Band structure of strongly correlated materials from the Dynamical Mean Field perspective
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Transcript of Band structure of strongly correlated materials from the Dynamical Mean Field perspective
Bonn, 2008
Band structure of strongly correlated materials from the Dynamical Mean Field perspective
K HauleRutgers University
Collaborators : J.H. Shim & Gabriel Kotliar, S. Savrasov
Outline
Dynamical Mean Field Theory in combination with band structure
LDA+DMFT results for 115 materials (CeIrIn5) Local Ce 4f - spectra and comparison to AIPES) Momentum resolved spectra and comparison to ARPES Optical conductivity Two hybridization gaps and its connection to optics Fermi surface in DMFT
Actinides Absence of magnetism in Pu and magnetic ordering in Cm
explained by DMFT Valence of correlates solids, example of Pu
References:•J.H. Shim, KH, and G. Kotliar, Science 318, 1618 (2007).•J.H. Shim, KH, and G. Kotliar, Nature 446, 513 (2007).
Standard theory of solidsStandard theory of solids
Band Theory: electrons as waves: Rigid band picture: En(k) versus k
Landau Fermi Liquid Theory applicable
Very powerful quantitative tools: LDA,LSDA,GWVery powerful quantitative tools: LDA,LSDA,GW
Predictions:
•total energies,
•stability of crystal phases
•optical transitions
M. Van SchilfgardeM. Van Schilfgarde
Fermi Liquid Theory does NOT work . Need new concepts to replace rigid bands picture!
Breakdown of the wave picture. Need to incorporate a real space perspective (Mott).
Non perturbative problem.
Strong correlation – Strong correlation –
Standard theory failsStandard theory fails
V2O3Ni2-xSex organics
Universality of the Mott transitionUniversality of the Mott transition
First order MITCritical point
Crossover: bad insulator to bad metal
1B HB model 1B HB model (DMFT):(DMFT): B
ad in
sula
tor
Bad metal1B HB model 1B HB model (plaquette):(plaquette):
Basic questions to addressBasic questions to address
How to computed spectroscopic quantities (single particle spectra, optical conductivity phonon dispersion…) from first principles?
How to relate various experiments into a unifying picture.
New concepts, new techniques….. DMFT maybe simplest approach to meet this challenge
atom solidHund’s rule, SO coupling, CFS
DMFT + electronic structure methodDMFT + electronic structure method
(G. Kotliar S. Savrasov K.H., V. Oudovenko O. Parcollet and C. Marianetti, RMP 2006).
Basic idea of DMFT+electronic structure method (LDA or GW): For less correlated orbitals (s,p): use LDA or GWFor correlated orbitals (f or d): add all local diagrams by solving QIM
observable of interestobservable of interest is the "local“is the "local“ Green's functionsGreen's functions (spectral (spectral function)function)
Currently Feasible approximations: LDA+DMFT:
LDA+DMFT
(G. Kotliar et.al., RMP 2006).
Variation gives st. eq.:
LDA functional ALL local diagrams
Generalized Q. impurity problem!
Exact Exact functionalfunctional of the of the local Green’s functionlocal Green’s function exists, its form exists, its form unknown!unknown!
DMFT + electronic structure methodDMFT + electronic structure method
obtained by DFT
Ce(4f) obtained by “impurity solution”Includes the collective excitations of the system
Self-energy is local in localized basis,in eigenbasis it is momentum dependent!
all bands are affected: have lifetimefractional weight
correlated orbitals
other “light” orbitals
hybridization
Dyson equation
General impurity problem
Diagrammatic expansion in terms of hybridization +Metropolis sampling over the diagrams
•Exact method: samples all diagrams!•Allows correct treatment of multiplets
K.H. Phys. Rev. B 75, 155113 (2007) ; P Werner, PRL (2007); N. Rubtsov PRB 72, 35122 (2005).
An exact impurity solver, continuous time QMC - expansion in terms of hybridization
Analytic impurity solvers (summing certain types of diagrams), expansion in terms of hybridization
K.H. Phys. Rev. B 64, 155111 (2001)
Fully dressed atomic propagators
hybridization
•Allows correct treatment of multiplets•Very precise at high and intermediate
frequencies and high to intermediate temperatures
Complementary to CTQMC (imaginary axis -> low energy)
“Bands” are not a good concept in DMFT!
