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Towards a Realistic DMFT based Theoretical Transport and Spectroscopy of Correlated Solids

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### Towards a Realistic DMFT based Theoretical Transport and Spectroscopy of Correlated Solids

G.Kotliar

Physics Department Center for Materials Theory

Rutgers University.

CRISMAT Caen October 30 (2007)

Outline

- 1]Introduction to correlated electrons and DMFT ideas. Central theme, localization-delocalization ! Thermoelectricity.
- 2] d’s Doped Titanites. Doping driven Mott transition.[G. Kotliar and G. Palsson PRLPRL 80, (1998), 4775]
- 3] 4f’s 115’s and the tale of multiple hybridization gaps.[K. Haule J. Shim G. Kotliar, Science Nov 1st 2007]
- 4] Conclusions

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

e

- Probability of removing an electron and transfering energy w=Ei-Ef, and momentum k

f(w) A(w, K) M2

n

n

Angle integrated spectral function

8

Georges Kotliar (1992)DMFT approximate quantum solid as atom in a medium

10

Self consistency for V and e

(GW) DFT+DMFT: determine H[k] and density and S self consitently from a functional

and obtain total energies.

12

Electronic structure problem: compute <r|G|r’> and <r|W|r’> given structure

Chitra and Kotliar PRB 62, 12715 (2000) PRB (2001)P.Sun and GK (2005) Zein et.al.PRL 96, 226403 (2006)). See also Bierman Aryasetiwan and Georges.

Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc .

Ir,>=|R, r,> Gloc=G(R r, R’ r’) dR,R’

“ Local” can mean a small cluster of sites or multiple unit cells. Cellular DMFT cluster DMFT.

DMFT mapping site or cluster of sites in a self consistent medium. Quantum impurity model, gives S and P.Need accurate impurity solvers..

Approximate the self energy of a subset “ uncorrelated electrons “ by dft Vxc(r)d(r,r’) replace W(w) by a static U acting only on the “correlated “ set, which we treat by DMFT.

LDA+DMFT . V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997) Review: G. Kotliar S. Savrasov K Haule O Parcollet V Oudovenko C. Marianetti RMP (2006)

Summary: part 1

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

- 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).

Thermoelectric Figure of Merit

- ZT = T S2 /kr

k = kel + kLattice

Best Case, Suppose kLattice = 0

ZT = T S2 /kel r

- Wiedemann-Franz law
- L0 = kel r/T= 2.4 x 10-8 V2/K2
- or ZT = S2/ L0
- which means that for ZT=1,
- or S > 156 mV/K

Basic Scale k/e 86 10-6 V /K

“Best” Thermoelectrics Among Mixed Valence Intermetallics(Physics Today, March 1997)

G. Mahan B. Sales J. Sharp

“State-of-the-art” Thermoelectric Materials. MRS Talk 2004 B. Sales ONRL

Outline

- 1]Introduction to correlated electrons and DMFT ideas. Central theme, localization-delocalization ! Thermoelectricity.
- 2]d’s Doped Titanites. Doping driven Mott transition.
- 3] 4f’s 115’s and the tale of multiple hybridization gaps.
- 4] Conclusions

(Tokura et. al. PRL 1993)A doped Mott insulator:La1-ySryO3

La+++ Ti+++ (O3)-- Mott insulator . (3d)1 one electron per site. x holes y electrons.

DMFT calculation U near the Mott transition, M. Rozenberg Zhang and GK PRB (1994)DMFT black dots.

Low T Fermi Liquid

DMFT analysis in limiting cases. Palsson and GK PRL (1998)

High T Localized “ particle-like” regime

Theory ? DD Sarma Barman Kajueter Kotliar EPL (1996 )

Even more spectacular, electron gas on SrTiO3 interface. Nature (2007)

Outline

- 1]Introduction to correlated electrons and DMFT ideas. Central theme, localization-delocalization ! Thermoelectricity.
- 2] d’s Doped Titanites. Doping driven Mott transition.
- 3] 4f’s 115’s and the tale of multiple hybridization gaps.
- 4] Conclusions

In

Ce

CeRhIn5: TN=3.8 K; 450 mJ/molK2CeCoIn5: Tc=2.3 K; 1000 mJ/molK2; CeIrIn5: Tc=0.4 K; 750 mJ/molK2CeMIn5 M=Co, Ir, Rh

out of plane

in-plane

In

Ce

In

Ce

In

Crystal structure of 115’s CeMIn5 M=Co, Ir, Rh

Tetragonal crystal structure

IrIn2 layer

3.27au

4 in plane In neighbors

3.3 au

CeIn3 layer

IrIn2 layer

8 out of plane in neighbors

Buildup of coherence in single impurity case

Very slow crossover!

coherent spectral weight

TK

T

Slow crossover more consistent with NP&F

coherent spectral weight

T

T*

T*

NP&F: Nakatsuji,Pines&Fisk, 2004

Buildup of lattice coherence

coherence peak

scattering rate

Crossover around 50K

Angle integrated photoemission

Expt Fujimori et al., PRB 73, 224517 (2006) P.R B 67, 144507 (2003).

