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Strongly Correlated Electron Materials: a Challenge for the 21 st Century

Strongly Correlated Electron Materials: a Challenge for the 21 st Century. Gabriel Kotliar and Center for Materials Theory. Colloquium Harvard University April 19 2010. $upport : NSF -DMR , DOE-Basic Energy Sciences, MURI, materials world network. 1. Outline.

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Strongly Correlated Electron Materials: a Challenge for the 21 st Century

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  1. Strongly Correlated Electron Materials: a Challenge for the 21st Century Gabriel Kotliar and Center for Materials Theory Colloquium Harvard University April 19 2010 $upport : NSF -DMR , DOE-Basic Energy Sciences, MURI, materials world network. 1

  2. Outline • Introduction to the problem of strongly correlated electron systems • Introduction to some ideas and techniques from dynamical mean field theory (DMFT) • Application to the most correlated element Pu • Application to copper oxides. • Outlook Collaborators. Rutgers, K. Haule, C. Weber, J. Shim T. Stanescu, M. Civelli Paris, M. Ferrero, A. Georges, L. DeLeo, P. Cornaglia, O Parcollet Sherbrooke, A.M. Tremblay B. Kyung D. Senechal $upport : NSF -DMR , DOE-Basic Energy Sciences, MURI, NSF materials world network. 1

  3. Standard Model of Solid State Physics • In many materials ( Cu, Au, …)electrons in solids behave as waves, quasiparticles [Sommerfeld Bloch] Metals conduct! • The Coulomb interactions renormalize to zero at low energies [ Landau] • Density functional theory in the LDA gives good estimates for the density and a good starting point for computing spectra [Kohn Sham ] • First order perturbation theory in the screened Coulomb interactions [GW ] is in good agreement with experiments. [Hedin] 2

  4. Different approach is needed for magnetic insulators • In other materials NiO, SmCo5 …Electron behave as particles • Solid as a collection of (open shell )atoms with localized electrons. • Interaction among the atom (exchange) order the atomic degrees of freedom (magnetic moments….) • Excitation spectra: multiplet theory, Hubbard bands, low wavelength collective fluctuations (spin waves) 3

  5. Strongly Correlated Materials • Not well described by either the fully localized picture (well separated atoms) or the the band picture (weakly interacting bloch waves). Materials for which the standard model of the solid state fails. Challenging non perturbativeproblem. • Continuous discovery of interesting material and phenomena that did not fit the standard model of solid state physics. Heavy fermions (early 80’s) , high temperature superconductors (late 80’s), other transition metal oxides (cobaltates, manganites, vanadates….) 90’s ..high Tc in FeAs based compounds (2009). Interesting Correlated Materials discovered by serendipity and the Edisonian approach 4

  6. Transition metal oxides transition metal ion Oxygen Correlated materials: oxides, simple recipe, rich behavior Transition metal ions Rare earth ions Cage : e.g 6 oxygen atoms (octahedra) or other ligands/geometry Actinides Transition metal (open shell ) Build crystal with this building block or build layers separated by spacers LixCoO2, NaxCoO2 Battery materials Thermoelectrics VO2 Room temperature MIT La1-xSrxMnO3 Colossal Magnetoresistance La1-xSrxCuO4 High temperature superconductor 5

  7. Probing the Electronic Structure:Photoemission A(k, w) A(k, w) Probability of removing an electron and transfering energy w=Ei-Ef, and momentum k f(w) A(w, K) M2 e w • Weak correlations • Strong correlation: fermi liquid parameters can’t be evaluated in perturbation theory or fermi liquid theory does not work. Angle integrated spectra 6

  8. Hubbard • Green A. Georges and G. Kotliar PRB 45, 6479 (1992). DMFT Collective field describing the localization delocalization phenomena DMFT self consistency condition

  9. Phase diagram :frustrated Hubbard model, integer filling M. Rozenberg G. Kotliar H. Kajuter G. Thomas PRL75, 105 (1995) Mott transition Coherence Incoherence Crossover Transfer of spectralweight T/W Quasiparticles +Hubbard bands Spectral functions 16

  10. DMFT meets electronic structure. LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Spectra=- Im G(k,w) DMFT Bands in a frequency dependent potential DMFT atom in a medium described Determine energy and and S self consistently from extremizing a functional . Savrasov and Kotliar PRB 69, 245101, (2001) Full self consistent implementation 9

  11. Dynamical Mean Field Theory DMFT • Describes the electron both in the itinerant (wave-like) and localized (particle-like) regimes and everything in between!. • Simpler reference systems to understand correlated solids [e.g. harmonic oscillator] • Impurity model non gaussian reference frame (dressed atom). Tools to think about correlated materials. Weiss field quantifies the notion of itineracy. Configuration histograms generalizes CI. • CDMFT Reference state cluser of sites ( links, triangles, plaquettes, etc. ) • Locality assumption v accurate at high T (recent comparisons with cold atoms expts. and exact numerical techniques) Kozik et. al.arXiv:0907.0863 Schneider et.al. Science, 322,1520(2008) . • Exact in the Metzner Vollhardt limit of infinite dimensions. 10

  12. Impurity solvers[ Recent advances, CTQMC Gull et. al. EPL 82, 57003 (2008), P. Werner et. al. Phys. Rev. Lett. 97, 076405 (2006) Bold CTQMC , OCA K. Haule arXiv:0907.0195 Phys. Rev. B 75, 155113 (2007) …..] • Breaks problems in two parts a) study of mean field states from b ) evaluation of their energies. • Compare different “ mean field states” of the system for the same value of parameters. Understand “mechanism” for ordering. • Qualitative lessons can be drawn from (a) applied to simple models. High temperature universality. • Low temperature, multiple ordered states. Detailed comparison experiments requires realistic implementions of electronic structure, e.g.LDA+DMFT. • Bridge between atomic information and physical properties.(Structure-Property relation ). • Theoretical spectroscopy. 11

  13. Localization Delocalization in Actinides Mott Transition Pu anomalies Large specific heat, large Pauli-like susceptility Large room temperature resistivity d Pu a a Lashley et. al. 2005 . 12

  14. The standard model of solids fails near Pu • Spin Density functional theory predicts that Pu , Am are magnetic, large orbital and spin moments. • Experiments (Lashley et. al. 2005, Heffner et al. (2006)): d Pu is non magnetic. No static or fluctuating moments. Susceptibility, specific heat in a field, neutron quasielastic and inelastic scattering, muon spin resonance. • Paramagnetic LDA underestimates Volume of dPu by 30% • Within LDA dPu has unstable phonon modes. • Thermodynamic and transport properties similar to other strongly correlated materials. 13 Pu strongly correlated paramagnetic metal

  15. 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) 14

  16. Gouder , Havela PRB 2002, 2003 alpa->delta volume collapse transition F0=4,F2=6.1 J. Shim K. Haule and G Kotliar Nature 446, 513 (2007). Photoemission 15

  17. Conclusions: elemental actinides. • Physical realization of the Mott localization-delocalization phenomena. • Successful testbed for the LDA+DMFT method, volume, phonon spectra. Plutonium is non magnetic Curium is magnetic with large moment. • Delta Pu, T* ~ 800K, mixed valent 20 % f6 80 % f5 strong coupling to the lattice. • Moment screening by valence fluctuations. s • Reconciles susceptibility, XAS, and specific heat expts. • Quasiparticlemultiplets, material specific fingerprints in the photoemisson spectra of Pu compounds. Prediction : temperature dependence . 16

  18. Smith-Kmetko phase diagram Conclusion: elemental actinides + Realism: electronic structure (band theory+atomic physics+ Coupling electrons to structure) 17

  19. Electron and Hole Doped Cuprates : Similar but Yet Different Apical oxygen NCCO : Robust AF Phase Comensurate Magnetism Lower Tc T^2 resistivity. Non monotonic angle dependence of SC order parameter ……… Review: Armitage Fournier Green (2009) 18

  20. Fundamental questions still unresolved • Important degrees of freedom at different energy scales [ dx2-y2, px py, pz, dz2 ] • Relevant effective hamiltonians, one band multiband, etc. • Mechanism of the superconductivity [ phonons, spin fluctuations, charge fluctuations, critical fluctuations, ….] • How to describe the underlying normal state which does not fit in the fermi liquid paradigm (reference system) experiments. • Difference among different families, electron vs hole doped. [ t, t’ fermi surfaces, ….] 19

  21. Single site DMFT (low T) look for AF and PM solutions for the parent compound LSCO NCCO LCO is a Mott charge transfer insulator. NCCO magnetism is responsible for gap ! 20

  22. Building phase diagram magnetization at T=0 vsd. Single site Two site 21

  23. Optical spectroscopy Theory (single site DMFT) C. Weber et. al. Onose et. al. PRB 69, 024504 (2004 ) Expt. Uchida et. al. PRB. 43, 7942 (1991) 22

  24. Optical Spectral Weights in LSCO and NCCO (up to 1.5 ev) [xx]Y. Onose et al., Phys. Rev. B, 69, 024504 (2004). [xx] S. Uchida et al., Phys. Rev. B 43, 7942 (1991). [xx]S. Lupi et al., Jour of superconductivity 17, 131 (2004). 23

  25. Include d-wave superconducting solutions. Four site DFMT T=0 Zero Temperature Moment 24

  26. Conclusions LDA+DMFT NCCO vs LSCO • Good agreement with many subtle experimental features in NCCO • even within single site DMFT. • No need to use x dependent values of the interaction U • In general, better modeling with DMFT (more sites, more orbitals etc ) better results. • Strength of correlations (as quantified by single site DMFT) the most fundamental difference between NCCO and LSCO compounds. • NCCO ( D < Dc2 )and LSCO (D > Dc2)straddle the ZaanenSawatsky Allen localization delocalization boundary. • Can be traced to the absence of apical oxygen in NCCO (structure property relation). • Introduces subtle differences in the metallization process and interplay of magnetism and superconductivity. 25

  27. Return to models, Hubbard, t-J Kinetic Energy Exchange Energy Plaquette DMFT: Lichtenstein and Kastnelson PRB (2000) T. Maier, et. al. 2001, Europhys. Lett. 56, 563. Sordi et.al. . arXiv:1002.2960 Civelli et. al. Phys. Rev. Lett. 100, 046402 (2008) Haule and Kotliar Phys. Rev. B 76, 104509 (2007) Real Space Link DMFT Ferrero et. al. Europhys. Lett. 85, 57009 (2009) Stanescu and Phillips P RB,69, 245104 (2004). Bath Momentum Space 1 2 26

  28. Link DMFT (high temperature results) Real Space Picture Momentum Space Picture 1+= 1/√2(|0, ↑> + | ↑, 0>) E=|0, 0> Fermi Liquid T=| ↑, ↑ > Holes in a sea of singlets Overdoped Underdoped S=1/√2( | ↑, ↓> -| ↑↓ >) 27

  29. Nodal Antinodal Dichotomy : Spectral Function A(k,ω→0) vs k K.M. Shen et.al. PRL 2004 LSCO Antinodal Region 2X2 CDMFT Civelli et.al. PRL 95 (2005) U=16 t=1, t’=-.3 , Hubbard model 28 Nodal Region

  30. Cluster DMFT charicature of something interersting • Plaquette and link DMFT share many similarities. • CDMFT finite T technique (finite range cumulants or self energies) . Underdoped region at low temperatures Fermi arcs turn into pockets. T. Stanescu and GK Phys. Rev. B 74, 125110 (2006) • Optimal doping. Maximum in the scattering rate. Surprising particle hole symmetry. Hidden QCP (K Haule and G. Kotliar PRB 76, 104509 (2007)or critical endpoint (Sordi et.al. 2010)) preempted by SC Tc. 29 Phase diagram from plaquette DMFT coherence scale extracted from (0,pi) cluster self energy.

  31. Superconductivity evolves continuously with doping Exp:Bi2212 with STM McElroy et.al. PRL 94, 197005 (2005) Anomalous superconductivity in the underdoped region. Coherence peaks decrease when gap increases. Non BCS. Tunneling : DOS Ratio AS/AN,. Theory K. Haule and GK. PRB (2007) Pushp et. al. arXiv:0906.0817 30

  32. Superexchange Mechanism . K. Haule and GK Phys. Rev. B 76, 104509 (2007). D.J. Scalapino and S.R. White, Phys. Rev. B 58, 8222 (1998). Reminiscent of PW Anderson RVG Science 235, 1196 (1987) and slave boson picture G. Kotliar and J. Liu P.RB 38,5412 (1988) 31

  33. Conclusion / Outlook • Correlated Electron Systems: fundamental questions, promising applications. Huge phase space • DMFT (simple ? ) framework to think about electrons in solids and compute their properties. • Many succesful applications to many materials for which cannot be treated with other techniques. • In use for many materials by many groups. Qualitative and quantitative system specific results gives us confidence in the method. Example of actinide series, cuprates. • Needed: progress in implementation! • DMFT treats local correlation well but ignores non linear interactions among long wavelength modes and topological defects and other fluctuation effects. Limited k dependence [Connection with other approaches to strong correlation] • Starting point for more sophisticated treatments [ including long wavelength modes and their interactions as in. Stat mech] • Needed : Fluctuations around DMFT 32