Frequency dependent complex object instead of “bands”
lifetime effectsquasiparticle “band” does not carry weight 1
DMFTDMFT
Spectral function is a good concept
DMFT is not a single impurity calculation
Auxiliary impurity problem:
High-temperature given mostly by LDA
low T: Impurity hybridization affected by the emerging coherence of the lattice
(collective phenomena)
Weiss field temperature dependent:
Feedback effect on makes the crossover from incoherent to coherent state very slow!
high T
low T
DMFT SCC:
CeIn3 CeCoIn5 CeRhIn5 CeIrIn5 PuCoG5 Na
Tc[K] 0.2K 2.3K 2.1K 0.4K 18.3K n/a
Tcrossover ~50K ~50K ~50K ~370K
Cv/T[mJ/molK^2] 1000 300 400 750 100 1
Phase diagram of CeIn3 and 115’s
N.D. Mathur et al., Nature (1998)
CeIn3
0
1
2
3
4
?SC
SCSC
X
0.50.50.5 IrRh CoCo
AFM
T* (
K)
CeCoIn5 CeRhIn5CeIrIn5 CeCoIn5
CeXIn5
layering
Tcrossover α Tc
Ce
In
Ir
CeIn
In
Crystal structure of 115’s
CeIn3 layer
IrIn2 layer
IrIn2 layer
Tetragonal crystal structure
4 in plane In neighbors
8 out of plane in neighbors
3.27au
3.3 au
Crossover scale ~50K
in-plane
out of plane
•Low temperature – Itinerant heavy bands
•High temperature Ce-4f local moments
ALM in DMFTSchweitzer&Czycholl,1991
Coherence crossover in experiment
•How does the crossover from localized moments to itinerant q.p. happen?
•How does the spectral
weight redistribute?
•How does the hybridization gap look like in momentum space?
?
k
A()
•Where in momentum space q.p. appear?
•What is the momentum dispersion of q.p.?
Issues for the system specific study
(e
Temperature dependence of the local Ce-4f spectra
•At low T, very narrow q.p. peak (width ~3meV)
•SO coupling splits q.p.: +-0.28eV
•Redistribution of weight up to very high frequency
SO
•At 300K, only Hubbard bands
J. H. Shim, KH, and G. Kotliar Science 318, 1618 (2007).
Broken symmetry (neglecting strong correlations) can give Hubbard bands, but not both Hubbard bands
And quasiparticles!
Very slow crossover!
T*
Buildup of coherence in single impurity case
TK
cohere
nt
spect
ral
weig
ht
T scattering rate
coherence peak
Buildup of coherence
Crossover around 50K
Slow crossover pointed out byS. Nakatsuji, D. Pines, and Z. FiskPhys. Rev. Lett. 92, 016401 (2004)
Consistency with the phenomenological approach of
NPF
Remarkable agreement with Y. Yang & D. Pines cond-mat/0711.0789!
Anom
alo
us
Hall
coeffi
cient
Fraction of itinerant heavy fluid
m* of the heavy fluid
ARPESFujimori, 2006 (T=10K)
Angle integrated photoemission vs DMFT
Very good agreement, but hard to see resonancein experiment: resonance very asymmetric in Ce ARPES is surface sensitive at 122eV
Angle integrated photoemission vs DMFT
ARPESFujimori, 2006
Nice agreement for the• Hubbard band position•SO split qp peak
Hard to see narrow resonance
in ARPES since very little weight
of q.p. is below Ef
Lower Hubbard band
T=10K T=300Kscattering rate~100meV
Fingerprint of spd’s due to hybridization
Not much weight
q.p. bandSO
Momentum resolved Ce-4f spectraAf(,k)
Hybridization gap
DMFT qp bands
LDA bands LDA bands DMFT qp bands
Quasiparticle bands
three bands, Zj=5/2~1/200
Momentum resolved total spectra A(,k)
Fujimori, 2003
LDA+DMFT at 10K ARPES, HE I, 15K
LDA f-bands [-0.5eV, 0.8eV] almostdisappear, only In-p bands remain
Most of weight transferred intothe UHB
Very heavy qp at Ef,hard to see in total spectra
Below -0.5eV: almost rigid downshift
Unlike in LDA+U, no new band at -2.5eV
Large lifetime of HBs -> similar to LDA(f-core)rather than LDA or LDA+U
Optical conductivity
Typical heavy fermion at low T:
Narrow Drude peak (narrow q.p. band)
Hybridization gap
k
Interband transitions across hybridization gap -> mid IR peak
CeCoIn5
no visible Drude peak
