Electronic Structure of Strongly Correlated Electron Materials A

Electronic Structure of Strongly Correlated Electron Materials: A Dynamical Mean Field Perspective. Kristjan Haule, Physics Department and Center for Materials Theory Rutgers University Collaborators: G. Kotliar, S. Savrasov, V. Oudovenko

Overview • Application of DMFT to real materials (Spectral density functional approach). Examples: – alpha to gamma transition in Ce, optics near the temperature driven Mott transition. – Mott transition in Americium under pressure • Extensions of DMFT to clusters. Examples: – Superconducting state in t-J the model – Optical conductivity of the t-J model

Universality of the Mott transition Crossover: bad insulator to bad metal Critical point First order MIT V 2 O 3 1 B HB model (DMFT): Ni 2 -x. Sex k organics

Coherence incoherence crossover in the 1 B HB model (DMFT) Phase diagram of the HM with partial frustration at half-filling M. Rozenberg et. al. , Phys. Rev. Lett. 75, 105 (1995).

Dynamical Mean Field Theory Basic idea of DMFT: reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition. Basic idea of Spectral density functional approach: instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission] mapping fermionic bath

DFT & DMFT from the unifying point of view Density functional theory observable of interest is the electron density Dynamical mean field theory: observable of interest is the local Green's function (on the lattice uniquely defined) exact BK functional mapping fermionic bath DMFT approximation

Spectral density functional theory G. Kotliar et. al. , cond-mat/0511085 observable of interest is the "local“ Green's functions (spectral function) Currently feasible approximations: LDA+DMFT and GW+DMFT: basic idea: sum-up all local diagrams for electrons in correlated orbitals LDA+U corresponds to LDA+DMFT when impurity is solved in the Hartree Fock approximation

Periodic table f 1 L=3, S=1/2 J=5/2 f 6 L=3, S=3 J=0

Cerium

Ce overview isostructural phase transition ends in a critical point at (T=600 K, P=2 GPa) (fcc) phase [ magnetic moment (Curie-Wiess law), large volume, stable high-T, low-p] (fcc) phase [ loss of magnetic moment (Pauli-para), smaller volume, stable low-T, high-p] with large volume collapse v/v 15 volumes exp. 28Å3 34. 4Å3 LDA 24. 7Å3 • Transition is 1. order • ends with CP LDA+U 35. 2Å3

LDA and LDA+U ferromagnetic volumes exp. 28Å3 34. 4Å3 LDA 24. 7Å3 LDA+U 35. 2Å3 f DOS total DOS

LDA+DMFT alpha DOS TK(exp)=1000 -2000 K

LDA+DMFT gamma DOS TK(exp)=60 -80 K

Photoemission&experiment • A. Mc Mahan K Held and R. Scalettar (2002) • K. Haule V. Udovenko and GK. (2003) Fenomenological Landau approach: Kondo volume colapse (J. W. Allen, R. M. Martin, 1982)

Optical conductivity + * + K. Haule, et. al. , Phys. Rev. Lett. 94, 036401 (2005) * J. W. van der Eb, A. B. Ku’zmenko, and D. van der Marel, Phys. Rev. Lett. 86, 3407 (2001)

Americium

Americium f 6 -> L=3, S=3, J=0 Mott Transition? "soft" phase f localized "hard" phase f bonding A. Lindbaum, S. Heathman, K. Litfin, and Y. Méresse, Phys. Rev. B 63, 214101 (2001) J. -C. Griveau, J. Rebizant, G. H. Lander, and G. Kotliar Phys. Rev. Lett. 94, 097002 (2005)

Am within LDA+DMFT Large multiple effects: F(0)=4. 5 e. V S. Y. Savrasov, K. Haule, and G. Kotliar Phys. Rev. Lett. 96, 036404 (2006) F(2)=8. 0 e. V F(4)=5. 4 e. V F(6)=4. 0 e. V

