization
Delocalization Transition.
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Pressure Driven Mott transition
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More recent work, organics, Limelette et. al.
PRL 91,061401 (2003)
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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.
Transfer of Spectral Weight. Zhang Rozenberg and
Kotliar 93
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• DMFT studies of elemental Plutonium

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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 mB S5/2 Ms5 mB
• Mtot0 L5, S5/2, J5/2,
• MtotMsmB gJ .7 mB
• Crystal fields G7 G8
• GGAU estimate ML-3.9 Mtot1.1 (Savrasov GK
  2000)
• This bit is quenches by the f and spd electrons
• Neutron Scattering in a field (Lander)

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Pu DMFT total energy vs Volume (Savrasov
Kotliar and Abrahams 2001)
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Double well structure and d Pu

• Qualitative explanation
  of negative thermal expansion
• Sensitivity to impurities which easily raise the
  energy of the a -like minimum.

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Generalized phase diagram
T
U/W
Structure, bands, orbitals
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Minimum in melting curve and divergence of the
compressibility at the Mott endpoint
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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 ionization
• and 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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• Alpha phase is also a correlated metal.
• It differs from delta in the relative weight of
  the resonance and the Hubbard band.
• Consistent with resistivity and specific heat
  measurements.

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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.
• Short distance behavior of the elastic moduli.

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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 - Ef Q ki - kf
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Expt. Wong et. al.
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Expts Wong et. al. Science 301. 1078 (2003)
Theory Dai et. al. Science 300, 953, (2003)
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Shear anisotropy. Expt. vs Theory

• C(C11-C12)/2 4.78 GPa C3.9 GPa
• C44 33.59 GPa C4433.0 GPa
• C44/C 7 Largest shear anisotropy in any
  element!
• C44/C 8.4

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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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Epsilon Plutonium.
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Phonon frequency (Thz ) vs q in epsilon Pu.
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Phonon entropy drives the epsilon delta phase
transition

• Epsilon is slightly more delocalized than delta,
  has SMALLER volume and lies at HIGHER energy than
  delta at T0. 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 following
  Drumont and Ackland et. al. PRB.65, 184104
  (2002) (and neglecting electronic entropy).
  TC 600 K.

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Phonons epsilon
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• Approaching the Mott transition from the
  localized side. Americium under pressure.

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Superconductivity among 5f elements
Localisation
1.4K
0.4K
0.9K
0.8K
52K
25K
52K
s/c
AF
FM
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Phase diagram (Lindbaum et. al. PRB 2003)
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Interesting fundamental questions.

• Closed shell system. Mott transition?
• Where does it occur? Interplay of spin orbit
  coupling and Coulomb interactions.
• Superconductivity (how does it depend on pressure
  ? Is it in the f or the spd system ? Does it
  correlated with the Mott transition ?)

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Insights into the anomalous 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. The Mott transition across the
  actinide series B. Johansson Phil Mag. 30,469
  (1974) concept has finally been worked out! .We
  learned how to think about this unusual situation
  using spectral functions.

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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, .

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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?

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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.
• They require, new concepts, new computational
  methods, new algorithms. System specific
  methods, DMFT and is being used in many other
  problems. International multidisciplinary effort
  Dresden, Trieste, Leiden, Trieste, Santa
  Barbara, Trieste ..

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Conclusions

• Methodology applicable to a large number of other
  problems, involving correlated electrons,
  thermoelectrics, batteries, optical devices,
  high temperature dilute magnetic
  semiconductors.
• Calculations can be refined in many ways,
  electronic structure calculations for correlated
  electrons research program, MINDLAB. Bring the
  method to the point, that we can start focusing
  in deviations from DMFT, isolate short and long
  wavelength physics.

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Acknowledgements Development of DMFT
Collaborators E. Abrahams,V. Anisimov, R.
Chitra, V. Dobrosavlevic, X. Dai, D. Fisher,
A. Georges, K. Haule H. Kajueter, W.Krauth, E.
Lange, A. Lichtenstein, G. Moeller, Y. Motome,
G. Palsson, A. Poteryaev, 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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Acknowledgements Development of DMFT
Collaborators E. Abrahams,V. Anisimov, G.
Biroli,C Bolech, M. Capone, R. Chitra, M.
Civelli, V. Dobrosavlevic, X. Dai, D. Fisher,
A. Georges, K. Haule, V Kancharla, H. Kajueter,
W.Krauth, E. Lange, A. Lichtenstein, G. Moeller,
Y. Motome, G. Palsson, O. Parcollet, A.
Poteryaev, M. Rozenberg, S. Savrasov, Q. Si, V.
Udovenko, I. Yang, X.Y. Zhang
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LDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
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Summary
Spectra
Method
E vs V
LDA
LDAU
DMFT
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More recent work, organics, Limelette et. al.
PRL 91,061401 (2003)
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Ising critical endpoint! In V2O3 P. Limelette
et.al. Science Vol 302,89 (2003).
67
Mott transition in layered organic conductors
S Lefebvre et al. cond-mat/0004455, Phys. Rev.
Lett. 85, 5420 (2000)
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Magnetic moment

• L5, S5/2, J5/2, MtotMsmB gJ .7 mB
• Crystal fields G7 G8
• GGAU estimate (Savrasov and Kotliar 2000)
  ML-3.9 Mtot1.1
• 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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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.

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Canonical Phase Diagram of the Localization
Delocalization Transition.