no sharp hybridization gap
F.P. Mena & D.Van der Marel, 2005
E.J. Singley & D.N Basov, 2002
second mid IR peakat 600 cm-1
first mid-IR peakat 250 cm-1
•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
Optical conductivity in LDA+DMFT
CeIn
In
Multiple hybridization gaps
300K
e V
10K
•Larger gap due to hybridization with out of plane In•Smaller gap due to hybridization with in-plane In
non-f spectra
Fermi surfaces of CeM In5 within LDA
Localized 4f:LaRhIn5, CeRhIn5
Shishido et al. (2002)
Itinerant 4f :CeCoIn5, CeIrIn5
Haga et al. (2001)
de Haas-van Alphen experiments
LDA (with f’s in valence) is reasonable for CeIrIn5
Haga et al. (2001)
Experiment LDA
Fermi surface changes under pressure in CeRhIn5
Fermi surface reconstruction at 2.34GPa Sudden jump of dHva frequencies Fermi surface is very similar on both sides, sl
ight increase of electron FS frequencies Reconstruction happens at the point of maxim
al Tc
Shishido, (2005)localized itinerant
We can not yet address FS change with pressure
We can study FS change with Temperature -
At high T, Ce-4f electrons are excluded from the FSAt low T, they are included in the FS
Electron fermi surfaces at (z=0)
LDA+DMFT (10 K)LDA LDA+DMFT (400 K)
X M
X
XX
M
MM
2 2
Slight decrease of the electron
FS with T
R A
R
RR
A
AA
3
a
3
LDA+DMFT (10 K)LDA LDA+DMFT (400 K)
Electron fermi surfaces at (z=)No a in DMFT!No a in Experiment!
Slight decrease of the electron
FS with T
LDA+DMFT (10 K)LDA LDA+DMFT (400 K)
X M
X
XX
M
MM
c
2 2
11
Electron fermi surfaces at (z=0)Slight decrease of the electron
FS with T
R A
R
RR
A
AA
c
2 2
LDA+DMFT (10 K)LDA LDA+DMFT (400 K)
Electron fermi surfaces at (z=)No c in DMFT!No c in Experiment!
Slight decrease of the electron
FS with T
LDA+DMFT (10 K)LDA LDA+DMFT (400 K)
X M
X
XX
M
MM
g h
Hole fermi surfaces at z=0
g h
Big change-> from small hole like to large electron like
1
Localization – delocalization transition
in Lanthanides and Actinides
Delocalized Localized
Electrical resistivity & specific heat
J. C. Lashley et al. PRB 72 054416 (2005)
Heavy ferm. in an element
closed shell Am
Itinerant
NO Magnetic moments in Pu!
Pauli-like from melting to lowest T
No curie Weiss up to 600K
Curium versus Plutonium
nf=6 -> J=0 closed shell
(j-j: 6 e- in 5/2 shell)(LS: L=3,S=3,J=0)
One hole in the f shell One more electron in the f shell
No magnetic moments,large massLarge specific heat, Many phases, small or large volume
Magnetic moments! (Curie-Weiss law at high T, Orders antiferromagnetically at low T) Small effective mass (small specific heat coefficient)Large volume
Standard theory of solids:DFT:
All Cm, Am, Pu are magnetic in LSDA/GGA LDA: Pu(m~5), Am (m~6) Cm (m~4)
Exp: Pu (m=0), Am (m=0) Cm (m~7.9)Non magnetic LDA/GGA predicts volume up to 30% off.In atomic limit, Am non-magnetic, but Pu magnetic with spin ~5B
Can LDA+DMFT account for anomalous properties of actinides?
Can it predict which material is magnetic and which is not?
Many proposals to explain why Pu is non magnetic: Mixed level model (O. Eriksson, A.V. Balatsky, and J.M. Wills) (5f)4 conf. +1itt. LDA+U, LDA+U+FLEX (Shick, Anisimov, Purovskii) (5f)6 conf.
Cannot account for anomalous transport and thermodynamics
-Plutonium
0
1
2
3
4
-6 -4 -2 0 2 4 6
DO
S (
stat
es/e
V)
Total DOS
f DOS
Curium
0
1
2
3
4
-6 -4 -2 0 2 4 6ENERGY (eV)
DO
S (
stat
es/e
V)
Total DOS f, J=5/2,jz>0f, J=5/2,jz<0 f, J=7/2,jz>0f, J=7/2,jz<0
Starting from magnetic solution, Curium develops antiferromagnetic long range order below Tc above Tc has large moment (~7.9 close to LS coupling)Plutonium dynamically restores symmetry -> becomes paramagnetic
J.H. Shim, K.H., G. Kotliar, Nature 446, 513 (2007).