Am within LDA+DMFT from J=0 to J=7/2 Comparisson with experiment V=V 0 Am I V=0. 76 V 0 Am III V=0. 63 V 0 Am IV nf=6. 2 nf=6 • “Soft” phase very different from g Ce not in local moment regime since J=0 (no entropy) • "Hard" phase similar to a Ce, Kondo physics due to hybridization, however, nf still far from Kondo regime Different from Sm! Exp: J. R. Naegele, L. Manes, J. C. Spirlet, and W. Müller Phys. Rev. Lett. 52, 1834 -1837 (1984) Theory: S. Y. Savrasov, K. Haule, and G. Kotliar Phys. Rev. Lett. 96, 036404 (2006)

Beyond single site DMFT What is missing in DMFT? • Momentum dependence of the self-energy m*/m=1/Z • Various orders: d-wave. SC, … • Variation of Z, m*, t on the Fermi surface • Non trivial insulator (frustrated magnets) • Non-local interactions (spin-spin, long range Columb, correlated hopping. . ) Present in DMFT: • Quantum time fluctuations Present in cluster DMFT: • Quantum time fluctuations • Spatially short range quantum fluctuations

The simplest model of high Tc’s t-J, PW Anderson Hubbard-Stratonovich ->(to keep some out-of-cluster quantum fluctuations) BK Functional, Exact cluster in k space cluster in real space

What can we learn from “small” Cluster-DMFT? Phase diagram t’=0

Insights into superconducting state (BCS/non-BCS)? BCS: upon pairing potential energy of electrons decreases, kinetic energy increases (cooper pairs propagate slower) Condensation energy is the difference non-BCS: kinetic energy decreases upon pairing (holes propagate easier in superconductor) J. E. Hirsch, Science, 295, 5563 (2226)

Optical conductivity optimally doped overdoped cond-mat/0601478 D van der Marel, Nature 425, 271 -274 (2003)

Optical weight, plasma frequency Weight bigger in SC, K decreases (non-BCS) ~1 e. V Bi 2212 Weight smaller in SC, K increases (BCS-like) D. van der Marel et. al. , in preparation

Hubbard versus t-J model Kinetic energy in Hubbard model: • Moving of holes • Excitations between Hubbard bands Hubbard model U Drude t 2/U Experiments Excitations into upper Hubbard band Kinetic energy in t-J model • Only moving of holes Drude J intraband interband transitions t-J model no-U ~1 e. V

Kinetic energy change Kinetic energy increases cluster-DMFT, cond-mat/0601478 Kinetic energy decreases Kinetic energy increases cond-mat/0503073 Phys Rev. B 72, 092504 (2005) Exchange energy decreases and gives largest contribution to condensation energy

Kinetic energy upon condensation underdoped overdoped J J electrons gain energy due to exchange energy holes gain kinetic energy (move faster) hole loose kinetic energy (move slower) J same as RVB (see P. W. Anderson Physica C, 341, 9 (2000), or slave boson mean field (P. Lee, Physica C, 317, 194 (1999) J BCS like

Optics mass and plasma frequency Extended Drude model • Within DMFT, optics mass is m*/m=1/Z and diverges at the Mott transition • Plasma frequency vanishes as 1/Z (Drude shrinks as Kondo peak shrinks) • In cluster-DMFT optics mass constant at low doping • Plasma frequency vanishes because the active (coherent) part of the Fermi surface shrinks line: cluster DMFT (cond-mat 0601478), symbols: Bi 2212, Van der Marel (in preparation)

Optimal doping: Powerlaws D. van der Marel et. al. , Nature 425, 271 (2003). cond-mat/0605149

Optimal doping: Coherence scale seems to vanish underdoped scattering at Tc optimally Tc overdoped

Local density of states of SC STM study Cluster DMFT 1 Larger doping 6 Smaller doping • V shaped gap (d-wave) • size of gap decreases with doping • CDMFT-optimall doping PH symmetric

41 me. V resonance • Resonance at 0. 16 t~48 me. V • Most pronounced at optimal doping • Second peak shifts with doping (at 0. 38~120 me. V opt. d. ) and changes below Tc – contribution to condensation energy local susceptibility YBa 2 Cu 3 O 6. 6 (Tc=62. 7 K) Pengcheng et. al. , Science 284, (1999)

Pseudoparticle insight N=4, S=0, K=0 N=4, S=1, K=(p, p) N=3, S=1/2, K=(p, 0) N=2, S=0, K=0 A(w) S’’(w) PH symmetry, Large t

Conclusions • • LDA+DMFT can describe interplay of lattice and electronic structure near Mott transition. Gives physical connection between spectra, lattice structure, optics, . . – Allows to study the Mott transition in open and closed shell cases. – In both Ce and Am single site LDA+DMFT gives the zeroth order picture – Am: Rich physics, mixed valence under pressure. 2 D models of high-Tc require cluster of sites. Some aspects of optimally doped, overdoped and slightly underdoped regime can be described with cluster DMFT on plaquette: – Evolution from kinetic energy saving to BCS kinetic energy cost mechanism – Optical mass approaches a constant at the Mott transition and plasma frequency vanishes – At optimal doping: Physical observables like optical conductivity and spin susceptibility show powerlaw behavior at intermediate frequencies, very large scattering rate – vanishing of coherence scale, PH symmetry is dynamically restored, 41 me. V resonance appears in spin response

LDA+DMFT implementation LDA DMFT SCC * * Impurity solver local in localized LMTO base Impurity problem (14 x 14):

Comparison of spectral weight cluster DMFT / Bi 2212 Spectral weight (kinetic energy) changes faster with T in overdoped system – larger coherence scale Carbone et. al. , in preparation

Partial DOS 4 f 5 d 6 s Z=0. 33

More complicated f systems • Hunds coupling is important when more than one electron in the correlated (f) orbital • Spin orbit coupling is very small in Ce, while it become important in heavier elements The complicated atom embedded into fermionic bath (with crystal fileds) is a serious chalange so solve! Coulomb interaction is diagonal in the base of total LSJ -> LS base while the SO coupling is diagonal in the j-base -> jj base Eigenbase of the atom depends on the strength of the Hund's couling and strength of the spin-orbit interaction

Classical theories Mott transition (B. Johansson, 1974): Hubbard model f electrons insulating changes and causes Mott tr. spd electrons pure spectators Anderson (impurity) model Kondo volume colapse (J. W. Allen, R. M. Martin, 1982): hybridization with spd electrons is crucial (Lavagna, Lacroix and Cyrot, 1982) changes → chnange of TK f electrons in local moment regime bath either constant or taken from LDA and rescaled Fenomenological Landau approach:

LDA+DMFT ab initio calculation bath for AIM is self-consistently determined contains tff and Vfd hopping Kondo volume colapse model resembles DMFT picture: Solution of the Anderson impurity model → Kondo physics Difference: with DMFT the lattice problem is solved (and therefore Δ must selfconsistently determined) while in KVC Δ is calculated for a fictious impurity (and needs to be rescaled to fit exp. ) In KVC scheme there is no feedback on spd bans, hence optics is not much affected.

An example Atomic physics of selected Actinides

optics mass and plasma w Basov, cond-mat/0509307

Two Site CDMFT in the 1 D Hubbard model M. Capone M. Civelli V. Kancharla C. Castellani and Kotliar, PRB 69, 195105 (2004)

Slave particle diagrammatic impurity solvers every atomic state represented with a unique pseudoparticle atomic eigenbase - full (atomic) base , where Luttinger Ward functional NCA general AIM: OCA ) TCA (

SUNCA vs QMC two band Hubbard model, Bethe lattice, U=4 D three band Hubbard model, Bethe lattice, U=5 D, T=0. 0625 D