Post on 21-Dec-2015
Dynamical Mean Field Theory (DMFT) of correlated solids.
G.Kotliar Physics Department Center for Materials Theory
Rutgers University.
Collaborators: K. Haule (Rutgers), S. Savrasov (UC Davis)
September 2-7 (2007) Magdalen College Oxford United Kingdom
Gordon Research Conference on Solid State Chemistry II
Outline
• 1]Introduction to DMFT ideas.• 2]Application to elemental actinides, what is
valence in a correlated solid ?• 3]Application to cobaltates, why are correlation
stronger near a band insulator than near a Mott insulator? [C. Delmas talk A.Maignan talk]
• Central theme, localization-delocalization !
• 4]Application to 115’s and the tale of multiple hybridization gaps. [F. Steglich talk]
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 ?
DMFT Spectral Function Photoemission and correlations
• Probability of removing an electron and transfering energy =Ei-Ef, and momentum k
f() A() M2
e
Angle integrated spectral Angle integrated spectral function function
( , ) ( )dkA k A 88
DMFT approximate quantum solid as atom in a medium † †
, ,
( )( )ij ij i j j i i ii j i
t c c c c U n n
† † † † †Anderson Imp 0 0 0 0 0 0 0
, , ,
( +c.c). H c A A A c c UcV c c c
1010
, ,
,
[ ] [ ]( )
[ ] [ ]spd sps spd f
f spd ff
H k H kt k
H k H k
æ ö÷ç ÷ç ÷ç ÷çè ø®
| 0 ,| , | , | | ... JLSJM g> > ¯> ¯> >®
(GW) DFT+DMFT: determine H[k] and density and(GW) DFT+DMFT: determine H[k] and density andself consitently from a functionalself consitently from a functional
and obtain total energies. and obtain total energies. 1212
[ ]*
11
( )( ) (
,)n n
n nk
i ii t k i
V VVa a
aaaa
ew m ww m ww e
-é ùê ú+ - = +Sê ú+ - - S- ë û
å å
1( , )
( ) ( )G k i
i t k i
Spectra=- Im G(k,)
Self consistency for V and
Summary: part 1
• Gabriel Kotliar and Dieter Vollhardt, Physics Today 57, 53 (2004).
• A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. of Mod. Phys. 68, 13-125 (1996).
• G. Kotliar, S. Savrasov, K. Haule, V. Oudovenko, O. Parcollet, and C. Marianetti, Rev. of Mod. Phys. 78, 000865 (2006).
Spectral function in DMFT analogous to density in DFT
Self consistent Impurity problem, natural language to describe localization/delocalization phenomena. combines atomic physics and band theory
Systematically improvable, cluster DMFT
Recent progress in implementation
after G. Lander, Science (2003)and Lashley et. al. PRB (2006).
Mott Transition
PuPu
Mott transition across the actinides. B. Johansson Phil Mag. 30,469 (1974)]
Pu phases: A. Lawson Los Alamos Science 26, (2000)
GGA LSDA predicts Pu to be magnetic with a large moment ( ~5 Bohr) . Experimentally Pu is not magnetic. [PRB 054416(2005). Valence of Pu is controversial.
DMFTPhononsinfccDMFTPhononsinfcc-Pu-Pu
C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)
Theory 34.56 33.03 26.81 3.88
Experiment 36.28 33.59 26.73 4.78
( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August 2003)
Curie-Weiss
Tc
PhotoemissionofActinidesalpa->deltavolumecollapsetransition
CuriumhaslargemagneticmomentandordersantifPudoesisnonmagnetic.
F0=4,F2=6.1
F0=4.5,F2=7.15
F0=4.5,F2=8.11
What is the valence in the late actinides ?
