The Mott transition in f electron systems, Pu, a dynamical mean field perspective. Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University. Collaborators: S. Savrasov (NJIT). ASR2002 Tokai Japan November 12-24 2002.

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The Mott transition in f electron systems, Pu, a dynamical mean field perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Collaborators: S. Savrasov (NJIT) ASR2002 Tokai Japan November 12-24 2002

Introduction: some Pu puzzles. Introduction to DMFT. Some qualitative insights from model Hamiltonian studies. Towards and understanding of elemental Pu. Conclusions Outline THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DFT in the LDA or GGA is a well established tool for the calculation of ground state properties. Many studies (Freeman, Koelling 1972)APW methods ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give an equilibrium volume of the d phaseIs 35% lower than experiment This is the largest discrepancy ever known in DFT based calculations. Plutonium Puzzles THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

LSDA predicts magnetic long range (Solovyev et.al.) Experimentally d Pu is not magnetic. If one treats the f electrons as part of the core LDA overestimates the volume by 30% DFT in GGA predicts correctly the volume of the a phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is usually taken as an indication that a Pu is a weakly correlated system U/W is not so different in alpha and delta DFT Studies THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

How to think about the alpha and delta phases and compute their physical properties? Why does delta have a negative thermal expansion? Why do minute amount of impurities stablize delta? Where does epsilon fit? Why is it smaller than delta? Plutonium puzzles. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Introduction: some Pu puzzles. Introduction to DMFT. Some qualitative insights from model Hamiltonian studies. Towards and understanding of elemental Pu. Conclusions Outline 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

Renormalizing the quartic term in the local impurity action. EDMFT. Taking several sites (clusters) as local entity. CDMFT Combining DMFT with other methods. LDA+DMFT, GWU. Extensions of DMFT. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Identify observable, A. Construct an exact functional of <A>=a, G [a] which is stationary at the physical value of a. Construct approximations to the functional G to perform practical calculations. Example: Density functional theory (Fukuda et. al.),density, LDA, GGA. Example: model DMFT. Observable: Local Greens function Gii (w). Exact functional G [Gii (w) ]. DMFT Approximation the functional keeping 2PI graphs DMFT: effective action point of view. R. Chitra and G. KotliarPhys. Rev. B 62, 12715 (2000), B63, 115110 (2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Effective action construction. Introduce local orbitals, caR(r-R), and local GF G(R,R)(i w) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for r(r) and G and performing a Legendre transformation, G[r(r),G(R,R)(iw)] Spectral Density Functionals for Electronic Structure Calculations, Sergej Savrasov and Gabriel Kotliar, cond-mat/0106308 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

The light, SP (or SPD) electrons are extended, well described by LDA The heavy, D (or F) electrons are localized,treat by DMFT. LDA already contains an average interaction of the heavy electrons, substract this out by shifting the heavy level (double counting term) The U matrix can be estimated from first principles (Gunnarson and Anisimov, McMahan et.al. Hybertsen et.al) of viewed as parameters Construct approximate functional which gives the LDA+DMFT equations.V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outer loop relax Edc G0 Impurity Solver Imp. Solver: Hartree-Fock G,S U SCC DMFT LDA+U THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod Phys 68,1 (1996) Kotliar, Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001). Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001). A. Lichtenstein M. Katsnelson and G. Kotliar condmat 0211076(2002) Review THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Introduction: some Pu puzzles. Introduction to DMFT. Some qualitative insights from model Hamiltonian studies. Towards and understanding of elemental Pu. Conclusions Outline THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Spectral Evolution at T=0 , half filling full frustration, Hubbard bands in the metal, transfer of spectral weight. X.Zhang M. Rozenberg G. Kotliar (PRL 1993) A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Coexistence regions between localized and delocalized spectral functions. k diverges at generic Mott endpoints Qualitative phase diagram in the U, T , m plane (two band Kotliar Murthy Rozenberg PRL (2002). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Introduction: some Pu puzzles. DMFT , qualitative aspects of the Mott transition in model Hamiltonians. DMFT as an electronic structure method. DMFT results for delta Pu, and some qualitative insights. Conclusions Outline THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Snapshots of the f electron Dominant configuration:(5f)5 Naïve view Lz=-3,-2,-1,0,1 ML=-5 mB S=5/2 Ms=5 mB Mtot=0 What is the dominant atomic configuration? Local moment? THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

L=5, S=5/2, J=5/2, Mtot=Ms=mB gJ =.7 mB Crystal fields G7 +G8 GGA+U estimate (Savrasov and Kotliar 2000) ML=-3.9 Mtot=1.1 This bit is quenched by Kondo effect of spd electrons [ DMFT treatment] Experimental consequence: neutrons large magnetic field induced form factor (G. Lander). How is the Magnetic moment quenched. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha). Is the natural consequence of the model Hamiltonian phase diagram once electronic structure is about to vary. Dynamical Mean Field View of Pu(Savrasov Kotliar and Abrahams, Nature 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Describes only the Hubbard bands. No QP states. Single well structure in the E vs V curve. (Savrasov and Kotliar PRL). Same if one uses a Hubbard one impurity solver. Comments on the HF static limit for Pu THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase. What drives this phase transition? A functional, that computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002) The delta –epsilon transition THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Compute the energy of the most unstable frozen mode. Transverse mode at ( 0,pi, pi) with polarization (0,1,-1). Extrapolate the form of the quartic interaction to the whole Brillouin zone. Carry out a self consistent Born approximation to obtain the restabilize phones. Recompute the entropy difference between delta and epsilon. Estimate the critical temperatures: 500-700 K , depending on the detials of the extrapolation. Epsilon plutonium. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Epsilon is slightly more metallic than delta, but it has a much larger phonon entropy than delta. At the phase transition the volume shrinks but the phonon entropy increases. Phonon entropy drives the epsilon delta phase transition THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Introduction: some Pu puzzles. DMFT , qualitative aspects of the Mott transition in model Hamiltonians. DMFT as an electronic structure method. DMFT results for delta Pu, and some qualitative insights into other phases. Conclusions Outline THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT produces non magnetic state, around a fluctuating (5f)^5 configuraton with correct volume the qualitative features of the photoemission spectra, and a double minima structure in the E vs V curve. Correlated view of the alpha and delta phases of Pu. Interplay of correlations and electron phonon interactions (delta-epsilon). Calculations can be refined. Conclusions THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phonons matter. Role of electronic entropy. In the making, new generation of DMFT programs, QMC with multiplets, full potential DMFT, frequency dependent U’s, multiplet effects , combination of DMFT with GW Other materials, Cerium and Yterbium compounds………… Conclusions THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Acknowledgements: Development of DMFT Collaborators: V. Anisimov, R. Chitra, V. Dobrosavlevic, D. Fisher, A. Georges, H. Kajueter, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, S. Pankov, M. Rozenberg,S. Murthy , S. Savrasov, Q. Si, V. Udovenko, X.Y. Zhang Funding: National Science Foundation. Department of Energy and LANL. Office of Naval Research. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS