Optical Properties of Strongly Correlated Electrons: A Dynamical Mean Field Approach

Optical Properties of Strongly Correlated Electrons: A Dynamical Mean Field Approach

G. Kotliar

Physics Department and Center for Materials Theory Rutgers University

THE STATE UNIVERSITY OF NEW JERSEY

RUTGERS

Outline Correlated Electrons and the Dynamical

Mean Field Theory (DMFT) framework. Restricted Sum Rules and Transfer of

Optical Spectral Weight. Optics near the temperature driven Mott

transition. The Cerium alpha-gamma transition, Mott

transition or Kondo collapse ? A perspective from optics.


RUTGERS

References, Collaborators.

DMFT: Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004).

Optical transfer or spectral weight near the Mott transition. M. Rozenberg G. Kotliar and H. Kajueter PRB 54, 8452, (1996).

DMFT Optics V. Udovenko S. Savrasov K. Haule and G. Kotliar Cond-matt 0209336.

Alpha-Gamma Cerium. K. Haule V. Udovenko S. Savrasov and G. Kotliar. Cond-matt 0403086.


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MAIN MESSAGE

DMFT is a working tool (under constant development).

Theory (DMFT) and experiments (optical conductivity) complement each other extraordinary well.

Interpretation. Predictions. Access to regimes that cannot be easily

reached in real materials.


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“Standard Model “. Kohn Sham reference system

2 / 2 ( ) KS kj kj kjV r y e y- Ñ + =

( ')( )[ ( )] ( ) ' [ ]

| ' | ( )

LDAxc

KS ext

ErV r r V r dr

r r r

drr r

dr= + +

-ò

2( ) ( ) | ( ) |kj

kj kjr f rr e y=å

Excellent starting point for computation of spectra in perturbation theory in screened Coulomb interaction GW. Bethe Salpeter equation for optics.


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“Standard Model” fails when Correlations localize the electron

Hubbard bands. One particle excitations: corresponding to adding or removing electrons. In solids they broaden by their incoherent motion (eg. Mott insulators NiO, CoO MnO….)H H H+ H H H motion of H+ forms the lower Hubbard band

H H H H- H H motion of H_ forms the upper Hubbard band

Optical conductivity, start from atomic physics and broaden the atomic transitions (on site processes). Transitions to neighboring atomic states (transitions between the Hubbard bands ). One needs a tool that treats quasiparticle bands and Hubbard bands on the same footing to contain the band and atomic limit. DMFT!


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Strong correlation anomalies

Metals with resistivities which exceed the Mott Ioffe Reggel limit.

Gigantic linear and non linear responses. Dramatic failure of DFT based

approximations (say DFT-GW) in predicting physical properties.

Breakdown of the rigid band picture.


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Transfer of optical spectral weight non local in frequency Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B Bucher et.al. Ce2Bi4Pt3 PRL 72, 522 (1994),

Neff depends on T


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Two roads for ab-initio calculation of electronic structure of strongly correlated materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions


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RESTRICTED SUM RULES

0( ) ,eff effd P J

iV

, ,eff eff effH J P

M. Rozenberg G. Kotliar and H. Kajueter PRB 54, 8452, (1996).

2

0( ) ,

ned P J

iV m

ApreciableT dependence found.

, ,H hamiltonian J electric current P polarization

Below energy

2

2

kk

k

nk


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RESTRICTED SUM RULES

0( ) ,eff effd P J

iV

, ,eff eff effH J P

M. Rozenberg G. Kotliar and H. Kajueter PRB 54, 8452, (1996).

2

0( ) ,

ned P J

iV m

ApreciableT dependence found.

, ,H hamiltonian J electric current P polarization

Below energy

2

2

kk

k

nk


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DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). First happy marriage of a technique from atomic physics and a technique band theory.

Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)


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Self-Consistency loop. S. Savrasov and G. Kotliar (2001) and

cond-matt 0308053

G0 G

Im puritySolver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å

2| ( ) | ( )k xc k LMTOV H ka ac r c- Ñ + =

DMFT

U

E

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å


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Spectral Density Functional : effective action construction G. Kotliar, and S. Savrasov, in New Theoretical approaches to strongly correlated systems, edited by A.M. Tsvelik, Kluwer Academic Publishers, 259 (2001); S. Y. Savrasov and G. Kotliar, Phys. Rev. B 69, 245101 (2004).)

DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation. DFT(r)]

Introduce local orbitals, R(r-R)orbitals, and local GF G(R,R)(i ) =

The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for (r) and G and performing a Legendre transformation, (r),G(R,R)(i)]

' ( )* ( , ')( ) ( ')R Rdr dr r G r r i r


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LDA+DMFT References

V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997).

A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988).

G. Kotliar, and S. Savrasov, in New Theoretical approaches to strongly correlated systems, edited by A. M. Tsvelik, Kluwer 259 (2001); S. Y. Savrasov and G.

Kotliar, Phys. Rev. B 69, 245101 (2004).


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LDA+DMFT Formalism.


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Optics formula

double poledouble pole

single pole

One divergence integrated out!


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Comments on LDA+DMFT

• Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing.

• Gives an approximate starting point, for perturbation theory in the non local part of the Coulomb interactions. [See for example, P. Sun and G. Kotliar PRL ].

• Good approximate starting point for optics.


RUTGERS

Outline Correlated Electrons and the Dynamical

Mean Field Theory (DMFT) framework. Restricted Sum Rules and Transfer of

Optical Spectral Weight. Optics near the temperature driven Mott

transition. The Cerium alpha-gamma transition, Mott

transition or Kondo collapse ? A perspective from optics.

Doping driven Mott transition in La1-x SrxTiO3. A perspective from the optical conductivity.


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Pressure Driven Mott transition


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Insights from DMFT Low temperature Ordered phases . Stability depends on chemistry and crystal structureHigh temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT. Role of magnetic frustration.


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Schematic DMFT phase diagram of a partially frustrated integered filled Hubbard model. M. J. Rozenberg, G. Kotliar, H. Kajueter, G. A. Thomas, D. H. Rapkine, J. M. Honig, and P. Metcalf, Phys. Rev. Lett. 75, 105, 1995


RUTGERS

Spectral Evolution at T=0 half filling full frustrationX.Zhang M. Rozenberg G. Kotliar (PRL 70,16661993)

Spectra of the strongly correlated metallic regime contains both quasiparticle-like and Hubbard band-like features.

Mott transition is driven by transfer of spectral weight.


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Evolution of the Spectral Function with Temperature

Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)


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Consequences for the optical conductivity Evidence for QP peak in V2O3 from optics.

M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)


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Anomalous transfer of spectral weight


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Anomalous transfer of optical spectral weight V2O3

:M Rozenberg G. Kotliar and H. Kajuter Phys. Rev. B 54, 8452 (1996).

M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)


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Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi 2000]


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Optical transfer of spectral weight , kappa organics. Eldridge, J., Kornelsen, K.,Wang, H.,Williams, J., Crouch, A., and Watkins, D., Sol. State. Comm., 79, 583 (1991).


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Epilogue, the search for a quasiparticle peak and its demise, photoemission, transport. Confirmation of the DMFT predictions

ARPES measurements on NiS2-xSexMatsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)

S.-K. Mo et al., Phys Rev. Lett. 90, 186403 (2003).

Limelette et. al. [Science] G. Kotliar [Science].


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Case study Cerium.

Study the alpha to gamma transition.

Test the approach, in a well studied setting.

Differentiate between the Kondo volume

collapse picture and the Mott transition picture.


