Transcript Correlated Electrons: A Dynamical Mean Field (DMFT

Understanding f-electron materials using
Dynamical Mean Field Theory
Gabriel Kotliar
and Center for Materials Theory
Solid State Seminar
U. Oregon January 15th 2010
$upport : NSF -DMR DOE-Basic Energy Sciences
Collaborators: K. Haule and J. Shim
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Outline
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•
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Introduction to Correlated Materials
Introduction to Dynamical Mean Field Theory
Applications to f electrons:
CeIrIn5
URu2Si2
Pu-Am-Cm
PuSe PuTe
Conclusions
Electrons in a Solid:the Standard Model
Landau Fermi Liquid Excitation spectrum of a Fermi system has the
same structure as the excitation spectrum of a perfect Fermi gas.
Bloch waves in a periodic potential
Rigid bands , optical transitions ,
thermodynamics, transport………
k en[k ]
n band index, e.g. s, p, d,,f
Kohn Sham Density Functional Theory
Etot[r ]
- Ñ 2 / 2 + VKS (r)[r ] y kn = en [k ]y kn
Static Mean Field Theory.
Kohn Sham Eigenvalues and Eigensates: Excellent starting point
for perturbation theory in the screened interactions (Hedin 1965)
2
GW= First order PT in screened Coulomb
interactions around LDA
• Quantum mechanical description of the states in metals and
semiconductors. Bloch waves. En(k).
• Inhomogenous systems. Doping. Theory of donors and
acceptors . Interfaces. p-n junctions. Transistors.
Integrated circuits computers.
Physical Insights into Materials -> Technology
3
Correlated materials: simple recipe
Transition metal oxides
transition metal ion
Oxygen
Cage : e.g 6 oxygen atoms (octahedra)
or other ligands/geometry
Transition metal inside
Build a microscopic crystal with this building block
Transition metal ions
Rare earth ions
Actinides
Layer the structure
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VO2
LixCoO2
Nax CoO2
YBa2Cu3O7
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Heavy Fermions: intermetallics containing 4f elements
Cerium, and 5f elements Uranium. Broad spd bands +
atomic f open shells.
How do we know that the electrons
are heavy ?
Heavy Fermion Metals
Coherence Incoherence
Crossover
-1 (emu/mol)-1
Magnetic
Oscillations
300
CeAl3
200
UBe13
100
0
0
100
T(K)
200
A Very Selected Class of HF
A signature
URu2Si2 problem ?
U
Ru
Si
Correlated Electron Systems Pose Basic
Questions in CMT: from atoms to solids
• How to describe
electron from localized
to itinerant ?
• How do the physical
properties evolve ?
• Non perturbative
techniques
Needed!! (Dynamical) mean field theory for this
problem ,
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Mean-Field : Classical vs Quantum
Classical case
-
å
J ij Si S j - hå Si
i, j
i
HMF = - heff So
Easy!!!
heff
Quantum case

 (t
i , j  ,
b
ij
  ij )(ci† c j  c†j ci )  U  ni  ni 
i
b
b
¶
†
m- D (t - t ')]cos (t ') + U ò no- no¯
ò ò cos (t )[ ¶ t + Hard!!!
0 0
but doable QMC, PT , ED , DMRG…….0
áS0 ñ= th[b heff ]
D (w)
m0 = áS0 ñHMF (heff,……………………………………...
)
Gos (iwn ) = - áco†s (iwn )cos (iwn )ñSMF (D )
heff =
å
Jij m j + h
G (iwn ) =
1
åk
1
[D (iwn ) - t (k ) + m]
Phys. (1995)
G (iwn )[D ]
j
• Prushke
T. et. al Adv.
• Georges Kotliar Krauth Rosenberg RMP
A. Georges, G. Kotliar (1992)
(1996) Kotliar et. al. RMP (2006)
Dynamical Mean Field Theory
• Describes the electron both in the itinerant
(wave-like) and localized (particle-like)
regimes and everything in between!.
• Follow different mean field states (phases)
Compare free energies.
• Non Gaussian reference frame for
correlated materials.
• Reference frame can be cluster of sites
CDMFT
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LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G.
Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997).
1
G(k , i ) 
i  H (k )  S(i )
Spectra=- Im G(k,)
U
®U
S
abcd
H (k )
®
®
æ0 0 ö
÷
çç
÷
çè0 S ff ÷
÷
ø
æH [k ]spd ,sps
çç
çè H [k ] f ,spd
| 0 > ,|- > ,|¯> ,|- ¯>
®
H [k ]spd , f ö
÷
÷
÷
÷
H [k ] ff ø
| LSJM J g... >
Determine energy and and S self consistently from extremizing a
functional . Savrasov and Kotliar PRB 69, 245101, (2001) Full self
consistent implementation
Gdft[r ] ¾ ¾
® Glda +
[Gloc, r ,U ]
dmft
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DMFT Concepts
Weiss Weiss field, collective
hybridizationfunction, quantifies the
degree of localization
ab
D (w) =
å
a
Valence Histograms. Describes
the history of the “atom” in the solid,
multiplets!
Functionals of density and
spectra give total energies
Gdft[r ] ¾ ¾
® Glda +
t[Gloc , r ,U ]
dmf
a *
b
a a
V V
w - ea
Photoemission Spectral functions and the
State of the Electron
Probability of removing
an electron and
transfering energy
=Ei-Ef, and
momentum k
f() A(, K) M2
e
A(k, )
A(k, )

a) Weak correlations
b) Strong correlation: FL parameters can’t be
evaluated in PT or FLT does not work.
Angle integrated spectra
 dkA(, k )  A()
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Qualitative Phase diagram :frustrated Hubbard model,
integer filling M. Rozenberg et.al. PRL,75, 105 (1995)
CONCEPT:
Quasiparticle bands,
T*, and Hubbard
bands
CONCEPT:
CONCEPT:
T/W
(orbitally
resolved)
spectral
function.
Transfer of
spectral weight.
(orbital
selective)
Mott
transition.
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Outline
• Introduction to Correlated Materials
• Introduction to Dynamical Mean Field Theory
•
•
•
•
•
•
Applications to f electrons:
CeIrIn5
Pu-Am-Cm
PuSe PuTe
URu2Si2
Conclusions
4f’s heavy fermions, 115’s, CeMIn5
M=Co, Ir, Rh
Ir
 CeRhIn5: TN=3.8 K;   450 mJ/molK2
In
 CeCoIn5: Tc=2.3 K;   1000 mJ/molK2;
 CeIrIn5: Tc=0.4 K;   750 mJ/molK2
Ce
out of plane
in-plane
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Optical conductivity in LDA+DMFT
Shim, HK Gotliar Science (2007)
D. Basov et.al.
K. Burch et.al.
•At 300K very broad Drude peak (e-e scattering, spd lifetime~0.1eV)
•At 10K:
•very narrow Drude peak
•First MI peak at 0.03eV~250cm-1
•Second MI peak at 0.07eV~600cm-1
eV
Structure Property Relation: Ce115’s
J. Shim et.
Optics and Multiple hybridization gaps al. Science
non-f spectra
10K
300K
In
Ce
In
•Larger gap due to hybridization with out of
plane In
•Smaller gap due to hybridization with inplane In
Difference between Co,Rh,Ir 115’s
Total and f DOS
f DOS
Ir
Co
more itinerant
“good” Fermi liquid
superconducting
Rh
more localized
magnetically ordered
Haule Yee and Kim arXiv:0907.0195
A signature
URu2Si2 problem ?
U
Ru
Si
Two Broken Symmetry Solutions
Weiss field
D (w)ab =
å
a
V a a *Va b
w - ea
Hidden Order
LMA
K. Haule and GK
Hidden order parameter
Valence Histogram
Paramagnetic phase
low lying singlets f^2
Order parameter:
Different orientation gives different
phases: “adiabatic continuity” explained.
Hexadecapole order testable by resonant X-ray
In the atomic limit:
Simplified toy model phase
diagram mean field theory
Mean field
Exp. by E. Hassinger et.al. PRL 77, 115117 (2008)
Orbitally resolved DOS
DMFT “STM” URu2Si2 T=20 K
Ru
Si
Si
U
Fano lineshape:
q~1.24, G~6.8meV, very similar to exp Davis
Lattice response
Localization Delocalization in Actinides
Mott Transition
 Pu
a
a
Modern understanding of this phenomenaDMFT.
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Total Energy as a function of volume for Pu
Moment is first reduced by orbital spin
moment compensation. The remaining
moment is screened by the spd and f
electrons
The f electron in phase is only slightly
more localized than in
the a-phase which has
larger spectral weight in
the quasiparticle peak
and smaller weight in the
Hubbard bands
(Savrasov, Kotliar, Abrahams, Nature ( 2001)
Non magnetic correlated state of fcc Pu.
Localization Delocalization in Actinides
Mott Transition
 Pu
a
a
Modern understanding of this phenomenaDMFT.
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The standard model of solids fails near Pu
• Spin Density functional theory: Pu , Am,
magnetic, large orbital and spin moments.
• Experiments (Lashley et. al. 2005, Heffner et al.
(2006)):  Pu is non magnetic. No static or
fluctuating moments. Susceptibility, specific heat
in a field, neutron quasielastic and inelastic
scattering, muon spin resonance…
•Paramagnetic LDA underestimates Volume of  Pu.
•Thermodynamic and transport properties similar to
strongly correlated materials.
•Plutonium: correlated paramagnetic metal.
DMFT Phonons in fcc -Pu
( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August 2003)
K.Haule J. Shim and GK Nature 446, 513 (2007)
Trends in Actinides
alpa->delta volume collapse transition
F0=4,F2=6.1
F0=4.5,F2=7.15
F0=4.5,F2=8.11
Curium has large magnetic moment and orders
antiferromagnetically Pu does is non magnetic.
Photoemission
Havela et. al. Phys. Rev. B
68, 085101 (2003)
What is the valence in the late actinides ?
Plutonium has an unusual form of MIXED VALENCE
Finding the f occupancyTobin et. al. PRB 72, 085109 2005 K. Moore and G. VanDerLaan
RMP (2009). Shim et. al. Europhysics Lett (2009)
LDA results
DMFT
results
Localization delocalization of f
electrons in compounds.
• Pu Chalcogenides [PuSe, PuS, PuTe]:
Pauli susceptibility, small gap in transport.
• Pu Pnictides [PuP, PuAs, PuSb], order
magnetically.
• Simple cubic NaCl structure
• Going from pnictides to
chalcogenides tunes the
degree of localization of the
Earlier work Shick et. al. Pourovski
f electron. et. al.
LDA+DMFT C. Yee Expts. T. Rurakiewicz et. al. PRB 70, 205103
PuTe: a 5f mixed valent semiconductor
PuSb: a local moment metal
Summary
• Correlated Electron Systems. Huge Phase Space.
Fundamental questions. Promising applications.
• DMFT reference frame to think about electrons in
solids. Quasiparticles Hubbard bands. Compare with
the standard model.
• Many succesful applications, some examples
illustrating a) the concepts, b) the role of realistic
modelling, and c) the connection between theory and
experiment and the role of theoretical spectroscopy.
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Conclusion:
• DMFT provides a surprisingly accurate
description of f electron systems.
• It’s physical content at very low
temperatures is that of a heavy Fermi
liquid in common with other methods but
asymptotia is hardly reached (and
relevant). Complete description of the
crossover.
• Variety and Universality.
Outlook
• “Locality “ as an alternative to Perturbation Theory.
• Needed: progress in implementation. e.g. full solution
of DMFT equations on a plaquette, robust
GW+DMFT ………….
• Fluctuation around DMFT.
• Interfaces, junctions, heterostructures………..
• Motterials, Materials,…….
• Towards rational material design with correlated
electrons systems
http://www.kitp.ucsb.edu/activities/auto/?id=970
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Looking for moments. Pu under (negative ) pressure. C
Marianetti, K Haule GK and M. Fluss
Phys. Rev. Lett. 101, 056403 (2008)
Conclusion: some general comments.
•DMFT approach. Can now start from the
material.
•Can start from high energies, high
temperatures, where the method (I believe
) is essentially exact, far from critical
points, provided that one starts from the
right “reference frame”.
•Spectral “fingerprints” and their chemical
origin.
•Still need better tools to analyze and
solve the DMFT equations.
•At lower temperatures, one has to study
different
broken symmetry states.
•At lower temperatures, one has to study
different broken symmetry states.
•Compare free energies, draw phase
diagram
•Beyond DMFT: Write effective low energy
theories that match the different regions of
the phase diagram.
Buildup of coherence
Very slow crossover!
coherent spectral weight
Buildup of coherence in single impurity case
coherence peak
T
TK
scattering rate
T*
Slow crossover compared to AIM
Crossover around 50K
Plutonium