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## Gabriel Kotliar Center for Materials Theory Rutgers University

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**The Mott transition in transition metal oxides and in**organic materials:a dynamical mean field theory (DMFT) perspective. Part 1+ 2 Gabriel Kotliar Center for Materials Theory Rutgers University THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**The Mott transition in f electron materials:a dynamical mean**field theory (DMFT) perspective. Part 3. Gabriel Kotliar Center for Materials Theory Rutgers University THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Outline**• Some comments on the role of DMFT in solid state physics problems and the strong correlation problem. (Part I) • Introduction to DMFT: cavity construction E-DMFT and cluster methods. (Part I) • Introduction to DMFT: functional method. (Part III) • Interfaces with electronic structure. DMFT as a first principles method. first principles approach GW-U method. LDA+DMFT. (Part III) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Outline**• The Mott transition problem. (Part I) • Predictions of single site DMFT, and experimental verification. Phase Diagram, Optics, Photoemission, Transport. (Part I) • Conclusions of Part I. • System specific studies of materials. LDA+DMFT. Some case studies. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Outline**• SYSTEM SPECIFIC STUDIES. • The Mott transition in kappa organics. A CDMFT study. (Part II) • The metal to insulator transition in Ti2O3.A CMDFT study. (Part II) • Itinerant Magnetism in Fe and Ni. (Part II). • Conclusions of Part II. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Outline**• The Mott transition in f electron systems. The role of the spd band. The coupling to the structure, volume collapse transitions. (Part III) • Case study: alpha-gamma transition in Cerium. Photoemission and Optical Spectroscopy. (Part III) • Case study: the phases of plutonium, photoemission, total energy and lattice vibrations. (Part III) • Case study: Americium under pressure. (Part III). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Weakly correlated electrons:band theory.**• Simple conceptual picture of the ground state, excitation spectra, transport properties of many systems (simple metals, semiconductors,….). • A methods for performing quantitative calculations. (Density functional theory, in various approximations+ perturbation theory in the Coulomb interactions). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**The electron in a solid: wave picture**Momentum Space , bands, k in Brillouin zone is good quantum number. Maximum metallic resistivity 200 mohm cm Landau Fermi liquid theory interactions renormalize away at low energy, simple band picture in effective field holds. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Standard Model of Solids**• Qualitative predictions: low temperature dependence of thermodynamics and transport. • Optical response, transition between the bands. • Qualitative predictions: filled bands give rise to insulting behavior. Compounds with odd number of electrons are metals. • Quantitative tools: Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy. Good starting point for perturbative calculation of spectra,eg. GW. Kinetic equations yield transport coefficients. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Kohn Sham reference system**Excellent starting point for computation of spectra in perturbation theory in screened Coulomb interaction GW. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Success story : Density Functional Linear Response**Tremendous progress in ab initio modelling of lattice dynamics & electron-phonon interactions has been achieved (Review: Baroni et.al, Rev. Mod. Phys, 73, 515, 2001) (Savrasov, PRB 1996)**LDA+GW: semiconducting gaps**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**The electron in a solid: particle picture.**• Array of hydrogen atoms is insulating if a>>aB. Mott: correlations localize the electron e_ e_ e_ e_ • Superexchange Think in real space , solid collection of atoms High T : local moments, Low T spin-orbital order ,RVB. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Mott : Correlations localize the electron**Low densities, electron behaves as a particle,use atomic physics, real space One particle excitations: Hubbard Atoms: sharp excitation lines corresponding to adding or removing electrons. In solids they broaden by their incoherent motion, Hubbard bands (eg. bandsNiO, 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 Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Localization vs Delocalization Strong Correlation Problem**• A large number of compounds with electrons in partially filled shells, are not close to the well understood limits (localized or itinerant). Non perturbative problem. • These systems display anomalous behavior (departure from the standard model of solids). • Neither LDA or LDA+U or Hartree Fock work well. • Dynamical Mean Field Theory: Simplest approach to electronic structure, which interpolates correctly between atoms and bands. Treats QP bands and Hubbard bands. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Strong correlation anomalies**• Metals with resistivities which exceed the Mott Ioffe Reggel limit. • Transfer of spectral weight which is non local in frequency. • Dramatic failure of DFT based approximations (say DFT-GW) in predicting physical properties. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Failure of the standard model : AnomalousResistivity:LiV2O4**Takagi et.al. PRL 2000 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Failure of the StandardModel: Anomalous Spectral Weight**Transfer Optical Conductivity Schlesinger et.al (1993) Neff depends on T THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**The competition of kinetic energy and Coulomb interactions,**is a central issue that needs to be resolved. • One needs a tool that treats quasiparticle bands and Hubbard bands on the same footing to contain the band and atomic limit. • The approach should allow to incorporate material specific information. • When the neither the band or the atomic description applies, a new reference point for thinking about correlated electrons is needed. • DMFT! THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Outline**• Some comments on the role of DMFT in solid state physics problems and the strong correlation problem. • Introduction to DMFT: cavity construction E-DMFT and cluster methods. • Introduction to DMFT: functional method. • Interfaces with electronic structure. A truly first principles approach GW-U method. LDA+DMFT. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**DMFT**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Single site DMFT Impurity cavity construction: A.**Georges, G. Kotliar, PRB, (1992)] Weissfield THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Extension to clusters. Cellular DMFT. C-DMFT. G.**Kotliar,S.Y. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) tˆ(K) is the hopping expressed in the superlattice notations. • Other cluster extensions (DCA, nested cluster schemes, PCMDFT ), causality issues, O. Parcollet, G. Biroli and GK cond-matt 0307587 (2003) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**More general DMFT loop**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Mean-Field : Classical vs Quantum**Classical case Quantum case A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Solving the DMFT equations**• Wide variety of computational tools (QMC,ED….)Analytical Methods • Extension to ordered states. • Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Methods of solution : some examples**• Iterative perturbation theory. A Georges and G Kotliar PRB 45, 6479 (1992). H Kajueter and G. Kotliar PRL (1996). Interpolative schemes (Oudovenko et.al.) • Exact diag schemes Rozenberg et. al. PRL 72, 2761 (1994)Krauth and Caffarel. PRL 72, 1545 (1994) • Projective method G Moeller et. al. PRL 74 2082 (1995). • NRG R. Bulla PRL 83, 136 (1999) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**QMC M. Jarrell, PRL 69 (1992) 168, Rozenberg Zhang Kotliar**PRL 69, 1236 (1992) ,A Georges and W Krauth PRL 69, 1240 (1992) M. Rozenberg PRB 55, 4855 (1987). • NCA Prushke et. al. (1993) . SUNCA K. Haule (2003). • Analytic approaches, slave bosons. • Analytic treatment near special points. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**How good is DMFT ?**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Single site DMFT is exact in the Limit of large lattice**coordination Metzner Vollhardt, 89 Muller-Hartmann 89 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**C-DMFT: test in one dimension. (Bolech, Kancharla GK PRB**2003) Gap vs U, Exact solution Lieb and Wu, Ovshinikov Nc=2 CDMFT vs Nc=1 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**N vs mu in one dimensional Hubbard model .Compare 2 site**cluster (in exact diag with Nb=8) vs exact Bethe Anzats, [M. Capone C. Castellani M.Civelli and GK (2003)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Comments on DMFT.**• Review of DMFT, technical tools for solving DMFT eqs. A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] • CDMFT , instead of studying finite systems with open or periodic boundary conditions, study a system in a medium. Connection with DMRG, infer the density matrix by using a Gaussian anzats, and the periodicity of the system. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**DMFT Impurity cavity construction**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Outline**• The Mott transition problem. • Predictions of DMFT, and experimental verification. Phase Diagram, Optics, Photoemission, Transport. • System specific studies of materials. LDA+DMFT. Some case studies. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**The Mott transition problem**• Universal and non universal aspects. • Frustration and the success of DMFT. In the phases without long range order, DMFT is valid if T > Jeff. Need frustration to supress it. When T < Jeff LRO sets in. If Tneel is to high it oblitarates the Mott phenomena. • t vs U fundamental competition and secondary instabilities. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**V2O3 under pressure or**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**NiSe2-xSx**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Mott transition in layered organic conductors S**Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Single site DMFT and expt.**• Single site DMFT study of the Mott transition, based on a study of the Hubbard model on frustrated lattices made several interesting qualitative predictions. • New experiments and reexamination of old ones give credence to that the local picture is quite good. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Pressure Driven Mott transition**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Insights from DMFT**• Low temperature Ordered phases . Stability depends on chemistry and crystal structure • High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT. Role of magnetic frustration. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Schematic DMFT phase diagram of a partially frustrated**integered filled Hubbard model. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Qualitative single site DMFT predictions.**• Spectra of the strongly correlated metallic regime contains both quasiparticle-like and Hubbard band-like features. • Mott transition is drive by transfer of spectral weight. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Spectral Evolution at T=0 half filling full frustration**X.Zhang M. Rozenberg G. Kotliar (PRL 1993) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Parallel development: Fujimori et.al**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**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) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Qualitative single site DMFT predictions: Optics**• Spectra of the strongly correlated metallic regime contains both quasiparticle-like and Hubbard band-like features. • Mott transition is drive by transfer of spectral weight. Consequences for optics. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Schematic DMFT phase diagram Hubbard model (partial**frustration). 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) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Anomalous transfer of spectral weight in v2O3**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS