The Mott transition across the actinide series and the double life of delta Plutonium.

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The Mott transition across the actinide series
and the double life of delta Plutonium.

• Gabriel Kotliar
• and Center for Materials Theory

Colloquium City College NY April 11 (2007)
upport NSF -DMR DOE-Basic Energy Sciences
Collaborators K. Haule and J. Shim Ref
Nature 446, 513, (2007)
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OUTLINE

• The challenge of strongly correlated electron
  systems.
• Late actinides experimental overview
• Introduction to Dynamical Mean Field Theory
• (DMFT).
• Theory of delta Pu
• Theory of Am and Cm
• Conclusions

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Electrons in a Solidthe Standard Model
Band Theory electrons as waves. Landau Fermi
Liquid Theory.
Rigid bands , optical transitions ,
thermodynamics, transport

• Quantitative Tools. Density Functional Theory
• Kohn Sham (1964)

Static Mean Field Theory.
Kohn Sham Eigenvalues and Eigensates Excellent
starting point for perturbation theory in the
screened interactions (Hedin 1965)
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4
Spectral Function Photoemission and correlations
e

• Probability of removing an electron and
  transfering energy wEi-Ef, and momentum k
• f(w) A(w, K) M2

1. Weak Correlation
2. Strong Correlation

n
n
Angle integrated spectral function
8
5
Strong Correlation Problemwhere the standard
model fails

• Fermi Liquid Theory works but parameters
  cant be computed in perturbation theory.
• Fermi Liquid Theory does NOT work . Need new
  concepts to replace of rigid bands !
• Partially filled d and f shells. Competition
  between kinetic and Coulomb interactions.
• Breakdown of the wave picture. Need to
  incorporate a real space perspective (Mott).
• Non perturbative problem.

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6
Strongly correlated systems

• Copper Oxides. High Temperature
  Superconductivity.
• Cobaltates Anomalous thermoelectricity.
• Manganites . Colossal magnetoresistance.
• Heavy Fermions. Huge quasiparticle masses.
• 2d Electron gases. Metal to insulator
  transitions.
• Lanthanides, Transition Metal Oxides,
  Multiferroics..

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Superconductivity among 5f elements
Localisation Delocalization
1.4K
0.4K
0.9K
0.8K
52K
25K
52K
s/c
AF
FM
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Localization Delocalization in Actinides
Mott Transition
d Pu
a
d
a
after G. Lander, Science (2003).
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Basic Questions

• How does the electron go from being localized to
  itinerant.
• How do the physical properties evolve.
• How to bridge between the microscopic information
  (atomic positions) and experimental measurements.
• New concepts, new techniques

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Phases of Pu (A. Lawson LANL)
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Approach the Mott point from the right Am under
pressure
Experimental Equation of State
(after Heathman et.al, PRL 2000)
Mott Transition?
Soft
Hard
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Small amounts of Ga stabilize the d phase (A.
Lawson LANL)
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Anomalous Resistivity
Maximum metallic resistivity
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Specific heat and susceptibility. Pu is non
magnetic
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Standard model fails for the late actinides

• Predicts all the phases of Pu and Am to be
  magnetic, with a large moment. (about 5 mB)
• Imposing paramagnetism, DFT fails to describe the
  volume of delta Pu by 25
• Many modfications have been attempted, to
  explain why Pu is non magnetic. Mixed level model
  (Wills et. al. ) Pu has (5f)4 conf.
• LDAU amf (Shick, Anisimov) Pu has (5f)5
• Cannot account for anomalous elastic properties,
  transport and thermodynamics

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DMFT A. Georges and G. Kotliar PRB 45, 6479
(1992). Happy marriage of atomic and band
physics. Extremize functional of A(w)
Extremize a functional of the local spectra.
Local self energy.
Reviews A. Georges G. Kotliar W. Krauth and M.
Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and
Dieter Vollhardt Physics Today 57,(2004). G.
Kotliar S. Savrasov K. Haule V. Oudovenko O.
Parcollet and C. Marianetti (RMP 2006).
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Dynamical Mean Field Theory

• Weiss field is a function. Multiple scales in
  strongly correlated materials.
• Exact large coordination (Metzner and Vollhardt
  89) , kinetic vs interaction energy
• Immediate extension to real materials DFTDMFT
• Functionals of density and spectra.
• Review Kotliar et. al. RMP (2006)

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18
.
T/W
Phase diagram of a Hubbard model with partial
frustration at integer filling. Rozenberg et.
al. PRL 1995 Evolution of the Local Spectra as a
function of U,and T. Mott transition driven by
transfer of spectral weight Zhang Rozenberg
Kotliar PRL (1993)..
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Volume Collapse Transitions relaxing the lattice
positions. Savrasov et. al.
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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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DMFT Phonons in fcc d-Pu
( Dai, Savrasov, Kotliar,Ledbetter, Migliori,
Abrahams, Science, 9 May 2003)
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(experiments from Wong et.al, Science, 22 August
2003)
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The DMFT-valence in the late actinides.
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Time scale of the fluctuations. Ef
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W110 2/3ltl.sgt and banching ratio
Moore and van der Laan, Ultramicroscopy 2007.
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2/3ltl.sgt in the late actinides DMFT results K.
Haule and J. Shim
See the expt. work of K. Moore G. Van der Laan
G. Haire M. Wall and A. Schartz Am H2
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alpa-gtdelta Photoemission
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Photoemission Spectra Shim. Haule,GK Nature
(2007)
alpa-gtdelta volume collapse transition
F04.5,F27.15
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F04,F26.1
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Photoemission and Mixed valence in Pu
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Conclusions

• Unique properties of Pu and Am under pressure
  result from a proximity of a localization
  delocalization transition. Rare form of mixed
  valence.
• DMFT provides a good start. Qualitative
  insights, some quantitative predictions into
  delta Pu. Other Pu phases.
• Meaningful interplay of theory and experiment.
  Key in condensed matter physics.

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Conclusions

• Pu is a unique strongly correlated element. It
  is one among many strongly correlated electron
  system, materials for which neither the standard
  model of solids, works well.
• They require, new concepts, new computational
  methods, new algorithms, DMFT provides all of the
  above, and is being used in many other problems.
• Many applications to othe problems exist, others
  are in progress, research opportunity in
  correlated materials.

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Approach the Mott point from the right Am under
pressure
Experimental Equation of State
(after Heathman et.al, PRL 2000)
Mott Transition?
Soft
Hard

• Density functional based electronic structure
  calculations
• Non magnetic LDA/GGA predicts volume 50 off.
• Magnetic GGA corrects most of error in volume but
  gives m6mB
• (Soderlind et.al., PRB 2000).
• Experimentally, Am has non magnetic f6 ground
  state with J0 (7F0)

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Am equation of state. LDADMFT.New acceleration
technique for solving DMFT equations S. Savrasov
K. Haule G. Kotliar cond-mat. 0507552 (2005)
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Mott transition in open (right) and closed (left)
shell systems. Superconductivity ? Localized
(5f)6 in L.S coupling or jj coupling ?
S
S
g T
Tc
Log2J1
???
Uc
J0
U
U
g 1/(Uc-U)
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Photoemission spectra using Hubbard I solver and
Sunca . Savrasov Haule and Kotliar cond-mat
0507552 PRL (2006) Hubbard bands width is
determined by multiplet splittings.
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Resistivity of Am under pressure. J. C. Griveau
et.al. PRL 94, 097002 (2005).
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Photomission Spectra of Am under pressure.
Sunca. Onset of mixed valence. Savrasov Haule
Kotliar (2005) PRL (2006)
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Conclusion Am

• Americium undergoes Mott transition under
  pressure. AmIII-AmIV boundary.
• Unusual superconductivity and resistivities.
• Theoretical clue mixed valent due to admixture of
  (5f) upon application of pressure.
• Realizes Mott transition from the insulating
  side, towards a close shell configuration..