  34. Realistic DMFT as a tool for material exploration Separates essential ingredients [e.g. phonons, orbitals, structure etc. ] responsible for an effect. Compare different “states” of the system for the same value of parameters.  Understand Mechanism for ordering. Bridge between atomic information and physical and spectroscopical properties. [Structure-Property relation  Design] • New arenas Interfaces, junctions heterostructures, artificial materials containing correlated electrons • Grand challenge: using theory and computation to accelerate discoveries in strongly correlated electron systems. 32

  35. Thank you for your attention!

  36. E Energy difference between the normal and superconducing state of the t-J model. K. Haule and GK PRB (2006) What is the Superconductivity Mechanism ?

  37. Optics and RESTRICTED SUM RULES [Ekin]n is only defined for T> Tc, while [Ekin]s exists only for T<Tc Experiment: use of this equation implies extrapolation. Theory : use of this equation implies of mean field picture to continue the normal state below Tc.

  38. . Spectral weight integrated up to 1 eV of the three BSCCO films. a) under-doped, Tc=70 K; b) ∼ optimally doped, Tc=80 K; c) overdoped, Tc=63 K; the fullsymbols are above Tc (integration from 0+), the open symbols below Tc, (integrationfrom 0, including th weight of the superfuid). H.J.A. Molegraaf et al., Science 295, 2239 (2002). A.F. Santander-Syro et al., Europhys. Lett. 62, 568 (2003). Cond-mat 0111539. G. Deutscher et. A. Santander-Syro and N. Bontemps. PRB 72, 092504(2005) .

  39. Even within the same scheme at low T. In some region of parameters of the Hamiltonian, there are at low temperature many many solutions to the DMFT equations with different broken symmetries and ever increasing unite cells. No unique approach to expand around DMFT. Difficulties Technical Issues • 2x2 cluster DMFT equations are considerably harder to solve and to interpret than single site DMFT. • Uniqueness: no unique formulation of cluster DMFT • Realistic implementations ARE demanding. Landscape of DMFT Solutions Problem

  40. Correlated Superconductivity • New concepts and techniques to treat highly incoherent normal state and the superconductivity that emerges from it. • Coherence incoherence crossover, lines of zeros, momentum space differentiation, …… • Proximity to the Mott transition, accounts for many observations in correlated superconductors. • Still further developments are needed to improve cluster dynamical mean field theories (k space resolution) • Separates local mean field effects from fluctuation effects, superconducting fluctuations[ Nernst region etc.] Fluctuations around mean field.

  41. T. Stanescu and GK Phys. Rev. B 74, 125110 (2006)Pseudogap state pockets + lines of zeros that screend them. Some similiarities with phenomenological approach developed around the same time. Yang Rice and Zhang PRB 73 174501 (2006). R. M. Konik, T. M. Rice, A. M. Tsvelik, Phys. Rev. Lett96, 086407 (2006).

  42. Superexchange Mechanism . K. Haule and GK Phys. Rev. B 76, 104509 (2007). D.J. Scalapino and S.R. White, Phys. Rev. B 58, 8222 (1998). Reminiscent of PW Anderson RVB Science 235, 1196 (1987) and slave boson picture G. Kotliar and J. Liu P.RB 38,5412 (1988) 31

  43. Superexchange Mechanism . K. Haule and GK Phys. Rev. B 76, 104509 (2007). D.J. Scalapino and S.R. White, Phys. Rev. B 58, 8222 (1998). Reminiscent of PW Anderson RVG Science 235, 1196 (1987) and slave boson picture G. Kotliar and J. Liu P.RB 38,5412 (1988) 31

  44. Superexchange Mechanism? . K. Haule and GK Phys. Rev. B 76, 104509 (2007). Reminiscent of PW Anderson RVB Science 235, 1196 (1987) and slave boson picture G. Kotliar and J. Liu P.RB 38,5412 (1988) Ex= Jij(< Si. Sj >s- < Si . Sj>n)/t How is the energy distributed in q and w ? D.J. Scalapino and S.R. White, Phys. Rev. B 58, 8222 (1998). Expts; Dai et.al.

  45. Uchida et. al. PRB. 43, 7942 (1991) Onose et. al. PRB 69, 024504 (2004)

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