Summary part 2
• Modern understanding of the Mott transition across the actinde series (B. Johanssen)
sheds light on the physics of actinides.• Important role of multiplets. Pu is non magnetic
and mixed valent element mixture of f6 and f5 • f electrons are localized in Cm f7
K. Haule and J. Shim Ref: Nature 446, 513, (2007)
DMFT study of Nax CoO2
Foo et.al. PRL 247001
CoO2NaCoO2
Assume Na patterns of Zandbergen et. al.PRB 70 024101
C. A. Marianetti and G. Kotliar Phys. Rev. Lett. 98, 176405 (2007) . A
DMFT calculations with and without disorder U=3 ev.
C. A. Marianetti and G. Kotliar Phys. Rev. Lett. 98, 176405 (2007)
x=.33 QP dispersion DMFT LDA
References: part 3
• C. Marianetti, G. Kotliar, and G. Ceder, Nature Materials 3, 627 - 631 (2004).
• C. A. Marianetti and G. Kotliar Phys. Rev. Lett. 98,176405 (2007)
• C. Marianetti, K. Haule and O Parcollet cond-mat (2007)
Alternative theory : low spin to high spin Khaliullin Phys. Rev. Lett. 96, 216404 (2006)
Summary part 3
• What is the minimal model of the cobaltes ? • t2g orbitals + binary potential a see which
results of the Li /Na vacancy .• Why are correlations stronger near a band
insulator than near a Mott insulator ?• U < Uc2 , hole moves in a restricted space
(where potential is low) and is strongly correlated.
• DMFT calculations account for the Curie Weiss phase and the Fermi liquid phase
Conclusion
• DMFT as a technique, makes contact with experiments, total energies, phonons, photoemission, ARPES,optics,…thermopower…..neutron scattering …..
• Concepts, atom in a quantum medium, Weiss field, local spectral function, A(), three peak structure,transfer of spectral weight , valence histogram, [bridges between atomic physics and band theory ]
• Under constant development, but already gives some exciting results.
• Actinides, phonons, role of multiplets, spectral signatures, Pu as mixed valent metal.
• Cobaltates, key role of inhomogeneities bringing correlations near a (correlated) insulator. DMFT treatment of an alloy.
Conclusions :chemistry brings out different aspects of localization delocalization physics.
•115’s delocalization transition as a function of T. Spectral function as a coherence order parameters. Multiple hybridization gaps.
Thanks!
• Acknowldegment. NSF-DMR. DOE-BES.
• Collaborators:K. Haule, C. Marianetti, J. Shim, and S. Savrasov
Conclusions :chemistry brings out different aspects of localization delocalization physics.
Ce
In
Ir
CeIn
In
Crystalstructureof115’sCeMIn5 M=Co, Ir, Rh
CeIn3layer
IrIn2layer
IrIn2layer
Tetragonalcrystalstructure
4inplaneInneighbors
8outofplaneinneighbors
3.27au
3.3 au
CeRhIn5: TN=3.8 K; 450 mJ/molK2 CeCoIn5: Tc=2.3 K; 1000 mJ/molK2;
CeIrIn5: Tc=0.4 K; 750 mJ/molK2
CeMIn5 M=Co, Ir, Rh
outofplane
in-plane
ARPESFujimori,2006
Angleintegratedphotoemission
Experimentalresolution~30meV,theorypredicts3meVbroadband
Surfacesensitiveat122eV
Very slow crossover!
T*
SlowcrossovermoreconsistentwithNP&F
T*
coherentspectral
weight
T
NP&F:Nakatsuji,Pines&Fisk,2004
Buildupofcoherenceinsingleimpuritycase
TK
coherentspectral
weight
T
scatteringrate
coherencepeak
Buildupofcoherence
Crossoveraround50K
•At300KverybroadDrudepeak(e-escattering,spdlifetime~0.1eV)•At10K:
•verynarrowDrudepeak•FirstMIpeakat0.03eV~250cm-1
•SecondMIpeakat0.07eV~600cm-1
OpticalconductivityinLDA+DMFT
CeIn
In
Multiplehybridizationgaps
300K
e V
10K
•Larger gap due to hybridization with out of plane In•Smaller gap due to hybridization with in-plane In
non-fspectra
Summary part 4
• 115’s model systems to study the evolution of the f electron as a function of temperature
• Multiple hybridization gaps in optics.
• Very different Ce-In hybridizations with In
out of plane being larger.
J. Shim K Haule and G.K. Submitted to Science. (2007).
PRL 80, (1998) GPalsson and GK Thermoelectricity near a Mott
transition La1-xSrxTiO3
ARPESFujimori,2006
AngleintegratedphotoemissionvsDMFT
Experimentalresolution~30meV,theorypredicts3meVbroadband
Surfacesensitiveat122eV
AngleintegratedphotoemissionvsDMFT
ARPESFujimori,2006
Niceagreementforthe•Hubbardbandposition•SOsplitqppeak
Hardtoseenarrowresonance
inARPESsinceverylittleweight
ofq.p.isbelowEf
LowerHubbardband
• . arXiv:0704.1065 [pdf] – Title: Precise Control of Band Filling in NaxCoO2 – Authors: Daisuke Yoshizumi, Yuji Muraoka,
Yoshihiko Okamoto, Yoko Kiuchi, Jun-Ichi Yamaura, Masahito Mochizuki, Masao Ogata, Zenji Hiroi
– Comments: 4 pages, 5 figures, submitted to J. Phys. Soc. Jpn
– Journal-ref: J. Phys. Soc. Jpn. 76 (2007) 063705 – Subjects: Strongly Correlated Electrons (cond-mat.str-
el)
1
10
1( ) ( )
( )n nn k nk
G i ii t i
w ww m w
-
-é ùê ú= +Sê ú- + - Sê úë ûå
DMFT Impurity cavity construction
1
10
1( ) ( )
V ( )n nk nk
D i ii
w ww
-
-é ùê ú= +Pê ú- Pê úë ûå
0
1 † 10 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ
†
0 0
( ) ( , ') ( ') ( , ') o o o oc Go c n n Ub b
s st t t t d t t ¯+òò
† †
, ,
( )( )ij ij i j j i i ii j i
t c c c c U n n
()
1 100 0 0( )[ ] ( ) [ ( ) ( ) ]n n n n Si G D i n i n iw w w w- -P = + á ñ
,ij i j
i j
V n n
0 0( , ')Do n nt t+
Functional formulation. Chitra and Kotliar Phys. Rev. B 62, 12715 (2000)
and Phys. Rev.B (2001) .
1 †1( ) ( , ') ( ') ( ) ( ) ( )
2Cx V x x x i x x xff f y y-+ +òò ò
†( ') ( )G x xy y=- < > ( ') ( ) ( ') ( )x x x x Wff ff< >- < >< >=
Ex. Ir>=|R, > Gloc=G(R, R ’) R,R’’
1 10
1 1[ , , , ] [ ] [ ] [ ] [ ] [ , ]
2 2C hartreeG W M P TrLn G M Tr G TrLn V P Tr P W E G W
Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc .
Sum of 2PI graphs[ , ] [ , , 0, 0]EDMFT loc loc nonloc nonlocG W G W G W
One can also view as an approximation to an exact Spetral Density Functional of Gloc and Wloc.
1 10
1 10
loc
loc
G
W
G M
V P1
1
( )
1
( )
lock
locq C
GH k
Wv q
M
P
0 0G V,,intM PLocal Impurity Model
Input: ,M P
Output: Self-Consistent Solution
Spectral Density Functional Theory withinLocal Dynamical Mean Field Approximation
,loc locG W 1 10
1 10
loc
loc
G
W
G M
V P1
1
( )
1
( )
lock
locq C
GH k
Wv q
M
P
0 0G V,,intM PLocal Impurity Model
Input: ,M P
Output: Self-Consistent Solution
Spectral Density Functional Theory withinLocal Dynamical Mean Field Approximation
•Full implementation in the context of a a one orbital model. P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002).
•After finishing the loop treat the graphs involving Gnonloc Wnonloc in perturbation theory. P.Sun and GK PRL (2004). Related work, Biermann Aersetiwan and Georges PRL 90,086402 (2003) .