-Plutonium
0
1
2
3
4
-6 -4 -2 0 2 4 6
DO
S (
stat
es/e
V)
Total DOS
f DOS
Curium
0
1
2
3
4
-6 -4 -2 0 2 4 6ENERGY (eV)
DO
S (
stat
es/e
V)
Total DOS f, J=5/2,jz>0f, J=5/2,jz<0 f, J=7/2,jz>0f, J=7/2,jz<0
Multiplet structure crucial for correct Tk in Pu (~800K)and reasonable Tc in Cm (~100K)
Without F2,F4,F6: Curium comes out paramagnetic heavy fermion Plutonium weakly correlated metal
Magnetization of Cm:
Curium
0.0
0.3
0.6
0.9
-6 -4 -2 0 2 4 6ENERGY (eV)
Pro
bab
ility
N =8
N =7
N =6
J=7/
2,g =
0
J=5,
g =0
J=6,
g =0
J=4,
g =0
J=3,
g =0
J=2,
g =0
J=5,
g =0
J=2,
g =0
J=1,
g =0
J=0,
g =0
J=6,
g =0
J=4,
g =0
J=3,
g =0
f
f
f
-Plutonium
0.0
0.3
0.6
Pro
bab
ility
N =6
N =5
N =4
JJ=
0,g =
0J=
1,g =
0J=
2,g =
0J=
3,g =
0J=
4,g =
0J=
5,g =
0
J=6,
g =1
J=4,
g =0
J=5,
g =0
J=2,
g =0
J=1,
g =0
J=2,
g =1
J=3,
g =1
J=5/
2, g
=0
J=7/
2,g =
0J=
9/2,
g =0
f
f
f
Valence histograms
Density matrix projected to the atomic eigenstates of the f-shell(Probability for atomic configurations)
f electron fluctuates
between theseatomic states on the time scale t~h/Tk
(femtoseconds)
One dominant atomic state – ground state of the atom
Pu partly f5 partly f6
Probabilities:
•5 electrons 80%
•6 electrons 20%
•4 electrons <1%
J.H. Shim, K. Haule, G. Kotliar, Nature 446, 513 (2007).
Gouder , Havela PRB
2002, 2003
Fingerprint of atomic multiplets - splitting of Kondo peak
Photoemission and valence in Pu
|ground state > = |a f5(spd)3>+ |b f6 (spd)2>
f5<->f6
f5->f4
f6->f7
Af(
)
approximate decomposition
DMFT can describe crossover from local moment regime to heavy fermion state in heavy fermions. The crossover is very slow.
Width of heavy quasiparticle bands is predicted to be only ~3meV. We predict a set of three heavy bands with their dispersion.
Mid-IR peak of the optical conductivity in 115’s is split due to presence of two type’s of hybridization
Ce moment is more coupled to out-of-plane In then in-plane In which explains the sensitivity of 115’s to
substitution of transition metal ion
DMFT predicts Pu to be nonmagnetic (heavy fermion like) and Cm to be magnetic
ConclusionsConclusions
Thank you!
Gradual decrease of electron FS
Most of FS parts show similar trend
Big change might be expected in the plane – small hole like FS pockets (g,h) merge into electron FS 1 (present in LDA-f-core but not in LDA)
Fermi surface a and c do not appear in DMFT results
Increasing temperature from 10K to 300K:
Fermi surfacesFermi surfaces
ARPES of CeIrIn5
Fujimori et al. (2006)
Ce 4f partial spectral functions
LDA+DMFT (10K) LDA+DMFT (400K)
Blue lines : LDA bands
Hole fermi surface at z=
R A
R
RR
A
AANo Fermi surfaces
LDA+DMFT (400 K)LDA+DMFT (10 K)LDA
dHva freq. and effective mass
Analytic impurity solvers (summing certain types of diagrams), expansion in terms of hybridization
K.H. Phys. Rev. B 64, 155111 (2001)
Fully dressed atomic propagators
hybridization
SUNCA
•Allows correct treatment of multiplets•Very precise at high and intermediate
frequencies and high to intermediate temperatures
Complementary to CTQMC (imaginary axis -> low energy)