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. Mott transition in the open shell case.
Heathman et. al. Science 309,110 (2006) Approach
the Mott transition from the right.
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Curium is magnetic Hurray et.al. Physica. B
(1980) 217
m2SL
LS coupling L0 S7 m7
jj coupling J7/2 m314
Expt monent . is closer to L S coupling
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K.Haule and J. Shim Trends in Actinides
alpa-gtdelta volume collapse transition
F04,F26.1
F04.5,F27.15
F04.5,F28.11
Curium has large magnetic moment and orders antif
Pu does is non magnetic.
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Conclusion

• DMFT conceptual framework to think about
  electrons in solids.
• Finite T Mott transition in 3d . Single site DMFT
  worked well!
• Ab-initio many body electronic structure of
  solids. Building theoretical spectroscopies.
• Frontier, cuprates, lower T, two dimensionality
  is a plaquette in a medium enough?
• Inhomogenous structure in correlated materials
• New renormalizaton group methods built around
  DMFT ?

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Conclusion

• A Few References
• A.Georges, G. K., W. Krauth and M. J. Rozenberg,
  Reviews of . Modern Physics 68, 13 (1996).
• G. K, S. Y. Savrasov, K. Haule, V. S. Oudovenko,
  O. Parcollet, C.A. Marianetti, RMP 78, 865-951,
  (2006).
• G. K and D. Vollhardt Physics Today, Vol 57, 53
  (2004).

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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 predictive
• power.
• 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

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Collaborators, Acknowledgements References

• 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.

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).

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Cluster DMFT removes limitations of single site
DMFT

• No k dependence of the self energy.
• No d-wave superconductivity.
• No Peierls dimerization.
• No (R)valence bonds.

Reviews Georges et.al. RMP(1996). Th. Maier et.
al. RMP (2005) Kotliar et. .al. RMP
(2006).
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Two Site Cellular DMFT (G.. Kotliar et.al. PRL
(2001)) in the 1D Hubbard model M.Capone
M.Civelli V. Kancharla C.Castellani and GK PRB
69,195105 (2004)T. D Stanescu and GK PRB (2006)
U/t4.
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Kohn Sham Eigenvalues and Eigensates Excellent
starting point for perturbation theory in the
screened interactions (Hedin 1965)
Self Energy
VanShilfgaarde (2005)
3
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Smith Kmeko Phase diagram. Minimum in melting
curve and divergence of the compressibility at
the Mott endpoint
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The enhancement of the specific heat, further
evidence for an open shell configuration,
presence of electronic entropy.
J. Lashley et.al. PRB(2005)
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Double well structure and d Pu

• Qualitative explanation of negative thermal
  expansionLawson, A. C., Roberts J. A., Martinez,
  B., and Richardson, J. W., Jr. Phil. Mag. B, 82,
  1837,(2002). G. Kotliar J.Low Temp. Physvol.126,
  1009 27. (2002)

F(T,V)FphononsFinvar
Natural consequence of the conclusions on the
model Hamiltonian level. We had two solutions at
the same U, one metallic and one insulating.
Relaxing the volume expands the insulator and
contract the metal.
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Invar model for Pu-Ga. Lawson et. Mag. Vol.
86, Nos. 1718, 1121 June 2006, 27132733(2006)
Data fits if the excited state has zero
stiffness.
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Dynamical Mean Field Theory. Cavity Construction.
A. Georges and G. Kotliar PRB 45, 6479 (1992).
A(w)
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A(w)
A. Georges, G. Kotliar (1992)
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Expt. Wong et. al.
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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.
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Localization Delocalization in Actinides
Mott Transition
d Pu
a
a
d
after G. Lander, Science (2003).
Modern understanding of this phenomena using
functional approach toDMFT. K Haule S.Savrasov J
Shim
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ltw110gt n7/2 4/3 n5/2nf n7/2 n5/2
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Spectral Function and Photoemission
e

• Probability of removing an electron and
  transfering energy wEi-Ef, and momentum k
• f(w) A(w, K) M2

n
n
Angle integrated spectral function
8
67
Kohn Sham Eigenvalues and Eigensates Excellent
starting point for perturbation theory in the
screened interactions (Hedin 1965)
Self Energy
Succesful description of the total energy and
the excitation spectra of a large number of
simple metals semiconductors and
insulators. Succesfully predicts semiconducting
gaps, phonon frequencies, resistivities, of
countless materials.

1. Weak Correlation
2. Strong Correlation

3
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W110 2/3ltl.sgt and banching ratio
See the expt. work of K. Moore G. Van der Laan
G. Haire M. Wall and A. Schartz Am H2