Towards a Realistic DMFT based Theoretical Transport and

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, localizationdelocalization ! Thermoelectricity. • 2] d’s Doped Titanites. Doping driven Mott transition. [G. Kotliar and G. Palsson PRLPRL 80, (1998), 4775] • 3] 4 f’s 115’s and the tale of multiple hybridization gaps. [K. Haule J. Shim G. Kotliar, Science Nov 1 st 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) M 2 n n Angle integrated spectral function 8

Georges Kotliar (1992) DMFT approximate quantum solid as atom in a medium 10

Spectra=- Im G(k, w) 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 Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc . Ir, >=|R, r, > Gloc=G(R r, R’ r’ ) d. R, R’ 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

“ Local” can mean a small cluster of sites or multiple unit cells. Cellular DMFT cluster 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 S 2 /kr k = kel + k. Lattice Best Case, Suppose k. Lattice = 0 ZT = T S 2 /kel r Wiedemann-Franz law L 0 = kel r/T= 2. 4 x 10 -8 V 2/K 2 or ZT = S 2/ L 0 which means that for ZT=1, or S > 156 m. V/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, localizationdelocalization ! Thermoelectricity. • 2] d’s Doped Titanites. Doping driven Mott transition. • 3] 4 f’s 115’s and the tale of multiple hybridization gaps. • 4] Conclusions

(Tokura et. al. PRL 1993)A doped Mott insulator: La 1 - Sry. O 3 y La+++ Ti+++ (O 3)-- Mott insulator . (3 d)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.

Hall Coefficient: expt. Tokura(1993) Theory Kajueter PRB (1996)

La. Sr. Ti. O 3 photoemission Fujimori et. al. expt. Theory Kajuter and GK

DMFT analysis in limiting cases. Palsson and GK PRL (1998) Low T Fermi Liquid High T Localized “ particle-like” regime

Theory : Palsson and Kotliar PRL 80, (1998), 4775 Expt. C. C. Hays PRB 90 (1999), 10367

PRB (2001) Theory ? DD Sarma Barman Kajueter Kotliar EPL (1996 ) Even more spectacular, electron gas on Sr. Ti. O 3 interface. Nature (2007)

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

Ce. MIn 5 M=Co, Ir, Rh Ce. Rh. In 5: TN=3. 8 K; 450 m. J/mol. K 2 Ir In Ce. Co. In 5: Tc=2. 3 K; 1000 m. J/mol. K 2; Ce. Ir. In 5: Tc=0. 4 K; 750 m. J/mol. K 2 Ce out of plane in-plane

Crystal structure of 115’s Ce. MIn 5 M=Co, Ir, Rh Tetragonal crystal structure Ir Ir. In 2 layer 3. 27 au In Ce Ce. In 3 layer Ir. In 2 layer 4 in plane In neighbors In 8 out of plane in neighbors 3. 3 au In Ce

Buildup of lattice coherent spectral weight Buildup of coherence in single impurity case T Very slow crossover! coherence peak TK scattering rate coherent spectral weight Slow crossover more consistent with NP&F T T* NP&F: Nakatsuji, Pines&Fisk, 2004 T* Crossover around 50 K

Angle integrated photoemission Expt Fujimori et al. , PRB 73, 224517 (2006) P. R B 67, 144507 (2003). Experimental resolution ~30 me. V Surface sensitivity at 122 ev , theory predicts 3 me. V broad band Theory: LDA+DMFT, impurity solvers SUNCA and CTQMC Shim Haule and GK (2007)

Momentum resolved total spectra tr. A(w, k) Most of weight transferred into the UHB LDA+DMFT at 10 K ARPES, HE I, 15 K LDA f-bands [-0. 5 e. V, 0. 8 e. V] almost disappear, only In-p bands remain Very heavy qp at Ef, hard to see in total spectra Below -0. 5 e. V: almost rigid downshift Unlike in LDA+U, no new band at -2. 5 e. V Fujimori, PRB Short lifetime of HBs -> similar to LDA(f-core) rather than LDA or LDA+U

Optical conductivity F. P. Mena & D. Van der Marel, 2005 Typical heavy fermion at low T: no visible Drude peak w k no sharp hybridization gap first mid-IR peak at 250 cm-1 Narrow Drude peak (narrow q. p. band) Hybridization gap Interband transitions across hybridization gap -> mid IR peak second mid IR peak at 600 cm-1 Ce. Co. In 5 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 300 K very broad Drude peak (e-e scattering, spd lifetime~0. 1 e. V) • At 10 K: • very narrow Drude peak • First MI peak at 0. 03 e. V~250 cm-1 • Second MI peak at 0. 07 e. V~600 cm-1

Multiple hybridization gaps e. V 10 K non-f spectra 300 K In Ce In • Larger gap due to hybridization with out of plane In • Smaller gap due to hybridization with in-plane In

Momentum resolved Ce-4 f spectra Af(w, k) Hybridization gap Fingerprint of spd’s due to hybridization q. p. band scattering rate~100 me. V SO T=10 K T=300 K Not much weight

Quasiparticle bands LDA bands DMFT qp bands three bands, Zj=5/2~1/200

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 1 st (2007).

Outline • 1]Introduction to correlated electrons and DMFT ideas. Central theme, localizationdelocalization ! Thermoelectricity. • 2] d’s Doped Titanites. Doping driven Mott transition. • 3] 4 f’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 ?

Na 0. 7 Co. O 2 (Terasaki). Good oxide thermoelectric K. Fujita et al. Jpn. J. Appl. Phys. 40 (2001) 4644

DMFT study of Nax Co. O 2

Foo et. al. PRL 247001

Co. O 2 Na. Co. O 2

Theoretical Issues: Na-Induced Correlations in Nax. Co. O 2 C. A. Marianetti and G. Kotliar Phys. Rev. Lett. 98, 176405 (2007) • What is the minimal model of the cobaltes ? • t 2 g 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 < Uc 2 , 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+DMFT C. Marianetti K. Haule and O Parcollet to appear in PRL

Fe. Sb 2 Bentien et. al.

Marcasite Structure Fe. Sb 2

Silicon. Treat all electrons as correlated. First order PT as impurity solver. [Cluster version of GW] LMTO basis set. F. 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 Coordination Sphere Self energy corrections beyond GW 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. • 4 f’s single site works reasonably well for the Ir 115. Quantum critical point : 2 site DMFT ? • 5 f’s Pu as a mixed valent metal. Cm RKKY metal. • 3 d’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 thesis of gunnar. Download the papers of petrovic and bentien Dowload paper by indranil paul. Download paper by gunnar. Download the stuff on latio 3 -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 Sb. F. Substittutions. Decrease in thermal conductivity. Tricks ?

Summary part 3 • What is the minimal model of the cobaltes ? • t 2 g 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 < Uc 2 , 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 a d Pu a d after G. Lander, Science (2003) and Lashley et. al. PRB (2006).

DMFT Phonons in fcc d-Pu C 11 (GPa) C 44 (GPa) C 12 (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)

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.

DMFT Phonons in fcc d-Pu C 11 (GPa) C 44 (GPa) C 12 (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)

Markert, and J. B. Goodenough REVIEW B VOLUME 60, NUMBER 14 1 OCTOBER 1999 -II