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Overview

Various phases :

isostructural phase transition (T=298K, P=0.7GPa)

(fcc) phase

[ magnetic moment

(Curie-Wiess law) ]

(fcc) phase

[ loss of magnetic

moment (Pauli-para) ]

with large

volume collapse

v/v 15

( -phase a 5.16 Å

-phase a 4.8 Å)

volumes exp. LDA LDA+U

28Å3 24.7Å3

34.4Å3 35.2Å3

-phase (localized): High T phaseCurie-Weiss law (localized magnetic moment),Large lattice constantTk around 60-80K

-phase (localized): High T phaseCurie-Weiss law (localized magnetic moment),Large lattice constantTk around 60-80K

-phase (delocalized:Kondo-

physics): Low T phaseLoss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic momentsmaller lattice constantTk around 1000-2000K

-phase (delocalized:Kondo-

physics): Low T phaseLoss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic momentsmaller lattice constantTk around 1000-2000K


RUTGERS

Qualitative Ideas.

Johanssen, Mott transition of the f electrons as a function of pressure. Ce alpha gamma transition. spd electrons are spectators.

Mathematical implementation, “metallic phase” treat spdf electrons by LDA, “insulating phase” put f electron in the core.

Allen and Martin. Kondo volume collapse picture. The dominant effect is the spd-f hybridization.


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Qualitative Ideas

“screened moment alpha phase” Kondo effect between spd and f takes place. “unscreend moment gamma phase” no Kondo effect (low Kondo temperature).

Mathematical implementation, Anderson impurity model in the Kondo limit suplemented with elastic terms. (precursor of DMFT ideas, but without self consistency condition).


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Photoemission&experiment

•A. Mc Mahan K Held and R. Scalettar (2002)

•K. Haule V. Udovenko and GK. (2003)


RUTGERS

X.Zhang M. Rozenberg G. Kotliar (PRL 1993) A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497

Unfortunately photoemission cannot decide between the Kondo collapse picture and the Mott transition picture.Evolution of the spectra as a function of U , half filling full frustration, Hubbard model!!!!


RUTGERS

Resolution: Turn to Optics!

Qualitative idea. The spd electrons have much larger velocities, so optics will be much more senstive to their behavior.

See if they are simple spectators (Mott transition picture ) or wether a Kondo binding unbinding takes pace (Kondo collapse picture).


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LDA and LDA+DMFT studies.K.Haule et. al.


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Optical Conductivity Temperature dependence.


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Origin of the features.


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Conclusion The anomalous temperature dependence and the formation of a pseudogap, suggests that

the Kondo collapse picture is closer to the truth for Cerium.

Possible experimental verification in Ce(ThLa) alloys.

Qualitative agreement with experiments, quantitative discrepancies. (see however J.Y. Rhee, X. Wang, B.N. Harmon, and D.W. Lynch, Phys. Rev. B 51, 17390 (1995) ).


RUTGERS

Conclusion Dynamical mean field theory, a first

principles approach to the computation of physical properties of correlated materials.

Tool under construction! Many improvements are possible.

Already giving interesting results. Violations of the restricted sum rule near the

temperature driven Mott transition of the order or 5 -10 %. Prediction of DMFT. Verified in experiments.


RUTGERS

Conclusion

Complementary tool to photoemission/inverse photoemission.

Experimental advantages. Ex. V2O3, Cerium.

Future work, investigate vertex corrections. Future work Where does the spectral weight

go ? Future work, study more materials.


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RUTGERS

La1-xSrx O3

Adding holes to a Mott insulator in three dimensions.

For very small doping,(x<.07) interesting spin and orbital order takes place, non universal physics and lattice distortions are important. Small energy scales, larger dopings more robust universal behavior.

Magnetic frustration. Good system to applyDMFT.


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Optical Conductivity


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Optical conductivity


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Realistic Computation of Optical Properties : La1-xSrxTiO3


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Conclusion

Reasonable agreement, between theory and experiments at both low and high energy.

The dependence of Neff on doping is due to the changes in the effective mass.


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(Tokura et. Al. 1993)A doped Mott insulator:LaxSr1-xO3


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DMFT calculation U near the Mott transition, Rozenberg et.al 94


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Hall Coefficient, electron like.


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La1-xSrxTiO3 photoemission


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Evolution of spectra with doping U=4


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Haule et. al.


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Optical Properties of Strongly Correlated Electrons: A Dynamical Mean Field Approach

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