EDMFT loop G. Kotliar and S. Savrasov in New Theoretical Approaches to Strongly Correlated G Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic Publishers. 259-301 . cond-mat/0208241 S. Y. Savrasov, G. Kotliar, Phys. Rev. B 69, 245101 (2004)
Anomalous Resistivity
2 ( )F Fe k k l
h
Maximum metallic resistivity 2
Fe k
h
TotalEnergyasafunctionofvolumeforTotalEnergyasafunctionofvolumeforPuPu (ev) vs (a.u. 27.2 ev)
(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.
iw
Zein (2005) Following Aryasetiwan Imada Georges Kotliar Bierman and
Lichtenstein. PRB 70 195104. (2004)
Pu
alpa->deltaPhotoemission
Gouder Havela Lande
PRB(2001)r
PhotoemissionSpectra[Shim.Haule,GKNature(2007)]
alpa->deltavolumecollapsetransition
F0=4,F2=6.1
F0=4.5,F2=7.15
2020
Resistivity of Am under pressure. J. C. Griveau Rebizant Lander and Kotliar PRL 94, 097002 (2005).
Photoemission spectra using Hubbard I solver [Lichtenstein and Katsnelson, PRB 57, 6884,(1998 ),
Svane cond-mat 0508311] and Sunca . [Savrasov Haule and Kotliar cond-mat 0507552] Hubbard bands width is
determined by multiplet splittings.
CeIrIn5
1( , ) (1/ ) Im[ ]
ˆˆ ˆ( ) ( )A k
H k E iab abw p
w w d=-
- - - S +
While DMFT aims at local quantities, one can derive non local information
Photomission Spectra of Am under pressure. Sunca. Onset of mixed valence. Savrasov Haule
Kotliar PRL (2005)
Am equation of state. LDA+DMFT.New acceleration technique for solving DMFT equations S. Savrasov
K. Haule G. Kotliar PRL (2006)
The “DMFT-valence” in the late actinides
Pu phases: A. Lawson Los Alamos Science 26, (2000)
GGA LSDA predicts Pu to be magnetic with a large moment ( ~5 Bohr) . Experimentally Pu is not magnetic. [Lashley et. al. cond-matt 0410634] PRB 054416(2005).
Approach the Mott transition from the left. (delocalized side).
DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer
filling
T/W
1414
Model Hamiltonians
• Tight binding form.
• Eliminate the “irrelevant” high energy degrees of freedom
• Add effective Coulomb interaction terms
Summary: part 1
LDA+DMFT Self-Consistency loop
G0 G
Im p u rityS o lver
S .C .C .
0( ) ( , , ) i
i
r T G r r i e w
w
r w+
= å
2( ) | ( ) |k xc kH k Va ac r c= - Ñ +
DMFT
U
-Edc
0( , , )HHi
HH
i
n T G r r i e w
w
w+
= å
DMFT Cavity Construction. Happy marriage of atomic and band physics.
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 Rev. Mod. Phys. 78, 865 (2006) . G. Kotliar and D . Vollhardt Physics 53 Today (2004)
1( , )
( )k
G k ii i
Extremize a functional of the local spectra. Local self energy.
A. Georges, G. Kotliar (1992)
( )wDlatt ( ,
1 G [ ]
( ) [( ) ])
[ ]n impn
n
ik ii
ktw m
ww+ + - S
DD
=
latt( ) G ([ [)] ] ,imp n nk
G i i kw wD D=å
[ ]ijij
jm mJth hb= +å
11
( ( )( )
( [))
][ ]
imp n
imp n
kn
G i
Gti
ik
w
ww -D
D
=+-
å
A(A())
1111
Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003
<l.s> in the late actinides [DMFT results: K. Haule and J. Shim ]
TotalEnergyasafunctionofvolumeforTotalEnergyasafunctionofvolumeforPuPuW (ev) vs (a.u. 27.2 ev)
(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.
iw
Nick Zein Following Aryasetiwan et. al. PRB 70 195104. (2004)
Moment is first reduced by orbital spin moment compensation. The
remaining moment is screened by the spd and f electrons
Double well structure and Pu Qualitative explanation of negative thermal expansion[Lawson, 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)]
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.