Experimental resolution ~30meV

Surface sensitivity at 122 ev ,

theory predicts 3meV broad band

Theory: LDA+DMFT, impurity solvers SUNCA and CTQMC Shim Haule and GK (2007)

Momentum resolved total spectra

trA(w,k)

Most of weight transferred into

the UHB

LDA f-bands [-0.5eV, 0.8eV] almost

disappear, only In-p bands remain

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

ARPES, HE I, 15K

LDA+DMFT at 10K

Fujimori, PRB

Short lifetime of HBs -> similar to LDA(f-core)

rather than LDA or LDA+U

k

first mid-IR peak

at 250 cm-1

CeCoIn5

Optical conductivity

F.P. Mena & D.Van der Marel, 2005

Typical heavy fermion at low T:

no visible Drude peak

no sharp

hybridization gap

Narrow Drude peak (narrow q.p. band)

Hybridization gap

second mid IR peak

at 600 cm-1

Interband transitions across

hybridization gap -> mid IR peak

E.J. Singley & D.N Basov, 2002

Optical conductivity in LDA+DMFT

Expts: F. P. Mena, D. van der Marel, J. L. Sarrao, PRB 72, 045119 (2005).

16. K. S. Burch et al., PRB 75, 054523 (2007).

17. E. J. Singley, D. N. Basov, E. D. Bauer, M. B. Maple, PRB 65, 161101(R) (2002).

- 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

In

eV

Ce

In

Multiple hybridization gaps

non-f spectra

300K

- Larger gap due to hybridization with out of plane In
- Smaller gap due to hybridization with in-plane In

Momentum resolved Ce-4f spectra

Af(w,k)

Hybridization gap

q.p. band

Fingerprint of spd’s due to hybridization

scattering rate~100meV

SO

Not much weight

T=10K

T=300K

Summary

- 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 Science Express November 1st (2007).

Outline

- 2] d’s Doped Titanites. Doping driven Mott transition.
- 3] 4f’s 115’s and the tale of multiple hybridization gaps.
- 4] Conclusions

Conclusion

- Strongly Correlated electrons, still fertile ground for discovery of new thermoelectrics.
- Theory has improved, DMFT! can it play now some role in assisting and guiding experimental discoveries ?

CoO2

NaCoO2

Theoretical Issues: Na-Induced Correlations in NaxCoO2 C. A. Marianetti and G. Kotliar Phys. Rev. Lett. 98, 176405 (2007)

- 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

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 LDA+DMFTC. Marianetti K. Haule and O Parcollet to appear in PRL

Silicon. Treat all electrons as correlated. First order PT as impurity solver. [Cluster version of GW] LMTO basis setF. Aryasetiawan and O. Gunnarson, Phys. Rev. B 49, 16 214 (1994).

Zein Savrasov and Kotliar PRL 96, 226403 (2006)

Locality of correlations Zein Savrasov and Kotliar PRL 96, 226403 (2006))

GW self energy for Si

Self energy corrections beyond GW

Coordination Sphere

Coordination Sphere

Conclusions

- DMFT as a technique, makes contact with experiments, total energies, phonons, photoemission,ARPES,optics,…
- Concepts “ cell “in a quantum medium, spectral function, temperature dependent electronic structure, transfer of spectral weight.
- Method under development, but already gives some exciting results.
- Ultimate goal, is to be able to focus on deviations from DMFT.

Conclusions

- Correlations in sp electrons (worse case ) require 3 coordination spheres.
- 4f’s single site works reasonably well for the Ir 115. Quantum critical point : 2 site DMFT ?
- 5f’s Pu as a mixed valent metal. Cm RKKY metal.
- 3d’s. High Tc. Nodal antinodal dichotomy, novel type of Mott transition. Two gap scenario in SC state ?

Thanks!!

Realistic DMFT: past succeses> and future perspectives for modelling electric> and thermal transport.

Download the thesis of gunnar.

Download the papers of petrovic and bentien

Dowload paper by indranil paul.

Download paper by gunnar.

Download the stuff on latio3-in particular goodenough

Paper.

The future

- Clear theoretical problems.
- The techniques used for titanites should apply to cobaltites and misfit cobaltates.
- Disorder. Electron eelctron interactions. Importance of detailed modelling.
- The techniques used for 115’s should be useful for SbF. Substittutions. Decrease in thermal conductivity. Tricks ?

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

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)

Conclusions :chemistry brings out different aspects of localization delocalization physics.

- 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.

- 115’s delocalization transition as a function of T. Spectral function as a coherence order parameters. Multiple hybridization gaps.

Mott transition across the actinides. B. Johansson Phil Mag. 30,469 (1974)]

Mott Transition

d Pu

a

d

a

after G. Lander, Science (2003)

and Lashley et. al. PRB (2006).

C44 (GPa)

C12 (GPa)

C'(GPa)

Theory

34.56

33.03

26.81

3.88

Experiment

36.28

33.59

26.73

4.78

DMFT Phonons in fcc d-Pu

( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)

Double well structure and d 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)]

F(T,V)=Fphonons+Finvar

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.

C44 (GPa)

C12 (GPa)

C'(GPa)

Theory

34.56

33.03

26.81

3.88

Experiment

36.28

33.59

26.73

4.78

DMFT Phonons in fcc d-Pu

( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)

C. C. Hays,* J.-S. Zhou, J. T. Markert, and J. B. GoodenoughREVIEW B VOLUME 60, NUMBER 14 1 OCTOBER 1999-II

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