F(T,V)=Fphonons+Finvar
Opticalconductivity
Typical heavy fermion at low T:
NarrowDrudepeak(narrowq.p.band)
Hybridizationgap
k
Interbandtransitionsacrosshybridizationgap->midIRpeak
CeCoIn5
novisibleDrudepeak
nosharphybridizationgap
F.P.Mena&D.VanderMarel,2005
E.J.Singley&D.NBasov,2002
secondmidIRpeakat600cm-1
firstmid-IRpeakat250cm-1
T=10K T=300Kscatteringrate~100meV
Fingerprintofspd’sduetohybridization
Notmuchweight
q.p. bandSO
MomentumresolvedCe-4fspectraAf(,k)
Hybridizationgap
DMFTqpbands
LDAbands LDAbands DMFTqpbands
Quasiparticlebands
threebands,Zj=5/2~1/200
Momentumresolvedtotalspectra A(,k)
Fujimori,2003
LDA+DMFTat10K ARPES,HEI,15K
LDAf-bands[-0.5eV,0.8eV]almostdisappear,onlyIn-pbandsremain
MostofweighttransferredintotheUHB
VeryheavyqpatEf,hardtoseeintotalspectra
Below-0.5eV:almostrigiddownshift
UnlikeinLDA+U,nonewbandat-2.5eV
LargelifetimeofHBs->similartoLDA(f-core)ratherthanLDAorLDA+U
DMFT : Evolution of the DOS.
( )A 1313
Approach Mott point from the right Am under pressureApproach Mott point from the right Am under pressure
ExperimentalEquationofState (after Heathman et.al, PRL 2000)
Mott Transition?“Soft”
“Hard”
J. C. Griveau et.al. PRL 94, 097002 (2005).
The “DMFT-valence” in the late actinides
From Atoms to Solids
Band (e.g. LDA)Atomic Physics (eg. CI) )
The “DMFT-valence” in the late actinides
Late actinide issues
• All the spin density functional studies of fcc Pu and Am , using either LDA or GGA, predict magnetic long range order with a large moment. Experimentally Pu and Am are not magnetic.
• If one treats the f electrons as part of the core LDA overestimates the volume by 30%
• Valence of Pu controversy. LDA+U Schick, Havela Lichtenstein et.al. Anisimov et.al.(5f)^6. Erickson and Wills (5f)^4.
CoO2
(e
TemperaturedependenceofthelocalCe-4fspectra
•AtlowT,verynarrowq.p.peak(width~3meV)
•SOcouplingsplitsq.p.:+-0.28eV
•Redistributionofweightbetweentheq.p.andtheupperHubbardband
SO
•At300K,onlyHubbardbands
• arXiv:0708.2528 [pdf] – Title: Synthesis and properties of CoO2, the x
= 0 end member of the LixCoO2 and NaxCoO2 systems
– Authors: T. Motohashi, Y. Katsumata, T. Ono, R. Kanno, M. Karppinen, H. Yamauchi
– Comments: 18 pages, 5 figures – Journal-ref: Chemistry of Materials,
cm0702464 (2007)
PhotoemissionGouder , Havela PRB
2002, 2003
CeCoIn5
CeRhIn
5
CeIrIn
5
Tc[K] 2.3K 2.1K@p>1.5GPa
0.4K
Cv/T[mJ/molK^2]
300 50 750
WhyCeIrIn5?
Phasediagramof115’s
5f elements: actinide series5f elements: actinide series
s/cs/c AFAF FMFM
Localisation Delocalization Localisation Delocalization
1.4K1.4K 0.40.4KK
0.9K0.9K 0.8K0.8K 52K52K 25K25K 52K52K
The “DMFT-valence” in the late actinidesHaule Shim Kotliar (2006)